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+{"text":"\n\n\n\n\n\n\n\n\n\n\n\n\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*{Acknowledgement}\nSupported by the National Key R\\&D Program of China under Grant 2019YFB1406500, National Natural Science Foundation of China (No.62025604, 62076213, 62006217). Open Project Program of State Key Laboratory of Virtual Reality Technology and Systems, Beihang University (No.VRLAB2021C06). The university development fund of the Chinese University of Hong Kong, Shenzhen under grant No. 01001810, and Tencent AI Lab Rhino-Bird Focused Research Program under grant No.JR202123. \n\\section{The Proposed Approach}\nWe propose a novel adversarial training framework by introducing the concept of ``learnable attack strategy\". \nWe first introduce the pipeline of our framework in Sec.~\\ref{sec:game} and then present our novel formulation of adversarial training in Sec.~\\ref{sec:minmax} and our proposed loss terms in Sec.~\\ref{sec:loss} followed by the proposed optimization algorithm in Sec.~\\ref{sec:optimize}. \n\\subsection{Pipeline of the Proposed Framework} \\label{sec:game}\nThe pipeline of our framework is shown in Fig.~\\ref{fig:santan1}. \nOur model is composed of a target network and a strategy network. \nThe former uses AEs for training to improve its robustness, whilst the latter generates attack strategies to create AEs to attack the target network. They are competitors. \n\n\\vspace{1mm}\n\\noindent \\textbf{Target Network.} \nThe target network is a convolutional network for image classification, denoted as $\\hat{y} = f_{\\mathbf{w}}(\\mathbf{x})$ where $\\hat{y}$ is  the estimation of the label, $\\mathbf{x}$ is an image, and $\\mathbf{w}$ are the parameters of the network.  \n\n\\vspace{1mm}\n\\noindent \\textbf{Strategy Network.} \nThe strategy network generates adversarial attack strategies to control the AE generation, which takes a sample as input and outputs a strategy. \nSince the strategy network updates gradually, it gives different strategies given the same sample as input according to the robustness of the target network at different training stages. \nThe architecture of the strategy network is illustrated in the \\textbf{supplementary material}.  Given an image, the strategy network outputs an attack strategy, \\textit{i.e.,} the configuration of how to perform the adversarial attack. \nLet $\\mathbf{a} = \\{a_1, a_2, ..., a_M\\} \\in \\mathcal{A}$ denote a strategy of which each element refers to an attack parameter. \n$\\mathcal{A}$ denotes the value space of strategy. \nParameter $a_m \\in \\{1,2,...,K_m\\}$ has $K_m$ options, which is encoded by a one-hot vector. \nThe meaning of each option differs in different attack parameters. \nFor example,  PGD attack~\\cite{madry2017towards} has three attack parameters, \\textit{i.e.,} the attack step size $\\alpha$, the attack iteration $I$, and the   maximal perturbation strength $\\varepsilon$. \nEach parameter has $K_m$ optional values to select, \\textit{e.g.,} the options for $\\alpha$ could be $\\{0.1, 0.2, 0.3, ...\\}$ and the options for $I$ could be $\\{1, 10, 20, ...\\}$.   \nA combination of the selected values for these attack parameters is an attack strategy. \nThe strategy is used to created AEs along with the target model. \nThe strategy network captures the conditional distribution  of a given $\\mathbf{x}$ and $\\boldsymbol{\\theta}$, $p(\\mathbf{a}|\\mathbf{x};\\boldsymbol{\\theta} )$,  where $\\mathbf{x}$ is the input image and $\\boldsymbol{\\theta}$ denotes the strategy network parameters. \n\n\n\n\\vspace{1mm}\n\\noindent \\textbf{Adversarial Example Generator.} Given a clean image, the process of the generation of AEs can be defined as:\n\\begin{align}\n    \\mathbf{x}_{adv}:= \\mathbf{x} + \\boldsymbol{\\delta} \\leftarrow  g(\\mathbf{x}, \\mathbf{a}, \\mathbf{w}) , \\label{Eq:adv} \n\\end{align}\nwhere $\\mathbf{x}$ is a clean image, $\\mathbf{x}_{adv}$ is its corresponding AEs, and $\\boldsymbol{\\delta}$ is the generated perturbation. $\\mathbf{a}$ is an attack strategy. $\\mathbf{w}$ represents the target network parameters, and $g(\\cdot)$ is the PGD attack. \nThe process is equivalent to solving the inner optimization problem of Eq.~(\\ref{Eq:sat}) given an attack strategy $\\mathbf{a}$, \\textit{i.e.,} finding the optimal perturbation to maximize the loss. \n\n\\subsection{Novel Formulation of Adversarial Training} \\label{sec:minmax}\nBy using Eq.~(\\ref{Eq:adv}) that represents the process of AE generation, the standard AT with a fixed attack strategy can be rewritten as: \n\\begin{align}\n  \\min_{\\mathbf{w}}  \\mathbb{E}_{(\\mathbf{x}, y) \\sim \\mathcal{D}}~  \\mathcal{L}(f_{\\mathbf{w}}(\\mathbf{x}_{adv}), y),  \\label{eq:oldAT}\n\\end{align}\nwhere $\\mathbf{x}_{adv} = g(\\mathbf{x}, \\mathbf{a}, \\mathbf{w})$ and $\\mathbf{a}$ is the hand-crafted attack strategy. \n$\\mathcal{D}$ is the training set. \n$\\mathcal{L}$ is the cross-entropy loss function which is used to measure the difference between the predicted label of the AE $\\mathbf{x}_{adv}$ and the ground truth $y$. \n\nDifferently, instead of using a hand-crafted sample-agnostic strategy, we use a strategy network to produce automatically generated sample-dependent strategies, \\textit{i.e.,} $p(\\mathbf{a}|\\mathbf{x};\\boldsymbol{\\theta})$.\nOur novel formulation for AT can be defined as: \n\\begin{align}\n  \\min_{\\mathbf{w}}  \\mathbb{E}_{(\\mathbf{x}, y) \\sim \\mathcal{D}}\n  [\\max_{\\boldsymbol{\\theta}}\n~\\mathbb{E}_{ \\mathbf{a} \\sim p(\\mathbf{a}| \\mathbf{x};\\boldsymbol{\\theta})}~ \\mathcal{L}(f_{\\mathbf{w}}(\\mathbf{x}_{adv}),y) ].\n\\label{eq:newAT}\n\\end{align}\nCompared to the standard AT, the most distinct difference lies in the generation of AEs, \\textit{i.e.,} $\\mathbf{x}_{adv}$ \\eqref{Eq:adv}. \nThe standard AT uses a hand-crafted sample-agnostic strategy $\\mathbf{a}$ to solve the inner optimization problem while we use a strategy network to produce the sample-dependent strategy by $p(\\mathbf{a}|\\mathbf{x};\\boldsymbol{\\theta})$, \\textit{i.e.,} our strategy is learnable. \nOur AE generation involves the parameters $\\boldsymbol{\\theta}$ of the strategy network, which leads that our loss being a function of the parameters of both networks.  \n\nComparing Eq.~(\\ref{eq:oldAT}) and Eq.~(\\ref{eq:newAT}), our formulation is a minimax problem and the inner optimization involves the parameters of the strategy network. \nFrom Eq.~(\\ref{eq:newAT}), it can be observed that the two networks compete with each other in minimizing or maximizing the same objective. \nThe target network learns to adjust its parameters to defend AEs generated by the attack strategies, while the strategy network learns to improve attack strategies according to the given samples to attack the target network.\nAt the beginning of the training phase, the target network is vulnerable, which a weak attack can fool. \nHence, the strategy network can easily generate effective attack strategies. The strategies could be diverse because both weak and strong attacks can succeed. \nAs the training process goes on, the target network becomes more robust. \nThe strategy network has to learn to generate attack strategies that create stronger AEs. \nTherefore, the gaming mechanism could boost the robustness of the target network gradually along with the improvement of the strategy network. \n\n\\subsection{The Proposed Loss Terms}\n\\label{sec:loss}\n\\vspace{1mm}\n\\noindent\\textbf{Loss of Evaluating Robustness.}\nTo guide the learning of the strategy network, we propose a new metric to evaluate attack strategy by using the robustness of the one-step updated version of the target model.\nSpecifically, an attack strategy $\\mathbf{a}$ is first used to create an AE $\\mathbf{x}_{adv}$ which is then used to adjust the parameters of the target model $\\mathbf{w}$ for one step through the first-order gradient descent. \nThe attack strategy is criticized to be effective if the updated target model $\\hat{\\mathbf{w}}$ can correctly predict labels for AEs $\\mathbf{x}_{adv}^{\\hat{\\mathbf{a}}}$ that generated by another attack strategy $\\hat{\\mathbf{a}}$, \\textit{e.g.,} PGD with the maximal perturbation strength of 8, iterative steps of $10$ and step size of $2$. \nThe loss function of evaluating robustness can be defined as: \n\\begin{align}\n   \\mathcal{L}_{2}(\\boldsymbol{\\theta}) \n&=-\\mathcal{L}(f( \\mathbf{x}_{adv}^{\\hat{\\mathbf{a}}},\\hat{\\mathbf{w}} ), y), \\label{eq:loss_robust}\n\\end{align}\nwhere $\\hat{\\mathbf{w}} = \\mathbf{w} - \\lambda \\nabla_\\mathbf{w} \\mathcal{L}_1|_{\\mathbf{x}_{adv}}$ is the parameters of the updated target network and $\\lambda$ is the step size. \n$\\mathcal{L}_1$ refers to the loss in Eq.~(\\ref{eq:newAT}), \\textit{i.e.,} $\\mathcal{L}_1(\\mathbf{w},\\boldsymbol{\\theta}):= \\mathcal{L}(f(\\mathbf{x}_{adv}, \\mathbf{w}), y)$. \n$\\mathbf{x}_{adv}$\nis created by the attack strategy $\\mathbf{a}$, which is to be evaluated. \n$\\mathbf{x}_{adv}^{\\hat{\\mathbf{a}}} :=  g(\\mathbf{x}, \\hat{\\mathbf{a}}, \\hat{\\mathbf{w}}) $ is the AE created by another attack strategy $\\hat{\\mathbf{a}}$, which is used to evaluate the robustness of the updated model $\\hat{\\mathbf{w}}$. \nPlease note that $\\mathcal{L}_2$ is used to evaluate the attack strategy and $\\mathbf{w}$ is treated as a variable here rather than parameters to optimize.  \nHence, the value of $\\mathbf{w}$ is used in Eq.~(\\ref{eq:loss_robust}), but the gradient of $\\mathcal{L}_2$ will not be backpropagated to update $\\mathbf{w}$ through $\\hat{\\mathbf{w}}$. \nEq.~(\\ref{eq:loss_robust}) indicates that a larger $\\mathcal{L}_2$ means the updated target model is more robust, \\textit{i.e.,} a better attack strategy. \n\n\\vspace{1mm}\n\\noindent\\textbf{Loss of Predicting Clean Samples.}\nA good attack strategy should not only improve the robustness of the target model but also maintain the performance of predicting clean samples, \\textit{i.e.,} clean accuracy. \nTo further provide guidance for learning the strategy network, we also consider the performance of the one-step updated target model in predicting clean samples. \nThe loss of evaluating the attack strategy can be defined as: \n\\begin{align}\n    \\mathcal{L}_3(\\boldsymbol{\\theta}) &= - \\mathcal{L}(f(\\mathbf{x}, \n    \\hat{\\mathbf{w}}), y),  \\label{eq:clean_acc}\n\\end{align}\nwhere $\\hat{\\mathbf{w}}$ is the same as that in Eq.~(\\ref{eq:loss_robust}), \\textit{i.e.,} the parameters of the one-step updated target model. \n$\\mathcal{L}_3$ is the function of $\\boldsymbol{\\theta}$ as the AE $\\mathbf{x}_{adv}$ involves computing $\\hat{\\mathbf{w}}$ and $\\mathbf{a}$ is the output of the strategy network. \nEq.~(\\ref{eq:clean_acc}) indicates that a larger $\\mathcal{L}_3$ means the updated target model has a lower loss in clean samples, \\textit{i.e.,} a better attack strategy. \n\n\\paragraph{Formal Formulation.} \nIncorporating the two proposed loss terms, our formulation for AT can be defined as: \n\\begin{equation}\n        \\begin{aligned}\n              & & &\\min_{\\mathbf{w}} \\mathbb{E}_{(\\mathbf{x}, y) \\sim \\mathcal{D}}\\left[\\max_{\\boldsymbol{\\theta}}~\\mathbb{E}_{ \\mathbf{a} \\sim p(\\mathbf{a}| \\mathbf{x};\\boldsymbol{\\theta})}~[\\mathcal{L}_1(\\mathbf{w},\\boldsymbol{\\theta})+\\right.\\\\ \n              & & & \\left.\\textcolor{white}{\\min_{\\mathbf{w}} \\mathbb{E}_{(\\mathbf{x}, y) \\sim \\mathcal{D}}[\\max_{\\boldsymbol{\\theta}}EEEE}\\alpha \\mathcal{L}_2(\\boldsymbol{\\theta}) + \\beta \\mathcal{L}_3(\\boldsymbol{\\theta})]\\right],\n        \\end{aligned}\n        \\label{eq:newAT_pro}\n    \\end{equation}\nwhere $\\mathcal{L}_1$ is a function of the parameters of both the target network and the strategy network while $\\mathcal{L}_2$ and $\\mathcal{L}_3$ involve the parameters of the strategy network. $\\alpha$ and $\\beta$ are the trade-off hyper-parameters of the two loss terms.\n\n\n\\subsection{Optimization} \\label{sec:optimize}\nWe propose an algorithm to alternatively optimize the parameters of the two networks. \nGiven $\\boldsymbol{\\theta}$, the subproblem of optimizing the target network can be defined as,\n\\begin{align}\n   \\min_{\\mathbf{w}} \\mathbb{E}_{(\\mathbf{x}, y) \\sim \\mathcal{D}}\n\\mathbb{E}_{ \\mathbf{a} \\sim p(\\mathbf{a}| \\mathbf{x};\\boldsymbol{\\theta})}\n[\\mathcal{L}_1(\\mathbf{w},\\boldsymbol{\\theta})]. \\label{Eq:subprob1}\n\\end{align}\nGiven a clean image, the strategy network generates a strategy distribution $p(\\mathbf{a}| \\mathbf{x};\\boldsymbol{\\theta})$, and we randomly sample a strategy from the conditional distribution. \nThe sampled strategy is used to generate AEs.  \nAfter collecting the AEs for a batch of samples, we can update the parameters of the target model through gradient descent, \\textit{i.e.,}\n\\begin{align}\n    \\label{eq:update_w}\n    \\mathbf{w}^{t+1}=\\mathbf{w}^{t}-\\eta_1 \\frac{1}{ N}  \\sum_{n=1}^{N} \\nabla_{\\mathbf{w}}\n\\mathcal{L}\\left(f(\\mathbf{x}_{adv}^n, \\mathbf{w}^t), y_{n}\\right),  \n\\end{align}\nwhere $N$ is the number of samples in a mini batch and $\\eta_1 $ is the learning rate. \n\nGiven $\\mathbf{w}$, the subproblem of optimizing the parameters of the strategy network can be written as,\n\\begin{align}\n    \\max_{\\boldsymbol{\\theta}} J(\\boldsymbol{\\theta}) , \n\\end{align}\nwhere $J(\\boldsymbol{\\theta}) := \\mathbb{E}_{(\\mathbf{x}, y) \\sim \\mathcal{D}}\n~\\mathbb{E}_{ \\mathbf{a} \\sim p(\\mathbf{a}| \\mathbf{x};\\boldsymbol{\\theta})}~ [\\mathcal{L}_1 + \\alpha \\mathcal{L}_2 + \\beta \\mathcal{L}_3]$. \nThe biggest challenge of this optimization problem is that the process of AE generation (see Eq.~(\\ref{Eq:adv})) is not differentiable, namely, the gradient can not be backpropagated to the attack strategy through the AEs. \nMoreover, there are some non-differentiable operations (\\emph{e.g.} choosing the iteration times) related to attack \\cite{peng2018jointly,ilyas2019adversarial}, which sets an obstacle to backpropagate the gradient to the strategy network. \n\nFollowing by the REINFORCE algorithm \\cite{williams1992simple}, we can compute the derivative of the objective function $J(\\boldsymbol{\\theta})$ with respect to the parameters $\\boldsymbol{\\theta}$ as:\n\\begin{align}\n    \\nabla_{\\boldsymbol{\\theta}} J(\\boldsymbol{\\theta})&= \\nabla_{\\boldsymbol{\\theta}} \\mathbb{E}_{(\\mathbf{x}, y) \\sim \\mathcal{D}}\n\\mathbb{E}_{ \\mathbf{a} \\sim p(\\mathbf{a}| \\mathbf{x};\\boldsymbol{\\theta})}\n\\left[ \\mathcal{L}_{0}\\right]  \\\\\n& =  \\mathbb{E}_{(\\mathbf{x}, y) \\sim \\mathcal{D}} \\int_{\\mathbf{a}} \\mathcal{L}_{0} \\cdot \\nabla_{\\boldsymbol{\\theta}}  p(\\mathbf{a}| \\mathbf{x};\\boldsymbol{\\theta}) d\\mathbf{a} \\nonumber \\\\\n& =  \\mathbb{E}_{(\\mathbf{x}, y) \\sim \\mathcal{D}} \\int_{\\mathbf{a}} \\mathcal{L}_{0} \\cdot   \np(\\mathbf{a}| \\mathbf{x};\\boldsymbol{\\theta}) \\nabla_{\\boldsymbol{\\theta}} \\log p(\\mathbf{a}| \\mathbf{x};\\boldsymbol{\\theta}) d\\mathbf{a} \\nonumber \\\\\n& = \\mathbb{E}_{(\\mathbf{x}, y) \\sim \\mathcal{D}}\n\\mathbb{E}_{ \\mathbf{a} \\sim p(\\mathbf{a}| \\mathbf{x};\\boldsymbol{\\theta})} [  \\mathcal{L}_{0} \\cdot \\nabla_{\\boldsymbol{\\theta}} \\log  p(\\mathbf{a}| \\mathbf{x};\\boldsymbol{\\theta}) ] \\nonumber ,\n\\end{align}\nwhere $  \\mathcal{L}_{0} =  \\mathcal{L}_{1} + \\alpha  \\mathcal{L}_{2} + \\beta  \\mathcal{L}_{3}$. \nSimilar to solving Eq.~(\\ref{Eq:subprob1}), we sample attack strategy from the conditional distribution of strategy to generate AEs. \nThe gradient with respect to the parameters\ncan be approximately computed as: \n\\begin{align}\n    \\nabla_{\\boldsymbol{\\theta}}J(\\boldsymbol{\\theta}) \\approx   \\frac{1}{N} \\sum_{n=1}^{N} \\mathcal{L}_{0}(\\mathbf{x}^n;\\boldsymbol{\\theta}) \\cdot \\nabla_{\\boldsymbol{\\theta}} \\log p_{\\boldsymbol{\\theta}} (\\mathbf{a}^n|\\mathbf{x}^n).\n\\end{align}\nThen, the parameters of the strategy network can be updated through gradient ascent, \\textit{i.e.,} \n\\begin{align}\n    \\label{eq:update_theta}\n    \\boldsymbol{\\theta}^{t+1} = \\boldsymbol{\\theta}^t + \\eta_2 \\nabla_{\\boldsymbol{\\theta}}J(\\boldsymbol{\\theta}^t), \n\\end{align}\nwhere $\\eta_2$ is the learning rate. And $\\boldsymbol{\\theta}$ and $\\mathbf{w}$ are updated iteratively. We update $\\mathbf{w}$ every k times of updating $\\boldsymbol{\\theta}$.\n\n\\subsection{Convergence Analysis}\nBased on \\eqref{eq:update_w} and \\eqref{eq:update_theta}, we have the following convergence result of the proposed adversarial training algorithm.\n\\vspace{-1mm}\n\\begin{restatable}{theorem}{convergence}\n\\label{thm:1}\nSuppose that the objective function $\\mathcal{L}_0=\\mathcal{L}_1+\\alpha \\mathcal{L}_2+\\beta \\mathcal{L}_3$ in \\eqref{eq:newAT_pro} satisfied the gradient Lipschitz conditions \\textit{w.r.t.} $\\boldsymbol{\\theta}$ and $\\mathbf{w}$, and $\\mathcal{L}_0$ is $\\mu$-strongly concave in $\\boldsymbol{\\Theta}$, the feasible set of $\\boldsymbol{\\theta}$.\nIf $\\mathbf{\\hat{x}}_{adv}(\\mathbf{x},\\mathbf{w})$ is a $\\sigma$-approximate solution of the $\\ell_{\\infty}$ ball with radius $\\epsilon$ constraint, the variance of the stochastic gradient is bounded by a constant $\\sigma^2>0$, and we set the learning rate of $\\mathbf{w}$ as \n\\begin{equation}\n    \\eta_1 = \\min\\left(\\frac{1}{L_0},\\ \\sqrt{\\frac{\\mathcal{L}_0(\\mathbf{w}^0)-\\underset{\\mathbf{w}}{\\min}\\ \\mathcal{L}_0(\\mathbf{w})}{\\sigma^2TL_0}}\\right),\n\\end{equation}\nwhere $L_0=L_{\\mathbf{w}\\boldsymbol{\\theta}}L_{\\boldsymbol{\\theta}\\mathbf{w}}\/\\mu+L_{\\mathbf{w}\\mathbf{w}}$ is the Lipschitz constants of $\\mathcal{L}_0$, it holds that\n\\begin{equation}\n    \\frac{1}{T}\\sum_{t=0}^{T-1}\\mathbb{E}\\big[\\|\\nabla \\mathcal{L}_0(\\mathbf{w}^t)\\|^2_2\\big]\\leq4\\sigma\\sqrt{\\frac{\\Delta L_0}{T}}+\\frac{5\\delta L^2_{\\mathbf{w}\\boldsymbol{\\theta}}}{\\mu},\n\\end{equation}\nwhere $T$ is the maximum adversarial training epoch number and $\\Delta=\\mathcal{L}_0(\\mathbf{w}^0)-\\underset{\\mathbf{w}}{\\min}\\ \\mathcal{L}_0(\\mathbf{w})$.\n\\end{restatable}\nThe detailed proof is presented in the \\textbf{supplementary material}. By Theorem 1, if the inner maximization process can obtain a $\\delta$-approximation of $\\mathbf{x}^*_{\\text{adv}}$, the proposed method \\textbf{LAS-AT} can archive a stationary point by  sub-linear rate with the precision $5\\delta L^2_{\\mathbf{w}\\boldsymbol{\\theta}}\/\\mu$. Moreover, if $5\\delta L^2_{\\mathbf{w}\\boldsymbol{\\theta}}\/\\mu$ is sufficient small, our method can find the desired robust model $\\mathbf{w}^T$ with a good approximation of $\\mathbf{x}^*_{\\text{adv}}$. \n\n\n\n \n\n\\section{Conclusion  and Discussion }\nWe propose a novel adversarial training framework by introducing the concept of ``learnable attack strategy\", which is composed of two competitors, \\textit{i.e.,} a target network and a strategy network. \nUnder the gaming mechanism, the strategy network learns to produce dynamic sample-dependent attack strategies according to the robustness of the target model for adversarial example generation, instead of using hand-crafted attack strategies. \nTo guide the learning of the strategy network, we also propose two loss terms that involve evaluating the robustness of the target network and predicting clean samples. \nExtensive experimental evaluations are performed on three benchmark databases to demonstrate the superiority of the proposed method.\n\n\n\n\n\\section{ Experiments}\nTo evaluate the proposed method, we conduct experiments on three databases, \\textit{i.e.,} CIFAR10 \\cite{krizhevsky2009learning}, CIFAR100 \\cite{krizhevsky2009learning}, and Tiny ImageNet \\cite{deng2009ImageNet}. The details of these databases are presented in the \\textbf{supplementary material}.\n\n\\input{table\/tb_ablation_k}\n\n\\input{table\/tb_cifar10_v5}\n\\input{table\/tb_cifar100_V5}\n\\subsection{Settings}\n\\label{Settings}\n\\vspace{1mm}\n\\noindent\\textbf{Competitive Methods.}\nTo evaluate the proposed method effectiveness in improving the robustness of a target model, we combine it with several state-of-the-art adversarial training methods and illustrate its performance improvements. \nWe choose not only the most popular methods such as the early stopping PGD-AT \\cite{rice2020overfitting} and TRADES\n\\cite{zhang2019theoretically} as base models but also the recently proposed method AWP \\cite{wu2020adversarial}.\nThe combinations of our method and these models are referred to as LAS-PGD-AT, LAS-TRADES, and LAS-AWP, respectively. \nNote that we use the \\textbf{same training settings} as the base models \\cite{rice2020overfitting, zhang2019theoretically, wu2020adversarial} to train our proposed models, including data splits and training losses. Then we compare the proposed LAS-PGD-AT, LAS-TRADES, and LAS-AWP with the following baselines: (1) PGD-AT~\\cite{rice2020overfitting}, (2) TRADES ~\\cite{zhang2019theoretically}, (3) SAT~\\cite{sitawarin2021sat}, (4) MART~\\cite{wang2019improving}, (5) FAT~\\cite{zhang2020attacks}, (6) GAIRAT~\\cite{zhang2020geometry}, (7) AWP ~\\cite{wu2020adversarial} and (8) LBGAT~\\cite{cui2021learnable}.\nMoreover, we compare our method with CAT \\cite{cai2018curriculum},  DART \\cite{wang2019convergence} and FAT~\\cite{zhang2020attacks}\nThey use different attack strategies at different training stages to conduct AT. Besides, we also compare our method with other state-of-the-art  hyper-parameter search methods \\cite{lin2019online, DBLP:conf\/iclr\/ZhangWZZ20} to evaluate our method.\n\n\n\n\n\\noindent\\textbf{Evaluation.}\\label{Evaluation}\nWe choose several adversarial attack methods to attack the trained models, including PGD~\\cite{madry2017towards},  C\\&W~\\cite{carlini2017towards}\nand AA \\cite{croce2020reliable} which consists of  APGD-CE \\cite{croce2020reliable}, APGD-DLR \\cite{croce2020reliable}, FAB \\cite{croce2019minimally} and Square \\cite{andriushchenko2019square}. \nFollowing the default setting of AT, the max  perturbation strength $\\epsilon$ is set to 8 for all attack methods under the $L_{\\infty}$. \nThe clean accuracy and robust accuracy are used as the evaluation metrics. \n\n\n\n\n\\noindent\\textbf{Implementation Details.}\nOn CIFAR-10 and CIFAR-100, we use ResNet18 \\cite{he2016deep} or WideResNet34-10(WRN34-10) \\cite{zagoruyko2016wide} as the target network. \nOn Tiny ImageNet, \nwe use PreActResNet18 \\cite{he2016identity} as the target model. \nFor all experiments, we train the target network for defense baselines, following their original papers.\nFor the training hyper-parameters of the target network of our method, we use the \\textbf{same setting} as the base models \\cite{rice2020overfitting, zhang2019theoretically, wu2020adversarial}.\nThe detailed settings are presented in the\n\\textbf{supplementary material}. \nFor the target network,\nwe adopt SGD momentum optimizer with a learning rate of 0.1, weight decay of $5 \\times 10^{-4}$.\nWe use ResNet18 as the backbone of the strategy network. \nFor the strategy network of our method, we adopt SGD momentum optimizer with a learning rate of $0.001$. The trade-off hyper-parameters $\\alpha$ and $\\beta$ are set to $2.0$ and $4.0$. \nThe range of the maximal perturbation strength is set from $3$ to $15$, the range of the attack step is set from $1$ to $6$, and the range of the attack iteration is set from $3$ to $15$. \n\\vspace{-1mm}\n\n\n\n\\subsection{Hyper-parameter Selection}\nThe hyper-parameter k controls the alternative update of $\\mathbf{w}$ and $\\boldsymbol{\\theta}$. We update $\\mathbf{w}$ every k times of updating $\\boldsymbol{\\theta}$. It \nnot only affects model robustness\nbut also affects model training efficiency. A hyper-parameter selection experiment with ResNet18 is conducted on CIFAR-10 to select the optimal hyper-parameter k. The results are shown in Table~\\ref{table:ablation_k}.\nThe training time of the proposed LAS-PGD-AT decreases along with the increase of parameter k. The more time the strategy network requires for training, the smaller the k is. When k = 40, the proposed LAS-PGD-AT achieves the best adversarial robustness.\nConsidering AT efficiency, we set k to 40. The selection of hyper-parameters  $\\alpha$ and $\\beta$  \n is presented in the \\textbf{supplementary material}.\n\n\\subsection{Comparisons with Other AT Methods}\nOur method is a plug-and-play component that can be combined with other AT methods to boost their robustness.\n\n\n\\input{table\/tb_tiny_v5}\n\\input{table\/tb_Comparison_AT_cifar100_v2}\n\\vspace{1mm}\n\\noindent \\textbf{Comparisons on CIFAR-10 and CIFAR-100.}\nThe results on CIFAR-10 and CIFAR-100 are shown in Table~\\ref{tb:cifar10} and Table~\\ref{tb:cifar100}. \nAnalyses are as follows. \nFirst, the three proposed models outperform their base models under most attack scenarios.\nIn a lot of cases, our method not only improves the robustness but also improves the clean accuracy of the base models though there is always a trade-off between accuracy and robustness. \nFor example, on CIFAR-10\nwhen using WRN34-10 as the target network, our method improves the clean accuracy of powerful AWP by about $2.2\\%$ and also improves the performance of AWP under PGD-10 attack and AA attack by about $2.1\\%$ and $1.62\\%$, respectively. Moreover, the proposed LAS-AWP achieves the best robustness performance under all attack scenarios.\nWe attribute the improvements to using automatically generated attack strategies instead of  hand-crafted ones. Second, on CIFAR-100, the proposed LAS-AWP not only achieves the highest accuracy on clean images but also achieves the best robustness performance under all attack scenarios.\nIn detail, our LAS-AWP outperforms the original AWP $4.5\\%$ and $1.9\\%$ on the clean accuracy and AA attack accuracy, respectively. Moreover, our LAS-AWP outperforms the powerful LBGAT under all attack scenarios. \n\\input{table\/tb_ablation_v5}\n\n\n\n\n\n\\input{table\/tb_CAT_DART}\n\n\\noindent \\textbf{Comparisons on Tiny ImageNet.}\nFollowing \\cite{lee2020adversarial}, we use PreActResNet18 \\cite{he2016identity} as the target model for evaluation on Tiny ImageNet. \nThe results are shown in Table~\\ref{tb:tiny}. \nAs Tiny ImageNet has more classes than CIFAR-10 and CIFAR-100, the defense of AEs is more challenging. \nOur method improves the  clean and adversarial robustness accuracy of the three base models. \n\n\\vspace{1mm}\n\\noindent \\textbf{Comparisons with state-of-the-art robustness model.}\nAuto Attack (AA) is a reliable and strong attack method to evaluate model robustness. It consists of three white-box attacks and a black-box attack. The details is introduced in Sec~\\ref{Related Work}. Under their leaderboard results~\\footnote{https:\/\/github.com\/fra31\/auto-attack}, on CIFAR-10, Gowal~\\emph{et al}.~\\cite{gowal2020uncovering} study the impact of hyper-parameters ( such as model weight averaging and model size ) on model robustness and adopt WideResNet70-16 (WRN-70-16) to conduct AT, which ranks the 1st  under AA attack without additional real or synthetic data. We also adopt WRN-70-16 for our method. LAS-AWP can boost the model robustness and achieve higher robustness accuracy.\nOn CIFAR-100, Cui~\\emph{et al}. train WideResNet34-20 (WRN-34-20) for LBGAT and achieves state-of-the-art robustness without additional real or synthetic data.\nWe also adopt WRN-34-20 for our method. LAS-AWP can also achieve higher robustness accuracy. The result is shown Table~\\ref{tb:Comparison_AT_cifar10}.\n\n\n\\subsection{Ablation Study}\nIn our formulation in Eq.~(\\ref{eq:newAT_pro}), besides the loss $\\mathcal{L}_1$, we propose two additional loss terms to guide the learning of the strategy network, \\textit{i.e.,}  the loss of evaluating robustness $\\mathcal{L}_2$ and the loss of predicting clean samples $\\mathcal{L}_3$. \nTo validate the effectiveness of each element in the objective function, we conduct ablation experiments with ResNet18 on CIFAR-10. \nWe train four LAS-PGD-AT models by using $\\mathcal{L}_1$, $\\mathcal{L}_1 \\& \\mathcal{L}_2$, $\\mathcal{L}_1 \\& \\mathcal{L}_3$, and $\\mathcal{L}_1 \\& \\mathcal{L}_2 \\& \\mathcal{L}_3$, respectively.\nThe trained models are attacked by a set of adversarial attack methods. \nThe results are shown in Table \\ref{table:table0}.\nThe classification accuracy is the evaluation metric. \n\\textit{Clean} represents using clean images for testing while other attack methods use AEs for testing.\n\nAnalyses are summarized as follows. \nFirst, when incorporating the loss $\\mathcal{L}_2$ only, the performance of robustness under all attacks improves while the clean accuracy slightly drops. \nWhen incorporating the loss $\\mathcal{L}_3$ only, the clean accuracy improves, but the performance of robustness under partial attacks slightly drops. \nThe results show that $\\mathcal{L}_2$ contributes more to improve the robustness and $\\mathcal{L}_3$ contributes more to improve the clean accuracy. \nSecond, using all losses achieves the best performance in robustness as well as the clean accuracy, which indicates that the two losses are compatible and combining them could remedy \nthe side effect of independent use. \n\n\n\n\n\n\\input{figtex\/fig_comparsion_HP}\n\\subsection{Performance Analysis}\n\n \\noindent \\textbf{Comparisons with hand-crafted attack strategy methods.} \nTo investigate the effectiveness of automatically generated attack strategies generated by our method, we compare our LAS-AT with some AT methods (CAT~\\cite{cai2018curriculum}, DART~\\cite{wang2019convergence} and FAT~\\cite{zhang2020attacks}) which adopt dynamic hand-crafted attack strategies for training. \nFor a fair comparison, we keep the training and evaluation setting as same as those used in FAT~\\cite{zhang2020attacks}. The details are presented in the \\textbf{supplementary material}. The result is shown in Table~\\ref{tb:CAT_DART}. Our method outperforms competing methods under all attacks. It indicates that compared with previous hand-crafted attack strategies, the proposed automatically generated attack strategies can achieve the greater robustness improvement.\n\n\n\n\\noindent \\textbf{Comparisons with hyper-parameter search methods.\n } \nWe compare the proposed method with other hyper-parameter search methods that include a classical hyper-parameter search method (random search) and two automatic  hyper-parameter search methods (OHL~\\cite{lin2019online} and AdvHP~\\cite{DBLP:conf\/iclr\/ZhangWZZ20}). For a fair comparison, the same hyper-parameters and search range that are used in our method (see Sec~\\ref{Settings}) are adopted for them. The detail settings are presented in the \\textbf{supplementary material}. The result is shown in Fig.~\\ref{fig:comparsion_HP}. It can be observed that our method achieves the best robustness performance\nunder all attack scenarios. The automatically generated attack strategies generated by our method are more suitable for AT.\n\n\n\n\\noindent \\textbf{Adversarial Training from Easy to Difficult.}\nTo investigate how LAS-AT works, we analyze the distribution of the strategy network's attack strategies at different training stages.\nExperiments using ResNet18 with LAS-PGD-AT are performed on the CIFAR-10 database. \nThe range of the maximal perturbation strength is set from 3 to 15. \nThe distribution evolution of the maximal perturbation strength during adversarial training is illustrated in Fig.~\\ref{fig:discuss}. \n\nAt the beginning of AT, the distribution covers all the optional values of the maximal perturbation strength. \nEach value has a chance to be selected, which ensures the diversity of AEs. \nAs the training process goes on, the percentage of small perturbation strengths decreases. \nAt the late stages, the distribution of the maximal perturbation strength is occupied by several large values. \nThis phenomenon indicates that the strategy network gradually increases the percentage of large perturbation strengths to generate strong AEs because the robustness of the target network is gradually boosted by training with the generated AEs. \nTherefore, it can be observed that under the gaming mechanism, our method starts training with diverse AEs when the target network is vulnerable, and then learns with more strong AEs at the late stages when the robustness of the target network improves.     \n CAT~\\cite{wang2019convergence}, DART~\\cite{wang2019convergence} and FAT~\\cite{zhang2020attacks} adopt hand-crafted strategies to use weak AEs at early stages and then use strong AEs at late stages. \nUnlike them, under our framework, the strategy network automatically generates strategies that determine the difficulty of AEs, according to the robustness of the target network at different stages. \n\n\\input{figtex\/fig_perturbation}\n\n\n\\section{Introduction}\n\n\n\n\n\nAlthough deep neural networks (DNNs) have achieved great success in academia and industry, they could be easily fooled by adversarial examples (AEs) \\cite{szegedy2013intriguing,goodfellow2014explaining}  generated via adding indistinguishable perturbations to benign images. Recently, many studies~\\cite{li2019nattack,dong2018boosting,wang2021enhancing,bai2021targeted,bai2020targeted,gu2021adversarial,jia2020adv} focus on generating AEs.\nIt has been proven that many real-world applications~\\cite{kurakin2016adversarial} of DNNs are vulnerable to AEs, such as image classification \\cite{goodfellow2014explaining,hirano2021universal}, object detection \\cite{wei2018transferable,jia2021effective}, \nneural machine translation \\cite{zou2019reinforced,huang2020reinforced}, \\emph{etc}. \nThe vulnerability of DNNs makes people pay attention to the safety of artificial intelligence and brings new challenges to the application of deep learning~\\cite{zheng2020pcal,yin2021adv,xu2020adversarial,gu2021effective,gu2021capsule}. Adversarial training (AT) \\cite{madry2017towards,wong2020fast,song2021regional,li2022semi} \nis considered as one of the most effective defense methods to \nimprove adversarial robustness by injecting AEs into the training procedure through a minimax formulation. \nUnder the minimax framework, the generation of AEs plays a key role in determining robustness. \n\\input{figtex\/Home}\n\nSeveral recent works improve the standard AT method from different perspectives. Although existing methods~\\cite{gowal2020uncovering,dai2021parameterizing,bai2021improving,cui2021learnable,rebuffi2021fixing,jia2021boosting} have made significant progress in improving robustness, they rarely explore the impact of attack strategy on adversarial training.\nFirst, as shown in Fig.~\\ref{fig:sat0}, most existing methods leverage a hand-crafted attack strategy to generate AEs by manually specifying the attack parameters, \\textit{e.g.,} PGD attack with the maximal perturbation of 8,  iteration of 10, and step size of 2. \nA hand-crafted attack strategy lacks flexibility and might limit the generalization performance. \nSecond, most methods use only one attack strategy. Though some works~\\cite{cai2018curriculum,wang2019convergence,zhang2020attacks} have realized that exploiting different attack strategies at different training stages could improve robustness, \\textit{i.e.,} using weak attacks at the early stages and strong attacks at the late stages, they use manually designed metrics to evaluate the difficulty of AEs and still use one strategy at each stage. However, they need much domain expertise, and the robustness improvement is limited. They use sample-agnostic attack strategies that are hand-crafted and independent of any information of specific samples. There exist statistical differences among samples, and attack strategy should be designed according to the information of the specific sample, \\textit{i.e.,} sample-dependent.\n\n\\input{figtex\/fig_pipeline}\n\nTo alleviate these issues, we propose a novel adversarial training framework by introducing the concept of ``learnable attack strategy\", \\textit{dubbed LAS-AT}, which learns to automatically produce sample-dependent attack strategies for AE generation  instead of using hand-crafted ones (see Fig.~\\ref{fig:las-at0}). \nOur framework consists of two networks, \\textit{i.e.,} a target network and a strategy network. \nThe former uses AEs for training to improve robustness, while the latter produces attack strategies to control the generation of AEs. \nThe two networks play a game where the target network learns to minimize the training loss of AEs while the strategy network learns to generate strategies to maximize the training loss. \nUnder such a gaming mechanism, at the early training stages, weak attacks can successfully attack the target network. \nAs the robustness improves, the strategy network learns to produce strategies to generate stronger attacks. \nUnlike~\\cite{cai2018curriculum} ~\\cite{wang2019convergence}, and ~\\cite{zhang2020attacks} that use designed metrics and hand-crafted attack strategies, we use the strategy network to automatically produce an attack strategy according to the given sample. \nAs the strategy network updates according to the robustness of the target model and the given sample, the strategy network figures out to produce different strategies accordingly at different stages, rather than setting up any manually designed metrics or strategies.  We propose two loss terms to guide the learning of the strategy network.\nOne evaluates the robustness of the target model updated with the AEs generated by the strategy. \nThe other evaluates how well the updated target model performs on clean samples.  \nOur main contributions are in three aspects:\n\\textbf{1)} We propose a novel adversarial training framework by introducing the concept of ``learnable attack strategy\", which learns to automatically produce sample-dependent attack strategies to generate AEs. \n    Our framework can be combined with other state-of-the-art methods as a plug-and-play component.\n\\textbf{2)} We propose two loss terms to guide the learning of the strategy network, which involve explicitly evaluating the robustness of the target model and the accuracy of clean samples.\n\\textbf{3)} We conduct  experiments and analyses on three databases to demonstrate the effectiveness of the proposed method. \n\n    \n\n    \n\n\\section{Related Work}\n\\paragraph{Adversarial Attack Methods.}\n\\label{Related Work}\nAs the vulnerability of deep learning models has been noticed~\\cite{szegedy2013intriguing}, many works studied the model's robustness and proposed a series of adversarial attack methods.  \nFast Gradient Sign Method (FGSM) \\cite{goodfellow2014explaining} was a classic adversarial attack method, which made use of the gradient of the model to generate AEs. \nMadry \\emph{et al}.~\\cite{madry2017towards} proposed a multi-step version of FGSM, called Projected Gradient Descent (PGD). \nTo solve the problem of parameter selection in FGSM, Moosavi-Dezfooli \\emph{et al}.~\\cite{moosavi2016deepfool} proposed a simple but accurate method, called Deepfool, to attack deep neural networks. \nIt generated AEs by using an iterative linearization of the classification model. \nCarlini-Wagner \\emph{et al}.~\\cite{carlini2017towards} proposed several powerful attack methods that could be widely used to evaluate the robustness of deep learning models.  \nMoreover, Croce \\emph{et al}.~\\cite{croce2020reliable} proposed two improved methods (APGD-CE, APGD-DLR) of the PGD-attack. \nThey did not need to choose a step size or alternate a loss function. \nAnd then they combined the proposed method with two complementary adversarial attack methods (FAB \\cite{croce2019minimally} and Square \\cite{andriushchenko2019square}) to evaluate the robustness, which was called AutoAttack (AA).\n\n\\vspace{-4mm}\n\\paragraph{Adversarial Training Defense Methods.}\nAdversarial training is an effective way to improve robustness by using AEs for training, such as ~\\cite{ kannan2018adversarial,roth2019adversarial, pang2020boosting, wang2020once, wu2020adversarial, wang2019improving,wang2021probabilistic}. \nThe standard adversarial training (AT) is formulated as\na minimax optimization problem in \\cite{madry2017towards}.\nThe objective function is defined as:\n\\begin{equation}\n\\min_{\\mathbf{w}} \\mathbb{E}_{(\\mathbf{x}, y) \\sim \\mathcal{D}}[\\max _{\\boldsymbol{\\delta} \\in \\Omega} \\mathcal{L}(f_{\\mathbf{w}}(\\mathbf{x} + \\boldsymbol{\\delta}), y)], \\label{Eq:sat}\n\\end{equation}\nwhere $\\mathcal{D}$ represents an underlying data distribution and $\\Omega$ represents the perturbation set. \n $\\mathbf{x}$ represents the example, $y$\nrepresents the corresponding label, and $\\boldsymbol{\\delta}$ represents the indistinguishable perturbation. \n$f_{\\mathbf{w}}(\\cdot)$ represents the target network and\n$\\mathcal{L}(f_{\\mathbf{w}}(\\mathbf{x}), y)$ represents the loss function of the target network.\nThe inner maximization problem of standard AT can be regarded as the attack strategy that guides the creation of AEs, which is the core to improve the model robustness. A training strategy is designed accordingly, which significantly improves the network's robustness. \nMadry \\emph{et al}. proposed the \nprime AT framework, PGD-AT~\\cite{\nmadry2017towards}, to improve the robustness.\nAnd  Rice \\emph{et al}. \nproposed a early stopping version~\\cite{rice2020overfitting} of PGD-AT, which gained a great improvement. Zhang \\emph{et al}.~\\cite{zhang2019theoretically} explored a trade-off between standard accuracy and adversarial robustness and proposed a defense method (TRADES) that can trade standard accuracy off against adversarial robustness. Wu \\emph{et al.}~\\cite{wu2020adversarial}  investigated the weight loss landscape and proposed an effective Adversarial Weight Perturbation (AWP) method to improve the robustness.\nCui \\emph{et al}.~\\cite{cui2021learnable} proposed to adapt the logits of one model trained on clean data to guide adversarial training (LBGAT). These AT methods adopted a fxed attack strategy to conduct AT. Some AT methods exploited different attack\nstrategies at different training stages to improve robustness. In detail, \n Cai \\emph{et al}.~\\cite{cai2018curriculum}\nadopted curriculum adversarial training (CAT) to improve model robustness.\nWang \\emph{et al}.~\\cite{wang2019convergence} designed a criterion to measure the convergence quality and proposed dynamic adversarial training\n(DART) to improve the robustness of the target model. Zhang \\emph{et al.}~\\cite{zhang2020attacks} proposed to search for the least adversarial data for AT, which could be called friendly adversarial training (FAT). \n\n\\section*{Detailed Proof}\n\nFirst we introduce some notations. Let $\\mathcal{L}_0:\\mathcal{X}\\times\\mathcal{Y}\\times\\mathcal{W}\\times\\boldsymbol{\\Theta}\\rightarrow\\mathbb{R}^+$ be the objective function in \\eqref{eq:newAT_pro} as\n\\begin{equation}\n    \\mathcal{L}_0 = \\mathcal{L}_1+\\alpha \\mathcal{L}_2+\\beta \\mathcal{L}_3.\n\\end{equation}\nWe define $\\boldsymbol{x}^*_{\\text{adv}}(\\boldsymbol{x},\\mathbf{w})$ as the optimal adversarial example generated by the strategy network\n\\begin{equation}\n    \\begin{aligned}\n        & \\boldsymbol{x}^*_{\\text{adv}}(\\boldsymbol{x},\\mathbf{w}) &=&\\ \\  \\underset{\\boldsymbol{\\theta}}{\\arg\\max}\\ g(\\boldsymbol{x},\\boldsymbol{a}(\\boldsymbol{\\theta}),\\mathbf{w})\\\\\n        & &=&\\ \\ \\underset{\\boldsymbol{\\theta}}{\\arg\\max}\\ \\mathbb{E}_{\\boldsymbol{a}\\sim p(\\boldsymbol{a}|\\boldsymbol{x},\\boldsymbol{\\theta})}[\\mathcal{L}_0],\n    \\end{aligned}\n\\end{equation}\nand $\\boldsymbol{\\hat{x}}_{\\text{adv}}(\\boldsymbol{x},\\mathbf{w})$ is a $\\delta$-approximate solution to $\\boldsymbol{x}^*_{\\text{adv}}(\\boldsymbol{x},\\mathbf{w})$. In addition, the full gradient of $\\mathcal{L}_0$ w.r.t $\\mathbf{w}$ is \n\\begin{equation}\n    \\label{eq:grad_1}\n    \\begin{aligned}\n        & \\nabla_{\\mathbf{w}} \\mathcal{L}_0(\\mathbf{w}) &=&\\ \\  \\frac{1}{N} \\sum_{i=n}^N \\nabla_{\\mathbf{w}} \\mathcal{L}^n_0\\\\\n        & &=&\\ \\ \\frac{1}{N} \\sum_{n=1}^N \\nabla_{\\mathbf{w}} \\mathcal{L}_0(\\boldsymbol{x}^*_{\\text{adv}}(\\boldsymbol{x}_n,\\mathbf{w}),\\mathbf{w}),\n    \\end{aligned}\n\\end{equation}\nwhere $\\boldsymbol{x}^*_{\\text{adv}}(\\boldsymbol{x}_n)$ is the optimal adversarial example for $\\boldsymbol{x}_n$. The stochastic gradient of $\\mathcal{L}_0$ w.r.t $\\mathbf{w}$ is \n\\begin{equation}\n \\label{eq:grad_2}\n    \\begin{aligned}\n        & \\nabla_{\\mathbf{w}}\\ell(\\mathbf{w}) &=&\\ \\  \\frac{1}{|\\mathcal{B}|}\\sum_{i=1}^{|\\mathcal{B}|} \\nabla_{\\mathbf{w}} \\mathcal{L}^i_0\\\\\n        & &=&\\ \\ \\frac{1}{|\\mathcal{B}|} \\sum_{n=1}^N \\nabla_{\\mathbf{w}} \\mathcal{L}_0(\\boldsymbol{x}^*_{\\text{adv}}(\\boldsymbol{x}_i,\\mathbf{w}),\\mathbf{w}).\n    \\end{aligned}\n\\end{equation}\nThen $\\nabla_{\\boldsymbol{\\theta}} \\mathcal{L}_0$ and $\\nabla_{\\boldsymbol{\\theta}}\\ell$ correspond to the full and stochastic gradients of $\\mathcal{L}_0$ w.r.t $\\boldsymbol{\\theta}$. Without lose of generality, we assume that\n\\begin{equation}\n    \\mathbb{E}[\\nabla_{\\mathbf{w}} \\ell(\\mathbf{w})] = \\nabla_{\\mathbf{w}}\\mathcal{L}_0(\\mathbf{w}).\n\\end{equation}\nWe note the approximate stochastic gradient as $\\nabla_{\\mathbf{w}}\\hat{\\ell}$:\n\\begin{equation}\n     \\label{eq:grad_3}\n    \\begin{aligned}\n        & \\nabla_{\\mathbf{w}}\\hat{\\ell}~(\\mathbf{w}) &=&\\ \\  \\frac{1}{|\\mathcal{B}|}\\sum_{i=1}^{|\\mathcal{B}|} \\nabla_{\\mathbf{w}} \\hat{\\mathcal{L}}^i_0\\\\\n        & &=&\\ \\ \\frac{1}{|\\mathcal{B}|} \\sum_{n=1}^N \\nabla_{\\mathbf{w}} \\mathcal{L}_0(\\hat{\\boldsymbol{x}}_{\\text{adv}}(\\boldsymbol{x}_i,\\mathbf{w}),\\mathbf{w}).\n    \\end{aligned}    \n\\end{equation}\nMoreover, the adversarial example $\\boldsymbol{x}_{\\text{adv}}(\\boldsymbol{x},\\mathbf{w})$ can be identified by a parameter $\\boldsymbol{\\theta}$ of the strategy network and the gradients like \\eqref{eq:grad_1}, \\eqref{eq:grad_2}, \\eqref{eq:grad_3} would be\n\\begin{equation}\n    \\begin{aligned}\n        & \\nabla_{\\mathbf{w}} \\mathcal{L}_0(\\boldsymbol{\\theta},\\mathbf{w}) &:=&\\ \\ \\nabla_{\\mathbf{w}} \\mathcal{L}_0(\\mathbf{w})\\\\\n        & \\nabla_{\\mathbf{w}}\\ell(\\boldsymbol{\\theta},\\mathbf{w}) &:=&\\ \\ \\nabla_{\\mathbf{w}}\\ell(\\mathbf{w})\\\\\n        & \\nabla_{\\mathbf{w}}\\hat{\\ell}(\\boldsymbol{\\theta},\\mathbf{w}) &:=&\\ \\ \\nabla_{\\mathbf{w}}\\hat{\\ell}~(\\mathbf{w}).\n    \\end{aligned}\n\\end{equation}\nThe corresponding gradients w.r.t $\\boldsymbol{\\theta}$ will be $\\nabla_{\\boldsymbol{\\theta}} \\mathcal{L}_0$, $\\nabla_{\\boldsymbol{\\theta}} \\ell$ and $\\nabla_{\\boldsymbol{\\theta}} \\hat{\\ell}$. As the $\\mathcal{L}_0$ in \\eqref{eq:newAT_pro} satisfies the Lipschitz gradient conditions, given $\\boldsymbol{x}_n\\in\\mathcal{X}$, it holds that\n\\begin{equation}\n    \\label{eq:lipschitz_grad}\n    \\begin{aligned}\n        & &&\\ \\ \\underset{\\boldsymbol{\\theta}}{\\sup}\\ \\|\\nabla_{\\mathbf{w}} \\mathcal{L}^n_0(\\boldsymbol{\\theta},\\mathbf{w})-\\nabla_{\\mathbf{w}} \\mathcal{L}^n_0(\\boldsymbol{\\theta},\\mathbf{w}')\\|_2\\\\\n        & &\\leq&\\ \\  L_{\\mathbf{w}\\mathbf{w}}\\|\\mathbf{w}-\\mathbf{w}'\\|_2\\\\[5pt]\n        & & &\\ \\ \\underset{\\mathbf{w}}{\\sup}\\ \\|\\nabla_{\\mathbf{w}} \\mathcal{L}^n_0(\\boldsymbol{\\theta},\\mathbf{w})-\\nabla_{\\mathbf{w}} \\mathcal{L}^n_0(\\boldsymbol{\\theta}',\\mathbf{w})\\|_2\\\\\n        & &\\leq&\\ \\ L_{\\mathbf{w}\\boldsymbol{\\theta}}\\|\\boldsymbol{\\theta}-\\boldsymbol{\\theta}'\\|_2\\\\[5pt]\n        & & &\\ \\ \\underset{\\boldsymbol{\\theta}}{\\sup}\\ \\|\\nabla_{\\boldsymbol{\\theta}} \\mathcal{L}^n_0(\\boldsymbol{\\theta},\\mathbf{w})-\\nabla_{\\boldsymbol{\\theta}} \\mathcal{L}^n_0(\\boldsymbol{\\theta},\\mathbf{w}')\\|_2\\\\\n        & &\\leq&\\ \\  L_{\\boldsymbol{\\theta}\\mathbf{w}}\\|\\mathbf{w}-\\mathbf{w}'\\|_2,\n    \\end{aligned}\n\\end{equation}\nwhere $L_{\\mathbf{w}\\mathbf{w}}$, $L_{\\mathbf{w}\\boldsymbol{\\theta}}$ and $L_{\\boldsymbol{\\theta}\\mathbf{w}}$ are positive constants. Furthermore, by the strongly-concavity of $\\mathcal{L}_0$ and given $\\boldsymbol{x}_n\\in\\mathcal{X}$, we know that for any $\\boldsymbol{\\theta}_1$ and $\\boldsymbol{\\theta}_2\\in\\boldsymbol{\\Theta}$,\n\\begin{equation}\n    \\begin{aligned}\n        & &&\\ \\ \\mathcal{L}^n_0(\\boldsymbol{\\theta}_1,\\mathbf{w})-\\mathcal{L}^n_0( \\boldsymbol{\\theta}_2,\\mathbf{w})\\\\[5pt]\n        & &\\leq&\\ \\ \\big\\langle\\nabla_{\\boldsymbol{\\theta}} \\mathcal{L}^n_0(\\boldsymbol{\\theta},\\mathbf{w}),\\boldsymbol{\\theta}_1-\\boldsymbol{\\theta}_2\\big\\rangle - \\frac{\\mu}{2}\\|\\boldsymbol{\\theta}_1-\\boldsymbol{\\theta}_2\\|_2^2.\n    \\end{aligned}\n\\end{equation}\nAs the variance of the stochastic gradient is bounded by $\\sigma^2>0$, it means that\n\\begin{equation}\n    \\mathbb{E}\\big[\\|\\nabla_{\\mathbf{w}} \\ell(\\mathbf{w})-\\nabla_{\\mathbf{w}} \\mathcal{L}_0(\\mathbf{w})\\|^2_2\\big]\\leq\\sigma^2.\n\\end{equation}\n\nTo prove the main result, we need the following two important lemmas.\n\\begin{lemma}\n\\label{lem:1}\nSuppose that $\\mathcal{L}_0$ in \\eqref{eq:newAT_pro} satisfies the Lipschitz gradient conditions as \\eqref{eq:lipschitz_grad} and $\\mathcal{L}_0$ is $\\mu$-strongly concave in $\\boldsymbol{\\Theta}$, we have $\\mathcal{L}_0$ is Lipschitz smooth with $L_0$ \n\\begin{equation}\n    L_0=\\frac{L_{\\mathbf{w}\\boldsymbol{\\theta}}L_{\\boldsymbol{\\theta}\\mathbf{w}}}{\\mu}+L_{\\mathbf{w}\\mathbf{w}}.\n\\end{equation}\nIt holds that\n\\begin{equation}\n    \\begin{aligned}\n    & \\mathcal{L}_0(\\mathbf{w}_1)\\leq\\mathcal{L}_0(\\mathbf{w}_2)&+&\\ \\ \\ \\left\\langle\\nabla_{\\mathbf{w}} \\mathcal{L}_0(\\mathbf{w}_2),\\mathbf{w}_1-\\mathbf{w}_2\\right\\rangle\\\\\n    & &+&\\ \\ \\ \\frac{L_0}{2}\\|\\mathbf{w}_1-\\mathbf{w}_2\\|^2_2,\n    \\end{aligned}\n\\end{equation}\nand\n\\begin{equation}\n    \\left\\|\\nabla_{\\mathbf{w}} \\mathcal{L}_0-\\nabla_{\\mathbf{w}} \\mathcal{L}_0(\\mathbf{w}_2)\\right\\|_2\\leq L_0\\|\\mathbf{w}_1-\\mathbf{w}_2\\|_2.\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nBy the strongly-concavity of $\\mathcal{L}_0$ and given $\\boldsymbol{x}_n\\in\\mathcal{X}$, for any $\\boldsymbol{\\theta}_1$, $\\boldsymbol{\\theta}_2$ and the corresponding $\\mathbf{w}_1$, $\\mathbf{w}_2$ , we have\n\\begin{equation}\n    \\label{eq:30}\n    \\begin{aligned}\n        & &&\\ \\ \\mathcal{L}^n_0(\\boldsymbol{\\theta}_1,\\mathbf{w}_2)-\\mathcal{L}^n_0( \\boldsymbol{\\theta}_2,\\mathbf{w}_2)\\\\[5pt]\n        & &\\leq&\\ \\ \\big\\langle\\nabla_{\\boldsymbol{\\theta}} \\mathcal{L}^n_0(\\boldsymbol{\\theta}_2,\\mathbf{w}_2),\\boldsymbol{\\theta}_1-\\boldsymbol{\\theta}_2\\big\\rangle - \\frac{\\mu}{2}\\|\\boldsymbol{\\theta}_1-\\boldsymbol{\\theta}_2\\|_2^2\\\\[3pt]\n        & &\\leq&\\ \\ -\\frac{\\mu}{2}\\|\\boldsymbol{\\theta}_1-\\boldsymbol{\\theta}_2\\|_2^2.\n    \\end{aligned}    \n\\end{equation}\nThe second inequality is true as \n$$\n    \\langle\\nabla_{\\boldsymbol{\\theta}} \\mathcal{L}^n_0(\\boldsymbol{\\theta}_2,\\mathbf{w}_2),\\boldsymbol{\\theta}_1-\\boldsymbol{\\theta}_2\\rangle\\leq 0.\n$$\nIn addition, we have\n\\begin{equation}\n    \\label{eq:31}\n    \\begin{aligned}\n        & &&\\ \\ \\mathcal{L}^n_0(\\boldsymbol{\\theta}_2,\\mathbf{w}_2)-\\mathcal{L}^n_0( \\boldsymbol{\\theta}_1,\\mathbf{w}_2)\\\\[5pt]\n        & &\\leq&\\ \\ \\big\\langle\\nabla_{\\boldsymbol{\\theta}} \\mathcal{L}^n_0(\\boldsymbol{\\theta}_1,\\mathbf{w}_2),\\boldsymbol{\\theta}_2-\\boldsymbol{\\theta}_1\\big\\rangle - \\frac{\\mu}{2}\\|\\boldsymbol{\\theta}_1-\\boldsymbol{\\theta}_2\\|_2^2\\\\[3pt]\n        & &\\leq&\\ \\ -\\frac{\\mu}{2}\\|\\boldsymbol{\\theta}_1-\\boldsymbol{\\theta}_2\\|_2^2.\n    \\end{aligned}  \n\\end{equation}\nCombining \\eqref{eq:30} and \\eqref{eq:31}, we have\n\\begin{equation}\n    \\label{eq:32}\n    \\begin{aligned}\n        & & &\\ \\ \\mu\\|\\boldsymbol{\\theta}_1-\\boldsymbol{\\theta}_2\\|^2_2\\\\[5pt]\n        & &\\leq&\\ \\ \\langle\\nabla_{\\boldsymbol{\\theta}} \\mathcal{L}^n_0(\\boldsymbol{\\theta}_1,\\mathbf{w}_2),\\boldsymbol{\\theta}_2-\\boldsymbol{\\theta}_1\\rangle\\\\[5pt]\n        & &\\leq&\\ \\ \\langle\\nabla_{\\boldsymbol{\\theta}} \\mathcal{L}^n_0(\\boldsymbol{\\theta}_1,\\mathbf{w}_2)-\\nabla_{\\boldsymbol{\\theta}} \\mathcal{L}^n_0(\\boldsymbol{\\theta}_1,\\mathbf{w}_1),\\boldsymbol{\\theta}_2-\\boldsymbol{\\theta}_1\\rangle\\\\[5pt]\n        & &\\leq&\\ \\ \\|\\nabla_{\\boldsymbol{\\theta}} \\mathcal{L}^n_0(\\boldsymbol{\\theta}_1,\\mathbf{w}_2)-\\nabla_{\\boldsymbol{\\theta}} \\mathcal{L}^n_0(\\boldsymbol{\\theta}_1,\\mathbf{w}_1)\\|_2\\|\\boldsymbol{\\theta}_2-\\boldsymbol{\\theta}_1\\|_2\\\\[5pt]\n        & &\\leq&\\ \\ L_{\\boldsymbol{\\theta}\\mathbf{w}}\\|\\mathbf{w}_2-\\mathbf{w}_1\\|_2\\|\\boldsymbol{\\theta}_2-\\boldsymbol{\\theta}_1\\|_2,\n    \\end{aligned}\n\\end{equation}\nwhere the second inequality holds as\n$$\n    \\langle\\nabla_{\\boldsymbol{\\theta}} \\mathcal{L}^n_0(\\boldsymbol{\\theta}_1,\\mathbf{w}_1),\\boldsymbol{\\theta}_2-\\boldsymbol{\\theta}_1\\rangle\\leq 0,\n$$\nthe third inequality follows from the Cauchy-Schwarz inequality, and the last one holds by the Lipschitz smoothness of the gradients of $\\mathcal{L}_0$ \\eqref{eq:lipschitz_grad}. \n\nFor any $n\\in[N]$, we have\n\\begin{equation}\n    \\begin{aligned}\n       & & &\\ \\ \\|\\nabla_{\\mathbf{w}} \\mathcal{L}^n_0(\\boldsymbol{\\theta}_1,\\mathbf{w}_1)-\\nabla_{\\mathbf{w}} \\mathcal{L}^n_0(\\boldsymbol{\\theta}_2,\\mathbf{w}_2)\\|_2\\\\[5pt]\n       & &\\leq&\\ \\ \\|\\nabla_{\\mathbf{w}} \\mathcal{L}^n_0(\\boldsymbol{\\theta}_1,\\mathbf{w}_1)-\\nabla_{\\mathbf{w}} \\mathcal{L}^n_0(\\boldsymbol{\\theta}_2,\\mathbf{w}_1)\\|_2\\\\[5pt]\n       & & &\\ \\ +\\|\\nabla_{\\mathbf{w}} \\mathcal{L}^n_0(\\boldsymbol{\\theta}_2,\\mathbf{w}_1)-\\nabla_{\\mathbf{w}} \\mathcal{L}^n_0(\\boldsymbol{\\theta}_2,\\mathbf{w}_2)\\|_2\\\\[5pt]\n       & &\\leq&\\ \\ L_{\\mathbf{w}\\boldsymbol{\\theta}}\\|\\boldsymbol{\\theta}_1-\\boldsymbol{\\theta}_2\\|_2+L_{\\mathbf{w}\\mathbf{w}}\\|\\mathbf{w}_1-\\mathbf{w}_2\\|_2\\\\[5pt]\n       & &=&\\ \\ \\left(\\frac{L_{\\mathbf{w}\\boldsymbol{\\theta}}L_{\\boldsymbol{\\theta}\\mathbf{w}}}{\\mu}+L_{\\mathbf{w}\\mathbf{w}}\\right)\\|\\mathbf{w}_1-\\mathbf{w}_2\\|_2,\n    \\end{aligned}\n\\end{equation}\nwhere the first inequality follows from the triangle inequality, and the second inequality holds due to \\eqref{eq:32} and the Lipschitz smoothness of the gradients of $\\mathcal{L}_0$ \\eqref{eq:lipschitz_grad}. By the definition of $\\mathcal{L}$, it holds that\n\\begin{equation}\n    \\begin{aligned}\n        & & &\\ \\ \\|\\nabla_{\\mathbf{w}} \\mathcal{L}_0(\\mathbf{w}_1)-\\nabla_{\\mathbf{w}} \\mathcal{L}_0(\\mathbf{w}_2)\\|_2\\\\[5pt]\n        & &=&\\ \\ \\left\\|\\frac{1}{N}\\sum_{n=1}^N \\left(\\nabla_{\\mathbf{w}} \\mathcal{L}^n_0(\\boldsymbol{\\theta}_1,\\mathbf{w}_1)-\\nabla_{\\mathbf{w}} \\mathcal{L}^n_0(\\boldsymbol{\\theta}_2,\\mathbf{w}_2)\\right)\\right\\|_2\\\\\n        & &\\leq&\\ \\ \\frac{1}{N}\\sum_{n=1}^N\\|\\nabla_{\\mathbf{w}} \\mathcal{L}^n_0(\\boldsymbol{\\theta}_1,\\mathbf{w}_1)-\\nabla_{\\mathbf{w}} \\mathcal{L}^n_0(\\boldsymbol{\\theta}_2,\\mathbf{w}_2)\\|_2\\\\\n        & &\\leq&\\ \\ \\left(\\frac{L_{\\mathbf{w}\\boldsymbol{\\theta}}L_{\\boldsymbol{\\theta}\\mathbf{w}}}{\\mu}+L_{\\mathbf{w}\\mathbf{w}}\\right)\\|\\mathbf{w}_1-\\mathbf{w}_2\\|_2.\n    \\end{aligned}\n\\end{equation}\nWith the definition of the Lipschitz smoothness, we complete the proof.\n\\end{proof}\n\n\\begin{lemma}\n\\label{lem:2}\nSuppose that $\\mathcal{L}_0$ in \\eqref{eq:newAT_pro} satisfies the Lipschitz gradient conditions as \\eqref{eq:lipschitz_grad} and $\\mathcal{L}_0$ is $\\mu$-strongly concave in $\\boldsymbol{\\Theta}$, the approximate stochastic gradient $\\nabla_{\\mathbf{w}}\\hat{\\ell}(\\mathbf{w})$ \\eqref{eq:grad_3} satisfies\n\\begin{equation}\n    \\|\\nabla_{\\mathbf{w}}\\hat{\\ell}~(\\mathbf{w})-\\nabla_{\\mathbf{w}}\\ell(\\mathbf{w})\\|_2\\leq L_{\\mathbf{w}\\boldsymbol{\\theta}}\\sqrt{\\frac{\\delta}{\\mu}},\n\\end{equation}\nwhere $\\boldsymbol{\\hat{x}}_{\\text{adv}}(\\boldsymbol{x},\\mathbf{w})$ is a $\\delta$-approximate solution to $\\boldsymbol{x}^*_{\\text{adv}}(\\boldsymbol{x},\\mathbf{w})$ with given $\\boldsymbol{x}\\in\\mathcal{X}$.\n\\end{lemma}\n\\begin{proof}\nBy the definitions of $\\nabla_{\\mathbf{w}}\\hat{\\ell}$ and $\\nabla_{\\mathbf{w}}\\ell$, we have\n\\begin{equation}\n    \\label{eq:36}\n    \\begin{aligned}\n    & &&\\ \\ \\|\\nabla_{\\mathbf{w}}\\hat{\\ell}~(\\mathbf{w})-\\nabla_{\\mathbf{w}}\\ell(\\mathbf{w})\\|_2\\\\[5pt]\n    & &=&\\ \\ \\left\\|\\frac{1}{|\\mathcal{B}|} \\sum_{i=1}^{|\\mathcal{B}|} (\\nabla_{\\mathbf{w}} \\mathcal{L}_0(\\boldsymbol{\\hat{x}}_{\\text{adv}}(\\boldsymbol{x}_i,\\mathbf{w}),\\mathbf{w})\\right.\\\\\n    & & &\\ \\ \\left.\\textcolor{white}{\\frac{1}{|\\mathcal{B}|} \\sum_{i=1}^{|\\mathcal{B}|}}-\\nabla_{\\mathbf{w}}\\mathcal{L}_0(\\boldsymbol{x}^*_{\\text{adv}}(\\boldsymbol{x}_i,\\mathbf{w}),\\mathbf{w}))\\right\\|_2\\\\\n    & &\\leq&\\ \\ \\frac{1}{|\\mathcal{B}|} \\sum_{n=1}^N \\left\\|\\nabla_{\\mathbf{w}} \\mathcal{L}_0(\\boldsymbol{\\hat{x}}_{\\text{adv}}(\\boldsymbol{x}_i,\\mathbf{w}),\\mathbf{w})\\right.\\\\\n    & & &\\ \\ \\textcolor{white}{\\frac{1}{|\\mathcal{B}|} \\sum_{i=1}^{|\\mathcal{B}|}}\\left.-\\nabla_{\\mathbf{w}}\\mathcal{L}_0(\\boldsymbol{x}^*_{\\text{adv}}(\\boldsymbol{x}_i,\\mathbf{w}),\\mathbf{w})\\right\\|_2\\\\[5pt]\n    & &\\leq&\\ \\ \\frac{1}{|\\mathcal{B}|} \\sum_{i=1}^{|\\mathcal{B}|} L_{\\mathbf{w}\\boldsymbol{\\theta}}\\|\\boldsymbol{\\hat{\\theta}}-\\boldsymbol{\\theta}^*\\|_2,\n    \\end{aligned}\n\\end{equation}\nwhere the second inequality follows from the triangle inequality, the third inequality holds due to the gradient Lipschitz condition, and $\\boldsymbol{\\hat{\\theta}}$ is the parameter of strategy network corresponding to $\\boldsymbol{\\hat{x}}_{\\text{adv}}(\\boldsymbol{x}_i,\\mathbf{w})$, $\\boldsymbol{\\theta}^*$ is similar.\n\nSince $\\boldsymbol{\\hat{x}}_{\\text{adv}}(\\boldsymbol{x}_i,\\mathbf{w})$ is a $\\delta$-approximate adversarial example generated by the strategy network, we have\n\\begin{equation}\n    \\label{eq:37}\n    \\left\\langle\\boldsymbol{\\theta}^*-\\boldsymbol{\\hat{\\theta}}, \\nabla_{\\boldsymbol{\\theta}} \\mathcal{L}_0(\\boldsymbol{\\hat{\\theta}},\\mathbf{w})\\right\\rangle\\leq \\delta.\n\\end{equation}\nIn addition, it holds that\n\\begin{equation}\n    \\label{eq:38}\n    \\left\\langle\\boldsymbol{\\hat{\\theta}}-\\boldsymbol{\\theta}^*, \\nabla_{\\boldsymbol{\\theta}} \\mathcal{L}_0(\\boldsymbol{\\theta}^*,\\mathbf{w})\\right\\rangle\\leq 0.\n\\end{equation}\nPutting \\eqref{eq:37} and \\eqref{eq:38} together gives birth to\n\\begin{equation}\n    \\label{eq:39}\n    \\left\\langle\\boldsymbol{\\hat{\\theta}}-\\boldsymbol{\\theta}^*,\\nabla_{\\boldsymbol{\\theta}} \\mathcal{L}_0(\\boldsymbol{\\theta}^*,\\mathbf{w})-\\nabla_{\\boldsymbol{\\theta}} \\mathcal{L}_0(\\boldsymbol{\\hat{\\theta}},\\mathbf{w})\\right\\rangle\\leq \\delta.\n\\end{equation}\nMoreover, by the strongly concavity of $\\mathcal{L}_0$ and \\eqref{eq:32}, we have\n\\begin{equation}\n    \\label{eq:40}\n    \\begin{aligned}\n        & & &\\ \\ \\mu\\|\\boldsymbol{\\theta}^*-\\boldsymbol{\\hat{\\theta}}\\|^2_2\\\\[2.5pt]\n        & &\\leq&\\ \\ \\langle\\nabla_{\\boldsymbol{\\theta}}\\mathcal{L}^n_0(\\boldsymbol{\\theta}^*,\\mathbf{w})-\\nabla_{\\boldsymbol{\\theta}} \\mathcal{L}^n_0(\\boldsymbol{\\hat{\\theta}},\\mathbf{w}),\\boldsymbol{\\hat{\\theta}}-\\boldsymbol{\\theta}^*\\rangle\\\\[5pt]\n        & &\\leq&\\ \\ \\delta.\n    \\end{aligned}\n\\end{equation}\nConsequently, it immediately yields\n\\begin{equation}\n    \\label{eq:41}\n    \\|\\boldsymbol{\\theta}^*-\\boldsymbol{\\hat{\\theta}}\\|_2\\leq\\sqrt{\\frac{\\delta}{\\mu}}.\n\\end{equation}\nSubstituting \\eqref{eq:41} into \\eqref{eq:36}, we complete the proof.\n\\end{proof}\n\n\\convergence*\n\\begin{proof}\nBy Lemma \\ref{lem:1}, we have\n\\begin{equation*}\n    \\begin{aligned}\n        & \\mathcal{L}_0(\\mathbf{w}^{t+1})&\\leq&\\ \\ \\mathcal{L}_0(\\mathbf{w}^t)+\\frac{L_0}{2}\\|\\mathbf{w}^{t+1}-\\mathbf{w}^{t}\\|^2_2\\\\[5pt]\n        & &+&\\ \\ \\left\\langle\\nabla_{\\mathbf{w}} \\mathcal{L}_0(\\mathbf{w}^t),\\mathbf{w}^{t+1}-\\mathbf{w}^{t}\\right\\rangle.\n    \\end{aligned}    \n\\end{equation*}\nDue to\n\\[\n    \\mathbf{w}^{t+1}=\\mathbf{w}^{t}-\\eta_t\\nabla_{\\mathbf{w}}\\hat{\\ell}(\\mathbf{w}^{t}),\n\\]\nit holds that\n\\begin{equation}\n    \\begin{aligned}\n        & & &\\ \\ \\mathcal{L}_0(\\mathbf{w}^{t+1})\\\\[5pt]\n        & &\\leq&\\ \\ \\mathcal{L}_0(\\mathbf{w}^t)-\\eta_t\\|\\nabla_{\\mathbf{w}}\\mathcal{L}_0(\\mathbf{w}^t)\\|_2^2+\\frac{L_0\\eta_t^2}{2}\\|\\nabla_{\\mathbf{w}}\\hat{\\ell}(\\mathbf{w}^{t})\\|_2^2\\\\[5pt]\n        & & &\\ \\ +\\eta_t\\langle\\nabla_{\\mathbf{w}}\\mathcal{L}_0(\\mathbf{w}^t),\\nabla_{\\mathbf{w}}\\mathcal{L}_0(\\mathbf{w}^t)-\\nabla_{\\mathbf{w}}\\hat{\\ell}(\\mathbf{w}^{t})\\rangle\\\\[5pt]\n        & &=&\\ \\ \\mathcal{L}_0(\\mathbf{w}^t)-\\eta_t\\left(1-\\frac{L_0\\eta_t}{2}\\right)\\|\\nabla_{\\mathbf{w}}\\mathcal{L}_0(\\mathbf{w}^t)\\|_2^2\\\\[5pt]\n        & & &\\ \\ +\\eta_t\\left(1-\\frac{L_0\\eta_t}{2}\\right)\\langle\\nabla_{\\mathbf{w}}\\mathcal{L}_0(\\mathbf{w}^t),\\nabla_{\\mathbf{w}}\\mathcal{L}_0(\\mathbf{w}^t)-\\nabla_{\\mathbf{w}}\\hat{\\ell}(\\mathbf{w}^{t})\\rangle\\\\[5pt]\n        & & &\\ \\ + \\frac{L_0\\eta_t^2}{2}\\|\\nabla_{\\mathbf{w}}\\hat{\\ell}(\\mathbf{w}^{t})-\\nabla_{\\mathbf{w}}\\mathcal{L}_0(\\mathbf{w}^t)\\|_2^2\\\\[5pt]\n        & &=&\\ \\ \\mathcal{L}_0(\\mathbf{w}^t)-\\eta_t\\left(1-\\frac{L_0\\eta_t}{2}\\right)\\|\\nabla_{\\mathbf{w}}\\mathcal{L}_0(\\mathbf{w}^t)\\|_2^2\\\\[5pt]\n        & & &\\ \\ +\\eta_t\\left(1-\\frac{L_0\\eta_t}{2}\\right)\\langle\\nabla_{\\mathbf{w}}\\mathcal{L}_0(\\mathbf{w}^t),\\nabla_{\\mathbf{w}}\\ell(\\mathbf{w}^{t})-\\nabla_{\\mathbf{w}}\\hat{\\ell}(\\mathbf{w}^{t})\\rangle\\\\[5pt]\n        & & &\\ \\ +\\eta_t\\left(1-\\frac{L_0\\eta_t}{2}\\right)\\langle\\nabla_{\\mathbf{w}}\\mathcal{L}_0(\\mathbf{w}^t),\\nabla_{\\mathbf{w}}\\mathcal{L}_0(\\mathbf{w}^t)-\\nabla_{\\mathbf{w}}\\ell(\\mathbf{w}^{t})\\rangle\\\\[5pt]\n        & & &\\ \\ +\\frac{L_0\\eta_t^2}{2}\\|\\nabla_{\\mathbf{w}}\\hat{\\ell}(\\mathbf{w}^{t})-\\nabla_{\\mathbf{w}}\\ell(\\mathbf{w}^{t})+\\nabla_{\\mathbf{w}}\\ell(\\mathbf{w}^{t})-\\nabla_{\\mathbf{w}}\\mathcal{L}_0(\\mathbf{w}^t)\\|_2^2\\\\\n        & &\\leq&\\ \\ \\mathcal{L}_0(\\mathbf{w}^t)-\\frac{\\eta_t}{2}\\left(1-\\frac{L_0\\eta_t}{2}\\right)\\|\\nabla_{\\mathbf{w}}\\mathcal{L}_0(\\mathbf{w}^t)\\|_2^2\\\\[5pt]\n        & & &\\ \\ +\\frac{\\eta_t}{2}\\left(1-\\frac{L_0\\eta_t}{2}\\right)\\|\\nabla_{\\mathbf{w}}\\ell(\\mathbf{w}^{t})-\\nabla_{\\mathbf{w}}\\hat{\\ell}(\\mathbf{w}^{t})\\|^2_2\\\\[5pt]\n        & & &\\ \\ +\\eta_t\\left(1+\\frac{L_0\\eta_t}{2}\\right)\\langle\\nabla_{\\mathbf{w}}\\mathcal{L}_0(\\mathbf{w}^t),\\nabla_{\\mathbf{w}}\\mathcal{L}_0(\\mathbf{w}^t)-\\nabla_{\\mathbf{w}}\\ell(\\mathbf{w}^{t})\\rangle\\\\[5pt]\n        & & &\\ \\ +L_0\\eta_t^2\\|\\nabla_{\\mathbf{w}}\\hat{\\ell}(\\mathbf{w}^{t})-\\nabla_{\\mathbf{w}}\\ell(\\mathbf{w}^{t})\\|_2^2\\\\[5pt]\n        & & &\\ \\ +L_0\\eta_t^2\\|\\nabla_{\\mathbf{w}}\\ell(\\mathbf{w}^{t})-\\nabla_{\\mathbf{w}}\\mathcal{L}_0(\\mathbf{w}^t)\\|_2^2\\\\\n    \\end{aligned}\n\\end{equation}\nTaking expectation on both sides of the above inequality conditioned on $\\mathbf{w}^t$, then we have\n\\begin{equation}\n    \\begin{aligned}\n        & & &\\ \\ \\mathbb{E}[\\mathcal{L}_0(\\mathbf{w}^{t+1})-\\mathcal{L}_0(\\mathbf{w}^{t})|\\mathbf{w}^t]\\\\[5pt]\n        & &\\leq&\\ \\ -\\frac{\\eta_t}{2}\\left(1-\\frac{L_0\\eta_t}{2}\\right)\\|\\nabla_{\\mathbf{w}}\\mathcal{L}_0(\\mathbf{w}^t)\\|_2^2\\\\[5pt]\n        & & &\\ \\ +\\frac{\\eta}{2}\\left(1+\\frac{3\\eta_tL_0}{2}\\right)\\frac{\\delta L^2_{\\mathbf{w}\\boldsymbol{\\theta}}}{\\mu}+L_0\\eta^2_t\\sigma^2.\n    \\end{aligned}\n\\end{equation}\nThen we do the telescope sum over $t=0,\\dots,T-1$, we obtain\n\\begin{equation}\n    \\begin{aligned}\n        & & &\\ \\ \\sum_{t=0}^{T-1}\\frac{\\eta_t}{2}\\left(1-\\frac{L_0\\eta_t}{2}\\right)\\mathbb{E}[\\|\\mathcal{L}_0(\\mathbf{w}^{t})\\|^2_2]\\\\[5pt]\n        & &\\leq&\\ \\ \\mathbb{E}[\\mathcal{L}_0(\\mathbf{w}^0)-\\mathcal{L}_0(\\mathbf{w}^T)]+L_0\\sum_{t=0}^{T-1}\\eta^2_t\\sigma^2\\\\[5pt]\n        & & &\\ \\ + \\sum_{t=0}^{T-1}\\frac{\\eta}{2}\\left(1+\\frac{3\\eta_tL_0}{2}\\right)\\frac{\\delta L^2_{\\mathbf{w}\\boldsymbol{\\theta}}}{\\mu}.\n    \\end{aligned}\n\\end{equation}\nChoosing $\\eta_t= \\eta_1$ as\n\\begin{equation}\n    \\eta_1 = \\min\\left(\\frac{1}{L_0},\\ \\sqrt{\\frac{\\mathcal{L}_0(\\mathbf{w}^0)-\\underset{\\mathbf{w}}{\\min}\\ \\mathcal{L}_0(\\mathbf{w})}{\\sigma^2TL_0}}\\right),\n\\end{equation}\nit holds that\n\\begin{equation}\n    \\frac{1}{T}\\sum_{t=0}^{T-1}\\mathbb{E}\\big[\\|\\nabla \\mathcal{L}_0(\\mathbf{w}^t)\\|^2_2\\big]\\leq4\\sigma\\sqrt{\\frac{\\Delta L_0}{T}}+\\frac{5\\delta L^2_{\\mathbf{w}\\boldsymbol{\\theta}}}{\\mu},\n\\end{equation}\nwhere $\\Delta=\\mathcal{L}_0(\\mathbf{w}^0)-\\underset{\\mathbf{w}}{\\min}\\ \\mathcal{L}_0(\\mathbf{w})$.\n\\end{proof}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Incompressible setting and results}\n\\label{sec:4.2.5}\nLet $B\\ss \\mathbb{R}^2$ be the unit ball. For $u_0\\in L^p(B,\\mathbb{R}^2)$ with $2\\le p<\\infty$ and $\\det\\ensuremath{\\nabla} u_0=1$ a.e. in $B$ we define\n\\[\n\\mathcal{A}_{u_0}^{p,c}:=\\{u\\in W^{1,p}(B,\\mathbb{R}^2): \\det\\ensuremath{\\nabla} u=1 \\;\\mb{a.e. in}\\;B ,\\;u-u_0\\in C_c^\\infty(B,\\mathbb{R}^2)\\}\n\\]\nand for every $u\\in \\mathcal{A}_{u_0}^{p,c}$ we define\n\\begin{equation}E(u)=\\int\\limits_B{f(x,\\ensuremath{\\nabla} u)\\;dx}\\label{eq:HFU.I.2.1},\\end{equation}\nwhere $f$ is given by\n\\[f(x,\\xi)=\\nu(x)|\\xi|^p \\]\nfor a.e. $x\\in B$ and $\\xi\\in \\mathbb{R}^{2\\ti2}.$ \nMoreover,  $\\nu\\in L^\\infty(B)$ is supposed to satisfy $\\nu(x)\\ge0$ a.e.\\! in $B.$ Since $\\nu$ is allowed to take on the value $0,$ the integrand could indeed disappear for some $x \\in B.$ Additionally, $f(x,\\ensuremath{\\cdot})$ is convex in its 2nd variable for a.e. $x\\in B.$ All of the above, is making $E$ a version of the $p-$Dirichlet functional. \\vspace{0.5cm}\n\nAs usual, we are interested in the  corresponding minimization problem\n\\begin{equation}\n\\inf\\limits_{u\\in\\mathcal{A}_{u_0}^{p,c}} E(u).\n\\label{eq:USPS:1.1}\n\\end{equation}\nNotice, that the missing uniformity might cause troubles guaranteeing the existence of a minimizer, however, this is without any consequence for this work.\nAssuming for now, that the situation is such that the minimum is indeed obtained and there are some corresponding minimizing maps or more generally corresponding stationary points of $E,$ which are defined as follows:\n\n\\begin{de}(Stationary point)\nWe say that $u$ is a stationary point of $E(\\cdot)$ if there exists a function $\\ensuremath{\\lambda}$, which we shall henceforth refer to as a pressure, belonging to $W^{1,1}(B)$ and such that\n\\begin{align}\\label{def:SP1} \\div( \\nabla_{\\xi}f(x,\\nabla u) + p \\ensuremath{\\lambda}(x) \\, \\textnormal{cof}\\; \\nabla u) = 0 \\quad \\mathrm{in} \\ \\mathcal{D}'(B).\n\\end{align}\n\\end{de}\n\n\n Now we can state the main result for the incompressible scenario. Recall, that for any vector $y=y_Re_R+y_\\th e_\\th\\in\\mathbb{R}^2$ we define its maximum norm via $|y|_{\\infty}:=\\max\\{y_R,y_\\th\\}.$\n\n\\begin{thm} [High frequency uniqueness] Let $2\\le p<\\infty,$ assume $u_0\\in L^p(B,\\mathbb{R}^2)$ to be the boundary conditions and let $u\\in \\mathcal{A}_{u_0}^{p,c}$ be a stationary point of $E,$ as given in \\eqref{def:SP1}. \nFurthermore, let $\\sigma(x):=\\sqrt{\\nu(x)|\\ensuremath{\\nabla} u(x)|^{p-2}}\\in L^{\\frac{4}{p-2}}(B)$ and assume that there exists $l\\in \\mathbb{N}$ s.t.\n\\begin{equation}|\\sigma,_\\th(x)|\\le l \\sigma(x) \\mb{for a.e.}\\;x\\in B\\label{eq:HFU.UC.G.01} \\end{equation}\nholds.\\\\\n\nThen the following statements are true:\n\ni) \\textbf{(purely high modes.)}  Suppose the corresponding pressure $\\ensuremath{\\lambda}$ exists and satisfies\n\\begin{equation}|\\ensuremath{\\nabla} \\ensuremath{\\lambda}(x)R|_{\\infty}\\le \\frac{n}{\\sqrt{2}}\\nu(x)|\\ensuremath{\\nabla} u|^{p-2}\\mb{for a.e.}\\;x\\in B\\label{eq:HFU.IC.G.02}\\end{equation}\nfor some $n\\in\\mathbb{N}.$\\\\\n\nThen $u$ is a minimizer of $E$ in the subclass \n\\[\\mathcal{F}_{n_*}^{p,\\sigma,c}=\\left\\{v\\in \\mathcal{A}_{u_0}^{p,c}|\\eta=v-u\\in W_0^{1,p}(B,\\mathbb{R}^2) \\;\\mb{and}\\; \\sigma\\eta=\\sum\\limits_{j\\ge n}(\\sigma\\eta)^{(j)} \\right\\},\\]\nwhere $n_*:=n+l.$\nMoreover, if there exists a constant $\\sigma_0>0$ s.t. $\\sigma(x)\\ge\\sigma_0>0$ for any $x\\in B$ and inequality \\eqref{eq:HFU.IC.G.02} is strictly satisfied on a non-trivial set, then $u$ is the unique minimizer in $\\mathcal{F}_{n_*}^{p,\\sigma,c}$.\\\\\n\nii) \\textbf{($0-$mode and high modes.)} Suppose the corresponding pressure $\\ensuremath{\\lambda}$ exists and satisfies\n\\begin{equation}|\\ensuremath{\\nabla} \\ensuremath{\\lambda}(x)R|_{\\infty}\\le \\frac{\\sqrt{3}m\\nu(x)|\\ensuremath{\\nabla} u|^{p-2}}{2\\sqrt{2}}\\mb{for a.e.}\\;x\\in B\\label{eq:HFU.IC.G.03}\\end{equation}\nfor some $m\\in\\mathbb{N}.$\\\\\nThen $u$ is a minimizer of $E$ in the subclass \n\\[\\mathcal{F}_{0,m_*}^{p,\\sigma,c}=\\left\\{v\\in \\mathcal{A}_{u_0}^{p,c}|\\eta=v-u\\in W_0^{1,p}(B,\\mathbb{R}^2) \\;\\mb{and}\\; \\sigma\\eta=(\\sigma\\eta)^{(0)}+\\sum\\limits_{j\\ge m_*}(\\sigma\\eta)^{(j)} \\right\\},\\]\nwhere $m_*=m+l.$\nMoreover, if there exists a constant $\\sigma_0>0$ s.t. $\\sigma(x)\\ge\\sigma_0>0$ for any $x\\in B$ and inequality \\eqref{eq:HFU.IC.G.03} is is strictly satisfied on a non-trivial set, then $u$ is the unique minimizer in $\\mathcal{F}_{0,m_*}^{p,\\sigma,c}$.\n\\label{thm1.1}\n\\end{thm}\\vspace{0.5cm}\n\n\\section{Compressible setting and results}\n For $u_0\\in L^p(B,\\mathbb{R}^2)$ with $2\\le p\\le\\infty$ we define the set of admissible maps by\n\\[\n\\mathcal{A}_{u_0}^p=\\{u\\in W^{1,p}(B,\\mathbb{R}^2): \\; u-u_0\\in C_c^\\infty(B,\\mathbb{R}^2)\\}.\n\\]\nHere we consider energies given by\n\\begin{equation}I(u)=\\int\\limits_B{\\Phi(x,\\ensuremath{\\nabla} u)\\;dx},\\label{eq:HFU.UC.G.0}\\end{equation}\nwhere the integrand is of the form\n\\begin{equation}\\Phi(x,\\xi)=\\frac{\\nu(x)}{p}|\\xi|^p+\\Psi(x,\\xi,\\det\\xi),\\label{eq:HFU.UC.GA}\\end{equation}\nNow we want $(\\xi,d)\\mapsto \\Psi(x,\\xi,d)$ to be convex for a.e.\\! $x \\in B,$ making $\\Psi(x,\\ensuremath{\\cdot})$ a convex representative of a polyconvex function a.e. in $B$. \nMoreover, the function $\\nu\\in L^\\infty(B),$ satisfying $\\nu(x)\\ge0$ a.e.\\! in $B,$ is ought to be optimal, in the sense, that there can be no term of the form  $a(x)|\\xi|^q$ for any $q\\ge p$ in $\\Psi.$ However, $\\Psi$ might be negative. Finally, we want $\\Phi$ to be of $p-$growth,  supposing that there exits $C\\in L^\\infty(B)$ with $C(x)\\ge0$ a.e. in $B$ s.t.\n\\[0\\le \\Phi(x,\\xi)\\le \\frac{C(x)}{p}(1+|\\xi|^p)\\;\\mb{for all} \\;\\xi\\in \\mathbb{R}^{2\\ti2}\\;\\mb{and a.e.} \\; x\\in B. \\]\nAll of the above combined guarantees that $x\\mapsto \\Phi(x,\\ensuremath{\\nabla} u(x))\\in L^1(B)$ for any $u\\in W^{1,p}(B,\\mathbb{R}^2)$ and hence the corresponding energy is finite. \\vspace{0.5cm}\n\nThe corresponding minimization problem\n\\begin{equation}\n\\inf\\limits_{u\\in\\mathcal{A}_{u_0}^p} I(u)\n\\label{eq:USPS:1.1}\n\\end{equation}\nmight lack a minimizer in general. Assuming, that such a minimizer\/stationary point of $I$ exists, motivates the following definition: \n\\begin{de}(Stationary point)\nWe say that $u\\in\\mathcal{A}_{u_0}^p$ is a stationary point of $I(\\cdot)$ if $u$ satisfies the Euler-Lagrange equation given by\n\\begin{align}\\label{def:SP.2}\n\\div( \\nu(x)|\\ensuremath{\\nabla} u|^{p-2}\\ensuremath{\\nabla} u+\\ensuremath{\\partial}_\\xi\\Psi(x,\\ensuremath{\\nabla} u,d_{\\ensuremath{\\nabla} u})+\\ensuremath{\\partial}_d\\Psi(x,\\ensuremath{\\nabla} u,d_{\\ensuremath{\\nabla} u})\\textnormal{cof}\\;\\ensuremath{\\nabla} u) = 0\\; \\mathrm{in} \\;\\mathcal{D}'(B).\n\\end{align}\n\\end{de} \\vspace{-0.3cm}\nNow we can give an analogous result for these types of integrands.\n\n \n\\begin{thm} [High frequency uniqueness] Let $B\\ss \\mathbb{R}^2$ be the unit ball. Furthermore, let $2\\le p\\le\\infty,$ assume $u_0\\in L^p(B,\\mathbb{R}^2)$ to be the boundary conditions and let $u\\in \\mathcal{A}_{u_0}^p$ be a stationary point of $I,$ as given in \\eqref{def:SP.2}. Furthermore, let $\\sigma(x):=\\sqrt{\\nu(x)|\\ensuremath{\\nabla} u(x)|^{p-2}}\\in L^{\\frac{4}{p-2}}(B)$ and assume that there exists $l\\in \\mathbb{N}$ s.t.\n\\begin{equation}|\\sigma,_\\th(x)|\\le l \\sigma(x) \\mb{for a.e.}\\;x\\in B\\label{eq:HFU.UC.G.01} \\end{equation}\nholds.\\\\\n\nThen the following statements are true:\n\ni) \\textbf{(purely high modes.)}  Assume there exists $n\\in\\mathbb{N}$ s.t.\n\\begin{equation}|\\ensuremath{\\nabla}_x \\ensuremath{\\partial}_d\\Psi(x,\\ensuremath{\\nabla} u(x),d_{\\ensuremath{\\nabla} u(x)})R|_{\\infty}\\le \\frac{n}{\\sqrt2}\\nu(x)|\\ensuremath{\\nabla} u|^{p-2}\\mb{for a.e.}\\;x\\in B.\\label{eq:HFU.UC.G.1} \\end{equation}\n\nThen $u$ is a minimizer of $E$ in the subclass \n\\[\\mathcal{F}_{n_*}^{p,\\sigma}=\\left\\{v\\in \\mathcal{A}_{u_0}^p|\\;\\eta=v-u\\in W_0^{1,p}(B,\\mathbb{R}^2) \\;\\mb{and}\\; \\sigma\\eta=\\sum\\limits_{j\\ge {n_*}}(\\sigma\\eta)^{(j)} \\right\\},\\]\nwhere $n_*:=n+l.$\nMoreover, if there exists a constant $\\sigma_0>0$ s.t. $\\sigma(x)\\ge\\sigma_0>0$ for any $x\\in B$ and inequality \\eqref{eq:HFU.UC.G.1} is strictly satisfied on a non-trivial set, then $u$ is the unique minimizer in $\\mathcal{F}_{n_*}^{p,\\sigma}$.\\\\\n\nii) \\textbf{($0-$mode and high modes.)} Assume there exists $m\\in\\mathbb{N}$ s.t.\n\\begin{equation}|\\ensuremath{\\nabla}_x \\ensuremath{\\partial}_d\\Psi(x,\\ensuremath{\\nabla} u(x),d_{\\ensuremath{\\nabla} u(x)})R|_{\\infty}\\le \\frac{\\sqrt{3}m\\nu(x)|\\ensuremath{\\nabla} u|^{p-2}}{2\\sqrt{2}}\\mb{for a.e.}\\;x\\in B.\\label{eq:HFU.UC.G.2}\\end{equation}\nThen $u$ is a minimizer of $E$ in the subclass \n\\[\\mathcal{F}_{0,m_*}^{p,\\sigma}=\\left\\{v\\in \\mathcal{A}_{u_0}^p|\\;\\eta=v-u\\in W_0^{1,p}(B,\\mathbb{R}^2) \\;\\mb{and}\\; \\sigma\\eta=(\\sigma\\eta)^{(0)}+\\sum\\limits_{j\\ge m_*}(\\sigma\\eta)^{(j)} \\right\\},\\]\nwhere $m_*:=m+l.$\nMoreover, if there exists a constant $\\sigma_0>0$ s.t. $\\sigma(x)\\ge\\sigma_0>0$ for any $x\\in B$ and inequality \\eqref{eq:HFU.UC.G.2} is strictly satisfied on a non-trivial set, then $u$ is the unique minimizer in $\\mathcal{F}_{0,m_*}^{p,\\sigma}$.\n\\label{thm:HFU.UC.G.1}\n\\end{thm}\n\n\nHere, we contribute to John Ball's objectives, as formulated in \\cite[\u00a7 2.6]{BallOP}, which tries to get an understanding of questions related to uniqueness in elastic situations. We recently started this discussion in \\cite{D1}, where we presented a uniqueness criteria, in the incompressible case\\footnote{Recall, that we call situations, where the energies remain finite either incompressible elastic, if the considered admissible maps must be measure-preserving, otherwise we call it compressible elastic. Moreover, we call a model (fully) non-linear elastic, if the considered integrand $f,$ ignoring any other dependencies, satisfies $f(\\xi)=+\\infty$ for any $\\xi\\in \\mathbb{R}^{n\\ti n}$ s.t. $\\det\\xi\\le 0$ and $f(\\xi)\\ensuremath{\\rightarrow}+\\infty$ if $\\det\\xi\\ensuremath{\\rightarrow} 0^+$ or $\\det\\xi\\ensuremath{\\rightarrow} +\\infty.$ Notice, that in the latter model some of the consider energies might be infinite.}, for quadratic uniformly convex integrands $f(x,\\xi)$ and subject to suitable boundary conditions. Here we go beyond the latter first by allowing the more general $p-$Dirichtlet integrands but also by discussing high-pressure situations, where uniqueness can only be guaranteed for a subclass of variations which consist of purely high-modes. We also provide the analogous results for polyconvex-type functionals, which are of $p-$growth, in the compressible case. The work, which is certainly most relevant to ours is the one by Sivaloganathan and Spector \\cite{SS18}. In particular, in \\cite[Theorem 4.2]{SS18} they consider the same functionals $I$, as given in (\\ref{eq:HFU.UC.G.0}-\\ref{eq:HFU.UC.GA}), however, on a suitable class of admissible maps satisfying the constraint $\\det\\ensuremath{\\nabla} u>0 \\;\\mb{a.e.}$ and subject to suitable boundary data. Assuming then that $u$ is a weak solution satisfying the corresponding equilibrium equation and the condition given by\n\n\\begin{equation}|\\ensuremath{\\partial}_d\\Psi(x,\\ensuremath{\\nabla} u,d_{\\ensuremath{\\nabla} u})R|\\le \\nu(x)|\\ensuremath{\\nabla} u|^{p-2}\\mb{for a.e.}\\;x\\in \\Omega.\n\\label{eq:Uni.SivSP.1}\n\\end{equation}\nThen $u$ must be a global minimizer of $I.$ Additionally, if the latter inequality is in some sense strictly satisfied then $u$ must be the unique one.  It is crucial to realise, that the criterion given by Sivaloganathan and Spector differs from ours. Indeed, \\eqref{eq:Uni.SivSP.1} involves a 1st order derivative of $\\Psi$ while in \\eqref{eq:HFU.UC.G.2} a 2nd order derivative is used.  \\\\\n \nA possible incomplete list of results regarding the compressible setting might look as follows: Initially, it is well known, that uniformly convex functionals possess unique global minimizers, see for instance \\cite[\u00a7 3.3]{Kl16}. Knorps and Stuart showed in \\cite{KS84}, that for a strongly quasiconvex integrand defined on a star-shaped domain and subject to linear boundary data $u_0=Ax,$ any $C^2-$stationary point needs to agree with $Ax$ everywhere. A generalisation can be found in \\cite{T03}. These results have been transferred to the incompressible by Shahrokhi-Dehkordi and Taheri  \\cite{ST10} and the fully non-linear case by Bevan \\cite{B11}. In \\cite{C14}, Cordero presented a uniqueness result guaranteeing a unique minimizer for strongly quasiconvex $C^2-$integrands if the given boundary data is smooth and small enough.\n Zhang \\cite{Z91} discusses situations, in in- and in compressible ones, where the considered energies are polyconvex and the under pure displacement boundary conditions and subject to that the Jacobian must be strictly positive a.e. Then the corresponding minimizer must agree with the solution of the corresponding Euler-Lagrange equation, which is highly non-trivial due to the weak spaces involved and the lack of compactness of the constraint.\n On the contrary, non-uniqueness for minimizers of strongly polyconvex functionals has been established by Spadaro in \\cite{S08}. However, these counterexamples rely highly on the fact that the determinant can take on negative values, which is neither possible in the incompressible nor in the NLE-stetting. John \\cite{J72} and Spector and Spector \\cite{SS19} obtain uniqueness of equilibrium solutions for small enough strains and under various boundary conditions. In sharp contrast, Post and Sivaloganathan \\cite{PS97} construct multiple equilibrium solutions in finite elasticity. \\\\\n\nA first treatment of uniqueness in incompressible elasticity can be found in \\cite[Section 6]{KS84}. Much research in the incompressible setting is concerned with the double covering problem, that is, given the Dirichlet Energy $\\mathbb{D}(\\xi)= |\\xi|^2\/2$ on the unit ball in $2d$ and subject to double covering boundary conditions given by $u_2=(\\cos(2\\th),\\sin(2\\th)),$ first considered by Ball \\cite{B77}.  Since then a lot of progress towards a solution has been made and partial results are available. For instance, Bevan \\cite{JB14} showed that $u_2$ is the unique global minimizer up to the first Fourier-mode. This paper is central for several reasons, firstly, it introduces the type of uniqueness argument we provide here and in \\cite{D1}. Moreover, it also contains the concept of high-frequency uniqueness. In \\cite{BeDe21} Bevan and Deane obtained that $u_2$ is the unique global minimizer for either purely inner \nor purely outer variations and local minimality is shown for a subclass of variations allowing a certain mixture of both types. In contrast, in \\cite{BeDe20}, equal energy stationary points of an inhomogeneous uniformly convex functional $(x,\\xi)\\mapsto f(x,\\xi)$ depending discontinuously on $x$ are constructed. It remains unknown for now if these stationary points are actually global minimizers.\\\\\n\n\nIn the fully non-linear case Bevan and Yan \\cite{BY07} show, that the famous BOP$-$map, constructed by Bauman, Owen and Phillips \\cite{BOP91MS}, is the unique global minimizer in a suitable sub-class of admissible maps. \\\\\n\nFinally, many papers address uniqueness questions, in situations where the reference domain agrees with an annulus, see for instance \\cite{J72,PS97,T09,MT17,MT19,BK19}.  \\\\\n\n\n\n\n\n\n  \n\n\n\n\n\n\\vspace{3mm}\n\n\\textbf{Plan of the paper:} After introducing the most important notation, which we will use in this paper, Then the proof of theorem \\ref{thm1.1} will be given in \u00a7.\\ref{sec:3}  followed by some important remarks in the incompressible situation. \u00a7.\\ref{sec.4} then discusses the compressible case instead and, in particular, a proof of theorem \\ref{thm:HFU.UC.G.1}.\\\\\n\n\\textbf{Notation:} For any $2\\times2-$matrix $A$ we define its cofactor by\n\\begin{equation}\n\\textnormal{cof}\\; A=\\begin{pmatrix}a_{22} &-a_{21}\\\\-a_{12} & a_{11}\\end{pmatrix}.\n\\label{eq:1.3}\n\\end{equation}\nMoreover, we will make use of the shorthand $d_A:=\\det A.$\\\\\n\nThe Fourier-representation for any $\\eta\\in C^\\infty(B,\\mathbb{R}^2)$ (For members of Sobolev- spaces one might approximate) is given by\n\\begin{align*}\n\\eta(x)=\\sum\\limits_{j\\ge0}\\eta^{(j)}(x),\\;\\mb{where}\\;\\eta^{(0)}(x)=\\frac{1}{2}A_0(R), \\;A_0(R)=\\frac{1}{2\\pi}\\int\\limits_{0}^{2\\pi}{\\eta(R,\\th)\\;d\\th}\\end{align*} \nand for any $ j\\ge1$ we have\n\\begin{align*}\n \\eta^{(j)}(x)=A_j(R)\\cos(j\\th)+B_j(R)\\sin(j\\th),\n \\end{align*}\n  where\n\\begin{align*}\nA_j(R)=\\frac{1}{2\\pi}\\int\\limits_{0}^{2\\pi}{\\eta(R,\\th)\\cos(j\\th)\\;d\\th}\\;\\mb{and}\\;\nB_j(R)=\\frac{1}{2\\pi}\\int\\limits_{0}^{2\\pi}{\\eta(R,\\th)\\sin(j\\th)\\;d\\th}.\n\\end{align*}\nFurther we use $\\tilde{\\eta}:=\\eta-\\eta^{(0)}.$\\\\\n\n\n\n\\section{The incompressible case}\nWe immediately start with the proof in the incompressible setting. The proof is obtained by comparing energies and gaining a lower bound by means of the ELE and a Poincar\u00e9-type inequality.\\\\\n\n\n\\label{sec:3}\n\\textbf{Proof of Theorem \\ref{thm1.1}:}\\\\\n\n\n\ni) Let $u\\in\\mathcal{A}_{u_0}^{p,c}$ be a stationary point of $E$ and let $v\\in\\mathcal{F}_{n_*}^{p,\\sigma,c}$ be arbitrary and set $\\eta:=v-u\\in W_0^{1,2}(B,\\mathbb{R}^2)$ and $ \\sigma\\eta=\\sum\\limits_{j\\ge {n_*}}(\\sigma\\eta)^{(j)}$ with $n_*=n+l$ assuming wlog. $\\eta\\in C_c^\\infty(B,\\mathbb{R}^2)$ and $\\sigma\\in C^\\infty(B).$\n\nWe start by the standard expansion\n\\begin{align}\nE(v)-E(u)=&\\int\\limits_B{\\nu(x)(|\\ensuremath{\\nabla} u+\\ensuremath{\\nabla} \\eta|^{p}-|\\ensuremath{\\nabla} u|^{p})\\;dx}&\\nonumber\\\\\n\\ge&\\frac{p}{2}\\int\\limits_B{\\nu(x)|\\ensuremath{\\nabla} u|^{p-2}|\\ensuremath{\\nabla}\\eta|^2+p\\nu(x)|\\ensuremath{\\nabla} u|^{p-2}\\ensuremath{\\nabla} u\\ensuremath{\\cdot}\\ensuremath{\\nabla}\\eta\\;dx},&\n\\label{eq:HFU.IC.G.04}\n\\end{align}\nwhere we used the following inequality\\footnote{see, \\cite[Prop A.1]{SS18}with $\\sigma=0.$}\\vspace{-5mm}\n\n\\begin{align}\\frac{1}{p}|b|^{p}\\ge\\frac{1}{p}|a|^{p}+|a|^{p-2}a(b-a)+\\frac{1}{2}|a|^{p-2}|b-a|^{2}.\\label{eq:3.1}\\end{align}\n\nHenceforth, we shall denote the rightmost term in \\eqref{eq:HFU.IC.G.04} by \\[H(u,\\eta):=\\int\\limits_B{p\\nu(x)|\\ensuremath{\\nabla} u|^{p-2}\\ensuremath{\\nabla} u\\ensuremath{\\cdot}\\ensuremath{\\nabla}\\eta\\;dx}.\\]\nRecall, the ELE \\eqref{def:SP1} is given by\n\\begin{align}\n\\int\\limits_B{p\\nu(x)|\\ensuremath{\\nabla} u|^{p-2}\\ensuremath{\\nabla} u\\ensuremath{\\cdot}\\ensuremath{\\nabla}\\eta\\;dx}=&-\\int\\limits_B{p\\ensuremath{\\lambda} \\textnormal{cof}\\;\\ensuremath{\\nabla} u\\ensuremath{\\cdot} \\ensuremath{\\nabla}\\eta\\;dx}\\;\\mb{for all}\\;\\eta\\in C_c^\\infty(B,\\mathbb{R}^2).\n\\label{eq:ELE.2}\n\\end{align}\nIn order for us, to control $H$ from below, we start by rewriting said term via the relation $\\det \\ensuremath{\\nabla} \\eta=-\\textnormal{cof}\\;\\ensuremath{\\nabla} u\\ensuremath{\\cdot}\\ensuremath{\\nabla}\\eta$ a.e. and lemma 2.1(v) of \\cite{D1} in the following way\n\\begin{align}\nH(u,\\eta)=&-\\frac{p}{2}\\int\\limits_B{R((\\textnormal{cof}\\;\\ensuremath{\\nabla}\\eta)\\ensuremath{\\nabla}\\ensuremath{\\lambda})\\ensuremath{\\cdot}\\eta\\;\\frac{dx}{R}}&\\label{eq:HFU.UC.G.21.a}\\\\\n=&-\\frac{p}{2}\\int\\limits_B(\\ensuremath{\\lambda},_RR e_R+ \\ensuremath{\\lambda},_\\th e_\\th)\\ensuremath{\\cdot}\\left[(\\tilde{\\eta}_1\\tilde{\\eta}_{2,\\th}-\\tilde{\\eta}_2\\tilde{\\eta}_{1,\\th})\\frac{e_R}{R}\\right.&\\nonumber\\\\\n&+\\left. (\\tilde{\\eta}_2\\eta_{1,R}-\\tilde{\\eta}_1\\eta_{2,R})e_\\th\\right]\\;\\frac{dx}{R}.&\\nonumber\n\\end{align}\nAn application of H\u00f6lder's inequality in $\\mathbb{R}^2$, that is, for any $y,z\\in \\mathbb{R}^2$ it holds $|y\\ensuremath{\\cdot} z|\\le|y|_{\\infty}|z|_{1},$  yields\n\\vspace{1mm}\n\\begin{small}\n\\begin{align}\nH(u,\\eta)\\ge&-\\frac{p}{2}\\int\\limits_B\\!|R\\ensuremath{\\nabla}\\ensuremath{\\lambda}(x)|_{\\infty}\\!\\ensuremath{\\cdot}\\!\\left[|\\eta_1\\eta_{2,\\th}-\\eta_2\\eta_{1,\\th}|\\frac{1}{R}+ |\\eta_2\\eta_{1,R}-\\eta_1\\eta_{2,R}|\\right]\\!\\frac{dx}{R}.&\n\\end{align}\n\\end{small}\nBy applying \\eqref{eq:HFU.UC.G.1} and Cauchy-Schwarz inequalities with weight $\\ensuremath{\\varepsilon}=\\frac{n}{\\sqrt{2}}$ we get \n\\begin{align*}\nH(u,\\eta)\\ge-\\frac{p}{2}\\frac{n}{\\sqrt{2}}&\\Big[\\frac{n}{\\sqrt{2}}\\|\\sigma\\eta_1\\|_{L^2(dx\/R^2)}^2+\\frac{n}{\\sqrt{2}}\\|\\sigma\\eta_2\\|_{L^2(dx\/R^2)}^2&\\\\\n&+\\frac{1}{\\sqrt{2}n}\\Big(\\|\\sigma\\eta_{1,\\th}\\|_{L^2(dx\/R^2)}^2+\\|\\sigma\\eta_{2,\\th}\\|_{L^2(dx\/R^2)}&\\\\\n&+\\|\\sigma\\eta_{1,R}\\|_{L^2(dx)}^2+\\|\\sigma\\eta_{2,R}\\|_{L^2(dx)}^2\\big)\\Big].&\n\\end{align*}\nUsing \\eqref{eq:HFU.FE.2} and collecting terms yields\n\\begin{align*}\nH(u,\\eta)\\ge&-\\frac{p}{2}\\left[\\|\\sigma\\eta,_\\th\\|_{L^2(B,\\mathbb{R}^2,\\frac{dx}{R^2})}^2+\\frac{1}{2}\\|\\sigma\\eta,_R\\|_{L^2(B,\\mathbb{R}^2,dx)}^2\\right]&\\nonumber\\\\\n\\ge&-\\frac{p}{2}\\int\\limits_B{\\nu(x)|\\ensuremath{\\nabla} u|^{p-2}|\\ensuremath{\\nabla}\\eta|^2\\;dx},&\\nonumber\n\\end{align*}\ncompleting the proof. For the last step we made used of the following version of the Fourier-estimate given by\n\\begin{equation} n^2\\int\\limits_B{\\sigma^2|\\eta|^2\\;\\frac{dx}{R^2}} \\le \\int\\limits_B{\\sigma^2|\\eta_{,\\th}|^2\\;dx}.\\label{eq:HFU.FE.2}\\end{equation}\nThis is indeed true, for this sake, first assume that  $\\sigma\\in C^\\infty(B).$ \nWe will make use of a Poincare type inequality, first established in \\cite{JB14},\n\\begin{equation}\\int\\limits_B{R^{-2}|\\xi_{,\\th}|^2\\;dx}\\ge N^2\\int\\limits_B{R^{-2}|\\xi|^2\\;dx},\\label{eq:Uni.HFU.1}\\end{equation}\nwhich holds true for any $\\xi\\in C^\\infty(B,\\mathbb{R}^2)$ if $\\xi$ only consists of Fourier-modes $N$ or higher.\\\\\n\nThen an application of \\eqref{eq:Uni.HFU.1}, the product rule, Minkowski's inequality and \\eqref{eq:HFU.UC.G.01} yields   \n\\begin{align} n_*\\|\\sigma\\eta\\|_{L^2\\left(\\frac{dx}{R^2}\\right)}\\le&\\|(\\sigma\\eta),_\\th\\|_{L^2\\left(\\frac{dx}{R^2}\\right)}&\\nonumber\\\\\n\\le& \\|\\sigma\\eta,_\\th\\|_{L^2\\left(\\frac{dx}{R^2}\\right)}+ \\|\\sigma,_\\th\\eta\\|_{L^2\\left(\\frac{dx}{R^2}\\right)}&\\nonumber\\\\\n\\le& \\|\\sigma\\ensuremath{\\nabla}\\eta\\|_{L^2(dx)}+l \\|\\sigma\\eta\\|_{L^2\\left(\\frac{dx}{R^2}\\right)}.&\n\\label{eq:HFU.FE.2a}\n\\end{align}\nAbsorbing the rightmost term of the latter expression into the LHS yields \\eqref{eq:HFU.FE.2}. Additionally, \\eqref{eq:HFU.FE.2a} justifies its own upgrade by remaining valid for any $\\sigma\\in L^2(B),$ satisfying \\eqref{eq:HFU.UC.G.01}.\\\\\n\nii) Let $u\\in\\mathcal{A}_{u_0}^{p,c}$ be a stationary point of $E$ and let $v\\in\\mathcal{F}_{0,m_*}^{p,\\sigma,c}$ be arbitrary and set $\\eta:=v-u\\in W_0^{1,2}(B,\\mathbb{R}^2)$ and $ \\sigma\\eta=(\\sigma\\eta)^{(0)}+\\sum\\limits_{j\\ge {m_*}}(\\sigma\\eta)^{(j)}$ with $m_*=m+l$ assuming wlog. $\\eta\\in C_c^\\infty(B,\\mathbb{R}^2)$ and $\\sigma\\in C^\\infty(B).$\n\nNotice, that\n\\begin{align}\nE(v)-E(u)\n\\ge&\\frac{p}{2}\\int\\limits_B{\\nu(x)|\\ensuremath{\\nabla} u|^{p-2}|\\ensuremath{\\nabla}\\eta|^2\\;dx}+H(u,\\eta),&\n\\label{eq:HFU.IC.G.05}\n\\end{align}\nremains valid with the same mixed term\n\\begin{equation}H(u,\\eta)=\\int\\limits_B{p\\nu(x)|\\ensuremath{\\nabla} u|^{p-2}\\ensuremath{\\nabla} u\\ensuremath{\\cdot}\\ensuremath{\\nabla}\\eta\\;dx}.\\label{eq:Uni.SPC.1}\\end{equation}\nFrom here on we have to argue more along the lines of the proof of \\cite[thm 1.2]{D1}. Again, by the ELE \\eqref{def:SP1}, the identity $\\det \\ensuremath{\\nabla} \\eta=-\\textnormal{cof}\\;\\ensuremath{\\nabla} u\\ensuremath{\\cdot}\\ensuremath{\\nabla}\\eta$ a.e. and by \\cite[lem 2.1.(vi)]{D1} we get\n\\[H(u,\\eta)=-p\\int\\limits_B{(\\textnormal{cof}\\;\\ensuremath{\\nabla}\\eta^{(0)}\\ensuremath{\\nabla}\\ensuremath{\\lambda}(x))\\ensuremath{\\cdot}\\tilde{\\eta}\\;dx}-p\\int\\limits_B{(\\textnormal{cof}\\;\\ensuremath{\\nabla}\\eta\\ensuremath{\\nabla}\\ensuremath{\\lambda}(x))\\ensuremath{\\cdot}\\tilde{\\eta}\\;dx}=:(I)+(II).\\]\nNow by noting that the $0-$mode is only a function of $R,$ we get\n\\begin{align*}\n(\\textnormal{cof}\\;\\ensuremath{\\nabla}\\eta^{(0)}\\ensuremath{\\nabla}\\ensuremath{\\lambda}(x))\\ensuremath{\\cdot}\\tilde{\\eta}=\\frac{\\ensuremath{\\lambda},_\\th}{R}(\\eta_{1,R}^{(0)}\\tilde{\\eta}_2-\\eta_{2,R}^{(0)}\\tilde{\\eta}_1).\n\\end{align*}\nInstead of just $\\ensuremath{\\lambda},_\\th$ on the right hand side of the latter equation we would like to have the full gradient of $\\ensuremath{\\lambda}.$ This can be achieved by using the basic relations $e_\\th\\ensuremath{\\cdot} e_\\th=1$ and  $e_R\\ensuremath{\\cdot} e_\\th=0$ to obtain \n\\begin{align*}\n(\\textnormal{cof}\\;\\ensuremath{\\nabla}\\eta^{(0)}\\ensuremath{\\nabla}\\ensuremath{\\lambda}(x))\\ensuremath{\\cdot}\\tilde{\\eta}=(\\ensuremath{\\lambda},_RR e_R+\\ensuremath{\\lambda},_\\th e_\\th)\\ensuremath{\\cdot}(\\eta_{1,R}^{(0)}\\tilde{\\eta}_2-\\eta_{2,R}^{(0)}\\tilde{\\eta}_1)\\frac{e_\\th}{R}.\n\\end{align*}\nArguing similarly for (II), and a short computation shows\n\\begin{align}\nH(u,\\eta)=&-p\\int\\limits_B(\\ensuremath{\\lambda},_RR e_R+\\ensuremath{\\lambda},_\\th e_\\th)\\ensuremath{\\cdot}\\left[(\\tilde{\\eta}_1\\tilde{\\eta}_{2,\\th}-\\tilde{\\eta}_2\\tilde{\\eta}_{1,\\th})\\frac{e_R}{R}\\right.&\\nonumber\\\\\n&+\\left. (\\tilde{\\eta}_2(\\eta_{1,R}^{(0)}+\\eta_{1,R})-\\tilde{\\eta}_1(\\eta_{2,R}^{(0)}+\\eta_{2,R}))e_\\th\\right]\\;\\frac{dx}{R}.&\n\\label{eq:Uni.SPC.2}\\end{align}\nBy H\u00f6lder's inequality in $\\mathbb{R}^2$ we get\\begin{align*} \nH(u,\\eta)\\ge&-p\\int\\limits_B|\\ensuremath{\\nabla} \\ensuremath{\\lambda}(x)R|_\\infty \\left[\\left|\\tilde{\\eta}_1\\tilde{\\eta}_{2,\\th}-\\tilde{\\eta}_2\\tilde{\\eta}_{1,\\th}\\right|\\frac{1}{R}\\right.&\\\\\n&+\\left.\\left|\\tilde{\\eta}_2(\\eta_{1,R}^{(0)}+\\eta_{1,R})-\\tilde{\\eta}_1(\\eta_{2,R}^{(0)}+\\eta_{2,R})\\right|\\right]\\;\\frac{dx}{R}.&\n\\end{align*}\nBy $|\\ensuremath{\\nabla} \\ensuremath{\\lambda}(x)R|_\\infty\\le\\frac{\\sqrt{3}m\\sigma^2(x)}{2\\sqrt{2}}$ and a weighted Cauchy-Schwarz inequality, we see\n\\begin{align*}\nH(u,\\eta)\\!\\ge&-\\frac{\\sqrt{3}mp}{4\\sqrt{2}}\\!\\left[2a\\|\\sigma\\tilde{\\eta}_1\\|_{L^2\\left(\\frac{dx}{R^2}\\right)}^2\\!+2a\\|\\sigma\\tilde{\\eta}_2\\|_{L^2\\left(\\frac{dx}{R^2}\\right)}^2\\!+\\frac{1}{a}\\|\\sigma\\eta_{2,R}^{(0)}+\\sigma\\eta_{2,R}\\|_{L^2(dx)}^2\\right.&\\\\\n&\\left.+\\frac{1}{a}\\|\\sigma\\tilde{\\eta}_{2,\\th}\\|_{L^2\\left(\\frac{dx}{R^2}\\right)}^2+\\frac{1}{a}\\|\\sigma\\eta_{1,R}^{(0)}+\\sigma\\eta_{1,R}\\|_{L^2(dx)}^2+\\frac{1}{a}\\|\\sigma\\tilde{\\eta}_{1,\\th}\\|_{L^2\\left(\\frac{dx}{R^2}\\right)}^2\\right].&\n\\end{align*}\n\nApplying the Cauchy-Schwarz inequality, inequality \\eqref{eq:HFU.FE.2}, and combining some of the norms yields\n\\begin{align*}\nH(u,\\eta)\\ge-\\frac{\\sqrt{3}mp}{4\\sqrt{2}}&\\left[\\left(\\frac{2a}{m^2}+\\frac{1}{a}\\right)\\|\\sigma\\tilde{\\eta},_\\th\\|_{L^2\\left(\\frac{dx}{R^2}\\right)}^2+\\frac{2}{a}\\|\\sigma\\eta,_R^{(0)}\\|_{L^2(dx)}^2\\right.&\\\\\n&\\left.+\\frac{2}{a}\\|\\sigma\\eta,_R\\|_{L^2(dx)}^2\\right].&\n\\end{align*}\nBy using $\\tilde{\\eta},_\\th=\\eta,_\\th$ and $\\|\\sigma\\eta,_R^{(0)}\\|_{L^2(dx)}^2\\le\\|\\sigma\\eta,_R\\|_{L^2(dx)}^2$ we obtain\n\\begin{align*}\nH(u,\\eta)\\ge&-\\frac{\\sqrt{3}mp}{4\\sqrt{2}}\\left[\\left(\\frac{2a}{m^2}+\\frac{1}{a}\\right)\\|\\sigma\\eta,_\\th\\|_{L^2\\left(\\frac{dx}{R^2}\\right)}^2+\\frac{4}{a}\\|\\sigma\\eta,_R\\|_{L^2(dx)}^2\\right].&\n\\end{align*}\nChoosing $a=\\frac{\\sqrt{3}m}{\\sqrt{2}}$ and combining the norms even further yields\n\\begin{align*}\nH(u,\\eta)\\ge-\\frac{p}{2}\\int\\limits_B{\\nu(x)|\\ensuremath{\\nabla} u|^{p-2}|\\ensuremath{\\nabla}\\eta|^2\\;dx},\n\\end{align*}\n completing the 2nd part of the proof.\n\\qed\n\n\\begin{re} \n 1. Notice, that \\eqref{eq:HFU.UC.G.01} is especially satisfied if $p=2$ and $\\nu(x)=\\nu(R).$ So condition \\eqref{eq:HFU.UC.G.01}\ncan be thought of as a natural extension of this fact to the case, where $p$ might be arbitrary and $\\sigma(x)$ depends on $x$ instead of $R.$\\\\\n2. Despite the fact that the sets $F_{n_*}^{p,\\sigma,c}$ and $F_{0,m_*}^{p,\\sigma,c}$ depend on $\\sigma$ it remains true that if $n_*=0$ or $m_*\\in\\{0,1\\}$ one gets uniqueness in the full class $\\mathcal{A}^{p,c}.$ Indeed, there are two cases to consider. Firstly, let $n_*=0$ or $m_*\\le1$ s.t. $n=0$ or $m=0.$ Then $\\ensuremath{\\nabla}\\ensuremath{\\lambda}\\equiv0$ and by  \\eqref{eq:HFU.IC.G.04} and \\eqref{eq:ELE.2} one obtains\n\\begin{align*}\nE(v)-E(u)\\ge\\int\\limits_B{\\frac{\\nu(x)}{2}|\\ensuremath{\\nabla} u|^{p-2}|\\ensuremath{\\nabla}\\eta|^2\\;dx},\\end{align*}\nimplying, that $u$ is a global minimizer in the full class $\\mathcal{A}^{p,c}.$ Assuming, additionally, that  $\\sigma(x)\\ge\\sigma_0>0$ for  a.e. $x\\in B$ then one can conclude that it needs to be the unique one. In the case when $m_*=1$ and $m=1,l=0,$ then  $\\sigma_{,\\th}=0$ and \\eqref{eq:HFU.FE.2} can be applied with $m_*=m=1$ completing the argument. \n\\label{re:1}\n\\end{re}\\vspace{0.5cm}\n\n\\section{Compressible uniqueness criterion}\nHere we outline the proof for the analogous compressible results, where the proof strategy remains the same.\\\\\n\n\\label{sec.4}\n\\textbf{Proof of Theorem \\ref{thm:HFU.UC.G.1}:}\\\\\ni) Let $u\\in\\mathcal{A}_{u_0}^p$ be a stationary point of $I$ and let $v\\in\\mathcal{F}_{n_*}^{p,\\sigma}$ be arbitrary and set $\\eta:=v-u\\in W_0^{1,2}(B,\\mathbb{R}^2)$ and $ \\sigma\\eta=\\sum\\limits_{j\\ge {n_*}}(\\sigma\\eta)^{(j)}$ with $n_*=n+l$ assuming wlog. $\\eta\\in C_c^\\infty(B,\\mathbb{R}^2)$ and $\\sigma\\in C^\\infty(B).$ Again, by the standard expansion, the subdifferential inequality for $\\Psi$, and inequality \\eqref{eq:3.1} we obtain\n\\begin{align}\nI(v)-I(u)=&\\int\\limits_B{\\frac{\\nu(x)}{p}(|\\ensuremath{\\nabla} u+\\ensuremath{\\nabla} \\eta|^{p}-|\\ensuremath{\\nabla} u|^{p})}&\\nonumber\\\\\n&+{\\Psi(x,\\ensuremath{\\nabla} u+\\ensuremath{\\nabla} \\eta,\\det\\ensuremath{\\nabla} u+\\ensuremath{\\nabla} \\eta)-\\Psi(x,\\ensuremath{\\nabla} u,\\det\\ensuremath{\\nabla} u)\\;dx}&\\nonumber\\\\\n\\ge&\\int\\limits_B{\\frac{\\nu(x)}{2}|\\ensuremath{\\nabla} u|^{p-2}|\\ensuremath{\\nabla}\\eta|^2+\\nu(x)|\\ensuremath{\\nabla} u|^{p-2}\\ensuremath{\\nabla} u\\ensuremath{\\cdot}\\ensuremath{\\nabla}\\eta}&\\nonumber\\\\\n&+{\\ensuremath{\\partial}_\\xi\\Psi(x,\\ensuremath{\\nabla} u,\\det\\ensuremath{\\nabla} u)\\ensuremath{\\cdot}\\ensuremath{\\nabla}\\eta+\\ensuremath{\\partial}_d\\Psi(x,\\ensuremath{\\nabla} u,\\det\\ensuremath{\\nabla} u)\\textnormal{cof}\\;\\ensuremath{\\nabla} u\\ensuremath{\\cdot}\\ensuremath{\\nabla}\\eta\\;dx}.&\\nonumber\n\\end{align}\n\nThen by applying the ELE \\eqref{def:SP.2} we get\n\\begin{align}\nI(v)-I(u)\\ge&\\int\\limits_B{\\frac{\\nu(x)}{2}|\\ensuremath{\\nabla} u|^{p-2}|\\ensuremath{\\nabla}\\eta|^2+\\ensuremath{\\partial}_d\\Psi(x,\\ensuremath{\\nabla} u,\\det\\ensuremath{\\nabla} u)d_{\\ensuremath{\\nabla}\\eta}\\;dx}.&\n\\label{eq:HFU.UC.G.21}\n\\end{align}\nReplacing $\\ensuremath{\\lambda}$ by $\\ensuremath{\\partial}_d\\Psi$ in the argument given in the 1st part of the proof of theorem \\ref{thm:HFU.UC.G.1}, see \u00a7.\\ref{sec:3}, completes the argument.\\\\\n\nii) If $\\sigma\\eta$ can be such that the first Fourier-mode $(\\sigma\\eta)^{(0)}\\not=0$ then \\eqref{eq:HFU.UC.G.21} does still hold, but one needs to conclude similarly to the 2nd part of the proof of theorem \\ref{thm:HFU.UC.G.1}.\\qed \n\\begin{re}\n1. Arguing as in Remark \\ref{re:1} in \u00a7.\\ref{sec:3}, assuming, additionally, that $\\sigma(x)\\ge\\sigma_0>0$ for a.e. $x\\in B$ then it remains true that if $n_*=0$ or $m_*\\in\\{0,1\\}$ one gets uniqueness in the full class $\\mathcal{A}^{p}.$\\\\\n2. It seems reasonable to believe, that similar uniqueness criteria could be given in many other elastic scenarios if the considered situation is such, that the describing functional decomposes into two parts, the main one and the perturbative one acting like the pressure. If the perturbation then is assumed to be small in some sense there might be a chance of obtaining uniqueness.\n\\end{re}\n\\vspace{0.5cm}\n\n\\textbf{Data Declaration}\nData sharing not applicable to this article as no datasets were generated or analysed during the current study.\\\\\n\n\\textbf{Acknowledgments}\n\nThe author is in deep dept to Jonathan J. Bevan, Bin Cheng, and Ali Taheri for vital discussions and suggestions, the fantastic people with the Department of Mathematics at the University of Surrey and the Engineering \\& Physical Sciences Research Council (EPRSC), which generously funded this work.\\\\\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe human brain can self-organize and coordinate different cognitive functions to flexibly adapt to complex and changing environments. A major challenge for Artificial Intelligence and computational neuroscience is integrating multi-scale biological principles to build biologically plausible brain-inspired intelligent models. As the third generation of neural networks~\\cite{Maass1997Networks}, Spiking Neural Networks (SNNs) are more biologically realistic at multiple scales, more biologically interpretable, more energy-efficient, and naturally more suitable for modeling various cognitive functions of the brain and creating biologically plausible AI. \n\n\\begin{figure*}[!htbp]\n\t\\centering\n\t\\includegraphics[scale=0.3]{.\/fig\/braincog.jpg}\n\t\\caption{The architecture of Brain-inspired Cognitive Intelligence Engine (BrainCog).}\n\t\\label{braincog}\n\\end{figure*}\n\nExisting neural simulators attempted to simulate elaborate biological neuron models or build large-scale neural network simulations, neural dynamics models, deep SNNs. Neuron~\\cite{carnevale2006neuron} focuses on simulating elaborate biological neuron models. NEST~\\cite{gewaltig2007nest} implements large-scale neural network simulations. Brian\/Brian2~\\cite{stimberg2019brian, goodman2009brian} provides an efficient and convenient tool for modeling spiking neural networks. Shallow SNNs implemented by Brian2 can realize unsupervised visual classification~\\cite{diehl2015unsupervised}. Further, BindsNET~\\cite{ hazan2018bindsnet} builds SNNs with coordination of various neurons and connections and incorporates multiple biological learning rules for training SNNs. Such learning SNNs can perform machine learning tasks, including simple supervised, unsupervised, and reinforcement learning. However, supporting more complex tasks would be great challenge for these SNNs frameworks, and there is a large gap in performance compared with traditional deep neural networks (DNNs). Deep SNNs trained by surrogate gradient or converting well-trained DNNs to SNNs have achieved great progress in the fields of speech recognition~\\cite{dominguez2018deep, loiselle2005exploration}, computer vision~\\cite{kim2020spiking, wu2018spatio}, and reinforcement learning~\\cite{tan2021strategy}. Motivated by this, SpikingJelly~\\cite{SpikingJelly} develops a deep learning SNN framework (trained by surrogate gradient or converting well-trained DNNs to SNNs), which integrates deep convolutional SNNs and various deep reinforcement learning SNNs for multiple benchmark tasks. Platforms as SpikingJelly are relatively more inspired by the field of deep learning and focuse on improving the performance of different tasks. They are currently lack of in-depth inspiration from the brain information processing mechanisms and may not aim at, hence short at simulating large scale functional brains.\n\nBrainPy~\\cite{Wang2021} excels at modeling, simulating, and analyzing the dynamics of brain-inspired neural networks from multiple perspectives, including neurons, synapses, and networks. While it focuses on computational neuroscience research, it fails to consider the learning and optimization of deep SNNs or the implementation of brain-inspired functions. SPAUN~\\cite{eliasmith2012large}, a large-scale brain function model consisting of 2.5 million simulated neurons and implemented by Nengo~\\cite{bekolay2014nengo}, integrates multiple brain areas and realizes multiple brain cognitive functions, including image recognition, working memory, question answering, reinforcement learning, and fluid reasoning. However, it is not suitable for solving challenging and complex AI tasks that can be handled by deep learning models. In summary, there is still a lack of open-source spiking neural network frameworks that could incorporate the ability of simulating brain structures, cognitive functions at large scale, while keep itself effective for creating complex and efficient AI models at the same time.\n\nConsidering the various limitations of existing frameworks mentioned above, in this paper, we present the Brain-inspired Cognitive Intelligence Engine (BrainCog), a spiking neural network based open source platform for both brain-inspired AI and brain simulation at multiple scales. As shown in Fig.~\\ref{braincog}, some basic modules (such as different types of neuron models, learning rules, encoding strategies, etc.) are provided as building blocks to construct different brain areas and neural circuits to implement brain-inspired cognitive functions. BrainCog is an easy-to-use framework that can flexibly replace different components according to various purposes and needs. BrainCog tries to achieve the vision ``the structure and mechanism are inspired by the brain, and the cognitive behaviors are similar to humans'' for Brain-inspired AI~\\cite{2016Retrospect}. BrainCog is developed based on the deep learning framework (currently it is based on PyTorch, while it is easy to migrate to other frameworks such as PaddlePaddle, TensorFlow, etc.). This approach aims at greatly facilitating researchers to quickly familiarize themselves with the platform and implement their own algorithms.\n\n\\subsection{Brain-inspired AI}\nBrainCog is aimed at providing infrastructural support for Brain-inspired AI. Currently, it provides cognitive functions components that can be classified into five categories: Perception and Learning, Decision Making, Motor Control, Knowledge Representation and Reasoning, and Social Cognition. These components collectively form neural circuits corresponding to 28 brain areas in the mammalian brains, as shown in Fig.~\\ref{born}. These brain-inspired AI models have been effectively validated on various supervised and unsupervised learning, deep reinforcement learning, and several complex brain-inspired cognitive tasks.  \n\n\n\\begin{figure}[!htbp]\n\t\\centering\n\t\\includegraphics[scale=0.35]{.\/fig\/born.jpg}\n\t\\caption{Multiple cognitive functions integrated in BrainCog, their related brain areas and neural circuits.}\n\t\\label{born}\n\\end{figure}\n\n\\subsubsection{Perception and Learning} \nBrainCog provides a variety of supervised and unsupervised methods for training spiking neural networks, such as the biologically-plausible Spike-Timing-Dependent Plasticity (STDP)~\\cite{bi1998synaptic}, the backpropagation based on surrogate gradients~\\cite{wuSpatioTemporalBackpropagationTraining2018,zhengGoingDeeperDirectlyTrained2021,fangIncorporatingLearnableMembrane2021}, and the conversion-based algorithms~\\cite{li2021free, han2020deep, han2020rmp}. In addition to high performance in common perception and learning process, it also shows strong adaptability in small samples and noisy scenarios. BrainCog also provides a multi-sensory integration framework for human-like concept learning~\\cite{wywFramework}. Inspired by quantum information theory, BrainCog provides a quantum superposition spiking neural network, which encodes complement information to neuronal spike trains with different frequency and phase~\\cite{SUN2021102880}. \n\n\\subsubsection{Decision Making}\nFor decision making, BrainCog provides multi-brain areas coordinated decision-making spiking neural network~\\cite{zhao2018brain}. The biologically inspired decision-making model implemented by BrainCog achieves human-like learning ability on the Flappy Bird game and supports UAVs' online decision-making tasks. In addition, BrainCog combines SNNs with deep reinforcement learning and provides the brain-inspired spiking deep Q network~\\cite{sun2022solving}.\n\n\\subsubsection{Motor Control}\nEmbodied cognition is crucial to realizing biologically plausible AI. As part of the embodied cognition modules, and inspired by the motor control mechanism of the brain, BrainCog provides a multi-brain areas coordinated robot motion spiking neural networks model, which includes premotor cortex (PMC), supplementary motor areas (SMA), basal ganglia and cerebellum functions. With proper mapping, the spiking motor network outputs can be used to control various robots.\n\n\\subsubsection{Knowledge Representation and Reasoning} \nBrainCog incorporates multiple neuroplasticity and population coding mechanisms for knowledge representation and reasoning. The brain-inspired music memory and stylistic composition model implements the knowledge representation and memory of note sequences and can generate music according to different styles~\\cite{LQ2020,LQ2021}. Sequence Production Spiking Neural Network (SPSNN) achieves the memory of the symbol sequence and can reconstruct the symbol sequence in the light of different rules~\\cite{fang2021spsnn}. Commonsense Knowledge Representation Graph SNN (CKR-GSNN) realizes the representation of commonsense knowledge through incorporating multi-scale neural plasticity and population coding mechanism into a graph SNN model~\\cite{KRRfang2022}.\nCausal Reasoning Spiking Neural Network (CRSNN) encodes the causal graph into a spiking neural network and realizes deductive reasoning tasks accordingly~\\cite{fang2021crsnn}.\n\n\\subsubsection{Social Cognition} \nBrainCog provides a brain-inspired social cognition model with biological plausibility. This model gives the agent a preliminary ability to perceive and understand itself and others and can enable the robots pass the Multi-Robots Mirror Self-Recognition Test~\\cite{zeng2018toward} and the AI Safety Risks Experiment~\\cite{zhao2022brain}. The former is a classic experiment of self-perception in social cognition, and the latter is a variation and application of the theory of mind experiment in social cognition.\n\\subsection{Brain Simulation}\n\n\\subsubsection{Brain Cognitive Function Simulation}\nTo demonstrate the capability of BrainCog for cognitive function simulation, we provide \\emph{Drosophila} decision-making and prefrontal cortex working memory function simulation~\\cite{zhao2020neural,zhangqian2021comparison}. For \\emph{Drosophila} nonlinear and linear decision-making simulation, BrainCog verifies the winner-takes-all behaviors of the nonlinear dopaminergic neuron\u2011GABAergic neuron\u2011mushroom body (DA\u2011GABA\u2011MB) circuit under a dilemma and obtains consistent conclusions with \\emph{Drosophila} biological experiments~\\cite{zhao2020neural}. For the working memory performance of the prefrontal cortex network implemented by BrainCog, we discover that using human neurons to replace rodent neurons without changing network structure can significantly improve the accuracy and completeness of an image memory task~\\cite{zhangqian2021comparison}, and this implies the evolution of the brains are not only on their structures, but also applies to single computational units such as neurons.\n\n\\subsubsection{Multi-scale Brain Structure Simulation} \nBrainCog provides simulations of brain structures at different scales, from microcircuits, and cortical columns, to whole-brain structure simulations. Anatomical and imaging multi-scale connectivity data is used to support whole-brain simulations from mouse brain, macaque brain to human brain at different scales.\n\n\n\n\\section{Basic Components}\nBrainCog provides essential and fundamental components to model biological and artificial intelligence. It includes various biological neuron models, learning rules, encoding strategies, and models of different brain areas. One can build brain-inspired SNN models through reusing and refining these building blocks. Expanding and refining the components and cognitive functions included in BrainCog is an ongoing effort. We believe this should be a continuous community effort, and we welcome researchers and practitioners to enrich and improve the work together in a complementary way.\n\n\\subsection{Neuron Models}\nBrainCog supports various models for spiking neurons, such as the following :\n\n(1) Integrate-and-Fire spiking neuron (IF)~\\cite{abbott1999lapicque}:\n\n\\begin{equation}\n\\frac{dV}{dt} = I\n\\end{equation}\n\n$I$ denotes the input current from the pre-synaptic neurons. Once the membrane potential reaches the threshold $V_{th}$, the neuron $j$ fires a spike~\\cite{abbott1999lapicque}.\n\n(2) Leaky Integrate-and-Fire spiking neuron (LIF)~\\cite{dayan2005theoretical}:\n\\begin{equation}\n \\tau\\frac{dV}{dt} = - V + I\n\\label{equa_lif1}\n\\end{equation}\n$\\tau=RC$ denote the time constant, $R$ and $C$ denotes the membrane resistance and capacitance respectively~\\cite{dayan2005theoretical}.  \n\n(3) The adaptive Exponential Integrate-and-Fire spiking neuron (aEIF)~\\cite{Fourcaud2003How,brette2005adaptive}:\n\n\\begin{equation}\n\\left\\{\n\\begin{aligned}\nC \\frac{d V}{d t}&=-g_{L}(V-E_{L})+g_{L} \\exp(\\frac{V-V_{t h}}{\\Delta T})+I-w \\\\\n\\tau_{w} \\frac{dw}{dt}&=a(V-E_L)-w\n\\end{aligned}\n\\right.\n\\end{equation}\n\nwhere $g_L$ is the leak conductance, $E_L$ is the leak reversal potential, $V_r$ is the reset potential, $\\Delta T$ is the slope factor, $I$ is the  background currents.  $\\tau_w$ is the adaptation time constant. When the membrane potential is greater than the threshold $V_{th}$, $V = V_{r}$, and $w = w +b$.  $a$ is the subthreshold adaptation, and $b$ is the spike-triggered adaptation~\\cite{Fourcaud2003How,brette2005adaptive}. \n\n(4) The Izhikevich spiking neuron~\\cite{izhikevich2003simple}:\n\\begin{equation}\n\\left\\{\n\\begin{aligned}\n\\frac{dV}{dt} &= 0.04V^2 + 5v + 140 - u + I \\\\\n\\frac{du}{dt} &= a(bv-u)\\\\\n\\end{aligned}\n\\right.\n\\end{equation}\nWhen the membrane potential is greater than the threshold:\n\\begin{equation}\n\\left\\{\n\\begin{aligned}\nV &= c \\\\\nu &= u + d\\\\\n\\end{aligned}\n\\right.\n\\end{equation}\n$u$ represents the membrane recovery variable, and $a, b, c, d$ are the dimensionless parameters~\\cite{izhikevich2003simple}. \n\n(5) The Hodgkin-Huxley spiking neuron (H-H)~\\cite{hodgkin1952quantitative}:\n\\begin{equation}\nI = C \\frac{dV}{dt} + \\overline{g}_K n^4 (V - V_K) + \\overline{g}_{Na} m^3 h (V - V_{Na}) + \\overline{g}_L (V - V_L)\n\\label{equa_hh}\n\\end{equation}\n\\begin{equation}\n\\left\\{\n\\begin{aligned}\n\\frac{dn}{dt} &= \\alpha_n(V)(1-n) - \\beta_n(V)n \\\\\n\\frac{dm}{dt} &= \\alpha_m(V)(1-m) - \\beta_m(V)m \\\\\n\\frac{dh}{dt} &= \\alpha_n(V)(1-h) - \\beta_n(V)h \\\\\n\\end{aligned}\n\\right.\n\\end{equation}\n\n\n$\\alpha_i$  and $\\beta_i$ are used to control the $i_{th}$ ion channel, $n$, $m$, $h$ are dimensionless probabilities between 0 and 1. $\\overline{g}_i$ is the maximal value of the conductance~\\cite{hodgkin1952quantitative}.\n\n\nThe H-H model shows elaborate modeling of biological neurons. In order to apply it more efficiently to AI tasks, BrainCog incorporates a simplified H-H model ($C=0.02\\mu F\/cm^2, V_r=0, V_{th}=60mV$), as illustrated in~\\cite{wang2016spiking}.\n\n\n\n\n\n\\subsection{Learning Rules}\nBrainCog provides various plasticity principles and rules to support biological plausible learning and inference, such as:\n\n(1) Hebbian learning theory~\\cite{amit1994correlations}:\n\\begin{equation}\n\\Delta w_{ij}^t=x_i^t x_j^t\n\\end{equation}\nwhere $w_{ij}^t$ means the $i$th synapse weight of $j$th neuron at the time $t$. $x_i^t$ is the input of $i$th synapse at time $t$. $x_j^t$ is the output of $j$th neuron at time $t$~\\cite{amit1994correlations}.\n\n(2) Spike-Timing-Dependent Plasticity (STDP)~\\cite{bi1998synaptic}:\n\n\\begin{equation}\n\t\\begin{split}\n\t\t&\\Delta w_{j}=\\sum\\limits_{f=1}^{N} \\sum\\limits_{n=1}^{N} W(t^{f}_{i}-t^{n}_{j})\\\\\n\t\t&W(\\Delta t)= \n\t\t\\left\\{\n\t\t\\begin{split}\n\t\tA^{+}e^{\\frac{-\\Delta t}{\\tau_{+}}}\\quad if\\; \\Delta t>0&\\\\ \n\t\t-A^{-}e^{\\frac{\\Delta t}{\\tau_{-}}}\\quad if\\;\\Delta t<0&\\\\ \n\t\t\\end{split}\n\t\t\\right.\n\t\\end{split}\n\t\\label{eq4}\n\\end{equation}\nwhere $\\Delta w_{j}$ is the modification of the synapse $j$, and $W(\\Delta t)$ is the STDP function. $t$ is the time of spike. $A^{+},A^{-}$ mean the modification degree of STDP. $\\tau_{+}$ and $\\tau_{-}$ denote the time constant~\\cite{bi1998synaptic}.\n\n(3) Bienenstock-Cooper-Munros theory (BCM)~\\cite{bienenstock1982theory}:\n\\begin{equation}\n\\Delta w=y\\left(y-\\theta_M \\right)x-\\epsilon w\n\\end{equation}\n\nWhere $x$ and $y$ denote the firing rates of pre-synaptic and post-synaptic neurons, respectively, threshold $\\theta_M$ is the average of historical activity of the post-synaptic neuron~\\cite{bienenstock1982theory}.\n\n(4) Short-term Synaptic Plasticity (STP)~\\cite{maass2002synapses}: \n\n\nShort-term plasticity is used to model the synaptic efficacy changes over time.\n\n\\begin{equation}\n{a_k} = {u_k} \\cdot {R_k}\n\\end{equation}\n\\begin{equation}\n{u_{k+1}} = U + {u_{k}}(1 - U)\\exp \\left( { - {{\\rm{\\Delta }}t_{k}}\/{\\tau _{{\\rm{fac}}}}} \\right)\n\\end{equation}\n\\begin{equation}\n{R_{k+1}} = 1 + \\left( {{R_{k}} - {u_{k}}{R_{k}} - 1} \\right)\\exp \\left( { - {{\\rm{\\Delta }}t_{k}}\/{\\tau _{{\\rm{rec}}}}} \\right)\n\\end{equation}\n\n$a$ denotes the synaptic weight, $U$ denotes the fraction of synaptic resources. $\\tau _{{\\rm{fac}}}$ and $\\tau _{{\\rm{rec}}}$ denotes the time constant for recovery from facilitation and depression. The variable $R_n$models the fraction of synaptic efficacy available for the (k)th spike,and $u_n R_n$ models the fraction of synaptic efficacy~\\cite{maass2002synapses}.\n\n\n(5) Reward-modulated Spike-Timing-Dependent Plasticity (R-STDP)~\\cite{Eugene2007}: \n\n\n\nR-STDP uses synaptic eligibility trace $e$ to store temporary information of STDP. The eligibility trace accumulates the STDP $\\Delta w_{STDP}$ and decays with a time constant $\\tau_e$~\\cite{Eugene2007}. \n\n\\begin{equation}\\label{rstdpe}\n\\Delta e=-\\frac{e}{\\tau _e}+\\Delta w_{STDP}\n\\end{equation}\n\nThen, synaptic weights are updated when a delayed reward $r$ is received, as Eq.~\\ref{rstdpw} shown~\\cite{Eugene2007}. \n\n\\begin{equation}\\label{rstdpw}\n\\Delta w=r*\\Delta e \n\\end{equation}\n\n\n\n\\subsection{Encoding Strategies}\nBrainCog supports a number of different encoding strategies to help encode the inputs to the spiking neural networks.\n\n\n(1) Rate Coding~\\cite{adrian1926impulses}:\n\nRate coding is mainly based on spike counting to ensure that the number of spikes issued in the time window corresponds to the real value. Poisson distribution can describe the number of random events occurring per unit of time, which corresponds to the firing rate~\\cite{adrian1926impulses}. Set $\\alpha \\sim \\mathbb{U}(0,1)$, the input can be encoded as\n\\begin{equation}\n\ts(t) =  \\begin{cases}\n\t   1,\\quad if \\quad x > \\alpha\\\\\n\t   0, \\quad else \n\t\\end{cases}\n\\end{equation}\n\n(2) Phase Coding~\\cite{kim2018deep}:\n\nThe idea of phase coding can be used to encode the analog quantity changing with time. The value of the analog quantity in a period can be represented by a spike time, and the change of the analog quantity in the whole time process can be represented by the spike train obtained by connecting all the periods.\nEach spike has a corresponding phase weighting under phase encoding, and generally, the pixel intensity is encoded as a 0\/1 input similar to binary encoding. $\\gg$ denotes the shift operation to the right, $K$ is the phase period~\\cite{kim2018deep}. Pixel $x$ is enlarged to $x^{\\prime}=x*(2^{K}-1)$ and shifted $k=K-1-(t \\mod K)$ to the right, where $\\mod$ is the  remainder operation. If the lowest bit is one, $s$ will be one at time $t$. $\\&$ means \nbit-wise AND operation.\n\\begin{equation}\n\ts(t) =  \\begin{cases} 1, \\quad if  \\quad (x^{\\prime} \\gg k) \\& 1 = 1\\\\\n\t0, \\quad else\n\t\\end{cases}\n\\end{equation}\n\n(3) Temporal Coding~\\cite{thorpe1996speed}:\n\nThe characteristic of the neuron spike is that the form of the spike is fixed, and there are only differences in quantity and time. The most common implementation is to express information regarding the timing of individual spike. The stronger the stimulus received, the earlier the spike generated~\\cite{rueckauer2018conversion}. Let the total simulation time be T, and the input $x$ of the neuron can be encoded as the spike at time $t^s$:\n\\begin{equation}\n\tt^{s} = T - \\operatorname{round}(Tx)\n\\end{equation}\n\n(4) Population Coding~\\cite{bohte2002error}:\n\nPopulation coding helps to solve the problem of the ambiguity of information carried by a single neuron. The ambiguity of information carried can be understood as: the original information is input into a neuron, which makes the network hard to distinguish some overlapping or similar related information~\\cite{quian2009extracting,grun2010analysis}.\nThe intuitive idea of population coding is to make different neurons to be with different sensitivity to different types of inputs, which is also reflected in biology. For example, rats' whiskers have different sensitivity to different directions~\\cite{quian2009extracting}. The inputs will be transformed into a spike train with a period by population coding. A classical population coding method is the neural information coding method based on the Gaussian tuning curve referred to Eq.~\\ref{gaus}. This method is more suitable when the amount of data is small, and the information is concentrated. A Gaussian neuron covers a range of analog quantities in the form of Gaussian function~\\cite{bohte2002error}, \\cite{li2021online}. Suppose that $m$ ($m > 2$) neurons are used to encode a variable x with a value range of $[I_{min}, I_{max}]$. $f(x)$ can be firing time or voltage.\n\n\\begin{equation}\n        f(x) = x^{-\\frac{(x-\\mu)^{2}}{2\\sigma^{2}}}\n\\label{gaus}\n\\end{equation}\n\nThe corresponding mean with adjustable parameters $\\beta$ and variance of the $ith$ ($i=1,2,..., m$) neuron are as follows:\n\n\\begin{equation}\n        \\mu = I_{min} + \\frac{2i-3}{2}\\frac{I_{max}-I_{min}}{m-2}\n\\end{equation}\n\n\\begin{equation}\n        \\sigma = \\frac{1}{\\beta}\\frac{I_{max}-I_{min}}{m-2}\n\\end{equation}\n\n\n\n\n\\subsection{Brain Areas Models}\n\nBrain-inspired models of several functional brain areas are constructed for BrainCog from different levels of abstraction.\n\n(1) Basal Ganglia (BG): \n\nBasal ganglia facilitates desired action selection and inhibit competing behavior (making winner-takes-all decisions)~\\cite{chakravarthy2010basal,redgrave1999basal}. It cooperates with PFC and the thalamus to realize the decision-making process in the brain~\\cite{parent1995functional}. BrainCog models the basal ganglia brain area, including excitatory and inhibitory connections among striatum, Globus pallidus internas (Gpi), Globus pallidus external (Gpe), and subthalamic nucleus (STN) of basal ganglia~\\cite{lanciego2012functional}, as shown in the orange areas of Fig.~\\ref{dmsnn}. The BG brain area component adopts the \\emph{LIF} neuron model in BrainCog, as well as the \\emph{STDP} learning rule and \\emph{CustomLinear} to build internal connections of the BG. Then, the BG brain area component can be used to build brain-inspired decision-making SNNs (see section 3.2.1 for detail).\n\n(2) Prefrontal Cortex (PFC):\n\nPFC is of significant importance when human high-level cognitive behaviors happen. In BrainCog, many cognitive tasks based on SNN are inspired by the mechanisms of the PFC~\\cite{bechara1998dissociation}, such as decision-making, working memory~\\cite{rao1999isodirectional,d2000prefrontals,lara2015role}, knowledge representation~\\cite{wood2003human}, theory of mind and music processing~\\cite{frewen2015healing}. Different circuits are involved to complete these cognitive tasks. In BrainCog, the data-driven PFC column model contains 6 layers and 16 types of neurons. The distribution of neurons, membrane parameters and connections of different types of neurons are all derived from existing biological experimental data. The PFC brain area component mainly employs the \\emph{LIF} neuron model to simulate the neural dynamics. The \\emph{STDP} and \\emph{R-STDP} learning rules are utilized to compute the weights between different neural circuits. \n\n(3) Primary Auditory Cortex (PAC):\n\nThe primary auditory cortex is responsible for analyzing sound features, memory, and the extraction of inter-sound relationships~\\cite{Koelsch2012}. This area exhibits a topographical map, which means neurons respond to their preferred sounds. In BrainCog, neurons in this area are simulated by the \\emph{LIF} model and organized as minicolumns to represent different frequencies of sounds. To store the ordered note sequences, the excitatory and inhibitory connections are updated by \\emph{STDP} learning rule.\n\n\n\n(4) Inferior Parietal Lobule (IPL):\n\nThe function of IPL is to realize motor-visual associative learning~\\cite{macuga2011selective}. The IPL consists of two subareas: IPLM (motor perception neurons in IPL) and IPLV (visual perception neurons in IPL). The IPLM receives information generated by self-motion from ventral premotor cortex (vPMC), and the IPLV receives information detected by vision from superior temporal sulcus (STS). The motor-visual associative learning is established according to the STDP mechanism and the spiking time difference of neurons in IPLM and IPLV.\n\n\n\n\n(5) Hippocampus (HPC):\n\nThe hippocampus is part of the limbic system and plays an essential role in the learning and memory processes of the human brain. Epilepsy patients with bilateral hippocampus removed (e.g. the patient H.M.) have symptoms of anterograde amnesia. They are unable to form new long-term declarative memories~\\cite{milner1998}. This case study has proved that the hippocampus is in the key process of converting short-term memory to long-term memory and plays a vital role~\\cite{smith1981role}. \n\nFurthermore, through electrophysiological means, it was found that the hippocampal region is also crucial for forming new concepts. Specific neurons in the hippocampus only respond selectively to specific concepts, completing the specific encoding between different concepts. Moreover, it was through the study of the hippocampus that neuroscientists discovered the STDP learning rule~\\cite{dan2004spike}, which further demonstrated the high plasticity of the hippocampus.\n\n\n(6) Insula:\n\nThe function of the Insula is to realize self-representation~\\cite{craig2009you}, that is, when the agent detects that the movement in the field of vision is generated by itself, the Insula is activated. The Insula receives information from IPLV and STS. The IPLV outputs the visual feedback information predicted according to its motion, and the STS outputs the motion information detected by vision. If both are consistent, the Insula will be activated.\n\n(7) Thalamus (ThA):\n\nStudies have shown that the thalamus is composed of a series of nuclei connected to different brain parts and heavily contributes to many brain processes. In BrainCog, this area is discussed from both anatomic and cognitive perspectives. Understanding the anatomical structure of the thalamus can help researchers to comprehend the mechanisms of the thalamus. Based on the essential and detailed anatomic thalamocortical data~\\cite{Izhikevichlargescale}, BrainCog reconstructs the thalamic structure by involving five types of neurons (including excitatory and inhibitory neurons) to simulate the neuronal dynamics and building the complex synaptic architecture according to the anatomic results. Inspired by the structure and function of the thalamus, the brain-inspired decision-making model implemented by BrainCog takes into account the transfer function of the thalamus and cooperates with the PFC and basal ganglia to realize multi-brain areas coordinated decision-making model.\n\n(8) Ventral Visual Pathway:\n\nCognitive neuroscience research has shown that the brain can receive external input and quickly recognize objects due to the hierarchical information processing of the ventral visual pathway. The ventral pathway is mainly composed of V1, V2, V4, IT, and other brain areas, which mainly process information such as object shape and color~\\cite{ishai1999distributed,kobatake1994neuronal}. These visual areas form connections through forward, feedback, and self-layer projections. The interaction of different visual areas enables humans to recognize visual objects. The primary visual cortex V1 is selective for simple edge features. With the transmission of information, high-level brain areas combine with lower-level receptive fields to form more complex large receptive fields to recognize more complex objects~\\cite{hubel1962receptive}. Inspired by the structure and function of the ventral visual pathway, BrainCog builds a deep forward SNN with layer-wise information abstraction and a feedforward and feedback interaction deep SNN. The performance is verified on several visual classification tasks. \n\n(9) Motor Cortex:\n\nThe control of biological motor function involves the cooperation of many brain areas. The extra circuits consisting of the PMC, cerebellum, and BA6 motor cortex area are primarily associated with motor control elicited by external stimuli such as visual, auditory, and tactual inputs. Internal motor circuits, including the basal ganglia and SMA, predominate in self-guided, learned movements~\\cite{geldberg1985supplementary,mushiake1991neuronal,gerloff1997stimulation}. Population activity of motor cortical neurons encodes movement direction. Each neuron has its preferred direction. The more consistent the target movement direction is with its preferred direction, the stronger the neuron activities are~\\cite{georgopoulos1995motor,kakei1999muscle}. The cerebellum receives input from motor-related cortical areas such as PMC, SMA, and the prefrontal cortex, which are important for the completion of fine movements, maintaining balance and coordination of movements~\\cite{strick2009cerebellum}. Inspired by the organization of the brain's motor cortex, we use spiking neurons to construct a motion control model, and apply it to the iCub robot, which enables the robot to play the piano according to music pieces.\n\n\\section{Brain-inspired AI}\nComputational units (different neuron models, learning rules, encoding strategies, brain area models, etc.) at multiple scales, provided by BrainCog serve as a foundation to develop functional networks. To enable BrainCog for Brain-inspired AI, cognitive function centric networks need to be built and provided as reusable functional building blocks to create more complex brain-inspired AI. This section introduces various functional building blocks developed based on BrainCog.\n\n\\subsection{Perception and Learning}\nIn this subsection, we will introduce the supervised and unsupervised perception and learning spiking neural networks based on the fundamental components of BrainCog. Inspired by the global feedback connections, the neural dynamics of spiking neurons, and the biologically plausible STDP learning rule, we improve the performance of the spiking neural networks. We show great adaption in the small training sample scenario. In addition, our model has shown excellent robustness in noisy scenarios by taking inspiration from the multi-component spiking neuron and the quantum mechanics. The burst spiking mechanism is used to help our converted SNNs with higher performance and lower latency. Based on this engine, we also present a human-like concept learning framework to generate representations with five types of perceptual strength information.\n\n\\textbf{1. Learning Model}\n\n\\emph{(1) SNN with global feedback connections}\n\n\nThe spiking neural network transmits information in discrete spike sequences, which is consistent with the information processing in the human brain~\\cite{Maass1997Networks}. The training of spiking neural networks has been widely concerned by researchers. Most researchers take inspiration from the mechanism of synaptic learning and updating between neurons in the human brain, and propose biologically plausible learning rules, such as Hebbian theory~\\cite{amit1994correlations}, STDP~\\cite{bi1998synaptic}, and STP~\\cite{zucker2002short}, which have been adopted into the training of spiking neural networks~\\cite{tavanaei2016bio,tavanaei2017multi,diehl2015unsupervised,falez2019multi,zhang2018plasticity,zhang2018brain}. However, most SNNs are based on feedforward structures, while the importance of the brain-inspired structures has been ignored. The anatomical and physiological evidences show that in addition to feedforward connections, numerous feedback connections exist in the brain, especially among sensory areas~\\cite{felleman1991distributed,sporns2004small}. The feedback connections will carry out the predictions from the top layer to cooperate with the local plasticity rules to formulate the learning and inference in the brain. \n\nHere, we introduce the global feedback connections and the local differential learning rule~\\cite{zhao2020glsnn} in the training of SNNs. \n\\begin{figure}[!htbp]\n\t\\centering\n\t\\includegraphics[scale=0.6]{.\/fig\/zdc_pipeline.pdf}\n\t\\caption{The feedforward and feedback pathway in the SNN model. The global feedback pathway propagates the target of the hidden layer, modified from~\\cite{zhao2020glsnn}.}\n\t\\label{snn}\n\\end{figure}\n\nWe use the LIF spiking neuron model in the BrainCog to simulate the dynamical process of the membrane potential $V(t)$ as shown in Eq.~\\ref{equa_lif1}. We use the mean firing rates $S_{l}$ of each layer to denote the representation of the $l_{th}$ layer in the forward pathway, and the corresponding target is denoted as $\\hat{S}_{l}$. Here we use the mean squared loss (MSE) as the final loss function.\n\n$\\hat{S}_{L-1}$ denotes the target of the penultimate layer, and is calculated as shown in Eq.~\\ref{equal_per}, $W_{L-1}$ denotes the forward weight between the $(L-1)_{th}$ layer and the $L_{th}$. $\\eta_t$ represents the learning rate of the target~\\cite{zhao2020glsnn}.\n\\begin{equation}\n\t\\hat{S}_{L-1} = S_{L-1} - \\eta_t\\Delta S = S_{L-1} - \\eta_t W_{L-1}^T(S_{out} - S^T)\n\t\\label{equal_per}\n\\end{equation}\n\nThe target of the other hidden layer can be obtained through the feedback connections:\n\n\\begin{equation}\n\t\\hat{S}_{l} = S_{l} - G_{l}(S_{out} - S^T)\n\t\\label{equal_all}\n\\end{equation}\n\nBy combining the feedforward representation and feedback target, we compute the local MSE loss. We can compute the local update of the parameters with the surrogate gradient. We have conducted experiments on the MNIST and Fashion-MNIST datasets, and achieved 98.23\\% and 89.68\\% test accuracy with three hidden layers. Each hidden layer is set with 800 neurons. The details are shown in Fig.~\\ref{zdc_result}.\n\\begin{figure}[!htbp]\n\t\\centering\n\t\\includegraphics[scale=0.35]{.\/fig\/result_zdc.pdf}\n\t\\caption{The test accuracy on MNIST and Fashion-MNIST datasets of the SNNs with global feedback connections.}\n\t\\label{zdc_result}\n\\end{figure}\n\n\n\\emph{(2) Biological-BP SNN}\n\nThe backpropagation algorithm is an efficient optimization method that is widely used in deep neural networks and has promoted the great success of deep learning. However, due to the non-smoothness of neurons in SNNs, the backpropagation optimization is difficult to be applied to train SNNs directly. To solve the above problem, Bengio et al.~\\cite{bengioEstimatingPropagatingGradients2013} proposed four gradient approximation methods, including Straight-Through Estimator to enable the application of the backpropagation algorithm to neural networks containing nonsmooth neurons. Wu et al.~\\cite{wuSpatioTemporalBackpropagationTraining2018} proposed the Spatio-Temporal Backpropagation (STBP) algorithm to train SNNs by using a differentiable function to approximate spiking neurons through a surrogate gradient method. Based on the spatio-temporal dynamic of SNNs, they achieved the backpropagation of SNNs in both time and space dimensions. Zheng et al.~\\cite{zhengGoingDeeperDirectlyTrained2021} proposed threshold-dependent batch normalization (tdBN) to improve the performance of deep SNNs. Fang et al.~\\cite{fangIncorporatingLearnableMembrane2021} proposed a parametric LIF (PLIF) model to further improve the performance of SNNs by optimizing the time constants of neurons during the training process. \n\nMost of these BP-based SNNs often simply regard SNN as a substitute for RNN, but ignore the dynamic of spiking neurons. To solve the problem of non-differentiable neuronal models, Bohte et al.~\\cite{bohte2011error} approximated the backpropagation process of neuronal models using the surrogate gradient method. However, this method results in gradient leakage and does not allow for proper credit assignment in both temporal and spatial dimensions. Also, due to the reset operation after the spikes are emitted, the errors in the backward process can not propagate across spikes. To solve the above problem, we propose Backpropagation with biologically plausible spatio-temporal adjustment~\\cite{shen2022backpropagation}, as shown in Fig.~\\ref{bpsta}, which can correctly assign credit according to the contribution of the neuron to the membrane potential at each moment.\n\n\\begin{figure}[!htbp]\n\t\\centering\n\t\\includegraphics[scale=0.48]{.\/fig\/shen_bpsta.pdf}\n\t\\caption{The forward and backward process of biological BP-based SNNs for BrainCog.}\n\t\\label{bpsta}\n\\end{figure}\n\nBased on LIF spiking neuron, the direct input encoding strategy, the MSE loss function and the surrogate gradient function supplied in BrainCog, we propose a Biologically Plausible Spatio-Temporal Adjustment (BPSTA) to help BP algorithm with more reasonable error adjustment in the spatial temporal dimension~\\cite{shen2022backpropagation}.  The algorithm realizes the reasonable adjustment of the gradient in the spatial dimension, avoids the unnecessary influence of the neurons that do not generate spikes on the weight update, and extracts more important features. By applying the temporal residual pathway, our algorithm helps the error to be transmitted across multiple spikes, and enhances the temporal dependency of the BP-based SNNs. Compared with SNNs and ANNs with the same structure that only used the BP algorithm, our model greatly improves the performance of SNNs on the DVS-CIFAR10 and DVS-Gesture datasets, while also greatly reducing the energy consumption and decay of SNNs, as shown in Tab.~\\ref{compare2}.\n\\begin{table}[!htbp]\n \\centering\n \\caption{The energy efficiency study. The former represents our method, the latter represents the baseline, adopted from~\\cite{shen2022backpropagation}.}\n \\begin{tabular}{cccc}\n  \\toprule[2pt]\n   Dataset   &  Accuracy    &  Firing-rate  &EE = $\\frac{E_{ANN}}{E_{SNN}}$ \\\\\n  \\midrule[2pt]\n   MNIST     &  99.58\\%\/99.42\\% & 0.082\/0.183 & 35.1x\/15.7x  \n  \\\\\n  \n    N-MNIST   &  99.61\\%\/99.32\\%  & 0.097\/0.176   & 29.6x\/16.3x      \\\\\n  \n   CIFAR10   &  92.33\\%\/89.49\\%  & 0.108\/0.214 & 26.6x\/13.4x    \\\\ \n  \n   DVS-Gesture  &  98.26\\%\/93.92\\%  & 0.083\/0.165   & 34.6x\/17.4x    \\\\ \n       \n        DVS-CIFAR10  &  77.76\\%\/71.40\\%  & 0.097\/0.177  & 29.5x\/16.2x    \\\\\n  \\bottomrule[2pt]\n \\end{tabular}\n \\label{compare2}\n\\end{table} \n\n\\emph{(3) Unsupervised STDP-based SNN}\n \nUnsupervised learning is an important cognitive function of the brain. The brain can complete the task of object recognition by summarizing the characteristics and features of objects. Unsupervised learning does not require explicit labels, but extracts the features of samples adaptively in learning process. Modeling of this ability of the brain is critical. There are multiple learning rules in the brain to accomplish various learning tasks. STDP is a widespread rule of synaptic weight modification in the brain. It updates the synaptic weights according to the temporal relationship of the pre- and post-synaptic spikes. Compared with the backpropagation algorithm widely used in DNN with gradient calculation, STDP is more biological plausible. However, unlike the backpropagation algorithm that relies on a large number of sample labels, STDP is a local optimization algorithm. Due to the lack of global information, the ability of self-organization and coordination between neurons is insufficient. In the SNN model, it will lead to disorder and uncertainty of spikes discharge, and it is difficult to achieve a stable release balance state of neurons. To this end, we design an unsupervised STDP-based spiking neural network model based on BrainCog, and bring unsupervised learning to BrainCog as a functional module.\nAs shown in Fig.~\\ref{frame}.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.48\\textwidth]{fig\/frame.pdf}\n\\caption{The framework of the unsupervised STDP-based spiking neural network model, which introduces the adaptive synaptic filter (ASF), the adaptive threshold balance (ATB), and the adaptive lateral inhibitory connection (ALIC) mechanisms to improve the information transmission and feature extraction of STDP-based SNNs. This figure is from \\cite{dong2022unsupervised}.}\n\\label{frame}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.49\\textwidth]{fig\/stp1.pdf}\n\\caption{(a) The adaptive synaptic filter and the adaptive threshold balance jointly regulate the neuron spikes balance of neuron. (b) The adaptive lateral inhibitory connection has different connections for different samples, preventing neurons from learning the same features. This figure is from~\\cite{dong2022unsupervised}.}\n\\label{stp}\n\\end{figure}\n\nTo solve the above problems, we introduced various adaptive mechanisms to improve the self-organization ability of the overall network. STP is another synaptic learning mechanism that exists in the brain. Inspired by STP, we designed an adaptive synaptic filter (ASF) that integrates input currents through nonlinear units, and an adaptive threshold balance (ATB) that dynamically changes the threshold of each neuron to avoid excessively high or low firing rates. The combination of the two controls the firing balance of neurons. As the Fig.~\\ref{stp} shows. We also address the problem of coordinating neurons within a single layer with an adaptive lateral inhibitory connection (ALIC). The mechanism have different connection structures for different input samples. Finally, in order to solve the problem of low efficiency of STDP training, we designed a sample temporal batch STDP. It combines the information between temporal and samples to uniformly update the synaptic weights, as shown by the following formula.\\begin{equation}\n\t\\begin{split}\n\t\t&\\frac{dw_{j}^{(t)}}{dt}^{+}=\\sum \\limits_{m=0}^{N_{batch}}\\sum \\limits_{n=0}^{T_{batch}}\\sum\\limits_{f=1}^{N} W(t^{f,m}_{i}-t^{n,m}_{j})\\\\\n\t\\end{split}\n\t\\label{eq5}\n\\end{equation} where $W(x)$ is the function of STDP, $N_{batch}$ is the batchsize of the input, $T_{batch}$ is the batch of time step, N is the number of neurons. We verified our model on MNIST and Fashion-MNIST, achieving 97.9\\% and 87.0\\% accuracy, respectively. To the best of our knowledge, these are the state-of-the-art results for unsupervised SNNs based on STDP.  \n\n\n\\textbf{2. Adaptability and Optimization}\n\n\\emph{(1) Quantum superposition inspired SNN}\n\n\nIn the microscopic size, quantum mechanics dominates the rules of operation of objects, which reveals the probabilistic and uncertainties of the world. New technologies based on quantum theory like quantum computation and quantum communication provide an alternative to information processing. Researches show that biological neurons spike at random and the brain can process information with huge parallel potential like quantum computing. \n\nInspired from this, we propose the Quantum Superposition Inspired Spiking Neural Network (QS-SNN)~\\cite{SUN2021102880}, complementing quantum image (CQIE) method to represent image in the form of quantum superposition state and then transform this state to spike trains with different phase. Spiking neural network with time differential convolution kernel (TCK) is used to do further classification shown in Fig.~\\ref{Fig:QSSNN}.\n\n\n\n\n\\begin{figure}[!htb]\n\t\\centering\n\t\\includegraphics[width=1.0\\linewidth]{fig\/QSSNN.pdf}  \n\t\\caption{Quantum superposition inspired spiking neural network, adopted from \\cite{SUN2021102880}.}\n\t\\label{Fig:QSSNN}\n\\end{figure}\n\n\n\\begin{figure}[!htb]\n\t\\centering\n\t\\includegraphics[width=1.0\\linewidth]{fig\/mnist_shift_result.pdf}  \n\t\\caption{Performance of QS-SNN on background MNIST inverse image, adopted from \\cite{SUN2021102880}.}\n\t\\label{Fig:mnist-res}\n\\end{figure}\n\nThe effort tries to incorporate the quantum superposition mechanism to SNNs as a new form of encoding strategy for BrainCog, and the model finally shows its capability on robustness for learning. The proposed QS-SNN model is tested on color inverted MNIST datasets. The background-inverted picture is encoded in the quantum superposition form as shown in Eq.~\\ref{QS-SNN} and~\\ref{QS-SNN_limitation}.\n\\begin{equation}\n\t\\mathinner{|I(\\theta)\\rangle}=\\frac{1}{2^n}\\sum\\limits_{i=0}^{2^{2n}-1}(cos(\\theta_{i})\\mathinner{|x_{i}\\rangle}+sin(\\theta_{i})\\mathinner{|\\bar{x}_{i}\\rangle})\\otimes\\mathinner{|i\\rangle}, \\\\\n\t\\label{QS-SNN}\n\\end{equation}\n\n\n\\begin{equation}\n\t\\theta_{i} \\in [0, \\frac{\\pi}{2}], i= 1, 2, 3, \\dots, 2^{2n}-1.\n\t\\label{QS-SNN_limitation}\n\\end{equation}\n\nSpike sequences of different frequencies and phases are generated from the picture information of the quantum superposition state. Furthermore we use two-compartment spiking neural networks to process these spike trains.\n\nWe compare the QS-SNN model with other convolutional models. The result in Fig.~\\ref{Fig:mnist-res} shows that our QS-SNN model overtakes other convolutional neural networks in recognizing background-inverted image tasks.\n\n\\emph{(2) Unsupervised SNN with adaptive learning rule and structure}\n\nBrain can accomplish specific tasks by adaptively learning to organize the features of a small number of samples. Few-shot learning is an important ability of the brain. In the above section, an unsupervised STDP-based spiking neural network is introduced~\\cite{dong2022unsupervised}. To better illustrate the power of our model on small sample training, we tested the model with small samples and find that this model has stronger small sample processing ability than ANN with similar structures, as shown in Tab.~\\ref{small}~\\cite{dong2022unsupervised}.\n\n\\begin{table}[h]\n\t\\caption{The performance of unsupervised SNN compared with ANN on MNIST dataset with different number of training samples~\\cite{dong2022unsupervised}.}\n\t\\centering\n\t\\resizebox{0.8\\linewidth}{!}{\n\t\t\\begin{tabular}{lrrrr}\n\t\t\t\\toprule \n\t\t\tsamples&200  &100&50&10 \\\\\n\t\t\t\\midrule\t\n\t\t\tANN&79.77\\%&71.40\\%&68.72\\%&47.12\\%\\\\\n\t\t\tOurs&81.45\\%&75.44\\%&72.88\\%&51.45\\%\\\\\n\t\t\t\\midrule\n\t\t\t&1.68\\%&4.04\\%&4.16\\%&4.33\\%\\\\\n\t\t\t\\bottomrule\n\t\t\\end{tabular}\n\t}\n\t\\label{small}\n\t\n\\end{table}\n\n\n\n\n\\emph{(3) Efficient and Accurate Conversion of SNNs}\n\n\tSNNs have attracted attention due to their biological plausibility, fast inference, and low energy consumption. However the training methods based on the plasticity~\\cite{zeng2017improving} and surrogate gradient algorithms~\\cite{wu2018spatio} need much memory and perform worse than ANN on large networks and complex datasets. For users of BrainCog, we are certain there will be clear need to use SNNs while keep the benifit from ANNs. As an efficient method, the conversion method combines the characteristics of backpropagation and low energy consumption and can achieve the same excellent performance as ANN with lower power consumption~\\cite{li2021free, han2020deep, han2020rmp}. However, the converted SNNs typically suffer from severe performance degradation and time delays. \n\t\n\t\\begin{figure}[htb]\n    \t\\centering\n    \t\\includegraphics[scale=0.3]{fig\/error.png}\n    \t\\caption{The conversion errors from IF neuron, time dimension, and MaxPooling, adopted from~\\cite{li2022efficient}.}\n    \t\\label{error}\n    \\end{figure}\n\n    We divide the performance loss to IF neurons, time dimension, and MaxPooling layer~\\cite{li2022efficient}, as shown in Fig.~\\ref{error}. In SNN, the neuron can only send one spike at most in each time step, so the maximum firing rate of the neuron is 1. After normalizing the weight of trained ANN, some activation values greater than 1 cannot be effectively represented by IF neurons. Therefore, the residual membrane potential in neurons will affect the performance of the conversion to some extent. In addition, since IF receives pre-synaptic neuron spikes for information transmission, the synaptic current received by the neuron at each time is unstable, but its sum can be approx to the converted value by increasing the simulation time. However, the total number of spikes is the time-varying extremum of the sum of the synaptic currents received. When the activation value corresponding to the IF neuron is negative and the total synaptic current received by the neuron in a short time exceeds the threshold, the neuron that should be resting will issue the spike, and the influence of the spike on the later layers cannot be eliminated by increasing the simulation time. Finally, in the conversion of the MaxPooling layer, the previous work has enabled spikes from the neuron with the maximum firing rate to pass through. However, due to the instability of synaptic current, neurons with the maximum firing rate are often not fixed, which makes the output of the converted MaxPooling layer usually larger.\n\n    \\begin{figure}[!h]\n    \t\\centering\n    \t\\includegraphics[scale=0.5]{fig\/conver.pdf}\n    \t\\caption{The conversion methods. (a) Burst spikes increase the upper limit of firing rates; (b) spike calibration corrects the effect of faulty spikes on conversion, and (c) LIPooling uses lateral inhibition mechanisms to achieve accurate conversion of the MaxPooling layers. Refined based on~\\cite{li2022efficient,li2022spike}.}\n    \t\\label{conver}\n    \\end{figure}\n\n    To solve the problem of the residual membrane potential of neurons, we introduce the burst mechanism, as shown in Fig.~\\ref{conver} (a), which enables neurons to send more than one spike between two time steps, depending on the current membrane potential. Once some neurons have residual information remaining, they can send spikes between two time steps. In this way, the firing rate of SNN can be increased, and the membrane potential remaining in the neuron can be transmitted to the neuron of the next layer. \n    \n    For classification problems, SNN only needs to ensure that the index of the maximum output is correct, but for more demanding conversion tasks, such as object detection, the solution of SIN problem is worth exploring. Note that the spikes emitted by hidden layer neurons are unstable, but the mean of its inter-spike interval distribution is related to its corresponding activation value~\\cite{li2022spike}. We monitor each neuron's spiking time in the forward propagation process and update its average inter-spike interval, as shown in Fig.~\\ref{conver} (b). Under a certain time allowance, the neuron that does not emit spikes will be determined to be Inactivated Neuron. Then, the twin weights emit spikes to suppress the influence of historical errors and calibrate the influence of wrong spikes to a certain extent to ensure accurate conversion.\n\n\n    Inspired by the lateral inhibition mechanism~\\cite{blakemore1970lateral}, we propose LIPooling for converting the maximum pooling layer, as shown in Fig.~\\ref{conver} (c). From the operation perspective, the inhibition of other neurons by the winner in LIPooling is -1, so the neuron with the largest firing rate at the current time step may not spike due to the inhibition of other neurons in history. From the output perspective, LIPooling sums up the output of all neurons during simulation. So the key is that LIPooling uses competition between neurons to get an accurate sum (equal to the actual maximum), instead of picking the winner.\n\n\\textbf{3. Multi-sensory Integration}\n\n\nOne can build SNN models through BrainCog to process different types of sensory inputs, while the human brain learns and makes decisions based on multi-sensory inputs. When information from various sensory inputs is combined, it can lead to increased perception, quicker response times, and better recognition. Hence, enabling BrainCog to process and integrate multi-sensory inputs are of vital importance.\n\nIn this section, we focus on concept learning with multi-sensory inputs. We present a multi-sensory concept learning framework based on BrainCog to generate integrated vectors with the multi-sensory representation of the concept.\n\nEmbodied theories, which emphasize that meaning is rooted in our sensory and experiential interactions with the environment, supports multi-sensory representations.\nBased on SNNs, we present a human-like framework to learn concepts which can generate integrated representations with five types of perceptual strength information~\\cite{wywFramework}.\nThe framework is developed with two distinct paradigms: Associate Merge (AM) and Independent Merge (IM), as Fig.~\\ref{MultisensoryIntegrationFramework} shows.\n\\begin{figure}[h!]\n  \\centering\n  \\includegraphics[width=8cm]{.\/fig\/MultisensoryIntegrationBrainCogNEW.png}\\\\\n  \\caption{The Framework of Concept Learning Based on SNNs with multi-sensory Inputs.}\n  \\label{MultisensoryIntegrationFramework}\n\\end{figure}\n\nIM is based on the widely accepted cognitive psychology premise that each type of sense for the concept is independent before integration~\\cite{wywFramework}.\nAs the input to the model, we will employ five common perceptual strength: visual, auditory, haptic, olfactory and gustatory.\nDuring the data preparation step, we min-max normalize all kinds of perceptual strength of the concept in the multi-sensory dataset so that each value of the vector is in $[0, 1]$.\nWe regard them as stimuli to the presynaptic neurons.\nIt's a 2-layer SNN model, with 5 neurons in the first layer matching the concept's 5 kinds of perceptual strength, and 1 neuron in the second layer representing the neural state following multi-sensory integration.\nIn this paradigm, we use perceptual strength based presynaptic Poisson neurons and LIF or Izhikevich as the postsynaptic neural model.\n\nThe weights between the neurons are $W^i= \\frac{g_i}{\\Sigma_i^n g_i} $ where  $g_i = \\frac{1}{\\sigma_i^2}$,$\\sigma_i^2$ is the variance of each kind of perceptual strength.\nWe convert the postsynaptic neuron's spiking train $S^{post}([0, T])$ in $[0, T]$ into integrated representations $B^{IM}([0,T])$ for the concept in this form:\n\\begin{equation}\n\\begin{aligned}\nB^{IM}([0,T]) &= [\\mathcal{T}(S^{post}((0, tol])), \\mathcal{T}(S^{post}((tol, 2*tol])), \\cdots , \\\\\n& \\mathcal{T}(S^{post}(((k-1)*tol, k*tol])) , \\cdots,  \\\\\n& \\mathcal{T}(S^{post}((\\lfloor \\frac{T}{tol} \\rfloor * tol, T]))]\n\\end{aligned}\n\\end{equation}\nHere if the interval has any spikes, the bit is 1. Otherwise it is 0, according to the $\\mathcal{T} (interval)$ function.\n\nThe AM paradigm presupposes that each kind of modality is associated before integration~\\cite{wywFramework}. \nIt includes 5 neurons, matching the concept's 5 distinct modal information sources. \nThey are linked with each other and are not self-connected. \nThe input spike trains to the network are generated using a Poisson event-generation algorithm based on perceptual strength.\nFor each concept, we turn the spike trains of these neurons into the ultimate integrated representations.\n\n\n\nThe weight value is defined by the correlation between each two modalities, i.e. $W = Corr(i, j)$, where $i, j \\in [A, G, H, O, V]$.\nWe convert the spike trains $S^i([0, T])$ of all neurons into binarycode $B^i([0,T])$ and conjoin them as the ultimate vector $B([0,T])$ as follows:\n\\begin{equation}\n\\begin{aligned}\nB^i([0,T]) &= [\\mathcal{T}(S^i((0, tol])), \\mathcal{T}(S^i((tol, 2*tol])), \\cdots , \\\\\n& \\mathcal{T}(S^i(((k-1)*tol, k*tol])) , \\cdots, \\\\\n& \\mathcal{T}(S^i((\\lfloor \\frac{T}{tol} \\rfloor * tol, T]))]\n\\end{aligned}\n\\end{equation}\n\n\\begin{equation}\n\\begin{aligned}\nB^{AM}([0,T]) &= [B^A([0,T]) \\oplus B^H([0,T]) \\oplus  B^G([0,T]) \\\\\n& \\oplus B^O([0,T]) \\oplus B^V([0,T])]\n\\end{aligned}\n\\end{equation}\n\n\nTo test our framework, we conducted experiments with three multi-sensory datasets (LC823~\\cite{wywLC423,wywLC400}, BBSR~\\cite{wywBBSR}, Lancaster40k~\\cite{wywLan40k}) for the IM and AM paradigms, respectively. \nWe used WordSim353~\\cite{wywWS353} and SCWS1994~\\cite{wywSCWS} as metrics~\\cite{wywFramework}. \nThe resuls show that integrated representations are closer to human beings than the original ones based on our framework, according to the overall results: 37 submodels outperformed a total of 48 tests for both AM and IM paradigm~\\cite{wywFramework}.\nMeanwhile, to compare the two paradigms, we introduce concept feature norms datasets which represent concepts with systematic and standardized feature descriptions. \nIn this study, we use the datasets McRae~\\cite{wywMcRae} and CSLB~\\cite{wywCSLB} as criteria.\nThe findings show that the IM paradigm performs better at multi-sensory integration for concepts with higher modality exclusivity. \nThe AM paradigm benefits the concept of uniform perceptual strength distribution. \nFurthermore, we present perceptual strength-free metrics to demonstrate that both paradigms of our framework have excellent generality~\\cite{wywFramework}.\n\n\n\\begin{figure}[h!]\n  \\centering\n  \\includegraphics[width=8cm]{.\/fig\/yuwei1.jpg}\\\\\n  \\caption{The Correlation Results Between Modality Exclusivity and Average of 3 Neighbors' Rankings.}\n  \\label{MEFS}\n\\end{figure}\n\n\n\n\n\n\\subsection{Decision Making}\nThis subsection introduces how BrainCog implements decision-making functions from the perspective of brain neural mechanism modeling and deep SNN-based reinforcement learning models. Using BrainCog, we build a multi-brain areas coordinated SNN model and a spiking deep Q-network to solve decision-making and control problems.\n\n\\textbf{1. Brain-inspired Decision-Making SNN}\n\n\nFor mammalian brain-inspired decision-making, we take inspiration from the PFC-BG-ThA-PMC neural circuit and build brain-inspired decision-making spiking neural network (BDM-SNN) model~\\cite{zhao2018brain} by BrainCog as shown in Fig. ~\\ref{dmsnn}. BDM-SNN contains the excitatory and inhibitory connections within the basal ganglia nuclei and direct, indirect, and hyperdirect pathways from the PFC to the BG~\\cite{frank2006anatomy,silkis2000cortico}. This BDM-SNN model incorporates biological neuron models (LIF and simplified H-H models), synaptic plasticity learning rules, and interactive connections among multi-brain areas developed by BrainCog. On this basis, we extend the dopamine (DA)-regulated BDM-SNN, which modulates synaptic learning for PFC-to-striatal direct and indirect pathways via dopamine. Different from the DA regulation method in~\\cite{zhao2018brain} which uses multiplication to modulate the specified connections, we improve it by introducing R-STDP~\\cite{Eugene2007} (from Eq.~\\ref{rstdpe} and Eq.~\\ref{rstdpw}) to modulate the PFC-to-striatal connections. \n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=8cm]{.\/fig\/bdmsnn.jpg}\n\\end{center}\n\\caption{The architecture of DA-regulated BDM-SNN, refined based on~\\cite{zhao2018brain}.}\\label{dmsnn}\n\\end{figure}\n\n\nThe BDM-SNN model implemented by BrainCog could perform different tasks, such as the Flappy Bird game and has the ability to support UAV online decision-making. For the Flappy Bird game, our method achieves a performance level similar to humans, stably passing the pipeline on the first try. Fig.~\\ref{fb}a illustrates the changes in the mean cumulative rewards for LIF and simplified H-H neurons while playing the game. The simplified H-H neuron achieves similar performances to that of LIF neurons. BDM-SNN with different neurons can quickly learn the correct rules and keep obtaining rewards. We also analyze the role of different ion channels in the simplified H-H model. From Fig.~\\ref{fb}b, we find that sodium and potassium ion channels have opposite effects on the neuronal membrane potential. Removing sodium ion channels will make the membrane potential decay, while the membrane potential rises faster and fires earlier when removing potassium ion channels. These results indicate that sodium ion channels can help increase the membrane potential, and potassium ion channels have the opposite effect. Experimental results also indicate that BDM-SNN with simplified H-H model that removes sodium ion channels fails to learn the Flappy Bird game.\n\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=8.8cm]{.\/fig\/hhfly.jpg}\n\\end{center}\n\\caption{(a) Experimental result of BrainCog based BDM-SNN on Flappy Bird. The y-axis is the mean of cumulative rewards. (b) Effects of different ion channels on membrane potential for simplified H-H model.}\\label{fb}\n\\end{figure}\n\nIn addition, for the UAV decision-making tasks in the real scene, our model could perform potential applications including flying over doors and windows and obstacle avoidance, which have been realized in~\\cite{zhao2018brain}. Users only need to divide the state space and action space according to different tasks, call the DA-regulated BDM-SNN decision-making model, and combine the UAV's action control instructions to complete the UAV's decision-making process.\n \n \nThis part of the work mainly draws on the neural structure and learning mechanism of brain decision-making and proposes the multi-brain areas coordinated decision-making spiking neural network constructed by BrainCog, and verifies the ability of reinforcement learning in different application scenarios. \n\n\n\\textbf{2. Spiking Deep Q Network with potential based layer normalization}\n\nDeep Q network is widely used for decision-making tasks, and it is required to have SNN based deep Q network for BrainCog so that it can be used for SNN based decision making. We propose potential-based layer normalization spiking deep Q network (PL-SDQN) model to combine SNN with deep reinforcement learning~\\cite{sun2022solving}. We use the LIF neuron model in BrainCog to simulate neurodynamics. Deep spiking neural networks are difficult to be applied to reinforcement learning tasks. On the one hand, it is due to the complexity of reinforcement learning task itself, on the other hand, it is challenging to train spiking neural networks and transmit spiking signal characteristics in deep layers. We find that the spiking deep Q network quickly dissipates the spiking signal in the convolutional layer. Inspired by how local environmental potentials influence brain neurons, we propose the potential-based layer normalization (pbLN) method. The $x_t$, the postsynaptic potential of convolution layers, are normalized as \n\\begin{equation}\n\t\\hat{x_t} = \\frac{x_t-\\bar{x}_t}{\\sqrt{\\sigma_{x_t}+\\epsilon}}\n\t\\label{Eq:norm}\n\\end{equation}\t\n\n\\begin{equation}\n\t\\bar{x}_t = \\frac{1}{H} \\sum_{i=1}^Hx_{t, i}\n\\end{equation}\t\n\n\\begin{equation}\n\t\\sigma_{x_t} = \\frac{1}{H}\\sqrt{\\sum_{i=1}^H(x_{t, i}-\\bar{x}_t)}\n\\end{equation}\n\n\nWe construct PL-SDQN model as shown in Fig.~\\ref{Fig:sdqn}. Atari game images are processed by spiking convolution network and pbLN method and input into a fully connected LIF neural network. The spiking output of PL-SDQN is weighted and summed to continue state-action values. \n\n\\begin{figure}[h!]\n\t\\centering\n\t\\includegraphics[width=8cm]{fig\/SDQN_struct.pdf}\n\t\\caption{The framework of PL-SDQN. Refined from~\\cite{sun2022solving}.}\n\t\\label{Fig:sdqn}\n\\end{figure}\n\n\n\nWe compared our model with the original ANN-based DQN model, and the results are shown in Fig.~\\ref{Fig:game_res}. It shows that our model achieved better performance compared with vanilla DQN model.\n\n\n\\begin{figure}[h!]\n\t\\centering\n\t\\includegraphics[width=8cm]{fig\/game_res_new_resize.pdf}\n\t\\caption{PL-SDQN performance on Atari games, adopted from~\\cite{sun2022solving}.}\n\t\\label{Fig:game_res}\n\\end{figure}\n\n\n\\subsection{Motor Control}\n\nNeuromorphic models on robot control can achieve more robust and energy efficient effects than conventional methods. Spiking neural networks have been used in robot control studies like navigation~\\cite{wang2014mobile} and robot arm control~\\cite{Tieck2018}. Inspired by the brain motor circuit, we construct a multi-brain areas coordinated SNN robot motor control model, to extend BrainCog to control various robots to model embodied intelligence.\n\nWe construct the brain-inspired motor control model with LIF neuron provided by BrainCog. The whole network model architecture is shown in Fig.~\\ref{Fig:motor}. The high-level motion information is produced by SMA and  PMC modules. As discussed above, the function of SMA is to process internal movement stimuli and is responsible for the planning and abstraction of advanced actions. The SMA model contains LIF neurons and receives input signals. One part of the output pulse of the SMA module stimulates the PMC module, and the other part is received by the BG module. The output of the BG module is used as a supplementary signal for action planning, and serves as the input to the PMC together with the SMA signal.\n\n\\begin{figure}[h!]\n\t\\centering\n\t\\includegraphics[width=8cm]{fig\/motor.pdf}\n\t\\caption{A spiking neural network for motor control based on BrainCog.}\n\t\\label{Fig:motor}\n\\end{figure}\n\n\nIn order to expand the dimension of neuron direction representation, we use neuron population coding to output the high-level action abstraction. Population-coded spiking neural network has been used for energy-efficient continuous control~\\cite{tang2021deep}, showing population coding can increase the ability of spiking neurons to represent precise continuous values. In our work, we are inspired by neural mechanisms of population encoding of motion directions in the brain and use LIF neuron groups to process the output spikes from PMC module. \n\nThe cerebellum plays an important role in motor coordination and fine regulation of movements. We built spiking neural network based cerebellum model to process the high-level motor control population embedding. The outputs of populations are fused to encode motor control information generated by high-level cortex area inputs to a three-layer cerebellum spiking neural network, including GCs, PCs and DCN modules. And the cerebellum takes pathway connections like DenseNet~\\cite{huang2017densely}. The DCN layer generates the final joint control outputs.\n\n\n\\subsection{Knowledge Representation and Reasoning}\nThis subsection shows how the BrainCog platform achieve the ability of knowledge representation and reasoning. Via neuroplasticity and population coding mechanisms, spiking neural networks acquire music and symbolic knowledge. Moreover, on this basis, cognitive tasks such as music generation, sequence production, deductive reasoning and inductive reasoning are realized.\n\n\n\\textbf{1. Music Memory and Stylistic Composition SNN}\n\n\nMusic is part of human nature. Listening to melodies involves sensory perception, personal memory, action, emotion, and even creative behaviors, etc~\\cite{Koelsch2012}. Music memory is a fundamental part of musical behaviors, and humans have strong abilities to store a sequence of notes in the brain. Learning and creating music are also essential processes. A musician engages his memory, emotion, musical knowledge and skills to write a beautiful melody. Actually, neuroscientists have found that many brain areas need to collaborate to complete the cognitive behaviors with music. Inspired by brain mechanisms, this paper focuses on the two key issues of music memory and composition, which are modeled by spiking neural networks based on the BrainCog platform.\n\n\\subsubsection{Musical Memory Spiking Neural Network} \\label{secA}\nA musical melody is composed of a sequence of notes. Pitch and duration are two essential attributes of a note. Scientists have found that the primary auditory cortex provides a tonotopic map to encode the pitches, which means that neurons in this region have their preferences of pitches~\\cite{Kalat2015}. Meanwhile, neural populations in the medial premotor cortex have the preferences of all the time intervals covered in hundreds of milliseconds~\\cite{merchant2013a}. Besides, researchers have emphasized the contribution of the hippocampus in sequence memory~\\cite{Fortin2002}. Inspired by these mechanisms, this work proposes a spiking neural model, which contains collaborated subnetworks to encode, store and retrieve the music melodies~\\cite{LQ2020}.\n\n\\emph{Encoding:} As is shown in Fig.~\\ref{01}, this work defines pitch subnetwork and duration subnetwork to encode pitches and durations of musical notes respectively. These two subnetworks are composed of numbers of minicolumns with different preferences. Synaptic connections with transmission delays exist between neurons from different layers. Besides, a cluster that represents the title of a musical melody is composed of numerous individual neurons. This cluster has the feedforward and feedback connections with pitch and duration subnetworks. Since the BrainCog platform supports various neural models, this work takes LIF model to simulate neural dynamics.\n\\begin{figure}[!htbp]\n    \\centering\n    \\includegraphics[scale=0.4]{.\/fig\/music1.png}\n    \\caption{The architecture of the music memory model, refined based on~\\cite{LQ2020}.}\n    \\label{01}\n\\end{figure}\n \n\\emph{Storing:} Based on the encoding process, as the notes input sequently, the neurons in pitch and duration subnetworks with different preferences respond to these sequential notes and fire orderly. Meanwhile, connections between these neurons are computed and updated by the STDP learning rule. It is important to indicate that the neurons and synapses are grown dynamically. Besides, synaptic connections between the title cluster and other two subnetworks are generated and updated by the STDP learning rule simultaneously. The details of note sequence memorizing can be found in our previous work~\\cite{LQ2020}.\n\n\\emph{Retrieving:} Given the title of a musical work, the ordered notes can be recalled accurately. Since the weights of connections are updated in storing process, neural activities in the title cluster lead to the excitations of neurons in pitch and duration subnetworks. Then, the notes are retrieved in order. We use a public corpus that contains 331 classical piano works~\\cite{Krueger2018} recorded by MIDI standard format to evaluate the model. The experiments have shown that our model can memorize and retrieve the melodies with an accuracy of 99\\%. The details of the experiments have been discussed in our previous work~\\cite{LQ2020}.\n\n\n\\subsubsection{Stylistic Composition Spiking Neural Network}\n\n\nHow to learn and make music are quite complex processes for humans. Scientists have found that the memory system and knowledge experience participate in human creative behaviors~\\cite{Dietrich2004}. Many brain areas like the prefrontal cortex are engaged in human creativity~\\cite{Jung2013}. However, the details of brain mechanisms are still unclear. Inspired by the current neuroscientific findings, the BrainCog introduces a spiking neural network for learning musical knowledge and creating melodies with different styles~\\cite{LQ2021}. \n\n\\begin{figure}[!htbp]\n    \\centering\n    \\includegraphics[scale=0.28]{.\/fig\/2.pdf}\n    \\caption{Stylistic composition model inspired by brain mechanisms. Refined based on~\\cite{LQ2021}.}\n    \\label{02}\n\\end{figure}\n\n\\emph{Musical Learning:} This work proposes a spiking neural model which is composed of a knowledge network and a sequence memory network. As is shown in Fig.~\\ref{02}, the knowledge network is designed as a hierarchical structure for encoding and learning musical knowledge. These layers store the genre (such as Baroque, Classical, and Romantic), the names of famous composers and the titles of musical pieces. Neurons in the upper layers project their synapses to the lower layers. The sequence memory network stores the ordered notes which have been discussed in section~\\ref{secA}. During the learning process, synaptic connections are also projected from the knowledge network to the sequence memory network. This work also takes LIF model which is supported by the BrainCog platform to simulate neural dynamics. Furthermore, all the connections are generated and updated dynamically by the STDP learning rule. \n\n\\begin{figure}[htbp]\n    \\centering\n    \\includegraphics[scale=0.6]{.\/fig\/3.pdf}\n    \\caption{A sample of a generated melody with Bach's characteristic.}\n    \\label{03}\n\\end{figure}\n\n\n\\emph{Musical Composition:} Based on the learning process, genre-based and composer-based melody compositions are discussed in this paper. Given the beginning notes and the length of the melody to be generated, the genre-based composition can produce a single-part melody with a specific genre style. This task is achieved by the neural circuits of genre cluster and sequential memory system. Similarly, the composer-based composition can produce melodies with composers' characters. The composer cluster and sequential memory system circuits contribute to this process~\\cite{LQ2021}. We also use a classical piano dataset including 331 musical works recorded by MIDI format~\\cite{Krueger2018} to train the model. Fig.~\\ref{03} shows a sample of the generated melody with Bach's style. The details of stylistic composition can be referred to our previous work~\\cite{LQ2021}.\n\n\n\nA total of 41 human listeners are invited to evaluate the quality of the generated melodies, and they are divided into two groups, one of which has a musical background. Experiments have shown that the pieces produced by the model have strong characteristics of different styles and some of them sound nice.\n\n\\textbf{2. Brain-Inspired Sequence Production SNN}\n\nSequence production is an essential function for AI applications. Components in BrainCog enable the community to build SNN models to handle this task. In this paper, we introduce the brain-inspired symbol sequences production spiking neural network (SPSNN) model that has been incorporated in BrainCog~\\cite{fang2021spsnn}. SPSNN incorporates multiple neuroscience mechanisms including Population Coding~\\cite{xie2022geometry}, STDP~\\cite{dan2004spike}, Reward-Modulated STDP~\\cite{fremaux2016neuromodulated}, and Chunking Mechanism~\\cite{pammi2004chunking}, mostly covered and provided by BrainCog. After reinforcement learning, the network can complete the memory of different sequences and production sequences according to different rules.\n\nFor Population Coding, this model utilizes populations of neurons to represent different symbols. The whole neural loop of SPSNN is divided into Working Memory Circuit, Reinforcement Learning Circuit, and Motor Neurons~\\cite{fang2021spsnn}, shown in Fig.~\\ref{SPSNN}. The Working Memory Circuit is mainly responsible for completing the memory of the sequence. The Reinforcement Learning Circuit is responsible for acquiring different rules during the reinforcement learning process. The Motor Neurons can be regarded as the network's output. \n\nIn the working process of the model, the Working Memory Circuit and the Reinforcement Learning Circuit cooperate to complete the memory and production of different sequences~\\cite{fang2021spsnn}. It is worth mentioning that with the increase of background noise, the recall accuracy of symbols at different positions in a sequence gradually decreases, and the overall change trend follows the \"U-shaped accuracy\", which is consistent with experiments in psychology and neuroscience~\\cite{jiang2018production}. The results are highly consistent due to the superposition of primacy and recency effects. Our model provides a possible explanation for both effects from a computational perspective.\n\\begin{figure}[!htbp]\n\\centering\n\\includegraphics [width=8.9cm]{.\/fig\/spsnnloop}\n\\caption{The architecture of SPSNN, adopted from~\\cite{fang2021spsnn}.}\n\\label{SPSNN}\n\\end{figure}\n\n\n\\textbf{3. Commonsense Knowledge Representation Graph SNN}\n\n\nCommonsense knowledge representation and reasoning are important  cornerstones  on the way to realize human-level general  AI~\\cite{minsky2007emotion}. In this module, we build Commonsense Knowledge Representation SNN(CKR-SNN) to explore whether SNN can complete these cognitive function.\n\n\n\nThe hippocampus plays a critical role in the formation of new knowledge memory~\\cite{schlichting2017hippocampus}. Inspired by the population coding mechanism found in hippocampus~\\cite{ramirez2013creating},  this module encodes the entities and relations of commonsense knowledge graph into different  populations of neurons. Via spiking timing-dependent plasticity (STDP) learning principle, the synaptic connections between neuron populations are formed after guiding the sequential firings of corresponding  neuron populations~\\cite{KRRfang2022}.\n\nAs Fig.~\\ref{GSNN} shows, neuron populations together constructed the giant graph spiking neural networks, which contain the commonsense knowledge. In this module, Commonsense Knowledge Representation SNN(CKR-SNN) represents a subset of Commonsense Knowledge Graph ConceptNet~\\cite{Conceptnet55}. After training, CKR-SNN can complete conceptual  knowledge generation and other cognitive tasks~\\cite{KRRfang2022}. \n\n\n\n\\begin{figure}\n\\centering\n\\includegraphics [width=8.9cm]{.\/fig\/GSNN}\n\\caption{Graph Spiking Neural Networks for Commonsense Representation, adopted from~\\cite{KRRfang2022}.}\n\\label{GSNN}\n\\end{figure}\n\n\n\n\n\n\n\n\n\\textbf{4. Causal Reasoning SNN}\n\nIn BrainCog, we constructed causal reasoning SNN, as an instance to verify the feasibility of spiking neural networks to realize deductive and inductive  reasoning. Specifically, Causal Reasoning Spiking Neural Network (CRSNN) module contains a brain-inspired causal reasoning spiking neural network model~\\cite{fang2021crsnn}.\n\n\nThis model explores how to encode a static causal graph into a spiking neural network and implement subsequent reasoning based on a spiking neural network. Inspired by the causal reasoning process of the human brain~\\cite{pearl2018book}, we try to explore how causal reasoning can be implemented based on spiking neural networks. The 3D model of CRSNN is shown in Fig.~\\ref{CRSNN}.\n\nInspired by neuroscience, the CRSNN module adopts the population coding mechanism and uses neuron populations to represent nodes and relationships in the causal graph. Each node indicates different events in the causal graph, as shown in Fig.~\\ref{CRSNN}. By giving current stimulation to different neuron populations in the spiking neural networks and combining the STDP learning rule~\\cite{dan2004spike}, CRSNN can encode the topology between different nodes in a causal graph into a spiking neural network. Furthermore, according to this network, CRSNN completes the subsequent deductive reasoning tasks.\n\nThen, by introducing an external evaluation function, we can grasp the specific reasoning path in the working process of the network according to the firing patterns of the model, which gives the CRSNN more interpretability compared to traditional ANN models~\\cite{fang2021crsnn}. \n\n\n\n\\begin{figure}\n\\centering\n\\includegraphics [width=8.8cm]{.\/fig\/cg3d}\n\\caption{CRSNN 3D model, adapted from~\\cite{fang2021crsnn}.}\n\\label{CRSNN}\n\\end{figure}\n\n\n\n\n\n\\subsection{Social Cognition}\n\nThe nature and neural correlates of social cognition is an advanced topic in cognitive neuroscience. In the field of artificial intelligence and robotics, there are few in-depth studies that take the neural correlation and brain mechanisms of biological social cognition seriously. Although the scientific understanding of biological social cognition is still in a preliminary stage~\\cite{zeng2018toward}, we integrate the biological findings of social cognition into a model to construct a brain-inspired model for social cognition to extend the functions of BrainCog.\n\nUnderstanding ourselves and other people is a prerequisite for social cognition. \n\nAn individual's perception of the world is realized through his own body, and the importance of  knowing oneself is the perception of self-body. Neuroscientific researches show that the inferior parietal lobule (IPL) is activated when the subjects see self-generated actions~\\cite{macuga2011selective} and their own faces~\\cite{sugiura2015neural}. Similar to the IPL, the Insula is activated in bodily ownership and self-recognition tasks~\\cite{craig2009you}. \n\nUnderstanding the mental states of others plays an important role for understanding other people. Theory of mind is an ability to distinguish between self and others and to infer others' mental states (such as desires, goals, beliefs, etc.) in the social context~\\cite{shamay-tsoory_dissociation_2007,sebastian_neural_2012,dennis2013cognitive}. This ability can help us reasonably infer other people's policies and goals. Inspired by this, we believe that applying theory of mind to the agent's decision-making process will improve the agent's inference of other agents, so as to take more reasonable actions. Neuroscientific researches~\\cite{abu-akel_neuroanatomical_2011,hartwright_multiple_2012,hartwright_special_2015, koster-hale_theory_2013} show that the brain areas related to theory of mind are mainly TPJ, part of PFC, ACC and IFG. The IPL contained in the TPJ is mainly used to represent self-relevant information, while the pSTS is used to represent information related to others. The insula, representing the abstract self, can be stimulated with self-related information~\\cite{zeng2018toward}. When theory of mind is going on, the IFG will suppress self-relevant information. Therefore, the TPJ will input other-relevant information into the PFC. The ACC evaluates the value of others' states, so as to help the PFC to infer others. The process of inferring others' goals or behaviors can be understood as simulating other people's decision-making~\\cite{suzuki_learning_2012}. Therefore, this process will be regulated by dopamine from substantia nigra compacta\/ventral tegmental area (Snc\/VTA).\n\n\n\n\n\n\n\n\nWith the neuron model and STDP function provided by the BrainCog framework, a brain-inspired social cognition model is constructed, as shown in Fig.~\\ref{BISC}.\n\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=8cm]{fig\/SC.png}\n\\end{center}\n\\caption{Brain-inspired social cognition model.}\n\\label{BISC}\n\\end{figure}\n\n\n\n\nThe brain-inspired social cognition model contains two pathways: the bodily self-perception pathway and the theory of mind pathway.\n\nThe bodily self-perception pathway (shown in Fig.~\\ref{Self}) consists of inferior parietal lobule spiking neural network (IPL-SNN) and Insula spiking neural network (Insula-SNN). The IPL-SNN realizes motor-visual associative learning. The Insula-SNN realizes the abstract representation of oneself, that is, when the detected movement's visual results match the expected results of its own movement, the Insula will be activated, and the robot considers that the moving part in the field of vision belongs to itself.\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=8cm]{fig\/Self.png}\n\\end{center}\n\\caption{The architecture and pathway of bodily self-perception in the brain-inspired social cognition model.}\n\\label{Self}\n\\end{figure}\n\nThe architecture of IPL-SNN and the process of motor-visual associative learning is shown in Fig.~\\ref{IPL}. The vPMC generates its own motion angle information, and the STS outputs the motion angle information detected by vision. According to the STDP mechanism and the spiking time difference of neurons in IPLM and IPLV, the motor-visual associative learning is established. \n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=8cm]{fig\/IPL.png}\n\\end{center}\n\\caption{Motor-visual associative learning in IPL, adapted from~\\cite{zeng2018toward}.}\n\\label{IPL}\n\\end{figure}\n\nThe architecture of Insula-SNN is shown in Fig.~\\ref{Insula}. The Insula receives angle information from IPLV and STS. After the motor-visual associative learning in IPL, IPLV outputs the visual feedback angle information predicted according to its own motion, and STS outputs the motion angle information detected by vision. If the two are consistent, the Insula will be activated.\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=8cm]{fig\/Insula.png}\n\\end{center}\n\\caption{The architecture of Insula-SNN.}\n\\label{Insula}\n\\end{figure}\n\n\n\nThe theory of mind pathway~\\cite{zhao2022brain} is mainly composed of three modules: the perspective taking module, the policy inference module, the action prediction module, and the state evaluation module (shown in Fig.~\\ref{ToM}).\n\n\\begin{figure}[!htbp]\n\\centering\n\\includegraphics[width=0.48\\textwidth]{.\/fig\/ToM1.jpg}\n\\caption{The architecture of theory of mind in the brain-inspired social cognition model, refined based on~\\cite{zhao2022brain}.}\n\\label{ToM}\n\\end{figure}\n\nThe perspective taking (also called self-perspective inhibition~\\cite{zeng2018toward}) module simulates the function of suppressing self-relevant information in the process of distinguish self and others. The information related to self can stimulate an representation of abstract self. When we infer others, the information related to self can be suppressed. Assuming that the agent knows the environment. When the agent with the theory of mind (ToM) infers the observation of others, it only needs to bring its own observation into the position of others. A matrix is used to represent the observed environment, where 1 indicates that the area can be observed at the location, and 0 indicates that the area cannot be observed at the location. Another matrix is used to represent the position of objects. The position which is occupied by objects is represented by 1, otherwise it is represented by 0. By taking the intersection of the two matrices, the estimation of others' states are obtained. According to the fact that the IFG helps the brain to suppress the performance of self-relevant information in the process of ToM, agent with the ToM will inhibit its own representation of states, and further infer the behavior of others by these estimation of others' states. In summary, the input of this module is the observation vector and the matrix of the environment. The output is the observation vector of others' perspective.\n\nThe dorsolateral prefrontal cortex (DLPFC) has the function of storing working memory and predicting others' behaviors. The action prediction module is used to simulate the function of the DLPFC. The input is formed by the observation value of others' state output by the perspective taking module. The module is a single layer spiking neural network. There is lateral suppression in the output layers. The network is trained by R-STDP. The source of reward is the difference between the predicted value and the real value. When the predicted value is consistent with the real value, the reward is positive. When the predicted value is not equal to the real value, the reward is negative.\n\nThe state evaluation module composed of a single layer spiking neural network simulates the function of the ACC brain area. The inputs of the module are the predicted state and the output is safe or unsafe.\n\nFinally, we conducted  two experiments to test the brain-inspired social cognition model.\n\n\\textbf{1. Multi-Robots Mirror Self-Recognition Test }\n\nThe mirror test is the most representative test of social cognition. Only a few animals passed the test, including chimpanzees~\\cite{RN826}, orangutans~\\cite{RN827}, bonobos~\\cite{RN828}, gorillas~\\cite{RN829,RN830}, Asiatic elephant~\\cite{RN831}, dolphins~\\cite{RN832}, orcas~\\cite{RN833}, macaque monkeys~\\cite{RN835}, etc. Based on the mirror test, we proposed the Multi-Robots Mirror Self-Recognition Test~\\cite{zeng2018toward}, in which three robots with identical appearance move their arms randomly in front of the mirror at the same time, and each robot needs to determine which mirror image belongs to it. The experiment includes training stage and test stage.\n\nThe training stage is shown in Fig.~\\ref{train}. Three blue robots with identical appearance move randomly in front of the mirror at the same time. Each robot establishes the motor-visual association according to the motion angle of its own arm and the angle of visual detection.\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=8cm]{fig\/mirrortest_train.jpg}\n\\end{center}\n\\caption{Training stage in multi-robots mirror self-recognition test, adapted from~\\cite{zeng2018toward}.}\n\\label{train}\n\\end{figure}\n\nThe test stage is shown in Fig.~\\ref{test}. In the test stage, the robot can predicts the visual feedback generated by its arm movement according to the training results. By comparing the similarity between the predicted visual feedback and the detected visual results, the robot can identify which mirror image belongs to it. \n\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=8cm]{fig\/mirrortest_test.jpg}\n\\end{center}\n\\caption{Test stage in multi-robots mirror self-recognition test, adapted from~\\cite{zeng2018toward}.}\n\\label{test}\n\\end{figure}\n\nIn the bodily self-perception pathway, the input is the angle of the robot's random motion and the angle detected by the robot's vision. After training and testing, the output is an image, which is the result of visual motion detection and the result of self motion prediction. The motion track corresponding to the red line in the visual motion detection result is generated by itself. The result is shown in Fig.~\\ref{result}.\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=8cm]{fig\/mirrortest_result.png}\n\\end{center}\n\\caption{The result of IPL-SNN.}\n\\label{result}\n\\end{figure}\n\n\\textbf{2. AI Safety Risks Experiment }\n\nAI safety risks experiment is shown in Fig.~\\ref{fig_sim}. After observing the behavior of the other two agents, the green agent can infer the behaviors of others by utilizing its ToM ability when safety risks may arise due to environmental changes. The experiment was conducted in the environments with some simulated types of safety risks (e.g., the intersection will block the view of agents and may cause agents to collide in the crossing).\n\nThe experiment shows that the agent can infer others when they have different perspectives. In the first two environments, the agent observes the movement of others, and in the third environment, the agent predicts others' actions. In the experiment, we verify the effectiveness of the model by taking the rescue behavior as the standard when other agents might be in danger. The experimental results show that the agent with the ToM can predict the danger of the other agent in a slightly changed environment after watching other agents move in the previous environments.\n\n\\begin{figure}[!htbp]\n\\centering\n\\subfloat[Green agent observes others' behaviors (example 1)]{\\includegraphics[height=0.65in]{.\/fig\/Figure6A.jpg}%\n\\label{1}}\n\\hfil\n\\subfloat[Green agent observes others' behaviors (example 2)]{\\includegraphics[height=0.65in]{.\/fig\/Figure6B.jpg}%\n\t\\label{2}}\n\\hfil\n\\subfloat[Test example 1 (with ToM)]{\\includegraphics[height=0.65in]{.\/fig\/Figure7A.jpg}%\n\t\\label{3}}\n\\hfil\n\\subfloat[Test example 2 (without ToM)]{\\includegraphics[height=0.65in]{.\/fig\/Figure7B.jpg}%\n\t\\label{4}}\n\\caption{Comparison diagram of experimental results. (a) Example 1. The green agent observes others' behaviors. (b) Example 2. The green agent observes others' behaviors. (c) The green agent with ToM can help other agents avoid risks. (d) The green agent without ToM is unable to help other agents avoid risks. Similar results can be found in~\\cite{zhao2022brain}.}\n\\label{fig_sim}\n\\end{figure}\n\n\\section{Brain Simulation}\nBrain simulation includes two parts: brain cognitive function simulation and multi-scale brain structure simulation. We incorporate as much published anatomical data as possible to simulate cognitive functions such as decision-making and working memory. Anatomical and imaging multi-scale connectivity data is used to make whole-brain simulations from mouse, macaque to human more biologically plausible.\n\n\\subsection{Brain Cognitive Function Simulation}\n\n\\textbf{1. \\emph{Drosophila}-inspired Decision-Making SNN}\n\n\\emph{Drosophila} decision-making consists of value-based nonlinear decision and perception-based linear decision, where the nonlinear decision could help to amplify the subtle distinction between conflicting cues and make winner-takes-all choices~\\cite{tang2001choice}. In this paper, the BrainCog framework is used to build \\emph{Drosophila} nonlinear and linear decision-making pathways as shown in Fig.~\\ref{pi}a-b. The entire model consists of a training phase and a testing phase as same as~\\cite{zhao2020neural}. In the training phase, a two-layer SNN with LIF neurons is trained by reward-modulated STDP, which combines local STDP synaptic plasticity with global dopamine regulation. The training phase learns the safe pattern (upright-green T) and the punished pattern (inverted-blue T)~\\cite{zhao2020neural}. Therefore, it is safe for green color and upright T shape factors, while blue color and inverted T shape are dangerous. \n\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=8cm]{.\/fig\/3.jpg}\n\\end{center}\n\\caption{(a) Linear Pathway. (b) Nonlinear Pathway. (c) Experiments for training and choice phases. (d) Experimental results of linear and nonlinear networks under the dilemma. The X-axis refers to the color density, and the Y-axis represents the PI values. Refined based on~\\cite{zhao2020neural}.}\n\\label{pi}\n\\end{figure}\n\nTwo cues (color and shape) are restructured during the testing phase, requiring linear and nonlinear pathways to make a choice between inverted-green T and upright-blue T, as shown in Fig.~\\ref{pi}c. The linear decision directly uses the knowledge acquired during the training phase to make decisions. The nonlinear network models the recurrent loop of the DA\u2011GABA\u2011MB circuit~\\cite{tang2001choice,zhang2007dopamine,zhou2019suppression}: KC activates the anterior posterior lateral (APL) neurons, which in turn releases GABA transmitter to inhibit the activity of KC. KC also provides mushroom body output neuron (MBON) with exciting input in order to generate behavioral choices. When faced with conflicting cues, the level of DA increases rapidly and produces mutual inhibition with APL, thereby producing a disinhibitory effect on KC. The excitatory connection between DA and MBON also helps speed up decision-making.\n\nTo verify the consistency of \\emph{drosophila}-inspired decision-making SNN with the conclusions from neuroscience~\\cite{tang2001choice}, we count the behavior paradigm of our model under different color intensities over a period of time. First, we run the network for 500 steps to count the time $t_1$ of selecting behavior 1 (avoiding) and the time $t_2$ of selecting behavior 2 (approaching). Then we calculate prefer index (PI) values under different color intensity: $PI=\\frac{\\left | t_1-t_2 \\right | }{\\left | t_1+t_2 \\right | } $. From Fig.~\\ref{pi}d, we find that nonlinear circuits could achieve a gain-gating effect to enhance relative salient cue and suppress less salient cue, thereby displaying the nonlinear sigmoid-shape curve~\\cite{zhao2020neural}. However, the linear network couldn't amplify the difference between conflicting cues, thus making an ambiguous choice (linear-shape curve)~\\cite{zhao2020neural}. This work proves that drawing on the neural mechanism and structure of the nonlinear and linear decision-making of the \\emph{Drosophila} brain, the brain-inspired computational model implemented by BrainCog could obtain consistent conclusions with the~\\emph{Drosophila} biological experiment~\\cite{tang2001choice}.\n\n\\textbf{2. PFC Working Memory }\n\n\nUnderstanding the detailed differences between the brains of humans and other species on multiple scales will help illuminate what makes us unique as a species~\\cite{zhang2021comparison}. The neocortex is associated with many cognitive functions such as working memory, attention and decision making~\\cite{miller2000prefontral,nieder2003coding,\nbishop2004prefrontal,koechlin2003architecture,\nwood2003human}. Based on the human brain neuron database of the Allen Institute for Brain Science, the key membrane parameters of human neurons are extracted \\footnote{http:\/\/alleninstitute.github.io\/AllenSDK\/cell\\_types.html}. Different types of human brain neuron models and rodent neuron models are established based on adaptive Exponential Integrate-and-Fire (aEIF) model~\\cite{Fourcaud2003How,brette2005adaptive}, which is supported by BrainCog.\n\nWe refined the model of a  single PFC proposed by Haas and colleagues \\footnote{http:\/\/senselab.med.yale.edu\/ModelDB\/}~\\cite{hass2016detailed}. Subsequently, a 6-layer PFC column model based on biometric parameters was established~\\cite{shapson2021connectomic}. The pyramidal cells and interneurons were proportionally distributed from the literature~\\cite{beaulieu1993numerical,defelipe2011evolution} and connected with different connection probabilities for different types of neurons based on previous studies~\\cite{gibson1999two,hass2016detailed,\ngao2003dopamine}. Firstly, the accuracy of information maintenance was tested on  rodent PFC network model.\n\n\n\\begin{figure}[!htbp]\n\\centering\n\\includegraphics[width=0.5\\textwidth]{.\/fig\/final.png}\n\\caption{Anatomical and network stimulation diagram. (a) The connection of a single PFC column. (b) The distribution proportion of different types of neurons in each column layer. (c) Network persistent activity performance. Refined based on~\\cite{zhangqian2021comparison}.}\n\\end{figure}\n\nKeeping the network structure and other parameters unchanged, only using human neurons to replace rodent neurons can significantly improve the accuracy and integrity of image output. From an evolutionary perspective, the lower membrane capacitance of human neurons facilitates firing. This change improves the efficiency of information transmission, which is consistent with the results of biological experiments~\\cite{eyal2016unique}. This data-driven PFC column model provides an effective simulation-validation platform to study other high-level cognitive functions~\\cite{2020Computational}.\n\n\\subsection{Multi-scale Brain Structure Simulation}\n\n\\textbf{1. Neural Circuit}\n\n\\subsubsection{Microcircuit}\nBrainCog implements a BDM-SNN model inspired by the decision-making neural circuit of PFC-BG-ThA-PMC in the mammalian brain (as shown in Fig.~\\ref{dm})~\\cite{zhao2018brain}. The BDM-SNN models the excitatory and inhibitory reciprocal connections between the basal ganglia nucleus~\\cite{lanciego2012functional}: (1) Excitatory connections: STN-Gpi, STN-Gpe. (2) Inhibitory connections: StrD1-Gpi, StrD2-Gpe, Gpe-Gpi, Gpe-STN. Direct pathway (PFC-StrD1), indirect pathway (PFC-StrD2), and hyperdirect pathway (PFC-STN) from PFC to BG are further constructed. The output from BG transmits an inhibitory connection to the thalamus and finally excites PMC~\\cite{parent1995functional}. In addition, excitatory connections are also formed between PFC and thalamus, and lateral inhibition exists in PMC. Such brain-inspired neural microcircuit, consisting of connections among different cortical and subcortical brain areas, and incorporating DA-regulated learning rules, enable human-like decision-making ability.\n\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=8cm]{.\/fig\/bdm.jpg}\n\\end{center}\n\\caption{The microcircuit of PFC-BG-ThA-PMC. Refined based on~\\cite{zhao2018brain}.}\n\\label{dm}\n\\end{figure}\n\n\\subsubsection{Cortical Column}\n\nA mammalian thalamocortical column is constructed in BrainCog, which is based on detailed anatomical data~\\cite{Izhikevichlargescale}. This column is made up of a six-layered cortical structure consisting of eight types of excitatory and nine types of inhibitory neurons. Thalamic neurons cover two types of excitatory neurons, inhibitory neurons and GABAergic neurons in the reticular thalamic nucleus (RTN). Neurons are simulated by \\emph{Izhikevich} model, which BrainCog applies to exhibit their specific spiking patterns depending on their different neural morphologies. For example, excitatory neurons (pyramidal and spiny stellate cells) always exhibit RS (Regular Spiking) or Bursting modes, while inhibitory neurons (basket and non-basket interneuron) are of FS (Fasting Spiking) or LTS (Low-threshold Spiking) patterns. Each neuron has a number of dendritic branches to accommodate a large number of synapses. The synaptic distribution and the microcircuits are reconstructed in BrainCog based on the previous works~\\cite{Izhikevichlargescale, Binzegger2004A}. Fig.~\\ref{minicolumn}(a) describes the details of the minicolumn. The column contains 1,000 neurons and more than 4,200,000 synapses. To understand the network further, we stimulate the spiny stellate cells in layer 4 to observe the activities of the whole network, Fig.~\\ref{minicolumn}(b) shows the result of neural activities after the cells in layer 4 receive the external stimulation. \n\n\\begin{figure}[!htbp]\n\\centering\n\\includegraphics[width=0.48\\textwidth]{.\/fig\/minicolumn.png}\n\\caption{The thalamocortical column. (a) shows the structure of the column and (b) describes the running activities of the unfold column when the neurons in Layer 4 receive the external stimulus.}\n\\label{minicolumn}\n\\end{figure}\n\n\\textbf{2. Mouse Brain}\n\n\nThe BrainCog mouse brain simulator is a spiking neural network model covering 213 brain areas of the mouse brain, which are classified according to the Allen Mouse Brain Connectivity Atlas~\\cite{richardson2003subthreshold}~\\footnote{http:\/\/connectivity.brain-map.org}. Each neuron was modeled by conductance-based spiking neuron model and simulated with a resolution of $dt$= 1 ms. A total of 6 types of neurons are included in this model, which are excitatory neurons (E), interneuron-basket cells (I-BC), interneuron-Matinotti cell (I-MC), thalamocortical relay neurons (TC), thalamic interneurons (TI) and thalamic reticular neurons (TRN).\n\n\n$$\nJ \\in\\{E, I\\underline{B} C, I\\underline{M} C, T C, T I, T R N\\}\n$$\n\nWe use the aEIF neuron model referring to previous work~\\cite{jiang2015principles,izhikevich2008large,tchumatchenko2014oscillations}, and obtain the parameters of this study, which are summarized in Tab.~\\ref{mouse1}.\n\n\\begin{table}[h]\n\t\\caption{Main parameters of different types of neuron models.}\n\t\\centering\n\t\\resizebox{0.8\\linewidth}{!}{\n\t\t\\begin{tabular}{lllllll}\n\t\t\t\\toprule \n\t\t &$V_{th,j}(mV)$ &$ V_{r,j}(mV)$ & $\\tau_{v,j}$ &$ \\tau_{w,j}$ &$ \\alpha _{j}$ & $\\beta_{ j} $\\\\\n\t\t\t\\midrule\t\n\t\t E & -50 & -110 & 100 & - & 0 & 0 \\\\\n\t\t\t\n\t\t\t\\midrule\n\t\t I-BC & -44 & -110 & 100 & 20 & -2 & 4.5 \\\\\n\t\t \\midrule\n\t\t  I-MC & -45 & -66 & 85 & 20 & -2 & 4.5 \\\\\n\t\t \n\t\t  \\midrule\n\t\t  TC & -50 & -60 & 200 & - & 0 & 0 \\\\\n\t\t   \\midrule\n\t\t    TI & -50 & -60 & 20 & 20 & -2 & 4.5 \\\\\n\t\t      \\midrule\n\t\t       TRN & -45 & -65 & 40 & 20 & -2 & 4.5 \\\\ \n\t\t\t\\bottomrule\n\t\t\\end{tabular}\n\t}\n\t\\label{mouse1}\n\t\n\\end{table}\n\n\nThe connections between brain areas are based on the quantitative anatomical dataset Allen Mouse Brain Connectivity Atlas. Methods for data generation have been previously described in~\\cite{oh2014mesoscale}. The proportions of the different types of neurons were adopted from a previous study~\\cite{izhikevich2008large, markram2004interneurons}.\n\nThe numbers of each type of neuron in the network are shown in Tab.~\\ref{mouse2}.\n\\begin{table}[h]\n\t\\caption{Number of different types of neurons in the BrainCog mouse brain simulator.}\n    \\centering\n    \\resizebox{0.8\\linewidth}{!}{\n\t\t\\begin{tabular}{lllllll}\n\t\t\t\\toprule \n        Neuron Type & E & I\\_BC & I\\_MC & TC & TI & TRN \\\\ \n        \t\\midrule\n        Neuron Number & 56100 & 14960 & 7480 & 1300 & 260 & 520 \\\\ \n        \t\\bottomrule\n    \\end{tabular}\n    }\n    \\label{mouse2}\n\\end{table}\n\nThe spontaneous discharge of the model without external stimulation is shown in Fig.~\\ref{figmouse1}.\n  \n\\begin{figure}\n\\centering\n\\includegraphics[width=0.48\\textwidth]{.\/fig\/rat.jpg}\n\\caption{Running of the BrainCog mouse brain simulator. The shining point is the spiking neuron at the time t and the point color represents the neuron belong to respective brain area.}\n\\label{figmouse1}\n\\end{figure}\nThis is an open platform, and both the parameters of the neuron model and the number of different types of neurons can be set flexibly.\n\n\\textbf{3. Macaque Brain}\n\nThe BrainCog macaque brain simulator is a large-scale spiking neural network model covering 383 brain areas~\\cite{Dharmendra2010}. We used the multi-scale connectome transformation method~\\cite{a2017Zhang} on the EGFP (enhanced green fluorescent protein) results~\\cite{Bakker2012, Chaudhuri2015ALC, Christine2010} to obtain the approximate amount of cells per region and the approximate number of synaptic connections between two connected regions~\\cite{Liu2016}. The final macaque model includes 1.21 billion spiking neurons and 1.3 trillion synapses, which is 1\/5 of a real macaque brain. \nSpecifically, the details of the brain micro-circuit are also considered in the simulation. The types of neurons in the micro-circuit include excitatory neurons (90 \\% of the neurons are of this type in the simulation) and inhibitory neurons (10\\% of the neurons are of this type in the simulation)~\\cite{Davis1979}. The spiking neuron follows Hodgkin\u2013Huxley model, which is supported by BrainCog. The running demo of the model is shown in Fig.~\\ref{mac}(a). To use the macaque model in the platform, the parameters of the neuron number in each region, the connectome power between regions, and the proportion between the excitatory and inhibitory neurons can be set flexibly.\n\n\\begin{figure}[!htbp]\n\\centering\n\\includegraphics[width=0.48\\textwidth]{.\/fig\/mac.jpg}\n\\caption{Running of the macaque brain (a) and the human brain (b) model. The shining point is the spiking neuron at the time t and the point color represents the region which the neuron belongs to.}\n\\label{mac}\n\\end{figure}\n\n\n\\textbf{4. Human Brain}\n\nThe BrainCog human brain simulator is built with the approach similiar to the BrainCog macaque brain. By using the EGFP results of the\nhuman brainnetom atlas~\\cite{Fan2016, Klein2012}, the BrainCog human brain simulator consists of 246 brain areas. It should be noted that since there is no directed human brain connectome available until the release of this paper, the BrainCog human brain simulator keeps bidirectional connections among brain areas. The details of the micro-circuit, including the excitatory neuron and the inhibitory neuron, are also considered. The final model (Fig.~\\ref{mac}(b) includes 0.86 billion spiking neurons and 2.5 trillion synapses, which is 1\/100 of a real human brain. To use this model in the platform, the neuron number per region, the connectome power, and the proportion between the excitatory and inhibitory neurons can be set flexibly.\nMorover, all the simulations were performed on distributed memory clusters~\\cite{Liu2016} at the super computing center affiliated to Institute of Automation, Chinese Academy of Sciences, Beijing, China. The cluster named the fat cluster is composed of 16 blade nodes and 2 \"fat\" computing nodes. In order to imporve the network communication efficiency and the simulation performance, the most connected areas were simulated in the 'fat' computing nodes to minimize the inter-node communications, while the other areas is randomly distributed in the blade nodes. The simulation shows the ability of the framework to deploy on the supercomputer or large-scale computer clusters.\n\n\\section{BORN: A spiking neural network driven Artificial Intelligence Engine based on BrainCog}\n\n\\begin{figure*}[htb]\n\\centering\n\\includegraphics[width=1\\textwidth]{.\/fig\/vision.jpg}\n\\caption{The functional framework and vision of BORN.}\n\\label{bornvision}\n\\end{figure*}\n\nBrainCog is designed to be an open source platform to enable the community to build spiking neural network based brain-inspired AI models and brain simulators. Based on the essential components developed for BrainCog, one can develop their own domain specific or general purpose AI engines. To further demonstrate how BrainCog can support building Brain-inspired AI engine, here we introduce BORN, an ongoing SNN driven Brain-inspired AI engine, ultimately designed for general purpose living AI. As shown in Fig.~\\ref{bornvision}, the high-level architecture of BORN is to integrate spatial and temporal plasticities to realize perception and learning, decision-making, motor control, working memory, long-term memory, attention and consciousness, emotion, knowledge representation and reasoning, social cognition and other brain cognitive functions. Spatial plasticity incorporates multi-scale neuroplasticity principles at micro, meso and macro scales. Temporal plasticity considers learning, developmental and evolutionary plasticity at different time scales. \n\n\nAs an essential component for BORN, we propose a developmental plasticity-inspired adaptive pruning (DPAP) model, that enables the complex deep SNNs and DNNs to gradually evolve into a brain-inspired efficient and compact structure, and eventually improves learning speed and accuracy in the extremely compressed networks~\\cite{Han2022Developmental}. The evolutionary process for the brain includes but not limited to searching for the proper connectome among different building blocks of the brain at multiple scales (e.g. neurons, microcircuits, brain areas). BioNAS for BORN uses brain-inspired neural architecture search to construct SNNs with diverse motifs in the brain and experimentally verify that SNNs with rich motif types perform better than plain feedforward SNNs~\\cite{shenBrainInspired2022}.\n\n\nHow the human brain selects and coordinates various learning methods to solve complex tasks is crucial for understanding human intelligence and inspiring future AI. BORN is dedicated to address critical research issues like this. The learning framework of BORN consists of multi-task continual learning, few-shot learning, multi-modal concept learning, online learning, lifelong learning, teaching-learning, and transfer learning, etc. \n\nTo demonstrate the ability and principles of BORN, we provide a relatively complex application on emotion dependent robotic music composition and playing. This application requires a humanoid robot perform music composition and playing depending on visual emotion recognition. The application requires BORN to provide cognitive functions such as visual emotion recognition, sequence learning and generation, knowledge representation and reasoning, and motor control, etc. This application of BORN starts with perception and learning, and ends with motor output. \n\nIt includes three modules implemented by BrainCog: the visual (emotion) recognition module, the emotion-dependent music composition module, and the robot music playing module. As shown in Fig.~\\ref{rmp}, the visual emotion recognition module enables robots to recognize the emotions expressed in images captured by the humanoid robot eyes. The emotion-dependent music composition module can generate music pieces according to various emotional inputs. When a picture is shown to the robot, the Visual Emotion Recognition network can firstly identify the emotions expressed in the picture, such as joy or sadness. The robot then selects or compose the music piece that best matches the emotions in the picture. And finally, with the help of the robot music playing module, the robot controls its arms and fingers in a series of movements, thus playing the music on the piano. Some details are introduced as follows:\n\n\\begin{figure*}[!htbp]\n\\centering\n\\includegraphics[width=1\\textwidth]{.\/fig\/rmp.jpg}\n\\caption{The procedure of multi-cognitive function coordinated emotion dependent music composition and playing by humanoid robot based on BORN.}\n\\label{rmp}\n\\end{figure*}\n\n\n\n\\emph{1) Visual Emotion Recognition: }\nFor emotion recognition, inspired by the ventral visual pathway, we construct a deep convolutional spiking neural network with LIF neuron model and surrogate gradient provided by BrainCog. The structure of the network is set as 32C3-32C3-MP-32C3-32C3-300-7. 32C3 means the output channel is set with 32, and the kernel size is set as 3. MP denotes the max pooling. The mean firing rate is used to make the final prediction. We use the Adam optimizer, and the mean square error loss. The initial learning rate is set with 0.001, and it will decay to 1\/10 of the previous value every 40 epochs, for a total of 100 epochs. We use the Emotion6 dataset~\\cite{peng2015mixed} to train and test our model. The Emotion6 dataset is composed of 6 emotions such as anger, disgust, fear, joy, sadness, surprise, and each type of emotion consists of 330 samples. On this basis, we extend the original Emotion6 dataset with exciting emotion which we collect online. 80\\% of the images are used as the training set, and the remaining 20\\% are used as the test set. \n\n\\emph{2) Emotion-dependent Music Composition: }\nListening to the the music can make us emotional, while, when people feel happy or sad, they always express their feeling with music. Amygdala plays a key role in human emotion. Inspired by this mechanism, we constructed a simple spiking neural network to simulate this important area to represent different types of emotions and learn the relationships with other brain areas related to music. As shown in Fig.~\\ref{rmp}, the amygdala network is composed of several LIF neurons supported through BrainCog, the connections from this cluster are projected to the musical sequence memory networks. \n\nDuring the learning process, amygdala, PFC, and the musical sequence memory networks cooperate with each other and form complex neural circuits. Here, connections are updated by the STDP learning rule. The dataset used here also contains 331 MIDI files of classical piano works~\\cite{Krueger2018}, and it is important to note that a part of these music works are labeled with different types of emotional categories (such as happy, depressed, passionate and beautiful). \n\nAt the generation process, given the beginning notes and specific emotion type, the model can generate a series of notes and form a melody with particular emotion finally.\n\n\\emph{3) Robot Music-Playing: }\nA humanoid robot iCub is used to validate the abilities of robotic music composition and playing depending on the result of visual emotion recognition. The iCub robot has a total of 53 degrees of freedoms throughout the body. In the piano playing task, we used 6 degrees of freedoms of the head, 3 degrees of freedoms of the torso, and 16 degrees of freedoms for each of the left and right arm (including the left and right hand). Besides, we mainly control the index fingers to press the keys; in the multi-fingered playing mode, we mainly control the thumbs, the index fingers, and the middle fingers to press the keys. During playing, the robot controls the movement of the hand in sequence according to the generated sequence of different musical notes, and presses the keys with corresponding fingers, thereby completing the performance. For each note to be played, the corresponding playing arm needs to complete the entire process of moving, waiting, pressing the key, holding, and releasing the key according to the beat. During the playing process, we also control the movements of the robot's head and the non-playing hand to match the performance.\n\nWe have constructed a multi-brain area coordinated robot motor control SNN model based on the brain motor control circuit. The SNN model is built with LIF neurons and implements SMA, PMC, basal ganglia and cerebellum functions. The  music notes is first processed by SMA, PMC and basal ganglia networks to generate high-level target movement directions, and the output of PMC is encoded by population neurons to target movement directions. The fusion of population codings of movement directions is further processed by the cerebellum model for low level motor control. The cerebellum SNN module consists of Granular Cells (GCs), Purkinje Cells (PCs) and Deep Cerebellar Nuclei (DCN), which implements the three level residual learning in motor control. The DCN network generates the final joint control outputs for the robot arms to perform the playing movement.\n\n\n\n\\section{Conclusion}\nBrainCog aims to provide a community based open source platform for developing spiking neural network based AI models and cognitive brain simulators. It integrates multi-scale biological plausible computational units and plasticity principles. Different from existing platforms, BrainCog incorporates and provides task ready SNN models for AI, and supports brain function and structure simulations at multiple scales. With the basic and functional components provided in the current version of BrainCog, we have shown how a variety of models and applications can already be implemented for both brain-inspired AI and brain simulations. Based on BrainCog, we are also committed to building BORN into a powerful SNN-based AI engine that incorporates multi-scale plasticity principles to realize brain-inspired cognitive functions towards human level. Powered by 9 years development of BrainCog modules, components and applications, and inspired by biological mechanisms and natural evolution, continuous efforts on BORN will enable it to be a general purpose AI engine. We have already started the efforts to extend BrainCog and BORN to support high-level cognition such as theory of mind~\\cite{zhao2022brain}, consciousness~\\cite{zeng2018toward}, and morality~\\cite{zhao2022brain}, and it definitely takes the world to build true and general purpose AI for human and ecology good. Join us on this explorations to create the future for human-AI symbiotic society. \n\\section*{Acknowledgements}\nThis work is supported by the National Key Research and Development Program (Grant No. 2020AAA0104305), the Strategic Priority Research Program of the Chinese Academy of Sciences (Grant No. XDB32070100).\n\n\n\n\n\n\\bibliographystyle{IEEEtran}\n\n\\section{Introduction}\nThe human brain can self-organize and coordinate different cognitive functions to flexibly adapt to complex and changing environments. A major challenge for Artificial Intelligence and computational neuroscience is integrating multi-scale biological principles to build biologically plausible brain-inspired intelligent models. As the third generation of neural networks~\\cite{Maass1997Networks}, Spiking Neural Networks (SNNs) are more biologically realistic at multiple scales, more biologically interpretable, more energy-efficient, and naturally more suitable for modeling various cognitive functions of the brain and creating biologically plausible AI. \n\n\\begin{figure*}[!htbp]\n\t\\centering\n\t\\includegraphics[scale=0.3]{.\/fig\/braincog.jpg}\n\t\\caption{The architecture of Brain-inspired Cognitive Intelligence Engine (BrainCog).}\n\t\\label{braincog}\n\\end{figure*}\n\nExisting neural simulators attempted to simulate elaborate biological neuron models or build large-scale neural network simulations, neural dynamics models, deep SNNs. Neuron~\\cite{carnevale2006neuron} focuses on simulating elaborate biological neuron models. NEST~\\cite{gewaltig2007nest} implements large-scale neural network simulations. Brian\/Brian2~\\cite{stimberg2019brian, goodman2009brian} provides an efficient and convenient tool for modeling spiking neural networks. Shallow SNNs implemented by Brian2 can realize unsupervised visual classification~\\cite{diehl2015unsupervised}. Further, BindsNET~\\cite{ hazan2018bindsnet} builds SNNs with coordination of various neurons and connections and incorporates multiple biological learning rules for training SNNs. Such learning SNNs can perform machine learning tasks, including simple supervised, unsupervised, and reinforcement learning. However, supporting more complex tasks would be great challenge for these SNNs frameworks, and there is a large gap in performance compared with traditional deep neural networks (DNNs). Deep SNNs trained by surrogate gradient or converting well-trained DNNs to SNNs have achieved great progress in the fields of speech recognition~\\cite{dominguez2018deep, loiselle2005exploration}, computer vision~\\cite{kim2020spiking, wu2018spatio}, and reinforcement learning~\\cite{tan2021strategy}. Motivated by this, SpikingJelly~\\cite{SpikingJelly} develops a deep learning SNN framework (trained by surrogate gradient or converting well-trained DNNs to SNNs), which integrates deep convolutional SNNs and various deep reinforcement learning SNNs for multiple benchmark tasks. Platforms as SpikingJelly are relatively more inspired by the field of deep learning and focuse on improving the performance of different tasks. They are currently lack of in-depth inspiration from the brain information processing mechanisms and may not aim at, hence short at simulating large scale functional brains.\n\nBrainPy~\\cite{Wang2021} excels at modeling, simulating, and analyzing the dynamics of brain-inspired neural networks from multiple perspectives, including neurons, synapses, and networks. While it focuses on computational neuroscience research, it fails to consider the learning and optimization of deep SNNs or the implementation of brain-inspired functions. SPAUN~\\cite{eliasmith2012large}, a large-scale brain function model consisting of 2.5 million simulated neurons and implemented by Nengo~\\cite{bekolay2014nengo}, integrates multiple brain areas and realizes multiple brain cognitive functions, including image recognition, working memory, question answering, reinforcement learning, and fluid reasoning. However, it is not suitable for solving challenging and complex AI tasks that can be handled by deep learning models. In summary, there is still a lack of open-source spiking neural network frameworks that could incorporate the ability of simulating brain structures, cognitive functions at large scale, while keep itself effective for creating complex and efficient AI models at the same time.\n\nConsidering the various limitations of existing frameworks mentioned above, in this paper, we present the Brain-inspired Cognitive Intelligence Engine (BrainCog), a spiking neural network based open source platform for both brain-inspired AI and brain simulation at multiple scales. As shown in Fig.~\\ref{braincog}, some basic modules (such as different types of neuron models, learning rules, encoding strategies, etc.) are provided as building blocks to construct different brain areas and neural circuits to implement brain-inspired cognitive functions. BrainCog is an easy-to-use framework that can flexibly replace different components according to various purposes and needs. BrainCog tries to achieve the vision ``the structure and mechanism are inspired by the brain, and the cognitive behaviors are similar to humans'' for Brain-inspired AI~\\cite{2016Retrospect}. BrainCog is developed based on the deep learning framework (currently it is based on PyTorch, while it is easy to migrate to other frameworks such as PaddlePaddle, TensorFlow, etc.). This approach aims at greatly facilitating researchers to quickly familiarize themselves with the platform and implement their own algorithms.\n\n\\subsection{Brain-inspired AI}\nBrainCog is aimed at providing infrastructural support for Brain-inspired AI. Currently, it provides cognitive functions components that can be classified into five categories: Perception and Learning, Decision Making, Motor Control, Knowledge Representation and Reasoning, and Social Cognition. These components collectively form neural circuits corresponding to 28 brain areas in the mammalian brains, as shown in Fig.~\\ref{born}. These brain-inspired AI models have been effectively validated on various supervised and unsupervised learning, deep reinforcement learning, and several complex brain-inspired cognitive tasks.  \n\n\n\\begin{figure}[!htbp]\n\t\\centering\n\t\\includegraphics[scale=0.35]{.\/fig\/born.jpg}\n\t\\caption{Multiple cognitive functions integrated in BrainCog, their related brain areas and neural circuits.}\n\t\\label{born}\n\\end{figure}\n\n\\subsubsection{Perception and Learning} \nBrainCog provides a variety of supervised and unsupervised methods for training spiking neural networks, such as the biologically-plausible Spike-Timing-Dependent Plasticity (STDP)~\\cite{bi1998synaptic}, the backpropagation based on surrogate gradients~\\cite{wuSpatioTemporalBackpropagationTraining2018,zhengGoingDeeperDirectlyTrained2021,fangIncorporatingLearnableMembrane2021}, and the conversion-based algorithms~\\cite{li2021free, han2020deep, han2020rmp}. In addition to high performance in common perception and learning process, it also shows strong adaptability in small samples and noisy scenarios. BrainCog also provides a multi-sensory integration framework for human-like concept learning~\\cite{wywFramework}. Inspired by quantum information theory, BrainCog provides a quantum superposition spiking neural network, which encodes complement information to neuronal spike trains with different frequency and phase~\\cite{SUN2021102880}. \n\n\\subsubsection{Decision Making}\nFor decision making, BrainCog provides multi-brain areas coordinated decision-making spiking neural network~\\cite{zhao2018brain}. The biologically inspired decision-making model implemented by BrainCog achieves human-like learning ability on the Flappy Bird game and supports UAVs' online decision-making tasks. In addition, BrainCog combines SNNs with deep reinforcement learning and provides the brain-inspired spiking deep Q network~\\cite{sun2022solving}.\n\n\\subsubsection{Motor Control}\nEmbodied cognition is crucial to realizing biologically plausible AI. As part of the embodied cognition modules, and inspired by the motor control mechanism of the brain, BrainCog provides a multi-brain areas coordinated robot motion spiking neural networks model, which includes premotor cortex (PMC), supplementary motor areas (SMA), basal ganglia and cerebellum functions. With proper mapping, the spiking motor network outputs can be used to control various robots.\n\n\\subsubsection{Knowledge Representation and Reasoning} \nBrainCog incorporates multiple neuroplasticity and population coding mechanisms for knowledge representation and reasoning. The brain-inspired music memory and stylistic composition model implements the knowledge representation and memory of note sequences and can generate music according to different styles~\\cite{LQ2020,LQ2021}. Sequence Production Spiking Neural Network (SPSNN) achieves the memory of the symbol sequence and can reconstruct the symbol sequence in the light of different rules~\\cite{fang2021spsnn}. Commonsense Knowledge Representation Graph SNN (CKR-GSNN) realizes the representation of commonsense knowledge through incorporating multi-scale neural plasticity and population coding mechanism into a graph SNN model~\\cite{KRRfang2022}.\nCausal Reasoning Spiking Neural Network (CRSNN) encodes the causal graph into a spiking neural network and realizes deductive reasoning tasks accordingly~\\cite{fang2021crsnn}.\n\n\\subsubsection{Social Cognition} \nBrainCog provides a brain-inspired social cognition model with biological plausibility. This model gives the agent a preliminary ability to perceive and understand itself and others and can enable the robots pass the Multi-Robots Mirror Self-Recognition Test~\\cite{zeng2018toward} and the AI Safety Risks Experiment~\\cite{zhao2022brain}. The former is a classic experiment of self-perception in social cognition, and the latter is a variation and application of the theory of mind experiment in social cognition.\n\\subsection{Brain Simulation}\n\n\\subsubsection{Brain Cognitive Function Simulation}\nTo demonstrate the capability of BrainCog for cognitive function simulation, we provide \\emph{Drosophila} decision-making and prefrontal cortex working memory function simulation~\\cite{zhao2020neural,zhangqian2021comparison}. For \\emph{Drosophila} nonlinear and linear decision-making simulation, BrainCog verifies the winner-takes-all behaviors of the nonlinear dopaminergic neuron\u2011GABAergic neuron\u2011mushroom body (DA\u2011GABA\u2011MB) circuit under a dilemma and obtains consistent conclusions with \\emph{Drosophila} biological experiments~\\cite{zhao2020neural}. For the working memory performance of the prefrontal cortex network implemented by BrainCog, we discover that using human neurons to replace rodent neurons without changing network structure can significantly improve the accuracy and completeness of an image memory task~\\cite{zhangqian2021comparison}, and this implies the evolution of the brains are not only on their structures, but also applies to single computational units such as neurons.\n\n\\subsubsection{Multi-scale Brain Structure Simulation} \nBrainCog provides simulations of brain structures at different scales, from microcircuits, and cortical columns, to whole-brain structure simulations. Anatomical and imaging multi-scale connectivity data is used to support whole-brain simulations from mouse brain, macaque brain to human brain at different scales.\n\n\n\n\\section{Basic Components}\nBrainCog provides essential and fundamental components to model biological and artificial intelligence. It includes various biological neuron models, learning rules, encoding strategies, and models of different brain areas. One can build brain-inspired SNN models through reusing and refining these building blocks. Expanding and refining the components and cognitive functions included in BrainCog is an ongoing effort. We believe this should be a continuous community effort, and we welcome researchers and practitioners to enrich and improve the work together in a complementary way.\n\n\\subsection{Neuron Models}\nBrainCog supports various models for spiking neurons, such as the following :\n\n(1) Integrate-and-Fire spiking neuron (IF)~\\cite{abbott1999lapicque}:\n\n\\begin{equation}\n\\frac{dV}{dt} = I\n\\end{equation}\n\n$I$ denotes the input current from the pre-synaptic neurons. Once the membrane potential reaches the threshold $V_{th}$, the neuron $j$ fires a spike~\\cite{abbott1999lapicque}.\n\n(2) Leaky Integrate-and-Fire spiking neuron (LIF)~\\cite{dayan2005theoretical}:\n\\begin{equation}\n \\tau\\frac{dV}{dt} = - V + I\n\\label{equa_lif1}\n\\end{equation}\n$\\tau=RC$ denote the time constant, $R$ and $C$ denotes the membrane resistance and capacitance respectively~\\cite{dayan2005theoretical}.  \n\n(3) The adaptive Exponential Integrate-and-Fire spiking neuron (aEIF)~\\cite{Fourcaud2003How,brette2005adaptive}:\n\n\\begin{equation}\n\\left\\{\n\\begin{aligned}\nC \\frac{d V}{d t}&=-g_{L}(V-E_{L})+g_{L} \\exp(\\frac{V-V_{t h}}{\\Delta T})+I-w \\\\\n\\tau_{w} \\frac{dw}{dt}&=a(V-E_L)-w\n\\end{aligned}\n\\right.\n\\end{equation}\n\nwhere $g_L$ is the leak conductance, $E_L$ is the leak reversal potential, $V_r$ is the reset potential, $\\Delta T$ is the slope factor, $I$ is the  background currents.  $\\tau_w$ is the adaptation time constant. When the membrane potential is greater than the threshold $V_{th}$, $V = V_{r}$, and $w = w +b$.  $a$ is the subthreshold adaptation, and $b$ is the spike-triggered adaptation~\\cite{Fourcaud2003How,brette2005adaptive}. \n\n(4) The Izhikevich spiking neuron~\\cite{izhikevich2003simple}:\n\\begin{equation}\n\\left\\{\n\\begin{aligned}\n\\frac{dV}{dt} &= 0.04V^2 + 5v + 140 - u + I \\\\\n\\frac{du}{dt} &= a(bv-u)\\\\\n\\end{aligned}\n\\right.\n\\end{equation}\nWhen the membrane potential is greater than the threshold:\n\\begin{equation}\n\\left\\{\n\\begin{aligned}\nV &= c \\\\\nu &= u + d\\\\\n\\end{aligned}\n\\right.\n\\end{equation}\n$u$ represents the membrane recovery variable, and $a, b, c, d$ are the dimensionless parameters~\\cite{izhikevich2003simple}. \n\n(5) The Hodgkin-Huxley spiking neuron (H-H)~\\cite{hodgkin1952quantitative}:\n\\begin{equation}\nI = C \\frac{dV}{dt} + \\overline{g}_K n^4 (V - V_K) + \\overline{g}_{Na} m^3 h (V - V_{Na}) + \\overline{g}_L (V - V_L)\n\\label{equa_hh}\n\\end{equation}\n\\begin{equation}\n\\left\\{\n\\begin{aligned}\n\\frac{dn}{dt} &= \\alpha_n(V)(1-n) - \\beta_n(V)n \\\\\n\\frac{dm}{dt} &= \\alpha_m(V)(1-m) - \\beta_m(V)m \\\\\n\\frac{dh}{dt} &= \\alpha_n(V)(1-h) - \\beta_n(V)h \\\\\n\\end{aligned}\n\\right.\n\\end{equation}\n\n\n$\\alpha_i$  and $\\beta_i$ are used to control the $i_{th}$ ion channel, $n$, $m$, $h$ are dimensionless probabilities between 0 and 1. $\\overline{g}_i$ is the maximal value of the conductance~\\cite{hodgkin1952quantitative}.\n\n\nThe H-H model shows elaborate modeling of biological neurons. In order to apply it more efficiently to AI tasks, BrainCog incorporates a simplified H-H model ($C=0.02\\mu F\/cm^2, V_r=0, V_{th}=60mV$), as illustrated in~\\cite{wang2016spiking}.\n\n\n\n\n\n\\subsection{Learning Rules}\nBrainCog provides various plasticity principles and rules to support biological plausible learning and inference, such as:\n\n(1) Hebbian learning theory~\\cite{amit1994correlations}:\n\\begin{equation}\n\\Delta w_{ij}^t=x_i^t x_j^t\n\\end{equation}\nwhere $w_{ij}^t$ means the $i$th synapse weight of $j$th neuron at the time $t$. $x_i^t$ is the input of $i$th synapse at time $t$. $x_j^t$ is the output of $j$th neuron at time $t$~\\cite{amit1994correlations}.\n\n(2) Spike-Timing-Dependent Plasticity (STDP)~\\cite{bi1998synaptic}:\n\n\\begin{equation}\n\t\\begin{split}\n\t\t&\\Delta w_{j}=\\sum\\limits_{f=1}^{N} \\sum\\limits_{n=1}^{N} W(t^{f}_{i}-t^{n}_{j})\\\\\n\t\t&W(\\Delta t)= \n\t\t\\left\\{\n\t\t\\begin{split}\n\t\tA^{+}e^{\\frac{-\\Delta t}{\\tau_{+}}}\\quad if\\; \\Delta t>0&\\\\ \n\t\t-A^{-}e^{\\frac{\\Delta t}{\\tau_{-}}}\\quad if\\;\\Delta t<0&\\\\ \n\t\t\\end{split}\n\t\t\\right.\n\t\\end{split}\n\t\\label{eq4}\n\\end{equation}\nwhere $\\Delta w_{j}$ is the modification of the synapse $j$, and $W(\\Delta t)$ is the STDP function. $t$ is the time of spike. $A^{+},A^{-}$ mean the modification degree of STDP. $\\tau_{+}$ and $\\tau_{-}$ denote the time constant~\\cite{bi1998synaptic}.\n\n(3) Bienenstock-Cooper-Munros theory (BCM)~\\cite{bienenstock1982theory}:\n\\begin{equation}\n\\Delta w=y\\left(y-\\theta_M \\right)x-\\epsilon w\n\\end{equation}\n\nWhere $x$ and $y$ denote the firing rates of pre-synaptic and post-synaptic neurons, respectively, threshold $\\theta_M$ is the average of historical activity of the post-synaptic neuron~\\cite{bienenstock1982theory}.\n\n(4) Short-term Synaptic Plasticity (STP)~\\cite{maass2002synapses}: \n\n\nShort-term plasticity is used to model the synaptic efficacy changes over time.\n\n\\begin{equation}\n{a_k} = {u_k} \\cdot {R_k}\n\\end{equation}\n\\begin{equation}\n{u_{k+1}} = U + {u_{k}}(1 - U)\\exp \\left( { - {{\\rm{\\Delta }}t_{k}}\/{\\tau _{{\\rm{fac}}}}} \\right)\n\\end{equation}\n\\begin{equation}\n{R_{k+1}} = 1 + \\left( {{R_{k}} - {u_{k}}{R_{k}} - 1} \\right)\\exp \\left( { - {{\\rm{\\Delta }}t_{k}}\/{\\tau _{{\\rm{rec}}}}} \\right)\n\\end{equation}\n\n$a$ denotes the synaptic weight, $U$ denotes the fraction of synaptic resources. $\\tau _{{\\rm{fac}}}$ and $\\tau _{{\\rm{rec}}}$ denotes the time constant for recovery from facilitation and depression. The variable $R_n$models the fraction of synaptic efficacy available for the (k)th spike,and $u_n R_n$ models the fraction of synaptic efficacy~\\cite{maass2002synapses}.\n\n\n(5) Reward-modulated Spike-Timing-Dependent Plasticity (R-STDP)~\\cite{Eugene2007}: \n\n\n\nR-STDP uses synaptic eligibility trace $e$ to store temporary information of STDP. The eligibility trace accumulates the STDP $\\Delta w_{STDP}$ and decays with a time constant $\\tau_e$~\\cite{Eugene2007}. \n\n\\begin{equation}\\label{rstdpe}\n\\Delta e=-\\frac{e}{\\tau _e}+\\Delta w_{STDP}\n\\end{equation}\n\nThen, synaptic weights are updated when a delayed reward $r$ is received, as Eq.~\\ref{rstdpw} shown~\\cite{Eugene2007}. \n\n\\begin{equation}\\label{rstdpw}\n\\Delta w=r*\\Delta e \n\\end{equation}\n\n\n\n\\subsection{Encoding Strategies}\nBrainCog supports a number of different encoding strategies to help encode the inputs to the spiking neural networks.\n\n\n(1) Rate Coding~\\cite{adrian1926impulses}:\n\nRate coding is mainly based on spike counting to ensure that the number of spikes issued in the time window corresponds to the real value. Poisson distribution can describe the number of random events occurring per unit of time, which corresponds to the firing rate~\\cite{adrian1926impulses}. Set $\\alpha \\sim \\mathbb{U}(0,1)$, the input can be encoded as\n\\begin{equation}\n\ts(t) =  \\begin{cases}\n\t   1,\\quad if \\quad x > \\alpha\\\\\n\t   0, \\quad else \n\t\\end{cases}\n\\end{equation}\n\n(2) Phase Coding~\\cite{kim2018deep}:\n\nThe idea of phase coding can be used to encode the analog quantity changing with time. The value of the analog quantity in a period can be represented by a spike time, and the change of the analog quantity in the whole time process can be represented by the spike train obtained by connecting all the periods.\nEach spike has a corresponding phase weighting under phase encoding, and generally, the pixel intensity is encoded as a 0\/1 input similar to binary encoding. $\\gg$ denotes the shift operation to the right, $K$ is the phase period~\\cite{kim2018deep}. Pixel $x$ is enlarged to $x^{\\prime}=x*(2^{K}-1)$ and shifted $k=K-1-(t \\mod K)$ to the right, where $\\mod$ is the  remainder operation. If the lowest bit is one, $s$ will be one at time $t$. $\\&$ means \nbit-wise AND operation.\n\\begin{equation}\n\ts(t) =  \\begin{cases} 1, \\quad if  \\quad (x^{\\prime} \\gg k) \\& 1 = 1\\\\\n\t0, \\quad else\n\t\\end{cases}\n\\end{equation}\n\n(3) Temporal Coding~\\cite{thorpe1996speed}:\n\nThe characteristic of the neuron spike is that the form of the spike is fixed, and there are only differences in quantity and time. The most common implementation is to express information regarding the timing of individual spike. The stronger the stimulus received, the earlier the spike generated~\\cite{rueckauer2018conversion}. Let the total simulation time be T, and the input $x$ of the neuron can be encoded as the spike at time $t^s$:\n\\begin{equation}\n\tt^{s} = T - \\operatorname{round}(Tx)\n\\end{equation}\n\n(4) Population Coding~\\cite{bohte2002error}:\n\nPopulation coding helps to solve the problem of the ambiguity of information carried by a single neuron. The ambiguity of information carried can be understood as: the original information is input into a neuron, which makes the network hard to distinguish some overlapping or similar related information~\\cite{quian2009extracting,grun2010analysis}.\nThe intuitive idea of population coding is to make different neurons to be with different sensitivity to different types of inputs, which is also reflected in biology. For example, rats' whiskers have different sensitivity to different directions~\\cite{quian2009extracting}. The inputs will be transformed into a spike train with a period by population coding. A classical population coding method is the neural information coding method based on the Gaussian tuning curve referred to Eq.~\\ref{gaus}. This method is more suitable when the amount of data is small, and the information is concentrated. A Gaussian neuron covers a range of analog quantities in the form of Gaussian function~\\cite{bohte2002error}, \\cite{li2021online}. Suppose that $m$ ($m > 2$) neurons are used to encode a variable x with a value range of $[I_{min}, I_{max}]$. $f(x)$ can be firing time or voltage.\n\n\\begin{equation}\n        f(x) = x^{-\\frac{(x-\\mu)^{2}}{2\\sigma^{2}}}\n\\label{gaus}\n\\end{equation}\n\nThe corresponding mean with adjustable parameters $\\beta$ and variance of the $ith$ ($i=1,2,..., m$) neuron are as follows:\n\n\\begin{equation}\n        \\mu = I_{min} + \\frac{2i-3}{2}\\frac{I_{max}-I_{min}}{m-2}\n\\end{equation}\n\n\\begin{equation}\n        \\sigma = \\frac{1}{\\beta}\\frac{I_{max}-I_{min}}{m-2}\n\\end{equation}\n\n\n\n\n\\subsection{Brain Areas Models}\n\nBrain-inspired models of several functional brain areas are constructed for BrainCog from different levels of abstraction.\n\n(1) Basal Ganglia (BG): \n\nBasal ganglia facilitates desired action selection and inhibit competing behavior (making winner-takes-all decisions)~\\cite{chakravarthy2010basal,redgrave1999basal}. It cooperates with PFC and the thalamus to realize the decision-making process in the brain~\\cite{parent1995functional}. BrainCog models the basal ganglia brain area, including excitatory and inhibitory connections among striatum, Globus pallidus internas (Gpi), Globus pallidus external (Gpe), and subthalamic nucleus (STN) of basal ganglia~\\cite{lanciego2012functional}, as shown in the orange areas of Fig.~\\ref{dmsnn}. The BG brain area component adopts the \\emph{LIF} neuron model in BrainCog, as well as the \\emph{STDP} learning rule and \\emph{CustomLinear} to build internal connections of the BG. Then, the BG brain area component can be used to build brain-inspired decision-making SNNs (see section 3.2.1 for detail).\n\n(2) Prefrontal Cortex (PFC):\n\nPFC is of significant importance when human high-level cognitive behaviors happen. In BrainCog, many cognitive tasks based on SNN are inspired by the mechanisms of the PFC~\\cite{bechara1998dissociation}, such as decision-making, working memory~\\cite{rao1999isodirectional,d2000prefrontals,lara2015role}, knowledge representation~\\cite{wood2003human}, theory of mind and music processing~\\cite{frewen2015healing}. Different circuits are involved to complete these cognitive tasks. In BrainCog, the data-driven PFC column model contains 6 layers and 16 types of neurons. The distribution of neurons, membrane parameters and connections of different types of neurons are all derived from existing biological experimental data. The PFC brain area component mainly employs the \\emph{LIF} neuron model to simulate the neural dynamics. The \\emph{STDP} and \\emph{R-STDP} learning rules are utilized to compute the weights between different neural circuits. \n\n(3) Primary Auditory Cortex (PAC):\n\nThe primary auditory cortex is responsible for analyzing sound features, memory, and the extraction of inter-sound relationships~\\cite{Koelsch2012}. This area exhibits a topographical map, which means neurons respond to their preferred sounds. In BrainCog, neurons in this area are simulated by the \\emph{LIF} model and organized as minicolumns to represent different frequencies of sounds. To store the ordered note sequences, the excitatory and inhibitory connections are updated by \\emph{STDP} learning rule.\n\n\n\n(4) Inferior Parietal Lobule (IPL):\n\nThe function of IPL is to realize motor-visual associative learning~\\cite{macuga2011selective}. The IPL consists of two subareas: IPLM (motor perception neurons in IPL) and IPLV (visual perception neurons in IPL). The IPLM receives information generated by self-motion from ventral premotor cortex (vPMC), and the IPLV receives information detected by vision from superior temporal sulcus (STS). The motor-visual associative learning is established according to the STDP mechanism and the spiking time difference of neurons in IPLM and IPLV.\n\n\n\n\n(5) Hippocampus (HPC):\n\nThe hippocampus is part of the limbic system and plays an essential role in the learning and memory processes of the human brain. Epilepsy patients with bilateral hippocampus removed (e.g. the patient H.M.) have symptoms of anterograde amnesia. They are unable to form new long-term declarative memories~\\cite{milner1998}. This case study has proved that the hippocampus is in the key process of converting short-term memory to long-term memory and plays a vital role~\\cite{smith1981role}. \n\nFurthermore, through electrophysiological means, it was found that the hippocampal region is also crucial for forming new concepts. Specific neurons in the hippocampus only respond selectively to specific concepts, completing the specific encoding between different concepts. Moreover, it was through the study of the hippocampus that neuroscientists discovered the STDP learning rule~\\cite{dan2004spike}, which further demonstrated the high plasticity of the hippocampus.\n\n\n(6) Insula:\n\nThe function of the Insula is to realize self-representation~\\cite{craig2009you}, that is, when the agent detects that the movement in the field of vision is generated by itself, the Insula is activated. The Insula receives information from IPLV and STS. The IPLV outputs the visual feedback information predicted according to its motion, and the STS outputs the motion information detected by vision. If both are consistent, the Insula will be activated.\n\n(7) Thalamus (ThA):\n\nStudies have shown that the thalamus is composed of a series of nuclei connected to different brain parts and heavily contributes to many brain processes. In BrainCog, this area is discussed from both anatomic and cognitive perspectives. Understanding the anatomical structure of the thalamus can help researchers to comprehend the mechanisms of the thalamus. Based on the essential and detailed anatomic thalamocortical data~\\cite{Izhikevichlargescale}, BrainCog reconstructs the thalamic structure by involving five types of neurons (including excitatory and inhibitory neurons) to simulate the neuronal dynamics and building the complex synaptic architecture according to the anatomic results. Inspired by the structure and function of the thalamus, the brain-inspired decision-making model implemented by BrainCog takes into account the transfer function of the thalamus and cooperates with the PFC and basal ganglia to realize multi-brain areas coordinated decision-making model.\n\n(8) Ventral Visual Pathway:\n\nCognitive neuroscience research has shown that the brain can receive external input and quickly recognize objects due to the hierarchical information processing of the ventral visual pathway. The ventral pathway is mainly composed of V1, V2, V4, IT, and other brain areas, which mainly process information such as object shape and color~\\cite{ishai1999distributed,kobatake1994neuronal}. These visual areas form connections through forward, feedback, and self-layer projections. The interaction of different visual areas enables humans to recognize visual objects. The primary visual cortex V1 is selective for simple edge features. With the transmission of information, high-level brain areas combine with lower-level receptive fields to form more complex large receptive fields to recognize more complex objects~\\cite{hubel1962receptive}. Inspired by the structure and function of the ventral visual pathway, BrainCog builds a deep forward SNN with layer-wise information abstraction and a feedforward and feedback interaction deep SNN. The performance is verified on several visual classification tasks. \n\n(9) Motor Cortex:\n\nThe control of biological motor function involves the cooperation of many brain areas. The extra circuits consisting of the PMC, cerebellum, and BA6 motor cortex area are primarily associated with motor control elicited by external stimuli such as visual, auditory, and tactual inputs. Internal motor circuits, including the basal ganglia and SMA, predominate in self-guided, learned movements~\\cite{geldberg1985supplementary,mushiake1991neuronal,gerloff1997stimulation}. Population activity of motor cortical neurons encodes movement direction. Each neuron has its preferred direction. The more consistent the target movement direction is with its preferred direction, the stronger the neuron activities are~\\cite{georgopoulos1995motor,kakei1999muscle}. The cerebellum receives input from motor-related cortical areas such as PMC, SMA, and the prefrontal cortex, which are important for the completion of fine movements, maintaining balance and coordination of movements~\\cite{strick2009cerebellum}. Inspired by the organization of the brain's motor cortex, we use spiking neurons to construct a motion control model, and apply it to the iCub robot, which enables the robot to play the piano according to music pieces.\n\n\\section{Brain-inspired AI}\nComputational units (different neuron models, learning rules, encoding strategies, brain area models, etc.) at multiple scales, provided by BrainCog serve as a foundation to develop functional networks. To enable BrainCog for Brain-inspired AI, cognitive function centric networks need to be built and provided as reusable functional building blocks to create more complex brain-inspired AI. This section introduces various functional building blocks developed based on BrainCog.\n\n\\subsection{Perception and Learning}\nIn this subsection, we will introduce the supervised and unsupervised perception and learning spiking neural networks based on the fundamental components of BrainCog. Inspired by the global feedback connections, the neural dynamics of spiking neurons, and the biologically plausible STDP learning rule, we improve the performance of the spiking neural networks. We show great adaption in the small training sample scenario. In addition, our model has shown excellent robustness in noisy scenarios by taking inspiration from the multi-component spiking neuron and the quantum mechanics. The burst spiking mechanism is used to help our converted SNNs with higher performance and lower latency. Based on this engine, we also present a human-like concept learning framework to generate representations with five types of perceptual strength information.\n\n\\textbf{1. Learning Model}\n\n\\emph{(1) SNN with global feedback connections}\n\n\nThe spiking neural network transmits information in discrete spike sequences, which is consistent with the information processing in the human brain~\\cite{Maass1997Networks}. The training of spiking neural networks has been widely concerned by researchers. Most researchers take inspiration from the mechanism of synaptic learning and updating between neurons in the human brain, and propose biologically plausible learning rules, such as Hebbian theory~\\cite{amit1994correlations}, STDP~\\cite{bi1998synaptic}, and STP~\\cite{zucker2002short}, which have been adopted into the training of spiking neural networks~\\cite{tavanaei2016bio,tavanaei2017multi,diehl2015unsupervised,falez2019multi,zhang2018plasticity,zhang2018brain}. However, most SNNs are based on feedforward structures, while the importance of the brain-inspired structures has been ignored. The anatomical and physiological evidences show that in addition to feedforward connections, numerous feedback connections exist in the brain, especially among sensory areas~\\cite{felleman1991distributed,sporns2004small}. The feedback connections will carry out the predictions from the top layer to cooperate with the local plasticity rules to formulate the learning and inference in the brain. \n\nHere, we introduce the global feedback connections and the local differential learning rule~\\cite{zhao2020glsnn} in the training of SNNs. \n\\begin{figure}[!htbp]\n\t\\centering\n\t\\includegraphics[scale=0.6]{.\/fig\/zdc_pipeline.pdf}\n\t\\caption{The feedforward and feedback pathway in the SNN model. The global feedback pathway propagates the target of the hidden layer, modified from~\\cite{zhao2020glsnn}.}\n\t\\label{snn}\n\\end{figure}\n\nWe use the LIF spiking neuron model in the BrainCog to simulate the dynamical process of the membrane potential $V(t)$ as shown in Eq.~\\ref{equa_lif1}. We use the mean firing rates $S_{l}$ of each layer to denote the representation of the $l_{th}$ layer in the forward pathway, and the corresponding target is denoted as $\\hat{S}_{l}$. Here we use the mean squared loss (MSE) as the final loss function.\n\n$\\hat{S}_{L-1}$ denotes the target of the penultimate layer, and is calculated as shown in Eq.~\\ref{equal_per}, $W_{L-1}$ denotes the forward weight between the $(L-1)_{th}$ layer and the $L_{th}$. $\\eta_t$ represents the learning rate of the target~\\cite{zhao2020glsnn}.\n\\begin{equation}\n\t\\hat{S}_{L-1} = S_{L-1} - \\eta_t\\Delta S = S_{L-1} - \\eta_t W_{L-1}^T(S_{out} - S^T)\n\t\\label{equal_per}\n\\end{equation}\n\nThe target of the other hidden layer can be obtained through the feedback connections:\n\n\\begin{equation}\n\t\\hat{S}_{l} = S_{l} - G_{l}(S_{out} - S^T)\n\t\\label{equal_all}\n\\end{equation}\n\nBy combining the feedforward representation and feedback target, we compute the local MSE loss. We can compute the local update of the parameters with the surrogate gradient. We have conducted experiments on the MNIST and Fashion-MNIST datasets, and achieved 98.23\\% and 89.68\\% test accuracy with three hidden layers. Each hidden layer is set with 800 neurons. The details are shown in Fig.~\\ref{zdc_result}.\n\\begin{figure}[!htbp]\n\t\\centering\n\t\\includegraphics[scale=0.35]{.\/fig\/result_zdc.pdf}\n\t\\caption{The test accuracy on MNIST and Fashion-MNIST datasets of the SNNs with global feedback connections.}\n\t\\label{zdc_result}\n\\end{figure}\n\n\n\\emph{(2) Biological-BP SNN}\n\nThe backpropagation algorithm is an efficient optimization method that is widely used in deep neural networks and has promoted the great success of deep learning. However, due to the non-smoothness of neurons in SNNs, the backpropagation optimization is difficult to be applied to train SNNs directly. To solve the above problem, Bengio et al.~\\cite{bengioEstimatingPropagatingGradients2013} proposed four gradient approximation methods, including Straight-Through Estimator to enable the application of the backpropagation algorithm to neural networks containing nonsmooth neurons. Wu et al.~\\cite{wuSpatioTemporalBackpropagationTraining2018} proposed the Spatio-Temporal Backpropagation (STBP) algorithm to train SNNs by using a differentiable function to approximate spiking neurons through a surrogate gradient method. Based on the spatio-temporal dynamic of SNNs, they achieved the backpropagation of SNNs in both time and space dimensions. Zheng et al.~\\cite{zhengGoingDeeperDirectlyTrained2021} proposed threshold-dependent batch normalization (tdBN) to improve the performance of deep SNNs. Fang et al.~\\cite{fangIncorporatingLearnableMembrane2021} proposed a parametric LIF (PLIF) model to further improve the performance of SNNs by optimizing the time constants of neurons during the training process. \n\nMost of these BP-based SNNs often simply regard SNN as a substitute for RNN, but ignore the dynamic of spiking neurons. To solve the problem of non-differentiable neuronal models, Bohte et al.~\\cite{bohte2011error} approximated the backpropagation process of neuronal models using the surrogate gradient method. However, this method results in gradient leakage and does not allow for proper credit assignment in both temporal and spatial dimensions. Also, due to the reset operation after the spikes are emitted, the errors in the backward process can not propagate across spikes. To solve the above problem, we propose Backpropagation with biologically plausible spatio-temporal adjustment~\\cite{shen2022backpropagation}, as shown in Fig.~\\ref{bpsta}, which can correctly assign credit according to the contribution of the neuron to the membrane potential at each moment.\n\n\\begin{figure}[!htbp]\n\t\\centering\n\t\\includegraphics[scale=0.48]{.\/fig\/shen_bpsta.pdf}\n\t\\caption{The forward and backward process of biological BP-based SNNs for BrainCog.}\n\t\\label{bpsta}\n\\end{figure}\n\nBased on LIF spiking neuron, the direct input encoding strategy, the MSE loss function and the surrogate gradient function supplied in BrainCog, we propose a Biologically Plausible Spatio-Temporal Adjustment (BPSTA) to help BP algorithm with more reasonable error adjustment in the spatial temporal dimension~\\cite{shen2022backpropagation}.  The algorithm realizes the reasonable adjustment of the gradient in the spatial dimension, avoids the unnecessary influence of the neurons that do not generate spikes on the weight update, and extracts more important features. By applying the temporal residual pathway, our algorithm helps the error to be transmitted across multiple spikes, and enhances the temporal dependency of the BP-based SNNs. Compared with SNNs and ANNs with the same structure that only used the BP algorithm, our model greatly improves the performance of SNNs on the DVS-CIFAR10 and DVS-Gesture datasets, while also greatly reducing the energy consumption and decay of SNNs, as shown in Tab.~\\ref{compare2}.\n\\begin{table}[!htbp]\n \\centering\n \\caption{The energy efficiency study. The former represents our method, the latter represents the baseline, adopted from~\\cite{shen2022backpropagation}.}\n \\begin{tabular}{cccc}\n  \\toprule[2pt]\n   Dataset   &  Accuracy    &  Firing-rate  &EE = $\\frac{E_{ANN}}{E_{SNN}}$ \\\\\n  \\midrule[2pt]\n   MNIST     &  99.58\\%\/99.42\\% & 0.082\/0.183 & 35.1x\/15.7x  \n  \\\\\n  \n    N-MNIST   &  99.61\\%\/99.32\\%  & 0.097\/0.176   & 29.6x\/16.3x      \\\\\n  \n   CIFAR10   &  92.33\\%\/89.49\\%  & 0.108\/0.214 & 26.6x\/13.4x    \\\\ \n  \n   DVS-Gesture  &  98.26\\%\/93.92\\%  & 0.083\/0.165   & 34.6x\/17.4x    \\\\ \n       \n        DVS-CIFAR10  &  77.76\\%\/71.40\\%  & 0.097\/0.177  & 29.5x\/16.2x    \\\\\n  \\bottomrule[2pt]\n \\end{tabular}\n \\label{compare2}\n\\end{table} \n\n\\emph{(3) Unsupervised STDP-based SNN}\n \nUnsupervised learning is an important cognitive function of the brain. The brain can complete the task of object recognition by summarizing the characteristics and features of objects. Unsupervised learning does not require explicit labels, but extracts the features of samples adaptively in learning process. Modeling of this ability of the brain is critical. There are multiple learning rules in the brain to accomplish various learning tasks. STDP is a widespread rule of synaptic weight modification in the brain. It updates the synaptic weights according to the temporal relationship of the pre- and post-synaptic spikes. Compared with the backpropagation algorithm widely used in DNN with gradient calculation, STDP is more biological plausible. However, unlike the backpropagation algorithm that relies on a large number of sample labels, STDP is a local optimization algorithm. Due to the lack of global information, the ability of self-organization and coordination between neurons is insufficient. In the SNN model, it will lead to disorder and uncertainty of spikes discharge, and it is difficult to achieve a stable release balance state of neurons. To this end, we design an unsupervised STDP-based spiking neural network model based on BrainCog, and bring unsupervised learning to BrainCog as a functional module.\nAs shown in Fig.~\\ref{frame}.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.48\\textwidth]{fig\/frame.pdf}\n\\caption{The framework of the unsupervised STDP-based spiking neural network model, which introduces the adaptive synaptic filter (ASF), the adaptive threshold balance (ATB), and the adaptive lateral inhibitory connection (ALIC) mechanisms to improve the information transmission and feature extraction of STDP-based SNNs. This figure is from \\cite{dong2022unsupervised}.}\n\\label{frame}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.49\\textwidth]{fig\/stp1.pdf}\n\\caption{(a) The adaptive synaptic filter and the adaptive threshold balance jointly regulate the neuron spikes balance of neuron. (b) The adaptive lateral inhibitory connection has different connections for different samples, preventing neurons from learning the same features. This figure is from~\\cite{dong2022unsupervised}.}\n\\label{stp}\n\\end{figure}\n\nTo solve the above problems, we introduced various adaptive mechanisms to improve the self-organization ability of the overall network. STP is another synaptic learning mechanism that exists in the brain. Inspired by STP, we designed an adaptive synaptic filter (ASF) that integrates input currents through nonlinear units, and an adaptive threshold balance (ATB) that dynamically changes the threshold of each neuron to avoid excessively high or low firing rates. The combination of the two controls the firing balance of neurons. As the Fig.~\\ref{stp} shows. We also address the problem of coordinating neurons within a single layer with an adaptive lateral inhibitory connection (ALIC). The mechanism have different connection structures for different input samples. Finally, in order to solve the problem of low efficiency of STDP training, we designed a sample temporal batch STDP. It combines the information between temporal and samples to uniformly update the synaptic weights, as shown by the following formula.\\begin{equation}\n\t\\begin{split}\n\t\t&\\frac{dw_{j}^{(t)}}{dt}^{+}=\\sum \\limits_{m=0}^{N_{batch}}\\sum \\limits_{n=0}^{T_{batch}}\\sum\\limits_{f=1}^{N} W(t^{f,m}_{i}-t^{n,m}_{j})\\\\\n\t\\end{split}\n\t\\label{eq5}\n\\end{equation} where $W(x)$ is the function of STDP, $N_{batch}$ is the batchsize of the input, $T_{batch}$ is the batch of time step, N is the number of neurons. We verified our model on MNIST and Fashion-MNIST, achieving 97.9\\% and 87.0\\% accuracy, respectively. To the best of our knowledge, these are the state-of-the-art results for unsupervised SNNs based on STDP.  \n\n\n\\textbf{2. Adaptability and Optimization}\n\n\\emph{(1) Quantum superposition inspired SNN}\n\n\nIn the microscopic size, quantum mechanics dominates the rules of operation of objects, which reveals the probabilistic and uncertainties of the world. New technologies based on quantum theory like quantum computation and quantum communication provide an alternative to information processing. Researches show that biological neurons spike at random and the brain can process information with huge parallel potential like quantum computing. \n\nInspired from this, we propose the Quantum Superposition Inspired Spiking Neural Network (QS-SNN)~\\cite{SUN2021102880}, complementing quantum image (CQIE) method to represent image in the form of quantum superposition state and then transform this state to spike trains with different phase. Spiking neural network with time differential convolution kernel (TCK) is used to do further classification shown in Fig.~\\ref{Fig:QSSNN}.\n\n\n\n\n\\begin{figure}[!htb]\n\t\\centering\n\t\\includegraphics[width=1.0\\linewidth]{fig\/QSSNN.pdf}  \n\t\\caption{Quantum superposition inspired spiking neural network, adopted from \\cite{SUN2021102880}.}\n\t\\label{Fig:QSSNN}\n\\end{figure}\n\n\n\\begin{figure}[!htb]\n\t\\centering\n\t\\includegraphics[width=1.0\\linewidth]{fig\/mnist_shift_result.pdf}  \n\t\\caption{Performance of QS-SNN on background MNIST inverse image, adopted from \\cite{SUN2021102880}.}\n\t\\label{Fig:mnist-res}\n\\end{figure}\n\nThe effort tries to incorporate the quantum superposition mechanism to SNNs as a new form of encoding strategy for BrainCog, and the model finally shows its capability on robustness for learning. The proposed QS-SNN model is tested on color inverted MNIST datasets. The background-inverted picture is encoded in the quantum superposition form as shown in Eq.~\\ref{QS-SNN} and~\\ref{QS-SNN_limitation}.\n\\begin{equation}\n\t\\mathinner{|I(\\theta)\\rangle}=\\frac{1}{2^n}\\sum\\limits_{i=0}^{2^{2n}-1}(cos(\\theta_{i})\\mathinner{|x_{i}\\rangle}+sin(\\theta_{i})\\mathinner{|\\bar{x}_{i}\\rangle})\\otimes\\mathinner{|i\\rangle}, \\\\\n\t\\label{QS-SNN}\n\\end{equation}\n\n\n\\begin{equation}\n\t\\theta_{i} \\in [0, \\frac{\\pi}{2}], i= 1, 2, 3, \\dots, 2^{2n}-1.\n\t\\label{QS-SNN_limitation}\n\\end{equation}\n\nSpike sequences of different frequencies and phases are generated from the picture information of the quantum superposition state. Furthermore we use two-compartment spiking neural networks to process these spike trains.\n\nWe compare the QS-SNN model with other convolutional models. The result in Fig.~\\ref{Fig:mnist-res} shows that our QS-SNN model overtakes other convolutional neural networks in recognizing background-inverted image tasks.\n\n\\emph{(2) Unsupervised SNN with adaptive learning rule and structure}\n\nBrain can accomplish specific tasks by adaptively learning to organize the features of a small number of samples. Few-shot learning is an important ability of the brain. In the above section, an unsupervised STDP-based spiking neural network is introduced~\\cite{dong2022unsupervised}. To better illustrate the power of our model on small sample training, we tested the model with small samples and find that this model has stronger small sample processing ability than ANN with similar structures, as shown in Tab.~\\ref{small}~\\cite{dong2022unsupervised}.\n\n\\begin{table}[h]\n\t\\caption{The performance of unsupervised SNN compared with ANN on MNIST dataset with different number of training samples~\\cite{dong2022unsupervised}.}\n\t\\centering\n\t\\resizebox{0.8\\linewidth}{!}{\n\t\t\\begin{tabular}{lrrrr}\n\t\t\t\\toprule \n\t\t\tsamples&200  &100&50&10 \\\\\n\t\t\t\\midrule\t\n\t\t\tANN&79.77\\%&71.40\\%&68.72\\%&47.12\\%\\\\\n\t\t\tOurs&81.45\\%&75.44\\%&72.88\\%&51.45\\%\\\\\n\t\t\t\\midrule\n\t\t\t&1.68\\%&4.04\\%&4.16\\%&4.33\\%\\\\\n\t\t\t\\bottomrule\n\t\t\\end{tabular}\n\t}\n\t\\label{small}\n\t\n\\end{table}\n\n\n\n\n\\emph{(3) Efficient and Accurate Conversion of SNNs}\n\n\tSNNs have attracted attention due to their biological plausibility, fast inference, and low energy consumption. However the training methods based on the plasticity~\\cite{zeng2017improving} and surrogate gradient algorithms~\\cite{wu2018spatio} need much memory and perform worse than ANN on large networks and complex datasets. For users of BrainCog, we are certain there will be clear need to use SNNs while keep the benifit from ANNs. As an efficient method, the conversion method combines the characteristics of backpropagation and low energy consumption and can achieve the same excellent performance as ANN with lower power consumption~\\cite{li2021free, han2020deep, han2020rmp}. However, the converted SNNs typically suffer from severe performance degradation and time delays. \n\t\n\t\\begin{figure}[htb]\n    \t\\centering\n    \t\\includegraphics[scale=0.3]{fig\/error.png}\n    \t\\caption{The conversion errors from IF neuron, time dimension, and MaxPooling, adopted from~\\cite{li2022efficient}.}\n    \t\\label{error}\n    \\end{figure}\n\n    We divide the performance loss to IF neurons, time dimension, and MaxPooling layer~\\cite{li2022efficient}, as shown in Fig.~\\ref{error}. In SNN, the neuron can only send one spike at most in each time step, so the maximum firing rate of the neuron is 1. After normalizing the weight of trained ANN, some activation values greater than 1 cannot be effectively represented by IF neurons. Therefore, the residual membrane potential in neurons will affect the performance of the conversion to some extent. In addition, since IF receives pre-synaptic neuron spikes for information transmission, the synaptic current received by the neuron at each time is unstable, but its sum can be approx to the converted value by increasing the simulation time. However, the total number of spikes is the time-varying extremum of the sum of the synaptic currents received. When the activation value corresponding to the IF neuron is negative and the total synaptic current received by the neuron in a short time exceeds the threshold, the neuron that should be resting will issue the spike, and the influence of the spike on the later layers cannot be eliminated by increasing the simulation time. Finally, in the conversion of the MaxPooling layer, the previous work has enabled spikes from the neuron with the maximum firing rate to pass through. However, due to the instability of synaptic current, neurons with the maximum firing rate are often not fixed, which makes the output of the converted MaxPooling layer usually larger.\n\n    \\begin{figure}[!h]\n    \t\\centering\n    \t\\includegraphics[scale=0.5]{fig\/conver.pdf}\n    \t\\caption{The conversion methods. (a) Burst spikes increase the upper limit of firing rates; (b) spike calibration corrects the effect of faulty spikes on conversion, and (c) LIPooling uses lateral inhibition mechanisms to achieve accurate conversion of the MaxPooling layers. Refined based on~\\cite{li2022efficient,li2022spike}.}\n    \t\\label{conver}\n    \\end{figure}\n\n    To solve the problem of the residual membrane potential of neurons, we introduce the burst mechanism, as shown in Fig.~\\ref{conver} (a), which enables neurons to send more than one spike between two time steps, depending on the current membrane potential. Once some neurons have residual information remaining, they can send spikes between two time steps. In this way, the firing rate of SNN can be increased, and the membrane potential remaining in the neuron can be transmitted to the neuron of the next layer. \n    \n    For classification problems, SNN only needs to ensure that the index of the maximum output is correct, but for more demanding conversion tasks, such as object detection, the solution of SIN problem is worth exploring. Note that the spikes emitted by hidden layer neurons are unstable, but the mean of its inter-spike interval distribution is related to its corresponding activation value~\\cite{li2022spike}. We monitor each neuron's spiking time in the forward propagation process and update its average inter-spike interval, as shown in Fig.~\\ref{conver} (b). Under a certain time allowance, the neuron that does not emit spikes will be determined to be Inactivated Neuron. Then, the twin weights emit spikes to suppress the influence of historical errors and calibrate the influence of wrong spikes to a certain extent to ensure accurate conversion.\n\n\n    Inspired by the lateral inhibition mechanism~\\cite{blakemore1970lateral}, we propose LIPooling for converting the maximum pooling layer, as shown in Fig.~\\ref{conver} (c). From the operation perspective, the inhibition of other neurons by the winner in LIPooling is -1, so the neuron with the largest firing rate at the current time step may not spike due to the inhibition of other neurons in history. From the output perspective, LIPooling sums up the output of all neurons during simulation. So the key is that LIPooling uses competition between neurons to get an accurate sum (equal to the actual maximum), instead of picking the winner.\n\n\\textbf{3. Multi-sensory Integration}\n\n\nOne can build SNN models through BrainCog to process different types of sensory inputs, while the human brain learns and makes decisions based on multi-sensory inputs. When information from various sensory inputs is combined, it can lead to increased perception, quicker response times, and better recognition. Hence, enabling BrainCog to process and integrate multi-sensory inputs are of vital importance.\n\nIn this section, we focus on concept learning with multi-sensory inputs. We present a multi-sensory concept learning framework based on BrainCog to generate integrated vectors with the multi-sensory representation of the concept.\n\nEmbodied theories, which emphasize that meaning is rooted in our sensory and experiential interactions with the environment, supports multi-sensory representations.\nBased on SNNs, we present a human-like framework to learn concepts which can generate integrated representations with five types of perceptual strength information~\\cite{wywFramework}.\nThe framework is developed with two distinct paradigms: Associate Merge (AM) and Independent Merge (IM), as Fig.~\\ref{MultisensoryIntegrationFramework} shows.\n\\begin{figure}[h!]\n  \\centering\n  \\includegraphics[width=8cm]{.\/fig\/MultisensoryIntegrationBrainCogNEW.png}\\\\\n  \\caption{The Framework of Concept Learning Based on SNNs with multi-sensory Inputs.}\n  \\label{MultisensoryIntegrationFramework}\n\\end{figure}\n\nIM is based on the widely accepted cognitive psychology premise that each type of sense for the concept is independent before integration~\\cite{wywFramework}.\nAs the input to the model, we will employ five common perceptual strength: visual, auditory, haptic, olfactory and gustatory.\nDuring the data preparation step, we min-max normalize all kinds of perceptual strength of the concept in the multi-sensory dataset so that each value of the vector is in $[0, 1]$.\nWe regard them as stimuli to the presynaptic neurons.\nIt's a 2-layer SNN model, with 5 neurons in the first layer matching the concept's 5 kinds of perceptual strength, and 1 neuron in the second layer representing the neural state following multi-sensory integration.\nIn this paradigm, we use perceptual strength based presynaptic Poisson neurons and LIF or Izhikevich as the postsynaptic neural model.\n\nThe weights between the neurons are $W^i= \\frac{g_i}{\\Sigma_i^n g_i} $ where  $g_i = \\frac{1}{\\sigma_i^2}$,$\\sigma_i^2$ is the variance of each kind of perceptual strength.\nWe convert the postsynaptic neuron's spiking train $S^{post}([0, T])$ in $[0, T]$ into integrated representations $B^{IM}([0,T])$ for the concept in this form:\n\\begin{equation}\n\\begin{aligned}\nB^{IM}([0,T]) &= [\\mathcal{T}(S^{post}((0, tol])), \\mathcal{T}(S^{post}((tol, 2*tol])), \\cdots , \\\\\n& \\mathcal{T}(S^{post}(((k-1)*tol, k*tol])) , \\cdots,  \\\\\n& \\mathcal{T}(S^{post}((\\lfloor \\frac{T}{tol} \\rfloor * tol, T]))]\n\\end{aligned}\n\\end{equation}\nHere if the interval has any spikes, the bit is 1. Otherwise it is 0, according to the $\\mathcal{T} (interval)$ function.\n\nThe AM paradigm presupposes that each kind of modality is associated before integration~\\cite{wywFramework}. \nIt includes 5 neurons, matching the concept's 5 distinct modal information sources. \nThey are linked with each other and are not self-connected. \nThe input spike trains to the network are generated using a Poisson event-generation algorithm based on perceptual strength.\nFor each concept, we turn the spike trains of these neurons into the ultimate integrated representations.\n\n\n\nThe weight value is defined by the correlation between each two modalities, i.e. $W = Corr(i, j)$, where $i, j \\in [A, G, H, O, V]$.\nWe convert the spike trains $S^i([0, T])$ of all neurons into binarycode $B^i([0,T])$ and conjoin them as the ultimate vector $B([0,T])$ as follows:\n\\begin{equation}\n\\begin{aligned}\nB^i([0,T]) &= [\\mathcal{T}(S^i((0, tol])), \\mathcal{T}(S^i((tol, 2*tol])), \\cdots , \\\\\n& \\mathcal{T}(S^i(((k-1)*tol, k*tol])) , \\cdots, \\\\\n& \\mathcal{T}(S^i((\\lfloor \\frac{T}{tol} \\rfloor * tol, T]))]\n\\end{aligned}\n\\end{equation}\n\n\\begin{equation}\n\\begin{aligned}\nB^{AM}([0,T]) &= [B^A([0,T]) \\oplus B^H([0,T]) \\oplus  B^G([0,T]) \\\\\n& \\oplus B^O([0,T]) \\oplus B^V([0,T])]\n\\end{aligned}\n\\end{equation}\n\n\nTo test our framework, we conducted experiments with three multi-sensory datasets (LC823~\\cite{wywLC423,wywLC400}, BBSR~\\cite{wywBBSR}, Lancaster40k~\\cite{wywLan40k}) for the IM and AM paradigms, respectively. \nWe used WordSim353~\\cite{wywWS353} and SCWS1994~\\cite{wywSCWS} as metrics~\\cite{wywFramework}. \nThe resuls show that integrated representations are closer to human beings than the original ones based on our framework, according to the overall results: 37 submodels outperformed a total of 48 tests for both AM and IM paradigm~\\cite{wywFramework}.\nMeanwhile, to compare the two paradigms, we introduce concept feature norms datasets which represent concepts with systematic and standardized feature descriptions. \nIn this study, we use the datasets McRae~\\cite{wywMcRae} and CSLB~\\cite{wywCSLB} as criteria.\nThe findings show that the IM paradigm performs better at multi-sensory integration for concepts with higher modality exclusivity. \nThe AM paradigm benefits the concept of uniform perceptual strength distribution. \nFurthermore, we present perceptual strength-free metrics to demonstrate that both paradigms of our framework have excellent generality~\\cite{wywFramework}.\n\n\n\\begin{figure}[h!]\n  \\centering\n  \\includegraphics[width=8cm]{.\/fig\/yuwei1.jpg}\\\\\n  \\caption{The Correlation Results Between Modality Exclusivity and Average of 3 Neighbors' Rankings.}\n  \\label{MEFS}\n\\end{figure}\n\n\n\n\n\n\\subsection{Decision Making}\nThis subsection introduces how BrainCog implements decision-making functions from the perspective of brain neural mechanism modeling and deep SNN-based reinforcement learning models. Using BrainCog, we build a multi-brain areas coordinated SNN model and a spiking deep Q-network to solve decision-making and control problems.\n\n\\textbf{1. Brain-inspired Decision-Making SNN}\n\n\nFor mammalian brain-inspired decision-making, we take inspiration from the PFC-BG-ThA-PMC neural circuit and build brain-inspired decision-making spiking neural network (BDM-SNN) model~\\cite{zhao2018brain} by BrainCog as shown in Fig. ~\\ref{dmsnn}. BDM-SNN contains the excitatory and inhibitory connections within the basal ganglia nuclei and direct, indirect, and hyperdirect pathways from the PFC to the BG~\\cite{frank2006anatomy,silkis2000cortico}. This BDM-SNN model incorporates biological neuron models (LIF and simplified H-H models), synaptic plasticity learning rules, and interactive connections among multi-brain areas developed by BrainCog. On this basis, we extend the dopamine (DA)-regulated BDM-SNN, which modulates synaptic learning for PFC-to-striatal direct and indirect pathways via dopamine. Different from the DA regulation method in~\\cite{zhao2018brain} which uses multiplication to modulate the specified connections, we improve it by introducing R-STDP~\\cite{Eugene2007} (from Eq.~\\ref{rstdpe} and Eq.~\\ref{rstdpw}) to modulate the PFC-to-striatal connections. \n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=8cm]{.\/fig\/bdmsnn.jpg}\n\\end{center}\n\\caption{The architecture of DA-regulated BDM-SNN, refined based on~\\cite{zhao2018brain}.}\\label{dmsnn}\n\\end{figure}\n\n\nThe BDM-SNN model implemented by BrainCog could perform different tasks, such as the Flappy Bird game and has the ability to support UAV online decision-making. For the Flappy Bird game, our method achieves a performance level similar to humans, stably passing the pipeline on the first try. Fig.~\\ref{fb}a illustrates the changes in the mean cumulative rewards for LIF and simplified H-H neurons while playing the game. The simplified H-H neuron achieves similar performances to that of LIF neurons. BDM-SNN with different neurons can quickly learn the correct rules and keep obtaining rewards. We also analyze the role of different ion channels in the simplified H-H model. From Fig.~\\ref{fb}b, we find that sodium and potassium ion channels have opposite effects on the neuronal membrane potential. Removing sodium ion channels will make the membrane potential decay, while the membrane potential rises faster and fires earlier when removing potassium ion channels. These results indicate that sodium ion channels can help increase the membrane potential, and potassium ion channels have the opposite effect. Experimental results also indicate that BDM-SNN with simplified H-H model that removes sodium ion channels fails to learn the Flappy Bird game.\n\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=8.8cm]{.\/fig\/hhfly.jpg}\n\\end{center}\n\\caption{(a) Experimental result of BrainCog based BDM-SNN on Flappy Bird. The y-axis is the mean of cumulative rewards. (b) Effects of different ion channels on membrane potential for simplified H-H model.}\\label{fb}\n\\end{figure}\n\nIn addition, for the UAV decision-making tasks in the real scene, our model could perform potential applications including flying over doors and windows and obstacle avoidance, which have been realized in~\\cite{zhao2018brain}. Users only need to divide the state space and action space according to different tasks, call the DA-regulated BDM-SNN decision-making model, and combine the UAV's action control instructions to complete the UAV's decision-making process.\n \n \nThis part of the work mainly draws on the neural structure and learning mechanism of brain decision-making and proposes the multi-brain areas coordinated decision-making spiking neural network constructed by BrainCog, and verifies the ability of reinforcement learning in different application scenarios. \n\n\n\\textbf{2. Spiking Deep Q Network with potential based layer normalization}\n\nDeep Q network is widely used for decision-making tasks, and it is required to have SNN based deep Q network for BrainCog so that it can be used for SNN based decision making. We propose potential-based layer normalization spiking deep Q network (PL-SDQN) model to combine SNN with deep reinforcement learning~\\cite{sun2022solving}. We use the LIF neuron model in BrainCog to simulate neurodynamics. Deep spiking neural networks are difficult to be applied to reinforcement learning tasks. On the one hand, it is due to the complexity of reinforcement learning task itself, on the other hand, it is challenging to train spiking neural networks and transmit spiking signal characteristics in deep layers. We find that the spiking deep Q network quickly dissipates the spiking signal in the convolutional layer. Inspired by how local environmental potentials influence brain neurons, we propose the potential-based layer normalization (pbLN) method. The $x_t$, the postsynaptic potential of convolution layers, are normalized as \n\\begin{equation}\n\t\\hat{x_t} = \\frac{x_t-\\bar{x}_t}{\\sqrt{\\sigma_{x_t}+\\epsilon}}\n\t\\label{Eq:norm}\n\\end{equation}\t\n\n\\begin{equation}\n\t\\bar{x}_t = \\frac{1}{H} \\sum_{i=1}^Hx_{t, i}\n\\end{equation}\t\n\n\\begin{equation}\n\t\\sigma_{x_t} = \\frac{1}{H}\\sqrt{\\sum_{i=1}^H(x_{t, i}-\\bar{x}_t)}\n\\end{equation}\n\n\nWe construct PL-SDQN model as shown in Fig.~\\ref{Fig:sdqn}. Atari game images are processed by spiking convolution network and pbLN method and input into a fully connected LIF neural network. The spiking output of PL-SDQN is weighted and summed to continue state-action values. \n\n\\begin{figure}[h!]\n\t\\centering\n\t\\includegraphics[width=8cm]{fig\/SDQN_struct.pdf}\n\t\\caption{The framework of PL-SDQN. Refined from~\\cite{sun2022solving}.}\n\t\\label{Fig:sdqn}\n\\end{figure}\n\n\n\nWe compared our model with the original ANN-based DQN model, and the results are shown in Fig.~\\ref{Fig:game_res}. It shows that our model achieved better performance compared with vanilla DQN model.\n\n\n\\begin{figure}[h!]\n\t\\centering\n\t\\includegraphics[width=8cm]{fig\/game_res_new_resize.pdf}\n\t\\caption{PL-SDQN performance on Atari games, adopted from~\\cite{sun2022solving}.}\n\t\\label{Fig:game_res}\n\\end{figure}\n\n\n\\subsection{Motor Control}\n\nNeuromorphic models on robot control can achieve more robust and energy efficient effects than conventional methods. Spiking neural networks have been used in robot control studies like navigation~\\cite{wang2014mobile} and robot arm control~\\cite{Tieck2018}. Inspired by the brain motor circuit, we construct a multi-brain areas coordinated SNN robot motor control model, to extend BrainCog to control various robots to model embodied intelligence.\n\nWe construct the brain-inspired motor control model with LIF neuron provided by BrainCog. The whole network model architecture is shown in Fig.~\\ref{Fig:motor}. The high-level motion information is produced by SMA and  PMC modules. As discussed above, the function of SMA is to process internal movement stimuli and is responsible for the planning and abstraction of advanced actions. The SMA model contains LIF neurons and receives input signals. One part of the output pulse of the SMA module stimulates the PMC module, and the other part is received by the BG module. The output of the BG module is used as a supplementary signal for action planning, and serves as the input to the PMC together with the SMA signal.\n\n\\begin{figure}[h!]\n\t\\centering\n\t\\includegraphics[width=8cm]{fig\/motor.pdf}\n\t\\caption{A spiking neural network for motor control based on BrainCog.}\n\t\\label{Fig:motor}\n\\end{figure}\n\n\nIn order to expand the dimension of neuron direction representation, we use neuron population coding to output the high-level action abstraction. Population-coded spiking neural network has been used for energy-efficient continuous control~\\cite{tang2021deep}, showing population coding can increase the ability of spiking neurons to represent precise continuous values. In our work, we are inspired by neural mechanisms of population encoding of motion directions in the brain and use LIF neuron groups to process the output spikes from PMC module. \n\nThe cerebellum plays an important role in motor coordination and fine regulation of movements. We built spiking neural network based cerebellum model to process the high-level motor control population embedding. The outputs of populations are fused to encode motor control information generated by high-level cortex area inputs to a three-layer cerebellum spiking neural network, including GCs, PCs and DCN modules. And the cerebellum takes pathway connections like DenseNet~\\cite{huang2017densely}. The DCN layer generates the final joint control outputs.\n\n\n\\subsection{Knowledge Representation and Reasoning}\nThis subsection shows how the BrainCog platform achieve the ability of knowledge representation and reasoning. Via neuroplasticity and population coding mechanisms, spiking neural networks acquire music and symbolic knowledge. Moreover, on this basis, cognitive tasks such as music generation, sequence production, deductive reasoning and inductive reasoning are realized.\n\n\n\\textbf{1. Music Memory and Stylistic Composition SNN}\n\n\nMusic is part of human nature. Listening to melodies involves sensory perception, personal memory, action, emotion, and even creative behaviors, etc~\\cite{Koelsch2012}. Music memory is a fundamental part of musical behaviors, and humans have strong abilities to store a sequence of notes in the brain. Learning and creating music are also essential processes. A musician engages his memory, emotion, musical knowledge and skills to write a beautiful melody. Actually, neuroscientists have found that many brain areas need to collaborate to complete the cognitive behaviors with music. Inspired by brain mechanisms, this paper focuses on the two key issues of music memory and composition, which are modeled by spiking neural networks based on the BrainCog platform.\n\n\\subsubsection{Musical Memory Spiking Neural Network} \\label{secA}\nA musical melody is composed of a sequence of notes. Pitch and duration are two essential attributes of a note. Scientists have found that the primary auditory cortex provides a tonotopic map to encode the pitches, which means that neurons in this region have their preferences of pitches~\\cite{Kalat2015}. Meanwhile, neural populations in the medial premotor cortex have the preferences of all the time intervals covered in hundreds of milliseconds~\\cite{merchant2013a}. Besides, researchers have emphasized the contribution of the hippocampus in sequence memory~\\cite{Fortin2002}. Inspired by these mechanisms, this work proposes a spiking neural model, which contains collaborated subnetworks to encode, store and retrieve the music melodies~\\cite{LQ2020}.\n\n\\emph{Encoding:} As is shown in Fig.~\\ref{01}, this work defines pitch subnetwork and duration subnetwork to encode pitches and durations of musical notes respectively. These two subnetworks are composed of numbers of minicolumns with different preferences. Synaptic connections with transmission delays exist between neurons from different layers. Besides, a cluster that represents the title of a musical melody is composed of numerous individual neurons. This cluster has the feedforward and feedback connections with pitch and duration subnetworks. Since the BrainCog platform supports various neural models, this work takes LIF model to simulate neural dynamics.\n\\begin{figure}[!htbp]\n    \\centering\n    \\includegraphics[scale=0.4]{.\/fig\/music1.png}\n    \\caption{The architecture of the music memory model, refined based on~\\cite{LQ2020}.}\n    \\label{01}\n\\end{figure}\n \n\\emph{Storing:} Based on the encoding process, as the notes input sequently, the neurons in pitch and duration subnetworks with different preferences respond to these sequential notes and fire orderly. Meanwhile, connections between these neurons are computed and updated by the STDP learning rule. It is important to indicate that the neurons and synapses are grown dynamically. Besides, synaptic connections between the title cluster and other two subnetworks are generated and updated by the STDP learning rule simultaneously. The details of note sequence memorizing can be found in our previous work~\\cite{LQ2020}.\n\n\\emph{Retrieving:} Given the title of a musical work, the ordered notes can be recalled accurately. Since the weights of connections are updated in storing process, neural activities in the title cluster lead to the excitations of neurons in pitch and duration subnetworks. Then, the notes are retrieved in order. We use a public corpus that contains 331 classical piano works~\\cite{Krueger2018} recorded by MIDI standard format to evaluate the model. The experiments have shown that our model can memorize and retrieve the melodies with an accuracy of 99\\%. The details of the experiments have been discussed in our previous work~\\cite{LQ2020}.\n\n\n\\subsubsection{Stylistic Composition Spiking Neural Network}\n\n\nHow to learn and make music are quite complex processes for humans. Scientists have found that the memory system and knowledge experience participate in human creative behaviors~\\cite{Dietrich2004}. Many brain areas like the prefrontal cortex are engaged in human creativity~\\cite{Jung2013}. However, the details of brain mechanisms are still unclear. Inspired by the current neuroscientific findings, the BrainCog introduces a spiking neural network for learning musical knowledge and creating melodies with different styles~\\cite{LQ2021}. \n\n\\begin{figure}[!htbp]\n    \\centering\n    \\includegraphics[scale=0.28]{.\/fig\/2.pdf}\n    \\caption{Stylistic composition model inspired by brain mechanisms. Refined based on~\\cite{LQ2021}.}\n    \\label{02}\n\\end{figure}\n\n\\emph{Musical Learning:} This work proposes a spiking neural model which is composed of a knowledge network and a sequence memory network. As is shown in Fig.~\\ref{02}, the knowledge network is designed as a hierarchical structure for encoding and learning musical knowledge. These layers store the genre (such as Baroque, Classical, and Romantic), the names of famous composers and the titles of musical pieces. Neurons in the upper layers project their synapses to the lower layers. The sequence memory network stores the ordered notes which have been discussed in section~\\ref{secA}. During the learning process, synaptic connections are also projected from the knowledge network to the sequence memory network. This work also takes LIF model which is supported by the BrainCog platform to simulate neural dynamics. Furthermore, all the connections are generated and updated dynamically by the STDP learning rule. \n\n\\begin{figure}[htbp]\n    \\centering\n    \\includegraphics[scale=0.6]{.\/fig\/3.pdf}\n    \\caption{A sample of a generated melody with Bach's characteristic.}\n    \\label{03}\n\\end{figure}\n\n\n\\emph{Musical Composition:} Based on the learning process, genre-based and composer-based melody compositions are discussed in this paper. Given the beginning notes and the length of the melody to be generated, the genre-based composition can produce a single-part melody with a specific genre style. This task is achieved by the neural circuits of genre cluster and sequential memory system. Similarly, the composer-based composition can produce melodies with composers' characters. The composer cluster and sequential memory system circuits contribute to this process~\\cite{LQ2021}. We also use a classical piano dataset including 331 musical works recorded by MIDI format~\\cite{Krueger2018} to train the model. Fig.~\\ref{03} shows a sample of the generated melody with Bach's style. The details of stylistic composition can be referred to our previous work~\\cite{LQ2021}.\n\n\n\nA total of 41 human listeners are invited to evaluate the quality of the generated melodies, and they are divided into two groups, one of which has a musical background. Experiments have shown that the pieces produced by the model have strong characteristics of different styles and some of them sound nice.\n\n\\textbf{2. Brain-Inspired Sequence Production SNN}\n\nSequence production is an essential function for AI applications. Components in BrainCog enable the community to build SNN models to handle this task. In this paper, we introduce the brain-inspired symbol sequences production spiking neural network (SPSNN) model that has been incorporated in BrainCog~\\cite{fang2021spsnn}. SPSNN incorporates multiple neuroscience mechanisms including Population Coding~\\cite{xie2022geometry}, STDP~\\cite{dan2004spike}, Reward-Modulated STDP~\\cite{fremaux2016neuromodulated}, and Chunking Mechanism~\\cite{pammi2004chunking}, mostly covered and provided by BrainCog. After reinforcement learning, the network can complete the memory of different sequences and production sequences according to different rules.\n\nFor Population Coding, this model utilizes populations of neurons to represent different symbols. The whole neural loop of SPSNN is divided into Working Memory Circuit, Reinforcement Learning Circuit, and Motor Neurons~\\cite{fang2021spsnn}, shown in Fig.~\\ref{SPSNN}. The Working Memory Circuit is mainly responsible for completing the memory of the sequence. The Reinforcement Learning Circuit is responsible for acquiring different rules during the reinforcement learning process. The Motor Neurons can be regarded as the network's output. \n\nIn the working process of the model, the Working Memory Circuit and the Reinforcement Learning Circuit cooperate to complete the memory and production of different sequences~\\cite{fang2021spsnn}. It is worth mentioning that with the increase of background noise, the recall accuracy of symbols at different positions in a sequence gradually decreases, and the overall change trend follows the \"U-shaped accuracy\", which is consistent with experiments in psychology and neuroscience~\\cite{jiang2018production}. The results are highly consistent due to the superposition of primacy and recency effects. Our model provides a possible explanation for both effects from a computational perspective.\n\\begin{figure}[!htbp]\n\\centering\n\\includegraphics [width=8.9cm]{.\/fig\/spsnnloop}\n\\caption{The architecture of SPSNN, adopted from~\\cite{fang2021spsnn}.}\n\\label{SPSNN}\n\\end{figure}\n\n\n\\textbf{3. Commonsense Knowledge Representation Graph SNN}\n\n\nCommonsense knowledge representation and reasoning are important  cornerstones  on the way to realize human-level general  AI~\\cite{minsky2007emotion}. In this module, we build Commonsense Knowledge Representation SNN(CKR-SNN) to explore whether SNN can complete these cognitive function.\n\n\n\nThe hippocampus plays a critical role in the formation of new knowledge memory~\\cite{schlichting2017hippocampus}. Inspired by the population coding mechanism found in hippocampus~\\cite{ramirez2013creating},  this module encodes the entities and relations of commonsense knowledge graph into different  populations of neurons. Via spiking timing-dependent plasticity (STDP) learning principle, the synaptic connections between neuron populations are formed after guiding the sequential firings of corresponding  neuron populations~\\cite{KRRfang2022}.\n\nAs Fig.~\\ref{GSNN} shows, neuron populations together constructed the giant graph spiking neural networks, which contain the commonsense knowledge. In this module, Commonsense Knowledge Representation SNN(CKR-SNN) represents a subset of Commonsense Knowledge Graph ConceptNet~\\cite{Conceptnet55}. After training, CKR-SNN can complete conceptual  knowledge generation and other cognitive tasks~\\cite{KRRfang2022}. \n\n\n\n\\begin{figure}\n\\centering\n\\includegraphics [width=8.9cm]{.\/fig\/GSNN}\n\\caption{Graph Spiking Neural Networks for Commonsense Representation, adopted from~\\cite{KRRfang2022}.}\n\\label{GSNN}\n\\end{figure}\n\n\n\n\n\n\n\n\n\\textbf{4. Causal Reasoning SNN}\n\nIn BrainCog, we constructed causal reasoning SNN, as an instance to verify the feasibility of spiking neural networks to realize deductive and inductive  reasoning. Specifically, Causal Reasoning Spiking Neural Network (CRSNN) module contains a brain-inspired causal reasoning spiking neural network model~\\cite{fang2021crsnn}.\n\n\nThis model explores how to encode a static causal graph into a spiking neural network and implement subsequent reasoning based on a spiking neural network. Inspired by the causal reasoning process of the human brain~\\cite{pearl2018book}, we try to explore how causal reasoning can be implemented based on spiking neural networks. The 3D model of CRSNN is shown in Fig.~\\ref{CRSNN}.\n\nInspired by neuroscience, the CRSNN module adopts the population coding mechanism and uses neuron populations to represent nodes and relationships in the causal graph. Each node indicates different events in the causal graph, as shown in Fig.~\\ref{CRSNN}. By giving current stimulation to different neuron populations in the spiking neural networks and combining the STDP learning rule~\\cite{dan2004spike}, CRSNN can encode the topology between different nodes in a causal graph into a spiking neural network. Furthermore, according to this network, CRSNN completes the subsequent deductive reasoning tasks.\n\nThen, by introducing an external evaluation function, we can grasp the specific reasoning path in the working process of the network according to the firing patterns of the model, which gives the CRSNN more interpretability compared to traditional ANN models~\\cite{fang2021crsnn}. \n\n\n\n\\begin{figure}\n\\centering\n\\includegraphics [width=8.8cm]{.\/fig\/cg3d}\n\\caption{CRSNN 3D model, adapted from~\\cite{fang2021crsnn}.}\n\\label{CRSNN}\n\\end{figure}\n\n\n\n\n\n\\subsection{Social Cognition}\n\nThe nature and neural correlates of social cognition is an advanced topic in cognitive neuroscience. In the field of artificial intelligence and robotics, there are few in-depth studies that take the neural correlation and brain mechanisms of biological social cognition seriously. Although the scientific understanding of biological social cognition is still in a preliminary stage~\\cite{zeng2018toward}, we integrate the biological findings of social cognition into a model to construct a brain-inspired model for social cognition to extend the functions of BrainCog.\n\nUnderstanding ourselves and other people is a prerequisite for social cognition. \n\nAn individual's perception of the world is realized through his own body, and the importance of  knowing oneself is the perception of self-body. Neuroscientific researches show that the inferior parietal lobule (IPL) is activated when the subjects see self-generated actions~\\cite{macuga2011selective} and their own faces~\\cite{sugiura2015neural}. Similar to the IPL, the Insula is activated in bodily ownership and self-recognition tasks~\\cite{craig2009you}. \n\nUnderstanding the mental states of others plays an important role for understanding other people. Theory of mind is an ability to distinguish between self and others and to infer others' mental states (such as desires, goals, beliefs, etc.) in the social context~\\cite{shamay-tsoory_dissociation_2007,sebastian_neural_2012,dennis2013cognitive}. This ability can help us reasonably infer other people's policies and goals. Inspired by this, we believe that applying theory of mind to the agent's decision-making process will improve the agent's inference of other agents, so as to take more reasonable actions. Neuroscientific researches~\\cite{abu-akel_neuroanatomical_2011,hartwright_multiple_2012,hartwright_special_2015, koster-hale_theory_2013} show that the brain areas related to theory of mind are mainly TPJ, part of PFC, ACC and IFG. The IPL contained in the TPJ is mainly used to represent self-relevant information, while the pSTS is used to represent information related to others. The insula, representing the abstract self, can be stimulated with self-related information~\\cite{zeng2018toward}. When theory of mind is going on, the IFG will suppress self-relevant information. Therefore, the TPJ will input other-relevant information into the PFC. The ACC evaluates the value of others' states, so as to help the PFC to infer others. The process of inferring others' goals or behaviors can be understood as simulating other people's decision-making~\\cite{suzuki_learning_2012}. Therefore, this process will be regulated by dopamine from substantia nigra compacta\/ventral tegmental area (Snc\/VTA).\n\n\n\n\n\n\n\n\nWith the neuron model and STDP function provided by the BrainCog framework, a brain-inspired social cognition model is constructed, as shown in Fig.~\\ref{BISC}.\n\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=8cm]{fig\/SC.png}\n\\end{center}\n\\caption{Brain-inspired social cognition model.}\n\\label{BISC}\n\\end{figure}\n\n\n\n\nThe brain-inspired social cognition model contains two pathways: the bodily self-perception pathway and the theory of mind pathway.\n\nThe bodily self-perception pathway (shown in Fig.~\\ref{Self}) consists of inferior parietal lobule spiking neural network (IPL-SNN) and Insula spiking neural network (Insula-SNN). The IPL-SNN realizes motor-visual associative learning. The Insula-SNN realizes the abstract representation of oneself, that is, when the detected movement's visual results match the expected results of its own movement, the Insula will be activated, and the robot considers that the moving part in the field of vision belongs to itself.\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=8cm]{fig\/Self.png}\n\\end{center}\n\\caption{The architecture and pathway of bodily self-perception in the brain-inspired social cognition model.}\n\\label{Self}\n\\end{figure}\n\nThe architecture of IPL-SNN and the process of motor-visual associative learning is shown in Fig.~\\ref{IPL}. The vPMC generates its own motion angle information, and the STS outputs the motion angle information detected by vision. According to the STDP mechanism and the spiking time difference of neurons in IPLM and IPLV, the motor-visual associative learning is established. \n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=8cm]{fig\/IPL.png}\n\\end{center}\n\\caption{Motor-visual associative learning in IPL, adapted from~\\cite{zeng2018toward}.}\n\\label{IPL}\n\\end{figure}\n\nThe architecture of Insula-SNN is shown in Fig.~\\ref{Insula}. The Insula receives angle information from IPLV and STS. After the motor-visual associative learning in IPL, IPLV outputs the visual feedback angle information predicted according to its own motion, and STS outputs the motion angle information detected by vision. If the two are consistent, the Insula will be activated.\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=8cm]{fig\/Insula.png}\n\\end{center}\n\\caption{The architecture of Insula-SNN.}\n\\label{Insula}\n\\end{figure}\n\n\n\nThe theory of mind pathway~\\cite{zhao2022brain} is mainly composed of three modules: the perspective taking module, the policy inference module, the action prediction module, and the state evaluation module (shown in Fig.~\\ref{ToM}).\n\n\\begin{figure}[!htbp]\n\\centering\n\\includegraphics[width=0.48\\textwidth]{.\/fig\/ToM1.jpg}\n\\caption{The architecture of theory of mind in the brain-inspired social cognition model, refined based on~\\cite{zhao2022brain}.}\n\\label{ToM}\n\\end{figure}\n\nThe perspective taking (also called self-perspective inhibition~\\cite{zeng2018toward}) module simulates the function of suppressing self-relevant information in the process of distinguish self and others. The information related to self can stimulate an representation of abstract self. When we infer others, the information related to self can be suppressed. Assuming that the agent knows the environment. When the agent with the theory of mind (ToM) infers the observation of others, it only needs to bring its own observation into the position of others. A matrix is used to represent the observed environment, where 1 indicates that the area can be observed at the location, and 0 indicates that the area cannot be observed at the location. Another matrix is used to represent the position of objects. The position which is occupied by objects is represented by 1, otherwise it is represented by 0. By taking the intersection of the two matrices, the estimation of others' states are obtained. According to the fact that the IFG helps the brain to suppress the performance of self-relevant information in the process of ToM, agent with the ToM will inhibit its own representation of states, and further infer the behavior of others by these estimation of others' states. In summary, the input of this module is the observation vector and the matrix of the environment. The output is the observation vector of others' perspective.\n\nThe dorsolateral prefrontal cortex (DLPFC) has the function of storing working memory and predicting others' behaviors. The action prediction module is used to simulate the function of the DLPFC. The input is formed by the observation value of others' state output by the perspective taking module. The module is a single layer spiking neural network. There is lateral suppression in the output layers. The network is trained by R-STDP. The source of reward is the difference between the predicted value and the real value. When the predicted value is consistent with the real value, the reward is positive. When the predicted value is not equal to the real value, the reward is negative.\n\nThe state evaluation module composed of a single layer spiking neural network simulates the function of the ACC brain area. The inputs of the module are the predicted state and the output is safe or unsafe.\n\nFinally, we conducted  two experiments to test the brain-inspired social cognition model.\n\n\\textbf{1. Multi-Robots Mirror Self-Recognition Test }\n\nThe mirror test is the most representative test of social cognition. Only a few animals passed the test, including chimpanzees~\\cite{RN826}, orangutans~\\cite{RN827}, bonobos~\\cite{RN828}, gorillas~\\cite{RN829,RN830}, Asiatic elephant~\\cite{RN831}, dolphins~\\cite{RN832}, orcas~\\cite{RN833}, macaque monkeys~\\cite{RN835}, etc. Based on the mirror test, we proposed the Multi-Robots Mirror Self-Recognition Test~\\cite{zeng2018toward}, in which three robots with identical appearance move their arms randomly in front of the mirror at the same time, and each robot needs to determine which mirror image belongs to it. The experiment includes training stage and test stage.\n\nThe training stage is shown in Fig.~\\ref{train}. Three blue robots with identical appearance move randomly in front of the mirror at the same time. Each robot establishes the motor-visual association according to the motion angle of its own arm and the angle of visual detection.\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=8cm]{fig\/mirrortest_train.jpg}\n\\end{center}\n\\caption{Training stage in multi-robots mirror self-recognition test, adapted from~\\cite{zeng2018toward}.}\n\\label{train}\n\\end{figure}\n\nThe test stage is shown in Fig.~\\ref{test}. In the test stage, the robot can predicts the visual feedback generated by its arm movement according to the training results. By comparing the similarity between the predicted visual feedback and the detected visual results, the robot can identify which mirror image belongs to it. \n\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=8cm]{fig\/mirrortest_test.jpg}\n\\end{center}\n\\caption{Test stage in multi-robots mirror self-recognition test, adapted from~\\cite{zeng2018toward}.}\n\\label{test}\n\\end{figure}\n\nIn the bodily self-perception pathway, the input is the angle of the robot's random motion and the angle detected by the robot's vision. After training and testing, the output is an image, which is the result of visual motion detection and the result of self motion prediction. The motion track corresponding to the red line in the visual motion detection result is generated by itself. The result is shown in Fig.~\\ref{result}.\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=8cm]{fig\/mirrortest_result.png}\n\\end{center}\n\\caption{The result of IPL-SNN.}\n\\label{result}\n\\end{figure}\n\n\\textbf{2. AI Safety Risks Experiment }\n\nAI safety risks experiment is shown in Fig.~\\ref{fig_sim}. After observing the behavior of the other two agents, the green agent can infer the behaviors of others by utilizing its ToM ability when safety risks may arise due to environmental changes. The experiment was conducted in the environments with some simulated types of safety risks (e.g., the intersection will block the view of agents and may cause agents to collide in the crossing).\n\nThe experiment shows that the agent can infer others when they have different perspectives. In the first two environments, the agent observes the movement of others, and in the third environment, the agent predicts others' actions. In the experiment, we verify the effectiveness of the model by taking the rescue behavior as the standard when other agents might be in danger. The experimental results show that the agent with the ToM can predict the danger of the other agent in a slightly changed environment after watching other agents move in the previous environments.\n\n\\begin{figure}[!htbp]\n\\centering\n\\subfloat[Green agent observes others' behaviors (example 1)]{\\includegraphics[height=0.65in]{.\/fig\/Figure6A.jpg}%\n\\label{1}}\n\\hfil\n\\subfloat[Green agent observes others' behaviors (example 2)]{\\includegraphics[height=0.65in]{.\/fig\/Figure6B.jpg}%\n\t\\label{2}}\n\\hfil\n\\subfloat[Test example 1 (with ToM)]{\\includegraphics[height=0.65in]{.\/fig\/Figure7A.jpg}%\n\t\\label{3}}\n\\hfil\n\\subfloat[Test example 2 (without ToM)]{\\includegraphics[height=0.65in]{.\/fig\/Figure7B.jpg}%\n\t\\label{4}}\n\\caption{Comparison diagram of experimental results. (a) Example 1. The green agent observes others' behaviors. (b) Example 2. The green agent observes others' behaviors. (c) The green agent with ToM can help other agents avoid risks. (d) The green agent without ToM is unable to help other agents avoid risks. Similar results can be found in~\\cite{zhao2022brain}.}\n\\label{fig_sim}\n\\end{figure}\n\n\\section{Brain Simulation}\nBrain simulation includes two parts: brain cognitive function simulation and multi-scale brain structure simulation. We incorporate as much published anatomical data as possible to simulate cognitive functions such as decision-making and working memory. Anatomical and imaging multi-scale connectivity data is used to make whole-brain simulations from mouse, macaque to human more biologically plausible.\n\n\\subsection{Brain Cognitive Function Simulation}\n\n\\textbf{1. \\emph{Drosophila}-inspired Decision-Making SNN}\n\n\\emph{Drosophila} decision-making consists of value-based nonlinear decision and perception-based linear decision, where the nonlinear decision could help to amplify the subtle distinction between conflicting cues and make winner-takes-all choices~\\cite{tang2001choice}. In this paper, the BrainCog framework is used to build \\emph{Drosophila} nonlinear and linear decision-making pathways as shown in Fig.~\\ref{pi}a-b. The entire model consists of a training phase and a testing phase as same as~\\cite{zhao2020neural}. In the training phase, a two-layer SNN with LIF neurons is trained by reward-modulated STDP, which combines local STDP synaptic plasticity with global dopamine regulation. The training phase learns the safe pattern (upright-green T) and the punished pattern (inverted-blue T)~\\cite{zhao2020neural}. Therefore, it is safe for green color and upright T shape factors, while blue color and inverted T shape are dangerous. \n\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=8cm]{.\/fig\/3.jpg}\n\\end{center}\n\\caption{(a) Linear Pathway. (b) Nonlinear Pathway. (c) Experiments for training and choice phases. (d) Experimental results of linear and nonlinear networks under the dilemma. The X-axis refers to the color density, and the Y-axis represents the PI values. Refined based on~\\cite{zhao2020neural}.}\n\\label{pi}\n\\end{figure}\n\nTwo cues (color and shape) are restructured during the testing phase, requiring linear and nonlinear pathways to make a choice between inverted-green T and upright-blue T, as shown in Fig.~\\ref{pi}c. The linear decision directly uses the knowledge acquired during the training phase to make decisions. The nonlinear network models the recurrent loop of the DA\u2011GABA\u2011MB circuit~\\cite{tang2001choice,zhang2007dopamine,zhou2019suppression}: KC activates the anterior posterior lateral (APL) neurons, which in turn releases GABA transmitter to inhibit the activity of KC. KC also provides mushroom body output neuron (MBON) with exciting input in order to generate behavioral choices. When faced with conflicting cues, the level of DA increases rapidly and produces mutual inhibition with APL, thereby producing a disinhibitory effect on KC. The excitatory connection between DA and MBON also helps speed up decision-making.\n\nTo verify the consistency of \\emph{drosophila}-inspired decision-making SNN with the conclusions from neuroscience~\\cite{tang2001choice}, we count the behavior paradigm of our model under different color intensities over a period of time. First, we run the network for 500 steps to count the time $t_1$ of selecting behavior 1 (avoiding) and the time $t_2$ of selecting behavior 2 (approaching). Then we calculate prefer index (PI) values under different color intensity: $PI=\\frac{\\left | t_1-t_2 \\right | }{\\left | t_1+t_2 \\right | } $. From Fig.~\\ref{pi}d, we find that nonlinear circuits could achieve a gain-gating effect to enhance relative salient cue and suppress less salient cue, thereby displaying the nonlinear sigmoid-shape curve~\\cite{zhao2020neural}. However, the linear network couldn't amplify the difference between conflicting cues, thus making an ambiguous choice (linear-shape curve)~\\cite{zhao2020neural}. This work proves that drawing on the neural mechanism and structure of the nonlinear and linear decision-making of the \\emph{Drosophila} brain, the brain-inspired computational model implemented by BrainCog could obtain consistent conclusions with the~\\emph{Drosophila} biological experiment~\\cite{tang2001choice}.\n\n\\textbf{2. PFC Working Memory }\n\n\nUnderstanding the detailed differences between the brains of humans and other species on multiple scales will help illuminate what makes us unique as a species~\\cite{zhang2021comparison}. The neocortex is associated with many cognitive functions such as working memory, attention and decision making~\\cite{miller2000prefontral,nieder2003coding,\nbishop2004prefrontal,koechlin2003architecture,\nwood2003human}. Based on the human brain neuron database of the Allen Institute for Brain Science, the key membrane parameters of human neurons are extracted \\footnote{http:\/\/alleninstitute.github.io\/AllenSDK\/cell\\_types.html}. Different types of human brain neuron models and rodent neuron models are established based on adaptive Exponential Integrate-and-Fire (aEIF) model~\\cite{Fourcaud2003How,brette2005adaptive}, which is supported by BrainCog.\n\nWe refined the model of a  single PFC proposed by Haas and colleagues \\footnote{http:\/\/senselab.med.yale.edu\/ModelDB\/}~\\cite{hass2016detailed}. Subsequently, a 6-layer PFC column model based on biometric parameters was established~\\cite{shapson2021connectomic}. The pyramidal cells and interneurons were proportionally distributed from the literature~\\cite{beaulieu1993numerical,defelipe2011evolution} and connected with different connection probabilities for different types of neurons based on previous studies~\\cite{gibson1999two,hass2016detailed,\ngao2003dopamine}. Firstly, the accuracy of information maintenance was tested on  rodent PFC network model.\n\n\n\\begin{figure}[!htbp]\n\\centering\n\\includegraphics[width=0.5\\textwidth]{.\/fig\/final.png}\n\\caption{Anatomical and network stimulation diagram. (a) The connection of a single PFC column. (b) The distribution proportion of different types of neurons in each column layer. (c) Network persistent activity performance. Refined based on~\\cite{zhangqian2021comparison}.}\n\\end{figure}\n\nKeeping the network structure and other parameters unchanged, only using human neurons to replace rodent neurons can significantly improve the accuracy and integrity of image output. From an evolutionary perspective, the lower membrane capacitance of human neurons facilitates firing. This change improves the efficiency of information transmission, which is consistent with the results of biological experiments~\\cite{eyal2016unique}. This data-driven PFC column model provides an effective simulation-validation platform to study other high-level cognitive functions~\\cite{2020Computational}.\n\n\\subsection{Multi-scale Brain Structure Simulation}\n\n\\textbf{1. Neural Circuit}\n\n\\subsubsection{Microcircuit}\nBrainCog implements a BDM-SNN model inspired by the decision-making neural circuit of PFC-BG-ThA-PMC in the mammalian brain (as shown in Fig.~\\ref{dm})~\\cite{zhao2018brain}. The BDM-SNN models the excitatory and inhibitory reciprocal connections between the basal ganglia nucleus~\\cite{lanciego2012functional}: (1) Excitatory connections: STN-Gpi, STN-Gpe. (2) Inhibitory connections: StrD1-Gpi, StrD2-Gpe, Gpe-Gpi, Gpe-STN. Direct pathway (PFC-StrD1), indirect pathway (PFC-StrD2), and hyperdirect pathway (PFC-STN) from PFC to BG are further constructed. The output from BG transmits an inhibitory connection to the thalamus and finally excites PMC~\\cite{parent1995functional}. In addition, excitatory connections are also formed between PFC and thalamus, and lateral inhibition exists in PMC. Such brain-inspired neural microcircuit, consisting of connections among different cortical and subcortical brain areas, and incorporating DA-regulated learning rules, enable human-like decision-making ability.\n\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=8cm]{.\/fig\/bdm.jpg}\n\\end{center}\n\\caption{The microcircuit of PFC-BG-ThA-PMC. Refined based on~\\cite{zhao2018brain}.}\n\\label{dm}\n\\end{figure}\n\n\\subsubsection{Cortical Column}\n\nA mammalian thalamocortical column is constructed in BrainCog, which is based on detailed anatomical data~\\cite{Izhikevichlargescale}. This column is made up of a six-layered cortical structure consisting of eight types of excitatory and nine types of inhibitory neurons. Thalamic neurons cover two types of excitatory neurons, inhibitory neurons and GABAergic neurons in the reticular thalamic nucleus (RTN). Neurons are simulated by \\emph{Izhikevich} model, which BrainCog applies to exhibit their specific spiking patterns depending on their different neural morphologies. For example, excitatory neurons (pyramidal and spiny stellate cells) always exhibit RS (Regular Spiking) or Bursting modes, while inhibitory neurons (basket and non-basket interneuron) are of FS (Fasting Spiking) or LTS (Low-threshold Spiking) patterns. Each neuron has a number of dendritic branches to accommodate a large number of synapses. The synaptic distribution and the microcircuits are reconstructed in BrainCog based on the previous works~\\cite{Izhikevichlargescale, Binzegger2004A}. Fig.~\\ref{minicolumn}(a) describes the details of the minicolumn. The column contains 1,000 neurons and more than 4,200,000 synapses. To understand the network further, we stimulate the spiny stellate cells in layer 4 to observe the activities of the whole network, Fig.~\\ref{minicolumn}(b) shows the result of neural activities after the cells in layer 4 receive the external stimulation. \n\n\\begin{figure}[!htbp]\n\\centering\n\\includegraphics[width=0.48\\textwidth]{.\/fig\/minicolumn.png}\n\\caption{The thalamocortical column. (a) shows the structure of the column and (b) describes the running activities of the unfold column when the neurons in Layer 4 receive the external stimulus.}\n\\label{minicolumn}\n\\end{figure}\n\n\\textbf{2. Mouse Brain}\n\n\nThe BrainCog mouse brain simulator is a spiking neural network model covering 213 brain areas of the mouse brain, which are classified according to the Allen Mouse Brain Connectivity Atlas~\\cite{richardson2003subthreshold}~\\footnote{http:\/\/connectivity.brain-map.org}. Each neuron was modeled by conductance-based spiking neuron model and simulated with a resolution of $dt$= 1 ms. A total of 6 types of neurons are included in this model, which are excitatory neurons (E), interneuron-basket cells (I-BC), interneuron-Matinotti cell (I-MC), thalamocortical relay neurons (TC), thalamic interneurons (TI) and thalamic reticular neurons (TRN).\n\n\n$$\nJ \\in\\{E, I\\underline{B} C, I\\underline{M} C, T C, T I, T R N\\}\n$$\n\nWe use the aEIF neuron model referring to previous work~\\cite{jiang2015principles,izhikevich2008large,tchumatchenko2014oscillations}, and obtain the parameters of this study, which are summarized in Tab.~\\ref{mouse1}.\n\n\\begin{table}[h]\n\t\\caption{Main parameters of different types of neuron models.}\n\t\\centering\n\t\\resizebox{0.8\\linewidth}{!}{\n\t\t\\begin{tabular}{lllllll}\n\t\t\t\\toprule \n\t\t &$V_{th,j}(mV)$ &$ V_{r,j}(mV)$ & $\\tau_{v,j}$ &$ \\tau_{w,j}$ &$ \\alpha _{j}$ & $\\beta_{ j} $\\\\\n\t\t\t\\midrule\t\n\t\t E & -50 & -110 & 100 & - & 0 & 0 \\\\\n\t\t\t\n\t\t\t\\midrule\n\t\t I-BC & -44 & -110 & 100 & 20 & -2 & 4.5 \\\\\n\t\t \\midrule\n\t\t  I-MC & -45 & -66 & 85 & 20 & -2 & 4.5 \\\\\n\t\t \n\t\t  \\midrule\n\t\t  TC & -50 & -60 & 200 & - & 0 & 0 \\\\\n\t\t   \\midrule\n\t\t    TI & -50 & -60 & 20 & 20 & -2 & 4.5 \\\\\n\t\t      \\midrule\n\t\t       TRN & -45 & -65 & 40 & 20 & -2 & 4.5 \\\\ \n\t\t\t\\bottomrule\n\t\t\\end{tabular}\n\t}\n\t\\label{mouse1}\n\t\n\\end{table}\n\n\nThe connections between brain areas are based on the quantitative anatomical dataset Allen Mouse Brain Connectivity Atlas. Methods for data generation have been previously described in~\\cite{oh2014mesoscale}. The proportions of the different types of neurons were adopted from a previous study~\\cite{izhikevich2008large, markram2004interneurons}.\n\nThe numbers of each type of neuron in the network are shown in Tab.~\\ref{mouse2}.\n\\begin{table}[h]\n\t\\caption{Number of different types of neurons in the BrainCog mouse brain simulator.}\n    \\centering\n    \\resizebox{0.8\\linewidth}{!}{\n\t\t\\begin{tabular}{lllllll}\n\t\t\t\\toprule \n        Neuron Type & E & I\\_BC & I\\_MC & TC & TI & TRN \\\\ \n        \t\\midrule\n        Neuron Number & 56100 & 14960 & 7480 & 1300 & 260 & 520 \\\\ \n        \t\\bottomrule\n    \\end{tabular}\n    }\n    \\label{mouse2}\n\\end{table}\n\nThe spontaneous discharge of the model without external stimulation is shown in Fig.~\\ref{figmouse1}.\n  \n\\begin{figure}\n\\centering\n\\includegraphics[width=0.48\\textwidth]{.\/fig\/rat.jpg}\n\\caption{Running of the BrainCog mouse brain simulator. The shining point is the spiking neuron at the time t and the point color represents the neuron belong to respective brain area.}\n\\label{figmouse1}\n\\end{figure}\nThis is an open platform, and both the parameters of the neuron model and the number of different types of neurons can be set flexibly.\n\n\\textbf{3. Macaque Brain}\n\nThe BrainCog macaque brain simulator is a large-scale spiking neural network model covering 383 brain areas~\\cite{Dharmendra2010}. We used the multi-scale connectome transformation method~\\cite{a2017Zhang} on the EGFP (enhanced green fluorescent protein) results~\\cite{Bakker2012, Chaudhuri2015ALC, Christine2010} to obtain the approximate amount of cells per region and the approximate number of synaptic connections between two connected regions~\\cite{Liu2016}. The final macaque model includes 1.21 billion spiking neurons and 1.3 trillion synapses, which is 1\/5 of a real macaque brain. \nSpecifically, the details of the brain micro-circuit are also considered in the simulation. The types of neurons in the micro-circuit include excitatory neurons (90 \\% of the neurons are of this type in the simulation) and inhibitory neurons (10\\% of the neurons are of this type in the simulation)~\\cite{Davis1979}. The spiking neuron follows Hodgkin\u2013Huxley model, which is supported by BrainCog. The running demo of the model is shown in Fig.~\\ref{mac}(a). To use the macaque model in the platform, the parameters of the neuron number in each region, the connectome power between regions, and the proportion between the excitatory and inhibitory neurons can be set flexibly.\n\n\\begin{figure}[!htbp]\n\\centering\n\\includegraphics[width=0.48\\textwidth]{.\/fig\/mac.jpg}\n\\caption{Running of the macaque brain (a) and the human brain (b) model. The shining point is the spiking neuron at the time t and the point color represents the region which the neuron belongs to.}\n\\label{mac}\n\\end{figure}\n\n\n\\textbf{4. Human Brain}\n\nThe BrainCog human brain simulator is built with the approach similiar to the BrainCog macaque brain. By using the EGFP results of the\nhuman brainnetom atlas~\\cite{Fan2016, Klein2012}, the BrainCog human brain simulator consists of 246 brain areas. It should be noted that since there is no directed human brain connectome available until the release of this paper, the BrainCog human brain simulator keeps bidirectional connections among brain areas. The details of the micro-circuit, including the excitatory neuron and the inhibitory neuron, are also considered. The final model (Fig.~\\ref{mac}(b) includes 0.86 billion spiking neurons and 2.5 trillion synapses, which is 1\/100 of a real human brain. To use this model in the platform, the neuron number per region, the connectome power, and the proportion between the excitatory and inhibitory neurons can be set flexibly.\nMorover, all the simulations were performed on distributed memory clusters~\\cite{Liu2016} at the super computing center affiliated to Institute of Automation, Chinese Academy of Sciences, Beijing, China. The cluster named the fat cluster is composed of 16 blade nodes and 2 \"fat\" computing nodes. In order to imporve the network communication efficiency and the simulation performance, the most connected areas were simulated in the 'fat' computing nodes to minimize the inter-node communications, while the other areas is randomly distributed in the blade nodes. The simulation shows the ability of the framework to deploy on the supercomputer or large-scale computer clusters.\n\n\\section{BORN: A spiking neural network driven Artificial Intelligence Engine based on BrainCog}\n\n\\begin{figure*}[htb]\n\\centering\n\\includegraphics[width=1\\textwidth]{.\/fig\/vision.jpg}\n\\caption{The functional framework and vision of BORN.}\n\\label{bornvision}\n\\end{figure*}\n\nBrainCog is designed to be an open source platform to enable the community to build spiking neural network based brain-inspired AI models and brain simulators. Based on the essential components developed for BrainCog, one can develop their own domain specific or general purpose AI engines. To further demonstrate how BrainCog can support building Brain-inspired AI engine, here we introduce BORN, an ongoing SNN driven Brain-inspired AI engine, ultimately designed for general purpose living AI. As shown in Fig.~\\ref{bornvision}, the high-level architecture of BORN is to integrate spatial and temporal plasticities to realize perception and learning, decision-making, motor control, working memory, long-term memory, attention and consciousness, emotion, knowledge representation and reasoning, social cognition and other brain cognitive functions. Spatial plasticity incorporates multi-scale neuroplasticity principles at micro, meso and macro scales. Temporal plasticity considers learning, developmental and evolutionary plasticity at different time scales. \n\n\nAs an essential component for BORN, we propose a developmental plasticity-inspired adaptive pruning (DPAP) model, that enables the complex deep SNNs and DNNs to gradually evolve into a brain-inspired efficient and compact structure, and eventually improves learning speed and accuracy in the extremely compressed networks~\\cite{Han2022Developmental}. The evolutionary process for the brain includes but not limited to searching for the proper connectome among different building blocks of the brain at multiple scales (e.g. neurons, microcircuits, brain areas). BioNAS for BORN uses brain-inspired neural architecture search to construct SNNs with diverse motifs in the brain and experimentally verify that SNNs with rich motif types perform better than plain feedforward SNNs~\\cite{shenBrainInspired2022}.\n\n\nHow the human brain selects and coordinates various learning methods to solve complex tasks is crucial for understanding human intelligence and inspiring future AI. BORN is dedicated to address critical research issues like this. The learning framework of BORN consists of multi-task continual learning, few-shot learning, multi-modal concept learning, online learning, lifelong learning, teaching-learning, and transfer learning, etc. \n\nTo demonstrate the ability and principles of BORN, we provide a relatively complex application on emotion dependent robotic music composition and playing. This application requires a humanoid robot perform music composition and playing depending on visual emotion recognition. The application requires BORN to provide cognitive functions such as visual emotion recognition, sequence learning and generation, knowledge representation and reasoning, and motor control, etc. This application of BORN starts with perception and learning, and ends with motor output. \n\nIt includes three modules implemented by BrainCog: the visual (emotion) recognition module, the emotion-dependent music composition module, and the robot music playing module. As shown in Fig.~\\ref{rmp}, the visual emotion recognition module enables robots to recognize the emotions expressed in images captured by the humanoid robot eyes. The emotion-dependent music composition module can generate music pieces according to various emotional inputs. When a picture is shown to the robot, the Visual Emotion Recognition network can firstly identify the emotions expressed in the picture, such as joy or sadness. The robot then selects or compose the music piece that best matches the emotions in the picture. And finally, with the help of the robot music playing module, the robot controls its arms and fingers in a series of movements, thus playing the music on the piano. Some details are introduced as follows:\n\n\\begin{figure*}[!htbp]\n\\centering\n\\includegraphics[width=1\\textwidth]{.\/fig\/rmp.jpg}\n\\caption{The procedure of multi-cognitive function coordinated emotion dependent music composition and playing by humanoid robot based on BORN.}\n\\label{rmp}\n\\end{figure*}\n\n\n\n\\emph{1) Visual Emotion Recognition: }\nFor emotion recognition, inspired by the ventral visual pathway, we construct a deep convolutional spiking neural network with LIF neuron model and surrogate gradient provided by BrainCog. The structure of the network is set as 32C3-32C3-MP-32C3-32C3-300-7. 32C3 means the output channel is set with 32, and the kernel size is set as 3. MP denotes the max pooling. The mean firing rate is used to make the final prediction. We use the Adam optimizer, and the mean square error loss. The initial learning rate is set with 0.001, and it will decay to 1\/10 of the previous value every 40 epochs, for a total of 100 epochs. We use the Emotion6 dataset~\\cite{peng2015mixed} to train and test our model. The Emotion6 dataset is composed of 6 emotions such as anger, disgust, fear, joy, sadness, surprise, and each type of emotion consists of 330 samples. On this basis, we extend the original Emotion6 dataset with exciting emotion which we collect online. 80\\% of the images are used as the training set, and the remaining 20\\% are used as the test set. \n\n\\emph{2) Emotion-dependent Music Composition: }\nListening to the the music can make us emotional, while, when people feel happy or sad, they always express their feeling with music. Amygdala plays a key role in human emotion. Inspired by this mechanism, we constructed a simple spiking neural network to simulate this important area to represent different types of emotions and learn the relationships with other brain areas related to music. As shown in Fig.~\\ref{rmp}, the amygdala network is composed of several LIF neurons supported through BrainCog, the connections from this cluster are projected to the musical sequence memory networks. \n\nDuring the learning process, amygdala, PFC, and the musical sequence memory networks cooperate with each other and form complex neural circuits. Here, connections are updated by the STDP learning rule. The dataset used here also contains 331 MIDI files of classical piano works~\\cite{Krueger2018}, and it is important to note that a part of these music works are labeled with different types of emotional categories (such as happy, depressed, passionate and beautiful). \n\nAt the generation process, given the beginning notes and specific emotion type, the model can generate a series of notes and form a melody with particular emotion finally.\n\n\\emph{3) Robot Music-Playing: }\nA humanoid robot iCub is used to validate the abilities of robotic music composition and playing depending on the result of visual emotion recognition. The iCub robot has a total of 53 degrees of freedoms throughout the body. In the piano playing task, we used 6 degrees of freedoms of the head, 3 degrees of freedoms of the torso, and 16 degrees of freedoms for each of the left and right arm (including the left and right hand). Besides, we mainly control the index fingers to press the keys; in the multi-fingered playing mode, we mainly control the thumbs, the index fingers, and the middle fingers to press the keys. During playing, the robot controls the movement of the hand in sequence according to the generated sequence of different musical notes, and presses the keys with corresponding fingers, thereby completing the performance. For each note to be played, the corresponding playing arm needs to complete the entire process of moving, waiting, pressing the key, holding, and releasing the key according to the beat. During the playing process, we also control the movements of the robot's head and the non-playing hand to match the performance.\n\nWe have constructed a multi-brain area coordinated robot motor control SNN model based on the brain motor control circuit. The SNN model is built with LIF neurons and implements SMA, PMC, basal ganglia and cerebellum functions. The  music notes is first processed by SMA, PMC and basal ganglia networks to generate high-level target movement directions, and the output of PMC is encoded by population neurons to target movement directions. The fusion of population codings of movement directions is further processed by the cerebellum model for low level motor control. The cerebellum SNN module consists of Granular Cells (GCs), Purkinje Cells (PCs) and Deep Cerebellar Nuclei (DCN), which implements the three level residual learning in motor control. The DCN network generates the final joint control outputs for the robot arms to perform the playing movement.\n\n\n\n\\section{Conclusion}\nBrainCog aims to provide a community based open source platform for developing spiking neural network based AI models and cognitive brain simulators. It integrates multi-scale biological plausible computational units and plasticity principles. Different from existing platforms, BrainCog incorporates and provides task ready SNN models for AI, and supports brain function and structure simulations at multiple scales. With the basic and functional components provided in the current version of BrainCog, we have shown how a variety of models and applications can already be implemented for both brain-inspired AI and brain simulations. Based on BrainCog, we are also committed to building BORN into a powerful SNN-based AI engine that incorporates multi-scale plasticity principles to realize brain-inspired cognitive functions towards human level. Powered by 9 years development of BrainCog modules, components and applications, and inspired by biological mechanisms and natural evolution, continuous efforts on BORN will enable it to be a general purpose AI engine. We have already started the efforts to extend BrainCog and BORN to support high-level cognition such as theory of mind~\\cite{zhao2022brain}, consciousness~\\cite{zeng2018toward}, and morality~\\cite{zhao2022brain}, and it definitely takes the world to build true and general purpose AI for human and ecology good. Join us on this explorations to create the future for human-AI symbiotic society. \n\\section*{Acknowledgements}\nThis work is supported by the National Key Research and Development Program (Grant No. 2020AAA0104305), the Strategic Priority Research Program of the Chinese Academy of Sciences (Grant No. XDB32070100).\n\n\n\n\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:INTRO}\n\\subsection{LISA Pathfinder, the LISA Technology Package, and ST7-DRS}\n\nThe Space Technology 7 Disturbance Reduction System (ST7-DRS) is a NASA technology demonstration payload hosted on the European Space Agency (ESA) LISA Pathfinder (LPF) spacecraft, which launched from Kourou, French Guiana on December 3, 2015 and operated until July 17th, 2017, when it was decommissioned by ESA.  The primary purpose of LPF was to validate key elements of the measurement concept for the Laser Interferometer Space Antenna (LISA), a planned space-based mission to observe gravitational waves in the millihertz band.  Specifically, LPF demonstrated that the technique of drag-free control could be employed to place a test mass in near-perfect free-fall~\\cite{LPF_PRL_2016,LPF_PRL_2018}. LISA will use three drag-free satellites, configured as an equilateral triangle with $\\sim 2.5$ million km arms, to detect spacetime strains caused by passing gravitational waves~\\cite{LISA_PROPOSAL_2017}.\n\nThe basic components of a drag-free system are the reference test mass, which resides inside the spacecraft but makes no physical contact with it; a metrology system that measures the position and attitude of the test mass relative to the spacecraft as an inertial sensor; a control system that determines what forces and torques to apply to the spacecraft, and possibly the test mass; and an actuation system that can apply forces and torques to the spacecraft and possibly the test mass.  In the case of LPF, European National Space Agencies provided the LISA Technology Package (LTP), which includes two test masses as part of the inertial sensor.  Each test mass has its own independent six-degree-of-freedom electrostatic metrology and control system.  LTP also includes an optical interferometer that measures the position and attitude of the test masses with respect to the spacecraft and each other much more precisely than the electrostatic system, but only along the axis that joins the two test masses as well as the tip and tilt angles orthogonal to that axis. Finally, the LTP includes systems to monitor and control the thermal, magnetic, and charge environment of the instrument.  The ESA-provided spacecraft included its own set of drag-free control laws and its own cold-gas micropropulsion system.  ESA's drag-free system was used for the majority of LPF's operations and achieved a striking level of performance, significantly exceeding the requirements set for LPF (which were deliberately relaxed from the LISA requirements) and meeting or exceeding the requirements for LISA itself~\\cite{LPF_PRL_2016,LPF_PRL_2018}. \n \nST7-DRS includes two main elements: an alternate set of drag-free control laws implemented on a separate computer, and an alternate micropropulsion system based on a novel colloidal microthruster technology~\\cite{Ziemer2006, Ziemer2008}. ST7-DRS provided the first demonstration of colloidal micropropulsion performance in space.   During phases of the LPF mission where ST7-DRS operated,  NASA's colloidal thrusters were used in place of ESA's cold-gas thrusters to move and orient the spacecraft, with the DRS control laws replacing the ESA control laws. For brief periods, NASA's colloid thrusters were also used as the actuators for ESA's drag-free system, replacing the cold gas thrusters, to show performance and the robust nature of the drag-free control laws and colloid microthruster technology. \n \nDuring ST7 operations, the LTP payload played the same role as during the ESA-led parts of the mission - providing information on the positions and attitudes of the test masses and applying forces and torques to the test masses, as requested by the DRS controllers. In this paper, we present an overview of the ST7-DRS operations, the measured performance of the ST7-DRS systems, and the implications for LISA.\n  \n\\subsection{History of ST7-DRS development and operations}\n \nInitiated in 2002 as part of NASA's New Millennium program,  ST7-DRS includes four subsystems: (1) The Integrated Avionics Unit (IAU), a computer based on the RAD750 processor; (2) Colloid Micro Newton Thrusters (CMNT), two clusters of four thrusters each; (3) Dynamic Control Software (DCS), a software subsystem which implements drag-free control algorithms and (4) Flight Software (FSW), a command and data handling software subsystem which processes commands and telemetry and hosts the DCS. The IAU was manufactured by Broadreach Engineering (Phoenix, AZ) and was delivered to NASA's Jet Propulsion Laboratory (JPL) for integrated testing in May, 2006. The CMNTs were manufactured by Busek (Natick, MA), put through acceptance and thermal testing in late 2007, and delivered to JPL with fully loaded propellant tanks in 2008. The DCS software was written at NASA's Goddard Spaceflight Center (GSFC) and the FSW was written at JPL, with initial versions both completed in March, 2006. The DRS completed a Pre-Ship Acceptance Review with ESA in June 2008 and was placed in storage until its delivery to Astrium UK, in Stevenage, England, for Assembly, Integration, Verification and Test (AIVT) in July 2009.\n\nDue to the the unexpectedly long duration between DRS delivery and LPF's launch, ST7 conducted shelf-life extension testing on the thruster propellant, materials, and microvalves in both 2010 and 2013, which qualified the system for launch in 2015 and serendipitously demonstrated a long storage lifetime (8 years on the ground and nearly 10 years total with on-orbit operations) that will be useful for LISA.  During the storage period, the thrusters were left on the spacecraft, fully loaded with propellant, with removable protective covers on each thruster head to prevent debris from entering the electrodes.  During this time, the spacecraft was kept in the integration and test facilities at Airbus Stevenage, UK with dynamic and thermal environmental testing occurring at IABG in Ottobrunn, Germany.  The thrusters were part of all spacecraft-level testing with at least annual inspections removing the protective covers, none of which showed any signs of propellant leakage or damage to the thruster electrodes.  The thrusters required no special handling or environmental control beyond the normal safeguards and environments used during typical spacecraft AIVT activities, and the protective covers were removed just before spacecraft encapsulation into the launch fairing.\n \nAfter launch, a composite of the LPF Science Module (SCM) and Propulsion Module (PRM) executed a series of orbit raising maneuvers culminating in a cruise to Earth-Sun L1. DRS, which was powered-off during launch, was turned on for initial commissioning January 2 - 10,  2016.  Because the PRM was still fastened to SCM, the DRS did not control the spacecraft  attitude in this commissioning, but the effects of the DRS thrusters were observed in the host spacecraft attitude control using on-board gyroscopes. LPF arrived on station at L1 on January 22, 2016 and the PRM was was discarded leaving the SCM. At this time, the ESA LISA Technology Package (LTP) was commissioned and began executing its primary mission on March 1st, 2016.  A second commissioning of DRS was conducted June 27 - July 8, 2016,  which included successful demonstrations of drag-free control. DRS operations were conducted over the next five months including 13 different experiments with occasional breaks for planned LTP station-keeping maneuvers or LTP experiments, as well as response to a number of anomalies on both DRS and LPF hardware.  The DRS anomalies are discussed in Section~\\ref{sec:ANOMALIES}. DRS completed its baseline mission on December 6, 2016. An extended mission to further characterize the thrusters and control system was requested and approved, and operated from Mar 17, 2017 to April 30, 2017.  The DRS was decommissioned as part of the LPF decommissioning process on July 13, 2017.  Key dates for DRS operations are summarized in Table \\ref{tab:dates}.    The complete ST7-DRS data set, along with tools for accessing it, is archived at \\url{https:\/\/heasarc.gsfc.nasa.gov\/docs\/lpf\/}. \n \n \n\\begin{table}\n\\begin{tabular}{|p{0.9in}|p{0.6in}|p{0.9in}|p{0.6in}|}\n\\hline\nEvent & Date & Event & Date \\\\ \\hline\nLPF Launch & 03 Dec '15 & Thruster-4 Anomaly & 27 Oct '16 \\\\ \\hline\nTransfer Phase Commissioning (9 days) & 02 Jan '16 & Start:Hybrid Propulsion & 29 Nov '16 \\\\ \\hline\nArrival at L1 & 22 Jan '16 & End:Primary Mission & 06 Dec '16 \\\\ \\hline\nExperiment Phase Commissioning (10 days) & 27 Jun '16 & Start:Extended Mission & 20 Mar '17 \\\\ \\hline\nCluster-2 DCIU Anomaly & 09 Jul '16 & End:Extended Mission  & 30 Apr '17 \\\\ \\hline\nStart:Primary Mission & 15 Aug '16 & Decommissioning Activities & 13 Jul '17 \\\\ \\hline\n\\end{tabular}\n\\caption{ Key dates for DRS operations}\\label{tab:dates}\n\\end{table}\n\\normalsize\n\n\\subsection{DRS Components and Interfaces}\n\nFigure~\\ref{fig:block_diagram} shows a block diagram of the DRS hardware and its major functional interfaces to the LPF spacecraft and the LTP instrument. The DRS consists of three distinct hardware units: the IAU and two Colloidal Micronewton Thruster Assemblies (CMTAs) with four thrusters each.  The IAU interfaces with the primary LPF computer, known as the On-Board Computer (OBC), and the OBC provides interfaces to the LTP instrument as well as other spacecraft systems such as the star tracker and communications systems. In drag-free operations when the DRS is in control of the spacecraft attitude, the LTP provides measurements of the position and attitude of the two test masses, which are processed by the OBC and sent to the IAU along with spacecraft attitude measurements derived from the LPF star trackers. This information is processed by the Dynamic Control System (DCS) software running on the IAU, which determines the appropriate forces and torques to apply to the spacecraft and the test masses. Test mass force\/torque commands are sent by the IAU to the OBC, which relays them to the GRS front-end electronics within the LTP.  Spacecraft force\/torque commands are decomposed into individual CMNT thrust commands, which are then sent to the CMNTs. \\footnote{After the anomaly experienced by CMTA2, the commands sent by the IAU to the CMTAs were actually low-level current and voltage commands that are functionally equivalent to thrust commands. See Sec.~\\ref{sec:ANOMALIES} on anomalies and recovery for more detail.}\n\nThe DRS is a single-string system, but with a redundant RS-422 communication interface between the IAU and the OBC, and redundant IAU DC\/DC power converters and thruster power switches.\nThe  redundant power busses are cross-strapped to each thruster cluster, which are single string. The A-Side power bus was the primary bus used during the mission.\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=\\columnwidth]{block_diagram.pdf}\n\\caption{Block diagram of the Disturbance Reduction System (DRS) and its interfaces with the LISA Pathfinder spacecraft and the LISA Technology Package instrument. Renderings of LPF and LTP courtesy of ESA\/Medialab.}\n\\label{fig:block_diagram}\n\\label{default}\n\\end{center}\n\\end{figure}\n\n\\section{The Colloidal Micro-Newton Thruster Assemblies}\n\\label{sec:CMNT}\n\\subsection{Components}\nColloid thrusters were selected to be developed by ST7-DRS because of their potential for extremely high precision thrust; extremely low noise; and a larger specific impulse compared to cold-gas systems ($\\sim240\\,$s vs $\\sim70\\,$s). Colloid thrusters are a type of electrospray propulsion, which operate by applying a high electric potential difference to charged liquid at the end of a hollow needle in such a way that a stream of tiny, charged droplets is emitted generating thrust. An advantage of this system is that the liquid colloidal propellant can be handled with a compact and lightweight propellent management system and requires no pressure vessels or high temperatures. The requirement for high-voltage power supplies is a disadvantage. Colloidal thrusters can be designed to operate in various thrust ranges according to the number of needles that are used in each thruster head. The ST7-DRS configuration, developed specifically for ST7s performance requirements by Busek, provides a thrust range  from 5 to 30 $\\mu$N per thruster (larger thrusts are achievable in diagnostic mode).\n\nDRS includes two Colloidal Micro-Newton Thruster Assemblies (CMTAs), each of which includes: 4 thruster heads, 4 propellant feed systems, 4 Power Processing Units (PPUs), 1 cathode, and 1 Digital Control Interface Unit (DCIU)~\\cite{Ziemer2008}. Figure \\ref{fig:cmnts} shows both a  block diagram for one of the 4-thruster assemblies. Each thruster head includes a manifold that feeds nine emitters in parallel, a heater to control propellant temperature and physical properties, and electrodes that extract and accelerate the propellant as charged droplets. The thruster heads are fed by independent bellows and microvalves (the feed system).  The propellant is the room temperature ionic liquid 1-ethyl-3methylimidazolium bis(triflouromethylsulfonyl)imide (EMI-Im)\nand is stored in four electrically isolated, stainless steel bellows, which use compressed constant-force springs to supply the four microvalves with propellant at approximately 1 atm of pressure. The propellant flow rate is controlled by a piezo-actuated microvalve. The thruster heads and feed system voltages are independently controlled through the PPUs, which are controlled, in turn by the DCIU. The DCIU has an on-board PROM (programmable read-only memory) that stores the thruster operating software and control algorithms. The DCIU has power, command, and telemetry interfaces to the IAU. The CMTA mass is 14.8 kg with $\\sim0.5\\,$kg of propellant distributed into each of the 4 thruster bellows. The nominal power consumptions of each CMTA is $\\sim17\\,$W. \n\nEach CMTA also includes one propellantless field emission cathode neutralizer, included to neutralize the emitted spray of charged droplets after they are accelerated, and so prevent spacecraft charging by the thrusters. The cathode neutralizers are fabricated from a carbon nanotube (CNT) base with an opposing gate electrode controlled by the DCIU. Each CNT is capable of producing 10 $\\mu$A to 1 mA  using extraction voltages of 250 to 800 V. The neutralizer was tested during the extended mission and produced the desired current. As expected, the measured spacecraft charging with respect to the test masses \\cite{LPF_CHARGE_PRD_2018} indicated that the induced spacecraft charge rate was larger in magnitude than and opposite in sign from the effect of the CMNTs, meaning the neutralizer was not necessary for maintaining spacecraft charge control. \n \n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=\\columnwidth]{cmnt-config.pdf}\n\\caption{CMNT propulsion system components and configuration. The Carbon nanotube cathode is not shown.\n \\label{fig:cmnts} }\n\\label{default}\n\\end{center}\n\\end{figure}\n\n\\subsection{Thrust model}\n\nThe thrust (T)  from each CMNT is approximated by~\\cite{ziemer2009performance, demmons2008st7}:\n\n\\begin{equation}\\label{eq:T=CIV}\t\nT =C_1\\,  I_B^{1.5} \\, V_B^{0.5} \\, ,\t\t\t\n\\end{equation}\nwhere $I_B$ is the total beam current from the $9$ emitters, $V_B$ is beam voltage, and $C_1$ is the thrust coefficient. $C_1$ depends mostly on physical properties of the propellant \n(viscosity, electrical conductivity, etc.) and also on the characteristics of the plume (beam divergence, charge-to-mass ratio distribution, etc.). In operation, the DCIU adjusts the the beam voltage (2-10 kV) and propellant flow rate for each thruster head to achieve the desired thrusts. The mass flow rate is not measured directly; instead the beam current $I_B$ is measured and controlled by actuating a piezo microvalve. $I_B$ is controlled to better than 1 nA  over the operating range of 2.25 to 5.4$\\mu$A, corresponding to a thrust resolution of  $\\le 0.01\\,\\mu$N. Independent, fine control of both the beam voltage and beam current allow for precise control of thrust to better than 0.1 $\\mu$N,  with $<$0.1 $\\mu\\textrm{N}\/\\sqrt{\\textrm{Hz}}$ thrust noise.\n\nThe value $C_1$ is temperature-dependent; $C_1$ decreases with increasing temperature. Thrust-stand measurements on Engineering Model (EM) units validated this model to a precision of $\\sim\\,$2\\%, consistent with the calibration of the thrust stand\\cite{demmons2008st7}. The best-fit value for $C_1$ under nominal beam voltage and current and with a  propellant temperature $T_{P} = 25^{\\circ}$C was $31.9\\,\\textrm{N}\\textrm{A}^{-3\/2}\\textrm{V}^{-1\/2}$.  A substantial portion of the DRS operations was utilized to perform calibration experiments to validate this model in flight as well as explore potential higher-fidelity models. These experiments and their results are discussed in section \\ref{sec:thustCal}.\n\n\\section{The Dynamic Control System (DCS)}\n\\label{sec:DCS}\nThe DRS control system maintains the attitude of the spacecraft, as well as the position of the test masses within their housings, both by moving the spacecraft via the CMNTs and by moving the test masses via electrostatic actuation. There were initially six DRS Mission Modes managed by the IAU: Standby, Attitude Control, Zero-G, Drag-Free Low Force, 18-DOF (Degree of Freedom) Transitional, and 18-DOF Mode. One additional mode,  the Zero-G Low Force was added for the extended DRS mission, to provide improved performance in the accelerometer mode, where some of the most sensitive thruster experiments were performed. \n\nStandby mode is used when IAU is powered on but no actuation commands are being generated by the control system, generally when the LTP controller is active.  Attitude Control Mode is used for the transition from LTP control to DRS Control.  In this mode, the DRS nulls spacecraft attitude errors and their rates using the thrusters, while electrostatically suspending the both test masses. In the Zero-G Mode, disturbance forces on the spacecraft, such as those from solar radiation pressure, are nulled out in a low-bandwidth loop that minimizes common-mode actuation on the test masses by applying forces on the spacecraft using the CMNTs. The Zero-G low force variant utilized the same control scheme but with the GRS actuation set to its high resolution \/ low force authority setting. In the Drag Free Low Force (DFLF) Mode the spacecraft's position is controlled via the CMNTs to follow the reference test mass (RTM, configurable to be either of the two LTP test masses) in all translational axes. Hence, it is the lowest mode in which drag-free flight of a single test mass is achieved. The 18-DOF Transitional is, as its name suggests, a transitional mode to get from DFLF to 18-DOF control of the spacecraft and the two test masses. In the 18-DOF mode, the DRS uses the thrusters to force the spacecraft to follow the RTM, i.e., to maintain the nominal gap of the RTM with respect to its housing along all three axes. The DRS uses the torque from the CMNTs to control the spacecraft attitude,  in the measurement band (1-30mHz), so that it follows: (a) the non-reference test mass (NTM) in the transverse directions (normal to the LTP axis); and (b) the relative attitude of the RTM about the sensitive axis. The orientations of both test masses are controlled via electrostatic suspension below the measurement bandwidth (MBW).  Further details on the spacecraft and test mass control design for each mode may be found in \\cite{Maghami2004,Hsu2004}. \n\nThe DRS baseline architecture made use of only the capacitive sensing measurements of the test mass positions and orientations, which was the configuration for both ST7 and LTP when the DRS design was consolidated.  After successfully commissioning DRS in this configuration, the system was modified to use the higher-precision interferometric data from LTP for the degrees of freedom where it was available. \n\n\\section{Characterization of Thruster Performance}\n\\label{sec:CMNTP}\nIn-flight characterization of the CMNT technology was a major goal of the ST7 mission. This section summarizes the experiments conducted during DRS operations and the top-level results. The CMNT properties investigated during the flight campaign include thrust range, response time, calibration, and thrust noise.  In general, two sources of data were available for these investigations. The first was internal thruster telemetry such as beam currents, beam voltages, valve voltages, temperatures, etc. These quantities can be used to estimate the CMNT thrust using physics-based models which were validated during the CMNT development with thrust stand measurements. The second source of data, which is unique to this flight test, was the rest of the Pathfinder spacecraft, in particular the test masses and interferometer of the LTP.  Measurements with LPF and LTP data allowed the CMNT thrust model to be independently validated and its exquisite sensitivity allowed thrust noise measurements in the LISA band at a level never achieved with ground-based thrust stands.  \n\n\\subsection{Functional Tests: Range and Response Time}\n\\label{sec:CMNTfunc}\nAs described in Table \\ref{tab:dates}, the CMNTs were commissioned in two phases in 2016.  The first phase, in January 2016, was conducted prior to separation of the propulsion module so that the CMNTs would be available to serve as a backup propulsion system for LPF, should the primary cold-gas system experience an anomaly after separation.  Figure \\ref{fig:cmnt_resp} shows a full-range response test  in which all 8 CMNTs are initially at their minimum thrust level of 5$\\,\\mu$N and then commanded to their maximum level of 30$\\,\\mu$N for 300$\\,$s before returning to 5$\\,\\mu$N.  The thrust command (same for all thrusters) is shown in black and the predicted thrust based on CMNT telemetry and the ground-validated model is shown in colored lines. This initial test demonstrated that all eight CMNTs could operate in their requisite 5-30$\\,\\mu$N  thrust range. With the propulsion module still attached to the spacecraft and the LTP instrument not yet commissioned, there was limited availability of precision data from the platform. However, telemetry from the propulsion module ACS thrusters and from the LPF star tracker showed force\/torque motions on the platform that were roughly consistent with thrust levels commanded to the CMNTs.\\\\\n\nBoth the thruster control law and the dynamics of the thruster head are expected to lead to a delayed response time, for which the design requirement was 10$\\,$s. As shown in Figure \\ref{fig:cmnt_resp}, 7 of the 8 CMNTs meet this requirement for the response time from 5$\\,\\mu$N to 30$\\,\\mu$N. The exception is CMNT\\#1, which has a response time of $\\sim 170\\,$s.  As discussed in Appendix A.1,  this delayed response is consistent with some obstruction in the CMNT\\#1 feed system.  The response time for 30$\\,\\mu$N to 5$\\,\\mu$N was slightly longer for all CMNTs and significantly longer for thrusters 1,6, and 7. This is likely due to the response of the piezo microvalve, which is controlled by a PID loop to maintain the desired current level. The valve actuator is encased in a potting compound that provides electrical isolation but also adds some mechanical compliance to the valve. In addition, there is some variation in the pre-load and piezo response from valve to valve that results in different flow response as a function of valve voltage. An examination of the valve voltage during this experiment reveals that while the voltage for valves 1,6, and 7 dropped to zero after the transition from commands of 30$\\,\\mu$N to 5$\\,\\mu$N, the electrometer still measured a slowly-declining current after the valve voltage reached zero.  This suggests that the piezo actuators in these valves were unable to close the valves to the desired position until the valve mechanically relaxed, after which time the piezo could begin to actuate again. \\\\\nFinally, CMNT\\#1 also shows an impulsive behavior known as `blipping', which is caused when the number of emitters actively flowing propellant in the CMNT head changes.  In standard operations, all 9 emitters should be expelling propellant for the full range of thrusts. However, if one emitter has a significantly increased hydraulic resistance due to an obstruction, it can periodically stop and start, leading to an abrupt change in the thrust. While efforts were made during the mission to improve the performance of CMNT\\#1 during the mission, neither the response time not the rate of `blipping' significantly improved. This is further discussed in Appendix A.1.   Note that there is also some evidence of `blipping' in in CMNT\\#7 in Figure \\ref{fig:cmnt_resp}, although at a much lower rate than for CMNT\\#1 and also only at the minimum thrust level.  The blipping behavior for CMNT\\#7 rapidly improved as commissioning proceeded and was not observed during the remainder of the mission, suggesting that the (presumably much smaller) obstruction that was responsible was cleared.  \n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=\\columnwidth]{CMNT_resp.pdf}\n\\caption{Full-range response test for all eight CMNTs conducted as part of initial thruster commissioning in January 2016 prior to separation of the LPF Propulsion Module. All 8 thrusters demonstrated the full thrust range, although CMNT\\#1 had an abnormally slow response time, perhaps due to some obstruction. In addition CMNT\\#1 exhibited a `blipping' mode consistent with one of the nine emitter tips cycling between a spraying and non-spraying state.  \\label{fig:cmnt_resp} }\n\\label{default}\n\\end{center}\n\\end{figure}\n\nAfter the propulsion module was successfully separated and the spacecraft was under control of the cold-gas micropropulsion, the CMNTs were placed in a safe mode for approximately 6 months of LTP operations. In July 2016, the second phase of CMNT commissioning was conducted to prepare for DRS operations. Figure \\ref{fig:cmnt_range} shows a thruster functional test in which each CMNT is successively ramped, in 5$\\,\\mu$N increments, from 5$\\,\\mu$N to 30$\\,\\mu$N and back.  The blue line shows the thrust command and the red data shows the estimated thrust  based on the CMNT telemetry and the ground-validated model. Again, 7 of the 8 CMNTs perform as designed, but CMNT\\#1 exhibits both the episodic `blipping' and an overall slow response time. We note that here the 30$\\,\\mu$N limit on the maximum thrust was set by the flight software. In the extended mission, the diagnostic mode of the CMNTs was used to manually command the current and voltage to demonstrate extended thrust range. CMNT\\#2 and CMNT\\#5 were stepped up to 40$\\,\\mu$N early in the extended mission and then CMNT\\#5 was stepped up to 50$\\,\\mu$N and CMNT\\#2  was stepped up to 60$\\,\\mu$N near the end of the extended mission. The CMNTs passed all of these extended-range tests without incident.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=\\columnwidth]{CMNT_range.pdf}\n\\caption{Thruster actuation test conducted during DRS commissioning in July 2016. Each of the eight CMNTs was successively cycled from 5$\\,\\mu$N to 30$\\,\\mu$N and back in 5$\\,\\mu$N steps. The achieved thrust estimated from the CMNT telemetry closely matches the thrust commands with the exception of CMNT\\#1, which exhibits both a slow response time and 'blipping' consistent with one of its nine emitters firing only intermittently. \\label{fig:cmnt_range} }\n\\label{default}\n\\end{center}\n\\end{figure}\n\n\n\\subsection{Thruster Calibration Measurements}\n\\label{sec:thustCal}\n\\vskip 0.2in\nAs mentioned above, the additional instrumentation on the Pathfinder spacecraft, in particular the LTP, allows use of the spacecraft as a low-noise thrust stand that can be used to calibrate the thrust model.  This section describes the design, analysis, and results of thruster calibration experiments carried out using this method.\n\n\\subsubsection{Experiment design}\nAll the calibration experiments had the following general form.  The thrust command to one of the eight CMNTs is modulated by some sinusoidal or square wave, which we refer to as the \\emph{injection waveform}, and each thruster's calibration constant is derived from the resulting modulations on i) the motion of the spacecraft  , ii) the motion of the two TMs, and iii) the electrostatic forces on the TMs.  The full set of injection waveforms that we used is summarized in Table \\ref{tab:T1}.  In each experiment, we cycle through each of the thrusters one at a time.\n\nWe chose to perform  the majority of the injections in accelerometer control mode, out of concern that drag-free control would lead to a complicated mixing of the injection signals across all thrusters simultaneously, as well as suppressing the modulation on the main thruster of interest.  Operating in accelerometer mode is very close to the standard operating mode in the ground-based thrust stand measurements.\n\nThe amplitude and frequency of the injections were selected to balance the needs of the system (low disturbance, slew rate limits, and available experiment time) against the needs of the analysis, as characterized by the expected signal to noise ratio (SNR). Our baseline approach was to perform the experiment in the DRS's high-force actuation mode, since it provides higher control authority and therefore should permit larger-amplitude injections without losing stability. Later in the mission, a set of injections was designed for the DRS's low-force actuation mode, which had a better-characterized calibration of the applied test mass forces and torques than the high-force mode.  \n\nEarly in DRS operations, each injection set was demonstrated on a subset of thrusters to assess the quality of the response and resulting analysis. Set 1, with the most gentle system response but longest duration, was used for initial checkout and sets 2 through 4 were used together to more rapidly characterize thruster performance. Set 5 utilized a square wave to measure the response at multiple Fourier frequencies and was utilized in some limited tests. In this paper, we present results from the `standard' suite of sets 2 through 4, which were used for the majority of the investigations in both baseline and extended DRS operations. \n\n\\begin{table}\n\\caption{Waveforms of thruster calibration experiments.\\label{tab:T1}}\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|}\n \\hline\nSet \\#  & Waveform &   Frequency &   Amplitude  &   Duration & $\\rho_{HF}$ & $\\rho_{LF}$  \\\\ \\hline\n1 & sine & 23 mHz & 1 $\\mu$ N & 5220 s & 200 & 500 \\\\ \\hline\n2 & sine  & 23 mHz & 3 $\\mu$ N &  696 s  & 200 & 500 \\\\ \\hline\n3 & sine & 29 mHz  & 5 $\\mu$ N &  552 s  & 200 & 800 \\\\ \\hline\n4 & sine  & 40 mHz  & 3 $\\mu$ N &  600 s & 66 & 400 \\\\ \\hline\n5 & square &  23 mHz & 3 $\\mu$ N & 696 s & ~200 & ~500 \\\\ \\hline\n \\end{tabular}\n\\end{center}\n\\label{default}\n\\end{table}\n\n\\subsubsection{Calibration results}\n\nThe basic analysis approach is to estimate the acceleration of the LISA Pathfinder spacecraft using the LTP data, and compare that with the thrust derived from the thruster's measured $V_B$ and $I_B$ and the thrust model, Eq.~(\\ref{eq:T=CIV}). As shown in Appendix B, using the average acceleration of the two test masses causes most of the rotating-frame effects to cancel out, making the results more robust against systematic errors.\n\nWe use a Markov-Chain Monte-Carlo method to estimate the maximum-likelihood gain and delay of each thruster, jointly fit across the three injection frequencies. Figure \\ref{fig:t1psd} shows an example fit for CMNT\\#5 for an injection with the TMs in the high-force mode. The injection signal is suppressed by a factor of roughly 20, although it is still visible in the residual. Figure \\ref{fig:EM8psd} shows a similar fit for CMNT\\#5 for an injection with the TMs in the low-force mode.  Here the injection signal is suppressed by a factor of roughly 100, and no residual is visible.\n\n Table \\ref{tab:C1} presents the best-fit value for the C1 coefficient in Eq.~(\\ref{eq:T=CIV}) for each thruster. These values are averaged over 4 measurements, except for CMNT\\#4 which was only averaged 3 times due to the propellant bridge (see Appendix A.3 for a discussion of this anomaly).  All of the results are in the range $29\\sim32\\,\\textrm{N}\\textrm{A}^{-3\/2}\\textrm{V}^{-1\/2}$, roughly consistent with the value of $31.9\\,\\textrm{N}\\textrm{A}^{-3\/2}\\textrm{V}^{-1\/2}$ derived from the ground tests \\cite{Ziemer2010}.\n\nThere is also a small but statistically-significant discrepancy between the experiments conducted in high-force and low-force modes, which suggests that there is some aspect of the calibration that has not been properly accounted for.  Note that there is no low -force mode measurement for CMNT\\#4, as this experiment was implemented after it was disabled.  We do not report the best-fit delays, since in addition to physical delays, they include relative delays between the DRS data packets containing the CMNT telemetry and the LTP data packets containing the LTP telemetry, and the latter are not physically relevant.  \n \n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=\\columnwidth]{t5a23residual.pdf}\n\\caption{Thrust spectrum as measured from LTP data (blue), thrust based on thrust model and best-fit C1 value(red), and residual (difference) between those two (green), for a 23 mHz injection in \\emph{high-force} actuation mode in CMNT\\#5. \\label{fig:t1psd} }\n\\label{default}\n\\end{center} \n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=\\columnwidth]{t5a23residualEM8.pdf}\n\\caption{Thrust spectrum as measured from LTP data (blue), thrust based on thrust model and best-fit C1 value(red), and residual (difference) between those two (green), for a 23 mHz injection in \\emph{low-force} actuation mode in CMNT\\#5.  \\label{fig:EM8psd} }\n\\label{default}\n\\end{center}\n\\end{figure}\n\n\\begin{table}\n\\caption{Summary of thruster calibration results. C1 values are in units of [$ N A^{-3\/2} V^{-1\/2}$].\\label{tab:C1}}\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline\nT\\# & T1 C1 & error & T1 (LF) C1 &  error  \\\\\n\\hline\n1 & 31.52 & 0.15 & 31.91 & 0.10 \\\\ \\hline\n2 & 30.18 & 0.10 & 30.97 & 0.10 \\\\ \\hline\n3 & 28.78 & 0.09 & 32.12 & 0.10 \\\\ \\hline\n4 & 29.92 & 0.09 & - & - \\\\ \\hline\n5 & 29.96 & 0.10 & 31.49  &  0.10 \\\\ \\hline\n6 & 30.02 & 0.10 & 30.90 &  0.10 \\\\ \\hline\n7 & 29.71 & 0.09 & 30.37 &  0.10 \\\\ \\hline\n8 & 29.86 & 0.10 & 30.53 & 0.10 \\\\ \\hline \n\\hline\n\\end{tabular}\n\\end{center}\n\\label{default}\n\\end{table}\n\n\\subsubsection{Temperature Dependence}\n\\label{sec:CMNT_T}\n\nThe $C1$ coefficient in the CMNT thrust model, Eq.~(\\ref{eq:T=CIV}), is expected to depend on the propellant temperature, which is set to $25\\,$C during most of the mission.  To validate this model, a campaign was undertaken to alter the temperature set point using onboard heaters, and measure the CMNT calibration using injection sets 2 through 4. Figure \\ref{fig:C1temp} shows the measured calibrations at temperatures of $15\\,$C, $20\\,$C, $25\\,$C, and $30\\,$C along with a linear fit for temperature dependance. Similar data obtained on the ground with thrust-stand measurements is included for comparison. Note that due to the CMNT\\#4 propellant bridge, there are no values for $30\\,$C for that thruster. Additionally, several other measurements were made in the initial calibration experiment at the nominal temperature of $25\\,$C, which are included for all thrusters. \n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=\\columnwidth]{C1vT.pdf}\n\\caption{ Comparison of measured and expected dependence of thruster coefficient on temperature. Legend entries show the best-fit linear coefficient for each CMNT and the ground test in units of $N A^{-3\/2} V^{-1\/2} C^{-1}$. \\label{fig:C1temp} }\n\\label{default}\n\\end{center}\n\\end{figure}\n\n\\subsection{Thruster noise performance}\n\\label{sec:CMNTnoise}\nThe DRS had a Level 1 performance requirement to demonstrate a spacecraft propulsion system with noise less than 0.1$\\, \\mu\\textrm{N}\/\\sqrt{\\textrm{Hz}}$ over a frequency range of 1 mHz to 30 mHz.   We use two different approaches to estimate the CMNT noise performance.  The first method uses the CMNT flight data and the thrust  model in Eq.~\\ref{eq:T=CIV}, using the calibration results for the $C1$ coefficients.  To estimate the intrinsic thruster noise apart from the required spacecraft control, we subtract the thrust commands.  This provides the thrust error. This represents a lower limit on the thrust noise, since additional effects not captured by the measured $I_B$ and $V_B$ values could produce additional noise. Some such effects, such as CMNT shot noise, are known to be well below the measured noise floor, but there is also the possibility of unmodeled noise. \n\nOur second approach for estimating the CMNT noise is to use the Pathfinder spacecraft as a thrust stand, as was done for the thruster calibration measurements.  In the measurement band, $1-30$mHz, thrust noise is expected to dominate the total budget of force noises on the spacecraft  .  Measuring the acceleration noise of the spacecraft should therefore be an effective way to estimate the thruster noise, and is formally an upper limit.  Unfortunately, this approach is complicated by the fact that after the anomaly experienced by CMNT\\#4 (see Appendix A), some portion of the cold-gas micropropulsion system was required to be active whenever the CMNT system was active. The thruster noise measured during these periods includes contributions from the cold-gas system as well as the CMNTs.\n\n\n\\subsubsection{CMNT Noise from internal telemetry}\n\\label{sec:IVnoise}\n\nThe light blue trace in Figure \\ref{fig:CMNT5_alias} shows the measured amplitude spectral density of the thrust noise for CMNT\\#5, using the CMNT flight data estimation.  This data, sampled at 1 Hz, comes from an 8-hour period on 2017-04-24, while in the 18DOF controller configuration. Below 100$\\,$mHz, it is a flat with a level of approximately 70$\\,\\textrm{nN}\/\\sqrt{\\textrm{Hz}}$.  Note that this individual thrust noise is somewhat better than the requirement. Additionally, around 250$\\,$mHz, well above the measurement band, the thrust error for CMNT\\#5 exhibits a sharp spectral line feature. The light red trace in Figure \\ref{fig:CMNT5_alias} shows the thrust error spectrum from a 200s period of 10Hz-sampled data taken just after the data for the blue trace.  In the red trace, the line is shifted to 750$\\,$mHz and the flat level at lower frequencies is reduced.  This already strongly suggests that the 250$\\,$mHz in the light-blue trace is actually a 750$\\,$mHz effect that is getting aliased down to  250$\\,$mHz in the 1Hz-sampled data.   The CMNT telemetry, which is delivered to the IAU at 10Hz, is typically decimated to 1Hz without the use of an anti-aliasing filter.  The rationale for this decision was that the CMNT thrust model depends non-linearly on the beam voltage and current and any averaging or other filtering operation applied to the current and voltage would not give the correct result for the thrust model. In addition, neither the current nor voltage telemetry was expected to have significant power above 0.5Hz. To confirm this interpretation, we fit a spectral model to the 10Hz data (dashed red line in Figure \\ref{fig:CMNT5_alias} and compute the aliased version of that spectrum, shown by the dashed blue line in Figure \\ref{fig:CMNT5_alias}).  Clearly, this reproduces the 250$\\,$mHz peak seen in the 1Hz data.  Based on this analysis, we estimate that the intrinsic noise of CMNT\\#5 is closer to 40$\\,\\textrm{nN}\/\\sqrt{\\textrm{Hz}}$ in the absence of aliasing. It is important to note that aliasing only affects the \\emph{telemetered} values of beam current and voltage. The on-board processing is done at the full $10\\,$Hz rate.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=\\columnwidth]{CMNT5_alias.pdf}\n\\caption{Measured thrust error (thrust command - modeled thrust) in CMNT\\#5 for an 8-hour period on 2017-04-24. The light blue trace is the full duration, sampled at 1Hz. The light red trace is for a adjacent 200s segment sampled at 10Hz. The dashed red line is a fit to the 10Hz spectrum and the dashed blue is a model of how that spectrum gets aliased by downsampling the data without using an anti-aliasing filter.}\n \\label{fig:CMNT5_alias} \n\\end{center}\n\\end{figure}\n\nThe source of the peak in the 10Hz data, which was not observed in ground testing, is suspected to be an oscillation in the thruster control loop which arose when the thrust control algorithm was moved from the DCIU to the IAU, following the DCIU anomaly described in Appendix A. This switch resulted in an additional delays--first for the current and voltage commands to travel from the IAU to the DCIU, and then for the current and voltage telemetry to travel from the DCIU back to the IAU. The additional round-trip delay is expected to be 3 clock cycles, or $\\sim$300$\\,$ms. To test this hypothesis a software simulation of the thrust controller and thruster response was performed using the flight thrust command data for CMNT\\#5 from the period in Figure \\ref{fig:CMNT5_alias}.  The green trace in Figure \\ref{fig:CMNTsim} shows the simulated thrust error, sampled at 10Hz, for the nominal case of zero delay between the $I_B$ and $V_B$ commands and the corresponding response. This would be the case for the original flight configuration in which the thrust control algorithm was implemented on the DCIU. The light red trace shows the simulated thrust error, sampled at 10Hz, for the case where a 300$\\,$ms delay is introduced between the command and response of the beam current and voltage.  This delay causes noise enhancement at  $\\sim$700$\\,$mHz by the control system, where the reduced control loop phase margin means that when the system attempts to suppress motion, the delayed commanded thrust actually mildly increases it instead, by the time the thrust change is enacted. The blue and yellow traces show the with- and without-delay signals, respectively, downsampled to 1Hz without anti-aliasing filters, as was done for nominal DRS operations. In both cases, this elevates the noise to roughly 50$\\,\\textrm{nN}\/\\sqrt{\\textrm{Hz}}$ and, for the delay case, produces a sharp peak near 300$\\,$mHz. \n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=\\columnwidth]{CMNT5_alias_sim.pdf}\n\\caption{Simulated thrust error (thrust command - modeled thrust) for CMNT\\#5 thrust commands using a software model of the thrust control algorithm. The green trace shows the expected thrust error in the baseline case of minimal delay between the commands and response of beam voltage and current. The light red trace shows the same signal when a 300ms delay is introduced in the beam voltage and current response. The blue and yellow traces show the with- and without-delay signals, respectively, downsampled to 1Hz without anti-aliasing filters}\n \\label{fig:CMNTsim} \n\\end{center}\n\\end{figure}\n\nAn expanded  8-thruster version of this simulation, including modeling of the commanding loop delays and flight software, a physically motivated model of the bubble-noise on thruster 1, and the decimation scheme for creating the 1 Hz data product, provided an estimate of the expected platform noise. The data produced by this model matched the available mission flight data, both the 10 Hz data and the heavily aliased 1 Hz data. The 10 Hz data provided over a sufficient duration gave an estimate of the noise in the required frequency band, without the aliasing effects. This system noise along a particular direction was computed using knowledge of the CMNT locations and orientations. Using this simulation of the noise floor from the 10 Hz data, of $\\sim$40$\\,\\textrm{nN}\/\\sqrt{\\textrm{Hz}}$ for thrusters $2-8$ and 74$\\,\\textrm{nN}\/\\sqrt{\\textrm{Hz}}$ for thruster 1, the estimated noise floor along spacecraft X,Y, and Z are $\\sim$70$\\,\\textrm{nN}\/\\sqrt{\\textrm{Hz}}$, $\\sim$87$\\,\\textrm{nN}\/\\sqrt{\\textrm{Hz}}$, and $\\sim$56$\\,\\textrm{nN}\/\\sqrt{\\textrm{Hz}}$.\n   \n\\subsubsection{CMNT Noise Estimated from Spacecraft Response}\n\\label{sec:CMNTnoiseSC}\nThe procedure for estimating the thrust noise from the spacecraft response is very similar to that used for calibrating the thrusters, described in section \\ref{sec:thustCal}, with the exceptions that no modulations are applied to the thrusters and that here we also analyze the Y- and Z- degrees of freedom.  This  analysis can be applied to any segment of data where injections are not present. Here we present results from four segments that are listed in Table \\ref{tab:segments}.  These segments were chosen to span various configurations of control systems and thrusters so as to better distinguish the contribution of the propulsion system to the overall spacecraft noise. Segment I represents the default configuration for LTP, with the ESA-provided DFACS in control of the spacecraft using the cold-gas thruster system. Segment II represents the design configuration for DRS, with the DCS in controlling the spacecraft using all 8 CMNTs (and the ESA-provided cold gas thruster system on standby). Segment III is from a brief `joint operations' campaign in which the DFACS controlled the spacecraft with the CMNTs and the cold gas system was on standby. Finally, Segment IV represents the DRS configuration after the CMNT\\#4 anomaly, with the DCS controlling the spacecraft using 7 CMNTs, and with the cold-gas system partially enabled as an out-of-loop static `crutch'. \n\n\\begin{table}\n\\caption{Experiments used to assess thruster noise performance from the spacecraft response. For each experiment a controller, either the ESA-provided DFACS or the NASA-provided DCS, controlled the spacecraft using a micropropulsion system, either the ESA-provided cold-gas (CGAS) or the NASA-provided CMNT. For segment IV, the DCS controlled the spacecraft with 7 CMNTs in-loop and the CGAS used as an out-of-loop `crutch'.\\label{tab:segments}}\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline\nSegment & Start Date & Duration & Controller & Propulsion \\\\\n\\hline\nI & 2016-9-28 & 238 ks& DFACS & CGAS \\\\ \\hline\nII & 2016-10-04 & 111 ks& DCS & CMNT \\\\ \\hline\nIII & 2016-10-06 & 124 ks& DFACS & CMNT \\\\ \\hline\nIV & 2017-04-21 & 236 ks& DCS & CMNT  \\\\ \n& & & & w\/ CGAS \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n \n Figures \\ref{fig:Fext_x}, \\ref{fig:Fext_y}, and \\ref{fig:Fext_z} show the estimated spacecraft force noise in the X-, Y-, and Z-directions for each of the four segments. The solid lines are amplitude spectral densities estimated using Welch's method of overlapped-averaged periodograms with 10,4,5, and 10 averages for segments I,II,II, and IV, respectively. For segment III, an impulse suspected to be from a micrometeoroid hit at 2016-10-07 9:51:12 UTC was excised from the data.  The solid points represent logrithmically-binned estimates with 1-sigma error bars. The solid dashed lines show a `requirement' based on an uncorrelated thrust noise of 0.1$\\,\\mu\\textrm{N}\/\\sqrt{\\textrm{Hz}}$ in each of the CMNTs projected into the spacecraft body frame using the thruster orientations. This corresponds to  160$\\,\\textrm{nN}\/\\sqrt{\\textrm{Hz}}$,  190$\\,\\textrm{nN}\/\\sqrt{\\textrm{Hz}}$, and 140$\\,\\textrm{nN}\/\\sqrt{\\textrm{Hz}}$ for X-, Y-, and Z-axes respectively.\n \n \\begin{figure}\n\\begin{center}\n\\includegraphics[width=\\columnwidth]{SCForce_X.pdf}\n\\caption{X-component of force on the spacecraft estimated using measured test mass dynamics and spacecraft mass properties. The four traces correspond to the segments in \\ref{tab:segments} and probe different thruster configurations. See text for discussion. }\n\\label{fig:Fext_x}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=\\columnwidth]{SCForce_Y.pdf}\n\\caption{Y-component of force on the spacecraft estimated using measured test mass dynamics and spacecraft mass properties. The four traces correspond to the segments in \\ref{tab:segments} and probe different thruster configurations. See text for discussion. }\n\\label{fig:Fext_y}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=\\columnwidth]{SCForce_Z.pdf}\n\\caption{Z-component of force on the spacecraft estimated using measured test mass dynamics and spacecraft mass properties. The four traces correspond to the segments in \\ref{tab:segments} and probe different thruster configurations. See text for discussion. }\n\\label{fig:Fext_z}\n\\end{center}\n\\end{figure}\n\n In general, the noise for the two CMNT-only configurations (II and III) are somewhat lower than the two including cold gas (I and IV). In addition to the overall higher noise level, the two cold-gas segments exhibit a set of narrow-line features at $\\sim1.5\\,$mHz and harmonics thereof.  Both of these effects are most pronounced in the Z-axis, possibly explained by a common-mode noise source in the cold gas system\\footnote{ Since all 6 Cold Gas thrusters thrust in the +Z (Sunward) direction with the same vector component, a common-mode noise will add coherently whereas correlated noise in X and Y would largely cancel when all 6 thrusters are active. Note that for the case of the `crutch' mode using only 4 of the 6 thrusters, the cancellation in X and Y no longer occurs.} Somewhat surprisingly, the CMNT noise under DFACS control (III) appears to be slightly lower than that for DCS (II). Upon inspection of the telemetry it was found that CMNT\\#1 was railed at the minimum thrust of 5$\\,\\mu$N due to a unoptimized thrust bias vector for this ad-hoc experiment. This inadvertently reduced the rate of `blipping' in CMNT\\#1, leading to a reduction in the overall noise. \n \n \\subsubsection{CMNT Noise Summary}\n \nIn both CMNT-only cases, the measured noise floor along the x-direction seems to be in the $100\\sim 300\\,\\textrm{nN}\/\\sqrt{\\textrm{Hz}}$ range, which is significantly higher than the noise predicted by the thrust telemetry in the absence of aliasing. Possible explanations for this include additional noise in the thrust system beyond what is inferred from the current and voltage noise, additional noise on the spacecraft platform, or noise on the test mass which is used as a reference. \n\n \\begin{table}\n\\caption{Comparison of measured spacecraft Force noise  and combined estimate from measured and estimated effects in the in $1\\sim30\\,$mHz band. Details of noise estimates can be found in Appendix C. \\label{tab:SCnoiseComp} }\n\\begin{center}\n\\begin{tabular}{|c|c|}\n\\hline\nEffect & Estimate nN$\/\\sqrt{\\textrm{Hz}}$ \\\\\n\\hline\nMeasured Noise & 120  \\\\\n\\hline\nTotal Estimate & 70  \\\\\n\\hline\nUnmodeled noise & 97  \\\\\n\\hline\n\\multicolumn{2}{|c|}{\\emph{CMNT noises}} \\\\\n\\hline\nIV noise (CMNT\\#1) & 41\\\\\nIV noise (CMNT\\#2-7) & 56 \\\\\nshot noise &  0.16 \\\\\nflutter  noise & 0.03 \\\\\n\\hline\n\\multicolumn{2}{|c|}{\\emph{S\/C noises}} \\\\\n\\hline\nSRP & 1.7 \\\\\nRadiometer Noise & 0.7 \\\\\nExt. B-fields & 0.01 \\\\\nMicrometeoroids & 0.5 \\\\\n\\hline\n\\multicolumn{2}{|c|}{\\emph{TM noises}} \\\\\n\\hline\nforce noise & $<$ 1 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\vspace{0cm}\n\\end{table}%\n\nTable \\ref{tab:SCnoiseComp} summarizes measurements and estimates of some of these effects for the x-axis of the spacecraft in the $1\\sim30\\,$mHz band. Details on how some of these effects were estimated can be found in Appendix C. Measured Noise is an approximate white noise level equivalent to the red trace (segment II) in Figure \\ref{fig:Fext_x}. Total Estimate is a uncorrelated sum of the remaining entries in the table, which represents the total amount of noise accounted for in our model. This is dominated by an estimate of the modeled thrust noise  without the presence of aliasing that is estimated from the simulations of current and voltage and resulting noise floor presented in section \\ref{sec:IVnoise} and Figure \\ref{fig:CMNTsim}. The table lists the contribution from CMNT\\#1, which has an elevated noise floor due to the blipping, as well as the sum of the rest of the thrusters. Unmodeled Noise is the size of the noise contribution, presumably uncorrelated with the modeled noise, that would be needed to be added to the model to match our measurements. This represents more than half of the measured noise, but at this time we are unable to account for the source of this effect. This suggests that designers of future low-disturbance platforms should take care when considering applications requiring force noise below $\\sim100\\,\\textrm{nN}\/\\sqrt{\\textrm{Hz}}$.\n\\\\\n\n\\section{DRS Performance}\n\\subsection{Operation of the Dynamic Control System}\nDRS operations were initialized with a handover sequence wherein control of the spacecraft and test masses was passed from the European DFACS control system to the DCS.  After initial capture, the DCS executed sequences to transition through the various control modes described in section \\ref{sec:DCS} in order to bring the instrument to the desired state for conducting experiments. During DRS operations, this procedure occurred roughly once per week after station-keeping maneuvers were performed under DFACS control. Figure~\\ref{fig:modeTransitions} shows the measured positions and attitudes as well as the commanded forces and torques of the test mass and spacecraft for a typical transition sequence from handover to the 18DOF science mode. Note that spacecraft forces do not include the bias levels of the thrusters, which were adjusted to provide a net DC force of 24$\\,\\mu$N in the $+Z$ direction to compensate for solar radiation pressure. After an initial transient caused by the handover sequence, the Attitude Only controller works to stabilize the spacecraft angular error. Both test masses are commanded to follow the spacecraft by applying appropriate forces and torques.  For the Zero-g mode, forces are applied to the spacecraft to minimize the z-axis forces on the test masses, thereby providing active compensation of the solar radiation pressure. In the drag-free mode, the spacecraft is commanded to follow the RTM along the linear DoFs, leading to an increase in the applied forces on the spacecraft and a decrease in both the position error and applied force on the RTM. The 18DOF mode, torques are applied to the spacecraft to further reduce the forces on the NTM in the transverse directions. In this particular sequence, the NTM experienced an impulsive disturbance approximately 2.5 hours after the transition into 18DOF mode that caused an excursion of the NTM angles and x position. The DCS compensated for this disturbance by applying appropriate torques and forces to the NTM. The DCS successfully executed dozens of mode transition sequences over the course of the baseline and extended mission, providing a robust platform with which to conduct experiments characterizing the CMNTs and other aspects of the spacecraft.\n\n\\begin{figure*}\\label{transitions}\n\\centering\n\\begin{tabular}{ccc}\n\n\\subf{\\includegraphics[width=0.32\\textwidth]{modes_xRTM.pdf}}\n     {RTM positions}\n&\n\\subf{\\includegraphics[width=0.32\\textwidth]{modes_xNTM.pdf}}\n     {NTM positions}\n &\n\\\\\n\\subf{\\includegraphics[width=0.32\\textwidth]{modes_aRTM.pdf}}\n     {RTM angles}\n&\n\\subf{\\includegraphics[width=0.32\\textwidth]{modes_aNTM.pdf}}\n     {NTM angles}\n &\n \\subf{\\includegraphics[width=0.32\\textwidth]{modes_aSC.pdf}}\n     {SC angles}\n\\\\\n\\subf{\\includegraphics[width=0.32\\textwidth]{modes_FRTM.pdf}}\n     {RTM forces}\n&\n\\subf{\\includegraphics[width=0.32\\textwidth]{modes_FNTM.pdf}}\n     {NTM forces}\n &\n \\subf{\\includegraphics[width=0.32\\textwidth]{modes_FSC.pdf}}\n     {SC forces}\n\\\\\n\\subf{\\includegraphics[width=0.32\\textwidth]{modes_NRTM.pdf}}\n     {RTM torques}\n&\n\\subf{\\includegraphics[width=0.32\\textwidth]{modes_NNTM.pdf}}\n     {NTM torques}\n &\n \\subf{\\includegraphics[width=0.32\\textwidth]{modes_NSC.pdf}}\n     {SC torques}\n\\\\\n\\end{tabular}\n\\caption{DCS behavior during a typical mode transition sequence from handover to the 18 degree-of-freedom science mode (18DOF). Plots show measured positions and angles of both the reference (RTM) and non-reference (NTM) test masses; angles of the spacecraft; and forces and torques applied to the RTM, NTM, and spacecraft . Note that for spacecraft forces, the thruster bias levels are set to provide a net force in the $+Z$ direction of 24$\\,\\mu$N. Time origin is 2016-10-02 14:00UTC.}\n\\label{fig:modeTransitions} \n\\vspace{2cm}\n\\end{figure*}\n\n\n\\subsection{Position Accuracy}\nThe spacecraft position error, or the precision with which the spacecraft position is maintained relative to the RTM, is an important requirement for the DCS. The ST7-DRS Level I requirement was for a position error amplitude spectral density of $S_{SCx}^{1\/2}\\leq10\\,\\textrm{nm}\/\\sqrt{\\textrm{Hz}}$ in the band $1\\,\\textrm{mHz}\\leq f \\leq 30\\,\\textrm{mHz}$. Figure~\\ref{fig:position_error_f} shows the measured  $S_{SCx}^{1\/2}$ for two different experiments: a 20.7 hr run in the drag-free low-force (DFLF) mode beginning on 2016-08-22 07:36 UTC and a 31.1 hr run in the 18 degree-of-freedom (18DOF) mode beginning on 2016-10-22 00:00 UTC.  Both control modes comfortably meet the requirement over the measurement band. At high frequencies, both traces show a spectrum that follows a roughly $f^{-2}$ power-law and has an amplitude that is consistent with a white force noise on the order of $\\sim0.1\\,\\mu\\textrm{N}\/\\sqrt{\\textrm{Hz}}$ and a spacecraft mass of 422$\\,$kg.  As discussed in section \\ref{sec:CMNTnoise}, thruster noise on the order of $\\sim0.1\\,\\mu\\textrm{N}\/\\sqrt{\\textrm{Hz}}$ is expected to dominate the spacecraft force noise budget. At $\\sim100\\,$mHz,  the drag-free controller begins compensating for this disturbance by applying force commands on the spacecraft , leading to a flattening of the position error spectrum. The DFLF controller has a slightly higher control bandwidth for the x-axis drag-free loop than the 18DOF controller, resulting in a slightly lower level of position noise within the control bandwidth.  At lower frequencies, additional gain in the drag-free loop further suppresses disturbances from the thrusters.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=\\columnwidth]{Spec_x1_zoom.pdf}\n\\caption{Amplitude spectral density of the measured RTM-Sspacecraft position along the x-direction for a 20.7 hr run in the  drag-free low-force (DFLF, red) mode beginning on 2016-08-22 and a 31.1 hr run in the 18 degree-of-freedom (18DOF, blue) mode beginning on 2016-10-22. The solid trace shows a linearly-binned spectral density computed using Welch's method with a frequency resolution of 25$\\,\\mu$Hz while the solid points show a logarithmically-binned estimate with 1-sigma error bars. The black dashed line is the Level I position error requirement for ST7-DRS.}\n \\label{fig:position_error_f} \n\\label{default}\n\\end{center}\n\\end{figure}\n\nFigure \\ref{fig:position_error_cdf} plots the measured cumulative probability distribution function for the spacecraft position error along x for the same two runs as are plotted in Figure \\ref{fig:position_error_f}. The distributions are well-approximated by a Gaussian distribution and have confidence intervals of (-1.0,+1.0)$\\,$nm for DFLF and (-2.1,+1.9)$\\,$nm for 18DOF.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=\\columnwidth]{cdf_x1.pdf}\n\\caption{Cumulative probability distribution for measured RTM-spacecraft position along the x-direction for the two runs in Figure \\ref{fig:position_error_f}. The 95\\% confidence intervals for the DFLF and 18DOF position errors are (-1.0,+1.0)$\\,$nm and (-2.1,+1.9)$\\,$nm respectively.}\n \\label{fig:position_error_cdf} \n\\label{default}\n\\end{center}\n\\end{figure}\n\n\\subsection{Differential Acceleration Measurements}\n\\label{sec:deltag}\n\\vskip 0.2in\nWhile the primary purpose of DRS operations was to validate the performance of both the drag-free control laws and the CMNT micropropulsion system, a small portion of the operations time, in both the prime and extended missions, was utilized to make differential acceleration measurements of the two test masses. This `$\\delta g$' measurement is the primary measurement reported by the LTP collaboration\\cite{LPF_PRL_2016, LPF_PRL_2018}.  To a leading approximation, one would not expect a change in the differential acceleration noise when either the control laws or the micropropulsion system was changed. As is extensively discussed in the LTP collaboration publications, the performance of the drag-free system is primarily determined by the physics of the sensor assembly and interferometric readout. For example, the minimum acceleration noise in the $1\\,\\textrm{mHz}\\,\\sim10\\,\\textrm{mHz}$ band is largely determined by the gas pressure around the test mass. The bulk of the LTP operations were composed of experiments to characterize and reduce these various couplings, leading to the improvement in performance from the initial\\cite{LPF_PRL_2016} to the final\\cite{LPF_PRL_2018} results. \n\nTo first order, a change in the control system does not affect $\\delta g$ measurements because the analysis used to construct the $\\delta g$ results includes both the error signal (motion of the test mass) as well as the control signal (forces on the test mass).  Second-order effects, such as larger sensitivities to calibration errors or actuation-cross talk, can be present. Indeed, early in the DRS operations, it was noticed that the measured $\\delta g$ for Fourier frequencies $\\mathrel{\\hbox{\\rlap{\\hbox{\\lower4pt\\hbox{$\\sim$}}}\\hbox{$>$}}} 30\\,\\textrm{mHz}$ was non-stationary and on average higher during DRS operations than during LTP operations. This was traced to the fact that the DRS suspension controller was initially tuned to be `softer' than the corresponding LTP controller, which resulted in larger RMS motion (but less actuation) of the suspended test mass. This larger motion caused an increase in the average sensing noise of the LTP interferometer, which had a known degradation in noise performance if the test masses were allowed to move appreciably from their nominal position.  Once this was understood, the DRS controllers were modified to be `stiffer' at high frequencies, thus reducing the motion of the suspended test mass and recovering the interferometric sensing noise performance observed in the LTP.\n\nOne also expects the $\\delta g$ results to depend only weakly on which micropropulsion system is used, since the differential nature of the measurement is specifically designed to reject disturbances on the spacecraft platform.  The most direct coupling of microproulsion noise is through test mass `stiffness', which represents the coupling between spacecraft position and force on the test mass, which is physically caused by all of the following: AC electric fields used to control the test masses, stray electrostatic and magnetic fields coupling to test mass charge, and the gradient of the gravitational field due to the spacecraft itself.  This coupling can be reduced by increasing the contribution from the actuation fields so that both test masses have the same stiffness, thus rejecting any coupling of spacecraft motion through the use of `matched' stiffness. In practice, the matched stiffness configuration was not extensively used in either LTP or DRS operations because the intrinsic stiffness of both test mass systems was significantly lower than requirements (and the spacecraft motion due to micropropulsion noise were within requirements).  Micropropulsion noise could also enter the $\\delta g$ measurement through more subtle effects such as rotating-frame effects. For example, the measured $\\delta g$ signal includes a centrifugal term that arises from the product of the low-frequency rotation of the spacecraft as well as the in-band attitude jitter of the spacecraft. The standard $\\delta g$ analysis uses a combination of star-tracker attitude data and test mass torque data to estimate this contribution and subtract it. It is possible that increased angular jitter caused by a noisier microproulsion system could result in a larger contribution that is more difficult to fully subtract.\n\nFigure \\ref{fig:delta-g-DRS} shows our estimate of the amplitude spectral density of $\\delta g$ for two DRS configurations as well as the `ultimate' LTP performance\\cite{LPF_PRL_2018} and the current estimated requirements for the LISA mission. These data were obtained using the same data analysis pipelines and tools \\cite{LTPDA} as used by the LTP collaboration in their major results papers\\cite{LPF_PRL_2016, LPF_PRL_2018}.  For all three segments, the solid trace represents the amplitude spectral density obtained using Welch's method of overlapped averaged periodograms of length 40$\\,$ks, with a Blackmann-Harris window applied. As was the case for the standard LTP analysis, the lowest reported frequency is the 4th bin (0.1$\\,$mHz). The solid points are logarithmically-binned estimates of the amplitude spectral density, including one-sigma error bars. For each time-series segment, the data is reduced in a series of steps that includes estimating the observed acceleration, correcting for the applied force on the non-reference test mass, correcting for stiffness as well as actuation and sensing cross talk, correcting for rotating-frame effects, and removing any impulsive `glitches'.  Each of these steps requires both a model of the underlying contribution to $\\delta g$ as well as a set of parameters corresponding to the state of the instrument. Again, to the greatest extent possible, our analysis used an identical set of models and parameters as the corresponding LTP analysis. Similarly, glitches were identified and removed using the same procedure as for the final LTP results. \n\nThe first DRS segment in Figure \\ref{fig:delta-g-DRS} (red trace, `initial DRS', data from segment II in Table \\ref{tab:segments}) represents the nominal configuration with the DCS controlling the test masses via the LTP and the spacecraft via all eight CMNTs.  Three glitches were identified in this segment, occurring at 2016-10-05 05:50 UTC, 2016-10-04 17:51 UTC, and 2016-10-04 18:07 UTC.  Each glitch was fit and removed using a double-exponential, as described in the final LTP paper\\cite{LPF_PRL_2018}. The second DRS segment (blue trace, `optimized DRS', data from segment IV in Table \\ref{tab:segments}) represents the optimized DRS performance, obtained after the system had been tuned but also after CMNT \\#4 had failed.  Again, after the failure of CMNT \\#4, the DCS was modified to control the spacecraft with seven CMNTs in closed-loop as well as an open-loop `crutch' provided by four of LPF's cold gas thrusters.  Two glitches, at 2017-04-23 03:25 UTC and 2017-04-23 13:22 UTC, were identified and removed. The LTP segment (orange trace, `ultimate LTP') is the February 2017 segment plotted in Figure 1 of the final LTP $\\delta g$ paper\\cite{LPF_PRL_2018}.  \n\nThe LISA requirements are the single test-mass acceleration noise requirement from the 2017 LISA Mission Proposal\\cite{LISA_PROPOSAL_2017}, but here multiplied by  $\\sqrt{2}$  to compensate for the fact that LPF makes a measurement of the differential noise between two test masses whereas the requirement is written for a single test mass.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=\\columnwidth]{deltag_DRS.pdf}\n\\caption{Measured residual differential acceleration between Pathfinder's two test masses ($\\delta g$) for DRS operations and comparison with LTP configuration. The red trace (initial DRS) is for an all-DRS configuration from early in the mission when all eight CMNTs were operating and the cold gas micropropulsion system was on standby (segment II in Table \\ref{tab:segments}). The blue trace (optimized DRS) is for a configuration with the DCS controlling the spacecraft using 7 of 8 CMNTs while four of the cold gas microthrusters provided an open-loop static force (segment IV in Table \\ref{tab:segments}). The orange trace is the ultimate published LTP performance\\cite{LPF_PRL_2018}.  For all three segments, the solid trace is a linearly-binned amplitude spectral density with a resolution of $250\\,\\mu$Hz while the solid markers represent logarithmically-binned estimates of the amplitude spectral density with one sigma error bars. The LISA requirements are the single test-mass acceleration requirement as expressed in the 2017 LISA Mission Proposal\\cite{LISA_PROPOSAL_2017} with a factor of $\\sqrt{2}$ applied to account for the fact that Pathfinder measures differential acceleration between two test masses. \\label{fig:delta-g-DRS} }\n\\label{default}\n\\end{center}\n\\end{figure}\n\nWhen comparing the three configurations in Fig. \\ref{fig:delta-g-DRS} it is useful to consider three different frequency regimes. At high frequencies ($f\\mathrel{\\hbox{\\rlap{\\hbox{\\lower4pt\\hbox{$\\sim$}}}\\hbox{$>$}}} 30\\,\\textrm{mHz}$), the three traces are quite similar, exhibiting a $f^2$ power-law behavior that is caused by white displacement noise in the LTP interferometric readout of the differential test mass position. The most notable differences are the presence of a line feature near 70$\\,$mHz in both the LTP and optimized DRS configurations as well as the fact that the initial DRS configuration is slightly lower than the other two. The elevated broad-band displacement noise is likely due to more significant misalignments of the test masses during the extended mission.  During the baseline mission, an extensive campaign was carried out to identify misalignments and actively correct for them by modifying the static offsets of the test mass positions and attitudes. This procedure was shown to significantly reduce the cross-coupling term in the $\\delta g$ analysis (see discussion in initial LTP paper\\cite{LPF_PRL_2016}). This adjustment was not repeated in the extended mission and it is likely that the offsets may have changed due to either deliberate changes in the spacecraft temperature or other effects such as creep and outgassing. The origin of the 70$\\,$mHz line feature is unknown, although the fact that it is not present in the initial DRS configuration, when the cold gas micropropulsion system was placed in standby mode, suggests that it may be related to the cold gas micropropulsion system in some way. \n\nIn the middle band between ($1\\,\\textrm{mHz}\\mathrel{\\hbox{\\rlap{\\hbox{\\lower4pt\\hbox{$\\sim$}}}\\hbox{$>$}}} f \\mathrel{\\hbox{\\rlap{\\hbox{\\lower4pt\\hbox{$\\sim$}}}\\hbox{$>$}}} 10\\,\\textrm{mHz}$), the three traces show clear differences, with the LTP trace presenting a nearly flat, feature-free noise floor of approximately 1.8$\\,\\textrm{fm}\\,\\textrm{s}^{-2} \/ \\sqrt{\\textrm{Hz}}$. The initial DRS trace is roughly two times higher at 10$\\,$mHz and rises slowly towards lower frequencies. The optimized DRS trace has a lower broad band noise floor than the initial DRS trace, around 3$\\,\\textrm{fm}\\,\\textrm{s}^{-2} \/ \\sqrt{\\textrm{Hz}}$.  This is still roughly a factor of two higher than the ultimate LTP case, although with the much shorter segment (236 ks vs. 1.15 Ms), the statistics are not as good. As mentioned above, the limiting noise in this band is expected to be gas pressure in the test mass enclosures. After an initial steady decrease in pressure in response to the opening of the vent duct to space, the GRS pressure was primarily controlled by setting the temperature of the GRS housing. The difference in temperatures between the optimized DRS (12.8$\\,^{\\circ}$C) and the optimized LTP (11.5$\\,^{\\circ}$C) is not large enough to account for the observed difference in noise,  although the higher noise floor in the initial DRS run, which occurred at a higher temperature (23.5$^{\\circ}$C) and several months earlier in the mission, may well be due to increased pressure in the housings.\n\nAt the low end of the measured frequency band ($0.1\\,\\textrm{mHz}\\mathrel{\\hbox{\\rlap{\\hbox{\\lower4pt\\hbox{$\\sim$}}}\\hbox{$>$}}} f \\mathrel{\\hbox{\\rlap{\\hbox{\\lower4pt\\hbox{$\\sim$}}}\\hbox{$>$}}} 1\\,\\textrm{mHz}$), all three traces in Fig. \\ref{fig:delta-g-DRS} show a rise with a slope of roughly $f^{-1}$, with their relative amplitudes similar to those in the mid-band. At the lower end of the band, the ultimate DRS noise is slightly higher than that of the ultimate LTP, although the statistics on the DRS measurement are poor. While the long-duration LTP results were able to demonstrate performance down to 20$\\,\\mu$Hz, a region with important astrophysical implications, the DRS data is not sufficiently long to make any measurements below 0.1$\\,$mHz. However, based on the available data it would appear that a drag-free system employing CMNT micropropulsion could meet all of the LISA science requirements while providing significant savings in mass to the flight system.\n\n\n\\section{Summary, Conclusions and Future Work}\n\\vskip 0.2in\n The ST7-DRS successfully demonstrated NASA developed drag-free control laws and colloid thrusters in space. Colloid thrusters were selected for the mission because of their potential to be used for future mission requiring low thrust noise ($\\le 0.1 \\mu$ N\/ $\\sqrt{Hz}$), high precision ($\\le 0.1  \\mu N$ steps) with a comparatively small amount volume and mass used for propellent. The ST7-DRS controlled the attitude the 424 kg ESA LISA Pathfinder Spacecraft for 103.4 days (~41 days of drag-free control) using less than 1 kg of propellant. The mission was also an example  of applying operational characteristics of colloid thrusters to a drag-free control application.  \n As a technology demonstration, ST7-DRS was a success, meeting its performance requirements. However, the anomalies experienced during the mission would not be acceptable had they occurred in the primary propulsion system of a long-duration mission. Also, the propellent volume must be increased for a several year mission. In the decade since the CMNT thrusters were completed in 2008, significant progress has been made to adapt the technology to a longer mission. The on-orbit performance observed on ST7 is also informing this development and also the planning for the verification and validation testing to demonstrate the improvements have been realized without introducing new problems.\nThe on-orbit data form ST7 is also being used to infer the performance of colloid thrusters in other applications. While ST7-DRS specifically implemented a drag-free control system, the data collected allows the design of other types of control system. The performance demonstrated on LPF can enable many applications needing ultra-precise pointing, formation flying or dynamic stability, including separated element interferometers, coronagraphs, very large aperture telescopes and fundamental physics experiments. In addition, the control modes and operational approach demonstrated on ST7 are an example as to how to hand over from a higher-noise control system to a very low noise system, and also demonstrated the response to impulse events (and anomalies) that can be used in planning for any of these applications. We look forward to the results of LISA and many other amazing science missions in the future. \n\n\n\n\\acknowledgments\n\n\nWe would like to acknowledge Dr. Landis Markley for insightful contributions to the design and development of the DRS and DCS.  We acknowledge the significant contributions of Jeff D\\'Agostino and Kathie Blackman of the Hammers Co., who implemented and tested the DCS algorithms in the FSW, and who supplied the corresponding simulators to JPL and ESA.  \nThe data was produced by the NASA Disturbance Reduction System payload, developed under the NASA New Millennium Program and hosted on the LISA Pathfinder mission, which was part of the space-science programme of the European Space Agency. The work of JPL authors was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under contract to the National Aeronautics and Space Administration.  Dr. Slutsky was supported by NASA through the CRESST II cooperative agreement CA 80GSFC17M0002.\nThe French contribution to LISA Pathfinder has been supported by the CNES (Accord Specific de projet CNES 1316634\/CNRS 103747), the CNRS, the Observatoire de Paris and the University Paris-Diderot. E.~Plagnol and H.~Inchausp\\'{e} would also like to acknowledge the financial support of the UnivEarthS Labex program at Sorbonne Paris Cit\\'{e} (ANR-10-LABX-0023 and ANR-11-IDEX-0005-02).\nThe Albert-Einstein-Institut acknowledges the support of the German Space Agency, DLR, in the development and operations of LISA Pathfinder. The work is supported by the Federal Ministry for Economic Affairs and Energy based on a resolution of the German Bundestag (FKZ 50OQ0501 and FKZ 50OQ1601). \nThe Italian contribution to LISA Pathfinder has been supported  by Agenzia Spaziale Italiana and Istituto Nazionale di Fisica Nucleare.\nThe Spanish contribution to LISA Pathfinder has been supported by contracts AYA2010-15709 (MICINN), ESP2013-47637-P, and ESP2015-67234-P (MINECO). M.~Nofrarias acknowledges support from Fundacion General CSIC (Programa ComFuturo). F.~Rivas acknowledges an FPI contract (MINECO).\nThe Swiss contribution to LISA Pathfinder was made possible by the support of the Swiss Space Office (SSO) via the PRODEX Programme of ESA. L.~Ferraioli is supported by the Swiss National Science Foundation.\nThe UK LISA Pathfinder groups wish to acknowledge support from the United Kingdom Space Agency (UKSA), the University of Glasgow, the University of Birmingham, Imperial College, and the Scottish Universities Physics Alliance (SUPA). @ 2018.  All rights reserved.\n\n\\section*{Appendix A: Anomalies experienced during ST7-DRS operations}\n\\label{sec:ANOMALIES}\n\nThe DRS experienced three significant anomalies during the mission that, while they affected operations, did not prevent the mission from meeting its objectives and performance requirements. First, CMNT\\#1 demonstrated a reduced maximum current and response time compared to acceptance and thermal vacuum testing prior to launch. One of its nine emitters also blipped on and off between 0.2-0.6 Hz, depending on current level, throughout both the primary and extended missions, significantly increasing thrust noise. Second, the DCIU PROM on cluster 2 suffered a partial memory failure in the location of the thruster control algorithm just after the instrument commissioning was complete, preventing direct thrust commands from being acted on without reseting the DCIU.  Third, near the end of the primary mission, CMNT\\#4 experienced a propellant bridge between the emitter and the extractor electrodes, effectively preventing further use of the thruster. This electrical short occurred after the mission performance requirements had been met (including 60 days of operation) and during experiments that were designed to prove performance at more stressful operating conditions. For all three anomalies, solutions or work-arounds were developed to enable operations and continue experiments up to the end of the extended mission.  The next three subsections provide more detail on all three anomalies.\n\n\\subsubsection*{CMNT\\#1 Performance Reduction} \n\\label{sec:anomaly1}\nCMNT\\#1 in-flight performance was not consistent with its performance in pre-delivery ground acceptance and thermal vacuum (TVAC) ground tests, where it met all response time and thrust noise requirements.  For example, in the TVAC tests, the response time required for CMNT\\#1 to increase thrust from 5 to 30 ${\\mu}N$ was 9.8 s, which was below the 100 s requirement.  In flight, CMNT\\#1 took 2 days longer than the other thrusters to fill the propellant feed system and initially turn on, indicating a significantly reduced maximum propellant flow rate.  After startup and bubble removal in a test during commissioning, it demonstrated a slower response time of about 170 s and a maximum thrust capability that was less than the other thrusters but still within requirements.  This increased response time characteristic continued throughout the mission but would gradually improve and nearly return to meeting the 100 s response time requirement after the microvalve was left open for a number of days.  Whenever CMNT\\#1 was deactivated and its microvalve closed, the response time would then decrease again.  Experiments during the extended mission showed that the response time increase was most closely related to how long CMNT\\#1's microvalve was closed.  This indicated a problem in the microvalve actuator or a non-constant and not complete blockage in the feed system upstream of the microvalve or in it's flow limiting orifice just upstream of the microvalve seat.  The net result was a requirement to ``prime'' CMNT\\#1 by running it in a diagnostic mode at constant current for 30 minutes prior to each start up to improve its response time.  The extra delay in CMNT\\#1's response also impacted thrust noise during periods of higher fluctuations in thrust commands (i.e. 18DOF).  Fortunately, the failure of CMNT\\#1 to meet this response time requirement did not significantly impact DRS operations during mode transitions or science mode, nor prevent the DRS as a whole from meeting its Level-1 performance requirements.\n\nIn flight, CMNT\\#1 also experienced beam current spikes of 160-240 nA every 1.75-4 seconds with the frequency and magnitude proportional to the current level throughout the entire mission. The size and high-speed characteristics of the current spikes were consistent with a single emitter turning on and off with $<10$\\% duty cycle - an effect colloquially referred to as `blipping'.  During tests designed to allow counting of the number of active emitters on each thruster and from direct measurements of thrust using the GRS, it was clear that CMNT\\#1 had one of its nine emitters mostly off. Analyses of these experiments and others suggest that the observed behavior was not due to a bubble but a blockage or constriction in one emitter with a constant hydraulic impedance that impeded the propellant flow rate significantly compared to the other eight emitters.  With a reduced flow rate in just one of the nine emitters, it was not possible to maintain a steady current. \n\nBoth of these performance reductions on CMNT\\#1 were observed to start at the same time, during startup, suggesting that they could be related and are both likely caused by an increase in hydraulic resistance; however, the location of the constrictions is not the same.  It should be noted that during startup, a large amount of current $>24 {\\mu}A$ was emitted for over 20 minutes, potentially indicating a bubble passing through the microvalve orifice, expanding by a factor of 10 in volume and pushing a significant amount of propellant out of one or more emitter.  This long-duration, high-current event occurred outside of daily real-time observations, fortunately terminating on its own, but potentially damaging one of the emitters in the process.  Follow-on ground tests have shown constrictive damage to emitters that experience abnormally high currents with a build-up of decomposed propellant products, which could explain CMNT\\#1's blipping emitter.  A bubble stuck in the microvalve orifice could also explain the reduced flow rate.  The cause of both performance issues with CMNT\\#1 continues to be under investigation.  Because the seven other thrusters and microvalves performed nearly the same in flight as on the ground, the CMNT\\#1 anomaly could be a yield issue that can be addressed by maturing the technology, improving the microvalve testing, screening, and filling processes and\/or flying redundant valves, as is common practice for primary propulsion systems on large science missions.\n\n\\subsection*{DCIU PROM Corruption}\n\\label{sec:anomalyDCIU}\nThe DCIU on cluster 2 experienced an anomaly on July 8, 2016, immediately after a successful instrument commissioning, resulting in a processor reset and CMTA 2 becoming disabled whenever it received a thrust command. Testing revealed that, while the Thrust Command Mode no longer functioned properly, the Diagnostic Mode and other control routines still functioned properly allowing diagnosis and an eventual work-around.  By design, the DCIU PROM and control software could not be updated on orbit, but fortunately the IAU flight software (FSW) was designed with a back-up thrust control algorithm using the DCIU's Diagnostic Mode in case there was a desire to modify it on orbit.  In this case, the IAU calculated the beam, extractor, and microvalve voltage commands directly from the DCS thrust requests and sent those commands instead of thrust commands to the DCIU.  The IAU FSW and operational sequences required updating and verification to make the embedded thrust control algorithm work properly that enabled this DCIU ``Pass-through Mode'', and operations continued on August 8, 2016. \n\nThe suspected cause of this anomaly is a single radiation event that permanently damaged part of cluster 2's PROM.  While the Pass-Through Mode was a work-around that solved the immediate issue, it also introduced a command and telemetry delay between thrust command processing and beam current and voltage commanding, execution, sensing, and feedback that did not exist when the DCIU had it's own internal control loop functioning properly.  This added delay was 1-2 realtime intervals (RTIs) or 0.1-0.2 seconds, depending on the exact timing between telemetry and command packets passing back and forth between the IAU, DCIUs and on-board computer. As discussed in section \\ref{sec:IVnoise}, this extra delay led to a limit-cycle oscillation in the thruster control loop. Fortunately, this limit cycle oscillation was confined to frequencies above the target performance bandwidth and did not impact mission performance. To prevent this anomaly in future missions, a more robust, radiation-hardened, and re-writable EEPROM for the DCIU is recommended.  \n \n\\subsection*{CMNT\\#4 Propellant Bridge}\n\\label{sec:anomaly4}\nNear the end of the primary mission and after all Level 1 mission requirements had been met, CMNT\\#4 developed an electrical short (An impedance of $200 M\\Omega $, which does constitute a ``short'' in the CMNT system) between the emitter and extractor electrodes due to a suspected propellant bridge, which rendered CMNT\\#4 effectively inoperable. The bridge occurred on Oct 27, 2016, after 1670 hours of operation on CMNT\\#4. While the exact location of the short on the emitters or extractor electrodes cannot be determined without access to the thruster, neither electrode was shorted to ground potential nor the accelerator electrode, indicating the short was not in the PPU.  The variable nature of the short impedance also indicated that it was an electrically conductive bridge of partially-polymerized propellant formed between the emitter and the extractor, likely after the porous extractor became saturated with propellant from normal and off-nominal operation.  These kind of propellant bridges have been observed on the ground previously due to poorly aligned emitters, saturated porous extractors, or large amounts of excess propellant near the emitter tip prior to operation.\n\nAt the time the short occurred, CMNT\\#4 was undergoing an experiment to verify the thrust performance model over a wide range of beam voltage and temperature conditions.  At the point of failure, the beam voltage was 4 kV and the thruster temperature was reduced to 20C compared to 6 kV and 25C that are nominal operating conditions.  Operating at a lower beam voltage widened the exhaust beam, increasing the flux of propellant at the edges of the beam onto the extractor electrode.  Operating at reduced temperatures increased the propellant's viscosity and impeded absorption and capillary action of the extractor's pores that are designed to soak-up excess propellant.  It is possible that operating in this off-nominal condition (which was still within the specified operational range) contributed to the failure.\n\nCareful analysis of the ground and in-flight data showed that CMNT\\#4 did experience significantly more operation time with bubble-driven flow during ground-based TVAC tests than the other thrusters, which could have increased the flux of propellant to the extractor, filling the pores.  In addition, because of another LPF anomaly unrelated to the DRS that occurred approximately a week before the CMNT\\#4 short, all the thrusters were shut down abruptly, which could have allowed some excess propellant to escape out of CMNT\\#4's emitters during a thermal transient that caused the propellant to expand without voltage on the electrodes.  The safer and normal version of the thruster shutdown procedure includes multiple steps to decrease the quantity of residual propellant in the emitters, reducing the risk of spraying during periods of prolonged shutdown. Finally, to preserve the stable thermal environment on the spacecraft, all CMNTs were left with PPUs enabled, default voltages on, and microvalves closed during standby mode and station keeping maneuvers, which was not originally specified during operations or tested on the ground.  While this kind of operation should not have caused any additional spray or flux to the extractor, on examining the data more carefully, a $~0.05 {\\mu}A$ level current was observed during most of this time in standby mode on CMNT\\#4, which was 10 times larger in terms of integrated current or total charge than any other thruster during these same periods, indicating low-level spraying between the electrodes that could have lead to premature saturation of the extractor.\n\nUnfortunately, the ST7 mission did not include a method of measuring the current to the extractor or accelerator electrodes, and since this was not measured except on early engineering model ground testing, it is difficult to quantify how thruster lifetime was impacted for this specific case. Determining how to prevent current flux to the extractor during normal operation, spraying between electrodes with default voltages on during standby mode, and monitoring any current to the extractor and accelerator electrodes, will be critical to the further development of this thruster technology, especially for missions with long lifetime requirements like LISA. Implementing redundant thruster heads will also be important for providing the required lifetimes, as is common practice for the primary propulsion system on large science missions. \n\nAfter the CMNT\\#4 anomaly and because the locations of the two CMTAs on only the spacecraft x-axis, having just seven operable thrusters gave insufficient rotational authority around the x-axis.  At the end of the nominal mission, with ESA's assistance, a hybrid ``crutch'' mode was developed to continue DRS operations using 4 of the LTP cold gas (CGAS) thrusters to provide a constant thrust bias that replaced what CMNT\\#4 would have normally provided. The colloid thruster and cold gas thruster thrust bias levels as well as new operational procedures and sequences were developed and validated on ground-based testbeds for this new operating mode. Both 4 and 2 CGAS thruster configurations were demonstrated on the ST7 testbed; however, the 4 CGAS thruster configuration was preferred to reduce the required colloidal thruster thrust bias levels. This hybrid operation was demonstrated, for all DCS modes, just before the final week of the primary mission,\nand it continued successfully through the extended mission. \n\n\\section*{Appendix B: Derivation of response to thrust injections}\n\\label{sec:mathAppendix}\nIn this section we present a more detailed derivation of the response of the Pathfinder spacecraft and LTP instrument to the thruster injections that were used to calibrate the CMNTs as described in \\ref{sec:thustCal} . \n\nLet $\\hat x_{bf}$, $\\hat y_{bf}$, $\\hat z_{bf}$ be unit vectors along the body-frame axes of the spacecraft ; let $\\vec r_{SC}$ be the position of the spacecraft CoM in some inertial frame; and let $\\vec r_{TM1}$ and $\\vec r_{TM2}$ be the positions (CoMs) of the two TMs in the same inertial frame. The $x1$  value measured by LTP is $\\big(\\vec r_{TM1} - \\vec r _{SC}\\big)\\cdot \\hat x_{bf}$, and similarly for $x2$.  \n\nApplying Newton's 2nd law to either $x1$ or $x2$, and restricting to the most-sensitive ($\\hat x_{bf}$) direction gives\n\\begin{eqnarray}\n&\\ddot x& \\equiv \\frac{d^2}{dt^2} \\big[\\big(\\vec r_{TM} -  \\vec r_{SC}\\big)\\cdot \\hat x_{bf}\\big] \\\\\n&=&   \\big[\\big(\\ddot{\\vec r}_{TM} -  \\ddot{\\vec r}_{SC}\\big)\\cdot \\hat{x}_{bf}\\big]  + 2 \\big[\\big( \\,\\dot{\\vec r}_{TM} -  \\dot{\\vec r}_{SC}\\big)\\cdot \\dot{\\hat x}_{bf}\\big] \\ \\ \\ \\ \\ \\    \\\\\n&+& \\big[\\big(\\vec r_{TM} -  \\vec r_{SC}\\big)\\cdot \\ddot{\\hat x}_{bf}\\big]  \\, \\label{basic} .\n\\end{eqnarray} \n\n\\noindent Re-arranging the above equation for $\\ddot x$ gives\n\\begin{eqnarray}\n&F^x_{SC} &= -M_{SC} \\ddot x  + \\frac{M_{SC}}{M_{TM} }F^x_{TM} \\label{b1} \\\\\n&+& 2 M_{SC} \\big[\\big( \\,\\dot{\\vec r}_{TM} -  \\dot{\\vec r}_{SC}\\big)\\cdot \\dot{\\hat x}_{bf}\\big]   \\label{b2} \\\\\n& +& M_{SC} \\big[\\big(\\vec r_{TM} -  \\vec r_{SC}\\big)\\cdot \\ddot{\\hat x}_{bf}\\big]  \\,  \\label{b3} \n\\end{eqnarray}\nwhere $F^x \\equiv \\vec F \\cdot \\hat x_{bf}$. The first term is Newton's 2nd law in the instrument frame while the later two terms account for the rotation of the frame.To evaluate the rotational frame terms, we define \n\\begin{equation}\n\\dot{\\hat x}_{bf} = \\vec \\Omega \\times \\hat x_{bf} \\, ,\n\\end{equation}\nwhere $\\vec \\Omega$ is the spacecraft 's instantaneous angular velocity, which also implies \n\\begin{eqnarray}\n\\ddot{\\hat x}_{bf} &=& \\dot{\\vec \\Omega} \\times \\hat x_{bf} +  \\vec \\Omega \\times \\dot{\\hat x}_{bf} \\, , \\\\\n&=& \\dot{\\vec \\Omega} \\times \\hat x_{bf} +  \\vec \\Omega \\times  \\big(\\vec \\Omega \\times \\hat x_{bf} \\big)  \\\\\n&=& \\dot{\\vec \\Omega} \\times \\hat x_{bf} + \\big(\\vec \\Omega \\cdot \\hat x_{bf}\\big) \\vec \\Omega \\, - \\, \\Omega^2 \\, \\hat x_{bf} \\label{c3} \\, .\n\\end{eqnarray}\n\nThe dynamical quantities can be separated into those dominated by the injection, including the TM actuation, and the rest, in order to estimate the sizes and timescales on which they are changing.  If one restricts attention to the thruster force at the injection frequency, then line (\\ref{b2}) is\nnegligible compared to line (\\ref{b3}).  Finally, $\\ddot{\\hat x}_{bf}$ in  (\\ref{b3}) is well approximated by the \n$\\dot{\\vec \\Omega} \\times \\hat x_{bf} $ term in line (\\ref{c3}), and terms quadratic in $\\Omega$ are negligible.  This implies the approximation\n\\begin{equation}\n\\dot{\\Omega}_i = (I^{-1})_{ij}N^j \\, .\n\\end{equation}\n\nIf we average the motion of $x1$ and $x2$, the largest rotational effects cancel.  The only rotational effects that do not cancel are proportional to $\\Delta z$, defined as the z-displacement of both TMs from the spacecraft CoM, which is on the order of 5$\\,$cm. Averaged over the two TMs, and restricting analysis to the injection frequency, all the rotating-frame effects can be approximated by \n\\begin{equation}\n- M_{SC} (\\Delta z) \\dot{\\Omega}_y \\, \n\\end{equation}\nor\n\\begin{equation}\n- M_{SC} (\\Delta z)\\big[(I^{-1})_{yx}N^x + (I^{-1})_{yy}N^y + (I^{-1})_{yz}N^z \\big]\\, . \n\\end{equation}\nOf these, the middle term is by far the largest, but it's simple to carry along the off-diagonal terms.\n\n If $F_i$ is the amplitude of the thrust from thruster i, then there are matrices $T^{xi}$ and $K^{xi}$ such that the x-component of the force and torque from thruster i are  $T^{xi}F_i$ and $K^{xi}F_i$. These matrices can be derived from the thruster positions and orientations. Plugging these into Eqs.~(\\ref{b1})--(\\ref{b3}) and re-arranging terms, we arrive at\n\n\\begin{equation}\nF_{i|x} =  \\frac{1}{2}\\big[ -M_{SC} (\\ddot x_1 + \\ddot x_2) + \\frac{M_{SC}}{M_{TM} }(F^x_{TM1} + F^x_{TM2})\\big] \/(1 + R^x_i)\\label{eq:Fresp}\n\\end{equation}\n\n\\noindent where the symbol $F_{i|x}$ denotes \"the force exerted by thruster $i$, as estimated from $x$ equation of motion\", \nand where the rotational correction term $R^x_i$ is given by\n\\begin{equation}\nR^x_i = (\\Delta z) M_{SC}  \\big[ (I^{-1})_{yx} K^{xi}+ (I^{-1})_{yy} K^{yi} + (I^{-1})_{yz} K^{zi} \\big] \/T^{xi} \\, .\\label{eq:Rcorrect}\n\\end{equation}\n\\\\\n\n\n\\section*{Appendix C: Estimates of Thruster and Platform Noise Contributions}\n\\label{sec:NoiseAppendix}\nThis appendix provides details of the estimates of contributions to the measured force noise on the spacecraft as listed in Table \\ref{tab:SCnoiseComp}. \n\n\\subsection*{CMNT Shot Noise}\nCMNT shot noise is an effect of the quantized nature of the electrospray thrust, which is composed of the momentum transfer from discrete droplets. Using a charge to mass ratio of 470 C$\/$kg, the current corresponds to a droplet rate of $3 * 10^{15}$ drops\/second with an associated shot noise of 0.16 nN when projected into the X direction.  \n\\subsection*{CMNT Flutter Noise}\nFlutter noise refers to variations in the thrust direction, which induce a thrust-dependent thrust noise $S_{flutter} = T\\cdot S_{cos\\alpha}$ where $\\alpha$ is the deviation of the thrust vector from its nominal direction. Ground measurements using a 2-D electrometer array measured $S_\\alpha < 10^{-3}\\,\\textrm{rad}\/\\sqrt{\\textrm{Hz}}$ in the relevant band, which gives $S_{flutter}\\sim 0.03\\,\\textrm{nN}\/\\sqrt{\\textrm{Hz}}$ at the maximum thrust of 30$\\,\\mu$N.\n \n \\subsection*{Solar Force Noise}\n Solar radiation pressure (SRP) produces a force noise along the x-axis that is described by \n \\begin{equation}\n S_{SRP,x} = F_{SRP}\\cdot S_{SRP}\\cdot\\bar{H}+F_{SRP}\\cdot S_H,\\label{eq:SRPx}\n \\end{equation}\n where $F_{SRP}$ is the DC force on the spacecraft due to the SRP, $S_{SRP}$ is the spectral density of the stability of the SRP, $H$ is the angle of the spacecraft about the y-axis and $S_H$ is the spectral density of variations in $H$.  The total DC force on the spacecraft in the anti-Sun direction ($-z$ axis) is measured at $\\sim24\\,\\mu$N. This includes contributions both from the direct solar radiation pressure as well as the differential thermal radiation from the warm sunward side of the spacecraft (radiometer effect). A rough order-of-magnitude estimate is that the the radiometer term is approximately 40\\% of the direct term. Hence $F_{SRP}\\sim17\\,\\mu$N. During DRS science operations, $\\bar{H}\\sim10^{-4}\\,$rad, and $S_H\\mathrel{\\hbox{\\rlap{\\hbox{\\lower4pt\\hbox{$\\sim$}}}\\hbox{$<$}}} 10^{-4}\\,\\textrm{rad}\/\\sqrt{\\textrm{Hz}}$.  Measurements of variations in solar flux give  estimates of $S_{SRP}\\mathrel{\\hbox{\\rlap{\\hbox{\\lower4pt\\hbox{$\\sim$}}}\\hbox{$<$}}} 10^{-3}\\,\/\\sqrt{\\textrm{Hz}}$. With these parameters, the second term in (\\ref{eq:SRPx}) dominates, with a contribution of $1.7\\,\\textrm{nN}\/\\sqrt{\\textrm{Hz}}$. \n \n \\subsection*{Radiometer Noise}\n The coupling of the radiometer noise, which is primarily along the z-axis, to motion in the x-axis is the same as for SRP.  Using the same logic as presented for the SRP above, the DC force along z is estimated as $F_{rad}\\sim7\\,\\mu$N and the noise along x due to angular jitter along H is $0.7\\,\\textrm{nN}\/\\sqrt{\\textrm{Hz}}$. Note that because the coupling mechanism for SRP and Radiometer noise is the same (spacecraft jitter in H), they will add \\emph{coherently}. This is taken into account for the noise summations in Table \\ref{tab:SCnoiseComp}.\n \n\\subsection*{Magnetic Field Noise}\n The size of the spacecraft is vastly smaller than the spatial length scale over which the interplanetary $\\vec B_{ip}$-field varies, so we treat $\\vec B_{ip}$ as spatially uniform over the spacecraft .   The interaction of $\\vec B_{ip}$ with the current $\\vec j$ in the spacecraft can torque the spacecraft , but it produces no net force. Note that time variations in $\\vec B_{ip}$ will cause extra currents to flow in the spacecraft , which create a B-field $\\vec B_{sc}$ that partially counteracts the changes in $\\vec B_{ip}$ -- this is magnetic shielding--and because magnetic permeability will vary across the spacecraft, the total field  $\\vec B_{ip} + \\vec B_{sc}$  {\\it can} have a significant spatial gradient. But this does not invalidate our earlier argument that there is no net force from the interaction $\\vec B_{ip}$ and the total $\\vec j$.\n\nThe interaction of $\\vec B_{ip}$ with the net charge on the spacecraft does produce a net force, which we now estimate. The photo-electric effect  \"kicks\" electrons off the surface of the solar panels.  Indeed, when the CMNTs (which generate thrust by accelerating positively charged droplets away from the spacecraft) are providing the thrust, these two effects largely cancel, and this is the mechanism that keeps the net charge on the spacecraft small.  The spacecraft potential adjusts until the net spacecraft charging rate averages over time to zero.  The spacecraft potential in equilibrium is $\\sim 100$V, or (in cgs units) $\\sim \\frac{1}{3}$statvolt.  From this, we can estimate the total charge on the spacecraft by $V \\sim Q\/R$.  Using $R \\sim 100$cm, we find $Q \\sim 33\\ $statcoulomb (or $\\approx 10^{-8}\\ $ Coulomb).    Using $\\vec F_{sc} = c^{-1}  Q \\,  \\vec v \\times \\vec B_{ip}$,  $v\/c \\sim 10^{-4}$, and $B_{ip} \\approx 3 \\times 10^{-4} \\big(\\frac{f}{1 mHz}\\big)^{-0.8} gauss\\, Hz^{-1\/2}$, we arrive at $F \\sim 10^{-2} nN Hz^{-1\/2} \\big(\\frac{T}{10^7 s}\\big)  \\big(\\frac{f}{1 mHz}\\big)^{-0.8} $  (where we have used the conversion $10^{-6}$dyne$ =  10^{-2}$ nN). \n\n\\subsection*{Micrometeoroid Impacts}\n\nThe LPF spacecraft occasionally encounters interplanetary dust particles which impart an impulsive momentum to the spacecraft. The size of these particles generally follow a power-law distribution with smaller size particles being more numerous than larger ones. For large impacts, the events can be identified and either subtracted or excised from the data, as was discussed in \\ref{sec:CMNTnoiseSC}. For smaller impacts, which are far more numerous, the impulsive momentum may not be recognized as a discrete signal and will instead average out to a force noise.  Using a sample of 44 impact events that were identified in a search of $\\sim180$ days of LPF data, a powerlaw estimate of the collision rate $R$, per transferred momentum (in units of $\\mu N s$), was estimated to be\\footnote{A full analysis of these impacts is the subject of a forthcoming paper}: \n\\begin{equation}\nR  =  1.5 \\times 10^{-6} \\, (\\bar p)^{-1.64} \\,  s^{-1}\\, ,\n\\end{equation}\nwhere we have defined the dimensionless momentum transfer $\\bar p \\equiv p\/p_0$, with $p_0 = 1 \\mu N s $.   Assume that in each collision, the momentum is deposited uniformly over some short time $\\delta t$.  (As long as $\\delta t$ is short compared to $ 33 s$ [$= 1\/(30 mHz)$], we shall see that $\\delta t$ drops out of the expression for the force noise spectral density in the measurement band.)  Since the collisions represent shot noise,  force-noise spectrum is some constant $S_0$ up to $f \\approx 1\/(2\\delta t)$ (and falls roughly as $f^{-2}$ at higher $f$), and  $S_0$ times $ (2\\delta t)^{-1}$ is the mean-square value of the force from collisions:\n\\begin{eqnarray}\nS_0 &=&  (2 \\delta t) (p_0)^2 \\int_0^{\\bar p_t} R(\\bar p) \\big(\\frac{\\bar p}{\\delta t}\\big)^2 (\\delta t)  \\ d \\bar p \\\\\n& = & 2.2 \\times 10^{-6} \\, \\bar p_t^{1.36} (\\mu N)^2\/Hz  \\, ,\n\\end{eqnarray}\nwhere $\\bar p_t$ is some threshold value, above which collisions are individually identified and removed from the data.\nUsing $\\bar p_t \\approx 0.5$, and dividing $S^{1\/2}_{0}$ by  $\\sqrt{3}$ to account for the fact that here we want only the x-component of the force noise, we arrive at\n\\begin{equation}\nS^{1\/2}_{0,x} \\approx  0.5 n N\\\/\\sqrt{Hz} \\, .\n\\end{equation}\n\n \n  \n \\subsection*{Test Mass Force Noise}\nTest mass force can by estimated by looking at the measured differential acceleration between the two test masses. As shown in section \\ref{sec:deltag}, this is at the $\\sim 3\\,\\textrm{fm}\\,\\textrm{s}^{-2}\/\\sqrt{\\textrm{Hz}}$ level, which would represent an equivalent spacecraft noise of $\\sim 1\\,\\textrm{pN}\/\\sqrt{\\textrm{Hz}}$.  A limitation of this estimate is that it is not sensitive to common-mode forces on the test masses, such as might be caused by time-varying magnetic fields. However, it seems unlikely that the couplings in each test mass would match sufficiently well to have a 6 order of magnitude difference between the absolute and differential effects. Assuming a worst-case common-mode rejection ratio of $10^3$ gives an upper limit of 1$\\,\\textrm{nN}\/\\sqrt{\\textrm{Hz}}$. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nSupernova SN1987A shows a ''pearl necklace'' and infrared emissions whose interpretation appears difficult (Bouchet et al. 2006, Lawrence et al. 2000, Thornhill \\& Ransom 2006). A source or these particularities may be simply spectroscopy.\n\nAssuming that the central object of SN 1987A is a neutron star heated by the accretion of a low density hydrogen cloud, the density of emitted energy is high, so that the transfers of energy are non-linear, strong. Such effects are usually observed with lasers.\n\nSection 2, recalls some of these effects, specifying the language and the notations.\n\nIn section 3 the generation of the pearl necklace appears as a trivial application of these effects.\n  \n\\section{Some high energy optical effects.}\n\\subsection{Superradiant emissions.}\nPlanck used the notion of electromagnetic modes introduced in the nineteenth century to study music instruments. For this application, the used modes were restricted to stationary systems, while the mathematical definition is wider: A mode is a ray in the real vector space of the solutions of a linear set of equations (that is, all solutions in a mode depend only on a single real parameter named amplitude of the solution). Happily, the  definition of the modes does not imply stationary and monochromatic conditions, because such conditions would require an infinite duration of the experiments.\n\nThe electromagnetic fields in the vacuum obey the linear Maxwell's equations up to very high frequencies. Schwarzschild and Fokker trick allows  introducing sources replaced by their advanced fields.\n\nPlanck mistook computing the additive term in his equation which gives the energy of an electromagnetic mode inside a blackbody at temperature $T$, defining also the temperature of a mode. Nernst (1916) found the right value $h\\nu\/2$, so that the correct mean energy in a monochromatic mode of frequence $\\nu$ writes $h\\nu(1\/(\\exp(h\\nu\/kT)-1)+1\/2)$.\n\nEinstein (1917) introduced the stimulated emission, but justifying the spontaneous emission, he did not remark that the electromagnetic field has a minimal mean value corresponding to $h\\nu\/2$ in a blackbody at 0K; laser experiments show that the spontaneous emission is a stimulated emission corresponding to the amplification of this minimal value.\n\nSuppose that only two molecular levels of energies $e_1$ and $e_2$ ($e_2>e_1$) are implied in a transition of energy $\\nu = (e_2-e_1)\/h$. The molecular temperature $T_{12}$ relative to this transition is deduced from the molecular populations $n_1$ and $n_2$ in these levels by $n_1\/n_2 = \\exp(h\\nu\/kT_{12})$.\n\nSuppose that a long cell is filled with this gas; a low temperature light beam entering into the cell increases the entropy of the system, being amplified, heated to reach temperature $T_{12}$. At the beginning, the amplification may be said a spontaneous emission, then it is induced, the energy of the beam increasing proportionally to this energy, that is exponentially. But the available molecular energy is limited, so that the temperature $T_{12}$ {} is decreased, usually strongly. Thus, spontaneous emissions in other directions are strongly decreased. This decrease resulting from a large excitation seems a paradox.\n\nIf a long path in a strongly excited medium is real, it is a superradiant emission; if it is virtual, using mirrors, it is a laser emission.\n\n\\subsection{The \\textquotedblleft Impulsive Stimulated Raman Scattering (ISRS).\\textquotedblright }\n\nIt is usually assumed that there is no interaction between light beams refracted by a transparent medium. Experiments and a regular theory (Giordmaine et al. (1968), Yan et al. (1985), Weiner et al. (1990), Dougherty et al. (1992), Dhar et al. (1994), ...) show that this assumption is wrong if light and matter verify conditions set by Lamb (1971). The interaction which increases the entropy of a set of refracted usual time-incoherent beams by frequency shifts, without blurring the images and the spectra, works well in atomic hydrogen in its first excited state. Planck's law and thermodynamics show that, usually, light is redshifted, and the energy it looses is transferred to the thermal background. There is no energy threshold of the energy of the beams, the ISRS becoming the Coherent Raman Effect on Incoherent Light (CREIL) at low intensities (Moret-Bailly 2005-6).\n\n\\section{Generation of the pearl necklace.}\n\\subsection{Absorption of light emitted by the neutron star.}\nAccreting hydrogen, the neutron star becomes extremely hot, at least at hot spots; thus it emits light mainly in the far ultraviolet.\n\nClose to the star, an atom is ionized; hydrogen and light element impurities lose all their electrons, producing a transparent plasma. Some heavier atoms may absorb light, but they re-emit energy in the UV, so that few energy is lost.\n\nAssuming a low density, a combination of protons and electrons requires a low temperature, say 20 000 K. Hydrogen is mainly produced in its ground state, but it absorbs Lyman lines, so that a lot of its states may be populated.\n\nHydrogen absorbs frequencies lower than Lyman limit only on the its spectral lines. In next subsection, we will see that a superradiant emission decreases the population of the excited states, so that few 2P hydrogen remains, able to shift the spectrum, until, paradoxically, the Lyman $\\alpha$ line is almost absorbed: When this happens, the superradiance decreases, so that more 2P hydrogen remains and shifts the light, the energy at the 2P frequency and at the frequencies of other absorption lines is renewed.\n\nFinally, the absorption at a Lyman line, which is strong, nearly complete, sweeps a wide band.\n\n\\medskip\nThe final result is an extremely strong absorption not only at the frequencies of the lines (eventually not Lyman), but on wide bands corresponding to a redshift which lasts as long as energy is shifted to the Lyman $\\alpha$ frequency. The energy lost by the redshifts, blueshifts the thermal background, that is heats it, simulating hot dust.\n\n\\subsection{Generation of an UV necklace.}\nPrevious subsection showed that the gas is strongly excited in a spherical shell centred on the neutron star, by a wide band absorption. Superradiant emissions may appear in directions for which the gas is thick, that is tangentially to the spheres. This system is very similar to a laser pumped transversally by a light-emitting diode of the same frequency. Although the gas is excited to eigenstates, the light emitted by the shell of excited gas makes a continuous spectrum because the column density of 2S hydrogen crossed from the emission point to outside is variable, producing variable redshifts. The frequency shift of the emitted lines prevents an absorption at the exciting (Lyman $\\alpha$ ...) frequency.\n\nConsider two thin, close spherical shells emitting lines whose local difference of frequencies is of the order of the linewidth; each line induces the emission of the other, so that their modes are bound; this binding extends to the whole superradiant band whose modes are, consequently, connected.\n\nThe strong absorptions and re-emissions in the shell of excited atomic hydrogen generate for each direction a bright ring in the UV. In this ring, the competition of the polychromatic modes selects a particular set of modes, producing, in the UV, bright spots. A nearly complete transfer of energy from the radial beams to the rings obeys thermodynamics because the solid angle on which a ring is seen is much larger than the solid angle on which the central engine could be seen.\n\n\\subsection{Generation of a visible necklace.}\nAround the shell of far UV absorbing hydrogen, the columns of UV light emitted to the Earth excite hydrogen and various atoms strongly, so that, in the direction of the Earth, the emissions of these atoms are superradiant and co-linear to the columns: the visible pearl necklace is generated.\n\n\\section{Conclusion}\nThe present paper neglects a large part of the complexity of the problen: the necklace is not a circle; can the other circles be generated by a similar process around two other stars ? If yes, why did they illuminate when the necklace appeared ?\n\nHowever, although partial, the present theory gives results using few hypothesis and regular physics. The explanation of the pearl necklace of SN1987A is rough, but it requires very few astrophysical hypothesis, its originality being the use of optical and spectroscopic properties more generally developed and well verified in laser technology. The visible necklace should come with a remaining UV spectrum in which absorptions corresponding to the lines observed in the visible may appear. Among other verifications, it could be a positive test of the present theory.\n\n\\section{Bibliography.}\n\n  Bouchet, P., E. Dwek, I. J. Danziger, R. G. Arendt, I. J. M. De Buizer, S. Park, N. B. Suntzeff, R. P. Kirshner, P. Challis, 2006, arxiv:astro-ph\/0601495\n\nDhar, L. , J. A. Rogers, \\& K. A. Nelson, 1994 {\\it Chem. Rev.} {\\bf 94}, 157\n \nDougherty, T. P., G. P. Wiederrecht, K. A. Nelson, M. H. Garrett, H. P. Jenssen \\& C. Warde,1992, {\\it Science} {\\bf 258,}, 770\n\nEinstein A., 1917, {\\it Phys. Z.}, {\\bf 18} 121\n\nGiordmaine, J. A. , M. A. Duguay \\& J. W.Hansen, 1968, {\\it IEEE J. Quantum Electron.}, {\\bf }4,252\n\nLawrence S. S.,B. E. Sugerman, P. Bouchet, A. P. S. Crotts, R. Uglesich \\& S. Heathcote, 2000, arxiv:astro-ph\/0004191\n\nMoret-Bailly, J., 2005, arxiv: physics\/0507141\n\nMoret-Bailly, J., 2006,  {\\it AIP Conference Proceedings}, 822, 226-238\n\nNernst W. , {\\it Verh. Deutsch. Phys. Ges}, {\\bf 18}, 83 (1916)\n\nThornhill,W. W., \\& C. J. Ransom 2006 {\\it 2006 IEEE International Conference on Plasma Science}, 369\n\nWeiner, A. M. , D. E. Leaird., G. P. Wiederrecht, \\& K. A. Nelson,  1990, {\\it Science}  {\\bf 247}, 1317\n\nYan, Y.-X. , E. B. Gamble Jr. \\& K. A. Nelson , 1985,  {\\it J. Chem Phys.}, {\\bf 83}, 5391\n\n\\end{document}\n\n \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\\label{sec:1}\n\nDetecting a very weak signal which is almost inaccessible within the \nclassical (i.e., non-quantum) regime is one of the most important subjects \nin quantum information science. \nA strong motivation to devise such an ultra-precise sensor stems from \nthe field of gravitational wave detection \n\\cite{BraginskyBook, Caves1980, Braginsky2008, Miao2012, Abbott2016}. \nIn fact, a variety of linear sensors composed of opto-mechanical oscillators \nhave been proposed \\cite{Milburn,Chen2013, Latune2013, Aspelmeyer2014}, \nand several experimental implementations of those systems in various scales \nhave been reported \n\\cite{Corbitt2007, Matsumoto2015, Underwood2015, Thompson2008, Verhagen2012}. \n\n\nIt is well known that in general a linear sensor is subjected to two types of \nfundamental noises, i.e., the {\\it back-action noise} and the {\\it shot noise}. \nAs a consequence, the measurement noise is lower bounded by the {\\it standard \nquantum limit (SQL)} \\cite{BraginskyBook, Caves1980}, which is mainly due \nto the presence of back-action noise. \nHence, high-precision detection of a weak signal requires us to devise a sensor \nthat evades the back-action noise and eventually beats the SQL; \ni.e., we need to have a sensor achieving {\\it back-action evasion (BAE)}. \nIn fact, many BAE methods have been developed especially in the field of \ngravitational wave detection, e.g., the variational measurement technique \n\\cite{Vyatchanin1995, Kimble2001, Khalili2007} or the quantum locking scheme \n\\cite{CourtyEurophys2003, CourtyPhysRev2003, Vitali2004}.  \nMoreover, towards more accurate detection, recently we find some high-level \napproaches to design a BAE sensor, based on those specific BAE methods. \nFor instance, Ref.~\\cite{Miao2014} provides a systematic comparison of \nseveral BAE methods and gives an optimal solution. \nAlso systems and control theoretical methods have been developed to synthesize \na BAE sensor for a specific opto-mechanical system \n\\cite{Tsang2010, YamamotoCFvsMF2014}; \nin particular, the synthesis is conducted by connecting an auxiliary system \nto a given plant system by {\\it direct-interaction} \\cite{Tsang2010} or \n{\\it coherent feedback} \\cite{YamamotoCFvsMF2014}. \n\n\nAlong this research direction, therefore, in this paper we set the goal to develop \na general systems and control theory for engineering a sensor achieving BAE, \nfor both the coherent feedback and the direct-interaction configurations. \nThe key tool used here is the {\\it geometric control theory} \n\\cite{Schumacher1980, Schumacher1982, Wonham1985, Marro2010, Otsuka2015}, \nwhich had been developed a long time ago. \nThis is indeed a beautiful theory providing a variety of controller design \nmethods for various purposes such as the non-interacting control and \nthe disturbance decoupling problem, but, to our best knowledge, it has not been \napplied to problems in quantum physics. \nActually in this paper we first demonstrate that the general synthesis problem of \na BAE sensor can be formulated and solved within the framework of geometric control theory, \nparticularly the above-mentioned disturbance decoupling problem. \\vspace{0.7mm}\\\\\n\\hspace{0.2cm}\nThis paper is organized as follows. \nSection~\\ref{sec:2} is devoted to some preliminaries including a review of the \ngeometric control theory, the general model of linear quantum systems, and \nthe idea of BAE. \nThen, in Section~\\ref{sec:3}, we provide the general theory for designing a \ncoherent feedback controller achieving BAE, and demonstrate an example for \nan opto-mechanical system. \nIn Section~\\ref{sec:4}, we discuss the case of direct interaction scheme, also \nbased on the geometric control theory. \nFinally, in Section~\\ref{sec:5}, for a realistic opto-mechanical system subjected \nto a thermal environment (the perfect BAE is impossible in this case), we provide \na convenient method to find an approximated BAE controller and show how \nmuch the designed controller can suppress the noise. \\vspace{0.7mm}\\\\\n\\hspace{0.2cm}\n{\\bf Notation:} \nFor a matrix $A=(a_{ij})$, $A^{\\top}$, $A^{\\dag}$, \nand $A^{\\sharp}$ represent the transpose, Hermitian conjugate, \nand element-wise complex conjugate of $A$, respectively; i.e., \n$A^{\\top}=(a_{ji})$,  $A^{\\dag}=(a_{ji}^*)$, and \n$A^{\\sharp}=(a_{ij}^*)=(A^{\\dag})^{\\top}$. \n$\\Re(a)$ and $\\Im(a)$ denote the real and imaginary parts of a complex \nnumber $a$. \n$O$ and $I_{n}$ denote the zero matrix and the $n \\times n$ identity matrix. \n$\\Kernel A$ and $\\Image A$ denote the kernel and the image of \na matrix $A$, i.e., $\\Kernel A=\\{x\\,|\\, Ax=0\\}$ and \n$\\Image A=\\{y\\,|\\, y=Ax, ~\\forall x\\}$. \n\n\n\n\n\n\\section{Preliminaries}\\label{sec:2}\n\n\n\n\\subsection{Geometric control theory for disturbance decoupling}\\label{sec:2-1}\n\nLet us consider the following {\\it classical} linear time-invariant system:\n\\begin{align}\n          \\frac{dx(t)}{dt}&=Ax(t)+Bu(t), ~~ \\nonumber\\\\\n          y(t)&=Cx(t)+Du(t),\n\\label{eq_linearsystem}\n\\end{align}\nwhere $x(t) \\in \\mathcal{X}:=\\mathbb{R}^{n}$ is a vector of system variables, \n$u(t) \\in \\mathcal{U}:=\\mathbb{R}^{m}$ and \n$y(t) \\in \\mathcal{Y}:=\\mathbb{R}^{l}$ are vectors of input and output, \nrespectively. \n$A, B, C$, and $D$ are real matrices. \nIn the Laplace domain, the input-output relation is represented by \n\\begin{align*}\n         Y(s)=\\Xi (s)U(s), ~~ \\Xi (s)=C(sI-A)^{-1}B+D, \n\\end{align*}\nwhere $U(s)$ and $Y(s)$ are the Laplace transforms of $u(t)$ and $y(t)$, \nrespectively. \n$\\Xi(s)$ is called the {\\it transfer function}. \nIn this subsection, we assume $D=0$. \n\n\n\nNow we describe the geometric control theory, for the disturbance \ndecoupling problem \\cite{Schumacher1980, Schumacher1982}. \nThe following {\\it invariant subspaces} play a key role in the theory. \n\n{\\bf Definition 1:}~\nLet $A : \\mathcal{X} \\rightarrow \\mathcal{X}$ be a linear map. \nThen, a subspace $\\mathcal{V} \\subseteq \\mathcal{X}$ is said \nto be $A$-{\\it invariant}, if $A\\mathcal{V} \\subseteq \\mathcal{V}$. \n\n{\\bf Definition 2:}~\nGiven a linear map $A : \\mathcal{X} \\rightarrow \\mathcal{X}$ \nand a subspace \\Image $B$ $\\subseteq \\mathcal{X}$, a subspace \n$\\mathcal{V} \\subseteq \\mathcal{X}$ is said to be $(A, B)$-{\\it invariant}, \nif $A\\mathcal{V} \\subseteq \\mathcal{V} \\oplus \\Image B$. \n\n{\\bf Definition 3:}~ \nGiven a linear map $A : \\mathcal{X} \\rightarrow \\mathcal{X}$ \nand a subspace \\Kernel $C$ $\\subseteq \\mathcal{X}$, a subspace \n$\\mathcal{V}\\subseteq \\mathcal{X}$ is said to be $(C, A)$-{\\it invariant}, \nif $A(\\mathcal{V} \\cap \\Kernel C) \\subseteq \\mathcal{V}$. \n\n{\\bf Definition 4:}~ \nAssume that $\\mathcal{V}_{1}$ is $(C, A)$-invariant, $\\mathcal{V}_{2}$ is \n$(A, B)$-invariant, and $\\mathcal{V}_{1} \\subseteq \\mathcal{V}_{2}$. \nThen, $(\\mathcal{V}_{1},\\, \\mathcal{V}_{2})$ is said to be a $(C,A,B)$-pair.\n\n\nFrom Definitions 2 and 3, we have the following two lemmas.\n\n\n{\\bf Lemma 1:}~ \n$\\mathcal{V} \\subseteq \\mathcal{X}$ is $(A, B)$-invariant if and only if \nthere exists a matrix $F$ such that\n$\n             F \\in \n             \\mathcal{F}(\\mathcal{V}) :=\n                   \\{ F : \\mathcal{X} \\rightarrow \\mathcal{U}\\,| \\, \n                          (A+BF)\\mathcal{V} \\subseteq \\mathcal{V} \\}. \n$\n\n\n{\\bf Lemma 2:}~\n$\\mathcal{V} \\subseteq \\mathcal{X}$ is $(C, A)$-invariant if and only if \nthere exists a matrix $G$ such that\n$\n              G \\in \n                \\mathcal{G}(\\mathcal{V}) := \n                  \\{ G : \\mathcal{Y} \\rightarrow \\mathcal{X} \\,| \\, \n                         (A+GC)\\mathcal{V} \\subseteq \\mathcal{V} \\}. \n$\n\n\nThe disturbance decoupling problem is described as follows. \nThe system of interest is represented, in an extended form of \nEq.~\\eqref{eq_linearsystem}, as \n\\begin{align*}\n         \\frac{dx(t)}{dt}&=Ax(t)+Bu(t)+Ed(t), ~\\\\\n         y(t)&=Cx(t), ~~\n         z(t)=Hx(t), \n\\end{align*}\nwhere $d(t)$ is the disturbance and $z(t)$ is the output to be regulated. \n$E$ and $H$ are real matrices. \nThe other output $y(t)$ may be used for constructing a feedback controller; \nsee Fig.~\\ref{fig_feedbackframework}. \nThe disturbance $d(t)$ can degrade the control performance evaluated on $z(t)$. \nThus it is desirable if we can modify the system structure by some means so \nthat eventually $d(t)$ dose not affect at all on $z(t)$\n\\footnote{This condition is satisfied if the transfer function from $d(s)$ to $z(s)$ is \nzero for all $s$, for the modified system. Or equivalently, the controllable subspace\nwith respect to $d(t)$ is contained in the unobservable subspace with respect to $z(t)$.}.\nThis control goal is called the disturbance decoupling. \nHere we describe a specific feedback control method to achieve this goal; \nnote that, as shown later, the direct-interaction method for linear quantum \nsystems can also be described within this framework. \nThe controller configuration is illustrated in Fig.~\\ref{fig_feedbackframework}; \nthat is, the system modification is carried out by combining an auxiliary \nsystem (controller) with the original system (plant), so that the whole \nclosed-loop system satisfies the disturbance decoupling condition. \nThe controller with variable \n${x}_{{\\scriptscriptstyle K}} \\in \\mathcal{X}_{{\\scriptscriptstyle K}}\n:=\\mathbb{R}^{n_{\\scriptscriptstyle k}}$ is assumed to take the following form: \n\\begin{align*}\n\\hspace{-0.1cm}\n     \\frac{dx_{{\\scriptscriptstyle K}}(t)}{dt}\n                   &=A_{\\scriptscriptstyle K}x_{{\\scriptscriptstyle K}}(t)\n                       +B_{\\scriptscriptstyle K}y(t), \\, \\\\\n      u(t)&=C_{\\scriptscriptstyle K}x_{{\\scriptscriptstyle K}}(t)\n                       +D_{\\scriptscriptstyle K}y(t), \n\\end{align*}\nwhere $A_{\\scriptscriptstyle K} : \\mathcal{X}_{{\\scriptscriptstyle K}} \\rightarrow \n\\mathcal{X}_{{\\scriptscriptstyle K}}$, \n$B_{\\scriptscriptstyle K} : \n\\mathcal{Y} \\rightarrow \\mathcal{X}_{{\\scriptscriptstyle K}}$,\n$C_{\\scriptscriptstyle K} : \n\\mathcal{X}_{{\\scriptscriptstyle K}} \\rightarrow \\mathcal{U}$,  \nand $D_{\\scriptscriptstyle K} : \\mathcal{Y} \\rightarrow \\mathcal{U}$ are real matrices. \n\\begin{figure}[!t]\n\\begin{center}\n\\includegraphics[width=4cm,clip]{Figure\/Fig1_feedbackframework.eps}\n\\caption{General configuration for the disturbance decoupling via a dynamical feedback controller.}\n\\label{fig_feedbackframework}\n\\end{center}\n\\end{figure}\nThen, the closed-loop system defined in the augmented space \n$\\mathcal{X}_{{\\scriptscriptstyle E}} := \\mathcal{X} \\oplus \\mathcal{X}_{{\\scriptscriptstyle K}}$ \nis given by \n\\begin{align}\n\\label{eq_extended_system}\n          \\frac{d}{dt}\\left[\\begin{array}{c}\n                x\\\\\n                x_{{\\scriptscriptstyle K}}\\\\\n          \\end{array}\\right] \n         &=\\left[\\begin{array}{cc}\n               A+BD_{\\scriptscriptstyle K}C & BC_{\\scriptscriptstyle K}\\\\\n               B_{\\scriptscriptstyle K}C & A_{\\scriptscriptstyle K}\\\\\n          \\end{array}\\right]\n          \\left[\\begin{array}{c}\n               x\\\\\n               x_{{\\scriptscriptstyle K}}\\\\\n          \\end{array}\\right] \n        +\\left[\\begin{array}{c}\n              E\\\\\n              O\\\\\n          \\end{array}\\right] d, ~ \\nonumber \\\\\n        z&=\\left[\\begin{array}{cc}\n              H & O\\\\\n         \\end{array}\\right]\n        \\left[\\begin{array}{c}\n              x\\\\\n              x_{{\\scriptscriptstyle K}}\\\\\n         \\end{array}\\right].\n\\end{align}\nThe control goal is to design $(A_{\\scriptscriptstyle K}, B_{\\scriptscriptstyle K}, \nC_{\\scriptscriptstyle K}, D_{\\scriptscriptstyle K})$ so that, in \nEq.~\\eqref{eq_extended_system}, the disturbance signal $d(t)$ dose not \nappear in the output $z(t)$: see the below footnote.\nHere, let us define \n\\begin{align}\n            A_{{\\scriptscriptstyle E}}=\\left[\\begin{array}{cc}\n                              A+BD_{\\scriptscriptstyle K}C & BC_{\\scriptscriptstyle K}\\\\\n                              B_{\\scriptscriptstyle K}C & A_{\\scriptscriptstyle K}\\\\\n                          \\end{array}\\right] , \n\\label{eq_extended_matrixAe}\n\\end{align}\n$\\mathcal{B}= \\Image B$, $\\mathcal{C}= \\Kernel C$, \n$\\mathcal{E}= \\Image E$, and $\\mathcal{H}= \\Kernel H$. \nThen, the following theorem gives the solvability condition for the disturbance \ndecoupling problem.\n\n\n{\\bf Theorem 1:}~\nFor the closed-loop system \\eqref{eq_extended_system}, the disturbance \ndecoupling problem via the dynamical feedback controller has a solution \nif and only if there exists a $(C,A,B)$-pair $(\\mathcal{V}_{1},\\mathcal{V}_{2})$ \nsatisfying \n\\begin{align}\n          \\mathcal{E} \\subseteq \\mathcal{V}_{1} \n                    \\subseteq \\mathcal{V}_{2} \\subseteq \\mathcal{H}.\n\\label{eq_DDP_condition}\n\\end{align}\n\nNote that this condition does not depend on the controller matrices to \nbe designed. \nThe following corollary can be used to check if the solvability condition \nis satisfied. \n\n\n{\\bf Corollary 1:}~ \nFor the closed-loop system \\eqref{eq_extended_system}, the disturbance \ndecoupling problem via the dynamical feedback controller has a solution \nif and only if\n\\begin{align}\n           \\mathcal{V}_{*}{\\scriptscriptstyle (\\mathcal{C}, \\mathcal{E})}\n              \\subseteq \\mathcal{V}^{*}{\\scriptscriptstyle (\\mathcal{B}, \\mathcal{H})}, \\nonumber\n\\end{align}\nwhere $\\mathcal{V}^{*}{\\scriptscriptstyle (\\mathcal{B}, \\mathcal{H})}$ is \nthe maximum element of $(A, B)$-invariant subspaces contained in $\\mathcal{H}$, \nand $\\mathcal{V}_{*}{\\scriptscriptstyle (\\mathcal{C}, \\mathcal{E})}$ is \nthe minimum element of $(C, A)$-invariant subspaces containing $\\mathcal{E}$. \nThese subspaces can be computed by the algorithms given in Appendix~A. \n\n\nOnce the solvability condition described above is satisfied, then we can \nexplicitly construct the controller matrices $(A_{\\scriptscriptstyle K}, \nB_{\\scriptscriptstyle K}, C_{\\scriptscriptstyle K}, D_{\\scriptscriptstyle K})$. \nThe following intersection and projection subspaces play a key role for this purpose; \nthat is, for a subspace $\\mathcal{V}_{{\\scriptscriptstyle E}} \\subseteq \n\\mathcal{X}_{{\\scriptscriptstyle E}}= \\mathcal{X} \\oplus \\mathcal{X}_{{\\scriptscriptstyle K}}$, \nlet us define\n\\begin{align*}\n             \\mathcal{V}_{{\\scriptscriptstyle I}}:&= \n                        \\left\\{x \\in \\mathcal{X}~\\Bigg| \\left[\\begin{array}{c}\n                                    x\\\\\n                                    O\\\\\n                               \\end{array}\\right] \n                \\in \\mathcal{V}_{{\\scriptscriptstyle E}} \\right\\}, ~ \\\\\n             \\mathcal{V}_{{\\scriptscriptstyle P}}:&= \n                        \\left\\{x \\in \\mathcal{X}~\\Bigg| \\left[\\begin{array}{c}\n                                    x\\\\  \n                                    x_{{\\scriptscriptstyle K}}\\\\\n                               \\end{array}\\right] \n                \\in \\mathcal{V}_{{\\scriptscriptstyle E}},~ \n                \\exists {x}_{{\\scriptscriptstyle K}} \\in \n                     \\mathcal{X}_{{\\scriptscriptstyle K}} \\right\\}.\n\\end{align*}\nThen, the following theorem is obtained:\n\n\n{\\bf Theorem 2:}~ \nSuppose that $(\\mathcal{V}_{1}, \\mathcal{V}_{2})$ is a $(C, A, B)$-pair. \nThen, there exist $F \\in \\mathcal{F}(\\mathcal{V}_{2})$, \n$G \\in \\mathcal{G}(\\mathcal{V}_{1})$,\nand $D_{\\scriptscriptstyle K} : \\mathcal{Y} \\rightarrow \\mathcal{U}$ \nsuch that $\\Kernel F_{0} \\supseteq \\mathcal{V}_{1}$ and \n$\\Image G_{0} \\subseteq \\mathcal{V}_{2}$ hold, where\n$F_{0}=F-D_{\\scriptscriptstyle K}C,~G_{0}=G-BD_{\\scriptscriptstyle K}$.\n\n\nMoreover, there exists $\\mathcal{X}_{{\\scriptscriptstyle K}}$ with \n${\\rm dim}\\,\\mathcal{X}_{{\\scriptscriptstyle K}}={\\rm dim}\\,\n\\mathcal{V}_{2} - {\\rm dim}\\,\\mathcal{V}_{1}$, and $A_{{\\scriptscriptstyle E}}$ \nhas an invariant subspace $\\mathcal{V}_{{\\scriptscriptstyle E}} \\subseteq \n\\mathcal{X}_{{\\scriptscriptstyle E}}$ such that $\\mathcal{V}_{1}=\\mathcal{V}_{{\\scriptscriptstyle I}}$ \nand $\\mathcal{V}_{2}=\\mathcal{V}_{{\\scriptscriptstyle P}}$. \nAlso, $(A_{\\scriptscriptstyle K}, B_{\\scriptscriptstyle K}, C_{\\scriptscriptstyle K})$ \nsatisfies  \n\\begin{align}\n\\label{eq_DFC_characterize}\n             &C_{\\scriptscriptstyle K}N=F_{0}, ~~ \\nonumber \\\\\n             &B_{\\scriptscriptstyle K}=-NG_{0}, ~~ \\nonumber \\\\\n             &A_{\\scriptscriptstyle K}N=N(A+BF_{0}+GC),\n\\end{align}\nwhere $N : \\mathcal{V}_{2} \\rightarrow \\mathcal{X}_{{\\scriptscriptstyle K}}$ \nis a linear map satisfying $\\Kernel N=\\mathcal{V}_{1}$.\n\n\nIn fact, under the condition given in Theorem 2, let us define the following \naugmented subspace $\\mathcal{V}_{{\\scriptscriptstyle E}} \\subseteq \\mathcal{X}_{{\\scriptscriptstyle E}}$:\n\\begin{align*}\n       \\mathcal{V}_{{\\scriptscriptstyle E}} :=& \\left\\{\n           \\left[\\begin{array}{c}\n                        x\\\\\n                        Nx\\\\\n           \\end{array}\\right]  \n       \\Bigg|~ x \\in \\mathcal{V}_{2} \\right\\}.\n\\end{align*} \nThen, $\\mathcal{V}_{1}=\\mathcal{V}_{{\\scriptscriptstyle I}}$ and \n$\\mathcal{V}_{2}=\\mathcal{V}_{{\\scriptscriptstyle P}}$ \nhold, and we have \n\\begin{align*}\n     A_{{\\scriptscriptstyle E}}\\left[\\begin{array}{c}\n                   x\\\\\n                   Nx\\\\\n          \\end{array}\\right]  \n     &=\\left[\\begin{array}{cc}\n                   A+BD_{\\scriptscriptstyle K}C & BC_{\\scriptscriptstyle K}\\\\\n                   B_{\\scriptscriptstyle K}C & A_{\\scriptscriptstyle K}\\\\\n          \\end{array}\\right]\n          \\left[\\begin{array}{c}\n                   x\\\\\n                   Nx\\\\\n          \\end{array}\\right]  \\nonumber\\\\\n     &=\\left[\\begin{array}{c}\n                   (A+BF)x\\\\\n                   N(A+BF)x\\\\\n          \\end{array}\\right] \n  \\in \\mathcal{V}_{{\\scriptscriptstyle E}},\n\\end{align*} \nimplying that $\\mathcal{V}_{{\\scriptscriptstyle E}}$ is actually $A_{{\\scriptscriptstyle E}}$-invariant. \nNow suppose that Theorem~1 holds, and let us take the $(C,A,B)$-pair \n$(\\mathcal{V}_{1}, \\mathcal{V}_{2})$ satisfying Eq.~\\eqref{eq_DDP_condition}. \nThen, together with the above result \n($A_{{\\scriptscriptstyle E}}\\mathcal{V}_{{\\scriptscriptstyle E}} \\subseteq \\mathcal{V}_{{\\scriptscriptstyle E}}$), \nwe have $\\Image [E^{\\top}~O]^{\\top} \\subseteq \\mathcal{V}_{{\\scriptscriptstyle E}} \n\\subseteq \\Kernel [H~O]$. \nThis implies that $d(t)$ must be contained in the unobservable subspace \nwith respect to $z(t)$, and thus the disturbance decoupling is realized.\n\n\n\n\n\n\\subsection{Linear quantum systems}\\label{sec:2-2}\n\n\nHere we describe a general linear quantum system composed of $n$ bosonic \nsubsystems. \nThe $j$-th mode can be modeled as a harmonic oscillator with the canonical \nconjugate pairs (or quadratures) $\\hat{q}_{j}$ and $\\hat{p}_{j}$ satisfying the \ncanonical commutation relation (CCR) \n$\\hat{q}_{j}\\hat{p}_{k}-\\hat{p}_{k}\\hat{q}_{j}=i \\delta_{jk}$. \nLet us define the vector of quadratures as \n$\\hat{x}=[\\hat{q}_{1}, \\hat{p}_{1}, \\ldots, \\hat{q}_{n}, \\hat{p}_{n}]^{\\top}$. \nThen, the CCRs are summarized as \n\\begin{align*} \n       &\\hat{x}\\hat{x}^{\\top}-(\\hat{x}\\hat{x}^{\\top})^{\\top} \n                = i \\Sigma_{n}, ~~\\\\\n       &\\Sigma_{n} \n                = {\\rm diag}\\{\\Sigma, \\ldots, \\Sigma\\},~~~\n        \\Sigma=\\left[\\begin{array}{cc}\n                            0 & 1 \\\\\n                          -1 & 0 \\\\\n                       \\end{array}\\right]. \n\\end{align*}\nNote that $\\Sigma_{n}$ is a $2n \\times 2n$ block diagonal matrix. \nThe linear quantum system is an open system coupled to $m$ environment \nfields via the interaction Hamiltonian \n$\\hat{H}_{\\rm int}=i\\sum_{j=1}^m (\\hat{L}_j \\hat{A}_j^*-\\hat{L}_j^* \\hat{A}_j)$, \nwhere $\\hat{A}_j(t)$ is the field annihilation operator satisfying \n$\\hat{A}_j(t)\\hat{A}_k^{*}(t')-\\hat{A}_k^{*}(t')\\hat{A}_j(t)=\\delta_{jk}\\delta(t-t')$. \nAlso $\\hat{L}_j$ is given by $\\hat{L}_j=c_j^\\top\\hat{x}$ with \n$c_j\\in{\\mathbb C}^{2n}$. \nIn addition, the system is driven by the Hamiltonian \n$\\hat{H}=\\hat{x}^\\top R\\hat{x}\/2$ with \n$R=R^\\top \\in \\mathbb{R}^{2n \\times 2n}$. \nThen, the Heisenberg equation of $\\hat{x}$ is given by\n\\begin{align}\n             \\frac{d\\hat{x}(t)}{dt}=A\\hat{x}(t)+\\sum_{j=1}^m B_j\\hat{W}_j(t), \n\\label{eq_LQS_dynamics}\n\\end{align}\nwhere $\\hat{W}_j(t)$ is defined by \n\\[\n        \\hat{W}_j\n        = \\left[\\begin{array}{cc}\n             \\hat{Q}_{j} \\\\\n             \\hat{P}_{j} \\\\\n           \\end{array}\\right]\n       = \\left[\\begin{array}{cc}\n             (\\hat{A}_j+\\hat{A}_j^*)\/\\sqrt{2} \\\\\n             (\\hat{A}_j-\\hat{A}_j^*)\/\\sqrt{2} i \\\\\n           \\end{array}\\right].\n\\]\nThe matrices are given by \n$A=\\Sigma_n(R + \\sum_{j=1}^m C_j^\\top\\Sigma C_j\/2)$ and \n$B_j=\\Sigma_n C_j^\\top\\Sigma$ \nwith $C_j=\\sqrt{2}[\\Re(c_j), \\Im(c_j)]^\\top \\in \\mathbb{R}^{2 \\times 2n}$. \nAlso, the instantaneous change of the field operator $\\hat{W}_j(t)$ via the \nsystem-field coupling is given by \n\\begin{align}\n           \\hat{W}_j^{\\rm out}(t)=C_j\\hat{x}(t)+\\hat{W}_j(t). \n\\label{eq_LQS_output}\n\\end{align}\nSummarizing, the linear quantum system is characterized by the dynamics \n\\eqref{eq_LQS_dynamics} and the output \\eqref{eq_LQS_output}, which are \nexactly of the same form as those in Eq.~\\eqref{eq_linearsystem} ($l=m$ in this case). \nHowever note that the system matrices have to satisfy the above-described \nspecial structure, which is equivalently converted to the following \n{\\it physical realizability condition} \\cite{James2008}: \n\\begin{equation}\n\\label{phys real condition}\n        A\\Sigma_n+\\Sigma_n A^{\\top}\n                 + \\sum_{j=1}^m B_j \\Sigma B_j^{\\top}=O,~\n        B_j=\\Sigma_n C_j^{\\top}\\Sigma. \n\\end{equation}\n\n\n\n\n\\subsection{Weak signal sensing, SQL, and BAE}\\label{sec:2-3}\n\n\nThe opto-mechanical oscillator illustrated in Fig.~\\ref{fig_opticalsensor} is a \nlinear quantum system, which serves as a sensor for a very weak signal. \nLet $\\hat{q}_{1}$ and $\\hat{p}_{1}$ be the oscillator's position and momentum \noperators, and $\\hat{a}_{2}=(\\hat{q}_{2}+i \\hat{p}_{2})\/\\sqrt{2}$ represents \nthe annihilation operator of the cavity mode. \nThe system Hamiltonian is given by \n$\\hat{H}=\\omega_{\\scriptscriptstyle m}(\\hat{q}_1^2+\\hat{p}_1^2)\/2\n-g\\hat{q}_1\\hat{q}_2$; that is, the oscillator's free evolution with resonant \nfrequency $\\omega_{\\scriptscriptstyle m}$ plus the linearized radiation \npressure interaction between the oscillator and the cavity field with coupling \nstrength $g$. \n\\begin{figure}[!t]\n \\begin{center}\n \\includegraphics[width=5.2cm,clip]{Figure\/Fig2_opticalsensor.eps}\n \\end{center} \n \\caption{Opto-mechanical system for weak signal sensing.}\n \\label{fig_opticalsensor}\n\\end{figure} \nThe system couples to an external probe field (thus $m=1$) via the coupling \noperator $\\hat{L}_1=\\sqrt{\\kappa}\\hat{a}_2$, with $\\kappa$ the coupling \nconstant between the cavity and probe fields. \nThe corresponding matrix $R$ and vector $c_1$ are then given by \n\\begin{align*}\n           R=\\left[\\begin{array}{cccc}\n                   \\omega_{\\scriptscriptstyle m} &0&-g&0\\\\\n                   0&\\omega_{\\scriptscriptstyle m} &0&0\\\\\n                  -g&0&0&0\\\\\n                  0&0&0&0\\\\\n                \\end{array}\\right],~~\n           c_{1}=\\sqrt{\\frac{\\kappa}{2}}\n                \\left[\\begin{array}{c}\n                   0 \\\\ 0 \\\\ 1 \\\\ i \\\\\n                \\end{array}\\right].\n\\end{align*}\nThe oscillator is driven by an unknown force $\\hat{f}(t)$ with coupling constant $\\gamma$ ; \nthen the vector of system variables $\\hat{x}=[\\hat{q}_1, \\hat{p}_1, \\hat{q}_2, \\hat{p}_2]^\\top$ satisfies \n\\begin{equation*}\n         \\frac{d\\hat{x}}{dt}\n                   =A\\hat{x}+B_1\\hat{W}_1+b \\hat{f},~~\n          \\hat{W}_{1}^{\\rm out}\n                   =C_1\\hat{x}+\\hat{W}_1,\n\\end{equation*}\nwhere\n\\begin{align}\n           A&=\\left[\\begin{array}{cccc}\n                      0&\\omega_{\\scriptscriptstyle m}&0&0\\\\\n                      -\\omega_{\\scriptscriptstyle m} &0 &g&0\\\\\n                      0&0&-\\kappa\/2 &0\\\\\n                      g&0&0&-\\kappa\/2 \\\\\n                  \\end{array}\\right], \\,\n           b=\\sqrt{\\gamma}\\left[\\begin{array}{c}\n                      0 \\\\\n                      1 \\\\\n                      0 \\\\\n                      0 \\\\\n               \\end{array}\\right], \\nonumber \\\\\n          C_1&=-B_1^{\\top}\n              =\\left[\\begin{array}{cccc}\n                     0 & 0 & \\sqrt{\\kappa} & 0\\\\\n                     0 & 0 & 0 & \\sqrt{\\kappa}\\\\\n                \\end{array}\\right], \\nonumber \\\\\n          \\hat{W}_{1}\n                 &=[\\hat{Q}_{1}, \\hat{P}_{1}]^{\\top}, ~~\n         \\hat{W}_{1}^{\\rm out}\n                  =[\\hat{Q}_{1}^{\\rm out}, \\hat{P}_{1}^{\\rm out}]^{\\top}.\n\\label{eq_matrix_originalplant}\n\\end{align}\nNote that we are in the rotating frame at the frequency of the probe field. \nThese equations indicate that the information about $\\hat{f}$ can be extracted \nby measuring $\\hat{P}_1^{\\rm out}$ by a homodyne detector. \nActually the measurement output in the Laplace domain is given by\n\\begin{align}\n          \\hat{P}_{1}^{\\rm out}(s)\n                =\\Xi_{f}(s) \\hat{f}(s)+\\Xi_{Q}(s) \\hat{Q}_{1}(s)\n                                             +\\Xi_{P}(s) \\hat{P}_{1}(s),\n\\label{eq_measurement_output}\n\\end{align}\nwhere $\\Xi_{f}$, $\\Xi_{Q}$, and $\\Xi_{P}$ are transfer functions given by \n\\begin{align*}\n\\hspace{-0.8cm}\n         \\Xi_{f}(s)&=\\frac{g\\omega_{\\scriptscriptstyle m}\\sqrt{\\gamma\\kappa}}\n                                       {(s^2+\\omega_{\\scriptscriptstyle m}^2)(s+\\kappa\/2)},  ~~\\\\\n         \\Xi_{Q}(s)&=-\\frac{g^2 \\omega_{\\scriptscriptstyle m}\\kappa}\n                                         {(s^2+\\omega_{\\scriptscriptstyle m}^2)(s+\\kappa\/2)^2}, ~~\n         \\Xi_{P}(s)=\\frac{s-\\kappa\/2}{s+\\kappa\/2}. \n\\end{align*}\nThus, $\\hat{P}_1^{\\rm out}$ certainly contains $\\hat{f}$. \nNote however that it is subjected to two noises. \nThe first one, $\\hat{Q}_{1}$, is the back-action noise, which is due to the \ninteraction between the oscillator and the cavity. \nThe second one, $\\hat{P}_{1}$, is the shot noise, which inevitably appears. \nNow, the normalized output is given by\n\\[\n             y_{1}(s)\n                    =\\frac{\\hat{P}_{1}^{\\rm out}(s)}{\\Xi_{f}(s)}\n                    =\\hat{f}(s) + \\frac{\\Xi_{Q}(s)}{\\Xi_{f}(s)}\\hat{Q}_{1}(s)\n                              + \\frac{\\Xi_{P}(s)}{\\Xi_{f}(s)}\\hat{P}_{1}(s),\n\\]\nand the normalized noise power spectral density of $y_{1}$ in the Fourier domain \n$(s=i\\omega)$ is calculated as follows:\n\\begin{align*}\n        &S(\\omega) =\\langle |y_{1}-\\hat{f}|^2 \\rangle\n                          =\\left| \\frac{\\Xi_{Q}}{\\Xi_{f}} \\right| ^2 \n                                 \\langle|\\hat{Q}_1|^2 \\rangle \n                           + \\left| \\frac{\\Xi_{P}}{\\Xi_{f}} \\right| ^2 \n                                 \\langle |\\hat{P}_1|^2 \\rangle  \\\\\n                           &\\geq 2\\sqrt{\\frac{|\\Xi_{Q}|^2|\\Xi_{P}|^2}{|\\Xi_{f}|^4} \n                                 \\langle |\\hat{Q}_1|^2 \\rangle \n                                 \\langle |\\hat{P}_1|^2 \\rangle }\n                          \\geq\\frac{| \\omega^2-\\omega_{\\scriptscriptstyle m}^2 | }\n                                              {\\gamma\\omega_{\\scriptscriptstyle m}}\n                           =S_{\\scriptscriptstyle \\rm SQL}(\\omega).\n\\end{align*}\nThe lower bound is called the SQL. \nNote that the last inequality is due to the Heisenberg uncertainty relation of the normalized noise power, i.e.,\n$\\langle|\\hat{Q}_1|^2 \\rangle \\langle|\\hat{P}_1|^2 \\rangle \\geq 1\/4$. \nHence, the essential reason why SQL appears is that $\\hat{P}_{1}^{\\rm out}$ \ncontains both the back-action noise $\\hat{Q}_1$ and the shot noise $\\hat{P}_1$. \nTherefore, toward the high-precision detection of $\\hat{f}$, we need BAE; \nthat is, the system structure should be modified by some means so that the \nback-action noise is completely evaded in the output signal (note that the shot noise can never be evaded). \nThe condition for BAE can be expressed in terms of the transfer function \nas follows \\cite{Tsang2010, YamamotoCFvsMF2014}; \ni.e., for the modified (controlled) sensor, the transfer function from \nthe back-action noise to the measurement output must satisfy \n\\begin{align}\n              \\Xi_{Q}(s)=0, ~~~\\forall s\n\\label{eq_perfectBAE1}.\n\\end{align}\nEquivalently, $\\hat{P}_{1}^{\\rm out}$ contains only the shot noise $\\hat{P}_{1}$; \nhence, in this case the signal to noise ratio can be further improved by \ninjecting a $\\hat{P}_{1}$-squeezed (meaning $\\langle|\\hat{P}_{1}|^2 \\rangle < 1\/2$) \nprobe field into the system. \n\n\n\n\n\n\\section{Coherent feedback control for back-action evasion}\\label{sec:3}\n\n\n\n\n\\subsection{Coherent and measurement-based feedback control}\\label{sec:3-1}\n\n\n\\begin{figure}[!t]\n\\begin{center}\n\\includegraphics[width=7.7cm,clip]{Figure\/Fig3_MFvsCF.eps}\n\\caption{General configurations of feedback control for a given plant quantum system: \n(a) measurement-based feedback and (b) coherent feedback.}\n\\label{fig_MFvsCF}\n\\end{center}\n\\end{figure}\n\n\nThere are two schemes for controlling a quantum system via feedback. \nThe first one is the {\\it measurement-based feedback} \n\\cite{Belavkin1999, Bouten2009, WisemanBook, KurtBook} illustrated in \nFig.~\\ref{fig_MFvsCF}~(a). \nIn this scheme, we measure the output fields and feed the measurement \nresults back to control the plant system. \nOn the other hand, in the {\\it coherent feedback} scheme \n\\cite{James2008,Wiseman1994, Yanagisawa-2003,Mabuchi2008, Iida2012} \nshown in Fig.~\\ref{fig_MFvsCF}~(b), the feedback loop dose not contain \nany measurement component and the plant system is controlled by another quantum system. \nRecently we find several works comparing the performance of these two schemes \n\\cite{Wiseman1994, NurdinLQG2009, Hamerly2012, Jacobs2014, Devoret2016}. \nIn particular, it was shown in \\cite{YamamotoCFvsMF2014} that there are \nsome control tasks that cannot be achieved by any measurement-based feedback \nbut can be done by a coherent one. \nMore specifically, those tasks are realizing BAE measurement, generating \na quantum non-demolished variable, and generating a decoherence-free \nsubsystem; \nin our case, of course, the first one is crucial. \nHence, here we aim to develop a theory for designing a coherent feedback \ncontroller such that the whole controlled system accomplishes BAE. \n\n\n\n\n\\subsection{Coherent feedback for BAE}\\label{sec:3-2}\n\n\nAs discussed in Section \\ref{sec:2-1}, the geometric control theory for \ndisturbance decoupling problem is formulated for the controlled system \nwith special structure \\eqref{eq_extended_system}; \nin particular, the coefficient matrix of the disturbance $d(t)$ is of the form $[E^\\top, O]^\\top$ \nand that of the state vector in the output $z(t)$ is $[H, O]$. \nHere we consider a class of coherent feedback configuration such that \nthe whole closed-loop system dynamics has this structure, in order for \nthe geometric control theory to be directly applicable. \n\n\n\nFirst, for the plant system given by Eqs.~\\eqref{eq_LQS_dynamics} and \n\\eqref{eq_LQS_output}, we assume that the system couples to all the probe fields  \nin the same way; i.e., \n\\begin{align}\n       B_{j}=B~~~\\forall j. \n\\label{eq_plant_symmetry}\n\\end{align}\nThis immediately leads to $C_{j}=C~\\forall j$. \nNext, as the controller, we take the following special linear quantum system \nwith $(m-1)$ input-output fields: \n\\begin{align}\n\\label{eq_dynamical_controller} \n          \\frac{d\\hat{x}_{\\scriptscriptstyle K}}{dt}\n                 &=A_{\\scriptscriptstyle K} \\hat{x}_{\\scriptscriptstyle K}\n                     +\\sum_{j=1}^{m-1} B_{{\\scriptscriptstyle K}} \\hat{w}_{j},  \\nonumber\\\\\n         \\hat{w}_{j}^{\\rm out}\n                 &=C_{{\\scriptscriptstyle K}} \\hat{x}_{\\scriptscriptstyle K}\n                               +\\hat{w}_{j}~~~(j=1,\\,2,\\, \\ldots, m-1),\n\\end{align}\nwhere the matrices $(A_{\\scriptscriptstyle K}, B_{\\scriptscriptstyle K}, \nC_{\\scriptscriptstyle K})$ satisfy the physical realizability condition \n\\eqref{phys real condition}. \nNote that, corresponding to the plant structure, we assumed that the controller \ncouples to all the fields in the same way, specified by $C_{\\scriptscriptstyle K}$. \nHere we emphasize that the number of channels, $m$, should be as small as \npossible from a viewpoint of implementation; \nhence in this paper let us consider the case $m=3$. \nNow, we consider the coherent feedback connection illustrated in \nFig.~\\ref{fig_3I3O_general}, i.e., \n\\begin{align*}\n               &\\hat{w}_{1}=S_{1}\\hat{W}_{1}^{\\rm out},~~ \n                   \\hat{w}_{2}=S_{2}\\hat{W}_{2}^{\\rm out}, ~~ \\\\\n                &\\hat{W}_{2}=T_1\\hat{w}_{1}^{\\rm out},~~ \n                   \\hat{W}_{3}=T_{2}\\hat{w}_{2}^{\\rm out},\n\\end{align*}\n\\begin{figure}[!t]\n\\begin{center}\n\\includegraphics[width=5cm,clip]{Figure\/Fig4_3I3O_general.eps}\n\\caption{Coherent feedback control of the 3 input-output plant system via the 2 input-output controller.}\n\\label{fig_3I3O_general}\n\\end{center}\n\\end{figure}\nwhere $S_{j}$ and $T_{j}$ are $2 \\times 2$ unitary matrices representing \nthe scattering process of the fields; \nrecall that the scattering process $\\hat{A}^{\\rm out}=e^{i\\theta}\\hat{A}$ \nwith $\\theta \\in \\mathbb{R}$ the phase shift can be represented in the \nquadrature form as \n\\begin{equation*}\n           \\left[\\begin{array}{cc}\n             \\hat{Q}^{\\rm out} \\\\\n             \\hat{P}^{\\rm out} \\\\\n           \\end{array}\\right]\n           = S(\\theta)\n              \\left[\\begin{array}{cc}\n                 \\hat{Q} \\\\\n                 \\hat{P} \\\\\n              \\end{array}\\right]\n            =\\left[\\begin{array}{cc}\n                   \\cos \\theta &-\\sin \\theta \\\\\n                    \\sin \\theta &\\cos \\theta \\\\\n               \\end{array}\\right]\n               \\left[\\begin{array}{cc}\n                 \\hat{Q} \\\\\n                 \\hat{P} \\\\\n              \\end{array}\\right].\n\\label{eq_phaseshifter}\n\\end{equation*}\n\n\n\nCombining the above equations, we find that the whole closed-loop system \nwith the augmented variable \n$\\hat{x}_{{\\scriptscriptstyle E}}=[\\hat{x}^{\\top},\\hat{x}_{\\scriptscriptstyle K}^{\\top}]^{\\top}$ \nis given by\n\\begin{align}\n\\label{eq_3I3Oclosed2}\n             \\frac{d\\hat{x}_{{\\scriptscriptstyle E}}}{dt}\n                                  &=A_{{\\scriptscriptstyle E}}\\hat{x}_{{\\scriptscriptstyle E}}\n                                              +B_{{\\scriptscriptstyle E}}\\hat{W}_{1}+b_{{\\scriptscriptstyle E}}\\hat{f}, \\nonumber\\\\\n             \\hat{W}_{3}^{\\rm out}\n                                  &=C_{{\\scriptscriptstyle E}}\\hat{x}_{{\\scriptscriptstyle E}}\n                                              +D_{{\\scriptscriptstyle E}}\\hat{W}_{1},\n\\end{align}\nwhere\n\\begin{align*}\nA_{{\\scriptscriptstyle E}}\n&=\\left[\\begin{array}{c}\nA+B\\{ T_{1}S_{1}+T_{2}S_{2}(T_{1}S_{1}+I_{2})\\} C  \\\\ [1.0ex]\nB_{\\scriptscriptstyle K}\\{ (I_{2}+S_{2}T_{1})S_{1}+S_{2}\\} C \n\\end{array}\\right. \\\\\n&~~~~~~~~~~~~~~~~~~~~~~~~~~~\\left. \\begin{array}{c}\n B\\{ T_{1}+T_{2}(I_{2}+S_{2}T_{1})\\} C_{\\scriptscriptstyle K} \\\\ [1.0ex]\n A_{\\scriptscriptstyle K}+B_{\\scriptscriptstyle K}S_{2}T_{1}C_{\\scriptscriptstyle K}\n\\end{array}\\right]. \\\\\nB_{{\\scriptscriptstyle E}}\n&=\\left[\\begin{array}{c}\n                            B(I_{2}+T_{1}S_{1}+T_{2}S_{2}T_{1}S_{1}) \\\\ [1.0ex]\n                            B_{{\\scriptscriptstyle K}}(I_{2}+S_{2}T_{1})S_{1} \\\\\n                        \\end{array}\\right] \\\\\nC_{{\\scriptscriptstyle E}}\n&=\\left[\\begin{array}{cc}\n                           (T_{2}S_{2}T_{1}S_{1}+T_{2}S_{2}+I_{2})C ~\n                           & ~T_{2}(S_{2}T_{1}+I_{2})C_{\\scriptscriptstyle K}\\\\\n                        \\end{array}\\right] \\\\\nD_{{\\scriptscriptstyle E}}&=T_{2}S_{2}T_{1}S_{1}, ~~\nb_{{\\scriptscriptstyle E}}=\\left[\\begin{array}{cc}\n                           b^{\\top} & O\\\\\n                       \\end{array}\\right]^{\\top}.\n\\end{align*}\nTherefore, the desired system structure of the form \\eqref{eq_extended_system} \nis realized if we take \n\\begin{align}\n       S_{2}T_{1}=-I_{2}.\n\\label{eq_special_structure1}\n\\end{align}\nIn addition, it is required that the back-action noise $\\hat{Q}_{1}$ \ndose not appear directly in $\\hat{P}_{3}^{\\rm out}$, which can be realized by \ntaking \n\\begin{align}\n        D_{{\\scriptscriptstyle E}}=-T_{2}S_{1}=\\pm I_{2}.\n\\label{eq_special_structure2}\n\\end{align}\nHere we set $S_j$ and $T_j$ to be the $\\pi\/2$-phase shifter \n(see Fig.~\\ref{fig_3I3OBAE}) to satisfy the above conditions \\eqref{eq_special_structure1} and \n\\eqref{eq_special_structure2};\n\\begin{align}\n\\label{eq_phaseshifter_matching}\n          S_{j}=T_{j}=S\n          =\\left[\\begin{array}{cc}\n             0 & -1 \\\\\n             1 & 0  \\\\\n           \\end{array}\\right] ~~(j=1,2). \n\\end{align}\nAs a consequence, we end up with \n\\begin{align}\n     &A_{{\\scriptscriptstyle E}}=\\left[\\begin{array}{cc}\n                       A-BC & BSC_{\\scriptscriptstyle K} \\\\ \n                       B_{\\scriptscriptstyle K}SC & \n                       A_{\\scriptscriptstyle K}-B_{\\scriptscriptstyle K}C_{\\scriptscriptstyle K} \n                    \\end{array}\\right], ~~\n       B_{{\\scriptscriptstyle E}}=\\left[\\begin{array}{c}\n                       B\\\\\n                       O \\\\\n                    \\end{array}\\right], \n\\nonumber \\\\\n     &\n      C_{{\\scriptscriptstyle E}}=\\left[\\begin{array}{cc}\n                    C & O \\\\\n                \\end{array}\\right], ~~\n      D_{{\\scriptscriptstyle E}}=I_{2}, ~~\n      b_{{\\scriptscriptstyle E}}=\\left[\\begin{array}{cc}\n                    b^{\\top} & O \\\\\n                \\end{array}\\right]^{\\top}.\n\\label{extended_matrices}\n\\end{align}\nThis is certainly of the form \\eqref{eq_extended_system} with \n$D_{\\scriptscriptstyle K}=-I_2$. \nHence, we can now directly apply the geometric control theory to design \na coherent feedback controller achieving BAE; \nthat is, our aim is to find $(A_{\\scriptscriptstyle K}, B_{\\scriptscriptstyle K}, \nC_{\\scriptscriptstyle K})$ such that, for the closed-loop system \n\\eqref{eq_3I3Oclosed2}, the back-action noise $\\hat{Q}_{1}$ (the first element \nof $\\hat W_1$) does not appear in the measurement output $\\hat P_3^{\\rm out}$ \n(the second element of $\\hat W_3^{\\rm out}$). \nNote that those matrices must satisfy the physical \nrealizability condition \\eqref{phys real condition}, and thus they cannot be \nfreely chosen. \nWe need to take into account this additional constraint when applying \nthe geometric control theory to determine the controller matrices.\n\n\n\n\n\n\\subsection{Coherent feedback realization of BAE in the opto-mechanical system}\\label{sec:3-3}\n\n\nHere we apply the coherent feedback scheme elaborated in Section~\\ref{sec:3-2} to \nthe opto-mechanical system studied in Section~\\ref{sec:2-3}. \nThe goal is, as mentioned before, to determine the controller matrices \n$(A_{\\scriptscriptstyle K}, B_{\\scriptscriptstyle K}, C_{\\scriptscriptstyle K})$ \nsuch that the closed-loop system achieves BAE. \nHere, we provide a step-by-step procedure to solve this problem; \nthe relationships of the class of controllers determined in each step is depicted \nin Fig.~\\ref{fig_controllerconstraint}. \n\n\\begin{figure}[!t]\n\\begin{center}\n\\includegraphics[width=7cm,clip]{Figure\/Fig5_3I3OBAE.eps}\n\\caption{\nCoherent feedback controlled system composed of the opto-mechanical oscillator, for realizing BAE. \nThe triangle represents the $\\pi\/2$-phase shifter corresponding to Eq.~\\eqref{eq_phaseshifter_matching}. \n}\n\\label{fig_3I3OBAE}\n\\end{center}\n\\end{figure}\n\n{\\bf (i)} \nFirst, to apply the geometric control theory developed above, we need to modify \nthe plant system so that it is a 3 input-output linear quantum system; \nhere we consider the plant composed of a mechanical oscillator and a 3-ports optical cavity, \nshown in Fig.~\\ref{fig_3I3OBAE}. \nAs assumed before, those ports have the same coupling constant $\\kappa$. \nIn this case the matrix $A$ given in Eq.~\\eqref{eq_matrix_originalplant} is \nreplaced by \n\\begin{align}\n           A=\\left[\\begin{array}{cccc}\n                     0&\\omega_{\\scriptscriptstyle m} &0&0\\\\\n                     -\\omega_{\\scriptscriptstyle m}& 0&g &0\\\\\n                     0&0&-3\\kappa\/2 &0\\\\\n                     g &0&0&-3\\kappa\/2 \\\\\n                \\end{array}\\right] .\n\\nonumber \n\\end{align} \nNow we focus only on the back-action noise $\\hat{Q}_{1}$ and the measurement output \n$\\hat{P}_{3}^{\\rm out}$; hence the closed-loop system \\eqref{eq_3I3Oclosed2} and\n\\eqref{extended_matrices}, which ignores the shot noise term in the dynamical \nequation, is given by \n\\begin{align*}\n\\hspace{-0.8cm}\n          \\frac{d\\hat{x}_{{\\scriptscriptstyle E}}}{dt}\n              &=\\left[\\begin{array}{cc}\n                       A-BC & BSC_{\\scriptscriptstyle K} \\\\ \n                       B_{\\scriptscriptstyle K}SC & \n                        A_{\\scriptscriptstyle K}\n                              -B_{\\scriptscriptstyle K}C_{\\scriptscriptstyle K} \n                    \\end{array}\\right]\n                      \\hat{x}_{{\\scriptscriptstyle E}}  \\\\\n&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + \\left[\\begin{array}{c}\n                        E \\\\\n                        O \\\\\n                    \\end{array}\\right] \\hat{Q}_{1}\n                + \\left[\\begin{array}{c}\n                        b\\\\\n                        O\\\\\n                   \\end{array}\\right] \\hat{f},  \\\\\n            \\hat{P}_{3}^{\\rm out}\n                 &=\\left[\\begin{array}{cc}\n                         H & O \\\\\n                      \\end{array}\\right] \\hat{x}_{{\\scriptscriptstyle E}}\n                    +\\hat{P}_{1}, \n\\end{align*}\nwhere $B=B_1$, $C=C_1$, and $b$ are given in \nEq.~\\eqref{eq_matrix_originalplant}, and \n\\begin{align*}\n      E=-\\sqrt{\\kappa}\\left[\\begin{array}{cccc}\n                        0&0&1&0 \\\\\n                    \\end{array}\\right]^{\\top},~~~ \n      H=\\sqrt{\\kappa}\\left[\\begin{array}{cccc}\n                        0&0&0&1 \\\\\n                    \\end{array}\\right].\n\\end{align*}\nThis system is certainly of the form \\eqref{eq_extended_system}, where now $D_{\\scriptscriptstyle K}=-I_2$.\n\n\n\n{\\bf (ii)} \nIn the next step we apply Theorem~1 to check if there exists a feedback \ncontroller such that the above closed-loop system achieves BAE; \nrecall that the necessary and sufficient condition is Eq.~\\eqref{eq_DDP_condition}, \ni.e., $\\mathcal{E} \\subseteq \\mathcal{V}_{1} \\subseteq \\mathcal{V}_{2} \n\\subseteq \\mathcal{H}$, where now \n\\begin{align*}\n       \\mathcal{E}&=\\Image E={\\rm span} \\left\\{\n                    \\left[\\begin{array}{c}\n                         0\\\\\n                         0\\\\\n                         1\\\\\n                         0\\\\\n                    \\end{array}\\right] \\right\\}, \\\\\n       \\mathcal{H}&=\\Kernel H={\\rm span} \\left\\{\n                    \\left[\\begin{array}{c}\n                         1\\\\\n                         0\\\\\n                         0\\\\\n                         0\\\\\n                    \\end{array}\\right] ,\n                    \\left[\\begin{array}{c}\n                         0\\\\\n                         1\\\\\n                         0\\\\\n                         0\\\\\n                    \\end{array}\\right] ,\n                    \\left[\\begin{array}{c}\n                         0\\\\\n                         0\\\\\n                         1\\\\\n                         0\\\\\n                    \\end{array}\\right] \\right\\}. \n\\end{align*}\nTo check if this solvability condition is satisfied, we use Corollary 1; \nfrom\n$\\mathcal{E} \\cap \\mathcal{C}=\\Image E \\cap \\Kernel C =\\phi$ and \n$\\mathcal{H} \\oplus \\mathcal{B} = \\Kernel H \\oplus \\Image B =\\mathbb{R}^{4}$,\nthe algorithms given in Appendix~A yield \n\\begin{align}\n       \\mathcal{V}_{*}{\\scriptscriptstyle (\\mathcal{C}, \\mathcal{E})}\n                    =\\mathcal{E},~~~\n       \\mathcal{V}^{*}{\\scriptscriptstyle (\\mathcal{B}, \\mathcal{H})}\n                    =\\mathcal{H}, \n\\label{eq_CABpairs}\n\\end{align}\nimplying that the condition in Corollary 1, i.e., \n$\\mathcal{V}_{*}{\\scriptscriptstyle (\\mathcal{C}, \\mathcal{E})} \\subseteq \n\\mathcal{V}^{*}{\\scriptscriptstyle (\\mathcal{B}, \\mathcal{H})}$, is satisfied. \nThus, we now see that the BAE problem is solvable, as long as there is no \nconstraint on the controller parameters. \n\n\n\nNext we aim to determine the controller matrices \n$(A_{\\scriptscriptstyle K}, B_{\\scriptscriptstyle K}, C_{\\scriptscriptstyle K})$, \nusing Theorem~2. First we set \n$\\mathcal{V}_{1}=\\mathcal{V}_{*}{\\scriptscriptstyle (\\mathcal{C}, \\mathcal{E})} = \\mathcal{E}$ and \n$\\mathcal{V}_{2}=\\mathcal{V}^{*}{\\scriptscriptstyle (\\mathcal{B}, \\mathcal{H})}= \\mathcal{H}$;\nnote that $(\\mathcal{V}_{1}, \\mathcal{V}_{2})$ is a $(C, A, B)$-pair.\nThen, from Theorem 2, there exists a feedback controller with dimension\n${\\rm dim}\\,\\mathcal{X}_{\\scriptscriptstyle K}={\\rm dim}\\,\\mathcal{V}_{2}-{\\rm dim}\\,\\mathcal{V}_{1}=2$. \nMoreover, noting again that $D_{\\scriptscriptstyle K}=-I_2$, there exist matrices \n$F \\in \\mathcal{F}(\\mathcal{V}_{2})$, $G \\in \\mathcal{G}(\\mathcal{V}_{1})$, \nand $N$ \nsuch that\n\\begin{align*}\n     \\Kernel F_{0}&=\\Kernel (F+C) \\supseteq \\mathcal{V}_{1}, \\\\\n     \\Image G_{0}&=\\Image (G+B) \\subseteq \\mathcal{V}_{2}, ~~\n     \\Kernel N=\\mathcal{V}_{1}. \n\\end{align*}\nThese conditions lead to\n\\begin{align*}\n\\hspace{-0.7cm}\n              F&=\\left[\\begin{array}{cccc}\n                        f_{\\scriptscriptstyle 11} & f_{\\scriptscriptstyle 12} \n                            & -\\sqrt{\\kappa} & f_{\\scriptscriptstyle 14} \\\\\n                        \\frac{g}{\\sqrt{\\kappa}}&0&0&f_{\\scriptscriptstyle 24} \\\\\n                     \\end{array}\\right], ~~\n              G=\\left[\\begin{array}{cc}\n                        0&g_{\\scriptscriptstyle 12}\\\\\n                        -\\frac{g}{\\sqrt{\\kappa}}&g_{\\scriptscriptstyle 22}\\\\\n                        g_{\\scriptscriptstyle 31}&g_{\\scriptscriptstyle 32}\\\\\n                        0&\\sqrt{\\kappa}\\\\\n                     \\end{array}\\right],  \\\\\n              N&=\\left[\\begin{array}{cccc}\n                        n_{\\scriptscriptstyle 11} & n_{\\scriptscriptstyle 12} \n                             & 0 & n_{\\scriptscriptstyle 14} \\\\\n                        n_{\\scriptscriptstyle 21} & n_{\\scriptscriptstyle 22}\n                             & 0 & n_{\\scriptscriptstyle 24}\\\\\n                     \\end{array}\\right], \n\\end{align*}\nwhere $f_{ij}, g_{ij}$, and $ n_{ij}$  are free parameters. Then the controller matrices \n$(A_{\\scriptscriptstyle K}, B_{\\scriptscriptstyle K}, C_{\\scriptscriptstyle K})$ \ncan be identified by Eq.~\\eqref{eq_DFC_characterize} with the above matrices $(F, G, N)$;\nspecifically, by substituting $C_{\\scriptscriptstyle K} \\to SC_{\\scriptscriptstyle K}$, \n$B_{\\scriptscriptstyle K} \\to B_{\\scriptscriptstyle K}S$, and \n$A_{\\scriptscriptstyle K} \\to A_{\\scriptscriptstyle K}-B_{\\scriptscriptstyle K}C_{\\scriptscriptstyle K}$\nin Eq.~\\eqref{eq_DFC_characterize}, we have \n\\begin{align*}\n      &SC_{\\scriptscriptstyle K}N=F+C,\\\\\n      &B_{\\scriptscriptstyle K}S=-N(G+B), \\\\\n      &(A_{\\scriptscriptstyle K}\n                        -B_{\\scriptscriptstyle K}C_{\\scriptscriptstyle K})N \n                                 = N(A+BF_{0}+GC),\n\\end{align*}\nwhich yield \n\\begin{align}\n\\label{controller matrices example 1}\n     &A_{\\scriptscriptstyle K}=N(A+BF_{0}+GC+G_{0}F_{0})N^{+}, \\nonumber \\\\\n     &B_{\\scriptscriptstyle K}=-NG_{0}\\Sigma, \\nonumber \\\\\n     &C_{\\scriptscriptstyle K}=\\Sigma F_{0}N^{+}, \n\\end{align}\nwhere $N^{+}$ is the right inverse to $N$, i.e., $NN^{+}=I_{2}$.\n\n\n{\\bf (iii)}\nNote again that the controller \\eqref{eq_dynamical_controller} has to satisfy \nthe physical realizability condition \\eqref{phys real condition}, which is now \n$A_{\\scriptscriptstyle K}\\Sigma+\\Sigma A_{\\scriptscriptstyle K}^{\\top}\n+2B_{\\scriptscriptstyle K}\\Sigma B_{\\scriptscriptstyle K}^{\\top}=O$ and \n$B_{\\scriptscriptstyle K}=\\Sigma C_{\\scriptscriptstyle K}^{\\top}\\Sigma$. \nThese constraints are represented in terms of the parameters as follows: \\vspace{-1mm}\n\\begin{align}\n\\label{controller parameter constraint}\n&f_{\\scriptscriptstyle 12}=-g_{\\scriptscriptstyle 12},~~\nf_{\\scriptscriptstyle 11}=g_{\\scriptscriptstyle 22},~~\nn_{\\scriptscriptstyle 11}n_{\\scriptscriptstyle 22}-n_{\\scriptscriptstyle 12}n_{\\scriptscriptstyle 21}=-1,~ \n\\nonumber \\\\\n&f_{\\scriptscriptstyle 12}n_{\\scriptscriptstyle 1}=f_{\\scriptscriptstyle 11}n_{\\scriptscriptstyle 2}-f_{\\scriptscriptstyle 14},~~\nf_{\\scriptscriptstyle 24}+\\sqrt{\\kappa}=\\frac{g}{\\sqrt{\\kappa}}n_{\\scriptscriptstyle 2}, \n\\nonumber \\\\\n& \\hspace{-0.15cm}\n\\left( \\frac{3}{2}\\kappa+\\sqrt{\\kappa}f_{\\scriptscriptstyle 24} \\right) n_{\\scriptscriptstyle 1}+\\omega_{\\scriptscriptstyle m}n_{\\scriptscriptstyle 2}\n=-\\sqrt{\\kappa}f_{\\scriptscriptstyle 11}, \n\\nonumber \\\\\n&\\omega_{\\scriptscriptstyle m}n_{\\scriptscriptstyle 1} - \\left ( \\frac{3}{2}\\kappa + \\sqrt{\\kappa}f_{\\scriptscriptstyle 24} \\right )n_{\\scriptscriptstyle 2}\n=\\sqrt{\\kappa}f_{\\scriptscriptstyle 12}, \n\\end{align}\nwhere \n$n_{\\scriptscriptstyle 1}=n_{\\scriptscriptstyle 11}n_{\\scriptscriptstyle 24}-\nn_{\\scriptscriptstyle 14}n_{\\scriptscriptstyle 21}$ and \n$n_{\\scriptscriptstyle 2}=n_{\\scriptscriptstyle 12}n_{\\scriptscriptstyle 24}-\nn_{\\scriptscriptstyle 14}n_{\\scriptscriptstyle 22}$. \nThis is one of our main results; \nthe linear controller \\eqref{eq_dynamical_controller} achieving BAE \nfor the opto-mechanical oscillator can be fully parametrized by \nEq.~\\eqref{controller matrices example 1} satisfying the condition \n\\eqref{controller parameter constraint}. \nWe emphasize that this full parametrization of the controller can be \nobtained thanks to the general problem formulation based on the geometric \ncontrol theory. \n\n\n\\begin{figure}[!t]\n\\begin{center}\n\\includegraphics[width=6.5cm,clip]{Figure\/Fig6_controllerconstraint.eps}\n\\caption{\nThe set of controllers satisfying the condition in each step. For the controller to be a quantum system, \nit must be included in the set (iii). In the set (iv), all the controllers are equivalent up to the phase shift. \n}\n\\label{fig_controllerconstraint}\n\\end{center}\n\\end{figure}\n{\\bf (iv)}\nIn practice, of course, we need to determine a concrete set of parameters to construct the controller. \nEspecially here let us consider a {\\it passive system}; \nthis is a static quantum system such as an empty optical cavity. \nThe main reason for choosing a passive system rather than a non-passive \n(or {\\it active}) one such as an optical parametric oscillator is that, due to the \nexternal pumping energy, the latter could become fragile and also its physical \nimplementation must be more involved compared to a passive system \\cite{Walls2008}. \nNow the condition for the system \n$(A_{\\scriptscriptstyle K}, \\,B_{\\scriptscriptstyle K}, \\,C_{\\scriptscriptstyle K})$ \nto be passive is given by \n$\\Sigma A_{\\scriptscriptstyle K}\\Sigma=-A_{\\scriptscriptstyle K}$ and \n$\\Sigma B_{\\scriptscriptstyle K}\\Sigma=-B_{\\scriptscriptstyle K}$; \nthe general result of this fact is given in Theorem~3 in Appendix~B. \nFrom these conditions, the system parameters are imposed to satisfy, in addition \nto Eq.~\\eqref{controller parameter constraint}, the following equalities: \n\\begin{align}\n\\label{passive para cond}\n    f_{\\scriptscriptstyle 12}=\\frac{g}{\\sqrt{\\kappa}}, ~~\n    f_{\\scriptscriptstyle 11}=0, ~~ \n    n_{\\scriptscriptstyle 11}=-n_{\\scriptscriptstyle 22}, ~~ \n    n_{\\scriptscriptstyle 12}=n_{\\scriptscriptstyle 21}.\n\\end{align}\nThere is still some freedom in determining $n_{ij}$, which however corresponds \nto simply the phase shift at the input-output ports of the controller, as indicated \nfrom Eq.~\\eqref{controller matrices example 1}. \nThus, the passive controller achieving BAE in this example is unique up to the \nphase shift. \nHere particularly we chose $n_{\\scriptscriptstyle 11}=1$ and \n$n_{\\scriptscriptstyle 12}=0$. \nThen the controller matrices \\eqref{controller matrices example 1} satisfying \nEqs.~\\eqref{controller parameter constraint} and \\eqref{passive para cond} \nare determined as \n\\begin{align*}\n        A_{\\scriptscriptstyle K}\n                    =\\left[\\begin{array}{cc}\n                         -\\frac{~g^2}{\\kappa} &-\\omega_{\\scriptscriptstyle m} \\\\\n                         \\omega_{\\scriptscriptstyle m} & -\\frac{~g^2}{\\kappa}  \\\\\n                      \\end{array}\\right],~~\n        C_{\\scriptscriptstyle K}=-B_{\\scriptscriptstyle K}^{\\top}\n                    =\\left[\\begin{array}{cc}\n                         \\frac{g}{\\sqrt{\\kappa}} & 0 \\\\\n                         0 & \\frac{g}{\\sqrt{\\kappa}} \\\\\n                      \\end{array}\\right]. \n\\end{align*}\nAs illustrated in Fig.~\\ref{fig_3I3OBAE}, the controller specified by these \nmatrices can be realized as a single-mode, 2-inputs and 2-outputs optical cavity \nwith decay rate $g^2\/\\kappa$ and detuning $-\\omega_{\\scriptscriptstyle m}$. \nIn other words, if we take the cavity with the following Hamiltonian and \nthe coupling operator ($\\hat{a}_{3}=(\\hat{q}_{3}+i \\hat{p}_{3})\/\\sqrt{2}$ is the \ncavity mode) \n\\begin{align}\n               \\hat{H}_{\\scriptscriptstyle K}\n                             &=\\Delta \\hat{a}^{\\ast}_{3}\\hat{a}_{3}\n                             =\\frac{\\Delta}{2}(\\hat{q}_{3}^2+\\hat{p}_{3}^2),  \\nonumber \\\\\n               \\hat{L}_{\\scriptscriptstyle K}\n                             &=\\sqrt{\\kappa_{\\scriptscriptstyle K}}\\hat{a}_{3}\n                             =\\sqrt{\\frac{\\kappa_{\\scriptscriptstyle K}}{2}}\n                                        (\\hat{q}_{3}+i \\hat{p}_{3}), \n\\label{eq_controller_H}\n\\end{align}\nthen to satisfy the BAE condition the controller parameters \n$(\\Delta, \\kappa_{\\scriptscriptstyle K})$ must satisfy \n\\begin{equation}\n\\label{perfect BAE condition example}\n                 \\Delta=-\\omega_{\\scriptscriptstyle m}, ~~~\n                 \\kappa_{\\scriptscriptstyle K}=g^2\/\\kappa. \n\\end{equation}\nSummarizing, the above-designed sensing system composed of the opto-mechanical \noscillator (plant) and the optical cavity (controller), which are combined via coherent \nfeedback, satisfies the BAE condition. \nHence, it can work as a high-precision detector of the force $\\hat{f}$ below the SQL, \nparticularly when the $\\hat P_1$-squeezed probe input field is used; \nthis fact will be demonstrated in Section~\\ref{sec:5}. \n\n\n\n\n\n\\section{Direct interaction scheme}\\label{sec:4}\n\n\n\\begin{figure}[!t]\n\\begin{center}\n\\includegraphics[width=8cm,clip]{Figure\/Fig7_directinteraction.eps}\n\\caption{(a) General configuration of direct interaction scheme. \n(b) Physical implementation of the passive direct interaction controller for the \nopto-mechanical oscillator.}\n\\label{fig_Mankei_scheme}\n\\end{center}\n\\end{figure}\n\n\nIn this section, we study another control scheme for achieving BAE. \nAs illustrated in Fig. \\ref{fig_Mankei_scheme}~(a), the controller in this case \nis directly connected to the plant, not through a coherent feedback; \nhence this scheme is called the {\\it direct interaction}. \nThe controller is characterized by the following two Hamiltonians: \n\\begin{align} \n\\label{eq_direct_intHamiltonian}\n          \\hat{H}_{\\scriptscriptstyle K}\n                 =\\frac{1}{2}\\hat{x}_{\\scriptscriptstyle K}^{\\top}\n                      R_{\\scriptscriptstyle K}\\hat{x}_{\\scriptscriptstyle K}, ~\n          \\hat{H}_{\\rm int}\n                 =\\frac{1}{2}(\\hat{x}^{\\top}R_1\\hat{x}_{\\scriptscriptstyle K}\n                            +\\hat{x}_{\\scriptscriptstyle K}^{\\top}R_2\\hat{x}),\n\\end{align}\nwhere $\\hat{x}_{\\scriptscriptstyle K}=[\\hat{q}^{\\scriptscriptstyle \\prime}_{1}, \n\\hat{p}^{\\scriptscriptstyle \\prime}_{1}, \\ldots , \n\\hat{q}^{\\scriptscriptstyle \\prime}_{n_{\\scriptscriptstyle k}}, \n\\hat{p}^{\\scriptscriptstyle \\prime}_{n_{\\scriptscriptstyle k}}]^{\\top}$ is the \nvector of controller variables with $n_{\\scriptscriptstyle k}$ the number of \nmodes of the controller. \n$\\hat{H}_{\\scriptscriptstyle K}$ is the controller's self Hamiltonian with \n$R_{\\scriptscriptstyle K}\\in \n\\mathbb{R}^{2n_{\\scriptscriptstyle k} \\times 2n_{\\scriptscriptstyle k}}$. \nAlso $\\hat{H}_{\\rm int}$ with \n$R_1 \\in \\mathbb{R}^{2n \\times 2n_{\\scriptscriptstyle k}}$, \n$R_2 \\in \\mathbb{R}^{2n_{\\scriptscriptstyle k} \\times 2n}$ represents the \ncoupling between the plant and the controller. \nNote that, for the Hamiltonians $\\hat{H}_{\\scriptscriptstyle K}$ and \n$\\hat{H}_{\\rm int}$ to be Hermitian, the matrices must satisfy \n$R_{\\scriptscriptstyle K}=R_{\\scriptscriptstyle K}^\\top$ and $R_1^\\top=R_2$; \nthese are the physical realizability conditions in the scenario of direct \ninteraction. \nIn particular, here we consider a plant system interacting with a single probe \nfield $\\hat W_1$, with coupling matrices $B_1=B$ and $C_1=C$. \nThen, the whole dynamics of the augmented system with variable \n$\\hat{x}_{{\\scriptscriptstyle E}}=[\\hat{x}^{\\top}, \\hat{x}_{\\scriptscriptstyle K}^{\\top}]^{\\top}$ \nis given by\n\\begin{align}\n\\label{direct whole system}\n             \\frac{d\\hat{x}_{{\\scriptscriptstyle E}}}{dt}\n                       &=A_{{\\scriptscriptstyle E}}\\hat{x}_{{\\scriptscriptstyle E}}\n                                            +B_{{\\scriptscriptstyle E}}\\hat{W}_{1}+b_{{\\scriptscriptstyle E}}\\hat{f},  \\nonumber \\\\\n             \\hat{W}_{1}^{\\rm out}&=C_{{\\scriptscriptstyle E}}\\hat{x}_{{\\scriptscriptstyle E}}+\\hat{W}_{1} ,\n\\end{align}\nwhere\n\\begin{align}\n\\label{eq_direct_matrixAe}\n       A_{{\\scriptscriptstyle E}}\n          &=\\left[\\begin{array}{cc}\n                 ~A ~~& \\Sigma_{n}R_1~~ \\\\\n                 \\Sigma_{n_{\\scriptscriptstyle k}}R_2 \n                              & \\Sigma_{n_{\\scriptscriptstyle k}}R_{\\scriptscriptstyle K}\\\\\n            \\end{array}\\right], ~~\n      B_{{\\scriptscriptstyle E}}=\\left[\\begin{array}{c}\n                       B\\\\\n                       O \\\\\n                    \\end{array}\\right],  \\nonumber  \\\\\n      C_{{\\scriptscriptstyle E}}&=\\left[\\begin{array}{cc}\n                       C & O  \\\\\n                \\end{array}\\right], ~~\n       b_{{\\scriptscriptstyle E}}=\\left[\\begin{array}{cc}\n                        b^{\\top} & O\\\\ \n                    \\end{array}\\right] ^{\\top}. \n\\end{align}\nNote that $B_{{\\scriptscriptstyle E}}$, $C_{{\\scriptscriptstyle E}}$, and $b_{{\\scriptscriptstyle E}}$ \nare the same matrices as those in Eq.~\\eqref{extended_matrices}. \nAlso, comparing the matrices \\eqref{eq_extended_matrixAe} and \n\\eqref{eq_direct_matrixAe}, we have that $D_{\\scriptscriptstyle K}=O$, which \nthus leads to $F=F_{0}$ and $G=G_{0}$ in Theorem~2. \nNow, again for the opto-mechanical system illustrated in Fig.~\\ref{fig_opticalsensor}, \nlet us aim to design the direct interaction controller, so that the whole system \n\\eqref{direct whole system} achieves BAE; \nthat is, the problem is to determine the matrices \n$(R_{\\scriptscriptstyle K}, R_1, R_2)$ so that the back-action noise $\\hat Q_1$ \ndoes not appear in the measurement output $\\hat P_1^{\\rm out}$. \nFor this purpose, we go through the same procedure as that taken in Section~\\ref{sec:3-3}.\n\n{\\bf (i)}\nBecause of the structure of the matrices $B_{{\\scriptscriptstyle E}}$ and $C_{{\\scriptscriptstyle E}}$, \nthe system is already of the form \\eqref{eq_extended_system}, \nwhere the geometric control theory is directly applicable. \n\n\n{\\bf (ii)}\nBecause we now focus on the same plant system as that in Section \\ref{sec:3-3}, \nthe same conclusion is obtained; \nthat is, the BAE problem is solvable as long as there is no constraint on the \ncontroller matrices $(R_{\\scriptscriptstyle K}, R_1, R_2)$. \n\n\nThe controller matrices can be determined in a similar way to Section \\ref{sec:3-3} as follows. \nFirst, because the $(C,A,B)$-pair $(\\mathcal{V}_{1}, \\mathcal{V}_{2})$ is the \nsame as before, it follows that ${\\rm dim} \\, \\mathcal{X}_{{\\scriptscriptstyle K}}=2$, i.e., \n$n_{\\scriptscriptstyle k}=1$. \nThen, from Theorem~2 with the fact that $F=F_{0}$ and $G=G_{0}$, \nwe find that the direct interaction controller can be parameterized as follows: \n\\begin{align}\n\\label{direct controller matrices example}\n       &R_{\\scriptscriptstyle K}=-\\Sigma N(A+BF+GC)N^{+}, \\nonumber  \\\\\n       &R_1=-\\Sigma_2 BFN^{+}, \\nonumber  \\\\\n       &R_2=\\Sigma NGC,\n\\end{align}\nThe matrices $F$, $G$, and $N$ satisfy\n$\\Kernel F \\supseteq \\mathcal{V}_{1},  \\Image G \\subseteq \\mathcal{V}_{2}$, and \n$\\Kernel N=\\mathcal{V}_{1}$, \nwhich lead to\n\\begin{align}\n\\label{eq_parameterization2}\n\\hspace{-0.8cm}\n              F&=\\left[\\begin{array}{cccc}\n                        f_{\\scriptscriptstyle 11}&f_{\\scriptscriptstyle 12}\n                               &0&f_{\\scriptscriptstyle 14}\\\\\n                        \\frac{g}{\\sqrt{\\kappa}}&0&0&f_{\\scriptscriptstyle 24}\\\\\n                     \\end{array}\\right], ~~\n              G=\\left[\\begin{array}{cc}\n                        0&g_{\\scriptscriptstyle 12}\\\\\n                        -\\frac{g}{\\sqrt{\\kappa}}&g_{\\scriptscriptstyle 22}\\\\\n                        g_{\\scriptscriptstyle 31}&g_{\\scriptscriptstyle 32}\\\\\n                        0&0\\\\\n                     \\end{array}\\right], \\nonumber  \\\\\n              N&=\\left[\\begin{array}{cccc}\n                        n_{\\scriptscriptstyle 11}&n_{\\scriptscriptstyle 12}\n                              &0&n_{\\scriptscriptstyle 14}\\\\\n                        n_{\\scriptscriptstyle 21}&n_{\\scriptscriptstyle 22}\n                              &0&n_{\\scriptscriptstyle 24}\\\\\n                     \\end{array}\\right],\n\\end{align}\nwhere $f_{ij}, g_{ij}$, and $ n_{ij}$ are free parameters. \n\n{\\bf (iii)}\nThe controller matrices have to satisfy the physical \nrealizability conditions $R_{\\scriptscriptstyle K}=R_{\\scriptscriptstyle K}^\\top$ \nand $R_1^\\top=R_2$; \nthese constraints impose the parameters to satisfy \n\\begin{align}\n\\label{direct controller parameter constraint}\n&f_{\\scriptscriptstyle 12}=-g_{\\scriptscriptstyle 12},~~\nf_{\\scriptscriptstyle 11}=g_{\\scriptscriptstyle 22}, ~~\nn_{\\scriptscriptstyle 11}n_{\\scriptscriptstyle 22}-n_{\\scriptscriptstyle 12}n_{\\scriptscriptstyle 21}=-1, \\nonumber \\\\\n&f_{\\scriptscriptstyle 12}n_{\\scriptscriptstyle 1}=f_{\\scriptscriptstyle 11}n_{\\scriptscriptstyle 2}-f_{\\scriptscriptstyle 14}, ~~ \nf_{\\scriptscriptstyle 24}=\\frac{g}{\\sqrt{\\kappa}}n_{\\scriptscriptstyle 2}, \n\\nonumber \\\\\n& \\hspace{-0.15cm}\n\\left( \\frac{\\kappa}{2}+\\sqrt{\\kappa}f_{\\scriptscriptstyle 24} \\right) \n      n_{\\scriptscriptstyle 1}+\\omega_{\\scriptscriptstyle m}n_{\\scriptscriptstyle 2}\n=-\\sqrt{\\kappa}f_{\\scriptscriptstyle 11}, ~~\\nonumber \\\\\n&\\omega_{\\scriptscriptstyle m}n_{\\scriptscriptstyle 1} - \\left ( \\frac{\\kappa}{2} + \\sqrt{\\kappa}f_{\\scriptscriptstyle 24} \\right )n_{\\scriptscriptstyle 2}\n=\\sqrt{\\kappa}f_{\\scriptscriptstyle 12}, \n\\end{align}\nwhere \n$n_{\\scriptscriptstyle 1}=n_{\\scriptscriptstyle 11}n_{\\scriptscriptstyle 24}-\nn_{\\scriptscriptstyle 14}n_{\\scriptscriptstyle 21}$ and \n$n_{\\scriptscriptstyle 2}=n_{\\scriptscriptstyle 12}n_{\\scriptscriptstyle 24}-\nn_{\\scriptscriptstyle 14}n_{\\scriptscriptstyle 22}$. \n\\\\\nEquations \\eqref{direct controller matrices example}, \n\\eqref{eq_parameterization2}, and \\eqref{direct controller parameter constraint} \nprovide the full parametrization of the direct interaction controller. \n\n\n{\\bf (iv)}\nTo specify a set of parameters, as in the case of Section \\ref{sec:3-3}, let us aim \nto design a passive controller. \nFrom Theorem~4 in Appendix~B, $R_{\\scriptscriptstyle K}$ and \n$R_2=R_1^\\top$ satisfy the condition \n$\\Sigma R_{\\scriptscriptstyle K}\\Sigma=-R_{\\scriptscriptstyle K}$ and \n$\\Sigma R_2 \\Sigma_{2}=-R_2$, which lead to the same equalities given in \nEq.~\\eqref{passive para cond}. \nThen, setting the parameters to be $n_{\\scriptscriptstyle 11}=1$ and $n_{\\scriptscriptstyle 12}=0$, \nwe can determine the matrices $R_{\\scriptscriptstyle K}$ and $R_2$ as follows: \n\\begin{align*}\n                R_{\\scriptscriptstyle K}\n                     =\\left[\\begin{array}{cc}\n                             -\\omega_{\\scriptscriptstyle m} &0\\\\\n                             0&-\\omega_{\\scriptscriptstyle m} \\\\\n                       \\end{array}\\right] , ~~\n                R_2=R_1^\\top\n                   =\\left[\\begin{array}{cccc}\n                             0&0&g&0 \\\\\n                             0&0&0&g \\\\\n                     \\end{array}\\right] . \n\\end{align*}\nThe controller specified by these matrices can be physically implemented \nas illustrated in Fig.~\\ref{fig_Mankei_scheme}~(b); that is, it is a single-mode \ndetuned cavity with Hamiltonian $\\hat{H}_{\\scriptscriptstyle K}\n= -\\omega_{\\scriptscriptstyle m}\\hat{a}^{\\ast}_{3}\\hat{a}_{3}$, which couples \nto the plant through a beam-splitter (BS) represented by\n$\\hat{H}_{\\rm int}=g(\\hat{a}_{3}\\hat{a}^{*}_{2}+\\hat{a}^{*}_{3}\\hat{a}_{2})$. \n\n\n\n{\\bf Remark:} \nWe can employ an active controller, as proposed in \\cite{Tsang2010}. \nIn this case the interaction Hamiltonian is given by \n$\\hat{H}_{\\rm int}=g_{\\scriptscriptstyle \\rm B}(\\hat{a}_{3}\\hat{a}^{*}_{2}\n+\\hat{a}^{*}_{3}\\hat{a}_{2})+g_{\\scriptscriptstyle \\rm D}(\\hat{a}_{3}\\hat{a}_{2}\n+\\hat{a}^{*}_{3}\\hat{a}^{*}_{2})$, \nwhile the system's self-Hamiltonian is the same as above; \n$\\hat{H}_{\\scriptscriptstyle K}\n= -\\omega_{\\scriptscriptstyle m}\\hat{a}^{\\ast}_{3}\\hat{a}_{3}$. \nThat is, the controller couples to the plant through a non-degenerate optical \nparametric amplification process in addition to the BS interaction. \nTo satisfy the BAE condition, the parameters must satisfy \n$g_{\\scriptscriptstyle \\rm B}+g_{\\scriptscriptstyle \\rm D}=g$. \nNote that this direct interaction controller can be specified, in the \nfull-parameterization \\eqref{direct controller matrices example}, \n\\eqref{eq_parameterization2}, and \\eqref{direct controller parameter constraint}, \nby \n\\[\n     f_{\\scriptscriptstyle 11}=f_{\\scriptscriptstyle 12}=f_{\\scriptscriptstyle 14}=0,~~\n     n_{\\scriptscriptstyle 11}=-n_{\\scriptscriptstyle 22}=1,~~ \n     n_{\\scriptscriptstyle 12}=n_{\\scriptscriptstyle 21}=0.\n\\]\n\n\n\n\n\n\\section{Approximate Back-Action Evasion}\\label{sec:5}\n\n\nWe have demonstrated in Sections \\ref{sec:3-3} and \\ref{sec:4} that the \nBAE condition can be achieved by engineering an appropriate auxiliary \nsystem and connecting it to the plant. \nHowever, in a practical situation, it cannot be expected to realize such perfect \nBAE due to several experimental imperfections. \nHence, in a realistic setup, we should modify our strategy for engineering \na sensor so that it would accomplish {\\it approximate BAE}. \nThen, looking back into Section \\ref{sec:2-3} where the BAE condition, \n$\\Xi_{Q}(s)=0~\\forall s$, was obtained, we are naturally led to consider \nthe following optimization problem to design an auxiliary system achieving \nthe approximate BAE: \n\\begin{align}\n\\label{eq_approximateBAE}\n             \\min \\Big\\|\\frac{\\Xi_{Q}(s)}{\\Xi_{f}(s)}\\Big\\|, \n\\end{align}\nwhere $\\|\\bullet\\|$ denotes a valid norm of a complex function. \nIn particular, in the field of robust control theory, the following $H_2$ norm \nand the $H_\\infty$ norm are often used \\cite{Zhou1996}: \n\\[\n        \\|\\Xi\\|_2 \n         = \\sqrt{\\frac{1}{2\\pi}\\int_{-\\infty}^{\\infty}|\\Xi(i\\omega )|^{2}d\\omega},~~~\n        \\|\\Xi\\|_\\infty\n              = \\max_{\\omega}|\\Xi(i\\omega )|. \n\\]\nThat is, the $H_2$ or $H_\\infty$ control theory provides a general procedure \nfor synthesizing a feedback controller that minimizes the above norm. \nIn this paper, we take the $H_2$ norm, mainly owing to the broadband \nnoise-reduction nature of the $H_2$ controller. \nThen, rather than pursuing an optimal quantum $H_2$ controller based \non the quantum $H_2$ control theory \\cite{NurdinLQG2009, Hamerly2012}, \nhere we take the following geometric-control-theoretical approach to solve the \nproblem \\eqref{eq_approximateBAE}. \nThat is, first we apply the method developed in Section \\ref{sec:3} or \n\\ref{sec:4} to the idealized system and obtain the controller achieving BAE; \nthen, in the practical setup containing some unwanted noise, we make a local \nmodification of the controller parameters obtained in the first step, to minimize \nthe cost $\\|\\Xi_{Q}(s)\/\\Xi_{f}(s)\\|_2$. \n\n\n\nAs a demonstration, here we consider the coherent feedback control for the \nopto-mechanical system studied in Section \\ref{sec:2-3}, which is now \nsubjected to the thermal noise $\\hat{f}_{\\rm th}$. \nFollowing the above-described policy, we employ the coherent feedback \ncontroller constructed for the idealized system that ignores $\\hat{f}_{\\rm th}$, \nleading to the controller given by Eqs.~\\eqref{eq_controller_H} and \n\\eqref{perfect BAE condition example}, illustrated in Fig.~\\ref{fig_3I3OBAE}. \nThe closed-loop system with variable $\\hat{x}_{{\\scriptscriptstyle E}}=\n[\\hat{x}^{\\top},\\hat{x}_{\\scriptscriptstyle K}^{\\top}]^{\\top}$, \nwhich now takes into account the realistic imperfections, \nthen obeys the following dynamics:\n\\begin{align}\n\\label{eq_stateout2}\n          \\frac{d\\hat{x}_{{\\scriptscriptstyle E}}}{dt}\n                     &=\\widetilde{A}_{{\\scriptscriptstyle E}}\\hat{x}_{{\\scriptscriptstyle E}}\n                            +B_{{\\scriptscriptstyle E}}\\hat{W}_{1}                        \n                            +b_{{\\scriptscriptstyle E}}(\\hat{f}_{\\rm th}+\\hat{f}),  \\nonumber \\\\\n          \\hat{W}_{3}^{\\rm out}\n                     &=C_{{\\scriptscriptstyle E}}\\hat{x}_{{\\scriptscriptstyle E}}+\\hat{W}_{1}, \n\\end{align}\nwhere\n\\[\n            \\widetilde{A}_{{\\scriptscriptstyle E}}\n                 =\\left[\\begin{array}{cccc|cc}\n                        0&\\omega_{\\scriptscriptstyle m} &0&0&0&0\\\\\n                        -\\omega_{\\scriptscriptstyle m} &-\\gamma &g&0&0&0\\\\\n                        0&0&-\\kappa\/2 &0&0& \\sqrt{\\kappa \\kappa_{\\scriptscriptstyle K}} \\\\\n                        g&0&0&-\\kappa\/2 &-\\sqrt{\\kappa \\kappa_{\\scriptscriptstyle K}}&0\\\\\n                        \\hline 0&0&0&\\sqrt{\\kappa \\kappa_{\\scriptscriptstyle K}}&0&\\Delta \\\\\n                        0&0&-\\sqrt{\\kappa \\kappa_{\\scriptscriptstyle K}}&0&-\\Delta &0\\\\\n                   \\end{array}\\right] .\n\n\\]\n$B_{{\\scriptscriptstyle E}}$, $C_{{\\scriptscriptstyle E}}$, and $b_{{\\scriptscriptstyle E}}$ are the same matrices \ngiven in Eq.~\\eqref{extended_matrices}. \n$\\hat{f}_{\\rm th}$ is the thermal noise satisfying \n$\\langle \\hat{f}_{\\rm th}(t)\\hat{f}_{\\rm th}(t') \\rangle \\simeq \\bar{n}\\delta (t-t')$,\nwhere $\\bar{n}$ is the mean phonon number at thermal equilibrium \n\\cite{Giovannetti2001, Wimmer2014}. \nNote that the damping effect appears in the $(2, 2)$ component of \n$\\widetilde{A}_{{\\scriptscriptstyle E}}$ due to the stochastic nature of $\\hat{f}_{\\rm th}$.\nAlso, again, $\\kappa_{\\scriptscriptstyle K}$ and $\\Delta$ are the \ndecay rate and the detuning of the controller cavity, respectively. \nIn the idealized setting where $\\hat{f}_{\\rm th}$ is negligible, the perfect BAE is achieved \nby choosing the parameters satisfying Eq.~\\eqref{perfect BAE condition example}. \nThe measurement output of this closed-loop system is, in the Laplace domain, \nrepresented by \n\\[\n         \\hat{P}_{3}^{\\rm out}(s)\n               =\\widetilde{\\Xi}_{f} (\\hat{f}_{\\rm th}(s)+\\hat{f}(s))\n                    +\\widetilde{\\Xi}_{Q}\\hat{Q}_{1}(s)+\\widetilde{\\Xi}_{P} \\hat{P}_{1}(s).\n\\]\nThe normalized noise power spectral density of \n$y_{3}(s)=\\hat{P}_{3}^{\\rm out}(s)\/\\widetilde{\\Xi}_{f}(s)$ is calculated as \n\\begin{align}\n          \\widetilde{S} (\\omega ) \n              &=\\langle | y_{3}(i\\omega )-\\hat{f}(i\\omega )|^2 \\rangle  \\nonumber \\\\\n              &=\\langle|\\hat{f}_{\\rm th}|^2 \\rangle \n                   + \\left| \\frac{\\widetilde{\\Xi}_{Q}}{\\widetilde{\\Xi}_{f}} \\right| ^2 \n                                  \\langle |\\hat{Q}_{1}|^2 \\rangle \n                   + \\left| \\frac{\\widetilde{\\Xi}_{P}}{\\widetilde{\\Xi}_{f}} \\right| ^2 \n                                  \\langle |\\hat{P}_{1}|^2 \\rangle.\n\\label{eq_noisepower_thermal}\n\\end{align}\nThe coefficient of the back-action noise is given by \n\\begin{align}\n       \\frac{\\widetilde{\\Xi}_{Q}(s)}\n                {\\widetilde{\\Xi}_{f}(s)}=\n          -\\frac{ \\sqrt{\\kappa}\\{ \\kappa \\kappa_{\\scriptscriptstyle K}\\Delta \n                         (s^2+\\gamma s + \\omega_{\\scriptscriptstyle m}^2 )\n                             + g^2\\omega_{\\scriptscriptstyle m} (s^2+\\Delta^2 )\\} }\n                    {g\\omega_{\\scriptscriptstyle m}\\sqrt{\\gamma}\n                         \\{ (s+\\kappa\/2)(s^2+\\Delta^2 ) \n                              +\\kappa \\kappa_{\\scriptscriptstyle K} s\\}   }\n\\label{eq_normalizedtrans_thermal}.\n\\end{align}\nOur goal is to find the optimal parameters $(\\kappa_{\\scriptscriptstyle K}, \\Delta)$ \nthat minimize the $H_2$ norm of the transfer function, \n$\\widetilde{\\Xi}_{Q}\/\\widetilde{\\Xi}_{f}$. \n\n\n\\begin{figure}[!t]\n\\begin{center}\n\\includegraphics[width=7cm,clip]{Figure\/Fig8_H2norm.eps}\n\\caption{\n$H_{2}$ norm $\\|\\widetilde{\\Xi}_{Q}\/\\widetilde{\\Xi}_{f}\\|_{2}$ \nversus the coupling constant $\\kappa_{\\scriptscriptstyle K}$ and \nthe detuning $\\Delta$.}\n\\label{fig_h2norm}\n\\end{center}\n\\end{figure}\n\n\nThe system parameters are taken as follows \\cite{Wimmer2014}: \n$\\omega_{\\scriptscriptstyle m}\/2\\pi=0.5$ {\\rm MHz}, \n$\\kappa\/2\\pi=1.0$ {\\rm MHz}, \n$\\gamma\/2\\pi=5.0$ {\\rm kHz}, \n$g\/2\\pi=0.3$ {\\rm MHz}, \n$\\bar{n} \\simeq 8.33 \\times 10^{2}$, \nand the effective mass is $1.0 \\times 10^{-12}$ {\\rm kg}. \nWe then have Fig.~\\ref{fig_h2norm}, showing \n$\\|\\widetilde{\\Xi}_{Q}\/\\widetilde{\\Xi}_{f}\\|_2$ as a function of \n$\\kappa_{\\scriptscriptstyle K}$ and $\\Delta$. \nThis figure shows that there exists a unique pair of \n$(\\kappa_{\\scriptscriptstyle K}^{\\rm opt}, \\Delta^{\\rm opt})$ that \nminimizes the norm, and they are given by \n$\\kappa_{\\scriptscriptstyle K}^{\\rm opt} \/2\\pi=0.093$ {\\rm MHz} and\n$\\Delta^{\\rm opt}\/2\\pi=-0.5$ {\\rm MHz},\nwhich are actually close to the ideal values \\eqref{perfect BAE condition example}. \nFig.~\\ref{fig_noisepower} shows the value of Eq.~\\eqref{eq_noisepower_thermal} \nwith these optimal parameters \n$(\\kappa_{\\scriptscriptstyle K}^{\\rm opt}, \\Delta^{\\rm opt})$, where \nthe noise floor $\\langle|\\hat{f}_{\\rm th}|^2 \\rangle$ is subtracted. \nThe solid black line represents the SQL, which is now given by \n\\begin{equation}\n         \\widetilde{S}_{\\scriptscriptstyle \\rm SQL} (\\omega ) \n               =\\frac{|(\\omega^2 - \\omega_{\\scriptscriptstyle m}^2)-i\\gamma\\omega|}\n                         {\\gamma \\omega_{\\scriptscriptstyle m}}.\n\\label{eq_SQL_practical}\n\\end{equation}\nThen the dot-dashed blue and dotted green lines indicate that, in the low frequency range, the \ncoherent feedback controller can suppress the noise below the SQL, while, by \ndefinition, the noise power of the autonomous (i.e., uncontrolled) plant system \nis above the SQL. \nMoreover, this effect can be enhanced by injecting a $\\hat{P}_{1}$-squeezed \nprobe field (meaning $\\langle |\\hat{Q}_{1}|^2 \\rangle=e^{r}\/2$ and \n$\\langle |\\hat{P}_{1}|^2 \\rangle=e^{-r}\/2$) into the system. \nIn fact the dashed red line in the figure illustrates the case $r=2$ (about 9 dB squeezing), \nshowing the significant reduction of the noise power. \n\n\n\n\\begin{figure}[!t]\n\\begin{center}\n\\includegraphics[width=8.5cm,clip]{Figure\/Fig9_noisepower.eps}\n\\caption{Normalized power spectral densities of the noise. \nThe black solid line represents the SQL \\eqref{eq_SQL_practical}, and \nthe dot-dashed blue line does the case without feedback. \nThe dotted green and dashed red lines show the cases for the feedback controlled system, \nwith coherent and squeezed probe field, respectively. }\n\\label{fig_noisepower}\n\\end{center}\n\\end{figure}\n\n\n\n \n\n\n\n\\section{Conclusion}\n\\label{sec:6}\n\n\nThe main contribution of this paper is in that it first provides the general \ntheory for constructing a back-action evading sensor for linear quantum \nsystems, based on the well-developed classical geometric control theory. \nThe power of the theory has been demonstrated by showing that, for the \ntypical opto-mechanical oscillator, a full parametrization of the auxiliary \ncoherent-feedback and direct interaction controller achieving BAE was \nderived, which contains the result of \\cite{Tsang2010}. \nNote that, although we have studied a simple example for the purpose of \ndemonstration, the real advantage of the theory developed in this paper \nwill appear when dealing with more complicated multi-mode systems such \nas an opto-mechanical system containing a membrane \n\\cite{Plenio2008, Meystre2008, Nakamura2016, Nielsen2016}. \nAnother contribution of this paper is to provide a general procedure for designing \nan approximate BAE sensor under realistic imperfections; \nthat is, an optimal approximate BAE system can be obtained by solving \nthe minimization problem of the transfer function from the back-action noise \nto the measurement output. \nWhile in Section~\\ref{sec:5} we have provided a simple approach based on \nthe geometric control theory for solving this problem, the $H_2$ or $H_\\infty$ control theory could be \nemployed for systematic design of an approximate BAE controller even \nfor the above-mentioned complicated system. \nThis is also an important future research direction of this work. \n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIn \\cite{KhR1} Khovanov and Rozansky $(KR)$ constructed for each $n >0$ a bigraded rational (co)homology theory categorifying the $\\mathfrak{sl}(n)$-link polynomial (the case $n=0$ is treated in \\cite{KhR2}). Their construction uses matrix factorizations with potential $x^{n+1}$ associated to certain planar graphs, and for $n=2$, the corresponding homology is equivalent to the Khovanov homology defined in \\cite{Kh1}. Gornik \\cite{G} carried out a deformation of the $KR$-theory with potential $x^{n+1} - (n+1)\\beta^n x$, for $\\beta \\in \\mathbb{C},$ and Rasmussen \\cite{Ras2} and Wu ~\\cite{W} investigated $KR$-homologies given by a general non-homogeneous monic potential with degree $n+1$ and complex coefficients. $KR$-construction can be generalized to give the \\textit{universal} matrix factorization link homologies for all $n >0,$ by working with a general homogeneous potential.  Recently, Mackaay and Vaz \\cite{MV2} worked out this generalization for $n=3,$ and proved that the universal rational $\\mathfrak{sl}(3)$-matrix factorization link homology is equivalent to the foam link homology in \\cite{MV1} tensored with $\\mathbb{Q}.$ \n\nIn \\cite{CC2} the author constructed the universal $\\mathfrak{sl}(2)$-link cohomology via foams modulo local relations, in the spirit of \\cite{BN} and \\cite{Kh2} (see also \\cite{CC1}). In this paper we introduce the universal rational Khovanov-Rozansky link cohomology for $n=2,$ and show that it is isomorphic to the foam $\\mathfrak{sl}(2)$-link cohomology in \\cite{CC2}, after both theories are tensored with appropriate rings. To obtain the universal $\\mathfrak{sl}(2)$-matrix factorization theory, we consider a potential $p$ that depends on two parameters $a$ and $h,$ and that satisfies $ \\partial p \/\\partial x = 3 (x^2 - hx - a).$\n\n\n$KR$-construction starts from a certain version of the calculus developed by Murakami, Ohtsuki and Yamada \\cite{MOY}, calculus which involves planar trivalent graphs. $KR$-graphs contain two types of edges, namely oriented edges and unoriented thick edges. For our purpose, we consider graphs that are obtained from the latter ones by ``erasing\" all unoriented thick edges. \n\n\n\\section {Webs and matrix factorizations}\\label{factorizations}\n\n\\subsection{The web space}\n\nA \\textit{web} with boundary $B$ is a planar graph with univalent and bivalent vertices. The univalent vertices correspond to boundary points, such as the boundary $\\partial{T}$ of a tangle, and bivalent vertices have either indegree 2 or outdegree 2. Specifically, the two arcs incident with a bivalent vertex are either oriented ``in'' or ``out'' as shown below: \n\\[ \\raisebox{-5pt}{\\includegraphics[height=.2in]{in.pdf}} \\quad \\text {or} \\quad \\raisebox{-5pt}{\\includegraphics[height=.2in]{out.pdf}}\\]\nWe call bivalent vertices \\textit{singular points}. A \\textit{closed web} is a web with empty boundary. We also allow webs with no bivalent vertices, thus oriented arcs or loops. We denote by $\\textit{Foams}(B)$ the category whose objects are web diagrams with boundary $B,$ and whose morphisms are \\textit{singular cobordisms}---called \\textit{foams}---between such webs, regarded up to boundary-preserving isotopies. As morphisms, we read cobordisms from bottom to top, and we compose them by stacking one on top the other. \n\nLet $L$ be a link in $S^3.$ We fix a generic planar diagram $D$ of $L$ and resolve each crossing in two ways, as in Figure~\\ref{fig:resolutions}. We refer to the diagram on the right as the \\textit{oriented resolution}, and to the one on the left as the \\textit{singular resolution}. We remark that the dotted lines should not be considered as edges---we prefer to draw them in order to record that there was a crossing before. \n\\begin{figure}[ht]\n$$\\xymatrix@C=20mm@R=1.5mm{\n  &  \\includegraphics[height=0.5in]{poscros.pdf} \\ar[ld]_0\\ar[rd]^1& \\\\\n\\includegraphics[height=0.5in]{singresol.pdf} & & \n\\includegraphics[height=0.5in]{orienresol.pdf} \\\\\n  & \\includegraphics[height=0.5in]{negcros.pdf} \\ar[ul]^1 \\ar[ur]_0&\n}$$\n\\caption{The two ways of resolving crossings}\n\\label{fig:resolutions}\n\\end{figure}\n\nAs webs have singular points, foams have \\textit{singular arcs} (or \\textit{singular circles}) where orientations disagree. Basic cobordisms, as those between the two different resolutions of a crossing are depicted in Figure~\\ref{fig:saddles}, where the arc colored red is a singular arc.\n\n\\begin{figure}[ht!]\n\\begin{center}\n\\includegraphics[height=.65in]{saddle.pdf}\\end{center}\n\\caption{Singular saddles}\n\\label{fig:saddles}\n\\end{figure}\n\nA diagram $\\Gamma$ obtained by resolving all crossings of $D$---in one of the two possible ways explaned above---is a disjoint union of closed webs. There is a unique way to assign a Laurent polynomial $\\brak{\\Gamma} \\in \\mathbb {Z} [q, q^{-1}]$ to each closed web $\\Gamma$ so that it  satisfies the skein relations explained in Figure~\\ref{fig:skein relations}. (We remark that these are exactly the graph skein relations for $n= 2$ given in~\\cite[Figure 3]{KhR1}, where all thick edges are erased.)\n\n\\begin{figure}[ht]\n$$\\xymatrix@R = 2mm{\n\\raisebox{-10pt}{\\includegraphics[height=.35in]{orienloop.pdf}}\\,\\,\\, = \\,\\,\\, q + q^{-1}\\hspace{1cm}\n\\raisebox{-30pt}{\\includegraphics[height=.9in]{removedweb1.pdf}}\\, = \\,\\,(q+ q^{-1})\\,\\raisebox{-30pt}{\\includegraphics[height=.9in]{removedweb2.pdf}} \\\\ \\raisebox{-15pt}{\\includegraphics[height=.5in]{singresclosed.pdf}} = \\raisebox{-15pt}{\\includegraphics[height=.5in]{arc.pdf}} \\hspace{2cm}\\raisebox{-15pt}{\\includegraphics[height=.5in]{2singresol.pdf}}\\,\\, =\\,\\, \\raisebox{-15pt}{\\includegraphics[height=.45in]{2arcs.pdf}} \\\\\n\\raisebox{-30pt}{\\includegraphics[height=.9in]{skein1.pdf}}\\,\\, +\\,\\, \\raisebox{-30pt}{\\includegraphics[height=.9in]{skein2.pdf}} \\,\\,=\\,\\, \\raisebox{-30pt}{\\includegraphics[height=.9in]{skein3.pdf}}\\,\\, + \\,\\,\\raisebox{-30pt}{\\includegraphics[height=.9in]{skein4.pdf}} \n}$$\n\\caption{Skein relations}\n\\label{fig:skein relations}\n\\end{figure}\n\nThe \\textit{bracket} of $D$ is defined by $\\brak{D} = \\sum_{\\Gamma} \\pm q^{\\alpha(\\Gamma)}\\brak{\\Gamma},$ where the sum is over all resolutions of $D$ and $\\alpha(\\Gamma)$ is determined by the rules in Figure~\\ref{fig:decomposition of crossings}.\n\n\\begin{figure}[ht]\n$$\\xymatrix@R=2mm{\n\\raisebox{-12pt}{\\includegraphics[height=0.4in]{poscros.pdf}}\\,\\, = \\,\\,q\\,\\,\\raisebox{-12pt} {\\includegraphics[height=0.4in]{orienresol.pdf}}\\,\\, -\\,\\, q^2\\,\\,\\raisebox{-12pt} {\\includegraphics[height=0.4in]{singresol.pdf}}\\\\\n\\raisebox{-12pt}{\\includegraphics[height=0.4in]{negcros.pdf}} \\,\\,=  \\,\\,q^{-1}\\,\\raisebox{-12pt} {\\includegraphics[height=0.4in]{orienresol.pdf}}\\, -\\,q^{-2} \\,\\raisebox{-12pt} {\\includegraphics[height=0.4in]{singresol.pdf}}\n}$$\n\\caption{Decomposition of crossings}\n\\label{fig:decomposition of crossings}\n\\end{figure}\n\nIt is well know that $\\brak{D_1} = \\brak{D_2}$ whenever $D_1$ and $D_2$ are related by a Reidemeister move, hence $\\brak{L}:= \\brak{D}$ is an invariant of the oriented link $L.$ Excluding the rightmost terms in Figure~\\ref{fig:decomposition of crossings}, we obtain the skein relation $$q^2 \\raisebox{-8pt}{\\includegraphics[height=.32in]{negcros.pdf}} - q^{-2} \\raisebox{-8pt}{\\includegraphics[height=.32in]{poscros.pdf}} = (q - q^{-1}) \\raisebox{-8pt}{\\includegraphics[height=.32in]{orienresol.pdf}}$$ which yields the quantum $sl(2)$-polynomial of the link $L$ (thus the unnormalized Jones polynomial---its categorification is introduced in~\\cite{Kh1}).\n\n\\subsection{From webs to matrix factorizations}\n\nWe mimic Khovanov's and Rozansky's work in ~\\cite{KhR1} for $n =2$ case, by replacing the polynomial $p(x) = x^3$ with $p(a, h, x) = x^3 -\\frac{3}{2} h x^2- 3ax,$ where $a$ and $h$ are formal variables. Note that $p(a, h, x)$ was chosen in such a way that $ \\frac{1}{3} \\partial p\/ \\partial x = x^2 -hx -a.$  We assume that the reader is somewhat familiar with~\\cite{KhR1}, and we only briefly recall some of its content.\n\nGiven a graded ring $R$ (we consider only polynomial rings) and a homogeneous element $\\omega \\in R,$ the category $\\textit{mf}_{\\omega}$ of graded matrix factorizations with potential $\\omega$ has objects  2-periodic chains\n\\[ M_0 \\stackrel{d_0}{\\rightarrow} M_1 \\stackrel{d_1}{\\rightarrow} M_0, \\] \nwhere $M_0, M_1$ are graded free $R$-modules and $d_0, d_1$ are homomorphisms such that $d_0d_1 = \\omega = d_1d_0.$ We call such an object an $(R, \\omega)$-factorization. Morphisms in $\\textit{mf}_{\\omega}$ are degree-preserving maps of $R$-modules $M_0 \\to N_0, M_1 \\to N_1$ that commute with differentials. In this paper, the morphisms $d_0$ and $d_1$ have degree $3$ (thus $\\deg(\\omega) = 6$) and a homotopy has degree $-3.$ \n\nThe category $\\textit{hmf}_\\omega$ of graded matrix factorizations up to chain homotopies has the same objects as $\\textit{mf}_\\omega$ but fewer morphisms, as homotopic morphisms in $\\textit{mf}_\\omega$ are declared the same in $\\textit{hmf}_\\omega.$ We denote by $\\{r\\}$ the grading shift up by $r,$ and by $\\brak{s}$ the cohomological shift functor for matrix factorizations. Specifically, $M \\brak{1}$ has the form $M_1 \\stackrel {-d_1}{\\rightarrow} M_0 \\stackrel{-d_0}{\\rightarrow} M_1.$ The cohomological shift functor for chain complexes is denoted by $[s].$ \n\nGiven a pair of elements $a_1, b_1 \\in R,$ we denote by $(a_1, b_1)$ the factorization with potential $a_1b_1$\n\\[R \\stackrel{a_1}{\\longrightarrow} R \\{3 -\\deg a \\} \\stackrel{b_1}{\\longrightarrow} R, \\]\nwhere $a_1$ and  $b_1$ act on $R$ by multiplication. The middle $R$ was shifted so that the differentials would have degree $3$. Given a finite set of pairs $(a_i, b_i), 1 \\leq i \\leq n$, we denote their tensor product over $R$ by $(\\textbf{a, b}) : = \\otimes_i(a_i, b_i)$, where $\\textbf{a} = (a_1, \\cdots, a_n)$ and $ \\textbf{b} = (b_1,\\cdots,  b_n).$ Note that $(\\textbf{a, b})$ is the factorization with potential $\\omega = \\sum_i a_i b_i,$ and we sometimes prefer to write it in the \\textit{Koszul matrix} form:\n\\[ (\\textbf{a, b}) = \\left ( \\begin{array}{cc} a_1 & b_1 \\\\ a_2 & b_2 \\\\ \\cdots & \\cdots \\\\ a_n & b_n \\end{array} \\right ). \\]\nLet $c \\in R$ and denote by $[ij]_c$ the \\textit{elementary row operation} which transforms\n\\[ \\left ( \\begin{array}{cc} a_i & b_i \\\\ a_j & b_j \\end{array} \\right ) \\stackrel{[ij]_c}{\\longrightarrow}  \\left ( \\begin{array}{cc} a_i+ca_j & b_i \\\\ a_j & b_j -cb_i\\end{array} \\right )\\]\n and leaves the remaining rows of the Koszul matrix unchanged. An elementary row operation corresponds to a change of the basis vector of the free $R$-module underlying the corresponding factorization, thus it takes a Koszul factorization $(\\textbf{a, b})$ to an isomorphic factorization in $\\textit{hmf}_\\omega.$\n\nAs noted by Rasmussen~\\cite{Ras2}, if $b_i$ and $b_j$ are relatively prime, we can apply a \\textit{twist} via the map $R \\to R$ which sends $x \\to kx,$ for some $k \\in R,$ and obtain an isomorphism of factorizations\n \\[ (a_i, b_i) \\otimes (a_j, b_j) \\cong (a_i + k b_j, b_i) \\otimes (a_j - kb_i, b_j).\\]\n\nUsing Koszul matrix form, the above isomorphism is written as\n \\[ \\left ( \\begin{array}{cc} a_i & b_i \\\\ a_j & b_j \\end{array} \\right ) \\stackrel{\\cong}{\\longrightarrow}  \\left ( \\begin{array}{cc} a_i + kb_j & b_i \\\\ a_j - kb_i & b_j \\end{array} \\right).\\]\n \nStarting with a link or tangle diagram $D,$ we put marks on each of its arcs. Resolving all of its crossings as explained in Figure~\\ref{fig:resolutions}, we obtain web diagrams $\\Gamma$ with marked arcs, and we denote by $m(\\Gamma)$ the set of all marks of $\\Gamma.$ We associate to each mark $i$ the polynomial $p(a, h, x_i) = x_i^3 -\\frac{3}{2}h x_i^2 -3ax_i.$ When working with a tangle diagram, each of its resolutions has internal and external marks, where the later ones are the boundary points $i \\in B$ of the tangle diagram. \n\nLet $R = \\mathbb{Q}[a,h, x_i], i \\in m(\\Gamma),$ be the polynomial ring with rational coefficients and variables $a, h$ and $x_i$ (over all marks $i$), and let $R'$ be its subring $\\mathbb{Q}[a,h, x_i], i \\in B.$ We introduce a grading on $R$ and $R'$ by letting $\\deg(x_i) = 2,  i \\in m(\\Gamma), \\deg(a) = 4$ and $\\deg(h) =2.$ To a web diagram $\\Gamma$ we assign a graded factorization $\\overline{C}(\\Gamma)$ with (degree 6) homogeneous potential $\\omega(\\Gamma) = \\sum_{i \\in B} o(i)p(a, h, x_i),$ where $o(i) \\in \\{\\pm1\\} $ are ``orientations'' of boundary points, given by the orientation of $\\Gamma$ at these points. \n\nTo an oriented arc $l$ between two neighboring marks $i, j$ and oriented from $i$ to $j$ we assign the factorization $\\overline{L}_i^j$ over the ring $R = \\mathbb{Q}[a, h, x_i, x_j],$ and with potential $$\\omega (\\overline{L}_i^j)= p (a, h, x_j) -p (a, h, x_i) = x_j^3 -x_i^3 -\\frac{3}{2} h (x_j^2 -x_i^2) -3a (x_j -x_i).$$\n \\[\\overline{L}_i^j \\co \\quad \\mathbb{Q}[a,h,x_i,x_j]\\stackrel{\\overline{\\pi}_{ij}}{\\longrightarrow} \\mathbb{Q}[a,h,x_i,x_j]\\{-1\\} \\stackrel{x_j-x_i}{\\longrightarrow}  \\mathbb{Q}[a,h,x_i,x_j] \\]\n\nwhere $\\overline{\\pi}_{ij} =\\displaystyle\\frac {\\omega(\\overline{L}_i^j)}{x_j-x_i} = x_i^2 + x_ix_j + x_j^2 -\\frac{3}{2}h(x_i + x_j) -3a$.\n\nTo an oriented circle with one mark $i$ we assign the factorization $\\overline{L}_i^i,$ which is the quotient of $\\overline{L}_i^j$ by the relation $x_j = x_i.$ We obtain a 2-periodic chain complex of $\\mathbb{Q}[a,h,x_i]$-modules\n  \\[\n\\mathbb{Q}[a,h,x_i]\\stackrel{\\overline{\\pi}_{ii}}{\\longrightarrow} \\mathbb{Q}[a,h,x_i]\\{-1\\} \\stackrel{0}{\\longrightarrow}  \\mathbb{Q}[a,h,x_i], \\] \nwhere $\\overline{\\pi}_{ii} = 3(x_i^2 - h x_i -a)$. This complex has cohomology only in degree 1, namely $$H^1(\\overline{L}_i^i) = \\mathbb{Q}[a,h,x_i]\/(x_i^2 -hx_i - a) \\{-1\\}, \\quad H^0(\\overline{L}_i^i) = 0.$$ \n\nLet $\\mathcal{A} : = \\mathbb{Q}[a, h, X]\/(X^2 - hX -a)$ and $\\iota \\co \\mathbb{Q}[a, h] \\to \\mathcal{A}$ be the inclusion map $\\iota(1) = 1.$ We identify $\\mathbb{Q}[a, h,x_i]\/(x_i^2 -hx_i- a)$ with $ \\mathcal{A}$ by taking $x_i^k \\in \\mathbb{Q}[a, h, x_i]\/(x_i^2 - hx_i -a)$ to $X^k \\in \\mathcal{A}.$ As a module over $\\mathbb{Q}[a, h], \\mathcal{A}$ is free with generators 1 and $X.$ We make $\\mathcal{A}$ graded by giving to 1 degree $-1$ and to $X$ degree 1.\n\nTo an oriented circle without marks we associate the 2-periodic  chain complex of $\\mathbb{Q}[a,h]$-modules $0 \\rightarrow \\mathcal{A} \\rightarrow 0$ and denote it $\\mathcal{A} \\brak{1},$ following \\cite{KhR1}. Note that $\\mathcal{A}\\brak{1} \\cong \\overline{L}_i^i$ as 2-periodic complexes of $\\mathbb{Q}[a,h]$-modules, up to homotopies. The isomorphism takes $X^i \\in \\mathcal{A}$ to $x^i \\in (\\overline{L}_i^i)^1,$ for $0 \\leq i \\leq 1,$ and graphically it consists of adding a mark to a circle with no marks.\n\nTo diagrams $\\Gamma^0$ and $\\Gamma^1$ as in Figure~\\ref{maps} we associate the factorizations $\\overline{C}(\\Gamma^0)$ and $\\overline{C}(\\Gamma^1)$ over the ring $R = \\mathbb{Q}[a, h, x_1, x_2, x_3, x_4]$ (note that $R = R'$):\n\\begin{align*}\n\\overline{C}(\\Gamma^0) &= (\\overline{\\pi}_{41}, x_1 -x_4) \\otimes (\\overline{\\pi}_{32}, x_2 -x_3) \\\\\n\\overline{C}(\\Gamma^1) &= (\\overline{u}_{1}, x_1 + x_2 -x_3 -x_4) \\otimes (\\overline{u}_{2}, x_1x_2 -x_3x_4)\n\\end{align*}\n with $\\omega(\\overline{C}(\\Gamma^0)) = p(a, h, x_1) + p(a, h, x_2) - p(a, h, x_3) - p(a, h, x_4) = \\omega(\\overline{C}(\\Gamma^1))$ and \n \\begin{align*} \\overline{u}_1 &= (x_1 + x_2)^2 + (x_1 + x_2)(x_3 + x_4) + (x_3 + x_4)^2 -3x_1x_2 -\\frac{3}{2} h( x_1+x_2+x_3 + x_4) -3a \\\\ \\overline{u}_2 &= -3(x_3+x_4) +3h.\\end{align*} \n Note that \\begin{align*} \\omega &= \\overline{\\pi}_{41}(x_1-x_4) + \\overline{\\pi}_{32}(x_2-x_3) \\\\ &= \\overline{u}_1\\cdot (x_1+x_2 -x_3 -x_4) + \\overline{u}_2 \\cdot (x_1x_2-x_3x_4).\\end{align*} Precisely we have\n \\[\n\\overline{C}(\\Gamma^0)\\co \\quad\n \\left( \\begin{array}{c}\n R \\\\ R\\{-2\\}\n \\end{array}\\right) \\stackrel{P_0}{\\longrightarrow}\n  \\left( \\begin{array}{c}\n  R\\{-1\\} \\\\ R\\{-1\\}\n  \\end{array} \\right) \\stackrel{P_1}{\\longrightarrow}\n  \\left(\\begin{array}{c}\n  R \\\\ R\\{-2\\} \\end{array} \\right)\n \\]\n \n \\[P_0 = \n \\left(\\begin{array}{cc}\n\\overline{\\pi}_{41}  & x_2-x_3 \\\\ \n \\overline{\\pi}_{32}  & x_4-x_1\\end{array}\n \\right),  \\quad P_1 = \n \\left(\\begin{array}{cc}\n x_1 - x_4 & x_2-x_3 \\\\ \n\\overline {\\pi}_{32}  & -\\overline{\\pi}_{41} \\end{array} \\right)\n \\]\n \n \\[ \\overline{C}(\\Gamma^1)\\co \\quad\n \\left( \\begin{array}{c}\n R\\{-1\\} \\\\ R\\{-1\\}\\end{array}\n \\right) \\stackrel{Q_0}{\\longrightarrow}\n \\left(\\begin{array}{c}\n R\\{-2\\} \\\\ R\\end{array}\n \\right) \\stackrel{Q_1}{\\longrightarrow} \\left(\n \\begin{array}{c} R\\{-1\\} \\\\ R\\{-1\\} \\end{array} \\right)\n \\]\n \n \\[Q_0 = \n \\left( \\begin{array}{cc}\n\\overline {u}_1  & x_1x_2 - x_3x_4 \\\\ \n\\overline {u}_2 & x_3 +  x_4- x_1 - x_2 \\end{array}\n \\right), \\quad Q_1 =\n \\left(\\begin{array}{cc}\n x_1 + x_2 - x_3 - x_4 & x_1 x_2 - x_3x_4 \\\\ \n \\overline{u}_2 & - \\overline{u}_1 \\end{array}\\right)\n \\]\n\n\\begin{figure}[ht]\n\\includegraphics[height=1in]{maps.pdf} \n\\caption{Webs $\\Gamma^0$ and $\\Gamma^1$}\n\\label{maps}\n\\end{figure}\n\nWritten in Koszul matrix form, $\\overline{C}(\\Gamma^0) = (\\mathbf{a}, \\mathbf{b})$ and $\\overline{C}(\\Gamma^1) = ( \\mathbf{c}, \\mathbf{d})\\{-1\\}$, where\n\\[(\\mathbf{a}, \\mathbf{b}) =\n \\left(\\begin{array}{cc}\n\\overline{\\pi}_{41}  & x_1-x_4 \\\\ \n\\overline{\\pi}_{32}  & x_2-x_3\\end{array} \n\\right) \\quad (\\mathbf{c}, \\mathbf{d}) = \n\\left( \\begin{array}{cc}\n\\overline{u}_1  & x_1+x_2-x_3-x_4 \\\\ \\overline{u}_2 & x_1x_2-x_3x_4\\end{array}\\right),\\]\nand where the shift $\\{-1\\}$ in $\\overline{C}(\\Gamma^1)$ is applied to the second row \n$$R \\stackrel {\\overline{u}_2}{\\longrightarrow} R\\{1\\} \\stackrel{x_1x_2-x_3x_4}{\\longrightarrow} R.$$\nWe shifted the degrees of $R$ so that each differential above has degree 3.\n\nFinally, we define the $\\overline{C}(\\Gamma)$ as the tensor product of $\\overline{C}(\\Gamma^1)$ over all singular resolution, of $\\overline{L}_i^j$ over all arcs $l,$ and of $\\mathcal{A} \\brak{1}$ over all loops with no mark. The tensor product is considered over appropriate rings, so that $\\overline{C}(\\Gamma)$ is a free module of finite rank over $R$, and we treat it as a graded factorization---with infinite rank---over the subring $R'.$ \n\n\\begin{lemma}\nGiven any web diagram $\\Gamma,$ its associated factorization $\\overline{C}(\\Gamma)$ lies in $\\textit{hmf}_\\omega.$ Moreover, if $\\Gamma'$ is obtained from $\\Gamma$ by placing a different collection of internal marks then there is a canonical isomorphism $\\overline{C}(\\Gamma') \\cong \\overline{C}(\\Gamma)$ in $\\textit{hmf}_\\omega.$\n\\end{lemma}\n\n\\begin{lemma} For any disjoint union of webs $\\Gamma_1 \\cup \\Gamma_2$ there is a canonical isomorphism in $\\textit{hmf}_\\omega,$ namely\n$\\overline{C}(\\Gamma_1 \\cup  \\Gamma_2) \\cong \\overline{C}( \\Gamma_1) \\otimes_{\\mathbb{Q}[a,h]}\\overline{C}( \\Gamma_2).$ In particular we have\n$$\\overline{C}(\\Gamma \\cup \\raisebox{-5pt}{\\includegraphics[height=.2in]{orienloop.pdf}}) \\cong \\overline{C}(\\Gamma)\\brak{1} \\otimes_{\\mathbb{Q}[a,h]} \\mathcal{A}.$$\n\\end{lemma}\n\nNote that multiplication by $\\partial_i \\omega : = \\partial \\omega \/ \\partial {x_i}$ endomorphism of $\\overline{C}(\\Gamma)$ is homotopic to zero. Moreover, multiplication by any polynomial in $(\\textbf{a}, \\textbf{b})$ induces a null-homotopic endomorphism of $\\overline{C}(\\Gamma)$ (see~\\cite[Proposition 2]{KhR1}).   \n\n\\textit{Excluding a variable.} An important tool introduced in~\\cite{KhR1} is the process of ``excluding a variable''. Supposed that $x_i$ is one of the generators of the polynomial ring$R$ and that $\\omega = \\sum a_i b_i \\in R',$ where $R = R'[x_i].$ We say that $x_i$ is an \\textit{internal} variable. Any $(R, \\omega)$-factorization $(\\textbf{a, b})$ restricts to an infinite rank factorization over $R'$, and we denote it by $(\\textbf{a, b})'.$ Suppose furthermore that for some $j,$ $b_j= x_i -\\alpha$ where $\\alpha \\in R'.$  Denote by $(\\textbf{a'}, \\textbf{b'})$ the factorization over $R'$ obtained from $(\\textbf{a, b})$ by removing the $j$-th row and substituting $\\alpha$ for $x_i$ everywhere in all other rows.\n \\begin{lemma}\n Factorizations $(\\textbf{a, b})'$ and $(\\textbf{a'}, \\textbf{b'})$ are isomorphic in the homotopy category of $(R', \\omega)$-factorizations. \\end{lemma}\n \n If $\\Gamma$ is a closed web, the potential $\\omega = 0$ and $\\overline{C}(\\Gamma)$ is a 2-periodic complex, thus we can take the cohomology of the corresponding complex. In particular, to the basic closed web with two vertices and with arcs labeled by $x_1$ and $x_2$ we assign the factorization over $\\mathbb{Q}[a, h]$ with trivial potential, which is the quotient of $\\overline{C}(\\Gamma^1)$ by the relations $x_1 = x_4$ and  $x_2 = x_3.$ We obtain a complex with homology only in degree zero:\n\\[ H^0(\\overline{C}(\\Gamma^1)\/_{x_1 = x_4,x_2 = x_3}) = \\mathbb{Q}[a,h,x_1, x_2]\/(\\overline{u}^*_1, \\overline{u}^*_2) \\{-1\\}, \\, \\text{where}\\]\n\\begin{align*}\n \\overline{u}^*_1 &= \\overline{u}_1\/ _{x_1=x_4, x_2 =x_3} = 3(x_1 + x_2)^2 -3x_1x_2 -3h(x_1 + x_2) -3a \\\\ \\overline{u}^*_2 &=  \\overline{u}_2\/ _{x_1=x_4, x_2 =x_3} = -3(x_1 + x_2 -h).\n\\end{align*}\nTherefore $H^0(\\raisebox{-8pt}{\\includegraphics[height=0.3in]{closed-web.pdf}}) \\cong \\mathbb{Q}[a,h,x_1, x_2]\/(x_1 + x_2 -h,\\, x_1x_2 + a) \\{-1 \\},$ or equivalently\n\\[ H^0(\\raisebox{-8pt}{\\includegraphics[height=0.3in]{closed-web.pdf}}) \\cong \\mathbb{Q}[a,h,x_1]\/(x_1^2 -h x_1- a) \\{-1 \\} \\cong \\mathcal{A} \\quad \\text{and} \\quad H^1(\\raisebox{-8pt}{\\includegraphics[height=0.3in]{closed-web.pdf}}) = 0.\\]\n\nSimilarly, the quotient of $\\overline{C}(\\Gamma^0)$ by the relations $x_1 = x_4$ and  $x_2 = x_3$ is a 2-complex with homology only in degree zero:\n\\[ H^0(\\overline{C}(\\Gamma^0)\/_{x_1 = x_4,x_2 = x_3}) = \\mathbb{Q}[a,h,x_1, x_2]\/(\\overline{\\pi}^*_{41}, \\overline{\\pi}^*_{32}) \\{-2\\}, \\, \\text{where}\\]\n\\begin{align*}\n \\overline{\\pi}^*_{41} &= \\overline{\\pi}_{41}\/ _{x_1=x_4, x_2 =x_3} = 3(x_1^2 -h x_1 - a) \\\\ \\overline{\\pi}^*_{32} &=  \\overline{u}_2\/ _{x_1=x_4, x_2 =x_3} = -3(x_2^2 -h x_2 -a).\n\\end{align*}\nTherefore $H^0(\\raisebox{-8pt}{\\includegraphics[height=0.3in]{orcircles.pdf}}) \\cong \\mathbb{Q}[a,h,x_1, x_2]\/(x_1^2 +h x_1 -a,\\, x_2^2 - hx_2 - a) \\{-2 \\}.$\n\nEquivalently,  $H^0(\\raisebox{-8pt}{\\includegraphics[height=0.3in]{orcircles.pdf}}) \\cong \\mathcal{A} \\otimes \\mathcal{A} \\quad \\text{and} \\quad H^1(\\raisebox{-8pt}{\\includegraphics[height=0.3in]{orcircles.pdf}}) = 0.$\n\n \\subsection{Maps $\\Lambda_0$ and $\\Lambda_1$}\\label{sec:maps}\nThe maps $\\Lambda_0$ and $\\Lambda_1$ between $\\overline{C}(\\Gamma^0)$ and $\\overline{C}(\\Gamma^1),$ where $\\Gamma^0$ and $\\Gamma^1$ are the web diagrams from Figure~\\ref{maps}, are given by a pair of $2 \\times 2$ matrices $(U_0, U_1)$ and $(V_0, V_1),$ respectively. \n\\[U_0 = \n\\left(\\begin{array}{cc}x_4 - x_2 & 0 \\\\ x_1 - x_2 + x_3 + 2x_4 -\\frac{3}{2} h & 1\\end{array}\\right), \\quad U_1 = \\left(\n\\begin{array}{cc}x_4 & -x_2 \\\\ -1 & 1\\end{array}\\right)\n\\]\n \n\\[V_0 = \n\\left(\\begin{array}{cc}1 & 0 \\\\ -x_1 + x_2 - x_3 - 2x_4 +\\frac{3}{2}h & x_4-x_2\\end{array}\\right), \\quad V_1 = \\left(\n\\begin{array}{cc}1 & x_2 \\\\ 1 & x_4\\end{array}\\right)\n\\]\n \nThe compositions $\\Lambda _0 \\Lambda_1$ and$\\Lambda_1 \\Lambda_0$ are homotopic to the multiplication by $x_4 - x_2$ endomorphism of $\\overline{C} (\\Gamma^0)$ and $\\overline{C}(\\Gamma^1),$ respectively. This is easily seen from the following relations:\n \\[U_0V_0 = U_1V_1 =(x_4 -x_2)\\id , \\quad  V_0U_0 = V_1U_1 = (x_4 -x_2) \\id.\\]\nThus $\\Lambda_1 \\circ \\Lambda_0 = m(x_4-x_2)$ and $\\Lambda_0 \\circ \\Lambda_1 = m(x_4-x_2)$. On the other hand, since the endomorphism of $\\overline{C}(\\Gamma^0)$---or $\\overline{C}(\\Gamma^1)$--- given by the multiplication by $x_1 + x_2 - x_3 -x_4$ is null-homotopic, we also have $\\Lambda_1 \\circ \\Lambda_0 =  m(x_1 - x_3)$ and  $\\Lambda_0 \\circ \\Lambda_1 = m(x_1 - x_3).$ Both maps $\\Lambda_0$ and $\\Lambda_1$ are maps of degree 1.\n\nConsidering the Koszul matrices for $\\overline{C}(\\Gamma^0)$ and $\\overline{C}(\\Gamma^1),$ we apply certain row transformation to each of them:\n\n$ \\overline{C}(\\Gamma^0): \\quad    \\left(\\begin{array}{cc}\n\\overline{\\pi}_{41}  & x_1-x_4 \\\\ \n\\overline{\\pi}_{32}  & x_2-x_3\\end{array} \\right) \\stackrel{[21]_{-1}}{\\longrightarrow}  \\left(\\begin{array}{cc}\n\\overline{\\pi}_{41}  & x_1 + x_2 -x_3 -x_4 \\\\ \n\\overline{\\pi}_{32} -\\overline{\\pi}_{41}  & x_2-x_3\\end{array} \\right) $ \n\n$ \\overline{C}(\\Gamma^1): \\quad    \\left(\\begin{array}{cc}\n\\overline{u}_1  & x_1 + x_2 - x_3 -x_4 \\\\ \n\\overline{u}_2  & x_1x_2-x_3x_4\\end{array} \\right) \\stackrel{[12]_{x_2}}{\\longrightarrow}  \\left(\\begin{array}{cc}\n\\overline{u}_1 + x_2 u_2 & x_1 + x_2 -x_3 -x_4 \\\\ \n\\overline{u}_2 &(x_4 - x_2)(x_2-x_3)\\end{array} \\right).$  \n\nWe apply further a twist to obtain \n\\[ \\overline{C}(\\Gamma^0) \\stackrel {\\cong}{\\longrightarrow} \\left(\\begin{array}{cc}\n\\overline{\\pi}_{41} -k (x_2-x_3)  & x_1 + x_2 -x_3 -x_4 \\\\ \n\\overline{\\pi}_{32} -\\overline{\\pi}_{41} + k (x_1 + x_2 -x_3 -x_4 ) & x_2-x_3\\end{array} \\right)\\]\n\n\\[\\overline{C}(\\Gamma^1) \\stackrel{\\cong}{\\longrightarrow} \\left(\\begin{array}{cc}\n\\overline{u}_1 + x_2 u_2 + 2 (x_4 - x_2)(x_2-x_3) & x_1 + x_2 -x_3 -x_4 \\\\ \n\\overline{u}_2 -2 (x_1 + x_2 -x_3 -x_4) & (x_4 - x_2)(x_2-x_3)\\end{array} \\right) \\]\n\nwhere $k = x_1 + x_2 + x_3 -\\frac{3}{2}h.$ An easy computation shows that the Koszul matrices above, hence $\\overline{C}(\\Gamma^0)$ and $\\overline{C}(\\Gamma^1)$, have the following equivalent forms \n\n$ \\overline{C}(\\Gamma^0) \\cong \\left(\\begin{array}{cc}\na_1  & x_1 + x_2 -x_3 -x_4 \\\\ \na_2 (x_4 -x_2) & x_2-x_3\\end{array} \\right) \\, \\text{and} \\,\\,  \\overline{C}(\\Gamma^1) \\cong \\left(\\begin{array}{cc}\na_1  & x_1 + x_2 -x_3 -x_4 \\\\ \na_2 & (x_4 -x_2)(x_2-x_3)\\end{array} \\right) $\n\nwhere $a_1 = \\overline{\\pi}_{41} - \\overline{\\pi}_{21} + \\overline{\\pi}_{31}$ and $a_2 = -2x_1 -2x_2 -x_3 -x_4 +3h.$ The first rows are identical and the second rows are related. Consider the \\textit{flip} homomorphisms $$\\psi_{x_4-x_2}\\co (a_2,  (x_4-x_2)(x_2-x_3)) \\longrightarrow (a_2(x_4-x_2), x_2-x_3)$$\n\\begin{displaymath} \n\\xymatrix @C=23mm@R=13mm{\nR \\ar [r]^{a_2} \\ar [d]_{ 1} & R \\ar [d]_{x_4-x_2} \\ar[r]^{(x_4-x_2)(x_2-x_3)} & R \\ar [d]_1 \\\\\nR \\ar [r]^{a_2(x_4-x_2)} & R \\ar [r]^{x_2-x_3}   & R}\n\\end{displaymath}\nand $ \\psi'_{x_4-x_2} \\co (a_2(x_4-x_2), x_2-x_3) \\longrightarrow (a_2, (x_4-x_2)(x_2-x_3))$\n\n\\begin{displaymath} \n\\xymatrix @C=23mm@R=13mm{\nR \\ar [r]^{a_2(x_4-x_2)} \\ar [d]_{ x_4-x_2} & R \\ar [d]_1 \\ar[r]^{x_2-x_3} & R \\ar [d]_{x_4-x_2} \\\\\nR \\ar [r]^{a_2} & R \\ar [r]^{(x_4-x_2)(x_2-x_3)}   & R}\n\\end{displaymath}\nUsing the equivalent Koszul matrices for $\\overline{C}(\\Gamma^0), \\overline{C}(\\Gamma^1),$ the following lemma follows. \n\\begin{lemma}\n$\\Lambda_0 = \\id \\otimes \\,\\psi'_{x_4-x_2}$ and $\\Lambda_1 = \\id \\otimes \\,\\psi_{x_4-x_2}.$\n\\end{lemma}\n\n\\subsection{Complexes of factorizations}\\label{sec:complexes}\nWe start with a generic diagram $D$ of a tangle $T$ with boundary points $B$, and put (at least) one mark on each segment bounded by two crossings. We let $m(D)$ be the set of all marks of $D,$ and consider the polynomial ring $R = \\mathbb{Q}[a, h, x_i]$ for all $i \\in m(D),$ and its subring $R' = \\mathbb{Q}[a, h, x_i]$ for all $i \\in B.$   \n\nWe associate to each crossing $p$ in $D$ a complex $\\overline{C}_p$ of matrix factorizations, as explained in Figure~\\ref{fig:crossings to chain complexes}, where the underlined objects are at the cohomological degree $0.$ \n  \\begin{figure}[ht]\n\\[ \\raisebox{-13pt}{\\includegraphics[height=.45in]{poscros.pdf}} \\,\\, = \\,\\, \\left[\\,\\, 0 \\longrightarrow \\raisebox{-13pt}{\\includegraphics[height=.45in]{singresol.pdf}}\\,\\,\\{2\\} \\longrightarrow \\underline{\\raisebox{-13pt}{\\includegraphics[height=.45in]{orienresol.pdf}}\\,\\, \\{1\\}}\\longrightarrow 0\\,\\, \\right ]\\]\n\n\\[ \\raisebox{-13pt}{\\includegraphics[height=.45in]{negcros.pdf}} \\,\\, = \\,\\, \\left[\\,\\, 0 \\longrightarrow \\underline{\\raisebox{-13pt}{\\includegraphics[height=.45in]{orienresol.pdf}}\\,\\,\\{-1\\}} \\longrightarrow\\raisebox{-13pt}{\\includegraphics[height=.45in]{singresol.pdf}}\\,\\, \\{-2\\}\\longrightarrow 0\\,\\, \\right ]\\]\n\\caption{Complex applied to a crossing}\n\\label{fig:crossings to chain complexes}\n\\end{figure}\n\nPrecisely, we have\n\\[ \\raisebox{-8pt}{\\includegraphics[height=.3in]{poscros.pdf}} = [\\,0 \\longrightarrow \\overline{C}(\\Gamma^1)\\{2\\} \\stackrel{\\Lambda_1}{\\longrightarrow} \\overline{C}(\\Gamma^0)\\{1\\} \\longrightarrow 0\\, ]\\]\n\\[ \\raisebox{-8pt}{\\includegraphics[height=.3in]{negcros.pdf}} = [\\, 0 \\longrightarrow \\overline{C}(\\Gamma^0)\\{-1\\} \\stackrel{\\Lambda_0}{\\longrightarrow} \\overline{C}(\\Gamma^1)\\{-2\\} \\longrightarrow 0\\,] \\]\nwhere $\\Gamma^0, \\Gamma^1$ are the oriented and the singular resolutions from Figure~\\ref{fig:resolutions}, and where the matrix factorization $\\overline{C}(\\Gamma^0)$ is at the cohomological degree $0.$\n\nWe associate to $D$ a complex of factorizations, which we denote it by $C(D),$  and which is the tensor product of $\\overline{C}_p,$ over all crossings $p$ in $D,$ of $\\overline{L}_i^j,$ over all arcs $i \\to j,$ and of $\\mathcal{A} \\brak{1},$ over all oriented loops in $D$ with no crossings and no marks. The tensoring is done over appropriate polynomial rings so that $C(D)$ is a free $R$-module of finite rank.\n\n$C(D)$ is a complex of graded ($R, \\omega$)-factorizations, where $\\omega = \\sum_{i \\in B} o(i)p(a, h, x_i).$ We regard $C(D)$ as an object in $K_\\omega : = Kom(\\textit{hmf}_\\omega),$ the homotopy category of complexes over $\\textit{hmf}_\\omega.$ Note that the differentials of $C(D)$ are grading-preserving. In Section \\ref{sec:invariance} we show that  the isomorphism class of $C(D)$ is a tangle invariant.\n\nIf $T$ is a link, the set $B$ of boundary points is empty, $R' = \\mathbb{Q}[a, h],$ and $C(D)$ is a complex of graded $\\mathbb{Q}[a, h]$-modules. \n\n\\subsection{Isomorphisms}\\label{sec:isomorphisms}\nWe show that $\\overline{C}(\\Gamma)$ mimics the skein relations of Figure~\\ref{fig:skein relations}. \n\\begin{proposition}\\label{isom 1}(\\textit{First Isomorphism}) There is an isomorphism in $hmf_\\omega$:\n\\[\\overline{C}(\\raisebox{-8pt}{\\includegraphics[height=.3in]{singresclosed.pdf}}\\,) \\cong \\overline{C}(\\raisebox{-8pt}{\\includegraphics[height=.3in]{arc.pdf}}\\,)\\brak{1}.\\]\n\\end{proposition}\n\n\\begin{proof} Consider the webs $\\Gamma$ and $\\Gamma_1$ given in Figure~\\ref{fig:direct sum decomposition 1-webs}. Factorizations $\\overline{C}(\\Gamma),\\overline{C}(\\Gamma_1)$ are $(R', \\omega)$-factorizations (the later has infinite rank over $R'$), where $R' = \\mathbb{Q}[a, h, x_2,x_3]$ and $\\omega = p(a, h, x_2) - p(a, h, x_3).$\n\\begin{figure}[h]\n\\[ \\Gamma = \\raisebox{-27pt}{\\includegraphics[height=.8in]{larc.pdf}} \\hspace{3cm} \\Gamma_1 = \\raisebox{-27pt}{\\includegraphics[height=.8in]{lsingresclosed.pdf}}\\]\n\\caption{Webs $\\Gamma$ and  $\\Gamma_1$}\n\\label{fig:direct sum decomposition 1-webs}\n\\end{figure}\n\nIn Koszul form, $\\overline{C}(\\Gamma_1) = \\left (\\begin{array}{cc}\\overline{u}'_1 & x_2-x_3 \\\\\n\\overline{u}'_2 & x_1(x_2 - x_3)  \\end{array} \\right)$ where $\\overline{u}'_1 = \\overline{u}_1(x_2, x_1,x_1, x_3)$ and $\\overline{u}'_2 = \\overline{u}_1(x_2, x_1,x_1, x_3).$ We apply the elementary row operation $[12]_{x_1}$ and obtain:\n$$\\left (\\begin{array}{cc}\\overline{u}'_1 & x_2-x_3 \\\\\n\\overline{u}'_2 & x_1(x_2 - x_3)  \\end{array} \\right) \\stackrel{\\cong}{\\longrightarrow} \\left (\\begin{array}{cc}\\overline{u}'_1+ x_1 \\overline{u}'_2 & x_2-x_3 \\\\\n\\overline{u}'_2 & 0  \\end{array} \\right)$$\nAn easy computation shows that $\\overline{u}'_1+ x_1 \\overline{u}'_2= \\overline{\\pi}_{23}$ and $\\overline{u}'_2 = -3(x_1+x_3) + 3h,$ thus we have $\\overline{C}(\\Gamma_1) \\brak{1}\\cong \\left (\\begin{array}{cc}\\overline{\\pi}_{23} & x_2-x_3 \\\\\n0 & -\\overline{u}'_2 \\end{array} \\right).$ Since $x_1$ is an internal variable, we can eliminate it by removing the row $(0 \\quad -\\overline{u}'_2) .$ Therefore, $\\overline{C}(\\Gamma_1) \\brak {1} \\cong (\\overline{\\pi}_{23}, \\, x_2-x_3 ) = \\overline{C}(\\Gamma).$ \\end{proof}\n\n\\begin{proposition}\\label{isom 2}(\\textit{Second Isomorphism})\nThere is an isomorphism in the category $hmf_{\\omega}$:\n\\[\\overline{C}(\\raisebox{-13pt}{\\includegraphics[height=.45in]{removedweb1.pdf}}) \\cong \\overline{C}(\\raisebox{-13pt}{\\includegraphics[height=.45in]{removedweb2.pdf}})\\{-1\\} \\oplus \\overline{C}(\\raisebox{-13pt}{\\includegraphics[height=.45in]{removedweb2.pdf}})\\{1\\}.\\]\n\\end{proposition}\n\n\\begin{proof} The proof is the same as that of the ``Direct sum decomposition II'' in~\\cite{KhR1}.\n\\end{proof}\n\n\\begin{proposition}\\label{isom 3}(\\textit{Third Isomorphism})\nThere is an isomorphism in the category $hmf_{\\omega}$:\n\\[\\overline{C}(\\, \\raisebox{-8pt}{\\includegraphics[height=.3in]{2singresol.pdf}}\\,) \\cong \\overline{C}(\\,\\raisebox{-5pt}{\\includegraphics[height=.25in]{2arcs.pdf}}\\,).\\]\n\\end{proposition}\n\\begin{proof} Consider the webs $\\Gamma$ and $\\Gamma'$ depicted in Figure~\\ref{fig:direct sum decomposition 3-webs}. Factorizations $\\overline{C}(\\Gamma),\\overline{C}(\\Gamma')$ are $(R', \\omega)$-factorizations, where $R' = \\mathbb{Q}[a, h, x_1, x_2, x_3, x_4]$ and $\\omega = p(a, h, x_1) - p(a, h, x_2) + p(a, h, x_3) - p(a, h, x_4).$\n\\begin{figure}[h]\n\\[ \\Gamma = \\raisebox{-20pt}{\\includegraphics[height=.6in]{l2singres.pdf}} \\hspace{3cm} \\Gamma' = \\raisebox{-20pt}{\\includegraphics[height=.6in]{larcsleftright.pdf}}\\]\n\\caption{Webs $\\Gamma$ and  $\\Gamma'$}\n\\label{fig:direct sum decomposition 3-webs}\n\\end{figure}\n\nIn Koszul form, $\\overline{C}(\\Gamma) = (\\textbf{a, b})\\{-2\\} = \\left(\\begin{array}{cc}\\overline{u}_1'  & x_1 + x_5 -x_4 -x_6 \\\\ \\overline {u}_2' & x_1x_5 -x_4x_6\\\\ \\overline{u}_1'' & x_3 + x_6 -x_2 -x_5 \\\\  \\overline{u}_2'' & x_3x_6 -x_2x_5  \\end{array} \\right )\\{-2\\},$ \n\nwhere $\\overline{u}_i' = \\overline{u}_i(x_1, x_5, x_6, x_4)$ and $\\overline{u}_i''= \\overline{u}_i(x_3, x_6, x_5, x_2), i = 1, 2.$ A shift of $\\{-1\\}$ corresponds to the second and fourth row. The potential $\\omega$ lives in $R',$ thus $x_5$ and $x_6$ are internal variables. Knowing that $\\overline{u}_2' = -3(x_4 + x_6) + 3h$ and $\\overline{u}_2''= -3(x_2 + x_5) + 3h,$ we can exclude $x_5, x_6$ by crossing out the second and fourth row and replacing $x_5 = h-x_2$ and $x_6 = h-x_4$ in the first and third row of $(\\textbf{a, b}).$ In particular, $\\overline{C}(\\Gamma)$ is isomorphic to the matrix factorization with Koszul form\n$$(\\textbf{c, d}) = \\left(\\begin{array}{cc}\\overline{u}_1(x_1, h -x_2, h-x_4, x_4) & x_1 -x_2 \\\\ \\overline{u}_1(x_3, h-x_4,h- x_2, x_2) & x_3 -x_4\\end{array} \\right). $$\n\nAn easy computations shows that $\\overline{u}_1(x_1, h -x_2,h- x_4, x_4) = \\overline{\\pi}_{21}$ and $\\overline{u}_1(x_3, h -x_3, h-x_2, x_2) = \\overline{\\pi}_{43}$, thus $\\{\\textbf{c, d}\\} = \\overline{C}(\\Gamma'),$ which implies that $\\overline{C}(\\Gamma) \\cong \\overline{C}(\\Gamma').$\n\\end{proof}\n\n\\begin{proposition}\\label{isom 4}(\\textit{Fourth Isomorphism})\nThere are isomorphisms in the category $hmf_{\\omega}$:\n\\[\\overline{C}(\\, \\raisebox{-15pt}{\\includegraphics[height=.5in]{4-isom-4.pdf}}\\,) \\cong \\overline{C}(\\,\\raisebox{-15pt}{\\includegraphics[height=.5in]{4-isom-1.pdf}}\\,) \\quad \\text{and} \\quad \\overline{C}(\\, \\raisebox{-15pt}{\\includegraphics[height=.5in]{4-isom-2.pdf}}\\,) \\cong \\overline{C}(\\,\\raisebox{-15pt}{\\includegraphics[height=.5in]{4-isom-3.pdf}}\\,).\\]\n\\end{proposition}\n\n\\begin{proof}\nConsider the webs $\\Gamma_1$ and $\\Gamma_2$ depicted in Figure~\\ref{fig:direct sum decomposition 4-webs}. This time, the potential has the form $\\omega = \\sum_{i \\in \\{1,2,3\\}}p(a, h, x_i) - \\sum_{j \\in \\{4,5,6\\}}p(a, h, x_j) \\in R',$ where  $R' = \\mathbb{Q}[a, h, x_1, x_2, x_3, x_4, x_5, x_6].$\n\\begin{figure}[h]\n\\[ \\Gamma_1 = \\raisebox{-42pt}{\\includegraphics[height=1.2in]{l4-isom-4.pdf}} \\hspace{3cm} \\Gamma_2 = \\raisebox{-42pt}{\\includegraphics[height=1.2in]{l4-isom-1.pdf}}\\]\n\\caption{Webs $\\Gamma_1$ and  $\\Gamma_2$}\n\\label{fig:direct sum decomposition 4-webs}\n\\end{figure}\n\nIn Koszul form, $\\overline{C}(\\Gamma_2) = \\left(\\begin{array}{cc}\\overline{u}_1'  & x_1 + x_2 -x_7 -x_8 \\\\ \\overline {u}_2' & x_1x_2 -x_7x_8\\\\ \\overline{u}_1'' & x_3 + x_8 -x_6 -x_9 \\\\  \\overline{u}_2'' & x_3x_8 -x_6x_9 \\\\  \\overline{u}_1''' & x_7 + x_9 -x_4 -x_5 \\\\  \\overline{u}_2''' & x_7x_9 -x_4x_5  \\end{array} \\right )\\{-3\\},$ \n\nwhere $\\overline{u}_i' = \\overline{u}_i(x_1, x_2, x_8, x_7), \\overline{u}_i''= \\overline{u}_i(x_8, x_3, x_6, x_9)$ and $\\overline{u}_i'''= \\overline{u}_i(x_7, x_9, x_5, x_4)$ for $i = 1, 2.$ A shift by $\\{-1\\}$ was applied to the rows of $\\overline{u}_2', \\overline{u}_2''$ and $\\overline{u}_2'''.$  Variables $x_7, x_8$ and $x_9$ are internal variables and we can use $\\overline{u}_2' = -3(x_7 + x_8) + 3h$ and $\\overline{u}_2''= -3(x_6 + x_9) + 3h$ to exclude $x_8$ and $x_9,$ by crossing out the second and fourth row and substituting in every other row $x_8 = h-x_7$ and $x_9 = h-x_6.$ To exclude the internal variable $x_7,$ we use the right-hand entry of the third row, which now has the form $-x_7 + x_3.$ After these operations, we obtain\n$$\\overline{C}(\\Gamma_2) \\cong \\left(\\begin{aligned}a_1 &= \\overline{u}_1(x_1, x_2 ,h-x_3, x_3) \\\\  a_2 &= \\overline{u}_1(x_3, h-x_6, x_5, x_4) \\\\ a_3 &= \\overline{u}_2(x_3, h-x_6, x_5, x_4) \\end{aligned}   \\quad \\begin{aligned} b_1 &= x_1 + x_2 -h \\\\ b_2 &= x_3 +h - x_6 -x_4 -x_5\\\\ b_3 &= x_3(h-x_6) - x_4 x_5 \\end{aligned} \\right )\\{-1\\}.$$\n We perform the row operation $[21]_{-1}$ and arrive at \n\\[ \\overline{C}(\\Gamma_2)\\{1\\} \\cong \\left(\\begin{array}{cc}a_1  & b_1 + b_2 \\\\ a_2 -a_1 & b_2 \\\\ a_3 & b_3 \\end{array} \\right).\\] \nNow we apply a twist to the left-hand entries of the first two rows above, for $k= \\frac{h}{2} -x_1 -x_2 -x_4 -x_5 + x_3 -x_6,$ to obtain\n \\[ \\left(\\begin{array}{cc}a_1+ kb_2  & b_1 + b_2 \\\\ a_2 -a_1 -k(b_1 + b_2) & b_2\\\\ a_3 & b_3 \\end{array} \\right). \\] \n Performing the row operation $[23]_{x_3 -x_6}$ we get\n \\[\\overline{C}(\\Gamma_2)\\{1\\} \\cong \\left(\\begin{array}{cc}a_1+ kb_2  & b_1 + b_2 \\\\ a_2 -a_1 -k(b_1 + b_2)+(x_3-x_6)a_3 & b_2\\\\ a_3 & b_3 - (x_3 -x_6)b_2 \\end{array} \\right ) =  \\left(\\begin{array}{cc}a'_1& b'_1 \\\\ a'_2& b'_2\\\\ a'_3 & b'_3 \\end{array} \\right).\\]\n The later Koszul factorization $(\\textbf{a}', \\textbf{b}')$ is isomorphic in $hmf_{\\omega}$ to $\\overline{C}(\\Gamma_2)\\{1\\},$ and we apply to it a twist with $k' =-\\frac{1}{3}$\n  \\[\\overline{C}(\\Gamma_2)\\{1\\} \\cong \\left(\\begin{array}{cc}a'_1& b'_1 \\\\ a'_2& b'_2 - \\frac{1}{3} a'_3\\\\ a'_3 & b'_3 + \\frac{1}{3} a'_2 \\end{array} \\right),\\]\nfollowed by the row operation $[23]_{-(x_3 - x_6 - x_4 - x_5)}$ \n \\[\\overline{C}(\\Gamma_2)\\{1\\} \\cong \\left(\\begin{array}{cc}a'_1& b'_1 \\\\ a'_2 -(x_3 -x_6 -x_4- x_5)a'_3& b'_2 - \\frac{1}{3} a'_3\\\\ a'_3 & b'_3 + \\frac{1}{3} a'_2 + (x_3 -x_6 -x_4- x_5)(b'_2 -\\frac{1}{3}a'_3) \\end{array}\\right).\\]\n \n Let's denote the previous Koszul matrix by $(\\textbf{a}'', \\textbf{b}'').$ Replacing the entries of $\\textbf{b}''$ we have\n\\[\\overline{C}(\\Gamma_2)\\{1\\} \\cong \\left(\\begin{array}{cc}a''_1& x_1+ x_2 +x_3 -x_4-x_5-x_6 \\\\ a''_2 & x_3-x_6\\\\ a''_3 & x_1x_2-x_4x_5 \\end{array} \\right).\\]\n Finally, we apply the row operation $[21]_1$ followed by a twist with $k'' = -\\frac{3}{2}h+x_1 + x_2 + 2x_4 + 2x_5 - x_3 + x_6,$ and we get\n \\begin{align*}\\overline{C}(\\Gamma_2)\\{1\\} &\\cong \\left(\\begin{array}{cc}a''_1& x_1+ x_2 -x_4-x_5 \\\\ a''_2 + a''_1 & x_3-x_6\\\\ a''_3 & x_1x_2-x_4x_5 \\end{array} \\right)\\\\\n  &\\cong  \\left(\\begin{array}{cc}a''_1 +k''(x_3-x_6) & x_1+ x_2 -x_4-x_5 \\\\ a''_2 + a''_1 - k''(x_1 + x_2 -x_4 -x_5) & x_3-x_6\\\\ a''_3 & x_1x_2-x_4x_5\\end{array}\\right).\\end{align*}\n Computing the entries in the left column above, one founds\n \\begin{align*}\\overline{C}(\\Gamma_2)\\{1\\} &\\cong \\left(\\begin{array}{cc}\\overline{u}_1(x_1, x_2, x_4, x_5) & x_1+ x_2 -x_4-x_5 \\\\ \\overline{\\pi}_{63} & x_3-x_6 \\\\ \\overline{u}_2(x_1 ,x_2, x_4, x_5) & x_1x_2-x_4x_5 \\end{array} \\right)\\\\\n &\\cong \\left(\\begin{array}{cc} \\overline{\\pi}_{63} & x_3-x_6 \\\\ \\overline{u}_1(x_1, x_2, x_4, x_5) & x_1+ x_2 -x_4-x_5 \\\\ \\overline{u}_2(x_1 ,x_2, x_4, x_5) & x_1x_2-x_4x_5 \\end{array} \\right) \\cong \\overline{C}(\\Gamma_1)\\{1\\}.\\end{align*}\nThe later matrix is the Koszul form of the factorization $\\overline{C}(\\Gamma_1)\\{1\\},$ thus $\\overline{C}(\\Gamma_2) \\cong \\overline{C}(\\Gamma_1).$ \n \\begin{figure}[h]\n\\[ \\Gamma_3 = \\raisebox{-42pt}{\\includegraphics[height=1.2in]{l4-isom-2.pdf}} \\hspace{3cm} \\Gamma_4 = \\raisebox{-42pt}{\\includegraphics[height=1.2in]{l4-isom-3.pdf}}\\]\n\\caption{Webs $\\Gamma_3$ and  $\\Gamma_4$}\n\\label{fig:direct sum decomposition 4bis-webs}\n\\end{figure}\n\nWith the labeling of the diagrams $\\Gamma_3, \\Gamma_4$ given in Figure~\\ref{fig:direct sum decomposition 4bis-webs}, the previous proof also implies that $\\overline{C}(\\Gamma_4) \\cong \\overline{C}(\\Gamma_3)$ in $hmf_\\omega.$\n\\end{proof}\n\n\\begin{definition}\nLet $\\Gamma$ be a closed web and $p(\\Gamma)$ be the mod 2 number of circles in the modification of $\\Gamma$ obtained by replacing all singular resolutions with the oriented resolution. Factorization $\\overline{C}(\\Gamma)$ is a 2-complex and has cohomology only in degree $p(\\Gamma).$ Define the cohomology groups of $\\overline{C}(\\Gamma)$ as\n\\[ \\overline{H}(\\Gamma) : = H^{p(\\Gamma)}(\\overline{C}(\\Gamma)). \\]\n\\end{definition}\n\n$\\overline{H}(\\Gamma)$ is a $\\mathbb{Z}$-graded module over $\\mathbb{Q}[a, h].$ The isomorphisms obtained in this section together with the fact that the skein relations in Figure~\\ref{fig:skein relations} determine the evaluation of $\\brak{\\Gamma},$ for any web $\\Gamma,$ imply the following result.\n\n\\begin{proposition}\nFor any closed web $\\Gamma,$ the graded dimension of $\\overline{H}(\\Gamma)$ is $\\brak {\\Gamma},$ namely\n\\[ \\sum _{j \\in \\mathbb{Z}} q^j \\rk_{\\mathbb{Q}[a,h]} \\overline{H}^j(\\Gamma) = \\brak{\\Gamma}.\\]\n\\end{proposition}\n\n\\begin{remark}\nNote that any resolution $\\Gamma$ of a link diagram consists of a disjoint union of closed webs $\\cup_k \\Gamma_k,$  and its ``homology'' satisfies $\\overline{H}(\\Gamma) \\cong \\mathcal{A}^k.$ In Section~\\ref{sec:TQFT} we show that $\\overline{H}$ can be regarded as a $(1+1)$--dimensional TQFT functor.\n\\end{remark}\n\n\n\n\\section{Invariance under Reidemeister moves}\\label{sec:invariance}\n\n\\begin{theorem}\\label{invariance}\nIf $D$ and $D'$ are two diagrams representing the same tangle $T,$ then complexes $C(D)$ and $C(D')$ are isomorphic in $K_{\\omega}.$\n\\end{theorem}\n\\begin{proof}\n\\textbf{Reidemeister I.}\nConsider diagrams $D$ and $\\Gamma$ that differ only in a circular region as in the figure below:\n$$D=\\raisebox{-11pt}{\\includegraphics[height=0.45in]{leftkink.pdf}}\\qquad\n\\Gamma=\\raisebox{-13pt}{\\includegraphics[height=0.45in]{arc.pdf}}$$\nThe complex $C(D)$ has the form\n$$0\\longrightarrow \\underline{\\overline{C}(\\Gamma_2) \\{-1\\}} \\stackrel{\\Lambda_0}{\\longrightarrow} \\overline{C}(\\Gamma_1) \\{-2\\} \\longrightarrow 0$$ \nwhere\n$$\\Gamma_2 = \\raisebox{-20pt}{\\includegraphics[height=.6in]{lreid1orien.pdf}}\\quad \\mbox{and} \\quad\\raisebox{-22pt}{\\includegraphics[height=0.65in]{lsingresclosed.pdf}} = \\Gamma_1$$\nLet $Y_1 \\subset \\overline{C}(\\Gamma_2)\\{-1\\}$ be the inclusion of $\\overline{C}(\\Gamma)\\brak{1}\\{-2\\}$ onto the first summand of $\\overline{C}(\\Gamma_2)\\{-1\\},$ and $Y_2 = f(\\overline {C}(\\Gamma)\\brak{1}\\{-2\\}) \\subset \\overline{C}(\\Gamma_2)\\{-1\\},$ where $f \\co \\overline {C}(\\Gamma)\\brak{1}\\{-2\\} \\to \\overline{C}(\\Gamma_2)\\{-1\\}, f = \\id \\otimes[ m(x_1) \\iota]$. Here we used that $\\overline{C}(\\Gamma_2) \\cong \\overline{C}(\\Gamma)\\brak{1} \\otimes_{\\mathbb{Q}[a,h]} \\mathcal{A}.$\n\nThus $\\overline{C}(\\Gamma_2)\\{-1\\} \\cong Y_1 \\oplus Y_2$ in $hmf_\\omega.$ On the other hand, from the First Isomorphism we know that $\\overline{C}(\\Gamma)\\{-2\\} \\cong \\overline{C}(\\Gamma) \\brak{1}\\{-2\\},$ therefore\n the complex $\\overline{C}(D)$ is isomorphic in $K_{\\omega}$ to the direct sum\n\\begin{align*} \n0 \\longrightarrow &\\underline{Y_2} \\longrightarrow 0 \\\\ \n0 \\longrightarrow &\\underline{Y_1} \\stackrel{\\cong}{\\longrightarrow} \\overline{C}(\\Gamma)\\brak{1}\\{-2\\}. \\end{align*} \nSince $Y_2 \\cong \\overline{C}(\\Gamma)$ and the second summand is contractible, it implies that $C(D) \\cong (\\Gamma),$ in the category $K_{\\omega}.$\n\nA similar approach is used to prove the invariance under Reidemeister I involving a positive kink.\n\n\\textbf{Reidemeister IIa.} Consider diagrams $D_1$ and $\\Gamma$ that differ only in a circular region as in the figure below:\n$$D_1=\\raisebox{-11pt}{\\includegraphics[height=0.45in]{tuckedsame.pdf}}\\qquad\n\\Gamma=\\raisebox{-13pt}{\\includegraphics[height=0.45in]{arcsup.pdf}}$$\n\nThe complex $C(D_1)$ has the form\n$$ 0\\longrightarrow \\overline{C}(\\Gamma_{00})\\{1\\} \\stackrel{(f_1, f_3)^t}{\\longrightarrow} \\underline{\\overline{C}(\\Gamma_{01}) \\oplus \\overline{C}(\\Gamma_{10})} \\stackrel{(f_2, - f_4)}{\\longrightarrow} \\overline{C}(\\Gamma_{11})\\{-1\\} \\longrightarrow 0,$$\nwhose objects are the matrix factorizations corresponding to the four resolutions of $D$ given in Figure~\\ref{fig:reidIIA}, with potential $\\omega = p(a, h, x_1) + p(a, h, x_2) - p(a, h, x_3)  - p(a, h, x_4).$\n\n\\begin{figure}[ht]\n\\raisebox{-11pt}{\\includegraphics[height=2.2in]{reidIIA.pdf}}\n\\caption{}\\label{fig:reidIIA}\n\\end{figure}\nUsing the Second Isomorphism and that the marking doesn't matter, we have  \n$$\\overline{C}(\\Gamma_{01}) \\cong \\overline{C}(\\Gamma^0)$$ $$ \\overline{C}(\\Gamma_{00})\\cong \\overline{C}(\\Gamma^1) \\cong \\overline{C}(\\Gamma_{11})$$ $$ \\overline{C}(\\Gamma_{10})\\cong \\overline{C}(\\Gamma^1)\\{1\\} \\oplus \\overline{C}(\\Gamma^1)\\{-1\\},$$\n\nwhere $\\Gamma^0$ and $\\Gamma^1$ are the diagrams from figure~\\ref{maps}. Therefore the complex $C(D_1)$ is isomorphic (in $K_{\\omega}$) to the complex\n$$ 0 \\longrightarrow \\overline{C}(\\Gamma^1)\\{1\\} \\stackrel{(f_1, f_{03}, f_{13})^t}{\\longrightarrow} \\underline{\\overline{C}(\\Gamma^0) \\oplus \\overline{C}(\\Gamma^1)\\{1\\} \\oplus \\overline{C}(\\Gamma^1) \\{-1\\}} \\stackrel{(f_2, -f_{04}, -f_{14})}{\\longrightarrow} \\overline{C}(\\Gamma^1)\\{-1\\} \\longrightarrow 0,$$\n\n(where $f_{03}, f_{13}$ and $ f_{04}, f_{14}$ are the components of $f_3$ and $f_4$, respectively, under the Second Isomorphism). The later complex decomposes into the direct sum of complexes\n\\begin{align*}\n0 \\longrightarrow & \\underline{\\overline{C}(\\Gamma^0)} \\longrightarrow 0 \\\\\n0\\longrightarrow \\overline{C}(\\Gamma^1)\\{1\\} \\stackrel{f_{03}}{\\longrightarrow} & \\underline{\\overline{C}(\\Gamma^1) \\{1\\}} \\longrightarrow 0 \\\\\n0 \\longrightarrow &\\underline{\\overline{C}(\\Gamma^1)\\{-1\\}} \\stackrel{f_{14}}{\\longrightarrow}  \\overline{C}(\\Gamma^1)\\{-1\\} \\longrightarrow 0.\\end{align*}\n\nThe last two complexes are contractible (this is because the only degree $0$ endomorphisms of $\\overline{C}(\\Gamma^1)$ are rational multiples of the identity endomorphism, thus $f_{03}$ and $f_{14}$ are isomorphisms). Moreover, $\\overline{C}(\\Gamma^0) \\cong \\overline{C}(\\Gamma)$ in $hmf_\\omega^{fd},$ and we conclude that  $C(D_1)$ and $C(\\Gamma)$ are isomorphic in $K_\\omega.$\n\n\\textbf{Reidemeister IIb.}\n$$D_2=\\raisebox{-7pt}{\\includegraphics[height=0.25in]{tuckedop.pdf}}\\qquad\n\\Gamma'=\\raisebox{-7pt}{\\includegraphics[height=0.25in]{arcsop.pdf}}$$\n\nThe complex of matrix factorizations $C(D_2)$ is an element of the category $K_{\\omega},$ where $\\omega = p(a, h, x_1) - p(a, h, x_2) + p(a, h, x_3) - p(a, h, x_4),$ and has the form\n$$ 0\\longrightarrow \\overline{C}(\\Gamma_{00})\\{1\\} \\stackrel{(f_1, f_3)^t}{\\longrightarrow} \\underline{\\overline{C}(\\Gamma_{01}) \\oplus \\overline{C}(\\Gamma_{10})} \\stackrel{(f_2, - f_4)}{\\longrightarrow} \\overline{C}(\\Gamma_{11})\\{-1\\} \\longrightarrow 0.$$\n\nThe resolutions of $D_2$ are given in Figure~\\ref{fig:reidIIB}. \n\\begin{figure}[ht]\n\\raisebox{-11pt}{\\includegraphics[height=2in]{reidIIB.pdf}}\n\\caption{}\\label{fig:reidIIB}\n\\end{figure}\n\nWe know that $ \\overline{C}(\\Gamma_{01}) \\cong \\overline{C}(\\Gamma'') \\brak{1} \\otimes _{\\mathbb{Q}[a,h]} \\mathcal{A} \\cong \\overline{C}(\\Gamma'') \\brak{1} \\{1\\} \\oplus \\overline{C}(\\Gamma'')\\brak{1} \\{-1\\},$\n$$\\overline{C}(\\Gamma_{00}) \\cong \\overline{C}(\\Gamma'') \\brak{1} \\cong \\overline{C}(\\Gamma_{11})\\quad \\mbox{and} \\quad \\overline{C}(\\Gamma_{10}) \\cong \\overline{C}(\\Gamma'),$$\nwhere $\\Gamma''$ is the diagram in Figure~\\ref{fig:Gamma-dprime}. Here we used the First Isomorphism and the Third Isomorphism, and that marking doesn't matter.\n\\begin{figure}[ht]\n\\raisebox{-11pt}{\\includegraphics[height=.9in]{larcsupdown.pdf}}\n\\caption{}\\label{fig:Gamma-dprime}\n\\end{figure}\n\nConsequently, $C(D_2)$ is isomorphic to the following complex\n$$ 0 \\longrightarrow  \\overline{C}(\\Gamma'') \\brak{1}\\{1\\} \\longrightarrow \\underline{\\overline{C}(\\Gamma'') \\brak{1} \\{1\\} \\oplus \\overline{C}(\\Gamma'') \\brak{1} \\{-1\\} \\oplus \\overline{C}(\\Gamma')} \\longrightarrow \\overline{C}(\\Gamma'') \\brak{1} \\{-1\\} \\longrightarrow 0,$$\nwhich decomposes into the direct sum of complexes\n\\begin{align*}\n0 \\longrightarrow &\\underline{\\overline{C}(\\Gamma')} \\longrightarrow 0 \\\\\n0 \\longrightarrow \\overline{C}(\\Gamma'')\\brak{1}\\{1\\} \\stackrel{\\cong}{\\longrightarrow} &\\underline{\\overline{C}(\\Gamma'')\\brak{1}\\{1\\}} \\longrightarrow 0 \\\\\n0 \\longrightarrow & \\underline{\\overline{C}(\\Gamma'')\\brak{1}\\{-1\\}} \\stackrel{\\cong}{\\longrightarrow} \\overline{C}(\\Gamma'')\\brak{1}\\{-1\\} \\longrightarrow 0.\\end{align*}\nThe last two are contractible, therefore $C(D_2) \\cong C(\\Gamma')$ in the category $K_{\\omega}.$\n\n\\textbf{Reidemeister III.} Given diagrams $D$ and $D'$ below, we show that complexes $C(D)$ and $C(D')$ are isomorphic by showing they are both isomorphic to the same third complex.\n\n$$D \\quad \\raisebox{-13pt}{\\includegraphics[height=.5in]{reid3left.pdf}} \\hspace{2cm} \\raisebox{-13pt}{\\includegraphics[height=.5in]{reid3right.pdf}} \\quad D'$$\n\nThe cube of resolutions corresponding to the diagram $D$ is given in Figure~\\ref{fig:reidIII-left}, and that of $D'$ in Figure~\\ref{fig:reidIII-right}.\n\\begin{figure}[ht]\n\\raisebox{-8pt}{\\includegraphics[height=3.5in]{reidIII-left.pdf}}\n\\caption{The cube of resolutions of $D$}\\label{fig:reidIII-left}\n\\end{figure}\n\n\\begin{figure}[ht]\n\\raisebox{-8pt}{\\includegraphics[height=3.5in]{reidIII-right.pdf}}\n\\caption{The cube of resolutions of $D'$}\\label{fig:reidIII-right}\n\\end{figure}\n\nThe complex $C(D)$ has the form\n\\[0 \\longrightarrow \\overline{C}(\\Gamma_{000}) \\{6\\} \\stackrel{d^{-3}}{\\longrightarrow} \\left ( \\begin{array}{c} \\overline{C}(\\Gamma_{001})\\{5\\} \\\\ \\overline{C}(\\Gamma_{010})\\{5\\} \\\\ \\overline{C}(\\Gamma_{100})\\{5\\} \\end{array}  \\right ) \\stackrel{d^{-2}}{\\longrightarrow} \\left ( \\begin{array}{c} \\overline{C}(\\Gamma_{011})\\{4\\} \\\\ \\overline{C}(\\Gamma_{101})\\{4\\} \\\\ \\overline{C}(\\Gamma_{110})\\{4\\} \\end{array}  \\right) \\stackrel{d^{-1}}{\\longrightarrow} \\underline{\\overline{C}(\\Gamma_{111})\\{3\\}} \\longrightarrow 0.\\]\n\nWe know that\n\\begin{align*}\n\\overline{C}(\\Gamma_{110})  & \\cong \\overline{C}(\\Gamma_{011}) \\quad (\\text{the two webs are isotopic})\\\\\n \\overline{C}(\\Gamma_{010}) &\\cong \\overline{C}(\\Gamma_{011}) \\{1\\} \\oplus \\overline{C}(\\Gamma_{011}) \\{-1\\} \\quad (\\text{by the Second Isomorphism})\\\\\n \\overline{C}(\\Gamma_{000}) &\\cong \\overline{C}(\\Gamma_{011}) \\quad (\\text{by the Fourth Isomorphism}).\\end{align*}\n \nThe complex $C(D')$ has the same form as $C(D).$ Moreover, similar isomorphisms of matrix factorizations as those above hold for resolutions of $D',$ with the remark that $\\Gamma_{ijk}$ in $C(D)$ should be replaced by $\\Gamma'_{ijk}$ in $C(D').$\n \n $C(D)$ is isomorphic to the following complex\n \\[0 \\longrightarrow \\overline{C}(\\Gamma_{011}) \\{6\\} \\stackrel{d^{-3}}{\\longrightarrow} \\left ( \\begin{array}{c} \\overline{C}(\\Gamma_{001})\\{5\\} \\\\ \\overline{C}(\\Gamma_{011})\\{6\\} \\\\ \\overline{C}(\\Gamma_{011})\\{4\\}\\\\ \\overline{C}(\\Gamma_{100})\\{5\\} \\end{array}  \\right ) \\stackrel{d^{-2}}{\\longrightarrow} \\left ( \\begin{array}{c} \\overline{C}(\\Gamma_{011})\\{4\\} \\\\ \\overline{C}(\\Gamma_{101})\\{4\\} \\\\ \\overline{C}(\\Gamma_{011})\\{4\\} \\end{array}  \\right) \\stackrel{d^{-1}}{\\longrightarrow} \\underline{\\overline{C}(\\Gamma_{111})\\{3\\}} \\longrightarrow 0,\\]\n but the later decomposes into contractible complexes of the form \n\\[0 \\longrightarrow \\overline{C}(\\Gamma_{011}) \\{6\\} \\stackrel{\\cong}{\\longrightarrow} \\overline{C}(\\Gamma_{011}) \\{6\\} \\longrightarrow 0 \\]\n\\[0 \\longrightarrow \\overline{C}(\\Gamma_{011}) \\{4\\} \\stackrel{\\cong}{\\longrightarrow} \\overline{C}(\\Gamma_{011}) \\{4\\} \\longrightarrow 0 \\]\nand the complex $\\mathcal{C}$\n\\[\\mathcal{C}: \\quad 0 \\longrightarrow \\left ( \\begin{array}{c} \\overline{C}(\\Gamma_{001})\\{5\\} \\\\ \\overline{C}(\\Gamma_{100})\\{5\\} \\end{array}\\right )  \\longrightarrow \\left ( \\begin{array}{c} \\overline{C}(\\Gamma_{011})\\{4\\} \\\\ \\overline{C}(\\Gamma_{101})\\{4\\} \\end{array}\\right) \\longrightarrow \\underline{\\overline{C}(\\Gamma_{111})\\{3\\}} \\longrightarrow 0.  \\]\nIn other words, complexes $C(D)$ and $\\mathcal{C}$ are isomorphic in $K_\\omega.$\n \nWe apply the same argument as in the case of $C(D)$ to conclude that $C(D')$ is isomorphic in $K_\\omega$ to the complex $\\mathcal{C'}$\n\\[\\mathcal{C'}: \\quad 0 \\longrightarrow \\left ( \\begin{array}{c} \\overline{C}(\\Gamma'_{001})\\{5\\} \\\\ \\overline{C}(\\Gamma'_{100})\\{5\\} \\end{array}\\right )  \\longrightarrow \\left ( \\begin{array}{c} \\overline{C}(\\Gamma'_{101})\\{4\\} \\\\ \\overline{C}(\\Gamma'_{011})\\{4\\} \\end{array}\\right) \\longrightarrow \\underline{\\overline{C}(\\Gamma'_{111})\\{3\\}} \\longrightarrow 0. \\]\nSince the web diagram $\\Gamma_{ijk}$ from $\\mathcal{C}$ and $\\Gamma'_{ijk}$ from $\\mathcal{C'}$ respectively are isotopic, their matrix factorizations are isomorphic in $hmf_{\\omega}.$ In particular $\\mathcal{C} \\cong \\mathcal {C'}$ in $K_\\omega,$ and the invariance under the third type of Reidemeister move follows. \n\\end{proof}\n\n\\begin{corollary}The isomorphism class of the object $C(D)$ in the category $K_\\omega$ is an invariant of the tangle $T.$\n\\end{corollary}\n\n\\textbf{The case of links.} When $T$ is a link $L,$ for any resolution $\\Gamma$ of $D,$ the corresponding factorization $\\overline{C}(\\Gamma)$ is a 2-periodic complex of graded $R' = \\mathbb{Q}[a,h]$-modules.  The category $K_\\omega$ is isomorphic to the category of finite-rank $\\mathbb{Z} \\oplus \\mathbb{Z} \\oplus \\mathbb{Z}_2$-graded $\\mathbb{Q}[a,h]$-modules. For each $\\Gamma,$ the homology groups of $\\overline{C}(\\Gamma)$ are nontrivial only in one degree, thus the cohomology of $C(D)$ is $\\mathbb{Z} \\oplus \\mathbb{Z}$-graded.\n\nWe denote by $KR_{a,h}(D)$ the universal Khovanov-Rozansky complex for $n=2.$ It is obtained from $C(D)$ by replacing each matrix factorization $\\overline{C}(\\Gamma)$ with its homology group $\\overline{H}(\\Gamma).$ Furthermore, we denote by $HKR_{a,h}$ the cohomology of $KR_{a,h}.$ Note that  $HKR_{a,h}$ is exactly the $\\mathbb{Z} \\oplus \\mathbb{Z}$-graded cohomology of $C(D).$ We have\n$$ HKR_{a,h}(D) : = \\bigoplus_{i, j \\in \\mathbb{Z}} H^{i,j}(C(D)). $$ \nIt follows from construction that the graded Euler characteristic of $HKR_{a,h}(D)$ is the polynomial $\\brak{L}.$ Specifically, the following holds\n$$ \\brak{L} = \\sum_{i,j \\in \\mathbb{Z}} (-1)^i q^j \\rk_{\\mathbb{Q}[a,h]} H^{i,j} (D).$$\n\n\n\\section{Understanding the differentials}\\label{sec:TQFT}\n\nThe ring $\\mathcal{A} = \\mathbb{Q}[a,h, X]\/(X^2 -hX - a)$ is commutative Frobenius with trace map\n $\\epsilon \\co \\mathcal{A} \\rightarrow \\mathbb{Q}[a, h], \\,\\,\\epsilon(1) = 0, \\,\\epsilon(X) = 1$.\n Multiplication $m \\co \\mathcal{A} \\otimes \\mathcal{A} \\rightarrow \\mathcal{A}$ and comultiplication $\\Delta \\co \\mathcal{A} \\rightarrow \\mathcal{A} \\otimes \\mathcal{A}$ are defined by\n$$ \\begin{cases}\n m(1 \\otimes X) =X, & m(X \\otimes 1) = X\\\\ \nm(1 \\otimes 1) =1, & m(X \\otimes X) =hX+ a\n \\end{cases},\\quad\n \\begin{cases}\n \\Delta(1) = 1 \\otimes X + X \\otimes 1-h1\\otimes 1\\\\ \n \\Delta(X) = X \\otimes X + a 1 \\otimes 1.\\end{cases} $$\n\nRecall that $\\mathcal{A} = \\brak {1, X}_{\\mathbb{Q}[a, h]}$ is graded with $\\deg(1) = -1$ and $\\deg(X) = 1.$ The trace $\\epsilon$ and unit $\\iota$ are maps of degree $-1,$ while multiplication and comultiplication are maps of degree 1.\n\nLet $\\Lambda_0^* \\co \\overline{H}(\\raisebox{-8pt}{\\includegraphics[height=0.3in]{orcircles.pdf}}) \\to \\overline{H}(\\raisebox{-8pt}{\\includegraphics[height=0.3in]{closed-web.pdf}})$ and $\\Lambda_1^*\\co \\overline{H}(\\raisebox{-8pt}{\\includegraphics[height=0.3in]{closed-web.pdf}}) \\to \\overline{H}(\\raisebox{-8pt}{\\includegraphics[height=0.3in]{orcircles.pdf}})$ be the maps induced by $\\Lambda_0$ and $\\Lambda_1,$ respectively, at the homology level of the corresponding factorizations.\n\nWe know that \n\\begin{align*}\n\\overline{C} (\\raisebox{-8pt}{\\includegraphics[height=0.3in]{closed-web.pdf}}) &\\stackrel {\\textit{hmf}_\\omega}{\\cong} \\overline{C}(\\raisebox{-8pt}{\\includegraphics[height=0.3in]{orcircle}}\\,)\\brak{1}, \\quad \\overline{H}(\\raisebox{-8pt}{\\includegraphics[height=0.3in]{orcircle}}\\,) \\cong \\mathcal{A}, \\\\\n\\overline{H}(\\raisebox{-8pt}{\\includegraphics[height=0.3in]{closed-web.pdf}}) &\\cong \\mathbb{Q}[a,h,x_1, x_2]\/(x_1 + x_2 -h,\\, x_1x_2 + a) \\{-1 \\} \\\\ & \\cong \\brak{1, x_1}_{\\mathbb{Q}[a,h]}\\{-1\\}, \\\\ \\overline{H}(\\raisebox{-8pt}{\\includegraphics[height=0.3in]{orcircles.pdf}}) &\\cong \\mathbb{Q}[a,h,x_1, x_2]\/(x_1^2 -hx_1 - a, \\, x_2^2 -hx_2 - a) \\{-2\\} \\\\\n&\\cong \\brak{1, x_1, h-x_2, x_1(h-x_2)}_{\\mathbb{Q}[a,h]} \\{-2\\}.\\end{align*}\n \nThere are $\\mathbb{Q}[a, h]$-module isomorphisms:\n\\begin{align} \\label{isomorphism f}\n \\overline{H}(\\raisebox{-8pt}{\\includegraphics[height=0.3in]{closed-web.pdf}}) \\stackrel{f}{\\longrightarrow} \\mathcal{A}, \\quad & f \\co \\begin{cases} 1 \\to 1\\\\ x_1 \\to X  \\end{cases} \\\\ \\label{isomorphism g}\n \\overline{H}(\\raisebox{-8pt}{\\includegraphics[height=0.3in]{orcircles.pdf}}) \\stackrel{g}{\\longrightarrow} \\mathcal{A} \\otimes \\mathcal{A}, \\quad & g\\co \\begin{cases}  1\\to 1\\otimes 1,\\,\\, x_1 \\to X \\otimes 1 \\\\ h-x_2 \\to 1 \\otimes X, \\,\\, x_1(h-x_2) \\to X\\otimes X.\\end{cases} \\end{align}\n \n \\begin{lemma}\n $\\Lambda_0^* = m$ and $\\Lambda_1^* = \\Delta.$\n \\end{lemma}\n \\begin{proof}\n We know that $\\Lambda_0\/(x_1 = x_4, x_2 = x_3) = \\id \\otimes \\,\\psi'_{x_1-x_2}$ and $\\Lambda_1\/(x_1 = x_4, x_2 = x_3)  = \\id \\otimes \\,\\psi_{x_1-x_2},$ that factorizations $\\overline{C}(\\Gamma^0)\/(x_1 = x_4, x_2 = x_3) = \\overline{C}(\\raisebox{-8pt}{\\includegraphics[height=0.3in]{orcircles.pdf}})$ and $\\overline{C}(\\Gamma^1)\/(x_1 = x_4, x_2 = x_3) = \\overline{C}(\\raisebox{-8pt}{\\includegraphics[height=0.3in]{closed-web.pdf}})$ have nontrivial homology only in degree 0. \n \n Applying the definition of the morphism $\\psi'_{x_1-x_2}$ we have \n \\begin{align*} \\Lambda^*_0(x_1) &= x_1, &\\Lambda^*_0(h-x_2) &= h-x_2 \\\\\n\\Lambda^*_0(1) &= 1, &\\Lambda^*_0 (x_1(h-x_2)) &= x_1(h-x_2).\\end{align*} But $h-x_2 = x_1$ and $x_1(h-x_2) = hx_1 + a$ in $\\overline{H}(\\raisebox{-8pt}{\\includegraphics[height=0.3in]{closed-web.pdf}}),$ thus using the isomorphisms \\ref{isomorphism f} and \\ref{isomorphism g} it follows that $\\Lambda_0^* = m.$ Similarly, using the definition of $\\psi_{x_1-x_2}$ we obtain\n \\begin{align*} \\Lambda^*_1(1) &= x_1-x_2 \\\\ \\Lambda^*_1(x_1) &= (x_1-x_2)x_1.\\end{align*} \n But $(x_1-x_2)x_1 =  hx_1 +a -x_1x_2 = a + x_1(h-x_2)$ in $\\overline{H}(\\raisebox{-8pt}{\\includegraphics[height=0.3in]{orcircles.pdf}}),$ and $x_1-x_2 = x_1 +(h -x_2) - h.$ By using again the isomorphisms \\ref{isomorphism f} and \\ref{isomorphism g} we obtain $\\Lambda^*_1 = \\Delta.$\n\\end{proof}\n\nIt is well known that the commutative Frobenius algebra $\\mathcal{A}$ gives rise to a TQFT functor---denoted here by $\\mathsf{F}$---from the category $OrCob$ of oriented $(1+1)$--dimensional cobordisms to the category $\\mathbb{Q}[a,h]$-Mod of graded $\\mathbb{Q}[a, h]$-modules and module homomorphisms. The functor assigns the ground ring $\\mathbb{Q}[a, h]$ to the empty 1-manifold, and $\\mathcal{A}^{\\otimes k}$ to the disjoint union of oriented $k$ circles. On the generating morphisms of $OrCob$, the functor associates the structure maps of the algebra $\\mathcal{A}.$ In  particular, $\\mathsf{F}(\\raisebox{-3pt}{\\includegraphics[height=0.15in]{cuplo.pdf}}) = \\iota,\\, \\mathsf{F}(\\raisebox{-3pt}{\\includegraphics[height=0.15in]{caplo.pdf}}) = \\epsilon,\\, \\mathsf{F}(\\raisebox{-3pt}{\\includegraphics[height=0.16in]{pair-of-pants.pdf}}) = m$ and\\, $\\mathsf{F}(\\raisebox{-3pt}{\\includegraphics[height=0.16in]{upsidedown-pants.pdf}}) = \\Delta$ (note that we read cobordisms from bottom to top). \n\nWe remark that the objects in $OrCob$ are oriented in such a way that a cobordism between two objects is ``properly'' oriented. For that, one needs to orient each nesting (concentric)  set of circles so that  orientations alternate, giving to the outermost circle the same orientation, for all nesting sets of circles. \n\nGiven a cobordism $S \\in OrCob,$ the homomorphism $\\mathsf{F}(S)$ has degree given by the formula $\\deg(S) = -\\chi(S),$ where $\\chi$ is the Euler charecteristic for $S.$ \n\nConsider a link diagram $D$ with its hypercube of resolutions and associated complex of matrix factorizations $C(D).$ Each resolution $\\Gamma$ of $D$ is a collection of closed webs with an even number of vertices and oriented loops. The homology group $\\overline{H}(\\Gamma)$ of the associated factorization $\\overline{C}(\\Gamma)$ satisfies $\\overline{H}(\\Gamma) \\cong \\mathcal{A}^k,$ where $k$ is the number of connected components of $\\Gamma.$ \n\nWe can replace each resolution of $D$ by a simplified one that has fewer number of vertices, via the First, Third and Fourth Isomorphisms. After this operation we are left with a complex of factorizations isomorphic in $K_\\omega$ to $C(D),$ in which each resolution is a disjoint union of basic closed webs (with exactly two vertices) and oriented loops. Finally, applying the First Isomorphism to all basic closed webs, the latter complex is isomorphic in $K_\\omega$ to a complex of factorizations whose underlying geometric objects are column vectors of nested oriented loops. Each nested set of loops is oriented in such a way that the outermost loop is oriented clockwise, by convention, and as we go inside of the nesting set of loops orientations alternate. In particular, the latter formal complex, call it $\\mathcal{C},$ is an object in the category of complexes over $OrCob.$ Applying the functor $\\mathsf{F}$ to $\\mathcal{C},$ we obtain an ordinary complex $\\mathsf{F}(\\mathcal{C})$ whose objects are $\\mathbb{Q}[a,h]$-modules and whose differentials are $\\mathbb{Q}[a,h]$-module homomorphisms. Moreover, $\\mathsf{F}(\\mathcal{C})$ is homotopy equivalent to $KR_{a,h}(D).$ \nConsequently, the following corollary is implied by the results of this section.\n\n\\begin{corollary}\nThe functor $\\overline{H}$ behaves in the same manner as $\\mathsf{F}$ does. \n\\end{corollary}\n\nRecalling the author's construction and results in~\\cite{CC2}, the previous Corollary implies that the underlying link cohomology and that introduced in~\\cite{CC2} are isomorphic, after tensoring them with appropriate rings. In particular, the functor $\\overline{H}$ is the same as the tautological functor $\\mathcal{F}$ in~\\cite{CC2} (at least when we restrict to the case of links). Moreover, this implies that the theory constructed here is functorial under link cobordisms, relative to boundaries.\n\n\\begin{corollary}\\label{cor:isomorphism}\nFor each oriented link $L$ there is an isomorphism \n$$HKR_{a,h}(L) \\otimes_{\\mathbb{Q}[a,h]} \\mathbb{Z}[i] \\cong \\mathcal{H}(L) \\otimes_{\\mathbb{Z}[i][a,h]} \\mathbb{Q}$$\nwhere $\\mathcal{H}$ is the cohomology link theory explored in~\\cite{CC2}. \\end{corollary}\n \n\\begin{corollary}\nGiven a link cobordism $C \\co L_1 \\to L_2,$ there is a well-defined induced map $HKR_{a,h}(C) \\co HKR_{a,h}(L_1) \\to HKR_{a,h}(L_2)$ between the associated cohomology groups.\n\\end{corollary}\n\n\n\\section{The algebra $\\overline{R}(\\Gamma)$}\n\nLet $\\Gamma$ be a resolution of a link $L$ and denote by $e(\\Gamma)$ the set of all edges in $\\Gamma.$ We introduce an algebra $\\overline{R}(\\Gamma)$ and exhibit the $\\overline{R}(\\Gamma)$ module structure of $\\overline{H}(\\Gamma).$\n\n\\begin{definition}\nWe define $\\overline{R}(\\Gamma) : = R\/(\\textbf{a}, \\textbf{b}),$ where $R = \\mathbb{Q}[a,h][X_i\\vert i \\in e(\\Gamma)]$ and $\\textbf{a}, \\textbf{b}$ are the polynomials that are used to define the factorization $\\overline{C}(\\Gamma).$\n\\end{definition}\n\n$\\overline{H}(\\Gamma)$ is an $\\overline{R}(\\Gamma)$ module, since multiplication by any polynomial in $(\\textbf{a},\\textbf{b})$ induces a null-homotopic endomorphisms of $\\overline{C}(\\Gamma).$\n\n\\begin{proposition}\\label{relations}\nThe algebra $\\overline{R}(\\Gamma)$ is spanned by generators $X_i,$ where $i\\in e(\\Gamma),$ subject to the following relations:\n\\begin{enumerate}\n\\item For every $X_i$ we have $X_i^2 = h X_i + a.$\n\\item For every singular resolution (of the form $\\Gamma^1$) in $\\Gamma,$ the generators $X_1, X_2, X_3, X_4$ satisfy $X_1 + X_2 = h = X_3 + X_4,$ and $X_1 X_2 = -a = X_3 X_4.$\n\\end{enumerate} \n\\end{proposition}\n\\begin{proof}\nFor each closed circle---with one mark $i$---in $\\Gamma,$ the expression $\\overline{\\pi}_{ii} = 3(x_i^2 - hx_i -a)$ lives in $(\\textbf{a},\\textbf{b}).$ For any other $X_i,$ we use that the multiplication by $\\partial_i \\omega= 3(x_i^2 -hx_i -a)$ induces a null-homotopic endomorphism of $\\overline{C}(\\Gamma).$ Thus $X_i^2 = h X_i + a$ holds for any $X_i.$ To prove the second statement we recall that for each singular resolution with arcs labeled as those of $\\Gamma^1,$ the polynomials $x_1 + x_2 -x_3 -x_4$ and  $x_1x_2 -x_3x_4$ are in $(\\textbf{a},\\textbf{b}),$ thus $X_1 + X_2 = X_3+X_4$ and $X_1X_2 = X_3X_4$ in  $\\overline{R}(\\Gamma),$ near each singular resolution of $\\Gamma.$ Moreover, since $\\overline{u}_2 = 3(x_3 +x_4) -3h$ and $\\overline{u}_1 = (x_1 + x_2)^2 + (x_1 + x_2)(x_3 + x_4) + (x_3 + x_4)^2 -3x_1x_2 -\\frac{3}{2} h( x_1+x_2+x_3 + x_4) -3a$ are also in $(\\textbf{a},\\textbf{b}),$ we have $X_1 + X_2 = h = X_3 + X_4,$ and $X_1 X_2 = -a = X_3 X_4.$\n\\end{proof}\n\n\\begin{remark}\nRelations given in Proposition~\\ref{relations} have a geometric interpretation via a TQFT with dots, where a dot stands for the multiplication by $X$ endomorphism of the algebra $\\mathcal{A}.$ Specifically, the relation $X_i^2 = h X_i + a$ translates into the following skein relation: a disk decorated by two dots equals a disk decorated by one dot times $h$ plus a disk times $a$ (this one is the relation (2D) in~\\cite{CC2}). Given two edges in $\\Gamma$ with labels $i$ and $j$ and which share a vertex, the relation $X_i + X_j = h$ gives a rule for exchanging dots between two neighboring facets of a foam, while relation $X_i X_j = -a$ means that if each of the two neighboring facets have a single dot, we can ``erase'' both dots and multiply the corresponding foam by $-a$ (see relations (ED) in~\\cite[Figure 7]{CC2}).\n\\end{remark}\n\nIt was proved in\\cite{CC2} that if one lets $a$ and $h$ be complex numbers and considers the polynomial $f(X) = X^2 -hX -a \\in \\mathbb{C}[X],$ the isomorphism class of the complex $\\mathfrak{sl}(2)$-foam cohomology $\\mathcal{H}(L, \\mathbb{C})$ is determined by the number of distinct roots of $f[X].$ Corollary~\\ref{cor:isomorphism} implies that this also holds for the complex matrix factorization cohomology $HKR_{a,h}(L, \\mathbb{C}).$ The results are as follows.\n\nIf $f(X) = (X-\\alpha)^2,$ for some $\\alpha \\in \\mathbb{C},$ there is an isomorphism between $HKR_{a,h}(L, \\mathbb{C})$ and the original $\\mathfrak{sl}(2)$ Khovanov-Rozansky cohomology over $\\mathbb{C}.$\n\nIf $f(X) = (X -\\alpha)(X-\\beta),$ for some $\\alpha, \\beta \\in \\mathbb{C},$ for each resolution $\\Gamma$ of a link $L,$ the cohomology $\\overline{H}(\\Gamma)$ is a free module of rank one over the complex algebra $\\overline{R}(\\Gamma).$ \n\n\\begin{proposition}\nFor any $n$-component link $L,$ the dimension of $HKR_{a,h}(L, \\mathbb{C})$ equals $2^n,$ and to each map $\\phi \\co \\{ \\text{components of L} \\} \\to S = \\{ \\alpha, \\beta \\}$ there exists a non-zero element $h_{\\phi} \\in HKR_{a,h}(L, \\mathbb{C})$ which lies in the cohomological degree\n$$ -2 \\sum_{\\substack {(u_1,u_2)\\in S \\times S \\\\ u_1 \\neq u_2}} lk(\\phi^{-1}(u_1), \\phi^{-1}(u_2)),$$\nand all $h_{\\phi}$ generate $HKR_{a,h}(L, \\mathbb{C}).$\n\\end{proposition}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Understanding and Improving Human-Machine Joint Action}\n\nHumans regularly make decisions with and alongside other humans. In what has come to be defined as {\\em  joint action}, humans coordinate with other humans to achieve shared goals, sculpting both their actions and their expectations about their partners during ongoing interaction (Sebanz et al. 2006; Knoblich et al. 2011; Pesquita et al. 2018). Humans now also regularly make decisions in partnership with computing machines in order to supplement their abilities to act, perceive, and decide (Pilarski et al. 2017). It is natural to expect that joint action with machine agents might be able to improve both work and play. In situations where someone is limited in their ability to perceive, remember, attend to, respond to, or process stimulus, a machine counterpart's specialized and complementary abilities to monitor, interpret, store, retrieve, synthesize, and relate information can potentially offset or even invert these limitations. Persistent computational processes that extend yet remain part of human cognition, perhaps best described in the words of the author Terry Pratchett as ``third thoughts,'' are evident in common tools like navigation software and calendar reminders.\n\nSpecifically with a focus on joint action, as opposed to more general forms of human-machine interaction, Moon et al. (2013), Bicho et al. (2010), Pilarski et al. (2013), Pezzulo et al. (2011), and others have provided compelling examples of fruitful human-machine coordination wherein a human and a machine work together and co-adapt in real-time joint action environments. One striking characteristic of many of these examples, and what separates them from other examples of human-machine interaction, is that they occur within {\\em peripersonal space} (Knoblich et al. 2011)---i.e., interaction is perceived by the human to unfold continuously in the region of physical space surrounding their body and upon which they can act. While the perception of spatial and temporal proximity between partners has been shown to significantly influence joint decision making (as reviewed by Knoblich et al. 2011), peripersonal joint action settings have to date received less attention than other settings for human-machine interaction. The study of how different machine learning approaches impact human-machine joint action is even less developed, but in our opinion equally important. Our present work therefore aims to extend the discussion on how a human decision maker (here termed a {\\em pilot}) and a machine learning assistant (termed a {\\em copilot}) can learn together and make decisions together in service of a shared goal. We do so by defining a virtual reality environment to probe real-time joint action between humans and learning machines, and, using this environment, contribute a human-machine interaction case study wherein we demonstrate the kinds of changes that might be expected as a pilot interacts with different machine learning copilots in a shared environment. \n\n\\begin{figure}[b]\n    \\centering\n    \\includegraphics[height=2in]{domain1}\n    \\includegraphics[height=2in]{domain2}      \n    \\includegraphics[height=2in]{domain3}  \n    \\caption{\\footnotesize The virtual-reality foraging environment used to explore human-agent learning and joint action, comprised of six equidistant fruit objects, background detail, and a repeated cycle of day and night illumination. The human participant (the {\\em pilot}) was able to move about within the virtual world and used their hand-held controllers to both harvest fruit in varying states of ripeness and train their machine-learning assistant (the {\\em copilot}).}\n    \\label{fig:world}\n\\end{figure}\n\n\\section{Virtual Reality Environment and Protocol}\n\nA single participant engaged in a foraging task over multiple experimental blocks. This foraging task was designed so as to embed a hard-to-learn sensorimotor skill within a superficially simple protocol. In each block, the participant was asked to interact with a simulated world and a machine assistant via a virtual reality (VR) headset and two hand-held controllers (HTC Vive with deluxe audio strap). The pilot was instructed that in each block they were to move through the world to collect objects, and that these objects would grant them ``points''; they were told that, during the experiment, their total points would be reflected in visual changes to the environment, and that they would receive a unique, momentary audio cue whenever they gained or lost points as a result of their actions (and that they could also be given different audio cues in situations where they might expect to gain or lose points). \n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=5in]{schematic.png}\n    \\caption{\\footnotesize Schematic showing the interactions available to both the human pilot and the machine copilot. Using the right and left hand controllers, respectively, the pilot was able to harvest fruit or train their copilot to estimate the value of hue\/saturation combinations, each action resulting in different sound cues as shown. On every time step, the copilot provided sound cues according to its learned predictions $V(h,s)$ and, for the bandit condition, its learned policy $\\pi(V)$.}\n    \\label{fig:protocol}\n    \\vspace{-1em}\n\\end{figure}\n\nLoosely inspired by the bee foraging example of Schultz et al. (1997), the virtual world presented to the pilot was a simple platform floating in space with six coloured balls (``fruit'') placed at equidistant locations around its perimeter (Fig. \\ref{fig:world}). To tease out behavioural changes in different conditions for pilot\/copilot joint action, over the course of a single trial the light level in the world varied from full sunlight (``day'') to regular periods of darkness lasting roughly 20s (``night'', c.f., Fig. \\ref{fig:results}c, bottom trace). While copilot perception was unaffected by light level, during twilight and night phases the pilot was by design unable to determine the colour of any objects in the environment, seeing only their general shape. The main mechanic of the world was that the pilot could collect fruit to either increase their points or, in some blocks, collect fruit in order to teach their copilot. \n\nTo create conditions for skill learning on the part of the pilot, the environment was designed with a confounding factor (ripeness) that related the appearance of the fruit as observed by the pilot to the reward structure of the task. Over the course of time, each fruit underwent a ``ripening cycle'' wherein it cycled through a progression of colours---hue and saturation levels---and reward that varied in relation to the time since the fruit's appearance. If a fruit was not contacted before a fixed time interval by the pilot, i.e., the end of the ripening cycle, it would disappear and no point gains or losses will be credited to the participant. A short, variable time after the participant collected a fruit or the fruit disappeared due to time, a new fruit would drop from above to the same position as the previous fruit. These new fruit were assigned a random fraction between 0\\% and 95\\% through their ripening cycle.  \n\nThe pilot started each trial in the centre of the platform, was able to move short distances and rotate in place, and could make contact with the fruit using either their left or right hand controllers (Fig. \\ref{fig:protocol}). Upon making contact with one of the fruit via their right-hand controller (``harvesting''), the ball would disappear and the pilot would hear an audio cue indicating that they either gained or lost points (one distinct sound for each case, with a small mote of light appearing in place of the fruit when points were gained). Upon contacting a ball with their left-hand controller (``teaching''), the fruit would disappear with no points gained or lost by the pilot, and they would hear a unique tone indicating that they had given information to their copilot (for blocks involving a copilot, otherwise the fruit disappeared and no sound played).\n\nHarvesting fruit was the only way the pilot could gain points. The points gained or lost by the participant for collecting a fruit was determined by a sinusoidal reward function that varied according to the time since the beginning of the ripening cycle (Fig. \\ref{fig:protocol}). All fruit in a given trial were assigned the same, randomly selected reward sinusoid (in terms of phase) and related hue\/saturation progression---i.e., all balls in a given trial would ripen in the same way, and would generate points in the same way, but these ripening mechanics would vary from trial to trial. The frequency of the reward sinusoid (i.e., number of reward maxima and minima until a fruit's disappearance), positive\/negative reward offset, length of the ripening cycle, and time range until reappearance (all akin to difficulty), were predetermined and held constant across all trials and blocks. Difficulty in terms of these parameters was empirically preselected and calibrated so as to provide the pilot with a challenging pattern-learning problem that was still solvable within a single trial.\n\nFollowing an extended period of familiarization with the navigation and control mechanics of the VR environment and the different interaction conditions, the participant (one of the co-authors for this pilot study) experienced three experimental blocks each consisting of three 180s trials; each trial utilized previously unseen ripening mechanics in terms of colour presentation and points phase. There was an approximately 20s break between trials.  The blocks related to the three different conditions (Fig. \\ref{fig:protocol}), presented in order, as follows:\n\n\n{\\bf Condition 1, No copilot (NoCP): } The pilot harvested fruit without any signalling or support from a machine copilot. Using the left, training hand to contact fruit had no effect, and did not provide any additional audio cues.\n\n\n{\\bf Condition 2, Copilot communication via Pavlovian control (Pav):} The pilot harvested fruit with support from a machine copilot that {\\em provided audio cues in a fixed way} based on its learned predictions. As described above, the pilot was able to train the copilot by contacting fruit with their left hand controller. Practically, this amounted to updating the value function of the copilot, denoted $V(h,s)$, according to the points value associated with a fruit's current hue $h$ and saturation level $s$ at the time of contact; updates were done via simple supervised learning.  At each point in time, the copilot queried its learned value function $V(h,s)$ with the current colour values of each of the six fruits, and, if the value of $V(h,s)$ was positive for a fruit, triggered an audio cue that was unique to that fruit---each fruit had a characteristic sound. In other words, feedback from the copilot to the human pilot was based on a pre-determined function that mapped learned predictions to specific actions (an example of Pavlovian control and communication, c.f., Modayil et al. (2014) and Parker et al. (2014)).\n\n{\\bf Condition 3, Copilot communication learned through trial and error (Bandit):}  The pilot harvested fruit with support from a machine copilot that {\\em provided audio cues in an adaptable way} based on collected points (reward) and its learned predictions. This condition was similar to Condition 2 in terms of how the pilot was able to train the copilot. However, instead of deterministically playing an audio cue for the pilot each time the copilot's prediction $V(h,s)$ for a given fruit was positive, the copilot was instead presented with a decision whether or not to play an audio cue for the pilot. The decision to play a cue was based on a stochastic policy $\\pi(V)$ that was updated as in a contextual bandit approach (Sutton and Barto, 2018) according to the points collected by the pilot if and when a fruit was harvested. In essence, if the copilot cued the user and this resulted in the pilot gaining points a short time later, it would reinforce the copilot's probability of playing a sound when it predicted a similar level of points in the future; should the pilot instead harvest the fruit and receive negative points, as when a fruit is harvested after the peak in its ripening cycle and\/or the pilot is consistently slow to react to the copilot's cue, the copilot would decrease its probability of playing a sound when it predicted that level of expected points. The copilot's control policy used learned predictions as state, c.f., prediction in human and robot motor control (Wolpert et al. 2001; Pilarski et al. 2013).\n\n\n\\section{Results and Discussion}\n\n\n\\begin{figure}[t]\n    \\centering\n     {\\bf(a)} \\includegraphics[height=1.4in]{results1.png} \\hfil\n     {\\bf(b)} \\includegraphics[height=1.4in]{results2.png}\\\\ \n {\\bf(c)} \\includegraphics[width=6.6in]{results3.png}\\\\\n    \\caption{ \\footnotesize Results from a single pilot participant for the three experimental conditions ({\\em NoCP, Pav,} and {\\em Bandit}). (a) Total score acquired by the pilot in each condition during day and night, as summed over all three trials in a block, along with the total score excluding any events with negative points (+); (b) total Bandit score with respect to fruit location to the front (F), middle (M) or back (B) with respect to the pilot's starting orientation; and (c) representative example of time-series data from the second trial for all three conditions, cross-plotting cumulative score, changes in score ($\\Delta Score$), light level, and bandit learning in terms of human teaching actions ($\\Delta V(h,s)$), copilot decisions to cue or not cue ({\\em up\/down ticks}), and post-cue updates to the copilot's policy as a result of pilot activity ($\\Delta \\pi$).}\n    \\label{fig:results}\n    \\vspace{-1em}\n\\end{figure}\n\nFigure \\ref{fig:results} presents the aggregate behaviour of the pilot in terms of total score over all three trials per condition, a fruit-by-fruit breakdown of total score, and a detailed presentation of time-series data from the second trial of the experiment. {As a key finding, we observed that interaction with different copilots (Pav and Bandit) led to different foraging behaviours on the part of the pilot during day and night, especially as compared to the no copilot (NoCP) condition.} Interactions with both the Pav and the Bandit copilot led to more foraging behaviour during night-time periods, as compared to the NoCP condition (Fig. \\ref{fig:results}c). Learning to interpret the communication from these copilots appeared to induce multiple foraging mistakes on the part of the pilot, especially during night-time phases (Fig. \\ref{fig:results}a,c) and less familiar fruit locations (Fig. \\ref{fig:results}b). Despite this, the total points collected in the absence of any mistakes (the score without any negative point value events, Fig. \\ref{fig:results}a+) suggest that collaboration with a policy-learning copilot could potentially lead to effective joint action once a good policy has been learned by both the pilot and the copilot. Teaching interactions ($\\Delta V(h,s)$, Fig. \\ref{fig:results}c) also provided a useful window into pilot skill learning. Broadly, the behaviour patterns observed in this preliminary study suggest that gradual addition of cues from a copilot, as in the Bandit policy-learning condition, is likely more appropriate than a strictly Pavlovian control approach. These initial results also indicate that there is room for more complex copilot architectures that can capture the appropriate timing of cues with respect to pilot activity (e.g., a pilot harvesting wrong fruit, or hesitation as per Moon et al. (2013)), and motivate more detailed study into the impact that pilot head position, gaze direction, light level, and other relevant signals have on a copilot's ability to generate good cues. Time delays and credit assignment matter in this joint-action setting and require further thought. \n\n{\\bf Conclusions:} This work demonstrated a complete (though straightforward) cycle of human-agent co-training and learned communication in a VR environment, where closing the loop between human learning (human learns then trains an agent regarding patterns in the world) and agent learning (agent learns to make predictions and provide cues that must be learned by the pilot) appears possible to realize even during brief interactions. The VR fruit foraging protocol presented in this work proved to be an interesting environment to study pilot-copilot interactions in detail, and allowed us to probe the way human-agent behaviour changed as we introduced copilots with different algorithmic capabilities.\n\n\\section*{References}\n\n\\footnotesize\n\n\nBicho, E., et al. (2011). Neuro-cognitive mechanisms of decision making in joint action: A human-robot interaction study. {\\em Human Movement Science} 30, 846--868.\n\nKnoblich, G., et al. (2011). Psychological research on joint action: Theory and data. In WDK2003 (Ed.), {\\em The Psychology of Learning and Motivation} (Vol. 54, pp. 59\u2013-101). Burlington: Academic Press.\n\nModayil, J., Sutton, R. S. (2014). Prediction driven behavior: Learning predictions that drive fixed responses. {\\em AAAI Wkshp. AI Rob}.\n\nMoon, A., et al. (2013). Design and impact of hesitation gestures during human-robot resource conflicts. {\\em Journal of Human-Robot Interaction} 2(3), 18--40.\n\nParker, A. S. R., et al. (2014). Using learned predictions as feedback to improve control and communication with an artificial limb: Preliminary findings. {\\em arXiv}:1408.1913 [cs.AI]\n\nPesquita, A., et al. (2018). Predictive joint-action model: A hierarchical predictive approach to human cooperation. {\\em Psychon. Bull. Rev.} 25, 1751--1769.\n\nPezzulo, G., Dindo, H. (2011). What should I do next? Using shared representations to solve interaction problems. {\\em Exp. Brain. Res.} 211, 613--630.\n\n\nPilarski, P. M., et al. (2013). Real-time prediction learning for the simultaneous actuation of multiple prosthetic joints. {\\em Proc. IEEE Int. Conf. Rehab. Robotics (ICORR)}, Seattle, USA, 1--8.\n\nPilarski, P. M., et al. (2017). Communicative capital for prosthetic agents. {\\em arXiv}:1711.03676 [cs.AI]\n\nSchultz, W., et al. (1997). A neural substrate of prediction and reward. {\\em Science} 275(5306), 1593--9. \n\nSebanz, N., et al. (2006). Joint action: Bodies and minds moving together. {\\em Trends. Cogn. Sci.} 10(2), 70--76.\n\nSutton, R. S., Barto, A. G. (2018). {\\em Reinforcement Learning: An Introduction}. Second Edition. Cambridge: MIT Press.\n\nWolpert, D. M., et al. (2001). Perspectives and problems in motor learning. {\\em Trends Cogn. Sci.} 5(11), 487--494.\n\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nExtreme values of certain spatio-temporal processes, such as wind speeds, are the main cause of severe damage in property, from electricity distribution grid to transport  and agricultural infrastructures. Accurate assessment of causal relationships between environmental processes and their effects on risk indicators, are highly important in risk analysis, which in return depends on sound inferential methods as well as on good quality informative data. Often, information on the relevant environmental processes comes from monitoring networks, as well as from numerical-physical models (simulators) that typically solve a large set of partial differential equations, capturing the essence of the physical process under study (Skamarock \\emph{et al.} 2008, Cardoso \\emph{et al.} 2013).  In general, monitoring networks are formed by a sparse set of stations, whose instrumentation are vulnerable to disruptions, resulting in data sets with many missing observations, whereas, simulated data from numerical simulators typically supply average yield of the process in grid cells of pre-specified dimensions, often at high resolutions, spanning large spatial domains, with no missing observations. However, simulated data typically mismatch and misaligned observed data, therefore calibrating it and bringing it in line with observed data may supply modellers with more reliable and richer sources of data. Data assimilation methods, namely combining data from multiple sources, are well known in environmental studies, with data often being used to generate initial boundary conditions for the numerical  simulators (Kalnay, 2003).  There is a very rich statistical literature on data assimilation and data fusion with the objective of enriching the information for inference (Fuentes and Raftery 2005, Berrocal \\emph{et al.} 2012, Zidek \\emph{et al.} 2012, Berrocal \\emph{et al.} 2014, McMillan \\emph{et al.} 2010). These statistical methods are often based on Bayesian hierarchical methods for space time data (Banerjee \\emph{et al.} 2004) and are constructed around the idea of relating the monitoring station data and the simulated data using spatial linear models with spatially varying coefficients (Berrocal 2019). Since these relations involve data measured at different spatial resolutions, the models often are called downscaler models (Berrocal \\emph{et al.} 2012). The principal objective of these downscaler models is to relate observations measured at different space resolutions using spatial linear models. However, as a by-product, these models can be used for calibrating one set of data as a function of the other, as it will be explained later.\n\nThe motivation behind this present work has its roots in a\nconsulting work done for  a major electricity producer and distributor.\n  The electricity grid constantly faces disruptions due to damages\nin the distribution system, with heavy economic losses. These damages and consequent disruptions occur due to a  combination of many\nfactors such as topography and  precipitation, however extreme winds and storms are the main cause of such damages.  Risk maps that indicate likely places of costly disruptions in electric grids are important decision support tools for administering the power grid and are particularly useful in deciding if   costly corrective actions should  be taken to improve\nstructures. It is natural that these risk maps should be based  primarily  on observed wind speeds among other factors and it was decided that daily maximum wind speeds should be used as proxy information.  Hence, such risk maps can be interpreted as vulnerability  maps of electricity grid to extreme wind speeds, expressed in terms of probability.  However generating such  maps depends on reliable wind data at fairly high spatial and temporal resolutions.\n\n     The data available for this particular study corresponds to simulated wind speeds from a simulator (The WRF model, version 3.1.1) at a regular grid of 81ksq grid cell size, obtained at 10 minutes interval from 2006-2013; however only daily maximum wind speed will be used.  Observed daily maximum wind speed is also available during the same period of time, from 117 stations in Portugal mainland, but missing observations reach to 90\\% in some stations. Only around one third of the stations have less than 30\\% missing observations. There is an additional challenge: although simulated and observed data are similar in the bulk\nof the distribution,  they quite often mismatch  at extreme values. Therefore,\n adequate methods of data\nfusion  and calibration, can be used to combine these two different sources of data and may   provide information which is more reliable from a spatial point of\nview and  produce more accurate probability maps showing the spatial\ndistribution of damage risks.\nSince electricity grid damages are ultimately caused by extreme wind speeds, ultimate aim should be  to develop  statistical methods for data fusion and calibration  that can extrapolate beyond the\nrange of observed data  into the  tails of a distribution.  However, in this manuscript,  we make a review of statistical fusion and calibration  methods  for the whole range of data. Calibration methods that extend beyond the range of data will be reported elsewhere.\nOur objective is to explore several methods to model the relationship between simulated and observed wind speeds at observation sites, so as to extrapolate this relationship in space at grid cell or county level resolution. In other words, more than imputing missing observations, we want to use simulated wind speeds for risk assessment, after being calibrated, i.e., brought in line with observed wind speeds.\n    After giving a brief description of standard calibration and data fusion methods to update simulated data based on the observed data, we will propose and describe in detail a specific conditional quantile matching calibration method  and show how our wind speed data can be calibrated using this method.  We also briefly explain how calibration can be extended specifically to data coming from the tails of simulated and observed data, using asymptotic models and methods suggested by extreme value theory.\n\nThe outline of the paper is as follows: In section \\ref{cali}, we give an\n    overview of statistical calibration methods. In section \\ref{NAV}, we report a new approach\n for calibration through a conditional quantile matching calibration method (Pereira \\emph{et al}., 2019), using an extended Generalized Pareto distribution (Naveau \\emph{et al.}, 2016) for the simulated and observed data, adequate for calibrating simultaneously  the bulk and the tails of the distribution.\nFinally, in section \\ref{wind},\n this method will be exemplified using a wind speed data. Further discussion and conclusions are in section \\ref{disc}.\n\n \n\n\\section{Statistical Calibration methods; an overview} \\label{cali}\n\nCalibration plays a crucial role in almost all areas of experimental sciences and can be defined in a nutshell as  \"the comparison between measurements - one of known magnitude or correctness made or set with one device and another measurement made in as similar a way as possible with a second device \" (Wikipedia).  When measurements are obtained under  random environments, then statistical methods and models have to be employed to compare such sets of measurements. In the simplest case of random experiments involving linear calibration,  linear regression models are used as models. The univariate calibration is then defined as inverse regression problem (Lavagnini and Magno, 2007). These models and consequent calibration techniques  are, inevitably, restricted to  uncorrelated repeated observations or dependent  Gaussian structures, simplifying immensely the problem (Aitchison and Dunsmore, 1976). However, more often than not, even in designed experiments, the response relationships are mostly nonlinear and therefore  Gaussian structures are hardly justifiable as models.  Nonlinear calibration then typically have to be formulated by conditional specification of distributions, and consequently  substantial amount of numerical integration and approximations are needed.\n\nLittle is known on calibration of environmental data sets displaying nonlinear, non-Gaussian structures in a spatio-temporal setting. In these cases, defining calibration through spatial linear models as inverse regression problems will oversimplify the structures and will not be adequate. Often calibrating simulated data based on observed data is  done by calibration of the simulator, namely the numerical-physical model, using Bayesian methods (Kennedy and O'Hagan, 2001,  and Wilkinson, 2010). These generic methods are based on the following general ideas: The simulator first is approximated by a linear parametric  emulator, and using Bayesian arguments,  data are used to convert prior knowledge on these model parameters to posterior distributions. The newly generated data from this approximate emulator  is  then treated as the calibrated data.\n\n\nThere are many statistical calibration methods for different purposes and based on different paradigms.  We can classify these methods into\n\\begin{enumerate}\n\\item[(i)] Quantile matching-based approaches,\n\\item[(ii)] Inverse regression,\n\\item[(iii)]  Simulator--emulator-based approaches,\n\\item[(iv)]  Data assimilation\/data fusion.\n\\end{enumerate}\n\nBefore describing these methods, we give here some basic notation.\n\n We denote by  $Y(s,t)$ and $X(s,t)$,  respectively the observed and simulated  wind speeds at location $s\\in \\mathbb{R}^2$ and time $t$.\nTo simplify notation, often  we will use $Y$ and $X$ for observed and simulated wind speeds when data are used without any space-time  reference. Typically  $X$ are  simulated over a regular grid, say  $B$,  often  represented by  points $s_B$ which correspond to  the centroid of the grid cells, whereas   $Y$ are observed in stations located at different spatial points $s$.\n\n\n\n\\subsection{Quantile matching-based approaches}\n\n\n For the time being, if we  ignore totally space-time variations and dependence structures, calibration can be seen as a\nsimple scaling making use of marginal distributions fitted respectively to $X$ and $Y$ (CDF transform method, Michelangeli \\emph{et al}, 2009).\n\nSuppose we have a set of $n$ observed $y_i$ and simulated $x_i$ , $i=1,...,n$ data. Let $F_Y$ and $F_X$ be, respectively, the distribution functions of $Y$ and $X$. Then the new calibrated (scaled) data $x_i^*$ is defined as\n \\begin{equation}\n x_i^*=F^{-1}_Y(F_X(x_i)), \\quad i= 1, \\ldots,n. \\label{qmb}\n\\end{equation}\n\n Since\n$$P(X^*\\le z)=F_Y(z),$$\ncalibrated data has the same distribution as the observed data. Note that if $F_X=F_Y$ then $x_i^*=x_i$. Figure \\ref{fig1} depicts the result of applying this calibration method  when $Y$ follows a Student distribution with 3 degrees of freedom, and $X$ follows a standard normal distribution.\n\n \\begin{figure}\n  \\centering\n  \\includegraphics[width=10cm]{qm.eps}\n  \\caption{Illustration of the quantile matching approach}\\label{fig1}\n\\end{figure}\n\n\n\n\n This calibration method depends on the marginal distributions of the random variables involved $Y$ and $X$ and hence it does not\nmake use of the expected strong dependence between the two sets of data.\n\n\nAn ideal calibration should involve the joint distribution of $Y$ and $X$ defined in some way. A possibility is the use of  a conditional quantile matching approach,  which will be described in section \\ref{NAV}.  Further, in the same  section,   we also introduce\nan extension  to  cover space-time non-homogeneity  by scaling (calibrating) the data from\n\\begin{equation}\n x^*(s,t)=F^{-1}_{Y(s,t)}(F_{X(s,t)}(x(s,t)),\\label{eqmb}\n\\end{equation}\nassuming  marginal distributions of $Y(s,t)$ and $X(s,t)$  for every $s$ and $t$.\n\n These distributions will be estimated by fitting them and considering the\n parameters  as smooth functions of spatially and temporarily varying covariates and space-time latent processes as in section \\ref{wind}.\n\n\n\n\n    \\subsection{Inverse regression}\n\n Calibration is usually seen as a method of adjusting the scale of a measurement instrument on the basis of an informative experiment. As such, it is seen as an inverse regression problem. However, there are several problems associated with this approach (see, e.g. Kang \\emph{et al.}, 2017).\n Aitchison and Dunsmore (1975) approach the problem from a Bayesian perspective by defining the \\textit{calibrative distribution}. See also (e.g. Racine-Poon, 1988, Osborne, 1991 and Muehleisen and Bergerson, 2016).\n\nAccording to Aitchison and Dunsmore's proposal, a parametric model is fitted to a random vector $(X,Y)$, where, e.g. $X$ is the random variable representing a  measurement obtained in a laboratory and $Y$ the random variable representing the measurement  obtained in a field experiment. This parametric model is defined through a conditional specification such that\n$$f_{(X,Y)}(x,y\\mid \\psi,\\theta)=f_{X\\mid Y}(x\\mid y,\\psi)f_Y(y\\mid \\theta).$$ There are two parameters involved, namely  the arrival parameter $\\psi$ and the structural parameter $\\theta$. A further assumption is that the initial sources of information are stochastically independent, so that\n$$p(\\psi,\\theta)=p(\\psi)p(\\theta).$$\n\n Now suppose that one has a future laboratory experiment resulting in a value $x_f$  and  that further experiments follow the same pattern of arrival as the original trials $(\\mathbf{x},\\mathbf{y})=(x_i,y_i, i=1,...,n)$. Now the data available is $(x_i,y_i, x_f), i=1,...,n$ and the unknowns are $(y_f,\\psi,\\theta)$. The objective is to obtain the predictive distribution (called in this case the \\emph{calibrative distribution}) for the corresponding field experiment $Y_f$, which is simply obtained by integrating out $\\psi$ and $\\theta$,\n\n$$p(y_f\\mid \\mathbf{x},\\mathbf{y},x_f)=\\int f_X(x_f\\mid y_f,\\psi)p(y_f,\\psi,\\theta\\mid x_f,\\mathbf{x},\\mathbf{y})d\\psi d\\theta  $$\n where\n $$p(y_f,\\psi,\\theta\\mid x_f,\\mathbf{x},\\mathbf{y}) \\propto f_X(x_f\\mid y_f,\\psi) f_Y(y_t\\mid \\theta)\\prod_{i=1}^{n} f_X(x_i\\mid y_i,\\psi)f_Y(y_i\\mid \\theta)p(\\psi)p(\\theta),$$\n assuming that the trial records are independent.\n\n\nGeneralizing to the  situation under study, without considering space and time dependence,\nfor a model $[X\\mid Y][Y]$, the objective is to obtain the distribution $$Y(s_0)\\mid x(s_0),x(s^*),y(s^*)$$ for an unknown $Y(s_0)$ based on the observed and simulated data $(x(s^*),y(s^*))$ on $N$ stations and the simulated value $x(s_0)$.\n\n\n    \\subsection{Simulator--emulator-based approaches}\n\n Kennedy and O'Hagan (2001)  describe calibration  as statistical postprocessing of simulator deterministic forecast.\n  They assume a computer model describing some physical system, that is a function of variable inputs $x$ that can be measured and calibration inputs $\\nu$ needed to run the model, but whose values are not\nknown in the experiment. The output of the computer model is then assumed to be some function of the inputs, say $\\eta(x,u)$. Observations from the field experiment are assumed to have been observed at $u = \\theta$ and, possibly at different values of $x$. The model for observations for known input variables $x$ is then assumed to be a function of the computer model output $\\eta(x,\\nu)$, of a\ntrue underlying process $\\xi(x)$  and some  model inadequacy described by $\\delta(x)$.\nThe objective is to estimate the calibration settings $\\theta$ consistent with the field experiments and the computer model.\n\nSigrist \\emph{et al.} (2015) give detailed description of  stochastic versions of space-time advection-diffusion PDE's and their solutions as models for emulators and describe a method of postprocessing simulated data.\n\nHowever  simulator--emulator-based approaches assume  detailed information of how emulators work in terms of a set of parameters, which is not the case in most situations related to climate models.\n\n\n \\subsection{Data fusion and calibration}\n\n\n  Data assimilation or data fusion methods are used to combine different sources of information in order to obtain more accurate results. A recent review on data assimilation is given in  Berrocal (2019) together with many references.\n\n\n  Among these methods, the interest lies in statistical approaches to data assimilation\/data fusion and\n  in particular to  hierarchical Bayesian models (HBM)  for combining monitoring data and computer model output.\n\n  There are basically two different approaches regarding these methods. The Bayesian melding proposed by Fuentes and Raftery (2005) assumes that there is a true latent point-level process $Z(s)$ (\\emph{a GP with spatially varying mean and non-stationary covariance function}) to which both the observed $Y(s)$ and simulated $X(s)$ processes are linked to.\n The observed values are governed by this latent process  with a random error and simulated values are  expressed as a linearly calibrated integral over a grid cell (scaled by the area of the cell) of the latent point-level process,\n $$Y(s)=Z(s)+\\epsilon(s) \\qquad X(s)=a(s) + b(s)Z(s) + \\delta(s),$$\nwhere $\\delta(s)$  explains the random deviation\nat location $s$ with respect to the underlying true process $Z(s)$.\n The aim is to obtain the posterior\npredictive distribution of the truth $Z$ at a new site $s^*$. However, the misalignment of the two processes involved brings computational difficulties in the implementation. Foley and Fuentes (2008) apply this type of modelling to hurricane surface wind prediction and\nMcMillan \\emph{et al.} (2010) propose a spatio-temporal extension of this Bayesian melding approach.\n\n\n  The other approach suggested by  Berrocal \\emph{et al.} (2010) is a Bayesian hierarchical downscaler model.  They consider a spatial linear model relating the monitoring station data and the computer model output, with spatially varying coefficients which are in turn modeled as Gaussian processes (GPs). These models offer the advantage of local calibration of the numerical model output without incurring in problems due to the dimensionality of the computer model output, since they are only fitted at the  grid cells where the monitoring stations reside. An extension to this downscaler model, by borrowing information from neighboring grid cells, was introduced by Berrocal \\emph{et al.} (2012).\n  The proposed approach, the downscaler model   for the observations and simulated data from the computer model is\n\n$$Y(s) =\\beta_0+\\beta_0(s)+\\beta_1+\\beta_1(s)x(B)+e(s),\\quad e(s)\\quad i.i.d.\\quad N(0, \\tau^2),$$ with   $B$ the grid cell containing $s$\n $$X(B)= \\mu + V(B) +\\eta(B), \\quad \\eta(B) \\quad i.i.d. \\quad N(0, \\sigma^2)$$ where $V(B)$ is a GP model with a ICAR structure (Rue and Held, 2005), and $\\beta_0(s)$ is modeled as a mean-zero GP with exponential covariance structure.\n\nA smoothed version is possible considering\n$$Y(s) =\\beta_0+\\beta_0(s)+\\beta_1(\\mu+V(B))+e(s),\\quad e(s)\\quad i.i.d. \\quad N(0, \\tau^2)$$ with   $B$ the grid cell containing $s$.\n\n The aim is to obtain the predictive distribution of $Y$ and its expected value at grid cell level.\n\nThey also considered a smoothed downscaler using spatially varying random weights and a\nspace-time extension.\n\n\n\\section{Calibration methods for bulk and tails}\\label{NAV}\n\nPereira \\emph{et al.} (2019) develop  a covariate-adjusted version of the  quantile matching-based approach as in (\\ref{qmb}) where the distributions of simulated and real data change along a covariate. At the same time they suggest a regression method\nthat simultaneously models the bulk and the (right) tail of  the distributions involved\n using the extended Generalized Pareto distribution  (EGPD) (Naveau \\emph{et al.}, 2016) as a model for both the simulated and observed data.\n\n Under fairly general conditions, according to the asymptotic theory of extremes, the generalized Pareto distribution (GPD) appears as a natural model for the right tail of a distribution, by focusing on the  excesses over a high but fixed threshold. Here, the choice of this threshold plays a very important role in inference, ignoring the part of the data that lie below this threshold. See, for example, Beirland \\emph{et al.} (2004). The EGPD modelling strategy suggested by Naveau \\emph{et al} (2016) avoids this selection problem, as we will see in next section.\n\n In what follows we propose an extension of this conditional quantile matching calibration for the bulk and tails, to spatial temporal data.\n\n\\subsection{ Naveau \\emph{et al} (2016) EGPD models}\nNaveau \\textit{et al.}~(2016) suggest an extension of Generalized Pareto  model tailored for both the bulk and tails, and---contrarily to most methods for extremes--- does not require a threshold to be selected.  The objective of this extension is to generate a new class of distributions with GPD type tails consistent with extreme value theory, but also flexible enough to model efficiently the main bulk of the observed data without complicated threshold selection procedures.\n\nLet $Y$ be a positive random variable with cumulative distribution function defined as:\n \\begin{equation*}\n  F_Y(y \\mid \\theta) = G\\bigg(H(y\\mid\\xi,\\sigma )\\bigg),\n  \\label{qr}\n\\end{equation*}\nwhere $G$ is a  function obeying some general assumptions (see Naveau \\emph{et al.}, 2016) and $H$ is the cumulative distribution function of a Generalized Pareto distribution (GPD), that is\n\\begin{equation*}\n  H(y\\mid\\xi,\\sigma ) =\n\\left\\{\n  \\begin{array}{ll}\n     1 - (1 + \\frac{\\xi}{\\sigma} y)_{+}^{-1\/\\xi}, & \\hbox{$\\xi  \\neq 0$.} \\\\\n    1 - \\exp(-\\frac{y}{\\sigma}), & \\hbox{$\\xi = 0$ .}\n  \\end{array}\n\\right.\n\\end{equation*}\n with $\\sigma>0$, and $y>0$ if $\\xi\\geq 0$ and $y<-\\frac{\\sigma}{\\xi}$ if $\\xi<0$. The parameter $\\sigma$ is a dispersion parameter while $\\xi$ is a shape parameter controlling the rate of decay of the right tail of a distribution (de Zea Bermudez and Kotz, 2010).\n\n  Naveau \\emph{et al.} (2016)  consider four  forms of $G(u)$ resulting in four different classes of distributions.\n\n\\subsection{Spatio temporal conditional quantile matching calibration for the bulk and tails}\nHere we  use one of the forms, namely, $G(u)=u^\\kappa$ where $\\kappa$ is a parameter controlling the shape of the lower tail, although the theory can be easily extended to any of the other forms of the $G$ function.\n\nLet us assume that both random variables  $X$  and  $Y$  are space-time dependent  and we want to calibrate $X$ based on $Y$. As in (\\ref{eqmb}) the calibrated data is given as\n\\begin{equation*}\n x^*(s,t)=F^{-1}_{Y(s,t)}(F_{X(s,t)}(x(s,t)),\n\\end{equation*}\nNow assume further that both random variables are distributed as an EGPD with different parameters. In order to better accommodate for the situation $\\xi<0$ we make a transformation $\\delta=-\\frac{\\sigma}{\\xi}$. Hence, for $\\xi_x\\neq 0$\n\\begin{equation}\nF_{X(s,t)}(x(s,t)\\mid \\delta_x(s,t), \\xi,\\kappa_x)=\n   \\left(1 - \\left(1 - \\frac{1}{\\delta_x(s,t)} x(s,t)\\right)_{+}^{-1\/\\xi_x}\\right)^{\\kappa_x},  \\label{fx}\n\\end{equation}\nand assuming as well $\\xi_y\\neq 0$\n\\begin{equation}\nF_{Y(s,t)}(y(s,t)\\mid \\delta_y(s,t), \\xi_y,\\kappa_y)=\n   \\left(1 - \\left(1 - \\frac{1}{\\delta_y(s,t)} y(s,t)\\right)_{+}^{-1\/\\xi_y}\\right)^{\\kappa_y}. \\label{fy}\n\\end{equation}\n\nAlthough it is assumed that these random variables are conditional independent, a dependence structure is introduced through the transformed space-time dependent parameters $\\delta_x, \\delta_y$  by modelling them as a function of a common random spatio-temporal process, in a Bayesian hierarchical modelling framework.\n\nAs an exemplification of this modelling strategy, in the next section, we will built a Bayesian hierarchical model for the wind speed data.\n\n\\section{Bayesian hierarchical model for  the wind speed data}\\label{wind1}\n A preliminary data analysis of the wind speed data used in this study, shows that observed and simulated data are consistent with the case $\\xi<0$ and hence, the distributions for $X$ and $Y$ will have an end-point characterized by the respective  parameter $\\delta$.\n\nWe assume that the observed data $\\{Y(s_i,t_j), i=1,...,N; j=1,...,T\\}$, with $N$ the number of stations with observed data in the  study period and $T$  the length of the time period,  follow a distribution as in (\\ref{fy}), where\n  $\\delta_y(i,j)\\sim Exp(\\lambda_y(i,j)) ,$ $\\delta_y(i,j)>\\max(y),$ i.e., follows a shifted exponential distribution with\n  $\\log(\\lambda_y(i,j))=\\beta_y+ W(s_i)+Z(t_j),$ and $W$ follows a  Multivariate Gaussian process, defined on the space,  $W\\sim MVN(0,\\tau_W\\Sigma_W)$.\nThe  matrix $\\Sigma_W$ has diagonal elements equal to 1 and off-diagonal elements, $\\Sigma_{i\\ell}=f(d_{i\\ell};\\alpha)$, where $f(.;.)$ is a function of  $d_{i\\ell}$,  the centroids' distance of every two stations $s_i$ and $s_\\ell$, and  $\\alpha$ a parameter representing the radius of the 'disc' centred at each  $s$. The  $\\tau_W$ is a precision parameter. For the temporal random process we assume a  random walk process of order 1, $Z\\sim MVN(0,\\tau_Z\\Sigma_Z)$,  where $\\tau_Z$ is a precision parameter and $\\Sigma_Z$ is a matrix with a structure reflecting the fact that any two increments $z_i-z_{i-1}$ are independent (Rue and Held, 2005).\n\n\n\nWe assume, as well, that the simulated data $\\{X(s_i,t_j), i=1,...,N_s; j=1,...,T\\}$ follow a distribution as in (\\ref{fx})\n with $N_s$ total number of stations, where the model for $\\delta$  shares the same latent processes $W$ and $Z$ with the model for the observed data, such that $\\delta_x(i,j)\\sim Exp(\\lambda_x(i,j)),$ $\\delta_x(i,j)>\\max(x),$  with\n $\\log(\\lambda_x(ij))=\\beta_x+ W(s_i)+Z(t_j).$\n\n\nTo complete the Bayesian hierarchical model we consider the following prior specification for the parameters and hyperparameters of the models\n\n  $\\beta_y,\\beta_x$ i.i.d. $N(0,0.01)$,\n\n  $\\kappa_y,\\kappa_x$ i.i.d. $Ga(0.05,0.05)$,\n\n  $\\xi_y, \\xi_x$ i.i.d. $U(-0.5,0)$,\n\n  $\\tau_W, \\tau_Z$ i.i.d. $Ga(1,0.1)$,\n\n  $\\alpha \\sim U(0.1,0.5)$.\n\nFinally, the calibrated values are obtained as the\nmean of the predictive distribution of $F^{-1}_Y(F_X(x(s_i,t_j))$\n at $s_i, i=1,...,N_s$ and time $t_j,j=1,...,T$.\n\n\n\n\n\n\n\n\\section {Application to wind speed data }\\label{wind}\n\nWe used observed and simulated wind speed data from the period 01\/01\/2013 to 28\/02\/2013, so $T=59$. There are $N=51$ stations where we have both observed and simulated daily maximum wind speeds. Additionally we have extra 66 stations with simulated values for the maximum daily wind speeds, so that $N_s=117$. In Figure \\ref{turkman:fig1} we depict the median of observed  and simulated wind speeds for the 51 stations together with the\n2.5\\% and 97.5\\% empirical quantiles (95\\% IQR).\n\n\n \\begin{figure}[!ht]\\centering\n\\includegraphics[width=10cm]{obs_simul.eps}\n\\caption{\\label{turkman:fig1} Median of observed  and simulated wind speeds for the 51 stations, and the 95\\% IQR wind speeds by station (dashed lines).}\n\\end{figure}\n\n\nThe model was implemented in \\texttt{R2OpenBUGS} (Sturtz \\emph{et al.} (2005). In Table \\ref{tab1} we show the summary statistics for the  marginal posterior distributions of the parameters of the model.\n\n We observe that the posterior mean of $\\kappa_y$ has a much smaller mean than the posterior mean of $\\kappa_x$ which is consistent with the fact that, in general, simulated data are shifted to the right in relation to the observed data, indicating the possible existence of some bias in the simulated data. The posterior mean of the precision (inverse of the variance) parameters for the space model ($\\tau_W$) and for the temporal model ($\\tau_Z$) suggest that time dependence is stronger than space dependence. The posterior mean for $\\beta_y$ is slightly smaller than the posterior mean for $\\beta_x$. This naturally contributes for higher values for $\\sigma_y(i,j)$ relatively to $\\sigma_x(i,j)$ and with greater dispersion, as it can be seen in  Figure  \\ref{boxplot} where we show daily boxplots  of the posterior means of the parameters $\\sigma(i,j),\\forall j$  for both models. In that figure it is marked two dates, 19 of January, a day where it was observed a storm with heavy winds (storm GONG, maximum observed wind 29.6m\/s), particularly in regions close to the littoral, and 14th of February, a very mild day all over the country (Valentine's day; maximum observed wind 8.20m\/s).  The variation observed along the days is consistent with the fact that on windy days the  maximum wind speed  along the stations varies much more than on mild days. Also the temporal dependence is clear in these pictures.\n\n\\begin{table}[ht]\n\\centering\n\\caption{Summary statistics  for the marginal posterior distributions} \\label{tab1}\n\\begin{tabular}{cccccccc}\n  \\hline\n & mean & standard deviation & 2.5\\% quantile & median & 97.5\\% quantile & min & max \\\\\n  \\hline\n$\\alpha$ & 0.451 & 0.042 & 0.346 & 0.462 & 0.499 & 0.241 & 0.500 \\\\\n $\\beta_y$ & -1.094 & 0.149 & -1.376 & -1.093 & -0.806 & -1.541 & -0.595 \\\\\n  $\\beta_x$ & -0.854 & 0.134 & -1.105 & -0.849 & -0.598 & -1.243 & -0.365 \\\\\n  $\\kappa_y$ & 5.312 & 0.197 & 4.936 & 5.310 & 5.701 & 4.675 & 5.936 \\\\\n  $\\kappa_x$ & 18.588 & 0.725 & 17.230 & 18.585 & 20.080 & 16.480 & 20.930 \\\\\n  $\\tau_W$ & 4.234 & 0.715 & 2.923 & 4.187 & 5.762 & 2.218 & 7.022 \\\\\n  $\\tau_Z$ & 0.396 & 0.095 & 0.241 & 0.384 & 0.618 & 0.191 & 0.850 \\\\\n  $\\xi_y$ & -0.070 & 0.002 & -0.074 & -0.070 & -0.067 & -0.077 & -0.065 \\\\\n  $\\xi_x$ & -0.081 & 0.001 & -0.084 & -0.081 & -0.078 & -0.085 & -0.076 \\\\\n   \\hline\n\\end{tabular}\n\\end{table}\n\n\n\\begin{figure}[!ht]\\centering\n\\includegraphics[width=7cm]{boxplot_sigmas.eps}\n\\includegraphics[width=7cm]{boxplot_sigmax.eps}\n\\caption{\\label{boxplot} Boxplot of the posterior means of $\\sigma_y(i,j)$ (left) and $\\sigma_x(i,j)$ (right) }\n\\end{figure}\n\n\n\n\nThese two days were studied, in particular, for exemplification of the conditional quantile calibration method proposed. For the purpose of exemplification of the results we represent in Figures \\ref{gong} and \\ref{val}, on the left, a kernel density estimation (considering all the stations) for the observed and simulated maximum wind speed on that day, together with  the mean of the predictive distribution of the calibrated data as defined in (\\ref{eqmb}). On the right side we represent the observed and simulated maximum wind speed on that day for each station, together with the mean of the predictive distribution for the calibrated data.\n\nWe observe that, on a storm day (Figure \\ref{gong}) the observed winds have longer tails than simulated winds. The calibration method was able to capture both tails of the distribution for the observed data, although it shifted the bulk of the distribution to the left.  Regarding a mild windy day (Figure \\ref{val}),  the distribution of the simulated data is shifted to the right relatively to the distribution of the observed data with longer tails, as it was observed in a preliminary study. This bias is corrected with the calibration method.\n\nIn Figures \\ref{mapa_gong} and \\ref{mapa_val} there is a spatial representation of the observed, simulated and calibrated values for each of these two days.\n\n\n\\begin{figure}[!ht]\\centering\n\\includegraphics[width=5.3cm]{density_full_GONG.eps}\n\\includegraphics[width=5.3cm]{GONG_stations.eps}\n\\caption{\\label{gong}   Kernel density estimation (left), observed and simulated maximum wind speed for each station, together with the mean of the predictive distribution for the calibrated data, for a storm day.}\n\\end{figure}\n\n\n\n\n\\begin{figure}[!ht]\\centering\n\\includegraphics[width=5.3cm]{density_full_14Fev.eps}\n\\includegraphics[width=5.3cm]{14Fev_stations.eps}\n\\caption{\\label{val}  Kernel density estimation (left), observed and simulated maximum wind speed for each station, together with the mean of the predictive distribution for the calibrated data, for a mild day.}\n\\end{figure}\n\n\n\\begin{figure}[!ht]\\centering\n\\includegraphics[width=10cm]{mapas_talk_at_14fev.eps}\n\\caption{\\label{mapa_gong} Storm day: observed, simulated and calibrated maximum wind speeds}\n\\end{figure}\n\n\\begin{figure}[!ht]\\centering\n\\includegraphics[width=10cm]{mapas_talk_at_gong2.eps}\n\\caption{\\label{mapa_val} Mild day: observed, simulated and calibrated maximum wind speeds }\n\\end{figure}\n\n\n\n\n\\section{Discussion and further extensions}\\label{disc}\n\nIn this article we discussed several possible ways of calibrating data obtained from a simulator based on observations at stations. We also  proposed and implemented a conditional quantile matching calibration (CQCM) using a space-time extended generalized Pareto distribution.\n\nThe  performance of the CQCM method was exemplified with two specific days, a storm day and a mild day. In both cases the calibrated data matched well the observed data on the tails, although on the storm day it did not capture well the bulk of the distribution. Ideally this method should be extended to the grid level, since the simulator produces data at a fine grid level and this is much more interesting if the objective is the construction of a risk map. However this extension is not trivial and some assumptions regarding the model structure have to be assumed.\n\n Damages in electricity grid are basically governed by extreme winds and primarily simulated and observed data coming from the right tail differ.\n     Hence adequate calibration methods must be specifically adapted to extreme observations coming from the right tails and\n     methods and models to be used in calibration should ideally be compatible with extreme value theory.\n    A range of approaches for characterising the extremal behaviour of spatial process  have been suggested and a brief comparison of these methods can be found in Tawn \\emph{et al.}~(2018).\n         Downscaling method described by Towe {\\it et al.}~(2017)--- based on the  conditional extremes process---is more suitable, with adequate modifications, to calibrate extreme simulated data based on observed wind speeds. Work on this approach is under progress.\n            \n\\section*{Acknowledgments}\nResearch partially financed by national funds through FCT -  Funda\\c{c}\\~{a}o para a Ci\\^{e}ncia e a Tecnologia, Portugal, under the projects  PTDC\/MAT-STA\/28649\/2017 and UIDB\/00006\/2020.\n\n\n\n\\References\n\n\n\\item[] Aitchison, J. and Dunsmore, I.~R. (1975).  \\emph{Statistical Prediction Analysis}. Cambridge: Cambridge University Press.\n\\item[] Beirland, J., Goegebeur, Y., Segers, J. and Teugels, J. (2004). \\emph{Statistics of\nExtremes: Theory and applications}. J Wiley, Chichester.\n\\item[] Banerjee, S., Carlin, B.~P. and Gelfand, A.~E. (2004). \\emph{Hierarchical Modeling and Analysis for Spatial Data}, Boca Raton, FL: Chapman and Hall.\n\\item[] Berrocal, V.~J. (2019). Data assimilation. In Gelfand AE, Fuentes M, Hoeting JA and Smith RL, \\emph{Handbook of Environmental and Ecological Statistics}, 133\\,--\\,151, Chapman and Hall\/CRC.\n\\item[] Berrocal, V.~J., Gelfand, A.~E. and Holland, D.~M. (2012).  Space-time data fusion under error in computer model output: An application to modeling air quality. \\emph{Biometrics}, {\\bf 68},\n      837\\,--\\,848.\n\\item[] Berrocal, V.~J., Gelfand, A.~E. and Holland, D.~M. (2014), Assessing Exceedance of Ozone Standards: A Space Time Downscaler for Fourth Highest Ozone Concentrations,\n\\emph{Environmetrics}, {\\bf 25}, 279\\,--\\,291.\n\\item[] Cardoso, R.~M., Soares, P.~M.~M, Miranda,P.~M.~A. and Belo-Perira, M. (2013), WRF High Resolution Simulation of Iberian Mean and Extreme Precipitation Climate, \\emph{International\nJournal of Climatology}, {\\bf33}, 2591\\,--\\,2608.\n\\item[] De Zea Bermudez, P., and S. Kotz. 2010. Parameter estimation of the generalized Pareto distribution\u2014Part I. \\emph{J. Stat. Plan. Inference}, {\\bf 140(6)}, 1353\\,\u2013-\\,1373.\n\\item[] Foley,K.M. and Fuentes, M.  (2008). A Statistical Framework to Combine Multivariate Spatial Data and Physical Models\nfor Hurricane Surface Wind Prediction. \\emph{Journal of Agricultural, Biological, and Environmental Statistics}, {\\bf 13},37\\,--\\,59.\n\\item[] Fuentes, M. and Raftery, A.~E. (2005).  Model evaluation and spatial interpolation by Bayesian\ncombination of observations with outputs from numerical models. \\emph{Biometrics}, {\\bf 61},\n      36\\,--\\,45.\n \n\\item[] Kennedy, M. and O'Hagan, A. (2001). Bayesian Calibration of Computer Models.  {\\it Journal of the Royal Statistical Society, Series B}, {\\bf 63},\n      425\\,--\\,464.\n\\item[] Kalnay (2003) \\emph{Atmospheric modeling, data assimilation and predictability}. Cambridge University Press.\n\\item[] Kang, P., Koo, C. and  Roh, H. (2017). Reversed inverse regression for the univariate linear calibration\nand its statistical properties derived using a new methodology. \\emph{International Journal of\nMetrology and Quality Engineering}, {\\bf 8 (28)}.\n\\item[] Lavagnini, I. and Magno, F.(2007). A statistical overview on univariate calibration, inverse regression, and detection limits: application to gas chromatography\/mass  spectrometry technique. \\emph{Mass Spectrometry  Techniques}, {\\bf 26}, 1\\,--\\,18\n\\item[] McMillan N.~J., Holland, D.~M., Morara, M., and Feng J. (2010). Combining numerical model output and particulate data using Bayesian space-time modeling. \\emph{Environmetrics},{\\bf 21}, 48\\,--\\,65.\n\\item[]  Michelangeli, P.~A., Vrac, M., and Loukos, H. (2009).\nProbabilistic downscaling approaches: Application to wind\ncumulative distribution functions. \\emph{Geophys. Res. Lett.}, {\\bf{36}},\n1\\,--\\,6.\n\\item[] Muehleisen, R.~T. and Bergerson,  J. (2016). Bayesian Calibration - What, Why And How. \\emph{International High Performance\nBuildings Conference}. Paper 167.\n\\item[] Naveau P., Huser R., Ribereau P. and Hannart A. (2016). Modeling jointly low, moderate, and heavy rainfall intensities\nwithout a threshold selection. \\emph{Water Resources Research}. 2753\\,--\\,2769.\n \\item[] Osborne, C. (1991). Statistical Calibration: A Review. \\emph{International Statistical Review \/ Revue Internationale De Statistique}, {\\bf 59}, 309\\,--\\,336.\n\\item[] Pereira, S., Pereira, P., de Carvalho, M. and de Zea Bermudez, P. (2019). Calibration of extreme values of simulated and real data. \\textit{Proceedings of International Workshop on Statistical Modelling 2019}.\n\\item[] Rue, H. and Held, L. (2005). \\emph{Gaussian Markov Random Fields: Theory and Applications}. Monographs on Statistics and Applied Probability, vol. 104. Ghapman\\& Hall: London.\n\\item[]  Racine-Poon, A. (1988). A Bayesian Approach to non-linear calibration problems. \\emph{JASA}, \\textbf{83}, 650\\,--\\,656.\n\\item[] Skamarock, W.C~., Klemp, J.B~., Dudhia, J., Gill, D.~O., Barker, D.~M., Duda,\nM.~G., Huang, X.~Y., Wang, W. and Powers, J.~G. (2008). A description of the\nadvanced research WRF version 3 . \\emph{NCAR Tech. Note TN-475\\_STR}.\n\\item[] Sigrist, F.,  K\\\"{u}nsch, H.R. and Stahel, W.A. (2015). Stochastic partial differential equation based modelling of large space-time data sets. \\emph{Journal of the Royal Statistical Society, Series B}, {\\bf 77},\n      3\\,--\\,33.\n\\item[] Sturtz, S.,  Ligges, U. and  Gelman, A. (2005). R2WinBUGS: A Package for Running WinBUGS from R. \\emph{Journal of Statistical Software},\n{\\bf 12}, 1\\,--\\,16.\n\\item[] Towe, R.P., Sherlock, E.F., Tawn, J.A., Jonathan, P. (2017). Statistical downscaling for future extreme wave heights in the North Sea.\n  \\emph{Annals of Applied Statistics}. {\\bf 11}, 2375\\,--\\,2403.\n\\item[] Wilkison, R.~D. (2010).  Bayesian Calibration of expensive multivariate computer experiments. \\emph{In Computational Methods for Large scale inverse problems and quantification of uncertainity}. J. Wiley and Sons.\n\\item[] Zidek, J.V., Le, N.~~D. and Liu, Z.  (2012). Combining data and simulated data for space-time fields: Application to ozone. \\emph{Environmental and Ecological Statistics}, {\\bf 19}, 37\\,--\\,56.\n\\endrefs\n\n\n\n\n\\end{document}\n\n\nSpicer \\etal (2011)\n\n\\References\n[] Spicer P E, Nijhoff F W and van der Kamp P H 2011 {\\it Nonlinearity}\n{\\bf 24} 2229\n\n[] Dorman L I 1975 {\\it Variations of Galactic Cosmic Rays}\n(Moscow: Moscow State University Press) p\u02dc103\n\n[] Caplar R and Kulisic P 1973 {\\it Proc. Int. Conf. on Nuclear\nPhysics (Munich)} vol\u02dc1 (Amsterdam: North-Holland\/American\nElsevier) p\u02dc517\n\\endrefs\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nShortly after the discovery of pulsars and the realization that they are\nrotating neutron stars, deformations of rotating neutron stars were proposed as\nsources of continuous gravitational radiation~\\cite{Shklovskii1969,\nOstriker1969, Ferrari1969, Melosh1969}; see~\\cite{Press1972} for an early\nreview.\nSearches for such radiation are an ongoing concern of the LIGO and Virgo\ngravitational wave detectors~\\cite{LIGO_psrs2010, LIGO_CasA, LIGO_Vela}; see~\\cite{Owen2009, Pitkin, Astone} for recent\nreviews.\nIt is thus of\ngreat interest to know the maximum\nquadrupolar deformation that a neutron star could sustain, in order to motivate\nfurther searches and help interpret upper limits or detections.\nIn the case of elastic (as opposed to magnetic) deformations, the main factor\ninfluencing the answer is whether the neutron star contains particles more\nexotic than neutrons~\\cite{OwenPRL, Owen2009}.\nHowever, the structure of the star also plays an important role.\n\nWhile there are\nrelativistic calculations of the quadrupole deformations due to magnetic\nfields (e.g.,~\\cite{IS,CFG,YKS, FR,CR}), all the computations involving elastic\ndeformations have used Newtonian gravity. Moreover, all but two of these computations\nhave used the integral expression obtained in the Cowling approximation (i.e.,\nneglecting the self-gravity of the perturbation) by\nUshomirsky, Cutler, and\nBildsten (UCB)~\\cite{UCB}; see \\cite{OwenPRL, Lin, KS, Horowitz}.\nHaskell, Jones, and Andersson (HJA)~\\cite{HJA} dropped the Cowling approximation using\na somewhat different formalism than UCB's; there is a further application of\ntheir results in~\\cite{Haskelletal}.\n\nWe improve these treatments by generalizing the UCB integral to relativistic\ngravity with no Cowling approximation. We also provide a\nsimilar generalization for the Newtonian no-Cowling case, as a warm-up. In\naddition to providing a simpler formalism for performing computations than the\nmore general Newtonian gravity treatment in HJA, the integrals we obtain allow us to verify that a\nmaximal uniform\nstrain continues to yield the maximum quadrupole deformation in the Newtonian\nand relativistic no-Cowling cases. (UCB showed this to be true for an arbitrary equation of state\nin the Newtonian Cowling approximation case; we are able to verify that it is\ntrue in the more general cases for each background stellar model we consider.) \n\nWe then apply our calculation to the standard case of quadrupoles supported by\nshearing the lattice of nuclei in the crust, as well as the cases where the\nquadrupole is supported by the hadron--quark mixed phase lattice in the core, or a crystalline color superconducting phase throughout a solid strange quark star. For the crustal\nquadrupoles, we calculate the shear modulus following HJA, using the equation\nof state (EOS) and composition results of Douchin and Haensel~\\cite{DH} and\nthe effective shear modulus calculated by Ogata and Ichimaru~\\cite{OI}. (There are recent\nimprovements to the Ogata and Ichimaru result~\\cite{HH, Baiko,BaikoCPP}, but these only reduce their\nshear modulus by $<10\\%$.) For the hadron--quark mixed phase, we use our recent calculations of the EOS and\nshear modulus~\\cite{J-MO1} for a variety of parameters. (We also consider the range of surface tensions for\nwhich the mixed phase is favored.) For crystalline quark matter, we use the shear modulus calculated by Mannarelli, Rajagopal, and Sharma~\\cite{MRS}, and the EOS given by\nKurkela, Romatschke, and Vuorinen~\\cite{KRV}.\n\nIn all cases, we use a breaking strain of $0.1$, comparable to that\ncalculated by Horowitz and Kadau~\\cite{HK} using molecular dynamics\nsimulations. (Hoffman and Heyl~\\cite{HoHe} have recently obtained\nvery similar values over more of parameter space.) This result is directly applicable to the crustal lattice, at least for the outer crust, above neutron drip (though see Chugunov and\nHorowitz~\\cite{ChHo} for caveats). We also\nfeel justified in applying it to the inner crust, as well as to the mixed phase and crystalline quark matter, since the primary source of\nthe high breaking strain appears to be the system's large pressure.\nBut one can apply our results to any breaking strain using the linear scaling of\nthe maximum quadrupole with breaking strain.\n\nIn our general relativistic calculation, we use the relativistic theory of elasticity given by Carter and\nQuintana~\\cite{CQ} and placed in a more modern guise by\nKarlovini and Samuelsson~\\cite{KaSaI}.\nHowever, all we need from it is the relativistic form of the elastic stress-energy tensor, which can be obtained by simple\ncovariance arguments, as noted by Schumaker and Thorne~\\cite{ST}. We also use the standard\n\\citet{TC} Regge-Wheeler gauge~\\cite{RW} formalism for perturbations\nof static relativistic stars, following\nHinderer's recent calculation~\\cite{Hinderer} of the quadrupole moment of a\ntidally deformed relativistic star (first discussed in~\\citet{FH}), and the classic calculation by \\citet{Ipser}.\n\nEven though we are interested in the gravitational radiation emitted by\nrotating stars, it is sufficient for us to calculate the static quadrupole deformation.\nAs discussed by\nIpser~\\cite{Ipser}, and then proved for more general situations by\nThorne~\\cite{Thorne}, this static quadrupole (obtained\nfrom the asymptotic\nform of the metric) can be inserted into the quadrupole formula to obtain the emitted\ngravitational radiation in the fully relativistic,\nslow-motion limit. [This\napproximation has uncontrolled remainders of order $(\\omega\/\\omega_K)^2$, where\n$\\omega$ and $\\omega_K$ are the star's angular velocity and its maximum---i.e.,\nKepler---angular velocity, respectively.\nThis ratio is $\\lesssim 10^{-2}$ for the pulsars for which LIGO has been able\nto beat the spin-down limit~\\cite{LIGO_psrs2010}.]\n\nWe shall generally show the gravitational constant $G$ and speed of light $c$\nexplicitly, though we shall take $G = c =1$ in most of Sec.~\\ref{GR}, only restoring them in our\nfinal expressions.\nThe relativistic calculation was aided by use of the computer algebra\nsystem {\\sc{Maple}}\nand the associated tensor manipulation package {\\sc{GRTensorII}}~\\cite{GRTensor}.\nWe used {\\sc{Mathematica}}~7 to perform numerical computations.\n\nThe paper is structured as follows: In Sec.~\\ref{Newt}, we review UCB's formalism\nand extend it by introducing a Green function to compute the maximum Newtonian quadrupole deformation without\nmaking the Cowling approximation. In Sec.~\\ref{GR}, we further generalize to\nthe fully relativistic case, and compare the various approximations for the maximum quadrupole.\nIn Sec.~\\ref{results2}, we show the maximum quadrupoles for three different cases: first crustal quadrupoles,\nthen hadron--quark hybrid quadrupoles, and finally solid strange quark star quadrupoles. We also describe\nthe modifications to our formalism needed to treat solid strange quark stars. We discuss all these\nresults in Sec.~\\ref{discussion}, and summarize and conclude in\nSec.~\\ref{concl2}.\nIn the Appendix, we show that the mixed phase is favored by global energy arguments even for surface tensions large enough that it is disfavored by local energy arguments.\n\n\\section{Newtonian calculation of the maximum quadrupole}\n\\label{Newt}\n\nWe first demonstrate how to compute the maximum Newtonian quadrupole without making the Cowling approximation. This provides\na warm-up before we tackle the full relativistic case, and also allows us to verify some of the\nstatements made by UCB and HJA. We use\nthe basic formalism of UCB, modeling the star as nonrotating, with the\nstress-energy tensor of a perfect fluid plus\nshear terms, and treating the shear contributions as a first-order\nperturbation of hydrostatic equilibrium.\nThis perturbative treatment should be quite a good approximation: The maximum shear stress to energy density ratio we consider in the crustal and hybrid star cases is $\\lesssim 0.05\\%$ (and the maximum shear stress to pressure ratio is\n$\\lesssim 0.3\\%$). (Here we have taken the shear stress to\nbe $\\mu\\bar{\\sigma}_\\mathrm{max}$, which is good up to factors of order unity.) And even in the case of solid strange quark stars, the\nmaximum shear stress to energy density ratio is still only at most $\\sim 0.2\\%$.\n[We have already discussed the effects of rotation in the relativistic case, above; UCB note at\nthe beginning of their Sec.~4 that rotation also only modifies the perturbative Newtonian results for the static deformations we and they consider at the $O([\\omega\/\\omega_K]^2)$ level.] \n\nIt is convenient to start by writing the quadrupole moment\nin terms of the surface value of the perturbation to the star's\nNewtonian potential. We start from UCB's definition of\n\\<\n\\label{Q22}\nQ_{22} := \\int_0^\\infty\\delta\\rho(r)r^4dr\n\\?\n[where the (Eulerian) density perturbation $\\delta\\rho$ and all similar\nperturbed quantities have only an $l = m = 2$ spherical harmonic component].\n[Note that this\nquadrupole moment differs by an overall constant from the one defined by\nThorne~\\cite{Thorne}---e.g., his Eq.~(5.27a).] We then recall that\nthe perturbed Poisson equation for the $l = 2$ part of the perturbed\ngravitational potential is\n\\<\\label{Poisson}\n(\\triangle_2\\delta\\Phi)(r) := \\frac{1}{r^2}[r^2\\delta\\Phi'(r)]' - \\frac{6}{r^2}\\delta\\Phi(r) = 4\\pi G\\delta\\rho\n\\?\n($\\triangle_2$ is the $l = 2$ radial part of the Laplacian), with boundary conditions of\n\\<\n\\label{PoissonBCs}\n\\delta\\Phi(0) = 0, \\qquad R\\delta\\Phi'(R) = -3\\delta\\Phi(R),\n\\?\nwhere $R$ is the radial coordinate of the star's surface.\n[See, e.g., Eqs.~(2.15) and (2.16) in \\cite{LMO}---their $\\Phi_{22}$ is our\n$\\delta\\Phi$. Note also that the primes denote derivatives with respect to $r$. Additionally, we shall\ncontinue to be inconsistent with our inclusion of the functional dependence of quantities---e.g., $\\delta\\rho$\ndepends upon $r$, even though we do not always indicate this explicitly. We will eventually stop displaying  $\\delta\\Phi$'s \nexplicit functional dependence on $r$, for instance.]\nIf we now substitute Eq.~\\eqref{Poisson} into Eq.~\\eqref{Q22} and integrate by\nparts using the boundary conditions~\\eqref{PoissonBCs}, we obtain\n\\<\\label{Q22N}\nQ_{22} = -\\frac{5R^3}{4\\pi G}\\delta\\Phi(R).\n\\?\nThis\nsort of expression is more commonly seen in the relativistic case, where it is\nnecessary to obtain the quadrupole in this manner by looking at the\nperturbation's asymptotic behavior---see the discussion in Sec.~\\ref{GR}.\n\nWe now wish to obtain an equation for $\\delta\\Phi$ in terms of the shear\nstresses. We follow UCB in decomposing the perturbed stress tensor as\n[see their Eqs.~(59) and (61)]\n\\<\\label{deltatau}\n\\begin{split}\n\\delta\\tau_{ab} &= -\\delta pY_{lm}g_{ab} + t_{rr}Y_{lm}(\\hat{r}_a\\hat{r}_b - e_{ab}\/2) + \nt_{r\\perp}f_{ab}\\\\\n&\\quad + t_\\Lambda(\\Lambda_{ab} + Y_{lm}e_{ab}\/2).\n\\end{split}\n\\? \nHere $\\delta p$ is the (Eulerian) pressure perturbation; $Y_{lm}$ is a spherical\nharmonic; $\\hat{r}_a$ is the radial unit vector; $t_{rr}$, $t_{r\\perp}$, and\n$t_\\Lambda$ are the components of the shear stresses; and\n$g_{ab}$ denotes the metric of flat, $3$-dimensional Euclidean space.\n(Following UCB, we will generally write out $l$ and $m$ explicitly, even\nthough we only consider $l = m = 2$ here.)\nAlso [Eqs.~(40) in UCB],\n\\begin{subequations}\n\\begin{align}\ne_{ab} &:= g_{ab} - \\hat{r}_a\\hat{r}_b,\n\\\\\nf_{ab} &:= 2r\\hat{r}_{(a}\\nabla_{b)}Y_{lm}\/\\beta,\n\\\\\n\\beta &:= \\sqrt{l(l+1)} = \\sqrt{6},\n\\\\\n\\label{Lambda}\n\\Lambda_{ab} &:= r^2\\nabla_a\\nabla_bY_{lm}\/\\beta^2 + f_{ab}\/\\beta.\n\\end{align}\n\\end{subequations}\n(We have corrected the dropped factor of $\\beta^{-1}$ multiplying $f_{ab}$ in\nUCB's definition of $\\Lambda_{ab}$---this was also noticed by HJA.)\nWe also have\n\\begin{equation}\n\\label{ts}\nt_{ab} = 2\\mu\\sigma_{ab},\n\\end{equation}\nwhere $\\mu$ is the shear modulus and $\\sigma_{ab}$ is the strain tensor.\n(This is a factor-of-$2$ correction to the expression in UCB, as noted in\n\\cite{OwenPRL}.)\nNow, a convenient expression can be obtained from the perturbed\nequation of hydrostatic equilibrium\n\\<\\label{hydro}\n\\nabla^a\\delta\\tau_{ab} = \\delta\\rho g(r)\\hat{r}_b + \\rho\\nabla_b\\delta\\Phi\n\\?\n($\\nabla_a$ denotes the flat-space covariant derivative), by\nsubstituting for $\\delta\\rho$ using the Poisson equation~\\eqref{Poisson}\nand projecting along $\\hat{r}^b$, yielding\n\\<\\label{deltaPhieq1}\n\\begin{split}\n\\frac{\\triangle_2\\delta\\Phi}{4\\pi G} + \\frac{\\rho}{g(r)}\\delta\\Phi' &= \\frac{\\hat{r}^b\\nabla^a\\delta\\tau_{ab}}{g(r)}\\\\\n&= \\frac{1}{g(r)}\\left[-\\delta p' + t_{rr}' + \\frac{3}{r}t_{rr} - \\frac{\\beta}{r}t_{r\\perp}\\right].\n\\end{split}\n\\?\nWe then project Eq.~\\eqref{hydro} along $\\nabla^bY_{lm}$ to express $\\delta p$ in terms of the shear stresses $t_{rr}$, $t_{r\\perp}$, and $t_\\Lambda$,\nalong with $\\rho$ and $\\delta\\Phi$, giving\n\\<\n\\delta p = -\\rho\\delta\\Phi - \\frac{t_{rr}}{2}  + \\frac{r}{\\beta}t_{r\\perp}' + \\frac{3}{\\beta}t_{r\\perp} + \\left(\\frac{1}{\\beta^2} - \\frac{1}{2}\\right)t_\\Lambda.\n\\?\nSubstituting this into Eq.~\\eqref{deltaPhieq1}, we thus obtain\n\\<\\label{deltaPhieq2}\n\\begin{split}\n\\triangle_2\\delta\\Phi - \\frac{4\\pi G}{g(r)}\\rho'\\delta\\Phi &=\n\\frac{4\\pi G}{g(r)}\\biggl[\\frac{3}{2}t_{rr}' - \\frac{4}{\\beta}t_{r\\perp}' - \\frac{r}{\\beta}t_{r\\perp}''\\\\\n&\\quad - \\left(\\frac{1}{\\beta^2} - \\frac{1}{2}\\right)t_\\Lambda' + \\frac{3}{r}t_{rr} - \\frac{\\beta}{r}t_{r\\perp}\\biggr].\n\\end{split}\n\\?\n\nWe now wish to obtain an integral expression for $Q_{22}$ that generalizes\nUCB's Eq.~(64) to the case where we do not make the Cowling approximation.\nWe shall do this by obtaining the Green function for the left-hand side of\nEq.~\\eqref{deltaPhieq2} and then integrating by parts.\nWe will be able to\ndiscard all of the boundary terms, since the\nstresses vanish at the star's surface (we assume that the shear modulus vanishes there) and the integrand vanishes at the star's center. We can obtain the\nGreen function using the standard Sturm-Liouville expression in terms of the solutions of the\nhomogeneous equation [e.g., Eq.~(10.103) in Arfken and Weber~\\cite{AW}~].\nWe obtain the appropriate solution to the homogeneous equation numerically\nfor a given background stellar model (EOS and mass). The equation for the Green function is\n[multiplying the left-hand side of Eq.~\\eqref{deltaPhieq2} by $r^2$ to improve its regularity]\n\\<\\label{Leq}\n\\begin{split}\n(\\mathcal{L}_N\\mathcal{G})(r,\\bar{r}) &:= \\frac{\\partial}{\\partial r}\\left[r^2\\frac{\\partial}{\\partial r}\\mathcal{G}(r,\\bar{r})\\right] - \\left[6 + \\frac{4\\pi G r^2}{g(r)}\\rho'\\right]\\mathcal{G}(r,\\bar{r})\\\\\n&\\,= \\delta(r - \\bar{r})\n\\end{split}\n\\?\n[$\\delta(r - \\bar{r})$ is the Dirac delta function], with boundary conditions (at the star's center and surface) of\n\\<\\label{Newt_BC}\n\\mathcal{G}(0,\\bar{r})=0, \\qquad R\\partial_1\\mathcal{G}(R,\\bar{r}) = -3\\mathcal{G}(R,\\bar{r}),\n\\?\nwhere $\\partial_1$ denotes a partial derivative taken with respect to the\nfirst ``slot'' of the function.\n\nIf we then write [using Eq.~\\eqref{Q22N}, the factor of $r^2$ from the Green function equation~\\eqref{Leq}, and the\nprefactor on the right-hand side of Eq.~\\eqref{deltaPhieq2}]\n\\begin{equation}\\label{GN}\nG_N} % Changed to save space; was {\\mathrm{Newt}(r) := -5R^3r^2\\mathcal{G}(R,r)\/g(r),\n\\end{equation}\nwe have\n\\begin{widetext}\n\\<\\label{Q22N1}\n\\begin{split}\nQ^N} % Changed to save space; was {\\mathrm{Newt}_{22} &= \\int_0^RG_N} % Changed to save space; was {\\mathrm{Newt}(r)\\left[\\frac{3}{2}t_{rr}' - \\frac{4}{\\beta}t_{r\\perp}' - \\frac{r}{\\beta}t_{r\\perp}'' -\n\\left(\\frac{1}{\\beta^2} - \\frac{1}{2}\\right)t_\\Lambda' + \\frac{3}{r}t_{rr} - \\frac{\\beta}{r}t_{r\\perp}\\right]dr\\\\\n&= -\\int_0^R\\biggl\\{\\left[\\frac{3}{2}G_N} % Changed to save space; was {\\mathrm{Newt}'(r) - \\frac{3}{r}G_N} % Changed to save space; was {\\mathrm{Newt}(r)\\right]t_{rr} + \\left[\\frac{r}{\\beta}G_N} % Changed to save space; was {\\mathrm{Newt}''(r) - \\frac{2}{\\beta}G_N} % Changed to save space; was {\\mathrm{Newt}'(r) + \\frac{\\beta}{r}G_N} % Changed to save space; was {\\mathrm{Newt}(r)\\right]t_{r\\perp\n+ \\left(\\frac{1}{2} - \\frac{1}{\\beta^2}\\right)G_N} % Changed to save space; was {\\mathrm{Newt}'(r)t_\\Lambda\\biggr\\}dr.\n\\end{split}\n\\?\n\\end{widetext}\nWe have freely integrated by parts in obtaining the second expression, noting that the boundary terms are zero since $G_N} % Changed to save space; was {\\mathrm{Newt}(r)$ vanishes sufficiently rapidly as $r \\to 0$ and\nthe stresses are zero at the surface of the star (since we assume that the shear modulus vanishes at the star's surface).\\footnote{We shall treat the case where the stresses do \\emph{not} vanish at the surface of the star when we consider solid strange quark stars in Sec.~\\ref{SQM_computation}. Also, note that HJA claim that UCB's\nexpression does not include distributional contributions due to sudden changes\nin the shear modulus. This is not the case---these are included due to UCB's\nintegration by parts (cf.\\ the definition of the distributional derivative). All that the UCB derivation requires is,\ne.g.,  that the shear modulus vanish outside of the crust, not that\nit do so continuously.}\nThis reduces to UCB's Eq.~(64) if we take the Cowling approximation\n\\begin{equation}\n\\label{CowlingGN}\nG_N} % Changed to save space; was {\\mathrm{Newt}(r) \\to r^4\/g(r),\n\\end{equation}\ncorresponding to dropping the second term on the left-hand side of\nEq.~\\eqref{deltaPhieq2}.\n\nTo obtain an analogue of the expression for the maximum quadrupole\ngiven in Eq.~(5) of Owen~\\cite{OwenPRL}, we note that\nUCB's argument about maximum uniform strain leading to the maximum quadrupole\nstill holds here for the stars we consider, since the coefficients of the stress\ncomponents in the integrand are all uniformly positive. (We have checked this numerically for each background\nstellar model we consider.)\nThe strain tensor components are\n\\begin{subequations}\n\\label{sNewt}\n\\begin{align}\n\\label{srr}\n\\sigma_{rr} &= (32\\pi\/15)^{1\/2}\\bar{\\sigma}_\\mathrm{max},\n\\\\\n\\label{srp}\n\\sigma_{r\\perp} &= (3\/2)^{1\/2}\\sigma_{rr},\n\\\\\n\\label{sL}\n\\sigma_\\Lambda &= 3\\sigma_{rr}\n\\end{align}\n\\end{subequations}\nin the case where the\nstar is maximally (and uniformly) strained---see Eqs.~(67) in UCB. The\nbreaking strain $\\bar{\\sigma}_\\mathrm{max}$ is given by the von Mises\nexpression,\n\\begin{equation}\n\\label{vonMises}\n\\sigma_{ab}\\sigma^{ab} = 2\\bar{\\sigma}_\\mathrm{max}^2.\n\\end{equation}\nIt thus\ncorresponds to assuming that the lattice yields when it has stored a certain maximum\nenergy density. We then have\n\\<\\label{Q22N2}\n\\frac{|Q_{22}^{\\mathrm{max}, N}|}{\\bar{\\sigma}_\\mathrm{max}} = \\sqrt{\\frac{32\\pi}{15}}\\int_0^R\\mu(r)\\left[rG_N} % Changed to save space; was {\\mathrm{Newt}''(r) + 3G_N} % Changed to save space; was {\\mathrm{Newt}'(r)\\right]dr.\n\\?\nThis reduces to Eq.~(5) in Owen~\\cite{OwenPRL} if we use the Cowling\napproximation~\\eqref{CowlingGN}.\n\nNote that there is no direct contribution from $\\rho'$ to $G_N} % Changed to save space; was {\\mathrm{Newt}''$ in the\nno-Cowling case, despite what one might expect from Eq.~\\eqref{Leq}:  Writing\n$\\bar{\\mathcal{G}}(r) := \\mathcal{G}(R,r)$ for notational simplicity, the $\\rho'$ contribution from\n\\<\n\\bar{\\mathcal{G}}''(r) = (2\/r)\\bar{\\mathcal{G}}'(r) + \n[6\/r^2 + 4\\pi G\\rho'(r)\/g(r)]\\bar{\\mathcal{G}}(r)\n\\?\nis exactly canceled by one from \n\\<\ng''(r) = 6Gm(r)\/r^4 - 8\\pi G\\rho(r)\/r + 4\\pi G\\rho'(r)\n\\?\nin\n\\<\n\\begin{split}\nG''_N} % Changed to save space; was {\\mathrm{Newt}(r) &= -5R^3r^2[\\bar{\\mathcal{G}}''(r)\/g(r) - \\bar{\\mathcal{G}}(r)g''(r)\/\\{g(r)\\}^2\\\\\n&\\quad + \\{\\text{terms with no }\\rho'\\}].\n\\end{split}\n\\?\nHowever, there is a direct contribution from\n$\\rho'$ to $G_N} % Changed to save space; was {\\mathrm{Newt}''$ (via $g''$) if we make the Cowling\napproximation [Eq.~\\eqref{CowlingGN}].\nWe shall see that this leads to a\nsignificant\ndifference in the resulting contributions to the quadrupole moment from regions\nof the star surrounding a sudden change in density (e.g., near the crust-core\ninterface, which will be relevant for the quadrupoles supported by crustal\nelasticity considered by UCB and others).\n\nNumerically, we compute $G_N} % Changed to save space; was {\\mathrm{Newt}$ using the standard expression for the Green function in terms of the\ntwo independent solutions to the homogeneous equation [see, e.g., Eq.~(10.103) in Arfken and Weber~\\cite{AW}~]. Since we are solely interested in the Green function evaluated at the star's surface, we can\neliminate one of the homogeneous solutions using the boundary conditions there, and only consider the\nhomogeneous solution that is regular at the origin, which we call $F$.\nIn terms of $F$, the Green function is given by\n\\<\\label{cG_F}\n\\mathcal{G}(R,r) = -\\frac{F(r)}{3RF(R) + R^2F'(R)}.\n\\?\nWe thus solve $\\mathcal{L}_N F = 0$ [with the operator $\\mathcal{L}_N$ given by Eq.~\\eqref{Leq}] with the boundary conditions\n$F(r_0) = 1$ and $F'(r_0) = 2\/r_0$, where $r_0$ is the small inner radius used in the solution of the\nOV equations, as discussed at the end of Sec.~\\ref{GR}. [These boundary conditions come from\nregularity at the origin, which implies that $F(r) = O(r^2)$ there.]\n\nOur Green function method for obtaining the maximum quadrupole numerically may\nseem more complicated than existing methods because it introduces extra steps.\nHowever this method is ideal for showing that maximum stress gives the maximum\nquadrupole and for seeing how much stresses at different radii contribute to\nthe total quadrupole.\nIt also appears to be the simplest way of dealing with any potential\ndistributional contributions from the derivatives of the shear modulus, since\nthey are automatically taken care of by the integration by parts.\n\n\\section{General relativistic calculation of the maximum quadrupole}\n\\label{GR}\n\nHere we compute the maximum quadrupole moment in general relativity, using the Regge-Wheeler\ngauge~\\cite{RW} relativistic stellar perturbation theory developed by \\citet{TC}, as in\nthe similar calculation of the tidal Love number of a relativistic star by\n\\citet{Hinderer}.\nWe\nstart by writing down the line element corresponding to a static, even-parity,\n$l = 2$ first-order perturbation of a static,\nspherical, relativistic star in the Regge-Wheeler gauge [cf.\\ Eq.~(14) in Hinderer~\\cite{Hinderer}~]:\n\\<\\label{ds2}\n\\begin{split}\nds^2 &= -[1 + H_0(r)Y_{lm}]f(r)dt^2 + [1 + H_2(r)Y_{lm}]h(r)dr^2\\\\\n&\\quad + [1 + K(r)Y_{lm}]r^2(d\\theta^2 + \\sin^2\\theta d\\phi^2).\n\\end{split}\n\\?\nHere we have used the notation of Wald~\\cite{Wald} for the background, so that $f$ and $h$ are the standard\nSchwarzschild functions for the unperturbed star, with $f = e^{2\\phi}$, where\n\\<\n\\phi'(r) = \\frac{m(r) + 4\\pi r^3 p}{r[r - 2m(r)]},\n\\?\nwith $\\phi(R) = \\log(1 - 2M\/R)\/2$, and\n\\<\nh(r) = \\left[1 - \\frac{2m(r)}{r}\\right]^{-1}.\n\\?\nIn these expressions,\n\\begin{equation}\nm(r) := 4\\pi \\int_0^r \\rho(\\bar{r})\\bar{r}^2d\\bar{r}.\n\\end{equation}\nAlso, recall that we write our spherical harmonics in terms of $l$ and $m$, following\nUCB, even though we specialized to $l = m = 2$, and that we are now taking $G = c = 1$.\n\nThe metric perturbation is determined by $H_0$, $H_2$, and $K$, which here are sourced by the\nperturbation to the star's stress-energy tensor. The appropriate stress-energy\ntensor can be obtained directly\nfrom the standard Newtonian expression~\\eqref{deltatau} by simple covariance arguments, as in\nSchumaker and Thorne~\\cite{ST}, or from the detailed relativistic elasticity\ntheory of Carter and Quintana~\\cite{CQ} [see their Eq.~(6.19); this is also given in Eq.~(128) of\nKarlovini and Samuelsson~\\cite{KaSaI}~]. All\nwe really need for our purposes is to note that the shear contribution is\ntracefree with respect to the background metric, so that we can use the obvious\ncovariant generalization of the decomposition given by UCB,\\footnote{Of course, this assumes that it is\npossible to obtain \\emph{any} symmetric tracefree tensor from the detailed relativistic expression, but---as would be expected (and can easily be seen from the expressions)---this is indeed the case, at least if one only works to first order in the perturbation, as we\ndo here.\nAlso, it is instructive to note that we do not need to know the specifics of the matter displacements that generate\nthe quadrupoles we consider, only that there is a tracefree contribution to the star's\nstress-energy tensor whose maximum value is given by the material's shear\nmodulus and von Mises breaking strain.} yielding \n\\<\\label{deltaT}\n\\begin{split}\n\\delta T_{ab} &= [\\delta\\rho\\hat{t}_a\\hat{t}_b +  \\delta p(g_{ab} + \\hat{t}_a\\hat{t}_b) - \nt_{rr}(\\hat{r}_a\\hat{r}_b - q_{ab}\/2)]Y_{lm}\\\\\n&\\quad - \nt_{r\\perp}f_{ab} - t_\\Lambda(\\tilde{\\Lambda}_{ab} + h^{1\/2}Y_{lm}q_{ab}\/2),\n\\end{split}\n\\?\nwith the full stress-energy tensor given by\n\\<\nT_a{}^b = \\rho\\hat{t}_a\\hat{t}^b + p(\\delta_a{}^b + \\hat{t}_a\\hat{t}^b) + \\delta T_a{}^b.\n\\?\nHere, indices now run over all four spacetime dimensions and $g_{ab}$ denotes\nthe background (spacetime) metric (which we use to raise and lower indices).\nAdditionally, we have introduced the background temporal and radial unit vectors $\\hat{t}_a$ and $\\hat{r}_a$;\n$q_{ab}$ is the induced metric on the unit $2$-sphere;\n$f_{ab} := 2r\\hat{r}_{(a}\\nabla_{b)}Y_{lm}\/\\beta$;\nand\n$\\tilde{\\Lambda}_{ab} :=  r^2h^{1\/2}\\nabla_a\\nabla_bY_{lm}\/\\beta^2 + f_{ab}\/\\beta$.\nHere $\\hat{r}_a$ and $\\nabla_a$ now have their curved-space meanings.\n\nOur $\\tilde{\\Lambda}_{ab}$\ndiffers from the Newtonian $\\Lambda_{ab}$ [from UCB, given in our Eq.~\\eqref{Lambda}] due to the insertion of $h^{1\/2}$. This insertion is necessary\nfor $\\tilde{\\Lambda}_{ab}$ to be transverse and orthogonal to $f_{ab}$ (with respect to the background spacetime metric).\nThe same logic leads to the introduction of the factor of $h^{1\/2}$ multiplying $q_{ab}$ in the $t_\\Lambda$ term in\nEq.~\\eqref{deltaT}; it is there so that the $t_\\Lambda$ term is orthogonal to the $t_{rr}$ term. We\nhave used UCB's convention\nfor the relative sign between the perfect fluid and shear portions of the stress-energy tensor, though we\nhave reversed the overall sign. (However, we used the UCB convention proper\nin Sec.~\\ref{Newt}.)\nThe factor of $h^{1\/2}$ in the coefficient of $t_\\Lambda$ leads to a factor\nof $h^{-1}$ in the\nstrain $\\sigma_\\Lambda$ that corresponds to the von Mises\nbreaking strain~\\eqref{vonMises}.\nWe thus replace the Newtonian Eq.~\\eqref{sL} with\n\\begin{equation}\n\\label{sLGR}\n\\sigma_\\Lambda = 3\\sigma_{rr}\/h,\n\\end{equation}\nleaving Eqs.~\\eqref{srr} and~\\eqref{srp} unchanged.\n\nOne can now obtain an equation for $H_0$ from the perturbed Einstein equations,\nas in Ipser~\\cite{Ipser}. (The other two metric functions, $H_2$ and $K$, can be expressed in terms of $H_0$;\nthese expressions are given by Ipser.) The concordance for notation\nis $\\nu = 2\\phi$, $e^\\nu = f$, $\\lambda = 2\\psi$, $e^\\lambda = h$, $\\rho_1 = -\\delta\\rho$, $p_1 = -\\delta p$,\n$\\mathfrak{P}_2 = t_{rr}$, $\\mathfrak{Q}_1 = h^{1\/2}t_{r\\perp}\/\\beta$, and $\\mathfrak{S} = h^{1\/2}t_\\Lambda\/\\beta^2$. Additionally, Ipser's $H_0$ is the\nnegative of ours. The relevant result is given in Ipser's Eqs.~(27)--(28), and\nis (in our notation)\n\\<\nH_0'' + \\left(\\frac{2}{r} + \\phi' -\\psi'\\right)H_0' + \\mathcal{P}(r)H_0= 8\\pi h^{1\/2}\\mathcal{S}(r),\n\\?\nwhere\n\\<\\label{cP}\n\\mathcal{P}(r) := 2\\phi'' + 2\\phi'\\left(\\frac{3}{r} - \\phi' -\\psi'\\right) + \\frac{2\\psi'}{r} - \\frac{\\beta^2}{r^2}h\n\\?\nand\n\\<\\label{cS}\n\\begin{split}\n\\mathcal{S}(r) &:=\nh^{1\/2}(\\delta\\rho + \\delta p - t_{rr}) +\n2\\biggl\\{(3 - r\\phi')\\frac{t_{r\\perp}}{\\beta} + r\\frac{t_{r\\perp}'}{\\beta}\\\\\n&\\,\\quad + [r^2\\phi'' + r\\phi'(5 - r\\phi') + r\\psi' -\\beta^2h\/2 + 1]\\frac{t_\\Lambda}{\\beta^2}\\\\\n&\\,\\quad + r^2\\phi'\\frac{t_\\Lambda'}{\\beta^2}\\biggr\\}\\\\\n&=: h^{1\/2}(\\delta\\rho + \\delta p) + \\mathcal{S}_{[t]}(r).\n\\end{split}\n\\?\nHere we have defined $\\psi := (1\/2)\\log h$ and written $\\mathcal{S}_{[t]}$\nfor the contributions from shear stresses. (The ``$=:$'' notation implies that the quantity\nbeing defined is on the right-hand side of the equality.)\n\nWe now wish to eliminate $\\delta\\rho$ and $\\delta p$ in favor of the shear stresses, as in the Newtonian calculation. We use the same projections of stress-energy conservation as in the Newtonian case (projecting onto the quantities defined by\nthe background spacetime, for simplicity) along with the Oppenheimer-Volkov\n(OV) equations, giving\n\\<\n\\begin{split}\n\\delta\\rho + \\delta p &= \\frac{1}{\\phi'}\\biggl[-\\frac{H_0'}{2}(\\rho + p)\n-\\delta p' + t_{rr}' + \\left(\\frac{3}{r} + \\phi'\\right)t_{rr}\\\\\n&\\quad - \\frac{\\beta}{r}h^{1\/2}t_{r\\perp}\\biggr]\n\\end{split}\n\\?\nand\n\\<\n\\begin{split}\n\\delta p &= -\\frac{H_0}{2}(\\rho + p) - \\frac{t_{rr}}{2} + \\frac{1}{\\beta h^{1\/2}}\\left[(3 + r\\phi')t_{r\\perp} + rt_{r\\perp}'\\right]\\\\\n&\\quad + h^{1\/2}\\left(\\frac{1}{\\beta^2} - \\frac{1}{2}\\right)t_\\Lambda.\n\\end{split}\n\\?\nUsing the second expression to substitute for $\\delta p'$ in the first, we have\n\\<\\label{drplusdp}\n\\begin{split}\n\\delta\\rho + \\delta p &= \\frac{1}{\\phi'}\\biggl\\{\\frac{H_0}{2}(\\rho' + p') + \\left[\\frac{3}{r} + \\phi'\\right]t_{rr} + \\frac{3}{2}t_{rr}'\\\\\n&\\quad - \\frac{1}{\\beta h^{1\/2}}\\biggl[\\left(\\frac{\\beta^2 h}{r} + \\phi' + r\\phi'' - \\psi'[3 + r\\phi'] \\right)t_{r\\perp}\\\\\n&\\quad + (4 + r[\\phi' - \\psi'])t_{r\\perp}' + rt_{r\\perp}''\\biggr] + \n\\left(\\frac{1}{2} - \\frac{1}{\\beta^2}\\right)\\\\\n&\\quad\\times h^{1\/2}(\\psi't_\\Lambda + t_\\Lambda')\\biggr\\}\\\\\n&=: \\frac{H_0}{2\\phi'}(\\rho' + p') + \\frac{\\mathcal{S}_{[\\delta\\rho,\\delta p]}(r)}{\\phi'}.\n\\end{split}\n\\?\nThe equation for $H_0$ thus becomes\n\\<\\label{cLGR}\n\\begin{split}\n(\\mathcal{L}_\\mathrm{GR} H_0)(r) &:= H_0'' + \\left(\\frac{2}{r} + \\phi' -\\psi'\\right)H_0'\\\\\n&\\,\\quad + \\left[\\mathcal{P}(r) - 4\\pi h\\frac{\\rho' + p'}{\\phi'}\\right]H_0\\\\\n&\\,= 8\\pi h^{1\/2}[h^{1\/2}\\mathcal{S}_{[\\delta\\rho,\\delta p]}(r)\/\\phi' + \\mathcal{S}_{[t]}(r)].\n\\end{split}\n\\?\n[$\\mathcal{P}(r)$ and $\\mathcal{S}_{[t]}(r)$ are given in Eqs.~\\eqref{cP} and~\\eqref{cS}, respectively.]\nAs expected, this\nreduces to Eq.~\\eqref{deltaPhieq2} in the Newtonian\nlimit [where we have $H_0 \\to 2\\delta\\Phi$ and $\\phi' \\to g(r)$].\n\nWe now want to write the equation for $H_0$ in Sturm-Liouville form in order to obtain its Green\nfunction easily. To do this, we note that the appropriate ``integrating factor'' (for the first\ntwo terms) is $r^2(f\/h)^{1\/2}$, which gives\n\\begin{multline}\n\\label{H0_S-L}\n[r^2(f\/h)^{1\/2}H_0']' + r^2(f\/h)^{1\/2}\\left[\\mathcal{P}(r) - 4\\pi h\\frac{\\rho' + p'}{\\phi'}\\right]H_0\\\\\n= 8\\pi r^2f^{1\/2}[h^{1\/2}\\mathcal{S}_{[\\delta\\rho,\\delta p]}(r)\/\\phi' + \\mathcal{S}_{[t]}(r)].\n\\end{multline}\nWe also need the boundary conditions, which are given by matching $H_0$ onto a\nvacuum solution at the surface of the star. The vacuum\nsolution that is regular at infinity is given by Eq.~(20) in Hinderer~\\cite{Hinderer} with $c_2 = 0$,\nviz.,\n\\<\\label{H0_BC}\n\\begin{split}\nH_0(R) &= c_1\\biggl[\\left(\\frac{2}{\\mathcal{C}} - 1\\right)\\frac{\\mathcal{C}^2\/2 + 3\\mathcal{C} - 3}{1 - \\mathcal{C}}\\\\\n&\\quad + \\frac{6}{\\mathcal{C}}\\left(1 - \\frac{1}{\\mathcal{C}}\\right)\\log\\left(1 - \\mathcal{C}\\right)\\biggr],\n\\end{split}\n\\?\nwhere we have evaluated this at the star's surface ($r = R$) and\ndefined the star's compactness\n\\<\n\\label{compactness}\n\\mathcal{C} := 2GM\/Rc^2\n\\?\n(now returning to showing factors of $G$ and $c$ explicitly).\nWe require that $H_0$\nand $H_0'$ be continuous at the star's surface. The value of $c_1$ obtained from\nthis matching of the internal and external solutions gives us \nthe quadrupole moment. If we use the quadrupole moment amplitude that reduces\nto the UCB integral [given in our Eq.~\\eqref{Q22}] in the Newtonian limit, we have\n\\<\\label{Q22rel}\nQ_{22} = \\frac{G^2}{c^4}\\frac{M^3c_1}{\\pi}.\n\\?\n[This expression comes from inserting a pure $l = m = 2$ density perturbation\ninto Eq.~(2) in Hinderer~\\cite{Hinderer}, contracting the free indices with\nunit position vectors, performing the angular integral, for\nwhich the expressions in Thorne~\\cite{Thorne} are useful,\nand noting that the result is $(8\\pi\/15)Y_{22}$ times\nour Eq.~\\eqref{Q22}. The\ngiven result then follows immediately from Hinderer's Eqs.~(7), (9), and (22);\nwe reverse the overall sign since we have reversed the UCB sign convention for the\nstress-energy tensor.]\n\nWe then have a Green function for $Q_{22}$ of \n\\<\\label{cG_GR}\n\\begin{split}\n\\mathcal{G}_\\mathrm{GR}(R,r) &= \\left(\\frac{2GM}{c^2}\\right)^3\\left(1 - \\frac{2GM}{Rc^2}\\right)^{-1}\\\\\n&\\quad\\times\\frac{\\mathcal{U}(r)}{c^2R^2[\\mathcal{U}'(R)H_0(R) - \\mathcal{U}(R)H_0'(R)]}\n\\end{split}\n\\?\n(including the overall factor of $8\\pi G\/c^4$ that\nmultiplies the source).\nHere $\\mathcal{U}$ is given by $\\mathcal{L}_\\mathrm{GR}\\mathcal{U} = 0$ [$\\mathcal{L}_\\mathrm{GR}$ is given in\nEq.~\\eqref{cLGR}], with boundary conditions $\\mathcal{U}(r_0) = 1$ and $\\mathcal{U}'(r_0) = 2\/r_0$.\n[Compare Eq.~(10.103) in Arfken and Weber~\\cite{AW}, as well as our Newtonian\nversion above.]\nAdditionally, $H_0(R)$ and $H_0'(R)$ are given by the boundary conditions~\\eqref{H0_BC}\nwith $c_1 \\to 1$. [One obtains this expression by first computing the Green function\nfor $H_0(R)$ following Arfken and Weber, then dividing through by the quantity in brackets in Eq.~\\eqref{H0_BC}\nto obtain $c_1$, and finally using Eq.~\\eqref{Q22rel} to obtain $Q_{22}$. We have also noted that\n$1\/f \\to h \\to 1\/(1 - 2GM\/Rc^2)$ at the star's surface.]\nWe thus define, for notational simplicity, two relativistic generalizations\nof $G_N} % Changed to save space; was {\\mathrm{Newt}(r)$: One,\n\\<\\label{GGR}\nG_\\mathrm{GR}(r) := \\frac{r^2(fh)^{1\/2}\\mathcal{G}_\\mathrm{GR}(R,r)}{\\phi'},\n\\?\nfor\nthe contributions from $\\mathcal{S}_{[\\delta\\rho,\\delta p]}$, and one,\n\\<\\label{GbGR}\n\\bar{G}_\\mathrm{GR}(r) := r^2f^{1\/2} \\mathcal{G}_\\mathrm{GR}(R,r),\n\\?\nfor the contributions\nfrom $\\mathcal{S}_{[t]}$. \n\nWith these definitions, the integral expression for the quadrupole in terms\nof the stresses and the\nstructure of the background star is\n\\<\\label{Q22GR}\n\\begin{split}\nQ_{22} &= \\int_0^R\\left[G_\\mathrm{GR}(r)\\mathcal{S}_{[\\delta\\rho,\\delta p]}(r)+\\bar{G}_\\mathrm{GR}(r)\\mathcal{S}_{[t]}(r)\\right]dr\\\\\n&= \\int_0^R(\\mathcal{C}_{rr}t_{rr} + \\mathcal{C}_{t\\perp}t_{r\\perp} + \\mathcal{C}_\\Lambda t_\\Lambda)dr,\n\\end{split}\n\\?\nwhere\n\\begin{subequations}\\label{cCs}\n\\begin{gather}\n\\mathcal{C}_{rr} := \\left(\\frac{3}{r} + \\phi'\\right)G_\\mathrm{GR}(r) - \\frac{3}{2}G_\\mathrm{GR}'(r) -\nh^{1\/2}\\bar{G}_\\mathrm{GR}(r),\\\\\n\\begin{split}\n\\mathcal{C}_{r\\perp} &:= -\\frac{\\beta h^{1\/2}}{r}G_\\mathrm{GR}(r) + \\frac{2 + r(\\phi' + \\psi')}{\\beta h^{1\/2}}G_\\mathrm{GR}'(r)\\\\\n&\\,\\quad - \\frac{r}{\\beta h^{1\/2}}G_\\mathrm{GR}''(r) + \\frac{4 - 2r\\phi'}{\\beta}\\bar{G}_\\mathrm{GR}(r) - 2\\frac{r}{\\beta}\\bar{G}'_\\mathrm{GR}(r),\n\\end{split}\\\\\n\\begin{split}\n\\mathcal{C}_\\Lambda &:= \\left(\\frac{1}{\\beta^2} - \\frac{1}{2}\\right)h^{1\/2}G_\\mathrm{GR}'(r)\\\\\n&\\,\\quad + \\frac{2r\\phi'(3 - r\\phi') + 2r\\psi' - \\beta^2h + 2}{\\beta^2}\\bar{G}_\\mathrm{GR}(r)\\\\\n&\\,\\quad - \\frac{2r^2\\phi'}{\\beta^2}\\bar{G}_\\mathrm{GR}'(r),\n\\end{split}\n\\end{gather}\n\\end{subequations}\nand we have integrated by parts twice to obtain the second equality in\nEq.~\\eqref{Q22GR}, using the same argument as in our Newtonian calculation.\n\nWe now look at the maximum quadrupole. This is still given by the uniformly\nmaximally strained case: We have checked numerically that the\ncoefficients of the three\nstress terms are always negative for all the background stars we consider. We thus\nhave a maximum quadrupole\ngiven by inserting Eqs.~\\eqref{ts}, \\eqref{srr}, \\eqref{srp}, and\n\\eqref{sLGR} into Eq.~\\eqref{Q22GR}, yielding\n\\begin{widetext}\n\\<\\label{Q22GR2}\n\\frac{|Q_{22}^\\mathrm{max, GR}|}{\\bar{\\sigma}_\\mathrm{max}} = \\sqrt{\\frac{32\\pi}{15}}\\int_0^R\\mu(r)\\biggl\\{\\left[\\frac{6}{r}(h^{1\/2} - 1) - 2\\phi'\\right]G_\\mathrm{GR}(r) + \\left[3 - \\frac{r}{h^{1\/2}}(\\phi' + \\psi')\\right]G_\\mathrm{GR}'(r\n+ \\frac{r}{h^{1\/2}}G_\\mathrm{GR}''(r) + \\mathcal{Q}^\\mathrm{stress}\\biggr\\}dr,\n\\?\nwhere\n\\<\n\\mathcal{Q}^\\mathrm{stress} := 2\\left[\\frac{r\\phi'(r\\phi' - 3) - r\\psi'  - 1}{h} + r\\phi' + h^{1\/2} + 1\\right]\\bar{G}_\\mathrm{GR}(r) + 2r\\left(\\frac{r\\phi'}{h} + 1\\right)\\bar{G}_\\mathrm{GR}'(r)\n\\?\n\\end{widetext}\nis the contribution from the stresses' own gravity. We have split it off both for\nease of notation and because it is negligible except for the\nmost massive and compact stars, as illustrated\nbelow. The  contributions from the density and pressure perturbations are so much larger due to the factor of $1\/\\phi'$ present in $G_\\mathrm{GR}$ [cf.\\ Eqs.~\\eqref{GGR} and~\\eqref{GbGR}]. It is easy to see that Eq.~\\eqref{Q22GR2} reduces to\nEq.~\\eqref{Q22N2} in the Newtonian limit, where $h\\to 1$,\nand we can neglect the contributions involving $\\phi'$, $\\psi'$, and $\\mathcal{Q}^\\mathrm{stress}$.\n\n\n\\begin{figure}[htb]\n\\begin{center}\n\\epsfig{file=Q22_integrands_SLy_h0_7e19.eps,width=8cm,clip=true}\n\\end{center}\n\\caption[$Q_{22}$ integrands for the SLy EOS and an\n$0.500\\,M_{\\odot}$ star]{\\label{Q22_integrands_SLy_2} The $Q_{22}$ integrands (without the factor of $\\mu\\bar{\\sigma}_\\mathrm{max}$) for the SLy EOS and an\n$0.500\\,M_{\\odot}$ star with a compactness of $0.12$.}\n\\end{figure}\n\n\\begin{figure}[htb]\n\\begin{center}\n\\epsfig{file=Q22_integrands_SLy_h0_2e20.eps,width=8cm,clip=true}\n\\end{center}\n\\caption[$Q_{22}$ integrands for the SLy EOS and a\n$1.40\\,M_{\\odot}$ star]{\\label{Q22_integrands_SLy_4} The $Q_{22}$ integrands (without the factor of $\\mu\\bar{\\sigma}_\\mathrm{max}$) for the SLy EOS and a\n$1.40\\,M_{\\odot}$ star with a compactness of $0.35$.}\n\\end{figure}\n\n\\begin{figure}[htb]\n\\begin{center}\n\\epsfig{file=Q22_integrands_SLy_h0_7e20.eps,width=8cm,clip=true}\n\\end{center}\n\\caption[$Q_{22}$ integrands for the SLy EOS and a maximum mass star]{\\label{Q22_integrands_SLy_6} The $Q_{22}$ integrands (without the factor of $\\mu\\bar{\\sigma}_\\mathrm{max}$) for the SLy EOS and a maximum mass,\n$2.05\\,M_{\\odot}$ star, with a compactness of $0.60$.}\n\\end{figure}\n\nWe now show how the relations between the different maximal-strain $Q_{22}$ Green functions\n[given by the integrands in Eqs.~\\eqref{Q22N2} and~\\eqref{Q22GR2} without the factors of\n$\\mu$ (but with the overall prefactor)] vary with\nEOS, as well as with the mass of the star for a given EOS.\nThis gives\nan indication of how much difference the various approximations make in different situations. We start with the unified SLy EOS~\\cite{DH}, obtained by Haensel\nand Potekhin~\\cite{HP}\n(using the table provided by\nthe Ioffe group~\\cite{HPY} at~\\cite{Ioffe}), which is a standard choice for making predictions about crustal\nquadrupoles (e.g., in Horowitz~\\cite{Horowitz}, HJA, and our Sec.~\\ref{results_crust}). Here we illustrate the\nchanges in the Green functions with mass for stars with masses ranging from $0.5\\,M_{\\odot}$ to the EOS's maximum mass of $2.05\\,M_{\\odot}$;\nsee Figs.~\\ref{Q22_integrands_SLy_2},  \\ref{Q22_integrands_SLy_4}, and \\ref{Q22_integrands_SLy_6}. (All three Green functions agree extremely closely for stars around the EOS's minimum mass of $0.094\\,M_{\\odot}$, so we do not show this case, particularly because such\nlow-mass neutron stars are of unclear astrophysical relevance.)\nThese stars' compactnesses [defined in Eq.~\\eqref{compactness}]\nrange from $0.12$ to\n$0.6$. Note that Fig.~\\ref{Q22_integrands_SLy_6} has a different vertical scale than the other two plots, due to the suppression of the quadrupole for massive, compact stars (discussed below). \n\n\\begin{figure}[htb]\n\\begin{center}\n\\epsfig{file=Q22_integrand_Hy1_h0_max_ratios.eps,width=8cm,clip=true}\n\\end{center}\n\\caption[Ratios of the $Q_{22}$ integrands for the Hy1 EOS and a maximum mass\nstar]{\\label{Q22_integrands_Hy1_ratios} Ratios of the $Q_{22}$ integrands\nwith the Newtonian Cowling approximation integrand for the Hy1 EOS and a maximum mass,\n$2.06\\,M_{\\odot}$ star, with a compactness of $0.49$. Note that the top and bottom plots have completely\nseparate vertical axis scalings.}\n\\end{figure}\n\nWe illustrate the ratios of the various $Q_{22}$ Green functions to the\nNewtonian Cowling approximation one for the maximum mass ($2.05\\,M_{\\odot}$) hybrid\nstar using the Hy1 EOS (see Table~I in~\\cite{J-MO1}) in Fig.~\\ref{Q22_integrands_Hy1_ratios}.\\footnote{As\ndiscussed in~\\cite{J-MO1}, for our low-density EOS, we use the same combination of\nthe Baym, Pethick, and Sutherland (BPS)~\\cite{BPS} EOS for $n_B < 0.001\\text{ fm}^{-3}$ and\nthe Negele and Vautherin~\\cite{NV} EOS for $0.001\\text{ fm}^{-3} < n_B < 0.08\\text{ fm}^{-3}$\nused by Lattimer and Prakash~\\cite{LP2001} ($n_B$ is the baryon number density).\nThese were obtained from the table provided by\nKurkela~\\emph{et al.}~\\cite{Kurkelaetal} at~\\cite{Kurkelaetal_URL}.\nBulk quantities of hybrid stars such as the mass and quadrupole moment (from core deformations) do not depend much on the precise choice of low-density EOS.} We see the overestimate of the Newtonian\nno Cowling approximation calculation for perturbations in the core, particularly compared with the \ngeneral relativistic (GR) version, and also see the\noverestimate of the Newtonian Cowling approximation version for crustal perturbations. (We do not make\nsome sort of similar plot for the solid strange quark star case, since the expressions for the maximum\nquadrupole in this case end up being rather different than the integrated-by-parts ones presented in the previous sections, as we shall see in Sec.~\\ref{SQM_computation}.)\n\nIn all these cases, we compute the stellar background fully relativistically, using\nthe OV equations and identifying the OV equations' Schwarzschild\nradial coordinate\nwith the Newtonian radial coordinate when necessary. We have used the enthalpy form of the OV\nequations given by Lindblom~\\cite{Lindblom} and implemented the inner boundary\ncondition by taking the star to have an inner core of radius $r_0 =\n100\\text{ cm}$, whose mass is given by $(4\/3)\\pi r_0^3\\epsilon_0$, where\n$\\epsilon_0$ is the energy density corresponding to the central enthalpy that\nparametrizes the solution. (The spike near the origin seen in the bottom plot in Fig.~\\ref{Q22_integrands_Hy1_ratios} is due to this implementation of the inner boundary condition and has a\nnegligible effect on the computed maximum quadrupoles.) In all cases, we have used {\\sc{Mathematica}}~7's\ndefault methods to solve the differential equations, find roots, etc. We have computed as many derivatives\nas possible analytically, to aid numerical accuracy, e.g., using the OV equations to substitute for derivatives of the pressure, and also using the Green function equations to\nexpress second derivatives of the Green functions in terms of the functions themselves and their first derivatives.\n\n\n\\section{Results}\n\\label{results2}\n\n\\subsection{Maximum $Q_{22}$ for crustal deformations}\n\\label{results_crust}\n\nHere we consider the maximum quadrupoles from elastic deformations of a\nnonaccreted\ncrust in three possible situations, following HJA. In particular, we use the SLy EOS\n(as do Horowitz~\\cite{Horowitz} and HJA, though they do not refer to it by that name) and\nimpose two comparison crustal thicknesses to ascertain how much this\naffects the maximum quadrupole. Here we use the\nsame rough\nmodel for the crust's shear modulus used by HJA. We\nalso consider the more detailed model for the shear modulus\nobtained using the crustal composition provided by Douchin and\nHaensel~\\cite{DH} (also used by Horowitz~\\cite{Horowitz} and HJA).\nHere the crust's thickness is fixed to the value given in that work. In this case,\nwe also consider a different high-density EOS that yields much less compact stars with larger crusts.\n\nSpecifically, the two comparison crustal thicknesses are given by taking the base of the\ncrust to occur at densities of $2.1\\times10^{14} \\text{ g cm}^{-3}$ (thick\ncrust, for comparison with UCB) or $1.6\\times10^{14} \\text{ g cm}^{-3}$ (thin\ncrust, following a suggestion by Haensel~\\cite{Haensel}), while Douchin and\nHaensel place the bottom of the crust at a density of\n$1.28\\times10^{14} \\text{ g cm}^{-3}$.\nFor the two comparison cases, we take the shear modulus to be\n$10^{16}\\text{ cm}^2\\text{ s}^{-2}$ times the star's density (in $\\text{g cm}^{-3}$). As\nillustrated in HJA's Fig.~2, this is an underestimate of $<50\\%$, except at the very extremes of the density range considered.\\footnote{Note that Fig.~3 in HJA is not in agreement with their Fig.~2. When we reproduce those figures, we\nfind that the ratio $\\mu\/\\rho$ is considerably closer to $10^{16}\\text{ cm}^2\\text{ s}^{-2}$ over all the density range than the trace shown in HJA's Fig.~3, so their approximation is better than it would appear from that figure.}\nWe plot the quadrupole moment and ellipticity for these two cases for masses between\n$\\sim 1.2\\,M_{\\odot}$ (around the minimum observed neutron star mass---see~\\cite{Lattimer_table})\nand the SLy EOS's maximum mass of $2.05\\,M_{\\odot}$ in Fig.~\\ref{Q22s_vs_M_SLy}.\n\n\\begin{figure}[htb]\n\\begin{center}\n\\epsfig{file=Q22s_vs_M_SLy_c16.eps,width=8cm,clip=true}\n\\end{center}\n\\caption{\\label{Q22s_vs_M_SLy} The Newtonian Cowling, Newtonian no\nCowling, and full relativistic (including stress contributions) values for the\nmaximum quadrupole deformations (and fiducial ellipticity) due to crustal\nstresses versus mass for two choices of crustal thickness. These are computed using the SLy EOS with the rough HJA recipe for the shear\nmodulus and a breaking strain of $0.1$.}\n\\end{figure}\n\nIn addition to the quadrupole moments, we also show the fiducial ellipticity\n$\\epsilon_\\mathrm{fid} = \\sqrt{8\\pi\/15}Q_{22}\/I_{zz}$\n[e.g., Eq.~(2) of~\\cite{OwenPRL}~].\nHere  $I_{zz}$ is the star's principal moment of\ninertia, for which we use the fiducial value of $I_{zz} = 10^{38} \\text{ kg m}^2 =\n10^{45} \\text{ g cm}^2$ used in the LIGO\/Virgo papers rather than the true value for a given mass and EOS,\nwhich can be greater by a factor of a few.\nWe do this for easy comparison with the observational\npapers, since they frequently quote\nresults in terms of this fiducial ellipticity instead of the quadrupole\nmoment, which is the quantity truly measured.\n\n\\emph{Nota bene} (N.B.):\\ We present these fiducial ellipticities \\emph{only} for comparison with LIGO\/Virgo\nresults, not to give\nany indication of the size of the deformation. While the true ellipticity\ngives a measure of the size of the deformation in the Newtonian case (up to ambiguities from the fact that the true density distribution is nonuniform), it does not do so in\nany obvious way in the relativistic case. Nevertheless, the relativistic shape of the star's surface can be obtained from its quadrupole deformation, as shown in~\\cite{NKJ-M_shape}. However, if one wished to know, for instance, how much the star is deformed as a function of radius, one would need to calculate this using a detailed relativistic theory of elasticity to relate the stresses to the matter displacements, as in Penner~\\emph{et al.}~\\cite{Penneretal}.\n\nIn the more detailed\ncase, we use the HJA version of the Ogata and Ichimaru~\\cite{OI} shear modulus,\ncombined with the Douchin and Haensel~\\cite{DH} results for the crust's\ncomposition.\nThis is [correcting a typo in HJA's Eq.~(20)],\n\\<\\label{mueff}\n\\mu_\\mathrm{eff} = 0.1194\\left(\\frac{4\\pi}{3}\\right)^{1\/3}\\left(\\frac{1-X_n}{A}n_b\\right)^{4\/3}(Ze)^2,\n\\?\nwhere $X_n$ is the fraction of neutrons outside of nuclei, $A$ and $Z$ are the\natomic and proton number of the nuclei, respectively, $n_b$ is the baryon number density, and\n$e$ is the fundamental charge.\n\nSince HJA's study, there have been a few improvements\nto the Ogata and Ichimaru result: Horowitz and Hughto~\\cite{HH} have computed\nthe effects of charge screening, finding a $\\sim 7\\%$ reduction in the shear\nmodulus. Baiko~\\cite{BaikoCPP} has also considered a relativistic model\nfor the electron polarizability and arrived at similar conclusions. Indeed, Baiko's results suggest that screening will yield an even smaller\ncorrection in the innermost portion of the crust, where the shear modulus is the largest, and the\nelectrons are the most relativistic, with a relativity parameter over an order of magnitude larger than the largest\nBaiko considers. (However, the ion charge numbers are also almost always somewhat greater than the largest Baiko considers,\nparticularly at the very innermost portion of the crust, which will tend to increase the effect.) \n\nBaiko~\\cite{Baiko} has also recently computed quantum corrections, and finds\nthat they reduce the shear modulus by up to $\\sim 18\\%$ in some regimes.\nHowever, in our case, the reduction will be much smaller, based on the scaling\nof $\\rho^{1\/6}\/(ZA^{2\/3})$ given near the end of Baiko's Sec.~6. Even though\nour densities are over an order of magnitude greater, the nuclei we\nconsider are also over an order of magnitude more massive than the ${}^{12}$C\ncomposition Baiko considers, so\nthe quantum mechanical effects end up being reduced by about an order of\nmagnitude from the number Baiko quotes.\nWe thus use the same Ogata and Ichimaru result used by HJA, noting that the\nresulting quadrupoles might be reduced by less than $10\\%$ due to charge screening\nand quantum effects---an error which is small compared to other\nuncertainties, such as crust thickness and the composition of dense matter. Indeed, there\nis a factor of $\\sim 2$ uncertainty in the shear modulus due to angle averaging (even\ndisregarding whether the implicit assumption of a polycrystalline structure for the crust is\nwarranted): As shown by\nHill~\\cite{Hill}, the Voigt average used by Ogata and Ichimaru is an upper bound on the\ntrue shear modulus of a polycrystal. A lower bound is given by the Reuss average (also\ndiscussed in Hill~\\cite{Hill}), for which the prefactor in Eq.~\\ref{mueff} would be\n$0.05106$. \n\nNote that there would be even\nfurther corrections to the shear modulus due to pasta phases (see~\\cite{PP}), but such phases are not present in the Douchin and Haensel model~\\cite{DH}. We also note that the Douchin and Haensel results only include the very\ninnermost portion of the outer crust. However, this lack of coverage has a negligible effect on the final\nresults for the quadrupoles, since the neglected region has at most half the radial extent of the inner crust and the shear modulus in this region is orders of magnitude below its maximum value at the bottom of the inner crust. We have checked this explicitly using the detailed calculations\nof the outer crust composition due to R{\\\"u}ster, Hempel, and Schaffner-Bielich~\\cite{RHS-B}, available at~\\cite{HempelURL}.\n\nWe plot the maximum quadrupole and ellipticity in the three approximations for the detailed shear modulus\nmodel in Fig.~\\ref{Q22s_vs_M_SLy_DH_HJA}. Here we show these for the SLy EOS proper, and also for\na high-density EOS that yields much less compact stars (and a crust that is $\\sim 2$ times as thick), and thus larger maximum quadrupoles. For the latter EOS, we have chosen (for simplicity) the LKR1 hybrid EOS from~\\cite{J-MO1}---the maximum compactnesses for the two EOSs are $0.60$ (SLy) and $0.43$ (LKR$1$). (We show the much larger quadrupoles that could be\nsupported by the mixed phase in the core for the LKR$1$ EOS in Fig.~\\ref{Q22_vs_M_EOSs}, but here just show the crustal quadrupoles using the Douchin and Haensel model for the crust.)\n\n\\begin{figure}[htb]\n\\begin{center}\n\\epsfig{file=Q22s_vs_M_crustal.eps,width=8cm,clip=true}\n\\end{center}\n\\caption{\\label{Q22s_vs_M_SLy_DH_HJA} The Newtonian Cowling, Newtonian no\nCowling, and full relativistic (including stress contributions) values for the\nmaximum quadrupole deformations (and fiducial ellipticity) due to crustal\nstresses versus mass, for the SLy EOS with the detailed Douchin and Haensel + Ogata and\nIchimaru model for the shear modulus and a breaking strain of $0.1$, plus the crustal quadrupoles for the LKR$1$ EOS with the same crustal model.\n}\n\\end{figure}\n\nIn all of these crustal results, in addition to the expected relativistic suppression of the quadrupole (which\nbecomes quite dramatic for compact, high-mass stars), we also find that the\nNewtonian Cowling approximation slightly overestimates the quadrupole (by\n$\\sim 25$--$50\\%$), as observed by HJA (though they found the overestimate to be\nconsiderably greater, around a factor of at least a few). This overestimate is due to the cancellation of\ncontributions from $\\rho'$ when one drops\nthe Cowling approximation (see the discussion at the end of Sec.~\\ref{Newt}).\nThe overall decrease in the maximum crustal quadrupole with mass is due primarily to the fact that the crust thins by a factor of $\\sim 4$ (SLy) or $\\sim 2$ (LKR1) in going from a $1\\,M_{\\odot}$ star to\nthe maximum mass star, though the quadrupole itself receives even further suppressions with mass due to relativistic\neffects and an increased gravitational field.\n\n\\subsection{Maximum $Q_{22}$ for hybrid stars}\n\\label{results_hybrid}\n\n\\begin{figure}[htb]\n\\begin{center}\n\\epsfig{file=Q22s_vs_M_Hy1_sigma80_mu_corr2.eps,width=8cm,clip=true}\n\\end{center}\n\\caption{\\label{Q22s_vs_M_Hy1_sigma80} The Newtonian Cowling, Newtonian no\nCowling, and full relativistic (including stress contributions) values for the\nmaximum quadrupole deformations (and fiducial ellipticity) of hybrid stars versus\nmass, using the Hy1 EOS with a surface tension of\n$\\sigma = 80\\text{ MeV fm}^{-2}$ and a breaking strain of $0.1$.}\n\\end{figure}\n\nHere we display the maximum quadrupole deformations as a\nfunction of stellar mass for each of the hybrid EOS parameter sets considered\nin~\\cite{J-MO1}. (N.B.: Most of the results from~\\cite{J-MO1} we use or refer to here were\ncorrected in the erratum to that paper.) We start by showing these values calculated in the various\napproximations using the Hy1 EOS (with a surface tension of\n$\\sigma = 80\\text{ MeV fm}^{-2}$; see Table~I in~\\cite{J-MO1}) in\nFig.~\\ref{Q22s_vs_M_Hy1_sigma80}, and then\nrestrict our attention to the relativistic results. (The relation between the results of the different approximations is roughly\nthe same for all the hybrid EOSs we consider.)\nHere the maximum quadrupoles increase with mass, since the volume of mixed phase increases with mass, and this is more than enough to offset the suppressions due to relativity and the increased gravitational field.\n\n\\begin{figure}[htb]\n\\begin{center}\n\\epsfig{file=Q22s_vs_M_Hy1_sigmas_mu_corr2.eps,width=8cm,clip=true}\n\\end{center}\n\\caption{\\label{Q22_vs_M_Hy1_sigmas} The full relativistic maximum quadrupole deformations (and fiducial ellipticity) of hybrid\nstars versus mass, using the Hy1 EOS with various surface tensions $\\sigma$\nand a breaking strain of $0.1$.}\n\\end{figure}\n\nWe also show how the maximum relativistic quadrupole\nvaries with the surface tension for the Hy1 EOS in\nFig.~\\ref{Q22_vs_M_Hy1_sigmas}.  The slightly larger quadrupoles for lower surface tensions\nat low masses are expected, due to a slightly larger shear modulus at low\npressures for lower surface tensions---see Fig.~10 in~\\cite{J-MO1}. In fact, despite differences of\nclose to an order of magnitude in the high-pressure shear modulus for the Hy$1$ EOS\nin going from a surface tension of $20\\text{ MeV fm}^{-2}$ to one of\n$80\\text{ MeV fm}^{-2}$ (see Fig.~10 in~\\cite{J-MO1}), the\ndifferences in the resulting maximum quadrupoles are at most\na factor of a few (for large masses). This is not unexpected: These quantities\nare dominated by the portions of the mixed phase further out in the star, where\nthe shear moduli have a much weaker dependence on the surface tension.\n(Additionally, the fact that larger surface tensions lead to smaller shear moduli at low\npressures helps to minimize the effect, though the maximum\nquadrupoles still increase with increasing surface tension for high masses, as\nexpected.) \n\n\\begin{figure}[htb]\n\\begin{center}\n\\epsfig{file=Q22s_vs_M_EOSs_mu_corr2.eps,width=8cm,clip=true}\n\\end{center}\n\\caption{\\label{Q22_vs_M_EOSs} The full relativistic maximum quadrupole deformations (and fiducial ellipticity) of hybrid\nstars versus mass, using the EOSs from Table~I in~\\cite{J-MO1}, all with a surface tension of\n$\\sigma = 80\\text{ MeV fm}^{-2}$\nand a breaking strain of $0.1$.}\n\\end{figure}\n\nFinally, we show the maximum quadrupoles for different\nhybrid EOSs in Fig.~\\ref{Q22_vs_M_EOSs}. (Note that these curves start somewhat above the minimum\nmasses for which the mixed phase is present, since we are mostly interested in the significantly larger\nmaximum quadrupoles possible for larger masses.) The\nconsiderable differences are due primarily to the substantial variations in the extent\nof the mixed phase in stable stars with EOS parameters as well as the\nEOS dependence of the stars'\ncompactnesses (see Table~I in~\\cite{J-MO1}), not to variations in the magnitude of the shear modulus for a\ngiven quark matter fraction (compared in Fig.~12 in~\\cite{J-MO1}). In particular, the LKR1 EOS produces stars with a very large region of mixed\nphase---up to $72.5\\%$ of the star's radius---and a (relatively) small maximum\ncompactness---only $0.433$. (Note that our quadrupole curve for\nthe LKR1 EOS ends slightly short of the EOS's maximum mass of $1.955\\,M_{\\odot}$,\nonly going to $1.948\\,M_{\\odot}$, due to problems with the numerics.)\n\nN.B.:\\ These maximum quadrupoles may all be overly optimistic. First, as was discussed in Sec.~\\ref{results_crust}, the averaging used to obtain the effective shear modulus only gives an upper bound on the true shear modulus of a\npolycrystal. (We do not quote results for the Reuss lower\nbound here, since it is only straightforward to obtain for the three-dimensional droplet phases. However, \nwe shall note that preliminary investigations, using the Reuss bound for the droplet phases, and the\nVoigt bound for the rest, give reductions in the maximum quadrupoles of up to $\\sim 5$ for lower masses.)\n\nSecond, the relatively large value we have chosen for the surface tension also increases the maximum\nquadrupoles, while recent calculations place the surface tension on the low side ($\\sim 10$--$30\\text{ MeV fm}^{-2}$)---see~\\cite{PKR} for the latest results. Nevertheless, as we show in the Appendix, the mixed phase is nevertheless favored by global energy\narguments even for these large surface tensions. The maximum quadrupoles are also affected by the\nmethod of EOS interpolation and the lattice contributions to the EOS, as is illustrated in the Appendix, though\nthe largest change is only $\\sim40\\%$ (at least for the LKR$1$ and Hy$1'$ EOSs, the two EOSs that yield the largest quadrupoles).\n\nNote that LIGO's current upper limits on fiducial ellipticity \nin the most interesting cases (the Crab pulsar, PSR~J0537--6910, and\nCas~A)~\\cite{LIGO_psrs2010, LIGO_CasA} are $\\sim 10^{-4}$, corresponding to a\nquadrupole moment of $\\sim 10^{41} \\text{ g cm}^2$.\nThe first hybrid star estimate by \\citet{OwenPRL} was an order of magnitude\nlower.\nThus our new results here show that current LIGO upper limits are interesting\nnot only for quark stars but also for hybrid stars, at least high-mass ones.\nIndeed, the most extreme case we consider, the LKR$1$ EOS with high surface tensions, gives maximum\nquadrupoles of a $\\text{few} \\times 10^{42}\\text{ g cm}^2$, which are above and therefore relevant to the\nlimits set by Virgo for the Vela pulsar~\\cite{LIGO_Vela}. \n\n\\subsection{Maximum $Q_{22}$ for crystalline color superconducting quark stars}\n\\label{SQM_computation}\n\nHere we consider stars made of crystalline color superconducting quark matter, for which the\nshear modulus has been estimated by Mannarelli, Rajagopal, and Sharma~\\cite{MRS}.\\footnote{This\nestimate is not angle averaged, but Mannarelli, Rajagopal, and Sharma's calculation has relatively large uncontrolled remainders, so we do not worry about the effects of angle averaging here.}\n[See Eq.~(1) in Haskell~\\emph{et al.}~\\cite{Haskelletal} for the expression in cgs units.] Such\nstars have also been treated (with varying degrees of sophistication) by Haskell~\\emph{et al.}~\\cite{Haskelletal}, \\citet{Lin}, and~\\citet{KS}. However, only Lin considers the case of a solid quark star, as we will do here, and does so using quite a rough model. (The others consider crystalline color superconducting\ncores in hybrid stars.)\n\nSince strange quark stars have a nonzero surface density---and solid quark stars have a nonzero\nsurface shear modulus, with the standard density-independent treatment of the superconducting\ngap---we have to make some changes to our previously obtained expressions\nin order to treat them.\n\nFirst, the outer boundary condition changes. The potential (in the Newtonian case) and metric\nperturbation (in the GR case) are no longer continuous at the star's surface, due to the presence\nof $\\rho'$ in both equations [see Eqs.~\\eqref{Leq} and~\\eqref{H0_S-L}]. As discussed in Hinderer~\\emph{et al.}~\\cite{Hindereretal} (following Damour and Nagar~\\cite{DN}), one can obtain the distributional contribution to the boundary conditions [Eqs.~\\eqref{Newt_BC} and~\\eqref{H0_BC}] using the usual procedure of integrating the defining\ndifferential equation over $[R-\\epsilon,R+\\epsilon]$ and taking the limit $\\epsilon\\searrow 0$.\nIn the Newtonian case, this gives [defining $\\rho_-$ as the density immediately inside the star's\nsurface and $R^-$ to mean evaluation at $R-\\epsilon$ in the limit $\\epsilon\\searrow 0$]\n\\<\n\\delta\\Phi'(R^-) = \\left[\\frac{4\\pi G}{g(R)}\\rho_- - \\frac{3}{R}\\right]\\delta\\Phi(R),\n\\?\nand in the GR case, we have (with $G = 1$)\n\\<\nH_0'(R^-) = H_{0,\\mathrm{old}}'(R) + \\frac{4\\pi h}{\\phi'(R)}\\rho_-H_0(R),\n\\?\nwhere $H_{0,\\mathrm{old}}'(R) $ is computed using Eq.~\\eqref{H0_BC}.\nWe thus make the replacement $3RF(R) \\to [3 - 4\\pi G \\rho_- R\/g(R)]RF(R)$ in the expression \nfor the Newtonian Green function [Eq.~\\eqref{cG_F}], and the replacement\n$H_0'(R) \\to H_{0,\\mathrm{old}}'(R) + 4\\pi h\\rho_-H_0(R)\/\\phi'(R)$ in the GR case [Eq.~\\eqref{cG_GR}].\nThese changes in the boundary conditions increase the maximum quadrupole by a factor of $\\lesssim 2$ in the example case considered below; the largest effect is for the least massive stars considered.\n\nSecond, we would have to keep the boundary terms at the outer boundary when integrating by parts to\nobtain the expressions for the maximum quadrupole, since the shear modulus no longer vanishes at the\nstar's surface. However, since here the shear modulus is smooth, it is numerically preferable not to perform any integration by parts, thus avoiding potential problems with large cancellations between the surface and integrated terms. In this case, the expressions for the quadrupole assuming the UCB maximum\nuniform strain are [cf.\\ Eqs.~\\eqref{Q22N2} and \\eqref{Q22GR2}]\n\\<\n\\label{Q22UCBs}\n\\frac{|Q_{22}^{\\text{UCB strain}, N}|}{\\bar{\\sigma}_\\mathrm{max}} = \\sqrt{\\frac{32\\pi}{15}}\\int_0^RG_N(r)[r\\mu''(r) - \\mu'(r)]dr\n\\?\nand\n\\<\n\\begin{split}\n\\frac{|Q_{22}^\\text{UCB strain, GR}|}{\\bar{\\sigma}_\\mathrm{max}} &= \\sqrt{\\frac{32\\pi}{15}}\\int_0^R\\bigl[G_\\mathrm{GR}(r)\\mathcal{I}_{[\\delta\\rho,\\delta p]}^\\mathrm{UCB}(r)\\\\\n&\\quad + \\bar{G}_\\mathrm{GR}(r)\\mathcal{I}_{[t]}^\\mathrm{UCB}(r)\\bigr]dr,\n\\end{split}\n\\?\nwhere\n\\begin{widetext}\n\\begin{subequations}\n\\begin{align}\n\\mathcal{I}_{[\\delta\\rho,\\delta p]}^\\mathrm{UCB}(r) &:= \\left[\\frac{6}{r}(h^{1\/2} - 1) - 2\\phi' + \\frac{r\\phi''  + \\phi'(1 - r\\psi') - \\psi'}{h^{1\/2}}\\right]\\mu(r) + \\left[\\frac{2 + r(\\phi' - \\psi')}{h^{1\/2}} - 3\\right]\\mu'(r) + \\frac{r\\mu''(r)}{h^{1\/2}},\\\\\n\\mathcal{I}_{[t]}^\\mathrm{UCB}(r) &:= 2\\left\\{\\frac{r\\phi'[r(\\phi' + 2\\psi') - 5] - r\\psi' - r^2\\phi'' - 1}{h} + r\\phi' + h^{1\/2}\\right\\}\\mu(r) - 2r\\left(\\frac{r\\phi'}{h} + 1\\right)\\mu'(r).\n\\end{align}\n\\end{subequations}\n\\end{widetext}\n\nHowever, these expressions will not actually yield the maximum quadrupole in this case, due to an important\ndifference between the cases where the shear modulus vanishes at the star's surface and those where it does not. It is simplest to see this in the\nNewtonian case for a star with a constant shear modulus: Since the UCB maximum strain expression~\\eqref{Q22UCBs} only\ndepends upon derivatives of the shear modulus, it\npredicts a \\emph{zero} maximum quadrupole, which seems absurd. One can, however, make a small\nadjustment to the form of the maximum strain one considers to yield a nonzero quadrupole in this case. This modification will also yield considerably larger maxima in the realistic case we\nconsider, as well, where the shear modulus is close to constant---it decreases by less than a factor of $2$ in going from\nthe star's center to its surface in the example case we consider below.\n\nSpecifically, in the case of a slowly varying shear modulus, with $\\mu(r)\\gg|r\\mu'(r)|,|r^2\\mu''(r)|$, appropriate for strange quark stars, we want the terms involving $\\mu$ itself to be\nlargest. The appropriate choice for the strain in this case is most readily apparent\nfrom inspection of the Newtonian expression for the maximum quadrupole in terms of the stress tensor components, Eq.~\\eqref{Q22N1}. We want the maximum contribution from the undifferentiated terms, which implies that we want $t_{rr}$ and $-t_{r\\perp}$ to be as large as possible. For $t_\\Lambda$, we note that since $\\mu'(r)<0$,\nwe also want $-t_\\Lambda$ to be as large as possible. Realizing that we can freely change the sign of any of\nthe $\\sigma_\\bullet$ that give maximum uniform strain [given for the Newtonian case in Eqs.~\\eqref{sNewt}; cf.\\ Eq.~(65) in UCB], we thus reverse the sign of $\\sigma_{r\\perp}$ and $\\sigma_\\Lambda$. [The same logic holds for the more involved GR case, as well, where\nthe appropriate expression for $\\sigma_\\Lambda$ will be the negative of Eq.~\\eqref{sLGR}.]\n\nThe resulting expressions for the putative maximum quadrupole in this case are thus\n\\<\\begin{split}\n\\frac{|Q_{22}^{\\text{mod.\\ strain}, N}|}{\\bar{\\sigma}_\\mathrm{max}} &= \\sqrt{\\frac{32\\pi}{15}}\\int_0^RG_N(r)\\biggl[\\frac{12}{r}\\mu(r) + 5\\mu'(r)\\\\\n&\\quad +r\\mu''(r)\\biggr]dr\n\\end{split}\n\\?\nand\n\\<\n\\begin{split}\n\\frac{|Q_{22}^{\\text{mod.\\ strain, GR}}|}{\\bar{\\sigma}_\\mathrm{max}} &=  \\sqrt{\\frac{32\\pi}{15}}\\int_0^R\\bigl[G_\\mathrm{GR}(r)\\mathcal{I}_{[\\delta\\rho,\\delta p]}^\\mathrm{mod}(r)\\\\\n&\\quad + \\bar{G}_\\mathrm{GR}(r)\\mathcal{I}_{[t]}^\\mathrm{mod}(r)\\bigr]dr,\\end{split}\n\\?\nwhere\n\\begin{widetext}\n\\begin{subequations}\n\\begin{align}\n\\mathcal{I}_{[\\delta\\rho,\\delta p]}^\\mathrm{mod}(r) &:= \\left[\\frac{6}{r}(h^{1\/2} + 1) + 2\\phi' + \\frac{r\\phi'' + \\phi'(1 - r\\psi') - \\psi'}{h^{1\/2}}\\right]\\mu(r) + \\left[\\frac{2 + r(\\phi' - \\psi')}{h^{1\/2}} + 3\\right]\\mu'(r) + \\frac{r\\mu''(r)}{h^{1\/2}},\\\\\n\\mathcal{I}_{[t]}^\\mathrm{mod}(r) &:= - 2\\left\\{\\frac{ r\\phi'[5- r(\\phi'+2\\psi')] + r\\psi' + r^2\\phi'' + 1}{h} - r\\phi' + h^{1\/2}\\right\\}\\mu(r)\n- 2r\\left(\\frac{r\\phi'}{h} + 1\\right)\\mu'(r).\n\\end{align}\n\\end{subequations}\n\n\\end{widetext}\nIn principle, these merely give a lower bound on the\nmaximum quadrupole, unlike the case in which the shear modulus vanishes below the surface, where there is a firm argument that maximum uniform strain maximizes the quadrupole. However, even if they do not give the absolute\nmaximum, they should be quite close for cases like the one we consider here, where the shear modulus varies quite slowly.\n\n\\begin{figure}[htb]\n\\begin{center}\n\\epsfig{file=Q22s_vs_M_SQM1_final.eps,width=8cm,clip=true}\n\\end{center}\n\\caption{\\label{Q22s_vs_M_SQM1} The Newtonian Cowling, Newtonian no\nCowling, and full relativistic (including stress contributions) values for the\nquadrupole deformations (and fiducial ellipticity) of maximally strained strange quark stars versus\nmass, using the EOS discussed in the text with a breaking strain of $0.1$. We show these both for the\nstandard UCB uniform maximum strain, and our modification that yields significantly larger quadrupoles in this case.}\n\\end{figure}\n\nApplying these expressions to a specific case, we use the strange quark matter EOS calculated by Kurkela,\nRomatschke, and Vuorinen (KRV)~\\cite{KRV}, generating an EOS for the parameter values of\ninterest using the {\\sc{Mathematica}} notebooks available at~\\cite{KRV_URL}.\nThe relevant parameters are the values of the $\\overline{\\mathrm{MS}}$ renormalization point,\n$\\Lambda_{\\overline{\\mathrm{MS}}}$, and the strange quark mass, $m_s$, both at a scale of $2$~GeV, along with the coefficient in the relation between the renormalization scale and the\nquark chemical potential, $X$, the color superconductivity gap parameter, $\\Delta$ (taken to be\nindependent of density),\\footnote{Note that $\\Delta$ enters the KRV EOS through a color flavor locked (CFL)\npressure term. This is not quite appropriate for the crystalline color superconducting phase we consider here, since it assumes that all the quarks pair, while only some of them pair in the crystalline phase. However, as\ndiscussed in Sec.~VI~B of~\\cite{Alford2008}, the condensation energy of the crystalline phases is easily\n$1\/3$ to $1\/2$ that of the CFL phase with zero strange quark mass, which is the pressure\ncontribution used by KRV. We have thus not altered this term in our calculations, since the contribution is\nalready approximate, in that it assumes a density-independent gap parameter.\nMoreover, we only consider a fairly low value of $\\Delta$,\nwhile Knippel and Sedrakian~\\cite{KS} suggest that the crystalline phase might be favored up to $\\Delta = 100$~MeV. Our EOS may thus simply correspond to a slightly larger value of $\\Delta$, which would \nincrease the maximum quadrupole, since the shear modulus scales as $\\Delta^2$.} and the\nminimal quark chemical potential at which strange quark matter exists, $\\mu_{q,\\mathrm{min}}$.\nWe consider the EOS obtained by choosing $\\Lambda_{\\overline{\\mathrm{MS}}} = 355$~MeV, $m_s = 70$~MeV, $X=4$, $\\Delta = 10$~MeV, and $\\mu_{q,\\mathrm{min}} = 280$~MeV.\nThis parameter set yields a maximum mass of $2.45\\,M_{\\odot}$, with a maximum compactness of $0.467$. \n\nThese parameter choices were generally inspired by those considered at~\\cite{KRV_URL}, though with a smaller value of $\\Delta$, to place us well within the crystalline\nsuperconducting regime. However, as Knippel and Sedrakian~\\cite{KS} suggest, the crystalline phase\ncould still be favored for considerably larger $\\Delta$s, up to $\\sim 100$~MeV, for the low-temperature\ncase relevant for neutron stars. We thus note that increasing $\\Delta$ decreases the maximum mass, and increases the\nmaximum quadrupole, though the latter is increased by considerably less than the na{\\\"\\i}ve scaling of $\\Delta^2$ one would\nexpect from the scaling of the shear modulus, likely due to the increased compactness of the stars with\nlarger $\\Delta$s: For $\\Delta = 100$~MeV, we have a maximum mass and compactness of $2.12\\,M_{\\odot}$ and $0.508$, respectively, and a maximum\nquadrupole of $\\sim 3.5\\times 10^{45}\\text{ g cm}^2$ for a $1.4\\,M_{\\odot}$ star, $\\sim 20$ times that for $\\Delta = 10$~MeV. However, one must bear in\nmind that our perturbative treatment starts to become questionable with such large gap parameters, for which the maximum shear stresses are more than $10\\%$ of the background's energy density. The uncontrolled remainders in the Mannarelli,\nRajagopal, and Sharma~\\cite{MRS} calculation of the shear modulus also increase as the gap parameter\nincreases.\n\nWe show the quadrupole for a maximally uniformly strained star in the three approximations (Newtonian\nCowling, Newtonian no Cowling, and GR) for both the UCB and modified maximum strain choices for this EOS in Fig.~\\ref{Q22s_vs_M_SQM1}. Here we have used a breaking strain of $0.1$, by the same high\npressure argument as in the mixed phase case. (While the very outermost portions of the star are at low\npressure, the parts that are at a lower pressure than the crustal case for which the $0.1$ breaking strain\nwas calculated make negligible\ncontributions to the quadrupole.)\n\nN.B.:\\ To obtain the EOS used for this figure, we made some slight modifications to the KRV {{\\sc{EoScalc}}} {\\sc{Mathematica}} notebook so that it would output particle number densities on a denser mesh for low strange quark chemical potentials. This then gave an EOS table with better low-pressure coverage than their default settings produced. We still needed to perform\nan extrapolation of the EOS to zero pressure, where we found that a linear extrapolation of the energy\ndensity and quark chemical potential in terms of the pressure using the lowest two entries of the table provided a good fit.\n(More involved approaches involving fitting to more points and\/or a quadratic extrapolation produce very similar results.)\n\nAdditionally, it is worth pointing out that the applying the KRV results to compact stars\npushes their second-order perturbative calculation towards the edge of its domain of validity. However,\nin our case, the smallest value of the quantum chromodynamics (QCD) renormalization scale we consider is\n$1.12$~GeV, at which value the QCD coupling constant is $\\sim 0.45$. Thus, the uncontrolled remainders in\nthe expansion are suppressed by at least a factor of $\\sim 0.1$. (While Rajagopal and Shuster~\\cite{RS} find that perturbative QCD calculations of the color superconducting gap are only reliable at energy scales of\n$\\gtrsim 10^5$~GeV, the specifics of this calculation are rather different from the calculation of the EOS we\nare considering here, where the gap is taken as an input parameter.) While it is unreasonable to expect\nthis calculation to be a truly accurate description of strange quark matter, it is not clear that any of the alternative descriptions of strange quark matter are \\emph{a priori} guaranteed to be a better description of the physics, given the very considerable uncertainties associated\nwith this phase of matter.\n\n\\section{Discussion}\n\\label{discussion}\n\nPrevious studies of the tidal and magnetic deformations of compact\nstars have found similar relativistic suppressions of quadrupole moments with\ncompactness.\nIn the tidal case, see the Love number computations in~\\cite{Hinderer, BP, DN, Hindereretal, PPL}.\nIn the case of magnetic deformations, the expected suppressions are seen in,\ne.g., \\cite{IS, CFG, YKS, FR}.\nIn fact, since the largest compactness considered in these latter papers is only\n$0.48$ (in~\\cite{IS}), one imagines that they overestimate the maximum quadrupoles\nby at least a factor of a few for more compact stars (for a fixed magnitude of magnetic\nfield).\n\nAs was argued by Damour and Nagar~\\cite{DN} in the tidal case, all these suppressions are primarily related to the ``no-hair'' property of black holes: The largest relativistic suppression we find comes from the boundary conditions\n[through the $H_0(R)$ and $H_0'(R)$ in the Green function's denominator---see Eq.~\\eqref{cG_GR}], where\none matches on to the external vacuum spacetime. For instance, for the SLy EOS's maximum\ncompactness of $0.6$, $H_0(R)$ and $H_0'(R)$ are $\\sim 3.5$ and $\\sim 6$ times their\nNewtonian values\n[which can be obtained from the first term of Eq.~(21) in Hinderer~\\cite{Hinderer}~].\nIn fact, these ratios go to infinity in the formal black hole limit, where the compactness\napproaches unity, as required by the\nno-hair property, and discussed by Damour and Nagar~\\cite{DN} (see\ntheir Secs.~IV~C and VII~A, but note that their definition for the compactness is half of ours). This implies that the stiffness of spherically symmetric\ncurved vacuum spacetime suppresses the quadrupole. The quadrupole is also\nsuppressed by a larger effective gravitational acceleration (given by $\\phi'$), which appears\nin the denominator of $G_\\mathrm{GR}$, replacing the Newtonian $g(r)$ [cf.\\ Eqs.~\\eqref{GN} and~\\eqref{GGR}]. (But\nrecall that we always compute the background stellar structure relativistically, so this\nlarger acceleration \\emph{only} affects the perturbation equations, and not, e.g., the thickness of\nthe crust for a given mass and EOS, which is the same in both the Newtonian and\nrelativistic calculations of the quadrupole.) \n\nOur results imply that nearly all of the Newtonian computations of quadrupoles due to elastic\ndeformations of relativistic stars overestimate the quadrupole moment, often by at\nleast a factor of a few. The only exceptions we have found are\nfor low-to-mid mass strange quark stars and for elastic stresses in the cores of neutron stars around $0.5\\,M_{\\odot}$. In\nboth of these cases, the Newtonian Cowling\napproximation is a slight underestimate for contributions to the quadrupole, though the Newtonian no Cowling\nversion is still an overestimate. See Fig.~\\ref{Q22_integrands_SLy_2} for an illustration in the core case;\nbut note that neutron stars with such low masses are not known to exist in nature.\nThe overestimate from performing a Newtonian Cowling approximation calculation \ncan be $\\sim 6$ for massive stars whose quadrupole is\nbeing generated by an elastic deformation near the\ncrust-core interface, as considered by UCB and others. This is due in part to the\nsudden changes in density at that interface entering directly through $g''$, as\ndiscussed at the end of Sec.~\\ref{Newt}.\n\nHowever, the calculations by Horowitz~\\cite{Horowitz} for crustal deformations of very low mass stars only receive negligible\ncorrections (of $\\lesssim5\\%$), since he considers\ncompactnesses of $\\sim 0.01$. In fact, one makes even\nsmaller errors in using the Cowling approximation to treat these stars,\nsince the changes in density in the crust (times $4\\pi G r^2$) are much smaller than the star's gravitational field there.\n\nNo neutron stars with such low masses have ever been observed (nor is there\na compelling mechanism for forming them). Nevertheless, Horowitz remarks that\ngravitational wave detection of\ngravitational waves from elastically deformed neutron stars will,\n\\emph{ceteris paribus}, be biased towards low(er) mass neutron stars, if one\nconsiders deformations generated by crustal stresses. This is an important \npoint, particularly when considering the astronomical interpretation of\ndetections (or even upper limits), and the results we present here make the\nbias against high-mass stars even stronger. (This bias also applies to solid\nquark stars, though there it is rather weak. It does \\emph{not} apply to hybrid\nstars, however, where it is high-mass stars that can sustain the largest quadrupoles.)\n\nOf course, one must remember that all of these values are maxima, assuming a\nmaximally strained star, while there is no reason, \\emph{a priori}, for a given star to be\nmaximally strained.\nMoreover, as UCB and HJA note, these calculations assume\nthat all the strain goes into the $l = m = 2$ perturbation, though strain in\nother modes (e.g., the $l = 2$, $m = 0$ mode due to rotation) can push the\nlattice closer to its breaking strain while not\nincreasing the $l = m = 2$ quadrupole.\n\n\\section{Conclusions and outlook}\n\\label{concl2}\n\nWe have presented a method for calculating the maximum elastic quadrupole\ndeformation of a relativistic star with a known shear modulus and breaking strain.\nWe then applied this method to stars whose elastic deformations are\nsupported by a shear modulus either from the Coulomb lattice of nuclei in the crust, a hadron--quark mixed phase in the core, or crystalline superconducting strange quark matter throughout the star. (In the last case, we have made the requisite changes to the method so that it is valid when the star has a nonzero surface\ndensity and the shear modulus does not vanish at the star's surface.) In all but the strange quark case, we find that the\nrelativistic quadrupole is suppressed, compared with the standard,\nNewtonian Cowling approximation calculation of the quadrupole, at least for\nstars with masses of $\\gtrsim 1\\,M_{\\odot}$ (corresponding to the observed masses\nof neutron stars) and the EOSs we have investigated. These suppressions can be\nup to $\\sim 4$ in the hybrid case, and\n$\\sim 6$ in the crustal case. In the strange quark star case, the Newtonian Cowling approximation\ncalculation slightly underestimates the quadrupole (by tens of percent) for low-to-standard mass stars, but is still an overestimate of $\\sim 2$ at higher\nmasses.\n\nThese suppressions strengthen the Horowitz~\\cite{Horowitz} argument that\nsearches for gravitational waves from elastically deformed neutron stars\nsupported by crustal stresses are biased towards lower-mass stars. The same argument also\napplies to strange quark stars, though there the suppressions with increasing mass are less\nsevere (and the maximum quadrupoles are all considerably larger). However,\nthis argument does not apply to quadrupole deformations of hybrid stars, since\nthe increase in the size of the region of mixed phase with increasing mass\ndominates the various suppressions.\n\nOur results also imply that many of the\nprevious calculations of elastic quadrupoles (e.g.,~\\cite{Lin, Haskelletal,\nKS, UCB, HJA}) will need their results revised downwards. (While we find much larger maximum quadrupoles for solid strange quark stars than did Lin~\\cite{Lin}, this is only because we assume a breaking\nstrain $10$ times that assumed by Lin. If we take the same $10^{-2}$ breaking strain as does Lin, then we find a suppression of a factor of a few, though this is very likely within the uncertainties of Lin's calculation,\nwhich assumed a uniform density, incompressible star with a uniform shear modulus.)\n\nIt is instructive to compare our results with the numbers quoted in Pitkin's review~\\cite{Pitkin}. All of these were obtained by Pitkin using scalings given in the aforementioned papers,\nsometimes updating to the Horowitz and Kadau~\\cite{HK} breaking strain, and provide a good overview of the standard Newtonian predictions.\nNone of our detailed calculations for maximum crustal quadrupoles approach the\nhigh values Pitkin obtained using UCB's fitting formula (as corrected by\n\\citet{OwenPRL}). However, our very largest hybrid star quadrupoles are an order of magnitude\nabove Pitkin's quoted maximum, even if one only assumes a breaking strain of $10^{-2}$, as does Pitkin. Additionally, our estimates for maximum solid quark star quadrupoles ($\\sim 10^{44}\\text{ g cm}^2$ for $1.4\\,M_{\\odot}$ stars) are considerably larger than the ones quoted by Pitkin (based on a different shear modulus model), even if we reduce them by an order of magnitude due to scaling the breaking strain to Pitkin's $10^{-2}$. In fact, they\nare in the same range as those Pitkin quotes for a model for crystalline superconducting hybrid stars (with an optimistic gap parameter $5$ times the one we used for solid quark stars, leading to a shear modulus $\\sim 40$ times our shear modulus's maximum value).\n\nEven with the relativistic suppressions, we obtain maximum quadrupole deformations of\n$\\text{a few}\\times10^{42}\\text{ g cm}^2$ \nin the hybrid case for a very stiff hadronic EOS, and\n$\\text{a few}\\times10^{41}\\text{ g cm}^2$ for more realistic cases. In both\nsituations, the largest maximum quadrupoles are given by the most massive stars. These\nvalues are proportional to the breaking strain and assume that the Horowitz and Kadau~\\cite{HK}\nbreaking strain of about $0.1$ is applicable to the mixed phase. Such large\nquadrupole deformations were previously thought only to be possible for solid\nquark stars (see~\\cite{OwenPRL, Lin, Haskelletal, KS}), or from crustal deformations in the very\nlow-mass neutron stars considered by Horowitz~\\cite{Horowitz}.  These large\ndeformations (corresponding to fiducial ellipticities of\n$\\text{a few}\\times10^{-3}$ in the extreme case, and\n$\\sim 5\\times10^{-4}$ in a more realistic case)\nwould be able to be\ndetected by current LIGO searches for gravitational waves from certain known\nneutron stars~\\cite{LIGO_psrs2010, LIGO_CasA, LIGO_Vela}. (However, we must note that there is no reason to assume that\nsuch isolated stars are anywhere near maximally strained, even neglecting the\nuncertainties in the description of their interiors.)\n\nThe prospects for\ncrustal quadrupoles are now somewhat less optimistic, and definitely favor\nlower-mass stars. However, for a canonical $1.4\\,M_{\\odot}$ neutron star, we find that the maximum\nrelativistic crustal quadrupole is in the range $\\sim\\text{(1--6)}\\times 10^{39}\\text{ g cm}^2$ [corresponding to\nfiducial ellipticities of $\\sim\\text{(1--8)}\\times 10^{-6}$], depending on the model used for the crust and the\nhigh-density EOS. (Note that the fully consistent Douchin and Haensel model with its associated high-density EOS yields the lowest numbers. Additionally, there is the possibility of a further reduction of up to $\\sim 2$ due to the angle averaging procedure used to obtain the shear modulus.) On the high side, these numbers are consistent\nwith those given previously for breaking strains of $0.1$ by Horowitz~\\cite{HK, Horowitz},\\footnote{But recall\nthat the results from Horowitz~\\cite{Horowitz} were obtained using the SLy EOS and crustal composition results, so they\nare the same as our Newtonian Cowling approximation SLy predictions, given in Fig.~\\ref{Q22s_vs_M_SLy_DH_HJA}, except $\\sim7\\%$ lower, since Horowitz is using the Horowitz and Hughto~\\cite{HH} result for the shear modulus. In the fully relativistic case, one requires a thicker crust than provided by the pure SLy results to obtain values for the maximum quadrupole comparable to those given by Horowitz.} though they are a factor of $\\sim 5$ lower than the maximum Pitkin~\\cite{Pitkin} obtained using scalings of previous results and the maximum\nvalue given by HJA (scaled to this breaking strain). For stars around $2\\,M_{\\odot}$, the relativistic suppressions\nlead to maximum quadrupoles that are nearly an order of magnitude smaller than those for a $1.4\\,M_{\\odot}$ star in the compact SLy case: ${\\sim\\text{(1--5)}}\\times 10^{38}\\text{ g cm}^2$ [corresponding to\nfiducial ellipticities of $\\sim\\text{(1--6)}\\times 10^{-7}$]; and even in the much less compact LKR$1$ case, there is a suppression of $\\sim 5$. Previous Newtonian studies (see Fig.~3 in~\\cite{Horowitz}) had only\nfound suppressions of around a factor of $4$, due to the thinning of the crust and the increase\nin Newtonian gravity with increasing mass. It will be interesting to consider further models for the crustal\ncomposition and EOS in this case, particularly the large\nsuite of crustal models including the pasta phases recently calculated by Newton, Gearheart, and Li~\\cite{NGL}. (See~\\cite{GNHL} for order-of-magnitude estimates of the maximum quadrupole for these models,\nillustrating the sensitive dependence on the slope of the symmetry energy.)\n\nOne can also compare these maximum elastic quadrupoles with those generated by an internal magnetic field. Here the values depend,\nof course, upon the equation of state, compactness, and---perhaps most crucially---magnetic field topology, as well as the quantity one\nchooses to use to measure the magnitude of the magnetic field. But\nsticking to order-of-magnitude numbers, and considering a canonical $1.4\\,M_{\\odot}$ neutron star, Frieben and\nRezzolla~\\cite{FR} show that a toroidal internal\nfield of $\\sim10^{15}$~G would generate a quadrupole of $\\sim 10^{39}$--$10^{40}\\text{ g cm}^2$, comparable to the\nmaxima we find for crustal quadrupoles. Similarly, quadrupoles of $\\sim 10^{41}$--$10^{42}\\text{ g cm}^2$, around the maxima we find for hybrid\nstars, could come from magnetic fields of $\\sim 10^{16}$~G, while the maximum quadrupoles of $\\sim 10^{44}\\text{ g cm}^2$ we find for crystalline strange quark stars could also be\ngenerated by magnetic fields of $\\sim 10^{17}$~G, close to the maximum allowed field strength. (But note that these magnetic deformations are all computed for ordinary, purely hadronic neutron stars. Additionally,\nthe quoted maximum elastic quadrupoles in the hybrid case are attained only for more massive stars than the $1.4\\,M_{\\odot}$\nstars for which we are quoting the magnetic deformation results.)\nThe quoted values for magnetic quadrupoles come from the fits given in Sec.~7 of Frieben and Rezzolla~\\cite{FR}, except for the final ones, which are obtained from inspection of their Fig.~5 and Table~3. All these values agree in order of magnitude with the\npredictions for the twisted torus topology given by Ciolfi, Ferrari, and Gualtieri~\\cite{CFG}, and with many other studies for various topologies cited in Frieben and Rezzolla~\\cite{FR}. But note that very recent calculations by Ciolfi and Rezzolla~\\cite{CR} show that the magnetic field required to obtain a given quadrupole deformation with the twisted torus topology could be reduced by about an order of magnitude if the toroidal contribution dominates.\n\nOne would also like to make relativistic calculations of the maximum energy that could be\nstored in an elastic deformation.\nThis would be useful in properly computing the available\nenergy for magnetar flares, for instance.\n(Using Newtonian scalings,  \\citet{CO} estimated that the hybrid case was especially\ninteresting compared to existing LIGO upper limits for gravitational wave emission from such flares.)\nThe basic expressions (at least in\nthe perfect fluid case) appear to be readily available in the literature\n(see, e.g.,~\\cite{Schutz2,DI}; \\cite{ST, Finn} give related results including\nelasticity). However, one cannot apply these directly to the crustal and hybrid cases, even in the Newtonian\nlimit, due to the\ndistributional nature of the density and pressure perturbations. Specifically, the\nsudden change in shear modulus at the phase transitions gives delta functions in the\nderivatives of the density and pressure perturbations. Since the energy expressions involve\nsquares of these derivatives, one would have to invoke some sort of regularization procedure, or apply a different\nmethod. Developing appropriate expressions for this case\nwill be the subject of future work.\n\nReturning to the quadrupoles, one might also want to consider the shape of the deformed star, particularly\nin the relativistic case---the ellipticity is already only a rough indicator of the shape of the\ndeformation in the Newtonian case---as has now been done in~\\cite{NKJ-M_shape}.\nBut the effects of the star's magnetic field are surely the most interesting to consider, from\nits influence on the lattices that support elastic deformations, to the changes to the boundary conditions at the star's surface from an external magnetic field (particularly for magnetars), to the internal magnetic field's own contribution to the star's deformation.\nOne might also want to consider the lattice's full elastic modulus tensor in this case, instead of simply assuming a polycrystalline structure and angle averaging to obtain an effective isotropic shear modulus, as was\ndone here. (And even if one assumes a polycrystalline structure, one could use more involved, sharper\nbounds on the shear modulus than the ones considered here---see~\\cite{WDO} for a classic review of such bounds.)\n\n\\acknowledgments\n\nWe wish to thank S.~Bernuzzi, D.~I.~Jones, A.~Maas, R.~O'Shaughnessy, and the anonymous referee for helpful suggestions.\nThis work was supported \nby NSF grants PHY-0855589 and PHY-1206027, the Eberly research funds of Penn State,  and the DFG SFB\/Transregio 7.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nMany modern particle accelerators are tuned to achieve as small transverse beam emittance as possible. This is due to the fact that most users demand for the highest beam brightness possible. Beam brightness is important for many accelerator applications, such as ultra-fast electron diffraction \\cite{doi:10.1098\/rsta.2005.1735}, free electron lasers \\cite{RevModPhys.88.015006} and the operation of new compact accelerator concepts in general (e.g. \\cite{England:2014bf,KARTNER201624,Giorgianni:2016ds,Nghiem:437316}). A common definition of brightness is \\cite{doi:10.1063\/1.860076}\n\\begin{equation}\n\tB = \\frac{\\eta I}{\\pi ^2 \\varepsilon_x \\varepsilon _y},\n\\end{equation}\nwhere $\\eta$ is a form factor close to unity, $I$ is the beam peak current and $\\varepsilon_{x,y}$ the horizontal and vertical transverse emittance respectively. Hence, in order to maximize $B$, transverse emittance has to be minimal. \n\nThere are multiple methods to characterize the transverse emittance. One of the most common techniques is the phase advance scan technique, where the transverse beam size is recorded on a screen vs.\\ the focusing strength of an upstream quadrupole or solenoid magnet \\cite{ReesRivkin:slac1984,Minty:2003,HACHMANN2016318}. The data can then be fitted based on the beam envelope equation. Space charge effects can be included to some extent \\cite{Hachmann:138874,PhysRevAccelBeams.20.013401}. Instead of scanning the focusing strength of a magnet, also multiple screens can be used to record the beam size vs.\\ the phase advance. Other -- potentially single-shot -- methods involve insertion of masks into the beamline, which then, subsequently, can be imaged on a downstream screen \\cite{Zhang:fermilab1996}. Coupled with advanced reconstruction algorithms these methods are capable of delivering reconstruction of the core 4D phase space \\cite{PhysRevAccelBeams.21.102802}. In this work we concentrate on the phase advance scan technique, as this is the easiest one to implement, only requiring standard beamline components.\n\nOne of the limitations of the phase advance scan technique is that there is no closed description of the beam envelope for space charge dominated beams \\cite{Hachmann:138874,PhysRevAccelBeams.20.013401}. It is therefore difficult to apply the method in this regime. Space charge dominated beams especially occur, for example, in the injector part of high-brightness electron sources, where the beam is still non-relativistic. In order to quantify whether a beam is space charge dominated, the so-called \\emph{laminarity parameter} $\\rho$ can be calculated \\cite{Ferrario:1982426}. This parameter represents the ratio between the space charge term and the emittance term of the beam envelope equation. It is given by\n\\begin{equation}\n \t\\rho = \\frac{I\\sigma^2}{2I_\\text{A}\\gamma \\varepsilon _n^2},\n \t\\label{eq:laminarity}\n \\end{equation}\nwhere $I$ is the peak current of the beam, $I_\\text{A} \\approx \\SI{17}{\\kilo\\ampere}$ is the Alfv\u00e9n current and $\\varepsilon _n = \\beta \\gamma \\varepsilon$ is the normalized emittance with the Lorentz factor $\\gamma$ and $\\beta = v\/c$. In case $\\rho \\gg 1$, the beam can be considered as space charge dominated (laminar flow regime). Otherwise the evolution of the beam envelope is dominated by the emittance pressure (thermal regime).\n\nIn this work we show in simulation that it is possible to successfully analyse phase advance scan data for $\\rho \\gg 1$ beams using a pre-trained fully connected neural network (FCNN). Subsequently, we apply the method to measured data. Machine learning and neural networks in particular have recently been used in the context of accelerators for various purposes. These include, among others, fault detection of machine components \\cite{Solopova:IPAC2019-TUXXPLM2}, machine stability optimization and analysis \\cite{PhysRevX.10.031039,PhysRevLett.121.044801,PhysRevAccelBeams.23.074601}, virtual diagnostics \\cite{Sanchez-Gonzalez2017,PhysRevAccelBeams.21.112802,Ratner:21}, beam quality optimization in plasma accelerators \\cite{PhysRevLett.126.174801,PhysRevLett.126.104801} and orders of magnitude speed-up in multiobjective optimization of accelerator parameters \\cite{PhysRevAccelBeams.23.044601}. Here, we focus on the analysis of otherwise difficult to interpret measurement data.\n\\section{Measurement Technique} \n\\label{sec:MeasurementTechnique}\nThe state of a single particle with respect to a given design, or reference trajectory, is usually defined by the 6D phase space vector\n\\begin{equation}\n\t\\mathbf{X} = (x, x^\\prime, y, y^\\prime, z, \\delta)^\\text{T},\n\\end{equation}\nwhere $x^\\prime = p_x\/p_z$ and $y^\\prime=p_y\/p_z$ are the horizontal and vertical divergence respectively, $x,y,z$ the distances of the particle from the reference trajectory and $\\delta = \\Delta p \/ p_0$ is the relative deviation of the particle's individual momentum from the reference momentum. $p_x, p_y, p_z$ are the three momentum components and $()^\\text{T}$ denotes the transpose of a matrix. We are interested in the evolution of the 4D transverse phase space vector\n\\begin{equation}\n\t \\mathbf{X^\\text{(tr)}} = (x, x^\\prime, y, y^\\prime)^\\text{T}.\n\\end{equation}\nAssuming negligible correlation between the evolution of the phase space coordinates in the $x$ and $y$ planes, it is possible to treat them separately, yielding the two sub-space vectors\n\\begin{equation}\n\t\\mathbf{x} = (x, x^\\prime)^\\text{T}, \\hspace{0.5cm}\\mathbf{y} = (y, y^\\prime)^\\text{T}.\n\\end{equation}\nA common framework to describe the evolution of a charged particle is linear beam optics, where only linear transformations of $\\mathbf{x}$ and $\\mathbf{y}$ are taken into account. Each beamline element is represented by a so-called transfer matrix defined by the relation\n\\begin{equation} \\label{eq:matrixtransformation}\n\t\\mathbf{x} = \\mathbf{M}\\cdot\\mathbf{x_0},\n\\end{equation}\nwhere $\\mathbf{x_0} = (x_0, x^\\prime_0)^\\text{T}$ is the initial phase space coordinate and\n\\begin{equation}\n\t\\mathbf{M} = \n\t\\begin{pmatrix}\n\t\tM_{11} & M_{12}\\\\\n\t\tM_{21} & M_{22}\n\t\\end{pmatrix}.\n\\end{equation}\nThe $2\\times2$ matrix for a simple drift is given by\n\\begin{equation}\n\t\\mathbf{M}_\\text{D}(s) = \n\t\\begin{pmatrix}\n\t\t1 & s\\\\\n\t\t0 & 1\n\t\\end{pmatrix},\n\\end{equation}\nwhere $s$ is the drift distance. Applying this matrix to $\\mathbf{x_0}$ would result in $x = x_0 + x^\\prime s, x^\\prime = x^\\prime_0$, as expected.\n\nIn an experiment only the rms beam size $\\sigma_x = \\sqrt{\\langle x^2 \\rangle}$ is accessible, where $\\langle \\rangle$ denotes the second central moment. Applying this to the general equation\n\\begin{equation}\n\tx = M_{11}x_0 + M_{12}x_0^\\prime\n\\end{equation}\nyields\n\\begin{equation}\n\t\\begin{split}\n\t\\sigma_x^2 = &M_{11}^2 \\sigma_{x,0}^2 \\\\\n\t&+ 2M_{11}M_{12}\\sigma_{x,0}(\\sigma_{x,0})^\\prime \\\\\n\t&+ M_{12}^2 \\left( \\frac{\\varepsilon _x^2}{\\sigma_{x,0}^2} + (\\sigma_{x,0})^{\\prime 2} \\right)\n\t\\end{split}\n\\label{eq:envelope}\n\\end{equation}\nwith $\\varepsilon_x = \\sqrt{\\langle x^2 \\rangle \\langle x^{\\prime 2} \\rangle - \\langle xx^\\prime \\rangle^2}$. Equation~\\ref{eq:envelope} is the so-called rms envelope equation, which can be used to determine the transverse rms emittance $\\varepsilon _x$ using a suitable (i.e. tunable) beam transformation $\\mathbf{M}$. Note that Eq.~\\ref{eq:envelope} does not take any space charge effects into account and is hence only valid in the $\\rho \\approx 1$ regime.\n\nFigure~\\ref{fig:ARES_sketch} shows a sketch of a potential measurement scenario. The elements and distances are chosen according to what is installed at the ARES electron linac at DESY, Hamburg \\cite{instruments5030028}. The transfer matrix of the double solenoid magnet can be written as\n\\begin{equation}\n\t\\begin{split}\n\t\t\\mathbf{M}_\\text{DS} &= \\mathbf{M}_\\text{TL} \\cdot \\mathbf{M}_\\text{D}(l_\\text{D}) \\cdot \\mathbf{M}_\\text{TL} \\\\\n\t\t&=\n\t\t\\begin{pmatrix}\n\t\t\t1 & 0 \\\\\n\t\t\t-\\frac{1}{f} & 1\n\t\t\\end{pmatrix} \n\t\t\\cdot\n\t\t\\begin{pmatrix}\n\t\t\t1 & l_\\text{D}\\\\\n\t\t\t0 & 1\n\t\t\\end{pmatrix}\n\t\t\\cdot\n\t\t \\begin{pmatrix}\n\t\t\t1 & 0 \\\\\n\t\t\t-\\frac{1}{f} & 1\n\t\t\\end{pmatrix}\\\\\n\t\t&=\n\t\t\\begin{pmatrix}\n\t\t\t1-l_\\text{D}\/f & l_\\text{D} \\\\\n\t\t\t(l_\\text{D} - 2f)\/f^2 & 1-l_\\text{D}\/f\n\t\t\\end{pmatrix}, \n\t\\end{split}\n\t\\label{eq:double_solenoid_matrix}\n\\end{equation}\nwhere $l_\\text{D}$ is the drift distance between the two single solenoids and $f$ the focal length of each solenoid. Here the approximation that $f$ is larger than the length of the solenoid was used, i.e. the thin lens approximation. The focal length of a solenoid is given by \\cite{doi:10.1119\/1.3129242}\n\\begin{equation}\n\tf(B_{z,\\text{max}}) = \\left[ \\left( \\frac{q}{2  \\overline p_z } \\right)^2 F_2\\right]^{-1}, \t\n\\end{equation}\nwhere $B_{z,\\text{max}}$ is the peak magnetic field, $q$ the particle charge and $\\overline p_z$ the average longitudinal beam momentum. $F_2 = \\int B_z^2 \\text{d}z \\propto B_{z,\\text{max}}^2$ is the second field integral of the on-axis magnetic field. By inserting the expression $\\mathbf{M}_\\text{D}(l_\\text{S}) \\cdot \\mathbf{M}_\\text{DS}$, where $l_\\text{S}$ is the drift between the solenoid and the screen, in Eq.~\\ref{eq:envelope}, it can be seen that now the $M_{ij}$ elements can be conveniently adjusted in the experiment as $B_{z,\\text{max}}$ is varied. The emittance at the position of the solenoid can thus be determined by fitting the recorded $\\sigma_{x,i}$ vs.\\ $B_{z,\\text{max},i}$ at the screen with Eq.~\\ref{eq:envelope}.\n\\begin{figure}[htbp]\n  \\centering\n  \\includegraphics[width=0.9\\columnwidth]{FIG_1.pdf}\n  \\caption{Schematic of a suitable beamline layout for transverse emittance measurements using the phase advance scan technique. The distances are based on what is installed at the ARES linac at DESY, Hamburg.}\n  \\label{fig:ARES_sketch}\n\\end{figure}\n\nIt is possible to include transverse space charge forces into the model to some extent. This is done by including a defocusing term in the drift between the focusing element and the screen. Considering a uniformly charged cylindrical bunch with radius $R$ and length $L$, the envelope equation then reads in differential form \\cite{PhysRevAccelBeams.20.013401}\n\\begin{equation}\n\t\\sigma _x ^{\\prime \\prime} - \\frac{P}{4\\sigma_x} G(\\xi, A) - \\frac{\\varepsilon _x ^2}{\\sigma_x ^3} = 0,\n\t\\label{eq:envelope_diff_eq_spch}\n\\end{equation}\nwhere $G(\\xi, A) \\in [0,1]$ is a form factor, which depends on the centered longitudinal intra-bunch coordinate $\\xi$ and the rest frame aspect ratio $A=R\/(\\gamma L)$. $P$ is the so-called generalized perveance given by\n\\begin{equation}\n\tP = \\frac{eQ}{2 \\pi \\epsilon_0 m_e c^2 L \\beta ^2\\gamma^3},\n\\end{equation}\nwhere $Q$ is the total charge of the bunch, $\\epsilon _0$ the vacuum permittivity and $m_e$ the electron rest mass. In principle Eq.~\\ref{eq:envelope_diff_eq_spch} can now be used to construct a similar fit model to Eq.~\\ref{eq:envelope}. There are a number of caveats to take into account, however:\n\\begin{itemize}\n\t\\item The model is only fully valid for a perfectly cylindrical bunch.\n\t\\item The form factor $G$ depends on the aspect ratio, which depends on the transverse beam size and bunch length, both not being constant in the experiment (especially in the over-focused part of the scan).\n\t\\item Since the concept of the envelope equation relies on the emittance being a constant of motion, non-linear space charge forces are intrinsically neglected in this approach.\n\t\\item The perveance term depends on the bunch length, which might not be accessible to sufficient precision in the experiment.\n\t\\item Equation~\\ref{eq:envelope_diff_eq_spch} cannot be solved analytically and has to be approximated by a polynomial series (see \\cite{Hachmann:138874} for a detailed description).\n\\end{itemize}\nAll of these caveats lead to the conclusion, that the envelope equation based data analysis method is not ideal in the $\\rho \\gg 1$ regime. \n\nIndependent of the value of $\\rho$, two more criteria need to be met in order to ensure an accurate fit result. Considering the third term of Eq.~\\ref{eq:envelope}, the purely mathematical criterion\n\\begin{equation}\n\t\\frac{\\varepsilon _x ^2}{\\sigma _{x,0}^2 \\cdot (\\sigma _{x,0})^{\\prime 2}} \\geq 0.01\n\t\\label{eq:fit_criterion_math}\n\\end{equation}\ncan be derived \\cite{Hachmann:138874}. This criterion ensures the numerical significance of $\\varepsilon _x$. The second criterion is based on the fact that the scan needs to include a minimum. The initial beam optics needs to be setup, such that a potential focus of the beam lies behind the screen used to measure the beamsize. At the same time, the focusing element needs to be strong enough to focus the beam onto the screen, which implies a constraint on the distance between focusing element and screen. Using Eq.~\\ref{eq:envelope}, these considerations can be summarized by the criterion\n\\begin{equation}\n\t-\\frac{\\sigma_{x,0} \\cdot (\\sigma_\\text{eff})^\\prime}{f_\\text{max}^2 \\cdot \\left((\\sigma_\\text{eff})^{\\prime 2} + \\frac{\\varepsilon_x ^2}{\\sigma_{x,0}^2}\\right)} \\leq l_\\text{S} \\leq \\frac{\\sigma _{x,0}^2}{2 \\varepsilon_x},\n\\end{equation}\nwhere $(\\sigma_\\text{eff})^{\\prime} = - (\\sigma _{x,0}\/f_\\text{max} - (\\sigma _{x,0})^\\prime)$. In case both of the two criteria are fulfilled, the emittance can be retrieved.\n\nBased on the aforementioned considerations, we propose using an alternative way to analyze phase advance scan data. Specifically, we propose using a pre-trained FCNN to overcome the problem of the incomplete fit model in $\\rho \\gg 1$ cases, as well as the criterion described by Eq.~\\ref{eq:fit_criterion_math}. To this end, we have performed a simulation study, which is presented in detail in the following sections. The resulting FCNN was then subsequently applied to real world data, as shown below.\n\\section{Simulation Study - Methodology}\n\\label{sec:Methodology}\nIt has been shown already in 1989 that neural networks with only one unbounded hidden layer can approximate any Borel measureable function from finite dimensional space to another to arbitrary precision \\cite{Cybenko1989,HORNIK1989359}. More recent research focuses on the expressiveness (approximation accuracy) of both depth (i.e. the number of hidden layers) and width (i.e. the number of artificial neurons in a layer) bounded networks. In \\cite{NIPS2017_32cbf687}, for example, the authors show that any Lebesgue integrable function $f: \\mathbb{R}^n \\rightarrow \\mathbb{R}$ on $n$-dimensional space can be approximated to arbitrary accuracy by a fully connected width-$(d_\\text{in}+4)$ ReLU network with respect to the $\\ell_1$ norm as a measure of approximation quality. In other words, the network represented by the transfer function $F$ satisfies $\\int _{\\mathbb{R}^n} |f(x) - F(x) | \\text{d}x < \\epsilon, \\forall \\epsilon > 0$ (see \\cite{NIPS2017_32cbf687}, Theorem 1). ReLU here refers to the the so-called Rectified Linear Unit neuron activation function, definded by $\\text{ReLU}(x) = \\text{max(0,x)}$ and $d_\\text{in}$ is the input dimensionality. This width boundary $w$ has since been refined and generalized for example in \\cite{hanin2018approximating,Hanin_2019} to be $d_\\text{in} + 1 \\leq w_\\text{min}(d_\\text{in},d_\\text{out}) \\leq d_\\text{in} + d_\\text{out}$, for functions of the form $f:[0,1]^{d_\\text{in}} \\rightarrow \\mathbb{R}^{d_\\text{out}}$, where $d_\\text{in}$ and $d_\\text{out}$ are the input and output dimensionality respectively. Limits on the depth can be estimated in specific cases in terms of the so-called modulus of continuity of $f$, given by $\\omega_f(\\varepsilon) = \\text{sup}\\{|f(x)-f(y)|||x-y|\\leq \\varepsilon\\}$, where $\\varepsilon$ is an arbitrarily small change in the argument of $f$. For continuous functions $f:[0,1]^{d_\\text{in}} \\rightarrow \\mathbb{R}_+$ the depth of a $d_\\text{in}+2$ wide network $\\mathcal N$ can, for example, be expressed as $\\text{depth}(\\mathcal N _\\varepsilon) = 2\\cdot d_\\text{in}! \/ \\omega_f(\\varepsilon)^{d_\\text{in}}$, cf. \\cite{hanin2018approximating}. We note that in practice the specific layout of a neural network is often determined experimentally, as the aforementioned boundaries are merely based on proofs of existence.\n\nIn this study, we aim to map the phase advance scan data to the normalized transverse emittance at the focusing element. Mathematically, this means we assume a connection of the scan data to the physical quantity of the form $f:\\mathbb{R}_+^{d_\\text{in}} \\rightarrow \\mathbb{R}_+$, where the dimensionality $d_\\text{in}$ is given by the number of scan data points. Note that, based on the knowledge of the problem, we can always map (normalize) the input data from $\\mathbb{R}_+^{d_\\text{in}}$ into $[0,1]^{d_\\text{in}}$. The function $f$ operates on the measure space $(\\mathbb{R}_+^{d_\\text{in}}, \\mathcal B, \\lambda)$ with the Borel-$\\sigma$-algebra $\\mathcal B$ and the Lebesgue measure $\\lambda$. It is hence measureable in the mathematical sense. In addition, we expect $f$ to be a continuous function based on the physical background of the problem. We can hence conclude that $f$ is Lebesgue integrable and suitable to be approximated for example by a width bounded ReLU network. \n\nTo validate this approach, we setup a simulation study based on the simple beamline layout shown in Fig.~\\ref{fig:ARES_sketch}. The main simulation study is split into three parts:\n\\begin{enumerate}\n\t\\item Building a large number of data sets (training, validation and test),\n\t\\item Training the FCNN and evaluation of the performance using the test data sets,\n\t\\item Comparison of the FCNN performance to the traditional fit method, as discussed above.\n\\end{enumerate}\nThe first step of the simulation study is to build a large number of data sets. Creating a data set consists of two steps:\n\\begin{itemize}\n\t\\item Numerical tracking of the beam from the cathode to the location of the solenoid,\n\t\\item Numerical simulation of the solenoid scan.\n\\end{itemize}\nFirst, the emittance at the solenoid position is determined by numerical tracking of the particles. In this step the solenoid field is set to zero. In addition to the emittance, other beam parameters, such as the beam size, divergence, or bunch length can be recorded as well. Then, the simulation domain is extended up to the position of the screen, which is used in the experiment to record the beam size vs.\\ the solenoid focusing strength. The experiment is then simulated for $M$ focusing strength settings. It is important to setup the scan range such that the resulting data includes the beam size minimum, i.e. the focus, as it carries most of the information about the emittance at the solenoid position \\cite{Hachmann:138874}. We use the well established code \\textsc{ASTRA} \\cite{ASTRAASpaceChar:LTSRiAsm}, which takes space charge effects into account. The beam size vs.\\ focusing strength scan data functions as the data set to be interpreted by the FCNN. Over the course of the study, a specific way to prepare the input data turned out to yield the best results. For each scan, $M\/2$ scan points centered around the minimum beam size are interleaved with the relative focusing strength difference $\\delta B_i = (B_i - B_\\text{foc})\/B_\\text{foc}$, where $B_\\text{foc}$ is the setting corresponding to the minimal beam size. The data set $S_\\text{in}$ is then of the form\n\\begin{equation}\n\tS_\\text{in} = [\\delta B_1, \\sigma_1, \\delta B_2, \\sigma_2, ..., \\delta B_{(M\/2)}, \\sigma_{(M\/2)}],\n\t\\label{eq:data_set_form}\n\\end{equation}\nwhere $\\sigma_i$ is the $i$th rms beam size. Each of these data sets is labeled with a set of important beam and simulation input parameters. These labels are then used to perform so-called supervised training of the FCNN. After the learning process, the FCNN is able to predict each of these parameters from given scan data, which is prepared according to Eq.~\\ref{eq:data_set_form}. Figure.~\\ref{fig:Methodology_1} summarizes how the data sets are created and what types of data they contain. \n\\begin{figure}[htbp]\n  \\centering\n  \\includegraphics[width=0.72\\columnwidth]{FIG_2.pdf}\n  \\caption{Diagram showing how the FCNN training, validation and test data is produced. $N$ is the total number of data sets. $M$ is the number of ASTRA simulations. $M_\\text{in}$ is the final number of data points to be fed to the FCNN. $M_\\text{out}$ is the length of the output vector, i.e. the number of predicted beam and simulation parameters. The data sets are split into three categories, where in this study the common split of $N_\\text{tra} = 0.6\\cdot N$, $N_\\text{val} = 0.2\\cdot N$ and $N_\\text{tes} = 0.2\\cdot N$ is used.}\n  \\label{fig:Methodology_1}\n\\end{figure}\n\nFor this particular study $N = 16066$ random data sets with $M=40$ were produced. Each data set differs in the three key \\textsc{ASTRA} input parameters \\emph{total charge}, \\emph{laser spot size} and \\emph{cathode emission time}. Table~\\ref{tab:data_set_ranges} summarizes the parameter ranges used for this study, which are losely based on typical settings at the ARES linac at the time. The parameters are varied according to a uniform distribution. The ARES S-band gun was simulated with an, at the time available, peak gradient of \\SI{65}{\\mega \\volt\/\\meter}, resulting in a final $\\gamma = 6.8$. Based on the parameter ranges shown in Table~\\ref{tab:data_set_ranges}, a convergence study in terms of required macro particles in the numerical simulation was performed. For the highest possible charge density, 10000 particles were found to be sufficient.\n\n\\begin{table}[htbp]\n   \\centering\n   \\caption{\\textsc{ASTRA} input parameter ranges for the data set production.}\n   \\begin{ruledtabular}\n   \\begin{tabular}{lc}\n   Parameter & Value \\\\\n   \\hline\n   Bunch charge & $[0.01, 2.1]\\,\\SI{}{\\pico\\coulomb}$\\\\\n   Laser spot size (flat top diameter) & $[240, 400]\\,\\SI{}{\\micro\\meter}$\\\\\n   Cathode emission time (rms) & $[60, 100]\\,\\SI{}{\\femto\\second}$\\\\\n   \\end{tabular}\n   \\end{ruledtabular}\n   \\label{tab:data_set_ranges}\n\\end{table}\n\nThe neural network was implemented using the TensorFlow framework \\cite{tensorflow2015-whitepaper}. The input layer has $M$ neurons, corresponding to the length of $S_\\text{in}$. Then one hidden layer with $M$ and two hidden layers with $M\/2$ neurons are added in order to capture non-linearities in the system. The system is then coupled to the output layer of size $M_\\text{out}$, which corresponds to the number of \\textsc{ASTRA} input and simulated beam parameters to be predicted by the network. The overall layout is hence\n\\begin{equation}\n\tM \\rightarrow [M]-[M\/2]-[M\/2] \\rightarrow M_\\text{out}.\n\\end{equation}\nThis particular layout was determined empirically. Each neuron is coupled to every neuron of the following layer, or in other words the layers are fully connected. The neurons are activated using the well established rectified linear activation function (ReLU) \\cite{pmlr-v15-glorot11a,ramachandran2017searching}. Training of the network is performed using a combination of the \\textsc{adam} and \\textsc{adagrad} gradient decent algorithm \\cite{ruder2017overview} with the mean squared error (\\textsc{MSE}) as the loss function. The available $N$ data sets are split into three categories. $N_\\text{tra}$ training sets, $N_\\text{val}$ validation sets and $N_\\text{tes}$ test sets. The training sets are used to adjust the neuron weights during the training procedure, while the performance of the network is judged after each so-called epoch based on the validation sets, which are not used during training. This is done to avoid overfitting the training data. An epoch refers to one forward and backward pass of the entire training data. Finally, the performance of the resulting model is determined using the test sets, which have not been part of the learning procedure at all. We use the common split of $N_\\text{tra} = 0.6\\cdot N$, $N_\\text{val} = 0.2\\cdot N$ and $N_\\text{tes} = 0.2\\cdot N$. The network was trained for $\\sim 10000$ epochs using \\textsc{adam} and another $\\sim 10000$ epochs using \\textsc{adagrad} \\footnote{In our study, the training procedure usually took $<\\SI{1}{\\hour}$ on an Apple M1 processor. Production of the training data sets, however, can take several days, depending on the available compute infrastructure. In order to speed-up data set production, we used the parallelized version of ASTRA}.\n\\section{Simulation Study - Data Set}\nBefore evaluating the prediction performance of the neural network, it is useful to inspect the training data set. Since we are especially interested in analyzing phase advance scan data for space charge dominated beams, the laminarity parameter $\\rho$ at the solenoid position was calculated for each data set (cf. Eq.~\\ref{eq:laminarity}). Figure~\\ref{fig:space_charge_dominance_train_data_sets} shows $\\rho$ vs.\\ the bunch charge. The color scale indicates the laser spot size on the cathode used in the particular simulation. In addition, the distribution of $\\rho$ across the whole data set is shown.\n\\begin{figure}[b]\n  \\centering\n  \\includegraphics[width=\\columnwidth]{FIG_3.pdf}\n  \\caption{Top: Scatter plot of the laminarity parameter $\\rho$ at the solenoid according to Eq.~\\ref{eq:laminarity} vs.\\ the bunch charge for all training data sets. $\\gamma = 6.8$ for this population. The color indicates the laser spot size on the cathode for each data set. Bottom: Distribution of the laminarity parameter.}\n  \\label{fig:space_charge_dominance_train_data_sets}\n\\end{figure}\nIt can be seen that all of the data sets lie in the $\\rho \\gg 1$, i.e. space charge dominated, regime ($\\rho_\\text{min} = 17.8$). Also, the higher the charge, the higher the value for $\\rho$, as expected. In addition, the color scale reveals that the smaller the laser spot size on the cathode, the higher the value for $\\rho$. The sensitivity of $\\rho$ on the laser spot size strongly depends on the bunch charge.\n\nAs noted above, the traditional fit method only works if Eq.~\\ref{eq:fit_criterion_math} is satisfied. Figure~\\ref{fig:feasibility_train_data_sets} shows the fit feasibility criterion for each data set, with the same color code as in Fig.~\\ref{fig:space_charge_dominance_train_data_sets}. None of the data sets satisfies the criterion, which leads to the expectation that the traditional fit method should not work well on the training data (and with that in reality for the ARES working point, which is the basis for the parameter space shown in Table~\\ref{tab:data_set_ranges}).\n\\begin{figure}[htbp]\n  \\centering\n  \\includegraphics[width=\\columnwidth]{FIG_4.pdf}\n  \\caption{Top: Scatter plot of the fit feasibility parameter according to Eq.~\\ref{eq:fit_criterion_math} vs.\\ the bunch charge for all training data sets. $\\gamma = 6.8$ for this population. The color indicates the laser spot size on the cathode for each data set. Bottom: Distribution of the fit feasibility parameter. \\emph{Note that the plot shows the values multiplied by 1000 for readability.}}\n  \\label{fig:feasibility_train_data_sets}\n\\end{figure}\n\\section{Results and Comparison} \n\\label{sec:ResultsAndComparison}\nIn this section the performance of the pre-trained FCNN is presented. We also compare its performance against the traditional fit method discussed above. In order to evaluate the performance of the FCNN, we try to predict the labels of the $N_\\text{tes}$ test data sets, which were not used in the supervised training procedure. The main goal of the study is to predict the transverse emittance, but since the labels include a number of other simulation and beam parameters, the FCNN also provides predictions of these. In order to better quantify the prediction performance for the different label components, the relative error between predicition and truth was calculated for each data set. This is shown in Fig.~\\ref{fig:error_dists_perf}. In addition to the error distributions, a radar plot visualizes the prediction performance in terms of number of data sets in \\SI{1}{\\percent}, \\SI{5}{\\percent} and \\SI{10}{\\percent} relative error intervals respectively (see Table~\\ref{tab:pred_perf_table} for the actual percentages). In the ideal case, the heptagon would be filled completely. Inspection of the results reveals that some quantities are predicted much better than others. Specifically, it can be seen that the cathode emission time is predicted particularly bad. This result is somewhat expected, however, because this quantity refers to the longitudinal phase space at emission time, which cannot directly be accessed via a transverse beam size measurement. All quantities, which refer to the transvere phase space at the solenoid show very good prediction performance with $<\\SI{5}{\\percent}$ error. The prediction performance of both bunch charge and bunch length at the solenoid needs to be considered in more detail. In case of the bunch charge, values $\\lesssim\\SI{0.5}{\\pico\\coulomb}$ are predicted much less accurately. This can be explained by the lack of significant space charge effects, which alter the shape of the beam size vs.\\ focusing strength curve, effectively leading to degeneracy w.r.t. the initial bunch charge. Despite being a quantity of the longitudinal phase space, the bunch length is generally predicted with an error $<\\SI{10}{\\percent}$. This is because here the bunch length is directly correlated with the bunch charge and hence space charge effects. The prediction performance decreases towards smaller bunch lengths, which can be explained by the fact that the bunch length at the solenoid actually increases with the initial bunch charge. Therefore the same argument applies as for the bunch charge.\n\\begin{figure*}[htbp]\n  \\centering\n  \\includegraphics[width=0.98\\textwidth]{FIG_5}\n  \\caption{Relative error distributions for all verification data sets and label components. The last plot is a radar plot showing the percentage of data sets within a \\SI{1}{\\percent} (red), \\SI{5}{\\percent} (black) and \\SI{10}{\\percent} (gray) relative error interval respectively. The dashed lines in the distribution plots refer to the same error intervals. $Q:$ Bunch charge, $\\sigma_{t,\\text{las}}:$ Emission time, $\\sigma_{x,\\text{las}}:$ Laser spot size, $\\sigma_{z,\\text{SOL}}:$ Bunch length at the solenoid, $\\sigma_{x^\\prime,\\text{SOL}}:$ Divergence at the solenoid, $\\sigma_{x,\\text{SOL}}:$ Beam size at the solenoid, $\\varepsilon_{\\text{n},x,\\text{SOL}}:$ Normalized emittance at the solenoid.}\n  \\label{fig:error_dists_perf}\n\\end{figure*}\n\nFrom the prediction results and the ground truth a mean relative error was calculated over the whole test data set, yielding a mean prediction error for the emittance at the solenoid of \\SI{1.0}{\\percent}. Beam size and divergence at the solenoid are predicted very accurately with an error less than \\SI{0.1}{\\percent}. In addition to the beam parameters at the solenoid position, \\textsc{ASTRA} input parameters were predicted. The laser spot size is predicted with an error down to \\SI{0.7}{\\percent}. Emission time and bunch charge are predicted with errors of \\SI{32.1}{\\percent} and \\SI{27.1}{\\percent} respectively. The fact that the laser spot size is predicted best out of the three input parameters is due to the fact that it has the strongest effect on the shape of the beam size vs.\\ focusing strength data, especially for low charges (linear dependence of the thermal emittance). These results, as well as the percentage of predictions within a \\SI{1}{\\percent}, \\SI{5}{\\percent} and \\SI{10}{\\percent} relative error interval are summarized in Table~\\ref{tab:pred_perf_table}.\n\n\\begin{table*}[htbp]\n   \\caption{Summary of the prediction performance for the different training label components. Mean error refers to the absolute prediction error and is calculated for the whole verification data set. The three remaining columns refer to the fraction of the verification data set within a certain prediction accuracy interval.}\n   \\begin{ruledtabular}\n   \\begin{tabular}{lcccc}\n       & Mean Error (\\SI{}{\\percent}) & $<$ \\SI{1}{\\percent} Error (\\SI{}{\\percent}) & $<$ \\SI{5}{\\percent} Error (\\SI{}{\\percent}) & $<$ \\SI{10}{\\percent} Error (\\SI{}{\\percent})\\\\\n      \\hline\n      Laser Spot Size & 0.71 & 76.59 & 99.84 & 100.00\\\\\n      Emission Time & 32.14 & 2.46 & 11.67 & 21.05\\\\\n      Bunch Charge & 27.05 & 10.24 & 52.74 & 76.74\\\\\n      Emittance at the Solenoid & 1.01 & 55.67 & 99.81 & 100.00\\\\\n      Beamsize at the Solenoid & 0.05 & 100.00 & 100.00 & 100.00\\\\\n      Divergence at the Solenoid & 0.05 & 100.00 & 100.00 & 100.00\\\\\n      Bunch Length at the Solenoid & 2.60 & 24.35 & 89.76 & 98.13\\\\\n   \\end{tabular}\n   \\end{ruledtabular}\n   \\label{tab:pred_perf_table}\n\\end{table*}\n\nThe main goal of the study is to find a better way to determine the transverse emittance from phase advance scan data in the $\\rho \\gg 1$ regime, as well as in regimes where Eq.~\\ref{eq:fit_criterion_math} does not hold. It is hence useful to evaluate the emittance prediction performance in form of the relative error versus these two quantities. Figure~\\ref{fig:error_perf_comp_Ideal} shows the result of this analysis using both the FCNN, as well as the traditional fit routine (cf. Sec.~\\ref{sec:MeasurementTechnique}). \n\nAs expected from Fig.~\\ref{fig:space_charge_dominance_train_data_sets} and Fig.~\\ref{fig:feasibility_train_data_sets}, the traditional fit yields inaccurate results across the whole data set. The FCNN, on the other hand, performs much better even for very high values of $\\rho$.\n\\begin{figure}[htbp]\n  \\centering\n  \\includegraphics[width=\\columnwidth]{FIG_6.pdf}\n  \\caption{Mean emittance FCNN prediction performance, as well as traditional fit results vs.\\ the laminarity parameter (top) and the fit feasibility criterion (bottom). The data is calculated for 100 equally sized laminarity parameter slices. $\\gamma = 6.8$ for this population. \\emph{Note that the bottom plot shows the fit criterion values multiplied by 1000 for readability.}}\n  \\label{fig:error_perf_comp_Ideal}\n\\end{figure}\n\\section{Extension to Measured Data}\nSo far, the neural network was trained and tested solely with ideal phase advance scan data. This means that the training, validation, as well as the test data sets contain only perfectly evenly spaced data points with flawless beam size values. In addition, all scans were simulated within the same focusing strength range. Although this approach yields very good prediction performance for simulated data, it cannot be applied to experimental data, for several reasons. First, in a measured data set it is not guaranteed that all data points are evenly spaced. It is also not guaranteed that the scan range corresponds to the trained one and that the measured values of the focusing strength are correct \\footnote{This issue is already somewhat alleviated by using $\\delta B_i$, as described in Sec.~\\ref{sec:Methodology}}. Finally, the measured beam sizes are subject to jitter and systematic measurement errors like resolution limitations.\n\nIn order to take all of this into account, the training procedure was modified. Each data set is now created with a slightly different focusing strength scan range. Since the number of scan points is kept constant, the spacing between data points is now slightly different every time, which might also be the case in reality. Furthermore, the first and last focusing setting were added to the range of predicted parameters ($\\rightarrow M_\\text{out} = 9$). Measurement errors were taken into account by generating noisy data sets from the ideal sets by adding normally distributed errors. Both relative and absolute errors on focusing strength and beam size were considered with magnitudes based on experience at ARES. From each data set, $N_\\text{err} = 100$ noisy sets with relative errors and $N_\\text{err}$ noisy sets with absolute errors were generated. In addition, the procedure was repeated, this time enforcing a \\SI{10}{\\micro\\meter} resolution limit on the beam sizes. Including the ideal data, the total number of data sets now increases to $N_\\text{tot} = 2(2N_\\text{err}+1)\\cdot N = 6458532$. To visualize the importance of training the FCNN with noisy data, we fed data sets with increasing artificial noise to both the network trained solely on ideal data, as well as one trained on noisy data. The results are shown in Fig.~\\ref{fig:noise_performance}. It can be seen that the population with less than \\SI{5}{\\percent} decreases significantly with noise level, if the FCNN is not trained on noisy data.\n\\begin{figure}[htbp]\n  \\centering\n  \\includegraphics[width=\\columnwidth]{FIG_7.pdf}\n  \\caption{Population with less than \\SI{5}{\\percent} prediction error on the emittance vs.\\ different artificial noise levels. The FCNN layout is the same in both shown cases. \\emph{Note that in order to enable a direct comparison, the FCNNs used here were trained on the data sets with fixed scan range and constant spacing. This explains the generally better performance compared to Table~\\ref{tab:pred_perf_table_2}.}}\n  \\label{fig:noise_performance}\n\\end{figure}\n\nWe performed the same general analysis for the new network, as described above, and saw the same overall behaviour. The results are summarized in Table~\\ref{tab:pred_perf_table_2}. Compared to the network based on ideal data, the performance is slightly worse, but still for the majority of data sets the emittance is predicted with less than $\\SI{5}{\\percent}$ error.\n\n\\begin{table*}[htbp]\n   \\caption{Summary of the prediction performance of the modified network for the different training label components. Mean error refers to the absolute prediction error and is calculated for the whole verification data set. The three remaining columns refer to the fraction of the verification data set within a certain prediction accuracy interval.}\n   \\begin{ruledtabular}\n   \\begin{tabular}{lcccc}\n       & Mean Error (\\SI{}{\\percent}) & $<$ \\SI{1}{\\percent} Error (\\SI{}{\\percent}) & $<$ \\SI{5}{\\percent} Error (\\SI{}{\\percent}) & $<$ \\SI{10}{\\percent} Error (\\SI{}{\\percent})\\\\\n      \\hline\n      Laser Spot Size & 2.76 & 22.25 & 87.85 & 98.83\\\\\n      Emission Time & 15.97 & 4.25 & 21.08 & 43.48\\\\\n      Bunch Charge & 14.13 & 6.44 & 30.89 & 67.20\\\\\n      Emittance at the Solenoid & 2.51 & 29.43 & 89.17 & 97.51\\\\\n      Beamsize at the Solenoid & 0.38 & 96.34 & 100.00 & 100.00\\\\\n      Divergence at the Solenoid & 0.38 & 96.63 & 100.00 & 100.00\\\\\n      Bunch Length at the Solenoid & 4.48 & 12.74 & 60.03 & 95.02\\\\\n   \\end{tabular}\n   \\end{ruledtabular}\n   \\label{tab:pred_perf_table_2}\n\\end{table*}\n\n\\section{Measurements at the ARES linac}\nAs a real world test, we conducted emittance measurements using the phase advance scan technique at the ARES linac at DESY. The layout of the measurement setup is shown in Fig.~\\ref{fig:ARES_sketch}. We took data for several bunch charges by adjusting an attenuator in the cathode laser beamline. The charge was measured both with a Faraday cup, which can be inserted into the beamline instead of the scintillating screen and a cavity based charge monitor $\\sim \\SI{0.7}{\\meter}$ downstream of the screen \\cite{Lipka:166172}. All measurements shown here were performed according to the procedure introduced in Sec.~\\ref{sec:MeasurementTechnique}. Transverse beam sizes were determined from camera images of a scintillating Ce:GAGG (Cerium doped Gadolinium Aluminium Gallium Garnet) screen. The spatial resolution of the system is specified to be $\\sim \\SI{10}{\\micro\\meter}$ \\cite{Wiebers:166567}.\n\nFigure~\\ref{fig:emittance_measurement_ARES} shows the emittance values obtained from the measurements using both the FCNN, as well as the traditional fit method. The data points are compared to an ASTRA simulation including space charge based on the machine settings at the day of the measurements, including uncertainty. It can be seen that the FCNN results are much closer to the expected values than the results obtained from the envelope equation fit. It is interesting to note that Fig.~\\ref{fig:emittance_measurement_ARES} reproduces the expected behaviour shown in Fig.~\\ref{fig:error_perf_comp_Ideal}, as the fit underestimates the emittance for charges $<\\SI{0.5}{\\pico\\coulomb}$ and overestimates them for higher charges. The FCNN result follows the ASTRA curve much closer, mostly staying within the uncertainty of the simulation.\n\\begin{figure}[htbp]\n  \\centering\n  \\includegraphics[width=\\columnwidth]{FIG_8.pdf}\n  \\caption{Emittance measurements conducted at the ARES linac at DESY using the phase advance scan technique for different bunch charges. The black dashed line denotes the mean ASTRA simulation result. The shaded areas correspond to the 1, 2 and 3\\,$\\sigma$ uncertainty of the simulation, based on 100 simulations with normally distributed input parameters.}\n  \\label{fig:emittance_measurement_ARES}\n\\end{figure}\n\nIn order to cross-check the obtained results, we performed additional emittance measurements using a grid mask based method, as described in \\cite{PhysRevAccelBeams.21.102802}, using the same machine setup. Three different grids were used for the measurements: An SPI G200TH TEM grid \\footnote{SPI G200TH TEM Grid, available at \\url{https:\/\/www.2spi.com\/item\/202hc-xa}, \\emph{last access: 22nd April 2022}}, an SPI G300 TEM grid \\footnote{SPI G300 TEM Grid, available at \\url{https:\/\/www.2spi.com\/item\/2030c-xa}, \\emph{last access: 22nd April 2022}} and a custom made pepper pot \\footnote{Custom made Tungsten pepper pot (laser drilled). Hole diameter: \\SI{15}{\\micro\\meter}, pitch: \\SI{85}{\\micro\\meter}, thickness: \\SI{20}{\\micro\\meter}}. Measurements were performed for two different charge settings, \\SI{1}{\\pico\\coulomb} and \\SI{2}{\\pico\\coulomb}. These values were deliberately chosen to be in the strongly space charge dominated regime, where the traditional fit yields particularly bad results. At ARES, the grid masks are installed at the same $z$-position as the screen used to record the phase advance scan data. This means that the emittance obtained from the grid measurements will always be different from the phase advance scan result, as the phase advance scan yields the emittance at the position of the focusing element. The emittance is furthermore expected to be different, because in order to image the grid, the beam needs to be focused slightly before the grid, which can lead to emittance growth. Nevertheless, it is still possible to compare the measured values to ASTRA simulations, which would show that the ARES setup depicted in Fig.~\\ref{fig:ARES_sketch} can be well simulated with ASTRA. This would validate the FCNN results indirectly. Figure~\\ref{fig:emittance_measurement_ARES_grid_sim} shows an ASTRA simulation of the grid measurement using a \\SI{1}{\\pico\\coulomb} bunch with $\\gamma = 6.8$. It can be seen that the emittance is strongly affected by focusing the beam down. The measurement results from all three grids, as well as the expected values from the ASTRA simulation are summarized in Table~\\ref{tab:grid_meas_comp}. The results are very close to the expected value in the \\SI{1}{\\pico\\coulomb} case. The measurement of the \\SI{2}{\\pico\\coulomb} beam shows a slightly higher than expected emittance value, which is in line with the high uncertainty of the high charge results shown in Fig.~\\ref{fig:emittance_measurement_ARES}. We hence conclude that ASTRA simulates the ARES beamline shown in Fig.~\\ref{fig:ARES_sketch} well.\n\\begin{figure}[htbp]\n  \\centering\n  \\includegraphics[width=\\columnwidth]{FIG_9.pdf}\n  \\caption{ASTRA simulation of the evolution of the transverse normalized emittance between gun and grid for a \\SI{1}{\\pico\\coulomb} bunch with $\\gamma = 6.8$. The orange dashed line marks the position of the solenoid, which is used to perform the phase advance scan measurements. The black dashed line denotes the mean ASTRA simulation result. The shaded areas correspond to the 1, 2 and 3\\,$\\sigma$ uncertainty of the simulation, based on 100 simulations with normally distributed input parameters. The blue dashed line shows the evolution of the beam size \u2013 the beam reaches its focus slightly before the grid.}\n  \\label{fig:emittance_measurement_ARES_grid_sim}\n\\end{figure}\n\\begin{table}[htbp]\n   \\caption{Summary of the grid based measurements, used to verify the FCNN results. The measurements are compared to the ASTRA simulation result.}\n   \\begin{ruledtabular}\n   \\begin{tabular}{lcc}\n      Method & $\\varepsilon _\\text{n}$ (\\SI{}{\\nano\\meter}) @ \\SI{1}{\\pico\\coulomb} & $\\varepsilon _\\text{n}$ (\\SI{}{\\nano\\meter}) @ \\SI{2}{\\pico\\coulomb}\\\\\n      \\hline\n      \\textbf{ASTRA Simulation} & $\\mathbf{105.5 \\pm 1.2}$ & $\\mathbf{152.9 \\pm 3.0}$ \\\\\n      \\hline\n      SPI G200TH TEM Grid & $107.4 \\pm 5.8$ & $165.4 \\pm 15.7$ \\\\\n      SPI G300 TEM Grid & $102.7 \\pm 3.8$ & $156.0 \\pm 15.6$ \\\\\n      Pepper Pot & $105.6 \\pm 11.9$ & $156.1 \\pm 27.4$ \\\\\n      \\textbf{Mean (Measurements)} & $\\mathbf{105.2 \\pm 8.0}$ & $\\mathbf{160.5 \\pm 20.3}$\n   \\end{tabular}\n   \\end{ruledtabular}\n   \\label{tab:grid_meas_comp}\n\\end{table}\n\\subsection{Prediction of other parameters}\nAs discussed above, the FCNN also predicts fixed machine parameters and other charge dependent beam parameters to varying accuracy (see Table~\\ref{tab:pred_perf_table_2}). Table~\\ref{tab:pred_fixed_params} summarizes the predicted fixed machine parameters in comparison to what was used in the experiment. It can be seen that both the mean laser spot size and pulse length are predicted to be larger than the expected values. Since measuring the laser spot size on the cathode directly is very difficult in the ARES setup, the predicted values fall within the uncertainty. As discussed in Sec.~\\ref{sec:ResultsAndComparison}, the prediction of the laser pulse length should be treated with caution. The solenoid scan range is predicted well within the uncertainties.\n\\begin{table}[htbp]\n   \\caption{Prediction results for fixed machine parameters.}\n   \\begin{ruledtabular}\n   \\begin{tabular}{lcc}\n      Parameter & Prediction & Experiment\\\\\n      \\hline\n      Laser Spot Size (\\SI{}{\\micro\\meter}) & $338.5 \\pm 0.5$ & $320 \\pm 30$ \\\\\n      Laser Pulse Length (\\SI{}{\\femto\\second}, rms) & $87.34 \\pm 0.04$ & $76 \\pm 8$ \\\\\n      Solenoid Field -- Start (mT) & $130.516 \\pm 0.001$ & $130.5 \\pm 0.1$ \\\\\n      Solenoid Field -- End (mT) & $150.783 \\pm 0.001$ & $150.7 \\pm 0.1$ \\\\\n   \\end{tabular}\n   \\end{ruledtabular}\n   \\label{tab:pred_fixed_params}\n\\end{table}\n\nFigure~\\ref{fig:pred_variable_params} shows the prediction results for the beam charge, as well as the charge dependent beam parameters \\emph{bunch length}, \\emph{beam size} and \\emph{beam divergence} at the solenoid. As in Fig.~\\ref{fig:emittance_measurement_ARES}, the data points are compared to an ASTRA simulation including space charge based on the machine settings at the day of the measurements, including uncertainty. It can be seen that the beam charge prediction fits the measured values well. Both beam size and beam divergence follow the ASTRA curve well, albeit at the lower end of the uncertainty, denoted by the shaded area. The bunch length follows a more linear charge dependence than expected from the ASTRA simulation, which might be attributed to either the not fully known temporal and spatial laser pulse shape at the cathode, as well as the overall prediction performance of parameters of the longitudinal phase space (see Sec.~\\ref{sec:ResultsAndComparison}). \n\nIn order to cross-validate the prediction results, an ASTRA simulation using the mean predicted laser spot size and pulse length (see Table~\\ref{tab:pred_fixed_params}) was performed. The results are shown in Fig.~\\ref{fig:pred_variable_params} as the blue dashed line. Indeed, a larger spot size and longer pulse length lead to results closer to the lower end of the uncertainty in all three cases, which can be explained by the reduced charge density. Remaining discrepancies might be explained by the not fully known temporal and spatial laser pulse shape at the cathode.\n\\begin{figure}[htbp]\n  \\centering\n  \\includegraphics[width=\\columnwidth]{FIG_10.pdf}\n  \\caption{Prediction results for other charge dependent beam parameters. The black dashed line denotes the mean ASTRA simulation result. The shaded areas correspond to the 1, 2 and 3\\,$\\sigma$ uncertainty of the simulation, based on 100 simulations with normally distributed input parameters. The blue dashed line corresponds to an ASTRA simulation using the mean predicted laser spot size and pulse length (see Table~\\ref{tab:pred_fixed_params}).}\n  \\label{fig:pred_variable_params}\n\\end{figure}\n\\section{Conclusion and Outlook}\nWe have shown in simulation that a pre-trained fully connected neural network can be used to predict the transverse emittance from phase advance scan data even in the $\\rho \\gg 1$ regime and in case a traditional envelope equation based fit is mathematically not feasible. We have optimized the network for real-world measurement data and achieved $< \\SI{5}{\\percent}$ error for the majority of the test data set population (\\SI{89.2}{\\percent}), resulting in a mean relative error of \\SI{2.5}{\\percent}. We have applied our method to measurements conducted at the ARES linac at DESY and compared the predictions to numerical simulations using the well benchmarked code ASTRA, as well as results obtained from the traditional fit method. As expected from the simulation study, the FCNN predictions are much closer to what is expected from the numerical simulation. We have furthermore cross-validated the results using additional emittance measurements based on a grid mask based method.\n\nIn addition to the transverse emittance, the network also predicts other key beam and machine parameters to varying accuracy. While quantities directly tied to the transverse phase space are predicted as accurate or better than the emittance, quantities tied to the longitudinal phase space, such as the bunch length, are predicted less accurate, as expected. It should be noted, that in our study the gun setting is not a variable in the process of training the FCNN. This means that for each gun setting (gradient and phase) a separate FCNN has to be trained. Inclusion of these two parameters could be part of a future study. Furthermore, difficult to directly access parameters, such as the thermal emittance could be added. In conclusion, we have demonstrated that pre-trained FCNNs can be a powerful tool for the analysis of previously difficult to interpret data sets.\n\\section{Model Availability}\nThe FCNN models are available from the corresponding author upon request in TensorFlow format.\n\\begin{acknowledgments}\nWe acknowledge support from DESY (Hamburg, Germany), a member of the Helmholtz Association HGF. Specifically, we would like to thank the DESY beam diagnostics group for their support during the measurement runs at ARES. We also thank the DESY Maxwell team for providing the compute resources we used to generate the training data sets in a reasonable time. Finally, we thank R.~Mayet (Halodi Robotics AS) for NN-related consulting.\n\\end{acknowledgments}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Introduction}\n\nIn a recent paper, Arditi et al.~\\cite{Arditi2016} stated that a proper patch model of population dynamics must obey a basic logical property: ''If two patches are linked by migration, and if the migration rate becomes infinite, the two patches become perfectly mixed among each other, and the system must behave as a one-patch model for the total population.'' To illustrate the issue, they studied the following model:\n\n\\begin{equation}\n\\label{eq:growth}\n\\begin{aligned}\n\\frac{dN_1}{dt} &= r_1N_1\\left(1-\\frac{N_1}{K_1}\\right)\n+\\beta\\left(N_2-N_1\\right) \\\\\n\\frac{dN_2}{dt} &= r_2N_2\\left(1-\\frac{N_2}{K_2}\\right)+\\beta\\left(N_1-N_2\\right),\n\\end{aligned}\n\\end{equation}\n\n\\noindent where $N_i$ with $i={1,2}$ being population size in patch $i$, $r_i$ the local intrinsic per capita growth rate in patch $i$, $K_i$ the local carrying capacity in patch $i$ and $\\beta$ the migration rate constant from and to any patch in the population. Note that each equation is the classical formula for logistic growth plus a term describing migration between patches.\n\nArditi et al.~\\cite{Arditi2016} found that the asymptotic dynamics of system \\eqref{eq:growth} in the case of perfect mixing (i.e. with $\\beta\\rightarrow \\infty$) is different from the asymptotic dynamics of the sum of the two populations in isolation (i.e. with $\\beta = 0$). In particular, they showed that the equilibrium population size of the system with perfect mixing is different (either larger or smaller) that the sum of equilibrium sizes of the isolated populations. In the limiting but plausible case that the local populations differed in the value of their carrying capacities $K_i$ but not in the values of $r_i$, merging two patches in a single one showed to be always detrimental for equilibrium population size.\n\nAlthough the analysis is correct, it is valid to ask whether the particular choice for describing migration in \\eqref{eq:growth} was the best one for studying such a general ecological phenomenon. Apparently, the choice for migration model in\\cite{Arditi2016} was made because of two main reasons: 1) this system was analyzed previously \\cite{Freedman77} \\cite{DeAngelis1979} \\cite{HOLT1985181} \\cite{hanski1999metapopulation} \\cite{DeAngelis20143087} \\cite{Arditi201545} thus it has some tradition within the ecological literature, and 2) Arditi et al.~\\cite{Arditi2016} considered this model as a ``natural way'' to represent a two-patch system with logistic growth.\n\nAll other things being equal, a well known and widely used model should be favored over its competitors. However this is only valid until we consider a model presenting some objective advantage (e.g. better match with empirical observations) without compromising any substantial aspects (such as number of parameters, mathematical tractability, etc.). In our opinion, model~\\eqref{eq:growth} is neither the most natural nor the best way to extend the logistic growth model to a two-patch scenario. Furthermore, we will show below that the paradoxical results reported by Arditi et al.~\\cite{Arditi2016} are only a consequence of using the specific model~\\eqref{eq:growth} and should not be considered to be a general fact.\n\n\\section*{A  more biologically plausible model}\n\nModel~\\eqref{eq:growth}, used in~\\cite{Arditi2016} to present the ``perfect mixing paradox'' contains as a key component a passive migration rate from patch $i$ to patch $j$, $\\beta(N_i-N_j)$. This formulation of passive migration rate assumes that there will be a positive flux of migrants from patch $i$ to patch $j$ whenever the absolute population size in patch $i$ is larger than the absolute population size in patch $j$, no matter the differences in patch size or quality. This means that, given equal patch quality, it is possible to have a flux of migrants from a path with greater absolute population size but with lesser population density (with a very large patch size) to a small and more dense patch which possesses a lower absolute population size (Fig.~\\ref{fig:fig1}a). This feature of model~\\eqref{eq:growth} represents an assumption of limited biological realism. Under the same scenario, a more reasonable assumption is that migrants should pass from the patch with higher population density (absolute population size divided by patch size) to the patch with lower population density (Fig.~\\ref{fig:fig1}b).\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=.6\\textwidth]{fig1.pdf}\n\\caption{\\label{fig:fig1}Graphical representation of the flux of migrants in a two-patch population dynamics model. a): biologically unrealistic assumption of model\\eqref{eq:growth}, where the net flux of migrants occurs from the less dense (with higher absolute population size but with a much larger patch size) to the denser patch (b): more realistic assumption, with migrant flux from the more dense to the less dense patch}\n\\end{figure}\n\n\n\nWe propose to re-evaluate the perfect mixing paradox using a slightly different system. This model is both amenable for analysis and contains a more realistic assumption about the direction of the net flux of migrants.\n\n\\begin{equation}\\label{eq:growth2}\n\\begin{aligned}\n\\frac{dN_1}{dt} &=& r_1N_1\\left(1-\\frac{N_1}{K_1}\\right)\n+\\beta\\left(\\frac{N_2}{K_2}-\\frac{N_1}{K_1}\\right) \\\\\n\\frac{dN_2}{dt} &=& r_2N_2\\left(1-\\frac{N_2}{K_2}\\right)\n+\\beta\\left(\\frac{N_1}{K_1}-\\frac{N_2}{K_2}\\right)\n\\end{aligned}\n\\end{equation}\n\n\nIn this model the flux of migrants is governed by the differences between the ratios $N_i\/K_i$. We will refer to the ratio $N_i\/K_i$ as the \\emph{saturation of patch $i$}, which is a balance between the local population size at time $t$ and the local equilibrium population size $K_i$. The value of $K_i$ depends on the quantity and quality of resources in patch $i$. The direction of the net flux of migrants in this model captures the intuition described in Fig.~\\ref{fig:fig1}b. As shown below, this model does not exhibit the paradox presented in~\\cite{Arditi2016}.\n\nFirst, note that in isolation (i.e., with $\\beta = 0$), the system converges to ${N_1}^* = K_1,\\ \\ {N_2}^* = K_2$. This equilibrium is the same as the one of model~\\eqref{eq:growth}.\nUsing the same reasoning used in~\\cite{Arditi2016}, if we assume perfect mixing of local populations (i.e. with $\\beta\\rightarrow\\infty$) in\nmodel~\\eqref{eq:growth2}, it can be shown that for all $t>0$\n\n\\begin{equation}\\label{ratios}\n\\begin{aligned}\n\\frac{N_1}{K_1}=\\frac{N_2}{K_2}\n\\end{aligned}\n\\end{equation}\n\n\\noindent and therefore, for calculating the saturation of both patches combined:\n\n\\begin{equation}\\label{ratios2}\n\\begin{aligned}\n\\frac{N_1+N_2}{K_1+K_2} = \\frac{N_1\\frac{K_1}{K_1}+N_1\\frac{K_2}{K_1}}{K_1+K_2} = \\frac{N_1}{K_1} = \\frac{N_2}{K_2}\n\\end{aligned}\n\\end{equation}\n\nThis shows that total population saturation under perfect mixing is equal to each of the local population saturations.\nNow, let us check whether the main paradoxical property presented in~\\cite{Arditi2016} holds for our model~\\eqref{eq:growth2}. This implies checking whether or not the long term total population size under perfect mixing is equal to total population size in isolation. Adding both equations of system~\\eqref{eq:growth2} and using the equalities~\\eqref{ratios2} which are valid for the perfect mixing scenario, yields: \n\n\\begin{align}\n\\frac{dN_T}{dt} = \\frac{dN_1}{dt} + \\frac{dN_2}{dt} &= (r_1 N_1+r_2 N_2)\\left(1-\\frac{N_T}{K_T}\\right) \\nonumber\\\\\n&= \\frac{r_1 K_1+r_2 K_2}{K_T}\\left(1-\\frac{N_T}{K_T}\\right)N_T \\nonumber \\\\\n&= \\bar{r}\\left(1-\\frac{N_T}{K_T}\\right)N_T\\label{our N_T}\n\\end{align}\n\n\\noindent where \n$N_T=N_1+N_2$,\\ \\ $K_T=K_1+K_2$ and $\\bar{r}=\\dfrac{r_1 K_1+r_2 K_2}{K_T}$.\n\nIt is clear that, at equilibrium, the total population size under perfect mixing (i.e. with $\\beta\\rightarrow\\infty$) is $K_T = K_1+K_2$. Thus, using the more realistic model~\\eqref{eq:growth2} resolves the main paradoxical behavior presented in~\\cite{Arditi2016} for mixed patches. Note also that in the logistic equation for $N_T$ the total intrinsic growth rate $\\bar{r}$ is the weighted average of the local intrinsic growth rates, with weights $K_1$ and $K_2$. In the case that the patches differ only in their intrinsic growth rates $r_i$ and do not differ in their carrying capacities (i.e., $K_1 = K_2$), the total intrinsic growth rate reduces to $\\bar{r}=(r_1+r_2)\/2$. Also, if $r_1=r_2$ then $\\bar{r}=r_1=r_2$.\n\nAnother issue presented by Arditi et al.~\\cite{Arditi2016} is what they call an ``apparent spatial dependency'' of the equation parameters when the dynamics of the total population is represented by the Verhulst equation. The undesirable model property in a multi-patch context is that the value of the self-interference coefficient in the quadratic term decreases with number of patches $S$:\n\n\\begin{equation}\\label{Verhulst1}\n\\frac{dN_T}{dt}=\\bar{r}N_T-\\frac{\\bar\\alpha}{S} {N_T}^2\n\\end{equation}\n\nTo solve this issue, Arditi et al.\\cite{Arditi2016} suggest to treat population size as density, in terms of mean population size per patch $\\bar{N}=N_T\/S$. When doing so, Eqn.\\eqref{Verhulst1} becomes\n\n\\begin{equation}\\label{Verhulst2}\n\\frac{d\\bar{N}}{dt}=\\bar{r}\\bar{N}-\\bar{\\alpha}\\bar{N}^2\n\\end{equation}\n\n\\noindent which follows the Verhulst equation. Thus, the form of the equation is invariant in the number of patches in the metapopulation system and their parameters ($\\bar{r}$ and $\\bar{\\alpha}$) are simply the average of the corresponding local patch parameter values.\n\nIn our case, and under the same reasoning, considering the average population  in $S$ well-mixed patches, $\\bar{N}=N_T\/S$, Eqn.~\\eqref{eq:growth2} becomes:\n\n\\begin{equation}\\label{invariant2}\n\\frac{d\\bar{N}}{dt}=\\bar{r}\\left(1-\\frac{\\bar{N}}{\\bar{K}}\\right)\\bar{N}\n\\end{equation}\n\n\\noindent with $\\bar{K}=K_T\/S$. That is, the carrying capacity of the average population is the average of the local carrying capacities. Like Eqn.~\\eqref{Verhulst2}, our Eqn.~\\eqref{invariant2} is also invariant in the number patches, and their parameters ($\\bar{r}$ and $\\bar{K}$) are the weighted and aritmetic means, respectively, of the corresponding local patch parameters. Therefore there is no reason to favor the Verhulst's logistic equation over the classical formulation with the familiar $r-K$ parameterization in a multi-patch context, as argued by Arditi et al.~\\cite{Arditi2016}.\n\n\\section*{Discussion}\n\nThe paper by Arditi et al.\\cite{Arditi2016} argued that the logistic equation, in its usual $r-K$ parameterization, presents some undesirable properties when used in a multiple patch context. These properties configure what those authors called the ''perfect mixing paradox.'' Arditi et al.~\\cite{Arditi2016} also showed that the Verhulst's formulation of the logistic growth model $\\sfrac{dN}{dt}=rN-\\alpha N^2 $ is less prone to these paradoxical features, as compared with the familiar Lotka formulation $\\sfrac{dN}{dt}=rN(1-\\sfrac{N}{K})$, when generalized to a multi-patch environment. They conclude, on the basis of the analysis of these models extended to a metapopulation context by including a specific migration function, that the Verhulst formulation should be favored over the Lotka one, and that the term ``carrying capacity'' is misleading and should be abandoned in favor of the more correct ``equilibrium density.''\n\nThe supposedly paradoxical behavior of the metapopulation version of the Lotka-Gause model rests, according to Arditi et al.~\\cite{Arditi2016}, on two main features that were exemplified considering a two-patch environment as a study case. The first undesirable property is that the total mixed population equilibrium $K_T$ is in general different from the sum of the equilibria in the isolated patches $K_1 + K_2$. This major shortcoming of the analyzed model led Arditi et al.~\\cite{Arditi2016} to state that using the term ``carrying capacity'' is incorrect except in specific contexts. The second undesirable feature is the parameter dependence on the number of patches in the system, exhibited by the Verhulst form of the logistic growth model for the total population size. However, when population size is expressed as mean (per patch) abundance the model parameters can be calculated as the average of the local parameters and do not depend on the number of patches. Nevertheless, Arditi et al.~\\cite{Arditi2016} claim that this scale-invariance is only exhibited by the Verhulst model and this gives it an advantage over the Lotka-Gause model.\n\nIn this paper, we show that the paradoxical behaviors presented by Arditi et al.~\\cite{Arditi2016} belong only to the specific variant of the Lotka-Gause model they analyzed. Also, we suggest that the model used by Arditi et al.~\\cite{Arditi2016} is not the best choice regarding biological realism. In fact, we present a model as simple as the one they used (two state-variables, five parameters) that is more realistic and is free of the alluded paradoxes exhibited by the Arditi's extensions to both the Lotka-Gause and the Verhulst logistic models. \n\nThe most remarkable advantage of our model \\eqref{eq:growth2} is that, unlike both logistic forms used by Arditi et al.~\\cite{Arditi2016} in their analysis, total population size at equilibrium of a perfectly mixed metapopulation is equal to the sum of local equilibria. This feature immediately invalidates the criticism posed over the meaning and usefulness of the carrying capacity term. In our model \\eqref{our N_T}, global intrinsic growth rate of the metapopulation is not the arithmetic average of  local growth rates but it is equal to the weighted average of the local growth rate parameters. This is very reasonable, since under perfect mixing among patches, the ratios $N_i\/K_i$ are equated while their absolute abundances are not. So, it is possible to have patches with contrasting amount of resources (e.g. space or nutrients) and therefore  with unequal population abundances, say $3$ individuals in patch $1$ and $1000$ individuals in patch $2$. Under this scenario, global intrinsic growth rate could not be the arithmetic mean of the local growth rates but it should be closer to the parameter value of the larger population. In the case of the Arditi's model, the absolute population abundances tend to be the same under perfect mixing and so the arithmetic mean and weighted mean are the same. Regarding the second issue stressed by Arditi et al.~\\cite{Arditi2016}, we showed that our model \\eqref{our N_T} does not suffer from a lack of scale-invariance and that the dynamics of the per patch mean size of the metapopulation is fully consistent with the well known logistic dynamics within a single patch.\n\nIn sum, we show here that the criticisms posed by Arditi et al.~\\cite{Arditi2016} to the familiar form of the logistic equation attributed to Lotka and Gause are only valid for the particular way in which those authors extended that equation to the multi-patch scenario. We also suggest that their model is not the best choice among other plausible models of the same complexity, and that their criticisms against the usefulness of the carrying capacity as a measure of patch size or richness is not well justified. However, the paper by Arditi et al. has the value of highlighting that modeling population, metapopulation or community dynamics requires more attention than is usually given to and that models should not be applied to any scenario without a rigorous theoretical analysis of their properties.\n\n\\section*{Acknowledgments}\nThis study was supported by FONDECYT grant 1150348.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec1}\n\\setcounter{equation}{0}\nA connected simply connected Lie group $N$\nwith center $Z$ is called {\\em square integrable} if it has unitary\nrepresentations $\\pi$ whose coefficients $f_{u,v}(x) = \n\\langle u, \\pi(x)v\\rangle$ satisfy $|f_{u,v}| \\in L^2(N\/Z)$.  \nC.C. Moore and the author worked out the structure and representation\ntheory of these groups \\cite{MW1973}.  If $N$ has one \nsuch square integrable representation then there is a certain polynomial\nfunction ${\\rm Pf}\\,(\\lambda)$ on the linear dual space $\\mathfrak{z}^*$ of the Lie algebra of\n$Z$ that is key to harmonic analysis on $N$.  Here ${\\rm Pf}\\,(\\lambda)$ is the\nPfaffian of the antisymmetric bilinear form on $\\mathfrak{n} \/ \\mathfrak{z}$ given by\n$b_\\lambda(x,y) = \\lambda([x,y])$.  The square integrable\nrepresentations of $N$ are the \n$\\pi_\\lambda$ where $\\lambda \\in \\mathfrak{z}^*$ with ${\\rm Pf}\\,(\\lambda) \\ne 0$,\nPlancherel  almost irreducible unitary representations of $N$ are square\nintegrable, and up to an explicit constant \n$|{\\rm Pf}\\,(\\lambda)|$ is the Plancherel density of the unitary\ndual $\\widehat{N}$ at $\\pi_\\lambda$.  \nThis theory has proved to have serious analytic consequences.  For example,\nfor most commutative nilmanifolds $G\/K$, i.e. Gelfand pairs $(G,K)$ \nwhere a nilpotent subgroup $N$ of $G$ acts transitively on $G\/K$, the\ngroup $N$ has square integrable representations \\cite{W2007}.\nAnd it is known just which maximal parabolic subgroups of semisimple Lie groups\nhave square integrable nilradical \\cite{W1979}.\n\\medskip\n\n\nIn \\cite{W2012} and \\cite{W2013} the theory of square integrable nilpotent\ngroups was extended to ``stepwise\nsquare integrable'' nilpotent groups.\nThey are the connected simply connected nilpotent Lie groups\nof (\\ref{setup}) just below.  We use $L$ and $\\mathfrak{l}$ to avoid conflict of\nnotation with the $M$ and $\\mathfrak{m}$ of minimal parabolic subgroups.\n\\begin{equation}\\label{setup}\n\\begin{aligned}\nN = &L_1L_2\\dots L_{m-1}L_m \\text{ where }\\\\\n &\\text{(a) each factor $L_r$ has unitary representations with coefficients in\n$L^2(L_r\/Z_r)$,} \\\\\n &\\text{(b) each } N_r := L_1L_2\\dots L_r \\text{ is a normal subgroup of } N\n   \\text{ with } N_r = N_{r-1}\\rtimes L_r \\text{ semidirect,}\\\\\n &\\text{(c) decompose }\\mathfrak{l}_r = \\mathfrak{z}_r + \\mathfrak{v}_r \\text{ and } \\mathfrak{n} = \\mathfrak{s} + \\mathfrak{v}\n        \\text{ as vector direct sums where } \\\\\n &\\phantom{XXXX}\\mathfrak{s} = \\oplus\\, \\mathfrak{z}_r \\text{ and } \\mathfrak{v} = \\oplus\\, \\mathfrak{v}_r;\n    \\text{ then } [\\mathfrak{l}_r,\\mathfrak{z}_s] = 0 \\text{ and } [\\mathfrak{l}_r,\\mathfrak{l}_s] \\subset \\mathfrak{v}\n        \\text{ for } r > s\\,.\n\\end{aligned}\n\\end{equation}\nDenote\n\\begin{equation}\\label{c-d}\n\\begin{aligned}\n&\\text{(a) }d_r = \\tfrac{1}{2}\\dim(\\mathfrak{l}_r\/\\mathfrak{z}_r) \\text{ so }\n        \\tfrac{1}{2} \\dim(\\mathfrak{n}\/\\mathfrak{s}) = d_1 + \\dots + d_m\\,,\n        \\text{ and } c = 2^{d_1 + \\dots + d_m} d_1! d_2! \\dots d_m!\\\\\n&\\text{(b) }b_{\\lambda_r}: (x,y) \\mapsto \\lambda([x,y])\n        \\text{ viewed as a bilinear form on } \\mathfrak{l}_r\/\\mathfrak{z}_r \\\\\n&\\text{(c) }S = Z_1Z_2\\dots Z_m = Z_1 \\times \\dots \\times Z_m \\text{ where } Z_r\n        \\text{ is the center of } L_r \\\\\n&\\text{(d) }{\\rm Pf}\\,: \\text{ polynomial } {\\rm Pf}\\,(\\lambda) = {\\rm Pf}\\,_{\\mathfrak{l}_1}(b_{\\lambda_1})\n        {\\rm Pf}\\,_{\\mathfrak{l}_2}(b_{\\lambda_2})\\dots {\\rm Pf}\\,_{\\mathfrak{l}_m}(b_{\\lambda_m}) \\text{ on } \n\t\\mathfrak{s}^* \\\\\n&\\text{(e) }\\mathfrak{t}^* = \\{\\lambda \\in \\mathfrak{s}^* \\mid {\\rm Pf}\\,(\\lambda) \\ne 0\\} \\\\\n&\\text{(f) } \\pi_\\lambda \\in \\widehat{N} \\text{ where } \\lambda \\in \\mathfrak{t}^*:\n    \\text{ irreducible unitary rep. of } N = L_1L_2\\dots L_m\n\\end{aligned}\n\\end{equation}\nThe basic result for these groups is\n\n\\begin{theorem}\\label{plancherel-general} {\\rm \\cite[Theorem 6.16]{W2013}}\nLet $N$ be a connected simply connected nilpotent Lie group that\nsatisfies {\\rm (\\ref{setup})}.  Then Plancherel measure for $N$ is\nconcentrated on $\\{\\pi_\\lambda \\mid \\lambda \\in \\mathfrak{t}^*\\}$.\nIf $\\lambda \\in \\mathfrak{t}^*$, and if $u$ and $v$ belong to the\nrepresentation space $\\mathcal{H}_{\\pi_\\lambda}$ of $\\pi_\\lambda$,  then\nthe coefficient $f_{u,v}(x) = \\langle u, \\pi_\\nu(x)v\\rangle$\nsatisfies\n\\begin{equation}\\label{sq-orthogrel}\n||f_{u,v}||^2_{L^2(N \/ S)} = \\frac{||u||^2||v||^2}{|{\\rm Pf}\\,(\\lambda)|}\\,.\n\\end{equation}\nThe distribution character $\\Theta_{\\pi_\\lambda}$ of $\\pi_{\\lambda}$ satisfies\n\\begin{equation}\\label{def-dist-char}\n\\Theta_{\\pi_\\lambda}(f) = c^{-1}|{\\rm Pf}\\,(\\lambda)|^{-1}\\int_{\\mathcal{O}(\\lambda)}\n        \\widehat{f_1}(\\xi)d\\nu_\\lambda(\\xi) \\text{ for } f \\in \\mathcal{C}(N)\n\\end{equation}\nwhere $\\mathcal{C}(N)$ is the Schwartz space, $f_1$ is the lift\n$f_1(\\xi) = f(\\exp(\\xi))$ of $f$ from $N$ to $\\mathfrak{n}$, \n$\\widehat{f_1}$ is its classical Fourier transform,\n$\\mathcal{O}(\\lambda)$ is the coadjoint orbit ${\\rm Ad}\\,^*(N)\\lambda = \\mathfrak{v}^* + \\lambda$,\n$c = 2^{d_1 + \\dots + d_m} d_1! d_2! \\dots d_m!$ as in {\\rm (\\ref{c-d}a)},\nand $d\\nu_\\lambda$ is the translate of normalized Lebesgue measure from\n$\\mathfrak{v}^*$ to ${\\rm Ad}\\,^*(N)\\lambda$.  The Fourier inversion formula on $N$ is\n\\begin{equation}\nf(x) = c\\int_{\\mathfrak{t}^*} \\Theta_{\\pi_\\lambda}(r_xf) |{\\rm Pf}\\,(\\lambda)|d\\lambda\n        \\text{ for } f \\in \\mathcal{C}(N).\n\\end{equation}\n\\end{theorem}\n\\begin{definition}\\label{stepwise2}\n{\\rm The representations $\\pi_\\lambda$ of (\\ref{c-d}(f)) are the\n{\\it stepwise square integrable} representations of $N$ relative to\nthe decomposition (\\ref{setup}).}\\hfill $\\diamondsuit$\n\\end{definition}\n\nOne of the main results of \\cite{W2012} and \\cite{W2013} is that nilradicals\nof minimal parabolic subgroups of finite dimensional real reductive Lie\ngroups are stepwise square integrable.  Even the\nsimplest case, the case of a minimal parabolic in $SL(n;\\mathbb{R})$, was a big\nimprovement over earlier results on the group of strictly upper triangular\nreal matrices.  Here we extend the construction of stepwise square integrable\nrepresentations to a class of locally nilpotent groups that are direct\nlimits in a manner that respects the basic setup (\\ref{setup}) of the\nfinite dimensional case, and we show how this applies to the nilradicals\nof direct limit minimal parabolic subgroups of the real and complex finitary\nreductive Lie groups, including $GL(\\infty;\\mathbb{F})$, $SL(\\infty;\\mathbb{F})$, $U(p,q;\\mathbb{F})$\nand $SU(p,q;\\mathbb{F})$ ($\\mathbb{F} = \\mathbb{R}, \\mathbb{C}$ or $\\mathbb{H}$ and $p+q = \\infty$), $Sp(\\infty;\\mathbb{F})$ \n($\\mathbb{F} = \\mathbb{R}$ or $\\mathbb{C}$), and $SO^*(2\\infty)$.\n\\medskip\n\nIn Section \\ref{sec2} we examine strict direct systems $\\{N_n, \\varphi_{m,n}\\}$\nof finite dimensional connected and simply connected nilpotent Lie groups\nthat satisfy (\\ref{setup}) in a manner that respects the maps\n$\\varphi_{m,n}: N_n \\to N_m$\\, $(m \\geqq n)$.  We show how this \nleads to sequences $\\{\\pi_{\\gamma_n}\\}$ of closely related stepwise square\nintegrable representations of the groups $N_n$\\,,  and then to their\nunitary representation limits $\\pi_\\gamma = \\varprojlim \\pi_{\\gamma_n}$\\,.\n\\medskip\n\nIn Section \\ref{sec3}\nwe prove stepwise Frobenius-Schur orthogonality relations and restriction\ntheorems for the coefficients of the representations $\\pi_{\\gamma_n}$\\,.\n\\medskip\n\nIn Section \\ref{sec4} we apply the tools of Section \\ref{sec3} to obtain\ninverse systems, by restriction, of the spaces\n$\\mathcal{A}(\\pi_{\\gamma_n})$ of coefficients of the representations $\\pi_{\\gamma_n}$\\,.\nThen we combine density of $\\mathcal{A}(\\pi_{\\gamma_n})$ in\n$\\mathcal{H}_{\\pi_{\\gamma_n}} \\widehat{\\otimes} \\mathcal{H}_{\\pi_{\\gamma_n}}^*$ with the\nrenormalization method of \\cite{W2010} to construct inverse systems,\nin the Hilbert space category, of the\n$\\mathcal{H}_{\\pi_{\\gamma_n}} \\widehat{\\otimes} \\mathcal{H}_{\\pi_{\\gamma_n}}^*$\\,.  These mirror the\ninverse systems of the $\\mathcal{A}(\\pi_{\\gamma_n})$, resulting in an\ninterpretation of the function space $\\mathcal{A}(\\pi_\\gamma) = \n\\varprojlim \\mathcal{A}(\\pi_{\\gamma_n})$ as\na dense subspace of the Hilbert space $\\mathcal{H}_{\\pi_\\gamma}\n\\widehat{\\otimes} \\mathcal{H}_{\\pi_\\gamma}^* =\n\\varprojlim \\mathcal{H}_{\\pi_{\\gamma_n}} \\widehat{\\otimes} \\mathcal{H}_{\\pi_{\\gamma_n}}^*$\\,.\nThis is somewhat analogous to the infinite dimensional\nPeter---Weyl Theorem of \\cite[Section 4]{W2009}.\n\\medskip\n\nIn Section \\ref{sec5} we set up the Schwartz space machinery that will allow us \nto carry over the somewhat abstract $\\mathcal{H}_{\\pi_\\gamma}\n\\widehat{\\otimes} \\mathcal{H}_{\\pi_\\gamma}^* =\n\\varprojlim \\mathcal{H}_{\\pi_{\\gamma_n}} \\widehat{\\otimes} \\mathcal{H}_{\\pi_{\\gamma_n}}^*$\nto an explicit Fourier inversion formulae.  This, incidentally, strengthens\nthe stepwise $L^2$ property for coefficients involving $C^\\infty$ vectors\nfrom $L^2$ to $L^1$.\n\\medskip\n\nIn Section \\ref{sec6} we work out that formula for the direct limit\ngroup $N = \\varinjlim N_n$\\,.  See Theorem \\ref{limit-inversion}.\n\\medskip\n\nIn Section \\ref{sec7} we discuss direct systems $\\{G_n , \\varphi_{m,n}\\}$\nof finite dimensional real reductive Lie groups, and conditions on their\nrestricted root systems $\\Delta(\\mathfrak{g}_n,\\mathfrak{a}_n)$, that lead to an appropriate \nlimit restricted root system\n$\\Delta(\\mathfrak{g},\\mathfrak{a}) = \\varprojlim \\Delta(\\mathfrak{g}_n,\\mathfrak{a}_n)$ of the Lie\nalgebra of $G = \\varinjlim \\{G_n , \\varphi_{m,n}\\}$.  That describes the\nstepwise square integrable structure of the nilradicals of minimal parabolic\nsubgroups.  \n\\medskip\n\nFinally, in Section \\ref{sec8}, we arrive at the goal of this paper,\nTheorem \\ref{inversion-for-ss-limits},\nan explicit Fourier inversion formula for the classical \ndirect limit of the nilradicals of those minimal parabolics. \n\\medskip\n\nI thank Michael Christ for useful discussions of Schwartz spaces related\nto the Heisenberg group.\n\n\\section{Alignment and Construction}\n\\label{sec2}\n\\setcounter{equation}{0}\nFor our direct limit considerations it will be necessary to adjust the\ndecompositions (\\ref{setup}) of the connected simply connected nilpotent\nLie groups $N_n$\\,.  This is so that the adjusted decompositions will\nfit together as $n$ increases.  We do that by reversing the indices and\nkeeping the $L_r$ constant as $n$ goes to infinity.  First, we suppose that\n\\begin{equation}\\label{nil-direct-system}\n\\{N_n\\} \\text{ is a strict direct system of connected simply connected \n\tnilpotent Lie groups,}\n\\end{equation}\nin other words  the connected simply connected nilpotent Lie groups\n$N_n$ have the property that $N_n$ is a closed analytic subgroup of $N_\\ell$\nfor all $\\ell \\geqq n$.  As usual, $Z_r$ denotes the center of $L_r$\\,.\n\t\tFor each $n$, we require that\n\\begin{equation}\\label{newsetup}\n\\begin{aligned}\nN_n = L_1&L_2\\cdots L_{m_n} \\text{ where }\\\\\n \\text{(a) }&L_r \\text{ is a closed analytic subgroup of } N_n \\text{ for }\n\t1 \\leqq r \\leqq m_n\\,; \\\\\n \\text{(b) }&\\text{each factor $L_r$ has unitary representations with \n\tcoefficients in $L^2(L_r\/Z_r)$;} \\\\\n \\text{(x) }&\\text{let $L_{p,q} = L_{p+1}L_{p+2}\\cdots L_q$  ($p < q$)\n\tand $N_{\\ell,n} = L_{m_\\ell +1}L_{m_\\ell +2}\\cdots L_{m_n}\n\t= L_{m_\\ell,m_n}$ ($\\ell < n$);}\\\\\n \\text{(c) }&\\text{each } N_{\\ell ,n} \n\t\\text{ is a normal subgroup of } N_n \n   \\text{ and } N_n = N_r \\ltimes N_{r+1,n} \\text{ semidirect product}; \\\\\n \\text{(d) }&\\text{decompose }\\mathfrak{l}_r = \\mathfrak{z}_r + \\mathfrak{v}_r \\text{ and } \n\t\\mathfrak{n}_n = \\mathfrak{s}_n + \\mathfrak{u}_n\n        \\text{ as vector space direct sums where } \\\\\n &\\mathfrak{s}_n = {\\bigoplus}_{r \\leqq m_n}\\, \\mathfrak{z}_r \\text{ and } \n\t\\mathfrak{u}_n = {\\bigoplus}_{r \\leqq m_n}\\, \\mathfrak{v}_r;\n    \\text{ then } [\\mathfrak{l}_r,\\mathfrak{z}_s] = 0 \\text{ and } [\\mathfrak{l}_r,\\mathfrak{l}_s] \\subset \\mathfrak{v}\n        \\text{ for } r < s\\,.\n\\end{aligned}\n\\end{equation}\nWith this setup we can follow the lines of the constructions in\n\\cite[Section 5]{W2013}.\n\\medskip\n\nWe have the Pfaffian polynomials on the $\\mathfrak{z}_r^*$ and on $\\mathfrak{s}_n^*$\nas follows.  Given $\\lambda_r \\in \\mathfrak{z}_r^*$,\nextended to an element of $\\mathfrak{l}_r^*$ by $\\lambda_r(\\mathfrak{v}_r) = 0$,\nwe have the antisymmetric bilinear form $b_{\\lambda_r}$ on $\\mathfrak{l}_r\/\\mathfrak{z}_r$\ndefined as usual by $b_{\\lambda_r}(x,y) = \\lambda_r([x,y])$, and\n${\\rm Pf}\\,_r(\\lambda_r)$ denotes its Pfaffian.  If $\\gamma_n = \\lambda_1 +\n\\cdots + \\lambda_{m_n} \\in \\mathfrak{s}_n^*$ with each $\\lambda_r \\in \\mathfrak{z}_r^*$, then\nwe have the product\n\\begin{equation}\nP_n(\\gamma_n) = {\\rm Pf}\\,_1(\\lambda_1){\\rm Pf}\\,_2(\\lambda_2) \\cdots {\\rm Pf}\\,_{m_n}(\\lambda_{m_n})\n\\end{equation}\nand the nonsingular set\n\\begin{equation}\n\\mathfrak{t}_n^* = \\{\\gamma_n \\in \\mathfrak{s}_n^* \\mid P_n(\\gamma_n) \\ne 0\\}.\n\\end{equation}\n\nRecall the construction (\\cite{W2013}) of stepwise square integrable \nrepresentations $\\pi_{\\gamma_n}$ of $N_n$\\,, where $\\gamma_n \\in \\mathfrak{t}_n^*$\\,,\nand where we adjust the indices to our situation.\nIf $m_n = 1$ then $\\pi_{\\gamma_n}$ is just the square integrable representation\n$\\pi_{\\lambda_1}$ of $L_1$ defined by $\\gamma_n = \\lambda_1$\\,.  Now let\n$m_n > 1$ and use $N_n = (L_1L_2\\dots L_{m_n-1}) \\ltimes L_{m_n}\n= L_{0,m_n-1} \\ltimes L_{m_n}$\\,.\nBy induction on $m_n$ we have the stepwise square integrable representation \n$\\pi_{\\lambda_1 + \\cdots + \\lambda_{m_n-1}}$ of $L_{0,m_n-1}$\\,, and we view it\nas a representation of $N_n$ whose kernel contains $L_{m_n}$\\,.  We also\nhave the square integrable representation $\\pi_{\\lambda_{m_n}}$ of $L_{m_n}$\\,.\nWrite $\\pi'_{\\lambda_{m_n}}$ for the extension of $\\pi_{\\lambda_{m_n}}$ to\na unitary representation of $N_n$ on the same Hilbert space \n$\\mathcal{H}_{\\pi_{\\lambda_{m_n}}}$ (the Mackey obstruction vanishes).  Now\n\\begin{equation}\\label{def-sq-reps}\n\\pi_{\\gamma_n} = \\pi_{\\lambda_1 + \\cdots + \\lambda_{m_n-1}} \\widehat{\\otimes}\n        \\pi'_{\\lambda_{m_n}}\\,.\n\\end{equation}\n\\medskip\n\nThe parameter space for our representations of the direct limit Lie group\n$N = \\varinjlim N_n$ will be\n\\begin{equation}\n\\mathfrak{t}^* = \\bigcup_{n > 0} \\left\\{\\gamma = \\sum \\gamma_\\ell \\in \\mathfrak{s}^* \\mid\n\t\\gamma_\\ell \\in \\mathfrak{t}_\\ell^* \\text{ for } \\ell \\leqq n\n\t\\text{ and } \\gamma_\\ell = 0 \\in \\mathfrak{s}_\\ell^*\\text{ for } \n\t\\ell > n\\right\\}\n\\text{ where } \\mathfrak{s}^* := \\sum_{\\ell > 0} \\mathfrak{s}_\\ell^* \\,.\n\\end{equation}\nThe representations $\\pi_\\gamma$ of $N$ are defined in a manner similar \nto that of (\\ref{def-sq-reps}).  Given $\\gamma = \\sum \\gamma_\\ell \\in \\mathfrak{t}^*$ \nwe have the index $n = n(\\gamma)$ defined by\n$\\gamma_\\ell \\in \\mathfrak{t}_\\ell^* \\text{ for } \\ell \\leqq n(\\gamma)\n        \\text{ and } \\gamma_\\ell = 0 \\in \\mathfrak{s}_\\ell^*\\text{ for }\n        \\ell > n(\\gamma)$.\n\\begin{equation}\nN = N_{n(\\gamma)} \\ltimes N_{n(\\gamma),\\infty} \\text{ semidirect product, where }\n\tN_{n(\\gamma),\\infty} = {\\prod}_{q > m_{n(\\gamma)}} L_q\\,.\n\\end{equation}\nIn particular the closed normal subgroup $N_{n(\\gamma),\\infty}$ satisfies\n$N_{n(\\gamma)} \\cong N\/N_{n(\\gamma),\\infty}$, and we denote\n\\begin{equation} \\label{def-sq-lim-reps}\n\\pi_\\gamma \\text{ is the lift to $N$ of the stepwise square integrable\nrepresentation } \\pi_{\\gamma_1 + \\cdots + \\gamma_{m_{n(\\gamma)}}} \\text{ of } \nN_{n(\\gamma)}\\,.\n\\end{equation}\nThe representation space of $\\pi_\\gamma$ is the projective (jointly continuous)\ntensor product\n\\begin{equation}\\label{gamma-decomp}\n\\mathcal{H}_{\\pi_\\gamma} = \\mathcal{H}_{\\pi_{\\gamma_1}} \\widehat{\\otimes} \n\t\\mathcal{H}_{\\pi_{\\gamma_2}}\n\t\\widehat{\\otimes} \\cdots \\widehat{\\otimes} \n\t\\mathcal{H}_{\\pi_{\\gamma_{n(\\gamma)}}}\n\\end{equation}\n\nThese representations $\\pi_\\gamma$ are the {\\sl limit stepwise square \nintegrable} representations of $N$.  We go on the see the extent to which\ntheir coefficients and characters imitate the properties of Theorem\n\\ref{plancherel-general}.\n\n\\section{Coefficient Functions}\n\\label{sec3}\n\\setcounter{equation}{0}\nLet $\\mathcal{H}_{\\pi_\\gamma}$ denote the representation space of $\\pi_\\gamma$ and\n$\\langle\\,\\,\\cdot\\, , \\,\\cdot\\,\\,\\rangle_{\\pi_\\gamma}$ the hermitian inner \nproduct on $\\mathcal{H}_{\\pi_\\gamma}$\\,.  Given $u, v \\in \\mathcal{H}_{\\pi_\\gamma}$ we have the\ncoefficient function on $N$ given by\n\\begin{equation}\nf_{\\pi_\\gamma,u,v}(g) = \\langle u, \\pi_\\gamma(g)v\\rangle_{\\pi_\\gamma}\\,.\n\\end{equation}\nWe use the standard $(r(x)f)(g) = f(gx)$ and $(\\ell(y)f)(g) = f(y^{-1}g)$.  \nThese right and left translations commute with each other.  They are \nwell defined on the $f_{\\pi_\\gamma,u,v}$ and satisfy\n\\begin{equation}\n\\ell(x) r(y): f_{\\pi_\\gamma,u,v} \\mapsto f_{\\pi_\\gamma, \\pi_\\gamma(x)u,\n\t\\pi_\\gamma(y)v}\\,.\n\\end{equation}\nBy our construction (\\ref{def-sq-lim-reps}), the value \n$f_{\\pi_\\gamma,u,v}(g)$ depends only on the coset $gN''_{n(\\gamma)}$.  In\nother words it really is a function on $N_{n(\\gamma)} \\cong N\/N''_{n(\\gamma)}$. \nFurther, $|f_{\\pi_\\gamma,u,v}(g)|$ depends only on the coset \n$gS_{n(\\gamma)}N''_{n(\\gamma)}$ where $S_{n(\\gamma)}$ is the quasicenter\n$Z_1Z_2\\cdots Z_{m_{n(\\gamma)}}$\nof $N_{n(\\gamma)} = L_1L_2\\cdots L_{m_{n(\\gamma)}}$\\,.\nBuilding on (\\ref{sq-orthogrel}), we have the following variation on the \nFrobenius-Schur orthogonality relations for finite groups:\n\\begin{proposition}\\label{sq-orthogrel-quo}\nLet $\\gamma \\in \\mathfrak{t}^*$ and $n = n(\\gamma)$.  Then\n$||f_{\\pi_\\gamma,u,v}||^2_{L^2(N \/ S_n N''_n)} \n\t= \\frac{||u||_{\\pi_\\gamma}^2||v||_{\\pi_\\gamma}^2}{|P_n(\\gamma)|}$\\,\\,.\n\\end{proposition}\n\n\\begin{proof}\nThis is an induction on $n$.  The case $n = 1$ is (\\ref{sq-orthogrel}).\nNow go from $n$ to $n+1$.  Express $N_{n+1} = N_n \\ltimes N_{n,n+1}$ where\n$$\nN_n = L_1L_2\\cdots L_{m_n} \\text{ and }\n\tN_{n,q} = L_{m_n+1}L_{m_n+2}\\cdots L_{m_q}\n\t\\text{ for } q > n.\n$$\nThen $S_{n+1} = S_n \\times S_{n,n+1}$ where the quasi-centers\n$$\nS_n = Z_1Z_2\\cdots Z_{m_n} \\text{ and }\n\tS_{n,q} = Z_{m_n+1}Z_{m_n+2}\\cdots Z_{m_q}\n\t\\text{ for } q > n.\n$$\nNow let $\\gamma_n \\in \\mathfrak{t}_n^*$ and $\\gamma_{n,n+1} \\in \\mathfrak{t}_{n,n+1}^*$\nwhere, as before, $\\mathfrak{t}^*$ is the nonzero set of the Pfaffian in $\\mathfrak{s}^*$.\nNote that $\\pi_{\\gamma_n} \\in \\widehat{N_n}$ and $\\pi_{\\gamma_{n,n+1}}\n\\in \\widehat{N_{n,n+1}}$ are stepwise square integrable.  Write\n$\\pi'_{\\gamma_{n,n+1}}$ for the extension of $\\pi_{\\gamma_{n,n+1}}$ from\n$N_{n,n+1}$ to $N_{n+1}$\\,.  Let\n$u,v \\in \\mathcal{H}_{\\pi_{\\gamma_n}}$ and $x,y \\in \\mathcal{H}_{\\pi_{\\gamma_{n,n+1}}}$\nso $u\\otimes x, v \\otimes y \\in \\mathcal{H}_{\\pi_{\\gamma_{n+1}}}$\\,.  Let\n$a$ run over $N_n$ and let $b$ run over $N_{n,n+1}$\\,.  Compute\n$$\n\\begin{aligned}\n&||f_{\\pi_{\\gamma_{n+1}},u\\otimes x,v\\otimes y}||^2_{L^2(N_{n+1}\/S_{n+1})} \\\\\n&\\phantom{XX}= \\int_{N_{n+1}\/S_{n+1}} |\\langle u\\otimes x, (\\pi_{\\gamma_n}\n\t\\widehat{\\otimes} \\pi'_{\\gamma_{n,n+1}})(ab)(v\\otimes y)\\rangle|^2\n\tda\\,db\\\\\n&\\phantom{XX}= \\int_{N_{n+1}\/S_{n+1}} |\\langle u\\otimes x,\n\t\\pi_{\\gamma_n}(a)\\pi'_{\\gamma_{n,n+1}}(b)(v) \\otimes\n\t\\pi_{\\gamma_{n,n+1}}(b)(y)\\rangle|^2 da\\,db \\\\\n&\\phantom{XX}= \\int_{N_{n,n+1}\/S_{n,n+1}} \\left ( \\int_{N_n\/S_n}\n\t|\\langle u, \\pi_{\\gamma_n}(a)\\pi'_{\\gamma_{n,n+1}}(b)(v)\\rangle |^2\n\tda \\right )\n\t\\cdot |\\langle x, \\pi_{\\gamma_{n,n+1}}(b)(y)\\rangle|^2 db \\\\\n&\\phantom{XX}= \\int_{N_{n,n+1}\/S_{n,n+1}} \n\t\\frac{||u||^2 ||\\pi'_{\\gamma_{n,n+1}}(b)(v)||^2}{|{\\rm Pf}\\,_n(\\gamma_n)|}\n\t|\\langle x, \\pi_{\\gamma_{n,n+1}}(b)(y)\\rangle|^2 db \\\\\n&\\phantom{XX}= \\int_{N_{n,n+1}\/S_{n,n+1}} \n        \\frac{||u||^2 ||v||^2}{|{\\rm Pf}\\,_n(\\gamma_n)|}\n        |\\langle x, \\pi_{\\gamma_{n,n+1}}(b)(y)\\rangle|^2 db \\\\\n&\\phantom{XX}= \\frac{||u||^2 ||v||^2}{|{\\rm Pf}\\,_n(\\gamma_n)|}\\cdot\n\t\\frac{||x||^2 ||y||^2}{|{\\rm Pf}\\,_{n,n+1}(\\gamma_{n,n+1})|} = \n\t\\frac{||u\\otimes x||^2 ||v \\otimes y||^2}{|{\\rm Pf}\\,_{n+1}(\\gamma_{n+1})|}.\n\\end{aligned}\n$$\nThe proposition follows.\n\\end{proof}\n\nIn the notation of the proof of Proposition \\ref{sq-orthogrel-quo},\n\\begin{equation}\\label{restr}\nf_{\\pi_{\\gamma_{n+1}},u\\otimes x,v\\otimes y}(a) \n        = \\langle u, \\pi_{\\gamma_n}(a)v\\rangle\\cdot \\langle x,y\\rangle\n        = \\langle x,y\\rangle f_{\\pi_{\\gamma_n},u,v}(a)\n        \\text{ for } a \\in N_n\\,.\n\\end{equation}\nIn other words, $f_{\\pi_{\\gamma_{n+1}},u\\otimes x,v\\otimes y}|_{N_n}\n= \\langle x,y\\rangle f_{\\pi_{\\gamma_n},u,v}$\\,.  In particular the case\nwhere $x = e = y$, where $e$ is a unit vector, is\n\\begin{equation}\\label{restr1}\nf_{\\pi_{\\gamma_{n+1}},u\\otimes e,v\\otimes e}|_{N_n} = f_{\\pi_{\\gamma_n},u,v}\n\\end{equation}\n\\medskip\n\nIterating this and combining it with Proposition \\ref{sq-orthogrel-quo}\nwe arrive at\n\n\\begin{proposition}\\label{rescale}\nLet $\\gamma \\in \\mathfrak{t}^*$ and $n = n(\\gamma)$.  Let $\\gamma' \\in \\mathfrak{t}^*$\nand $n' = n(\\gamma')$ with $n' > n$ and $\\gamma'|_{\\mathfrak{s}_n} = \\gamma$.  Then \n$\\pi_{\\gamma'}|_{N_n}$ is an infinite multiple of $\\pi_\\gamma$.\nSplit $\\mathcal{H}_{\\pi_{\\gamma'}} = \\mathcal{H}_{\\pi_\\gamma} \\widehat{\\otimes}\\mathcal{H}''$ where\n$\\mathcal{H}'' = \\mathcal{H}_{\\pi_{\\gamma'_{n+1}}} \\widehat{\\otimes} \\cdots \\widehat{\\otimes}\n\\mathcal{H}_{\\pi_{\\gamma'_{n'}}}$ in the notation of {\\rm (\\ref{gamma-decomp})}.  \nChoose a unit vector $e \\in \\mathcal{H}''$.  Then\n\\begin{equation}\\label{unitfactor}\n\\mathcal{H}_{\\pi_{\\gamma}} \\hookrightarrow \\mathcal{H}_{\\pi_{\\gamma'}} \\text{ by }\nv \\mapsto v\\otimes e\n\\end{equation}\nis a well defined $N_n$--equivariant isometric injection.  If\n$u, v \\in \\mathcal{H}_{\\pi_\\gamma}$ then\n\\begin{equation}\\label{rescale1}\n||f_{\\pi_{\\gamma'},u\\otimes e,v\\otimes e}||^2_{L^2(\n\tN \/ S_{n'} N''_{n'})} \n\t= \\frac{|P_n(\\gamma)|}{|P_{n'}(\\gamma')|}\n\t||f_{\\pi_\\gamma,u,v}||^2_{L^2(N \/ S_n N''_n)}\\,.\n\\end{equation}\n\\end{proposition}\n\n\\section{Hilbert Space Limits}\n\\label{sec4}\n\\setcounter{equation}{0}\nNow we combine the restriction maps of Section \\ref{sec3}.\nLet $\\gamma \\in \\mathfrak{t}^*$ and $n = n(\\gamma)$.  Then $\\gamma$ defines a\nunitary character\n\\begin{equation}\\label{defrelchar}\n\\zeta_\\gamma = \\exp(2\\pi i \\gamma) \\text{ by } \\zeta_\\gamma(\\exp(\\xi)y) = \n\te^{2\\pi i \\gamma(\\xi)} \\text{ where } \\xi \\in \\mathfrak{s}_n \\text{ and }\n\ty \\in N''_n\\,.\n\\end{equation}\nThat defines the Hilbert space\n\\begin{equation}\\label{defrelHilbert}\n\\begin{aligned}\nL^2&(N\/S_nN''_n,\\zeta_\\gamma) = \\\\\n&\\{f:N \\to \\mathbb{C} \\mid f(gx) = \\zeta_\\gamma(x)^{-1}\n\tf(g) \\text{ and } |f| \\in L^2(N \/ S_n N''_n) \\text{ for all }\n\tg \\in N \\text{ and } x \\in N''_n\\}.\n\\end{aligned}\n\\end{equation}\nThe finite linear combinations of the coefficients \n$f_{\\pi_\\gamma, u, v}$ (where $u, v \\in \\mathcal{H}_{\\pi_\\gamma}$) form a dense\nsubspace of $L^2(N\/S_nN''_n,\\zeta_n$), and that gives an\n$N \\times N$ equivariant Hilbert space isomorphism\n\\begin{equation}\nL^2(N\/S_nN''_n,\\zeta_\\gamma) \\cong \n\t\\mathcal{H}_{\\pi_\\gamma} \\widehat{\\otimes} \\mathcal{H}_{\\pi_\\gamma}^*.\n\\end{equation}\n\\medskip\n\nWe know that the stepwise square integrable group $N_n = N\/N_n''$ satisfies\n\\begin{equation}\nL^2(N_n) = L^2(N\/N_n'') \n= \\int_{\\gamma \\in \\mathfrak{t}^*\\,\\,  and\\,\\, n(\\gamma) = n}\n   (\\mathcal{H}_{\\pi_\\gamma} \\widehat{\\otimes} \\mathcal{H}_{\\pi_\\gamma}^*) |P_n(\\gamma)|d\\gamma.\n\\end{equation}\nIn brief, that expands the functions on $N$ that depend only on the first\n$m(n)$ factors in $N = N_1N_2N_3\\cdots$\\,.  To expand the functions that\ndepend on more factors, say the first $m(n')$ factors in the notation of\nProposition \\ref{rescale}, we would like to inject\n$$L^2(N\/N_n'') = \n  \\int_{\\gamma \\in \\mathfrak{t}_n^*} L^2(N\/S_nN''_n,\\zeta_\\gamma)|P_n(\\gamma)|d\\gamma\n$$\ninto\n$$ \nL^2(N\/N_{n'}'') = \n  \\int_{\\gamma' \\in \\mathfrak{t}_{n'}^*}L^2(N\/S_{n'}N''_{n'},\\zeta_{\\gamma'})\n\t|P_{n'}(\\gamma')|d\\gamma'\n$$\nusing the renormalizations of (\\ref{rescale1}).  However, $\\gamma$ has\nmany extensions $\\gamma'$ with the given $n(\\gamma') = n'$, so this will\nnot work directly.  But we can take the orthogonal projections dual to\nthe injections of (\\ref{rescale1}) and form an inverse system of Hilbert\nspaces.  \n\\medskip\n\nTo start, if $u, v \\in \\mathcal{H}_{\\pi_\\gamma}$ and $x, y \\in \\mathcal{H}''$, using\n(\\ref{restr}) and Proposition \\ref{rescale}, \n\\begin{equation}\\label{rescale2}\np_{\\gamma',\\gamma}: f_{\\pi_{\\gamma'},u\\otimes x,v\\otimes y} \\mapsto\n\\langle x, y \\rangle \\left |\\frac{P_n(\\gamma)}{P_{n'}(\\gamma')}\\right |^{1\/2} \nf_{\\pi_\\gamma,u,v}\n\\end{equation}\nis the orthogonal projection dual to the isometric inclusion (\\ref{unitfactor}).\nSince $\\gamma$ is the restriction of $\\gamma'$ from $\\mathfrak{s}_{n(\\gamma')}$ to\n$\\mathfrak{s}_{n(\\gamma)}$ we can reformulate (\\ref{rescale2}) as\n\\begin{equation}\\label{rescale3}\np_{\\gamma',n} : f_{\\pi_{\\gamma'},u\\otimes x,v\\otimes y} \\mapsto\n\t\\langle x, y \\rangle \n\t\\left | \\frac{P_n(\\gamma'|_{\\mathfrak{s}_n})}{P_{n'}(\\gamma')} \\right |^{1\/2}\n\tf_{\\pi_{\\gamma'|_{\\mathfrak{s}_n},u,v}} \\text{ where } n = n(\\gamma).\n\\end{equation}\nThe maps $p_{\\gamma,n}$ of (\\ref{rescale3}) sum to a Hilbert space projection,\nessentially restriction of coefficients,\n\\begin{equation}\\label{rescale4}\np_{n',n} = \\left ( \\int_{\\gamma'\\in\\mathfrak{s}^*_{n'}}p_{\\gamma',n}\\,d\\gamma'\\right ) : \n\tL^2(N_{n'}) \\to\n\tL^2(N_n) \\text{ where } n = n(\\gamma'|_{\\mathfrak{s}_n})\n\t\\text{ and } n' = n(\\gamma') \\geqq n.\n\\end{equation}\nThe maps $p_{n',n}$ of (\\ref{rescale4}) define an inverse system in the\ncategory of Hilbert spaces and partial isometries:\n\\begin{equation}\\label{rescale5}\nL^2(N_1) \\overset{p_{2,1}}{\\longleftarrow} L^2(N_2) \n\t\\overset{p_{3,2}}{\\longleftarrow}  L^2(N_3) \n\t\\overset{p_{4,3}}{\\longleftarrow} \\,\\, ... \\,\\,\\,\\longleftarrow\\,\n\tL^2(N)\n\\end{equation}\nwhere the projective limit $L^2(N) := \\varprojlim \\{L^2(N_n),p_{n',n}\\}$ \nis taken in the category\nof Hilbert spaces and partial isometries.  We now have the Hilbert space\n\\begin{equation}\\label{rescale6}\nL^2(N) := \\varprojlim \\{L^2(N_n), p_{n',n}\\}.\n\\end{equation}\n\n\\section{The Schwartz Spaces}\n\\label{sec5}\n\\setcounter{equation}{0}\nIn order to refine (\\ref{rescale6}) to a Fourier\ninversion formula we must first\nmake it more explicit.  The span $\\mathcal{A}(\\pi_{\\gamma_n})$ of the \ncoefficients of the representation $\\pi_{\\gamma_n}$ is dense in\nthe space of functions on $N_n$ given by\n$\\mathcal{H}_{\\pi_{\\gamma_n}} \\widehat{\\otimes} \\mathcal{H}_{\\pi_{\\gamma_n}}$\\,. \nThe idea in the background here is to realize Schwartz class functions\nas wave packets\n$\nf(a) = \\int_{\\mathfrak{s}_n^*} \\varphi(\\gamma_n) f_{\\pi_{\\gamma_n},u(\\gamma_n),\nv(\\gamma_n)}(a)d\\gamma_n\n$\nwhere $\\varphi$ is a Schwartz class function on $\\mathfrak{s}_n$ and where \n$u(\\gamma_n)$ and $v(\\gamma_n)$ are fields of \n$C^\\infty$ unit vectors in the $\\mathcal{H}_{\\pi_{\\gamma_n}}$.\nMore concretely we show that the coefficient\n$f_{\\pi_{\\gamma_n,u,v}}$ belongs to an appropriate Schwartz space (and thus\nan appropriate $L^1$ space) when $u$ and $v$ are $C^\\infty$ vectors for\n$\\pi_{\\gamma_n}$\\,.\n\\medskip\n\nWe first collect some standard facts from Kirillov theory concerning the \nanalog of the Schr\\\" odinger representation of the Heisenberg group.\nLet $L$ be a connected simply connected nilpotent Lie group that has\nsquare integrable representations.  $Z$ is the center of $L$, and\n$\\lambda \\in \\mathfrak{z}^*$ with ${\\rm Pf}\\,_\\mathfrak{l}(\\lambda) \\ne 0$.  Let $\\mathfrak{p}$ and $\\mathfrak{q}$ \nbe totally real polarizations for $\\lambda$, $\\mathfrak{p} = \\mathfrak{z} + \\mathfrak{a}$ and\n$\\mathfrak{q} = \\mathfrak{z} + \\mathfrak{b}$, and suppose that we chose them so that \n$b_\\lambda(x,y) = \\lambda([x,y])$ gives  a nondegenerate pairing of\n$\\mathfrak{a}$ with $\\mathfrak{b}$.  In this setting, the square integrable representation\n$\\pi_\\lambda$ of $L$ is ${\\rm Ind\\,}_P^N(\\exp(2\\pi i \\lambda)$, and it\nrepresents $L$ on $L^2(N\/P) = L^2(B)$.  Further, here $\\pi_\\lambda$\nmaps the universal enveloping algebra $\\mathcal{U}(\\mathfrak{l})$ onto the set of all\npolynomial (in linear coordinates from $\\exp: \\mathfrak{b} \\to B$) differential operators\non $B$.  In particular,\n\\begin{lemma}\\label{sch-vectors-sqint}\nThe $C^\\infty$ vectors for the representation $\\pi_\\lambda$ are the\nSchwartz class functions on $B$.  In other words if $p$ and $q$\nare polynomials on $B$\\,, if $D$ is a constant coefficient differential\noperator on $B$\\,, and if $u: B \\to \\mathbb{C}$ is a $C^\\infty$ vector\nfor $\\pi_\\lambda$\\,, then $|q(x)p(D)u|$ is bounded.\n\\end{lemma}\n\nIn order to extend this to stepwise square integrable representations we\nmust take into account the problem that $S_n$ need not be central in $N_n$\\,.\nWe do this by decomposing \n\\begin{equation}\\label{decomp_n_n}\nN_n \\simeq L_1 \\times \\cdots \\times L_{m(n)}\n\\end{equation}\nwhere $\\simeq$ is the measure preserving real analytic diffeomorphism given\nby the polynomial map\n\\begin{equation}\\label{decomp_exp_n}\n\\exp':\\mathfrak{n}_n \\to N_n \\text{ by } \\exp'(\\xi_1 + \\dots + \\xi_{m(n)})\n        = \\exp(\\xi_1)\\exp(\\xi_2)\\cdots\\exp(\\xi_{m(n)})\n\\text{ where each } \\xi_r \\in \\mathfrak{l}_r\\,.\n\\end{equation}\nUsing the part of (\\ref{newsetup}d) that says $[\\mathfrak{l}_r,\\mathfrak{z}_s] = 0$ for $r < s$\nthe decomposition (\\ref{decomp_n_n}) gives us\n\\begin{equation}\\label{decomp_n_n_mod_s}\n\\begin{aligned}\nN_n\/S_n &= \\{x_{m(n)}\\cdots x_2x_1Z_{m(n)}\\cdots Z_2Z_1\\mid x_r \\in L_r\\}\\\\\n&= \\{x_{m(n)}Z_{m(n)} \\cdots x_2Z_2x_1Z_1\\mid x_r \\in L_r\\}\n= (L_{m(n)}\/Z_{m(n)}) \\times \\cdots \\times (L_1\/S_1) \\\\\n&\\simeq (L_1\/S_1) \\times \\cdots \\times (L_{m(n)}\/Z_{m(n)}).\n\\end{aligned}\n\\end{equation}\nNow let $\\gamma_n = \\lambda_1 + \\cdots + \\lambda_{m(n)} \\in \\mathfrak{t}^*_n$\\,.\nLet $\\mathfrak{p}_r$ and $\\mathfrak{q}_r$ be totally real polarizations on $\\mathfrak{l}_r$ for\n$\\lambda_r$\\,, paired as above by $b_{\\lambda_r}$\\,.  We do not claim\nthat $\\mathfrak{p} = \\sum \\mathfrak{p}_r$ and $\\mathfrak{q} = \\sum \\mathfrak{q}_r$ are polarizations on $\\mathfrak{n}_n$\nfor $\\gamma_n$\\, (we don't know that they are algebras), but still\n$\\mathfrak{p}_r = \\mathfrak{z}_r + \\mathfrak{a}_r$ and $\\mathfrak{q}_r = \\mathfrak{z}_r + \\mathfrak{b}_r$ where $b_{\\lambda_r}$ \npairs $\\mathfrak{a}_r$ with $\\mathfrak{b}_r$\\,, so $b_{\\gamma_n}$ is\na nondegenerate pairing of $\\mathfrak{a} = \\sum \\mathfrak{a}_r$ with $\\mathfrak{b} = \\sum \\mathfrak{b}_r$\\,.\nNow the stepwise square integrable representation $\\pi_{\\gamma_n}$ of $N_n$\nis realized on $L^2(B)$ where $B = \\exp'(\\mathfrak{b})$ in the notation of\n(\\ref{decomp_exp_n}).  Again, in this setting, $\\pi_{\\gamma_n}$\nmaps the universal enveloping algebra of $\\mathfrak{n}_n$ onto the set of all\npolynomial (in linear coordinates from $\\exp': \\mathfrak{b} \\to B$) \ndifferential operators on $B$.  This extends Lemma \\ref{sch-vectors-sqint}\nto\n\\begin{lemma}\\label{sch-vectors}\nIdentify $B = \\exp'(\\mathfrak{b})$ with the real vector space $\\mathfrak{b}$.\nThe $C^\\infty$ vectors for the representation $\\pi_{\\gamma_n}$ are the\nSchwartz class functions on $B$.  In other words if $p$ and $q$\nare polynomials on $B$\\,, if $D$ is a constant coefficient differential\noperator on $B$\\,, and if $u: B \\to \\mathbb{C}$ is a $C^\\infty$ vector\nfor $\\pi_{\\gamma_n}$\\,, then $|q(x)p(D)u|$ is bounded.\n\\end{lemma}\n\nNow consider the Schwartz space analog of the definition (\\ref{defrelHilbert}).\nWe define the {\\em relative Schwartz space} $\\mathcal{C}(N\/S_nN''_n,\\zeta_\\gamma)\n= \\mathcal{C}(N_n\/S_n,\\zeta_{\\gamma_n})$ to be\n\\begin{equation}\\label{defrelSchwartz}\n\\begin{aligned}\n\\text{ all } f\\in &C^\\infty(N) \\text{ such that}\\\\\n\t& f(xs) = \\zeta_\\gamma(s)^{-1} f(x) \\text{ for all }\n\t\tx \\in N_n \\text{ and } s \\in S_n\\,,\\text{ and } \\\\\n\t& |q(x)p(D)f| \\text{ is bounded for all polynomials } \n        p,q \\text{ on } N_n\/S_n \\text{ and all } D \\in \\mathcal{U}(\\mathfrak{n}_n).\n\\end{aligned}\n\\end{equation}\nIt is a nuclear Fr\\' echet space and is dense in $L^2(N\/S_nN''_n,\\zeta_\\gamma)\n= L^2(N_n\/S_n,\\zeta_{\\gamma_n})$.  \n\\medskip\n\nWe define \n$C_c^\\infty(N\/S_nN''_n,\\zeta_\\gamma)\n= C_c^\\infty(N_n\/S_n,\\zeta_{\\gamma_n})$ as the space of functions\n$f \\in C^\\infty(N)$ such that $f(xs) = \\zeta_\\gamma(s)^{-1} f(x)$ for all\n$x \\in N_n \\text{ and } s \\in S_n$\\,, whose absolute values are compactly\nsupported modulo $S_nN''_n$ for $C_c^\\infty(N\/S_nN''_n,\\zeta_\\gamma)$,\nmodulo $S_n$ for $C_c^\\infty(N_n\/S_n,\\zeta_{\\gamma_n})$. It is\ndense in the corresponding Schwartz space.  Thus we have the expected\ncontinuous inclusions $C_c^\\infty \\hookrightarrow \\mathcal{C} \\hookrightarrow L^2$\nwith dense images.\n\n\\begin{theorem}\\label{coef-sch}\nLet $u$ and $v$ be $C^\\infty$ vectors for the stepwise square integrable\nrepresentation $\\pi_{\\gamma_n}$ of $N_n$\\,.  Define $\\zeta_\\gamma$ and \n$\\zeta_{\\gamma_n}$\nas in {\\rm (\\ref{defrelchar})}, and $A = \\exp'(\\mathfrak{a})$ and $B = \\exp'(\\mathfrak{b})$\nas in the discussion following {\\rm(\\ref{decomp_n_n_mod_s})}.  \nThen the coefficient function\n$f_{\\pi_{\\gamma_n},u,v}$ belongs to the relative Schwartz space\n$\\mathcal{C}(N\/S_nN''_n,\\zeta_\\gamma) = \\mathcal{C}(N_n\/S_n,\\zeta_{\\gamma_n})$\\,.\n\\end{theorem}\n\n\\begin{proof}\nWrite $f_{u,v}$ for $f_{\\pi_{\\gamma_n},u,v}$ and $\\pi$ for \n$\\pi_{\\gamma_n}$\\,.  So $f_{u,v}(x) = \\langle u, \\pi(x)v\\rangle$.\nThe left\/right action of the enveloping algebra is $Df_{u,v}E = \nf_{\\pi(D)u,\\pi(E)v}$.  View $u \\in \\mathcal{C}(A)$ and $v \\in \\mathcal{C}(B)$.\nHere $\\pi(D)u$ is the image of $u$ under the (arbitrary) polynomial\ndifferential operator $\\pi(D)$ on $A$ and $\\pi(E)v$ is the\nimage of $v$ under the (arbitrary) polynomial differential operator\n$\\pi(D)$ on $B$.  Together they give the image of $f_{u,v}$ under\nthe polynomial differential operator $\\pi(D) \\otimes \\pi(E)$\non $A \\times B = N_n\/S_n$\\,.  Every polynomial differential operator\non $A \\times B$ is a finite sum of such operators $\\pi(D) \\otimes \\pi(E)$.\nSince coefficients are bounded, here\n$|f_{\\pi(D)u,\\pi(E)v}(x)| \\leqq ||\\pi(D)u||\\cdot ||\\pi(E)v||$, and\nsince $f_{\\pi(D)u,\\pi(E)v}(xs) = \\zeta(s)^{-1}f_{\\pi(D)u,\\pi(E)v}(x)$,\nthe coefficient $f_{u,v} \\in \\mathcal{C}(N_n\/S_n,\\zeta_{\\gamma_n})$.\n\\end{proof}\n\n\\begin{corollary}\\label{l1-coef1}\nLet $u$ and $v$ be $C^\\infty$ vectors for the stepwise square integrable\nrepresentation $\\pi_{\\gamma_n}$ of $N_n$\\,.  Then the coefficient function\n$f_{\\pi_{\\gamma_n},u,v} \\in L^1(N_n\/S_n,\\zeta_{\\gamma_n})$\\,.\n\\end{corollary}\n\n\\begin{corollary}\\label{l1-coef2}\nLet $L$ be a connected simply connected nilpotent Lie group, $Z$ its center, \nand $\\pi$\na square integrable representation of $L$.  Let $\\zeta \\in \\widehat{Z}$\nsuch that $\\pi|_Z$ is a multiple of $\\zeta$.  Let $u$ and $v$ be $C^\\infty$\nvectors for $\\pi$.  Then the coefficient $f_{\\pi,u,v} \\in\nL^1(L\/Z,\\zeta)$.\n\\end{corollary}\n\nAny norm $|\\xi|$ on $\\mathfrak{n}_n$ carries over to a norm $|\\exp(\\xi)| := |\\xi|$\non $N_n$.  We have the standard Schwartz space $\\mathcal{C}(N_n)$, given by the \nseminorms $\\nu_{k,D,E}(f) = \\sup_{x \\in N_n} |(1 + |x|^2)^k (DfE)(x)|$\nwhere $k$ is a positive integer and $D, E \\in \\mathcal{U}(\\mathfrak{n}_n)$ acting on the\nleft and right.  Since $\\exp: \\mathfrak{n}_n \\to N_n$ is a polynomial diffeomorphism\nit gives a topological isomorphism of $\\mathcal{C}(N_n)$ onto the classical\nSchwartz space $\\mathcal{C}(\\mathfrak{n}_n)$.  Fourier transform and\ninverse Fourier transform of Schwartz class functions on $S_n$.\n\n\\begin{remark}\\label{make-relative}\n{\\rm If $\\gamma_n \\in \\mathfrak{s}_n^*$ and $f \\in \\mathcal{C}(N_n)$ define\n$f_{\\gamma_n}(x) = \\int_{S_n} f(xs)\\zeta_{\\gamma_n}(s)ds$.  Then\n$f_{\\gamma_n} \\in \\mathcal{C}(N_n\/S_n,\\zeta_{\\gamma_n})$.\nLet $z \\in S_n$\\,.  Since $S_n$ is commutative, \n$$f_{\\gamma_n}(xz) = \\int_{S_n} f(xzs)\\zeta_{\\gamma_n}(s)ds\n= \\int_{S_n} f(xsz)\\zeta_{\\gamma_n}(s)ds\n= \\int_{S_n} f(xs)\\zeta_{\\gamma_n}(z^{-1}s)ds\n= \\zeta_{\\gamma_n}(z)^{-1}f_{\\gamma_n}(x).\n$$\nGiven $x \\in N_n$ we view $f_{\\gamma_n}(x)$ as a function on $\\mathfrak{s}_n^*$\nby $\\varphi_x(\\gamma_n) := f_{\\gamma_n}(x)$.  Note that $\\varphi_x$ is\n(a multiple of) the Fourier transform of the left translate\n$(\\ell(x^{-1})f)|_{S_n}$\\,, say $\\mathcal{F}_{S_n}(\\ell(x^{-1})f)|_{S_n}$.  \nThe inverse Fourier $\\mathcal{F}_{S_n}^{-1}(\\varphi_x)$ transform reconstructs\n$f$ from the $f_{\\gamma_n}$\\,.   Each of the $f_{\\gamma_n}$ is a limit\n(in $\\mathcal{C}(N_n\/S_n,\\zeta_{\\gamma_n})$) of finite linear combinations of\ncoefficient functions $f_{\\pi_{\\gamma_n}, u,v}$\\,.  Thus every\n$f \\in \\mathcal{C}(N_n)$ is approximated (in $\\mathcal{C}(N_n)$) by wave packets of\ncoefficient functions of stepwise square integrable representations. }\n\\hfill $\\diamondsuit$\n\\end{remark}\n\nProceeding as in Section \\ref{sec4} let $n' \\geqq n$ and consider\n$\\gamma_{n'} \\in \\mathfrak{t}^*_{n'}$ with $\\gamma_{n'}|_{\\mathfrak{s}_n} = \\gamma_n$.  \nFor brevity write $\\gamma = \\gamma_n$ and $\\gamma' = \\gamma_{n'}$\\,.\nWe reformulate\n(\\ref{rescale4}) through (\\ref{rescale6}) for the Schwartz spaces.\n\\begin{equation}\\label{rescale4s}\nq_{n',n}: \\mathcal{C}(N_{n'}) \\to \\mathcal{C}(N_n) \\text{ by } f \\mapsto f|_{N_n}\\,. \n\\end{equation}\nThe maps $q_{n',n}$ of (\\ref{rescale4s}) define an inverse system in the\ncategory of complete locally convex topological vector spaces\n\\begin{equation}\\label{rescale5s}\n\\mathcal{C}(N_1) \\overset{q_{2,1}}{\\longleftarrow} \\mathcal{C}(N_2) \n        \\overset{q_{3,2}}{\\longleftarrow}  \\mathcal{C}(N_3) \n        \\overset{q_{4,3}}{\\longleftarrow}  ... \n\\end{equation}\nWe define the projective limit\n\\begin{equation}\\label{rescale6s}\n\\mathcal{C}(N) := \\varprojlim \\{\\mathcal{C}(N_n),q_{n',n}\\}\n\\end{equation}\nto be the Schwartz space of $N = \\varinjlim N_n$\\,.  This is dual to the \nconstruction of \\cite[(2.20)]{W2010}.  Now we relate it to (\\ref{rescale6}).\nWe scale the natural injections to maps\n\\begin{equation}\\label{rescalecl1}\nr_{n,\\gamma}: \\mathcal{C}(N_n\/S_n,\\zeta_\\gamma) \\to L^2(N_n\/S_n,\\zeta_\\gamma)\n\\text{ by } f \\mapsto |{\\rm Pf}\\,_{\\mathfrak{n}_n}(\\gamma)|^{1\/2}f\\,.\n\\end{equation}\nThey sum to maps\n\\begin{equation}\\label{rescalecl2}\nr_n = \\left ( \\int_{\\mathfrak{s}_n^*} r_{n,\\gamma}\\, d\\gamma\\right ) : \n\t\\mathcal{C}(N_n) \\to L^2(N_n)\n\\end{equation}\nthat are equivariant for the maps $p_{n',n}$ and $q_{n',n}$\\,.  The \narguments leading to \\cite[Proposition 2.22]{W2010} can be dualized\nfrom direct limits to projective limits.  Thus, dual to\n\\cite[Proposition 2.22]{W2010},\n\\begin{proposition}\\label{compare}\nThe maps $r_n$ of {\\rm (\\ref{rescalecl2})} satisfy\n$p_{n',n}\\cdot r_{n'} = r_n\\cdot q_{n',n}$ for $n' \\geqq n$ and send the\ninverse system $\\{\\mathcal{C}(N_n),q_{n',n}\\}$ into the inverse system\n$\\{L^2(N_n),p_{n',n}\\}$.  That defines a continuous $N$--equivariant injection\n$$\nr: \\mathcal{C}(N) \\to L^2(N)\n$$\nwith dense image.  In particular $r$ defines a pre Hilbert space structure on\n$\\mathcal{C}(N)$ with completion isometric to $L^2(N)$.\n\\end{proposition}\n\n\\section{Fourier Inversion for the Limit Group}\n\\label{sec6}\n\\setcounter{equation}{0}\nIn this section we apply the material of Section \\ref{sec5} to extend the\nFourier inversion portion of Theorem \\ref{plancherel-general}\nfrom the $N_n$ to the limit group $N = \\varinjlim N_n$\\,.\nTo set this up recall that\n\\begin{itemize}\n\\item $\\mathfrak{t}^* = \\varprojlim \\mathfrak{t}^*_n$ consists of all\ncollections $\\gamma = (\\gamma_n)$ where each $\\gamma_n \\in \\mathfrak{t}_n^*$ and\nif $n' \\geqq n$ then $\\gamma_{n'}|_{\\mathfrak{s}_n} = \\gamma_n$\\,.  \n\\item given $\\gamma = (\\gamma_n) \\in \\mathfrak{t}^*$ the limit representation\n$\\pi_\\gamma = \\varprojlim \\pi_{\\gamma_n}$ is constructed as in \nSection \\ref{sec2},\n\\item The distribution character $\\Theta_{\\pi_{\\gamma_n}}$ are given by\n(\\ref{def-dist-char}), and\n\\item $\\mathcal{C}(N) = \\varprojlim \\mathcal{C}(N_n)$ consists of all collections\n$f = (f_n)$ where each $f_n \\in \\mathcal{C}(N_n)$ and if $n' \\geqq n$ then\n$f_{n'}|_{N_n} = f_n$\\,.\n\\end{itemize}\nThen the limit Fourier inversion formula is\n\\begin{theorem}\\label{limit-inversion}\nSuppose that $N = \\varinjlim N_n$ where $\\{N_n\\}$ satisfies \n{\\rm (\\ref{newsetup})}.  Then the Plancherel measure for $N$ is\nconcentrated on $\\mathfrak{t}^*$.   Let $f = (f_n) \\in \\mathcal{C}(N)$ and $x \\in N$.\nThen $x \\in N_n$ for some $n$ and\n\\begin{equation}\\label{lim-inv-formula}\nf(x) = c_n\\int_{\\mathfrak{t}_n^*} \\Theta_{\\pi_{\\gamma_n}}(r_xf) \n\t|{\\rm Pf}\\,_{\\mathfrak{n}_n}({\\gamma_n})|d\\gamma_n\n\\end{equation}\nwhere $c_n = 2^{d_1 + \\dots + d_m} d_1! d_2! \\dots d_m!$ as in {\\rm (\\ref{c-d}a)}\nand $m$ is the number of factors $L_r$ in $N_n$. \n\\end{theorem}\n\n\\begin{proof}  Apply Theorem \\ref{plancherel-general} to $N_n$.  That gives\n$f(x) = f_n(x) =\nc_n\\int_{\\mathfrak{t}_n^*} \\Theta_{\\pi_{\\gamma_n}}(r_xf) |{\\rm Pf}\\,_{\\mathfrak{n}_n}({\\gamma_n})|\\,\nd\\gamma_n$\\,.\n\\end{proof}\n\n\\begin{remark}{\\rm\nA Plancherel Formula of the sort $||f||_{L^2}^2 = \\int ||\\pi(f)||^2_{HS}\\, d\\pi$\nusually is somewhat easier than a Fourier inversion formula.  This \nin part is \nbecause it usually is easier to prove that operators $\\pi(f)$ are\nHilbert-Schmidt than to prove that (for appropriate functions $f$)\nthey are of trace class.  Thus one might expect that a formula \n$||f||^2_{L^2(N)} = \\lim c_n'\\int_{\\mathfrak{t}_n^*}\n||\\pi_{\\gamma_n}(f|_{N_n})||^2_{HS} |{\\rm Pf}\\,_{\\mathfrak{n}_n}({\\gamma_n})|\\,d\\gamma_n$\nwould be easier to prove than (\\ref{lim-inv-formula}).  But it is not\nclear how to relate the Hilbert--Schmidt norms to the limit process, because\nwe have have not yet found an appropriate form of the Frobenius--Schur \northogonality relations.  Thus the ``less delicate'' Plancherel Formula remains\nproblematical.  } \\hfill $\\diamondsuit$\n\\end{remark}\n\n\\section{Nilradicals of Parabolics in Finite Dimensional Groups}\n\\label{sec7}\n\\setcounter{equation}{0}\nIn Section \\ref{sec8} we will specialize our results to nilradicals \nof minimal parabolic subgroups\nof finitary real reductive Lie groups such as the infinite special and general\nlinear groups and the infinite real, complex and quaternionic unitary groups.\nIn order to do that, in this section we review the relevant restricted\nroot structure that gives the finite dimensional case, reversing some\nof the enumerations used in \\cite{W2013} to be appropriate for our direct limit\nsystems.\n\\medskip\n\nLet $G$ be a finite dimensional connected \nreal reductive Lie group.  We recall some structural results\non its minimal parabolic subgroups, some standard and some from \\cite{W2013}.\n\\medskip\n\nFix an Iwasawa decomposition $G = KAN$.  Write $\\mathfrak{k}$ for the Lie \nalgebra of $K$, $\\mathfrak{a}$ for the Lie algebra of $A$, and $\\mathfrak{n}$ for the\nLie algebra of $N$.  Complete $\\mathfrak{a}$ to a Cartan subalgebra $\\mathfrak{h}$ of $\\mathfrak{g}$.\nThen $\\mathfrak{h} = \\mathfrak{t} + \\mathfrak{a}$ with $\\mathfrak{t} = \\mathfrak{h} \\cap \\mathfrak{k}$.  Now we have root systems\n\\begin{itemize}\n\\item $\\Delta(\\mathfrak{g}_\\mathbb{C},\\mathfrak{h}_\\mathbb{C})$: roots of $\\mathfrak{g}_\\mathbb{C}$ relative to $\\mathfrak{h}_\\mathbb{C}$ \n(ordinary roots), and\n\n\\item $\\Delta(\\mathfrak{g},\\mathfrak{a})$: roots of $\\mathfrak{g}$ relative to $\\mathfrak{a}$ (restricted roots). \n\n\\item $\\Delta_0(\\mathfrak{g},\\mathfrak{a}) = \\{\\gamma \\in \\Delta(\\mathfrak{g},\\mathfrak{a}) \\mid \n\t2\\gamma \\notin \\Delta(\\mathfrak{g},\\mathfrak{a})\\}$ (nonmultipliable restricted roots).\n\\end{itemize}\nSometimes we will identify a restricted root\n$\\gamma = \\alpha|_\\mathfrak{a}$, $\\alpha \\in \\Delta(\\mathfrak{g}_\\mathbb{C},\\mathfrak{h}_\\mathbb{C})$ and \n$\\alpha |_\\mathfrak{a} \\ne 0$, with the set \n\\begin{equation}\\label{resrootset}\n[\\gamma] := \n\\{\\alpha' \\in \\Delta(\\mathfrak{g}_\\mathbb{C},\\mathfrak{h}_\\mathbb{C}) \\mid \\alpha'|_\\mathfrak{a} = \\alpha|_\\mathfrak{a}\\}\n\\end{equation}\nof all roots that restrict to it.  Further, \n$\\Delta(\\mathfrak{g},\\mathfrak{a})$ and $\\Delta_0(\\mathfrak{g},\\mathfrak{a})$ are root \nsystems in the usual sense.  Any positive system \n$\\Delta^+(\\mathfrak{g}_\\mathbb{C},\\mathfrak{h}_\\mathbb{C}) \\subset \\Delta(\\mathfrak{g}_\\mathbb{C},\\mathfrak{h}_\\mathbb{C})$ defines positive \nsystems\n\\begin{itemize}\n\\item $\\Delta^+(\\mathfrak{g},\\mathfrak{a}) = \\{\\alpha|_\\mathfrak{a} \\mid \\alpha \\in \n\\Delta^+(\\mathfrak{g}_\\mathbb{C},\\mathfrak{h}_\\mathbb{C}) \n\\text{ and } \\alpha|_\\mathfrak{a} \\ne 0\\}$ and $\\Delta_0^+(\\mathfrak{g},\\mathfrak{a}) =\n\\Delta_0(\\mathfrak{g},\\mathfrak{a}) \\cap \\Delta^+(\\mathfrak{g},\\mathfrak{a})$.\n\\end{itemize}\n\\noindent We can (and do) choose $\\Delta^+(\\mathfrak{g},\\mathfrak{h})$ so that \n\\begin{itemize}\n\\item$\\mathfrak{n}$ is the sum of the positive restricted root spaces and\n\\item if $\\alpha \\in \\Delta(\\mathfrak{g}_\\mathbb{C},\\mathfrak{h}_\\mathbb{C})$ and $\\alpha|_\\mathfrak{a} \\in\n\\Delta^+(\\mathfrak{g},\\mathfrak{a})$ then $\\alpha \\in \\Delta^+(\\mathfrak{g}_\\mathbb{C},\\mathfrak{h}_\\mathbb{C})$.\n\\end{itemize}\n\\medskip\n\nRecall that two roots are {\\em strongly orthogonal} if their sum and their\ndifference are not roots.  Then they are orthogonal.  We define\n\\begin{equation}\\label{cascade}\n\\begin{aligned}\n&\\beta'_1 \\in \\Delta^+(\\mathfrak{g},\\mathfrak{a}) \\text{ is a maximal positive restricted root\nand }\\\\\n& \\beta'_{r+1} \\in \\Delta^+(\\mathfrak{g},\\mathfrak{a}) \\text{ is a maximum among the roots of }\n\\Delta^+(\\mathfrak{g},\\mathfrak{a}) \\text{ orthogonal to all } \\beta'_i \\text{ with } i \\leqq r\n\\end{aligned}\n\\end{equation}\nThen the $\\beta'_r$ are mutually strongly orthogonal.  \nNote that each $\\beta'_r \\in \\Delta_0^+(\\mathfrak{g},\\mathfrak{a})$.\nThis is the Kostant cascade coming down from the maximal root.  Denote\n\\begin{equation}\\label{wrong-numbering}\n\\{\\beta'_1, \\dots , \\beta'_m\\}: \\text{ the set of strongly orthogonal roots\nconstructed in (\\ref{cascade}).}\n\\end{equation}\nThe enumeration (\\ref{wrong-numbering}) is not appropriate for the direct\nlimit process, but we need it for some of the lemmas below.  For direct \nlimit considerations we will use the reversed ordering\n\\begin{equation}\\label{right-numbering}\n\\beta_r = \\beta'_{m-r+1} \\text{, so the ordered sets }\n\\{\\beta_1, \\dots , \\beta_m\\} = \\{\\beta'_m, \\dots , \\beta'_1\\}.\n\\end{equation}\n\nFor $1\\leqq r \\leqq m$ define \n\\begin{equation}\\label{layers}\n\\begin{aligned}\n&\\Delta^+_m = \\{\\alpha \\in \\Delta^+(\\mathfrak{g},\\mathfrak{a}) \\mid \\beta_m - \\alpha \\in \\Delta^+(\\mathfrak{g},\\mathfrak{a})\\} \n\\text{ and }\\\\\n&\\Delta^+_{m-r-1} = \\{\\alpha \\in \\Delta^+(\\mathfrak{g},\\mathfrak{a}) \\setminus \n\t(\\Delta^+_m \\cup \\dots \\cup \\Delta^+_{m-r})\n\t\\mid \\beta_{m-r-1} - \\alpha \\in \\Delta^+(\\mathfrak{g},\\mathfrak{a})\\}.\n\\end{aligned}\n\\end{equation} \n\n\\begin{lemma} \\label{fill-out} {\\rm \\cite[Lemma 6.3]{W2013}}\nIf $\\alpha \\in \\Delta^+(\\mathfrak{g},\\mathfrak{a})$ then either \n$\\alpha \\in \\{\\beta_1, \\dots , \\beta_m\\}$\nor $\\alpha$ belongs to exactly one of the sets $\\Delta^+_r$\\,.\nIn particular the Lie algebra $\\mathfrak{n}$ of $N$ is the\nvector space direct sum of its subspaces\n\\begin{equation}\\label{def-m}\n\\mathfrak{l}_r = \\mathfrak{g}_{\\beta_r} + {\\sum}_{\\Delta^+_r}\\, \\mathfrak{g}_\\alpha \n\\text{ for } 1\\leqq r\\leqq m\n\\end{equation}\n\\end{lemma}\n\n\\begin{lemma}\\label{layers2}{\\rm \\cite[Lemma 6.4]{W2013}}\nThe set $\\Delta^+_r\\cup \\{\\beta_r\\}  \n= \\{\\alpha \\in \\Delta^+ \\mid \\alpha \\perp \\beta_i \\text{ for } i > r\n\\text{ and } \\langle \\alpha, \\beta_r\\rangle > 0\\}.$\nIn particular, $[\\mathfrak{l}_r,\\mathfrak{l}_s] \\subset \\mathfrak{l}_t$ where $t = \\max\\{r,s\\}$.\nThus $\\mathfrak{n}$ has an increasing foliation based on the ideals\n\\begin{equation}\\label{def-filtration}\n\\mathfrak{l}_{r,m} = \\mathfrak{l}_{r+1} + \\dots + \\mathfrak{l}_m \\text{ for } 0 \\leqq r < m\n\\end{equation}\nwith a corresponding group level decomposition by normal subgroups \n$L_{r,m}$ where\n\\begin{equation}\\label{def-filtration-group}\nN = L_{0,m} = L_1L_2\\dots L_m \\text{ and } L_{r,m} = L_{r+1} \\ltimes N_{r+1,m}\n\t\\text{ for } 0 \\leqq r < m.\n\\end{equation}\n\\end{lemma}\n\nThe structure of $\\Delta^+_r$, and later of $\\mathfrak{l}_r$, is exhibited by a \nparticular Weyl group element of $\\Delta(\\mathfrak{g},\\mathfrak{a})$ and the negative of\nthat Weyl group element.  Denote\n\\begin{equation}\\label{beta-reflect}\ns_{\\beta_r} \\text{ is the Weyl group reflection in } \\beta_r\n\\text{ and } \\sigma_r: \\Delta(\\mathfrak{g},\\mathfrak{a}) \\to \\Delta(\\mathfrak{g},\\mathfrak{a}) \\text{ by }\n\\sigma_r(\\alpha) = -s_{\\beta_r}(\\alpha).\n\\end{equation}\nHere $\\sigma_r(\\beta_s) = -\\beta_s$ for $s \\ne r$, $+\\beta_s$ if $s = r$.\nIf $\\alpha \\in \\Delta^+_r$ we still have $\\sigma_r(\\alpha) \\perp \\beta_i$\nfor $i > r$ and $\\langle \\sigma_r(\\alpha), \\beta_r\\rangle > 0$.  If\n$\\sigma_r(\\alpha)$ is negative then $\\beta_r - \\sigma_r(\\alpha) > \\beta_r$\ncontradicting the maximality property of $\\beta_{m-r+1}$.  Thus, using \nLemma \\ref{layers2}, $\\sigma_r(\\Delta^+_r) = \\Delta^+_r$.\nThis divides each $\\Delta^+_r$ into pairs:\n\n\\begin{lemma} \\label{layers-nilpotent}{\\rm \\cite[Lemma 6.8]{W2013}}\nIf $\\alpha \\in \\Delta^+_r$ then $\\alpha + \\sigma_r(\\alpha) = \\beta_r$.\n{\\rm (}Of course it is possible that \n$\\alpha = \\sigma_r(\\alpha) = \\tfrac{1}{2}\\beta_r$ when \n$\\tfrac{1}{2}\\beta_r$ is a root.{\\rm ).}\nIf $\\alpha, \\alpha' \\in \\Delta^+_r$ and $\\alpha + \\alpha' \\in \\Delta(\\mathfrak{g},\\mathfrak{a})$\nthen $\\alpha + \\alpha' = \\beta_r$\\,.\n\\end{lemma}\n\nIt comes out of Lemmas \\ref{fill-out} and \\ref{layers2} that the \ndecompositions of (\\ref{layers}), (\\ref{def-m}) and\n(\\ref{def-filtration}) satisfy (\\ref{newsetup}), so\nTheorem \\ref{plancherel-general}\napplies to nilradicals of minimal parabolic subgroups.  In other words,\nas in Theorem \\ref{plancherel-general},\n\\begin{theorem}\\label{iwasawa-layers}{\\rm \\cite[Theorem 6.16]{W2013}}\nLet $G$ be a real reductive Lie group, $G = KAN$ an Iwasawa\ndecomposition, $\\mathfrak{l}_r$ and $\\mathfrak{n}_r$ the subalgebras of $\\mathfrak{n}$ defined in \n{\\rm (\\ref{def-m})} and {\\rm (\\ref{def-filtration})},\nand $L_r$ and $N_r$ the corresponding analytic subgroups of $N$.  \nThen the $L_r$ and $N_r$ satisfy {\\rm (\\ref{newsetup})}.  In particular,\nPlancherel measure for $N$ is\nconcentrated on $\\{\\pi_\\lambda \\mid \\lambda \\in \\mathfrak{t}^*\\}$.\nIf $\\lambda \\in \\mathfrak{t}^*$, and if $u$ and $v$ belong to the\nrepresentation space $\\mathcal{H}_{\\pi_\\lambda}$ of $\\pi_\\lambda$,  then\nthe coefficient $f_{u,v}(x) = \\langle u, \\pi_\\lambda(x)v\\rangle$\nsatisfies\n\\begin{equation}\n||f_{u,v}||^2_{L^2(N \/ S)} = \\frac{||u||^2||v||^2}{|{\\rm Pf}\\,(\\lambda)|}\\,.\n\\end{equation}\nThe distribution character $\\Theta_{\\pi_\\lambda}$ of $\\pi_{\\lambda}$ satisfies\n\\begin{equation}\n\\Theta_{\\pi_\\lambda}(f) = c^{-1}|{\\rm Pf}\\,(\\lambda)|^{-1}\\int_{\\mathcal{O}(\\lambda)}\n        \\widehat{f_1}(\\xi)d\\nu_\\lambda(\\xi) \\text{ for } f \\in \\mathcal{C}(N)\n\\end{equation}\nwhere $\\mathcal{C}(N)$ is the Schwartz space, $f_1$ is the lift\n$f_1(\\xi) = f(\\exp(\\xi))$, $\\widehat{f_1}$ is its classical Fourier transform,\n$\\mathcal{O}(\\lambda)$ is the coadjoint orbit ${\\rm Ad}\\,^*(N)\\lambda = \\mathfrak{v}^* + \\lambda$,\n$c = 2^{d_1 + \\dots + d_m} d_1! d_2! \\dots d_m!$ as in {\\rm (\\ref{c-d}a)},\nand $d\\nu_\\lambda$ is the translate of normalized Lebesgue measure from\n$\\mathfrak{v}^*$ to ${\\rm Ad}\\,^*(N)\\lambda$.  The Fourier inversion formula on $N$ is\n\\begin{equation}\nf(x) = c\\int_{\\mathfrak{t}^*} \\Theta_{\\pi_\\lambda}(r_xf) |{\\rm Pf}\\,(\\lambda)|d\\lambda\n        \\text{ for } f \\in \\mathcal{C}(N).\n\\end{equation}\n\\end{theorem}\n\n\\section{Nilradicals of Parabolics in Infinite Dimensional Groups}\n\\label{sec8}\n\\setcounter{equation}{0}\nWe now look at the classical real forms of the three classical simple locally \nfinite countable--dimensional Lie\nalgebras $\\mathfrak{g}_\\mathbb{C} = \\varinjlim \\mathfrak{g}_{n,\\mathbb{C}}$, and their real forms\n$\\mathfrak{g}_\\mathbb{R}$.  The Lie algebras $\\mathfrak{g}_\\mathbb{C}$ are the classical direct limits,\n$\\mathfrak{sl}(\\infty,\\mathbb{C}) = \\varinjlim \\mathfrak{sl}(n;\\mathbb{C})$,\n$\\mathfrak{so}(\\infty,\\mathbb{C}) = \\varinjlim \\mathfrak{so}(2n;\\mathbb{C}) = \\varinjlim \\mathfrak{so}(2n+1;\\mathbb{C})$, and\n$\\mathfrak{sp}(\\infty,\\mathbb{C}) = \\varinjlim \\mathfrak{sp}(n;\\mathbb{C})$,\nwhere the direct systems are\ngiven by the inclusions of the form\n$A \\mapsto (\\begin{smallmatrix} A & 0 \\\\ 0 & 0 \\end{smallmatrix} )$\nor $A \\mapsto \\left (\\begin{smallmatrix} 0 & 0 & 0 \\\\ 0 & A & 0 \\\\ 0 & 0 & 0 \n\\end{smallmatrix} \\right )$.\nWe often consider the locally reductive algebra\n$\\mathfrak{gl}(\\infty;\\mathbb{C}) = \\varinjlim \\mathfrak{gl}(n;\\mathbb{C})$ along with $\\mathfrak{sl}(\\infty;\\mathbb{C})$.\n\\medskip\n\nLet $G_n$ be a real (this includes complex) simple Lie group of classical \ntype and real rank $n$.  We have just described it as sitting in a direct \nsystem $\\{G_n\\}$ of Lie algebras in the same series.  \nSet $G = \\varinjlim G_n$ as above.  Then we have coherent Iwasawa\ndecompositions $G_n = K_nA_nN_n$ with $K_n \\subset K_\\ell$, \n$A_n \\subset A_\\ell$ and $N_n \\subset N_\\ell$ for $\\ell \\geqq n$.  We need\nto do this so that the direct limit respects the restricted root structures,\nin particular the strongly orthogonal root structures, \nof the $N_n$\\,.   To do that we enumerate the set \n$\\Psi_n = \\Psi(\\mathfrak{g}_n, \\mathfrak{h}_n)$ of nonmultipliable simple restricted\nroots so that, in the Dynkin diagram, for type $A$ we spread from the \ncenter of the diagram.  For types $B$, $C$ and $D$ \n$\\psi_1$ is the \\textit{right} endpoint,\nIn other words for $\\ell \\geqq n$ $\\Psi_\\ell$\nis constructed from $\\Psi_n$ adding simple roots to the \\textit{left} end\nof their Dynkin diagrams.  Thus\n\\begin{equation}\\label{rootorderA}\n\\begin{aligned} \n&\\begin{tabular}{|c|l|c|}\\hline\n$\\Psi_\\ell \\text{ type } A_{2\\ell+1}$ &\n\\setlength{\\unitlength}{.4 mm}\n\\begin{picture}(180,18)\n\\put(10,2){\\circle{2}}\n\\put(5,5){$\\psi_{-\\ell}$}\n\\put(11,2){\\line(1,0){13}}\n\\put(27,2){\\circle*{1}}\n\\put(30,2){\\circle*{1}}\n\\put(33,2){\\circle*{1}}\n\\put(36,2){\\line(1,0){13}}\n\\put(50,2){\\circle{2}}\n\\put(45,5){$\\psi_{-n}$}\n\\put(51,2){\\line(1,0){13}}\n\\put(67,2){\\circle*{1}}\n\\put(70,2){\\circle*{1}}\n\\put(73,2){\\circle*{1}}\n\\put(76,2){\\line(1,0){13}}\n\\put(90,2){\\circle{2}}\n\\put(87,5){$\\psi_0$}\n\\put(91,2){\\line(1,0){13}}\n\\put(107,2){\\circle*{1}}\n\\put(110,2){\\circle*{1}}\n\\put(113,2){\\circle*{1}}\n\\put(116,2){\\line(1,0){13}}\n\\put(130,2){\\circle{2}}\n\\put(128,5){$\\psi_n$}\n\\put(131,2){\\line(1,0){13}}\n\\put(147,2){\\circle*{1}}\n\\put(150,2){\\circle*{1}}\n\\put(153,2){\\circle*{1}}\n\\put(156,2){\\line(1,0){13}}\n\\put(170,2){\\circle{2}}\n\\put(167,5){$\\psi_\\ell$}\n\\end{picture}\n&$\\ell \\geqq n \\geqq 0$\n\\\\\n\\hline\n\\end{tabular}\\\\\n&\\begin{tabular}{|c|l|c|}\\hline\n\\setlength{\\unitlength}{.4 mm}\n$\\Psi_\\ell \\text{ type } A_{2\\ell}\\phantom{i.}$ &\n\\setlength{\\unitlength}{.4 mm}\n\\begin{picture}(180,18)\n\\put(1,2){\\circle{2}}\n\\put(-4,5){$\\psi_{-\\ell}$}\n\\put(2,2){\\line(1,0){13}}\n\\put(18,2){\\circle*{1}}\n\\put(21,2){\\circle*{1}}\n\\put(24,2){\\circle*{1}}\n\\put(27,2){\\line(1,0){13}}\n\\put(41,2){\\circle{2}}\n\\put(36,5){$\\psi_{-n}$}\n\\put(42,2){\\line(1,0){13}}\n\\put(58,2){\\circle*{1}}\n\\put(61,2){\\circle*{1}}\n\\put(64,2){\\circle*{1}}\n\\put(67,2){\\line(1,0){13}}\n\\put(81,2){\\circle{2}}\n\\put(78,5){$\\psi_{-1}$}\n\\put(82,2){\\line(1,0){13}}\n\\put(96,2){\\circle{2}}\n\\put(93,5){$\\psi_1$}\n\\put(97,2){\\line(1,0){13}}\n\\put(113,2){\\circle*{1}}\n\\put(116,2){\\circle*{1}}\n\\put(119,2){\\circle*{1}}\n\\put(122,2){\\line(1,0){13}}\n\\put(136,2){\\circle{2}}\n\\put(134,5){$\\psi_n$}\n\\put(137,2){\\line(1,0){13}}\n\\put(153,2){\\circle*{1}}\n\\put(156,2){\\circle*{1}}\n\\put(159,2){\\circle*{1}}\n\\put(162,2){\\line(1,0){13}}\n\\put(176,2){\\circle{2}}\n\\put(173,5){$\\psi_\\ell$}\n\\end{picture}\n&$\\ell \\geqq n \\geqq 1$\n\\\\\n\\hline\n\\end{tabular}\n\\end{aligned}\n\\end{equation}\n\n\\begin{equation}\\label{rootorderBCD}\n\\begin{aligned}\n&\\begin{tabular}{|c|l|c|}\\hline\n$\\Psi_\\ell \\text{ type } B_\\ell$&\n\\setlength{\\unitlength}{.5 mm}\n\\begin{picture}(155,13)\n\\put(5,2){\\circle{2}}\n\\put(2,5){$\\psi_{\\ell}$}\n\\put(6,2){\\line(1,0){13}}\n\\put(24,2){\\circle*{1}}\n\\put(27,2){\\circle*{1}}\n\\put(30,2){\\circle*{1}}\n\\put(34,2){\\line(1,0){13}}\n\\put(48,2){\\circle{2}}\n\\put(45,5){$\\psi_n$}\n\\put(49,2){\\line(1,0){23}}\n\\put(73,2){\\circle{2}}\n\\put(70,5){$\\psi_{n-1}$}\n\\put(74,2){\\line(1,0){13}}\n\\put(93,2){\\circle*{1}}\n\\put(96,2){\\circle*{1}}\n\\put(99,2){\\circle*{1}}\n\\put(104,2){\\line(1,0){13}}\n\\put(118,2){\\circle{2}}\n\\put(115,5){$\\psi_2$}\n\\put(119,2.5){\\line(1,0){23}}\n\\put(119,1.5){\\line(1,0){23}}\n\\put(143,2){\\circle*{2}}\n\\put(140,5){$\\psi_1$}\n\\end{picture}\n&$\\ell\\geqq n \\geqq 2$\\\\\n\\hline\n\\end{tabular} \\\\\n&\\begin{tabular}{|c|l|c|}\\hline\n$\\Psi_\\ell \\text{ type } C_\\ell$ &\n\\setlength{\\unitlength}{.5 mm}\n\\begin{picture}(155,13)\n\\put(5,2){\\circle*{2}}\n\\put(2,5){$\\psi_{\\ell}$}\n\\put(6,2){\\line(1,0){13}}\n\\put(24,2){\\circle*{1}}\n\\put(27,2){\\circle*{1}}\n\\put(30,2){\\circle*{1}}\n\\put(34,2){\\line(1,0){13}}\n\\put(48,2){\\circle*{2}}\n\\put(45,5){$\\psi_n$}\n\\put(49,2){\\line(1,0){23}}\n\\put(73,2){\\circle*{2}}\n\\put(70,5){$\\psi_{n-1}$}\n\\put(74,2){\\line(1,0){13}}\n\\put(93,2){\\circle*{1}}\n\\put(96,2){\\circle*{1}}\n\\put(99,2){\\circle*{1}}\n\\put(104,2){\\line(1,0){13}}\n\\put(118,2){\\circle*{2}}\n\\put(115,5){$\\psi_2$}\n\\put(119,2.5){\\line(1,0){23}}\n\\put(119,1.5){\\line(1,0){23}}\n\\put(143,2){\\circle{2}}\n\\put(140,5){$\\psi_1$}\n\\end{picture}\n& $\\ell\\geqq n \\geqq 3$\n\\\\\n\\hline\n\\end{tabular}\\\\\n&\\begin{tabular}{|c|l|c|}\\hline\n$\\Psi_\\ell \\text{ type } D_\\ell$ &\n\\setlength{\\unitlength}{.5 mm}\n\\begin{picture}(155,20)\n\\put(5,9){\\circle{2}}\n\\put(2,12){$\\psi_{\\ell}$}\n\\put(6,9){\\line(1,0){13}}\n\\put(24,9){\\circle*{1}}\n\\put(27,9){\\circle*{1}}\n\\put(30,9){\\circle*{1}}\n\\put(34,9){\\line(1,0){13}}\n\\put(48,9){\\circle{2}}\n\\put(45,12){$\\psi_n$}\n\\put(49,9){\\line(1,0){23}}\n\\put(73,9){\\circle{2}}\n\\put(70,12){$\\psi_{n-1}$}\n\\put(74,9){\\line(1,0){13}}\n\\put(93,9){\\circle*{1}}\n\\put(96,9){\\circle*{1}}\n\\put(99,9){\\circle*{1}}\n\\put(104,9){\\line(1,0){13}}\n\\put(118,9){\\circle{2}}\n\\put(113,12){$\\psi_3$}\n\\put(119,8.5){\\line(2,-1){13}}\n\\put(133,2){\\circle{2}}\n\\put(136,0){$\\psi_1$}\n\\put(119,9.5){\\line(2,1){13}}\n\\put(133,16){\\circle{2}}\n\\put(136,14){$\\psi_2$}\n\\end{picture}\n& $\\ell\\geqq n \\geqq 4$\n\\\\\n\\hline\n\\end{tabular}\n\\end{aligned}\n\\end{equation}\nWe describe this by saying that $G_\\ell$ {\\em propagates} $G_n$\\,.\nFor types $B$, $C$ and $D$ this is the same as the notion of propagation in\n\\cite{OW2011} and \\cite{OW2014}, but for type $A$ is it s bit different.\nWith the simple root enumeration of (\\ref{rootorderA}) and (\\ref{rootorderBCD})\nthe set $\\{\\beta_1, \\dots , \\beta_m\\}$ of strongly orthogonal positive \nrestricted roots of (\\ref{right-numbering}) is\n\\medskip\n\ntype $A_{2n+1}$: $m = n+1$; $\\beta_1 = \\psi_0$\\,; \n\t$\\beta_2 = \\psi_{-1} + \\psi_0 + \\psi_1$\\,; $\\cdots$ ;\\,\n\t$\\beta_r = \\psi_{-r+1} + \\beta_{r-1} + \\psi_{r-1}$\\,; $\\cdots$\n\\medskip\n\ntype $A_{2n}$: $m = n$; $\\beta_1 = \\psi_{-1} + \\psi_1$\\,;\n\t$\\beta_2 = \\psi_{-2} + \\psi_{-1} + \\psi_1 + \\psi_2$\\,; $\\cdots$ ;\\, \n\t$\\beta_r = \\psi_{-r} + \\beta_{r-1} + \\psi_r$\\,; $\\cdots$\n\\medskip\n\ntype $B_{2n+1}$: $m = 2n+1$; $\\beta_1 = \\psi_1\\,;\n\t\\beta_2 = \\psi_3$ and $\\beta_3 = 2(\\psi_1 + \\psi_2)+\\psi_3$\\,;\n\t$\\cdots$ ; \\hfill\\newline\n\t\\phantom{XXXXXXXXXXXXXXXXXXXXXXx}$\\beta_{2r} = \\psi_{2r+1}$ and\n\t$\\beta_{2r+1} = 2(\\psi_1 + \\dots \\psi_{2r})+\\psi_{2r+1}$\\,;\\,\\,$\\cdots$\n\\medskip\n\ntype $B_{2n}$: $m = 2n$; $\\beta_1 = \\psi_2$ and $\\beta_2 = 2\\psi_1 + \\psi_2$\\,; \n   $\\beta_3 = \\psi_4$ and $\\beta_4 = 2(\\psi_1 + \\psi_2 + \\psi_3) + \\psi_4$\\,;\n   $\\cdots$ ; \\hfill\\newline\n   \\phantom{XXXXXXXXXXXXXi}$\\beta_{2r+1} = \\psi_{2r-1}$ and\n\t$\\beta_{2r} = 2(\\psi_1 + \\dots \\psi_{2r-1}) + \\psi_{2r}$\\,;\\,\\,$\\cdots$\n\\medskip\n\ntype $C_n$: $m = n$; $\\beta_1 = \\psi_1$\\,; $\\beta_2 = \\psi_1 + 2\\psi_2$\\,;\n\t\\,\\,$\\cdots$\\,; $\\beta_r = \\psi_1 + 2(\\psi_2 + \\dots + \\psi_r)$\\,;\n\t\\,\\,$\\cdots$\n\\medskip\n\ntype $D_{2n+1}$: $m = 2n$; $\\beta_1 = \\psi_3$\\,; \n\t$\\beta_2 = \\psi_1 + \\psi_2 + \\psi_3$\\,; \\hfill\\newline\n\t\\phantom{XXXXXXXXXXXXXXi} $\\beta_3 = \\psi_5$ and \n\t\t$\\beta_4 = \\psi_1 + \\psi_2 + 2(\\psi_3 +\n\t\t\\psi_4) + \\psi_5$\\,; $\\cdots$ ; \\hfill\\newline\n        \\phantom{XXXXXXXXXXXXXXi} $\\beta_{2r-1} = \\psi_{2r+1}$ and \n\t$\\beta_{2r} = \\psi_1 + \\psi_2 \n\t  + 2(\\psi_3 + \\dots + \\psi_{2r}) + \\psi_{2r+1}$\\,;\\,\\,$\\cdots$\n\\medskip\n\ntype $D_{2n}$: $m = 2n$; $\\beta_1 = \\psi_1$\\,; $\\beta_2 = \\psi_2$\\,;\n\t$\\beta_3 = \\psi_4$ and $\\beta_4 = \\psi_1 + \\psi_2 + 2\\psi_3 + \\psi_4$\\;\n\t\\hfill\\newline \\phantom{XXXXXXXXXXXXXXXXXXXXXXXXi}\n\t$\\beta_5 = \\psi_6$ and $\\beta_6 = \\psi_1 + \\psi_2 + \n\t\t2(\\psi_3 + \\psi_4 + \\psi_5) + \\psi_6$\\,; $\\cdots$ ; \n\t\\hfill\\newline \\phantom{XXXXXXXXXXXXXXXXXXXXXXXXi}\n\t$\\beta_{2r-1} = \\psi_{2r}$ and $\\beta_{2r} = \\psi_1 + \\psi_2 + \n\t\t2(\\psi_3+ \\dots + \\psi_{2r-1})+\\psi_{2r}$\\,;\\,\\,$\\cdots$\n\\smallskip\n\nIn order to simplify use of these constructions we denote\n\\begin{definition}\\label{well-aligned}{\\rm\nLet $G = \\varinjlim G_n$ be a classical simple locally finite \ncountable dimensional Lie group.  Possibly passing to a cofinal subsequence\nsuppose that we have coherent Iwasawa decompositions $G_n = K_n A_n N_n$ \nsuch that $G_\\ell$ propagates $G_n$ for\n$\\ell \\geqq n$.  Then, again possibly passing to a cofinal subsequence,\nwe can assume that all of the nonmultipliable restricted root \nsystems $\\Delta_0(\\mathfrak{g}_n,\\mathfrak{a}_n)$ are of the same type $A_{2n+1}$,\n$A_{2n}$, $B_{2n+1}$, $B_{2n}$, $C_n$, $D_{2n+1}$ or $D_{2n}$.  Then \nwe will say that the direct system $\\{G_n\\}$ is} well--aligned.\n\\hfill $\\diamondsuit$\n\\end{definition}\n\nThe condition that $\\{G_n\\}$ be well--aligned is exactly what we need for\n$\\{N_n\\}$ to satisfy (\\ref{newsetup}), and given $G$ we have a realization\n$G = \\varinjlim G_n$ for which $\\{G_n\\}$ is well--aligned.  In summary,\n\\begin{theorem}\\label{inversion-for-ss-limits}\nLet $G$ be a classical connected countable dimensional real reductive Lie\ngroup.  Express $G = \\varinjlim G_n$ with $\\{G_n\\}$ well--aligned.\nThen $\\{N_n\\}$ satisfies {\\rm (\\ref{newsetup})}.  In particular\n{\\rm Theorem \\ref{iwasawa-layers}} holds for the \nmaximal locally unipotent subgroup $N = \\varinjlim N_n$ of $G$.\n\\end{theorem}\n\n\\begin{remark}{\\rm\nIn Theorem \\ref{inversion-for-ss-limits} the possibilities for $G$ are\nthe finite dimensional simple Lie groups and the infinite dimensional\n$SL(\\infty;\\mathbb{C})$, $SO(\\infty;\\mathbb{C})$, $Sp(\\infty;\\mathbb{C})$, $SL(\\infty;\\mathbb{R})$,\n$SL(\\infty;\\mathbb{H})$, $SU(\\infty,q)$ with $q \\leqq \\infty$, $SO(\\infty,q)$\nwith $q \\leqq \\infty$, $Sp(\\infty,q)$ with $q \\leqq \\infty$, \n$Sp(\\infty;\\mathbb{R})$ and $SO^*(2\\infty)$.  Further, the normalizer\n$P = MAN$ of $N$ in $G$ is a classical minimal parabolic subgroup\n$\\varinjlim (P_n = M_nA_nN_n)$ where $P_n$ is the minimal\nparabolic in $G_n$ that is the normalizer of $N_n$\\,.}\n\\hfill $\\diamondsuit$\n\\end{remark} \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\n\nContinuous logic is an extension of classical first order logic used to study the model theory of structures based on metric spaces.  In this paper, we use continuous logic as presented in \\cite{N} and \\cite{BU} to study the model theory of $\\R$-trees.  \n\nAn $\\R$-tree is a metric space $T$ such that for any two points $a,b \\in T$ there is a unique arc in $T$ from $a$ to $b$, and that arc is a geodesic segment (i.e., an isometric copy of some closed interval in \n$\\R$).  These spaces arise naturally in geometric group theory, for example: the asymptotic cone of a hyperbolic finitely generated group is an $\\R$-tree.\n\nAn $\\R$-tree may be unbounded, while the existing full treatments of continuous model theory are restricted to bounded structures. With this in mind, we consider pointed trees, choose a\nreal number $r>0$, and axiomatize the theory $\\mathbb{R}\\mathrm{T}_r$ of pointed $\\R$-trees of\nradius at most $r$ in a suitable continuous signature. \n\nWe then define the notion of \\textit{richly branching} and axiomatize the theory\n$\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ of the class of richly branching pointed $\\R$-trees with radius $r$.  We prove that the models of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ are exactly the existentially closed models of $\\mathbb{R}\\mathrm{T}_r$; thus $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ is the model companion of $\\mathbb{R}\\mathrm{T}_r$.  Next, we investigate some model theoretic properties of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$, showing that it is complete and has quantifier elimination.  In particular, that means $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ is the model completion\nof $\\mathbb{R}\\mathrm{T}_r$.  Further, we prove that $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ is stable but not superstable and identify its model-theoretic independence relation. We characterize the principal types of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$, and show that this theory has no atomic model. Finally, we show that $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ is highly non-categorical.  In fact, for any density character this theory has the maximum possible number of pairwise non-isomorphic models; indeed, the models we construct are pairwise non-homeomorphic. We also give examples of richly branching $\\R$-trees which come from the literature, including some that will be familiar to geometric group theorists.\n\nIn the remainder of this introduction we detail the contents of each section of this paper:\n\nIn Sections 2 and 3 we provide background concerning $\\R$-trees and continuous logic, respectively.  In Section 4 we specify a continuous signature $L$ suitable for the class of pointed $\\R$-trees of radius at most $r$, and axiomatize this class of $L$-structures; the theory of the class is denoted $\\mathbb{R}\\mathrm{T}_r$.\n\nIn Section 5 we discuss definability of certain sets and functions in $\\mathbb{R}\\mathrm{T}_r$.  In Section 6 we show \n$\\mathbb{R}\\mathrm{T}_r$ has amalgamation over substructures.  This plays an important role in many of the primary results in this paper.\n\nIn Section 7 we introduce the class of richly branching pointed $\\R$-trees with radius $r$ and axiomatize this class.  The associated theory is denoted $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$.  We then show that $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ is the model companion of $\\mathbb{R}\\mathrm{T}_r$.  The theory $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ is the main object of study in this paper.\n\nIn Sections 8 and 9 we verify the main model-theoretic properties of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$.  We show that this theory is complete and admits quantifier elimination.  We characterize its types over sets of parameters and use this to show $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ is $\\kappa$-stable if and only if $\\kappa = \\kappa^\\omega$; hence this theory is strictly stable (stable but not superstable).  We also show that $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ is not a small theory; indeed, its space of $2$-types over $\\emptyset$ has metric density character $2^\\omega$.  (The space of $1$-types over $\\emptyset$ is isometric to the real interval $[0,r]$ with the usual absolute value metric.)  We give a simple geometric characterization of the independence relation of \n$\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$.  Finally, we show that non-algebraic types have built-in canonical bases (\\textit{i.e.}, these bases are sets of ordinary elements in models of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ and do not require the introduction of  imaginaries).\n\nIn Section 10 we discuss some models of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ that have been constructed within the theory of $\\R$-trees \\cite{DP1,DP2} and some other models that arise in geometric group theory.\n\nIn Section 11 we show that $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ has very few isolated $n$-types over $\\emptyset$ and conclude that it has no atomic model (equivalently, it has no prime model).  Then we use amalgamation constructions to build large families of models of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ and to characterize its isolated types over $\\emptyset$.  For each infinite cardinal $\\kappa$, we show there are  $2^\\kappa$ many pairwise non-isomorphic models of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ of density character $\\kappa$.  This is the maximum possible number of models, and the models we construct are, in fact, pairwise non-homeomorphic. \n\nIn Section 12 we briefly discuss how the results in this paper could be obtained\nfor the full class of pointed $\\R$-trees (\\textit{i.e.}, without imposing a boundedness\nrequirement).\n\n\\section{$\\R$-trees}\nIn this section we give some background concerning $\\R$-trees.\n\n\\begin{definition}\nA \\emph{geodesic segment} in a metric space $M$ is the image of an\nisometric embedding $\\gamma \\colon [0,r] \\to M$ for some $r \\geq 0$.\nWe say that such a geodesic segment is \\emph{ from $\\gamma(0)$ to $\\gamma(r)$}.  A metric space $M$ is called\n\\emph{geodesic} if for every $a,b \\in M$ there is at least one\ngeodesic segment in $M$ from $a$ to $b$.\n\\end{definition}\n\n   \\begin{fact}\n   \\label{midptfact}\n   \n   A complete metric space $M$ is a geodesic space if and only if for any two points $x,y\\in M$ there\n   exists a midpoint $z$ between $x$ and $y$. That is, there exists $z$ such that:\n   $$d(x,z)=\\frac{d(x,y)}{2}, \\text{ and } d(y,z)=\\frac{d(x,y)}{2}.$$\n   \\end{fact}\n\\begin{proof}\nSee \\cite[Chapter I, 1.4]{BH}.\n\\end{proof}\n\n\\begin{definition}\nAn {\\it $\\R$-tree} is a metric space $M$ such that for any two\npoints $a,b \\in M$ there is a unique arc from $a$ to $b$, and that arc is a geodesic\nsegment.\n\\end{definition}\n\nIn an $\\R$-tree, $[a,b]$ denotes the unique geodesic segment from $a$ to $b$.\nSince metric structures are required to be based on complete metric spaces,\nit is a helpful fact that the completion of an $\\R$-tree is an $\\R$-tree\n(see \\cite[Lemma 2.4.14]{C}).\n\nLet $M$ be an $\\R$-tree and $a\\in M$.  Call the connected components of\n$M\\setminus\\{a\\}$ {\\it branches} at $a$.  Let the \\emph{degree} of a point\n$a\\in M$ be the cardinal number of branches at $a$.  If there are three or more\nbranches at $a\\in M$, then we call $a$ a \\emph{branch point}.\nThe {\\it height} of a branch $\\beta$ at $a$ is $\\sup\\{d(a,x)|x\\in \\beta\\}$ if that\nsupremum exists, and is $\\infty$ otherwise.  A \\emph{subtree} of $M$ is any\nsubspace of $M$ that is itself an $\\R$-tree.\nA \\emph{ray} in an $\\R$-tree is an isometric copy of $\\R^{\\geq 0}$.  If $a\\in M$,\na \\emph{ray at $a$} is a ray so that the image of $0$ under the isometric embedding of $\\R^{\\geq0}$\ninto $M$ is $a$.\n\nThe following lemmas and definitions collect some straightforward facts about $\\R$-trees\nused in this paper. For helpful pictures and more facts about $\\R$-trees see \\cite{C}.\n\n\\begin{lemma}\n\\label{branch}\nIf $M$ is an $\\R$-tree and $a,b,c \\in M$, then\n\\begin{enumerate}\n\\item $d(a,b)+d(b,c)=d(a,c)+2 \\dist(b,[a,c]).$\n\\item $b\\in [a,c]$ if and only if $d(a,c)=d(a,b)+d(b,c)$.\n\\item For $b$ distinct from $a$ and $c$, we have $b\\in [a,c]$ if and only if $a$ and $c$ are in different branches at $b$.\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\nStatement (1) follows from \\cite[Lemma 2.1.2]{C},\nStatement (2) is proved in \\cite[Lemma 1.2.2]{C} and\nStatement (3) comes from \\cite[Lemma 2.2.2]{C}.\n\\end{proof}\n\n\\begin{lemma}\n\\label{!geo}\nIf $M$ is an $\\R$-tree and $E_1$, $E_2$ are disjoint, closed, non-empty subtrees of $M$,\nthen there exists a unique shortest geodesic segment $[u, v]$ with $u\\in E_1$ and $v\\in E_2$.  Moreover, for all $b\\in E_1$ and $c\\in E_2$, the geodesic segment from $b$ to $c$ must contain $[u, v]$.\n\\end{lemma}\n\n\\begin{proof}\nThis is \\cite[Lemma 2.1.9]{C}\n\\end{proof}\n\nThe preceding lemma directly implies the following fact, used often in this paper:\nGiven an $\\R$-tree $M$, a closed subtree $E$ and a point $a\\in M$,\nthere is a unique point $e$ in $E$ closest to $a$, so that $\\dist(a, E)=d(a, e)$,\nand for any point $b$ in $E$, $e$ is on the segment $[a,b].$\n\n\n\\begin{definition}\n\\label{piecewise segment}\nLet $x_0, x_1, x_2, ..., x_n$ be points in an $\\R$-tree $M$ and\n$\\gamma \\colon [0, d(x_0, x_n)] \\rightarrow M$  the isometric embedding\nwith $\\gamma(0)=x_0$ and $\\gamma(d(x_0, x_n))=x_n$ that has image\nequal to the geodesic segment $[x_0, x_n]$. If for each $i=0,...,n$\nwe have $x_i=\\gamma(a_i)$ where $0=a_0\\leq a_1\\leq ...\\leq a_n=d(x_0, x_n)$,\nthen we write\n$[x_0, x_n]=[x_0, x_1, ...., x_n]$, and call $[x_0, x_1, ...., x_n]$ a \\emph{piecewise segment}.\n\nIn other words, if $x_0, x_1,..., x_n$ are elements of $[x_0, x_n]$ listed\nin increasing order of distance from $x_0$, then we  write $[x_0, x_n]=[x_0, x_1, ..., x_n]$\nfor this \\emph{piecewise segment.}\nNote that we also know\n$[x_1, x_n]=\\bigcup_{i=1}^{n-1} [x_i, x_{i+1}]$, and  by Lemma 2.1.4 in \\cite{C} we have that\n$[x_1, x_n]=[x_1,x_2,...,x_n]$ if and only if $d(x_1, x_n)=\\sum_{i=1}^{n-1} d(x_i, x_{i+1})$.\n\n\\end{definition}\n\n\\begin{lemma}\n\\label{differentclosestpoints}\nLet $E$ be a closed subtree of $M$ and let $a,b\\in M$. Let $e_a\\in E$ be the\nunique closest point to $a$, and let $e_b\\in E$ be the unique closest point to $b$. If $e_a\\not=e_b$, then $$d(a,b)=d(a, e_a)+d(e_a, e_b)+d(b, e_b).$$\nThat is, $[a, e_a, e_b, b]$ is a piecewise segment\n\\end{lemma}\n\\begin{proof}\nFollows from Lemma \\ref{branch} and Lemma \\ref{!geo}.\n\\end{proof}\n\n\n\n\\begin{definition}[{{\\it Gromov product\\\/}}]\n\\label{gromovproduct}\nFor a metric space $M$ and $x,y,w \\in M$, define\n$$(x \\cdot y)_w=\\frac{1}{2}[d(x,w)+d(y,w)-d(x,y)].$$\n\\end{definition}\n\nIt follows easily from Lemma \\ref{branch} and Lemma \\ref{!geo} that in an $\\R$-tree, the Gromov product $(x\\cdot y)_w$ computes the distance from $w$ to the geodesic segment $[x,y]$.\n\n\\begin{definition}\n\\label{defhyperbolic}\nLet $\\delta>0$. A metric space $M$ is {\\it $\\delta$-hyperbolic} if,\nfor all $ x,y,z,w\\in M$\n$$\\min\\{(x\\cdot z)_w, (y\\cdot z)_w\\}-\\delta \\leq (x\\cdot y)_w.$$\nA metric space is {\\it 0-hyperbolic} if it is $\\delta$-hyperbolic for all $\\delta>0$.\n\\end{definition}\n\nGiven a metric space $X$, a subset $Z$ of $X$ and $\\delta>0$, we\ndenote the set $\\{x\\in X\\mid \\dist(x, Z)\\leq \\delta\\}$ by\n$Z^\\delta$.\n\n\n\n\\begin{lemma}\nIf $M$ is a geodesic metric space, then $M$ is $\\delta$-hyperbolic\nif and only if, given any $a,b,c \\in M$ and geodesic segments\n$[a,b], [b,c],$ and $[c,a]$, the segment $[a,b]$ is contained in\n$([b,c]\\cup[c,a])^\\delta$. \n\\end{lemma}\n\\begin{proof}\nThis is \\cite[Proposition 1.22]{BH}.\n\\end{proof}\n\n\\begin{lemma}\n\\label{0hyp}\nAny $\\R$-tree is $0$-hyperbolic. Moreover, any $0$-hyperbolic metric space\nembeds isometrically in an $\\R$-tree.\n\\end{lemma}\n\n\\begin{proof}\nThis is \\cite[Proposition 6.13]{R}.\n\\end{proof}\n\n\\begin{definition}\n\\label{Ypoint}\nGiven $a, b, c$ in an $\\R$-tree, there is a unique point $Y$ so that $[a,b]\\cap[a,c]=[a,Y]$.\nIn \\cite{C} after Lemma 2.1.2, they show that this $Y$ is also the unique point so that\n$[b,a]\\cap[b,c]=[b, Y]$ and $[c,a]\\cap[c,b]=[c, Y]$, and that in fact,\n$\\{Y\\}=[a,b]\\cap[b,c]\\cap[a,c]$.\nWe denote this point by $Y(a,b,c)$, or simply by $Y$ when $a, b, c$ are understood.\n\n\\end{definition}\n\n\n\n\n\n\n\\begin{definition}\nIf $A \\subseteq M$ is a subset of the $\\R$-tree $M$, let $E_A$ denote the smallest subtree containing $A$.\nWe call this the \\emph{$\\R$-tree spanned by $A$.}\nNote that\n$$E_A=\\bigcup\\{[a_1,a_2]\\mid a_1, a_2\\in A\\}.$$\nThe closure $\\overline{E_A}$ of $E_A$ is the smallest closed subtree containing $A$.\n\\end{definition}\n\\begin{definition}\nAn $\\R$-tree $M$ is {\\it finitely spanned}\nif there exists a finite subset $A\\subseteq  M$ such that $M=E_A$.\n\\end{definition}\n\nThat an $\\R$-tree is $0$-hyperbolic tells us that for any 3 points $a, b, c$\nthe segment $[a,b]$ is contained in $[b,c]\\cup [c,a]$. Thus, either $a, b, c$ are in a single piecewise segment contained in $[b,c]\\cup [c,a]$, or the subtree spanned by $a, b, c$\nis comprised of the segments $[a, Y], [b,Y]$ and $[c, Y]$, which share only the point\n$Y=Y(a,b,c)$, making that subtree ``Y-shaped\" and explaining the terminology defined above.\nPoints $a, b, c$ are arranged along a piecewise segment in some order if and only\nif $Y(a,b,c)$ is the one of $a, b, c$ that is between the other two. (See \\cite[2.1.2]{C}.)\n\n\n\\begin{definition}\nLet $M$ be an $\\R$-tree.  If $c\\in M$ is such that there do \\emph{not}\nexist $a,b\\in M\\setminus \\{c\\}$ with\n$c\\in [a,b]$, then $c$ is called an {\\it endpoint} of $M$.\nEquivalently, an endpoint is a point with degree one.\n\\end{definition}\n\nNote that if $A$ is finite, $E_A=\\overline{E_A}$ and $E_A$ is complete with\nfinite diameter. \nIf an $\\R$-tree $M$ is finitely spanned, then there is a unique smallest\nset of elements of $M$ that spans $M$, namely the endpoints of $M$.\n\n\n\\begin{lemma}\n\\label{mingenset}\nIf an $\\R$-tree $M$ is finitely spanned and $C$ is the set of endpoints of $M$, then\n\n\\begin{enumerate}\n    \\item if $B$ spans $M$, then $C\\subseteq B$;\n    \\item the set $C$ spans $M$.\n\\end{enumerate}\nThus, $C$ is the unique smallest set that spans $M$.\n\\end{lemma}\n\\begin{proof}\nLet $M$ be a finitely spanned $\\R$-tree. Let $D$ be the diameter of $M$.\nLet $B$ be a set that spans $M$.\\\\\nProof  of (1): Assume there is an endpoint $c\\in M$ not contained in $B$.\nThen there must exist $a, b\\in B$ such that $c\\in [a,b]$.  But, this is a contradiction\nbecause $c$ is an endpoint.\\\\\nProof of (2): Let $a\\in M$.\nLet $S_a$ be the set of all segments $[b, c]\\subseteq M$ such that $a\\in [b,c]$\nand order $S_a$ by inclusion.\nThis is a partial ordering on $S_a$.\nLet $\\{[b_i,c_i]\\mid i\\in \\alpha\\}$ be a chain in this partial ordering, where $\\alpha$ is a cardinal.\nLet $I$ be the closure of $\\bigcup_{i\\in \\alpha}[b_i, c_i]$.  Then $I$ is a geodesic segment in $M$.\nClearly $a\\in I$, and the length of $I$ is at most $D$. Therefore $I\\in S_a$, and\n$I$ is an upper bound for the chain. The chain was arbitrary, so any chain has an upper bound. Therefore, by Zorn's Lemma\nthere exists a maximal element of $S_a$.  Let $[b_a,c_a]$ denote such a maximal element.\nThe elements $b_a$ and $c_a$ must be endpoints of $M$.  Say, for instance,\nthat $b_a$ is not an endpoint.  Then there exist $e, f\\in M$ such that\n$b_a\\in[e,f]$, and either $[e,c_a]$ or $[f,c_a]$ will contain $[b_a, c_a]$.\nThis would mean $[b_a, c_a]$ was not maximal in $S_a$.\nTherefore, for each $a\\in M$, there exist endpoints\n$b_a$ and $c_a$ so that $a\\in [b_a, c_a]$.\nSo, $M$ is spanned by the set of its endpoints, and this spanning\nset is as small as possible by (1).\n\\end{proof}\n\n\n\n\\section{Some continuous model theory}\n\nWe investigate the model theory of $\\R$-trees using\ncontinuous logic for metric structures as presented in \\cite{N}\nand \\cite{BU}. In this section we summarize a few facts about model\ncompanions that are not discussed in those papers.\nFor the rest of this section we fix a continuous first\norder language $L$.\n\nAs explained in \\cite[Section 3]{N}, in continuous model theory it is required that \\emph{structures} and \\emph{models} are metrically complete.  However, formulas and conditions are evaluated more generally in \\emph{pre-structures}, as explained in \\cite[Definition 3.3]{N}. Further, it is shown in \\cite[Theorem 3.7]{N} that the completion of a prestructure is an elementary extension.  In this paper we use notation of the form $\\mathcal{M} \\models \\theta$ only when $\\mathcal{M}$ is a structure; in other words, $M$ must be metrically complete.\n\n  Next, some reminders about saturation in the continuous logic setting.\n  A set $\\Sigma(x_1,...,x_n)$ of $L$-conditions (with free variables\n  among $x_1,...,x_n$) is called {\\it satisfiable in $\\mathcal{M}$} if\n  there exist $a_1,...,a_n$ in $\\mathcal{M}$ such that $\\mathcal{M}\\models E[a_1,...,a_n]$\n  for every $E(x_1,...,x_n)\\in \\Sigma$.\n  Let $\\kappa$\n  be a cardinal. A model $\\mathcal{M}$ of $T$ is called {\\it $\\kappa$-saturated} if\n  for any set of parameters $A\\subseteq M$ with cardinality $<\\kappa$\n  and any set $\\Sigma(x_1,...,x_n)$ of $L(A)$-conditions, if every finite subset\n  of $\\Sigma(x_1,...,x_n)$ is satisfiable in $(\\mathcal{M},a)_{a\\in A}$, then the entire set $\\Sigma(x_1,...,x_n)$\n  is satisfiable in $(\\mathcal{M},a)_{a\\in A}$.\n \n  \\begin{proposition}\n  For any countably incomplete ultrafilter $U$ on $I$, the $U$-ultraproduct\n  of a family of $L$-structures $(\\mathcal{M}_i\\mid i\\in I)$ is $\\omega_1$-saturated.\n  \\end{proposition}\n\n\\begin{proof}\nSee \\cite[Proposition 7.6]{N}.\n\\end{proof}\n  Note that any non-principal ultrafilter on $\\N$ is countably incomplete.\n  \n  \\begin{proposition}\n  For any cardinal $\\kappa$, any $L$-structure $\\mathcal{M}$ has a $\\kappa$-saturated elementary extension.\n  \\end{proposition}\n\n\\begin{proof}\nSee \\cite[Proposition 7.10]{N}.\n\\end{proof}\n  Saturated structures have many useful properties.\n  For example, in an $\\omega$-saturated structure\n  all quantifiers are realized exactly.\n  The next proposition captures this idea.  \n\n  \\begin{proposition}\n  \\label{exactquantifiers}\n  Let $\\mathcal{M}$ be an $L$-structure and suppose $E(x_1,...,x_m)$ is the $L$-condition\n  $$(Q^1_{y_1}...Q^n_{y_n}\\phi(x_1,...,x_m, y_1,...,y_n))=0$$\n  where each $Q^i$ is either $\\inf$ or $\\sup$ and $\\phi$ is quantifier free.\n  Let $\\mathcal{E}(x_1,...,x_m)$ be the mathematical statement\n  $$\\widetilde{Q}^1_{y_1}...\\widetilde{Q}^n_{y_n}(\\phi(x_1,...,x_m, y_1,...,y_n)=0)$$\n  where each $\\widetilde{Q}^i$ is $\\exists y_i$ if $Q^i_{y_i}$ is $\\inf_{y_i}$ and is\n  $\\forall y_i$ if $Q^i_{y_i}$ is $\\sup_{y_i}$.\n  If $\\mathcal{M}$ is $\\omega$-saturated, then for any elements $a_1,...,a_m$ of $M$,\n  we have\n  $\\mathcal{M}\\models E[a_1,...,a_m]$ if and only if $\\mathcal{E}(a_1,...,a_n)$ is true in $M$.\n  \\end{proposition}\n\\begin{proof}\nSee \\cite[Proposition 7.7]{N}.\n\\end{proof}\n\n\n\\begin{definition}\nAn \\emph{$\\inf$-formula} of $L$ is a formula of the form\n$$\\inf_{s_1,y_1}...\\inf_{s_n,y_n}\\phi(x_1,...,x_k,y_1,...,y_n)$$\nwhere $\\phi(x_1,...,x_k, y_1,...,y_n)$ is quantifier-free.\n\\end{definition}\nA $\\sup$-formula of $L$ is defined similarly. These $\\sup$-formulas\nare the universal formulas in continuous logic. For an $L$-theory $T$, we use the\nnotation $T_\\forall$ for the set of universal sentences ($\\sup$-formulas with no free variables) \nimplied by the theory $T$.  Note that, as in classical first order logic, $T_\\forall$\nis the theory of the class of $L$-substructures of models of $T$.\n\n\\begin{definition}\nLet $T$ be an $L$-theory and suppose $\\mathcal{M} \\models T$. We say $\\mathcal{M}$ is\nan {\\it existentially closed (e.c.)} model of $T$ if, for any\n$\\inf$-formula $\\psi(x_1,\\dots,x_m)$,\nany $a_1,...,a_k \\in M$, and any $\\mathcal{N} \\models T$ that is an extension of $\\mathcal{M}$,\nwe have $\\psi^\\mathcal{N}(a_1,\\dots,a_m) = \\psi^\\mathcal{M}(a_1,\\dots,a_m)$.\n\\end{definition}\n\nAn $L$-theory $T$ is {\\it model complete} if any embedding between models\nof $T$ is an elementary embedding.\n\n\\begin{proposition}\n\\label{allec}\nThe $L$-theory $T$ is model complete if and only if\nevery model of $T$ is an existentially closed model of $T$.\n\\end{proposition}\n\n\\begin{proof}\nThis is Robinson's Criterion for model completeness.  The proof given in\n\\cite[Theorem 8.3.1]{Ho} for classical first order logic can easily be adapted to the continuous setting.\n\\end{proof}\n\nIn \\cite[Appendix A]{IN} there is some further discussion of $\\inf$- and $\\sup$-formulas and of model completeness.\n\n\\begin{definition}\n\\label{defnmodelcompanion} Let $T$ be an $L$-theory. A {\\it model\ncompanion} of $T$ is an $L$-theory $S$ such that:\n\\begin{itemize}\n\\item every model of $S$ embeds in a model of $T$;\n\\item every model of $T$ embeds in a model of $S$;\n\\item $S$ is model complete.\n\\end{itemize}\n\\end{definition}\n\nNote that the first two criteria in this definition together\nare equivalent to the statement $S_{\\forall}=T_{\\forall}$.\nAs in classical first order logic, if a theory has a model\ncompanion, then that model companion is unique (up to equivalence of\ntheories).\n\nRecall that a theory $T$ is \\emph{inductive} if whenever $\\Lambda$ is a linearly ordered set\nand $(\\mathcal{M}_\\lambda\\mid\\lambda \\in \\Lambda)$ is a chain of\nmodels of $T$, then the completion of the union of $(\\mathcal{M}_\\lambda\\mid\\lambda \\in \\Lambda)$ is a model of $T$.\n\n\\begin{proposition}\n\\label{axiomatizeec}\nLet $T$ be an inductive $L$-theory\nand let $\\mathcal{K}$ be the class of existentially closed models of $T$.\nIf there exists an $L$-theory $S$ so that $\\mathcal{K}=\\Mod(S)$, then\n$S$ is the model companion of $T$.\n\\end{proposition}\n\\begin{proof}\nThe proof from \\cite[Theorem 8.3.6]{Ho} can be adapted to the continuous setting.\n\\end{proof}\n\nWe say the $L$-theory $T$ has \\emph{amalgamation over substructures} if\nfor any substructures  $\\mathcal{M}_0$, $\\mathcal{M}_1$ and $\\mathcal{M}_2$ of models of $T$ and\nembeddings $f_1\\colon \\mathcal{M}_0\\rightarrow \\mathcal{M}_1$, $f_2\\colon \\mathcal{M}_0\\rightarrow \\mathcal{M}_2$,\nthere exists a model $\\mathcal{N}$ of $T$  and embeddings\n$g_i\\colon \\mathcal{M}_i\\rightarrow \\mathcal{N}$ such that $g_1\\circ f_1=g_2\\circ f_2$.\n\n\n\n\\begin{proposition}\n\\label{modelcompanionplusamalg}\nLet $T_1$ and $T_2$ be $L$-theories such that $T_2$\nis the model companion of $T_1$.  Assume $T_1$ has amalgamation over substructures.\nThen $T_2$ has quantifier elimination.\n\\end{proposition}\n\n\\begin{proof}\nThe corresponding result in classical first order logic is the\nequivalence of (a) and (d) in \\cite[Theorem 8.4.1]{Ho}.  The proof\ngiven there can be adapted to the continuous setting.\n\\end{proof}\n\n\\endinput\n\n\n\\section{The theory of pointed $\\R$-trees with radius at most $r$}\n\\label{theory of R-trees}\nIn this section we first present the continuous\nsignature used in this paper to study $\\R$-trees.  We then give\naxioms for the theory $\\mathbb{R}\\mathrm{T}_r$ of $\\R$-trees with radius $\\leq r$.\n\nLet $r>0$ be a real number. Define the signature $L_r:=\\{p\\}$ where $p$ is a \nconstant symbol and specify\nthat the metric symbol $d$ has values which lie in the interval $[0,2r]$.\nAny pointed metric space $(M, p)$ with radius $\\leq r$ naturally\ngives rise to an $L_r$-prestructure $\\mathcal{M}=(M,d,p)$, in which $d$ is a metric;\n$\\mathcal{M}$ is an $L_r$-structure if the metric space involved is\nmetrically complete.\n\nNext we define a set of axioms $\\mathbb{R}\\mathrm{T}_r$ for the class of complete,\npointed $\\R$-trees of radius $\\leq r$. \nRecall the connective\n$\\mathbin{\\mathpalette\\dotminussym{}}\\colon [0,\\infty)\\times[0,\\infty) \\rightarrow [0,\\infty)$\ndefined by $x\\mathbin{\\mathpalette\\dotminussym{}} y=\\max\\{x-y,0\\}$.\n\n\n\\begin{definition}\n\\label{rtreetheory}\nLet $\\mathbb{R}\\mathrm{T}_r$ be the $L_r$-theory consisting of the following conditions:\n\\begin{enumerate}\n\n    \\item \\label{boundax}\n  \\hfil$\\sup_{x} d(x,p) \\leq r$;\\vspace{0.25cm}\n\n    \\item \\label{midptax} \n    \\hfil{$\\sup_{x}\\sup_{y}\\inf_{z}\\max\\{\n    \\big|d(x,z)-\\frac12 d(x,y) \\big|,\\ \\ \\big| d(y,z)-\\frac12 d(x,y) \\big|\\}=0$;}\\vspace{0.25cm}\n\n    \\item \\label{hypax} \n    \\hfil{$\\sup_{x}\\sup_{y}\\sup_{z}\\sup_{w}\n    \\big( \\min\\{(x\\cdot z)_w, (y\\cdot z)_w\\}\\mathbin{\\mathpalette\\dotminussym{}} (x\\cdot y)_w\\big)=0$.}\n\n\\end{enumerate}\n\\end{definition}\n\nThe next lemma shows that the class of complete pointed $\\R$-trees of\nradius $\\leq r$ is axiomatized by $\\mathbb{R}\\mathrm{T}_r$.\n\n\n\\begin{lemma}\n\\label{modelsofT}\nThe models of $\\mathbb{R}\\mathrm{T}_r$ are exactly the complete, pointed $\\R$-trees of radius $\\leq r$. \n\\end{lemma}\n\\begin{proof}\nFirst we assume $\\mathcal{M}\\models\\mathbb{R}\\mathrm{T}_r$.  Then $(M, d,p)$ is a complete, pointed\nmetric space. Axiom (\\ref{boundax}) guarantees that $M$ has radius $\\leq r$. \nAxiom (\\ref{midptax}) implies that for any $x,y\\in M$ and\nany $\\epsilon >0$ there is $z\\in M$ such that $d(x,z)$ and $d(y,z)$ are within $\\epsilon$\nof $\\frac{1}{2}d(x,y)$. Iterating this process, we \ndefine a map from the set of dyadic rational numbers in $[0,1]$ into $M$ so that $f(0)=x$ and $f(1)=y$.\nBy letting $\\epsilon$ approach $0$ fast enough, we make this map uniformly continuous, so that\nits completion will be a path from $x$ to $y$ in $M$.\nTherefore, $M$ is path-connected. \nUsing Definition \\ref{defhyperbolic}, Axiom (\\ref{hypax}) implies that $M$ is a $0$-hyperbolic metric space.\nBy \\cite[Lemma 2.4.13]{C}, any connected $0$-hyperbolic metric space\nis an $\\R$-tree. Therefore, $\\mathcal{M}$ is a pointed $\\R$-tree with radius $\\leq r$.\nThat a complete, pointed $\\R$-tree with radius $\\leq r$ is a model of $\\mathbb{R}\\mathrm{T}_r$ is clear.\n\\end{proof}\n\n\\begin{remark}\nStructures in continuous logic are required to be metrically complete, while\nin general, $\\R$-trees are not complete.\nA pointed $\\R$-tree $M$ with radius $\\leq r$ can naturally be viewed as an\n$L_r$-prestructure, which is an $L_r$-structure iff it is complete (since the\npseudometric on the prestructure is actually a metric). If $M$ is not complete,\nthen its metric completion is known to be an $\\R$-tree (see \\cite[Lemma 2.4.14]{C}).\nFurther, the completion of a prestructure is known to be an elementary extension,\nand therefore the prestructure and its completion are completely equivalent from a \nmodel-theoretic perspective. (See \\cite[pages 15--17]{N}.)\nNote that this also means any two pointed $\\R$-trees of radius $\\leq r$ that have the\nsame metric completion are indistinguishable from a model-theoretic\nperspective, and that a metrically complete $\\R$-tree can be identified model-theoretically \nwith any of\nits dense sub-prestructures.  (However, those metric sub-prestructures are not\nnecessarily $\\R$-trees. In particular, they are not necessarily geodesic spaces.)\n\\end{remark}\n\n\\bigskip\nWe close this section by noting a property of $\\mathbb{R}\\mathrm{T}_r$ that will be used later.\n\n\\begin{lemma}\n\\label{chainofmodels}\nThe theory $\\mathbb{R}\\mathrm{T}_r$ is inductive. That is, the completion of the union\nof an arbitrary chain of models of $\\mathbb{R}\\mathrm{T}_r$ is a model of $\\mathbb{R}\\mathrm{T}_r$.\n\\end{lemma}\n\\begin{proof}\nThe proof of \\cite[Lemma 2.1.14]{C} can be modified to show that the\nunion of an arbitrary chain of pointed $\\R$-trees is again a pointed $\\R$-tree.\nAlso, the completion of an $\\R$-tree is an $\\R$-tree.\nSince basepoints are preserved by embeddings\nof models, the radius of the underlying pointed $\\R$-tree for the union of a chain is most $r$. \n\nAlternatively, note that $\\mathbb{R}\\mathrm{T}_r$ is an $\\forall \\exists$-theory, and therefore the\nclass of its models is closed under completions of unions of chains.\n\\end{proof}\n\n\\section{Some Definability in $\\mathbb{R}\\mathrm{T}_r$}\n\\label{definable}\n\nWe now discuss the notion of \\emph{definability} for subsets of and functions\non the underlying $\\R$-tree $M$ of a model $\\mathcal{M}$ of $\\mathbb{R}\\mathrm{T}_r$.  For background\non definable predicates, sets and functions see \\cite{N}.\nThe first result shows that every closed ball centered at the base point\nis uniformly quantifier-free $0$-definable in models of $\\mathbb{R}\\mathrm{T}_r$.\n\n\\begin{lemma}\n\\label{geodballs}\nLet $s\\in [0,r]$. Let $\\phi(x)$ be the quantifier-free formula $d(x,p)\\mathbin{\\mathpalette\\dotminussym{}} s$.  Suppose $\\mathcal{M}\\models \\mathbb{R}\\mathrm{T}_r$ and $(M,d,p)$ is the underlying $\\R$-tree of $\\mathcal{M}$. Then the closed ball\n$B_s(p)\\subseteq M$ is uniformly $0$-definable.  Indeed, for $x\\in M$ we have $\\dist(x, B_s(p))=\\phi(x)^\\mathcal{M}$.\n\\end{lemma}\n\n\\begin{proof}\nIt suffices to show that $\\phi^\\mathcal{M}(x)$ is equal to the distance function\n$\\dist(x,B_s(p))$.\nFor $x\\in M$ we know $\\phi^\\mathcal{M}(x)=0$ if and only if $d(x,p)\\leq s$,\ni.e. if and only if $x\\in B_s(p)$.\nNow, let $x\\notin B_s(p)$. Then $\\phi^\\mathcal{M}(x)=d(x,p)-s$.\nLet $\\gamma$ be a geodesic segment from $p$ to $x$ with $\\gamma(0)=p$.\nThen\n$$\\dist(x,B_s(p))\\leq d(x,\\gamma(s))=d(x,p)-d(p, \\gamma(s))=d(x,p)-s=\\phi^\\mathcal{M}(x).$$\n\nNow toward a contradiction, assume $d(x,p)-s >\\dist(x, B_s(p)).$\nThen there exists a point in $c\\in B_s(p)$ with $d(c,x)<d(x,p)-s$.  This implies,\n$$d(x,p)\\leq d(p,c)+d(c,x)\\leq s+d(c,x)<s+d(x,p)-s=d(x,p)$$\nso $d(x,p)<d(x,p)$,\na contradiction. Therefore, $\\dist(x, B_s(p))=\\phi(x)^\\mathcal{M}.$\n\\end{proof}\n\nNote that the preceding lemma holds in any metric structure where the underlying\nspace is a pointed geodesic space.  Moreover, it can be easily adapted to show that\nany closed ball is definable over its center.  \n\nNext, we present a short discussion of some specific definable functions,\npoints and sets in models of $\\mathbb{R}\\mathrm{T}_r$.\n\nLet $M$ be an $\\R$-tree and for $s\\in [0,1]$ define the function\n$\\nu_s\\colon M\\times M\\rightarrow M$ by:\n$\\nu_s(x_1,x_2)=$ the point in $[x_1,x_2]$ with distance $sd(x_1, x_2)$ from $x_1$\nand distance $(1-s)d(x_1, x_2)$ from $x_2$. When $s=\\frac{1}{2}$ we refer to the function $\\nu_{1\/2}$ as the definable\nmidpoint function.  \n\n\\begin{lemma}\n\\label{definablemidpoints}\nLet $s\\in [0,1]$. The function $\\nu_s\\colon M\\times M \\rightarrow M$\nis uniformly $0$-definable in models $\\mathcal{M}$ of $\\mathbb{R}\\mathrm{T}_r$ via a quantifier-free formula.\n\\end{lemma}\n\\begin{proof}\nLet $\\psi$ be the formula\n$$\\max\\{d(x_1,y)\\mathbin{\\mathpalette\\dotminussym{}} sd(x_1,x_2), d(x_2,y)\\mathbin{\\mathpalette\\dotminussym{}} (1-s)d(x_1,x_2)\\}.$$\nLet $\\mathcal{M}\\models \\mathbb{R}\\mathrm{T}_r.$\nIn $\\mathcal{M}$, the distance $d(\\nu_s(x_1,x_2),y)$ is equal to $\\psi^\\mathcal{M}(x_1,x_2,y).$\nSo the function $\\nu_s$ is $0$-definable via this quantifier-free formula in any model of $\\mathbb{R}\\mathrm{T}_r$.\n\\end{proof}\n\n\nIn Definition \\ref{Ypoint} we defined $Y=Y(a,b,c)$, the unique point so that\n$\\{Y\\}=[a,b]\\cap[b,c]\\cap[a,c]$. Next we show that $Y$ is a definable function.\n\n\n\n\\begin{theorem}\nThe function $Y\\colon M^3\\rightarrow M$ \nthat for inputs $x_1, x_2, x_3\\in M$ returns $Y(x_1, x_2, x_3)$ is uniformly\n$0$-definable in models $\\mathcal{M}$ of $\\mathbb{R}\\mathrm{T}_r$ via a quantifier-free formula.\n\n\\end{theorem}\n\\begin{proof}\nRecall that in an $\\R$-tree, the Gromov\nproduct $(a \\cdot b)_x$ is equal to the distance from $x$ to the segment $[a,b]$,\nand this distance is realized by a unique closest point on $[a,b]$\n\nLet\n$$\\phi(x_1, x_2, x_3, x)=\\max\\left\\{(x_1\\cdot x_2)_x, (x_1\\cdot x_3)_x, (x_2\\cdot  x_3)_x\\right\\}.$$\nLet $\\mathcal{M}\\models \\mathbb{R}\\mathrm{T}_r$ and $x_1, x_2, x_3\\in M$. We will show that in $\\mathcal{M}$, the distance $d( x, Y(x_1, x_2, x_3))=\\phi^\\mathcal{M}(x_1, x_2, x_3, x)$. \nLet $E$ be the closed subtree of $M$ spanned by $x_1, x_2$ and $x_3$. Abbreviate $Y(x_1, x_2, x_3)$ by $Y$.\nFor $x\\in M$, let $z$ be the unique point closest to $x$ in $E$. Then we have $d(x, Y)=d(x, z)+d(z, Y)$. The point $z$ must lie on at least one of $[x_1, Y]$, $[x_2, Y]$ or $[x_3, Y]$.\nWithout loss of generality, assume $z\\in [x_1, Y]=[x_1, x_3]\\cap[x_1, x_2]$. \n\nSince $z$ is closest in $E$ to $x$  and $z\\in [x_1, x_3]\\cap[x_1, x_2]$ we know $d(x, z)=(x_1\\cdot x_2)_x=(x_1\\cdot x_3)_x$.\nThat $z$ is closest in $E$ to $x$ also implies $(x_2\\cdot x_3)_x\\geq d(x,z)$. \nThis makes $(x_2\\cdot x_3)_x\\geq(x_1\\cdot x_2)_x=(x_1\\cdot x_3)_x$.\nThus, $\\phi^{\\mathcal{M}}(x_1, x_2, x_3, x)=(x_2\\cdot x_3)_x$.\n\nSince $z\\in [x_1, Y]$, the point $Y$ is the closest point to $z$ on $[x_2, x_3]$, and therefore $Y$ is the closest point to $x$ on $[x_2, x_3].$ It follows that, $d(x, Y)=(x_2\\cdot x_3)_x$.\nWe conclude that $\\phi^{\\mathcal{M}}(x_1, x_2, x_3, x)=d(x,Y).$\n\n\n\\end{proof}\n\n\n\\section{Amalgamation} \n\nNext, we discuss amalgamation for the $L_r$-theory $\\mathbb{R}\\mathrm{T}_r$.\nThe following result from \\cite{C} discussing amalgamating over points in $\\R$-trees is then extended to amalgamation of $\\R$-trees over subtrees. This leads to a proof of amalgamation over substructures for $\\mathbb{R}\\mathrm{T}_r$.\n\n\\begin{lemma}\n\\label{chiswell add to points}\nLet $(X,d)$ be an $\\R$-tree and let $\\{(X_i, d_i)\\mid i\\in I\\}$ be a family of $\\R$-trees\nsuch that\n$X_i\\cap X=\\{x_i\\}$ for all $i\\in I$.\n$X_i\\cap X_j= \\{x_i\\}$ if $x_i=x_j$ and $X_i\\cap X_j=\\emptyset$ otherwise.\nDefine $N=(\\bigcup_{i\\in I} X_i)\\cup M$ and define $\\hat d\\colon N\\times N \\rightarrow \\R$ by \n\\begin{itemize}\n\\item On $X\\times X$, $\\hat d=d$, and on $X_i\\times X_i$, $\\hat d=d_i$\n\\item If $x\\in X_i$ and $x'\\in X_j$, with $i\\not= j$, then\n$\\hat d(x,x')=\\hat d(x',x)=d_i(x,x_i)+d(x_i, x_j)+d_j(x_j, x')$\n\\item If $x\\in X_i$ and $x'\\in X$, then $\\hat d(x,x')=\\hat d(x',x)=d_i(x,x_i)+d(x_i, x')$\n\\end{itemize}\nThen $(N,\\hat d)$ is an $\\R$-tree.\n\\end{lemma}\n\\begin{proof}\nSee \\cite[Lemma 2.1.13]{C}\n\\end{proof}\n\nWe next apply this lemma to prove $\\R$-trees can be amalgamated over subtrees.\n\n\\begin{lemma}\n\\label{rtreeamalg}\nGiven $\\R$-trees $M_0$, $M_1$ and $M_2$ and isometric embeddings\n$f_i\\colon M_0\\rightarrow M_i$ for $i=1, 2$ there exists an $\\R$-tree $N$\nand isometric embeddings $g_i\\colon M_i\\rightarrow N$ such that $g_1\\circ f_1=g_2\\circ f_2$.\n\\end{lemma}\n\n\\begin{proof}\nWithout loss of generality, we may assume that $M_0\\subseteq M_1$ and $M_0\\subseteq M_2$,\nthat $M_1\\cap M_2=M_0$, and that the $f_i$ are inclusion maps.\nLet $\\{x_k\\mid k\\in |M_0|\\}$ be a list of the elements of $M_0$.\nFor each $k\\in |M_0|$ let $\\alpha_k$ be the cardinality of the set of branches at $x_k$,\nand let $A_k=\\{B_v\\mid v\\in \\alpha_k\\}$ be a list of the distinct branches at $x_k$ in $M_2$.\nThen let $I=\\{(k, v)\\mid k\\in|M_0|, v\\in \\alpha_k\\},$\nand define $\\beta_i=B_v\\cup \\{x_k\\}$ for each $i=(k,v)\\in I$.\nThus, for each $i\\in I$ we get a distinct subtree $\\beta_i$ of $M_2$\nsuch that $(\\bigcup_{i\\in I} \\beta_i)=M_2$. \nIt is straightforward to show that the $\\R$-tree $(M_1, d_1)$ and the family\n$\\{\\beta_i\\mid i\\in I\\}$ fit the hypotheses of Lemma \\ref{chiswell add to points}.\nThus $N=(\\bigcup_{i\\in I} \\beta_i)\\cup M_1$ with the metric $\\hat d$ is an $\\R$-tree, and clearly $N=M_2\\cup M_1$. Define $g_i$ to be the inclusion of $M_i$ in $N$. Then $g_1\\circ f_1=g_2\\circ f_2$ is clear.\n\\end{proof}\n\n\nTo apply these results to models of $\\mathbb{R}\\mathrm{T}_r$, we begin by\nproving any subset of a model of $\\mathbb{R}\\mathrm{T}_r$ gives rise to a unique model of $\\mathbb{R}\\mathrm{T}_r$.\nIn particular, any substructure extends to a unique model.\n\n\\begin{lemma}\n\\label{substructureofrtree}\nLet $\\mathcal{N}\\models \\mathbb{R}\\mathrm{T}_r$. Any subset $A\\subseteq N$ extends to a model $\\mathcal{M}\\models \\mathbb{R}\\mathrm{T}_r$\nsuch that any embedding of $A$ into $\\mathcal{W}\\models \\mathbb{R}\\mathrm{T}_r$ extends to an embedding of $\\mathcal{M}$\ninto $\\mathcal{W}$.\n\\end{lemma}\n\\begin{proof}\nLet $(N,d,p)$ be the underlying $\\R$-tree of the model $\\mathcal{N}$, and let $A\\subseteq N$.   \nDefine the structure $\\mathcal{M}=(M, d,p)$ by $M=\\overline{E_{A\\cup \\{p\\}}}$ with the metric and base point from $(N,d,p)$.\nThen $M$ is the smallest closed subtree of $N$ containing $A\\cup\\{p\\}$.\nIt is straightforward to show that $\\mathcal{M}\\models \\mathbb{R}\\mathrm{T}_r$ and is the desired extension of $A$.\n\\end{proof}\n\n\n\n\\begin{theorem}\n\\label{amalg}\nThe $L_r$-theory $\\mathbb{R}\\mathrm{T}_r$ has amalgamation over substructures.\nThat is, if  $\\mathcal{M}_0$, $\\mathcal{M}_1$ and $\\mathcal{M}_2$\nare substructures of models of $\\mathbb{R}\\mathrm{T}_r$ and\n$f_1\\colon \\mathcal{M}_0\\rightarrow \\mathcal{M}_1$, $f_2\\colon \\mathcal{M}_0\\rightarrow \\mathcal{M}_2$ are embeddings,\nthen there exists a model $\\mathcal{N}$ of $\\mathbb{R}\\mathrm{T}_r$  and embeddings\n$g_i\\colon \\mathcal{M}_i\\rightarrow \\mathcal{N}$ such that $g_1\\circ f_1=g_2\\circ f_2$.\n\\end{theorem}\n\\begin{proof}\nFirst, by Lemma \\ref{substructureofrtree} we may assume that $\\mathcal{M}_0$, $\\mathcal{M}_1$ and $\\mathcal{M}_2$\nare models of $\\mathbb{R}\\mathrm{T}_r$ with underlying complete, pointed $\\R$-trees $(M_i, d_i, p_i)$.\nFurther, we assume that $M_0\\subseteq M_i$ and $p_0=p_i$ for $i=1,2$\nand that the $f_i$ are inclusion maps.\n\nWe use Lemma \\ref{rtreeamalg} to construct the $\\R$-tree $(N, d)$ \nand isometric embeddings $g_i\\colon M_i\\rightarrow N$ such that $g_1\\circ f_1=g_2\\circ f_2$.\nRecall that in this construction, $N=M_1\\cup M_2$.\nDefine the base point of $N$ to be $q=p_1=p_2=p_0$.\nSince every point in each $M_i$ has distance $\\leq r$ from $p_i=q$, we conclude that the pointed $\\R$-tree $(N,d,q)$ has radius $\\leq r$.\nLet $\\mathcal{N}=(N,d,q)$ be the corresponding $L_r$-structure. \nThen $\\mathcal{N}$ is a model of $\\mathbb{R}\\mathrm{T}_r$, and since $g_i$ are isometric embeddings\npreserving the base point, they give rise to embeddings of $L_r$-structures as required.\n\\end{proof}\n\n\\endinput\n\n\\section{The model companion of $\\mathbb{R}\\mathrm{T}_r$}\n\\label{model companion}\n\nIn this section we define what it means for a pointed $\\R$-tree of radius $\\leq r$\nto be {\\it richly branching.}\nWe then show that the theory of richly branching pointed $\\R$-trees\nwith radius $r$ is the model companion of $\\mathbb{R}\\mathrm{T}_r$.\n\n\n\\begin{definition}\n\\label{richlybranching}\n\nA pointed $\\R$-tree $(M, d, p)$ of radius $\\leq r$ is {\\it richly branching} if the set\n$$B=\\{b\\in M\\mid \\text{ at } b \\text{ there are at least 3 branches of height } \\geq r-d(p, b)\\}$$\nis dense in $M$.\n\\end{definition}\n\n\\begin{remark}\n\\label{general richly branching}\nThe preceding definition and arguments below apply specifically to pointed $\\R$-trees with radius $\\leq r.$  For general $\\R$-trees we define:\n\\emph{an $\\R$-tree $M$ is {\\it richly branching} if the set of points at which there are at least 3 branches of infinite height is dense in $M$.} Note that an $\\R$-tree that is richly branching in this sense must be unbounded.\n\nTo help illustrate this idea, consider a segment $[x,y]$ where $x\\not=y$ in a richly branching $\\R$-tree $M$. \\emph{There must exist a point $z$ in the open segment $(x,y)$ such that there is a branch of infinite height at $z$ containing neither $x$ nor $y$.} (We leave the proof as an exercise for the reader.) Indeed, working inductively, it is easy to see that this property implies that the set of such points $z$ is dense in $(x,y)$.  Therefore, $(x,y)$ contains infinitely many points $z$ at each of which there are at least 3 branches of infinite height.\n\nIf we start with an unbounded richly branching $\\R$-tree $M$, assign an arbitrary basepoint $p$ and select $r>0$, we can make a richly branching $\\R$-tree with radius $\\leq r$ as in Definition \\ref{richlybranching} by taking the closed ball of radius $r$ with center $p$ in $M$. Note that this yields an $L_r$-prestructure\nand the completion is an $L_r$-structure.\n\nConversely, suppose $M$ is an $\\R$-tree and $p$ is any point in $M$.  If the closed ball of radius $r$ with center $p$ in $M$ is richly branching in the sense of \\ref{richlybranching} for an unbounded set of $r>0$, then $M$ is richly branching as a general $\\R$-tree.\n(See Lemma \\ref{branches of sufficient height} below.) \\end{remark}\n\n\n\n\n\nNext we give axioms for the class of complete, richly branching, pointed $\\R$-trees with radius $r$.\n\n\\begin{definition}\n\\label{rbrtaxioms}\nDefine ${\\psi}(x)$ to be the $L_r$-formula\n$$\\inf_{y_1}\\inf_{y_2}\\inf_{y_3}\\max\\left\\{\\max_{i=1,2,3}\\{|d(x,y_i)-(r-d(p,x))|\\},\n\\max_{1\\leq i<j \\leq 3}\\{ d(x,y_i)+d(x,y_j)-d(y_i,y_j) \\}\\right\\}$$\nand let $\\phi=\\displaystyle \\sup_{x} {\\psi}(x)$.\n\\end{definition}\n\n\n\\begin{definition}\n\\label{rbrttheory}\nLet $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r=\\mathbb{R}\\mathrm{T}_r\\cup \\{\\phi=0\\}$.\n\\end{definition}\n\nTo help parse these axioms and picture what they mean, note that\n$d(x,y_i)+d(x,y_j)-d(y_i,y_j)=2(y_i\\cdot y_j)_x$ so in an $\\R-$tree\n$d(x,y_i)+d(x,y_j)-d(y_i,y_j)=2\\dist(x, [y_i, y_j])$. Consider the case when the infima are realized exactly\nin $M$ (e.g. if $M$ an $\\omega$-saturated model.) In this setting, $\\phi=0$ means that for any element $a\\in M$\nwith $d(p, a)<r$, there exist $b_1, b_2, b_3$ each on a separate branch at $a$ and each\nwith distance $r-d(p,a)$ from $a$.\nWe know $b_1, b_2, b_3$ are on distinct branches at $a$  because $2\\dist(a, [b_i, b_j])=0$ for each $1\\leq i<j\\leq 3$. This makes $a$ a branch point.\nIn a general model where the infima are not necessarily realized, an approximate version\nof this is true; given an element $a$ there must be a branch point within $\\epsilon$ of $a$, as we now show.\n\n\n\\begin{lemma}\n\\label{branching point within epsilon}\nLet $\\mathcal{M}\\models \\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ with underlying pointed $\\R$-tree $(M, d, p)$ and let $a\\in M$. Let $h\\in \\R$ so that $0<h<r-d(p,a)$. For any $\\epsilon>0$ there exists a point\n$b\\in M$ so that $d(a,b)<\\epsilon$ and there are 3 branches at $b$,\neach with height at least $h.$ \n\\end{lemma}\n\\begin{proof}\nLet $l=r-d(p, a)$, and assume $0<4\\epsilon< l-h$.\nSince $\\phi^{\\mathcal{M}}=0$, for $a\\in M$ we know $\\psi(a)^{\\mathcal{M}}=0$.\nThus there exist points $c_1, c_2, c_3\\in M$ so that \n$$|d(a,c_i)-l|<\\epsilon \\text{ and } d(a,c_i)+d(a,c_j)-d(c_i,c_j)<\\epsilon. $$\nIt follows that for $i\\not=j$ $$d(c_i, c_j)>d(a, c_i)+d(a, c_j)-\\epsilon >2l-3\\epsilon>0$$ meaning $c_1, c_2, c_3$ must be distinct. Moreover, $c_1, c_2$\nand $c_3$ cannot lie along a single geodesic segment.\nTo see this assume $[c_i, c_k]=[c_i, c_j, c_k]$  for $\\{i,j,k\\}\\in\\{1, 2, 3\\}$. Then $d(c_i, c_k)=d(c_i, c_j)+d(c_j, c_k)$, implying\n$$d(c_i, c_k)=d(c_i, c_j)+d(c_j, c_k)>4l-6\\epsilon.$$\nHowever, \n$$d(c_i, c_k)\\leq d(a, c_i)+d(a, c_k)<2l+2\\epsilon.$$\nThese inequalities imply $4\\epsilon >l>l-h$, a contradiction.  So, each of\n$c_1, c_2, c_3$ lies on\na different branch at $Y(c_1, c_2, c_3)$.\nThus, there are at least 3 branches at $Y(c_1, c_2, c_3)$.  \n\nNext, let $q=Y(a, c_1, c_2)$,  $r=Y(a, c_2, c_3)$ and $s=Y(a, c_1, c_3)$.\nWe proceed under the assumption that $d(a, q)\\leq d(a, r)\\leq d(a, s)$, or in other\nwords, $\\dist(a, [c_1, c_2])\\leq \\dist(a, [c_2, c_3])\\leq \\dist (a, [c_1, c_3])$.\nThe other possible cases proceed similarly.\nNote that $M$ being $0$-hyperbolic now implies $\\dist(a, [c_1, c_2])=\\dist(a, [c_2, c_3]$).\nLemma 2.1.6 in \\cite{C} and the subsequent discussion\nyield that $q=r$, so the point closest to $a$  on $[c_1, c_2]$ and the point\nclosest to $a$ on $[c_2, c_3]$ are the same point and $s=Y(c_1, c_2, c_3)$.\nBy Lemma \\ref{branch} $2\\dist(a, [c_1, c_3])=d(a,c_1)+d(a,c_3)-d(c_1,c_3)$.\nThus, $d(a, s)=\\dist(a, [c_1, c_3])<\\frac{\\epsilon}{2}<\\epsilon$.\nSo, $s=Y(c_1, c_2, c_3)$ has distance $<\\epsilon$ from $a$.\nLastly, for $i=1, 2, 3$ we have\n$$d(s, c_i)\\geq d(a, c_i)-d(a,s)>l-\\epsilon-\\frac{\\epsilon}{2}$$\n\n$$=l-\\frac{3\\epsilon}{2}>l-4\\epsilon>h.$$\nSo, $b=Y(c_1, c_2, c_3)$ works.\n\\end{proof}\n\nThe next lemma shows that any branch at any point in a model of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$\nmust have maximum possible height with that height realized by a point with distance $r$ from $p$.  This also\nimplies every model of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ has radius equal to $r$.\n\n\\begin{lemma}\n\\label{branches of sufficient height}\nLet $\\mathcal{M}\\models \\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ and let $a\\in M$.\nIn any branch $\\beta$ at $a$, there exists at least one point $b$ so that $d(p, b)=r$. \n\\end{lemma}\n\\begin{proof}\nLet $a\\in M$ and $\\beta$ a branch at $a$. Let $\\delta=r-d(p, a)$. \nWe first assume $p\\notin \\beta$ (including the case where $a=p$.) \nIn what follows, by iterating the use of Lemma \\ref{branching point within epsilon},\nwe build a sequence $b_1, b_2, b_3,....$ of points in $\\beta$\nso that for any $k\\in \\N$, $[p, a, b_1, ..., b_k]$ is a geodesic segment, and\n$|r-d(p, b_i)|=r-d(p, b_i)<\\frac{\\delta}{2^i}$ \n\n\nLet $c\\in \\beta$. So, $d(a,c)>0$. If $c$ is such that $r-d(p, c)<\\frac{\\delta}{2}$, then\nlet $c=b_1$. If $r-d(p,c)\\geq \\frac{\\delta}{2}$, then let $0<\\epsilon<\\frac{d(a, c)}{2}$.\nUse Lemma \\ref{branching point within epsilon} to find $c'$ so that $d(c, c')<\\epsilon$, \nand so that there are at least 3 branches at $c'$ with height at least\n$(1-\\frac{d(a,c)}{2\\delta})(r-d(p, c)).$  So, $|d(a, c)-d(a, c')|<\\epsilon$, and since $\\epsilon<\\frac{d(a, c)}{2}$ we know $c'\\in \\beta$.  Then find a point $b_1$\non a branch at $c'$ other than the one containing $a$. This makes $[p, a, c', b_1]$ a piecewise segment.  Select\n$b_1$ so that $d(c', b_1)=(1-\\frac{d(a,c)}{\\delta})(r-d(p,c))\\geq (1-\\frac{d(a,c)}{\\delta})\\frac{\\delta}{2}=\\frac{\\delta}{2}-\\frac{d(a,c)}{2}$.\nThen\n$$d(a, b_1)=d(a, c')+d(c', b_1)>(d(a, c)-\\epsilon)+d(c', b_1)$$\n$$>d(a,c)-\\frac{d(a,c)}{2}+\\frac{\\delta}{2}-\\frac{d(a,c)}{2}\n=\\frac{\\delta}{2}.$$\nThus $d(a, b_1)>\\frac{\\delta}{2}$. So, $r-d(p, b_1)=r-(d(p, a)+d(a, b_1))=(r-d(p,a))-d(a, b_1)\n=\\delta-d(a, b_1)<\\delta-\\frac{\\delta}{2}=\\frac{\\delta}{2}.$\n\n\nOnce we have $b_i$, find $b_{i+1}$ in a manner analogous to the argument above,\nso that $|r-d(p, b_i)|=r-d(p, b_i)<\\frac{\\delta}{2^i}.$ Then,\ngiven an index $i$, for any $j>i$\nwe have $d(b_i, b_j)=d(p, b_j)-d(p, b_i)<r-(r-\\frac{\\delta}{2^i})=\\frac{\\delta}{2^i}$.\nIt follows that the sequence $(b_n)$ is Cauchy.\nLet $b$ be the limit of the sequence $(b_i)$.  \n\n\nClaim: $b_i\\in [p, b]$ for all $i\\in \\N$. If not, let $i\\in \\N$ be such that $b_i\\notin [p, b]$.\nThen for all $j\\geq i$, $b_j\\notin [p, b]$ because $[p, b_i, b_j]$ is a geodesic segment.\nLet $j>i$ and let $q$ be the closest point to $b$ on $[p, b_i, b_j]$, i.e. $q=Y(p, b_j, b)$. \nEither $q\\in [p, b_i]$ or $q\\in [b_i, b_j]$.  If $q\\in [b_i, b_j]$, then $[p, q]=[p, b_i, q]$\nimplying $b_i\\in [p, b]=[p, b_i, q, b]$, a contradiction. \nThus, $q\\in [p, b_i]$, and $[p, q, b_i, b_j]$ is a geodesic segment.\nUsing these facts and our choice of $q$ we conclude\n$$d(b, b_j)=d(b, q)+d(q, b_j)=d(b, q)+d(q, b_i)+d(b_i, b_j)\\geq d(b, q)+d(q, b_i)=d(b, b_i).$$\nWe have demonstrated that  $d(b, b_j)\\geq d(b, b_i)>0$ for all $j>i$, contradicting\nthat $b$ is the limit of the sequence $(b_i)$.\n\nBecause $b_i\\in [p, b]$, we know, $r\\geq d(p, b)\\geq d(p, b_i)>r-\\frac{\\delta}{2^i}$  for any $i\\in \\N$.\nTherefore, $d(p, b)=r$. And $b\\in \\beta$, since otherwise it would be on a different branch\nat $a$ than $b_1$. This would make $[b_1, a, b]$ a geodesic segment and contradict\nthat $[p,b]=[p, a, b_1, b]$ is a geodesic segment.\n\n\nNow, assume $p\\in \\beta$. Then $[p,a)\\subseteq \\beta$.\nUsing Lemma \\ref{branching point within epsilon}\nwe find a point $a'\\in [p, a)$ with 3 branches at $a'$.\nSelect $\\beta'$ a branch\nat $a'$ so that $p\\notin \\beta'$ and $a\\notin \\beta'$. The latter guarantees $\\beta'\\subseteq \\beta$. Then apply Case I to $a'$.\n\n\n\\end{proof}\n\n\n\nThe following theorem shows that complete richly branching pointed $\\R$-trees\nwith radius $r$ form an elementary class.\n\n\\begin{theorem}\n\\label{rbrt}\nThe models of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ are exactly the complete, richly branching $\\R$-trees with radius $r$.\n\\end{theorem}\n\\begin{proof}\nLet $(M,d,p)$ be a complete, richly branching pointed $\\R$-tree with radius $r$ and let $\\mathcal{M}$\nbe the corresponding $L_r$-structure. Clearly $\\mathcal{M}\\models \\mathbb{R}\\mathrm{T}_r$, and it remains\nto verify that $\\phi^\\mathcal{M}=0.$\nLet $a\\in M$. If $d(p, a)=r$, then let $c_1=c_2=c_3=a$\nand note that these witness $\\psi^\\mathcal{M}(a)=0$. \nIf $d(p, a)<r$, let $\\epsilon >0$  be such that $\\frac{\\epsilon}{2}<r-d(p,a)$.\nSince $M$ is richly branching, the set $B$ from Definition \\ref{richlybranching}\nis dense in $M$.  So, there exists $b\\in B$ so that $d(a,b)<\\frac{\\epsilon}{2}.$\nThen since $M$ is complete, there exist $c_1, c_2, c_3\\in M$\nthat are each distance $r-d(b, p)$ from $b$, \nand each of $c_1, c_2, c_3$ is on a different branch at $b$.\n\nThe triangle inequality in $M$ implies $|d(a, c_i)-d(b, c_i)|<\\frac{\\epsilon}{2}$ for\nall $i=1, 2, 3$,\nand $|d(p, a)-d(p, b)|\\leq d(a,b)<\\frac{\\epsilon}{2}$.\nIt follows directly that $|(r-d(p,b))-(r-d(p,a))|<\\frac{\\epsilon}{2}$.\n\n\nClaim: $|d(a,c_i)-(r-d(p,a))|<\\epsilon$ for each $i=1, 2, 3$.\nLet $i$ be one of $1,2$ or $3$. By the triangle inequality in $\\R$\n\\begin{align*}\n|d(a,c_i)-(r-d(p,a))|&\\leq|d(a, c_i)-d(b, c_i)|+|d(b,c_i)-(r-d(p,a))|\\\\\n&=|d(a, c_i)-d(b, c_i)|+|(r-d(p,b))-(r-d(p,a))|\\\\\n&<\\frac{\\epsilon}{2}+\\frac{\\epsilon}{2}=\\epsilon\\\\\n\\end{align*}\n\nClaim: $|d(a,c_i)+d(a,c_j)-d(c_i, c_j)|<\\epsilon$ for each $i \\neq j$ in $\\{1,2,3\\}$.\n\nWe know $d(c_i, c_j)=d(c_i, b)+d(c_j, b)$ \nbecause $c_1, c_2, c_3$ are on different branches at $b$.\nSince $|d(a, c_i)-d(b, c_i)|<\\frac{\\epsilon}{2}$\nfor each $i$, we conclude that \n\\begin{align*}\n|d(a, c_i)+d(a, c_j)-d(c_i, c_j)|&=|d(a, c_i)+d(a, c_j)-(d(b, c_i)+d(b, c_j))|\\\\\n&=|(d(a, c_i)-d(b, c_i))+(d(a, c_j)-d(b, c_j))|\\\\\n&\\leq|d(a, c_i)-d(b, c_i)|+|d(a, c_j)-d(b, c_j)|<\\epsilon.\n\\end{align*}\nSo, for each $\\epsilon$, there are $c_1, c_2$ and $c_3$ making\n$$\\max\\left\\{\\max_{i=1, 2, 3}\\{|d(a,c_i)-(r-d(p,a))|\\}, \\max_{1\\leq i<j \\leq 3}\\{ |(d(a,c_i)+d(a,c_j))-d(c_i,c_j)| \\}\\right\\}<\\epsilon.$$\nThus, $\\psi(a)^\\mathcal{M}=0$, and since $a$ was arbitrary, we conclude $\\phi^\\mathcal{M}=0$.\n\n\nNow, assume $\\mathcal{M}\\models \\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$, so $(M, d, p)$ is clearly a complete, \npointed $\\R$-tree. Let $a\\in M$ and $\\epsilon>0$.\nBy Lemma \\ref{branching point within epsilon}\nwe may find $b\\in M$ so that $d(a, b)<\\epsilon$ and there are at least 3\ndistinct branches at $b$.\nBy Lemma \\ref{branches of sufficient height}\neach branch at $b$ contains a point with distance $r$ from $p$,\nso the height of each branch is at least $r-d(p, b)$. It also follows from \\ref{branches\nof sufficient height} that $M$ has radius $r$.\nSince $a\\in M$ and $\\epsilon>0$ were arbitrary, we conclude the\nset $B$ of points with at least 3 branches of height $\\geq r-d(p, b)$\nis dense in $M$.  Therefore, $M$ is richly branching.\n\\end{proof}\n\n\nNext, we turn to a series of lemmas that are needed for our proof that the theory \n$\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ is the model companion of $\\mathbb{R}\\mathrm{T}_r$.  (See Theorem \\ref{modcomp}.)  \n\n\n\\begin{lemma}\\label{ec implies richly branching}\nEvery existentially closed model of $\\mathbb{R}\\mathrm{T}_r$ is a model of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$.\n\\end{lemma}\n\n\\begin{proof}\nLet $\\mathcal{M} \\models \\mathbb{R}\\mathrm{T}_r$ be existentially closed with underlying $\\R$-tree $(M,d,p)$.\nConsider $a \\in M$. If $r-d(p,a)=0$, then ${\\psi}^{\\mathcal{M}}(a)=0$\nis true: make $y_1=y_2=y_3=a$.\nWhen $r-d(p,a)>0$ using Lemma \\ref{chiswell add to points}\n we may construct an extension $\\mathcal{N} \\models \\mathbb{R}\\mathrm{T}_r$ of $\\mathcal{M}$\nwith underlying $\\R$-tree $(N,d,p)$ such that $N$\nhas at least 3 branches of height $r-d(p,a)$ at $a$.\nThen in $N$ there exist $c_1, c_2$\nand $c_3$, each on a different branch at $a$ and each with distance $r-d(p,a)$ from\n$a$.  It follows that ${\\psi}^{\\mathcal{N}}(a)=0$.\nSince $\\mathcal{M}$ is existentially closed ${\\psi}^{\\mathcal{N}}(a)={\\psi}^{\\mathcal{M}}(a)=0$.\nOur choice of $a \\in M$ was arbitrary, implying $\\phi^\\mathcal{M}=0$.\nIt follows that $\\mathcal{M}$ is a model of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$.\n\\end{proof}\n\nThe next lemma connects $\\kappa$-saturation with the number\nof branches at every interior point in a richly branching $\\R$-tree.\nNote that for points on the boundary where $d(p, a)=r$, there is always\nexactly one branch at $a$, namely the branch containing $p$. \nOtherwise, there would be $b\\in M$ on a different branch at $a$,\nmaking $d(p, b)=d(p, a)+d(a,b)>r$.\nThe converse of Lemma \\ref{saturatedrbrt} is also true, and this\ncharacterization of saturation is Theorem \\ref{kappasaturated=kappabranching},\nwhich will be proved later, once we have shown that $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ admits quantifier elimination.\n\n\n\\begin{lemma}\n\\label{saturatedrbrt}\nLet $\\mathcal{M}\\models \\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$. Let $\\kappa$ be an infinite cardinal.\nIf $\\mathcal{M}$ is $\\kappa$-saturated, then $(M,d,p)$ has at least $\\kappa$ many branches at every point $a$ such that $d(p,a)<r$.\n\\end{lemma}\n\\begin{proof}\nLet $a\\in M$ with $d(p,a)<r$. Assume toward a contradiction that there are exactly $\\alpha$-many distinct branches at $a$ where $\\alpha<\\kappa.$\nBy \\ref{rbrt}, $(M, d, p)$ is a complete, pointed richly branching $\\R$-tree.\nIndex the branches at $a$ by $i<\\alpha$,\nand by Lemma \\ref{branches of sufficient height} on each of these branches\ndesignate a point $b_i$ such that $d(a, b_i)=r-d(p,a)$. \n\nLet $A=\\{a\\}\\cup \\{b_i\\mid i<\\alpha\\}$\nand note that $A$ has cardinality less than $\\kappa$. Define\n$$\\Sigma=\\{|d(a, x)-r-d(p,a)|=0\\}\\cup\\{|d(b_i, x)-\\left(d(b_i, a)+d(a, x)\\right)|=0\\mid i<\\alpha\\}.$$\nIt is straightforward to show $\\Sigma$ is finitely satisfiable.\nSo, by $\\kappa$-saturation, there exists $b\\in M$ that satisfies all of these conditions.\nBy Lemma \\ref{branch}, $b$ is on a different branch out of $a$ from each $b_i$,\ncontradicting that every branch at $a$ was represented by one of the $b_i$.\n\\end{proof}\n\n\n\\begin{lemma}\n\\label{embed}\n Let $\\kappa$ be an infinite cardinal. Assume $\\mathcal{M}$ is a $\\kappa$-saturated model of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ with underlying $\\R$-tree $(M,d,p)$.\nLet $K$ be a non-empty, finitely spanned $\\R$-tree. Designate a base point $q$ in $K$.\n\nFor any $e\\in M$ such that $r-d(p, e)\\geq \\sup\\{d(q, x)\\mid x\\in K\\}$\nand any collection $\\{\\beta_i \\mid i< \\alpha\\}$ of branches at $e$ with cardinality\n$\\alpha<\\kappa$, there exists an isometric embedding $f$ of $K$\ninto $M$ such that $f(q)=e$ and $f(K)\\cap \\beta_i=\\{e\\}$ for all $i<\\alpha$.\n\\end{lemma}\n\\begin{proof}\nBy Lemma \\ref{saturatedrbrt},\n$(M,d,p)$ has at least $\\kappa$ many branches at each point $a$ satisfying $d(p,a)<r$. By Lemma \\ref{mingenset}, there is\na minimal set that spans $K$, namely, the set of endpoints of $K$.\nProceed by induction on the size of this minimal spanning set,\nbuilding up the embedding at each step using the fact that there are $\\kappa$-many branches of sufficient height at every interior point.\nThe restriction that $r-d(p, e)\\geq \\sup\\{d(q, x)\\mid x\\in K\\}$\nkeeps the image of the embedding inside $M$. \n\\end{proof}\n\n\\begin{theorem}\n\\label{modcomp}\nThe $L_r$-theory $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ is the model companion of $\\mathbb{R}\\mathrm{T}_r$.\n\\end{theorem}\n\n\\begin{proof}\nSince $\\mathbb{R}\\mathrm{T}_r$ is an inductive theory, by Lemma \\ref{axiomatizeec} it suffices\nto show that the models of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ are exactly the existentially closed models of $\\mathbb{R}\\mathrm{T}_r$.\nBy Lemma \\ref{ec implies richly branching} we know every existentially closed model\nof $\\mathbb{R}\\mathrm{T}_r$ is a model of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$.\nIt remains to show that every model of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ is an existentially closed model\nof $\\mathbb{R}\\mathrm{T}_r$.  Let $\\mathcal{M}\\models \\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r.$ Let $\\mathcal{N}\\models \\mathbb{R}\\mathrm{T}_r$ be an\nextension of $\\mathcal{M}$. We may assume $\\mathcal{M}$ and $\\mathcal{N}$ are\n$\\omega_1$-saturated. (This is because we may consider the structure\n$\\big(\\mathcal{M},\\mathcal{N}, \\iota \\big)$ where $\\iota$ is the embedding\nfrom $\\mathcal{M}$ to $\\mathcal{N}$, and take an $\\omega_1$-saturated extension of\nthat structure. If we can verify the definition of existentially\nclosed in that setting, it will be true of $\\mathcal{M}$ and $\\mathcal{N}$.) Let $(M,d,p)$ and $(N,d,p)$\nbe the underlying pointed $\\R$-trees for $\\mathcal{M}$ and $\\mathcal{N}$ respectively.\n\nLet $a=a_1,...,a_k\\in M$, and without loss of generality assume basepoint $p$ is\namong $a_1,...,a_k$.\nWe claim that for any $b_1,...,b_l\\in N$, there exist $c_1,...,c_l\\in M$\nso that $d(b_i, b_j)=d(c_i, c_j)$ for all $i,j\\in \\{1,...,l\\}$ and $d(a_i, b_j)=d(a_i, c_j)$\nfor all $i\\in \\{1,...,k\\}$ and $j\\in \\{1,...,l\\}$.\\\\\nTo prove this claim, let  $b_1,...,b_l\\in N$, and\nlet $E_a\\subseteq M$ be the subtree spanned by $a=a_1,...,a_k$.\nNote that $E_a=\\overline{E_a}$.\nDefine an equivalence relation on $b_1,...,b_l$ by: $b_i\\sim b_j$\nif $b_i$ and $b_j$ have the same closest point in $E_a$. Let $A_1,...,A_m$\nbe the equivalence classes of this equivalence relation, and for $v=1,...,m$\nlet $e_v$ be the unique closest point in $E_a$ common to the members of $A_v$.\n\nFor each $v=1,...,m$\nlet $K_v$ be the $\\R$-tree spanned by $A_v\\cup\\{e_v\\}$ in $N$, with base point $e_v$.\nNote that each $K_v$ is closed and for $u\\not=v$, $K_u\\cap K_v=\\emptyset$.\nSince $(N, d, p)$ has radius $r$ and $p\\in E_a$ and $e_v$ is the unique closest point to $x$ in $E_a$,\nLemma \\ref{branch} gives\n$d(p, e_v)+d(e_v, x)=d(p, x)\\leq r$ for each $x\\in K_v$. Therefore\n$d(e_v, x)\\leq r-d(p, e_v)$ for each $x\\in K_v$. Thus, $\\sup\\{d(e_v, x)\\mid x\\in K_v\\}\\leq r-d(p, e_v).$\nNow, for each $v=1,...,m$, by Lemma \\ref{embed}, there is an isometric embedding\n$f_v\\colon K_v\\rightarrow M$ sending $e_v$ to $e_v$\nsuch that $f_v(K_v)$ does not intersect $E_a$ except at $e_v$.  Note that $e_v$ is the\nunique closest point in $E_a$ for every point in $f_v(K_v)$.\n\nLet $f$ be the union of the functions $f_v$ for $v=1,...,m$. If $b_i$ and $b_j$\nare both in $A_v$, then $d(b_i, b_j)=d(f_v(b_i), f_v(b_j))=d(f(b_i), f(b_j))$.\nIf $b_i$ and $b_j$ are in $A_{u}\\not=A_{v}$ respectively, then using Lemma\n\\ref{differentclosestpoints}\n\\begin{align*}\nd(b_i, b_j)&=d(b_i, e_u)+d(e_u, e_v)+d(e_v, b_j)\\\\\n&= d(f_u(b_i), f_u(e_u))+d(e_u, e_v)+d(f_v(e_v), f_v(b_j))\\\\\n&= d(f(b_i), e_u)+d(e_u, e_v)+d(e_v, f(b_j))\\\\\n&=d(f(b_i), f(b_j)).\n\\end{align*}\nTherefore the function $f$ is an isometric embedding from\n$\\displaystyle\\bigcup_{v=1}^m K_v$ to $M$.\\\\\nLet $c_j=f(b_j)$ for all $j\\in\\{1,...,l\\}$.\nThen clearly $d(b_i, b_j)=d(f(b_i), f(b_j))=d(c_i, c_j)$ for all $i,j\\in \\{1,...,l\\}$.\nNow let $i\\in\\{1,...,k\\}$ and $j\\in \\{1,...,l\\}$.  Let $e_v$ be the closest point to $b_i$\nin $E_a$.  Then\n\\begin{align*}\nd(a_i,b_j)&=d(a_i, e_v)+d(e_v, b_j)\\\\\n&=d(a_i, e_v)+d(f(e_v),f(b_i))\\\\\n&=d(a_i, e_v)+d(e_v, c_j)=d(a_i, c_j).\n\\end{align*}\nThus, the claim is true.\n\nThe values of quantifier free\nformulas in $\\mathcal{M}$ are determined by distances in $M$. (This can be shown\nusing induction on the definition of quantifier free formula, since connectives\nare continuous functions on atomic formulas, which in this case are all of the\nform $d(t_1,t_2)$ for terms $t_1, t_2$.)\nSo, the preceding claim implies that for any quantifier free formula\n$\\phi(x_1,...,x_k, y_1,...,y_l)$\nand any $b_1,...,b_l\\in \\mathcal{N}$,\nthere exist $c_1,...,c_l\\in \\mathcal{M}$ so that\n$$\\phi(a_1,...,a_k,b_1,...,b_l)^\\mathcal{N}=\\phi(a_1,...,a_k,c_1,...,c_l)^\\mathcal{N}=\\phi(a_1,...,a_k,c_1,...,c_l)^\\mathcal{M}.$$\nThen standard arguments about infima imply\n$$\\inf_{y_1}...\\inf_{y_l}\\phi(a_1,...,a_k, y_1,...,y_l)^\\mathcal{M}=\\inf_{y_1}...\\inf_{y_l}\\phi(a_1,...,a_k, y_1,...,y_l)^\\mathcal{N}.$$\nTherefore, $\\mathcal{M}$ is an existentially closed model of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$.\n\\end{proof}\n\n\n\\section{Properties of the theory $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$}\n\\label{propertiesrbrt}\nIn this section we show that $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ has\nquantifier elimination and is complete and stable, but not superstable.  We characterize types, and show that\nthe space of 2-types over the empty set has metric density $2^\\omega$. We also characterize definable closure and algebraic closure in models of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$. In the rest of the paper, for convenience of notation, whenever we take a set of parameters $A$ in a model $\\mathcal{M}=(M, d, p)$ we assume $p\\in A$.\n\n\\begin{lemma}\n\\label{qe}\nThe $L_r$-theory $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ has quantifier elimination.\n\\end{lemma}\n\\begin{proof}\nBy Theorem \\ref{modcomp}, Theorem \\ref{amalg} and Proposition \\ref{modelcompanionplusamalg}.\n\\end{proof}\n\n\\begin{corollary}\nThe $L_r$-theory $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ is complete.\n\\end{corollary}\n\\begin{proof}\nIn any model of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ we may embed the structure consisting of just the base point.\nThis fact together with quantifier elimination implies that $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ is complete.\n\\end{proof}\n\nWe now turn our attention to a discussion of types and type spaces.\nLet $\\mathcal{M}\\models \\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$. Recall that if $b=b_1,...,b_n$ is tuple of elements in $\\mathcal{M}$, $\\tp_\\mathcal{M}(b\/A)$ is the complete $n$-type of $b$ over $A$ in $\\mathcal{M}$.\nThe space of all $n$-types in models of a theory $T$ is denoted $S_n(T)$. The space\nof $n$-types over $A$ is denoted $S_n(T_A)$ or $S_n(A)$ when the theory $T$ is\nclear from context. If $q$ is an $n$-type and $a\\in \\mathcal{M}$ is a tuple such that $\\mathcal{M}\\models q(a)$ we say $a$ realizes $q$, and write $a\\models q$. If $\\phi(x_1,..., x_n)$ is an $L_r$-formula,\nwe write $\\phi(x_1,..., x_n)^q$ for the value of this formula specified by the type $q$. \n\n\n\nThe following lemma gives criteria for when two $n$-tuples have the same type.\n\n\\begin{lemma}\n\\label{determinetypes}\nLet $\\mathcal{M}\\models \\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$. Fix $A\\subseteq M$.\n\\begin{enumerate}\n\\item Let $b,c \\in M$. Then $\\tp_\\mathcal{M}(b\/A)=\\tp_\\mathcal{M}(c\/A)$ if and only if\n$b$ and $c$ have the same unique closest point $e\\in \\overline{E_{A}}$ and\n$d(b,e)=d(c,e)$.\n\\item Let $b=(b_1,\\dots,b_n)$ and $c=(c_1,\\dots,c_n)$ be tuples in $M$. \nThen $\\tp_\\mathcal{M}(b\/A)=\\tp_\\mathcal{M}(c\/A)$ if and only if $\\tp_\\mathcal{M}(b_i\/A)=\\tp_\\mathcal{M}(c_i\/A)$\nfor all $1 \\leq i \\leq n$ and $d(b_i,b_j)=d(c_i,c_j)$ for all $1 \\leq i,j \\leq n$.\n\\item Let $b=(b_1,\\dots,b_n)$ and $c=(c_1,\\dots,c_n)$ be tuples in $M$. Let \n$E_b$ be the subtree of $M$ spanned by $A\\cup b$, and similarly define $E_c$.\nThen $\\tp_\\mathcal{M}(b\/A)=\\tp_\\mathcal{M}(c\/A)$ if and only if there is an isometry $f\\colon E_b\\rightarrow E_c$\nthat fixes $A$ (or equivalently $\\overline{E_A}$) and satisfies $f(b_i)=c_i$ for all $i=1,...,n$.\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof} \nTo show (1), first assume $\\tp_\\mathcal{M}(b\/A)=\\tp_\\mathcal{M}(c\/A)$.  This implies\n$d(b,a)=d(c,a)$ for all $a\\in \\overline{E_{A}}$, which means $b$ and $c$ must\nhave the same unique closest point  $e\\in \\overline{E_{A}}$.\nMoreover, $d(b, e)$ must equal $d(c,e)$.\nFor the other direction, assume $b$ and $c$ have the same\nunique closest point  $e\\in  \\overline{E_{A}}$ and that $d(b, e)=d(c,e)$.\nSince $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ has quantifier elimination,  and the values quantifier-free formulas\nare determined by the values of atomic formulas, it suffices to show\n$d(a, b)=d(a,c)$ for all $a\\in A$ and $d(p,b)=d(p,c)$.  This follows easily\nfrom our assumptions using Lemma \\ref{branch}, since for any $a\\in A$,\nthe point $e$ must be on both $[a, b]$ and $[a,c]$.\n\\newline\nStatement (2) follows from part (1) and the fact that $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ admits quantifier elimination.\nnote that $\\tp_\\mathcal{M}(b_i\/A)=\\tp_\\mathcal{M}(c_i\/A)$\nfor all $1 \\leq i \\leq n$, if and only if $d(b_i, a)=d(c_i, a)$ for all $a\\in A.$ \nAgain, we use that the values of quantifier-free formulas are determined\nby the values of the atomic formulas. The atomic formulas involved are\nall of the form $d(t_1, t_2)$ with terms $t_1$ and $t_2$ involving the variables\n$x_1,...,x_n$ and\/or parameters from $A$.\n\\newline\nStatement (3) follows from (2) and the fact that an isometry between subsets\nof an $\\R$-tree extends uniquely in the obvious way to an isometry\nof the subtrees spanned by those sets.\n\\end{proof}\n\nLemma \\ref{determinetypes} tells us that a type $q\\in S_n(A)$ is uniquely determined\nby the choice of closest points $e_1,..., e_n$ in $\\overline{E_A}$,  the distances \nto these points, and the pairwise distances between the elements of a tuple realizing the type.\nMoreover, given a realization $b_1,..., b_n$ of an $n$-type $q$, the points $b_1, ..., b_n, e_1,..., e_n$\nform a finite $0$-hyperbolic metric space where $d(b_i, e_i)\\leq r-d(p, e_i).$\nThus, $b_1,...,b_n, e_1,...,e_n$ span an $\\R$-tree that extends (perhaps trivially) the $\\R$-tree spanned by $e_1,...,e_n$ in which all points have distance at most $r$ from $p$.\n\n\nIn fact, for any choice of finitely many points $e_1,...,e_n$ in $\\overline{E_A}$\nand values for the formulas $d(x_i, e_i)$ and $d(x_i, x_j)$ satisfying\nconditions guaranteeing $0$-hyperbolicity and boundedness, there exists a unique type\n$q\\in S_n(A)$ realizing this information. \nWe prove this in detail in Lemma \\ref{n types over A} below.\n\n\nIn Definitions \\ref{defhyperbolic} and \\ref{0hyp} we defined the notion of\na $0$-hyperbolic metric space. Definition \\ref{four point condition} is another condition\nthat, in a metric space, is equivalent to $0$-hyperbolicity.  This version is convenient for our description of types.\n\n\\begin{definition}\\label{four point condition}\nLet $X$ be a set, and $d\\colon X\\times X\\rightarrow \\R^{\\geq 0}$ a\nfunction. If for all $x, y, z, t\\in X$\n$$d(x, y)+d(z, t)\\leq \\max\\{d(x, z)+d(y, t), d(y, z)+d(x, t)\\}$$\nwe say $d$ satisfies the \\emph{4-point condition} on $X$.\n\\end{definition}\n\n\n\\begin{lemma}\n\\label{four point condition implies triangle}\nA metric space $(X, d)$ is $0$-hyperbolic if and only if $d$ satisfies the 4-point condition on $X$.\nMoreover, if $X$ is a non-empty set and $d\\colon X\\times X\\to \\R$ satisfies the\nfour point condition on $X$, and satisfies $d(x,y)=0$ if and only if $x=y$ and $d(y, x)=d(x, y)$, then the triangle inequality follows.\n\\end{lemma}\n\\begin{proof}\nFor the first statement, see page 13 in \\cite{C}. For the second statement, let $z=t$ in the 4-point condition.\n\\end{proof}\n\n\n\nIn the next lemma, we focus on $n$-types realized by $n$ distinct\nelements. It is straightforward to extend to the case where the elements realizing the type may not be distinct.\n\n\n\\begin{lemma}\\label{n types over A}\nLet $\\mathcal{M}\\models \\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ and $A\\subseteq M$. Let $e_1,..., e_n\\in\\overline{E_A}$.\nLet $\\Sigma$ be a set of $L$-conditions over $A$\n$$\\Sigma=\\left\\{d(x_i, x_j)=\\rho_{i,j}, \\, d(x_i, e_j)=d(e_i, e_j)^{\\mathcal{M}}+s_i, \\, d(e_i, e_j)=d(e_i, e_j)^{\\mathcal{M}}\\mid 1\\leq i,j\\leq n\\right\\}$$\nwhere\n\\begin{itemize}\n\\item for each $1\\leq i\\leq n$,  $s_i \\in [0, r-d(p, e_i)^{\\mathcal{M}}]\\subseteq \\R$\n\\item $\\{\\rho_{i,j}\\in \\R^{\\geq 0}\\mid 1\\leq  i,j\\leq n\\}$ are such that \n$\\rho_{i,i}=0$ and $\\rho_{i, j}=\\rho_{j, i}\\not=0$ for all $1\\leq i, j, \\leq n\n\n\\item the values of $d$ specified by $\\Sigma$ satisfy the 4-point condition on the set of symbols $\\{e_1,..., e_n, x_1,..., x_n\\}$.\n\\end{itemize}\n\nThere exists a unique type $q\\in S_n(A)$ so that $\\Sigma\\subseteq q$. \n\\end{lemma}\n\n\n\n\\begin{proof}\nLet $\\kappa> |\\overline{E_A}|$ and let $\\mathcal{N}$ be a $\\kappa$-saturated elementary extension of $\\mathcal{M}$.\nLet $X=\\{x_1,..., x_n\\}$ and note that by Lemma \\ref{four point condition implies triangle}, $d$ is a metric on $X\\cup \\{e_1,..., e_n\\}$.\nDefine an equivalence relation on $X$ by $x_i\\sim x_j$ if and only if $d(e_i, e_j)^{\\mathcal{M}}=0.$\nLet $C_1,..., C_k$ be the equivalence classes. Each of these equivalence classes\ncorresponds to exactly one of the $e_1,..., e_n$.\nFor each $C_l$, $1\\leq l\\leq k$ let $e_l$ be the corresponding element of $\\overline{E_A}$. Then, with the values of $d$ specified in $\\Sigma$, the set $C_l\\cup\\{e_l\\}$\nis a finite $0$-hyperbolic metric space. Thus it spans an $\\R$-tree. Call that $\\R$-tree $K_l$.\nApply Lemma \\ref{embed} to $K_l$ with basepoint $q=e_l$ to embed $K_l$ in $N$\nso that the image of $K_l$ intersects $\\overline{E_A}$ only at $e_l$.\nThat the conditions of Lemma \\ref{embed} on $K_l$ are satisfied follows\nfrom the conditions on the values of $d$ specified in $\\Sigma$. Let $f_l\\colon K_l\\rightarrow N$\ndenote this isometric embedding.\n\nFor each $l$ and each $i$ where $x_i\\in C_l$, let $b_i=f_l(x_i)$. Note that for\n$i\\not=j$ we know $b_i\\not= b_j$, because $d(b_i, b_j)=\\rho_{i,j}\\not=0$.\nLet $q$ be the type\nof $b_1,...,b_n$ over $A$ in $\\mathcal{N}$. \nAn straightforward argument like that in the proof of Theorem \\ref{modcomp} now shows that\n$\\Sigma\\subseteq q$. Uniqueness follows from Lemma \\ref{determinetypes}.\n\\end{proof}\n\n\nLemmas \\ref{determinetypes} and \\ref{n types over A} tell us that types over $A$ correspond\n(up to isometry fixing $\\overline{E_A}$) to\nfinitely-spanned $\\R$-trees with radius at most $r$ that extend\nfinitely-spanned subtrees of $\\overline{E_A}$. \nNext, we give some results about the type spaces as metric spaces.\nFirst, a reminder of the metric on types.\n\nLet $\\mathcal{M}\\models \\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ such that every type in $S_n(\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r)$ is realized in $\\mathcal{M}$ for each $n\\geq 1$.\nAs defined in \\cite{N}, the $d$-metric on $n$-types over the empty set is:\n$$d(q,s)=\\inf_{a\\models q, b\\models s}\\max_{i=1,...,n} d(a_i, b_i)$$\nwhere $a=a_1,...,a_n$ and $b=b_1,...,b_n$ are tuples in $\\mathcal{M}$..\nNote that by compactness, the infimum in the definition is actually realized. This definition can be extended in the obvious way to spaces of types over parameters.\nIn what follows, we will call a choice of a particular $a\\models q$ and $b\\models s$\na  \\emph{configuration.}  \n\n\n\\begin{lemma}\n\\label{how many types}\n\n\\begin{enumerate}\n\\item The space $S_1(\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r)$ of $1$-types over the empty set is in\nbijective correspondence with the interval $[0,r]$. In fact, with the $d$-metric\non types, it is isometric to $[0, r]$.\n\n\\item If $A$ is a set of parameters, then $S_1(A)$ is \nin bijective correspondence with the set of ordered pairs\n$$\\{(e,s)\\mid e\\in \\overline{E_A} \\text{ and } s\\in [0,r-d(p,e)]\\subseteq \\R\\}.$$\n\\item Given $t,u\\in S_1(A)$ and $e_t$, $e_u$ the unique\nclosest points in $\\overline{E_A}$ to realizations of $t$, $u$ respectively,\nwe have 2 cases.\n\\begin{enumerate}\n\\item  If $e_t\\not=e_u$, then\n$d(t, u)= \\dist(x, \\overline{E_A})^t+d(e_t, e_u)+\\dist(x, \\overline{E_A})^u$.\n\\item If $e_t=e_u$, then $d(t,u)=|d(x,p)^t-d(x,p)^u|.$\n\\end{enumerate}\n\\end{enumerate}\n\\end{lemma}\n\n\n\\begin{proof}\nTo prove statement (1), define the function $f\\colon S_1(\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r)\\rightarrow [0,r]$\nby $f(q)=d(p, b)$ for any $q\\in S_1(\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r)$ and any $b\\models q$. This function\nis well defined because the value of the formula $d(p,x)$ will be the same for any\nrealization of the type $q$.\nThe function $f$ is surjective because given $s\\in [0,r]$, in any model of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ there\nis always at least one point with distance $s$ from $p$ by Lemmas \\ref{branching point within epsilon} and \\ref{branches of sufficient height}.\nGiven two 1-types $t, u \\in S_1(\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r)$\na configuration minimizing the distance between realizations $b\\models t$\nand $c\\models u$ is the one where $b$ and $c$ are arranged along a piecewise segment.\nIf $d(p,c)\\geq d(p,b)$ then this segment is of the form $[p, b, c]$, and it is of the form\n$[p, c, b]$ otherwise.  This configuration makes\n$d(b, c)=|d(p, b)-d(p, c)|$ which, by the triangle inequality, is the least possible value of $d(b,c)$.\nThus, $d(t, u)=d(b, c)=|d(p, b)-d(p, c)|=|f(t)-f(u)|$, showing $f$ is an isometry.\n\nStatement (2) is a consequence of Lemmas \\ref{determinetypes} and \\ref{n types over A}.\nTo prove (3) first assume $e_t\\not=e_u$. Then for any realizations $b\\models t$ and $c\\models u$,\nby Lemma \\ref{differentclosestpoints}, $[b, e_t, e_u, c]$ is a piecewise segment, and the conclusion\nfollows.  If $e_t=e_u$, then as in the proof of (1), a minimizing configuration will have\n$b\\models t$ and $c\\models u$ along a geodesic segment, and the conclusion follows.\n\\end{proof}\n\nIt becomes complex to give precise description, as in the preceding lemma, of the metric on $S_n(A)$ over the theory $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ from the data describing the types. This complexity\nonly increases with $n$. Accordingly, we limit ourselves to giving (in the next result) a precise\nstatement of the metric density of the space of 2-types over $\\emptyset$. Its proof illustrates\nsome of the ideas needed to understand these metric spaces more completely.\n\n\\begin{proposition}\nThe space of $2$-types over the empty set has metric density character equal to $2^\\omega$.\n\\end{proposition}\n\\begin{proof}\nLet $\\mathcal{M}\\models \\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ such that every type in $S_n(\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r)$ is realized in $\\mathcal{M}$ for each $n\\geq 1$.\nFor each $s\\in [\\frac{r}{2}, r]$, choose a pair of points $a_s, b_s\\in M$ so that\n$d(p, a_s)=2s$, $d(p, b_s)=2s$ and $d(a_s, b_s)=2s$. Let $Y_s=Y(a_s, b_s, p)$. \nWith our choice of distances, $Y_s$ is not equal to $a_s, b_s$ or $p$, so $p, a_s, b_s$ are not\narranged along a piecewise segment.\nThe subtree spanned by $p, a_s, b_s$ consists of 3 branches at $Y_s$,\neach with length $s$, where $p, a_s, b_s$ are the endpoints of the 3 branches.\nFor any $a_s', b_s'\\models \\tp(a_s, b_s)$ in $\\mathcal{M}$,\nthe subtree spanned by $p, a_s', b_s'$ is isometric to the one spanned by\n$p, a_s, b_s$ via an isometry fixing $p$ and matching $a_s$ and $a_s'$, and $b_s$ with $b_s'$.\n\nClaim: If $s,t \\in [\\frac{r}{2}, r]$ and $s>t$, then $d(\\tp(a_s,b_s), \\tp(a_t,b_t))\\geq 2s.$\\newline\nTo determine the distance between these types, we consider\npossible configurations of points $a_s',b_s'\\models \\tp(a_s, b_s)$\nand $a_t',b_t'\\models \\tp(a_t, b_t)$.\nIt is straightforward to see that in a minimizing configuration,\nwe must put $Y_t'$ and $Y_s'$ on the same branch at $p$.\nSpecifically, we must have $Y_t'\\in [p,Y_s']$ making $d(Y_t', Y_s')=s-t$. \n\nSince $a_t', b_t'$ must be on separate branches at $Y_t'$, we know that\nat most one of $a_t'$ or $b_t'$ is on the same branch at $Y_t'$ as $Y_s'$.\nAssume $a_t'$ is on a different branch at $Y_t'$ from $Y_s'$. \nThen $[a_t', Y_t', Y_s', a_s']$ is a piecewise segment, and thus\n\n\\begin{align*}\nd(a_t', a_s')\n&=d(a_t', Y_t')+d(Y_t', Y_s')+d(Y_s', a_s')\\\\  \n&=t+(s-t)+s=2s.\n\\end{align*}\nIf, instead, it is $b_t'$ on a different branch at $Y_t'$ from $Y_s'$,\nthen we get $d(b_t', b_s')=2s$. \nTherefore, $\\max\\{d(a_s', a_t'), d(b_s', b_t')\\}\\geq 2s$ for any configuration.\nSince $s\\in [\\frac{r}{2}, r]$, we know $2s\\geq r$.\nThus, we can make a set of $2$-types, one for each $s\\in [\\frac{r}{2}, r]$,\nso that the distance  between any two of them is $\\geq r$. \nThis implies that the space of $2$-types has metric density character at least $2^\\omega$.  It is straightforward to show using Lemma \\ref{determinetypes} that the cardinality of the space of $2$-types over the empty set is $2^\\omega$, and therefore the metric density character must be exactly\nequal to $2^\\omega$.\n\n\\end{proof}\n\n\n\nNext, we show that the definable closure and the algebraic closure\nof a set of parameters $A$ are the same, and equal to the closed subtree spanned\nby $A$.\n\n\\begin{proposition}\n\\label{L: dcl and acl characterized}\nLet $\\mathcal{M}\\models \\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ with $(M,d,p)$ as its underlying $\\R$-tree.\nLet $A\\subseteq M$ be a non-empty set of parameters. Then $\\dcl(A)=\\acl(A)=\\overline{E_A}$.\n\\end{proposition}\n\\begin{proof}\nIf $a,b\\in A$, then by Lemma \\ref{definablemidpoints} any point in $[a,b]$ is in $\\dcl(A)$. Then $E_A\\subseteq \\dcl(A)$ because $E_A$ is the union of all such geodesic segments.\nSince $\\dcl(A)$ is closed, $\\overline{E_A}\\subseteq \\dcl(A)$. Combined with the fact that\n$\\dcl(A)\\subseteq \\acl(A)$, this gives $\\overline{E_A}\\subseteq \\dcl(A)\\subseteq \\acl(A)$.\nIt remains to show $\\acl(A)\\subseteq \\overline{E_A}$, which we do in the contrapositive. \n\nIf $c\\notin \\overline{E_A}$, let $e$ be the unique closest point to $c$\nin $\\overline{E_A}$. This $e$ exists by Lemma \\ref{!geo}.\nLet $B$ be the branch at $e$ that contains $c$.\nUsing Lemma \\ref{chiswell add to points} we construct\nan extension $\\mathcal{N}\\models \\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ of $\\mathcal{M}$ which adds an infinite number of branches at $e$, each of which is isometric to $B$.   \nBy Lemma \\ref{determinetypes}, on each of these branches is a realization of $\\tp(c\/A)$. The distance between any two such realizations is $2d(e, c)>0$. This gives us a non-compact set of realizations. Thus, $c\\notin \\acl(A)$. \n\\end{proof}\n\n\n\nThe next result gives a characterization of $\\kappa$-saturated $\\R$-trees, completing the result\npromised in Section \\ref{model companion}.\n\\begin{theorem}\n\\label{kappasaturated=kappabranching}\nLet $\\mathcal{M}\\models \\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$.\nLet $\\kappa$ be an infinite cardinal.\nThen $\\mathcal{M}$ is $\\kappa$-saturated if and only if $M$ has at least $\\kappa$ many branches\nat every point $a\\in M$ where $d(p, a)<r$.\n\\end{theorem}\n\\begin{proof}\nThe forward direction is Lemma \\ref{saturatedrbrt}.\nNow, assume $M$ has $\\kappa$-many\nbranches at every point $a\\in M$ where $d(p, a)<r$, and recall\nthat by Lemma \\ref{branches of sufficient height} each of these branches\ncontains at least one point with distance $r$ from $p$.\nLet $A\\subseteq \\mathcal{M}$ have cardinality less than $\\kappa$, and\nwithout loss of generality assume basepoint $p$ is in $A$.\n\nLet $q$ be a $1$-type over $A$.\nBy Lemma \\ref{determinetypes}, this type is determined by a\nclosest point $e\\in \\overline{E_A}$ and a distance $0\\leq s\\leq r-d(p, e)$\nwhere $s$ is the distance between $\\overline{E_A}$ and any realization of $q$.\nIf $s=0$, then the type\n$q$ is satisfied by the point $e\\in \\overline{E_A}$. Now, assume $s>0$.\nBecause there are $\\kappa>2|A|$ branches at $e$, it follows from the definition of $\\overline{E_A}$ that\nthere are branches at $e$ in $M$ that do not intersect $\\overline{E_A}$ except at $e$. On one of these branches take $b$ with distance $s$ from $e$.\nThis $b$ satisfies the type $q$.  Therefore, all $1$-types over $A$ are realized in $\\mathcal{M}$, implying that $\\mathcal{M}$ is $\\kappa$-saturated.\n\\end{proof}\n\nTo finish this section we show $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ is stable, but not superstable. (\\textit{i.e.}, it is strictly stable.)\n\n\\begin{theorem}\n\\label{rbrtstability}\nThe theory $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ is stable.\nIndeed when $\\kappa$ is an infinite cardinal, $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ is $\\kappa$-stable\nif and only if $\\kappa$ satisfies $\\kappa^\\omega = \\kappa$.\n\\end{theorem}\n\\begin{proof}\nLet $\\kappa$ be an infinite cardinal.\nLet $\\mathcal{M}\\models \\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ be $\\kappa^{+}$-saturated, with underlying $\\R$-tree $(M,d,p)$.\\\\\nFirst, assume $\\kappa=\\kappa^\\omega$.\nLet $|A|=\\kappa$. Then\n$$|E_A|\\leq |A\\times A|2^\\omega=\\kappa^2 2^\\omega\n\\leq \\kappa^\\omega 2^\\omega=\\kappa^\\omega=\\kappa.$$\nThus, $|E_A|=\\kappa$.\nWe count possible $1$-types using Lemma \\ref{how many types}, showing that \\\\\n$$|S_1(A)|\\leq |\\overline{E_A}\\times [0,1]|=|\\overline{E_A}|2^\\omega\\leq\n|E_A|^\\omega 2^\\omega=\\kappa^\\omega 2^\\omega=\\kappa^\\omega=\\kappa.$$\nThus the $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ is $\\kappa$-stable.\n\nFor the other direction, assume $\\kappa<\\kappa^\\omega$. We construct, via a tree construction, a subset $A$ of $M$ with $|A|=\\kappa$ and\n$|\\overline{E_A}|=\\kappa^\\omega$.\nAt Step 1 choose $\\kappa$-many points $(a_i\\mid i<\\kappa)$ on distinct branches at $p$, each with distance $\\frac{r}{4}$ from $p$.\nWe can do this since there are at least $\\kappa$ branches of sufficient height at every point in $M$ by Theorem \\ref{kappasaturated=kappabranching} and Lemma \\ref{branches of sufficient height}. Note that we only need $\\kappa$-saturation\nto guarantee $\\kappa$-many branches.  \nAt Step 2, for each $a_{i}$ we choose $\\kappa$-many points on distinct branches at $a_i$,\neach with distance $\\frac{r}{8}$ from $a_i$, and distance $\\frac{3r}{8}$ from $p$. \nWe can index all of these points by $(a_{i,j}\\mid i,j<\\kappa)$.\nAt Step $n$ for $n\\geq 2$, we have already designated points $a_{i_1, i_2,...,i_{n-1}}$ each\nof which has distance\n$\\sum_{k=1}^{n-1} \\frac{r}{2^{k+1}}$ from $p$. At each of these points choose $\\kappa$-many\npoints on distinct branches at $a_{i_1, i_2,...,i_{n-1}}$,\neach with distance $\\frac{r}{2^{n+1}}$ from $a_{i_1, i_2,...,i_{n-1}}$, and distance $\\sum_{k=1}^{n} \\frac{r}{2^{k+1}}$ from $p$. \nWe can index all of these points by $(a_{i_1, i_2,...,i_n}\\mid i_1,...,i_n<\\kappa)$. \nLet $A=\\{p\\}\\bigcup_{k=1}^\\infty (a_{i_1,...,i_k}\\mid i_1,...,i_k<\\kappa)$. If we associate $p$\nwith the empty sequence, then the elements of $A$ are in 1-1 correspondence with $\\kappa^{<\\omega}$.\nSo, the cardinality of $A$ is $|\\kappa^{<\\omega}|=\\kappa$.\n\nNow, for each function $f\\colon \\omega \\rightarrow \\kappa$ with $f(0)=0$ there is a unique\nsequence $(b_n)$ of elements of $A$ with $b_0=p$ and $b_n=a_{f(1),...,f(n)}$.\nNote that $b_n$ has distance $\\frac{r}{2^{n+1}}$ from $b_{n-1}$,\nmaking $(b_n)$ a Cauchy sequence. Since $M$ is complete, $b_n$ must converge to a limit with distance $\\frac{r}{2}$ from $p$.\\\\\nLet $f$ and $g$ be two distinct functions from $\\omega$ to $\\kappa$, and let $(b_n)$ and $(c_n)$\nbe their respective associated sequences.   Let $m$ be the first index at which $f(m)$ and $g(m)$ \ndisagree (note that $m\\not=0$). Then $b_m=a_{f(1),...,f(m)}$ and $c_m=a_{g(1),...,g(m)}$\nare on different branches out of \n$a_{f(1),...,f(m-1)}=b_{m-1}=c_{m-1}$. Moreover, for all $k\\geq m$, $b_k$ is in the same branch at\n$b_{m-1}$ as $b_m$ and likewise for the sequence $c_m$.\nThus, the limits of these two sequences must be in\ndifferent branches at $b_{m-1}=c_{m-1}$, so these limits are distinct points.\nThere are $\\kappa^\\omega$-many different functions from $\\omega$ to $\\kappa$\nwith $f(0)=0$.  Thus, $|\\overline{E_A}|=\\kappa^\\omega$.\n\nFor each limit $e\\in \\overline{E_A}$ constructed above,\nchoose $b_e$ on a branch at $e$ that intersects $\\overline{E_A}$\nonly at $e$ with $d(e,b_e)=\\frac{r}{4}$.  We may always\nfind such a branch, because there are $\\kappa$-many branches at the limit $e$, but\nonly one of those branches intersects $\\overline{E_A}$ at any point other than $e$.\nThe branch will be of sufficient height by Lemma \\ref{branches of sufficient height}.\nThe set of such points $b_e$ has\ncardinality $\\kappa^\\omega$, and for any limits $e\\not= f$ in $\\overline{E_A}$\nit is straightforward to show $d(\\tp(b_e\/A), \\tp(b_f\/A))\\geq \\frac{r}{2}$.\nSince $\\kappa^\\omega>\\kappa$, this implies the theory is not $\\kappa$-stable.\n\\end{proof}\n\n\n\\section{The independence relation for $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$}\n\nIn this section we characterize the model theoretic independence relation of\n$\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ and show that in models of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$, types have canonical\nbases that are easily-described sets of ordinary (not imaginary) elements.\n\nLet $\\kappa$ be a cardinal so that $\\kappa=\\kappa^\\omega$ and $\\kappa >2^\\omega$.\nIn this section let $U$ be a $\\kappa$-universal domain for $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$.\nA subset of $U$ is \\emph{small} if its cardinality is $<\\kappa$.\n\\begin{definition}\n\\label{starindependence}\nLet $A,B$ and $C$ be small\nsubsets of $U$.\nSay A is ${}^*$-independent from B over C, denoted $A\\ind[*]_C B$,\nif and only if for all $a\\in A$ we have $\\dist(a,\\overline{E_{B\\cup C}})=\\dist(a,\\overline{E_C})$.\n\\end{definition}\n{\\bf Note:} In what follows, we will abbreviate unions such as $B\\cup C$ as $BC$.\n\n\\begin{lemma}\n$A\\ind[*]_C B$\nif and only if for all $a\\in A$ the closest point to $a$ in\n$\\overline{E_{BC}}$ is the same as the closest point to $a$ in $\\overline{E_C}$.\n\\end{lemma}\n\\begin{proof}\nAssume $A\\ind[*]_C B$.\nTake an arbitrary $a\\in A$. Let $e_1$ be the unique closest point to $a$ in $\\overline{E_{BC}}$\nand $e_2$ the unique closest point to $a$ in $\\overline{E_C}$.\nWe assumed $\\dist(a,\\overline{E_{BC}})=\\dist(a,\\overline{E_C})$, which implies $d(a, e_1)=d(a, e_2)$.\nSince $e_2\\in \\overline{E_C}\\subseteq \\overline{E_{BC}}$, we know $e_1\\in [a,e_2]$ by Lemma \\ref{!geo}.\nTherefore, $e_1=e_2$. Since $a$ was arbitrary, we know this holds for all $a\\in A$.\nFor the other direction, assume for all $a\\in A$ the closest point to $a$ in $\\overline{E_{BC}}$\nis the closest point to $a$ in $\\overline{E_C}$. Then clearly\n$\\dist(a, \\overline{E_{BC}})=\\dist(a, \\overline{E_C})$ for all $a\\in A$.\n\\end{proof}\n\n\\begin{theorem}\n\\label{starindependencetheorem}\nThe relation $\\ind[*]$ is the model theoretic\nindependence relation for $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$.  Moreover, types over arbitrary sets\nof parameters are stationary.\n\\end{theorem}\n\\begin{proof}\nWe will show $\\ind[*]$ satisfies all the properties of a stable independence relation\non a universal domain of a stable theory as given in \\cite[Theorem 14.12]{N}.\nThen by \\cite[Theorem 14.14]{N} we know $\\ind[*]$ is the\nmodel theoretic independence relation for the stable theory $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$.\n\n\\noindent{\\bf (1) Invariance under automorphisms}\n\nAny automorphism $\\sigma$ satisfies $\\sigma(\\overline{E_A})=\\overline{E_{\\sigma(A)}}$\nand is distance preserving.\n\n\\noindent{\\bf (2) Symmetry:} if $A\\ind[*]_C B$, then $B\\ind[*]_C A$.\n\nAssume $A\\ind[*]_C B$.\nThis means for all $a\\in A$ we have that the closest point in $\\overline{E_{BC}}$\nto $a$ is $e_a\\in \\overline{E_C}$.\nThus, by Lemma \\ref{branch}, for any $a\\in A$, for any $y\\in \\overline{E_{BC}}$ we have\n$[a, y]\\cap \\overline{E_C}\\not=\\emptyset$. It follows that\nfor any $x\\in \\overline{E_A}$, for any $y\\in \\overline{E_{BC}}$\nthere exists a point of $\\overline{E_C}$ on $[x,y]$.\nLet $b\\in B$. Then for any $x\\in \\overline{E_A}$ there is a point of $\\overline{E_C}$\non $[x,b]$. It follows that the closest point in $\\overline{E_{AC}}$ to any $b\\in B$ is in $\\overline{E_C}$.\n\n\\noindent{\\bf (3) Transitivity:} $A\\ind[*]_C BD$ if and only if $A\\ind[*]_C B$ and $A\\ind[*]_{BC} D$.\n\nWe know\n$$E_C\\subseteq E_{BC}\\subseteq E_{BCD}$$\nwhich implies\n$$\\dist(a, \\overline{E_C})\\geq \\dist(a, \\overline{E_{BC}})\\geq \\dist(a, \\overline{E_{BCD}}).$$\nTherefore $\\dist(a, \\overline{E_{BCD}})=\\dist(a, \\overline{E_C})$ if and only if\n$$\\dist(a, \\overline{E_{BC}})=\\dist(a, \\overline{E_C}) \\text{ and }\n\\dist(a, \\overline{E_{BCD}})=\\dist(a, \\overline{E_{BC}}).$$\nHence\n$$A\\ind[*]_C BD \\text{ if and only if } A\\ind[*]_C B \\text{ and } A\\ind[*]_{BC} D.$$\n\n\\noindent{\\bf (4) Finite character:} $A\\ind[*]_C B$ if and only if $a\\ind[*]_CB$ for all finite tuples $a\\in A$.\n\nThis is clear from the definition.\n\n\n\\noindent{\\bf (5) Extension:} for all $A, B, C$ we can find $A'$ such that\ntp$(A\/C)=$ tp$(A'\/C)$ and $A'\\ind[*]_C B$.\n\nBy finite character and compactness, it suffices to show this statement when $A$ is\na finite tuple.  Let $e\\in \\overline{E_C}$ be the unique point closest to $\\overline{E_A}=E_A$.\nLet $\\beta<\\kappa$ be the cardinality of $\\overline{E_B}$.  Then there\nare at most $\\beta$ branches in $\\overline{E_B}$ at any point of $\\overline{E_B}$.\nSince $A$ is finite, use Lemma \\ref{embed} to embed a copy of $\\overline{E_A}=E_A$ on branches\nat $e$ that do not intersect $\\overline{E_B}$ except at $e$.\nThe image of $A$ under this embedding gives us $A'$.\n\n\n\\noindent{\\bf (6) Local Character:} if $a=a_1,...,a_m$ is a finite tuple,\nthere is a countable $B_0\\subseteq B$ such that $a\\ind[*]_{B_0}B$.\n\nLet $e_i$ be the closest point of $\\overline{E_B}$ to $a_i$ for $i=1,...,m$. Let $B_i$\nbe a countable subset of $B$ such that $e_i$ is an element of $\\overline{E_{B_i}}$.\nLet $B_0=\\bigcup_i^m B_i$.\n\n\\noindent{\\bf  (7) Stationarity (over arbitrary sets of parameters):} \nif $\\tp(A\/C)=\\tp(A'\/C)$, $A\\ind[*]_C B$, and $A'\\ind[*]_C B$,\nthen\n$\\tp(A\/BC)=\\tp(A'\/BC)$,\nwhere $C$ is a small submodel of $U$.\n\nBy quantifier elimination, $\\tp(A\/BC)$ is determined by $\\{\\tp(a\/BC)\\mid a\\in A\\}$ plus\nthe information $\\{d(a_1, a_2)\\mid a_1, a_2\\in A\\}$. These distances\n$\\{d(a_1, a_2)\\mid a_1, a_2\\in A\\}$ are fixed by $\\tp(A\/C)$.\nThus, it suffices to show the conclusion in the case when $A=\\{a\\}$ and $A'=\\{a'\\}$.\nIf $a$ or $a'$ is in $C$ the conclusion is obvious, so assume $a, a' \\notin C$.\nThe type of $a$ (or $a'$) over $BC$ is determined by two parameters,\nthe unique point in $\\overline{E_{BC}}$ that is closest to $a$, and the distance from $a$\nto that point.  Since $a\\ind[*]_C B$, it follows that the closest point in $\\overline{E_C}$\nto $a$ is the same as the closest point in $\\overline{E_{BC}}$ to $a$,\nand the same is true for $a'$.\nSince tp$(a\/C)=$tp$(a'\/C)$, we know $a$ and $a'$ have the same\nclosest point $e$ in $\\overline{E_C}$ and $d(a, e)=d(a',e)$.  Since $e$ is also the closest point\nin $\\overline{E_{BC}}$ to $a$ and $a'$, we know that tp$(a\/BC)=$tp$(a'\/BC)$\nby Lemma \\ref{determinetypes}.\n\\end{proof}\n\n\\noindent \\textbf{Canonical Bases} \\medskip\n\nA canonical base of a stationary type is a minimal set of parameters over which that type is\ndefinable.  However, to avoid a discussion of definable types, we here use an\nequivalent definition of canonical base, as given in \\cite{IAW}.  As in that paper, we\nhere take advantage of the fact (Lemma \\ref{L: dcl and acl characterized} and part (7)\nof the proof of Theorem \\ref{starindependencetheorem}) that every type over an\narbitrary set of parameters is stationary.\n\nFor stable theories in general, canonical bases exist as sets of imaginary elements, \nhowever, in models of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$, they are sets of ordinary elements.  That is,\nthe theory has built-in canonical bases.  Indeed, in this setting they are very simple.\n\nFor sets $A\\subseteq B\\subseteq U$,\nand $q\\in S_n(A)$ we say $q'\\in S_n(B)$ is a \\emph{non-forking extension} of $q$ if\n$b\\models q'$\nimplies $b\\models q$ and $b\\ind_A B$.\nBy the definition of independence, the condition $b\\ind_A B$ implies that the points\n$e_1,...,e_n$ in $\\overline{E_A}$ closest to $b_1,...,b_n$ respectively must\nalso be the closest points to $b_1,...,b_n$ in $\\overline{E_B}$.\nBecause $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ is stable and all types are stationary, non-forking extensions are unique.\nDenote the unique non-forking extension of $q$ to the set $B$ by $q\\upharpoonright^B$. \nGiven a type $q$ over a set $A\\subseteq U$ and an automorphism $f$ of $U$,\n$f(q)$ denotes the set of $L_r$-conditions over $f(A)$ corresponding to the\nconditions in $q$, where each parameter $a\\in A$ is replaced by its image $f(a)$.\n\n\\begin{definition}[{\\cite[Definition 6.1]{IAW}}]\nA \\emph{canonical base} $Cb(q\/A)$ for a type $q\\in S_n(A)$ is a subset $C\\subseteq U$ such that\nfor every automorphism $f\\in \\Aut(U)$, we have:\n$q\\upharpoonright^U= f(q)\\upharpoonright^U$ if and only if $f$ fixes each member of $C$.\n\\end{definition}\n\nThe following result describes canonical bases in $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$. \n\n\\begin{theorem}\nLet $b=(b_1,...,b_n)\\in U^n$ and $A\\subseteq U$ a set of parameters. Let $q \\in S_n(A)$\nbe the type over $A$ of the tuple $b$. Then a canonical base of $q$ is given by the set\n$\\{e_i \\mid 1\\leq i\\leq n\\}$, where $e_i\\in \\overline{E_A}$ is the closest point to $b_i$ in $\\overline{E_A}$.\nNote that this set depends only on $q$.\n\\end{theorem}\n\\begin{proof}\nLet $b$,  $A\\subseteq U$ and $q \\in S_n(A)$\nbe as described in the statement of the theorem. Let $C=\\{e_i \\mid 1\\leq i\\leq n\\}$\nwhere $e_i\\in \\overline{E_A}$ is the closest point to $b_i$ in $\\overline{E_A}$.\nFirst, assume $f$ is an automorphism of $U$ fixing $C$ pointwise.\nLet $c=(c_1,...,c_n)$ be a realization of $q\\upharpoonright^U$ (in some extension of $U$).\nThen $c\\models q$ and $c\\ind_{A} U$. To show $f(q)\\upharpoonright^U$=$q\\upharpoonright^U$\nit suffices to show that $c\\models f(q)$ and $c\\ind_{f(A)} U$, because\nthen $q\\upharpoonright^U$ is the unique non-forking extension of $f(q)$ to $U$.\nBy Lemma \\ref{determinetypes}, an $n$-type over a set $A$ is determined by the values it assigns to the formulas\n$d(x_i, x_j)$ and $d(x_i, a)$ for $a\\in A$.  Note that in $f(q)$, the parameter-free\n$L_r$-conditions are the same as in the type $q$. So,\nfor example, $d(x_i, x_j)$ must have the same value in $f(q)$ as in $q$.  \nThus, to show that $c\\models f(q)$ we just need to show\nthat $d(c_i,a)=d(c_i, f(a))$ for all $a\\in A$ and $i\\in \\{1,...,n\\}.$\n\nWe know $c\\models q$. Thus Lemma \\ref{determinetypes} implies that $e_i$ must be the closest\npoint to $c_i$ in $\\overline{E_A}$.\nAlso, $c\\ind_A U$ implies that $e_i$ is also the closest point in $U$ to $c_i$.\nSince $f(\\overline{E_A})\\subseteq U$ we know the closest point in $f(\\overline{E_A})$ to $c_i$ is $e_i=f(e_i)$.\nTherefore by Lemmas \\ref{!geo} and \\ref{branch}\nwe know $d(c_i, a)=d(c_i, e_i)+d(e_i, a)$ and $d(c_i, f(a))=d(c_i, e_i)+d(e_i, f(a))$ for any $a\\in A$. Thus, \n\\begin{eqnarray*}\nd(c_i, a)&=&d(c_i, e_i)+d(e_i, a)\\\\\n&= &d(c_i, e_i)+d(f(e_i), f(a))\\\\\n&=&d(c_i, e_i)+d(e_i, f(a))=d(c_i, f(a))\n\\end{eqnarray*}\nestablishing that $c\\models f(q)$.\n\nSince $f$ is an isometry, clearly $f(e_i)=e_i$ is the closest point to $c_i$ in $f(\\overline{E_A})$.\nThe closed subtree $f(\\overline{E_A})$ is equal to the closed subtree $\\overline{E_{f(A)}}$\nsince $f([a,b])=[f(a), f(b)]$ for all $a, b\\in U$.\nThis implies that $c\\ind_{f(A)} U$.   We conclude that $f(q)\\upharpoonright^U$=$q\\upharpoonright^U$.\n\n\n\nFor the other direction, assume $f$ is an automorphism of $U$\nthat does not fix all of the elements of $C$. Without loss of generality,\nassume $f(e_1)\\not=e_1$. Let $(c_1,...,c_n)\\models q\\upharpoonright^U$.\nThen the closest point in $U$ to $c_1$ is $e_1$, which is also the closest point to $c_1$ in $\\overline{E_A}$.\nThen, since $f(e_1)\\in U$, the point $e_1$ must be\non the geodesic segment joinging $f(e_1)$ and $c_1$, so $d(f(e_1), c_1)=d(f(e_1), e_1)+d(e_1,c_1)$.\nThus, $d(f(e_1), e_1)=d(f(e_1), c_1)-d(e_1, c_1)$.\nSince $d(f(e_1), e_1)\\not=0$, then $d(f(e_1), c_1)\\not=d(e_1, c_1)$.\nBut, $d(e_1, c_1)$ is the value of the formula $d(e_1, x_1)$ in $q\\upharpoonright^U$,\nand by the definition of $f(q)$, the value of $d(f(e_1), x_1)$ in $f(q)$ must equal the value of\n$d(e_1, x_1)$ in $q$. So, the $L$-condition $|d(f(e_1), x_1)-d(e_1, c_1)|=0$ is in the type $f(q)$,\nand therefore in the type $f(q)\\upharpoonright^U$.\nThus, the tuple $c=(c_1,...,c_n)$ cannot be a realization of $f(q)\\upharpoonright^U$, and\ntherefore $q\\upharpoonright^U\\not= f(q)\\upharpoonright^U$\n\\end{proof}\n\n\\endinput\n\n\n\\section{Models of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$: Examples}\n\nIn this section we discuss examples of models of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ from the literature.  Our first examples\ncome from the explicitly described universal $\\R$-trees that are treated in \\cite{DP2}.  We show that\nthey give exactly the (fully) saturated models of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$.  Our second examples come from\nasymptotic cones of hyperbolic finitely generated groups.  They give exactly the saturated\nmodel of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ of density $2^\\omega$.\n\nWe begin with a lemma about the density of a $\\kappa$-saturated model.\n\n\\begin{lemma}\n\\label{kappabranch}\n\\label{saturateddensity}\nLet $\\mathcal{M}\\models \\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ with underlying $\\R$-tree $(M,d,p)$, and let $\\kappa$\nbe an infinite cardinal.\n\\newline\n(1) If there exists $a\\in M$ with degree $\\kappa$, then\nthe density character of $M$ is at least $\\kappa$.\n\\newline\n(2) If  $\\mathcal{M}$ is $\\kappa$-saturated, then the density character of $\\mathcal{M}$ \nis at least $\\kappa^\\omega$.\n\\end{lemma}\n\\begin{proof}\n(1) If $d(p,a)=r$, then there is a single branch at $a$, namely the branch\ncontaining $p$. So, we must have $d(p, a)<r$.\nUsing Lemma \\ref{branches of sufficient height} and taking points on different branches at $a$\neach with distance $r$ from $p$, we find a collection of\n$\\kappa$-many points such that the distance between any two of them is $2(r-d(p,a))$.\nThus, the density character of $M$ must be at least $\\kappa$.\n\n(2) By Theorem \\ref{kappasaturated=kappabranching}, at each point in the \nunderlying $\\R$-tree of $M$ of $\\mathcal{M}$ there are at least $\\kappa$-many branches.\nThe tree construction from Theorem \\ref{rbrtstability} then yields at least\n$\\kappa^\\omega$-many\ndistinct points with pairwise distances at least $\\frac{r}{2}$.\n\\end{proof}\n\n\n\n\n\n\\noindent \\textbf{Universal $\\R$-trees} \\medskip\n\nNext, we include the description of the universal $\\R$-trees from \n\\cite{DP2} and relate them to saturated models of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$.\n\n\\begin{definition}[{See \\cite{MNO}}]\nLet $\\mu$ be a cardinal. An $\\R$-tree $M$ is called {\\it $\\mu$-universal}\nif, for any $\\R$-tree $N$ with $\\leq \\mu$ branches at every point, there\nis an isometric embedding of $N$ into $M$.\n\\end{definition}\n\n\\begin{example}[{See \\cite[Lemma 2.1.1]{DP2}}]\n\\label{universalrtrees}\nFor each $\\mu \\geq 2$ let $C_{\\mu}$ be\na set with cardinality $\\mu$ if $\\mu$ is infinite, and cardinality\n$\\mu-1$ if $\\mu$ is finite. Consider the set of functions from intervals in $\\R$ to $C_{\\mu}$\n\n$$f\\colon (-\\infty, \\rho_f)\\to C_{\\mu}$$\nwhere $f$ is such that\n\\begin{enumerate}\n\\item there exists $\\tau_f\\leq \\rho_f$ so that $f=0$ for all $t\\in (-\\infty, \\tau_f).$\n\\item The function $f$ is piecewise constant from the right. That is, for\nany $t\\in (-\\infty, \\rho_f)$ there exists $\\delta>0$ so\nthat $f$ is constant on $[t, t+\\delta]$.\n\\end{enumerate}\n\nOn this set of functions define the metric\n$$d(f, g)=(\\rho_f-s)+(\\rho_g-s) \\text{ where } s=\\sup\\{t\\mid f(t')=g(t') \\, \\forall t'<t\\}.$$\n\nThe authors of \\cite{DP2} show that $A_\\mu$ is a complete $\\R$-tree with $\\mu$-many branches of infinite height at every point. They also show that $A_\\mu$ is the unique (up to isometry) $\\R$-tree with $\\mu$ branches at each point and that it is homogeneous and $\\mu$-universal.\nIn our terminology, for $\\mu\\geq 3$ the space $A_\\mu$ is a complete, unbounded richly branching $\\R$-tree.\nLet $\\mathcal{M}_\\mu$ be the model of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ with underlying space equal to\nthe closed $r$-ball of $A_\\mu$ with an arbitrary choice of base point. \nFor infinite $\\mu$, Theorem \\ref{kappasaturated=kappabranching} shows $\\mathcal{M}_\\mu$\nis a $\\mu$-saturated model of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$.\n\n\nClassical model theory suggests that since $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ is complete and $\\kappa$-stable\nexactly when $\\kappa=\\kappa^\\omega$, there should be a unique saturated model\nof $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ with density $\\kappa$ exactly when $\\kappa=\\kappa^\\omega$.\nWe verify here that these unique saturated models are exactly those described in the preceding\nparagraph\nfor $\\mu=\\mu^\\omega$.\n\n\nSince $\\mathcal{M}_\\mu$ is $\\mu$-saturated, $\\mathcal{M}_\\mu$ (and $A_\\mu$) have density character at least $\\mu^\\omega$\nby Lemma \\ref{saturateddensity}.\nAn examination of the construction reveals that the cardinality of $A_\\mu$ is\nat most $\\mu^\\omega$. Conditions (1) and (2) in Example \\ref{universalrtrees}\nimply that each $f\\in A_{\\mu}$ can only change values at (at most) countably many points.\nMore specifically, we can recursively build a sequence $\\{t_n\\}_{n\\in \\N}$\nrecording where $f$ changes value as follows. For $f\\in A_{\\mu}$, let $t_0=\\tau_f$.\n\nGiven $t_n$, let $t_{n+1}=\\sup\\{t\\in (-\\infty, \\rho_f)\\mid f(t)=f(t_n)\\}$\nif this supremum is $<\\rho_f$. If $\\sup\\{t\\in (-\\infty, \\rho_f)\\mid f(t)=f(t_n)\\}=\\rho_f$,\nthen set $t_{n+i}=\\rho_f$ for all $i\\in \\N$.\n\\begin{itemize}\n\\item if $t_n=t_{n+1}$, then $t_{n+i}=t_n=\\rho_f$ for all $i\\in \\N$\n\\item $f$ is constant on $[t_n, t_{n+1})$ for each $n\\in \\N$\n\\end{itemize}\nFor each $n\\in \\N$, let $\\alpha_n\\in C_{\\mu}$\nbe the value of $f$ on $[t_n, t_{n+1})$.\nThis gives us a sequence $\\{\\alpha_n\\}$ of elements of $C_{\\mu}$.\nThe sequences $\\{t_n\\}$ and  $\\{\\alpha_n\\}$ determine $f$.\nThus, the cardinality of $A_{\\mu}$ is at most $2^\\omega \\cdot \\mu^\\omega=\\mu^\\omega$.\nIt follows that the metric density of $A_{\\mu}$ is exactly $\\mu^\\omega$.\nThus, in the case that $\\mu=\\mu^\\omega$, the model $\\mathcal{M}_{\\mu}$ is a\nsaturated model of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$, and uniqueness follows from the uniqueness\nof $A_{\\mu}$.  \nIf $\\mu<\\mu^\\omega$, then there is no $\\mu$-saturated model\nwith density $\\mu$, by Part (2) of Lemma \\ref{saturateddensity}.\n\nIn the case that $\\mu=\\mu^\\omega$, we outline an alternative\nmodel-theoretic argument that $A_{\\mu}$ is the unique complete $\\R$-tree\nwith $\\mu$ branches at every point. \nGiven any two complete $\\R$-trees $M_1$ and $M_2$ with $\\mu$-many branches every point,\nselect a basepoint in each.\nThen for each $r>0$ the closed $r$-balls in $M_1$ and $M_2$ (centered at their respective basepoints) are saturated models of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$.   Hence those $r$-balls are isomorphic\nby the fact that saturated models of a complete theory with the same density are isomorphic.\nA back-and-forth argument can be used to build an isomorphism\nfrom $M$ to $N$, where\neach time we extend the partial isomorphism we take its distance from the basepoint\ninto account and work in a large enough closed $r$-ball.\n\n\n\n\n\\end{example}\n\n\\noindent \\textbf{Asymptotic Cones} \\medskip\n\nA finitely generated group is $\\emph{hyperbolic}$ if its Cayley graph is a $\\delta$-hyperbolic\nmetric space for some $\\delta>0$.\nA \\emph{non-elementary} hyperbolic group is one that has no cyclic subgroup of finite index.\n\n\n\\begin{definition}\nLet $(M, d, p)$ be a metric space. Let $U$ be a non-principal ultrafilter on $\\N$ and let\n$(\\nu_m)_{m\\in \\N}$ be a sequence of positive integers\nsuch that $\\lim_{m\\to \\infty} \\nu_m=\\infty$.\nThe asymptotic cone of $(M, d, p)$ with respect to $(\\nu_m)_{m\\in \\N}$ and $U$ is\nthe ultraproduct of pointed metric spaces $\\prod_U (M, \\frac{d}{\\nu_m}, p)$. \nDenote this asymptotic cone by $\\mathrm{Con}_{U, (\\nu_m)}(M, d,p)$.  Elements of\n$\\mathrm{Con}_{U, (\\nu_m)}(M, d,p)$ are denoted $[a_n]$ where $a_n\\in M$ for each $n$.\n\\end{definition}\nThere are broader versions of this definition that allow, for example, a different\nchoice of base point in each factor. Keeping the same base point is sufficient\nfor our discussion.\n\n\n\\begin{example}\nAn asymptotic cone $\\mathrm{Con}_{U, (\\nu_m)}(G)$ of a finitely generated\ngroup is defined to be the asymptotic cone of its Cayley graph with base point $e$ and some designated word metric on $G$.\nIt is a fact that any asymptotic cone of a hyperbolic group is an $\\R$-tree and is homogeneous (see \\cite{DW} or \\cite{D}).\nIn fact, in the case of a non-elementary hyperbolic group, all asymptotic cones are homogeneous with $2^\\omega$ branches at every point (see \\cite[Proposition 3.A.7]{D}) and are thus are isometric to $A_{2^\\omega}$ from Example \\ref{universalrtrees}.   The next result gives\na proof of this fact.\n\\end{example}\n\n\n\\begin{fact}\\label{switch gen sets}\nSay $B$ and $C$ are both generating sets for the hyperbolic group $G$ and $U$ a non-principal ultrafilter.\nThe word metrics $d_B$ and $d_C$ are Lipschitz equivalent (and the corresponding\nCayley graphs are quasi-isometric.) It follows that the asymptotic cones $\\mathrm{Con}_{U, (\\nu_m)}(G, d_B, e)$ and $\\mathrm{Con}_{U, (\\nu_m)}(G, d_C, e)$ are homeomorphic. \n\\end{fact}\n\n\\begin{lemma}\nLet $G$ be a non-elementary hyperbolic group. Let $U$ be a non-principal ultrafilter and let\n$\\{\\nu_m\\}_{m\\in \\N}$ be a sequence of positive integers such that $\\lim_{m\\to \\infty} \\nu_m=\\infty$.\nThen for any generating set $C\\subseteq G$ the asymptotic cone $\\mathrm{Con}_{U, (\\nu_m)}(G, d_C, e)$  is a homogeneous richly branching $\\R$-tree with $2^\\omega$-many branches at every point.\n\\end{lemma}\n\\begin{proof}\nSince $G$ is finitely generated, we know that $G$ is countable.\nTherefore, any asymptotic cone of $G$ has cardinality (and therefore density)\nat most $2^\\omega$. By Lemma \\ref{kappabranch} we conclude that any point in the cone\ncan have at most $2^\\omega$ branches.\nWe also know $\\mathrm{Con}_{U, (\\nu_m)}(G, d_C, e)$ is a homogeneous $\\R$-tree.\nSince it is non-elementary $G$ contains a free subgroup $F$ with 2 generators (See \\cite{BH}).\nBy Fact \\ref{switch gen sets} we may assume without loss of generality that\nthe generators of $F$ are also generators\nof $G$ and that $C$ is a minimal set of generators. In $F$,\nfor each $m\\in \\N$, we can find a finite set $C_m$ such that $d(e,a)\\geq \\nu_m$ for any $a\\in C_m$\nand so that $(a\\cdot b)_e\\leq \\sqrt{\\nu_m}$ for distinct $a, b\\in C_m$.\nThe Cayley graph of $F$ is a subgraph of the Cayley graph of $G$, and the Cayley graph of a free group is an $\\R$-tree, thus we use $\\R$-tree terminology to describe how to find $C_m.$\n\nGiven $m\\in \\N^{>0}$ let $n_m$ be the largest integer so that $n_m\\leq \\sqrt{\\nu_m}$.\nNote that $n_m\\to\\infty$. There are $4\\cdot3^{{n_m}-1}$ elements of $F$ with distance $n_m$ from $e$. Let $c$ denote such an element. There are 3 branches at $c$ in the Cayley graph of $F$ that do not not contain $e$.\nOn each of these branches, choose a point $a$ with $d(a, e)\\geq \\nu_m$.\nRepeat this process for each $c$ with distance $n_m$ from $e$.\nLet $C_m$ be the collection of all the points $a$. Then $|C_m|=4\\cdot3^{n_m}$, and since $n_m\\to \\infty$\nwe know $|C_m|\\to\\infty$. For any distinct $a, b\\in C_m$, the distance from $e$ to $[a,b]$\nis at most $n_m\\leq \\sqrt{\\nu_m}$, because $[a,b]$ will always contain at least one of the points\nin $F$ with distance $n_m$ from $e$.\n\n\nSince $|C_m|\\to \\infty$ and $U$ is a non-principal ultrafilter on $\\N$,\nwe know that $\\displaystyle \\Pi_m C_m\/U$ is a set of points in the asymptotic cone with cardinality $2^\\omega$.\nMoreover, in the asymptotic cone $d([a_m],[e])=\\lim_U \\frac{d_C(a_m, e)}{\\nu_m}\\geq 1$.\nSince $(a_m\\cdot b_m)_e\\leq \\sqrt{\\nu_m}$, we know the distance from\n $[e]$ to the geodesic segment connecting $[a_m]$ and $[b_m]$ is\n$$([a_m]\\cdot [b_m])_{[e]}=\\lim_U \\frac{(a_m\\cdot b_m)_e}{\\nu_m}=0.$$\nThus, $[e]\\in \\left[[a_m],[b_m]\\right]$, putting $[a_m]$ and $[b_m]$ on separate branches\nat $[e]$.  This gives us $2^\\omega$ many distinct\nbranches at $[e]$ in $\\mathrm{Con}_{U, (\\nu_m)}(G, d_C, e)$. Therefore,\nthere must be $2^\\omega$ many branches at every point in the cone.\n\\end{proof}\n\n\n\\begin{corollary}\nLet $G$ be a non-elementary hyperbolic group.\nLet $\\mathcal{M}$ be the model of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ with underlying space\nequal to the closed $r$-ball of  $\\mathrm{Con}_{U, (\\nu_m)}(G, d_C, e)$.\nThen $\\mathcal{M}$ is the unique saturated model of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ of density $2^\\omega$.\n\\end{corollary}\n\\begin{proof}\nBy the preceding lemma and Lemma \\ref{kappabranch}, we know\n$\\mathcal{M}$ has density $2^\\omega$ and by Lemma \\ref{kappasaturated=kappabranching} $\\mathcal{M}$ is $2^\\omega$-saturated.\n\\end{proof}\n\n\n\\section{Models of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$: Constructions and non-categoricity}\\label{noncat}\n\nIn this section we show that $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ has the maximum number of\nmodels of density character $\\kappa$ for every infinite cardinal $\\kappa$.\nIndeed, for each $\\kappa$ we construct a family of $2^\\kappa$-many such models such that\nno two members of the family are  homeomorphic.  (Two models \nof $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ are homeomorphic if their underlying $\\R$-trees are homeomorphic \nby a map that takes base point to base point.  Note that non-homeomorphic models\nof $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ are necessarily non-isomorphic.)  First we treat \nseparable models, and the amalgamation techniques used\nin that case also allow us to characterize the principal types of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ and\nto show that this theory has no atomic model.  Then we use simple amalgamation\nconstructions to handle nonseparable models.\n\n\\begin{lemma}\n\\label{separabletreewitharbbranching}\nLet $S$ be a non-empty set of integers, each of which is $\\geq 3$.\nThere exists a separable richly branching $\\R$-tree $M$ such that\n\\begin{enumerate}\n\\item for each $k\\in S$ the set $\\{x\\in M\\mid x \\text{ has degree }k\\}$ is dense in $M$\n\\item given a branch point $x\\in M$ the degree of $x$ is an element of $S$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nLet $(k_j\\mid j\\in \\N)$\nbe a sequence such that every element of $S$ appears infinitely many times in the sequence,\nand every term of the sequence is an element of $S$.\nWe construct an increasing sequence $\\mathcal{N}_0\\subseteq \\mathcal{N}_1\\subseteq ... \\subseteq \\mathcal{N}_j... $ of \nseparable $\\R$-trees as follows.\n\nLet $N_0$ be the $\\R$-tree $\\R$ with base point $0$.\nLet $A_0$ be a countable, dense subset of $N_0$. \nUse Lemma \\ref{chiswell add to points} to add $k_0-2$ distinct rays (copies of $\\R^{\\geq 0}$) at each\npoint in $A_0$, bringing the number of branches of infinite length at each point in $A_0$ up to $k_0$.\nCall the resulting $\\R$-tree $N_1$. Note that $N_0\\subseteq N_1$.\nThe $\\R$-tree $N_1$ is separable, since it is a countable union of separable spaces.\nNote also that all the points in $N_1\\setminus A_0$ only have 2 branches,\nand it is straightforward to show $N_1\\setminus A_0$ is uncountable and dense in $N_1$.\n\nOnce $N_j$ has been constructed, to construct $N_{j+1}$\nlet $\\displaystyle A_{j}\\subset N_j\\setminus (\\cup_{i=0}^{j-1} A_j)$ be a countable, dense subset of $N_j$. \nThis is possible since $\\displaystyle  N_j\\setminus (\\cup_{i=0}^{j-1} A_j)$ is dense in $N_j$.\nUse Lemma \\ref{chiswell add to points} to add $k_j-2$ rays at each\npoint in $A_j$, bringing the number of branches of infinite length at each point in $A_j$ up to $k_j$.\nThe resulting $\\R$-tree is $N_{j+1}$.\nNote that $N_{j+1}$ is separable, since it is a countable union of separable spaces.\nNote also that all the points in $\\displaystyle N_{j+1}\\setminus \\cup_{j=0}^{j} A_j$ still only have 2 branches,\nand that this set is uncountable and dense in $N_{j+1}$.\nLastly, it is clear that given $x\\in N_{j+1}$ the number of branches at $x$ must be\neither $2$ (in which case $x$ is not a ``branch point\") or one of $\\{k_0, ..., k_j\\}$.\nThis is because at the $j$th step, the only points at which we add rays are those in $A_j$,\nand then in subsequent steps we do not add rays at any of those points.\n\nLet $\\displaystyle M=\\cup_{j\\in\\N} N_j$ be the union of this countable chain of separable $\\R$-trees. Then $M$ is a separable $\\R$-tree (see \\cite[Lemma 2.1.14]{C}.)\nSince for each $N_j$ the number of branches at each branch point is an element\nof $S$, this will also be true in $M$.\nLet $k\\in S$. Let $\\displaystyle J(k)=\\{j\\in \\N\\mid k_j=k\\}$. By how we chose the sequence $(k_j)$ the set $J(k)$\nis infinite.  The set of points in $M$ which have exactly $k$ branches is $\\cup_{j\\in J(k)} A_j$.\nWe will show this set is dense in $M$.\nLet $x\\in M$. Let $j_x\\in \\N$ be the smallest integer such that $x\\in N_{j_x}$. Let $j^*\\in J(k)$ be\nsuch that $j^*>j_x$. We know that $A_{j^*}$ is dense in $N_{j^*}$, and that $\\displaystyle x\\in N_{j_x}\\subseteq N_{j^*}$.\nTherefore, there are points in $\\displaystyle A_{j^*}\\subseteq  \\cup_{j\\in J(k)} A_j$ arbitrarily close to $x$.\nOur choice of $x\\in M$ was arbitrary, therefore $\\displaystyle \\cup_{j\\in J(k)} A_j$ is dense in $M$.\nBecause we chose a non-empty $S$ with members all $\\geq 3$\nthe set of branch points with at least 3 branches of infinite length is dense in $M$. Therefore, $M$\nis a richly branching $\\R$-tree.\n\\end{proof}\n\n\\begin{remark}\nIn the preceding proof, we did not use the fact that we are considering homeomorphisms\nof \\emph{pointed} topological spaces. The $\\R$-trees constructed in Lemma \\ref{separabletreewitharbbranching}\nare in fact non-homeomorphic even when we are not required to preserve the base point.\n\\end{remark}\n\n\\begin{remark}\nThe $\\R$-trees constructed in Lemma \\ref{separabletreewitharbbranching} all have\nat least 2 branches at every point. It is straightforward to modify this construction so that\nthere are some points with degree 1 as well. For example, begin the construction with the\n$\\R$-tree $[0, 1]$, and always exclude $0$ from the set of points where rays are added. This\nwill result in a richly branching tree where we know there is a single branch at that point.\nOne could also start with a richly branching tree and ``trim\" away all but one branch at some points.\n\\end{remark}\n\n\\begin{theorem}  \n\\label{number of separable models}\nThere exist $2^{\\omega}$-many pairwise non-homeomorphic\n(hence non-isomorphic) separable models of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$.\n\\end{theorem}\n\n\\begin{proof}\nAny homeomorphism $g$ between models $\\mathcal{M}$ and $\\mathcal{N}$ of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ is a homeomorphism\non the underlying $\\R$-trees which must preserve branching.  In particular, given $n\\in \\N^{\\geq 3}$,\nif there is a point with degree $n$ in $M$, then there must be a point with degree $n$ in $N$.\nChoose two different subsets $S$ and $S'$ of $\\N^{\\geq 3}$, and construct\na richly branching tree for each as in Lemma \\ref{separabletreewitharbbranching}. Let\n$\\mathcal{M}$ and $\\mathcal{M}'$ be the models of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ based on the completions of their closed $r$-balls, respectively.\n(Note that taking the completion here can only add points of degree 1.)\nIt follows that $\\mathcal{M}$ and $\\mathcal{M}'$ cannot be homeomorphic.\nSince there are $2^\\omega$-many different such sets $S$,\nthere are $2^\\omega$-many different non-homeomorphic, separable models of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$.\n\\end{proof}\n\n\n\nRecall that given a continuous theory $T$, a type $q\\in S_n(T)$ is \\emph{principal}\nif for every model $\\mathcal{M}$ of $T$, the set $q(\\mathcal{M})$ of realizations of $q$ in $\\mathcal{M}$ is definable\nover the empty set.  As in classical first order logic, given a complete theory in a countable signature, there is\na Ryll-Nardzewski theorem stating the equivalence between $\\omega$-categoricity and the fact\nthat every type is principal.  (See \\cite[Theorem 12.2]{N}.)   Furthermore, a type $q$ is principal\nif and only if $q$ is realized in every model of $T$.  (See \\cite[Theorem 12.6]{N}.)\n\nTheorem \\ref{number of separable models} obviously implies that $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ is not $\\omega$-categorical, \nand thus not every type of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ is principal.   Our next result gives a characterization of the principal  types in $S_n(\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r)$. In particular,\na principal type is the type of a tuple of points that all lie along a single piecewise segment\nwith $p$ as an endpoint. Thus, there are very few of them. \nAs a consequence, we\nconclude that $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ does not have a prime model (equivalently, does not have an atomic model, one in which only principal types are realized).\n\nFor a clear and comprehensive treatment of separable models in continuous model theory,\nwe refer the reader to Section 1 in \\cite{BUdfinite}.  Note that where we and \\cite{N} have the word \\emph{principal}, the authors of \\cite{BUdfinite} use \\emph{isolated,} which is now the standard terminology.\nIn (\\cite[Theorem 1.11]{BUdfinite}) they prove an omitting types theorem, and as a corollary (\\cite[Corollary 1.13]{BUdfinite}) show that principal types can be omitted.\nFurther, it follows from \\cite[Definition 1.7]{BUdfinite} and properties of definable sets in continuous model theory, that every principal type is realized in every model, and this is implicit in the discussion following that definition.\n\n\\begin{theorem}\n\\label{principal types}\nLet $q\\in S_n(\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r)$. The following are equivalent\n\\begin{enumerate}\n\\item The type $q$ is principal.\n\\item If $j\\in\\{1,...,n\\}$ is such that $d(p, x_j)^q\\geq d(p, x_i)^q$ for all $i\\in \\{1,..,n\\}$,\nthen $d(p, x_j)^q=d(p, x_i)^q+d(x_i, x_j)^q$ for all $i\\in \\{1,...,n\\}$.\n\\item For any model $\\mathcal{M}$ and $b_1,..., b_n\\in M$ that realize $q$,\nif $j\\in\\{1,...,n\\}$ is such that $d(p, b_j)\\geq d(p, b_i)$ for all $i\\in \\{1,..,n\\}$,\nthen $b_i$ is on the segment $[p, b_j]$ for all $i\\in \\{1,...,n\\}$. \n\\item  For any model $\\mathcal{M}$ and $b_1,..., b_n\\in M$ that realize $q$,  the points\n$p, b_1,...,b_n$ are arranged (in some order) along a piecewise segment with $p$ as\nan endpoint.\n\\end{enumerate}\n\n\\end{theorem}\n\\begin{proof}\nThat (2) and (3) are equivalent follows from Lemma \\ref{branch}.\nThat (3) and (4) are equivalent follows from the definition of a piecewise segment.\nWe show (1) implies (3) by proving the contrapositive.\nLet $\\mathcal{M}\\models \\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ and $b_1, ..., b_n\\in M$ realizing $q$ and let\n$j\\in\\{1,...,n\\}$ be such that $d(p, b_j)\\geq d(p, b_i)$ for all $i\\in \\{1,..,n\\}$.\nAssume there exists $i\\in \\{1,...,n\\}$ so that $b_i$ is not on $[p, b_j]$.\n\nThen let $e$ be the closest point to $b_i$ on the\nsegment $[p, b_j]$. If $e=b_k$ for some $k\\in \\{1,...,n\\}$, with $b_k\\not=b_i$, then\n$$d(p, b_i)=d(p, e)+d(e, b_i)=d(p, b_k)+d(b_k, b_i)>d(p, b_k).$$\nThe last inequality gives us a contradiction.\nThis leaves two possibilities.\n\nCase 1: Assume $e=p$. Then $b_i$ and $b_j$ are on different branches at $p$.\nUsing techniques as in\n\\ref{separabletreewitharbbranching} we can construct $\\mathcal{N}\\models \\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ so that there\nis only a single branch at $p$. Then clearly $q$ is not realized in $\\mathcal{N}$.\n\nCase 2: Assume $e\\not=p$.\nThen $p$ and $b_j$ are on different branches at $e$,\nand Lemma \\ref{!geo} implies that $e\\in [p, b_j]$ and $e\\in [b_i, b_j]$. So,\n$p$ and $b_j$ are on different branches at $e$, as are $b_i$ and $b_j$.\nThus, there are at least 3 branches at $e$ in the subtree spanned\nby $p, b_1,...,b_n$. \nLet $\\alpha=d(p, e)$, which is a value determined\nby the type $q$.\\\\\nBuild a model $\\mathcal{N}$ so that there are no branching points at distance\n$\\alpha$ from $p$. Then $q$ is not realized in $\\mathcal{N}$, implying that $q$ is not\na principal type.\n\nLastly, we show (2) implies (1).\nLet $q\\in S_n(\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r)$ be a type satisfying the condition in (2)\nand let $j\\in \\{1,...,n\\}$ be such that $d(p, x_j)^q\\geq d(p, x_i)^q$ for all $i\\in \\{1,..,n\\}$.\nTake any model $\\mathcal{M}\\models \\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$. Then there is at least one branch\nat the base point $p$, and by Lemma \\ref{branches of sufficient height} along this branch we can find a point $b$ so that $d(p, b)=d(p, x_j)^q$. Let $b=b_j$. We can find realizations\nof the other $x_i$'s at appropriate distances along the segment $[p,b_j]$, and prove that\n$b_1,...,b_n$ realize $q$ by Lemma \\ref{determinetypes}.\nThus, if $q$ is a type satisfying the conditions in (2), we know $q$ is a principal type.\n\\end{proof}\n\n\\begin{corollary}\nThe $L$-theory $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ has no prime model.\n\\end{corollary}\n\\begin{proof}\nAssume $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ has a prime model $\\mathcal{M}$ with underlying $\\R$-tree\n$(M, d,p)$. Then $\\mathcal{M}$ is atomic and the type\nof any tuple $b_1,...,b_n$ must be principal. By the preceding theorem, this means\nthat for any pair $b_1, b_2$ in $M$, either $b_1\\in [p, b_2]$ or $b_2\\in [p, b_1]$.\nIt follows that $M$ consists of a piecewise segment with endpoint $p$.  In particular, \n$\\mathcal{M}$ is not richly branching, which is a contradiction.\n\\end{proof}\n\nWe finish this section by showing that when $\\kappa$ is uncountable,\nthen the number of different models of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ having density character\nequal to $\\kappa$ is also the maximum possible, namely $2^\\kappa$.\nAs in the case $\\kappa = \\omega$, which was treated in the first part of this\nsection, we produce large sets of models that are not only non-isomorphic,\nbut in fact have underlying $\\R$-trees which are non-homeomorphic\n(as pointed topological spaces).\n\nWe will carry out the construction by induction on $\\kappa$, and we begin with a useful lemma. \n\\begin{lemma}\n\\label{getting many nonseparable models}\nLet $\\kappa$ be an uncountable cardinal.  \nThe following conditions are equivalent:\n\\newline\n(1) The number of non-homeomorphic models of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ of density character $\\leq \\kappa$ is at least $\\kappa$.\n\\newline\n(2) The number of non-homeomorphic models of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ of density character $\\leq \\kappa$ that have just one branch at the base point is at least $\\kappa$.\n\\newline\n(3) The number of non-homeomorphic models of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ of density character $= \\kappa$ is $2^\\kappa$.\n\\end{lemma}\n\n\\begin{proof}\nLet $\\kappa$ be an uncountable cardinal. Clearly, (3) implies (1).\nTo show (2) implies (3), assume (2) and  let $(B_\\alpha \\mid \\alpha <  \\kappa)$ be a list of the pairwise non-homeomorphic models of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$, each with density character $\\leq \\kappa$ and exactly one branch at the base point.  Given a subset $S\\subseteq \\kappa$ of cardinality $= \\kappa$, take the collection of $B_\\alpha$\nfor $\\alpha\\in S$ and glue them all together at their base points\nusing Theorem \\ref{amalg}.  Call this amalgam $M_S$,\nand let $p$ be the point in $M_S$ at which the $B_\\alpha$ are all glued together. \nMake $p$ the base point of the $L_r$-structure $\\mathcal{M}_S$.   Note that the density character of $\\mathcal{M}_S$ is exactly $\\kappa$, since each branch of \nits underlying $\\R$-tree $M_S$ at $p$ has density $\\leq \\kappa$\nand height $r$, and there are exactly $\\kappa$ many branches at $p$.\nMoreover, it is easy to check that $\\mathcal{M}_S$ is a model of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$.\n\nBy this construction, if $B$ ranges over the branches of $M_S$ at $p$, the\nhomeomorphism type of $B \\cup \\{p\\}$ (with $p$ as distinguished element)\nranges bijectively over the homeomorphism types of $B_\\alpha$ (also with $p$ as distinguished element) as $\\alpha$ ranges over $S$.  It follows that the homeomorphism type of $\\mathcal{M}_S$ determines $S$.\nTherefore the family $\\{ \\mathcal{M}_S \\mid S \\subseteq \\kappa \\text{ and } S \\text{ has cardinality } = \\kappa \\}$\nverifies condition (3), since $\\kappa$ has $2^\\kappa$ many subsets of cardinality $= \\kappa$.\n\nFinally, we prove that (1) implies (2).\nFor each model $\\mathcal{M}$ of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$, let $b(\\mathcal{M})$ denote the number of branches of $M$ at its base point; we take this to be a positive integer or $\\infty$, where $b(\\mathcal{M}) = \\infty$ means that there are infinitely many branches.  Since $\\kappa$ is uncountable, condition (1) yields a class $\\mathcal{K}$ of at least $\\kappa$ many non-homeomorphic models of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$, each of density character $\\leq \\kappa$, such that $b(\\mathcal{M})$ has a constant value $b$ as $\\mathcal{M}$ ranges over $\\mathcal{K}$.  We may assume $b \\neq 1$, since otherwise condition (2) is satisfied by the models in $\\mathcal{K}$.\n\nLet $a$ be an integer $\\geq 3$ that is different from $b+1$.  ( Note $a\\not=b+1$ is  automatically true if $b$ is $\\infty$.)  Using a method similar to that in the proof of Lemma \\ref{separabletreewitharbbranching}, we may take $\\mathcal{N}=(N, d, p)$ to be a separable model of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ with the following properties: (1) for all $x$ in $N$, the number of branches in $N$ at $x$ is $1$ or $2$ or $a$; (2) $N$ has a single branch at its base point $p$; and (3) \n$N$ has a single branch at some point $y$, where $d^{\\mathcal{N}}(p, y)=\\frac{r}{2}$.\nTo get single branches at 2 points with a given distance as required here,\ninstead of starting the construction with\nthe $\\R$-tree $\\R$ as in the proof of Lemma \\ref{separabletreewitharbbranching}, start with the interval $[0, \\frac{r}{2}]\\subseteq \\R$, and in\nsubsequent steps always exclude $0$ and $\\frac{r}{2}$ from the sets of points where rays\nare added. \n\n\nNow consider an arbitrary $\\mathcal{M} \\in \\mathcal{K}$, and denote the base point of $\\mathcal{M}$ by $q$.\nScale the metric on $\\mathcal{M}$ down by a factor of 2, resulting in an $L_r$-structure with a radius of \n$\\frac{r}{2}$.\nWe construct a larger $\\R$-tree $\\mathcal{M}^*$ by amalgamating the scaled-down $\\mathcal{M}$ and $\\mathcal{N}$ in the way that identifies $q$ and $y$; we will denote this point of $M^*$ by $qy$.  We take the base point of $\\mathcal{M}^*$ to be the base point $p$ of $\\mathcal{N}$.  The radius of  $\\mathcal{M}^*$ is $r$, because the radius of $\\mathcal{N}$ was $r$, and by our amalgamation construction, for every point $x\\in M^*$ residing in the scaled down copy of $M$, \n$$d^{\\mathcal{M}^*}(p,x)=d^{\\mathcal{M}^*}(p, qy)+d^{\\mathcal{M}^*}(qy, x)=\\frac{r}{2}+\\frac{d^{\\mathcal{M}}(y, x)}{2}$$\nwhich attains its maximum value $r$ as $x$ ranges over the scaled down copy of $M$.\n\nIt is straightforward to check that $\\mathcal{M}^*$ is a model of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$, has density $\\leq \\kappa$, and has a single branch at its base point.  Note that the branches of $M^*$ at the amalgamated point $qy$ consist of the branches of $q$ in $M$ together with the tree that results from $N$ by removing $y$.  In particular, this means that $M^*$ has $b+1$ many branches at $qy$.\n\nWe claim that the class $\\mathcal{K}^* = \\{ \\mathcal{M}^* \\mid \\mathcal{M} \\in \\mathcal{K} \\}$ verifies condition (3); it remains only to show that no two members of this class are homeomorphic.  (Recall that the homeomorphisms we consider must take base point to base point.)  The key to this is the fact that the point $qy$ can be topologically identified in $\\mathcal{M}^*$, given that we know the base point $p$.  To do this, note first that the segment $X=[p,qy)$ in $\\mathcal{M}^*$ is identical to the segment $[p,y)$ in $\\mathcal{N}$, and every point in $X$ has the same number of branches in $\\mathcal{M}^*$ as in \n$\\mathcal{N}$. Therefore every point $x$ of $X$ has  $1$, $2$, or $a$ many branches in $\\mathcal{M}^*$, and thus the number of branches at $x$ is different from the number of branches at $qy$.  From this we conclude that for any $\\mathcal{M}_1,\\mathcal{M}_2 \\in \\mathcal{K}$, any homeomorphism of $\\mathcal{M}_1^*$ onto $\\mathcal{M}_2^*$ that takes base point to base point must map the scaled version of $\\mathcal{M}_1$ onto the scaled version of $\\mathcal{M}_2$.  Since this can only happen when $\\mathcal{M}_1 = \\mathcal{M}_2$, by assumption on $\\mathcal{K}$, we conclude that $\\mathcal{M}_1^* = \\mathcal{M}_2^*$, as desired.\n\\end{proof}\n\n\n\\begin{theorem}\nLet $\\kappa$ be an uncountable cardinal. The number of non-homeomorphic models of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ of density character equal to $\\kappa$ is $2^\\kappa$.\n\\end{theorem}\n\n\\begin{proof}\nWe assume that $\\sigma$ is the least uncountable cardinal at which there are strictly fewer than $2^\\sigma$ many non-homeomorphic models of density character equal to $\\sigma$, and derive a contradiction.  Using Theorem \\ref{number of separable models}, we see that condition (1) in Theorem \\ref{getting many nonseparable models} holds when $\\kappa = \\omega_1$; condition (3) in that result yields that $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ has $2^{\\omega_1}$ many non-homeomorphic models of density character equal to $\\omega_1$.  Thus $\\sigma > \\omega_1$.  Now suppose $\\sigma$ is a successor cardinal; say it is the next cardinal bigger than $\\lambda$, which must be uncountable.  Our choice of $\\sigma$ ensures that there must be $2^\\lambda \\geq \\sigma$ many non-homeomorphic models of density character $\\lambda$.  Applying Lemma \\ref{getting many nonseparable models} with $\\kappa = \\sigma$ gives a contradiction; indeed, we have verified condition (1), while condition (3) is false.\nSo $\\sigma$ must be a limit cardinal.  Let $\\tau$ be the number of non-homeomorphic models of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ that have density character $\\leq \\sigma$; our treatment of $\\omega_1$ shows that $\\tau$ is uncountable.  Furthermore, Lemma \\ref{getting many nonseparable models} applied to $\\kappa = \\sigma$ yields $\\tau < \\sigma$.   Our choice of $\\sigma$ ensures that there are $2^ \\tau > \\tau$ many non-homeomorphic models of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ that have density character $\\tau$, contradicting the definition of $\\tau$.\n\\end{proof}\n\n\\section{Unbounded $\\R$-trees}\n\nAs noted in the Introduction, we have chosen to treat \\emph{bounded} pointed $\\R$-trees\nin this paper, because many of the model-theoretic ideas and tools we need from continuous\nfirst order logic are only documented for bounded metric structures in the literature.\n\nHowever, it would certainly be a natural research topic to study the model theory of unbounded\n(\\textit{i.e.}, not necessarily bounded) pointed $\\R$-trees.  The most immediately available setting\nfor doing this would be to consider a pointed $\\R$-tree $(M,p)$ as a many-sorted metric structure\nin which each sort is one of the (closed) bounded balls of $(M,p)$ (centered at $p$), and the union\nof the family of distinguished balls is all of $M$.  \nEverything done in this paper can easily be carried over to that setting.  The disadvantages of\ndoing so are the technical awkwardness of the many-sorted framework and the need for imposing\nan arbitrary family of radii for the bounded balls into which the full tree is stratified.\n\nIt is certainly more mathematically natural to consider pointed $\\R$-trees on their own, without\nimposing a many-sorted stratification.  There are suitable\nlogics for doing model theory with such unbounded structures.  For example, a version of continuous first\norder logic for unbounded metric structures is described in \\cite{BY}.  Also, a logic based\non \\emph{positive bounded formulas} and an associated concept of \\emph{approximate satisfaction}\nis presented in Section 6 of \\cite{DI}.  However, for neither of these approaches are the ideas and tools\nof model theory developed as we need them in this paper.\n\nIn each of these three available settings for treating arbitrary pointed $\\R$-trees, the arguments in this paper\ncan be used easily to demonstrate: (1) the class of pointed $\\R$-trees is axiomatizable and (2) for each $r>0$, the \nball $\\{ x \\mid d(x,p) \\leq r \\}$ is a definable set (over $\\emptyset$, uniformly in all pointed $\\R$-trees).  \nTogether with what is developed in \\cite{N}, \\cite{BY}, and \\cite{DI}, this quickly yields that the model theoretic \nframeworks for pointed $\\R$-trees provided by these three settings are completely equivalent.  \nIn particular, this approach yields a model completion \nfor the theory of pointed $\\R$-trees whose models are exactly the\nrichly branching $\\R$-trees (\\textit{i.e.}, the complete pointed $\\R$-trees described in Remark \\ref{general richly branching}).  Furthermore, this\nmodel completion has suitably stated versions of all the properties of $\\mathrm{r}\\mathrm{b}\\mathbb{R}\\mathrm{T}_r$ that are proved in this paper.\n\n\\endinput\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Introduction}\n\nThe commutative division algebras constructed by Dickson \\cite{Dic} yield proper semifields of even dimension over finite fields. They have been subsequently studied in many papers, for example in \\cite{Bur}, \\cite{Bur2}, \\cite{HTW}, \\cite{Thompson19}. Knuth recognised that Dickson's commutative division algebras also appear as a special case of another family of semifields \\cite{Knu}: A subalgebra $L$ of a division algebra $S$ is called a \\textit{weak nucleus} if $x(yz)-(xy)z=0$, whenever two of $x,y,z$ lie in $L$.  Semifields which are quadratic over a weak nucleus are split into two cases; Case I semifields contain Dickson's construction as the only commutative semifields of this type. Due to this, Case I semifields are also called \\textit{generalized Dickson semifields}. Their construction is as follows: given a finite field $K=GF(p^n)$ for some odd prime $p$, define a multiplication on $K\\oplus K$ by $$(u,v)(x,y)=(uv+c\\alpha(v)\\beta(y), \\sigma(u)y+vx),$$ for some automorphisms $\\alpha$, $\\beta$, $\\sigma$ of $K$ not all the identity automorphism and $c\\in K\\setminus K^2$. This construction produces a proper semifield containing $p^{2n}$ elements. Further work on semifields quadratic over a weak nucleus was done in \\cite{Ganley} and \\cite{Cohen}.\\\\\nIn this paper, we define a doubling process which generalizes Knuth's construction in \\cite{Knu}: for a central simple associative algebra $D\/F$ or finite field extension $K\/F$, we define a multiplication on the $F$-vector space $D\\oplus D$ (resp. $K\\oplus K$) as $$(u,v)(x,y)=(ux+c\\sigma_1(v)\\sigma_2(y),\\sigma_3(u)y+v\\sigma_4(x))$$ for some $c\\in D^{\\times}$ and $\\sigma_i\\in Aut_F(D)$ for $i=1,2,3,4$ (resp. $c\\in K^{\\times}$ and $\\sigma_i\\in Aut_F(K)$). This yields an algebra of dimension $2dim_F(D)$ or $2[K:F]$ over $F$. \nOver finite fields, our construction yields examples of some Hughes-Kleinfeld, Knuth and Sandler semifields (for example, see \\cite{Cor}) and all generalized Dickson and commutative Dickson semifields \\cite{Knu}\\cite{Dic}. Hughes-Kleinfeld, Knuth and Sandler semifield constructions were studied over arbitrary base fields in \\cite{BrownSteelePump}. Dickson's commutative semifield construction was introduced over finite fields in \\cite{Dic} and considered over any base field of characteristic not 2 when $K$ is a finite cyclic extension in \\cite{Bur}. This was generalized to a doubling of any finite field extension and central simple algebras in \\cite{Thompson19}.\\\\\nAfter preliminary results and definitions, we define a doubling process for both a central simple algebra $D\/F$ and a finite field extension $K\/F$; we recover the multiplication used in Knuth's construction of generalized Dickson semifields when $\\sigma_4=id$. We find criteria for them to be division algebras. We then determine the nucleus and commutator of these algebras and examine both isomorphisms and automorphisms. The results of this paper are part of the author's PhD thesis written under the supervision of Dr S. Pumpl\\\"{u}n.\n\n\\section{Definitions and preliminary results}\n\nIn this paper, let $F$ be a field. We define an $F$-algebra $A$ as a finite dimensional $F$-vector space equipped with a (not necessarily associative) bilinear map $A\\times A\\to A$ which is the multiplication of the algebra. $A$ is a \\textit{division algebra} if for all nonzero $a\\in A$ the maps $L_a:A\\to A$, $x\\mapsto ax$, and $R_a:A\\to A$, $x\\mapsto xa$, are bijective maps. As $A$ is finite dimensional, $A$ is a division algebra if and only if there are no zero divisors \\cite{Sch}.\\\\\nThe \\textit{associator} of $x,y,z\\in A$ is defined to be $[x,y,z]:=(xy)z-x(yz).$ Define the \\textit{left, middle and right nuclei} of $A$ as $\\text{Nuc}_l(A):=\\lbrace x\\in A \\mid [x,A,A]=0\\rbrace,$ $\\text{Nuc}_m(A):=\\lbrace x\\in A \\mid [A,x,A]=0\\rbrace,$ and $\\text{Nuc}_r(A):=\\lbrace x\\in A \\mid [A,A,x]=0\\rbrace.$ The left, middle and right nuclei are associative subalgebras of $A$. Their intersection $\\text{Nuc}(A):=\\lbrace x\\in A \\mid [x,A,A]=[A,x,A]=[A,A,x]=0\\rbrace$ is the \\textit{nucleus} of $A$. The \\textit{commutator} of $A$ is the set of elements which commute with every other element, $Comm(A):=\\lbrace x\\in A\\mid xy=yx \\:\\forall y\\in A\\rbrace.$ The \\textit{center} of $A$ is given by the intersection of $\\text{Nuc}(A)$ and $Comm(A)$, $Z(A):=\\lbrace x\\in \\text{Nuc}(A)\\mid xy=yx\\: \\forall y\\in A\\rbrace.$ For two algebras $A$ and $B$, any isomorphism $f:A\\to B$ maps $\\text{Nuc}_l(A)$ isomorphically onto $\\text{Nuc}_l(B)$ (similarly for the middle and right nuclei).\\\\\nAn algebra $A$ is \\textit{unital} if there exists an element $1_A\\in A$ such that $x1_A=1_Ax=x$ for all $x\\in A$. \nA form $N:A \\to F$ is called \\textit{multiplicative} if $N(xy)=N(x)N(y)$ for all $x,y\\in A$ and \\textit{nondegenerate} if we have $N(x)=0$ if and only if $x=0$. Note that if $N:A\\to F$ is a nondegenerate multiplicative form and $A$ is a unital algebra, it follows that $N(1_A)=1_F$. Every central simple algebra admits a uniquely determined nondegenerate multiplicative form, called the \\textit{norm} of the algebra.\\\\\n\n\n\n\n\\section{A doubling process which generalizes Knuth's construction}\\label{GCDD_Field}\n\n\nLet $D$ be a central simple associative division algebra over $F$ with nondegenerate multiplicative norm form $N_{D\/F}:D\\to F$. Given $\\sigma_i\\in Aut_F(D)$ for $i=1,2,3,4$ and $c\\in D^{\\times}$, define a multiplication on the $F$-vector space $D\\oplus D$ by $$(u,v)(x,y)=(ux+c\\sigma_1(v)\\sigma_2(y),\\sigma_3(u)y+v\\sigma_4(x)).$$ We denote the $F$-vector space endowed with this multiplication by $\\text{Cay}(D,c,\\sigma_1,\\sigma_2,\\sigma_3,\\sigma_4)$. We can also define an analogous multiplication on $K\\oplus K$ for a finite field extension $K\/F$ for some $c\\in K^{\\times}$ and $\\sigma_i\\in Aut_F(K)$. We similarly denote these algebras by $\\text{Cay}(K,c,\\sigma_1,\\sigma_2,\\sigma_3,\\sigma_4).$ This yields unital $F$-algebras of dimension $2dim_F(D)$ and $2[K:F]$ respectively. When $\\sigma_4=id$, our multiplication is identical to the one used in the construction of generalized Dickson semifields. For every subalgebra $E\\subset D$ such that $c\\in E^{\\times}$ and $\\sigma_i\\mid_E=\\phi_i\\in Aut_F(E)$ for $i=1,2,3,4$, it is clear that $\\text{Cay}(E,c,\\phi_1,\\phi_2,\\phi_3,\\phi_4)$ is a subalgebra of $\\text{Cay}(D,c,\\sigma_1,\\sigma_2,\\sigma_3,\\sigma_4)$.\n\n\\begin{theorem}\\label{KDivision Algebras} \n\\begin{enumerate}[(i)]\n\\item If $N_{D\/F}(c)\\not\\in N_{D\/F}(D^{\\times})^2$, $\\text{Cay}(D,c,\\sigma_1,\\sigma_2,\\sigma_3,\\sigma_4)$ is a division algebra.\n\\item If $K$ is separable over $F$ and $N_{K\/F}(c)\\not\\in N_{K\/F}(K^{\\times})^2$, then $\\text{Cay}(K,c,\\sigma_1,\\sigma_2,\\sigma_3,\\sigma_4)$ is a division algebra.\n\\end{enumerate}\n\\end{theorem}\n\\begin{proof}\n\n(i) Suppose $(0,0)=(u,v)(x,y)$ for some $u,v,x,y\\in D$ such that $(u,v)\\neq(0,0)\\neq(x,y)$. This is equivalent to \\begin{align}\nux+c\\sigma_1(v)\\sigma_2(y)=&0,\\label{Div1K}\\\\\n\\sigma_3(u)y+v\\sigma_4(x)=&0.\\label{Div2K}\n\\end{align}\nAssume $y=0$. Then by (\\ref{Div1K}), $ux=0$, so $u=0$ or $x=0$ as $D$ is a division algebra. As $(x,y)\\neq (0,0)$, we must have $x\\neq 0$ so $u=0$. Then by (\\ref{Div2K}), $v\\sigma_4(x)=0$ which implies $v=0 \\mbox{ or } x=0$. This is a contradiction, thus it follows that $y\\neq 0$. By (\\ref{Div2K}), $v\\sigma_4(x)=-\\sigma_3(u)y.$ Let $N=N_{D\/F}:D\\to F$. Taking norms of both sides, we have \\begin{align*}\n&N(v)N(x)=-N(u)N(y)\\\\\n\\implies &N(u)= -N(v)N(x)N(y)^{-1},\n\\end{align*}\nsince $y\\neq 0$. Substituting this result into (\\ref{Div1K}) implies \\begin{align}\n0=&N(u)N(x)+N(c)N(v)N(y)\\nonumber\\\\\n=&(-N(v)N(x)N(y)^{-1})N(x)+N(c)N(v)N(y) \\nonumber\\\\\n=&N(v)[(N(x)N(y)^{-1})^2-N(c)]. \\label{Div3K}\n\\end{align}\nIf $N(v)=0$, then $v=0$ so by (\\ref{Div1K}) $ux=0$ implies $x=0$ (else $(u,v)=(0,0)$).  Thus (\\ref{Div3K}) implies $N(c)=0\\not\\in F^{\\times},$ which cannot happen as $c\\neq 0$.\nThus we must have $N(v)\\neq 0$ and $(N(x)N(y)^{-1})^2=N(c)$. Thus we conclude $N(c)\\in N(D^{\\times})^2$.\\\\\n(ii) The proof follows analogously as in (i); we require $K$ to be separable over $F$ so that $N_{K\/F}(\\sigma(x))=N_{K\/F}(x)$ for all $\\sigma\\in Aut_F(K)$ and $x\\in K$.\n\\end{proof}\n\n\\begin{remark}\nIf $F=\\mathbb{F}_{p^s}$ and $K=\\mathbb{F}_{p^r}$ is a finite extension of $F$, then $Aut_F(K)$ is cyclic of order $r\/s$ and is generated by $\\phi^s$, where $\\phi$ is defined by the Frobenius automorphism $\\phi(x)=x^p$ for all $x\\in K$. Then $A=\\text{Cay}(K,c,\\sigma_1,\\sigma_2,\\sigma_3,\\sigma_4)$ is a division algebra if and only if $c$ is not a square in $K$. The proof of this is analogous to the one given in \\cite[p. 53]{Knu}.\n\\end{remark}\n\n\n\\subsection{Commutator and nuclei}\n\nUnless otherwise stated, we will write $A_D=\\text{Cay}(D,c,\\sigma_1,\\sigma_2,\\sigma_3,\\sigma_4)$ and $A_K=\\text{Cay}(K,c,\\sigma_1,\\sigma_2,\\sigma_3,\\sigma_4)$.\n\n\\begin{proposition}\\label{Commutator of Cay(K)} If $\\sigma_1=\\sigma_2$ and $\\sigma_3=\\sigma_4$, $Comm(A_D)=F\\oplus F$ and $A_K$ is commutative. Otherwise, $Comm(A_D)=F$ and $Comm(A_K)=\\lbrace (u,0)\\mid \\sigma_3(u)=\\sigma_4(u)\\rbrace\\subseteq K.$\n\\end{proposition}\n\\begin{proof}\nWe compute this only for $A_D$ as the computations for $A_K$ follow analogously. By definition, $(u,v)\\in Comm(A_D)$ if and only if for all $x,y\\in D$, $(u,v)(x,y)=(x,y)(u,v).$ This is equivalent to \\begin{align*}\nux+c\\sigma_1(v)\\sigma_2(y)=&xu+c\\sigma_1(y)\\sigma_2(v),\\\\\n\\sigma_3(u)y+v\\sigma_4(x)=&\\sigma_3(x)v+y\\sigma_4(u),\n\\end{align*}\nfor all $x,y\\in D$. The first equation implies $u\\in F$ and either $\\sigma_1=\\sigma_2$ or $v=0$ . Additionally, the second equation implies $v\\in F$ and $\\sigma_3=\\sigma_4$ or $v=0$. The result follows immediately.\n\\end{proof}\n\n\\begin{proposition}\\label{Cay(B) left nuc}\n\\begin{enumerate}[(i)]\n\\item Suppose that at least one of the following holds:\n\\begin{itemize}\n\\item $\\sigma_2\\circ\\sigma_4\\neq id$,\n\\item $\\sigma_1\\circ\\sigma_4\\neq \\sigma_2\\circ\\sigma_3$,\n\\item $\\sigma_4\\circ\\sigma_1\\neq \\sigma_3\\circ\\sigma_2$.\n\\end{itemize}\nThen $\\text{Nuc}_l(A_D)=\\lbrace x\\in D\\mid \\sigma_1\\circ\\sigma_3(x)=c^{-1}xc\\rbrace\\subset D$ and $\\text{Nuc}_l(A_K)=\\text{Fix}(\\sigma_1\\circ\\sigma_3)\\subset K$.\n\n\\item Suppose that at least one of the following holds:\n\\begin{itemize}\n\\item there exists some $x\\in D$ (resp. $K$) such that $\\sigma_1\\circ\\sigma_3(x)\\neq c^{-1}xc$,\n\\item $\\sigma_2\\circ\\sigma_4\\neq id$,\n\\item for all $v\\in D$, there exists some $x\\in D$ (resp. $K$) such that $\\sigma_3(c)\\sigma_3\\circ\\sigma_1(x)\\sigma_3\\circ\\sigma_2(v)\\neq x\\sigma_4(c)\\sigma_4\\circ\\sigma_1(v)$.\n\\end{itemize}\nThen $\\text{Nuc}_m(A)=\\text{Fix}(\\sigma_3^{-1}\\circ\\sigma_2^{-1}\\circ\\sigma_1\\circ\\sigma_4)$ for both $A=A_D$ and $A=A_K$.\n\n\\item Suppose that at least one of the following holds:\n\\begin{itemize}\n\\item  there exists some $x\\in D$ (resp. $K$) such that $\\sigma_1\\circ\\sigma_3(x)\\neq c^{-1}xc$,\n\\item $\\sigma_1\\circ\\sigma_4\\neq\\sigma_2\\circ\\sigma_3$,\n\\item there exists some $x\\in D$ (resp. $K$) such that $\\sigma_3(c)\\sigma_3\\circ\\sigma_1(x)\\neq x\\sigma_4(c)$.\n\\end{itemize}\nThen $\\text{Nuc}_r(A)=\\text{Fix}(\\sigma_2\\circ\\sigma_4)$ for both $A=A_D$ and $A=A_K$. \n\\end{enumerate}\n\\end{proposition}\n\\begin{proof}\nWe show the proof for (i) since (ii) and (iii) follow analagously. First consider all elements of the form $(k,0)$ for $k\\in D$. Then $(k,0)\\in \\text{Nuc}_l(A_D)$ if and only if we have $((k,0)(u,v))(x,y)=(k,0)((u,v)(x,y))$ for all $u,v,x,y\\in D$. Computing this directly, we obtain the equations \\begin{align*}\nkux+c\\sigma_1(\\sigma_3(k)v)\\sigma_2(y)=&kux+kc\\sigma_1(v)\\sigma_2(y),\\\\\n\\sigma_3(ku)y+\\sigma_3(k)v\\sigma_4(x)=&\\sigma_3(k)\\sigma_3(u)y+\\sigma_3(k)v\\sigma_4(x).\n\\end{align*}\nThese hold for all $u,v,x,y\\in D$ if and only if $c\\sigma_1\\circ\\sigma_3(k)=kc$, i.e. we have $\\sigma_1\\circ\\sigma_3(k)=c^{-1}kc$. The same calculations yield that this holds for all $u,v,x,y\\in D$ if and only if $\\sigma_1\\circ\\sigma_3(k)=k$.\\\\\nTo show that there are no other elements in the left nucleus, it suffices to check that there are no elements of the form $(0,m)$, $m\\in D$, in $\\text{Nuc}_l(A_D)$. This is because the associator is linear in the first component:\n$[(k,m),(u,v),(x,y)]=[(k,0),(u,v),(x,y)]+[(0,m),(u,v),(x,y)].$ If $(0,m)\\in \\text{Nuc}_l(A_D)$, then for all $u,v,x,y\\in D$ we have $((0,m)(u,v))(x,y)=(0,m)((u,v)(x,y)).$ This holds for all $u,v,x,y\\in D$ if and only if $$c\\sigma_1(m)[\\sigma_2(v)x+\\sigma_1(\\sigma_4(u))\\sigma_2(y)]=c\\sigma_1(m)[\\sigma_2(v)\\sigma_2(\\sigma_4(x))+\\sigma_2(\\sigma_3(u))\\sigma_2(y)],$$ $$m\\sigma_3(c\\sigma_2(v))y=m\\sigma_4(c\\sigma_1(v)\\sigma_2(y)).$$ In order for this to be satisfied for all $u,v,x,y\\in D$, we have either $m=0$ or all the following must hold:\\begin{itemize}\n\\item $\\sigma_2\\circ\\sigma_4=id$,\n\\item $\\sigma_1\\circ\\sigma_4=\\sigma_2\\circ\\sigma_3$,\n\\item $\\sigma_4\\circ\\sigma_1=\\sigma_3\\circ\\sigma_2$.\n\\end{itemize}\nIf $m\\neq 0$, this contradicts the assumptions we made, so this yields $m=0$. The same argument also gives $m=0$ in the field case.\n\\end{proof}\n\n\n\\begin{corollary} $A_K$ is associative if and only if $A_K=\\text{Cay}(K,c,\\sigma,\\tau,\\sigma^{-1},\\tau^{-1})$ for some $\\tau,\\sigma\\in Aut_F(K)$ such that $(\\sigma\\circ\\tau)^2=id$ and $c\\in \\text{Fix}(\\sigma\\circ\\tau)$.\n\\end{corollary}\n\n\nAs the center of $A$ is defined as $Z(A)=Comm(A)\\cap \\text{Nuc}_l(A)\\cap \\text{Nuc}_m(A)\\cap \\text{Nuc}_r(A),$ we see that $Z(A_K)\\subset K$ unless $\\sigma_1=\\sigma_2=\\sigma$ and $\\sigma_3=\\sigma_4=\\sigma^{-1}$. If $A_K=\\text{Cay}(K,c,\\sigma,\\sigma,\\sigma^{-1},\\sigma^{-1})$ for some $\\sigma\\in Aut_F(K)$, then $A_K$ is a commutative, associative algebra.\n\n\n\\subsection{Isomorphisms}\n\n\\begin{theorem}\\label{ConstructingIsomorphismsDtoD'}\nLet $D$ and $D'$ be two central simple $F$-algebras (respectively, $K$ and $L$ finite field extensions of $F$) and $g,h:D\\to D'$ be two $F$-algebra isomorphisms. Let $A_D=\\text{Cay}(D,c,\\sigma_1,\\sigma_2,\\sigma_3,\\sigma_4)$ and $B_{D'}=\\text{Cay}(D',g(c)b^2,\\phi_1,\\phi_2,\\phi_3,\\phi_4)$ for some $b\\in F^{\\times}$ (resp. $A_K=\\text{Cay}(K,c,\\sigma_1,\\sigma_2,\\sigma_3,\\sigma_4)$ and $B_L=\\text{Cay}(L,g(c)\\phi_1(b)\\phi_2(b),\\phi_1,\\phi_2,\\phi_3,\\phi_4)$ for some $b\\in K^{\\times}$). If \n\\begin{align}\n\\phi_i&=g\\circ\\sigma_i\\circ h^{-1} \\mbox{ for } i=1,2, \\label{Equation 1}\\\\\n\\phi_i&=h\\circ\\sigma_i\\circ g^{-1} \\mbox{ for } i=3,4, \\label{Equation 2}\n\\end{align}\nthen the map $G:A\\to B,$  $G(u,v)=(g(u),h(v)b^{-1})$ defines an $F$-algebra isomorphism.\n\\end{theorem}\n\\begin{proof}\nWe show the proof in the central simple algebra case. It follows analogously when we take field extensions $K$ and $L$. Clearly $G$ is $F$-linear, additive and bijective. It only remains to show that $G$ is multiplicative; that is, $G((u,v)(x,y))=G(u,v)G(x,y)$ for all $u,v,x,y\\in D$. First we have \\begin{align*}\nG(u,v)G(x,y)=&(g(u),h(v)b^{-1})(g(x),h(y)b^{-1})\\\\\n=&(g(u)g(x)+g(c)b^2\\phi_1(h(v)b^{-1})\\phi_2(h(y)b^{-1}),\\phi_3(g(u))h(y)b^{-1}+h(v)b^{-1}\\phi_4(g(x)))\\\\\n=&(g(ux)+g(c)\\phi_1(h(v))\\phi_2(h(y)),[\\phi_3(g(u))h(y)+h(v)\\phi_4(g(x))]b^{-1}).\n\\end{align*}\nIt similarly follows that \\begin{align*}\nG((u,v)(x,y))=&G(ux+c\\sigma_1(v)\\sigma_2(y),\\sigma_3(u)y+v\\sigma_4(x))\\\\\n=&(g(ux+c\\sigma_1(v)\\sigma_2(y)),h(\\sigma_3(u)y+v\\sigma_4(x))b^{-1})\\\\\n=&(g(ux)+g(c)g(\\sigma_1(v))g(\\sigma_2(y)),[h(\\sigma_3(u))h(y)+h(v)h(\\sigma_4(x))]b^{-1}).\n\\end{align*}\nBy (\\ref{Equation 1}) and (\\ref{Equation 2}), we obtain equality and thus $G$ is an $F$-algebra isomorphism.\n\\end{proof}\n\n\n\n\\begin{corollary}\\label{ConstructingIsomorphismsKtoK}\nLet $g,h\\in Aut_F(D)$ (resp. $Aut_F(K)$) and $b\\in F^{\\times}$ (resp. $b\\in K^{\\times})$. Let $B_D=\\text{Cay}(D,g(c)b^2,\\phi_1,\\phi_2,\\phi_3,\\phi_4)$ (resp. $B_K=\\text{Cay}(K,g(c)\\phi_1(b)\\phi_2(b),\\phi_1,\\phi_2,\\phi_3,\\phi_4)$ for some $b\\in K^{\\times}$). If \n\\begin{align*}\n\\phi_i&=g\\circ\\sigma_i\\circ h^{-1} \\mbox{ for } i=1,2,\\\\\n\\phi_i&=h\\circ\\sigma_i\\circ g^{-1} \\mbox{ for } i=3,4,\n\\end{align*}\nthen the map $G:A\\to B,\\quad G(u,v)=(g(u),h(v)b^{-1})$ defines an $F$-algebra isomorphism.\n\\end{corollary}\n\n\\begin{corollary} Every generalised Dickson algebra $A_D=\\text{Cay}(D,c,\\sigma_1,\\sigma_2,\\sigma_3,\\sigma_4)$ is isomorphic to an algebra of the form $\\text{Cay}(D,c,\\sigma_1',\\sigma_2',\\sigma_3', id)$ (analogously for the algebras $A_K$).\n\\end{corollary}\n\\begin{proof}\nConsider the map $G:D\\oplus D\\to D\\oplus D$ defined by $G(u,v)=(u,\\sigma_4^{-1}(v))$. By Theorem \\ref{ConstructingIsomorphismsDtoD'}, this yields the isomorphism $\\text{Cay}(D,c,\\sigma_1,\\sigma_2,\\sigma_3,\\sigma_4)\\cong \\text{Cay}(D,c,\\sigma_1\\circ\\sigma_4,\\sigma_2\\circ\\sigma_4,\\sigma_4^{-1}\\circ\\sigma_3,id).$\n\\end{proof}\n\n\\begin{remark} If $Comm(A_D)\\neq F$ or $Comm(A_K)\\not\\subset K$, then $\\sigma_1=\\sigma_2$ and $\\sigma_3=\\sigma_4$ by Lemma \\ref{Commutator of Cay(K)}. Via the map $G(u,v)=(u,\\sigma_3^{-1}(v))$, Corollary \\ref{ConstructingIsomorphismsKtoK} yields that every such algebra is isomorphic to the generalisation of commutative Dickson algebras as defined in \\cite{Thompson19}.\n\\end{remark}\n\nIn certain cases, the maps defined in Theorem \\ref{ConstructingIsomorphismsDtoD'} and Corollary \\ref{ConstructingIsomorphismsKtoK} are the only possible isomorphisms between two algebras constructed via our generalised Cayley-Dickson doubling:\n\n\\begin{theorem}\\label{IsomorphismRestrictToKtoL}\nLet $A_K=\\text{Cay}(K,c,\\sigma_1,\\sigma_2,\\sigma_3,\\sigma_4)$ and $B_L=\\text{Cay}(L,c',\\phi_1,\\phi_2,\\phi_3,\\phi_4)$. Suppose that $G:A_K\\to B_L$ is an isomorphism that restricts to an isomorphism $g:K\\to L$. Then $G$ is of the form $G(x,y)=(g(x),h(y)b)$ for some isomorphism $h:K\\to L$ such that $\\phi_i\\circ h=g\\circ\\sigma_i$ for $i=1,2$ and $\\phi_i\\circ g=h\\circ\\sigma_i$ for $i=3,4$ and some $b\\in L^{\\times}$ such that $g(c)=c'\\phi_1(b)\\phi_2(b).$\n\\end{theorem}\n\\begin{proof}\nSuppose $G$ is an isomorphism from $A_K$ to $B_L$ such that $G\\!\\mid_K=g:K\\to L$ is an isomorphism. Then for all $x\\in K$, we have $G(x,0)=(g(x),0).$ Let $G(0,1)=(a,b)$ for some $a,b\\in L$. As $G$ is multiplicative, this yields\\begin{align*}\nG(x,y)=&G(x,0)+G(\\sigma_3^{-1}(y),0)G(0,1)\\\\\n=&(g(x),0)+(g(\\sigma_3^{-1}(y)),0)(a,b)\\\\\n=&(g(x)+g(\\sigma_3^{-1}(y))a,\\phi_3(g(\\sigma_3^{-1}(y)))b),\n\\end{align*}\nand \n\\begin{align*}\nG(x,y)=&G(x,0)+G(0,1)G(\\sigma_4^{-1}(y),0)\\\\\n=&(g(x),0)+(a,b)(g(\\sigma_4^{-1}(y)),0)\\\\\n=&(g(x)+g(\\sigma_4^{-1}(y))a,b\\phi_4(g(\\sigma_4^{-1}(y)))).\n\\end{align*}\nIt follows that either $\\phi_3\\circ g\\circ\\sigma_3^{-1}=\\phi_4\\circ g\\circ\\sigma_4^{-1}$ or $b=0$. However, if $b=0$ this would imply that $G$ was not surjective, which is a contradiction to the assumption that $G$ is an isomorphism. Thus it follows that $\\phi_3\\circ g\\circ\\sigma_3^{-1}=\\phi_4\\circ g\\circ\\sigma_4^{-1}.$ Additionally, we have either $g\\circ\\sigma_3^{-1}=g\\circ\\sigma_4^{-1}$ or $a=0$.\\\\\nConsider $G((0,1)^2)=G(0,1)^2$. This gives $(a^2+c'\\phi_1(b)\\phi_2(b),\\phi_3(a)b+b\\phi_4(a))=(g(c),0).$ As we have established that $b\\neq 0$, this implies that $\\phi_3(a)=-\\phi_4(a).$ If $a\\neq 0$, we obtain $g\\circ\\sigma_3^{-1}=g\\circ\\sigma_4^{-1}$. Substituting this into the condition $\\phi_3\\circ g\\circ\\sigma_3^{-1}=\\phi_4\\circ g\\circ\\sigma_4^{-1}$, we conclude that $\\phi_3=\\phi_4$. This contradicts $\\phi_3(a)=-\\phi_4(a)$. Thus we must in fact have $a=0$ and $G(x,y)=(g(x),h(y)b)$ where $h=\\phi_3\\circ g\\circ\\sigma_3^{-1}$ and $g(c)=c'\\phi_1(b)\\phi_2(b)$. Computing $G(u,v)G(x,y)=G((u,v)(x,y))$ gives the remaining conditions.\n\\end{proof}\n\nThis proof does not hold when we consider the algebras $A_D$, as we rely heavily on the commutativity of $K$.\n\n\\begin{corollary}\\label{IsomorphismRestrictToK}\n Suppose that $G:A_K\\to B_K$ is an isomorphism that restricts to an automorphism $g$ of $K$. Then $G$ is of the form $G(x,y)=(g(x),h(y)b)$ for $g,h\\in Aut_F(K)$ such that $\\phi_i\\circ h=g\\circ\\sigma_i$ for $i=1,2$ and $\\phi_i\\circ g=h\\circ\\sigma_i$ for $i=3,4$ and some $b\\in K^{\\times}$ such that $g(c)=c'\\phi_1(b)\\phi_2(b).$\n\\end{corollary}\n\nIf $\\text{Nuc}_l(A)=\\text{Nuc}_l(B)=K$, all isomorphisms from $A\\to B$ must restrict to an automorphism of $K$; similar considerations are true for restrictions to the middle and right nuclei. It follows that we can determine precisely when two such algebras are isomorphic by Corollary \\ref{IsomorphismRestrictToK}.\n\n\\begin{corollary}\\label{IsomorphismRestrictToKCommutes}\nSuppose that $G:A_K\\to B_K$ is an isomorphism that restricts to an automorphism $g$ of $K$. If $\\sigma_i=\\phi_i=id$ for any $i=1,2,3,4$, $G$ must be of the form $$G(x,y)=(g(x),g(y)b)$$ for $g\\in Aut_F(K)$ such that $\\phi_i\\circ g=g\\circ\\sigma_i$ for $i=1,2$ and $\\sigma_i\\circ g=g\\circ\\phi_i$ for $i=3,4$ and some $b\\in K^{\\times}$ such that $g(c)=c'\\phi_1(b)\\phi_2(b).$\n\\end{corollary}\n\\begin{proof}\nFrom Theorem \\ref{IsomorphismRestrictToK}, we see that $\\phi_i\\circ h=g\\circ\\sigma_i.$ If $\\phi_i=\\sigma_i=id$ for some $i=1,2,3,4$, we conclude that $g=h$ and the result follows.\n\\end{proof}\n\n\\begin{corollary}\\label{IsomorphismRestrictToKN}\nSuppose that $G:A_K\\to B_K$ is an isomorphism that restricts to an automorphism of $K$. If $K$ is a separable extension of $F$, we must have $N_{K\/F}(cc'^{-1})=N_{K\/F}(b^2)$ for some $b\\in K^{\\times}$.\n\\end{corollary}\n\\begin{proof}\nSuppose $G:A_K\\to B_K$ is an isomorphism that restricts to an automorphism of $K$. By Theorem \\ref{IsomorphismRestrictToK}, we have $g(c)=c'\\phi_1(b)\\phi_2(b).$ Applying norms to both side, we obtain $$N_{K\/F}(g(c))=N_{K\/F}(c'\\phi_1(b)\\phi_2(b)).$$ As $K$ is a separable extension of $F$, it follows that $N_{K\/F}(g(x))=N_{K\/F}(x)$ for all $x\\in K$, $g\\in Aut_F(K)$. This yields $N_{K\/F}(c)=N_{K\/F}(c'b^2).$ As $c'\\in K^{\\times}$ and $N_{K\/F}$ is multiplicative, we conclude that $N_{K\/F}(cc'^{-1})=N_{K\/F}(b^2)$.\n\\end{proof}\n\n\\begin{example} Let $F=\\mathbb{Q}_p$ ($p\\neq 2$) and $K$ be a separable extension of $\\mathbb{Q}_p$. It is well known that $(\\mathbb{Q}_p^{\\times})^2\/\\mathbb{Q}_p=\\lbrace [1],[u],[p],[up]\\rbrace$ for some $u\\in\\mathbb{Z}_p\\setminus \\mathbb{Z}_p^2$. If $N_{K\/F}(c)$ and $N_{K\/F}(c')$ do not lie in the same coset of $(\\mathbb{Q}_p^{\\times})^2\/\\mathbb{Q}_p$, there does not exist an isomorphism that restricts to $K$ such that $\\text{Cay}(K,c,\\sigma_1,\\sigma_2,\\sigma_3,\\sigma_4)\\cong \\text{Cay}(K,c',\\phi_1,\\phi_2,\\phi_3,\\phi_4)$ by Corollary \\ref{IsomorphismRestrictToKN}.\n\\end{example}\n\n\n\\subsection{Automorphisms}\n\n\\begin{theorem}\\label{CSA Construction Automorphisms}\nLet $g,h\\in Aut_F(D)$ (resp. $Aut_F(K)$) such that $g\\circ f=f\\circ h$ for $f=\\sigma_1,\\sigma_2,\\sigma_3^{-1},\\sigma_4^{-1}$ and let $b\\in F^{\\times}$ (resp. $b\\in K^{\\times}$) such that $g(c)=b^2c$ (resp. $g(c)=\\sigma_1(b)\\sigma_2(b)c$). Then the map $G:A\\to A$ defined by $G(u,v)=(g(u), h(v)b)$ is an automorphism of $A_D$ (resp. $A_K$).\n\\end{theorem}\nThis is easily checked via some long calculations.\n\n\\begin{theorem}\\label{GeneralAutomorphisms}\nSuppose that at least one of $\\text{Nuc}_l(A_K)$, $\\text{Nuc}_m(A_K)$, $\\text{Nuc}_r(A_K)$ is equal to $K$. Then $G:A_K\\to A_K$ is an automorphism of $A_K$ if and only if $G$ has the form stated in Theorem \\ref{CSA Construction Automorphisms}.\n\\end{theorem}\n \\begin{proof}\n Let $A=A_K$. Suppose $G\\in Aut_F(A)$ and $\\text{Nuc}_l(A)=K$. As automorphisms preserve the nuclei of an algebra, $G$ restricted to $\\text{Nuc}_l(A)$ must be an automorphism of $K$; that is, $G\\mid_K= g\\in Aut_F(K)$ and so we have $G(x,0)=( g(x),0)$ for all $x\\in K$.\\\\\nIf $\\text{Nuc}_l(A)\\neq K$, by our assumptions one of $\\text{Nuc}_m(A)$ or $\\text{Nuc}_r(A)$ are equal to $K$. In either case, we can use an identical argument by restricting $G$ to $\\text{Nuc}_m(A)$ or $\\text{Nuc}_r(A)$ respectively. As automorphisms preserve the nuclei of an algebra, $G$ restricted to $\\text{Nuc}_m(A)$ (respectively $\\text{Nuc}_r(A)$) must be an automorphism of $K$. Let $G(0,1)=(a,b)$ for some $a,b\\in K$. Then \\begin{align*}\n G(x,y)=&G(x,0)+G(\\sigma_3^{-1}(y),0)G(0,1)\\\\\n=&( g(x)+ g\\sigma_3^{-1}(y)a,\\sigma_3 g\\sigma_3^{-1}(y)b),\n \\end{align*}\nand also \\begin{align*}\n G(x,y)=&G(x,0)+G(0,1)G(\\sigma_4^{-1}(y),0)\\\\\n=&( g(x)+ g\\sigma_4^{-1}(y)a,\\sigma_4 g\\sigma_4^{-1}(y)b)\n\\end{align*}\nfor all $x,y\\in K$. Hence we must have $ g\\sigma_3^{-1}(y)a= g\\sigma_4^{-1}(y)a$ for all $y\\in K$, which implies either $\\sigma_3=\\sigma_4$ or $a=0$. Additionally we have $\\sigma_3 g\\sigma_3^{-1}(y)b=\\sigma_4 g\\sigma_4^{-1}(y)b$. If $b=0$, this would imply $G(x,y)=( g(x)+ g\\sigma_4^{-1}(y)a,0)$, which is a contradiction as it implies $G$ is not surjective. Thus we must in fact have $\\sigma_3 g\\sigma_3^{-1}(y)=\\sigma_4 g\\sigma_4^{-1}(y)$ for all $y\\in K$.\\\\\nNow we consider $G((0,1)^2)=G(0,1)^2$. This gives $(a,b)(a,b)=( g(c),0),$ which implies \\begin{align*}\na^2+c\\sigma_1(b)\\sigma_2(b)=& g(c),\\\\\n\\sigma_3(a)b+b\\sigma_4(a)=& 0.\n\\end{align*}\nIf $\\sigma_3\\neq\\sigma_4$, we already know that $a=0$. On the other hand if $\\sigma_3=\\sigma_4$, we obtain $2\\sigma_3(a)b=0$. As $K$ has characteristic not 2 and $b\\neq 0$, this implies $a=0$. In either case, we obtain $c\\sigma_1(b)\\sigma_2(b)= g(c)$ and $G(u,v)=( g(u), h(v)b),$ where $ h=\\sigma_3\\circ g\\circ\\sigma_3^{-1}=\\sigma_4\\circ g\\circ\\sigma_4^{-1}.$ We note that this definition of $ h$ implies that $ h\\circ\\sigma_3=\\sigma_3\\circ g$ and $ h\\circ\\sigma_4=\\sigma_4\\circ g$.\\\\\nFinally we consider $G(u,v)G(x,y)=G((u,v)(x,y))$ for all $u,v,x,y\\in K$. We obtain $( g(u), h(v)b)( g(x), h(y)b)=( g(uv+c\\sigma_1(v)\\sigma_2(y)), h(\\sigma_3(u)y+v\\sigma_4(x))b)$ which gives the equations \\begin{align*}\nc\\sigma_1( h(v)b)\\sigma_2( h(y)b)=& g(c) g(\\sigma_1(v)\\sigma_2(y)),\\\\\n\\sigma_3( g(u)) h(y)b+ h(y)\\sigma_4( g(x))b=& h(\\sigma_3(u)y+v\\sigma_4(x))b.\n\\end{align*}\n As $ h\\circ\\sigma_3=\\sigma_3\\circ g$ and $ h\\circ\\sigma_4=\\sigma_4\\circ g$, the second equation holds for all $u,v,x,y\\in K$. Substituting $ g(c)=c\\sigma_1(b)\\sigma_2(b)$ into the first equation, we obtain $\\sigma_1(h(v))\\sigma_2(h(y))= g(\\sigma_1(v)) g(\\sigma_2(y))$ for all $v,y\\in K$. This implies $\\sigma_1\\circ h= g\\circ\\sigma_1$ and $\\sigma_2\\circ h= g\\circ\\sigma_2$. Hence if $G$ is an automorphism of $A$ we must have $G(u,v)=( g(u), h(v)b)$ for some $ g, h\\in Aut_F(K)$, such that $ g\\circ f=f\\circ h$ for $f=\\sigma_1,\\sigma_2,\\sigma_3^{-1},\\sigma_4^{-1}$ and some $b\\in K^{\\times}$, such that $ g(c)=\\sigma_1(b)\\sigma_2(b)c.$\n\\end{proof}\n\n \n\\begin{corollary}\\label{KDoublingAutGroup}\nSuppose that at least one of $\\text{Nuc}_l(A_K)$, $\\text{Nuc}_m(A_K)$, $\\text{Nuc}_r(A_K)$ is equal to $K$ and $Aut_F(K)=\\langle\\sigma\\rangle$. Then $G:A_K\\to A_K$ is an automorphism of $A_K$ if and only if $G(u,v)=(\\sigma^i(u),\\sigma^i(v)b)$ for some $i\\in \\mathbb{Z}$ and $b\\in K^{\\times}$ satisfying $\\sigma^i(c)=c\\sigma^{\\alpha_2}(b)\\sigma^{\\beta_2}(b).$ \n\\end{corollary}\n\n\nIn the case when doubling a central simple algebra, we obtain a partial generalisation of Theorem \\ref{GeneralAutomorphisms}:\n\n\\begin{lemma}\\label{CSAAutomorphismsScalars} Let $G\\in Aut(A_D)$ be such that $G\\!\\mid_D=g\\in Aut_F(D)$. Then there must exist some $a,b\\in D$, $b\\neq 0$, such that for all $y\\in D$, $$a g\\circ\\sigma_4^{-1}(y)=g\\circ\\sigma_3^{-1}(y)a,$$ $$b\\sigma_4\\circ g\\circ\\sigma_4^{-1}(y)=\\sigma_3\\circ g\\circ\\sigma_3^{-1}(y)b.$$\n\\end{lemma}\n\\begin{proof}\nSuppose $G\\mid_D=g\\in Aut_F(D)$. Then for all $x\\in D$, we obtain $G(x,0)=(g(x),0)$. Let $G(0,1)=(a,b)$ for some $a,b\\in D$. It now follows that \\begin{align*}\n G(x,y)=&G(x,0)+G(\\sigma_3^{-1}(y),0)G(0,1)\\\\\n=&( g(x)+ g\\circ\\sigma_3^{-1}(y)a,\\sigma_3\\circ g\\circ\\sigma_3^{-1}(y)b),\n \\end{align*}\nand also \\begin{align*}\n G(x,y)=&G(x,0)+G(0,1)G(\\sigma_4^{-1}(y),0)\\\\\n\n=&( g(x)+ ag\\circ\\sigma_4^{-1}(y),b\\sigma_4\\circ g\\circ\\sigma_4^{-1}(y)).\n\\end{align*}\nSetting these two equivalent expressions for $G(x,y)$ equal to each other yields the result. Note that if $b=0$, $G$ would no longer be surjective, which would contradict our assumption that $G\\in Aut(A_d)$. \n\\end{proof}\n\n\\begin{theorem}\\label{CSAAutomorphisms3=4} Let $G\\in Aut(A_D)$ be such that $G\\!\\mid_D=g\\in Aut_F(D)$. If $\\sigma_3=\\sigma_4$, then $G:A_D\\to A_D$ must have the form as stated in Theorem \\ref{CSA Construction Automorphisms}.\n\\end{theorem}\n\\begin{proof}\nSuppose $G\\!\\mid_D=g\\in Aut_F(D)$. Substituting $\\sigma_3=\\sigma_4$ into Lemma \\ref{CSAAutomorphismsScalars}, we see that $G(0,1)=(a,b)$ for some $a,b\\in D$ such that $$a g\\circ\\sigma_3^{-1}(y)=g\\circ\\sigma_3^{-1}(y)a,$$ $$b\\sigma_3\\circ g\\circ\\sigma_3^{-1}(y)=\\sigma_3\\circ g\\circ\\sigma_3^{-1}(y)b.$$ This is satisfied for all $y\\in D$ if and only if $a,b\\in F$ and so $G(x,y)=(g(x)+g\\circ\\sigma_3^{-1}(y)a,\\sigma_3\\circ g\\circ\\sigma_3^{-1}(y)b).$ The remainder of this proof follows almost exactly the same to Theorem \\ref{GeneralAutomorphisms}:\n\nNow we consider $G((0,1)^2)=G(0,1)^2$. This gives $(a,b)(a,b)=( g(c),0),$ which implies \\begin{align*}\na^2+c\\sigma_1(b)\\sigma_2(b)=& g(c)\\\\\n\\sigma_3(a)b+b\\sigma_4(a)=0.\n\\end{align*}\nAs $a,b\\in F$, the second equation is equivalent to $2ab=0$. As $F$ has characteristic not 2, this implies $a=0$ or $b=0$. If $b=0$, $G$ would not be surjective, which contradicts our assumption that $G$ is an isomorphism. Thus we must have $a=0$ and so we obtain $g(c)=cb^2$ and $G(u,v)=( g(u), h(v)b),$ where $ h=\\sigma_3\\circ g\\circ\\sigma_3^{-1}.$ We note that this definition of $ h$ implies that $ h\\circ\\sigma_3=\\sigma_3\\circ g$.\\\\\nFinally we consider $G(u,v)G(x,y)=G((u,v)(x,y))$ for all $u,v,x,y\\in D$. We obtain $( g(u), h(v)b)( g(x), h(y)b)=( g(uv+c\\sigma_1(v)\\sigma_2(y)), h(\\sigma_3(u)y+v\\sigma_4(x))b),$ which gives the equations \\begin{align*}\nc\\sigma_1( h(v)b)\\sigma_2( h(y)b)=& g(c) g(\\sigma_1(v)\\sigma_2(y)),\\\\\n\\sigma_3( g(u)) h(y)b+ h(y)\\sigma_4( g(x))b=& h(\\sigma_3(u)y+v\\sigma_4(x))b.\n\\end{align*} \nAfter substituting $cb^2=g(c)$, we conclude that this is satisfied for all $x,y,u,v\\in D$ if and only if we have $\\sigma_1\\circ h=g\\circ \\sigma_1$ and $\\sigma_2\\circ h=g\\circ \\sigma_2$.\n\\end{proof}\n\n\n\n\n\n\n\n\n\\printbibliography\n\n\\end{document}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{introduction}\\label{sec1}\n\n\nIn standard cosmology, the Copernican principle is applied i.e., it is assumed\nthat we are not living at a privileged position in the universe. \nCombining the Copernican principle with the isotropy of \nthe Cosmic Microwave Background (CMB) radiation, \nleads to the conclusion that our universe is well described by the\nhomogeneous and isotropic universe model. \nIn the framework  of the homogeneous and isotropic universe, \nthe observational data of the luminosity distances of Type Ia supernovae (SNIa) \nindicates an acceleration of the cosmic volume expansion of our universe.  \nThe acceleration of the cosmic volume expansion in the homogeneous and \nisotropic universe implies the existence of \nso-called dark energy that acts as a source of a repulsive gravitational force\nif we assume general relativity at cosmological scales.\nAt present, there is no theory that can naturally explain the origin of dark energy, \nand it seems worth investigating alternative scenarios. \nIn order to do this, we have to discard general relativity or the homogeneity \nassumption. \n\nInhomogeneous cosmological models without dark energy\nhave been proposed independently \nby Tomita~\\cite{Tomita:1999qn,Tomita:2000jj,Tomita:2001gh} \nand C\\'el\\'erier~\\cite{Celerier:1999hp}. \nIn C\\'el\\'erier's model, the observer is located at the symmetry center \nof a very large spherical void which can explain the SNIa observations. \nSince the observer is located at\na special position in the universe, we call this model \na ``the non-Copernican universe model\"  in this paper. \nThe common assumption of non-Copernican universe models \nis that an observer is located \nin the vicinity of the symmetry center,\nwhich explains the fact that the CMB radiation is observed to be isotropic. \nThe most common way to describe  \nnon-Copernican universe models \nis to use the Lema\\^{\\i}tre-Tolman-Bondi(LTB) solution for the Einstein equations, \nwhich describes the motion of spherically symmetric dust.  \n\nSNIa observations in non-Copernican universe models have been\nstudied by many researchers~\\cite{Tomita:1999qn,Tomita:2000jj,Tomita:2001gh,\nCelerier:1999hp,Goodwin:1999ej,Clifton:2008hv,\nIguchi:2001sq,Vanderveld:2006rb,Yoo:2008su,Celerier:2009sv,Kolb:2009hn,Yoo:2010qn}, \nand it has been proven that the distance-redshift relation \nin the $\\Lambda$CDM universe can be reproduced using LTB universe \nmodels~\\cite{Mustapha:1998jb,Yoo:2008su,Celerier:2009sv,Kolb:2009hn}. \nNon-Copernican universe models have been tested by other observations including \nthe CMB acoustic \npeaks~\\cite{Alnes:2005rw,Alexander:2007xx,Zibin:2008vk,\nGarciaBellido:2008nz,Yoo:2010qy,Clarkson:2010ej,\nMarra:2010pg,Moss:2010jx,Biswas:2010xm,Marra:2011ct,Nadathur:2010zm}, \nthe radial baryon acoustic oscillation scales~\\cite{Zibin:2008vk,GarciaBellido:2008yq,Zumalacarregui:2012pq},\nthe kinematic Suniyaev-Zeldovich \neffect~\\cite{Bull:2011wi,GarciaBellido:2008gd,Yoo:2010ad,Zhang:2010fa,Moss:2011ze} \nand others~\\cite{Alnes:2006pf,Alnes:2006uk,Bolejko:2005fp,Dunsby:2010ts,\nEnqvist:2009hn,Enqvist:2006cg,Goto:2011ru,Kodama:2010gr,\nQuartin:2009xr,Regis:2010iq,Romano:2009mr,Romano:2010nc,Romano:2011mx,Tanimoto:2009mz,\nUzan:2008qp,Yoo:2010hi,Zibin:2011ma}.\nAlthough these observations have imposed restrictions on these models, \nthey have not yet ruled out the models. It is not easy to confirm whether our universe \nfollows the Copernican principle. \n\n\nIn this paper, we focus on the evolution of structures such as clusters of galaxies \nand super-clusters in the non-Copernican universe in the matter \ndominant era which is well described by the LTB solution.  \nIt is expected that observations of the large-scale structures and \ntheir evolution can be used to test the non-Copernican universe model, \nsince the evolution of the anisotropic perturbations reflects the tidal force\nin the background spacetime. \nHowever, the evolution of perturbations in  \nthe LTB solution has not yet been fully studied. \nThis is because the isometries in the LTB spacetime are less than \nin the homogeneous and isotropic universe. \nAlthough master equations for perturbations for \ngeneral spherically symmetric spacetimes \nhave been derived a long time ago~\\cite{Gerlach:1980}\n(see also Ref.~\\cite{Clarkson:2009sc} for the LTB background), \nthese equations for the LTB solution cannot be reduced \nto ordinary differential equations, in general.\nThis is a very different situation from the case of the homogeneous \nand isotropic universe. \n\nRecently, Alonso et al.~\\cite{Alonso:2010zv} performed \nnumerical simulations for non-Copernican models including only cold dark matter. \nThey studied the perturbed Einstein-deSitter universe with \ntwo kinds of perturbations: one forms a spherical void, \nand the other is a non-spherical perturbation with a power spectrum with a random phase \nGaussian probability distribution. \nThey followed the growth of these perturbations using Newtonian $N$-body simulations.\nHowever, in order to confirm the validity of the tnumerical simulations, analytic complementary\nstudies are necessary. \nSome authors~\\cite{Moss:2010jx,Zibin:2008vj,Dunsby:2010ts} have studied perturbations\nby using a ``silent approximation'' that neglects the magnetic part of the Weyl tensor.\nHere, we should note that the magnetic part of the Weyl tensor usually plays an important role \neven in Newtonian situations~\\cite{Bertschinger:1994nc}. \nHence, in this paper, we propose another complementary analytic approach.  \n\nIn many non-Copernican models, \nthe void structure becomes nonlinear at the present time. \nHowever, Enqvist et al.~\\cite{Enqvist:2009hn} pointed out that \nthe void inhomogeneity remains in a quasi-linear regime \n$\\sim \\mathcal{O}(0.1)$ inside a past light-cone of an observer at the center of the void. \nActually, they considered a linear perturbation in \nthe Einstein-deSitter universe that is consistent with the SNIa data, \nand showed that the fraction of the spherically symmetric \nlinear perturbation does not exceed 30\\% inside the past light-cone. \nThis result implies that non-Copernican LTB cosmological models compatible with \nthe observed distance-redshift relation \nmay be studied by perturbation theory for the homogeneous and \nisotropic universe filled with dust at least for the inside of \nthe past light-cone of the central observer. \n\nIn this paper, we investigate the growth of perturbations in the \nnon-Copernican universe models by applying the above idea. \nIt is rather difficult to analyze the evolution of anisotropic perturbations \nin the non-Copernican LTB universe model, while it is much easier to study the evolution of \nnon-linear perturbations in the homogeneous and isotropic universe model \nby successive approximation. \nWe adopt the latter approach. We introduce two-parameter perturbations with \nsmall expansion parameters $\\kappa$ and $\\epsilon$\nin a homogeneous and isotropic dust universe. \nThe limit $\\epsilon\\rightarrow0$ leads to the exact \nLTB solution, if we take all orders of $\\kappa$ into account. By contrast, the limit \n$\\kappa\\rightarrow0$ with $0<\\epsilon\\ll1$ leads to the homogeneous and \nisotropic universe with small anisotropic perturbations. \nThen, in order to see the effect of the void structure \non the evolution of the anisotropic perturbations, we study the non-linear \neffects up to the order of $\\kappa\\epsilon$, following Ref.~\\cite{Tomita:1967}. \n\nThis paper is organized as follows. \nIn \\S~\\ref{sec2}, we derive the equations for \nperturbations parametrized by $\\kappa$ and $\\epsilon$ \nin the homogeneous and isotropic dust universe and obtain general solutions \nup to order $\\kappa\\epsilon$. \nIn \\S~\\ref{sec3}, by fixing the initial conditions, \nwe calculate the angular power spectrum of the density perturbations. \nIn \\S~\\ref{sec4}, we analyze the growth of the perturbations \nby using the angular growth rate. \n\\S~\\ref{sec5} is devoted to a summary and discussion.\n\n\nIn this paper, we use the geometrized units in which \nthe speed of light and Newton's gravitational constant are one, respectively. \nThe Latin indices denote the spatial components, whereas the Greek indices represent\nthe spacetime components. \n \n\\section{two-parameter perturbations in a homogeneous and isotropic dust universe}\n\\label{sec2}\n\\subsection{Perturbations with two kinds of parameters}\nAs mentioned, we study the perturbations in the homogeneous and isotropic universe which is \noften called the Friedmann-Lema\\^{\\i}tre-Robertson-Walker (FLRW) universe. \nSince the structure formation begins after the universe has begun to be dominated by non-relativistic matter, \nit is sufficient for our purpose to consider the universe model filled with dust. \nUsing the spherical polar coordinates for 3-dimensional space, the line-element is given by\n\\begin{eqnarray}\n d\\bar{s}^2 &=& -dt^2+a^2(t)\\left(\\frac{dr^2}{1-Kr^2}+r^2d\\Omega^2\\right) =:-dt^2+a^2(t)\\gamma_{ij}dx^idx^j,\n \\end{eqnarray}\nwhere $a(t)$ is the scale factor which will be determined by the Einstein equations, \n$K$ is constant, $d\\Omega^2$ is the 2-dimensional round metric,  \nand, for later convenience, we have defined the background conformal 3-metric $\\gamma_{ij}$. \nThe constant $K$ has the same sign as that of the curvature of the 3-dimensional \nspace specified by $t=$constant.\nThe stress-energy tensor of dust is given by\n\\begin{equation}\n \\bar T^{\\mu \\nu}= \\bar{\\rho }(t)\\bar{u}^\\mu \\bar{u}^\\nu ,\n\\end{equation}\nwhere $\\bar{\\rho}(t)$ is the energy density, \nand $\\bar{u}^\\mu$ is the 4-velocity whose components are \ngiven by $\\bar{u}^\\mu=(1,0,0,0)$. \nThe Einstein equations for the FLRW universe are  \n\\begin{eqnarray}\n \\left(\\frac{\\dot{a}}{a}\\right)^2=\\frac{8\\pi}{3}\\bar{\\rho }-\\frac{K}{a^2} \\quad{\\rm and}\\quad\n \\frac{\\ddot{a}}{a}=-\\frac{4\\pi}{3}\\bar{\\rho},\n \\label{bac1}\n\\end{eqnarray}\nwhere a dot denotes differentiation with respect to $t$. \n\nSince our main interest is the evolution of density contrasts \nand their correlations, \nwe consider only scalar perturbations in the FLRW universe. \nAs mentioned in the previous section, \nwe introduce two small independent non-negative parameters, $\\kappa $ and $\\epsilon $. \nThe limit $\\epsilon\\rightarrow0$ leads to the exact \nLTB solution, if we take all the orders of $\\kappa$ into account. By contrast, the limit \n$\\kappa\\rightarrow0$ with $0<\\epsilon\\ll1$ leads to the homogeneous and \nisotropic universe with small anisotropic perturbations. \n\nThen, by choosing the synchronous comoving gauge, \nthe line element of the perturbed spacetime can be written in the form\n\\begin{eqnarray}\n ds^2\n&=&\n -dt^2+a^2(t)\\sum_{N=0}\\kappa^N\\biggl[\nl^{(N)}_{\\parallel}(t,r)\\frac{dr^2}{1-Kr^2}+l^{(N)}_{\\bot }(t,r)r^2d\\Omega^2 \\cr\n&+&\\epsilon\\Big(A^{(N+1)}(t,r,{\\bf \\Omega})\\gamma _{ij}\n+\\mathcal{D}_i\\mathcal{D}_jB^{(N+1)}(t,r, {\\bf \\Omega})\\Big)dx^idx^j \\cr\n&+&{\\cal O}(\\epsilon^2)\\biggr],\n\\label{ds2sec}\n\\end{eqnarray}\nwhere $l_{\\parallel}^{(0)}=l_{\\bot}^{(0)}=1$, $\\bf \\Omega=(\\theta,\\phi)$ \nare the polar and azimuthal angles, and $\\mathcal{D}_i$ denotes the covariant derivative \nwith respect to $\\gamma_{ij}$. \nThe perturbed stress-energy tensor is given by\n\\begin{equation}\n T^{\\mu \\nu }\n =\n \\bar{\\rho }(t)\\bar{u}^\\mu \\bar{u}^\\nu \n\\sum_{N=0}\\kappa^N\\left[\\Delta ^{(N)}(t,r)+\\epsilon \\delta ^{(N+1)}(t,r,{\\bf \\Omega})\n+{\\cal O}(\\epsilon^2)\\right],\n \\label{tmnsec}\n\\end{equation}\nwhere $\\Delta^{(0)}=1$. \nIf we wish to study the evolution of the perturbed FLRW universe \nwith the same accuracy as the linear perturbation analysis for the LTB solution, \nwe should take all orders of $\\kappa$ and the first order with respect to $\\epsilon$.  \nHowever, if $\\kappa$ is much smaller than unity, \nit will be possible to evaluate the evolution of the anisotropic perturbations in the \nLTB solution by studying up to the first order with respect to $\\kappa$.\nIn this approximation,\nthe effect of the void structure on the evolution of anisotropic perturbations appears \nat order $\\kappa\\epsilon$. \n\n\\subsection{First order perturbations}\nWe expand the perturbation variables in terms of the\nspherical harmonic functions $Y_{\\ell m}({\\bf \\Omega})$.\nThe Einstein equations of order $\\kappa$ correspond to the equations for the \nperturbations of $\\ell=0$ mode: \n\\begin{eqnarray}\n \\ddot{\\Delta}^{(1)}+2H\\dot{\\Delta}^{(1)}-4\\pi \\bar{\\rho}\\Delta^{(1)} &=&0,\n \\label{kap1}\n \\\\\n \\dot{l}^{(1)}_{\\parallel}-(r\\dot{l}^{(1)}_\\bot )^{'}&=&0,\n \\label{kap2}\n \\\\\n \\dot{l}^{(1)}_{\\parallel}+2\\dot{l}^{(1)}_\\bot &=&-2\\dot{\\Delta}^{(1)},\n \\label{kap3}\n\\end{eqnarray}\nwhere \n\\begin{equation}\nH:=\\frac{\\dot{a}}{a}, \n\\end{equation}\nand a dash denotes a partial differentiation with respect to $r$. \nThe Einstein equations of order $\\epsilon$ lead to the equations for the \nperturbations of $\\ell>0$ modes, and we obtain\n\\begin{eqnarray}\n \\dot{A}_{\\ell m}^{(1)}(t,r)-K\\dot{B}_{\\ell m}^{(1)}(t,r)&=&0,\n \\label{eps2}\n \\\\\n (1-Kr^2)\\dot{B}_{\\ell m}^{(1)}{}''(t,r)\n +\\biggl[\\frac{2(1-Kr^2)}{r}-Kr\\biggr]\\dot{B}_{\\ell m}^{(1)}{}'(t,r)&& \\cr\n +\\biggl[3K-\\frac{\\ell (\\ell +1)}{r^2}\\biggr]\\dot{B}_{\\ell m}^{(1)}(t,r) \n &=&-2\\dot{\\delta}_{\\ell m}^{(1)}(t,r),\n \\label{eps3} \\\\\n \\ddot{\\delta}_{\\ell m}^{(1)}(t,r)+2H\\dot{\\delta}_{\\ell m}^{(1)}(t,r)-4\\pi \\bar{\\rho }\\delta_{\\ell m}^{(1)}(t,r)&=&0,\n \\label{eps1}\n\\end{eqnarray}\nwhere we have used the eigenvalue equation\n\\begin{eqnarray}\n & & \n \\left(\n \\frac{\\partial^2}{\\partial\\theta ^2}+\\frac{\\cos \\theta }{\\sin \\theta }\\frac{\\partial}{\\partial\\theta}\n  +\\frac{1}{\\sin^2\\theta }\\frac{\\partial^2}{\\partial\\phi ^2}\n \\right)\n Y_{\\ell m}({\\bf \\Omega})=-\\ell (\\ell +1)Y_{\\ell m}({\\bf \\Omega}).\n \\label{eigen}\n\\end{eqnarray}\nWe note that the equations of order $\\epsilon$ and $\\kappa$ \ndecouple with each other. \nFor later convenience, we also show the equation for $B(t,r,{\\bf \\Omega})$ before deriving  \nEq.~\\eqref{eps3} by the spherical harmonics expansion:\n\\begin{eqnarray}\n & &\n \\big(\\mathcal{D}^i\\mathcal{D}_i+3K\\big)\\dot{B}^{(1)}(t,r,{\\bf \\Omega})\n =-2\\dot{\\delta}^{(1)}(t,r,{\\bf \\Omega}). \n \\label{eps4}\n\\end{eqnarray}\n\nGeneral solutions for Eqs.~\\eqref{kap1} and \\eqref{eps1} are given by\n\\begin{eqnarray}\n \\Delta ^{(1)}(t,r)\n &=&\n D^+(t)\\Delta_+^{(\\rm i)}(r)+D^-(t)\\Delta_-^{(\\rm i)}(r),\n \\label{Del1}\n \\\\\n \\delta_{\\ell m}^{(1)}(t,r)\n &=&\n D^+(t)\\delta^{(\\rm i)+}_{\\ell m}(r)+D^-(t)\\delta^{(\\rm i)-}_{\\ell m}(r),\n \\label{del1}\n\\end{eqnarray}\nwhere \n\\begin{eqnarray}\n & & \n D^+(t)=H\\int^{a(t)}\\frac{da}{a^3H^3}\\quad{\\rm and}\\quad D^-(t)=H, \n \\label{grow1}\n\\end{eqnarray}\nand \n$\\Delta^{(\\rm i)}_{\\pm }$ and $\\delta^{(\\rm i)\\pm }_{\\ell m}$ stand for initial values.\n$D^+$ and $D^-$ represent the growing and decaying modes, respectively.\n\n\\subsection{The perturbations of order $\\kappa\\epsilon$}\nAs already mentioned, we are interested in \nthe effect of the void structure on the evolution of \nanisotropic linear perturbation, and this effect first appears at \norder $\\kappa\\epsilon$. Hence we shall focus on the perturbations of this order. \n\nThe perturbations of order $\\kappa\\epsilon$ correspond to the second order \nperturbations in the FLRW universe model. From the second-order Einstein equations together with \nthe background and the linearized Einstein equations,\nwe obtain the evolution equations for the expansion coefficients of $\\delta^{(2)}(t,r,{\\bf \\Omega})$ \nwith respect to $Y_{\\ell m}({\\bf \\Omega})$ as follows:\n\\begin{eqnarray}\n & &\n \\ddot{\\delta}^{(2)}_{\\ell m}(t,r)+2H\\dot{\\delta}^{(2)}_{\\ell m}(t,r)-4\\pi \\bar{\\rho }(t)\\delta ^{(2)}_{\\ell m}(t,r)\n =\n S_{\\ell m}(t,r),\n \\label{second1}\n\\end{eqnarray}\nwhere the source term $S_{\\ell m}(t,r)$ is given by\n\\begin{eqnarray}\n S_{\\ell m}(t,r)\n &=&\n \\left(\\dot{l}^{(1)}_\\bot -\\dot{l}^{(1)}_{||}\\right)\\left(K-\\frac{\\ell (\\ell +1)}{2r^2}\\right)\\dot{B}_{\\ell m}^{(1)}\n +\\left(\\dot{l}^{(1)}_\\bot -\\dot{l}^{(1)}_{||}\\right)\\frac{(1-Kr^2)}{r}\\dot{B}_{\\ell m}^{(1)}{}'\n \\nonumber \\\\\n & &\n +\\left(2\\dot{\\Delta}^{(1)}-\\dot{l}^{(1)}_{||}\\right)\\dot{\\delta}^{(1)}_{\\ell m}\n +8\\pi \\bar{\\rho}\\Delta ^{(1)}\\delta ^{(1)}_{\\ell m}. \n\\end{eqnarray}\nBy solving Eq.~\\eqref{second1}, we obtain \n\\begin{eqnarray}\n \\delta _{\\ell m}^{(2)}(t,r)\n =\n\\int ^t_{t_{\\rm i}}S_{\\ell m}(s,r)\\left(\\frac{D^-(t)D^+(s)-D^+(t)D^-(s)}{W(s)}\\right)ds\n\\end{eqnarray}\nwhere $W(s)$ is the Wronskian given by $W(s)=D^+(s)\\dot{D}^-(s)-\\dot{D}^+(s)D^-(s)$, \nand homogeneous solutions have been absorbed in $\\delta_{\\ell m}^{(1)}$. \nThen, we obtain the anisotropic linear density contrast $\\delta_{\\ell m}$  \nin the LTB solution as\n\\begin{eqnarray}\n \\delta_{\\ell m}(t,r)\n &=&\n \\epsilon \\delta^{(1)}_{\\ell m}(t,r)\n +\\kappa \\epsilon\n \\int ^t_{t_{\\rm i}}S_{\\ell m}(s,r)\\left(\\frac{D^-(t)D^+(s)-D^+(t)D^-(s)}{W(s)}\\right)ds \\cr\n &+&{\\cal O}(\\kappa^2\\epsilon),\n \\label{dens2}\n\\end{eqnarray}\nwhere $t_{\\rm i}$ is the initial time.\n\n\nHereafter, we neglect the decaying modes of order $\\epsilon$. \nBy using Eqs.~\\eqref{eps3} and \\eqref{del1}, $\\dot{B}_{\\ell m}^{(1)}$ is \nwritten in the form \n\\begin{eqnarray}\n \\dot{B}_{\\ell m}^{(1)}(t,r)=\\dot{D}^+(t)B^{(\\rm i)+}_{\\ell m}(r),\n \\label{B1}\n\\end{eqnarray}\nwhere $B^{(\\rm i)+}_{\\ell m}(r)$ is the initial value.\nThen, the anisotropic density contrast \\eqref{dens2} is rewritten as\n\\begin{eqnarray}\n \\delta _{\\ell m}(t,r)\n &=&\n \\epsilon D^+(t)\\delta^{(\\rm i)+}_{\\ell m}(r) \\cr\n &+&\\kappa \\epsilon \\left[T_1(t,r)\\delta^{(\\rm i)+}_{\\ell m}(r)\n +T_2(t,r,\\ell )B^{(\\rm i)+}_{\\ell m}(r)\n + T_3(t,r)B^{(\\rm i)+}_{\\ell m}{}'(r)\n \\right] \\cr\n &+&{\\cal O}(\\kappa^2\\epsilon),\n \\label{dens3}\n\\end{eqnarray}\nwhere $T_1$, $T_2$ and $T_3$ are defined by\n\\begin{eqnarray}\n T_1(t,r)\n &:=&\n \\int ^t_{t_{\\rm i}}ds\\left(\\frac{D^-(t)D^+(s)-D^+(t)D^-(s)}{W(s)}\\right)\n \\nonumber \\\\\n & &\n \\times\n \\left[\\dot{D}^+(s)\\left(2\\dot{\\Delta}^{(1)}(s,r)-\\dot{l}^{(1)}_{||}(s,r)\\right)\n +D^+(s)\\times 8\\pi \\bar{\\rho}(s)\\Delta ^{(1)}(s,r)\n \\right],\n \\\\\n T_2(t,r,\\ell )\n &:=&\n \\int ^t_{t_{\\rm i}}ds\\left(\\frac{D^-(t)D^+(s)-D^+(t)D^-(s)}{W(s)}\\right)\n \\nonumber \\\\\n & &\n \\times\n \\dot{D}^+(s)\n \\left(\\dot{l}^{(1)}_\\bot(s,r) -\\dot{l}^{(1)}_{||}(s,r)\\right)\\left(K-\\frac{\\ell (\\ell +1)}{2r^2}\\right),\n \\\\\n T_3(t,r)\n &:=&\n \\int ^t_{t_{\\rm i}}ds\\left(\\frac{D^-(t)D^+(s)-D^+(t)D^-(s)}{W(s)}\\right)\n \\nonumber \\\\\n & &\n \\times\n \\dot{D}^+(s)\\left(\\dot{l}^{(1)}_\\bot (s,r)-\\dot{l}^{(1)}_{||}(s,r)\\right)\\frac{1-Kr^2}{r}.\n\\end{eqnarray}\n\n\\section{Angular power spectrum and angular growth rate}\\label{sec3}\nIn the previous section, we derived the growing solutions for density contrasts. \nOnce we have specified the isotropic linear perturbations \n$l^{(1)}_{||}$, $l^{(1)}_{\\bot }$, $\\Delta ^{(1)}$ and \nthe initial anisotropic inhomogeneities \n$\\delta^{(\\rm i)+}_{\\ell m}$, $B^{(\\rm i)+}_{\\ell m}$, \nwe obtain the density contrasts $\\delta_{\\ell m}$ by Eq.~\\eqref{dens3}. \nIn this section, \nwe derive the explicit form of the angular power spectrum of \nthe density perturbation $\\delta_{\\ell m}$ \nfor a given set of initial conditions in terms of the standard power spectrum. \nThen, we define the angular growth rate by using the angular power spectrum. \nHereafter, in order to determine the perturbations of order $\\kappa$,\nwe refer to the non-Copernican LTB universe model with uniform big-bang time (see Appendix A).\n\n\n\\subsection{Initial power spectrum of density contrast} \nBy virtue of the uniform big-bang time (see Appendix A),\nthe present non-Copernican LTB universe \napproaches the homogeneous and isotropic universe as time goes back.  \nHence, it is reasonable to assume that the initial conditions for the anisotropic \nperturbations are the same as in the case of the FLRW universe.\nThen, the initial power spectrum of the density contrast can be \nexpressed as follows: \n\\begin{eqnarray}\n \\langle\\delta ^{(1)*}(t_{\\rm i},{\\bf k})\\delta ^{(1)}(t_{\\rm i},{\\bf k}')\\rangle\n =(2\\pi)^3\\delta_{\\rm D}^3({\\bf k}-{\\bf k}')P(t_{\\rm i},k),\n \\label{pow1}\n\\end{eqnarray}\nwhere $\\delta_{\\rm D}$ is the Dirac's delta function, \n$t_{\\rm i}$ represents some sufficiently early time already introduced in Eq.~\\eqref{dens2},\nand the Fourier transform of the density contrast is defined by\n\\begin{eqnarray}\n \\delta^{(1)} (t,{\\bf k})=\\int d^3x \\delta^{(1)} (t,{\\bf x})e^{-i{\\bf k\\cdot x}}.\n\\end{eqnarray}\nIf we choose the initial time after recombination, \nthe matter power spectrum including baryons and cold dark matter can be written as\n\\begin{eqnarray}\n P(t_{\\rm i},k) &=& [D^+(t_{\\rm i})]^2P(k), \\nonumber \\\\\n P(k)&=&A_0k^nT^2(k),\n \\label{pow2}\n\\end{eqnarray}\nwhere $A_0$ is a positive constant which represents the amplitude for perturbations on large scales, \n$n$ is constant, and $T(k)$ is the matter transfer function.\nIn this paper, we assume the Harrison-Zel'dovich spectrum $n=1$. \nAs for the transfer function,\nwe adopt the fitting formula developed by Eisenstein \\& Hu~\\cite{Eisenstein:1997ik} \n(see Appendix B). \n\n\\subsection{Angular power spectrum and angular growth rate}\nIn order to observationally study the evolution of perturbations in the non-Copernican universe,\nwe need to specify the observable quantities by using the density contrast \\eqref{dens3}.\nThe simplest quantity that we can currently calculate is the angular power spectrum.\nWe define the angular power spectrum of the density contrast by \n\\begin{eqnarray}\n C_\\ell (t,r)=\n \\frac{r^2}{2\\ell +1}\\sum_{m=-\\ell}^\\ell \\langle\\delta^{*}_{\\ell m}(t,r)\\delta_{\\ell m}(t,r)\\rangle,\n \\label{CL1}\n\\end{eqnarray}\nwhere $*$ denotes the complex conjugate.\n\nHereafter, we focus on the non-spherical perturbations whose wavelengths \nare much smaller than the spatial curvature radius $(k\\gg \\sqrt{|K|})$. \nThen, from Eq. \\eqref{eps4}, we have\n\\begin{eqnarray}\n k^2\\dot{B}^{(1)}(t,{\\bf k})\\simeq 2\\dot{\\delta}^{(1)}(t,{\\bf k}). \n \\label{fou}\n\\end{eqnarray}\nThe initial values $\\delta^{(\\rm i)}_{\\ell m}(r)$ and $B^{(\\rm i)}_{\\ell m}(r)$ \nwhich appear in Eqs.~\\eqref{B1} and \\eqref{dens3} \ncan be written using the Fourier transform of the initial density contrast $\\delta^{(1)}(t_{\\rm i},{\\bf k})$ as\n\\begin{eqnarray}\n \\delta^{(\\rm i)+}_{\\ell m}(r)\n &=&\n \\Big[\\frac{1}{D^+(t_{\\rm i})}\\Big](4\\pi i^\\ell )\\int \\frac{d^3k}{(2\\pi)^3}\\delta^{(1)} (t_{\\rm i},{\\bf k})j_\\ell (kr)Y^*_{\\ell m}\n ({\\bf \\Omega}_{\\rm k}),\n \\label{de2}\n \\\\\n B^{(\\rm i)+}_{\\ell m}(r)\n &=&\n \\Big[\\frac{1}{D^+(t_{\\rm i})}\\Big](4\\pi i^\\ell )\\int \\frac{d^3k}{(2\\pi)^3}\\delta^{(1)} (t_{\\rm i},{\\bf k})j_\\ell (kr)\n Y^*_{\\ell m}({\\bf \\Omega}_{\\rm k})\\left(\\frac{2}{k^2}\\right),\n \\label{B2}\n\\end{eqnarray}\nwhere ${\\bf \\Omega}_{\\rm k}$ denotes the polar and azimuthal angles in the Fourier space, and \nwe have used the relation between the expansion coefficient with respect to  \n$Y_{\\ell m}({\\bf \\Omega}_{\\rm k})$ and the Fourier transform\n\\begin{eqnarray}\n \\phi_{\\ell m}(t,r)\n =\n (4\\pi i^\\ell )\\int \\frac{d^3k}{(2\\pi)^3}\\phi (t,{\\bf k})j_\\ell (kr)Y^*_{\\ell m}({\\bf \\Omega}_{\\rm k}),\n \\label{phi2}\n\\end{eqnarray}\nand Eq.~\\eqref{fou}. \nUsing Eqs.~\\eqref{pow1}, \\eqref{pow2}, \\eqref{de2} and \\eqref{B2}, \nthe angular power spectrum of the density contrast \\eqref{dens3} \ncan be rewritten in the following form: \n\\begin{eqnarray}\nC_\\ell (t,r)&=&\n\\epsilon^2{D^+}^2(t)K_1(\\ell ,r)+\\kappa \\epsilon^2 \\Big[2D^+(t)T_1(t,r)K_1(\\ell ,r)\n\\nonumber \\\\\n& &\n+2D^+(t)\\tilde{T}_{2}(t,r,\\ell )K_2(\\ell ,r)+2D^+(t)\\tilde{T}_{3}(t,r)K_3(\\ell ,r)\n\\Big]+{\\cal O}(\\kappa^2\\epsilon^2),\n\\label{CL4}\n\\end{eqnarray}\nwhere \n\\begin{eqnarray}\n K_1(\\ell ,r)\n &=&\n \\left(\\frac{2}{\\pi }\\right)\\int _0^\\infty dk P(k)(kr)^2j_\\ell ^2(kr),\n \\nonumber \\\\\n K_2(\\ell ,r)\n &=&\n \\left(\\frac{2}{\\pi }\\right)\\int _0^\\infty dk P(k)(kr)^2j_\\ell ^2(kr)\\left(\\frac{2}{k^2}\\right),\n \\nonumber \\\\\n K_3(\\ell ,r)\n &=&\n \\left(\\frac{2}{\\pi }\\right)\\int _0^\\infty dk P(k)(kr)^2j_\\ell (kr)j_\\ell ^{'}(kr)\\left(\\frac{2}{k^2}\\right),\n\\end{eqnarray}\nand $\\tilde{T}_2$ and $\\tilde{T}_3$ are defined \nas the short wavelength approximation $(k\\gg \\sqrt{|K|})$ of \n$T_2$ and $T_3$ by \n\\begin{eqnarray}\n \\tilde{T}_2(t,r,\\ell )\n &=&\n \\int ^t_{t_{\\rm i}}ds\\left(\\frac{D^-(t)D^+(s)-D^+(t)D^-(s)}{W(s)}\\right)\n \\nonumber \\\\\n & &\n \\times\n \\dot{D}^+(s)\n \\left(\\dot{l}^{(1)}_\\bot(s,r) -\\dot{l}^{(1)}_{\\parallel}(s,r)\\right)\\left(-\\frac{\\ell (\\ell +1)}{2r^2}\\right),\n \\\\\n \\tilde{T}_3(t,r)\n &=&\n \\int ^t_{t_{\\rm i}}ds\\left(\\frac{D^-(t)D^+(s)-D^+(t)D^-(s)}{W(s)}\\right)\n \\nonumber \\\\\n & &\n \\times\n \\dot{D}^+(s)\\left(\\dot{l}^{(1)}_\\bot (s,r)-\\dot{l}^{(1)}_{||}(s,r)\\right)\\left(\\frac{1}{r}\\right).\n\\end{eqnarray} \nOnce the initial density power spectrum $P(t_{\\rm i},k)$ is specified, \nwe can calculate the angular power spectrum $C_\\ell (t,r)$ by using Eq.~\\eqref{CL4}.\n\nTo investigate the growth rates of the perturbations, \nwe define the angular growing factor by \n\\begin{eqnarray}\n D_\\ell(t,r)=\\left[\\frac{C_\\ell(t,r)}{C_\\ell(t_{\\rm i},r)}\\right]^{1\/2}. \n \\label{eff1}\n\\end{eqnarray}\nIt is easy to see that the angular growing factor $D_\\ell(t,r)$ is equal to $D^+(t)\/D^+(t_{\\rm i})$ \nup to order $\\epsilon$. \n\nBasically, an observer can see the cosmological structures on his\/her past light cone \nby observations through electromagnetic \nradiation\\footnote{The past light cone of an observer at the event $p$ \nis defined by the boundary of the causal past of $p$, which is usually \ndenoted by $\\dot{J}^-(p)$ in general relativity. \nStrictly speaking, the observer can see the inside of the light cone \nthrough a congruence of the light rays which have experienced caustics caused by gravitational \nlens effects or scattering due to electromagnetic interactions in the real universe.}\nand hence, in the case of the non-Copernican universe model, \nit is useful to consider quantities on the light cone of an observer who stays \nat the symmetry center of the void at present. \nHereafter, for simplicity, we call the observer who stays at the symmetry center of \nthe void at present ``the central observer\", and the past light cone of the central observer \nis denoted by $\\Sigma_{\\rm lc}$. \nThe past light cone $\\Sigma_{\\rm lc}$ is generated by the past-directed outgoing \nradial null geodesics $k^\\mu=(dt\/d\\lambda,dr\/d\\lambda,0,0)$, where $\\lambda$ is \nthe affine parameter. The cosmological redshift $z$ is defined by\n\\begin{equation}\nz=\\frac{dt\/d\\lambda}{(dt\/d\\lambda)_0}-1,\n\\end{equation}\nwhere $dt\/d\\lambda$ and $(dt\/d\\lambda)_0$ are the value at the time of the emission of \na photon and that at the time of the detection of the photon by the central observer, respectively. \nBy using the cosmological redshift $z$ instead of \nthe affine parameter $\\lambda$, the geodesic equations for the generator of the \npast light cone $\\Sigma_{\\rm lc}$ up to order $\\kappa$ are given by\n\\begin{eqnarray}\n\\frac{dr}{dz}&=&\\frac{\\sqrt{1-Kr^2}}{(1+z)aH}\\left[1-\\frac{\\kappa}{2}\n\\left(l_\\parallel+\\frac{1}{H}\\dot{l}_\\parallel\\right)\\right], \\\\\n\\frac{dt}{dz}&=&-\\frac{1}{(1+z)H}\\left(1-\\frac{\\kappa}{2H}\\dot{l}_\\parallel\\right). \n\\end{eqnarray}\nWe denote the solution of the above equations by $t=t_{\\rm lc}(z)$ and $r=r_{\\rm lc}(z)$.  \n\nThen, using the angular growing factor, we define the \nangular growth rate on the past light cone $\\Sigma_{\\rm lc}$ as \na function of redshift $z$ as follows:\n\\begin{eqnarray}\n f_\\ell(z)=- \\frac{d [\\ln D_\\ell(t_{\\rm lc}(z),r_{\\rm lc}(z))]}{d \\ln(1+z)}. \n \\label{eff2}\n\\end{eqnarray}\nHere, we note that the angular growing factor and the angular growth rate \ndo not depend on the amplitude $A_0$ in Eq.~\\eqref{pow2}. \nWe also note that the angular growth rate up to order $\\epsilon$ agrees with the growth rate \nusually used in the linear perturbation theory of the FLRW universe, \n $d ({\\rm ln} D^+)\/d ({\\rm ln} a)$. \n\n\n\\section{evolution of density perturbations in the Clarkson-Regis model}\\label{sec4}\n\\subsection{Linearized Clarkson-Regis model}\nIn order to determine the perturbations of order $\\kappa$, \nwe use the non-Copernican LTB universe model \ngiven by Clarkson and Regis~\\cite{Clarkson:2010ej} \n(see Appendix A), which we call the Clarkson and Regis~(CR) model. \nWe shall study the evolution of linear anisotropic perturbations \nin the CR model by using the second order perturbation theory of the FLRW universe filled with dust. \nIn order to approximate the CR model by the linearly perturbed FLRW \nuniverse filled with dust, we must first specify the background FLRW universe.  \nHere, we determine the background FLRW universe so that \nthe cosmological density parameter of the background is equal to \n$0.242$, which is equal to the value of the density parameter function $\\Omega_{\\rm M}(r)$\nat the symmetry center of  the CR model (see Eq.~\\eqref{CR1}).\n\nWe define the ``density contrast'' of the CR model as\n\\begin{eqnarray}\n & &\n \\Delta^{(\\rm CR)}(t,r)= \\frac{\\rho^{(\\rm CR)}(t,r)-\\rho^{\\rm (CR)}(t,0)}{\\rho^{\\rm (CR)}(t,0)},\n \\label{Del2}\n\\end{eqnarray}\nwhere $\\rho^{(\\rm CR)}$ is the energy density of \nthe CR model. The density contrasts $\\Delta^{(\\rm CR)}$'s on three constant time \nhypersurfaces are depicted in Fig.~\\ref{density_timeconst} as functions of $r$. \n\\begin{figure}[htbp]\n \\begin{center}\n \\includegraphics[width=10cm,clip]{density_timeconst.eps}\n \\end{center}\n \\caption{\nDensity contrasts on the spacelike hypersurfaces for\n$t=t_{\\rm lc}(100)$, $t=t_{\\rm lc}(1)$ \nand $t=t_0$, as functions of $r$.\n}\n \\label{density_timeconst}\n\\end{figure}\nHere, we have used the cosmological redshift $z$ to specify each constant \ntime hypersurface given by $t=t_{\\rm lc}(z)$.\nWe can see that the void structure grows with time. \nSince the big-bang time is uniform, there is only the growing mode in the CR model. \nThe void size is about $12{\\rm Gpc}$, and the vicinity of the center \nis locally the dust filled FLRW model \nwith the cosmological density parameter $\\Omega_{\\rm M}=0.242$, whereas \nthe asymptotic region is almost the same as the dust filled FLRW model with $\\Omega_{\\rm M}=0.7$.\nThe Hubble parameter at the center is $H_{0}=74{\\rm kms^{-1}Mpc^{-1}}$. \n\nBy using the density contrast $\\Delta^{(\\rm CR)}$, \nwe give the initial conditions for the isotropic linear density contrast \n$\\Delta ^{(\\rm i)}_{\\pm }$ in Eq.~\\eqref{Del1} as follows.\nAs mentioned, since the CR model has only the growing mode,  \nwe should set $\\Delta ^{(\\rm i)}_{-}(r)=0$. By contrast, \n$\\Delta_+^{({\\rm i})}(r)$ is determined by the assumption that  \nthe density contrast $\\Delta^{(1)}$ exactly agrees \nwith that of the CR model at the initial time $\\Delta^{\\rm (CR)}(t_{\\rm i},r)$, i.e.,\n\\begin{eqnarray}\n \\Delta ^{(\\rm i)}_+(r)=\\frac{\\Delta^{(\\rm CR)}(t_{\\rm i},r)}{D^+(t_{\\rm i})},\n\\end{eqnarray}\nwhere the initial time is determined by $t_{\\rm i}=t_{\\rm lc}(1000)$. \nOnce the initial condition for the density contrast is given, \nwe obtain all perturbations of order $\\kappa$ \nby solving the perturbation equations up to the corresponding order, and as a result, \nthe linearized CR model is obtained. \n\nTo evaluate the accuracy of the linear approximation,\nwe plot $\\Delta^{(\\rm CR)}$ and $\\Delta^{(1)}$ on the past light cone $\\Sigma_{\\rm lc}$ \nin Fig.~\\ref{density_redshift}.\n\\begin{figure}[htbp]\n \\begin{center}\n \\includegraphics[width=10cm,clip]{density_redshift.eps}\n \\end{center}\n \\caption{\n Exact density contrast $\\Delta^{(\\rm CR)}$ and \n that of the linearized CR model $\\Delta^{(1)}$ on the past light cone $\\Sigma_{\\rm lc}$,\n plotted as functions of the redshift $z$.\n }\n \\label{density_redshift}\n\\end{figure}\nThe relative error between the exact and linearized CR models is less than $30\\%$\non the light cone $\\Sigma_{\\rm lc}$. \nInside the past light cone $\\Sigma_{\\rm lc}$, \nthe error is smaller than that on the past light cone $\\Sigma_{\\rm lc}$,\nsince the CR model has only the growing mode. \nThere is no qualitative difference between the exact and the linearized CR models,\nand thus we may see the qualitative behavior of linear perturbations in the CR model by  \nthe perturbative analysis of the FLRW universe based on the linearized CR model. \n\n\\subsection{Evolution of density contrasts in the CR model}\nLet us consider the evolution of the anisotropic density contrasts in the CR model. \nBy using the angular power spectrum $C_\\ell(t,r)$ given by Eq.~\\eqref{CL4} and \nthe transfer function $T(k)$ given in appendix~\\ref{appendixB},\nwe depict the angular growing factors $D_\\ell(t,r)$'s defined by Eq.~\\eqref{eff1} \nat each comoving distance as functions of $t$ in Fig.~\\ref{growthfactor_CR}. \n\n\\begin{figure}[htbp] \n \\begin{center}\n\\includegraphics[width=10cm,clip]{growthfactor_CR.eps}\n \\end{center}\n \\caption{\nAngular growing factors $D_\\ell$'s in the CR model at \n$r=40{\\rm Gpc}$ (dotted line), $r=4{\\rm Gpc}$ (dashed line) and $r=0$ (dot-dashed line)\ndepicted as functions of $t$. \nThe present time is $H_0t_0=0.83$. \nWe choose $\\ell$ so that $\\tilde{k}=0.5 {\\rm Mpc}^{-1}$.\n}\n \\label{growthfactor_CR}\n\\end{figure}\n\\begin{figure}[htbp]\n \\begin{center}\n \\includegraphics[width=10cm,clip]{growthfactor_FLRW.eps}\n \\end{center}\n \\caption{\nAngular growing factors $D_\\ell$'s in the dust filled FLRW universe models \nwith $\\Omega_{\\rm M}=0.242$ and $\\Omega_{\\rm M}=0.7$, together with that  \nfor the CR model. \n}\n \\label{growthfactor_FLRW}\n\\end{figure}\n\nHere, we introduce a useful quantity defined by  \n\\begin{equation}\n{\\tilde k}:=\\frac{\\ell}{r}~.\n\\end{equation} \nNote that $\\tilde k$ is equal to the comoving wave number of the mode \n$\\ell$ at a distance $r$ in the flat sky approximation \n(see e.g., Ref.~\\cite{Lyth:textbook}).\nIn Fig.~\\ref{growthfactor_CR}, the present time is $H_0t_0=0.83$, and $\\ell$ is \nchosen so that $\\tilde{k}=0.5 {\\rm Mpc}^{-1}$.  \nThis choice of $\\ell$ shows us the evolution of perturbations with the size of a cluster of galaxies, i.e.,  \n$2\\pi \/\\tilde{k}  \\sim 10 {\\rm Mpc}$.\n\nWe can see from Fig.~\\ref{growthfactor_CR} that \nthe larger the comoving distance of a perturbation from the symmetry center, \nthe faster the growth of the perturbation. \nThis result may be explained by the fact that the energy density of the CR model \nis a monotonically increasing function of $r$, since the growth rates of \nperturbations in the FLRW universe is a monotonically increasing function of $\\Omega_{\\rm M}$. \nWe also depict $D_\\ell$ of the FLRW universe models   \nwith $\\Omega_{\\rm M}=0.242$ and $0.7$, respectively,  \ntogether with that of the CR model in Fig.~\\ref{growthfactor_FLRW}. \nWe can see from this figure that $D_\\ell$ of the FLRW universe with \n$\\Omega_{\\rm M}=0.242$ agrees with  $D_\\ell(t,r=0)$ of the CR model. \nWe note that $D_\\ell$ in the FLRW universe with $\\Omega_{\\rm M}=0.7$ \ndoes not agree with that far from the void ($r=40{\\rm Gpc}$) in the CR  model.\nThis result might not be real, but rather could be an error caused by using \nthe linearized CR model, since the error due to the linear\napproximation becomes larger for larger $r$.  \n\n\\begin{figure}[htbp]\n \\begin{center}\n \\includegraphics[trim = 5 110 5 110 ,width=14cm,clip]{kdepend.eps}\n \\end{center}\n \\caption{\n  Angular growing factors $D_\\ell(t,r=4{\\rm Gpc})$ in the CR model for\n  $\\tilde{k}=0.01{\\rm Mpc^{-1}}$ (solid line), $0.1{\\rm Mpc^{-1}}$ (dashed line) \n  and $1{\\rm Mpc^{-1}}$ (dot-dashed line), depicted \n  as functions of $t$.\n  The right panel shows a close-up of the $0.7<H_0t<0.71$}.\n \\label{kdepend}\n\\end{figure}\n\nWe have also investigated the dependence of $D_\\ell$ \non $\\tilde{k}$. \nThe angular growing factors $D_\\ell(t,r=4{\\rm Gpc})$'s \nwith various values of $\\ell$, or equivalently, \n$\\tilde{k}$ are depicted as functions of $t$ in Fig.~\\ref{kdepend}.\nWe find that the dependence of $D_\\ell(t,r=4{\\rm Gpc})$ on \n$\\tilde{k}$ is very small in the case of the CR model.\n\nNext, the angular growth rate $f_\\ell$ defined by \\eqref{eff2} is plotted as \na function of $z$ for the CR model, together with those of  \nthe dust filled FLRW with $\\Omega_{\\rm M}=0.242$ and $\\Omega_{\\rm M}=0.7$ \nand the flat $\\rm {\\Lambda CDM}$ with $\\Omega_{\\rm M}=0.28$\nin Figs.~$\\ref{growthrate_high}$ and $\\ref{growthrate_low}$.\n\\begin{figure}[htbp]\n \\begin{center}\n \\includegraphics[width=10cm,clip]{growthrate_high.eps}\n \\end{center}\n \\caption{Angular growth rate $f_\\ell$ in the CR model (solid line), \ntogether with those of \nthe dust filled FLRW universe with $\\Omega_{\\rm M}=0.242$ (dot-dashed line) and \n$\\Omega_{\\rm M}=0.7$ (dashed line)\nand the flat $\\rm {\\Lambda CDM}$ universe with $\\Omega_{\\rm M}=0.28$ (dotted line)\nas a function of $z$. $\\ell$ is chosen so that $\\tilde{k}=0.5 {\\rm Mpc}^{-1}$.\n}\n \\label{growthrate_high}\n\\end{figure}\n\\begin{figure}[htbp]\n \\begin{center}\n \\includegraphics[width=10cm,clip]{growthrate_low.eps}\n \\end{center}\n \\caption{\nClose-up of the region $0<z<1$ in Fig.~\\ref{growthrate_high}.\n}\n \\label{growthrate_low}\n\\end{figure}\nFigure~\\ref{growthrate_low} shows a close-up of the region $0<z<1$ in \nFig.~\\ref{growthrate_high}.  \nHere, $\\ell$ is also chosen so that $\\tilde{k}=0.5 {\\rm Mpc}^{-1}$.\nFrom Fig.~\\ref{growthrate_high},\nwe can see that the value of $f_\\ell$ of these models approach\nthe value of the Einstein de-Sitter universe ($f_\\ell=1$) in the high redshift domain. \nWe can see from Fig.~\\ref{growthrate_low} that $f_\\ell$ at the central observer $z=0$ of the CR model \nagrees with the value of the dust filled FLRW universe with $\\Omega_{\\rm M}=0.242$. \nWe can also see that $f_\\ell$ in the CR model \nis significantly different from those in homogeneous and \nisotropic universes for redshift $0<z\\lesssim 1$.\nThis result implies that, if we can somehow observe $f_\\ell$,  \nthe observational data may give a strong constraint on non-Copernican universe models.\n\n\\section{summary and discussion}\\label{sec5}\nWe have studied the evolution of anisotropic density perturbations \nin a large spherical void model, \nwhich is a non-Copernican universe model based on the LTB solution.  \nAs is well known, the analysis of anisotropic linear perturbations of the LTB solution \nis much harder than that of the homogeneous and isotropic universe. \nTherefore, we have not tried to directly solve linear perturbations in the LTB model. \nInstead, we have studied linear perturbations in a linearized LTB model.\nThe linearized LTB model is obtained by regarding the inhomogeneities \nin the LTB model as isotropic linear perturbations in the dust filled FLRW universe.  \nHence, our method is relevant for the LTB model only if the isotropic inhomogeneities is so small \nthat they can be treated as linear perturbations on a dust filled FLRW universe.\n\nAssuming the uniform big-bang time, \nwe have given the initial conditions for anisotropic perturbations, which are the same as \nthose of the FLRW universe model. \nThen, we calculated the angular power spectrum of the anisotropic density contrast\ntaking into account the tidal effect of isotropic inhomogeneity, \nand further we defined the angular growing factor and an angular growth rate \nusing the angular power spectrum. \n\nIn \\S~\\ref{sec4}, \nwe calculated the angular growing factor using the\nlinearized Clarkson-Regis (CR) model, which has the uniform big-bang time.\nFrom the behavior of the angular growing factor,\nwe found that the speed of growth of a perturbation is a monotonically increasing \nfunction of the comoving distance from the center of the void. \nBecause of this property, the angular growth rate in the CR model differs from \nthat in the dust filled FLRW universe even for low redshift $(z<1)$.\nSo, if we can observe the angular power spectrum of \nthe density contrasts and can observe their angular growth rate,\nthe observational data will strongly restrict non-Copernican universe models.\n\nGuzzo et al. and Blake et al. have presented the observational results on the \ngrowth rate of the density perturbation as a function of the redshift $z$ \non our past light cone~\\cite{Guzzo:2008ac,Blake:2011rj}. \nHere, we should note that their observations are a distortion of the correlation function \nof galaxies in redshift space and they derived the growth rate from it \nassuming the homogeneous and isotropic background universe. \nHence, our present result regarding the angular growth rate does not correspond to \ntheir ``observational results on the growth rate''. \nTo calculate the redshift space distortion of the galaxy distribution \nin non-Copernican universe models,\nwe have to consider not only the growth in the transverse \ndirection but also the radial direction. \nThis is a future work, and we shall present the results elsewhere. \n\nWe expect that the formula developed here \ncould be applied to many inhomogeneous cosmological models,\nsince we have only assumed that the isotropic \ninhomogeneities are small inside the past light cone $\\Sigma_{\\rm lc}$. \nWe also expect that the formula could be potentially used for \ncalculating the integrated Sachs-Wolfe effect, \nbaryon acoustic oscillations, the shape of the 3D power spectrum and so on. \n\n\\section*{Acknowledgments}\n\nWe are grateful to H.~Ishihara and colleagues in the astrophysics and \ngravity group of Osaka City University for their useful and helpful \ndiscussion and criticism. \nRN and CY are supported by a Grant-in-Aid through the\nJapan Society for the Promotion of Science (JSPS).\nThis work was partially supported by a JSPS Grant-in-aid for Scientific Research (C) (No.\\ 21540276). \n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Funding Information}\n\n\\textbf{Funding.} Russian Foundation for Basic Research (16-02-00762) and Government of Russian Federation, Grant 074-U01; I.B. thanks German Research Foundation (DFG) (project BA 4156\/4-1); Nieders. Vorab (project ZN3061).\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\n\nTraining deep reinforcement learning agents requires significant time and computational resources. This is especially true in rich, virtual, 3D dimensional environments with large state spaces such as the game, Minecraft (See Figure~\\ref{fig:minecraft}). Latent states must be inferred from noisy, pixel-level data and high amounts of perceptual aliasing confound interpretation of the underlying state. \n{\\em Perceptual aliasing} occurs when many states share nearly identical visual features. \nFigure~\\ref{fig:aliasing} shows an example of an area with a lot of perceptual aliasing.\nA consequence of perceptual aliasing is that deep reinforcement learning agents struggle to learn the relationship between visual features and the utility of particular actions when those features are present---sometimes one action must be taken and sometimes another action must be taken. \nThis can cause the agent to fail to converge on a policy and\/or to require additional exploration of areas of high aliasing to learn better feature representations.\nMany environments, including for games and those used in machine learning research (c.f., \\cite{zhiyu}) seed an environment with distinguishing features, such as different wall colors or structural markers.\nThese interventions in the environment to aide agent learning are artificial.\nBecause of these factors, it is non-trivial to train a reinforcement learning agent to perform optimally in increasingly realistic environments with high amounts of perceptual aliasing. \n\nHuman intervention can help speed up reinforcement learning by assisting a deep reinforcement learning (DRL) agent when its confidence in its learned policy is low.\n{\\em Interactive machine learning} (IML) \\cite{iml} seeks to improve upon traditional machine learning algorithms by allowing humans to play a direct role in training. Human collaborators provide demonstrations of correct behavior, providing action advice before a policy action is taken, or providing critique after the agent acts.\nDemonstrations are very effective but can often be costly, difficult, or time-consuming to produce and record.\nAction advice and critique allow a human teacher to monitor the agent's learning from remote and provide feedback that biases it to better learning outcomes.\nStudies \\cite{krening} show that non-experts prefer giving action advice over critique. \nHowever, a majority of studies on advice and critique giving are conducted in grid-world environments. An agent must navigate through a 2D grid in which the agent's $(x,y)$ location is a given feature.\n3D virtual environments add a layer of complexity by requiring an agent to learn features from pixel-level input that correspond to utility toward a goal and\nmake agents susceptible to perceptual aliasing.\n\nIn this paper, we attempt to understand how Learning from Advice (LfA) algorithms perform in 3D environments with perceptual aliasing.\nWe adapt two LfA algorithms to the Minecraft environment and train them to perform an item delivery quest in a map with high perceptual aliasing.\nThe {\\em Feedback Arbitration}~\\cite{zhiyu} algorithm is a straightforward technique for incorporating human action advice into reinforcement learning in which the agent accepts action advice whenever it is provided by a human teacher and the agent has low confidence in its own policy.\nThe {\\em Newtonian Action Advice} algorithm accepts human action advice whenever provided and assumes any piece of advice remains applicable for a period of time afterward even if no other advice is given.\n\nEffective IML requires balancing the learning performance of the agent against the cognitive load demand on the human teacher.\nThe teacher may give intermittent advice because they are fatigued or distracted.\nSimilarly, error, confusion, or distraction may degrade advice accuracy.\nIn order to run controlled studies assessing different rates of teacher response and teacher accuracy, we construct an {\\em oracle}, a synthetic teacher with perfect environment information. \nThe oracle can be parameterized with regard to (a)~how often to provide action advice to a reinforcement learning agent and (b)~how often to give incorrect advice. \n\nOur contributions show that high frequency advice is superior to low frequency advice and greatly superior to baseline reinforcement learning. Further, we show that even significantly inaccurate advice is better than no advice. \nExperimental results show that Newtonian Action Advice uses advice feedback more efficiently allowing it to mitigate time spent in aliased states.\n\n\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.75\\columnwidth]{malmopip}\n\\caption{A questing environment in Minecraft. The agent's view is shown in the inlay.}\n\\label{fig:minecraft}\n\\end{figure}\n \n\n \n\n\n\n\n\\section{Background and Related Work}\n\nReinforcement Learning is proven to be effective at learning agent control policies in 2D and 3D game environments including Atari~\\cite{drqn}, navigation in Minecraft environments~\\cite{liu,oh,singh}, Doom~\\cite{doom}, Quake II~\\cite{ctf}, DOTA 2, and StarCraft II.\nMost game environments are designed to have low perceptual aliasing to make it easier on human players even though the real world can often have high perceptual aliasing.\n3D computer game environments provide a stepping stone to application of reinforcement learning in real world environments~\\cite{laird,ctf} but sometimes overlook this aspect of the real world.\n\nPerceptual aliasing is objectively higher when an agent is closely facing a wall due to repeated block textures in Minecraft.\nWalls and structure boundaries are exemplars for aliased states because it guarantees the texture is repeated elsewhere in the maze. \nWalls and structures with significant height can also block the agent's view of distant but unique visual landmarks that it might otherwise learn to orient itself.\nAnother example of perceptual aliasing occurs in longer stretches of maze-like environments.\nTo a mazerunner, every point along the path between two hedges, a corridor, or a tunnel can look similar.\n(Figure~\\ref{fig:minecraft} shows a short hedge maze).\nOften there are non-perceptual features that afford better task performance in these situations that reinforcement learning agents don't explicitly model, such as length of time traveling straight.\nConventional deep reinforcement learning algorithms must spend more time exploring high-aliasing parts of the environment.\n\n\\begin{figure}[t]\n\\centering{\n\\includegraphics[width=0.75\\columnwidth]{alias}}\n\\caption{An example of perceptual aliasing in Minecraft\n}\n\\label{fig:aliasing}\n\\end{figure}\n\nHuman feedback in IML can take different forms. \n{\\em Learning from Demonstration} (LfD) allows humans to directly provide examples of proper behavior \\cite{argall}. \nFrom demonstrations, an agent can learn the policy directly, learn to explore more effectively \\cite{griffith}, or learn a reward function from which to reconstruct a policy \\cite{abbeel}. \nHowever, it is not always feasible to provide demonstrations. \n{\\em Learning from Critique} (LfC) allows human teachers to indicate that the agent is doing well or not doing well in order to bias the agent toward certain outcomes. Learning from Critique can also include human indication of preferences over variations in agent behavior \\cite{critique}. \n{\\em Learning from Advice} (LfA) is similar to Learning from Critique, except the human teacher proactively advises the agent on the actions it should take instead of retroactively rewarding or punishing the agent for an action it already took.\nSince studies show that non-expert human teachers prefer giving action advice over critique~\\cite{krening} we focus on Learning from Action. \n\nWe build upon a rich set of research exploring human feedback in reinforcement learning~\\cite{griffith,krening,thomaz,isbell,stone,littman}. A majority of this work assesses the algorithms in 2D grid-world environments where the agent's $(x,y)$ location is a given feature. Some do address 3D \\cite{Imperfect}, more simplistic 3D environments \\cite{zhiyu}, or augment the agent with non-visual state information \\cite{abel}. Overall there are still open questions about the scalability of these algorithms in 3D environments\n\n\n\n\n\n\nWe build off two LfA algorithms in our exploration of perceptual aliasing in 3D virtual environments.\nThe first, {\\em Feedback Arbitration}~\\cite{zhiyu} is based on a standard deep reinforcement learning technique called {\\em Deep $Q$ Networks} (DQN).\nDeep $Q$ Networks \\cite{mnih2} use a convolutional neural network to learn visual features in the state that correspond to the utility of different actions---called a $q$-value.\nA Feedback Arbitration agent alternates between exploring the environment, exploiting it's $Q$ network, and listening to a human teacher depending on (a) its confidence in its own $Q$ network and (b) it's learned confidence in the human trainer.\nFeedback Arbitration was tested in very simple 3D grid-world environment with landmarking features which helped lower perceptual aliasing.\n\nThe second, the {\\em Newtonian Action Advice} algorithm~\\cite{krening} is built on tabular $Q$-learning, where a table of $q$-values is stored for every state and action. Krening et al.~\\shortcite{KreningSubjective} found that human trainers perceived a Newtonian Action Advice agent as more intelligent, more transparent, more performant and less frustrating than a standard $q$-learning agent.\nThe standard $Q$-learning algorithm is enhanced such that if a human trainer provides action advice the agent takes the advice depending on its confidence in its $q$-table.  \nIf there is no action advice, the agent continues to follow the last received advice for $t$ additional timesteps. \nThere is no neural component to NAA and requires an enumerable state space; as such it is not directly scalable to 3D environments.\nWe extend this approach to work with a DQN instead of a $q$-table, allowing it to operate in 3D graphical environments.\n\n\n\n\n\n\n\\section{The Minecraft Environment}\n\n\nMinecraft is a 3D game in which players can move, explore and build. \nMinecraft allows for continuous movement, is partially observable, and is mostly deterministic and dynamic.\nWhile Minecraft uses a graphical aesthetic of blocks and thus simpler than photo-realistic graphical environments, it easily accommodates large, easily modifiable, open worlds. \nThis makes Minecraft a valuable stepping stone to more real environments;\nother 3D game environments tend to have smaller state spaces.\n{\\em Project Malmo} \\cite{malmo} provides an API for reinforcement learning agents to control virtual characters within a Minecraft map.\n\nWhereas many games and many research projects using Minecraft seek to reduce perceptual aliasing, we designed a 20x20 Minecraft level to have a mixture of high perceptual aliasing and low perceptual aliasing. \nSee Figure~\\ref{fig:minecraft}.\nHigh walls around the border of the map and a hedge corridor create areas of high perceptual aliasing. \nThree colored buildings and a stationary non-player character provide areas of low perceptual aliasing.\nThe map is mirrored across the x-axis, the center area is identical in each cardinal direction and the majority of surfaces share repeating textures.\nThe map schematic is shown in Figure~\\ref{fig:map}.\n\nThe agent must interact with the environment to learn a policy to optimally reach the non-player character on the far side of the map.\nThe task approximates the common situation found in many 3D role-playing games where the player must fund a non-player character to deliver an item or receive a quest. \n\nFor purposes of this paper, only 4 discrete actions are permitted: move forward, turn left, turn right and a complete 180-degree turn. \nThe agent reward signal is as follows. \nThe agent receives $-0.5$ points for each action taken and $+15000$ for reaching the non-player character on the far side of the map.\nThere are two intermediate rewards of \n $+1500$ given when the agent reaches a $x$-coordinate corresponding to the entrance of the hedge corridor and when the agent reaches an $x$-coordinate corresponding to the exit from the hedge corridor.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.65\\columnwidth]{malmomap}\n\\caption{Environment map with 3D structure and key points labeled.}\n\\label{fig:map}\n\\end{figure}\n\n\n\n\n\n\n\\section{Reinforcement Learning Preliminaries}\n\n\n\n\nA Markov Decision Process \\cite{SuttonBarto}, MDP, is a model used to describe a potentially stochastic sequential decision making problem. \nAn MDP can be expressed as a tuple $\\langle S,A,P,R,\\gamma\\rangle$ which contains:\n\\begin{itemize}\n\\item A set of possible world states $\\sset$\n\\item A set of possible agent actions $\\aset$\n\\item A transition function $\\Tdef$ expressing the probability $Pr(s_{t+1} \\mid s_{t}, a_{t})$ of transitioning to state $s_{t+1}$ from state $s_t$ when action $a_t$ is performed.  \n\\item A reward function $R(s,a): \\sset \\bigtimes \\aset \\rightarrow \\mathbb{R}$\n\\item A discount factor $\\gamma \\in [0,1]$ that prioritizes immediate rewards over longer-term rewards\n\\end{itemize}\n\nThe solution to a MDP or POMDP is a policy $\\pi:\\sset \\rightarrow a$, a function that dictates the best action an agent can take in any world state in order to maximize future rewards. \nReinforcement Learning is a technique that learns an optimal policy for a MDP\nwhen the transition function is not given.\n\n$Q$-Learning \\cite{qlearning} is an approach to reinforcement learning in which an agent uses trial-and-error in a simulated environment to converge on the utilities of performing certain actions in certain states, $Q(s,a)$.\n$Q(s,a)$ is iteratively updated in each learning phase as follows:\n\\begin{equation}\n\\label{bellman}\n\\begin{split}\nQ(s_t,a_t)=&Q(s_t,a_t)+\\\\\n& \\alpha[R(s_{t+1})+\\gamma\\underset{a'}{\\max}\\: Q(s_{t+1},a')-Q(s,a)]\n\\end{split}\n\\end{equation}\nwhere $t$ is the previous time step and $\\alpha$ is the learning rate.\nThe agent's policy is thus $a=argmax_{a'} Q(s,a')$.\n\nFor environments in which the states cannot be enumerated and maintaining a $q$-table is not possible, Deep Q-Learning \\cite{dql} learns a deep neural network to approximate $Q(s,a)$.\nThis is typically done by training a multi-layered convolutional neural network (CNN) to identify features that are predictive of state-action utilities. This is also known as a deep $Q$ network (DQN) \\cite{mnih2}.\nTo train a DQN, the neural network attempts to predict $Q(s,a)$ given the current pixel array and the action just performed; the loss is computed as the difference between the current prediction and the one-step Bellman update in Equation~\\ref{bellman}.\n\n\n\n\n\n    \n    \n\n\n    \n\n\n\n\n\n\n\n\n\n\n\n\\section{Methods}\n\n\nTwo action-advice augmented reinforcement learning algorithms are used.\nThe first is based on the {\\em Feedback Arbitration} (FA) algorithm~\\cite{zhiyu}. \nAn addition to the standard reinforcement learning training loop, an {\\em arbiter}, continually assesses feedback consistency and quality versus the confidence it has in its learned policy.\nFeedback Arbitration is already designed to work in 3D virtual environments similar to Minecraft though on much simpler maps.\nThe second is the {\\em Newtonian Action Advice} (NAA) algorithm~\\cite{krening}. \nNewtonian Action Advice was mainly assessed in 2D environments and uses tabular $q$-learning. \nTo focus on the effects of perceptual aliasing, the mechanism for measuring human trainer consistency in FA is not used; only the confidence check\nis implemented. \nThese algorithms and our modifications for perceptual aliasing experiments are described in further detail in the next sections.\n\nA synthetic oracle is used in place of a human to overcome large sample size requirements, advice frequency variability and accuracy variability.\nThese dimensions cannot be controlled for using human trainers, however see \\cite{krening} for human-subject experiments in grid worlds. \nThe synthetic oracle has ``perfect'' knowledge of maze environment and always gives optimal advice unless explicitly tuned to have lower accuracy, in which case it gives random advice some percentage of the time. \nAnother parameter determines how often the oracle provides advice.\n\n\\subsection{The Feedback Arbitration Agent}\n\nThe Feedback Arbitration (FA) agent is a standard DQN agent with the following updates. \nWe assume that the oracle provides advice intermittently. \nAction advice is queued up in a pending advice array. Queued advice instances dequeue after some time.\nThe agent either (a)~explores using a random action, (b)~exploits its $Q$-network by picking the action it believes has the highest utility in the current state, or (c)~consults the pending advice array.\nThe agent chooses a random action according to the standard $\\epsilon$-greedy strategy.\nOtherwise, it measures its confidence in its $Q$-network and, if the confidence is low, chooses to consult the action advice instead.\nThen the $Q$-network confidence score is computed as a cost function such that low confidence incurs a high cost:\n\n\\smallskip\n\\begin{equation}\nrelativeCost = \\frac{-1}\n     {\\ln\\sqrt(\\frac{\\min_{a\\in A(s)} L_a}\n     {L_{\\mathtt{max}}})-1}\n\\end{equation}\n\\smallskip\n\n\\noindent\nwhere $L_a$ is the loss value for the predicted activation of action $a$ in the current state  and $L_{\\mathtt{max}}$ is the highest loss observed by the DQN thus far.\nIf $relativeCost(l) <= 0.25$ the agent acts using policy output. \nOtherwise, the agent utilizes the pending action advice.\n\n\n\n\\subsection{The Newtonian Action Advice Agent}\n\nNewtonian Action Advice (NAA) relies on the same arbitration mechanism as FA but adds one key feature. \nInstead of atomic advice events, NAA persists advice input across timesteps. \nA ``friction''  \nparameter specifies a count of timesteps \nduring which the action advice is recited by the agent back to itself.\nIt is shown \\cite{krening} that methods which persist human feedback can significantly reduce the cognitive load on human collaborators. \n\nThe original implementation of NAA implements a confidence decay function for discounting advice as a function of time since it was given.\nThat decay function was not replicated herein. \n\nNAA was originally designed for grid-worlds where there is a one-to-one correspondence between actions and advice.\nIn a grid world advice is given as {\\em up}, {\\em down}, {\\em left}, or {\\em right} and that exactly corresponds to the actions the agent can perform. \nIn Minecraft, the agent's actions are {\\em walk forward}, {\\em turn left}, {\\em turn right}, and {\\em turn around}.\nNewtonian action advice does not make sense for turn actions---it rarely makes sense to continue turning once advised to do it once.\nIn our implementation of NAA, advice is provided in the form of movement in the cardinal directions: {\\em north}, {\\em south}, {\\em east}, or {\\em west}.\nThe agent converts this to instructions relative to its orientation; \nif the agent is not oriented to face the cardinal direction of the movement action advice, it turns to that orientation and then moves forward for $friction$ time steps.\n\n\n\n\n\n\n\\section{Experiments}\n\nA deep $Q$-network architecture is shared by the Feedback Arbitration and Newtonian Action Advice agents. \nA four-layer convolutional neural network consumes state $s$ as a matrix consisting of the 50 most recent frames of the game. \nPixel values are single channel (greyscale) to constrain model memory requirements. \nAfter each convolutional layer we employ a $3x3$ kernel, $relu$ activation, and batch normalization.\nWe add a max pooling layer after each batch normalization. \nFinal dense layers output a probability distribution across 4 different action types. \nWe set the discount factor (gamma) to 0.95. \nWe use the Adam optimizer.\nWe use RMS logarithmic error as the loss function.\n\n\nBecause human oracle behavior is hypothesized to vary significantly, we simulate oracle advice with a synthetic oracle that is parameterized along two dimensions:\n\\begin{itemize}\n\\item {\\em Frequency of advice}, which can be high (advice given ~5\\% of the time) or low (advice given ~1\\% of the time)\n\\item {\\em Oracle accuracy}, which can be high (100\\% accurate) or low (50\\% accuracy)\n\\end{itemize}\nThis establishes four experimental conditions: high-frequency high-accuracy (HFHA), high-frequency low-accuracy (HFLA), low-frequency high-accuracy (LFHA), and low-frequency low-accuracy (LFLA).\nOur experiments thus pinpoint four possible points on two dimensions that a human oracle might inhabit. \nThey were chosen to be extreme to make it easier to observe how our agent behaves under differing potential human oracle behaviors.\nIn addition, a baseline RL agent using the same DQN architecture is included.\nThe baseline RL agent ignores all oracle advice.\n\n\n\n\n\n\\begin{figure}[t]\n\\centering{\n\\includegraphics[width=0.9\\columnwidth]{all-max}}\n\\caption{Agent performance for all techniques as a 10-episode moving average of the max score across training runs across all 4 conditions.}\n\\label{fig:results-max}\n\\end{figure}\n\n\\begin{figure}[t]\n\\centering{\n\\includegraphics[width=0.9\\columnwidth]{all-mean}}\n\\caption{Agent performance for all techniques as a 10-episode moving average of the mean score across training runs across all 4 conditions.}\n\\label{fig:results-mean}\n\\end{figure}\n\n\\begin{figure}[t]\n\\centering{\n\\includegraphics[width=0.9\\columnwidth]{hfhalfla.png}}\n\\caption{Mean performance of FA vs NAA when varying accuracy and frequency of advice.}\n\\label{fig:results-varied}\n\\end{figure}\n\n\n\nFor each episode, the agent terminates after 1,000 actions (turns inclusive), or when successfully reaching the objective.\nEach agent (RL, FA, NAA) was trained for at least 250 episodes under each of our four synthetic oracle conditions. \nThis was repeated for at least 5 sessions per technique. \n\nIn initial testing, the baseline RL agent struggled to converge when the terminating action count was set to 1,000. \nWe therefore allow the RL agent to terminate after 1,500 actions to compensate, receiving 50\\% more observations than the other techniques. \n\nThe NAA agent is given a very conservative friction parameter, $friction=2$.\n\n\n\\section{Results}\n\n\n\nWe measure agent performance as the score achieved, which is identical to reward received.\nFigure~\\ref{fig:results-max} shows the maximum reward achieved plotted against training episode.\nFigure~\\ref{fig:results-mean} shows the mean reward achieved plotted against training episode.\nAs expected, the FA and NAA agents converge much faster than the baseline RL.\nNAA converges faster than FA. \nThis latter result is likely due to the intuition built into NAA that human trainer advice is intended to be continuous unless superseded.\n\n\n\n\n\n\n\n\n\\begin{figure*}[t]\n\\centering{\n\\includegraphics[width=0.85\\textwidth]{heatmapresize}}\n\\caption{Mean accumulated actions across techniques and the KL Divergence between FA and NAA.}\n\\label{fig:heat-map}\n\\end{figure*}\n\n\n\\subsection{Frequency and Accuracy Variation}\n\nFour different conditions are established to assess FA and NAA sensitivity to varying advice frequency and oracle accuracy.\nResults are shown in Figure~\\ref{fig:results-varied}.\nAs expected, agents that receive high-accuracy advice outperform agents that receive low-accuracy advice. \nLikewise, agents that receive high-frequency advice outperform agents that receive advice less often.\nNAA  dominates FA under the ideal circumstances of high advice frequency and high oracle accuracy.\n\nNo significant difference was observed in per-episode mean advice count for high-frequency (HF) conditions ($M_{NAA}$= 73, $M_{FA}$= 76). \nPer-episode, NAA exhibited a slightly higher reliance on advice across all conditions combined ($M_{NAA}$= 57, $M_{FA}$= 42).\nBoth agents received the same numbers of advice, but only access the advice when policy confidence is low.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nOne critique of reinforcement learning in games is that the policy is over specialized to a specific map.\nWe conducted an additional experiment in which we trained an agent for 250 episodes on the map in Figure~\\ref{fig:map}---long enough for NAA and FA agents to converge.\nWe then continued training on the map rotated 90-degrees.\nThe NAA agent was able to re-converge in less than 40 episodes, demonstrating that filters within its $Q$-network were general enough to transfer.\n\n\\subsection{Robustness to Aliasing}\n\nWe hypothesize that Feedback Arbitration and Newtonian Action Advice will be less susceptible to perceptual aliasing in the environment. \nOur environment is specifically designed to have locations with high and low perceptual aliasing. \nHigh aliasing in particular happens around the walls of the map and in the hedge corridor in the middle.\n\n\n\nWe can inspect the different agents' robustness to aliasing by compiling a heatmap in which we count the number of times each agent visits a particular location in the map. \nIn Figure~\\ref{fig:heat-map} the first three heatmaps show location visit frequency for the baseline RL agent, FA agent, and NAA agent, respectively.\nThe baseline RL agent spends the most time near the start and in the first half of the hedge corridor, suggesting it does struggle with this area of high perceptual aliasing.\nIt also appears to trace along the walls of the hedge corridor, spending a lot of time in similar looking states.\nThe FA agent shows a similar pattern, though it does manage to converge on a policy whereas the baseline RL still struggles after hundreds of episodes.\n\nThe heatmap for the NAA agent is qualitatively different.\nIts visits to locations within the hedge corridor are much more uniform, suggesting that it navigates the high perceptual aliasing area with less difficulty. It also spends much less time fixated on walls.\nConsequently, it spends a larger portion of its overall training time exploring locations near the goal. \nThe right-most image shows the difference between the NAA agent's heatmap and the FA agent's heatmap;\nthe bluer the location, the more likely that location will be visited by the NAA agent instead of the FA agent.\nThe difference was computed using Kullback-Leibler Divergence between matrices containing the FA and NAA action counts.\n\n\n\n\n\n\n\n\n\n\n\\subsection{Discussion}\n\nWe conclude that Newtonian Action Advice improves agent learning performance inside the high perceptual aliasing areas.\nNot only does the NAA agent converge on a policy that obtains high reward faster than the FA agent, but qualitatively the NAA agent spends a larger fraction of its training time moving decisively through areas of high perceptual aliasing.\nThis is achieved despite the fact that \nboth FA and NAA receive the same advice, with the same oracle accuracy, and at the same frequency of advice. \nNAA is able to use this advice more effectively by assuming any advice remains valid for a period of time (2 timesteps in the experiments in this paper).\nCoping with aliasing requires the agent leverage long-term memory to learn that it is okay to see nearly indistinguishable perceptual features for a number of time steps.\nThe oracle is acting as a memory aide until the learned policy overcomes confusion due to aliasing, which manifests itself as random exploration due to policy error.\n\nThe FA agent and NAA agent receive the same rate of advice from the synthetic oracle. \nHowever, the FA agent accessed the advice less often.\nOn first glance it seems FA might require less advice. \nHowever, we hypothesize that FA finds advice in high-aliasing situations less useful and is less reliant on oracle advice.\nThat is, it gains high confidence in its policy when tracing along the edges of the maze. Oracle advice to move into the middle of the hedge corridor does not help it disambiguate error in its $Q$-network.\nThe NAA agent, on the other hand, uses the given advice more often---confidence in its policy can be low due to aliasing though the recital of advice helps it cross highly aliased areas so that it ultimately converges on a robust policy.\n\n\n\\section{Conclusions}\n\nPerceptual aliasing frequently occurs in real world environments, whereas computer games may often be designed to reduce aliasing. \nAs demonstrated in this paper, it is relatively straightforward to create a task which is extremely difficult for a deep reinforcement learning agent to learn to solve. \nWhile we experiment with standard deep $Q$-networks, one can extrapolate that this will be true even for more sophisticated reinforcement learning algorithms. \nLearning from Advice (LfA) leverages the knowledge of a human teacher to overcome inefficiencies in learning due to perceptual aliasing. \nIt is challenging to keep cognitive load low enough such that the human does not suffer from degraded human factors when interacting with the agent.\nA human teacher may not be able to engage with the training agent and may give incorrect advice -- agents cant and must be robust to this.\nWe show that the Newtonian Action Advice LfA agent can be adapted to 3D environments and converges on a robust policy quickly despite the presence of perceptual aliasing, suggesting that it is using the oracle feedback more effectively than alternative techniques investigated.\n\n\n\n\n\n \\section*{Acknowledgements}\n \n This material is based upon work supported by the Office of Naval Research (ONR) under Grant \\#N00014-14-1-0003.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIn this paper, we consider the problem of designing computationally efficient\nand truthful mechanism for multi-parameter mechanism design problems in\nthe Bayesian setting. \n\nSuppose a major Internet search service provider wants to sell multiple\nadvertisement slots to a number of companies. From the history of previous \ntransactions, we can estimate a prior distribution of each company's\nvaluation of the advertisement slots. What mechanism shall\nthe search service provider use to obtain good social welfare, or good revenue?\n\nThis is a typical multi-parameter mechanism design problem.\nIn general, we consider the scenario in which a principal wants to sell \na number of different services to multiple \nheterogeneous strategic agents subject to some feasibility constraints (e.g.\ntotal cost of providing these services must not exceed the budget), so\nthat some desired objective (e.g. social welfare, revenue, residual surplus) \nis achieved. If we interpret this as simply a combinatorial optimization\nproblem, then there exists approximation algorithms for many of these problems.\nAnd the approximation ratios of many of these algorithms are tight subject\nto certain computational complexity assumptions. However, if we wants to \ndesign protocols of allocations and setting prices in order to\nachieve the desired objective in the equilibrium strategic behavior of the \nagents, we usually have much worse approximation ratio.\nTherefore, it is natural to ask the following question:\n\\begin{quote}\n\t{\\em Can we convert any algorithm into a truthful mechanism while \n\tpreserving the performance, say, social welfare?}\n\\end{quote}\n\nUnfortunately, from previous work we learn that this is impossible for\nsome problems. Papadimitriou et al. \n\\cite{papadimitriou2008hardness} showed the first significant gap between\nthe performance of deterministic algorithms and deterministic truthful\nmechanisms via the Combinatorial Public Project problem.\n\n\\paragraph{Bayesian setting.}\n\nThe standard game theoretic model for incomplete information is the \nBayesian setting, in which the agent valuations \nare drawn from a publicly \nknown distribution. The standard solution concept in this setting is \n{\\em Bayesian-Nash Equilibrium}. In a Bayesian-Nash equilibrium, each player \nmaximizes its expected payoff by following the strategy profile\ngiven the prior distribution of the agent valuations.\n\n\\medskip\n\nIn this paper, we will consider multi-parameter welfare-preserving\nalgorithm\/mechanism reductions \nin the Bayesian setting, and weaken truthfulness\nconstraint from Incentive Compatibility (IC) to Bayesian Incentive \nCompatibility (BIC), which means truth telling is the equilibrium strategy\nover random choice of the mechanism as well as the random realization of \nthe other agent valuations. In many real world applications\nsuch as online auctions, AdWords auctions, spectrum auctions etc., the\navailability of data of past transactions make it possible to obtain\ngood estimation of the prior distribution of the agent valuations.\nThus, revisiting the algorithm\/mechanism reduction problem \nin the Bayesian setting is of both theoretical and practical importance.\n\nHartline and Lucier \\cite{hartline2009bayesian} studied \nthis problem in the single-parameter\nsetting. They showed a brilliant black-box reduction from any approximation \nalgorithm\nto BIC mechanism that preserves the performance with respect to social welfare\nmaximization. In this paper, we prove that similar reduction also exists for\nthe realm of multi-parameter mechanism design for social welfare! Moreover,\nwe can also obtain BIC mechanism for revenue or residual surplus via some\nvariants of our black-box reduction.\n\n\\paragraph{Our results and technique.} \nOur main result is a black-box reduction \nthat converts algorithms into BIC mechanisms with essentially \nthe same social welfare for arbitrary multi-parameter mechanism design problem\nin the Bayesian setting. More concretely, given an \nalgorithm $\\mathcal{A}$ that provides $SW^\\mathcal{A}$ social welfare,\nthe reduction provides a mechanism that gives \n$SW^\\mathcal{A} - \\epsilon$ social welfare and is $\\epsilon$-BIC.\nThe running time is polynomial in the input size and $1\/\\epsilon$. \nThis resolves an open problem in \\cite{hartline2009bayesian}. The key idea is \nto decouple the reported valuations and the input valuations for the algorithm $\\mathcal{A}$. \nWhen the reported valuations are $v_1, v_2, \\dots, \nv_n$, we will manipulate the valuations via some carefully designed \nintermediate algorithms $\\mathcal{B}_1, \\dots, \\mathcal{B}_n$, and use allocation\n$\\mathcal{A}(\\mathcal{B}_1(v_1), \\dots, \\mathcal{B}_n(v_n))$. \nWe prove that there exist intermediate algorithms\n$\\mathcal{B}_1, \\dots, \\mathcal{B}_n$ so that there are prices that achieve BIC.\nUnder certain conditions, the marginal loss factor in social welfare can be made multiplicative.\n\nAs an application of this reduction, we get a $(\\frac{1}{2} - \\epsilon)$-approximate and \n$\\epsilon v_{max}$-BIC mechanism for social welfare maximization in \ncombinatorial auctions with sub-additive agents. \nFor the more restricted case of fractionally\nsub-additive agents, we obtain $(1 - \\frac{1}{e} - \\epsilon)$-approximate\nmechanism. \n\n\\paragraph{Related work.}\nThe problem of maximizing social welfare against strategic agents is one of\nthe oldest and most famous problems in the area of mechanism design. It has\nbeen extensively studied by the economists in both Bayesian and prior-free\nsetting without considering computational power constraint. The celebrated\nVCG mechanism \\cite{clarke1971multipart, groves1973incentives,\nvickrey1961counterspeculation} which guarantees optimal social welfare and\nincentive compatibility is one of the most exciting results \nin this domain. However, implementing the VCG mechanism is NP-hard\nin general. This is one of the reasons that VCG mechanism is rarely used in\npractice despite of its lovely theoretical features.\n\nIn the past decade, computer scientists introduced many novel techniques in\nthe prior-free setting to design computationally efficient mechanisms\nthat provide incentive compatibility and\/or good approximation to\noptimal social welfare for various families of valuation functions.\n\nOn the one hand, \nDobzinski, Nisan and Schapira \\cite{dobzinski2006truthful} proposed\npoly-time mechanisms which achieved $\\Omega(1 \/ \\sqrt{n})$-approximation for\ngeneral agents and $\\Omega(1 \/ \\log^2 n)$-approximation for sub-modular \nagnets. Dobzinski \\cite{dobzinski-two} later proposed a truthful mechanism \nwhich achieved an improved\n$\\widetilde{\\Omega}(1 \/ \\log n)$-approximation for a strictly broader class of\nsub-additive agents.\n\nOn the other hand, \nif we focus on the algorithmic problem of maximizing social welfare\nassuming all valuations are truthfully revealed, then the algorithm by\nDobzinski, Nisan and Schapira \\cite{dobzinski2005approximation} \ngave $\\Omega(1 \/ \\sqrt{n})$-approximation for\ngeneral case and $\\Omega(1 \/ \\log n)$-approximation for sub-additive agents.\nThe latter approximation ratio is later improved to $\\frac{1}{2}$\nfor sub-additive agents \\cite{feige2006maximizing} and $(1 - \\frac{1}{e})$ for\nthe more restricted  class of fractionally sub-additive agents \n\\cite{dobzinski-two,feige2006approximation}.\n\nThe above results suggest that there exists a gap between the performance of\nthe best poly-time algorithms and that of the best poly-time and incentive \ncompatible mechanism.\nAs an effort to study the relation between designing algorithms and\ndesigning truthful mechanisms with good approximation ratio, Lavi and Swamy\n\\cite{lavi2005truthful} proposed a meta-mechanism that converted\nstrong rounding algorithms for the standard LP of social welfare maximization\ninto IC mechanisms. However,\ntheir approach required the rounding algorithm to work for arbitrary valuation\nfunctions. This requirement prevents their technique to get good\napproximation beyond cases of general valuations and additive valuations \n(via a different linear program). But the more interesting classes of \nvaluations (e.g. sub-additive valuations and sub-modular valuations) lies between these two\nextremes. Another notable attempt on reducing IC mechanism design to algorithm design\nis the very recent work by Dughmi and Roughgarden~\\cite{dughmi2010black}. They proved that\nfor any packing problem that admitted an FPTAS, there was an IC mechanism that was also an FPTAS.\n\nMost of the previous effort from computer scientists has focused on the\nprior-free setting. Until very recently, there has been a few work\nthat brought more and more Bayesian analysis into the field of algorithmic\nmechanism design. Hartline\nand Lucier \\cite{hartline2009bayesian} gave a black-box reduction that\nconverted any Bayesian approximation algorithm into a Bayesian incentive\ncompatible mechanism that preserved \nsocial welfare in the single parameter domain.\nBhattacharya et al. \\cite{bhattacharya2009budget} studied the revenue\nmaximization problem for auctioning heterogeneous items when the valuations\nof the agents were additive. Their result gave constant approximation in\nthe Bayesian setting even when the agents had public known budget constraints.\nChawla et al. \\cite{chawla2009multi} considered the revenue maximization \nproblem in the multi-dimensional multi-unit auctions. \nThey introduced mechanism that\ngave constant approximation in various settings via sequential posted pricing.\n\nFinally, in concurrent and independent work, Hartline et al. \\cite{jason2010} \nstudy the relation of algorithm and mechanism in Bayesian setting and propose\nsimilar reduction. In the discrete support setting that is considered \nin this paper, they use essentially the same reduction. However,\ntheir work achieves perfectly BIC instead of $\\epsilon$-BIC. They also extend\nthe reduction to the more general continuous support setting.\n\n\\section{Preliminaries}\n\n\\subsection{Notations.}\n\nWe use $\\{x_i\\}_{1 \\leq i \\leq n}$ to denote an array of \nsize $n$. We also use the natural extension of this notation for \nmulti-dimensional arrays. \nWe will use bold font $\\bm{x}$ to denote a vector $(x_1, \\dots, x_n)$.\nWe let $\\Delta(S)$ denote the set of distributions over the elements in a set \n$S$. For a random variable $x$, we let $\\e{}{x}$\ndenote its expectation and let $\\sd{}{x}$ denote its standard deviation. \nWe use subscripts to represent the random choices over which we consider\nthe expectation and variance. For instance, $\\e{y \\sim F}{x}$\nis the expectation of $x$ when $y$ is drawn from distribution\n$F$. We sometimes use $\\e{y}{x}$ for short when the distribution $F$ is \nclear from the context.\n\n\\subsection{Model and definitions.}\n\nIn this section, we will formally introduce the model in this \npaper. We study the general multi-parameter mechanism design problems. In a\nmulti-parameter mechanism design problem, a principal wants to sell a set of \nservices to multiple heterogeneous agents in order to optimize the desired \nobjective (e.g. social welfare, revenue, residual surplus, etc.). \nA Bayesian multi-parameter mechanism design problem with $n$ agents is defined \nby a tuple $\\langle \\bm{I}, \\bm{J}, \\bm{V}, \\bm{F} \\rangle$. \n\n\\begin{itemize}\n\t\\item $\\bm{I} = (I_1, \\dots, I_n)$:  \n\t\tThe set of services that the principal wants to sell to the agents. \n\n\t\tSince we can impose arbitrary feasibility constraints on the \n\t\tallocations, \n\t\twe assume without loss of generality that the services are \n\t\tpartitioned into $n$ disjoint sets $I_1, \\dots, I_n$ so that the \n\t\tservices in $I_i$ only aim for agent $i$, and each agent $i$ is \n\t\tinterested in any one of these services.\n\n\t\\item $\\bm{J} \\subseteq I_1 \\times \\dots \\times I_n$: \n\t\tThe set of feasible allocations.\n\t\t\n\t\\item $\\bm{V} = V_1 \\times \\dots \\times V_n$:\n\t\tThe space of agent valuations.\n\t\t\n\t\tWe let $V_i \\subseteq \\mathbb{R}^{I_i}$ denote\n\t\tthe set of possible valuations of agent $i$. \n\t\tWe let $v_{max}$ denote the maximal valuation, that is,\n\t\t$v_{max} = \\max_{i, v \\in V_i, S \\in I_i} v(S)$ \n\t\\item $\\bm{F} = F_1 \\times F_2 \\times \\dots \\times F_n$: The\n\t\tjoint prior distribution of the agent valuations.\n\t\t\n\t\tWe assume the prior distribution is a product distribution.\n\t\tWe let $F_i \\in \\Delta(V_i)$ denote the prior distribution\n\t\tof the valuation of agent $i$. In this paper, \n\t\twe only consider distributions with finite and polynomially large\n\t\tsupport. We will assume without \n\t\tloss of generality that the support of each distribution $F_i$ is \n\t\t$\\{v^1_i, \\dots, v^\\ell_i\\}$. Suppose $v_i \\sim F_i$, We will let\n\t\t$f_i(t)$ denote the probability that $v_i = v^t_i$.\n\\end{itemize}\n\nFor example, in the {\\em combinatorial auction}\nproblem with $n$ agents and $m$ items, we let $[m] = \\{1, 2, \\dots, m\\}$\ndenote the set of items. The set of services for each agent $i$ is\nthe set of all subsets of items, that is, $I_i = 2^{[m]}$, $1 \\leq i \\leq n$.\nThe set of feasible allocations is\n$$\\bm{J} = \\{(S_1, \\dots, S_n) : S_i \\in I_i, S_i \\cap S_j = \\emptyset\\}\n\\enspace.$$\n\nThe set of valuations, $V_i$, is the set of mappings from subset of items\n$I_i$ to $\\mathbb{R}_+$ that are monotone ($v_i(S) \\leq v_i(T)$ for $S \\subseteq T$)\nand normalized ($v_i(\\emptyset) = 0$).\nWe usually assume that the valuations in $\\bigcup_i V_i$ satisfies certain \nproperties, e.g. sub-additivity, sub-modularity, etc. \n\n\\paragraph{Algorithm.} An algorithm for a multi-parameter mechanism design \nproblem $\\langle \\bm{I}, \\bm{J}, \\bm{V}, \\bm{F}\\rangle$ is a protocol (may\nor may not be randomized) that\ntakes a realization of agent valuations $\\bm{v} \\in \\bm{V}$ as input, and\noutputs a feasible allocation $\\bm{S} \\in \\bm{J}$. \n\n\\paragraph{Mechanism.} A mechanism is an interactive protocol (may or may not\nbe randomized) between the \nprincipal and the agents so that the principal can retrieve information from\nthe agents (presumably via their bids), and determine an allocation of \nservices $\\bm{S} \\in \\bm{J}$ and a collection of prices \n$\\bm{p} = (p_1, \\dots, p_n)$. The extra challenge for mechanism design,\ncompared to algorithm design, is to retrieve genuine valuations from the\nagents and handle their strategic behavior.\n\nFor each $1 \\leq i \\leq n$, we will assume the prior distribution $F_i$ is \n{public known}. But the actual realization $v_i \\sim F_i$ is {\\em private}\ninformation of agent $i$. Each agent $i$ aims to maximizes the quasi-linear\nutility $v_i(S_i) - p_i$, where $S_i$ is the service it gets and $p_i$ is\nthe price. Thus, the agents may not reveal their genuine valuations if\nmanipulating their bids strategically can increase their utility.\n\n\\paragraph{Objectives.}\nWe will consider\nthree different objectives: social welfare, revenue, and residual surplus.\nThe expected {\\em social welfare} of a mechanism $\\mathcal{M}$ is\n$$SW^\\mathcal{M} = \\e{\\bm{v} \\sim \\bm{F}, (\\bm{S}, \\bm{p}) \\sim \\mathcal{M}(\\bm{v})}\n{\\sum_{i = 1}^n v_i(S_i)} \\enspace.$$\n\nSimilarly, we will let $SW^\\mathcal{A}$ denote the expected\nsocial welfare of an algorithm $\\mathcal{A}$.\n\n\\begin{definition}\n\tAn algorithm $\\mathcal{A}$ is $\\alpha$-approximate in social welfare for \n\ta multi-parameter mechanism design problem \n\t$\\langle \\bm{I}, \\bm{J}, \\bm{V}, \\bm{F} \\rangle$, if \n\t$SW^\\mathcal{A} \\geq \\alpha \\, \\mathsf{OPT}$.\n\\end{definition}\n\nThe expected {\\em revenue} of a mechanism is\n$$R^\\mathcal{M} = \\e{\\bm{v} \\sim \\bm{F}, (\\bm{S}, \\bm{p}) \\sim \\mathcal{M}(\\bm{v})}\n{\\sum_{i = 1}^n p_i} \\enspace.$$\n\nThe last objective, {\\em residual surplus}, was recently proposed by Hartline\nand Roughgarden \\cite{hartline2008optimal} as an alternative objective in\nthe flavour of social welfare. In the residual surplus maximization problem, \nthe principal aims to maximize the sum of the agents' utilities instead of \nthe sum of their valuations. The expected {\\em residual surplus} is\n$$RS^\\mathcal{M} = \\e{\\bm{v} \\sim \\bm{F}, (\\bm{S}, \\bm{p}) \\sim \\mathcal{M}(\\bm{v})}\n{\\sum_{i = 1}^n \\left(v_i(S) - p_i\\right)} \\enspace.$$\n\nWe will let $\\mathsf{OPT}$ denote the optimal social welfare, that is, $\\mathsf{OPT}\n= \\max_\\mathcal{M} SW^\\mathcal{M}$. Since both revenue and residual surplus are \nupper-bounded by social welfare. We will use $\\mathsf{OPT}$ as our benchmark \nfor all three objectives.\n\n\\paragraph{Solution concepts.}\n\nIdeally, a mechanism shall provide incentive for the agents to reveal their\nvaluations truthfully. In this section, we will formalize this requirement\nby introducing the game-theoretical solution concepts that we use in this \npaper.\n\n\\begin{definition}\n\tA mechanism $\\mathcal{M}$ is {\\em Bayesian incentive compatible (BIC)} if\n\tfor each agent $i$ and any two valuations $v_i, \\widetilde{v}_i \\in V_i$, \n\twe have\n\t$$\\e{\\bm{v}_{-i},(\\bm{S},\\bm{p}) \\sim \\mathcal{M}(v_i,\\bm{v}_{-i})}\n\t{v_i(S_i)-p_i} \\geq \\e{\\bm{v}_{-i},(\\bm{S},\\bm{p}) \\sim \n\t\\mathcal{M}(\\widetilde{v}_i, \\bm{v}_{-i})} {v_i(S_i)-p_i} \\enspace.$$\n\\end{definition}\n\n\\begin{definition}\n\tA mechanism $\\mathcal{M}$ is {\\em $\\epsilon$-Bayesian Incentive Compatible\n\t($\\epsilon$-BIC)} if for any agent $i$ and any two valuations\n\t$v_i, \\widetilde{v}_i \\in V_i$,\n\t$$\\e{\\bm{v}_{-i}, (\\bm{S}, \\bm{p}) \\sim \n\t\\mathcal{M}(v_i, \\bm{v}_{-i})}{v_i(S_i) - p_i} \\geq \n\t\\e{\\bm{v}_{-i}, (\\bm{S}, \\bm{p}) \\sim \n\t\\mathcal{M}(\\widetilde{v}_i, \\bm{v}_{-i})}{v_i(S_i) - p_i} \n\t- \\epsilon \\enspace.$$\n\\end{definition}\n\nOther than the above constraints of incentive compatibility, the mechanism \nshall also guarantee that the agents always get non-negative utility.\nOtherwise, the agents may choose not to participate in the \nmechanism. This is known as the {\\em individual rationality} constraint.\n\n\\begin{definition}\n\tA mechanism $\\mathcal{M}$ is {\\em individually rational (IR)}\n\tif for any realization $\\bm{v}$ of agent valuations, and any allocation\n\t$\\bm{S}$ and prices $\\bm{p}$ by the mechanism, we always have that \n\t$v_i(S_i) - p_i \\geq 0$ for all agent $i$,\n\\end{definition}\n\n\\section{Characterization of BIC mechanisms}\n\\label{sec:charbic}\n\nIn this section, we will introduce a \nnon-trivial characterization of BIC multi-parameter mechanisms via a novel\nconnection between BIC mechanisms and envy-free prices. This characterization\ninspires our reduction in the next section. \n\n\\subsection{Fractional assignment problem.}\n\nWe will first introduce the fractional assignment problems,\nwhich will play a critical role in the results of this paper,\nand a useful lemma about envy-free prices in fractional assignment problems.\n\nIn order to distinguish the notations for fractional assignment problems and \nthose for the mechanism design problems, we will use superscripts instead\nof subscripts to specify different entries of a vector for the fractional\nassignment problems. For instance, we will use\n$x^s$ to denote the $s^{th}$ entry of a vector $\\bm{x}$.\n\nLet us consider a market with $\\ell$ buyers and $m$ infinitely divisible \nproducts. \nEach buyer $s$ has a non-negative demand $\\alpha^s$, which denotes the maximal \namount of products the buyer will buy. Each product $t$ has a non-negative \nsupply $\\beta^t$, which denotes the available amount of this product in the \nmarket. For each buyer $s$ and each product $t$, we let $w^{st}$ denote the\nnon-negative value of buyer $s$ of product $t$. \n\nThe goal is to set prices for the products and to assign the products to the \nbuyers subject to the demand and supply constraints. Thus, a solution \n$(\\bm{x}, \\bm{p})$ to the fractional assignment problem \nconsists of a collection of prices $\\bm{p} = (p^1, \\dots, p^\\ell)$ \nand a feasible allocation $\\bm{x} = \\{x^{st}\\}_{1 \\leq s \\leq \\ell, 1 \\leq t \n\\leq m}$ in the polytope:\n$$\\left\\{\\bm{x} : \\forall s, \\sum_{t = 1}^m x^{st} \\leq \\alpha^s; \\,\n\\forall t, \\sum_{s = 1}^\\ell x^{st} \\leq \\beta^t; \\,\n\\bm{x} \\geq 0\\right\\} \\enspace,$$\nwhere $x^{st}$ denotes the amount of product $t$ that is \nassigned to buyer $s$.\n\n\\begin{definition}\n\tA solution $(\\bm{x}, \\bm{p})$ is {\\em envy-free} if for any $x^{st} > 0$, \n\tthen $t$ is a product that maximizes the quasi-linear utility of agent \n\t$s$, and the utility for agent $s$ is non-negative. That is, \n\t\\begin{equation}\n\t\t\\label{eq:envyfree}\n\t\t\\forall s, t : x^{st} > 0 \\Rightarrow \n\t\tw^{st} - p^t = \\max_k \\{w^{sk} - p^k\\} \\geq 0 \\enspace.\n\t\\end{equation}\n\\end{definition}\n\n\\begin{definition}\n\tAn allocation $\\bm{x}$ is {\\em market-clearing}\n\tif all demand constraints and supply constraints hold with equality\n\tThat is, \n\t$$\\forall 1 \\leq s \\leq \\ell: \\sum_{t=1}^m x^{st} = \\alpha^s ~,~\n\t\\forall 1 \\leq t \\leq m: \\sum_{s=1}^\\ell x^{st} = \\beta^t \\enspace.$$\n\\end{definition}\n\nThe social welfare maximization problem for a fractional assignment problem\nis characterized by the following linear program (P) and its dual (D).\n\\begin{align*}\n\t\\text{{\\bf (P)} Maximize} & ~ ~ \\Sigma_{s = 1}^\\ell \n\t\\Sigma_{t = 1}^m x^{st} w^{st} & & \\text{s.t.} &\n\t\\text{{\\bf (D)} Minimize} & ~ ~ \\Sigma_{s = 1}^\\ell \\alpha^s u^s + \n\t\\Sigma_{t = 1}^m \\beta^t p^t & & \\text{s.t.} \\\\\n\t\\Sigma_{t = 1}^m x^{st} & \\leq \\alpha^s & & \\forall s & \n\tu^s + p^t & \\geq w^{st} & & \\forall s, t \\\\\n\t\\Sigma_{s = 1}^\\ell x^{st} & \\leq \\beta^t & & \\forall t &\n\tu^s & \\geq 0 & & \\forall s \\\\\n\tx^{st} & \\geq 0 & & \\forall s, t &\n\tp^t & \\geq 0 & & \\forall t\n\\end{align*}\n\n\\begin{align*}\n\\end{align*}\n\nWe will introduce two useful lemmas about the connection between \nenvy-free prices and social welfare maximization for fractional\nassignment problems. These lemmas were known in different forms in the\neconomics literature \\cite{gul1999walrasian}.\n\n\\begin{lemma}\n\t\\label{lemma:assignment1}\n\tIf there exist envy-free prices $\\bm{p}$ for a market-clearing\n\tallocation $\\bm{x}$, then $\\bm{x}$ maximizes the social welfare,\n\tthat is, $\\bm{x} \\cdot \\bm{w} = \\max_{\\bm{z}} \\bm{z} \\cdot \\bm{w}$.\n\\end{lemma}\n\n\\begin{proof}\n\tSuppose there exist envy-free prices $\\bm{p}$ for \n\tan allocation $\\bm{x}$. Let $u^s = \\max_t \\left\\{w^{st}-p^t\\right\\}$. \n\tWe have that $u^s + p^t \\geq w^{st}$ for all $s, t$. So \n\t$(\\bm{u}, \\bm{p})$ is a feasible solution for the dual LP.\n\n\tMoreover, by definition of envy-freeness, we have\n\t$$\\forall s, t : x^{st} > 0 \\Rightarrow u^s = w^{st} - p^t \\enspace.$$\n\t\n\tTherefore, we get that\n\t$$\\sum_{s=1}^\\ell \\sum_{t=1}^m x^{st} w^{st} = \n\t\\sum_{s=1}^\\ell \\sum_{t=1}^m x^{st} (u^s + p^t) \n\t= \\sum_{s=1}^\\ell \\alpha^s u^s + \\sum_t \\beta^t p^t \\enspace.$$\n\n\tThe last equality holds because $\\bm{x}$ is market clearing.\n\tNotice that $\\bm{x}$ is a feasible solution to the primal LP. \n\tBy duality theorem, we get that the allocation $\\bm{x}$ maximizes \n\tthe social welfare for the fractional assignment problem.\n\\end{proof}\n \n\\begin{lemma}\n\t\\label{lemma:assignment2}\n\tIf an allocation $\\bm{x}$ maximizes the social welfare, then there exist\n\tenvy-free prices $\\bm{p}$ for the fractional assignment problem.\n\\end{lemma}\n\n\\begin{proof}\n\tSuppose the allocation $\\bm{x}$ maximizes the social\n\twelfare. Let $(\\bm{u}, \\bm{p})$ be an optimal solution to the dual\n\tLP. By complementary slackness we get that $x^{st} > 0$ only if the\n\tcorresponding dual constraint is tight, that is, $u^s + p^t = w^{st}$.\n\tTherefore, $x^{st} > 0$ implies that $w^{st} - p^t = u^s \\geq \n\tw^{sk} - p^k$ for all $k$. Thus $\\bm{p}$ is a collection of envy-free \n\tprices for the allocation $\\bm{x}$ in this fractional assignment problem.\n\\end{proof}\n\nNote that the above proof also gives a poly-time algorithm for\nfinding the welfare maximizing allocation $\\bm{x}$ and the corresponding\nenvy-free prices $\\bm{p}$ by solving the primal and dual LPs.\nMoreover, we also get that the envy-free prices $\\bm{p}$ satisfy a weak \nuniqueness in the sense that it must be part of an optimal solution for the \ndual LP.\n\n\\begin{corollary}\n\tThere exists a poly-time algorithm that computes \n\tthe welfare-maximizing market-clearing allocation and \n\tthe envy-free prices. \n\\end{corollary}\n\n\\subsection{Characterizing BIC via envy-free prices.}\n\nWe first introduce some notations that will simplify our discussion.\nGiven a mechanism $\\mathcal{M}$ for a multi-parameter mechanism design problem\n$\\langle \\bm{I}, \\bm{J}, \\bm{V}, \\bm{F} \\rangle$, we will consider the\nexpected values and expected prices for each agent when it choose a specific \nstrategy (each strategy corresponds to reporting a specific valuation).\n\nAssuming the other agents report their valuations truthfully, \nagent $i$'s expected value of the service it gets, when the \ngenuine valuation is $v^s_i$ and the reported valuation is $v^t_i$, is\n$$w^{st}_i = \\e{\\bm{v}_{-i},(\\bm{S},\\bm{p})\\sim\\mathcal{M}(v^t_i,\\bm{v}_{-i})}{\nv^s_i(S_i)} \\enspace.$$\n\nSimilarly, we let $p^i_t$ denote the expected price the mechanism would charge\nto agent $i$ if its reported valuation is $v^t_i$, that is, \n$$p^t_i = \\e{\\bm{v}_{-i},(\\bm{S},\\bm{p})\\sim\\mathcal{M}(v^t_i,\\bm{v}_{-i})}{p_i} \n\\enspace.$$\n\nBy the definition of BIC and IR, the mechanism $\\mathcal{M}$ is BIC and IR if and \nonly if for any $1 \\leq i \\leq n$ and $1 \\leq s \\leq \\ell$,\n\\begin{equation}\n\tw^{ss}_i - p^s_i = \\max_t \\{w^{st}_i - p^t_i \\} \\geq 0\n\t\\enspace. \\label{eq:bic}\n\\end{equation}\n\nThe above equation \\eqref{eq:bic} is similar to \nequation \\eqref{eq:envyfree} in the definition of envy-freeness in fractional\nassignment problem. In fact,\nthe key observation is that the above BIC condition is equivalent to the \nenvy-free condition for a set of properly chosen fractional assignment \nproblems.\n\n\\paragraph{Induced assignment problems.}\n\nFor each agent $i$, we will consider the following {\\em induced assignment \nproblem}. \nWe consider $\\ell$ virtual buyers with demands $f_i(1), \\dots, f_i(\\ell)$ \nrespectively, and $\\ell$ virtual products with supplies $f_i(1), \\dots, \nf_i(\\ell)$ respectively. For each virtual buyer $s$ and each virtual product\n$t$, let virtual buyer $s$ has value $w^{st}_i$ on virtual product $t$.\nWe will refer to this fractional assignment problem the {\\em induced\nassignment problem} of agent $i$.\n\n\\medskip\n\nLet us consider the {\\em identity allocation} $\\bm{x}_i$ defined as follows:\n$$x^{st}_i = \\left\\{\\begin{aligned}\n\t& f_i(s) & & \\text{, if } s = t \\enspace,\\\\\n\t& 0 & & \\text{, otherwise.}\n\\end{aligned}\\right.$$ \n\nWe can easily verify that \na collection of prices $\\bm{p}_i = (p^1_i, \\dots, p^\\ell_i)$ satisfies\nconstraint \\eqref{eq:bic} if and only if $\\bm{p}_i$ satisfies the envy-free\ncondition \\eqref{eq:envyfree} of the induced assignment problem of agent $i$\nwith respect to the above identity allocation.\nHence, we have the following connection between BIC mechanism and the\nenvy-free prices of the induced assignment problems.\n\n\\begin{lemma}[Characterization Lemma~\\cite{rochet1987necessary}]\n\t\\label{lemma:characterization}\n\tA mechanism $\\mathcal{M}$ is BIC if and only if in the induced assignment \n\tproblem of each agent $i$ the identity \n\tallocation $\\bm{x}_i = \\{x^{st}_i\\}_{1 \\leq s,t \\leq \\ell}$ \n\tmaximizes the social welfare, and $\\bm{p}_i = (p^1_i, \\dots, p^\\ell_i)$ \n\tare chosen to be the corresponding envy-free prices.\n\\end{lemma}\n\n\\paragraph{Comparing with Myerson's characterization.}\n\nSuppose the problem falls into the single-parameter domain. Each\nvaluation $v^s_i$ is represented by a single non-negative real number. With\na little abuse of notation, we let $v^s_i$ denote this value. Without loss\nof generality, we assume that $v^1_i > \\dots > v^\\ell_i$. We let\n$y^t_i$ denote the probability that agent $i$ would be served if the reported\nvalue was $v^t_i$. The values $\\bm{w}_i$ in the fractional assignment\nproblems of agent $i$ are $w^{st}_i = v^s_i y^t_i$ for $1 \\leq s, t \\leq \\ell$.\nMyerson's famous characterization \\cite{myerson1981optimal}\nof truthfulness in single-parameter domain\nimplied that the mechanism is BIC if and only if $y^1_i \\geq\n\\dots \\geq y^\\ell_i$. Indeed, due to rearrangement inequality, the identity \nallocation $\\bm{x}_i$ maximizes the social welfare if and only if \n$y^1_i \\geq \\dots \\geq y^\\ell_i$. Thus, the characterization lemma implies\nMyerson's characterization in the single-parameter domain.\n\n\\section{Reduction for social welfare}\n\\label{sec:socialwelfare}\n\nLemma \\ref{lemma:characterization} suggests an interesting connection between \nBIC and envy-free prices for the induced assignment problems. \nHence, given an algorithm $\\mathcal{A}$, we will manipulate the allocation by \n$\\mathcal{A}$ based on this connection in order to make \nit satisfy the condition in Lemma \\ref{lemma:characterization}.\n\n\\subsection{Main ideas.}\n\nLet us first briefly convey two key ideas in the construction of the\nwelfare-preserving reduction.\n\nThe first idea is to decouple the reported agent valuations and the \ninput agent valuations for algorithm $\\mathcal{A}$. More\nconcretely, we will introduce $n$ intermediate algorithm $\\mathcal{B}_1, \\dots,\n\\mathcal{B}_n$. Each $\\mathcal{B}_i$ will take the reported valuation $v'_i$ as\ninput, then output a valuation $\\widetilde{v}_i$ that may or may not equals \n$v'_i$. Then, we will use algorithm $\\mathcal{A}$ to compute the allocation $\\bm{S}$\nfor agent valuations $\\widetilde{v}_1, \\dots, \\widetilde{v}_n$, \nand allocate services according to $\\bm{S}$. \n\nIf we revisit the values \n$\\widetilde{\\bm{w}}_i$ in the induced assignment problem of agent $i$ after\nthis manipulation, we will get that for any $1 \\leq s, t \\leq \\ell$,\n$$\\widetilde{w}^{st}_i = \\e{\\bm{v}_{-i}, \\widetilde{\\bm{v}} \\sim \n\\mathcal{B}(v^t_i, \\bm{v}_{-i}), \\bm{S} \\sim \\mathcal{A}(\\widetilde{\\bm{v}})}\n{v^s_i(S_i)} \\enspace.$$\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=.45\\textwidth]{reduction.pdf}\n\t\\caption{High-level picture of the reduction for social welfare.\n\t$\\mathcal{B}_i$'s are intermediate algorithms for manipulating the \n\tinput of algorithm $\\mathcal{A}$.\n\t$\\widetilde{v}_i$'s are the reported valuations. $v'_i$'s are the \n\tmanipulated input valuations for algorithm $\\mathcal{A}$. $\\bm{S}$ is\n\tthe final allocation.}\n\t\\label{fig:reduction}\n\\end{figure}\n\nBy Lemma \\ref{lemma:characterization}, we need to choose $\\mathcal{B}_i$'s carefully, \nso that the identity allocations in the manipulated assignment problems are\nwelfare-maximizing allocations. However, from the above equation we can see \nthat by using $\\mathcal{B}_i$ to manipulate the $i^{th}$ valuation, we may change\nnot only the structure of the induced assignment problem of agent $i$, but the\nstructure of the induced assignment problems of other agents as well. Hence,\nwe need to handle such correlation among the induced assignment problems when\nwe choose intermediate algorithms $\\mathcal{B}_1, \\dots, \\mathcal{B}_n$.\n\nThe idea that handles this correlation is to impose an extra constraint on\neach intermediate algorithm $\\mathcal{B}_i$: if the input valuation $v'_i$ is \ndrawn from the distribution $F_i$, then the output valuation \n$\\widetilde{v}_i$ also follows the same distribution, that is, \nfor all $1 \\leq i \\leq n$ and $1 \\leq t \\leq \\ell$,\n\\begin{equation}\n\t\\label{eq:interconstraint}\n\t\\pr{v'_i \\sim F_i, \\widetilde{v}_i \\sim \\mathcal{B}_i(v'_i)}{\\widetilde{v}_i =\n\tv^t_i} = f_i(t) \\enspace.\n\\end{equation}\n\t\nWith this extra constraint, the values $\\widetilde{\\bm{w}}_i$ after the \nmanipulation in the induced assignment problem of agent $i$ becomes\n\\begin{eqnarray*}\n\t\\widetilde{w}^{st}_i & = & \\e{\\bm{v}_{-i} \\sim F_{-i}, \n\t\\widetilde{\\bm{v}}\\sim\\mathcal{B}(v^t_i, \\bm{v}_{-i}), \\bm{S} \\sim \n\t\\mathcal{A}(\\widetilde{\\bm{v}})}{v^s_i(S_i)} \\\\\n\t& = & \\e{\\widetilde{\\bm{v}}_{-i} \\sim F_{-i}, \\widetilde{v}_i \\sim \n\t\\mathcal{B}_i(v^t_i), \\bm{S} \\sim \\mathcal{A}(\\widetilde{\\bm{v}})}{v^s_i(S_i)} \\\\\n\t& = & \\e{\\bm{v}_{-i} \\sim F_{-i}, \\widetilde{v}_i \\sim \n\t\\mathcal{B}_i(v^t_i), \\bm{S} \\sim \\mathcal{A}(\\widetilde{v}_i, \\bm{v}_{-i})}\n\t{v^s_i(S_i)} \\enspace.\n\\end{eqnarray*}\n\nThus, from the Bayesian viewpoint of agent $i$, the intermediate algorithms\n$\\mathcal{B}_{-i}$ of other agents are transparent. This property enables us to \nmanipulate the structure of each assignment problem separately.\n\n\\subsection{Black-box reduction.}\n\nGiven an algorithm $\\mathcal{A}$, the black-box reduction for social welfare will \nconvert algorithm $\\mathcal{A}$ into the following mechanism $\\mathcal{M}_\\mathcal{A}$:\n\n\\begin{enumerate}\n\t\\item For each agent $i$, $1 \\leq i \\leq n$ \n\t\t\\hspace*{\\fill} {\\bf (Pre-computation)}\n\t\t\\begin{enumerate}\n\t\t\t\\item Estimate the values\n\t\t\t\t$\\bm{w}_i = \\{w^{st}_i\\}_{1 \\leq s, t \\leq \\ell}$ \n\t\t\t\tof the induced assignment problem of $i$ with respect to \n\t\t\t\talgorithm $\\mathcal{A}$. Let \n\t\t\t\t$\\hat{\\bm{w}}_i = \\{\\hat{w}^{st}_i\\}_{1 \\leq s, t\\leq \\ell}$\n\t\t\t\tdenote the estimated values.\n\t\t\t\\item Find the social welfare maximizing allocation \n\t\t\t\t$\\bm{x}_i = \\{x^{st}_i\\}_{1 \\leq s, t \\leq \\ell}$\n\t\t\t\tand the corresponding envy-free prices \n\t\t\t\t$\\bm{p}_i = (p^1_i, \\dots, p^\\ell_i)$ for the induced \n\t\t\t\tassignment problem of agent $i$ with estimated values.\n\t\t\\end{enumerate}\n\t\\item Manipulate the valuations with intermediate algorithms\n\t\t$\\mathcal{B}_i$, $1 \\leq i \\leq n$, as follows:\n\t\t\\hspace*{\\fill} {\\bf (Decoupling)}\n\t\t\\begin{enumerate}\n\t\t\t\\item[] Suppose the reported valuation of agent $i$ is\n\t\t\t\t$v'_i = v^s_i$, $1 \\leq i \\leq n$.\n\t\t\t\tLet $\\widetilde{v}_i = \\mathcal{B}_i(v'_i) = v^t_i$ \n\t\t\t\twith probability $x^{st}_i \/ f_i(s)$ for $1 \\leq t \\leq \\ell.$\n\t\t\\end{enumerate}\n\t\\item Allocate services according to $\\mathcal{A}(\\widetilde{\\bm{v}})$. \n\t\t\\hspace*{\\fill} {\\bf (Allocation)}\n\t\t\\begin{enumerate}\n\t\t\t\\item Let $\\bm{S} = (S_1, \\dots, S_n)$ denote the allocation by\n\t\t\t\talgorithm $\\mathcal{A}$ with input $\\widetilde{\\bm{v}}$.\n\t\t\t\\item For each agent $i$, suppose the reported valuation is \n\t\t\t\t$v'_i = v^s_i$ and the manipulated valuation is \n\t\t\t\t$\\widetilde{v}_i = v^t_i$, charge agent $i$ with price \n\t\t\t\t$p^t_i v^s_i(S_i) \/ \\hat{w}^{st}_i$.\n\t\t\\end{enumerate}\n\\end{enumerate}\n\nThe following theorem\nstates that this reduction produces BIC while preserving the performance\nwith respect to social welfare.\n\n\\begin{theorem}\n\t\\label{thm:welfarereduction}\n\tSuppose $\\mathcal{A}$ is an algorithm for a multi-parameter mechanism design\n\tproblem $\\langle \\bm{I}, \\bm{J}, \\bm{V}, \\bm{F} \\rangle$.\n\t\\begin{enumerate}\n\t\t\\item If the estimated values $\\hat{\\bm{w}}_i$ are \n\t\t\taccurate, then mechanism $\\mathcal{M}_\\mathcal{A}$ is BIC, IR, and \n\t\t\tguarantees at least $SW^\\mathcal{A}$ of social welfare. \n\t\t\\item If the estimated values $\\hat{\\bm{w}}_i$ satisfy that for any \n\t\t\t$1 \\leq s, t \\leq \\ell$, \n\t\t\t$\\hat{w}^{st}_i \\in [(1 - \\epsilon) w^{st}_i, \n\t\t\t(1 + \\epsilon) w^{st}_i]$, \n\t\t\tthen mechanism $\\mathcal{M}_\\mathcal{A}$ is $4 \\epsilon v_{max}$-BIC, IR, \n\t\t\tand guarantees at least $(1 - 2\\epsilon) \\cdot SW^\\mathcal{A}$ of \n\t\t\tsocial welfare.\n\t\t\\item If the estimated values $\\hat{\\bm{w}}_i$ \n\t\t\tsatisfy that for any $1 \\leq s, t \\leq \\ell$,\n\t\t\t$\\hat{w}^{st}_i \\in [w^{st}_i - \\epsilon, \n\t\t\tw^{st}_i + \\epsilon]$, then mechanism $\\mathcal{M}_\\mathcal{A}$ is \n\t\t\t$4 \\epsilon$-BIC, IR, and guarantees at least \n\t\t\t$SW^\\mathcal{A} - 2 n \\epsilon$ of social welfare.\n\t\\end{enumerate}\n\\end{theorem}\n\nLet us illustrate the proof of part 1. The proofs of the other two parts are \ntedious and simple calculations along the same line. We will omit these proofs\nin this extended abstract.\n\n\\begin{proof}\n\tWe consider the case when the estimated values $\\hat{\\bm{w}}_i$ \n\tare accurate,\n\tthat is, $\\hat{w}^{st}_i = w^{st}_i$ for all $1 \\leq i \\leq n$ and $1 \\leq s, t \\leq \\ell$.\n\n\t\\paragraph{Individual rationality.}\n\tBy our choice of envy-free prices, we have that \n\t$p^t_i \\leq w^{st}_i$ for all $1 \\leq i \\leq n$ and $1 \\leq s, t \\leq \\ell$.\n\tThus, we always guarantee\n\t$$p^t_i \\, \\frac{v^s_i(S_i)}{w^{st}_i} \\leq v^s_i(S_i) \\enspace.$$ \n\n\tSo the mechanism $\\mathcal{M}_\\mathcal{A}$ that we get from the reduction\n\talways provides non-negative utilities for the agents.\n\tEssentially the same proof also shows IR for part 2 and 3.\t\n\t\n\t\\paragraph{Bayesian incentive compatibility.}\n\tWe will first show that the intermediate algorithms in the decoupling\n\tstep of the reduction satisfy constraint \\eqref{eq:interconstraint}.\n\tLet $\\bm{x}_i$ denote the social welfare maximizing allocation\n\tthat the reduction finds\n\tfor the induced assignment problem of agent $i$ for $1 \\leq i \\leq n$.\n\tNote that these social welfare maximizing allocations are market-clearing. \n\tWe have that if the reported valuation $v'_i$ follows the distribution \n\t$F_i$, then the distribution of the manipulated valuation \n\t$\\widetilde{v}_i$ satisfies that\n\t$$\\pr{}{\\widetilde{v}_i = v^t_i} = \\sum_{s=1}^\\ell \\pr{}{v'_i = v^s_i} \\,\n\t\\pr{}{\\widetilde{v}_i = v^t_i : v'_i = v^s_i} \n\t= \\sum_{s=1}^\\ell f_i(s) \\, \\frac{x^{st}_i}{f_i(s)} \n\t= \\sum_{s=1}^\\ell x^{st}_i = f_i(t) \\enspace.$$\n\n\tIndeed, the intermediate algorithms satisfy constraint\n\t\\eqref{eq:interconstraint}. Thus, for each $1 \\leq i \\leq n$ \n\tthe intermediate algorithm $\\mathcal{B}_i$\n\tonly changes the structure of induced assignment problem of agent $i$\n\tand leaves the induced assignment problems of other agents untouched.\n\t\n\tNext, we will verify that in each of the manipulated assignment \n\tproblem, the identity allocation maximizes the social welfare and the\n\tprices are the corresponding envy-free prices.\n\n\tFor each agent $i$,\n\twe let $\\widetilde{\\bm{w}}_i = \n\t\\{\\widetilde{w}^{st}_i\\}_{1 \\leq s, t \\leq \\ell}$\n\tand $\\widetilde{\\bm{p}}_i = \n\t(\\widetilde{p}^1_i, \\dots, \\widetilde{p}^\\ell_i)$\n\tdenote the values and the expected prices of the virtual products \n\trespectively in\n\tthe manipulated assignment problem of agent $i$. We have that for any \n\t$1 \\leq r, s \\leq \\ell$, \n\t\\begin{align*}\n\t\t\\widetilde{w}^{rs}_i & = \n\t\t\\sum_{t=1}^\\ell \\pr{}{\\widetilde{v}_i = v^t_i} \\e{\\bm{v}_{-i}, \n\t\t\\bm{S} \\sim \\mathcal{A}(v^t_i, \\bm{v}_{-i})}{v^r_i(S_i)} \n\t\t= \\sum_{t=1}^\\ell \\frac{x^{st}_i}{f_i(s)} \\, w^{rt}_i \\enspace, \\\\\n\t\t\\widetilde{p}^s_i \n\t\t& = \\sum_{t=1}^\\ell \\pr{}{\\widetilde{v}_i = v^t_i} \n\t\t\\e{\\bm{v}_{-i}, \\bm{S} \\sim \\mathcal{A}(v^t_i, \\bm{v}_{-i})}\n\t\t{p^t_i \\frac{v^s_i(S_i)}{w^{rs}_i}} \n\t\t= \\sum_{t=1}^\\ell \\frac{x^{st}_i}{f_i(s)} \\, p^t_i\\enspace.\n\t\\end{align*}\n\n\tThus, in the manipulated assignment problem of agent $i$,\n\tthe utility of the virtual buyer $r$ of the virtual product $s$,\n\t$1 \\leq r, s \\leq \\ell$, is \n\t$$\\widetilde{w}^{rs}_i - \\widetilde{p}^s_i = \\sum_{t=1}^\\ell\n\t\\frac{x^{st}_i}{f_i(s)} \\, (w^{rt}_i - p^t_i) \n\t\\leq \\sum_{t=1}^\\ell \\frac{x^{st}_i}{f_i(s)} \\, \\max_k \n\t\\{w^{rk}_i - p^k_i\\} = \\max_k \\{w^{rk}_i - p^k_i\\} \\enspace.$$\n\n\tSince $\\bm{p}_i$ are chosen to be the envy-free prices, we have that\n\t$x^{rt}_i > 0$ only if $w^{rt}_i - p^t_i = \\max_k \\{w^{rk}_i - p^k_i\\}$.\n\tHence, when agent $i$ reports its valuation truthfully, that is, $r = s$,\n\tthe above inequality holds with equality. \n\tSo the $\\widetilde{p}_i$ are envy-free\n\tprices with respect to the identity allocation $\\widetilde{\\bm{x}}_i$\n\tof the manipulated assignment problem of agent $i$. By Lemma \n\t\\ref{lemma:assignment1}\n\twe know the allocation $\\widetilde{\\bm{x}}_i$ maximizes the social welfare.\n\tThus, mechanism $\\mathcal{M}_\\mathcal{A}$ is BIC according to Lemma \n\t\\ref{lemma:characterization}.\n\t\n\t\\paragraph{Social welfare.} The expected social welfare for this \n\tmechanism is $\\sum_{i=1}^n \\sum_{s=1}^\\ell \\sum_{t=1}^\\ell\n\tx^{st}_i w^{st}_i$. Since for any $1 \\leq i \\leq n$\n\tthe allocation $\\bm{x}_i$ maximizes the social welfare for the \n\tinduced assignment problem of agent $i$, the social welfare of $\\bm{x}_i$\n\tis at least as large as that of the identity allocation, that is,\n\t$$\\forall i : \\sum_{s=1}^\\ell \\sum_{t=1}^\\ell x^{st}_i w^{st}_i \\geq \n\t\\sum_{s=1}^\\ell f_i(s) w^{ss}_i \n\t= \\e{\\bm{v} \\sim \\bm{F}, \\bm{S} \\sim \\mathcal{A}(\\bm{v})}{v_i(S_i)} \n\t\\enspace.$$\n\n\tThus, we have that\n\t$$SW^{\\mathcal{M}_\\mathcal{A}} =\n\t\\sum_{i=1}^n \\sum_{s, t=1}^\\ell x^{st}_i w^{st}_i\n\t\\geq \\sum_{i=1}^n \\e{\\bm{v} \\sim \\bm{F}, \\bm{S} \\sim \\mathcal{A}(\\bm{v})}\n\t{v_i(S_i)} = \\e{\\bm{v} \\sim \\bm{F}, \\bm{S} \\sim \\mathcal{A}(\\bm{v})}\n\t{\\sum_{i=1}^n v_i(S_i)} = SW^\\mathcal{A} \\enspace.$$\n\\end{proof}\n\n\\subsection{Estimating values by sampling.}\n\nThere is still one technical issue that we need to settle in the reduction.\nIn this section, we will briefly discuss how to use the standard sampling\ntechnique to obtain good estimated values of \n$\\bm{w}_i = \\{w^{st}_i\\}_{1 \\leq s, t \\leq \\ell}$ \nfor the induced assignment problem of agent $i$ for $1 \\leq i \\leq n$. \n\nBy definition, $w^{st}_i$ is the expectation of a random\nvariable $v^s_i(S_i)$, where $S_i$ is the allocated service given by \n$\\mathcal{A}$ over random realization of the valuations $\\bm{v}_{-i}$ of other agents\nand random coin flips of the algorithm. Hence, if the standard deviation \nof $v^s_i(S_i)$ is not too large compared to its mean (no more than a \npolynomial factor), then we can draw polynomially many independent samples and \ntake the average value as our estimated value. \nMore concretely, the sampling algorithm proceeds as follows.\n\\begin{enumerate}\n\t\\item Draw $N = 4 \\, c^2 \\log(n \\ell^2 \/\\epsilon) \/ \\epsilon^2$ independent \n\t\tsamples of $\\bm{v} \\sim \\bm{F}$ conditioned on that the valuation\n\t\tof agent $i$ is $v^t_i$, where \n\t\t$$c = \\frac{\\sd{\\bm{v}_{-i}, \\bm{S} \\sim \\mathcal{A}(v^t_i, \\bm{v}_{-i})}\n\t\t{v^s_i(S_i)}}\n\t\t{\\e{\\bm{v}_{-i}, \\bm{S} \\sim \\mathcal{A}(v^t_i, \\bm{v}_{-i})}{v^s_i(S_i)}}\n\t\t\\enspace.$$\n\t\tLet $\\bm{v}^1, \\dots, \\bm{v}^N$ denote these $N$ sample.\n\t\\item Use algorithm $\\mathcal{A}$ to compute an allocation $\\bm{S}^k \\sim \n\t\t\\mathcal{A}(\\bm{v}^k)$ for each sample $\\bm{v}^k$, $1 \\leq k \\leq N$.\n\t\\item Let $\\hat{w}^{st}_i$ be the average of $v^s_i(S^k_i)$, $1 \\leq k\n\t\t\\leq N$.\n\\end{enumerate}\n\n\\begin{lemma}\n\t\\label{lemma:relative}\n\tThe estimated values $\\hat{\\bm{w}}_i$, $1 \\leq i \\leq n$, by the above\n\tsampling procedure satisfy for any $1 \\leq i \\leq n$ and $1 \\leq s, t \\leq \n\t\\ell$,\n\t$$\\hat{w}^{st}_i \\in \\left[(1 - \\epsilon) w^{st}_i,\n\t(1 + \\epsilon) w^{st}_i\\right]$$\n\twith probability at least $1 - \\epsilon$.\n\\end{lemma}\n\n\\begin{proof}\n\tWe shall have that \n\t\\begin{align*}\n\t\t\\e{}{\\hat{w}^{st}_i} & = \\e{\\bm{v}_{-i}, \\bm{S} \\sim \n\t\t\\mathcal{A}(v^t_i,\\bm{v}_{-i})}{v^s_i(S_i))} = w^{st}_i \\enspace, \\\\\n\t\t\\sd{}{\\hat{w}^{st}_i} & = \\frac{1}{\\sqrt{N}} \\, \\sd{\\bm{v}_{-i}, \\bm{S} \n\t\t\\sim \\mathcal{A}(v^t_i,\\bm{v}_{-i})}{v^s_i(S_i)} \n\t\t= \\frac{c}{\\sqrt{N}} \\, \\e{}{\\hat{w}^{st}_i}\n\t\t= \\frac{c}{\\sqrt{N}} \\, w^{st}_i \\enspace.\n\t\\end{align*}\n\t\n\tBy Chernoff-Hoeffding bound, we get\n\t\\begin{eqnarray*}\n\t\t\\pr{}{\\left|\\hat{w}^{st}_i - w^{st}_i\\right| > \n\t\t\\epsilon w^{st}_i} \n\t\t& = & \\pr{}{\\left|\\hat{w}^{st}_i - \\e{}{\\hat{w}^{st}_i}\\right| > \n\t\t\\frac{\\epsilon \\sqrt{N}}{c} \\, \\sd{}{\\hat{w}^{st}_i}} \\\\\n\t\t& = & \\pr{}{\\left|\\hat{w}^{st}_i - \\e{}{\\hat{w}^{st}_i}\\right| > \n\t\t2 \\, \\sqrt{\\log{(n \\ell^2\/\\epsilon)}} \\, \\sd{}{\\hat{w}^{st}_i}} \\\\\n\t\t& \\leq & e^{- \\log{(n \\ell^2 \/\\epsilon)}} = \\frac{\\epsilon}{n \\ell^2}\n\t\t\\enspace.\n\t\\end{eqnarray*}\n\n\tSince we only need to estimate $n \\ell^2$ values,\n\tby union bound we get that with probability at least $1 - \\epsilon$\n\tthe estimated value $\\hat{w}^{st}_i$ is within\n\t$\\epsilon$ relative error compared to $w^{st}_i$ for all\n\t$1 \\leq i \\leq n$ and $1 \\leq s, t \\leq \\ell$.\n\\end{proof}\n\nThus, if the allocation algorithm $\\mathcal{A}$ admits $SW^\\mathcal{A}$ social welfare and\nthe ratio $c$ is only polynomially large, then by part 2 of Theorem \n\\ref{thm:welfarereduction} we get that mechanism $\\mathcal{M}_\\mathcal{A}$ gives\n$(1-3\\epsilon) \\cdot SW^\\mathcal{A}$ social welfare and is $4\\epsilon v_{max}$-BIC.\nThe running time is polynomial in the input size and $1 \/ \\epsilon$, assuming\na black-box call to algorithm $\\mathcal{A}$ can be done in a single step. \nIn other words, we get a FPTAS reduction.\n\n\\medskip\n\nThe next lemma gives an alternative bound of the sampling error with respect\nto absolute error.\n\n\\begin{lemma}\n\t\\label{lemma:absolute}\n\tIf we draw $N' = 4\\log(n\\ell^2\/\\epsilon)\/\\epsilon^2$ independent samples, \n\tthen with probability at least $1-\\epsilon$ the estimated values \n\t$\\hat{w}^{st}_i \\in [w^{st}_i - \\epsilon v_{max}, \n\tw^{st}_i + \\epsilon v_{max}]$ for all $1 \\leq i \\leq n$ and $1 \\leq s, t\n\t\\leq \\ell$.\n\\end{lemma}\n\n\\begin{proof}\n\tIn this case, we have \n\t\\begin{align*}\n\t\t\\e{}{\\hat{w}^{st}_i} & = \\e{\\bm{v}_{-i}, \\bm{S} \\sim \n\t\t\\mathcal{A}(v^t_i,\\bm{v}_{-i})}{v^s_i(S_i))} = w^{st}_i \\enspace, \\\\\n\t\t\\sd{}{\\hat{w}^{st}_i} & = \\frac{1}{\\sqrt{N'}} \\, \\sd{\\bm{v}_{-i}, \n\t\t\\bm{S}\\sim\\mathcal{A}(v^t_i,\\bm{v}_{-i})}{v^s_i(S_i)} \n\t\t\\leq \\frac{1}{\\sqrt{N'}}\\max_{S_i} \\, v^s_i(S_i) \\leq \\frac{1}\n\t\t{\\sqrt{N'}} \\, v_{max}\n\t\t\\enspace.\n\t\\end{align*}\n\t\n\tBy Chernoff bound we get that\n\t\\begin{eqnarray*}\n\t\t\\pr{}{\\left|\\hat{w}^{st}_i - w^{st}_i\\right| > \n\t\t\\epsilon v_{max}}\n\t\t& \\leq & \\pr{}{\\left|\\hat{w}^{st}_i - \\e{}{\\hat{w}^{st}_i}\\right| > \n\t\t\\frac{\\epsilon}{\\sqrt{N'}} \\sd{}{\\hat{w}^{st}_i}} \\\\\n\t\t& = & \\pr{}{\\left|\\hat{w}^{st}_i - \\e{}{\\hat{w}^{st}_i}\\right| > \n\t\t2 \\, \\sqrt{\\log{(n \\ell^2\/\\epsilon)}} \\, \\sd{}{\\hat{w}^{st}_i}} \\\\\n\t\t& \\leq & e^{- \\log{(n \\ell^2 \/\\epsilon)}} = \\frac{\\epsilon}{n \\ell^2}\n\t\t\\enspace.\n\t\\end{eqnarray*}\n\n\tBy union bound, we have $\\hat{w}^{st}_i \\in [w^{st}_i - \\epsilon v_{max},\n\tw^{st}_i + \\epsilon v_{max}]$ for all $1 \\leq i \\leq n$ and $1 \\leq s, t\n\t\\leq \\ell$.\n\\end{proof}\n\nSuppose the ratio $v_{max} \/ SW^\\mathcal{A}$ is upper bounded by a polynomial of \nthe input size.\nThen, if we choose $\\epsilon = \\delta \\, SW^\\mathcal{A} \/ 2n v_{max}$ in the above\nlemma, we will get that\n$$\\left|\\hat{w}^{st}_i - w^{st}_i\\right| < \\delta \\, SW^\\mathcal{A} \/ 2n \\enspace.$$\n\nBy part 3 of Theorem {\\ref{thm:welfarereduction} we obtain that\nmechanism $\\mathcal{M}_\\mathcal{A}$ provides at least $(1 - \\delta) SW^\\mathcal{A}$ of social\nwelfare and is $4 \\epsilon$-BIC and IR.\nThe running time is polynomial in the input size and $1 \/ \\delta$.\n\n\\section{Reductions for revenue and residual surplus}\n\nIn the reduction for social welfare in the previous section, \nwe only consider market-clearing allocations in the induced assignment \nproblems. This is because for any agent \n$i$, we want to make sure that the intermediate algorithm $\\mathcal{B}_i$ \nis transparent to all agents except agent $i$. If we restrict ourselves to \nmarket-clearing allocations, we do not know any way to get reasonable bounds \non revenue and residual surplus.\n\nHowever, if we focus on an important sub-class of multi-parameter mechanism\ndesign problems that includes the combinatorial auction problem and its special\ncases, then we have some flexibility in choosing the allocations for the \ninduced assignment problem and obtain theoretical bounds on revenue and\nresidual surplus. More concretely, we will consider mechanism design problems\nthat are {\\em downward-closed}. We let $\\phi$ denote the null service so that\nallocating $\\phi$ to an agent implies that agent is not served, that is,\n$v_i(\\phi) = 0$ for all $1 \\leq i \\leq n$. \n\n\\begin{definition}\n\tA multi-parameter mechanism design problem \n\t$\\langle \\bm{I}, \\bm{J}, \\bm{V}, \\bm{F} \\rangle$\n\tis {\\em downward-closed} if for any \n\tfeasible allocation $\\bm{S} = (S_1, \\dots, S_n) \\in \\bm{J}$ and any\n\t$1 \\leq i \\leq n$, the allocation \n\t$(S_1, \\dots, S_{i-1}, \\phi, S_{i+1}, \\dots, S_n)$ is also feasible.\n\\end{definition}\n\nWe let $\\delta = \\min \\{f_i(s) : 1 \\leq i \\leq n, 1 \\leq s \\leq \\ell, \nf_i(s) > 0\\}$ denote the granularity of the prior distributions. We will \nprove the following result.\n\n\\begin{theorem}\n\t\\label{thm:otherreduction}\n\tFor any algorithm $\\mathcal{A}$, there is a mechanism that is IR, BIC, and \n\tprovides at least $\\Omega(SW^\\mathcal{A} \/ \\log(1 \/ \\delta))$ of revenue \n\t(residual surplus).\n\\end{theorem}\n\n\\subsection{Meta-reduction.}\n\nWe will first introduce a meta-reduction scheme based on algorithms that\ncompute envy-free solutions for fractional assignment problems.\nSuppose $\\mathcal{C}$ is an algorithm that computes envy-free solutions\n$(\\bm{x}, \\bm{p})$ for any given fractional assignment problem.\nLet $\\mathcal{A}$ be an algorithm for a multi-parameter mechanism design problem\n$\\langle \\bm{I}, \\bm{J}, \\bm{V}, \\bm{F} \\rangle$.\nWe will convert algorithm $\\mathcal{A}$ into to a mechanism $\\mathcal{M}^\\mathcal{C}_\\mathcal{A}$:\n\n\\begin{enumerate}\n\t\\item For each agent $i$ \\hspace*{\\fill} {\\bf (Pre-computation)}\n\t\t\\begin{enumerate}\n\t\t\t\\item Estimate the values \n\t\t\t\t$\\bm{w}_i = \\{w^{st}_i\\}_{1 \\leq s, t \\leq \\ell}$ \n\t\t\t\tfor the induced assignment problem of agent $i$\n\t\t\t\twith respect to $\\mathcal{A}$. Let $\\hat{\\bm{w}}_i = \n\t\t\t\t\\{\\hat{w}^{st}_i\\}_{1 \\leq s, t \\leq \\ell}$ denote the\n\t\t\t\testimated values.\n\t\t\t\\item Use $\\mathcal{C}$ to solve the induced assignment problems \n\t\t\t\twith estimated values. Let\n\t\t\t\t$(\\bm{x}_i, \\bm{p}_i)$ denote the solution by $\\mathcal{C}$ for\n\t\t\t\tthe induce assignment problem of agent $i$.\n\t\t\t\\item Let $y^t_i = f_i(t) - \\sum_{s=1}^\\ell x^{st}_i$ denote the \n\t\t\t\tunallocated supply of virtual product $t$ in solution \n\t\t\t\t$(\\bm{x}_i, \\bm{p}_i)$ for all $1 \\leq i \\leq n$ and \n\t\t\t\t$1 \\leq t \\leq \\ell$. \n\t\t\t\\item Let $y_i = \\sum_{t=1}^\\ell y^t_i$ \n\t\t\t\tdenote the total amount of unallocated virtual products \n\t\t\t\tin $(\\bm{x}_i, \\bm{p}_i)$ for all $1 \\leq i \\leq n$.\n\t\t\\end{enumerate}\n\t\\item Manipulate the valuations with intermediate algorithm $\\mathcal{B}_i$,\n\t\t$1 \\leq i \\leq n$, as follows:\n\t\t\\hspace*{\\fill} {\\bf (Decoupling)}\n\t\t\\begin{enumerate}\n\t\t\t\\item Suppose the reported valuation of agent $i$ is $v'_i = \n\t\t\t\tv^s_i$.\n\t\t\t\\item Let $\\widetilde{v}_i = \\mathcal{B}_i(v'_i) = v^t_i$ with \n\t\t\t\tprobability $x^i_{st} \/ f_i(s)$ for $1 \\leq t \\leq \\ell$.\n\t\t\t\\item With probability $1 - \\sum_t x^{st}_i \/ f_i(s)$, the\n\t\t\t\tmanipulated valuation $\\widetilde{v}_i$ is unspecified \n\t\t\t\tin the previous step. In this\n\t\t\t\tcase, let $\\widetilde{v}_i = v^t_i$ with probability \n\t\t\t\t$y^t_i \/ y_i$ for $1 \\leq t \\leq \\ell$.\n\t\t\\end{enumerate}\n\t\\item Allocate services as follows: \\hspace*{\\fill} {\\bf (Allocation)}\n\t\t\\begin{enumerate}\n\t\t\t\\item Compute a tentative allocation\n\t\t\t\t$\\widetilde{\\bm{S}} = \n\t\t\t\t(\\widetilde{S}_1, \\dots, \\widetilde{S}_n) = \n\t\t\t\t\\mathcal{A}(\\widetilde{\\bm{v}})$.\n\t\t\t\\item For each agent $i$, let $S_i = \\widetilde{S}_i$ if the\n\t\t\t\tmanipulated valuation $\\widetilde{v}_i$ is specified in\n\t\t\t\tstep 2b). Let $S_i = \\phi$ otherwise. Allocate services\n\t\t\t\taccording to $\\bm{S}$.\n\t\t\t\\item For each agent $i$, suppose the reported valuation is \n\t\t\t\t$v'_i = v^s_i$ and the manipulated valuation is \n\t\t\t\t$\\widetilde{v}_i = v^t_i$, charge agent $i$ with price \n\t\t\t\t$p^t_i v^s_i(S_i) \/ \\hat{w}^{st}_i$.\n\t\t\\end{enumerate}\n\\end{enumerate}\n\nThe following theorem states the above meta-reduction scheme converts \nalgorithms into IR and BIC mechanisms.\n\n\\begin{theorem}\n\tSuppose the algorithm $\\mathcal{C}$ always provides envy-free solutions.\n\t\\begin{enumerate}\n\t\t\\item If the estimated values $\\hat{\\bm{w}}_i$\n\t\t\tare accurate, then mechanism $\\mathcal{M}^\\mathcal{C}_\\mathcal{A}$ is IR and BIC.\n\t\t\\item If the estimated values $\\hat{\\bm{w}}_i$ satisfy that for\n\t\t\tany $1 \\leq s, t \\leq \\ell$, $\\hat{w}^{st}_i \\in [(1 - \\epsilon) \n\t\t\tw^{st}_i, (1 + \\epsilon) w^{st}_i]$, then $\\mathcal{M}^\\mathcal{C}_\\mathcal{A}$ is \n\t\t\tIR and $4 \\epsilon v_{max}$-BIC.\n\t\t\\item If the estimated values $\\hat{\\bm{w}}_i$ satisfy that\n\t\t\tfor any $1 \\leq s, t \\leq \\ell$, $\\hat{w}^{st}_i \\in \n\t\t\t[w^{st}_i - \\epsilon, w^{st}_i + \\epsilon]$, then \n\t\t\t$\\mathcal{M}^\\mathcal{C}_\\mathcal{A}$ is IR and $4 \\epsilon$-BIC.\n\t\\end{enumerate}\n\\end{theorem}\n\n\\begin{proof}\n\tLet us outline the proof for part 1. Proofs of the other two parts are\n\tcalculations along the same line.\n\n\tNote that $p^t_i \\leq w^{st}_i$ for all $1 \\leq i \\leq n$ and \n\t$1 \\leq s, t \\leq \\ell$.\n\tThe mechanism is IR because for any $1 \\leq i \\leq n$ and \n\t$1 \\leq s \\leq \\ell$ the utility for an agent $i$ with valuation $v^s_i$ \n\tin any realization is\n\t$$v^s_i(S_i) - p^t_i \\frac{v^s_i(S_i)}{w^{st}_i} \\geq 0 \\enspace.$$\n\t\n\tNext, we will show that mechanism $\\mathcal{M}^\\mathcal{C}_\\mathcal{A}$ is BIC.\n\tWe first verify that the intermediate algorithms $\\mathcal{B}_i$,\n\t$1 \\leq i \\leq n$, satisfy the constraint \\eqref{eq:interconstraint}. \n\tFor any agent $i$, if its valuation $v_i$\n\tis drawn from distribution $F_i$, then the probability that the\n\tmanipulated valuation $\\widetilde{v}_i = \\mathcal{B}_i(v_i) = v^t_i$ is \n\t\\begin{eqnarray*}\n\t\t\\sum_{s=1}^\\ell f_i(s) \\, \\left[\\frac{x^{st}_i}{f_i(s)} + \n\t\t\\left(1 - \n\t\t\\sum_{r=1}^\\ell \\frac{x^{sr}_i}{f_i(s)} \\right) \\, \\frac{y^t_i}{y_i}\n\t\t\\right]\n\t\t& = & \\sum_{s=1}^\\ell x^{st}_i + \\left( \\sum_{s=1}^\\ell f_i(s) - \n\t\t\\sum_{s=1}^\\ell \\sum_{r=1}^\\ell x^{sr}_i \\right) \\, \\frac{y^t_i}{y_i} \\\\\n\t\t& = & \\sum_{s=1}^\\ell x^{st}_i + \\left( \\sum_{r=1}^\\ell f_i(r) - \n\t\t\\sum_{r=1}^\\ell \\sum_{s=1}^\\ell x^{sr}_i \\right) \\, \\frac{y^t_i}{y_i} \\\\\n\t\t& = & \\sum_{s=1}^\\ell x^{st}_i + \\sum_{r=1}^\\ell \\left( f_i(r) - \n\t\t\\sum_{s=1}^\\ell x^i_{sr} \\right) \\, \\frac{y^t_i}{y_i} \\\\\n\t\t& = & \\sum_{s=1}^\\ell x^{st}_i + \\sum_{r=1}^\\ell y^r_i \\,\n\t\t\\frac{y^t_i}{y_i} = \\sum_{s=1}^\\ell x^{st}_i + y^t_i \n\t\t= f_i(t) \\enspace.\n\t\\end{eqnarray*}\n\n \tThus, we get that for each agent $i$, the intermediate algorithms \n\t$\\mathcal{B}_j$, $1 \\leq j \\leq n$ and $j \\neq i$, are transparent to it.\n\tSo the expected value of agent $i$ of the service it gets, when\n\tits genuine valuation is $v_i = v^s_i$ and the manipulate valuation, is \n\t$\\widetilde{v}_i = v^t_i$ is exactly \n\t$$w^{st}_i = \\e{\\bm{v}_{-i}, \\bm{S} \\sim \\mathcal{A}(v^t_i, \\bm{v}_{-i})}\n\t{v^s_i(S_i)} \\enspace.$$ \n\n\tHence, the expected value of agent $i$ of the servie it gets, when its\n\tgenuine valuation is $v_i = v^s_i$ and the reported valuation is $v'_i\n\t= v^t_i$, is \n\t$$\\widetilde{w}^{st}_i = \n\t\\sum_{r=1}^\\ell \\frac{x^{tr}_i}{f_i(t)} \\, w^{sr}_i \\enspace.$$\n\t\n\tAnd the expected price for agent $i$ when the reported valuation is \n\t$v'_i = v^t_i$ is \n\t$$\\widetilde{p}^t_i = \n\t\\sum_{r=1}^\\ell \\frac{x^{tr}_i}{f_i(t)} \\, \\e{\\bm{v}_{-i}, \\bm{S} \\sim \n\t\\mathcal{A}(v^r_i, \\bm{v}_{-i})}{p^r_i \\, \\frac{v^t_i(S_i)}{w^{tr}_i}} \n\t= \\sum_{r=1}^\\ell \\frac{x^{tr}_i}{f_i(t)} p^r_i \\enspace.$$\n\n\tThus, the the expected utility of agent $i$, when its genuine valuation\n\tis $v_i = v^s_i$ and its reported valuation is $v'_i = v^t_i$, is \n\t$$\\widetilde{w}^{st}_i - \\widetilde{p}^t_i = \\sum_{r=1}^\\ell\n\t\\frac{x^{tr}_i}{f_i(t)} \\, (w^{sr}_i - p^r_i) \n\t\\leq \\sum_{r=1}^\\ell \\frac{x^{tr}_i}{f_i(t)} \\, \\max_k \n\t\\{w^{sk}_i - p^k_i\\} = \\max_k \\{w^{sk}_i - p^k_i\\} \\enspace.$$\n\n\tSince $\\bm{p}_i$ are chosen to be the envy-free prices, we have that\n\t$x^{sr}_i > 0$ only if $w^{sr}_i - p^r_i = \\max_k \\{w^{sk}_i - p^k_i\\}$.\n\tHence, when agent $i$ reports its valuation truthfully, that is, $s = t$,\n\tthe above inequality if tight. Moreover, the above utility is always\n\tnon-negative. So mechanism $\\mathcal{M}^\\mathcal{C}_\\mathcal{A}$ is BIC.\n\\end{proof}\n\nMoreover, the revenue and residual surplus of mechanism \n$\\mathcal{M}^\\mathcal{C}_\\mathcal{A}$ \nis related to the social welfare and revenue of the induced assignment \nproblems as stated in following proposition.\n\n\\begin{proposition}\n\tThe expected revenue (residual surplus) of the mechanism \n\t$\\mathcal{M}^\\mathcal{C}_\\mathcal{A}$ equals the sum of the revenue (residual surplus) \n\tof the manipulated assignment problems.\n\\end{proposition}\n\nBy choosing proper allocation algorithm $\\mathcal{C}$, we can obtain\ntheoretical bounds for the revenue or residual surplus in the manipulated\ninduced assignment problems and thus theoretical bounds for mechanism\n$\\mathcal{M}^\\mathcal{C}_\\mathcal{A}$.\n\n\\subsection{Assignment algorithms.}\n\nIn this section, we will introduce two algorithms for computing envy-free \nsolutions for the induced assignment problems. These two algorithms\nprovides theoretical bounds for revenue and residual surplus.\n\n\\paragraph{Revenue.} \nThe first algorithm provides revenue that is a $\\Omega(1\/\\log(1\/\\delta))$\nfraction of $SW^\\mathcal{A}$, the social welfare by algorithm $\\mathcal{A}$. The idea is\nto introduce proper reserve prices to the induced assignment problems by\nredundant virtual buyers. This is inspired by the techniques by\nGuruswami et al.~\\cite{guruswami2005profit}.\nFor the induced assignment problem of agent $i$, $1 \\leq i \\leq n$, \nthe assignment algorithm $\\mathcal{C}_R$ for revenue maximization proceeds as\nfollows:\n\\begin{enumerate}\n\t\\item Find the social welfare maximizing allocation $\\bm{x}_i = \n\t\t\\{x^{st}_i\\}_{1 \\leq s, t \\leq \\ell}$.\n\t\\item Suppose $u_{max}$ is the maximal valuation \n\t\tamong the virtual buyer-product pair $(s, t)$ with non-zero $x^{st}_i$,\n\t\tthat is, \n\t\t$$u_{max} = \\max \\{w^{st}_i : 1 \\leq s, t \\leq \\ell, x^{st}_i > 0\\}\n\t\t\\enspace.$$\n\t\\item Recall that $\\delta = \\min \\{f_i(t) : 1 \\leq i \\leq n, 1 \\leq t \\leq\n\t\t\\ell, f_i(t) > 0\\}$ denotes the granularity of the prior distribution. \n\t\tFor $1 \\leq k \\leq \\log(1 \/ \\delta)$:\n\t\t\\begin{enumerate}\n\t\t\t\\item Consider the following variant\n\t\t\t\tof the induced assignment instance of agent $i$: \\\\\n\t\t\t\tFor each virtual product $1 \\leq t \\leq \\ell$, add a \n\t\t\t\tdummy virtual buyer with demand $1 + \\delta$ and value \n\t\t\t\t$u_k = u_{max} \/ 2^k$ for virtual product $t$\n\t\t\t\tand value $0$ for other virtual products.\n\t\t\t\\item Find social welfare maximizing allocation $\\bm{x}_{ik}$ and \n\t\t\t\tenvy-free prices $\\bm{p}_{ik}$ for this variant.\n\t\t\t\\item Let $(\\hat{\\bm{x}}_{ik}, \\hat{\\bm{p}}_{ik})$ be the projection\n\t\t\t\tof $(\\bm{x}_{ik}, \\bm{p}_{ik})$ on the original induced \n\t\t\t\tassignment problem of agent $i$, that is, for any \n\t\t\t\t$1 \\leq s, t \\leq \\ell$,\n\t\t\t\t$\\hat{x}^{st}_{ik} = x^{st}_{ik}$, $\\hat{p}^t_{ik} = p^t_{ik}$.\n\t\t\\end{enumerate}\n\t\\item Return the $(\\hat{\\bm{x}}_{ik}, \\hat{\\bm{p}}_{ik})$, \n\t\t$1 \\leq k \\leq \\log(1\/\\delta)$, with best revenue.\n\\end{enumerate}\n\n\\begin{lemma}\n\tAssignment algorithm $\\mathcal{C}_R$ always return an envy-free solution \n\t$(\\bm{x}, \\bm{p})$. The revenue is at least a\n\t$\\Omega(1\/\\log(1\/\\delta))$ fraction of the optimal social welfare of\n\tthe assignment problem.\n\\end{lemma}\n\n\\begin{proof}\n\tThe envy-freeness follows from the fact that \n\t$(\\hat{\\bm{x}}_{ik}, \\hat{\\bm{p}}_{ik})$, $1 \\leq k \\leq \\log(1\/\\delta)$,\n\tare projections of envy-free solutions and thus are also envy-free.\n\t\n\tNow we consider the revenue by $\\mathcal{C}_R$. We let $r_k$ denote\n\tthe revenue by solution $(\\hat{\\bm{x}}_{ik}, \\hat{\\bm{p}}_{ik})$.\n\tNote that in $(\\hat{\\bm{x}}_{ik}, \\hat{\\bm{p}}_{ik})$, all prices are at \n\tleast $u_k$ and the amount of virtual products that are sold is at least\n\t$\\sum_{s, t : w^{st}_i \\geq u_k} x^{st}_i$. Hence, we have \n\t$$r_k \\geq w_k \\sum_{s, t : w_{st} \\geq u_k} x^{st}_i \\enspace.$$\n\n\tNote that if we extend the definition of $u_k$ and let $u_k = u_{max} \/ \n\t2^k$ for all non-negative integer $k$, then we have\n\t\\begin{eqnarray}\n\t\t\\sum_{k=1}^\\infty u_k \\sum_{s, t : w^{st}_i \\geq u_k} x^{st}_i \n\t\t\\notag \n\t\t& = & \\sum_{k=1}^\\infty (u_{k - 1} - u_k) \n\t\t\\sum_{s, t : w^{st}_i \\geq u_k} x^{st}_i \\notag \\\\\n\t\t& = & \\sum_{s=1}^\\ell \\sum_{t=1}^\\ell x^{st}_i \n\t\t\\sum_{k : w^{st}_i \\geq u_k} (u_{k-1} - u_k) \\notag \\\\\n\t\t& = & \\sum_{s=1}^\\ell \\sum_{t=1}^\\ell x^{st}_i \n\t\t\\max_k \\{u_{k-1} : w^{st}_i \\geq u_k\\} \\notag \\\\\n\t\t& \\geq & \\sum_{s, t} x^{st}_i w^{st}_i \\enspace. \\label{eq:rev1}\n\t\\end{eqnarray}\n\n\tOn the other hand, the contribution of the tail is small compared to\n\tthe social welfare.\n\t\\begin{equation}\n\t\t\\label{eq:rev2}\n\t\t\\sum_{k=\\log(1\/\\delta)+1}^{\\infty} u_k \n\t\t\\sum_{s, t : w^{st}_i \\geq u_k} x^{st}_i \n\t\t\\leq \\sum_{k = \\log(1\/\\delta) + 1}^{\\infty} w_k \n\t\t\\leq \\frac{\\delta w_{max}}{2}  \n\t\t\\leq \\frac{\\sum_{s, t} x^{st}_i w^{st}_i}{2} \\enspace. \n\t\\end{equation}\n\n\tThe last inequality holds because allocating the most valuable \n\tvirtual product the one of the virtual buyer is a feasible allocation.\n\tHence, consider the difference of the above formulas, \n\t$\\eqref{eq:rev1} - \\eqref{eq:rev2}$, and we get that\n\t$$\\sum_{k = 1}^{\\log(1\/\\delta)} r_k \\geq \\sum_{k = 1}^{\\log(1\/\\delta)} u_k\n\t\\sum_{s, t : w_{st} \\geq u_k} x^{st}_i \\geq \n\t\\frac{\\sum_{s, t} x^{st}_i w^{st}_i}{2} \\enspace. $$\n\n\tThus, by pigeon-hole-principle at least one of the assignment \n\t$(\\hat{\\bm{x}}_{ik}, \\hat{\\bm{p}}_{ik})$ provides \n\trevenue that is at least a $1\/2\\log(1\/\\delta)$ fraction of \n\tthe social welfare.\n\\end{proof}\n\nThe above lemma leads to the following results for revenue maximization.\n\n\\begin{proposition}\n\tSuppose the social welfare given by allocation algorithm $\\mathcal{A}$ is\n\t$SW^\\mathcal{A}$, the mechanism $\\mathcal{M}^{\\mathcal{C}_R}_\\mathcal{A}$ guarantees at least\n\t$\\Omega(SW^\\mathcal{A} \/ \\log(1\/\\delta))$ of revenue.\n\\end{proposition}\n\n\\paragraph{Complementary lower bound.}\nThe approximation ratio with respect to $SW^\\mathcal{A}$ is tight due to the \nfollowing example. Consider the auction problem with only one agent and one\nitem. Suppose with probability $1 \/ 2^k$ the agent has value $2^k$ for the item \nfor $k = 1, 2, \\dots, \\log(1 \/ \\delta)$. And with probability $\\delta$, the\nagent has value $0$ for the item. In this example, the granularity\nof the prior distribution is $\\delta$. \nThe optimal social welfare is \n$\\sum_{k=1}^{\\log(1\/\\delta)} \\frac{1}{2^k} \\, 2^k = \\log(1\/\\delta)$. \nBut no BIC mechanism can achieve revenue better than $1$.\n\n\\paragraph{Residual surplus.} \nWe turn to the residual surplus maximization problem. Note that revenue\nand residual surplus are symmetric in the induced assignment problems. \nWe will use the following assignment algorithm $\\mathcal{C}_{RS}$\nbased on the same idea we use for the revenue maximization algorithm.\n\nThe residual surplus maximizing envy-free algorithm $\\mathcal{C}_{RS}$ is as\nfollows:\n\\begin{enumerate}\n\t\\item Find the social welfare maximizing allocation $\\bm{x}_i = \n\t\t\\{x^{st}_i\\}_{1 \\leq s, t \\leq \\ell}$.\n\t\\item Suppose $u_{max}$ is the maximal valuation \n\t\tamong the virtual buyer-product pair $(s, t)$ with non-zero $x^{st}_i$,\n\t\tthat is, \n\t\t$$u_{max} = \\max \\{w^{st}_i : 1 \\leq s, t \\leq \\ell, x^{st}_i > 0\\}\n\t\t\\enspace.$$\n\t\\item Recall that $\\delta = \\min \\{f_i(t) : 1 \\leq i \\leq n, 1 \\leq t \\leq\n\t\t\\ell, f_i(t) > 0\\}$ denotes the granularity of the prior distribution. \n\t\tFor $1 \\leq k \\leq \\log(1 \/ \\delta)$:\n\t\t\\begin{enumerate}\n\t\t\t\\item Consider the following variant\n\t\t\t\tof the induced assignment instance of agent $i$: \\\\\n\t\t\t\tFor each virtual buyer $1 \\leq t \\leq \\ell$, add a \n\t\t\t\tdummy virtual product with demand $1 + \\delta$ and value \n\t\t\t\t$u_k = u_{max} \/ 2^k$ for virtual buyer $t$\n\t\t\t\tand value $0$ for other virtual buyer.\n\t\t\t\\item Find social welfare maximizing allocation $\\bm{x}_{ik}$ and \n\t\t\t\tenvy-free prices $\\bm{p}_{ik}$ for this variant.\n\t\t\t\\item Let $(\\hat{\\bm{x}}_{ik}, \\hat{\\bm{p}}_{ik})$ be the projection\n\t\t\t\tof $(\\bm{x}_{ik}, \\bm{p}_{ik})$ on the original induced \n\t\t\t\tassignment problem of agent $i$, that is, for any \n\t\t\t\t$1 \\leq s, t \\leq \\ell$,\n\t\t\t\t$$\\hat{x}^{st}_{ik} = x^{st}_{ik} \\quad , \\quad \n\t\t\t\t\\hat{p}^t_{ik} = p^t_{ik} \\enspace.$$\n\t\t\\end{enumerate}\n\t\\item Return the $(\\hat{\\bm{x}}_{ik}, \\hat{\\bm{p}}_{ik})$, \n\t\t$1 \\leq k \\leq \\log(1\/\\delta)$, with best revenue.\n\\end{enumerate}\n\nThe proofs of the following lemma and theorem is almost identical to the \nrevenue maximization part so we omit the proofs here.\n\n\\begin{lemma}\n\tAssignment algorithm $\\mathcal{C}_{RS}$ always return an envy-free solution \n\t$(\\bm{x}, \\bm{p})$. The residual surplus is at least a\n\t$\\Omega(1\/\\log(1\/\\delta))$ fraction of the optimal social welfare of\n\tthe assignment problem.\n\\end{lemma}\n\n\\begin{proposition}\n\tSuppose the social welfare given by allocation algorithm $\\mathcal{A}$ is\n\t$SW^\\mathcal{A}$, the mechanism $\\mathcal{M}^{\\mathcal{C}_{RS}}_\\mathcal{A}$ guarantees at least\n\t$\\Omega(SW^\\mathcal{A} \/ \\log(1\/\\delta))$ of residual surplus.\n\\end{proposition}\n\n\\section{Application in combinatorial auctions}\n\nIn this section we will briefly illustrates how to use the reduction for social \nwelfare in this paper to derive a combinatorial auction mechanism that matches \nthe best algorithmic result. \n\n\\paragraph{Combinatorial auctions.} In the combinatorial auctions, we consider\na principal with $m$ items (exactly one copy of each item) and $n$ agents. \nEach agent has a private valuation $v_i \\sim F_i$ for subsets of items. \nThe goal is to design a protocol to \nallocate the items and to charge prices to the agents.\n\n\\medskip\n\nWe will show the following corollaries of our reduction for social welfare.\n\n\\begin{corollary}\n\tFor combinatorial auctions with sub-additive (or fractionally sub-additive) \n\tagents where the prior distributions have finite and poly-size \n\tsupports, there is a $\\left(\\frac{1}{2} - \\epsilon\\right)$-approximate\n\t(or $\\left(1- \\frac{1}{e}-\\epsilon\\right)$-approximate respectively),\n\t$\\epsilon v_{max}$-BIC, and IR mechanism for social welfare maximization.\n\tThe running time is polynomial in the input size and $1 \/ \\epsilon$. \n\\end{corollary}\n\n\\paragraph{Algorithm.}\n\nWe will consider a variant of the LP-based algorithms by Feige \n\\cite{feige2006maximizing} and use the reduction for social welfare to\nconvert it into an IR and $\\epsilon v_{max}$-BIC mechanism. More concretely,\nwe will consider the Bayesian version of the standard \nsocial welfare maximization linear program (CA):\n\\begin{align*}\n\t\\text{Maximize} ~ ~ \\sum_i \\sum_t \\sum_S & ~ f_i(t) \\, v^t_i(S) \\,\n\tx_{i, t, S} & & \\text{s.t.} \\\\\n\t\\sum_i \\sum_t \\sum_{S : j \\in S} f_i(t) \\,x_{i, t, S} & ~ \\leq ~ 1 & &\n\t\\forall j \\\\\n\t\\sum_S x_{i, t, S} & ~ \\leq ~ 1 & & \\forall i, t \\\\\n\tx_{i, t, S} & ~ \\geq ~ 0 & & \\forall i, t, S\n\\end{align*}\n\nIn this LP, $x_{i, t, S}$ denote the probability that agent $i$ is allocated \nwith a subset of items $S$ conditioned on its valuation is $v^t_i$.\nThis LP can be solved in polynomial time by the standard primal dual \ntechnique via demand queries. See \\cite{dobzinski2005approximation} for more\ndetails. We let $LP^*$ denote the optimal value of this LP.\nMoreover, for any basic feasible optimal solution of the above LP,\nthere are at most $nm\\ell$ non-zero entries since there are only $nm\\ell$\nnon-trivial constraints. Hence, we have the following lemma:\n\n\\begin{lemma}\n\tIn poly-time we can find an optimal solution $\\bm{x}^*$ to (CA) with at \n\tmost $nm\\ell$ non-zero entries.\n\\end{lemma}\n\nNext, we will filter this solution $\\bm{x}^*$ by removing insignificant \nentries. We let $\\hat{x}_{i, t, S} = x^*_{i, t, S} < \\epsilon \/ n m \\ell$.\nNote that $LP^* \\geq f_i(t) v^t_i(S)$ for any $i$, $t$, and $S$ since always\nallocating subset $S$ to agent $i$ is a feasible allocation. We get that\n$\\hat{\\bm{x}}$ is a feasible solution to (CA) with value at least \n$(1 - \\epsilon) LP^*$.\n\nThen, we will use the rounding algorithms by Feige \\cite{feige2006maximizing}\nto get a $\\frac{1}{2}$-rounding\nfor sub-additive agents and a $\\left(1-\\frac{1}{e}\\right)$-rounding for\nfractionally sub-additive agents: \n\\begin{enumerate}\n\t\\item Allocate a tentative subset of items\n\t\t$\\widetilde{S}_i$ to each agent $i$, $1 \\leq i \\leq n$, according\n\t\tto the reported valuation $v'_i = v^t_i$ and \n\t\t$\\hat{x}_{i, t, \\widetilde{S}_i}$.\n\t\\item Resolve conflicts properly by choosing $S_i \\subseteq \n\t\t\\widetilde{S}_i$ so that $\\bm{S} = (S_1, \\dots, S_n)$ is a feasible\n\t\tallocation.\n\\end{enumerate}\n\nBy extending Feige's original proof, we can show that there is a randomized\nalgorithm for choosing $\\bm{S}$ such that for sub-additive agents, we have:\n\\begin{equation}\n\t\\label{eq:suba}\n\t\\e{\\bm{v}_{-i}, \\bm{S}}{v_i(S_i)} \\geq \\frac{1}{2} \\, v_i(\\widetilde{S}_i)\n\t\\enspace.\n\\end{equation}\n\nAnd for fractionally sub-additive agents, we have:\n\\begin{equation}\n\t\\label{eq:fracsuba}\n\t\\e{\\bm{v}_{-i}, \\bm{S}}{v_i(S_i)} \\geq \n\t\\left(1 - \\frac{1}{e}\\right) \\, v_i(\\widetilde{S}_i)\n\t\\enspace.\n\\end{equation}\n\nWe will omit the proof in this extended abstract.\nWe denote the above algorithm as $\\mathcal{A}$. Then, $\\mathcal{A}$ provides \n$\\left(\\frac{1}{2} - \\epsilon\\right)$-approximation for sub-additive agents \nand $\\left(1 - \\frac{1}{e} - \\epsilon\\right)$-approximation for \nfractionally sub-additive agents.\n\n\\paragraph{Estimating values.}\n\nBy Theorem \\ref{thm:welfarereduction} and \\ref{thm:otherreduction}, we only\nneed to show how to estimate the values $\\bm{w}_i$, $1 \\leq i \\leq n$, for\nthe induced assignment problem of agent $i$ efficiently. Further, by\nLemma \\ref{lemma:relative}, we can efficiently estimate the values \n$\\bm{w}_i = \\{w^{st}_i\\}_{1 \\leq s, t \\leq \\ell}$, $1 \\leq i \\leq n$, if\nthe following lemma holds.\n\n\\begin{lemma}\n\tFor any $1 \\leq i \\leq n$, and any $1 \\leq s, t \\leq \\ell$,\n\t$$\\frac{\\sd{\\bm{v}_{-i}, \\bm{S}\\sim\\mathcal{A}(v^t_i, \\bm{v}_{-i})}{v^s_i(S_i)}}\n\t{\\e{\\bm{v}_{-i}, \\bm{S}\\sim\\mathcal{A}(v^t_i, \\bm{v}_{-i})}{v^s_i(S_i)}}\n\t\\leq \\sqrt{\\frac{4nm\\ell}{\\epsilon}} \\enspace.$$\n\\end{lemma}\n\n\\begin{proof}\n\tBy inequalities \\eqref{eq:suba} and \\eqref{eq:fracsuba}, we get that\n\tconditioned on $\\widetilde{S}_i$ being chosen as the tentative set,\n\t\\begin{equation*}\n\t\t{\\e{\\bm{v}_{-i}, \\bm{S}\\sim\\mathcal{A}(v^t_i, \\bm{v}_{-i})}\n\t\t{v^s_i(S_i) : \\widetilde{S}_i}}\n\t\t\\geq \\frac{1}{2} \\, v^s_i\\left(\\widetilde{S}_i\\right) \\enspace.\n\t\\end{equation*}\n\n\tWe also have that\t\n\t$$\\sd{\\bm{v}_{-i}, \\bm{S}\\sim\\mathcal{A}(v^t_i, \\bm{v}_{-i})}\n\t{v^s_i(S_i) : \\widetilde{S}_i} \\leq \\max \\left\\{v^s_i(S_i) : \n\t\\widetilde{S}_i\\right\\} \\leq v^s_i(\\widetilde{S}_i) \\enspace.$$\n\n\tHence, \n\t\\begin{align*}\n\t\t\\sd{\\bm{v}_{-i}, \\bm{S}\\sim\\mathcal{A}(v^t_i, \\bm{v}_{-i})}\n\t\t{v^s_i(S_i)}^2\n\t\t= \\, & \\sum_{\\widetilde{S}_i}\n\t\t\\hat{x}_{i, t, \\widetilde{S}_i} \\,\n\t\t\\sd{\\bm{v}_{-i}, \\bm{S}\\sim\\mathcal{A}(v^t_i, \\bm{v}_{-i})}\n\t\t{v^s_i(S_i) : \\widetilde{S}_i}^2 \\\\\n\t\t\\leq \\, & \\sum_{\\widetilde{S}_i}\n\t\t\\hat{x}_{i, t, \\widetilde{S}_i} \\, v^s_i(\\widetilde{S}_i)^2 \\\\\n\t\t\\leq \\, & \\frac{1}{\\min \\left\\{\\hat{x}_{i, t, \\widetilde{S}_i} > 0\n\t\t\\right\\}} \\, \\left(\\sum_{i, t, \\widetilde{S}_i}\n\t\t\\hat{x}_{i, t, \\widetilde{S}_i} \\, v^s_i(\\widetilde{S}_i)\\right)^2 \\\\\n\t\t\\leq \\, & \\frac{nm\\ell}{\\epsilon} \\, \\left(\\sum_{\\widetilde{S}_i}\n\t\t\\hat{x}_{i, t, \\widetilde{S}_i} \\,\n\t\t\\e{\\bm{v}_{-i}, \\bm{S}\\sim\\mathcal{A}(v^t_i, \\bm{v}_{-i})}\n\t\t{v^s_i(S_i) : \\widetilde{S}_i}\\right)^2 \\\\\n\t\t\\leq \\, & \\frac{4nm\\ell}{\\epsilon} \\, \n\t\t\\e{\\bm{v}_{-i}, \\bm{S}\\sim\\mathcal{A}(v^t_i, \\bm{v}_{-i})}{v^s_i(S_i)}^2 \n\t\t\\enspace.\n\t\\end{align*}\n\\end{proof}\n\n\n\\bibliographystyle{alpha}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\label{introduction}Introduction}\nWhen the mass of a quark is significantly larger than the quantum chromodynamics (QCD) scale parameter, $\\Lambda_{QCD} \\sim 250$~MeV, we categorize it as a heavy quark \\cite{Behnke:2015qja}. The production of heavy quarks in photoproduction ($\\gamma p$) and deep inelastic scattering (DIS) of ${e^\\pm}p$ was one of the main tasks at HERA. The only heavy quarks kinematically accessible at HERA were beauty and charm quarks, and investigation of the impact of charm quark cross section H1-ZEUS combined data \\cite{Abramowicz:1900rp}  on simultaneous determination of parton distribution functions and the strong coupling, $\\alpha_s(M^2_Z)$, is the main topic of this analysis. In deep inelastic ${e^\\pm}p$ scattering we can approximate the ratio of photon couplings corresponding to a heavy quark, $Q_h, h=b, c$, by\n\\begin{equation} \\label{eq:hfrac}\nf(h) \\sim \\frac{Q_h^2}{\\Sigma{Q_q^2}} ,\n\\end{equation}\nwhere $Q_h=\\frac{1}{3},~\\frac{2}{3}$ are the beauty and charm electric charges, respectively, and $Q_q$~, with $q=u,d,s,c,b$~, represent the kinematically accessible quark flavours. \n\n Now, for the charm quark we have\n\\begin{equation} \\label{eq:cfrac}\nf(c) \\sim \\frac{Q_c^2}{Q_d^2+Q_u^2+Q_s^2+Q_c^2+Q_b^2}\n        = \\frac{4}{11} \\simeq 0.36~.\n\\end{equation}\nFrom Eq. \\ref{eq:cfrac}~ we see that up to 36 percent of the cross sections at HERA originate from processes with charm quarks in the final state. This is our main motivation to investigate the impact of only charm quarks on simultaneous determination of parton distribution functions or their uncertainties and the strong coupling, $\\alpha_s(M^2_Z)$.\n\nThe ratio $f(c) \\simeq 0.36~$ implies that charm quarks are an integral part of the quark-antiquark sea within the proton. On the other hand, the proton has no net charm flavour number, which in turn implies that the charm quarks within the proton can only arise in pairs of $c\\overline{c}$. Since the charm-quark mass is about $1.5$ GeV~, at the low-energy limit the $c\\overline{c}$ pairs are considerably heavier than that to have a contribution within the proton. \n\n Although consideration of so-called intrinsic charm (IC) \\cite{Brodsky:1980pb} may alter this simple view of the heavy flavour content of the proton, at present there is no evidence for the existence of such a contribution from HERA data. Therefore, in this analysis the charm quarks within the proton are as usual considered  as virtual quarks, which in turn arise as fluctuations of probing the gluon content of the proton.\n \n The charm PDFs play an important role in hadronic collisions and cause  photons to emerge from hard parton-parton interactions in association with one or more charm quark jets. Clearly, to study and analyse these processes, we need the charm PDFs, which in turn have sizeable uncertainties. A series of experimental measurements involving charm (or beauty) and photon final states have recently been published by the CDF and D0 Collaborations \\cite{Abazov:2012ea,Abazov:2009de,D0:2012gw,Abazov:2014hoa,Aaltonen:2009wc,Aaltonen:2013ama}. \n \n \n As we noted, the charm quark mass is about $1.5$ GeV~, whereas the QCD scale is about $\\Lambda_{QCD} \\sim 0.25$ GeV~, so it is reasonable to treat the charm quark mass as a hard scale in perturbative quantum chromodynamics (pQCD) and investigate the charm mass effect in pQCD. Accordingly, in this study we use the full HERA run I and II combined data \\cite{Abramowicz:2015mha} as a new measurements of inclusive deep inelastic scattering cross sections for our base data set and then we investigate, simultaneously, the impact of charm quark cross section H1-ZEUS combined data \\cite{Abramowicz:1900rp} on the central value of the PDFs and determination of the strong coupling, $\\alpha_s(M^2_Z)$.  \n  \n  Although the charm quark mass is large compared to the QCD scale, it is small with respect to other pQCD scales, such as the transverse momentum of a quark or a jet, $p_T$, and the virtuality of the photon, $Q^2$. This smallness leads to the logarithmic correction terms, $\\sim [\\alpha_s \\ln(p^2_{T}\/m_c^2)]^n$ and $\\sim [\\alpha_s \\ln(Q^2\/m_c^2)]^n$~, corresponding to $p_T$ and $Q^2$, respectively.\nAt present, the order of magnitude and treatment of these correction terms are open questions.\n\n The outline of this  paper is as follows. In Section~2, we describe the theoretical framework of our study and discuss the reduced cross sections. We introduce the data set which we use in this QCD-analysis and discuss our methodology in Section~3. In Section~4, the impact of charm quark cross section H1-ZEUS combined data on QCD fit quality is discussed. We explain the impact of charm production data on PDFs and $\\alpha_s(M^2_Z)$ in Section~5. We present our results in Section~6 and conclude with a summary in Section~7.\n\n\\section{\\label{dis}Cross sections and parton distributions}\nIn perturbative quantum chromodynamics, the deep inelastic scattering of ${e^\\pm}p$, at the centre-of-mass energies up to $\\sqrt{s} \\simeq 320\\,$GeV at HERA, plays a central role in probing the structure of the proton, as a sea of strongly interacting quarks and gluons. For neutral current (NC) interactions, the reduced cross sections can be expressed in terms of the generalized structure functions as:\n\\begin{eqnarray}\n   \\sigma_{r,NC}^{{\\pm}}&=&   \\frac{d^2\\sigma_{NC}^{e^{\\pm} p}}{d{x_{\\rm Bj}}dQ^2} \\frac{Q^4 x_{\\rm Bj}}{2\\pi \\alpha^2 Y_+}                                               \\nonumber\\\\\n    &=&  \\tilde{F_2} \\mp \\frac{Y_-}{Y_+} x\\tilde{F_3} -\\frac{y^2}{Y_+} \\tilde{F_{\\rm L}}~,\n    \\label{eq:NC}\n\\end{eqnarray} \nwhere  $Y_{\\pm} = 1 \\pm (1-y)^2$ and $\\alpha$ is the fine-structure constant which is defined \nat zero momentum transfer. The generalized structure functions $\\tilde F_2$, $\\tilde F_L$ and $\\tilde F_3$ can be expressed as linear combinations of the proton structure functions $F^{\\gamma}_2, F^{\\gamma Z}_2, F^{\\gamma Z}_3, F^Z_2$ and $F^Z_3$ as follows:\n\\begin{eqnarray} \\label{strf}                                                   \n \\tilde F_2 &=& F_2 - \\kappa_Z v_e  \\cdot F_2^{\\gamma Z} +                      \n  \\kappa_Z^2 (v_e^2 + a_e^2 ) \\cdot F_2^Z~, \\nonumber \\\\   \n \\tilde F_L &=& F_{\\rm L} - \\kappa_Z v_e  \\cdot F_{\\rm L}^{\\gamma Z} +                      \n  \\kappa_Z^2 (v_e^2 + a_e^2 ) \\cdot F_{\\rm L}^Z~, \\nonumber \\\\                     \n x\\tilde F_3 &=&  - \\kappa_Z a_e  \\cdot xF_3^{\\gamma Z} +                    \n  \\kappa_Z^2 \\cdot 2 v_e a_e  \\cdot xF_3^Z~,                                   \n\\end{eqnarray}\nwhere $v_e$ and $a_e$ are the vector and axial-vector weak couplings of the electron to the $Z$ boson, and $\\kappa_Z(Q^2)$ is defined as\n\\begin{eqnarray}\n\\kappa_Z(Q^2) &=&   \\frac{Q^2}{(Q^2+M_Z^2)(4\\sin^2 \\theta_W \\cos^2 \\theta_W)}.\n\\end{eqnarray}\nThis analysis is based on xFitter, an open source QCD framework \n\\cite{xFitter} which is an update of the former HERAFitter package \\cite{ Sapronov}. The values of the $Z$-boson mass and the electroweak mixing angle are $M_Z=91.1876$~GeV and $\\sin^2 \\theta_W=0.23127$, respectively, and electroweak effects have been treated only at leading order (LO).\n\n In the range of low values of $Q^2$, $Q^2\\ll M_Z^2$, the $Z$ boson exchange contribution may be ignored and then the reduced NC DIS cross sections can be expressed by \n\\begin{eqnarray}\n   \\sigma_{r,NC}^{{\\pm}}&= &   F_2  - \\frac{y^2}{Y_{+}}  F_L~.\n    \\label{eq:LOWNC}\n\\end{eqnarray}\n\n Similarly, the reduced charged current (CC) deep inelastic $e^{\\pm}p$ scattering cross sections may be expressed as follows:\n \n\\begin{eqnarray}\n\\sigma_{r,CC}^{\\pm} &=&\\frac{2\\pi x_{\\rm Bj}}{G^2_F} \\left[\\frac{M^2_W+Q^2}{M^2_W}\\right]^2 \\frac{d^2\\sigma_{CC}^{e^{\\pm} p}}{d{x_{\\rm Bj}}dQ^2} \\nonumber \\\\\n&= &  \\frac{Y_{+}}{2}  W_2^{\\pm} \\mp \\frac{Y_{-}}{2}x  W_3^{\\pm} - \\frac{y^2}{2} W_L^{\\pm}~~,\n\\end{eqnarray}\nwhere $\\tilde W_2^{\\pm}$, $\\tilde W_3^{\\pm}$ and $\\tilde W_L^{\\pm}$  are  another set of structure functions and $G_F$ is the Fermi constant, which is related to the weak coupling constant $g$ and electromagnetic coupling constant $e$ by:\n\\begin{eqnarray}\nG^2_F = \\frac{e^2}{4\\sqrt{2}{\\sin ^2\\theta_W}M^2_W} = \\frac{g^2}{4M_W}.\n\\end{eqnarray}\nThe values of the Fermi constant and $W$-boson mass in the xFitter QCD framework \\cite{xFitter} are: $G_F=1.16638\\times 10^{-5} $~GeV$^{-2}$\nand $M_W=80.385$~GeV.\n\n In the quark parton model (QPM), the sums and differences\nof quark and anti-quark distributions, depending on the \ncharge of the lepton beam, can be represented by $W_2^\\pm$, $xW_3^\\pm$ structure functions, respectively, and $W_{\\rm L}^\\pm = 0$~:\n\\begin{eqnarray}\n \\label{WSF}\n    W_2^{+}  \\approx  x\\overline{U}+xD\\,,\\hspace{0.05cm} ~~~~~~~\n    W_2^{-}  \\approx  xU+x\\overline{D}\\,,\\hspace{0.05cm} \\nonumber \\\\   \n   xW_3^{+}  \\approx   xD-x\\overline{U}\\,,\\hspace{0.05cm} ~~~~~~~\n   xW_3^{-}  \\approx  xU-x\\overline{D}\\,.\n\\end{eqnarray}\nAccording to Eq.~\\ref{WSF}~ we have:\n\n\\begin{eqnarray}\n\\sigma_{r,CC}^{+} \\approx  x\\overline{U}+ (1-y)^2xD \\nonumber \\\\\n\\sigma_{r,CC}^{-} \\approx  xU +(1-y)^2 x\\overline{D}.\n\\end{eqnarray}\nNow it is possible to determine both the valence-quark distributions, $xu_v$ and $xd_v$, and the combined sea-quark distributions, $x\\overline{U}$ and $x\\overline{D}$, by combination of NC and CC measurements.\n\n In analogy to the inclusive NC deep inelastic $e^{\\pm}p$ cross section, the reduced cross sections for charm-quark production, $\\sigma_{red}^{C\\bar{C}}$,  can be expressed by\n\\begin{eqnarray}\n\t\\sigma_{red}^{C\\bar{C}} &=& \n\t          \\frac{d\\sigma^{C\\bar{C}}(e^{\\pm} p)}{d{x_{\\rm Bj}} \\, dQ^2} \\cdot \\frac{x_{\\rm Bj} \\, Q^4}{2 \\pi \\alpha^2 Y_{+}} \\nonumber \\\\ \n\t          &=&F_2^{C\\bar C} \\mp \\frac{Y_{-}}{Y_{+}}x  F_3^{C\\bar C} - \\frac{y^2}{Y_{+}}  F_L^{C\\bar C}~, \n    \\label{eq:NCheavy}\n\\end{eqnarray}\t\nwhere $Y_{\\pm} = (1 \\pm (1-y)^2)$, $\\alpha$ is the electromagnetic coupling constant, and $F_2^{C\\bar{C}}$, $xF_3^{C\\bar{C}}$ and $F_L^{C\\bar{C}}$ are charm-quark contributions to the \ninclusive structure functions $F_2$, $xF_3$ and $F_L$, respectively.\n\n In the kinematic region at HERA, the $F_2^{C\\bar{C}}$ structure function makes a dominant contribution. The $xF_3^{C\\bar{C}}$ structure function contributes only from $Z^0$ exchange and $\\gamma Z^0$, which implies that for the $Q^{2} \\ll M_{Z}^{2}$ region, this contribution may be ignored. Finally, the contribution of longitudinal charm-quark structure function, $F_L^{C\\bar{C}}$, is suppressed only for the $y^2 \\ll 1$ region which can be a few percent in the kinematic region accessible at HERA and therefore cannot be ignored.\n \n Therefore, neglecting the $xF_3^{C\\bar C}$ structure function contribution, the reduced charm-quark cross section, $\\sigma_{red}^{ C\\bar C}$, for both positron and electron beams, can be expressed by  \n\\begin{eqnarray}\n   \\sigma_{red}^{ C\\bar C}&=&   \\frac{d^2\\sigma^{C\\bar C}(e^{\\pm}p)}{dxdQ^2} \\frac{xQ^4}{2\\pi\\alpha^2 Y_{+}} \\nonumber\\\\\n  &= &  F_2^{C\\bar C}  - \\frac{y^2}{Y_{+}}  F_L^{C\\bar C}~. \n    \\label{eq:NCheavylast}\n\\end{eqnarray}\nAccordingly, at high $y$, the reduced charm-quark cross section, $\\sigma_{red}^{ C\\bar C}$, and the $F_2^{C\\bar C}$ structure function only differ by a small $F_L^{C\\bar C}$ contribution \\cite{Daum:1996ec}.\n\n In the QPM, the structure functions depend only on the $Q^2$ \nvariable and then they can be directly related to the PDFs. In QCD, however, and especially when heavy flavour           production is included, the structure functions depend on both $x$ and $Q^2$ variables, \\cite{Sjostrand:2001yu,Aktas:2006hy,DeRoeck:2011na,Ball:2010de,Aaron:2011gp,Tung:2006tb,Aaron:2009aa,Engelen:1998rf,CooperSarkar:2012tx,Frixione:1993yw,Marchesini:1991ch,Jung:1993gf,Sjostrand:1985ys,Sjostrand:1986hx,Lonnblad:1992tz,Kuraev:1976ge,Ciafaloni:1987ur,Jung:2000hk,Beneke:1994rs,Agashe:2014kda,Schmidt:2012az,Alekhin:2010sv,Gao:2013wwa,Martin:1998sq,Pumplin:2002vw,Chekanov:2002pv}. In Section \\ref{methodology}, based on our methodology, we extract the PDFs as functions of $x$ and $Q^2$ variables, using full HERA run I and II combined data, with and without the charm cross section H1-ZEUS combined measurements data set included.\n\n\\section{\\label{methodology}Data Set and Methodology}         \nIn this paper, we use two different data sets: the full HERA run I and II combined NC and CC DIS $e^{\\pm}p$ scattering cross sections \\cite{Abramowicz:2015mha}, and the charm production reduced cross section measurements data \\cite{Abramowicz:1900rp}. In our analysis, we choose the full HERA run I and II combined data as our base data set, and then we investigate the impact of charm production reduced cross section data on simultaneous determination of PDFs and the strong coupling, $\\alpha_s(M^2_Z)$~ in the Thorne-Roberts (RT) and Thorne-Roberts optimal (RTOPT) schemes. The kinematic ranges for these two data sets have been reported in Ref.~\\cite{Vafaee:2017nze}.\n\n We use xFitter \\cite{xFitter}, version 1.2.2, as our QCD fit framework. Using the QCDNUM package \\cite{Botje:2010ay}, version 17-01\/12, we evolved the parton distribution functions and $\\alpha_s(M^2_Z)$. In the evolution of PDFs and $\\alpha_s(M^2_Z)$, we set our theory type based on the DGLAP collinear evolution equations \\cite{DGLAP} and make several fits at leading order and next-to-leading order in the RT and RTOPT schemes.\n \n The RT scheme is a General Mass-Variable Flavour Number Scheme (GM-VFNS). Really, the RT scheme was designed to provide a smooth transition from the massive FFN scheme at low scales $Q^2 \\sim m_h^2$ to the massless ZM-VFNS scheme at high scales $Q^2 \\gg m_h^2$. However, the connection is not unique. A GM-VFNS can be defined by demanding equivalence of the $n_f = n$ (FFNS) and $n_f = n + 1$ flavour (ZM-VFNS) descriptions above the transition\npoint for the new parton distributions. Of course they are by definition identical below this point, at all orders. The RT scheme has two different variants: RT standard and RT optimal, with a smoother transition across the heavy flavour threshold region. A review of the two different schemes has been given in Ref.~\\cite{Vafaee:2017nze}.\n \n  To investigate the impact of charm production reduced cross section data,  we need to use the heavy flavour scheme in our analysis. Different theoretical groups use various heavy flavour schemes. For example, some theory groups such as CT10 \\cite{Lai:2010vv}, ABKM09 \\cite{Alekhin:2009vn}, and NNPDF2.1 \\cite{Ball:2008by,Mironov:2009uv} used  \nS-ACOT \\cite{Collins:1998rz}, FFNS \\cite{Martin:2006qz} and  FONLL \\cite{Forte:2010ta}, respectively and some other groups such as MSTW08 \\cite{Martin:2009iq} and HERAPDF1.5\/2.0 \\cite{Aaron:2009aa} used the RT (also referred to as TR) standard and optimal heavy flavour schemes \\cite{Thorne:2006qt,Thorne:2012az} to calculate the reduced charm cross sections in DIS. On the other hand, to include heavy flavour contributions, the perturbative QCD scales $\\mu_f^2$ and $\\mu_r^2$  play a central rule. Some groups such as CT10 \\cite{Lai:2010vv} and ABKM09  \\cite{Alekhin:2009vn}  choose  $\\mu_f^2 = \\mu_r^2=Q^2+m_C^2$ and $\\mu_f^2 = \\mu_r^2=Q^2+4m_C^2$  respectively, where $m_C$ denotes the pole mass of the charm quark, whereas the NNPDF2.1 \\cite{Ball:2008by,Mironov:2009uv}, HERAPDF1.5 \\cite{Aaron:2009aa} and  MSTW08 \\cite{Martin:2009iq} groups use $\\mu_f = \\mu_r=Q$ in their heavy quark QCD approach.\n  \n  To include the heavy-flavor contributions, we use both RT and RTOPT schemes,  and choose $\\mu_f = \\mu_r=Q$~ as the perturbative quantum chromodynamics scale for the pole mass of the charm quark $m_c=1.5 \\pm 0.15$ GeV.\n  \n The last step in our QCD analysis is the minimization procedure. In this regard, we use the standard MINUIT minimization package~\\cite{James:1975dr}, as a powerful program for minimization, correlations and parameter errors. \n   \n   In order to minimize the influence of higher twist contributions we use kinematic cuts. In the various DIS analyses, different kinds of kinematic cuts should be applied. In this analysis we imposed a kinematic cut $Q^2$=3.5 GeV$^2$  to omit all data with $Q^2$ less than this value. The cuts on the kinematic coverage of the DIS data have been made for values of $Q^2$ between $Q^2=0.045$\\,GeV$^2$ and $Q^2=50000$\\,GeV$^2$ and values of $x_{\\rm Bj}$ \nbetween $x_{\\rm Bj}=6\\times10^{-7}$ and $x_{\\rm Bj}=0.65$. The cuts on $Q^2$  not only significantly increase the number of data points available to constrain PDFs, but also allow access to a greater range of kinematics, which in turn lead to reduced PDF uncertainties, especially at higher values of $x$.\n\n In this analysis, based on the HERAPDF approach \\cite{Abramowicz:2015mha}, we generically parameterized the PDFs of the proton, $xf(x)$, at the initial scale of the QCD evolution $Q^2_0= 1.9$ GeV$^2$ as\n\\begin{equation}\n xf(x) = A x^{B} (1-x)^{C} (1 + D x + E x^2)~~,\n\\label{eqn:pdf}\n\\end{equation}\nwhere in the infinite momentum frame, $x$ is the fraction of the proton's momentum taken by the struck parton.\n  \nTo determine the normalization constants $A$ for the valence and gluon distributions, we use the QCD number and momentum sum rules. Using this functional form, Eq. \\ref{eqn:pdf} leads to the following independent combinations of parton distribution functions:\n\\begin{eqnarray}\n\\label{eq:xgpar}\nxg(x) &=   & A_g x^{B_g} (1-x)^{C_g} - A_g' x^{B_g'} (1-x)^{C_g'}  ,  \\\\\n\\label{eq:xuvpar}\nxu_v(x) &=  & A_{u_v} x^{B_{u_v}}  (1-x)^{C_{u_v}}\\left(1+E_{u_v}x^2 \\right) , \\\\\n\\label{eq:xdvpar}\nxd_v(x) &=  & A_{d_v} x^{B_{d_v}}  (1-x)^{C_{d_v}} , \\\\\n\\label{eq:xubarpar}\nx\\bar{U}(x) &=  & A_{\\bar{U}} x^{B_{\\bar{U}}} (1-x)^{C_{\\bar{U}}}\\left(1+D_{\\bar{U}}x\\right) , \\\\\n\\label{eq:xdbarpar}\nx\\bar{D}(x) &= & A_{\\bar{D}} x^{B_{\\bar{D}}} (1-x)^{C_{\\bar{D}}} .\n\\end{eqnarray}\nwhere $xg(x)$ is the gluon distribution, $xu_{{v}}(x)$ and $xd_{{v}}(x)$ are the valence-quark distributions, and $x\\bar{U}(x)$ and $x\\bar{D}(x)$ are the $u$-type and $d$-type anti-quark distributions, which are identical to the sea-quark distributions. A review of HERAPDF functional form, including some more details, can be found in Ref.~\\cite{Vafaee:2017nze}.\n\n\\section{\\label{qcdfitquality}Impact of Charm Production Data on the QCD fit quality }\nWe now investigate the impact of the charm cross section H1-ZEUS combined measurements on simultaneous determination of PDFs and $\\alpha_s(M^2_Z)$. We also explain how adding these data improve the uncertainty of PDFs, reducing the error bars of some parton distributions, especially gluon distributions and some of their ratios, when the HERA run I and II combined inclusive DIS $e^{\\pm}p$ scattering cross sections data are chosen as a ``BASE''.\nTo investigate the fit quality, we use the $\\chi^2$ definition as reported in Ref.~\\cite{Vafaee:2017nze}.\n\n\n For HERA run I and II combined inclusive DIS $e^{\\pm}p$ scattering cross sections and the charm cross section H1-ZEUS combined measurements, the number of data points are 1307 and 42, respectively. Accordingly, the total number of data points for BASE and BASE plus charm, which we refer to sometimes as ``TOTAL'', are 1307 and 1349, respectively.  In various configurations, the $Q^2 \\ge 1.5$\\,GeV$^2$ range was covered by the HERA run I and II combined data \\cite{Abramowicz:2015mha}. The MINUIT parameters are sensitive to the $Q^2_{min}$ value, so to get a convergent fit result we set $Q^2_{min}=3.5$ GeV$^2$, as suggested in~Ref \\cite{Abramowicz:2015mha}. Clearly, this cut on $Q^2$  omits all data with $Q^2$ less than $Q^2_{min}=3.5$~GeV$^2$ and therefore, reduces the total number of data points from 1307 and 1349 to 1145 and 1192 for the BASE and TOTAL data sets, respectively. Now, based on Table~\\ref{tab:data}, we can present our QCD fit quality as follows:\n\nfor the RT scheme:\n\\begin{eqnarray}\n\\noindent\\centerline{ $\\chi^2_{TOTAL}$ \/ dof = $\\frac{1335}{1131}=1.180~~$for BASE~,} \\label{lobase}\\\\\n\\noindent\\centerline{ $\\chi^2_{TOTAL}$ \/ dof = $\\bf \\frac{1389}{1178}=1.179 ~~$for TOTAL~,} \\label{lototal}\n\\end{eqnarray} \nand for the RTOPT scheme:\n\\begin{eqnarray}\n\\noindent\\centerline{ $\\chi^2_{TOTAL}$ \/ dof = $\\frac{1331}{1131}=1.176~~$for BASE~,} \\label{nlobase}\\\\\n\\noindent\\centerline{ $\\chi^2_{TOTAL}$ \/ dof = $\\bf \\frac{1378}{1178}=1.169 ~~$for TOTAL~,} \\label{nlototal}\n\\end{eqnarray}\nwhere dof refers to the $\\chi^2$ per degrees of freedom and is defined as the number of data points minus the number of free parameters.\nAs we can  see from Eqs.~(19-22), we obtain four different values of $\\chi^2_{\\rm TOTAL}$\/dof, corresponding to four different fits, which in turn imply four different fit-qualities in some PDFs. Now, according to the relative change of $\\chi^2$, which is defined by $\\frac{\\chi^2_{\\rm RT}-\\chi^2_{\\rm RT\\;OPT}}{\\chi^2_{\\rm RT}}$ and the numerical results of Eqs.~(19-22), we see that in going from the RT scheme to the RT OPT scheme, we get $\\sim 0.4$~\\% and $\\sim 0.9$~\\%~improvement in the fit quality, without and with the charm flavour contribution, respectively. Clearly these differences in fit quality imply a significant reduction of some PDF uncertainties, especially for gluon distributions, as we will explain in the next section.\n\n\\begin{table}[h]\n\\begin{center}\n\\begin{tabular}{|l|c|c|c|c|}\n\\hline\n\\hline\n{ \\bf Order} & \\multicolumn{4}{c|}{ {\\bf NLO} }    \\\\ \\hline\n { \\bf Experiment} & {$~~~~$RT BASE$~~~~$} & {RT TOTAL} & { $~~~$RTOPT BASE$~~~$} & { RTOPT TOTAL} \\\\ \\hline\n  HERA I+II CC $e^{+}p$ \\cite{Abramowicz:2015mha} & 44 \/ 39& 45 \/ 39& 44 \/ 39& 44 \/ 39 \\\\ \n  HERA I+II CC $e^{-}p$ \\cite{Abramowicz:2015mha} & 49 \/ 42& 49 \/ 42& 50 \/ 42& 49 \/ 42 \\\\ \n  HERA I+II NC $e^{-}p$ \\cite{Abramowicz:2015mha} & 221 \/ 159& 221 \/ 159& 221 \/ 159& 221 \/ 159 \\\\ \n  HERA I+II NC $e^{+}p$ 460 \\cite{Abramowicz:2015mha} & 208 \/ 204& 209 \/ 204& 210 \/ 204& 210 \/ 204 \\\\ \n  HERA I+II NC $e^{+}p$ 575 \\cite{Abramowicz:2015mha} & 213 \/ 254& 213 \/ 254& 212 \/ 254& 212 \/ 254 \\\\\n  HERA I+II NC $e^{+}p$ 820 \\cite{Abramowicz:2015mha} & 66 \/ 70& 66 \/ 70& 65 \/ 70& 66 \/ 70 \\\\ \n  HERA I+II NC $e^{+}p$ 920 \\cite{Abramowicz:2015mha} & 422 \/ 377& 424 \/ 377& 418 \/ 377& 419 \/ 377 \\\\ \\hline\n {Charm H1-ZEUS} \\cite{Abramowicz:1900rp} & - & 40 \/ 47& - & 38 \/ 47 \\\\ \\hline\n { Correlated ${\\bf \\chi^2}$} & 111& 122& 111& 119 \\\\\\hline\n{\\bf {Total $\\bf{\\chi^2}$ \/ dof}}  & ${\\bf \\frac{1335}{1131}}$ &  ${\\bf \\frac{1389}{1178}}$  & ${\\bf \\frac{1331}{1131}}$ &  ${\\bf \\frac{1378}{1178}}$  \\\\ \\hline\n\\hline\n    \\end{tabular}\n\\vspace{-0.0cm}\n\\caption{\\label{tab:data}{Data sets used in our NLO QCD analysis, with corresponding partial $\\chi^2$ per data point for each data set, including  $\\chi^2$ per degrees of freedom (dof) for the RT and RT OPT schemes.}}\n\\vspace{-0.4cm}\n\\end{center}\n\\end{table}\n\n\\section{\\label{impapdfalphs}Impact of Charm Production Data on PDFs and $\\alpha_s(M^2_Z)$ }\nNow, we present the impact of charm cross section H1-ZEUS combined measurements data on simultaneous determination of PDFs and $\\alpha_s(M^2_Z)$ in the RT and RTOPT schemes and for two separate scenarios. In the first scenario we fix $\\alpha_s(M^2_Z)$ to 0.117 and develop our QCD fit analysis based on only  14 unknown free parameters, according to Eqs.~(14-18). Although in this scenario the value of $\\chi^2_{TOTAL}$ \/ dof is reduced, according to Eqs.~(19-22), from 1.180 to 1.169,  we find nothing to show the impact of charm cross section H1-ZEUS combined measurements data on the PDFs. In the second scenario we consider the strong coupling $\\alpha_s(M^2_Z)$ as an extra free parameter and refit our analysis, but this time with 15 unknown free parameters. Based on the second scenario, not only do we obtain $\\sim 0.4$~\\% and $\\sim 0.9$~\\%~improvement in the fit quality, without and with the charm flavour contribution, respectively, the same as the first scenario, but we also clearly find the impact of charm on the PDFs, especially on the gluon distribution. Some more details about the central role of the strong coupling in pQCD have been reported in Ref.~\\cite{Vafaee:2017nze}.\n\n In Tables \\ref{tab:pa1} and  \\ref{tab:pa2}, we present next-to-leading order numerical values of parameters and their uncertainties for  the \n$xu_v$, $xd_v$, sea and gluon PDFs at the input scale of $Q^2_0 = 1.9$~GeV$^2$ for the two different scenarios.\n\n\\begin{table}[h]\n\\begin{center}\n\\begin{tabular}{|l|c|c|c|c|}\n\\hline\n\\hline\n \\multicolumn{5}{|c|}{ {\\bf First Scenario: The Strong Coupling, $\\alpha_s(M^2_Z)$, is Fixed} }    \\\\ \\hline\n { \\bf Parameter} & {$~~~~$RT BASE$~~~~$} & { $~~~$RT TOTAL$~~~$} & {RTOPT BASE} & {RTOPT TOTAL} \\\\ \\hline\n  ${B_{u_v}}$ & $0.723 \\pm 0.046$& $0.723 \\pm 0.044$& $0.730 \\pm 0.042$& $0.726 \\pm 0.039$ \\\\ \n  ${C_{u_v}}$ & $4.841 \\pm 0.088$& $4.833 \\pm 0.087$& $4.827 \\pm 0.086$& $4.823 \\pm 0.084$ \\\\ \n  $E_{u_v}$ & $13.6 \\pm 2.6$& $13.5 \\pm 2.5$& $13.1 \\pm 2.3$& $13.2 \\pm 2.1$ \\\\ \\hline\n  ${B_{d_v}}$ & $0.818 \\pm 0.095$& $0.826 \\pm 0.094$& $0.825 \\pm 0.094$& $0.822 \\pm 0.095$ \\\\ \n  $C_{d_v}$ & $4.16 \\pm 0.40$& $4.18 \\pm 0.39$& $4.21 \\pm 0.39$& $4.19 \\pm 0.38$ \\\\ \\hline\n  $C_{\\bar{U}}$ & $8.91 \\pm 0.81$& $8.72 \\pm 0.78$& $8.90 \\pm 0.81$& $8.70 \\pm 0.80$ \\\\ \n  $D_{\\bar{U}}$ & $17.7 \\pm 3.3$& $16.4 \\pm 3.0$& $17.6 \\pm 3.3$& $16.5 \\pm 3.2$ \\\\ \n  $A_{\\bar{D}}$ & $0.158 \\pm 0.011$& $0.160 \\pm 0.011$& $0.1561 \\pm 0.0098$& $0.1594 \\pm 0.0099$ \\\\ \n  $B_{\\bar{D}}$ & $-0.1682 \\pm 0.0082$& $-0.1666 \\pm 0.0080$& $-0.1760 \\pm 0.0074$& $-0.1732 \\pm 0.0073$ \\\\ \n  $C_{\\bar{D}}$ & $4.2 \\pm 1.3$& $4.5 \\pm 1.3$& $4.0 \\pm 1.2$& $4.3 \\pm 1.3$ \\\\\\hline\n  $B_g$ & $-0.11 \\pm 0.16$& $-0.12 \\pm 0.15$& $-0.07 \\pm 0.13$& $-0.09 \\pm 0.12$ \\\\ \n  $C_g$ & $11.2 \\pm 1.7$& $10.7 \\pm 1.4$& $12.3 \\pm 1.7$& $11.8 \\pm 1.5$  \\\\ \n  $A_g'$ &  $2.1 \\pm 1.5$& $1.9 \\pm 1.2$& $2.9 \\pm 1.6$& $2.7 \\pm 1.1$  \\\\ \n  ${B_g'}$ & $-0.206 \\pm 0.079$& $-0.216 \\pm 0.075$& $-0.142 \\pm 0.083$& $-0.162 \\pm 0.098$ \\\\  \n  \\hline \n  {${ \\alpha_s(M^2_Z)}$} & $ 0.1176$ & $ 0.1176$ & $ 0.1176$ & $0.1176$ \\\\ \\hline\n\\hline\n    \\end{tabular}\n\\vspace{-0.0cm}\n\\caption{\\label{tab:pa1}{ {The NLO numerical values of parameters and their uncertainties for the $xu_v$, $xd_v$, $x\\bar u$, $x\\bar d$, $x\\bar s$ and $xg$ PDFs at the initial scale of $Q^2_0 = 1.9$~GeV$^2$ in the first scenario, where the strong coupling, $\\alpha_s(M_Z^2)$, is fixed to $0.117$.}}}\n\\vspace{-0.4cm}\n\\end{center}\n\\end{table}\n\n\\begin{table}[h]\n\\begin{center}\n\\begin{tabular}{|l|c|c|c|c|}\n\\hline\n\\hline\n \\multicolumn{5}{|c|}{ {\\bf Second Scenario: The Strong Coupling, $\\alpha_s(M^2_Z)$, is Free} }    \\\\ \\hline\n { \\bf Parameter} & {$~~~~$RT BASE$~~~~$} & { $~~~$RT TOTAL$~~~$} & {RTOPT BASE} & {RTOPT TOTAL} \\\\ \\hline\n  ${B_{u_v}}$ & $0.712 \\pm 0.047$& $0.725 \\pm 0.048$& $0.710 \\pm 0.045$& $0.709 \\pm 0.043$ \\\\ \n  ${C_{u_v}}$ & $4.88 \\pm 0.12$& $4.83 \\pm 0.12$& $4.89 \\pm 0.11$& $4.87 \\pm 0.10$ \\\\ \n  $E_{u_v}$ & $13.9 \\pm 2.3$& $13.4 \\pm 2.2$& $13.7 \\pm 2.2$& $13.7 \\pm 2.3$ \\\\ \\hline\n  ${B_{d_v}}$ & $0.811 \\pm 0.094$& $0.826 \\pm 0.096$& $0.812 \\pm 0.093$& $0.813 \\pm 0.091$ \\\\ \n  $C_{d_v}$ &  $4.18 \\pm 0.38$& $4.17 \\pm 0.39$& $4.24 \\pm 0.38$& $4.22 \\pm 0.39$ \\\\ \\hline\n  $C_{\\bar{U}}$ & $9.09 \\pm 0.89$& $8.70 \\pm 0.90$& $9.21 \\pm 0.87$& $8.97 \\pm 0.84$ \\\\ \n  $D_{\\bar{U}}$ & $18.5 \\pm 3.9$& $16.3 \\pm 3.7$& $19.2 \\pm 3.9$& $17.8 \\pm 3.5$ \\\\ \n  $A_{\\bar{D}}$ & $0.160 \\pm 0.012$& $0.160 \\pm 0.011$& $0.158 \\pm 0.010$& $0.1610 \\pm 0.0100$ \\\\ \n  $B_{\\bar{D}}$ & $-0.1657 \\pm 0.0099$& $-0.167 \\pm 0.010$& $-0.1728 \\pm 0.0083$& $-0.1704 \\pm 0.0079$  \\\\ \n  $C_{\\bar{D}}$ & $4.4 \\pm 1.4$& $4.5 \\pm 1.4$& $4.4 \\pm 1.3$& $4.6 \\pm 1.3$ \\\\\\hline\n  $B_g$ & $-0.13 \\pm 0.11$& $-0.12 \\pm 0.12$& $-0.10 \\pm 0.12$& $-0.12 \\pm 0.10$ \\\\ \n  $C_g$ & $11.8 \\pm 2.2$& $10.6 \\pm 2.0$& $13.5 \\pm 2.3$& $12.7 \\pm 2.3$  \\\\ \n  $A_g'$ & $2.3 \\pm 1.1$& $1.87 \\pm 0.84$& $3.4 \\pm 1.6$& $3.0 \\pm 1.6$ \\\\ \n  ${B_g'}$ & $-0.217 \\pm 0.074$& $-0.215 \\pm 0.073$& $-0.164 \\pm 0.096$& $-0.182 \\pm 0.074$ \\\\  \n  \\hline \n  {${ \\alpha_s(M^2_Z)}$} & $0.1161 \\pm 0.0037$& $0.1178 \\pm 0.0038$& $0.1151 \\pm 0.0032$& $0.1154 \\pm 0.0028$ \\\\ \\hline\n\\hline\n    \\end{tabular}\n\\vspace{-0.0cm}\n\\caption{\\label{tab:pa2}{ {The NLO numerical values of parameters and their uncertainties for  the \n$xu_v$, $xd_v$, $x\\bar u$, $x\\bar d$, $x\\bar s$ and $xg$ PDFs at the initial scale of $Q^2_0 = 1.9$~GeV$^2$ in the second scenario, where the strong coupling, $\\alpha_s(M_Z^2)$, is taken as an extra free parameter.}}}\n\\vspace{-0.4cm}\n\\end{center}\n\\end{table}\n\n\n According to the numerical results in Table~\\ref{tab:pa2}, when we add the charm cross section H1-ZEUS combined measurements data to the  HERA run I and II combined data, the numerical value of $\\alpha_s(M^2_Z)$ changes from $0.1161 \\pm 0.0037$ to $0.1178 \\pm 0.0038$  and from $0.1151 \\pm 0.0032$ to $0.1154 \\pm 0.0028$, for the RT and RTOPT schemes, without and with charm flavour data included, respectively.  If we compare our results for  $\\alpha_s(M^2_Z)$ for RT TOTAL and RTOPT TOTAL with the world average value, $\\alpha_s(M^2_Z)=0.1185 \\pm 0.0006$, which was recently reported by the PDG \\cite{Agashe:2014kda}, we  find a good agreement with the world average value. Of course, it should be noted, since the PDG value of $\\alpha_s(M^2_Z)$ is extracted by global fits to a variety of experimental data, it has a much smaller uncertainty. In other words, although our QCD analysis has been performed based on only two data sets, our numerical results for the strong coupling are in good agreement with the world average value. Also, these values of strong coupling show the impact of the RT and RTOPT schemes on the determination of $\\alpha_s(M^2_Z)$, when considered as an extra free parameter.  \n\n\n\\clearpage\n\\section{\\label{Results}Results}\nAccording to Table~\\ref{tab:f1}, in going from the RT scheme to the RTOPT scheme, we get $\\sim 0.4$~\\% and $\\sim 0.9$~\\%~improvement in the fit quality, without and with the charm flavour contributions included, respectively. Also, according to Table~\\ref{tab:f2}, in going from the RT scheme to the RTOPT scheme, we get $\\sim 0.9$~\\% and $\\sim 2.0$~\\%~improvement in the  $\\alpha_s(M^2_Z)$ value, without and with the charm flavour contributions respectively. \n\\begin{table}[h]\n\\begin{center}\n\\begin{tabular}{|l|c|c|c|c|}\n\\hline\n\\hline\n \\multicolumn{3}{|c|}{ {\\bf First Scenario: The Strong Coupling, $\\alpha_s(M^2_Z)$ is Fixed} }    \\\\ \\hline\n {Scheme} & $\\chi^2_{\\rm TOTAL}\/dof$ & ${\\alpha_s(M^2_Z)}$ \\\\ \\hline \n  {RT BASE} & {${ 1335\/1131}$}& $0.1176$  \\\\\n  {RT TOTAL} & {${1389\/1178}$}& $0.1176$  \\\\\n  {RTOPT BASE} & {${1331\/1131}$}& $0.1176$  \\\\ \n  {RTOPT TOTAL} & {${1378\/1178}$}& $0.1176$  \\\\ \\hline\n\\hline\n    \\end{tabular}\n\\vspace{-0.0cm}\n\\caption{\\label{tab:f1}{ Comparison of the numerical values of $\\frac{\\chi^2_{\\rm TOTAL}}{dof}$ for the RT and RTOPT schemes in the first scenario, where the strong coupling, $\\alpha_s(M_Z^2)$, is fixed to $0.117$. RTOPT TOTAL has the best fit quality, as an impact of adding charm cross section H1-ZEUS combined measurements data to HERA I and II combined data. }}\n\\vspace{-0.4cm}\n\\end{center}\n\\end{table} \n\n\\begin{table}[h]\n\\begin{center}\n\\begin{tabular}{|l|c|c|c|c|}\n\\hline\n\\hline\n \\multicolumn{3}{|c|}{ {\\bf Second Scenario: The Strong Coupling, $\\alpha_s(M^2_Z)$ is Free} }    \\\\ \\hline\n {Scheme} & $\\chi^2_{\\rm TOTAL}\/dof$ & ${\\alpha_s(M^2_Z)}$ \\\\ \\hline \n  {RT BASE}  & {${1335\/1130}$}& $0.1161 \\pm 0.0037$  \\\\\n  {RT TOTAL} & {${1389\/1177}$}& $0.1178 \\pm 0.0038$  \\\\\n  {RTOPT BASE} & {${1330\/1130}$}& $0.1151 \\pm 0.0032$  \\\\  \n  {RTOPT TOTAL} & {${1377\/1177}$}& $0.1154 \\pm 0.0028$ \\\\ \\hline\n\\hline\n    \\end{tabular}\n\\vspace{-0.0cm}\n\\caption{\\label{tab:f2}{Comparison of the numerical values of $\\frac{\\chi^2_{\\rm TOTAL}}{dof}$ and ${\\alpha_s(M^2_Z)}$ for the RT and RTOPT schemes in the second scenario, where the strong coupling, $\\alpha_s(M_Z^2)$, is taken as an extra free parameter. RTOPT TOTAL has the best fit quality and improvement in oupling, $\\alpha_s(M^2_Z)$, as an impact of adding charm cross section H1-ZEUS combined measurements data to HERA I and II combined data.}}\n\\vspace{-0.4cm}\n\\end{center}\n\\end{table} \n\n\n In Fig.~\\ref{fig:1},  we illustrate the consistency of HERA measurements of the reduced deep inelastic  $e^{\\pm}p$  scattering  cross sections data \\cite{Abramowicz:2015mha} and charm production reduced cross section measurements data \\cite{Abramowicz:1900rp} with the theory predictions as a function of $x$ and for different values of $Q^2$. According to our QCD analysis, we have good agreement between the theoretical and experimental data. The uncertainties on the cross sections in Fig.~\\ref{fig:1} are obtained using Hessian error propagation. The corresponding $\\frac{\\chi^2_{\\rm TOTAL}}{dof}$ values for each of the data sets in Fig.~\\ref{fig:1} are listed in Table~\\ref{tab:data}. \n\n\\begin{figure*}\n\\includegraphics[width=0.19\\textwidth]{.\/fig\/f1}\n\\includegraphics[width=0.19\\textwidth]{.\/fig\/f2}\n\\includegraphics[width=0.19\\textwidth]{.\/fig\/f3}\n\\includegraphics[width=0.19\\textwidth]{.\/fig\/f4}\n\\includegraphics[width=0.19\\textwidth]{.\/fig\/f5}\n\n\\includegraphics[width=0.19\\textwidth]{.\/fig\/f6}\n\\includegraphics[width=0.19\\textwidth]{.\/fig\/f7}\n\\includegraphics[width=0.19\\textwidth]{.\/fig\/f8}\n\\includegraphics[width=0.19\\textwidth]{.\/fig\/f9}\n\\includegraphics[width=0.19\\textwidth]{.\/fig\/f10}\n\n\\includegraphics[width=0.19\\textwidth]{.\/fig\/f11}\n\\includegraphics[width=0.19\\textwidth]{.\/fig\/f12}\n\\includegraphics[width=0.19\\textwidth]{.\/fig\/f13}\n\\includegraphics[width=0.19\\textwidth]{.\/fig\/f14}\n\\includegraphics[width=0.19\\textwidth]{.\/fig\/f15}\n\n\\includegraphics[width=0.19\\textwidth]{.\/fig\/f16}\n\\includegraphics[width=0.19\\textwidth]{.\/fig\/f17}\n\\includegraphics[width=0.19\\textwidth]{.\/fig\/f18}\n\\includegraphics[width=0.19\\textwidth]{.\/fig\/f19}\n\\includegraphics[width=0.19\\textwidth]{.\/fig\/f20}\n\n\\includegraphics[width=0.19\\textwidth]{.\/fig\/f21}\n\\includegraphics[width=0.19\\textwidth]{.\/fig\/f22}\n\\includegraphics[width=0.19\\textwidth]{.\/fig\/f23}\n\\includegraphics[width=0.19\\textwidth]{.\/fig\/f24}\n\\includegraphics[width=0.19\\textwidth]{.\/fig\/f25}\n\n\\includegraphics[width=0.19\\textwidth]{.\/fig\/f26}\n\\includegraphics[width=0.19\\textwidth]{.\/fig\/f27}\n\\includegraphics[width=0.19\\textwidth]{.\/fig\/f28}\n\\includegraphics[width=0.19\\textwidth]{.\/fig\/f29}\n\\includegraphics[width=0.19\\textwidth]{.\/fig\/f30}\n\n\\caption{Illustrations of the consistency of HERA combined measurements of the reduced DIS  $e^{\\pm}p$ data \\cite{Abramowicz:2015mha} and the theory predictions as a function of $x$ and for different values of $Q^2$.}\n\\label{fig:1}\n\\end{figure*}\n\n\n\n The impact of charm cross section H1-ZEUS combined measurements data on HERA I and II combined data for gluon distribution functions are shown in Figs. \\ref{fig:2} and \\ref{fig:3}, at the starting value of $Q_0^2$ = 1.9~GeV$^2$ and $Q^2$ = 3, 4 and 5~GeV$^2$, in the RT and RTOPT schemes and for two separate scenarios. Clearly, in the first scenario, where the strong coupling $\\alpha_s(M_Z^2)$ is fixed, we find no impact from adding charm H1-ZEUS combined data to the HERA I and II combined data. In the second scenario, however, where we consider the strong coupling $\\alpha_s(M_Z^2)$ as an extra free parameter, we clearly find the impact of adding charm H1-ZEUS combined data to the HERA I and II combined data.\n  \n\\begin{figure*}\n\\includegraphics[width=0.23\\textwidth]{.\/fig\/ng1}\n\\includegraphics[width=0.23\\textwidth]{.\/fig\/ng2}\n\\includegraphics[width=0.23\\textwidth]{.\/fig\/ng3}\n\\includegraphics[width=0.23\\textwidth]{.\/fig\/ng4}\n\n\\includegraphics[width=0.23\\textwidth]{.\/fig\/g1}\n\\includegraphics[width=0.23\\textwidth]{.\/fig\/g2}\n\\includegraphics[width=0.23\\textwidth]{.\/fig\/g3}\n\\includegraphics[width=0.23\\textwidth]{.\/fig\/g5}\n\n\\caption{The gluon PDFs as extracted for the RT scheme in two separate scenarios. These distributions are plotted at the starting value of $Q_0^2$ = 1.9~GeV$^2$ and $Q^2$ = 3, 4 and 5~GeV$^2$, as a function of $x$. The upper four  diagrams correspond to the first scenario, where the strong coupling, $\\alpha_s(M_Z^2)$, is fixed and we find no impact of adding charm H1-ZEUS combined data to the HERA I and II combined data. The lower four diagrams  correspond to the second scenario, where we consider the strong coupling, $\\alpha_s(M_Z^2)$, as an extra free parameter, clearly revealing the impact of adding charm H1-ZEUS combined data to the HERA I and II combined data.}\n\\label{fig:2}\n\\end{figure*}\n\n\\begin{figure*}\n\\includegraphics[width=0.23\\textwidth]{.\/fig\/ng5}\n\\includegraphics[width=0.23\\textwidth]{.\/fig\/ng6}\n\\includegraphics[width=0.23\\textwidth]{.\/fig\/ng7}\n\\includegraphics[width=0.23\\textwidth]{.\/fig\/ng8}\n\n\\includegraphics[width=0.23\\textwidth]{.\/fig\/g5}\n\\includegraphics[width=0.23\\textwidth]{.\/fig\/g6}\n\\includegraphics[width=0.23\\textwidth]{.\/fig\/g7}\n\\includegraphics[width=0.23\\textwidth]{.\/fig\/g8}\n\n\n\\caption{The gluon PDFs as extracted for the RTOPT scheme in two separate scenarios, at the starting value of $Q_0^2$ = 1.9~GeV$^2$ and $Q^2$ = 3, 4 and 5~GeV$^2$, as a function of $x$. The impact of adding charm data can be seen only in the four lower diagrams, where the strong coupling, $\\alpha_s(M_Z^2)$, is considered as an extra free parameter.}\n\\label{fig:3}\n\\end{figure*}\n\n\n The partial gluon distribution functions are shown in Figs. \\ref{fig:4} and \\ref{fig:5}, at $Q^2$ = 1.9, 3, 5 and 10~GeV$^2$ in the RT and RTOPT schemes and for two separate scenarios. The impact of adding charm H1-ZEUS combined data to the HERA I and II combined data can be seen only in the second scenario, where the strong coupling, $\\alpha_s(M_Z^2)$, is considered as an extra free parameter.  \n\n\n\n\\begin{figure*}\n\\includegraphics[width=0.23\\textwidth]{.\/fig\/ngr1}\n\\includegraphics[width=0.23\\textwidth]{.\/fig\/ngr2}\n\\includegraphics[width=0.23\\textwidth]{.\/fig\/ngr3}\n\\includegraphics[width=0.23\\textwidth]{.\/fig\/ngr4}\n\n\\includegraphics[width=0.23\\textwidth]{.\/fig\/gr1}\n\\includegraphics[width=0.23\\textwidth]{.\/fig\/gr2}\n\\includegraphics[width=0.23\\textwidth]{.\/fig\/gr3}\n\\includegraphics[width=0.23\\textwidth]{.\/fig\/gr4}\n\n\\caption{The partial gluon PDFs as extracted for the RT scheme in two separate scenarios. These PDFs are plotted at the starting value of $Q_0^2$ = 1.9~GeV$^2$ and $Q^2$ = 3, 5 and 10~GeV$^2$, as a function of $x$. The upper four diagrams are based on a fixed strong coupling, and do not show the impact of adding charm flavour. The lower four diagrams, based on the second scenario, where $\\alpha_s(M_Z^2)$ is considered as an extra free parameter, clearly show this impact. }\n\\label{fig:4}\n\\end{figure*}\n\n\\begin{figure*}\n\\includegraphics[width=0.23\\textwidth]{.\/fig\/ngr5}\n\\includegraphics[width=0.23\\textwidth]{.\/fig\/ngr6}\n\\includegraphics[width=0.23\\textwidth]{.\/fig\/ngr7}\n\\includegraphics[width=0.23\\textwidth]{.\/fig\/ngr8}\n\n\\includegraphics[width=0.23\\textwidth]{.\/fig\/gr5}\n\\includegraphics[width=0.23\\textwidth]{.\/fig\/gr6}\n\\includegraphics[width=0.23\\textwidth]{.\/fig\/gr7}\n\\includegraphics[width=0.23\\textwidth]{.\/fig\/gr8}\n\n\n\\caption{The partial of gluon PDFs as extracted for the RTOPT scheme, at the starting value of $Q_0^2$ = 1.9~GeV$^2$ and $Q^2$ = 3, 5 and 10~GeV$^2$, as a function of $x$. The impact of adding charm cross section H1-ZEUS combined  data can bee seen in the four lower diagrams, which have been plotted based on the second scenario.}\n\\label{fig:5}\n\\end{figure*}\n\nThe total sea quark $\\Sigma$-PDFs are defined by $\\Sigma=2x(\\bar u+\\bar d+\\bar s+\\bar c)$. In Figs.~\\ref{fig:6} and \\ref{fig:7}  \nwe plot the partial ratio of gluon distributions over the $\\Sigma$-PDF to show the impact of adding charm cross section H1-ZEUS combined  data to HERA I and II combined data, at $Q^2$ = 4, 5, 100 and 10000~GeV$^2$ in the RT and RTOPT schemes and for the two different scenarios. Clearly, these impacts can be seen only in the second scenario.\n\n\\begin{figure*}\n\\includegraphics[width=0.23\\textwidth]{.\/fig\/ngo1}\n\\includegraphics[width=0.23\\textwidth]{.\/fig\/ngo2}\n\\includegraphics[width=0.23\\textwidth]{.\/fig\/ngo3}\n\\includegraphics[width=0.23\\textwidth]{.\/fig\/ngo4}\n\n\\includegraphics[width=0.23\\textwidth]{.\/fig\/go1}\n\\includegraphics[width=0.23\\textwidth]{.\/fig\/go2}\n\\includegraphics[width=0.23\\textwidth]{.\/fig\/go3}\n\\includegraphics[width=0.23\\textwidth]{.\/fig\/go4}\n\n\\caption{The partial ratio of gluon distributions over $\\Sigma$-PDFs, as extracted for the RT scheme in two separate scenarios, at $Q^2$ = 4, 5, 100 and 1000~GeV$^2$, as a function of $x$. Only the four lower diagrams, corresponding to second scenario, show the impact of charm flavour on the PDFs.}\n\\label{fig:6}\n\\end{figure*}\n\n\\begin{figure*}\n\\includegraphics[width=0.23\\textwidth]{.\/fig\/ngo5}\n\\includegraphics[width=0.23\\textwidth]{.\/fig\/ngo6}\n\\includegraphics[width=0.23\\textwidth]{.\/fig\/ngo7}\n\\includegraphics[width=0.23\\textwidth]{.\/fig\/ngo8}\n\n\\includegraphics[width=0.23\\textwidth]{.\/fig\/go5}\n\\includegraphics[width=0.23\\textwidth]{.\/fig\/go6}\n\\includegraphics[width=0.23\\textwidth]{.\/fig\/go7}\n\\includegraphics[width=0.23\\textwidth]{.\/fig\/go8}\n\n\n\\caption{The partial ratio of gluon distributions over $\\Sigma$-PDFs, as extracted for the RTOPT scheme. The upper four diagrams correspond to the first scenario, while the lower four diagrams correspond to the second scenario and clearly show the impact of charm flavour data on the PDFs. }\n\\label{fig:7}\n\\end{figure*}\n\n\\clearpage\n\\section{\\label{Summary}Summary}\nUp to 36 percent of the cross sections at HERA originate from processes with charm quarks in the final state. In this QCD analysis we investigated the simultaneous impact of charm quark cross section H1-ZEUS combined data on the PDFs and on the determination of the strong coupling.\n\nWe chose the full HERA run I and II  DIS charged and neutral current data as a base data set and developed our QCD analysis at next-to-leading order in both RT and RTOPT schemes and for two separate scenarios using the HERAPDF parametrization form.\n\nThe sensitivity of PDF uncertainties to reduced charm quark cross section H1-ZEUS combined data at next-to-leading order, especially when in our second scenario we take the strong coupling, $\\alpha_s(M^2_Z)$, as an extra free parameter, is reported in this QCD analysis.\n\n This analysis shows a dramatic reduction of some PDF uncertainties and good agreement of the strong coupling constant, $\\alpha_s(M^2_Z)$, with the world average value, when the reduced charm quark cross section H1-ZEUS combined data are included.\n \n As we mentioned, the strong coupling, $\\alpha_s(M^2_Z)$, plays a central role in the pQCD factorization theorem and the result of this QCD-analysis emphasis on its dramatic correlation with the PDFs reveals the impact of the charm flavour contribution.\n \n According our QCD analysis, in going from the RT scheme to the RTOPT scheme, we get $\\sim 0.4$~\\% and $\\sim 0.9$~\\%~improvement in the fit quality, without and with the charm flavour contribution, respectively.\n Also, we show that in going from the RT scheme to the RTOPT scheme, we get $\\sim 0.9$~\\% and $\\sim 2.0$~\\%~improvement in the strong coupling value, without and with the charm flavour contribution, respectively.\n\n A standard LHAPDF library file of this QCD analysis at next-to-leading order is available and can be obtained from the author via e-mail.\n \n    \n\\clearpage\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\n\nIn this work we raise  one generic issue with Maxwell equations in curved space-time. Specifically,\nwe focus on the case of pure electrostatic systems in the noninertial reference frame with a globally time-independent metric tensor with $g_{0i}\\neq0$.\nEvidently, for an observer in a flat space-time and inertial frame the charges in motion do generate both \nelectric and magnetic fields. The issue is what are the fields seen by an observer in the noninertial frame  comoving with the\nelectric charges, so that to this observer  the electric current is a vanishing one. Will he see pure\nelectric field or shall there be some trace of the noninertial motion in the form of a residual magnetic field?\nFrom the point of view of Einstein's general relativity theory such a magnetic field if exists would be \nof geometric origin and will have a peculiar property of not to be subjected to screening by the\nconventional magnetic shields.\n\n\nA focus of this paper will be on the special case of noninertial motion of practical interest: magnetic fields\nin electrostatic systems residing at rest in the curved space-time of the rotating body.\nOur principal statement is  that the full system of Maxwell  equations entails a non-vanishing \nmagnetic field despite zero currents. In terms of the\nmetric properties of the relevant space-time, the effect is due to the nonvanishing off-diagonal components \nof the metric tensor $g_{0i}, \\ i=1,2,3$. In cases of practical interest, this geometric magnetic field is driven by the Earth rotation.\n\n\n\nOur primary motivation was  the recent interest in the impact of the\ngravity induced spin rotations in the ultimate sensitivity searches for\nthe electric dipole moments (EDM) of neutrons and of charged particles in the storage ring experiments \\cite{Silenko2007}-\\cite{Abusaif2019}.  The effect of the Earth\nrotation in the neutron EDM experiments has already received much attention \\cite{Baker2007},  \\cite{Lamoreaux2007},\n\\cite{Serebrov2015}.  The impact of the\ngeometric magnetic field has not been discussed before. As we shall see, this field has a\nstrong dependence on the configuration of the static electric field.   In the  geometry  of the neutron EDM experiments, the geometric\nmagnetic field is a nonuniform one with the nonvanishing gradient. Such a gradient of the magnetic\nfields is of particular concern in the comagnetometry which is crucial in the neutron EDM experiments (\\cite{Abel2019} and references therein) . To this end, the salient feature of the geometric magnetic field is that it changes a sign when the external electric field is flipped. Consequently,\ninteraction of the magnetic moment of the particle with the geometric magnetic field will generate a false signal of EDM. Numerically, the false EDM is still below the sensitivity of the current neutron EDM experiments, but will be in the\nballpark of the ultimate sensitivity neutron EDM experiments aiming atvimproving the existing upper bound by  one order in magnitude and beyond \\cite{Chupp2019}.\n\n\nThe further presentation is organized as follows. A treatment of physics in the noninertial frames demands for a consistent use of Einsten's general relativity (GR) formalism. In  Section II we invoke GR to derive the results for the geometry of the curved spfce-time and dynamics of spinning charged particles in such spaces. \nIn particular, since the space-time is curved, one needs to elucidate the meaning of the  \nmagnetic field. Specifically, we check that the magnetic  field excerts on a charged particle a force which rotates its velocity ${\\bf v}$ (defined in a certain orthonormal basis) preserving its magnitude, and induces the conventional precession of its magnetic moment. Trivial though may they sound, these constraints are important, when the nonvanishing geometrical magnetic field is present.\n\n\nA subject of Section III is a derivation of the geometric magnetic field in the pure electrostatic systems at rest in the noninertial reference frames. We demonstrate that the geometric magnetic field is a salient feature of Maxwell equations in the stationary metrics with $g_{0i}\\neq0$. In the simplest case of the laboratory residing on the rotating Earth, a connection between the geometric magnetic field, which is an axial vector, and the polar external electric field, contains the angular velocity of  Earth's rotation, so that the parity conservation is warranted.  \n\nIn Section IV we comment on three examples of the geometric magnetic field. The most interesting case at present is a search for the EDM of neutrons via precession of the neutron spin in the external electric field. Here the geometric magnetic field happens to have a finite gradient along the electric field and gives rise to the false EDM signal. Although tiny, this false EDM is in the ballpark of future ultimate precision neutron EDM experiments. The second example is the geometric magnetic field in all electric storage rings. Such a ring is discussed as a dedicated proton EDM machine \\cite{Abusaif2019,Anastopoulos2016}. In this case, in contrast to the constant residual magnetic field of the Earth, the geometric magnetic moment has a quadrupole-like behaviour along the circumference of the storage ring. Finally, for the sake of academic completeness, we report an exact solution for the uniformly charged sphere at rest on Earth's surface. \n\nThe major points of this study are summarized in the Conclusions. The Appendix provides a brief survey of the GR formalism our derivations are based upon.\n\n\n \n  \n\\section{Maxwell equations and particle and spin dynamics}\n\n\n\n\\subsection{ Maxwell  equations}\n\n\n The local\ncoordinates are denoted as $x^{\\mu},\\,\\mu=0,1,2,3=(0,i)$, $i=1,2,3$. \nLet the electromagnetic field (2-form) be expressed through 4-potential $A_{\\mu}$ (1-form) in local coordinates as:\n\\begin{gather}\nF_{\\mu\\nu}=\\partial_{\\mu}A_{\\nu}-\\partial_{\\nu}A_{\\mu}.\n\\label{M10}\n\\end{gather}\nThen the homogeneous Maxwell equations\n\\begin{gather}\n\\varepsilon_{\\mu\\nu\\lambda\\rho}\\partial_{\\nu}F_{\\lambda\\rho}=0\n\\label{M20}\n\\end{gather} \nare satisfied automatically. The field (\\ref{M10}) is a holonomic one. \n\nWe have also  the inhomogeneous Maxwell equations in the local coordinates:\n\\begin{gather}\n\\frac{1}{\\sqrt{-g}}\\partial_{\\nu}\\left(\\sqrt{-g}F^{\\nu\\mu}\\right)=4\\pi J^{\\mu},  \n\\nonumber \\\\ \ng\\equiv\\det g_{\\mu\\nu},  \\quad F^{\\mu\\nu}=g^{\\mu\\lambda}g^{\\nu\\rho}F_{\\lambda\\rho}.\n\\label{M30}\n\\end{gather} \n\n\nThe electric and magnetic fields in ONB are defined by the usual rules (see Appendix D):\n\\begin{gather}\n{\\bf E}^{\\alpha}=F^{\\alpha 0}=e^{\\alpha}_{\\mu}e^0_{\\nu}F^{\\mu\\nu}=e^{\\alpha}_ie^0_0F^{i0}+e^{\\alpha}_ie^0_jF^{ij},\n\\label{M40}\n\\end{gather}\n\\begin{gather}\n\\varepsilon_{\\alpha\\beta\\gamma}{\\bf H}^{\\gamma}=-F^{\\alpha\\beta}=-e^{\\alpha}_{\\mu}e^{\\beta}_{\\nu}F^{\\mu\\nu},\n\\quad \\varepsilon_{123}=1.\n\\label{M50}\n\\end{gather}\n\n\n\n\n\n\\subsection{The particle and spin dynamics}\n\n\n\nLet $\\mathop{\\rm d}\\nolimits s\\equiv \\sqrt{g_{\\mu\\nu}\\mathop{\\rm d}\\nolimits x^{\\mu}\\mathop{\\rm d}\\nolimits x^{\\nu}}$ be infinitely small interval  when particle is moving\nalong its  world line. We use the standard  notations $D X^{\\mu}\/\\mathop{\\rm d}\\nolimits s\\equiv u^{\\nu}\\nabla_{\\nu}X^{\\mu}$,\nwhere $u^{\\nu}\\equiv \\mathop{\\rm d}\\nolimits x^{\\nu}\/\\mathop{\\rm d}\\nolimits s$, so that $g_{\\mu\\nu}u^{\\mu}u^{\\nu}=1$. According to the definition and\nEqs. (\\ref{G65}),  (\\ref{G70}) we have\n\\begin{gather}\n\\frac{D X^a}{\\mathop{\\rm d}\\nolimits s}=\\frac{\\mathop{\\rm d}\\nolimits X^a}{\\mathop{\\rm d}\\nolimits s}+\\left(\\gamma^a_{b\\mu}\\frac{\\mathop{\\rm d}\\nolimits x^{\\mu}}{\\mathop{\\rm d}\\nolimits s}\\right)X^b=\\frac{\\mathop{\\rm d}\\nolimits X^a}{\\mathop{\\rm d}\\nolimits s}+\n\\left(\\gamma^a_{bc}u^c\\right)X^b.\n\\label{D5}\n\\end{gather}\nThen the equation of motion of a charged particle in the local coordinates has the form\n\\begin{gather}\nmc\\frac{D u^{\\mu}}{\\mathop{\\rm d}\\nolimits s}=\\frac{q}{c}F^{\\mu\\nu}u_{\\nu}.\n\\label{D10}\n\\end{gather}\n\nEvidently, inside the \"freely falling elevator\" in the  Riemann normal coordinates (\\ref{G160}),\nthe dynamic equations are of the same form as in a flat space-time with Cartesian coordinates:\n\\begin{gather}\n\\frac{\\mathop{\\rm d}\\nolimits \\gamma}{\\mathop{\\rm d}\\nolimits t}=\\frac{q}{mc}{\\bf E}{\\boldsymbol\\beta},\n\\nonumber \\\\\n\\frac{\\mathop{\\rm d}\\nolimits(\\gamma{\\boldsymbol\\beta}^{\\alpha})}{\\mathop{\\rm d}\\nolimits t}=\n\\frac{q}{mc}\n\\left({\\bf E}+[{\\boldsymbol\\beta},\\,{\\bf H}]\\right)^{\\alpha},\n\\nonumber \\\\\nu^a=\\gamma(1,\\,{\\boldsymbol\\beta}),  \\quad {\\boldsymbol\\beta}\\equiv{\\bf v}\/c, \\quad\n\\mathop{\\rm d}\\nolimits s=\\frac{c}{\\gamma}\\mathop{\\rm d}\\nolimits t.\n\\label{D20N}\n\\end{gather}\nHere $\\mathop{\\rm d}\\nolimits t$ is the proper time interval in the laboratory frame of reference in which the particle moves with \nthe velocity ${\\bf v}$.  In the general case \n\\begin{gather}\nu^a=e^a_{\\mu}\\frac{\\mathop{\\rm d}\\nolimits x^{\\mu}}{\\mathop{\\rm d}\\nolimits s}.\n\\label{D22N}\n\\end{gather}\n\n\n A complete description\nof spin precession of relativistic particle with MDM and EDM in the gravity field is found in  works \\cite{Orlov2012}-\\cite{Vergeles2019}.\nLet ${\\bf S}$ be the polarization vector in the particle rest frame. In the Riemann coordinates, the precession angular velocity ${\\bf \\Omega}$ equals\n\\cite{Frenkel1926}-\\cite{Fukuyama2013}\n\\begin{gather}\n\\frac{\\mathop{\\rm d}\\nolimits{\\bf S}}{\\mathop{\\rm d}\\nolimits t}=\\big[{\\boldsymbol\\Omega},\\,{\\bf S}\\big], \n\\nonumber \\\\ \n{\\boldsymbol\\Omega}=\\frac{q}{mc}\\Bigg\\{-\\left(G+\\frac{1}{\\gamma}\\right){\\bf H}+\n\\frac{\\gamma G({\\bf H}{\\boldsymbol\\beta})}{\\gamma+1}{\\boldsymbol\\beta}+\n\\left(G+\\frac{1}{\\gamma+1}\\right)[{\\boldsymbol\\beta},\\,{\\bf E}]+\n\\nonumber \\\\\n+\\eta_{EDM}\\left(-{\\bf E}+\\frac{\\gamma}{\\gamma+1}({\\bf E}{\\boldsymbol\\beta}){\\boldsymbol\\beta}+\n[{\\bf H},\\,{\\boldsymbol\\beta}] \\right)\\Bigg\\}.\n\\label{D30N}\n\\end{gather}\nHere\n\\begin{gather}\n\\mu=(G+1)\\frac{q\\hbar}{2mc}, \\quad d=\\eta_{EDM}\\frac{q\\hbar}{2mc}\n\\nonumber\n\\end{gather}\nare MDM and EDM, correspondingly.\n\n\n\n\nA formulation of the spin precession problem demands the orthonormal basis (ONB).\n It is important that the fields ${\\bf E}$ and ${\\bf H}$ in different ONB differ only by the Lorentz transformation. \nHowever, when describing dynamics in arbitrary ONB, a connection appears in the equations.\n As a result, Eqs. (\\ref{D20N}) take the following form:\n\\begin{gather}\n\\frac{\\mathop{\\rm d}\\nolimits \\gamma}{\\mathop{\\rm d}\\nolimits t}=\\frac{q}{mc}{\\bf E}{\\boldsymbol\\beta}+c(\\gamma_{\\alpha 0c}u^c){\\boldsymbol\\beta}^{\\alpha},\n\\nonumber \\\\\n\\frac{\\mathop{\\rm d}\\nolimits(\\gamma{\\boldsymbol\\beta}^{\\alpha})}{\\mathop{\\rm d}\\nolimits t}=\n\\frac{q}{mc}\n\\left({\\bf E}^{\\alpha}+[{\\boldsymbol\\beta},\\,{\\bf H}]^{\\alpha}\\right)+\nc\\Big((\\gamma_{\\alpha 0c}u^c)+(\\gamma_{\\alpha\\beta c}u^c){\\boldsymbol\\beta}^{\\beta}\\Big).\n\\label{D20}\n\\end{gather}\n\nSince the polarisation vector ${\\bf S}$ is defined in the particle rest frame, the explicit expression for a contribution of the connection to the precession frequency (\\ref{D30N}) for a relativistic particle is too  lengthy to be reproduced here \\cite{Vergeles2019}, we  only cite a result for the nonrelativistic case:\n\\begin{gather}\n{\\boldsymbol\\Omega}^{\\alpha}=-\\frac{q}{mc}\\bigg\\{\\big(G+1\\big){\\bf H}+\n\\eta_{EDM}{\\boldsymbol E}\\bigg\\}^{\\alpha}\n-\\frac{c}{2}\\varepsilon_{\\alpha\\beta\\rho}\n\\gamma_{\\beta\\rho0}.\n\\label{D30}\n\\end{gather}\n\n\n\nThe general conclusion from this brief discussion is \nthat the fields ${\\bf E}$ and ${\\bf H}$ in ONB defined according to\n(\\ref{M40}) and (\\ref{M50}) possess all the dynamic properties of the electric and magnetic fields, correspondingly.\nWe use hereafter the ONB (\\ref{G50}) which is a physically sensible one. \n\n\n\n\\section{Magnetic field in the pure electrostatic system in the nonertial frame }\n\n\n\nLet's consider a noninertial space-time with the stationary metric in the framework of Einstein's general theory of relativity. The stationarity means that all components of the metric\ntensor $g_{\\mu\\nu}(x)$  are time-independent, i.e. $(\\partial\/\\partial x^0)g_{\\mu\\nu}=0$.\nA generic feature of the correspontent metric is nonvanishig off-diagonal elements,\n\\begin{gather}\ng_{0i}\\neq 0.\n\\label{intr10}\n\\end{gather} \nThe outlined coordinate frame is denoted as $K$. In the cases of practical interest treated in Section IV the \nreference frame K is the laboratory frame used in the\ndescription of terrestrial experiments. We use also the inertial reference frame of distant stars $K'$.\nThe  gravitating body and the corresponding frame $K$ rotate w.r.t. to the frame $K'$ with the  angular velocity ${\\boldsymbol\\omega}$. Our electrostatic system resides at rest on the rotating body and  \nthe corresponding electric 4-current, as defined in the frame $K$, is the stationary one:\n\\begin{gather}\n(\\partial\/\\partial x^0)J^{\\mu}(x)=0, \\quad J^0\\neq 0, \\quad J^i=0.\n\\label{intr20}\n\\end{gather}\n\nIn this Section we derive the main result of the paper: {\\it For the stationary metric} (\\ref{intr10})  {\\it in the case of stationary electric 4-current} (\\ref{intr20})  {\\it the Maxwell equations can not be satisfied with the vanishing magnetic field} (see below Eq. (\\ref{H60n})).\n\n\n\nA proof of this property proceeds as follows. Note that any antisymmetric field in the three-dimensional space $\\sqrt{-g}F^{ij}=-\\sqrt{-g}F^{ji}$ can be represented as\n\\begin{gather}\n\\sqrt{-g}F^{ij}=\\varepsilon_{ijk}\\partial_k\\psi+\\left(\\partial_i{\\cal A}_j-\\partial_j{\\cal A}_i\\right).\n\\label{H10n}\n\\end{gather}\nThe decomposition (\\ref{H10n}) would be unique in three-dimensional Euclidean space for fields decreasing at infinity. As well known, in the rotating frames one faces the formal issue of the horizon. However, in all the cases of practical interest, the charge and current distributions are localized way inside the horizon radius. Consequently, the decomposition (\\ref{H10n}) is well defined and unique, and $J_i=0$ entails ${\\cal A}_i=0$.\n\n\n\n\n\nAccording to the rules (\\ref{G64}), the field definition (\\ref{M50}) and  the fact that $\\tilde{e}^i_0=0$, we obtain:\n\\begin{gather}\nF^{ij}=\\tilde{e}^i_a\\tilde{e}^j_bF^{ab}=-\\varepsilon_{\\alpha\\beta\\gamma}\n\\tilde{e}^i_{\\alpha}\\tilde{e}^j_{\\beta}{\\bf H}^{\\gamma}=-\\frac{e^0_0}{\\sqrt{-g}}\n\\varepsilon_{ijk}e^{\\alpha}_k{\\bf H}^{\\alpha}.\n\\label{H20n}\n\\end{gather}\nHere we  used one of  the relations (\\ref{H25n}).\n Making use of (\\ref{H10n}) (with ${\\cal A}_i=0$) in (\\ref{H20n}) yields\n\\begin{gather}\n{\\bf H}^{\\alpha}=-(e^0_0)^{-1}\\tilde{e}^i_{\\alpha}\\partial_i\\psi.\n\\label{H30n}\n\\end{gather}\n\n\nNow we turn to the  homogeneous Maxwell equations (\\ref{M20}).\nWe express the  holonomic field $F_{\\mu \\nu}$  in terms of the electric  and  magnetic fields:\n\\begin{gather}\nF_{i0}=e^a_ie^b_0F_{ab}=e^0_0e^{\\alpha}_iF_{\\alpha0}=\n-e^0_0e^{\\alpha}_i{\\bf E}^{\\alpha}=\\partial_iA_0, \\quad\n \\mbox{or}\n \\quad {\\bf E}^{\\alpha}=-\\frac{\\tilde{e}^i_{\\alpha}}{e^0_0}F_{i0},\n\\label{H40n}\n\\end{gather} \n\\begin{gather}\nF_{ij}=e^a_ie^b_jF_{ab}=-\\varepsilon_{\\alpha\\beta\\gamma}e^{\\alpha}_ie^{\\beta}_j{\\bf H}^{\\gamma}-\n\\left(e^0_je^{\\alpha}_i-e^0_ie^{\\alpha}_j\\right){\\bf E}^{\\alpha}.\n\\label{H50n}\n\\end{gather} \nEq. (\\ref{M20}) with $\\mu=0$ imply the identity $\\varepsilon_{ijk}\\partial_kF_{ij}=0$. Therefore,\napplying the operator $\\varepsilon_{ijk}\\partial_k$ to the Eq. (\\ref{H50n}), using (\\ref{H30n})\nand the identity $\\varepsilon_{ijk}\\partial_kF_{i0}=-\\varepsilon_{ijk}\\partial_k\\left(\ne^0_0e^{\\alpha}_i{\\bf E}^{\\alpha}\\right)=0$ (see (\\ref{H40n})), we\n obtain the equation \n\\begin{gather} \n\\partial_i\\bigg(\\sqrt{-g}\\left(e^0_0\\right)^{-2}g^{ij}\\partial_j\\psi\\bigg)=\n-\\varepsilon_{ijk}e^0_0e^{\\alpha}_i{\\bf E}^{\\alpha}\\partial_k\\left(\\frac{e^0_j}{e^0_0}\\right).\n\\label{H60n}\n\\end{gather} \nRecall that  $e^0_j=g_{0j}\/\\sqrt{g_{00}}\\neq0$  according to (\\ref{G130}) and (\\ref{intr10}). Therefore\nif ${\\bf E}\\neq0$, then (\\ref{H60n}) entails  $\\partial_j\\psi\\neq0$ and,\naccording to Eq. (\\ref{H30n}),  ${\\bf H}\\neq0$ as well. We shall refer to the field (\\ref{H30n}) as the geometric magnetic field $ {\\bf H}_{\\boldsymbol\\omega}$. As we shall see, the geometric magnetic field has its origin in the rotation of the frame $K$, hence the subscript ${\\boldsymbol\\omega}$.\n\n\nNext we look into the equation for the electric field.\nLet's  express $F^{i0}$ in terms of physical fields:\n\\begin{gather} \nF^{i0}=\\tilde{e}^i_a\\tilde{e}^0_bF^{ab}=\\tilde{e}^0_0\\tilde{e}^i_{\\alpha}{\\bf E}^{\\alpha}-\n\\varepsilon_{\\alpha\\beta\\gamma}\\tilde{e}^i_{\\alpha}\\tilde{e}^0_{\\beta}{\\bf H}^{\\gamma}\n=\\frac{1}{e^0_0}\\tilde{e}^i_{\\alpha}{\\bf E}^{\\alpha}-\\frac{1}{\\sqrt{-g}e^0_0}\\varepsilon_{ijk}e^0_j\\partial_k\\psi.\n\\label{H70n}\n\\end{gather} \nThe substitution of the right-hand side of (\\ref{H70n}) into Eq. (\\ref{M30}) with $\\mu=0$ leads to \n\\begin{gather}\n\\partial_i\\left(\\frac{\\sqrt{-g}}{e^0_0}\\tilde{e}^i_{\\alpha}{\\bf E}^{\\alpha}\\right)-\n\\varepsilon_{ijk}\\partial_i\\left(\\frac{e^0_j}{e^0_0}\\right)\\cdot\\partial_k\\psi=\n4\\pi\\sqrt{-g}J^0.\n\\label{H80n}\n\\end{gather}\nHere we have used representation (\\ref{H30n}) and one of the relations (\\ref{H25n}).\n\nThe system of equations  (\\ref{H30n}), (\\ref{H60n}) and (\\ref{H80n})  is complete  and exact. The electric field ${\\bf E}$  is expressed in terms of the potential $A_0$ with the help of relation (\\ref{H40n}).\n\nTo be specific, we describe the curved space-time of the rotating body by the Kerr  metric described in Appendix B.  We report here the iterative solutions for the electromagnetic potentials $A_{0}$ and $\\psi$ and the corresponding fields, treating the angular velocity of the rotating body  ${\\boldsymbol{\\omega}}$ and its gravitational radius $r_g$ as small parameters. Making use of (\\ref{G120})-(\\ref{G150}), we rewrite Eqs.  (\\ref{H30n}), (\\ref{H60n}) and (\\ref{H80n}) keeping the terms of the order of   ${\\boldsymbol{\\omega}}$,\n$r_g$, $r_g{\\boldsymbol{\\omega}}$, ${\\boldsymbol{\\omega}}\\otimes{\\boldsymbol{\\omega}}$. \n\n There emerges a simple hierarchy of corrections in powers of ${\\boldsymbol{\\omega}}$. We are interested in the electrostatic system with the initial potential $A_0^{(0)}$ and the corresponding Minkowski space defined electric field  ${\\boldsymbol{E}}^{(0)}=-\\nabla A_0^{(0)}$. Concerning the geometric magnetic field, the crucial points are Eq. (B3) for the off diagonal $g_{0i}$, Eq. (B5) for $e^0_i$ and Eq. (B6) for $\\tilde{e}^0_{\\alpha}$. The three related  quantities are all proportional to the angular velocity of the rotating body.  Consequently  $e^0_j\/e^0_0=\\mathop{\\rm O}\\nolimits({\\boldsymbol\\omega})$ and, according to (\\ref{H60n}), the expansion for the magnetic potential $\\psi$ starts with the linear term $\\psi^{(1)}=\\mathop{\\rm O}\\nolimits({\\boldsymbol{\\omega}})$. \n \t\n Now we proceed to the electric field. According to the expansion of the Kerr metric in Appendix B, we have\n\\begin{gather}\n\\frac{\\sqrt{-g}\\tilde{e}^i_{\\alpha}}{e^0_0}=\\left(1+\\frac{r_g}{R}\\right)\\delta^i_{\\alpha}\n+\\left(\\frac{[{\\boldsymbol\\omega},\\,{\\bf R}]^2}{2c^2}\\delta^i_{\\alpha}-\n\\frac{[{\\boldsymbol\\omega},\\,{\\bf R}]^i[{\\boldsymbol\\omega},\\,{\\bf R}]^{\\alpha}}{2c^2}\\right)\\, ,\n\\label{H90n}\n\\end{gather}\nwhich does not contain the linear term.  Then, equations (\\ref{H80n}) and (\\ref{H90n}) tell that the electric field  acquires the first correction only to the second order in ${\\boldsymbol{\\omega}}$, i.e., $ {\\bf E}^{(1)}=0, \\,{\\bf E}^{(2)}\\neq 0$. In the due turn,  Eq. (\\ref{H60n})  guarantees that the quadratic correction to the magnetic potential vanishes: $\\psi^{(2)}=0$. \n\nGoing back to the magnetic potential $\\psi$, we make use of \n\\begin{gather}\n\\frac{\\tilde{e}^i_{\\alpha}}{e^0_0}=\\delta^i_{\\alpha}+\n\\left(\\frac{[{\\boldsymbol\\omega},\\,{\\bf R}]^2}{2c^2}\\delta^i_{\\alpha}-\n\\frac{[{\\boldsymbol\\omega},\\,{\\bf R}]^i[{\\boldsymbol\\omega},\\,{\\bf R}]^{\\alpha}}{2c^2}\\right).\n\\label{H100n}\n\\end{gather} \nand invoke the relations \n\\begin{gather}\n\\frac{\\sqrt{-g}}{\\left(e^0_0\\right)^2}g^{ij}=-\\left(1+\\frac{r_g}{R}\\right)\\delta^{ij}, \\quad\ne^0_0e^{\\alpha}_i=\\delta^{\\alpha}_i+\\mathop{\\rm O}\\nolimits({\\boldsymbol{\\omega}}\\otimes{\\boldsymbol{\\omega}}), \n\\nonumber \\\\\n\\frac{e^0_j}{e^0_0}=-\\Bigg\\{1+\\frac{r_g}{R}\\left(2-\\frac{I R^2_{\\oplus}}{R^2}\\right)\\Bigg\\}\n\\frac{[{\\boldsymbol\\omega},\\,{\\bf R}]^j}{c},\n\\label{H120n}\n\\end{gather} \nThen Eq. (\\ref{H60n}) takes the final form \n\\begin{gather}\n\\partial_i\\bigg\\{\\left(1+\\frac{r_g}{R}\\right)\\partial_i\\psi\\bigg\\}\n\\nonumber \\\\\n=\\varepsilon_{ijk}{\\bf E}^{(0)i}\n\\partial_j\\Bigg\\{\\left[1+\\frac{r_g}{R}\\left(2-\\frac{I R^2_{\\oplus}}{R^2}\\right)\\right]\\frac{[{\\boldsymbol\\omega},\\,{\\bf R}]^k}{c}\\Bigg\\}.\n\\label{H130n}\n\\end{gather} \n\nA departure of the metric of the curved space-time from the Minkowski one changes the relationship between the potentail $A_{0}$ and the charge distribution and the relationship between the electric field ${\\bf E}^{(2)}$ and the gradient of $A_0^{(2)}$. Although these second order corrections are unlikely to be of any practical significance in the terrestrial laboratories, we cite them for the sake of completeness. To the zeroth order in ${\\bf \\omega}$, but keeping the terms linear in $r_g$, we have\n\\begin{gather}\n-\\nabla\\bigg\\{\\left(1+\\frac{r_g}{R}\\right)\\nabla A^{(0)}_0\\bigg\\}=4\\pi\\left(1+\\frac{r_g}{R}\\right)J^0,.\n\\label{H160n}\n\\end{gather}\nAfter some algebra, Eq. (\\ref{H80n}) with the account for  Eqs. (\\ref{H90n}), (\\ref{H120n}) yields the equation for the second order correction to the scalar potential,  \n\\begin{gather}\n-\\Delta A^{(2)}_0=\\frac{1}{c^2}\\nabla\\bigg\\{[{\\boldsymbol\\omega},{\\bf R}]^2\\nabla A^{(0)}_0-\n\\left([{\\boldsymbol\\omega},{\\bf R}]\\nabla A_0^{(0)}\\right)[{\\boldsymbol\\omega},{\\bf R}]\\bigg\\}\n-\\frac{2}{c}({\\boldsymbol\\omega}\\nabla\\psi).\n\\label{H170n}\n\\end{gather}\nIn this derivation, to the desired accuracy, we  made use of $e^0_j\/e^0_0=-[{\\boldsymbol\\omega},\\,{\\bf R}]\/c$. \nNote how the second order correction to the electric potential couplles to the first order potential for the geometric magnetic field. \n\nFinally, upon using  Eq. (\\ref{H100n}) and gathering together all second order corrections, the Eq. (\\ref{H40n}) yields the second order correction to the electric field \n\\begin{gather}\n{\\bf E}^{(2)}=-\\nabla A_0^{(2)}-\\frac{[{\\boldsymbol\\omega},\\,{\\bf R}]^2}{2c^2}\\nabla A_0^{(0)}\n+\\left([{\\boldsymbol\\omega},\\,{\\bf R}]\\nabla A_0^{(0)}\\right)\\frac{[{\\boldsymbol\\omega},\\,{\\bf R}]}{2c^2}.\n\\label{H150n}\n\\end{gather} \nApart from the gradient of the second order scalar potential, here emerge corrections quadratic in the angular velocity ${\\bf \\omega}$. The system of equations  (\\ref{H130n})-(\\ref{H150n}) is complete  to the desired accuracy. \n \nThe rotating body of the practical interest is the Earth. It is the case of weak gravity. On the terrestrial surface  \n$r_g\/R_{\\oplus}\\sim 10^{-9}$  and $\\omega R_{\\oplus} \\sim 1.5\\cdot 10^{-6}$. Hence we keep the terms $\\mathop{\\rm O}\\nolimits(|{\\boldsymbol\\omega}|)$ and neglect the former corrections. In this approximation\nequations  (\\ref{G120})-(\\ref{G150}) simplify to  \n\\begin{gather}\ng_{00}=g^{00}=1, \\quad g_{0i}=g^{0i}=-\\frac{\\big[{\\boldsymbol\\omega},\\,{\\bf R}]^i}{c},\n\\nonumber \\\\\ng_{ij}=g^{ij}=-\\delta^{ij}, \\quad g=-1,\n\\nonumber \\\\\ne^0_0=1, \\quad e^0_i=-\\frac{\\big[{\\boldsymbol\\omega},\\,{\\bf R}\\big]^i}{c},\n\\quad e^{\\alpha}_i=\\delta^{\\alpha}_i, \\quad e^{\\alpha}_0=0,\n\\nonumber \\\\\n\\tilde{e}^0_0=1, \\quad \\tilde{e}^0_{\\alpha}=\\frac{\\big[{\\boldsymbol\\omega},\\,{\\bf R}\\big]^{\\alpha}}{c},\n\\quad \\tilde{e}_{\\alpha}^i=\\delta_{\\alpha}^i, \\quad \\tilde{e}^i_0=0,\n\\nonumber \\\\\n\\frac{c}{2}\\varepsilon_{\\alpha\\beta\\rho}\\gamma_{\\beta\\rho 0}={\\boldsymbol\\omega}^{\\alpha}.\n\\label{H90}\n\\end{gather}\nThe second order corrections to the electric potential and electric field can be neglected and we have the familiar ${\\bf E}=-\\nabla A_0$ and  the Poisson equation $\\mathop{\\rm div}\\nolimits{\\bf E}=-\\Delta A_0=4\\pi J^0$, while the Poisson equation for the potential of the geometric magnetic field takes a simple form, \n\\begin{gather}\n\\Delta\\psi=\\frac{2}{c}({\\boldsymbol\\omega}{\\bf E})\\,,\n\\label{H140}\n\\end{gather}\nto be used in the subsequent analysis of terrestrial experiments.\n\n\\section{Manifestations of the geometric magnetic field}\n\n\n\n\\subsection{False EDM signal in the neutron EDM experiments }\n\n\n\n\n\nHere we comment on the possible implications of the geometrical magnetic field for the neutron EDM experiments. The fundamental observable is the change of the Larmor precession frequency \n\\begin{equation}\nf_n =\\frac{1}{\\pi\\hbar}|\\mu_n{\\bf B} +d_n{\\bf E}| \\label{eq:Larmor}\n\\end{equation}\nfrom the parallel to antiparallel uniform fields. The EDM is extracted from the frequency shift\n\\begin{equation} \nd_n = \\frac{\\pi \\hbar \\Delta f}{2|{\\bf E}|} . \n\\label{eq:EDM extraction}\n\\end{equation}\nThe implicit assumption is that flipping the electric field does not change the magnetic one. Our point is that this is not the case with the geometric magnetic field. \n\n\nIn practice the electric field is generated in the plane capacitor with the gap much smaller than the plate size. The relevant solution of the one-dimensional problem for the geometric magnetic field proceeds as follows. In the gap in between the plates we have \n\\begin{gather}\n{\\bf E}_0=(0,\\,0,\\,{\\cal E}_0)=-\\nabla A_0, \\quad A_0=-{\\cal E}_0z.\n\\label{IIiE10}\n\\end{gather}\nNow we solve the Poisson equation (\\ref{H140}) for the magnetic potential, representing the electric field through $A_0(z)$ and integrating by parts:\n\\begin{gather}\n\\psi(z)=\\frac{2{\\boldsymbol\\omega}_z}{c}\\int\\mathop{\\rm d}\\nolimits z'\\Delta^{-1}(z-z'){\\cal E}_0\n=-\\frac{2{\\boldsymbol\\omega}_z}{c}\\int\\mathop{\\rm d}\\nolimits z'\\frac{\\mathop{\\rm d}\\nolimits}{\\mathop{\\rm d}\\nolimits z}\\Delta^{-1}(z-z')A_0(z'),\n\\label{IIiE20}\n\\end{gather}\nwhere $\\Delta^{-1}(z)=|z|\/2$ is the inverse to the Laplace operator. One more differentiation yields \\begin{gather}\n{\\bf H}_{\\boldsymbol\\omega}=-\\nabla\\psi(z)\n=\\left(0,\\,0,\\,\\frac{2{\\boldsymbol\\omega}_z}{c}A_0(z)\\right)=\n\\left(0,\\,0,\\,-\\frac{2{\\boldsymbol\\omega}_z{\\cal E}_0}{c}z\\right)=\n-\\frac{2{\\boldsymbol\\omega}_zz}{c}{\\bf E}_0.\n\\label{IIiE30}\n\\end{gather}\nThe geometric  field ${\\bf H}_{\\boldsymbol\\omega}$ is parallel to the external electric field ${\\bf E}_0$. Its salient feature is the nonvanishing constant gradient\n\\begin{equation}\n\\frac{\\mathop{\\rm d}\\nolimits{\\bf H}_{\\boldsymbol\\omega}}{\\mathop{\\rm d}\\nolimits z} = -\\frac{2{\\boldsymbol\\omega}_z}{c}{\\bf E}_0 \n\\label{eq:Gradient}\n\\end{equation}\nAt the mid plane the geometric field vanishes: ${\\bf H}_{\\boldsymbol\\omega}(0)=0$.\n\nThe crucial feature of the neutron EDM experiments is the comagnetometry: one measures the neutron spin precession frequency with respect to that of the mercury comagnetometer. The mercury atoms are evenly distributed in the volume of the neutron storage cell and the average geometric magnetic field acting on the mercury comagnetometer vanishes: $\\langle {\\bf H}_{\\boldsymbol\\omega}^{(Hg)}\\rangle =\n{\\bf H}_{\\boldsymbol\\omega}(0) =0$.\nThe centre of mass of neutrons differs from that of the mercury by the  offset  $\\langle z \\rangle$, what entails the nonvanishing average geometric magnetic field acting on the magnetic moment of neutrons\n\\begin{equation}\n{\\bf H}_{\\boldsymbol\\omega}^{(n)} = -\\frac{2\\langle z\\rangle{\\boldsymbol\\omega}_z}{c}{\\bf E}_0 .\n\\label{eq:HgeomNeutron}\n\\end{equation}\nThe most important point is that this geometric field changes the sign when the electric field is flipped. The net effect is that the apparent EDM of neutrons, $d_n^{obs}$, as given by the procedure (\\ref{eq:EDM extraction}), will acquire the false component, $d_{n}^{obs} = d_n + d_{false}$, where \n\\begin{equation}\nd_{false} = -\\frac{2\\langle z \\rangle{\\boldsymbol\\omega}_z}  {c}\\mu_n\\, . \\label{eq:dFalse}\n\\end{equation}\nIn the experiment \\cite{Pendlebury2015} the neutron center of mass offset was $\\langle z \\rangle \\simeq 2.8$mm, the more recent experiment reports $\\langle z \\rangle \\simeq 3.9$mm. Taking the former, we find $d_{false} \\approx 2.5\\times  10^{-28}$ e$\\cdot$cm. It is still way below the recently reported best upper bound on the neutron EDM, $d_n = (0.0\\pm 1.1_{stat} \\pm 0.2_{sys})\\times 10^{-26}$e$\\cdot$cm   \n \\cite{Afach2020}, but can become sizeable in the next generation of the neutron EDM experiments aiming at $d_n < 10^{27}$ e$\\cdot$cm  \\cite{Chupp2019}. With the neutron storage cell of height 12 cm, the geometric magnetic field induced spread of the false EDM within the ensemble of stored neutrons can be as large as \n \\begin{equation}\n \\Delta d_{false} = \\pm \\frac{h{\\bf \\omega}_z}{c}\\mu_n \\simeq \\pm 5\\times 10^{-27} {\\text e}\\cdot\\text{cm} \\, . \\label{eq:dFalseSpread}\n \\end{equation}\n \n\n\n\n\n\n\n\\subsection{Geometric  magnetic field as a background in all electric proton  EDM storage rings}\n\nThe principal idea of searches for the proton EDM in the all electric ring, run at the so-called magic energy, is to eliminate the magnetic field acting on the proton magnetic moment. Then the sole rotation of the proton spin will be due to interaction of its EDM with the radial electric field that confines protons in the storage ring, and very ambitious sensitivity  to the proton EDM,\n\\begin{gather}\n\\eta_{EDM}\\sim10^{-15}\\, ,\n\\label{IIE}\n\\end{gather}\nis in sight \\cite{Anastopoulos2016,Abusaif2019}.  Here we comment on implications of the geometric magnetic field for such proton EDM experiments. \n\nThe storage ring is a cylinder capacitor with the gap $d$\nwhich is  much smaller compared to the height of cylinders $h$, which in its turn is  much smaller than radii of cylinders $r_{1,2}=\\rho\\mp d\/2$. In view of $d\\ll h\\ll\\rho$ we neglect the dependence on the vertical coordinate and have the two-dimensional geometry. The beam trajectory is in the midplane of the storage ring at the orbit radius $|{\\bf r}|=\\rho$. The electric field in the gap is given by \n\\begin{gather}\n{\\bf E}_0=-{\\cal E}_0\\frac{\\rho{\\bf r}}{r^2} = -\\nabla A_{0}(r), \\quad\\quad A_0(r) = {\\cal E}_0 \\rho \\ln\\frac{r}{\\rho}\\, .\n\\label{IIE10}\n\\end{gather}\n\nIt is instructive to start with the storage ring located on the North pole. From the viewpoint of distant observer in the reference frame K', the ring rotates with the Earth's rotation angular velocity ${\\bf \\omega}$. The static charges on the two rotating cylinders produce the opposite currents and generate in the gap the magnetic fields of the same sign and magnitude. The net result is the magnetic field\n\\begin{gather}\n{\\bf H}'_{\\boldsymbol\\omega}({\\bf r})=\\frac{1}{c}\\big[{\\bf v}({\\bf r}),\\,{\\bf E}_0({\\bf r})\\big].\n\\label{IIE20}\n\\end{gather}\nAt first sight, it introduces an asymmetry between the  clockwise and anticlockwise beams in the all electric storage ring, parasitic from the viewpoint of searches for the proton EDM. However, it is basically the motional magnetic field and, to the experimenter in the terrestrial laboratory K, it vanishes entirely. Such an exact cancellation only holds at the North and South poles, and at an  arbitrary  latitude it does not work. In the generic case, the result  (\\ref{IIE20}) for the magnetic field  suggests the small parameter $\\eta_{\\boldsymbol\\omega}={|{\\boldsymbol\\omega}|\\rho}\/{c}$, similar to that appearing in Eq. (\\ref{eq:dFalse}). For the storage ring of radius $\\rho \\sim 50$m we have \n\\begin{gather}\n\\eta_{\\boldsymbol\\omega}=\\frac{|{\\boldsymbol\\omega}|\\rho}{c}\\sim 10^{-11}.\n\\label{IIE30}\n\\end{gather}\nwhich is some four orders in magnitude larger than the target value $\\eta_{EDM}\\sim 10^{-15}$.\n\nNow we solve for the geometric magnetic field following the formalism of Section III. The electric field (\\ref{IIE10}) suggests  for the magnetic potential $\\psi$ the Ansatz\n\\begin{gather}\n\\psi=f(r)\\cdot({\\boldsymbol\\omega_t}{\\bf r})\\, ,\n\\label{IIE40}\n\\end{gather}\nwhere ${\\boldsymbol\\omega}_t$ is a projection of the Earth's angular velocity onto the ring plane. A generic solution to Eq. (\\ref{H140}) is\n\\begin{gather}\nf(r)=-\\frac{{\\cal E}_0\\rho}{c}\\left(\\ln\\frac{r}{\\rho}+\\zeta\\right) = -\\frac{A_o(r)}{c} - \\frac{{\\cal E}_0\\rho}{c}\\zeta,\n\\label{IIE70}\n\\end{gather}\nand \n\\begin{gather}\n{\\bf H}_{\\boldsymbol\\omega}^i= \\frac{2{\\boldsymbol\\omega}_t^j}{c}A_{ij}({\\bf r}) =\\Bigg\\{\\frac{A_0}{c}\\delta_{ij}\n+\\frac{{\\cal E}_0\\rho}{c}\\bigg[\\left(\\zeta-1\/2\\right)\\delta_{ij}\n+\\frac12(\\delta_{ij}-2{\\bf n}_i{\\bf n}_j)\\bigg]\\Bigg\\}\n{\\boldsymbol \\omega}_t^j,\n\\label{IIE80}\n\\end{gather}\nwhere ${\\bf n}= {\\bf r}\/r$ .\n\nThe constant $\\zeta$ is fixed by the boundary condition that the electric potential $A_{0}$ vanishes rapidly beyond the capacitor, so that in the integral representation for $\\psi$ one can perform the integration by parts:\n\\begin{gather}\n\\psi({\\bf r})=\\frac{2{\\boldsymbol\\omega}_t^j}{c}\\int\\mathop{\\rm d}\\nolimits^{(2)}y\\,\\Delta^{-1}({\\bf r}-{\\bf y}){\\bf E}^j_0({\\bf y})\n=-\\frac{2{\\boldsymbol\\omega}_t^j}{c}\\int\\mathop{\\rm d}\\nolimits^{(2)}y\\,\\partial_j\\Delta^{-1}({\\bf r}-{\\bf y})A_0(\\bf y), ,\n\\label{IIE90}\n\\end{gather}\nwhere $\\Delta^{-1}({\\bf r})=(1\/2\\pi)\\ln|{\\bf r}|$ is the inverse to the Laplace operator. The resulting equation for the symmetric matrix $A_{ij}({\\bf r})$\n\\begin{gather}\nA_{ij}({\\bf r})=\\int\\mathop{\\rm d}\\nolimits^{(2)}y\\,\\partial_i\\partial_j\\Delta^{-1}({\\bf r}-{\\bf y})A_0({\\bf y}).\n\\label{IIE100}\n\\end{gather}\nentails \n\\begin{gather}\n\\mathop{\\rm tr}\\nolimits A({\\bf r})=A_0({\\bf r}).\n\\label{IIE110}\n\\end{gather}\nHence the expansion of $A_{ij}$ into irreducible tensor structures is of the form  \n\\begin{gather}\nA_{ij}({\\bf r})=\\frac{1}{2}\\delta_{ij}A_0({\\bf r}) + \\sigma({\\bf r})(\\delta_{ij}-\n2{\\bf n}^i{\\bf n}^j) \\, ,\n\\label{IIE120}\n\\end{gather}\nand a comparison to (\\ref{IIE80}) gives immediately\n\\begin{gather}\n\\sigma({\\bf r}) = \\frac{1}{4}{\\cal E}_0\\rho, \\quad \\zeta = \\frac{1}{2}.\n\\label{IIE130}\n\\end{gather}\nOur final result for the geometric magnetic field in the gap of the storage ring is\n\\begin{gather}\n{\\bf H}_{\\boldsymbol\\omega}^i=\\frac{{\\cal E}_0\\rho}{c}\\Bigg\\{\\ln\\left(\\frac{r}{\\rho}\\right)\\cdot\\delta_{ij}+\n\\left(\\frac12\\delta_{ij}-{\\bf n}_i{\\bf n}_j\\right)\\Bigg\\}\n{\\boldsymbol \\omega}_t^j\\simeq  \\frac{{\\cal E}_0\\rho}{2c}\n\\left(\\delta_{ij}-2{\\bf n}_i{\\bf n}_j\\right){\\boldsymbol \\omega}_t^j,\n\\label{IIE130}\n\\end{gather}\nwhere in the last step we neglected $|\\log(r\/\\rho)| < d\/(2\\rho) \\ll 1$.\n\n\nThe background magnetic fields are of prime concern to the planned searches for the proton EDM in the all electric magic storage rings, for a detailed  discussion see the recent monographic document by the CPEDM (Charged Particles EDM) collaboration \\cite{Abusaif2019}. Important virtue of the all electric rings is a cancellation of many systematic effects when one compares spin rotations of simultaneously stored clockwise (CW) and anticlockwise (ACW) rotating protons. The magnetic Lorentz forces split the orbits of the CW and ACW beams. As discussed extensively in \\cite{Abusaif2019}, the modern techniques allow a very strong, but as yet incomplete,  screening of the Earth's magnetic field.  Despite much work, reported in \\cite{Abusaif2019}, the analysis of the magnetic imperfection effects is still in the formative stage.\n\nThe Earth's magnetic field ${\\bf H}_\\oplus$ and the geometric magnetic field ${\\bf H}_{\\bf \\omega}$ do differ markedly. In contrast to the Earth's magnetic field, the  geometric one is not subject to screening by magnetic shields. On the scale of the storage ring, the Earth's magnetic field can be regarded as a uniform one and has the constant projection onto the ring plane. It is pointing along the (magnetic) meridian, which we take for the y-axis : ${\\bf H}_\\oplus ^t = (0, H_\\oplus ^t )$. In contrast to that, the geometric magnetic field has the quadrupole-like behaviour along the particle orbit, ${\\bf H}_{\\bf \\omega} = H_{\\bf \\omega}(\\sin 2\\theta, \\cos 2\\theta) $. The angular position of the particle in a ring,  $\\theta$, is defined by ${\\bf n} = (\\cos \\theta, -\\sin \\theta)$. \n\nIn the all  electric magnetic rings the most dangerous ones are the radial magnetic fields. In the above two cases they are  equal to  $H_\\oplus^{(r)} = ({\\bf n}\\cdot {\\bf H}_\\oplus) = -H_\\oplus \\sin \\theta $  and $H_{\\bf \\omega}^{(r)} = ({\\bf n}\\cdot {\\bf H}_{\\bf \\omega}) = H_{\\bf \\omega} \\sin \\theta $\\, \nAccording to \\cite{Abusaif2019}, to the first approximation the rotation of the proton spin is proportional to the one-turn integral  $\\oint d\\theta H_\\oplus^{(r)} $. .Obviously, both the Earth's  and  geometric magnetic fields share the property \n\\begin{equation}\n\\oint d\\theta H_\\oplus^{(r)}  = \\oint d\\theta H_{\\bf \\omega}^{(r)} =0\\, . \\label{eq:EDMlike}\n\\end{equation}\nThe argument of Ref.  \\cite{Abusaif2019} about vanishing false EDM signal from is then applicable to the geometric magnetic field as well.\n\nThe above consideration is somewhat naive and must be complemented by a consistent  treatment of the spin-orbit coupling, though. Namely, this cancellation of the false EDM effect might become incomplete because of the orbit distortions which are very much distinct in the two cases.  To be on the safe side,  one needs a dedicated analysis of the false spin rotations with simultaneous allowance for the orbit distortions. Furthremore, one needs to pay an attention to  a possible cross talk between the impact of the geometric magnetic field and the residual Earth's magnetic field. It is an important complex issue on its own to be addressed to in the future,  it  goes beyond the scope of the present communication. \n\n\n\\subsection{Geometric magnetic field of the conducting charged sphere}\n\nFor the sake of completeness, we comment on the charged sphere at rest in the  rotating system $K$. Inside the sphere we have ${\\bf E}=0$  and $A_0=\\mathop{\\rm const}\\nolimits$ thereof: \n\\begin{gather}\n{\\bf E}({\\bf r})=\\left\\{\n\\begin{array}{rl}\n\\dfrac{Q{\\bf r}}{r^3},  &  \\mbox{for} \\quad r>a, \\\\  [4mm]\n0, & \\mbox{for} \\quad r<a.\n\\end{array}  \\right.\n\\label{IE10}\n\\end{gather}\nHere $Q$ is the charge of the sphere, and the  radius-vector ${\\bf r}=0$ at the centre of the  sphere. \n\nAccording to (\\ref{H140})\n\\begin{gather}\n\\psi({\\bf r})=-\\frac{\\boldsymbol\\omega}{2\\pi c}\\int\\mathop{\\rm d}\\nolimits^{(3)}x\\frac{1}{|{\\bf r}-{\\bf x}|}{\\bf E}({\\bf x})=\n-\\frac{Q\\cdot ({\\boldsymbol\\omega}{\\bf r})}{3c}\\left\\{\n\\begin{array}{rl}\n\\dfrac{3}{r}-\\dfrac{a^2}{r^3},  &  \\mbox{for} \\quad r>a, \\\\  [4mm]\n\\dfrac{2}{a}, & \\mbox{for} \\quad r<a\\,,\n\\end{array}  \\right.\n\\label{IE30}\n\\end{gather}\nand \n\\begin{widetext}\n\\begin{gather}\n{\\bf H}_{\\boldsymbol\\omega}({\\bf r})=\\left\\{\n\\begin{array}{rl}\n\\dfrac{1}{r^3}\\Big\\{3({\\boldsymbol\\mu}{\\bf n}){\\bf n}-{\\boldsymbol\\mu}\\Big\\}+\n\\dfrac{1}{c}\\big[{\\bf E}({\\bf r}),\\,[{\\boldsymbol\\omega},\\,{\\bf r}]\\big], \\quad {\\bf E}({\\bf r})=\\dfrac{Q{\\bf r}}{r^3} &  \\mbox{for} \\quad r>a, \\\\ [4mm]\n\\dfrac{2Q{\\boldsymbol\\omega}}{3ca}, & \\mbox{for} \\quad r<a.\n\\end{array}  \\right.\n\\label{IE40}\n\\end{gather}\n\\end{widetext}\nHere ${\\bf n}={\\bf r}\/{r}$ and\n\\begin{gather}\n{\\boldsymbol\\mu}=\\frac{Qa^2}{3c}{\\boldsymbol\\omega}.\n\\nonumber\n\\end{gather}\nis the geometric magnetic moment  of  the conducting charged sphere, induced by the Earth's rotation. The second term in (\\ref{IE40}) is the\nfamiliar motional magnetic field in the rotating frame $K$ which is entailed by the electric field in inertial frame $K'$.\n\n\n\n\n\n\n\n\n\n\\section{Conclusions}\n\nWe have shown that in the pure electrostatic systems at rest on the rotating bodies there can exist a geometric magnetic field. From the general relativity point of view, it originates from the nonvanishing off-diagonal elements $g_{0i}$ of the metric tensor which are proportional to the angular velocity of rotation of the gravitating body. We presented several specific examples of the geometric field for different configurations of the static electric field on the rotating body described by the Kerr metric. In the configuration of experimental setups used in the terrestrial searches for the EDM of neutrons, the geometric magnetic field changes the sign when the electric field is flipped. Consequently, its interaction with the magnetic moment of the neutron can imitate the neutron EDM and that can become a sizable background in the next generation of the neutron EDM experiments. We found a fairly large background geometric magnetic field in all electric magic storage rings considered a dedicated machine for searches of the proton EDM. The symmetry properties  of the geometric magnetic field suggest strong cancellations of its contribution to the proton spin rotations. Stil, its impact on the signal of EDM remains an open issue - here one needs a dedicated analysis with full allowance for the spin-orbit dynamics in the storage ring. \n \n\n\n\n\\begin{acknowledgments}\n\nWe are grateful to A.Ya. Maltsev, A.A. Starobinsky and  S.S. Vergeles for valuable comments and discussions.\nThis work was carried out as a part of the State Program 0033-2019-0005.\n\n\n\n\\end{acknowledgments}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Methods}  \\label{sec:Methods}\n\n{\\noindent \\textbf{Foreshock disturbance properties.}}  Since previous simulation\\cite{riquelme11a} and observational\\cite{oka06a, wilsoniii12c} studies found that magnetosonic-whistler waves\\cite{wilsoniii16a} -- right-hand polarized electromagnetic plasma waves, with density fluctuating in phase with $\\lvert \\mathbf{B}{\\scriptstyle_{o}} \\rvert$ -- can accelerate particles to $>$1 keV, we limited the types of foreshock disturbances to those having a magnetosonic-whistler nature.  We further limited our search to disturbances occurring multiple times per day, those with nonlinear properties, and those capable of forming shock waves, all properties associated with energizing particles.  Therefore, the three foreshock disturbances we examined, short large-amplitude magnetic structures\\cite{lucek08a, mann94a, scholer03b, schwartz92a, wilsoniii13b, wilsoniii16a}, hot flow anomalies\\cite{eastwood08a, omidi07a, omidi13a, omidi14c, schwartz85a, zhang10a, zhang13a}, and foreshock bubbles\\cite{archer15a, omidi10a, turner13a}, are all produced by the interaction between the incident solar wind and suprathermal ($>$100 eV to 100s of keV) ions\\cite{burgess12a, burgess13a, wilsoniii16a}.  Short large-amplitude magnetic structures are short duration ($\\sim$few to 10s of seconds), nonlinear large amplitude ($\\delta B\/B$ $>$ 2), monolithic ``magnetic pulsations'' with spatial scales of $\\sim$1000 km that can exhibit a soliton-like behavior (i.e., large amplitude fluctuations are fast and spatially narrow)\\cite{schwartz92a, mann94a}.  Both hot flow anomalies and foreshock bubbles are localized rarefaction regions surrounded by compression regions that are effectively ``carved out'' by an accumulation of suprathermal ions along a discontinuity in the interplanetary magnetic field.  The difference is that the compression regions for hot flow anomalies are centered on the discontinuity and the discontinuity must interact with the Earth's bow shock, whereas foreshock bubbles form upstream of the discontinuity and the discontinuity need not interact with the Earth's bow shock\\cite{omidi07a, turner13a}.  Both hot flow anomalies and foreshock bubbles are several Earth radii in scale (i.e., $>$10,000 km).\n\n{\\noindent \\textbf{Exclusion of solar source.}}  We first eliminated interplanetary shocks as a possibility by examining the \\emph{Wind} shock database at the Harvard Smithsonian for Astrophysics (Online at \\textcolor{Blue}{\\seqsplit{http:\/\/themis.ssl.berkeley.edu\/index.shtml}}) finding no interplanetary shocks during any of the energetic electron enhancements.\n\n\\indent  We next examine the radio data from the \\emph{Wind} and STEREO spacecraft (Extended Data Fig. \\ref{fig:WindandSTEREOWAVESRadio}).  There are no clear radio bursts or any evidence of significant radio activity on the sun during any of the four THEMIS foreshock passes.  For comparative purposes, we include a date with clear solar radio bursts (Extended Data Figs \\ref{fig:WindandSTEREOWAVESRadio}\\textbf{e}, \\ref{fig:WindandSTEREOWAVESRadio}\\textbf{j}, and \\ref{fig:WindandSTEREOWAVESRadio}\\textbf{o}).  The enhanced radio intensity near $\\sim$200 kHz (Figs \\ref{fig:WindandSTEREOWAVESRadio}\\textbf{f}--\\ref{fig:WindandSTEREOWAVESRadio}\\textbf{i}) is most likely auroral kilometric radiation\\cite{ergun98b}, which would have no effect on particle observations by THEMIS.  Examination of the solid state telescope particle data from \\emph{Wind}\\cite{lin95a} (freely available on CDAWeb, see \\textbf{Data availability} below) \\textcolor{Red}{show} no significant energetic electron enhancements during any of the enhancements observed by THEMIS.  Finally, the electron data \\textcolor{Red}{show} no evidence of forward energy dispersion (i.e., higher energies arrive before lower due to a time-of-flight effect) characteristic solar energetic electrons\\cite{ergun98e} (Fig. \\ref{fig:exampleTIFPs} and Extended Data Figs \\ref{fig:THEMISFBPADExample}\\textbf{k}--\\ref{fig:THEMISFBPADExample}\\textbf{n}).  Note that the energetic ions exhibit slightly larger anisotropies than the electrons with anti-earthward intensities generally dominating (see Extended Data Fig. \\ref{fig:THEMISFBPADExample}).\n\n{\\noindent \\textbf{Exclusion of Earth's bow shock as source.}}  The most common shock acceleration mechanisms cited are diffusive shock acceleration\\cite{malkov01a}, shock drift acceleration\\cite{anagnostopoulos98a, park13a}, and the ``fast Fermi'' mechanism\\textcolor{Red}{\\cite{wu84b, leroy84a}}.  However, for electron acceleration at the Earth's bow shock we can rule out these mechanisms for the following reasons.  First, though diffusive shock acceleration predicts an isotropic particle distribution, it is more efficient for quasi-parallel ($\\theta{\\scriptstyle_{Bn}}$ $<$ 45$^{\\circ}$) shocks with pre-existing upstream electromagnetic fluctuations and the efficiency increases with particle kinetic energy\\cite{caprioli14a, park15a}.  This mechanism also cannot energize electrons below $\\sim$100 keV because their Larmor radii are smaller than the gradient scale lengths of the shock \\textcolor{Red}{ramp\\cite{hobara10a, mazelle10a}} and upstream electromagnetic fluctuations\\cite{wilsoniii16a}.  Further, this mechanism predicts an inverse energy dispersion (i.e., lower energies enhance first)\\cite{anagnostopoulos86a, sarris87a}, which is not observed in the energetic electron enhancements (Fig. \\ref{fig:exampleTIFPs} and Extended Data Fig. \\ref{fig:THEMISFBPADExample}).  Thus, diffusive shock acceleration is generally ignored as a mechanism for energizing electrons from thermal to relativistic energies at the Earth's bow shock.\n\n\\indent  Second, shock drift acceleration predicts anisotropic velocity distributions, perpendicular downstream of shock and field-aligned far upstream of the Earth's bow shock\\cite{anagnostopoulos94b, anagnostopoulos09a}, neither of which are observed (Fig. \\ref{fig:examplePADs} and Extended Data Fig. \\ref{fig:THEMISFBPADExample}).  The mechanism efficiency decreases with increasing ratio of shock speed to $\\cos{\\theta{\\scriptstyle_{Bn}}}$ relative to the particle thermal energy because this increases the minimum energy threshold requirement\\cite{guo14b}, where each interaction with the shock can produce energy gains of factors $>$10 for strong quasi-perpendicular shocks\\cite{ball01a}.  However, any electron distribution observed far upstream (e.g., near the foreshock disturbances) would be highly anisotropic along the magnetic field streaming away from the shock as previously observed\\cite{anderson79a, anderson81a}, inconsistent with the isotropic distributions we observe (Fig. \\ref{fig:examplePADs} and Extended Data Fig. \\ref{fig:THEMISFBPADExample}).  Thus, we can rule out shock drift acceleration at the Earth's bow shock as a source.\n\n\\indent  Third, fast Fermi acceleration assumes electrons undergo a single adiabatic reflection -- particle conserves its magnetic moment, $\\mu$ $=$ $m{\\scriptstyle_{e}} v{\\scriptstyle_{\\perp}}^{2}\/2 B{\\scriptstyle_{o}}$ $\\sim$ constant, during the reflection, where $m{\\scriptstyle_{e}}$ is the electron mass and $v{\\scriptstyle_{\\perp}}$ is the speed perpendicular to $\\mathbf{B}{\\scriptstyle_{o}}$ -- and gains energy proportional to the shock speed divided by $\\cos{\\theta{\\scriptstyle_{Bn}}}$\\cite{savoini10a, wu84b}.  Previous studies\\textcolor{Red}{\\cite{anderson79a, anderson81a, leroy84a}} proposed this mechanism as an explanation for the ``thin sheets'' of highly anisotropic (i.e., field-aligned streaming away from Earth's bow shock) energetic electrons.  To satisfy the condition $\\mu$ $\\sim$ constant, the magnetic gradient scale length must be larger than the particle Larmor radius.  Further, for significant electron energy gains this mechanism requires either very large shock speeds compared to typical electron thermal speeds (i.e., $\\sim$1500--3000 km\/s) or $\\theta{\\scriptstyle_{Bn}}$ $\\gtrsim$ 88$^{\\circ}$.  Since the Earth's bow shock is very slow (i.e., typically $\\sim$100--500 km\/s), the Larmor radii of electrons $\\gtrsim$ few hundred eV are comparable to the shock ramp \\textcolor{Red}{thickness\\cite{hobara10a, mazelle10a}}, and this mechanism can only energize electrons with large pitch-angles, fast Fermi acceleration is not expected to produce energies beyond several 10s of keV\\cite{gosling89b, sarris85a, savoini10a, wu84b}.  We thus rule out the Earth's bow shock as the source of these energetic electron enhancements.\n\n{\\noindent \\textbf{Exclusion of Earth's magnetosphere as source.}}  As discussed in the main article, the observed energetic electron distributions are not highly anisotropic along the magnetic field streaming away from the Earth as previously reported in studies arguing for a magnetospheric source\\cite{anagnostopoulos94b, krimigis78a, kronberg11a, sarris76a, sarris87a}.  These studies observed enhanced geomagnetic activity in association with bursts of energetic electrons, where they argued that substorms -- a fundamental mode of the terrestrial magnetosphere resulting in magnetospheric circulation\/flows and enhanced auroral activity\\cite{angelopoulos08a} -- led to an increased rate of magnetospheric ``leakage.''  During substorms geostationary spacecraft can observe intense and rapid changes in 10s of keV to MeV electron fluxes\\cite{angelopoulos08a}.  One measure of substorm activity can be given by the well known AE indices\\cite{mann08a}.  These previous studies defined enhanced geomagnetic activity as an AE index $>$ 200 nT.\n\n\\indent  Extended Data Fig. \\ref{fig:THEMISAEIndices} shows the AE indices with color-coded bars indicating the time ranges for the example foreshock disturbances in Fig. \\ref{fig:exampleTIFPs}.  The AE index (Extended Data Figs \\ref{fig:THEMISAEIndices}\\textbf{q}--\\ref{fig:THEMISAEIndices}\\textbf{t}) was $<$ 200 nT for $>$1 hour prior to and during the three example disturbances with energetic electron enhancements (Figs \\ref{fig:exampleTIFPs}\\textbf{m}--\\ref{fig:exampleTIFPs}\\textbf{o}).  Thus, we can rule out a magnetospheric source due to the inconsistent pitch-angle distributions and lack of enhanced geomagnetic activity.\n\n\\indent  \\textcolor{Red}{One previous study\\cite{paschmann88a} did find nearly isotropic energetic electrons ($\\sim$70--200 keV) within hot flow anomalies.  However, the authors explicitly state these are not a suprathermal tail of the thermal electrons and that they originated from the magnetosphere.  There are several important differences with our results:  (1) we observe a single power-law from 100s of eV to $\\geq$140 keV in many enhancements, suggesting a common acceleration mechanism;  (2) we always observe ions above $\\sim$10 keV with every foreshock disturbance;  (3) we observe most electron enhancements (8\/10) within short large-amplitude magnetic structures and foreshock bubbles, both of which are disconnected from the bow shock; and (4) the ``magnetic bottle'' model proposed to explain isotropy in the previous study\\cite{paschmann88a} would not work for short large-amplitude magnetic structures because they contract as they evolve, not expand.}\n\n{\\noindent \\textbf{Instrument details.}}  Quasi-static (i.e., finite gain from zero up to Nyquist frequency) vector magnetic field measurements ($\\mathbf{B}{\\scriptstyle_{o}}$) were obtained using the fluxgate magnetometer\\cite{auster08a} at 4 and 128 samples per second.  The data are presented in units of nanotesla [nT] in the geocentric solar ecliptic coordinate basis.\n\n\\indent  Particle data are stored as velocity distribution functions covering $4 \\ \\pi$ steradian over an energy range defined by instrument design.  The electrostatic analyzers\\cite{mcfadden08a, mcfadden08b}, onboard each THEMIS\\cite{angelopoulos08a} spacecraft, detect particles using anodes placed behind microchannel plate detectors and cover an energy range of few eV to over 25 keV.  The number and placement of the anodes determines the poloidal\/latitudinal angular resolution, which is usually $\\Delta \\theta$ $\\sim$ 22.5$^{\\circ}$ for both the electron and ion electrostatic analyzers (Note that $\\Delta \\theta$ can be as low as $\\sim$5$^{\\circ}$ in some ion instrument modes.).  The azimuthal\/longitudinal resolution, $\\Delta \\phi$, is limited by the spacecraft spin rate and instrument design and mode of operation, but is generally $\\sim$11.25$^{\\circ}$.  The energy resolution, $\\Delta E\/E$, is defined by the instrument design and mode of operation but is generally $\\sim$20\\% for both electrostatic analyzers.\n\n\\indent  At higher energies (i.e., $\\sim$30 keV to over 500 keV), data from the electron and ion solid state telescopes\\cite{ni11b, turner13b} onboard each THEMIS spacecraft were used.  Each detector is comprised of two identical telescopes mounted at different angles on the side of the spacecraft body\\cite{angelopoulos08a}.  The angular and energy resolution is usually $\\Delta \\theta$ $\\sim$ 30$^{\\circ}$--40$^{\\circ}$, $\\Delta \\phi$ $\\sim$ 22.5$^{\\circ}$, and $\\Delta E\/E$ $\\sim$ $\\sim$ 30\\%.  For instance, the $\\sim$293 keV energy bin actually includes energetic electrons with $\\sim$249--337 keV energies.\n\n\\indent  All particle data presented herein, except for the high energy ions, \\textcolor{Red}{were} taken \\textcolor{Red}{while in} burst \\textcolor{Red}{mode,} which has a time resolution equal to the spacecraft spin period (i.e., $\\sim$3 seconds).  Even though the high energy ion data were measured in a different mode, the time resolution is still the spin period for the intervals presented.\n\n{\\noindent \\textbf{Unit conversion.}}  The raw data are measured in units of counts, which correspond to the number of events with a pulse height exceeding a defined threshold specific to each instrument.  Conversion to intensity and\/or phase space density requires knowledge of the instruments efficiency\\cite{bordoni71a, goruganthu84a}, deadtime\\cite{meeks08a, schecker92a}, accumulation time, and optical geometric factor\\cite{curtis89a, paschmann98a, wuest07b}.\n\n\\indent  Particle intensity is defined as:  number of particles, per unit area, per unit solid angle, per unit time, per unit energy (e.g., \\# cm$^{-2}$ s$^{-1}$ sr$^{-1}$ eV$^{-1}$).  This unit is not a Lorentz invariant, thus it requires one taking into account the Compton-Getting effect.  Phase space density is defined as:  number of particles, per unit spatial volume, per unit ``velocity volume'' (e.g., \\# s$^{+3}$ cm$^{-3}$ km$^{-3}$).  This unit is a Lorentz invariant under conditions when phase space is incompressible (i.e., when Liouville's theorem reduces to $df\/dt$ $=$ 0), which is true for most cases in in situ space observations\\cite{paschmann98a}.\n\n\\indent  The exact details of the unit conversion can be found in the THEMIS calibration software, called SPEDAS, found at: \\textcolor{Blue}{\\seqsplit{http:\/\/themis.ssl.berkeley.edu\/index.shtml}}.\n\n{\\noindent \\textbf{Reference frames and coordinate systems.}}  All particle data shown herein has been transformed from the spacecraft to the bulk flow reference frame using a relativistically correct Lorentz transformation.  The distributions were converted into units of phase space density prior to any frame transformation, thus any anisotropies due to the Compton-Getting effect\\cite{compton35a, ipavich74a} have been removed.\n\n\\noindent  Reference frame transformations were performed through the following steps:\n\\begin{enumerate}[itemsep=0pt,parsep=0pt,topsep=0pt]\n  \\item  bulk flow velocities were determined from the first velocity moment of the low energy ($<$30 keV) ion velocity distributions\\cite{wilsoniii14a};\n  \\item  all particle distributions were converted from raw counts to phase space density;\n  \\item  particle distributions were then transformed into the bulk flow rest frame using a standard Lorentz velocity transformation;\n  \\item  the particle distributions shown in units of intensity were converted to phase space density prior to the transformation and then back to intensity.\n\\end{enumerate}\n\n\\indent  Direction-dependent spectra, as opposed to omnidirectional averages (i.e., average over all solid angles), called pitch-angle distributions were calculated through the following steps:\n\\begin{enumerate}[itemsep=0pt,parsep=0pt,topsep=0pt]\n  \\item  particle distributions were transformed into the bulk flow rest frame as described above;\n  \\item  construct particle velocity unit vector for the $m^{th}$ particle distribution, $\\hat{\\mathbf{v}}{\\scriptstyle_{i,j,k}}^{m}$, for the $k^{th}$ energy bin from the $i^{th}$ latitude and $j^{th}$ longitude detector look directions described in \\textbf{Instrument details} above;\n  \\item  define the projection angle, $\\alpha{\\scriptstyle_{i,j,k}}^{m}$, between $\\hat{\\mathbf{v}}{\\scriptstyle_{i,j,k}}^{m}$ and the respective orientation unit vector, $\\hat{\\mathbf{u}}^{m}$ (e.g., $\\mathbf{B}{\\scriptstyle_{o}}$\/$\\lvert \\mathbf{B}{\\scriptstyle_{o}} \\rvert$), at the measurement time of the $m^{th}$ particle distribution, where $\\alpha{\\scriptstyle_{i,j,k}}^{m}$ $=$ $\\cos^{-1}{\\left( \\hat{\\mathbf{v}}{\\scriptstyle_{i,j,k}}^{m} \\cdot \\hat{\\mathbf{u}}^{m} \\right)}$; and\n  \\item  define three cuts for the $m^{th}$ particle distribution by averaging data within $\\pm$22.5$^{\\circ}$ of the parallel, perpendicular, and anti-parallel directions defined by $\\hat{\\mathbf{u}}^{m}$.\n\\end{enumerate}\n\n\\noindent  The two relevant directions about which we oriented the particle distributions are the local quasi-static magnetic field vector, $\\mathbf{B}{\\scriptstyle_{o}}$, and the spacecraft-to-Earth unit vector, $\\hat{\\mathbf{e}}{\\scriptstyle_{SC}}$, at the time of each distribution.  Note that the formal definition of a pitch-angle distribution is constructed only with respect to $\\mathbf{B}{\\scriptstyle_{o}}$ but we use the term here for both orientations for brevity.\n\n\\indent  The high time resolution equivalent of the above algorithm involves only a few differences described below.\n\\begin{enumerate}[itemsep=0pt,parsep=0pt,topsep=0pt]\n  \\item  Instead of using a single $\\hat{\\mathbf{u}}^{m}$ for the $m^{th}$ particle distribution, we define $\\hat{\\mathbf{u}}{\\scriptstyle_{i,j,k}}^{m}$ for the $k^{th}$ energy bin from the $i^{th}$ latitude and $j^{th}$ longitude detector look directions of the $m^{th}$ particle distribution.\n  \\item  Now the pitch-angles are defined as $\\alpha{\\scriptstyle_{i,j,k}}^{m}$ $=$ $\\cos^{-1}{\\left( \\hat{\\mathbf{v}}{\\scriptstyle_{i,j,k}}^{m} \\cdot \\hat{\\mathbf{u}}{\\scriptstyle_{i,j,k}}^{m} \\right)}$.\n\\end{enumerate}\n\n\\noindent  The result is a pitch-angle distribution with fewer aliasing effects due to the use of a single $\\hat{\\mathbf{u}}^{m}$ averaged over the duration of the $m^{th}$ particle distribution (Extended Data Fig. \\ref{fig:exampleCompareLTHTBo}).\n\n\\indent  The exact details of the rotations and frame transformations can be found in the additional analysis software at:  \\textcolor{Blue}{\\seqsplit{https:\/\/github.com\/lynnbwilsoniii\/{wind\\_3dp\\_pros}}}.\n\n{\\noindent \\textbf{Particle data presentation.}}  All particle distributions are presented in the bulk flow rest frame in a physically significant coordinate basis, e.g., magnetic field-aligned coordinates.  We define the bulk flow reference frame as described above (in \\textbf{Reference frames and coordinate systems}).  The energy ranges listed above (in \\textbf{Instrument details}) are the measured midpoint kinetic energies in the spacecraft frame of reference.  Energy values will change under any Lorentz transformation.  Thus, the energies for the ion electrostatic analyzer data are not constant in time and show a large variability owing to the large variability of the bulk flow near foreshock disturbances.  In contrast, the low energy electron data above $\\sim$50 eV and all solid state telescope data suffer little energy change under these Lorentz transformations, thus can be approximated as constant in time.  Therefore, we show the low and high energy electron and high energy ion data as stacked line plots vs. time where each line corresponds to a different energy and the low energy ion data are presented as a dynamic energy spectrogram of energy vs. time with the color scale indicating the particle intensity.  Note that while the electron solid state telescope can measure electrons to $>$400 keV, for only the 2008-08-19 event in Fig. \\ref{fig:exampleTIFPs} were significant fluxes observed in energy bins $>$140 keV.\n\n{\\noindent \\textbf{Data availability.}}  The THEMIS data used in this paper is publicly available at: \\\\\n\\textcolor{Blue}{\\seqsplit{http:\/\/themis.ssl.berkeley.edu\/index.shtml}}.\n\n\\noindent  The \\emph{Wind} and STEREO radio data were taken from the S\/WAVES website at: \\\\ \n\\textcolor{Blue}{\\seqsplit{http:\/\/swaves.gsfc.nasa.gov\/data\\_access.html}}.\n\n\\noindent  Solar wind data was taken from the OMNI data products found on CDAWeb at: \\\\\n\\textcolor{Blue}{\\seqsplit{http:\/\/cdaweb.gsfc.nasa.gov}}.\n\n\\noindent  The \\emph{Wind} interplanetary shock list can be found at: \\\\\n\\textcolor{Blue}{\\seqsplit{https:\/\/www.cfa.harvard.edu\/shocks\/wi\\_data\/}}.\n\n{\\noindent \\textbf{Code availability.}}  The THEMIS instrument calibration software, called SPEDAS, can be found at: \\\\ \n\\textcolor{Blue}{\\seqsplit{http:\/\/themis.ssl.berkeley.edu\/index.shtml}};\n\n\\noindent and additional analysis software can be found at: \\\\\n\\textcolor{Blue}{\\seqsplit{https:\/\/github.com\/lynnbwilsoniii\/{wind\\_3dp\\_pros}}}.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nTaking the Yang-Mills model as the matter part of the equations of Einstein,\none finds a rich spectrum of stationary regular and non-regular solutions.\nThis in contrast\nwith the vacuum solution and Einstein-Maxwell(EM) solutions, where the Schwarzschild\nrespectively the Reissner-Nordstr{\\\"o}m(RN) solution are the only static black hole\nsolutions. In the Einstein-Yang-Mills(EYM) theory one finds not only the embedding of the\nAbelian RN black hole, but in addition there are genuine non-Abelian coloured black\nholes. Their co-existence gives rise to the violation of the no-hair conjecture,\nsince they carry the same (magnetic) charge. For an overview, see Volkov\nand Gal'tsov~\\cite{volk99}.\nAfter the discovery by Bartnik and McKinnon(BMK)~\\cite{bart88}\nof these non-Abelian particle-like and later the non-RN (hairy) black hole solutions\nof the SU(2) EYM theory~\\cite{smol96}, throughout instability\ninvestigations where done by several authors~\\cite{strau190,lav95,strau290,zhou91,breit97}.\nIt turns out that under spherically symmetric perturbations the BMK solution\nis unstable. The nth BMK solution has in fact 2n unstable modes,\ncomparable with the flat space-time 'sphaleron' solution. In the static case,\nit was recently conjectured~\\cite{mav98} that there are hairy black holes\nin the SU(N) EYM-theories with topological instabilities.\nIn order to analyze the stability of the solutions, one usually linearized\nthe field equations~\\cite{strau190,strau290}.\nWe conjectured that the conventional linear analysis is in fact inadequate\nto be applied to the situation where a singularity is formed~\\cite{Slag398}.\nOne better can apply the so-called multiple-scale or two-timing method,\ndeveloped by Choquet-Bruhat~\\cite{choq169,choq277,taub80}. This method\nis particularly useful for constructing uniformly valid approximations to\nsolutions of perturbation problems~\\cite{slag191}.\nAt the threshold of black hole\nformation it is found numerically~\\cite{chop193,chop296,garf97,bizon96}\nthat there is a critical parameter\nwhose value separates solutions containing black holes\nfrom those which do not. For a critical value one observes self-similarity:\nit produces itself (echoes) on progressively finer scales (Choptuik-scaling).\nIt is evident that the collapsing ball of field energy will produce\ngravitational waves, which will be coupled to the  YM-field perturbations.\nDue to the accumulation of echoes, the curvature diverges.\nClose related to this critical behavior is the comparable irregularities which\ncan be found in the Einstein-Skyrme(ES) model and the Einstein-Yang-Mills-Higgs(EYMH)\nmodel. In the latter model, space-like hypersurfaces develop for a critical\nvalue of $\\alpha =M_W\/gM_{pl}$ two distinct regions separated by a long throat,\nwith $M_W$ the YM scale and $M_{pl}=1\/\\sqrt{4\\pi G}$.\n\nThe solutions mentioned above need not be spherically symmetric. Since the\npioneering work of Stachel~\\cite{Sta66}, we know that stationary\ncylindrical symmetric rotating models possess intriguing features.\nFirst, the evolution of the 'classical' cylindrical gravitational wave-solution\nin the vacuum situation can be described in terms of two effects: the usual\ncylindrical reflections and a rotation of the polarization vector of the waves,\nan effect comparable with the Faraday rotation~\\cite{Piran85}.\nSecondly, the time-dependent solutions can be related to the cosmic-string\nsolution interacting with gravitational waves~\\cite{Xan186}.\nUsually the cosmic string solution, with its famous conical structure\n,is found in the U(1)-gauge Einstein-Higgs system.\nIt is remarkable that the cylindrical rotating vacuum solution possesses the\nsame conical structure. Moreover, it is asymptotically flat and free\nof singularities.\nThe physical interpretation can nicely be formulated using the Belinsky-Zakharov\nmethod of integrating the Einstein equations with the help of the Ernst\npotentials~\\cite{Eco88}. Using the C-energy description, one can conclude that the\nsolution represents solitonic gravitational waves interacting with a\nstraight-line cosmic string, where the asymptotic angle deficit depends\non the C-energy of the gravitational waves.\nWhen  a Maxwell field is incorporated, the physical interpretation of the solutions\nis not easily formulated. It is hard to believe that in the electrovac solution\nthe electro-magnetic waves do not contribute to the angle deficit~\\cite{Xan287}.\nMoreover, the mass per unit length of the cosmic string, which is a constant\nin the Xanthopoulos solution, will be determined by the energy-momentum tensor\nof the non-vacuum situation. In stead of 'electrifying' or 'magnetifying' the\nvacuum solutions via the Ernst formulation, one can better consider the coupled\nsystem of Einstein equations and matter field equations.\n\nHere we consider the Einstein-Yang-Mills system on a cylindrical symmetric\nrotating space-time. This situation con formally be obtained from the\naxially symmetric situation by complex substitution of $t\\to iz, z\\to it$~\\cite{kram80}.\nIn the non-rotating cylindrically symmetric situation, it was numerically found\n~\\cite{slag299} that the singular behavior at finite distance of the core as found\nin the supermassive situation, will be pushed to infinity for some critical values\nof the gauge coupling constant and one of the YM components.\n\nWe will use a (2+1)+1 reduction scheme~\\cite{Mae80}\nin order to obtain the suitable parameterization for the Yang-Mills potential.\nIn fact one replaces the YM field by a 2D Abelian one-form, a 2D Kaluza-Klein\ntwo-form, a 2D matrix-valued scalar and a 3D Higgs field~\\cite{gal98}.\n\nIn the Abelian counterpart model it is known that the space-time around\n(spinning) gauge strings could have, besides the usual angle deficit feature,\nexotic properties, such as the violation of causality~\\cite{Jen92,Sol92},\ntime-delay effects~\\cite{Har88}, helical structure\nof time~\\cite{Des92,Let95,Maz86} and frame-dragging~\\cite{Bon91}.\nA class of approximate solutions of the coupled Einstein-scalar-gauge field equations\non the ($r,z$)-plane of an axially symmetric space-time was found~\\cite{slag695}.\nSo a question addressed in this paper will be whether the string features will be\nmaintained in the EYM system.\nIn the general $(r,z)$-dependent situation, a nice overview was given by\nIslam~\\cite{Islam85}. One can proof via the Ernst formulation that for any solution\nof the stationary\nEinstein equations one can generate a corresponding solution of the EM equations.\nIn fact, one introduces two complex potentials, for which one finds two second-order\nelliptic differential equations.\nStarting with the Kerr solution, it is possible to use this correspondence to find\na solution of the EM equations leading to the Kerr-Newman solution.\nThis 'Ernst-route' fails in the EYM system, simply by the fact that one\nof the metric components does not decouple from the other ones.\nIt is also known that the electrovac staticity theorem~\\cite{heus93} does not generalize to the EYM\nsystem. In fact, spinning EYM systems must be electrically charged. So another question\naddressed in this paper is whether electrically and magnetically charged solution will exist\nin the EYM system.\nFinally, this is not the complete story: the investigations can be extended to the\nmodel where the interior solution is  properly matched onto the exterior\nsolution of the string~\\cite{slag496}. This is currently under study by the author.\n\nThe plan of this paper is as follows. In Sec. II we derive the field equation\non an cylindrical symmetric space time. In Sec. III we present the numerical solution.\nIn section IV we present a approximate solution using the multiple-scale method and\nin Sec. V we summarize and analyse our results.\n\n\\section{The field equations of the Einstein Yang-Mills system}\n\nConsider the Lagrangian of the SU(2) EYM system\n\\begin{equation}\nS=\\int d^4x\\sqrt{-g}\\Bigl[\\frac{{\\cal R}}{16\\pi G}-\n\\frac{1}{4}{\\cal F}_{\\mu\\nu}^a{\\cal F}^{\\mu\\nu a}\\Bigr],\n\\end{equation}\nwith the YM field strength\n\\begin{equation}\n{\\cal F}_{\\mu\\nu}^a =\\partial_\\mu A_\\nu^a-\\partial_\\nu\nA_\\mu^a+g\\epsilon^{abc}A_\\mu^bA_\\nu^c,\n\\end{equation}\ng the gauge coupling constant, G Newton's constant, $A_\\mu^a$ the gauge potential,\nand ${\\cal R}$ the curvature scalar. The field equations then become\n\\begin{equation}\nG_{\\mu\\nu}=8\\pi G {\\cal T}_{\\mu\\nu},\n\\end{equation}\n\\begin{equation}\n{\\cal D}_\\mu {\\cal F}^{\\mu\\nu a}=0,\n\\end{equation}\nwith ${\\cal T}$ the energy-momentum tensor\n\\begin{equation}\n{\\cal T}_{\\mu\\nu}={\\cal F}_{\\mu\\lambda}^a{\\cal F}_\\nu^{\\lambda a}\n-\\frac{1}{4}g_{\\mu\\nu}{\\cal F}_{\\alpha\\beta}^a {\\cal F}^{\\alpha\\beta a},\n\\end{equation}\nand ${\\cal D}$ the gauge-covariant derivative,\n\\begin{equation}\n{\\cal D}_\\alpha{\\cal F}\n_{\\mu\\nu}^a\\equiv \\nabla_\\alpha{\\cal F}_{\\mu\\nu}^a+g\\epsilon^{abc}A_\\alpha^b{\\cal F}\n_{\\mu\\nu}^c.\n\\end{equation}\nLet us consider the cylindrical space time\n\\begin{equation}\nds^2=-\\Omega^2(dt^2-dr^2)+f(dz+\\omega d\\varphi)^2+\\frac{r^2}{f}d\\varphi ^2,\n\\end{equation}\nwhere $\\Omega, f$ and $\\omega $ are functions of t and r.\nThe problem is, which ansatz one needs for the YM field.\nIt is often argued that Einsteins gravity on a three dimensional\nspace time shows some similarity with Chern Simons theory with the Poincar\\'e\ngroup being the underlying gauge group. So when trying to find the dimensional\nreduction for the stationary axially symmetric EYM system, one can be guided by\nthe results found by using the (2+1)+1 reduction scheme suggested by Maeda, Sasaki,\nNakamura and Miyama~\\cite{Mae80}, and later worked out by Gal'tsov~\\cite{gal98}.\nOne writes in general\n\\begin{equation}\nds^2=-e^\\psi (dt+v_idx^i)^2+e^{2\\phi-\\psi}(d\\varphi +k_adx^a)^2+g_{ab}dx^adx^b,\n\\end{equation}\nwhere $v_idx^i=\\omega (d\\varphi +k_adx^a)+\\nu_adx^a, k_a$ an one-form, generating\na field strength  $\\kappa_{ab}=\\partial_ak_b-\\partial_bk_a$. The one-form $\\nu_a$ give rise to the\n2D field strength $\\omega_{ab}=\\partial_a\\nu_b-\\partial_b\\nu_a+\\omega\\kappa_{ab}$(a,b=1,2).\nThe YM potential is then written as\n\\begin{equation}\nA_\\mu dx^\\mu ={\\bf a}_adx^a +{\\bf \\Phi}(dt+\\nu_idx^i)+{\\bf \\Psi}(d\\varphi +k_adx^a)\n\\end{equation}\nHere ${\\bf a}$ is the dynamical part of the YM field, parameterized by a 2D complex\nAbelian one-form, ${\\bf \\Phi}$ a 3D Higgs field and ${\\bf \\Psi}$ a matrix-valued\nscalar. In our simplified case of metric (7), we have $\\nu_1=-\\omega k_1, k_2=\\nu_2 =0$ and\nthe YM potential can be as\n\\begin{equation}\nA_\\mu ={\\bf \\Phi}(dz+\\omega d\\varphi )+{\\bf \\Psi}(d\\varphi -k_1dt) +{\\bf a}_adx^a\n\\end{equation}\nThe YM-part of the Lagrangian can then be written as\n\n\\begin{eqnarray}\n{\\cal L}_{YM}=\\Omega^2r\\Bigl[\\Bigl\\{f_{ab}f^{ab}+\\frac{2f}{r^2}(D_a{\\bf\\Psi} +{\\bf\\Phi}\n\\partial_a\\omega -a_a^{'})(D^a{\\bf\\Psi}+{\\bf\\Phi}\\partial^a\\omega\n -a^{' a}) \\Bigr\\} \\cr +\\frac{2}{f}\\Bigl\\{D_a{\\bf\\Phi}D^a{\\bf\\Phi}\n +\\frac{f}{r^2}({\\bf\\Phi}^{'}+g[{\\bf\\Phi},{\\bf\\Psi}])^2\\Bigr\\}\\Bigl],\n\\end{eqnarray}\nwith\n\\begin{equation}\nD_a{\\bf\\Psi}=\\partial_a{\\bf\\Psi}+g[a_a,{\\bf\\Psi}]-k_a{\\bf\\Psi}^{'},\n\\end{equation}\n$f_{ab}$ the 2D field strength\n\\begin{equation}\nf_{ab}=\\partial_a a_b-\\partial_b a_a +g[a_a,a_b]+a_a^{'}k_b-a_b^{'}k_a+{\\bf \\Phi}\\omega_{ab}+\n{\\bf \\Psi}k_{ab},\n\\end{equation}\nand a prime denoting the partial derivative with respect to $\\varphi$.\n\nLet us consider the following parameterization\n\\begin{eqnarray}\na_1=A_2(t,r)\\tau_\\varphi ,\\quad &&a_2=A_1(t,r)\\tau_\\varphi,\\quad\n{\\bf\\Psi}=W_1(t,r)\\tau_r +(W_2(t,r)-1)\\tau_z,\\cr\n&& {\\bf\\Phi}=\\Phi_1(t,r)\\tau_r+\\Phi_2(t,r)\\tau_z,\n\\end{eqnarray}\nwith $\\tau_i$ the usual set of orthonormal vectors.\nFrom the condition $T_{tt}-T_{rr}=0$, we obtain\n\\begin{equation}\n\\partial_t A_1=\\partial_r A_2,\\quad  W_2=\\frac{\\Phi_2 W_1}{\\Phi_1}+\\frac{g-1}{g}.\n\\end{equation}\n\nFrom the Einstein equations and the YM equations we obtain the set of\npartial differential equations for $\\Omega , f , \\omega , \\Phi_1 , \\Phi_2$\nand $W_1$.\nIt follows  from a combination of the YM equations, that $A_1$\nand $A_2$ can be expressed in $\\Phi_1$ and $\\Phi_2$:\n\\begin{eqnarray}\nA_1 =\\frac{\\Phi_1\\partial_r\\Phi_2 -\\Phi_2\\partial_r\\Phi_1}{g(\\Phi_1^2+\\Phi_2^2)},\n\\qquad A_2 =\\frac{\\Phi_1\\partial_t\\Phi_2-\\Phi_2\\partial_t\\Phi_1}{g(\\Phi_1^2+\\Phi_2^2)}\n\\end{eqnarray}\nFor the gauge $\\partial_t A_2=\\partial_r A_1$\nand hence $\\partial_t^2 A_i-\\partial_r^2 A_i =0$, we obtain from the Einstein equations Eq.(3)\n\\begin{eqnarray}\n\\partial_t^2\\Omega -\\partial_r^2\\Omega +\\frac{f^2\\Omega}{4r^2}\\Bigl[(\\partial_t\\omega )^2\n-(\\partial_r\\omega)^2\\Bigr]+\\frac{1}{\\Omega}\\Bigl[(\\partial_r\\Omega )^2\n-(\\partial_t\\Omega)^2\\Bigr]+\n\\frac{\\Omega}{4f^2}\\Bigl[(\\partial_t f)^2-(\\partial_r f)^2\\Bigr]\n+ \\frac{\\Omega}{2fr}\\partial_r f=0,\n\\end{eqnarray}\n\\begin{eqnarray}\n\\partial_t^2 f -\\partial_r^2 f +\\frac{1}{f}\\Bigr[(\\partial_r f)^2-(\\partial_t f)^2\n\\Bigr]+\\frac{f^3}{r^2}\\Bigl[(\\partial_r\\omega)^2-(\\partial_t\\omega)^2\\Bigr]\n-\\frac{1}{r}\\partial_r f = \\cr \\frac{8\\pi G}{r^2\\Phi_1^4}\\Bigl[(\\Phi_1^4r^2-f^2\\Phi_2^2W^2)(\n(\\partial_r\\Phi_1)^2-(\\partial_t\\Phi_1)^2) +(\\Phi_1^4r^2-f^2\\Phi_1^2W^2)((\\partial_r\\Phi_2)^2\n-(\\partial_t\\Phi_2)^2) \\cr +f^2\\Phi_1^2(\\Phi_1^2+\\Phi_2^2)\\Bigl((\\partial_tW)^2-(\\partial_rW)^2\n+\\Phi_1^2\\Bigl((\\partial_t\\omega)^2-(\\partial_r\\omega)^2\\Bigr)\\Bigr) \\cr\n+2f^2\\Phi_1^3(\\Phi_1^2+\\Phi_2^2)(\\partial_t\\omega\\partial_tW-\\partial_r\\omega\\partial_rW)\n+2f^2\\Phi_2^2\\Phi_1W(\\partial_rW\\partial_r\\Phi_1-\\partial_tW\\partial_t\\Phi_1)\\cr\n+2f^2\\Phi_1^3\\Phi_2W(\\partial_t\\Phi_2\\partial_t\\omega -\\partial_r\\Phi_2\\partial_r\\omega )\n+2f^2\\Phi_1^2\\Phi_2W(\\partial_tW\\partial_t\\Phi_2-\\partial_rW\\partial_r\\Phi_2)\\cr\n+2f^2\\Phi_1\\Phi_2W^2(\\partial_r\\Phi_2\\partial_r\\Phi_1-\\partial_t\\Phi_2\\partial_t\\Phi_1)\n+2f^2\\Phi_2^2\\Phi_1^2W(\\partial_r\\Phi_1\\partial_r\\omega-\\partial_t\\Phi_1\\partial_t\\omega )\\cr\n+g\\Phi_1^2(\\Phi_1^2r^2-f^2W^2)\\Bigl(2A_1(\\Phi_2\\partial_r\\Phi_1-\\Phi_1\\partial_r\\Phi_2)\n+2A_2(\\Phi_1\\partial_t\\Phi_2-\\Phi_2\\partial_t\\Phi_1)+g(\\Phi_1^2+\\Phi_2^2)(A_1^2-A_2^2)\\Bigl)\\Bigl]\n\\end{eqnarray}\n\\begin{eqnarray}\n\\partial_t^2\\omega -\\partial_r^2\\omega +\\frac{2}{f}(\\partial_tf\\partial_t\\omega\n-\\partial_rf\\partial_r\\omega )+\\frac{1}{r}\\partial_r\\omega\n=\\frac{16\\pi G}{f\\Phi_1^2}\\Bigl[W\\Phi_1 ((\\partial_r\\Phi_2)^2-(\\partial_t\\Phi_2)^2)\\cr\n+\\Phi_2W(\\partial_t\\Phi_2\\partial_t\\Phi_1-\\partial_r\\Phi_2\\partial_r\\Phi_1)\n+\\Phi_2\\Phi_1^2(\\partial_r\\Phi_2\\partial_r\\omega -\\partial_t\\Phi_2\\partial_t\\omega )\n+\\Phi_2\\Phi_1(\\partial_r\\Phi_2\\partial_rW-\\partial_t\\Phi_2\\partial_tW) \\cr\n+\\Phi_1^2(\\partial_r\\Phi_1\\partial_rW-\\partial_t\\Phi_1\\partial_tW)\n+\\Phi_1^3(\\partial_r\\Phi_1\\partial_r\\omega-\\partial_t\\Phi_1\\partial_t\\omega ) \\cr\n+g^2\\Phi_1W(\\Phi_1^2+\\Phi_2^2)(A_1^2-A_2^2)\n+2g\\Phi_1^2W(A_2\\partial_t\\Phi_2-A_1\\partial_r\\Phi_2)+2g\\Phi_1\\Phi_2W(A_1\\partial_r\\Phi_1\n-A_2\\partial_t\\Phi_1)\\Bigr],\n\\end{eqnarray}\nand from the YM equations Eq. (4)\n\\begin{eqnarray}\n\\partial_t^2\\Phi_1-\\partial_r^2\\Phi_1-\\frac{1}{r}\\partial_r\\Phi_1+\\frac{f^2\\Phi_1}{r^2}\n\\Bigl((\\partial_r\\omega)^2-(\\partial_t\\omega)^2\\Bigl)\n+\\frac{1}{f}(\\partial_rf\\partial_r\\Phi_1-\\partial_tf\\partial_t\\Phi_1) \\cr\n+\\frac{f^2}{r^2}(\\partial_r\\omega\\partial_rW-\\partial_t\\omega\\partial_tW)\n+\\frac{gf^2\\Phi_2W}{\\Phi_1r^2}(A_1\\partial_r\\omega - A_2\\partial_t\\omega )\\cr\n+\\frac{g\\Phi_2}{f}(A_1\\partial_rf-A_2\\partial_tf)\n+2g(A_2\\partial_t\\Phi_2-A_1\\partial_r\\Phi_2)-\\frac{gA_1\\Phi_2}{r}+g^2\\Phi_1(A_1^2-A_2^2)=0\n\\end{eqnarray}\n\\begin{eqnarray}\n\\partial_t^2\\Phi_2-\\partial_r^2\\Phi_2 -\\frac{1}{r}\\partial_r\\Phi_2+\\frac{f^2\\Phi_2}{r^2}\n\\Bigl((\\partial_r\\omega)^2-(\\partial_t\\omega)^2\\Bigr)\n+\\frac{1}{f}(\\partial_rf\\partial_r\\Phi_2-\\partial_tf\\partial_t\\Phi_2)\n+\\frac{f^2\\Phi_2}{r^2\\Phi_1}(\\partial_r\\omega\\partial_rW-\\partial_t\\omega\\partial_tW)\\cr\n+\\frac{f^2\\Phi_2W}{r^2\\Phi_1^2}(\\partial_t\\omega\\partial_t\\Phi_1-\\partial_r\\omega\\partial_r\\Phi_1)\n+\\frac{f^2W}{r^2\\Phi_1}(\\partial_r\\omega\\partial_r\\Phi_2-\\partial_t\\omega\\partial_t\\Phi_2)\n+2g(A_1\\partial_r\\Phi_1-A_2\\partial_t\\Phi_1) \\cr +\\frac{g\\Phi_1}{f}(A_2\\partial_tf-A_1\\partial_rf)\n+\\frac{gf^2W}{r^2}(A_2\\partial_t\\omega-A_1\\partial_r\\omega)+g^2\\Phi_2(A_1^2-A_2^2)\n+\\frac{gA_1\\Phi_1}{r}=0\n\\end{eqnarray}\n\\begin{eqnarray}\n\\partial_t^2 W_1-\\partial_r^2 W_1 +\\frac{1}{r}\\partial_rW\n+\\frac{1}{f}(\\partial_tf\\partial_tW-\\partial_rf\\partial_rW)\n+\\frac{\\Phi_1}{f}(\\partial_rf\\partial_r\\omega -\\partial_tf\\partial_t\\omega ) \\cr\n+\\partial_t\\omega\\partial_t\\Phi_1-\\partial_r\\omega\\partial_r\\Phi_1\n+\\frac{2g\\Phi_2}{\\Phi_1}(A_2\\partial_tW-A_1\\partial_rW)\n+g\\Phi_2(A_2\\partial_t\\omega-A_1\\partial_r\\omega) \\cr\n+\\frac{2gW\\Phi_2}{\\Phi_1^2}(A_1\\partial_r\\Phi_1-A_2\\partial_t\\Phi_1)+\n\\frac{2gW}{\\Phi_1}(A_2\\partial_t\\Phi_2-A_1\\partial_r\\Phi_2)  +\\frac{gW\\Phi_2}{f\\Phi_1}\n(A_2\\partial_tf-A_1\\partial_rf)\\cr +g^2W(A_1^2-A_2^2)+\\frac{gA_1\\Phi_2W}{r\\Phi_1}\n-\\frac{16\\pi G}{f}\\Bigl[W\\Bigl((\\partial_t\\Phi_2)^2-(\\partial_r\\Phi_2)^2\\Bigr)\\cr\n+\\Phi_1(\\partial_t\\Phi_1\\partial_tW-\\partial_r\\Phi_1\\partial_rW)\n+\\Phi_1^2(\\partial_t\\omega\\partial_t\\Phi_1-\\partial_r\\omega\\partial_r\\Phi_1) \\cr\n+\\Phi_2(\\partial_t\\Phi_2\\partial_tW-\\partial_r\\Phi_2\\partial_rW)\n+\\frac{W\\Phi_2}{\\Phi_1}(\\partial_r\\Phi_2\\partial_r\\Phi_1-\\partial_t\\Phi_2\\partial_t\\Phi_1)\n+\\Phi_1\\Phi_2(\\partial_t\\Phi_2\\partial_t\\omega-\\partial_r\\Phi_2\\partial_r\\omega) \\cr\n+2gW\\Phi_1(A_1\\partial_r\\Phi_2-A_2\\partial_t\\Phi_2)\n+2gW\\Phi_2(A_2\\partial_t\\Phi_1-A_1\\partial_r\\Phi_1)+g^2\\Phi_1^2W(\\Phi_1^2+\\Phi_2^2)(A_2^2-A_1^2)\\Bigr]=0.\n\\end{eqnarray}\n\n\\section{Numerical solution}\nIn the vacuum situation, the behavior of the cylindrical wave solution is\nwell established~\\cite{Bon294,Tom89}. One can have reflection of incoming\ninto outgoing waves with the same polarization. Further, there is\na non-linear effect describing the interaction between the two different\npolarization $+$ and $\\times$ modes. If an\noutgoing cylindrical wave is linear polarized, its polarization vector\nrotates as it propagates. This effect  is often called 'Faraday' rotation and is given by\n\\begin{equation}\n\\tan\\theta_A=\\frac{A_x}{A_+},\\quad \\tan\\theta_B=\\frac{B_x}{B_+}\n\\end{equation}\nwith\n\\begin{equation}\nA_{+,x}=\\frac{1}{f}(\\partial_tf\\pm\\partial_rf),\\quad B_{+,x}=\\frac{f}{r}(\\partial_t\\omega\n\\pm\\partial_r\\omega)\n\\end{equation}\n\n\nA wave emitted from the axis of symmetry behaves like a shock\nfront and is given by\n\\begin{equation}\nf=w(a^2+1)\/(a^2+w^2),\\quad \\omega=ra(1-w^2)\/w(a^2+1),\n\\quad \\Omega^2 f=b\\sqrt{r}\/\\sqrt{t^2-r^2},\n\\end{equation}\nwith $w=(t-\\sqrt{t^2-r^2})\/r$ and a and b arbitrary constants.\nOne can generate a two-soliton solution and it is found that near the axis\n$\\log (\\Omega^2 f)$, which also describes the 'C-energy',\ndoes not approach zero. So some matter field\nmust act as a possible source of the two-soliton field. This curious\nconclusion can also be formulated in context of the cosmic string with mass\nunit length $m=p\/(2p-2)$, with p a constant originating from the static solution\n$f=r^{2p}, \\Omega =ar^{p^2-p}$, which on his turn is related to the\nangle deficit of the string, given by\n\\begin{equation}\n\\delta\\phi=2\\pi(1-\\frac{1}{\\sqrt{\\Omega^2f}})\n\\end{equation}\nWhen a 'Rosen'-pulse $\\ln f =\\int_0^{t-r}f(\\beta)d\\beta \/ \\sqrt{(t-\\beta)^2-r^2}$ is emitted,\n$\\log (\\Omega^2 f)$ changes by an amount $-k^2$, where k depends solely on $f(\\beta )$.\nSo after the pulse, one has a different string, which let Marder~\\cite{mar58}\nto conclude that the mass per unit length has decreased by an amount\n$\\sim k^2$. Because $\\log (\\Omega^2f)$ is related to the 'C'-energy, this confirms that\nthe waves carry away energy and are evidently generated by a source on the\naxis. So a continuous change in the parameter p  may produce suddenly\nand inexplicable change in physical meaning.\n\n\nFurther, it was emphasized in the vacuum case that there is a reflection of the outgoing\nwave into an ingoing wave and visa versa.\n\n\nFrom the Einstein-Maxwell case~\\cite{Xan186,Xan287} it is known that there exist\nstable open cosmic strings coupled with gravity and electromagnetism.\nWhen gravitational and electromagnetic waves are generated,the angle deficit\nis increased asymptotically and near the axis independent of the coupling\nof the Maxwell field to gravity.\nFurther, it was emphasized that the electromagnetic waves do contribute\nto the rotation of the spacetime by effecting the 'dragging' of the Killing field\n$\\frac{\\partial}{\\partial z}$, but not to the angle deficit.\n\n\nIn our EYM case we solved Eq.(17)-(22) for  the initial values\n\\begin{eqnarray}\n\\Omega(0,r)=C_1r^{p^2-p},\\quad f(0,r)=r^{2p}, \\quad \\Phi_1(0,r)=1-e^{-C_2r},\n\\quad \\Phi_2(0,r)=e^{-C_3r}-1\n\\end{eqnarray}\nFor  $W_1$ we will take as initial value a node number 1\nor node number 2 function, i.e., $W_1(0,r)=-\\tanh(r-1)$, or $W_1 (0,r)=1-e^{-C_4r^2}$.\nHere are $C_i$ some constants.\n\nWe used a 100x100 grid and applied the method of lines with cubic Hermite polynomials.\nThe roundoff error remained below 0.001.\nIn figure 1 we took  $\\omega(0,r)=\\frac{3.5r}{r^2+1}$ and for $W_1$ the\nnode number 2 function. We plotted respectively: $\\Omega$, f, $\\omega$, $\\Phi_1$,\n$W_1$, $\\Phi_2$, the angle deficit, the gravitational puls wave $\\Omega^2f$, $g_{\\varphi\\varphi}$,\nand $W_2$.\nIn figure 2 we took a different initial $\\Omega$ (p=0.1 in stead of 0.25).\nWe obtain a solitonlike solution. The behavior depends  on the initial mass\nper unit length of the string, just as in the vacuum case.\nIn figure 3 we took the node number 1 function for $W_1$. We observe that the angle deficit\nchanges significantly as well as the gravitational outgoing pulse. So the Yang-Mills waves\nhas an impact on the gravitational waves. In figure 4 we took p=0.1 in stead of\np=0.25. We see that the behavior of the solution changes significantly: the angle deficit\nclose to the r-axis increases, while the gravitational puls 'hangs' close to the r-axes.\n\nIn figure 5 we plotted a long-time run for a different initial $\\omega$: an outgoing wave\nis reflected into an ingoing wave.\n\nAn important question is if the solution admits gauge-charges.\nThe gauge-charges $Q_E,Q_M$ (electric and magnetic) are given by\n\n\\begin{eqnarray}\nQ_E=\\frac{1}{4\\pi}\\oint_S\\vert ^*F\\vert,\\quad Q_M=\\frac{1}{4\\pi}\\oint_S\\vert F\\vert,\n\\end{eqnarray}\nwith F the YM field strength, $^*F$ the dual tensor and the integration over\na two-sphere at spatial infinity.\nIn our situation, we obtain for the charges\n\\begin{eqnarray}\nQ_E=\\int r\\Bigl\\{\\Bigl[\\partial_t(W_1+\\omega\\Phi_1)+\ngA_2(W_2-1+\\omega\\Phi_2)+A_2\\Bigr]\\tau_r +\\Bigl[\\partial_t(W_2+\\omega\\Phi_2)\n-gA_2(W_1+\\omega\\Phi_1)\\Bigr]\\tau_z\\Bigr\\}dr\n\\end{eqnarray}\n\\begin{eqnarray}\nQ_M=\\int\\frac{r}{\\Omega^2} \\Bigl\\{\\Bigl[\\frac{1}{f}(\\partial_r\\Phi_1+gA_1\\Phi_2\n-\\frac{f\\omega}{r^2}(\\Phi_1\\partial_r\\omega+\\partial_rW_1+A_1+gA_1(W_2-1))\\tau_r\\Bigr] \\cr\n+\\Bigl[\\frac{1}{f}(\\partial_r\\Phi_2-gA_1\\Phi_1)-\\frac{f\\omega}{r^2}(\\partial_rW_2+\\Phi_2\\partial_r\\omega-gA_1W_1)\\Bigr]\\tau_z)\\Bigr\\}dr,\n\\end{eqnarray}\nwhere we used Eq. (15). So in general the solution will possess gauge charges.\n\n\\section{An approximate wave solution to first order}\n\nIn the appendix we presented the multiple-scale method for the EYM system.\nOn the metric Eq. (7) we have for the tetrad component:\n$l_\\mu =(-1,1,0,0)$ and\n$l^\\mu =\\frac{1}{\\Omega^2}(1,1,0,0)$, which implies that the divergence of the\nnull congruence $l^\\mu$ becomes\n\\begin{equation}\n\\nabla_\\mu l^\\mu =\\frac{1}{\\Omega^2 r},\n\\end{equation}\nas it should be.\nFrom Eq. (A10) we obtain $B_0^a=-B_1^a$, or, in the notation of Eq. (14),\n$\\dot A_1^{(1)}=-\\dot A_2^{(1)}$.\nFrom Eq. (A8) we obtain\n\\begin{eqnarray}\n\\dot h_{01}=-\\frac{1}{2}(\\dot h_{00}+\\dot h_{11}),\\quad\n\\dot h_{12}=-\\dot h_{02},\\quad \\dot h_{03}=-\\dot h_{13},\n\\quad \\dot h_{23}=-\\frac{\\bar g^{22}\\dot h_{22}+\\bar g^{33}\\dot h_{33}}{2\\bar g^{23}}.\n\\end{eqnarray}\nFirst of all we have just as Eq. (15),\n\\begin{eqnarray}\n\\partial_t\\bar A_1=\\partial_r\\bar A_2,\\quad \\bar W_2=\\frac{\\bar \\Phi_2\\bar W_1}{\\bar \\Phi_1}\n+1-\\frac{1}{g}\n\\end{eqnarray}\nFor simplicity we will take $\\Phi_2=0$ and try to find an approximate wave solution to second order.\nFrom the propagation equations Eq. (A14) and (A18) we obtain\n\\begin{eqnarray}\n\\bar\\Phi_1\\partial_\\varphi\\dot \\omega^{(1)}+\\partial_\\varphi\\dot W_1^{(1)}=0,\n\\end{eqnarray}\n\\begin{eqnarray}\n\\partial_\\varphi\\dot W_2^{(1)}=0.\n\\end{eqnarray}\nSo we will take $\\dot\\omega^{(1)}$ and $\\dot W_2^{(1)}$ independent of $\\varphi$.\nFurther,\n\\begin{eqnarray}\n\\partial_t\\dot A_1^{(1)}+\\partial_r\\dot A_1^{(1)}\n=\\ddot A_1^{(2)}+\\ddot A_2^{(2)}-\\frac{g\\bar f\\bar\\Omega^2}{r^2}\\bar W_1\\dot W_2^{(1)},\n\\end{eqnarray}\n\n\\begin{eqnarray}\n\\partial_t\\dot\\Phi_1^{(1)}+\\partial_r\\dot\\Phi_1^{(1)}=-\\frac{\\dot\\Phi_1^{(1)}}{2r}\n+\\frac{\\dot f^{(1)}}{2\\bar f}(\\partial_t\\bar\\Phi_1 +\\partial_r\\bar\\Phi_1 )\n+\\frac{\\dot\\Phi_1^{(1)}}{2\\bar f}(\\partial_t\\bar f +\\partial_r\\bar f )\n\\cr +\\frac{\\bar f^2}{r^2}(\\dot\\omega^{(1)}\\bar\\Phi_1+\\frac{1}{2}\\dot W_1^{(1)})\n(\\partial_t\\bar\\omega +\\partial_r\\bar\\omega )\n+\\frac{\\bar f^2\\dot\\omega^{(1)}}{2r^2}(\\partial_t\\bar W_1+\\partial_r\\bar W_1),\n\\end{eqnarray}\n\\begin{eqnarray}\n\\partial_t\\dot W_2^{(1)}+\\partial_r\\dot W_2^{(1)}=-\\frac{1}{2\\bar f}\\dot W_2^{(1)}\n(\\partial_t\\bar f+\\partial_r\\bar f)+\\frac{1}{2r}\\dot W_2^{(1)}+g(\\bar A_1+\\bar A_2)\n\\Bigl(\\dot W_1^{(1)}+\\frac{\\bar\\Phi_1\\dot\\omega^{(1)}}{2}+\\frac{\\dot f^{(1)}\\bar W_1}{2\\bar f}\n\\Bigr)\n\\end{eqnarray}\n\n\\begin{eqnarray}\n\\partial_t\\dot W_1^{(1)}+\\partial_r\\dot W_1^{(1)}=\\frac{1}{2r}\\dot W_1^{(1)}\n-\\frac{1}{2\\bar f} (\\dot W_1^{(1)} +\\bar\\Phi_1\\dot\\omega^{(1)})\n(\\partial_t\\bar f+\\partial_r\\bar f)\n-\\bar\\Phi_1(\\partial_t\\dot\\omega^{(1)}+\\partial_r\\dot\\omega^{(1)})-\n\\frac{\\dot f^{(1)}}{2\\bar f}(\\partial_r\\bar W_1+\\partial_t\\bar W_1)\\cr\n-\\frac{1}{2}(\\partial_t\\bar\\omega+\\partial_r\\bar\\omega)(\\dot\\Phi_1^{(1)}+\\frac{\\bar\\Phi_1\n\\dot f^{(1)}}{\\bar f})-\\frac{\\dot\\omega^{(1)}}{2}(\\partial_t\\bar\\Phi_1+\\partial_r\\bar\\Phi_1)\n+\\frac{\\bar\\Phi_1\\dot\\omega^{(1)}}{2r}-g\\dot W_2^{(1)}(\\bar A_1+\\bar A_2)\n\\end{eqnarray}\n\\begin{eqnarray}\n\\partial_t\\dot\\Omega^{(1)}+\\partial_r\\dot\\Omega^{(1)}=-\\frac{\\bar\\Omega\\dot f^{(1)}}{4\\bar f^2}\n(\\partial_t\\bar f+\\partial_r\\bar f)+\\frac{\\dot\\Omega^{(1)}}{\\bar\\Omega}\n(\\partial_t\\bar\\Omega +\\partial_r\\bar\\Omega)+\\frac{\\bar\\Omega\\dot f^{(1)}}{4r\\bar f}\n+\\frac{1}{4\\bar\\Omega}(\\ddot k_{00}+\\ddot k_{11}+2\\ddot k_{01})\\cr\n-\\frac{8\\pi G \\bar\\Omega \\bar f}{r^2}\\Bigl[\\frac{\\dot\\Phi_1^{(1)}}{\\bar f^2}\n(\\partial_t\\bar\\Phi_1+\\partial_r\\bar\\Phi_1)\n+(\\dot\\omega^{(1)}\\bar\\Phi_1+\\dot W_1^{(1)})\\Bigl(\\partial_t\\bar W_1+\\partial_r\\bar W_1\n+\\bar\\Phi_1(\\partial_t\\bar\\omega\n+\\partial_r\\bar\\omega)\\Bigr)-g\\dot W_2^{(1)}\\bar W_1(\\bar A_1+\\bar A_2)\\Bigr]\n\\end{eqnarray}\n\n\\begin{eqnarray}\n\\partial_t\\dot f^{(1)}+\\partial_r\\dot f^{(1)}=\\frac{\\dot f^{(1)}}{\\bar f}(\\partial_t \\bar f\n+\\partial_r\\bar f)-\\frac{\\dot f^{(1)}}{2r}+\\frac{\\bar f^3(\\partial_t\\bar\\omega +\n\\partial_r\\bar\\omega )}{r^2}\\dot\\omega^{(1)}-\\frac{8\\pi G\\bar f^2}{r^2}\\Bigl((\n\\dot\\omega^{(1)}\\bar\\Phi_1+\\dot W_1^{(1)})(\\partial_t\\bar W_1+\\partial_r\\bar W_1) \\cr\n+(\\bar\\Phi_1^2\\dot\\omega^{(1)}+\\bar\\Phi_1\\dot W_1^{(1)})(\\partial_t\\bar\\omega\n+\\partial_r\\bar\\omega)\n+\\frac{3r^2\\dot\\Phi_1^{(1)}}{\\bar f^2}(\\partial_t\\bar\\Phi_1+\\partial_r\\bar\\Phi_1)\\Bigr),\n\\end{eqnarray}\nand\n\\begin{eqnarray}\n\\partial_t\\dot\\omega^{(1)}+\\partial_r\\dot\\omega^{(1)}=\\frac{1}{2r}\\dot\\omega^{(1)}\n-\\frac{\\dot f^{(1)}}{\\bar f}(\\partial_t\\bar\\omega +\\partial_r\\bar\\omega)\n-\\frac{\\dot\\omega^{(1)}}{\\bar f}(\\partial_t\\bar f+\\partial_r\\bar f)\n-\\frac{8\\pi G}{\\bar f}\\Bigl(\\bar\\Phi_1\\dot\\Phi_1^{(1)}(\\partial_t\\bar\\omega\n+\\partial_r\\bar\\omega)\\cr +(\\bar\\Phi_1\\dot\\omega^{(1)}+\\dot W_1^{(1)})(\\partial_t\\bar\\Phi_1\n+\\partial_r\\bar\\Phi_1)+\\dot\\Phi_1^{(1)}(\\partial_t\\bar W_1+\\partial_r\\bar W_1)\n-g\\bar\\Phi_1\\dot W_2^{(1)}(\\bar A_1+\\bar A_2)\\Bigr)\n\\end{eqnarray}\n\n\nThese perturbation equations are linear differential equations of first order in $\\dot h_{\\mu\\nu}$\nand $\\dot B_\\mu^a$ and contain in general  the term  $\\ddot k_{\\mu\\nu}$ and $\\ddot C_\\mu^a$\nIn our special case, we see appear in Eq. (38) a  second order term $\\ddot k_{\\mu\\nu}$ and in\nEq. (34) a term $\\ddot A_i^{(2)}$.\nThe remaining perturbation\nequations will also contain second order term like $\\ddot W_1^{(2)}$.\n\n\n\nFor the background variables $\\bar\\Phi_1, \\bar W_1$ and $\\bar W_2$ we use Eq.(A13).\nThe resulting equations are similar to the equations found in section 2 for $\\Phi_2=0$.\nFrom Eq. (A17) we obtain equations for the background variables $\\bar\\Omega , \\bar f$,\nand $\\bar\\omega$ with back-reaction terms, for example,\n\\begin{eqnarray}\n\\partial_t^2\\bar\\Omega -\\partial_r^2\\bar\\Omega = -\\frac{1}{r}\\partial_r\\bar\\Omega\n+\\frac{1}{\\bar\\Omega}\\Bigl((\\partial_t\\bar\\Omega)^2-(\\partial_r\\bar\\Omega)^2\\Bigr)\n+\\frac{\\bar\\Omega(\\partial_r\\bar f)^2}{2\\bar f^2}-\\frac{\\bar\\Omega\\partial_r\\bar f}{\\bar fr}\n+\\frac{\\bar f^2\\bar\\Omega(\\partial_r\\bar\\omega)^2}{2r^2} \\cr +\\frac{\\bar\\Omega}{2\\tau \\bar f^2r^2}\n\\int[\\bar f^4(\\dot\\omega^{(1)})^2+r^2(\\dot f^{(1)})^2]d\\xi\n+\\frac{8\\pi G\\bar\\Omega}{\\bar fr^2\\tau}\\int\\Bigl[\\bar f^2(\\dot W_1^{(1)}+\\bar\\Phi_1\n\\dot\\omega^{(1)})^2+r^2(\\dot\\Phi_1^{(1)})^2+\\bar f^2(\\dot W_2^{(1)})^2\\Bigr]d\\xi \\cr\n+\\frac{4\\pi G\\bar\\Omega \\bar f}{r^2}\\Bigl[(\\partial_t\\bar W_1+\\bar\\Phi_1\\partial_t\\bar\\omega)^2\n+(\\partial_r\\bar W_1+\\bar\\Phi_1\\partial_r\\bar\\omega)^2+\\frac{r^2}{\\bar f^2}(\n\\partial_r\\bar\\Phi_1^2+\\partial_t\\bar\\Phi_1^2)\\Bigr].\n\\end{eqnarray}\n\nSo we see that a first order HF gravitational and Yang-Mills wave emitted from the\nstring, will change the metric background component $\\bar\\Omega$, just as in\nthe vacuum situation for pure gravitational waves. Moreover, there will be a distortion of the\nshape of the wave during its propagation, which does not occur in the pure Einstein case.\nIn general, this is due to the fact that second order terms appear in the perturbation equations of\n$\\dot h_{\\mu\\nu}$ and $\\dot B_\\mu ^a$.\nNow let us consider, for example, the equation for $\\dot W_2^{(1)}$, Eq. (36). For $A_1=-A_2$, we can solve\n$\\dot W_2^{(1)}$ for $\\bar f=e^{2p}$. For example for $p=1\/4$ we have\n\\begin{equation}\n\\dot W_2^{(1)}=\\sqrt{r}B(t-r)e^{-2\/3r^{1.5}},\n\\end{equation}\nwhich approaches to zero for large r. Here is $B(t-r)$ an arbitrary function of $(t-r)$.\nTogether with Eq.(31) we then have to first order\n\\begin{eqnarray}\n\\dot W_2^{(1)}=1-1\/g+\\sqrt{r}B(t-r)e^{-2\/3r^{1.5}}+...,\n\\end{eqnarray}\nwhich is bounded for large r. One can proceed in the same way for the other first order\nperturbations by imposing some trial solutions for the background variables~\\cite{Slag398}\n\n\\section{Conclusions}\n\nNon-Abelian gravity-coupled solitons and black holes could have\nplayed an important role in gauge theories of elementary particles in the early stage of\nthe universe. Besides the well-known spherically symmetric solutions, axially symmetric\nsolutions are of interest due to the fact that they admit cosmic strings, which could have\nplayed the role of 'seeds' for the large-scale structure of clusters in the universe.\nIt is quite evident that the solution found here is a radiating one and there is an\ninteraction of the solitonlike gravitational waves and the Yang-Mills waves with the\nrotating string. This can be seen from figure 5 where the angle deficit  and $\\Omega$\nis significantly changed by the pulses. This can also be seen from Eq. (41), where the\nterm on the right-hand side represents the back-reaction of the waves on $\\Omega$.\nThe wave-like solutions found in this model is significantly different from the electrovac\nsolution found in the Einstein-Maxwell model.\nThe intervening electro-magnetic waves did not contribute to the angle\ndeficit asymptotically, while in our model the magnetic components $W_i$ has an impact on the\nangle deficit for large r (compare figures 1 and 3). In particularly the number of times the\n$W_i$ crosses the r-axis influences the gravitational wave puls.\nFurther, for a different mass per unit length of the string, a different behavior\nof the solution is obtained (compare figures 3 and 4).\n\nIn order to conclude if the solution is topological stable, we must solve the perturbation\nequations to higher order.\nMoreover, a combination of the results of a throughout numerical investigation with the\nmultiple-scale method pushed to higher orders, will lead to further understanding\nof the rich structure of the EYM system on a rotating space time. Specially the question\nif a electrically and magnetically charged solution ('dyon') of the rotating cosmic string\nexists.\nFurther, bursts of high-frequency gravitational waves might be detectable by the planned\ngravitational wave detectors LIGO\/VIRGO and LISA, even for strings at GUT scale~\\cite{dam00}.\n\n\nThe issues mentioned above are currently under study by the author.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nSince the seminal work of Mumford on the moduli space $\\mathcal{M}_g$ of smooth curves with $g \\geq 2$ and its compactification $\\overline{\\mathcal{M}}_g$, significant progress has been made in understanding its higher-dimensional analog, namely stable varieties. On a first approximation, those are $\\mathbb{Q}$-Gorenstein varieties with relatively mild singularities, and such that $K_X$ is ample.\nA complete and satisfactory moduli theory for stable varieties of any dimension has been settled due to the work of several mathematicians (see \\cites{KSB, Kol90, Ale94, Vie95, Kol13-mod, Kol13, HK04, AH11, kol08, HX13, kol1, HMX }).\nFurthermore, in this setup, it is well understood what families should be parametrized by the moduli functor.\n\nA natural generalization of $\\mathcal{M}_g$ is given by the moduli space of $n$-pointed curves and its compactification, denoted by $\\mathcal{M}_{g,n}$ and $\\overline{\\mathcal{M}}_{g,n}$, respectively.\nA first attempt to generalize the notion of stable pointed curve is to consider mildly singular pairs $(X,D)$ (specifically, \\emph{semi-log canonical} pairs) where $D$ is a reduced divisor, such that $K_X+D$ is ample.\nThis approach has been worked out Alexeev in dimension 2 \\cites{Ale94, Ale96}, and combining the efforts of several mathematicians (see \\cites{ Ale94, Kol13, kol08, HX13, KP17, kol1, HMX, kol19s }), it has been generalized in higher dimensions.\n\nWhile a stable variety is $\\mathbb{Q}$-Gorenstein, a stable pair might not be, so if one is interested in $\\mathbb{Q}$-Gorenstein pairs, the aforementioned formalism might not give the desired moduli space.\nIn fact, one cannot simply consider the $\\qq$-Gorenstein locus of the moduli of stable pairs, as this could be not proper.\nTherefore, the main goal of this paper is to construct a \\emph{proper} moduli space parametrizing $\\mathbb{Q}$-Gorenstein \\emph{pairs} $(X,D)$.\nAs in the classical case, we require our pairs to have semi-log canonical singularities, but relax the ampleness condition on $K_X+D$,\nrequiring only that $K_X+(1-\\epsilon)D$ is ample for $0< \\epsilon \\ll 1$.\nWe also impose a numerical condition on the intersection number $(K_X + tD)^{\\dim(X)}$, analogous to the genus condition for curves, which we encode via a polynomial function $p(t)$.\nThis condition guarantees the boundedness of our moduli problem.\nThese choices are encoded in the notion of \\emph{$p$-stable pair} (Defintion \\ref{def_p-pair}) and lead to the following\n\\begin{theorem} \\label{thm intro 1}\nFix an integer $n\\in \\mathbb{N}$ and a polynomial $p(t) \\in \\mathbb{Q}[t]$.\nAssume that assumption (A) holds.\nThen, there is a proper Deligne--Mumford stack $\\mathscr{F}_{n,p,1}$, with projective coarse moduli space, parametrizing $p$-stable ($\\mathbb{Q}$-Gorenstein) pairs $(X;D)$ of dimension $n$ and with polynomial $p(t)$ and reduced boundary.\n\\end{theorem}\nAssumption (A) is a technical assumption (which is vacuous in dimension $\\le 3$ and for the irreducible components generically parametrizing klt pairs) defined in \\S \\ref{term.subs}.\nIn the statement of Theorem \\ref{thm intro 1}, the subscript 1 in $\\mathscr{F}_{n,p,1}$ means that $D$ is a reduced divisor.\n\nWhile it is very natural to consider the divisor $D$ with its reduced structure, the framework of the Minimal Model Program highlights the importance of the use of fractional coefficients for the divisor $D$.\nFor example, in the case of curves, one can replace $\\overline{\\mathcal{M}}_{g,n}$ with a weighted version, where the markings can attain any fractional value in $(0,1]$.\nThis was accomplished by Hassett in \\cite{Has03}, and this approach leads to different compactifications of\n$\\mathcal{M}_{g,n}$.\n\nIt turns out, however, that the construction of the higher dimensional analogs of these weighted moduli spaces is very delicate.\nAmong the many difficulties, the definition of a suitable notion of family of boundary divisors represents a major problem, see \\cite{kol1}*{Ch. 4}. \nNevertheless, over the past decade, there has been significant progress in the development of a moduli theory for higher dimensional stable pairs (see \\cites{ Ale94, Kol13, kol08, HX13, KP17, kol1, HMX }), and this last missing piece has finally been settled by Koll\\'ar in \\cite{kol19s}.\nIn \\emph{loc. cit.} a rather subtle refinement of the flatness condition is introduced, leading to the ultimate notion of family of divisors, that in turn gives a satisfactory treatment of a moduli functor of families of stable pairs, admitting divisors with arbitrary coefficients.\n\nThe difficulty in defining a suitable notion of family of boundary divisors is due to the fact that, in general, a deformation of a pair $(X,D)$ cannot be reduced to a deformation of the total space $X$ and a deformation of the divisor $D$.\nThis na\\\"ive expectation is only true if the coefficients of $D$ are all strictly greater than $\\frac{1}{2}$ (see \\cite{Kol14}), in which case a deformation of $(X,D)$ over a base curve induces a flat deformation of $D$.\nOn the other hand, if smaller coefficients are allowed, the situation becomes much more subtle.\nFor instance, as showed by Hassett (see \\S \\ref{hassett example} for details), by allowing coefficient $\\frac{1}{2}$, it is possible to define a family of stable surface pairs $(\\mathcal{X},\\mathcal{D}) \\rar \\mathbb A ^1$ such that the flat limit of $\\mathcal{D}$ acquires an embedded point along the special fiber.\n\nOne advantage of our formalism is that it allows overcoming the aforementioned difficulties.\nThat is, our setup easily generalizes to the case of any fractional coefficients, preventing the above-mentioned pathologies of families of divisors, and leading to a more transparent definition of the moduli functor in Definition \\ref{Def:functor}.\nIndeed, we generalize $\\mathscr{F}_{n,p,1}$ to an analogous moduli functor $\\mathscr{F}_{n,p,I}$ in Definition \\ref{Def:functor}, allowing arbitrary coefficients on the divisor.\nThe main feature of Definition \\ref{Def:functor} is that the family of boundaries is characterized as a flat, proper, and relatively $S_1$ morphism over the base, thus significantly simplifying the definition of the classical moduli functor.\nThe $S_1$ condition prevents the existence of embedded points, while the polynomial $p(t)$ controls the relevant invariants of the varieties, preventing them from jumping as in Hassett's example. This leads to the following generalization of Theorem \\ref{thm intro 1}:\n\n\\begin{theorem} \\label{thm intro 1 on steroids}\nFix an integer $n\\in \\mathbb{N}$, a finite subset $I\\subseteq (0,1] \\cap \\mathbb{Q}$, and a polynomial $p(t) \\in \\mathbb{Q}[t]$.\nAssume that $I$ is closed under sum: that is, if $a,b \\in I$ and $a+b \\leq 1$, then $a+b \\in I$.\nAssume that assumption (A) holds.\nThen there is a proper Deligne--Mumford stack $\\mathscr{F}_{n,p,I}$ with projective coarse space, parametrizing $p$-stable pairs of dimension $n$ and coefficients in $I$.\n\\end{theorem}\n\n\\begin{Remark}\nIn the proofs of Theorem \\ref{thm intro 1} and Theorem \\ref{thm intro 1 on steroids}, assumption (A) is needed to guarantee that, for every $p$-stable pair $(X;D)$ in $\\mathscr{F}_{n,p,I}$, $K_X+D$ is semi-ample (equivalently, $(X;D)$ admits a good minimal model).\nBy \\cite{HX13}*{Theorem 1.1} and \\cite{HMX}*{Theorem 1.2}, if the general element of an irreducible component of $\\mathscr{F}_{n,p,I}$ is normal and admits a good minimal model, then so does every element of such component.\nIn particular, by the basepoint-free theorem \\cite{KM98}*{Theorem 3.3}, we have that assumption (A) is satisfied for the components of $\\mathscr{F}_{n,p,I}$ that generically parametrize klt pairs.\nThus, Theorem \\ref{thm intro 1} and Theorem \\ref{thm intro 1 on steroids} hold unconditionally for the subfunctor that parametrizes $p$-stable pairs that are generically klt.\n\\end{Remark}\n\nIn \\S\\ref{section morphism} we relate our moduli functor with the moduli functor of stable pairs, and as the main application of the theory of $p$-stable pairs, we obtain a simpler proof of the projectivity of the moduli space of stable pairs, originally proved by Kov\\'acs and Patakfalvi \\cite{KP17}.\n\n\\begin{theorem}[Corollary \\ref{cor proj moduli}] \\label{thm intro 2}\nConsider a proper DM stack $ \\mathcal{K}_{n,v,I}$ which satisfies the following two conditions:\n\\begin{enumerate}\n    \\item for every normal scheme $S$, the data of a morphism $f \\colon S\\to \\mathcal{K}_{n,v,I}$ is equivalent to a stable family of pairs $q \\colon (\\mathcal{Y},\\mathcal{D})\\to B$ with fibers of dimension $n$, volume $v$ and coefficients in $I$; and\n    \\item there is $m_0\\in \\mathbb{N}$ such that, for every $k \\in \\mathbb{N}$, there is a line bundle $\\mathcal{L}_k$ on $\\mathcal{K}_{n,v,I}$ such that, for every morphism $f$ as above, $f^*\\mathcal{L}_k \\cong \\det(q_*(\\omega_{\\mathcal{Y}\/B}^{[km_0]}(km_0\\mathcal{D})))$.\n\\end{enumerate}\nThen, the coarse moduli space of $\\mathcal{K}_{n,v,I}$ is projective. \n\\end{theorem}\n\nAs observed in \\cite{KP17}*{\\S 1.1}, the approach of \\cite{Has03} for proving projectivity of these moduli spaces cannot be adapted to higher dimensions, as certain sheaves are no longer functorial with respect to base change.\nFor this reason, Kov\\'acs and Patakfalvi develop a refinement of Koll\\'ar's ampleness lemma.\nOn the other hand, in the setup of $p$-stable pairs, all the needed sheaves remain functorial with respect to base change.\nIndeed, flatness and the $S_1$ condition guarantee that all notions of pull-back agree.\nThus, we can follow Hassett's strategy and directly apply \\cite{Kol90}.\nIn this way, the projectivity part of Theorem \\ref{thm intro 1 on steroids} is established;\nthen, to deduce Theorem \\ref{thm intro 2}, it suffices to show that the moduli space of $p$-stable pairs naturally admits a \\emph{finite} morphism to the moduli space of stable pairs, and the needed polarization descends with this morphism.\nFinally, we remark that Theorem \\ref{thm intro 2} holds independently of assumption (A), as every stable pair is the stable model of a $p$-stable pair.\n\n\\subsection{Structure of the paper}\n\\label{struttura}\nThe first part of this work is devoted to developing the notion of $p$-stable pair and extending several statements from pairs to $p$-stable pairs.\nIn particular, in \\S \\ref{preliminaries} we set the key definitions and properties, while in \\S \\ref{section_boundedness} we extend the boundedness results of \\cite{HMX18} to the context of $p$-stable pairs.\nIn doing so, a key technique is to relate a $p$-stable pair $(X,D)$ to its ample model $\\operatorname{Proj} \\bigoplus_{m\\ge 0} H^0(X,\\mathcal{O}_{X}(m(K_X + \\Delta +D))$, which is a stable pair.\nIn the framework of nef and big log canonical divisors, the abundance conjecture is fully known only in dimension up to 3 \\cite{KMM94}, while in higher dimensions we only know the klt case by the basepoint-free theorem.\nTherefore, some technical statements are phrased conditionally to the existence of the needed ample models;\nthe specific condition is referred to as assumption (A), which is defined in \\S \\ref{term.subs}.\nThis also justifies the presence of assumption (A) in Theorem \\ref{thm intro 1} and Theorem \\ref{thm intro 1 on steroids}.\nOn the other hand, any stable pair is the ample model of a $p$-stable pair.\nThus, the application to Theorem \\ref{thm intro 2} holds unconditionally.\n\nIn \\S \\ref{section functor} and \\S \\ref{section_properness}, we analyze the moduli functor $\\mathscr{F}_{n,p,I}$.\nIn particular, in \\S \\ref{section functor} we show that $\\mathscr{F}_{n,p,I}$, which a priori is only a category fibered in groupoids over $\\mathrm{Sch}\/k$, is a Deligne--Mumford stack.\nThen, in \\S \\ref{section_properness} we show that $\\mathscr{F}_{n,p,I}$ is proper.\nThus, \\S \\ref{section functor} and \\S \\ref{section_properness} settle Theorem \\ref{thm intro 1 on steroids}, except for the projectivity part.\n\nIn \\S \\ref{section morphism}, we make a connection between $\\mathscr{F}_{n,p,I}$ and the moduli space of stable pairs.\nIn particular, we show that every irreducible component of the moduli space of stable pairs is dominated by a component of $\\mathscr{F}_{n,p,I}$ (for a suitable choice of invariants), and this morphism is finite.\n\nIn \\S \\ref{section projectivity ksba} we conclude our work.\nWe use the ampleness lemma to show that the coarse space of $\\mathscr{F}_{n,p,I}$ is projective, and we use the results from \\S \\ref{section morphism} to descend the analogous statement to the moduli space of stable pairs.\n\n\\subsection*{Acknowledgements} We thank Jarod Alper, Dori Bejleri, Christopher Hacon, S\\'andor Kov\\'acs, Joaqu\\'in Moraga, and Zsolt Patakfalvi for helpful discussions.\n\n\n\n\\section{Preliminaries} \\label{preliminaries}\n\n\\subsection{Terminology and conventions}\n\\label{term.subs}\nThroughout this paper, we will work over the field of complex numbers.\nFor the standard notions in the Minimal Model Program (MMP) that are not addressed explicitly, we direct the reader to the terminology and the conventions of~\\cite{KM98}.\nSimilarly, for the relevant notions regarding non-normal varieties, we direct the reader to \\cite{Kol13}.\nA variety will be an integral separated scheme of finite type over $\\mathbb{C}$.\n\nThroughout the paper, when we say ``assume that assumption (A) holds'' we mean \"assume that one of the two following conditions hold\": \n\\begin{itemize}\n    \\item[(i)] the dimension of all the varieties appearing as general fibers of the families\/morphisms is at most 3; or\n    \\item[(ii)]  if $(X,D)$ is an lc pair such that $K_X + (1-\\epsilon)D$ is ample for $0<\\epsilon \\ll 1$, then $K_X + D$ is semi-ample.\n\\end{itemize}\n\n\\begin{Remark}\nWe remark that assumption (A) is a weakening of the Abundance Conjecture, which is known up to dimension 3.\nMoreover, assumption (A) is known to hold for klt pairs $(X,D)$ (it is the basepoint-free theorem).\nFinally, a similar statement holds if $D$ (rather than $K_X + D$) contains an ample divisor: it is proved in \\cite{Hu17} that if $D\\ge A$ where $A$ is an ample $\\mathbb{R}$-divisor, if $K_X + D $ is nef it is also semi-ample.\n\\end{Remark}\n\n\\subsection{Contractions}\nA \\emph{contraction} is a projective morphism $f\\colon X \\rar Z$ of quasi-projective varieties with $f_\\ast  \\O X. = \\O Z.$. \nIf $X$ is normal, then so is $Z$.\n\n\\subsection{Divisors}\nLet $\\mathbb{K}$ denote $\\mathbb{Z}$, $\\qq$, or $\\mathbb{R}$. We say that $D$ is a \\emph{$\\mathbb{K}$-divisor} on a variety $X$ if we can write $D = \\sum \\subs i=1. ^n d_i P_i$ where $d_i \\in \\mathbb{K}$, $n \\in \\mathbb{N}$ and the $P_i$ are prime Weil divisors on $X$ for all $i=1, \\ldots, n$. \nWe say that $D$ is $\\mathbb{K}$-Cartier if it can be written as a $\\mathbb{K}$-linear combination of $\\mathbb{Z}$-divisors that are Cartier.\nThe \\textit{support} of a $\\mathbb{K}$-divisor $D=\\sum_{i=1}^n d_iP_i$ is the union of the prime divisors appearing in the formal sum $\\mathrm{Supp}(D)= \\sum_{i=1}^n P_i$.\n\nIn all of the above, if $\\mathbb{K}= \\mathbb{Z}$, we will systematically drop it from the notation.\n\nGiven a prime divisor $P$ in the support of $D$, we will denote by $\\mu_P (D)$ the coefficient of $P$ in $D$.\nGiven a divisor $D = \\sum \\mu_{P_i}(D) P_i$ on a normal variety $X$, and a morphism $\\pi \\colon X \\to Z$, we define\n\\[\nD^v \\coloneqq \\sum_{\\pi(P_i) \\subsetneqq Z} \\mu_{P_i}(D) P_i, \\\nD^h \\coloneqq \\sum_{\\pi(P_i) = Z} \\mu_{P_i}(D) P_i.\n\\]\n\nLet $D_1$ and $D_2$ be divisors on $X$.\nWe write $D_1 \\sim_{\\mathbb K ,Z} D_2$ if there is a $\\mathbb{K}$-Cartier divisor $L$ on $Z$ such that $D_1 - D_2 \\sim _{\\mathbb K}f^\\ast L$.\nEquivalently, we may also write $D_1 \\sim_{\\mathbb{K}} D_2\/Z$, or $D_1 \\sim_{\\mathbb{K}} D_2$ over $Z$.\nIf $\\mathbb{K}=\\mathbb{Z}$, we omit it from the notation.\nSimilarly, if $Z= \\mathrm{Spec}(k)$, where $k$ is the ground field, we omit $Z$ from the notation.\n\n\nLet $\\pi \\colon X \\rar Z$ be a projective morphism of normal varieties.\nLet $D_1$ and $D_2$ be two $\\mathbb{K}$-divisors on $X$.\nWe say that $D_1$ and $D_2$ are numerically equivalent over $Z$, and write $D_1 \\equiv D_2\/Z$, if $D_1 . C = D_2 . C$ for every curve $C \\subset X$ such that $\\pi(C)$ is a point.\nIn case the setup is clear, we just write $D_1 \\equiv D_2$, omitting the notation $\/Z$.\n\n\\subsection{Non-normal varieties and pairs} There are two important generalizations of the notion of normal variety.\n\\begin{Def}\nAn $S_2$ scheme is called \\emph{demi-normal} if its codimension 1 points are either regular or nodal.\\end{Def}\nRoughly speaking, the notion of demi-normal schemes allows extending the notion of log canonical singularities to non-normal varieties, allowing for a generalization of the notion of stable curve to higher dimensions.\nWe refer to \\cite{kol1}*{\\S 1.8} for more details.\n\n\\begin{Def}A finite morphism of schemes $X' \\rar X$ is called a \\emph{partial seminormalization} if $X'$ is reduced and, for every point $x \\in X$,  the induced map $k(x) \\rar k(\\mathrm{red}(g^{-1}(x)))$ is an isomorphism.\nThere exists a unique maximal partial seminormalization, which is called \\emph{the seminormalization} of $X$.\nA scheme is called \\emph{seminormal} if the seminormalization is an isomorphism.\\end{Def}\n\n\nThis is an auxiliary lemma, it is probably well known. We include it for completeness.\nWe refer to \\cite{Kol13}*{\\S 10.2} for the details about seminormality.\n\n\\begin{Lemma}\\label{lemma:connected:fivers:vs:cohomol:connected:fibers}\nLet $p \\colon X\\to Y$ be a proper surjective morphism with connected fibers, with $Y$ seminormal and $X$ reduced.\nThen $p_*\\mathcal{O}_X = \\mathcal{O}_Y$.\n\\end{Lemma}\n\n\\begin{Remark}\\label{remark:cohom:connected:implies:one:can:take:global:sections:after:pull:back}\nObserve that, in situation of Lemma \\ref{lemma:connected:fivers:vs:cohomol:connected:fibers}, by the projection formula it follows that, for any line bundle $L$ on $Y$, we have $H^0(Y,L) =H^0(X,p^*L)$.\n\\end{Remark}\n\n\\begin{proof}\nWe can take the Stein factorization $X\\xrightarrow{q} Z\\xrightarrow{a} Y$.\nSince $X$ is reduced, then $Z$ is reduced.\nObserve that $q_*\\mathcal{O}_X = \\mathcal{O}_Z$, so the desired result follows if we can show that $a$ is an isomorphism.\n\nSince the composition $p=a \\circ q$ has connected fibers and $q \\colon X\\to Z$ is surjective, then $a \\colon Z\\to Y$ has connected fibers.\nSince $a$ is finite, it is injective.\nIt is also surjective since $p$ is surjective, so $a$ is a bijection.\nThen, we can take the seminormalization $Z^{sn}\\to Z$ and consider the composition $Z^{sn}\\to Z \\to Y$.\nThis is a bijective morphism since it is a composition of bijections, and it is proper since $a$ is proper and $Z^{sn}\\to Z$ is proper.\nThen, the composition $Z^{sn} \\to Y$ is an isomorphism, since the source and the target are seminormal.\n\nNow, on the topological space given by $Z$ we have the following morphsims of sheaves:\n$$\\mathcal{O}_Y\\xrightarrow{a^\\#} \\mathcal{O}_Z\\xrightarrow{b} \\mathcal{O}_{Z^{sn}}.$$\nThe composition is surjective, so $b$ is surjective. Moreover, since $Z$ is reduced, the map $\\mathcal{O}_Z\\xrightarrow{b} \\mathcal{O}_{Z^{sn}}$ is also injective. Then it is an isomorphism, so $Z\\cong Z^{sn}\\cong Y$.\n\\end{proof}\n\n\\begin{Lemma}\\label{lemma:canonical:model:of:slc:pairs:has:connected:fibers}\nLet $(X,D_X)$ and $(Y,D_Y)$ be semi-log canonical pairs, and let $(X^n,D_X^n+\\Delta_X)$ and $(Y^n,D_Y^n+\\Delta_Y)$ denote the respective normalizations.\nAssume we have the following diagram:\n\\[\n\\xymatrix{\\Delta_X^n \\ar[r] \\ar[d]_\\gamma & (X^n, D_X^n+\\Delta_X)\\ar[r]^{\\quad n_X} \\ar[d]^\\alpha & (X,D_X)\\ar[d]^\\beta \\\\\\Delta^n_Y \\ar[r] & (Y^n, D^n_Y+\\Delta_Y) \\ar[r]^{\\quad n_Y} & (Y,D_Y)}\n\\]\nwhere the following properties are satisfied:\n\\begin{itemize}\n    \\item $\\alpha$ induces a bijection between the irreducible components of $X^n$ and $Y^n$ and it is birational on each irreducible component; and\n    \\item the fibers of $\\alpha$ and $\\gamma$ are connected.\n\\end{itemize}\nThen the fibers of $\\beta$ are connected.\n\nFurthermore, if we assume that\n\\begin{itemize}\n    \\item $\\gamma$ induces a bijection between the irreducible components of $\\Delta_X^n$ and $\\Delta_Y^n$ and it is birational on each such component; and\n    \\item $\\alpha^*(\\K Y^n. + D^n_Y + \\Delta_Y) = \\K X^n. + D^n_X + \\Delta_X$.\n\\end{itemize}\nThen, we have $\\beta^*(K_{Y} +D_Y) =  K_{X} +D_X$.\n\\end{Lemma}\n\n\\begin{proof}\nLet $\\tau_X$ be the natural involution induced on $\\Delta_X^n$.\nSince $n_X$ is determined as the quotien by a finite equivalence relation, by \\cite{Kol13}*{Lemma 9.10}, we can write $\\mathcal{O}_X$ as the kernel of the following morphism:\n\\begin{equation} \\label{eq_ses_normalization}\n0\\to \\mathcal{O}_X \\to (n_{X})_*\\mathcal{O}_{X^n} \\xrightarrow{g_X} (n_{X})_*\\mathcal{O}_{\\Delta^n_X},\n\\end{equation}\nwhere $g_X$ is the difference between the pull-back to $\\Delta^n_X$ and the pull-back to $\\Delta^n_X$ composed with $\\tau_X$.\nWe push forward the sequence \\eqref{eq_ses_normalization} via $\\beta$ and we get \n$$\n0 \\to \\beta_*  \\mathcal{O}_X \\to (\\beta \\circ n_X)_*  \\mathcal{O}_{X^n} \\xrightarrow{\\beta_*g}(\\beta \\circ {n_X})_* \\mathcal{O}_{\\Delta^n_X}.\n$$\nSince both $\\alpha$ and $\\gamma$ have connected fibers and the diagram in the statement commutes, the sequence above becomes \n$$\n0\\to \\beta_* \\mathcal{O}_X \\to (n_{Y})_*\\mathcal{O}_{Y^n} \\xrightarrow{g_Y} (n_{Y})_*\\mathcal{O}_{\\Delta^n_Y},\n$$\nwhere $g_Y$ is the difference between the pull-back to $\\Delta^n_Y$ and the pull-back to $\\Delta^n_Y$ composed with the involution $\\tau_Y$.\nSince $\\beta_*$ is left exact, we conclude that $\\beta_*\\mathcal{O}_X = \\mathcal{O}_Y$, as we can again apply \\cite{Kol13}*{Lemma 9.10}; thus, $\\beta$ has connected fibers.\n\nNow, we are left with proving that $\\beta^*(K_Y + D_Y) = K_X + D$.\nAssume the additional hypotheses in the statement\nIt suffices to check that these two divisors agree in codimension 1.\nBy the additional assumptions, $\\beta$ is an isomorphism generically on the double loci of $X$ and $Y$.\nThus, we are left with checking that the desired statement holds in a neighborhood of the generic point of any closed subscheme of codimension one not contained in the double locus.\nThen, the desired statement follows since the normalization $Y^n\\to Y$ is an isomorphism on the generic point of every divisor $E$ intersecting the smooth locus of $Y$, and $\\alpha^*(K_{Y^n} +D^n_Y+\\Delta_Y) =  K_{X^n} +D_X^n+\\Delta_X$.\n\\end{proof}\n\n\n\\subsection{Divisorial sheaves}\\label{divisorial sheaves} We begin this subsection with the following definition:\n\n\\begin{Def}Let $X$ be a scheme.\nA sheaf $\\mathfrak F$ on $X$ is called \\emph{divisorial sheaf} if it is $S_2$ and there is a closed subscheme $Z \\subset X$ of codimension at least 2 such that $\\mathfrak F | \\subs X \\setminus Z.$ is locally free of rank 1.\\end{Def}\n\n\\begin{Def}[\\cite{HK04}*{Section 3}]Let $f:X\\to S $ a flat morphism of schemes, and $\\mathcal{E}$ a coherent sheaf on $X$. We say that $\\mathcal{E}$ is relatively $S_n$ if it is flat over $S$ and its restriction to each fiber is $S_n.$\n\\end{Def}\n\nNow, let $X$ be a demi-normal scheme, and let $\\mathrm{Weil}^*(X)$ denote the subgroup of $\\mathrm{Weil}(X)$ generated by the prime divisors that are not contained in the conductor of $X$.\nThen, there is an identification between $\\mathrm{Weil}^*(X)\/\\sim$ and the group of divisorial sheaves, where $\\sim$ denotes linear equivalence.\nThe identification is defined as follows.\nBy the definition of $\\mathrm{Weil}^*(X)$ and the demi-normality of $X$, for every element $B \\in \\mathrm{Weil}^*(X)$,there is a closed subset $Z \\subset X$ of codimension at least 2 such that $B| \\subs X \\setminus Z.$ is a Cartier divisor.\nThen, the corresponding divisorial sheaf is defined as $j_* \\O X \\setminus Z. (B| \\subs X \\setminus Z.)$, where we have $j \\colon X \\setminus Z \\rar X$.\n\nConsider a flat family $f \\colon X\\to B$ with $S_2$ fibers, and assume that there is an open subset $i \\colon U\\subseteq X$ such that $U_b$ is big for every $b\\in B$.\nThen, for every locally free sheaf $\\mathcal{G}$ on $U$, we can consider $i_*\\mathcal{G}$ on $X$.\nFrom \\cite{HK04}*{Corollary 3.7}, this is a reflexive sheaf on $X$ (a priori, it is not $S_2$ relatively to $B$, see \\S \\ref{hassett example}).\n\nIf $X$ is demi-normal, its canonical sheaf is a divisorial sheaf, as $X$ is Gorenstein in codimension 1.\nBy the above identification, we can then choose a canonical divisor $K_X$ such that $\\O X. (K_X) \\cong \\omega_X$.\nObserve that this construction can be carried out in families.\nIndeed, if $f \\colon X\\to B$ is a flat morphism with demi-normal fibers of dimension $n$, there is an open locus $i \\colon U\\subseteq X$ that has codimension 2 along each fiber, on which $f$ is Gorenstein.\nThen we can define $\\omega_{X\/B} \\coloneqq i_*\\omega_{U\/B}$.\nObserve that this agrees with the $(-n)$-th cohomology of the relative dualizing complex.\nIndeed, the latter is $S_2$ (see \\cite{LN18}*{Section 5}).\n\nLet $X$ be demi-normal, and consider two divisorial sheaves $L_1$ and $L_2$.\nThen, their \\emph{reflexive tensor product} is defined as $L_1 \\hat{\\otimes} L_2 \\coloneqq (L_1 \\otimes L_2)^{**}$ and it is a divisorial sheaf itself.\nIf we have $L_1 \\cong \\O X. (D_1)$ and $L_2 \\cong \\O X. (D_2)$, we have $L_1 \\hat{\\otimes} L_2 \\cong \\O X. (D_1+D_2)$.\nThe \\emph{$m$-fold reflexive power $L^{[m]}$} is defined as the $m$-fold self reflexive tensor product.\n\nFor more details, we refer to \\cite{kol1}*{\\S 3.3}, \\cite{Kol13}*{5.6} and \\cite{HK04}.\n\n\n\\subsection{Boundedness}\nLet $\\mathfrak{D}$ be a set of projective pairs.\nThen, we say that $\\mathfrak{D}$ is {\\it log bounded} (resp. {\\it log birationally bounded}) if there exist a variety $\\mathcal{X}$, a reduced divisor $\\mathcal{E}$ on $\\mathcal{X}$, and a projective morphism $\\pi \\colon \\mathcal{X} \\rar T$, where $T$ is of finite type, such that $\\mathcal{E}$ does not contain any fiber of $\\pi$, and, for every $(X,B) \\in \\mathfrak{D}$, there are a closed point $t \\in T$ and a morphism (resp. a birational map) $f_t \\colon \\mathcal{X}_t \\rar X$ inducing an isomorphism $(X,\\Supp(B)) \\cong (\\mathcal{X}_t,\\mathcal{E}_t)$ (resp. such that $\\Supp(\\mathcal B _t)$ contains the strict transform of $\\Supp(B)$ and all the $f_t$ exceptional divisors).\n\nA set of projective pairs $\\mathfrak{D}$ is said to be {\\it strongly log bounded} if there is a quasi-projective pair $(\\mathcal{X},\\mathcal{B})$ and a projective morphism $\\pi \\colon \\mathcal{X} \\rar T$, where $T$ is of finite type, such that $\\Supp(\\mathcal{B})$ does not contain any fiber of $\\pi$, and for every $(X,B) \\in \\mathfrak D$, there is a closed point $t \\in T$ and an isomorphism $f \\colon X \\rar \\mathcal{X}_t$ such that $f_*B=\\mathcal{B}_t$.\n\nA set of projective pairs $\\mathfrak{D}$ is {\\it effectively log bounded} if it is strongly log bounded and we may choose a bounding pair $\\pi \\colon (\\mathcal{X},\\mathcal{B}) \\rar T$ such that, for every closed point $t \\in T$, we have $(\\mathcal{X}_t,\\mathcal{B}_t) \\in \\mathfrak D$.\n\n\\subsection{Index of a set}\nGiven a finite subset $I \\subseteq \\mathbb{Q}^n$, we define the \\emph{index} of $I$ to be the smallest positive rational number $r$ such that $rI\\subseteq \\mathbb{Z}^n$.\n\n\\subsection{Stable pairs and $p$-stable pairs} \\label{p-pairs section}\nLet $(X,D)$ denote a projective semi-log canonical pair, where $D$ has coefficients in $\\qq$.\nWe say that $(X,D)$ is a \\emph{stable pair} if $K_X + D$ is ample.\n\\begin{Def}\\label{def_p-pair}\nConsider a polynomial $p(t)\\in \\mathbb{Q}[t]$ and a set $I\\subseteq (0,1]$.\nA \\emph{$p$-pair} with polynomial $p(t)$ and coefficients in $I$ is the datum of a semi-log canonical pair $(X,\\Delta)$ and a $\\qq$-Cartier $\\qq$-divisor $D$ on $X$ satisfying the following properties:\n\\begin{itemize}\n    \\item $(X,D+\\Delta)$ is semi-log canonical;\n    \\item $p(t) = (K_X+tD+\\Delta)^{\\dim(X)}$; and\n    \\item the coefficients of $D$ and $\\Delta$ are in $I$.\n\\end{itemize}\nIf moreover there is $\\epsilon_0>0$ such that for every $0<\\epsilon \\le\\epsilon_0$ the pair $(X,(1-\\epsilon)D+\\Delta)$ is stable, the $p$-pair $(X,\\Delta;D)$ is called a \\emph{$p$-stable pair}. Finally, if $\\mathcal{A} \\coloneqq \\bigoplus_{m\\ge 0} H^0(X,\\mathcal{O}_{X}(md(K_X + \\Delta +D))$ is finitely generated, where $d$ is the Cartier index of $K_X + \\Delta + D$, we will call $\\operatorname{Proj}(A)$ the \\emph{ample model} of $(X,\\Delta; D)$.\\end{Def}\n\nFor brevity, we denote the datum of a $p$-pair by $(X,\\Delta;D)$, where the polynomial $p(t)$ and the set of coefficients $I$ are omitted in the notation.\nIn case $\\Delta=0$, we then write $(X;D)$.\n\\begin{Remark}\nFor example, if $(X,0;D)$ is an slc $p$-stable pair, and $\\nu:X^n\\to X$ is the normalization of $X$ with conductor $\\Delta$, then $(X^n,\\Delta;\\nu_*^{-1}D)$ is a $p$-stable pair.\n\\end{Remark}\n\n\\begin{Notation}\\label{notation_Dsc}\nIf $I$ is a finite set and $r$ is its index, given a $p$-pair $(X,\\Delta;D)$ with polynomial $p(t)$ and coefficients in $I$ we denote by $D^{sc}$ as the subscheme of $X$ defined by the reflexive sheaf of ideals $\\O X.(-rD)$.\n\\end{Notation}\n\n\n\\begin{Remark}\\label{remark_KX+D+delta_is_nef}\nObserve that if $(X,\\Delta;D)$ is a $p$-stable pair, the $\\qq$-divisor $K_X+D+\\Delta$ is nef since it is limit of ample $\\qq$-divisors.\nFurthermore, $K_X+D+\\Delta$ is big, as it is the sum of an ample $\\qq$-divisor and an effective $\\qq$-divisor.\n\\end{Remark}\n\n\\begin{Lemma} \\label{lemma_convex_combination}\nLet $X$ be a projective variety, and let $D_1$ and $D_2$ be two nef $\\mathbb{R}$-Cartier $\\mathbb{R}$-divisors.\nAssume that for some $t \\in (0,1)$, the divisor $tD_1 + (1-t)D_2$ is ample.\nThen, $cD_1 + (1-c)D_2$ is ample for every $c \\in (0,1)$.\n\\end{Lemma}\n\n\\begin{proof}\nArguing by contradiciton, there exists $c_0 \\in (0,1)$ such that $c_0D_1+(1-c_0)D_2$ is not ample.\nNotice that, as $D_1$ and $D_2$ are nef, then so is $c_0D_1+(1-c_0)D_2$.\nThen, by Kleiman's criterion (see \\cite{urb}*{Proposition 4.4} for a reference including $\\mathbb{R}$-divisors), there exists a class $\\gamma \\in \\overline{NE}(X) \\setminus \\{ 0 \\}$ such that $(c_0D_1+(1-c_0)D_2).\\gamma = 0$.\nAs $D_1$ and $D_2$ are nef, we have $D_1 . \\gamma \\geq 0$ and $D_2 . \\gamma \\geq 0$.\nIf $D_1 . \\gamma = D_2 . \\gamma =0$, it follows that $(tD_1+(1-t)D_2).\\gamma =0$, and $(tD_1+(1-t)D_2)$ is not ample by Kleiman's criterion.\nThen, $D_i . \\gamma>0$ for some $i=1,2$.\nWithout loss of generality, we may assume $D_1 . \\gamma > 0$.\nThen, as $(c_0D_1+(1-c_0)D_2).\\gamma = 0$, it follows that $D_2 . \\gamma <0$, which is impossible, as $D_2$ is nef.\n\\end{proof}\n\n\\begin{Lemma} \\label{ample model p-pair}\nLet $(X,\\Delta;D)$ be a $p$-stable pair, and assume that $f \\colon X \\rar Y$ is its ample model.\nThen, $\\mathrm{Ex}(f) \\subset \\operatorname{Supp}(D)$.\n\\end{Lemma}\n\n\\begin{proof}\nBy assumption, there exists $\\epsilon>0$ such that $(X,(1-\\epsilon)D + \\Delta)$ is a stable pair.\nThus, if $C$ is an irreducible curve that is not contained in $\\operatorname{Supp}(D)$, we have that\n\\[\n(\\K X. + D + \\Delta).C = (\\K X. + (1-\\epsilon)D + \\Delta).C + \\epsilon D .C >  0,\n\\]\nsince the first summand is positive by ampleness and the second is non-negative as $C$ is not contained in $\\operatorname{Supp}(D)$.\nThus, the curves contracted by $f$ are contained in $\\operatorname{Supp}(D)$ and the claim follows.\n\\end{proof}\nThe following lemma is the main technical tool that we will use in Sections \\ref{section_boundedness} and \\ref{section_properness}:\n\\begin{Lemma}\\label{lemma_bound_epsilon_lc_case}\nLet $n$ and $M$ be natural numbers.\nThen, there exists $\\epsilon_0 > 0$, only depending on $n$ and $M$, such that the following holds.\nLet $(Y,\\Delta_Y;D_Y)$ be a normal $p$-stable pair of dimension $n$, and assume that $K_Y+D_Y+\\Delta_Y$ induces a morphism $\\pi \\colon Y \\rar X$ to the canonical model $(X,D_X+\\Delta_X)$ of $(Y,D_Y+\\Delta_Y)$.\nFurther assume that the Cartier index of $K_X + D_X + \\Delta_X$ is less than $M$.\nThen, for every $0 < \\epsilon < \\epsilon_0$, the pair $(Y,(1-\\epsilon)D_Y+\\Delta_Y)$ is stable.\n\\end{Lemma}\n\n\\begin{proof}\nBy definition of $p$-stable pair and of canonical model,\nfor every $\\pi$-exceptional curve $C$, we have $(K_Y+D_Y+\\Delta_Y).C=0$ and $(K_Y+(1-\\epsilon)D_Y+\\Delta_Y).C>0$ for $0 < \\epsilon \\ll 1$.\nFor every such curve $C$, the function $t\\mapsto (K_Y+tD_Y+\\Delta_Y).C$ is linear and not identically 0, so it has at most one zero. \nIn particular, for every $\\epsilon >0$ we have \n\\begin{equation} \\label{eq_intersection0}\n(K_Y+(1-\\epsilon)D_Y+\\Delta_Y).C>0.    \n\\end{equation}\nHence, if we choose $\\epsilon =1$, we have\n\\begin{equation} \\label{eq_intersection1}\n(K_Y+\\Delta_Y).C>0\n\\end{equation}\nfor every $\\pi$-exceptional curve.\n\nBy the same argument, for every curve $C'$ such that $(K_Y + \\Delta_Y).C' \\geq 0$, it follows that\n\\begin{equation} \\label{eq_mori_cone}\n    (K_Y  + (1-\\eta) D_Y + \\Delta_Y).C' >0\n\\end{equation}\nfor every $\\eta \\in (0,1)$.\nNow, define\n\\[\na\\coloneqq \\frac{1}{2M},\\quad \\epsilon_0 \\coloneqq \\displaystyle{\\frac{a}{2n+a+1}}.\n\\]\n\n\n\nSince $K_X+D_X+\\Delta_X$ is ample with Cartier index bounded by $M$, for every curve $\\Gamma$ that is not $\\pi$-exceptional,\nby the projection formula, we have\n\\begin{equation} \\label{eq_intersection2}\n\\frac{1}{2M} = a<(K_X+D_X+\\Delta_X).p_*\\Gamma = (K_Y+D_Y+\\Delta_Y).\\Gamma.\n\\end{equation}\nFrom \\cite{Fuj11}*{Theorem 1.4}, for every $(K_Y+\\Delta_Y)$-negative extremal ray $R$, we may find a curve $\\Gamma$ generating $R$ such that\n\\begin{equation} \\label{eq_bdd_negative_curves}\n-2n\\le (K_Y+\\Delta_Y).\\Gamma<0.\n\\end{equation}\nBy \\eqref{eq_intersection1}, any such curve is not $\\pi$-exceptional.\nThen, by \\eqref{eq_intersection2} and \\eqref{eq_bdd_negative_curves}, for any such $\\Gamma$, we have\n\\begin{equation} \\label{eq_extremal_curves}\n    (K_Y + (1-\\epsilon_0)D_Y + \\Delta_Y).\\Gamma= (1-\\epsilon_0)(K_Y + D_Y + \\Delta_Y).\\Gamma + \\epsilon_0 (K_Y + \\Delta_Y).\\Gamma>(1-\\epsilon_0)a-2n\\epsilon_0 > 0.\n\\end{equation}\n\n\nNow, consider the cone of curves $\\overline{NE}(Y)$ and its decomposition given by the cone theorem associated to the pair $(Y,\\Delta_Y)$ \\cite{Fuj11}*{Theorem 1.4}.\nThen, by \\eqref{eq_mori_cone}, we have that $K_Y + (1-\\epsilon_0)D_Y + \\Delta_Y$ is positive on $\\overline{NE}(Y)_{K_Y + \\Delta_Y \\geq 0}$.\nThus, every $(K_Y + (1-\\epsilon_0)D_Y + \\Delta_Y)$-negative extremal ray is also a $(K_Y +  \\Delta_Y)$-negative extremal ray.\nThen, by \\eqref{eq_extremal_curves}, we have that $(K_Y + (1-\\epsilon_0)D_Y + \\Delta_Y)$ is positive on $R$.\nThus, by the cone theorem \\cite{Fuj11}*{Theorem 1.4}, the log canonical pair $(Y,(1-\\epsilon_0)D_Y + \\Delta_Y)$ has no negative extremal rays; thus, $K_Y + (1-\\epsilon_0)D_Y + \\Delta_Y$ is nef.\n\nNow, by the definition of $p$-stable pair, some convex combination of $K_Y+(1-\\epsilon_0)D_Y+\\Delta_Y$ and $K_Y+D_Y+\\Delta_Y$ is ample.\nThen, the claim follows by Lemma \\ref{lemma_convex_combination}.\n\\end{proof}\n\n\\begin{Lemma}\\label{lemma_bound_for_epsilon}\nFix an integer $n\\in \\mathbb{N}$, a volume $v\\in \\mathbb{Q}_{>0}$, and a finite subset $I\\subseteq (0,1] \\cap \\mathbb{Q}$.\nAssume that assumption (A) holds.\nThere is $0<\\epsilon_0$, only depending only on $n$ and $v$, such that, for every $p$-stable pair $(Y,\\Delta_Y;D_Y)$ of dimension $n$, coefficients in $I$, and polynomial $p(t)$ with $p(1) = v$, the pair $(Y,(1-\\epsilon)D_Y+\\Delta_Y)$ is stable for every $0<\\epsilon < \\epsilon_0$.\n\\end{Lemma}\n\n\\begin{proof}\nWithout loss of generality, we may assume from now on that $1 \\in I$.\nLet $Y^n\\to Y$ be the normalization of $Y$.\nLet $(Y^n,\\Delta_Y^n)$ be the pair induced by $(Y,\\Delta_Y)$, and let $D_Y^n$ denote the pull-back of $D_Y$ to $Y^n$.\nNotice that $Y^n$ possibly has more than one connected component.\nThen, we have:\n\\begin{enumerate}\n     \\item $(Y^n,D_Y^n+\\Delta_Y^n)$ has still coefficients in $I$;\n     \\item $K_{Y^n} + D_Y^n+\\Delta_Y^n$ is nef and big; and\n     \\item $(Y^n,\\Delta_Y^n;D_Y^n)$ is a $p$-stable pair.\n\\end{enumerate}\nFrom \\cite{HMX14}*{Theorem 1.3},\nthere are finitely many possibilities to write $v$ as $v = \\sum v_i$, where $v_i>0$ is the volume of a log canonical pair of general type with coefficients in $I$.\nSo from (1), there is a finite set $\\mathscr{S}=\\{v_1,...,v_k\\}$ such that the volume of each connected component of $Y^n$ is in $\\mathscr{S}$. \n\nBy assumption (A), if $(Z,D)$ is a log canonical pair with $K_Z+D$ big and nef, there is a canonical model $(Z^c,D^c)$ of $(Z,D)$, and by construction $\\vol(Z,D) = \\vol(Z^c,D^c)$.\nIn particular, it follows from (2) that the irreducible components of $(Y^n,D_Y^n+\\Delta_Y^n)$ admit a canonical model, and the volumes of these models are contained in $\\mathcal{S}$ as well.\nFrom \\cite{HMX}, there is an $M>0$ such that every stable pair with coefficients in $I$, dimension $n$, and volume in the finite set $\\mathscr{S}$ has Cartier index less than $M$.\nSince we can check ampleness after passing to the normalization, and the Cartier indexes of each irreducible component of the normalization are bounded, the thesis follows from Lemma \\ref{lemma_bound_epsilon_lc_case}.\n\\end{proof}\n\n\n\\begin{Lemma} \\label{lift stable pair slc}\nLet $(Y,D_Y)$ be a stable pair.\nThen, there exists a $p$-stable pair $(X;D)$ having $(Y,D_Y)$ as ample model.\n\\end{Lemma}\nObserve in particular that from Lemma \\ref{ample model p-pair}, $X$ is normal if and only if $Y$ is normal.\n\n\\begin{proof}\nWe consider the semi-canonical modification $\\pi \\colon X \\rar Y$ of $(Y,0)$, in the sense of \\cite{Fuj15}.\nWe observe that, in order to consider a semi-canonical modification, $K_Y$ does not need to be $\\qq$-Cartier.\nSuch modification exists by \\cite{Fuj15}*{Theorem 1.1} and the fact that $Y$ demi-normal.\nBy \\cite{Fuj15}*{Definition 2.6}, $\\pi \\colon X \\rar Y$ has the following properties:\n\\begin{itemize}\n    \\item $\\pi$ is an isomorphism around every generic point of the double locus of $X$;\n    \\item this procedure is compatible with taking the normalizations of $X$ and $Y$, see \\cite{Fuj15}*{Lemma 3.7.(2)}.\n    In particular, $X$ is normal if so is $Y$, and, in general, $\\pi$ establishes a bijection between irreducible components of $X$ and $Y$; and\n    \\item $K_X$ is $\\qq$-Cartier and $\\pi$-ample.\n\\end{itemize}\nNow, set $K_X + D = \\pi^*(K_Y+D)$.\nSince $K_Y+D$ is ample, for $0 < \\epsilon \\ll 1$, we have that $\\epsilon K_X + (1-\\epsilon(K_X+D)=K_X + (1-\\epsilon)D$ is ample.\nSince both $K_X$ and $K_X+D$ are $\\qq$-Cartier, then so is $D$.\nLastly, as $X$ and $(Y,D_Y)$ are semi-log canonical, to conclude it suffices to show that $D \\geq 0$.\n\nNow, let $(X^\\nu,D^\\nu + \\Delta^\\nu)$ and $(Y^\\nu,D_Y^\\nu + \\Delta^\\nu_Y)$ denote the respective normalizations, where $\\Delta^\\nu$ and $\\Delta_Y^\\nu$ denote the double loci.\nThus, it suffices to show that $D^\\nu \\geq 0$.\nSince this can be checked by considering one irreducible component of $X^\\nu$ at the time, by abusing notation, we may assume that $X^\\nu$ and $Y^\\nu$ are irreducible.\nBy construction, we have\n\\[\n\\K X^\\nu. + (1-\\epsilon)D^\\nu + \\Delta^\\nu \\sim_{\\qq}-\\epsilon D^\\nu \\sim_{\\qq} \\epsilon(\\K X^\\nu. + \\Delta^\\nu) \/Y^\\nu,\n\\]\nand $\\K X^\\nu. + \\Delta^\\nu$ is relatively ample over $Y^\\nu$.\nThen, since we have $D^\\nu_Y \\geq 0$, by the negativity lemma \\cite{KM98}*{Lemma 3.39}, it follows that $D^\\nu \\geq 0$.\nThis concludes the proof.\n\\end{proof}\n\n\\subsection{Families of pairs} \nWe now recall the main definitions of families of pairs from \\cite{kol1}*{Chapter 4} and \\cite{kol19s}. A \\emph{family of pairs} $f \\colon (X,D) \\rar S$ over a reduced base $S$ is the datum of a morphism $f \\colon X \\rar S$ and an effective $\\qq$-divisor $D$ on $X$, such that the following conditions hold:\n\n\\begin{itemize}\n    \\item $f$ is flat with reduced fibers of pure dimension $n$;\n    \\item the fibers of $\\Supp(D) \\rar S$ are either empty or of pure dimension $n-1$; and\n    \\item $f$ is smooth at the generic points of $X_s \\cap \\Supp(D)$ for every $s \\in S$.\n\\end{itemize}\n\nFurthermore, we say that a family of pairs is \\emph{well defined} if it also satisfies the following property:\n\n\\begin{itemize}\n    \\item $mD$ is Cartier locally around the generic point of each irreducible component of $X_s \\cap \\Supp(D)$ for every $s \\in S$, where $m \\in \\mathbb{N}$ is a sufficiently divisible natural number clearing the denominators of $D$.\n\\end{itemize}\n\nThis latter condition guarantees that $mD$ is Cartier on a big open set $U \\subset X$ with the property that $U \\cap X_s$ is a big open set of $X_s$ for every $s \\in S$.\nThis guarantees that we have a well-defined notion of pull-back of $D$ under any possible base change $S' \\rar S$, as we can pull back $mD|_U$, take its closure in $X \\times_S S'$, and then divide the coefficients by $m$.\n\nThere is a more general definition of families of divisors, over possibly non-reduced bases due to K\\'ollar in \\cite{kol19s}. We will not report it here since we will not need it, we refer the interested reader to \\emph{loc. cit.}\n\nA well defined family of pairs $f \\colon (X,D) \\rar S$ over a reduced base $S$ is called \\emph{locally stable} if, for every base change $g \\colon (X_C,D_C) \\rar C$ where $C$ is the spectrum of a DVR with closed point $0$, $(X_C,D_C+X_0)$ is a semi-log canonical pair. \nThen, a family $f \\colon (X,D) \\rar S$ is called \\emph{stable} if it is locally stable, $f$ is proper, and $\\K X\/S. + D$ is $f$-ample.\n\n\n\\subsection{Families of $p$-pairs}\n\nLet us fix a positive integer $n$, a polynomial $p(t)\\in \\mathbb{Q}[t]$, and a finite set of coefficients $I \\subset (0,1] \\cap \\qq$.\nLet $r$ be the index of $I$.\n\n\\begin{Def}\\label{Def:functor}\nLet $\\mathscr{F}_{n,p,I}$ be the category fibered in groupoids over $\\operatorname{Sch}\/k$ whose fibers over a scheme $B$  consists of:\n\\begin{itemize}\n    \\item a flat and proper morphism $f \\colon \\mathcal{X}\\to B$ of relative dimension $n$;\n    \\item a flat and proper morphism $\\mathcal{D}\\to B$ of relative dimension $n-1$ and relatively $S_1$;\n    \\item a closed embedding $i \\colon \\mathcal{D}\\to \\mathcal{X}$ over $B$; and\n    \\item for every point $p\\in B$, the fiber $(\\mathcal{X}_b;\\frac{1}{r}\\mathcal{D}_b) $ is $p$-stable with polynomial $p(t)$ and coefficients in $I$.\n\\end{itemize}\nFor every morphism $B'\\to B$, we denote by  $j_{B'} \\colon \\mathcal{X}\\times_B B'\\to \\mathcal{X}$ the second projection, and by $\\mathcal{X}'$ the fiber product $\\mathcal{X}\\times_B B'$.\nSimilarly, for every point $p\\in B$, we denote by $\\mathcal{D}_b \\coloneqq \\operatorname{Spec}(k(b))\\times_B\\mathcal{D}$.\nIf we denote by $\\mathcal{I}_\\mathcal{D}$ the ideal sheaf of $\\mathcal{D}$, we require that for every $B'\\to B$ and every $m$, the data above is such that\n\\begin{enumerate}\n    \\item the natural morphism $(j_{B'}^*(\\mathcal{I}_\\mathcal{D}^{[m]}))\\to (j_{B'}^*(\\mathcal{I}_\\mathcal{D}))^{[m]}$ is an isomorphism; and\n    \\item the natural morphism $(j_{B'}^*(\\omega_{\\mathcal{X}\/B}^{[m]}))\\to (\\omega_{\\mathcal{X}'\/B'})^{[m]}$ is an isomorphism. \n\\end{enumerate}\nWe will denote by $(\\mathcal{X};\\mathcal{D})\\to B$ an object of $\\mathscr{F}_{n,p,I}$ over $B$, and we will call it a \\emph{family of p-stable pairs}.\n\\end{Def}\n\n\\begin{Def}\nWe will call \\emph{$p$-stable morphism} the data of a flat and proper morphisms $f \\colon \\mathcal{X}\\to B$ and a closed embedding $i \\colon \\mathcal{D}\\to \\mathcal{X}$ that satisfies the four bullet points of Definition \\ref{Def:functor}.\nIf there is no ambiguity we will still denote it with $(\\mathcal{X};\\mathcal{D})\\to B$.\n\\end{Def}\n\nConditions as (1) and (2) in Definition \\ref{Def:functor} are usually referred to as \\emph{Koll\\'ar's condition}. \nNow, we add a series of remarks and technical statements that are relevant in this context.\nWe keep the notation of Definition \\ref{Def:functor}.\n\\begin{Remark}\nObserve that, with the notation of Definition \\ref{Def:functor}, $(\\frac{1}{r}\\mathcal{D}_b)^{sc} = \\mathcal{D}_b$.\n\\end{Remark}\n\\begin{Remark} \\label{remark_ideal_flat}\nThe ideal sheaf $\\mathcal{I}_\\mathcal{D}$ is flat over $B$.\nThis follows from the fact that $\\O \\mathcal{X}.$ and $\\O \\mathcal{D}.$ are flat, by considering the associated long exact sequence of $\\operatorname{Tor}$.\n\\end{Remark}\n\n\\begin{Remark}\nSince $f$ is flat and by condition (3) its fibers are $S_2$, $f$ is relatively $S_2$.\nThen, since $\\mathcal{D}\\to B$ is flat and relatively $S_1$, it follows from \\cite{Kol13}*{Corollary 2.61} that $\\mathcal{I}_\\mathcal{D}$ is relatively $S_2$.\n\\end{Remark}\n\n\\begin{Remark}\nBy condition (3), there exists an open subset $V\\subseteq \\mathcal{X}$ whose restriction to any fiber is a big open subset, such that $\\mathcal{D}_b$ is a Cartier divisor on $V_b$ for every $b \\in B$.\nThen, by flatness, we may apply \\cite{stacks-project}*{Tag 062Y}, and we conclude that $\\mathcal{D}$ is Cartier along $V$.\nThen, by Remark \\ref{remark_ideal_flat} and \\cite{HK04}*{Proposition 3.5}, $\\O \\mathcal{X}.$ and $\\mathcal{I}_\\mathcal{D}$ are reflexive.\nIn particular, we have $(\\mathcal{I}_\\mathcal{D})^{[1]} = \\mathcal{I}_\\mathcal{D}$, and $\\mathcal{I}_\\mathcal{D}$ satisfies the conditions in \\cite{kol1}*{Definition 3.51}.\n\\end{Remark}\n\n\\begin{Remark}\nObserve that in Definition \\ref{Def:functor}, the divisor $\\mathcal{D}_b$ in point (3) was defined via a fiber product.\nHowever, according to our conventions (see \\S \\ref{divisorial sheaves}) it should also correspond to a Weil divisor on $\\mathcal{X}_b$.\nThis is true since $\\mathcal{D}\\to B$ is relatively $S_1$, so $\\mathcal{D}_b$ is $S_1$ so its ideal sheaf in $\\mathcal{X}_b$ is $S_2$ from \\cite{Kol13}*{Corollary 2.61}. \n\\end{Remark}\n\n\n\n\n\\begin{Remark}\nThe sheaves $\\mathcal{I}_\\mathcal{D}^{[m]}$ are ideal sheaves of $\\mathcal{O}_\\mathcal{X}$.\nIndeed, we can again consider an open subset $V\\subseteq \\mathcal{X}$ whose restriction to any fiber is a big open subset and such that $\\mathcal{D}$ is a Cartier divisor on $V$.\nThen, if we denote by $i \\colon V\\to \\mathcal{X}$ the inclusion of $V$, by \\cite{HK04}*{Corollary 3.7} we have\n\\[\n\\mathcal{I}_\\mathcal{D}^{[m]} = i_*(\\mathcal{O}_V(-m\\mathcal{D}|_{V})).\n\\]\nThen, the inclusion \n$\\mathcal{O}_V(-m\\mathcal{D}|_{V})\\hookrightarrow \\mathcal{O}_V$ can be pushed forward via $i$, to have an inclusion $\\mathcal{I}_\\mathcal{D}^{[m]}\\hookrightarrow i_*(\\mathcal{O}_V)=\\mathcal{O}_\\mathcal{X}$, where the last equality follows from \\cite{HK04}*{Proposition 3.5} since $\\mathcal{X}\\to B$ is $S_2$.\n\\end{Remark}\n\\begin{Notation}\\label{Notation_mD_for_pstable_fam}\nIf $f:(\\mathcal{X};\\mathcal{D})\\to B$ is a family of p-stable pairs, we denote by $m\\mathcal{D}$ the closed subscheme of $\\mathcal{X}$ with ideal sheaf $\\mathcal{I}^{[m]}_\\mathcal{D}$. \n\\end{Notation}\n\n\n\\begin{Remark}\\label{remark:bc:implies:flat}\nBy \\cite{AH11}*{Proposition 5.1.4}, the sheaves $\\mathcal{I}_\\mathcal{D}^{[m]}$ and $\\omega_{\\mathcal{X}\/B}^{[m]}$ are flat over $B$ for every $m\\in \\mathbb{Z}$.\n\\end{Remark}\n\n\\begin{Lemma}\nThe morphism $m\\mathcal{D}\\to B$ is flat with $S_1$ fibers (i.e., the fibers have no embedded points).\\end{Lemma}\n\\begin{proof}\nTo check that $m\\mathcal{D}\\to B$ is flat it suffices to check that for every closed point $\\operatorname{Spec}(k(b))\\to B$ we have $\\operatorname{Tor}^1(k(b),\\mathcal{O}_{m\\mathcal{D}}) = 0$. We pull back the exact sequence\n\\[\n0\\to \\mathcal{I}_{\\mathcal{D}}^{[m]}\\to \\mathcal{O}_{\\mathcal{X}} \\to \\mathcal{O}_{m\\mathcal{D}}\\to 0\n\\]\nvia $j_{b} \\colon \\mathcal{X}_b\\to \\mathcal{X}$, and we obtain\n\\[\n0 = \\operatorname{Tor}^1(k(b),\\mathcal{O}_{\\mathcal{X}}) \\to \\operatorname{Tor}^1(k(b),\\mathcal{O}_{m\\mathcal{D}}) \\to j_{b}^*\\mathcal{I}_{\\mathcal{D}}^{[m]}\\to \\mathcal{O}_{\\mathcal{X}_{b}} \\to \\mathcal{O}_{(m\\mathcal{D})_{b}}\\to 0.\n\\]\nHowever, $j_{b}^*\\mathcal{I}_{\\mathcal{D}}^{[m]} \\cong (j_{b}^*\\mathcal{I}_{\\mathcal{D}})^{[m]}$, so in particular it is a torsion free sheaf of rank 1.\nThen, the map $j_{b}^*\\mathcal{I}_{\\mathcal{D}}^{[m]}\\to \\mathcal{O}_{\\mathcal{X}_{b}}$ is injective, so $\\operatorname{Tor}^1(k(b),\\mathcal{O}_{m\\mathcal{D}})=0$ as desired.\n\nFinally, $\\mathcal{I}_{\\mathcal{D}_p}^{[m]}$ is $S_2$ and $\\mathcal{O}_{\\mathcal{X}_p}$ is $S_2$ from condition (2), so $\\mathcal{O}_{(m\\mathcal{D})_p}$ is $S_1$ from \\cite{Kol13}*{Corollary 2.61}.\\end{proof}\n\n\\begin{Remark}\\label{remark_for_m_big_enough_kollar_comndition_guarantees_cartier}\nThere is an $m>0$, which does not depend on $B$, such that $\\omega_{\\mathcal{X}\/B}^{[m]}$ and $\\mathcal{I}_\\mathcal{D}^{[m]}$ are locally free on $\\mathcal{X}$.\nIndeed, we will prove in Theorem \\ref{thm_boundedness} that there is an $m$ such that for every fiber $(X;D)$ of $f$, the sheaves $\\omega_X^{[m]}$ and $\\mathcal{O}_X(-mD)$ are Cartier. Since our family is bounded (see Theorem \\ref{thm_boundedness}), we can choose such an $m$ which does not depend on the basis $B$.\nThen, by conditions (1) and (2) and Remark \\ref{remark:bc:implies:flat}, we may apply  \\cite{stacks-project}*{Tag 00MH}, which implies that $\\omega_{\\mathcal{X}\/B}^{[m]}$ and $\\mathcal{I}_\\mathcal{D}^{[m]}$ are locally free since they restrict to locally free sheaves along each fiber.\n\\end{Remark}\n\nNow, we specify the morphisms in the fibered category $\\mathscr{F}_{n,p,I}$ over a morphism $f \\colon T\\to B.$\nLet $\\alpha = ((\\mathcal{Y},\\mathcal{D}_\\mathcal{Y})\\to T)$ be an element of $\\mathscr{F}_{n,p,I}(T)$, and let $\\beta = ((\\mathcal{X};\\mathcal{D}_\\mathcal{X})\\to B)$ be an element of $\\mathscr{F}_{n,p,I}(B)$.\nAn arrow $\\alpha \\to \\beta$ is the data of two morphisms $(g,h)$ which fit in a diagram like the one below, where all the squares are fibered diagrams:\n\\[\n\\xymatrix{\\mathcal{D}_\\mathcal{Y}\\ar[r]^g\\ar[d] & \\mathcal{D}\\ar[d] \\\\ \\mathcal{Y}\\ar[d] \\ar[r]^h & \\mathcal{X}\\ar[d] \\\\ T\\ar[r]^f & B.}\n\\]\n\n\\begin{Oss}\nObserve that the only morphisms over the identity $\\operatorname{Id} \\colon B\\to B$ are isomorphisms.\nThus, $\\mathscr{F}_{n,p,I}$ is fibered in groupoids.\n\\end{Oss}\n\n\\subsection{Hassett's example} \\label{hassett example}\nIn this subsection, we present a well-known example due to Hassett that is helpful to keep in mind to navigate the rest\nof the paper.\nSee also \\cite{KP17}*{Subsection 1.2} or \\cite{kol1}. \n\nConsider the DVR $R = \\operatorname{Spec}(k[t]_{(t)})$, let $\\eta$ (resp. $p$) be the generic (resp. closed) point of $\\operatorname{Spec}(R)$, and let $\\mathcal{X}=\\mathbb{P}^1\\times \\mathbb{P}^1 \\times \\operatorname{Spec}(R)$.\nConsider $C$ a smooth member of $\\mathcal{O}_{\\mathbb{P}^1\\times \\mathbb{P}^1}(1,2)$, and let $\\mathcal{Y}$ be the blow-up of $C$ in the special fiber of $\\mathcal{X}$.\nThen, if we compose the blow-down $\\mathcal{Y}\\to \\mathcal{X}$ with the projection $\\mathcal{X}\\to \\operatorname{Spec}(R)$, we get a family of surfaces $\\mathcal{Y}\\to \\operatorname{Spec}(R)$ where the generic fiber is a copy of $\\mathbb{P}^1\\times \\mathbb{P}^1$, while the special fiber is a surface with two irreducible components.\nOne irreducible component of the special fiber is isomorphic to $\\mathbb{P}^1\\times \\mathbb{P}^1$\n(the proper transform of the special fiber of $\\mathcal{X}\\to \\operatorname{Spec}(R)$), and the other one is the exceptional divisor $F$.\nThe surface $F$ is the projectivization of the normal bundle of $C$.\nSince $C\\cong \\mathbb{P}^1$ and the normal bundle of $C$ in $\\mathcal{X}$ is isomorphic to $\\mathcal{O}_{\\mathbb{P}^1}(0) \\oplus \\mathcal{O}_{\\mathbb{P}^1}(4)$, we have that $F$ is isomorphic to the Hirzebruch surface $\\mathbb{F}_4$.\nWe denote by $\\Delta\\subseteq \\mathbb{F}_4$ the preimage of the double locus of the central fiber on $\\mathbb{F}_4$.\n\nWe consider a divisor on $\\mathbb{P}^1\\times \\mathbb{P}^1$ consisting of five irreducible components, three general members of $\\mathcal{O}_{\\mathbb{P}^1\\times \\mathbb{P}^1}(1,2)$ (which we denote by $C_1, C_2, C_3$), $C$, and a smooth member $G$ of $\\mathcal{O}_{\\mathbb{P}^1\\times \\mathbb{P}^1}(2,0)$.\nWe consider a deformation of $C_1+C_2+C_3+C + D$ in $\\mathcal{X}$ given by the trivial deformation of $C$ and $D$, and we deform $C_i$ to $C$ for every $i$.\nWe denote by $\\mathcal{D}_\\mathcal{X}$ the total space of this deformation, and by $\\mathcal{D}$ its proper transform in $\\mathcal{Y}$.\nMore explicitly, if $C$ is the zero locus of a global section $g\\in H^0(\\mathcal{O}_{\\mathbb{P}^1\\times \\mathbb{P}^1}(C))$ and $\\varphi_1, \\varphi_2, \\varphi_3$ are generic sections of $H^0(\\mathcal{O}_{\\mathbb{P}^1\\times \\mathbb{P}^1}(C))$, the deformation we consider is $\\mathcal{D}_\\mathcal{X} = V(g(t\\varphi_1 - g)(t\\varphi_2 - g)(t\\varphi_3 - g))$.\n\n\\begin{center}\n\\includegraphics[scale=0.22]{Y_and_div.jpg}\\\\ Figure 1: the family $(\\mathcal{Y},c\\mathcal{D})\\to \\operatorname{Spec}(R).$\n\\end{center}\n\nNow, we construct the ample model of $(\\mathcal{Y},c\\mathcal{D})\\to \\operatorname{Spec}(R)$, with the coefficient $c$ in a neighbourhood of $\\frac{1}{2}$.\nFirst, we introduce some notations.\nWe denote $K_{\\mathcal{Y}\/\\operatorname{Spec}(R)}+c\\mathcal{D}$ by $\\mathcal{L}(c)$, the irreducible component of the central fiber isomorphic to $\\mathbb{P}^1\\times \\mathbb{P}^1$ by $\\mathscr{P}$, and the two rulings of $\\mathscr{P}$ by $f_1$ and $f_2$, with the convention that $C = f_1+2f_2$.\nSince the generic fiber $(\\mathcal{Y},c\\mathcal{D})_\\eta$ is stable for every $c$ with $|c-\\frac{1}{2}| \\ll 1$, we just need to control the intersection pairings on the special fiber.\n\nLet $c=\\frac{1}{2} + \\epsilon$, where $|\\epsilon| \\ll 1$.\n\\begin{center}\n\\includegraphics[scale=0.20]{special_fiber.jpg} \\\\ Figure 2: special fiber of the family $(\\mathcal{Y},c\\mathcal{D})\\to \\operatorname{Spec}(R)$.\n\\end{center}\nOne can check that:\n\\begin{enumerate}\n\\item when $\\epsilon > 0$, the divisor $\\mathcal{L}(\\frac{1}{2} + \\epsilon)$ is nef.\nIt is ample on $F$, positive on $f_2$ and 0 on $f_1$;\n\\item when $\\epsilon = 0$, the divisor $\\mathcal{L}(\\frac{1}{2})$ is nef.\nIt is 0 on $\\mathscr{P}$ and on $\\Delta$; and\n\\item when $\\epsilon < 0$, the divisor $\\mathcal{L}(\\frac{1}{2}+\\epsilon)$ is  not nef.\nOn $\\mathscr{P}$, it is negative on $f_2$ and 0 on $f_1$, while on $F $ it is negative on $\\Delta$.\n\\end{enumerate}\nTherefore, we can explicitly describe the special fiber of the ample model of $(\\mathcal{Y},c\\mathcal{D})$ over $\\operatorname{Spec}(R)$. We denote by $(\\mathcal{Z}^+,c\\mathcal{D}^+)$ (resp. $(\\mathcal{Z}^0, c\\mathcal{D}^0)$, $(\\mathcal{Z}^-,c\\mathcal{D}^-)$) the ample model of $(\\mathcal{Y},c\\mathcal{D})\\to \\operatorname{Spec}(R)$ when $0<\\epsilon\\ll 1$ (resp. $\\epsilon=0$, $-1 \\ll \\epsilon < 0$):\n\\begin{enumerate}\n\\item when $\\epsilon > 0$, to construct the canonical model we contract the ruling $f_1$.\nSince via this contraction the map $C\\subseteq \\mathbb{P}^1\\times \\mathbb{P}^1 \\to \\mathbb{P}^1$ is a $2:1$ ramified cover of $\\mathbb{P}^1$, the special fiber is the push-out of the following diagram:\n$$\\xymatrix{\\mathbb{P}^1\\cong \\Delta \\ar[r] \\ar[d]^{2:1} \\ar[r] & F\\ar[d]_\\pi \\\\ \\mathbb{P}^1\\ar[r] & \\mathcal{Z}^+_p}$$\n\\item when $\\epsilon = 0$, to construct the canonical model, we contract $\\mathscr{P}$ and $\\Delta$.\nThe special fiber is isomorphic to $\\mathbb{F}_4$ with the section $\\Delta$ contracted, which is the projectivization of the cone over a rational quartic curve; and\n\\item When $\\epsilon < 0$, we perform a divisorial contraction to make the divisor nef.\nWe contract the ruling $f_2$, and the special fiber is $\\mathbb{F}_4$.\n\\end{enumerate}\nIn particular, there are morphisms $\\pi^+ \\colon \\mathbb{F}_4\\to \\mathcal{Z}^+_p$,  $\\pi^0 \\colon \\mathbb{F}_4\\to \\mathcal{Z}^0_p$ and $\\pi^- \\colon \\mathbb{F}_4\\to \\mathcal{Z}^-_p$ from $\\mathbb{F}_4$ to the special fibers of $\\mathcal{Z}^+$, $\\mathcal{Z}^0$ and $\\mathcal{Z}^-$.\nThe divisors $\\mathcal{D}^+_p$, $\\mathcal{D}^0_p$ and $\\mathcal{D}^-_p$ can be described as follows.\nFirst, recall that $\\mathbb{F}_4$ is the projectivization of a vector bundle on $\\mathbb{P}^1$, so it has a relative $\\mathcal{O}_{\\mathbb{F}_4}(1)$.\nWe denote a generic section of $\\mathcal{O}_{\\mathbb{F}_4}(1)$ by $h$.\nThen, the following holds:\n\n\\begin{enumerate}\n\\item when $\\epsilon > 0$, the divisor $\\mathcal{D}^+_p$ is the image via $\\pi^+$ of four generic fibers of $F$, together with a divisor linearly equivalent to $4h$. All the components have coefficient $\\frac{1}{2} + \\epsilon$;\n\\item when $\\epsilon = 0$, the divisor $\\mathcal{D}^0_p$ is the image via $\\pi^0$ of four generic fibers with a divisor linearly equivalent to $4h$.\nAll the components have coefficient $\\frac{1}{2}$; and\n\\item when $\\epsilon < 0$, the divisor $\\mathcal{D}^-_p$ consists of four generic fibers with a divisor linearly equivalent to $4h$, four generic fibers, and $\\Delta$.\nAll the components have coefficient $\\frac{1}{2} + \\epsilon$, with the exception of $\\Delta$, which has coefficient $1+2\\epsilon$.\n\\end{enumerate}\n\n\nRecall now that the flat limit of $\\mathcal{D}_{\\eta}\\subseteq \\mathcal{Z}^0_\\eta$ in $\\mathcal{Z}^0_p$ is \\emph{not} $S_1$, since it has an embedded point (see \\cite{KP17}*{Subsection 1.2}). However, $(\\mathcal{Z}^-;\\mathcal{D}^-)\\to \\operatorname{Spec}(R)$ is a $p$-stable morphism with coefficients $I = \\{\\frac{1}{2}, 1\\}$ (see Proposition \\ref{prop:flat:limit:is:S1}), so in particular the flat limit of $\\mathcal{D}_\\eta$ in $\\mathcal{Z}_0^-$ does \\emph{not} have an embedded point on the special fiber.\n\n\n\\subsection{$p$-stable morphisms with constant part}\nFor proving that $\\mathscr{F}_{n,p,I}$ is bounded, it will be useful to introduce the following definition.\n\n\\begin{Def}\nAssume that $S$ is reduced.\nA \\emph{locally $p$-stable morphism with constant part} $f \\colon (\\mathcal{X},\\Omega;\\mathcal{D}) \\rar S$ over $S$ and with coefficients in $I$ is the datum of a proper morphism $f \\colon \\mathcal{X} \\rar S$, a closed subscheme $\\mathcal{D}$ on $\\mathcal{X}$, and an effective $\\qq$-divisor $\\Omega$ on $X$, such that the following conditions hold:\n\\begin{itemize}\n    \\item $f \\colon (X,\\Omega) \\rar S$ is a proper, locally stable family of pairs;\n    \\item $\\mathcal{D} \\rar S$ is flat and relatively $S_1$ (namely, with no embedded points) with fibers of pure dimension $n-1$; and\n    \\item for every $s \\in S$, there is a $p$-pair $(X,\\Delta;D)$  with coefficients in $I$ such that $X=\\mathcal{X}_s$, $\\Delta=\\Omega_s$, and $D^{sc}=\\mathcal{D}_s$.\n\\end{itemize}\nFurthermore, if we have a polynomial $p(t) \\in \\qq[t]$ and for every $s \\in S$, $(\\mathcal{X}_s;\\mathcal{D}_s)$ has polynomial $p(t)$, we say that it is a family of $p$-pairs with polynomial $p(t)$.\n\\end{Def}\n\n\\begin{Notation}\nWe say that a locally $p$-stable morphism with constant part (with coefficients in $I$ and polynomial $p(t)$) is a \\emph{$p$-stable morphism with constant part} (\\emph{with coefficients in $I$ and polynomial $p(t)$}) if $\\K \\mathcal{X}\/S. + \\frac{1}{r} \\mathcal{D}$ (respectively, $\\K \\mathcal{X}\/S. + \\frac{1}{r} \\mathcal{D} + \\Omega$) is $f$-ample.\nHere, recall the $r$ denotes the index of $I$.\n\\end{Notation}\n\n\\begin{Prop} \\label{prop_well_defined_p-pairs}\nLet us fix a set of coefficients $I \\subset (0,1] \\cap \\qq$, a polynomial $p$ and an integer $n$.\nLet $r$ denote the index of $I$.\nLet $f \\colon (\\mathcal{X},\\Omega;\\mathcal{D}) \\rar S$ be a $p$-stable morphism with constant part $\\Omega$ over a reduced base $S$.\nThen $f \\colon (\\mathcal{X}, \\frac{1}{r}\\mathcal{D}+\\Omega) \\rar S$ is a well defined family of pairs.\n\\end{Prop}\n\\begin{proof}\nBy definition $\\mathcal{D} \\rar S$ is flat with fibers of pure dimension $n-1$.\nThus, it follows that the fibers of $\\mathrm{Supp}(\\mathcal{D} + \\Omega) \\rar S$ are either empty or of pure dimension $n-1$.\nThus, to show that $f \\colon (\\mathcal{X}, \\frac{1}{r}\\mathcal{D} + \\Omega) \\rar S$ is a family of pairs, we are left with showing that $f$ is smooth at the generic points of $X_s \\cap \\operatorname{Supp}(\\mathcal{D} + \\Omega)$ for every $s \\in S$.\n\nBy assumption, this is the case for all the generic points of $X_s \\cap \\operatorname{Supp}(\\mathcal{D} + \\Omega)$ arising from $\\Omega$.\nThus, we may focus on the contribution of $\\mathcal{D}$.\nBut then, since each fiber $(\\mathcal{X}_s,\\Omega_s;\\mathcal{D}_s)$ is a $p$-pair, it follows that the generic points of $\\mathcal{D}_s$ are contained in the smooth locus of $X_s$ by the semi-log canonical condition.\nThus, $f \\colon (\\mathcal{X}, \\frac{1}{r}\\mathcal{D} + \\Omega) \\rar S$ is a family of pairs.\n\nTo conclude, we need to show that the family is well defined.\nTo this end, it suffices we focus on $\\mathcal{D}$, as $\\Omega$ satisfied the needed conditions by definition.\nAs argued in \\cite{kol1}*{Definition 3.66}, there is a big open subset $U \\subset \\mathcal{X}$ such that every point of $U$ is either smooth or nodal, and $U \\cap \\mathcal{X}_s$ has codimension at least 2 in $\\mathcal{X}_s$ for every $s \\in S$.\nFor every $s \\in S$, $\\mathcal{D}_s$ does not contain any irreducible component of the double locus.\nThus, the intersection between $\\mathcal{D}$ and $U$ has codimension at least 2 in every fiber.\nLet $V$ denote the open set obtained by removing this intersection from $U$.\nThen, as the scheme theoretic restriction $\\mathcal{D}_s$ is an integral Weil divisor, it is a Carter divisor along $V_s$.\nThen, by \\cite{stacks-project}*{Tag 062Y}, $\\mathcal{D}$ is a Cartier divisor along $V$.\nThus, the claim follows.\n\\end{proof}\n\n\\begin{Prop} \\label{prop_stable_family_p-pairs}\nFix an integer $n\\in \\mathbb{N}$, a polynomial $p(t)\\in \\mathbb{Q}[t]$, and a finite subset $I\\subseteq (0,1] \\cap \\mathbb{Q}$.\nAssume that assumption (A) holds, and let $\\epsilon_0>0$ be as in Lemma \\ref{lemma_bound_for_epsilon}.\nLet $f \\colon (\\mathcal{X},\\Omega;\\mathcal{D}) \\rar S$ be a $p$-stable morphism with constant part, coefficients in $I$, and polynomial $p(t)$, over a reduced base $S$.\nThen $f \\colon (\\mathcal{X}, \\frac{1-\\epsilon}{r}\\mathcal{D} + \\Omega) \\rar S$ is a stable family of pairs for every $0 < \\epsilon < \\epsilon_0$ rational.\nIn particular, $\\mathcal{D}$ is $\\qq$-Cartier.\n\\end{Prop}\n\n\\begin{proof}\nFix $\\epsilon$ as in the statement.\nThen, by Proposition \\ref{prop_well_defined_p-pairs}, $f \\colon (\\mathcal{X}, \\frac{1-\\epsilon}{r}\\mathcal{D} + \\Omega) \\rar S$ is a well defined family of pairs.\nBy assumption, the self-intersection $(\\K \\mathcal{X}_s. + \\frac{1-\\epsilon}{r}\\mathcal{D}_s + \\Omega_s)^n$ is independent of $s \\in S$, as it is $p(1-\\epsilon)$.\nSince $f \\colon (\\mathcal{X}, \\frac{1-\\epsilon}{r}\\mathcal{D} + \\Omega) \\rar S$ is a well defined family of pairs, we may find a big open subset $U \\subset \\mathcal{X}$ such that $U_s$ has codimension at least 2 for every $s \\in S$ and $\\K \\mathcal{X}\/S. + \\frac{1-\\epsilon}{r}\\mathcal{D} + \\Omega$ is $\\qq$-Cartier along $U$.\nThen, $\\K \\mathcal{X}\/S. + \\frac{1-\\epsilon}{r}\\mathcal{D} + \\Omega$ is $\\qq$-Cartier by \\cite{kol1}*{Theorem 5.7}.\nThe claim follows by \\cite{kol1}*{Theorem-Definition 4.45} and the fact that the argument was independent of $\\epsilon \\in (0,\\epsilon_0) \\cap \\qq$.\n\\end{proof}\n\n\n\\subsection{Existence of good minimal models}\nLet $(X,\\Delta)$ be a log canonical pair, and let $f \\colon X \\rar S$ be a projective morphism such that $\\K X. + \\Delta$ is $f$-pseudo-effective.\nThen, it is expected that $X$ admits a good minimal model over $S$.\nThat is, $X$ admits a birational contraction $\\phi \\colon (X,\\Delta) \\dashrightarrow (X',\\Delta')$ over $S$ to a log canonical pair $(X',\\Delta')$, such that $\\Delta'$ is the push-forward of $\\Delta$ to $X'$, $\\phi$ is $(K_X+\\Delta)$-negative, and $\\K X'. + \\Delta'$ is semi-ample over $S$.\n\nHere, we collect a technical statement that shows the existence of relative good minimal models under certain assumptions.\n\n\\begin{Lemma} \\label{lemma minimal models}\nLet $(X,\\Delta)$ be a log canonical pair, and let $f \\colon X \\rar S$ be a projective morphism such that $\\K X. + \\Delta$ is $f$-pseudo-effective.\nAssume that $f$ has relative dimension at most 3 and that every log canonical center of $(X,\\Delta)$ dominates $S$.\nThen, up to extracting some log canonical places of $(X,\\Delta)$, $(X,\\Delta)$ admits a relative good minimal model over $S$.\nFurthermore, if $\\K X. + \\Delta$ is nef over $S$, then it is semi-ample.\n\\end{Lemma}\n\n\\begin{Remark}\nThe assumption on the relative dimension in Lemma \\ref{lemma minimal models} is only needed to guarantee the existence of a good minimal model (possibly after extracting some log canonical center).\nIf the existence of a good minimal model (possibly after extracting some log canonical places) is known by other means, then the nefness of $K_X+\\Delta$ still implies its semi-ampleness, regardless of the relative dimension of $f$.\n\\end{Remark}\n\n\\begin{proof}\nLet $\\pi \\colon X' \\rar X$ be a log resolution of $(X,\\Delta)$, and let $\\pi^*(K_X+\\Delta)= \\K X'. + \\Delta' + E' + \\Gamma' - F'$.\nHere $\\Delta'$ denotes the strict transform of $\\Delta$, the divisors $E'$, $\\Gamma'$, and $F'$ are effective, $\\pi$-exceptional, and share no common components.\nFurthermore, $\\Gamma'$ is reduced, while the coefficients of $E'$ are in $(0,1)$.\nLet $\\Xi'$ denote the reduced $\\pi$-exceptional divisor, and fix a rational number $0 < \\epsilon \\ll 1$.\nThen, $(X',\\Delta' + E' + \\Gamma' + \\epsilon (\\Xi' - \\Gamma'))$ is dlt, and it has the same pluricanonical ring as $(X,\\Delta)$.\nFurthermore, by the addition of $\\epsilon (\\Xi' - \\Gamma')$, every $\\pi$-exceptional divisor that is not in $\\Gamma$ is in the relative stable base locus of $\\K X'. +\\Delta' + E' + \\Gamma' + \\epsilon (\\Xi' - \\Gamma')$.\nFinally, by assumption and the choice of $\\epsilon$, every log canonical center of $(X',\\Delta' + E' + \\Gamma' + \\epsilon (\\Xi' - \\Gamma'))$ dominates $S$.\n\nSince the fibers have dimension at most three, by \\cite{KMM94} and \\cite{HMX18}*{Theorem 1.9.1}, it follows that $(X',\\Delta' + E' + \\Gamma' + \\epsilon (\\Xi' - \\Gamma'))$ has a relative good minimal model over a non-empty open subset $U \\subseteq S$.\nThen, as by assumption there are no vertical log canonical centers, it follows from \\cite{HX13}*{Theorem 1.1} that $(X',\\Delta' + E' + \\Gamma' + \\epsilon (\\Xi' - \\Gamma'))$ has a relative good minimal model over $S$.\nSince every $\\pi$-exceptional divisor that is not in $\\Gamma'$ is in the relative stable base locus, any such divisor is contracted on the minimal model.\nThis shows the first part of the claim.\n\nNow, assume that $K_X + \\Delta$ is relatively nef.\nThen, $(X,\\Delta)$ is a relative weak log canonical model for $(X',\\Delta' + E' + \\Gamma' + \\epsilon (\\Xi' - \\Gamma'))$ in the sense of \\cite{HMX}.\nThen, we conclude by \\cite{HMX}*{Lemma 2.9.1} that $(X,\\Delta)$ is a relatively semi-ample model.\nNotice that \\cite{HMX}*{Lemma 2.9.1} is phrased for projective pairs.\nOn the other hand, by \\cite{HX13}*{Corollary 1.2}, one can first take a projective closure of $\\tilde X$ over a compactification $\\overline{S}$ of $S$, and take a projective relative good minimal model.\nThen, by adding the pull-back of some divisor on $\\overline{S}$, we can regard the relative good minimal model as a projective minimal model.\nThus, it follows that $K_{X} + \\Delta$ is relatively semi-ample over $S$, and it defines a morphism to the relative ample model.\n\\end{proof}\n\n\n\n\\section{Boundedness}\\label{section_boundedness}\nThe goal of this section is to prove that, if we fix a set of coefficients $I$, a polynomial $p$ and a dimension $n$, the corresponding set of $p$-stable pairs is effectively log-bounded. \n\\begin{Lemma} \\label{lemma_nef_generalizes}\nLet $f \\colon Y \\rar B$ be a flat and proper morphism, and let $L$ be a line bundle on $Y$.\nThen, the locus of points $b \\in B$ such that $L_b$ is nef is closed under generalization.\n\\end{Lemma}\n\n\\begin{proof}\nWithout loss of generality, we may assume that $B=\\operatorname{Spec}(R)$ and $R$ is a DVR.\nLet also $0$ (resp. $\\eta$) be the closed (resp. generic) point of $\\operatorname{Spec}(R)$.\n\nAssume by contradiction that $L$ is nef on the special fiber, but it is not nef on the generic one.\nThen, there is an irreducible curve $C'_\\eta\\subseteq Y_\\eta$ that is negative for $L_\\eta$.\nLet $C_\\eta \\to C_\\eta'$ denote the normalization of $C'_\\eta$.\nBy semistable reduction, up to replacing $\\operatorname{Spec}(R)$ with a ramified cover, we can compactify  $C_\\eta\\to Y_\\eta$ to a family of curves $C\\to \\operatorname{Spec}(R)$ where the special fiber is nodal and maps to $Y$.\nThe situation is described by the following diagram:\n\\[\n\\xymatrix{C_\\eta \\ar[r] \\ar[d] & C \\ar[d] \\\\ Y_\\eta \\ar[r] \\ar[d] & Y \\ar[d] \\\\ \\eta \\ar[r] & \\operatorname{Spec}(R).}\n\\]\nLet $G$ denote the pull-back of $L$ to $C$.\nBy assumption, $G_\\eta$ has negative degree, while $G_0$ has non-negative degree on each irreducible component of $C_0$.\nThen we can consider the cohomology groups $H^1(C_\\eta, G_{\\eta}^{\\otimes n})$ and $H^1(C_0,G_0^{\\otimes n})$.\nBy Riemann--Roch, $H^1(C_\\eta, G_{\\eta}^{\\otimes n})$ grows linearly in $n$ and is unbounded, while $H^1(C_0,G_0^{\\otimes n})$ is bounded.\nThis contradicts the upper-semicontinuity theorems for cohomology and base change.\n\\end{proof}\n\n\n\\begin{Prop}\\label{prop:nef:open}\nFix an integer $n\\in \\mathbb{N}$.\nAssume that assumption (A) holds.\nConsider a locally stable family of stable pairs of relative dimension $n$ $(\\mathcal{X},\\mathcal{D})\\to B$ over a reduced scheme $B$, with $\\mathcal{D}$ being $\\mathbb{Q}$-Cartier.\nAssume that there is an $0<\\epsilon_0 < 1$ such that  $(\\mathcal{X},(1-\\epsilon_0)\\mathcal{D})\\to B$ is stable.\nThen, the set of points $p\\in B$ such that $K_{\\mathcal{X}\/B} + \\mathcal{D}$ is nef is open.\n\\end{Prop}\n\n\\begin{proof}\nIt is well known that a subset is open if and only if it is constructible and stable under generalization, see \\cite{stacks-project}*{Tag 0542}.\nBy Lemma \\ref{lemma_nef_generalizes}, the set of interest is stable under generalization.\nThus, we are left with showing that it is constructible.\n\nRecall that, if $f \\colon B'\\to B$ is a morphism between schemes of finite type, $f(B')$ is a constructible subset of $B$.\nIn particular, we can replace $B$ with a resolution, and assume that $B$ is smooth and irreducible.\n\n{\\bf Step 1:} In this step, we reduce the proof to the case when $\\mathcal{X}$ and its general fiber are normal.\n\nIf $\\nu \\colon Z^n\\to Z$ is the normalization of a scheme and $L$ is a line bundle on $Z$, then $L$ is nef if and only if $\\nu^*L$ is nef.\nTo reduce to the case when $\\mathcal{X}$ is normal, we will produce a simultaneous normalization of the family $\\mathcal{X} \\to B$.\nThis will be achieved by stratifying $B$.\n\n\nWe begin by considering the generic point $\\eta\\in B$, and we spread out the normalization $\\mathcal{Y}'$ of $\\mathcal{X}_\\eta$, together with its conductor $\\Gamma_{\\mathcal{Y}}$, along an open subset $U$ of $B$ to get a pair $(\\mathcal{Y},\\mathcal{D}_\\mathcal{Y} + \\Omega_\\mathcal{Y} +\\Gamma_{\\mathcal{Y}})$.\nUp to shrinking $U$, we can assume that $(\\mathcal{Y},\\mathcal{D}_\\mathcal{Y} + \\Omega_\\mathcal{Y} +\\Gamma_{\\mathcal{Y}}) \\to U$ is the normalization fiber by fiber, with conductor $\\Gamma_{\n\\mathcal{Y}}$.\nWe proceed then by Noetherian induction.\nAt the end of this procedure, up to stratifying $B$, we get a family of (possibly with multiple irreducible components) log canonical pairs $(\\mathcal{Y},\\mathcal{D}_\\mathcal{Y} + \\Omega_\\mathcal{Y} +\\Gamma_{\\mathcal{Y}}) \\to B$.\nFurthermore, we assume that $B$ is smooth: this can be achieved by resolving $B$ and pulling back the family, or by stratifying $B$.\n\nIn the above procedure, it may be that, over an irreducible component of $B$, the family $\\mathcal{Y} \\rar B$ is not irreducible.\nOn the other hand, by construction, each irreducible component of $(\\mathcal{Y},\\mathcal{D}_\\mathcal{Y} + \\Omega_\\mathcal{Y} +\\Gamma_{\\mathcal{Y}}) \\to B$ provides a family of irreducible log canonical pairs.\nThen,\n$(K_{\\mathcal{X}\/B} + \\mathcal{D} + \\Omega)|_{{\\mathcal{X}}_p}$ is nef if and only if so is its pull-back to each irreducible component of $\\mathcal{Y}_p$.\nSince $\\mathcal{Y} \\rar B$ has finitely many irreducible components, the set \n\\[\n\\{p \\in B : (K_{\\mathcal{X}\/B} + \\mathcal{D} + \\Omega)|_{{\\mathcal{X}}_p}\\text{ is nef}\\}\n\\]\nis locally closed if and only if so are the sets corresponding to the same property for each irreducible component of $\\mathcal{Y} \\rar B$.\nThus, in the following, we may assume that $\\mathcal{Y} \\rar B$ is a morphism with connected fibers.\n\n{\\bf Step 2:} We conclude the proof by addressing the case when $\\mathcal{X} \\rar B$ has normal fibers.\nWe will use \\cite{HMX}*{Theorem 1.9.1} to finish the proof.\n\nConsider an irreducible component, which we will still denote by $\\pi \\colon (\\mathcal{Y},  \\mathcal{D} + \\Omega_\\mathcal{Y} + \\Gamma_\\mathcal{Y}) \\rightarrow B$,\nand assume that  for a point $0\\in B$, the divisor $K_{\\mathcal{Y} _0}+ \\mathcal{D} _{0} + \\Omega_0 + \\Gamma_0$ is nef.\nIn the following, we will perform several stratifications of $B$.\nWe will always assume that $0$ (or some other member of our moduli problem, which will be then denoted by $0$) is still a point in the new stratum, and we will disregard the strata not containing members of our moduli problem. \n\nUp to a stratification (which we omit in the notation), we may assume that $\\pi$ admits a log resolution in family.\nCall it $\\rho \\colon (\\mathcal{Y}',  \\mathcal{D}' + \\Omega' + \\Gamma' + \\mathcal{F}) \\rightarrow B$, where $\\mathcal{F}$ is the reduced exceptional divisor, and the remaining divisors are the strict transforms of $\\mathcal{D}_\\mathcal{Y}$, $\\Omega_\\mathcal{Y}$ and $\\Gamma_\\mathcal{Y}$, respectively.\nThen, $(\\mathcal{Y}_0,  \\mathcal{D}_0 + \\Omega_0 + \\Gamma_0)$ is a minimal model for $(\\mathcal{Y}'_0,  \\mathcal{D}'_0 + \\Omega'_0 + \\Gamma'_0 + \\mathcal{F}_0)$.\n\nBy our assumptions on the existence of good minimal models for projective pairs, $(\\mathcal{Y}_0,  \\mathcal{D}_0 + \\Omega_0 + \\Gamma_0)$ is a good minimal model for $(\\mathcal{Y}'_0,  \\mathcal{D}'_0 + \\Omega'_0 + \\Gamma'_0 + \\mathcal{F}_0)$.\nThus, by \\cite{HMX18}*{Theorem 1.9.1}, $\\rho \\colon (\\mathcal{Y}',  \\mathcal{D}' + \\Omega' + \\Gamma' + \\mathcal{F}) \\rightarrow B$ admits a relative good minimal model over $B$, and it admits a relative ample model, which computes the ample model fiberwise.\nBy construction, this is the relative ample model of $\\pi \\colon (\\mathcal{Y},  \\mathcal{D}_\\mathcal{Y} + \\Omega_\\mathcal{Y} + \\Gamma_\\mathcal{Y}) \\rightarrow B$.\nCall this model $\\sigma \\colon (\\mathcal{Y}^c,  \\mathcal{D}^c + \\Omega^c + \\Gamma^c) \\rightarrow B$.\n\nLet $p \\colon \\mathcal{W} \\rightarrow \\mathcal{Y}$ and $q \\colon \\mathcal{W} \\rightarrow \\mathcal{Y}^c$ be a common resolution.\nUp to a stratification, we may assume that this induces a resolution fiberwise, and the exceptional divisors of $p$ and $q$ are flat over $B$.\nIn particular, every fiber $\\mathcal{W}_0$ intersects all the exceptional divisors of $p $ and $q$.\nThen, we have\n\\begin{equation} \\label{eq_discr_constructible}\np^*(K_{\\mathcal{Y}} + \\mathcal{D}_\\mathcal{Y} + \\Omega_\\mathcal{Y} + \\Gamma_\\mathcal{Y}) \\geq q^*(K_{\\mathcal{Y}^c} + \\mathcal{D}^c + \\Omega^c + \\Gamma^c).\n\\end{equation}\nOn the other hand, $K_{\\mathcal{Y}_0} + \\mathcal{D}_0 + \\Omega_0 + \\Gamma_0$ is the pull-back of $K_{\\mathcal{Y}^c_0} + \\mathcal{D}^c_0 + \\Omega^c_0 + \\Gamma_0^c$ under the contraction $\\mathcal{Y}_0 \\rightarrow \\mathcal{Y}^c_0$, so\n\\begin{equation} \\label{eq_dirsc2_constructible}\n(p^*(K_{\\mathcal{Y}} + \\mathcal{D}_\\mathcal{Y} + \\Omega_\\mathcal{Y} + \\Gamma_\\mathcal{Y}))|_{\\mathcal{W}_0} = (q^*(K_{\\mathcal{Y}^c} + \\mathcal{D}^c + \\Omega^c + \\Gamma^c))|_{\\mathcal{W}_0} .\n\\end{equation}\nFrom how the resolution $\\mathcal{W}$ and the stratification of $B$ are constructed, the fiber $\\mathcal{W}_0$ intersects all the $p$ and $q$ exceptional divisors.\nIn particular, by \\eqref{eq_dirsc2_constructible}, \\eqref{eq_discr_constructible} has to be an equality.\nThus, $K_{\\mathcal{Y}} + \\mathcal{D}_\\mathcal{Y} + \\Omega_\\mathcal{Y} + \\Gamma_\\mathcal{Y}$ is relatively nef over the stratum of $0 \\in B$.\nThis shows that\n\\[\n\\{p \\in B : (K_{\\mathcal{X}\/B} + \\mathcal{D} + \\Delta)|_{{\\mathcal{X}}_p}\\text{ is nef}\\}.\n\\]\nis locally closed and it concludes the proof.\n\\end{proof}\n\n\n\\begin{theorem}\\label{thm_boundedness}\nFix an integer $n\\in \\mathbb{N}$, a finite subset $I\\subseteq (0,1] \\cap \\mathbb{Q}$, and a polynomial $p(t) \\in \\mathbb{Q}[t]$.\nAssume that assumption (A) holds.\nThen, the set of $p$-stable pairs $(X,\\Delta;D)$ of dimension $n$, polynomial $p(t)$ and coefficients in $I$ is effectively log bounded.\n\\end{theorem}\n\\begin{proof}\nWe proceed in several steps.\n\n{\\bf Step 1:} In this step, we show that the $p$-stable pairs of interest are log bounded.\n\nFrom Lemma \\ref{lemma_bound_for_epsilon}, there is an $\\epsilon_0>0$ such that, for every $p$-stable pair $(X,\\Delta;D)$ as in the statement, $(X,(1-\\epsilon_0)D + \\Delta)$ is a stable pair.\nThen, from \\cite{HMX}, there is a bounding family $(\\mathcal{X},\\mathcal{E})\\to B$ of stable pairs of volume $p(1-\\epsilon_0)$, coefficients in the finite set $(1-\\epsilon_0)I\\cup I \\cup \\{1\\}$ and dimension $n$.\n\n{\\bf Step 2:} In this step, we show that the $p$-stable of pairs of interest are strongly log bounded.\nFurthermore, we may choose the family to be locally stable.\n\nSince the set of coefficients involved is finite, up to taking finitely many copies of the family, we may find divisors $\\mathcal{D}$ and $\\Omega$ supported on $\\mathcal{E}$ such that $(1-\\epsilon_0)\\mathcal{D}$ restricts to $(1-\\epsilon_0)D$ fiberwise, and $\\Omega$ restricts to $\\Delta$ fiberwise.\nBy \\cite{kol1}*{Lemma 4.48}, up to replacing $B$ with a finite disjoint union of locally closed subsets, we can further assume that both $(\\mathcal{X},(1-\\epsilon)\\mathcal{D} + \\Omega)\\to B$ and\n$(\\mathcal{X},\\mathcal{D} + \\Omega)\\to B$ are locally stable.\nIn particular, $K_{\\mathcal{X}\/B} + t\\mathcal{D} + \\Omega$ is $\\mathbb{Q}$-Cartier for any $t$.\nFurthermore, by flatness, up to disregarding some irreducible components of $B$, we can assume that for every $b\\in B,$ $p(t) = (K_{\\mathcal{X}\/B} + t\\mathcal{D} + \\Omega)^{\\dim(X)}$.\nFinally, up to stratifying $B$, we may assume that each irreducible component of $B$ is smooth;\nin particular, $K_B$ is well defined, and it follows that the pairs are strongly log bounded. \n\n{\\bf Step 3:} In this step we finish the proof.\n\nBy construction, for the choice of $t = 1-\\epsilon_0$, $K_{\\mathcal{X}\/B} + (1-\\epsilon_0)\\mathcal{D} + \\Omega$ is ample on the general fibers.\nThus, up to removing some proper closed subset of $B$, we may assume that $(\\mathcal{X},(1-\\epsilon_0)\\mathcal{D} + \\Omega)\\to B$ is a stable family.\nThus, to conclude the proof it suffices to use Proposition \\ref{prop:nef:open}, which guarantees that the set $\n\\{p \\in B : (K_{\\mathcal{X}\/B} + \\mathcal{D} + \\Omega)|_{{\\mathcal{X}}_p}\\text{ is nef}\\}$ is open.\n\\end{proof}\n\n\n\n\n\\section{The moduli functor}\n\\label{section functor}\n\n\nThe goal of this section is to prove that $\\mathscr{F}_{n,p,I}$ is an algebraic stack. We begin by the following proposition:\n\n\n\\begin{Prop}\nThe fibered category $\\mathscr{F}_{n,p,I}$ is a stack.\n\\end{Prop}\n\n\\begin{proof}\nSince our argument follows the same strategy in \\cite{Alp21}*{Proposition 1.4.6}, we only sketch the salient steps here.\nThe role that in \\emph{loc. cit.} is the one of $\\omega_\\mathscr{C}^{\\otimes 3}$, for us is $L \\coloneqq \\mathcal{O}_{X}(m(K_X + (1-\\epsilon)D))$, where $\\epsilon$ and $m$ are chosen such that $L$ is very ample with $h^i(X,L)=0$ for $i \\geq 1$ and such that $H^0(X,L)\\to H^0(D^{sc},L|_{D^{sc}})$ is surjective.\nThese $m$ and $\\epsilon$ can be chosen uniformly by Theorem \\ref{thm_boundedness}. \n\nThe fact that isomorphisms are a sheaf in the \\'etale topology of $\\mathscr{F}_{n,p,I}$ follows from descent as in \\cite{Alp21}*{Proposition 1.4.6}.\n\nFor proving that $\\mathscr{F}_{n,p,I}$ satisfies descent, we begin by the following observation.\nConsider an object $f \\colon (\\mathcal{X},\\mathcal{D})\\to B$ of $\\mathscr{F}_{n,p,I}(B)$, and pick $m$ and $\\epsilon$ as before. \nThen, up to replacing $m$ with some uniform multiple, from Remark \\ref{remark_for_m_big_enough_kollar_comndition_guarantees_cartier} and from cohomology and base change, we can assume that $\\mathcal{G} \\coloneqq \\omega_{\\mathcal{X}\/B}^{[m]}\\otimes \\mathcal{I}_\\mathcal{D}^{[-\\frac{m(1-\\epsilon)}{r}]}$ is relatively very ample, and $f_*\\mathcal{G}$ is a vector bundle over $B$.\nIndeed, by the deformation invariance of $\\chi(\\mathcal{X}_b,\\mathcal{G}_b)$ and the vanishing of $H^i(X,L)$ for $i \\geq 1$, it follows that the sections of $H^0(\\mathcal{X}_b,\\mathcal{G}_b)$ are deformation invariant, and hence $f_*\\mathcal{G}$ is a vector bundle.\nThen, on every affine open trivializing $f_*\\mathcal{G}$, the morphism $\\mathcal{X} \\rar \\mathbb{P}(f_*\\mathcal{G})$ can be identified with $\\mathcal{X} \\rar B \\times \\mathbb{P} (H^0(\\mathcal{X}_b,\\mathcal{G}_b))$, and the latter is an embedding as it is an embedding over $B$ fiber by fiber, by Nakayama's lemma.\nThis gives an embedding $\\mathcal{X}\\hookrightarrow \\mathbb{P}(f_*\\mathcal{G})$, and composing it with $\\mathcal{D}\\hookrightarrow \\mathcal{X}$ an embedding $\\mathcal{D}\\hookrightarrow \\mathbb{P}(f_*\\mathcal{G})$.\nNow the proof is analogous to the one in \\cite{Alp21}*{Proposition 1.4.6}.\n\\end{proof}\n\n\n\n\n\n\\begin{theorem}\nFix an integer $n\\in \\mathbb{N}$, a finite subset $I\\subseteq (0,1] \\cap \\mathbb{Q}$, and a polynomial $p(t) \\in \\mathbb{Q}[t]$.\nAssume that assumption (A) holds.\nThen $\\mathscr{F}_{n,p,I}$ is an algebraic stack.\n\\end{theorem}\n\n\\begin{proof}Let $r$ denote the index of $I$.\nWe will proceed in several steps.\n\n\n{\\bf Step 1:} In this step, we fix some invariants and consider a suitable Hilbert scheme parametrizing (among others) the total spaces of the $p$-pairs of interest.\n\nBy Lemma \\ref{lemma_bound_for_epsilon}, we may find a rational number $\\epsilon \\in (0,1)$ such that $K_X+(1-\\epsilon)D$ is ample for every $p$-stable pair $(X;D)$ with polynomial $p(t)$ and coefficients in $I$.\nWithout loss of generality, we may assume that $\\epsilon = \\frac{1}{k}$ for a suitable $k \\in \\mathbb{N}$.\nBy Theorem \\ref{thm_boundedness}, we may consider a family $\\pi \\colon (\\mathcal{X};\\mathcal{D})\\to B$ of $p$-stable pairs with coefficients in $I$ and of dimension $n$ that is effectively log bounding for our moduli problem.\nBy stratification of $B$, we may assume that $B$ is smooth, $\\pi$ and $\\pi|_{\\mathcal{D}}$ are flat, and that $\\pi|_{\\mathcal{D}}$ has $S_1$ fibers.\nFurthermore, by Proposition \\ref{prop_stable_family_p-pairs}, $\\pi$ also induces a family of stable pairs.\nIn particular, we can also regard $\\mathcal{D}$ as a divisor, not only as a subscheme, and we will be free to take its multiples.\nSimilarly, any natural multiple $k\\mathcal{D}$ can be regarded as a subscheme, by considering the vanishing locus of $\\O \\mathcal{X}.(-k\\mathcal{D})$.\nThen, there is $m>0$ such that, for every $p$-stable pair $(X;D)$ in our moduli problem, we have:\n\\begin{enumerate}\n    \\item $mD$ and $m(1-\\epsilon)D$ are Cartier;\n    \\item $mK_X$ is Cartier; and\n    \\item $m(K_X+(1-\\epsilon)D)$ is a very ample line bundle that embeds $X$ into $\\mathbb{P}(H^0(X,\\mathcal{O}_X (m(K_X+(1-\\epsilon)D)))$.\n\\end{enumerate}\nFurthermore, up to taking a multiple, we may assume that $\\frac{m}{r}$ and $\\frac{m(1-\\epsilon)}{r}$ are integers, and\n\\begin{enumerate}\n    \\item $\\frac{m}{r} \\mathcal{D}$ and $\\frac{m(1-\\epsilon)}{r} \\mathcal{D}$ are Cartier;\n    \\item $mK_{\\mathcal{X}\/B}$ is Cartier; and\n    \\item $m(K_{\\mathcal{X}\/B}+\\frac{1-\\epsilon}{r}\\mathcal{D})$ is a relatively very ample line bundle.\n\\end{enumerate}\n\nBy boundedness and upper semi-continuity of the space of global sections, $h^0(X,\\mathcal{O}_X (m(K_X+(1-\\epsilon)D))$ attains finitely many values.\nThen, we have finitely many polynomials $q_1,\\ldots,q_l$ such that the fibers of $\\pi$ have Hilbert polynomial $q_i$ for some $i$, for the relatively ample line bundle $\\mathcal{O}_\\mathcal{X}(m(\\K \\mathcal{X}\/B.+\\frac{1-\\epsilon}{r}\\mathcal{D}))$.\nWe consider a union of Hilbert schemes for the polynomials $q_i$, and we denote such a union with $\\mathscr{H}_0$.\nOver $\\mathscr{H}_0$, we have a universal family $f_0 \\colon \\mathscr{X}\\subseteq \\mathscr{H}_0\\times \\mathbb{P}^N\\to \\mathscr{H}_0$, where the fibers are closed subschemes of $\\mathbb{P}^N$ with Hilbert polynomial $q_i$ for some $i$.\nHere, we observe that $N$ may actually attain finitely many distinct values, as we are assuming that each $X$ is embedded with a full linear series; on the other hand, we will work on one Hilbert scheme at the time, thus, by abusing notation, we will simply write $N$.\n\n{\\bf Step 2:} In this step, we highlight the strategy for the construction of the moduli functor.\n\nWe will construct our moduli functor as a subfunctor of a suitable relative Hilbert scheme, modulo the action of $\\mathrm{PGL}_{N+1}$.\nFor this reason, we will shrink $\\mathscr{H}_0$ to cut the locus of interest for our moduli problem.\nIn doing so, we have to guarantee that this locus is locally closed and has a well-defined scheme structure.\nIf we shrink to an open subset, there is no ambiguity in the scheme structure.\nOn the other hand, if we need to consider a closed or locally closed subset, we need to show this choice has a well-defined scheme structure, which will be functorial in nature.\nFinally, we need to guarantee that, at each step, the locus we consider is invariant under the action of $\\mathrm{PGL}_{N+1}$.\n\n{\\bf Step 3:} In this step, we cut the locus parametrizing demi-normal schemes.\n\nSince being $S_2$ is an open condition for flat and proper families \\cite{EGAIV}*{Theorem 12.2.1}, and since small deformations of nodes are either nodes or regular points, up to shrinking $\\mathscr{H}_0$ we may assume that the fibers of $f_0$ are $S_2$ and nodal in codimension one.\nThat is, the fibers are demi-normal.\nSince being demi-normal is independent of the embedding into projective space, this locus is stable under the action of $\\mathrm{PGL}_{N+1}$.\n\n{\\bf Step 4:} In this step, we cut the locus parametrizing varieties embedded with a full linear series.\n\nLet $(X;D)$ be a $p$-pair in our moduli problem, and let $I_X$ denote its ideal sheaf in $\\pr N.$.\nThen, by assumption we have $\\O {\\mathbb P ^N}.(1)|_X \\cong \\O X. (m(\\K X. + (1-\\epsilon)D))$ and the higher cohomologies of both sheaves vanish.\nThus, if we consider the short exact sequence\n\\[\n0 \\rar I_X \\otimes \\O {\\mathbb P ^N}.(1) \\rar \\O {\\mathbb P ^N}.(1) \\rar \\O {\\mathbb P ^N}.(1)|_X \\rar 0,\n\\]\nit provides the following long exact sequence of cohomology groups\n\\[\n0 \\rar H^0(\\pr N.,I_X \\otimes \\O {\\mathbb P ^N}.(1)) \\rar H^0(\\pr N.,\\O {\\mathbb P ^N}.(1)) \\rar H^0(X,\\O {\\mathbb P ^N}.(1)|_X) \\rar H^1(\\pr N.,I_X \\otimes \\O {\\mathbb P ^N}.(1))=0,\n\\]\nwhere the vanishing of $H^1(\\pr N.,I_X \\otimes \\O {\\mathbb P ^N}.(1))$ follows from the surjectivity of the map $H^0(\\pr N.,\\O {\\mathbb P ^N}.(1)) \\rar H^0(X,\\O {\\mathbb P ^N}.(1)|_X)$ and the vanishing of the higher order cohomologies of $\\O {\\mathbb P ^N}.(1)|_X$.\nBy definition of Hilbert scheme, $\\mathscr{X} \\rar \\mathscr{H}_0$ is flat, and the subschemes parametrized correspond to a flat quotient sheaf of $\\O \\mathscr{X}.$.\nThus, as we have a short exact sequence of sheaves where the last two terms are flat over the base, then so is the first term, which is the family of ideal sheaves of the subschemes of interest.\nThus, upper semi-continuity of the dimension of cohomology groups applies, and we may shrink $\\mathscr{H} _0$ to the open locus parametrizing varieties $Y$ with $H^1(\\pr N.,I_Y \\otimes \\O {\\mathbb P ^N}.(1))=0$.\nThis guarantees that, for every such variety $Y$, any automorphism of $Y$ preserving $\\O {\\mathbb P ^N}.(1)|_Y$ is induced by an automorphism of $\\pr N.$.\nSince the vanishing of $H^1(\\pr N.,I_Y \\otimes \\O {\\mathbb P ^N}.(1))=0$ is invariant under the action of $\\mathrm{PGL}_{N+1}$, then so is the locus we cut.\n\n{\\bf Step 5:} In this step, we introduce a relative Hilbert scheme, in order to parametrize the boundaries of the $p$-pairs of interest.\n\nProceeding as in Step 1, for every $p$-pair $(X;D)$ in our moduli problem, we may consider the Hilbert polynomial of $rD$ with respect to $\\mathcal{O}_X(m(\\K X.+(1-\\epsilon)D))$.\nHere, recall that $\\mathcal{D}$ corresponds to $rD$ fiberwise, hence the choice of Hilbert polynomial for $rD$ rather than for $D$.\nBy effective log boundedness and generic flatness of $\\mathcal{D}$, there exist finitely many such polynomials.\nAs before, we will deal with one Hilbert polynomial at the time, and omit this choice from the notation.\n\nNow, $f$ is projective over $\\mathscr{H}_0$.\nIn particular, if we pull back the ample line bundle $\\mathcal{O}_{\\mathscr{H}_0\\times \\mathbb{P}^N}(1)$ to get a relatively very ample line bundle $\\mathscr{G}$ on $\\mathscr{X}$, we can take the relative Hilbert scheme for the morphism $f$, the line bundle $\\mathscr{G}$, and the polynomial determined by $\\mathcal{D}$ (see \\cite{ACH11}*{Chapter IX}).\nThis gives a scheme $\\mathscr{H}_1\\to \\mathscr{H}_0$, together with an universal family $\\mathscr{D}\\subseteq \\mathscr{X}_1 \\coloneqq \\mathscr{X}\\times_{\\mathscr{H}_0} \\mathscr{H}_1$.\nThen, as $\\mathscr{G}$ corresponds to $\\mathcal{O}_X(m(\\K X.+(1-\\epsilon)D))$ on the elements of our moduli problem, every $p$-pair of interest appears as a fiber of this family.\n\n\n\n{\\bf Step 6:} In this step, we shrink $\\mathscr{H}_1$ to an open subset such that $\\mathscr{D} \\cap \\mathrm{Sing}(\\mathscr{X}_1 \\rar \\mathscr{H}_1)$ has codimension at least 2 along each fiber and such that the ideal sheaf $\\mathscr{I}_\\mathscr{D}$ of $\\mathscr{D}$ is relatively $S_2$.\n\nFor every $p$-pair $(X;D)$, $\\operatorname{Supp}(D)$ does not contain any component of the double locus of $X$.\nThus, by upper semi-continuity of the dimension of the fibers of a morphism, we may shrink $\\mathscr{H}_1$ to an open subset such that $\\mathscr{D} \\cap \\mathrm{Sing}(\\mathscr{X}_1 \\rar \\mathscr{H}_1)$ has codimension at least 2 along each fiber.\n\nNow, by Step 3, all the fibers of $\\mathscr{X}_1 \\rar \\mathscr{H}_1$ are demi-normal.\nThus, we may find an open subset $V \\subset \\mathscr{X}_1$ such that the following properties hold:\n\\begin{enumerate}\n    \\item for every $s \\in \\mathscr{H}_1$, $V_s$ is a big open set in $\\mathscr{X}_{1,p}$;\n    \\item $\\mathscr{D} \\cap \\mathrm{Sing}(\\mathscr{X}_1 \\rar \\mathscr{H}_1) \\subset \\mathscr{X}_1 \\setminus V$; and\n    \\item the fibers of $V \\rar \\mathscr{H}_1$ have at worst nodal singularities.\n\\end{enumerate}\nThen, $V \\rar \\mathscr{H}_1$ is a Gorenstein morphism, so, by \\cite{stacks-project}*{Tag 0C08}, $\\omega_{\\mathscr{X}_1\/\\mathscr{H}_1}$ is an invertible sheaf along $V$.\nFurthermore, by \\cite{stacks-project}*{Tag 062Y}, the ideal sheaf $\\mathscr{I}_\\mathscr{D}$ is locally free along $V$.\nFinally, up to shrinking $\\mathscr{H}_1$, we may assume that $\\mathscr{I}_\\mathscr{D}$ is relatively $S_2$.\n\nIndeed, $\\mathscr{X}_1$ is flat over $\\mathscr{H}_1$, as $\\mathscr{X}$ is flat over $\\mathscr{H}_0$, and $\\mathscr{D}$ is flat over $\\mathscr{H}_1$ by definition of relative Hilbert scheme.\nThus, $\\mathscr{I}_\\mathscr{D}$ is flat over $\\mathscr{H}_1$, as it is the kernel of a surjection of flat sheaves.\nThen, we conclude, as being $S_2$ is an open condition for flat and proper families \\cite{EGAIV}*{Theorem 12.2.1}.\nNotice that, by \\cite{HK04}*{Proposition 3.5}, we have $\\mathscr{I}_\\mathscr{D}=\\mathscr{I}_\\mathscr{D}^{[1]}$.\n\nNotice that in this step we shrank $\\mathscr{H}_1$ twice, and both times the process is invariant under the natural action of $\\mathrm{PGL}_{N+1}$, as the locus is characterized by properties of the fibers.\n\n{\\bf Step 7:} In this step, we cut $\\mathscr{H}_1$ to a locally closed subset $\\mathscr{H}_2$ to ensure that $\\mathscr{I}_\\mathscr{D}^{[n]}$ and $\\omega_{\\mathscr{X}_1\/\\mathscr{H}_1}^{[n]}$ are flat and $S_2$ over $\\mathscr{H}_1$ for $0 \\leq n \\leq m$, and that $\\mathscr{I}_\\mathscr{D}^{[\\frac{m(1-\\epsilon)}{r}]}$, $\\mathscr{I}_\\mathscr{D}^{[\\frac{m}{r}]}$, and $\\omega_{\\mathscr{X}_1\/sH_1}^{[m]}$ are invertible sheaves.\n\nNow, we consider the sheaves $\\omega_{\\mathscr{X}_1\/\\mathscr{H}_1}^{[n]}$ and $\\mathscr{I}_\\mathscr{D}^{[n]}$ for every $0\\le n \\le m$.\nThese are defined as \n$i_*(\\omega_{\\mathscr{X}_1 \/ \\mathscr{H}_1}|_V^{\\otimes n})$ and\n$i_*(\\mathscr{I}_\\mathscr{D}|_V^{\\otimes n})$, respectively, where $V$ is the open set defined in Step 6 and $i \\colon V\\to \\mathscr{X}_1$ is the natural inclusion.\n\nThere is a stratification into functorial locally closed subsets of $\\mathscr{H}_1$, which we denote by $C_i\\subseteq \\mathscr{H}_1$, where the above sheaves are flat over $C_i$ (see \\cite{Mum16}*{Lecture 8}).\nNotice that, as the stratification is functorial, it is respected by the natural action of $\\mathrm{PGL}_{N+1}$.\nThen, as at the end of Step 6, up to replacing the $C_i$ with an open subset, we may assume that the above sheaves are relatively $S_2$.\nHence, the above sheaves are reflexive by \\cite{HK04}*{Proposition 3.5}.\nAgain by \\cite{HK04}*{Corollary 3.8} and the existence of the big open set $V$, the formation of the above sheaves commutes with base change for every $0 \\leq n \\leq m$.\nFurthermore, by Remark \\ref{remark_for_m_big_enough_kollar_comndition_guarantees_cartier}, the sheaves $\\mathscr{I}_\\mathscr{D}^{[\\frac{m(1-\\epsilon)}{r}]}$, $\\mathscr{I}_\\mathscr{D}^{[\\frac{m}{r}]}$,  and $\\omega_{\\mathscr{X}_1\/sH_1}^{[m]}$ are invertible over each component $C_i$.\n\n\nConsidering the union of the $C_i$'s produces another base $\\mathscr{H}_2$, with a family $f_2 \\colon \\mathscr{X}_2 \\to \\mathscr{H}_2$ and a closed subset $\\mathscr{D}_2 \\subset \\mathscr{X}_2$ as the one over $\\mathscr{H}_1$, but such that $\\mathscr{I}_{\\mathscr{D}_2}^{[\\frac{m(1-\\epsilon)}{r}]}$, $\\mathscr{I}_{\\mathscr{D}_2}^{[\\frac{m}{r}]}$, and $\\omega_{\\mathscr{X}_2\/\\mathscr{H}_2}^{[m]}$ are Cartier and the formation of $\\mathscr{I}_{\\mathscr{D}_2}^{[n]}$ and $\\omega_{\\mathscr{X}_2\/\\mathscr{H}_2}^{[n]}$ commutes with base change for every $0\\leq n \\leq m$.\nThen, the formation of $\\mathscr{I}_{\\mathscr{D}_2}^{[n]}$ and $\\omega_{\\mathscr{X}_2\/\\mathscr{H}_2}^{[n]}$ commutes with base change for every $n$, since we can write $n=km+b$ for $0\\le b <m$, and we have\n$\\mathscr{I}_{\\mathscr{D}_2}^{[n]}=\\mathscr{I}_{\\mathscr{D}_2}^{[b]} \\otimes (\\mathscr{I}_{\\mathscr{D}_2}^{[\\frac{m}{r}]})^{\\otimes kr}$ and $\\omega_{\\mathscr{X}_2\/\\mathscr{H}_2}^{[n]}=\\omega_{\\mathscr{X}_2\/\\mathscr{H}_2}^{[b]} \\otimes (\\omega_{\\mathscr{X}_2\/\\mathscr{H}_2}^{[m]})^{\\otimes k}$.\n\n{\\bf Step 8:} In this step, we shrink $\\mathscr{H}_2$ to an open subset parametrizing semi-log canonical pairs.\n\n\nBy the reductions in the previous steps, for every $k \\geq 1$, the ideal sheaf $\\mathscr{I}_{\\mathscr{D}_2}^{[k]}$ determines a family of generically Cartier divisors, in the sense of \\cite{kol1}*{Definition 4.25}.\nIndeed, this is obtained by dualizing the inclusion $\\mathscr{I}_{\\mathscr{D}_2}^{[k]} \\rar \\O \\mathscr{X}_2.$, as observed in \\cite{kol1}*{Definition 4.25}.\nThen, as observed in \\cite{kol1}*{4.25.5}, a family of generically Cartier divisors induces a well-defined family of divisors.\nAbusing notation, we denote by $\\mathscr{D}_2$ the family of divisors corresponding to $\\mathscr{I}_{\\mathscr{D}_2}$.\n\nNow, recall that taking the reduced structure does not change the topology.\nThus, in the remainder of this step, we may assume that $\\mathscr{H}_2$ is reduced.\nThen, $(\\mathscr{X}_2,\\frac{1}{r}\\mathscr{D}_2) \\rar \\mathscr{H}_2$ is a well defined family of pairs, and by \\cite{kol1}*{Corollary 4.49}, there is an open locus where the fibers of $(\\mathscr{X}_2,\\frac{1}{r}\\mathscr{D}_2) \\rar \\mathscr{H}_2$ are semi-log canonical.\nSince we shrunk $\\mathscr{H}_2$ according to a property of the fibers, this respects the natural action of $\\mathrm{PGL}_{N+1}$.\nAlso, since we took an open subset of $\\mathscr{H}_2$, this choice is not affected by considering the reduced structure of $\\mathscr{H}_2$.\n\n{\\bf Step 9:} In this step, we shrink $\\mathscr{H}_2$ to an open subset parametrizing the $p$-stable pairs in our moduli problem.\n\nRecall that taking the reduced structure does not change the topology.\nThus, in this step, we may take the reduced structure of $\\mathscr{H}_2$ if necessary.\nThen, by Proposition \\ref{prop:nef:open}, up to shrinking $\\mathscr{H}_2$ and possibly discarding some components, we can assume that $\\omega_{\\mathscr{X}_2\/\\mathscr{H}_2}^{[m]}\\otimes \\mathscr{I}_{\\mathscr{D}_2}^{[-\\frac{m}{r}]}$ is nef along each fiber.\nSimilarly, since being ample is an open condition, we can assume that $\\omega_{\\mathscr{X}_2\/\\mathscr{H}_2}^{[m]} \\otimes \\mathscr{I}_{\\mathscr{D}_2}^{[-\\frac{m(1-\\epsilon)}{r}]}$ is relatively ample.\nNow, recall that we have chosen $\\epsilon = \\frac{1}{k}$ for a chosen fixed $k \\in \\mathbb N$.\n\nNotice that, the two former conditions guarantee that, for every $j \\geq k$, we have that $\\omega_{\\mathscr{X}_2\/\\mathscr{H}_2}^{[rjm]} \\otimes \\mathscr{I}_{\\mathscr{D}_2}^{[-m(j-1)]}$ is relatively ample, as we have $mj > m(j-1) \\geq mj(1-\\frac{1}{k})$.\nThen, we may fix $n+1$ such values $j_1,\\ldots,j_{n+1}$ and disregard all components of $\\mathscr{H}_2$ but the ones where, over $s \\in \\mathscr{H}_2$, $(\\omega_{\\mathscr{X}_2\/\\mathscr{H}_2}^{[rj_im]} \\otimes \\mathscr{I}_{\\mathscr{D}_2}^{[-m(j_i-1)]})^n$ has prescribed value.\nBy flatness, these self-intersections are locally constant, and thus this condition is open.\nFor each $i$, the self-intersection is prescribed by $p(1-\\frac{1}{j_i})$, up to the rescaling factor given by $rj_im$.\nSince we are prescribing $n+1$ values of a polynomial of degree $n$, this guarantees that all the fibers correspond to $p$-stable pairs $(X;D)$ with $(K_X+tD)^n=p(t)$.\n\nAs in the previous steps, we shrunk $\\mathscr{H}_2$ according to properties of the fibers of $\\mathscr{X}_2 \\rar \\mathscr{H}_2$, thus this preserves the natural action of $\\mathrm{PGL}_{N+1}$.\nAlso, since we took an open subset of $\\mathscr{H}_2$, this choice is not affected by considering the reduced structure of $\\mathscr{H}_2$.\n\n\n{\\bf Step 10:} In this step, we cut $\\mathscr{H}_2$ to a closed subset $\\mathscr{H}_3$ to ensure that the natural polarization $\\O \\mathscr{H}_3 \\times \\mathbb{P}^N.(1)$ coincides with $\\omega_{\\mathscr{X}_3\/\\mathscr{H}_3}^{[m]} \\otimes \\mathscr{I}_{\\mathscr{D}_3}^{[-\\frac{m(1-\\epsilon)}{r}]}$.\n\n\nBy construction, for every $p$-stable pair $(X;D)$ in our moduli problem, we have $\\omega_{\\mathscr{X}_3\/\\mathscr{H}_3}^{[m]} \\otimes \\mathscr{I}_3^{-[\\frac{m(1-\\epsilon)}{r}]}|_X \\sim \\O X. (m(\\K X. + (1-\\epsilon)D)) \\sim \\O \\mathbb P ^N .(1)|_X$.\nSince $\\omega_{\\mathscr{X}_3\/\\mathscr{H}_3}^{[m]} \\otimes \\mathscr{I}_3^{-[\\frac{m(1-\\epsilon)}{r}]}$ is a line bundle and the natural polarization of $\\mathscr{H}_2$ coming from the original choice of Hilbert scheme restricts to $\\O \\mathbb P ^ N. (1)$ fiberwise, by \\cite{Vie95}*{Lemma 1.19} there is a locally closed subscheme $\\mathscr{H}_3$ where $\\omega_{\\mathscr{X}_3\/\\mathscr{H}_3}^{[m]} \\otimes \\mathscr{I}_3^{-[\\frac{m(1-\\epsilon)}{r}]}$ is linearly equivalent to the natural polarization of $\\mathscr{X}_2 \\rar \\mathscr{H}_2$ over $\\mathscr{H}_3$.\nFurthermore, this subscheme is functorial in nature, it is preserved by the natural action of $\\mathrm{PGL}_{N+1}$.\n\n{\\bf Step 11:} In this step, we show that there is an isomorphism $\\mathscr{F}_{n,p,I}\\cong [\\mathscr{H}_3\/\\operatorname{PGL}_{N+1}] $.\n\nOur argument follows closely \\cite{Alp21}*{Theorem 2.1.11}.\nFirst, observe that from its construction, over $\\mathscr{H}_3$ there is a stable family of $p$-stable pairs.\nThis gives a morphism $\\mathscr{H}_3 \\to \\mathscr{F}_{n,p,I}$, and if we forget the embedding into $\\mathbb{P}^N$, this descends to a morphism $\\Phi^{pre} \\colon [\\mathscr{H}_3\/\\operatorname{PGL}_{N+1}]^{pre} \\to \\mathscr{F}_{n,p,I}$, where the superscript pre stands for prestack (see \\cite{Alp21}*{Definition 1.3.12}).\nThis induces a map $\\Phi \\colon [\\mathscr{H}_3\/\\operatorname{PGL}_{N+1}] \\to \\mathscr{F}_{n,p,I}$, which we now show is an isomorphism.\n\nTo show it is fully faithful, as in \\cite{Alp21}, it suffices to check that $\\Phi^{pre}$ is fully faithful.\nBut $\\Phi^{pre}$ is fully faithful since any isomorphism between two families of $p$-stable\npairs $\\pi \\colon (\\mathscr{Y};\\mathscr{D})\\to B$ and $\\pi \\colon (\\mathscr{Y}',\\mathscr{D}')\\to B$ over $B$ sends $\\mathscr{L} \\coloneqq \\omega_{\\mathscr{Y}\/B}^{[m]} \\otimes \\mathscr{I}_{\\mathscr{Y}}^{-[\\frac{m(1-\\epsilon)}{r}]}$ to $\\mathscr{L}' \\coloneqq\\omega_{\\mathscr{Y}'\/B}^{[m]} \\otimes \\mathscr{I}_{\\mathscr{Y}'}^{-[\\frac{m(1-\\epsilon)}{r}]}$, where we denoted by $ \\mathscr{I}_{\\mathscr{Y}}$ (resp. $\\mathscr{I}_{\\mathscr{Y}'}$) the ideal sheaves of $\\mathscr{D}$ (res. $\\mathscr{D}'$) in $\\mathscr{Y}$ (resp. $\\mathscr{Y}'$). This induces an unique isomorphism $\\mathbb{P}(\\pi_*\\mathscr{L} )\\cong \\mathbb{P}(\\pi'_*\\mathscr{L}')$ which sends $\\mathscr{Y}$ to $\\mathscr{Y}'$.\n\nSince $\\Phi$ is a morphism of stacks, also essential surjectivity can be checked locally on $B$.\nIn particular, it suffices to check that if $\\pi \\colon (\\mathscr{Y};\\mathscr{D})\\to B$ is a family of $p$-stable pairs such that $\\pi_*\\mathscr{L}$ is free, then the morphism $B\\to \\mathscr{F}_{n,p,I}$ lifts to a morphism $B\\to \\mathscr{H}_3$.\nThis follows since if we pick an isomorphism $\\mathbb{P} (\\pi_*\\mathscr{L})\\cong \\mathbb{P}^N \\times B$ then\n$\\mathscr{Y}, \\mathscr{D}\\subseteq \\mathbb{P} (\\pi_*\\mathscr{L}) = \\mathbb{P}^N\\times B$, and then from the functorial properties of $\\mathscr{H}_3$ it induces a morphism $B\\to \\mathscr{H}_3$.\n\\end{proof}\n\\begin{Remark}\nObserve that the stack $\\mathscr{F}_{n,p,I}$ is in fact Deligne--Mumford.\nIndeed, since we are working over a field of characteristic 0, it suffices to show that the automorphisms of the objects over the points are finite. But this follows since an automorphism of a $p$-pair $(X;D)$ induces an automorphism of the stable pair $(X,(1-\\epsilon)D)$, and those are finite from \\cite{KP17}*{Proposition 5.5}.\n\\end{Remark}\n\n\\section{Properness of $\\mathscr{F}_{n,p,I}$}\\label{section_properness}\nThe goal of this section is to prove that $\\mathscr{F}_{n,p,I}$ is proper.\nIn particular, since in the definition of a $p$-stable pair $(X;D)$ there are prescribed conditions on the scheme-theoretic structure of $D$.\nThen, when proving that a moduli functor for $p$-pairs satisfies the valuative criterion for properness, one needs to check that these scheme-theoretic properties are preserved.\nIt is convenient to check that the flat limit of $D^{sc}$ (recall that $D^{sc}$ was introduced in Notation \\ref{notation_Dsc}) is $S_1$.\nThis will be the content of the next proposition.\n\n\\begin{Prop}\\label{prop:flat:limit:is:S1}\nLet $\\operatorname{Spec}(R)$ be the spectrum of a DVR with generic point $\\eta$ and closed point $p$.\nConsider a locally stable family $(X,D)\\to \\operatorname{Spec}(R)$ such that $D$ is $\\mathbb{Q}$-Cartier.\nThen, for every $m\\in \\mathbb{N}$, the ideal sheaf $\\mathcal{O}_X(-mD)$ is $S_3$ on every point $x\\in X_p$.\nIn particular:\n\\begin{enumerate}\n    \\item the restriction $\\mathcal{O}_X(-mD)|_{X_p}$ is $S_2$; and\n    \\item if we denote by $mD$ the closed subscheme of $X$ with ideal sheaf $\\mathcal{O}_X(-mD)$, then $(mD)|_{X_p}$ is $S_1$.\n\\end{enumerate}\n\\end{Prop}\n\n\\begin{proof}\nFirst, observe that since $X\\to \\operatorname{Spec}(R)$ is locally stable, $x \\in X_p$ cannot be a log canonical center for $X$ (see \\cite{kol1}*{Proposition 2.13}).\nThe statement is local, so up to shrinking $X$, we can assume that $X$ is affine, and, since $D$ is $\\mathbb{Q}$-Cartier, $\\mathcal{O}_X(m_0D) \\cong \\mathcal{O}_X$ for a certain $m_0 \\in \\mathbb{N}$.\nThen, for every $m \\in \\mathbb{N}$, $mD\\sim_\\bQ0$.\nIn particular, if we apply \\cite{Kol13}*{Theorem 7.20} where, with the notations of \\emph{loc. cit.}, we take $\\Delta'=0$, we conclude that $\\mathcal{O}_X(-mD)$ is $S_3$.\n\nNow, we denote by $\\pi$ the pull-back to $X$ of a uniformizer on $\\operatorname{Spec}(R)$.\nSince $\\mathcal{O}_X(-mD)$ is a subsheaf of $\\mathcal{O}_X$ and $\\pi$ is not a zero divisor on $\\mathcal{O}_X$, it is not a zero divisor on $\\mathcal{O}_X(-mD)$.\nThen, \\begin{center}$(\\mathcal{O}_X(-mD))|_{X_p}$ is $S_2$ by \\cite{KM98}*{Proposition 5.3}.\\end{center}\n\nNotice that $\\mathcal{O}_X$ is $S_3$ since it is a flat and proper family of $S_2$ schemes over a smooth base.\nThen, by \\cite{Kol13}*{Corollary 2.61}, $\\mathcal{O}_{mD}$ is $S_2$.\nIn particular, it is $S_1$, so its generic points are the only associated points.\nHence, if we denote by $mD$ the closed subscheme of $X$ with ideal sheaf $\\mathcal{O}_X(-mD)$, then $mD\\to \\operatorname{Spec}(R)$ is flat.\nThen, if we pull back the exact sequence\n\\[\n0\\to \\mathcal{O}_X(-mD)\\to \\mathcal{O}_X\\to \\mathcal{O}_{mD}\\to 0\n\\]\nto $X_p$, the sequence remains exact and we get\n\\[\n0\\to (\\mathcal{O}_X(-mD))|_{X_p}\\to (\\mathcal{O}_X)|_{X_p}\\to (\\mathcal{O}_{mD})|_{X_p}\\to 0.\n\\]\nThe desired result follows again by \\cite{Kol13}*{Corollary 2.61}.\n\\end{proof}\n\n\\begin{Prop} \\label{properness_lc_case}\nFix an integer $n\\in \\mathbb{N}$, a finite subset $I\\subseteq (0,1] \\cap \\mathbb{Q}$, and a polynomial $p(t) \\in \\mathbb{Q}[t]$.\nAssume that $I$ is closed under sum: that is, if $a,b \\in I$ and $a+b \\leq 1$, then $a+b \\in I$.\nAssume that assumption (A) holds.\nLet $C$ be a smooth affine curve, let $0 \\in C$ be a distinguished closed point, and let $U\\coloneqq C \\setminus \\{ 0 \\}$.\nLet $(\\mathcal{X}_U, \\Delta_U; \\mathcal{D}_U) \\rightarrow U$ be a $p$-stable morphism of relative dimension $n$, polynomial $p(t)$, coefficients in $I$, and constant part $\\Delta_U$, which is reduced.\nFurther, assume that the geometric generic fiber is normal.\nThen, up to a finite base change $B \\rar C$, the family can be filled with a unique $p$-stable pair of dimension $n$, with polynomial $p(t)$ and coefficients in $I$.\n\\end{Prop}\n\n\\begin{proof}\nWe proceed in several steps.\nIn the following, $r$ will denote the index of $I$.\n\n{\\bf Step 1:} In this step, we show that the family of pairs $(\\mathcal{X}_U,\\frac{1}{r}\\mathcal{D}_U + \\Delta_U) \\rar U$ admits a relative ample model $(\\mathcal{Y}_U, \\frac{1}{r}\\mathcal{D} _{U,\\mathcal{Y}} + \\Delta_{U, \\mathcal{Y}})$ over $U$.\n\nBy Proposition \\ref{prop_stable_family_p-pairs} and the fact that $C$ is smooth, it follows that $(\\mathcal{X} _U, \\frac{1}{r}\\mathcal{D}_U + \\Delta_U)$ is a pair.\nFurthermore, by inversion of adjunction, this pair is log canonical.\nWe need to argue that $(\\mathcal{X} _U, \\frac{1}{r}\\mathcal{D}_U + \\Delta_U)$ admits a morphism to its relative ample model $(\\mathcal{Y}_U, \\frac{1}{r}\\mathcal{D} _{U,\\mathcal{Y}} + \\Delta_{U, \\mathcal{Y}})$ over $U$.\nIf we assume case (ii) of assumption (A), this is clear.\nSo, assume that $n \\leq 3$.\nThen, the claim follows by Lemma \\ref{lemma minimal models}.\n\n{\\bf Step 2:} In this step, we show that, up to a base change, $(\\mathcal{Y}_U, \\frac{1}{r}\\mathcal{D} _{U,\\mathcal{Y}} + \\Delta_{U,\\mathcal{Y}}) \\rar U$ can be compactified to a family of stable pairs.\nFurthermore, this model is dominated by a $\\qq$-factorial log canonical model.\n\nBy construction, $(\\mathcal{Y}_U, \\frac{1}{r}\\mathcal{D} _{U,\\mathcal{Y}} + \\Delta_{U,\\mathcal{Y}}) \\rar U$ is a family of stable pairs.\nThen, by \\cite{HX13}*{Corollary 1.5}, up to a finite base change, which we omit in the notation, we may assume that $(\\mathcal{Y} _U, \\frac{1}{r}\\mathcal{D} _{U,\\mathcal{Y}} + \\Delta_{U,\\mathcal{Y}}) \\rightarrow U$ admits a compactification $(\\mathcal{Y}, \\frac{1}{r}\\mathcal{D} _{\\mathcal{Y}} + \\Delta_{\\mathcal{Y}}) \\rightarrow C$ to a stable family.\nFurthermore, arguing as in \\cite{HX13}*{proof of Corollary 1.4}, by semi-stable reduction, we\nmay assume that $(\\mathcal{Y}, \\frac{1}{r}\\mathcal{D} _{\\mathcal{Y}} + \\Delta_{\\mathcal{Y}}+\\mathcal{Y}_0) \\rightarrow C$ admits a $\\mathbb{Q}$-factorial log canonical model $(\\mathcal{X}', \\frac{1}{r}\\mathcal{D}' + \\Delta'+\\mathcal{X}'_0) \\rightarrow (\\mathcal{Y}, \\frac{1}{r}\\mathcal{D} _{\\mathcal{Y}} + \\Delta_{\\mathcal{Y}}+\\mathcal{Y}_0)$, where the special fiber $\\mathcal{X}'_0$ is reduced, and such that $K_{\\mathcal{X}'}+\\frac{1}{r}\\mathcal{D}' + \\Delta'+\\mathcal{X}'_0$ is the pull-back of $K_{\\mathcal{Y}}+ \\frac{1}{r}\\mathcal{D} _{\\mathcal{Y}} + \\Delta_{\\mathcal{Y}}+\\mathcal{Y}_0$, where $\\Delta'$ is the proper transform of $\\Delta_\\mathcal{Y}$.\nFurthermore, by increasing the coefficients of the divisors of log discrepancy 1 on the initial semi-stable model, we may assume that all the divisors extracted in $\\mathcal{X}'$ have log discrepancy in $[0,1)$ with respect to $(\\mathcal{Y},\\frac{1}{r}\\mathcal{D}_\\mathcal{Y} + \\Delta_\\mathcal{Y})$.\nIn particular, all the exceptional divisors that are horizontal over $U$ appear in the support of $\\frac{1}{r}\\mathcal{D}'$.\n\n{\\bf Step 3:} In this step, we show that we may choose the $\\qq$-factorial log canonical model $(\\mathcal{X}', \\frac{1}{r}\\mathcal{D}' + \\Delta'+\\mathcal{X}'_0) \\rightarrow (\\mathcal{Y}, \\frac{1}{r}\\mathcal{D} _{\\mathcal{Y}} + \\Delta_{\\mathcal{Y}}+\\mathcal{Y}_0)$ so that $\\mathcal{X}'_U \\dashrightarrow \\mathcal{X}_U$ is a rational contraction over $U$.\n\n\nSince $K_{\\mathcal{X}_U}+\\frac{1-\\epsilon}{r}\\mathcal{D}_U + \\Delta_U$ is ample over $U$ for $0 < \\epsilon \\ll 1$, the divisors contracted by $\\mathcal{X}_U \\rightarrow \\mathcal{Y}_U$ are contained in the support of $\\mathcal{D}$.\nIn particular, they have log discrepancy in $[0,1)$.\nThen, from \\cite{Mor19}*{Theorem 1}, we can produce a morphism $\\mathcal{X}''\\to  \\mathcal{X}'$ extracting all the exceptional divisors of $\\pi \\colon \\mathcal{X}_U \\rightarrow \\mathcal{Y}_U$.\nObserve that $\\mathcal{X}''$ remains $\\mathbb{Q}$-factorial: by \\cite{Mor19}*{Theorem 1}, all the extracted divisors are $\\mathbb{Q}$-Cartier.\nIf a divisor is not exceptional for $\\pi$, we can write it as the difference of its push-pull to $\\mathcal{X}'$ and the exceptional divisors for $\\pi$.\nAll the divisors extracted dominate $U$, as their generic point is disjoint from the special fiber over 0.\nThus, the irreducible components of the special fiber remain unchanged, and, in particular, it is still reduced.\nThe claim then follows by replacing $\\mathcal{X}' $ with $\\mathcal{X}''$.\n\n\nFor clarity, we can represent the morphisms so far constructed with the following diagram\n\\[\n\\begin{tikzcd}\n\t& {\\mathcal{X}'_U} & {\\mathcal{X}'} \\\\\n\t{\\mathcal{X}_U} & {\\mathcal{Y}_U} & {\\mathcal{Y}} \\\\\n\tU && C\n\t\\arrow[hook, from=3-1, to=3-3]\n\t\\arrow[hook, from=2-2, to=2-3]\n\t\\arrow[from=2-2, to=3-1]\n\t\\arrow[from=2-3, to=3-3]\n\t\\arrow[from=1-3, to=2-3]\n\t\\arrow[hook, from=1-2, to=1-3]\n\t\\arrow[from=1-2, to=2-2]\n\t\\arrow[from=2-1, to=3-1]\n\t\\arrow[from=2-1, to=2-2]\n\t\\arrow[dashed, from=1-2, to=2-1]\n\\end{tikzcd}\n\\]\n\n{\\bf Step 4:} In this step, we construct a new model $\\mathcal{X}$, which will be then showed to be the sought compactification of $\\mathcal{X}_U$.\n\nSince $\\mathcal{X}'_U \\dashrightarrow \\mathcal{X}_U$ is a birational contraction and $\\mathcal{D}_U$ is $\\mathbb{Q}$-Cartier, the rational pull-back of $\\mathcal{D}_U$ to $\\mathcal{X}'_U$ is well defined.\nWe denote by $\\mathcal{B}'$ its closure in $\\mathcal{X}'$.\nBy construction, we have that $\\frac{1}{r}\\mathcal{D} ' + \\Delta'\\geq 0$ and\n\\begin{equation} \\label{eq_lin_eq}\nK_{\\mathcal{X}'} + \\frac{1}{r}\\mathcal{D} ' + \\Delta' - \\epsilon \\mathcal{B}' \\sim_{\\mathbb{Q},\\mathcal{Y}} -\\epsilon \\mathcal{B}',\n\\end{equation}\nfor any $\\epsilon \\in \\mathbb{Q}$.\nAs observed at the end of Step 2, $\\operatorname{Supp}(\\frac{1}{r}\\mathcal{D}')$ contains all the divisors that are exceptional for $\\mathcal{X}' \\rar \\mathcal{Y}$.\nThen, as $\\mathcal{X}'_U \\dashrightarrow \\mathcal{X}_U$ is a rational contraction, it follows that $\\operatorname{Supp}(\\mathcal{B}') \\subset \\operatorname{Supp}(\\frac{1}{r}\\mathcal{D}')$.\nThus, for $0< \\epsilon \\ll 1$, the the divisor $ \\frac{1}{r}\\mathcal{D} ' + \\Delta'-\\epsilon \\mathcal{B}'$ is  effective, and hence $({\\mathcal{X}'} , \\frac{1}{r}\\mathcal{D} ' + \\Delta'-\\epsilon \\mathcal{B}')$ is a log canonical pair.\nNow, we want to define $\\mathcal{X}$ as its relative ample model over $\\mathcal{Y}$.\nFor this definition to be well-posed, we need to check that such an ample model exists.\n\n{\\bf Step 5:} In this step, we show that the definition of $\\mathcal{X}$ is well-posed.\n\n\nFirst, observe that since $(\\mathcal{X}', \\frac{1}{r}\\mathcal{D}' + \\Delta'+\\mathcal{X}'_0)$ is log canonical, there are no log canonical centers of $(\\mathcal{X}', \\frac{1}{r}\\mathcal{D}' + \\Delta'-\\epsilon \\mathcal{B}')$ contained in $\\mathcal{X}_0'$.\nTherefore, up to replacing $C$ with an open neighborhood of $0\\in C$, we may assume that all the log canonical centers are horizontal.\nThen, we may apply Lemma \\ref{lemma minimal models}, which guarantees the existence of a relative good minimal model, up to further extracting some log canonical places.\nIn particular, the relative ample model over $\\mathcal{Y}$ exists, and by definition, it coincides with $\\mathcal{X}$.\nWe let $\\mathcal{D}$, $\\Delta$, and $\\mathcal{B}$ be the push-forwards of $\\mathcal{D}'$, $\\Delta'$, and $\\mathcal{B}'$ to $\\mathcal{X}$, respectively. Observe also that from \\eqref{eq_lin_eq}, $\\mathcal{X}$ is independent of $\\epsilon$, as long as $\\epsilon>0$ and $\\epsilon \\in \\mathbb{Q}$, as it is the relative Proj of the same graded algebra, only with shifted degrees. Moreover, $\\operatorname{Supp}(\\frac{1}{r}\\mathcal{D}+\\Delta)$ does not contain any irreducible component of the special fiber, since $(\\mathcal{X},\\frac{1}{r}\\mathcal{D} + \\Delta + \\mathcal{X}_0)$ is log canonical.\n\n{\\bf Step 6:} In this step, we show that $\\mathcal{X} \\times_C U = \\mathcal{X}_U$ and that $\\mathcal{D}\\times_C U  = \\mathcal{D}_U$.\nIn particular, we have $\\mathcal{B} = \\mathcal{D}$.\n\nRecall that the rational map $\\mathcal{X}'_U \\dashrightarrow \\mathcal{X}_U$ is a contraction.\nThus, we can define the rational pull-back of $K_{\\mathcal{X}_U} + \\frac{1-\\epsilon}{r}\\mathcal{D}_U+\\Delta_U$ to $\\mathcal{X}'_U$.\nBy definition of $\\mathcal{B}'$, this pull-back agrees with $K_{\\mathcal{X}'_U}+\\frac{1}{r}\\mathcal{D}'_U+\\Delta'_U -\\epsilon \\mathcal{B}'_U$.\nThus, the relative ample model of $(\\mathcal{X}'_U,\\frac{1}{r}\\mathcal{D}'_U+\\Delta'_U -\\epsilon \\mathcal{B}'_U)$ over $U$ is $(\\mathcal{X}_U,\\frac{1-\\epsilon}{r}\\mathcal{D}_U+\\Delta_U)$.\nThen, since $K_{\\mathcal{X}'}+\\frac{1}{r}\\mathcal{D}'+\\Delta'$ is the pull-back of an ample divisor on $\\mathcal{Y}$, and the model $\\mathcal{X}$ is independent of $\\epsilon>0$, it follows that, if we choose $0<\\epsilon \\ll 1$, the relative ample model of $(\\mathcal{X}'_U,\\frac{1}{r}\\mathcal{D}'_U+\\Delta'_U-\\epsilon \\mathcal{B}'_U)$ over $\\mathcal{Y}$ is a relative ample model over $C$.\nThis shows that $\\mathcal{X} \\times_C U = \\mathcal{X}_U$ and therefore $\\mathcal{D}\\times_C U  = \\mathcal{D}_U$.\n\nSince $\\mathcal{B}'$ is defined as the closure of the rational pull-back of $\\mathcal{D}_U$ to $\\mathcal{X}'_U$, all of its components are horizontal over $U$.\nIn particular, to check the equality $\\mathcal{B}= \\mathcal{D}$, it suffices to show it holds over $U$.\nBut then, this is immediate, by the definition of $\\mathcal{B}'$ and the fact that $\\mathcal{X} \\times_C U = \\mathcal{X}_U$.\n\n{\\bf Step 7:} In this step, we show that $(\\mathcal{X},\\Delta;\\frac{1}{r}\\mathcal{D})$ is a $p$-pair and that the central fiber of $(\\mathcal{X},\\Delta;\\frac{1}{r}\\mathcal{D})$ is a $p$-stable pair with polynomial $p(t)$ and coefficients in $I$.\n\n\nRecall that $\\mathcal{X}$ is independent of $\\epsilon$, as long as $\\epsilon>0$ and $\\epsilon \\in \\mathbb{Q}$.\nIn particular, $\\mathcal{B}$ is $\\mathbb{Q}$-Cartier on $\\mathcal{X}$.\nThen, as $\\K \\mathcal{X}. + \\frac{1}{r}\\mathcal{D} + \\Delta - \\epsilon \\mathcal{B}$ is $\\mathbb{Q}$-Cartier for $0 < \\epsilon \\ll 1$ rational and $\\mathcal{B}=\\mathcal{D}$, it follows that $\\K \\mathcal{X}. + \\frac{1}{r}\\mathcal{D} + \\Delta$ is $\\mathbb{Q}$-Cartier.\nAs $(\\mathcal{X},\\frac{1}{r}\\mathcal{D} + \\Delta)$ is log canonical by construction, it follows that $(\\mathcal{X},\\Delta;\\frac{1}{r}\\mathcal{D})$ is a $p$-pair.\n\n\nBy construction and by adjunction, $(\\mathcal{X}_0,\\frac{1}{r}\\mathcal{D}_0 + \\Delta_0)$ is semi-log canonical.\nFurthermore, $\\K \\mathcal{X}_0. + \\frac{1-\\epsilon}{r}\\mathcal{D}_0 + \\Delta_0$ is ample for $0 < \\epsilon \\ll 1$.\nSince $\\mathcal{D}$ and $K_{\\mathcal{X}} + \\frac{1}{r}\\mathcal{D} +\\Delta$ are $\\mathbb{Q}$-Cartier, the self-intersection $(K_{\\mathcal{X}_c} + \\frac{1}{r}\\mathcal{D}_c +\\Delta_c-\\epsilon \\mathcal{D}_c)^{{\\rm dim}(\\mathcal{X}_0)}$ is well-defined for every $\\epsilon$ and $c\\in C$, and does not depend on $c \\in C$.\nSince the general fiber is $p$-stable with polynomial $p(t)$, we have $(K_{\\mathcal{X}_0} + \\frac{1}{r}\\mathcal{D}_0 +\\Delta_0-\\epsilon \\mathcal{D}_0)^{{\\rm dim}(\\mathcal{X}_0)} = p(1-\\epsilon)$.\nThe coefficients of $\\frac{1}{r}\\mathcal{D}_0$ are still in $I$,  since $I$ is closed under addition.\n\n{\\bf Step 8:} In this step, we show that $(\\mathcal{X}, \\Delta;\\mathcal{D}) \\rar C$ is a $p$-stable morphism with constant part $\\Delta$.\n\nBy construction, the fibers are proper.\nSince the base is a curve and every divisor is horizontal, all the morphisms are flat of the appropriate relative dimension.\nBy Step 7, every fiber is a $p$-stable pair.\nThen, as $\\mathcal{D}$ is $\\qq$-Cartier, every fiber of $(\\mathcal{X},\\Delta) \\rar C$ is semi-log canonical.\nThus, by \\cite{kol1}*{Definition-Theorem 4.45}, the morphism $(\\mathcal{X},\\Delta) \\rar C$ is locally stable.\n\n{\\bf Step 9:} In this step, we show that the limit is unique.\n\nFrom Theorem \\ref{thm_boundedness}, there exists $\\epsilon_0 > 0$ such that, for every $p$-stable pair $(Z,B,\\Gamma)$ with coefficients in $I$ and polynomial $p(t)$, $(Z,(1-\\epsilon_0)B + \\Gamma)$ is a stable pair.\nThen, the claim follows from the separatedness of stable morphisms \\cite{kol1}*{2.45}.\n\\end{proof}\n\n\n\n\\begin{theorem}\\label{Thm_properness}\nFix an integer $n\\in \\mathbb{N}$, a finite subset $I\\subseteq (0,1] \\cap \\mathbb{Q}$, and a polynomial $p(t) \\in \\mathbb{Q}[t]$.\nAssume that $I$ is closed under sum: that is, if $a,b \\in I$ and $a+b \\leq 1$, then $a+b \\in I$.\nAssume that assumption (A) holds.\nLet $C$ be a smooth affine curve, let $0 \\in C$ be a distinguished closed point, and let $U\\coloneqq C \\setminus \\{ 0 \\}$.\nLet $(\\mathcal{X}_U, \\Delta_U; \\mathcal{D}_U) \\rightarrow U$ be a $p$-stable morphism of dimension $n$, polynomial $p(t)$, coefficients in $I$, and constant part $\\Delta_U$.\nThen, up to a finite base change $B \\rar C$, the family can be filled with a unique $p$-stable pair of dimension $n$, with polynomial $p(t)$ and coefficients in $I$.\n\\end{theorem}\n\n\n\\begin{proof}\nBy Proposition \\ref{properness_lc_case}, we may assume that the geometric generic fiber is not normal.\nIn the following, $r$ will denote the index of $I$.\n\nLet $(\\overline \\mathcal{X}_U, \\frac{1}{r}\\overline \\mathcal{D}_U + \\overline \\Delta_U)$ denote the normalization of $(\\mathcal{X}_U, \\frac{1}{r}\\mathcal{D}_U + \\Delta_U)$, where $\\overline{\\Delta}_U$ also includes the conductor with coefficient 1.\nBy assumption, $\\mathcal{D}_U$ is $\\mathbb{Q}$-Cartier.\nThen, by \\cite{Kol13}*{Corollary 5.39}, $\\overline{\\mathcal{D}}_U$ is $\\mathbb{Q}$-Cartier.\nThen, since $\\K \\mathcal{X}_U. + \\frac{1-\\epsilon}{r}\\mathcal{D}_U + \\Delta_U$ is ample over $U$, it follows that $(\\overline \\mathcal{X} _U, \\overline \\Delta_U; \\overline \\mathcal{D} _U)$ is $p$-stable morphism with constant part $\\overline \\Delta_U$, where the polynomial on each connected component depends on the original choice of $p$.\n\nThen, as $\\overline{\\mathcal{X}}_U$ has finitely many connected components, by Proposition \\ref{properness_lc_case}, there is a finite base change $B \\rar C$ such that the family can be filled with a unique $p$-stable pair.\nTo simplify the notation, we omit the base change $B \\rar C$, and we assume that the filling is realized over $C$ itself.\nWe denote this family by $(\\overline \\mathcal{X},  \\overline \\Delta; \\overline \\mathcal{D}) \\rar C$.\nThen, we may find $0 < \\epsilon \\ll 1$ such that $(\\overline \\mathcal{X}, \\frac{1-\\epsilon}{r} \\overline \\mathcal{D}  + \\overline \\Delta) \\rar C$ is a stable morphism.\nBy \\cite{kol1}*{Corollary 2.56}, also $(\\mathcal{X} _U, \\frac{1-\\epsilon}{r} \\mathcal{D} _U + \\Delta_U)$ admits a compactification $(\\mathcal{X} , \\frac{1-\\epsilon}{r} \\mathcal{D} + \\Delta)$ over $C$ obtained by gluing $\\overline{\\mathcal{X}}$ along some components of $\\overline{\\Delta}$.\nBy \\cite{Kol13}*{Corollary 5.39}, the divisor $\\mathcal{D}$ is $\\mathbb{Q}$-Cartier.\nThus, we have that $\\mathcal{D}_0$ is $\\mathbb{Q}$-Cartier, as needed.\nSimilarly, the coefficients of $\\frac{1}{r}\\mathcal{D}_0+\\Delta_0$ are in $I$, by construction.\n\nTo conclude, we need to show that $(\\K \\mathcal{X}_0. + \\frac{t}{r}\\mathcal{D}_0 + \\Delta_0)^n=p(t)$.\nThis follows from flatness over the base $C$, as we have\n\\[\n(\\K \\mathcal{X}_0. + \\frac{t}{r}\\mathcal{D}_0 + \\Delta_0)^n = (\\K \\mathcal{X}_c. + \\frac{t}{r}\\mathcal{D}_c + \\Delta_c)^n = p(t),\n\\]\nwhere $c \\in C \\setminus \\{ 0 \\}$.\nThis concludes the proof.\n\\end{proof}\n\n\\begin{Cor}Fix an integer $n\\in \\mathbb{N}$, a finite subset $I\\subseteq (0,1] \\cap \\mathbb{Q}$, and a polynomial $p(t) \\in \\mathbb{Q}[t]$.\nAssume that $I$ is closed under sum: that is, if $a,b \\in I$ and $a+b \\leq 1$, then $a+b \\in I$.\nAssume that assumption (A) holds.\nThen the algebraic stack $\\mathscr{F}_{n,p,I}$ is proper.\n\\end{Cor}\n\\begin{proof}\nIt suffices to check that it satisfies the valuative criterion for properness. So assume that we have a family of $p$-stable pairs $f_\\eta \\colon (X;D)\\to \\eta$ over the generic point $\\eta$ of the spectrum of a DVR $R$, and we need to fill in the central fiber up to replacing $\\operatorname{Spec}(R)$ with a ramified cover of it. Theorem \\ref{Thm_properness} guarantees the existence and uniqueness of a p-stable morphism $f \\colon (\\mathcal{X};\\mathcal{D})\\to \\operatorname{Spec}(R)$ extending $f_\\eta$, up to a ramified cover of $\\operatorname{Spec}(R)$. We need to check that $f$ satisfies conditions (1) and (2) of Definition \\ref{Def:functor}. Condition (1) follows from Proposition \\ref{prop:flat:limit:is:S1} and \\cite{HK04}*{Corollary 3.8}. Condition (2) follows from \\cite{kol1}*{Proposition 2.76}.\n\\end{proof}\n\n\\section{Morphism from the $p$-moduli to the moduli of stable pairs}\n\\label{section morphism}\nGiven a pair $(X,D)$ with $K_X+D$ semi-ample and big, one can take its ample model. The goal of this subsection is to show that for $p$-pairs, one can take the ample model in families.\n\n\n\\begin{Lemma}\\label{Lemma:slc:canonical:model:for:p:pairs}\nFix an integer $n\\in \\mathbb{N}$, a finite subset $I\\subseteq (0,1] \\cap \\mathbb{Q}$, and a polynomial $p(t) \\in \\mathbb{Q}[t]$.\nLet $r$ denote the index of $I$.\nConsider a $p$-stable morphism with coefficients in $I$ and polynomial $p(t)$ over a smooth scheme $U$:\n$p \\colon (\\mathcal{X};\\mathcal{D})\\to U.$\nAssume that assumption (A) holds.\nThen, there is a stable family of pairs $(\\mathcal{Y},\\frac{1}{r}\\mathcal{D}_\\mathcal{Y})\\to U$ such that:\n\\begin{enumerate}\n    \\item there is a contraction $\\pi \\colon \\mathcal{X} \\to \\mathcal{Y}$; and\n    \\item we have $\\pi^* (\\K \\mathcal{Y}\/U. + \\frac{1}{r}\\mathcal{D}_\\mathcal{Y}) = \\K \\mathcal{X} \/U. + \\frac{1}{r}\\mathcal{D}$.\n\\end{enumerate}\n\\end{Lemma}\n\\begin{proof}\nBy Proposition \\ref{prop_stable_family_p-pairs} and the fact that $U$ is smooth, it follows that $(\\mathcal{X}, \\frac{1}{r}\\mathcal{D})$ is a pair.\nFurthermore, by inversion of adjunction, this pair is semi-log canonical.\n\n\nFirst, we assume that $\\mathcal{X}$ is normal.\nThen, by Lemma \\ref{lemma minimal models} and our assumptions, the canonical model of $(\\mathcal{X},\\frac{1}{r}\\mathcal{D})$ over $U$ exists.\nSince $U$ is smooth, this canonical model gives a family of stable pairs by \\cite{kol1}*{Corollary 4.86}, and the result follows from how canonical models are constructed.\n\nWe now treat the case where $\\mathcal{X}$ is not normal.\nFirst, we normalize $\\mathcal{X}^n\\to \\mathcal{X}$ to get $(\\mathcal{X}^n,\\Delta;\\mathcal{D}') $.\nAs argued in the previous case, we can construct the canonical model of $(\\mathcal{X}^n,\\Delta;\\mathcal{D}') $ over $U$.\nAs above, this gives a family of stable pairs $(\\mathcal{Y}',\\frac{1}{r}\\mathcal{D}_\\mathcal{Y}'+\\Delta_\\mathcal{Y})\\to U$.\nFrom Koll\\'ar's gluing theory (see \\cite{Kol13}*{Chapter 5}), there is an involution $\\tau \\colon \\Delta^n \\to \\Delta^n$ that fixes the different.\nWe first show that this involution descends onto $\\Delta_\\mathcal{Y}^n$.\n\nRecall that, by Lemma \\ref{ample model p-pair}, the map $\\mathcal{X}^n\\to \\mathcal{Y}'$ does not contract any component of $\\Delta$.\nIn particular, for every irreducible component $F\\subseteq \\Delta^n$, there is an irreducible component $F_Y\\subseteq \\Delta_\\mathcal{Y}^n$ birational to it, and we have the following diagram:\n\n$$\\xymatrix{\\Delta^n\\ar[r]^f \\ar[d] & \\Delta^n_\\mathcal{Y} \\ar[d] \\\\ \\mathcal{X} \\ar[r] & \\mathcal{Y}.}$$\nObserve now that\n\\begin{center}\n    {\\bf ($\\ast$)} {\\it a curve $C \\subset \\Delta^n$ gets contracted by $f$ if and only if $(K_{\\Delta^n} + \\operatorname{Diff}_{\\Delta^n}(\\frac{1}{r}\\mathcal{D} ' + \\Delta)).C = 0$.}\n\\end{center}\nIn particular, since the involution $\\tau$ preserves the different, it preserves all the curves that are contracted by $f$.\nHence, the involution descends to an involution $\\tau_\\mathcal{Y}$ on $\\Delta_\\mathcal{Y}^n$.\n\nWe prove that $\\tau_\\mathcal{Y}$ preserves the different.\nIndeed, by Lemma \\ref{ample model p-pair}, the only divisors contracted by $f$ are contained in $\\operatorname{Supp}(\\mathcal{D}')$, so the morphism $f$ is an isomorphism generically around each divisor not contained in $\\operatorname{Supp}(\\mathcal{D})$.\nIn particular, since the computation of the different is local,\n\\[\nf_*(\\operatorname{Diff}_{\\Delta^n}(\\frac{1}{r}\\mathcal{D}' + \\Delta)) = \\operatorname{Diff}_{\\Delta^n_\\mathcal{Y}}(\\frac{1}{r}\\mathcal{D}'_\\mathcal{Y} + \\Delta_\\mathcal{Y}).\n\\]\nSince $\\tau$ preserves the different on $\\Delta^n$, $\\tau_\\mathcal{Y}$ preserves the different on $\\Delta_\\mathcal{Y}^n$.\n\nThen from \\cite{kol1}*{Chapter 5}, we can glue $\\mathcal{Y}'$ to get an semi-log canonical pair $(\\mathcal{Y},\\frac{1}{r}\\mathcal{D}_\\mathcal{Y})$.\nNow, recall that $\\mathcal{Y}$ is a geometric quotient (see \\cite{Kol13}*{Theorem 5.32} and the proof of \\cite{Kol13}*{Corollary 5.33}), so for any morphism $\\phi \\colon \\mathcal{Y}^n\\to Z$ such that $\\phi|_{\\Delta_\\mathcal{Y}^n} = \\tau \\circ \\phi|_{\\Delta_\\mathcal{Y}^n} $, there is a unique morphism $\\mathcal{Y}\\to Z$ which fits in the obvious commutative diagram.\nTherefore, we have a morphism $\\mathcal{Y}\\to U$, and applying this result to $\\mathcal{X}^n$ and $\\mathcal{X}$, we obtain a morphism $\\mathcal{X}\\to \\mathcal{Y}$.\nNow, by Lemma \\ref{ample model p-pair}, we can apply Lemma \\ref{lemma:canonical:model:of:slc:pairs:has:connected:fibers}, and the desired result follows.\n\\end{proof}\n\n\n\n\n\\begin{theorem}\\label{thm_morphism_our_stack_to_kollars_one_for_reduced_bases}\nFix an integer $n\\in \\mathbb{N}$, a finite subset $I\\subseteq (0,1] \\cap \\mathbb{Q}$, and a polynomial $p(t) \\in \\mathbb{Q}[t]$.\nLet $r$ denote the index of $I$.\nAssume that assumption (A) holds.\nLet $(\\mathcal{X};\\mathcal{D})\\to B$ be a $p$-stable morphism of dimension $n$, with coefficients in $I$, and polynomial $p(t)$, over a reduced connected base $B$, and assume that there is an open dense subscheme $U\\subseteq B$ with:\n\\begin{enumerate}\n    \\item a stable family of pairs $(\\mathcal{Y}_U,\\frac{1}{r}\\mathcal{D}_{\\mathcal{Y},U})\\to U$ of relative dimension $n$; and\n    \\item a contraction $\\pi_U \\colon \\mathcal{X}_U\\to \\mathcal{Y}_U $ such that $\\pi^*(K_{\\mathcal{Y}_U\/U} + \\frac{1}{r}\\mathcal{D}_{\\mathcal{Y},U}) = K_{\\mathcal{X}_U\/U} + \\frac{1}{r}\\mathcal{D}_{U}$.\n\\end{enumerate}\nThen, there there is an $m>0$ such that for every $d$ and every $b\\in B$ we have $$p_*(\\mathcal{O}_{\\mathcal{X}}(m(K_{\\mathcal{X}\/B}+\\frac{1}{r}\\mathcal{D}))^{\\otimes d})\\otimes k(b) = H^0(\\mathcal{X}_b, \\mathcal{O}_{\\mathcal{X}_b}(m(K_{\\mathcal{X}_b}+\\frac{1}{r}\\mathcal{D}_b))^{\\otimes d}) .$$\n\nMoreover, if we define $\\mathcal{Y} \\coloneqq \\operatorname{Proj}(\\bigoplus_d p_*\\mathcal{O}_{\\mathcal{X}}(m(K_{\\mathcal{X}\/B}+\\frac{1}{r}\\mathcal{D}))^{\\otimes d})$, then:\n\\begin{itemize}\n    \\item there is a unique family of divisors $\\mathcal{D}_\\mathcal{Y}$ such that the pair $q: (\\mathcal{Y},\\frac{1}{r}\\mathcal{D}_\\mathcal{Y}) \\to B $ is a stable family extending $q_U$;\n    \\item there is a contraction $\\pi \\colon \\mathcal{X}\\to \\mathcal{Y}$ over $B$ that extends $\\pi_U$; and\n    \\item $\\pi^*(K_{\\mathcal{Y}\/B} + \\frac{1}{r}\\mathcal{D}_{\\mathcal{Y}}) = K_{\\mathcal{X}\/B} + \\frac{1}{r}\\mathcal{D}$.\n\\end{itemize}\n\\end{theorem}\n\\begin{Remark}\nObserve that since $K_{\\mathcal{Y}_U\/U} + \\frac{1}{r}\\mathcal{D}_{\\mathcal{Y},U}$ is $\\mathbb{Q}$-Cartier, we can define its pull-back as a $\\mathbb{Q}$-Cartier divisor. \n\\end{Remark}\n\n\n\\begin{proof}\nWe proceed in several steps.\n\n{\\bf Step 1:} In this step, we make some preliminary considerations and set some notation.\n\nBy Proposition \\ref{prop_stable_family_p-pairs}, we have that $\\K \\mathcal{X}\/B. + \\frac{1}{r}\\mathcal{D}$ and $\\mathcal{D}$ are $\\qq$-Cartier.\nThen, observe that for every $s,t\\in U$, the volumes of the pairs $(\\mathcal{Y}_s,\\frac{1}{r}\\mathcal{D}_{\\mathcal{Y},s})$ and $(\\mathcal{Y}_t,\\frac{1}{r}\\mathcal{D}_{\\mathcal{Y},t})$ agree.\nIndeed, since $(\\mathcal{X};\\mathcal{D})\\to B$ is a $p$-stable morphism, $\\K \\mathcal{X}_s.+ \\frac{1}{r}\\mathcal{D}_s$ and $\\K \\mathcal{X}_t.+ \\frac{1}{r}\\mathcal{D}_t$ are nef.\nThus, their volumes are computed by the $n$-fold self-intersection, which is independent of $s,t \\in S$.\nBut from condition (2) the morphisms $\\pi_s \\colon \\mathcal{X}_s\\to \\mathcal{Y}_s$ and $\\pi_t \\colon \\mathcal{X}_t\\to \\mathcal{Y}_t$ have connected\nfibers, and we have $\\pi_s^*(\\K \\mathcal{Y}_s. + \\frac{1}{r}\\mathcal{D} \\subs \\mathcal{Y},s.)=\\K \\mathcal{X}_s. +\\frac{1}{r} \\mathcal{D}_s$ and $\\pi_t^*(\\K \\mathcal{Y}_t. +\\frac{1}{r} \\mathcal{D} \\subs \\mathcal{Y},t.)=\\K \\mathcal{X}_t. +\\frac{1}{r} \\mathcal{D}_t$.\nThen, by Remark \\ref{remark:cohom:connected:implies:one:can:take:global:sections:after:pull:back}, the volumes of $\\K \\mathcal{Y}_s. + \\frac{1}{r}\\mathcal{D} \\subs \\mathcal{Y},s.$ (resp. $\\K \\mathcal{Y}_t. + \\frac{1}{r}\\mathcal{D} \\subs \\mathcal{Y},t.$) and $\\K \\mathcal{X}_s. +\\frac{1}{r} \\mathcal{D}_s$ (resp. $\\K \\mathcal{X}_t. + \\frac{1}{r}\\mathcal{D}_t$) agree.\nLet then $v$ be the volume of any fiber of $q_U$ and let $k$ be a natural number such that $r$ divides $k$ and, for every stable pair $(Y,D)$ of dimension $n$, volume $v$ and coefficients in $I$, the line bundle $\\mathcal{O}_Y(k(K_Y + D))$ is very ample and the higher cohomologies of all of its natural multiples vanish.\nNotice that $k$ exists by \\cite{HMX18}*{Theorem 1.2.2}.\nThen, we set $\\mathcal{L} \\coloneqq  \\O \\mathcal{X}. (k(K_{\\mathcal{X}\/B} + \\frac{1}{r}\\mathcal{D}))$.\nUp to replacing $k$ with a multiple, we may further assume that $\\mathcal{L}$ is Cartier.\n\n{\\bf Step 2:} In this step, we show that the theorem holds if $B$ is a smooth curve.\n\nFrom Lemma \\ref{Lemma:slc:canonical:model:for:p:pairs}, we can construct a family of stable pairs $(\\mathcal{Z},\\frac{1}{r}\\mathcal{D}_\\mathcal{Z})\\to B$ with a contraction $\\phi \\colon \\mathcal{X}\\to \\mathcal{Z}$.\nFirst, observe that $(\\mathcal{Z}_U,\\frac{1}{r}\\mathcal{D}_{\\mathcal{Z},U})\\cong (\\mathcal{Y}_U,\\frac{1}{r}\\mathcal{D}_{\\mathcal{Y},U})$ over $U$.\nIndeed, consider the reflexive sheaves $\\mathcal{L} _\\mathcal{Z} \\coloneqq \\O \\mathcal{Z}. (k(\\K \\mathcal{Z}. + \\frac{1}{r}\\mathcal{D} _\\mathcal{Z}))$ and $\\mathcal{L} _{\\mathcal{Y},U} \\coloneqq \\O \\mathcal{Y}_U. (k(\\K \\mathcal{Y}_U. + \\frac{1}{r}\\mathcal{D} _{\\mathcal{Y},U}))$.\nBy construction, the fibers of $(\\mathcal{Z},\\frac{1}{r}\\mathcal{D}_\\mathcal{Z}) \\rar B$ and $(\\mathcal{Y}_U,\\frac{1}{r}\\mathcal{D} \\subs \\mathcal{Y},U.) \\rar U$ belong to the moduli problem of stable pairs with volume $v$ and coefficients in the finite set $I$.\nThus, by \\cite{kol1}*{Theorem 5.7.(4)} and the choice of $k$, $\\mathcal{L} _\\mathcal{Z}$ and $\\mathcal{L} _{\\mathcal{Y},U}$ are line bundles.\nIn order to apply \\cite{kol1}*{Theorem 5.7.(4)}, notice that we know that $\\mathcal{L} _\\mathcal{Z}$ and $\\mathcal{L} _{\\mathcal{Y},U}$ are line bundles away from the exceptional locus of $\\phi$ and $\\pi_U$, which are big open subsets restricting to big open subsets fiberwise.\n\nSince $\\pi_U$ and $\\phi_U$ have connected fibers, we have\n\\[\n\\pi_U^*\\mathcal{L}_{\\mathcal{Y},U} \\cong \\mathcal{L}_U \\cong \\phi^*_U\\mathcal{L}_{\\mathcal{Z},U}.\n\\]\nBut both $\\pi_U$ and $\\phi_U$ have connected fibers so, by the projection formula, for every $m \\geq 1$, we have\n\\[\nH^0(\\mathcal{Y}_U,\\mathcal{L}_{\\mathcal{Y},U}^{\\otimes m}) = H^0(\\mathcal{X}_U,\\pi^*_U\\mathcal{L}_{\\mathcal{Y},U}^{\\otimes m}) =H^0(\\mathcal{X}_U,\\mathcal{L}_{U}^{\\otimes m}) = H^0(\\mathcal{X}_U,\\phi^*_U\\mathcal{L}_{\\mathcal{Z},U}^{\\otimes m})  =  H^0(\\mathcal{Z}_U,\\mathcal{L}_{\\mathcal{Z},U}^{\\otimes m}).\n\\]\nBut $\\mathcal{L}_{\\mathcal{Z},U}$ and $\\mathcal{L}_{\\mathcal{Y},U}$ are ample over $U$, so $\\mathcal{Z}_U$ and $\\mathcal{Y}_U$ are isomorphic as $U$-schemes, as they are the relative Proj of the same sheaf of graded algebras.\nIn particular, the three final claims of the theorem hold if we consider the family $(\\mathcal{Z},\\frac{1}{r}\\mathcal{D}_\\mathcal{Z})$.\n\nThus, we are left with proving that for every $d$ and every $b\\in B$, we have $p_*(\\mathcal{L}^{\\otimes d})\\otimes k(b) = H^0(\\mathcal{X}_b, \\mathcal{L}^{\\otimes d}_b) $.\nBut since $\\phi^*\\mathcal{L}_\\mathcal{Z} = \\mathcal{L}$, we have $\\phi_b^*(\\mathcal{L}_{\\mathcal{Z},b}) = \\mathcal{L}_{b}$.\nThus, by Remark \\ref{remark:cohom:connected:implies:one:can:take:global:sections:after:pull:back}, we have $h^0(\\mathcal{X}_b,\\mathcal{L} _b^{\\otimes m}) = h^0(\\mathcal{Z}_b,\\mathcal{L} \\subs \\mathcal{Z},b.^{\\otimes m})$ for all $m \\geq 1$.\nThen, the latter is locally constant from the assumptions on $k$, as by the vanishing of the higher cohomologies we have $h^0(\\mathcal{Z}_b,\\mathcal{L} \\subs \\mathcal{Z},b.^{\\otimes m})= \\chi(\\mathcal{Z}_b,\\mathcal{L} \\subs \\mathcal{Z},b.^{\\otimes m})$, and the Euler characteristic of $\\mathcal{L} \\subs \\mathcal{Z},b.^{\\otimes m}$ is independent of $b \\in B$. Now the desired statement follows from \\cite{Mum74}*{Corollary 2, page 50}.\n\n{\\bf Step 3:} In this step, we return to the general case and we show that for every $m$, the morphism $B\\ni b\\mapsto h^0(\\mathcal{X}_b,\\mathcal{L}|_{\\mathcal{X}_b}^{\\otimes m})$ is constant and the algebra $\\bigoplus_m H^0(\\mathcal{X}_b,\\mathcal{L}|_{\\mathcal{X}_b}^{\\otimes m})$ is finitely generated.\n\n\nObserve that the claim holds for every $p\\in U$. Indeed, the following diagram commutes:\n$$\\xymatrix{\\mathcal{X}_b \\ar[r] \\ar[d]_{\\pi_b} & \\mathcal{X}_U \\ar[d] \\\\ \\mathcal{Y}_b \\ar[r] & \\mathcal{Y}_U}$$\nwhich from point (2) guarantees $\\pi_b^*(\\mathcal{L}_{\\mathcal{Y},b}) = \\mathcal{L}_{b}$.\nThen, in this case, we can conclude as at the end of Step 2.\nFurthermore, the finite generation follows from the fact that $\\mathcal{L}_{b}$ is very ample and $\\mathcal{Y}_b$ is the Proj of its associated graded ring.\n\nNow, we treat the case when $b \\not \\in U$.\nConsider a smooth curve $C$ with a map $C\\to B$.\nAssume that the generic point of $C$ maps into $U$.\nNotice that any point $b \\in B$ is contained in the image of such a curve.\nThen, for any point $s\\in C$, we have the following diagram:\n$$\\xymatrix{\\mathcal{X}_s\\ar[r] \\ar[d] & \\mathcal{X}_C \\ar[r] \\ar[d] & \\mathcal{X} \\ar[d] \\\\ \\{s\\}\\ar[r] & C \\ar[r] & B.}$$\nSince both squares are fibered squares, the big rectangle is a fibered square.\nIn particular, since we want to prove that $B\\ni b\\mapsto h^0(\\mathcal{X}_b,\\mathcal{L}|_{\\mathcal{X}_b}^{\\otimes m})$\nis constant, and since we know it is constant as long as $b\\in U$, it suffices to check that for every such $C$ the functions $C\\ni s\\mapsto h^0(\\mathcal{X}_s,\\mathcal{L}|_{\\mathcal{X}_s}^{\\otimes m})$ are constant.\nNow, this follows by Step 2.\nSimilarly, the finite generation of $\\bigoplus_m H^0(\\mathcal{X}_s,\\mathcal{L}|_{\\mathcal{X}_s}^{\\otimes m})$ follows from Step 2.\n\n\n{\\bf Step 4:} In this step, we construct the model $\\mathcal{Y}$ and the morphism $\\pi$.\n\nBy Step 3 and cohomology and base change (see \\cite{Mum74}), the sheaves $p_*(\\mathcal{L}^{\\otimes m})$ commute with the restriction to points. In particular, the algebra $\\mathcal{A}  \\coloneqq  \\bigoplus_{m} p_*(\\mathcal{L}^{\\otimes m})$ is finitely generated since it is finitely generated when restricted to every point $b\\in B$.\nSo we can consider $\\mathcal{Y} \\coloneqq \\operatorname{Proj}_B(\\mathcal{A})$.\n\nAs observed in Step 3, the pluri-sections of $\\mathcal{L}|_{\\mathcal{X} b}$ are deformation invariant.\nAs $\\mathcal{L}|_{\\mathcal{X} b}$ is semi-ample for every $b \\in B$ by our assumptions, it follows that $\\mathcal{L}$ is relatively semi-ample.\nIn particular, we have a morphism $\\mathcal{X} \\rar \\mathcal{Y}$.\nFurthermore, this implies the equality $\\mathcal{Y}_b=\\mathrm{Proj}(\\bigoplus_m H^0(\\mathcal{X}_b,\\mathcal{L}_b^{\\otimes m}))$ for every $b \\in B$.\n\nAs already discussed, this construction commutes with base change.\nIn particular, for every $b\\in B$, for checking properties of $\\mathcal{Y}_b$ we can consider a smooth curve $C\\to B$ which sends the generic point to $U$ and the special one to $b$, and first pull back $\\mathcal{Y}$ to $C$ and then restrict it to $b$:\n$$\\xymatrix{\\mathcal{Y}_b\\ar[r] \\ar[d] & \\mathcal{Y}_C \\ar[r] \\ar[d] & \\mathcal{Y} \\ar[d] \\\\ \\{b\\}\\ar[r] & C \\ar[r] & B.}$$\nThe advantage is that now we can apply the results of Step 2 to $\\mathcal{Y}_C$.\n\nSince we have $\\mathcal{L}_U \\cong \\pi_U^* \\mathcal{L}_{\\mathcal{Y},U}$ and $\\mathcal{L}_{\\mathcal{Y},U}$ is relatively ample over $U$, it follows that $\\mathcal{Y} \\times_B U = \\mathcal{Y}_U$ and that $\\pi$ extends $\\pi_U$.\nFurthermore, since $B$ is reduced, the construction commutes with base change, and each fiber is reduced, it follows that $\\mathcal{Y}$ is reduced.\n\n{\\bf Step 5:} In this step, we show that $\\pi$ is a contraction and we construct the divisor $\\mathcal{D}_\\mathcal{Y}$.\n\nWe first prove that $\\pi$ is a contraction. We denote by $V\\subseteq \\mathcal{Y}$ the locus where $\\pi^{-1}(V)\\to V$ is an isomorphism. Then\nit follows from Lemma \\ref{ample model p-pair}:\n\\begin{center}\n    {\\bf ($\\ast \\ast$)} {\\it for every fiber $\\mathcal{Y}_b$, the complement of $V_b \\coloneqq V \\cap \\mathcal{Y}_b$ has codimension at least 2 in $\\mathcal{Y}_b$ and it does not contain any irreducible component of the conductor of $\\mathcal{Y}_b$.}\n\\end{center}\nConsider now the inclusion $i:\\pi^{-1}(V)\\to \\mathcal{X}$, which induces the injective map $0\\to \\mathcal{O}_\\mathcal{X} \\to i_*\\mathcal{O}_{\\pi^{-1}(V)}$. We can push this sequence forward via $\\pi$ and we obtain $0\\to \\pi_*\\mathcal{O}_\\mathcal{X} \\to \\pi_*i_*\\mathcal{O}_{\\pi^{-1}(V)}$. But $\\pi\\circ i :\\pi^{-1}(V)\\to \\mathcal{Y}$ is the inclusion $j: V\\hookrightarrow Y$, so $\\pi_*i_*\\mathcal{O}_{\\pi^{-1}(V)} = j_*j^*\\mathcal{O}_\\mathcal{Y}$ is reflexive from \\cite{HK04}*{Corollary 3.7}, and it is isomorphic to $\\mathcal{O}_{\\mathcal{Y}}$ from \\cite{HK04}*{Proposition 3.6.2}. In particular, this gives an injective map $\\pi_*\\mathcal{O}_\\mathcal{X}\\to \\mathcal{O}_\\mathcal{Y}$. One can check that this is the inverse of the canonical morphism $\\mathcal{O}_\\mathcal{Y}\\to \\pi_*\\mathcal{O}_\\mathcal{X}$, so in particular the latter is an isomorphism.\n\nConsider the ideal sheaf $\\mathcal{I}$ of $\\mathcal{D}$, and consider the inclusion\n\\[\n0\\to \\mathcal{I} \\to \\mathcal{O}_\\mathcal{X}.\n\\]\nThen, as $\\pi$ is a contraction, if we push it forward via $\\pi$, we get\n\\[0\\to \\pi_*\\mathcal{I} \\to \\mathcal{O}_\\mathcal{Y}.\n\\]\nIn particular, $\\pi_* \\mathcal{I}$ is an ideal sheaf on $\\mathcal{Y}$.\nWe denote by $\\mathcal{S}\\subseteq \\mathcal{Y}$ the closed subscheme with ideal sheaf $\\pi_*\\mathcal{I}$.\n\nA priori, $\\mathcal{S}$ may not be pure dimensional, however, consider the intersection between $\\pi^{-1}(V)$ and the locus in $\\mathcal{X}$ where $\\mathcal{D}$ is $\\mathbb{Q}$-Cartier: we denote this locus with $W$.\nObserve that, since $\\pi^{-1}(V)\\to V$ is an isomorphism, we can identify $W$ with a subset of $\\mathcal{Y}$.\nMoreover, since $\\mathcal{D}$ is Cartier on codimension one point of $\\mathcal{X}$ and $V$ is a big open subset, the locus $W$ contains all the codimension one points of $\\mathcal{Y}_b$ for every $b$.\nThen, we consider $\\mathcal{S} \\cap W$, and we define $\\mathcal{D}_\\mathcal{Y}$ to be the closure of $\\mathcal{S} \\cap W$ in $\\mathcal{Y}$.\n\n\n{\\bf Step 6:} In this step, we show that $q \\colon (\\mathcal{Y},\\frac{1}{r}\\mathcal{D}_\\mathcal{Y}) \\rar B$ is a well defined family of pairs.\n\nBy construction, it is immediate that $\\mathcal{D}_\\mathcal{Y}$ is a relative Mumford divisor in the sense of \\cite{kol19s}*{Definition 1}. Indeed, the three conditions of \\cite{kol19s}*{Definition 1} are now clear, since they hold on $\\mathcal{X}$.\n\nThus, we just need to check that $q \\colon \\mathcal{Y}\\to B$ is flat.\nBy pulling back $\\mathcal{Y}\\to B$ along a smooth curve through $U$, it follows from Step 1 that all the fibers of $q$ are reduced and equidimensional.\nThus, $q$ is an equidimensional morphism with reduced fibers over a reduced base, so \\cite{kol1}*{Lemma 10.48} applies.\n\n{\\bf Step 7:} In this step, we show that $q \\colon (\\mathcal{Y},\\frac{1}{r}\\mathcal{D}_\\mathcal{Y}) \\rar B$ is a stable family of pairs and $\\pi^*(K_{\\mathcal{Y}\/B} + \\frac{1}{r}\\mathcal{D}_{\\mathcal{Y}}) = K_{\\mathcal{X}\/B} + \\frac{1}{r}\\mathcal{D}$.\n\nSince $q \\colon (\\mathcal{Y},\\frac{1}{r}\\mathcal{D}_\\mathcal{Y}) \\rar B$ is a well defined family of pairs and its fibers belong to a prescribed moduli problem for stable pairs, we can argue as in the proof of Proposition \\ref{prop_stable_family_p-pairs} to conclude that $K_{\\mathcal{Y}\/B} + \\frac{1}{r}\\mathcal{D}_{\\mathcal{Y}}$ is $\\mathbb Q$-Cartier.\nIn particular, $q \\colon (\\mathcal{Y},\\frac{1}{r}\\mathcal{D}_\\mathcal{Y}) \\rar B$ is a stable family of pairs.\n\nBy construction, we have that $K_{\\mathcal{Y}\/B} + \\frac{1}{r}\\mathcal{D}_{\\mathcal{Y}} =\\pi_*( K_{\\mathcal{X}\/B} + \\frac{1}{r}\\mathcal{D})$.\nFurthermore, we have that the equality $\\pi^*(K_{\\mathcal{Y}\/B} + \\frac{1}{r}\\mathcal{D}_{\\mathcal{Y}}) = K_{\\mathcal{X}\/B} + \\frac{1}{r}\\mathcal{D}$ holds over $U$.\nThen, by construction, all the exceptional divisors of $\\pi$ dominate $B$, as they are contained in the support of $\\mathcal{D}$.\nThus, as $K_{\\mathcal{Y}\/B} + \\frac{1}{r}\\mathcal{D}_{\\mathcal{Y}}$ is $\\mathbb Q$-Cartier and so $\\pi^*(K_{\\mathcal{Y}\/B} + \\frac{1}{r}\\mathcal{D}_{\\mathcal{Y}})$ is well defined, it follows that $\\pi^*(K_{\\mathcal{Y}\/B} + \\frac{1}{r}\\mathcal{D}_{\\mathcal{Y}}) = K_{\\mathcal{X}\/B} + \\frac{1}{r}\\mathcal{D}$.\n\\end{proof}\n\n\\begin{Lemma}\\label{lemma:finitedness:paper:zsolt}\nWith the notation and assumptions of Theorem \\ref{thm_morphism_our_stack_to_kollars_one_for_reduced_bases}, assume that $B$ is an affine curve and that there is a stable pair $(Y,D_Y)$ such that $(\\mathcal{Y},\\frac{1}{r}\\mathcal{D}_\\mathcal{Y})\\cong (Y\\times B,D_Y\\times B)$.\nThen there are finitely many isomorphism classes of $p$-stable pairs in the fibers of $(\\mathcal{X};\\mathcal{D})\\to B$.\n\\end{Lemma}\n\\begin{proof}\nThe proof is analogous to the proof of \\cite{ABIP}*{Claim 6.2}, we summarize here the most salient steps of the argument.\n\n\\begin{bf}Step 1:\\end{bf} Using Koll\\'ar's gluing theory and the fact that stable pairs have finitely many isomorphisms, up to normalizing and disregarding finitely many points on $B$, we can assume that $\\mathcal{X}$ (and therefore also $\\mathcal{Y}$) is normal.\nThis is achieved in \\cite{ABIP}*{Lemma 6.5}.\nIn particular, $(Y\\times B,D_Y\\times B)$ is the canonical model of $(\\mathcal{X},\\frac{1}{r}\\mathcal{D})$.\n\n\\begin{bf}Step 2:\\end{bf} We observe that the divisors contracted by $\\mathcal{X}\\to Y\\times B$ have negative discrepancies and can be extracted by a log resolution of the form $Z\\times B\\to Y\\times B$, where $Z\\to Y$ is a log resolution.\nThis is achieved in \\cite{ABIP}*{Proposition 6.13}.\n\n\n\\begin{bf}Step 3:\\end{bf} To conclude, we observe that all the fibers of $(\\mathcal{X};\\mathcal{D})\\to B$ are isomorphic in codimension 2.\n\nBut two \\emph{stable} pairs $(X_1, D_1)$ and $(X_2, D_2)$ which are isomorphic in codimension 2 must be isomorphic. Indeed, if $U$ is the open subset where they agree, and $L_1$ (resp. $L_2$) is the log-canonical divisor $K_{X_1} + D_1$ (resp. $K_{X_2} + D_2$) then $$H^0(X_1, L_1^{[m]}) = H^0(U, L_1^{[m]}) = H^0(U, L_2^{[m]}) =  H^0(X_2, L_2^{[m]}).$$\nTherefore $X_1$ and $X_2$ are $\\mathrm{Proj}$ of the same graded algebra, so they are all isomorphic.\n\\end{proof}\n\n\\section{Projectivity of the moduli of stable pairs}\n\\label{section projectivity ksba}\nThe goal of this section is to provide a different proof of the projectivity of the moduli of stable pairs, established in \\cite{KP17}, using $p$-pairs.\n\\begin{Lemma}\\label{Lemma_there_is_a_section_vanishing_on_the_divisor}\nWith the notation of Theorem \\ref{thm_morphism_our_stack_to_kollars_one_for_reduced_bases}, we will denote by $p\\colon (\\mathcal{X};\\mathcal{D})\\to B$ the $p$-stable family, and by $q:(\\mathcal{Y},\\frac{1}{r}\\mathcal{D}_\\mathcal{Y})\\to B$ the resulting stable family of Theorem \\ref{thm_morphism_our_stack_to_kollars_one_for_reduced_bases}.\nAssume this $p$-stable morphism is a $p$-stable family, and let $m_0$ be the smallest positive integer such that $m_0\\frac{1}{r}\\mathcal{D}$ is Cartier.\nThen, there is $k_0$ dividing $m_0$ such that, for every $k=\\ell k_0$ positive multiple of $k_0$, the sheaf  $\\mathcal{L}_{\\mathcal{Y}} \\coloneqq \\mathcal{O}_\\mathcal{Y}(k(K_{\\mathcal{Y}\/B} + \\frac{1}{r}\\mathcal{D}_\\mathcal{Y}))$ satisfies the following properties:\n\\begin{enumerate}[(a)]\n    \\item $\\mathcal{L}_\\mathcal{Y}$ is Cartier;\n    \\item $R^iq_*(\\mathcal{L}_\\mathcal{Y}^{\\otimes j}) = 0$ for every $i>0$ and $j>0$;\n    \\item $\\mathcal{L}_\\mathcal{Y}$ gives an embedding $\\mathcal{Y}\\hookrightarrow \\mathbb{P}(q_*\\mathcal{L}_\\mathcal{Y})$;\n    \\item $\\mathcal{O}_\\mathcal{X}(k(K_{\\mathcal{X}\/B} + \\frac{1}{r}\\mathcal{D}_\\mathcal{Y}) - \\frac{m_0}{r}\\mathcal{D})$ is relatively ample, and\n    \\item for every $b\\in B$, there is a section $s\\in H^0(\\mathcal{Y}_b, \\mathcal{L}_{\\mathcal{Y},b})$ such that $V(s)$ has codimension 1 in each irreducible component of $\\mathcal{Y}_b$ and the scheme-theoretic image of $m_0\\mathcal{D}\\to \\mathcal{Y}$ restricted to $\\mathcal{Y}_b$ is contained in $V(s)$.\n\\end{enumerate}\n\\end{Lemma}\n\n\\begin{Remark}\nObserve that the definition of $m_0\\frac{1}{r}\\mathcal{D}$ is given in Notation \\ref{Notation_mD_for_pstable_fam}.\nIn our case, we can still define $m_0\\frac{1}{r}\\mathcal{D}$ even if $(\\mathcal{X};\\mathcal{D})\\to B$ was a $p$-stable morphism instead, since the base $B$ is reduced, and Lemma \\ref{Lemma_there_is_a_section_vanishing_on_the_divisor} would go through verbatim.\nSince we will use Lemma \\ref{Lemma_there_is_a_section_vanishing_on_the_divisor} only in the case in which $(\\mathcal{X};\\mathcal{D})\\to B$ is a $p$-stable family, for simplicity we stick with the family case.\n\\end{Remark}\n\n\\begin{proof}\nBy construction, $K_{\\mathcal{Y}\/B} + \\frac{1}{r}\\mathcal{D}_\\mathcal{Y}$ is relatively ample.\nThus, for a sufficiently divisible $k_0$, properties (a) to (d) are satisfied for $k=k_0$.\n\nThen, we can achieve (a) to (d) since if $\\mathcal{O}_\\mathcal{Y}(k_0(K_{\\mathcal{Y}\/B} + \\frac{1}{r}\\mathcal{D}_\\mathcal{Y}))$ satisfies any condition between (a) to (e), then for every $m>0$ also $\\mathcal{O}_\\mathcal{Y}(mk_0(K_{\\mathcal{Y}\/B} + \\frac{1}{r}\\mathcal{D}_\\mathcal{Y}))$ satisfies the same condition.\nWe only need to check that up to choosing $k_0$ divisible enough, also (e) holds.\n\nLet $\\mathcal{G} \\coloneqq \\mathcal{O}_\\mathcal{Y}(k_0(K_{\\mathcal{Y}\/B} + \\frac{1}{r}\\mathcal{D}_\\mathcal{Y}))$ for a $k_0$ which satisfies (a)-(d), and we need to show that\n$\\mathcal{G}^{\\otimes m}$ satisfies (a)-(e) for some $m\\gg 0$.\nFirst observe that from cohomology and base change and point (b), for every $b\\in B$ we have $q_*(\\mathcal{G}^{\\otimes m})\\otimes k(b) = H^0(\\mathcal{Y}_b,\\mathcal{G}_{|\\mathcal{Y}_b}^{\\otimes m})$.\n\nLet $\\mathcal{Z}\\subseteq \\mathcal{Y}$ be the scheme-theoretic image of $\\pi_{|\\frac{m_0}{r}\\mathcal{D}}$.\nUp to replacing $B$ with a locally closed stratification $B'\\to B$, we can assume that $\\mathcal{Z}' \\coloneqq \\mathcal{Z}\\times_B B'\\to B'$ is flat, and $q_*\\mathcal{G}$ is free.\nThen, we define $\\mathcal{Y}' \\coloneqq \\mathcal{Y}\\times_B B'$, the first projection $\\pi_1 \\colon \\mathcal{Y}'\\to \\mathcal{Y}$, the second projection $q' \\colon \\mathcal{Y}'\\to B'$, and $\\mathcal{G}' \\coloneqq \\pi_1^*\\mathcal{G}$.\nThen, consider the exact sequence \n\\[\n0\\to \\mathcal{I}\\to \\mathcal{O}_{\\mathcal{Y}'}\\to \\mathcal{O}_{\\mathcal{Z}'} \\to 0.\n\\]\nWe twist the exact sequence above by $\\mathcal{G}'^{\\otimes m}$ and we obtain \n\\[\n0\\to \\mathcal{I}\\otimes \\mathcal{G}'^{\\otimes m} \\to  \\mathcal{G}'^{\\otimes m}\\to \\mathcal{O}_{\\mathcal{Z}'}\\otimes \\mathcal{G}'^{\\otimes m} \\to 0.\n\\]\nSince $\\mathcal{Z}'\\to B'$ is flat, also $\\mathcal{I}$ is flat over $B'$. Therefore since $\\mathcal{G}'$ is relatively ample, up to choosing $m$ big enough, the following is an exact sequence on $B'$:\n\n\\[\n0\\to q'_*(\\mathcal{I}\\otimes \\mathcal{G}'^{\\otimes m})\\to  q'_*(\\mathcal{G}'^{\\otimes m})\\to q'_*(\\mathcal{O}_{\\mathcal{Z}'}\\otimes \\mathcal{G}'^{\\otimes m}) \\to 0.\n\\]\nBy \\cite{ACH11}*{Corollary 4.5 (iii)}, there is $n_0$ such that for every $n\\ge n_0$, the multiplication map\n\\[\n\\operatorname{Sym}^a(q'_*\\mathcal{G}')\\otimes q'_*(\\mathcal{I}\\otimes \\mathcal{G}'^{\\otimes m}) \\to q'_*(\\mathcal{I}\\otimes \\mathcal{G}'^{\\otimes m+a})\n\\]\nis surjective. Then for every $b\\in B'$, also \n$$\\operatorname{Sym}^a(q'_*\\mathcal{G}')\\otimes q'_*(\\mathcal{I}\\otimes \\mathcal{G}'^{\\otimes n}) \\otimes k(b) \\cong( \\operatorname{Sym}^a(q'_*\\mathcal{G}')\\otimes k(b) )\\otimes_{k(b)}( q'_*(\\mathcal{I}\\otimes \\mathcal{G}'^{\\otimes n}) \\otimes k(b)) \\to q'_*(\\mathcal{I}\\otimes \\mathcal{G}'^{\\otimes n+a})\\otimes k(b)$$\nis surjective.\nFrom \\cite{ACH11}*{Corollary 4.5 (vi)} (or cohomology and base change) we have:\n\\[\n\\operatorname{Sym}^a(q'_*\\mathcal{G}')\\otimes k(b) \\cong H^0(\\mathcal{O}_{\\mathbb{P}^N}(a)),\n\\]\n\\[ q'_*(\\mathcal{I}\\otimes \\mathcal{G}'^{\\otimes n}) \\otimes k(b) \\cong H^0(\\mathcal{Y}_b,(\\mathcal{I}\\otimes \\mathcal{G}'^{\\otimes n})_{|\\mathcal{Y}_b}) = H^0(\\mathcal{Y}_b,\\mathcal{I}_{|\\mathcal{Y}_b }(n))\\text{ } \\text{ }\\text{ and }\n\\]\n\\[\nq'_*(\\mathcal{I}\\otimes \\mathcal{G}'^{\\otimes n+s}) \\otimes k(b) \\cong H^0(\\mathcal{Y}_b,(\\mathcal{I}\\otimes \\mathcal{G}'^{\\otimes n+s})_{|\\mathcal{Y}_b}) = H^0(\\mathcal{Y}_b,\\mathcal{I}_{|\\mathcal{Y}_b }(n+s)).\n\\]\nIn other terms, for every $b\\in B$ the ideal $\\mathcal{I}_{\\mathcal{Y}_b}$ is generated in degree $n_0$. Then for every $b\\in B$ there is a section $s\\in H^0(\\mathcal{Y}_b, \\mathcal{L}_{\\mathcal{Y},b})$ (and therefore also a section $s^{\\otimes m}\\in H^0(\\mathcal{Y}_b, \\mathcal{L}_{\\mathcal{Y},b}^{\\otimes m})$) such that $V(s)$ does not contain any irreducible component of $\\mathcal{Y}_b$ and the scheme-theoretic image of $\\frac{m_0}{r}\\mathcal{D}\\to \\mathcal{Y}$ restricted to $\\mathcal{Y}_b$ is contained in $V(s)$.\n\\end{proof}\n\n\\begin{Cor}\\label{corollary_the_section_that_vanishes_on_the_scheme_theoretic_image_of_m0\/rD_can_be_lifted_on_X}\nWith the notation and assumptions of Lemma \\ref{Lemma_there_is_a_section_vanishing_on_the_divisor}, for every $b\\in B$ and every $a>0$, there is a global section $t$ of $\\mathcal{L}_{|\\mathcal{X}_b}^{\\otimes a}$ that is not zero on the generic points of $\\mathcal{X}_b$ but its maps to 0 via the restriction map $H^0(\\mathcal{X}_b, \\mathcal{L}_{|\\mathcal{X}_b}^{\\otimes a})\\to H^0(\\frac{am_0}{r}\\mathcal{D}_b, \\mathcal{L}^{\\otimes a}_{|\\frac{am_0}{r}\\mathcal{D}_b})$.\n\\end{Cor}\n\n\\begin{proof}\nFrom Lemma \\ref{Lemma_there_is_a_section_vanishing_on_the_divisor}, for every $b \\in B$, there is a section $s$ that does not vanish on the generic points of $\\mathcal{Y}_b$ but vanishes on the restriction to $\\mathcal{Y}_b$ of the scheme theoretic image of $\\frac{m_0}{r}\\mathcal{D}$.\nFrom point (b) of  Lemma \\ref{Lemma_there_is_a_section_vanishing_on_the_divisor}, $q_*\\mathcal{L}_\\mathcal{Y}$ is a vector bundle on $B$, so there is an open subset $b\\in U\\subseteq B$ such that $(q_*\\mathcal{L}_\\mathcal{Y})_{|U}$ is free.\nThen, we can extend the section $s$ to a global section $s'\\in H^0(U,q_*(\\mathcal{L}_\\mathcal{Y})_{|U}) = H^0(U,(q_U)_*(\\mathcal{L}_{\\mathcal{Y},U}))$.\nBut $\\pi$ is a contraction, so $\\pi_{U} \\colon \\mathcal{X}_U\\to \\mathcal{Y}_U$ is still a contraction, and $\\pi_{U}^*\\mathcal{L}_{\\mathcal{Y},U} = \\mathcal{L}_U$.\nIn particular, consider the section $t\\in H^0(\\mathcal{X}_U,\\mathcal{L}_U)$ that is the pull-back of $s'$.\nThen, $t$ does not vanish on the generic points of $\\mathcal{X}_b$, as by Lemma \\ref{ample model p-pair} those are in bijection with the generic points of $\\mathcal{Y}_b$.\nMoreover, it vanishes along $\\frac{m_0}{r}\\mathcal{D}_b$: this is a local computation. Indeed, if we replace $\\mathcal{Y}$ and $\\mathcal{X}$ with appropriate open subsets $\\operatorname{Spec}(A)$, $\\operatorname{Spec}(B)$; and $f$ is the generator for the ideal sheaf of $\\frac{m_0}{r}\\mathcal{D}$, we have the following commutative diagram:\n$$\\xymatrix{B\\otimes k(b)\/f\\otimes 1 & B\/f \\ar[l] \\\\ B\\otimes k(b) \\ar[u] & t\\in B  \\ar[u] \\ar[l] \\\\ A\\otimes k(b) \\ar[u] & s\\in A. \\ar[u] \\ar[l]}$$\nSince the image of $s$ vanishes in $B\\otimes k(b)\/f\\otimes 1 $, so does the image of $t$. \n\nTherefore for every $a>0$, the section $t^{\\otimes a}\\in H^0(\\mathcal{X}_U,\\mathcal{L}_U^{\\otimes a})$ will vanish along $\\frac{am_0}{r}\\mathcal{D}_b$.\nIn other terms, if we look at the exact sequence\n\\[\n0 \\to H^0(\\mathcal{X}_b, \\mathcal{L}_{|\\mathcal{X}_b}^{\\otimes a}\\otimes \\mathcal{I}_{\\mathcal{D}_b}^{[\\frac{am_0}{r}]})\\to H^0(\\mathcal{X}_b, \\mathcal{L}_{|\\mathcal{X}_b}^{\\otimes a})\\xrightarrow{\\psi} H^0(\\frac{am_0}{r}\\mathcal{D}_b, \\mathcal{L}^{\\otimes a}_{|\\frac{am_0}{r}\\mathcal{D}_b}),\n\\]\nwe have an element of $H^0(\\mathcal{X}_b, \\mathcal{L}_{|\\mathcal{X}_b}^{\\otimes a})$ (namely, the restriction of $t^{\\otimes a}$ to $\\mathcal{X}_b$) that does not vanish along the generic points of $\\mathcal{X}_b$ but maps to 0 via $\\psi$.\n\\end{proof}\n\n\n\\begin{Lemma}\\label{lemma_lc_restricted_to_mD_pushes_forward_to_a_vector_bundle}\nWith the notation and assumptions of Lemma \\ref{Lemma_there_is_a_section_vanishing_on_the_divisor}, we will denote by $p\\colon (\\mathcal{X};\\mathcal{D})\\to B$ the $p$-stable family, and with $q \\colon (\\mathcal{Y},\\frac{1}{r}\\mathcal{D}_\\mathcal{Y})\\to B$ the resulting stable family of Theorem \\ref{thm_morphism_our_stack_to_kollars_one_for_reduced_bases}.\nLet $k$ be a multiple of $k_0$, where $k_0$ is as in Lemma \\ref{Lemma_there_is_a_section_vanishing_on_the_divisor}.\nAlso, set $\\mathcal{L} \\coloneqq \\mathcal{O}_\\mathcal{X}(k(K_{\\mathcal{X}\/B} + \\frac{1}{r}\\mathcal{D}))$.\n\nThen, there is an $a_0$ such that, for every $a\\ge a_0$, the sheaf $p_*(\\mathcal{L}_{|\\frac{am_0}{r}\\mathcal{D}}^{\\otimes a})$ is a vector bundle, and fits in an exact sequence as follows: \n\\[\n0\\to p_*(\\mathcal{O}_\\mathcal{X}(ka(K_{\\mathcal{X}\/B} + \\frac{1}{r}\\mathcal{D}) - \\frac{am_0}{r}\\mathcal{D})) \\to p_*( \\mathcal{L}^{\\otimes a}) \\to p_*(\\mathcal{L}_{|\\frac{am_0}{r}\\mathcal{D}}^{\\otimes a})\\to 0.\n\\]\n\\end{Lemma}\n\n\n\n\n\\begin{proof}\nFirst, from relative Serre's vanishing and Lemma \\ref{Lemma_there_is_a_section_vanishing_on_the_divisor} (b) and (d), there is an integer $a_0$ such that for every $b\\in B$ and $a \\geq a_0$, we have\n    \\begin{itemize}\n        \\item $H^i(\\mathcal{O}_{\\mathcal{X}_b}(ak(K_{\\mathcal{X}_b} + \\frac{1}{r}\\mathcal{D}_b) - \\frac{am_0}{r}\\mathcal{D}_b)=0$ for $i>0$; and\n        \\item $H^i(\\mathcal{O}_{\\mathcal{Y}_b}(ak(K_{\\mathcal{Y}_b} + \\frac{1}{r}\\mathcal{D}_{\\mathcal{Y},b})))=0$ for $i>0$.\n    \\end{itemize}\n\n\n\nFix a positive integer $a>a_0$.\nWe begin by considering the following exact sequence, where $\\mathcal{J}$ is the locally free ideal sheaf $\\mathcal{O}_\\mathcal{X}\\left(-\\frac{m_0}{r}\\mathcal{D}\\right) $:\n$$0\\to \\mathcal{L}^{\\otimes a}\\otimes \\mathcal{J}^{\\otimes a} \\to \\mathcal{L}^{\\otimes a} \\to \\mathcal{L}_{|\\frac{am_0}{r}\\mathcal{D}}^{\\otimes a}\\to 0.$$\nWe push it forward via $p$, and from the first bullet point and from cohomology and base change,\n$R^1p_*(\\mathcal{L}^{\\otimes a}\\otimes \\mathcal{J}^{\\otimes a}) = 0$, so we have:\n\\begin{equation} \\label{eq_ses}\n    0\\to p_*(\\mathcal{L}^{\\otimes a}\\otimes \\mathcal{J}^{\\otimes a} ) \\to p_*( \\mathcal{L}^{\\otimes a}) \\to p_*(\\mathcal{L}_{|\\frac{m_0a}{r}\\mathcal{D}}^{\\otimes a})\\to 0.\n\\end{equation}\n\nRecall that there is a contraction $\\pi \\colon \\mathcal{X}\\to \\mathcal{Y}$ such that, if we denote $\\mathcal{L}_\\mathcal{Y} \\coloneqq \\mathcal{O}_{\\mathcal{Y}}(k(K_{\\mathcal{Y}\/B} + \\frac{1}{r}\\mathcal{D}_\\mathcal{Y}))$, then  $\\pi^*(\\mathcal{L}_\\mathcal{Y}) = \\mathcal{L}$.\nIn particular\n\\begin{equation} \\label{eq_push_fwd}\n    p_*(\\mathcal{L}^{\\otimes a}) = q_*\\pi_* (\\mathcal{L}^{\\otimes a}) = q_*(\\mathcal{L}_\\mathcal{Y}^{\\otimes a}).\n\\end{equation}\n\nNow, pick $b\\in B$. From the first bullet point and from cohomology and base-change, we have \\begin{equation} \\label{eq_13}\n    p_*(\\mathcal{L}^{\\otimes a}\\otimes \\mathcal{J}^{\\otimes a} )\\otimes k(b)\\cong H^0(\\mathcal{X}_b, (\\mathcal{L}^{\\otimes a}\\otimes \\mathcal{J}^{\\otimes a})_{|\\mathcal{X}_b})  \\cong H^0(\\mathcal{X}_b,\\mathcal{O}_{\\mathcal{X}_b}(ak(K_{\\mathcal{X}_b} + \\frac{1}{r}\\mathcal{D}_b) - \\frac{am_0}{r}\\mathcal{D}_b))\n\\end{equation}\nand from the second bullet point, equation \\eqref{eq_push_fwd} and the fact that $\\pi_b\\colon \\mathcal{X}_b\\to \\mathcal{Y}_b$ is a contraction,\n\n\\begin{equation} \\label{eq_14}\n    p_*(\\mathcal{L}^{\\otimes a})\\otimes k(b)\\cong H^0(\\mathcal{Y}_b, \\mathcal{L}^{\\otimes a}_{\\mathcal{Y}, b}) \\cong  H^0(\\mathcal{Y}_b,\\mathcal{O}_{\\mathcal{Y}_b}(ak(K_{\\mathcal{Y}_b} + \\frac{1}{r}\\mathcal{D}_b)))  \\cong H^0(\\mathcal{X}_b, \\mathcal{L}^{\\otimes a}_{|\\mathcal{X}_b}).\n\\end{equation}\n\n\nNow, we tensor the sequence \\eqref{eq_ses} by $k(b)$, and we obtain \n\\[\n     p_*(\\mathcal{L}^{\\otimes a}\\otimes \\mathcal{J}^{\\otimes a})\\otimes k(b) \\to p_*( \\mathcal{L}^{\\otimes a})\\otimes k(b) \\to p_*(\\mathcal{L}_{|\\frac{m_0a}{r}\\mathcal{D}}^{\\otimes a})\\otimes k(b)\\to 0.\n\\]\nFrom the natural adjunction between pull-back and push-forward, we have the following commutative diagram:\n\\[\n\\xymatrix{ & p_*(\\mathcal{L}^{\\otimes a}\\otimes \\mathcal{J}^{\\otimes a})\\otimes k(b) \\ar[d]_{\\cong \\text{ from \\eqref{eq_13}}}\\ar[r]^-\\phi & p_*( \\mathcal{L}^{\\otimes a})\\otimes k(b) \\ar[r] \\ar[d]^{\\cong \\text{ from \\eqref{eq_14}}} &p_*(\\mathcal{L}_{|\\frac{m_0a}{r}\\mathcal{D}}^{\\otimes a})\\otimes k(b)\\ar[r] \\ar[d] & 0\n\\\\ 0 \\ar[r] & H^0( \\mathcal{X}_b, (\\mathcal{L}^{\\otimes a}\\otimes \\mathcal{J}^{\\otimes a})_{|\\mathcal{X}_b})\\ar[r] & H^0(\\mathcal{X}_b, \\mathcal{L}^{\\otimes a}_{|\\mathcal{X}_b}) \\ar[r] & H^0(\\frac{m_0a}{r}\\mathcal{D}, (\\mathcal{L}^{\\otimes a})_{|\\frac{m_0a}{r}\\mathcal{D}_b}) \\ar[r]^-{(\\ast)}& 0}\n\\]\nObserve that the exactness of the map $(\\ast)$ follows from the first bullet point.\nThen, from diagram chasing, also $\\phi$ is injective.\nIn particular,\n\\[\n\\dim_{k(b)}(p_*(\\mathcal{L}_{|\\frac{m_0a}{r}\\mathcal{D}}^{\\otimes a})\\otimes k(b)) = \\dim_{k(b)}(H^0(\\mathcal{X}_b, \\mathcal{L}^{\\otimes a}_{|\\mathcal{X}_b})) - \\dim_{k(b)}(H^0( \\mathcal{X}_b, (\\mathcal{L}^{\\otimes a}\\otimes \\mathcal{J}^{\\otimes a})_{|\\mathcal{X}_b})).\n\\]\nHowever the right-hand side does not depend on $b\\in B$. Therefore also the left-hand side does not depend on $b\\in B$, so $p_*(\\mathcal{L}_{|\\frac{m_0a}{r}\\mathcal{D}}^{\\otimes a})$ is a vector bundle.\n\\end{proof}\n\n\\begin{Cor}\\label{cor_determinant}\nWith the notation and assumptions of Lemma \\ref{lemma_lc_restricted_to_mD_pushes_forward_to_a_vector_bundle}, there is an isomorphism\n\\[\n\\det p_*(\\mathcal{L}^{\\otimes a}) \\cong \\det (p_*(\\mathcal{O}_{\\mathcal{X}}(ak(K_{\\mathcal{X}\/B} + \\frac{1}{r}\\mathcal{D}) - \\frac{am_0}{r}\\mathcal{D})))\\otimes \\det( p_*(\\mathcal{L}_{|\\frac{m_0a}{r}\\mathcal{D}}^{\\otimes a})).\n\\]\n\\end{Cor}\n\n\\begin{theorem}\\label{thm:main:step:proj}\nWith the notation and assumptions of Lemma \\ref{lemma_lc_restricted_to_mD_pushes_forward_to_a_vector_bundle}, further assume that the map $B\\to \\mathscr{F}_{n,p,I}$ given by the $p$-stable family $(\\mathcal{X};\\mathcal{D})\\to B$ is finite.\nThen for $a$ divisible enough, the line bundle $\\det p_*(\\mathcal{L}^{\\otimes a})$ is ample.\n\\end{theorem}\n\n\\begin{proof}\nWe plan on using Koll\\'ar's Ampleness Lemma, see \\cite{Kol90}.\nBy Corollary \\ref{cor_determinant}, it suffices to show that $\\det p_*(\\mathcal{L}^{\\otimes a}) \\otimes \\det (p_*(\\mathcal{O}_\\mathcal{X}(ak(K_{\\mathcal{X}\/B} + \\frac{1}{r}(1-\\frac{m_0}{k})\\mathcal{D}))))\\otimes \\det( p_*(\\mathcal{L}_{|\\frac{m_0a}{r}\\mathcal{D}}^{\\otimes a}))$ is ample.\n\nConsider the vector bundles\n\\[\n\\mathcal{Q}_a \\coloneqq p_*(\\mathcal{L}^{\\otimes a}) \\oplus p_*(\\mathcal{O}_\\mathcal{X}(ak(K_{\\mathcal{X}\/B} + \\frac{1}{r}(1-\\frac{m_0}{k})\\mathcal{D})))) \\oplus p_*(\\mathcal{L}_{|\\frac{m_0a}{r}\\mathcal{D}}^{\\otimes a})\\text{ }\\text{ }\\text{ }\\text{ and }\n\\]\n\\[\n\\mathcal{W}_{b,c} \\coloneqq \\operatorname{Sym}^b( p_*(\\mathcal{L}^{\\otimes c})) \\oplus \\operatorname{Sym}^b(\\mathcal{O}_\\mathcal{X}(ak(K_{\\mathcal{X}\/B} + \\frac{1}{r}(1-\\frac{m_0}{k})\\mathcal{D})) \\oplus \\operatorname{Sym}^b( p_*(\\mathcal{L}^{\\otimes c}))\n\\]\nfor appropriate choices of $a$, $b$, and $c$.\n\nWhen $a = bc$, there is a morphism $\\mathcal{W}_{b,c}\\to \\mathcal{Q}_{bc}$ given by the sum of the multiplication morphisms\n\\[\n\\operatorname{Sym}^b( p_*(\\mathcal{L}^{\\otimes c})) \\xrightarrow{\\alpha} p_*(\\mathcal{L}^{\\otimes bc}),\\quad \\operatorname{Sym}^b( p_*(\\mathcal{O}_\\mathcal{X}(ck(K_{\\mathcal{X}\/B} + \\frac{1}{r}(1-\\frac{m_0}{k})\\mathcal{D}))) \\xrightarrow{\\beta} p_*(\\mathcal{O}_\\mathcal{X}(bck(K_{\\mathcal{X}\/B} + \\frac{1}{r}(1-\\frac{m_0}{k})\\mathcal{D})),\n\\]\nand the composition of $\\alpha$ with the surjection in the exact sequence of Lemma \\ref{lemma_lc_restricted_to_mD_pushes_forward_to_a_vector_bundle}, denoted by\n\\[\n\\operatorname{Sym}^b( p_*(\\mathcal{L}^{\\otimes c})) \\xrightarrow{\\gamma} p_*(\\mathcal{L}_{|\\frac{m_0bc}{r}\\mathcal{D}}^{\\otimes bc}).\n\\]\n\nFirst, observe that for $c$ divisible enough, the vector bundles $p_*(\\mathcal{L}^{\\otimes c})$ and $  p_*(\\mathcal{O}_\\mathcal{X}(ck(K_{\\mathcal{X}\/B} + \\frac{1}{r}(1-\\frac{m_0}{k})\\mathcal{D})))$ are nef. Indeed, from the assumptions of Lemma \\ref{lemma_lc_restricted_to_mD_pushes_forward_to_a_vector_bundle}, the formation of $p_*(\\mathcal{L}^{\\otimes c})$ and $  p_*(\\mathcal{O}_\\mathcal{X}(ck(K_{\\mathcal{X}\/B} + \\frac{1}{r}(1-\\frac{m_0}{k})\\mathcal{D})))$ commutes with base change, so we can assume that the base $B$ is a smooth curve.\nThen the statement follows from \\cite{Fuj18}*{Theorem 1.11}.\nTherefore, their symmetric powers and sum are nef, so $\\mathcal{W}_{b,c}$ is nef for $c$ divisible enough.\n\nSince $\\mathcal{L}_{\\mathcal{Y}}$ is relatively ample and $q_*(\\mathcal{L}_{\\mathcal{Y}}^{\\otimes m}) \\cong p_*(\\mathcal{L}^{\\otimes m})$, there is a $b_0$ such that, for every $b\\ge b_0$, the map $\\alpha$ is surjective, and for the same reason also $\\beta$ is surjective (see  \\cite{ACH11}*{Corollary 4.5}).\nThen, also $\\gamma$ is surjective since it is the composition of surjective morphisms.\nTherefore we have a surjection $\\Phi:\\mathcal{W}_{b,c}\\to \\mathcal{Q}_{bc}$.\nWe denote by $G$ the structure group of $\\mathcal{W}_{b,c}$.\nThis gives a map of sets\n\\[\n\\Psi \\colon |B|\\to |[\\operatorname{Gr}(w,q)\/G]|,\n\\]\nwhere $w$ (resp. $q$) is the rank of $\\mathcal{W}_{b,c}$ (resp. $\\mathcal{Q}_{bc}$), and where for a stack $\\mathcal{Z}$ we denote by $|\\mathcal{Z}|$ its associated topological space.\nIf we show that $\\Psi$ has finite fibers, then the theorem follows from \\cite{Kol90}*{Ampleness Lemma, 3.9}.\n\nConsider two points $x_1$, $x_2$ that map to the same point via $\\Psi$.\nOver the point $x_1$ we have the surjection $\\Phi_{x_1} \\colon \\mathcal{W}_{b,c}\\otimes k(x_1)\\to \\mathcal{Q}_{bc}\\otimes k(x_1)$, and similarly we have one denoted by $\\Phi_{x_2}$ over $x_2$.\nChoose two isomorphisms\n\\[\n\\tau_1 \\colon H^0(\\mathbb{P}^{N}, \\mathcal{O}_{\\mathbb{P}^{N}}(1)) \\to p_*(\\mathcal{L}^{\\otimes c})\\otimes k(x_1), \\text{ and }\n\\]\n\\[\n\\tau_1' \\colon H^0(\\mathbb{P}^{M}, \\mathcal{O}_{\\mathbb{P}^{M}}(1)) \\to  p_*(\\mathcal{O}_\\mathcal{X}(ck(K_{\\mathcal{X}\/B} + \\frac{1}{r}(1-\\frac{M}{k})\\mathcal{D})))  \\otimes k(x_1).\n\\]\nSimilarly we define  $\\tau_2$ and $\\tau_2'$ for the same isomorphisms over $x_2$.\nThis gives an isomorphism\n\\[\nH^0(\\mathbb{P}^{N}, \\mathcal{O}_{\\mathbb{P}^{N}}(b)) \\oplus H^0(\\mathbb{P}^{M}, \\mathcal{O}_{\\mathbb{P}^{M}}(b)) \\oplus H^0(\\mathbb{P}^{N}, \\mathcal{O}_{\\mathbb{P}^{N}}(b)) \\xrightarrow{\\operatorname{Sym}^b(\\tau_1) \\oplus \\operatorname{Sym}^b(\\tau_1') \\oplus \\operatorname{Sym}^b(\\tau_1)} \\mathcal{W}_{b,c}\\otimes k(x_1).\n\\]\nSince $x_1$ and $x_2$ map to the same point via $\\Psi$, using the identifications above, there is an element $g\\in G$ such that $g\\operatorname{Ker}( \\Phi_{x_1}) =\\operatorname{Ker}(\\Phi_{x_2}) $.\n\nIn particular, we can choose a basis for $\\mathcal{W}_{b,c}\\otimes k(x_1)$ by first choosing a basis for each summand of the left-hand side, choosing the same basis for the first and third summand.\nThen, since $\\mathcal{W}_{b,c}$ is a direct sum of vector bundles, in this basis the linear transformation $g$ is block diagonal:\n\\[\ng = \\begin{bmatrix}A & 0 & 0 \\\\ 0 & B & 0 \\\\ 0 & 0 & A\\end{bmatrix}.\n\\]\nIn particular, $g$ will send $\\operatorname{Ker}(\\alpha_{x_1})$ (resp. $\\operatorname{Ker}(\\gamma_{x_1})$) to $\\operatorname{Ker}(\\alpha_{x_2})$ (resp. $\\operatorname{Ker}(\\gamma_{x_2})$).\nBut the kernel of $\\alpha_{x_1}$ corresponds to the symmetric functions of degree $b$ that vanish on a subvariety of $\\mathbb{P}^{n_1}$ isomorphic to $\\mathcal{Y}_{x_1}$, which we will still denote by $\\mathcal{Y}_{x_1}$.\nFrom \\cite{ACH11}*{Corollary 4.5}, up to choosing $b_0$ big enough, this kernel generates a graded ideal that corresponds to a subvariety isomorphic to $\\mathcal{Y}_{x_1}$.\nThe same conclusion holds for $\\operatorname{Ker}(\\alpha_{x_2})$, since there is a linear transformation (given by a block of the matrix $A$) that induces an isomorphism $\\operatorname{Ker}(\\alpha_{x_1})\\to \\operatorname{Ker}(\\alpha_{x_2})$.\nSo, in particular, this linear transformation induces a map of projective spaces, that gives an isomorphism $h \\colon \\mathcal{Y}_{x_1}\\to \\mathcal{Y}_{x_2}$.\n\nBy Corollary \\ref{corollary_the_section_that_vanishes_on_the_scheme_theoretic_image_of_m0\/rD_can_be_lifted_on_X}, $\\operatorname{Ker}(\\gamma_{x_1})$ contains a function that does not vanish along the generic points of the irreducible components of $\\mathcal{Y}_{x_1}$ but, if pulled back via $\\pi_{x_1} \\colon \\mathcal{X}_{x_1}\\to \\mathcal{Y}_{x_1}$, it vanishes along $\\frac{m_0a}{r}\\mathcal{D}_{x_1}$.\nTherefore, the zero locus of the polynomials in $\\operatorname{Ker}(\\gamma_{x_1})$ has codimension 1 in $\\mathcal{Y}_{x_1}$ and the union of its irreducible components of codimension 1, which we denote by $\\Gamma_{x_1}$, contains $\\operatorname{Supp}(\\frac{1}{r}\\mathcal{D}_{\\mathcal{Y},x_1})$.\nSince $\\Gamma_{x_1}$ has finitely many irreducible components of codimension one and the coefficients of $\\mathcal{D}_{\\mathcal{Y},x_1}$ are in the finite set $rI$, the divisor $\\mathcal{D}_{\\mathcal{Y},x_1}$ is determined up to finitely many possible choices of prime divisors and coefficients. \n\nA similar conclusion holds by replacing $x_1$ with $x_2$. Since in the description of $g$ the block at position (1,1) is the same as the block at position (3,3), we also know that $h(\\Gamma_{x_1}) = \\Gamma_{x_2}$.\n\nTherefore the fiber of $\\Psi(x_1)$ corresponds to stable pairs $(Y,D)$ in our moduli problem where $Y\\cong \\mathcal{Y}_{x_1}$, $\\Supp(D)\\subseteq \\Supp(\\Gamma_{x_1})$.\nBut there are finitely many such subvarieties, and since $B\\to \\mathscr{F}_{n,p,I}$ is finite by our assumptions, the fiber $\\Psi(x_1)$ has to be finite.\n\\end{proof}\n\n\\begin{Cor} \\label{cor proj moduli}\nConsider a proper DM stack $ \\mathcal{K}_{n,v,I}$ which satisfies the following two conditions:\n\\begin{enumerate}\n    \\item for every normal scheme $S$, the data of a morphism $f \\colon S\\to \\mathcal{K}_{n,v,I}$ is equivalent to a stable family of pairs $q \\colon (\\mathcal{Y},\\mathcal{D})\\to B$ with fibers of dimension $n$, volume $v$ and coefficients in $I$; and\n    \\item there is $m_0\\in \\mathbb{N}$ such that, for every $k \\in \\mathbb{N}$, there is a line bundle $\\mathcal{L}_k$ on $\\mathcal{K}_{n,v,I}$ such that, for every morphism $f$ as above, $f^*\\mathcal{L}_k \\cong \\det(q_*(\\omega_{\\mathcal{Y}\/B}^{[km_0]}(km_0\\mathcal{D})))$.\n\\end{enumerate}\nThen, for $m_0$ divisible enough, $\\mathcal{L}_k$ descends to an ample line bundle on the coarse moduli space of $\\mathcal{K}_{n,v,I}$ for every $k\\ge 1$. In particular, the coarse moduli space of $\\mathcal{K}_{n,v,I}$ is projective. \n\\end{Cor}\n\n\\begin{Remark}\nAs a concrete example of Corollary \\ref{cor proj moduli}, one can consider $\\mathcal{K}_{n,v,I}$ to be any moduli space of KSB-stable pairs $\\mathscr{K}$, such that, if we denote by $\\pi \\colon (\\mathscr{Y},\\mathscr{D})\\to \\mathscr{K}$ its universal family, then $\\omega_{\\mathscr{Y}\/\\mathscr{K}}^{[m_0]}(m_0\\mathscr{D})$ is $\\pi$-ample for $m_0$ divisible enough.\nIndeed, in this case, by cohomology and base change $\\pi_*\\omega_{\\mathscr{Y}\/\\mathscr{K}}^{[m_0]}(m_0\\mathscr{D})$ is a vector bundle for $m_0$ divisible enough, its formation commutes with base change from cohomology and base change, and the formation of the determinant commutes with base change as well.\n\\end{Remark}\n\n\\begin{proof}\nThe argument is divided into two steps. We denote by $K_{n,v,I}$ the coarse moduli space of $\\mathcal{K}_{n,v,I}$, and let $X$ be an irreducible component of it.\n\n\\begin{bf}Step 1.\\end{bf} There is a $p$-stable morphism $(\\mathcal{X};\\mathcal{D})\\to B$ which satisfies the following conditions:\n\\begin{enumerate}\n    \\item $B$ is normal and projective,\n    \\item The map $B\\to \\mathscr{F}_{n,p,I}$ is finite,\n    \\item There is a dense open subset $U\\subseteq B$ and a stable family $(\\mathcal{Y}_U,\\mathcal{D}_{\\mathcal{Y}, U})$ which satisfies the assumptions of Theorem \\ref{thm_morphism_our_stack_to_kollars_one_for_reduced_bases}, and\n    \\item The family $q:(\\mathcal{Y}_U,\\mathcal{D}_{\\mathcal{Y}, U})\\to U$ induces a map $U\\to \\mathcal{K}_{n,v,I} \\to K_{n,v,I}$ which dominates $X$.\n\\end{enumerate}\n\nConsider the generic point of $X$, and consider the stable pair $(\\mathcal{Z}_\\eta, \\mathcal{D}_{\\mathcal{Z},\\eta})$ it corresponds to. By Lemma \\ref{lift stable pair slc}, there is the spectrum of a field $\\operatorname{Spec}(\\mathbb{F})$ and a $p$-stable family over it $f:(\\mathcal{X}_\\eta; \\mathcal{D}_\\eta) \\to \\operatorname{Spec}(\\mathbb{F})$ such that the\nample model of $(\\mathcal{X}_\\eta; \\mathcal{D}_\\eta)$ is $(\\mathcal{Z}_\\eta, \\mathcal{D}_{\\mathcal{Z},\\eta})$. The family $f$ induces a morphism $\\operatorname{Spec}(\\mathbb{F}) \\to \\mathscr{F}_{n,p,I}$, and we can take $\\mathcal{F}$ to be the closure of its image. This is still a proper DM stack, it admits a finite and surjective cover by a scheme $B'\\to \\mathcal{F}$ with $B'$ a normal and \\emph{projective} scheme using \\cite{LMB18}*{Theorem 16.6}, Chow's lemma and potentially normalizing. Up to replacing $\\mathbb{F}$ with a finite cover of it, we can lift $\\operatorname{Spec}(\\mathbb{F})\\to \\mathcal{F}$ to $\\operatorname{Spec}(\\mathbb{F})\\to B'$, and consider $B$ an irreducible component of $B$ containing the image of $\\operatorname{Spec}(\\mathbb{F})\\to B'$.\n\nThe morphism $B\\to \\mathscr{F}_{n,p,I}$ induces a $p$-stable family $(\\mathcal{X};\\mathcal{D})\\to B$, and by construction its generic fiber admits an ample model.\nThen such an ample model can be spread out: there is an open subset $U\\subseteq B$ and a family $(\\mathcal{Y}_U,\\mathcal{D}_{\\mathcal{Y},U})$ as in Theorem \\ref{thm_morphism_our_stack_to_kollars_one_for_reduced_bases}, and the generic fiber $(\\mathcal{Y}_U,\\mathcal{D}_{\\mathcal{Y},U})\\to U$ is isomorphic to $(\\mathcal{Z}_\\eta, \\mathcal{D}_{\\mathcal{Z},\\eta})$. Therefore the image of the corresponding map $U\\to \\mathcal{K}_{n,v,I}\\to K_{n,v,I}$ contains the generic point of $X$, and since both $U$ and $X$ are irreducible, $U\\to X$ is dominant.\n\n\\begin{bf}Step 2.\\end{bf} The map $U\\to \\mathcal{K}_{n,v,I}$ extends to a finite map $\\Phi:B\\to \\mathcal{K}_{n,v,I}$.\n\nThe extension follows from Theorem \\ref{thm_morphism_our_stack_to_kollars_one_for_reduced_bases}. To prove that $\\Phi$ is finite, we can use Lemma \\ref{lemma:finitedness:paper:zsolt}. Indeed, if it was not finite, there would be a curve contained in a fiber of $\\Phi$. But then, up to replacing $C$ with an open subset of it, there would be a stable pair $(Y,D_Y)$ and a $p$-stable family $g:(\\mathcal{X};\\mathcal{D})\\to C$ as in Lemma \\ref{lemma:finitedness:paper:zsolt}. Therefore there would be finitely many isomorphism types of $p$-stable pairs in the fibers of $g$: this contradicts point (2) above.\n\n\\begin{bf}End of the proof.\\end{bf} From Theorem \\ref{thm:main:step:proj}, up to replacing $m_0$ with a multiple (which depends only on $X$), the line bundle $\\Phi^*\\mathcal{L}_k$ is ample. But a multiple of $\\mathcal{L}_k$ descends to a line bundle on $K_{n,v,I}$. In other terms, there is a line bundle $G$ on $K_{n,v,I}$ whose pull-back is $\\mathcal{L}_k^{\\otimes c}$ for a certain $c>0$. Therefore $G_{|X}$ is ample since it is ample once pulled-back via the \\emph{finite} map $B\\to X$. But if a line bundle is ample when restricted to each irreducible component, it is ample.\n\\end{proof}\n\\begin{Remark}\nWe remark that for the proof of Corollary \\ref{cor proj moduli} we do not need assumption (A).\nIndeed, with the notation of the proof of Corollary \\ref{cor proj moduli}, we just need assumption (A) to hold for the pairs parametrized by $\\mathscr{F}$.\nBut for the generic points of $\\mathscr{F}$, assumption (A) holds by construction, whereas for the other points of $\\mathscr{F}$ it holds from \\cite{HX13}*{Section 7}.\n\\end{Remark}\n\n\\begin{Cor}\\label{cor_projectivity_our_moduli} The stack $\\mathscr{F}_{n,p,I}$ admits a projective coarse moduli space.\n\\end{Cor}\n\\begin{proof}\nWe will denote with $F_{n,p,I}$ the coarse moduli space of $\\mathscr{F}_{n,p,I}$, which exists from Keel-Mori's theorem, and with $F_{n,p,I}^n$ the one of the normalization $\\mathscr{F}_{n,p,I}^n$ of $\\mathscr{F}_{n,p,I}$.\n\nConsider $\\epsilon_0$ such that $K_X+(1-\\epsilon_0)D$ is ample for every $p$-pair $(X;D)$ parametrized by $\\mathscr{F}_{n,p,I}$. Recall that such an\n$\\epsilon_0$ exists from Lemma \\ref{lemma_bound_for_epsilon} (or from boundedness). In particular, if we denote with $(\\mathscr{X};\\mathscr{D})\\to \\mathscr{F}_{n,p,I}$ the universal family, for $m$ divisibie enough the formation of $\\mathscr{G}:= \\det(p_*(\\mathcal{O}_{\\mathscr{X}}(m(K_{\\mathscr{X}\/\\mathscr{F}_{n,p,I}} + \\frac{1-\\epsilon}{r}\\mathscr{D}))))$ commutes with base change and an high enough power of $\\mathscr{G}$ descends to a line bundle $G$ on $F_{n,p,I}$. Therefore to prove that $G$ is ample, it suffices to replace $F_{n,p,I}$ (resp. $G$) with $F_{n,p,I}^n$ (resp. the pull-back $G^n$ of $G$ via $F_{n,p,I}^n \\to F_{n,p,I}$).\n\nConsider a proper DM stack $\\mathcal{K}_{n,p(1-\\epsilon_0),I}$ as in Corollary \\ref{cor proj moduli}. When $D$ has coefficient $1-\\epsilon_0$ the pairs parametrized by $\\mathscr{F}_{n,p,I}^n$ are stable of volume $p(1-\\epsilon_0)$, so we have a map $\\mathscr{F}_{n,p,I}^n\\to \\mathcal{K}_{n,p(1-\\epsilon_0),I}$ which is finite, as different points of $\\mathscr{F}_{n,p,I}$ parametrize different $p$-pairs, and the normalization is a finite morphism. From Corollary \\ref{cor proj moduli}, the formation of $\\mathcal{L}_k$ commutes with base change, so $\\mathcal{L}_k$ pulls back to $G^n$. Then $G^n$ is ample as it is the pull-back of an ample line bundle via a finite morphism.\\end{proof}\n\n\n\n\\bibliographystyle{amsalpha}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{Sec:Intro}\n\nLet $S_n$ be the set of permutations of $[n]=\\{1,\\ldots,n\\}$, which we write as words in one-line notation. Given $\\tau=\\tau_1\\cdots\\tau_m\\in S_m$, we say a permutation $\\sigma=\\sigma_1\\cdots\\sigma_n$ \\emph{contains the pattern} $\\tau$ if there exist indices $i_1<\\cdots<i_m$ in $[n]$ such that for all $j,k\\in[m]$, we have $\\sigma_{i_j}<\\sigma_{i_k}$ if and only if $\\tau_j<\\tau_k$. We say $\\sigma$ \\emph{avoids} $\\tau$ if it does not contain $\\tau$. Let $\\operatorname{Av}_n(\\tau^{(1)},\\tau^{(2)},\\ldots)$ be the set of permutations avoiding the patterns $\\tau^{(1)},\\tau^{(2)},\\ldots$ (this list of patterns could be finite or infinite). \n\nThe study of pattern avoidance in permutations, which began with Knuth's analysis of permutations that are sortable via a stack \\cite{Knuth}, has grown into a large, thriving area of research in combinatorics \\cite{Bona,Kitaev,Linton}. Recently, there has been a great amount of interest in pattern avoidance in other combinatorial objects such as words, inversion sequences, set partitions, and trees \\cite{Albert, Anderson, Baxter, Bevan, Branden, Burstein2, Corteel, Dairyko, Daly, DefantEnumeration, DefantFertilityWilf, DefantPoset, DefantClass, DefantKravitz, DefantSupertrees, Godbole, Levin, Mansour1, Mansour2, Martinez, Rowland, Yakoubov}. The papers \\cite{Anderson2, Bevan, DefantPoset, Levin, Yakoubov} consider the following general type of problem. Let $\\mathcal P$ be an $n$-element poset, and suppose we are given some canonical total ordering of the elements of $\\mathcal P$. We can view each linear extension of $\\mathcal P$ as a bijective labeling of the elements of $\\mathcal P$ with the elements of $[n]$. If we read these labels in the canonical order, we obtain a permutation. We can then ask how many linear extensions give rise to permutations that avoid certain patterns. An alternative, yet essentially equivalent, formulation of this problem is to consider a canonical \\emph{labeling} of $\\mathcal P$ and then view each linear extension of $\\mathcal P$ as an ordering. Again, each linear extension gives rise to a permutation, so we can ask the same enumerative questions about pattern avoidance. \n\nAs a first example, we consider the problems introduced in \\cite{Levin}. A \\emph{complete $k$-ary tree} is a rooted tree in which each vertex that is not in the penultimate or the last level has exactly $k$ children and all vertices in the last level are as far left as possible. We make the convention that trees grow up from their roots (as in real life). A \\emph{$k$-ary heap} is a complete $k$-ary tree whose vertices are bijectively labeled with the elements of $[n]$ so that each vertex is given a label that is smaller than the labels of its children. Associating complete $k$-ary trees with posets in the natural way, one can view a $k$-ary heap as a linear extension of the underlying complete $k$-ary tree. We endow each complete $k$-ary tree with the breadth-first ordering (traversing the levels from bottom to top with each level traversed from left to right). Reading the labels in a $k$-ary heap in this order yields a permutation in $S_n$, which we call the permutation \\emph{associated to} the $k$-ary heap. For example, the left image in Figure \\ref{Fig1} shows the unique complete binary tree with $12$ vertices. The right image shows a binary heap obtained by labeling this tree. The permutation associated to this binary heap is $1\\,3\\,2\\,4\\,6\\,9\\,7\\,12\\,5\\,11\\,8\\,10$. In Section \\ref{Sec:Heaps}, we prove a conjecture from \\cite{Levin} concerning binary heaps whose associated permutations avoid the pattern $321$. In fact, we will prove a much more general result. \n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=.8\\linewidth]{LinearExtensionsPIC1}\n\\end{center}  \n\\caption{The complete binary tree with $12$ vertices (left) and a $12$-vertex binary heap (right). Reading the labels of this heap in the breadth-first order yields the associated permutation $1\\,3\\,2\\,4\\,6\\,9\\,7\\,12\\,5\\,11\\,8\\,10$.}\n\\end{figure}\\label{Fig1}\n\nThe second type of problem we consider deals with the rectangular posets $\\EN_{s,t}$ studied in \\cite{Anderson2}. The poset $\\EN_{s,t}$ has $st$ elements that are labeled in a canonical fashion. The easiest way to define these posets is via examples. The Hasse diagrams of $\\EN_{3,2}$, $\\EN_{3,5}$, and $\\EN_{4,3}$, along with their canonical labelings, are shown in Figure \\ref{Fig2}. Each linear extension of $\\EN_{s,t}$ can be viewed as an ordering of the labels, which is a permutation in $S_{st}$. For example, the linear extensions of $\\EN_{3,2}$ correspond to the permutations $531642,536142,536412,563142$, and $563412$. In Section \\ref{Sec:Rectangular}, we prove several of the conjectures that the authors of \\cite{Anderson2} posed about pattern-avoiding linear extensions of these posets. \n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=.65\\linewidth]{LinearExtensionsPIC2}\n\\end{center}  \n\\caption{The Hasse diagrams of three rectangular posets. The numbers show the canonical labelings.}\n\\end{figure}\\label{Fig2}\n\n\\section{Pattern-Avoiding $k$-ary Heaps}\\label{Sec:Heaps}\n\nLet $\\mathcal H_n^k(\\tau^{(1)},\\tau^{(2)},\\ldots)$ denote the set of $n$-vertex $k$-ary heaps whose associated permutations avoid \\linebreak the patterns $\\tau^{(1)},\\tau^{(2)},\\ldots$. The authors of \\cite{Levin} found formulas for $|\\mathcal H_n^k(\\tau^{(1)},\\ldots,\\tau^{(r)})|$ for every set $\\{\\tau^{(1)},\\ldots,\\tau^{(r)}\\}\\subseteq S_3$ except the singleton set $\\{321\\}$. In fact, they were unable to explicitly enumerate binary heaps avoiding $321$. They did, however, compute $|\\mathcal H_n^2(321)|$ for $1\\leq n\\leq 31$ and prove that \\[2^{n-1}<|\\mathcal H_n^2(321)|<4^n.\\] Their data led them to conjecture that\\footnote{The actual conjecture was written as ``$|\\mathcal H_n^2(321)|\\sim c^n$ for some $c\\in(3.66,4)$,\" but it is clear from context that the authors really meant what is written in \\eqref{Eq1}.} \n\\begin{equation}\\label{Eq1}\n\\lim_{n\\to\\infty}|\\mathcal H_n^2(321)|^{1\/n}\\text{ exists and is in the interval }(3.66,4].\n\\end{equation} We will see that this conjecture follows as a special consequence of the main theorem of this section. To state this theorem, we need one additional piece of terminology. \n\nGiven $\\lambda=\\lambda_1\\cdots\\lambda_\\ell\\in S_\\ell$ and $\\mu=\\mu_1\\cdots\\mu_m\\in S_m$, the \\emph{direct sum} of $\\lambda$ and $\\mu$, denoted $\\lambda\\oplus\\mu$, is the permutation in $S_{\\ell+m}$ obtained by ``placing $\\mu$ above and to the right of $\\lambda$.\" More formally, the $i^\\text{th}$ entry of $\\lambda\\oplus\\mu$ is \\[(\\lambda\\oplus\\mu)_i=\\begin{cases} \\lambda_i & \\mbox{if } 1\\leq i\\leq \\ell; \\\\ \\mu_{i-\\ell}+\\ell & \\mbox{if } \\ell+1\\leq i\\leq \\ell+m. \\end{cases}\\] A permutation is called \\emph{sum indecomposable} if it cannot be written as the direct sum of two shorter permutations. \n\n\\begin{theorem}\\label{Thm1}\nFix an integer $k\\geq 2$ and a nonempty sequence of permutation patterns $\\tau^{(1)},\\tau^{(2)},\\ldots$. If the permutations $\\tau^{(1)},\\tau^{(2)},\\ldots$ are all sum indecomposable, then \\[\\lim_{n\\to\\infty}|\\operatorname{Av}_n(\\tau^{(1)},\\tau^{(2)},\\ldots)|^{1\/n}=\\lim_{n\\to\\infty}|\\mathcal H_n^k(\\tau^{(1)},\\tau^{(2)},\\ldots)|^{1\/n}.\\]\n\\end{theorem}\n\n\\begin{remark}\\label{Rem1}\nHow do we know that the limits in Theorem \\ref{Thm1} actually exist? Because the permutations $\\tau^{(1)},\\tau^{(2)},\\ldots$ are sum indecomposable, we have an injective map $S_\\ell\\times S_m\\hookrightarrow S_{\\ell+m}$ given by $(\\lambda,\\mu)\\mapsto\\lambda\\oplus\\mu$ for all $\\ell,m\\geq 1$. This shows that the sequence $(|\\operatorname{Av}_n(\\tau^{(1)},\\tau^{(2)},\\ldots)|)_{n\\geq 1}$ is supermultiplicative, so the limit on the left-hand side exists by Fekete's lemma. We also have an injective map $\\mathcal H_n^k(\\tau^{(1)},\\tau^{(2)},\\ldots)\\hookrightarrow\\operatorname{Av}_n(\\tau^{(1)},\\tau^{(2)},\\ldots)$ sending each $k$-ary heap to its associated permutation. This shows that $\\limsup\\limits_{n\\to\\infty}|\\mathcal H_n^k(\\tau^{(1)},\\tau^{(2)},\\ldots)|^{1\/n}\\leq\\lim\\limits_{n\\to\\infty}|\\operatorname{Av}_n(\\tau^{(1)},\\tau^{(2)},\\ldots)|^{1\/n}$. We will see in the proof of Theorem \\ref{Thm1} that $\\lim\\limits_{n\\to\\infty}|\\operatorname{Av}_n(\\tau^{(1)},\\tau^{(2)},\\ldots)|^{1\/n}\\leq\\liminf\\limits_{n\\to\\infty}|\\mathcal H_n^k(\\tau^{(1)},\\tau^{(2)},\\ldots)|^{1\/n}$, so the limit on the right-hand side exists as well. It is interesting to note that this right-hand limit does not depend on $k$. \\hfill$\\lozenge$ \n\\end{remark}\n\n\nLet $C_n=\\frac{1}{n+1}{2n\\choose n}$ denote the $n^\\text{th}$ Catalan number. It is well known that $|\\operatorname{Av}_n(321)|=C_n$. This implies that $\\lim\\limits_{n\\to\\infty}|\\operatorname{Av}_n(321)|^{1\/n}=4$. Consequently, we can specialize Theorem \\ref{Thm1} to obtain the following result. In the special case in which $k=2$, this settles the conjecture from \\cite{Levin} that is stated in \\eqref{Eq1}. \n\n\\begin{corollary}\\label{Cor2}\nIf $k\\geq 2$, then $\\displaystyle\\lim_{n\\to\\infty}|\\mathcal H_n^k(321)|^{1\/n}=4$.\n\\end{corollary}\n\n\\begin{proof}[of Theorem \\ref{Thm1}]\nFix an integer $k\\geq 2$ and a sequence of sum indecomposable permutation patterns $\\tau^{(1)},\\tau^{(2)},\\ldots$. As mentioned in Remark \\ref{Rem1}, we must prove that \\[\\lim_{n\\to\\infty}|\\operatorname{Av}_n(\\tau^{(1)},\\tau^{(2)},\\ldots)|^{1\/n}\\leq\\liminf_{n\\to\\infty}|\\mathcal H_n^k(\\tau^{(1)},\\tau^{(2)},\\ldots)|^{1\/n}.\\] To ease \nnotation, let \\[a(n)=|\\operatorname{Av}_n(\\tau^{(1)},\\tau^{(2)},\\ldots)|^{1\/n},\\quad b(n)=|\\mathcal H_n^k(\\tau^{(1)},\\tau^{(2)},\\ldots)|^{1\/n},\\] \\[\\quad L_a=\\lim_{n\\to\\infty}a(n),\\quad L_b=\\liminf_{n\\to\\infty}b(n).\\] The proof is trivial if \none of the patterns $\\tau^{(i)}$ is the permutation $1\\in S_1$, so we may assume otherwise. Because the patterns $\\tau^{(1)},\\tau^{(2)},\\ldots$ are sum indecomposable, the identity permutation $123\\cdots n$ is in $\\operatorname{Av}_n(\\tau^{(1)},\\tau^{(2)},\\ldots)$ and is the permutation associated to an element of $\\mathcal H_n^k(\\tau^{(1)},\\tau^{(2)},\\ldots)$. Thus, $L_a,L_b\\geq 1$. Fix $\\varepsilon\\in(0,1)$, and let $N\\geq 1$ be such that $a(n-\\left\\lfloor(n-1)\/k\\right\\rfloor)>L_a-\\varepsilon$ and $b(\\left\\lfloor(n-1)\/k\\right\\rfloor)\\!>L_b-\\varepsilon$ for every $n\\geq N$. Now choose $n\\geq N$ such that $b(n)<L_b+\\varepsilon$, and put $m=\\left\\lfloor(n-1)\/k\\right\\rfloor$. \n\n\\noindent {\\bf Claim:} If $\\lambda\\in S_m$ is a permutation associated to a $k$-ary heap in $\\mathcal H_m^k(\\tau^{(1)},\\tau^{(2)},\\ldots)$ and \\linebreak $\\mu\\in\\operatorname{Av}_{n-m}(\\tau^{(1)},\\tau^{(2)},\\ldots)$, then $\\lambda\\oplus\\mu$ is a permutation associated to a $k$-ary heap in $\\mathcal H_n^k(\\tau^{(1)},\\tau^{(2)},\\ldots)$. \n\nLet us see how this claim implies Theorem \\ref{Thm1}. Suppose by way of contradiction that $L_b<L_a$. Assuming the claim, we have \\[(L_b+\\varepsilon)^n>b(n)^n=|\\mathcal H_n^k(\\tau^{(1)},\\tau^{(2)},\\ldots)|\\geq|\\mathcal H_m^k(\\tau^{(1)},\\tau^{(2)},\\ldots)|\\cdot|\\operatorname{Av}_{n-m}(\\tau^{(1)},\\tau^{(2)},\\ldots)|\\] \\[=b(m)^ma(n-m)^{n-m}>(L_b-\\varepsilon)^m(L_a-\\varepsilon)^{n-m}.\\] Since $m<n\/2$ and $L_b<L_a$, we have $(L_b-\\varepsilon)^m(L_a-\\varepsilon)^{n-m}>(L_b-\\varepsilon)^{n\/2}(L_a-\\varepsilon)^{n\/2}$. Hence, $(L_b+\\varepsilon)^2>(L_b-\\varepsilon)(L_a-\\varepsilon)$. Letting $\\varepsilon$ tend to $0$ shows that $L_b\\geq L_a$, contradicting our assumption that $L_b<L_a$. This completes the proof of Theorem \\ref{Thm1} assuming the claim. \n\nNow, let $\\lambda$ and $\\mu$ be as in the claim. Let $T$ be the complete $k$-ary tree with $n$ vertices, and let $v_i$ be the vertex of $T$ that appears $i^\\text{th}$ in the\nbreadth-first ordering. It is straightforward to check that the vertices $v_{m+1},\\ldots,v_n$ are incomparable (meaning none is a descendant of another). It follows that $\\lambda\\oplus\\mu$ is the permutation associated to some $k$-ary heap in $\\mathcal H_n^k$. Because $\\lambda$ and $\\mu$ avoid the sum indecomposable permutations $\\tau^{(1)},\\tau^{(2)},\\ldots$, their direct sum $\\lambda\\oplus\\mu$ must also avoid these permutations. This completes the proof of the claim. \n\\end{proof}\n\n\\section{Rectangular Posets}\\label{Sec:Rectangular}\n\nAn \\emph{inversion} of a permutation $\\pi=\\pi_1\\cdots\\pi_n\\in S_n$ is a pair $(i,j)$ such that $i<j$ and $\\pi_i>\\pi_j$. Let $\\inv(\\pi)$ denote the number of inversions of $\\pi$. Following \\cite{Anderson2}, we let $\\EN_{s,t}(\\tau)(q)=\\sum_{\\pi\\in\\EN_{s,t}(\\tau)}q^{\\inv(\\pi)}$, where $\\EN_{s,t}(\\tau)$ is the set of linear extensions of $\\EN_{s,t}$ (viewed as permutations of the labels of $\\EN_{s,t}$) that avoid the pattern $\\tau$. The following three theorems were stated as conjectures in \\cite{Anderson2}\\footnote{Technically speaking, Theorem \\ref{Thm2} was stated incorrectly in that article.}.\n  \n\\begin{theorem}\\label{Thm2}\nFor all $t\\geq 1$, we have \\[\\EN_{3,t}(1243)(q)=\\frac{q^{3(t^2-t+1)}(1-q^{2t-1}-2q^{2t}+q^{3t-1}+q^{3t})}{(1-q)(1-q^2)}.\\]\n\\end{theorem}  \n\n\\begin{theorem}\\label{Thm3}\nFor all $s\\geq 1$, we have \\[\\EN_{s,2}(2143)(q)=q^{(2s-1)(s-1)}(1+q)^{s-1}.\\]\n\\end{theorem}\n  \n\\begin{theorem}\\label{Thm4}\nFor all $s\\geq 1$, we have \\[\\EN_{s,3}(2143)(q)=q^{9{s\\choose 2}}F_s(1\/q),\\] where $F_s(r)$ is defined by $F_0(r)=F_1(r)=1$ and $F_s(r)=(1+r+2r^2)F_{s-1}(r)+r^3F_{s-2}(r)$ for $s\\geq 2$. \n\\end{theorem}\n\n\\begin{proof}[of Theorem \\ref{Thm2}]\nIt is easy to verify this theorem when $t\\in\\{1,2\\}$, so assume $t\\geq 3$. Let $\\pi=\\pi_1\\cdots\\pi_n$ be a linear extension of $\\EN_{3,t}(1243)$ (viewed as a permutation of the labels). Note that $2t$ appears before $2$ in $\\pi$ because, otherwise, the entries $1,2,2t,t$ would form a $1243$ pattern. Similarly, $3t$ must appear before $t+2$, lest the entries $t+1,t+2,3t,2t$ form a $1243$ pattern. It follows that if we remove the entries $1$ and $t+1$ from $\\pi$, then we will be left with the permutation \\[(2t+1)(2t+2)\\cdots(3t)(t+2)(t+3)\\cdots(2t)23\\cdots t.\\] Let $i,j$ be such that $\\pi_{i+1}=t+1$ and $\\pi_{j+1}=1$. We can easily check that the possibilities for $i$ and $j$ are given by $i\\in\\{1,\\ldots,t\\}$ and $j\\in\\{i+1,\\ldots,2t\\}$. We find that \\[\\EN_{3,t}(1243)(q)=\\sum_{i=1}^t\\sum_{j=i+1}^{2t}q^{3t^2-3t+i+j}.\\] This can easily be rewritten as $\\dfrac{q^{3(t^2-t+1)}(1-q^{2t-1}-2q^{2t}+q^{3t-1}+q^{3t})}{(1-q)(1-q^2)}$.  \n\\end{proof}\n\n\\begin{proof}[of Theorem \\ref{Thm3}]\nLet $S$ be the collection of subsets of $\\{1,3,5,\\ldots,2s-3\\}$, and define $\\eta\\colon\\EN_{s,2}(2143)\\to S$ by \\[\\eta(\\pi)=\\{i\\in\\{1,3,5,\\ldots,2s-3\\}\\colon i+3\\text{ appears before }i\\text{ in }\\pi\\}.\\] The map $\\eta$ is a bijection. It is now easy to check that \\[\\EN_{s,2}(2143)(q)=\\sum_{X\\subseteq\\{1,3,5,\\ldots,2s-3\\}}q^{(2s-1)(s-1)+|X|}=q^{(2s-1)(s-1)}(1+q)^{s-1}. \\qedhere\\] \n\\end{proof}\n\n\\begin{proof}[of Theorem \\ref{Thm4}]\nLet $H_\\ell$ be the set of $\\pi=\\pi_1\\cdots\\pi_{3\\ell}\\in\\EN_{\\ell,3}(2143)$ such that $\\pi_2=3\\ell-1$. Let $H_\\ell(q)=\\sum_{\\pi\\in H_\\ell}q^{\\inv(\\pi)}$. Fix $s\\geq 2$, and let $J(r_1,\\ldots,r_k)$ be the set of permutations $\\pi=\\pi_1\\cdots\\pi_{3s}\\in\\EN_{s,3}(2143)$ such that $\\pi_i=r_i$ for all $i\\in\\{1,\\ldots,k\\}$. One can check that the sets \\[J(3s-2,3s-1,3s),\\hspace{.5cm}J(3s-2,3s-1,3s-5,3s),\\hspace{.5cm}J(3s-2,3s-5,3s-1,3s),\\] \\[J(3s-2,3s-5,3s-1,3s-4,3s),\\hspace{.5cm}\\text{and}\\hspace{.5cm}J(3s-2,3s-1,3s-5,3s-4,3s)\\] partition $\\EN_{s,3}(2143)$. Call these sets $J_1,J_2,J_3,J_4$, and $J_5$, respectively. \n\nThe operation that consists of removing the entries $3s-2,3s-1$, and $3s$ from a permutation establishes bijections $J_1\\to\\EN_{s-1,3}(2143)$, $J_2\\to\\EN_{s-1,3}(2143)$, $J_3\\to\\EN_{s-1,3}(2143)$, $J_4\\to H_{s-1}$, and $J_5\\to H_{s-1}$. After taking into account the number of inversions that are removed by each of these bijections, we obtain the identities \\[\\sum_{\\pi\\in J_1}q^{\\text{inv}(\\pi)}=q^{9(s-1)}\\EN_{s-1,3}(2143)(q),\\quad\\sum_{\\pi\\in J_2}q^{\\text{inv}(\\pi)}=q^{9(s-1)-1}\\EN_{s-1,3}(2143)(q),\\] \n\\[\\sum_{\\pi\\in J_3}q^{\\text{inv}(\\pi)}=q^{9(s-1)-2}\\EN_{s-1,3}(2143)(q),\\quad\\sum_{\\pi\\in J_4}q^{\\text{inv}(\\pi)}=q^{9(s-1)-3}H_{s-1}(q),\\] and \n\\[\\sum_{\\pi\\in J_5}q^{\\text{inv}(\\pi)}=q^{9(s-1)-2}H_{s-1}(q).\\] This yields \n\\begin{equation}\\label{Eq2}\n\\EN_{s,3}(2143)(q)=q^{9(s-1)-2}(1+q+q^2)\\EN_{s-1,3}(2143)(q)+q^{9(s-1)-3}(1+q)H_{s-1}(q).\n\\end{equation}\nThe sets $J_1,J_2,J_5$ partition $H_s$, so \n\\begin{equation}\\label{Eq3}\nH_s(q)=q^{9(s-1)-1}(1+q)\\EN_{s-1,3}(2143)(q)+q^{9(s-1)-2}H_{s-1}(q).\n\\end{equation}\nSolving the recurrence relations \\eqref{Eq2} and \\eqref{Eq3} subject to the initial conditions $\\EN_{1,3}(2143)(q)=H_1(q)=1$, we obtain the desired identity $\\EN_{s,3}(2143)(q)=q^{9{s\\choose 2}}F_s(1\/q)$. \n\\end{proof}\n\nThe article \\cite{Anderson2} also poses several conjectures concerning the polynomials $F_s(q)$ and various OEIS sequences \\cite{OEIS}. We settle many of these conjectures\\footnote{We also correct some typos made in the original statements of these conjectures.} in the following theorem. Since our focus is on the combinatorics of pattern-avoiding linear extensions and not these specific polynomials, we omit some details from the proof. In what follows, let $[q^r]G(q)$ denote the coefficient of $q^r$ in the Laurent series $G(q)$.\n\n\\begin{theorem}\\label{Thm5}\nDefine the polynomials $F_s(r)$ by $F_0(r)=F_1(r)=1$ and $F_s(r)=(1+r+2r^2)F_{s-1}(r)+r^3F_{s-2}(r)$ for $s\\geq 2$. For $s\\geq 2$, \n\\begin{itemize}\n\\item the values of $[q^3]F_s(q)$ are given by OEIS sequence A134465;\n\\item the values of $[q^{2s-2}]F_s(q)$ are given by OEIS sequence A098156; \n\\item the values of $[q^{s-1}]F_s(q)$ are given by OEIS sequence A116914;  \n\\item the values of $[q^s]F_s(q)$ are given by OEIS sequence A072547;\n\\item the values of $[q^{s+1}]F_s(q)$ are given by OEIS sequence A002054;  \n\\item the values of $[q^{s+2}]F_s(q)$ are given by OEIS sequence A127531.\n\\end{itemize}\n\\end{theorem} \n\n\\begin{proof}\nLet $A(x,q)=\\sum_{s\\geq 0}F_s(q)x^s$. The recurrence for $F_s(q)$ translates into the identity \n\\begin{equation}\\label{Eq4}\nA(x,q)=\\frac{1-(q+2q^2)x}{1-(1+q+2q^2)x-q^3x^2}.\n\\end{equation}\nComputing $\\dfrac{1}{6}\\dfrac{\\partial^3}{\\partial q^3}A(x,q)$ proves the first bullet point. The remainder of the proof makes use of the method of diagonals, which is discussed in Section 6.3 of \\cite{Stanley}.\n\nIf we view $qA(x\/q^2,q)$ as a function of the complex variable $q$, then \\[\\sum_{s\\geq 0}([q^{2s-2}]F_s(q))x^s=[q^{-1}](qA(x\/q^2,q))=\\frac{1}{2\\pi i}\\int_{|q|=\\rho}qA(x\/q^2,q)\\,dq,\\] where $\\rho>0$ is sufficiently small and the integral is taken over the circle of radius $\\rho$ centered at the origin. By the Residue Theorem, this is \\[\\sum_{j=1}^r\\Res_{q=u_j(x)}(qA(x\/q^2,q)),\\] where $u_1(x),\\ldots,u_r(x)$ are the singularities of $qA(x\/q^2,q)$ (viewed as functions of $x$) that tend to $0$ as $x\\to 0$. We can explicitly compute that $r=2$ and that \\[u_1(x)=\\frac{x(1+x)+(1-x)\\sqrt{x(4+x)}}{2(1-2x)}\\quad\\text{and}\\quad u_2(x)=\\frac{x(1+x)-(1-x)\\sqrt{x(4+x)}}{2(1-2x)}.\\] Let $U(x,q)=q^2(x-q(1-2x))$ and $V(x,q)=x+qx(1+x)-q^2(1-2x)$ so that $qA(x\/q^2,q)=U(x,q)\/V(x,q)$. Let $V_q(x,q)=\\dfrac{\\partial}{\\partial q}V(x,q)$. Since $u_1(x)$ and $u_2(x)$ are simple poles of $qA(x\/q^2,q)$, we find that \\[\\Res_{q=u_j(x)}(qA(x\/q^2,q))=\\frac{U(x,u_j(x))}{V_q(x,u_j(x))}\\quad\\text{for }j\\in\\{1,2\\}.\\] We have \\[\\sum_{j=1}^2\\Res_{q=u_j(x)}(qA(x\/q^2,q))=\\sum_{j=1}^2\\frac{U(x,u_j(x))}{V_q(x,u_j(x))}=\\frac{x(1-2x+x^2+x^3)}{(1-2x)^2},\\] and this proves the second bullet point. \n\nTo prove the third, fourth, fifth, and sixth bullet points, we choose an integer $\\ell\\leq 2$ and view \\linebreak $q^{-\\ell-1}A(x\/q,q)$ as a complex function of the variable $q$. As above, we have \\[\\sum_{s\\geq 0}([q^{s+\\ell}]F_s(q))x^s=[q^{-1}](q^{-\\ell-1}A(x\/q,q))=\\sum_{j=1}^t\\Res_{q=v_j(x)}(q^{-\\ell-1}A(x\/q,q)),\\] where $v_1(x),\\ldots,v_t(x)$ are the singularities of $q^{-\\ell-1}A(x\/q,q)$ that tend to $0$ as $x\\to 0$. Let $Y(x,q)=-1+x+2qx$ and $Z(x,q)=x-(1-x)q+x(2+x)q^2$ so that $q^{-\\ell-1}A(x\/q,q)=q^{-\\ell}Y(x,q)\/Z(x,q)$. If $\\ell\\leq 0$, then the only singularity of $q^{-\\ell-1}A(x\/q,q)$ that tends to $0$ as $x\\to 0$ is \\[v_1(x)=\\dfrac{1-x-\\sqrt{1-2x-7x^2-4x^3}}{2x(2+x)}.\\] If $\\ell\\in\\{1,2\\}$, then there is one other singularity, which is $v_2(x)=0$. One can check that \\[\\Res_{q=0}(q^{-2}A(x\/q,q))=1-x^{-1}\\quad\\text{\\and}\\quad\\Res_{q=0}(q^{-3}A(x\/q,q))=1+2x^{-1}-x^{-2}.\\] Note that in each of these expressions, the coefficient of $x^s$ is $0$ for every $s\\geq 2$. It follows that for all integers $\\ell\\leq 2$ and $s\\geq 2$, the coefficient of $x^s$ in $\\sum_{s\\geq 0}([q^{s+\\ell}]F_s(q))x^s$ agrees with the coefficient of $x^s$ in $\\Res_{q=v_1(x)}(q^{-\\ell-1}A(x\/q,q))$. We have \\[\\Res_{q=v_1(x)}(q^{-\\ell-1}A(x\/q,q))=\\frac{v_1(x)^{-\\ell}Y(x,v_1(x))}{Z_q(x,v_1(x))},\\] where $Z_q(x,q)=\\dfrac{\\partial}{\\partial q}Z(x,q)$. When $\\ell\\in\\{-1,0,1,2\\}$, we can explicitly compute and simplify these expressions in order to obtain proofs of the last four bullet points. \n\\end{proof}\n\n\\section{Acknowledgments}\nThe author thanks the anonymous referees for helpful comments. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\n\\section{Introduction}\n\nOne of the most compelling problems facing astrophysics in the early\n21$^{st}$ century is to understand the early Universe, and\nin particular, the epoch known as the\nCosmic Dawn \\citep{Astro2010}, when starlight from\nthe first generations of stars reionized the neutral intergalactic\nmedium, and when the structure of the Universe was set into the general\npatterns that still prevail today.\nRecent studies have shown that this reionization is complete\naround redshift $z \\sim 6$ \\citep{Fan06}.  \nThe Universe at such high redshifts is challenging to study, but progress has been\nrapid in recent years, with redshift records being broken repeatedly\nthrough a variety of techniques \\citep[e.g.][]{Tanvir09,Bouwens10a,Cucciara11}\ninvolving observations of high-$z$ quasars, galaxies, and gamma-ray\nbursts (GRBs).\n\nEach of these probes of the high-$z$ universe has strengths and weaknesses.\nQuasars are extremely luminous and can be bright sources even at \nhigh redshift, but the high UV and X-ray flux from the central engine\nstrongly affects the properties of the host galaxy, making it atypical.  Furthermore,\nthe density of quasars drops rapidly at $z>6$.\nNormal\ngalaxies are being found, possibly at very high redshifts, in the Hubble Ultra\nDeep Field \\citep[e.g.][]{Bouwens10a} through photometric dropouts.  These\nphotometric redshifts are not entirely reliable, as the dropouts can also be\nproduced by dust extinction.  More importantly, the galaxies being\nfound represent the bright end of the luminosity distribution, and\neven so, are so faint that the spectroscopy needed to probe their gas\nis beyond the capabilities of current instrumentation.\nGRBs, on the other hand, probe star formation regions in all types of\ngalaxies, and are so intrinsically bright that they can be used to\nprobe the interstellar medium (gas and dust content) of host galaxies\ntoo faint to see, while the duration of the burst is short enough so\nthat only gas in the immediate vicinity of the central engine is\ndisturbed by the strong UV and X-ray emission.  However, the\nshort duration means that observations of GRBs must be carried out\nrapidly, before the GRB fades too much.\n\n\\begin{figure}[tbp]\n\\resizebox{\\hsize}{!}{\\includegraphics[clip=true]{janus.eps}}\n\\caption{\n\\footnotesize\n{\\it JANUS}\\ design concept.  The mission features an X-ray Coded Aperture\nTelescope (XCAT; gold-colored array of 10 modules mounted on the\nsatellite top deck) to find GRBs and X-ray transients, and a Near\nInfrared Telescope (NIRT) that measures their redshifts with a\nlow-resolution objective prism spectrometer.\n}\n\\label{fig:JANUS}\n\\end{figure}\n\nThe use of GRBs as probes of distant galaxies has blossomed in the\npast six years \\cite[e.g.][]{Prochaska06,Vreeswijk07,Prochaska07,Tumlinson07,Tejos08,Prochaska08,Chen09,DElia09,Levesque10b,Levesque10a}, \nsince the launch of the \\swift\\ satellite\n\\citep{Gehrels04_Swift}, which provides rapid arcsecond positions of GRBs.\nStill, after six years of operations, \\swift\\ has found only 3 GRBs\nwith spectroscopic redshifts exceeding 6 (Table~\\ref{tbl:GRBs}).\n\\begin{table*}\n\\caption{Time delays to obtain redshift measurements for high-$z$ \\swift\\ GRBs}\n\\label{tbl:GRBs}\n\\begin{center}\n\\begin{tabular}{lccccl}\n\\hline\n\\\\\nGRB & $z$ & Type$^a$ & $T_P$ & $T_S$ & References \\\\\n\\hline\n\\\\\n050904 & 6.295 & S & 10 hr & 3.5 days & \\citet{Cusumano06_GRB050904,\n  Tagliaferri05}, \\\\\n             &           &           &         &      & \\qquad \\citet{Kawai06} \\\\\n060116 & 6.6 & P & 41 hr & N\/A & GCN Circ. 4545 \\\\\n080913 & 6.695 & S & 10 hr & 11 hr & \\citet{Greiner09} \\\\\n090423 & 8.2 & S & 7 hr & 24 hr &  \\citet{Tanvir09} \\\\\n090429B & 9.4 & P & 2.5 hr & N\/A & \\cite{Cucciara11} \\\\\n\\hline \\\\\n\\multicolumn{6}{l}{$^a$P = photometric redshift, S = spectroscopic\n  redshift} \\\\\n\\multicolumn{6}{l}{$T_P$ = delay time to obtain photometric redshift;\n  $T_S$ = delay time to obtain spectroscopic redshift}\n\\end{tabular}\n\\end{center}\n\\end{table*}\nThe paucity of high-$z$ bursts found to date probably stems from\nseveral factors, \none of which is almost certainly the long\ndelay in obtaining the first indication of a high-$z$ event, which is\ntypically many hours (Table~\\ref{tbl:GRBs}).  The key to finding more high-$z$ bursts is to\ndesign a satellite optimized to detect high-$z$ GRBs and to measure\ntheir redshifts within minutes.  Here we discuss the optimization of\nan instrument, the X-ray Coded Aperture Telescope (XCAT), designed to\nfind bright high-$z$ GRBs.\n\n\\section{The X-ray Coded Aperture Telescope on the Joint Astrophysics Nascent Universe Satellite}\n\n\\begin{figure}[tbp]\n\\resizebox{\\hsize}{!}{\\includegraphics[angle=90,bb=10 100 495 670, clip=true]{racusin_fig3.eps}}\n\\caption{\n\\footnotesize\nPrompt emission spectra for GRB\\,080319B ($z =0.937$) for three time\nintervals, plotted as $F_\\nu$ in units of (keV cm$^{-2}$ s$^{-1}$ kev$^{-1}$).\nAlthough there is strong spectral evolution, the spectra from all\nthree intervals are well-fit by a Band function (solid curves) with peak energy of a\nfew hundred keV.  From \\citet{Racusin08}.\n}\n\\label{fig:Band}\n\\end{figure}\n\n\\begin{figure}[tbp]\n\\resizebox{\\hsize}{!}{\\includegraphics[clip=true]{BATSE_break_energy_distr_Preece2000ApjS.eps}}\n\\caption{\n\\footnotesize\nDistribution of peak energies for BATSE bursts, from \\citet{Preece00}.  \n}\n\\label{fig:BATSE}\n\\end{figure}\n \nThe Joint Astrophysics Nascent Universe Satellite\n\\citep[JANUS;][see Fig.~\\ref{fig:JANUS}]{Burrows10_JANUS} is a proposed NASA Explorer mission\noptimized to study the Cosmic Dawn by finding high-$z$ GRBs and\nquasars and measuring their redshifts within minutes on-board.\nIt has two main instruments: an X-ray Coded Aperture Telescope (XCAT) \\citep{Falcone10_XCAT}\nand a Near Infrared Telescope (NIRT).  Here we discuss results of a\ntrade study made for the XCAT instrument to maximize the number of\nbright high-$z$ bursts that we can discover.\n\nTraditionally, GRB detectors have operated in the hard\nX-ray to gamma-ray energy range, though {\\it HETE-2} and {\\it\n  Beppo-SAX} both had soft X-ray capabilities.  From previous studies,\nwe know that the prompt emission from GRBs is typically characterized\nas a ``Band function'' \\citep{Band93}, an empirical function with a low\nenergy power law that transitions smoothly to a high energy power law,\nwith a peak energy ($E_p$, the peak in the $\\nu F_\\nu$ spectrum) \nnear the transition point.  An example of time-resolved spectra of the\nprompt emission from a bright burst at $z \\sim 1$ is\nshown in Figure~\\ref{fig:Band}.   The distribution of BATSE peak \nenergies is shown in Figure~\\ref{fig:BATSE}: roughly 60\\% of the bursts\nhave 100~keV $<E_p<$ 600~keV, with the most probable value at $E_p \\sim 220$~keV.\n\n\n At high redshifts, the GRB prompt spectrum is shifted down in energy\nby a factor of $(1+z)$.  We can therefore expect that high redshift\nbursts will have peak energies in the tens of keV rather than the hundreds\nof keV.  Below $E_p$ the photon spectrum is typically $I(E) \\propto E^{-1}$\n(photon index of $\\Gamma \\sim -1$). \nAbove the peak energy, the photon spectrum drops off rapidly with energy\n(typically as $E^{-4.5}$), and observations at energies above $E_p$\nhave poor sensitivity.  \nSearches for high $z$ bursts should therefore concentrate on \nthe region below about 30~keV, where\nthe number of photons per unit energy interval is high.\nBecause the spectrum favors the soft X-ray bandpass, and because soft\nX-rays are much easier to detect than hard X-rays or $\\gamma$-rays, a\nsmall, lightweight soft X-ray instrument can have better\nsensitivity to high-$z$ GRBs than a much larger, heavier hard X-ray or $\\gamma$-ray\ninstrument, and is better suited to small missions in the SMEX or EX class.  \n\n\\subsection{Wide-Field X-Ray Imaging}\n\n\\begin{figure}[tbp]\n\\resizebox{\\hsize}{!}{\\includegraphics[clip=true]{lobster_optics.eps}}\n\\caption{\n\\footnotesize\nCartoon of lobster-eye optics.  X-rays are reflecting at grazing\nincidence from planar surfaces arranged in a spherical pattern (a\nmicrochannel plate optic in this case).  The\nthick yellow line represents the focal surface.\n}\n\\label{fig:lobster_optics}\n\\end{figure}\n\nTechniques available for performing wide angle X-ray transient surveys are severely restricted.\nLarge fields of view can be monitored with scanning collimator\ninstruments (such as the {\\it Uhuru} satellite or the {\\it MAXI}\nmission), but these typically have poor position determination\naccuracy (of order tens of arcminutes) that is insufficient to\nlocalize GRBs and other transients well enough to identify\noptical counterparts.  They also typically miss short transients like\nmost GRBs.\nThere are no materials with sufficient transparency and refractive power available to make\nconventional lenses such as those used in optical cameras.\nLarge fields of view are feasible using multilayer-coated mirrors at\nnormal incidence \\citep[e.g.][]{Smith90}, which work by\nconstructive interference of X-rays from layers of alternating\nhigh-and-low Z materials, but these mirrors have very narrow bandpasses and\ncannot be used for broad-band studies.\nBroad-band X-rays can only be reflected at grazing incidence, making the\nconstruction of conventional wide field optics virtually impossible. \nThe relatively large grazing angles needed for wide fields of view\nalso limit the bandpass, since the maximum grazing angle is inversely\nproportional to the photon energy.  The standard astronomical X-ray\ntelescope uses a Wolter I design \\citep{Wolter52b,Wolter52a} that \ntypically has fields of view less than a degree in diameter, with\nstrong vignetting near the edges of the field of view.\n\nThese problems can be solved through the use of lobster-eye optics,\nusing a curved grazing incidence optic in which each segment\nconcentrates light from a different direction on the sky, but all reflections\nare at grazing incidence (Figure~\\ref{fig:lobster_optics}).  Such designs have been under development\nfor more than a decade, utilizing micropore optics or other systems of\nreflectors \\citep[e.g.][]{Angel79,Fraser02,Pearson03,Sveda09,Tichy09,Putkunz09}.\nLobster-eye designs have several disadvantages, including a complex Point\nSpread Function (PSF; see Figure~\\ref{fig:lobster} or\n\\citet{Hudec04}), \nsmall bandpass due to the reflective optics,\na complex energy-dependent effective area curve (e.g.,\nFigure~\\ref{fig:XCAT_EA}), and relatively\nheavy optics.  Nevertheless, the sensitivity to point sources is\nsuperior to coded apertures (discussed below).\nLobster-eye designs, however, have limited flight heritage.\n\n\n\\begin{figure}[tbp]\n\\resizebox{\\hsize}{!}{\\includegraphics[clip=true]{lobster_image.eps}}\n\\caption{\n\\footnotesize\nSimulation of an observation of a $10^\\circ \\times 10^\\circ$\nregion of the LMC with the {\\it LOBSTER-ISS} instrument (1 day exposure).  \nFrom \\citet{Fraser02}.  The square pore optics produce a cruciform PSF.\n}\n\\label{fig:lobster}\n\\end{figure}\n\n\nFor continuous, wide-field-of-view,\nbroadband, X-ray imaging with arcminute position determination, \nonly coded aperture imaging has\nsignificant flight history.  Since 1972, approximately two dozen\nimaging coded aperture astronomical telescope designs\nhave flown in space or on balloons (see \nhttp:\/\/astrophysics.gsfc.nasa.gov\/cai). \nMost instruments designed to detect GRBs have used 1-D or 2-D coded\naperture masks, using techniques developed and discussed\nby a number of authors including \\citet{Dicke68,Fenimore78,Proctor79}\nand \\cite{Caroli87}.\n\\begin{figure}[tbp]\n\\resizebox{\\hsize}{!}{\\includegraphics[clip=true]{coded_optics.eps}}\n\\caption{\n\\footnotesize\nCartoon of coded aperture ``optics''.  X-rays are either transmitted\nor absorbed by cells in the mask (black) before impinging on the\ndetector plane (yellow).  We use a 2-D random mask pattern to\nsuppress ghost images.\n}\n\\label{fig:coded_mask}\n\\end{figure}\nCoded aperture masks work as multiple pinhole cameras or shadow\nmasks (Figure~\\ref{fig:coded_mask}).  Each point source in the field of view casts a shadow of the\nmask pattern onto the detectors.  As the angle of the source varies on\nthe sky, the shadow on the detectors is offset accordingly.  By\ndeconvolving the detector event pattern with the mask pattern, an\nimage of the sky can be obtained.  Two dimensional coded apertures\nprovide more robust positions on the sky than perpendicular 1-D coded\napertures (which can sometimes provide only a 1-D position and which\nsometimes have ghost images).\nThe 2D coded mask, combined with sophisticated triggering algorithms,\ngives the \\swift\\ BAT instrument an extremely low false burst rate of just a few percent.\nCoded aperture instruments are in general less sensitive than\nlobster-eye instruments because the photons are not concentrated into\nimages, and because every source and all diffuse emission contribute\nnoise to every pixel on the detector.  Nevertheless, coded aperture instruments have\nadvantages in terms of broad-band performance and extremely wide\nfields of view for a given mass.\n\nThree designs were considered in detail in our trade studies for {\\it JANUS}: a\nlobster-eye design \nsimilar to the {\\it LOBSTER-ISS} instrument proposed a decade ago for flight\non the International Space Station \\citep{Fraser02}; a monolithic\ncoded aperture design similar to the Swift BAT, with a single large mask imaging onto a single\nfocal plane detector array; and a modular coded aperture design, in which each module pointed\nin a different direction (X-ray Coded Aperture Telescope, or XCAT).\nWe found that sensitivity limits for monolithic and modular coded\naperture instruments are similar for on-axis sources, but that the\nmodular design performs better at the edges of the field of view (less\nvignetting) and provided more design flexibility for accommodating\nmultiple instruments on a small platform.\nWe then compared the lobster-eye concept to the modular coded\naperture. \nInstrument parameters for the two design concepts we considered are given in\nTable~\\ref{tbl:comp}, and effective area curves are shown in \nFigure~\\ref{fig:XCAT_EA}.  \nInstrument mass and volume constraints limited the size of both\ninstruments; the lobster-eye design considered here is larger (larger\nfocal length) and more massive than XCAT, and covers only 7\\% of\nthe solid angle of the XCAT instrument, but is much more sensitive.\n\n\\begin{table*}\n\\caption{Comparison of coded aperture and lobster-eye designs}\n\\label{tbl:comp}\n\\begin{center}\n\\begin{tabular}{lll}\n\\hline\n\\\\\nInstrument & XCAT (coded aperture) & Lobster-eye \\\\\n\\hline \\\\\nEnergy Range (keV) & 0.5--20 & 0.3--3 \\\\\nImaging technique & 2-D coded mask & Lobster-eye grazing incidence \\\\\nFOV (sr) & 3.9 & 0.26 \\\\\nMask\/Optics area (cm$^2$) & 1690 & 1600 \\\\\nFocal length & 158 mm & $>350$ mm \\\\\nAngular resolution (arcmin) & 6.3 & 4 \\\\\nFocal Plane Area (cm$^2$) & 147 & 400 \\\\\nDetector type & H2RG HyViSI CMOS & GEM \\\\\nPower & 70 W & 51 W \\\\\nMass & 57 kg & 95 kg \\\\\nLimiting sensitivity in \\\\\n\\qquad 30 s (mCrabs) & 240 (6.5$\\sigma$) & 2.7 (5$\\sigma$) \\\\\n\\# GRBs in 2 years with $5<z<12$ \\\\\n\\qquad and $S_x > 3 \\times 10^{-7}$ erg cm$^{-2}$ & 48 & 11 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table*} \n\n\\begin{figure*}[t!]\n\\resizebox{\\hsize}{!}{\n\\includegraphics[width=3.2in,clip=true]{XCAT_EA_curve.eps}\n\\hfill\n\\includegraphics[width=2.8in,clip=true]{LOBSTER-ISS_EA_curve.eps}\n}\n\\caption{\\footnotesize\nLeft: XCAT grasp and effective area curves for a single module.  Right: LOBSTER-ISS\neffective area curve \\citep[from][]{Fraser02}.  The dotted curve shows\nthe effective area for the central true focus, while the solid curve\nincludes area scattered into the cruciform arms.\nThe XCAT effective area curve is much broader and\nsimpler because the coded mask cells are either transparent or opaque\nover most of the energy range of the detector, so the curve is\ndominated by the detector quantum efficiency.  The LOBSTER-ISS curve has\nstrong features caused by the grazing incidence reflections off of the\nNi-coated optic, and has a much narrower bandpass.\n}\n\\label{fig:XCAT_EA}\n\\end{figure*}\n\n\n\\section{Simulations}\n\n\\subsection{JANUS Mission Simulation}\n\nOur detailed simulation began with a simulation of the mission\ngeometry for {\\it JANUS}.\nThe {\\it JANUS}\\ mission will spend most of its time surveying nearly 1\/2\nof the high-latitude sky.  The {\\it JANUS}\\ team produced a list of fields covering the\nnorthern and southern galactic caps, and developed an algorithm to\nselect fields for observation.  A detailed mission simulation was\nconstructed that included realistic orbital parameters and spacecraft\ndesign parameters, and calculated the position of the spacecraft every\nminute for a two year mission.  At each one minute time step, the\nspacecraft field of view was compared against observing constraints\n(keep-out zones around the Sun, Moon, and Earth).  When one of these\nconstraints impinged on the field of view, a new target was selected\nthat would be observable for at least ten minutes, and the spacecraft\nslewed to the new target (using realistic slew times based on a\ndetailed model of the spacecraft's attitude control system).  The\nsimulation correctly accounted for time spent in the South Atlantic\nAnomaly (where the instruments cannot observe due to high background\nrates), and included interruptions to the mission timeline for GRBs\nthat were injected at random times using the expected GRB event rate.\nThe mission simulation demonstrated that {\\it JANUS}\\ can complete its\nquasar survey in a two year mission with considerable margin, and\nserved as the basis for our detailed simulation of the XCAT instrument performance.\n\n\n\n\\subsection{XCAT GRB Simulation} \n\nTo predict\n{\\it JANUS}\\ burst rates as a function of redshift, we\ncarried out a rigorous simulation of XCAT performance\non-orbit, making use of\nthe {\\it JANUS}\\ mission simulation to\nprovide a realistic history of live-time periods\nand satellite pointings (including relative Earth\nposition, to account for partial occultation of\nthe XCAT FOV). To generate an appropriate\npopulation of GRBs for detection, we used the\nbest-fit luminosity and redshift distributions\nfrom \\citet{Wanderman10}, as updated\non the web to reflect the latest Swift results. We\ndrew burst redshifts ($0<z<30$) and luminosities\n($L_{peak} \\ge 10^{50}$~erg~s$^{-1}$) from these distributions\nat Poissonian intervals according to the known\nall-sky rate, placed the burst at a random position\non the sky, and compared to the FOVs of\nthe 10 XCAT modules (from the mission simulation). If the burst was in view\nof one or more modules, then a simulated burst\nlight curve (with Poisson noise) was generated\nfor each viewing module -- including contributions\nfrom all point X-ray sources in the module's\nFOV along with the known diffuse X-ray\nand particle backgrounds -- and was fed to the\nXCAT triggering software. If the burst resulted in a trigger,\nthe properties of the burst were recorded.  These successful triggers\nformed the basis for our predicted redshift distribution.\nTo minimize Poisson uncertainties due\nto the Monte Carlo nature of the simulation,\nwe carried out 20 runs of 400 days (on-orbit)\nduration each.\n\n\\begin{table*}\n\\caption{Redshift Distribution of Simulated {\\it JANUS}\\ GRBs (2 year mission)}\n\\label{tbl:zdist}\n\\begin{center}\n\\begin{tabular}{ccccccccc}\n\\hline \\\\\n$z$ & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \\\\\n$f(>z)$ & 12\\% &  6.4\\% & 3.4\\% & 1.8\\% & 0.9\\% & 0.4\\% & 0.1\\% & 0.16\\% \\\\\n$N(>z)$ & 61 & 33 & 17 & 9 & 5 & 2 & 1 & 1 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table*}\n\nBurst light curves were simulated by rescaling\nthe Swift BAT (15--150 keV) light curves of\nbursts with known redshifts to the (new and\ndistinct) redshift and peak luminosity of the\nsimulated burst, and extrapolating\nthe observed burst spectrum to the lower energy\nrange of XCAT. (For simulated high-$z$ bursts\nthis extrapolation is minimal, since the median\nredshift for the Swift burst sample is only $z=2.3$).\nWe have developed a library of 111 Swift BAT\nlight curves for this purpose.\n\nOn the basis of the simulations, we predict a\nmean burst detection rate of 0.7 GRBs per day\nwith XCAT.  \nThe distribution of these bursts on the sky is shown in Figure~\\ref{fig:GRB_map}.\nThe redshift distribution of detected bursts is given in\nTable~\\ref{tbl:zdist} and is shown in Figure~\\ref{fig:zdist}.\nBeyond $z=12$ the Ly$\\alpha$ break\nredshifts beyond the NIRT bandpass so that a\nredshift measurement is not possible. \nThe XCAT design will detect 61 bursts with $z>5$ in a 2 year mission.\n\n\\begin{figure}[tbp]\n\\resizebox{\\hsize}{!}{\\includegraphics[clip=true]{xcat_2yr.eps}}\n\\caption{\n\\footnotesize\nDistribution of simulated GRBs on the sky for the XCAT instrument\ndesign (2 year mission).  \nBlack diamonds indicate the high-$z$ GRBs.\nRed crosses show the location of the 50\nbrightest X-ray\nsources seen by the {\\it RXTE} All-Sky Monitor (ASM); \nthe 263 sources detected by the ASM during the week of 7\nFebruary 2009 were included as background sources in the\nsimulation.   (Cross size indicates relative source brightness.)\n}\n\\label{fig:GRB_map}\n\\end{figure}\n\n\\begin{figure}[tbp]\n\\resizebox{\\hsize}{!}{\\includegraphics[clip=true]{frac_v_z.eps}}\n\\caption{\n\\footnotesize\nRedshift distribution of GRBs.  The black line shows the distribution\nof the parent population from \\citet{Wanderman10}.  The red line shows\nthe results of our simulation for the XCAT instrument.  The cyan line\nshows the distribution for the \\swift\\ BAT instrument.  The dark blue\nline shows the distribution for BATSE.  We expect XCAT to detect\na significantly higher fraction of high-$z$ GRBs than \\swift; this,\ncombined with rapid redshift measurements for every high-$z$ GRB\ndetected by XCAT, will result in about 30 GRBs per year with $z>5$.\n}\n\\label{fig:zdist}\n\\end{figure}\n\n\\subsection{Lobster-Eye Simulation}\n\nThe GRBs generated during the detailed XCAT simulation were also used\nas input to a simulation of the lobster-eye design.  The\ninstrument parameters used for the simulation are shown in\nTable~\\ref{tbl:comp}.\nWe found that the lobster-eye instrument detected about the same total number of bursts\nin spite of its small field of view, because the improved sensitivity\nallowed it to detect every GRB that went off in its field of view.\nAlthough the total number of bursts is similar, the distribution of\npeak flux is very different from that of the XCAT instrument\n(Figure~\\ref{fig:flux_distr}).\n\\begin{figure}[tbp]\n\\resizebox{\\hsize}{!}{\\includegraphics[clip=true]{GRB_flux_distribution.eps}}\n\\caption{\n\\footnotesize\nDistribution of peak fluxes of high-redshift GRBs ($5 < z < 12$) \ndetected by XCAT (black) and the lobster-eye design considered here\n(red).  The two instruments detect similar numbers of high-$z$ GRBs, but the\nXCAT distribution is much brighter.\n}\n\\label{fig:flux_distr}\n\\end{figure}\n\n\n\n\\subsection{Optical \/ Gamma-ray Correlation}\n\nThis difference in peak flux distributions of the high redshift burst\nsamples detected by XCAT and by our lobster-eye implementation \nis quite significant.  Previous\nstudies have shown that the brightness of the optical afterglow is\ncorrelated with the brightness of the prompt $\\gamma$-ray emission\n\\citep{Gehrels08,Nysewander09,Kann10}.  Although there is considerable\nscatter (perhaps due to dust extinction affecting the optical\nbrightness), the trend is that brighter GRBs (measured by peak flux,\ntotal fluence, or total isotropic energy output) have brighter optical\nafterglows. \nBright afterglows are essential for detailed high resolution studies of\nthe host galaxies of the bursts.\n {\\it Therefore, in order to use high redshift bursts to\nprobe galaxies at the Cosmic Dawn, one should optimize the burst\ndetection to maximize the number of bright  bursts.}\nWe find that the XCAT design will detect 24 high-$z$ GRBs per year at\nfluence $S_x > 3 \\times 10^{-7}$ erg cm$^{-2}$, while the lobster-eye design\ndetects only 6 per year.\n\n\n\\section{Conclusions}\n\nOur detailed simulations confirmed our ``back-of-the-envelope''\ndetermination that high-$z$ GRBs are best detected in the soft X-ray\nband, by which we mean energies of roughly 0.5-20 keV.  In this energy\nband, the best options for wide-field transient detectors are\nlobster-eye and coded aperture designs.\nThe lobster eye\ndesign is more sensitive and will find more faint X-ray transients,\nbut its field of view is too small to find many rare, bright, high-$z$\nGRBs that can be used to probe galaxies in the era of the Cosmic Dawn\nwith high resolution spectroscopy followup.  The XCAT coded mask\ndesign was selected for the {\\it JANUS}\\ mission because it is optimized to\nfind these high-$z$ bursts, and therefore provides the best solution\nto the prime {\\it JANUS}\\ science goals of probing the Cosmic Dawn with\nGRBs and quasars.\n \n\n\\bibliographystyle{aa}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe rapid progress in ultra-intense laser technology \\cite{lasertechnology}\ntriggered the interest in properties of relativistic plasmas\nconsisting mainly of electrons ($e^-$), positrons $e^+$ and photons ($\\gamma$). \nFor instance, in \\cite{Rafelski1,Munshi_bk,Rafelski2,Ruffini}\nvarious processes, including thermal (kinetic) and chemical equilibration,\nhave been considered for a hot charge-symmetric electron-positron-photon \n($e^- e^+ \\gamma$) plasma,\nwhile the kinetics of a temporarily created\nelectron-positron plasma has been analyzed in \\cite{Blaschke}. \nHot plasmas, may be charge-symmetric or asymmetric \nor with ion load, have interesting features \\cite{Thoma}, e.g.\\\nwith respect to collective excitation modes (including the van Hove singularity),\nastrophysical situations and secondary particle (such as muon or pion)\nproduction. Furthermore, an optically thick $e^- e^+ \\gamma$ plasma represents\nthe Abelian analog of the non-Abelian quark-gluon plasma \\cite{QM} currently\ninvestigated at large-scale accelerators.\n\nThe scenario for droplet creation can be visualized in the following way: \nSuppose that the radiation\npressure of counter-propagating ultra-intense laser beams on a thin foil\nis so high that a substantial compression accompanied by sizeable\npair creation is achieved. Estimates for such a process have been presented \nin \\cite{Shen_MtV}. Extrapolating these estimates one may envisage the\nidealized scenario of a hot and optically thick $e^- e^+ \\gamma$ plasma,\nwith temperatures exceeding the 1 MeV scale.  \nOptically thick $e^- e^+ \\gamma$ plasmas play an important role in the \ngamma ray bursts phenomena \\cite{Ruffini2}.\n\nIn \\cite{Munshi_bk} it has been pointed out that an $e^- e^+ \\gamma$ plasma droplet\nat initial temperatures $T$ much larger than the electron mass $m$ emits an intense\ngamma ray burst. However, the expansion dynamics was considered in\n\\cite{Munshi_bk} in a schematic manner. Nevertheless, \nthe time structure and the spectrum of the burst depend crucially on\nthe expansion dynamics of the plasma droplet driven by the thermodynamic\npressure gradients. \nIt is necessary to consider a realistic expansion dynamics as well as the particle \nproduction rates in the relativistic framework for a proper understanding\nof $e^- e^+ \\gamma$ plasmas produced in the laboratory and astrophysical sites\nsuch as gravitational collapse. \nThe aim of the present paper is to quantify the droplet\nexpansion dynamics by means of the relativistic hydrodynamics. This provides \na realistic evaluation of the time evolution of the temperature profile and\nvelocity profile. The latter one is important for properly accounting for\nthe blue\/red shift effects of the emitted photons.\n\nAnother avenue towards the creation of $e^- e^+$ plasma droplets has been\nenvisaged in \\cite{Blaschke} as a result of the dynamical Schwinger\nprocess. Due to the short (sub cycle) time scales the thermalization\nand the collective dynamics are not an issue. Instead, the emergence\nof $e^- e^+$ annihilation into $\\gamma$ pairs may be employed as diagnostic\ntool.   \n\nOur paper is organized as follows. In section \\ref{sec2} we consider the spherically\nsymmetric expansion of plasma droplets within the relativistic hydrodynamics.\nThis input is used in section \\ref{sec3} to estimate the spectrum of emitted photons.\nSection \\ref{sec4} gives an account of the spectral emissivity for a warm plasma with \na temperature in the order of the electron mass and focusses on the onset \nof the broadening of the 511 keV line when achieving higher temperatures.\nSection \\ref{sec5} is devoted to the summary.\nAppendix \\ref{appA} presents the expansion dynamics with planar geometry. \n\n\\section{Relativistic Hydrodynamics\\label{sec2}}\n\nA sufficiently large $e^- e^+ \\gamma$ plasma droplet can be dealt with by\nmeans of relativistic hydrodynamics. The equations of motion imply local\nenergy and momentum conservation expressed as\n\\begin{equation}\n{T^{\\mu \\nu}}_{; \\mu} = 0,\n\\label{eq.1}\n\\end{equation}\nwhere $\\mu, \\nu = 0, \\dots, 3$ denote Lorentz indices and a colon stands\nfor the covariant derivative. For an isotropic medium (gas or fluid or plasma)\nthe energy-momentum tensor reads\n\\begin{equation}\nT^{\\mu \\nu} = (e + p) u^\\mu u^\\nu - p g^{\\mu \\nu} + \\cdots.\n\\label{eq.2}\n\\end{equation}\nHere, the four-velocity is $u^\\mu$ with normalization $u^\\mu u_\\mu = 1$\nand $g^{\\mu \\nu}$ denotes the metric tensor,\n${\\rm diag}(g^{\\mu \\nu}) = (1, -1, -1, -1)$ in Cartesian coordinates.\nThe energy density is $e$, and $p$ stands for the thermodynamic pressure.\nThe ellipsis $\\cdots$ in (\\ref{eq.2}) denote dissipative terms being\nproportional to transport coefficients, such as shear and bulk viscosities,\nfor instance. Neglecting the latter ones, (\\ref{eq.1}, \\ref{eq.2}) can be\nsolved once an equation of state is given. For $T > m$, the equation of state\nof a relativistic ideal gas can be employed, that is $e = 3p$.\nWe consider a charge-symmetric plasma with zero net density. Accordingly,\nthe chemical potential is zero, too.\nThe pressure and temperature for our equation of state are related via\n$p = d_{eff} \\frac{\\pi^2}{90} T^4$\n(we employ units with $\\hbar = c = 1$)\nwith $d_{eff} = 2 + \\frac78 4$ for photons (first term) and electrons and positrons\nin equilibrium. This is the leading-order term in an expansion in\npowers of $m\/T$.\n\nSpecial symmetries simplify (\\ref{eq.1}) tremendously. We consider here\nspherically symmetric droplets (planar slab geometry is considered in appendix A).\nThen (\\ref{eq.1}) degenerates to two coupled equations, one for the pressure\ngradient and one for the radial velocity gradient. These equations can\nbe solved by the methods of characteristics (cf.~\\cite{Baym} for details). \nNumerical results are exhibited in Fig.~1.\n\n\\begin{figure}[!htbp]\n\\begin{center}\n\\includegraphics[width=0.49\\columnwidth]{Tr_3D.pdf}\n\\includegraphics[width=0.49\\columnwidth]{vr_3D.pdf}\n\\caption{\\label{fig.2}\nTemperature (left) and velocity (right) profiles at various time instants for the\nspherically symmetric expansion.} \n\\end{center}  \n\\end{figure} \n\nThe solution for the given equation of state can be represented for scaled\nquantities, $T\/T_0$ and $r\/R$, where $T_0$ and $R$ stand for the initial\ntemperature and radius, respectively, of the $e^- e^+ \\gamma$ droplet. \nWe assume initially homogenous droplets,\ni.e.\\ $T(t = 0, r) = T_0$ for $r < R$ and $T(t = 0, r) = 0$ for $r > R$.\nThe initial radial profile is $v(t = 0, r) = 0$ for $r < R$.\nInitially, a rarefaction wave travels towards the center with the velocity of\nsound $v_s = c \/ \\sqrt{3}$\n(corresponding to a Riemann type solution) and becomes reflected then,\ncausing the non-trivial radial temperature profile with somewhat lower\ntemperature in the center.\nWe allow the temperature evolution\nuntil $\\langle T \\rangle \\sim 0.1 T_0$. For late times, the temperature\ndrops further, but our assumed equation of state \nas well the photon rate below based on the perturbative estimates \ndo not longer apply.\nThese late times until disassembly are not important for the\nhard photon emission off the expanding droplet.\nThe scaled time to cool from $T_0$ to $\\langle T \\rangle \\sim 0.1 T_0$\nis about $ct = 3.5 R$.\nThat means, for instance, a droplet of initial radius $R = 2$ nm and $T_0 = 10$ MeV\nwith energy content 13 kJ expands within the relaxation time\nfor pairs and photon chemical equilibration,\n0.025 fs, and cools to $\\langle T \\rangle \\sim 1$ MeV.\n\nThe radial velocity profiles in Fig.~1 expose the rapid expansion of the cool\nsurface layers nearly at velocity of light.\nIn particular, due to the rapid expansion, \nthe droplet cools faster that what was assumed in the\nschematic model in \\cite{Munshi_bk}. This is reflected in the photon\nemission considered in the next section. \n\n\\section{Photon emission\\label{sec3}}\n\nHaving at our disposal the realistic temperature and velocity profiles,\nwe can now proceed to calculate the photon emission. It should be emphasized\nthat the leakage of energy via photon emission is not properly accounted\nfor in (\\ref{eq.1}, \\ref{eq.2})\n(cf.~\\cite{HW} for hydrodynamic schemes with energy leakage).\nIt turns out however that the emitted energy during the considered evolution\nis small in comparison with the internal and kinetic energies so that\na consistent hydrodynamic treatment with radiation transport analog\nto stellar core collapse simulations is not needed.\n\nWe employ the emission rate from thermo-field theory \\cite{Moore}\n(cf.\\ figure 1 in \\cite{Munshi_bk})\n\\begin{eqnarray} \\label{eq.3}\n\\omega \\frac{dN}{d^4x d^3 k} &=& \n\\frac{2\\alpha m^2_\\infty f_{+1}(\\omega)}{(2\\pi)^3} \n\\left [\\ln \\left (\\frac{T}{m_\\infty}\\right )+\\frac{1}{2} \\ln \\left ( \n\\frac{2 \\omega}{T} \\right )\n\\right .  \\nonumber  \\\\\n&+&\\left . C_{22}\\left(\\frac{\\omega}{T}\\right) \\ \n+  \\ C_{\\rm{b}}\\left(\\frac{\\omega}{T}\\right) \n+  C_{\\rm{a}}\\left ( \\frac{\\omega}{T}\\right) \\right ],\n\\label{local}\n\\end{eqnarray}\nwhere $f_{+1}(\\omega) = (\\exp (\\omega\/T)+1)^{-1}$ \ndenotes the Fermi distribution function, \nand the dynamically generated asymptotic mass of the electron is \n$ m^2_\\infty = 2 m^2_{\\rm{th}}$.\nThe thermal mass squared is determined by $m^2_{\\rm{th}}=e^2T^2\/8$ \nwith $e^2=4\\pi\\alpha$ and the $C$ coefficients are given explicitly \nin \\cite{Munshi_bk}.\n$\\hat E_\\gamma = u^\\mu k^\\nu g_{\\mu \\nu}$ \ntakes properly into account the red\/blue shifts\nvia the scalar product\nof the medium's four-velocity $u^\\mu(t, \\vec x)$ and the photon's four-momentum \n$k^\\nu = (\\omega, \\vec k)$\nThe rate includes $2 \\leftrightarrow 2$ Compton and annihilation processes\n(first three terms), bremsstrahlung processes $(C_b)$ and off-shell pair annihilations\nprocesses ($C_a$). \n\nThe space-time integrated emission spectrum includes only such photons\nwith a mean free path $\\lambda_\\gamma$ being larger larger than the \ndistance to the cool droplet surface in a given direction.\nThe mean free path is locally determined by $\\lambda_\\gamma = \\Gamma_\\gamma^{-1}$\nwith the damping rate\n$\\Gamma_\\gamma(\\omega) = \\frac{(2\\pi)^3}{4} \\ e^{\\omega \/ T} \\ \\frac{dN}{d^4xd^3k}$\nwhich depends in turn on the photon energy.\n\nThe resulting energy-weighted photon spectrum is exhibited in Fig.~\\ref{fig.3}.\nSimilar to the schematic model \\cite{Munshi_bk} it extends beyond 50 MeV with\nnotable strength. In contrast to \\cite{Munshi_bk}, however, the maximum of the\nspectrum is an order of magnitude smaller. The origin of this difference may be\ntraced back to the improved description of the droplet dynamics\nwhich makes the droplet cooling considerably faster.\nThe hard tail of the photon spectrum is emitted at early times and does not change \nat later times. In contrast, the softer part with $\\omega \\sim 10$ MeV\ngets significant contributions at later times. Comparing the spectra\nat $t = 4$ nm\/c and $t = 7$ nm\/c, one concludes that the very\nlate radiation contribution for $T < 1$ MeV will contribute to photon\nenergies $\\omega < 5$ MeV.    \n\n\\begin{figure}[!htbp]\n\\begin{center}\n\\includegraphics[width=0.75\\columnwidth]{wdNdw_v2.pdf}\n\\caption{\\label{fig.3}\nEnergy-weighted spectrum of emitted photons as a function of the energy\nfor various times for $T > 1$ MeV.}\n\\end{center}  \n\\end{figure} \n\n\\section{Broadening of the 511 keV Line\\label{sec4}}\n\nThe photon emission rate (\\ref{eq.3}) applies for $T > m$, i.e. a hot\nplasma. In a warm plasma with $T \\sim m$ or $T < m$, one expects\na significant contribution from the annihilation process\n$e^- e^+ \\to \\gamma \\gamma$. The corresponding inclusive photon rate \nfor this process is obtained by the kinetic theory estimate \n\\begin{eqnarray}\n\\omega \\frac{dR}{d^4x \\, d^3 k} &=&\n\\frac{{\\cal N}}{(2\\pi)^7 16 \\omega}\n\\int_{4 m^2}^\\infty \\frac{ds}{s}\n\\int_{t_{min}}^{t_{max}} dt \\,\n\\vert {\\cal M} (s,t) \\vert^2\n\\int_{\\nu_{min}}^\\infty d\\nu \\, (1 + f_{-1}(\\nu-\\omega)) \\nonumber \\\\\n&\\times&  \\int_{E_{min}}^{E_{max}} dE\n\\frac{f_{+1}(E) \\, f_{+1} (\\nu - E)}{\\sqrt{(E_{max} - E)(E - E_{min})}}\n\\label{mass_rate}\n\\end{eqnarray}\nwith ${\\cal N} = 4$ and\n\\begin{eqnarray}\nt_{{min}\\atop{max}} &=& m^2 - \\frac12 s \\left( 1 \\pm \\sqrt{1 - \\frac{4 m^2}{s}} \\right), \\\\\n\\nu_{min} &=& \\omega + \\frac{s}{4 \\omega}, \\quad t' = t -m^2,\\\\\nE_{{min}\\atop{max}} &=& \\frac{-B \\pm \\sqrt{B^2 - 4 AC}}{2A},\\\\\nA &=& - s^2, \\quad B = 2s (\\omega s + 2 \\omega t' - \\nu t'),\\\\\nC &=& s t' (s+t') - \\omega^2 s (s- 4 m^2) \\nonumber\\\\\n&-& 2\\omega \\nu s t' - \\nu^2 t'^2\n- 4 \\omega \\nu m^2 s + s^2 m^2\n\\end{eqnarray}\nalong with the additional kinematical restriction\n$-4s^2 t' (s+t') \\left(1+s m^2\/((s+t')t')\\right) > 0$. \n${\\cal M}(s,t)$ is the lowest-order invariant matrix element for the\nprocess $e^- e^+ \\to \\gamma \\gamma$ (cf.~\\cite{Rafelski2}),\n\\begin{equation}\n\\vert {\\cal M} (s,t) \\vert^2 = 32 \\pi^2 \\alpha^2 \\left(\n\\frac{m^2 - u}{m^2 -s} +\n\\frac{m^2 - t}{m^2 - u} +\n4 \\frac{m^2}{m^2 -t} +\n4 \\frac{m^2}{m^2 -t} -\n4 \\left[ \\frac{m^2}{m^2 -t} + \\frac{m^2}{m^2 -u} \\right]^2 \\right)\n\\end{equation}\nwith $u = 2 m^2 - s -t$,\nand the distribution functions are\n$f_{\\cal S}(y) = (\\exp\\{y\/T\\} + {\\cal S})^{-1}$.  \nFor $T \\gg m$, corresponding to $m\\to 0$, one recovers the form presented in \n\\cite{gamma_rate}; the rate scales $\\propto T^2$. \nIn the opposite region $\\omega \\gg T$, where the Boltzmann approximation\nfor the $f_{+1}$ terms is justified \\cite{gamma_rate}, one finds\n\\begin{eqnarray}\n\\label{Boltzmann}\n\\omega \\frac{dR}{d^4x \\, d^3 k} &=&\n\\frac{{\\cal N}}{2 (2\\pi)^5} \\frac{T}{\\omega} {\\rm e}^{-\\omega\/T} \\nonumber \\\\\n& \\times &\\int_{4 m^2}^\\infty \\frac{ds}{s} \\, (s-4m^2)^2 \\, \\sigma (s) \\, \n\\ln (1-{\\rm e}^{-s\/(4 \\omega T)} )^{-1},\n\\end{eqnarray}\nwhere the total cross section $\\sigma (s)$ follows from\nintegrating $d \\sigma \/ dt =  \\vert {\\cal M} \\vert^2 \/ (16 \\pi (s - 4 m^2)^2)$\nwithin $t_{min} - t_{max}$.\nDue to the interplay of the factors ${\\rm e}^{-\\omega\/T}$ and\n$\\ln (1-{\\rm e}^{-s\/(4 \\omega T)} )^{-1}$ in (\\ref{Boltzmann})\nthe rate has a sharp maximum\nat $\\omega = m$ for $T \\ll m$. This is the 511 keV annihilation line,\nat very low temperatures,\nwhich broadens rapidly with increasing temperature. \nThe high-frequency tail, $\\omega \\gg max(T,m)$, behaves as \n$\\omega dR\/d^3k \\propto {\\rm e}^{- \\omega \/ T}$.\nThe full width at half maximum is \n$\\Delta \\omega = \\sqrt{-4 m T \\ln \\frac12}$, for $T \\ll m$,\nwith maximum $\\propto {\\rm e }^{-2m\/T}$.\n\nThe rate (\\ref{mass_rate}) is displayed in Fig.~\\ref{fig.A3} for a sequence of temperatures.\nWith increasing temperature the relative kinetic energy of the annihilating\nelectron-positron pairs increases and, consequently, the photon energy\nincreases in the medium's rest frame and becomes larger than $m$. This\nbroadening of the 511 keV annihilation line, once isolated from other \nprocesses with continuum radiation such as the Compton and the various\nbremsstrahlungs, may serve as a measure \nof the achieved temperature in a warm $e^- e^+ \\gamma$ plasma.\nAt $T > 0.1 m$, the broadened annihilation line changes gradually into a smooth\ncontinuum with typical exponential tail. For $T > m$, the maximum of the\nemission rate is shifted to $\\omega > m$. The annihilation is then part\nof the complete rate (\\ref{eq.3}).\n\nThe presented annihilation rates apply also to the transiently created\n$e^- e^+$ pairs by the dynamical Schwinger process, as envisaged in \\cite{Blaschke},\nwhen approximating the distribution functions by thermal ones. \n\n\\begin{figure}[!htbp]\n\\begin{center}\n\\includegraphics[width=0.75\\columnwidth]{rate_rescale.pdf}\n\\caption{\\label{fig.A3}\nPhoton emission rate from $e^- e^+$ pair annihilation \nas a function of photon energy for various values of the temperature.\nNote the scale factors in the legend. \nThe vertical dashed line indicates the position of 511 keV line.} \n\\end{center}  \n\\end{figure} \n\n\\section{Summary and Discussion\\label{sec5}}\n\nIn summary we consider the expansion dynamics of hot\nplasma droplets which can be created with next-generation\nultra-intense laser beams. We present here the dynamics of charge-symmetric and\nequilibrium $e^- e^+ \\gamma$ plasma droplets by means of relativistic hydrodynamics.\nOur results support the estimate in \\cite{Munshi_bk} pointing to gamma flashes\nemitted off the rapidly exploding droplets. \n\nClearly, our considered scenario is still idealized. The surplus of electrons\nover positrons needs to be considered in future analysis. This can be\naccommodated by a finite chemical potential steering the asymmetry of electrons \nand positrons. Furthermore, at given temperature, implying kinetic equilibrium,\nthe positrons may not be chemically equilibrated. This requires an additional\ndynamical variable, e.g.\\ a positron fugacity supplemented by appropriate\nrate equations. Also, ion impurities may have an impact on the plasma dynamics.\n\nAs indicated by the largely differing expansion time scales for spherical and plane\nexpansion patterns, the flow symmetry may play an important role. To clarify\nthat issue, more simulations\nare needed along with realistic initial conditions. The lessons learnt\nfrom the investigation of the quark-gluon plasma \\cite{QM}, however, indicate\nthat it is useful to develop parallel with experiments the theoretical description\nafter first initializing considerations. \n\nFinally, we mention the broadening of the 511 keV line as useful tool for\nthermometry of small electron-positron-photon droplets produced in\nthe laboratory and in astrophysical sites.  \n\n\\acknowledgements\nThe authors thank T.E. Cowan and R. Sauerbrey for inspiring discussions.\nMGM is thankful to FZ Dresden-Rossendorf for support during the \ncourse of this work.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{sec:Intro. in stress-energy tensor}\n\n\nThe properties of quantum states of physical interest in the Kerr background has been broadly investigated by studying the\nscalar field. The study of the quantized electromagnetic field in this background has been studied\nmuch less (notably, in ~\\cite{ar:CCH}, ~\\cite{ar:F&Z'85} and ~\\cite{ar:Bolash&Frolov}). \nIn this paper we perform the canonical quantization of the electromagnetic field in the Kerr background and calculate\nthe renormalized expectation value of the stress-energy tensor (abbreviated as RSET) when the electromagnetic field is in various states of physical interest.\nWe analytically calculate new expressions which are \nsymmetric under parity for the expectation value of the stress-energy tensor in these physical states.\nThese expressions correct those given in Candelas, Chrzanowski and Howard ~\\cite{ar:CCH}, hereafter referred to as CCH,\nwhich are not\nsymmetric under the parity operation even when the state is\nand even though the parity operation is a symmetry of both the Kerr metric and Maxwell's equations.\nWe also numerically calculate differences between two states of the RSET.\nWe show that the difference in the RSET between a Hartle-Hawking-type state and the past Boulware state\nis exactly (minus) thermal near the horizon, unlike a result in CCH.\nWe also study the rate of rotation of this difference, for which\nthere is no unanimous consensus in the literature,\nand find that it is rigidly rotating with the horizon to at least second order in the distance from the horizon.\n\n\nThis paper is organized as follows.\nWe outline the classical theory of the electromagnetic field on the Kerr background in Section \\ref{sec:classical}.\nIn Section \\ref{sec:CCH} we lay out a canonical quantization of the electromagnetic potential and field on this background. \nIn the following section we give a description of the main physical quantum states.\nSection \\ref{sec:theta->pi-theta Symmetry} is split into two subsections. \nIn the first one we derive expressions for the expectation value of the electromagnetic stress-energy tensor when the field\nis in these states. \nWe also show that similar expressions given by CCH result in RSETs\nwhich are not symmetric under the parity operation.\nIn the second subsection we give a physical interpretation of the various sets of terms appearing in \nthese expectation values.\nIn Section \\ref{sec:luminosity} we calculate the luminosity of the Kerr black hole in the past Boulware and past\nUnruh states for the spin-1 case. \nIn the last section we study the behaviour close to the horizon of the difference in the RSET between a Hartle-Hawking-type state defined in CCH and\nthe past Boulware state.\n\n\nThe method we followed to solve the radial Teukolsky equation is described in Appendix \\ref{sec:radial func.},\nwhereas we solved the angular equation as described in ~\\cite{ar:Casals&Ott'04}. \nFurther details can be found in ~\\cite{th:CasalsPhD}.\nIn Appendix \\ref{sec:asympts. close to r_+} we extend to the spin-1 case an asymptotic analysis in ~\\cite{ar:Candelas'80} for \nthe radial solutions close to the horizon in the scalar case.\n\nWe use  geometrized Planck units: $c=G=\\hbar=1$,\nand follow the sign conventions of Misner, Thorne and Wheeler ~\\cite{bk:M&T&W}.      \nAll figures have been obtained for the values: $a=0.95M$ and $M=1$;\nthe Boyer-Lindquist radius of the event horizon for such values of the black hole\nparameters is $r_+\\simeq 1.3122$.\n\n\n\\section{Classical theory}  \\label{sec:classical}\n\nThe Kerr metric corresponding to a black hole of mass $M$ and intrinsic angular \nmomentum $a$ per unit mass as viewed from infinity is, in the Boyer-Lindquist co-ordinate system  $\\{t,r,\\theta,\\phi\\}$, given by\n\\begin{equation} \\label{eq:K-N metric}\n\\begin{aligned}\n\\d{s}^2\n&=-\\frac{\\Delta}{\\Sigma}\\left(\\d{t}-a\\sin^2\\theta\\d{\\phi}\\right)^2+\\frac{\\sin^2\\theta}{\\Sigma}\\left[(r^2+a^2)\\d{\\phi}-a\\d{t}\\right]^2+\n\\frac{\\Sigma}{\\Delta}\\d{r}^2+\\Sigma\\d{\\theta}^2\n\\end{aligned}\n\\end{equation}\nwhere $\\Sigma\\equiv r^2+a^2\\cos^2\\theta$ and $\\Delta \\equiv r^2-2Mr+a^2$.\nThe Kerr metric has co-ordinate singularities at $\\Delta=0$, which has for solutions $r_{\\pm}=M\\pm\\sqrt{M^2-a^2}$.\nThe hypersurface $r=r_+$ is the outer event horizon.\nThe surface gravity on the outer[inner] horizon is $\\kappa_+[\\kappa_-]$, given by\n$\\kappa_{\\pm}\\equiv (r_{\\pm}-r_{\\mp})\/(2(r_{\\pm}^2+a^2))$.\n\nThe Kerr space-time is stationary and axially symmetric;\nthe two Killing vectors associated with these two symmetries are \n$\\vec{k}_{\\mathscr{I}}\\equiv \\partial \/ \\partial t$ and $\\vec{k}_{\\circlearrowleft}\\equiv \\partial\/\\partial\\phi$ \nrespectively.\nWe can construct another Killing vector as\n$\\vec{k}_{\\mathscr{H}}\\equiv \\vec{k}_{\\mathscr{I}}+\\Omega_+\\vec{k}_{\\circlearrowleft}$\nwith $\\Omega_+\\equiv a\/(r_+^2+a^2)$.\nThe surface where the Killing vector $\\vec{k}_{\\mathscr{I}}$ becomes a null vector \nis called the stationary limit surface. \nThe vector $\\vec{k}_{\\mathscr{I}}$ is timelike outside this surface and is spacelike in\nthe region between the event horizon and the stationary limit surface. \nThis region is called the ergosphere.\nThe surface where the Killing vector $\\vec{k}_{\\mathscr{H}}$ becomes null is called the speed-of-light surface.\nThe vector $\\vec{k}_{\\mathscr{H}}$ is timelike between the event horizon and the speed-of-light surface, and it is spacelike \noutside it.\n\nAn observer who moves along a world line of constant $r$ and $\\theta$ with an angular velocity \n$\\Omega\\equiv\\d{\\phi}\/\\d{t}$\nrelative to the asymptotic rest frame has a tetrad $\\{\\vec{e}_{(t)},\\vec{e}_{(r)},\\vec{e}_{(\\theta)},\\vec{e}_{(\\phi)}\\}$ \nassociated with him. Such an observer sees no local change in the geometry and is therefore considered a stationary\nobserver relative to the local geometry. If his angular velocity is zero, and therefore he moves along a world line of \nconstant $r$, $\\phi$ and $\\theta$, he is a static observer (SO) relative to radial infinity. A SO moves along the integral\ncurves of $\\vec{k}_{\\mathscr{I}}$.        \nIf we require $\\vec{e}_{(r)}$ and $\\vec{e}_{(\\theta)}$ to be parallel to \n$\\partial\/\\partial r$ and $\\partial\/\\partial \\theta$ \nrespectively, we then find that the two other\nvectors in the tetrad of a stationary observer are given by        \n\\begin{equation} \\label{eq:tetrad of stationary obs.}\n\\begin{aligned}\n\\vec{e}_{(t)}&=\\frac{1}{\\sqrt{\\left|g_{tt}+2\\Omega g_{t\\phi}+\\Omega^2g_{\\phi\\phi}\\right|}}\\left(\\pardiff{}{t}+\\Omega\\pardiff{}{\\phi}\\right) \\\\\n\\vec{e}_{(\\phi)}&=\\frac{1}{\\sqrt{\\left|g_{tt}+2\\Omega g_{t\\phi}+\\Omega^2g_{\\phi\\phi}\\right|}}\\frac{1}{\\sqrt{g^2_{t\\phi}-g_{tt}g_{\\phi\\phi}}}\n\\left[-(g_{t\\phi}+\\Omega g_{\\phi\\phi})\\pardiff{}{t}+(g_{tt}+\\Omega g_{t\\phi})\\pardiff{}{\\phi}\\right]\n\\end{aligned}\n\\end{equation}\nThe quantity $p_{\\phi}\\equiv \\vec{p}\\vec{k}_{\\circlearrowleft}=p_{\\alpha}k_{\\circlearrowleft}^{\\alpha}$, where $\\vec{p}$ is the 4-momentum of a certain observer, is the component\nof angular momentum of that observer along the black hole's spin axis.\nThis quantity is conserved for geodesic observers.\nThe only stationary observers for whom this quantity is zero are those with an angular velocity $\\Omega_{\\text{ZAMO}}=-g_{t\\phi}\/g_{\\phi\\phi}$.\nThese observers are called zero angular momentum observers (ZAMO).\nThese observers are the closest analogue to the static observers in Schwarzschild space-time on the Kerr-Newman space-time in the\nsense that their 4-velocity is orthogonal to $\\vec{k}_{\\circlearrowleft}$, the hypersurfaces of constant $t$. \nThe angular velocity $\\Omega_{\\text{ZAMO}}$ of the ZAMOs as they approach the horizon tends to $\\Omega_+$, which can therefore be\ninterpreted as the angular velocity of the horizon. The stationary observers whose angular velocity is constant and equal to $\\Omega_+$ are observers\nthat follow integral curves of $\\vec{k}_{\\mathscr{H}}$ and are called rigidly rotating observers (RRO).       \nThe last orthonormal tetrad that is of interest to us is the Carter orthonormal tetrad ~\\cite{ar:Carter'68b}, \nwhich corresponds to a stationary observer (\\ref{eq:tetrad of stationary obs.}) with angular velocity $\\Omega_{\\text{CARTER}}=a\/(r^2+a^2)$.\n\nThe Newman-Penrose (NP) formalism (~\\cite{ar:N&P'62}) is based on four null basis vectors: $\\{\\vec{l},\\vec{n},\\vec{m},\\vec{m^*}\\}$.\nIn this paper we will use the Kinnersley tetrad:\n\\begin{subequations} \\label{eq:def. null Kinnersley tetrad}\n\\begin{align}\n\\vec{l}&=\\frac{1}{\\Delta}\\left[(r^2+a^2)\\pardiff{}{t}+\\Delta\\pardiff{}{r}+a\\pardiff{}{\\phi}\\right]\\\\\n\\vec{n}&=\\frac{1}{2\\Sigma}\\left[(r^2+a^2)\\pardiff{}{t}-\\Delta\\pardiff{}{r}+a\\pardiff{}{\\phi}\\right]\\\\\n\\vec{m}&=\\frac{-\\rho^*}{\\sqrt 2}\\left[ia\\sin\\theta\\pardiff{}{t}+\\pardiff{}{\\theta}+i\\mathop{\\mathrm{cosec}}\\theta\\pardiff{}{\\phi}\\right]\n\\end{align}\n\\end{subequations}\nwhere $\\rho=-1\/(r-ia\\cos\\theta)$ is one of the spin coefficients of the Kerr metric in the Kinnersley tetrad.\nThe other spin coefficients and NP scalars can be found in ~\\cite{bk:Chandr}. Of interest to us are the \nNP Maxwell scalars: $\\phi_{-1} \\equiv F_{\\vec{l}\\vec{m}}$, $\\phi_{0} \\equiv \\left(F_{\\vec{l}\\vec{n}}+F_{\\vec{m^*}\\vec{m}}\\right)\/2$\nand $\\phi_{+1} \\equiv F_{\\vec{m^*}\\vec{n}}$, where $F_{\\alpha \\beta}\\equiv A_{\\beta ;\\alpha}-A_{\\alpha;\\beta}$ is the anti-symmetric Maxwell tensor\nand $A_{\\alpha}$ is the electromagnetic potential.\nFrom these equations, one can obtain the operator $K_{h}^{\\mu}$ which maps the potential onto the NP scalars, i.e., \n\\begin{equation}\\label{eq:phi as func. of potential with K}\n\\phi_{h}=K_{h}^{\\mu}A_{\\mu}\n\\end{equation}\nThe explicit form of the operator $K_{h}^{\\mu}$ can be found in ~\\cite{ar:Chrzan'75}.\n\nIt is easy to see that the effect of the parity operator $\\mathcal{P}:(\\theta,\\phi)\\to(\\pi-\\theta,\\phi+\\pi)$ on the NP Kinnersley tetrad \nis\n\\begin{equation} \\label{eq:parity op. on NP objs.}\n\\mathcal{P}\\vec{l}=\\vec{l}, \\qquad\\qquad \\mathcal{P}\\vec{n}=\\vec{n},  \\qquad\\qquad \\mathcal{P}\\vec{m}=-\\vec{m}^*\n\\end{equation}\nIn the limit $r\\rightarrow+\\infty$, the Kinnersley tetrad becomes\n\\begin{equation} \\label{eq:Kinnersley tetrad, r->inf}\n\\begin{aligned}\\vec{l}&\\rightarrow -\\hat{\\vec{e}}_t+\\hat{\\vec{e}}_r, & \n\\qquad \n\\vec{n}&\\rightarrow -\\frac{1}{2}\\left(\\hat{\\vec{e}}_t+\\hat{\\vec{e}}_r\\right), & \n\\qquad \n\\vec{m}&\\rightarrow +\\frac{1}{\\sqrt{2}}\\left(\\hat{\\vec{e}}_{\\theta}+i\\hat{\\vec{e}}_{\\phi}\\right) & \n\\qquad (r\\rightarrow+\\infty) \n\\end{aligned}\n\\end{equation}\nin terms of the usual unit polar vectors \n$\\{\\hat{\\vec{e}}_t,\\hat{\\vec{e}}_r,\\hat{\\vec{e}}_{\\theta},\\hat{\\vec{e}}_{\\phi}\\}$\nin flat space-time.\n\nBy making use of the NP formalism, Teukolsky (~\\cite{ar:Teuk'72}, ~\\cite{ar:Teuk'73}) decoupled the equations describing the \nspin-0,-1\/2,-1 and -2 linear perturbations in a general Type D vacuum background.\nHe further showed that, using the Kinnersley tetrad, some of the decoupled equations are separable in Boyer-Lindquist co-ordinates in the Kerr background.\nWe will refer to the second order ordinary differential equations for the variables $r$ and $\\theta$ resulting from this separation as the \nradial and angular Teukolsky equations respectively.\nThe terminology we will use is the following.\nA bullet $\\bullet$ (and a primed bullet $\\bullet'$) superscript indicates either `in', `up', `down' or `out' modes, which are standard but, for clarity, are\ndefined in Appendix \\ref{sec:radial func.}.\nCorrespondingly, the symbol $\\omega^{\\bullet}$ is defined as being equal to $\\omega$ when it is part of an expression\ncontaining `in' modes and it is equal to $\\tilde{\\omega}$ when the expression contains `up' modes.\nThe separability of the equations describing the spin-1 perturbations implies that the electromagnetic potential\ncan be expressed as a Fourier mode sum as\n\\begin{subequations}\n\\begin{align}\nA_{\\mu}^{\\bullet}&=\\int_{-\\infty}^{+\\infty}\\d{\\omega}\\sum_{l=|h|}^{+\\infty}\\sum_{m=-l}^{+l}\\sum_{P=\\pm 1}a_{lm\\omega P}^{\\bullet}\\ {}_{lm\\omega P}A_{\\mu}^{\\bullet}\n\\label{eq:Fourier expansion for A} \\\\\n{}_{lm\\omega P}A_{\\mu}^{\\bullet}&\\equiv {}_{lm\\omega}A_{\\mu}^{\\bullet}+P\\mathcal{P}{}_{lm\\omega}A_{\\mu}^{\\bullet} \\label{eq: def. of lmwPA_mu}\n\\end{align}\n\\end{subequations}\nwhere $a_{lm\\omega P}^{\\bullet}$ are the coefficients of the Fourier series.\nThe parameter $l$ labels the eigenvalues of the angular Teukolsky equation.\nThe parameter $P$ is summed in (\\ref{eq:Fourier expansion for A}) over the values $+1$ and $-1$, corresponding \nto two linearly independent polarization states for the potential, as we shall see in Section \\ref{sec:theta->pi-theta Symmetry}.\nThe decomposition of the potential into eigenstates of the parity operator $\\mathcal{P}$ is the natural choice because of the \ninvariance of the Kerr metric under this operation. \n\nChrzanowski ~\\cite{ar:Chrzan'75} was the first author to give\nanalytic expressions, using Teukolsky's results,  for the linear electromagnetic and gravitational perturbation potentials in the Kerr background. \nHe obtained, for the electromagnetic potential modes:\n\\begin{subequations} \\label{eq:tableIChrzan.}\n\\begin{align}\n\\begin{split}\n&{}_{l-m-\\omega P}A{}^{\\text{in}}_{\\mu}\n=\n\\\\\n&=\\Big\\{\n\\left[l_{\\mu}(\\delta^*+2\\beta^*+\\tau^*)-m^{*}_{\\mu}(D+2\\epsilon^*+\\rho^*)\\right]\n{}_{-1}R_{lm\\omega }^{\\text{in}}(r){}_{+1}Z_{lm\\omega }(\\theta,\\phi)e^{-i\\omega t}\n+\\\\&+P\n\\left[l_{\\mu}(\\delta+2\\beta+\\tau)-m_{\\mu}(D+2\\epsilon+\\rho)\\right]\n{}_{-1}R_{lm\\omega }^{\\text{in}}(r){}_{-1}Z_{lm\\omega }(\\theta,\\phi)e^{-i\\omega t}\n\\Big\\}\n\\end{split} \\label{eq:A_1, tableIChrzan.} \\\\\n\\begin{split}\n&{}_{l-m-\\omega P}A{}^{\\text{up}}_{\\mu}\n=\n\\\\\n&=\\Big\\{\n\\rho^{* -2}\\left[-n_{\\mu}(\\delta-2\\alpha^*+\\pi^*)+m_{\\mu}(\\Delta-2\\gamma^*+\\mu^*)\\right] \n{}_{+1}R_{lm\\omega }^{\\text{up}}(r){}_{-1}Z_{lm\\omega }(\\theta,\\phi)e^{-i\\omega t}\n+\\\\&+P\n\\rho^{* -2}\\left[-n_{\\mu}(\\delta^*-2\\alpha+\\pi)+m^{*}_{\\mu}(\\Delta-2\\gamma+\\mu)\\right] \n{}_{+1}R_{lm\\omega }^{\\text{up}}(r){}_{+1}Z_{lm\\omega }(\\theta,\\phi)e^{-i\\omega t}\n\\Big\\}\n\\end{split}\n\\label{eq:A1, tableIChrzan.}\n\\end{align}\n\\end{subequations}\nwhere\n\\begin{equation}\n{}_{h}Z_{lm\\omega}(\\theta,\\phi)\\equiv \\frac{(-1)^{m+1}}{\\sqrt{2\\pi}}{}_{h}S_{lm\\omega}(\\theta)e^{+im\\phi}.\n\\end{equation}\nThe only difference here with Chrzanowski's original expressions \nis an overall change in the sign of $m$ and $\\omega$, \njustified by the sum over $m$ and integration over $\\omega$.\n\nIt is immediate from the radial and angular Teukolsky equations that their respective solutions ${}_{h}R_{lm\\omega}(r)$\nand ${}_{h}S_{lm\\omega}(\\theta)$ satisfy the following symmetries:\n\\begin{subequations} \\label{eq:R symms.}\n\\begin{align}\n&{}_{h}R_{lm\\omega}(r)=\\Delta^{-h }{}_{-h }R_{lm\\omega}^*(r)   \\label{eq:R symm.->cc,-s} \\\\\n&{}_{h}R_{lm\\omega}(r)={}_{h}R_{l-m-\\omega}^*(r)  \\label{eq:R symm.->cc,-m,-w}\n\\end{align}\n\\end{subequations}\nand\n\\begin{subequations} \\label{eq: S symms}\n\\begin{align}\n&{}_{h}S_{lm\\omega}(\\theta)=(-1)^{l+m}{}_{-h}S_{lm\\omega}(\\pi-\\theta)   \\label{eq:S symm.->pi-t,-s} \\\\\n&{}_{h}S_{lm\\omega}(\\theta)=(-1)^{l+h }{}_{h}S_{l-m-\\omega}(\\pi-\\theta)  \\label{eq:S symm.->pi-t,-m,-w}\\\\\n&{}_{h}S_{lm\\omega}(\\theta)=(-1)^{h+m}{}_{-h}S_{l-m-\\omega}(\\theta)     \\label{eq:S symm.->-s,-m,-w}\n\\end{align}\n\\end{subequations}\n\nNote the symmetry property\n\\begin{equation}\n\\mathcal{P}{}_{lm\\omega}A_{\\mu}^{\\bullet}=(-1)^{l+m}{}_{l-m-\\omega}A_{\\mu}^{\\bullet *}\n\\end{equation}\nwhich is easily obtained from (\\ref{eq:R symm.->cc,-m,-w}), (\\ref{eq:S symm.->-s,-m,-w}) and (\\ref{eq:tableIChrzan.}).\nIt then follows that the Fourier sum coefficients must satisfy the following condition so that the potential remains real:\n\\begin{equation} \\label{eq: c.c. of lmwPa}\na_{lm\\omega P}^{\\bullet *}=(-1)^{l+m}Pa_{l-m-\\omega P}^{\\bullet}\n\\end{equation}\n\nThe Maxwell scalars $\\phi_{h}^{\\bullet}$ can be decomposed in a Fourier mode sum analogous to that in (\\ref{eq:Fourier expansion for A}).\nFollowing ~\\cite{ar:CMSR} and ~\\cite{bk:Chandr}, the Maxwell scalars modes can be expressed as\n\\begin{equation} \\label{eq:phi_0\/2(in\/up)}\n\\begin{aligned}\n{}_{l-m-\\omega}\\phi_{-1}^{\\text{in}}&=-\\frac{1}{2}{}_{+1}R^{\\text{in}}_{lm\\omega}{}_{+1}Z_{lm\\omega}e^{-i\\omega t} \\\\ \n{}_{l-m-\\omega}\\phi_{+1}^{\\text{in}}&=-\\frac{{}_{1}B_{lm\\omega}}{2}\\rho^2{}_{-1}R^{\\text{in}}_{lm\\omega}{}_{-1}Z_{lm\\omega}e^{-i\\omega t} \\\\\n{}_{l-m-\\omega}\\phi_{-1}^{\\text{up}}&=-\\frac{{}_{1}B_{lm\\omega}}{2}{}_{+1}R^{\\text{up}}_{lm\\omega}{}_{+1}Z_{lm\\omega}e^{-i\\omega t}      \\\\\n{}_{l-m-\\omega}\\phi_{+1}^{\\text{up}}&=-\\frac{{}_{1}B_{lm\\omega}^2}{2}\\rho^2{}_{-1}R^{\\text{up}}_{lm\\omega}{}_{-1}Z_{lm\\omega}e^{-i\\omega t}\n\\end{aligned}\n\\end{equation}\nand \n\\begin{equation} \\label{eq:phi0(ch)}\n\\begin{aligned}\n{}_{l-m-\\omega}\\phi_{0}^{\\text{in}}&=-\\frac{\\rho^2}{2^{3\/2}{}_{1}B_{lm\\omega}}\\left[\\left(\\rho^{-1}\\mathcal{D}_0^{\\dagger}+1\\right)\n\\mathcal{L}^{\\dagger}_{1}+ia\\sin\\theta\\mathcal{D}_0^{\\dagger}\\right]\\Delta{}_{+1}R_{lm\\omega}^{\\text{in}}{}_{-1}Z_{lm\\omega}e^{-i\\omega t}= \\\\\n&=\\frac{\\rho^2}{\\sqrt{2}}\\left[\\left(\\rho^{-1}\\mathcal{D}_0+1\\right)\n\\mathcal{L}_{1}+ia\\sin\\theta\\mathcal{D}_0\\right]{}_{-1}R_{lm\\omega}^{\\text{in}}{}_{+1}Z_{lm\\omega}e^{-i\\omega t} \\\\\n{}_{l-m-\\omega}\\phi_{0}^{\\text{up}}&=-\\frac{\\rho^2}{2^{3\/2}}\\left[\\left(\\rho^{-1}\\mathcal{D}_0^{\\dagger}+1\\right)\n\\mathcal{L}^{\\dagger}_{1}+ia\\sin\\theta\\mathcal{D}_0^{\\dagger}\\right]\\Delta{}_{+1}R_{lm\\omega}^{\\text{up}}{}_{-1}Z_{lm\\omega}e^{-i\\omega t}= \\\\\n&=\\frac{\\rho^2{}_{1}B_{lm\\omega}}{\\sqrt{2}}\\left[\\left(\\rho^{-1}\\mathcal{D}_0+1\\right)\n\\mathcal{L}_{1}+ia\\sin\\theta\\mathcal{D}_0\\right]{}_{-1}R_{lm\\omega}^{\\text{up}}{}_{+1}Z_{lm\\omega}e^{-i\\omega t}\n\\end{aligned}\n\\end{equation}\nwhere ${}_1B_{lm\\omega}^2\\equiv {}_{-1}\\lambda_{lm\\omega}^2+4ma\\omega-4a^2\\omega^2$ and \n${}_{h}\\lambda_{lm\\omega}$ is the eigenvalue of the angular Teukolsky equation.\nWe are using the definitions of the operators\n\\begin{equation} \\label{eq:def. L_n}\n\\mathcal{L}_{n}^\n{ \\topbott{}{\\dagger}} \\equiv \\partial_{\\theta} \\pm \\mathcal{Q}+n\\cot\\theta  \n\\qquad \\text{and} \\qquad \n\\mathcal{D}_{n}^\n{ \\topbott{}{\\dagger}} \\equiv \\partial_{r} \\mp \\frac{iK}{\\Delta}+2n\\frac{r-M}{\\Delta}    \n\\end{equation}\nwhere $\\mathcal{Q}\\equiv -a\\omega\\sin\\theta+m\/\\sin\\theta$ and $K\\equiv (r^2+a^2)\\omega-am$.\nWe use the convention that the upper and lower symbols inside braces go \nwith the upper and lower signs in the equation.\n\nThe asymptotic behaviour of the NP Maxwell scalars, separately for outgoing and ingoing waves, in the limit $r\\to +\\infty$ is \n\\begin{equation} \\label{eq:peeling th}\n\\begin{aligned}\n{}_{lm\\omega}\\phi_{+1}&\\sim r^{-1}e^{+i\\omega r},     &\\quad r^{-3}e^{-i\\omega r}&\\quad (r\\rightarrow +\\infty) \\\\\n{}_{lm\\omega}\\phi_{0}&\\sim r^{-2}e^{+i\\omega r},     &\\quad r^{-2}e^{-i\\omega r}&\\quad (r\\rightarrow +\\infty) \\\\\n{}_{lm\\omega}\\phi_{-1}&\\sim r^{-3}e^{+i\\omega r},     &\\quad r^{-1}e^{-i\\omega r}&\\quad (r\\rightarrow +\\infty)\n\\end{aligned}\n\\end{equation}\nIt is therefore the scalar $\\phi_{+1[-1]}$ the one with the asymptotically dominant behaviour \nfor the upgoing[ingoing] waves. The above asymptotic behaviour (\\ref{eq:peeling th}) was originally obtained by\nNewman and Penrose ~\\cite{ar:N&P'62} and is commonly referred to as the peeling theorem. \n\nThe Teukolsky-Starobinski\\u{\\i} identities relate solutions, radial or angular,\nof opposite helicity $h$. These identities are (~\\cite{ar:Teuk&Press'74}):\n\\begin{subequations} \\label{eq:Teuk-Starob. ids.}\n\\begin{align}\n\\mathcal{D}_0\\mathcal{D}_0{}_{-1}R_{lm\\omega}&=\n\\frac{1}{2}{}_{+1}R_{lm\\omega} & \\mathcal{L}_0\\mathcal{L}_1{}_{+1}S_{lm\\omega}&={}_1B_{lm\\omega}{}_{-1}S_{lm\\omega} \\label{eq:a)Teuk-Starob. ids.} \\\\\n\\Delta\\mathcal{D}^{\\dagger}_0\\mathcal{D}^{\\dagger}_0\\Delta{}_{+1}R_{lm\\omega}&=2{}_1B_{lm\\omega}^2{}_{-1}R_{lm\\omega} & \n\\mathcal{L}^{\\dagger}_0\\mathcal{L}^{\\dagger}_1{}_{-1}S_{lm\\omega}&={}_1B_{lm\\omega}{}_{+1}S_{lm\\omega} \\label{eq:b)Teuk-Starob. ids.}\n\\end{align}\n\\end{subequations}\n\n\n\nThe angular Teukolsky equation has singular points at $x=\\pm 1$, where $x\\equiv \\cos\\theta$.\nBy using the Frobenius method it can be found that the solution that is regular at both \nboundary points $x=+1$ and $-1$ is given by                            \n\\begin{equation} \\label{eq:asympt. S for x->+\/-1}\n{}_{h}S_{lm\\omega}(x)=(1-x)^{|m+h |\/2}(1+x)^{|m-h |\/2}{}_{h}y_{lm\\omega}(x)\n\\end{equation}\nand the function \n${}_{h}y_{lm\\omega}(x)$ behaves close to the boundary points as\n\\begin{equation} \\label{eq:asympt. y for x->+\/-1}\n{}_{h}y_{lm\\omega}(x)=\\sum_{n=0}^{\\infty} {{}_{h}\\topbott{a}{b}_{n,lm\\omega}(1\\mp x)^n}    \\quad   \\text{for} \\quad x\\to \\pm1            \n\\end{equation}\n\nThe solutions ${}_{h}Z_{lm\\omega}$ are called the spin-weighted spheroidal harmonics (SWSH).\nIn the spherical limit $a=0$, the SWSH reduce to the spin-weighted spherical harmonics ${}_{h}Y_{lm}$.\nWe give here two properties (~\\cite{ar:J&McL&Ott'91}, ~\\cite{th:McL'90} and ~\\cite{ar:J&McL&Ott'95}) \nsatisfied by the spin-weighted spherical harmonics which we will need later on:\n\\begin{equation} \\label{eq:eq.2aJ,McL,Ott'91}\n\\sum_{m=-l}^l {}_{-1}Y_{lm}(\\theta,\\phi){}_{+1}Y_{lm}^*(\\theta,\\phi)=0\n\\end{equation}\nand the ``addition theorem''  \n\\begin{equation} \\label{eq:eq.B6J,McL,Ott'95}\n\\sum_{m=-l}^l {}_{h}Y_{lm}(\\theta,\\phi){}_{h}Y^*_{lm}(\\theta',\\phi')=\\frac{2l+1}{4\\pi}P_l(\\cos\\gamma)\n\\end{equation}\nwhere $\\gamma$ is defined by $\\cos\\gamma\\equiv \\cos\\theta\\cos\\theta'+\\sin\\theta\\sin\\theta'\\cos(\\phi-\\phi')$.\n\nThe classical, electromagnetic stress-energy tensor can be expressed\nin terms of the NP Maxwell scalars as\n\\begin{equation} \\label{eq:stress tensor, spin 1}\n\\begin{aligned}\nT_{\\mu\\nu}&=\\Big\\{\\phi_{-1}\\phi_{-1}^*n_{\\mu}n_{\\nu}+2\\phi_0\\phi_0^*\\left[l_{(\\mu}n_{\\nu)}+\nm_{(\\mu}m_{\\nu)}^*\\right]+\\phi_{+1}\\phi_{+1}^*l_{\\mu}l_{\\nu}-\n\\\\ \n&\n-4\\phi_0\\phi_{-1}^*n_{(\\mu}m_{\\nu)}-4\\phi_{+1}\\phi_0^*l_{(\\mu}m_{\\nu)}+2\\phi_{+1}\\phi_{-1}^*m_{\\mu}m_{\\nu}\\Big\\}+c.c.\n\\end{aligned}\n\\end{equation}\nin the case of absence of a charge current.\nBy virtue of the Maxwell field equations, the stress-energy tensor \n(\\ref{eq:stress tensor, spin 1})\nsatisfies the conservation equation $T^{\\mu\\nu}{}_{;\\mu}=0$.\n\n\nNote that it follows from (\\ref{eq:parity op. on NP objs.}) that all the pairs of null tetrad vectors  \nappearing in the different terms in (\\ref{eq:stress tensor, spin 1}) remain invariant under\nthe parity operation, except for the ones that have a factor containing $\\phi_0$ together with either $\\phi_{-1}$ or $\\phi_{+1}$, which\nchange sign. That is, a pair of null \nvectors $\\vec{e}_{(a)}\\vec{e}_{(b)}$ \nappearing in (\\ref{eq:stress tensor, spin 1}) with a factor\n$\\phi_{h}\\phi_{h'}^*$ changes under the parity operation as\n\\begin{equation}  \\label{eq:pairs of vects. in stress tensor under parity}\n\\mathcal{P}\\left(\\vec{e}_{(a)}\\vec{e}_{(b)}\\right)=(-1)^{h+h'}\\vec{e}_{(a)}\\vec{e}_{(b)}\n\\end{equation}\n\nThe variable $\\Gamma$ refers to either the potential components $A_{\\mu}$ or the NP scalars $\\phi_{h}$.\nThe Fourier series expansion for either the potential components or the NP Maxwell scalars may then be expressed as\n\\begin{equation} \\label{eq:Fourier series of Gamma field}\n\\Gamma^{\\bullet}= \\sum_{lmP} \\int_{-\\infty}^{+\\infty}\\d{\\omega^{\\bullet}}a_{lm\\omega P}^{\\bullet}{}_{lm\\omega P}\\Gamma^{\\bullet}\n\\end{equation}\nwhich may be re-arranged as\n\\begin{equation} \\label{eq:Fourier series of Gamma field for pos.freq.}\n\\Gamma^{\\bullet}=\\sum_{lmP} \\int_{0}^{+\\infty}\\d{\\omega^{\\bullet}}\\left(a_{lm\\omega P}^{\\bullet}{}_{lm\\omega P}\\Gamma^{\\bullet}+\n(-1)^{l+m}Pa_{lm\\omega P}^{\\bullet *}{}_{l-m-\\omega P}\\Gamma^{\\bullet}\\right)\n\\end{equation}\n\nWe can now use the symmetry relations\n\\begin{subequations} \\label{eq: symms. of lmwA_mu and lmwphi_i}\n\\begin{align}\n\\mathcal{P}{}_{lm\\omega P}A_{\\mu}^{\\bullet}&=(-1)^{l+m}{}_{l-m-\\omega P}A_{\\mu}^{\\bullet *}=\nP{}_{lm\\omega P}A_{\\mu}^{\\bullet} \\label{eq: symm. of lmwA_mu}\n\\\\\n\\mathcal{P}{}_{lm\\omega}\\phi_{h}^{\\bullet}&=(-1)^{l+m+1+h}{}_{l-m-\\omega}\\phi_{h}^{\\bullet *} \\label{eq: symm. of lmwphi_i}\n\\end{align}\n\\end{subequations}\n\nThe equations above for $\\Gamma$ are equally valid for $A_{\\mu}$ and $\\phi_{h}$. In particular, we only need to \napply the operator $K^{\\mu}_{h}$ to an equation for $A_{\\mu}$ in order to obtain the \ncorresponding equation for $\\phi_{h}$. However the last step in (\\ref{eq: symm. of lmwA_mu}) has no \nequivalent for  $\\phi_{h}$ in (\\ref{eq: symm. of lmwphi_i}).  \nThe reason is that\n\\begin{equation} \\label{eq:parity term is pure gauge}\nK^{\\mu}_{h}\\ \\mathcal{P}{}_{lm\\omega}A_{\\mu}^{\\bullet} \\propto \nK^{\\mu}_{h}\\ {}_{l-m-\\omega}A_{\\mu}^{\\bullet *} \\equiv 0\n\\end{equation}\nas can be checked; that is, this term is pure gauge. \nHence the fact that\n${}_{lm\\omega}\\phi_{h}^{\\bullet}=K^{\\mu}_{h}\\ {}_{lm\\omega P}A_{\\mu}^{\\bullet}$ \ndoes not actually depend on $P$ which is why $P$ is not a subindex of the NP scalar modes. \nThe potential is real whereas the field \ncomponents are not, as seen in equations (\\ref{eq:classical mode expansion for A(in)_mu}) \nand (\\ref{eq:classical mode expansion for phi(in)_i}) below. \nFrom (\\ref{eq:Fourier series of Gamma field for pos.freq.}) and using (\\ref{eq: symm. of lmwA_mu}) and \n(\\ref{eq: symm. of lmwphi_i}) for the potential and the field respectively, we have\n\\begin{equation} \\label{eq:classical mode expansion for A(in)_mu}\n\\begin{aligned}\nA_{\\mu}^{\\bullet}\n&=\\sum_{lmP} \\int_{0}^{+\\infty}\\d{\\omega^{\\bullet}}\\left(a_{lm\\omega P}^{\\bullet}{}_{lm\\omega P}A_{\\mu}^{\\bullet}+\nPa_{lm\\omega P}^{\\bullet *}\\mathcal{P}{}_{lm\\omega P}A_{\\mu}^{\\bullet *}\\right)=            \\\\    \n&=\\sum_{lmP} \\int_{0}^{+\\infty}\\d{\\omega^{\\bullet}}\\left(a_{lm\\omega P}^{\\bullet}{}_{lm\\omega P}A_{\\mu}^{\\bullet}+\na_{lm\\omega P}^{\\bullet *}{}_{lm\\omega P}A_{\\mu}^{\\bullet *}\\right)\n\\end{aligned}\n\\end{equation}\nand\n\\begin{equation} \\label{eq:classical mode expansion for phi(in)_i}\n\\phi_{h}^{\\bullet}=\\sum_{lmP} \\int_{0}^{+\\infty}\\d{\\omega^{\\bullet}}\\left(a_{lm\\omega P}^{\\bullet}{}_{lm\\omega}\\phi_{h}^{\\bullet}+\n(-1)^{h+1}Pa_{lm\\omega P}^{\\bullet *}\\mathcal{P}{}_{lm\\omega}\\phi_{h}^{\\bullet *}\\right)\n\\end{equation}\n\n\n\n\\section{Quantization of the electromagnetic potential\/field}  \\label{sec:CCH}\n\nThe abundance in the literature of the quantization of the scalar field in a curved background is in sharp contrast\nwith the scarce treatment of the quantization of the electromagnetic -or gravitational- field in such a background.\nCCH did quantize both the electromagnetic and the\ngravitational fields in the Kerr background. They used a canonical quantization method, which is the one\nwe have chosen to use in this paper.\n\nWe quantize the field by promoting $a_{lm\\omega P}^{\\bullet}$ and $a_{lm\\omega P}^{\\bullet *}$ to \noperators $\\hat{a}_{lm\\omega P}^{\\bullet}$ and $\\hat{a}_{lm\\omega P}^{\\bullet \\dagger}$ respectively.\nCohen and Kegeles ~\\cite{ar:Coh&Keg'74} and Wald ~\\cite{ar:Wald'78} have shown\nthat the theory may be expressed in terms of one\nsingle NP complex scalar, which represents the two radiative degrees of freedom of the electromagnetic perturbations. \nIf we introduce expansion (\\ref{eq:classical mode expansion for phi(in)_i}) \nfor the NP scalars into $T^{00}$ given by (\\ref{eq:stress tensor, spin 1}), we then obtain a hamiltonian which is a superposition of \nindependent harmonic oscillator hamiltonians, one for each mode of the electromagnetic field.\nFrom the standard quantization of the harmonic oscillator, we know that the operators \n$\\hat{a}_{lm\\omega P}^{\\bullet}$ and $\\hat{a}_{lm\\omega P}^{\\bullet \\dagger}$ must\nsatisfy the commutation relations:\n\\begin{equation} \\label{eq:commut. rlns. for a,a_dagger}\n\\begin{aligned}\n\\left[\\hat{a}_{lm\\omega P}^{\\bullet},\\hat{a}_{l'm'\\omega'P'}^{\\bullet \\dagger}\\right]=\\delta(\\omega-\\omega')\\delta_{ll'}\\delta_{mm'}\\delta_{PP'} \\\\\n\\Big[\\hat{a}_{lm\\omega P}^{\\bullet},\\hat{a}_{l'm'\\omega'P'}^{\\bullet}\\Big]=\\left[\\hat{a}_{lm\\omega P}^{\\bullet \\dagger},\\hat{a}_{l'm'\\omega'P'}^{\\bullet \\dagger}\\right]=0\n\\end{aligned}\n\\end{equation}\n\nThese commutation relations are satisfied provided that the orthonormality conditions\n\\begin{equation} \\label{eq:orthonormality conds. for potential}\n\\begin{aligned}\n\\left\\langle {}_{lm\\omega P}A^{\\bullet}_{\\alpha},{}_{l'm'\\omega'P'}A^{\\bullet'}_{\\alpha}\\right\\rangle_{\\mathcal{S}}&=\n\\delta_{\\bullet \\bullet'}\\delta_{ll'}\\delta_{mm'}\\delta(\\omega-\\omega')\\delta_{PP'} \\\\\n\\left\\langle {}_{lm\\omega P}A^{\\bullet *}_{\\alpha},{}_{l'm'\\omega'P'}A^{\\bullet'}_{\\alpha}\\right\\rangle_{\\mathcal{S}}&=0\n\\end{aligned}\n\\end{equation}\nare satisfied, where $\\mathcal{S}$ is any complete Cauchy hypersurface for the outer region of the space-time\nand where the Klein-Gordon inner product is taken as \n\\begin{equation} \\label{eq:def. inner prod.}\n\\left\\langle \\psi_{\\alpha},\\varphi_{\\alpha}\\right\\rangle_{\\mathcal{S}}=\ni\\int_{\\mathcal{S}}\\d^3{\\Sigma}^{\\mu}\\left(\\psi^{\\alpha *}\\nabla_{\\mu}\\varphi_{\\alpha}-\\varphi^{\\alpha}\\nabla_{\\mu}\\psi^*_{\\alpha}+\n\\varphi_{\\mu}\\nabla_{\\alpha}\\psi^{\\alpha *}-\\psi^*_{\\mu}\\nabla_{\\alpha}\\varphi^{\\alpha}\\right)\n\\end{equation}\nThe inner product (\\ref{eq:def. inner prod.}) has the same form as the one taken by CCH. \nHowever, CCH give an expression for the stress-energy tensor which includes a factor $4\\pi$ in (\\ref{eq:stress tensor, spin 1}),\ncorresponding to unrationalized units. If unrationalized units are used, then a factor $4\\pi$ should also be included in the\ninner product (\\ref{eq:def. inner prod.}). \n\nConstants of normalization are to be included in front of the radial functions so that the potential modes (\\ref{eq:tableIChrzan.}) satisfy\nthe orthonormality conditions (\\ref{eq:orthonormality conds. for potential}) given the asymptotic behaviour of the radial functions in (\\ref{eq:R_in\/up}).\nWe find that the constants of normalization are given by:\n\\begin{subequations}  \\label{eq:normalization consts.}\n\\begin{align}\n|N_{-1}^{\\text{in}}|^2&=\\frac{1}{2^5\\omega^3\\pi} \\label{eq:N_1(in) normalization const.} \\\\\n|N_{+1}^{\\text{up}}|^2&=\\frac{1}{2^3\\pi|\\EuFrak{N}|^2\\tilde{\\omega}(r_+^2+a^2)} \\label{eq:N1(up) normalization const.}\\\\\n|N_{-1}^{\\text{up}}|^2&=|N_{+1}^{\\text{up}}|^2\\left|\\frac{{}_{-1}R^{\\text{up,inc}}_{lm\\omega}}{{}_{+1}R^{\\text{up,inc}}_{lm\\omega}}\\right|^2=\n\\frac{\\tilde{\\omega}(r_+^2+a^2)}{2\\pi {}_1B_{lm\\omega}^4} \\label{eq:N_1(up) normalization const.}\n\\end{align}\n\\end{subequations}\nwhere $\\EuFrak{N} \\equiv iK_++(r_+-r_-)\/2$ and $K_+\\equiv K(r_+)$.\nWe have chosen ${}_{-1}R^{\\text{in,inc}}_{lm\\omega}=1$ and ${}_{+1}R^{\\text{up,inc}}_{lm\\omega}=1$ in equations (\\ref{eq:N_1(in) normalization const.})\nand (\\ref{eq:N1(up) normalization const.}) respectively.\nThe constant of normalization $|N_{-1}^{\\text{up}}|$ is calculated as indicated with the use of (\\ref{eq:R1 coeffs from R_1's}). It is \ntherefore the constant of normalization that corresponds to using the radial function (\\ref{eq:R_up}) when setting ${}_{-1}R^{\\text{up,inc}}_{lm\\omega}=1$,\nwhich is the actual normalization we have used in the numerical calculation of the `up' solutions.\nThe NP scalars are therefore assumed to include the constants of normalization (\\ref{eq:normalization consts.}). \nThat is, the NP scalar modes ${}_{lm\\omega}\\phi_{h}^{\\bullet}$ are to be calculated from expressions \n(\\ref{eq:phi_0\/2(in\/up)}) and (\\ref{eq:phi0(ch)}) with the inclusion of the appropriate constant of normalization (\\ref{eq:normalization consts.}),\nwhile the radial functions remain unaltered.\n\nIt can be checked that the set of modes $\\{{}_{lm\\omega P}A_{\\mu}^{\\text{in}},{}_{lm\\omega P}A_{\\mu}^{\\text{up}}\\}$\nforms a complete set of orthonormal solutions to the Maxwell equations in the outer region of the Kerr space-time.\nSimilarly, it can be checked that $\\{{}_{lm\\omega P}A_{\\mu}^{\\text{out}},{}_{lm\\omega P}A_{\\mu}^{\\text{down}}\\}$ also\nform a complete set.\nWe may expand the electromagnetic potential by using the\ncomplete set of solutions $\\{{}_{lm\\omega P}A_{\\mu}^{\\text{in}},{}_{lm\\omega P}A_{\\mu}^{\\text{up}}\\}$ and then quantize it as:\n\\begin{equation} \\label{eq:quantum mode expansion for A in in\/up modes}\n\\begin{aligned}\n\\hat{A}_{\\mu}&\n=\\sum_{lmP} \\int_{0}^{+\\infty}\\d{\\omega}\n\\left(\\hat{a}_{lm\\omega P}^{\\text{in}}{}_{lm\\omega P}A_{\\mu}^{\\text{in}}+\n\\hat{a}_{lm\\omega P}^{\\text{in} \\dagger}{}_{lm\\omega P}A_{\\mu}^{\\text{in} *}\\right)+\\\\ \n&+\\sum_{lmP} \\int_{0}^{+\\infty}\\d{\\tilde{\\omega}}\n\\left(\\hat{a}_{lm\\omega P}^{\\text{up}}{}_{lm\\omega P}A_{\\mu}^{\\text{up}}+\n\\hat{a}_{lm\\omega P}^{\\text{up} \\dagger}{}_{lm\\omega P}A_{\\mu}^{\\text{up} *}\\right)\n\\end{aligned}\n\\end{equation}\n\nAlternatively, we could proceed exactly in the same manner but using the complete set of solutions\n$\\{{}_{lm\\omega P}A_{\\mu}^{\\text{out}},{}_{lm\\omega P}A_{\\mu}^{\\text{down}}\\}$ instead.\nThe result is then:\n\\begin{equation} \\label{eq:quantum mode expansion for A in out\/dn modes}\n\\begin{aligned}\n\\hat{A}_{\\mu}&=\n\\sum_{lmP} \\int_{0}^{+\\infty}\\d{\\omega}\\left(\\hat{a}_{lm\\omega P}^{\\text{out}}{}_{lm\\omega P}A_{\\mu}^{\\text{out}}+\n\\hat{a}_{lm\\omega P}^{\\text{out} \\dagger}{}_{lm\\omega P}A_{\\mu}^{\\text{out} *}\\right)+\\\\ \n&+\\sum_{lmP} \\int_{0}^{+\\infty}\\d{\\tilde{\\omega}}\\left(\\hat{a}_{lm\\omega P}^{\\text{down}}{}_{lm\\omega P}A_{\\mu}^{\\text{down}}+\n\\hat{a}_{lm\\omega P}^{\\text{down} \\dagger}{}_{lm\\omega P}A_{\\mu}^{\\text{down} *}\\right)\n\\end{aligned}\n\\end{equation}\n\nThe asymptotic behaviour in terms of the advanced $v$ and retarded $u$ time co-ordinates of the electromagnetic\npotential and NP scalars for the `in' and `up' modes is the same as the one exhibited by the modes (\\ref{eq:X_in\/up as func. of u,v}).\nThe same applies to the asymptotic behaviour of the \n`out' and `down' modes exhibited in (\\ref{eq:X_out\/down as func. of u,v}).\nAccordingly, the operators $\\hat{a}_{lm\\omega P}^{\\text{in} \\dagger}$, $\\hat{a}_{lm\\omega P}^{\\text{up} \\dagger}$, \n$\\hat{a}_{lm\\omega P}^{\\text{out} \\dagger}$ and $\\hat{a}_{lm\\omega P}^{\\text{down} \\dagger}$ are creation operators of particles incident\nfrom past null infinity $\\mathcal{I}^-$, past horizon $\\mathcal{H}^-$, future null infinity $\\mathcal{I}^+$ and \nfuture horizon $\\mathcal{H}^+$ respectively. \n\nSince the `in' and `out' modes are only defined for $\\omega$ non-negative, they have non-negative energy as measured by\nan observer following the integral curve of $\\vec{k}_{\\mathscr{I}}$, by virtue of (\\ref{eq:hamiltonians on in\/up modes}).\nSimilarly, the `up' and `down' modes, defined for $\\tilde{\\omega}$ non-negative, have non-negative energy with respect\nto observers following the integral curve of $\\vec{k}_{\\mathscr{H}}$.\n\nWe may now construct the stress-energy tensor operator from either the potential operator \n(\\ref{eq:quantum mode expansion for A in in\/up modes}) or (\\ref{eq:quantum mode expansion for A in out\/dn modes}).\nIt is well-known that the stress-energy tensor as an operator does not have a well-defined meaning. It suffers from ultra-violet divergences \nand its expectation value when the field is in a certain state $\\ket{\\Psi}$ must be renormalized. \nThere are several techniques for renormalization.\nThe point-splitting technique consists in starting from each quadratic term in the stress-energy tensor and\ntemporarily displacing one field point, thus forming the object $\\vac{\\hat{T}_{\\alpha\\beta}(x,x')}{\\Psi}$, \nwhich is finite. \nSpecific divergent terms, gathered in the bitensor $T^{\\text{div}}_{\\alpha\\beta}(x,x')$, \nwhich are purely geometric and thus independent of the quantum state, are then subtracted from $\\vac{\\hat{T}_{\\alpha\\beta}(x,x')}{\\Psi}$.\nThe end result is obtained by finally bringing the separated points together:\n\\begin{equation}\n\\vac[ren]{\\hat{T}_{\\alpha\\beta}(x)}{\\Psi}=\\lim_{x'\\to x}\\left(\\vac{\\hat{T}_{\\alpha\\beta}(x,x')}{\\Psi}-T^{\\text{div}}_{\\alpha\\beta}(x,x')\\right)\n\\end{equation}\nIt is this renormalized expectation value of the stress-energy tensor (RSET)\nthat is the source in Einstein's field equations in the semiclassical theory.\nChristensen ~\\cite{ar:Christ'78} has explicitly calculated the divergent terms $T^{\\text{div}}_{\\alpha\\beta}$ by using covariant geodesic point separation. \nJensen, McLaughlin and Ottewill ~\\cite{ar:J&McL&Ott'88} calculated a linearly divergent term for the spin-1 case, \nwhich was not explicitly given by Christensen. The reason being that this term does not have to be included when an average \nis taken over the covariant derivative of the biscalar of geodetic interval $\\sigma^{\\mu}$ and $-\\sigma^{\\mu}$, \nas performed by Christensen.\n\nBefore we start a description of the various physical states of the field, we give an important result found by \nUnruh ~\\cite{ar:Unruh'76} and further established by ~\\cite{ar:Brown&Ott'83} and ~\\cite{ar:Grove&Ott'83}. The result is that a `particle detector'\nwill react to states of the field which have positive frequency with respect to the detector's proper time. \nIf a certain observer $\\mathcal{A}$ makes measurements relative to a certain vacuum state $\\ket{\\Xi}$, then \nhe or she measures a stress tensor \n\\begin{equation}   \\label{eq:stress for gral. obs.&vac.}\n\\vac[\\mathcal{A}]{\\hat{T}_{\\alpha\\beta}}{\\Psi}=\\vac{\\hat{T}_{\\alpha\\beta}}{\\Psi}-\\vac{\\hat{T}_{\\alpha\\beta}}{\\Xi}\n\\end{equation}\nwhen the field is in a certain state $\\ket{\\Psi}$.\n\n\n\\section{Vacua states} \\label{sec:vac. states}\n\nThe properties of the various states described in this section have been obtained in the literature for the scalar case,\nexcept where explicitly indicated otherwise. \n\nThe Boulware vacuum state, denoted by $\\ket{B}$, is defined in Schwarzschild space-time as the vacuum that corresponds \nto quantizing the field with normal modes that have all positive frequency with respect to the space-time's hypersurface-orthogonal\ntimelike killing vector $\\vec{k}_{\\mathscr{I}}$. \nThis state respects the isometries of Schwarzschild space-time.\nSince it is the static observers SO the ones that move along integral curves\nof $\\vec{k}_{\\mathscr{I}}$, from Unruh's result stated in the previous section it follows that these observers will make\nmeasurements relative to the Boulware vacuum $\\ket{B}$. \nCandelas ~\\cite{ar:Candelas'80}, based on conjectures made previously by Christensen and Fulling ~\\cite{ar:Christ&Fulling'77},\nhas found that the RSET when the scalar field is in the Boulware vacuum is zero at both $\\mathcal{I}^-$ and $\\mathcal{I}^+$\nin the Schwarzschild space-time. \nCandelas also found that the RSET, close to the horizon, when the field is in the Boulware vacuum diverges and corresponds to the\nabsence from the vacuum of black-body radiation at the black hole temperature appropriately red-shifted. \nThe Boulware vacuum is therefore irregular at \n$\\mathcal{H}^-$ and $\\mathcal{H}^+$.\n\nIn Schwarzschild space-time the Boulware vacuum may be associated with the field expansion in terms of either the\n`in' and `up' modes or the `out' and `down', both pairs of sets of complete modes defining the same vacuum $\\ket{B}$.\nWe can perform a similar expansion for the electromagnetic potential in the Kerr space-time.\nThe past Boulware state is defined by\n\\begin{equation} \\label{eq:a^in\/up on B}\n\\begin{aligned}\n\\hat{a}_{lm\\omega P}^{\\text{in}}\\ket{B^-}&=0  \\\\\n\\hat{a}_{lm\\omega P}^{\\text{up}}\\ket{B^-}&=0\n\\end{aligned}\n\\end{equation}\ncorresponding to an absence of particles at $\\mathcal{H}^-$ and $\\mathcal{I}^-$. \nWe can also define the future Boulware state, as that state which is empty at $\\mathcal{I}^+$ and $\\mathcal{H}^+$: \n\\begin{equation} \\label{eq:a^out\/down on B}\n\\begin{aligned}\n\\hat{a}_{lm\\omega P}^{\\text{out}}\\ket{B^+}&=0  \\\\\n\\hat{a}_{lm\\omega P}^{\\text{down}}\\ket{B^+}&=0\n\\end{aligned}\n\\end{equation}\nThe Bogolubov transformation between the pair of operators $\\hat{a}_{lm\\omega P}^{\\text{in}}$ and $\\hat{a}_{lm\\omega P}^{\\text{up}}$\nand the pair $\\hat{a}_{lm\\omega P}^{\\text{down}}$ and $\\hat{a}_{lm\\omega P}^{\\text{out}}$ is non-trivial: \nthe expression for $\\hat{a}_{lm\\omega P}^{\\text{in} \\dagger}\\left[\\hat{a}_{lm\\omega P}^{\\text{up} \\dagger}\\right]$ \nin terms of `out' and `down' operators contains $\\hat{a}_{l,-m,-\\omega,P}^{\\text{down} \\dagger}\\left[\\hat{a}_{l,-m,-\\omega,P}^{\\text{out} \\dagger}\\right]$\nfor modes in the superradiant regime.   \nThis implies that the past Boulware state contains both outgoing and downgoing superradiant particles, \nand is therefore not empty at $\\mathcal{I}^+$ and $\\mathcal{H}^+$.\nThis flux of particles out to $\\mathcal{I}^+$ corresponds to the Starobinski\\u{\\i}-Unruh effect. \nSimilarly, the future Boulware state contains ingoing and upgoing superradiant particles, and is therefore not empty \nat $\\mathcal{I}^-$ and $\\mathcal{H}^-$. \nAs $\\hat{a}_{lm\\omega P}^{\\text{up}}$, when expressed in terms of `out' and `down' operators, contains \nthe creator operator $\\hat{a}_{l,-m,-\\omega,P}^{\\text{out} \\dagger}$, it is not possible to construct a state which is empty at both \n$\\mathcal{I}^-$ and $\\mathcal{I}^+$, unlike the situation in the Schwarzschild space-time.\n\nFrom the definitions (\\ref{eq:a^in\/up on B}) and (\\ref{eq:a^out\/down on B}) together with the relations (\\ref{NP scalars in\/up->out\/down}),\nthe past and future Boulware states are obtainable\none from the other under the transformation $(t,\\phi)\\to (-t,-\\phi)$.\nBecause the two states are not equivalent, it follows that neither is invariant under this symmetry of the Kerr space-time.\n\nThe defining features of a Hartle-Hawking state (~\\cite{ar:Hartle&Hawk'76})  is that it possesses the symmetries\nof the space-time and that it is regular everywhere, including on both the past and the future event horizons.\nKay and Wald ~\\cite{ar:Kay&Wald'91} have proven that for any globally hyperbolic space-time which has a Killing field with a bifurcate Killing \nhorizon there can be at most one state with the above features. \nKay and Wald have further shown that for the Kerr space-time this state does not exist. \nRindler and Schwarzschild space-times are covered by Kay and Wald's theorem;\nin Rindler space-time this state is clearly the Minkowski vacuum. \n\nIn Schwarzschild space-time the state $\\ket{H}$ corresponds to quantizing the\nfield with upgoing normal modes which on $\\mathcal{H}^-$ have positive frequency with respect to the Kruskal co-ordinate $U\\equiv -e^{-\\kappa_+u}$\nand with ingoing normal modes which on $\\mathcal{H}^+$ have positive frequency with respect to the Kruskal co-ordinate $V\\equiv e^{\\kappa_+v}$. \nCandelas ~\\cite{ar:Candelas'80} showed that the state defined in this manner is regular on both the past and \nfuture horizons. He also found that the RSET at infinity when the field is in the $\\ket{H}$ state corresponds \nto that of a bath of black body radiation at the black hole temperature $T_H=\\kappa_+\/(2\\pi)$. The Hartle-Hawking state models a black\nhole in (unstable) thermal equilibrium with an infinite distribution at the Hawking temperature.\n\nFrom the above results and from the previous section we know that $\\vac{\\hat{T}_{\\alpha\\beta}}{H-B}$ is thermal both for\n$r\\rightarrow r_+$ and for $r\\rightarrow +\\infty$. Christensen and Fulling conjectured that this is the case everywhere.\nHowever, Jensen, McLaughlin and Ottewill \\cite{ar:J&McL&Ott'92} numerically showed that $\\vac{\\hat{T}_{\\alpha\\beta}}{H-B}$\ndeviates from isotropic, thermal form as one moves away from the horizon.\n\nCandelas showed that the RSET close to the horizon when the field is in the $\\ket{B}$ state diverges\nlike minus the stress tensor of black body radiation at the black hole temperature, and that it must tend  \nto $-\\vac[SO]{\\hat{T}_{\\alpha\\beta}}{H}$, due to (\\ref{eq:stress for gral. obs.&vac.}) and to the regularity of $\\ket{H}$. \nAnalogously, Unruh ~\\cite{ar:Unruh'76} showed that in flat space-time and when the field is in the Minkowski vacuum, \na Rindler observer RO will also see a bath of black body radiation at the Hawking temperature of a black hole \nwith surface gravity $\\kappa_+=a\\alpha$, where $a$ is the RO's acceleration and $\\alpha$ is the lapse function in Rindler space.  \n\nFrolov and Thorne ~\\cite{ar:F&T'89} defined a new ``Hartle-Hawking'' state $\\ket{FT}$ invariant under the symmetries of the Kerr space-time\nby using a variant of the $\\eta$ formalism (which employs non-standard commutation relations for the creation and annihilation operators). They\nproved that the RSET when the field is in the $\\ket{FT}$ state is finite at the horizon but that, at least\nfor arbitrarily slow rotation, it is equal to the stress tensor of a thermal distribution at the Hawking\ntemperature rigidly rotating with the horizon. This suggests that it becomes irregular wherever $\\vec{k}_{\\mathscr{H}}$ is not timelike,\nthat is on and outside the speed-of-light surface.         \nOttewill and Winstanley ~\\cite{ar:Ott&Winst'00}, however, proved that although $\\ket{FT}$ has a Feynman propagator\nwith the correct properties for regularity on the horizons, its two-point function is actually pathological almost everywhere, not just outside the\nspeed-of-light surface. Only at the axis of symmetry, where all the modes in the two-point function for the scalar field \nare evaluated for $\\tilde{\\omega}=\\omega$ (i.e., $m=0$), it does not suffer from this pathology. \n\nFrolov and Thorne claim that close to the horizon ZAMOs make measurements relative to an unspecified Boulware vacuum.\nThey also claim that, when the field is in the state $\\ket{FT}$, ZAMOs measure close to the horizon \na stress tensor equal to that of a thermal distribution at the Hawking temperature rigidly rotating with the horizon. \n\nDuffy ~\\cite{th:GavPhD} modified the Kerr space-time by introducing a mirror and constructed a state $\\ket{H_{\\mathcal{M}}}$\nfor the scalar field that is invariant under the isometries of the modified space-time.\nHe then showed that $\\ket{H_{\\mathcal{M}}}$ is regular everywhere in the modified space-time if, and only if, the mirror\nremoves the region outside the speed-of-light surface. \nHe constructed another state, $\\ket{B_{\\mathcal{M}}}$, invariant under the isometries of the modified space-time and empty on both the past\nand future horizons. \nThis is the state that RROs make measurements relative to in the modified space-time.\nHe also numerically showed that when the field is in the $\\ket{H_{\\mathcal{M}}}$ state the stress tensor measured by a RRO\nis, close to the horizon, that of a thermal distribution at the Hawking temperature rigidly rotating with the horizon.\n\nCCH defined a new Hartle-Hawking-type state, which we will hereafter denote by $\\ket{CCH^-}$. \nThis state is obtained by thermalizing the `in' and `up' modes with respect to their natural energy.\nOttewill and Winstanley ~\\cite{ar:Ott&Winst'00} showed that this state is, however, not invariant under the symmetry transformation \n$(t,\\phi)\\to (-t,-\\phi)$ of the space-time.\nThey further argued that the RSET when the scalar field is in the $\\ket{CCH^-}$ state is regular on the future horizon but irregular on the past horizon.\nWe must note that these results were derived in ~\\cite{ar:Ott&Winst'00} based on a stress-energy tensor for which the $t\\theta$- and $\\phi\\theta$-components\nare identically zero. We shall see in Section \\ref{sec:luminosity} that although this is indeed the case for the scalar field, which is the case\nthey considered, this is most probably not true for the electromagnetic field.\nA similar state could be constructed by applying the transformation $(t,\\phi)\\to (-t,-\\phi)$ to the state $\\ket{CCH^-}$. \nThis state, suitably named $\\ket{CCH^+}$, would then be irregular on the future horizon and regular on the past horizon.\nIn Section \\ref{sec:RRO} we will investigate\nthe form close to the horizon of $\\vac[ren]{\\hat{T}^{\\mu}{}_{\\nu}}{CCH^--B^-}$ for electromagnetism.\n\nFinally, Unruh ~\\cite{ar:Unruh'76} constructed a state in the Schwarzschild space-time \nby expanding the\nscalar field in modes that are positive frequency with respect to the proper time $t$ of \ninertial observers in $\\mathcal{I^-}$ and modes that are positive frequency with respect to\nthe proper time of inertial observers close to $\\mathcal{H^-}$.           \nIt is possible to construct a state in the Kerr space-time with the same positive-frequency mode definitions.\nWe call this state the past Unruh state, $\\ket{U^-}$.\n\n\n\n\n\\section{Expectation value of the stress tensor} \\label{sec:theta->pi-theta Symmetry} \n\n\n\n\\subsection{Analytic expressions} \\label{subsec:new expressions}\n\nIn this subsection we derive new, analytic expressions for the expectation value of the stress-energy tensor\nwhen the electromagnetic field is in quantum states of interest in the Kerr space-time.\nIt is well-known that there is an operator-ordering ambiguity in the transition from classical to quantum theory:\nthe classical term $\\phi_{h}\\phi_{h'}^*$ should be quantized to\nthe symmetrized form \n$\\left(\\hat{\\phi}_{h}\\hat{\\phi}_{h'}^{\\dagger}+\\hat{\\phi}_{h'}^{\\dagger}\\hat{\\phi}_{h}\\right)\/2$.\nWe will calculate separately the expectation value of the two quadratic terms in the symmetrized form.\n\nIt is straight-forward to check that in the past Boulware state we have\n\\begin{equation} \\label{eq:phi(in)_iphi(in,dagger)_j on vac.0}\n\\begin{aligned}\n\\bra{B^-}\n\\hat{\\phi}_{h} \\hat{\\phi}_{h'}^{\\dagger}\n\\ket{B^-}\n=&\n\\sum_{lmP}\\Bigg(\\int_{0}^{+\\infty}\\d{\\tilde{\\omega}}\\ {}_{lm\\omega}\\phi^{\\text{up}}_{h}\\ {}_{lm\\omega}\\phi_{h'}^{\\text{up} *}\n+\\int_{0}^{+\\infty}\\d{\\omega}\\ {}_{lm\\omega}\\phi^{\\text{in}}_{h}\\ {}_{lm\\omega}\\phi_{h'}^{\\text{in} *}\\Bigg)\n=\n\\\\=\n&(-1)^{h+h'}\\mathcal{P}\\left(\\bra{B^-}\n\\hat{\\phi}_{h'}^{\\dagger} \\hat{\\phi}_{h}\n\\ket{B^-}\\right)\n^*\n\\end{aligned}\n\\end{equation}\n\nThe sign $(-1)^{h+h'}$ is precisely the same sign appearing in (\\ref{eq:pairs of vects. in stress tensor under parity}). \nThis implies that if the quadratic terms in the expression (\\ref{eq:stress tensor, spin 1})\nare quantum-mechanically symmetrized when promoting the NP scalars to operators, \nthen the expectation value in the state $\\ket{B^-}$ of the stress-energy tensor will be invariant under parity.\n\nIn order to calculate the expectation value of the quadratic terms \n$\\hat{\\phi}_{h}\\hat{\\phi}_{h'}^{\\dagger}$ and\n$\\hat{\\phi}_{h'}^{\\dagger}\\hat{\\phi}_{h}$ in the past Unruh state we are going to\nmake use of the expression calculated in ~\\cite{ar:F&T'89} which gives the past Unruh state in terms of the past Boulware state:\n\\begin{equation} \\label{eq:U in terms of B}\n\\ket{U^-}=\\prod_{lm\\tilde{\\omega}P}C_{lm\\omega P}\\exp\\left(e^{-\\pi\\tilde{\\omega}\/\\kappa_+}\\hat{a}_{lm\\omega P}^{\\text{up} \\dagger}\n\\hat{a}_{lm\\omega P}^{\\text{up'} \\dagger}\\right)\\ket{B^-}\n\\end{equation}\nwhere $C_{lm\\omega P}$ are normalization constants and $\\hat{a}_{lm\\omega P}^{\\text{up'} \\dagger}$ are\ncreation operators in the left hand region of the extended Kerr space-time.\nWe will also make use of the following expression in ~\\cite{ar:Schum&Caves'85}:\n\\begin{equation} \\label{eq:eq3.66Schum&Caves'85}\n\\begin{aligned}\n&S(r,\\phi)=\\\\\n&=(\\cosh r)^{-1}\\exp\\left(-\\hat{a}^{\\dagger}_+\\hat{a}^{\\dagger}_-e^{2i\\phi}\\tanh r\\right)\n\\exp\\left(-(\\hat{a}^{\\dagger}_+\\hat{a}_{+}+\\hat{a}^{\\dagger}_-\\hat{a}_-)\\ln(\\cosh r)\\right)\\exp\\left(\\hat{a}_+\\hat{a}_{-}e^{-2i\\phi}\\tanh r\\right)\n\\end{aligned}\n\\end{equation}\nwhere $S(r,\\phi)$ is the two-mode squeeze operator\n\\begin{equation} \\label{eq:S op.} \nS(r,\\phi)=\\exp\\left(r(\\hat{a}_+\\hat{a}_{-}e^{-2i\\phi}-\\hat{a}^{\\dagger}_+\\hat{a}^{\\dagger}_-e^{2i\\phi})\\right)\n\\end{equation}\nand the independent operators $\\hat{a}_+$ and $\\hat{a}_{-}$ satisfy the standard commutation relations.\nBy using (\\ref{eq:eq3.66Schum&Caves'85}) and (\\ref{eq:S op.}) we can re-express (\\ref{eq:U in terms of B}) as\n\\begin{equation}\n\\ket{U^-}=\\exp{\\left\\{\\sum_{lmP}\\int_0^{\\infty}\\d{\\tilde{\\omega}}\\Big[\\ln C_{lm\\omega P}+\\ln(\\cosh r_{\\tilde{\\omega}})\\Big]\\right\\}}e^{-\\hat{A}}\\ket{B^-}\n\\end{equation}\nwith\n\\begin{equation}\n\\hat{A}\\equiv\\sum_{lmP}\\int_0^{\\infty}\\d{\\tilde{\\omega}}\\  \nr_{\\tilde{\\omega}}\\left(\\hat{a}_{lm\\omega P}^{\\text{up} \\dagger}\\hat{a}_{lm\\omega P}^{\\text{up'} \\dagger}-\n\\hat{a}_{lm\\omega P}^{\\text{up}}\\hat{a}_{lm\\omega P}^{\\text{up'}}\\right)\n\\qquad \\text{and}\\qquad\nr_{\\tilde{\\omega}}\\equiv -\\tanh^{-1} \\left(e^{-\\pi\\tilde{\\omega}\/\\kappa_+}\\right).\n\\end{equation}\nSince $\\left(e^{\\hat{A}}\\right)^{\\dagger}=e^{-\\hat{A}}$, \nthe normalization $\\left\\langle\\right.U^- \\left|\\right.U^-\\left.\\right\\rangle=1$ implies \n\\begin{equation}\n\\exp{\\left\\{\\sum_{lmP}\\int_0^{\\infty}\\d{\\tilde{\\omega}}\\Big[\\ln C_{lm\\omega P}+\\ln C^*_{lm\\omega P}+2\\ln(\\cosh r_{\\tilde{\\omega}})\\Big]\\right\\}}=1\n\\end{equation}\n\nUsing now the Baker-Campbell-Hausdorff equation (~\\cite{bk:Louisell}) we find that\n\\begin{equation} \\label{}\ne^{\\hat{A}}\\hat{a}_{lm\\omega P}^{\\text{up}}e^{-\\hat{A}}=\n\\hat{a}_{lm\\omega P}^{\\text{up}}\\cosh r_{\\tilde{\\omega}}+\\hat{a}_{lm\\omega P}^{\\text{up'} \\dagger}\\sinh r_{\\tilde{\\omega}}\n\\end{equation}\nand finally\n\\begin{subequations} \\label{eq:a_dagger*a on unruh}\n\\begin{align}\n\\bra{U^-}\\hat{a}_{lm\\omega P}^{\\text{up} \\dagger}\\hat{a}_{l'm'\\omega'P'}^{\\text{up}}\\ket{U^-}&=\n\\frac{1}{2}\\left[\\coth \\left(\\frac{\\pi\\tilde{\\omega}}{\\kappa_+}\\right)-1\\right]\\delta(\\omega-\\omega')\\delta_{ll'}\\delta_{mm'}\\delta_{PP'} \\\\\n\\bra{U^-}\\hat{a}_{lm\\omega P}^{\\text{up} \\dagger}\\hat{a}_{l'm'\\omega'P'}^{\\text{up} \\dagger}\\ket{U^-}&=0\n=\\bra{U^-}\\hat{a}_{lm\\omega P}^{\\text{up}}\\hat{a}_{l'm'\\omega'P'}^{\\text{up}}\\ket{U^-}\n\\end{align}\n\\end{subequations}\nIt immediately follows that\n\\begin{equation}  \\label{eq:phi*phi^dagger `up' on U-}\n\\begin{aligned}\n&\\bra{U^-}\\hat{\\phi}_{h}\\hat{\\phi}_{h'}^{\\dagger}\\ket{U^-}=  \n\\\\ &=\n\\sum_{lmP}\\Bigg(\\frac{1}{2}\\int_0^{\\infty}\\d{\\tilde{\\omega}}\n\\begin{aligned}[t]\n\\bigg\\{\n&\n\\left[{}_{lm\\omega}\\phi_{h}^{\\text{up}}{}_{lm\\omega}\\phi_{h'}^{\\text{up} *}\n+(-1)^{h+h'}\\mathcal{P}({}_{lm\\omega}\\phi_{h}^{\\text{up} *}{}_{lm\\omega}\\phi_{h'}^{\\text{up}})\\right]\n\\coth\\left(\\frac{\\pi\\tilde{\\omega}}{\\kappa_+}\\right)+\n\\\\ +&\n\\left[{}_{lm\\omega}\\phi_{h}^{\\text{up}}{}_{lm\\omega}\\phi_{h'}^{\\text{up} *}-\n(-1)^{h+h'}\\mathcal{P}({}_{lm\\omega}\\phi_{h}^{\\text{up} *}{}_{lm\\omega}\\phi_{h'}^{\\text{up}})\\right]\n\\bigg\\}\n+ \n\\end{aligned}\n\\\\ &\\qquad\\qquad+\n\\int_0^{\\infty}\\d{\\omega}\\ {}_{lm\\omega}\\phi_{h}^{\\text{in}}\\ {}_{lm\\omega}\\phi_{h'}^{\\text{in} *}\n\\Bigg)=\n\\\\ & =\n(-1)^{h+h'}\\mathcal{P}\\left(\\bra{U^-}\\hat{\\phi}_{h'}^{\\dagger}\\hat{\\phi}_{h}\\ket{U^-}\\right)^*\n\\end{aligned}\n\\end{equation}\nNote the minus sign in the second term in (\\ref{eq:phi*phi^dagger `up' on U-}). Its presence may seem a bit surprising at first but,\nas we shall now see, it is precisely this sign that causes the expectation value of the stress-energy tensor in the\npast Unruh state to adopt a more familiar form by having all `up' terms multiplied by a $\\coth$ factor. \nThis is already clear from looking at (\\ref{eq:phi*phi^dagger `up' on U-}) and realizing that\nwhen quantum-symmetrizing the classical expression $\\phi_{h}^{\\text{up}}\\phi_{h'}^{\\text{up} *}$ \nthe terms without a $\\coth$ factor will cancel out.\n\nThe following identities are therefore immediately satisfied\n\\begin{equation}\n\\begin{aligned}\n\\vac{\\left[\\hat{\\phi}_{h}^{\\bullet},\\hat{\\phi}_{h'}^{\\bullet \\dagger}\\right]}{U^-}&=\n\\vac{\\left[\\hat{\\phi}_{h}^{\\bullet},\\hat{\\phi}_{h'}^{\\bullet \\dagger}\\right]}{B^-}=\\\\\n&=\\sum_{lmP}\\int_0^{\\infty}\\d{\\omega^{\\bullet}}\n\\left[{}_{lm\\omega}\\phi_{h}^{\\bullet}{}_{lm\\omega}\\phi_{h'}^{\\bullet *}\n-(-1)^{h+h'}\\mathcal{P}\\left({}_{lm\\omega}\\phi_{h}^{\\bullet *}{}_{lm\\omega}\\phi_{h'}^{\\bullet}\\right)\\right]\n\\end{aligned}\n\\end{equation}\nand hence\n\\begin{equation}\n\\vac{\\left[\\hat{\\phi}_{h}^{\\bullet},\\hat{\\phi}_{h'}^{\\bullet \\dagger}\\right]}{U^--B^-}=0\n\\end{equation}\nas it should be.\n\nWhen the classical term $\\left(\\phi_{h}\\phi_{h'}^*+c.c.\\right)$ is quantized to\nthe symmetrized term $\\left(\\hat{\\phi}_{h}\\hat{\\phi}_{h'}^{\\dagger}+\\hat{\\phi}_{h'}^{\\dagger}\\hat{\\phi}_{h}\\right)\/2+h.c.$,\nwhere the symbol $`h.c.'$ stands for hermitian conjugate,\nwe obtain the following real, parity-invariant expressions in the past Boulware and past Unruh states:\n\\begin{subequations}\n\\begin{align}\n\\begin{split}\n&\\vac{\\frac{\\hat{\\phi}_{h}\\hat{\\phi}_{h'}^{\\dagger}+\\hat{\\phi}_{h'}^{\\dagger}\\hat{\\phi}_{h}}{2}+h.c.}{B^-}= \n\\\\ &=\n\\frac{1}{2}\\sum_{lmP}\\bigg(\\int_0^{\\infty}\\d{\\tilde{\\omega}}\n\\left[\n{}_{lm\\omega}\\phi_{h}^{\\text{up}}{}_{lm\\omega}\\phi_{h'}^{\\text{up} *}+\n(-1)^{h+h'}\\mathcal{P}({}_{lm\\omega}\\phi_{h}^{\\text{up}}{}_{lm\\omega}\\phi_{h'}^{\\text{up} *})\\right]+\n\\\\ &\n\\qquad \\quad\\ \\\n+\n\\int_0^{\\infty}\\d{\\omega}\n\\left[{}_{lm\\omega}\\phi_{h}^{\\text{in}}{}_{lm\\omega}\\phi_{h'}^{\\text{in} *}+\n(-1)^{h+h'}\\mathcal{P}({}_{lm\\omega}\\phi_{h}^{\\text{in}}{}_{lm\\omega}\\phi_{h'}^{\\text{in} *})\\right]\n\\bigg)+c.c.\n\\end{split}\\\\\n\\begin{split}\n&\\vac{\\frac{\\hat{\\phi}_{h}\\hat{\\phi}_{h'}^{\\dagger}+\\hat{\\phi}_{h'}^{\\dagger}\\hat{\\phi}_{h}}{2}+h.c.}{U^-}= \n\\\\ &=\n\\frac{1}{2}\\sum_{lmP}\\bigg(\\int_0^{\\infty}\\d{\\tilde{\\omega}}\n\\left[{}_{lm\\omega}\\phi_{h}^{\\text{up}}{}_{lm\\omega}\\phi_{h'}^{\\text{up} *}\n+(-1)^{h+h'}\\mathcal{P}({}_{lm\\omega}\\phi_{h}^{\\text{up}}{}_{lm\\omega}\\phi_{h'}^{\\text{up} *})\\right]\n\\coth\\left(\\frac{\\pi\\tilde{\\omega}}{\\kappa}\\right)+\n\\\\ &\n\\qquad \\quad\\ \\\n+\n\\int_0^{\\infty}\\d{\\omega}\n\\left[{}_{lm\\omega}\\phi_{h}^{\\text{in}}{}_{lm\\omega}\\phi_{h'}^{\\text{in} *}+\n(-1)^{h+h'}\\mathcal{P}({}_{lm\\omega}\\phi_{h}^{\\text{in}}{}_{lm\\omega}\\phi_{h'}^{\\text{in} *})\\right]\n\\bigg)+c.c.\n\\end{split}\n\\end{align}\n\\end{subequations}\n\nNote that in the above expressions we have been able to complex conjugate the mode functions that are operated on by $\\mathcal{P}$\nbecause of the existence of the $+c.c.$ terms.\nThis immediately leads to the following real, parity-invariant expressions for the stress-energy tensor in\nthe past Boulware and past Unruh states:\n\\begin{subequations} \\label{eq:corrected stress tensor for s=1 on B-,U-}\n\\begin{align}\n\\begin{split}\n&\\expct{\\hat{T}_{\\mu\\nu}}{B^-}= \\\\\n&=\\frac{1}{2}\\sum_{lmP}\\left(\n\\int_0^{\\infty}\\d{\\tilde{\\omega}}\\, \n\\Big\\{T_{\\mu\\nu}\n\\left[{}_{lm\\omega}\\phi_{h}^{\\text{up}},{}_{lm\\omega}\\phi_{h}^{\\text{up} *}\\right]\n+(-1)^{\\vartheta}\\mathcal{P}\\left(T_{\\mu\\nu}\n\\left[{}_{lm\\omega}\\phi_{h}^{\\text{up}},{}_{lm\\omega}\\phi_{h}^{\\text{up} *}\\right]\n\\right)\\Big\\}\n+  \\right. \\\\\n& \n\\qquad \\quad \\ \\\n\\left. +\\int_0^{\\infty}\\d{\\omega}\\, \n\\Big\\{T_{\\mu\\nu}\n\\left[{}_{lm\\omega}\\phi_{h}^{\\text{in}},{}_{lm\\omega}\\phi_{h}^{\\text{in} *}\\right]\n+(-1)^{\\vartheta}\\mathcal{P}\\left(T_{\\mu\\nu}\n\\left[{}_{lm\\omega}\\phi_{h}^{\\text{in}},{}_{lm\\omega}\\phi_{h}^{\\text{in} *}\\right]\n\\right)\\Big\\}\n\\right)   \n\\end{split} \\label{eq:corrected stress tensor for s=1 on B-} \\\\ \\begin{split}\n&\\expct{\\hat{T}_{\\mu\\nu}}{U^-}= \n\\frac{1}{2}\\sum_{lmP}\n\\\\&\n\\left(\n\\int_0^{\\infty}\\d{\\tilde{\\omega}}\\, \n\\coth\\left(\\frac{\\pi\\tilde{\\omega}}{\\kappa_+}\\right)\n\\Big\\{T_{\\mu\\nu}\n\\left[{}_{lm\\omega}\\phi_{h}^{\\text{up}},{}_{lm\\omega}\\phi_{h}^{\\text{up} *}\\right]\n+(-1)^{\\vartheta}\\mathcal{P}\\left(T_{\\mu\\nu}\n\\left[{}_{lm\\omega}\\phi_{h}^{\\text{up}},{}_{lm\\omega}\\phi_{h}^{\\text{up} *}\\right]\n\\right)\\Big\\}\n+  \\right. \\\\\n& \n\\qquad \\qquad \\quad\n\\left. +\\int_0^{\\infty}\\d{\\omega}\\, \n\\Big\\{T_{\\mu\\nu}\n\\left[{}_{lm\\omega}\\phi_{h}^{\\text{in}},{}_{lm\\omega}\\phi_{h}^{\\text{in} *}\\right]\n+(-1)^{\\vartheta}\\mathcal{P}\\left(T_{\\mu\\nu}\n\\left[{}_{lm\\omega}\\phi_{h}^{\\text{in}},{}_{lm\\omega}\\phi_{h}^{\\text{in} *}\\right]\n\\right)\\Big\\}\n\\right) \\end{split} \\label{eq:corrected stress tensor for s=1 on U-}\n\\end{align}\n\\end{subequations}\nWe use the obvious notation that $T_{\\mu\\nu}\\left[{}_{lm\\omega}\\phi_{h}^{\\bullet},{}_{lm\\omega}\\phi_{h}^{\\bullet *}\\right]$ \ndenotes the general expression for the stress-energy tensor (\\ref{eq:stress tensor, spin 1}) where the scalars \n$\\phi_{h}$ have been replaced by the modes ${}_{lm\\omega}\\phi_{h}^{\\bullet}$. \nWe will also use the symbol ${}_{lm\\omega}T^{\\bullet}_{\\mu\\nu}$ to refer to \n$T_{\\mu\\nu}\\left[{}_{lm\\omega}\\phi_{h}^{\\bullet},{}_{lm\\omega}\\phi_{h}^{\\bullet *}\\right]$.\nThe variable $\\vartheta$ is defined so that $(-1)^{\\vartheta}$ is equal to -1 if one\nindex of the component of the stress tensor is $\\theta$ and the other one is not, and it is equal to +1 otherwise.\nNote that the sign $(-1)^{\\vartheta}$ appears in the above expressions instead of $(-1)^{h+h'}$ by virtue of the change under the parity operation \nof the coefficients of the quadratic field operators that appear in the expression for the stress-energy tensor, as seen in\n(\\ref{eq:pairs of vects. in stress tensor under parity}).\n\nTo our knowledge, these important expressions for the electromagnetic field in the Kerr space-time have only been given so far by CCH.\nTheir expressions, however, differ from ours in that they do not include the terms explicitly containing the operator $\\mathcal{P}$.\nAs a result, the expressions for the expectation value of the stress tensor that CCH give are not symmetric under the parity operation \n$\\mathcal{P}:(\\theta,\\phi)\\to (\\pi-\\theta,\\phi+\\pi)$,\neven though this is a symmetry of both the Kerr metric and the Maxwell field equations.\n\nWe initially used CCH's expressions in our numerical calculations and the numerical results we\nobtained were clearly not symmetric under the parity operation.\nWe also showed analytically that at least for one particular instance the RSET is not symmetric under $\\mathcal{P}$.\nWe outline here the derivation.\n\nFrom equation (\\ref{eq:asympt. y for x->+\/-1}) \nand the two equivalent expressions for ${}_{lm\\omega}\\phi_{0}$ given in (\\ref{eq:phi0(ch)}) it follows that\n $|{}_{lm\\omega}\\phi_{0}|^2$ is only non-zero at $\\theta=0,\\pi$ if $m=0$ and \nits value, for the `up' modes, in that case is\n\\begin{equation} \\label{eq: phi1^2 0->pi}  \n\\begin{aligned}\n&\\abs{{}_{l,m=0,\\omega}\\phi_{0}^{\\text{up}}(r,\\theta=0)}^2-\\abs{{}_{l,m=0,\\omega}\\phi_{0}^{\\text{up}}(r,\\theta=\\pi)}^2= \\\\\n&=\\frac{-4ia|N^{\\text{up}}_{+1}|^2W[{}_{+1}R,{}_{-1}R^{*}]^{\\text{up}}_{l,m=0,\\omega}({}_{-1}a_{n=0,l,m=0,\\omega}\\ {}_{+1}a_{n=0,l,m=0,\\omega})}\n{(r^2+a^2)^2{}_1B_{l,m=0,\\omega}^2}\n\\end{aligned}\n\\end{equation}\nBy virtue of the property (\\ref{eq: wronskian in=-up}),\nthe corresponding value for the `in' modes is obtained by merely changing the sign.\n\nSince $T^{\\text{div}}_{\\mu\\nu}$ is a purely geometrical object and the metric is invariant under $\\mathcal{P}$, this\ndivergent stress tensor must also be invariant under $\\mathcal{P}$. \nFrom this property together with equations (\\ref{eq:stress tensor, spin 1}) and (\\ref{eq: phi1^2 0->pi})\nwe find that CCH's expressions for the expectation value of the stress tensor yield:\n\\begin{equation} \\label{eq:T_thetatheta 0->pi,past Unruh}\n\\begin{aligned}\n&\\vac[ren]{\\hat{T}_{\\theta\\theta}(r,\\theta=0)}{U^-}-\\vac[ren]{\\hat{T}_{\\theta\\theta}(r,\\theta=\\pi)}{U^-}= \\\\\n&=\\sum_{l=0}^{\\infty}\n\\int_{0}^{\\infty}\\d{\\omega}\\left[\\coth\\left(\\frac{\\pi \\omega}{\\kappa}\\right)-1\\right]\n\\times \\\\ & \\qquad \\quad \\times\n\\frac{2^3ia|N^{\\text{up}}_{+1}|^2W[{}_{+1}R,{}_{-1}R^{*}]^{\\text{up}}_{l,m=0,\\omega}\\ {}_{+1}a_{n=0,l,m=0,\\omega}^2}\n{(r^2+a^2)^2{}_1B_{l,m=0,\\omega}^2}\n\\sqrt{\\frac{{}_{-1}\\lambda_{l,m=0,\\omega}+2a\\omega}{{}_{-1}\\lambda_{l,m=0,\\omega}-2a\\omega}}\n\\end{aligned}\n\\end{equation}\nwhere we have made use of the Teukolsky-Starobinski\\u{\\i} identities (\\ref{eq:Teuk-Starob. ids.})\nto relate the SWSH for $h=+1$ with the one for $h=-1$ in the limit $x\\to +1$.\n\nIt is easy to check that \n$iW[{}_{+1}R,{}_{-1}R^{*}]^{\\text{up}}_{l,m=0,\\omega}=\n{}_1B^2_{l,m=0,\\omega}\\abs{{}_{-1}R^{\\text{up,tra}}_{l,m=0,\\omega}}^2\/\\omega \\ge 0$ \nas long as \n$\\omega \\ge 0$, and therefore\nthe integrand in (\\ref{eq:T_thetatheta 0->pi,past Unruh}) is non-negative for $\\omega \\ge 0$.\nIt follows that\n$\\vac[ren]{\\hat{T}_{\\theta\\theta}}{U^-}$ is not symmetric under $\\mathcal{P}$ at the axis \nwhen CCH's expressions are used.\n\nOn the Schwarzschild background, CCH's expressions for the expectation value of the electromagnetic field are straight-forwardly invariant under parity:\nwhen $a=0$, the angular mode function does not depend on $\\omega$ \nand the radial mode function does not depend on $m$. Therefore, applying the \ntransformation $(\\theta \\to \\pi -\\theta)$ on any mode in their expressions\nis equivalent to performing $(m \\to -m)$ only, by virtue of the relations (\\ref{eq:R symm.->cc,-m,-w})\nand (\\ref{eq:S symm.->pi-t,-m,-w}) and the reality of the stress tensor.\nThe presence of the sum $\\sum_{m=-l}^{l}$ in their expressions\nguarantees their invariance under $\\mathcal{P}$.\nSuch a straight-forward reasoning does not follow in the Kerr background because applying the parity operation on \nCCH's expressions implies a change in the sign of $\\omega$ as well as in the sign of $m$.\nWhereas the sum over $m$ is symmetric with respect to $m=0$, the integration over the frequency is not symmetric\nwith respect to $\\omega=0$. \n\nCCH's expressions are, however, valid when applied to the scalar field case in the Kerr background.\nBy comparing the scalar field's expectation values and CCH's expectation values with the symmetrized \nversions (\\ref{eq:corrected stress tensor for s=1 on B-,U-}) for the states $\\ket{B^-}$ and $\\ket{U^-}$, \nwe can give analogous symmetrized versions for the states $\\ket{FT}$ (not included in CCH as this state was only defined later) and $\\ket{CCH^-}$:\n\\begin{subequations} \\label{eq:stress tensor for s=1 on FT and CCH^-}\n\\begin{align}\n\\begin{split}\n&\\expct{\\hat{T}_{\\mu\\nu}}{FT}= \n\\frac{1}{2}\\sum_{lmP}\n\\\\ &\n\\left(\n\\int_0^{\\infty}\\d{\\tilde{\\omega}}\\, \n\\coth\\left(\\frac{\\pi\\tilde{\\omega}}{\\kappa_+}\\right)\n\\Big\\{T_{\\mu\\nu}\n\\left[{}_{lm\\omega}\\phi_{h}^{\\text{up}},{}_{lm\\omega}\\phi_{h}^{\\text{up} *}\\right]\n+(-1)^{\\vartheta}\\mathcal{P}\\left(T_{\\mu\\nu}\n\\left[{}_{lm\\omega}\\phi_{h}^{\\text{up}},{}_{lm\\omega}\\phi_{h}^{\\text{up} *}\\right]\n\\right)\\Big\\}\n+ \\right.  \\\\\n& \\left. +\\int_0^{\\infty}\\d{\\omega}\\, \n\\coth\\left(\\frac{\\pi\\tilde{\\omega}}{\\kappa_+}\\right)\n\\Big\\{T_{\\mu\\nu}\n\\left[{}_{lm\\omega}\\phi_{h}^{\\text{in}},{}_{lm\\omega}\\phi_{h}^{\\text{in} *}\\right]\n+(-1)^{\\vartheta}\\mathcal{P}\\left(T_{\\mu\\nu}\n\\left[{}_{lm\\omega}\\phi_{h}^{\\text{in}},{}_{lm\\omega}\\phi_{h}^{\\text{in} *}\\right]\n\\right)\\Big\\}\n\\right) \n\\end{split} \\label{eq:corrected stress tensor for s=1 on CCH^-}\n\\\\\n\\begin{split}\n&\\expct{\\hat{T}_{\\mu\\nu}}{CCH^-}= \n\\frac{1}{2}\\sum_{lmP}\n\\\\ &\n\\left(\n\\int_0^{\\infty}\\d{\\tilde{\\omega}}\\, \n\\coth\\left(\\frac{\\pi\\tilde{\\omega}}{\\kappa_+}\\right)\n\\Big\\{T_{\\mu\\nu}\n\\left[{}_{lm\\omega}\\phi_{h}^{\\text{up}},{}_{lm\\omega}\\phi_{h}^{\\text{up} *}\\right]\n+(-1)^{\\vartheta}\\mathcal{P}\\left(T_{\\mu\\nu}\n\\left[{}_{lm\\omega}\\phi_{h}^{\\text{up}},{}_{lm\\omega}\\phi_{h}^{\\text{up} *}\\right]\n\\right)\\Big\\}\n+ \\right.  \\\\\n& \\left. +\\int_0^{\\infty}\\d{\\omega}\\, \n\\coth\\left(\\frac{\\pi\\omega}{\\kappa_+}\\right)\n\\Big\\{T_{\\mu\\nu}\n\\left[{}_{lm\\omega}\\phi_{h}^{\\text{in}},{}_{lm\\omega}\\phi_{h}^{\\text{in} *}\\right]\n+(-1)^{\\vartheta}\\mathcal{P}\\left(T_{\\mu\\nu}\n\\left[{}_{lm\\omega}\\phi_{h}^{\\text{in}},{}_{lm\\omega}\\phi_{h}^{\\text{in} *}\\right]\n\\right)\\Big\\}\n\\right) \n\\end{split} \\label{eq:stress tensor for s=1 on FT}\n\\end{align}\n\\end{subequations}\n\n\n\n\n\n\\subsection{Polarization} \\label{sec:polarization}\n\n\nIn this subsection we will give a physical interpretation of the non-parity term and the parity term appearing in the\nexpressions (\\ref{eq:corrected stress tensor for s=1 on B-,U-}) and (\\ref{eq:stress tensor for s=1 on FT and CCH^-})\nfor the expectation value of the stress-energy tensor in different states.\nWe denote by parity term in a certain expression a term that explicitly contains the parity operator $\\mathcal{P}$, and by\nnon-parity term one in the same expression that does not explicitly contain this operator.\n\nIt is clear from the classical expression (\\ref{eq:A1, tableIChrzan.}) for the `upgoing gauge' potential ${}_{lm\\omega}A^{\\text{up} \\mu}$ \nthat this potential only contains the null vectors $\\vec{n}$ and $\\vec{m}$.\nThe parity term $P\\mathcal{P}{}_{lm\\omega}A^{\\text{up} \\mu}$ in (\\ref{eq: def. of lmwPA_mu}), \nbecause of the transformations (\\ref{eq:parity op. on NP objs.}) of the null base under parity, \ncontains the vectors $\\vec{n}$ and $\\vec{m}^*$. \nThe potential ${}_{lm\\omega P}A^{\\text{up} \\mu}$ therefore contains the vectors $\\vec{n}$, $\\vec{m}$ and $\\vec{m}^*$.\nHowever, it is well-known (~\\cite{bk:Weinberg}) that only two of them are physically significant at radial infinity, where the space-time is flat. \n\nIt is in the limit for large $r$ that the physical meaning of the various vectors becomes clear.\nWe know that in flat space-time an electric field mode of positive frequency \nwith radial and timelike exponential behaviour $e^{-i\\omega(t-r)}$ (which we shall see is the asymptotic behaviour in our case)\nthat is proportional to the vector \n$\\left(\\hat{\\vec{e}}_{\\theta}+i\\hat{\\vec{e}}_{\\phi}\\right)[\\left(\\hat{\\vec{e}}_{\\theta}-i\\hat{\\vec{e}}_{\\phi}\\right)]$\npossesses a negative[positive] angular momentum and we thus say that it is negatively[positively] polarized.\nIf the mode is instead of negative frequency, the sign of the angular momentum changes and then an electric field mode\nwith radial and timelike exponential behaviour $e^{-i\\omega(t-r)}$\nproportional to $\\left(\\hat{\\vec{e}}_{\\theta}+i\\hat{\\vec{e}}_{\\phi}\\right)[\\left(\\hat{\\vec{e}}_{\\theta}-i\\hat{\\vec{e}}_{\\phi}\\right)]$\nis said to be positively[negatively] polarized.\nTherefore, according to (\\ref{eq:Kinnersley tetrad, r->inf}), \nan electric and a magnetic field modes of positive frequency that are proportional to the vector $\\vec{m}[\\vec{m}^*]$ correspond, \nin the flat space limit, to a negative[positive] polarization, whereas the vectors $\\vec{l}$ and $\\vec{n}$ correspond both to neutral polarization.\n\nWe know from (\\ref{eq:parity term is pure gauge}) that the parity term is\npure gauge and therefore the contribution to the NP scalars from the term with $\\vec{m}^*$ in the potential \nis zero. Only the terms with $\\vec{n}$ and $\\vec{m}$ in the `upgoing' potential contribute to the NP scalars. \nIn particular, it is immediate from expressions (\\ref{eq:phi as func. of potential with K}) \nfor the Maxwell scalars that only the term in the potential ${}_{lm\\omega P}A^{\\text{up}}_{\\mu}$ that contains the vector\n$\\vec{m}$ contributes to ${}_{lm\\omega}\\phi^{\\text{up}}_{+1}$ whereas only the term with $\\vec{n}$ contributes\nto ${}_{lm\\omega}\\phi^{\\text{up}}_{-1}$. Both, terms with $\\vec{n}$ and terms with $\\vec{m}$, contribute to ${}_{lm\\omega}\\phi^{\\text{up}}_{0}$.\nThis implies that the positive-frequency modes ${}_{lm\\omega}\\phi^{\\text{up}}_{-1}$, ${}_{lm\\omega}\\phi^{\\text{up}}_{+1}$ and \n${}_{lm\\omega}\\phi^{\\text{up}}_{0}$ are obtained from terms in the potential that, in the flat space limit, are\nneutrally-, negatively- and both neutrally- and negatively- polarized respectively. \nThe parity term in the potential in (\\ref{eq: def. of lmwPA_mu}), \nwhich is of the opposite polarization to that of the non-parity term, \ndoes not contribute to any NP scalar mode because it is pure gauge. \nThe opposite polarization to that of the non-parity term in the potential contributes to the NP scalars\nthrough the negative-frequency\nmodes when the integration is over all frequencies, as in (\\ref{eq:Fourier series of Gamma field}). \nIn the expression (\\ref{eq:classical mode expansion for A(in)_mu}) for the potential\nor (\\ref{eq:classical mode expansion for phi(in)_i}) for the NP scalars, \nin which we have rid of the negative-frequency modes, the opposite polarization appears\nvia the complex-conjugate term or the parity-term respectively.\n\nEven though all three Maxwell scalars appear in the classical expression for the electromagnetic stress tensor, \ndue to their different asymptotic behaviour (\\ref{eq:peeling th}) for large $r$, \nthe terms in the stress tensor (\\ref{eq:stress tensor, spin 1}) with ${}_{lm\\omega}\\phi^{\\text{up}}_{+1}$ predominate in this limit.\nThat is, the radiation field components of the stress tensor $T^{\\text{up}}_{\\mu\\nu}$ are calculated in the flat space limit \nfrom modes in the potential (\\ref{eq:Fourier series of Gamma field}) which for positive[negative] frequency \ncorrespond to a negative[positive] polarization. \nNote that the complex-conjugation of NP scalars in the stress tensor does not change the polarization \nof the field since it is merely a consequence of the fact that the null tetrad contains complex vectors, and does\nnot imply the complex-conjugation of the tensor field components $F_{\\mu\\nu}$.\n\nWe give here expressions for the `upgoing' field modes in the limit for large $r$, where the space-time is flat.\nWe wish, however, to obtain expressions for the fields that are real mode by mode. \nWe will therefore not calculate them from the potential modes ${}_{lm\\omega P}A^{\\text{up}}_{\\mu}$\nin (\\ref{eq:tableIChrzan.}), since the transformation $(m,\\omega)\\to (-m,-\\omega)$ has been applied to\nthe parity term in (\\ref{eq: def. of lmwPA_mu}). \nEven though the potential $A^{\\text{up}}_{\\mu}$ is obviously real, the potential modes ${}_{lm\\omega P}A^{\\text{up}}_{\\mu}$ are not.\nInstead, we will calculate field modes from the potential modes: ${}_{lm\\omega}A^{\\text{up}}_{\\mu}+{}_{lm\\omega}A^{\\text{up} *}_{\\mu}$.\nThe result is:\n\\begin{equation}  \\label{eq:E_up,B_up, r->inf}\n\\begin{aligned}\n&{}_{lm\\omega P}\\vec{E}^{\\text{up}}\\rightarrow\n\\frac{\\omega^2 |N^{\\text{up}}_{+1}|}{\\sqrt{2}r}\n\\times \\\\&\\times\n\\left[{}_{-1}Y_{l-m-\\omega}\\ {}_{+1}R^{\\text{up,tra}}_{l-m-\\omega}e^{+i\\omega (t-r)}\\left(\\hat{\\vec{e}}_{\\theta}+i\\hat{\\vec{e}}_{\\phi}\\right)+\n{}_{-1}Y_{l-m-\\omega}^*\\ {}_{+1}R^{\\text{up,tra} *}_{l-m-\\omega}e^{-i\\omega (t-r)}\\left(\\hat{\\vec{e}}_{\\theta}-i\\hat{\\vec{e}}_{\\phi}\\right)\\right]\n,\n\\\\\n&{}_{lm\\omega P}\\vec{B}^{\\text{up}}\\rightarrow\n\\frac{-\\omega^2 i|N^{\\text{up}}_{+1}|}{\\sqrt{2}r}\n\\times \\\\&\\times\n\\left[{}_{-1}Y_{l-m-\\omega}\\ {}_{+1}R^{\\text{up,tra}}_{l-m-\\omega}e^{+i\\omega (t-r)}\\left(\\hat{\\vec{e}}_{\\theta}+i\\hat{\\vec{e}}_{\\phi}\\right)-\n{}_{-1}Y_{l-m-\\omega}^*\\ {}_{+1}R^{\\text{up,tra} *}_{l-m-\\omega}e^{-i\\omega (t-r)}\\left(\\hat{\\vec{e}}_{\\theta}-i\\hat{\\vec{e}}_{\\phi}\\right)\\right]\n, \\\\\n&\\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \n\\quad (r\\rightarrow+\\infty)\n\\end{aligned}\n\\end{equation}\nwhere we have used the fact that in flat space we can replace ${}_{h}Z_{lm\\omega}$ by ${}_{h}Y_{lm\\omega}$.\nIt is worth noting that even though the second term in either $({}_{lm\\omega}A^{\\text{up}}_{\\mu}+{}_{lm\\omega}A^{\\text{up} *}_{\\mu})$ \nor (\\ref{eq: def. of lmwPA_mu})\ndoes not contribute to the NP scalars it does contribute to the fields. \nThe reason is that the NP scalars must be calculated from real fields. \nIt does not make physical sense to consider the contribution to the NP scalars from a non-real field, such as the second\nterm in the above expressions for the field. We say that this term is `pure gauge' in the sense that it does not contribute\nto the NP scalars even if it does contribute to the physical fields so as to make them real. \nThe large-$r$ asymptotics for the NP Maxwell scalars in terms of the electric and magnetic fields are easily obtained:\n\\begin{equation}  \\label{eq:phi_i, r->inf}\n\\begin{aligned}\n\\phi_{-1}&\\rightarrow -\\frac{1}{\\sqrt{2}}\\left(\\vec{E}+\\vec{B}i\\right)\\left(\\hat{\\vec{e}}_{\\theta}+i\\hat{\\vec{e}}_{\\phi}\\right)  & (r\\rightarrow+\\infty)\\\\\n\\phi_{0}&\\rightarrow \\frac{1}{2}\\left(\\vec{E}+\\vec{B}i\\right)\\hat{\\vec{e}}_{r}  & (r\\rightarrow+\\infty) \\\\\n\\phi_{+1}&\\rightarrow \\frac{1}{2\\sqrt{2}}\\left(\\vec{E}+\\vec{B}i\\right)\\left(\\hat{\\vec{e}}_{\\theta}-i\\hat{\\vec{e}}_{\\phi}\\right)  & (r\\rightarrow+\\infty)\n\\end{aligned}\n\\end{equation}\nIt is clear that the parity and non-parity terms correspond to opposite polarizations for\nthe fields (\\ref{eq:E_up,B_up, r->inf}) (and also for the potential). \nIt is also clear that the only contribution to $\\phi^{\\text{up}}_{+1}$ \nfrom the `upgoing' electric and magnetic fields comes from the non-parity term. \nTo leading order in $r$ for the electric and magnetic fields both $\\phi^{\\text{up}}_{-1}$ and $\\phi^{\\text{up}}_{0}$ vanish, \nin agreement with (\\ref{eq:peeling th}).\nTo next order in $r$, expressions (\\ref{eq:E_up,B_up, r->inf}) and (\\ref{eq:phi_i, r->inf}) must be calculated to include lower\norder terms and it is therefore not valid to conclude from them that the only contribution to $\\phi^{\\text{up}}_{-1}$ and $\\phi^{\\text{up}}_{0}$ \ncomes from positively- and neutrally- polarized terms respectively. \nWe have indeed seen in the beginning of this subsection that this is not the case. \n\nThe reasoning used so far for the `upgoing gauge' potential can be applied in the same manner to the `ingoing gauge' potential ${}_{lm\\omega}A^{\\text{in}}_{\\mu}$.\nIn this case, the potential contains one term with the vector $\\vec{l}$, which is the only one that contributes to\n${}_{lm\\omega}\\phi^{\\text{in}}_{+1}$, and one term with the vector $\\vec{m}^*$, which is the only one that contributes to\n${}_{lm\\omega}\\phi^{\\text{in}}_{-1}$. Like in the `upgoing gauge' case, both terms contribute to ${}_{lm\\omega}\\phi^{\\text{up}}_{0}$,\nand the parity term (containing $\\vec{l}$ and $\\vec{m}$) does not contribute to any of the NP scalars.\nThe scalar ${}_{lm\\omega}\\phi^{\\text{in}}_{-1}$ is the one that diminishes more slowly in the limit for large $r$.\nThe radiative components of the classical stress tensor $T^{\\text{in}}_{\\mu\\nu}$ is thus calculated in the flat space limit from modes in the potential \n(\\ref{eq:Fourier series of Gamma field}) that for positive[negative] frequency correspond to positive[negative] polarization.\n\nSo far in this subsection we have looked at the physical meaning of the different terms in classical expressions only.\nWe are now in a position to understand the physical meaning of the terms in the quantum field theory expressions.\nThe positive frequency modes in (\\ref{eq:Fourier series of Gamma field}) correspond to the non-parity term\nin the expression (\\ref{eq:classical mode expansion for phi(in)_i}) for the NP scalar and, ultimately, \ngive rise to the non-parity term in the expectation value of the stress tensor (\\ref{eq:corrected stress tensor for s=1 on B-,U-}).\nSimilarly, the negative frequency modes in (\\ref{eq:Fourier series of Gamma field}) give rise to the \nparity term in the NP scalars and the parity term in the expectation value of the stress tensor.\nWe therefore reach the conclusion that the non-parity terms in expressions (\\ref{eq:corrected stress tensor for s=1 on B-,U-})\nfor the expectation value of the stress tensor \ncorrespond in the flat space limit to one specific polarization (negative in the `up' case and positive in the `in' case)\nand that the corresponding parity terms in the same expressions correspond to the opposite polarization.\nBoth the contribution from the positive-polarization terms and from the negative-polarization terms are\nseparately real, as it should be.\nWe also know, from the end of Subsection \\ref{subsec:new expressions}, that in the spherically-symmetrical case $a=0$, \nthe contribution to the expectation value of the stress tensor from the positive-polarization terms \nis identical to the one from the negative-polarization terms, as one would expect.\n\nThe notable exception to this picture are the `up' superradiant modes. \nIndeed, these modes have a sign of $\\omega$ opposite to the non-superradiant modes in the same term in the expectation\nvalue, whether the parity term or the non-parity term. The polarization of the `up' superradiant modes\nis therefore the opposite to the non-superradiant modes in the same term, that is, it is positive if part of the non-parity\nterm and negative if part of the parity term. Note, however, that the `in' superradiant modes\nhave the same sign of $\\omega$ (positive), and therefore the same polarization, as the non-superradiant modes\nin the same term in the expectation value\n\nWhen CCH only include non-parity terms in their expressions for the expectation value of the stress tensor\nthey are only including one polarization and leaving out the other one for the `in' modes. For the `up' modes, \nthey are only including one polarization for the non-superradiant modes and the opposite polarization for the superradiant modes.\nIn particular, when subtracting the expectation value of the stress tensor \nin the past Boulware state from the one in the past Unruh state, only `up' modes are needed.\nNeglecting the parity terms is in this case equivalent to neglecting positive polarization non-superradiant modes\nas well as negative polarization superradiant modes. \nThat is the case in the calculation of $\\vac[ren]{\\hat{T}^{\\mu}{}_{\\nu}}{B^-}$ close to the horizon in Section \\ref{sec:RRO} \nbut,\nas explained in that section, in this limit\nthe non-parity and the parity terms coincide. \n\nIt is interesting to group the terms with the same polarization in the expectation value of the stress energy tensor.\nOf course, in the case of the difference between the states $\\ket{CCH^-}$ and $\\ket{U^-}$, which only has contribution from the `in' modes,\nthe sum of the positive polarization terms coincides with the direct evaluation of CCH's expressions.\nThe negative polarization contribution can be obtained by applying the transformation $x\\to -x$.\nWe include the plots of the tensor components corresponding to the fluxes of energy and angular momentum \nfrom the positive polarization terms in Figures \\ref{fig:deltaTtr_cch_u_past}--\\ref{fig:Ttphi_cch_u_past}.\nThe evaluation of the positive polarization contribution to the difference in the expectation value of the stress energy tensor \nbetween the states $\\ket{U^-}$ and $\\ket{B^-}$ requires carefully adding the contribution of the superradiant modes to the appropriate\npolarization. We calculated the positive polarization contribution to these differences of \nexpectation values and plot them in Figures \\ref{fig:Ttr_polpos_u_b}--\\ref{fig:Ttphi_polpos_u_b}.\nThe corresponding negative polarization contribution is, again, obtained by applying the transformation $x\\to -x$.\nThe interest of these graphs lies in the region far from the horizon. Close to the horizon \nthe irregularity of the state $\\ket{B^-}$ dominates and thermality guarantees symmetry with respect to the equator.\n\n\\begin{figure}[H]\n\\rotatebox{90}\n\\centering\n\\includegraphics*[width=70mm,angle=270]{.\/deltaTtr_cch_u_past_spher_last_r100to130.ps}\n\\caption{Positive polarization terms of $\\frac{1}{4\\pi}\\Delta\\vac{\\hat{T}_{tr}}{CCH^--U^-}$}\n\\label{fig:deltaTtr_cch_u_past}\n\\includegraphics*[width=70mm,angle=270]{.\/Ttphi_cch_u_past_spher_last_r100to110.ps}\n\\caption{Positive polarization terms of $\\frac{1}{4\\pi}\\vac{\\hat{T}_{t\\phi}}{CCH^--U^-}$}\n\\label{fig:Ttphi_cch_u_past}\n\\end{figure}\n\n\n\\begin{figure}[H]\n\\rotatebox{90}\n\\centering\n\\includegraphics*[width=70mm,angle=270]{.\/Ttr_polpos_u_b_spher_last_spher_last_r200to250.ps}\n\\caption{Positive polarization terms of $\\frac{1}{4\\pi}\\vac{\\hat{T}_{tr}}{U^--B^-}$}\n\\label{fig:Ttr_polpos_u_b}\n\\includegraphics*[width=70mm,angle=270]{.\/Ttphi_polpos_u_b_spher_last_r200to250.ps}\n\\caption{Positive polarization terms of $\\frac{1}{4\\pi}\\vac{\\hat{T}_{t\\phi}}{U^--B^-}$}\n\\label{fig:Ttphi_polpos_u_b}\n\\end{figure}\n\n\n\n\n\\section{Luminosity} \\label{sec:luminosity}\n\n\nLet $E^{\\text{(inc)}}$ and $E^{\\text{(ref)}}$ denote, respectively, the energy incident and the energy reflected\nby the black hole at infinity. \nLet $E^{\\text{(tra)}}$ denote the energy going down across the event-horizon of the black hole. \nThe reflection coefficient $\\mathbb{R}_{lm\\omega}$  and the transmission coefficient $\\mathbb{T}_{lm\\omega}$ \nof an incoming wave mode are then defined as the following flux ratios:\n\\begin{equation}\n\\begin{aligned}\n\\mathbb{R}_{lm\\omega}&\\equiv \\frac{\\d E^{\\text{(ref)}}_{lm\\omega}\/\\d{t}}{\\d E^{\\text{(inc)}}_{lm\\omega}\/\\d{t}}, \\qquad\n\\mathbb{T}_{lm\\omega}&\\equiv \\frac{\\d E^{\\text{(tra)}}_{lm\\omega}\/\\d{t}}{\\d E^{\\text{(inc)}}_{lm\\omega}\/\\d{t}}\n\\end{aligned}\n\\end{equation}\n\nThe wronskian relations for the solutions of the radial Teukolsky equation \ncorrespond to the conservation of energy law that equates the net flux of energy coming in\nfrom infinity to the net flux of energy going down into the black hole:\n\\begin{equation} \\label{eq:conserv. of energy law}\n1-\\mathbb{R}_{lm\\omega}=\\mathbb{T}_{lm\\omega}=1-\\left|A^{\\text{in}}_{lm\\omega}\\right|^2=\\frac{-i}{2^4\\omega^3}W[{}_{+1}R,{}_{-1}R^{*}]^{\\text{in}}_{lm\\omega}\n\\end{equation}\nwhere we have made use of the relations (\\ref{eq:R1 coeffs from R_1's}) and (\\ref{eq:R_1 coeffs from X's}).\nFor superradiant wave modes the fractional gain or loss of energy of an incoming wave mode,\ni.e., $\\mathbb{R}_{lm\\omega}=|A^{\\text{in}}_{lm\\omega}|^2$, is greater than one while\nthe transmission coefficient $\\mathbb{T}_{lm\\omega}$ is negative.\n\nThe conservation equations $\\nabla_{\\nu}T_{\\mu}{}^{\\nu}=0$ can alternatively be written ~\\cite{bk:Dirac} as\n\\begin{equation} \\label{eq:compact conserv.eqs.}\n\\partial_{\\nu}\\left(T_{\\mu}{}^{\\nu}\\sqrt{-g}\\right)=\\frac{1}{2}\\sqrt{-g}\\left(\\partial_{\\mu}g_{\\alpha\\beta}\\right)T^{\\alpha\\beta}\n\\end{equation}\nAssuming that the stress-energy tensor is independent of $t$ and $\\phi$, like the Kerr metric, the $\\mu=t$ and \n$\\mu=\\phi$ components of equations (\\ref{eq:compact conserv.eqs.}) can be integrated over $r$ to yield:\n\\begin{subequations} \\label{eq:integrated conserv.eqs., T_tr,Ttphi}\n\\begin{align}\nT_{tr}&=\\frac{K(\\theta)}{\\Delta}-\\frac{1}{\\Delta\\sin\\theta}\\partial_{\\theta}\\left(\\sin\\theta\\int_{r_+}^r\\d{r'}T_{t\\theta}\\right) \n\\label{eq:integrated conserv.eqs., T_tr} \\\\\nT_{\\phi r}&=\\frac{L(\\theta)}{\\Delta}-\\frac{1}{\\Delta\\sin\\theta}\\partial_{\\theta}\\left(\\sin\\theta\\int_{r_+}^r\\d{r'}T_{\\phi\\theta}\\right) \n\\label{eq:integrated conserv.eqs., Ttphi}\n\\end{align}\n\\end{subequations}\nwhere $K(\\theta)$ and $L(\\theta)$ are arbitrary functions.\nThe luminosity when the field is in the state $\\ket{\\Psi}$ is given by\n\\begin{equation} \\label{eq:def. dM\/dt}\n\\left. \\diff{M}{t}\\right|_{\\Psi}=\\Delta\\int_S\\d{\\Omega}\\vac[ren]{\\hat{T}_{tr}}{\\Psi}\n\\end{equation}\nwhere the surface $S$ can be any surface of constant $t$ and $r$. \nIn the forthcoming the subindex $A$ refers to either $t$ or $\\phi$ and the subindex $X$ to either $r$ or $\\theta$.\n\nWe will now compare some spin-1 results with the corresponding spin-0 results. \nThe following results for spin-0 are proven in part by Frolov and Thorne ~\\cite{ar:F&T'89} and extended by Ottewill and Winstanley ~\\cite{ar:Ott&Winst'00}:\n\\begin{equation} \\label{eq:T_Ar for s=0}\n\\begin{aligned}\n\\vac[ren]{\\hat{T}_{A\\theta}}{\\Psi} &=0, \\quad &\n\\Delta \\vac[ren]{\\hat{T}_{tr}}{\\Psi}&=K(\\theta), \\quad  &\n\\Delta \\vac[ren]{\\hat{T}_{\\phi r}}{\\Psi}&=L(\\theta) \n\\quad \\text{for} \\quad s=0\n\\end{aligned}\n\\end{equation}\nwhere $\\ket{\\Psi}$ is any state among $\\ket{B^\\pm}$, $\\ket{U^-}$, $\\ket{CCH^-}$ or $\\ket{FT}$. \nThe last two equations in (\\ref{eq:T_Ar for s=0}) are a direct consequence of the first one and of (\\ref{eq:integrated conserv.eqs., T_tr,Ttphi}).\nIt may indeed be calculated directly from the expression for the spin-0 stress-energy tensor \nthat all the radial dependence of $\\Delta {}_{lm\\omega}T_{tr}$ can be expressed as a radial wronskian. \nIt can also be checked that ${}_{lm\\omega}T_{tr}^{\\text{in}}=-{}_{lm\\omega}T_{tr}^{\\text{up}}$ for spin-0 so that \nthe only contribution to the luminosity in the past Boulware vacuum comes from the superradiant modes:\n\\begin{equation}\n\\Delta \\vac[ren]{\\hat{T}_{tr}}{B^-}=-2\\sum_{l=1}^{\\infty}\\sum_{m=1}^{l}\\int_0^{m\\Omega_+}\\d{\\omega}\\Delta {}_{lm\\omega}T_{tr}^{\\text{up}} \\qquad \\text{for} \\quad s=0\n\\end{equation}  \n\nIt is immediately apparent that for the spin-1 case the task of proving analytically \nwhether $\\Delta T_{tr}$ is constant in $r$ or not is not as straight-forward as for spin-0. \nWhen using expressions (\\ref{eq:corrected stress tensor for s=1 on B-,U-}) for the expectation value of the stress-energy tensor for a spin-1 field,\nwe are interested in the following calculation: \n\\begin{equation} \\label{eq:Delta(Ttr+Ttr(t->pi-t))}\n\\begin{aligned}\n&\\Delta \\left({}_{lm\\omega}T_{tr}^{\\text{up}}+\\mathcal{P}{}_{lm\\omega}T_{tr}^{\\text{up}}\\right)=\n\\\\&=\n\\frac{\\mathbb{T}_{lm\\omega}}{4\\pi^2\\Sigma}\\Big\\{-\\omega\\Sigma\\left({}_{-1}S_{lm\\omega}^2+{}_{+1}S_{lm\\omega}^2\\right)+ \n\\frac{a^3\\cos\\theta\\sin^2\\theta}{\\Sigma}\\left({}_{-1}S_{lm\\omega}^2-{}_{+1}S_{lm\\omega}^2\\right)+\n\\\\ &+\na\\sin\\theta\\left({}_{-1}S_{lm\\omega}\\partial_{\\theta}{}_{-1}S_{lm\\omega}-{}_{+1}S_{lm\\omega}\\partial_{\\theta}{}_{+1}S_{lm\\omega}\\right)\\Big\\}\n\\end{aligned}\n\\end{equation}\nwhere we have made use of the symmetry (\\ref{eq:S symm.->pi-t,-s}) in order to relate terms that contain\n$({}_{-1}S_{lm\\omega}^2+\\mathcal{P}{}_{-1}S_{lm\\omega}^2)$ to terms that contain $({}_{+1}S_{lm\\omega}^2+\\mathcal{P}{}_{+1}S_{lm\\omega}^2)$.\nThe corresponding result for the `in' modes is equal to (\\ref{eq:Delta(Ttr+Ttr(t->pi-t))}) with a change of sign, by virtue of\n(\\ref{eq:conserv. of energy law}) and the property (\\ref{eq: wronskian in=-up}).\nWe can get rid of the derivatives in (\\ref{eq:Delta(Ttr+Ttr(t->pi-t))}) by using the Teukolsky-Starobinski\\u{\\i} identities\nand the angular Teukolsky equation. But even when doing that, it is not possible to express the last term in (\\ref{eq:Delta(Ttr+Ttr(t->pi-t))})\nin terms of ${}_{-1}S_{lm\\omega}^2$ and ${}_{+1}S_{lm\\omega}^2$ only. \nAs a matter of fact, when evaluated at the axis of symmetry $\\theta=0$ or $\\pi$, only one term containing ${}_{-1}S_{lm\\omega}^2$ and one\nterm containing ${}_{+1}S_{lm\\omega}^2$ appear, neither of which is constant in $r$.\nIt is therefore apparent that if we wish to prove that $\\Delta \\left({}_{lm\\omega}T_{tr}^{\\text{up}}+\\mathcal{P}{}_{lm\\omega}T_{tr}^{\\text{up}}\\right)$\nis constant in $r$, or otherwise, we must then somehow relate ${}_{-1}S_{lm\\omega}^2$ to ${}_{+1}S_{lm\\omega}^2$.\nIt follows from the symmetries (\\ref{eq: S symms}) that we can only relate at the same \npoint a term whose spherical dependence is ${}_{-1}S_{lm\\omega}^2$ to a term whose spherical dependence is ${}_{+1}S_{lm\\omega}^2$ \nby applying the transformation $(m,\\omega)\\to (-m,-\\omega)$ to one\nof them. The change in sign of $m$ can be overturned due to the symmetric sum in $m$ in the Fourier sums \nin (\\ref{eq:corrected stress tensor for s=1 on B-,U-}) and (\\ref{eq:stress tensor for s=1 on FT and CCH^-}).\nThe change in sign of $\\omega$, however, is a problem when trying to relate a term with \n${}_{-1}S_{lm\\omega}^2$ to a term with ${}_{+1}S_{lm\\omega}^2$\ndue to the non-symmetric nature under $(m,\\omega)\\to (-m,-\\omega)$ of the integrals over $\\omega$ or $\\tilde{\\omega}$ \nfor all states involved in (\\ref{eq:corrected stress tensor for s=1 on B-,U-}) and (\\ref{eq:stress tensor for s=1 on FT and CCH^-}). \nIt follows that $\\Delta \\left({}_{lm\\omega}T_{tr}+\\mathcal{P}{}_{lm\\omega}T_{tr}\\right)$ is not\nconstant in $r$ and therefore neither is $\\Delta\\vac[ren]{\\hat{T}_{tr}}{B^-}$ nor $\\Delta\\vac[ren]{\\hat{T}_{tr}}{U^-}$. \nSimilarly, a calculation of $\\left({}_{lm\\omega}T_{t\\theta}+\\mathcal{P}{}_{lm\\omega}T_{t\\theta}\\right)$\nshows that it is not zero and therefore neither $\\vac[ren]{\\hat{T}_{t\\theta}}{B^-}$ nor $\\vac[ren]{\\hat{T}_{t\\theta}}{U^-}$\nare zero.\n\nIndeed, Graphs \\ref{fig:deltaSymmTtr_u_b}--\\ref{fig:delta2SymmTthetaphi_cch_u_past} numerically corroborate the above \nconclusions.\nGraphs \\ref{fig:deltaSymmTtr_u_b} and \\ref{fig:deltaSymmTtr_cch_u_past} show that neither \n$\\Delta\\vac{\\hat{T}_{tr}}{U^--B^-}$ nor $\\Delta\\vac{\\hat{T}_{tr}}{CCH^--U^-}$ are constant in $r$.\nGraphs \\ref{fig:delta2SymmTttheta_u_b}--\\ref{fig:delta2SymmTthetaphi_cch_u_past} show that neither \n$\\vac[ren]{\\hat{T}_{A\\theta}}{U^--B^-}$ nor $\\vac[ren]{\\hat{T}_{A\\theta}}{CCH^--U^-}$ are zero.\nGraphs \\ref{fig:deltaSymmTrphi_cch_u_past}--\\ref{fig:deltaSymmTrphi_u_b}, however, seem to indicate that\nboth $\\Delta\\vac{\\hat{T}_{r\\phi}}{CCH^--U^-}$ and $\\Delta\\vac{\\hat{T}_{r\\phi}}{U^--B^-}$ might actually be constant in $r$.\n\nNote that if instead of using expressions (\\ref{eq:corrected stress tensor for s=1 on B-,U-}) we use\nCCH's expressions for the expectation value of the stress tensor, we encounter the same difficulty when trying to prove\nwhether $T_{t\\theta}$ is zero and whether $\\Delta T_{tr}$ is constant in $r$. \n\nIn the case $a=0$, since the spin-weighted spherical harmonics ${}_{h}Y_{lm}$ do not depend on $\\omega$, we only need\na change in the sign of $m$ to relate terms with $|{}_{+1}Y_{lm}|^2$ to terms with $|{}_{-1}Y_{lm}|^2$ in $\\Delta {}_{lm\\omega}T_{tr}$. \nIndeed, use of (\\ref{eq:eq.B6J,McL,Ott'95}) allows us\nto prove that $\\sum_m \\Delta {}_{lm\\omega}T_{tr}$ is constant in $r$ and that \n$\\sum_m {}_{lm\\omega}T_{tr}^{\\text{up}}=-\\sum_m {}_{lm\\omega}T_{tr}^{\\text{in}}$ in the Schwarzschild background.  \nA numerical investigation of the luminosity and components of the RSET for spin-1 in the past Unruh state is presented in ~\\cite{ar:J&McL&Ott'91}.\n\nThe solution to this deadlock for the spin-1 case in the Kerr background consists in integrating over the solid angle. \nThis allows us to relate a term with\n$\\int\\d{\\Omega}{}_{-1}S_{lm\\omega}^2$ to a term with $\\int\\d{\\Omega}{}_{+1}S_{lm\\omega}^2$, \nwhen both types of terms appear in $\\int\\d{\\Omega}\\Delta {}_{lm\\omega}T_{tr}$ .\nThis is in accord with the fact that if we integrate the conservation\nequation (\\ref{eq:integrated conserv.eqs., T_tr}) over the solid angle we immediately obtain that\n\\begin{equation}\n\\int\\d{\\Omega}\\Delta T_{tr}=\\int\\d{\\Omega} K(\\theta)=const.\n\\end{equation}\nIndeed, we found that\n\\begin{equation} \\label{eq:DeltaTtr integrated over solid angle}\n\\int\\d{\\Omega} \\Delta {}_{lm\\omega}T_{tr}^{\\text{up}}=-\\int\\d{\\Omega} \\Delta {}_{lm\\omega}T_{tr}^{\\text{in}}=\\frac{-1}{4\\pi}\\omega\\mathbb{T}_{lm\\omega}\n\\end{equation}\nwhere we have included the constants of normalization (\\ref{eq:normalization consts.}).\nWe can now give simple expressions for the luminosity when the electromagnetic field is in the past Boulware state and in the past Unruh state:\n\\begin{subequations} \\label{eq:dM\/dt for B-,U-}\n\\begin{align}\n\\left.\\diff{M}{t}\\right|_{B^-}&=\n\\frac{1}{2\\pi}\\sum_{l=1}^{\\infty}\\sum_{m=1}^{+l}\\sum_{P=\\pm1}\\int_0^{m\\Omega_+}\\d{\\omega}\\omega\\mathbb{T}_{lm\\omega} \\label{eq:dM\/dt for B-}\n\\\\\n\\left.\\diff{M}{t}\\right|_{U^-}&=\n\\frac{1}{2\\pi}\\sum_{l=1}^{\\infty}\\sum_{m=-l}^{+l}\\sum_{P=\\pm1}\\int_0^{\\infty}\\d{\\omega}\\frac{\\omega\\mathbb{T}_{lm\\omega}}{e^{2\\pi\\tilde{\\omega}\/\\kappa}-1}\n\\label{eq:dM\/dt for U-}\n\\end{align}\n\\end{subequations}\nin agreement with Page's ~\\cite{ar:PageII'76}. \nThe former is a manifestation of the Starobinski\\u{\\i}-Unruh radiation and the latter of the Hawking radiation.\nSince only superradiant modes are being included in the Starobinski\\u{\\i}-Unruh radiation (\\ref{eq:dM\/dt for B-}) and \nthe transmission coefficient $\\mathbb{T}_{lm\\omega}$ is negative for these modes, there is a constant outflow of energy from the black hole\nwhen the field is in the past Boulware state.\n\nWe numerically evaluated (\\ref{eq:dM\/dt for B-,U-}) for the case $a=0.95M$.\nThe results, compared against values in the literature are:\n\\begin{subequations} \\label{eq:dM\/dt for B- for a=0.95,M=1}\n\\begin{align}\nM^2\\left.\\diff{M}{t}\\right|_{B^-}&=-4.750*10^{-4}    && (\\text{spin-1})          \\label{eq:num. dM\/dt in B-, s=1}\\\\\nM^2\\left.\\diff{M}{t}\\right|_{B^-}&=-5.01*10^{-5}      && (\\text{spin-0, Duffy})   \\label{eq:Gav. num. dM\/dt in B-, s=0} \n\\end{align}\n\\end{subequations}\nin the past Boulware state, and\n\\begin{subequations} \\label{eq:dM\/dt for U- for a=0.95,M=1}\n\\begin{align}\nM^2\\left.\\diff{M}{t}\\right|_{U^-}&=-1.1714*10^{-3}    && (\\text{spin-1})                \\label{eq:num. dM\/dt in U-, s=1} \\\\\nM^2\\left.\\diff{M}{t}\\right|_{U^-}&=-1.18*10^{-3}      && (\\text{spin-1, Page})   \\label{eq:Page num. dM\/dt in U-, s=1} \n\\end{align}\n\\end{subequations}\nin the past Unruh state.\nThe value (\\ref{eq:Gav. num. dM\/dt in B-, s=0}) for the scalar field is calculated by Duffy ~\\cite{th:GavPhD} and \nwe have calculated (\\ref{eq:Page num. dM\/dt in U-, s=1}) from splining Page's ~\\cite{ar:PageII'76} numerical results. Both\nof them have also been calculated for $a=0.95M$.\n\n\n\n\\begin{figure}[H]\n\\rotatebox{90}\n\\centering\n\\includegraphics*[width=70mm,angle=270]{.\/deltaSymmTtr_u_b_spher_last_r120to140.ps} \n\\caption{$\\frac{1}{4\\pi}\\Delta\\vac{\\hat{T}_{tr}}{U^--B^-}$}    \\label{fig:deltaSymmTtr_u_b}\n\\includegraphics*[width=70mm,angle=270]{.\/deltaSymmTtr_cch_u_past_spher_last_r100to130.ps} \n\\caption{$\\frac{1}{4\\pi}\\Delta\\vac{\\hat{T}_{tr}}{CCH^--U^-}$}    \\label{fig:deltaSymmTtr_cch_u_past}\n\\end{figure}\n\n\n\\begin{figure}[H]\n\\rotatebox{90}\n\\centering\n\\includegraphics*[width=70mm,angle=270]{.\/deltaSymmTrphi_cch_u_past_spher_last_r100to140.ps} \n\\caption{$\\frac{1}{4\\pi}\\Delta\\vac{\\hat{T}_{r\\phi}}{CCH^--U^-}$}    \\label{fig:deltaSymmTrphi_cch_u_past}\n\\includegraphics*[width=70mm,angle=270]{.\/deltaSymmTrphi_u_b_spher_last_r100to140.ps} \n\\caption{$\\frac{1}{4\\pi}\\Delta\\vac{\\hat{T}_{r\\phi}}{U^--B^-}$}    \\label{fig:deltaSymmTrphi_u_b}\n\\end{figure}\n\n\n\\begin{figure}[H]\n\\rotatebox{90}\n\\centering\n\\includegraphics*[width=70mm,angle=270]{.\/delta2SymmTttheta_u_b_spher_last_r100to140.ps} \n\\caption{$\\frac{1}{4\\pi}\\Delta^2\\vac{\\hat{T}_{t\\theta}}{U^--B^-}$.\nNote that the viewing angle in this and the following graphs is different from that of the previous graphs \nin order to make more visible the region far from the horizon.}    \\label{fig:delta2SymmTttheta_u_b}\n\\includegraphics*[width=70mm,angle=270]{.\/delta2SymmTttheta_cch_u_past_spher_last_r90to130.ps} \n\\caption{$\\frac{1}{4\\pi}\\Delta^2\\vac{\\hat{T}_{t\\theta}}{CCH^--U^-}$}    \\label{fig:delta2SymmTttheta_cch_u_past}\n\\end{figure}\n\n\\begin{figure}[H]\n\\rotatebox{90}\n\\centering\n\\includegraphics*[width=70mm,angle=270]{.\/delta2SymmTthetaphi_u_b_spher_last_r90to130.ps} \n\\caption{$\\frac{1}{4\\pi}\\Delta^2\\vac{\\hat{T}_{\\theta\\phi}}{U^--B^-}$}    \\label{fig:delta2SymmTthetaphi_u_b}\n\\includegraphics*[width=70mm,angle=270]{.\/delta2SymmTthetaphi_cch_u_past_spher_last_r90to130.ps} \n\\caption{$\\frac{1}{4\\pi}\\Delta^2\\vac{\\hat{T}_{\\theta\\phi}}{CCH^--U^-}$}    \\label{fig:delta2SymmTthetaphi_cch_u_past}\n\\end{figure}\n\n\n\\section{RSET close to the horizon in the Boulware vacuum} \\label{sec:RRO}\n\nCandelas and Deutsch ~\\cite{ar:Cand&Deutsch'77} consider flat space-time in the presence of an accelerating barrier\nwith acceleration $a_B^{-1}$. They then calculate the spin-1 RSET in the tetrad of an accelerating\nobserver RO with local acceleration $\\xi^{-1}$. In the limit $\\xi\/a_B\\rightarrow \\infty$ \nthe vacuum state above the accelerating mirror approximates the Fulling vacuum $\\ket{F}$.\nThe result is\n\\begin{equation} \\label{eq:stress tensor in Fulling in RO tetrad}\n\\begin{aligned}\n\\vac[ren]{T^{\\bar{\\mu}}{}_{\\bar{\\nu}}}{F}\n\\sim& -\\frac{1}{\\pi^2\\xi^4}\\int_0^{\\infty}\\d{x}\n\\frac{x^3+x}{e^{2\\pi x}-1}\\text{diag}\\left(-1,\\frac{1}{3},\\frac{1}{3},\\frac{1}{3}\\right)= \\\\\n&=\\frac{-11}{240\\pi^2\\xi^4}\\text{diag}\\left(-1,\\frac{1}{3},\\frac{1}{3},\\frac{1}{3}\\right)  \\qquad (\\xi\/a_B\\rightarrow \\infty)\n\\end{aligned}\n\\end{equation}\nwhere the bars on the indices indicate RO tetrad.  \nExpression (\\ref{eq:stress tensor in Fulling in RO tetrad}) is equivalent to minus the stress-energy tensor for thermal radiation\nat a temperature of $(2\\pi\\xi)^{-1}$.\nAs mentioned in Section \\ref{sec:vac. states}, in the Schwarzschild space-time, analogously to (\\ref{eq:stress tensor in Fulling in RO tetrad}) in flat space,\nthe RSET close to the horizon when the field is in the Boulware vacuum diverges like minus the stress tensor of black body radiation at the black hole temperature. \nIt is therefore reasonable to expect that if there existed a\nstate in Kerr with the defining features that the Boulware vacuum possesses in Schwarzschild, then\nthe RSET close to the horizon\nwhen the field were in this vacuum, would \ndiverge like minus the stress tensor of black body radiation at the black hole temperature rotating with the horizon.\nThe past Boulware vacuum is not invariant under $(t,\\phi)$ reversal because of the existence of the Starobinski\\u{\\i}-Unruh radiation.\nHowever, the stress tensor components $tr$ and $r\\phi$, which correspond to the Starobinski\\u{\\i}-Unruh radiation, are expected \n(from Section \\ref{sec:luminosity}) to have a divergence of one lower leading order than that of the diagonal components as the horizon is approached.\nIt is with this understanding that we say that a state is isotropic at the horizon and that, in particular, the past Boulware vacuum might be isotropic.\nIt is obvious that to next order in $\\Delta$ the past Boulware vacuum cannot be isotropic, \nbut $\\vac[ren]{\\hat{T}^{\\mu}{}_{\\nu}}{CCH^--B^-}$\nmight be since $\\ket{CCH^-}$ is not invariant under $(t,\\phi)$ reversal either.\n\nCCH claim that the RSET of the electromagnetic field in the past Boulware vacuum close to the horizon \ndiffers from that of minus the stress-energy tensor of a thermal distribution rotating at the angular \nvelocity of a Carter observer by a factor which is a function of $\\theta$.\nIn the present section, we will show that CCH's result is due to a flawed assumption in the asymptotic behaviour of the SWSH. \nWe will show this by re-calculating their result using the assumptions we believe they used. \nThe numerical results back up the fact that the mentioned RSET is (minus) thermal at the horizon.\n\nWe also mentioned in Section \\ref{sec:vac. states} that Frolov and Thorne claim that close to the horizon ZAMOs measure a thermal \nstress tensor which is rigidly rotating with the horizon when the field is in the $\\ket{FT}$ state.\nThat is, they argue that $\\vac[ren]{\\hat{T}^{\\mu}{}_{\\nu}}{FT-B}$ (where $\\ket{B}$ is an unspecified Boulware-type state)\nis thermal close to the horizon, and isotropic in the frame of a RRO.\nDuffy, in turn, shows that close to the horizon RROs measure a thermal state which is rigidly\nrotating with the horizon when the field is in the $\\ket{H_{\\mathcal{M}}}$ state in the Kerr space-time \nmodified with the introduction of a mirror. \nThat is to say, close to the horizon $\\vac[ren]{\\hat{T}^{\\mu}{}_{\\nu}}{H_{\\mathcal{M}}-B_{\\mathcal{M}}}$ is \nthermal and isotropic in the frame of a RRO.\nFinally, as mentioned above, CCH claim that $\\vac[ren]{\\hat{T}^{\\mu}{}_{\\nu}}{CCH^--B^-}$ at the horizon differs by a factor from a thermal \ndistribution isotropic in the Carter tetrad. \nOf course, the angular velocity at the horizon of a Carter observer, a RRO and a ZAMO is $\\Omega=\\Omega_+$ for them all,\nso that CCH's result\ndoes not actually distinguish between these observers.\n\nOttewill and Winstanley ~\\cite{ar:Ott&Winst'00Lett} have proved that if a certain stress-energy tensor is thermal and rigidly-rotating with\nthe horizon everywhere, then it is divergent on the speed-of-light surface in the Boyer-Lindquist co-ordinates,\nwhich are regular on this surface. \nThis implies that if $\\vac[ren]{\\hat{T}^{\\mu}{}_{\\nu}}{CCH^--B^-}$ were thermal and\nrigidly-rotating with the horizon everywhere then the state $\\ket{CCH^-}$ would have to be irregular on the speed-of-light surface.\nIn the present section we will numerically investigate the rate of rotation of the thermal distribution in question.\n\nThe stress-energy tensor of a spin-1 thermal distribution at the Hawking temperature rigidly rotating with the horizon is given by\n\\begin{equation} \\label{eq:thermal RR stress tensor}\nT^{\\text{(th,RR)} \\mu}{}_{\\nu}=\\frac{11T^4\\pi^2}{45}\\left[\\delta^{\\mu}_{\\nu}-4\\frac{\\chi^{\\mu}\\chi_{\\nu}}\n{\\chi^{\\rho}\\chi_{\\rho}}\\right]\n\\end{equation}\nwhere\n\\begin{equation} \\label{eq:def. T}\nT \\equiv \\frac{T_H}{\\sqrt{-\\chi^{\\rho}\\chi_{\\rho}}}\n\\end{equation}\nis the local temperature. Note that this stress-energy tensor is obviously isotropic in the frame of a RRO, but it is not\nin the rigidly-rotating co-ordinate system $\\{t_+,r,\\theta,\\phi_+\\}$, where $\\phi_+\\equiv \\phi-\\Omega_+t$ and $t_+\\equiv t$,\nwhich is not adapted to a RRO.\n\nIn primed co-ordinates, which are adapted to a RRO, the rigidly-rotating thermal stress tensor becomes\n\\begin{equation} \\label{eq:thermal RR stress tensor in RRO coords.}\nT^{\\text{(th,RR)} \\mu'}{}_{\\nu'}=\\frac{11(r_+-r_-)^4}{2^8\\cdot 3^2\\cdot 5\\pi^2}\\frac{1}{\\Delta^2\\Sigma^2}\\text{diag}\\left(-1,\\frac{1}{3},\\frac{1}{3},\\frac{1}{3}\\right)\n\\end{equation}\nin the Kerr space-time.\n\nCCH calculate an expression for the RSET close to the horizon when the electromagnetic field is in the \npast Boulware state. They make the assumption that the RSET close to the horizon when the field \nis in this state is more irregular than when it is in the $\\ket{CCH^-}$ state and therefore approximate\n\\begin{equation} \\label{eq:RSET in B^- at horizon}\n\\vac[ren]{\\hat{T}^{\\mu}{}_{\\nu}}{B^-}\\sim\\vac[ren]{\\hat{T}^{\\mu}{}_{\\nu}}{B^-}-\\vac[ren]{\\hat{T}^{\\mu}{}_{\\nu}}{CCH^-}=\n\\vac{\\hat{T}^{\\mu}{}_{\\nu}}{B^-}-\\vac{\\hat{T}^{\\mu}{}_{\\nu}}{CCH^-} \\qquad (r\\rightarrow r_+)\n\\end{equation}\nOne can then use the expressions for the expectation value of the stress tensor, and it is clear that only the\n`up' modes are involved in the calculation.\nTheir result, when the components of the stress tensor are put in the Carter orthonormal\ntetrad is:\n\\begin{equation}   \\label{eq:CCH's RSET in B^- in Carter, for r->r_+}\n\\begin{aligned}\n\\vac[ren]{\\hat{T}^{\\hat{\\mu}}{}_{\\hat{\\nu}}}{B^-} &\\sim\n-\\frac{8M^3r_+}{\\pi^2\\Delta^2\\Sigma}\\int_0^{\\infty}\\d{\\tilde{\\omega}}\n\\frac{\\tilde{\\omega}\\left(\\tilde{\\omega}^2+\\kappa^2\\right)}{e^{2\\pi\\tilde{\\omega}\/\\kappa}-1}\\text{diag}\\left(-1,\\frac{1}{3},\\frac{1}{3},\\frac{1}{3}\\right)=\\\\\n&=-\\frac{1}{r_+}\\frac{11(r_+-r_-)^4}{2^8\\cdot 3^2\\cdot 5\\pi^2}\\frac{1}{\\Delta^2\\Sigma (2Mr_+)}\\text{diag}\\left(-1,\\frac{1}{3},\\frac{1}{3},\\frac{1}{3}\\right)\n\\end{aligned}\n\\end{equation}\nwhere the hats on the indices indicate adaptation to the Carter orthonormal tetrad.\nThis expression and the expected result, minus \n(\\ref{eq:thermal RR stress tensor in RRO coords.}), differ in a factor of $r_+(2Mr_+)\/\\Sigma$.\nWe proceed to reproduce CCH's expression to explain\nthis disagreement.\n\nWe believe that \nCCH followed Candelas ~\\cite{ar:Candelas'80} method for spin-0 to obtain asymptotic expansions \nfor the radial solutions for spin-1 close to the horizon. This is the method that we develop for spin-1 in Appendix\n\\ref{sec:asympts. close to r_+}. Armed with the asymptotics of that appendix, we can proceed to calculate \nthe different components of the stress-energy tensor. In order to do that,\nwe are first going to separately calculate the asymptotic expressions for the various terms that occur in \nthe classical stress-energy tensor (\\ref{eq:stress tensor, spin 1}).\n\nAs mentioned in Appendix \\ref{sec:asympts. close to r_+}, for the asymptotic behaviour we are seeking here \nwe can replace the spin-weighted spheroidal harmonics ${}_{h}Z_{lm\\omega}$ by the spin-weighted spherical \nharmonics ${}_{h}Y_{lm}$. \nNote that this implies that the result of this asymptotic analysis is the same whether it is \nequations (\\ref{eq:corrected stress tensor for s=1 on B-,U-}) and (\\ref{eq:stress tensor for s=1 on FT and CCH^-})\nor CCH's expressions for the expectation value of the stress tensor that are used.\nWe can make use of\n(\\ref{eq:eq.2aJ,McL,Ott'91}), which immediately leads to\n\\begin{equation} \\label{eq:phi_iphi_j,i<>j,approx l->inf,r->rplus}\n\\sum_{m=-l}^l {}_{lm\\omega}\\phi_{h}^{\\text{up}}{}_{lm\\omega}\\phi_{h'}^{\\text{up} *} \n\\rightarrow 0 \n\\qquad (l\\rightarrow +\\infty, r \\rightarrow r_+) \n\\qquad \\text{when } h\\neq h'\n\\end{equation}\n\nThe asymptotic calculation of the term $\\abs{{}_{lm\\omega}\\phi_{0}^{\\text{up}}}^2$ requires a more careful treatment.\nWe observe in Appendix \\ref{sec:asympts. close to r_+} that the large-$l$ modes dominate the Fourier series for the\n`up' radial solution close to the horizon.\nWe can therefore replace $\\sum_{l=0}^{\\infty}$ with $\\int_{0}^{\\infty} \\d{l}$  \nand use the fact that of the two independent variables $\\tilde{\\omega}$ and $m$,\n${}_{h}R^{\\text{up}}_{lm\\omega}$ depends only on $\\tilde{\\omega}$ in the limit $(l\\rightarrow +\\infty, r \\rightarrow r_+)$,\nwhereas ${}_{h}Y_{lm}$ depends only on $m$.\nFrom (\\ref{eq:phi0(ch)}) and using\nequations (\\ref{eq:R1 `up' approx l->inf,r->rplus;compact version}),\n(\\ref{eq:Ddagger Delta R1 `up' approx l->inf,r->rplus}), (\\ref{eq:eq.B6J,McL,Ott'95}),  (\\ref{eq:eq.2aJ,McL,Ott'91}) \nand other properties of the spin-weighted spheroidal harmonics that can be found in ~\\cite{ar:J&McL&Ott'95}\nwe then obtain that\n\\begin{equation} \\label{eq:phi1^2 approx l->inf,r->rplus;2nd step}\n\\sum_{l,m,P}\\abs{{}_{lm\\omega}\\phi_{0}^{\\text{up}}}^2 \\sim\n\\frac{1}{2^3\\pi\\Sigma}\n\\int_{0}^{\\infty} \\d{l}\nl^3\n|N^{\\text{up}}_{+1}|^2 \\abs{\\mathcal{D}_0^{\\dagger}(\\Delta{}_{+1}R^{\\text{up}}_{lm\\omega})}^2\n\\qquad \\quad  (r \\rightarrow r_+) \n\\end{equation}\nWe substitute (\\ref{eq:def.A_s}), (\\ref{eq:def.D}), and \n(\\ref{eq:Ddagger Delta R+1 `up' approx l->inf,r->rplus})\nin the above equation and approximate ${}_1B_{lm\\omega} \\sim l^2$, which is valid in the limit of large $l$.\nThe next integral, found in ~\\cite{bk:GR}, is needed:\n\\begin{equation} \\label{eq:eq.6.576(4)G&R,lambda=-3}\n\\int_{0}^{\\infty} \\d{l} l^3 K_{iq}^2(2lx^{1\/2})=\\frac{q^2(1+q^2)\\abs{\\Gamma(iq)}^2}{3\\cdot 2^4x^2}\n\\end{equation}\nWe finally obtain \n\\begin{equation} \\label{eq:phi1^2 approx l->inf,r->rplus}\n\\begin{aligned}\n\\sum_{l,m,P}\\abs{{}_{lm\\omega}\\phi_{0}^{\\text{up}}}^2 \\sim\n\\frac{Mr_+\\tilde{\\omega}\\abs{\\EuFrak{N}}^2}{6\\pi^2\\Sigma \\Delta^2}\n\\qquad \\quad  \\text{($r \\rightarrow r_+$)} \n\\end{aligned}\n\\end{equation}\n\nThe other terms in the expression for the stress-energy tensor can be obtained in a similar manner,\nbut they are easier to calculate. We will therefore only give the final results:\n\\begin{subequations} \\label{eq:phi0^2,phi2^2 approx l->inf,r->rplus}\n\\begin{align}\n\\sum_{l,m,P}\\abs{{}_{lm\\omega}\\phi_{-1}^{\\text{up}}}^2 & \\sim\n\\frac{2Mr_+\\tilde{\\omega}\\abs{\\EuFrak{N}}^2}{3\\pi^2 \\Delta^3}\n&\n(r \\rightarrow r_+)\n\\label{eq:phi0^2 approx l->inf,r->rplus} \\\\ \n\\sum_{l,m,P}\\abs{{}_{lm\\omega}\\phi_{+1}^{\\text{up}}}^2 & \\sim\n\\frac{Mr_+\\tilde{\\omega}\\abs{\\EuFrak{N}}^2}{6\\pi^2 \\Sigma^2 \\Delta}\n&\n(r \\rightarrow r_+)  \\label{eq:phi2^2 approx l->inf,r->rplus}\n\\end{align}\n\\end{subequations}\n\nWe can now use equations (\\ref{eq:phi_iphi_j,i<>j,approx l->inf,r->rplus}), (\\ref{eq:phi1^2 approx l->inf,r->rplus}) \nand (\\ref{eq:phi0^2,phi2^2 approx l->inf,r->rplus}) together with the quantum expressions (\\ref{eq:RSET in B^- at horizon})\nand the expectation value of the stress tensor to reproduce equation 3.7 in CCH. We obtain\n\\begin{equation} \\label{eq:eq.3.7CCH;mine}\n\\begin{aligned}\n&\\vac[ren]{\\hat{T}^{\\mu}{}_{\\nu}}{B^-} \\sim \\vac{\\hat{T}^{\\mu}{}_{\\nu}}{B^--CCH^-} \\sim\n\\frac{-8M^3r_+^3}{3\\pi^2\\Delta^2\\Sigma^2}\\int_0^\\infty \n\\frac{\\d{\\tilde{\\omega}}\\tilde{\\omega}(\\tilde{\\omega}^2+\\kappa^2)}{e^{2\\pi\\tilde{\\omega}\/\\kappa}-1}\n\\times\n\\\\\n&\n\\times\n\\left(\n\\begin{array}{cccc}\n-3(r_+^2+a^2)-a^2\\sin^2\\theta & 0 & 0 & 4a\\sin^2\\theta(r_+^2+a^2)  \\\\\n0 & \\Sigma & 0 & 0 \\\\\n0 & 0 & \\Sigma & 0 \\\\\n-4a & 0 & 0 & (r_+^2+a^2)+3a^2\\sin^2\\theta\n\\end{array}\n\\right)\n\\qquad \\quad \n(r \\rightarrow r_+)\n\\end{aligned}\n\\end{equation}\nin Boyer-Lindquist co-ordinates.\nThis is exactly equation 3.7 in CCH except for the fact that (\\ref{eq:eq.3.7CCH;mine}) contains a factor $r_+^3$ instead\nof a $r_+$ in CCH. We believe that the discrepancy is due to a typographical error in CCH \nsince otherwise the stress-energy tensor would not have the correct units.\nWe have also checked that equation (\\ref{eq:eq.3.7CCH;mine}), when the tensor indices are adapted\nto the Carter orthonormal tetrad, produces the result (\\ref{eq:CCH's RSET in B^- in Carter, for r->r_+}) above.\nAgain, the discrepancy with respect to (\\ref{eq:CCH's RSET in B^- in Carter, for r->r_+}) is only \nin the power of $r_+$.\nIt seems that, despite the dicrepancy in the power of $r_+$, this is the method that CCH used to calculate\ntheir expression (\\ref{eq:CCH's RSET in B^- in Carter, for r->r_+}).\nHowever, as we point out in Appendix \\ref{sec:asympts. close to r_+}, this asymptotic analysis is only valid when both $\\omega$ and $m$ are kept bounded\nsince otherwise we would not be able to replace the spin-weighted spheroidal harmonics by the spin-weighted\nspherical harmonics. In the analysis we have just carried out $\\omega$ and $m$ do not both remain bounded in general.\nThe only points in the Kerr space-time where both remain bounded are the points along the axis $\\theta=0$ or $\\pi$ since, \nthere, the Newman-Penrose scalars ${}_{lm\\omega}\\phi_{h}$ are only non-zero for $m=\\pm1,0$ and thus $m$ is bounded. \nThe frequency $\\omega$ is then also kept bounded because the factor in the integrand diminishes\nexponentially with $\\tilde{\\omega}$ and thus the contribution is only important when $\\tilde{\\omega}$ is bounded.\nEquations (\\ref{eq:eq.3.7CCH;mine}) and (\\ref{eq:CCH's RSET in B^- in Carter, for r->r_+}) are therefore\nonly valid at the axis. An asymptotic behaviour of the `up' radial solutions uniform both in $l$\nand $\\tilde{\\omega}$ is required.\n\nAnother issue is the fact that the state $\\ket{CCH^-}$ has been used in (\\ref{eq:RSET in B^- at horizon}) as a Hartle-Hawking\nstate, regular on both the past and future horizons. \nWe know from Kay and Wald's work that there exists no such state on the Kerr space-time satisfying its isommetries.\nSince $\\ket{CCH^-}$ is not invariant under $(t,\\phi)$ reversal it is not covered by Kay and Wald's result and thus it might\nbe regular on both $\\mathcal{H^-}$ and $\\mathcal{H^+}$.\nWe saw that Ottewill and Winstanley ~\\cite{ar:Ott&Winst'00} argued that in the scalar case this state is \n irregular on $\\mathcal{H^-}$ and regular on $\\mathcal{H^+}$.\nEven if that were also the case for spin-1, using $\\ket{CCH^-}$ in the preceding calculation could still be acceptable if the divergence\nof $\\ket{CCH^-}$ close to $r_+$ is of a smaller order than that of $\\ket{B^-}$. \nIn the Schwarzschild background Candelas has shown that the Unruh state is irregular on $\\mathcal{H^-}$, regular on $\\mathcal{H^+}$ and\nthat the order of its divergence close to $r_+$ is smaller than that of the Boulware state. \nIt is therefore reasonable to expect that the order of the divergence of $\\ket{CCH^-}$ close to $r_+$ in the Kerr background\nis smaller than that of the past Boulware state.\nIndeed, our numerical data indicate that the approximation in (\\ref{eq:RSET in B^- at horizon}) is correct.\n\nGraphs \\ref{fig:delta2Ttt_cch_b_past}--\\ref{fig:delta2Tphiphi_cch_b_past}\nshow that the RSET when the field is in the past Boulware vacuum approaches a thermal distribution rotating\nwith the horizon rather than CCH's result (\\ref{eq:eq.3.7CCH;mine}). \nThe red lines in the graphs correspond to the thermal stress tensor (\\ref{eq:thermal RR stress tensor}) \nrotating with the horizon located at $r=r_+\\simeq 1.3122$.\nThe black lines are also located at $r=r_+$ and correspond to CCH's result (\\ref{eq:eq.3.7CCH;mine}).\nIt can be seen in the graphs that as $r$ becomes closer to the horizon, $\\vac[ren]{\\hat{T}{}_{\\mu\\nu}}{CCH^--B^-}$ approaches \nthe thermal stress tensor (\\ref{eq:thermal RR stress tensor}) (red line) rather than CCH's \ncorrected equation (\\ref{eq:eq.3.7CCH;mine}) (black line). At the poles, however, it can be straight-forwardly checked\nanalytically that the two coincide, as expected. Only for the $rr$-component, which is the only component that diverges like\n$O(\\Delta^{-3})$ close to the horizon, we did not seem to be able to obtain a clear plot, which we do not include.\n\nWithin the range of $r$ considered in Graphs \\ref{fig:delta2Ttt_cch_b_past}--\\ref{fig:delta2Tphiphi_cch_b_past}\n(except \\ref{fig:delta2Tthetatheta_u_b_cch_b_past})\nfor the difference between the states $\\ket{CCH^-}$ and $\\ket{B^-}$ of the various expectation values, \nthe corresponding plots for the difference between the states $\\ket{U^-}$ and $\\ket{B^-}$ are identical. \nThis is the expected behaviour since for small radius $r$ the `up' modes dominate in these RSETs.\nGraph \\ref{fig:delta2Tthetatheta_u_b_cch_b_past} includes the two differences \nfor the $\\theta\\theta$-component of the stress-energy tensor up to a value of $r$ large enough so that the two differences become clearly distinct.\n\nIn following with the notation used in (\\ref{eq:tetrad of stationary obs.}) and the one used so far for tensor components \nin Boyer-Lindquist co-ordinates, we use the obvious\nnotation of `$(\\alpha\\beta)$-component' to refer to the stress-energy tensor component $T{}_{\\mu\\nu}e_{(\\alpha)}{}^{\\mu}e_{(\\beta)}{}^{\\nu}$\nin the tetrad of a stationary observer.\nSince the angular velocities of a RRO, ZAMO\nand Carter observer all equal $\\Omega_+$ at the horizon, each one of the diagonal components of a stress tensor for a thermal\ndistribution will be the same \nin any of the three tetrads adapted to these observers.\nThe $(r\\theta)$-component will also be the same \nin any of the three tetrads\nsince the tetrad vectors $\\vec{e}_{(r)}$ and $\\vec{e}_{(\\theta)}$ do not depend\non the rate of rotation. The $(t\\phi)$-component,\nhowever, vanishes to leading order for the radial functions as $r\\rightarrow r_+$.\nTo the next leading order for the radial functions this component does depend on the rate of rotation of \nthe stationary observer that the tetrad is adapted to.\nGraphs \\ref{fig:deltaTtplusphiplus_cch_b_past}--\\ref{fig:deltaTtplusphiplus_cch_b_past_surfs} for $\\vac[ren]{\\hat{T}^{}_{t_+\\phi_+}}{CCH^--B^-}$ \nshow that the rate of rotation of the thermal distribution\napproaches, to next order in $\\Delta$, that of a RRO, rather than that of a ZAMO or a Carter observer.\nThis result tallies with Duffy~\\cite{th:GavPhD}'s results for the spin-0 case in the Kerr space-time \nmodified with a mirror when the field is in the $\\ket{H_{\\mathcal{M}}}$ state. \nHe also numerically shows that $\\vac[ren]{\\hat{T}{}_{\\mu\\nu}}{U^--B^-}$ is, close to the horizon and for the scalar field, \nthermal and rotating at the rate of a RRO to $O(\\Delta)$ in the angular frequency. \nWe calculated and plotted $\\vac[ren]{\\hat{T}_{t_+\\phi_+}}{U^--B^-}$ and \nit fully coincided with $\\vac[ren]{\\hat{T}_{t_+\\phi_+}}{CCH^--B^-}$ in the region of Graphs \n\\ref{fig:deltaTtplusphiplus_cch_b_past}--\\ref{fig:deltaTtplusphiplus_cch_b_past_surfs}, which is why we do not include them.\nWe conclude that the rate of rotation close to the horizon for the difference between the states $\\ket{U^-}$ and $\\ket{B^-}$ is also\nthat of a RRO, in agreement with Duffy's results.\n\nAn alternative technique for investigating what is the rate of rotation of the thermal distribution at the horizon\nis as follows. We find what is the frequency $\\Omega=\\Omega_{\\text{ZEFO}}$ of rotation of the tetrad frame \n(\\ref{eq:tetrad of stationary obs.}) such that $T_{(t\\phi)}=0$, where\nthe term ZEFO stands for zero energy flux observer. The answer is\n\\begin{equation} \\label{eq:omega_ZEFO}\n\\Omega_{\\text{ZEFO}}=\\frac{-2C}{B+\\sqrt{B^2-4AC}}\n\\end{equation}\nwhere\n\\begin{equation} \\label{eq:def. of A,B,C for ZEFO}\n\\begin{aligned}\nA&=g_{\\phi\\phi}T_{t\\phi}-g_{t\\phi}T_{\\phi\\phi} \\\\\nB&=g_{\\phi\\phi}T_{tt}-g_{tt}T_{\\phi\\phi}\\\\\nC&=g_{t\\phi}T_{tt}-g_{tt}T_{t\\phi}\n\\end{aligned}\n\\end{equation}\nWe then plot $\\Omega_{\\text{ZEFO}}$ where $T_{\\mu\\nu}$ is replaced by \n$\\vac[ren]{\\hat{T}_{\\mu\\nu}}{CCH^--B^-}$ in (\\ref{eq:def. of A,B,C for ZEFO}). This plot is compared against that of the angular velocities \nof a RRO, ZAMO and Carter observer in Figure \\ref{fig:freqCarterZAMO_omega_hor_cch_b_past}.\nWe also plotted $\\Omega_{\\text{ZEFO}}$ where $T_{\\mu\\nu}$ is replaced by \n$\\vac[ren]{\\hat{T}_{\\mu\\nu}}{U^--B^-}$ and it fully coincided with the corresponding one for $\\vac[ren]{\\hat{T}_{\\mu\\nu}}{CCH^--B^-}$\nin the region of Figure \\ref{fig:freqCarterZAMO_omega_hor_cch_b_past}.\n\nGraphs \\ref{fig:Tthetatheta_cch_u_past_u_b_spher_last_r100r150_n}\nshow the behaviour of the various modes as the horizon is approached. \nThe behaviour of the `up' modes close to the horizon are explained by the \nhorizon asymptotics developed in Appendix \\ref{sec:asympts. close to r_+}.\n\n\n\n\n\\begin{figure}[H]\n\\rotatebox{90}\n\\centering\n\\includegraphics*[width=70mm,angle=270]{.\/delta2SymmTtt_cch_b_past_spher_last_r90to100.ps} \n\\caption{$\\frac{1}{4\\pi}\\Delta^2\\vac{\\hat{T}_{tt}}{CCH^--B^-}$}    \\label{fig:delta2Ttt_cch_b_past}\n\\includegraphics*[width=70mm,angle=270]{.\/delta2SymmTtphi_cch_b_past_spher_last_r97to108.ps}\n\\caption{$\\frac{1}{4\\pi}\\Delta^2\\vac{\\hat{T}_{t\\phi}}{CCH^--B^-}$}             \\label{fig:delta2Ttphi_cch_b_past}\n\\end{figure}\n\n\n\\begin{figure}[H]\n\\rotatebox{90}\n\\centering\n\\includegraphics*[width=70mm,angle=270]{.\/delta2SymmTthetatheta_cch_b_past_spher_last_r98to110.ps}\n\\caption{$\\frac{1}{4\\pi}\\Delta^2\\vac{\\hat{T}_{\\theta\\theta}}{CCH^--B^-}$}      \\label{fig:delta2Tthetatheta_cch_b_past_bis}\n\\includegraphics*[width=70mm,angle=270]{.\/delta2SymmTthetatheta_u_b_cch_b_past_spher_last_r98to140.ps} \n\\caption{$\\frac{1}{4\\pi}\\Delta^2\\vac{\\hat{T}_{\\theta\\theta}}{CCH^--B^-}$ and \n$\\frac{1}{4\\pi}\\Delta^2\\vac{\\hat{T}_{\\theta\\theta}}{U^--B^-}$(orange)}    \\label{fig:delta2Tthetatheta_u_b_cch_b_past}\n\\end{figure}\n\n\n\\begin{figure}[H]\n\\rotatebox{90}\n\\centering\n\\includegraphics*[width=70mm,angle=270]{.\/delta2SymmTphiphi_cch_b_past_spher_last_r93to110.ps} \n\\caption{$\\frac{1}{4\\pi}\\Delta^2\\vac{\\hat{T}_{\\phi\\phi}}{CCH^--B^-}$}          \\label{fig:delta2Tphiphi_cch_b_past}\n\\end{figure}\n\n\n\n\\begin{figure}[H]\n\\rotatebox{90}\n\\centering\n\\includegraphics*[width=80mm,angle=270]{.\/deltaSymmTtplusphiplus_cch_b_past_spher_last_r95to115.ps} \n\\caption{$\\frac{1}{4\\pi}\\Delta\\vac{\\hat{T}_{t_+\\phi_+}}{CCH^--B^-}$, $\\frac{1}{4\\pi}\\Delta T^{\\text{(th,RR)}}_{t_+\\phi_+}$ (red), \n$\\frac{1}{4\\pi}\\Delta T^{\\text{(th,ZAMO)}}_{t_+\\phi_+}$ (blue)\nand $\\frac{1}{4\\pi}\\Delta T^{\\text{(th,Carter)}}_{t_+\\phi_+}$ (cyan).}           \\label{fig:deltaTtplusphiplus_cch_b_past}\n\\includegraphics*[width=80mm,,angle=270]{.\/deltaSymmTtplusphiplus_cch_b_past_spher_last_r97to120_surfs.ps} \n\\caption{$\\frac{1}{4\\pi}\\Delta\\vac{\\hat{T}_{t_+\\phi_+}}{CCH^--B^-}$, $\\frac{1}{4\\pi}\\Delta T^{\\text{(th,RR)}}_{t_+\\phi_+}$ (red), \n$\\frac{1}{4\\pi}\\Delta T^{\\text{(th,ZAMO)}}_{t_+\\phi_+}$ (blue)\nand $\\frac{1}{4\\pi}\\Delta T^{\\text{(th,Carter)}}_{t_+\\phi_+}$ (yellow). The light blue and brown surfaces correspond to the speed-of-light and \nthe static limit surfaces respectively.}                     \\label{fig:deltaTtplusphiplus_cch_b_past_surfs}\n\\end{figure}\n\n\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics*[width=70mm,angle=270]{.\/plot_freqCarterZAMO_omega_hor_cch_b_past_1to125.ps} \\\\\n$(\\Omega_{\\text{ZEFO}}-\\Omega_+)$ \\\\\n\\includegraphics*[width=70mm,angle=270]{.\/plot_freqCarterZAMO_omega_hor_Delta_cch_b_past_r120to140.ps} \\\\\n$\\left(\\Omega_{\\text{ZEFO}}-\\Omega_+\\right)\/\\Delta$ \\\\\n\\caption{Plots of $(\\Omega_{\\text{ZEFO}}-\\Omega_+)$ and  $\\left(\\Omega_{\\text{ZEFO}}-\\Omega_+\\right)\/\\Delta$ (dark surfaces) where \n$T_{\\mu\\nu}$ is replaced by $\\vac[ren]{\\hat{T}^{}_{\\mu\\nu}}{CCH^--B^-}$ in (\\ref{eq:omega_ZEFO}), together with the corresponding\nplots with the angular velocities of a RRO (red), ZAMO (magenta) and Carter observer (yellow).\n}\n\\label{fig:freqCarterZAMO_omega_hor_cch_b_past}\n\\end{figure}\n\n\n\n\\begin{figure}[H]\n\\centering\n\\begin{tabular}{cc}\n\\includegraphics*[height=65mm,width=75mm,angle=270]{.\/SymmTthetatheta_cch_u_past_spher_last_r100_n.ps} &\n\\includegraphics*[height=65mm,width=75mm,angle=270]{.\/SymmTthetatheta_cch_u_past_spher_last_r150_n.ps} \\\\\n$\\frac{1}{4\\pi}\\vac{\\hat{T}_{\\theta\\theta}(r \\simeq 1.36,\\theta)}{CCH^--U^-}$ & $\\frac{1}{4\\pi}\\vac{\\hat{T}_{\\theta\\theta}(r \\simeq 10.26,\\theta)}{CCH^--U^-}$\\\\\n\\includegraphics*[height=65mm,width=75mm,angle=270]{.\/SymmTthetatheta_u_b_spher_last_r100_n.ps} &\n\\includegraphics*[height=65mm,width=75mm,angle=270]{.\/SymmTthetatheta_u_b_spher_last_r150_n.ps} \\\\\n$\\frac{1}{4\\pi}\\vac{\\hat{T}_{\\theta\\theta}(r \\simeq 1.36,\\theta)}{U^-B^-}$    & $\\frac{1}{4\\pi}\\vac{\\hat{T}_{\\theta\\theta}(r \\simeq 10.26,\\theta)}{U^-B^-}$\\\\\n\\end{tabular}\n\\caption{\nThe sum over $l$ has not been performed. For each value of $l$ the sum $\\sum_{m=-l}^{l}$ has been performed.\nIn the case of $\\vac{\\hat{T}_{\\theta\\theta}}{CCH^--U^-}$ the low-$l$ modes clearly dominate close to the horizon.\nOn the other hand, the high-$l$ modes dominate close to the horizon in the case of $\\vac{\\hat{T}_{\\theta\\theta}}{U^--B^-}$.\n}\n\\label{fig:Tthetatheta_cch_u_past_u_b_spher_last_r100r150_n}\n\\end{figure}\n\n\n\n\n\n\\newpage\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\n\\subsection{Baselines and Variants}\n\nWe  compare  several variants  of  our  approach to  some  of  the baselines  we\nintroduced  in the  related  work  section. The  baselines  we  consider are  as\nfollows:\n \\begin{itemize}\n  \\setlength{\\itemsep}{1pt}\n  \\setlength{\\parskip}{0pt}\n  \\setlength{\\parsep}{0pt}\n   \\item  \\texttt{SWEEP}. As  discussed  in Section~\\ref{sec:related:seqMF},  it\n     involves  sequential  coordinate  descent~\\cite{Bishop06} and  is  not\n     always computationally tractable for large problems.\n   \\item  \\texttt{ADHOC}.   As  discussed in  Section~\\ref{sec:parallel-mf},  it\n     performs parallel updates with the {\\it ad hoc} damping parameter $\\eta$ of\n     Eq.~\\ref{eq:related:parallel-update} chosen manually. \n   \\item      \\texttt{FULL-PARALLEL}.        As      also       discussed      in\n     Section~\\ref{sec:parallel-mf}, it  relies on the inference described in~\\cite{Kraehenbuehl13}. \n     For example, the popular \\texttt{densecrf} framework~\\cite{Kraehenbuehl11} uses this approach.  \n\\end{itemize}\nWe compare to these the following variants of our approach:\n\\vspace{-0.3cm}\n\\begin{itemize}\n  \\setlength{\\itemsep}{1pt}\n  \\setlength{\\parskip}{0pt}\n  \\setlength{\\parsep}{0pt}\n\n  \\item \\texttt{OURS-FIXED}. Damping  occurs in the space  of natural parameters\n    instead  of mean  ones as  described in  Section~\\ref{sec:fixedStep}. \n  \\item \\texttt{OURS-ADAPTIVE}. Adaptative  and anisotropic damping in the\n    space       of      natural       parameters      as       described      in\n    Section~\\ref{sec:method:qs}. \n  \\item \\texttt{OURS-MOMENTUM}. Similar to  \\texttt{OURS-ADAPTIVE}, but using the\n    momentum  method  instead of  ordinary  gradient  descent, as  described  in\n    Section~\\ref{sec:momentum}.  We use  the  same parameter  value $\\gamma_1  =\n    0.95$ for all datasets.\n  \\item \\texttt{OURS-ADAM}. Similar to \\texttt{OURS-ADAPTIVE} but using the ADAM\n    method    instead of ordinary gradient as  described   in\n    Section~\\ref{sec:adam}. We  use the same  parameters as in  the original\n    publication~\\cite{Kingma14}, $\\gamma_1  = 0.99$, $\\gamma_2 =  0.999$ and $\\epsilon =$1E-8 for\n    all datasets. \n\\end{itemize}\nAll   four   methods    involve   a  parameter    $\\eta = \\frac{1}{1 + d}$,   defined   in\nEq.~\\ref{eq:fixedStepSize} for \\texttt{OURS-FIXED}, Eq.~\\ref{eq:method:hessian} for\n\\texttt{OURS-ADAPTIVE}, Eq.~\\ref{eq:momentumUpdate}   for   \\texttt{OURS-MOMENTUM}\nand Eq.~\\ref{eq:adamStepSize} for \\texttt{OURS-ADAM}. Additionally, in\nSection~\\ref{sec:eval:results} and Fig.~\\ref{fi:eval:benchmarks:steps} we demonstrate that our\nmethod is less sensitive to the choice of this parameter than its competitors. \n\n\n\\subsection{Experimental Setup}\n\nWe   evaluated  all   these    methods   first   on    a   set    of   standardized\nbenchmarks~\\cite{Frostig14}: \\texttt{DBN}, \ncontaining 108 instances of deep belief networks, \\texttt{GRID},\ncontaining 21 instances of two-dimensional grids, and \\texttt{SEG},\ncontaining 100 instances of segmentation problems, where each instance is\nrepresented as a binary pairwise random field. \n\n\nWe then consider three realistic Computer Vision tasks that all involve minimizing a\nfunctional of the form given in Eq.~\\ref{eq:related:mf-objective}. We describe them below.\n\n\n\\vspace{-0.35cm}\n\\paragraph{Characters  Inpainting} We consider character inpainting, formulated as a\nbinary pairwise random field, Decision Tree Fields (DTF,~\\cite{Nowozin11}). \nThe dataset contains 100 test instances of occluded characters, and the goal is to restore the\noccluded part, as shown in the last row of Fig.~\\ref{fig:intro:teaser}.   \nWe use pre-computed potentials provided by the authors of~\\cite{Nowozin11}. Note, that this model\nconsists of data-driven potentials, and includes both short and long-range interactions, which\nmakes it particularly interesting from the optimization perspective.  \n\n\\vspace{-0.35cm}\n\\paragraph{People  Detection} \nWe consider detecting upright people in a multi-camera settings, using the Probabilistic Occupancy Map\napproach (POM,~\\cite{Fleuret08a}), that relies on a random field with high-order repulsive\npotentials, which models background subtraction signal given the presences of people in the\nenvironment.  \nWe evaluate it on the ISSIA~\\cite{DOrazio09} dataset, which contains 3000 frames of a\nfootball game, captured by 6 cameras located on two sides of the field. The original\nwork~\\cite{Fleuret08a} does not explicitly mention it, but the publicly available  \nimplementation uses the~\\texttt{ADHOC} damping method. \nWe implement all our methods and remaining baselines directly in this code of~\\cite{Fleuret08a}. \\comment{In\norder to make the  it tractable, we had to introduce an additional data structure, denoting this modified\nalgorithm \\texttt{SWEEP}. }\n\n\\vspace{-0.35cm}\n\\paragraph{Semantic Segmentation} \nWe consider semantic segmentation on PASCAL VOC 2012 dataset~\\cite{Pascal-voc-2012},\nwhich defines 20 object classes and 1 backround class. We based our evaluation on DeepLab-CRF\nmodel~\\cite{Chen15b}, which is currently one of the best-performing methods. This\nmodel uses CNNs to obtain unary potentials, and then employs \\texttt{densecrf}\nof~\\cite{Kraehenbuehl13} with dense pairwise potentials. However, this basic CRF model does not  \ncontain any strong repulsive terms, and thus we expect \\texttt{densecrf}'s standard inference,\n\\texttt{FULL-PARALLEL}, to work well. To improve performance, we additionally introduced\nco-occurrence potentials~\\cite{Vineet14}, which, as we will show, violate the conditions assumed in\n\\texttt{densecrf}, but can still be successfully handled by our method. Intuitively, these\nco-occurrence terms put priors on the sets of classes that can appear together. We made minor\nmodifications of \\texttt{densecrf} to support both our inference and co-occurrence potentials. \n\n\n\n\n\n\\subsection{Comparative Results}\n\\label{sec:eval:results}\n\nIn order to understand how the methods behave in practical settings, when the available\ncomputational time is limited, we evaluate all methods for several computational \\textit{budgets}. \nThe shortest budget corresponds to the early-stopping scenario after few iterations, the longest one\nroughly models the time until convergence, and the middle one is around 20-30\\% of the longest.\n\n\\vspace{-0.3cm}\n\\paragraph{Benchmarks}\n\n\\begin{table*}[hbt!]\n\\small\n\\begin{center}\n\\begin{tabular}{l?ccc?ccc?ccc}\n& \\multicolumn{3}{c?}{\\sttt{DBN}} \n& \\multicolumn{3}{c?}{\\sttt{GRID}} \n& \\multicolumn{3}{c}{\\sttt{SEG}} \\\\\\hline\nmethod\n&  0.05s &  0.30s &  1.00s \n&  0.05s &  0.30s &  1.00s \n&  0.05s &  0.30s &  1.00s \\\\\n\\hline\n\\sttt{SWEEP}         \n&    -112.94 &   -2088.07 &   -2138.13 \n&   -5540.59 &  -16675.55 &  -18592.26\n&      78.81 &      75.50 &      75.50 \n\\\\                       \n\\sttt{FULL-PARALLEL} \n&   -1952.52 &   -1951.54 &   -1942.86 \n&   -2564.39 &   -2777.33 &   -2439.08\n&      75.66 &      75.66 &      75.66 \n\\\\\n\\sttt{ADHOC}         \n&   -2047.31 &   -2047.31 &   -2047.31 \n& -18345.42 &  -18348.80 &  -18349.03\n&     76.10 &      75.66 &      75.66 \n\\\\                           \n\\hline\n\\sttt{OURS-FIXED}        \n&   -2081.91 &   -2081.91 &   -2081.91 \n&  -18213.81 &  -18219.42 &  -18219.45 \n&      77.17 &      75.61 &      75.61 \n\\\\\n\\sttt{OURS-ADAPTIVE}            \n&   -2125.48 &   -2130.61 &   -2130.61 \n& -18245.93 &  -18252.48 &  -18252.48\n& 77.68 &      75.64 &      75.61\n\\\\\n\\sttt{OURS-MOMENTUM}      \n&   \\textbf{-2260.98} &   \\textbf{-2362.14} &   \\textbf{-2374.51}\n& -18143.48 &  \\textbf{-19074.45} &  \\textbf{-19184.37}\n&      74.35 &      73.75 &      73.75 \n\\\\\n\\sttt{OURS-ADAM}          \n&   -2107.98 &   -2107.93 &   -2107.93 \n&  \\textbf{-18617.06} &  -18732.59 &  -18740.36\n&      \\textbf{72.37} &      \\textbf{72.32} &      \\textbf{72.32} \\\\\n\\hline\n\\end{tabular}\n\\caption{Results for KL minimization for three benchmark datasets~\\cite{Frostig14}: \\texttt{DBN} (deep belief\n  networks), \\texttt{GRID} (two-dimensional grids), \\texttt{SEG} (binary segmentation). All the\n  numbers are KL divergence (lower is better) averaged over the instances.} \n\\vspace{-0.5cm}\n\\label{tab:eval:benchmark}\n\\end{center}\n\\end{table*}\n\n\n\n\n\\begin{figure*}[ht!]\n\\begin{center}\n\\begin{tabular}{c@{}c@{}c@{}c@{}c@{}c}\n\\setlength{\\tabcolsep}{0pt}\n\\includegraphics[trim=10mm 10mm 10mm 10mm, clip, width=0.16\\textwidth]{{steps_dbn10.00}.pdf} & \n\\includegraphics[trim=10mm 10mm 10mm 10mm, clip, width=0.16\\textwidth]{{steps_dbn30.00}.pdf} &\n\\includegraphics[trim=10mm 10mm 10mm 10mm, clip, width=0.16\\textwidth]{{steps_grid10.00}.pdf} &\n\\includegraphics[trim=10mm 10mm 10mm 10mm, clip, width=0.16\\textwidth]{{steps_grid30.00}.pdf} &\n\\includegraphics[trim=10mm 10mm 10mm 10mm, clip, width=0.16\\textwidth]{{steps_seg10.00}.pdf} &\n\\includegraphics[trim=10mm 10mm 10mm 10mm, clip, width=0.16\\textwidth]{{steps_seg30.00}.pdf} \\\\ \n  \\sttt{ DBN(0.05s)} & \\sttt{ DBN(0.3s)} &\n  \\sttt{ GRID(0.05s)} & \\sttt{ GRID(0.3s)} &\n  \\sttt{ SEG(0.05s)} & \\sttt{ SEG(0.3s)} \\\\\n\\end{tabular}\n\\end{center}\n\\vspace{-0.2cm}\n   \\caption{Insensitivity of~\\texttt{OURS-FIXED} (red) and~\\texttt{OURS-ADAPTIVE} (dashed red) vs\n     \\texttt{ADHOC} (blue) to the damping parameter \\small{$\\eta = \\frac{1}{1+d}$}. We report\n     KL-divergence (lower is better) vs the value of the parameter, both in log-space.} \n\\label{fi:eval:benchmarks:steps}\n\\vspace{-0.3cm}\n\\end{figure*}\n\n\n\\begin{figure*}[ht!]\n  \\begin{center}\n\n    \\begin{tabular}{c@{}c@{}c}\n \\includegraphics[trim=0mm 2mm 11mm 10mm, clip,\n      width=0.3\\textwidth]{dtf_convergence.pdf}  & \n \\includegraphics[trim=2mm 2mm 11mm 10mm, clip, width=0.3\\textwidth]{soccer_convergence.pdf}\n      & \n \\includegraphics[trim=0mm 2mm 11mm 10mm, clip, width=0.3\\textwidth]{voc2012_convergence.pdf}\n      \\\\              \n (a) Characters Inpainting  &\n (b) People Detection &\n (c) Semantic Segmentation \\\\             \n\n    \\end{tabular}\n  \\end{center}\n  \\vspace{-0.2cm}\n  \\caption{Convergence results. (a) \\texttt{OURS-ADAM} and \\texttt{OURS-MOMENTUM} converge very fast\n    to a much better minima. \n    (b) \\texttt{OURS-FIXED} outperforms \\texttt{ADHOC} both in terms of speed of convergence and the\n    value of the objective.\n    (c) \\texttt{OURS-ADAM} and \\texttt{OURS-FIXED} show the best performance. The former converges a bit\n    slower, but in the end provide slightly better minima. \\texttt{ADHOC} for this dataset converges\n    rather fast, but fails to find a better optima.}  \n\\vspace{-0.3cm}\n \\label{fi:eval:convergence}\n \\end{figure*}\n\n\n\n\n\\begin{table*}[!hbt]\n\\small\n\\vspace{-0.25cm}\n\\begin{center}\n\\begin{tabular}{l?C{1.3cm}C{0.9cm}?C{1.3cm}C{0.9cm}?C{1.3cm}C{0.9cm}}\n & \\multicolumn{2}{c?}{0.05s}  & \\multicolumn{2}{c?}{0.3s} & \\multicolumn{2}{c}{3s} \\\\\\hline\nmethod & \\sttt{KL} & \\sttt{PA} & \\sttt{KL} & \\sttt{PA} & \\sttt{KL} & \\sttt{PA}\\\\\\hline\n\\sttt{SWEEP} & -6342.56  & 54.57  & -25233.54 & 58.38 & -49519.33 & 62.50 \\\\\n\\sttt{FULL-PARALLEL} & \\textbf{-49516.98} & 60.99  & -49519.27 & 62.00 & -49519.33 & 62.05 \\\\\n\\sttt{ADHOC} & -49514.27 & 61.46  & -49520.09 & 62.15 & -49520.20 & 62.17 \\\\\\hline\n\\sttt{OURS-FIXED} & -49505.59 &     60.99 & -49520.33 &    62.26 & -49521.71  &    62.35 \\\\\n\\sttt{OURS-ADAPTIVE} & -49503.43&    60.93 & -49520.14 &    62.32 & -49522.49  &    62.60 \\\\\n\\sttt{OURS-MOMENTUM} & -49513.57 &    63.69 & -49536.67 &    65.26 & -49540.76   &    65.95 \\\\\n\\sttt{OURS-ADAM} & -49516.02 & \\textbf{65.36}  & \\textbf{-49538.84} & \\textbf{67.03} & \\textbf{-49544.58} & \\textbf{67.12} \\\\\\hline\n\\end{tabular}\n\\end{center}\n\\vspace{-0.2cm}\n\\caption{Results for characters inpainting\n  problem~\\cite{Nowozin11} based on DTFs. \\texttt{PA} is the pixel accuracy for the occluded\n  region (bigger is better). Our methods outperform the baselines by a margin of 3-5\\%. Since\n  \\texttt{FULL-PARALLEL} is not damped, it gets to low KL-divergence value quickly, however the\n  actual solution is significantly worse. }  \n\\label{tab:eval:dtf}\n\n\\vspace{-0.25cm}\n\\begin{center}\n\\begin{tabular}{l?C{1.3cm}C{0.9cm}?C{1.3cm}C{0.9cm}?C{1.3cm}C{0.9cm}}\n & \\multicolumn{2}{c?}{0.5s}  & \\multicolumn{2}{c?}{1.3s} & \\multicolumn{2}{c}{5s} \\\\\\hline\nmethod & \\sttt{KL}&\\sttt{MODA}&\\sttt{KL}&\\sttt{MODA}& \\sttt{KL}& \\sttt{MODA}\\\\ \n\\hline\n\\sttt{SWEEP}   & 1865.43 & \\textbf{0.630}  & 1795.66 & 0.656 & 1795.60 & 0.656\\\\\n\\sttt{FULL-PARALLEL} & 2573.79 & 0.000  &2573.79 & 0.000 & 8500.90 & 0.030\\\\\n\\sttt{ADHOC} & 2573.79 & 0.308 & 1760.02  & 0.781 & 1753.71 & \\textbf{0.829} \\\\\\hline\n\\sttt{OURS-FIXED} &  \\textbf{1783.63} & 0.626 & \\textbf{1754.55}  & \\textbf{0.802} &  \\textbf{1753.63} & \\textbf{0.829}\\\\\n\\sttt{OURS-MOMENTUM} & 1931.36 & 0.040  & 1797.19  & 0.650 &  1753.83 & 0.826\\\\\n\\sttt{OURS-ADAM} &  2008.52 & 0.021  & 1813.66  & 0.501 & 1754.52 & 0.824\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\vspace{-0.2cm}\n\\caption{Results for people detection task~\\cite{DOrazio09} based on\n  POM~\\cite{Fleuret08a}. \\texttt{OURS-FIXED} outperforms the baselines and adaptive methods. This\n  means that this problem does not require more sophisticated parameter exploration techniques. }  \n\\label{tab:eval:pom-soccer}\n\n\n\\vspace{-0.25cm}\n\\begin{center}\n\\begin{tabular}{l?C{1.3cm}C{0.9cm}?C{1.3cm}C{0.9cm}?C{1.3cm}C{0.9cm}}\n & \\multicolumn{2}{c?}{5s}  & \\multicolumn{2}{c?}{15s} & \\multicolumn{2}{c}{50s} \\\\\\hline\nmethod & \\sttt{KL} & \\sttt{I\/U} & \\sttt{KL} & \\sttt{I\/U}  & \\sttt{KL} & \\sttt{I\/U}\n  \\\\\\hline \n\n\\sttt{FULL-PARALLEL [o]} &$-$ & 67.18 &$-$ & 67.70 & $-$ & 68.00 \\\\\n\\sttt{OURS-ADAM [o]} & $-$ & 66.45 & $-$ & 67.50 & $-$ & 68.07 \\\\\n\\hline\\hline                                                                              \n\\sttt{FULL-PARALLEL} & \\textbf{-3129799} & 67.21 & -3134437 & 67.72 & -3133010 & 68.01 \\\\\n\\sttt{ADHOC}         & -3129469          & 67.19 & -3134557 & 67.73 & -3136865 & 68.04 \\\\\\hline\n\\sttt{OURS-FIXED}    & -3100079          & \\textbf{67.76} & \\textbf{-3135225} & \\textbf{68.18} \n& -3138206 & 68.44 \\\\ \n\\sttt{OURS-MOMENTUM}     & -3060405          & 66.20 & -3128121 & 67.39 & -3136543 & 68.18 \\\\\n\\sttt{OURS-ADAM}     & -3091787          & 67.08 & -3131624 & 68.02 & \\textbf{-3138335} & \\textbf{68.47} \\\\\n\n\\end{tabular}\n\\end{center}\n\\vspace{-0.2cm}\n\\caption{Results for semantic segmentation problem~\\cite{Pascal-voc-2012} based on\n  DeepLab-CRF~\\cite{Chen15b}. For all the budgets, our method obtains better segmentation\n  accuracy. Again, \\texttt{FULL-PARALLEL} obtains lower KL faster, with a price of reduced\n  performance. \nOn the top, we provide results for the original DeepLab-CRF model without co-occurrence potentials (denoted by [o]), for which the KL divergence has therefore a different meaning and is not shown.}  \n\\label{tab:eval:segmentation-voc2012}\n\\vspace{-0.4cm}\n\n\\end{table*}\n\nQuantitative results are given in Table~\\ref{tab:eval:benchmark}. Our methods systematically\noutperform the~\\texttt{ADHOC} damping \nmethod. The~\\texttt{SWEEP} method usually provides good performance, but is generally slow due\nto its sequential nature. \n\nFig.~\\ref{fi:eval:benchmarks:steps} shows that our methods are less sensitive to\ndamping parameter changes than \\texttt{ADHOC}.  \nIn Fig.~\\ref{fi:eval:benchmarks:steps}, the vertical orange lines corresponds to the choice of the\ndamping parameter according to $d=L$, \nwhich can be computed directly by the power-method. Interstingly, for the\n\\texttt{GRID} dataset, which includes strong repulsive \npotentials, algorithms do not produce reasonable results when no damping is applied. On the other\nhand, for the segmentation task, \\texttt{SEG}, all the algorithms work well even without damping, in\naccordance with the results of~\\cite{Kraehenbuehl13} or Section~\\ref{sec:fixedStep}. \n\n\\vspace{-0.5cm}\n\\paragraph{Characters Inpainting}\nQuantitative results in terms of average pixel accuracies and KL-divergences are given in\nTable~\\ref{tab:eval:dtf} and Fig.~\\ref{fi:eval:convergence}~(a). Our method, especially when used with \nmore advanced gradient descent schemes, outperforms all the baselines. \\texttt{SWEEP} shows\nrelatively good performance, but does not scale as well in terms of the running time. See the bottom\nrow of Fig.~\\ref{fig:intro:teaser} for an example of a result.\n\n\\vspace{-0.5cm}\n\\paragraph{People Detection}\nQuantitative results, presented in Table~\\ref{tab:eval:pom-soccer} and\nFig.~\\ref{fi:eval:convergence}~(b), demonstrate that our method with a fixed step size,\n\\texttt{OURS-FIXED}, brings both faster convergence and better performance. Thanks to our\noptimization scheme, the time required to get a Multiple Object Detection \nAccuracy(MODA,~\\cite{Bernardin08}) within 3\\% of the value at convergence is reduced by a factor of\ntwo. This can be of big practical importance for surveillance applications of the\nalgorithm~\\cite{BenShitrit14,Bagautdinov15}, in which it is required to run in\nreal-time. \\texttt{SWEEP} exhibits much worse performance than our parallel method because \nof its greedy behaviour.  \n\n\\vspace{-0.5cm}\n\\paragraph{Semantic Segmentation}\nQuantitative results are presented in Table~\\ref{tab:eval:segmentation-voc2012} and\nFig.~\\ref{fi:eval:convergence}~(c). We observe that a similar oscillation issue as noted\nby~\\cite{Vineet14} starts happening when the \\texttt{FULL-PARALLEL} method is used in conjunction\nwith co-ocurrence potentials, producing even worse results than without those.  \nUsing our convergent inference method fixes oscillations and provides an\nimprovement of 0.5\\% in the average Intersection over Union measure (\\texttt{I\/U}) compared to the\nbasic method without co-occurrence. This is a significant improvement that would be sufficient to\nincrease the position of an algorithm by 2 or 3 places in the official\nranking~\\cite{Pascal-voc-2012}. What it represents is a big improvement in performance, as the ones\nshown in Fig~\\ref{fig:intro:teaser}, for at least 30-40 images out of total 1449. Note also,\nthat we obtain this improvement with minimal changes in the original code. By contrast,\nauthors~\\cite{Chen15b} get similar or smaller improvements by significantly augmenting the training\nset or by exploiting multi-scale features, which leads to additional computational burden. \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Proximal Gradient for Mean-Field Inference}\n\\label{sec:PGD}\n\nIn   our   approach    to   minimizing   the   variational    free   energy   of\nEq.~\\ref{eq:related:mf-objective}, we treat $\\mathcal{E}$ as the function $f$ of\nEq.~\\ref{eq:proxUpdate}  and  the negative  entropy  $-\\mH$  as the  regularizer\n$g$.  This choice  stems from  the  fact that  $-\\mH$ is  separable,  and\ntherefore, can be minimized in parallel in \nEq.~\\ref{eq:proxUpdate}, without using a first order approximation.\nAlso, $-\\mH$ being  the regularizer  $g$ means\nthat  we  do  not  need  to look at  its  derivatives  with  respect  to  the\nmean-parameters, which are not well behaved when they approach zero. We then define\n\n\\begin{equation}\n  \\Psi^t(\\bq,\\bq^t) = \\sum  \\limits_i \\sum \\limits_l d^t_{i,l} q_{i,l} \\log \\dfrac{q_{i,l}}{q_{i,l}^t}\n = \\bD^t \\odot \\KL(\\bq || \\bq^t) \\; , \\label{eq:psiQ}\n\\end{equation}\nwhere $\\KL$ is  the non-negative KL-divergence, which is a  natural choice for a\ndistance  between distributions.   $\\bD^t$ is  a diagonal  matrix with  positive\ndiagonal elements $d^t_{i,l}$s, which we  introduce to allow for anisotropic scaling\nof the proximal KL-divergence term. As will be discussed below,  different choices of\nthe $d^t_{i,l}$s  yield different  variants of our  algorithms.  Note  however that,\n$\\Psi^t$ is a valid proximal function.\n\\comment{whatever their values, $\\Psi^t$ is a valid proximal function as long as they are\nall non-negative since $\\Psi(\\bq,\\bq^t)=0$ if and only if $\\bq = \\bq^t$.}\n\nThe update of Eq.~\\ref{eq:proxUpdate} then becomes\n\\begin{equation}\n\\label{eq:method:proxUpdateOurs}\n\\bq^{t+1} = \\argmin_{\\bq \\in \\mM} \\{\n\\langle  \\bq,  \\nabla  \\mE(\\bq^t)  \\rangle  - \\mH(\\bq)  +  \\bD^t  \\odot  \\KL(\\bq\n||\\bq^{t}) \\}  \\; .\n\\end{equation}\nThis  computation  can  be  performed   independently  for  each  index  $i  \\in\n\\{1,\\dots,N\\}$. Furthermore, we prove in  the supplementary material that it can\nbe done in closed form and can be written as\n\\begin{eqnarray}\nq_{i,l}^{t+1} & \\propto \\exp[ & \\eta^t_{i,l} \\cdot \\mathbf{E}_{Q(\\bX  \/  X_i=l ; \\bq)}  \\big[ \\log  P(\\bX  | \\bI)  \\big] \n \\label{eq:method:update_mean}  \\\\\n   & & + (1-\\eta^t_{i,l}) \\cdot \\log q_{i,l}^t ] \\; , \\nonumber\n\\end{eqnarray}\nwhere  $\\eta^t_{i,l} =  \\dfrac{1}{1+d^t_{i,l}}$.  Eq~\\ref{eq:method:update_mean}  can be\nrewritten as\n\\begin{equation}\n\\label{eq:method:update_natural} \n\\theta_{i,l}^{t+1} = \\eta^t_{i,l} \\cdot \\theta_{i,l}^{\\star} + (1-\\eta^t_{i,l}) \\cdot \\theta_{i,l}^{t} \\; \\; ,  \n\\end{equation}\nwhere $\\theta_{i,l}^{\\star} = -\\mathbf{E}_{Q(\\bX \/  X_i ; \\bq)} \\big[ \\log P(\\bX | \\bI)\n  \\big]$ now is a natural parameter, like those of Eq.~\\ref{eq:naturalParam}. In\nother   words,    we   have    replaced   the    heuristic   update    rule   of\nEq.~\\ref{eq:related:parallel-update}  in  the  space  of mean  parameters  by  a\nprincipled  one  in the  space  of  natural ones.  As  we  will see,  this\nyields performance and convergence improvements in most cases. \nAs for the stopping criteria, one can define one based on the value of the objective, or, in\npractice, run inference for a fixed number of iterations.\n\n\n\n\\subsection{Fixed Step Size}\n\\label{sec:fixedStep}\n\nThe simplest way to  instantiate our algorithm is to fix  all the $d^t_{i,l}$s of\nEq.~\\ref{eq:psiQ} to the same value $d$ and to write\n\\begin{equation}\n  \\forall t  \\; ,  \\; \\bD^t =  \\bD =  d \\mathbb{I} \\;  \\Rightarrow \\;  \\forall t,i,l,\n  \\eta^t_{i,l} = \\dfrac{1}{1+d} \\; ,\n  \\label{eq:fixedStepSize}\n\\end{equation}\nwhere  $\\eta^t_{i,l}$   plays  the  same   role  as   the  damping  factor of\n  Eq.~\\ref{eq:related:parallel-update}. \\comment{In fact, the whole update rule is similar\n    to the one of Eq.~\\ref{eq:related:parallel-update}, except for the fact that\n    the damping is performed in the  space of natural parameters instead of mean\n    ones. } We  now show that  this is guaranteed  to converge when  the proximal\n    term is given enough weight.\n\nIn  our mean-field  settings,  $\\mE(\\bq)$   is  a  polynomial  function   of  the\nmean-parameters vector $\\bq$.  Therefore, one can always find some positive real\nnumber $L$ such that $\\mE$ is {\\it L-Lipschitz gradient}, which means\n\\begin{equation}\n \\forall x,y \\in \\mM \\; , \\; \\|\\nabla \\mE(x) - \\nabla \\mE(y)\\|_2 \\leq L \\|x-y \\|_2 \\; .\n\\end{equation}\nIn the  supplementary material,  we prove  that this  property implies  that our\nproximal gradient descent scheme is guaranteed  to converge for any fixed matrix\n$D = d \\mathbb{I}$ such that $d  >L $.\n\\comment{\n\n  \\begin{theorem}[Convergence]\nLet us assume that the $\\mE(q)$ is L-Lipschitz gradient for some real constant $L$. \nOur \n  \\end{theorem} \n  \n  \\begin{proof}\n  The proof is provided in the supplementary material.\n  \\end{proof}\n}\n\nIntuitively, when updating the value  of $\\bq^{t}$ to $\\bq^{t+1}$, the magnitude\nof  the   gradient  change   controlled  and thus  the   coordinate-wise  optimum\n$\\theta_{i,l}^{\\star} = - \\nabla \\mE(\\bq^t)_{i,l}$ will also be changing smoothly\nacross  iterations. As a  result,  $L$   is  the  key  value to understand\n  oscillations.  In practice, our goal is to find its smallest possible value to\n  allow steps as large as possible while guaranteeing convergence.\n\nIn the pairwise case, the Hessian of the objective function is a constant matrix, which\nwe call potential matrix. Therefore, the highest eigenvalue  of the potential matrix  is a valid\nLipschitz constant and efficient methods  allow to compute it for moderately sized problems.\n\nIn fact, the  convergence result presented in~\\cite{Kraehenbuehl13} is strongly related to this.\nNamely, assuming that the potential matrix is negative semi-definite, is equivalent to assuming that\n$L<0$ in our formulation. This directly corresponds to the concavity assumptions on the potentials\nin~\\cite{Kraehenbuehl13}. Therefore, under  the assumptions   of~\\cite{Kraehenbuehl13},\nour algorithm leads to $\\eta=1$, corresponding  to  the fully-parallel  update  procedure.\nIn that sense, our procedure is a generalization of the one proposed by~\\cite{Kraehenbuehl13}. \n\nIn the non-pairwise case, the Hessian is not constant, and the calculation of the Lipschitz constant\nis not trivial. For each specific problem, bounds should be derived using the particular\nshape of the CRF at hand. \n\n\n\n\\subsection{Adaptive Step Size}\n\\label{sec:method:qs}\n\nNote that the Hessian of the KL-proximal term is diagonal with\n\\begin{equation}\n\\label{eq:method:hessian0} \n\\dfrac{\\partial^2 \\bD^t \\cdot \n\\KL(\\bq ||\\bq^{t})}{\\partial q_{i,l}^2}|_{\\bq=\\bq^{t}}    = \\dfrac{d^t_{i,l}}{q^{t}_{i,l}}\\; .\n\\end{equation}\nTherefore, when  some of  the $q_{i,l}$s  get close  to 0,  the elements  of the\nHessian may become very  large, especially when using a  constant value for the\n$d^t_{i,l}$ as  suggested above.   When that  happens, the  local KL-approximation\nremains a valid  upper bound of the  objective function, but not  a tight enough\none,  which results  in step  sizes  that are  too small  for fast  convergence.\n\nThis can  be reduced  by choosing  a matrix  $\\bD^t$ that  compensates for\nthis.    A    simple   way   to   do    this   would   be   to   scale the $d_{i,l}^t $ proportionally to \n$\\max(q_{i,0},\\dots,q_{i,L_i-1})$  to start compensating for diagonal terms.  However,  this  method is  still  sub-optimal\nbecause  it ignores  the fact  that  all our  variables lie  inside the  simplex\n$\\mM$. A better\nalternative is to bound from below the proximal term by a quadratic function, but on $\\mM$\nrather than on $ \\mathbb{R}^n$. \n\nIn this paper, we only apply this method to the binary case, for which we set\n\\begin{equation}\n\\label{eq:method:hessian} \nd^t_{i,0} =d^t_{i,1}= q^t_{i,0}q^t_{i,1} \\cdot d \\; ,\n\\end{equation}\nwere $d$ is an additional parameter that should be set close to $L$. Extending this approach\nto the multi-label case will be a topic for future work. In Section~\\ref{sec:momentum}, we provide a\ndifferent alternative to performing adaptive anisotropic updates in all settings. \n\nIntuitively, when the current parameters are close to the\nborders of the simplex, the mean parameters are less sensititive to natural parameters, which, therefore,\nneed less damping.\nWe demonstrate in our experiments that it provides a way to choose the step size without \ntuning. \n\n\\subsection{Momentum}\n\\label{sec:momentum}\n\nOur approach can easily be extended  to incorporate techniques that are known to\nspeed-up  gradient descent  and help to avoid local  minima, such  as the  classic\nmomentum     method~\\cite{Polyak64}     or      the     more     recent     ADAM\ntechnique~\\cite{Kingma14}.\nThe  momentum  method  involves  averaging the  gradients  of  the  objective\n $f(\\bx)$ over the iterations in a \\textit{momentum} vector $\\bmm$ and use it as the direction for the update instead of simply following the current gradient. \nTo  integrate it into our  framework, we replace\nthe gradient  $\\nabla \\mE$ in Eq.~\\ref{eq:method:proxUpdateOurs}  by its rolling exponentially weighted\naverage $\\bmm$ computed as\n\\begin{eqnarray}\n\\bmm^{t+1} & = & \\gamma_1 \\bmm^t + (1 - \\gamma_1) \\nabla \\mE(\\bq^t) \\; , \n\\end{eqnarray}\nwith the exponential decay parameter $\\gamma_1 \\in [0;1]$. This substitution brings the following update rule\n\\begin{eqnarray}\n\\label{eq:momentumUpdate}\n\\theta_{i,l}^{t+1} & = & \\eta \\cdot m_{i,l}^t+ (1-\\eta) \\cdot \\theta_{i,l}^{t}  \\; .\n\\end{eqnarray}\nWe will refer to this approach as \\texttt{OURS-MOMENTUM}.\n\n\\subsection{ADAM}\n\\label{sec:adam}\n\nThe ADAM method~\\cite{Kingma14} has become very popular in deep learning. Our\nframework  makes  it   easy  to  use  for  mean-field inference   as  well  by\nappropriately choosing the matrix $\\bD^t$ at each step and combining it with the momentum technique. \n\nWe define the averaged second moment vector $\\bv$ of the natural gradient as\n\\begin{eqnarray}\n{v}^{t+1}_{i,l} & = &\\gamma_2 [\\theta^t_{i,l} +\\nabla \\mE(\\bq^t)_{i,l}]^2 + (1 - \\gamma_2) {v}^t_{i,l} \\; , \n\\end{eqnarray}\nwhere $\\bv$ is initialised to a strictly positive value and $\\gamma_2 \\in [0;1]$ is an exponential\nmemory parameter for $\\bv$. \n\nThen, the $\\bD^t$ matrix is defined through each of its diagonal entries as\n\\begin{equation}\n  d^t_{i,l} = \\sqrt{{v}^{t+1}_{i,l}}d +\\epsilon - 1 \\;,\n  \\label{eq:adamStepSize}\n\\end{equation}\nwhere $\\epsilon$ is a fixed parameters and $d$ controls the damping. We will refer to this method as \\texttt{OURS-ADAM}.  \n\nIntuitively it is good at exploring parameter space thanks to a form of auto-annealing\nof the gradient. The natural gradient $\\btheta_t + \\nabla \\mE(\\bq^t)$ is zero at a local minimum of\nthe objective function~\\cite{Hoffman13}. Therefore, close to a minimum, the proximal term $\\bD^t$\nbecomes small, thus allowing more exploration of the space. On the other hand, after a long period\nof exploration with large natural gradients, more damping will tend to make the algorithm converge. \n\n\n\\subsection{Conditional Random Fields}\n\nLet $\\bX = \\left(X_1,  \\ldots, X_N\\right)$ represent hidden variables and $\\bm{I}$ represent\nobserved variables.  For example, for  semantic segmentation, the\n$X_i$s  are taken to be variables representing semantic classes of $N$ pixels,\n  and  $\\bm{I}$  represents the  observed  image  evidence.\n\n\nA Conditional Random Field (CRF) models the relationship between  $\\bX$ and $\\bI$\nin terms of the posterior distribution\n\\begin{equation}\n\\vspace{-0.1cm}\nP(\\bX \\mid \\bI) = \\exp\\left(\\sum_{ c \\subset\\{1,\\dots,N\\} }\\bm{\\phi}_c(\\bX_c \\mid \\bI) - \\log Z\n  (\\bI) \\right)\\; ,\n\\label{eq:related:mrf}\n\\end{equation}\nwhere $\\bm{\\phi}_c(.)$  are non-negative functions  known as potentials  and $\\log\nZ(\\bI)$ is the  log-partition function. It is  a constant that we  will omit for\nsimplicity since  we are  mostly concerned by estimating  values of  $\\bX$ that\nmaximize $P(\\bX \\mid \\bI)$.\n\nThis model is often further simplified  by only considering unary and pairwise terms:\n\\begin{equation}\n\\vspace{-0.1cm}\nP(\\bX \\mid  \\bI) \\propto  \\exp \\left( \\sum_{i}  \\phi_i(X_i, I_i) +  \\sum_{(i,j)} \\phi_{ij}(X_i,\n  X_j) \\right).\n \\label{eq:related:pairwise-mrf}\n\\end{equation}\n\n\\subsection{Mean-Field Inference}\n\nTypically, one  wants either to estimate  the posterior $P(\\bX|\\bI)$ or  to find\nthe vector  $\\hat{\\bX}$ that maximizes $P(\\bX|\\bI)$,  which is known as  the MAP\nassignment. Unfortunately, even\nfor the  simplified formulation  of Eq.~\\ref{eq:related:pairwise-mrf},  both are\nintractable for realistic sizes of $\\bX$. As\na  result, many  approaches  settle for  approximate  solutions.  These  include\nsampling methods, such as Gibbs sampling~\\cite{Gelfand90}, and deterministic ones\nsuch as mean-field variational inference~\\cite{Winn2005}, belief\npropagation~\\cite{Murphy99,Minka01,Kolmogorov15}, and others~\\cite{Boykov01b,Gorelick2014}. \n\nNote that, mean-field  methods have  been  shown  to combine the advantages of good\nconvergence  guarantees~\\cite{Bishop06}, flexibility with respect to the potential functions that\ncan be handled~\\cite{Saito12}, and potential for parallelization~\\cite{Kraehenbuehl13}. \nAs a  result, they have become  very popular  in our field. Furthermore, they  have recently been\nshown to yield state-of-the-art performance for several Computer Vision\ntasks~\\cite{Saito12,Vineet14,Chen15b,Zheng15}. \n\nMean-field involves introducing a distribution $Q$ of the factorized form\n\\begin{equation}\n\\vspace{-0.1cm}\n  Q(\\bX=(x_1,\\ldots,x_N) ; \\bq) = \\prod_{i = 1}^{N} Q_i(X_i=x_i ; \\bq_i)\\; , \n\\label{eq:related:q-factorized}\n\\end{equation}\nwhere $Q_i( \\, . \\,  ; \\bq_i)$ is a categorical distribution with mean\nparameters $\\bq_i$. That is,\n\\begin{equation}\n\\label{eq:related:definition_q}\n\\forall l, \\, Q_i(X_i = l ; \\bq_i) = q_{i,l},\n\\end{equation}\nwith $\\bq$ on the simplex $\\mM$ such that $\\forall i,l, \\; 0 \\leq q_{i,l} \\leq 1$ and $\\forall i,\n\\sum_l q_{i,l}=1$. \n\n$Q$ is then used to approximate $P(\\bX \\mid \\bI)$ by minimizing the KL-divergence:\n\\begin{equation}\n\\KL(Q || P) = \\sum_{\\bm{x}} Q(\\bX = \\bx ; \\bq) \\log \\frac{Q(\\bX = \\bx ; \\bq)} {P(\\bX = \\bx \\mid \\bI)}\\; .\n\\label{eq:related:kl}\n\\end{equation}\nIn  some  cases,  this  approximation  is the  desired  final\nresult. In others, one seeks a MAP assignment. To this end, a standard method is\nto select the assignment that maximizes the \\textit{approximate} posterior $Q(\\bX ; \\bq)$,\nwhich is  equivalent to rounding  when the  $X_i$s are Bernoulli  variables.  An\nalternative approach is to draw samples from $Q(\\bX ; \\bq)$.\n\n\n\nWhen minimizing the KL-divergence of  Eq.~\\ref{eq:related:kl}, $Q(\\bX ; \\bq)$ can be\nreparameterized in terms of its {\\it natural} parameters defined as follows. For\neach variable $X_i$ and label $l$,  we take the natural parameter $\\theta_{i,l}$\nto be such that\n\\begin{equation}\n  Q(X_i = l ; \\bq_i) = q_{i,l} \\propto \\exp[-\\theta_{i,l}] .\n  \\label{eq:naturalParam}\n\\end{equation}\nAs  we  will see  below,  this  typically yields simpler  notations  and implementations.\n\n\\vspace{-0.3cm}\n\\subsubsection{Sweep Mean-Field Inference}\n\\label{sec:related:seqMF}\n\nSolving the minimization problem of Eq.~\\ref{eq:related:kl} is equivalent~\\cite{Bishop06} to minimizing\n\\begin{equation}\n\\mathcal{F}(\\bq) = \n\\underbrace{-\\mathbf{E}_{Q(\\bm{X} ; \\bq)}[\\log P (\\bX \\mid \\bI)]}_{\\mathcal{E}(\\bq)} +\n\\underbrace{\\mathbf{E}_{Q(\\bm{X} ; \\bq)}[\\log Q(\\bm{X} ; \\bq)]}_{-\\mH(\\bq)},\n\\label{eq:related:mf-objective}\n\\end{equation}\nwith  respect to  $\\bq$ on the simplex $\\mM$. $\\mF(.)$  is  sometimes   called  the\nvariational free energy. Its first term is the expectation of the energy under\n$Q(\\bX ; \\bq)$, and its second term is the negative entropy, which acts as a regularizer.\n\nOne  can  minimize $\\mathcal{F}(\\bq)$  by  iteratively  updating each  $q_{i,l}$  in\nsequence while  keeping the  others fixed. Each update involves setting $q_{i,l}$ to\n\\begin{equation}\nq^{\\star}_{i,l} \\propto \\exp\\left[\\mathbf{E}_{Q(\\bX \/ X_i ; \\bq)} \\big[ \\log P(\\bX \\mid \\bI) \\big] \\right].\n\\label{eq:related:sweep-update}\n\\end{equation}\n\n This coordinate descent procedure, which we will call \\texttt{SWEEP}, is guaranteed to\n converge to a local minimum of $\\mF$~\\cite{Bishop06}.\n However, it tends to be very slow for realistic image  sizes and impractical for many Computer\n Vision problems~\\cite{Vineet14,Kraehenbuehl13}. Namely, in the case of dense random fields, it involves re-computing a large number of expectations\n(one per factor adjacent to the variable)  \nafter each sequential update. Filter-based mean-field inference~\\cite{Kraehenbuehl11} attempts to reduce\nthe complexity of these updates, but it effectively performs parallel updates, which we will describe\nbelow. \n\n\\vspace{-0.3cm}\n\\subsubsection{Parallel Mean-Field Inference}\n\\label{sec:parallel-mf}\n\nTo   obtain  reasonable   efficiency  in   practice,  Computer   Vision\npractitioners often perform the  updates of Eq.~\\ref{eq:related:sweep-update} in\nparallel as opposed to sequentially. Not only does it avoid having to reevaluate\na large number of factors after each update, it also allows the use\nof vectorized instructions and GPUs, both of which can have a dramatic impact on\nthe computation speed.\n\n\nUnfortunately, these parallel updates invalidate the convergence guarantees and\nin practice often lead to undesirable oscillations in the objective. Several approaches  to remedying this problem\nhave been proposed, which we review below.\n\n\\vspace{-0.3cm}\n\\paragraph{Damping}\n\nA  natural   way  to  improve   convergence  is   to  replace  the   updates  of\nEq.~\\ref{eq:related:sweep-update} by a damped version, expressed as\n\\begin{equation}\nq_{i,l}^{t+1} = (1-\\eta) \\cdot q_{i,l}^{t} + \\eta \\cdot q_{i,l}^{\\star} \\; ,\n\\label{eq:related:parallel-update}\n\\end{equation}\nwhere  $t$ denotes  the current  iteration, $q_{i,l}^{\\star}$  is the  result of\nsolving  the  optimization  problem  of  Eq.~\\ref{eq:related:sweep-update},  and\n$\\eta$ is a  heuristically chosen damping parameter. This damping is  explicitly mentioned in\npapers     such    as~\\cite{Campbell13,Sun13,Frostig14}.     In~\\cite{Vineet14},\nconvergence  issues are  mentioned  and  a  damping parameter  is\nprovided  in the  publicly available  code.  \nSimilarly,  in~\\cite{Fleuret08a}, the  algorithm relies  on\nmean-field  optimization with  repulsive terms.   The  need for  damping is  not\nexplicitly discussed in the paper, but  the publicly available code also includes\na damping. \n\nDamping  delivers satisfactory  results  in  many cases,  but  does not  formally\nguarantee convergence.   It may fail  if the  parameter $\\eta$ is  not carefully\nchosen, and sometimes  changed at different stages of the  optimization.  In all\nthe approaches that we are aware of,  this is done heuristically.  We will refer\nto this type of methods as \\texttt{ADHOC}.\n\n\\vspace{-0.3cm}\n\\paragraph{Concave potentials}\n\nA principled way to address the convergence issue for the pairwise random fields is offered\nin~\\cite{Kraehenbuehl13}, and we refer to the corresponding algorithm as\n\\texttt{FULL-PARALLEL}. However, authors restrict their potentials $\\phi_{ij}$ of\nEq.~\\ref{eq:related:pairwise-mrf} to be concave, which in some \ncases is reasonable, but as we  will show in Section~\\ref{sec:evaluation}, many Computer Vision models\nviolate this requirement. By contrast, our approach is similarly principled but without\nadditional constraints. In practice it works for higher-order, or, equivalently, non-pairwise \npotentials. \n\n\n\\subsection{Proximal Gradient Descent}\n\\label{sec:related:PGD}\n\nLet $F$  be a generic objective  function of the form  $F(\\bx)=f(\\bx) + g(\\bx)$,\nwhere $g$  is a  regularizer, and  $\\bx_t$ is  the value  of the\noptimized  variable  at iteration  $t$  of  a minimization  procedure on a constraint set $\\mX$.  Proximal\ngradient descent, also known as composite mirror-descent \\cite{Duchi10}, is \nan iterative method that relies on the update rule\n\\begin{equation}\n\\bx^{t+1} = \\argmin_{\\bx \\in \\mathcal{X}}\\{ \\langle \\bx, \\nabla f(\\bx^t) \\rangle + g(\\bx) + \\lambda\n\\Psi(\\bx,\\bx^t) \\} \\; \\; , \n  \\label{eq:proxUpdate}\n\\end{equation}\nwhere   $\\Psi$  is   a  non-negative   \\textit{proximal}  function   that  satisfies\n$\\Psi(\\bx,\\bx^t) =  0$ if and only  if $\\bx =  \\bx^t$, and $\\lambda$ is  a scalar\nparameter. $g$ contains the terms of the objective function that do not need to\nbe approximated to the first order,\nwhile still allowing efficient computation of\nupdate of Eq.~\\ref{eq:proxUpdate}.\n  $\\Psi$  can be understood as  a distance function that  accounts for\nthe geometry of $\\mX$~\\cite{Teboulle92} while also  making it possible to compute the\nupdate of Eq.~\\ref{eq:proxUpdate} efficiently. $\\lambda$  can then be thought of\nas the inverse  of the step size. \\comment{Intuitively, solving  the minimization problem\nof Eq.~\\ref{eq:proxUpdate}  amounts to minimizing the  first-order approximation\nof  $F(\\bx)$, while  taking  into account  the composite  structure of  the\nfunction and keeping the next  estimate $\\bx^{t+1}$ close  to $\\bx^t$.   }\n\nAs  shown  in  Section~\\ref{sec:PGD}, our  algorithm  is  a version  of\nproximal gradient  descent in  which $\\Psi$  is based  on the  KL-divergence and\nallows  automated step-size  adaptation as  the optimization  progresses. \\comment{It  is\nrelated to the AdaGrad method~\\cite{Duchi11}, which is a gradient descent technique that uses\nadaptive proximal terms, and has been extensively used for Deep Learning purposes.\nHowever, since  AdaGrad uses a Euclidean norm as its proximal function, is not well suited for\nprobability distributions. } Recently, a variational approach  that also relies on  the KL-divergence\nas  the proximal function has been  proposed~\\cite{Emti15}. This paper explores  the\nconnection between the KL-proximal  method and the Stochastic Variational\nInference~\\cite{Amari98,Hoffman13}.  However,  it does  not allow for  step size\nadaptation, which often yields better performance, as we demonstrate in our\nexperiments and is not directly applied to the mean-field setting.\n\n\n\\section{Introduction}\n\\label{sec:intro}\n\\input{introduction}\n\n\\section{Background and Related Work}\n\\label{sec:related}\n\\input{related}\n\n\\section{Method}\n\\label{sec:method}\n\\input{method}\n\n\\section{Experimental Evaluation}\n\\label{sec:evaluation}\n\\input{evaluation}\n\n\\section{Discussion and Future Work}\n\\label{sec:discussion}\n\\input{discussion}\n\n{\\small\n  \\bibliographystyle{ieee}\n  ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n   In the \\emph{nucleated instability\\\/} (also called core\n   instability) hypothesis of giant planet\n   formation, a critical mass for static core  envelope\n   protoplanets has been found. Mizuno (\\cite{mizuno}) determined\n   the critical mass of the core to be about $12 \\,M_\\oplus$\n   ($M_\\oplus=5.975 \\times 10^{27}\\,\\mathrm{g}$ is the Earth mass), which\n   is independent of the outer boundary\n   conditions and therefore independent of the location in the\n   solar nebula. This critical value for the core mass corresponds\n   closely to the cores of today's giant planets.\n\n   Although no hydrodynamical study has been available many workers\n   conjectured that a collapse or rapid contraction will ensue\n   after accumulating the critical mass. The main motivation for\n   this article\n   is to investigate the stability of the static envelope at the\n   critical mass. With this aim the local, linear stability of static\n   radiative gas  spheres is investigated on the basis of Baker's\n   (\\cite{baker}) standard one-zone model. \n\n   Phenomena similar to the ones described above for giant planet\n   formation have been found in hydrodynamical models concerning\n   star formation where protostellar cores explode\n   (Tscharnuter \\cite{tscharnuter}, Balluch \\cite{balluch}),\n   whereas earlier studies found quasi-steady collapse flows. The\n   similarities in the (micro)physics, i.e., constitutive relations of\n   protostellar cores and protogiant planets serve as a further\n   motivation for this study.\n\n\n\\section{Baker's standard one-zone model}\n\n   \\begin{figure*}\n   \\centering\n  \n  \n  \n   \\caption{Adiabatic exponent $\\Gamma_1$.\n               $\\Gamma_1$ is plotted as a function of\n               $\\lg$ internal energy $\\mathrm{[erg\\,g^{-1}]}$ and $\\lg$\n               density $\\mathrm{[g\\,cm^{-3}]}$.}\n              \\label{FigGam}%\n    \\end{figure*}\n   In this section the one-zone model of Baker (\\cite{baker}),\n   originally used to study the Cephe{\\\"{\\i}}d pulsation mechanism, will\n   be briefly reviewed. The resulting stability criteria will be\n   rewritten in terms of local state variables, local timescales and\n   constitutive relations.\n\n   Baker (\\cite{baker}) investigates the stability of thin layers in\n   self-gravitating,\n   spherical gas clouds with the following properties:\n   \\begin{itemize}\n      \\item hydrostatic equilibrium,\n      \\item thermal equilibrium,\n      \\item energy transport by grey radiation diffusion.\n   \\end{itemize}\n   For the one-zone-model Baker obtains necessary conditions\n   for dynamical, secular and vibrational (or pulsational)\n   stability (Eqs.\\ (34a,\\,b,\\,c) in Baker \\cite{baker}). Using Baker's\n   notation:\n   \\[\n      \\begin{array}{lp{0.8\\linewidth}}\n         M_{r}  & mass internal to the radius $r$     \\\\\n         m               & mass of the zone                    \\\\\n         r_0             & unperturbed zone radius             \\\\\n         \\rho_0          & unperturbed density in the zone     \\\\\n         T_0             & unperturbed temperature in the zone \\\\\n         L_{r0}          & unperturbed luminosity              \\\\\n         E_{\\mathrm{th}} & thermal energy of the zone\n      \\end{array}\n   \\]\n\\noindent\n   and with the definitions of the \\emph{local cooling time\\\/}\n   (see Fig.~\\ref{FigGam})\n   \\begin{equation}\n      \\tau_{\\mathrm{co}} = \\frac{E_{\\mathrm{th}}}{L_{r0}} \\,,\n   \\end{equation}\n   and the \\emph{local free-fall time}\n   \\begin{equation}\n      \\tau_{\\mathrm{ff}} =\n         \\sqrt{ \\frac{3 \\pi}{32 G} \\frac{4\\pi r_0^3}{3 M_{\\mathrm{r}}}\n}\\,,\n   \\end{equation}\n   Baker's $K$ and $\\sigma_0$ have the following form:\n   \\begin{eqnarray}\n      \\sigma_0 & = & \\frac{\\pi}{\\sqrt{8}}\n                     \\frac{1}{ \\tau_{\\mathrm{ff}}} \\\\\n      K        & = & \\frac{\\sqrt{32}}{\\pi} \\frac{1}{\\delta}\n                        \\frac{ \\tau_{\\mathrm{ff}} }\n                             { \\tau_{\\mathrm{co}} }\\,;\n   \\end{eqnarray}\n   where $ E_{\\mathrm{th}} \\approx m (P_0\/{\\rho_0})$ has been used and\n   \\begin{equation}\n   \\begin{array}{l}\n      \\delta = - \\left(\n                    \\frac{ \\partial \\ln \\rho }{ \\partial \\ln T }\n                 \\right)_P \\\\\n      e=mc^2\n   \\end{array}\n   \\end{equation}\n   is a thermodynamical quantity which is of order $1$ and equal to $1$\n   for nonreacting mixtures of classical perfect gases. The physical\n   meaning of $ \\sigma_0 $ and $K$ is clearly visible in the equations\n   above. $\\sigma_0$ represents a frequency of the order one per\n   free-fall time. $K$ is proportional to the ratio of the free-fall\n   time and the cooling time. Substituting into Baker's criteria, using\n   thermodynamic identities and definitions of thermodynamic quantities,\n   \\begin{displaymath}\n      \\Gamma_1      = \\left( \\frac{ \\partial \\ln P}{ \\partial\\ln \\rho}\n                           \\right)_{S}    \\, , \\;\n      \\chi^{}_\\rho  = \\left( \\frac{ \\partial \\ln P}{ \\partial\\ln \\rho}\n                           \\right)_{T}    \\, , \\;\n      \\kappa^{}_{P} = \\left( \\frac{ \\partial \\ln \\kappa}{ \\partial\\ln P}\n                           \\right)_{T}\n   \\end{displaymath}\n   \\begin{displaymath}\n      \\nabla_{\\mathrm{ad}} = \\left( \\frac{ \\partial \\ln T}\n                             { \\partial\\ln P} \\right)_{S} \\, , \\;\n      \\chi^{}_T       = \\left( \\frac{ \\partial \\ln P}\n                             { \\partial\\ln T} \\right)_{\\rho} \\, , \\;\n      \\kappa^{}_{T}   = \\left( \\frac{ \\partial \\ln \\kappa}\n                             { \\partial\\ln T} \\right)_{T}\n   \\end{displaymath}\n   one obtains, after some pages of algebra, the conditions for\n   \\emph{stability\\\/} given\n   below:\n   \\begin{eqnarray}\n      \\frac{\\pi^2}{8} \\frac{1}{\\tau_{\\mathrm{ff}}^2}\n                ( 3 \\Gamma_1 - 4 )\n         & > & 0 \\label{ZSDynSta} \\\\\n      \\frac{\\pi^2}{\\tau_{\\mathrm{co}}\n                   \\tau_{\\mathrm{ff}}^2}\n                   \\Gamma_1 \\nabla_{\\mathrm{ad}}\n                   \\left[ \\frac{ 1- 3\/4 \\chi^{}_\\rho }{ \\chi^{}_T }\n                          ( \\kappa^{}_T - 4 )\n                        + \\kappa^{}_P + 1\n                   \\right]\n        & > & 0 \\label{ZSSecSta} \\\\\n     \\frac{\\pi^2}{4} \\frac{3}{\\tau_{ \\mathrm{co} }\n                              \\tau_{ \\mathrm{ff} }^2\n                             }\n         \\Gamma_1^2 \\, \\nabla_{\\mathrm{ad}} \\left[\n                                   4 \\nabla_{\\mathrm{ad}}\n                                   - ( \\nabla_{\\mathrm{ad}} \\kappa^{}_T\n                                     + \\kappa^{}_P\n                                     )\n                                   - \\frac{4}{3 \\Gamma_1}\n                                \\right]\n        & > & 0   \\label{ZSVibSta}\n   \\end{eqnarray}\n   For a physical discussion of the stability criteria see Baker\n   (\\cite{baker}) or Cox (\\cite{cox}).\n\n   We observe that these criteria for dynamical, secular and\n   vibrational stability, respectively, can be factorized into\n   \\begin{enumerate}\n      \\item a factor containing local timescales only,\n      \\item a factor containing only constitutive relations and\n         their derivatives.\n   \\end{enumerate}\n   The first factors, depending on only timescales, are positive\n   by definition. The signs of the left hand sides of the\n   inequalities~(\\ref{ZSDynSta}), (\\ref{ZSSecSta}) and (\\ref{ZSVibSta})\n   therefore depend exclusively on the second factors containing\n   the constitutive relations. Since they depend only\n   on state variables, the stability criteria themselves are \\emph{\n   functions of the thermodynamic state in the local zone}. The\n   one-zone stability can therefore be determined\n   from a simple equation of state, given for example, as a function\n   of density and\n   temperature. Once the microphysics, i.e.\\ the thermodynamics\n   and opacities (see Table~\\ref{KapSou}), are specified (in practice\n   by specifying a chemical composition) the one-zone stability can\n   be inferred if the thermodynamic state is specified.\n   The zone -- or in\n   other words the layer -- will be stable or unstable in\n   whatever object it is imbedded as long as it satisfies the\n   one-zone-model assumptions. Only the specific growth rates\n   (depending upon the time scales) will be different for layers\n   in different objects.\n\n   \\begin{table}\n      \\caption[]{Opacity sources.}\n         \\label{KapSou}\n     $$ \n         \\begin{array}{p{0.5\\linewidth}l}\n            \\hline\n            \\noalign{\\smallskip}\n            Source      &  T \/ {[\\mathrm{K}]} \\\\\n            \\noalign{\\smallskip}\n            \\hline\n            \\noalign{\\smallskip}\n            Yorke 1979, Yorke 1980a & \\leq 1700^{\\mathrm{a}}     \\\\\n            Kr\\\"ugel 1971           & 1700 \\leq T \\leq 5000 \\\\\n            Cox \\& Stewart 1969     & 5000 \\leq             \\\\\n            \\noalign{\\smallskip}\n            \\hline\n         \\end{array}\n     $$ \n   \\end{table}\n   We will now write down the sign (and therefore stability)\n   determining parts of the left-hand sides of the inequalities\n   (\\ref{ZSDynSta}), (\\ref{ZSSecSta}) and (\\ref{ZSVibSta}) and thereby\n   obtain \\emph{stability equations of state}.\n\n   The sign determining part of inequality~(\\ref{ZSDynSta}) is\n   $3\\Gamma_1 - 4$ and it reduces to the\n   criterion for dynamical stability\n   \\begin{equation}\n     \\Gamma_1 > \\frac{4}{3}\\,\\cdot\n   \\end{equation}\n   Stability of the thermodynamical equilibrium demands\n   \\begin{equation}\n      \\chi^{}_\\rho > 0, \\;\\;  c_v > 0\\, ,\n   \\end{equation}\n   and\n   \\begin{equation}\n      \\chi^{}_T > 0\n   \\end{equation}\n   holds for a wide range of physical situations.\n   With\n   \\begin{eqnarray}\n      \\Gamma_3 - 1 = \\frac{P}{\\rho T} \\frac{\\chi^{}_T}{c_v}&>&0\\\\\n      \\Gamma_1     = \\chi_\\rho^{} + \\chi_T^{} (\\Gamma_3 -1)&>&0\\\\\n      \\nabla_{\\mathrm{ad}}  = \\frac{\\Gamma_3 - 1}{\\Gamma_1}         &>&0\n   \\end{eqnarray}\n   we find the sign determining terms in inequalities~(\\ref{ZSSecSta})\n   and (\\ref{ZSVibSta}) respectively and obtain the following form\n   of the criteria for dynamical, secular and vibrational\n   \\emph{stability}, respectively:\n   \\begin{eqnarray}\n      3 \\Gamma_1 - 4 =: S_{\\mathrm{dyn}}      > & 0 & \\label{DynSta}  \\\\\n      \\frac{ 1- 3\/4 \\chi^{}_\\rho }{ \\chi^{}_T } ( \\kappa^{}_T - 4 )\n         + \\kappa^{}_P + 1 =: S_{\\mathrm{sec}} > & 0 & \\label{SecSta} \\\\\n      4 \\nabla_{\\mathrm{ad}} - (\\nabla_{\\mathrm{ad}} \\kappa^{}_T\n                             + \\kappa^{}_P)\n                             - \\frac{4}{3 \\Gamma_1} =: S_{\\mathrm{vib}}\n                                      > & 0\\,.& \\label{VibSta}\n   \\end{eqnarray}\n   The constitutive relations are to be evaluated for the\n   unperturbed thermodynamic state (say $(\\rho_0, T_0)$) of the zone.\n   We see that the one-zone stability of the layer depends only on\n   the constitutive relations $\\Gamma_1$,\n   $\\nabla_{\\mathrm{ad}}$, $\\chi_T^{},\\,\\chi_\\rho^{}$,\n   $\\kappa_P^{},\\,\\kappa_T^{}$.\n   These depend only on the unperturbed\n   thermodynamical state of the layer. Therefore the above relations\n   define the one-zone-stability equations of state\n   $S_{\\mathrm{dyn}},\\,S_{\\mathrm{sec}}$\n   and $S_{\\mathrm{vib}}$. See Fig.~\\ref{FigVibStab} for a picture of\n   $S_{\\mathrm{vib}}$. Regions of secular instability are\n   listed in Table~1.\n\n   \\begin{figure}\n   \\centering\n  \n      \\caption{Vibrational stability equation of state\n               $S_{\\mathrm{vib}}(\\lg e, \\lg \\rho)$.\n               $>0$ means vibrational stability.\n              }\n         \\label{FigVibStab}\n   \\end{figure}\n\n\\section{Conclusions}\n\n   \\begin{enumerate}\n      \\item The conditions for the stability of static, radiative\n         layers in gas spheres, as described by Baker's (\\cite{baker})\n         standard one-zone model, can be expressed as stability\n         equations of state. These stability equations of state depend\n         only on the local thermodynamic state of the layer.\n      \\item If the constitutive relations -- equations of state and\n         Rosseland mean opacities -- are specified, the stability\n         equations of state can be evaluated without specifying\n         properties of the layer.\n      \\item For solar composition gas the $\\kappa$-mechanism is\n         working in the regions of the ice and dust features\n         in the opacities, the $\\mathrm{H}_2$ dissociation and the\n         combined H, first He ionization zone, as\n         indicated by vibrational instability. These regions\n         of instability are much larger in extent and degree of\n         instability than the second He ionization zone\n         that drives the Cephe{\\\"\\i}d pulsations.\n   \\end{enumerate}\n\n\\begin{acknowledgements}\n      Part of this work was supported by the German\n      \\emph{Deut\\-sche For\\-schungs\\-ge\\-mein\\-schaft, DFG\\\/} project\n      number Ts~17\/2--1.\n\\end{acknowledgements}\n\n\n\n\n\\section{Introduction}\n\nGalaxy clusters are fundamental tools in modern cosmology. Their distribution in number and mass as a function of redshift is an indicator of the background cosmological properties of the Universe [{\\it e.g.} \\cite{2016A&A...594A..24P,2017arXiv170800697S} and references therein]. For this reason the detection of statistically complete samples of these objects is of considerable importance for their use as cosmological probes.\n\nGalaxy clusters are complex objects containing a hot intracluster gas (ICM), amounting to about 15\\% of their mass, and galaxies, amounting to about 5\\%, lying in a dark matter (DM) potential well which represents about 80\\% of the total mass. Clusters emit at several wavelengths and so are detectable by different means. X-ray catalogs [{\\it e.g.} REFLEX \\citep{2004A&A...425..367B}] are constructed using the X-ray emission from the hot plasma of the ICM. The same ICM can be detected through the Sunyaev-Zel'dovich (SZ) effect \\citep{1972CoASP...4..173S}. Recently, the Planck \\citep{2016A&A...594A..27P,2011A&A...536A...8P}, ACT \\citep{2013JCAP...07..008H} and SPT \\citep{2015ApJS..216...27B} produced several catalogs of SZ clusters. In total, they contain more than $1700$ identified SZ-selected clusters up to $z=1.5$ (\\url{http:\/\/szcluster-db.ias.u-psud.fr\/}). \nOptical catalogs [{\\it e.g.} redMaPPer SDSS \\citep{2014ApJ...785..104R, 2016ApJS..224....1R}] use the information from the emission of the stellar component to detect clusters. There are several ways to identify clusters in optical observations. redMaPPer is a red-sequence cluster finder optimized for large photometric surveys like SDSS (see \\cite{2015ApJS..219...12A} for recent releases). For another red-sequence based detection algorithm, see \\cite{2017arXiv170504331L}. There exist also cluster detection techniques based on matching filter algorithms for photometric optical\/infrared (IR) data [{\\it e.g.} BCF \\citep{2012MNRAS.420.1167A} and AMICO \\citep{2017arXiv170503029B}]. In general, those algorithms try to match a template for the object under detection to the distribution of sources. For clusters, the template is usually a combination of the spatial distribution {\\it e.g.}, Gaussian, NFW \\cite{1996ApJ...462..563N} or $\\beta$-model \\cite{1976A&A....49..137C} profile and the luminosity function of the galaxy members.\n\nThe luminosity function (LF) is a common tool to characterize the distribution of luminous matter in the Universe [{\\it e.g.} \\cite{2003MNRAS.342..725D,2016MNRAS.459.3998L,2017AJ....153..189L} and references therein]. It depends on the environment, and its evolution with redshift can be used to trace the changes in the galaxy population. The exact shape of the LF is still debated, in particular regarding the faint end [{\\it e.g.} \\cite{2005ApJ...633..122H,2009A&A...508.1217Z,2011MNRAS.414.2771D,2015A&A...581A..11M} and references therein]. There is general agreement that the bright end is described by a Schechter function \\citep{1976ApJ...203..297S}. At optical and infrared wavelengths, it is usually determined from photometry by statistical means, either in a single band or using color information. In particular, the infrared $K$ band provides a good approximation of the underlying stellar mass function \\citep{1996A&A...312..397G,2001ApJ...550..212B,2018MNRAS.473..776K}. Because the bright end is always poorly sampled in a single cluster, individual LF of different objects can be combined in a composite luminosity function \\citep[][and references therein]{2003MNRAS.342..725D}.\n\nIn this paper, we aim at characterizing the IR luminosity function of cluster members using the Wide-field Infrared Survey Explorer (WISE) data \\citep{2010AJ....140.1868W}. WISE is an infrared space telescope which surveyed the all-sky in four mid-infrared bands at $3.4$, $4.6$, $12$ and $22 \\ \\mu m$. By completely scanning the sky twice, it took images of three-quarter of a billion objects, ranging from galaxies to stars and asteroids (\\url{https:\/\/www.nasa.gov\/mission_pages\/WISE\/mission\/index.html}). We stress here that our main purpose is to provide a description of how cluster galaxies appear in WISE, so that the cluster galaxy LF we obtain may be used in the future in matched-filter detection techniques applied to WISE data. Numerous other studies of galaxy clusters focused on their IR properties using the WISE data to detect new clusters. The MaDCoWS project \\citep[][and references therein]{2014ApJS..213...25S} aims at detecting high-redshift massive galaxy clusters by combining WISE and SDSS data and using spectroscopic observations to confirm their results. \\cite{2017AstL...43..507B} extended the Planck cluster catalog by identifying clusters combining the $3.4 \\ \\mu m$ band of WISE and the SDSS data, using the algorithm described in \\cite{2015AstL...41..167B}. For rich ($\\lambda > 40$, where $\\lambda$ is the richness parameter defined in Section \\ref{data}) redMaPPer clusters, the algorithm detects IR emission in WISE images convolved with $\\beta$-models and then identifies the brightest central galaxy and the red sequence using SDSS data. In both cases, no information on the luminosity function is used. Indeed, the cluster members luminosity function has remained mostly unexplored in WISE despite its potential importance as an ingredient in cluster detection techniques based on matched filtering. By using WISE data, we are facing several issues (e.g. lack of redshifts; difficulty in identifying cluster members and the Brightest Cluster Galaxy; contamination by foreground galaxies, AGNs, etc.). Therefore, our approach is to choose a  large sample of known clusters with identified members from the redMaPPer catalog and to characterize their IR properties in WISE through the LF. This provides us with a reasonably representative LF to be used in the future cluster detections. WISE and SDSS use different magnitude systems (VEGA and AB, respectively), but this is not going to affect our results because we only use SDSS objects to crossmatch the WISE catalog.\n\nThe paper is organized as follows: in Section \\ref{data} we describe the data, the sample selection and the matching procedure. The construction of the WISE composite luminosity function and the parametrization we use to fit it are presented in Section \\ref{WISE_LF}. In Section \\ref{results} we describe the our results. We discuss their impact and draw our conclusions in Section \\ref{conclusions}. Throughout the paper, we use the following cosmological parameters: $H_{0}=70 \\ {\\rm km s^{-1} Mpc^{-1}}$, $\\Omega_{m0}=0.3$ and $\\Omega_{\\Lambda}=0.7$.\n   \n\n\\section{Data} \\label{data}\n\nThe first step to build the luminosity function is to identify cluster members in WISE. \n\n\\subsection{Cluster samples}\n\nWe start from the redMaPPer DR8 cluster catalog of the SDSS survey (\\url{http:\/\/risa.stanford.edu\/redmapper\/}). It contains $26311$ groups and clusters with redshifts ranging from $0.08$ to $0.6$ detected using a photometric red-sequence cluster finder algorithm \\citep{2016ApJS..224....1R}. Every galaxy is associated a redMaPPer photometric redshift probability $p_{mem}$ \\citep{2014ApJ...785..104R}. The cluster richness $\\lambda$ is defined as the sum of the membership probabilities over all galaxies within a scale-radius. The spectroscopic probability $p_{spec}$ is defined by comparing the membership probability with the measured spectroscopic membership rate \\citep{2015MNRAS.453...38R}. The only other quantities we are taking from the SDSS catalog, in addition to the above mentioned ones, are the celestial coordinates of cluster members. \n\nGiven the redshift distribution of the redMaPPer clusters (see Figure \\ref{rozo_redshift}), we consider nine redshift bins of width $\\Delta z=0.05$, from $z=0.10-0.15$ up to $z=0.50-0.55$. We define a reference cluster sample constituted by $100$ randomly selected massive clusters in each redshift bin by setting the richness limit to $\\lambda > 45$.\n\n\n\\begin{figure}\n\\centering\n\\hbox{\n \\includegraphics[width=0.45\\textwidth]{rozo_redshift.png}\n}\n\\hbox{\n}\n\\caption{Redshift distribution of redMaPPer SDSS clusters.}\n\\label{rozo_redshift}\n\\end{figure}\n\nWe consider also a second sample from which we excluded the most massive clusters by randomly choosing $100$ objects with $45 < \\lambda < 90$, roughly corresponding to a mass range from $2.6$ to $6.4 \\times 10^{14} h^{-1} {\\rm M_{\\odot}}$ \\citep{2017MNRAS.466.3103S,2017arXiv170701907M}. This control sample will allow us to check that our results are not driven by the most massive clusters. In both cluster samples, the average richness increases with redshift, but as expected the effect is less pronounced in sample 2, when we do not consider very rich systems (see Fig. \\ref{richness}).\n\n\n\\begin{figure}\n\\centering\n\\hbox{\n \\includegraphics[width=0.45\\textwidth]{richness.png}\n}\n\\hbox{\n}\n\\caption{Mean (full line) and median (dashed line) richness as a function of redshift for the $\\lambda > 45$ (blue) and $45 < \\lambda < 90$ (green) samples. Error bars mark one standard deviation.}\n\\label{richness}\n\\end{figure}\n\n\n\\subsection{Galaxy members}\n\nFor each cluster in the two samples, we search for the WISE counterparts of the galaxy members. The redMaPPer catalog provides us with the galaxy members in SDSS. For every member in the redMaPPer catalog with spectroscopic probability $p_{spec} > 0.5$ we compute the celestial coordinate distance and search the closest counterpart in the WISE galaxy catalog. Using $p_{spec}$ instead of $p_{mem}$ is a more conservative choice, because there are few members with $p_{mem}>0.5$ and $p_{spec}<0.5$, but not the reverse, so by choosing $p_{spec}>0.5$ we also ensure that $p_{mem}>0.5$. We checked that the cross-match completeness did not change from $p_{spec}>0.9$ to $p_{spec}>0.5$, so we choose this value in order to increase the statistics of the galaxy members in each cluster. We retain as matched members only WISE sources that have a distance to the SDSS source $< 0.5 \\ {\\rm arcsec}$. The point spread function (PSF) of WISE is $6$\", while the PSF of SDSS is $1.4$\". This can generate blending of different SDSS sources into a single WISE source. For this reason, we allow multiple assoctiations of different SDSS sources to a single WISE source, but not the contrary. We stress however that the frequency of these multiple matches is rare, and this should not affect our results. For the spatially matched WISE sources, we consider the magnitude measured with profile-fitting photometry, noted $mpro$  (see \\url{http:\/\/wise2.ipac.caltech.edu\/docs\/release\/allsky\/expsup\/sec2_2a.html#w1mpro}). For our study, we are mainly interested in the magnitudes in the $w1$ band of WISE, corresponding to $3.4 \\ \\mu m$, since it is the most sensitive to star emission from galaxies. All the same, we consider in addition the $w2$ band at $4.6 \\ \\mu m$, which is also sensitive to hot dust. We further set a last constraint to the cross-match between SDSS cluster members and WISE sources. The errors on both $w1$ and $w2$ magnitudes should be greater than $0$. This ensures us that we have a measured magnitude in both bands and not only an upper limit. We do not perform any color selection, but we check {\\it a posteriori} that the contamination by quasar objects is low by verifying that sources with $w1 - w2 > 0.4$ represent only a few percent of the matched members \\citep[cfr.][]{2016A&A...592A..25K}.\n\nFor each cluster from the two samples, we can define a completeness as the ratio between the number of retained matched WISE sources and the initial number of redMaPPer SDSS members (with $p_{spec} > 0.5$). While it is important to bear in mind that this ratio can depend on luminosity, we however describe the behaviour of the obtained ``effective\" completeness. We find that it is a decreasing function of redshift independent of the richness of the sample. Indeed, as can be seen from Fig. \\ref{completeness}, the evolution of the effective completeness with redshift is identical for the two cluster samples: it decreases linearly from approximately $80 \\%$ at $z=0.10$ to $40 \\%$ at $z=0.35$, and then remains constant. So at $z>0.3$ we lose more than half of the initial redMaPPer members, but this fraction is independent of the richness of the sample, which is significantly different at those redshifts. \n\n\n\\begin{figure}\n\\centering\n\\hbox{\n \\includegraphics[width=0.45\\textwidth]{completeness.png}\n}\n\\hbox{\n}\n\\caption{Mean crossmatch completeness as a function of redshift for the $\\lambda > 45$ (blue) and $45 < \\lambda < 90$ (green) samples. Error bars mark one standard deviation.}\n\\label{completeness}\n\\end{figure}\n\n\n\\section{The WISE composite luminosity function}\\label{WISE_LF}\n\nFollowing \\cite{2003MNRAS.342..725D}, we construct the apparent $w1$ and $w2$ composite luminosity function  \\citep{1989MNRAS.237..799C} for the WISE counterparts of SDSS cluster members. The number of galaxies in the $j$-th magnitude bin of the composite LF, $N_{cj}$, is defined as:\n\n\\begin{equation}\nN_{cj}=\\frac{N_{c0}}{m_j}\\sum_{i}\\frac{N_{ij}}{N_{i0}} \\ ,\n\\label{Ncj}\n\\end{equation}\n\n\\noindent where $N_{ij}$ is the number of galaxies in the $j$-th bin of the $i$-th cluster LF, $N_{i0}$ is the normalization used for the $i$-th cluster LF (number of galaxies brighter than $w1_{norm}$), $m_j$ is the number of clusters contributing to the $j$-th bin, and $N_{c0}$ is the sum of all the normalizations. The latter is given by:\n\n\\begin{equation}\nN_{c0}=\\sum_{i}N_{i0} \\ .\n\\end{equation}\n\n\\noindent The errors in $N_{cj}$ are computed according to\n\n\\begin{equation}\n\\delta N_{cj} = \\frac{N_{c0}}{m_j} \\left[ \\sum_{i} \\left( \\frac{\\delta N_{ij}}{N_{i0}} \\right)^2  \\right]^{1\/2} \\ ,\n\\end{equation}\n\n\\noindent where $\\delta N_{cj}$ and $\\delta N_{ij}$ are the errors in the $j$-th LF bin for the composite and $i$-th cluster, respectively. In our case, we assign Poissonian errors to $N_{ij}$, so that $\\delta N_{ij} = \\sqrt{N_{ij}}$ and\n\n\\begin{equation}\n\\delta N_{cj} = \\frac{N_{c0}}{m_j} \\left( \\sum_{i} \\frac{N_{ij}}{N_{i0}^2}  \\right)^{1\/2} \\ .\n\\label{sigma}\n\\end{equation}\n\n\nAs discussed in the introduction, by using WISE data alone we do not have any information on the redshifts of the sources. For this reason, and with the aim of detecting clusters without having any a priori knowledge of their redshift, we consider here apparent magnitudes. We will briefly discuss absolute magnitudes in Appendix \\ref{app_absolute}. For the same reason, we cannot correct the effective completeness for redshift effects as in \\cite{2003MNRAS.342..725D}. In order to compute the number of galaxies in the $j$-th magnitude bin of the composite LF, we consider a magnitude bin width of $0.5 \\ {\\rm mag}$. This is large enough to contain several galaxies within each bin and small enough to sample sufficiently well the LF by allowing a sufficient number of bins. For the normalization, we set $w1_{\\mathrm{norm}} = 17$ and $w2_{\\mathrm{norm}} = 16$. Those values are close to the limiting magnitude of WISE, but the precise choice has not a crucial impact on our final results. For example, the shape of the bright end composite LF for $w1$ does not change for $w1_{\\mathrm{norm}}$ in the range $[16-18]$.\n\nOur objective is not to explore in detail the physical properties of the WISE LF of the redMaPPer SDSS clusters but rather to characterize the composite LF, in particular its shape, in order to apply it as a template for blind detection of clusters in WISE. We therefore fit the normalized composite LF, i.e. the composite LF divided by the number of clusters used to build it, $N_{cj,norm}$. This is similar, in principle, to what is done in \\cite{2015A&A...581A..11M} and leads to a LF which is representative of the average cluster.\nThe WISE composite luminosity function increases with increasing magnitudes, up to a point when it starts decreasing. This feature was already evident in the SDSS composite luminosity function. It reflects the incompleteness of the original data and it is not due to the crossmatching procedure. In Figure \\ref{composite_LF_w1} we show the obtained composite LFs (points with error bars) of the reference sample for the $w1$ band, up to the point where they reach their maximum before decreasing.\n\nWe now fit the obtained composite LF. Different fitting functions have been proposed in the literature for the IR luminosity function [{\\it e.g.} \\cite{2005ApJ...632..169L,2006A&A...445...29P,2006MNRAS.370.1159B,2017arXiv170207829L} and references therein]. We found that the Schechter function was not returning a good fit to our data. For this reason, in our study, we consider a double power law \\citep{2006MNRAS.370.1159B,2010A&A...514A...6G} which is known to provide a good description of the cluster members IR luminosity function: \n\n\\begin{eqnarray}\n\\phi (L) dL &=& \\left( \\frac{\\phi^{*}}{L^{*}} \\right) \\left( \\frac{L}{L^{*}} \\right)^{1-\\alpha} dL \\ \\ \\ \\ \\ \\ \\ (L<L^{*}) \\nonumber \\\\ \n\\phi (L) dL &=& \\left( \\frac{\\phi^{*}}{L^{*}} \\right) \\left( \\frac{L}{L^{*}} \\right)^{1-\\beta} dL \\ \\ \\ \\ \\ \\ \\ (L>L^{*}) \\end{eqnarray}\n\n\\noindent leading to\n\n\\begin{eqnarray} \n\\phi (M) dM &=& 0.4 \\ \\ln 10 \\ \\phi^{*} \\ 10^{0.4 (2-\\alpha) (M^{*}-M)} dM \\ \\ \\ \\ \\ \\ \\ (M>M^{*}) \\nonumber \\\\\n\\phi (M) dM &=& 0.4 \\ \\ln 10 \\ \\phi^{*} \\ 10^{0.4 (2-\\beta) (M^{*}-M)} dM \\ \\ \\ \\ \\ \\ \\ (M<M^{*}) \\ ,\n\\label{double_power_law_m}\n\\end{eqnarray}\n\n\\noindent where $\\alpha$ is the slope of the faint end and $\\beta$ is the slope of the bright end. The other two parameters of the fit are the normalization $\\phi^{*}$ and the transition magnitude between the two power laws $M^{*}$.\n\nIn each redshift bin, we fit the composite LF up to the magnitude where it reaches its maximum, and do not consider data at fainter magnitudes. We evaluate the quality of the fit by computing the reduced $\\chi^2_{\\mathrm{red}}$ of the fit as\n\n\\begin{equation}\n\\chi^2_{\\mathrm{red}} = \\frac{1}{N_{\\mathrm{bin}}-N_{\\mathrm{par}}} \\sum_{N_{\\mathrm{bin}}} {\\frac{({\\mathrm{data}}-{\\mathrm{fit}})^2}{\\sigma^2}} \\ ,\n\\end{equation}\n\n\\noindent where $\\sigma$ is given by equation (\\ref{sigma}) and $N_{par}=4$.\n\nWe noted that the fit is sensitive to the choice of the initial parameters, so we sample the parameter space with an iterative process, starting from a default set of initial values (namely, $\\phi^*_{ini} = max(N_{cj,norm})$, $M^*_{ini} = w(\\phi^*_{ini})$, $\\alpha_{ini} = 2$ and $\\beta_{ini} = 2$). For the first iteration, we use the best-fit parameters as starting values for the new fit. Then, we compare the $\\chi^2_{\\mathrm{red}}$ of the two fits and choose the parameters giving the best $\\chi^2_{\\mathrm{red}}$ as initial parameters for the following iteration. We perform $10^4$ iterations and take the final results as best-fit parameters. We note that in general the fit converges after a number of iterations which is orders of magnitude smaller than the total number of iteration. For every redshift bin, we independently repeat the iteration $10$ times to avoid any bias in the fit.\n\n\n\n\\section{Results} \\label{results}\n\nFigure \\ref{composite_LF_w1} displays the obtained composite LFs for $w1$ magnitudes of our reference sample. It is immediately evident that there is a break in the slope at a given magnitude. For each redshift bin, we provide in Tables \\ref{w1_17_45-500}, \\ref{w1_17_45-90}, \\ref{w2_16_45-500} and \\ref{w2_16_45-90} the best-fit parameters and associated errors for the $w1$ and $w2$ composite luminosity function of the two cluster samples. In the following, we will not discuss the normalization $\\phi^{*}$ but rather focus on the shape parameters of the composite LF.\n\n\\begin{table*}\n\\begin{center}\n\\caption{Best-fit parameters for the double power law, equation (\\ref{double_power_law_m}), fit to the normalized apparent $w1$ composite LF for WISE data (with $w1_{norm} = 17$) for our reference sample ($\\lambda>45$). \\label{w1_17_45-500}}\n\\begin{tabular}{cccccccccc}\n\\hline\n\\hline\n$z$ & $\\phi^{*}$ & $\\sigma_{\\phi^{*}}$ & $\\alpha$ & $\\sigma_{\\alpha}$ & $\\beta$ & $\\sigma_{\\beta}$ & $M^{*}$ & $\\sigma_{M^{*}}$ & $\\chi^2_{red}$ \\\\\n\\hline\n$0.10-0.15$ & $15.01$ & $0.77$ & $2.08$ & $0.08$ & $3.53$ & $0.06$ & $14.42$ & $0.05$ & $2.26$ \\\\\n\\hline\n$0.15-0.20$ & $14.67$ & $0.82$ & $2.17$ & $0.08$ & $3.53$ & $0.06$ & $14.92$ & $0.05$ & $2.22$ \\\\ \n\\hline\n$0.20-0.25$ & $13.39$ & $0.76$ & $2.45$ & $0.09$ & $3.49$ & $0.09$ & $15.17$ & $0.08$ & $2.36$ \\\\ \n\\hline\n$0.25-0.30$ & $9.51$ & $1.40$ & $2.89$ & $0.10$ & $3.57$ & $0.14$ & $15.16$ & $0.15$ & $0.95$ \\\\ \n\\hline\n$0.30-0.35$ & $12.67$ & $0.57$ & $2.02$ & $0.09$ & $3.52$ & $0.09$ & $15.54$ & $0.05$ & $0.83$ \\\\ \n\\hline\n$0.35-0.40$ & $10.79$ & $0.64$ & $2.33$ & $0.09$ & $3.76$ & $0.12$ & $15.55$ & $0.06$ & $1.54$ \\\\ \n\\hline\n$0.40-0.45$ & $9.09$ & $0.64$ & $2.46$ & $0.11$ & $3.56$ & $0.17$ & $15.73$ & $0.11$ & $0.62$ \\\\ \n\\hline\n$0.45-0.50$ & $7.38$ & $--$ & $2.68$ & $--$ & $3.50$ & $0.10$ & $15.75$ & $--$ & $0.84$ \\\\ \n\\hline\n$0.50-0.55$ & $6.33$ & $--$ & $2.70$ & $--$ & $3.52$ & $0.11$ & $15.75$ & $--$ & $2.47$ \\\\ \n\\hline\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table*}\n\n\\begin{table*}\n\\begin{center}\n\\caption{Best-fit parameters for the double power law, equation (\\ref{double_power_law_m}), fit to the normalized apparent $w1$ composite LF for WISE data (with $w1_{norm} = 17$) for our control sample ($45<\\lambda<90$). \\label{w1_17_45-90}}\n\\begin{tabular}{cccccccccc}\n\\hline\n\\hline\n$z$ & $\\phi^{*}$ & $\\sigma_{\\phi^{*}}$ & $\\alpha$ & $\\sigma_{\\alpha}$ & $\\beta$ & $\\sigma_{\\beta}$ & $M^{*}$ & $\\sigma_{M^{*}}$ & $\\chi^2_{red}$ \\\\\n\\hline\n$0.10-0.15$ & $13.49$ & $0.76$ & $2.13$ & $0.09$ & $3.50$ & $0.06$ & $14.41$ & $0.05$ & $1.67$ \\\\\n\\hline\n$0.15-0.20$ & $13.13$ & $0.74$ & $2.18$ & $0.09$ & $3.46$ & $0.06$ & $14.95$ & $0.05$ & $1.50$ \\\\\n\\hline\n$0.20-0.25$ & $11.40$ & $0.70$ & $2.45$ & $0.09$ & $3.36$ & $0.10$ & $15.22$ & $0.10$ & $2.12$ \\\\\n\\hline\n$0.25-0.30$ & $6.78$ & $1.06$ & $2.88$ & $0.11$ & $3.60$ & $0.15$ & $15.02$ & $0.14$ & $1.96$ \\\\\n\\hline\n$0.30-0.35$ & $10.31$ & $0.53$ & $2.02$ & $0.10$ & $3.52$ & $0.10$ & $15.52$ & $0.06$ & $0.67$ \\\\\n\\hline\n$0.35-0.40$ & $7.71$ & $0.46$ & $2.17$ & $0.12$ & $3.39$ & $0.13$ & $15.59$ & $0.08$ & $0.36$ \\\\\n\\hline\n$0.40-0.45$ & $6.68$ & $--$ & $2.25$ & $--$ & $3.44$ & $0.09$ & $15.75$ & $--$ & $1.57$\\\\ \n\\hline\n$0.45-0.50$ & $5.15$ & $--$ & $2.43$ & $--$ & $3.25$ & $0.10$ & $15.75$ & $--$ & $0.26$ \\\\ \n\\hline\n$0.50-0.55$ & $3.90$ & $--$ & $2.63$ & $--$ & $3.17$ & $0.14$ & $15.75$ & $--$ & $2.13$ \\\\ \n\\hline\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table*}\n\n\\begin{table*}\n\\begin{center}\n\\caption{Best-fit parameters for the double power law, equation (\\ref{double_power_law_m}), fit to the normalized apparent $w2$ composite LF for WISE data (with $w2_{norm} = 16$) for our reference sample ($\\lambda>45$). \\label{w2_16_45-500}}\n\\begin{tabular}{cccccccccc}\n\\hline\n\\hline\n$z$ & $\\phi^{*}$ & $\\sigma_{\\phi^{*}}$ & $\\alpha$ & $\\sigma_{\\alpha}$ & $\\beta$ & $\\sigma_{\\beta}$ & $M^{*}$ & $\\sigma_{M^{*}}$ & $\\chi^2_{red}$ \\\\\n\\hline\n$0.10-0.15$ & $12.79$ & $0.64$ & $2.26$ & $0.09$ & $3.65$ & $0.09$ & $14.07$ & $0.06$ & $0.78$ \\\\\n\\hline\n$0.15-0.20$ & $11.48$ & $0.82$ & $2.40$ & $0.09$ & $3.61$ & $0.09$ & $14.43$ & $0.06$ & $1.48$ \\\\ \n\\hline\n$0.20-0.25$ & $14.74$ & $0.68$ & $2.04$ & $0.09$ & $3.45$ & $0.07$ & $15.00$ & $0.05$ & $0.09$ \\\\ \n\\hline\n$0.25-0.30$ & $12.28$ & $0.58$ & $2.16$ & $0.09$ & $3.58$ & $0.09$ & $15.08$ & $0.06$ & $0.86$ \\\\ \n\\hline\n$0.30-0.35$ & $9.59$ & $0.67$ & $2.44$ & $0.10$ & $3.46$ & $0.13$ & $15.14$ & $0.10$ & $1.15$ \\\\ \n\\hline\n$0.35-0.40$ & $8.25$ & $0.87$ & $2.65$ & $0.11$ & $3.58$ & $0.18$ & $15.23$ & $0.14$ & $1.05$ \\\\ \n\\hline\n$0.40-0.45$ & $9.30$ & $0.47$ & $2.11$ & $0.11$ & $3.49$ & $0.12$ & $15.62$ & $0.07$ & $0.25$ \\\\ \n\\hline\n$0.45-0.50$ & $7.78$ & $0.44$ & $2.25$ & $0.12$ & $3.40$ & $0.17$ & $15.73$ & $0.12$ & $1.03$ \\\\\n\\hline\n$0.50-0.55$ & $6.34$ & $--$ & $2.63$ & $--$ & $3.34$ & $0.10$ & $15.75$ & $--$ & $1.45$ \\\\ \n\\hline\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table*}\n\n\n\\begin{table*}\n\\begin{center}\n\\caption{Best-fit parameters for the double power law, equation (\\ref{double_power_law_m}), fit to the normalized apparent $w1$ composite LF for WISE data (with $w2_{norm} = 16$) for our control sample ($\\lambda>45$). \\label{w2_16_45-90}}\n\\begin{tabular}{cccccccccc}\n\\hline\n\\hline\n$z$ & $\\phi^{*}$ & $\\sigma_{\\phi^{*}}$ & $\\alpha$ & $\\sigma_{\\alpha}$ & $\\beta$ & $\\sigma_{\\beta}$ & $M^{*}$ & $\\sigma_{M^{*}}$ & $\\chi^2_{red}$ \\\\\n\\hline\n$0.10-0.15$ & $11.57$ & $0.62$ & $2.28$ & $0.09$ & $3.60$ & $0.10$ & $14.06$ & $0.06$ & $0.68$ \\\\\n\\hline\n$0.15-0.20$ & $10.22$ & $0.77$ & $2.42$ & $0.09$ & $3.53$ & $0.09$ & $14.44$ & $0.07$ & $0.86$ \\\\\n\\hline\n$0.20-0.25$ & $12.13$ & $0.61$ & $2.03$ & $0.09$ & $3.37$ & $0.07$ & $14.98$ & $0.05$ & $1.19$ \\\\\n\\hline\n$0.25-0.30$ & $9.49$ & $0.54$ & $2.19$ & $0.10$ & $3.46$ & $0.10$ & $15.06$ & $0.07$ & $0.61$ \\\\\n\\hline\n$0.30-0.35$ & $8.24$ & $0.52$ & $2.33$ & $0.11$ & $3.42$ & $0.14$ & $15.17$ & $0.10$ & $1.31$ \\\\\n\\hline\n$0.35-0.40$ & $5.96$ & $--$ & $2.51$ & $--$ & $3.27$ & $0.09$ & $15.25$ & $--$ & $0.40$ \\\\\n\\hline\n$0.40-0.45$ & $6.53$ & $0.41$ & $2.09$ & $0.13$ & $3.30$ & $0.14$ & $15.60$ & $0.09$ & $0.82$ \\\\ \n\\hline\n$0.45-0.50$ & $5.41$ & $--$ & $2.14$ & $--$ & $3.12$ & $0.09$ & $15.75$ & $--$ & $0.32$ \\\\ \n\\hline\n$0.50-0.55$ & $4.13$ & $--$ & $2.42$ & $--$ & $3.01$ & $0.11$ & $15.75$ & $--$ & $0.98$ \\\\ \n\\hline\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table*}\n\n\n\\begin{figure}\n\\centering\n\\hbox{\n \\includegraphics[width=0.45\\textwidth]{composite_LF_w1_log.png}\n}\n\\hbox{\n}\n\\caption{Normalized composite luminosity function (points with error bars) $N_{cj,norm}$, as defined in Section \\ref{WISE_LF}, in the nine redshift bins, for our reference sample, $\\lambda > 45$, in the $w1$ band. Full lines show the best-fit of the double power law formula, equation (\\ref{double_power_law_m}), to the data.}\n\\label{composite_LF_w1}\n\\end{figure}\n\n\nWe first focus on the apparent magnitude $M^{*}$ marking the transition between the two regimes of the double power-law composite LF. For our reference cluster sample, we note that the best-fit value of $M^{*}$ for the $w2$ LF are 0.5 magnitudes brighter than the best-fit values for the $w1$ LF. We also note that the best-fit values of $M^{*}$ for the $w1$ LF are the same, within the error bars, when considering the reference sample ($\\lambda > 45$) and the sample without the richest clusters ($45 < \\lambda < 90$), and the same is true for $w2$. This indicates that $M^{*}$ is no biased by the most massive clusters. Finally as can be seen from Fig. \\ref{w_app_star}, the best-fit $M^{*}$ increases with redshift, with high-redshift clusters showing the transition between the two LF regimes one magnitude fainter than low-redshift clusters. We note here, but discuss in Appendix \\ref{app_absolute}, that this is not just a redshift effect.\n\nWe now turn to the slopes of the composite LF. The obtained best-fits values are summarized in Tables \\ref{w1_17_45-500}, \\ref{w1_17_45-90}, \\ref{w2_16_45-500} and \\ref{w2_16_45-90} and displayed in Fig. \\ref{slopes}. The left panel shows the results for $\\beta$, i.e. the slope of the power law in the bright end of the composite LF up to $M^{*}$. Let us start by considering the reference sample in the $w1$ band. The values for $\\beta$ range between $3.49$ and $3.76$. If we fit the redshift evolution of $\\beta$ with a constant [$\\beta(z) = \\beta_0$] we get a value of $3.54$, while if we fit it with a linear relation [$\\beta(z) = \\beta_0 + s z$] we get a result compatible with no evolution, and the two $\\chi^2$ are similar, as summarized in Table \\ref{beta_evolution}. So we can conclude that the bright end slope of the composite LF of our reference sample is constant with redshift. If we now move to the second sample, the one that does not contain the richest clusters, we see that, while the fit with a constant still provides an acceptable $\\chi^2$, the linear relation provides a better fit, with a negative slope incompatible with no redshift evolution. A similar result is found when considering the two samples in the $w2$ band. In both samples, the evolution with redshift is stronger for $w2$ than for $w1$. Most importantly, the value of $\\beta$ for the cluster composite LF significantly differs from the average value for the field. We checked this by evaluating the field composite LF for $1 \\ {\\rm deg}^2$ patches of the sky in the same regions where the clusters of the reference sample are located, and by repeating the procedure in other $10$ random patches of the sky of the same size, for a total of $910$ field patches. We do this simply by evaluating the composite LF of all the sources in the field satisfying the selection criteria we used for the cluster members. The slope of the power law for the field varies within $2.8$ and $3.1$, with no significant scatter and no evolution with redshift. This result clearly shows that the cluster LF of our reference sample shows a characteristic bright end in the WISE $w1$ band, steeper than the one of the field, with a constant slope of $3.5$. Even when considering the second sample of clusters, despite the fact that the redshift evolution of $\\beta$ is better described by a decreasing linear function, the bright-end slope is incompatible with the one of the field.\n\n\n\\begin{table*}\n\\begin{center}\n\\caption{Best-fit parameters for the constant and linear function fit to the evolution of the bright-end slope $\\beta$ with redshift. \\label{beta_evolution}}\n\\begin{tabular}{cccccccccccc}\n\\hline\n\\hline\nBand & $\\lambda$ & & \\multicolumn{3}{c}{Constant} & & \\multicolumn{5}{c}{Linear} \\\\\n\\multicolumn{2}{c}{} & & $\\beta_{0}$ & $\\sigma_{\\beta_{0}}$ & $\\chi^2$ & & $\\beta_{0}$ & $\\sigma_{\\beta_{0}}$ & $s$ & $\\sigma_s$ & $\\chi^2$ \\\\\n\\hline\n$w1$ & $\\lambda > 45$ & & $3.54$ & $0.03$ & $0.50$ & & $3.52$ & $0.07$ & $+0.06$ & $0.22$ & $0.56$ \\\\\n\\hline\n$w1$ & $45 < \\lambda < 90$ & & $3.43$ & $0.03$ & $1.37$ & & $3.56$ & $0.07$ & $-0.49$ & $0.22$ & $0.87$ \\\\\n\\hline\n$w2$ & $\\lambda > 45$ & & $3.51$ & $0.03$ & $1.08$ & & $3.68$ & $0.08$ & $-0.59$ & $0.27$ & $0.52$ \\\\\n\\hline\n$w2$ & $45 < \\lambda < 90$ & & $3.34$ & $0.03$ & $3.60$ & & $3.74$ & $0.08$ & $-1.28$ & $0.25$ & $0.47$ \\\\\n\\hline\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table*}\n\n\nThe right panel of Fig. \\ref{slopes} shows the best-fit values for the faint-end slope, $\\alpha$, of the composite LF. The values for $\\alpha$ range from $2$ to $3$ and are significantly different from the ones for $\\beta$, indicating that there is indeed a transition in the shape of the LF for cluster members, that justify the use of the double power law equation \\ref{double_power_law_m}. This is not the case for the field, where the value for $\\alpha$ is similar to $\\beta$, around $3$, meaning that the field is well described by a single power law. Even if it is difficult to find a trend with redshift of $\\alpha$ (but we note here that the value is similar for the two samples, indicating that the average richness has no major impact on these results, and is different for $w1$ and $w2$), it is the change itself in the slope that can be used as another indicator to identify cluster members in WISE. Indeed, in addition to the different bright-end slope, the composite LF for cluster members also has a different shape with respect to the one for the field.\n\n\n\\begin{figure}\n\\centering\n\\hbox{\n \\includegraphics[width=0.45\\textwidth]{w_app_star.png}\n}\n\\hbox{\n}\n\\caption{Evolution with redshift of the best-fit apparent $M^{*}$ for $w1$ ($\\lambda>45$) [blue dashed-line], $w1$ ($45<\\lambda<90$) [green dashed-line], $w2$ ($\\lambda>45$) [red dashed-line] and $w2$ ($45<\\lambda<90$) [magenta dashed-line]. Error bars mark one standard deviation.}\n\\label{w_app_star}\n\\end{figure}\n\n\\begin{figure*}\n\\centering\n\\hbox{\n \\includegraphics[width=0.45\\textwidth]{beta.png}\n \\includegraphics[width=0.45\\textwidth]{alpha.png}\n}\n\\hbox{\n}\n\\caption{Evolution with redshift of the best-fit bright $\\beta$ (left panel) and faint $\\alpha$ (right panel) end slopes for $w1$ ($\\lambda>45$) [blue dashed-lines], $w1$ ($45<\\lambda<90$) [green dashed-lines], $w2$ ($\\lambda>45$) [red dashed-lines] and $w2$ ($45<\\lambda<90$) [magenta dashed-line]. The black horizontal lines in the left panel mark the best-fit value of $\\beta$ for a $1 \\ {\\rm deg}^2$ field centered on the same clusters and for a random field of the same size. Error bars mark one standard deviation. The full lines in the left panel show the best-fit of the linear evolution of $\\beta$ with redshift in the three cases.}\n\\label{slopes}\n\\end{figure*}\n\n\n\n\\section{Summary and Conclusions} \\label{conclusions}\n\nIn this work, we build an IR composite luminosity function for the apparent magnitude of cluster members using WISE data. We identify the cluster members by matching the position of the WISE sources with the cluster members in the redMaPPer DR8 SDSS catalog. We consider $9$ redshift bins containing $100$ clusters each, in two different samples. The reference sample contains clusters with $\\lambda > 45$, while the control sample contains clusters with $45 < \\lambda < 90$. For the two samples, we consider both $w1$ and $w2$ apparent magnitudes. We fit the normalized composite LF with a double power law and provide best-fit parameters for the transition magnitude $M^{*}$ and the bright and faint-end slopes $\\beta$ and $\\alpha$, respectively. From our analysis we can draw the following conclusions:\n\n\\begin{enumerate}\n\\item The composite LF for cluster members in WISE is characterized by a value of the bright-end slope $\\beta$ which is significantly steeper than the one for the field. For clusters in the reference sample, $\\beta$ in the $w1$ band ranges from $3.49$ to $3.76$ at different redshifts, and it is compatible with a constant value of $3.54$. For clusters in the control sample, $\\beta$ shows a mild negative trend with increasing redshift, with values ranging from $3.17$ to $3.60$. On the other end, the field is described by a slope between $2.8$ and $3.1$, which is significantly different from the one of clusters, independently on the richness of the sample. This is a clear signature that can be used to identify cluster members in WISE data.\n \\item The faint end slope $\\alpha$ of the members composite LF has a value between $2.0$ and $3.0$, remarkably different from $\\beta$. This is an indication of the existence of a transition between the bright end and the faint end that is not present in the field, where the value for $\\alpha$ is similar to $\\beta$. The trend with redshift is less clear than in the bright end, but the existence of this transition itself can be used as a secondary characterization when searching for clusters in the WISE data. \n\\item The transition apparent magnitude $M^{*}$ is increasing (shifting towards fainter luminosities) with increasing redshift. This means that, even if the global shape of the composite LF (and in particular the bright end) is the same at all redshifts, there is a shift in the transition magnitude. This is not just a redshift effect, as we discuss in Appendix \\ref{app_absolute}. This fact can be used as an a posteriori additional information to constrain the redshift of clusters detected with WISE data.\n\\end{enumerate}\n      \n\nWith the composite LF discussed in this work, we provide an ingredient for detecting clusters in the WISE survey with matched-filter algorithms that will be implemented in subsequent studies.\n   \n\n\\begin{acknowledgements}\nThe authors thank Alexandre Beelen, Herv\\'e Dole and Guillaume Hurier for fruitful discussion.\nC.D.B. also thanks Carlotta Gruppioni, Elena Zucca and Micol Bolzonella for useful discussion on infrared astronomy.\nThis work was supported by a grant from R\\'egion Ile-de-France DIM-ACAV. This research made use of the SZ-Cluster Database operated by the Integrated Data and Operation Center (IDOC) at the Institut d'Astrophysique Spatiale (IAS) under contract with CNES and CNRS. It also used data products from the Wide-field Infrared Survey Explorer, which is a joint project of the University of California, Los Angeles, and the Jet Propulsion Laboratory\/California Institute of Technology, funded by the National Aeronautics and Space Administration.  This research made use of Astropy, a community-developed core Python package for Astronomy \\citep{2013A&A...558A..33A}, and Matplotlib library \\citep{Hunter:2007}.\n\n\\end{acknowledgements}\n\n\n\n\\bibliographystyle{aa}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Supplementary Lemmas for the proof of Theorem \\ref{thm:regression}}\n\\label{sec:supp_B}\n\\subsection{Proof of Lemma \\ref{bandwidth}}\n\\begin{proof}\nFirst we establish the fact that $\\theta_0^s \\to \\theta_0$. Note that for all $n$, we have: \n$$\n\\mathbb{M}^s(\\theta_0^s) \\le \\mathbb{M}^s(\\theta_0) \n$$\nTaking $\\limsup$ on the both side we have: \n$$\n\\limsup_{n \\to \\infty} \\mathbb{M}^s(\\theta_0^s) \\le \\mathbb{M}(\\theta_0) \\,.\n$$\nNow using Lemme \\ref{lem:uniform_smooth} we have: \n$$\n\\limsup_{n \\to \\infty} \\mathbb{M}^s(\\theta_0^s) = \\limsup_{n \\to \\infty} \\left[\\mathbb{M}^s(\\theta_0^s) - \\mathbb{M}(\\theta_0^s) + \\mathbb{M}(\\theta_0^s)\\right] = \\limsup_{n \\to \\infty} \\mathbb{M}(\\theta_0^s) \\,.\n$$\nwhich implies $\\limsup_{n \\to \\infty} \\mathbb{M}(\\theta_0^s) \\le \\mathbb{M}(\\theta_0)$ and from the continuity of $\\mathbb{M}(\\theta)$ and $\\theta_0$ being its unique minimizer, we conclude the proof. Now, using Lemma \\ref{lem:pop_curv_nonsmooth} and Lemma \\ref{lem:uniform_smooth} we further obtain: \n\\begin{align}\n   u_- d^2(\\theta_0^s, \\theta_0) & \\le \\mathbb{M}(\\theta_0^s) - \\mathbb{M}(\\theta_0) \\notag \\\\\n    & = \\mathbb{M}(\\theta_0^s) - \\mathbb{M}^s(\\theta^s_0) + \\underset{\\le 0}{\\underline{\\mathbb{M}^s(\\theta_0^s) - \\mathbb{M}^s(\\theta_0)}} + \\mathbb{M}^s(\\theta_0) - \\mathbb{M}(\\theta_0) \\notag \\\\\n    \\label{eq:est_dist_bound} & \\le \\sup_{\\theta \\in \\Theta}\\left|\\mathbb{M}^s(\\theta) - \\mathbb{M}(\\theta)\\right| \\le K_1 \\sigma_n \\,. \n\\end{align}\nNote that we neeed consistency of $\\theta_0^s$ here as the lower bound in Lemma \\ref{lem:pop_curv_nonsmooth} is only valid in a neighborhood around $\\theta_0$. As $\\theta_0^s$ is the minimizer of $\\mathbb{M}^s(\\theta)$, from the first order condition we have: \n\\begin{align}\n    \\label{eq:beta_grad}\\nabla_{\\beta}\\mathbb{M}^s_n(\\theta_0^s) & = -2\\mathbb{E}\\left[X(Y - X^{\\top}\\beta_0^s)\\right] + 2\\mathbb{E} \\left\\{\\left[X_iX_i^{\\top}\\delta_0^s\\right] K\\left(\\frac{Q_i^{\\top}\\psi_0^s}{\\sigma_n}\\right)\\right\\}  = 0 \\\\\n    \\label{eq:delta_grad}\\nabla_{\\delta}\\mathbb{M}^s_n(\\theta_0^s) & = \\mathbb{E} \\left\\{\\left[-2X_i\\left(Y_i - X_i^{\\top}\\beta_0^s\\right) + 2X_iX_i^{\\top}\\delta_0^s\\right] K\\left(\\frac{Q_i^{\\top}\\psi_0^s}{\\sigma_n}\\right)\\right\\} = 0\\\\\n    \\label{eq:psi_grad}\\nabla_{\\psi}\\mathbb{M}^s_n(\\theta_0^s) & = \\frac{1}{\\sigma_n}\\mathbb{E} \\left\\{\\left[-2\\left(Y_i - X_i^{\\top}\\beta_0^s\\right)X_i^{\\top}\\delta_0^s + (X_i^{\\top}\\delta_0^s)^2\\right]\\tilde Q_i K'\\left(\\frac{Q_i^{\\top}\\psi_0^s}{\\sigma_n}\\right)\\right\\} = 0\n\\end{align}\nWe first show that $(\\tilde \\psi^s_0 - \\tilde \\psi_0)\/\\sigma_n \\to 0$ by \\emph{reductio ab absurdum}. From equation \\eqref{eq:est_dist_bound}, we know $\\|\\psi_0^s - \\psi_0\\|\/\\sigma_n = O(1)$. Hence it has a convergent subsequent $\\psi^s_{0, n_k}$, where $(\\tilde \\psi^s_{0, n_k} - \\tilde \\psi_0)\/\\sigma_n \\to h$. If we can prove that $h = 0$, then we establish every subsequence of $\\|\\psi_0^s - \\psi_0\\|\/\\sigma_n$ has a further subsequence which converges to $0$ which further implies $\\|\\psi_0^s - \\psi_0\\|\/\\sigma_n$ converges to $0$. To save some notations, we prove that if $(\\psi_0^s - \\psi_0)\/\\sigma_n \\to h$ then $h = 0$. We start with equation \\eqref{eq:psi_grad}. Define $\\tilde \\eta = (\\tilde \\psi^s_0 - \\tilde \\psi_0)\/\\sigma_n = (\\psi_0^s - \\psi_0)\/\\sigma_n$ where $\\tilde \\psi$ is all the co-ordinates of $\\psi$ except the first one, as the first co-ordinate of $\\psi$ is always assumed to be $1$ for identifiability purpose. \n\\allowdisplaybreaks\n\\begin{align}\n    0 & =  \\frac{1}{\\sigma_n}\\mathbb{E} \\left\\{\\left[-2\\left(Y_i - X_i^{\\top}\\beta_0^s\\right)X_i^{\\top}\\delta_0^s + (X_i^{\\top}\\delta_0^s)^2\\right]\\tilde Q_i K'\\left(\\frac{Q_i^{\\top}\\psi_0^s}{\\sigma_n}\\right)\\right\\}  \\notag \\\\\n    & =  \\frac{1}{\\sigma_n}\\mathbb{E}\\left[\\left( -2\\delta_0^s XX^{\\top}(\\beta_0 - \\beta^s_0) -2\\delta_0^s XX^{\\top}\\delta_0\\mathds{1}_{Q^{\\top}\\delta_0 > 0} + (X_i^{\\top}\\delta_0^s)^2\\right)\\tilde QK'\\left(\\frac{Q^{\\top}\\psi^s_0}{\\sigma_n}\\right)\\right] \\notag \\\\\n     & =  \\frac{1}{\\sigma_n}\\mathbb{E}\\left[\\left( -2\\delta_0^s XX^{\\top}(\\beta_0 - \\beta^s_0) -2\\delta_0^s XX^{\\top}(\\delta_0 - \\delta_0^s)\n     \\mathds{1}_{Q^{\\top}\\delta_0 > 0} \\right. \\right. \\notag \\\\\n     & \\hspace{10em} \\left. \\left. + (X_i^{\\top}\\delta_0^s)^2\\left(1 - 2\\mathds{1}_{Q^{\\top}\\delta_0 > 0}\\right)\\right)\\tilde QK'\\left(\\frac{Q^{\\top}\\psi^s_0}{\\sigma_n}\\right)\\right] \\notag \\\\\n    & =  \\frac{-2}{\\sigma_n}\\mathbb{E}\\left[\\left(\\delta_0^{s^{\\top}} g(Q)(\\beta_0 - \\beta^s_0)\\right)\\tilde QK'\\left(\\frac{Q^{\\top}\\psi^s_0}{\\sigma_n}\\right)\\right] \\notag \\\\\n    & \\qquad \\qquad \\qquad -  \\frac{2}{\\sigma_n} \\mathbb{E}\\left[\\left(\\delta_0^{s^{\\top}}g(Q)(\\delta_0 - \\delta^s_0)\\right)\\tilde QK'\\left(\\frac{Q^{\\top}\\psi^s_0}{\\sigma_n}\\right)\\mathds{1}_{Q^{\\top}\\delta_0 > 0}\\right] \\notag \\\\\n    & \\hspace{15em} +   \\frac{1}{\\sigma_n}\\mathbb{E}\\left[\\left(\\delta_0^{s^{\\top}}g(Q)\\delta^s_0\\right)\\tilde QK'\\left(\\frac{Q^{\\top}\\psi^s_0}{\\sigma_n}\\right)\\left(1 - 2\\mathds{1}_{Q^{\\top}\\delta_0 > 0}\\right)\\right] \\notag \\\\\n     & = -\\underbrace{\\frac{2}{\\sigma_n}\\mathbb{E}\\left[\\left(\\delta_0^{s^{\\top}} g(Q)(\\beta_0 - \\beta^s_0)\\right)\\tilde QK'\\left(\\frac{Q^{\\top}\\psi^s_0}{\\sigma_n}\\right)\\right]}_{T_1} \\notag \\\\\n     & \\qquad \\qquad -\\underbrace{\\frac{2}{\\sigma_n} \\mathbb{E}\\left[\\left(\\delta_0^{s^{\\top}}g(Q)(\\delta_0 - \\delta^s_0)\\right)\\tilde QK'\\left(\\frac{Q^{\\top}\\psi^s_0}{\\sigma_n}\\right)\\mathds{1}_{Q^{\\top}\\delta_0 > 0}\\right]}_{T_2} \\notag \\\\\n     \\label{eq:pop_est_conv_1} & \\qquad \\qquad \\qquad  + \\underbrace{\\frac{1}{\\sigma_n}\\mathbb{E}\\left[\\left(\\delta_0{\\top}g(Q)\\delta_0\\right)\\tilde QK'\\left(\\frac{Q^{\\top}\\psi^s_0}{\\sigma_n}\\right)\\left(1 - 2\\mathds{1}_{Q^{\\top}\\delta_0 > 0}\\right)\\right]}_{T_3} \\notag \\\\\n     & \\qquad \\qquad \\qquad \\qquad  + \\underbrace{\\frac{2}{\\sigma_n}\\mathbb{E}\\left[\\left((\\delta_0 - \\delta_0^s)^{\\top}g(Q)\\delta_0\\right)\\tilde QK'\\left(\\frac{Q^{\\top}\\psi^s_0}{\\sigma_n}\\right)\\left(1 - 2\\mathds{1}_{Q^{\\top}\\delta_0 > 0}\\right)\\right]}_{T_4} \\notag \\\\\n & = T_1 + T_2 + T_3 + T_4\n \\end{align}\nAs mentioned earlier, there is a bijection between $(Q_1, \\tilde Q)$ and $(Q^{\\top}\\psi_0, \\tilde Q)$. The map of one side is obvious. The other side is also trivial as the first coordinate of $\\psi_0$ is 1, which makes $Q^{\\top}\\psi_0 = Q_1 + \\tilde Q^{\\top}\\tilde \\psi_0$:  \n$$\n(Q^{\\top}\\psi_0, \\tilde Q) \\mapsto (Q^{\\top}\\psi_0 - \\tilde Q^{\\top}\\tilde \\psi_0, \\tilde Q) \\,.\n$$\nWe first show that $T_1, T_2$ and $T_4$ are $o(1)$. Towards that end first note that: \n\\begin{align*}\n|T_1| & \\le \\frac{2}{\\sigma_n}\\mathbb{E}\\left[\\|g(Q)\\|_{op} \\ \\|\\tilde Q\\| \\ \\left|K'\\left(\\frac{Q^{\\top}\\psi^s_0}{\\sigma_n}\\right)\\right|\\right]\\|\\delta_0^s\\|\\|\\beta_0 - \\beta_0^s\\| \\\\\n|T_2| & \\le \\frac{2}{\\sigma_n} \\mathbb{E}\\left[\\|g(Q)\\|_{op} \\ \\|\\tilde Q\\| \\ \\left|K'\\left(\\frac{Q^{\\top}\\psi^s_0}{\\sigma_n}\\right)\\right|\\right]\\|\\delta_0^s\\|\\|\\delta_0 - \\delta_0^s\\| \\\\\n|T_4| & \\le \\frac{2}{\\sigma_n} \\mathbb{E}\\left[\\|g(Q)\\|_{op} \\ \\|\\tilde Q\\| \\ \\left|K'\\left(\\frac{Q^{\\top}\\psi^s_0}{\\sigma_n}\\right)\\right|\\right]\\|\\delta_0^s\\|\\|\\delta_0 - \\delta_0^s\\|\n\\end{align*}\nFrom the above bounds, it is immediate that to show that above terms are $o(1)$ all we need to show is: \n$$\n \\frac{1}{\\sigma_n}\\mathbb{E}\\left[\\|g(Q)\\|_{op} \\ \\|\\tilde Q\\| \\ \\left|K'\\left(\\frac{Q^{\\top}\\psi^s_0}{\\sigma_n}\\right)\\right|\\right] = O(1) \\,.\n$$\nTowards that direction, define $\\eta = (\\tilde \\psi_0^s - \\tilde \\psi_0)\/\\sigma_n$: \n\\begin{align*}\n& \\frac{1}{\\sigma_n}\\mathbb{E}\\left[\\|g(Q)\\|_{op} \\ \\|\\tilde Q\\| \\ \\left|K'\\left(\\frac{Q^{\\top}\\psi^s_0}{\\sigma_n}\\right)\\right|\\right] \\\\\n& \\le c_+  \\frac{1}{\\sigma_n}\\mathbb{E}\\left[\\|\\tilde Q\\| \\ \\left|K'\\left(\\frac{Q^{\\top}\\psi^s_0}{\\sigma_n}\\right)\\right|\\right]  \\\\\n& = c_+ \\frac{1}{\\sigma_n}\\int \\int \\|\\tilde q\\| \\left|K'\\left(\\frac{t}{\\sigma_n} + \\tilde q^{\\top}\\eta \\right)\\right| f_0\\left(t \\mid \\tilde q\\right) f(\\tilde q) \\ dt \\ d\\tilde q \\\\\n& = c_+ \\int \\int \\|\\tilde q\\| \\left|K'\\left(t + \\tilde q^{\\top}\\eta \\right)\\right| f_0\\left(\\sigma_n t \\mid \\tilde q\\right) f(\\tilde q) \\ dt \\ d\\tilde q \\\\\n& =  c_+ \\int  \\|\\tilde q\\| f_0\\left(0 \\mid \\tilde q\\right) \\int \\left|K'\\left(t + \\tilde q^{\\top}\\eta \\right)\\right| \\ dt  \\ f(\\tilde q) \\ d\\tilde q + R_1 \\\\\n& = c_+ \\int \\left|K'\\left(t\\right)\\right| dt \\ \\mathbb{E}\\left[\\|\\tilde Q\\| f_0(0 \\mid \\tilde Q)\\right] + R_1 = O(1) + R_1 \\,.\n\\end{align*}\nTherefore, all it remains to show is $R_1$ is also $O(1)$ (or of smaller order): \n\\begin{align*}\n|R_1| & = \\left|c_+ \\int \\int \\|\\tilde q\\| \\left|K'\\left(t + \\tilde q^{\\top}\\eta \\right)\\right| \\left(f_0\\left(\\sigma_n t \\mid \\tilde q\\right) - f_0(0 \\mid \\tilde q) \\right)f(\\tilde q) \\ dt \\ d\\tilde q\\right| \\\\\n& \\le c_+ F_+ \\sigma_n \\int \\|\\tilde q\\| \\int_{-\\infty}^{\\infty} |t|\\left|K'\\left(t + \\tilde q^{\\top}\\eta \\right)\\right| \\ dt \\ f(\\tilde q) \\ d\\tilde q \\\\\n& = c_+ F_+ \\sigma_n \\int \\|\\tilde q\\| \\int_{-\\infty}^{\\infty} |t - q^{\\top}\\eta|\\left|K'\\left(t\\right)\\right| \\ dt \\ f(\\tilde q) \\ d\\tilde q \\\\\n& \\le c_+ F_+ \\sigma_n \\left[\\int \\|\\tilde q\\| \\int_{-\\infty}^{\\infty} |t|\\left|K'\\left(t\\right)\\right| \\ dt \\ f(\\tilde q) \\ d\\tilde q  + \\int \\|\\tilde q\\|^2\\|\\eta\\| \\int_{-\\infty}^{\\infty}\\left|K'\\left(t\\right)\\right| \\ dt \\ f(\\tilde q) \\ d\\tilde q\\right] \\\\\n& = c_+ F_+ \\sigma_n \\left[\\left(\\int_{-\\infty}^{\\infty} |t|\\left|K'\\left(t\\right)\\right| \\ dt\\right) \\times \\mathbb{E}[\\|\\tilde Q\\|] +  \\left(\\int_{-\\infty}^{\\infty}\\left|K'\\left(t\\right)\\right| \\ dt\\right) \\times \\|\\eta\\| \\ \\mathbb{E}[\\|\\tilde Q\\|^2]\\right] \\\\\n& = O(\\sigma_n) = o(1) \\,.\n\\end{align*}\nThis completes the proof.  For $T_3$, the limit is non-degenerate which can be calculated as follows: \n\\begin{align*}\nT_3 &= \\frac{1}{\\sigma_n}\\mathbb{E}\\left[\\left(\\delta_0{\\top}g(Q)\\delta_0\\right)\\tilde QK'\\left(\\frac{Q^{\\top}\\psi^s_0}{\\sigma_n}\\right)\\left(1 - 2\\mathds{1}_{Q^{\\top}\\delta_0 > 0}\\right)\\right] \\\\\n& =  \\frac{1}{\\sigma_n} \\int \\int \\left(\\delta_0{\\top}g(t - \\tilde q^{\\top}\\tilde \\psi_0, \\tilde q)\\delta_0\\right)\\tilde q K'\\left(\\frac{t}{\\sigma_n} + \\tilde q^{\\top} \\eta\\right)\\left(1 - 2\\mathds{1}_{t > 0}\\right) \\ f_0(t \\mid \\tilde q) \\ f(\\tilde q) \\ dt \\ d\\tilde q \\\\\n& = \\int \\int \\left(\\delta_0{\\top}g(\\sigma_n t - \\tilde q^{\\top}\\tilde \\psi_0, \\tilde q)\\delta_0\\right)\\tilde q K'\\left(t + \\tilde q^{\\top} \\eta\\right)\\left(1 - 2\\mathds{1}_{t > 0}\\right) \\ f_0(\\sigma_n t \\mid \\tilde q) \\ f(\\tilde q) \\ dt \\ d\\tilde q \\\\\n& = \\int \\int \\left(\\delta_0{\\top}g(- \\tilde q^{\\top}\\tilde \\psi_0, \\tilde q)\\delta_0\\right)\\tilde q K'\\left(t + \\tilde q^{\\top} \\eta\\right)\\left(1 - 2\\mathds{1}_{t > 0}\\right) \\ f_0(0 \\mid \\tilde q) \\ f(\\tilde q) \\ dt \\ d\\tilde q + R \\\\\n& = \\int  \\left(\\delta_0{\\top}g(- \\tilde q^{\\top}\\tilde \\psi_0, \\tilde q)\\delta_0\\right)\\tilde q f_0(0 \\mid \\tilde q) \\left[\\int_{-\\infty}^0 K'\\left(t + \\tilde q^{\\top} \\eta\\right) \\ dt - \\int_0^\\infty K'\\left(t + \\tilde q^{\\top}\\tilde \\eta\\right) \\ dt \\right] \\ f(\\tilde q) \\ d\\tilde q + R \\\\\n&=  \\int  \\left(\\delta_0{\\top}g(- \\tilde q^{\\top}\\tilde \\psi_0, \\tilde q)\\delta_0\\right)\\tilde q f_0(0 \\mid \\tilde q)\\left(2K\\left(\\tilde q^{\\top}\\eta\\right) - 1\\right) \\ f(\\tilde q) \\ d\\tilde q + R  \\\\\n& =   \\mathbb{E}\\left[\\tilde Q f(0 \\mid \\tilde Q) \\left(\\delta_0^{\\top}g(- \\tilde Q^{\\top}\\tilde \\psi_0, \\tilde Q)\\delta_0\\right)\\left(2K(\\tilde Q^{\\top} \\eta)- 1\\right)\\right] + R \n\\end{align*} \nThat the remainder $R$ is $o(1)$ again follows by similar calculation as before and hence skipped. Therefore we have when $\\eta = (\\tilde \\psi_0^s - \\psi_0)\/\\sigma_n \\to h$: \n$$\nT_3 \\overset{n \\to \\infty}{\\longrightarrow}  \\mathbb{E}\\left[\\tilde Q f(0 \\mid \\tilde Q) \\left(\\delta_0^{\\top}g(- \\tilde Q^{\\top}\\tilde \\psi_0, \\tilde Q)\\delta_0\\right)\\left(2K(\\tilde Q^{\\top}h)- 1\\right)\\right]  \\,,\n$$\nwhich along with equation \\eqref{eq:pop_est_conv_1} implies: \n$$\n\\mathbb{E}\\left[\\tilde Q f(0 \\mid \\tilde Q) \\left(\\delta_0^{\\top}g(- \\tilde Q^{\\top}\\tilde \\psi_0, \\tilde Q)\\delta_0\\right)\\left(2K(\\tilde Q^{\\top}h)- 1\\right)\\right]  = 0 \\,.\n$$\nTaking inner product with respect to $h$ on both side of the above equation we obtain: \n$$\n\\mathbb{E}\\left[\\tilde Q^{\\top}h f(0 \\mid \\tilde Q) \\left(\\delta_0^{\\top}g(- \\tilde Q^{\\top}\\tilde \\psi_0, \\tilde Q)\\delta_0\\right)\\left(2K(\\tilde Q^{\\top}h)- 1\\right)\\right]  = 0\n$$\nNow from the symmetry of our Kernel $K$ we have $\\left(\\delta_0^{\\top}g(- \\tilde Q^{\\top}\\tilde \\psi_0, \\tilde Q)\n\\delta_0\\right)\\tilde Q^{\\top}h f(0 \\mid \\tilde Q) (2K(\\tilde Q^{\\top}\\tilde h) - 1) \\ge 0$ almost surely. As the expectation is $0$, we further deduce that $\\tilde Q^{\\top}h f(0 \\mid \\tilde Q) (2K(\\tilde Q^{\\top}\\tilde h)-1) = 0$ almost surely, which further implies $h = 0$. \n\\\\\\\\\n\\noindent\nWe next prove that $(\\beta_0 - \\beta^s_0)\/\\sqrt{\\sigma_n} \\to 0$ and $(\\delta_0 - \\delta^s_0)\/\\sqrt{\\sigma_n} \\to 0$ using equations\\eqref{eq:beta_grad} and \\eqref{eq:delta_grad}. We start with equation \\eqref{eq:beta_grad}: \n\\begin{align}\n    0 & = -\\mathbb{E}\\left[X(Y - X^{\\top}\\beta_0^s)\\right] + \\mathbb{E} \\left\\{\\left[X_iX_i^{\\top}\\delta_0^s\\right] K\\left(\\frac{Q_i^{\\top}\\psi_0^s}{\\sigma_n}\\right)\\right\\}  \\notag \\\\\n    & = -\\mathbb{E}\\left[XX^{\\top}(\\beta_0 - \\beta_0^s)\\right] - \\mathbb{E}[XX^{\\top}\\delta_0\\mathds{1}_{Q^{\\top}\\psi_0 > 0}] + \\mathbb{E} \\left[ g(Q)K\\left(\\frac{Q_i^{\\top}\\psi_0^s}{\\sigma_n}\\right)\\right]\\delta_0^s \\notag \\\\\n    & = -\\Sigma_X(\\beta_0 - \\beta_0^s) -\\mathbb{E}\\left[g(Q)\\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right](\\delta_0 - \\delta_0^s) + \\mathbb{E} \\left[g(Q)\\left\\{K\\left(\\frac{Q_i^{\\top}\\psi_0^s}{\\sigma_n}\\right) - \\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right\\}\\right]\\delta_0^s \\notag \\\\\n     \\label{eq:deriv1} & = \\Sigma_X\\frac{(\\beta_0^2 - \\beta_0)}{\\sigma_n}  + \\mathbb{E}\\left[g(Q)\\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right]\\frac{(\\delta_0^2 - \\delta_0)}{\\sigma_n} + \\frac{1}{\\sigma_n}\\mathbb{E} \\left[g(Q)\\left\\{K\\left(\\frac{Q_i^{\\top}\\psi_0^s}{\\sigma_n}\\right) - \\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right\\}\\right]\\delta_0^s \\notag \\\\  \n     & = \\left( \\mathbb{E}\\left[g(Q)\\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right]\\right)^{-1}\\Sigma_X \\frac{(\\beta_0^2 - \\beta_0)}{\\sigma_n}  + \\frac{\\delta_0^s - \\delta_0}{\\sigma_n}  \\notag \\\\\n     & \\qquad \\qquad \\qquad \\qquad +  \\left( \\mathbb{E}\\left[g(Q)\\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right]\\right)^{-1} \\frac{1}{\\sigma_n}\\mathbb{E} \\left[g(Q)\\left\\{K\\left(\\frac{Q_i^{\\top}\\psi_0^s}{\\sigma_n}\\right) - \\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right\\}\\right]\\delta_0^s \n\\end{align}\nFrom equation \\eqref{eq:delta_grad} we have:\n\\begin{align}\n    0 & = \\mathbb{E} \\left\\{\\left[-X\\left(Y - X^{\\top}\\beta_0^s\\right) + XX^{\\top}\\delta_0^s\\right] K\\left(\\frac{Q^{\\top}\\psi_0^s}{\\sigma_n}\\right)\\right\\} \\notag \\\\\n    & = -\\mathbb{E}\\left[g(Q)K\\left(\\frac{Q^{\\top}\\psi_0^s}{\\sigma_n}\\right)\\right](\\beta_0 - \\beta_0^s) -  \\mathbb{E}\\left[g(Q)K\\left(\\frac{Q^{\\top}\\psi_0^s}{\\sigma_n}\\right)\\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right]\\delta_0 \\notag \\\\\n    & \\hspace{20em}+  \\mathbb{E}\\left[g(Q)K\\left(\\frac{Q^{\\top}\\psi_0^s}{\\sigma_n}\\right)\\right]\\delta_0^s \\notag \\\\\n     & = -\\mathbb{E}\\left[g(Q)K\\left(\\frac{Q^{\\top}\\psi_0^s}{\\sigma_n}\\right)\\right](\\beta_0 - \\beta_0^s) -  \\mathbb{E}\\left[g(Q)K\\left(\\frac{Q^{\\top}\\psi_0^s}{\\sigma_n}\\right)\\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right](\\delta_0 - \\delta_0^s) \\notag \\\\\n     & \\hspace{20em} +   \\mathbb{E}\\left[g(Q)K\\left(\\frac{Q^{\\top}\\psi_0^s}{\\sigma_n}\\right)\\left(1 - \\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right)\\right]\\delta_0^s  \\notag \\\\\n        & = \\mathbb{E}\\left[g(Q)K\\left(\\frac{Q^{\\top}\\psi_0^s}{\\sigma_n}\\right)\\right]\\frac{(\\beta_0^s - \\beta_0)}{\\sigma_n} + \\mathbb{E}\\left[g(Q)K\\left(\\frac{Q^{\\top}\\psi_0^s}{\\sigma_n}\\right)\\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right]\\frac{(\\delta^s_0 - \\delta_0)}{\\sigma_n} \\notag \\\\\n     \\label{eq:deriv2} & \\hspace{20em} +   \\frac{1}{\\sigma_n}\\mathbb{E}\\left[g(Q)K\\left(\\frac{Q^{\\top}\\psi_0^s}{\\sigma_n}\\right)\\left(1 - \\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right)\\right]\\delta_0^s  \\notag \\\\ \n     & = \\left( \\mathbb{E}\\left[g(Q)K\\left(\\frac{Q^{\\top}\\psi_0^s}{\\sigma_n}\\right)\\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right]\\right)^{-1}\\mathbb{E}\\left[g(Q)K\\left(\\frac{Q^{\\top}\\psi_0^s}{\\sigma_n}\\right)\\right]\\frac{(\\beta_0^s - \\beta_0)}{\\sigma_n} + \\frac{(\\delta^s_0 - \\delta_0)}{\\sigma_n} \\notag \\\\\n     & \\qquad \\qquad \\qquad + \\left( \\mathbb{E}\\left[g(Q)K\\left(\\frac{Q^{\\top}\\psi_0^s}{\\sigma_n}\\right)\\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right]\\right)^{-1} \\frac{1}{\\sigma_n}\\mathbb{E}\\left[g(Q)K\\left(\\frac{Q^{\\top}\\psi_0^s}{\\sigma_n}\\right)\\left(1 - \\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right)\\right]\\delta_0^s \n     \\end{align}\nSubtracting equation \\eqref{eq:deriv2} from \\eqref{eq:deriv1} we obtain: \n$$\n0 = A_n \\frac{(\\beta_0^s - \\beta_0)}{\\sigma_n} + b_n \\,,\n$$\ni.e. \n$$\n\\lim_{n \\to \\infty}  \\frac{(\\beta_0^s - \\beta_0)}{\\sigma_n} = \\lim_{n \\to \\infty} -A_n^{-1}b_n \\,.\n$$\nwhere: \n\\begin{align*}\nA_n & =  \\left( \\mathbb{E}\\left[g(Q)\\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right]\\right)^{-1}\\Sigma_X   \\\\\n& \\qquad \\qquad  - \\left( \\mathbb{E}\\left[g(Q)K\\left(\\frac{Q^{\\top}\\psi_0^s}{\\sigma_n}\\right)\\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right]\\right)^{-1}\\mathbb{E}\\left[g(Q)K\\left(\\frac{Q^{\\top}\\psi_0^s}{\\sigma_n}\\right)\\right] \\\\\nb_n & =  \\left( \\mathbb{E}\\left[g(Q)\\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right]\\right)^{-1} \\frac{1}{\\sigma_n}\\mathbb{E} \\left[g(Q)\\left\\{K\\left(\\frac{Q_i^{\\top}\\psi_0^s}{\\sigma_n}\\right) - \\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right\\}\\right]\\delta_0^s  \\\\\n& \\qquad  - \\left( \\mathbb{E}\\left[g(Q)K\\left(\\frac{Q^{\\top}\\psi_0^s}{\\sigma_n}\\right)\\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right]\\right)^{-1} \\frac{1}{\\sigma_n}\\mathbb{E}\\left[g(Q)K\\left(\\frac{Q^{\\top}\\psi_0^s}{\\sigma_n}\\right)\\left(1 - \\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right)\\right]\\delta_0^s \n\\end{align*}\nIt is immediate via DCT that as $n \\to \\infty$: \n\\begin{align}\n     \\label{eq:limit_3} \\mathbb{E}\\left[g(Q)K\\left(\\frac{Q^{\\top}\\psi_0^s}{\\sigma_n}\\right)\\right] & \\longrightarrow \\mathbb{E}\\left[g(Q)\\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right] \\,. \\\\\n     \\label{eq:limit_4} \\mathbb{E}\\left[g(Q)K\\left(\\frac{Q^{\\top}\\psi_0^s}{\\sigma_n}\\right)\\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right] & \\longrightarrow \\mathbb{E}\\left[g(Q)\\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right] \\,.\n\\end{align}\nFrom equation \\eqref{eq:limit_3} and \\eqref{eq:limit_4} it is immediate that: \n\\begin{align*}\n\\lim_{n \\to \\infty} A_n  & =  \\left( \\mathbb{E}\\left[g(Q)\\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right]\\right)^{-1}\\Sigma_X - I \\\\\n& = \\left( \\mathbb{E}\\left[g(Q)\\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right]\\right)^{-1}\\left( \\mathbb{E}\\left[g(Q)\\mathds{1}_{Q^{\\top}\\psi_0 \\le 0}\\right]\\right) := A\\,.\n\\end{align*}\nNext observe that: \n\\begin{align}\n    & \\frac{1}{\\sigma_n} \\mathbb{E}\\left[g(Q)\\left\\{K\\left(\\frac{Q^{\\top}\\psi_0^s}{\\sigma_n}\\right) - \\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right\\}\\right] \\notag \\\\\n    & = \\frac{1}{\\sigma_n} \\mathbb{E}\\left[g(Q)\\left\\{K\\left(\\frac{Q^{\\top}\\psi_0}{\\sigma_n} + \\tilde Q^{\\top}\\tilde \\eta\\right) - \\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right\\}\\right] \\notag \\\\\n    & = \\int_{\\mathbb{R}^{p-1}} \\int_{-\\infty}^{\\infty} g(\\sigma_nt - \\tilde q^{\\top}\\tilde \\psi_0, \\tilde q)\\left[K\\left(t + \\tilde q^{\\top}\\tilde \\eta\\right) - \\mathds{1}_{t > 0}\\right] f(\\sigma_n t \\mid \\tilde q) \\ dt \\ f(\\tilde q) \\ d\\tilde q  \\notag \\\\\n    \\label{eq:limit_1} & \\longrightarrow \\mathbb{E}\\left[g(-\\tilde Q^{\\top}\\tilde \\psi_0, \\tilde Q)f(0 \\mid \\tilde Q)\\right] \\cancelto{0}{\\int_{-\\infty}^{\\infty} \\left[K\\left(t\\right) - \\mathds{1}_{t > 0}\\right] \\ dt} \\,. \n\\end{align}\nSimilar calculation yields: \n\\begin{align}\n    \\label{eq:limit_2} & \\frac{1}{\\sigma_n} \\mathbb{E}\\left[g(Q)K\\left(\\frac{Q^{\\top}\\psi_0^s}{\\sigma_n}\\right)\\left(1 - \\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right)\\right] \\notag \\\\\n    & \\longrightarrow \\mathbb{E}[g(-\\tilde Q^{\\top}\\tilde \\psi_0, \\tilde Q)f_0(0 \\mid \\tilde Q)]\\int_{-\\infty}^{\\infty} \\left[K\\left(t\\right)\\mathds{1}_{t \\le 0}\\right] \\ dt \\,.\n\\end{align}\nCombining equation \\eqref{eq:limit_1} and \\eqref{eq:limit_2} we conclude: \n\\begin{align*}\n\\lim_{n \\to \\infty} b_n &= \\left( \\mathbb{E}\\left[g(Q)\\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right]\\right)^{-1} \\mathbb{E}[g(-\\tilde Q^{\\top}\\tilde \\psi_0, \\tilde Q)\\delta_0f_0(0 \\mid \\tilde Q)]\\int_{-\\infty}^{\\infty} \\left[K\\left(t\\right)\\mathds{1}_{t \\le 0}\\right] \\ dt  \\\\\n& := b \\,.\n\\end{align*}\nwhich further implies, \n$$\n\\lim_{n \\to \\infty}  \\frac{(\\beta_0^s - \\beta_0)}{\\sigma_n} = -A^{-1}b \\implies (\\beta_0^s - \\beta_0) = o(\\sqrt{\\sigma_n})\\,,\n$$\nand by similar calculations: \n$$\n(\\delta_0^s - \\delta_0) = o(\\sqrt{\\sigma_n}) \\,.\n$$\nThis completes the proof. \n\\end{proof}\n\n\n\n\n\n\n\\subsection{Proof of Lemma \\ref{lem:pop_curv_nonsmooth}}\n\\begin{proof}\nFrom the definition of $M(\\theta)$ it is immediate that $\\mathbb{M}(\\theta_0) = \\mathbb{E}[{\\epsilon}^2] = \\sigma^2$. For any general $\\theta$: \n\\begin{align*}\n    \\mathbb{M}(\\theta) & = \\mathbb{E}\\left[\\left(Y - X^{\\top}\\left(\\beta + \\delta\\mathds{1}_{Q^{\\top}\\psi > 0}\\right)\\right)^2\\right] \\\\\n    & = \\sigma^2 + \\mathbb{E}\\left[\\left( X^{\\top}\\left(\\beta + \\delta\\mathds{1}_{Q^{\\top}\\psi > 0} - \\beta_0 - \\delta_0\\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right)\\right)^2\\right] \\\\\n    & \\ge \\sigma^2 + c_- \\mathbb{E}_Q\\left[\\left\\|\\beta - \\beta_0 + \\delta\\mathds{1}_{Q^{\\top}\\psi > 0}- \\delta_0\\mathds{1}_{Q^{\\top}\\psi_0 > 0} \\right\\|^2\\right]\n\\end{align*}\nThis immediately implies: \n$$\n\\mathbb{M}(\\theta) - \\mathbb{M}(\\theta_0) \\ge c_- \\mathbb{E}\\left[\\left\\|\\beta - \\beta_0 + \\delta\\mathds{1}_{Q^{\\top}\\psi > 0}- \\delta_0\\mathds{1}_{Q^{\\top}\\psi_0 > 0} \\right\\|^2\\right] \\,.\n$$\n\n\\noindent\nFor notational simplicity, define $p_{\\psi} = \\mathbb{P}(Q^{\\top}\\psi > 0)$. Expanding the RHS we have: \n\\begin{align}\n    & \\mathbb{E}\\left[\\left\\|\\beta - \\beta_0 + \\delta\\mathds{1}_{Q^{\\top}\\psi > 0}- \\delta_0\\mathds{1}_{Q^{\\top}\\psi_0 > 0} \\right\\|^2\\right] \\notag \\\\\n    & = \\|\\beta - \\beta_0\\|^2 +  2(\\beta - \\beta_0)^{\\top}\\mathbb{E}\\left[\\delta\\mathds{1}_{Q^{\\top}\\psi > 0}- \\delta_0\\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right] + \\mathbb{E}\\left[\\left\\|\\delta\\mathds{1}_{Q^{\\top}\\psi > 0}- \\delta_0\\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right\\|^2\\right] \\notag  \\\\\n     & = \\|\\beta - \\beta_0\\|^2 +  2(\\beta - \\beta_0)^{\\top}\\mathbb{E}\\left[\\delta\\mathds{1}_{Q^{\\top}\\psi > 0}-\\delta\\mathds{1}_{Q^{\\top}\\psi_0 > 0} + \\delta\\mathds{1}_{Q^{\\top}\\psi_0 > 0} - \\delta_0\\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right] \\notag  \\\\\n     & \\qquad \\qquad \\qquad \\qquad \\qquad+ \\mathbb{E}\\left[\\left\\|\\delta\\mathds{1}_{Q^{\\top}\\psi > 0}-\\delta\\mathds{1}_{Q^{\\top}\\psi_0 > 0} + \\delta\\mathds{1}_{Q^{\\top}\\psi_0 > 0} - \\delta_0\\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right\\|^2\\right] \\notag  \\\\\n    & = \\|\\beta - \\beta_0\\|^2 + 2(\\beta - \\beta_0)^{\\top}(\\delta - \\delta_0)p_{\\psi_0} +  \\|\\delta - \\delta_0\\|^2 p_{\\psi_0} \\notag  \\\\\n    & \\qquad \\qquad \\qquad +  2(\\beta - \\beta_0)^{\\top}\\delta\\left(p_{\\psi} - p_{\\psi_0}\\right) + \\|\\delta\\|^2 \\mathbb{P}\\left(\\text{sign}(Q^{\\top}\\psi) \\neq \\text{sign}(Q^{\\top}\\psi_0)\\right)  \\notag  \\\\\n    \\label{eq:nsb1} & \\qquad \\qquad \\qquad \\qquad \\qquad - 2\\delta^{\\top}(\\delta - \\delta_0)\\mathbb{P}\\left(Q^{\\top}\\psi_0 > 0, Q^{\\top}\\psi < 0\\right) \n    \\end{align}\nUsing the fact that $2ab \\ge (a^2\/c) + cb^2$ for any constant $c$ we have: \n\\begin{align*}\n& \\|\\beta - \\beta_0\\|^2 + 2(\\beta - \\beta_0)^{\\top}(\\delta - \\delta_0)p_{\\psi_0} +  \\|\\delta - \\delta_0\\|^2 p_{\\psi_0} \\\\\n& \\ge \\|\\beta - \\beta_0\\|^2  + \\|\\delta - \\delta_0\\|^2 p_{\\psi_0}  -  \\frac{\\|\\beta - \\beta_0\\|^2 p_{\\psi_0}}{c} - c \\|\\delta - \\delta_0\\|^2 p_{\\psi_0} \\\\\n& = \\|\\beta - \\beta_0\\|^2\\left(1 - \\frac{p_{\\psi_0}}{c}\\right) +  \\|\\delta - \\delta_0\\|^2  p_{\\psi_0} (1 - c) \\,.\n\\end{align*}\nfor any $c$. To make the RHS non-negative we pick $p_{\\psi_0} < c < 1$ and concludes that: \n\\begin{equation}\n\\label{eq:nsb2}\n \\|\\beta - \\beta_0\\|^2 + 2(\\beta - \\beta_0)^{\\top}(\\delta - \\delta_0)p_{\\psi_0} +  \\|\\delta - \\delta_0\\|^2 p_{\\psi_0} \\gtrsim \\left( \\|\\beta - \\beta_0\\|^2 + \\|\\delta - \\delta_0\\|^2\\right) \\,.\n\\end{equation}\nFor the last 3 summands of RHS of equation \\eqref{eq:nsb1}:     \n\\begin{align}\n& 2(\\beta - \\beta_0)^{\\top}\\delta\\left(p_{\\psi} - p_{\\psi_0}\\right) + \\|\\delta\\|^2 \\mathbb{P}\\left(\\text{sign}(Q^{\\top}\\psi) \\neq \\text{sign}(Q^{\\top}\\psi_0)\\right)  \\notag  \\\\\n & \\qquad \\qquad - 2\\delta^{\\top}(\\delta - \\delta_0)\\mathbb{P}\\left(Q^{\\top}\\psi_0 > 0, Q^{\\top}\\psi < 0\\right) \\notag \\\\\n & = 2(\\beta - \\beta_0)^{\\top}\\delta \\mathbb{P}\\left(Q^{\\top}\\psi > 0, Q^{\\top}\\psi_0 < 0\\right) - 2(\\beta - \\beta_0)^{\\top}\\delta \\mathbb{P}\\left(Q^{\\top}\\psi < 0, Q^{\\top}\\psi_0 > 0\\right) \\notag \\\\\n  & \\qquad \\qquad + |\\delta\\|^2 \\mathbb{P}\\left(\\text{sign}(Q^{\\top}\\psi) \\neq \\text{sign}(Q^{\\top}\\psi_0)\\right) - 2\\delta^{\\top}(\\delta - \\delta_0)\\mathbb{P}\\left(Q^{\\top}\\psi_0 > 0, Q^{\\top}\\psi < 0\\right) \\notag \\\\\n  & = \\left[\\|\\delta\\|^2 - 2(\\beta - \\beta_0)^{\\top}\\delta - 2\\delta^{\\top}(\\delta - \\delta_0)\\right]\\mathbb{P}\\left(Q^{\\top}\\psi_0 > 0, Q^{\\top}\\psi < 0\\right) \\notag \\\\\n  & \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad + \\left[\\|\\delta\\|^2 + 2(\\beta - \\beta_0)^{\\top}\\delta\\right]\\mathbb{P}\\left(Q^{\\top}\\psi > 0, Q^{\\top}\\psi_0 < 0\\right) \\notag \\\\\n    & = \\left[\\|\\delta_0\\|^2 - 2(\\beta - \\beta_0)^{\\top}(\\delta - \\delta_0) -  2(\\beta - \\beta_0)^{\\top}\\delta_0 - \\|\\delta - \\delta_0\\|^2\\right]\\mathbb{P}\\left(Q^{\\top}\\psi_0 > 0, Q^{\\top}\\psi < 0\\right) \\notag \\\\\n  & \\qquad   + \\left[\\|\\delta_0\\|^2 + \\|\\delta - \\delta_0\\|^2 + 2(\\delta - \\delta_0)^{\\top}\\delta_0 + 2(\\beta - \\beta_0)^{\\top}(\\delta - \\delta_0) +  2(\\beta - \\beta_0)^{\\top}\\delta_0\\right]\\mathbb{P}\\left(Q^{\\top}\\psi > 0, Q^{\\top}\\psi_0 < 0\\right) \\notag \\\\\n  & \\ge  \\left[\\|\\delta_0\\|^2 - 2\\|\\beta - \\beta_0\\|\\|\\delta - \\delta_0\\| -  2\\|\\beta - \\beta_0\\|\\|\\delta_0\\| - \\|\\delta - \\delta_0\\|^2\\right]\\mathbb{P}\\left(Q^{\\top}\\psi_0 > 0, Q^{\\top}\\psi < 0\\right) \\notag \\\\\n  & \\qquad   + \\left[\\|\\delta_0\\|^2 + \\|\\delta - \\delta_0\\|^2 + 2\\|\\delta - \\delta_0\\|\\|\\delta_0\\| + 2\\|\\beta - \\beta_0\\|\\|\\delta - \\delta_0\\| +  2\\|\\beta - \\beta_0\\|\\|\\delta_0\\|\\right]\\mathbb{P}\\left(Q^{\\top}\\psi > 0, Q^{\\top}\\psi_0 < 0\\right) \\notag \\\\\n\\label{eq:nsb3} & \\gtrsim \\|\\delta_0\\|^2 \\mathbb{P}\\left(\\text{sign}(Q^{\\top}\\psi) \\neq \\text{sign}(Q^{\\top}\\psi_0)\\right)   \\gtrsim \\|\\psi - \\psi_0\\| \\hspace{0.2in} [\\text{By Assumption }\\ref{eq:assm}]\\,.\n\\end{align}\nCombining equation \\eqref{eq:nsb2} and \\eqref{eq:nsb3} we complete the proof of lower bound. The upper bound is relatively easier: note that by our previous calculation: \n\\begin{align*}\n    \\mathbb{M}(\\theta) - \\mathbb{M}(\\theta_0) & =  \\mathbb{E}\\left[\\left( X^{\\top}\\left(\\beta + \\delta\\mathds{1}_{Q^{\\top}\\psi > 0} - \\beta_0 - \\delta_0\\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right)\\right)^2\\right] \\\\\n    & \\le  c_+\\mathbb{E}\\left[\\left\\|\\beta - \\beta_0 + \\delta\\mathds{1}_{Q^{\\top}\\psi > 0}- \\delta_0\\mathds{1}_{Q^{\\top}\\psi_0 > 0} \\right\\|^2\\right] \\\\\n    & = c_+\\mathbb{E}\\left[\\left\\|\\beta - \\beta_0 + \\delta\\mathds{1}_{Q^{\\top}\\psi > 0}- \\delta\\mathds{1}_{Q^{\\top}\\psi_0 > 0} + \\delta\\mathds{1}_{Q^{\\top}\\psi_0 > 0} - \\delta_0\\mathds{1}_{Q^{\\top}\\psi_0 > 0} \\right\\|^2\\right] \\\\\n    & \\lesssim \\left[\\|\\beta - \\beta_0\\|^2 + \\|\\delta - \\delta_0\\|^2 + \\mathbb{P}\\left(\\text{sign}(Q^{\\top}\\psi) \\neq \\text{sign}(Q^{\\top}\\psi_0)\\right)\\right] \\\\\n    & \\lesssim \\left[\\|\\beta - \\beta_0\\|^2 + \\|\\delta - \\delta_0\\|^2 + \\|\\psi - \\psi_0\\|\\right] \\,.\n\\end{align*}\nThis completes the entire proof.\n\\end{proof}\n\n\n\n\n\\subsection{Proof of Lemma \\ref{lem:uniform_smooth}}\n\\begin{proof}\nThe difference of the two losses: \n\\begin{align*}\n\\left|\\mathbb{M}^s(\\theta) - \\mathbb{M}(\\theta)\\right| & = \\left|\\mathbb{E}\\left[\\left\\{-2\\left(Y_i - X^{\\top}\\beta\\right)X^{\\top}\\delta + (X^{\\top}\\delta)^2\\right\\}\\left(K\\left(\\frac{Q^{\\top}\\psi}{\\sigma_n}\\right) - \\mathds{1}_{Q^{\\top}\\psi > 0}\\right)\\right]\\right| \\\\\n& \\le \\mathbb{E}\\left[\\left|-2\\left(Y_i - X^{\\top}\\beta\\right)X^{\\top}\\delta + (X^{\\top}\\delta)^2\\right|\\left|K\\left(\\frac{Q^{\\top}\\psi}{\\sigma_n}\\right) - \\mathds{1}_{Q^{\\top}\\psi > 0}\\right|\\right] \\\\\n& := \\mathbb{E}\\left[m(Q)\\left|K\\left(\\frac{Q^{\\top}\\psi}{\\sigma_n}\\right) - \\mathds{1}_{Q^{\\top}\\psi > 0}\\right|\\right] \n\\end{align*}\nwhere $m(Q) = \\mathbb{E}\\left[\\left|-2\\left(Y_i - X^{\\top}\\beta\\right)X^{\\top}\\delta + (X^{\\top}\\delta)^2\\right| \\mid Q\\right]$. This function can be bounded as follows: \n\\begin{align*}\nm(Q) & = \\mathbb{E}\\left[\\left|-2\\left(Y_i - X^{\\top}\\beta\\right)X^{\\top}\\delta + (X^{\\top}\\delta)^2\\right| \\mid Q\\right] \\\\\n& \\le \\mathbb{E}[ (X^{\\top}\\delta)^2 \\mid Q] + 2\\mathbb{E}\\left[\\left|(\\beta - \\beta_0)^{\\top}XX^{\\top}\\delta\\right|\\right] + 2\\mathbb{E}\\left[\\left|\\delta_0^{\\top}XX^{\\top}\\delta\\right|\\right] \\\\\n& \\le c_+\\left(\\|\\delta\\|^2 + 2\\|\\beta - \\beta_0\\|\\|\\delta\\| + 2\\|\\delta\\|\\|\\delta_0\\|\\right) \\lesssim 1 \\,,\n\\end{align*}\nas our parameter space is compact. For the rest of the calculation define $\\eta = (\\tilde \\psi - \\tilde \\psi_0)\/\\sigma_n$. The definition of $\\eta$ may be changed from proof to proof, but it will be clear from the context. Therefore we have: \n\\begin{align*}\n\\left|\\mathbb{M}^s(\\theta) - \\mathbb{M}(\\theta)\\right| & \\lesssim \\mathbb{E}\\left[\\left|K\\left(\\frac{Q^{\\top}\\psi}{\\sigma_n}\\right) - \\mathds{1}_{Q^{\\top}\\psi > 0}\\right|\\right]  \\\\\n& =  \\mathbb{E}\\left[\\left| \\mathds{1}\\left(\\frac{Q^{\\top}\\psi_0}{\\sigma_n} + \\eta^{\\top}\\tilde{Q} \\ge 0\\right) - K\\left(\\frac{Q^{\\top}\\psi_0}{\\sigma_n} + \\eta^{\\top}\\tilde{Q}\\right)\\right|\\right]  \\\\\n& = \\sigma_n \\int_{\\mathbb{R}^{p-1}} \\int_{-\\infty}^{\\infty}  \\left | \\mathds{1}\\left(t  \\ge 0\\right) - K\\left(t \\right)\\right | f_0(\\sigma_n (t-\\eta^{\\top}\\tilde{q}) | \\tilde{q}) \\ dt \\ dP(\\tilde{q}) \\\\\n& \\le f_+  \\sigma_n  \\int_{-\\infty}^{\\infty}  \\left | \\mathds{1}\\left(t  \\ge 0\\right) - K\\left(t \\right)\\right | \\ dt \\lesssim \\sigma_n \\,.\n\\end{align*}\nwhere the integral over $t$ is finite follows from the definition of the kernel. This completes the proof. \n\\end{proof}\n\n\n\n\n\n\n\n\n\\subsection{Proof of Lemma \\ref{lem:pop_smooth_curvarture}}\n\\begin{proof}\nFirst note that we can write: \n\\begin{align}\n    & \\mathbb{M}^s(\\theta) - \\mathbb{M}^s(\\theta_0^s) \\notag \\\\\n    & = \\underbrace{\\mathbb{M}^s(\\theta) - \\mathbb{M}(\\theta)}_{\\ge -K_1\\sigma_n} + \\underbrace{\\mathbb{M}(\\theta) - \\mathbb{M}(\\theta_0)}_{\\underbrace{\\ge u_- d^2(\\theta, \\theta_0)}_{\\ge \\frac{u_-}{2} d^2(\\theta, \\theta_0^s) - u_-\\sigma_n }} + \\underbrace{\\mathbb{M}(\\theta_0) - \\mathbb{M}(\\theta_0^s)}_{\\ge - u_+ d^2(\\theta_0, \\theta_0^s) \\ge - u_+\\sigma_n} + \\underbrace{\\mathbb{M}(\\theta_0^s) - \\mathbb{M}^s(\\theta_0^s)}_{\\ge - K_1 \\sigma_n} \\notag \\\\\n    & \\ge \\frac{u_-}{2}d^2(\\theta, \\theta_0^s) - (2K_1 + \\xi)\\sigma_n \\notag \\\\\n    & \\ge \\frac{u_-}{2}\\left[\\|\\beta - \\beta^s_0\\|^2 + \\|\\delta - \\delta^s_0\\|^2 + \\|\\psi - \\psi^s_0\\|\\right] - (2K_1 + \\xi)\\sigma_n \\notag \\\\ \n    & \\ge \\left[\\frac{u_-}{2}\\left(\\|\\beta - \\beta^s_0\\|^2 + \\|\\delta - \\delta^s_0\\|^2\\right) + \\frac{u_-}{4}\\|\\psi - \\psi^s_0\\|\\right]\\mathds{1}_{\\|\\psi - \\psi^s_0\\| > \\frac{4(2K_1 + \\xi)}{u_-}\\sigma_n} \\notag \\\\\n    \\label{eq:lower_curv_smooth} & \\gtrsim  \\left[\\|\\beta - \\beta^s_0\\|^2 + \\|\\delta - \\delta^s_0\\|^2 + \\|\\psi - \\psi^s_0\\|\\right]\\mathds{1}_{\\|\\psi - \\psi^s_0\\| > \\frac{4(2K_1 + \\xi)}{u_-}\\sigma_n}\n\\end{align}\nwhere $\\xi$ can be taken as close to $0$ as possible. Henceforth we set $\\mathcal{K} = 4(2K_1 + \\xi)\/u_-$. For the other part of the curvature (i.e. when $\\|\\psi - \\psi_0^s\\| \\le \\mathcal{K} \\sigma_n$) we start with a two step Taylor expansion of the smoothed loss function: \n\\begin{align*}\n    \\mathbb{M}^s(\\theta) - \\mathbb{M}^s(\\theta_0^s) = \\frac12 (\\theta_0 - \\theta^0_s)^{\\top}\\nabla^2 \\mathbb{M}^s(\\theta^*)(\\theta_0 - \\theta^0_s) \n\\end{align*}\nRecall the definition of $\\mathbb{M}^s(\\theta)$: \n$$\n\\mathbb{M}^s_n(\\theta) = \\mathbb{E}\\left(Y - X^{\\top}\\beta\\right)^2 + \\mathbb{E} \\left\\{\\left[-2\\left(Y_i - X_i^{\\top}\\beta\\right)X_i^{\\top}\\delta + (X_i^{\\top}\\delta)^2\\right] K\\left(\\frac{Q_i^{\\top}\\psi}{\\sigma_n}\\right)\\right\\}\n$$\nThe partial derivates of $\\mathbb{M}^s(\\theta)$ with respect to $(\\beta, \\delta, \\psi)$ was derived in equation \\eqref{eq:beta_grad} - \\eqref{eq:psi_grad}. From there, we calculate the hessian of $\\mathbb{M}^s(\\theta)$: \n\\begin{align*}\n    \\nabla_{\\beta\\beta}\\mathbb{M}^s(\\theta) & = 2\\Sigma_X \\\\\n    \\nabla_{\\delta\\delta}\\mathbb{M}^s(\\theta) & = 2 \\mathbb{E}\\left[g(Q)K\\left(\\frac{Q_i^{\\top}\\psi}{\\sigma_n}\\right)\\right] = 2 \\mathbb{E}\\left[g(Q)K\\left(\\frac{Q_i^{\\top}\\psi_0 }{\\sigma_n} + \\tilde Q^{\\top}\\tilde \\eta\\right)\\right] \\\\\n    \\nabla_{\\psi\\psi} \\mathbb{M}^s(\\theta) & = \\frac{1}{\\sigma_n^2}\\mathbb{E} \\left\\{\\left[-2\\left(Y_i - X_i^{\\top}\\beta\\right)X_i^{\\top}\\delta + (X_i^{\\top}\\delta)^2\\right]\\tilde Q_i\\tilde Q_i^{\\top} K''\\left(\\frac{Q_i^{\\top}\\psi_0 }{\\sigma_n} + \\tilde Q^{\\top}\\tilde \\eta\\right)\\right\\} \\\\\n    \\nabla_{\\beta \\delta}\\mathbb{M}^s(\\theta) & = 2 \\mathbb{E}\\left[g(Q)K\\left(\\frac{Q_i^{\\top}\\psi}{\\sigma_n}\\right)\\right] = 2 \\mathbb{E}\\left[g(Q)K\\left(\\frac{Q_i^{\\top}\\psi_0 }{\\sigma_n} + \\tilde Q^{\\top}\\tilde \\eta\\right)\\right] \\\\\n    \\nabla_{\\beta \\psi}\\mathbb{M}^s(\\theta) & = \\frac{2}{\\sigma_n}\\mathbb{E}\\left(g(Q)\\delta\\tilde Q^{\\top}K'\\left(\\frac{Q_i^{\\top}\\psi_0 }{\\sigma_n} + \\tilde Q^{\\top}\\tilde \\eta\\right)\\right)  \\\\\n    \\nabla_{\\delta \\psi} \\mathbb{M}^s(\\theta) & = \\frac{2}{\\sigma_n}\\mathbb{E} \\left\\{\\left[-X_i\\left(Y_i - X_i^{\\top}\\beta\\right) + X_iX_i^{\\top}\\delta\\right]\\tilde Q_i^{\\top} K'\\left(\\frac{Q_i^{\\top}\\psi_0 }{\\sigma_n} + \\tilde Q^{\\top}\\tilde \\eta\\right)\\right\\} \\,.\n\\end{align*}\nwhere we use $\\tilde \\eta$ for a generic notation for $(\\tilde \\psi - \\tilde \\psi_0)\/\\sigma_n$. For notational simplicity, we define $\\gamma = (\\beta, \\delta)$ and $\\nabla^2\\mathbb{M}^{s, \\gamma}(\\theta)$, $\\nabla^2\\mathbb{M}^{s, \\gamma \\psi}(\\theta), \\nabla^2\\mathbb{M}^{s, \\psi \\psi}(\\theta)$ to be corresponding blocks of the hessian matrix. We have: \n\\begin{align}\n      \\mathbb{M}^s(\\theta) - \\mathbb{M}^s(\\theta_0^s) & = \\frac12 (\\theta - \\theta^0_s)^{\\top}\\nabla^2 \\mathbb{M}^s(\\theta^*)(\\theta - \\theta^0_s)  \\notag \\\\\n      & = \\frac12 (\\gamma - \\gamma_0^s)^{\\top}\\nabla^2 \\mathbb{M}^{s, \\gamma}(\\theta^*)(\\gamma - \\gamma^0_s) + (\\gamma - \\gamma_0^s)^{\\top}\\nabla^2 \\mathbb{M}^{s, \\gamma \\psi}(\\theta^*)(\\psi - \\psi^0_s) \\notag \\\\\n      & \\qquad \\qquad \\qquad \\qquad + \\frac12(\\psi - \\psi_0^s)^{\\top}\\nabla^2 \\mathbb{M}^{s, \\psi \\psi}(\\theta^*)(\\psi - \\psi^0_s)  \\notag \\\\\n      \\label{eq:hessian_1} & := \\frac12 \\left(T_1 + 2T_2 + T_3\\right)\n\\end{align}\nNote that we can write: \n\\begin{align*}\n    T_1 & = (\\gamma - \\gamma_0^s)^{\\top}\\nabla^2 \\mathbb{M}^{s, \\gamma}(\\tilde \\theta)(\\gamma - \\gamma^0_s) \\\\\n    & = (\\gamma - \\gamma_0^s)^{\\top}\\nabla^2 \\mathbb{M}^{s, \\gamma}(\\theta_0)(\\gamma - \\gamma^0_s)  + (\\gamma - \\gamma_0^s)^{\\top}\\left[\\nabla^2 \\mathbb{M}^{s, \\gamma}(\\tilde \\theta) - \\nabla^2 \\mathbb{M}^{s, \\gamma}(\\theta_0)\\right](\\gamma - \\gamma^0_s) \n\\end{align*}\nThe operator norm of the difference of two hessians can be bounded as: \n$$\n\\left\\|\\nabla^2 \\mathbb{M}^{s, \\gamma}(\\theta^*) - \\nabla^2 \\mathbb{M}^{s, \\gamma}(\\theta_0)\\right\\|_{op}  = O(\\sigma_n) \\,.\n$$\nfor any $\\theta^*$ in a neighborhood of $\\theta_0^s$ with $\\|\\psi - \\psi_0^s\\| \\le \\mathcal{K} \\sigma_n$. To prove this note that for any $\\theta$: \n$$\n\\nabla^2 \\mathbb{M}^{s, \\gamma}(\\theta^*) - \\nabla^2 \\mathbb{M}^{s, \\gamma}(\\theta_0) = 2\\begin{pmatrix}0 & A \\\\\nA & A\\end{pmatrix} = \\begin{pmatrix}0 & 1 \\\\ 1 & 1\\end{pmatrix} \\otimes A \n$$\nwhere: \n$$\nA = \\mathbb{E}\\left[g(Q)K\\left(\\frac{Q_i^{\\top}\\psi}{\\sigma_n}\\right)\\right] - \\mathbb{E}\\left[g(Q)K\\left(\\frac{Q_i^{\\top}\\psi_0 }{\\sigma_n}\\right)\\right]\n$$\nTherefore it is enough to show $\\|A\\|_{op} = O(\\sigma_n)$. Towards that direction: \n\\begin{align*}\nA & = \\mathbb{E}\\left[g(Q)K\\left(\\frac{Q_i^{\\top}\\psi}{\\sigma_n}\\right)\\right] - \\mathbb{E}\\left[g(Q)K\\left(\\frac{Q_i^{\\top}\\psi_0 }{\\sigma_n}\\right)\\right] \\\\\n& = \\sigma_n \\int \\int g(\\sigma_n t - \\tilde q^{\\top}\\tilde \\psi_0)\\left(K(t + \\tilde q^{\\top}\\eta) - K(t) \\right) f_0(\\sigma_n t \\mid \\tilde q) \\ f(\\tilde q) \\ dt  \\ d\\tilde q \\\\\n& = \\sigma_n \\left[\\int \\int g(- \\tilde q^{\\top}\\tilde \\psi_0)\\left(K(t + \\tilde q^{\\top}\\eta) - K(t) \\right) f_0(0 \\mid \\tilde q) \\ f(\\tilde q) \\ dt  \\ d\\tilde q + R \\right] \\\\\n& = \\sigma_n \\left[\\int \\int g(- \\tilde q^{\\top}\\tilde \\psi_0)f_0(0 \\mid \\tilde q) \\int_t^{t + \\tilde q^{\\top}\\eta}K'(s) \\ ds \\ f(\\tilde q) \\ dt  \\ d\\tilde q + R \\right] \\\\\n& = \\sigma_n \\left[\\int g(- \\tilde q^{\\top}\\tilde \\psi_0)f_0(0 \\mid \\tilde q) \\int_{-\\infty}^{\\infty}K'(s) \\int_{s-\\tilde q^{\\top}\\eta}^s \\ dt \\ ds \\ f(\\tilde q)\\ d\\tilde q + R \\right] \\\\\n& = \\sigma_n \\left[\\int g(- \\tilde q^{\\top}\\tilde \\psi_0)f_0(0 \\mid \\tilde q)\\tilde q^{\\top}\\eta  \\ f(\\tilde q)\\ d\\tilde q + R \\right] \\\\\n& = \\sigma_n \\left[\\mathbb{E}\\left[g(- \\tilde Q^{\\top}\\tilde \\psi_0, \\tilde Q)f_0(0 \\mid \\tilde Q)\\tilde Q^{\\top}\\eta\\right] + R \\right]\n\\end{align*}\nusing the fact that $\\left\\|\\mathbb{E}\\left[g(- \\tilde Q^{\\top}\\tilde \\psi_0, \\tilde Q)f_0(0 \\mid \\tilde Q)\\tilde Q^{\\top}\\eta\\right]\\right\\|_{op} = O(1)$ and $\\|R\\|_{op} = O(\\sigma_n)$ we conclude the claim. From the above claim we conclude: \n\\begin{equation}\n    \\label{eq:hessian_gamma}\n    T_1 = (\\gamma - \\gamma_0^s)^{\\top}\\nabla^2 \\mathbb{M}^{s, \\gamma}(\\theta^*)(\\gamma - \\gamma^s_0) \\ge \\|\\gamma - \\gamma^s_0\\|^2(1 - O(\\sigma_n)) \\ge \\frac12 \\|\\gamma - \\gamma_0^s\\|^2\n\\end{equation}\nfor all large $n$. \n\\\\\\\\\n\\noindent \nWe next deal with the cross term $T_2$ in equation \\eqref{eq:hessian_1}. Towards that end first note that: \n\\begin{align*}\n    & \\frac{1}{\\sigma_n}\\mathbb{E}\\left((g(Q)\\delta)\\tilde Q^{\\top}K'\\left(\\frac{Q_i^{\\top}\\psi_0 }{\\sigma_n} + \\tilde Q^{\\top}\\eta^*\\right)\\right) \\\\\n    & = \\int_{\\mathbb{R}^{(p-1)}}\\left[ \\int_{-\\infty}^{\\infty} \\left(g\\left(\\sigma_nt - \\tilde q^{\\top}\\tilde \\psi_0, \\tilde q\\right)\\delta\\right) K'\\left(t + \\tilde q^{\\top}\\eta^*\\right) f_0(\\sigma_n t \\mid \\tilde q) \\ dt\\right] \\tilde q^{\\top} \\ f(\\tilde q) \\ d\\tilde q  \\\\\n     & = \\int_{\\mathbb{R}^{(p-1)}}\\left[ \\int_{-\\infty}^{\\infty} \\left(g\\left(- \\tilde q^{\\top}\\tilde \\psi_0, \\tilde q\\right)\\delta\\right) K'\\left(t + \\tilde q^{\\top}\\eta^*\\right) f_0(0 \\mid \\tilde q) \\ dt\\right] \\tilde q^{\\top} \\ f(\\tilde q) \\ d\\tilde q  + R_1\\\\\n    & = \\mathbb{E}\\left[\\left(g\\left( - \\tilde Q^{\\top}\\tilde \\psi_0, \\tilde Q\\right)\\delta\\right)\\tilde Q^{\\top}f_0(0 \\mid \\tilde Q)\\right] + R_1\n\\end{align*}\nwhere the remainder term $R_1$ can be further decomposed $R_1 = R_{11} + R_{12} + R_{13}$ with: \n\\begin{align*}\n    \\left\\|R_{11}\\right\\|  & = \\left\\|\\int_{\\mathbb{R}^{(p-1)}}\\left[ \\int_{-\\infty}^{\\infty} \\left(g\\left(- \\tilde q^{\\top}\\tilde \\psi_0, \\tilde q\\right)\\delta\\right) K'\\left(t + \\tilde q^{\\top}\\eta^*\\right) (f_0(\\sigma_nt\\mid \\tilde q) -  f_0(0 \\mid \\tilde q)) \\ dt\\right] \\tilde q^{\\top} \\ f(\\tilde q) \\ d\\tilde q\\right\\| \\\\\n    & \\le \\left\\|\\int_{\\mathbb{R}^{(p-1)}}\\left[ \\int_{-\\infty}^{\\infty} \\left\\|g\\left(- \\tilde q^{\\top}\\tilde \\psi_0, \\tilde q\\right)\\right\\|_{op}\\|\\delta\\| \\left|K'\\left(t + \\tilde q^{\\top}\\eta^*\\right)\\right| \\left|f_0(\\sigma_nt\\mid \\tilde q) -  f_0(0 \\mid \\tilde q)\\right| \\ dt\\right] \\left|\\tilde q\\right| \\ f(\\tilde q) \\ d\\tilde q\\right\\| \\\\\n    & \\le \\sigma_n  \\dot{f}^+  c_+ \\|\\delta\\| \\int_{\\mathbb{R}^{(p-1)}} \\|\\tilde q\\|  \\int_{-\\infty}^{\\infty} |t| \\left|K'\\left(t + \\tilde q^{\\top}\\eta^*\\right)\\right| \\ dt \\ f(\\tilde q) \\ d\\tilde q \\\\\n    & \\le \\sigma_n  \\dot{f}^+  c_+ \\|\\delta\\| \\int_{\\mathbb{R}^{(p-1)}} \\|\\tilde q\\|  \\int_{-\\infty}^{\\infty} |t -  \\tilde q^{\\top}\\eta^*| \\left|K'\\left(t\\right)\\right| \\ dt \\ f(\\tilde q) \\ d\\tilde q \\\\\n    & \\le \\sigma_n  \\dot{f}^+  c_+ \\|\\delta\\| \\left[\\int_{\\mathbb{R}^{(p-1)}} \\|\\tilde q\\|  \\int_{-\\infty}^{\\infty} |t| \\left|K'\\left(t\\right)\\right| \\ dt \\ f(\\tilde q) \\ d\\tilde q \\right. \\\\\n    & \\qquad \\qquad \\qquad \\left. + \\int_{\\mathbb{R}^{(p-1)}} \\|\\tilde q\\|^2 \\|\\eta^*\\|  \\int_{-\\infty}^{\\infty} |K'(t)| \\ dt \\ f(\\tilde q) \\ d\\tilde q\\right] \\\\\n     & \\le \\sigma_n  \\dot{f}^+  c_+ \\|\\delta\\| \\left[\\int_{\\mathbb{R}^{(p-1)}} \\|\\tilde q\\|  \\int_{-\\infty}^{\\infty} |t| \\left|K'\\left(t\\right)\\right| \\ dt \\ f(\\tilde q) \\ d\\tilde q + \\mathcal{K}\\int_{\\mathbb{R}^{(p-1)}} \\|\\tilde q\\|^2 \\int_{-\\infty}^{\\infty} |K'(t)| \\ dt \\ f(\\tilde q) \\ d\\tilde q\\right] \\\\\n        & \\lesssim \\sigma_n \\,.\n\\end{align*}\nwhere the last bound follows from our assumptions using the fact that: \n\\begin{align*}\n    & \\|R_{12}\\| \\\\\n    &= \\left\\|\\int_{\\mathbb{R}^{(p-1)}}\\left[ \\int_{-\\infty}^{\\infty} \\left(\\left(g\\left(\\sigma_n t- \\tilde q^{\\top}\\tilde \\psi_0, \\tilde q\\right) - g\\left(- \\tilde q^{\\top}\\tilde \\psi_0, \\tilde q\\right)\\right)\\delta\\right) K'\\left(t + \\tilde q^{\\top} \\eta^*\\right)  f_0(0 \\mid \\tilde q) \\ dt\\right] \\tilde q^{\\top} \\ f(\\tilde q) \\ d\\tilde q\\right\\| \\\\\n    & \\le \\int \\|\\tilde q\\|\\|\\delta\\|f_0(0 \\mid \\tilde q) \\int_{-\\infty}^{\\infty} \\left\\|g\\left(\\sigma_n t- \\tilde q^{\\top}\\tilde \\psi_0, \\tilde q\\right) - g\\left(- \\tilde q^{\\top}\\tilde \\psi_0, \\tilde q\\right) \\right\\|_{op}\\left|K'\\left(t + \\tilde q^{\\top} \\eta^*\\right)\\right| \\ dt \\ f(\\tilde q) \\ d\\tilde q \\\\\n    & \\le \\dot{c}_+ \\sigma_n \\int \\|\\tilde q\\|\\|\\delta\\|f_0(0 \\mid \\tilde q)\\dot \\int_{-\\infty}^{\\infty} |t| \\left|K'\\left(t + \\tilde q^{\\top}\\tilde \\eta\\right)\\right| \\ dt \\ f(\\tilde q) \\ d\\tilde q \\hspace{0.2in} [\\text{Assumption }\\ref{eq:assm}]\\\\\n    &  \\lesssim \\sigma_n \\,.\n\\end{align*}\nThe other remainder term $R_{13}$ is the higher order term and can be shown to be $O(\\sigma_n^2)$ using same techniques. This implies for all large $n$: \n\\begin{align*}\n    \\left\\|\\nabla_{\\beta \\psi}\\mathbb{M}^s(\\theta)\\right\\|_{op} & = O(1) \\,.\n\\end{align*}\nand similar calculation yields $ \\left\\|\\nabla_{\\delta \\psi}\\mathbb{M}^s(\\theta)\\right\\|_{op} = O(1)$. Using this we have: \n\\begin{align}\n    T_2 & = (\\gamma - \\gamma_0^s)^{\\top}\\nabla^2 \\mathbb{M}^{s, \\gamma \\psi}(\\tilde \\theta)(\\psi - \\psi^0_s) \\notag \\\\\n    & = (\\beta - \\beta_0^s)^{\\top}\\nabla_{\\beta \\psi}^2 \\mathbb{M}^{s}(\\tilde \\theta)(\\psi - \\psi^0_s) + (\\delta - \\delta_0^s)^{\\top}\\nabla_{\\delta \\psi}^2 \\mathbb{M}^{s}(\\tilde \\theta)(\\psi - \\psi^0_s) \\notag \\\\\n    & \\ge - C\\left[\\|\\beta - \\beta_0^s\\| + \\|\\delta - \\delta_0^s\\| \\right]\\|\\psi - \\psi^0_s\\| \\notag \\\\\n    & \\ge -C \\sqrt{\\sigma_n}\\left[\\|\\beta - \\beta_0^s\\| + \\|\\delta - \\delta_0^s\\| \\right]\\frac{\\|\\psi - \\psi^0_s\\| }{\\sqrt{\\sigma_n}}  \\notag \\\\\n    \\label{eq:hessian_cross} & \\gtrsim - \\sqrt{\\sigma_n}\\left(\\|\\beta - \\beta_0^s\\|^2 + \\|\\delta - \\delta_0^s\\|^2 +\\frac{\\|\\psi - \\psi^0_s\\|^2 }{\\sigma_n} \\right)\n\\end{align}\nNow for $T_3$ note that: \n\\allowdisplaybreaks\n\\begin{align*}\n& \\sigma_n \\nabla_{\\psi\\psi} \\mathbb{M}^s_n(\\theta) \\\\\n& = \\frac{1}{\\sigma_n}\\mathbb{E} \\left\\{\\left[-2\\left(Y_i - X_i^{\\top}\\beta\\right)X_i^{\\top}\\delta + (X_i^{\\top}\\delta)^2\\right]\\tilde Q_i\\tilde Q_i^{\\top} K''\\left(\\frac{Q_i^{\\top}\\psi_0 }{\\sigma_n} + \\tilde Q^{\\top}\\tilde \\eta\\right)\\right\\} \\\\\n& = \\frac{1}{\\sigma_n}\\mathbb{E} \\left\\{\\left[-2\\left(Y_i - X_i^{\\top}\\beta\\right)X_i^{\\top}\\delta \\right]\\tilde Q_i\\tilde Q_i^{\\top} K''\\left(\\frac{Q_i^{\\top}\\psi_0 }{\\sigma_n} + \\tilde Q^{\\top}\\tilde \\eta\\right)\\right\\}  \\\\\n& \\qquad \\qquad \\qquad + \\frac{1}{\\sigma_n}\\mathbb{E} \\left\\{(\\delta^{\\top}g(Q) \\delta)\\tilde Q_i\\tilde Q_i^{\\top} K''\\left(\\frac{Q_i^{\\top}\\psi_0 }{\\sigma_n} + \\tilde Q^{\\top}\\tilde \\eta\\right)\\right\\}  \\\\\n& = \\frac{1}{\\sigma_n}\\mathbb{E} \\left\\{\\left[-2 X_i^{\\top}\\left(\\beta_0 -\\beta\\right)X_i^{\\top}\\delta - 2(X_i^{\\top}\\delta_0)(X_i^{\\top}\\delta)\\mathds{1}_{Q_i^{\\top}\\psi_0 > 0}\\right]\\tilde Q_i\\tilde Q_i^{\\top} K''\\left(\\frac{Q_i^{\\top}\\psi_0 }{\\sigma_n} + \\tilde Q^{\\top}\\tilde \\eta\\right)\\right\\}  \\\\\n& \\qquad \\qquad \\qquad + \\frac{1}{\\sigma_n}\\mathbb{E} \\left\\{(\\delta^{\\top}g(Q) \\delta)\\tilde Q_i\\tilde Q_i^{\\top} K''\\left(\\frac{Q_i^{\\top}\\psi_0 }{\\sigma_n} + \\tilde Q^{\\top}\\tilde \\eta\\right)\\right\\}  \\\\\n& = \\frac{-2}{\\sigma_n}\\mathbb{E} \\left\\{((\\beta_0 - \\beta)^{\\top}g(Q) \\delta)\\tilde Q_i\\tilde Q_i^{\\top} K''\\left(\\frac{Q_i^{\\top}\\psi_0 }{\\sigma_n} + \\tilde Q^{\\top}\\tilde \\eta\\right)\\right\\} \\\\\n& \\qquad \\qquad \\qquad + \\frac{-2}{\\sigma_n}\\mathbb{E} \\left\\{(\\delta_0^{\\top}g(Q) \\delta)\\tilde Q_i\\tilde Q_i^{\\top} K''\\left(\\frac{Q_i^{\\top}\\psi_0 }{\\sigma_n} + \\tilde Q^{\\top}\\tilde \\eta\\right)\\mathds{1}_{Q_i^{\\top}\\psi_0 > 0}\\right\\} \\\\\n& \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad + \\frac{1}{\\sigma_n}\\mathbb{E} \\left\\{(\\delta^{\\top}g(Q) \\delta)\\tilde Q_i\\tilde Q_i^{\\top} K''\\left(\\frac{Q_i^{\\top}\\psi_0 }{\\sigma_n} + \\tilde Q^{\\top}\\tilde \\eta\\right)\\right\\}  \\\\\n& = \\underbrace{\\frac{-2}{\\sigma_n}\\mathbb{E} \\left\\{((\\beta_0 - \\beta)^{\\top}g(Q)\\delta)\\tilde Q_i\\tilde Q_i^{\\top} K''\\left(\\frac{Q_i^{\\top}\\psi_0 }{\\sigma_n} + \\tilde Q^{\\top}\\tilde \\eta\\right)\\right\\}}_{M_1} \\\\\n& \\qquad \\qquad \\qquad + \\underbrace{\\frac{-2}{\\sigma_n}\\mathbb{E} \\left\\{(\\delta_0^{\\top}g(Q) \\delta_0)\\tilde Q_i\\tilde Q_i^{\\top} K''\\left(\\frac{Q_i^{\\top}\\psi_0 }{\\sigma_n} + \\tilde Q^{\\top}\\tilde \\eta\\right)\\mathds{1}_{Q_i^{\\top}\\psi_0 > 0}\\right\\}}_{M_2} \\\\\n& \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad +\n\\underbrace{\\frac{-2}{\\sigma_n}\\mathbb{E} \\left\\{(\\delta_0^{\\top} g(Q) (\\delta - \\delta_0))\\tilde Q_i\\tilde Q_i^{\\top} K''\\left(\\frac{Q_i^{\\top}\\psi_0 }{\\sigma_n} + \\tilde Q^{\\top}\\tilde \\eta\\right)\\mathds{1}_{Q_i^{\\top}\\psi_0 > 0}\\right\\}}_{M_3} \\\\\n& \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad + \\underbrace{\\frac{1}{\\sigma_n}\\mathbb{E} \\left\\{(\\delta^{\\top}g(Q) \\delta)\\tilde Q_i\\tilde Q_i^{\\top} K''\\left(\\frac{Q_i^{\\top}\\psi_0 }{\\sigma_n} + \\tilde Q^{\\top}\\tilde \\eta\\right)\\right\\}}_{M_4}  \\\\\n& := M_1 + M_2 + M_3 + M_4\n\\end{align*}\nWe next show that $M_1$ and $M_4$ are $O(\\sigma_n)$. Towards that end note that for any two vectors $v_1, v_2$: \n\\begin{align*}\n    & \\frac{1}{\\sigma_n}\\mathbb{E} \\left\\{(v_1^{\\top}g(Q)v_2)\\tilde Q_i\\tilde Q_i^{\\top} K''\\left(\\frac{Q_i^{\\top}\\psi_0 }{\\sigma_n} + \\tilde Q^{\\top}\\tilde \\eta\\right)\\right\\} \\\\\n    & = \\int \\tilde q \\tilde q^{\\top} \\int_{-\\infty}^{\\infty}(v_1^{\\top}g(\\sigma_nt - \\tilde q^{\\top}\\tilde \\eta, \\tilde q)v_2) K''(t + \\tilde q^{\\top}\\tilde \\eta) f(\\sigma_nt \\mid \\tilde q) \\ dt \\ f(\\tilde q) \\ d\\tilde q \\\\\n    & = \\int \\tilde q \\tilde q^{\\top} (v_1^{\\top}g( - \\tilde q^{\\top}\\tilde \\eta, \\tilde q)v_2)f(0 \\mid \\tilde q) f(\\tilde q) \\ d\\tilde q \\cancelto{0}{\\int_{-\\infty}^{\\infty} K''(t) \\ dt}  + R = R\n\\end{align*}\nas $\\int K''(t) \\ dt = 0$ follows from our choice of kernel $K(x) = \\Phi(x)$. Similar calculation as in the case of analyzing the remainder of $T_2$ yields $\\|R\\|_{op} = O(\\sigma_n)$.\n\\noindent\nThis immediately implies $\\|M_1\\|_{op} = O(\\sigma_n)$ and $\\|M_4\\|_{op} = O(\\sigma_n)$. Now for $M_2$: \n\\begin{align}\nM_2 & = \\frac{-2}{\\sigma_n}\\mathbb{E} \\left\\{(\\delta_0^{\\top}g(Q) \\delta_0)\\tilde Q_i\\tilde Q_i^{\\top} K''\\left(\\frac{Q_i^{\\top}\\psi_0 }{\\sigma_n} + \\tilde Q^{\\top}\\tilde \\eta\\right)\\mathds{1}_{Q_i^{\\top}\\psi_0 > 0}\\right\\} \\notag \\\\\n& = -2\\int \\int (\\delta_0^{\\top}g(\\sigma_n t - \\tilde q^{\\top}\\tilde \\psi_0) \\delta_0)\\tilde q\\tilde q^{\\top} K''\\left(t + \\tilde q^{\\top}\\eta^*\\right)\\mathds{1}_{t > 0} f_0(\\sigma_n t \\mid \\tilde q) \\ dt f(\\tilde q) \\ d\\tilde q \\notag \\\\\n& = -2\\int (\\delta_0^{\\top}g(- \\tilde q^{\\top}\\tilde \\psi_0) \\delta_0)\\tilde q\\tilde q^{\\top}  f_0(0 \\mid \\tilde q)  \\int_{-\\infty}^{\\infty} K''\\left(t + \\tilde q^{\\top}\\eta^*\\right)\\mathds{1}_{t > 0} \\ dt f(\\tilde q) \\ d\\tilde q  + R \\notag \\\\\n\\label{eq:M_2_double_deriv} & = 2\\mathbb{E}\\left[(\\delta_0^{\\top}g(- \\tilde Q^{\\top}\\tilde \\psi_0) \\delta_0)\\tilde \nQ\\tilde Q^{\\top}  f_0(0 \\mid \\tilde Q) K'(\\tilde Q^{\\top}\\eta^*)\\right] + R \n\\end{align}\nwhere the remainder term R is $O_p(\\sigma_n)$ can be established as follows: \n\\begin{align*}\nR & = -2\\left[\\int \\int (\\delta_0^{\\top}g(\\sigma_n t - \\tilde q^{\\top}\\tilde \\psi_0) \\delta_0)\\tilde q\\tilde q^{\\top} K''\\left(t + \\tilde q^{\\top}\\eta^*\\right)\\mathds{1}_{t > 0} f_0(\\sigma_n t \\mid \\tilde q) \\ dt f(\\tilde q) \\ d\\tilde q \\right. \\\\\n& \\qquad \\qquad - \\left. \\int (\\delta_0^{\\top}g(- \\tilde q^{\\top}\\tilde \\psi_0) \\delta_0)\\tilde q\\tilde q^{\\top}  f_0(0 \\mid \\tilde q)  \\int_{-\\infty}^{\\infty} K''\\left(t + \\tilde q^{\\top}\\eta^*\\right)\\mathds{1}_{t > 0} \\ dt f(\\tilde q) \\ d\\tilde q \\right] \\\\\n& = -2\\left\\{\\left[\\int \\int (\\delta_0^{\\top}g(\\sigma_n t - \\tilde q^{\\top}\\tilde \\psi_0) \\delta_0)\\tilde q\\tilde q^{\\top} K''\\left(t + \\tilde q^{\\top}\\eta^*\\right)\\mathds{1}_{t > 0} f_0(\\sigma_n t \\mid \\tilde q) \\ dt f(\\tilde q) \\ d\\tilde q \\right. \\right. \\\\\n& \\qquad \\qquad - \\left. \\left. \\int (\\delta_0^{\\top}g(\\sigma_n t - \\tilde q^{\\top}\\tilde \\psi_0) \\delta_0) \\tilde q\\tilde q^{\\top}  f_0(0 \\mid \\tilde q)  \\int_{-\\infty}^{\\infty} K''\\left(t + \\tilde q^{\\top}\\eta^*\\right)\\mathds{1}_{t > 0} \\ dt f(\\tilde q) \\ d\\tilde q \\right] \\right. \\\\\n& \\left. +  \\left[\\int (\\delta_0^{\\top}g(\\sigma_n t - \\tilde q^{\\top}\\tilde \\psi_0) \\delta_0)\\tilde q\\tilde q^{\\top}  f_0(0 \\mid \\tilde q)  \\int_{-\\infty}^{\\infty} K''\\left(t + \\tilde q^{\\top}\\eta^*\\right)\\mathds{1}_{t > 0} \\ dt f(\\tilde q) \\ d\\tilde q \\right. \\right. \\\\\n& \\qquad \\qquad \\left. \\left. -\\int (\\delta_0^{\\top}g(- \\tilde q^{\\top}\\tilde \\psi_0) \\delta_0)\\tilde q\\tilde q^{\\top}  f_0(0 \\mid \\tilde q)  \\int_{-\\infty}^{\\infty} K''\\left(t + \\tilde q^{\\top}\\eta^*\\right)\\mathds{1}_{t > 0} \\ dt f(\\tilde q) \\ d\\tilde q \\right]\\right\\} \\\\\n& = -2(R_1 + R_2) \\,.\n\\end{align*}\nFor $R_1$: \n\\begin{align*}\n\\left\\|R_1\\right\\|_{op} & =  \\left\\|\\left[\\int \\int (\\delta_0^{\\top}g(\\sigma_n t - \\tilde q^{\\top}\\tilde \\psi_0) \\delta_0)\\tilde q\\tilde q^{\\top} K''\\left(t + \\tilde q^{\\top}\\eta^*\\right)\\mathds{1}_{t > 0} f_0(\\sigma_n t \\mid \\tilde q) \\ dt f(\\tilde q) \\ d\\tilde q \\right. \\right. \\,.\\\\\n& \\qquad \\qquad - \\left. \\left. \\int (\\delta_0^{\\top}g(\\sigma_n t - \\tilde q^{\\top}\\tilde \\psi_0) \\delta_0)\\tilde q\\tilde q^{\\top}  f_0(0 \\mid \\tilde q)  \\int_{-\\infty}^{\\infty} K''\\left(t + \\tilde q^{\\top}\\eta^*\\right)\\mathds{1}_{t > 0} \\ dt f(\\tilde q) \\ d\\tilde q \\right] \\right\\|_{op} \\\\\n& \\le c_+ \\int \\int \\|\\tilde q\\|^2 |K''\\left(t + \\tilde q^{\\top}\\eta^*\\right)| |f_0(\\sigma_n t \\mid \\tilde q) -f_0(0\\mid \\tilde q)| \\ dt \\ f(\\tilde q) \\ d\\tilde q  \\\\\n& \\le c_+ F_+\\sigma_n  \\int \\|\\tilde q\\|^2 \\int |t| |K''\\left(t + \\tilde q^{\\top}\\eta^*\\right)| \\ dt \\ f(\\tilde q) \\ d\\tilde q \\\\\n& = c_+ F_+\\sigma_n  \\int \\|\\tilde q \\|^2 \\int |t -  \\tilde q^{\\top}\\eta^*| |K''\\left(t\\right)| \\ dt \\ f(\\tilde q) \\ d\\tilde q \\\\\n& \\le c_+ F_+ \\sigma_n \\left[\\mathbb{E}[\\|\\tilde Q\\|^2]\\int |t||K''(t)| \\ dt + \\|\\eta^*\\|\\mathbb{E}[\\|\\tilde Q\\|^3]\\int |K''(t)| \\ dt\\right] = O(\\sigma_n) \\,.\n\\end{align*}\nand similarly for $R_2$: \n\\begin{align*}\n\\|R_2\\|_{op} & = \\left\\|\\left[\\int (\\delta_0^{\\top}g(\\sigma_n t - \\tilde q^{\\top}\\tilde \\psi_0) \\delta_0)\\tilde q\\tilde q^{\\top}  f_0(0 \\mid \\tilde q)  \\int_{-\\infty}^{\\infty} K''\\left(t + \\tilde q^{\\top}\\eta^*\\right)\\mathds{1}_{t > 0} \\ dt f(\\tilde q) \\ d\\tilde q \\right. \\right. \\\\\n& \\qquad \\qquad \\left. \\left. -\\int (\\delta_0^{\\top}g(- \\tilde q^{\\top}\\tilde \\psi_0) \\delta_0)\\tilde q\\tilde q^{\\top}  f_0(0 \\mid \\tilde q)  \\int_{-\\infty}^{\\infty} K''\\left(t + \\tilde q^{\\top}\\eta^*\\right)\\mathds{1}_{t > 0} \\ dt f(\\tilde q) \\ d\\tilde q \\right]\\right\\|_{op} \\\\\n& \\le F_+ \\|\\delta_0\\|^2 \\int \\left\\|g(\\sigma_n t - \\tilde q^{\\top}\\tilde \\psi_0)  - g( - \\tilde q^{\\top}\\tilde \\psi_0) \\right\\|_{op} \\|\\tilde q\\|^2 \\int_{-\\infty}^{\\infty} |K''\\left(t + \\tilde q^{\\top}\\eta^*\\right)| \\ dt \\\\\n& \\le G_+ F_+ \\sigma_n \\int \\|\\tilde q\\|^2 \\int_{-\\infty}^{\\infty} |t||K''\\left(t + \\tilde q^{\\top}\\eta^*\\right)| \\ dt = O(\\sigma_n) \\,.\n\\end{align*}\nTherefore from \\eqref{eq:M_2_double_deriv} we conclude: \n\\begin{equation}\nM_2 = 2\\mathbb{E}\\left[(\\delta_0^{\\top}g(- \\tilde Q^{\\top}\\tilde \\psi_0) \\delta_0)\\tilde \nQ\\tilde Q^{\\top}  f_0(0 \\mid \\tilde Q) K'(\\tilde Q^{\\top}\\eta^*)\\right] + O(\\sigma_n) \\,.\n\\end{equation}\nSimilar calculation for $M_3$ yields: \n\\begin{equation*}\nM_3 = 2\\mathbb{E}\\left[(\\delta_0^{\\top}g(- \\tilde Q^{\\top}\\tilde \\psi_0)(\\delta - \\delta_0))\\tilde \nQ\\tilde Q^{\\top}  f_0(0 \\mid \\tilde Q) K'(\\tilde Q^{\\top}\\eta^*)\\right] + O(\\sigma_n) \\,.\n\\end{equation*}\ni.e. \n\\begin{equation}\n\\|M_3\\|_{op} \\le c_+ \\mathbb{E}\\left[\\|\\tilde Q\\|^2f_0(0 \\mid \\tilde Q) K'(\\tilde Q^{\\top}\\eta^*)\\right]\\|\\delta_0\\| \\|\\delta - \\delta_0\\| \\,.\n\\end{equation}\nNow we claim that for any $\\mathcal{K} < \\infty$, $\\lambda_{\\min} (M_2) > 0$ for all $\\|\\eta^*\\| \\le \\mathcal{K}$. Towards that end, define a function $\\lambda:B_{\\mathbb{R}^{2d}}(1) \\times B_{\\mathbb{R}^{2d}}(\\mathcal{K}) \\to \\mathbb{R}_+$ as: \n$$\n\\lambda: (v, \\eta) \\mapsto 2\\mathbb{E}\\left[(\\delta_0^{\\top}g(- \\tilde Q^{\\top}\\tilde \\psi_0) \\delta_0) \n\\left(v^{\\top}\\tilde Q\\right) ^2 f_0(0 \\mid \\tilde Q) K'(\\tilde Q^{\\top}\\eta)\\right]\n$$\nClearly $\\lambda \\ge 0$ and is continuous on a compact set. Hence its infimum must be attained. Suppose the infimum is $0$, i.e. there exists $(v^*, \\eta^*)$ such that: \n$$\n\\mathbb{E}\\left[(\\delta_0^{\\top}g(- \\tilde Q^{\\top}\\tilde \\psi_0) \\delta_0) \n\\left(v^{*^{\\top}}\\tilde Q\\right) ^2 f_0(0 \\mid \\tilde Q) K'(\\tilde Q^{\\top}\\eta^*)\\right] = 0 \\,.\n$$\nas $\\lambda_{\\min}(g(\\dot)) \\ge c_+$, we must have $\\left(v^{*^{\\top}}\\tilde Q\\right) ^2 f_0(0 \\mid \\tilde Q) K'(\\tilde Q^{\\top}\\eta^*) = 0$ almost surely. But from our assumption, $\\left(v^{*^{\\top}}\\tilde Q\\right) ^2 > 0$ and $K'(\\tilde Q^{\\top}\\eta^*) > 0$ almost surely, which implies $f_0(0 \\mid \\tilde q) = 0$ almost surely, which is a contradiction. Hence there exists $\\lambda_-$ such that: \n$$\n\\lambda_{\\min} (M_2)  \\ge \\lambda_- > 0 \\ \\ \\forall \\ \\ \\|\\psi - \\psi_0^s\\| \\le \\mathcal{K} \\,.\n$$\nHence we have: \n$$\n\\lambda_{\\min}\\left(\\sigma_n \\nabla_{\\psi \\psi}\\mathbb{M}^2(\\theta)\\right) \\ge \\frac{\\lambda_-}{2}(1 - O(\\sigma_n)) \n$$\nfor all theta such that $d_*(\\theta, \\theta_0^s) \\le {\\epsilon} \\,.$ \n\\begin{align}\n\\label{eq:hessian_psi}\n    & \\frac{1}{\\sigma_n}(\\psi - \\psi_0^s)^{\\top}\\sigma_n \\nabla^{\\psi \\psi}\\mathbb{M}^s(\\tilde \\theta) (\\psi - \\psi^0) \\gtrsim \\frac{\\|\\psi - \\psi^s_0\\|^2}{\\sigma_n} \\left(1- O(\\sigma_n)\\right)  \n\\end{align}\nFrom equation \\eqref{eq:hessian_gamma}, \\eqref{eq:hessian_cross} and \\eqref{eq:hessian_psi} we have: \n\\begin{align*}\n& \\frac12 (\\theta_0 - \\theta^0_s)^{\\top}\\nabla^2 \\mathbb{M}^s(\\theta^*)(\\theta_0 - \\theta^0_s) \\\\\n& \\qquad \\qquad \\gtrsim \\left[\\|\\beta - \\beta^s_0\\|^2 + \\|\\gamma - \\gamma^s_0\\|^2 + \\frac{\\|\\psi - \\psi^s_0\\|^2}{\\sigma_n}\\right]\\mathds{1}_{\\|\\psi - \\psi_0^s\\| \\le \\mathcal{K} \\sigma_n}  \\,.\n\\end{align*}\nThis, along with equation \\eqref{eq:lower_curv_smooth} concludes the proof.  \n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\\subsection{Proof of Lemma \\ref{asymp-normality}}\nWe start by proving that analogues of Lemma 2 of \\cite{seo2007smoothed}: we show that: \n\\begin{align*}\n\\lim_{n \\to \\infty} \\mathbb{E}\\left[ \\sqrt{n\\sigma_n}\\nabla \\mathbb{M}_n^{s, \\psi}(\\theta_0)\\right] & = 0 \\\\\n\\lim_{n \\to \\infty} {\\sf var}\\left[ \\sqrt{n\\sigma_n}\\nabla \\mathbb{M}_n^{s, \\psi}(\\theta_0)\\right] & = V^{\\psi}\n\\end{align*}\nfor some matrix $V^{\\psi}$ which will be specified later in the proof. To prove the limit of the expectation: \n\\begin{align*}\n&  \\mathbb{E}\\left[ \\sqrt{n\\sigma_n}\\nabla \\mathbb{M}_n^{s, \\psi}(\\theta_0)\\right] \\\\\n& =  \\sqrt{\\frac{n}{\\sigma_n}}\\mathbb{E}\\left[\\left\\{(Y - X^{\\top}(\\beta_0 + \\delta_0))^2 - (Y - X^{\\top}\\beta_0)^2\\right\\}\\tilde Q K'\\left(\\frac{Q^{\\top}\\psi_0}{\\sigma_n}\\right)\\right] \\\\\n& =  \\sqrt{\\frac{n}{\\sigma_n}}\\mathbb{E}\\left[\\left(\\delta_0^{\\top}g(Q)\\delta_0\\right)\\left(1 - 2\\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right)\\tilde Q K'\\left(\\frac{Q^{\\top}\\psi_0}{\\sigma_n}\\right)\\right] \\\\\n& = \\sqrt{\\frac{n}{\\sigma_n}} \\times \\sigma_n \\int \\int \\left(\\delta_0^{\\top}g(\\sigma_nt - \\tilde q^{\\top}\\tilde \\psi_0, \\tilde q)\\delta_0\\right)\\left(1 - 2\\mathds{1}_{t > 0}\\right)\\tilde q K'\\left(t\\right) \\ f_0(\\sigma_n t \\mid \\tilde q) f (\\tilde q) \\ dt \\ d\\tilde q \\\\\n& = \\sqrt{n\\sigma_n} \\left[\\int \\tilde q \\left(\\delta_0^{\\top}g(- \\tilde q^{\\top}\\tilde \\psi_0, \\tilde q)\\delta_0\\right)f_0(0 \\mid \\tilde q) \\cancelto{0}{\\left(\\int_{-\\infty}^{\\infty} \\left(1 - 2\\mathds{1}_{t > 0}\\right)K'\\left(t\\right)  \\ dt\\right)} f (\\tilde q) d\\tilde q  + O(\\sigma_n)\\right] \\\\\n& = O(\\sqrt{n\\sigma_n^3}) = o(1) \\,.\n\\end{align*}\nFor the variance part: \n\\begin{align*}\n& {\\sf var}\\left[ \\sqrt{n\\sigma_n}\\nabla \\mathbb{M}_n^{s, \\psi}(\\theta_0)\\right] \\\\\n& = \\frac{1}{\\sigma_n}{\\sf var}\\left(\\left\\{(Y - X^{\\top}(\\beta_0 + \\delta_0))^2 - (Y - X^{\\top}\\beta_0)^2\\right\\}\\tilde Q K'\\left(\\frac{Q^{\\top}\\psi_0}{\\sigma_n}\\right)\\right) \\\\\n& = \\frac{1}{\\sigma_n}\\mathbb{E}\\left(\\left\\{(Y - X^{\\top}(\\beta_0 + \\delta_0))^2 - (Y - X^{\\top}\\beta_0)^2\\right\\}^2 \\tilde Q\\tilde Q^{\\top} \\left(K'\\left(\\frac{Q^{\\top}\\psi_0}{\\sigma_n}\\right)\\right)^2\\right) \\\\\n& \\qquad \\qquad + \\frac{1}{\\sigma_n}\\mathbb{E}^{\\otimes 2}\\left[\\left\\{(Y - X^{\\top}(\\beta_0 + \\delta_0))^2 - (Y - X^{\\top}\\beta_0)^2\\right\\}\\tilde Q K'\\left(\\frac{Q^{\\top}\\psi_0}{\\sigma_n}\\right)\\right]\n\\end{align*}\nThe outer product of the expectation (the second term of the above summand) is $o(1)$ which follows from our previous analysis of the expectation term. For the second moment: \n\\begin{align*}\n& \\frac{1}{\\sigma_n}\\mathbb{E}\\left(\\left\\{(Y - X^{\\top}(\\beta_0 + \\delta_0))^2 - (Y - X^{\\top}\\beta_0)^2\\right\\}^2 \\tilde Q\\tilde Q^{\\top} \\left(K'\\left(\\frac{Q^{\\top}\\psi_0}{\\sigma_n}\\right)\\right)^2\\right) \\\\\n& = \\frac{1}{\\sigma_n}\\mathbb{E}\\left(\\left\\{(X^{\\top}\\delta_0)^2(1 - 2\\mathds{1}_{Q^{\\top}\\psi_0 > 0}) -2{\\epsilon} (X^{\\top}\\delta_0)\\right\\}^2 \\tilde Q\\tilde Q^{\\top} \\left(K'\\left(\\frac{Q^{\\top}\\psi_0}{\\sigma_n}\\right)\\right)^2\\right) \\\\\n& = \\frac{1}{\\sigma_n}\\left[\\mathbb{E}\\left((X^{\\top}\\delta_0)^4 \\tilde Q\\tilde Q^{\\top} \\left(K'\\left(\\frac{Q^{\\top}\\psi_0}{\\sigma_n}\\right)\\right)^2\\right) + 4\\sigma_{\\epsilon}^2\\mathbb{E}\\left((X^{\\top}\\delta_0)^2 \\tilde Q\\tilde Q^{\\top} \\left(K'\\left(\\frac{Q^{\\top}\\psi_0}{\\sigma_n}\\right)\\right)^2\\right)  \\right] \\\\\n& \\longrightarrow \\left(\\int_{-\\infty}^{\\infty}(K'(t))^2 \\ dt\\right)\\left[\\mathbb{E}\\left(g_{4, \\delta_0}(-\\tilde Q^{\\top}\\tilde \\psi_0, \\tilde Q)\\tilde Q\\tilde Q^{\\top}f_0(0 \\mid \\tilde Q)\\right) \\right. \\\\\n& \\hspace{10em}+ \\left. 4\\sigma_{\\epsilon}^2\\mathbb{E}\\left(\\delta_0^{\\top}g(-\\tilde Q^{\\top}\\tilde \\psi_0, \\tilde Q)\\delta_0 \\tilde Q\\tilde Q^{\\top}f_0(0 \\mid \\tilde Q)\\right)\\right] \\\\\n& := 2V^{\\psi} \\,.\n\\end{align*}\nFinally using Lemma 6 of \\cite{horowitz1992smoothed} we conclude that $ \\sqrt{n\\sigma_n}\\nabla \\mathbb{M}_n^{s, \\psi}(\\theta_0) \\implies \\mathcal{N}(0, V^{\\psi})$. \n\\\\\\\\\n\\noindent\nWe next prove that $ \\sqrt{n}\\nabla \\mathbb{M}_n^{s, \\gamma}(\\theta_0)$ to normal distribution. This is a simple application of CLT along with bounding some remainder terms which are asymptotically negligible. The gradients are: \n\\begin{align*}\n\\sqrt{n}\\begin{pmatrix} \\nabla_{\\beta}\\mathbb{M}^s_n(\\theta_0^s) \\\\ \\nabla_{\\delta}\\mathbb{M}^s_n(\\theta_0^s) \\end{pmatrix} & = 2\\sqrt{n}\\begin{pmatrix}\\frac1n \\sum_i X_i(X_i^{\\top}\\beta_0 - Y_i)+ \\frac1n \\sum_i X_iX_i^{\\top}\\delta_0 K\\left(\\frac{Q_i^{\\top}\\psi_0}{\\sigma_n}\\right)  \\\\ \n\\frac1n \\sum_i \\left[X_i(X_i^{\\top}\\beta_0 + X_i^{\\top}\\delta_0 - Y_i)\\right] K\\left(\\frac{Q_i^{\\top}\\psi_0^s}{\\sigma_n}\\right) \\end{pmatrix} \\\\\n& = 2\\begin{pmatrix}  -\\frac{1}{\\sqrt{n}} \\sum_i X_i {\\epsilon}_i + \\frac{1}{\\sqrt{n}} \\sum_i X_iX_i^{\\top}\\delta_0 \\left(K\\left(\\frac{Q_i^{\\top}\\psi_0}{\\sigma_n}\\right) - \\mathds{1}_{Q_i^{\\top}\\psi_0 > 0}\\right)  \\\\  -\\frac{1}{\\sqrt{n}} \\sum_i X_i {\\epsilon}_iK\\left(\\frac{Q_i^{\\top}\\psi_0}{\\sigma_n}\\right)  + \\frac{1}{\\sqrt{n}} \\sum_i X_iX_i^{\\top}\\delta_0K\\left(\\frac{Q_i^{\\top}\\psi_0}{\\sigma_n}\\right)\\mathds{1}_{Q_i^{\\top}\\psi_0 \\le 0} \n\\end{pmatrix}\\\\\n& = 2\\begin{pmatrix}  -\\frac{1}{\\sqrt{n}} \\sum_i X_i {\\epsilon}_i + R_1  \\\\  -\\frac{1 }{\\sqrt{n}} \\sum_i X_i {\\epsilon}_i\\mathbf{1}_{Q_i^{\\top}\\psi_0 > 0}  +R_2 \n\\end{pmatrix}\n\\end{align*}\nThat $(1\/\\sqrt{n})\\sum_i X_i {\\epsilon}_i$ converges to normal distribution follows from a simple application of CLT. Therefore, once we prove that $R_1$ and $R_2$ are $o_p(1)$ we have: \n$$\n\\sqrt{n} \\nabla_{\\gamma}\\mathbb{M}^s_n(\\theta_0^s) \\overset{\\mathscr{L}}{\\implies} \\mathcal{N}\\left(0, 4V^{\\gamma}\\right)\n$$\nwhere: \n\\begin{equation}\n\\label{eq:def_v_gamma}\nV^{\\gamma} = \\sigma_{\\epsilon}^2 \\begin{pmatrix}\\mathbb{E}\\left[XX^{\\top}\\right] & \\mathbb{E}\\left[XX^{\\top}\\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right] \\\\\n\\mathbb{E}\\left[XX^{\\top}\\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right] & \\mathbb{E}\\left[XX^{\\top}\\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right] \\end{pmatrix} \\,.\n\\end{equation}\nTo complete the proof we now show that $R_1$ and $R_2$ are $o_p(1)$. For $R_1$, we show that $\\mathbb{E}[R_1] \\to 0$ and ${\\sf var}(R_1) \\to 0$.  For the expectation part: \n\\begin{align*}\n & \\mathbb{E}[R_1] \\\\\n & = \\sqrt{n}\\mathbb{E}\\left[XX^{\\top}\\delta_0 \\left(K\\left(\\frac{Q^{\\top}\\psi_0}{\\sigma_n}\\right) - \\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right)\\right] \\\\\n & = \\sqrt{n}\\delta_0^{\\top}\\mathbb{E}\\left[g(Q) \\left(K\\left(\\frac{Q^{\\top}\\psi_0}{\\sigma_n}\\right) - \\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right)\\right] \\\\\n & = \\sqrt{n}\\int_{\\mathbb{R}^{p-1}} \\int_{-\\infty}^{\\infty} \\delta_0^{\\top}g\\left(t-\\tilde q^{\\top}\\tilde \\psi_0, \\tilde q\\right)\\left(\\mathds{1}_{t > 0} - K\\left(\\frac{t}{\\sigma_n}\\right)\\right)f_0(t \\mid \\tilde q) f(\\tilde q) \\ dt \\ d\\tilde q \\\\\n  & = \\sqrt{n}\\sigma_n \\int_{\\mathbb{R}^{p-1}} \\int_{-\\infty}^{\\infty} \\delta_0^{\\top}g\\left(\\sigma_n z-\\tilde q^{\\top}\\tilde \\psi_0, \\tilde q\\right)\\left(\\mathds{1}_{z > 0} - K\\left(z\\right)\\right)f_0(\\sigma_n z \\mid \\tilde q) f(\\tilde q) \\ dz \\ d\\tilde q \\\\\n  & =  \\sqrt{n}\\sigma_n \\left[\\int_{\\mathbb{R}^{p-1}}\\delta_0^{\\top}g\\left(-\\tilde q^{\\top}\\tilde \\psi_0, \\tilde q\\right) f_0(0 \\mid \\tilde q) f(\\tilde q)  \\ d\\tilde q \\cancelto{0}{\\left[\\int_{-\\infty}^{\\infty}  \\left(\\mathds{1}_{z > 0} - K\\left(z\\right)\\right)\\ dz\\right]}  + O(\\sigma_n) \\right] \\\\\n  & = O(\\sqrt{n}\\sigma_n^2) = o(1) \\,.\n\\end{align*}\nFor the variance part: \n\\begin{align*}\n& {\\sf var}(R_1) \\\\\n& = {\\sf var}\\left(XX^{\\top}\\delta_0 \\left(K\\left(\\frac{Q^{\\top}\\psi_0}{\\sigma_n}\\right) - \\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right)\\right) \\\\\n& \\le \\mathbb{E}\\left[\\|X\\|^2 \\delta_0^{\\top}XX^{\\top}\\delta_0 \\left(K\\left(\\frac{Q^{\\top}\\psi_0}{\\sigma_n}\\right) - \\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right)^2\\right] \\\\\n& = O(\\sigma_n ) = o(1) \\,.\n\\end{align*}\nThis shows that ${\\sf var}(R_1) = o(1)$ and this establishes $R_1 = o_p(1)$. The proof for $R_2$ is similar and hence skipped for brevity. \n\\\\\\\\\nOur next step is to prove that $\\sqrt{n\\sigma_n}\\nabla_{\\psi}\\mathbb{M}^s_n(\\theta_0^s)$ and $\\sqrt{n}\\nabla \\mathbb{M}^{s, \\gamma}_n(\\theta_0^s)$ are asymptotically uncorrelated. Towards that end, first note that: \n\\begin{align*}\n& \\mathbb{E}\\left[X(X^{\\top}\\beta_0 - Y) + XX^{\\top}\\delta_0 K\\left(\\frac{Q^{\\top}\\psi_0}{\\sigma_n}\\right) \\right] \\\\\n& = \\mathbb{E}\\left[XX^{\\top}\\delta_0\\left(K\\left(\\frac{Q^{\\top}\\psi_0}{\\sigma_n}\\right) - \\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right)\\right] \\\\\n& = \\mathbb{E}\\left[g(Q)\\delta_0\\left(K\\left(\\frac{Q^{\\top}\\psi_0}{\\sigma_n}\\right) - \\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right)\\right] \\\\\n& = \\sigma_n \\int \\int g(\\sigma_n t - \\tilde q^{\\top}\\tilde \\psi_0, \\tilde q)(K(t) - \\mathds{1}_{t>0})f_0(\\sigma_n t \\mid \\tilde q) f(\\tilde q) \\ dt \\ d\\tilde q  \\\\\n& = \\sigma_n \\int g(- \\tilde q^{\\top}\\tilde \\psi_0, \\tilde q)\\cancelto{0}{\\int_{-\\infty}^{\\infty} (K(t) - \\mathds{1}_{t>0}) \\ dt} \\ f_0(0 \\mid \\tilde q) f(\\tilde q) \\ dt \\ d\\tilde q + O(\\sigma_n^2) \\\\\n& = O(\\sigma_n^2) \\,.\n\\end{align*}\nAlso, it follows from the proof of $\\mathbb{E}\\left[\\sqrt{n\\sigma_n}\\nabla_\\psi \\mathbb{M}_n^s(\\theta_0)\\right] \\to 0$ we have: \n$$\n\\mathbb{E}\\left[\\left\\{(Y - X^{\\top}(\\beta_0 + \\delta_0))^2 - (Y - X^{\\top}\\beta_0)^2\\right\\}\\tilde Q K'\\left(\\frac{Q^{\\top}\\psi_0}{\\sigma_n}\\right)\\right] = O(\\sigma_n^2) \\,.\n$$\nFinally note that: \n\\begin{align*}\n& \\mathbb{E}\\left[\\left(\\left\\{(Y - X^{\\top}(\\beta_0 + \\delta_0))^2 - (Y - X^{\\top}\\beta_0)^2\\right\\}\\tilde Q K'\\left(\\frac{Q^{\\top}\\psi_0}{\\sigma_n}\\right)\\right) \\times \\right. \\\\\n& \\qquad \\qquad \\qquad \\qquad \\qquad \\left. \\left(X(X^{\\top}\\beta_0 - Y) + XX^{\\top}\\delta_0 K\\left(\\frac{Q^{\\top}\\psi_0}{\\sigma_n}\\right)\\right)^{\\top}\\right] \\\\\n& = \\mathbb{E}\\left[\\left(\\left\\{(X^{\\top}\\delta_0)^2(1 - 2\\mathds{1}_{Q^{\\top}\\psi_0 > 0}) - 2{\\epsilon} X^{\\top}\\delta_0\\right\\}\\tilde QK'\\left(\\frac{Q^{\\top}\\psi_0}{\\sigma_n}\\right)\\right) \\right. \\\\\n& \\qquad \\qquad \\qquad \\qquad \\qquad \\left. \\times \\left\\{XX^{\\top}\\delta_0\\left(K\\left(\\frac{Q^{\\top}\\psi_0}{\\sigma_n}\\right) - \\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right) - X{\\epsilon} \\right\\}\\right] \\\\\n& = \\mathbb{E}\\left[\\left((X^{\\top}\\delta_0)^2(1 - 2\\mathds{1}_{Q^{\\top}\\psi_0 > 0})\\tilde QK'\\left(\\frac{Q^{\\top}\\psi_0}{\\sigma_n}\\right)\\right) \\right. \\\\\n& \\qquad \\qquad \\qquad  \\left. \\times \\left(XX^{\\top}\\delta_0\\left(K\\left(\\frac{Q^{\\top}\\psi_0}{\\sigma_n}\\right) - \\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right)\\right)^{\\top}\\right] \\\\\n& \\qquad \\qquad + 2\\sigma^2_{\\epsilon} \\mathbb{E}\\left[XX^{\\top}\\delta_0\\tilde Q^{\\top}K'\\left(\\frac{Q^{\\top}\\psi_0}{\\sigma_n}\\right)\\right] \\\\\n&= O(\\sigma_n ) \\,.\n\\end{align*}\nNow getting back to the covariance: \n\\begin{align*}\n& \\mathbb{E}\\left[\\left(\\sqrt{n\\sigma_n}\\nabla_{\\psi}\\mathbb{M}^s_n(\\theta_0)\\right)\\left(\\sqrt{n}\\nabla_\\beta \\mathbb{M}^s_n(\\theta_0)\\right)^{\\top}\\right] \\\\\n& = \\frac{1}{\\sqrt{\\sigma_n}}\\mathbb{E}\\left[\\left(\\left\\{(Y - X^{\\top}(\\beta_0 + \\delta_0))^2 - (Y - X^{\\top}\\beta_0)^2\\right\\}\\tilde Q K'\\left(\\frac{Q^{\\top}\\psi_0}{\\sigma_n}\\right)\\right) \\times \\right. \\\\\n& \\qquad \\qquad \\qquad \\qquad \\qquad \\left. \\left(X(X^{\\top}\\beta_0 - Y) + XX^{\\top}\\delta_0 K\\left(\\frac{Q^{\\top}\\psi_0}{\\sigma_n}\\right)\\right)^{\\top}\\right] \\\\\n& \\qquad \\qquad + \\frac{n-1}{\\sqrt{\\sigma_n}}\\left[\\mathbb{E}\\left[\\left\\{(Y - X^{\\top}(\\beta_0 + \\delta_0))^2 - (Y - X^{\\top}\\beta_0)^2\\right\\}\\tilde Q K'\\left(\\frac{Q^{\\top}\\psi_0}{\\sigma_n}\\right)\\right] \\right. \\\\\n& \\qquad \\qquad \\qquad \\qquad \\times \\left. \\left(\\mathbb{E}\\left[X(X^{\\top}\\beta_0 - Y) + XX^{\\top}\\delta_0 K\\left(\\frac{Q^{\\top}\\psi_0}{\\sigma_n}\\right) \\right]\\right)^{\\top}\\right] \\\\\n& = \\frac{1}{\\sqrt{\\sigma_n}} \\times O(\\sigma_n) + \\frac{n-1}{\\sqrt{\\sigma_n}} \\times O(\\sigma_n^4) = o(1) \\,.\n\\end{align*}\nThe proof for $\\mathbb{E}\\left[\\left(\\sqrt{n\\sigma_n}\\nabla_{\\psi}\\mathbb{M}^s_n(\\theta_0)\\right)\\left(\\sqrt{n}\\nabla_\\delta \\mathbb{M}^s_n(\\theta_0)\\right)^{\\top}\\right]$ is similar and hence skipped. This completes the proof. \n\n\n\n\\subsection{Proof of Lemma \\ref{conv-prob}}\nTo prove first note that by simple application of law of large number (and using the fact that $\\|\\psi^* - \\psi_0\\|\/\\sigma_n = o_p(1)$ we have: \n\\begin{align*}\n\\nabla^2 \\mathbb{M}_n^{s, \\gamma}(\\theta^*) & = 2\\begin{pmatrix}\\frac{1}{n}\\sum_i X_i X_i^{\\top} & \\frac{1}{n}\\sum_i X_i X_i^{\\top}K\\left(\\frac{Q_i^{\\top}\\psi^*}{\\sigma_n}\\right) \\\\ \\frac{1}{n}\\sum_i X_i X_i^{\\top}K\\left(\\frac{Q_i^{\\top}\\psi^*}{\\sigma_n}\\right) & \\frac{1}{n}\\sum_i X_i X_i^{\\top}K\\left(\\frac{Q_i^{\\top}\\psi^*}{\\sigma_n}\\right)\n\\end{pmatrix} \\\\\n& \\overset{p}{\\longrightarrow} 2 \\begin{pmatrix}\\mathbb{E}\\left[XX^{\\top}\\right] & \\mathbb{E}\\left[XX^{\\top}\\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right] \\\\ \\mathbb{E}\\left[XX^{\\top}\\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right] & \\mathbb{E}\\left[XX^{\\top}\\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right] \\end{pmatrix} := 2Q^{\\gamma}\n\\end{align*}\nThe proof of the fact that $\\sqrt{\\sigma_n}\\nabla^2_{\\psi \\gamma}\\mathbb{M}_n^s(\\theta^*) = o_p(1)$ is same as the proof of Lemma 5 of \\cite{seo2007smoothed} and hence skipped. Finally the proof of the fact that \n$$\n\\sigma_n \\nabla^2_{\\psi \\psi}\\mathbb{M}_n^s(\\theta^*) \\overset{p}{\\longrightarrow} 2Q^{\\psi}\\,.\n$$\nfor some non-negative definite matrix $Q$. The proof is similar to that of Lemma 6 of \\cite{seo2007smoothed}, using which we conclude the proof with:  \n$$\nQ^{\\psi} = \\left(\\int_{-\\infty}^{\\infty} -\\text{sign}(t) K''(t) \\ dt\\right) \\times \\mathbb{E}\\left[\\delta_0^{\\top} g\\left(-\\tilde Q^{\\top}\\tilde \\psi_0, \\tilde Q\\right)\\delta_0 \\tilde Q \\tilde Q^{\\top} f_0(0 \\mid \\tilde Q)\\right]  \\,.\n$$\nThis completes the proof. So we have established: \n\\begin{align*}\n\\sqrt{n}\\left(\\hat \\gamma^s  - \\gamma_0\\right) & \\overset{\\mathscr{L}}{\\implies} \\mathcal{N}\\left(0, \\left(Q^\\gamma\\right)^{-1}V^\\gamma \\left(Q^\\gamma\\right)^{-1}\\right) \\,, \\\\\n\\sqrt{\\frac{n}{\\sigma_n}}\\left(\\hat \\psi^s  - \\psi_0\\right) & \\overset{\\mathscr{L}}{\\implies} \\mathcal{N}\\left(0, \\left(Q^\\psi\\right)^{-1}V^\\psi \\left(Q^\\psi\\right)^{-1}\\right) \\,.\n\\end{align*}\nand they are asymptotically uncorrelated. \n\n\n\n\n\n\n\n\n\n\n\n\\section{Proof of Theorem \\ref{thm:binary}}\n\\label{sec:supp_classification}\nIn this section, we present the details of the binary response model, the assumptions, a roadmap of the proof and then finally prove Theorem \\ref{thm:binary}.\n\\noindent \n\\begin{assumption}\n\\label{as:distribution}\nThe below assumptions pertain to the parameter space and the distribution of $Q$:\n\\begin{enumerate}\n\\item  The parameter space $\\Theta$ is a compact subset of $\\mathbb{R}^p$. \n\\item  The support of the distribution of $Q$ contains an open subset around origin of $\\mathbb{R}^p$ and the distribution of $Q_1$ conditional on $\\tilde{Q} = (Q_2, \\dots, Q_p)$ has, almost surely, everywhere positive density with respect to Lebesgue measure. \n\\end{enumerate}\n\\end{assumption}\n\n\n\n\\noindent \nFor notational convenience, define the following: \n\\begin{enumerate}\n\\item Define $f_{\\psi} (\\cdot | \\tilde{Q})$ to the conditional density of $Q^{\\top}\\psi$ given $\\tilde{Q}$ for $\\theta \\in \\Theta$. Note that the following relation holds: $$f_{\\theta}(\\cdot |\\tilde{Q}) = f_{Q_1}(\\cdot - \\tilde{\\psi}^{\\top}\\tilde{Q} | \\tilde{Q}) \\,.$$ where we define $f_{Q_1}(\\cdot | \\tilde X)$ is the conditional density of $Q_1$ given $\\tilde Q$. \n\\item Define $f_0(\\cdot | \\tilde{Q}) = f_{\\psi_0}(\\cdot | \\tilde{Q})$ where $\\psi_0$ is the unique minimizer of the population score function $M(\\psi)$. \n\\item Define $f_{\\tilde Q}(\\cdot)$ to be the marginal density of $\\tilde Q$. \n\\end{enumerate}\n\n\n\\noindent\nThe rest of the assumptions are as follows: \n\\begin{assumption}\n\\label{as:differentiability}\n$f_0(y|\\tilde{Q})$ is at-least once continuously differentiable almost surely for all $\\tilde{Q}$. Also assume that there exists $\\delta$ and $t$ such that $$\\inf_{|y| \\le \\delta} f_0(y|\\tilde{Q}) \\ge t$$ for all $\\tilde{Q}$ almost surely. \n\\end{assumption}\nThis assumption can be relaxed in the sense that one can allow the lower bound $t$ to depend on $\\tilde{Q}$, provided that some further assumptions are imposed on $\\mathbb{E}(t(\\tilde{Q}))$. As this does not add anything of significance to the import of this paper, we use Assumption \\ref{as:differentiability} to simplify certain calculations. \n\n\n\n\\begin{assumption}\n\\label{as:density_bound}\nDefine $m\\left(\\tilde{Q}\\right) = \\sup_{t}f_{X_1}(t | \\tilde{Q}) = \\sup_{\\theta} \\sup_{t}f_{\\theta}(t | \\tilde{Q})$. Assume that $\\mathbb{E}\\left(m\\left(\\tilde{Q}\\right)^2\\right) < \\infty$. \n\\end{assumption}\n\n\n\n\n\n\n\n\\begin{assumption}\n\\label{as:derivative_bound}\nDefine $h(\\tilde{Q}) = \\sup_{t} f_0'(t | \\tilde{Q})$. Assume that $\\mathbb{E}\\left(h^2\\left(\\tilde{Q}\\right)\\right) < \\infty$.  \n\\end{assumption}\n\\begin{assumption}\n\\label{as:eigenval_bound}\nAssume that $f_{\\tilde{Q}}(0) > 0$ and also that the minimum eigenvalue of $\\mathbb{E}\\left(\\tilde{Q}\\tilde{Q}^{\\top}f_0(0|\\tilde{Q})\\right) > 0$. \n\\end{assumption}\n\n\n\n\n\n\\subsection{Sufficient conditions for above assumptions }\nWe now demonstrate some sufficient conditions for the above assumptions to hold. If the support of $Q$ is compact and both $f_1(\\cdot | \\tilde Q)$ and $f'_1(\\cdot | \\tilde Q)$ are uniformly bounded in $\\tilde Q$, then Assumptions $(\\ref{as:distribution}, \\ \\ref{as:differentiability}, \\ \\ref{as:density_bound},\\ \\ref{as:derivative_bound})$ follow immediately. The first part of Assumption \\ref{as:eigenval_bound}, i.e. the assumption $f_{\\tilde{Q}}(0) > 0$ is also fairly general and satisfied by many standard probability distributions. The second part of Assumption \\ref{as:eigenval_bound} is satisfied when $f_0(0|\\tilde{Q})$ has some lower bound independent of $\\tilde{Q}$ and $\\tilde{Q}$ has non-singular dispersion matrix. \n\n\n\n\n\nBelow we state our main theorem. In the next section, we first provide a roadmap of our proof and then fill in the corresponding details. For the rest of the paper, \\emph{we choose our bandwidth $\\sigma_n$ to satisfy $\\frac{\\log{n}}{n \\sigma_n} \\rightarrow 0$}.  \n\n\n\\noindent\n\\begin{remark}\nAs our procedure requires the weaker condition $(\\log{n})\/(n \\sigma_n) \\rightarrow 0$, it is easy to see from the above Theorem that the rate of convergence can be almost as fast as $n\/\\sqrt{\\log{n}}$. \n\\end{remark}\n\\begin{remark}\nOur analysis remains valid in presence of an intercept term. Assume, without loss of generality, that the second co-ordinate of $Q$ is $1$ and let $\\tilde{Q} = (Q_3, \\dots, Q_p)$. It is not difficult to check that all our calculations go through under this new definition of $\\tilde Q$. We, however, avoid this scenario for simplicity of exposition. \n\\end{remark}\n\\vspace{0.2in}\n\\noindent\n{\\bf Proof sketch: }We now provide a roadmap of the proof of Theorem \\ref{thm:binary} in this paragraph while the elaborate technical derivations in the later part. \nDefine the following: $$T_n(\\psi) = \\nabla \\mathbb{M}_n^s(\\psi)= -\\frac{1}{n\\sigma_n}\\sum_{i=1}^n (Y_i - \\gamma)K'\\left(\\frac{Q_i^{\\top}\\psi}{\\sigma_n}\\right)\\tilde{Q}_i$$ $$Q_n(\\psi) = \\nabla^2 \\mathbb{M}_n^s(\\psi) = -\\frac{1}{n\\sigma_n^2}\\sum_{i=1}^n (Y_i - \\gamma)K''\\left(\\frac{Q_i^{\\top}\\psi}{\\sigma_n}\\right)\\tilde{Q}_i\\tilde{Q}_i^{\\top}$$ As $\\hat{\\psi}^s$ minimizes $\\mathbb{M}^s_n(\\psi)$ we have $T_n(\\hat{\\psi}^s) = 0$. Using one step Taylor expansion we have:\n\\allowdisplaybreaks \n\\begin{align*}\nT_n(\\hat{\\psi}^s) = T_n(\\psi_0) + Q_n(\\psi^*_n)\\left(\\hat{\\psi}^s - \\psi_0\\right) = 0\n\\end{align*}\nor: \n\\begin{equation}\n\\label{eq:main_eq}  \\sqrt{n\/\\sigma_n}\\left(\\hat{\\psi}^s - \\psi_0\\right) = -\\left(\\sigma_nQ_n(\\psi^*_n)\\right)^{-1}\\sqrt{n\\sigma_n}T_n(\\psi_0) \n\\end{equation}\nfor some intermediate point $\\psi^*_n$ between $\\hat \\psi^s$ and $\\psi_0$. The following lemma establishes the asymptotic properties of $T_n(\\psi_0)$: \n\\begin{lemma}[Asymptotic Normality of $T_n$]\n\\label{asymp-normality}\n\\label{asymp-normality}\nIf $n\\sigma_n^{3} \\rightarrow \\lambda$, then \n$$\n\\sqrt{n \\sigma_n} T_n(\\psi_0) \\Rightarrow \\mathcal{N}(\\mu, \\Sigma)\n$$\nwhere \n$$\\mu = -\\sqrt{\\lambda}\\frac{\\beta_0 - \\alpha_0}{2}\\left[\\int_{-1}^{1} K'\\left(t\\right)|t| \\ dt \\right] \\int_{\\mathbb{R}^{p-1}}\\tilde{Q} f'(0 | \\tilde{Q}) \\ dP(\\tilde{Q})\n$$ \nand \n$$\\Sigma = \\left[a_1 \\int_{-1}^{0} \\left(K'\\left(t\\right)\\right)^2  \\ dt + a_2 \\int_{0}^{1} \\left(K'\\left(t\\right)\\right)^2 \\ dt \\right]\\int_{\\mathbb{R}^{p-1}}\\tilde{Q}\\tilde{Q}^{\\top} f(0|\\tilde{Q})  \\ dP(\\tilde{Q}) \\,.\n$$ \nHere $a_1 = (1 - \\gamma)^2 \\alpha_0 + \\gamma^2 (1-\\alpha_0), a_2 = (1 - \\gamma)^2 \\beta_0 + \\gamma^2 (1-\\beta_0)$ and $\\alpha_0, \\beta_0, \\gamma$ are model parameters defined around equation \\eqref{eq:new_loss}. \n\\end{lemma}\n\\noindent\nIn the case that $n \\sigma_n^3 \\rightarrow 0$, which, holds when $n\\sigma_n \\rightarrow 0$ as assumed prior to the statement of the theorem, $\\lambda = 0$ and we have: \n$$\\sqrt{n \\sigma_n} T_n(\\psi_0) \\rightarrow \\mathcal{N}(0, \\Sigma) \\,.$$ \nNext, we analyze the convergence of $Q_n(\\psi^*_n)^{-1}$ which is stated in the following lemma: \n\\begin{lemma}[Convergence in Probability of $Q_n$]\n\\label{conv-prob}\nUnder Assumptions (\\ref{as:distribution} - \\ref{as:eigenval_bound}), for any random sequence $\\breve{\\psi}_n$ such that $\\|\\breve{\\psi}_n - \\psi_0\\|\/\\sigma_n \\overset{P} \\rightarrow 0$, \n$$\n\\sigma_n Q_n(\\breve{\\psi}_n) \\overset{P} \\rightarrow Q = \\frac{\\beta_0 - \\alpha_0}{2}\\left(\\int_{-1}^{1} -K''\\left(t \\right)\\text{sign}(t) \\ dt\\right) \\ \\mathbb{E}\\left(\\tilde{Q}\\tilde{Q}^{\\top} f(0 |\\tilde{Q})\\right) \\,.\n$$\n\\end{lemma}\nIt will be shown later that the condition $\\|\\breve{\\psi}_n - \\psi_0\\|\/\\sigma_n \\overset{P} \\rightarrow 0$ needed in Lemma \\ref{conv-prob} holds for the (random) sequence $\\psi^*_n$. Then, combining Lemma \\ref{asymp-normality} and Lemma \\ref{conv-prob} we conclude from equation \\ref{eq:main_eq} that: \n$$\n\\sqrt{n\/\\sigma_n} \\left(\\hat{\\psi}^s - \\psi_0\\right) \\Rightarrow N(0, Q^{-1}\\Sigma Q^{-1}) \\,.\n$$ \nThis concludes the proof of the our Theorem \\ref{thm:binary} with $\\Gamma = Q^{-1}\\Sigma Q^{-1}$. \n\\newline\n\\newline\nObserve that, to show $\\left\\|\\psi^*_n - \\psi_0 \\right\\| = o_P(\\sigma_n)$, it suffices to to prove that $\\left\\|\\hat \\psi^s - \\psi_0 \\right\\| = o_P(\\sigma_n)$. Towards that direction, we have following lemma: \n\n\\begin{lemma}[Rate of convergence]\n\\label{lem:rate}\nUnder Assumptions (\\ref{as:distribution} - \\ref{as:eigenval_bound}), \n$$\nn^{2\/3}\\sigma_n^{-1\/3}  d^2_n\\left(\\hat \\psi^s, \\psi_0^s\\right) = O_P(1) \\,,\n$$ \nwhere \n$$\nd_n\\left(\\psi, \\psi_0^s\\right) = \\sqrt{\\left[\\frac{\\|\\psi - \\psi_0^s\\|^2}{\\sigma_n} \\mathds{1}(\\|\\psi - \\psi_0^s\\| \\le \\mathcal{K}\\sigma_n) + \\|\\psi - \\psi_0^s\\|  \\mathds{1}(\\|\\psi - \\psi_0^s\\| \\ge \\mathcal{K}\\sigma_n)\\right]}\n$$\nfor some specific constant $\\mathcal{K}$. (This constant will be mentioned precisely in the proof). \n\\end{lemma}\n\n\\noindent\nThe lemma immediately leads to the following corollary: \n\n\\begin{corollary}\n\\label{rate-cor}\nIf $n\\sigma_n \\rightarrow \\infty$ then $\\|\\hat \\psi^s - \\psi_0^s\\|\/\\sigma_n  \\overset{P}  \\longrightarrow 0$.\n\\end{corollary}\n\n\\noindent\nFinally, to establish $\\|\\hat \\psi^s - \\psi_0\\|\/\\sigma_n \\overset{P} \\rightarrow 0$, all we need is that $\\|\\psi_0^s - \\psi_0\\|\/\\sigma_n \\rightarrow 0$ as demonstrated in the following lemma:\n\n\\begin{lemma}[Convergence of population minimizer]\n\\label{bandwidth}\nFor any sequence of $\\sigma_n \\rightarrow 0$, we have: $\\|\\psi_0^s - \\psi_0\\|\/\\sigma_n \\rightarrow 0$. \n\\end{lemma}\n\n\\noindent\nHence the final roadmap is the following: Using Lemma \\ref{bandwidth} and Corollary \\ref{rate-cor} we establish that $\\|\\hat \\psi^s - \\psi_0\\|\/\\sigma_n \\rightarrow 0$ if $n\\sigma_n \\rightarrow \\infty$. This, in turn, enables us to prove that $\\sigma_n Q_n(\\psi^*_n) \\overset{P} \\rightarrow Q$,which, along with Lemma \\ref{asymp-normality}, establishes the main theorem. \n\n\\begin{remark}\n\\label{rem:gamma}\nIn the above analysis, we have assumed knowledge of $\\gamma$ in between $(\\alpha_0, \\beta_0)$. However, all our calculations go through if we replace $\\gamma$ by its estimate (say $\\bar Y$) with more tedious book-keeping. One way to simplify the calculations is to split the data into two halves, estimate $\\gamma$ (via $\\bar Y$) from the first half and then use it as a proxy for $\\gamma$ in the second half of the data to estimate $\\psi_0$. As this procedure does not add anything of interest to the core idea of our proof, we refrain from doing so here.  \n\\end{remark}\n\n\\subsection{Variant of quadratic loss function}\n\\label{loss_func_eq}\nIn this sub-section we argue why the loss function in \\eqref{eq:new_loss} is a variant of the quadratic loss function for any $\\gamma \\in (\\alpha_0, \\beta_0)$. Assume that we know $\\alpha_0, \\beta_0$ and seek to estimate $\\psi_0$. We start with an expansion of the quadratic loss function: \n\\begin{align*}\n& \\mathbb{E}\\left(Y - \\alpha_0\\mathds{1}_{Q^{\\top}\\psi \\le 0} - \\beta_0 \\mathds{1}_{Q^{\\top}\\psi > 0}\\right)^2  \\\\\n& = \\mathbb{E}\\left(\\mathbb{E}\\left(Y - \\alpha_0\\mathds{1}_{Q^{\\top}\\psi \\le 0} - \\beta_0 \\mathds{1}_{Q^{\\top}\\psi > 0}\\right)^2 \\ | X\\right) \\\\\n& = \\mathbb{E}_{Q}\\left(\\mathbb{E}\\left( Y^2 \\mid Q \\right) \\right) + \\mathbb{E}_{Q}\\left(\\alpha_0\\mathds{1}_{Q^{\\top}\\psi \\le 0} + \\beta_0 \\mathds{1}_{Q^{\\top}\\psi > 0}\\right)^2 \\\\\n& \\qquad \\qquad \\qquad -2 \\mathbb{E}_{Q}\\left(\\left(\\alpha_0\\mathds{1}_{Q^{\\top}\\psi \\le 0} + \\beta_0 \\mathds{1}_{Q^{\\top}\\psi > 0}\\right) \\mathbb{E}(Y \\mid Q)\\right) \\\\\n& = \\mathbb{E}_Q\\left(\\mathbb{E}\\left( Y \\mid Q \\right) \\right) + \\mathbb{E}_Q\\left(\\alpha_0\\mathds{1}_{Q^{\\top}\\psi \\le 0} + \\beta_0 \\mathds{1}_{Q^{\\top}\\psi > 0}\\right)^2 \\\\\n& \\qquad \\qquad \\qquad -2 \\mathbb{E}_Q\\left(\\left(\\alpha_0\\mathds{1}_{Q^{\\top}\\psi \\le 0} + \\beta_0 \\mathds{1}_{Q^{\\top}\\psi > 0}\\right) \\mathbb{E}(Y \\mid Q)\\right) \\\\\n\\end{align*}\nSince the first summand is just $\\mathbb{E} Y$, it is irrelevant to the minimization. A cursory inspection shows that it suffices to minimize\n\\begin{align}\n& \\mathbb{E}\\left(\\left(\\alpha_0\\mathds{1}_{Q^{\\top}\\psi \\le 0} + \\beta_0 \\mathds{1}_{Q^{\\top}\\psi > 0}\\right) - \\mathbb{E}(Y \\mid Q)\\right)^2 \\notag\\\\\n\\label{eq:lse_1} & = (\\beta_0 - \\alpha_0)^2 \\P\\left(\\text{sign}(Q^{\\top}\\psi) \\neq \\text{sign}(Q^{\\top}\\psi_0)\\right)\n\\end{align}\nOn the other hand the loss we are considering is $\\mathbb{E}\\left((Y - \\gamma)\\mathds{1}_{Q^{\\top}\\psi \\le 0}\\right)$: \n\\begin{align}\n\\label{eq:lse_2} \\mathbb{E}\\left((Y - \\gamma)\\mathds{1}_{Q^{\\top}\\psi \\le 0}\\right) & = (\\beta_0 - \\gamma)\\P(Q^{\\top}\\psi_0 > 0 , Q^{\\top}\\psi \\le 0) \\notag \\\\\n& \\hspace{10em}+ (\\alpha_0 - \\gamma)\\P(Q^{\\top}\\psi_0 \\le 0, Q^{\\top}\\psi \\le 0)\\,,\n\\end{align}\nwhich can be rewritten as: \n\\begin{align*}\n& (\\alpha_0 - \\gamma)\\P(X^{\\top} \\psi_0 \\leq 0) + (\\beta_0 - \\gamma)\\,\\P(X^{\\top} \\psi_0 > 0, X^{\\top} \\psi \\leq 0) \\\\\n& \\qquad \\qquad \\qquad + (\\gamma - \\alpha_0)\\,P (X^{\\top} \\psi_0 \\leq 0, X^{\\top} \\psi > 0) \\,.\n\\end{align*}\nBy Assumption \\ref{as:distribution}, for $\\psi \\neq \\psi_0$, $\\P\\left(\\text{sign}(Q^{\\top}\\psi) \\neq \\text{sign}(Q^{\\top}\\psi_0)\\right) > 0$. As an easy consequence, equation \\eqref{eq:lse_1} is uniquely minimized at $\\psi = \\psi_0$. To see that the same is true for \\eqref{eq:lse_2} when $\\gamma \\in (\\alpha_0, \\beta_0)$, note that the first summand in the equation does not depend on $\\psi$, that the second and third summands are both non-negative and that at least one of these must be positive under Assumption \\ref{as:distribution}. \n\\subsection{Linear curvature of the population score function}\nBefore going into the proofs of the Lemmas and the Theorem, we argue that the population score function $M(\\psi)$ has linear curvature near $\\psi_0$, which is useful in proving Lemma \\ref{lem:rate}. We begin with the following observation: \n\\begin{lemma}[Curvature of population risk]\n\\label{lem:linear_curvature}\nUnder Assumption \\ref{as:differentiability} we have: $$u_- \\|\\psi - \\psi_0\\|_2 \\le \\mathbb{M}(\\psi) - \\mathbb{M}(\\psi_0) \\le u_+ \\|\\psi - \\psi_0\\|_2$$ for some constants $0 < u_- < u_+ < \\infty$, for all $\\psi \\in \\psi$. \n\\end{lemma}\n\\begin{proof}\nFirst, we show that \n$$\n\\mathbb{M}(\\psi) - \\mathbb{M}(\\psi_0) = \\frac{(\\beta_0 - \\alpha_0)}{2} \\P(\\text{sign}(Q^{\\top}\\psi) \\neq X^{\\top}(\\psi_0))\n$$ which follows from the calculation below:\n\\begin{align*}\n& \\mathbb{M}(\\psi) - \\mathbb{M}(\\psi_0)  \\\\\n& = \\mathbb{E}\\left((Y - \\gamma)\\mathds{1}(Q^{\\top}\\psi \\le 0)\\right) - \\mathbb{E}\\left((Y - \\gamma)\\mathds{1}(Q^{\\top}\\psi_0 \\le 0)\\right) \\\\\n& = \\frac{\\beta_0 - \\alpha_0}{2} \\mathbb{E}\\left(\\left\\{\\mathds{1}(Q^{\\top}\\psi \\le 0) - \\mathds{1}(Q^{\\top}\\psi_0 \\le 0)\\right\\}\\left\\{\\mathds{1}(Q^{\\top}\\psi_0 \\ge 0) - \\mathds{1}(Q^{\\top}\\psi_0 \\le 0)\\right\\}\\right) \\\\\n& = \\frac{\\beta_0 - \\alpha_0}{2} \\mathbb{E}\\left(\\left\\{\\mathds{1}(Q^{\\top}\\psi \\le 0, Q^{\\top}\\psi_0 \\ge 0) - \\mathds{1}(Q^{\\top}\\psi \\le 0, Q^{\\top}\\psi_0 \\le 0) + \\mathds{1}(Q^{\\top}\\psi_0 \\le 0)\\right\\}\\right) \\\\\n& = \\frac{\\beta_0 - \\alpha_0}{2} \\mathbb{E}\\left(\\left\\{\\mathds{1}(Q^{\\top}\\psi \\le 0, Q^{\\top}\\psi_0 \\ge 0) + \\mathds{1}(Q^{\\top}\\psi \\ge 0, Q^{\\top}\\psi_0 \\le 0)\\right\\}\\right) \\\\\n& = \\frac{\\beta_0 - \\alpha_0}{2} \\P(\\text{sign}(Q^{\\top}\\psi) \\neq \\text{sign}(Q^{\\top}\\psi_0)) \\,.\n\\end{align*}\nWe now analyze the probability of the wedge shaped region, the region between the two hyperplanes $Q^{\\top}\\psi = 0$ and $Q^{\\top}\\psi_0 = 0$. Note that, \n\\allowdisplaybreaks\n\\begin{align}\n& \\P(Q^{\\top}\\psi > 0 > Q^{\\top}\\psi_0) \\notag\\\\\n& = \\P(-\\tilde{Q}^{\\top}\\tilde{\\psi} < X_1 < -\\tilde{Q}^{\\top}\\tilde{\\psi}_0) \\notag\\\\\n\\label{lin1} & = \\mathbb{E}\\left[\\left(F_{X_1 | \\tilde{Q}}\\left(-\\tilde{Q}^{\\top}\\tilde{\\psi}_0\\right) - F_{X_1 | \\tilde{Q}}\\left(-\\tilde{Q}^{\\top}\\tilde{\\psi}\\right)\\right)\\mathds{1}\\left(\\tilde{Q}^{\\top}\\tilde{\\psi}_0 \\le \\tilde{Q}^{\\top}\\tilde{\\psi}\\right)\\right]\n\\end{align}\nA similar calculation yields\n\\allowdisplaybreaks\n\\begin{align}\n\\label{lin2}  \\P(Q^{\\top}\\psi < 0 < Q^{\\top}\\psi_0) & = \\mathbb{E}\\left[\\left(F_{X_1 | \\tilde{Q}}\\left(-\\tilde{Q}^{\\top}\\tilde{\\psi}\\right) - F_{X_1 | \\tilde{Q}}\\left(-\\tilde{Q}^{\\top}\\tilde{\\psi}_0\\right)\\right)\\mathds{1}\\left(\\tilde{Q}^{\\top}\\tilde{\\psi}_0 \\ge \\tilde{Q}^{\\top}\\tilde{\\psi}\\right)\\right]\n\\end{align}\nAdding both sides of equation \\ref{lin1} and \\ref{lin2} we get: \n\\begin{equation}\n\\label{wedge_expression}\n\\P(\\text{sign}(Q^{\\top}\\psi) \\neq \\text{sign}(Q^{\\top}\\psi_0)) =  \\mathbb{E}\\left[\\left|F_{X_1 | \\tilde{Q}}\\left(-\\tilde{Q}^{\\top}\\tilde{\\psi}\\right) - F_{X_1 | \\tilde{Q}}\\left(-\\tilde{Q}^{\\top}\\tilde{\\psi}_0\\right)\\right|\\right]\n\\end{equation}\nDefine $\\psi_{\\max} = \\sup_{\\psi \\in \\psi}\\|\\psi\\|$, which is finite by Assumption \\ref{as:distribution}. Below, we establish the lower bound:\n\\allowdisplaybreaks\n\\begin{align*}\n& \\P(\\text{sign}(Q^{\\top}\\psi) \\neq \\text{sign}(Q^{\\top}\\psi_0)) \\notag\\\\\n& =  \\mathbb{E}\\left[\\left|F_{X_1 | \\tilde{Q}}\\left(-\\tilde{Q}^{\\top}\\tilde{\\psi}\\right) - F_{X_1 | \\tilde{Q}}\\left(-\\tilde{Q}^{\\top}\\tilde{\\psi}_0\\right)\\right|\\right] \\\\\n& \\ge  \\mathbb{E}\\left[\\left|F_{X_1 | \\tilde{Q}}\\left(-\\tilde{Q}^{\\top}\\tilde{\\psi}\\right) - F_{X_1 | \\tilde{Q}}\\left(-\\tilde{Q}^{\\top}\\tilde{\\psi}_0\\right)\\right|\\mathds{1}\\left(\\left|\\tilde{Q}^{\\top}\\tilde{\\psi}\\right| \\vee \\left| \\tilde{Q}^{\\top}\\tilde{\\psi}_0\\right| \\le \\delta\\right)\\right] \\hspace{0.2in} [\\delta \\ \\text{as in Assumption \\ref{as:differentiability}}]\\\\\n& \\ge  \\mathbb{E}\\left[\\left|F_{X_1 | \\tilde{Q}}\\left(-\\tilde{Q}^{\\top}\\tilde{\\psi}\\right) - F_{X_1 | \\tilde{Q}}\\left(-\\tilde{Q}^{\\top}\\tilde{\\psi}_0\\right)\\right|\\mathds{1}\\left(\\|\\tilde{Q}\\| \\le \\delta\/\\psi_{\\max}\\right)\\right] \\\\\n& \\ge  t \\mathbb{E}\\left[\\left| \\tilde{Q}^{\\top}(\\psi - \\psi_0)\\right| \\mathds{1}\\left(\\|\\tilde{Q}\\| \\le \\delta\/\\psi_{\\max}\\right)\\right] \\\\\n& = t \\|\\psi - \\psi_0\\| \\,\\mathbb{E}\\left[\\left| \\tilde{Q}^{\\top}\\frac{(\\psi - \\psi_0)}{\\|\\psi - \\psi_0\\|}\\right| \\mathds{1}\\left(\\|\\tilde{Q}\\| \\le \\delta\/\\psi_{\\max}\\right)\\right] \\\\\n& \\ge t\\|\\psi - \\psi_0\\| \\inf_{\\gamma \\in S^{p-1}}\\mathbb{E}\\left[\\left| \\tilde{Q}^{\\top}\\gamma\\right| \\mathds{1}\\left(\\|\\tilde{Q}\\| \\le \\delta\/\\psi_{\\max}\\right)\\right] \\\\\n& = u_-\\|\\psi - \\psi_0\\| \\,.\n\\end{align*} \nAt the very end, we have used the fact that $$\\inf_{\\gamma \\in S^{p-1}}\\mathbb{E}\\left[\\left| \\tilde{Q}^{\\top}\\gamma\\right| \\mathds{1}\\left(\\|\\tilde{Q}\\| \\le \\delta\/\\psi_{\\max}\\right)\\right] > 0$$ To prove this, assume that the infimum is 0. Then, there exists $\\gamma_0 \\in S^{p-1}$ such that \n$$\\mathbb{E}\\left[\\left| \\tilde{Q}^{\\top}\\gamma_0\\right| \\mathds{1}\\left(\\|\\tilde{Q}\\| \\le \\delta\/\\psi_{\\max}\\right)\\right] = 0 \\,,$$ \nas the above function continuous in $\\gamma$ and any continuous function on a compact set attains its infimum. Hence, $\\left|\\tilde{Q}^{\\top}\\gamma_0 \\right| = 0$ for all $\\|\\tilde{Q}\\| \\le \\delta\/\\psi_{\\max}$, which implies that $\\tilde{Q}$ does not have full support, violating Assumption  \\ref{as:distribution} (2). This gives a contradiction.\n\\\\\\\\\n\\noindent\nEstablishing the upper bound is relatively easier. Going back to equation \\eqref{wedge_expression}, we have: \n\\begin{align*}\n& \\P(\\text{sign}(Q^{\\top}\\psi) \\neq \\text{sign}(Q^{\\top}\\psi_0)) \\notag\\\\\n& =  \\mathbb{E}\\left[\\left|F_{Q_1 | \\tilde{Q}}\\left(-\\tilde{Q}^{\\top}\\tilde{\\psi}\\right) - F_{Q_1 | \\tilde{Q}}\\left(-\\tilde{Q}^{\\top}\\tilde{\\psi}_0\\right)\\right|\\right] \\\\\n& \\le  \\mathbb{E}\\left[m(\\tilde Q) \\, \\|Q\\| \\,\\|\\psi- \\psi_0\\|\\right] \\hspace{0.2in} [m(\\cdot) \\ \\text{is defined in Assumption \\ref{as:density_bound}}]\\\\\n& \\le u_+ \\|\\psi - \\psi_0\\| \\,,\n\\end{align*}\nas $ \\mathbb{E}\\left[m(\\tilde Q) \\|Q\\|\\right] < \\infty$  by Assumption \\ref{as:density_bound} and the sub-Gaussianity of $\\tilde X$. \n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\\subsection{Proof of Lemma \\ref{asymp-normality}}\n\\begin{proof}\nWe first prove that under our assumptions $\\sigma_n^{-1} \\mathbb{E}(T_n(\\psi_0)) \\overset{n \\to \\infty}\\longrightarrow A$ where $$A = -\\frac{\\beta_0 - \\alpha_0}{2!}\\left[\\int_{-\\infty}^{\\infty} K'\\left(t\\right)|t| \\ dt \\right] \\int_{\\mathbb{R}^{p-1}}\\tilde{Q}f_0'(0 | \\tilde{Q}) \\ dP(\\tilde{Q})$$ The proof is based on Taylor expansion of the conditional density: \n\\allowdisplaybreaks\n\\begin{align*}\n& \\sigma_n^{-1} \\mathbb{E}(T_n(\\psi_0)) \\\\\n& =  -\\sigma_n^{-2}\\mathbb{E}\\left((Y - \\gamma)K'\\left(\\frac{Q^{\\top}\\psi_0}{\\sigma_n}\\right)\\tilde{Q}\\right) \\\\\n& =  -\\frac{\\beta_0 - \\alpha_0}{2}\\sigma_n^{-2}\\mathbb{E}\\left(K'\\left(\\frac{Q^{\\top}\\psi_0}{\\sigma_n}\\right)\\tilde{Q}(\\mathds{1}(Q^{\\top}\\psi_0 \\ge 0) - \\mathds{1}(Q^{\\top}\\psi_0 \\le 0))\\right) \\\\\n& =  -\\frac{\\beta_0 - \\alpha_0}{2}\\sigma_n^{-2}\\int_{\\mathbb{R}^{p-1}}\\tilde{Q}\\left[\\int_{0}^{\\infty} K'\\left(\\frac{z}{\\sigma_n}\\right)f_0(z|\\tilde{Q}) \\ dz - \\int_{-\\infty}^{0} K'\\left(\\frac{z}{\\sigma_n}\\right)f_0(z|\\tilde{Q}) \\ dz \\right] \\ dP(\\tilde{Q}) \\\\\n& =  -\\frac{\\beta_0 - \\alpha_0}{2}\\sigma_n^{-1}\\int_{\\mathbb{R}^{p-1}}\\tilde{Q}\\left[\\int_{0}^{\\infty} K'\\left(t\\right)f_0(\\sigma_n t|\\tilde{Q}) \\ dt - \\int_{-\\infty}^{0} K'\\left(t\\right)f_0(\\sigma_n t |\\tilde{Q}) \\ dt \\right] \\ dP(\\tilde{Q}) \\\\\n& =  -\\frac{\\beta_0 - \\alpha_0}{2}\\sigma_n^{-1}\\left[\\int_{\\mathbb{R}^{p-1}}\\tilde{Q}\\left[\\int_{0}^{\\infty} K'\\left(t\\right)f_0(0|\\tilde{Q}) \\ dt - \\int_{-\\infty}^{0} K'\\left(t\\right)f_0(0 |\\tilde{Q}) \\ dt \\right] \\ dP(\\tilde{Q}) \\right. \\\\ \n& \\qquad \\qquad \\qquad + \\left. \\int_{\\mathbb{R}^{p-1}}\\sigma_n \\left[\\int_{0}^{\\infty} K'\\left(t\\right)tf_0'(\\lambda \\sigma_n t|\\tilde{Q}) \\ dt - \\int_{-\\infty}^{0} K'\\left(t\\right) t f_0'(\\lambda \\sigma_n t |\\tilde{Q}) \\ dt \\right] \\ dP(\\tilde{Q}) \\right] \\hspace{0.2in} [0 < \\lambda < 1]\\\\ \n& = -\\frac{\\beta_0 - \\alpha_0}{2}\\int_{\\mathbb{R}^{p-1}}\\tilde{Q}\\left[\\int_{0}^{\\infty} k\\left(t\\right)tf_0'(\\lambda \\sigma_n t|\\tilde{Q}) \\ dz - \\int_{-\\infty}^{0} k\\left(t\\right)tf_0'(\\lambda \\sigma_nt |\\tilde{Q}) \\ dz \\right] \\ dP(\\tilde{Q})\\\\\n& \\underset{n \\rightarrow \\infty} \\longrightarrow -\\frac{\\beta_0 - \\alpha_0}{2}\\left[\\int_{-\\infty}^{\\infty} k\\left(t\\right)|t| \\ dt \\right] \\int_{\\mathbb{R}^{p-1}}\\tilde{Q}f_0'(0 | \\tilde{Q}) \\ dP(\\tilde{Q})\n\\end{align*}\nNext, we prove that $\\mbox{Var}\\left(\\sqrt{n\\sigma_n}T_n(\\psi_0)\\right)\\longrightarrow \\Sigma$ as $n \\rightarrow \\infty$, where $\\Sigma$ is as defined in Lemma \\ref{asymp-normality}. Note that: \n\\allowdisplaybreaks\n\\begin{align*}\n\\mbox{Var}\\left(\\sqrt{n\\sigma_n}T_n(\\psi_0)\\right) & = \\sigma_n \\mathbb{E}\\left((Y - \\gamma)^2\\left(K'\\left(\\frac{Q^{\\top}\\psi_0}{\\sigma_n}\\right)^2\\frac{\\tilde{Q}\\tilde{Q}^{\\top}}{\\sigma_n^2}\\right)\\right) - \\sigma_n \\mathbb{E}(T_n(\\psi_0))\\mathbb{E}(T_n(\\psi_0))^{\\top}\n\\end{align*}\nAs $\\sigma_n^{-1}\\mathbb{E}(T_n(\\psi_0)) \\rightarrow A$, we can conclude that $\\sigma_n \\mathbb{E}(T_n(\\psi_0))\\mathbb{E}(T_n(\\psi_0))^{\\top} \\rightarrow 0$. \nDefine $a_1 = (1 - \\gamma)^2 \\alpha_0 + \\gamma^2 (1-\\alpha_0), a_2 = (1 - \\gamma)^2 \\beta_0 + \\gamma^2 (1-\\beta_0)$. For the first summand: \n\\allowdisplaybreaks\n\\begin{align*}\n& \\sigma_n \\mathbb{E}\\left((Y - \\gamma)^2\\left(K^{'^2}\\left(\\frac{Q^{\\top}\\psi_0}{\\sigma_n}\\right)\\frac{\\tilde{Q}\\tilde{Q}^{\\top}}{\\sigma_n^2}\\right)\\right) \\\\\n& = \\frac{1}{\\sigma_n} \\int_{\\mathbb{R}^{p-1}}\\tilde{Q}\\tilde{Q}^{\\top} \\left[a_1 \\int_{-\\infty}^{0} K^{'^2}\\left(\\frac{z}{\\sigma_n}\\right) f(z|\\tilde{Q}) \\ dz \\right. \\notag \\\\ & \\left. \\qquad \\qquad \\qquad + a_2 \\int_{0}^{\\infty}K^{'^2}\\left(\\frac{z}{\\sigma_n}\\right) f(z|\\tilde{Q}) \\ dz \\right] \\ dP(\\tilde{Q})\\\\\n& =  \\int_{\\mathbb{R}^{p-1}}\\tilde{Q}\\tilde{Q}^{\\top} \\left[a_1 \\int_{-\\infty}^{0} K^{'^2}\\left(t\\right)f(\\sigma_n t|\\tilde{Q}) \\ dt + a_2 \\int_{0}^{\\infty} K^{'^2}\\left(t\\right) f(\\sigma_n t |\\tilde{Q}) \\ dt \\right] \\ dP(\\tilde{Q}) \\\\\n& =  \\int_{\\mathbb{R}^{p-1}}\\tilde{Q}\\tilde{Q}^{\\top} \\left[a_1 \\int_{-\\infty}^{0} K^{'^2}\\left(t\\right)f(\\sigma_n t|\\tilde{Q}) \\ dt + a_2 \\int_{0}^{\\infty} K^{'^2}\\left(t\\right) f(\\sigma_n t |\\tilde{Q}) \\ dt \\right] \\ dP(\\tilde{Q}) \\\\\n& \\underset{n \\rightarrow \\infty} \\longrightarrow \\left[a_1 \\int_{-\\infty}^{0} K^{'^2}\\left(t\\right)  \\ dt + a_2 \\int_{0}^{\\infty} K^{'^2}\\left(t\\right) \\ dt \\right]\\int_{\\mathbb{R}^{p-1}}\\tilde{Q}\\tilde{Q}^{\\top} f(0|\\tilde{Q})  \\ dP(\\tilde{Q}) \\ \\ \\overset{\\Delta} = \\Sigma \\, . \n\\end{align*}\nFinally, suppose $n \\sigma_n^{3} \\rightarrow \\lambda$. Define $W_n = \\sqrt{n\\sigma_n}\\left[T_n(\\psi) - \\mathbb{E}(T_n(\\psi))\\right]$. Using Lemma 6 of Horowitz \\cite{horowitz1992smoothed}, it is easily established that $W_n \\Rightarrow N(0, \\Sigma)$. Also, we have: \n\\allowdisplaybreaks \n\\begin{align*}\n\\sqrt{n\\sigma_n}\\mathbb{E}(T_n(\\psi_0)) = \\sqrt{n\\sigma_n^{3}}\\sigma_n^{-1}\\mathbb{E}(T_n(\\psi_0) & \\rightarrow \\sqrt{\\lambda}A = \\mu\n\\end{align*}\nAs $\\sqrt{n\\sigma_n}T_n(\\psi_0) = W_n + \\sqrt{n\\sigma_n}\\mathbb{E}(T_n(\\psi_0))$, we conclude that $\\sqrt{n\\sigma_n} T_n(\\psi_0) \\Rightarrow N(\\mu, \\Sigma)$.\n\\end{proof}\n\\subsection{Proof of Lemma \\ref{conv-prob}}\n\\begin{proof}\nLet $\\epsilon_n \\downarrow 0$ be a sequence such that $\\P(\\|\\breve{\\psi}_n - \\psi_0\\| \\le \\epsilon_n \\sigma_n) \\rightarrow 1$. Define $\\Psi_n = \\{\\psi: \\|\\psi - \\psi_0\\| \\le \\epsilon_n \\sigma_n\\}$. We show that $$\\sup_{\\psi \\in \\psi_n} \\|\\sigma_n Q_n(\\psi) - Q\\|_F \\overset{P} \\to 0$$ where $\\|\\cdot\\|_F$ denotes the Frobenius norm of a matrix. Sometimes, we omit the subscript $F$ when there is no ambiguity. Define $\\mathcal{G}_n$ to be collection of functions: \n$$\n\\mathcal{G}_n= \\left\\{g_{\\psi}(y, q) = -\\frac{1}{\\sigma_n}(y - \\gamma)\\tilde q\\tilde q^{\\top} \\left(K''\\left(\\frac{q^{\\top}\\psi}{\\sigma_n}\\right) - K''\\left(\\frac{q^{\\top}\\psi_0}{\\sigma_n}\\right)\\right), \\psi \\in \\Psi_n \\right\\}\n$$\nThat the function class $\\mathcal{G}_n$ has bounded uniform entropy integral (BUEI) is immediate from the fact that the function $Q \\to Q^{\\top}\\psi$ has finite VC dimension (as the hyperplanes has finite VC dimension) and it does change upon constant scaling. Therefore $Q \\mapsto Q^{\\top}\\psi\/\\sigma_n$ also has finite VC dimension which does not depend on n and hence BUEI. As composition with a monotone function and multiplication with constant (parameter free) functions or multiplication of two BUEI class of functions keeps BUEI property, we conclude that $\\mathcal{G}_n$ has BUEI. \nWe first expand the expression in two terms:\n\\allowdisplaybreaks\n\\begin{align*}\n\\sup_{\\psi \\in \\psi_n} \\|\\sigma_n Q_n(\\psi) - Q\\| & \\le \\sup_{\\psi \\in \\psi_n} \\|\\sigma_n Q_n(\\psi) - \\mathbb{E}(\\sigma_n Q_n(\\psi))\\| + \\sup_{\\psi \\in \\psi_n} \\| \\mathbb{E}(\\sigma_n Q_n(\\psi)) - Q\\| \\\\ \n& = \\|(\\mathbb{P}_n - P)\\|_{\\mathcal{G}_n} + \\sup_{\\psi \\in \\psi_n}\\| \\mathbb{E}(\\sigma_n Q_n(\\psi)) - Q\\| \\\\\n& = T_{1,n} + T_{2,n} \\hspace{0.3in} \\,. [\\text{Say}]\n\\end{align*}\n\n\n\\vspace{0.2in}\n\\noindent\nThat $T_{1,n} \\overset{P} \\to 0$ follows from uniform law of large number of a BUEI class (e.g. combining Theorem 2.4.1 and Theorem 2.6.7 of \\cite{vdvw96}). \nFor uniform convergence of the second summand $T_{n,2}$, define $\\chi_n = \\{\\tilde{Q}: \\|\\tilde{Q}\\| \\le 1\/\\sqrt{\\epsilon_n}\\}$. Then $\\chi_n \\uparrow \\mathbb{R}^{p-1}$. Also for any $\\psi \\in \\Psi_n$, if we define $\\gamma_n \\equiv \\gamma_n(\\psi) = (\\psi - \\psi_0)\/\\sigma_n$, then $|\\tilde \\gamma_n^{\\top}\\tilde{Q}| \\le \\sqrt{\\epsilon_n}$ for all $n$ and for all $\\psi \\in \\Psi_n, \\tilde{Q} \\in \\chi_n$. Now, \n\\allowdisplaybreaks\n\\begin{align*}\n& \\sup_{\\psi \\in \\psi_n}\\| \\mathbb{E}(\\sigma_n Q_n(\\psi)) - Q\\| \\notag \\\\\n&\\qquad \\qquad  = \\sup_{\\psi \\in \\psi_n}\\| (\\mathbb{E}(\\sigma_n Q_n(\\psi)\\mathds{1}(\\chi_n))-Q_1) +  (\\mathbb{E}(\\sigma_n Q_n(\\psi)\\mathds{1}(\\chi_n^c))-Q_2)\\|\n\\end{align*}\nwhere $$Q_1 =  \\frac{\\beta_0 - \\alpha_0}{2}\\left(\\int_{-\\infty}^{\\infty} -K''\\left(t \\right)\\text{sign}(t) \\ dt\\right) \\ \\mathbb{E}\\left(\\tilde{Q}\\tilde{Q}^{\\top} f_0(0 |\\tilde{Q})\\mathds{1}(\\chi_n) \\right)$$ $$Q_2 =  \\frac{\\beta_0 - \\alpha_0}{2}\\left(\\int_{-\\infty}^{\\infty} -K''\\left(t \\right)\\text{sign}(t) \\ dt\\right) \\ \\mathbb{E}\\left(\\tilde{Q}\\tilde{Q}^{\\top} f(0 |\\tilde{Q})\\mathds{1}(X_n^c) \\right) \\,.$$\nNote that \n\\allowdisplaybreaks\n\\begin{flalign}\n& \\|\\mathbb{E}(\\sigma_n Q_n(\\psi)\\mathds{1}(\\chi_n)) - Q_1\\| \\notag\\\\\n& =\\left\\| \\frac{\\beta_0 - \\alpha_0}{2}\\left[\\int_{\\chi_n} \\tilde{Q}\\tilde{Q}^{\\top} \\left[\\int_{-\\infty}^{\\tilde{Q}^{\\top}\\gamma_n} K''\\left(t \\right) f_0(\\sigma_n (t-\\tilde{Q}^{\\top}\\gamma_n) |\\tilde{Q}) \\ dt \\right. \\right. \\right. \\notag \\\\\n& \\left. \\left. \\left. \\qquad \\qquad - \\int_{\\tilde{Q}^{\\top}\\gamma_n}^{\\infty} K''\\left(t\\right) f_0(\\sigma_n (t - \\tilde{Q}^{\\top}\\gamma_n) | \\tilde{Q}) \\ dt \\right]dP(\\tilde{Q})\\right]\\right. \\notag\\\\ & \\left. \\qquad \\qquad \\qquad - \\frac{\\beta_0 - \\alpha_0}{2}\\left[\\int_{\\chi_n} \\tilde{Q}\\tilde{Q}^{\\top} f(0 |\\tilde{Q})\\left[\\int_{-\\infty}^{0} K''\\left(t \\right) \\ dt - \\int_{0}^{\\infty} K''\\left(t\\right) \\ dt \\right]dP(\\tilde{Q})\\right] \\right \\|\\notag\\\\\n& =\\left \\| \\frac{\\beta_0 - \\alpha_0}{2}\\left[\\int_{\\chi_n} \\tilde{Q}\\tilde{Q}^{\\top} \\left[\\int_{-\\infty}^{\\tilde{Q}^{\\top}\\gamma_n} K'''\\left(t \\right) (f_0(\\sigma_n (t-\\tilde{Q}^{\\top}\\gamma_n) |\\tilde{Q})-f_0(0 | \\tilde{Q})) \\ dt \\right. \\right. \\right.\\notag\\\\& \\qquad \\qquad- \\left. \\left. \\left. \\int_{\\tilde{Q}^{\\top}\\gamma_n}^{\\infty} K''\\left(t\\right) (f_0(\\sigma_n (t - \\tilde{Q}^{\\top}\\gamma_n) | \\tilde{Q}) - f_0(0 | \\tilde{Q})) \\ dt \\right]dP(\\tilde{Q})\\right]\\right. \\notag\\\\ & \\qquad \\qquad \\qquad + \\left. \\frac{\\beta_0 - \\alpha_0}{2}\\left[\\int_{\\chi_n} \\tilde{Q}\\tilde{Q}^{\\top} f_0(0 |\\tilde{Q}) \\left[\\int_{-\\infty}^{\\tilde{Q}^{\\top}\\gamma_n} K''\\left(t \\right) \\ dt - \\int_{-\\infty}^{0} K''\\left(t \\right) \\ dt \\right. \\right. \\right. \\notag \\\\ \n& \\qquad \\qquad \\qquad \\qquad \\left. \\left. \\left. + \\int_{\\tilde{Q}^{\\top}\\gamma_n}^{\\infty} K''\\left(t \\right) \\ dt - \\int_{0}^{\\infty} K''\\left(t\\right) \\ dt \\right]dP(\\tilde{Q})\\right] \\right \\|\\notag\\\\\n& \\le \\frac{\\beta_0 - \\alpha_0}{2}\\sigma_n  \\int_{\\chi_n}\\|\\tilde{Q}\\tilde{Q}^{\\top}\\|h(\\tilde{Q})\\int_{-\\infty}^{\\infty}|K''(t)||t - \\gamma_n^{\\top}\\tilde{Q}| \\ dt \\ dP(\\tilde{Q}) \\notag\\\\ & \\qquad \\qquad + \\frac{\\beta_0 - \\alpha_0}{2}  \\int_{\\chi_n}\\|\\tilde{Q}\\tilde{Q}^{\\top}\\| f_0(0 | \\tilde{Q}) \\left[\\left| \\int_{-\\infty}^{\\tilde{Q}^{\\top}\\gamma_n} K''\\left(t \\right) \\ dt - \\int_{-\\infty}^{0} K''\\left(t \\right) \\ dt \\right| \\right. \\notag \\\\ & \\left. \\qquad \\qquad \\qquad + \\left| \\int_{\\tilde{Q}^{\\top}\\gamma_n}^{\\infty} K''\\left(t \\right) \\ dt - \\int_{0}^{\\infty} K''\\left(t\\right) \\ dt \\right|\\right] \\ dP(\\tilde{Q})\\notag\\\\\n&  \\le \\frac{\\beta_0 - \\alpha_0}{2}\\left[\\sigma_n  \\int_{\\chi_n}\\|\\tilde{Q}\\tilde{Q}^{\\top}\\|h(\\tilde{Q})\\int_{-\\infty}^{\\infty}|K''(t)||t - \\gamma_n^{\\top}\\tilde{Q}| \\ dt \\ dP(\\tilde{Q}) \\right. \\notag \\\\ \n& \\left. \\qquad \\qquad \\qquad  + 2\\int_{\\chi_n}\\|\\tilde{Q}\\tilde{Q}^{\\top}\\| f_0(0 | \\tilde{Q}) (K'(0) - K'(\\gamma_n^{\\top}\\tilde{Q})) \\ dP(\\tilde{Q})\\right]\\notag \\\\\n\\label{cp1}&\\rightarrow 0 \\hspace{0.3in} [\\text{As} \\ n \\rightarrow \\infty] \\,,\n\\end{flalign}\nby DCT and Assumptions \\ref{as:distribution} and \\ref{as:derivative_bound}. For the second part: \n\\allowdisplaybreaks\n\\begin{align}\n& \\|\\mathbb{E}(\\sigma_n Q_n(\\psi)\\mathds{1}(\\chi_n^c)) - Q_2\\|\\notag\\\\\n& =\\left\\| \\frac{\\beta_0 - \\alpha_0}{2}\\left[\\int_{\\chi_n^c} \\tilde{Q}\\tilde{Q}^{\\top} \\left[\\int_{-\\infty}^{\\tilde{Q}^{\\top}\\gamma_n} K''\\left(t \\right) f_0(\\sigma_n (t-\\tilde{Q}^{\\top}\\gamma_n) |\\tilde{Q}) \\ dt \\right. \\right. \\right. \\notag \\\\ \n& \\left. \\left. \\left. \\qquad \\qquad - \\int_{\\tilde{Q}^{\\top}\\gamma_n}^{\\infty} K''\\left(t\\right) f_0(\\sigma_n (t - \\tilde{Q}^{\\top}\\gamma_n) | \\tilde{Q}) \\ dt \\right]dP(\\tilde{Q})\\right]\\right. \\notag\\\\ &  \\left. \\qquad \\qquad \\qquad -\\frac{\\beta_0 - \\alpha_0}{2}\\left[\\int_{\\chi_n^c} \\tilde{Q}\\tilde{Q}^{\\top} f_0(0 |\\tilde{Q})\\left[\\int_{-\\infty}^{0} K''\\left(t \\right) \\ dt - \\int_{0}^{\\infty} K''\\left(t\\right) \\ dt \\right]dP(\\tilde{Q})\\right] \\right \\|\\notag\\\\\n& \\le \\frac{\\beta_0 - \\alpha_0}{2} \\int_{\\infty}^{\\infty} |K''(t)| \\ dt \\int_{\\chi_n^c} \\|\\tilde{Q}\\tilde{Q}^{\\top}\\|(m(\\tilde{Q}) + f_0(0|\\tilde{Q})) \\ dP(\\tilde{Q}) \\notag\\\\\n\\label{cp2} & \\rightarrow 0 \\hspace{0.3in} [\\text{As} \\ n \\rightarrow \\infty] \\,,\n\\end{align}\nagain by DCT and Assumptions \\ref{as:distribution} and \\ref{as:density_bound}. Combining equations \\ref{cp1} and \\ref{cp2}, we conclude the proof. \n\\end{proof}\n\n\n\\subsection{Proof of Lemma \\ref{bandwidth}}\nHere we prove that $\\|\\psi^s_0 - \\psi_0\\|\/\\sigma_n \\rightarrow 0$ where $\\psi^s_0$ is the minimizer of $\\mathbb{M}^s(\\psi)$ and $\\psi_0$ is the minimizer of $M(\\psi)$. \n\\begin{proof}\nDefine $\\eta = (\\psi^s_0 -  \\psi_0)\/\\sigma_n$. At first we show that, $\\|\\tilde \\eta\\|_2$ is $O(1)$, i.e. there exists some constant $\\Omega_1$ such that $\\|\\tilde \\eta\\|_2 \\le \\Omega_1$ for all $n$: \n\\begin{align*}\n\\|\\psi^s_0 - \\psi_0\\|_2 & \\le \\frac{1}{u_-} \\left(\\mathbb{M}(\\psi_n) - \\mathbb{M}(\\psi_0)\\right)  \\hspace{0.2in} [\\text{Follows from Lemma} \\ \\ref{lem:linear_curvature}]\\\\\n& \\le \\frac{1}{u_-} \\left(\\mathbb{M}(\\psi_n) - \\mathbb{M}^s(\\psi_n) + \\mathbb{M}^s(\\psi_n) - \\mathbb{M}^s(\\psi_0) + \\mathbb{M}^s(\\psi_0) - \\mathbb{M}(\\psi_0)\\right)  \\\\\n& \\le  \\frac{1}{u_-} \\left(\\mathbb{M}(\\psi_n) - \\mathbb{M}^s(\\psi_n) + \\mathbb{M}^s(\\psi_0) - M(\\psi_0)\\right)  \\hspace{0.2in} [\\because \\mathbb{M}^s(\\psi_n) - \\mathbb{M}^s(\\psi_0) \\le 0]\\\\\n& \\le \\frac{2K_1}{u_-}\\sigma_n \\hspace{0.2in} [\\text{from equation} \\ \\eqref{eq:lin_bound_1}]\n\\end{align*}\n\n\\noindent\nAs $\\psi^s_0$ minimizes $\\mathbb{M}^s(\\psi)$: \n$$\\nabla \\mathbb{M}^s(\\psi^s_0) = -\\mathbb{E}\\left((Y-\\gamma)\\tilde{Q}K'\\left(\\frac{Q^{\\top}\\psi^0_s}{\\sigma_n}\\right)\\right) = 0$$\nHence:\n\\begin{align*}\n0 &= \\mathbb{E}\\left((Y-\\gamma)\\tilde{Q}K'\\left(\\frac{Q^{\\top}\\psi_0^s}{\\sigma_n}\\right)\\right) \\\\\n& = \\frac{(\\beta_0 - \\alpha_0)}{2} \\mathbb{E}\\left(\\tilde{Q}K'\\left(\\frac{Q^{\\top}\\psi_0^s}{\\sigma_n}\\right)\\left\\{\\mathds{1}(Q^{\\top}\\psi_0 \\ge 0) -\\mathds{1}(Q^{\\top}\\psi_0 < 0)\\right\\}\\right) \\\\\n& = \\frac{(\\beta_0 - \\alpha_0)}{2} \\mathbb{E}\\left(\\tilde{Q}K'\\left(\\frac{Q^{\\top}\\psi_0}{\\sigma_n} + \\tilde{\\eta}^{\\top} \\tilde{Q}\\right)\\left\\{\\mathds{1}(Q^{\\top}\\psi_0 \\ge 0) -\\mathds{1}(Q^{\\top}\\psi_0 < 0)\\right\\}\\right) \\\\\n& = \\frac{(\\beta_0 - \\alpha_0)}{2} \\left[\\int_{\\mathbb{R}^{p-1}}\\tilde{Q} \\int_0^{\\infty} K'\\left(\\frac{z}{\\sigma_n} + \\tilde{\\eta}^{\\top} \\tilde{Q}\\right) \\ f_0(z|\\tilde{Q}) \\ dz \\ dP(\\tilde{Q})\\right. \\\\\n& \\qquad \\qquad \\qquad \\qquad \\qquad \\left. - \\int_{\\mathbb{R}^{p-1}}\\tilde{Q} \\int_{-\\infty}^0 K'\\left(\\frac{z}{\\sigma_n} + \\tilde{\\eta}^{\\top} \\tilde{Q}\\right) \\ f_0(z|\\tilde{Q}) \\ dz \\ dP(\\tilde{Q})\\right] \\\\\n& =\\sigma_n \\frac{(\\beta_0 - \\alpha_0)}{2} \\left[\\int_{\\mathbb{R}^{p-1}}\\tilde{Q} \\int_0^{\\infty} K'\\left(t + \\tilde{\\eta}^{\\top} \\tilde{Q}\\right) \\ f_0(\\sigma_n t|\\tilde{Q}) \\ dt \\ dP(\\tilde{Q})\\right. \\\\\n& \\qquad \\qquad \\qquad \\qquad \\qquad \\left. - \\int_{\\mathbb{R}^{p-1}}\\tilde{Q} \\int_{-\\infty}^0 K'\\left(t + \\tilde{\\eta}^{\\top} \\tilde{Q}\\right) \\ f_0(\\sigma_n t|\\tilde{Q}) \\ dz \\ dP(\\tilde{Q})\\right] \n\\end{align*}\nAs $\\sigma_n\\frac{(\\beta_0 - \\alpha_0)}{2} > 0$, we can forget about it and continue. Also, as we have proved $\\|\\tilde \\eta\\| = O(1)$, there exists a subsequence $\\eta_{n_k}$ and a point $c \\in \\mathbb{R}^{p-1}$ such that $\\eta_{n_k} \\rightarrow c$. Along that sub-sequence we have:  \n\\begin{align*}\n0 & = \\left[\\int_{\\mathbb{R}^{p-1}}\\tilde{Q} \\int_0^{\\infty} K'\\left(t + \\tilde{\\eta}_{n_k}^{\\top} \\tilde{Q}\\right) \\ f_0(\\sigma_{n_k} t|\\tilde{Q}) \\ dt \\ dP(\\tilde{Q})\\right. \\\\\n& \\qquad \\qquad \\qquad \\qquad \\qquad \\left. - \\int_{\\mathbb{R}^{p-1}}\\tilde{Q} \\int_{-\\infty}^0 K'\\left(t + \\tilde{\\eta}_{n_k}^{\\top} \\tilde{Q}\\right) \\ f_0(\\sigma_{n_k} t|\\tilde{Q}) \\ dt \\ dP(\\tilde{Q})\\right] \n\\end{align*}\nTaking limits on both sides and applying DCT (which is permissible by DCT) we conclude: \n\\begin{align*}\n0 & = \\left[\\int_{\\mathbb{R}^{p-1}}\\tilde{Q} \\int_0^{\\infty} K'\\left(t +c^{\\top} \\tilde{Q}\\right) \\ f_0(0|\\tilde{Q}) \\ dt \\ dP(\\tilde{Q})\\right. \\\\\n& \\qquad \\qquad \\qquad \\qquad \\qquad \\left. - \\int_{\\mathbb{R}^{p-1}}\\tilde{Q} \\int_{-\\infty}^0 K'\\left(t + c^{\\top} \\tilde{Q}\\right) \\ f_0(0|\\tilde{Q}) \\ dt \\ dP(\\tilde{Q})\\right]  \\\\\n& = \\left[\\int_{\\mathbb{R}^{p-1}}\\tilde{Q}  \\ f_0(0|\\tilde{Q}) \\int_{c^{\\top} \\tilde{Q}}^{\\infty} K'\\left(t\\right) \\ dt \\ dP(\\tilde{Q})\\right. \\\\\n& \\qquad \\qquad \\qquad \\qquad \\qquad \\left. - \\int_{\\mathbb{R}^{p-1}}\\tilde{Q}\\ f_0(0|\\tilde{Q})  \\int_{-\\infty}^{c^{\\top} \\tilde{Q}} K'\\left(t \\right) \\ dt \\ dP(\\tilde{Q})\\right]  \\\\\n& = \\left[\\int_{\\mathbb{R}^{p-1}}\\tilde{Q}  \\ f_0(0|\\tilde{Q}) \\left[1 - K(c^{\\top} \\tilde{Q})\\right] \\ dt \\ dP(\\tilde{Q})\\right. \\\\\n&  \\qquad \\qquad \\qquad \\qquad \\qquad\\left. - \\int_{\\mathbb{R}^{p-1}}\\tilde{Q}\\ f_0(0|\\tilde{Q}) K(c^{\\top} \\tilde{Q}) \\ dt \\ dP(\\tilde{Q})\\right]  \\\\\n& =   \\mathbb{E}\\left(\\tilde{Q} \\left(2K(c^{\\top} \\tilde{Q}) - 1\\right)f_0(0|\\tilde{Q})\\right) \\,.\n\\end{align*}\nNow, taking the inner-products of both sides with respect to $c$, we get: \n\\begin{equation}\n\\label{eq:zero_eq}\n\\mathbb{E}\\left(c^{\\top}\\tilde{Q} \\left(2K(c^{\\top} \\tilde{Q}) - 1\\right)f_0(0|\\tilde{Q})\\right) = 0 \\,.\n\\end{equation}\nBy our assumption that $K$ is symmetric kernel and that $K(t) > 0$ for all $t \\in (-1, 1)$, we easily conclude that $c^{\\top}\\tilde{Q} \\left(2K(c^{\\top} \\tilde{Q}) - 1\\right) \\ge 0$ almost surely in $\\tilde{Q}$ with equality iff $c^{\\top}X = 0$, which is not possible unless $c = 0$. Hence we conclude that $c = 0$. This shows that any convergent subsequence of $\\eta_n$ converges to $0$, which completes the proof. \n\\end{proof}\n\n\n\n\\subsection{Proof of Lemma \\ref{lem:rate}}\n\\begin{proof}\nTo obtain the rate of convergence of our kernel smoothed estimator we use Theorem 3.4.1 of \\cite{vdvw96}: There are three key ingredients that one needs to take care of if in order to apply this theorem:   \n\\begin{enumerate}\n\\item Consistency of the estimator (otherwise the conditions of the theorem needs to be valid for all $\\eta$). \n\\item The curvature of the population score function near its minimizer.\n\\item A bound on the modulus of continuity in a vicinity of the minimizer of the population score function. \n\\end{enumerate}\nBelow, we establish the curvature of the population score function (item 2 above) globally, thereby obviating the need to establish consistency separately. Recall that the population score function was defined as: \n$$\n\\mathbb{M}^s(\\psi) = \\mathbb{E}\\left((Y - \\gamma)\\left(1 - K\\left(\\frac{Q^{\\top}\\psi}{\\sigma_n}\\right)\\right)\\right)\n$$ \nand our estimator $\\hat{\\psi}_n$ is the argmin of the corresponding sample version. Consider the set of functions $\\mathcal{H}_n = \\left\\{h_{\\psi}: h_{\\psi}(q,y) = (y - \\gamma)\\left(1 - K\\left(\\frac{q^{\\top}\\psi}{\\sigma_n}\\right)\\right)\\right\\}$. Next, we argue that $\\mathcal{H}_n$ is a VC class of functions with fixed VC dimension. We know that the function $\\{(q,y) \\mapsto q^{\\top}\\psi\/\\sigma_n: \\psi \\in \\psi\\}$ has fixed VC dimension (i.e. not depending on $n$). Now, as a finite dimensional VC class of functions composed with a fixed monotone function or multiplied by a fixed function still remains a finite dimensional VC class, we conclude that $\\mathcal{H}_n$ is a fixed dimensional VC class of functions with bounded envelope (as the functions considered here are bounded by 1). \n\nNow, we establish a lower bound on the curvature of the population score function $\\mathbb{M}^s(\\psi)$ near its minimizer $\\psi_n$: \n$$\n\\mathbb{M}^s(\\psi) - \\mathbb{M}^s(\\psi_n) \\gtrsim d^2_n(\\psi, \\psi_n)$$ where $$d_n(\\psi, \\psi_n) = \\sqrt{\\frac{\\|\\psi - \\psi_n\\|^2}{\\sigma_n} \\mathds{1}\\left(\\|\\psi - \\psi_n\\| \\le \\mathcal{K}\\sigma_n\\right) + \\|\\psi - \\psi_n\\|\\mathds{1}\\left(\\|\\psi - \\psi_n\\| > \\mathcal{K}\\sigma_n\\right)}\n$$ for some constant $\\mathcal{K} > 0$. The intuition behind this compound structure is following: When $\\psi$ is in $\\sigma_n$ neighborhood of $\\psi_n$, $\\mathbb{M}^s(\\psi)$ behaves like a smooth quadratic function, but when it is away from the truth, $\\mathbb{M}^s(\\psi)$ starts resembling $M(\\psi)$ which induces the linear curvature. \n\\\\\\\\\n\\noindent\nFor the linear part, we first establish that $|\\mathbb{M}(\\psi) - \\mathbb{M}^s(\\psi)| = O(\\sigma_n)$ uniformly for all $\\psi$. Define $\\eta = (\\psi - \\psi_0)\/\\sigma_n$:\n\\allowdisplaybreaks\n\\begin{align}\n& |\\mathbb{M}(\\psi) - \\mathbb{M}^s(\\psi)| \\notag \\\\\n& \\le \\mathbb{E}\\left(\\left | \\mathds{1}(Q^{\\top}\\psi \\ge 0) - K\\left(\\frac{Q^{\\top}\\psi}{\\sigma_n}\\right)\\right | \\right) \\notag\\\\\n& = \\mathbb{E}\\left(\\left | \\mathds{1}\\left(\\frac{Q^{\\top}\\psi_0}{\\sigma_n} + \\eta^{\\top}\\tilde{Q} \\ge 0\\right) - K\\left(\\frac{Q^{\\top}\\psi_0}{\\sigma_n} + \\eta^{\\top}\\tilde{Q}\\right)\\right | \\right) \\notag \\\\\n& = \\sigma_n \\int_{\\mathbb{R}^{p-1}} \\int_{-\\infty}^{\\infty} \\left | \\mathds{1}\\left(t + \\eta^{\\top}\\tilde{Q} \\ge 0\\right) - K\\left(t + \\eta^{\\top}\\tilde{Q}\\right)\\right | f_0(\\sigma_n t | \\tilde{Q}) \\ dt \\ dP(\\tilde{Q}) \\notag\\\\\n& = \\sigma_n \\int_{\\mathbb{R}^{p-1}} \\int_{-\\infty}^{\\infty} \\left | \\mathds{1}\\left(t  \\ge 0\\right) - K\\left(t \\right)\\right | f_0(\\sigma_n (t-\\eta^{\\top}\\tilde{Q}) | \\tilde{Q}) \\ dt \\ dP(\\tilde{Q}) \\notag \\\\\n& = \\sigma_n \\int_{\\mathbb{R}^{p-1}} m(\\tilde{Q})\\int_{-\\infty}^{\\infty} \\left | \\mathds{1}\\left(t  \\ge 0\\right) - K\\left(t \\right)\\right |  \\ dt \\ dP(\\tilde{Q}) \\notag\\\\\n& = \\sigma_n \\mathbb{E}(m(\\tilde{Q})) \\int_{-\\infty}^{\\infty} \\left | \\mathds{1}\\left(t  \\ge 0\\right) - K\\left(t \\right)\\right |  \\ dt \\notag \\\\\n\\label{eq:lin_bound_1} & \\le K_1 \\sigma_n \\mathbb{E}(m(\\tilde{Q})) < \\infty \\hspace{0.3in}  [\\text{by Assumption \\ref{as:density_bound}}] \\,.\n\\end{align}\nHere, the constant $K_1$ is $\\mathbb{E}(m(\\tilde{Q}))  \\left[\\int_{-1}^{1}\\left | \\mathds{1}\\left(t  \\ge 0\\right) - K\\left(t \\right)\\right |  \\ dt \\right]$ which does not depend on $\\psi$, hence the bound is uniform over $\\psi$. Next: \n\\begin{align*}\n\\mathbb{M}^s(\\psi) - \\mathbb{M}^s(\\psi_0^s) & = \\mathbb{M}^s(\\psi) - \\mathbb{M}(\\psi) + \\mathbb{M}(\\psi) - \\mathbb{M}(\\psi_0) \\\\\n& \\qquad \\qquad + \\mathbb{M}(\\psi_0) - \\mathbb{M}(\\psi_0^s) + \\mathbb{M}(\\psi_0^s) -\\mathbb{M}^s(\\psi_0^s) \\\\ \n& = T_1 + T_2 + T_3 + T_4\n\\end{align*}\n\\noindent\nWe bound each summand separately: \n\\begin{enumerate}\n\\item $T_1 = \\mathbb{M}^s(\\psi) - \\mathbb{M}(\\psi) \\ge -K_1 \\sigma_n$ by equation \\ref{eq:lin_bound_1}\\, \n\\item $T_2 = \\mathbb{M}(\\psi) - \\mathbb{M}(\\psi_0) \\ge u_-\\|\\psi - \\psi_0\\|$ by Lemma \\ref{lem:linear_curvature}\\,\n\\item $T_3 = \\mathbb{M}(\\psi_0) - \\mathbb{M}(\\psi_0^s) \\ge -u_+\\|\\psi_0^s - \\psi_0\\|  \\ge  -\\epsilon_1 \\sigma_n$ where one can take $\\epsilon_1$ as small as possible, as we have established $\\|\\psi_0^s - \\psi_0\\|\/\\sigma_n \\rightarrow 0$. This follows by Lemma \\ref{lem:linear_curvature} along with Lemma \\ref{bandwidth}\\,  \n\\item $T_4 = \\mathbb{M}(\\psi_0^s) -\\mathbb{M}^s(\\psi_0^s) \\ge -K_1 \\sigma_n$ by equation \\ref{eq:lin_bound_1}. \n\\end{enumerate}\nCombining, we have \n\\allowdisplaybreaks\n\\begin{align*}\n\\mathbb{M}^s(\\psi) - \\mathbb{M}^s(\\psi_0^s) & \\ge u_-\\|\\psi - \\psi_0\\| -(2K_1 + \\epsilon_1) \\sigma_n \\\\\n& \\ge ( u_-\/2)\\|\\psi - \\psi_0\\| \\hspace{0.2in} \\left[\\text{If} \\ \\|\\psi - \\psi_0\\| \\ge \\frac{2(2K_1 + \\epsilon_1)}{u_-}\\sigma_n\\right] \\\\\n& \\ge ( u_-\/4)\\|\\psi - \\psi_0^s\\| \n\\end{align*}\nwhere the last inequality holds for all large $n$ as proved in Lemma \\ref{bandwidth}. Using Lemma \\ref{bandwidth} again, we conclude that for any pair of positive constants $(\\epsilon_1, \\epsilon_2)$:  \n$$\\|\\psi - \\psi_0^s\\| \\ge \\left(\\frac{2(2K_1 + \\epsilon_1)}{u_-}+\\epsilon_2\\right)\\sigma_n \\Rightarrow \\|\\psi - \\psi_0\\| \\ge \\frac{2(2K_1 + \\epsilon_1)}{u_-}\\sigma_n$$ for all large $n$, which implies: \n\\begin{align}\n& \\mathbb{M}^s(\\psi) - \\mathbb{M}^s(\\psi_0^s) \\notag \\\\\n& \\ge (u_-\/4) \\|\\psi - \\psi_0^s\\| \\mathds{1}\\left(\\|\\psi - \\psi_0^s\\| \\ge \\left(\\frac{2(2K_1 + \\epsilon_1)}{u_-}+\\epsilon_2\\right)\\sigma_n \\right) \\notag \\\\\n\\label{lb2} & \\ge (u_-\/4) \\|\\psi - \\psi_0^s\\| \\mathds{1}\\left(\\frac{\\|\\psi - \\psi_0^s\\|}{\\sigma_n} \\ge \\left(\\frac{7K_1}{u_-}\\right) \\right)  \\hspace{0.2in} [\\text{for appropriate specifications of} \\ \\epsilon_1, \\epsilon_2] \\notag \\\\\n& :=  (u_-\/4) \\|\\psi - \\psi_0^s\\| \\mathds{1}\\left(\\frac{\\|\\psi - \\psi_0^s\\|}{\\sigma_n} \\ge \\mathcal{K} \\right)\n\\end{align}\n\n\\noindent\nIn the next part, we find the lower bound when $\\|\\psi - \\psi^0_s\\| \\le \\mathcal{K} \\sigma_n$. For the quadratic curvature, we perform a two step Taylor expansion: Define $\\eta = (\\psi - \\psi_0)\/\\sigma_n$. We have: \n\\allowdisplaybreaks \n\\begin{align}\n& \\nabla^2\\mathbb{M}^s(\\psi) \\notag\\\\\n& = \\frac{\\beta_0 - \\alpha_0}{2}\\frac{1}{\\sigma_n^2} \\mathbb{E}\\left(\\tilde{Q}\\tilde{Q}^{\\top} K''\\left(\\frac{Q^{\\top}\\psi}{\\sigma_n}\\right)\\left\\{\\mathds{1}(Q^{\\top}\\psi_0 \\le 0) - \\mathds{1}(Q^{\\top}\\psi_0 \\ge 0)\\right\\}\\right) \\notag\\\\\n& = \\frac{\\beta_0 - \\alpha_0}{2}\\frac{1}{\\sigma_n^2} \\mathbb{E}\\left(\\tilde{Q}\\tilde{Q}^{\\top} K''\\left(\\frac{Q^{\\top}\\psi_0}{\\sigma_n} + \\tilde{Q}^{\\top}\\tilde \\eta \\right)\\left\\{\\mathds{1}(Q^{\\top}\\psi_0 \\le 0) - \\mathds{1}(Q^{\\top}\\psi_0 \\ge 0)\\right\\}\\right) \\notag\\\\\n& = \\frac{\\beta_0 - \\alpha_0}{2}\\frac{1}{\\sigma_n^2} \\mathbb{E}\\left[\\tilde{Q}\\tilde{Q}^{\\top} \\left[\\int_{-\\infty}^{0} K''\\left(\\frac{z}{\\sigma_n} + \\tilde{Q}^{\\top}\\tilde \\eta \\right) f_0(z |\\tilde{Q}) \\ dz \\right. \\right. \\notag \\\\ \n& \\left. \\left. \\qquad \\qquad \\qquad \\qquad -\\int_{0}^{\\infty} K''\\left(\\frac{z}{\\sigma_n} + \\tilde{Q}^{\\top}\\tilde \\eta \\right) f_0(z | \\tilde{Q}) \\ dz \\right]\\right] \\notag\\\\\n& = \\frac{\\beta_0 - \\alpha_0}{2}\\frac{1}{\\sigma_n} \\mathbb{E}\\left[\\tilde{Q}\\tilde{Q}^{\\top} \\left[\\int_{-\\infty}^{0} K''\\left(t+ \\tilde{Q}^{\\top}\\tilde \\eta \\right) f_0(\\sigma_n t |\\tilde{Q}) \\ dt \\right. \\right. \\notag \\\\\n& \\left. \\left. \\qquad \\qquad \\qquad \\qquad - \\int_{0}^{\\infty} K''\\left(t + \\tilde{Q}^{\\top}\\tilde \\eta \\right) f_0(\\sigma_n t | \\tilde{Q}) \\ dt \\right]\\right] \\notag\\\\\n& = \\frac{\\beta_0 - \\alpha_0}{2}\\frac{1}{\\sigma_n} \\mathbb{E}\\left[\\tilde{Q}\\tilde{Q}^{\\top} f_0(0| \\tilde{Q})\\left[\\int_{-\\infty}^{0} K''\\left(t+ \\tilde{Q}^{\\top}\\tilde \\eta \\right) \\ dt \\right. \\right. \\notag \\\\\n& \\left. \\left. \\qquad \\qquad \\qquad \\qquad - \\int_{0}^{\\infty} K''\\left(t + \\tilde{Q}^{\\top}\\tilde \\eta \\right) \\ dt \\right]\\right] + R \\notag\\\\\n\\label{eq:quad_eq_1} & =(\\beta_0 - \\alpha_0)\\frac{1}{\\sigma_n}\\mathbb{E}\\left[\\tilde{Q}\\tilde{Q}^{\\top} f_0(0| \\tilde{Q})K'(\\tilde{Q}^{\\top}\\tilde \\eta)\\right] + R \\,.\n\\end{align}\nAs we want a lower bound on the set $\\|\\psi - \\psi^0_s\\| \\le \\mathcal{K} \\sigma_n$, we have $\\|\\eta\\| \\le \\mathcal{K}$. For the rest of the analysis, define \n\\begin{align*}\n\\Lambda: (v_1, v_2) \\mapsto  \\inf_{\\|v_1\\| = 1, \\|v_2\\| \\le \\mathcal{K}} \\mathbb{E}_{\\tilde X}\\left[|v_1^{\\top}\\tilde{Q}|^2 f(0|\\tilde{Q})K'(\\tilde{Q}^{\\top}v_2) \\right]\n\\end{align*}\nClearly $\\Lambda \\ge 0$ and continuous on a compact set, hence its infimum is attained. Suppose $\\Lambda(v_1, v_2) = 0$ for some $v_1, v_2$. Then we have: \n\\begin{align*}\n\\mathbb{E}\\left[|v_1^{\\top}\\tilde{Q}|^2 f(0|\\tilde{Q})K'(\\tilde{Q}^{\\top}v_2)  \\right] = 0 \\,,\n\\end{align*}\nwhich further implies $|\\tilde v_1^{\\top}\\tilde X| = 0$ almost surely and violates Assumption \\ref{as:eigenval_bound}. Hence, our claim is demonstrated. On the other hand, for the remainder term of equation \\eqref{eq:quad_eq_1}: \nfix $\\nu \\in S^{p-1}$. Then: \n\\allowdisplaybreaks\n\\begin{align}\n& \\left| \\nu^{\\top} R \\nu \\right| \\notag \\\\\n& = \\left|\\frac{1}{\\sigma_n} \\mathbb{E}\\left[\\left(\\nu^{\\top}\\tilde{Q}\\right)^2 \\left[\\int_{-\\infty}^{0} K''\\left(t+ \\tilde{Q}^{\\top}\\tilde \\eta \\right) (f_0(\\sigma_n t |\\tilde{Q}) - f_0(0|\\tilde{Q})) \\ dt  \\right. \\right. \\right. \\notag \\\\\n& \\qquad \\qquad \\qquad \\qquad \\left. \\left. \\left. - \\int_{0}^{\\infty} K''\\left(t + \\tilde{Q}^{\\top}\\tilde \\eta \\right) (f_0(\\sigma_n t |\\tilde{Q}) - f_0(0|\\tilde{Q})) \\ dt \\right]\\right]\\right| \\notag\\\\\n& \\le  \\mathbb{E} \\left[\\left(\\nu^{\\top}\\tilde{Q}\\right)^2h(\\tilde{Q}) \\int_{-\\infty}^{\\infty} \\left|K''\\left(t+ \\tilde{Q}^{\\top}\\tilde \\eta \\right)\\right| |t| \\ dt\\right] \\notag\\\\\n& \\le  \\mathbb{E} \\left[\\left(\\nu^{\\top}\\tilde{Q}\\right)^2h(\\tilde{Q}) \\int_{-1}^{1} \\left|K''\\left(t\\right)\\right| |t - \\tilde{Q}^{\\top}\\tilde \\eta | \\ dt\\right] \\notag\\\\\n\\label{eq:quad_eq_3} & \\le  \\mathbb{E} \\left[\\left(\\nu^{\\top}\\tilde{Q}\\right)^2h(\\tilde{Q})(1+ \\|\\tilde{Q}\\|\/2\\kappa) \\int_{-1}^{1} \\left|K''\\left(t\\right)\\right| \\ dt\\right] = C_1 \\hspace{0.2in} [\\text{say}]\n\\end{align}\nby Assumption \\ref{as:distribution} and Assumption \\ref{as:derivative_bound}. By a two-step Taylor expansion, we have: \n\\begin{align*}\n\\mathbb{M}^s(\\psi) - \\mathbb{M}^s(\\psi_0^s) & =  \\frac12 (\\psi - \\psi_0^s)^{\\top} \\nabla^2\\mathbb{M}^s(\\psi^*_n) (\\psi - \\psi_0^s)  \\\\\n& \\ge \\left(\\min_{\\|v_1\\| = 1, \\|v_2 \\| \\le \\mathcal{K}} \\Lambda(v_1, v_2)\\right) \\frac{\\|\\psi - \\psi_0^s\\|^2}{2\\sigma_n} - \\frac{C_1\\sigma_n}{2} \\, \\frac{\\|\\psi - \\psi_0^s\\|^2_2}{\\sigma_n} \\\\\n& \\gtrsim \\frac{\\|\\psi - \\psi_0^s\\|^2_2}{\\sigma_n} \\,\n\\end{align*}\nThis concludes the proof of the curvature. \n\\\\\\\\\n\\noindent \nFinally, we bound the modulus of continuity:\n$$\\mathbb{E}\\left(\\sup_{d_n(\\psi, \\psi_0^s) \\le \\delta} \\left|(\\mathbb{M}^s_n-\\mathbb{M}^s)(\\psi) - (\\mathbb{M}^s_n-\\mathbb{M}^s)(\\psi_n)\\right|\\right) \\,.$$ \nThe proof is similar to that of Lemma \\ref{lem:rate_smooth} and therefore we sketch the main steps briefly. Define the estimating function $f_\\psi$ as: \n$$\nf_\\psi(Y, Q) = (Y - \\gamma)\\left(1 - K\\left(\\frac{Q^{\\top}\\psi}{\\sigma_n}\\right)\\right) \n$$\nand the collection of functions $\\mathcal{F}_\\zeta = \\{f_\\psi - f_{\\psi_0^n}: d_n(\\psi, \\psi_0^s) \\le \\delta\\}$. That $\\mathcal{F}_\\zeta$ has finite VC dimension follows from the same argument used to show $\\mathcal{G}_n$ has finite VC dimension in the proof of Lemma \\ref{conv-prob}. Now to bound modulus of continuity, we use Lemma 2.14.1 of \\cite{vdvw96}, which implies: \n$$\n\\sqrt{n}\\mathbb{E}\\left(\\sup_{d_n(\\psi, \\psi_0^s) \\le \\delta} \\left|(\\mathbb{M}^s_n-\\mathbb{M}^s)(\\psi) - (\\mathbb{M}^s_n-\\mathbb{M}^s)(\\psi_n)\\right|\\right)  \\lesssim \\mathcal{J}(1, \\mathcal{F}_\\zeta) \\sqrt{PF_\\zeta^2}\n$$\nwhere $F_\\zeta(Y, Q)$ is the envelope of $\\mathcal{F}_\\zeta$ defined as: \n\\begin{align*}\nF_\\zeta(Y, Q) & = \\sup_{d_*(\\psi, \\psi_0^s) \\le \\zeta}\\left|(Y - \\gamma)\\left(K\\left(\\frac{Q^{\\top}\\psi^s_0}{\\sigma_n}\\right)-K\\left(\\frac{Q^{\\top}\\psi}{\\sigma_n}\\right)\\right)\\right| \\\\\n& = \\left|(Y - \\gamma)\\right| \\sup_{d_*(\\psi, \\psi_0^s) \\le \\zeta} \\left|\\left(K\\left(\\frac{Q^{\\top}\\psi^s_0}{\\sigma_n}\\right)-K\\left(\\frac{Q^{\\top}\\psi}{\\sigma_n}\\right)\\right)\\right|\n\\end{align*}\nand $\\mathcal{J}(1, \\mathcal{F}_\\zeta)$ is the entropy integral which can be bounded above by a constant independent of $n$ as the class $\\mathcal{F}_\\zeta$ has finite VC dimension. As in the proof of Lemma \\ref{lem:rate_smooth}, we here consider two separate cases: (1) $\\zeta \\le \\sqrt{\\mathcal{K} \\sigma_n}$ and (2) $\\zeta > \\sqrt{\\mathcal{K} \\sigma_n}$. In the first case, we have $\\sup_{d_n(\\psi, \\psi_0^s) \\le \\zeta} \\|\\psi_ - \\psi_0^s\\| = \\zeta \\sqrt{\\sigma_n}$. This further implies: \n\\begin{align*}\n    & \\sup_{d_*(\\psi, \\psi_0^s) \\le \\zeta} \\left|\\left\\{K\\left(\\frac{Q^{\\top}\\psi_0^s}{\\sigma_n}\\right) -  K\\left(\\frac{Q^{\\top}\\psi}{\\sigma_n}\\right)\\right\\}\\right|^2 \\\\\n    & \\le \\max\\left\\{\\left|\\left\\{K\\left(\\frac{Q^{\\top}\\psi_0^s}{\\sigma_n}\\right) -  K\\left(\\frac{Q^{\\top}\\psi_0^s}{\\sigma_n} + \\|\\tilde Q\\|\\frac{\\zeta}{\\sqrt{\\sigma_n}}\\right)\\right\\}\\right|^2, \\right. \\\\\n    & \\qquad \\qquad \\qquad \\qquad \\left. \\left|\\left\\{K\\left(\\frac{Q^{\\top}\\psi_0^s}{\\sigma_n}\\right) -  K\\left(\\frac{Q^{\\top}\\psi_0^s}{\\sigma_n} - \\|\\tilde Q\\|\\frac{\\zeta}{\\sqrt{\\sigma_n}}\\right)\\right\\}\\right|^2\\right\\} \\\\\n    & := \\max\\{T_1, T_2\\} \\,.\n\\end{align*}\nTherefore to bound $\\mathbb{E}[F_\\zeta^2(Y, Q)]$ is equivalent to bounding both $\\mathbb{E}[(Y- \\gamma)^2 T_1]$ and $\\mathbb{E}[(Y - \\gamma)^2 T_2]$ separately, which, in turn equivalent to bound $\\mathbb{E}[T_1]$ and $\\mathbb{E}[T_2]$, as $|Y - \\gamma| \\le 1$. These bounds follows from similar calculation as of Lemma \\ref{lem:rate_smooth}, hence skipped. Finally we have in this case, $$\n\\mathbb{E}[F_\\zeta^2(Y, Q)] \\lesssim \\zeta \\sqrt{\\sigma_n} \\,.\n$$ \nThe other case, when $\\zeta > \\sqrt{\\mathcal{K} \\sigma_n}$ also follows by similar calculation of Lemma \\ref{lem:rate_smooth}, which yields: \n$$\n\\mathbb{E}[F_\\zeta^2(Y, Q)] \\lesssim \\zeta^2 \\,.\n$$\n\n\n\n\n\n\n\n\n\n\n\n\n\\noindent\nUsing this in the maximal inequality yields: \n\\begin{align*}\n\\sqrt{n}\\mathbb{E}\\left(\\sup_{d_n(\\psi, \\psi_0) \\le \\delta} \\left|\\mathbb{M}_n(\\psi - \\psi_n) - \\mathbb{M}^s(\\psi - \\psi_n)\\right|\\right) & \\lesssim \\sqrt{\\zeta}\\sigma^{1\/4}_n\\mathds{1}_{\\zeta \\le \\sqrt{\\mathcal{K} \\sigma_n}} +  \\zeta \\mathds{1}_{\\zeta > \\sqrt{\\mathcal{K} \\sigma_n}} \\\\\n& := \\phi_n(\\zeta) \\,\n\\end{align*}\nThis implies (following the same argument as of Lemma \\ref{lem:rate_smooth}): \n$$\nn^{2\/3}\\sigma_n^{-1\/3}d^2(\\hat \\psi^s, \\psi_0^s) = O_p(1) \\,.\n$$\nNow as $n^{2\/3}\\sigma_n^{-1\/3} \\gg \\sigma_n^{-1}$, we have: \n$$\n\\frac{1}{\\sigma_n}d_n^2(\\hat \\psi^s, \\psi_0^s) = o_p(1) \\,.\n$$\nwhich further indicates\n\\begin{align}\n\\label{rate1} & n^{2\/3}\\sigma_n^{-1\/3}\\left[\\frac{\\|\\hat \\psi^s - \\psi_0^s\\|^2}{\\sigma_n} \\mathds{1}(\\|\\hat \\psi^s - \\psi_0^s\\| \\le \\mathcal{K}\\sigma_n) \\right. \\notag \\\\\n& \\qquad \\qquad \\qquad \\left. + \\|\\hat \\psi^s - \\psi_0^s\\|  \\mathds{1}(\\|\\hat \\psi^s - \\psi_0^s\\|\\ge \\mathcal{K}\\sigma_n)\\right] = O_P(1)\n\\end{align}\nThis implies: \n\\begin{enumerate}\n\\item $\\frac{n^{2\/3}}{\\sigma_n^{4\/3}}\\|\\hat \\psi^s - \\psi_0^s\\| \\mathds{1}(\\|\\hat \\psi^s - \\psi_0^s\\|\\le \\mathcal{K}\\sigma_n) = O_P(1)$\n\\item $\\frac{n^{2\/3}}{\\sigma_n^{1\/3}}\\|\\hat \\psi^s - \\psi_0^s\\| \\mathds{1}(\\|\\hat \\psi^s - \\psi_0^s\\| \\ge \\mathcal{K}\\sigma_n) = O_P(1)$\n\\end{enumerate}\nTherefore: \n\\begin{align*}\n& \\frac{n^{2\/3}}{\\sigma_n^{4\/3}}\\|\\hat \\psi^s - \\psi_0^s\\| \\mathds{1}(\\|\\hat \\psi^s - \\psi_0^s\\| \\le \\mathcal{K}\\sigma_n) \\\\\n& \\qquad \\qquad \\qquad + \\frac{n^{2\/3}}{\\sigma_n^{1\/3}}\\|\\hat \\psi^s - \\psi_0^s\\| \\mathds{1}(\\|\\hat \\psi^s - \\psi_0^s\\| \\ge \\mathcal{K}\\sigma_n) = O_p(1) \\,.\n\\end{align*}\ni.e. \n$$\n\\left(\\frac{n^{2\/3}}{\\sigma_n^{4\/3}} \\wedge  \\frac{n^{2\/3}}{\\sigma_n^{1\/3}}\\right)\\|\\hat \\psi^s - \\psi_0^s\\| = O_p(1) \\,.\n$$\nNow $(n^{2\/3}\/\\sigma_n^{4\/3} \\gg 1\/\\sigma_n$ as long as $n^{2\/3} \\gg \\sigma_n^{1\/3}$ which is obviously true. On the other hand, $n^{2\/3}\/\\sigma_n^{1\/3} \\gg 1\/\\sigma_n$ iff $n\\sigma_n \\gg 1$ which is also true as per our assumption. Therefore we have: \n$$\n\\frac{\\|\\hat \\psi^s - \\psi_0^s\\|}{\\sigma_n} = O_p(1) \\,.\n$$\nThis completes the proof. \n\n\\end{proof}\n\n\\section{Proof of Theorem \\ref{thm:regression}}\n\n\\section{Appendix}\nIn this section, we present the proof of Lemma \\ref{lem:rate_smooth}, which lies at the heart of our refined analysis of the smoothed change plane estimator. Proofs of the other lemmas and our results for the binary response model are available in the Appendix \\ref{sec:supp_B}.  \n\\subsection{Proof of Lemma \\ref{lem:rate_smooth}}\n\n\n\\begin{proof}\nThe proof of Lemma \\ref{lem:rate_smooth} is quite long, hence we further break it into few more lemmas. \n\\begin{lemma}\n\\label{lem:pop_curv_nonsmooth}\nUnder Assumption \\eqref{eq:assm}, there exists $u_- , u_+ > 0$ such that: \n$$\nu_- d^2(\\theta, \\theta_0) \\le \\mathbb{M}(\\theta) - \\mathbb{M}(\\theta_0) \\le u_+ d^2(\\theta, \\theta_0) \\,,\n$$\nfor $\\theta$ in a (non-srinking) neighborhood of $\\theta_0$, where: \n$$\nd(\\theta, \\theta_0) := \\sqrt{\\|\\beta - \\beta_0\\|^2 + \\|\\delta - \\delta_0\\|^2 + \\|\\psi - \\psi_0\\|} \\,.\n$$\n\\end{lemma}\n\n\n\n\n\n\\begin{lemma}\n\\label{lem:uniform_smooth}\nUnder Assumption \\ref{eq:assm} the smoothed loss function $\\mathbb{M}^s(\\theta)$ is uniformly close to the non-smoothed loss function $\\mathbb{M}(\\theta)$: \n$$\n\\sup_{\\theta \\in \\Theta}\\left|\\mathbb{M}^s(\\theta) - \\mathbb{M}(\\theta)\\right| \\le K_1 \\sigma_n \\,,\n$$ \nfor some constant $K_1$. \n\\end{lemma}\n\n\n\n\n\n\n\n\n\n\\begin{lemma}\n\\label{lem:pop_smooth_curvarture}\nUnder certain assumptions: \n\\begin{align*}\n\\mathbb{M}^s(\\theta) - \\mathbb{M}^s(\\theta_0^s) & \\gtrsim \\|\\beta - \\beta_0^s\\|^2 + \\|\\delta - \\delta_0^s\\|^2 \\\\\n& \\qquad \\qquad + \\frac{\\|\\psi - \\psi_0^s\\|^2}{\\sigma_n} \\mathds{1}_{\\|\\psi - \\psi_0^s\\| \\le \\mathcal{K}\\sigma_n} + \\|\\psi - \\psi_0^s\\| \\mathds{1}_{\\|\\psi - \\psi_0^s\\| > \\mathcal{K}\\sigma_n} \\\\\\\n& := d_*^2(\\theta, \\theta_0^s) \\,.\n\\end{align*}\nfor some constant $\\mathcal{K}$ and for all $\\theta$ in a neighborhood of $\\theta_0$, which does not change with $n$. \n\\end{lemma}\n\nThe proofs of the three lemmas above can be found in Appendix \\ref{sec:supp_B}. We next move to the proof of Lemma \\ref{lem:rate_smooth}. In Lemma \\ref{lem:pop_smooth_curvarture} we have established the curvature of the smooth loss function $\\mathbb{M}^s(\\theta)$ around $\\theta_0^s$. To determine the rate of convergence of $\\hat \\theta^s$ to $\\theta_0^s$, we further need an upper bound on the modulus of continuity of our loss function. Towards that end, first recall that our loss function is:\n$$\nf_{\\theta}(Y, X, Q) =  \\left(Y - X^{\\top}\\beta\\right)^2 +  \\left[-2\\left(Y - X^{\\top}\\beta\\right)X^{\\top}\\delta + (X^{\\top}\\delta)^2\\right] K\\left(\\frac{Q^{\\top}\\psi}{\\sigma_n}\\right)\n$$\nThe centered loss function can be written as: \n\\begin{align}\n    & f_{\\theta}(Y, X, Q) - f_{\\theta_0^s}(Y, X, Q) \\notag \\\\\n    & = \\left(Y - X^{\\top}\\beta\\right)^2 +  \\left[-2\\left(Y - X^{\\top}\\beta\\right)X^{\\top}\\delta + (X^{\\top}\\delta)^2\\right] K\\left(\\frac{Q^{\\top}\\psi}{\\sigma_n}\\right) \\notag \\\\\n    & \\qquad \\qquad \\qquad \\qquad - \\left(Y - X^{\\top}\\beta_0^s\\right)^2 -  \\left[-2\\left(Y - X^{\\top}\\beta_0^s\\right)X^{\\top}\\delta_0^s + (X^{\\top}\\delta_0^s)^2\\right] K\\left(\\frac{Q^{\\top}\\psi_0^s}{\\sigma_n}\\right) \\notag \\\\\n    & = \\left(Y - X^{\\top}\\beta\\right)^2 +  \\left[-2\\left(Y - X^{\\top}\\beta\\right)X^{\\top}\\delta + (X^{\\top}\\delta)^2\\right] K\\left(\\frac{Q^{\\top}\\psi}{\\sigma_n}\\right) \\notag \\\\\n    & \\qquad \\qquad \\qquad \\qquad - \\left(Y - X^{\\top}\\beta_0^s\\right)^2 -  \\left[-2\\left(Y - X^{\\top}\\beta_0^s\\right)X^{\\top}\\delta_0^s + (X^{\\top}\\delta_0^s)^2\\right] K\\left(\\frac{Q^{\\top}\\psi}{\\sigma_n}\\right) \\notag \\\\\n    & \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad - \\left[-2\\left(Y - X^{\\top}\\beta_0^s\\right)X^{\\top}\\delta_0^s + (X^{\\top}\\delta_0^s)^2\\right] \\left\\{K\\left(\\frac{Q^{\\top}\\psi_0^s}{\\sigma_n}\\right) -  K\\left(\\frac{Q^{\\top}\\psi}{\\sigma_n}\\right)\\right\\} \\notag \\\\\n    & = \\underbrace{\\left(Y - X^{\\top}\\beta\\right)^2 - \\left(Y - X^{\\top}\\beta_0^s\\right)^2}_{M_1}  \\notag \\\\\n    & \\qquad + \\underbrace{\\left\\{ \\left[-2\\left(Y - X^{\\top}\\beta\\right)X^{\\top}\\delta + (X^{\\top}\\delta)^2\\right]  -  \\left[-2\\left(Y - X^{\\top}\\beta_0^s\\right)X^{\\top}\\delta_0^s + (X^{\\top}\\delta_0^s)^2\\right]\\right\\} K\\left(\\frac{Q^{\\top}\\psi}{\\sigma_n}\\right)}_{M_2} \\notag \\\\\n    & \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad - \\underbrace{\\left[-2\\left(Y - X^{\\top}\\beta_0^s\\right)X^{\\top}\\delta_0^s + (X^{\\top}\\delta_0^s)^2\\right] \\left\\{K\\left(\\frac{Q^{\\top}\\psi_0^s}{\\sigma_n}\\right) -  K\\left(\\frac{Q^{\\top}\\psi}{\\sigma_n}\\right)\\right\\}}_{M_3} \\notag \\\\\n    \\label{eq:expand_f} & := M_1 + M_2 + M_3\n\\end{align}\nFor the rest of the analysis, fix $\\zeta > 0$ and consider the collection of functions $\\mathcal{F}_{\\zeta}$ which is defined as: \n$$\n\\mathcal{F}_{\\zeta} = \\left\\{f_\\theta - f_{\\theta^s}: d_*(\\theta, \\theta^s) \\le \\zeta\\right\\} \\,.\n$$\nFirst note that $\\mathcal{F}_\\zeta$ has bounded uniform entropy integral (henceforth BUEI) over $\\zeta$. To establish this, it is enough to argue that the collection $\\mathcal{F} = \\{ f_\\theta : \\theta \\in \\Theta\\}$ is BUEI. Note that the functions $X \\mapsto X^{\\top}\\beta$ has VC dimension $p$ and so is the map $X \\mapsto X^{\\top}(\\beta + \\delta)$. Therefore the functions $(X, Y) \\mapsto (Y - X^{\\top}(\\beta + \\delta))^2 - (Y - X^{\\top}\\beta)^2$ is also BUEI, as composition with monotone function (here $x^2$) and taking difference keeps this property. Further by the hyperplane $Q \\mapsto Q^{\\top}\\psi$ also has finite dimension (only depends on the dimension of $Q$) and the VC dimension does not change by scaling it with $\\sigma_n$. Therefore the functions $Q \\mapsto Q^{\\top}\\psi\/sigma_n$ has same VC dimension as $Q \\mapsto Q^{\\top}\\psi$ which is independent of $n$. Again, as composition of monotone function keeps BUEI property, the functions $Q \\mapsto K(Q^{\\top}\\psi\/\\sigma_n)$ is also BUEI. As the product of two BUEI class is BUEI, we conclude that $\\mathcal{F}$ (and hence $\\mathcal{F}_\\zeta$) is BUEI. \n\\\\\\\\\n\\noindent\nNow to bound the modulus of continuity we use Lemma 2.14.1 of \\cite{vdvw96}: \n\\begin{equation*}\n\\label{eq:moc_bound}\n\\sqrt{n}\\mathbb{E}\\left[\\sup_{\\theta: d_*(\\theta, \\theta_0^s) \\le \\zeta} \\left|\\left(\\mathbb{P}_n - P\\right)\\left(f_\\theta - f_{\\theta_0^s}\\right)\\right|\\right] \\lesssim \\mathcal{J}(1, \\mathcal{F}_\\zeta) \\sqrt{\\mathbb{E}\\left[F_{\\zeta}^2(X, Y, Q)\\right]}\n\\end{equation*}\nwhere $F_\\zeta$ is some envelope function of $\\mathcal{F}_\\zeta$. As the function class $\\mathcal{F}_\\zeta$ has bounded entropy integral, $\\mathcal{J}(1, \\mathcal{F}_\\zeta) $ can be bounded above by some constant independent of $n$. We next calculate the order of the envelope function $F_\\zeta$. Recall that, by definition of envelope function is: \n$$\nF_{\\zeta}(X, Y, Q) \\ge \\sup_{\\theta: d_*(\\theta, \\theta_0^s) \\le \\zeta} \\left| f_{\\theta} - f_{\\theta_0}\\right| \\,.\n$$\nand we can write $f_\\theta - f_{\\theta_0^s} = M_1 + M_2 + M_3$ which follows from equation \\eqref{eq:expand_f}. Therefore, to find the order of the envelope function, it is enough to find the order of bounds of $M_1, M_2, M_3$ over the set $d_*(\\theta, \\theta_0^s) \\le \\zeta$. We start with $M_1$: \n\\begin{align}\n    \\sup_{d_*(\\theta, \\theta_0^s) \\le \\zeta}|M_1| & = \\sup_{d_*(\\theta, \\theta_0^s) \\le \\delta}\\left|\\left(Y - X^{\\top}\\beta\\right)^2 - \\left(Y - X^{\\top}\\beta_0^s\\right)^2\\right| \\notag \\\\\n    & = \\sup_{d_*(\\theta, \\theta_0^s) \\le \\zeta} \\left|2YX^{\\top}(\\beta_0^s - \\beta) + (X^{\\top}\\beta)^2 - (X^{\\top}\\beta_0^S)^2\\right| \\notag \\\\\n    & \\le \\sup_{d_*(\\theta, \\theta_0^s) \\le \\zeta} \\|\\beta - \\beta_0^s\\| \\left[2|Y|\\|X\\| + (\\|\\beta_0^s\\| + \\zeta)\\|X\\|^2\\right] \\notag \\\\\n    \\label{eq:env_1} & \\le  \\zeta\\left[2|Y|\\|X\\| + (\\|\\beta_0^s\\| + \\zeta)\\|X\\|^2\\right] := F_{1, \\zeta}(X, Y, Q) \\hspace{0.1in} [\\text{Envelope function of }M_1]\n\\end{align}\nand the second term: \n\\allowdisplaybreaks\n\\begin{align}\n    & \\sup_{d_*(\\theta, \\theta_0^s) \\le \\zeta} |M_2| \\notag \\\\\n    & = \\sup_{d_*(\\theta, \\theta_0^s) \\le \\zeta} \\left|\\left\\{\\left[-2\\left(Y - X^{\\top}\\beta\\right)X^{\\top}\\delta + (X^{\\top}\\delta)^2\\right] \\right. \\right. \\notag \\\\\n    & \\qquad \\qquad \\qquad \\qquad \\left. \\left. -  \\left[-2\\left(Y - X^{\\top}\\beta_0^s\\right)X^{\\top}\\delta_0^s + (X^{\\top}\\delta_0^s)^2\\right]\\right\\}\\right|K\\left(\\frac{Q^{\\top}\\psi}{\\sigma_n}\\right) \\notag \\\\\n    & \\le \\sup_{d_*(\\theta, \\theta_0^s) \\le \\zeta} \\left|\\left\\{\\left[-2\\left(Y - X^{\\top}\\beta\\right)X^{\\top}\\delta + (X^{\\top}\\delta)^2\\right] \\right. \\right. \\notag \\\\\n    & \\qquad \\qquad \\qquad \\qquad \\left. \\left. -  \\left[-2\\left(Y - X^{\\top}\\beta_0^s\\right)X^{\\top}\\delta_0^s + (X^{\\top}\\delta_0^s)^2\\right]\\right\\}\\right| \\notag \\\\\n    & = \\sup_{d_*(\\theta, \\theta_0^s) \\le \\zeta} \\left|\\left\\{\\left[2Y(X^{\\top}\\delta_0^s - X^{\\top}\\delta) + 2[(X^{\\top}\\beta)(X^{\\top}\\delta) \\right. \\right. \\right.  \\notag \\\\\n    & \\qquad \\qquad \\qquad \\qquad \\left. \\left. \\left.  - (X^{\\top}\\beta_0^s)(X^{\\top}\\delta_0^s)] + (X^{\\top}\\delta)^2 - (X^{\\top}\\delta_0^s)^2\\right]\\right\\}\\right| \\notag \\\\\n    & \\le \\sup_{d_*(\\theta, \\theta_0^s) \\le \\zeta} \\left\\{\\|\\delta - \\delta_0^s\\|2|Y|\\|X\\| + 2\\|\\beta - \\beta_0\\|\\|X\\|\\|\\delta\\| \\right. \\notag \\\\\n    & \\qquad \\qquad \\qquad \\qquad \\left. + 2\\|\\delta - \\delta_0^s\\|\\|X\\|\\|\\beta_0^s\\| + 2\\|X\\|\\|\\delta + \\delta_0^s\\|\\|\\delta - \\delta_0^s\\|\\right\\} \\notag \\\\ \\notag \\\\\n    & \\le \\zeta \\left[2|Y|\\|X\\| + 2\\|X\\|(\\|\\delta_0^s\\| + \\|\\zeta\\|) + 2\\|X\\|\\|\\beta_0^s\\| + 2\\|X\\|(\\|\\delta_0^s\\| + \\zeta)\\right] \\notag \\\\ \n    \\label{eq:env_2}& = \\zeta \\times 2\\|X\\|\\left[2|Y| + 2(\\|\\delta_0^s\\| + \\|\\zeta\\|) + \\|\\beta_0^s\\|\\right] := F_{2, \\zeta}(X, Y, Q) \\hspace{0.1in} [\\text{Envelope function of }M_2]\n    \\end{align}\nFor the third term, note that: \n\\begin{align*}\n& \\sup_{d_*(\\theta, \\theta_0^s) \\le \\zeta} |M_3| \\\\\n& \\le \\left|\\left[-2\\left(Y - X^{\\top}\\beta_0^s\\right)X^{\\top}\\delta_0^s + (X^{\\top}\\delta_0^s)^2\\right]\\right| \\times  \\sup_{d_*(\\theta, \\theta_0^s) \\le \\zeta} \\left|\\left\\{K\\left(\\frac{Q^{\\top}\\psi_0^s}{\\sigma_n}\\right) -  K\\left(\\frac{Q^{\\top}\\psi}{\\sigma_n}\\right)\\right\\}\\right| \\\\\n& := F_{3, \\zeta} (X, Y, Q)\n\\end{align*}\nHenceforth, we define the envelope function to be $F_\\zeta = F_{\\zeta, 1} + F_{\\zeta, 2} + F_{\\zeta, 3}$. Hence we have by triangle inequality: \n$$\n\\sqrt{\\mathbb{E}\\left[F_{\\zeta}^2(X, Y, Q)\\right]} \\le \\sum_{i=1}^3 \\sqrt{\\mathbb{E}\\left[F_{i, \\zeta}^2(X, Y, Q)\\right]}\n$$\nFrom equation \\eqref{eq:env_1} and \\eqref{eq:env_2} we have: \n\\begin{equation}\n\\label{eq:moc_bound_2}\n\\sqrt{\\mathbb{E}\\left[F_{1, \\zeta}^2(X, Y, Q)\\right]} + \\sqrt{\\mathbb{E}\\left[F_{2, \\zeta}^2(X, Y, Q)\\right]} \\lesssim \\zeta \\,.\n\\end{equation}\nFor $F_{3, \\zeta}$, first note that: \n\\begin{align*}\n    & \\mathbb{E}\\left[\\left|\\left[-2\\left(Y - X^{\\top}\\beta_0^s\\right)X^{\\top}\\delta_0^s + (X^{\\top}\\delta_0^s)^2\\right]\\right|^2 \\mid Q\\right] \\\\\n    & \\le 8\\mathbb{E}\\left[\\left(Y - X^{\\top}\\beta_0^s\\right)^2(X^{\\top}\\delta_0)^2 \\mid Q\\right] + 2\\mathbb{E}[(X^{\\top}\\delta_0^s)^4 \\mid Q] \\\\\n    & \\le \\left\\{8\\|\\beta - \\beta_0^s\\|^2\\|\\delta_0\\|^2 + 8\\|\\delta_0\\|^4 + 2\\|\\delta_0^s\\|^4\\right\\}m_4(Q) \\,.\n\\end{align*}\nwhere $m_4(Q)$ is defined in Assumption \\ref{eq:assm}. In this part, we have to tackle the dichotomous behavior of $\\psi$ around $\\psi_0^s$ carefully. Henceforth define $d_*^2(\\psi, \\psi_0^s)$ as: \n\\begin{align*}\n    d_*^2(\\psi, \\psi_0^s) =  & \\frac{\\|\\psi - \\psi_0^s\\|^2}{\\sigma_n}\\mathds{1}_{\\|\\psi - \\psi_0^s\\| \\le \\mathcal{K}\\sigma_n} + \\|\\psi - \\psi_0^s\\|\\mathds{1}_{\\|\\psi - \\psi_0^s\\| > \\mathcal{K}\\sigma_n} \n\\end{align*}\nThis is a slight abuse of notation, but the reader should think of it as the part of $\\psi$ in $d_*^2(\\theta, \\theta_0^s)$. Define $B_{\\zeta}(\\psi_0^s)$ to be set of all $\\psi$'s such that $d^2_*(\\psi, \\psi_0^s) \\le \\zeta^2$. We can decompose $B_{\\zeta}(\\psi_0^s)$ as a disjoint union of two sets: \n\\begin{align*}\n    B_{\\zeta, 1}(\\psi_0^s) & = \\left\\{\\psi: d^2_*(\\psi, \\psi_0^s) \\le \\zeta^2, \\|\\psi - \\psi_0^s\\| \\le \\mathcal{K}\\sigma_n\\right\\} \\\\\n    & = \\left\\{\\psi:\\frac{\\|\\psi - \\psi_0^s\\|^2}{\\sigma_n} \\le \\zeta^2, \\|\\psi - \\psi_0^s\\| \\le \\mathcal{K}\\sigma_n\\right\\}  \\\\\n    & = \\left\\{\\psi:\\|\\psi - \\psi_0^s\\| \\le \\zeta \\sqrt{\\sigma_n}, \\|\\psi - \\psi_0^s\\| \\le \\mathcal{K}\\sigma_n\\right\\}  \\\\\\\\\n    B_{\\zeta, 2}(\\psi_0^s) & = \\left\\{\\psi: d^2_*(\\psi, \\psi_0^s) \\le \\zeta^2, \\|\\psi - \\psi_0^s\\| > \\mathcal{K}\\sigma_n\\right\\} \\\\\n    & = \\left\\{\\psi: \\|\\psi - \\psi_0^s\\| \\le \\zeta^2, \\|\\psi - \\psi_0^s\\| > \\mathcal{K}\\sigma_n\\right\\} \n\\end{align*}\nAssume $\\mathcal{K} > 1$. The case where $\\mathcal{K} < 1$ follows from similar calculations and hence skipped for brevity. Consider the following two cases: \n\\\\\\\\\n\\noindent\n{\\bf Case 1: }Suppose $\\zeta \\le \\sqrt{\\mathcal{K}\\sigma_n}$. Then $B_{\\zeta, 2} = \\phi$. Also as $\\mathcal{K} > 1$, we have: $\\zeta\\sqrt{\\sigma_n} \\le \\mathcal{K}\\sigma_n$. Hence we have: \n$$\n\\sup_{d_*^2(\\psi, \\psi_0^s) \\le \\zeta^2}\\|\\psi - \\psi_0^s\\| = \\sup_{B_{\\zeta, 1}}\\|\\psi - \\psi_0^s\\| = \\zeta\\sqrt{\\sigma_n} \\,.\n$$\nThis implies: \n\\begin{align*}\n    & \\sup_{d_*(\\theta, \\theta_0^s) \\le \\zeta} \\left|\\left\\{K\\left(\\frac{Q^{\\top}\\psi_0^s}{\\sigma_n}\\right) -  K\\left(\\frac{Q^{\\top}\\psi}{\\sigma_n}\\right)\\right\\}\\right|^2 \\\\\n    & \\le \\max\\left\\{\\left|\\left\\{K\\left(\\frac{Q^{\\top}\\psi_0^s}{\\sigma_n}\\right) -  K\\left(\\frac{Q^{\\top}\\psi_0^s}{\\sigma_n} + \\|\\tilde Q\\|\\frac{\\zeta}{\\sqrt{\\sigma_n}}\\right)\\right\\}\\right|^2, \\right. \\\\\n    & \\qquad \\qquad \\qquad \\left. \\left|\\left\\{K\\left(\\frac{Q^{\\top}\\psi_0^s}{\\sigma_n}\\right) -  K\\left(\\frac{Q^{\\top}\\psi_0^s}{\\sigma_n} - \\|\\tilde Q\\|\\frac{\\zeta}{\\sqrt{\\sigma_n}}\\right)\\right\\}\\right|^2\\right\\} \\\\\n    & := \\max\\{T_1, T_2\\} \\,.\n\\end{align*}\nTherefore we have: \n$$\n\\mathbb{E}\\left[F^2_{3, \\zeta}(X, Y, Q)\\right] \\le   \\mathbb{E}[m_4(Q) T_1] +   \\mathbb{E}[m_4(Q) T_2] \\,.\n$$\nNow: \n\\begin{align}\n    & \\mathbb{E}[m_4(Q) T_1] \\notag \\\\\n    & = \\mathbb{E}\\left[m_4(Q) \\left|\\left\\{K\\left(\\frac{Q^{\\top}\\psi_0^s}{\\sigma_n}\\right) -  K\\left(\\frac{Q^{\\top}\\psi_0^s}{\\sigma_n} + \\|\\tilde Q\\|\\frac{\\zeta}{\\sqrt{\\sigma_n}}\\right)\\right\\}\\right|^2\\right] \\notag \\\\\n    & = \\sigma_n \\int_{\\mathbb{R}^{p-1}}\\int_{-\\infty}^{\\infty} m_4(\\sigma_nt - \\tilde q^{\\top}\\tilde \\psi_0^s, \\tilde q)  \\left|K\\left(t\\right) -  K\\left(t + \\|\\tilde q\\|\\frac{\\zeta}{\\sqrt{\\sigma_n}}\\right)\\right|^2 \\ f_s(\\sigma_nt\\mid \\tilde q) \\ dt \\ f(\\tilde q) \\ d\\tilde q \\notag\\\\\n    & \\le \\sigma_n \\int_{\\mathbb{R}^{p-1}}\\int_{-\\infty}^{\\infty} m_4(\\sigma_nt - \\tilde q^{\\top}\\tilde \\psi_0^s, \\tilde q) \\left|K\\left(t\\right) -  K\\left(t + \\|\\tilde q\\|\\frac{\\zeta}{\\sqrt{\\sigma_n}}\\right)\\right| \\ f_s(\\sigma_nt\\mid \\tilde q) \\ dt \\ f(\\tilde q) \\ d\\tilde q \\notag \\\\\n    & = \\sigma_n \\int_{\\mathbb{R}^{p-1}}\\int_{-\\infty}^{\\infty}m_4(\\sigma_nt - \\tilde q^{\\top}\\tilde \\psi_0^s, \\tilde q) \\int_{t}^{t + \\|\\tilde q\\|\\frac{\\zeta}{\\sqrt{\\sigma_n}}} K'(s) \\ ds \\ f_s(\\sigma_nt\\mid \\tilde q) \\ dt \\ f(\\tilde q) \\ d\\tilde q \\notag \\\\\n    & = \\sigma_n \\int_{\\mathbb{R}^{p-1}}\\int_{-\\infty}^{\\infty}K'(s) \\int_{s- \\|\\tilde q\\|\\frac{\\zeta}{\\sqrt{\\sigma_n}}}^s m_4(\\sigma_nt - \\tilde q^{\\top}\\tilde \\psi_0^s, \\tilde q) f_s(\\sigma_nt\\mid \\tilde q) \\ dt \\ ds \n    \\ f(\\tilde q) \\ d\\tilde q \\notag \\\\\n    & = \\zeta \\sqrt{\\sigma_n} \\mathbb{E}[\\|\\tilde Q\\|m_4(-\\tilde Q^{\\top}\\psi_0^s, \\tilde Q)f_s(0 \\mid \\tilde Q)] + R \\notag \n\\end{align}\nwhere as before we split $R$ into three parts $R = R_1 + R_2 + R_3$. \n\\begin{align}\n    \\left|R_1\\right| & = \\left|\\sigma_n \\int_{\\mathbb{R}^{p-1}}\\int_{-\\infty}^{\\infty}K'(s) \\int_{s- \\|\\tilde q\\|\\frac{\\zeta}{\\sqrt{\\sigma_n}}}^s m_4(- \\tilde q^{\\top}\\tilde \\psi_0^s, \\tilde q) (f_s(\\sigma_nt\\mid \\tilde q) - f_s(0 \\mid \\tilde q)) \\ dt \\ ds \\ f(\\tilde q) \\ d\\tilde q\\right| \\notag \\\\\n   \\label{eq:r1_env_1} & \\le \\sigma_n^2 \\int_{\\mathbb{R}^{p-1}}m_4(- \\tilde q^{\\top}\\tilde \\psi_0^s, \\tilde q)\\dot f_s(\\tilde q) \\int_{-\\infty}^{\\infty}K'(s) \\int_{s- \\|\\tilde q\\|\\frac{\\zeta}{\\sqrt{\\sigma_n}}}^s |t| dt\\ ds \\ f(\\tilde q) \\ d\\tilde q \n    \\end{align}\nWe next calculate the inner integral (involving $(s,t)$) of equation \\eqref{eq:r1_env_1}: \n\\begin{align*}\n& \\int_{-\\infty}^{\\infty}K'(s) \\int_{s- \\|\\tilde q\\|\\frac{\\zeta}{\\sqrt{\\sigma_n}}}^s |t| dt\\ ds \\\\\n& =\\left(\\int_{-\\infty}^0 + \\int_0^{\\|\\tilde q\\|\\frac{\\zeta}{\\sqrt{\\sigma_n}}} + \\int_{\\|\\tilde q\\|\\frac{\\zeta}{\\sqrt{\\sigma_n}}}^{\\infty}\\right)K'(s)  \\int_{s- \\|\\tilde q\\|\\frac{\\zeta}{\\sqrt{\\sigma_n}}}^s |t| dt\\ ds \\\\\n& = \\frac12\\int_{-\\infty}^0 K'(s)\\left[\\left(s- \\|\\tilde q\\|\\frac{\\zeta}{\\sqrt{\\sigma_n}}\\right)^2 - s^2\\right] \\ ds  +  \\frac12\\int_0^{\\|\\tilde q\\|\\frac{\\zeta}{\\sqrt{\\sigma_n}}} K'(s)\\left[\\left(s- \\|\\tilde q\\|\\frac{\\zeta}{\\sqrt{\\sigma_n}}\\right)^2 + s^2\\right] \\ ds \\\\\n& \\qquad \\qquad \\qquad \\qquad + \\frac12 \\int_{\\|\\tilde q\\|\\frac{\\zeta}{\\sqrt{\\sigma_n}}}^{\\infty}K'(s) \\left[s^2 - \\left(s- \\|\\tilde q\\|\\frac{\\zeta}{\\sqrt{\\sigma_n}}\\right)^2\\right] \\ ds\\\\\n& = -\\|\\tilde q\\|\\frac{\\zeta}{\\sqrt{\\sigma_n}}  \\int_{-\\infty}^0 K'(s) s  \\ ds + \\|\\tilde q\\|^2\\frac{\\zeta^2}{2\\sigma_n}    \\int_{-\\infty}^0 K'(s)  \\ ds + \\int_0^{\\|\\tilde q\\|\\frac{\\zeta}{\\sqrt{\\sigma_n}}} s^2K'(s) \\ ds \\\\ \n& \\qquad \\qquad  -\\|\\tilde q\\|\\frac{\\zeta}{\\sqrt{\\sigma_n}} \\int_0^{\\|\\tilde q\\|\\frac{\\zeta}{\\sqrt{\\sigma_n}}} sK'(s) \\ ds +  \\|\\tilde q\\|^2\\frac{\\zeta^2}{2\\sigma_n} \\int_0^{\\|\\tilde q\\|\\frac{\\zeta}{\\sqrt{\\sigma_n}}} K'(s) \\ ds \\\\\n& \\qquad \\qquad \\qquad + \\|\\tilde q\\|\\frac{\\zeta}{\\sqrt{\\sigma_n}} \\int_{\\|\\tilde q\\|\\frac{\\zeta}{\\sqrt{\\sigma_n}}}^{\\infty} sK'(s) \\ ds -  \\|\\tilde q\\|^2\\frac{\\zeta^2}{2\\sigma_n}  \\int_{\\|\\tilde q\\|\\frac{\\zeta}{\\sqrt{\\sigma_n}}}^{\\infty} K'(s) \\ ds \\\\\n&  =   \\|\\tilde q\\|^2\\frac{\\zeta^2}{2\\sigma_n}\\left[2K\\left(\\|\\tilde q\\|\\frac{\\zeta}{\\sqrt{\\sigma_n}}\\right) - 1\\right] + \\|\\tilde q\\|\\frac{\\zeta}{\\sqrt{\\sigma_n}} \\left[ -\\int_{-\\infty}^0 K'(s) s  \\ ds -  \\right. \\\\\n& \\qquad \\qquad \\left. \\int_0^{\\|\\tilde q\\|\\frac{\\zeta}{\\sqrt{\\sigma_n}}} K'(s)s \\ ds + \\int_{\\|\\tilde q\\|\\frac{\\zeta}{\\sqrt{\\sigma_n}}}^{\\infty} sK'(s) \\ ds\\right]  + \\int_0^{\\|\\tilde q\\|\\frac{\\zeta}{\\sqrt{\\sigma_n}}} s^2K'(s) \\ ds \\\\\n&  =   \\|\\tilde q\\|^2\\frac{\\zeta^2}{\\sigma_n}\\left[K\\left(\\|\\tilde q\\|\\frac{\\zeta}{\\sqrt{\\sigma_n}}\\right) - K(0)\\right] + \\|\\tilde q\\|\\frac{\\zeta}{\\sqrt{\\sigma_n}} \\left[ -\\int_{-\\infty}^{-\\|\\tilde q\\|\\frac{\\zeta}{\\sqrt{\\sigma_n}}} K'(s) s  \\ ds + \\int_{\\|\\tilde q\\|\\frac{\\zeta}{\\sqrt{\\sigma_n}}}^{\\infty} sK'(s) \\ ds\\right] \\\\\n& \\qquad \\qquad + \\int_0^{\\|\\tilde q\\|\\frac{\\zeta}{\\sqrt{\\sigma_n}}} s^2K'(s) \\ ds \\\\\n&  =   \\|\\tilde q\\|^2\\frac{\\zeta^2}{\\sigma_n}\\left[K\\left(\\|\\tilde q\\|\\frac{\\zeta}{\\sqrt{\\sigma_n}}\\right) - K(0)\\right] + \\|\\tilde q\\|\\frac{\\zeta}{\\sqrt{\\sigma_n}}\\int_{-\\infty}^{\\infty} K'(s)|s|\\mathds{1}_{|s| \\ge \\|\\tilde q\\|\\frac{\\zeta}{\\sqrt{\\sigma_n}}}  \\ ds + \\int_0^{\\|\\tilde q\\|\\frac{\\zeta}{\\sqrt{\\sigma_n}}} s^2K'(s) \\ ds \\\\\n& \\le \\dot{K}_+ \\|\\tilde q\\|^3\\frac{\\zeta^3}{\\sigma^{3\/2}_n} + \\|\\tilde q\\|\\frac{\\zeta}{\\sqrt{\\sigma_n}} \\int_{-\\infty}^{\\infty} K'(s)|s|  \\ ds + \\|\\tilde q\\|^2\\frac{\\zeta^2}{\\sigma_n}\\left(K\\left(\\|\\tilde q\\|\\frac{\\zeta}{\\sqrt{\\sigma_n}}\\right) - K(0)\\right) \\\\\n& \\lesssim \\|\\tilde q\\|^3\\frac{\\zeta^3}{\\sigma^{3\/2}_n} + \\|\\tilde q\\|\\frac{\\zeta}{\\sqrt{\\sigma_n}}  \n\\end{align*}\nPutting this bound in equation \\eqref{eq:r1_env_1} we obtain: \n    \\begin{align*}\n    |R_1| & \\le \\frac{\\sigma_n^2}{2}  \\int_{\\mathbb{R}^{p-1}}m_4(- \\tilde q^{\\top}\\tilde \\psi_0^s, \\tilde q)\\dot f_s(\\tilde q) \\left(\\|\\tilde q\\|^3\\frac{\\zeta^3}{\\sigma^{3\/2}_n} + \\|\\tilde q\\|\\frac{\\zeta}{\\sqrt{\\sigma_n}}\\right) \\ f(\\tilde q) \\ d\\tilde q \\\\\n    & \\le \\frac{\\zeta^3}{2\\sqrt{\\sigma_n}} \\mathbb{E}\\left[m_4(- \\tilde Q^{\\top}\\tilde \\psi_0^s, \\tilde Q)\\dot f_s(\\tilde Q)\\|\\tilde Q\\|^3\\right] + \\frac{\\zeta \\sqrt{\\sigma_n}}{2} \\mathbb{E}\\left[m_4(- \\tilde Q^{\\top}\\tilde \\psi_0^s, \\tilde Q)\\dot f_s(\\tilde Q)\\|\\tilde Q\\|\\right] \n\\end{align*}\nand \n\\begin{align*}\n    & \\left|R_2\\right| \\\\\n    & = \\left|\\sigma_n \\int_{\\mathbb{R}^{p-1}}\\int_{-\\infty}^{\\infty}K'(s) \\int_{s- \\|\\tilde q\\|\\frac{\\zeta}{\\sqrt{\\sigma_n}}}^s \\left(m_4(\\sigma_n t - \\tilde q^{\\top}\\tilde \\psi_0^s, \\tilde q) - m_4( - \\tilde q^{\\top}\\tilde \\psi_0^s, \\tilde q)\\right)f_s(0 \\mid \\tilde q) \\ dt \\ ds \\ f(\\tilde q) \\ d\\tilde q\\right| \\\\\n    & \\le \\sigma_n^2 \\int_{\\mathbb{R}^{p-1}}\\dot m_4( \\tilde q)f_s(0 \\mid \\tilde q) \\int_{-\\infty}^{\\infty}K'(s) \\int_{s- \\|\\tilde q\\|\\frac{\\zeta}{\\sqrt{\\sigma_n}}}^s |t| dt\\ ds \\ f(\\tilde q) \\ d\\tilde q \\\\\n    & \\le \\sigma_n^2 \\int_{\\mathbb{R}^{p-1}}\\dot m_4( \\tilde q)f_s(0 \\mid \\tilde q) \\left(\\|\\tilde q\\|^3\\frac{\\zeta^3}{\\sigma^{3\/2}_n} + \\|\\tilde q\\|\\frac{\\zeta}{\\sqrt{\\sigma_n}}\\right) \\ f(\\tilde q) \\ d\\tilde q \\\\\n    & = \\zeta \\sigma_n^{3\/2} \\mathbb{E}\\left[\\dot m_4( \\tilde Q)f_s(0 \\mid \\tilde Q)\\|\\tilde Q\\|\\right] + \\zeta^3 \\sqrt{\\sigma_n} \\mathbb{E}\\left[\\dot m_4( \\tilde Q)f_s(0 \\mid \\tilde Q)\\|\\tilde Q\\|^3\\right]\n\\end{align*}\nThe third residual $R_3$ is even higher order term and hence skipped. It is immediate that the order of the remainders are equal to or smaller than $\\zeta \\sqrt{\\sigma_n}$ which implies: \n$$\n\\mathbb{E}[m_4(Q)T_1] \\lesssim \\zeta\\sqrt{\\sigma_n} \\,.\n$$\nThe calculation for $T_2$ is similar and hence skipped for brevity. Combining conclusions for $T_1$ and $T_2$ we conclude when $\\zeta \\le \\sqrt{\\mathcal{K} \\sigma_n}$: \n\\begin{align}\n& \\mathbb{E}\\left[F^2_{3, \\zeta}(X, Y, Q)\\right] \\notag \\\\\n    & \\mathbb{E}\\left[\\left|\\left[-2\\left(Y - X^{\\top}\\beta_0^s\\right)X^{\\top}\\delta_0^s + (X^{\\top}\\delta_0^s)^2\\right]\\right|^2 \\times  \\sup_{d_*(\\theta, \\theta_0^s) \\le \\zeta} \\left|\\left\\{K\\left(\\frac{Q^{\\top}\\psi_0^s}{\\sigma_n}\\right) -  K\\left(\\frac{Q^{\\top}\\psi}{\\sigma_n}\\right)\\right\\}\\right|^2\\right] \\notag \\\\\n    & \\lesssim \\mathbb{E}\\left[m_4(Q)\\sup_{d_*(\\theta, \\theta_0^s) \\le \\zeta} \\left|\\left\\{K\\left(\\frac{Q^{\\top}\\psi_0^s}{\\sigma_n}\\right) -  K\\left(\\frac{Q^{\\top}\\psi}{\\sigma_n}\\right)\\right\\}\\right|^2\\right] \\notag \\\\\n    \\label{eq:env_3} & \\lesssim \\zeta \\sqrt{\\sigma_n} \\,.\n\\end{align}\n\\\\\n\\noindent\n{\\bf Case 2: } Now consider $\\zeta > \\sqrt{\\mathcal{K} \\sigma_n}$. Then it is immediate that: \n$$\n\\sup_{d_*^2(\\psi, \\psi^s_0) \\le \\zeta^2} \\|\\psi - \\psi^s_0\\| = \\zeta^2 \\,.\n$$\nUsing this we have: \n\\begin{align}\n    & \\mathbb{E}[m_4(Q) T_1] \\notag \\\\\n    & = \\mathbb{E}\\left[m_4(Q)\\left|\\left\\{K\\left(\\frac{Q^{\\top}\\psi_0^s}{\\sigma_n}\\right) -  K\\left(\\frac{Q^{\\top}\\psi_0^s}{\\sigma_n} + \\|\\tilde Q\\|\\frac{\\zeta^2}{\\sqrt{\\sigma_n}}\\right)\\right\\}\\right|^2\\right] \\notag \\\\\n    & = \\sigma_n \\int_{\\mathbb{R}^{p-1}}\\int_{-\\infty}^{\\infty} m_4(\\sigma_nt - \\tilde q^{\\top}\\tilde \\psi_0^s, \\tilde q)  \\left|K\\left(t\\right) -  K\\left(t + \\|\\tilde q\\|\\frac{\\zeta^2}{\\sigma_n}\\right)\\right|^2 \\ f_s(\\sigma_nt\\mid \\tilde q) \\ dt \\ f(\\tilde q) \\ d\\tilde q \\notag \\\\\n    & \\le \\sigma_n \\int_{\\mathbb{R}^{p-1}}\\int_{-\\infty}^{\\infty} m_4(\\sigma_nt - \\tilde q^{\\top}\\tilde \\psi_0^s, \\tilde q)  \\left|K\\left(t\\right) -  K\\left(t + \\|\\tilde q\\|\\frac{\\zeta^2}{\\sigma_n}\\right)\\right| \\ f_s(\\sigma_nt\\mid \\tilde q) \\ dt \\ f(\\tilde q) \\ d\\tilde q \\notag \\\\\n     & \\le \\sigma_n \\int_{\\mathbb{R}^{p-1}}\\int_{-\\infty}^{\\infty} m_4(\\sigma_nt - \\tilde q^{\\top}\\tilde \\psi_0^s, \\tilde q)\\|\\tilde q\\|\\frac{\\zeta^2}{\\sigma_n} \\ f_s(\\sigma_nt\\mid \\tilde q) \\ dt \\ f(\\tilde q) \\ d\\tilde q \\notag \\\\\n      & = \\zeta^2 \\int_{\\mathbb{R}^{p-1}}m_4(- \\tilde q^{\\top}\\tilde \\psi_0^s, \\tilde q)  f_s(0 \\mid \\tilde q)\\|\\tilde q\\| \\ f(\\tilde q) \\ d\\tilde q + R \\notag\\\\\n    & \\le \\zeta^2  \\mathbb{E}\\left[\\|\\tilde Q\\|m_4\\left(- \\tilde Q^{\\top}\\tilde \\psi_0^s, \\tilde Q\\right)  f_s(0 \\mid \\tilde Q)\\right]  + R \\notag \n\\end{align}\nThe analysis of the remainder term is similar and if is of higher order. This concludes when $\\zeta > \\sqrt{K\\sigma_n}$: \n\\begin{align}\n& \\mathbb{E}\\left[F^2_{3, \\zeta}(X, Y, Q)\\right] \\notag \\\\\n    & \\mathbb{E}\\left[\\left|\\left[-2\\left(Y - X^{\\top}\\beta_0^s\\right)X^{\\top}\\delta_0^s + (X^{\\top}\\delta_0^s)^2\\right]\\right|^2 \\times  \\sup_{d_*(\\theta, \\theta_0^s) \\le \\zeta} \\left|\\left\\{K\\left(\\frac{Q^{\\top}\\psi_0^s}{\\sigma_n}\\right) -  K\\left(\\frac{Q^{\\top}\\psi}{\\sigma_n}\\right)\\right\\}\\right|^2\\right] \\notag \\\\\n    & \\lesssim \\mathbb{E}\\left[m_4(Q)\\sup_{d_*(\\theta, \\theta_0^s) \\le \\zeta} \\left|\\left\\{K\\left(\\frac{Q^{\\top}\\psi_0^s}{\\sigma_n}\\right) -  K\\left(\\frac{Q^{\\top}\\psi}{\\sigma_n}\\right)\\right\\}\\right|^2\\right] \\notag \\\\\n    \\label{eq:env_4}  & \\lesssim \\zeta^2\n\\end{align}\nCombining \\eqref{eq:env_3}, \\eqref{eq:env_4} with equation \\eqref{eq:moc_bound_2} we have:\n\\begin{align*}\n\\sqrt{n}\\mathbb{E}\\left[\\sup_{\\theta: d_*(\\theta, \\theta_0^s) \\le \\zeta} \\left|\\left(\\mathbb{P}_n - P\\right)\\left(f_\\theta - f_{\\theta_0^s}\\right)\\right|\\right] & \\lesssim  \\sqrt{\\zeta}\\sigma_n^{1\/4}\\mathds{1}_{\\zeta \\le \\sqrt{\\mathcal{K}\\sigma_n}} + \\zeta \\mathds{1}_{\\zeta > \\sqrt{\\mathcal{K} \\sigma_n}} \\\\\n& := \\phi_n(\\zeta) \\,.\n\\end{align*}\nHence to obtain rate we have to solve $r_n^2 \\phi_n(1\/r_n) \\le \\sqrt{n}$, i.e. (ignoring $\\mathcal{K}$ as this does not affect the rate)\n$$\nr_n^{3\/2}\\sigma_n^{1\/4}\\mathds{1}_{r_n \\ge \\sigma_n^{-1\/2}} + r_n \\mathds{1}_{r_n \\le \\sigma_n^{-1\/2}} \\le \\sqrt{n} \\,.\n$$\nNow if $r_n \\le \\sigma_n^{-1\/2}$ then $r_n = \\sqrt{n}$ which implies $\\sqrt{n} \\le \\sigma_n^{-1\/2}$ i.e. $n\\sigma_n \\to 0$ and hence contradiction. On the other hand, if $r_n \\ge \\sigma_n^{-1\/2}$ then $r_n = n^{1\/3}\\sigma_n^{-1\/6}$. This implies $n^{1\/3}\\sigma_n^{-1\/6} \\ge \\sigma_n^{-1\/2}$, i.e. $n^{1\/3} \\ge \\sigma_n^{-1\/3}$, i.e. $n\\sigma_n \\to \\infty$ which is okay. This implies: \n$$\nn^{2\/3}\\sigma_n^{-1\/3}d^2(\\hat \\theta^s, \\theta_0^s) = O_p(1) \\,.\n$$\nNow as $n^{2\/3}\\sigma_n^{-1\/3} \\gg \\sigma_n^{-1}$, we have: \n$$\n\\frac{1}{\\sigma_n}d^2(\\hat \\theta^s, \\theta_0^s) = o_p(1) \\,.\n$$\nwhich further indicates $\\|\\hat \\psi^s - \\psi_0^s\\|\/\\sigma_n = o_p(1)$. This, along with the fact that $\\|\\psi_0^s - \\psi_0\\|\/\\sigma_n = o(1)$ (from Lemma \\ref{bandwidth}), establishes that $\\|\\hat \\psi_0^s - \\psi_0\\|\/\\sigma_n = o_p(1)$. This completes the proof. \n\\end{proof}\n\n\n\n\n\\section{Real data analysis}\n\\label{sec:real_data}\nWe illustrate our method using cross-country data on pollution (carbon-dioxide), income and urbanization obtained from the World Development Indicators (WDI), World Bank. The Environmental Kuznets Curve hypothesis (EKC henceforth), a popular and ongoing area of research in environmental economics, posits that at an initial stage of economic development pollution increases with economic growth, and then diminishes when society's priorities change, leading to an inverted U-shaped relation between income (measured via real GDP per capita) and pollution. The hypothesis has led to numerous empirical papers (i) testing the hypothesis (whether the relation is inverted U-shaped for countries\/regions of interest in the sample), (ii) exploring the threshold level of income at which pollution starts falling, as well as (iii) examining the countries\/regions which belong to the upward rising part versus the downward sloping part of the inverted U-shape, if at all. The studies have been performed using US state level data or cross-country data (e.g. \\cite{shafik1992economic}, \\cite{millimet2003environmental}, \\cite{aldy2005environmental}, \\cite{lee2019nonparametric},\\cite{boubellouta2021cross}, \\cite{list1999environmental}, \\cite{grossman1995economic}, \\cite{bertinelli2005environmental}, \\cite{azomahou2006economic}, \\cite{taskin2000searching} to name a few). While some of these papers have found evidence in favor of the EKC hypothesis (inverted U-shaped income-pollution relation), others have found evidence against it (monotonically increasing or other shapes for the relation). The results often depend on countries\/regions in the sample, period of analysis, as well as the pollutant studied.\n\\\\\\\\\n\\noindent\nWhile income-pollution remains the focal point of most EKC studies, several of them have also included urban agglomeration (UA) or some other measures of urbanization as an important control variable especially while investigating carbon emissions.\\footnote {Although income growth is connected to urbanization, countries are heterogenous and follow different growth paths due to their varying geographical structures, population densities, infrastructures, ownerships of resources making a case for using urbanization as another control covariate in the income-pollution study. The income growth paths of oil rich UAE, manufacturing based China, serviced based Singapore, low population density Canada (with vast land) are all different.} (see for example, \\cite{shafik1992economic}, \\cite{boubellouta2021cross}and \\cite{liang2019urbanization}). The theory of ecological economics posits potentially varying effects of increased urbanization on pollution\u2013 (i) urbanization leading to more pollution (due to its close links with sanitations, dense transportations, and proximities to polluting manufacturing industries), (ii) urbanization potentially leading to less pollution based on 'compact city theory' (see \\cite{burton2000compact}, \\cite{capello2000beyond}, \\cite{sadorsky2014effect}) that explains the potential benefits of increased urbanization in terms of economies of scale (for example, replacing dependence on automobiles with large scale subway systems, using multi-storied buildings instead of single unit houses, keeping more open green space). \\cite{liddle2010age}, using 17 developed countries, find a positive and significant effect of urbanization on pollution. On the contrary, using a set of 69 countries \\cite{sharma2011determinants} find a negative and significant effect of urbanization on pollution while \\cite{du2012economic} find an insignificant effect of urbanization on carbon emission. Using various empirical strategies \\cite{sadorsky2014effect} conclude that the positive and negative effects of urbanization on carbon pollution may cancel out depending on the countries involved often leaving insignificant effects on pollution. They also note that many countries are yet to achieve a sizeable level of urbanization which presumably explains why many empirical works using less developed countries find insignificant effect of urbanization.  In summary, based on the existing literature, both the relationship between urbanization and pollution as well as the relationship between income and pollution appear to depend largely on the set of countries considered in the sample. This motivates us to use UA along with income in our change plane model for analyzing carbon-dioxide emission to plausibly separate the countries into two regimes. \n\\\\\\\\\n\\noindent\nFollowing the broad literature we use pollution emission per capita (carbon-dioxide measured in metric tons per capita) as the dependent variable and real GDP per capita (measured in 2010 US dollars), its square (as is done commonly in the EKC literature) and a popular measure of urbanization, namely urban agglomeration  (UA)\\footnote{The exact definition can be found in the World Development Indicators database from the World Bank website.} as covariates (in our notation $X$) in our regression. In light of the preceding discussions we fit a change plane model comprising real GDP per capita and UA (in our notation $Q$). To summarize the setup, we use the continuous response model as described in equation \\eqref{eq:regression_main_eqn}, i.e \n\\begin{align*}\nY_i & = X_i^{\\top}\\beta_0 + X_i^{\\top}\\delta_0\\mathds{1}_{Q_i^{\\top}\\psi_0 > 0} + {\\epsilon}_i \\\\\n& = X_i^{\\top}\\beta_0\\mathds{1}_{Q_i^{\\top}\\psi_0 \\le 0} + X_i^{\\top}(\\beta_0 + \\delta_0)\\mathds{1}_{Q_i^{\\top}\\psi_0 > 0} + {\\epsilon}_i\n\\end{align*}\nwith the per capita $CO_2$ emission in metric ton as $Y$, per capita GDP, square of per capita GDP and UA as $X$ (hence $X \\in \\mathbb{R}^3$) and finally, per capita GDP and UA as $Q$ (hence $Q \\in \\mathbb{R}^2$). Observe that $\\beta_0$ represents the regression coefficients corresponding to the countries with $Q_i^{\\top}\\psi_0 \\le 0$ (henceforth denoted by Group 1) and $(\\beta_0+ \\delta_0)$ represents the regression coefficients corresponding to the countries with $Q_i^{\\top}\\psi_0 \\ge 0$ (henceforth denoted by Group 2). As per our convention, in the interests of identifiability we assume $\\psi_{0, 1} = 1$, where $\\psi_{0,1}$ is the change plane parameter corresponding to per capita GDP. Therefore the only change plane coefficient to be estimated is $\\psi_{0, 2}$, the change plane coefficient for UA. For numerical stability, we divide per capita GDP by $10^{-4}$ (consequently square of per capital GDP is scaled by $10^{-8}$)\\footnote{This scaling helps in the numerical stability of the gradient descent algorithm used to optimize the least squares criterion.}.  After some pre-processing (i.e. removing rows consisting of NA and countries with $100\\%$ UA) we estimate the coefficients $(\\beta_0, \\delta_0, \\psi_0)$ of our model based on data from 115 countries with $\\sigma_n = 0.05$ and test the significance of the various coefficients using the methodologies described in Section \\ref{sec:inference}. We present our findings in Table \\ref{tab:ekc_coeff}.  \n\\begin{table}[!h]\n    \\centering\n    \\begin{tabular}{|c||c||c|}\n    \\hline\n    Coefficients & Estimated values & p-values \\\\\n    \\hline \\hline \n      $\\beta_{0, 1}$ (\\text{RGDPPC for Group 1})  & 6.98555060 & 4.961452e-10   \\\\\n       $\\beta_{0, 2}$ (\\text{squared RGDPPC for Group 1})  &  -0.43425991 & 7.136484e-02 \\\\\n       $\\beta_{0, 3}$ (\\text{UA for Group 1}) & -0.02613813  & 1.066065e-01\n\\\\\n       $\\beta_{0, 1} + \\delta_{0, 1}$ (\\text{RGDPPC for Group 2}) & 2.0563337 & 0.000000e+00\\\\\n      $\\beta_{0, 2} + \\delta_{0, 2}$ (\\text{squared RGDPPC for Group 2}) &  -0.1866490 & 4.912843e-04   \\\\\n       $\\beta_{0, 3} + \\delta_{0, 3}$ (\\text{UA for Group 2}) & 0.1403171& 1.329788e-05 \\\\\n       $\\psi_{0,2 }$  (\\text{Change plane coeff for UA}) & -0.07061785 & 0.000000e+00\\\\\n         \\hline\n    \\end{tabular}\n    \\caption{Table of the estimated regression and change plane coefficients along with their p-values.}\n    \\label{tab:ekc_coeff}\n\\end{table}\n\\\\\\\\\n\\noindent\nFrom the above analysis, we find that GDP has significantly positive effect on pollution for both groups of countries. The effect of its squared term is negative for both groups; but the effect is significant for Group-2 consisting of mostly high income countries whereas its effect is insignificant (at the 5\\% level) for the Group-1 countries (consisting of mostly low or middle income and few high income countries). Thus, not surprisingly, we find evidence in favor of EKC for the developed countries, but not for the mixed group.  Notably, Group-1 consists of a mixed set of countries like Angola, Sudan, Senegal, India, China, Israel, UAE etc., whereas Group-1 consists of rich and developed countries like Canada, USA, UK, France, Germany etc. The urban variable, on the other hand, is seen to have insignificant effect on Group-1 which is in keeping with \\cite{du2012economic}, \\cite{sadorsky2014effect}. Many of them are yet to achieve substantial urbanization and this is more true for our sample period \\footnote{We use 6 years average from 2010-2015 for GDP and pollution measures. Such averaging is in accordance with the cross-sectional empirical literature using cross-country\/regional data and helps avoid business cycle fluctuations in GDP. It also minimizes the impacts of outlier events such as the financial crisis or great recession period.  The years that we have chosen are ones for which we could find data for the largest number of countries.}. In contrast, UA has a positive and significant effect on Group-2 (developed) countries which is consistent with the findings of \\cite{liddle2010age}, for example. Note that UA plays a crucial role in dividing the countries into different regimes, as the estimated value of $\\psi_{0,2}$ is significant. Thus, we are able to partition countries into two regimes: a mostly rich and a mixed group. \n\\\\\\\\\n\\noindent\nNote that many underdeveloped countries and poorer regions of emerging countries are still swamped with greenhouse gas emissions from burning coal, cow dung etc., and usage of poor exhaust systems in houses and for transport. This is more true for rural and semi-urban areas of developing countries. So even while being less urbanized compared to developed nations, their overall pollution load is high (due to inefficient energy usage and higher dependence on fossil fuels as pointed out above) and rising with income and they are yet to reach the descending part of the inverted U-shape for the income-pollution relation. On the contrary, for countries in Group-2, the adoption of more efficient energy and exhaust systems are common in households and transportations in general, leading to eventually decreasing pollution with increasing income (supporting EKC). Both the results are in line with the existing EKC literature. Additionally we find that the countries in Group 2 are yet to achieve 'compact city' and green urbanization. This is a stylized fact that is confirmed by the positive and significant effect of UA on pollution in our analysis.  \n\\\\\\\\\n\\noindent\nThere are many future potential applications of our method in economics. Similar analyses can be performed for other pollutants (such as sulfur emission, electrical waste\/e-waste, nitrogen pollution etc.). While income\/GDP remains a common, indeed the most crucial variable in pollution studies, other covariates (including change plane defining variables) may vary, depending on the pollutant of interest. Another potential application can be that of identifying the determinants of family health expenses in household survey data. Families are often asked about their health expenses incurred in the past one year. An interesting case in point may be household surveys collected in India where one finds numerous (large) joint families with several children and old people residing in the same household and most families are uninsured. It is often seen that health expenditure increases with income with a major factor being the costs associated with regularly performed preventative medical examinations which are affordable only once a certain income level is reached. The important covariates here are per capita family income, family wealth, `dependency ratio' (number of children and old to the total number of people in the family) and the binary indicator of any history of major illness\/hospitalizations in the family in the past year. Family income per capita and history of major illness are natural candidate covariates for defining the change plane. \n\n\n\\section{Binary response model}\n\\label{sec:classification_analysis}\nRecall our binary response model in equation \\eqref{eq:classification_eqn}. To estimate $\\psi_0$, we resort to the following  loss (without smoothing): \n\\begin{equation}\n\\label{eq:new_loss}\n\\mathbb{M}(\\psi) = \\mathbb{E}\\left((Y - \\gamma)\\mathds{1}(Q^{\\top}\\psi \\le 0)\\right)\\end{equation}\nwith $\\gamma \\in (\\alpha_0, \\beta_0)$, which can be viewed as a variant of the square error loss function: \n$$\n\\mathbb{M}(\\alpha, \\beta, \\psi) = \\mathbb{E}\\left(\\left(Y - \\alpha\\mathds{1}(Q^{\\top}\\psi < 0) - \\beta\\mathds{1}(Q^{\\top}\\psi > 0)\\right)^2\\right)\\,.\n$$\nWe establish the connection between these losses in sub-section \\ref{loss_func_eq}. It is easy to prove that under fairly mild conditions (discussed later) \n$\\psi_0 = {\\arg\\min}_{\\psi \\in \\Theta}\\mathbb{M}(\\psi)$, uniquely. Under the standard classification paradigm, when we know a priori that \n$\\alpha_0 < 1\/2 < \\beta_0$, we can take $\\gamma = 1\/2$, and in the absence of this constraint, $\\bar{Y}$, which converges to some $\\gamma$ between $\\alpha_0$ and $\\beta_0$, may be substituted in the loss function. In the rest of the paper, we confine ourselves to a known $\\gamma$, and for technical simplicity, we take $\\gamma = \\frac{(\\beta_0 + \\alpha_0)}{2}$, but this assumption can be removed with more mathematical book-keeping. Thus, $\\psi_0$ is estimated by:  \n\\begin{equation}\n\\label{non-smooth-score} \n\\hat \\psi = {\\arg\\min}_{\\psi \\in \\Theta} \\mathbb{M}_n(\\psi) = {\\arg\\min}_{\\psi \\in \\Theta} \\frac{1}{n}\\sum_{\\i=1}^n (Y_i - \\gamma)\\mathds{1}(Q_i^{\\top}\\psi \\le 0)\\,.\n\\end{equation} We resort to a smooth approximation of the indicator function in \n\\eqref{non-smooth-score} using a distribution kernel with suitable bandwidth. The smoothed version of the population score function then becomes: \n\\begin{equation}\n\\label{eq:kernel_smoothed_pop_score}\n\\mathbb{M}^s(\\psi) = \\mathbb{E}\\left((Y - \\gamma)\\left(1-K\\left(\\frac{Q^{\\top}\\psi}{\\sigma_n}\\right)\\right)\\right)\n\\end{equation}\nwhere as in the continuous response model, we use $K(x) = \\Phi(x)$, and the corresponding empirical version is: \n\\begin{equation}\n\\label{eq:kernel_smoothed_emp_score}\n\\mathbb{M}^s_n(\\psi) = \\frac{1}{n}\\sum_{i=1}^n \\left((Y_i - \\gamma)\\left(1-K\\left(\\frac{Q_i^{\\top}\\psi}{\\sigma_n}\\right)\\right)\\right)\n\\end{equation}\nDefine $\\hat{\\psi}^s$ and $\\psi_0^s$ to be the minimizer of the smoothed version of the empirical (equation \\eqref{eq:kernel_smoothed_emp_score}) and population score (equation \\eqref{eq:kernel_smoothed_pop_score}) function respectively. Here we only consider the choice of bandwidth $n\\sigma_n \\to \\infty$ and $n\\sigma_n^2 \\to 0$. Analogous to Theorem \\ref{thm:regression} we prove the following result for binary response model: \n\\begin{theorem}\n\\label{thm:binary}\nUnder Assumptions (\\ref{as:distribution} - \\ref{as:eigenval_bound}): \n$$\n\\sqrt{\\frac{n}{\\sigma_n}}\\left(\\hat{\\psi}_n - \\psi_0\\right) \\Rightarrow N(0, \\Gamma) \\,,\n$$ \nfor some non-stochastic matrix $\\Gamma$, which will be defined explicitly in the proof. \n\\end{theorem}\nWe have therefore established that in the regime $n\\sigma_n \\to \\infty$ and $n\\sigma_n^2 \\to 0$, it is possible to attain asymptotic normality using a smoothed estimator for binary response model. \n\n\n\n\n\n\\section{Inferential methods}\n\\label{sec:inference}\nWe draw inferences on $(\\beta_0, \\delta_0, \\psi_0)$ by resorting to similar techniques as in \\cite{seo2007smoothed}. For the continuous response model, we need consistent estimators of $V^{\\gamma}, Q^{\\gamma}, V^{\\psi}, Q^{\\psi}$ (see Lemma \\ref{conv-prob} for the definitions) for hypothesis testing. By virtue of the aforementioned Lemma, we can estimate $Q^{\\gamma}$ and $Q^{\\psi}$ as follows: \n\\begin{align*}\n\\hat Q^{\\gamma} & = \\nabla^2_{\\gamma} \\mathbb{M}_n^s(\\hat \\theta) \\,, \\\\ \n\\hat Q^{\\psi} & = \\sigma_n \\nabla^2_{\\psi} \\mathbb{M}_n^s(\\hat \\theta) \\,.\n\\end{align*}\nThe consistency of the above estimators is established in the proof of Lemma \\ref{conv-prob}. For the other two parameters $V^{\\gamma}, V^{\\psi}$ we use the following estimators: \n\\begin{align*}\n\\hat V^{\\psi} & = \\frac{1}{n\\sigma_n^2}\\sum_{i=1}^n\\left(\\left(Y_i - X_i^{\\top}(\\hat \\beta + \\hat \\delta)\\right)^2 - \\left(Y_i- X_i^{\\top}\\hat \\beta\\right)^2\\right)^2\\tilde Q_i \\tilde Q_i^{\\top}\\left(K'\\left(\\frac{Q_i^{\\top}\\hat \\psi}{\\sigma_n}\\right)\\right)^2 \\\\\n\\hat V^{\\gamma} & = \\hat \\sigma^2_{\\epsilon} \\begin{pmatrix} \\frac{1}{n}X_iX_i^{\\top} & \\frac{1}{n}X_iX_i^{\\top}\\mathds{1}_{Q_i^{\\top}\\hat \\psi > 0} \\\\ \\frac{1}{n}X_iX_i^{\\top}\\mathds{1}_{Q_i^{\\top}\\hat \\psi > 0} & \\frac{1}{n}X_iX_i^{\\top}\\mathds{1}_{Q_i^{\\top}\\hat \\psi > 0} \\end{pmatrix}\n\\end{align*}\nwhere $\\hat \\sigma^2_{\\epsilon}$ can be obtained as $(1\/n)(Y_i - X_i^{\\top}\\hat \\beta - X_i^{\\top}\\hat \\delta \\mathds{1}(Q_i^{\\top}\\hat \\psi > 0))^2$, i.e. the residual sum of squares. The explicit value of $V_\\gamma$ (as derived in equation \\eqref{eq:def_v_gamma} in the proof Lemma \\ref{asymp-normality}) is: \n$$\nV^{\\gamma} = \\sigma_{\\epsilon}^2 \\begin{pmatrix}\\mathbb{E}\\left[XX^{\\top}\\right] & \\mathbb{E}\\left[XX^{\\top}\\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right] \\\\\n\\mathbb{E}\\left[XX^{\\top}\\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right] & \\mathbb{E}\\left[XX^{\\top}\\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right] \\end{pmatrix} \n$$ \nTherefore, the consistency of $\\hat V_\\gamma$ is immediate from the law of large numbers. The consistency of $\\hat V^{\\psi}$ follows via arguments similar to those employed in proving Lemma \\ref{conv-prob} but under somewhat more stringent moment conditions: in particular, we need $\\mathbb{E}[\\|X\\|^8] < \\infty$ and $\\mathbb{E}[(X^{\\top}\\delta_0)^k \\mid Q]$ to be Lipschitz functions over $Q$ for $1 \\le k \\le 8$. The inferential techniques for the classification model are similar and hence skipped, to avoid repetition. \n\n\n\n\n\n\n\n\\section{Proof of Theorem \\ref{thm:regression}}\n\n\\section{Appendix}\nIn this section, we present the proof of Lemma \\ref{lem:rate_smooth}, which lies at the heart of our refined analysis of the smoothed change plane estimator. Proofs of the other lemmas and our results for the binary response model are available in the Appendix \\ref{sec:supp_B}.  \n\\subsection{Proof of Lemma \\ref{lem:rate_smooth}}\n\n\n\\begin{proof}\nThe proof of Lemma \\ref{lem:rate_smooth} is quite long, hence we further break it into few more lemmas. \n\\begin{lemma}\n\\label{lem:pop_curv_nonsmooth}\nUnder Assumption \\eqref{eq:assm}, there exists $u_- , u_+ > 0$ such that: \n$$\nu_- d^2(\\theta, \\theta_0) \\le \\mathbb{M}(\\theta) - \\mathbb{M}(\\theta_0) \\le u_+ d^2(\\theta, \\theta_0) \\,,\n$$\nfor $\\theta$ in a (non-srinking) neighborhood of $\\theta_0$, where: \n$$\nd(\\theta, \\theta_0) := \\sqrt{\\|\\beta - \\beta_0\\|^2 + \\|\\delta - \\delta_0\\|^2 + \\|\\psi - \\psi_0\\|} \\,.\n$$\n\\end{lemma}\n\n\n\n\n\n\\begin{lemma}\n\\label{lem:uniform_smooth}\nUnder Assumption \\ref{eq:assm} the smoothed loss function $\\mathbb{M}^s(\\theta)$ is uniformly close to the non-smoothed loss function $\\mathbb{M}(\\theta)$: \n$$\n\\sup_{\\theta \\in \\Theta}\\left|\\mathbb{M}^s(\\theta) - \\mathbb{M}(\\theta)\\right| \\le K_1 \\sigma_n \\,,\n$$ \nfor some constant $K_1$. \n\\end{lemma}\n\n\n\n\n\n\n\n\n\n\\begin{lemma}\n\\label{lem:pop_smooth_curvarture}\nUnder certain assumptions: \n\\begin{align*}\n\\mathbb{M}^s(\\theta) - \\mathbb{M}^s(\\theta_0^s) & \\gtrsim \\|\\beta - \\beta_0^s\\|^2 + \\|\\delta - \\delta_0^s\\|^2 \\\\\n& \\qquad \\qquad + \\frac{\\|\\psi - \\psi_0^s\\|^2}{\\sigma_n} \\mathds{1}_{\\|\\psi - \\psi_0^s\\| \\le \\mathcal{K}\\sigma_n} + \\|\\psi - \\psi_0^s\\| \\mathds{1}_{\\|\\psi - \\psi_0^s\\| > \\mathcal{K}\\sigma_n} \\\\\\\n& := d_*^2(\\theta, \\theta_0^s) \\,.\n\\end{align*}\nfor some constant $\\mathcal{K}$ and for all $\\theta$ in a neighborhood of $\\theta_0$, which does not change with $n$. \n\\end{lemma}\n\nThe proofs of the three lemmas above can be found in Appendix \\ref{sec:supp_B}. We next move to the proof of Lemma \\ref{lem:rate_smooth}. In Lemma \\ref{lem:pop_smooth_curvarture} we have established the curvature of the smooth loss function $\\mathbb{M}^s(\\theta)$ around $\\theta_0^s$. To determine the rate of convergence of $\\hat \\theta^s$ to $\\theta_0^s$, we further need an upper bound on the modulus of continuity of our loss function. Towards that end, first recall that our loss function is:\n$$\nf_{\\theta}(Y, X, Q) =  \\left(Y - X^{\\top}\\beta\\right)^2 +  \\left[-2\\left(Y - X^{\\top}\\beta\\right)X^{\\top}\\delta + (X^{\\top}\\delta)^2\\right] K\\left(\\frac{Q^{\\top}\\psi}{\\sigma_n}\\right)\n$$\nThe centered loss function can be written as: \n\\begin{align}\n    & f_{\\theta}(Y, X, Q) - f_{\\theta_0^s}(Y, X, Q) \\notag \\\\\n    & = \\left(Y - X^{\\top}\\beta\\right)^2 +  \\left[-2\\left(Y - X^{\\top}\\beta\\right)X^{\\top}\\delta + (X^{\\top}\\delta)^2\\right] K\\left(\\frac{Q^{\\top}\\psi}{\\sigma_n}\\right) \\notag \\\\\n    & \\qquad \\qquad \\qquad \\qquad - \\left(Y - X^{\\top}\\beta_0^s\\right)^2 -  \\left[-2\\left(Y - X^{\\top}\\beta_0^s\\right)X^{\\top}\\delta_0^s + (X^{\\top}\\delta_0^s)^2\\right] K\\left(\\frac{Q^{\\top}\\psi_0^s}{\\sigma_n}\\right) \\notag \\\\\n    & = \\left(Y - X^{\\top}\\beta\\right)^2 +  \\left[-2\\left(Y - X^{\\top}\\beta\\right)X^{\\top}\\delta + (X^{\\top}\\delta)^2\\right] K\\left(\\frac{Q^{\\top}\\psi}{\\sigma_n}\\right) \\notag \\\\\n    & \\qquad \\qquad \\qquad \\qquad - \\left(Y - X^{\\top}\\beta_0^s\\right)^2 -  \\left[-2\\left(Y - X^{\\top}\\beta_0^s\\right)X^{\\top}\\delta_0^s + (X^{\\top}\\delta_0^s)^2\\right] K\\left(\\frac{Q^{\\top}\\psi}{\\sigma_n}\\right) \\notag \\\\\n    & \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad - \\left[-2\\left(Y - X^{\\top}\\beta_0^s\\right)X^{\\top}\\delta_0^s + (X^{\\top}\\delta_0^s)^2\\right] \\left\\{K\\left(\\frac{Q^{\\top}\\psi_0^s}{\\sigma_n}\\right) -  K\\left(\\frac{Q^{\\top}\\psi}{\\sigma_n}\\right)\\right\\} \\notag \\\\\n    & = \\underbrace{\\left(Y - X^{\\top}\\beta\\right)^2 - \\left(Y - X^{\\top}\\beta_0^s\\right)^2}_{M_1}  \\notag \\\\\n    & \\qquad + \\underbrace{\\left\\{ \\left[-2\\left(Y - X^{\\top}\\beta\\right)X^{\\top}\\delta + (X^{\\top}\\delta)^2\\right]  -  \\left[-2\\left(Y - X^{\\top}\\beta_0^s\\right)X^{\\top}\\delta_0^s + (X^{\\top}\\delta_0^s)^2\\right]\\right\\} K\\left(\\frac{Q^{\\top}\\psi}{\\sigma_n}\\right)}_{M_2} \\notag \\\\\n    & \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad - \\underbrace{\\left[-2\\left(Y - X^{\\top}\\beta_0^s\\right)X^{\\top}\\delta_0^s + (X^{\\top}\\delta_0^s)^2\\right] \\left\\{K\\left(\\frac{Q^{\\top}\\psi_0^s}{\\sigma_n}\\right) -  K\\left(\\frac{Q^{\\top}\\psi}{\\sigma_n}\\right)\\right\\}}_{M_3} \\notag \\\\\n    \\label{eq:expand_f} & := M_1 + M_2 + M_3\n\\end{align}\nFor the rest of the analysis, fix $\\zeta > 0$ and consider the collection of functions $\\mathcal{F}_{\\zeta}$ which is defined as: \n$$\n\\mathcal{F}_{\\zeta} = \\left\\{f_\\theta - f_{\\theta^s}: d_*(\\theta, \\theta^s) \\le \\zeta\\right\\} \\,.\n$$\nFirst note that $\\mathcal{F}_\\zeta$ has bounded uniform entropy integral (henceforth BUEI) over $\\zeta$. To establish this, it is enough to argue that the collection $\\mathcal{F} = \\{ f_\\theta : \\theta \\in \\Theta\\}$ is BUEI. Note that the functions $X \\mapsto X^{\\top}\\beta$ has VC dimension $p$ and so is the map $X \\mapsto X^{\\top}(\\beta + \\delta)$. Therefore the functions $(X, Y) \\mapsto (Y - X^{\\top}(\\beta + \\delta))^2 - (Y - X^{\\top}\\beta)^2$ is also BUEI, as composition with monotone function (here $x^2$) and taking difference keeps this property. Further by the hyperplane $Q \\mapsto Q^{\\top}\\psi$ also has finite dimension (only depends on the dimension of $Q$) and the VC dimension does not change by scaling it with $\\sigma_n$. Therefore the functions $Q \\mapsto Q^{\\top}\\psi\/sigma_n$ has same VC dimension as $Q \\mapsto Q^{\\top}\\psi$ which is independent of $n$. Again, as composition of monotone function keeps BUEI property, the functions $Q \\mapsto K(Q^{\\top}\\psi\/\\sigma_n)$ is also BUEI. As the product of two BUEI class is BUEI, we conclude that $\\mathcal{F}$ (and hence $\\mathcal{F}_\\zeta$) is BUEI. \n\\\\\\\\\n\\noindent\nNow to bound the modulus of continuity we use Lemma 2.14.1 of \\cite{vdvw96}: \n\\begin{equation*}\n\\label{eq:moc_bound}\n\\sqrt{n}\\mathbb{E}\\left[\\sup_{\\theta: d_*(\\theta, \\theta_0^s) \\le \\zeta} \\left|\\left(\\mathbb{P}_n - P\\right)\\left(f_\\theta - f_{\\theta_0^s}\\right)\\right|\\right] \\lesssim \\mathcal{J}(1, \\mathcal{F}_\\zeta) \\sqrt{\\mathbb{E}\\left[F_{\\zeta}^2(X, Y, Q)\\right]}\n\\end{equation*}\nwhere $F_\\zeta$ is some envelope function of $\\mathcal{F}_\\zeta$. As the function class $\\mathcal{F}_\\zeta$ has bounded entropy integral, $\\mathcal{J}(1, \\mathcal{F}_\\zeta) $ can be bounded above by some constant independent of $n$. We next calculate the order of the envelope function $F_\\zeta$. Recall that, by definition of envelope function is: \n$$\nF_{\\zeta}(X, Y, Q) \\ge \\sup_{\\theta: d_*(\\theta, \\theta_0^s) \\le \\zeta} \\left| f_{\\theta} - f_{\\theta_0}\\right| \\,.\n$$\nand we can write $f_\\theta - f_{\\theta_0^s} = M_1 + M_2 + M_3$ which follows from equation \\eqref{eq:expand_f}. Therefore, to find the order of the envelope function, it is enough to find the order of bounds of $M_1, M_2, M_3$ over the set $d_*(\\theta, \\theta_0^s) \\le \\zeta$. We start with $M_1$: \n\\begin{align}\n    \\sup_{d_*(\\theta, \\theta_0^s) \\le \\zeta}|M_1| & = \\sup_{d_*(\\theta, \\theta_0^s) \\le \\delta}\\left|\\left(Y - X^{\\top}\\beta\\right)^2 - \\left(Y - X^{\\top}\\beta_0^s\\right)^2\\right| \\notag \\\\\n    & = \\sup_{d_*(\\theta, \\theta_0^s) \\le \\zeta} \\left|2YX^{\\top}(\\beta_0^s - \\beta) + (X^{\\top}\\beta)^2 - (X^{\\top}\\beta_0^S)^2\\right| \\notag \\\\\n    & \\le \\sup_{d_*(\\theta, \\theta_0^s) \\le \\zeta} \\|\\beta - \\beta_0^s\\| \\left[2|Y|\\|X\\| + (\\|\\beta_0^s\\| + \\zeta)\\|X\\|^2\\right] \\notag \\\\\n    \\label{eq:env_1} & \\le  \\zeta\\left[2|Y|\\|X\\| + (\\|\\beta_0^s\\| + \\zeta)\\|X\\|^2\\right] := F_{1, \\zeta}(X, Y, Q) \\hspace{0.1in} [\\text{Envelope function of }M_1]\n\\end{align}\nand the second term: \n\\allowdisplaybreaks\n\\begin{align}\n    & \\sup_{d_*(\\theta, \\theta_0^s) \\le \\zeta} |M_2| \\notag \\\\\n    & = \\sup_{d_*(\\theta, \\theta_0^s) \\le \\zeta} \\left|\\left\\{\\left[-2\\left(Y - X^{\\top}\\beta\\right)X^{\\top}\\delta + (X^{\\top}\\delta)^2\\right] \\right. \\right. \\notag \\\\\n    & \\qquad \\qquad \\qquad \\qquad \\left. \\left. -  \\left[-2\\left(Y - X^{\\top}\\beta_0^s\\right)X^{\\top}\\delta_0^s + (X^{\\top}\\delta_0^s)^2\\right]\\right\\}\\right|K\\left(\\frac{Q^{\\top}\\psi}{\\sigma_n}\\right) \\notag \\\\\n    & \\le \\sup_{d_*(\\theta, \\theta_0^s) \\le \\zeta} \\left|\\left\\{\\left[-2\\left(Y - X^{\\top}\\beta\\right)X^{\\top}\\delta + (X^{\\top}\\delta)^2\\right] \\right. \\right. \\notag \\\\\n    & \\qquad \\qquad \\qquad \\qquad \\left. \\left. -  \\left[-2\\left(Y - X^{\\top}\\beta_0^s\\right)X^{\\top}\\delta_0^s + (X^{\\top}\\delta_0^s)^2\\right]\\right\\}\\right| \\notag \\\\\n    & = \\sup_{d_*(\\theta, \\theta_0^s) \\le \\zeta} \\left|\\left\\{\\left[2Y(X^{\\top}\\delta_0^s - X^{\\top}\\delta) + 2[(X^{\\top}\\beta)(X^{\\top}\\delta) \\right. \\right. \\right.  \\notag \\\\\n    & \\qquad \\qquad \\qquad \\qquad \\left. \\left. \\left.  - (X^{\\top}\\beta_0^s)(X^{\\top}\\delta_0^s)] + (X^{\\top}\\delta)^2 - (X^{\\top}\\delta_0^s)^2\\right]\\right\\}\\right| \\notag \\\\\n    & \\le \\sup_{d_*(\\theta, \\theta_0^s) \\le \\zeta} \\left\\{\\|\\delta - \\delta_0^s\\|2|Y|\\|X\\| + 2\\|\\beta - \\beta_0\\|\\|X\\|\\|\\delta\\| \\right. \\notag \\\\\n    & \\qquad \\qquad \\qquad \\qquad \\left. + 2\\|\\delta - \\delta_0^s\\|\\|X\\|\\|\\beta_0^s\\| + 2\\|X\\|\\|\\delta + \\delta_0^s\\|\\|\\delta - \\delta_0^s\\|\\right\\} \\notag \\\\ \\notag \\\\\n    & \\le \\zeta \\left[2|Y|\\|X\\| + 2\\|X\\|(\\|\\delta_0^s\\| + \\|\\zeta\\|) + 2\\|X\\|\\|\\beta_0^s\\| + 2\\|X\\|(\\|\\delta_0^s\\| + \\zeta)\\right] \\notag \\\\ \n    \\label{eq:env_2}& = \\zeta \\times 2\\|X\\|\\left[2|Y| + 2(\\|\\delta_0^s\\| + \\|\\zeta\\|) + \\|\\beta_0^s\\|\\right] := F_{2, \\zeta}(X, Y, Q) \\hspace{0.1in} [\\text{Envelope function of }M_2]\n    \\end{align}\nFor the third term, note that: \n\\begin{align*}\n& \\sup_{d_*(\\theta, \\theta_0^s) \\le \\zeta} |M_3| \\\\\n& \\le \\left|\\left[-2\\left(Y - X^{\\top}\\beta_0^s\\right)X^{\\top}\\delta_0^s + (X^{\\top}\\delta_0^s)^2\\right]\\right| \\times  \\sup_{d_*(\\theta, \\theta_0^s) \\le \\zeta} \\left|\\left\\{K\\left(\\frac{Q^{\\top}\\psi_0^s}{\\sigma_n}\\right) -  K\\left(\\frac{Q^{\\top}\\psi}{\\sigma_n}\\right)\\right\\}\\right| \\\\\n& := F_{3, \\zeta} (X, Y, Q)\n\\end{align*}\nHenceforth, we define the envelope function to be $F_\\zeta = F_{\\zeta, 1} + F_{\\zeta, 2} + F_{\\zeta, 3}$. Hence we have by triangle inequality: \n$$\n\\sqrt{\\mathbb{E}\\left[F_{\\zeta}^2(X, Y, Q)\\right]} \\le \\sum_{i=1}^3 \\sqrt{\\mathbb{E}\\left[F_{i, \\zeta}^2(X, Y, Q)\\right]}\n$$\nFrom equation \\eqref{eq:env_1} and \\eqref{eq:env_2} we have: \n\\begin{equation}\n\\label{eq:moc_bound_2}\n\\sqrt{\\mathbb{E}\\left[F_{1, \\zeta}^2(X, Y, Q)\\right]} + \\sqrt{\\mathbb{E}\\left[F_{2, \\zeta}^2(X, Y, Q)\\right]} \\lesssim \\zeta \\,.\n\\end{equation}\nFor $F_{3, \\zeta}$, first note that: \n\\begin{align*}\n    & \\mathbb{E}\\left[\\left|\\left[-2\\left(Y - X^{\\top}\\beta_0^s\\right)X^{\\top}\\delta_0^s + (X^{\\top}\\delta_0^s)^2\\right]\\right|^2 \\mid Q\\right] \\\\\n    & \\le 8\\mathbb{E}\\left[\\left(Y - X^{\\top}\\beta_0^s\\right)^2(X^{\\top}\\delta_0)^2 \\mid Q\\right] + 2\\mathbb{E}[(X^{\\top}\\delta_0^s)^4 \\mid Q] \\\\\n    & \\le \\left\\{8\\|\\beta - \\beta_0^s\\|^2\\|\\delta_0\\|^2 + 8\\|\\delta_0\\|^4 + 2\\|\\delta_0^s\\|^4\\right\\}m_4(Q) \\,.\n\\end{align*}\nwhere $m_4(Q)$ is defined in Assumption \\ref{eq:assm}. In this part, we have to tackle the dichotomous behavior of $\\psi$ around $\\psi_0^s$ carefully. Henceforth define $d_*^2(\\psi, \\psi_0^s)$ as: \n\\begin{align*}\n    d_*^2(\\psi, \\psi_0^s) =  & \\frac{\\|\\psi - \\psi_0^s\\|^2}{\\sigma_n}\\mathds{1}_{\\|\\psi - \\psi_0^s\\| \\le \\mathcal{K}\\sigma_n} + \\|\\psi - \\psi_0^s\\|\\mathds{1}_{\\|\\psi - \\psi_0^s\\| > \\mathcal{K}\\sigma_n} \n\\end{align*}\nThis is a slight abuse of notation, but the reader should think of it as the part of $\\psi$ in $d_*^2(\\theta, \\theta_0^s)$. Define $B_{\\zeta}(\\psi_0^s)$ to be set of all $\\psi$'s such that $d^2_*(\\psi, \\psi_0^s) \\le \\zeta^2$. We can decompose $B_{\\zeta}(\\psi_0^s)$ as a disjoint union of two sets: \n\\begin{align*}\n    B_{\\zeta, 1}(\\psi_0^s) & = \\left\\{\\psi: d^2_*(\\psi, \\psi_0^s) \\le \\zeta^2, \\|\\psi - \\psi_0^s\\| \\le \\mathcal{K}\\sigma_n\\right\\} \\\\\n    & = \\left\\{\\psi:\\frac{\\|\\psi - \\psi_0^s\\|^2}{\\sigma_n} \\le \\zeta^2, \\|\\psi - \\psi_0^s\\| \\le \\mathcal{K}\\sigma_n\\right\\}  \\\\\n    & = \\left\\{\\psi:\\|\\psi - \\psi_0^s\\| \\le \\zeta \\sqrt{\\sigma_n}, \\|\\psi - \\psi_0^s\\| \\le \\mathcal{K}\\sigma_n\\right\\}  \\\\\\\\\n    B_{\\zeta, 2}(\\psi_0^s) & = \\left\\{\\psi: d^2_*(\\psi, \\psi_0^s) \\le \\zeta^2, \\|\\psi - \\psi_0^s\\| > \\mathcal{K}\\sigma_n\\right\\} \\\\\n    & = \\left\\{\\psi: \\|\\psi - \\psi_0^s\\| \\le \\zeta^2, \\|\\psi - \\psi_0^s\\| > \\mathcal{K}\\sigma_n\\right\\} \n\\end{align*}\nAssume $\\mathcal{K} > 1$. The case where $\\mathcal{K} < 1$ follows from similar calculations and hence skipped for brevity. Consider the following two cases: \n\\\\\\\\\n\\noindent\n{\\bf Case 1: }Suppose $\\zeta \\le \\sqrt{\\mathcal{K}\\sigma_n}$. Then $B_{\\zeta, 2} = \\phi$. Also as $\\mathcal{K} > 1$, we have: $\\zeta\\sqrt{\\sigma_n} \\le \\mathcal{K}\\sigma_n$. Hence we have: \n$$\n\\sup_{d_*^2(\\psi, \\psi_0^s) \\le \\zeta^2}\\|\\psi - \\psi_0^s\\| = \\sup_{B_{\\zeta, 1}}\\|\\psi - \\psi_0^s\\| = \\zeta\\sqrt{\\sigma_n} \\,.\n$$\nThis implies: \n\\begin{align*}\n    & \\sup_{d_*(\\theta, \\theta_0^s) \\le \\zeta} \\left|\\left\\{K\\left(\\frac{Q^{\\top}\\psi_0^s}{\\sigma_n}\\right) -  K\\left(\\frac{Q^{\\top}\\psi}{\\sigma_n}\\right)\\right\\}\\right|^2 \\\\\n    & \\le \\max\\left\\{\\left|\\left\\{K\\left(\\frac{Q^{\\top}\\psi_0^s}{\\sigma_n}\\right) -  K\\left(\\frac{Q^{\\top}\\psi_0^s}{\\sigma_n} + \\|\\tilde Q\\|\\frac{\\zeta}{\\sqrt{\\sigma_n}}\\right)\\right\\}\\right|^2, \\right. \\\\\n    & \\qquad \\qquad \\qquad \\left. \\left|\\left\\{K\\left(\\frac{Q^{\\top}\\psi_0^s}{\\sigma_n}\\right) -  K\\left(\\frac{Q^{\\top}\\psi_0^s}{\\sigma_n} - \\|\\tilde Q\\|\\frac{\\zeta}{\\sqrt{\\sigma_n}}\\right)\\right\\}\\right|^2\\right\\} \\\\\n    & := \\max\\{T_1, T_2\\} \\,.\n\\end{align*}\nTherefore we have: \n$$\n\\mathbb{E}\\left[F^2_{3, \\zeta}(X, Y, Q)\\right] \\le   \\mathbb{E}[m_4(Q) T_1] +   \\mathbb{E}[m_4(Q) T_2] \\,.\n$$\nNow: \n\\begin{align}\n    & \\mathbb{E}[m_4(Q) T_1] \\notag \\\\\n    & = \\mathbb{E}\\left[m_4(Q) \\left|\\left\\{K\\left(\\frac{Q^{\\top}\\psi_0^s}{\\sigma_n}\\right) -  K\\left(\\frac{Q^{\\top}\\psi_0^s}{\\sigma_n} + \\|\\tilde Q\\|\\frac{\\zeta}{\\sqrt{\\sigma_n}}\\right)\\right\\}\\right|^2\\right] \\notag \\\\\n    & = \\sigma_n \\int_{\\mathbb{R}^{p-1}}\\int_{-\\infty}^{\\infty} m_4(\\sigma_nt - \\tilde q^{\\top}\\tilde \\psi_0^s, \\tilde q)  \\left|K\\left(t\\right) -  K\\left(t + \\|\\tilde q\\|\\frac{\\zeta}{\\sqrt{\\sigma_n}}\\right)\\right|^2 \\ f_s(\\sigma_nt\\mid \\tilde q) \\ dt \\ f(\\tilde q) \\ d\\tilde q \\notag\\\\\n    & \\le \\sigma_n \\int_{\\mathbb{R}^{p-1}}\\int_{-\\infty}^{\\infty} m_4(\\sigma_nt - \\tilde q^{\\top}\\tilde \\psi_0^s, \\tilde q) \\left|K\\left(t\\right) -  K\\left(t + \\|\\tilde q\\|\\frac{\\zeta}{\\sqrt{\\sigma_n}}\\right)\\right| \\ f_s(\\sigma_nt\\mid \\tilde q) \\ dt \\ f(\\tilde q) \\ d\\tilde q \\notag \\\\\n    & = \\sigma_n \\int_{\\mathbb{R}^{p-1}}\\int_{-\\infty}^{\\infty}m_4(\\sigma_nt - \\tilde q^{\\top}\\tilde \\psi_0^s, \\tilde q) \\int_{t}^{t + \\|\\tilde q\\|\\frac{\\zeta}{\\sqrt{\\sigma_n}}} K'(s) \\ ds \\ f_s(\\sigma_nt\\mid \\tilde q) \\ dt \\ f(\\tilde q) \\ d\\tilde q \\notag \\\\\n    & = \\sigma_n \\int_{\\mathbb{R}^{p-1}}\\int_{-\\infty}^{\\infty}K'(s) \\int_{s- \\|\\tilde q\\|\\frac{\\zeta}{\\sqrt{\\sigma_n}}}^s m_4(\\sigma_nt - \\tilde q^{\\top}\\tilde \\psi_0^s, \\tilde q) f_s(\\sigma_nt\\mid \\tilde q) \\ dt \\ ds \n    \\ f(\\tilde q) \\ d\\tilde q \\notag \\\\\n    & = \\zeta \\sqrt{\\sigma_n} \\mathbb{E}[\\|\\tilde Q\\|m_4(-\\tilde Q^{\\top}\\psi_0^s, \\tilde Q)f_s(0 \\mid \\tilde Q)] + R \\notag \n\\end{align}\nwhere as before we split $R$ into three parts $R = R_1 + R_2 + R_3$. \n\\begin{align}\n    \\left|R_1\\right| & = \\left|\\sigma_n \\int_{\\mathbb{R}^{p-1}}\\int_{-\\infty}^{\\infty}K'(s) \\int_{s- \\|\\tilde q\\|\\frac{\\zeta}{\\sqrt{\\sigma_n}}}^s m_4(- \\tilde q^{\\top}\\tilde \\psi_0^s, \\tilde q) (f_s(\\sigma_nt\\mid \\tilde q) - f_s(0 \\mid \\tilde q)) \\ dt \\ ds \\ f(\\tilde q) \\ d\\tilde q\\right| \\notag \\\\\n   \\label{eq:r1_env_1} & \\le \\sigma_n^2 \\int_{\\mathbb{R}^{p-1}}m_4(- \\tilde q^{\\top}\\tilde \\psi_0^s, \\tilde q)\\dot f_s(\\tilde q) \\int_{-\\infty}^{\\infty}K'(s) \\int_{s- \\|\\tilde q\\|\\frac{\\zeta}{\\sqrt{\\sigma_n}}}^s |t| dt\\ ds \\ f(\\tilde q) \\ d\\tilde q \n    \\end{align}\nWe next calculate the inner integral (involving $(s,t)$) of equation \\eqref{eq:r1_env_1}: \n\\begin{align*}\n& \\int_{-\\infty}^{\\infty}K'(s) \\int_{s- \\|\\tilde q\\|\\frac{\\zeta}{\\sqrt{\\sigma_n}}}^s |t| dt\\ ds \\\\\n& =\\left(\\int_{-\\infty}^0 + \\int_0^{\\|\\tilde q\\|\\frac{\\zeta}{\\sqrt{\\sigma_n}}} + \\int_{\\|\\tilde q\\|\\frac{\\zeta}{\\sqrt{\\sigma_n}}}^{\\infty}\\right)K'(s)  \\int_{s- \\|\\tilde q\\|\\frac{\\zeta}{\\sqrt{\\sigma_n}}}^s |t| dt\\ ds \\\\\n& = \\frac12\\int_{-\\infty}^0 K'(s)\\left[\\left(s- \\|\\tilde q\\|\\frac{\\zeta}{\\sqrt{\\sigma_n}}\\right)^2 - s^2\\right] \\ ds  +  \\frac12\\int_0^{\\|\\tilde q\\|\\frac{\\zeta}{\\sqrt{\\sigma_n}}} K'(s)\\left[\\left(s- \\|\\tilde q\\|\\frac{\\zeta}{\\sqrt{\\sigma_n}}\\right)^2 + s^2\\right] \\ ds \\\\\n& \\qquad \\qquad \\qquad \\qquad + \\frac12 \\int_{\\|\\tilde q\\|\\frac{\\zeta}{\\sqrt{\\sigma_n}}}^{\\infty}K'(s) \\left[s^2 - \\left(s- \\|\\tilde q\\|\\frac{\\zeta}{\\sqrt{\\sigma_n}}\\right)^2\\right] \\ ds\\\\\n& = -\\|\\tilde q\\|\\frac{\\zeta}{\\sqrt{\\sigma_n}}  \\int_{-\\infty}^0 K'(s) s  \\ ds + \\|\\tilde q\\|^2\\frac{\\zeta^2}{2\\sigma_n}    \\int_{-\\infty}^0 K'(s)  \\ ds + \\int_0^{\\|\\tilde q\\|\\frac{\\zeta}{\\sqrt{\\sigma_n}}} s^2K'(s) \\ ds \\\\ \n& \\qquad \\qquad  -\\|\\tilde q\\|\\frac{\\zeta}{\\sqrt{\\sigma_n}} \\int_0^{\\|\\tilde q\\|\\frac{\\zeta}{\\sqrt{\\sigma_n}}} sK'(s) \\ ds +  \\|\\tilde q\\|^2\\frac{\\zeta^2}{2\\sigma_n} \\int_0^{\\|\\tilde q\\|\\frac{\\zeta}{\\sqrt{\\sigma_n}}} K'(s) \\ ds \\\\\n& \\qquad \\qquad \\qquad + \\|\\tilde q\\|\\frac{\\zeta}{\\sqrt{\\sigma_n}} \\int_{\\|\\tilde q\\|\\frac{\\zeta}{\\sqrt{\\sigma_n}}}^{\\infty} sK'(s) \\ ds -  \\|\\tilde q\\|^2\\frac{\\zeta^2}{2\\sigma_n}  \\int_{\\|\\tilde q\\|\\frac{\\zeta}{\\sqrt{\\sigma_n}}}^{\\infty} K'(s) \\ ds \\\\\n&  =   \\|\\tilde q\\|^2\\frac{\\zeta^2}{2\\sigma_n}\\left[2K\\left(\\|\\tilde q\\|\\frac{\\zeta}{\\sqrt{\\sigma_n}}\\right) - 1\\right] + \\|\\tilde q\\|\\frac{\\zeta}{\\sqrt{\\sigma_n}} \\left[ -\\int_{-\\infty}^0 K'(s) s  \\ ds -  \\right. \\\\\n& \\qquad \\qquad \\left. \\int_0^{\\|\\tilde q\\|\\frac{\\zeta}{\\sqrt{\\sigma_n}}} K'(s)s \\ ds + \\int_{\\|\\tilde q\\|\\frac{\\zeta}{\\sqrt{\\sigma_n}}}^{\\infty} sK'(s) \\ ds\\right]  + \\int_0^{\\|\\tilde q\\|\\frac{\\zeta}{\\sqrt{\\sigma_n}}} s^2K'(s) \\ ds \\\\\n&  =   \\|\\tilde q\\|^2\\frac{\\zeta^2}{\\sigma_n}\\left[K\\left(\\|\\tilde q\\|\\frac{\\zeta}{\\sqrt{\\sigma_n}}\\right) - K(0)\\right] + \\|\\tilde q\\|\\frac{\\zeta}{\\sqrt{\\sigma_n}} \\left[ -\\int_{-\\infty}^{-\\|\\tilde q\\|\\frac{\\zeta}{\\sqrt{\\sigma_n}}} K'(s) s  \\ ds + \\int_{\\|\\tilde q\\|\\frac{\\zeta}{\\sqrt{\\sigma_n}}}^{\\infty} sK'(s) \\ ds\\right] \\\\\n& \\qquad \\qquad + \\int_0^{\\|\\tilde q\\|\\frac{\\zeta}{\\sqrt{\\sigma_n}}} s^2K'(s) \\ ds \\\\\n&  =   \\|\\tilde q\\|^2\\frac{\\zeta^2}{\\sigma_n}\\left[K\\left(\\|\\tilde q\\|\\frac{\\zeta}{\\sqrt{\\sigma_n}}\\right) - K(0)\\right] + \\|\\tilde q\\|\\frac{\\zeta}{\\sqrt{\\sigma_n}}\\int_{-\\infty}^{\\infty} K'(s)|s|\\mathds{1}_{|s| \\ge \\|\\tilde q\\|\\frac{\\zeta}{\\sqrt{\\sigma_n}}}  \\ ds + \\int_0^{\\|\\tilde q\\|\\frac{\\zeta}{\\sqrt{\\sigma_n}}} s^2K'(s) \\ ds \\\\\n& \\le \\dot{K}_+ \\|\\tilde q\\|^3\\frac{\\zeta^3}{\\sigma^{3\/2}_n} + \\|\\tilde q\\|\\frac{\\zeta}{\\sqrt{\\sigma_n}} \\int_{-\\infty}^{\\infty} K'(s)|s|  \\ ds + \\|\\tilde q\\|^2\\frac{\\zeta^2}{\\sigma_n}\\left(K\\left(\\|\\tilde q\\|\\frac{\\zeta}{\\sqrt{\\sigma_n}}\\right) - K(0)\\right) \\\\\n& \\lesssim \\|\\tilde q\\|^3\\frac{\\zeta^3}{\\sigma^{3\/2}_n} + \\|\\tilde q\\|\\frac{\\zeta}{\\sqrt{\\sigma_n}}  \n\\end{align*}\nPutting this bound in equation \\eqref{eq:r1_env_1} we obtain: \n    \\begin{align*}\n    |R_1| & \\le \\frac{\\sigma_n^2}{2}  \\int_{\\mathbb{R}^{p-1}}m_4(- \\tilde q^{\\top}\\tilde \\psi_0^s, \\tilde q)\\dot f_s(\\tilde q) \\left(\\|\\tilde q\\|^3\\frac{\\zeta^3}{\\sigma^{3\/2}_n} + \\|\\tilde q\\|\\frac{\\zeta}{\\sqrt{\\sigma_n}}\\right) \\ f(\\tilde q) \\ d\\tilde q \\\\\n    & \\le \\frac{\\zeta^3}{2\\sqrt{\\sigma_n}} \\mathbb{E}\\left[m_4(- \\tilde Q^{\\top}\\tilde \\psi_0^s, \\tilde Q)\\dot f_s(\\tilde Q)\\|\\tilde Q\\|^3\\right] + \\frac{\\zeta \\sqrt{\\sigma_n}}{2} \\mathbb{E}\\left[m_4(- \\tilde Q^{\\top}\\tilde \\psi_0^s, \\tilde Q)\\dot f_s(\\tilde Q)\\|\\tilde Q\\|\\right] \n\\end{align*}\nand \n\\begin{align*}\n    & \\left|R_2\\right| \\\\\n    & = \\left|\\sigma_n \\int_{\\mathbb{R}^{p-1}}\\int_{-\\infty}^{\\infty}K'(s) \\int_{s- \\|\\tilde q\\|\\frac{\\zeta}{\\sqrt{\\sigma_n}}}^s \\left(m_4(\\sigma_n t - \\tilde q^{\\top}\\tilde \\psi_0^s, \\tilde q) - m_4( - \\tilde q^{\\top}\\tilde \\psi_0^s, \\tilde q)\\right)f_s(0 \\mid \\tilde q) \\ dt \\ ds \\ f(\\tilde q) \\ d\\tilde q\\right| \\\\\n    & \\le \\sigma_n^2 \\int_{\\mathbb{R}^{p-1}}\\dot m_4( \\tilde q)f_s(0 \\mid \\tilde q) \\int_{-\\infty}^{\\infty}K'(s) \\int_{s- \\|\\tilde q\\|\\frac{\\zeta}{\\sqrt{\\sigma_n}}}^s |t| dt\\ ds \\ f(\\tilde q) \\ d\\tilde q \\\\\n    & \\le \\sigma_n^2 \\int_{\\mathbb{R}^{p-1}}\\dot m_4( \\tilde q)f_s(0 \\mid \\tilde q) \\left(\\|\\tilde q\\|^3\\frac{\\zeta^3}{\\sigma^{3\/2}_n} + \\|\\tilde q\\|\\frac{\\zeta}{\\sqrt{\\sigma_n}}\\right) \\ f(\\tilde q) \\ d\\tilde q \\\\\n    & = \\zeta \\sigma_n^{3\/2} \\mathbb{E}\\left[\\dot m_4( \\tilde Q)f_s(0 \\mid \\tilde Q)\\|\\tilde Q\\|\\right] + \\zeta^3 \\sqrt{\\sigma_n} \\mathbb{E}\\left[\\dot m_4( \\tilde Q)f_s(0 \\mid \\tilde Q)\\|\\tilde Q\\|^3\\right]\n\\end{align*}\nThe third residual $R_3$ is even higher order term and hence skipped. It is immediate that the order of the remainders are equal to or smaller than $\\zeta \\sqrt{\\sigma_n}$ which implies: \n$$\n\\mathbb{E}[m_4(Q)T_1] \\lesssim \\zeta\\sqrt{\\sigma_n} \\,.\n$$\nThe calculation for $T_2$ is similar and hence skipped for brevity. Combining conclusions for $T_1$ and $T_2$ we conclude when $\\zeta \\le \\sqrt{\\mathcal{K} \\sigma_n}$: \n\\begin{align}\n& \\mathbb{E}\\left[F^2_{3, \\zeta}(X, Y, Q)\\right] \\notag \\\\\n    & \\mathbb{E}\\left[\\left|\\left[-2\\left(Y - X^{\\top}\\beta_0^s\\right)X^{\\top}\\delta_0^s + (X^{\\top}\\delta_0^s)^2\\right]\\right|^2 \\times  \\sup_{d_*(\\theta, \\theta_0^s) \\le \\zeta} \\left|\\left\\{K\\left(\\frac{Q^{\\top}\\psi_0^s}{\\sigma_n}\\right) -  K\\left(\\frac{Q^{\\top}\\psi}{\\sigma_n}\\right)\\right\\}\\right|^2\\right] \\notag \\\\\n    & \\lesssim \\mathbb{E}\\left[m_4(Q)\\sup_{d_*(\\theta, \\theta_0^s) \\le \\zeta} \\left|\\left\\{K\\left(\\frac{Q^{\\top}\\psi_0^s}{\\sigma_n}\\right) -  K\\left(\\frac{Q^{\\top}\\psi}{\\sigma_n}\\right)\\right\\}\\right|^2\\right] \\notag \\\\\n    \\label{eq:env_3} & \\lesssim \\zeta \\sqrt{\\sigma_n} \\,.\n\\end{align}\n\\\\\n\\noindent\n{\\bf Case 2: } Now consider $\\zeta > \\sqrt{\\mathcal{K} \\sigma_n}$. Then it is immediate that: \n$$\n\\sup_{d_*^2(\\psi, \\psi^s_0) \\le \\zeta^2} \\|\\psi - \\psi^s_0\\| = \\zeta^2 \\,.\n$$\nUsing this we have: \n\\begin{align}\n    & \\mathbb{E}[m_4(Q) T_1] \\notag \\\\\n    & = \\mathbb{E}\\left[m_4(Q)\\left|\\left\\{K\\left(\\frac{Q^{\\top}\\psi_0^s}{\\sigma_n}\\right) -  K\\left(\\frac{Q^{\\top}\\psi_0^s}{\\sigma_n} + \\|\\tilde Q\\|\\frac{\\zeta^2}{\\sqrt{\\sigma_n}}\\right)\\right\\}\\right|^2\\right] \\notag \\\\\n    & = \\sigma_n \\int_{\\mathbb{R}^{p-1}}\\int_{-\\infty}^{\\infty} m_4(\\sigma_nt - \\tilde q^{\\top}\\tilde \\psi_0^s, \\tilde q)  \\left|K\\left(t\\right) -  K\\left(t + \\|\\tilde q\\|\\frac{\\zeta^2}{\\sigma_n}\\right)\\right|^2 \\ f_s(\\sigma_nt\\mid \\tilde q) \\ dt \\ f(\\tilde q) \\ d\\tilde q \\notag \\\\\n    & \\le \\sigma_n \\int_{\\mathbb{R}^{p-1}}\\int_{-\\infty}^{\\infty} m_4(\\sigma_nt - \\tilde q^{\\top}\\tilde \\psi_0^s, \\tilde q)  \\left|K\\left(t\\right) -  K\\left(t + \\|\\tilde q\\|\\frac{\\zeta^2}{\\sigma_n}\\right)\\right| \\ f_s(\\sigma_nt\\mid \\tilde q) \\ dt \\ f(\\tilde q) \\ d\\tilde q \\notag \\\\\n     & \\le \\sigma_n \\int_{\\mathbb{R}^{p-1}}\\int_{-\\infty}^{\\infty} m_4(\\sigma_nt - \\tilde q^{\\top}\\tilde \\psi_0^s, \\tilde q)\\|\\tilde q\\|\\frac{\\zeta^2}{\\sigma_n} \\ f_s(\\sigma_nt\\mid \\tilde q) \\ dt \\ f(\\tilde q) \\ d\\tilde q \\notag \\\\\n      & = \\zeta^2 \\int_{\\mathbb{R}^{p-1}}m_4(- \\tilde q^{\\top}\\tilde \\psi_0^s, \\tilde q)  f_s(0 \\mid \\tilde q)\\|\\tilde q\\| \\ f(\\tilde q) \\ d\\tilde q + R \\notag\\\\\n    & \\le \\zeta^2  \\mathbb{E}\\left[\\|\\tilde Q\\|m_4\\left(- \\tilde Q^{\\top}\\tilde \\psi_0^s, \\tilde Q\\right)  f_s(0 \\mid \\tilde Q)\\right]  + R \\notag \n\\end{align}\nThe analysis of the remainder term is similar and if is of higher order. This concludes when $\\zeta > \\sqrt{K\\sigma_n}$: \n\\begin{align}\n& \\mathbb{E}\\left[F^2_{3, \\zeta}(X, Y, Q)\\right] \\notag \\\\\n    & \\mathbb{E}\\left[\\left|\\left[-2\\left(Y - X^{\\top}\\beta_0^s\\right)X^{\\top}\\delta_0^s + (X^{\\top}\\delta_0^s)^2\\right]\\right|^2 \\times  \\sup_{d_*(\\theta, \\theta_0^s) \\le \\zeta} \\left|\\left\\{K\\left(\\frac{Q^{\\top}\\psi_0^s}{\\sigma_n}\\right) -  K\\left(\\frac{Q^{\\top}\\psi}{\\sigma_n}\\right)\\right\\}\\right|^2\\right] \\notag \\\\\n    & \\lesssim \\mathbb{E}\\left[m_4(Q)\\sup_{d_*(\\theta, \\theta_0^s) \\le \\zeta} \\left|\\left\\{K\\left(\\frac{Q^{\\top}\\psi_0^s}{\\sigma_n}\\right) -  K\\left(\\frac{Q^{\\top}\\psi}{\\sigma_n}\\right)\\right\\}\\right|^2\\right] \\notag \\\\\n    \\label{eq:env_4}  & \\lesssim \\zeta^2\n\\end{align}\nCombining \\eqref{eq:env_3}, \\eqref{eq:env_4} with equation \\eqref{eq:moc_bound_2} we have:\n\\begin{align*}\n\\sqrt{n}\\mathbb{E}\\left[\\sup_{\\theta: d_*(\\theta, \\theta_0^s) \\le \\zeta} \\left|\\left(\\mathbb{P}_n - P\\right)\\left(f_\\theta - f_{\\theta_0^s}\\right)\\right|\\right] & \\lesssim  \\sqrt{\\zeta}\\sigma_n^{1\/4}\\mathds{1}_{\\zeta \\le \\sqrt{\\mathcal{K}\\sigma_n}} + \\zeta \\mathds{1}_{\\zeta > \\sqrt{\\mathcal{K} \\sigma_n}} \\\\\n& := \\phi_n(\\zeta) \\,.\n\\end{align*}\nHence to obtain rate we have to solve $r_n^2 \\phi_n(1\/r_n) \\le \\sqrt{n}$, i.e. (ignoring $\\mathcal{K}$ as this does not affect the rate)\n$$\nr_n^{3\/2}\\sigma_n^{1\/4}\\mathds{1}_{r_n \\ge \\sigma_n^{-1\/2}} + r_n \\mathds{1}_{r_n \\le \\sigma_n^{-1\/2}} \\le \\sqrt{n} \\,.\n$$\nNow if $r_n \\le \\sigma_n^{-1\/2}$ then $r_n = \\sqrt{n}$ which implies $\\sqrt{n} \\le \\sigma_n^{-1\/2}$ i.e. $n\\sigma_n \\to 0$ and hence contradiction. On the other hand, if $r_n \\ge \\sigma_n^{-1\/2}$ then $r_n = n^{1\/3}\\sigma_n^{-1\/6}$. This implies $n^{1\/3}\\sigma_n^{-1\/6} \\ge \\sigma_n^{-1\/2}$, i.e. $n^{1\/3} \\ge \\sigma_n^{-1\/3}$, i.e. $n\\sigma_n \\to \\infty$ which is okay. This implies: \n$$\nn^{2\/3}\\sigma_n^{-1\/3}d^2(\\hat \\theta^s, \\theta_0^s) = O_p(1) \\,.\n$$\nNow as $n^{2\/3}\\sigma_n^{-1\/3} \\gg \\sigma_n^{-1}$, we have: \n$$\n\\frac{1}{\\sigma_n}d^2(\\hat \\theta^s, \\theta_0^s) = o_p(1) \\,.\n$$\nwhich further indicates $\\|\\hat \\psi^s - \\psi_0^s\\|\/\\sigma_n = o_p(1)$. This, along with the fact that $\\|\\psi_0^s - \\psi_0\\|\/\\sigma_n = o(1)$ (from Lemma \\ref{bandwidth}), establishes that $\\|\\hat \\psi_0^s - \\psi_0\\|\/\\sigma_n = o_p(1)$. This completes the proof. \n\\end{proof}\n\n\n\n\n\\section{Supplementary Lemmas for the proof of Theorem \\ref{thm:regression}}\n\\label{sec:supp_B}\n\\subsection{Proof of Lemma \\ref{bandwidth}}\n\\begin{proof}\nFirst we establish the fact that $\\theta_0^s \\to \\theta_0$. Note that for all $n$, we have: \n$$\n\\mathbb{M}^s(\\theta_0^s) \\le \\mathbb{M}^s(\\theta_0) \n$$\nTaking $\\limsup$ on the both side we have: \n$$\n\\limsup_{n \\to \\infty} \\mathbb{M}^s(\\theta_0^s) \\le \\mathbb{M}(\\theta_0) \\,.\n$$\nNow using Lemme \\ref{lem:uniform_smooth} we have: \n$$\n\\limsup_{n \\to \\infty} \\mathbb{M}^s(\\theta_0^s) = \\limsup_{n \\to \\infty} \\left[\\mathbb{M}^s(\\theta_0^s) - \\mathbb{M}(\\theta_0^s) + \\mathbb{M}(\\theta_0^s)\\right] = \\limsup_{n \\to \\infty} \\mathbb{M}(\\theta_0^s) \\,.\n$$\nwhich implies $\\limsup_{n \\to \\infty} \\mathbb{M}(\\theta_0^s) \\le \\mathbb{M}(\\theta_0)$ and from the continuity of $\\mathbb{M}(\\theta)$ and $\\theta_0$ being its unique minimizer, we conclude the proof. Now, using Lemma \\ref{lem:pop_curv_nonsmooth} and Lemma \\ref{lem:uniform_smooth} we further obtain: \n\\begin{align}\n   u_- d^2(\\theta_0^s, \\theta_0) & \\le \\mathbb{M}(\\theta_0^s) - \\mathbb{M}(\\theta_0) \\notag \\\\\n    & = \\mathbb{M}(\\theta_0^s) - \\mathbb{M}^s(\\theta^s_0) + \\underset{\\le 0}{\\underline{\\mathbb{M}^s(\\theta_0^s) - \\mathbb{M}^s(\\theta_0)}} + \\mathbb{M}^s(\\theta_0) - \\mathbb{M}(\\theta_0) \\notag \\\\\n    \\label{eq:est_dist_bound} & \\le \\sup_{\\theta \\in \\Theta}\\left|\\mathbb{M}^s(\\theta) - \\mathbb{M}(\\theta)\\right| \\le K_1 \\sigma_n \\,. \n\\end{align}\nNote that we neeed consistency of $\\theta_0^s$ here as the lower bound in Lemma \\ref{lem:pop_curv_nonsmooth} is only valid in a neighborhood around $\\theta_0$. As $\\theta_0^s$ is the minimizer of $\\mathbb{M}^s(\\theta)$, from the first order condition we have: \n\\begin{align}\n    \\label{eq:beta_grad}\\nabla_{\\beta}\\mathbb{M}^s_n(\\theta_0^s) & = -2\\mathbb{E}\\left[X(Y - X^{\\top}\\beta_0^s)\\right] + 2\\mathbb{E} \\left\\{\\left[X_iX_i^{\\top}\\delta_0^s\\right] K\\left(\\frac{Q_i^{\\top}\\psi_0^s}{\\sigma_n}\\right)\\right\\}  = 0 \\\\\n    \\label{eq:delta_grad}\\nabla_{\\delta}\\mathbb{M}^s_n(\\theta_0^s) & = \\mathbb{E} \\left\\{\\left[-2X_i\\left(Y_i - X_i^{\\top}\\beta_0^s\\right) + 2X_iX_i^{\\top}\\delta_0^s\\right] K\\left(\\frac{Q_i^{\\top}\\psi_0^s}{\\sigma_n}\\right)\\right\\} = 0\\\\\n    \\label{eq:psi_grad}\\nabla_{\\psi}\\mathbb{M}^s_n(\\theta_0^s) & = \\frac{1}{\\sigma_n}\\mathbb{E} \\left\\{\\left[-2\\left(Y_i - X_i^{\\top}\\beta_0^s\\right)X_i^{\\top}\\delta_0^s + (X_i^{\\top}\\delta_0^s)^2\\right]\\tilde Q_i K'\\left(\\frac{Q_i^{\\top}\\psi_0^s}{\\sigma_n}\\right)\\right\\} = 0\n\\end{align}\nWe first show that $(\\tilde \\psi^s_0 - \\tilde \\psi_0)\/\\sigma_n \\to 0$ by \\emph{reductio ab absurdum}. From equation \\eqref{eq:est_dist_bound}, we know $\\|\\psi_0^s - \\psi_0\\|\/\\sigma_n = O(1)$. Hence it has a convergent subsequent $\\psi^s_{0, n_k}$, where $(\\tilde \\psi^s_{0, n_k} - \\tilde \\psi_0)\/\\sigma_n \\to h$. If we can prove that $h = 0$, then we establish every subsequence of $\\|\\psi_0^s - \\psi_0\\|\/\\sigma_n$ has a further subsequence which converges to $0$ which further implies $\\|\\psi_0^s - \\psi_0\\|\/\\sigma_n$ converges to $0$. To save some notations, we prove that if $(\\psi_0^s - \\psi_0)\/\\sigma_n \\to h$ then $h = 0$. We start with equation \\eqref{eq:psi_grad}. Define $\\tilde \\eta = (\\tilde \\psi^s_0 - \\tilde \\psi_0)\/\\sigma_n = (\\psi_0^s - \\psi_0)\/\\sigma_n$ where $\\tilde \\psi$ is all the co-ordinates of $\\psi$ except the first one, as the first co-ordinate of $\\psi$ is always assumed to be $1$ for identifiability purpose. \n\\allowdisplaybreaks\n\\begin{align}\n    0 & =  \\frac{1}{\\sigma_n}\\mathbb{E} \\left\\{\\left[-2\\left(Y_i - X_i^{\\top}\\beta_0^s\\right)X_i^{\\top}\\delta_0^s + (X_i^{\\top}\\delta_0^s)^2\\right]\\tilde Q_i K'\\left(\\frac{Q_i^{\\top}\\psi_0^s}{\\sigma_n}\\right)\\right\\}  \\notag \\\\\n    & =  \\frac{1}{\\sigma_n}\\mathbb{E}\\left[\\left( -2\\delta_0^s XX^{\\top}(\\beta_0 - \\beta^s_0) -2\\delta_0^s XX^{\\top}\\delta_0\\mathds{1}_{Q^{\\top}\\delta_0 > 0} + (X_i^{\\top}\\delta_0^s)^2\\right)\\tilde QK'\\left(\\frac{Q^{\\top}\\psi^s_0}{\\sigma_n}\\right)\\right] \\notag \\\\\n     & =  \\frac{1}{\\sigma_n}\\mathbb{E}\\left[\\left( -2\\delta_0^s XX^{\\top}(\\beta_0 - \\beta^s_0) -2\\delta_0^s XX^{\\top}(\\delta_0 - \\delta_0^s)\n     \\mathds{1}_{Q^{\\top}\\delta_0 > 0} \\right. \\right. \\notag \\\\\n     & \\hspace{10em} \\left. \\left. + (X_i^{\\top}\\delta_0^s)^2\\left(1 - 2\\mathds{1}_{Q^{\\top}\\delta_0 > 0}\\right)\\right)\\tilde QK'\\left(\\frac{Q^{\\top}\\psi^s_0}{\\sigma_n}\\right)\\right] \\notag \\\\\n    & =  \\frac{-2}{\\sigma_n}\\mathbb{E}\\left[\\left(\\delta_0^{s^{\\top}} g(Q)(\\beta_0 - \\beta^s_0)\\right)\\tilde QK'\\left(\\frac{Q^{\\top}\\psi^s_0}{\\sigma_n}\\right)\\right] \\notag \\\\\n    & \\qquad \\qquad \\qquad -  \\frac{2}{\\sigma_n} \\mathbb{E}\\left[\\left(\\delta_0^{s^{\\top}}g(Q)(\\delta_0 - \\delta^s_0)\\right)\\tilde QK'\\left(\\frac{Q^{\\top}\\psi^s_0}{\\sigma_n}\\right)\\mathds{1}_{Q^{\\top}\\delta_0 > 0}\\right] \\notag \\\\\n    & \\hspace{15em} +   \\frac{1}{\\sigma_n}\\mathbb{E}\\left[\\left(\\delta_0^{s^{\\top}}g(Q)\\delta^s_0\\right)\\tilde QK'\\left(\\frac{Q^{\\top}\\psi^s_0}{\\sigma_n}\\right)\\left(1 - 2\\mathds{1}_{Q^{\\top}\\delta_0 > 0}\\right)\\right] \\notag \\\\\n     & = -\\underbrace{\\frac{2}{\\sigma_n}\\mathbb{E}\\left[\\left(\\delta_0^{s^{\\top}} g(Q)(\\beta_0 - \\beta^s_0)\\right)\\tilde QK'\\left(\\frac{Q^{\\top}\\psi^s_0}{\\sigma_n}\\right)\\right]}_{T_1} \\notag \\\\\n     & \\qquad \\qquad -\\underbrace{\\frac{2}{\\sigma_n} \\mathbb{E}\\left[\\left(\\delta_0^{s^{\\top}}g(Q)(\\delta_0 - \\delta^s_0)\\right)\\tilde QK'\\left(\\frac{Q^{\\top}\\psi^s_0}{\\sigma_n}\\right)\\mathds{1}_{Q^{\\top}\\delta_0 > 0}\\right]}_{T_2} \\notag \\\\\n     \\label{eq:pop_est_conv_1} & \\qquad \\qquad \\qquad  + \\underbrace{\\frac{1}{\\sigma_n}\\mathbb{E}\\left[\\left(\\delta_0{\\top}g(Q)\\delta_0\\right)\\tilde QK'\\left(\\frac{Q^{\\top}\\psi^s_0}{\\sigma_n}\\right)\\left(1 - 2\\mathds{1}_{Q^{\\top}\\delta_0 > 0}\\right)\\right]}_{T_3} \\notag \\\\\n     & \\qquad \\qquad \\qquad \\qquad  + \\underbrace{\\frac{2}{\\sigma_n}\\mathbb{E}\\left[\\left((\\delta_0 - \\delta_0^s)^{\\top}g(Q)\\delta_0\\right)\\tilde QK'\\left(\\frac{Q^{\\top}\\psi^s_0}{\\sigma_n}\\right)\\left(1 - 2\\mathds{1}_{Q^{\\top}\\delta_0 > 0}\\right)\\right]}_{T_4} \\notag \\\\\n & = T_1 + T_2 + T_3 + T_4\n \\end{align}\nAs mentioned earlier, there is a bijection between $(Q_1, \\tilde Q)$ and $(Q^{\\top}\\psi_0, \\tilde Q)$. The map of one side is obvious. The other side is also trivial as the first coordinate of $\\psi_0$ is 1, which makes $Q^{\\top}\\psi_0 = Q_1 + \\tilde Q^{\\top}\\tilde \\psi_0$:  \n$$\n(Q^{\\top}\\psi_0, \\tilde Q) \\mapsto (Q^{\\top}\\psi_0 - \\tilde Q^{\\top}\\tilde \\psi_0, \\tilde Q) \\,.\n$$\nWe first show that $T_1, T_2$ and $T_4$ are $o(1)$. Towards that end first note that: \n\\begin{align*}\n|T_1| & \\le \\frac{2}{\\sigma_n}\\mathbb{E}\\left[\\|g(Q)\\|_{op} \\ \\|\\tilde Q\\| \\ \\left|K'\\left(\\frac{Q^{\\top}\\psi^s_0}{\\sigma_n}\\right)\\right|\\right]\\|\\delta_0^s\\|\\|\\beta_0 - \\beta_0^s\\| \\\\\n|T_2| & \\le \\frac{2}{\\sigma_n} \\mathbb{E}\\left[\\|g(Q)\\|_{op} \\ \\|\\tilde Q\\| \\ \\left|K'\\left(\\frac{Q^{\\top}\\psi^s_0}{\\sigma_n}\\right)\\right|\\right]\\|\\delta_0^s\\|\\|\\delta_0 - \\delta_0^s\\| \\\\\n|T_4| & \\le \\frac{2}{\\sigma_n} \\mathbb{E}\\left[\\|g(Q)\\|_{op} \\ \\|\\tilde Q\\| \\ \\left|K'\\left(\\frac{Q^{\\top}\\psi^s_0}{\\sigma_n}\\right)\\right|\\right]\\|\\delta_0^s\\|\\|\\delta_0 - \\delta_0^s\\|\n\\end{align*}\nFrom the above bounds, it is immediate that to show that above terms are $o(1)$ all we need to show is: \n$$\n \\frac{1}{\\sigma_n}\\mathbb{E}\\left[\\|g(Q)\\|_{op} \\ \\|\\tilde Q\\| \\ \\left|K'\\left(\\frac{Q^{\\top}\\psi^s_0}{\\sigma_n}\\right)\\right|\\right] = O(1) \\,.\n$$\nTowards that direction, define $\\eta = (\\tilde \\psi_0^s - \\tilde \\psi_0)\/\\sigma_n$: \n\\begin{align*}\n& \\frac{1}{\\sigma_n}\\mathbb{E}\\left[\\|g(Q)\\|_{op} \\ \\|\\tilde Q\\| \\ \\left|K'\\left(\\frac{Q^{\\top}\\psi^s_0}{\\sigma_n}\\right)\\right|\\right] \\\\\n& \\le c_+  \\frac{1}{\\sigma_n}\\mathbb{E}\\left[\\|\\tilde Q\\| \\ \\left|K'\\left(\\frac{Q^{\\top}\\psi^s_0}{\\sigma_n}\\right)\\right|\\right]  \\\\\n& = c_+ \\frac{1}{\\sigma_n}\\int \\int \\|\\tilde q\\| \\left|K'\\left(\\frac{t}{\\sigma_n} + \\tilde q^{\\top}\\eta \\right)\\right| f_0\\left(t \\mid \\tilde q\\right) f(\\tilde q) \\ dt \\ d\\tilde q \\\\\n& = c_+ \\int \\int \\|\\tilde q\\| \\left|K'\\left(t + \\tilde q^{\\top}\\eta \\right)\\right| f_0\\left(\\sigma_n t \\mid \\tilde q\\right) f(\\tilde q) \\ dt \\ d\\tilde q \\\\\n& =  c_+ \\int  \\|\\tilde q\\| f_0\\left(0 \\mid \\tilde q\\right) \\int \\left|K'\\left(t + \\tilde q^{\\top}\\eta \\right)\\right| \\ dt  \\ f(\\tilde q) \\ d\\tilde q + R_1 \\\\\n& = c_+ \\int \\left|K'\\left(t\\right)\\right| dt \\ \\mathbb{E}\\left[\\|\\tilde Q\\| f_0(0 \\mid \\tilde Q)\\right] + R_1 = O(1) + R_1 \\,.\n\\end{align*}\nTherefore, all it remains to show is $R_1$ is also $O(1)$ (or of smaller order): \n\\begin{align*}\n|R_1| & = \\left|c_+ \\int \\int \\|\\tilde q\\| \\left|K'\\left(t + \\tilde q^{\\top}\\eta \\right)\\right| \\left(f_0\\left(\\sigma_n t \\mid \\tilde q\\right) - f_0(0 \\mid \\tilde q) \\right)f(\\tilde q) \\ dt \\ d\\tilde q\\right| \\\\\n& \\le c_+ F_+ \\sigma_n \\int \\|\\tilde q\\| \\int_{-\\infty}^{\\infty} |t|\\left|K'\\left(t + \\tilde q^{\\top}\\eta \\right)\\right| \\ dt \\ f(\\tilde q) \\ d\\tilde q \\\\\n& = c_+ F_+ \\sigma_n \\int \\|\\tilde q\\| \\int_{-\\infty}^{\\infty} |t - q^{\\top}\\eta|\\left|K'\\left(t\\right)\\right| \\ dt \\ f(\\tilde q) \\ d\\tilde q \\\\\n& \\le c_+ F_+ \\sigma_n \\left[\\int \\|\\tilde q\\| \\int_{-\\infty}^{\\infty} |t|\\left|K'\\left(t\\right)\\right| \\ dt \\ f(\\tilde q) \\ d\\tilde q  + \\int \\|\\tilde q\\|^2\\|\\eta\\| \\int_{-\\infty}^{\\infty}\\left|K'\\left(t\\right)\\right| \\ dt \\ f(\\tilde q) \\ d\\tilde q\\right] \\\\\n& = c_+ F_+ \\sigma_n \\left[\\left(\\int_{-\\infty}^{\\infty} |t|\\left|K'\\left(t\\right)\\right| \\ dt\\right) \\times \\mathbb{E}[\\|\\tilde Q\\|] +  \\left(\\int_{-\\infty}^{\\infty}\\left|K'\\left(t\\right)\\right| \\ dt\\right) \\times \\|\\eta\\| \\ \\mathbb{E}[\\|\\tilde Q\\|^2]\\right] \\\\\n& = O(\\sigma_n) = o(1) \\,.\n\\end{align*}\nThis completes the proof.  For $T_3$, the limit is non-degenerate which can be calculated as follows: \n\\begin{align*}\nT_3 &= \\frac{1}{\\sigma_n}\\mathbb{E}\\left[\\left(\\delta_0{\\top}g(Q)\\delta_0\\right)\\tilde QK'\\left(\\frac{Q^{\\top}\\psi^s_0}{\\sigma_n}\\right)\\left(1 - 2\\mathds{1}_{Q^{\\top}\\delta_0 > 0}\\right)\\right] \\\\\n& =  \\frac{1}{\\sigma_n} \\int \\int \\left(\\delta_0{\\top}g(t - \\tilde q^{\\top}\\tilde \\psi_0, \\tilde q)\\delta_0\\right)\\tilde q K'\\left(\\frac{t}{\\sigma_n} + \\tilde q^{\\top} \\eta\\right)\\left(1 - 2\\mathds{1}_{t > 0}\\right) \\ f_0(t \\mid \\tilde q) \\ f(\\tilde q) \\ dt \\ d\\tilde q \\\\\n& = \\int \\int \\left(\\delta_0{\\top}g(\\sigma_n t - \\tilde q^{\\top}\\tilde \\psi_0, \\tilde q)\\delta_0\\right)\\tilde q K'\\left(t + \\tilde q^{\\top} \\eta\\right)\\left(1 - 2\\mathds{1}_{t > 0}\\right) \\ f_0(\\sigma_n t \\mid \\tilde q) \\ f(\\tilde q) \\ dt \\ d\\tilde q \\\\\n& = \\int \\int \\left(\\delta_0{\\top}g(- \\tilde q^{\\top}\\tilde \\psi_0, \\tilde q)\\delta_0\\right)\\tilde q K'\\left(t + \\tilde q^{\\top} \\eta\\right)\\left(1 - 2\\mathds{1}_{t > 0}\\right) \\ f_0(0 \\mid \\tilde q) \\ f(\\tilde q) \\ dt \\ d\\tilde q + R \\\\\n& = \\int  \\left(\\delta_0{\\top}g(- \\tilde q^{\\top}\\tilde \\psi_0, \\tilde q)\\delta_0\\right)\\tilde q f_0(0 \\mid \\tilde q) \\left[\\int_{-\\infty}^0 K'\\left(t + \\tilde q^{\\top} \\eta\\right) \\ dt - \\int_0^\\infty K'\\left(t + \\tilde q^{\\top}\\tilde \\eta\\right) \\ dt \\right] \\ f(\\tilde q) \\ d\\tilde q + R \\\\\n&=  \\int  \\left(\\delta_0{\\top}g(- \\tilde q^{\\top}\\tilde \\psi_0, \\tilde q)\\delta_0\\right)\\tilde q f_0(0 \\mid \\tilde q)\\left(2K\\left(\\tilde q^{\\top}\\eta\\right) - 1\\right) \\ f(\\tilde q) \\ d\\tilde q + R  \\\\\n& =   \\mathbb{E}\\left[\\tilde Q f(0 \\mid \\tilde Q) \\left(\\delta_0^{\\top}g(- \\tilde Q^{\\top}\\tilde \\psi_0, \\tilde Q)\\delta_0\\right)\\left(2K(\\tilde Q^{\\top} \\eta)- 1\\right)\\right] + R \n\\end{align*} \nThat the remainder $R$ is $o(1)$ again follows by similar calculation as before and hence skipped. Therefore we have when $\\eta = (\\tilde \\psi_0^s - \\psi_0)\/\\sigma_n \\to h$: \n$$\nT_3 \\overset{n \\to \\infty}{\\longrightarrow}  \\mathbb{E}\\left[\\tilde Q f(0 \\mid \\tilde Q) \\left(\\delta_0^{\\top}g(- \\tilde Q^{\\top}\\tilde \\psi_0, \\tilde Q)\\delta_0\\right)\\left(2K(\\tilde Q^{\\top}h)- 1\\right)\\right]  \\,,\n$$\nwhich along with equation \\eqref{eq:pop_est_conv_1} implies: \n$$\n\\mathbb{E}\\left[\\tilde Q f(0 \\mid \\tilde Q) \\left(\\delta_0^{\\top}g(- \\tilde Q^{\\top}\\tilde \\psi_0, \\tilde Q)\\delta_0\\right)\\left(2K(\\tilde Q^{\\top}h)- 1\\right)\\right]  = 0 \\,.\n$$\nTaking inner product with respect to $h$ on both side of the above equation we obtain: \n$$\n\\mathbb{E}\\left[\\tilde Q^{\\top}h f(0 \\mid \\tilde Q) \\left(\\delta_0^{\\top}g(- \\tilde Q^{\\top}\\tilde \\psi_0, \\tilde Q)\\delta_0\\right)\\left(2K(\\tilde Q^{\\top}h)- 1\\right)\\right]  = 0\n$$\nNow from the symmetry of our Kernel $K$ we have $\\left(\\delta_0^{\\top}g(- \\tilde Q^{\\top}\\tilde \\psi_0, \\tilde Q)\n\\delta_0\\right)\\tilde Q^{\\top}h f(0 \\mid \\tilde Q) (2K(\\tilde Q^{\\top}\\tilde h) - 1) \\ge 0$ almost surely. As the expectation is $0$, we further deduce that $\\tilde Q^{\\top}h f(0 \\mid \\tilde Q) (2K(\\tilde Q^{\\top}\\tilde h)-1) = 0$ almost surely, which further implies $h = 0$. \n\\\\\\\\\n\\noindent\nWe next prove that $(\\beta_0 - \\beta^s_0)\/\\sqrt{\\sigma_n} \\to 0$ and $(\\delta_0 - \\delta^s_0)\/\\sqrt{\\sigma_n} \\to 0$ using equations\\eqref{eq:beta_grad} and \\eqref{eq:delta_grad}. We start with equation \\eqref{eq:beta_grad}: \n\\begin{align}\n    0 & = -\\mathbb{E}\\left[X(Y - X^{\\top}\\beta_0^s)\\right] + \\mathbb{E} \\left\\{\\left[X_iX_i^{\\top}\\delta_0^s\\right] K\\left(\\frac{Q_i^{\\top}\\psi_0^s}{\\sigma_n}\\right)\\right\\}  \\notag \\\\\n    & = -\\mathbb{E}\\left[XX^{\\top}(\\beta_0 - \\beta_0^s)\\right] - \\mathbb{E}[XX^{\\top}\\delta_0\\mathds{1}_{Q^{\\top}\\psi_0 > 0}] + \\mathbb{E} \\left[ g(Q)K\\left(\\frac{Q_i^{\\top}\\psi_0^s}{\\sigma_n}\\right)\\right]\\delta_0^s \\notag \\\\\n    & = -\\Sigma_X(\\beta_0 - \\beta_0^s) -\\mathbb{E}\\left[g(Q)\\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right](\\delta_0 - \\delta_0^s) + \\mathbb{E} \\left[g(Q)\\left\\{K\\left(\\frac{Q_i^{\\top}\\psi_0^s}{\\sigma_n}\\right) - \\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right\\}\\right]\\delta_0^s \\notag \\\\\n     \\label{eq:deriv1} & = \\Sigma_X\\frac{(\\beta_0^2 - \\beta_0)}{\\sigma_n}  + \\mathbb{E}\\left[g(Q)\\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right]\\frac{(\\delta_0^2 - \\delta_0)}{\\sigma_n} + \\frac{1}{\\sigma_n}\\mathbb{E} \\left[g(Q)\\left\\{K\\left(\\frac{Q_i^{\\top}\\psi_0^s}{\\sigma_n}\\right) - \\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right\\}\\right]\\delta_0^s \\notag \\\\  \n     & = \\left( \\mathbb{E}\\left[g(Q)\\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right]\\right)^{-1}\\Sigma_X \\frac{(\\beta_0^2 - \\beta_0)}{\\sigma_n}  + \\frac{\\delta_0^s - \\delta_0}{\\sigma_n}  \\notag \\\\\n     & \\qquad \\qquad \\qquad \\qquad +  \\left( \\mathbb{E}\\left[g(Q)\\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right]\\right)^{-1} \\frac{1}{\\sigma_n}\\mathbb{E} \\left[g(Q)\\left\\{K\\left(\\frac{Q_i^{\\top}\\psi_0^s}{\\sigma_n}\\right) - \\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right\\}\\right]\\delta_0^s \n\\end{align}\nFrom equation \\eqref{eq:delta_grad} we have:\n\\begin{align}\n    0 & = \\mathbb{E} \\left\\{\\left[-X\\left(Y - X^{\\top}\\beta_0^s\\right) + XX^{\\top}\\delta_0^s\\right] K\\left(\\frac{Q^{\\top}\\psi_0^s}{\\sigma_n}\\right)\\right\\} \\notag \\\\\n    & = -\\mathbb{E}\\left[g(Q)K\\left(\\frac{Q^{\\top}\\psi_0^s}{\\sigma_n}\\right)\\right](\\beta_0 - \\beta_0^s) -  \\mathbb{E}\\left[g(Q)K\\left(\\frac{Q^{\\top}\\psi_0^s}{\\sigma_n}\\right)\\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right]\\delta_0 \\notag \\\\\n    & \\hspace{20em}+  \\mathbb{E}\\left[g(Q)K\\left(\\frac{Q^{\\top}\\psi_0^s}{\\sigma_n}\\right)\\right]\\delta_0^s \\notag \\\\\n     & = -\\mathbb{E}\\left[g(Q)K\\left(\\frac{Q^{\\top}\\psi_0^s}{\\sigma_n}\\right)\\right](\\beta_0 - \\beta_0^s) -  \\mathbb{E}\\left[g(Q)K\\left(\\frac{Q^{\\top}\\psi_0^s}{\\sigma_n}\\right)\\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right](\\delta_0 - \\delta_0^s) \\notag \\\\\n     & \\hspace{20em} +   \\mathbb{E}\\left[g(Q)K\\left(\\frac{Q^{\\top}\\psi_0^s}{\\sigma_n}\\right)\\left(1 - \\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right)\\right]\\delta_0^s  \\notag \\\\\n        & = \\mathbb{E}\\left[g(Q)K\\left(\\frac{Q^{\\top}\\psi_0^s}{\\sigma_n}\\right)\\right]\\frac{(\\beta_0^s - \\beta_0)}{\\sigma_n} + \\mathbb{E}\\left[g(Q)K\\left(\\frac{Q^{\\top}\\psi_0^s}{\\sigma_n}\\right)\\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right]\\frac{(\\delta^s_0 - \\delta_0)}{\\sigma_n} \\notag \\\\\n     \\label{eq:deriv2} & \\hspace{20em} +   \\frac{1}{\\sigma_n}\\mathbb{E}\\left[g(Q)K\\left(\\frac{Q^{\\top}\\psi_0^s}{\\sigma_n}\\right)\\left(1 - \\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right)\\right]\\delta_0^s  \\notag \\\\ \n     & = \\left( \\mathbb{E}\\left[g(Q)K\\left(\\frac{Q^{\\top}\\psi_0^s}{\\sigma_n}\\right)\\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right]\\right)^{-1}\\mathbb{E}\\left[g(Q)K\\left(\\frac{Q^{\\top}\\psi_0^s}{\\sigma_n}\\right)\\right]\\frac{(\\beta_0^s - \\beta_0)}{\\sigma_n} + \\frac{(\\delta^s_0 - \\delta_0)}{\\sigma_n} \\notag \\\\\n     & \\qquad \\qquad \\qquad + \\left( \\mathbb{E}\\left[g(Q)K\\left(\\frac{Q^{\\top}\\psi_0^s}{\\sigma_n}\\right)\\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right]\\right)^{-1} \\frac{1}{\\sigma_n}\\mathbb{E}\\left[g(Q)K\\left(\\frac{Q^{\\top}\\psi_0^s}{\\sigma_n}\\right)\\left(1 - \\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right)\\right]\\delta_0^s \n     \\end{align}\nSubtracting equation \\eqref{eq:deriv2} from \\eqref{eq:deriv1} we obtain: \n$$\n0 = A_n \\frac{(\\beta_0^s - \\beta_0)}{\\sigma_n} + b_n \\,,\n$$\ni.e. \n$$\n\\lim_{n \\to \\infty}  \\frac{(\\beta_0^s - \\beta_0)}{\\sigma_n} = \\lim_{n \\to \\infty} -A_n^{-1}b_n \\,.\n$$\nwhere: \n\\begin{align*}\nA_n & =  \\left( \\mathbb{E}\\left[g(Q)\\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right]\\right)^{-1}\\Sigma_X   \\\\\n& \\qquad \\qquad  - \\left( \\mathbb{E}\\left[g(Q)K\\left(\\frac{Q^{\\top}\\psi_0^s}{\\sigma_n}\\right)\\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right]\\right)^{-1}\\mathbb{E}\\left[g(Q)K\\left(\\frac{Q^{\\top}\\psi_0^s}{\\sigma_n}\\right)\\right] \\\\\nb_n & =  \\left( \\mathbb{E}\\left[g(Q)\\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right]\\right)^{-1} \\frac{1}{\\sigma_n}\\mathbb{E} \\left[g(Q)\\left\\{K\\left(\\frac{Q_i^{\\top}\\psi_0^s}{\\sigma_n}\\right) - \\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right\\}\\right]\\delta_0^s  \\\\\n& \\qquad  - \\left( \\mathbb{E}\\left[g(Q)K\\left(\\frac{Q^{\\top}\\psi_0^s}{\\sigma_n}\\right)\\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right]\\right)^{-1} \\frac{1}{\\sigma_n}\\mathbb{E}\\left[g(Q)K\\left(\\frac{Q^{\\top}\\psi_0^s}{\\sigma_n}\\right)\\left(1 - \\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right)\\right]\\delta_0^s \n\\end{align*}\nIt is immediate via DCT that as $n \\to \\infty$: \n\\begin{align}\n     \\label{eq:limit_3} \\mathbb{E}\\left[g(Q)K\\left(\\frac{Q^{\\top}\\psi_0^s}{\\sigma_n}\\right)\\right] & \\longrightarrow \\mathbb{E}\\left[g(Q)\\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right] \\,. \\\\\n     \\label{eq:limit_4} \\mathbb{E}\\left[g(Q)K\\left(\\frac{Q^{\\top}\\psi_0^s}{\\sigma_n}\\right)\\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right] & \\longrightarrow \\mathbb{E}\\left[g(Q)\\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right] \\,.\n\\end{align}\nFrom equation \\eqref{eq:limit_3} and \\eqref{eq:limit_4} it is immediate that: \n\\begin{align*}\n\\lim_{n \\to \\infty} A_n  & =  \\left( \\mathbb{E}\\left[g(Q)\\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right]\\right)^{-1}\\Sigma_X - I \\\\\n& = \\left( \\mathbb{E}\\left[g(Q)\\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right]\\right)^{-1}\\left( \\mathbb{E}\\left[g(Q)\\mathds{1}_{Q^{\\top}\\psi_0 \\le 0}\\right]\\right) := A\\,.\n\\end{align*}\nNext observe that: \n\\begin{align}\n    & \\frac{1}{\\sigma_n} \\mathbb{E}\\left[g(Q)\\left\\{K\\left(\\frac{Q^{\\top}\\psi_0^s}{\\sigma_n}\\right) - \\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right\\}\\right] \\notag \\\\\n    & = \\frac{1}{\\sigma_n} \\mathbb{E}\\left[g(Q)\\left\\{K\\left(\\frac{Q^{\\top}\\psi_0}{\\sigma_n} + \\tilde Q^{\\top}\\tilde \\eta\\right) - \\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right\\}\\right] \\notag \\\\\n    & = \\int_{\\mathbb{R}^{p-1}} \\int_{-\\infty}^{\\infty} g(\\sigma_nt - \\tilde q^{\\top}\\tilde \\psi_0, \\tilde q)\\left[K\\left(t + \\tilde q^{\\top}\\tilde \\eta\\right) - \\mathds{1}_{t > 0}\\right] f(\\sigma_n t \\mid \\tilde q) \\ dt \\ f(\\tilde q) \\ d\\tilde q  \\notag \\\\\n    \\label{eq:limit_1} & \\longrightarrow \\mathbb{E}\\left[g(-\\tilde Q^{\\top}\\tilde \\psi_0, \\tilde Q)f(0 \\mid \\tilde Q)\\right] \\cancelto{0}{\\int_{-\\infty}^{\\infty} \\left[K\\left(t\\right) - \\mathds{1}_{t > 0}\\right] \\ dt} \\,. \n\\end{align}\nSimilar calculation yields: \n\\begin{align}\n    \\label{eq:limit_2} & \\frac{1}{\\sigma_n} \\mathbb{E}\\left[g(Q)K\\left(\\frac{Q^{\\top}\\psi_0^s}{\\sigma_n}\\right)\\left(1 - \\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right)\\right] \\notag \\\\\n    & \\longrightarrow \\mathbb{E}[g(-\\tilde Q^{\\top}\\tilde \\psi_0, \\tilde Q)f_0(0 \\mid \\tilde Q)]\\int_{-\\infty}^{\\infty} \\left[K\\left(t\\right)\\mathds{1}_{t \\le 0}\\right] \\ dt \\,.\n\\end{align}\nCombining equation \\eqref{eq:limit_1} and \\eqref{eq:limit_2} we conclude: \n\\begin{align*}\n\\lim_{n \\to \\infty} b_n &= \\left( \\mathbb{E}\\left[g(Q)\\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right]\\right)^{-1} \\mathbb{E}[g(-\\tilde Q^{\\top}\\tilde \\psi_0, \\tilde Q)\\delta_0f_0(0 \\mid \\tilde Q)]\\int_{-\\infty}^{\\infty} \\left[K\\left(t\\right)\\mathds{1}_{t \\le 0}\\right] \\ dt  \\\\\n& := b \\,.\n\\end{align*}\nwhich further implies, \n$$\n\\lim_{n \\to \\infty}  \\frac{(\\beta_0^s - \\beta_0)}{\\sigma_n} = -A^{-1}b \\implies (\\beta_0^s - \\beta_0) = o(\\sqrt{\\sigma_n})\\,,\n$$\nand by similar calculations: \n$$\n(\\delta_0^s - \\delta_0) = o(\\sqrt{\\sigma_n}) \\,.\n$$\nThis completes the proof. \n\\end{proof}\n\n\n\n\n\n\n\\subsection{Proof of Lemma \\ref{lem:pop_curv_nonsmooth}}\n\\begin{proof}\nFrom the definition of $M(\\theta)$ it is immediate that $\\mathbb{M}(\\theta_0) = \\mathbb{E}[{\\epsilon}^2] = \\sigma^2$. For any general $\\theta$: \n\\begin{align*}\n    \\mathbb{M}(\\theta) & = \\mathbb{E}\\left[\\left(Y - X^{\\top}\\left(\\beta + \\delta\\mathds{1}_{Q^{\\top}\\psi > 0}\\right)\\right)^2\\right] \\\\\n    & = \\sigma^2 + \\mathbb{E}\\left[\\left( X^{\\top}\\left(\\beta + \\delta\\mathds{1}_{Q^{\\top}\\psi > 0} - \\beta_0 - \\delta_0\\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right)\\right)^2\\right] \\\\\n    & \\ge \\sigma^2 + c_- \\mathbb{E}_Q\\left[\\left\\|\\beta - \\beta_0 + \\delta\\mathds{1}_{Q^{\\top}\\psi > 0}- \\delta_0\\mathds{1}_{Q^{\\top}\\psi_0 > 0} \\right\\|^2\\right]\n\\end{align*}\nThis immediately implies: \n$$\n\\mathbb{M}(\\theta) - \\mathbb{M}(\\theta_0) \\ge c_- \\mathbb{E}\\left[\\left\\|\\beta - \\beta_0 + \\delta\\mathds{1}_{Q^{\\top}\\psi > 0}- \\delta_0\\mathds{1}_{Q^{\\top}\\psi_0 > 0} \\right\\|^2\\right] \\,.\n$$\n\n\\noindent\nFor notational simplicity, define $p_{\\psi} = \\mathbb{P}(Q^{\\top}\\psi > 0)$. Expanding the RHS we have: \n\\begin{align}\n    & \\mathbb{E}\\left[\\left\\|\\beta - \\beta_0 + \\delta\\mathds{1}_{Q^{\\top}\\psi > 0}- \\delta_0\\mathds{1}_{Q^{\\top}\\psi_0 > 0} \\right\\|^2\\right] \\notag \\\\\n    & = \\|\\beta - \\beta_0\\|^2 +  2(\\beta - \\beta_0)^{\\top}\\mathbb{E}\\left[\\delta\\mathds{1}_{Q^{\\top}\\psi > 0}- \\delta_0\\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right] + \\mathbb{E}\\left[\\left\\|\\delta\\mathds{1}_{Q^{\\top}\\psi > 0}- \\delta_0\\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right\\|^2\\right] \\notag  \\\\\n     & = \\|\\beta - \\beta_0\\|^2 +  2(\\beta - \\beta_0)^{\\top}\\mathbb{E}\\left[\\delta\\mathds{1}_{Q^{\\top}\\psi > 0}-\\delta\\mathds{1}_{Q^{\\top}\\psi_0 > 0} + \\delta\\mathds{1}_{Q^{\\top}\\psi_0 > 0} - \\delta_0\\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right] \\notag  \\\\\n     & \\qquad \\qquad \\qquad \\qquad \\qquad+ \\mathbb{E}\\left[\\left\\|\\delta\\mathds{1}_{Q^{\\top}\\psi > 0}-\\delta\\mathds{1}_{Q^{\\top}\\psi_0 > 0} + \\delta\\mathds{1}_{Q^{\\top}\\psi_0 > 0} - \\delta_0\\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right\\|^2\\right] \\notag  \\\\\n    & = \\|\\beta - \\beta_0\\|^2 + 2(\\beta - \\beta_0)^{\\top}(\\delta - \\delta_0)p_{\\psi_0} +  \\|\\delta - \\delta_0\\|^2 p_{\\psi_0} \\notag  \\\\\n    & \\qquad \\qquad \\qquad +  2(\\beta - \\beta_0)^{\\top}\\delta\\left(p_{\\psi} - p_{\\psi_0}\\right) + \\|\\delta\\|^2 \\mathbb{P}\\left(\\text{sign}(Q^{\\top}\\psi) \\neq \\text{sign}(Q^{\\top}\\psi_0)\\right)  \\notag  \\\\\n    \\label{eq:nsb1} & \\qquad \\qquad \\qquad \\qquad \\qquad - 2\\delta^{\\top}(\\delta - \\delta_0)\\mathbb{P}\\left(Q^{\\top}\\psi_0 > 0, Q^{\\top}\\psi < 0\\right) \n    \\end{align}\nUsing the fact that $2ab \\ge (a^2\/c) + cb^2$ for any constant $c$ we have: \n\\begin{align*}\n& \\|\\beta - \\beta_0\\|^2 + 2(\\beta - \\beta_0)^{\\top}(\\delta - \\delta_0)p_{\\psi_0} +  \\|\\delta - \\delta_0\\|^2 p_{\\psi_0} \\\\\n& \\ge \\|\\beta - \\beta_0\\|^2  + \\|\\delta - \\delta_0\\|^2 p_{\\psi_0}  -  \\frac{\\|\\beta - \\beta_0\\|^2 p_{\\psi_0}}{c} - c \\|\\delta - \\delta_0\\|^2 p_{\\psi_0} \\\\\n& = \\|\\beta - \\beta_0\\|^2\\left(1 - \\frac{p_{\\psi_0}}{c}\\right) +  \\|\\delta - \\delta_0\\|^2  p_{\\psi_0} (1 - c) \\,.\n\\end{align*}\nfor any $c$. To make the RHS non-negative we pick $p_{\\psi_0} < c < 1$ and concludes that: \n\\begin{equation}\n\\label{eq:nsb2}\n \\|\\beta - \\beta_0\\|^2 + 2(\\beta - \\beta_0)^{\\top}(\\delta - \\delta_0)p_{\\psi_0} +  \\|\\delta - \\delta_0\\|^2 p_{\\psi_0} \\gtrsim \\left( \\|\\beta - \\beta_0\\|^2 + \\|\\delta - \\delta_0\\|^2\\right) \\,.\n\\end{equation}\nFor the last 3 summands of RHS of equation \\eqref{eq:nsb1}:     \n\\begin{align}\n& 2(\\beta - \\beta_0)^{\\top}\\delta\\left(p_{\\psi} - p_{\\psi_0}\\right) + \\|\\delta\\|^2 \\mathbb{P}\\left(\\text{sign}(Q^{\\top}\\psi) \\neq \\text{sign}(Q^{\\top}\\psi_0)\\right)  \\notag  \\\\\n & \\qquad \\qquad - 2\\delta^{\\top}(\\delta - \\delta_0)\\mathbb{P}\\left(Q^{\\top}\\psi_0 > 0, Q^{\\top}\\psi < 0\\right) \\notag \\\\\n & = 2(\\beta - \\beta_0)^{\\top}\\delta \\mathbb{P}\\left(Q^{\\top}\\psi > 0, Q^{\\top}\\psi_0 < 0\\right) - 2(\\beta - \\beta_0)^{\\top}\\delta \\mathbb{P}\\left(Q^{\\top}\\psi < 0, Q^{\\top}\\psi_0 > 0\\right) \\notag \\\\\n  & \\qquad \\qquad + |\\delta\\|^2 \\mathbb{P}\\left(\\text{sign}(Q^{\\top}\\psi) \\neq \\text{sign}(Q^{\\top}\\psi_0)\\right) - 2\\delta^{\\top}(\\delta - \\delta_0)\\mathbb{P}\\left(Q^{\\top}\\psi_0 > 0, Q^{\\top}\\psi < 0\\right) \\notag \\\\\n  & = \\left[\\|\\delta\\|^2 - 2(\\beta - \\beta_0)^{\\top}\\delta - 2\\delta^{\\top}(\\delta - \\delta_0)\\right]\\mathbb{P}\\left(Q^{\\top}\\psi_0 > 0, Q^{\\top}\\psi < 0\\right) \\notag \\\\\n  & \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad + \\left[\\|\\delta\\|^2 + 2(\\beta - \\beta_0)^{\\top}\\delta\\right]\\mathbb{P}\\left(Q^{\\top}\\psi > 0, Q^{\\top}\\psi_0 < 0\\right) \\notag \\\\\n    & = \\left[\\|\\delta_0\\|^2 - 2(\\beta - \\beta_0)^{\\top}(\\delta - \\delta_0) -  2(\\beta - \\beta_0)^{\\top}\\delta_0 - \\|\\delta - \\delta_0\\|^2\\right]\\mathbb{P}\\left(Q^{\\top}\\psi_0 > 0, Q^{\\top}\\psi < 0\\right) \\notag \\\\\n  & \\qquad   + \\left[\\|\\delta_0\\|^2 + \\|\\delta - \\delta_0\\|^2 + 2(\\delta - \\delta_0)^{\\top}\\delta_0 + 2(\\beta - \\beta_0)^{\\top}(\\delta - \\delta_0) +  2(\\beta - \\beta_0)^{\\top}\\delta_0\\right]\\mathbb{P}\\left(Q^{\\top}\\psi > 0, Q^{\\top}\\psi_0 < 0\\right) \\notag \\\\\n  & \\ge  \\left[\\|\\delta_0\\|^2 - 2\\|\\beta - \\beta_0\\|\\|\\delta - \\delta_0\\| -  2\\|\\beta - \\beta_0\\|\\|\\delta_0\\| - \\|\\delta - \\delta_0\\|^2\\right]\\mathbb{P}\\left(Q^{\\top}\\psi_0 > 0, Q^{\\top}\\psi < 0\\right) \\notag \\\\\n  & \\qquad   + \\left[\\|\\delta_0\\|^2 + \\|\\delta - \\delta_0\\|^2 + 2\\|\\delta - \\delta_0\\|\\|\\delta_0\\| + 2\\|\\beta - \\beta_0\\|\\|\\delta - \\delta_0\\| +  2\\|\\beta - \\beta_0\\|\\|\\delta_0\\|\\right]\\mathbb{P}\\left(Q^{\\top}\\psi > 0, Q^{\\top}\\psi_0 < 0\\right) \\notag \\\\\n\\label{eq:nsb3} & \\gtrsim \\|\\delta_0\\|^2 \\mathbb{P}\\left(\\text{sign}(Q^{\\top}\\psi) \\neq \\text{sign}(Q^{\\top}\\psi_0)\\right)   \\gtrsim \\|\\psi - \\psi_0\\| \\hspace{0.2in} [\\text{By Assumption }\\ref{eq:assm}]\\,.\n\\end{align}\nCombining equation \\eqref{eq:nsb2} and \\eqref{eq:nsb3} we complete the proof of lower bound. The upper bound is relatively easier: note that by our previous calculation: \n\\begin{align*}\n    \\mathbb{M}(\\theta) - \\mathbb{M}(\\theta_0) & =  \\mathbb{E}\\left[\\left( X^{\\top}\\left(\\beta + \\delta\\mathds{1}_{Q^{\\top}\\psi > 0} - \\beta_0 - \\delta_0\\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right)\\right)^2\\right] \\\\\n    & \\le  c_+\\mathbb{E}\\left[\\left\\|\\beta - \\beta_0 + \\delta\\mathds{1}_{Q^{\\top}\\psi > 0}- \\delta_0\\mathds{1}_{Q^{\\top}\\psi_0 > 0} \\right\\|^2\\right] \\\\\n    & = c_+\\mathbb{E}\\left[\\left\\|\\beta - \\beta_0 + \\delta\\mathds{1}_{Q^{\\top}\\psi > 0}- \\delta\\mathds{1}_{Q^{\\top}\\psi_0 > 0} + \\delta\\mathds{1}_{Q^{\\top}\\psi_0 > 0} - \\delta_0\\mathds{1}_{Q^{\\top}\\psi_0 > 0} \\right\\|^2\\right] \\\\\n    & \\lesssim \\left[\\|\\beta - \\beta_0\\|^2 + \\|\\delta - \\delta_0\\|^2 + \\mathbb{P}\\left(\\text{sign}(Q^{\\top}\\psi) \\neq \\text{sign}(Q^{\\top}\\psi_0)\\right)\\right] \\\\\n    & \\lesssim \\left[\\|\\beta - \\beta_0\\|^2 + \\|\\delta - \\delta_0\\|^2 + \\|\\psi - \\psi_0\\|\\right] \\,.\n\\end{align*}\nThis completes the entire proof.\n\\end{proof}\n\n\n\n\n\\subsection{Proof of Lemma \\ref{lem:uniform_smooth}}\n\\begin{proof}\nThe difference of the two losses: \n\\begin{align*}\n\\left|\\mathbb{M}^s(\\theta) - \\mathbb{M}(\\theta)\\right| & = \\left|\\mathbb{E}\\left[\\left\\{-2\\left(Y_i - X^{\\top}\\beta\\right)X^{\\top}\\delta + (X^{\\top}\\delta)^2\\right\\}\\left(K\\left(\\frac{Q^{\\top}\\psi}{\\sigma_n}\\right) - \\mathds{1}_{Q^{\\top}\\psi > 0}\\right)\\right]\\right| \\\\\n& \\le \\mathbb{E}\\left[\\left|-2\\left(Y_i - X^{\\top}\\beta\\right)X^{\\top}\\delta + (X^{\\top}\\delta)^2\\right|\\left|K\\left(\\frac{Q^{\\top}\\psi}{\\sigma_n}\\right) - \\mathds{1}_{Q^{\\top}\\psi > 0}\\right|\\right] \\\\\n& := \\mathbb{E}\\left[m(Q)\\left|K\\left(\\frac{Q^{\\top}\\psi}{\\sigma_n}\\right) - \\mathds{1}_{Q^{\\top}\\psi > 0}\\right|\\right] \n\\end{align*}\nwhere $m(Q) = \\mathbb{E}\\left[\\left|-2\\left(Y_i - X^{\\top}\\beta\\right)X^{\\top}\\delta + (X^{\\top}\\delta)^2\\right| \\mid Q\\right]$. This function can be bounded as follows: \n\\begin{align*}\nm(Q) & = \\mathbb{E}\\left[\\left|-2\\left(Y_i - X^{\\top}\\beta\\right)X^{\\top}\\delta + (X^{\\top}\\delta)^2\\right| \\mid Q\\right] \\\\\n& \\le \\mathbb{E}[ (X^{\\top}\\delta)^2 \\mid Q] + 2\\mathbb{E}\\left[\\left|(\\beta - \\beta_0)^{\\top}XX^{\\top}\\delta\\right|\\right] + 2\\mathbb{E}\\left[\\left|\\delta_0^{\\top}XX^{\\top}\\delta\\right|\\right] \\\\\n& \\le c_+\\left(\\|\\delta\\|^2 + 2\\|\\beta - \\beta_0\\|\\|\\delta\\| + 2\\|\\delta\\|\\|\\delta_0\\|\\right) \\lesssim 1 \\,,\n\\end{align*}\nas our parameter space is compact. For the rest of the calculation define $\\eta = (\\tilde \\psi - \\tilde \\psi_0)\/\\sigma_n$. The definition of $\\eta$ may be changed from proof to proof, but it will be clear from the context. Therefore we have: \n\\begin{align*}\n\\left|\\mathbb{M}^s(\\theta) - \\mathbb{M}(\\theta)\\right| & \\lesssim \\mathbb{E}\\left[\\left|K\\left(\\frac{Q^{\\top}\\psi}{\\sigma_n}\\right) - \\mathds{1}_{Q^{\\top}\\psi > 0}\\right|\\right]  \\\\\n& =  \\mathbb{E}\\left[\\left| \\mathds{1}\\left(\\frac{Q^{\\top}\\psi_0}{\\sigma_n} + \\eta^{\\top}\\tilde{Q} \\ge 0\\right) - K\\left(\\frac{Q^{\\top}\\psi_0}{\\sigma_n} + \\eta^{\\top}\\tilde{Q}\\right)\\right|\\right]  \\\\\n& = \\sigma_n \\int_{\\mathbb{R}^{p-1}} \\int_{-\\infty}^{\\infty}  \\left | \\mathds{1}\\left(t  \\ge 0\\right) - K\\left(t \\right)\\right | f_0(\\sigma_n (t-\\eta^{\\top}\\tilde{q}) | \\tilde{q}) \\ dt \\ dP(\\tilde{q}) \\\\\n& \\le f_+  \\sigma_n  \\int_{-\\infty}^{\\infty}  \\left | \\mathds{1}\\left(t  \\ge 0\\right) - K\\left(t \\right)\\right | \\ dt \\lesssim \\sigma_n \\,.\n\\end{align*}\nwhere the integral over $t$ is finite follows from the definition of the kernel. This completes the proof. \n\\end{proof}\n\n\n\n\n\n\n\n\n\\subsection{Proof of Lemma \\ref{lem:pop_smooth_curvarture}}\n\\begin{proof}\nFirst note that we can write: \n\\begin{align}\n    & \\mathbb{M}^s(\\theta) - \\mathbb{M}^s(\\theta_0^s) \\notag \\\\\n    & = \\underbrace{\\mathbb{M}^s(\\theta) - \\mathbb{M}(\\theta)}_{\\ge -K_1\\sigma_n} + \\underbrace{\\mathbb{M}(\\theta) - \\mathbb{M}(\\theta_0)}_{\\underbrace{\\ge u_- d^2(\\theta, \\theta_0)}_{\\ge \\frac{u_-}{2} d^2(\\theta, \\theta_0^s) - u_-\\sigma_n }} + \\underbrace{\\mathbb{M}(\\theta_0) - \\mathbb{M}(\\theta_0^s)}_{\\ge - u_+ d^2(\\theta_0, \\theta_0^s) \\ge - u_+\\sigma_n} + \\underbrace{\\mathbb{M}(\\theta_0^s) - \\mathbb{M}^s(\\theta_0^s)}_{\\ge - K_1 \\sigma_n} \\notag \\\\\n    & \\ge \\frac{u_-}{2}d^2(\\theta, \\theta_0^s) - (2K_1 + \\xi)\\sigma_n \\notag \\\\\n    & \\ge \\frac{u_-}{2}\\left[\\|\\beta - \\beta^s_0\\|^2 + \\|\\delta - \\delta^s_0\\|^2 + \\|\\psi - \\psi^s_0\\|\\right] - (2K_1 + \\xi)\\sigma_n \\notag \\\\ \n    & \\ge \\left[\\frac{u_-}{2}\\left(\\|\\beta - \\beta^s_0\\|^2 + \\|\\delta - \\delta^s_0\\|^2\\right) + \\frac{u_-}{4}\\|\\psi - \\psi^s_0\\|\\right]\\mathds{1}_{\\|\\psi - \\psi^s_0\\| > \\frac{4(2K_1 + \\xi)}{u_-}\\sigma_n} \\notag \\\\\n    \\label{eq:lower_curv_smooth} & \\gtrsim  \\left[\\|\\beta - \\beta^s_0\\|^2 + \\|\\delta - \\delta^s_0\\|^2 + \\|\\psi - \\psi^s_0\\|\\right]\\mathds{1}_{\\|\\psi - \\psi^s_0\\| > \\frac{4(2K_1 + \\xi)}{u_-}\\sigma_n}\n\\end{align}\nwhere $\\xi$ can be taken as close to $0$ as possible. Henceforth we set $\\mathcal{K} = 4(2K_1 + \\xi)\/u_-$. For the other part of the curvature (i.e. when $\\|\\psi - \\psi_0^s\\| \\le \\mathcal{K} \\sigma_n$) we start with a two step Taylor expansion of the smoothed loss function: \n\\begin{align*}\n    \\mathbb{M}^s(\\theta) - \\mathbb{M}^s(\\theta_0^s) = \\frac12 (\\theta_0 - \\theta^0_s)^{\\top}\\nabla^2 \\mathbb{M}^s(\\theta^*)(\\theta_0 - \\theta^0_s) \n\\end{align*}\nRecall the definition of $\\mathbb{M}^s(\\theta)$: \n$$\n\\mathbb{M}^s_n(\\theta) = \\mathbb{E}\\left(Y - X^{\\top}\\beta\\right)^2 + \\mathbb{E} \\left\\{\\left[-2\\left(Y_i - X_i^{\\top}\\beta\\right)X_i^{\\top}\\delta + (X_i^{\\top}\\delta)^2\\right] K\\left(\\frac{Q_i^{\\top}\\psi}{\\sigma_n}\\right)\\right\\}\n$$\nThe partial derivates of $\\mathbb{M}^s(\\theta)$ with respect to $(\\beta, \\delta, \\psi)$ was derived in equation \\eqref{eq:beta_grad} - \\eqref{eq:psi_grad}. From there, we calculate the hessian of $\\mathbb{M}^s(\\theta)$: \n\\begin{align*}\n    \\nabla_{\\beta\\beta}\\mathbb{M}^s(\\theta) & = 2\\Sigma_X \\\\\n    \\nabla_{\\delta\\delta}\\mathbb{M}^s(\\theta) & = 2 \\mathbb{E}\\left[g(Q)K\\left(\\frac{Q_i^{\\top}\\psi}{\\sigma_n}\\right)\\right] = 2 \\mathbb{E}\\left[g(Q)K\\left(\\frac{Q_i^{\\top}\\psi_0 }{\\sigma_n} + \\tilde Q^{\\top}\\tilde \\eta\\right)\\right] \\\\\n    \\nabla_{\\psi\\psi} \\mathbb{M}^s(\\theta) & = \\frac{1}{\\sigma_n^2}\\mathbb{E} \\left\\{\\left[-2\\left(Y_i - X_i^{\\top}\\beta\\right)X_i^{\\top}\\delta + (X_i^{\\top}\\delta)^2\\right]\\tilde Q_i\\tilde Q_i^{\\top} K''\\left(\\frac{Q_i^{\\top}\\psi_0 }{\\sigma_n} + \\tilde Q^{\\top}\\tilde \\eta\\right)\\right\\} \\\\\n    \\nabla_{\\beta \\delta}\\mathbb{M}^s(\\theta) & = 2 \\mathbb{E}\\left[g(Q)K\\left(\\frac{Q_i^{\\top}\\psi}{\\sigma_n}\\right)\\right] = 2 \\mathbb{E}\\left[g(Q)K\\left(\\frac{Q_i^{\\top}\\psi_0 }{\\sigma_n} + \\tilde Q^{\\top}\\tilde \\eta\\right)\\right] \\\\\n    \\nabla_{\\beta \\psi}\\mathbb{M}^s(\\theta) & = \\frac{2}{\\sigma_n}\\mathbb{E}\\left(g(Q)\\delta\\tilde Q^{\\top}K'\\left(\\frac{Q_i^{\\top}\\psi_0 }{\\sigma_n} + \\tilde Q^{\\top}\\tilde \\eta\\right)\\right)  \\\\\n    \\nabla_{\\delta \\psi} \\mathbb{M}^s(\\theta) & = \\frac{2}{\\sigma_n}\\mathbb{E} \\left\\{\\left[-X_i\\left(Y_i - X_i^{\\top}\\beta\\right) + X_iX_i^{\\top}\\delta\\right]\\tilde Q_i^{\\top} K'\\left(\\frac{Q_i^{\\top}\\psi_0 }{\\sigma_n} + \\tilde Q^{\\top}\\tilde \\eta\\right)\\right\\} \\,.\n\\end{align*}\nwhere we use $\\tilde \\eta$ for a generic notation for $(\\tilde \\psi - \\tilde \\psi_0)\/\\sigma_n$. For notational simplicity, we define $\\gamma = (\\beta, \\delta)$ and $\\nabla^2\\mathbb{M}^{s, \\gamma}(\\theta)$, $\\nabla^2\\mathbb{M}^{s, \\gamma \\psi}(\\theta), \\nabla^2\\mathbb{M}^{s, \\psi \\psi}(\\theta)$ to be corresponding blocks of the hessian matrix. We have: \n\\begin{align}\n      \\mathbb{M}^s(\\theta) - \\mathbb{M}^s(\\theta_0^s) & = \\frac12 (\\theta - \\theta^0_s)^{\\top}\\nabla^2 \\mathbb{M}^s(\\theta^*)(\\theta - \\theta^0_s)  \\notag \\\\\n      & = \\frac12 (\\gamma - \\gamma_0^s)^{\\top}\\nabla^2 \\mathbb{M}^{s, \\gamma}(\\theta^*)(\\gamma - \\gamma^0_s) + (\\gamma - \\gamma_0^s)^{\\top}\\nabla^2 \\mathbb{M}^{s, \\gamma \\psi}(\\theta^*)(\\psi - \\psi^0_s) \\notag \\\\\n      & \\qquad \\qquad \\qquad \\qquad + \\frac12(\\psi - \\psi_0^s)^{\\top}\\nabla^2 \\mathbb{M}^{s, \\psi \\psi}(\\theta^*)(\\psi - \\psi^0_s)  \\notag \\\\\n      \\label{eq:hessian_1} & := \\frac12 \\left(T_1 + 2T_2 + T_3\\right)\n\\end{align}\nNote that we can write: \n\\begin{align*}\n    T_1 & = (\\gamma - \\gamma_0^s)^{\\top}\\nabla^2 \\mathbb{M}^{s, \\gamma}(\\tilde \\theta)(\\gamma - \\gamma^0_s) \\\\\n    & = (\\gamma - \\gamma_0^s)^{\\top}\\nabla^2 \\mathbb{M}^{s, \\gamma}(\\theta_0)(\\gamma - \\gamma^0_s)  + (\\gamma - \\gamma_0^s)^{\\top}\\left[\\nabla^2 \\mathbb{M}^{s, \\gamma}(\\tilde \\theta) - \\nabla^2 \\mathbb{M}^{s, \\gamma}(\\theta_0)\\right](\\gamma - \\gamma^0_s) \n\\end{align*}\nThe operator norm of the difference of two hessians can be bounded as: \n$$\n\\left\\|\\nabla^2 \\mathbb{M}^{s, \\gamma}(\\theta^*) - \\nabla^2 \\mathbb{M}^{s, \\gamma}(\\theta_0)\\right\\|_{op}  = O(\\sigma_n) \\,.\n$$\nfor any $\\theta^*$ in a neighborhood of $\\theta_0^s$ with $\\|\\psi - \\psi_0^s\\| \\le \\mathcal{K} \\sigma_n$. To prove this note that for any $\\theta$: \n$$\n\\nabla^2 \\mathbb{M}^{s, \\gamma}(\\theta^*) - \\nabla^2 \\mathbb{M}^{s, \\gamma}(\\theta_0) = 2\\begin{pmatrix}0 & A \\\\\nA & A\\end{pmatrix} = \\begin{pmatrix}0 & 1 \\\\ 1 & 1\\end{pmatrix} \\otimes A \n$$\nwhere: \n$$\nA = \\mathbb{E}\\left[g(Q)K\\left(\\frac{Q_i^{\\top}\\psi}{\\sigma_n}\\right)\\right] - \\mathbb{E}\\left[g(Q)K\\left(\\frac{Q_i^{\\top}\\psi_0 }{\\sigma_n}\\right)\\right]\n$$\nTherefore it is enough to show $\\|A\\|_{op} = O(\\sigma_n)$. Towards that direction: \n\\begin{align*}\nA & = \\mathbb{E}\\left[g(Q)K\\left(\\frac{Q_i^{\\top}\\psi}{\\sigma_n}\\right)\\right] - \\mathbb{E}\\left[g(Q)K\\left(\\frac{Q_i^{\\top}\\psi_0 }{\\sigma_n}\\right)\\right] \\\\\n& = \\sigma_n \\int \\int g(\\sigma_n t - \\tilde q^{\\top}\\tilde \\psi_0)\\left(K(t + \\tilde q^{\\top}\\eta) - K(t) \\right) f_0(\\sigma_n t \\mid \\tilde q) \\ f(\\tilde q) \\ dt  \\ d\\tilde q \\\\\n& = \\sigma_n \\left[\\int \\int g(- \\tilde q^{\\top}\\tilde \\psi_0)\\left(K(t + \\tilde q^{\\top}\\eta) - K(t) \\right) f_0(0 \\mid \\tilde q) \\ f(\\tilde q) \\ dt  \\ d\\tilde q + R \\right] \\\\\n& = \\sigma_n \\left[\\int \\int g(- \\tilde q^{\\top}\\tilde \\psi_0)f_0(0 \\mid \\tilde q) \\int_t^{t + \\tilde q^{\\top}\\eta}K'(s) \\ ds \\ f(\\tilde q) \\ dt  \\ d\\tilde q + R \\right] \\\\\n& = \\sigma_n \\left[\\int g(- \\tilde q^{\\top}\\tilde \\psi_0)f_0(0 \\mid \\tilde q) \\int_{-\\infty}^{\\infty}K'(s) \\int_{s-\\tilde q^{\\top}\\eta}^s \\ dt \\ ds \\ f(\\tilde q)\\ d\\tilde q + R \\right] \\\\\n& = \\sigma_n \\left[\\int g(- \\tilde q^{\\top}\\tilde \\psi_0)f_0(0 \\mid \\tilde q)\\tilde q^{\\top}\\eta  \\ f(\\tilde q)\\ d\\tilde q + R \\right] \\\\\n& = \\sigma_n \\left[\\mathbb{E}\\left[g(- \\tilde Q^{\\top}\\tilde \\psi_0, \\tilde Q)f_0(0 \\mid \\tilde Q)\\tilde Q^{\\top}\\eta\\right] + R \\right]\n\\end{align*}\nusing the fact that $\\left\\|\\mathbb{E}\\left[g(- \\tilde Q^{\\top}\\tilde \\psi_0, \\tilde Q)f_0(0 \\mid \\tilde Q)\\tilde Q^{\\top}\\eta\\right]\\right\\|_{op} = O(1)$ and $\\|R\\|_{op} = O(\\sigma_n)$ we conclude the claim. From the above claim we conclude: \n\\begin{equation}\n    \\label{eq:hessian_gamma}\n    T_1 = (\\gamma - \\gamma_0^s)^{\\top}\\nabla^2 \\mathbb{M}^{s, \\gamma}(\\theta^*)(\\gamma - \\gamma^s_0) \\ge \\|\\gamma - \\gamma^s_0\\|^2(1 - O(\\sigma_n)) \\ge \\frac12 \\|\\gamma - \\gamma_0^s\\|^2\n\\end{equation}\nfor all large $n$. \n\\\\\\\\\n\\noindent \nWe next deal with the cross term $T_2$ in equation \\eqref{eq:hessian_1}. Towards that end first note that: \n\\begin{align*}\n    & \\frac{1}{\\sigma_n}\\mathbb{E}\\left((g(Q)\\delta)\\tilde Q^{\\top}K'\\left(\\frac{Q_i^{\\top}\\psi_0 }{\\sigma_n} + \\tilde Q^{\\top}\\eta^*\\right)\\right) \\\\\n    & = \\int_{\\mathbb{R}^{(p-1)}}\\left[ \\int_{-\\infty}^{\\infty} \\left(g\\left(\\sigma_nt - \\tilde q^{\\top}\\tilde \\psi_0, \\tilde q\\right)\\delta\\right) K'\\left(t + \\tilde q^{\\top}\\eta^*\\right) f_0(\\sigma_n t \\mid \\tilde q) \\ dt\\right] \\tilde q^{\\top} \\ f(\\tilde q) \\ d\\tilde q  \\\\\n     & = \\int_{\\mathbb{R}^{(p-1)}}\\left[ \\int_{-\\infty}^{\\infty} \\left(g\\left(- \\tilde q^{\\top}\\tilde \\psi_0, \\tilde q\\right)\\delta\\right) K'\\left(t + \\tilde q^{\\top}\\eta^*\\right) f_0(0 \\mid \\tilde q) \\ dt\\right] \\tilde q^{\\top} \\ f(\\tilde q) \\ d\\tilde q  + R_1\\\\\n    & = \\mathbb{E}\\left[\\left(g\\left( - \\tilde Q^{\\top}\\tilde \\psi_0, \\tilde Q\\right)\\delta\\right)\\tilde Q^{\\top}f_0(0 \\mid \\tilde Q)\\right] + R_1\n\\end{align*}\nwhere the remainder term $R_1$ can be further decomposed $R_1 = R_{11} + R_{12} + R_{13}$ with: \n\\begin{align*}\n    \\left\\|R_{11}\\right\\|  & = \\left\\|\\int_{\\mathbb{R}^{(p-1)}}\\left[ \\int_{-\\infty}^{\\infty} \\left(g\\left(- \\tilde q^{\\top}\\tilde \\psi_0, \\tilde q\\right)\\delta\\right) K'\\left(t + \\tilde q^{\\top}\\eta^*\\right) (f_0(\\sigma_nt\\mid \\tilde q) -  f_0(0 \\mid \\tilde q)) \\ dt\\right] \\tilde q^{\\top} \\ f(\\tilde q) \\ d\\tilde q\\right\\| \\\\\n    & \\le \\left\\|\\int_{\\mathbb{R}^{(p-1)}}\\left[ \\int_{-\\infty}^{\\infty} \\left\\|g\\left(- \\tilde q^{\\top}\\tilde \\psi_0, \\tilde q\\right)\\right\\|_{op}\\|\\delta\\| \\left|K'\\left(t + \\tilde q^{\\top}\\eta^*\\right)\\right| \\left|f_0(\\sigma_nt\\mid \\tilde q) -  f_0(0 \\mid \\tilde q)\\right| \\ dt\\right] \\left|\\tilde q\\right| \\ f(\\tilde q) \\ d\\tilde q\\right\\| \\\\\n    & \\le \\sigma_n  \\dot{f}^+  c_+ \\|\\delta\\| \\int_{\\mathbb{R}^{(p-1)}} \\|\\tilde q\\|  \\int_{-\\infty}^{\\infty} |t| \\left|K'\\left(t + \\tilde q^{\\top}\\eta^*\\right)\\right| \\ dt \\ f(\\tilde q) \\ d\\tilde q \\\\\n    & \\le \\sigma_n  \\dot{f}^+  c_+ \\|\\delta\\| \\int_{\\mathbb{R}^{(p-1)}} \\|\\tilde q\\|  \\int_{-\\infty}^{\\infty} |t -  \\tilde q^{\\top}\\eta^*| \\left|K'\\left(t\\right)\\right| \\ dt \\ f(\\tilde q) \\ d\\tilde q \\\\\n    & \\le \\sigma_n  \\dot{f}^+  c_+ \\|\\delta\\| \\left[\\int_{\\mathbb{R}^{(p-1)}} \\|\\tilde q\\|  \\int_{-\\infty}^{\\infty} |t| \\left|K'\\left(t\\right)\\right| \\ dt \\ f(\\tilde q) \\ d\\tilde q \\right. \\\\\n    & \\qquad \\qquad \\qquad \\left. + \\int_{\\mathbb{R}^{(p-1)}} \\|\\tilde q\\|^2 \\|\\eta^*\\|  \\int_{-\\infty}^{\\infty} |K'(t)| \\ dt \\ f(\\tilde q) \\ d\\tilde q\\right] \\\\\n     & \\le \\sigma_n  \\dot{f}^+  c_+ \\|\\delta\\| \\left[\\int_{\\mathbb{R}^{(p-1)}} \\|\\tilde q\\|  \\int_{-\\infty}^{\\infty} |t| \\left|K'\\left(t\\right)\\right| \\ dt \\ f(\\tilde q) \\ d\\tilde q + \\mathcal{K}\\int_{\\mathbb{R}^{(p-1)}} \\|\\tilde q\\|^2 \\int_{-\\infty}^{\\infty} |K'(t)| \\ dt \\ f(\\tilde q) \\ d\\tilde q\\right] \\\\\n        & \\lesssim \\sigma_n \\,.\n\\end{align*}\nwhere the last bound follows from our assumptions using the fact that: \n\\begin{align*}\n    & \\|R_{12}\\| \\\\\n    &= \\left\\|\\int_{\\mathbb{R}^{(p-1)}}\\left[ \\int_{-\\infty}^{\\infty} \\left(\\left(g\\left(\\sigma_n t- \\tilde q^{\\top}\\tilde \\psi_0, \\tilde q\\right) - g\\left(- \\tilde q^{\\top}\\tilde \\psi_0, \\tilde q\\right)\\right)\\delta\\right) K'\\left(t + \\tilde q^{\\top} \\eta^*\\right)  f_0(0 \\mid \\tilde q) \\ dt\\right] \\tilde q^{\\top} \\ f(\\tilde q) \\ d\\tilde q\\right\\| \\\\\n    & \\le \\int \\|\\tilde q\\|\\|\\delta\\|f_0(0 \\mid \\tilde q) \\int_{-\\infty}^{\\infty} \\left\\|g\\left(\\sigma_n t- \\tilde q^{\\top}\\tilde \\psi_0, \\tilde q\\right) - g\\left(- \\tilde q^{\\top}\\tilde \\psi_0, \\tilde q\\right) \\right\\|_{op}\\left|K'\\left(t + \\tilde q^{\\top} \\eta^*\\right)\\right| \\ dt \\ f(\\tilde q) \\ d\\tilde q \\\\\n    & \\le \\dot{c}_+ \\sigma_n \\int \\|\\tilde q\\|\\|\\delta\\|f_0(0 \\mid \\tilde q)\\dot \\int_{-\\infty}^{\\infty} |t| \\left|K'\\left(t + \\tilde q^{\\top}\\tilde \\eta\\right)\\right| \\ dt \\ f(\\tilde q) \\ d\\tilde q \\hspace{0.2in} [\\text{Assumption }\\ref{eq:assm}]\\\\\n    &  \\lesssim \\sigma_n \\,.\n\\end{align*}\nThe other remainder term $R_{13}$ is the higher order term and can be shown to be $O(\\sigma_n^2)$ using same techniques. This implies for all large $n$: \n\\begin{align*}\n    \\left\\|\\nabla_{\\beta \\psi}\\mathbb{M}^s(\\theta)\\right\\|_{op} & = O(1) \\,.\n\\end{align*}\nand similar calculation yields $ \\left\\|\\nabla_{\\delta \\psi}\\mathbb{M}^s(\\theta)\\right\\|_{op} = O(1)$. Using this we have: \n\\begin{align}\n    T_2 & = (\\gamma - \\gamma_0^s)^{\\top}\\nabla^2 \\mathbb{M}^{s, \\gamma \\psi}(\\tilde \\theta)(\\psi - \\psi^0_s) \\notag \\\\\n    & = (\\beta - \\beta_0^s)^{\\top}\\nabla_{\\beta \\psi}^2 \\mathbb{M}^{s}(\\tilde \\theta)(\\psi - \\psi^0_s) + (\\delta - \\delta_0^s)^{\\top}\\nabla_{\\delta \\psi}^2 \\mathbb{M}^{s}(\\tilde \\theta)(\\psi - \\psi^0_s) \\notag \\\\\n    & \\ge - C\\left[\\|\\beta - \\beta_0^s\\| + \\|\\delta - \\delta_0^s\\| \\right]\\|\\psi - \\psi^0_s\\| \\notag \\\\\n    & \\ge -C \\sqrt{\\sigma_n}\\left[\\|\\beta - \\beta_0^s\\| + \\|\\delta - \\delta_0^s\\| \\right]\\frac{\\|\\psi - \\psi^0_s\\| }{\\sqrt{\\sigma_n}}  \\notag \\\\\n    \\label{eq:hessian_cross} & \\gtrsim - \\sqrt{\\sigma_n}\\left(\\|\\beta - \\beta_0^s\\|^2 + \\|\\delta - \\delta_0^s\\|^2 +\\frac{\\|\\psi - \\psi^0_s\\|^2 }{\\sigma_n} \\right)\n\\end{align}\nNow for $T_3$ note that: \n\\allowdisplaybreaks\n\\begin{align*}\n& \\sigma_n \\nabla_{\\psi\\psi} \\mathbb{M}^s_n(\\theta) \\\\\n& = \\frac{1}{\\sigma_n}\\mathbb{E} \\left\\{\\left[-2\\left(Y_i - X_i^{\\top}\\beta\\right)X_i^{\\top}\\delta + (X_i^{\\top}\\delta)^2\\right]\\tilde Q_i\\tilde Q_i^{\\top} K''\\left(\\frac{Q_i^{\\top}\\psi_0 }{\\sigma_n} + \\tilde Q^{\\top}\\tilde \\eta\\right)\\right\\} \\\\\n& = \\frac{1}{\\sigma_n}\\mathbb{E} \\left\\{\\left[-2\\left(Y_i - X_i^{\\top}\\beta\\right)X_i^{\\top}\\delta \\right]\\tilde Q_i\\tilde Q_i^{\\top} K''\\left(\\frac{Q_i^{\\top}\\psi_0 }{\\sigma_n} + \\tilde Q^{\\top}\\tilde \\eta\\right)\\right\\}  \\\\\n& \\qquad \\qquad \\qquad + \\frac{1}{\\sigma_n}\\mathbb{E} \\left\\{(\\delta^{\\top}g(Q) \\delta)\\tilde Q_i\\tilde Q_i^{\\top} K''\\left(\\frac{Q_i^{\\top}\\psi_0 }{\\sigma_n} + \\tilde Q^{\\top}\\tilde \\eta\\right)\\right\\}  \\\\\n& = \\frac{1}{\\sigma_n}\\mathbb{E} \\left\\{\\left[-2 X_i^{\\top}\\left(\\beta_0 -\\beta\\right)X_i^{\\top}\\delta - 2(X_i^{\\top}\\delta_0)(X_i^{\\top}\\delta)\\mathds{1}_{Q_i^{\\top}\\psi_0 > 0}\\right]\\tilde Q_i\\tilde Q_i^{\\top} K''\\left(\\frac{Q_i^{\\top}\\psi_0 }{\\sigma_n} + \\tilde Q^{\\top}\\tilde \\eta\\right)\\right\\}  \\\\\n& \\qquad \\qquad \\qquad + \\frac{1}{\\sigma_n}\\mathbb{E} \\left\\{(\\delta^{\\top}g(Q) \\delta)\\tilde Q_i\\tilde Q_i^{\\top} K''\\left(\\frac{Q_i^{\\top}\\psi_0 }{\\sigma_n} + \\tilde Q^{\\top}\\tilde \\eta\\right)\\right\\}  \\\\\n& = \\frac{-2}{\\sigma_n}\\mathbb{E} \\left\\{((\\beta_0 - \\beta)^{\\top}g(Q) \\delta)\\tilde Q_i\\tilde Q_i^{\\top} K''\\left(\\frac{Q_i^{\\top}\\psi_0 }{\\sigma_n} + \\tilde Q^{\\top}\\tilde \\eta\\right)\\right\\} \\\\\n& \\qquad \\qquad \\qquad + \\frac{-2}{\\sigma_n}\\mathbb{E} \\left\\{(\\delta_0^{\\top}g(Q) \\delta)\\tilde Q_i\\tilde Q_i^{\\top} K''\\left(\\frac{Q_i^{\\top}\\psi_0 }{\\sigma_n} + \\tilde Q^{\\top}\\tilde \\eta\\right)\\mathds{1}_{Q_i^{\\top}\\psi_0 > 0}\\right\\} \\\\\n& \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad + \\frac{1}{\\sigma_n}\\mathbb{E} \\left\\{(\\delta^{\\top}g(Q) \\delta)\\tilde Q_i\\tilde Q_i^{\\top} K''\\left(\\frac{Q_i^{\\top}\\psi_0 }{\\sigma_n} + \\tilde Q^{\\top}\\tilde \\eta\\right)\\right\\}  \\\\\n& = \\underbrace{\\frac{-2}{\\sigma_n}\\mathbb{E} \\left\\{((\\beta_0 - \\beta)^{\\top}g(Q)\\delta)\\tilde Q_i\\tilde Q_i^{\\top} K''\\left(\\frac{Q_i^{\\top}\\psi_0 }{\\sigma_n} + \\tilde Q^{\\top}\\tilde \\eta\\right)\\right\\}}_{M_1} \\\\\n& \\qquad \\qquad \\qquad + \\underbrace{\\frac{-2}{\\sigma_n}\\mathbb{E} \\left\\{(\\delta_0^{\\top}g(Q) \\delta_0)\\tilde Q_i\\tilde Q_i^{\\top} K''\\left(\\frac{Q_i^{\\top}\\psi_0 }{\\sigma_n} + \\tilde Q^{\\top}\\tilde \\eta\\right)\\mathds{1}_{Q_i^{\\top}\\psi_0 > 0}\\right\\}}_{M_2} \\\\\n& \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad +\n\\underbrace{\\frac{-2}{\\sigma_n}\\mathbb{E} \\left\\{(\\delta_0^{\\top} g(Q) (\\delta - \\delta_0))\\tilde Q_i\\tilde Q_i^{\\top} K''\\left(\\frac{Q_i^{\\top}\\psi_0 }{\\sigma_n} + \\tilde Q^{\\top}\\tilde \\eta\\right)\\mathds{1}_{Q_i^{\\top}\\psi_0 > 0}\\right\\}}_{M_3} \\\\\n& \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad + \\underbrace{\\frac{1}{\\sigma_n}\\mathbb{E} \\left\\{(\\delta^{\\top}g(Q) \\delta)\\tilde Q_i\\tilde Q_i^{\\top} K''\\left(\\frac{Q_i^{\\top}\\psi_0 }{\\sigma_n} + \\tilde Q^{\\top}\\tilde \\eta\\right)\\right\\}}_{M_4}  \\\\\n& := M_1 + M_2 + M_3 + M_4\n\\end{align*}\nWe next show that $M_1$ and $M_4$ are $O(\\sigma_n)$. Towards that end note that for any two vectors $v_1, v_2$: \n\\begin{align*}\n    & \\frac{1}{\\sigma_n}\\mathbb{E} \\left\\{(v_1^{\\top}g(Q)v_2)\\tilde Q_i\\tilde Q_i^{\\top} K''\\left(\\frac{Q_i^{\\top}\\psi_0 }{\\sigma_n} + \\tilde Q^{\\top}\\tilde \\eta\\right)\\right\\} \\\\\n    & = \\int \\tilde q \\tilde q^{\\top} \\int_{-\\infty}^{\\infty}(v_1^{\\top}g(\\sigma_nt - \\tilde q^{\\top}\\tilde \\eta, \\tilde q)v_2) K''(t + \\tilde q^{\\top}\\tilde \\eta) f(\\sigma_nt \\mid \\tilde q) \\ dt \\ f(\\tilde q) \\ d\\tilde q \\\\\n    & = \\int \\tilde q \\tilde q^{\\top} (v_1^{\\top}g( - \\tilde q^{\\top}\\tilde \\eta, \\tilde q)v_2)f(0 \\mid \\tilde q) f(\\tilde q) \\ d\\tilde q \\cancelto{0}{\\int_{-\\infty}^{\\infty} K''(t) \\ dt}  + R = R\n\\end{align*}\nas $\\int K''(t) \\ dt = 0$ follows from our choice of kernel $K(x) = \\Phi(x)$. Similar calculation as in the case of analyzing the remainder of $T_2$ yields $\\|R\\|_{op} = O(\\sigma_n)$.\n\\noindent\nThis immediately implies $\\|M_1\\|_{op} = O(\\sigma_n)$ and $\\|M_4\\|_{op} = O(\\sigma_n)$. Now for $M_2$: \n\\begin{align}\nM_2 & = \\frac{-2}{\\sigma_n}\\mathbb{E} \\left\\{(\\delta_0^{\\top}g(Q) \\delta_0)\\tilde Q_i\\tilde Q_i^{\\top} K''\\left(\\frac{Q_i^{\\top}\\psi_0 }{\\sigma_n} + \\tilde Q^{\\top}\\tilde \\eta\\right)\\mathds{1}_{Q_i^{\\top}\\psi_0 > 0}\\right\\} \\notag \\\\\n& = -2\\int \\int (\\delta_0^{\\top}g(\\sigma_n t - \\tilde q^{\\top}\\tilde \\psi_0) \\delta_0)\\tilde q\\tilde q^{\\top} K''\\left(t + \\tilde q^{\\top}\\eta^*\\right)\\mathds{1}_{t > 0} f_0(\\sigma_n t \\mid \\tilde q) \\ dt f(\\tilde q) \\ d\\tilde q \\notag \\\\\n& = -2\\int (\\delta_0^{\\top}g(- \\tilde q^{\\top}\\tilde \\psi_0) \\delta_0)\\tilde q\\tilde q^{\\top}  f_0(0 \\mid \\tilde q)  \\int_{-\\infty}^{\\infty} K''\\left(t + \\tilde q^{\\top}\\eta^*\\right)\\mathds{1}_{t > 0} \\ dt f(\\tilde q) \\ d\\tilde q  + R \\notag \\\\\n\\label{eq:M_2_double_deriv} & = 2\\mathbb{E}\\left[(\\delta_0^{\\top}g(- \\tilde Q^{\\top}\\tilde \\psi_0) \\delta_0)\\tilde \nQ\\tilde Q^{\\top}  f_0(0 \\mid \\tilde Q) K'(\\tilde Q^{\\top}\\eta^*)\\right] + R \n\\end{align}\nwhere the remainder term R is $O_p(\\sigma_n)$ can be established as follows: \n\\begin{align*}\nR & = -2\\left[\\int \\int (\\delta_0^{\\top}g(\\sigma_n t - \\tilde q^{\\top}\\tilde \\psi_0) \\delta_0)\\tilde q\\tilde q^{\\top} K''\\left(t + \\tilde q^{\\top}\\eta^*\\right)\\mathds{1}_{t > 0} f_0(\\sigma_n t \\mid \\tilde q) \\ dt f(\\tilde q) \\ d\\tilde q \\right. \\\\\n& \\qquad \\qquad - \\left. \\int (\\delta_0^{\\top}g(- \\tilde q^{\\top}\\tilde \\psi_0) \\delta_0)\\tilde q\\tilde q^{\\top}  f_0(0 \\mid \\tilde q)  \\int_{-\\infty}^{\\infty} K''\\left(t + \\tilde q^{\\top}\\eta^*\\right)\\mathds{1}_{t > 0} \\ dt f(\\tilde q) \\ d\\tilde q \\right] \\\\\n& = -2\\left\\{\\left[\\int \\int (\\delta_0^{\\top}g(\\sigma_n t - \\tilde q^{\\top}\\tilde \\psi_0) \\delta_0)\\tilde q\\tilde q^{\\top} K''\\left(t + \\tilde q^{\\top}\\eta^*\\right)\\mathds{1}_{t > 0} f_0(\\sigma_n t \\mid \\tilde q) \\ dt f(\\tilde q) \\ d\\tilde q \\right. \\right. \\\\\n& \\qquad \\qquad - \\left. \\left. \\int (\\delta_0^{\\top}g(\\sigma_n t - \\tilde q^{\\top}\\tilde \\psi_0) \\delta_0) \\tilde q\\tilde q^{\\top}  f_0(0 \\mid \\tilde q)  \\int_{-\\infty}^{\\infty} K''\\left(t + \\tilde q^{\\top}\\eta^*\\right)\\mathds{1}_{t > 0} \\ dt f(\\tilde q) \\ d\\tilde q \\right] \\right. \\\\\n& \\left. +  \\left[\\int (\\delta_0^{\\top}g(\\sigma_n t - \\tilde q^{\\top}\\tilde \\psi_0) \\delta_0)\\tilde q\\tilde q^{\\top}  f_0(0 \\mid \\tilde q)  \\int_{-\\infty}^{\\infty} K''\\left(t + \\tilde q^{\\top}\\eta^*\\right)\\mathds{1}_{t > 0} \\ dt f(\\tilde q) \\ d\\tilde q \\right. \\right. \\\\\n& \\qquad \\qquad \\left. \\left. -\\int (\\delta_0^{\\top}g(- \\tilde q^{\\top}\\tilde \\psi_0) \\delta_0)\\tilde q\\tilde q^{\\top}  f_0(0 \\mid \\tilde q)  \\int_{-\\infty}^{\\infty} K''\\left(t + \\tilde q^{\\top}\\eta^*\\right)\\mathds{1}_{t > 0} \\ dt f(\\tilde q) \\ d\\tilde q \\right]\\right\\} \\\\\n& = -2(R_1 + R_2) \\,.\n\\end{align*}\nFor $R_1$: \n\\begin{align*}\n\\left\\|R_1\\right\\|_{op} & =  \\left\\|\\left[\\int \\int (\\delta_0^{\\top}g(\\sigma_n t - \\tilde q^{\\top}\\tilde \\psi_0) \\delta_0)\\tilde q\\tilde q^{\\top} K''\\left(t + \\tilde q^{\\top}\\eta^*\\right)\\mathds{1}_{t > 0} f_0(\\sigma_n t \\mid \\tilde q) \\ dt f(\\tilde q) \\ d\\tilde q \\right. \\right. \\,.\\\\\n& \\qquad \\qquad - \\left. \\left. \\int (\\delta_0^{\\top}g(\\sigma_n t - \\tilde q^{\\top}\\tilde \\psi_0) \\delta_0)\\tilde q\\tilde q^{\\top}  f_0(0 \\mid \\tilde q)  \\int_{-\\infty}^{\\infty} K''\\left(t + \\tilde q^{\\top}\\eta^*\\right)\\mathds{1}_{t > 0} \\ dt f(\\tilde q) \\ d\\tilde q \\right] \\right\\|_{op} \\\\\n& \\le c_+ \\int \\int \\|\\tilde q\\|^2 |K''\\left(t + \\tilde q^{\\top}\\eta^*\\right)| |f_0(\\sigma_n t \\mid \\tilde q) -f_0(0\\mid \\tilde q)| \\ dt \\ f(\\tilde q) \\ d\\tilde q  \\\\\n& \\le c_+ F_+\\sigma_n  \\int \\|\\tilde q\\|^2 \\int |t| |K''\\left(t + \\tilde q^{\\top}\\eta^*\\right)| \\ dt \\ f(\\tilde q) \\ d\\tilde q \\\\\n& = c_+ F_+\\sigma_n  \\int \\|\\tilde q \\|^2 \\int |t -  \\tilde q^{\\top}\\eta^*| |K''\\left(t\\right)| \\ dt \\ f(\\tilde q) \\ d\\tilde q \\\\\n& \\le c_+ F_+ \\sigma_n \\left[\\mathbb{E}[\\|\\tilde Q\\|^2]\\int |t||K''(t)| \\ dt + \\|\\eta^*\\|\\mathbb{E}[\\|\\tilde Q\\|^3]\\int |K''(t)| \\ dt\\right] = O(\\sigma_n) \\,.\n\\end{align*}\nand similarly for $R_2$: \n\\begin{align*}\n\\|R_2\\|_{op} & = \\left\\|\\left[\\int (\\delta_0^{\\top}g(\\sigma_n t - \\tilde q^{\\top}\\tilde \\psi_0) \\delta_0)\\tilde q\\tilde q^{\\top}  f_0(0 \\mid \\tilde q)  \\int_{-\\infty}^{\\infty} K''\\left(t + \\tilde q^{\\top}\\eta^*\\right)\\mathds{1}_{t > 0} \\ dt f(\\tilde q) \\ d\\tilde q \\right. \\right. \\\\\n& \\qquad \\qquad \\left. \\left. -\\int (\\delta_0^{\\top}g(- \\tilde q^{\\top}\\tilde \\psi_0) \\delta_0)\\tilde q\\tilde q^{\\top}  f_0(0 \\mid \\tilde q)  \\int_{-\\infty}^{\\infty} K''\\left(t + \\tilde q^{\\top}\\eta^*\\right)\\mathds{1}_{t > 0} \\ dt f(\\tilde q) \\ d\\tilde q \\right]\\right\\|_{op} \\\\\n& \\le F_+ \\|\\delta_0\\|^2 \\int \\left\\|g(\\sigma_n t - \\tilde q^{\\top}\\tilde \\psi_0)  - g( - \\tilde q^{\\top}\\tilde \\psi_0) \\right\\|_{op} \\|\\tilde q\\|^2 \\int_{-\\infty}^{\\infty} |K''\\left(t + \\tilde q^{\\top}\\eta^*\\right)| \\ dt \\\\\n& \\le G_+ F_+ \\sigma_n \\int \\|\\tilde q\\|^2 \\int_{-\\infty}^{\\infty} |t||K''\\left(t + \\tilde q^{\\top}\\eta^*\\right)| \\ dt = O(\\sigma_n) \\,.\n\\end{align*}\nTherefore from \\eqref{eq:M_2_double_deriv} we conclude: \n\\begin{equation}\nM_2 = 2\\mathbb{E}\\left[(\\delta_0^{\\top}g(- \\tilde Q^{\\top}\\tilde \\psi_0) \\delta_0)\\tilde \nQ\\tilde Q^{\\top}  f_0(0 \\mid \\tilde Q) K'(\\tilde Q^{\\top}\\eta^*)\\right] + O(\\sigma_n) \\,.\n\\end{equation}\nSimilar calculation for $M_3$ yields: \n\\begin{equation*}\nM_3 = 2\\mathbb{E}\\left[(\\delta_0^{\\top}g(- \\tilde Q^{\\top}\\tilde \\psi_0)(\\delta - \\delta_0))\\tilde \nQ\\tilde Q^{\\top}  f_0(0 \\mid \\tilde Q) K'(\\tilde Q^{\\top}\\eta^*)\\right] + O(\\sigma_n) \\,.\n\\end{equation*}\ni.e. \n\\begin{equation}\n\\|M_3\\|_{op} \\le c_+ \\mathbb{E}\\left[\\|\\tilde Q\\|^2f_0(0 \\mid \\tilde Q) K'(\\tilde Q^{\\top}\\eta^*)\\right]\\|\\delta_0\\| \\|\\delta - \\delta_0\\| \\,.\n\\end{equation}\nNow we claim that for any $\\mathcal{K} < \\infty$, $\\lambda_{\\min} (M_2) > 0$ for all $\\|\\eta^*\\| \\le \\mathcal{K}$. Towards that end, define a function $\\lambda:B_{\\mathbb{R}^{2d}}(1) \\times B_{\\mathbb{R}^{2d}}(\\mathcal{K}) \\to \\mathbb{R}_+$ as: \n$$\n\\lambda: (v, \\eta) \\mapsto 2\\mathbb{E}\\left[(\\delta_0^{\\top}g(- \\tilde Q^{\\top}\\tilde \\psi_0) \\delta_0) \n\\left(v^{\\top}\\tilde Q\\right) ^2 f_0(0 \\mid \\tilde Q) K'(\\tilde Q^{\\top}\\eta)\\right]\n$$\nClearly $\\lambda \\ge 0$ and is continuous on a compact set. Hence its infimum must be attained. Suppose the infimum is $0$, i.e. there exists $(v^*, \\eta^*)$ such that: \n$$\n\\mathbb{E}\\left[(\\delta_0^{\\top}g(- \\tilde Q^{\\top}\\tilde \\psi_0) \\delta_0) \n\\left(v^{*^{\\top}}\\tilde Q\\right) ^2 f_0(0 \\mid \\tilde Q) K'(\\tilde Q^{\\top}\\eta^*)\\right] = 0 \\,.\n$$\nas $\\lambda_{\\min}(g(\\dot)) \\ge c_+$, we must have $\\left(v^{*^{\\top}}\\tilde Q\\right) ^2 f_0(0 \\mid \\tilde Q) K'(\\tilde Q^{\\top}\\eta^*) = 0$ almost surely. But from our assumption, $\\left(v^{*^{\\top}}\\tilde Q\\right) ^2 > 0$ and $K'(\\tilde Q^{\\top}\\eta^*) > 0$ almost surely, which implies $f_0(0 \\mid \\tilde q) = 0$ almost surely, which is a contradiction. Hence there exists $\\lambda_-$ such that: \n$$\n\\lambda_{\\min} (M_2)  \\ge \\lambda_- > 0 \\ \\ \\forall \\ \\ \\|\\psi - \\psi_0^s\\| \\le \\mathcal{K} \\,.\n$$\nHence we have: \n$$\n\\lambda_{\\min}\\left(\\sigma_n \\nabla_{\\psi \\psi}\\mathbb{M}^2(\\theta)\\right) \\ge \\frac{\\lambda_-}{2}(1 - O(\\sigma_n)) \n$$\nfor all theta such that $d_*(\\theta, \\theta_0^s) \\le {\\epsilon} \\,.$ \n\\begin{align}\n\\label{eq:hessian_psi}\n    & \\frac{1}{\\sigma_n}(\\psi - \\psi_0^s)^{\\top}\\sigma_n \\nabla^{\\psi \\psi}\\mathbb{M}^s(\\tilde \\theta) (\\psi - \\psi^0) \\gtrsim \\frac{\\|\\psi - \\psi^s_0\\|^2}{\\sigma_n} \\left(1- O(\\sigma_n)\\right)  \n\\end{align}\nFrom equation \\eqref{eq:hessian_gamma}, \\eqref{eq:hessian_cross} and \\eqref{eq:hessian_psi} we have: \n\\begin{align*}\n& \\frac12 (\\theta_0 - \\theta^0_s)^{\\top}\\nabla^2 \\mathbb{M}^s(\\theta^*)(\\theta_0 - \\theta^0_s) \\\\\n& \\qquad \\qquad \\gtrsim \\left[\\|\\beta - \\beta^s_0\\|^2 + \\|\\gamma - \\gamma^s_0\\|^2 + \\frac{\\|\\psi - \\psi^s_0\\|^2}{\\sigma_n}\\right]\\mathds{1}_{\\|\\psi - \\psi_0^s\\| \\le \\mathcal{K} \\sigma_n}  \\,.\n\\end{align*}\nThis, along with equation \\eqref{eq:lower_curv_smooth} concludes the proof.  \n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\\subsection{Proof of Lemma \\ref{asymp-normality}}\nWe start by proving that analogues of Lemma 2 of \\cite{seo2007smoothed}: we show that: \n\\begin{align*}\n\\lim_{n \\to \\infty} \\mathbb{E}\\left[ \\sqrt{n\\sigma_n}\\nabla \\mathbb{M}_n^{s, \\psi}(\\theta_0)\\right] & = 0 \\\\\n\\lim_{n \\to \\infty} {\\sf var}\\left[ \\sqrt{n\\sigma_n}\\nabla \\mathbb{M}_n^{s, \\psi}(\\theta_0)\\right] & = V^{\\psi}\n\\end{align*}\nfor some matrix $V^{\\psi}$ which will be specified later in the proof. To prove the limit of the expectation: \n\\begin{align*}\n&  \\mathbb{E}\\left[ \\sqrt{n\\sigma_n}\\nabla \\mathbb{M}_n^{s, \\psi}(\\theta_0)\\right] \\\\\n& =  \\sqrt{\\frac{n}{\\sigma_n}}\\mathbb{E}\\left[\\left\\{(Y - X^{\\top}(\\beta_0 + \\delta_0))^2 - (Y - X^{\\top}\\beta_0)^2\\right\\}\\tilde Q K'\\left(\\frac{Q^{\\top}\\psi_0}{\\sigma_n}\\right)\\right] \\\\\n& =  \\sqrt{\\frac{n}{\\sigma_n}}\\mathbb{E}\\left[\\left(\\delta_0^{\\top}g(Q)\\delta_0\\right)\\left(1 - 2\\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right)\\tilde Q K'\\left(\\frac{Q^{\\top}\\psi_0}{\\sigma_n}\\right)\\right] \\\\\n& = \\sqrt{\\frac{n}{\\sigma_n}} \\times \\sigma_n \\int \\int \\left(\\delta_0^{\\top}g(\\sigma_nt - \\tilde q^{\\top}\\tilde \\psi_0, \\tilde q)\\delta_0\\right)\\left(1 - 2\\mathds{1}_{t > 0}\\right)\\tilde q K'\\left(t\\right) \\ f_0(\\sigma_n t \\mid \\tilde q) f (\\tilde q) \\ dt \\ d\\tilde q \\\\\n& = \\sqrt{n\\sigma_n} \\left[\\int \\tilde q \\left(\\delta_0^{\\top}g(- \\tilde q^{\\top}\\tilde \\psi_0, \\tilde q)\\delta_0\\right)f_0(0 \\mid \\tilde q) \\cancelto{0}{\\left(\\int_{-\\infty}^{\\infty} \\left(1 - 2\\mathds{1}_{t > 0}\\right)K'\\left(t\\right)  \\ dt\\right)} f (\\tilde q) d\\tilde q  + O(\\sigma_n)\\right] \\\\\n& = O(\\sqrt{n\\sigma_n^3}) = o(1) \\,.\n\\end{align*}\nFor the variance part: \n\\begin{align*}\n& {\\sf var}\\left[ \\sqrt{n\\sigma_n}\\nabla \\mathbb{M}_n^{s, \\psi}(\\theta_0)\\right] \\\\\n& = \\frac{1}{\\sigma_n}{\\sf var}\\left(\\left\\{(Y - X^{\\top}(\\beta_0 + \\delta_0))^2 - (Y - X^{\\top}\\beta_0)^2\\right\\}\\tilde Q K'\\left(\\frac{Q^{\\top}\\psi_0}{\\sigma_n}\\right)\\right) \\\\\n& = \\frac{1}{\\sigma_n}\\mathbb{E}\\left(\\left\\{(Y - X^{\\top}(\\beta_0 + \\delta_0))^2 - (Y - X^{\\top}\\beta_0)^2\\right\\}^2 \\tilde Q\\tilde Q^{\\top} \\left(K'\\left(\\frac{Q^{\\top}\\psi_0}{\\sigma_n}\\right)\\right)^2\\right) \\\\\n& \\qquad \\qquad + \\frac{1}{\\sigma_n}\\mathbb{E}^{\\otimes 2}\\left[\\left\\{(Y - X^{\\top}(\\beta_0 + \\delta_0))^2 - (Y - X^{\\top}\\beta_0)^2\\right\\}\\tilde Q K'\\left(\\frac{Q^{\\top}\\psi_0}{\\sigma_n}\\right)\\right]\n\\end{align*}\nThe outer product of the expectation (the second term of the above summand) is $o(1)$ which follows from our previous analysis of the expectation term. For the second moment: \n\\begin{align*}\n& \\frac{1}{\\sigma_n}\\mathbb{E}\\left(\\left\\{(Y - X^{\\top}(\\beta_0 + \\delta_0))^2 - (Y - X^{\\top}\\beta_0)^2\\right\\}^2 \\tilde Q\\tilde Q^{\\top} \\left(K'\\left(\\frac{Q^{\\top}\\psi_0}{\\sigma_n}\\right)\\right)^2\\right) \\\\\n& = \\frac{1}{\\sigma_n}\\mathbb{E}\\left(\\left\\{(X^{\\top}\\delta_0)^2(1 - 2\\mathds{1}_{Q^{\\top}\\psi_0 > 0}) -2{\\epsilon} (X^{\\top}\\delta_0)\\right\\}^2 \\tilde Q\\tilde Q^{\\top} \\left(K'\\left(\\frac{Q^{\\top}\\psi_0}{\\sigma_n}\\right)\\right)^2\\right) \\\\\n& = \\frac{1}{\\sigma_n}\\left[\\mathbb{E}\\left((X^{\\top}\\delta_0)^4 \\tilde Q\\tilde Q^{\\top} \\left(K'\\left(\\frac{Q^{\\top}\\psi_0}{\\sigma_n}\\right)\\right)^2\\right) + 4\\sigma_{\\epsilon}^2\\mathbb{E}\\left((X^{\\top}\\delta_0)^2 \\tilde Q\\tilde Q^{\\top} \\left(K'\\left(\\frac{Q^{\\top}\\psi_0}{\\sigma_n}\\right)\\right)^2\\right)  \\right] \\\\\n& \\longrightarrow \\left(\\int_{-\\infty}^{\\infty}(K'(t))^2 \\ dt\\right)\\left[\\mathbb{E}\\left(g_{4, \\delta_0}(-\\tilde Q^{\\top}\\tilde \\psi_0, \\tilde Q)\\tilde Q\\tilde Q^{\\top}f_0(0 \\mid \\tilde Q)\\right) \\right. \\\\\n& \\hspace{10em}+ \\left. 4\\sigma_{\\epsilon}^2\\mathbb{E}\\left(\\delta_0^{\\top}g(-\\tilde Q^{\\top}\\tilde \\psi_0, \\tilde Q)\\delta_0 \\tilde Q\\tilde Q^{\\top}f_0(0 \\mid \\tilde Q)\\right)\\right] \\\\\n& := 2V^{\\psi} \\,.\n\\end{align*}\nFinally using Lemma 6 of \\cite{horowitz1992smoothed} we conclude that $ \\sqrt{n\\sigma_n}\\nabla \\mathbb{M}_n^{s, \\psi}(\\theta_0) \\implies \\mathcal{N}(0, V^{\\psi})$. \n\\\\\\\\\n\\noindent\nWe next prove that $ \\sqrt{n}\\nabla \\mathbb{M}_n^{s, \\gamma}(\\theta_0)$ to normal distribution. This is a simple application of CLT along with bounding some remainder terms which are asymptotically negligible. The gradients are: \n\\begin{align*}\n\\sqrt{n}\\begin{pmatrix} \\nabla_{\\beta}\\mathbb{M}^s_n(\\theta_0^s) \\\\ \\nabla_{\\delta}\\mathbb{M}^s_n(\\theta_0^s) \\end{pmatrix} & = 2\\sqrt{n}\\begin{pmatrix}\\frac1n \\sum_i X_i(X_i^{\\top}\\beta_0 - Y_i)+ \\frac1n \\sum_i X_iX_i^{\\top}\\delta_0 K\\left(\\frac{Q_i^{\\top}\\psi_0}{\\sigma_n}\\right)  \\\\ \n\\frac1n \\sum_i \\left[X_i(X_i^{\\top}\\beta_0 + X_i^{\\top}\\delta_0 - Y_i)\\right] K\\left(\\frac{Q_i^{\\top}\\psi_0^s}{\\sigma_n}\\right) \\end{pmatrix} \\\\\n& = 2\\begin{pmatrix}  -\\frac{1}{\\sqrt{n}} \\sum_i X_i {\\epsilon}_i + \\frac{1}{\\sqrt{n}} \\sum_i X_iX_i^{\\top}\\delta_0 \\left(K\\left(\\frac{Q_i^{\\top}\\psi_0}{\\sigma_n}\\right) - \\mathds{1}_{Q_i^{\\top}\\psi_0 > 0}\\right)  \\\\  -\\frac{1}{\\sqrt{n}} \\sum_i X_i {\\epsilon}_iK\\left(\\frac{Q_i^{\\top}\\psi_0}{\\sigma_n}\\right)  + \\frac{1}{\\sqrt{n}} \\sum_i X_iX_i^{\\top}\\delta_0K\\left(\\frac{Q_i^{\\top}\\psi_0}{\\sigma_n}\\right)\\mathds{1}_{Q_i^{\\top}\\psi_0 \\le 0} \n\\end{pmatrix}\\\\\n& = 2\\begin{pmatrix}  -\\frac{1}{\\sqrt{n}} \\sum_i X_i {\\epsilon}_i + R_1  \\\\  -\\frac{1 }{\\sqrt{n}} \\sum_i X_i {\\epsilon}_i\\mathbf{1}_{Q_i^{\\top}\\psi_0 > 0}  +R_2 \n\\end{pmatrix}\n\\end{align*}\nThat $(1\/\\sqrt{n})\\sum_i X_i {\\epsilon}_i$ converges to normal distribution follows from a simple application of CLT. Therefore, once we prove that $R_1$ and $R_2$ are $o_p(1)$ we have: \n$$\n\\sqrt{n} \\nabla_{\\gamma}\\mathbb{M}^s_n(\\theta_0^s) \\overset{\\mathscr{L}}{\\implies} \\mathcal{N}\\left(0, 4V^{\\gamma}\\right)\n$$\nwhere: \n\\begin{equation}\n\\label{eq:def_v_gamma}\nV^{\\gamma} = \\sigma_{\\epsilon}^2 \\begin{pmatrix}\\mathbb{E}\\left[XX^{\\top}\\right] & \\mathbb{E}\\left[XX^{\\top}\\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right] \\\\\n\\mathbb{E}\\left[XX^{\\top}\\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right] & \\mathbb{E}\\left[XX^{\\top}\\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right] \\end{pmatrix} \\,.\n\\end{equation}\nTo complete the proof we now show that $R_1$ and $R_2$ are $o_p(1)$. For $R_1$, we show that $\\mathbb{E}[R_1] \\to 0$ and ${\\sf var}(R_1) \\to 0$.  For the expectation part: \n\\begin{align*}\n & \\mathbb{E}[R_1] \\\\\n & = \\sqrt{n}\\mathbb{E}\\left[XX^{\\top}\\delta_0 \\left(K\\left(\\frac{Q^{\\top}\\psi_0}{\\sigma_n}\\right) - \\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right)\\right] \\\\\n & = \\sqrt{n}\\delta_0^{\\top}\\mathbb{E}\\left[g(Q) \\left(K\\left(\\frac{Q^{\\top}\\psi_0}{\\sigma_n}\\right) - \\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right)\\right] \\\\\n & = \\sqrt{n}\\int_{\\mathbb{R}^{p-1}} \\int_{-\\infty}^{\\infty} \\delta_0^{\\top}g\\left(t-\\tilde q^{\\top}\\tilde \\psi_0, \\tilde q\\right)\\left(\\mathds{1}_{t > 0} - K\\left(\\frac{t}{\\sigma_n}\\right)\\right)f_0(t \\mid \\tilde q) f(\\tilde q) \\ dt \\ d\\tilde q \\\\\n  & = \\sqrt{n}\\sigma_n \\int_{\\mathbb{R}^{p-1}} \\int_{-\\infty}^{\\infty} \\delta_0^{\\top}g\\left(\\sigma_n z-\\tilde q^{\\top}\\tilde \\psi_0, \\tilde q\\right)\\left(\\mathds{1}_{z > 0} - K\\left(z\\right)\\right)f_0(\\sigma_n z \\mid \\tilde q) f(\\tilde q) \\ dz \\ d\\tilde q \\\\\n  & =  \\sqrt{n}\\sigma_n \\left[\\int_{\\mathbb{R}^{p-1}}\\delta_0^{\\top}g\\left(-\\tilde q^{\\top}\\tilde \\psi_0, \\tilde q\\right) f_0(0 \\mid \\tilde q) f(\\tilde q)  \\ d\\tilde q \\cancelto{0}{\\left[\\int_{-\\infty}^{\\infty}  \\left(\\mathds{1}_{z > 0} - K\\left(z\\right)\\right)\\ dz\\right]}  + O(\\sigma_n) \\right] \\\\\n  & = O(\\sqrt{n}\\sigma_n^2) = o(1) \\,.\n\\end{align*}\nFor the variance part: \n\\begin{align*}\n& {\\sf var}(R_1) \\\\\n& = {\\sf var}\\left(XX^{\\top}\\delta_0 \\left(K\\left(\\frac{Q^{\\top}\\psi_0}{\\sigma_n}\\right) - \\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right)\\right) \\\\\n& \\le \\mathbb{E}\\left[\\|X\\|^2 \\delta_0^{\\top}XX^{\\top}\\delta_0 \\left(K\\left(\\frac{Q^{\\top}\\psi_0}{\\sigma_n}\\right) - \\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right)^2\\right] \\\\\n& = O(\\sigma_n ) = o(1) \\,.\n\\end{align*}\nThis shows that ${\\sf var}(R_1) = o(1)$ and this establishes $R_1 = o_p(1)$. The proof for $R_2$ is similar and hence skipped for brevity. \n\\\\\\\\\nOur next step is to prove that $\\sqrt{n\\sigma_n}\\nabla_{\\psi}\\mathbb{M}^s_n(\\theta_0^s)$ and $\\sqrt{n}\\nabla \\mathbb{M}^{s, \\gamma}_n(\\theta_0^s)$ are asymptotically uncorrelated. Towards that end, first note that: \n\\begin{align*}\n& \\mathbb{E}\\left[X(X^{\\top}\\beta_0 - Y) + XX^{\\top}\\delta_0 K\\left(\\frac{Q^{\\top}\\psi_0}{\\sigma_n}\\right) \\right] \\\\\n& = \\mathbb{E}\\left[XX^{\\top}\\delta_0\\left(K\\left(\\frac{Q^{\\top}\\psi_0}{\\sigma_n}\\right) - \\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right)\\right] \\\\\n& = \\mathbb{E}\\left[g(Q)\\delta_0\\left(K\\left(\\frac{Q^{\\top}\\psi_0}{\\sigma_n}\\right) - \\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right)\\right] \\\\\n& = \\sigma_n \\int \\int g(\\sigma_n t - \\tilde q^{\\top}\\tilde \\psi_0, \\tilde q)(K(t) - \\mathds{1}_{t>0})f_0(\\sigma_n t \\mid \\tilde q) f(\\tilde q) \\ dt \\ d\\tilde q  \\\\\n& = \\sigma_n \\int g(- \\tilde q^{\\top}\\tilde \\psi_0, \\tilde q)\\cancelto{0}{\\int_{-\\infty}^{\\infty} (K(t) - \\mathds{1}_{t>0}) \\ dt} \\ f_0(0 \\mid \\tilde q) f(\\tilde q) \\ dt \\ d\\tilde q + O(\\sigma_n^2) \\\\\n& = O(\\sigma_n^2) \\,.\n\\end{align*}\nAlso, it follows from the proof of $\\mathbb{E}\\left[\\sqrt{n\\sigma_n}\\nabla_\\psi \\mathbb{M}_n^s(\\theta_0)\\right] \\to 0$ we have: \n$$\n\\mathbb{E}\\left[\\left\\{(Y - X^{\\top}(\\beta_0 + \\delta_0))^2 - (Y - X^{\\top}\\beta_0)^2\\right\\}\\tilde Q K'\\left(\\frac{Q^{\\top}\\psi_0}{\\sigma_n}\\right)\\right] = O(\\sigma_n^2) \\,.\n$$\nFinally note that: \n\\begin{align*}\n& \\mathbb{E}\\left[\\left(\\left\\{(Y - X^{\\top}(\\beta_0 + \\delta_0))^2 - (Y - X^{\\top}\\beta_0)^2\\right\\}\\tilde Q K'\\left(\\frac{Q^{\\top}\\psi_0}{\\sigma_n}\\right)\\right) \\times \\right. \\\\\n& \\qquad \\qquad \\qquad \\qquad \\qquad \\left. \\left(X(X^{\\top}\\beta_0 - Y) + XX^{\\top}\\delta_0 K\\left(\\frac{Q^{\\top}\\psi_0}{\\sigma_n}\\right)\\right)^{\\top}\\right] \\\\\n& = \\mathbb{E}\\left[\\left(\\left\\{(X^{\\top}\\delta_0)^2(1 - 2\\mathds{1}_{Q^{\\top}\\psi_0 > 0}) - 2{\\epsilon} X^{\\top}\\delta_0\\right\\}\\tilde QK'\\left(\\frac{Q^{\\top}\\psi_0}{\\sigma_n}\\right)\\right) \\right. \\\\\n& \\qquad \\qquad \\qquad \\qquad \\qquad \\left. \\times \\left\\{XX^{\\top}\\delta_0\\left(K\\left(\\frac{Q^{\\top}\\psi_0}{\\sigma_n}\\right) - \\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right) - X{\\epsilon} \\right\\}\\right] \\\\\n& = \\mathbb{E}\\left[\\left((X^{\\top}\\delta_0)^2(1 - 2\\mathds{1}_{Q^{\\top}\\psi_0 > 0})\\tilde QK'\\left(\\frac{Q^{\\top}\\psi_0}{\\sigma_n}\\right)\\right) \\right. \\\\\n& \\qquad \\qquad \\qquad  \\left. \\times \\left(XX^{\\top}\\delta_0\\left(K\\left(\\frac{Q^{\\top}\\psi_0}{\\sigma_n}\\right) - \\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right)\\right)^{\\top}\\right] \\\\\n& \\qquad \\qquad + 2\\sigma^2_{\\epsilon} \\mathbb{E}\\left[XX^{\\top}\\delta_0\\tilde Q^{\\top}K'\\left(\\frac{Q^{\\top}\\psi_0}{\\sigma_n}\\right)\\right] \\\\\n&= O(\\sigma_n ) \\,.\n\\end{align*}\nNow getting back to the covariance: \n\\begin{align*}\n& \\mathbb{E}\\left[\\left(\\sqrt{n\\sigma_n}\\nabla_{\\psi}\\mathbb{M}^s_n(\\theta_0)\\right)\\left(\\sqrt{n}\\nabla_\\beta \\mathbb{M}^s_n(\\theta_0)\\right)^{\\top}\\right] \\\\\n& = \\frac{1}{\\sqrt{\\sigma_n}}\\mathbb{E}\\left[\\left(\\left\\{(Y - X^{\\top}(\\beta_0 + \\delta_0))^2 - (Y - X^{\\top}\\beta_0)^2\\right\\}\\tilde Q K'\\left(\\frac{Q^{\\top}\\psi_0}{\\sigma_n}\\right)\\right) \\times \\right. \\\\\n& \\qquad \\qquad \\qquad \\qquad \\qquad \\left. \\left(X(X^{\\top}\\beta_0 - Y) + XX^{\\top}\\delta_0 K\\left(\\frac{Q^{\\top}\\psi_0}{\\sigma_n}\\right)\\right)^{\\top}\\right] \\\\\n& \\qquad \\qquad + \\frac{n-1}{\\sqrt{\\sigma_n}}\\left[\\mathbb{E}\\left[\\left\\{(Y - X^{\\top}(\\beta_0 + \\delta_0))^2 - (Y - X^{\\top}\\beta_0)^2\\right\\}\\tilde Q K'\\left(\\frac{Q^{\\top}\\psi_0}{\\sigma_n}\\right)\\right] \\right. \\\\\n& \\qquad \\qquad \\qquad \\qquad \\times \\left. \\left(\\mathbb{E}\\left[X(X^{\\top}\\beta_0 - Y) + XX^{\\top}\\delta_0 K\\left(\\frac{Q^{\\top}\\psi_0}{\\sigma_n}\\right) \\right]\\right)^{\\top}\\right] \\\\\n& = \\frac{1}{\\sqrt{\\sigma_n}} \\times O(\\sigma_n) + \\frac{n-1}{\\sqrt{\\sigma_n}} \\times O(\\sigma_n^4) = o(1) \\,.\n\\end{align*}\nThe proof for $\\mathbb{E}\\left[\\left(\\sqrt{n\\sigma_n}\\nabla_{\\psi}\\mathbb{M}^s_n(\\theta_0)\\right)\\left(\\sqrt{n}\\nabla_\\delta \\mathbb{M}^s_n(\\theta_0)\\right)^{\\top}\\right]$ is similar and hence skipped. This completes the proof. \n\n\n\n\\subsection{Proof of Lemma \\ref{conv-prob}}\nTo prove first note that by simple application of law of large number (and using the fact that $\\|\\psi^* - \\psi_0\\|\/\\sigma_n = o_p(1)$ we have: \n\\begin{align*}\n\\nabla^2 \\mathbb{M}_n^{s, \\gamma}(\\theta^*) & = 2\\begin{pmatrix}\\frac{1}{n}\\sum_i X_i X_i^{\\top} & \\frac{1}{n}\\sum_i X_i X_i^{\\top}K\\left(\\frac{Q_i^{\\top}\\psi^*}{\\sigma_n}\\right) \\\\ \\frac{1}{n}\\sum_i X_i X_i^{\\top}K\\left(\\frac{Q_i^{\\top}\\psi^*}{\\sigma_n}\\right) & \\frac{1}{n}\\sum_i X_i X_i^{\\top}K\\left(\\frac{Q_i^{\\top}\\psi^*}{\\sigma_n}\\right)\n\\end{pmatrix} \\\\\n& \\overset{p}{\\longrightarrow} 2 \\begin{pmatrix}\\mathbb{E}\\left[XX^{\\top}\\right] & \\mathbb{E}\\left[XX^{\\top}\\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right] \\\\ \\mathbb{E}\\left[XX^{\\top}\\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right] & \\mathbb{E}\\left[XX^{\\top}\\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right] \\end{pmatrix} := 2Q^{\\gamma}\n\\end{align*}\nThe proof of the fact that $\\sqrt{\\sigma_n}\\nabla^2_{\\psi \\gamma}\\mathbb{M}_n^s(\\theta^*) = o_p(1)$ is same as the proof of Lemma 5 of \\cite{seo2007smoothed} and hence skipped. Finally the proof of the fact that \n$$\n\\sigma_n \\nabla^2_{\\psi \\psi}\\mathbb{M}_n^s(\\theta^*) \\overset{p}{\\longrightarrow} 2Q^{\\psi}\\,.\n$$\nfor some non-negative definite matrix $Q$. The proof is similar to that of Lemma 6 of \\cite{seo2007smoothed}, using which we conclude the proof with:  \n$$\nQ^{\\psi} = \\left(\\int_{-\\infty}^{\\infty} -\\text{sign}(t) K''(t) \\ dt\\right) \\times \\mathbb{E}\\left[\\delta_0^{\\top} g\\left(-\\tilde Q^{\\top}\\tilde \\psi_0, \\tilde Q\\right)\\delta_0 \\tilde Q \\tilde Q^{\\top} f_0(0 \\mid \\tilde Q)\\right]  \\,.\n$$\nThis completes the proof. So we have established: \n\\begin{align*}\n\\sqrt{n}\\left(\\hat \\gamma^s  - \\gamma_0\\right) & \\overset{\\mathscr{L}}{\\implies} \\mathcal{N}\\left(0, \\left(Q^\\gamma\\right)^{-1}V^\\gamma \\left(Q^\\gamma\\right)^{-1}\\right) \\,, \\\\\n\\sqrt{\\frac{n}{\\sigma_n}}\\left(\\hat \\psi^s  - \\psi_0\\right) & \\overset{\\mathscr{L}}{\\implies} \\mathcal{N}\\left(0, \\left(Q^\\psi\\right)^{-1}V^\\psi \\left(Q^\\psi\\right)^{-1}\\right) \\,.\n\\end{align*}\nand they are asymptotically uncorrelated. \n\n\n\n\n\n\n\n\n\n\n\n\\section{Proof of Theorem \\ref{thm:binary}}\n\\label{sec:supp_classification}\nIn this section, we present the details of the binary response model, the assumptions, a roadmap of the proof and then finally prove Theorem \\ref{thm:binary}.\n\\noindent \n\\begin{assumption}\n\\label{as:distribution}\nThe below assumptions pertain to the parameter space and the distribution of $Q$:\n\\begin{enumerate}\n\\item  The parameter space $\\Theta$ is a compact subset of $\\mathbb{R}^p$. \n\\item  The support of the distribution of $Q$ contains an open subset around origin of $\\mathbb{R}^p$ and the distribution of $Q_1$ conditional on $\\tilde{Q} = (Q_2, \\dots, Q_p)$ has, almost surely, everywhere positive density with respect to Lebesgue measure. \n\\end{enumerate}\n\\end{assumption}\n\n\n\n\\noindent \nFor notational convenience, define the following: \n\\begin{enumerate}\n\\item Define $f_{\\psi} (\\cdot | \\tilde{Q})$ to the conditional density of $Q^{\\top}\\psi$ given $\\tilde{Q}$ for $\\theta \\in \\Theta$. Note that the following relation holds: $$f_{\\theta}(\\cdot |\\tilde{Q}) = f_{Q_1}(\\cdot - \\tilde{\\psi}^{\\top}\\tilde{Q} | \\tilde{Q}) \\,.$$ where we define $f_{Q_1}(\\cdot | \\tilde X)$ is the conditional density of $Q_1$ given $\\tilde Q$. \n\\item Define $f_0(\\cdot | \\tilde{Q}) = f_{\\psi_0}(\\cdot | \\tilde{Q})$ where $\\psi_0$ is the unique minimizer of the population score function $M(\\psi)$. \n\\item Define $f_{\\tilde Q}(\\cdot)$ to be the marginal density of $\\tilde Q$. \n\\end{enumerate}\n\n\n\\noindent\nThe rest of the assumptions are as follows: \n\\begin{assumption}\n\\label{as:differentiability}\n$f_0(y|\\tilde{Q})$ is at-least once continuously differentiable almost surely for all $\\tilde{Q}$. Also assume that there exists $\\delta$ and $t$ such that $$\\inf_{|y| \\le \\delta} f_0(y|\\tilde{Q}) \\ge t$$ for all $\\tilde{Q}$ almost surely. \n\\end{assumption}\nThis assumption can be relaxed in the sense that one can allow the lower bound $t$ to depend on $\\tilde{Q}$, provided that some further assumptions are imposed on $\\mathbb{E}(t(\\tilde{Q}))$. As this does not add anything of significance to the import of this paper, we use Assumption \\ref{as:differentiability} to simplify certain calculations. \n\n\n\n\\begin{assumption}\n\\label{as:density_bound}\nDefine $m\\left(\\tilde{Q}\\right) = \\sup_{t}f_{X_1}(t | \\tilde{Q}) = \\sup_{\\theta} \\sup_{t}f_{\\theta}(t | \\tilde{Q})$. Assume that $\\mathbb{E}\\left(m\\left(\\tilde{Q}\\right)^2\\right) < \\infty$. \n\\end{assumption}\n\n\n\n\n\n\n\n\\begin{assumption}\n\\label{as:derivative_bound}\nDefine $h(\\tilde{Q}) = \\sup_{t} f_0'(t | \\tilde{Q})$. Assume that $\\mathbb{E}\\left(h^2\\left(\\tilde{Q}\\right)\\right) < \\infty$.  \n\\end{assumption}\n\\begin{assumption}\n\\label{as:eigenval_bound}\nAssume that $f_{\\tilde{Q}}(0) > 0$ and also that the minimum eigenvalue of $\\mathbb{E}\\left(\\tilde{Q}\\tilde{Q}^{\\top}f_0(0|\\tilde{Q})\\right) > 0$. \n\\end{assumption}\n\n\n\n\n\n\\subsection{Sufficient conditions for above assumptions }\nWe now demonstrate some sufficient conditions for the above assumptions to hold. If the support of $Q$ is compact and both $f_1(\\cdot | \\tilde Q)$ and $f'_1(\\cdot | \\tilde Q)$ are uniformly bounded in $\\tilde Q$, then Assumptions $(\\ref{as:distribution}, \\ \\ref{as:differentiability}, \\ \\ref{as:density_bound},\\ \\ref{as:derivative_bound})$ follow immediately. The first part of Assumption \\ref{as:eigenval_bound}, i.e. the assumption $f_{\\tilde{Q}}(0) > 0$ is also fairly general and satisfied by many standard probability distributions. The second part of Assumption \\ref{as:eigenval_bound} is satisfied when $f_0(0|\\tilde{Q})$ has some lower bound independent of $\\tilde{Q}$ and $\\tilde{Q}$ has non-singular dispersion matrix. \n\n\n\n\n\nBelow we state our main theorem. In the next section, we first provide a roadmap of our proof and then fill in the corresponding details. For the rest of the paper, \\emph{we choose our bandwidth $\\sigma_n$ to satisfy $\\frac{\\log{n}}{n \\sigma_n} \\rightarrow 0$}.  \n\n\n\\noindent\n\\begin{remark}\nAs our procedure requires the weaker condition $(\\log{n})\/(n \\sigma_n) \\rightarrow 0$, it is easy to see from the above Theorem that the rate of convergence can be almost as fast as $n\/\\sqrt{\\log{n}}$. \n\\end{remark}\n\\begin{remark}\nOur analysis remains valid in presence of an intercept term. Assume, without loss of generality, that the second co-ordinate of $Q$ is $1$ and let $\\tilde{Q} = (Q_3, \\dots, Q_p)$. It is not difficult to check that all our calculations go through under this new definition of $\\tilde Q$. We, however, avoid this scenario for simplicity of exposition. \n\\end{remark}\n\\vspace{0.2in}\n\\noindent\n{\\bf Proof sketch: }We now provide a roadmap of the proof of Theorem \\ref{thm:binary} in this paragraph while the elaborate technical derivations in the later part. \nDefine the following: $$T_n(\\psi) = \\nabla \\mathbb{M}_n^s(\\psi)= -\\frac{1}{n\\sigma_n}\\sum_{i=1}^n (Y_i - \\gamma)K'\\left(\\frac{Q_i^{\\top}\\psi}{\\sigma_n}\\right)\\tilde{Q}_i$$ $$Q_n(\\psi) = \\nabla^2 \\mathbb{M}_n^s(\\psi) = -\\frac{1}{n\\sigma_n^2}\\sum_{i=1}^n (Y_i - \\gamma)K''\\left(\\frac{Q_i^{\\top}\\psi}{\\sigma_n}\\right)\\tilde{Q}_i\\tilde{Q}_i^{\\top}$$ As $\\hat{\\psi}^s$ minimizes $\\mathbb{M}^s_n(\\psi)$ we have $T_n(\\hat{\\psi}^s) = 0$. Using one step Taylor expansion we have:\n\\allowdisplaybreaks \n\\begin{align*}\nT_n(\\hat{\\psi}^s) = T_n(\\psi_0) + Q_n(\\psi^*_n)\\left(\\hat{\\psi}^s - \\psi_0\\right) = 0\n\\end{align*}\nor: \n\\begin{equation}\n\\label{eq:main_eq}  \\sqrt{n\/\\sigma_n}\\left(\\hat{\\psi}^s - \\psi_0\\right) = -\\left(\\sigma_nQ_n(\\psi^*_n)\\right)^{-1}\\sqrt{n\\sigma_n}T_n(\\psi_0) \n\\end{equation}\nfor some intermediate point $\\psi^*_n$ between $\\hat \\psi^s$ and $\\psi_0$. The following lemma establishes the asymptotic properties of $T_n(\\psi_0)$: \n\\begin{lemma}[Asymptotic Normality of $T_n$]\n\\label{asymp-normality}\n\\label{asymp-normality}\nIf $n\\sigma_n^{3} \\rightarrow \\lambda$, then \n$$\n\\sqrt{n \\sigma_n} T_n(\\psi_0) \\Rightarrow \\mathcal{N}(\\mu, \\Sigma)\n$$\nwhere \n$$\\mu = -\\sqrt{\\lambda}\\frac{\\beta_0 - \\alpha_0}{2}\\left[\\int_{-1}^{1} K'\\left(t\\right)|t| \\ dt \\right] \\int_{\\mathbb{R}^{p-1}}\\tilde{Q} f'(0 | \\tilde{Q}) \\ dP(\\tilde{Q})\n$$ \nand \n$$\\Sigma = \\left[a_1 \\int_{-1}^{0} \\left(K'\\left(t\\right)\\right)^2  \\ dt + a_2 \\int_{0}^{1} \\left(K'\\left(t\\right)\\right)^2 \\ dt \\right]\\int_{\\mathbb{R}^{p-1}}\\tilde{Q}\\tilde{Q}^{\\top} f(0|\\tilde{Q})  \\ dP(\\tilde{Q}) \\,.\n$$ \nHere $a_1 = (1 - \\gamma)^2 \\alpha_0 + \\gamma^2 (1-\\alpha_0), a_2 = (1 - \\gamma)^2 \\beta_0 + \\gamma^2 (1-\\beta_0)$ and $\\alpha_0, \\beta_0, \\gamma$ are model parameters defined around equation \\eqref{eq:new_loss}. \n\\end{lemma}\n\\noindent\nIn the case that $n \\sigma_n^3 \\rightarrow 0$, which, holds when $n\\sigma_n \\rightarrow 0$ as assumed prior to the statement of the theorem, $\\lambda = 0$ and we have: \n$$\\sqrt{n \\sigma_n} T_n(\\psi_0) \\rightarrow \\mathcal{N}(0, \\Sigma) \\,.$$ \nNext, we analyze the convergence of $Q_n(\\psi^*_n)^{-1}$ which is stated in the following lemma: \n\\begin{lemma}[Convergence in Probability of $Q_n$]\n\\label{conv-prob}\nUnder Assumptions (\\ref{as:distribution} - \\ref{as:eigenval_bound}), for any random sequence $\\breve{\\psi}_n$ such that $\\|\\breve{\\psi}_n - \\psi_0\\|\/\\sigma_n \\overset{P} \\rightarrow 0$, \n$$\n\\sigma_n Q_n(\\breve{\\psi}_n) \\overset{P} \\rightarrow Q = \\frac{\\beta_0 - \\alpha_0}{2}\\left(\\int_{-1}^{1} -K''\\left(t \\right)\\text{sign}(t) \\ dt\\right) \\ \\mathbb{E}\\left(\\tilde{Q}\\tilde{Q}^{\\top} f(0 |\\tilde{Q})\\right) \\,.\n$$\n\\end{lemma}\nIt will be shown later that the condition $\\|\\breve{\\psi}_n - \\psi_0\\|\/\\sigma_n \\overset{P} \\rightarrow 0$ needed in Lemma \\ref{conv-prob} holds for the (random) sequence $\\psi^*_n$. Then, combining Lemma \\ref{asymp-normality} and Lemma \\ref{conv-prob} we conclude from equation \\ref{eq:main_eq} that: \n$$\n\\sqrt{n\/\\sigma_n} \\left(\\hat{\\psi}^s - \\psi_0\\right) \\Rightarrow N(0, Q^{-1}\\Sigma Q^{-1}) \\,.\n$$ \nThis concludes the proof of the our Theorem \\ref{thm:binary} with $\\Gamma = Q^{-1}\\Sigma Q^{-1}$. \n\\newline\n\\newline\nObserve that, to show $\\left\\|\\psi^*_n - \\psi_0 \\right\\| = o_P(\\sigma_n)$, it suffices to to prove that $\\left\\|\\hat \\psi^s - \\psi_0 \\right\\| = o_P(\\sigma_n)$. Towards that direction, we have following lemma: \n\n\\begin{lemma}[Rate of convergence]\n\\label{lem:rate}\nUnder Assumptions (\\ref{as:distribution} - \\ref{as:eigenval_bound}), \n$$\nn^{2\/3}\\sigma_n^{-1\/3}  d^2_n\\left(\\hat \\psi^s, \\psi_0^s\\right) = O_P(1) \\,,\n$$ \nwhere \n$$\nd_n\\left(\\psi, \\psi_0^s\\right) = \\sqrt{\\left[\\frac{\\|\\psi - \\psi_0^s\\|^2}{\\sigma_n} \\mathds{1}(\\|\\psi - \\psi_0^s\\| \\le \\mathcal{K}\\sigma_n) + \\|\\psi - \\psi_0^s\\|  \\mathds{1}(\\|\\psi - \\psi_0^s\\| \\ge \\mathcal{K}\\sigma_n)\\right]}\n$$\nfor some specific constant $\\mathcal{K}$. (This constant will be mentioned precisely in the proof). \n\\end{lemma}\n\n\\noindent\nThe lemma immediately leads to the following corollary: \n\n\\begin{corollary}\n\\label{rate-cor}\nIf $n\\sigma_n \\rightarrow \\infty$ then $\\|\\hat \\psi^s - \\psi_0^s\\|\/\\sigma_n  \\overset{P}  \\longrightarrow 0$.\n\\end{corollary}\n\n\\noindent\nFinally, to establish $\\|\\hat \\psi^s - \\psi_0\\|\/\\sigma_n \\overset{P} \\rightarrow 0$, all we need is that $\\|\\psi_0^s - \\psi_0\\|\/\\sigma_n \\rightarrow 0$ as demonstrated in the following lemma:\n\n\\begin{lemma}[Convergence of population minimizer]\n\\label{bandwidth}\nFor any sequence of $\\sigma_n \\rightarrow 0$, we have: $\\|\\psi_0^s - \\psi_0\\|\/\\sigma_n \\rightarrow 0$. \n\\end{lemma}\n\n\\noindent\nHence the final roadmap is the following: Using Lemma \\ref{bandwidth} and Corollary \\ref{rate-cor} we establish that $\\|\\hat \\psi^s - \\psi_0\\|\/\\sigma_n \\rightarrow 0$ if $n\\sigma_n \\rightarrow \\infty$. This, in turn, enables us to prove that $\\sigma_n Q_n(\\psi^*_n) \\overset{P} \\rightarrow Q$,which, along with Lemma \\ref{asymp-normality}, establishes the main theorem. \n\n\\begin{remark}\n\\label{rem:gamma}\nIn the above analysis, we have assumed knowledge of $\\gamma$ in between $(\\alpha_0, \\beta_0)$. However, all our calculations go through if we replace $\\gamma$ by its estimate (say $\\bar Y$) with more tedious book-keeping. One way to simplify the calculations is to split the data into two halves, estimate $\\gamma$ (via $\\bar Y$) from the first half and then use it as a proxy for $\\gamma$ in the second half of the data to estimate $\\psi_0$. As this procedure does not add anything of interest to the core idea of our proof, we refrain from doing so here.  \n\\end{remark}\n\n\\subsection{Variant of quadratic loss function}\n\\label{loss_func_eq}\nIn this sub-section we argue why the loss function in \\eqref{eq:new_loss} is a variant of the quadratic loss function for any $\\gamma \\in (\\alpha_0, \\beta_0)$. Assume that we know $\\alpha_0, \\beta_0$ and seek to estimate $\\psi_0$. We start with an expansion of the quadratic loss function: \n\\begin{align*}\n& \\mathbb{E}\\left(Y - \\alpha_0\\mathds{1}_{Q^{\\top}\\psi \\le 0} - \\beta_0 \\mathds{1}_{Q^{\\top}\\psi > 0}\\right)^2  \\\\\n& = \\mathbb{E}\\left(\\mathbb{E}\\left(Y - \\alpha_0\\mathds{1}_{Q^{\\top}\\psi \\le 0} - \\beta_0 \\mathds{1}_{Q^{\\top}\\psi > 0}\\right)^2 \\ | X\\right) \\\\\n& = \\mathbb{E}_{Q}\\left(\\mathbb{E}\\left( Y^2 \\mid Q \\right) \\right) + \\mathbb{E}_{Q}\\left(\\alpha_0\\mathds{1}_{Q^{\\top}\\psi \\le 0} + \\beta_0 \\mathds{1}_{Q^{\\top}\\psi > 0}\\right)^2 \\\\\n& \\qquad \\qquad \\qquad -2 \\mathbb{E}_{Q}\\left(\\left(\\alpha_0\\mathds{1}_{Q^{\\top}\\psi \\le 0} + \\beta_0 \\mathds{1}_{Q^{\\top}\\psi > 0}\\right) \\mathbb{E}(Y \\mid Q)\\right) \\\\\n& = \\mathbb{E}_Q\\left(\\mathbb{E}\\left( Y \\mid Q \\right) \\right) + \\mathbb{E}_Q\\left(\\alpha_0\\mathds{1}_{Q^{\\top}\\psi \\le 0} + \\beta_0 \\mathds{1}_{Q^{\\top}\\psi > 0}\\right)^2 \\\\\n& \\qquad \\qquad \\qquad -2 \\mathbb{E}_Q\\left(\\left(\\alpha_0\\mathds{1}_{Q^{\\top}\\psi \\le 0} + \\beta_0 \\mathds{1}_{Q^{\\top}\\psi > 0}\\right) \\mathbb{E}(Y \\mid Q)\\right) \\\\\n\\end{align*}\nSince the first summand is just $\\mathbb{E} Y$, it is irrelevant to the minimization. A cursory inspection shows that it suffices to minimize\n\\begin{align}\n& \\mathbb{E}\\left(\\left(\\alpha_0\\mathds{1}_{Q^{\\top}\\psi \\le 0} + \\beta_0 \\mathds{1}_{Q^{\\top}\\psi > 0}\\right) - \\mathbb{E}(Y \\mid Q)\\right)^2 \\notag\\\\\n\\label{eq:lse_1} & = (\\beta_0 - \\alpha_0)^2 \\P\\left(\\text{sign}(Q^{\\top}\\psi) \\neq \\text{sign}(Q^{\\top}\\psi_0)\\right)\n\\end{align}\nOn the other hand the loss we are considering is $\\mathbb{E}\\left((Y - \\gamma)\\mathds{1}_{Q^{\\top}\\psi \\le 0}\\right)$: \n\\begin{align}\n\\label{eq:lse_2} \\mathbb{E}\\left((Y - \\gamma)\\mathds{1}_{Q^{\\top}\\psi \\le 0}\\right) & = (\\beta_0 - \\gamma)\\P(Q^{\\top}\\psi_0 > 0 , Q^{\\top}\\psi \\le 0) \\notag \\\\\n& \\hspace{10em}+ (\\alpha_0 - \\gamma)\\P(Q^{\\top}\\psi_0 \\le 0, Q^{\\top}\\psi \\le 0)\\,,\n\\end{align}\nwhich can be rewritten as: \n\\begin{align*}\n& (\\alpha_0 - \\gamma)\\P(X^{\\top} \\psi_0 \\leq 0) + (\\beta_0 - \\gamma)\\,\\P(X^{\\top} \\psi_0 > 0, X^{\\top} \\psi \\leq 0) \\\\\n& \\qquad \\qquad \\qquad + (\\gamma - \\alpha_0)\\,P (X^{\\top} \\psi_0 \\leq 0, X^{\\top} \\psi > 0) \\,.\n\\end{align*}\nBy Assumption \\ref{as:distribution}, for $\\psi \\neq \\psi_0$, $\\P\\left(\\text{sign}(Q^{\\top}\\psi) \\neq \\text{sign}(Q^{\\top}\\psi_0)\\right) > 0$. As an easy consequence, equation \\eqref{eq:lse_1} is uniquely minimized at $\\psi = \\psi_0$. To see that the same is true for \\eqref{eq:lse_2} when $\\gamma \\in (\\alpha_0, \\beta_0)$, note that the first summand in the equation does not depend on $\\psi$, that the second and third summands are both non-negative and that at least one of these must be positive under Assumption \\ref{as:distribution}. \n\\subsection{Linear curvature of the population score function}\nBefore going into the proofs of the Lemmas and the Theorem, we argue that the population score function $M(\\psi)$ has linear curvature near $\\psi_0$, which is useful in proving Lemma \\ref{lem:rate}. We begin with the following observation: \n\\begin{lemma}[Curvature of population risk]\n\\label{lem:linear_curvature}\nUnder Assumption \\ref{as:differentiability} we have: $$u_- \\|\\psi - \\psi_0\\|_2 \\le \\mathbb{M}(\\psi) - \\mathbb{M}(\\psi_0) \\le u_+ \\|\\psi - \\psi_0\\|_2$$ for some constants $0 < u_- < u_+ < \\infty$, for all $\\psi \\in \\psi$. \n\\end{lemma}\n\\begin{proof}\nFirst, we show that \n$$\n\\mathbb{M}(\\psi) - \\mathbb{M}(\\psi_0) = \\frac{(\\beta_0 - \\alpha_0)}{2} \\P(\\text{sign}(Q^{\\top}\\psi) \\neq X^{\\top}(\\psi_0))\n$$ which follows from the calculation below:\n\\begin{align*}\n& \\mathbb{M}(\\psi) - \\mathbb{M}(\\psi_0)  \\\\\n& = \\mathbb{E}\\left((Y - \\gamma)\\mathds{1}(Q^{\\top}\\psi \\le 0)\\right) - \\mathbb{E}\\left((Y - \\gamma)\\mathds{1}(Q^{\\top}\\psi_0 \\le 0)\\right) \\\\\n& = \\frac{\\beta_0 - \\alpha_0}{2} \\mathbb{E}\\left(\\left\\{\\mathds{1}(Q^{\\top}\\psi \\le 0) - \\mathds{1}(Q^{\\top}\\psi_0 \\le 0)\\right\\}\\left\\{\\mathds{1}(Q^{\\top}\\psi_0 \\ge 0) - \\mathds{1}(Q^{\\top}\\psi_0 \\le 0)\\right\\}\\right) \\\\\n& = \\frac{\\beta_0 - \\alpha_0}{2} \\mathbb{E}\\left(\\left\\{\\mathds{1}(Q^{\\top}\\psi \\le 0, Q^{\\top}\\psi_0 \\ge 0) - \\mathds{1}(Q^{\\top}\\psi \\le 0, Q^{\\top}\\psi_0 \\le 0) + \\mathds{1}(Q^{\\top}\\psi_0 \\le 0)\\right\\}\\right) \\\\\n& = \\frac{\\beta_0 - \\alpha_0}{2} \\mathbb{E}\\left(\\left\\{\\mathds{1}(Q^{\\top}\\psi \\le 0, Q^{\\top}\\psi_0 \\ge 0) + \\mathds{1}(Q^{\\top}\\psi \\ge 0, Q^{\\top}\\psi_0 \\le 0)\\right\\}\\right) \\\\\n& = \\frac{\\beta_0 - \\alpha_0}{2} \\P(\\text{sign}(Q^{\\top}\\psi) \\neq \\text{sign}(Q^{\\top}\\psi_0)) \\,.\n\\end{align*}\nWe now analyze the probability of the wedge shaped region, the region between the two hyperplanes $Q^{\\top}\\psi = 0$ and $Q^{\\top}\\psi_0 = 0$. Note that, \n\\allowdisplaybreaks\n\\begin{align}\n& \\P(Q^{\\top}\\psi > 0 > Q^{\\top}\\psi_0) \\notag\\\\\n& = \\P(-\\tilde{Q}^{\\top}\\tilde{\\psi} < X_1 < -\\tilde{Q}^{\\top}\\tilde{\\psi}_0) \\notag\\\\\n\\label{lin1} & = \\mathbb{E}\\left[\\left(F_{X_1 | \\tilde{Q}}\\left(-\\tilde{Q}^{\\top}\\tilde{\\psi}_0\\right) - F_{X_1 | \\tilde{Q}}\\left(-\\tilde{Q}^{\\top}\\tilde{\\psi}\\right)\\right)\\mathds{1}\\left(\\tilde{Q}^{\\top}\\tilde{\\psi}_0 \\le \\tilde{Q}^{\\top}\\tilde{\\psi}\\right)\\right]\n\\end{align}\nA similar calculation yields\n\\allowdisplaybreaks\n\\begin{align}\n\\label{lin2}  \\P(Q^{\\top}\\psi < 0 < Q^{\\top}\\psi_0) & = \\mathbb{E}\\left[\\left(F_{X_1 | \\tilde{Q}}\\left(-\\tilde{Q}^{\\top}\\tilde{\\psi}\\right) - F_{X_1 | \\tilde{Q}}\\left(-\\tilde{Q}^{\\top}\\tilde{\\psi}_0\\right)\\right)\\mathds{1}\\left(\\tilde{Q}^{\\top}\\tilde{\\psi}_0 \\ge \\tilde{Q}^{\\top}\\tilde{\\psi}\\right)\\right]\n\\end{align}\nAdding both sides of equation \\ref{lin1} and \\ref{lin2} we get: \n\\begin{equation}\n\\label{wedge_expression}\n\\P(\\text{sign}(Q^{\\top}\\psi) \\neq \\text{sign}(Q^{\\top}\\psi_0)) =  \\mathbb{E}\\left[\\left|F_{X_1 | \\tilde{Q}}\\left(-\\tilde{Q}^{\\top}\\tilde{\\psi}\\right) - F_{X_1 | \\tilde{Q}}\\left(-\\tilde{Q}^{\\top}\\tilde{\\psi}_0\\right)\\right|\\right]\n\\end{equation}\nDefine $\\psi_{\\max} = \\sup_{\\psi \\in \\psi}\\|\\psi\\|$, which is finite by Assumption \\ref{as:distribution}. Below, we establish the lower bound:\n\\allowdisplaybreaks\n\\begin{align*}\n& \\P(\\text{sign}(Q^{\\top}\\psi) \\neq \\text{sign}(Q^{\\top}\\psi_0)) \\notag\\\\\n& =  \\mathbb{E}\\left[\\left|F_{X_1 | \\tilde{Q}}\\left(-\\tilde{Q}^{\\top}\\tilde{\\psi}\\right) - F_{X_1 | \\tilde{Q}}\\left(-\\tilde{Q}^{\\top}\\tilde{\\psi}_0\\right)\\right|\\right] \\\\\n& \\ge  \\mathbb{E}\\left[\\left|F_{X_1 | \\tilde{Q}}\\left(-\\tilde{Q}^{\\top}\\tilde{\\psi}\\right) - F_{X_1 | \\tilde{Q}}\\left(-\\tilde{Q}^{\\top}\\tilde{\\psi}_0\\right)\\right|\\mathds{1}\\left(\\left|\\tilde{Q}^{\\top}\\tilde{\\psi}\\right| \\vee \\left| \\tilde{Q}^{\\top}\\tilde{\\psi}_0\\right| \\le \\delta\\right)\\right] \\hspace{0.2in} [\\delta \\ \\text{as in Assumption \\ref{as:differentiability}}]\\\\\n& \\ge  \\mathbb{E}\\left[\\left|F_{X_1 | \\tilde{Q}}\\left(-\\tilde{Q}^{\\top}\\tilde{\\psi}\\right) - F_{X_1 | \\tilde{Q}}\\left(-\\tilde{Q}^{\\top}\\tilde{\\psi}_0\\right)\\right|\\mathds{1}\\left(\\|\\tilde{Q}\\| \\le \\delta\/\\psi_{\\max}\\right)\\right] \\\\\n& \\ge  t \\mathbb{E}\\left[\\left| \\tilde{Q}^{\\top}(\\psi - \\psi_0)\\right| \\mathds{1}\\left(\\|\\tilde{Q}\\| \\le \\delta\/\\psi_{\\max}\\right)\\right] \\\\\n& = t \\|\\psi - \\psi_0\\| \\,\\mathbb{E}\\left[\\left| \\tilde{Q}^{\\top}\\frac{(\\psi - \\psi_0)}{\\|\\psi - \\psi_0\\|}\\right| \\mathds{1}\\left(\\|\\tilde{Q}\\| \\le \\delta\/\\psi_{\\max}\\right)\\right] \\\\\n& \\ge t\\|\\psi - \\psi_0\\| \\inf_{\\gamma \\in S^{p-1}}\\mathbb{E}\\left[\\left| \\tilde{Q}^{\\top}\\gamma\\right| \\mathds{1}\\left(\\|\\tilde{Q}\\| \\le \\delta\/\\psi_{\\max}\\right)\\right] \\\\\n& = u_-\\|\\psi - \\psi_0\\| \\,.\n\\end{align*} \nAt the very end, we have used the fact that $$\\inf_{\\gamma \\in S^{p-1}}\\mathbb{E}\\left[\\left| \\tilde{Q}^{\\top}\\gamma\\right| \\mathds{1}\\left(\\|\\tilde{Q}\\| \\le \\delta\/\\psi_{\\max}\\right)\\right] > 0$$ To prove this, assume that the infimum is 0. Then, there exists $\\gamma_0 \\in S^{p-1}$ such that \n$$\\mathbb{E}\\left[\\left| \\tilde{Q}^{\\top}\\gamma_0\\right| \\mathds{1}\\left(\\|\\tilde{Q}\\| \\le \\delta\/\\psi_{\\max}\\right)\\right] = 0 \\,,$$ \nas the above function continuous in $\\gamma$ and any continuous function on a compact set attains its infimum. Hence, $\\left|\\tilde{Q}^{\\top}\\gamma_0 \\right| = 0$ for all $\\|\\tilde{Q}\\| \\le \\delta\/\\psi_{\\max}$, which implies that $\\tilde{Q}$ does not have full support, violating Assumption  \\ref{as:distribution} (2). This gives a contradiction.\n\\\\\\\\\n\\noindent\nEstablishing the upper bound is relatively easier. Going back to equation \\eqref{wedge_expression}, we have: \n\\begin{align*}\n& \\P(\\text{sign}(Q^{\\top}\\psi) \\neq \\text{sign}(Q^{\\top}\\psi_0)) \\notag\\\\\n& =  \\mathbb{E}\\left[\\left|F_{Q_1 | \\tilde{Q}}\\left(-\\tilde{Q}^{\\top}\\tilde{\\psi}\\right) - F_{Q_1 | \\tilde{Q}}\\left(-\\tilde{Q}^{\\top}\\tilde{\\psi}_0\\right)\\right|\\right] \\\\\n& \\le  \\mathbb{E}\\left[m(\\tilde Q) \\, \\|Q\\| \\,\\|\\psi- \\psi_0\\|\\right] \\hspace{0.2in} [m(\\cdot) \\ \\text{is defined in Assumption \\ref{as:density_bound}}]\\\\\n& \\le u_+ \\|\\psi - \\psi_0\\| \\,,\n\\end{align*}\nas $ \\mathbb{E}\\left[m(\\tilde Q) \\|Q\\|\\right] < \\infty$  by Assumption \\ref{as:density_bound} and the sub-Gaussianity of $\\tilde X$. \n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\\subsection{Proof of Lemma \\ref{asymp-normality}}\n\\begin{proof}\nWe first prove that under our assumptions $\\sigma_n^{-1} \\mathbb{E}(T_n(\\psi_0)) \\overset{n \\to \\infty}\\longrightarrow A$ where $$A = -\\frac{\\beta_0 - \\alpha_0}{2!}\\left[\\int_{-\\infty}^{\\infty} K'\\left(t\\right)|t| \\ dt \\right] \\int_{\\mathbb{R}^{p-1}}\\tilde{Q}f_0'(0 | \\tilde{Q}) \\ dP(\\tilde{Q})$$ The proof is based on Taylor expansion of the conditional density: \n\\allowdisplaybreaks\n\\begin{align*}\n& \\sigma_n^{-1} \\mathbb{E}(T_n(\\psi_0)) \\\\\n& =  -\\sigma_n^{-2}\\mathbb{E}\\left((Y - \\gamma)K'\\left(\\frac{Q^{\\top}\\psi_0}{\\sigma_n}\\right)\\tilde{Q}\\right) \\\\\n& =  -\\frac{\\beta_0 - \\alpha_0}{2}\\sigma_n^{-2}\\mathbb{E}\\left(K'\\left(\\frac{Q^{\\top}\\psi_0}{\\sigma_n}\\right)\\tilde{Q}(\\mathds{1}(Q^{\\top}\\psi_0 \\ge 0) - \\mathds{1}(Q^{\\top}\\psi_0 \\le 0))\\right) \\\\\n& =  -\\frac{\\beta_0 - \\alpha_0}{2}\\sigma_n^{-2}\\int_{\\mathbb{R}^{p-1}}\\tilde{Q}\\left[\\int_{0}^{\\infty} K'\\left(\\frac{z}{\\sigma_n}\\right)f_0(z|\\tilde{Q}) \\ dz - \\int_{-\\infty}^{0} K'\\left(\\frac{z}{\\sigma_n}\\right)f_0(z|\\tilde{Q}) \\ dz \\right] \\ dP(\\tilde{Q}) \\\\\n& =  -\\frac{\\beta_0 - \\alpha_0}{2}\\sigma_n^{-1}\\int_{\\mathbb{R}^{p-1}}\\tilde{Q}\\left[\\int_{0}^{\\infty} K'\\left(t\\right)f_0(\\sigma_n t|\\tilde{Q}) \\ dt - \\int_{-\\infty}^{0} K'\\left(t\\right)f_0(\\sigma_n t |\\tilde{Q}) \\ dt \\right] \\ dP(\\tilde{Q}) \\\\\n& =  -\\frac{\\beta_0 - \\alpha_0}{2}\\sigma_n^{-1}\\left[\\int_{\\mathbb{R}^{p-1}}\\tilde{Q}\\left[\\int_{0}^{\\infty} K'\\left(t\\right)f_0(0|\\tilde{Q}) \\ dt - \\int_{-\\infty}^{0} K'\\left(t\\right)f_0(0 |\\tilde{Q}) \\ dt \\right] \\ dP(\\tilde{Q}) \\right. \\\\ \n& \\qquad \\qquad \\qquad + \\left. \\int_{\\mathbb{R}^{p-1}}\\sigma_n \\left[\\int_{0}^{\\infty} K'\\left(t\\right)tf_0'(\\lambda \\sigma_n t|\\tilde{Q}) \\ dt - \\int_{-\\infty}^{0} K'\\left(t\\right) t f_0'(\\lambda \\sigma_n t |\\tilde{Q}) \\ dt \\right] \\ dP(\\tilde{Q}) \\right] \\hspace{0.2in} [0 < \\lambda < 1]\\\\ \n& = -\\frac{\\beta_0 - \\alpha_0}{2}\\int_{\\mathbb{R}^{p-1}}\\tilde{Q}\\left[\\int_{0}^{\\infty} k\\left(t\\right)tf_0'(\\lambda \\sigma_n t|\\tilde{Q}) \\ dz - \\int_{-\\infty}^{0} k\\left(t\\right)tf_0'(\\lambda \\sigma_nt |\\tilde{Q}) \\ dz \\right] \\ dP(\\tilde{Q})\\\\\n& \\underset{n \\rightarrow \\infty} \\longrightarrow -\\frac{\\beta_0 - \\alpha_0}{2}\\left[\\int_{-\\infty}^{\\infty} k\\left(t\\right)|t| \\ dt \\right] \\int_{\\mathbb{R}^{p-1}}\\tilde{Q}f_0'(0 | \\tilde{Q}) \\ dP(\\tilde{Q})\n\\end{align*}\nNext, we prove that $\\mbox{Var}\\left(\\sqrt{n\\sigma_n}T_n(\\psi_0)\\right)\\longrightarrow \\Sigma$ as $n \\rightarrow \\infty$, where $\\Sigma$ is as defined in Lemma \\ref{asymp-normality}. Note that: \n\\allowdisplaybreaks\n\\begin{align*}\n\\mbox{Var}\\left(\\sqrt{n\\sigma_n}T_n(\\psi_0)\\right) & = \\sigma_n \\mathbb{E}\\left((Y - \\gamma)^2\\left(K'\\left(\\frac{Q^{\\top}\\psi_0}{\\sigma_n}\\right)^2\\frac{\\tilde{Q}\\tilde{Q}^{\\top}}{\\sigma_n^2}\\right)\\right) - \\sigma_n \\mathbb{E}(T_n(\\psi_0))\\mathbb{E}(T_n(\\psi_0))^{\\top}\n\\end{align*}\nAs $\\sigma_n^{-1}\\mathbb{E}(T_n(\\psi_0)) \\rightarrow A$, we can conclude that $\\sigma_n \\mathbb{E}(T_n(\\psi_0))\\mathbb{E}(T_n(\\psi_0))^{\\top} \\rightarrow 0$. \nDefine $a_1 = (1 - \\gamma)^2 \\alpha_0 + \\gamma^2 (1-\\alpha_0), a_2 = (1 - \\gamma)^2 \\beta_0 + \\gamma^2 (1-\\beta_0)$. For the first summand: \n\\allowdisplaybreaks\n\\begin{align*}\n& \\sigma_n \\mathbb{E}\\left((Y - \\gamma)^2\\left(K^{'^2}\\left(\\frac{Q^{\\top}\\psi_0}{\\sigma_n}\\right)\\frac{\\tilde{Q}\\tilde{Q}^{\\top}}{\\sigma_n^2}\\right)\\right) \\\\\n& = \\frac{1}{\\sigma_n} \\int_{\\mathbb{R}^{p-1}}\\tilde{Q}\\tilde{Q}^{\\top} \\left[a_1 \\int_{-\\infty}^{0} K^{'^2}\\left(\\frac{z}{\\sigma_n}\\right) f(z|\\tilde{Q}) \\ dz \\right. \\notag \\\\ & \\left. \\qquad \\qquad \\qquad + a_2 \\int_{0}^{\\infty}K^{'^2}\\left(\\frac{z}{\\sigma_n}\\right) f(z|\\tilde{Q}) \\ dz \\right] \\ dP(\\tilde{Q})\\\\\n& =  \\int_{\\mathbb{R}^{p-1}}\\tilde{Q}\\tilde{Q}^{\\top} \\left[a_1 \\int_{-\\infty}^{0} K^{'^2}\\left(t\\right)f(\\sigma_n t|\\tilde{Q}) \\ dt + a_2 \\int_{0}^{\\infty} K^{'^2}\\left(t\\right) f(\\sigma_n t |\\tilde{Q}) \\ dt \\right] \\ dP(\\tilde{Q}) \\\\\n& =  \\int_{\\mathbb{R}^{p-1}}\\tilde{Q}\\tilde{Q}^{\\top} \\left[a_1 \\int_{-\\infty}^{0} K^{'^2}\\left(t\\right)f(\\sigma_n t|\\tilde{Q}) \\ dt + a_2 \\int_{0}^{\\infty} K^{'^2}\\left(t\\right) f(\\sigma_n t |\\tilde{Q}) \\ dt \\right] \\ dP(\\tilde{Q}) \\\\\n& \\underset{n \\rightarrow \\infty} \\longrightarrow \\left[a_1 \\int_{-\\infty}^{0} K^{'^2}\\left(t\\right)  \\ dt + a_2 \\int_{0}^{\\infty} K^{'^2}\\left(t\\right) \\ dt \\right]\\int_{\\mathbb{R}^{p-1}}\\tilde{Q}\\tilde{Q}^{\\top} f(0|\\tilde{Q})  \\ dP(\\tilde{Q}) \\ \\ \\overset{\\Delta} = \\Sigma \\, . \n\\end{align*}\nFinally, suppose $n \\sigma_n^{3} \\rightarrow \\lambda$. Define $W_n = \\sqrt{n\\sigma_n}\\left[T_n(\\psi) - \\mathbb{E}(T_n(\\psi))\\right]$. Using Lemma 6 of Horowitz \\cite{horowitz1992smoothed}, it is easily established that $W_n \\Rightarrow N(0, \\Sigma)$. Also, we have: \n\\allowdisplaybreaks \n\\begin{align*}\n\\sqrt{n\\sigma_n}\\mathbb{E}(T_n(\\psi_0)) = \\sqrt{n\\sigma_n^{3}}\\sigma_n^{-1}\\mathbb{E}(T_n(\\psi_0) & \\rightarrow \\sqrt{\\lambda}A = \\mu\n\\end{align*}\nAs $\\sqrt{n\\sigma_n}T_n(\\psi_0) = W_n + \\sqrt{n\\sigma_n}\\mathbb{E}(T_n(\\psi_0))$, we conclude that $\\sqrt{n\\sigma_n} T_n(\\psi_0) \\Rightarrow N(\\mu, \\Sigma)$.\n\\end{proof}\n\\subsection{Proof of Lemma \\ref{conv-prob}}\n\\begin{proof}\nLet $\\epsilon_n \\downarrow 0$ be a sequence such that $\\P(\\|\\breve{\\psi}_n - \\psi_0\\| \\le \\epsilon_n \\sigma_n) \\rightarrow 1$. Define $\\Psi_n = \\{\\psi: \\|\\psi - \\psi_0\\| \\le \\epsilon_n \\sigma_n\\}$. We show that $$\\sup_{\\psi \\in \\psi_n} \\|\\sigma_n Q_n(\\psi) - Q\\|_F \\overset{P} \\to 0$$ where $\\|\\cdot\\|_F$ denotes the Frobenius norm of a matrix. Sometimes, we omit the subscript $F$ when there is no ambiguity. Define $\\mathcal{G}_n$ to be collection of functions: \n$$\n\\mathcal{G}_n= \\left\\{g_{\\psi}(y, q) = -\\frac{1}{\\sigma_n}(y - \\gamma)\\tilde q\\tilde q^{\\top} \\left(K''\\left(\\frac{q^{\\top}\\psi}{\\sigma_n}\\right) - K''\\left(\\frac{q^{\\top}\\psi_0}{\\sigma_n}\\right)\\right), \\psi \\in \\Psi_n \\right\\}\n$$\nThat the function class $\\mathcal{G}_n$ has bounded uniform entropy integral (BUEI) is immediate from the fact that the function $Q \\to Q^{\\top}\\psi$ has finite VC dimension (as the hyperplanes has finite VC dimension) and it does change upon constant scaling. Therefore $Q \\mapsto Q^{\\top}\\psi\/\\sigma_n$ also has finite VC dimension which does not depend on n and hence BUEI. As composition with a monotone function and multiplication with constant (parameter free) functions or multiplication of two BUEI class of functions keeps BUEI property, we conclude that $\\mathcal{G}_n$ has BUEI. \nWe first expand the expression in two terms:\n\\allowdisplaybreaks\n\\begin{align*}\n\\sup_{\\psi \\in \\psi_n} \\|\\sigma_n Q_n(\\psi) - Q\\| & \\le \\sup_{\\psi \\in \\psi_n} \\|\\sigma_n Q_n(\\psi) - \\mathbb{E}(\\sigma_n Q_n(\\psi))\\| + \\sup_{\\psi \\in \\psi_n} \\| \\mathbb{E}(\\sigma_n Q_n(\\psi)) - Q\\| \\\\ \n& = \\|(\\mathbb{P}_n - P)\\|_{\\mathcal{G}_n} + \\sup_{\\psi \\in \\psi_n}\\| \\mathbb{E}(\\sigma_n Q_n(\\psi)) - Q\\| \\\\\n& = T_{1,n} + T_{2,n} \\hspace{0.3in} \\,. [\\text{Say}]\n\\end{align*}\n\n\n\\vspace{0.2in}\n\\noindent\nThat $T_{1,n} \\overset{P} \\to 0$ follows from uniform law of large number of a BUEI class (e.g. combining Theorem 2.4.1 and Theorem 2.6.7 of \\cite{vdvw96}). \nFor uniform convergence of the second summand $T_{n,2}$, define $\\chi_n = \\{\\tilde{Q}: \\|\\tilde{Q}\\| \\le 1\/\\sqrt{\\epsilon_n}\\}$. Then $\\chi_n \\uparrow \\mathbb{R}^{p-1}$. Also for any $\\psi \\in \\Psi_n$, if we define $\\gamma_n \\equiv \\gamma_n(\\psi) = (\\psi - \\psi_0)\/\\sigma_n$, then $|\\tilde \\gamma_n^{\\top}\\tilde{Q}| \\le \\sqrt{\\epsilon_n}$ for all $n$ and for all $\\psi \\in \\Psi_n, \\tilde{Q} \\in \\chi_n$. Now, \n\\allowdisplaybreaks\n\\begin{align*}\n& \\sup_{\\psi \\in \\psi_n}\\| \\mathbb{E}(\\sigma_n Q_n(\\psi)) - Q\\| \\notag \\\\\n&\\qquad \\qquad  = \\sup_{\\psi \\in \\psi_n}\\| (\\mathbb{E}(\\sigma_n Q_n(\\psi)\\mathds{1}(\\chi_n))-Q_1) +  (\\mathbb{E}(\\sigma_n Q_n(\\psi)\\mathds{1}(\\chi_n^c))-Q_2)\\|\n\\end{align*}\nwhere $$Q_1 =  \\frac{\\beta_0 - \\alpha_0}{2}\\left(\\int_{-\\infty}^{\\infty} -K''\\left(t \\right)\\text{sign}(t) \\ dt\\right) \\ \\mathbb{E}\\left(\\tilde{Q}\\tilde{Q}^{\\top} f_0(0 |\\tilde{Q})\\mathds{1}(\\chi_n) \\right)$$ $$Q_2 =  \\frac{\\beta_0 - \\alpha_0}{2}\\left(\\int_{-\\infty}^{\\infty} -K''\\left(t \\right)\\text{sign}(t) \\ dt\\right) \\ \\mathbb{E}\\left(\\tilde{Q}\\tilde{Q}^{\\top} f(0 |\\tilde{Q})\\mathds{1}(X_n^c) \\right) \\,.$$\nNote that \n\\allowdisplaybreaks\n\\begin{flalign}\n& \\|\\mathbb{E}(\\sigma_n Q_n(\\psi)\\mathds{1}(\\chi_n)) - Q_1\\| \\notag\\\\\n& =\\left\\| \\frac{\\beta_0 - \\alpha_0}{2}\\left[\\int_{\\chi_n} \\tilde{Q}\\tilde{Q}^{\\top} \\left[\\int_{-\\infty}^{\\tilde{Q}^{\\top}\\gamma_n} K''\\left(t \\right) f_0(\\sigma_n (t-\\tilde{Q}^{\\top}\\gamma_n) |\\tilde{Q}) \\ dt \\right. \\right. \\right. \\notag \\\\\n& \\left. \\left. \\left. \\qquad \\qquad - \\int_{\\tilde{Q}^{\\top}\\gamma_n}^{\\infty} K''\\left(t\\right) f_0(\\sigma_n (t - \\tilde{Q}^{\\top}\\gamma_n) | \\tilde{Q}) \\ dt \\right]dP(\\tilde{Q})\\right]\\right. \\notag\\\\ & \\left. \\qquad \\qquad \\qquad - \\frac{\\beta_0 - \\alpha_0}{2}\\left[\\int_{\\chi_n} \\tilde{Q}\\tilde{Q}^{\\top} f(0 |\\tilde{Q})\\left[\\int_{-\\infty}^{0} K''\\left(t \\right) \\ dt - \\int_{0}^{\\infty} K''\\left(t\\right) \\ dt \\right]dP(\\tilde{Q})\\right] \\right \\|\\notag\\\\\n& =\\left \\| \\frac{\\beta_0 - \\alpha_0}{2}\\left[\\int_{\\chi_n} \\tilde{Q}\\tilde{Q}^{\\top} \\left[\\int_{-\\infty}^{\\tilde{Q}^{\\top}\\gamma_n} K'''\\left(t \\right) (f_0(\\sigma_n (t-\\tilde{Q}^{\\top}\\gamma_n) |\\tilde{Q})-f_0(0 | \\tilde{Q})) \\ dt \\right. \\right. \\right.\\notag\\\\& \\qquad \\qquad- \\left. \\left. \\left. \\int_{\\tilde{Q}^{\\top}\\gamma_n}^{\\infty} K''\\left(t\\right) (f_0(\\sigma_n (t - \\tilde{Q}^{\\top}\\gamma_n) | \\tilde{Q}) - f_0(0 | \\tilde{Q})) \\ dt \\right]dP(\\tilde{Q})\\right]\\right. \\notag\\\\ & \\qquad \\qquad \\qquad + \\left. \\frac{\\beta_0 - \\alpha_0}{2}\\left[\\int_{\\chi_n} \\tilde{Q}\\tilde{Q}^{\\top} f_0(0 |\\tilde{Q}) \\left[\\int_{-\\infty}^{\\tilde{Q}^{\\top}\\gamma_n} K''\\left(t \\right) \\ dt - \\int_{-\\infty}^{0} K''\\left(t \\right) \\ dt \\right. \\right. \\right. \\notag \\\\ \n& \\qquad \\qquad \\qquad \\qquad \\left. \\left. \\left. + \\int_{\\tilde{Q}^{\\top}\\gamma_n}^{\\infty} K''\\left(t \\right) \\ dt - \\int_{0}^{\\infty} K''\\left(t\\right) \\ dt \\right]dP(\\tilde{Q})\\right] \\right \\|\\notag\\\\\n& \\le \\frac{\\beta_0 - \\alpha_0}{2}\\sigma_n  \\int_{\\chi_n}\\|\\tilde{Q}\\tilde{Q}^{\\top}\\|h(\\tilde{Q})\\int_{-\\infty}^{\\infty}|K''(t)||t - \\gamma_n^{\\top}\\tilde{Q}| \\ dt \\ dP(\\tilde{Q}) \\notag\\\\ & \\qquad \\qquad + \\frac{\\beta_0 - \\alpha_0}{2}  \\int_{\\chi_n}\\|\\tilde{Q}\\tilde{Q}^{\\top}\\| f_0(0 | \\tilde{Q}) \\left[\\left| \\int_{-\\infty}^{\\tilde{Q}^{\\top}\\gamma_n} K''\\left(t \\right) \\ dt - \\int_{-\\infty}^{0} K''\\left(t \\right) \\ dt \\right| \\right. \\notag \\\\ & \\left. \\qquad \\qquad \\qquad + \\left| \\int_{\\tilde{Q}^{\\top}\\gamma_n}^{\\infty} K''\\left(t \\right) \\ dt - \\int_{0}^{\\infty} K''\\left(t\\right) \\ dt \\right|\\right] \\ dP(\\tilde{Q})\\notag\\\\\n&  \\le \\frac{\\beta_0 - \\alpha_0}{2}\\left[\\sigma_n  \\int_{\\chi_n}\\|\\tilde{Q}\\tilde{Q}^{\\top}\\|h(\\tilde{Q})\\int_{-\\infty}^{\\infty}|K''(t)||t - \\gamma_n^{\\top}\\tilde{Q}| \\ dt \\ dP(\\tilde{Q}) \\right. \\notag \\\\ \n& \\left. \\qquad \\qquad \\qquad  + 2\\int_{\\chi_n}\\|\\tilde{Q}\\tilde{Q}^{\\top}\\| f_0(0 | \\tilde{Q}) (K'(0) - K'(\\gamma_n^{\\top}\\tilde{Q})) \\ dP(\\tilde{Q})\\right]\\notag \\\\\n\\label{cp1}&\\rightarrow 0 \\hspace{0.3in} [\\text{As} \\ n \\rightarrow \\infty] \\,,\n\\end{flalign}\nby DCT and Assumptions \\ref{as:distribution} and \\ref{as:derivative_bound}. For the second part: \n\\allowdisplaybreaks\n\\begin{align}\n& \\|\\mathbb{E}(\\sigma_n Q_n(\\psi)\\mathds{1}(\\chi_n^c)) - Q_2\\|\\notag\\\\\n& =\\left\\| \\frac{\\beta_0 - \\alpha_0}{2}\\left[\\int_{\\chi_n^c} \\tilde{Q}\\tilde{Q}^{\\top} \\left[\\int_{-\\infty}^{\\tilde{Q}^{\\top}\\gamma_n} K''\\left(t \\right) f_0(\\sigma_n (t-\\tilde{Q}^{\\top}\\gamma_n) |\\tilde{Q}) \\ dt \\right. \\right. \\right. \\notag \\\\ \n& \\left. \\left. \\left. \\qquad \\qquad - \\int_{\\tilde{Q}^{\\top}\\gamma_n}^{\\infty} K''\\left(t\\right) f_0(\\sigma_n (t - \\tilde{Q}^{\\top}\\gamma_n) | \\tilde{Q}) \\ dt \\right]dP(\\tilde{Q})\\right]\\right. \\notag\\\\ &  \\left. \\qquad \\qquad \\qquad -\\frac{\\beta_0 - \\alpha_0}{2}\\left[\\int_{\\chi_n^c} \\tilde{Q}\\tilde{Q}^{\\top} f_0(0 |\\tilde{Q})\\left[\\int_{-\\infty}^{0} K''\\left(t \\right) \\ dt - \\int_{0}^{\\infty} K''\\left(t\\right) \\ dt \\right]dP(\\tilde{Q})\\right] \\right \\|\\notag\\\\\n& \\le \\frac{\\beta_0 - \\alpha_0}{2} \\int_{\\infty}^{\\infty} |K''(t)| \\ dt \\int_{\\chi_n^c} \\|\\tilde{Q}\\tilde{Q}^{\\top}\\|(m(\\tilde{Q}) + f_0(0|\\tilde{Q})) \\ dP(\\tilde{Q}) \\notag\\\\\n\\label{cp2} & \\rightarrow 0 \\hspace{0.3in} [\\text{As} \\ n \\rightarrow \\infty] \\,,\n\\end{align}\nagain by DCT and Assumptions \\ref{as:distribution} and \\ref{as:density_bound}. Combining equations \\ref{cp1} and \\ref{cp2}, we conclude the proof. \n\\end{proof}\n\n\n\\subsection{Proof of Lemma \\ref{bandwidth}}\nHere we prove that $\\|\\psi^s_0 - \\psi_0\\|\/\\sigma_n \\rightarrow 0$ where $\\psi^s_0$ is the minimizer of $\\mathbb{M}^s(\\psi)$ and $\\psi_0$ is the minimizer of $M(\\psi)$. \n\\begin{proof}\nDefine $\\eta = (\\psi^s_0 -  \\psi_0)\/\\sigma_n$. At first we show that, $\\|\\tilde \\eta\\|_2$ is $O(1)$, i.e. there exists some constant $\\Omega_1$ such that $\\|\\tilde \\eta\\|_2 \\le \\Omega_1$ for all $n$: \n\\begin{align*}\n\\|\\psi^s_0 - \\psi_0\\|_2 & \\le \\frac{1}{u_-} \\left(\\mathbb{M}(\\psi_n) - \\mathbb{M}(\\psi_0)\\right)  \\hspace{0.2in} [\\text{Follows from Lemma} \\ \\ref{lem:linear_curvature}]\\\\\n& \\le \\frac{1}{u_-} \\left(\\mathbb{M}(\\psi_n) - \\mathbb{M}^s(\\psi_n) + \\mathbb{M}^s(\\psi_n) - \\mathbb{M}^s(\\psi_0) + \\mathbb{M}^s(\\psi_0) - \\mathbb{M}(\\psi_0)\\right)  \\\\\n& \\le  \\frac{1}{u_-} \\left(\\mathbb{M}(\\psi_n) - \\mathbb{M}^s(\\psi_n) + \\mathbb{M}^s(\\psi_0) - M(\\psi_0)\\right)  \\hspace{0.2in} [\\because \\mathbb{M}^s(\\psi_n) - \\mathbb{M}^s(\\psi_0) \\le 0]\\\\\n& \\le \\frac{2K_1}{u_-}\\sigma_n \\hspace{0.2in} [\\text{from equation} \\ \\eqref{eq:lin_bound_1}]\n\\end{align*}\n\n\\noindent\nAs $\\psi^s_0$ minimizes $\\mathbb{M}^s(\\psi)$: \n$$\\nabla \\mathbb{M}^s(\\psi^s_0) = -\\mathbb{E}\\left((Y-\\gamma)\\tilde{Q}K'\\left(\\frac{Q^{\\top}\\psi^0_s}{\\sigma_n}\\right)\\right) = 0$$\nHence:\n\\begin{align*}\n0 &= \\mathbb{E}\\left((Y-\\gamma)\\tilde{Q}K'\\left(\\frac{Q^{\\top}\\psi_0^s}{\\sigma_n}\\right)\\right) \\\\\n& = \\frac{(\\beta_0 - \\alpha_0)}{2} \\mathbb{E}\\left(\\tilde{Q}K'\\left(\\frac{Q^{\\top}\\psi_0^s}{\\sigma_n}\\right)\\left\\{\\mathds{1}(Q^{\\top}\\psi_0 \\ge 0) -\\mathds{1}(Q^{\\top}\\psi_0 < 0)\\right\\}\\right) \\\\\n& = \\frac{(\\beta_0 - \\alpha_0)}{2} \\mathbb{E}\\left(\\tilde{Q}K'\\left(\\frac{Q^{\\top}\\psi_0}{\\sigma_n} + \\tilde{\\eta}^{\\top} \\tilde{Q}\\right)\\left\\{\\mathds{1}(Q^{\\top}\\psi_0 \\ge 0) -\\mathds{1}(Q^{\\top}\\psi_0 < 0)\\right\\}\\right) \\\\\n& = \\frac{(\\beta_0 - \\alpha_0)}{2} \\left[\\int_{\\mathbb{R}^{p-1}}\\tilde{Q} \\int_0^{\\infty} K'\\left(\\frac{z}{\\sigma_n} + \\tilde{\\eta}^{\\top} \\tilde{Q}\\right) \\ f_0(z|\\tilde{Q}) \\ dz \\ dP(\\tilde{Q})\\right. \\\\\n& \\qquad \\qquad \\qquad \\qquad \\qquad \\left. - \\int_{\\mathbb{R}^{p-1}}\\tilde{Q} \\int_{-\\infty}^0 K'\\left(\\frac{z}{\\sigma_n} + \\tilde{\\eta}^{\\top} \\tilde{Q}\\right) \\ f_0(z|\\tilde{Q}) \\ dz \\ dP(\\tilde{Q})\\right] \\\\\n& =\\sigma_n \\frac{(\\beta_0 - \\alpha_0)}{2} \\left[\\int_{\\mathbb{R}^{p-1}}\\tilde{Q} \\int_0^{\\infty} K'\\left(t + \\tilde{\\eta}^{\\top} \\tilde{Q}\\right) \\ f_0(\\sigma_n t|\\tilde{Q}) \\ dt \\ dP(\\tilde{Q})\\right. \\\\\n& \\qquad \\qquad \\qquad \\qquad \\qquad \\left. - \\int_{\\mathbb{R}^{p-1}}\\tilde{Q} \\int_{-\\infty}^0 K'\\left(t + \\tilde{\\eta}^{\\top} \\tilde{Q}\\right) \\ f_0(\\sigma_n t|\\tilde{Q}) \\ dz \\ dP(\\tilde{Q})\\right] \n\\end{align*}\nAs $\\sigma_n\\frac{(\\beta_0 - \\alpha_0)}{2} > 0$, we can forget about it and continue. Also, as we have proved $\\|\\tilde \\eta\\| = O(1)$, there exists a subsequence $\\eta_{n_k}$ and a point $c \\in \\mathbb{R}^{p-1}$ such that $\\eta_{n_k} \\rightarrow c$. Along that sub-sequence we have:  \n\\begin{align*}\n0 & = \\left[\\int_{\\mathbb{R}^{p-1}}\\tilde{Q} \\int_0^{\\infty} K'\\left(t + \\tilde{\\eta}_{n_k}^{\\top} \\tilde{Q}\\right) \\ f_0(\\sigma_{n_k} t|\\tilde{Q}) \\ dt \\ dP(\\tilde{Q})\\right. \\\\\n& \\qquad \\qquad \\qquad \\qquad \\qquad \\left. - \\int_{\\mathbb{R}^{p-1}}\\tilde{Q} \\int_{-\\infty}^0 K'\\left(t + \\tilde{\\eta}_{n_k}^{\\top} \\tilde{Q}\\right) \\ f_0(\\sigma_{n_k} t|\\tilde{Q}) \\ dt \\ dP(\\tilde{Q})\\right] \n\\end{align*}\nTaking limits on both sides and applying DCT (which is permissible by DCT) we conclude: \n\\begin{align*}\n0 & = \\left[\\int_{\\mathbb{R}^{p-1}}\\tilde{Q} \\int_0^{\\infty} K'\\left(t +c^{\\top} \\tilde{Q}\\right) \\ f_0(0|\\tilde{Q}) \\ dt \\ dP(\\tilde{Q})\\right. \\\\\n& \\qquad \\qquad \\qquad \\qquad \\qquad \\left. - \\int_{\\mathbb{R}^{p-1}}\\tilde{Q} \\int_{-\\infty}^0 K'\\left(t + c^{\\top} \\tilde{Q}\\right) \\ f_0(0|\\tilde{Q}) \\ dt \\ dP(\\tilde{Q})\\right]  \\\\\n& = \\left[\\int_{\\mathbb{R}^{p-1}}\\tilde{Q}  \\ f_0(0|\\tilde{Q}) \\int_{c^{\\top} \\tilde{Q}}^{\\infty} K'\\left(t\\right) \\ dt \\ dP(\\tilde{Q})\\right. \\\\\n& \\qquad \\qquad \\qquad \\qquad \\qquad \\left. - \\int_{\\mathbb{R}^{p-1}}\\tilde{Q}\\ f_0(0|\\tilde{Q})  \\int_{-\\infty}^{c^{\\top} \\tilde{Q}} K'\\left(t \\right) \\ dt \\ dP(\\tilde{Q})\\right]  \\\\\n& = \\left[\\int_{\\mathbb{R}^{p-1}}\\tilde{Q}  \\ f_0(0|\\tilde{Q}) \\left[1 - K(c^{\\top} \\tilde{Q})\\right] \\ dt \\ dP(\\tilde{Q})\\right. \\\\\n&  \\qquad \\qquad \\qquad \\qquad \\qquad\\left. - \\int_{\\mathbb{R}^{p-1}}\\tilde{Q}\\ f_0(0|\\tilde{Q}) K(c^{\\top} \\tilde{Q}) \\ dt \\ dP(\\tilde{Q})\\right]  \\\\\n& =   \\mathbb{E}\\left(\\tilde{Q} \\left(2K(c^{\\top} \\tilde{Q}) - 1\\right)f_0(0|\\tilde{Q})\\right) \\,.\n\\end{align*}\nNow, taking the inner-products of both sides with respect to $c$, we get: \n\\begin{equation}\n\\label{eq:zero_eq}\n\\mathbb{E}\\left(c^{\\top}\\tilde{Q} \\left(2K(c^{\\top} \\tilde{Q}) - 1\\right)f_0(0|\\tilde{Q})\\right) = 0 \\,.\n\\end{equation}\nBy our assumption that $K$ is symmetric kernel and that $K(t) > 0$ for all $t \\in (-1, 1)$, we easily conclude that $c^{\\top}\\tilde{Q} \\left(2K(c^{\\top} \\tilde{Q}) - 1\\right) \\ge 0$ almost surely in $\\tilde{Q}$ with equality iff $c^{\\top}X = 0$, which is not possible unless $c = 0$. Hence we conclude that $c = 0$. This shows that any convergent subsequence of $\\eta_n$ converges to $0$, which completes the proof. \n\\end{proof}\n\n\n\n\\subsection{Proof of Lemma \\ref{lem:rate}}\n\\begin{proof}\nTo obtain the rate of convergence of our kernel smoothed estimator we use Theorem 3.4.1 of \\cite{vdvw96}: There are three key ingredients that one needs to take care of if in order to apply this theorem:   \n\\begin{enumerate}\n\\item Consistency of the estimator (otherwise the conditions of the theorem needs to be valid for all $\\eta$). \n\\item The curvature of the population score function near its minimizer.\n\\item A bound on the modulus of continuity in a vicinity of the minimizer of the population score function. \n\\end{enumerate}\nBelow, we establish the curvature of the population score function (item 2 above) globally, thereby obviating the need to establish consistency separately. Recall that the population score function was defined as: \n$$\n\\mathbb{M}^s(\\psi) = \\mathbb{E}\\left((Y - \\gamma)\\left(1 - K\\left(\\frac{Q^{\\top}\\psi}{\\sigma_n}\\right)\\right)\\right)\n$$ \nand our estimator $\\hat{\\psi}_n$ is the argmin of the corresponding sample version. Consider the set of functions $\\mathcal{H}_n = \\left\\{h_{\\psi}: h_{\\psi}(q,y) = (y - \\gamma)\\left(1 - K\\left(\\frac{q^{\\top}\\psi}{\\sigma_n}\\right)\\right)\\right\\}$. Next, we argue that $\\mathcal{H}_n$ is a VC class of functions with fixed VC dimension. We know that the function $\\{(q,y) \\mapsto q^{\\top}\\psi\/\\sigma_n: \\psi \\in \\psi\\}$ has fixed VC dimension (i.e. not depending on $n$). Now, as a finite dimensional VC class of functions composed with a fixed monotone function or multiplied by a fixed function still remains a finite dimensional VC class, we conclude that $\\mathcal{H}_n$ is a fixed dimensional VC class of functions with bounded envelope (as the functions considered here are bounded by 1). \n\nNow, we establish a lower bound on the curvature of the population score function $\\mathbb{M}^s(\\psi)$ near its minimizer $\\psi_n$: \n$$\n\\mathbb{M}^s(\\psi) - \\mathbb{M}^s(\\psi_n) \\gtrsim d^2_n(\\psi, \\psi_n)$$ where $$d_n(\\psi, \\psi_n) = \\sqrt{\\frac{\\|\\psi - \\psi_n\\|^2}{\\sigma_n} \\mathds{1}\\left(\\|\\psi - \\psi_n\\| \\le \\mathcal{K}\\sigma_n\\right) + \\|\\psi - \\psi_n\\|\\mathds{1}\\left(\\|\\psi - \\psi_n\\| > \\mathcal{K}\\sigma_n\\right)}\n$$ for some constant $\\mathcal{K} > 0$. The intuition behind this compound structure is following: When $\\psi$ is in $\\sigma_n$ neighborhood of $\\psi_n$, $\\mathbb{M}^s(\\psi)$ behaves like a smooth quadratic function, but when it is away from the truth, $\\mathbb{M}^s(\\psi)$ starts resembling $M(\\psi)$ which induces the linear curvature. \n\\\\\\\\\n\\noindent\nFor the linear part, we first establish that $|\\mathbb{M}(\\psi) - \\mathbb{M}^s(\\psi)| = O(\\sigma_n)$ uniformly for all $\\psi$. Define $\\eta = (\\psi - \\psi_0)\/\\sigma_n$:\n\\allowdisplaybreaks\n\\begin{align}\n& |\\mathbb{M}(\\psi) - \\mathbb{M}^s(\\psi)| \\notag \\\\\n& \\le \\mathbb{E}\\left(\\left | \\mathds{1}(Q^{\\top}\\psi \\ge 0) - K\\left(\\frac{Q^{\\top}\\psi}{\\sigma_n}\\right)\\right | \\right) \\notag\\\\\n& = \\mathbb{E}\\left(\\left | \\mathds{1}\\left(\\frac{Q^{\\top}\\psi_0}{\\sigma_n} + \\eta^{\\top}\\tilde{Q} \\ge 0\\right) - K\\left(\\frac{Q^{\\top}\\psi_0}{\\sigma_n} + \\eta^{\\top}\\tilde{Q}\\right)\\right | \\right) \\notag \\\\\n& = \\sigma_n \\int_{\\mathbb{R}^{p-1}} \\int_{-\\infty}^{\\infty} \\left | \\mathds{1}\\left(t + \\eta^{\\top}\\tilde{Q} \\ge 0\\right) - K\\left(t + \\eta^{\\top}\\tilde{Q}\\right)\\right | f_0(\\sigma_n t | \\tilde{Q}) \\ dt \\ dP(\\tilde{Q}) \\notag\\\\\n& = \\sigma_n \\int_{\\mathbb{R}^{p-1}} \\int_{-\\infty}^{\\infty} \\left | \\mathds{1}\\left(t  \\ge 0\\right) - K\\left(t \\right)\\right | f_0(\\sigma_n (t-\\eta^{\\top}\\tilde{Q}) | \\tilde{Q}) \\ dt \\ dP(\\tilde{Q}) \\notag \\\\\n& = \\sigma_n \\int_{\\mathbb{R}^{p-1}} m(\\tilde{Q})\\int_{-\\infty}^{\\infty} \\left | \\mathds{1}\\left(t  \\ge 0\\right) - K\\left(t \\right)\\right |  \\ dt \\ dP(\\tilde{Q}) \\notag\\\\\n& = \\sigma_n \\mathbb{E}(m(\\tilde{Q})) \\int_{-\\infty}^{\\infty} \\left | \\mathds{1}\\left(t  \\ge 0\\right) - K\\left(t \\right)\\right |  \\ dt \\notag \\\\\n\\label{eq:lin_bound_1} & \\le K_1 \\sigma_n \\mathbb{E}(m(\\tilde{Q})) < \\infty \\hspace{0.3in}  [\\text{by Assumption \\ref{as:density_bound}}] \\,.\n\\end{align}\nHere, the constant $K_1$ is $\\mathbb{E}(m(\\tilde{Q}))  \\left[\\int_{-1}^{1}\\left | \\mathds{1}\\left(t  \\ge 0\\right) - K\\left(t \\right)\\right |  \\ dt \\right]$ which does not depend on $\\psi$, hence the bound is uniform over $\\psi$. Next: \n\\begin{align*}\n\\mathbb{M}^s(\\psi) - \\mathbb{M}^s(\\psi_0^s) & = \\mathbb{M}^s(\\psi) - \\mathbb{M}(\\psi) + \\mathbb{M}(\\psi) - \\mathbb{M}(\\psi_0) \\\\\n& \\qquad \\qquad + \\mathbb{M}(\\psi_0) - \\mathbb{M}(\\psi_0^s) + \\mathbb{M}(\\psi_0^s) -\\mathbb{M}^s(\\psi_0^s) \\\\ \n& = T_1 + T_2 + T_3 + T_4\n\\end{align*}\n\\noindent\nWe bound each summand separately: \n\\begin{enumerate}\n\\item $T_1 = \\mathbb{M}^s(\\psi) - \\mathbb{M}(\\psi) \\ge -K_1 \\sigma_n$ by equation \\ref{eq:lin_bound_1}\\, \n\\item $T_2 = \\mathbb{M}(\\psi) - \\mathbb{M}(\\psi_0) \\ge u_-\\|\\psi - \\psi_0\\|$ by Lemma \\ref{lem:linear_curvature}\\,\n\\item $T_3 = \\mathbb{M}(\\psi_0) - \\mathbb{M}(\\psi_0^s) \\ge -u_+\\|\\psi_0^s - \\psi_0\\|  \\ge  -\\epsilon_1 \\sigma_n$ where one can take $\\epsilon_1$ as small as possible, as we have established $\\|\\psi_0^s - \\psi_0\\|\/\\sigma_n \\rightarrow 0$. This follows by Lemma \\ref{lem:linear_curvature} along with Lemma \\ref{bandwidth}\\,  \n\\item $T_4 = \\mathbb{M}(\\psi_0^s) -\\mathbb{M}^s(\\psi_0^s) \\ge -K_1 \\sigma_n$ by equation \\ref{eq:lin_bound_1}. \n\\end{enumerate}\nCombining, we have \n\\allowdisplaybreaks\n\\begin{align*}\n\\mathbb{M}^s(\\psi) - \\mathbb{M}^s(\\psi_0^s) & \\ge u_-\\|\\psi - \\psi_0\\| -(2K_1 + \\epsilon_1) \\sigma_n \\\\\n& \\ge ( u_-\/2)\\|\\psi - \\psi_0\\| \\hspace{0.2in} \\left[\\text{If} \\ \\|\\psi - \\psi_0\\| \\ge \\frac{2(2K_1 + \\epsilon_1)}{u_-}\\sigma_n\\right] \\\\\n& \\ge ( u_-\/4)\\|\\psi - \\psi_0^s\\| \n\\end{align*}\nwhere the last inequality holds for all large $n$ as proved in Lemma \\ref{bandwidth}. Using Lemma \\ref{bandwidth} again, we conclude that for any pair of positive constants $(\\epsilon_1, \\epsilon_2)$:  \n$$\\|\\psi - \\psi_0^s\\| \\ge \\left(\\frac{2(2K_1 + \\epsilon_1)}{u_-}+\\epsilon_2\\right)\\sigma_n \\Rightarrow \\|\\psi - \\psi_0\\| \\ge \\frac{2(2K_1 + \\epsilon_1)}{u_-}\\sigma_n$$ for all large $n$, which implies: \n\\begin{align}\n& \\mathbb{M}^s(\\psi) - \\mathbb{M}^s(\\psi_0^s) \\notag \\\\\n& \\ge (u_-\/4) \\|\\psi - \\psi_0^s\\| \\mathds{1}\\left(\\|\\psi - \\psi_0^s\\| \\ge \\left(\\frac{2(2K_1 + \\epsilon_1)}{u_-}+\\epsilon_2\\right)\\sigma_n \\right) \\notag \\\\\n\\label{lb2} & \\ge (u_-\/4) \\|\\psi - \\psi_0^s\\| \\mathds{1}\\left(\\frac{\\|\\psi - \\psi_0^s\\|}{\\sigma_n} \\ge \\left(\\frac{7K_1}{u_-}\\right) \\right)  \\hspace{0.2in} [\\text{for appropriate specifications of} \\ \\epsilon_1, \\epsilon_2] \\notag \\\\\n& :=  (u_-\/4) \\|\\psi - \\psi_0^s\\| \\mathds{1}\\left(\\frac{\\|\\psi - \\psi_0^s\\|}{\\sigma_n} \\ge \\mathcal{K} \\right)\n\\end{align}\n\n\\noindent\nIn the next part, we find the lower bound when $\\|\\psi - \\psi^0_s\\| \\le \\mathcal{K} \\sigma_n$. For the quadratic curvature, we perform a two step Taylor expansion: Define $\\eta = (\\psi - \\psi_0)\/\\sigma_n$. We have: \n\\allowdisplaybreaks \n\\begin{align}\n& \\nabla^2\\mathbb{M}^s(\\psi) \\notag\\\\\n& = \\frac{\\beta_0 - \\alpha_0}{2}\\frac{1}{\\sigma_n^2} \\mathbb{E}\\left(\\tilde{Q}\\tilde{Q}^{\\top} K''\\left(\\frac{Q^{\\top}\\psi}{\\sigma_n}\\right)\\left\\{\\mathds{1}(Q^{\\top}\\psi_0 \\le 0) - \\mathds{1}(Q^{\\top}\\psi_0 \\ge 0)\\right\\}\\right) \\notag\\\\\n& = \\frac{\\beta_0 - \\alpha_0}{2}\\frac{1}{\\sigma_n^2} \\mathbb{E}\\left(\\tilde{Q}\\tilde{Q}^{\\top} K''\\left(\\frac{Q^{\\top}\\psi_0}{\\sigma_n} + \\tilde{Q}^{\\top}\\tilde \\eta \\right)\\left\\{\\mathds{1}(Q^{\\top}\\psi_0 \\le 0) - \\mathds{1}(Q^{\\top}\\psi_0 \\ge 0)\\right\\}\\right) \\notag\\\\\n& = \\frac{\\beta_0 - \\alpha_0}{2}\\frac{1}{\\sigma_n^2} \\mathbb{E}\\left[\\tilde{Q}\\tilde{Q}^{\\top} \\left[\\int_{-\\infty}^{0} K''\\left(\\frac{z}{\\sigma_n} + \\tilde{Q}^{\\top}\\tilde \\eta \\right) f_0(z |\\tilde{Q}) \\ dz \\right. \\right. \\notag \\\\ \n& \\left. \\left. \\qquad \\qquad \\qquad \\qquad -\\int_{0}^{\\infty} K''\\left(\\frac{z}{\\sigma_n} + \\tilde{Q}^{\\top}\\tilde \\eta \\right) f_0(z | \\tilde{Q}) \\ dz \\right]\\right] \\notag\\\\\n& = \\frac{\\beta_0 - \\alpha_0}{2}\\frac{1}{\\sigma_n} \\mathbb{E}\\left[\\tilde{Q}\\tilde{Q}^{\\top} \\left[\\int_{-\\infty}^{0} K''\\left(t+ \\tilde{Q}^{\\top}\\tilde \\eta \\right) f_0(\\sigma_n t |\\tilde{Q}) \\ dt \\right. \\right. \\notag \\\\\n& \\left. \\left. \\qquad \\qquad \\qquad \\qquad - \\int_{0}^{\\infty} K''\\left(t + \\tilde{Q}^{\\top}\\tilde \\eta \\right) f_0(\\sigma_n t | \\tilde{Q}) \\ dt \\right]\\right] \\notag\\\\\n& = \\frac{\\beta_0 - \\alpha_0}{2}\\frac{1}{\\sigma_n} \\mathbb{E}\\left[\\tilde{Q}\\tilde{Q}^{\\top} f_0(0| \\tilde{Q})\\left[\\int_{-\\infty}^{0} K''\\left(t+ \\tilde{Q}^{\\top}\\tilde \\eta \\right) \\ dt \\right. \\right. \\notag \\\\\n& \\left. \\left. \\qquad \\qquad \\qquad \\qquad - \\int_{0}^{\\infty} K''\\left(t + \\tilde{Q}^{\\top}\\tilde \\eta \\right) \\ dt \\right]\\right] + R \\notag\\\\\n\\label{eq:quad_eq_1} & =(\\beta_0 - \\alpha_0)\\frac{1}{\\sigma_n}\\mathbb{E}\\left[\\tilde{Q}\\tilde{Q}^{\\top} f_0(0| \\tilde{Q})K'(\\tilde{Q}^{\\top}\\tilde \\eta)\\right] + R \\,.\n\\end{align}\nAs we want a lower bound on the set $\\|\\psi - \\psi^0_s\\| \\le \\mathcal{K} \\sigma_n$, we have $\\|\\eta\\| \\le \\mathcal{K}$. For the rest of the analysis, define \n\\begin{align*}\n\\Lambda: (v_1, v_2) \\mapsto  \\inf_{\\|v_1\\| = 1, \\|v_2\\| \\le \\mathcal{K}} \\mathbb{E}_{\\tilde X}\\left[|v_1^{\\top}\\tilde{Q}|^2 f(0|\\tilde{Q})K'(\\tilde{Q}^{\\top}v_2) \\right]\n\\end{align*}\nClearly $\\Lambda \\ge 0$ and continuous on a compact set, hence its infimum is attained. Suppose $\\Lambda(v_1, v_2) = 0$ for some $v_1, v_2$. Then we have: \n\\begin{align*}\n\\mathbb{E}\\left[|v_1^{\\top}\\tilde{Q}|^2 f(0|\\tilde{Q})K'(\\tilde{Q}^{\\top}v_2)  \\right] = 0 \\,,\n\\end{align*}\nwhich further implies $|\\tilde v_1^{\\top}\\tilde X| = 0$ almost surely and violates Assumption \\ref{as:eigenval_bound}. Hence, our claim is demonstrated. On the other hand, for the remainder term of equation \\eqref{eq:quad_eq_1}: \nfix $\\nu \\in S^{p-1}$. Then: \n\\allowdisplaybreaks\n\\begin{align}\n& \\left| \\nu^{\\top} R \\nu \\right| \\notag \\\\\n& = \\left|\\frac{1}{\\sigma_n} \\mathbb{E}\\left[\\left(\\nu^{\\top}\\tilde{Q}\\right)^2 \\left[\\int_{-\\infty}^{0} K''\\left(t+ \\tilde{Q}^{\\top}\\tilde \\eta \\right) (f_0(\\sigma_n t |\\tilde{Q}) - f_0(0|\\tilde{Q})) \\ dt  \\right. \\right. \\right. \\notag \\\\\n& \\qquad \\qquad \\qquad \\qquad \\left. \\left. \\left. - \\int_{0}^{\\infty} K''\\left(t + \\tilde{Q}^{\\top}\\tilde \\eta \\right) (f_0(\\sigma_n t |\\tilde{Q}) - f_0(0|\\tilde{Q})) \\ dt \\right]\\right]\\right| \\notag\\\\\n& \\le  \\mathbb{E} \\left[\\left(\\nu^{\\top}\\tilde{Q}\\right)^2h(\\tilde{Q}) \\int_{-\\infty}^{\\infty} \\left|K''\\left(t+ \\tilde{Q}^{\\top}\\tilde \\eta \\right)\\right| |t| \\ dt\\right] \\notag\\\\\n& \\le  \\mathbb{E} \\left[\\left(\\nu^{\\top}\\tilde{Q}\\right)^2h(\\tilde{Q}) \\int_{-1}^{1} \\left|K''\\left(t\\right)\\right| |t - \\tilde{Q}^{\\top}\\tilde \\eta | \\ dt\\right] \\notag\\\\\n\\label{eq:quad_eq_3} & \\le  \\mathbb{E} \\left[\\left(\\nu^{\\top}\\tilde{Q}\\right)^2h(\\tilde{Q})(1+ \\|\\tilde{Q}\\|\/2\\kappa) \\int_{-1}^{1} \\left|K''\\left(t\\right)\\right| \\ dt\\right] = C_1 \\hspace{0.2in} [\\text{say}]\n\\end{align}\nby Assumption \\ref{as:distribution} and Assumption \\ref{as:derivative_bound}. By a two-step Taylor expansion, we have: \n\\begin{align*}\n\\mathbb{M}^s(\\psi) - \\mathbb{M}^s(\\psi_0^s) & =  \\frac12 (\\psi - \\psi_0^s)^{\\top} \\nabla^2\\mathbb{M}^s(\\psi^*_n) (\\psi - \\psi_0^s)  \\\\\n& \\ge \\left(\\min_{\\|v_1\\| = 1, \\|v_2 \\| \\le \\mathcal{K}} \\Lambda(v_1, v_2)\\right) \\frac{\\|\\psi - \\psi_0^s\\|^2}{2\\sigma_n} - \\frac{C_1\\sigma_n}{2} \\, \\frac{\\|\\psi - \\psi_0^s\\|^2_2}{\\sigma_n} \\\\\n& \\gtrsim \\frac{\\|\\psi - \\psi_0^s\\|^2_2}{\\sigma_n} \\,\n\\end{align*}\nThis concludes the proof of the curvature. \n\\\\\\\\\n\\noindent \nFinally, we bound the modulus of continuity:\n$$\\mathbb{E}\\left(\\sup_{d_n(\\psi, \\psi_0^s) \\le \\delta} \\left|(\\mathbb{M}^s_n-\\mathbb{M}^s)(\\psi) - (\\mathbb{M}^s_n-\\mathbb{M}^s)(\\psi_n)\\right|\\right) \\,.$$ \nThe proof is similar to that of Lemma \\ref{lem:rate_smooth} and therefore we sketch the main steps briefly. Define the estimating function $f_\\psi$ as: \n$$\nf_\\psi(Y, Q) = (Y - \\gamma)\\left(1 - K\\left(\\frac{Q^{\\top}\\psi}{\\sigma_n}\\right)\\right) \n$$\nand the collection of functions $\\mathcal{F}_\\zeta = \\{f_\\psi - f_{\\psi_0^n}: d_n(\\psi, \\psi_0^s) \\le \\delta\\}$. That $\\mathcal{F}_\\zeta$ has finite VC dimension follows from the same argument used to show $\\mathcal{G}_n$ has finite VC dimension in the proof of Lemma \\ref{conv-prob}. Now to bound modulus of continuity, we use Lemma 2.14.1 of \\cite{vdvw96}, which implies: \n$$\n\\sqrt{n}\\mathbb{E}\\left(\\sup_{d_n(\\psi, \\psi_0^s) \\le \\delta} \\left|(\\mathbb{M}^s_n-\\mathbb{M}^s)(\\psi) - (\\mathbb{M}^s_n-\\mathbb{M}^s)(\\psi_n)\\right|\\right)  \\lesssim \\mathcal{J}(1, \\mathcal{F}_\\zeta) \\sqrt{PF_\\zeta^2}\n$$\nwhere $F_\\zeta(Y, Q)$ is the envelope of $\\mathcal{F}_\\zeta$ defined as: \n\\begin{align*}\nF_\\zeta(Y, Q) & = \\sup_{d_*(\\psi, \\psi_0^s) \\le \\zeta}\\left|(Y - \\gamma)\\left(K\\left(\\frac{Q^{\\top}\\psi^s_0}{\\sigma_n}\\right)-K\\left(\\frac{Q^{\\top}\\psi}{\\sigma_n}\\right)\\right)\\right| \\\\\n& = \\left|(Y - \\gamma)\\right| \\sup_{d_*(\\psi, \\psi_0^s) \\le \\zeta} \\left|\\left(K\\left(\\frac{Q^{\\top}\\psi^s_0}{\\sigma_n}\\right)-K\\left(\\frac{Q^{\\top}\\psi}{\\sigma_n}\\right)\\right)\\right|\n\\end{align*}\nand $\\mathcal{J}(1, \\mathcal{F}_\\zeta)$ is the entropy integral which can be bounded above by a constant independent of $n$ as the class $\\mathcal{F}_\\zeta$ has finite VC dimension. As in the proof of Lemma \\ref{lem:rate_smooth}, we here consider two separate cases: (1) $\\zeta \\le \\sqrt{\\mathcal{K} \\sigma_n}$ and (2) $\\zeta > \\sqrt{\\mathcal{K} \\sigma_n}$. In the first case, we have $\\sup_{d_n(\\psi, \\psi_0^s) \\le \\zeta} \\|\\psi_ - \\psi_0^s\\| = \\zeta \\sqrt{\\sigma_n}$. This further implies: \n\\begin{align*}\n    & \\sup_{d_*(\\psi, \\psi_0^s) \\le \\zeta} \\left|\\left\\{K\\left(\\frac{Q^{\\top}\\psi_0^s}{\\sigma_n}\\right) -  K\\left(\\frac{Q^{\\top}\\psi}{\\sigma_n}\\right)\\right\\}\\right|^2 \\\\\n    & \\le \\max\\left\\{\\left|\\left\\{K\\left(\\frac{Q^{\\top}\\psi_0^s}{\\sigma_n}\\right) -  K\\left(\\frac{Q^{\\top}\\psi_0^s}{\\sigma_n} + \\|\\tilde Q\\|\\frac{\\zeta}{\\sqrt{\\sigma_n}}\\right)\\right\\}\\right|^2, \\right. \\\\\n    & \\qquad \\qquad \\qquad \\qquad \\left. \\left|\\left\\{K\\left(\\frac{Q^{\\top}\\psi_0^s}{\\sigma_n}\\right) -  K\\left(\\frac{Q^{\\top}\\psi_0^s}{\\sigma_n} - \\|\\tilde Q\\|\\frac{\\zeta}{\\sqrt{\\sigma_n}}\\right)\\right\\}\\right|^2\\right\\} \\\\\n    & := \\max\\{T_1, T_2\\} \\,.\n\\end{align*}\nTherefore to bound $\\mathbb{E}[F_\\zeta^2(Y, Q)]$ is equivalent to bounding both $\\mathbb{E}[(Y- \\gamma)^2 T_1]$ and $\\mathbb{E}[(Y - \\gamma)^2 T_2]$ separately, which, in turn equivalent to bound $\\mathbb{E}[T_1]$ and $\\mathbb{E}[T_2]$, as $|Y - \\gamma| \\le 1$. These bounds follows from similar calculation as of Lemma \\ref{lem:rate_smooth}, hence skipped. Finally we have in this case, $$\n\\mathbb{E}[F_\\zeta^2(Y, Q)] \\lesssim \\zeta \\sqrt{\\sigma_n} \\,.\n$$ \nThe other case, when $\\zeta > \\sqrt{\\mathcal{K} \\sigma_n}$ also follows by similar calculation of Lemma \\ref{lem:rate_smooth}, which yields: \n$$\n\\mathbb{E}[F_\\zeta^2(Y, Q)] \\lesssim \\zeta^2 \\,.\n$$\n\n\n\n\n\n\n\n\n\n\n\n\n\\noindent\nUsing this in the maximal inequality yields: \n\\begin{align*}\n\\sqrt{n}\\mathbb{E}\\left(\\sup_{d_n(\\psi, \\psi_0) \\le \\delta} \\left|\\mathbb{M}_n(\\psi - \\psi_n) - \\mathbb{M}^s(\\psi - \\psi_n)\\right|\\right) & \\lesssim \\sqrt{\\zeta}\\sigma^{1\/4}_n\\mathds{1}_{\\zeta \\le \\sqrt{\\mathcal{K} \\sigma_n}} +  \\zeta \\mathds{1}_{\\zeta > \\sqrt{\\mathcal{K} \\sigma_n}} \\\\\n& := \\phi_n(\\zeta) \\,\n\\end{align*}\nThis implies (following the same argument as of Lemma \\ref{lem:rate_smooth}): \n$$\nn^{2\/3}\\sigma_n^{-1\/3}d^2(\\hat \\psi^s, \\psi_0^s) = O_p(1) \\,.\n$$\nNow as $n^{2\/3}\\sigma_n^{-1\/3} \\gg \\sigma_n^{-1}$, we have: \n$$\n\\frac{1}{\\sigma_n}d_n^2(\\hat \\psi^s, \\psi_0^s) = o_p(1) \\,.\n$$\nwhich further indicates\n\\begin{align}\n\\label{rate1} & n^{2\/3}\\sigma_n^{-1\/3}\\left[\\frac{\\|\\hat \\psi^s - \\psi_0^s\\|^2}{\\sigma_n} \\mathds{1}(\\|\\hat \\psi^s - \\psi_0^s\\| \\le \\mathcal{K}\\sigma_n) \\right. \\notag \\\\\n& \\qquad \\qquad \\qquad \\left. + \\|\\hat \\psi^s - \\psi_0^s\\|  \\mathds{1}(\\|\\hat \\psi^s - \\psi_0^s\\|\\ge \\mathcal{K}\\sigma_n)\\right] = O_P(1)\n\\end{align}\nThis implies: \n\\begin{enumerate}\n\\item $\\frac{n^{2\/3}}{\\sigma_n^{4\/3}}\\|\\hat \\psi^s - \\psi_0^s\\| \\mathds{1}(\\|\\hat \\psi^s - \\psi_0^s\\|\\le \\mathcal{K}\\sigma_n) = O_P(1)$\n\\item $\\frac{n^{2\/3}}{\\sigma_n^{1\/3}}\\|\\hat \\psi^s - \\psi_0^s\\| \\mathds{1}(\\|\\hat \\psi^s - \\psi_0^s\\| \\ge \\mathcal{K}\\sigma_n) = O_P(1)$\n\\end{enumerate}\nTherefore: \n\\begin{align*}\n& \\frac{n^{2\/3}}{\\sigma_n^{4\/3}}\\|\\hat \\psi^s - \\psi_0^s\\| \\mathds{1}(\\|\\hat \\psi^s - \\psi_0^s\\| \\le \\mathcal{K}\\sigma_n) \\\\\n& \\qquad \\qquad \\qquad + \\frac{n^{2\/3}}{\\sigma_n^{1\/3}}\\|\\hat \\psi^s - \\psi_0^s\\| \\mathds{1}(\\|\\hat \\psi^s - \\psi_0^s\\| \\ge \\mathcal{K}\\sigma_n) = O_p(1) \\,.\n\\end{align*}\ni.e. \n$$\n\\left(\\frac{n^{2\/3}}{\\sigma_n^{4\/3}} \\wedge  \\frac{n^{2\/3}}{\\sigma_n^{1\/3}}\\right)\\|\\hat \\psi^s - \\psi_0^s\\| = O_p(1) \\,.\n$$\nNow $(n^{2\/3}\/\\sigma_n^{4\/3} \\gg 1\/\\sigma_n$ as long as $n^{2\/3} \\gg \\sigma_n^{1\/3}$ which is obviously true. On the other hand, $n^{2\/3}\/\\sigma_n^{1\/3} \\gg 1\/\\sigma_n$ iff $n\\sigma_n \\gg 1$ which is also true as per our assumption. Therefore we have: \n$$\n\\frac{\\|\\hat \\psi^s - \\psi_0^s\\|}{\\sigma_n} = O_p(1) \\,.\n$$\nThis completes the proof. \n\n\\end{proof}\n\n\n\\section{Methodology and Theory for Continuous Response Model}\n\\label{sec:theory_regression}\nIn this section we present our analysis for the continuous response model. Without smoothing, the original estimating equation is: \n$$\nf_{\\beta, \\delta, \\psi}(Y, X, Q) = \\left(Y - X^{\\top}\\beta - X^{\\top}\\delta\\mathds{1}_{Q^{\\top}\\psi > 0}\\right)^2  \n$$\nand we estimate the parameters as: \n\\begin{align}\n\\label{eq:ls_estimator}\n\\left(\\hat \\beta^{LS}, \\hat \\delta^{LS}, \\hat \\psi^{LS}\\right) & = {\\arg\\min}_{(\\beta, \\delta, \\psi) \\in \\Theta} \\mathbb{P}_n f_{\\beta, \\delta, \\psi} \\notag \\\\\n& :=  {\\arg\\min}_{(\\beta, \\delta, \\psi) \\in \\Theta}\\mathbb{M}_n(\\beta, \\delta, \\psi)\\,.\n\\end{align}\nwhere $\\mathbb{P}_n$ is empirical measure based on i.i.d. observations $\\{(X_i, Y_i, Q_i)\\}_{i=1}^n$ and $\\Theta$ is the parameter space. Henceforth, we assume $\\Theta$ is a compact subset of dimension ${\\bbR}^{2p+d}$. We also define $\\theta = (\\beta, \\delta, \\psi)$, i.e. all the parameters together as a vector and by $\\theta_0$ is used to denote the true parameter vector $(\\beta_0, \\delta_0, \\psi_0)$. Some modification of equation \\eqref{eq:ls_estimator} leads to the following:  \n\\begin{align*}\n(\\hat \\beta^{LS}, \\hat \\delta^{LS}, \\hat \\psi^{LS}) & = {\\arg\\min}_{\\beta, \\delta, \\psi}  \\sum_{i=1}^n \\left(Y_i - X_i^{\\top}\\beta - X_i^{\\top}\\delta\\mathds{1}_{Q_i^{\\top}\\psi > 0}\\right)^2 \\\\ \n& = {\\arg\\min}_{\\beta, \\delta, \\psi}  \\sum_{i=1}^n \\left[\\left(Y_i - X_i^{\\top}\\beta\\right)^2\\mathds{1}_{Q_i^{\\top}\\psi_0 \\le 0} \\right. \\\\\n&  \\hspace{14em} \\left. + \\left(Y_i - X_i^{\\top}\\beta - X_i^{\\top}\\delta\\right)^2\\mathds{1}_{Q_i^{\\top}\\psi > 0} \\right] \\\\\n& = {\\arg\\min}_{\\beta, \\delta, \\psi}  \\sum_{i=1}^n \\left[\\left(Y_i - X_i^{\\top}\\beta\\right)^2 + \\left\\{\\left(Y_i - X_i^{\\top}\\beta - X_i^{\\top}\\delta\\right)^2 \\right. \\right. \\\\\n& \\hspace{17em} \\left. \\left. - \\left(Y_i - X_i^{\\top}\\beta\\right)^2\\right\\}\\mathds{1}_{Q_i^{\\top}\\psi > 0} \\right] \n\\end{align*}\nTypical empirical process calculations yield under mild conditions: \n$$\n\\|\\hat \\beta^{LS} - \\beta_0\\|^2 + \\|\\hat \\delta^{LS} - \\delta_0\\|^2 + \\|\\hat \\psi^{LS} - \\psi_0 \\|_2 = O_p(n^{-1})\n$$\nbut inference is difficult as the limit distribution is unknown, and in any case, would be a highly non-standard distribution. Recall that even in the one-dimensional change point model with fixed jump size, the least squares change point estimator converges at rate $n$ to the truth with a non-standard limit distribution, namely a minimizer of a two-sided compound Poisson process (see \\cite{lan2009change} for more details). To obtain a computable estimator with tractable limiting distribution, we resort to a smooth approximation of the indicator function in \\eqref{eq:ls_estimator} using a distribution kernel with suitable bandwidth, i.e we replace $\\mathds{1}_{Q_i^{\\top}\\psi > 0}$ by $K(Q_i^{\\top}\\psi\/\\sigma_n)$ for some appropriate distribution function $K$ and bandwidth $\\sigma_n$, i.e. \n\\begin{align*}\n(\\hat \\beta^S, \\hat \\delta^S, \\hat \\psi^S) & = {\\arg\\min}_{\\beta, \\delta, \\psi} \\left\\{ \\frac1n \\sum_{i=1}^n \\left[\\left(Y_i - X_i^{\\top}\\beta\\right)^2 + \\left\\{\\left(Y_i - X_i^{\\top}\\beta - X_i^{\\top}\\delta\\right)^2 \\right. \\right. \\right. \\\\\n& \\hspace{15em} \\left. \\left. \\left. - \\left(Y_i - X_i^{\\top}\\beta\\right)^2\\right\\}K\\left(\\frac{Q_i^{\\top}\\psi}{\\sigma_n}\\right) \\right] \\right\\} \\\\\n& = {\\arg\\min}_{(\\beta, \\delta, \\psi) \\in \\Theta} \\mathbb{P}_n f^s_{(\\beta, \\delta, \\psi)}(X, Y, Q) \\\\\n& := {\\arg\\min}_{\\theta \\in \\Theta} \\mathbb{M}^s_n(\\theta) \\,.\n\\end{align*}\nDefine $\\mathbb{M}$ (resp. $\\mathbb{M}^s$) to be the population counterpart of $\\mathbb{M}_n$ and $\\mathbb{M}_n^s$ respectively which are defined as: \n\\begin{align*}\n\\mathbb{M}(\\theta) & = \\mathbb{E}\\left(Y - X^{\\top}\\beta\\right)^2 + \\mathbb{E}\\left(\\left[-2\\left(Y_i - X^{\\top}\\beta\\right)X^{\\top}\\delta + (X^{\\top}\\delta)^2\\right] \\mathds{1}_{Q^{\\top}\\psi > 0}\\right) \\,, \\\\\n\\mathbb{M}^s(\\theta) & = \\mathbb{E}\\left[(Y - X^{\\top}\\beta)^2 + \\left\\{-2(Y-X^{\\top}\\beta)(X^{\\top}\\delta) + (X^{\\top}\\delta)^2\\right\\}K\\left(\\frac{Q^{\\top}\\psi}{\\sigma_n}\\right)\\right] \\,.\n\\end{align*}\nAs noted in the proof of \\textcolor{blue}{Seo and Linton}, the assumption $\\log{n}\/n\\sigma_n^2 \\to 0$ was only used to show: \n$$\n\\frac{\\left\\|\\hat \\psi^s - \\psi_0\\right\\|}{\\sigma_n} = o_p(1) \\,.\n$$\nIn this paper, we show that one can achieve the same conclusion as long as $n\\sigma_n \\to \\infty$.  The rest of the proof for the normality is similar to that of \\cite{seo2007smoothed}, we will present it briefly for the ease the readers. The proof is quite long and technical, therefore we break the proof into several lemmas. We, first, list our assumptions: \n\\begin{assumption}\n\\label{eq:assm}\n\\begin{enumerate}\n\\item Define $f_\\psi(\\cdot \\mid \\tilde Q)$ to be the conditional distribution of $Q^{\\top}\\psi$ given $\\tilde Q$. (In particular we will denote by $f_0(\\cdot \\mid \\tilde q)$ to be conditional distribution of $Q^{\\top}\\psi_0$ given $\\tilde Q$ and $f_s(\\cdot \\mid \\tilde q)$ to be the conditional distribution of $Q^{\\top}\\psi_0^s$ given $\\tilde Q$. Assume that there exists $F_+$ such that $\\sup_t f_0(t | \\tilde Q) \\le F_+$ almost surely on $\\tilde Q$ and for all $\\psi$ in a neighborhood of $\\psi_0$ (in particular for $\\psi_0^s$). Further assume that $f_\\psi$ is differentiable and the derivative is bounded by $F_+$ for all $\\psi$ in a neighborhood of $\\psi_0$ (again in particular for $\\psi_0^s$).\n\\vspace{0.1in}\n\\item Define $g(Q) = {\\sf var}(X \\mid Q)$. There exists $c_-$ and $c_+$ such that $c_- \\le \\lambda_{\\min}(g(Q)) \\le \\lambda_{\\max}(g(Q)) \\le c_+$ almost surely. Also assume that $g$ is a Lipschitz with constant $G_+$ with respect to $Q$. \n\\vspace{0.1in}\n\\item There exists $p_+ < \\infty$ and $p_- > 0, r > 0$ such that: \n$$\np_- \\|\\psi - \\psi_0\\| \\le \\mathbb{P}\\left(\\text{sign}\\left(Q^{\\top}\\psi\\right) \\neq \\text{sign}\\left(Q^{\\top}\\psi_0\\right)\\right) \\le p_+ \\|\\psi - \\psi_0\\| \\,,\n$$\nfor all $\\psi$ such that $\\|\\psi - \\psi_0\\| \\le r$. \n\\vspace{0.1in}\n\\item For all $\\psi$ in the parameter space $0 < \\mathbb{P}\\left(Q^{\\top}\\psi > 0\\right) < 1$. \n\\vspace{0.1in} \n\\item Define $m_2(Q) = \\mathbb{E}\\left[\\|X\\|^2 \\mid Q\\right]$ and $m_4(Q) = \\mathbb{E}\\left[\\|X\\|^4 \\mid Q\\right]$. Assume $m_2, m_4$ are bounded Lipschitz function of $Q$.  \n\\end{enumerate}\n\\end{assumption}\n\n\n\\subsection{Sufficient conditions for above assumptions }\nWe now demonstrate some sufficient conditions for the above assumptions to hold. The first condition is essentially a condition on the conditional density of the first co-ordinate of $Q$ given all other co-ordinates. If this conditional density is bounded and has bounded derivative, then first assumption is satisfied. This condition is satisfied in fair generality. The second assumption implies that the conditional distribution of X given Q has variance in all the direction over all $Q$. This is also very weak condition, as is satisfied for example if X and Q and independent (with $X$ has non-degenerate covariance matrix) or $(X, Q)$ are jointly normally distributed to name a few. This condition can further be weaken by assuming that the maximum and minimum eigenvalues of $\\mathbb{E}[g(Q)]$ are bounded away from $\\infty$ and $0$ respectively but it requires more tedious book-keeping. The third assumption is satisfied as long as as $Q^{\\top}\\psi$ has non-zero density near origin, while the fourth assumption merely states that the support of  $Q$ is not confined to one side of the hyperplane for any hyperplane and a simple sufficient condition for this is $Q$ has continuous density with non-zero value at the origin. The last assumption is analogous to the second assumption for the conditional fourth moment which is also satisfied in fair generality. \n\\\\\\\\\n\\noindent\n{\\bf Kernel function and bandwidth: } We take $K(x) = \\Phi(x)$ (distribution of standard normal random variable) for our analysis. For the bandwidth we assume $n\\sigma_n^2 \\to 0$ and $n \\sigma_n \\to \\infty$ as the other case, (i.e. $n\\sigma_n^2 \\to \\infty$) is already established in \\cite{seo2007smoothed}. \n\\\\\\\\\n\\noindent\nBased on Assumption \\ref{eq:assm} and our choice of kernel and bandwidth we establish the following theorem: \n\\begin{theorem}\n\\label{thm:regression}\nUnder Assumption \\ref{eq:assm} and the above choice of kernel and bandwidth we have: \n$$\n\\sqrt{n}\\begin{pmatrix}\\begin{pmatrix} \\hat \\beta^s \\\\ \\hat \\delta^s \\end{pmatrix} - \\begin{pmatrix} \\beta_0 \\\\ \\delta_0 \\end{pmatrix} \\end{pmatrix} \\overset{\\mathscr{L}}{\\implies} \\mathcal{N}(0, \\Sigma_{\\beta, \\delta})\n$$\nand \n$$\n\\sqrt{n\/\\sigma_n} \\left(\\hat \\psi^s - \\psi_0\\right) \\overset{\\mathscr{L}}{\\implies} \\mathcal{N}(0, \\Sigma_\\psi) \\,,\n$$\nfor matrices $\\Sigma_{\\beta, \\delta}$ and $\\Sigma_\\psi$ mentioned explicitly in the proof. Moreover they are asymptotically independent. \n\\end{theorem}\nThe proof of the theorem is relatively long, so we break it into several lemmas. We  provide a roadmap of the proof in this section while the elaborate technical derivations of the supporting lemmas can be found in Appendix. Let $\\nabla \\mathbb{M}_n^s(\\theta)$ and $\\nabla^2 \\mathbb{M}_n^s(\\theta)$ be the gradient and Hessian of $\\mathbb{M}_n^s(\\theta)$ with respect to $\\theta$.  As $\\hat \\theta^s$ minimizes $\\mathbb{M}_n^s(\\theta)$, we have from the first order condition, $\\nabla \\mathbb{M}_n^s(\\hat \\theta^s) = 0$. Using one step Taylor expansion we have:\n\\allowdisplaybreaks \n\\begin{align*}\n\\label{eq:taylor_first}\n\\nabla \\mathbb{M}_n^s(\\hat \\theta^s) = \\nabla \\mathbb{M}_n^s(\\theta_0) + \\nabla^2 \\mathbb{M}_n^s(\\theta^*)\\left(\\hat \\theta^s - \\theta_0\\right) = 0\n\\end{align*}\ni.e.\n\\begin{equation}\n\\label{eq:main_eq}  \n\\left(\\hat{\\theta}^s - \\theta_0\\right) = -\\left(\\nabla^2 \\mathbb{M}_n^s(\\theta^*)\\right)^{-1} \\nabla \\mathbb{M}_n^s(\\theta_0)\n\\end{equation}\nfor some intermediate point $\\theta^*$ between $\\hat \\theta^s$ and $\\theta_0$. \nFollowing the notation of \\cite{seo2007smoothed}, define a diagonal matrix $D_n$ of dimension $2p + d$ with first $2p$ elements being 1 and the last $d$ elements being $\\sqrt{\\sigma_n}$. \n we can write: \n\\begin{align}\n\\sqrt{n}D_n^{-1}(\\hat \\theta^s - \\theta_0) & = - \\sqrt{n}D_n^{-1}\\nabla^2\\mathbb{M}_n^s(\\theta^*)^{-1}\\nabla \\mathbb{M}_n^s(\\theta_0)  \\notag \\\\\n\\label{eq:taylor_main} & = \\begin{pmatrix} \\nabla^2\\mathbb{M}_n^{s, \\gamma}(\\theta^*) & \\sqrt{\\sigma_n}\\nabla^2\\mathbb{M}_n^{s, \\gamma \\psi}(\\theta^*) \\\\\n\\sqrt{\\sigma_n}\\nabla^2\\mathbb{M}_n^{s, \\gamma \\psi}(\\theta^*) & \\sigma_n\\nabla^2\\mathbb{M}_n^{s, \\psi}(\\theta^*)\\end{pmatrix}^{-1}\\begin{pmatrix} \\sqrt{n}\\nabla \\mathbb{M}_n^{s, \\gamma}(\\theta_0) \\\\ \\sqrt{n\\sigma_n}\\nabla \\mathbb{M}_n^{s, \\psi}(\\theta_0)\\end{pmatrix}\n\\end{align}\nwhere $\\gamma = (\\beta, \\delta) \\in {\\bbR}^{2p}$. The following lemma establishes the asymptotic properties of $\\nabla \\mathbb{M}_n^s(\\theta_0)$: \n\\begin{lemma}[Asymptotic Normality of $\\nabla \\mathbb{M}_n^s(\\theta_0)$]\n\\label{asymp-normality}\n\\label{asymp-normality}\nUnder assumption \\ref{eq:assm} we have: \n\\begin{align*}\n\\sqrt{n}\\nabla \\mathbb{M}_n^{s, \\gamma}(\\theta_0) \\implies \\mathcal{N}\\left(0, 4V^{\\gamma}\\right) \\,,\\\\\n\\sqrt{n\\sigma_n}\\nabla \\mathbb{M}_n^{s, \\psi}(\\theta_0) \\implies \\mathcal{N}\\left(0, V^{\\psi}\\right) \\,.\n\\end{align*} \nfor some n.n.d. matrices $V^{\\gamma}$ and $V^{\\psi}$ which is mentioned explicitly in the proof. Further more $\\sqrt{n}\\nabla \\mathbb{M}_n^{s, \\gamma}(\\theta_0)$ and $\\sqrt{n\\sigma_n}\\nabla \\mathbb{M}_n^{s, \\psi}(\\theta_0)$ are asymptotically independent. \n\\end{lemma}\n\\noindent\nNext, we analyze the convergence of $\\nabla^2 \\mathbb{M}_n^s(\\theta^*)$ which is stated in the following lemma: \n\\begin{lemma}[Convergence in Probability of $\\nabla^s \\mathbb{M}_n^s(\\theta^*)$]\n\\label{conv-prob}\nUnder Assumption \\eqref{eq:assm}, for any random sequence $\\breve{\\theta} = \\left(\\breve{\\beta}, \\breve{\\delta}, \\breve{\\psi}\\right)$ such that $\\breve{\\beta} \\overset{p}{\\to} \\beta_0, \\breve{\\delta} \\overset{p}{\\to} \\delta_0, \\|\\breve{\\psi} - \\psi_0\\|\/\\sigma_n \\overset{P} \\rightarrow 0$, we have: \n\\begin{align*}\n\\nabla^2_{\\gamma} \\mathbb{M}_n^s(\\breve{\\theta}) & \\overset{p}{\\longrightarrow} 2Q^{\\gamma} \\,, \\\\\n\\sqrt{\\sigma_n}\\nabla^2_{\\psi \\gamma} \\mathbb{M}_n^s(\\breve{\\theta}) & \\overset{p}{\\longrightarrow} 0 \\,, \\\\\n\\sigma_n \\nabla^2_{\\psi} \\mathbb{M}_n^s(\\breve{\\theta}) & \\overset{p}{\\longrightarrow} Q^{\\psi} \\,.\n\\end{align*}\nfor some matrices $Q^{\\gamma}, Q^{\\psi}$ mentioned explicitly in the proof. This, along with equation \\eqref{eq:taylor_main}, establishes: \n\\begin{align*}\n\\sqrt{n}\\left(\\hat \\gamma^s - \\gamma_0\\right) & \\overset{\\mathscr{L}}{\\implies} \\mathcal{N}\\left(0, Q^{\\gamma^{-1}}V^{\\gamma}Q^{\\gamma^{-1}}\\right) \\,, \\\\\n\\sqrt{n\/\\sigma_n}\\left(\\hat \\psi^s - \\psi_0\\right) & \\overset{\\mathscr{L}}{\\implies} \\mathcal{N}\\left(0, Q^{\\psi^{-1}}V^{\\psi}Q^{\\psi^{-1}}\\right) \\,.\n\\end{align*}\nwhere as before $\\hat \\gamma^s = (\\hat \\beta^s, \\hat \\delta^s)$. \n\\end{lemma}\nIt will be shown later that the condition $\\|\\breve{\\psi}_n - \\psi_0\\|\/\\sigma_n \\overset{P} \\rightarrow 0$ needed in Lemma \\ref{conv-prob} holds for the (random) sequence $\\psi^*$, the intermediate point in the Taylor expansion. Then, combining Lemma \\ref{asymp-normality} and Lemma \\ref{conv-prob} we conclude the proof of Theorem \\ref{thm:regression}.\nObserve that, to show $\\left\\|\\psi^* - \\psi_0 \\right\\| = o_P(\\sigma_n)$, it suffices to to prove that $\\left\\|\\hat \\psi^s - \\psi_0 \\right\\| = o_P(\\sigma_n)$. Towards that direction, we have following lemma: \n\n\\begin{lemma}[Rate of convergence]\n\\label{lem:rate_smooth}\nUnder Assumption \\ref{eq:assm} and our choice of kernel and bandwidth, \n$$\nn^{2\/3}\\sigma_n^{-1\/3} d^2_*\\left(\\hat \\theta^s, \\theta_0^s\\right) = O_P(1) \\,,\n$$\nwhere \n\\begin{align*}\nd_*^2(\\theta, \\theta_0^s) & =  \\|\\beta - \\beta_0^s\\|^2 + \\|\\delta - \\delta_0^s\\|^2 \\\\\n& \\qquad \\qquad + \\frac{\\|\\psi - \\psi_0^s\\|^2}{\\sigma_n} \\mathds{1}_{\\|\\psi - \\psi_0^s\\| \\le \\mathcal{K}\\sigma_n} + \\|\\psi - \\psi_0^s\\| \\mathds{1}_{\\|\\psi - \\psi_0^s\\| > \\mathcal{K}\\sigma_n}  \\,.\n\\end{align*}\nfor some specific constant $\\mathcal{K}$. (This constant will be mentioned precisely in the proof). Hence as $n\\sigma_n \\to \\infty$, we have $n^{2\/3}\\sigma_n^{-1\/3} \\gg \\sigma_n^{-1}$ which implies $\\|\\hat \\psi^s - \\psi_0^s\\|\/\\sigma_n  \\overset{P}  \\longrightarrow 0 \\,.$\n\\end{lemma}\n\\noindent\nThe above lemma establishes $\\|\\hat \\psi^s - \\psi_0^s\\|\/\\sigma_n = o_p(1)$ but our goal is to show that $\\|\\hat \\psi^s - \\psi_0\\|\/\\sigma_n = o_p(1)$. Therefore, we further need $\\|\\psi^s_0 - \\psi_0\\|\/\\sigma_n \\rightarrow 0$ which is demonstrated in the following lemma:\n\n\\begin{lemma}[Convergence of population minimizer]\n\\label{bandwidth}\nUnder Assumption \\ref{eq:assm} and our choice of kernel and bandwidth, we have: $\\|\\psi^s_0 - \\psi_0\\|\/\\sigma_n \\rightarrow 0$. \n\\end{lemma}\n\n\\noindent\nHence the final roadmap is the following: Using Lemma \\ref{bandwidth} and Lemma \\ref{lem:rate_smooth} we establish that $\\|\\hat \\psi^s - \\psi_0\\|\/\\sigma_n = o_p(1)$ if $n\\sigma_n \\to 0$. This, in turn, enables us to prove Lemma \\ref{conv-prob}, i.e. $\\sigma_n \\nabla^2 \\mathbb{M}_n^s(\\theta^*) \\overset{P} \\rightarrow Q$,which, along with Lemma \\ref{asymp-normality}, establishes the main theorem. \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Introduction}\nThe simple linear regression model assumes a uniform linear relationship between the covariate and the response, in the sense that the regression parameter $\\beta$ is the same over the entire covariate domain. In practice, the situation can be more complicated: for instance, the regression parameter may differ from sub-population to sub-population within a large (super-) population. Some common techniques to account for such heterogeneity include mixed linear models, introducing an interaction effect, or fitting different models among each sub-population which corresponds to a supervised classification setting where the true groups (sub-populations) are \\emph{a priori known}. \n\t\\newline\n\t\\newline\n\t\\indent A more difficult scenario arises when the sub-populations are unknown, in which case regression and classification must happen simultaneously. Consider the scenario where the conditional mean of $Y_i$ given $X_i$ is different for different unknown sub-groups. A  well-studied treatment of this problem -- the so-called change point problem -- considers a simple thresholding model where membership in a sub-group is determined by whether a real-valued observable $X$ falls to the left or right of an unknown parameter $\\gamma$. More recently, there has been work for multi-dimensional covariates, namely when the membership is determined by which side a random vector $X$ falls with respect to an hyperplane with unknown normal vector $\\theta_0$. A concrete example appears in \\cite{wei2014latent} who extend the linear thresholding model due to  \\cite{kang2011new} to general dimensions: \n\t\\begin{eqnarray}\\label{eq:weimodel}\n\tY=\\mu_1\\cdot 1_{X^{\\top}\\theta_0\\geq 0}+\\mu_2\\cdot 1_{X^{\\top}\\theta_0<0}+\\varepsilon\\,,\n\t\\end{eqnarray}\n\tand studied computational algorithms and consistency of the same. This model and others with similar structure, called \\emph{change plane models}, are useful in various fields of research, e.g. modeling treatment effect heterogeneity in drug treatment (\\cite{imai2013estimating}), modeling sociological data on voting and employment (\\cite{imai2013estimating}), or cross country growth regressions in econometrics \n(\\cite{seo2007smoothed}).\n\t\\newline\n\t\\newline\n\\indent Other aspects of this model have also been investigated. \\cite{fan2017change} examined the change plane model from the statistical testing point of view, with the null hypothesis being the absence of a separating hyperplane. They proposed a test statistic, studied its asymptotic distribution and provided sample size recommendations for achieving target values of power. \\cite{li2018multi} extended the change point detection problem in the multi-dimensional setup by considering the case where $X^{\\top}\\theta_0$ forms a multiple change point data sequence. \n\nThe key difficultly with change plane type models is the inherent discontinuity in the optimization criteria involved where the parameter of interest appears as an argument to some indicator function, rendering the optimization extremely hard. To alleviate this, one option is to kernel smooth the indicator function, an approach that was adopted by Seo and Linton \\cite{seo2007smoothed} in a version of the change-plane problem, motivated by earlier results of Horowitz \\cite{horowitz1992smoothed} that dealt with a smoothed version of the maximum score estimator. Their model has an additive structure of the form:\n\\[Y_t = \\beta^{\\top}X_t + \\delta^{\\top} \\tilde{X}_t \\mathds{1}_{Q_t^{\\top} \\boldmath \\psi > 0} + \\epsilon_t \\,,\\]\nwhere $\\psi$ is the (fixed) change-plane parameter, and $t$ can be viewed as a time index. Under a set of assumptions on the model (Assumptions 1 and 2 of their paper), they showed asymptotic normality of their estimator of $\\psi$ obtained by minimizing a smoothed least squares criterion\nthat uses a differentiable distribution function $\\mathcal{K}$. The rate of convergence of $\\hat{\\psi}$ to the truth was shown to be $\\sqrt{n\/\\sigma_n}$ where $\\sigma_n$ was the bandwidth parameter used to smooth the least squares function. As noted in their Remark 3, under the special case of i.i.d. observations, their requirement that $\\log n\/(n \\sigma_n^2) \\rightarrow 0$ translates to a maximal convergence rate of $n^{3\/4}$ up to a logarithmic factor. The work of \\cite{li2018multi} who considered multiple parallel change planes (determined by a fixed dimensional normal vector) and high dimensional linear models in the regions between consecutive hyperplanes also builds partly upon the methods of \\cite{seo2007smoothed} and obtains the same (almost) $n^{3\/4}$ rate for the normal vector (as can be seen by putting Condition 6 in their paper in conjunction with the conclusion of Theorem 3). \n\\\\\\\\\n\nWhile it is established that the condition $n\\sigma_n^2 \\to \\infty$ is sufficient (upto a log factor) for achieving asymptotic normality of the smoothed estimator, there is no result in the existing literature to ascertain whether its necessity. Intuitively speaking, the necessary condition for asymptotic normality ought to be $n \\sigma_n \\to 0$, as this will ensure a growing number of observations in a $\\sigma_n$ neighborhood around the true hyperplane, allowing the central limit theorem to kick in. In this paper we \\emph{bridge this gap} by proving that asymptotic normality of the smoothed change point estimator is, in fact, achievable with $n \\sigma_n \\to \\infty$. \nThis implies that the best possible rate of convergence of the smoothed estimator can be arbitrarily close to $n^{-1}$, the minimax optimal rate of estimation for this problem. To demonstrate this, we focus on two change plane estimation problems, one with a continuous and another with a binary response. The continuous response model we analyze here is the following: \n\\begin{equation}\n\\label{eq:regression_main_eqn}\nY_i = \\beta_0^{\\top}X_i + \\delta_0^{\\top}X_i\\mathds{1}_{Q_i^{\\top}\\psi_0 > 0} + {\\epsilon}_i \\,.\n\\end{equation}\nfor i.i.d. observations $\\{(X_i, Y_i, Q_i\\}_{i=1}^n$, where the zero-mean transitory shocks ${\\epsilon}_i \\rotatebox[origin=c]{90}{$\\models$} (X_i, Q_i)$. Our calculation can be easily extended to the case when the covariates on the either side of the change hyperplane are different and $\\mathbb{E}[{\\epsilon} \\mid X, Q] = 0$ with more tedious bookkeeping. As this generalization adds little of interest, conceptually, to our proof, we posit the simpler model for ease of understanding.\nAs the parameter $\\psi_0$ is only identifiable upto its norm, we assume that the first co-ordinate is $1$ (along the lines of \\cite{seo2007smoothed}) which removes one degree of freedom and makes the parameter identifiable. \n\\\\\\\\\nTo illustrate that a similar phenomenon transpires with binary response, we also study a canonical version of such a model which can be briefly described as follows: The covariate $Q \\sim P$ where $P$ is distribution on $\\mathbb{R}^d$ and the conditional distribution of $Y$ given $Q$ is modeled as follows: \n\\begin{equation}\n\\label{eq:classification_eqn}\nP(Y=1|Q) = \\alpha_0 \\mathds{1}(Q^{\\top}\\psi_0 \\le 0) + \\beta_0\\mathds{1}(Q^{\\top}\\psi_0 > 0)\n\\end{equation}\nfor some parameters $\\alpha_0, \\beta_0\\in (0,1)$ and $\\psi_0\\in\\mathbb{R}^d$ (with first co-ordinate being one for identifiability issue as for the continuous response model), the latter being of primary interest for estimation. \nThis model is identifiable up to a permutation of $(\\alpha_0, \\beta_0)$, so we further assume $\\alpha_0 < \\beta_0$. For both models, we show that $\\sqrt{n\/\\sigma_n}(\\hat \\psi - \\psi_0)$ converges to zero-mean normal distribution as long as $n \\sigma_n \\to \\infty$ but the calculations for the binary model are completely relegated to Appendix \\ref{sec:supp_classification}. \n\\\\\\\\\n{\\bf Organization of the paper:} The rest of the paper is organized as follows: In Section \\ref{sec:theory_regression} we present the methodology, the statement of the asymptotic distributions and a sketch of the proof for the continuous response model \\eqref{eq:regression_main_eqn}. In Section \\ref{sec:classification_analysis} we briefly describe the binary response model \\eqref{eq:classification_eqn} and related assumptions, whilst the details can be found in the supplementary document. In Section \\ref{sec:simulation} we present some simulation results, both for the binary and the continuous response models to study the effect of the bandwidth on the quality of the normal approximation in finite samples. In Section \\ref{sec:real_data}, we present a real data analysis where we analyze the effect of  income and urbanization on the $CO_2$ emission in different countries. \n\\\\\\\\\n{\\bf Notations: } Before delving into the technical details, we first setup some notations here. We assume from now on, $X \\in {\\bbR}^p$ and $Q \\in {\\bbR}^d$. For any vector $v$ we define by $\\tilde v$ as the vector with all the co-ordinates expect the first one. We denote by $K$ the kernel function used to smooth the indicator function. For any matrix $A$, we denote by $\\|A\\|_2$ (or $\\|A\\|_F$) as its Frobenious norm and $\\|A\\|_{op}$ as its operator norm. For any vector, $\\| \\cdot \\|_2$ denotes its $\\ell_2$ norm.  \n\n\n\n\n\n\n\\input{regression.tex}\n\\input{classification.tex}\n\n\\section{Simulation studies}\n\\label{sec:simulation}\nIn this section, we present some simulation results to analyse the effect of the choice of $\\sigma_n$ on the finite sample approximation of asymptotic normality, i.e. Berry-Essen type bounds. If we choose a smaller sigma, the rate of convergence is accelerated but the normal approximation error at smaller sample sizes will be higher, as we don't have enough observations in the vicinity of the change hyperplane for the CLT to kick in. This problem is alleviated by choosing $\\sigma_n$ larger, but this, on the other hand, compromises the convergence rate. Ideally, a Berry-Essen type of bound will quantify this, but this will require a different set of techniques and is left as an open problem. In our simulations, we generate data from following setup: \n\\begin{enumerate}\n\\item Set $N = 50000, p = 3, \\alpha_0 = 0.25, \\beta = 0.75$ and some $\\theta_0 \\in \\mathbb{R}^p$ with first co-ordinate $ = 1$. \n\\item Generate $X_1, \\dots, X_n \\sim \\mathcal{N}(0, I_p)$. \n\\item Generate $Y_i \\sim \\textbf{Bernoulli}\\left(\\alpha_0\\mathds{1}_{X_i^{\\top}\\theta_0 \\le 0} + \\beta_0 \\mathds{1}_{X_i^{\\top}\\theta_0 > 0}\\right)$. \n\\item Estimate $\\hat \\theta$ by minimizing $\\mathbb{M}_n(\\theta)$ (replacing $\\gamma$ by $\\bar Y$) based on $\\{(X_i, Y_i)\\}_{i=1}^n$ for different choices of $\\sigma_n$. \n\\end{enumerate}\nWe repeat Step 2 - Step 4 a hundred times to obtain $\\hat \\theta_1, \\dots, \\hat \\theta_{100}$. Define $s_n$ to be the standard deviation of $\\{\\hat \\theta_i\\}_{i=1}^{100}$. Figures ref{fig:co2} and \\ref{fig:co3} show the qqplots of $\\tilde \\theta_i = (\\hat \\theta_i - \\theta_0)\/s_n$ against the standard normal for four different choices of $\\sigma_n = n^{-0.6}, n^{-0.7}, n^{-0.8}, n^{-0.9}$. \n\\begin{figure}\n\\centering \n\\includegraphics[scale=0.4]{Coordinate_2}\n\\caption{In this figure, we present qqplot for estimating second co-ordinate of $\\theta_0$ with different choices of $\\sigma_n$ mentioned at the top of each plots.}\n\\label{fig:co2}\n\\end{figure}\n\\begin{figure}\n\\centering \n\\includegraphics[scale=0.4]{Coordinate_3}\n\\caption{In this figure, we present qqplot for estimating third co-ordinate of $\\theta_0$ with different choices of $\\sigma_n$ mentioned at the top of each plots.}\n\\label{fig:co3}\n\\end{figure}\nIt is evident that smaller value of $\\sigma_n$ yield a poor normal approximation. Although our theory shows that asymptotic normality holds as long as $n\\sigma_n \\to \\infty$, in practice we recommend choosing $\\sigma_n$ such that $n\\sigma_n \\ge 30$ for the central limit of theorem to take effect. \n\n\n\n\\begin{comment}\n\\section{Real data analysis}\n\\label{sec:real_data}\nWe illustrate our method using cross-country data on pollution (carbon-di-oxide), income and urbanization obtained from the World Development Indicators (WDI), World Bank (website?). The Environmental Kuznets Curve hypothesis (EKC henceforth), a popular and ongoing area of research in environmental economics, posits that at an initial stage of economic development pollution increases with economic growth, and then diminishes when society's priorities change, leading to an inverted U-shaped relation between income (measured via real GDP per capita) and pollution. The hypothesis has led to numerous empirical papers (i) testing the hypothesis (whether the relation is inverted U-shaped for countries\/regions of interest in the sample), (ii) exploring the threshold level of income (change point) at which pollution starts falling, as well as (iii) examining the countries\/regions which belong to the upward rising part versus the downward sloping part of the inverted U-shape, if at all. The studies have been performed using US state level data or cross-country data (e.g. \\cite{shafik1992economic}, \\cite{millimet2003environmental}, \\cite{aldy2005environmental}, \\cite{lee2019nonparametric},\\cite{boubellouta2021cross}, \\cite{list1999environmental}, \\cite{grossman1995economic}, \\cite{bertinelli2005environmental}, \\cite{azomahou2006economic}, \\cite{taskin2000searching} to name a few) and most have found strong evidence in favor of the EKC hypothesis. While regressing pollution emission per capita on income and its squared term, most studies find a significantly positive effect of income and a significantly negative effect of its quadratic term on pollution, thus concluding in favor of the inverted U-shapedness of the relation.\n\\\\\\\\\n\\noindent\nWhile income-pollution remains the focal point of most EKC studies, several of them have also included urban agglomeration (UA) or some other measures of urbanization as an important control variable especially while investigating carbon emissions.\\footnote{Although income growth is connected to urbanization, they are different due to varying geographical area, population density, and infrastructure of the countries. Also, different countries follow different income growth paths \u2013 labor intensive manufacturing, technology based manufacturing, human capital based servicing, technology based service sector, or natural resource (oil) based growth owing to their differences in terms of location, ownership of natural resources and capital.} (see for example, \\cite{shafik1992economic}, \\cite{boubellouta2021cross}, \\cite{liang2019urbanization}). The ecological economics literature finds mixed evidence in this regard \u2013 (i) urbanization leading to more pollution (due to its close links with sanitation or dense transportations issues and proximities to polluting manufacturing industries), (ii) urbanization leading to less pollution (explained by 'compact city theory'). The 'compact city theory' (see \\cite{burton2000compact}, \\cite{capello2000beyond}, \\cite{sadorsky2014effect}) explains the benefits of increased urbanization in terms of economies of scale (for example, replacing dependence on automobiles with subway systems, the use of costlier but improved and green technology for basic infrastructure etc). \\cite{cole2004examining}, using a set of 86 countries, and \\cite{liddle2010age}, using 17 developed countries, find a positive and significant effect of urbanization on pollution. On the contrary, using a set of 69 countries \\cite{sharma2011determinants} find a negative and significant effect of urbanization on pollution while \\cite{du2012economic} find an insignificant effect of urbanization on carbon emission in China. Using various empirical strategies \\cite{sadorsky2014effect} conclude that the positive and negative effects of urbanization on carbon pollution may cancel out depending on the countries involved, often leaving insignificant effects on pollution.  In summary, based on the existing literature, the relationship between urbanization and pollution relation appears to depend largely on the set of countries considered in the sample. This motivates us to use UA along with income in our change plane model for analyzing carbon-dioxide emission to plausibly separate the countries into two regimes. \n\\\\\\\\\n\\noindent\nFollowing the broad literature we use pollution emission per capita (carbon-dioxide measured in metric tons per capita) as the dependent variable and real GDP per capita (measured in 2010 US dollars), its square (as is done commonly in the EKC literature) and a popular measure of urbanization, namely urban agglomeration  (UA)\\footnote{The exact definition can be found in WDI website} as covariates (in our notation $X$) in our regression. In light of the preceding discussions we fit a change plane model comprising real GDP per capita and UA (in our notation $Q$). To summarize the setup, we use the continuous response model as described in equation \\eqref{eq:regression_main_eqn}, i.e \n\\begin{align*}\nY_i & = X_i^{\\top}\\beta_0 + X_i^{\\top}\\delta_0\\mathds{1}_{Q_i^{\\top}\\psi_0 > 0} + {\\epsilon}_i \\\\\n& = X_i^{\\top}\\beta_0\\mathds{1}_{Q_i^{\\top}\\psi_0 \\le 0} + X_i^{\\top}(\\beta_0 + \\delta_0)\\mathds{1}_{Q_i^{\\top}\\psi_0 > 0} + {\\epsilon}_i\n\\end{align*}\nwith the per capita $CO_2$ emission in metric ton as $Y$, per capita GDP, square of per capita GDP and UA as $X$ (hence $X \\in \\mathbb{R}^3$) and finally, per capita GDP and UA as $Q$ (hence $Q \\in \\mathbb{R}^2$). Observe that $\\beta_0$ represents the regression coefficients corresponding to the countries with $Q_i^{\\top}\\psi_0 \\le 0$ (henceforth denoted by Group 1) and $(\\beta_0+ \\delta_0)$ represents the regression coefficients corresponding to the countries with $Q_i^{\\top}\\psi_0 \\ge 0$ (henceforth denoted by Group 2). As per our convention, in the interests of identifiability we assume $\\psi_{0, 1} = 1$, where $\\psi_{0,1}$ is the change plane parameter corresponding to per capita GDP. Therefore the only change plane coefficient to be estimated is $\\psi_{0, 2}$, the change plane coefficient for UA. For numerical stability, we divide per capita GDP by $10^{-4}$ (consequently square of per capital GDP is scaled by $10^{-8}$)\\footnote{This scaling helps in the numerical stability of the gradient descent algorithm used to optimize the least squares criterion.}.  After some pre-processing (i.e. removing rows consisting of NA and countries with $100\\%$ UA) we estimate the coefficients $(\\beta_0, \\delta_0, \\psi_0)$ of our model based on data from 115 countries with $\\sigma_n = 0.05$ and test the significance of the various coefficients using the methodologies described in Section \\ref{sec:inference}. We present our findings in Table \\ref{tab:ekc_coeff}.  \n\\begin{table}[!h]\n    \\centering\n    \\begin{tabular}{|c||c||c|}\n    \\hline\n    Coefficients & Estimated values & p-values \\\\\n    \\hline \\hline \n      $\\beta_{0, 1}$ (\\text{RGDPPC for Group 1})  & 6.98555060 & 4.961452e-10   \\\\\n       $\\beta_{0, 2}$ (\\text{squared RGDPPC for Group 1})  &  -0.43425991 & 7.136484e-02 \\\\\n       $\\beta_{0, 3}$ (\\text{UA for Group 1}) & -0.02613813  & 1.066065e-01\n\\\\\n       $\\beta_{0, 1} + \\delta_{0, 1}$ (\\text{RGDPPC for Group 2}) & 2.0563337 & 0.000000e+00\\\\\n      $\\beta_{0, 2} + \\delta_{0, 2}$ (\\text{squared RGDPPC for Group 2}) &  -0.1866490 & 4.912843e-04   \\\\\n       $\\beta_{0, 3} + \\delta_{0, 3}$ (\\text{UA for Group 2}) & 0.1403171& 1.329788e-05 \\\\\n       $\\psi_{0,2 }$  (\\text{Change plane coeff for UA}) & -0.07061785 & 0.000000e+00\\\\\n         \\hline\n    \\end{tabular}\n    \\caption{Table of the estimated regression and change plane coefficients along with their p-values.}\n    \\label{tab:ekc_coeff}\n\\end{table}\n\\\\\\\\\n\\noindent\nFrom the above analysis, we find that both GDP and its square have statistically significant effects on carbon pollution for both groups of countries with their expected signs (positive for GDP and negative for its square term), supporting the inverted U-shaped income-pollution nexus and being in line with most papers in the EKC literature.  The urban variable, on the other hand, is seen to have insignificant effect on Group 1 countries  (less developed and emerging) which is in keeping with \\cite{du2012economic}, \\cite{sadorsky2014effect}. However, the urban variable seems to have a positive and significant impact on Group-2 countries which is in line with \\cite{liddle2010age} for example. Note that many of the group-1 countries are yet to experience sizeable levels of urbanization compared to the other group and this is even truer for our sample period.\\footnote{We use 6 years average from 2010-2015 for GDP and pollution measures. Such averaging is in accordance with the cross-sectional empirical literature using cross-country\/regional data and helps avoid business cycle fluctuations in GDP. It also minimizes the impacts of outlier events such as the financial crisis or great recession period.  The years that have we have chosen are ones for which we could find data for the largest number of countries.}Further, note that UA plays a crucial role in dividing the countries into different regimes, as the estimated value of $\\psi_{0,2}$ is significant.   \n\\\\\\\\\n\\noindent\nThere are many future potential applications of our method in economics. Similar exercises can be followed for other pollutants (such as sulfur emission, electrical waste\/e-waste, nitrogen pollution etc.). While income\/GDP remains a common, indeed the most crucial variable in the pollution study, other covariates (including change plane defining variables) may vary, depending on the pollutant of interest. One may also be interested in the determinants of health expenses in the household survey data. Often the families are asked about their health expenses incurred in the past one year. An interesting case in point may be household surveys collected in India where there are many large joint families with several children and old people live in the same household and most families are uninsured. It is often seen that health expenses increases with income as it includes expenses on regularly performed preventative medical examinations which are affordable only beyond an income level is reached. The important covariates are per capita family income, family wealth, 'dependency ratio' (number of children and old to the total number of people in the family) and binary indicator of any history of major illness\/hospitalizations in the family in the past year. Family income per capita and history of major illness can potentially define the change plane.\n\\end{comment}\n\n\n\\input{Real_data_analysis.tex}\n\\section{Conclusion}\n\\label{sec:conclusion}\nIn this paper we have established that under some mild assumptions the kernel-smoothed change plane estimator is asymptotically normal with near optimal rate $n^{-1}$. To the best of our knowledge, the state of the art result in this genre of problems is due to \\cite{seo2007smoothed}, where they demonstrate a best possible rate about $n^{-3\/4}$ for i.i.d. data. The main difference between their approach and ours is mainly the proof of Lemma \\ref{bandwidth}. Our techniques are based upon modern empirical process theory which allow us to consider much smaller bandwidths $\\sigma_n$ compared to those in \\cite{seo2007smoothed}, who appear to require larger values to achieve the result, possibly owing to their reliance on the techniques developed in \\cite{horowitz1992smoothed}. Although we have established it is possible to have asymptotic normality with really small bandwidths, we believe that the finite sample approximation (e.g. Berry-Essen bound) to normality could be poor, which is also evident from our simulation. \n\n\n\n\n\n\n\\section{Real data analysis}\n\\label{sec:real_data}\nWe illustrate our method using cross-country data on pollution (carbon-dioxide), income and urbanization obtained from the World Development Indicators (WDI), World Bank. The Environmental Kuznets Curve hypothesis (EKC henceforth), a popular and ongoing area of research in environmental economics, posits that at an initial stage of economic development pollution increases with economic growth, and then diminishes when society's priorities change, leading to an inverted U-shaped relation between income (measured via real GDP per capita) and pollution. The hypothesis has led to numerous empirical papers (i) testing the hypothesis (whether the relation is inverted U-shaped for countries\/regions of interest in the sample), (ii) exploring the threshold level of income at which pollution starts falling, as well as (iii) examining the countries\/regions which belong to the upward rising part versus the downward sloping part of the inverted U-shape, if at all. The studies have been performed using US state level data or cross-country data (e.g. \\cite{shafik1992economic}, \\cite{millimet2003environmental}, \\cite{aldy2005environmental}, \\cite{lee2019nonparametric},\\cite{boubellouta2021cross}, \\cite{list1999environmental}, \\cite{grossman1995economic}, \\cite{bertinelli2005environmental}, \\cite{azomahou2006economic}, \\cite{taskin2000searching} to name a few). While some of these papers have found evidence in favor of the EKC hypothesis (inverted U-shaped income-pollution relation), others have found evidence against it (monotonically increasing or other shapes for the relation). The results often depend on countries\/regions in the sample, period of analysis, as well as the pollutant studied.\n\\\\\\\\\n\\noindent\nWhile income-pollution remains the focal point of most EKC studies, several of them have also included urban agglomeration (UA) or some other measures of urbanization as an important control variable especially while investigating carbon emissions.\\footnote {Although income growth is connected to urbanization, countries are heterogenous and follow different growth paths due to their varying geographical structures, population densities, infrastructures, ownerships of resources making a case for using urbanization as another control covariate in the income-pollution study. The income growth paths of oil rich UAE, manufacturing based China, serviced based Singapore, low population density Canada (with vast land) are all different.} (see for example, \\cite{shafik1992economic}, \\cite{boubellouta2021cross}and \\cite{liang2019urbanization}). The theory of ecological economics posits potentially varying effects of increased urbanization on pollution\u2013 (i) urbanization leading to more pollution (due to its close links with sanitations, dense transportations, and proximities to polluting manufacturing industries), (ii) urbanization potentially leading to less pollution based on 'compact city theory' (see \\cite{burton2000compact}, \\cite{capello2000beyond}, \\cite{sadorsky2014effect}) that explains the potential benefits of increased urbanization in terms of economies of scale (for example, replacing dependence on automobiles with large scale subway systems, using multi-storied buildings instead of single unit houses, keeping more open green space). \\cite{liddle2010age}, using 17 developed countries, find a positive and significant effect of urbanization on pollution. On the contrary, using a set of 69 countries \\cite{sharma2011determinants} find a negative and significant effect of urbanization on pollution while \\cite{du2012economic} find an insignificant effect of urbanization on carbon emission. Using various empirical strategies \\cite{sadorsky2014effect} conclude that the positive and negative effects of urbanization on carbon pollution may cancel out depending on the countries involved often leaving insignificant effects on pollution. They also note that many countries are yet to achieve a sizeable level of urbanization which presumably explains why many empirical works using less developed countries find insignificant effect of urbanization.  In summary, based on the existing literature, both the relationship between urbanization and pollution as well as the relationship between income and pollution appear to depend largely on the set of countries considered in the sample. This motivates us to use UA along with income in our change plane model for analyzing carbon-dioxide emission to plausibly separate the countries into two regimes. \n\\\\\\\\\n\\noindent\nFollowing the broad literature we use pollution emission per capita (carbon-dioxide measured in metric tons per capita) as the dependent variable and real GDP per capita (measured in 2010 US dollars), its square (as is done commonly in the EKC literature) and a popular measure of urbanization, namely urban agglomeration  (UA)\\footnote{The exact definition can be found in the World Development Indicators database from the World Bank website.} as covariates (in our notation $X$) in our regression. In light of the preceding discussions we fit a change plane model comprising real GDP per capita and UA (in our notation $Q$). To summarize the setup, we use the continuous response model as described in equation \\eqref{eq:regression_main_eqn}, i.e \n\\begin{align*}\nY_i & = X_i^{\\top}\\beta_0 + X_i^{\\top}\\delta_0\\mathds{1}_{Q_i^{\\top}\\psi_0 > 0} + {\\epsilon}_i \\\\\n& = X_i^{\\top}\\beta_0\\mathds{1}_{Q_i^{\\top}\\psi_0 \\le 0} + X_i^{\\top}(\\beta_0 + \\delta_0)\\mathds{1}_{Q_i^{\\top}\\psi_0 > 0} + {\\epsilon}_i\n\\end{align*}\nwith the per capita $CO_2$ emission in metric ton as $Y$, per capita GDP, square of per capita GDP and UA as $X$ (hence $X \\in \\mathbb{R}^3$) and finally, per capita GDP and UA as $Q$ (hence $Q \\in \\mathbb{R}^2$). Observe that $\\beta_0$ represents the regression coefficients corresponding to the countries with $Q_i^{\\top}\\psi_0 \\le 0$ (henceforth denoted by Group 1) and $(\\beta_0+ \\delta_0)$ represents the regression coefficients corresponding to the countries with $Q_i^{\\top}\\psi_0 \\ge 0$ (henceforth denoted by Group 2). As per our convention, in the interests of identifiability we assume $\\psi_{0, 1} = 1$, where $\\psi_{0,1}$ is the change plane parameter corresponding to per capita GDP. Therefore the only change plane coefficient to be estimated is $\\psi_{0, 2}$, the change plane coefficient for UA. For numerical stability, we divide per capita GDP by $10^{-4}$ (consequently square of per capital GDP is scaled by $10^{-8}$)\\footnote{This scaling helps in the numerical stability of the gradient descent algorithm used to optimize the least squares criterion.}.  After some pre-processing (i.e. removing rows consisting of NA and countries with $100\\%$ UA) we estimate the coefficients $(\\beta_0, \\delta_0, \\psi_0)$ of our model based on data from 115 countries with $\\sigma_n = 0.05$ and test the significance of the various coefficients using the methodologies described in Section \\ref{sec:inference}. We present our findings in Table \\ref{tab:ekc_coeff}.  \n\\begin{table}[!h]\n    \\centering\n    \\begin{tabular}{|c||c||c|}\n    \\hline\n    Coefficients & Estimated values & p-values \\\\\n    \\hline \\hline \n      $\\beta_{0, 1}$ (\\text{RGDPPC for Group 1})  & 6.98555060 & 4.961452e-10   \\\\\n       $\\beta_{0, 2}$ (\\text{squared RGDPPC for Group 1})  &  -0.43425991 & 7.136484e-02 \\\\\n       $\\beta_{0, 3}$ (\\text{UA for Group 1}) & -0.02613813  & 1.066065e-01\n\\\\\n       $\\beta_{0, 1} + \\delta_{0, 1}$ (\\text{RGDPPC for Group 2}) & 2.0563337 & 0.000000e+00\\\\\n      $\\beta_{0, 2} + \\delta_{0, 2}$ (\\text{squared RGDPPC for Group 2}) &  -0.1866490 & 4.912843e-04   \\\\\n       $\\beta_{0, 3} + \\delta_{0, 3}$ (\\text{UA for Group 2}) & 0.1403171& 1.329788e-05 \\\\\n       $\\psi_{0,2 }$  (\\text{Change plane coeff for UA}) & -0.07061785 & 0.000000e+00\\\\\n         \\hline\n    \\end{tabular}\n    \\caption{Table of the estimated regression and change plane coefficients along with their p-values.}\n    \\label{tab:ekc_coeff}\n\\end{table}\n\\\\\\\\\n\\noindent\nFrom the above analysis, we find that GDP has significantly positive effect on pollution for both groups of countries. The effect of its squared term is negative for both groups; but the effect is significant for Group-2 consisting of mostly high income countries whereas its effect is insignificant (at the 5\\% level) for the Group-1 countries (consisting of mostly low or middle income and few high income countries). Thus, not surprisingly, we find evidence in favor of EKC for the developed countries, but not for the mixed group.  Notably, Group-1 consists of a mixed set of countries like Angola, Sudan, Senegal, India, China, Israel, UAE etc., whereas Group-1 consists of rich and developed countries like Canada, USA, UK, France, Germany etc. The urban variable, on the other hand, is seen to have insignificant effect on Group-1 which is in keeping with \\cite{du2012economic}, \\cite{sadorsky2014effect}. Many of them are yet to achieve substantial urbanization and this is more true for our sample period \\footnote{We use 6 years average from 2010-2015 for GDP and pollution measures. Such averaging is in accordance with the cross-sectional empirical literature using cross-country\/regional data and helps avoid business cycle fluctuations in GDP. It also minimizes the impacts of outlier events such as the financial crisis or great recession period.  The years that we have chosen are ones for which we could find data for the largest number of countries.}. In contrast, UA has a positive and significant effect on Group-2 (developed) countries which is consistent with the findings of \\cite{liddle2010age}, for example. Note that UA plays a crucial role in dividing the countries into different regimes, as the estimated value of $\\psi_{0,2}$ is significant. Thus, we are able to partition countries into two regimes: a mostly rich and a mixed group. \n\\\\\\\\\n\\noindent\nNote that many underdeveloped countries and poorer regions of emerging countries are still swamped with greenhouse gas emissions from burning coal, cow dung etc., and usage of poor exhaust systems in houses and for transport. This is more true for rural and semi-urban areas of developing countries. So even while being less urbanized compared to developed nations, their overall pollution load is high (due to inefficient energy usage and higher dependence on fossil fuels as pointed out above) and rising with income and they are yet to reach the descending part of the inverted U-shape for the income-pollution relation. On the contrary, for countries in Group-2, the adoption of more efficient energy and exhaust systems are common in households and transportations in general, leading to eventually decreasing pollution with increasing income (supporting EKC). Both the results are in line with the existing EKC literature. Additionally we find that the countries in Group 2 are yet to achieve 'compact city' and green urbanization. This is a stylized fact that is confirmed by the positive and significant effect of UA on pollution in our analysis.  \n\\\\\\\\\n\\noindent\nThere are many future potential applications of our method in economics. Similar analyses can be performed for other pollutants (such as sulfur emission, electrical waste\/e-waste, nitrogen pollution etc.). While income\/GDP remains a common, indeed the most crucial variable in pollution studies, other covariates (including change plane defining variables) may vary, depending on the pollutant of interest. Another potential application can be that of identifying the determinants of family health expenses in household survey data. Families are often asked about their health expenses incurred in the past one year. An interesting case in point may be household surveys collected in India where one finds numerous (large) joint families with several children and old people residing in the same household and most families are uninsured. It is often seen that health expenditure increases with income with a major factor being the costs associated with regularly performed preventative medical examinations which are affordable only once a certain income level is reached. The important covariates here are per capita family income, family wealth, `dependency ratio' (number of children and old to the total number of people in the family) and the binary indicator of any history of major illness\/hospitalizations in the family in the past year. Family income per capita and history of major illness are natural candidate covariates for defining the change plane. \n\n\\section{Binary response model}\n\\label{sec:classification_analysis}\nRecall our binary response model in equation \\eqref{eq:classification_eqn}. To estimate $\\psi_0$, we resort to the following  loss (without smoothing): \n\\begin{equation}\n\\label{eq:new_loss}\n\\mathbb{M}(\\psi) = \\mathbb{E}\\left((Y - \\gamma)\\mathds{1}(Q^{\\top}\\psi \\le 0)\\right)\\end{equation}\nwith $\\gamma \\in (\\alpha_0, \\beta_0)$, which can be viewed as a variant of the square error loss function: \n$$\n\\mathbb{M}(\\alpha, \\beta, \\psi) = \\mathbb{E}\\left(\\left(Y - \\alpha\\mathds{1}(Q^{\\top}\\psi < 0) - \\beta\\mathds{1}(Q^{\\top}\\psi > 0)\\right)^2\\right)\\,.\n$$\nWe establish the connection between these losses in sub-section \\ref{loss_func_eq}. It is easy to prove that under fairly mild conditions (discussed later) \n$\\psi_0 = {\\arg\\min}_{\\psi \\in \\Theta}\\mathbb{M}(\\psi)$, uniquely. Under the standard classification paradigm, when we know a priori that \n$\\alpha_0 < 1\/2 < \\beta_0$, we can take $\\gamma = 1\/2$, and in the absence of this constraint, $\\bar{Y}$, which converges to some $\\gamma$ between $\\alpha_0$ and $\\beta_0$, may be substituted in the loss function. In the rest of the paper, we confine ourselves to a known $\\gamma$, and for technical simplicity, we take $\\gamma = \\frac{(\\beta_0 + \\alpha_0)}{2}$, but this assumption can be removed with more mathematical book-keeping. Thus, $\\psi_0$ is estimated by:  \n\\begin{equation}\n\\label{non-smooth-score} \n\\hat \\psi = {\\arg\\min}_{\\psi \\in \\Theta} \\mathbb{M}_n(\\psi) = {\\arg\\min}_{\\psi \\in \\Theta} \\frac{1}{n}\\sum_{\\i=1}^n (Y_i - \\gamma)\\mathds{1}(Q_i^{\\top}\\psi \\le 0)\\,.\n\\end{equation} We resort to a smooth approximation of the indicator function in \n\\eqref{non-smooth-score} using a distribution kernel with suitable bandwidth. The smoothed version of the population score function then becomes: \n\\begin{equation}\n\\label{eq:kernel_smoothed_pop_score}\n\\mathbb{M}^s(\\psi) = \\mathbb{E}\\left((Y - \\gamma)\\left(1-K\\left(\\frac{Q^{\\top}\\psi}{\\sigma_n}\\right)\\right)\\right)\n\\end{equation}\nwhere as in the continuous response model, we use $K(x) = \\Phi(x)$, and the corresponding empirical version is: \n\\begin{equation}\n\\label{eq:kernel_smoothed_emp_score}\n\\mathbb{M}^s_n(\\psi) = \\frac{1}{n}\\sum_{i=1}^n \\left((Y_i - \\gamma)\\left(1-K\\left(\\frac{Q_i^{\\top}\\psi}{\\sigma_n}\\right)\\right)\\right)\n\\end{equation}\nDefine $\\hat{\\psi}^s$ and $\\psi_0^s$ to be the minimizer of the smoothed version of the empirical (equation \\eqref{eq:kernel_smoothed_emp_score}) and population score (equation \\eqref{eq:kernel_smoothed_pop_score}) function respectively. Here we only consider the choice of bandwidth $n\\sigma_n \\to \\infty$ and $n\\sigma_n^2 \\to 0$. Analogous to Theorem \\ref{thm:regression} we prove the following result for binary response model: \n\\begin{theorem}\n\\label{thm:binary}\nUnder Assumptions (\\ref{as:distribution} - \\ref{as:eigenval_bound}): \n$$\n\\sqrt{\\frac{n}{\\sigma_n}}\\left(\\hat{\\psi}_n - \\psi_0\\right) \\Rightarrow N(0, \\Gamma) \\,,\n$$ \nfor some non-stochastic matrix $\\Gamma$, which will be defined explicitly in the proof. \n\\end{theorem}\nWe have therefore established that in the regime $n\\sigma_n \\to \\infty$ and $n\\sigma_n^2 \\to 0$, it is possible to attain asymptotic normality using a smoothed estimator for binary response model. \n\n\n\n\n\n\\section{Inferential methods}\n\\label{sec:inference}\nWe draw inferences on $(\\beta_0, \\delta_0, \\psi_0)$ by resorting to similar techniques as in \\cite{seo2007smoothed}. For the continuous response model, we need consistent estimators of $V^{\\gamma}, Q^{\\gamma}, V^{\\psi}, Q^{\\psi}$ (see Lemma \\ref{conv-prob} for the definitions) for hypothesis testing. By virtue of the aforementioned Lemma, we can estimate $Q^{\\gamma}$ and $Q^{\\psi}$ as follows: \n\\begin{align*}\n\\hat Q^{\\gamma} & = \\nabla^2_{\\gamma} \\mathbb{M}_n^s(\\hat \\theta) \\,, \\\\ \n\\hat Q^{\\psi} & = \\sigma_n \\nabla^2_{\\psi} \\mathbb{M}_n^s(\\hat \\theta) \\,.\n\\end{align*}\nThe consistency of the above estimators is established in the proof of Lemma \\ref{conv-prob}. For the other two parameters $V^{\\gamma}, V^{\\psi}$ we use the following estimators: \n\\begin{align*}\n\\hat V^{\\psi} & = \\frac{1}{n\\sigma_n^2}\\sum_{i=1}^n\\left(\\left(Y_i - X_i^{\\top}(\\hat \\beta + \\hat \\delta)\\right)^2 - \\left(Y_i- X_i^{\\top}\\hat \\beta\\right)^2\\right)^2\\tilde Q_i \\tilde Q_i^{\\top}\\left(K'\\left(\\frac{Q_i^{\\top}\\hat \\psi}{\\sigma_n}\\right)\\right)^2 \\\\\n\\hat V^{\\gamma} & = \\hat \\sigma^2_{\\epsilon} \\begin{pmatrix} \\frac{1}{n}X_iX_i^{\\top} & \\frac{1}{n}X_iX_i^{\\top}\\mathds{1}_{Q_i^{\\top}\\hat \\psi > 0} \\\\ \\frac{1}{n}X_iX_i^{\\top}\\mathds{1}_{Q_i^{\\top}\\hat \\psi > 0} & \\frac{1}{n}X_iX_i^{\\top}\\mathds{1}_{Q_i^{\\top}\\hat \\psi > 0} \\end{pmatrix}\n\\end{align*}\nwhere $\\hat \\sigma^2_{\\epsilon}$ can be obtained as $(1\/n)(Y_i - X_i^{\\top}\\hat \\beta - X_i^{\\top}\\hat \\delta \\mathds{1}(Q_i^{\\top}\\hat \\psi > 0))^2$, i.e. the residual sum of squares. The explicit value of $V_\\gamma$ (as derived in equation \\eqref{eq:def_v_gamma} in the proof Lemma \\ref{asymp-normality}) is: \n$$\nV^{\\gamma} = \\sigma_{\\epsilon}^2 \\begin{pmatrix}\\mathbb{E}\\left[XX^{\\top}\\right] & \\mathbb{E}\\left[XX^{\\top}\\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right] \\\\\n\\mathbb{E}\\left[XX^{\\top}\\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right] & \\mathbb{E}\\left[XX^{\\top}\\mathds{1}_{Q^{\\top}\\psi_0 > 0}\\right] \\end{pmatrix} \n$$ \nTherefore, the consistency of $\\hat V_\\gamma$ is immediate from the law of large numbers. The consistency of $\\hat V^{\\psi}$ follows via arguments similar to those employed in proving Lemma \\ref{conv-prob} but under somewhat more stringent moment conditions: in particular, we need $\\mathbb{E}[\\|X\\|^8] < \\infty$ and $\\mathbb{E}[(X^{\\top}\\delta_0)^k \\mid Q]$ to be Lipschitz functions over $Q$ for $1 \\le k \\le 8$. The inferential techniques for the classification model are similar and hence skipped, to avoid repetition. \n\n\n\n\n\n\n\n\\section{Introduction}\nThe simple linear regression model assumes a uniform linear relationship between the covariate and the response, in the sense that the regression parameter $\\beta$ is the same over the entire covariate domain. In practice, the situation can be more complicated: for instance, the regression parameter may differ from sub-population to sub-population within a large (super-) population. Some common techniques to account for such heterogeneity include mixed linear models, introducing an interaction effect, or fitting different models among each sub-population which corresponds to a supervised classification setting where the true groups (sub-populations) are \\emph{a priori known}. \n\t\\newline\n\t\\newline\n\t\\indent A more difficult scenario arises when the sub-populations are unknown, in which case regression and classification must happen simultaneously. Consider the scenario where the conditional mean of $Y_i$ given $X_i$ is different for different unknown sub-groups. A  well-studied treatment of this problem -- the so-called change point problem -- considers a simple thresholding model where membership in a sub-group is determined by whether a real-valued observable $X$ falls to the left or right of an unknown parameter $\\gamma$. More recently, there has been work for multi-dimensional covariates, namely when the membership is determined by which side a random vector $X$ falls with respect to an hyperplane with unknown normal vector $\\theta_0$. A concrete example appears in \\cite{wei2014latent} who extend the linear thresholding model due to  \\cite{kang2011new} to general dimensions: \n\t\\begin{eqnarray}\\label{eq:weimodel}\n\tY=\\mu_1\\cdot 1_{X^{\\top}\\theta_0\\geq 0}+\\mu_2\\cdot 1_{X^{\\top}\\theta_0<0}+\\varepsilon\\,,\n\t\\end{eqnarray}\n\tand studied computational algorithms and consistency of the same. This model and others with similar structure, called \\emph{change plane models}, are useful in various fields of research, e.g. modeling treatment effect heterogeneity in drug treatment (\\cite{imai2013estimating}), modeling sociological data on voting and employment (\\cite{imai2013estimating}), or cross country growth regressions in econometrics \n(\\cite{seo2007smoothed}).\n\t\\newline\n\t\\newline\n\\indent Other aspects of this model have also been investigated. \\cite{fan2017change} examined the change plane model from the statistical testing point of view, with the null hypothesis being the absence of a separating hyperplane. They proposed a test statistic, studied its asymptotic distribution and provided sample size recommendations for achieving target values of power. \\cite{li2018multi} extended the change point detection problem in the multi-dimensional setup by considering the case where $X^{\\top}\\theta_0$ forms a multiple change point data sequence. \n\nThe key difficultly with change plane type models is the inherent discontinuity in the optimization criteria involved where the parameter of interest appears as an argument to some indicator function, rendering the optimization extremely hard. To alleviate this, one option is to kernel smooth the indicator function, an approach that was adopted by Seo and Linton \\cite{seo2007smoothed} in a version of the change-plane problem, motivated by earlier results of Horowitz \\cite{horowitz1992smoothed} that dealt with a smoothed version of the maximum score estimator. Their model has an additive structure of the form:\n\\[Y_t = \\beta^{\\top}X_t + \\delta^{\\top} \\tilde{X}_t \\mathds{1}_{Q_t^{\\top} \\boldmath \\psi > 0} + \\epsilon_t \\,,\\]\nwhere $\\psi$ is the (fixed) change-plane parameter, and $t$ can be viewed as a time index. Under a set of assumptions on the model (Assumptions 1 and 2 of their paper), they showed asymptotic normality of their estimator of $\\psi$ obtained by minimizing a smoothed least squares criterion\nthat uses a differentiable distribution function $\\mathcal{K}$. The rate of convergence of $\\hat{\\psi}$ to the truth was shown to be $\\sqrt{n\/\\sigma_n}$ where $\\sigma_n$ was the bandwidth parameter used to smooth the least squares function. As noted in their Remark 3, under the special case of i.i.d. observations, their requirement that $\\log n\/(n \\sigma_n^2) \\rightarrow 0$ translates to a maximal convergence rate of $n^{3\/4}$ up to a logarithmic factor. The work of \\cite{li2018multi} who considered multiple parallel change planes (determined by a fixed dimensional normal vector) and high dimensional linear models in the regions between consecutive hyperplanes also builds partly upon the methods of \\cite{seo2007smoothed} and obtains the same (almost) $n^{3\/4}$ rate for the normal vector (as can be seen by putting Condition 6 in their paper in conjunction with the conclusion of Theorem 3). \n\\\\\\\\\n\nWhile it is established that the condition $n\\sigma_n^2 \\to \\infty$ is sufficient (upto a log factor) for achieving asymptotic normality of the smoothed estimator, there is no result in the existing literature to ascertain whether its necessity. Intuitively speaking, the necessary condition for asymptotic normality ought to be $n \\sigma_n \\to 0$, as this will ensure a growing number of observations in a $\\sigma_n$ neighborhood around the true hyperplane, allowing the central limit theorem to kick in. In this paper we \\emph{bridge this gap} by proving that asymptotic normality of the smoothed change point estimator is, in fact, achievable with $n \\sigma_n \\to \\infty$. \nThis implies that the best possible rate of convergence of the smoothed estimator can be arbitrarily close to $n^{-1}$, the minimax optimal rate of estimation for this problem. To demonstrate this, we focus on two change plane estimation problems, one with a continuous and another with a binary response. The continuous response model we analyze here is the following: \n\\begin{equation}\n\\label{eq:regression_main_eqn}\nY_i = \\beta_0^{\\top}X_i + \\delta_0^{\\top}X_i\\mathds{1}_{Q_i^{\\top}\\psi_0 > 0} + {\\epsilon}_i \\,.\n\\end{equation}\nfor i.i.d. observations $\\{(X_i, Y_i, Q_i\\}_{i=1}^n$, where the zero-mean transitory shocks ${\\epsilon}_i \\rotatebox[origin=c]{90}{$\\models$} (X_i, Q_i)$. Our calculation can be easily extended to the case when the covariates on the either side of the change hyperplane are different and $\\mathbb{E}[{\\epsilon} \\mid X, Q] = 0$ with more tedious bookkeeping. As this generalization adds little of interest, conceptually, to our proof, we posit the simpler model for ease of understanding.\nAs the parameter $\\psi_0$ is only identifiable upto its norm, we assume that the first co-ordinate is $1$ (along the lines of \\cite{seo2007smoothed}) which removes one degree of freedom and makes the parameter identifiable. \n\\\\\\\\\nTo illustrate that a similar phenomenon transpires with binary response, we also study a canonical version of such a model which can be briefly described as follows: The covariate $Q \\sim P$ where $P$ is distribution on $\\mathbb{R}^d$ and the conditional distribution of $Y$ given $Q$ is modeled as follows: \n\\begin{equation}\n\\label{eq:classification_eqn}\nP(Y=1|Q) = \\alpha_0 \\mathds{1}(Q^{\\top}\\psi_0 \\le 0) + \\beta_0\\mathds{1}(Q^{\\top}\\psi_0 > 0)\n\\end{equation}\nfor some parameters $\\alpha_0, \\beta_0\\in (0,1)$ and $\\psi_0\\in\\mathbb{R}^d$ (with first co-ordinate being one for identifiability issue as for the continuous response model), the latter being of primary interest for estimation. \nThis model is identifiable up to a permutation of $(\\alpha_0, \\beta_0)$, so we further assume $\\alpha_0 < \\beta_0$. For both models, we show that $\\sqrt{n\/\\sigma_n}(\\hat \\psi - \\psi_0)$ converges to zero-mean normal distribution as long as $n \\sigma_n \\to \\infty$ but the calculations for the binary model are completely relegated to Appendix \\ref{sec:supp_classification}. \n\\\\\\\\\n{\\bf Organization of the paper:} The rest of the paper is organized as follows: In Section \\ref{sec:theory_regression} we present the methodology, the statement of the asymptotic distributions and a sketch of the proof for the continuous response model \\eqref{eq:regression_main_eqn}. In Section \\ref{sec:classification_analysis} we briefly describe the binary response model \\eqref{eq:classification_eqn} and related assumptions, whilst the details can be found in the supplementary document. In Section \\ref{sec:simulation} we present some simulation results, both for the binary and the continuous response models to study the effect of the bandwidth on the quality of the normal approximation in finite samples. In Section \\ref{sec:real_data}, we present a real data analysis where we analyze the effect of  income and urbanization on the $CO_2$ emission in different countries. \n\\\\\\\\\n{\\bf Notations: } Before delving into the technical details, we first setup some notations here. We assume from now on, $X \\in {\\bbR}^p$ and $Q \\in {\\bbR}^d$. For any vector $v$ we define by $\\tilde v$ as the vector with all the co-ordinates expect the first one. We denote by $K$ the kernel function used to smooth the indicator function. For any matrix $A$, we denote by $\\|A\\|_2$ (or $\\|A\\|_F$) as its Frobenious norm and $\\|A\\|_{op}$ as its operator norm. For any vector, $\\| \\cdot \\|_2$ denotes its $\\ell_2$ norm.  \n\n\n\n\n\n\n\\input{regression.tex}\n\\input{classification.tex}\n\n\\section{Simulation studies}\n\\label{sec:simulation}\nIn this section, we present some simulation results to analyse the effect of the choice of $\\sigma_n$ on the finite sample approximation of asymptotic normality, i.e. Berry-Essen type bounds. If we choose a smaller sigma, the rate of convergence is accelerated but the normal approximation error at smaller sample sizes will be higher, as we don't have enough observations in the vicinity of the change hyperplane for the CLT to kick in. This problem is alleviated by choosing $\\sigma_n$ larger, but this, on the other hand, compromises the convergence rate. Ideally, a Berry-Essen type of bound will quantify this, but this will require a different set of techniques and is left as an open problem. In our simulations, we generate data from following setup: \n\\begin{enumerate}\n\\item Set $N = 50000, p = 3, \\alpha_0 = 0.25, \\beta = 0.75$ and some $\\theta_0 \\in \\mathbb{R}^p$ with first co-ordinate $ = 1$. \n\\item Generate $X_1, \\dots, X_n \\sim \\mathcal{N}(0, I_p)$. \n\\item Generate $Y_i \\sim \\textbf{Bernoulli}\\left(\\alpha_0\\mathds{1}_{X_i^{\\top}\\theta_0 \\le 0} + \\beta_0 \\mathds{1}_{X_i^{\\top}\\theta_0 > 0}\\right)$. \n\\item Estimate $\\hat \\theta$ by minimizing $\\mathbb{M}_n(\\theta)$ (replacing $\\gamma$ by $\\bar Y$) based on $\\{(X_i, Y_i)\\}_{i=1}^n$ for different choices of $\\sigma_n$. \n\\end{enumerate}\nWe repeat Step 2 - Step 4 a hundred times to obtain $\\hat \\theta_1, \\dots, \\hat \\theta_{100}$. Define $s_n$ to be the standard deviation of $\\{\\hat \\theta_i\\}_{i=1}^{100}$. Figures ref{fig:co2} and \\ref{fig:co3} show the qqplots of $\\tilde \\theta_i = (\\hat \\theta_i - \\theta_0)\/s_n$ against the standard normal for four different choices of $\\sigma_n = n^{-0.6}, n^{-0.7}, n^{-0.8}, n^{-0.9}$. \n\\begin{figure}\n\\centering \n\\includegraphics[scale=0.4]{Coordinate_2}\n\\caption{In this figure, we present qqplot for estimating second co-ordinate of $\\theta_0$ with different choices of $\\sigma_n$ mentioned at the top of each plots.}\n\\label{fig:co2}\n\\end{figure}\n\\begin{figure}\n\\centering \n\\includegraphics[scale=0.4]{Coordinate_3}\n\\caption{In this figure, we present qqplot for estimating third co-ordinate of $\\theta_0$ with different choices of $\\sigma_n$ mentioned at the top of each plots.}\n\\label{fig:co3}\n\\end{figure}\nIt is evident that smaller value of $\\sigma_n$ yield a poor normal approximation. Although our theory shows that asymptotic normality holds as long as $n\\sigma_n \\to \\infty$, in practice we recommend choosing $\\sigma_n$ such that $n\\sigma_n \\ge 30$ for the central limit of theorem to take effect. \n\n\n\n\\begin{comment}\n\\section{Real data analysis}\n\\label{sec:real_data}\nWe illustrate our method using cross-country data on pollution (carbon-di-oxide), income and urbanization obtained from the World Development Indicators (WDI), World Bank (website?). The Environmental Kuznets Curve hypothesis (EKC henceforth), a popular and ongoing area of research in environmental economics, posits that at an initial stage of economic development pollution increases with economic growth, and then diminishes when society's priorities change, leading to an inverted U-shaped relation between income (measured via real GDP per capita) and pollution. The hypothesis has led to numerous empirical papers (i) testing the hypothesis (whether the relation is inverted U-shaped for countries\/regions of interest in the sample), (ii) exploring the threshold level of income (change point) at which pollution starts falling, as well as (iii) examining the countries\/regions which belong to the upward rising part versus the downward sloping part of the inverted U-shape, if at all. The studies have been performed using US state level data or cross-country data (e.g. \\cite{shafik1992economic}, \\cite{millimet2003environmental}, \\cite{aldy2005environmental}, \\cite{lee2019nonparametric},\\cite{boubellouta2021cross}, \\cite{list1999environmental}, \\cite{grossman1995economic}, \\cite{bertinelli2005environmental}, \\cite{azomahou2006economic}, \\cite{taskin2000searching} to name a few) and most have found strong evidence in favor of the EKC hypothesis. While regressing pollution emission per capita on income and its squared term, most studies find a significantly positive effect of income and a significantly negative effect of its quadratic term on pollution, thus concluding in favor of the inverted U-shapedness of the relation.\n\\\\\\\\\n\\noindent\nWhile income-pollution remains the focal point of most EKC studies, several of them have also included urban agglomeration (UA) or some other measures of urbanization as an important control variable especially while investigating carbon emissions.\\footnote{Although income growth is connected to urbanization, they are different due to varying geographical area, population density, and infrastructure of the countries. Also, different countries follow different income growth paths \u2013 labor intensive manufacturing, technology based manufacturing, human capital based servicing, technology based service sector, or natural resource (oil) based growth owing to their differences in terms of location, ownership of natural resources and capital.} (see for example, \\cite{shafik1992economic}, \\cite{boubellouta2021cross}, \\cite{liang2019urbanization}). The ecological economics literature finds mixed evidence in this regard \u2013 (i) urbanization leading to more pollution (due to its close links with sanitation or dense transportations issues and proximities to polluting manufacturing industries), (ii) urbanization leading to less pollution (explained by 'compact city theory'). The 'compact city theory' (see \\cite{burton2000compact}, \\cite{capello2000beyond}, \\cite{sadorsky2014effect}) explains the benefits of increased urbanization in terms of economies of scale (for example, replacing dependence on automobiles with subway systems, the use of costlier but improved and green technology for basic infrastructure etc). \\cite{cole2004examining}, using a set of 86 countries, and \\cite{liddle2010age}, using 17 developed countries, find a positive and significant effect of urbanization on pollution. On the contrary, using a set of 69 countries \\cite{sharma2011determinants} find a negative and significant effect of urbanization on pollution while \\cite{du2012economic} find an insignificant effect of urbanization on carbon emission in China. Using various empirical strategies \\cite{sadorsky2014effect} conclude that the positive and negative effects of urbanization on carbon pollution may cancel out depending on the countries involved, often leaving insignificant effects on pollution.  In summary, based on the existing literature, the relationship between urbanization and pollution relation appears to depend largely on the set of countries considered in the sample. This motivates us to use UA along with income in our change plane model for analyzing carbon-dioxide emission to plausibly separate the countries into two regimes. \n\\\\\\\\\n\\noindent\nFollowing the broad literature we use pollution emission per capita (carbon-dioxide measured in metric tons per capita) as the dependent variable and real GDP per capita (measured in 2010 US dollars), its square (as is done commonly in the EKC literature) and a popular measure of urbanization, namely urban agglomeration  (UA)\\footnote{The exact definition can be found in WDI website} as covariates (in our notation $X$) in our regression. In light of the preceding discussions we fit a change plane model comprising real GDP per capita and UA (in our notation $Q$). To summarize the setup, we use the continuous response model as described in equation \\eqref{eq:regression_main_eqn}, i.e \n\\begin{align*}\nY_i & = X_i^{\\top}\\beta_0 + X_i^{\\top}\\delta_0\\mathds{1}_{Q_i^{\\top}\\psi_0 > 0} + {\\epsilon}_i \\\\\n& = X_i^{\\top}\\beta_0\\mathds{1}_{Q_i^{\\top}\\psi_0 \\le 0} + X_i^{\\top}(\\beta_0 + \\delta_0)\\mathds{1}_{Q_i^{\\top}\\psi_0 > 0} + {\\epsilon}_i\n\\end{align*}\nwith the per capita $CO_2$ emission in metric ton as $Y$, per capita GDP, square of per capita GDP and UA as $X$ (hence $X \\in \\mathbb{R}^3$) and finally, per capita GDP and UA as $Q$ (hence $Q \\in \\mathbb{R}^2$). Observe that $\\beta_0$ represents the regression coefficients corresponding to the countries with $Q_i^{\\top}\\psi_0 \\le 0$ (henceforth denoted by Group 1) and $(\\beta_0+ \\delta_0)$ represents the regression coefficients corresponding to the countries with $Q_i^{\\top}\\psi_0 \\ge 0$ (henceforth denoted by Group 2). As per our convention, in the interests of identifiability we assume $\\psi_{0, 1} = 1$, where $\\psi_{0,1}$ is the change plane parameter corresponding to per capita GDP. Therefore the only change plane coefficient to be estimated is $\\psi_{0, 2}$, the change plane coefficient for UA. For numerical stability, we divide per capita GDP by $10^{-4}$ (consequently square of per capital GDP is scaled by $10^{-8}$)\\footnote{This scaling helps in the numerical stability of the gradient descent algorithm used to optimize the least squares criterion.}.  After some pre-processing (i.e. removing rows consisting of NA and countries with $100\\%$ UA) we estimate the coefficients $(\\beta_0, \\delta_0, \\psi_0)$ of our model based on data from 115 countries with $\\sigma_n = 0.05$ and test the significance of the various coefficients using the methodologies described in Section \\ref{sec:inference}. We present our findings in Table \\ref{tab:ekc_coeff}.  \n\\begin{table}[!h]\n    \\centering\n    \\begin{tabular}{|c||c||c|}\n    \\hline\n    Coefficients & Estimated values & p-values \\\\\n    \\hline \\hline \n      $\\beta_{0, 1}$ (\\text{RGDPPC for Group 1})  & 6.98555060 & 4.961452e-10   \\\\\n       $\\beta_{0, 2}$ (\\text{squared RGDPPC for Group 1})  &  -0.43425991 & 7.136484e-02 \\\\\n       $\\beta_{0, 3}$ (\\text{UA for Group 1}) & -0.02613813  & 1.066065e-01\n\\\\\n       $\\beta_{0, 1} + \\delta_{0, 1}$ (\\text{RGDPPC for Group 2}) & 2.0563337 & 0.000000e+00\\\\\n      $\\beta_{0, 2} + \\delta_{0, 2}$ (\\text{squared RGDPPC for Group 2}) &  -0.1866490 & 4.912843e-04   \\\\\n       $\\beta_{0, 3} + \\delta_{0, 3}$ (\\text{UA for Group 2}) & 0.1403171& 1.329788e-05 \\\\\n       $\\psi_{0,2 }$  (\\text{Change plane coeff for UA}) & -0.07061785 & 0.000000e+00\\\\\n         \\hline\n    \\end{tabular}\n    \\caption{Table of the estimated regression and change plane coefficients along with their p-values.}\n    \\label{tab:ekc_coeff}\n\\end{table}\n\\\\\\\\\n\\noindent\nFrom the above analysis, we find that both GDP and its square have statistically significant effects on carbon pollution for both groups of countries with their expected signs (positive for GDP and negative for its square term), supporting the inverted U-shaped income-pollution nexus and being in line with most papers in the EKC literature.  The urban variable, on the other hand, is seen to have insignificant effect on Group 1 countries  (less developed and emerging) which is in keeping with \\cite{du2012economic}, \\cite{sadorsky2014effect}. However, the urban variable seems to have a positive and significant impact on Group-2 countries which is in line with \\cite{liddle2010age} for example. Note that many of the group-1 countries are yet to experience sizeable levels of urbanization compared to the other group and this is even truer for our sample period.\\footnote{We use 6 years average from 2010-2015 for GDP and pollution measures. Such averaging is in accordance with the cross-sectional empirical literature using cross-country\/regional data and helps avoid business cycle fluctuations in GDP. It also minimizes the impacts of outlier events such as the financial crisis or great recession period.  The years that have we have chosen are ones for which we could find data for the largest number of countries.}Further, note that UA plays a crucial role in dividing the countries into different regimes, as the estimated value of $\\psi_{0,2}$ is significant.   \n\\\\\\\\\n\\noindent\nThere are many future potential applications of our method in economics. Similar exercises can be followed for other pollutants (such as sulfur emission, electrical waste\/e-waste, nitrogen pollution etc.). While income\/GDP remains a common, indeed the most crucial variable in the pollution study, other covariates (including change plane defining variables) may vary, depending on the pollutant of interest. One may also be interested in the determinants of health expenses in the household survey data. Often the families are asked about their health expenses incurred in the past one year. An interesting case in point may be household surveys collected in India where there are many large joint families with several children and old people live in the same household and most families are uninsured. It is often seen that health expenses increases with income as it includes expenses on regularly performed preventative medical examinations which are affordable only beyond an income level is reached. The important covariates are per capita family income, family wealth, 'dependency ratio' (number of children and old to the total number of people in the family) and binary indicator of any history of major illness\/hospitalizations in the family in the past year. Family income per capita and history of major illness can potentially define the change plane.\n\\end{comment}\n\n\n\\input{Real_data_analysis.tex}\n\\section{Conclusion}\n\\label{sec:conclusion}\nIn this paper we have established that under some mild assumptions the kernel-smoothed change plane estimator is asymptotically normal with near optimal rate $n^{-1}$. To the best of our knowledge, the state of the art result in this genre of problems is due to \\cite{seo2007smoothed}, where they demonstrate a best possible rate about $n^{-3\/4}$ for i.i.d. data. The main difference between their approach and ours is mainly the proof of Lemma \\ref{bandwidth}. Our techniques are based upon modern empirical process theory which allow us to consider much smaller bandwidths $\\sigma_n$ compared to those in \\cite{seo2007smoothed}, who appear to require larger values to achieve the result, possibly owing to their reliance on the techniques developed in \\cite{horowitz1992smoothed}. Although we have established it is possible to have asymptotic normality with really small bandwidths, we believe that the finite sample approximation (e.g. Berry-Essen bound) to normality could be poor, which is also evident from our simulation. \n\n\n\n\n\n\n\\section{Methodology and Theory for Continuous Response Model}\n\\label{sec:theory_regression}\nIn this section we present our analysis for the continuous response model. Without smoothing, the original estimating equation is: \n$$\nf_{\\beta, \\delta, \\psi}(Y, X, Q) = \\left(Y - X^{\\top}\\beta - X^{\\top}\\delta\\mathds{1}_{Q^{\\top}\\psi > 0}\\right)^2  \n$$\nand we estimate the parameters as: \n\\begin{align}\n\\label{eq:ls_estimator}\n\\left(\\hat \\beta^{LS}, \\hat \\delta^{LS}, \\hat \\psi^{LS}\\right) & = {\\arg\\min}_{(\\beta, \\delta, \\psi) \\in \\Theta} \\mathbb{P}_n f_{\\beta, \\delta, \\psi} \\notag \\\\\n& :=  {\\arg\\min}_{(\\beta, \\delta, \\psi) \\in \\Theta}\\mathbb{M}_n(\\beta, \\delta, \\psi)\\,.\n\\end{align}\nwhere $\\mathbb{P}_n$ is empirical measure based on i.i.d. observations $\\{(X_i, Y_i, Q_i)\\}_{i=1}^n$ and $\\Theta$ is the parameter space. Henceforth, we assume $\\Theta$ is a compact subset of dimension ${\\bbR}^{2p+d}$. We also define $\\theta = (\\beta, \\delta, \\psi)$, i.e. all the parameters together as a vector and by $\\theta_0$ is used to denote the true parameter vector $(\\beta_0, \\delta_0, \\psi_0)$. Some modification of equation \\eqref{eq:ls_estimator} leads to the following:  \n\\begin{align*}\n(\\hat \\beta^{LS}, \\hat \\delta^{LS}, \\hat \\psi^{LS}) & = {\\arg\\min}_{\\beta, \\delta, \\psi}  \\sum_{i=1}^n \\left(Y_i - X_i^{\\top}\\beta - X_i^{\\top}\\delta\\mathds{1}_{Q_i^{\\top}\\psi > 0}\\right)^2 \\\\ \n& = {\\arg\\min}_{\\beta, \\delta, \\psi}  \\sum_{i=1}^n \\left[\\left(Y_i - X_i^{\\top}\\beta\\right)^2\\mathds{1}_{Q_i^{\\top}\\psi_0 \\le 0} \\right. \\\\\n&  \\hspace{14em} \\left. + \\left(Y_i - X_i^{\\top}\\beta - X_i^{\\top}\\delta\\right)^2\\mathds{1}_{Q_i^{\\top}\\psi > 0} \\right] \\\\\n& = {\\arg\\min}_{\\beta, \\delta, \\psi}  \\sum_{i=1}^n \\left[\\left(Y_i - X_i^{\\top}\\beta\\right)^2 + \\left\\{\\left(Y_i - X_i^{\\top}\\beta - X_i^{\\top}\\delta\\right)^2 \\right. \\right. \\\\\n& \\hspace{17em} \\left. \\left. - \\left(Y_i - X_i^{\\top}\\beta\\right)^2\\right\\}\\mathds{1}_{Q_i^{\\top}\\psi > 0} \\right] \n\\end{align*}\nTypical empirical process calculations yield under mild conditions: \n$$\n\\|\\hat \\beta^{LS} - \\beta_0\\|^2 + \\|\\hat \\delta^{LS} - \\delta_0\\|^2 + \\|\\hat \\psi^{LS} - \\psi_0 \\|_2 = O_p(n^{-1})\n$$\nbut inference is difficult as the limit distribution is unknown, and in any case, would be a highly non-standard distribution. Recall that even in the one-dimensional change point model with fixed jump size, the least squares change point estimator converges at rate $n$ to the truth with a non-standard limit distribution, namely a minimizer of a two-sided compound Poisson process (see \\cite{lan2009change} for more details). To obtain a computable estimator with tractable limiting distribution, we resort to a smooth approximation of the indicator function in \\eqref{eq:ls_estimator} using a distribution kernel with suitable bandwidth, i.e we replace $\\mathds{1}_{Q_i^{\\top}\\psi > 0}$ by $K(Q_i^{\\top}\\psi\/\\sigma_n)$ for some appropriate distribution function $K$ and bandwidth $\\sigma_n$, i.e. \n\\begin{align*}\n(\\hat \\beta^S, \\hat \\delta^S, \\hat \\psi^S) & = {\\arg\\min}_{\\beta, \\delta, \\psi} \\left\\{ \\frac1n \\sum_{i=1}^n \\left[\\left(Y_i - X_i^{\\top}\\beta\\right)^2 + \\left\\{\\left(Y_i - X_i^{\\top}\\beta - X_i^{\\top}\\delta\\right)^2 \\right. \\right. \\right. \\\\\n& \\hspace{15em} \\left. \\left. \\left. - \\left(Y_i - X_i^{\\top}\\beta\\right)^2\\right\\}K\\left(\\frac{Q_i^{\\top}\\psi}{\\sigma_n}\\right) \\right] \\right\\} \\\\\n& = {\\arg\\min}_{(\\beta, \\delta, \\psi) \\in \\Theta} \\mathbb{P}_n f^s_{(\\beta, \\delta, \\psi)}(X, Y, Q) \\\\\n& := {\\arg\\min}_{\\theta \\in \\Theta} \\mathbb{M}^s_n(\\theta) \\,.\n\\end{align*}\nDefine $\\mathbb{M}$ (resp. $\\mathbb{M}^s$) to be the population counterpart of $\\mathbb{M}_n$ and $\\mathbb{M}_n^s$ respectively which are defined as: \n\\begin{align*}\n\\mathbb{M}(\\theta) & = \\mathbb{E}\\left(Y - X^{\\top}\\beta\\right)^2 + \\mathbb{E}\\left(\\left[-2\\left(Y_i - X^{\\top}\\beta\\right)X^{\\top}\\delta + (X^{\\top}\\delta)^2\\right] \\mathds{1}_{Q^{\\top}\\psi > 0}\\right) \\,, \\\\\n\\mathbb{M}^s(\\theta) & = \\mathbb{E}\\left[(Y - X^{\\top}\\beta)^2 + \\left\\{-2(Y-X^{\\top}\\beta)(X^{\\top}\\delta) + (X^{\\top}\\delta)^2\\right\\}K\\left(\\frac{Q^{\\top}\\psi}{\\sigma_n}\\right)\\right] \\,.\n\\end{align*}\nAs noted in the proof of \\textcolor{blue}{Seo and Linton}, the assumption $\\log{n}\/n\\sigma_n^2 \\to 0$ was only used to show: \n$$\n\\frac{\\left\\|\\hat \\psi^s - \\psi_0\\right\\|}{\\sigma_n} = o_p(1) \\,.\n$$\nIn this paper, we show that one can achieve the same conclusion as long as $n\\sigma_n \\to \\infty$.  The rest of the proof for the normality is similar to that of \\cite{seo2007smoothed}, we will present it briefly for the ease the readers. The proof is quite long and technical, therefore we break the proof into several lemmas. We, first, list our assumptions: \n\\begin{assumption}\n\\label{eq:assm}\n\\begin{enumerate}\n\\item Define $f_\\psi(\\cdot \\mid \\tilde Q)$ to be the conditional distribution of $Q^{\\top}\\psi$ given $\\tilde Q$. (In particular we will denote by $f_0(\\cdot \\mid \\tilde q)$ to be conditional distribution of $Q^{\\top}\\psi_0$ given $\\tilde Q$ and $f_s(\\cdot \\mid \\tilde q)$ to be the conditional distribution of $Q^{\\top}\\psi_0^s$ given $\\tilde Q$. Assume that there exists $F_+$ such that $\\sup_t f_0(t | \\tilde Q) \\le F_+$ almost surely on $\\tilde Q$ and for all $\\psi$ in a neighborhood of $\\psi_0$ (in particular for $\\psi_0^s$). Further assume that $f_\\psi$ is differentiable and the derivative is bounded by $F_+$ for all $\\psi$ in a neighborhood of $\\psi_0$ (again in particular for $\\psi_0^s$).\n\\vspace{0.1in}\n\\item Define $g(Q) = {\\sf var}(X \\mid Q)$. There exists $c_-$ and $c_+$ such that $c_- \\le \\lambda_{\\min}(g(Q)) \\le \\lambda_{\\max}(g(Q)) \\le c_+$ almost surely. Also assume that $g$ is a Lipschitz with constant $G_+$ with respect to $Q$. \n\\vspace{0.1in}\n\\item There exists $p_+ < \\infty$ and $p_- > 0, r > 0$ such that: \n$$\np_- \\|\\psi - \\psi_0\\| \\le \\mathbb{P}\\left(\\text{sign}\\left(Q^{\\top}\\psi\\right) \\neq \\text{sign}\\left(Q^{\\top}\\psi_0\\right)\\right) \\le p_+ \\|\\psi - \\psi_0\\| \\,,\n$$\nfor all $\\psi$ such that $\\|\\psi - \\psi_0\\| \\le r$. \n\\vspace{0.1in}\n\\item For all $\\psi$ in the parameter space $0 < \\mathbb{P}\\left(Q^{\\top}\\psi > 0\\right) < 1$. \n\\vspace{0.1in} \n\\item Define $m_2(Q) = \\mathbb{E}\\left[\\|X\\|^2 \\mid Q\\right]$ and $m_4(Q) = \\mathbb{E}\\left[\\|X\\|^4 \\mid Q\\right]$. Assume $m_2, m_4$ are bounded Lipschitz function of $Q$.  \n\\end{enumerate}\n\\end{assumption}\n\n\n\\subsection{Sufficient conditions for above assumptions }\nWe now demonstrate some sufficient conditions for the above assumptions to hold. The first condition is essentially a condition on the conditional density of the first co-ordinate of $Q$ given all other co-ordinates. If this conditional density is bounded and has bounded derivative, then first assumption is satisfied. This condition is satisfied in fair generality. The second assumption implies that the conditional distribution of X given Q has variance in all the direction over all $Q$. This is also very weak condition, as is satisfied for example if X and Q and independent (with $X$ has non-degenerate covariance matrix) or $(X, Q)$ are jointly normally distributed to name a few. This condition can further be weaken by assuming that the maximum and minimum eigenvalues of $\\mathbb{E}[g(Q)]$ are bounded away from $\\infty$ and $0$ respectively but it requires more tedious book-keeping. The third assumption is satisfied as long as as $Q^{\\top}\\psi$ has non-zero density near origin, while the fourth assumption merely states that the support of  $Q$ is not confined to one side of the hyperplane for any hyperplane and a simple sufficient condition for this is $Q$ has continuous density with non-zero value at the origin. The last assumption is analogous to the second assumption for the conditional fourth moment which is also satisfied in fair generality. \n\\\\\\\\\n\\noindent\n{\\bf Kernel function and bandwidth: } We take $K(x) = \\Phi(x)$ (distribution of standard normal random variable) for our analysis. For the bandwidth we assume $n\\sigma_n^2 \\to 0$ and $n \\sigma_n \\to \\infty$ as the other case, (i.e. $n\\sigma_n^2 \\to \\infty$) is already established in \\cite{seo2007smoothed}. \n\\\\\\\\\n\\noindent\nBased on Assumption \\ref{eq:assm} and our choice of kernel and bandwidth we establish the following theorem: \n\\begin{theorem}\n\\label{thm:regression}\nUnder Assumption \\ref{eq:assm} and the above choice of kernel and bandwidth we have: \n$$\n\\sqrt{n}\\begin{pmatrix}\\begin{pmatrix} \\hat \\beta^s \\\\ \\hat \\delta^s \\end{pmatrix} - \\begin{pmatrix} \\beta_0 \\\\ \\delta_0 \\end{pmatrix} \\end{pmatrix} \\overset{\\mathscr{L}}{\\implies} \\mathcal{N}(0, \\Sigma_{\\beta, \\delta})\n$$\nand \n$$\n\\sqrt{n\/\\sigma_n} \\left(\\hat \\psi^s - \\psi_0\\right) \\overset{\\mathscr{L}}{\\implies} \\mathcal{N}(0, \\Sigma_\\psi) \\,,\n$$\nfor matrices $\\Sigma_{\\beta, \\delta}$ and $\\Sigma_\\psi$ mentioned explicitly in the proof. Moreover they are asymptotically independent. \n\\end{theorem}\nThe proof of the theorem is relatively long, so we break it into several lemmas. We  provide a roadmap of the proof in this section while the elaborate technical derivations of the supporting lemmas can be found in Appendix. Let $\\nabla \\mathbb{M}_n^s(\\theta)$ and $\\nabla^2 \\mathbb{M}_n^s(\\theta)$ be the gradient and Hessian of $\\mathbb{M}_n^s(\\theta)$ with respect to $\\theta$.  As $\\hat \\theta^s$ minimizes $\\mathbb{M}_n^s(\\theta)$, we have from the first order condition, $\\nabla \\mathbb{M}_n^s(\\hat \\theta^s) = 0$. Using one step Taylor expansion we have:\n\\allowdisplaybreaks \n\\begin{align*}\n\\label{eq:taylor_first}\n\\nabla \\mathbb{M}_n^s(\\hat \\theta^s) = \\nabla \\mathbb{M}_n^s(\\theta_0) + \\nabla^2 \\mathbb{M}_n^s(\\theta^*)\\left(\\hat \\theta^s - \\theta_0\\right) = 0\n\\end{align*}\ni.e.\n\\begin{equation}\n\\label{eq:main_eq}  \n\\left(\\hat{\\theta}^s - \\theta_0\\right) = -\\left(\\nabla^2 \\mathbb{M}_n^s(\\theta^*)\\right)^{-1} \\nabla \\mathbb{M}_n^s(\\theta_0)\n\\end{equation}\nfor some intermediate point $\\theta^*$ between $\\hat \\theta^s$ and $\\theta_0$. \nFollowing the notation of \\cite{seo2007smoothed}, define a diagonal matrix $D_n$ of dimension $2p + d$ with first $2p$ elements being 1 and the last $d$ elements being $\\sqrt{\\sigma_n}$. \n we can write: \n\\begin{align}\n\\sqrt{n}D_n^{-1}(\\hat \\theta^s - \\theta_0) & = - \\sqrt{n}D_n^{-1}\\nabla^2\\mathbb{M}_n^s(\\theta^*)^{-1}\\nabla \\mathbb{M}_n^s(\\theta_0)  \\notag \\\\\n\\label{eq:taylor_main} & = \\begin{pmatrix} \\nabla^2\\mathbb{M}_n^{s, \\gamma}(\\theta^*) & \\sqrt{\\sigma_n}\\nabla^2\\mathbb{M}_n^{s, \\gamma \\psi}(\\theta^*) \\\\\n\\sqrt{\\sigma_n}\\nabla^2\\mathbb{M}_n^{s, \\gamma \\psi}(\\theta^*) & \\sigma_n\\nabla^2\\mathbb{M}_n^{s, \\psi}(\\theta^*)\\end{pmatrix}^{-1}\\begin{pmatrix} \\sqrt{n}\\nabla \\mathbb{M}_n^{s, \\gamma}(\\theta_0) \\\\ \\sqrt{n\\sigma_n}\\nabla \\mathbb{M}_n^{s, \\psi}(\\theta_0)\\end{pmatrix}\n\\end{align}\nwhere $\\gamma = (\\beta, \\delta) \\in {\\bbR}^{2p}$. The following lemma establishes the asymptotic properties of $\\nabla \\mathbb{M}_n^s(\\theta_0)$: \n\\begin{lemma}[Asymptotic Normality of $\\nabla \\mathbb{M}_n^s(\\theta_0)$]\n\\label{asymp-normality}\n\\label{asymp-normality}\nUnder assumption \\ref{eq:assm} we have: \n\\begin{align*}\n\\sqrt{n}\\nabla \\mathbb{M}_n^{s, \\gamma}(\\theta_0) \\implies \\mathcal{N}\\left(0, 4V^{\\gamma}\\right) \\,,\\\\\n\\sqrt{n\\sigma_n}\\nabla \\mathbb{M}_n^{s, \\psi}(\\theta_0) \\implies \\mathcal{N}\\left(0, V^{\\psi}\\right) \\,.\n\\end{align*} \nfor some n.n.d. matrices $V^{\\gamma}$ and $V^{\\psi}$ which is mentioned explicitly in the proof. Further more $\\sqrt{n}\\nabla \\mathbb{M}_n^{s, \\gamma}(\\theta_0)$ and $\\sqrt{n\\sigma_n}\\nabla \\mathbb{M}_n^{s, \\psi}(\\theta_0)$ are asymptotically independent. \n\\end{lemma}\n\\noindent\nNext, we analyze the convergence of $\\nabla^2 \\mathbb{M}_n^s(\\theta^*)$ which is stated in the following lemma: \n\\begin{lemma}[Convergence in Probability of $\\nabla^s \\mathbb{M}_n^s(\\theta^*)$]\n\\label{conv-prob}\nUnder Assumption \\eqref{eq:assm}, for any random sequence $\\breve{\\theta} = \\left(\\breve{\\beta}, \\breve{\\delta}, \\breve{\\psi}\\right)$ such that $\\breve{\\beta} \\overset{p}{\\to} \\beta_0, \\breve{\\delta} \\overset{p}{\\to} \\delta_0, \\|\\breve{\\psi} - \\psi_0\\|\/\\sigma_n \\overset{P} \\rightarrow 0$, we have: \n\\begin{align*}\n\\nabla^2_{\\gamma} \\mathbb{M}_n^s(\\breve{\\theta}) & \\overset{p}{\\longrightarrow} 2Q^{\\gamma} \\,, \\\\\n\\sqrt{\\sigma_n}\\nabla^2_{\\psi \\gamma} \\mathbb{M}_n^s(\\breve{\\theta}) & \\overset{p}{\\longrightarrow} 0 \\,, \\\\\n\\sigma_n \\nabla^2_{\\psi} \\mathbb{M}_n^s(\\breve{\\theta}) & \\overset{p}{\\longrightarrow} Q^{\\psi} \\,.\n\\end{align*}\nfor some matrices $Q^{\\gamma}, Q^{\\psi}$ mentioned explicitly in the proof. This, along with equation \\eqref{eq:taylor_main}, establishes: \n\\begin{align*}\n\\sqrt{n}\\left(\\hat \\gamma^s - \\gamma_0\\right) & \\overset{\\mathscr{L}}{\\implies} \\mathcal{N}\\left(0, Q^{\\gamma^{-1}}V^{\\gamma}Q^{\\gamma^{-1}}\\right) \\,, \\\\\n\\sqrt{n\/\\sigma_n}\\left(\\hat \\psi^s - \\psi_0\\right) & \\overset{\\mathscr{L}}{\\implies} \\mathcal{N}\\left(0, Q^{\\psi^{-1}}V^{\\psi}Q^{\\psi^{-1}}\\right) \\,.\n\\end{align*}\nwhere as before $\\hat \\gamma^s = (\\hat \\beta^s, \\hat \\delta^s)$. \n\\end{lemma}\nIt will be shown later that the condition $\\|\\breve{\\psi}_n - \\psi_0\\|\/\\sigma_n \\overset{P} \\rightarrow 0$ needed in Lemma \\ref{conv-prob} holds for the (random) sequence $\\psi^*$, the intermediate point in the Taylor expansion. Then, combining Lemma \\ref{asymp-normality} and Lemma \\ref{conv-prob} we conclude the proof of Theorem \\ref{thm:regression}.\nObserve that, to show $\\left\\|\\psi^* - \\psi_0 \\right\\| = o_P(\\sigma_n)$, it suffices to to prove that $\\left\\|\\hat \\psi^s - \\psi_0 \\right\\| = o_P(\\sigma_n)$. Towards that direction, we have following lemma: \n\n\\begin{lemma}[Rate of convergence]\n\\label{lem:rate_smooth}\nUnder Assumption \\ref{eq:assm} and our choice of kernel and bandwidth, \n$$\nn^{2\/3}\\sigma_n^{-1\/3} d^2_*\\left(\\hat \\theta^s, \\theta_0^s\\right) = O_P(1) \\,,\n$$\nwhere \n\\begin{align*}\nd_*^2(\\theta, \\theta_0^s) & =  \\|\\beta - \\beta_0^s\\|^2 + \\|\\delta - \\delta_0^s\\|^2 \\\\\n& \\qquad \\qquad + \\frac{\\|\\psi - \\psi_0^s\\|^2}{\\sigma_n} \\mathds{1}_{\\|\\psi - \\psi_0^s\\| \\le \\mathcal{K}\\sigma_n} + \\|\\psi - \\psi_0^s\\| \\mathds{1}_{\\|\\psi - \\psi_0^s\\| > \\mathcal{K}\\sigma_n}  \\,.\n\\end{align*}\nfor some specific constant $\\mathcal{K}$. (This constant will be mentioned precisely in the proof). Hence as $n\\sigma_n \\to \\infty$, we have $n^{2\/3}\\sigma_n^{-1\/3} \\gg \\sigma_n^{-1}$ which implies $\\|\\hat \\psi^s - \\psi_0^s\\|\/\\sigma_n  \\overset{P}  \\longrightarrow 0 \\,.$\n\\end{lemma}\n\\noindent\nThe above lemma establishes $\\|\\hat \\psi^s - \\psi_0^s\\|\/\\sigma_n = o_p(1)$ but our goal is to show that $\\|\\hat \\psi^s - \\psi_0\\|\/\\sigma_n = o_p(1)$. Therefore, we further need $\\|\\psi^s_0 - \\psi_0\\|\/\\sigma_n \\rightarrow 0$ which is demonstrated in the following lemma:\n\n\\begin{lemma}[Convergence of population minimizer]\n\\label{bandwidth}\nUnder Assumption \\ref{eq:assm} and our choice of kernel and bandwidth, we have: $\\|\\psi^s_0 - \\psi_0\\|\/\\sigma_n \\rightarrow 0$. \n\\end{lemma}\n\n\\noindent\nHence the final roadmap is the following: Using Lemma \\ref{bandwidth} and Lemma \\ref{lem:rate_smooth} we establish that $\\|\\hat \\psi^s - \\psi_0\\|\/\\sigma_n = o_p(1)$ if $n\\sigma_n \\to 0$. This, in turn, enables us to prove Lemma \\ref{conv-prob}, i.e. $\\sigma_n \\nabla^2 \\mathbb{M}_n^s(\\theta^*) \\overset{P} \\rightarrow Q$,which, along with Lemma \\ref{asymp-normality}, establishes the main theorem. \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nModeling the long-timescale behavior of complex dynamical systems is a fundamental task in the physical sciences. In principle, molecular dynamics (MD) simulations allow us to probe the spatiotemporal details of molecular processes, but the so-called sampling problem severely limits their usefulness in practice. This sampling problem comes from the fact that a typical free energy landscape consists of many metastable states separated by free energy barriers much higher than the thermal energy $\\kT$. Therefore, on the timescale one can simulate, barrier crossings are rare events, and the system remains kinetically trapped in a single metastable state.\n\nOne way to alleviate the sampling problem is to employ enhanced sampling methods~\\cite{abrams2014enhanced,valsson2016enhancing}. In particular, one class of such methods works by identifying a few critical slow degrees of freedom, commonly referred to as collective variables (CVs), and then enhancing their fluctuations by introducing an external bias potential~\\cite{valsson2016enhancing,yang2019enhanced,bussi2020using}. The performance of CV-based enhanced sampling methods depends heavily on the quality of the CVs. Effective CVs should discriminate between the relevant metastable states and include most of the slow degrees of freedom~\\cite{noe2017collective}. Typically, the CVs are selected manually by using physical and chemical intuition. Within the enhanced sampling community, numerous generally applicable CVs~\\cite{abrams2014enhanced,pietrucci_strategies_2017,rydzewski2017ligand} have been developed and implemented in open-source codes~\\cite{Fiorin_2013,tribello2014plumed,Sidky_2018}. However, despite immense progress in devising CVs, it may be far from trivial to find a set of CVs that quantify all the essential characteristics of a molecular system.\n\nMachine learning (ML) techniques, in particular dimensionality reduction or representation learning methods~\\cite{murdoch2019definitions,xie2020representation}, provide a possible solution to this problem by automatically finding or constructing the CVs directly from the simulation data~\\cite{wang2020machine,noe2020machine,Gkeka2020mlffcgv,sidky2020machine}. Such dimensionality reduction methods typically work in a high-dimensional feature space (e.g., distances, dihedral angles, or more intricate functions~\\cite{geiger2013neural,rogal2019neural, musil2021physicsinspired}) instead of directly using the microscopic coordinates, as this is much more efficient. Dimensionality reduction may employ linear or nonlinear transformations, e.g., diffusion map~\\cite{coifman2005geometric,coifman2006diffusion,nadler2006diffusion,coifman2008diffusion}, stochastic neighbor embedding (SNE)~\\cite{hinton2002stochastic,maaten2008visualizing,maaten2009learning}, sketch-map~\\cite{ceriotti2011simplifying,tribello2012using}, and UMAP~\\cite{mcinnes2018umap}. In the recent years, there has been a growing interest in performing nonlinear dimensionality reduction with deep neural networks (NNs) to provide parametric embeddings. Inspired by the seminal work of Ma and Dinner~\\cite{Ma2005autorc}, several such techniques recently applied to finding CVs include variational autoencoders~\\cite{chen2018molecular,hernandez2018variational,ribeiro2018reweighted,chen2018collective}, time-lagged autoencoders~\\cite{wehmeyer2018time}, symplectic flows~\\cite{li2020neural}, stochastic kinetic embedding~\\cite{zhang2018unfolding}, and encoder-map~\\cite{Lemke2019EncMap}.\n\nThis work proposes a novel technique called multiscale reweighted stochastic embedding (MRSE) that unifies dimensionality reduction via deep NNs and enhanced sampling methods. The method constructs a low-dimensional representation of CVs by learning a parametric embedding from a high-dimensional feature space to a low-dimensional latent space. Our work builds upon various SNE methods~\\cite{hinton2002stochastic,maaten2008visualizing,maaten2009learning,van2014accelerating}. We introduce several new aspects to SNE that makes MRSE particularly suitable for enhanced sampling simulations:\n\\begin{enumerate}\n  \\item A weight-tempered random sampling as a landmark selection scheme to obtain training data sets that strike a balance between equilibrium representation and capturing important metastable states lying higher in free energy.\n  \\item Multiscale representation of the high-dimensional feature space via a Gaussian mixture probability model.\n  \\item Reweighting procedure to account for the sampling of the training data from a biased probability distribution.\n\\end{enumerate}\n\nWe note that the overall objective of our research is to employ MRSE within an enhanced sampling scheme and improve the learned CVs iteratively. However, we focus mainly on the learning procedure for training data from enhanced sampling simulations in this work. Therefore, to eliminate the influence of possible incomplete sampling, we employ idealistic sampling conditions that are generally not achievable in practice~\\cite{pant2020statistical}. To gauge the performance of the learning procedure and the quality of the resulting embeddings, we apply MRSE to three model systems (the M\\\"uller-Brown potential, alanine dipeptide, and alanine tetrapeptide) and provide a thorough analysis of the results.\n\n\n\n\n\n\\section{Methods}\n\\label{sec:methods}\n\\subsection{Collective Variable Based Enhanced Sampling}\n\\label{sec:cv_based_methods}\nWe start by giving a theoretical background on CV-based enhanced sampling methods. We consider a molecular system, described by microscopic coordinates $\\mathbf{R}$ and a potential energy function $U(\\mathbf{R})$, which we want to study using MD or Monte Carlo simulations. Without loss of generality, we limit our discussion to the canonical ensemble (NVT). At equilibrium, the microscopic coordinates follow the Boltzmann distribution, $P(\\mathbf{R}) = \\e^{-\\beta U(\\mathbf{R})}\/\\int \\d\\mathbf{R} \\,\\e^{-\\beta U(\\mathbf{R})}$, where $\\beta = (k_{\\mathrm{B}}T)^{-1}$ is the inverse of the thermal energy.\n\nIn CV-based enhanced sampling methods, we identify a small set of coarse-grained order parameters that correspond to the essential slow degrees of freedom, referred to as CVs. The CVs are defined as $\\mathbf{s}(\\mathbf{R}) = [s_1(\\mathbf{R}), s_2(\\mathbf{R}), \\ldots, s_d(\\mathbf{R})]$, where $d$ is the number of CVs (i.e., the dimension of the CV space), and the dependence on $\\mathbf{R}$ can be either explicit or implicit. Having defined the CVs, we obtain their equilibrium marginal distribution by integrating out all other degrees of freedom:\n\\begin{align}\n\\label{eq:ps}\n  P(\\mathbf{s}) =\n    \\int \\d\\mathbf{R} \\, \\delta\n      \\left[\n        \\mathbf{s} - \\mathbf{s}(\\mathbf{R})\n      \\right]\n    P(\\mathbf{R}),\n\\end{align}\nwhere $\\delta[\\cdot]$ is the Dirac delta function. The integral in eq~\\ref{eq:ps} is equivalent to $\\big< \\delta[\\mathbf{s}-\\mathbf{s(R)}] \\big>$, where $\\left<\\cdot\\right>$ denotes an ensemble average. Up to an unimportant constant, the free energy surface (FES) is given by $F(\\mathbf{s})= -\\beta^{-1} \\log P(\\mathbf{s})$. In systems plagued by sampling problems, the FES consists of many metastable states separated by free energy barriers much larger than the thermal energy $k_{\\mathrm{B}}T$. Therefore, on the timescales we can simulate, the system stays kinetically trapped and is unable to explore the full CV space. In other words, barrier crossings between metastable states are rare events.\n\nCV-based enhanced sampling methods overcome the sampling problem by introducing an external bias potential $V(\\mathbf{s}(\\mathbf{R}))$ acting in CV space. This leads to sampling according to a biased distribution $P_{V}(\\mathbf{R}) = \\e^{-\\beta \\left[U(\\mathbf{R})+V(\\mathbf{s}(\\mathbf{R})) \\right]}\/\\int \\d\\mathbf{R} \\,\\e^{-\\beta \\left[U(\\mathbf{R})+V(\\mathbf{s}(\\mathbf{R})) \\right]}$.  We can trace this idea of non-Boltzmann sampling back to the seminal work by Torrie and Valleau published in 1977~\\cite{torrie1977nonphysical}. Most CV-based methods adaptively construct the bias potential on-the-fly during the simulation to reduce free energy barriers or even completely flatten them. At convergence, the CVs follow a biased distribution:\n\\begin{equation}\n  P_{V}(\\mathbf{s}) =\n  \\int \\d\\mathbf{R} \\, \\delta\n    \\left[\n      \\mathbf{s} - \\mathbf{s}(\\mathbf{R})\n    \\right]\n  P_{V}(\\mathbf{R}) =\n  \\frac{ \\e^{ -\\beta\\left[ F(\\mathbf{s}) + V(\\mathbf{s}) \\right] } }\n    {\\int\\d\\mathbf{s} \\, \\e^{ -\\beta\\left[ F(\\mathbf{s}) + V(\\mathbf{s}) \\right] } },\n\\end{equation}\nthat is easier to sample. CV-based methods differ in how they construct the bias potential and which kind of biased CV sampling they obtain at convergence. A non-exhaustive list of modern CV-based enhanced sampling techniques includes multiple windows umbrella sampling~\\cite{Kastner2011umbreallsampling}, adaptive biasing force~\\cite{Darve-JCP-2001,Comer2015_TheAdaptiveBiasing,Lesage2016_Smoothed}, Gaussian-mixture umbrella sampling~\\cite{Maragakis-JPCB-2009}, metadynamics~\\cite{laio2002escaping,barducci2008well,valsson2016enhancing}, variationally enhanced sampling~\\cite{valsson2014variational,Valsson2020_VES}, on-the-fly probability-enhanced sampling (OPES)~\\cite{Invernizzi2020opus,invernizzi2020unified}, and ATLAS~\\cite{gilberti2020atlas}. In the following, we focus on well-tempered metadynamics (WT-MetaD)~\\cite{barducci2008well,valsson2016enhancing}. However, we can use MRSE with almost any CV-based enhanced sampling approach.\n\nIn WT-MetaD, the time-dependent bias potential is constructed by periodically depositing repulsive Gaussian kernels at the current location in CV space. Based on the previously deposited bias, the Gaussian height is scaled such that it gradually decreases over time~\\cite{barducci2008well}. In the long-time limit, the Gaussian height goes to zero. As has been proven~\\cite{PhysRevLett.112.240602}, the bias potential at convergence is related to the free energy by:\n\\begin{equation}\n  \\label{eq:wt-bias_infty}\n  V(\\bs,t\\to\\infty) = -\\left( 1-\\frac{1}{\\gamma} \\right) F(\\bs),\n\\end{equation}\nand we obtain a so-called well-tempered distribution for the CVs:\n\\begin{equation}\n  \\label{eq:wt-pv}\n  P_{V}(\\mathbf{s}) = \\frac{ \\left[ P(\\mathbf{s}) \\right]^{1\/\\gamma} }\n    {\\int\\d\\mathbf{s}\\, \\left[ P(\\mathbf{s}) \\right]^{1\/\\gamma}},\n\\end{equation}\nwhere $\\gamma>1$ is a parameter called bias factor that determines how much we enhance CV fluctuations. The limit $\\gamma\\to 1$ corresponds to the unbiased ensemble, while the limit $\\gamma\\to\\infty$ corresponds to conventional (non-well-tempered) metadynamics~\\cite{laio2002escaping}. If we take the logarithm of both sides of eq~\\ref{eq:wt-pv}, we can see that sampling the well-tempered distribution is equivalent to sampling an effective FES, $F_{\\gamma}(\\bs) = F(\\bs)\/\\gamma$, where the barriers of the original FES are reduced by a factor of $\\gamma$. In general, one should select a bias factor $\\gamma$ such that effective free energy barriers are on the order of the thermal energy $k_{\\mathrm{B}}T$.\n\nDue to the external bias potential, each microscopic configuration $\\mathbf{R}$ carries an additional statistical weight $w(\\mathbf{R})$ that needs to be taken into account when calculating equilibrium properties. For a static bias potential, the weight is time-independent and given by $w(\\mathbf{R})=\\e^{\\beta V(\\mathbf{s}(\\mathbf{R}))}$. In WT-MetaD, however, we need to take into account the time-dependence of the bias potential, and thus, the weight is modified in the following way:\n\\begin{equation}\n  \\label{eq:weight-wtm}\n  w(\\mathbf{R},t)=\\exp[\\beta \\tilde{V}( \\mathbf{s}(\\mathbf{R}), t )],\n\\end{equation}\nwhere $\\tilde{V}(\\mathbf{s}(\\mathbf{R}),t)=V(\\mathbf{s}(\\mathbf{R}),t)-c(t)$ is the relative bias potential modified by introducing $c(t)$, a time-dependent constant that can be calculated from the bias potential at time $t$ as~\\cite{tiwary_rewt,valsson2016enhancing}:\n\\begin{equation}\n  \\label{eq:coft}\n  c(t)=\\frac{1}{\\beta}\\log{\n  \\frac{\\int \\d\\mathbf{s}\\,\n  \\exp\\left[\n  \\frac{\\gamma}{\\gamma-1} \\beta V(\\mathbf{s},t)\n  \\right]}\n  {\\int \\d\\mathbf{s}\\,\n  \\exp\\left[\n  \\frac{1}{\\gamma-1} \\beta V(\\mathbf{s},t)\n  \\right]}}.\n\\end{equation}\nThere are also other ways to reweight WT-MetaD simulations~\\cite{bonomi_rewt,Branduardi-JCTC-2012,Giberti_2019,Sch_fer_2020}.\n\nIn MD simulations, we do not only need to know the values of the CVs but also their derivatives with respect to the microscopic coordinates, $\\nabla_{\\mathbf{R}} \\, \\mathbf{s}(\\mathbf{R})$. The derivatives are needed to calculate the biasing force $-\\nabla_{\\mathbf{R}} \\, V(\\mathbf{s}(\\mathbf{R})) = -\\partial_{\\mathbf{s}} V(\\mathbf{s}) \\cdot\\nabla_{\\mathbf{R}} \\, \\mathbf{s}(\\mathbf{R})$. In practice, however, the CVs might not depend directly on $\\mathbf{R}$, but rather indirectly through a set of some other input variables (e.g., features). We can even define a CV that is a chain of multiple variables that depend sequentially on each other. In such cases, it is sufficient to know the derivatives of the CVs with respect to the input variables, as we can obtain the total derivatives via the chain rule. In codes implementing CVs and enhanced sampling methods~\\cite{Fiorin_2013,tribello2014plumed,Sidky_2018}, like \\textsc{plumed}~\\cite{tribello2014plumed,plumed-nest}, the handling of the chain rule is done automatically. Thus, when implementing a new CV, we only need to calculate its values and derivatives with respect to the input variables.\n\nHaving provided the basics of CV-based enhanced sampling simulations, we now introduce our method for learning CVs.\n\n\n\n\n\n\\subsection{Multiscale Reweighted Stochastic Embedding (MRSE)}\n\\label{sec:mrse}\nThe basis of our method is the $t$-distributed variant of stochastic neighbor embedding ($t$-SNE)~\\cite{maaten2008visualizing}, a dimensionality reduction algorithm for visualizing high-dimensional data, for instance, generated by unbiased MD simulations~\\cite{rydzewski2016machine,zhou2018t,spiwok2020time,fleetwood2021identification}. We introduce here a parametric and multiscale variant of SNE aimed at learning CVs from atomistic simulations. In particular, we focus on using the method within enhanced sampling simulations, where we need to consider biased simulation data. We refer to this method as multiscale reweighted stochastic embedding or MRSE.\n\nWe consider a high-dimensional feature space, $\\bx=[x_1, \\dots, x_k]$, of dimension $k$. The features could be distances, dihedral angles, or some more complex functions~\\cite{geiger2013neural,rogal2019neural,musil2021physicsinspired}, which depend on the microscopic coordinates. We introduce a parametric embedding function $f_\\bt(\\bx)=\\bs(\\bx)$, that depends on parameters $\\bt$, to map from the high-dimensional feature space to the low-dimensional latent space (i.e., the CV space), $\\bs=[s_1, \\dots, s_d]$, of dimension $d$. From a molecular simulation, we collect $N$ observations (or simply samples) of the features, $[\\bx_1, \\dots, \\bx_N]^T$, that we use as training data. Using these definitions, the problem of finding a low-dimensional set of CVs amounts to using the training data to find an optimal parametrization for the embedding function given a nonlinear ML model. We can then use the embedding as CVs and project any point in feature space to CV space.\n\nIn SNE methods, this problem is approached by taking the training data and modeling the pairwise probability distributions for distances in the feature and latent space. To establish the notation, we write the pairwise probability distributions as ${\\bf M}=(p_{ij})$ and $\\sfQ=(q_{ij})$, where $1\\leq i,j \\leq N$, for the feature and the latent space, respectively. For the pairwise probability distribution ${\\bf M}$ ($\\sfQ$), the interpretation of a single element $p_{ij}$ ($q_{ij}$) is that higher the value, higher is the probability of picking $\\bx_j$ ($\\bs_j$) as a neighbor of $\\bx_i$ ($\\bs_i$). The mapping from the feature space to the latent space is then varied by adjusting the parameters $\\bt$ to minimize a loss function that measures the statistical difference between the two pairwise probability distributions. In the following, we explicitly introduce the pairwise probability distributions and the loss function used in MRSE.\n\n\n\n\n\n\n\\subsubsection{Feature Pairwise Probability Distribution}\n\\label{sec:feature_distribution}\nWe model the feature pairwise probability distribution for a pair of samples $\\bx_i$ and $\\bx_j$ from the training data as a discrete Gaussian mixture. Each term in the mixture is a Gaussian kernel:\n\\begin{equation}\n  K_{\\varepsilon_i}(\\bx_i,\\bx_j)=\\exp\\left(-\\varepsilon_i\\|\\bx_i-\\bx_j\\|^2_2\\right)\n\\end{equation}\nthat is characterized by a scale parameter $\\varepsilon_i$ associated to feature sample $\\bx_i$. A scale parameter is defined as $\\varepsilon_i=1\/(2\\sigma^2_i)$, where $\\sigma_i$ is the standard deviation (i.e., bandwidth) of the Gaussian kernel. Because $\\varepsilon_i \\neq \\varepsilon_j$, the kernels are not symmetric. To measure the distance between data points, we employ the Euclidean distance $\\|\\cdot\\|_2$ as an appropriate metric for representing high-dimensional data on a low-dimensional manifold~\\cite{globerson2007euclidean}. Then, a pair $\\bx_i$ and $\\bx_j$ of points close to each other, as measured by the Euclidean distance, have a high probability of being neighbors.\n\n\\begin{figure}[htp]\n  \\includegraphics[width=0.5\\columnwidth]{fig-probabilities.pdf}\n  \\caption{Schematic representation depicting how MRSE (and $t$-SNE) preserves the local structure of high-dimensional data. The pairwise probability distributions are represented by Gaussian kernels in the high-dimensional feature space and by the $t$-distribution kernels in the low-dimensional latent space. The minimization of the Kullback-Leibler divergence between the pairwise probability distributions enforces similar feature samples close to each other and separates dissimilar feature samples in the latent space. As the difference between the distributions fulfills $\\Delta'>\\Delta$, MRSE is likely to group close-by points into metastable states that are well separated.}\n  \\label{fig:probabilities}\n\\end{figure}\n\nFor training data obtained from an enhanced sampling simulation, we need to correct the feature pairwise probability distribution because each feature sample $\\bx$ has an associated statistical weight $w(\\bx)$. To this aim, we introduce a reweighted Gaussian kernel as:\n\\begin{equation}\n  \\label{eq:reweighted_kernel}\n  \\tilde{K}_{\\varepsilon_{i}}(\\bx_i, \\bx_j) =\n    r(\\bx_i, \\bx_j) K_{\\varepsilon_{i}}(\\bx_i,\\bx_j),\n\\end{equation}\nwhere $r(\\bx_i, \\bx_j)=\\sqrt{w(\\bx_i)w(\\bx_j)}$ is a pairwise reweighting factor. As noted previously, the exact expression for the weights depends on the enhanced sampling method used. For training data from an unbiased simulation, or if we do not incorporate the weights into the training, all the weights are equal to one and $r(\\bx_i, \\bx_j) \\equiv 1$ for $1 \\leq i,j \\leq N$.\n\nA reweighted pairwise probability distribution for the feature space is then written as:\n\\begin{equation}\n\\label{eq:ppp}\n  \\sfP=\n    \\Big(\n      p^{\\boldsymbol{\\varepsilon}}_{ij}\n    \\Big)_{1\\leq i,j \\leq N}\n    ~~\\text{and}~~\n  p^{\\boldsymbol{\\varepsilon}}_{ij}=\n    \\frac{\\tilde{K}_{\\varepsilon_{i}}(\\bx_i, \\bx_j)}\n      {\\sum_{k} \\tilde{K}_{\\varepsilon_{i}}(\\bx_i, \\bx_k)},\n\\end{equation}\nwith $p^{\\boldsymbol{\\varepsilon}}_{ii}=0$. This equation represents the reweighted pairwise probability of features $\\bx_i$ and $\\bx_j$ for a given set of scale parameters $\\boldsymbol{\\varepsilon}=[\\varepsilon_1,\\varepsilon_2, \\dots, \\varepsilon_N]$, where each scale parameter is assigned to a row of the matrix $\\sfP$. The pairwise probabilities $p^{\\boldsymbol{\\varepsilon}}_{ij}$ are not symmetric due to the different values of the scale parameters ($\\varepsilon_i \\neq \\varepsilon_j$), which is in contrast to $t$-SNE, where the symmetry of the feature pairwise probability distribution is enforced~\\cite{maaten2008visualizing}.\n\nAs explained in Section~\\ref{sec:multiscale_rep} below, the multiscale feature pairwise probability distribution ${\\bf M}$ is written as a mixture of such pairwise probability distributions, each with a different set of scale parameters. In the next section, we describe how to calculate the scale parameters for the probability distribution given by eq~\\ref{eq:ppp}.\n\n\n\n\n\n\\subsubsection{Entropy of the Reweighted Feature Probability Distribution}\n\\label{sec:entropies}\nThe scale parameters $\\boldsymbol{\\varepsilon}$ used for the reweighted Gaussian kernels in eq~\\ref{eq:ppp} are positive scaling factors that need to be optimized to obtain a proper density estimation of the underlying data. We have that $\\varepsilon_i=1\/(2\\sigma^2_i)$, where $\\sigma_i$ is the standard deviation (i.e., bandwidth) of the Gaussian kernel. Therefore, we want a smaller $\\sigma_i$ in dense regions and a larger $\\sigma_i$ in sparse regions. To achieve this task, we define the Shannon entropy of the $i$th Gaussian probability as:\n\\begin{equation}\n  \\label{eq:information_entropy}\n  H(\\bx_i)=-\\sum_j p_{ij}^{\\varepsilon_i} \\log p_{ij}^{\\varepsilon_i},\n\\end{equation}\nwhere the term $p_{ij}^{\\varepsilon_i}$ refers to matrix elements from the $i$th row of $\\sfP$ as eq~\\ref{eq:information_entropy} is solved for each row independently. We can write $p_{ij}^{\\varepsilon_i} = \\frac{1}{\\bar{p}_i} \\tilde{K}_{\\varepsilon_{i}}(\\bx_i, \\bx_j)$ where $\\bar{p}_i=\\sum_k \\tilde{K}_{\\varepsilon_{i}}(\\bx_i, \\bx_k)$ is a row-wise normalization constant.\n\nInserting $p_{ij}^{\\varepsilon_i}$ from eq~\\ref{eq:ppp} leads to the following expression:\n\\begin{align}\n  \\label{eq:ent}\n  H(\\bx_i) = \\log\\bar{p}_i\n    &+ \\frac{\\varepsilon_{i}}{ \\bar{p}_i }\n       \\sum_j \\tilde{K}_{\\varepsilon_{i}}(\\bx_i, \\bx_j) \\|\\bx_i-\\bx_j\\|_2^2 \\nonumber\\\\\n    & \\underbrace{ -\\frac{1}{ \\bar{p}_i }\\sum_j \\tilde{K}_{\\varepsilon_{i}}(\\bx_i, \\bx_j) \\log r(\\bx_i,\\bx_j) }_{ H_V(\\bx_i) },\n\\end{align}\nwhere $H_V(\\bx_i)$ is a correction term due to the reweighting factor $r(\\bx_i,\\bx_j)$ introduced in eq~\\ref{eq:reweighted_kernel}. The reweighting factor is included also in the other two terms through $\\tilde{K}_{\\varepsilon_{i}}(\\bx_i, \\bx_j)$. For weights of exponential form, like in WT-MetaD (eq~\\ref{eq:weight-wtm}), we have $w(\\bx_i)=\\e^{\\beta V(\\bx_i)}$, and the correction term $H_V(\\bx_i)$ further reduces to:\n\\begin{equation}\n  \\label{eq:entropy_expl}\n  H_V(\\bx_i)=-\\frac{\\beta}{2}\n  \\left(\n    \\frac{1}{\\bar{p}_i}\n    \\sum_j \\tilde{K}_{\\varepsilon_{i}}(\\bx_i, \\bx_j) V(\\bx_i) + V(\\bx_j)\n  \\right).\n\\end{equation}\nFor the derivation of eq~\\ref{eq:ent} and eq~\\ref{eq:entropy_expl}, see Section~S1 in the Supporting Information (SI).\n\nFor an unbiased simulation, or if we do not incorporate the weights into the training, is $r(\\bx_i,\\bx_j)\\equiv 1$ for $1\\leq i,j \\leq N$ and the correction term $H_V(\\bx_i)$ vanishes. Equation~\\ref{eq:ent} then becomes $H(\\bx_i) = \\log\\bar{p}_i + \\frac{\\varepsilon_{i}}{ \\bar{p}_i }\\sum_j {K}_{\\varepsilon_{i}}(\\bx_i, \\bx_j) \\|\\bx_i-\\bx_j\\|_2^2$.\n\nWe use eq~\\ref{eq:ent} to define an objective function for an optimization procedure that fits the Gaussian kernel to the data by adjusting the scale parameter so that $H(\\bx_i)$ is approximately $\\log_2 PP$ (i.e., $\\min_{\\varepsilon_{i}} \\left[H(\\bx_i)-\\log_2 PP\\right]$). Here $PP$ is a model parameter that represents the perplexity of a discrete probability distribution. Perplexity is defined as an exponential of the Shannon entropy, $PP = 2^H$, and measures the quality of predictions for a probability distribution~\\cite{cover2006elements}. We can view the perplexity as the effective number of neighbors in a manifold~\\cite{maaten2008visualizing,maaten2009learning}. To find the optimal values of the scale parameters, we perform the optimization using a binary search separately for each row of $\\sfP$ (eq~\\ref{eq:ppp}).\n\n\n\n\n\n\\subsubsection{Multiscale Representation}\n\\label{sec:multiscale_rep}\nAs suggested in the work of Hinton and Roweis~\\cite{hinton2002stochastic}, the feature probability distribution can be extended to a mixture, as done in refs~\\citenum{lee2014multiscale,de2018perplexity,crecchi2020perplexity}. To this aim, for a given value of the perplexity $PP$, we find the optimal set of scale parameters $\\boldsymbol{\\varepsilon}^{PP}$ using eq~\\ref{eq:ent}. We do this for multiple values of the perplexity, $PP_l=2^{L_{PP}-l+1}$, where $l$ goes from 0 to $L_{PP}={\\lfloor \\log N \\rfloor}$-2, and $N$ is the size of the training data set. We then write the probabilities $p_{ij}$ as an average over the different reweighted feature pairwise probability distributions:\n\\begin{equation}\n  {\\bf M}=\\Big(p_{ij}\\Big)_{1\\leq i,j \\leq N}\n  ~~\\text{and}~~\n  p_{ij} = \\frac{1}{N_{PP}} \\sum^{L_{PP}}_{l=0} p^{\\boldsymbol{\\varepsilon}^{PP_l}}_{ij},\n\\end{equation}\nwhere $N_{PP}$ is the number of perplexities. Therefore, by taking $p_{ij}$ as a Gaussian mixture over different perplexities, we obtain a multiscale representation of the feature probability distribution ${\\bf M}$, without the need of setting perplexity by the user.\n\n\n\n\n\n\\subsubsection{Latent Pairwise Probability Distribution}\n\\label{sec:latent_distribution}\nA known issue in many dimensionality reduction methods, including SNE, is the so-called ``crowding problem''~\\cite{sammon1969nonlinear,hinton2002stochastic}, which is caused partly by the curse of dimensionality~\\cite{marimont1979nearest}. In the context of enhanced sampling, the crowding problem would lead to the definition of CVs that inadequately discriminate between metastable states due to highly localized kernel functions in the latent space. As shown in Figure~\\ref{fig:probabilities}, if we change from a Gaussian kernel to a more heavy-tailed kernel for the latent space probability distribution, like a $t$-distribution kernel, we enforce that close-by data points are grouped while far-away data points are separated.\n\nTherefore, for the pairwise probability distribution in the latent space, we use a one-dimensional heavy-tailed $t$-distribution, which is the same as in $t$-SNE. We set:\n\\begin{equation}\n\\label{eq:Q}\n\\sfQ=\\Big(q_{ij}\\Big)_{1\\leq i,j \\leq N}\n~~\\text{and}~~\n  q_{ij}=\\frac{\\left(1+\\|\\bs_i-\\bs_j\\|^2_2\\right)^{-1}}{\\sum_{k}%\n    \\left(1+\\|\\bs_i-\\bs_k\\|^2_2\\right)^{-1}},\n\\end{equation}\nwhere $q_{ii}=0$ and the latent variables (i.e., the CVs) are obtained via the embedding function, e.g., $\\bs_i = f_\\bt(\\bx_i)$.\n\n\n\n\n\n\\subsubsection{Minimization of Loss Function}\n\\label{sec:kl_div}\nFor the loss function to be minimized during the training procedure, we use the Kullback-Leibler (KL) divergence $\\kldiv$ to measure the statistical distance between the pairwise probability distributions ${\\bf M}$ and $\\sfQ$~\\cite{kullback1951information}. The loss function $L$ for a data batch is defined as:\n\\begin{align}\n\\label{eq:kl}\n  \\kldiv = \\frac{1}{N_b}\\sum_{i=1}^{N_b}\\sum_{\\substack{j=1 \\\\ i \\neq j}}^{N_b} p_{ij}\\log\\left(\\frac{p_{ij}}{q_{ij}}\\right),\n\\end{align}\nwhere $\\kldiv \\geq 0$ with equality only when ${\\bf M}=\\sfQ$, and we split the training data into $B$ batches of size $N_b$. We show the derivation of the loss function for the full set of $N$ training data points in Section~S2 in the SI.\n\n\\begin{figure}[htp]\n  \\includegraphics[width=0.4\\columnwidth]{fig-nn.pdf}\n  \\caption{Neural network used to model the parametric embedding function $f_\\bt(\\bx)$. The input features $\\bx$, $\\dim(\\bx)=k$ are fed into the NN to generate the output CVs $\\bs$, $\\dim(\\bs)=d$. The parameters $\\bt$ represent the weights and biases of NN. The input layer is shown in blue, and the output layer is depicted in red. The hidden layers (gray) use dropout and leaky ReLU activations.}\n  \\label{fig:nn}\n\\end{figure}\n\nFor the parametric embedding function $f_\\bt(\\bx)$, we employ a deep NN (see Figure~\\ref{fig:nn}). After minimizing the loss function, we can use the parametric NN embedding function to project any given point in feature space to the latent space without rerunning the training procedure. Therefore, we can use the embedding as CVs, $\\bs(\\bx)=f_\\bt(\\bx)$. The derivatives of $f_\\bt(\\bx)$ with respect to $\\bx$ are obtained using backpropagation. Using the chain rule, we can then calculate the derivatives of $\\bs(\\bx)$ with respect to the microscopic coordinates $\\mathbf{R}$, which is needed to calculate the biasing force in an enhanced sampling simulation.\n\n\n\n\n\n\\subsection{Weight-Tempered Random Sampling of Landmarks}\n\\label{sec:wtrs}\nA common way to reduce the size of a training set is to employ a landmark selection scheme before performing a dimensionality reduction~\\cite{ceriotti2013demonstrating,long2019landmark,tribello2019using_a,tribello2019using_b}. The idea is to select a subset of the feature samples (i.e., landmarks) representing the underlying characteristics of the simulation data.\n\nWe can achieve this by selecting the landmarks randomly or with some given frequency in an unbiased simulation. If the unbiased simulation has sufficiently sampled phase space or if we use an enhanced sampling method that preserves the equilibrium distribution, like parallel tempering (PT)~\\cite{parallel_tempering}, the landmarks represent the equilibrium Boltzmann distribution. However, such a selection of landmarks might give an inadequate representation of transient metastable states lying higher in free energy, as they are rarely observed in unbiased simulations sampling the equilibrium distribution.\n\nFor simulation data resulting from an enhanced sampling simulation, we need to account for sampling from a biased distribution when selecting the landmarks. Thus, we take the statistical weights $w(\\bR)$ into account within the landmark selection scheme. Ideally, we want the landmarks obtained from the biased simulation to strike a balance between an equilibrium representation and capturing higher-lying metastable states. Inspired by well-tempered farthest-point sampling (WT-FPS)~\\cite{ceriotti2013demonstrating} (see Section~S3 in the SI), we achieve this by proposing a simple landmark selection scheme appropriate for enhanced sampling simulations that we call weight-tempered random sampling.\n\nIn weight-tempered random sampling, we start by modifying the underlying data density by rescaling the statistical weights of the feature samples as $w(\\bR) \\rightarrow [w(\\bR)]^{1\/\\alpha}$. Here, $\\alpha \\geq 1$ is a tempering parameter similar in a spirit to the bias factor $\\gamma$ in the well-tempered distribution (eq~\\ref{eq:wt-pv}). Next, we randomly sample landmarks according to the rescaled weights. This procedure results in landmarks distributed according to the following probability distribution:\n\\begin{equation}\n  P_\\alpha(\\bx) =\n  \\frac\n  {\\int\\d\\bR \\, \\left[w(\\bR)\\right]^{1\/\\alpha} \\delta\\left[\\bx-\\bx(\\bR)\\right] P_V(\\bR)}\n  {\\int\\d\\bR \\, \\left[w(\\bR)\\right]^{1\/\\alpha} P_V(\\bR)},\n\\label{eq:wt-random-sampling}\n\\end{equation}\nwhich we can rewrite as a biased ensemble average:\n\\begin{equation}\n  P_\\alpha(\\bx) =\n  \\frac\n  {\\Big< [w(\\bR)]^{1\/\\alpha} \\delta[\\bx - \\bx(\\bR)] \\Big>_V}\n  {\\Big< [w(\\bR)]^{1\/\\alpha} \\Big>_V}.\n\\label{eq:wt-palpha}\n\\end{equation}\nSimilar weight transformations have been used for treating weights degeneracy in importance sampling~\\cite{koblents2015population}.\n\nFor $\\alpha=1$, we recover weighted random sampling~\\cite{bortz1975new}, where we sample landmarks according to their unscaled weights $w(\\bR)$. As we can see from eq~\\ref{eq:wt-random-sampling}, this should, in principle, give an equilibrium representation of landmarks, $P_{\\alpha=1}(\\bx)=P(\\bx)$. By employing $\\alpha>1$, we gradually start to ignore the underlying weights when sampling the landmarks and enhance the representation of metastable states lying higher in free energy. In the limit of $\\alpha\\to\\infty$, we ignore the weights (i.e., all are equal to unity) and sample the landmarks randomly so that their distribution should be equal to the biased feature distribution sampled under the influence of the bias potential, $P_{\\alpha\\to\\infty}(\\bx)=P_{V}(\\bx)$. Therefore, the tempering parameter $\\alpha$ allows us to tune the landmark selection between these two limits of equilibrium and biased representation. Using $\\alpha>1$ that is not too large, we can obtain a landmark selection that makes a trade-off between an equilibrium representation and capturing higher-lying metastable states.\n\nTo understand better the effect of the tempering parameter $\\alpha$, we can look at how the landmarks are distributed in the space of the biased CVs for the well-tempered case (eq~\\ref{eq:wt-pv}). As shown in Section~S4 in the SI, we obtain:\n\\begin{equation}\n  P_\\alpha(\\bs) = \\frac{ \\left[P(\\bs)\\right]^{1\/\\tilde{\\alpha} }}\n    { \\int\\d\\bs\\; \\left[P(\\bs)\\right]^{1\/\\tilde{\\alpha} }},\n    \\label{eq:effective_alpha_pdf}\n\\end{equation}\nwhere we introduce an effective tempering parameter $\\tilde{\\alpha}$ as:\n\\begin{equation}\n  \\tilde{\\alpha}=\\left( \\frac{1}{\\alpha} - \\frac{1}{\\alpha\\gamma} + \\frac{1}{\\gamma} \\right)^{-1} = \\frac{\\gamma \\alpha} {\\gamma + \\alpha - 1}\n  \\label{eq:effective_alpha}\n\\end{equation}\nthat is unity for $\\alpha=1$ and goes to $\\gamma$ in the limit $\\alpha\\to\\infty$. Thus, the effect of $\\alpha$ is to broaden the CV distribution of the selected landmarks. In Figure~\\ref{fig:ala-wt-eff-alpha}, we show how the effective tempering parameter $\\tilde{\\alpha}$ depends on $\\alpha$ for typical bias factor values $\\gamma$.\n\n\\begin{figure}[htp]\n  \\includegraphics[width=0.5\\columnwidth]{fig-ala1-effective-alpha.pdf}\n  \\caption{The effective tempering parameter $\\tilde{\\alpha}$ in the weight-tempered random sampling landmark selection scheme.}\n  \\label{fig:ala-wt-eff-alpha}\n\\end{figure}\n\nThe effect of $\\alpha$ on the landmark feature distribution $P_\\alpha(\\bx)$ is harder to gauge as we cannot write the biased feature distribution $P_V(\\bx)$ as a closed-form expression. In particular, for the well-tempered case, $P_V(\\bx)$ is not given by $\\propto [P(\\bx)]^{1\/\\gamma}$, as the features are generally not fully correlated to the biased CVs~\\cite{gil2015enhanced}. The correlation of the features with biased CVs will vary greatly, also within the selected feature set. For example, for features uncorrelated to the biased CVs, the biased distribution is nearly the same as the unbiased distribution. Consequently, the effect of tempering parameter $\\alpha$ for a given feature will depend on the correlation with the biased CVs. In Section~\\ref{sec:ala1_results}, we will show examples of this issue.\n\n\n\n\n\n\\subsection{Implementation}\n\\label{sec:implementation}\nWe implement the MRSE method and the weight-tempered random sampling landmark selection method in an additional module called \\texttt{LowLearner} in a development version (2.7.0-dev) of the open-source \\textsc{plumed}~\\cite{tribello2014plumed,plumed-nest} enhanced sampling plugin. The implementation is available openly at Zenodo~\\cite{mrse-dataset} (DOI: \\href{https:\/\/zenodo.org\/record\/4756093}{\\texttt{10.5281\/zenodo.4756093}}) and from the \\textsc{plumed} NEST~\\cite{plumed-nest} under \\texttt{plumID:21.023} at \\url{https:\/\/www.plumed-nest.org\/eggs\/21\/023\/}. We use the LibTorch~\\cite{pytorch} library (PyTorch C++ API, git commit \\texttt{89d6e88} used to obtain the results in this paper) that allows us to perform immediate execution of dynamic tensor computations with automatic differentiation~\\cite{paszke2017automatic}.\n\n\n\n\n\n\\section{Computational Details}\n\\label{sec:comp_details}\n\\subsection{Model Systems}\n\\label{sec:model_systems}\nWe consider three different model systems to evaluate the performance of the MRSE approach: the M\\\"uller-Brown Potential, alanine dipeptide, and alanine tetrapeptide. We use WT-MetaD simulations to generate biased simulation data sets used to train the MRSE embeddings for all systems. We also run unbiased simulation data sets for alanine di- and tetrapeptide by performing PT simulations that ensure proper sampling of the equilibrium distribution.\n\n\n\n\n\n\\subsubsection{M\\\"uller-Brown Potential}\n\\label{sec:mb_details}\nWe consider the dynamics of a single particle moving on the two-dimensional M\\\"uller-Brown potential~\\cite{MuellerBrown_1979}, $U(x, y)=\\sum_{j} A_{j}\\e^{p_{j}(x,y)}$, where $p_{j}(x,y)=a_{j}(x-x_{0,j})^2 + b_{j}(x-x_{0,j})(y-y_{0,j}) + c_{j} (y-y_{0,j})^2$, $x, y$ are the particle coordinates, and $\\mathbf{A}, \\mathbf{a}, \\mathbf{b}, \\mathbf{c}, \\mathbf{x}_{0}$ and $\\mathbf{y}_{0}$ are the parameters of the potential given by $\\mathbf{A}=(-40, -20, -34, 3)$, $\\mathbf{a}=(-1, -1, 6.5, 0.7)$, $\\mathbf{b}=(0,0,11,0.6)$, $\\mathbf{c}=(-10,-10,-6.5,-0.7)$, $\\mathbf{x}_{0}=(1,0,-0.5,-1$), and $\\mathbf{y}_{0}=(0,0.5,1.5,1)$. Note that the $\\mathbf{A}$ parameters are not the same as in ref~\\citenum{MuellerBrown_1979} as we scale the potential to reduce the height of the barrier by a factor of 5. The FES as a function of the coordinates $x$ and $y$ is given directly by the potential, $F(x,y)=U(x,y)$. We employ rescaled units such that $k_{\\mathrm B}=1$. We use the \\texttt{pesmd} code from \\textsc{plumed}~\\cite{tribello2014plumed,plumed-nest} to simulate the system at a temperature of $T=1$ using a Langevin thermostat~\\cite{bussi_accurate_2007} with a friction coefficient of 10 and employ a time step of 0.005. At this temperature, the potential has a barrier of around 20 $k_{\\mathrm B}T$ between its two states and thus is a rare event system.\n\nFor the WT-MetaD simulations, we take $x$ and $y$ as CVs. We use different bias factors values (3, 4, 5, and 7), an initial Gaussian height of 1.2, a Gaussian width of 0.1 for both CVs, and deposit Gaussians every 200 steps. We calculate $c(t)$ (eq~\\ref{eq:coft}), needed for the weights, every time a Gaussian is added using a grid of $500^2$ over the domain $[-5,5]^2$. We run the WT-MetaD simulations for a total time of $2 \\times 10^7$ steps. We skip the first 20\\% of the runs (up to step $4 \\times 10^6$) to ensure that we avoid the period at the beginning of the simulations where the weights might be unreliable due to rapid changes of the bias potential. For the remaining part, we normalize the weights such that they lie in the range 0 to 1 to avoid numerical issues.\n\nWe employ features saved every 1600 steps for the landmark selection data sets, yielding a total of $10^4$ samples. From these data sets, we then use weight-tempered random sampling with $\\alpha=2$ to select 2000 landmarks that we use as training data to generate the MRSE embeddings.\n\nFor the embeddings, we use the coordinates $x$ and $y$ as input features ($k=2$), while the number of output CVs is also 2 ($d=2$). We do not standardize or preprocess the input features.\n\n\n\n\n\n\\subsubsection{Alanine Dipeptide}\n\\label{sec:ala1_details}\nWe perform the alanine dipeptide (Ace-Ala-Nme) simulations using the \\textsc{gromacs} 2019.2 code~\\cite{gromacs} patched with a development version of the \\textsc{plumed} plugin~\\cite{tribello2014plumed,plumed-nest}. We use the Amber99-SB force field~\\cite{Hornak2006a}, and a time step of 2 fs. We perform the simulations in the canonical ensemble using the stochastic velocity rescaling thermostat~\\cite{bussi2007canonical} with a relaxation time of 0.1 fs. We constrain hydrogen bonds using LINCS~\\cite{hess2008p}. The simulations are performed in vacuum without periodic boundary conditions. We employ no cut-offs for electrostatic and non-bonded van der Waals interactions.\n\nWe employ 4 replicas with temperatures distributed geometrically in the range 300 K to 800 K (300.0 K, 416.0 K, 576.9 K, 800.0 K) for the PT simulation. We attempt exchanges between neighboring replicas every 10 ps. We run the PT simulation for 100 ns per replica. We only use the 300 K replica for analysis.\n\nWe perform the WT-MetaD simulations at 300 K using the backbone dihedral angles $\\Phi$ and $\\Psi$ as CVs and employ different values for the bias factor (2, 3, 5, and 10). We use an initial Gaussian height of 1.2 kJ\/mol, a Gaussian width of 0.2 rad for both CVs, and deposit Gaussians every 1 ps.  We calculate $c(t)$ (eq~\\ref{eq:coft}) every time a Gaussian is added (i.e., every 1 ps) employing a grid of $500^2$ over the domain $[-\\pi,\\pi]^2$. We run the WT-MetaD simulations for 100 ns. We skip the first 20 ns of the runs (i.e., first 20\\%) to ensure that we avoid the period at the beginning of the simulations where the weights might be unreliable due to rapid changes in the bias potential. For the remaining part, we normalize the weights such that they lie in the range 0 to 1 to avoid numerical issues.\n\nFor the landmark selection data sets, we employ features saved every 1 ps, which results in data sets of $8 \\times 10^4$ and $1 \\times 10^5$ samples for the WT-MetaD and PT simulations, respectively. We select 4000 landmarks for the training from these data sets, using weighted random sampling for the PT simulation and weight-tempered random sampling for the WT-MetaD simulations ($\\alpha=2$ unless otherwise specified).\n\nFor the embeddings, we use 21 heavy atoms pairwise distances as input features ($k=21$) and the number of output CVs as 2 ($d=2$). To obtain an impartial selection of features, we start with all 45 heavy atoms pairwise distances. Then, to avoid unimportant features, we automatically check for low variance features and remove all distances with a variance below $2 \\times 10^{-4}$ nm$^{2}$ from the training set (see Section~S9 in the SI). This procedure removes 24 distances and leaves 21 distances for the embeddings (both training and projections). We standardize remaining distances individually such that their mean is zero and their standard deviation is one.\n\n\n\n\n\n\\subsubsection{Alanine Tetrapeptide}\n\\label{sec:ala3_details}\nWe perform simulations of alanine tetrapeptide (Ace-Ala$_3$-Nme) in vacuum using the \\textsc{gromacs} 2019.2 code~\\cite{gromacs} and a development version of the \\textsc{plumed} plugin~\\cite{tribello2014plumed,plumed-nest}. We use the same MD setup and parameters as for the alanine dipeptide system, e.g., the Amber99-SB force field~\\cite{Hornak2006a}, see Section~\\ref{sec:ala1_details} for further details.\n\nFor the PT simulation, we employ 8 replicas with temperatures ranging from 300 K to 1000 K according to a geometric distribution (300.0 K, 356.4 K, 424.3 K, 502.6 K, 596.9 K, 708.9 K, 842.0 K, 1000.0 K). We attempt exchanges between neighboring replicas every 10 ps. We simulate each replica for 100 ns. We only use the 300 K replica for analysis.\n\nWe perform the WT-MetaD simulation at 300 K using the backbone dihedral angles $\\Phi_1$, $\\Phi_2$, and $\\Phi_3$ as CVs and a bias factor of 5. We use an initial Gaussian height of 1.2 kJ\/mol, a Gaussian width of 0.2 rad, and deposit Gaussians every 1 ps. We run the WT-MetaD simulation for 200 ns. We calculate $c(t)$ every 50 ps using a grid of $200^3$ over the domain $[-\\pi,\\pi]^3$. We skip the first 40 ns of the run (i.e., first 20\\%) to ensure that we avoid the period at the beginning of the simulation where the weights are not equilibrated. We normalize the weights such that they lie in the range 0 to 1.\n\nFor the landmark selection data sets, we employ features saved every 2 ps for the WT-MetaD simulation and every 1 ps for the PT simulation. This results in data sets of $8 \\times 10^4$ and $1 \\times 10^5$ samples for the WT-MetaD and PT simulations, respectively. We select 4000 landmarks for the training from these data sets, using weighted random sampling for the PT simulation and weight-tempered random sampling with $\\alpha=2$ for the WT-MetaD simulations.\n\nFor the embeddings, we use sines and cosines of the dihedral angles $(\\Phi_1,\\Psi_1,\\Phi_2,\\Psi_2,\\Phi_3,\\Psi_3)$ as input features ($k=12$), and the number of output CVs is 2 ($d=2$). We do not standardize or preprocess the input features further.\n\n\n\n\n\n\\subsection{Neural Network Architecture}\n\\label{sec:nn_architecture}\nFor the NN, we use the same size and number of layers as in the work of van der Maaten and Hinton~\\cite{maaten2009learning,hinton2006reducing}. The NN consists of an input layer with a size equal to the dimension of the feature space $k$, followed by three hidden layers of sizes $h_1=500$, $h_2=500$, and $h_3=2000$, and an output layer with a size equal to the dimension of the latent space $d$.\n\nTo allow for any output value, we do not wrap the output layer within an activation function. Moreover, for all hidden layers, we employ leaky rectified linear units (leaky ReLU)~\\cite{maas2013rectifier} with a leaky parameter set to $0.2$. Each hidden layer is followed by a dropout layer~\\cite{dropout} (dropout probability $p=0.1$). For the details regarding the architecture of NNs, see Table~\\ref{tab:hyperparams}.\n\n\n\n\n\n\\subsection{Training Procedure}\n\\label{sec:training_details}\nWe shuffle the training data sets and divide them into batches of size 500. We initialize all trainable weights of the NNs with the Glorot normal scheme~\\cite{glorot2010understanding} using the gain value calculated for leaky ReLU. The bias parameters of the NNs are initialized with 0.005.\n\nWe minimize the loss function given by eq~\\ref{eq:kl} using the Adam optimizer~\\cite{kingma2014adam} with AMSGrad~\\cite{Reddi2019}, where we use learning rate $\\eta=10^{-3}$, and momenta $\\beta_1=0.9$ and $\\beta_2=0.999$. We also employ a standard L2 regularization term on the trainable network parameters in the form of weight decay set to $10^{-4}$. We perform the training for 100 epochs in all cases. The loss function learning curves for the systems considered here are shown in Section~S7 in the SI.\n\nWe report all hyperparameters used to obtain the results in this work in Table~\\ref{tab:hyperparams}. For reproducibility purposes, we also list the random seeds used while launching the training runs (the seed affects both the landmark selection and the shuffling of the landmarks during the training).\n\n\\begin{table*}[htp!]\n  \\centering\\footnotesize\n  \\caption{Hyperparameters used to obtain the results reported in this paper.}\n  \\begin{tabular}{|l|l|l|l|}\n  \\hline\n  Hyperparameter      & M\\\"uller-Brown         & Alanine dipeptide       & Alanine tetrapeptide      \\\\\n  \\hline\n  Features            & $x$ and $y$            & Heavy atom distances    & Dihedral angles (cos\/sin) \\\\\n  NN architecture     & [2, 500, 500, 2000, 2] & [21, 500, 500, 2000, 2] & [12, 500, 500, 2000, 2]   \\\\\n  Optimizer           & Adam (AMSGrad)         & Adam (AMSGrad)          & Adam (AMSGrad)            \\\\\n  Number of landmarks & $N=2000$               & $N=4000$                & $N=4000$                  \\\\\n  Batch size          & $N_b=500$              & $N_b=500$               & $N_b=500$                 \\\\\n  Training iterations & 100                    & 100                     & 100                       \\\\\n  Learning rate       & $\\eta=10^{-3}$         & $\\eta=10^{-3}$          & $\\eta=10^{-3}$            \\\\\n  Seed                & 111                    & 111 (SI: 222, 333)      & 111                       \\\\\n  Leaky parameter     & 0.2                    & 0.2                     & 0.2                       \\\\\n  Dropout             & $p=0.1$                & $p=0.1$                 & $p=0.1$                   \\\\\n  Weight decay        & $10^{-4}$              & $10^{-4}$               & $10^{-4}$                 \\\\\n  $\\beta_1,\\beta_2$   & 0.9 and 0.999          & 0.9 and 0.999           & 0.9 and 0.999             \\\\\n  \\hline\n  \\end{tabular}\n  \\label{tab:hyperparams}\n\\end{table*}\n\n\n\n\n\n\\subsection{Kernel Density Estimation}\n\\label{sec:kde}\nWe calculate FESs for the trained MRSE embeddings using kernel density estimation (KDE) with Gaussian kernels. We employ a grid of $200^2$ for the FES figures. We choose the bandwidths for each simulation data set by first estimating them using Silverman's rule and then adjusting the bandwidths by comparing the KDE FES to an FES obtained with a discrete histogram. We show a representative comparison between KDE and discrete FESs in Section~S6 in the SI. We employ reweighting for FESs from WT-MetaD simulation data where we weigh each Gaussian KDE kernel by the statistical weight $w(\\mathbf{R})$ of the given data point.\n\n\n\n\n\n\\subsection{Data Availability}\n\\label{sec:data}\nThe data supporting the results of this study are openly available at Zenodo~\\cite{mrse-dataset} (DOI: \\href{https:\/\/zenodo.org\/record\/4756093}{\\texttt{10.5281\/zenodo.4756093}}). \\textsc{plumed} input files and scripts required to replicate the results presented in the main text are available from the \\textsc{plumed} NEST~\\cite{plumed-nest} under \\texttt{plumID:21.023} at \\url{https:\/\/www.plumed-nest.org\/eggs\/21\/023\/}.\n\n\n\n\n\n\\section{Results}\n\\label{sec:results}\n\\subsection{M\\\"uller-Brown Potential}\n\\label{sec:mb_results}\nWe start by considering a single particle moving on the two-dimensional M\\\"uller-Brown potential shown in Figure~\\ref{fig:mb-emb}(a). We use this system as a simple test to check if the MRSE method can preserve the topography of the FES in the absence of any dimensionality reduction when performing a mapping with a relatively large NN.\n\n\\begin{figure}[htp]\n  \\includegraphics[width=0.8\\columnwidth]{fig-mb-emb.png}\n  \\caption{Results for the M\\\"uller-Brown potential. FESs for MRSE embeddings obtained from the WT-MetaD simulation ($\\gamma=5$). We show MRSE embeddings obtained with (b) and without (c) incorporating weights into the training via a reweighted feature pairwise probability distribution (see eq~\\ref{eq:reweighted_kernel}). The units for the MRSE embeddings are arbitrary and only shown as a visual guide. To facilitate comparison, we post-process the MRSE embeddings using the Procrustes algorithm to find an optimal rotation that best aligns with the original coordinates $x$ and $y$, see text.}\n  \\label{fig:mb-emb}\n\\end{figure}\n\nWe train the MRSE embeddings on simulation data sets obtained from WT-MetaD simulations using the coordinates $x$ and $y$ as CVs. Here, we show only the results obtained with bias factor $\\gamma=5$, while the results for other values are shown in Section~S8 in the SI. The MRSE embeddings can be freely rotated and overall rotation is largely determined by the random seed used to generate the embeddings. Therefore, to facilitate comparison, we show here results obtained using the Procrustes algorithm to find an optimal rotation of the MRSE embeddings that best aligns with the original coordinates $x$ and $y$. The original non-rotated embeddings are shown in Section~S8 in the SI. We present the FESs obtained with the MRSE embeddings in Figure~\\ref{fig:mb-emb}(b-c). We can see that the embeddings preserve the topography of the FESs very well and demonstrate a fine separation of metastable states, both when we incorporate the weights into the training through eq~\\ref{eq:reweighted_kernel} (panel b), and when we do not (panel c).\n\\begin{figure}[htp]\n  \\includegraphics[width=0.4\\columnwidth]{fig-mb-vs.pdf}\n  \\caption{Results for the M\\\"uller-Brown potential. We show how the MRSE embeddings map the coordinates $x$ and $y$ by plotting the normalized coordinates $x$ and $y$ versus the normalized MRSE CVs. The MRSE embeddings are trained using data from a WT-MetaD simulation with $\\gamma=5$, and obtained with (red) and without (blue) incorporating weights into the training via a reweighted feature pairwise probability distribution (see eq~\\ref{eq:reweighted_kernel}). To facilitate comparison, we post-process the MRSE embeddings using the Procrustes algorithm to find an optimal rotation that best aligns with the original coordinates $x$ and $y$, see text.}\n  \\label{fig:mb-vs}\n\\end{figure}\n\nTo quantify the difference between the $x$ and $y$ coordinates and the CVs found by MRSE, we normalize all coordinates and plot CV$_1$ as a function of $x$ and CV$_2$ as a function of $y$. In Figure~\\ref{fig:mb-vs}, we can see that the points lie along the identity line, which shows that both MRSE embeddings preserve well the original coordinates of the MB system. In other words, the embeddings maintain the normalized distances between points. We analyze this aspect in a detailed manner for a high-dimensional set of features in Section~\\ref{sec:ala1_results}.\n\n\n\n\n\n\\subsection{Alanine Dipeptide}\n\\label{sec:ala1_results}\nNext, we consider alanine dipeptide in vacuum, a small system often used to benchmark free energy and enhanced sampling methods. The free energy landscape of the system is described by the backbone $(\\Phi,\\Psi)$ dihedral angles. Generally, the $(\\Phi,\\Psi)$ angles are taken as CVs for biasing, as we do here to generate the training data set. However, for this particular setup in vacuum, it is sufficient to bias $\\Phi$ to drive the sampling between states as $\\Psi$ is a fast CV compared to $\\Phi$. We can see in Figure~\\ref{fig:ala-mol} that three metastable states characterize the FES. The C$7_{\\mathrm{eq}}$ and C$5$ states are separated only by a small barrier of around 1--2 $k_{\\mathrm{B}}T$, so transitions between these two states are frequent. The C$7_{\\mathrm{ax}}$ state lies higher in free energy (i.e., is less probable to sample), and is separated by a high barrier of around 14 $k_{\\mathrm{B}}T$ from the other two states, so transitions from C$7_{\\mathrm{eq}}$\/C$5$ to C$7_{\\mathrm{ax}}$ are rare.\n\\begin{figure}[htp]\n  \\includegraphics[width=0.5\\linewidth]{fig-ala1-mol.pdf}\n  \\caption{Results for alanine dipeptide in vacuum at 300 K. (a) The free energy landscape $F(\\Phi,\\Psi)$ from the PT simulation. The metastable states C$7_{\\mathrm{eq}}$, C5, and C$7_{\\mathrm{ax}}$ are shown. (b) The molecular structure of alanine dipeptide with the dihedral angles $\\Phi$ and $\\Psi$ indicated.}\n  \\label{fig:ala-mol}\n\\end{figure}\n\nFor the MRSE embeddings, we do not use the $(\\Phi,\\Psi)$ angles as input features, but rather a set of 21 heavy atom pairwise distances that we impartially select as described in Section~\\ref{sec:ala1_details}. Using only the pairwise distances as input features makes the exercise of learning CVs more challenging as the $\\Phi$ and $\\Psi$ angles cannot be represented as linear combinations of the interatomic distances. We can assess the quality of our results by examining how well the MRSE embeddings preserve the topography of the FES on local and global scales. However, before presenting the MRSE embeddings, let us consider the landmark selection, which we find crucial to our protocol to construct embeddings accurately.\n\nAs discussed in Section~\\ref{sec:wtrs}, we need to have a landmark selection scheme that takes into account the weights of the configurations and gives a balanced selection that ideally is close to the equilibrium distribution but represents all metastable states of the system, also the higher-lying ones. We devise for this task a method called weight-tempered random sampling. This method has a tempering parameter $\\alpha$ that allows us to interpolate between an equilibrium and a biased representation of landmarks (see eq~\\ref{eq:wt-random-sampling}).\n\n\\begin{figure}[htp]\n  \\includegraphics[width=0.4\\columnwidth]{fig-ala1-wt-sampling.pdf}\n  \\caption{Results for alanine dipeptide in vacuum at 300 K. The effect of the tempering parameter $\\alpha$ in the weight-tempered random sampling landmark selection scheme for a WT-MetaD simulation ($\\gamma=5$) biasing $(\\Phi, \\Psi)$. Marginal landmark distributions for two examples of features (i.e., heavy atom distances)) from the feature set that are (a) correlated and (b) uncorrelated with the biased CVs. The units are nm.}\n  \\label{fig:ala-wt-sampling}\n\\end{figure}\n\nThe effect of the tempering parameter $\\alpha$ on the landmark feature distribution $P_\\alpha(\\bx)$ will depend on the correlation of the features with the biased CVs. The correlation will vary greatly, also within the selected feature set. In Figure~\\ref{fig:ala-wt-sampling}, we show the marginal distributions for two examples from the feature set. For a feature correlated with the biased CVs, the biasing enhances the fluctuations, and we observe a significant difference between the equilibrium distribution and the biased one, as expected. In this case, the effect of introducing $\\alpha$ is to interpolate between these two limits. On the other hand, for a feature not correlated to the biased CVs, the equilibrium and biased distribution are almost the same, and $\\alpha$ does not affect the distribution of this feature.\n\nIn Figure~\\ref{fig:ala-weights}, we show the results from the landmark selection for one of the WT-MetaD simulations ($\\gamma=5$). In the top row, we show how the selected landmarks are distributed in the CV space. In the bottom row, we show the effective FES of selected landmarks projected on the $\\Psi$ dihedral angle.\n\nFor $\\alpha=1$, equivalent to weighted random sampling~\\cite{tribello2019using_b}, we can see that we get a worse representation of the C$7_{\\mathrm{ax}}$ state as compared to the other states. We can understand this issue by considering the weights of configurations in the C$7_{\\mathrm{ax}}$ that are are considerably smaller than the weights from the other states. As shown in Section~S10 in the SI, using the $\\alpha=1$ landmarks results in an MRSE embedding close to the equilibrium PT embedding (shown in Figure~\\ref{fig:ala-embeddings}(a) below), but has a worse separation of the metastable states as compared to other embeddings.\n\nOn the other hand, if we use $\\alpha=2$, we obtain a much more balanced landmark selection that is relatively close to the equilibrium distribution but has a sufficient representation of the C$7_{\\mathrm{ax}}$ state. Using larger values of $\\alpha$ renders a selection closer to the sampling from the underlying biased simulation, with more features higher in free energy. We observe that using $\\alpha=2$ gives the best MRSE embedding. In contrast, higher values of $\\alpha$ result in worse embeddings characterized by an inadequate mapping of the C$7_{\\mathrm{ax}}$ state, as can be seen in Section~S12 in the SI. Therefore, in the following, we use a value of $\\alpha=2$ for the tempering parameter in the landmark selection. This value corresponds to an effective landmark CV distribution broadening of $\\tilde{\\alpha} \\approx 1.67$ (see eqs~\\ref{eq:effective_alpha_pdf} and~\\ref{eq:effective_alpha}).\n\n\\begin{figure}[htp]\n  \\includegraphics[width=0.9\\columnwidth]{fig-ala-weights.pdf}\n  \\caption{Results for alanine dipeptide in vacuum at 300 K. Weight-tempered random sampling as a landmark selection scheme for a WT-MetaD simulation ($\\gamma=5$) biasing $(\\Phi, \\Psi)$. (a) In the first two panels, we show the reference FES in the $(\\Phi, \\Psi)$ space and the points sampled during the simulations. In the subsequent panels, we present the 4000 landmarks selected for different values of the $\\alpha$ parameter. (b) In the bottom row, we show the results projected on $\\Phi$, where the reference FES is shown in light blue. The projections (black) are calculated as a negative logarithm of the histogram of the selected landmarks.}\n  \\label{fig:ala-weights}\n\\end{figure}\n\nThese landmark selection results underline the importance of having a balanced selection of landmarks that is close to the equilibrium distribution and gives a proper representation of all metastable states, but excludes points from unimportant higher-lying free energy regions.  The exact value of $\\alpha$ that achieves such optimal selection will depend on the underlying free energy landscape.\n\nIn Section~S11 in the SI, we show results obtained using WT-FPS for the landmark selection (see Section~S3 in the SI for a description of WT-FPS). We can observe that the WT-MetaD embeddings obtained using WT-FPS with $\\alpha=2$  are similar to the WT-MetaD embeddings shown in Figure~\\ref{fig:ala-embeddings} below. Thus, for small values of the tempering parameter, both methods give similar results.\n\nHaving established how to perform the landmark selection, we now consider the results for MRSE embeddings obtained on unbiased and biased simulation data at 300 K. The unbiased simulation data comes from a PT simulation that accurately captures the equilibrium distribution within each replica~\\cite{parallel_tempering}. Therefore, for the 300 K replica used for the analysis and training, we obtain the equilibrium populations of the different metastable states while not capturing the higher-lying and transition state regions. In principle, we could also include simulation data from the higher-lying replica into the training by considering statistical weights to account for the temperature difference, but this would defeat the purpose of using the PT to generate unbiased simulation data that does not require reweighting. We refer to the embedding trained on the PT simulation data as the PT embedding. The biased simulation data comes from WT-MetaD simulations where we bias the ($\\Phi$, $\\Psi$) angles. We refer to these embeddings as the WT-MetaD embeddings.\n\nIn the WT-MetaD simulations, we use bias factors from 2 to 10 to generate training data sets representing a biased distribution that progressively goes from a distribution closer to the equilibrium one to more flatter distribution as we increase $\\gamma$ (see eq~\\ref{eq:wt-pv}). In this way, we can test how the MRSE training and reweighting procedure works when handling simulation data obtained under different biasing strengths.\n\nFor the WT-MetaD training data sets, we also investigate the effect of not incorporating the weight into the training via a reweighted feature pairwise probability distribution (i.e., all weights equal to unity in eq~\\ref{eq:reweighted_kernel}). In this case, only the weight-tempered random sampling landmark selection takes the weights into account. In the following, we refer to these WT-MetaD embeddings as without reweighting or not-reweighted.\n\nTo be consistent and allow for a fair comparison between embeddings, we evaluate all the trained WT-MetaD embeddings on the unbiased PT simulation data and use the resulting projections to perform analysis and generate FESs. This procedure is possible as both the unbiased PT and the biased WT-MetaD simulations sample all metastable states of alanine dipeptide (i.e., the WT-MetaD simulations do not sample metastable states that the PT simulation does not).\n\n\\begin{figure}[htp]\n  \\includegraphics[width=0.9\\columnwidth]{fig-ala-clusters.pdf}\n  \\caption{Results for alanine dipeptide in vacuum at 300 K. Clustering of the PT simulation data for the different embeddings. The results show how the embeddings map the metastable states. The data points are colored accordingly to their cluster. The first panel shows the metastable state clusters in the $(\\Phi,\\Psi)$ space. The second panel shows the results for the PT embedding. The third and fourth panels show the results for a representative case of a WT-MetaD embedding ($\\gamma=5$), obtained with and without incorporating weights into the training via a reweighted feature probability distribution (see eq~\\ref{eq:reweighted_kernel}), respectively. For the details about clustering~\\cite{scikit-learn}, see Section~S5 in the SI. The units for the MRSE embeddings are arbitrary and only shown as a visual guide.}\n  \\label{fig:ala-cluster}\n\\end{figure}\n\nTo establish that the MRSE embeddings correctly map the metastable states, we start by considering the clustering results in Figure~\\ref{fig:ala-cluster}. We can see that the PT embedding (second panel) preserves the topography of the FES and correctly maps all the important metastable states. We can say the same for the reweighted (third panel) and not-reweighted (fourth panel) embeddings. Thus, the embeddings map both the local and global characteristics of the FES accurately. Next, we consider the MRSE embeddings for the different bias factors.\n\n\\begin{figure}[htp]\n  \\includegraphics[width=0.8\\linewidth]{fig-ala-embeddings.pdf}\n  \\caption{Results for alanine dipeptide in vacuum at 300 K. MRSE embeddings trained on unbiased and biased simulation data. (a) The free energy landscape $F(\\Phi,\\Psi)$ from the PT simulation. The metastable states C$7_{\\mathrm{eq}}$, C5, and C$7_{\\mathrm{ax}}$ are shown. (b) The FES for the MRSE embedding trained using the PT simulation data. (c) The FESs for the MRSE embeddings trained using the WT-MetaD simulation data. We show results obtained from the simulations using different bias factors $\\gamma$. We show WT-MetaD embeddings obtained with (top row) and without (bottom row) incorporating weights into the training via a reweighted feature pairwise probability distribution (see eq~\\ref{eq:reweighted_kernel}). We obtain all the FESs by calculating the embeddings on the PT simulation data and using kernel density estimation as described in Section~\\ref{sec:kde}. The units for the MRSE embeddings are arbitrary and only shown as a visual guide.}\n  \\label{fig:ala-embeddings}\n\\end{figure}\n\nIn Figure~\\ref{fig:ala-embeddings}, we show the FESs for the different embeddings along with the FES for the $\\Phi$ and $\\Psi$ dihedral angles. For the reweighted WT-MetaD embeddings (top row of panel c), we can observe that all the embeddings are of consistent quality and exhibit a clear separation of the metastable states. In contrast, we can see that the not-reweighted WT-MetaD embeddings (bottom row of panel c) have a slightly worse separation of the metastable states. Thus, we can conclude that incorporating the weights into the training via a reweighted feature pairwise probability distribution (see eq~\\ref{eq:reweighted_kernel}) improves the visual quality of the embeddings for this system.\n\n\\begin{figure}[htp]\n  \\includegraphics[width=0.5\\columnwidth]{fig-ala-fed.pdf}\n  \\caption{Results for alanine dipeptide in vacuum at 300 K. Free energy differences between metastable states for the FESs of the embeddings shown in Figure~\\ref{fig:ala-embeddings}. We show the reference values from the $F(\\Phi, \\Psi)$ FES obtained from the PT simulation at 300 K as horizontal gray lines. The results for the reweighted embeddings are shown as red crosses, while the results for the not-reweighted embeddings are shown as blue dots. The results for the PT embedding are shown as green plus symbols.}\n  \\label{fig:ala-fed}\n\\end{figure}\n\nTo further check the quality of the embeddings, we calculate the free energy difference between metastable states as $\\Delta F_{\\text{A,B}}=-\\frac{1}{\\beta}\\log(\\int_{\\text{A}}\\d\\bs\\,\\e^{-\\beta F(\\bs)} \/ \\int_{\\text{B}}\\d\\bs\\,\\e^{-\\beta F(\\bs)})$, where the integration domains are the regions in CV space corresponding to the states A and B, respectively. This equation is only valid if the CVs correctly discriminate between the different metastable states. For the MRSE embeddings, we can thus identify the integration regions for the different metastable states in the FES and calculate the free energy differences. Reference values can be obtained by integrating the $F(\\Phi,\\Psi)$ FES from the PT simulation. A deviation from a reference value would indicate that an embedding does not correctly map the density of the metastable states. In Figure~\\ref{fig:ala-fed}, we show the free energy differences for all the MRSE embeddings. All free energy differences obtained with the MRSE embeddings agree with the reference values within a 0.1 $\\kT$ difference for both reweighted and not-reweighted WT-MetaD embeddings. For bias factors larger than 3, we can observe that the reweighted embeddings perform distinctly better than the not-reweighted ones.\n\n\\begin{figure}[htp]\n  \\includegraphics[width=\\linewidth]{fig-ala-distances.pdf}\n  \\caption{Results for alanine dipeptide in vacuum at 300 K. The joint probability density functions for the pairwise distances in the high-dimensional feature space and the low-dimensional latent space for the embeddings shown in Figure~\\ref{fig:ala-embeddings}. We show the results for the (a) PT and (b) WT-MetaD embeddings (evaluated on the PT simulation data). These histograms show the similarities between distances in the feature and latent spaces. For an embedding that preserves distances accurately, the density would lie on the identity line $y=x$ (shown as a black line). We normalize the distances to lie in the range 0 to 1.}\n  \\label{fig:ala-distances}\n\\end{figure}\n\nAs a final test of the MRSE embeddings for this system, we follow the approach used by Tribello and Gasparotto~\\cite{tribello2019using_a,tribello2019using_b}. We calculate the pairwise distances between points in the high-dimensional feature space and the corresponding pairwise distances between points in the low-dimensional latent (i.e., CV) space given by the embeddings. We then calculate the joint probability density function of the distances using histogramming. The joint probability density should be concentrated on the identity line if an embedding preserves distances accurately. However, this only is valid for the MRSE embeddings constructed without incorporating the weights into the training, since for this case, there are no additional constraints besides geometry.\n\nAs we can see in Figure~\\ref{fig:ala-distances}, the joint density is concentrated close to the identity line for most cases. For the reweighted WT-MetaD embeddings (panel b), the density for the distances in the middle range slightly deviates from the identity line in contrast to the not-reweighted embeddings. This deviation is due to additional constraints on the latent space. In the reweighted cases, apart from the Euclidean distances, we also include the statistical weights into the construction of the feature pairwise probability distribution. Consequently, having landmarks with low weights in the feature space decreases the probability of being neighbors to these landmarks in the latent space. Therefore, the deviation from the identity line must be higher for the reweighted embeddings.\n\nSummarizing the results in this section, we can observe that MRSE can construct embeddings, both from unbiased and biased simulation data, that correctly describe the local and global characteristics of the free energy landscape of alanine dipeptide. For the biased WT-MetaD simulation data, we have investigated the effect of not including the weights in the training of the MRSE embeddings. Then only the landmark selection takes the weights into account. The not-reweighted embeddings are similar or slightly worse than the reweighted ones. We can explain the slight difference between the reweighted and not-reweighted embeddings by that the weight-tempered random sampling does the primary reweighting. Nevertheless, we can conclude that incorporating the weights into the training is beneficial for the alanine dipeptide test case.\n\n\n\n\n\n\\subsection{Alanine Tetrapeptide}\n\\label{sec:ala3_results}\nAs the last example, we consider alanine tetrapeptide, a commonly used test system for enhanced sampling methods~\\cite{valsson2015well,tiwary2016spectral,mccarty2017variational,yang2018refining,bonati2019neural,Invernizzi2020opus,gilberti2020atlas}. Alanine tetrapeptide is a considerably more challenging test case than alanine dipeptide. Its free energy landscape consists of many metastable states, most of which are high in free energy and thus difficult to capture in an unbiased simulation. We anticipate that we can only obtain an embedding that accurately separates all of the metastable states by using training data from an enhanced sampling simulation, which better captures higher-lying metastable states. Thus, the system is a good test case to evaluate the performance of the MRSE method and the reweighting procedure.\n\n\\begin{figure}[htp]\n  \\includegraphics[width=\\columnwidth]{fig-ala3-system.png}\n  \\caption{Results for alanine tetrapeptide in vacuum at 300 K. (a) The conditional FESs (eq~\\ref{eq:conditional_fe}), obtained from the WT-MetaD simulation, shown as a function of $\\Phi_1$ and $\\Phi_2$ for two minima of $\\Phi_3$ labeled as A and B. We denote the ten metastable states as $s_1$ to $s_{10}$. (b) The alanine tetrapeptide system with the backbone dihedral angles $\\mathbf{\\Phi} \\equiv (\\Phi_1,\\Phi_2,\\Phi_3)$ and $\\mathbf{\\Psi} \\equiv (\\Psi_1,\\Psi_2,\\Psi_3)$ that we use as the input features for the MRSE embeddings. (c) The free energy profile $F(\\Phi_3)$, obtained from the WT-MetaD simulation, with the two minima A and B. The grey shaded area indicates the areas integrated over in eq~\\ref{eq:conditional_fe}. The FESs are obtained using kernel density estimation as described in Section~\\ref{sec:kde}.}\n  \\label{fig:ala3-system}\n\\end{figure}\n\nAs it is often customary~\\cite{valsson2015well,tiwary2016spectral,Invernizzi2020opus,gilberti2020atlas}, we consider the backbone dihedral angles $\\mathbf{\\Phi} \\equiv (\\Phi_1,\\Phi_2,\\Phi_3)$ and $\\mathbf{\\Psi} \\equiv (\\Psi_1,\\Psi_2,\\Psi_3)$ that characterize the configurational landscape of alanine tetrapeptide. We show the dihedral angles in Figure~\\ref{fig:ala3-system}(b). For this particular setup in vacuum, it is sufficient to use $\\mathbf{\\Phi}$ to describe the free energy landscape and separate the metastable states, as $\\mathbf{\\Psi}$ are fast CVs in comparison to $\\mathbf{\\Phi}$~\\cite{valsson2015well,Invernizzi2020opus}. To generate biased simulation data, we perform a WT-MetaD simulation using the $\\mathbf{\\Phi}$ angles as CVs and a bias factor $\\gamma=5$. Moreover, we perform a PT simulation and employ the 300 K replica to obtain unbiased simulation data. As before, the embeddings obtained by training on these simulation data sets are denoted as WT-MetaD and PT embeddings, respectively. As before, we also consider a WT-MetaD embedding, denoted as not-reweighted, where we do not include the weights into the construction of the feature pairwise probability distribution.\n\nTo verify the quality of the sampling and the accuracy of the FESs, we compare the results obtained from the WT-MetaD and PT simulations to results from bias-exchange metadynamics simulations~\\cite{piana2007bias} using $\\mathbf{\\Phi}$ and $\\mathbf{\\Psi}$ as CVs (see Section~S13 in the SI). Comparing the free energy profiles for $\\mathbf{\\Phi}$ obtained with different methods (Figure~S12 in the SI), and keeping in mind that the 300 K replica from the PT simulation only describes well the lower-lying metastable states, we find that all simulations are in good agreement. Therefore, we conclude that the WT-MetaD and PT simulations are converged.\n\nTo show the results from the three-dimensional CV space on a two-dimensional surface, we consider a conditional FES where the landscape is given as a function of $\\Phi_1$ and $\\Phi_2$ conditioned on values of $\\Phi_3$ being in one of the two distinct minima shown in Figure~\\ref{fig:ala3-system}(c). We label these minima as A and B. We define the conditional FES as:\n\\begin{equation}\n  \\label{eq:conditional_fe}\n  F(\\Phi_1,\\Phi_2 | \\Phi_3 \\in S)=-\\frac{1}{\\beta}\\log\\int_{S} \\mathrm{d}\\Phi_3 \\, \\e^{-\\beta F(\\mathbf{\\Phi})},\n\\end{equation}\nwhere $F(\\mathbf{\\Phi})$ is the FES obtained from the WT-MetaD simulation (aligned such that its minimum is at zero), $S$ is either the A or B minima, and we integrate over the regions indicated by the gray areas in Figure~\\ref{fig:ala3-system}(c). We show the two conditional FESs in Figure~\\ref{fig:ala3-system}(a). Through a visual inspection of Figure~\\ref{fig:ala3-system}, we can identify ten different metastable states, denoted as $s_1$ to $s_{10}$. Three of the states, $s_{5}$, $s_{7}$, and $s_{8}$, are sampled properly in the 300 K replica of the PT simulation, and thus we consider them as the equilibrium metastable states. The rest of the metastable states are located higher in free energy and only sampled accurately in the WT-MetaD simulation. The number of the metastable states observed in Figure~\\ref{fig:ala3-system}(a) is in agreement with a recent study of Giberti et al.~\\cite{gilberti2020atlas}.\n\nWe can judge the quality of the MRSE embeddings based on whether they can correctly capture the metastable states in only two dimensions. As input features for the MRSE embeddings, we use sines and cosines of backbone dihedral angles $\\mathbf{\\Phi}$ and $\\mathbf{\\Psi}$ (12 features in total), instead of heavy atom distances as we do in the previous section for alanine dipeptide. We use weight-tempered random sampling with $\\alpha=2$ to select landmarks for the training of the WT-MetaD embeddings.\n\nWe show the PT and WT-MetaD embeddings in Figure~\\ref{fig:ala3-emb}. We can see that the PT embedding in Figure~\\ref{fig:ala3-emb}(a) is able to accurately describe the equilibrium metastable states (i.e., $s_5$, $s_7$, and $s_8$). However, as expected, the PT embedding cannot describe all ten metastable states, as the 300 K replica in the PT simulation rarely samples the higher-lying states.\n\\begin{figure}[htp]\n  \\includegraphics[width=0.9\\columnwidth]{fig-ala3-emb.png}\n  \\caption{Results for alanine tetrapeptide in vacuum at 300 K. FESs for the MRSE embeddings trained on the unbiased and biased simulation data. (a) The PT embedding trained and evaluated on the PT simulation data. (b-c) The WT-MetaD embeddings trained and evaluated on the WT-MetaD simulation data. The WT-MetaD embeddings are obtained without (b) and with (c) incorporating weights into the training via a reweighted feature pairwise probability distribution (see eq~\\ref{eq:reweighted_kernel}). The FESs are obtained using kernel density estimation as described in Section~\\ref{sec:kde}. The state labels in the FESs correspond to the labeling used in Figure~\\ref{fig:ala3-system}(a). The embeddings are rescaled so that the equilibrium states are of similar size. The units for the MRSE embeddings are arbitrary and thus not shown.}\n  \\label{fig:ala3-emb}\n\\end{figure}\n\nIn contrast, we can see that the WT-MetaD embeddings in Figure~\\ref{fig:ala3-emb}(b-c) capture accurately all ten metastable states. By visual inspection of the simulation data, we can assign state labels for the embeddings in Figure~\\ref{fig:ala3-emb}, corresponding to the states labeled in Figure~\\ref{fig:ala3-system}(a). One interesting aspect of the MRSE embeddings in Figure~\\ref{fig:ala3-emb} is that they similarly map the equilibrium states, even if we obtain the embeddings from different simulation data sets (PT and WT-MetaD). This similarity underlines the consistency of our approach. The fact that both the reweighted and not-reweighted WT-MetaD embeddings capture all ten states suggests we could use both embeddings as CVs for biasing.\n\nHowever, we can observe that the reweighted embedding has a better visual separation of the states. For example, we can see this for the separation between $s_9$ and $s_{10}$. Furthermore, we can see that the reweighted embedding separates the states from the A and B regions better than the not-reweighted embedding. In the reweighted embedding, states $s_1$ to $s_4$ are close to each other and separated from states $s_5$--$s_{10}$ as indicated by line drawn in Figure~\\ref{fig:ala3-emb}(c). Therefore, we can conclude that the reweighted WT-MetaD embedding is of better quality and better represents distances between metastable states for this system. These results show that we need to employ a reweighted feature pairwise probability distribution for more complex systems.\n\n\n\n\n\n\\section{Discussion and Conclusions}\n\\label{sec:discussion_conclusions}\nWe present multiscale reweighted stochastic embedding, a general framework that unifies enhanced sampling and machine learning for constructing collective variables. MRSE builds on top of ideas from stochastic neighbor embedding methods~\\cite{hinton2002stochastic,maaten2008visualizing,maaten2009learning,van2014accelerating}. We introduce several advancements to SNE methods that make MRSE suitable for constructing CVs from biased data obtained from enhanced sampling simulations.\n\nWe show that this method can construct CVs automatically by learning a mapping from a high-dimensional feature space to a low-dimensional latent space via a deep neural network. We can use the trained NN to project any given point in feature space to CV space without rerunning the training procedure. Furthermore, we can obtain the derivatives of the learned CVs with respect to the input features and bias the CVs within an enhanced sampling simulation. In future work, we will use this property by employing MRSE within an enhanced sampling scheme where the CVs are iteratively improved~\\cite{zhang2018unfolding,chen2018collective,ribeiro2018reweighted}.\n\nIn this work, we focus entirely on the training of the embeddings, using training data sets obtained from both unbiased simulation and biased simulation employing different biasing strengths (i.e., bias factors in WT-MetaD). As the ``garbage in, garbage out'' adage applies to ML (a model is only as good as training data), to eliminate the influence of incomplete sampling, we employ idealistic sampling conditions that are not always achievable in practice~\\cite{pant2020statistical}. In future work, we will need to consider how MRSE performs under less ideal sampling conditions. One possible option to address this issue is to generate multiple embeddings by running independent training attempts and score them using the maximum caliber principle, as suggested in ref~\\citenum{pant2020statistical}.\n\nThe choice of the input features depends on the physical system under study. In this work, we use conventional features, i.e., microscopic coordinates, distances, and dihedral angles, as they are a natural choice for the model systems considered here. In general, the features can be complicated functions of the microscopic coordinates~\\cite{musil2021physicsinspired}. For example, symmetry functions have been used as input features in studies of phase transformations in crystalline systems~\\cite{geiger2013neural,rogal2019neural}. Additionally, features may be correlated or simply redundant. See ref~\\citenum{dy2004feature} for a general outline of feature selection in unsupervised learning. We will explore the usage of more intricate input features and modern feature selection methods~\\cite{ravindra2020automatic,cersonsky2020improving} for MRSE embeddings in future work.\n\nOne of the issues with using kernel-based dimensionality reduction methods, such as diffusion maps~\\cite{coifman2008diffusion} or SNE methods~\\cite{hinton2002stochastic}, is that the user needs to select the bandwidths (i.e., the scale parameters $\\boldsymbol{\\varepsilon}$) when using the Gaussian kernels. In $t$-SNE~\\cite{maaten2008visualizing,maaten2009learning}, the Gaussian bandwidths are optimized by fitting to a parameter called perplexity. We can view the perplexity as the effective number of neighbors in a manifold~\\cite{maaten2008visualizing,maaten2009learning}. However, this only redirects the issue as the user still needs to select the perplexity parameter~\\cite{wattenberg2016how}. Larger perplexity values lead to a larger number of nearest neighbors and an embedding less sensitive to small topographic structures in the data. Conversely, lower perplexity values lead to fewer neighbors and ignore global information in favor of the local environment. However, what if several length scales characterize the data? In this case, it is impossible to represent the density of the data with a single set of bandwidths, so viewing multiple embeddings obtained with different perplexity values is quite common~\\cite{wattenberg2016how}.\n\nIn MRSE, we circumvent the issue of selecting the Gaussian bandwidths or the perplexity value by employing a multiscale representation of feature space. Instead of a single Gaussian kernel, we use a Gaussian mixture where each term has its bandwidths optimized for a different perplexity value. We perform this procedure in an automated way by employing a range of perplexity values representing several length scales. This mixture representation allows describing both the local and global characteristics of the underlying data topography. The multiscale nature of MRSE makes the method particularly suitable for tackling complex systems, where the free energy landscape consists of several metastable states of different sizes and shapes. However, as we have seen in Section~\\ref{sec:ala3_results}, also model systems may exhibit such complex behavior.\n\nEmploying nonlinear dimensionality reduction methods is particularly problematic when considering training data obtained from enhanced sampling simulations. In this case, the feature samples are drawn from a biased probability distribution, and each feature sample carries a statistical weight that we need to take into account. In MRSE, we take the weights into account when selecting the representative feature samples (i.e., landmarks) for the training. For this, we introduce a weight-tempered selection scheme that allows us to obtain landmarks that strike a balance between equilibrium distribution and capturing important metastable states lying higher in free energy. This weight-tempered random sampling method depends on a tempering parameter $\\alpha$ that allows us to tune between obtaining equilibrium and biased distribution of landmarks. This parameter is case-dependent and similar in spirit to the bias factor $\\gamma$ in WT-MetaD. Generally, $\\alpha$ should be selected so that every crucial metastable state is densely populated. However, $\\alpha$ should not be too large, as it may result in including feature samples from unimportant higher-lying free energy regions.\n\nThe weight-tempered random sampling algorithm is inspired by and bears a close resemblance to the well-tempered farthest-point sampling (WT-FPS) landmark selection algorithm, introduced by Ceriotti et al.~\\cite{ceriotti2013demonstrating}. For small values of the tempering parameter $\\alpha$, both methods give similar results as discussed in Section~\\ref{sec:ala1_results}. The main difference between the algorithms lies in the limit $\\alpha \\to \\infty$. In weight-tempered random sampling, we obtain a landmark distribution that is the same as the biased distribution from the enhanced sampling simulation. On the other hand, WT-FPS results in landmarks that are sampled uniformly distributed from the simulation data set. Due to usage of FPS~\\cite{Hochbaum1985_Abestpo} in the initial stage, WT-FPS is computationally more expensive. Thus, as we are interested in a landmark selection obtained using smaller values of $\\alpha$ and do not want uniformly distributed landmarks, we prefer weight-tempered random sampling.\n\nThe landmarks obtained with weight-tempered random sampling still carry statistical weights that can vary considerably. Thus, we also incorporate the weights into the training by employing a reweighted feature pairwise probability distribution. To test the effect of this reweighting, we constructed MRSE embeddings without including the weights in the training. Then, we only take the weights into account during the landmark selection. For alanine dipeptide, the reweighted MRSE embeddings are more consistent and slightly better than the not-reweighted ones. For the more challenging alanine tetrapeptide case, both the reweighted and not-reweighted embeddings capture all the metastable states. However, we can observe that the reweighted embedding has a better visual separation of states. Thus, we can conclude from these two systems that employing a reweighted feature pairwise probability distribution is beneficial or even essential, especially when considering more complex systems. Nevertheless, this is an issue that we need to consider further in future work.\n\nFinally, we have implemented the MRSE method and weight-tempered random sampling in the open-source \\textsc{plumed} library for enhanced sampling and free energy computation~\\cite{tribello2014plumed,plumed-nest}. Having MRSE integrated into \\textsc{plumed} is of significant advantage. We can use MRSE with the most popular MD codes and learn CVs in postprocessing and on the fly during a molecular simulation. Furthermore, we can employ the learned CVs with the various CV-based enhanced sampling methods implemented in \\textsc{plumed}. We will make our code publicly available under an open-source license by contributing it as a module called \\texttt{LowLearner} to the official \\textsc{plumed} repository in the future. In the meantime, we release an initial implementation of \\texttt{LowLearner} with our data. The archive of our data is openly available at Zenodo~\\cite{mrse-dataset} (DOI: \\href{https:\/\/zenodo.org\/record\/4756093}{\\texttt{10.5281\/zenodo.4756093}}). \\textsc{plumed} input files and scripts required to replicate the results are available from the \\textsc{plumed} NEST~\\cite{plumed-nest} under \\texttt{plumID:21.023} at \\url{https:\/\/www.plumed-nest.org\/eggs\/21\/023\/}.\n\n\n\n\n\n\\section*{Acknowledgments}\nWe want to thank Ming Chen (UC Berkeley) and Gareth Tribello (Queen's University Belfast) for valuable discussions, and Robinson Cortes-Huerto, Oleksandra Kukharenko, and Joseph F. Rudzinski (Max Planck Institute for Polymer Research) for carefully reading over an initial draft of the manuscript. JR gratefully acknowledges financial support from the Foundation for Polish Science (FNP). We acknowledge using the MPCDF (Max Planck Computing \\& Data Facility) DataShare.\n\n\n\n\n\n\\section*{Associated Content}\nThe Supporting Information is available free of charge at \\url{https:\/\/pubs.acs.org\/doi\/xxx\/yyy}.\n\n\\begin{adjustwidth}{2cm}{}\n(S1) Entropy of the reweighted feature pairwise probability distribution;\n(S2) Kullback-Leibler divergence loss for a full set of training data;\n(S3) Description of well-tempered farthest-point sampling (WT-FPS);\n(S4) Effective landmark CV distribution for weight-tempered random sampling;\n(S5) Details about the clustering used in Figure~7.\n(S6) Bandwidth values for kernel density estimation;\n(S7) Loss function learning curves;\n(S8) Additional embeddings for the M\\\"uller-Brown potential;\n(S9) Feature preprocessing in the alanine dipeptide system;\n(S10) Alanine dipeptide embeddings for different values of $\\alpha$ in weight-tempered random sampling;\n(S11) Alanine dipeptide embeddings for $\\alpha=2$ in WT-FPS;\n(S12) Alanine dipeptide embeddings for different random seed values;\n(S13) Convergence of alanine tetrapeptide simulations;\n\\end{adjustwidth}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec1}\nThe extent of social media users consists of billions of people from all around the world. Initially, social media was developed with the object in mind to help connect with family and friends online. It was used to share everyday things, events, interests, and news within closed circles of family and friends. These were personal events, e.g. birthdays, weddings, vacations, graduation ceremonies, and going out. After the usability of social media was discovered, it soon caught the attention of individuals and companies that started using social media to reach more customers. Soon after, the trend became a global phenomenon where people started connecting worldwide based on their common interests. The influence of social media on people's lives and attitudes has been widely studied and established in many different perspectives \\cite{intro1}, \\cite{intro2}.\n\nAlthough social media is a broad term, it mainly refers to Facebook, Twitter, Reddit, Instagram, and YouTube. Some social media platforms allow users to post text, photos and videos. At the same time, many other social media applications have limited options and restrictions for sharing the type of content. YouTube allows users to post videos, while Instagram only allows users to share videos and photos. 4.66 billion active internet users worldwide, and 4.2 billion users are active on social media. As of the first quarter of 2020, Facebook has 2.6 billion monthly active users globally, making it the most extensive social media network globally. Twitter is one of the leading social media with 397 million users worldwide, which is becoming increasingly prominent during events and an essential tool in politics \\cite{Statista2021.}. Another study \\cite{Tnewsmed} shows that Twitter is an effective and fast way of sharing news and developing stories. This trend has continued to grow since the last decade as the internet has become widespread.\n\nHowever, the use of social media has become more complex in the last decade. It became a broader phenomenon because of the involvement of multiple stakeholders such as companies, groups, and other organizations. Many lobbying and public relations firms got on board and started targeting social audiences to change people's perspectives and influence their decisions. Mostly these campaigns are related to a particular individual or a company. A similar process happens in the public sphere, where people rally against or support their target. It played a significant role in different outcomes, affecting countries, people, and eventually the world. One such example is ``Arab Spring\" \\cite{arabspring}, which is an event that started in Tunisia and spread among other regional countries. Another example of good and bad events in political spheres of the UK and US is given in the study that uses Twitter to evaluate the perceived impact on users \\cite{goodbadevents}.\n\nThe term ``event\" implies typically a change which is an occurrence bounded by time and space. In the context of social media, an event can be happening on the ground or online. Different mediums can broadcast events happenings on the ground while people participate in the event through social media discussion. These kinds of events can be referred to as hybrid events. An example of such a hybrid event can be a volcano eruption where people participate in the event using online discussions on social media while it is happening on the ground. While some events solely happen online, such as gaming, marketing, and learning events. Events can be communicated in text, photos and videos across social media platforms. Many events can happen on social media platforms simultaneously, providing beneficial and prosperous information. It provides information about the event itself, but it also reveals sentiments and opinions of the general public and the direction where the events are evolving shortly. This quick interaction of users and transmission of information makes it a dynamic process that sometimes proves hard to follow the latest development, making it a challenging task.\n\nEvents also have time dimensions; it is equally important to determine the time of an event after its detection. For example, an event may have occurred in the past, happened in the present, or planned to occur in the future. Based on that, further steps would be taken accordingly as per the requirements of the situation. The events occurring on social media may directly impact the personal or social life of the man\/woman. The past event can tell us people's opinions and other factors; current events can be a great source of developing a story, while future events can help us prepare in advance. The study \\cite{eventtime} reviews the existing research for the detection of disaster events and classifies them in three dimensions early warning and event detection, post-disaster, and damage assessment. \n\nThe recent example of violence in Bangladesh can explain the link between social media with real life. On Wednesday, 15 October 2021, clashes were sparked by videos and allegations that spread across social media that a Qur'an, the Muslim holy book, had been placed on the knee of a statue of the Hindu god Hanuman. The violence continued in the following days, which resulted in the deaths of 7 people, with about 150 people injured; more than 80 special shrines set up for the Hindu festival were attacked. This case shows social media's severe and robust effect on our daily lives and ground situation \\cite{bngldshviolnc}. This violence was termed as ``worst communal violence in years\" by New York Times. Similar episodes of violence are becoming a norm in India since the right of ring-wing politics. If there is the detection of events occurring on social media in advance, which alerts possible coming hazards, it can be countered in anticipation, significantly reducing the reaction immobilization of state forces while maximizing the protection of people at risk.\n\nEvent detection has been long addressed in the Topic Detection, and Tracking (TDT) in academia \\cite{onewdet}. It mainly focuses on finding and following events in a stream of broadcast news stories shared by social media posts. Event Detection (ED) is further divided into two categories depending on the type of its task; New Event Detection (NED) and Retrospective Event Detection (RED) \\cite{redev}. NED focuses on detecting a newly occurred event from online text streams, while RED aims to discover strange events from offline historical data. Often event detection is associated with identifying the first story on topics of interest through constant monitoring of social media and news streams. Other related fields of research are associated with event detection, such as; event tracking, event summarization, and event prediction. Event tracking is related to the development of some events over time. Event summarization outlines an event from the given data, while the event forecasts the next event within a current event sequence. These topics are part of the Topic Detection and Tracking (TDT) field.\n\nEvent detection is a vast research field, and there are various requirements and challenges for each task. Various terms have been used to address different events, making it complex to navigate the literature, sometimes adding confusion. We propose relevant events based on their characteristics under the umbrella term ``Dangerous Events\" (DE). Section 2 defines the term ``Dangerous Events\" and the main grouping criteria. In order to clarify the idea of different dangerous event types, we employ SocialNetCrawler \\cite{sncrwlr} for extracting tweets that are given as examples for each type.\n\nThe rest of the paper is organized as follows: Section II presents the definition related to dangerous events in social media. Section 3 discusses event detection and related terms; Section 4 is dedicated to approaches and techniques used for event detection. Section 5 presents the warning and countering measures of dangerous events and challenges. Section 6 lists the challenges, and open problems and the paper is finally concluded in Section 7 with some interesting future research directions.\n\n\\section{Dangerous Events}\\label{sec2}\nAccording to Merriam-Webster \\cite{merrwebdang}, the word ``dangerous\" means involving possible injury, pain, harm, or loss characterized by danger. In that context, we define a dangerous event as the event that poses any danger to an individual, group, or society. This danger can come in many shapes and intensities. The objective is to draw a fine line between normal, harmless, unpleasant, and extreme, abnormal and harmful events. Less sensitive, unpleasant, and disliked events do not compel the person to feel threatened. While, in the case of dangerous events, the person will feel fearful, unsafe, and threatened. This provides the objective to approach the term ``event\" in a broader sense to address the common element of all such events. The details of dangers can always be discussed in detail, providing the necessity of the situation; for example, a natural disaster proceeds urgent hate speech. In other words, the first one requires an immediate response with no time to lose, while the latter can allow some time to take action. \n\nDangerous events can be anomalies, novelty, outliers, and extreme. These terms can be used to refer to positive or negative meanings. However, Not all anomalies, novelties, and extremes are dangerous, but all dangerous events fulfil one or all of those conditions (extreme, anomaly, novelty). Authors in \\cite{extremsentilax} proposed an unsupervised approach to detect extreme sentiments on social media. It shows that Positive Extreme sentiments can be detected and differentiated from everyday positive sentiments. Therefore, it may be concluded that extreme negative sentiments are likely to turn into dangerous events.\n\nGrouping and defining dangerous events based on their characteristics is another challenging task, and it can help address the issue of approaching different types of dangerous events by narrowing it down to specific details. We will define three broad categories of dangerous events with commonality among them.\n\\begin{enumerate}\n\\item Scenario-based Dangerous Events\n\\item Sentiment-based Dangerous Events\n\\item Action-based Dangerous Events\n\\end{enumerate}\n\nFigure \\ref{figde} gives the depiction of dangerous events and their categories. In the following subsections, we will outline the definition for each type of dangerous event.\n\n\\begin{figure}[h]%\n\\centering\n\\includegraphics[width=0.9\\textwidth]{deflw.png}\n\\caption{Dangerous Events and it's categories.}\\label{figde}\n\\end{figure} \n \n\\subsection{Scenario-based Dangerous Events}\nWe refer to the word ``scenario\" as the development of events. These events are unplanned and unscripted, and most of the time, they occur naturally. Some planned events can also turn into surprising scenarios. For example, a peaceful protest can turn into a riot, like in 2020 where a peaceful protest against corona restrictions in Germany turned into an ugly situation when the rally was hijacked by right-wing extremists, which ended up storming Parliament building and exhibiting right-wing symbols and slogans \\cite{germanparl}. \n\nDetecting and tracking natural disasters on social media have been investigated intensively, and studies \\cite{sc-based1} have proposed different methods to identify those disasters by various means. The aim of these studies has been mainly to tap into the potential of social media to get the latest updated information provided by social media users in real-time and identify the areas where assistance is required. These categories are considered scenario-based dangerous events in this paper, including earthquakes, force majeure, hurricanes, floods, tornadoes, volcano eruptions, and tsunamis. Although each calamity's nature is different, the role of social media in such events provides a joint base to approach them as scenario-based dangerous events. A supposed example of scenario-based danger is obtained using the crawler tool SocialNetCrawler, which can be accessed using the link\\footnote{\\url{http:\/\/sncrawler.di.ubi.pt\/}}:\n\n\\textit{``@politicususa BREAKING: \nScientists predict a tsunami will hit \nWashington, DC on 1\/18\/2020\nWe Are Marching in DC\u2026 https:\/\/t.co\/3af4ZhyV3J\"}\n\n\\subsection{Sentiment-based Dangerous Events}\nSentiment Analysis (SA), also known as Opinion Mining (OM), is the process of extracting people's opinions, feelings, attitudes, and perceptions on different topics, products, and services. The sentiment analysis task can be viewed as a text classification problem as the process involves several operations that ultimately classify whether a particular text expresses positive or negative sentiment \\cite{sncamb}. For example, A micro-blogging website like Twitter is beneficial for predicting the index of emerging epidemics. These are platforms where users can share their feelings which can be processed to generate vital information related to many areas such as healthcare, elections, reviews, illnesses, and others. Previous research suggests that understanding user behaviour, especially regarding the feelings expressed during elections, can indicate the outcome of elections \\cite{saelctn}. \n\nSentiments can be positive and negative, but for defining sentiment-based dangerous events, the applicable sentiments are negatives and, in some instances, negative extremes. Online radicalization can be attributed to this threat related to extreme negative sentiments towards certain people, countries, and governments. Such extreme negative sentiments can result in protests, online abuse, and social unrest. Detecting these events can help reduce its impact by allowing the concerned parties to counter beforehand. A hypothetical example of sentiment-based dangerous example of a tweet obtained using SocialNetCrawler is given below:\n\n\\textit{``RT @Lrihendry: When Trump is elected in 2020, I'm outta here. \nIt's a hate-filled sewer. \nIt is nearly impossible to watch the hateful at\u2026\"}\n\n\n\\subsection{Action-based Dangerous Events}\nThe action involves human indulgence in an event. Various types of actions happen on the ground that can be detected using social media. Actions can be of many types, but we point out actions that are causing harm, loss, or threat to any entity, which again shares the common attribute of negativity and is highly similar to previously defined types of dangerous events. Some examples of Action-based dangerous events can be prison breaks, terrorist attacks, military conflicts, shootings, etc. Several studies have been published focusing on one or more types of such actions-based events. The study \\cite{antifas} focuses on anti-fascist accounts on Twitter to detect acts of violence, vandalism, de-platforming, and harassment of political speakers by Antifa. An assumed example of action-based example acquired using SocialNetCrawler is given below:\n\n\\textit{``RT @KaitMarieox: This deranged leftist and LGBT activist named Keaton Hill assaulted and threatened to kill @FJtheDeuce, a black conservati\u2026''}\n\n\\section{Event Detection methods and techniques}\n\nEvent Detection has been a popular topic in the research community. Several methods and techniques have been proposed to detect events depending on different requirements. These methods directly depend on the type of task and the data available. As such, they were detecting events from image data is undoubtedly different from text data. However, in this work, we only refer to event detection techniques related to text data, particularly data obtained from social media platforms. \n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.80\\textwidth]{edtyp.png}\n\\caption{Classification of ED methods.} \\label{evmethds}\n\\end{figure}\n\nEvent detection methods and techniques revolve around a few basic approaches. Two approaches that are being used in event detection are document-pivot and feature-pivot. What differs in these approaches is mainly the clustering approach, the way documents are used to feature vectors, and the similarity metric used to identify if the two documents represent the same event or not. Another approach is the topic modelling approach, primarily based on probabilistic models.\n\nIt originates from the Topic Detection and Tracking task (TDT) field and can be seen as a clustering issue. \\textbf{Document-pivot approach} detects events by clustering documents based on document similarity as given in Figure \\ref{doc:fig}. Documents are compared using cosine similarity with Tf-IDF (term frequency-inverse document frequency) representations, while a Locality Sensitive Hashing (LSH) \\cite{lcltyhashing} scheme is utilized to retrieve the best match rapidly. \n \\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.8\\textwidth]{dpi.png}\n\\caption{Event Detection using Document-pivot approach \\cite{31}. \\label{doc:fig} }\n\\end{figure}\n\nThis technique was initially proposed for the analysis of timestamped document streams. The bursty activity is considered an event that makes some of the text features more prominent. The features can be keywords, entities and phrases. \\textbf{Feature-pivot Approach} clusters together with terms based on the pattern they occur as shown in the Figure \\ref{featurebsd}. A study \\cite{featurenaive} uses a Naive Bayes classifier to learn the selected features such as keywords to identify civil unrest and protests and accordingly predict the event days.\n\n \\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.8\\textwidth]{fdp.png}\n\\caption{Event Detection using Feature-pivot approach \\cite{31}. \\label{featurebsd}}\n\\end{figure}\n\n\\textbf{Topic modelling approaches} are based on probabilistic models which detect events in social media documents in a similar way that topic models identify\nlatent topics in text documents. In the beginning, topic models depended on word occurrence, where the text corpora were given as a mixture of words with latent model topics and the set of identified topics were given as documents. Latent Dirichlet Allocation (LDA) \\cite{jelodar2019latent} is the most known probabilistic topic modelling technique. It is a hierarchical Bayesian model where a topic distribution is supposed to have a sparse Dirichlet prior. The model is shown in the Figure. \\ref{ldafig}, where \u03b1 is the parameter of the Dirichlet before the per-document topic distribution \u03b8 and \u03c6 is the word distribution for a topic. K represents the number of topics, M represents the document number, while N gives the number of words in a document. If the word W is the only observable variable, the learning of topics, word probabilities per topic, and the topic mixture of each document is tackled as a problem of Bayesian inference solved by Gibbs sampling.\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.4\\textwidth]{LDAtm.png}\n\\caption{LDA - A common topic modeling technique \\cite{31}. } \\label{ldafig}\n\\end{figure}\n\nMany methods are proposed for the detection of events. This event detection (ED) methods are mainly categorized under the category of supervised and unsupervised, as shown in Figure \\ref{evmethds}. Supervised methods include support vector machine (SVM), Conditional random field (CRF), Decision tree (DT), Naive Bayes (NB) and others. At the same time, the unsupervised approaches include query-based, statistical-based, probabilistic based, clustering-based, and graph-based.\n\n\\subsection{Event Detection Datasets}\nDue to the growth of the internet and related technologies, the research in event detection has experienced significant interest and effort. However, the benchmark datasets for event detection witnessed slow progress. This can be attributed to the complexity and costliness of annotating events that require human input. There are a handful number of datasets available that covers event detection. These datasets are mostly limited to the small size of data and very restricted types of events. They address specific domains based on certain features. This also raises issues while using a data-hungry deep learning model and typically requires balanced data for each class. Some of these datasets are briefed in the following paragraphs. Table \\ref{tab:edds} gives the comparison of the discussed datasets and knowledge bases.\n\nMAVEN \\cite{wang2020MAVEN} which stands for MAssive eVENt detection dataset, offers a general domain event detection dataset manually annotated by humans. It uses English Wikipedia and FrameNet (Baker et al., 1998) documents for building the dataset. It contains 111,611 various events and 118,732 events mentioned. The authors claim this to be the largest available human-annotated event detection dataset. There are 164 different types of events, representing a much wider range of public domain events. The event types are grouped under five top-level types: action, change, scenario, sentiment, and possession. \n\nEventWiki \\cite{ge2018eventwiki} is a knowledge base of events, which consists of 21,275 events containing 95 types of significant events collected from Wikipedia. EventWiki gives four kinds of information: event type,\nevent info-box, event summary, and full-text description. Authors claim to be the first knowledge base of significant events, whereas most knowledge bases focus on static entities such as people, locations, and organizations.\n\nThe EventKG \\cite{Abdollahi2020EventKGClickAD} is a multilingual1 resource incorporating event-centric information extracted from several large-scale knowledge graphs such as Wikidata, DBpedia and YAGO, as well as less structured sources such as the Wikipedia Current Events Portal and Wikipedia event lists in 15 languages. It contains\ndetails of more than 1,200,000 events in nine languages. Supported languages include; English, French, German, Italian, Russian, Portuguese, Spanish, Dutch, Polish, Norwegian, Romanian, Croatian, Slovene, Bulgarian and Danish.\n\nEVIN \\cite{EVIN} which stands for EVents In News, describes a method that can extract events from a news corpus and organize them in relevant classes. It contains 453 classes of event types and 24,348 events extracted from f 300,000 heterogeneous news articles. The news articles used in this work are from a highly diverse set of newspapers and other online news providers (e.g., http:\/\/aljazeera.net\/,\nhttp:\/\/www.independent.co.uk, http:\/\/www.irishtimes.com,\netc.). These news articles were crawled from the external links mentioned on Wikipedia pages while ignoring the content of Wikipedia pages to get the articles from the original website source.\n\n\n\\begin{table}[h!]\n  \\begin{center}\n    \\resizebox{\\textwidth}{!}{%\n    \\begin{tabular}{lllllll}\n    \\toprule\n        \\textbf{Dataset} & \\textbf{Events} & \\textbf{Event types} & \\textbf{Document Source}& \\textbf{Language}& \\textbf{Year} &\\textbf{Reference}  \\\\\n         \\midrule\n            MAVEN & 111, 611 & 164 & English Wikipedia \\& FrameNet  & English & 2020 & \\cite{wang2020MAVEN} \\\\\n            EventWiki & 21,275 & 94 & English WIkipedia & English& 2018 & \\cite{ge2018eventwiki} \\\\\n            EventKG & 1,200,000 & undefined & DBpedia \\& YAGO... & Multilingual(9) & 2020 & \\cite{Abdollahi2020EventKGClickAD} \\\\\n            EVIN &  24,348 & 453\n & news corpus &English &2014 & \\cite{EVIN}\\\\\n        \\bottomrule\n    \\end{tabular}} \\caption{Comparison of Related Event Detection Datasets }     \\label{tab:edds}\n  \\end{center}\n\\end{table}\n\n\\subsection{Supervised Methods}\nSupervised methods are expensive and lengthy as they require labels and training, and this becomes difficult for more extensive data where the cost of training the model is higher and time-consuming. Some of the supervised methods for event detection are discussed below. \n\n\\textbf{Support Vector Machines (SVM):}\nSupport vector machines are based on the principle of minimizing structural risks \\cite{statlrnth} of computer learning theory. Minimizing structural risks\nis to find an assumption h for which we can guarantee the lowest true error. The real error in h is the probability that h will make an error in a sample test selected at random. An upper limit can be used to connect the true error of a hypothesis h with the error of h in the training set and the complexity of H (measured by VC-Dimension), the space of hypotheses which contains h \\cite{statlrnth}. The supporting vector machines find the hypothesis h, which (approximately) minimize this limit on the true error by controlling effectively and efficiently the VC dimension of H.\\cite{txtcatsvm}\n\nIt has been confirmed in many works that SVM is one of the most efficient algorithms for text classification. The accuracy of 87\\% was achieved to classify the traffic or nontraffic events on Twitter. It was able to identify valuable information regarding traffic events through Twitter \\cite{incdetsm}. SVM\ncombination with incremental clustering technique was applied to detect social and real-world events from photos posted on Flicker site \\cite{smedgrphmdl}.\\\\\n\n\\textbf{Conditional Random Fields (CRF):}The CRFs is an essential type of machine learning model developed based on the Maximum Entropy Markov Model (MEMM). It was first proposed by Lafferty et al. (2001) as probabilistic models to segment and label sequence data, inherit the advantages of the previous models, increase their efficiency, overcome their defects, and solve more practical problems \\cite{crfprbmdl}.\nA conditional Random Field (CRF) classifier was learned to extract the artist name and location of music events from a corpus of tweets \\cite{edinsmfeeds}. \n\n\n\\textbf{Decision Tree (DT):}\nDecision tree learning is a supervised machine learning technique for producing a decision tree from training data. A decision tree is also referred to as a classification tree or a reduction tree, and it is a predictive model which draws from observations about an item to conclusions about its target value. In the tree structure, leaves represent classifications (also referred to as labels), non-leaf nodes are features, and branches represent conjunctions of features that lead to the classification \\cite{artanlyzsft}.\nA decision tree classifier called gradient boosted was used to anticipate whether the tweets consist of an event concerning the target entity or not. \n\n\n\\textbf{Na\u00efve Bayes (NB):}\nNa\u00efve Bayes is a simple learning algorithm that uses the Bayes rule and a strong assumption that the attributes are conditionally independent if the class is given. Although this independence assumption is often violated in practice, na\u00efve Bayes often provides competitive accuracy. Its computational efficiency and many other distinctive features result in na\u00efve Bayes being extensively applied in practice.\n\nNa\u00efve Bayes gives a procedure for using the information in sample data to determine the posterior probability P(y\\textbar x) of each class y, given an object x. Once we have such estimates, they can be then used for classification or other decision support applications. \\cite{enclypmlnaivebyes}\n\n\\subsection{Unsupervised Methods}\nThe unsupervised method does not usually require training or target labels. However, they can depend on specific rules based on the model and requirements. The unsupervised methods being used for event detection are discussed below. Many unsupervised methods are developed by scientists who are grouped into different categories that are described in the following subsections. \n\n\\textbf{Query Based Methods:}\nQuery-based methods are based on queries and simple rules to identify planned rules from multiple websites. e.g., YouTube, Flicker, Twitter. An event's temporal and spatial information was extracted and then used to inquire about other social media websites to obtain relevant information.\\cite{qureybsd} \nThe query-based method requires predefined keywords if there are many keywords to avoid unimportant events.\n\n\\textbf{Statistical Based Methods:}\nDifferent researchers under this category introduced many methods. \nFor example, the average frequency of unigrams was calculated to find the significant unigrams (keywords) and combine those unigrams to illustrate the trending events. \\cite{ unsuptopkextrct} The attempt was made to detect the hot events by identifying burst features (i.e., unigram) during different time windows. Each unigram bursty feature signal was then converted into a frequency domain. They were using Discrete Fourier Transformation (DFT). However, DFT was not able to detect the period when there is a burst which is very important in ED process\\cite{twitternewsrepo}. \n\n\\textbf{Wavelet Transformation(WT):} Another technique called Wavelet Transformation (WT) was introduced to assign signals to each unigram feature. WT technique is different from DFT in term of isolating time and frequency and provide better results\\cite{edintwitter}. \nA new framework was proposed that integrated different unsupervised techniques. For example, LDA, NER, bipartite graph clustering algorithm based on relation and centrality scores to discover hidden events and extract their essential information such as time, location, and people that have been involved \\cite{edsocialweb}.\n\n\\textbf{Named Entity Relation(NER):} Named Entity Relation (NER) identifies increasing weights for the proper noun features. A proposed technique applied tweet segmentation to get the sentences containing more phrasing words instead of unigrams. Later, they computed the TFIDF of these sentences and user frequency and increased weights for the proper noun features identified by Named Entity Relation (NER). Li et al. (2012a)first applied tweets classified them using K-Nearest Neighbor (KNN) to identify the events from tweets published by Singapore users\\cite{twitterbsded}.\n\nWeiler et al. (2014) \\cite{eventidentity} used shifts of terms computed by Inverse Document Frequency (IDF) over a simple sliding window model to detect events and trace their evolution. Petrovi\u0107 et al. ( 2010) \\cite{fsdtwitter} modified and used Locality Sensitive Hashing (LSH) to perform First Story Detection (FSD) task on Twitter.\n\n\\textbf{Probabilistic Based Methods:}\nLatent Dirichlet Allocation (LDA) and Probabilistic Latent Semantic Indexing (PLSI) are topic modelling methods that are being used for event detection. In LDA, each document has many topics, and it is supposed to have a group of topics for each document. The model is shown in Figure \\ref{lda}.\n\nLDA worked well with news articles and academic abstracts, but it fell short for small texts. However, the LDA model has been improved by adding tweet pooling schemes and automatic labelling. Pooling schemes include basic scheme, author scheme, burst term scheme, temporal scheme, and hashtag scheme tweets published under the same hashtag. The experiment results proved that the hashtag scheme produced the best clusters results \\cite{ldatpc}. However, LDA defines the number of topics and terms per topic in advance, inefficiently implementing it over social media. \n \n \\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.8\\textwidth]{LDA.png}\n\\caption{Topic Modeling in LDA \\cite{srvyedtxtstrm}. } \\label{lda}\n\\end{figure}\n \n\\textbf{Clustering-Based Method:}\nClustering-based methods mainly rely on selecting the most informative features, which contribute to event detection, unlike supervised methods, which need labelled data for prediction. It contributes to detecting events more accurately.\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.8\\textwidth]{cluster.png}\n\\caption{Clustering-Based method \\cite{srvyedtxtstrm}. } \n\\end{figure}\n\nMany clustering-based methods exist for text data, and K-means is a famous clustering algorithm. A novel dual-level clustering was proposed to detect events based on news representation with time2vec \\cite{dualclstr}. Clustering-based methods have been employed in various ways and other techniques such as NER, TFIDF and others in different tasks, but the ideal clustering technique is still yet to come.\n\n\\textbf{Graph-Based Methods:}\nGraph-based methods consist of nodes\/vertices representing entities and edges representing the relationship between the nodes. Valuable information can be extracted from these graphs by grouping a set of nodes based on the set of edges. Each generated group is called a cluster\/graph structure, and it is also known as community, cluster or module. The links between different nodes are called intra-edges. Meanwhile, links that connect different communities are called inter-edges.\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.8\\textwidth]{graph.png}\n\\caption{Graph-based clustering method \\cite{srvyedtxtstrm}. } \\end{figure}\n\n\\subsection{Semi-Supervised Methods}\nSemi-supervised learning combines both supervised and unsupervised learning methods. Typically, a small number of labelled and largely unlabeled data is used for training purposes. Sometimes they are also referred to as the hybrid method. If there is a vast number of unlabeled data combined with insufficient labelled data, it can affect the classification accuracy. It is also referred to as imbalanced training data.\n\nSimilarly, if there is no labelled data for a particular class, the classification can become inefficient and accurate. Some of the semi-supervised methods include self-training, generative models and graph-based methods. A semi-supervised algorithm based on tolerance roughest and ensemble learning is recommended for such kinds of problems \\cite{ensamblern}. The missing class is extracted by approximation from the dataset and used as the labelled sample. The ensemble classifier iteratively builds the margin between positive and negative classes to estimate negative data further since negative data is mixed with the positive data. Therefore, classification is done without training samples by applying a hybrid approach, and it saves the cost of getting labelled data manually, especially for larger datasets.\n\n\\section{Discussion}\nThis section discusses different works related to event detection that are categorized under the types proposed earlier in this work. The types of events are scenario-based, sentiment-based, and action-based dangerous events. Each work is described in this section and its event type and technique. Furthermore, this section also discusses the research related to event prediction. Table \\ref{detable} illustrates  different type of events detection from social media.\n\\subsection{Detection of Different\/Dangerous Events on Social Media}\nNourbakhsh et al. \\cite{nourbakhsh2017breaking} address natural and artificial disasters on social media. They identified events from local news sources that may become global breaking news within the next 24 hours. They used Reuters News Tracer, a real-time news detection and verification engine. It uses a fixed sphere decoding (FSD) algorithm to detect breaking stories in real-time from Twitter. Each event is shown as a cluster of tweets engaging with that story. By considering different data features, they applied SGD and SVM classifier that detects breaking disasters from postings of local authorities and local news outlets.\n\nSakaki et al. \\cite{equake2010} leverage Twitter for detecting earthquake occurrence promptly. They propose a method to scrutinize the real-time interaction of earthquakes events and, similar to detect a target event. Semantic analyses were deployed on tweets to classify them into positive and negative classes. The target for classification is two keywords; earthquake or shaking, which are also addressed as query words. Total of 597 positive samples of tweets that report earthquake occurrence is used as training data. They also implemented filtering methods to identify the location, and an application called the earthquake reporting system in Japan.\n\nLiu et al. \\cite{crisisbert} aims for crisis events. They propose a state-of-the-art attention-based deep neural networks model called CrisisBERT to embed and classify crisis events. It consists of two phases which are crisis detection and crisis recognition. In addition, another model for embedding tweets is also introduced. The experiments are conducted on C6 and C36 datasets. According to the authors, these models surpass state-of-the-art performance for detection and recognition problems by up to 8.2\\% and 25.0\\%, respectively.\n\nArchie et al. \\cite{hurrearth} proposed an unsupervised approach for the detection of sub-events in major natural disasters. Firstly, noun-verb pairs and phrases are extracted from tweets as an important sub-event prospect. In the next stage, the semantic embedding of extracted noun-verb pairs and phrases is calculated and then ranked against a crisis-specific ontology called management of Crisis (MOAC) ontology. After filtering these obtained candidate sub-events, clusters are formed, and top-ranked clusters describe the highly important sub-events. The experiments are conducted on Hurricane Harvey and the 2015 Nepal Earthquake datasets. According to the authors, the approach outperforms the current state-of-the-art sub-event identification from social media data.\n\nForests fire have become a global phenomenon due to rising droughts and increasing temperatures which is often attributed to global warming and climate change. The work \\cite{firehaze} tests the usefulness of social media to support disaster management. However, the primary data for dealing with such incidents come from NASA satellite imagery. The authors use GPS-stamped tweets posted during 2014 from Sumatra Island, Indonesia, which experiences many haze events. Twitter has proven to be a valuable resource during such events, confirmed by performing analysis on the dataset. Furthermore, the authors also announced the development of a tool for disaster management. \n\nHuang et al. \\cite{emergemcyweibo} focuses on emergency events. They consider the various type of events under the term ``emergency events\". It includes infectious disease, explosions, typhoons, hurricanes, earthquakes, floods], tsunamis, wildfires, and nuclear disasters. The model must automatically identify the attribute information 3W (What, When, and Where) of emergency events to respond in time. Their proposed solution contains three phases, the classification phase, the extraction phase, and the clustering phase, and it is based on the Similarity-Based Emergency Event Detection (SBEED) framework. The experiment is done using the Weibo dataset. Different classification models such as KNN, Decision Trees, Na\u00efve Bayes, Linear SVC (RBF), and Text-CNN are used in the classification phase. Secondly, time and location are extracted from the classification obtained. Lastly, an unsupervised dynamical text clustering algorithm is deployed to cluster events depending on the text-similarity of type, time and location information. The authors claim superiority of the proposed framework having good performance and high timeliness that can be described what emergency, and when and where it happened.\n  \nPais et al. present an unsupervised approach to detect extreme sentiments on social networks. Online wings of radical groups use social media to study human sentiments engaging with uncensored content to recruit them. They use people who show sympathy for their cause to further promote their radical and extreme ideology. The authors developed a prototype system composed of two components, i.e., Extreme Sentiment Generator (ESG) and Extreme Sentiment Classifier (ESC). ESG is a statistical method used to generate a standard lexical resource called ExtremesentiLex that contains only extreme positive and negative terms. This lexicon is then embedded to ESC and tested on five different datasets. ESC finds posts with extremely negative and positive sentiments in these datasets. The result verifies that the posts that were previously classified as negatives or positives are, in fact, extremely negatives or positives in most cases.\n \nCOVID-19 pandemic has forced people to change their lifestyles. Lockdown further pushed people to use social media to express their opinions and feelings. It provides a good source for studying users' topics, emotions, and attitudes discussed during the pandemic. The authors of work \\cite{covsent} collected two massive COVID-19 datasets from Twitter and Instagram. They explore data with different aspects, including sentiment analysis, topics detection, emotions, and geo-temporal. Topic modelling on these datasets with distinct sentiment types (negative, neutral, positive) shows spikes on specific periods. Sentiment analysis detects spikes on specific periods and identifies what topics led to those spikes attributed to economy, politics, health, social, and tourism. Results showed that COVID-19 affected significant countries and experienced a shift in public opinion. Much of their attention was towards China. This study can be very beneficial to read people's behaviour as an aftermath; Chinese people living in those countries also faced discrimination and even violence because of the Covid-19 linked with China.\n \nPlaza-del-Arco et al. \\cite{hateoffense} investigate the link of hate speech and offensive language(HOF) with relevant concepts. Hate speech targets a person or group as a negative opinion, and it is related to sentiment analysis and emotion analysis as it causes anger and fear inside the person experiencing it. The approach consists of three phases and is based on multi-task learning (MTL). The setup is based on BERT, a transformer-based encoder pre-trained on a large English corpus. Four sequence classification heads are added to the encoder, and the model is fine-tuned for multi-class classification tasks. The sentiment classification task categorizes tweets into positive and negative categories, while emotion classification classifies tweets into different emotion categories (anger, disgust, fear, joy, sadness, surprise, enthusiasm, fun, hate, neutral, love, boredom, relief, none). The offence target is categorized as an individual, group, and unmentioned to others. Final classification detects HOF and classifies tweets into HOF and non-HOF. \n \nKong et al. \\cite{farrightextreme} explore a method that explains how extreme views creep into online posts. Qualitative analysis is applied to make ontology using Wikibase. It proceeded from the vocabulary of annotations such as the opinions expressed in topics and labelled data collected from three online social networking platforms (Facebook, Twitter, and Youtube). In the next stage, a dataset was created using keyword search. The labelled dataset is then expanded to using a looped machine learning algorithm. Two detailed case studies are outlined with observations of problematic online speech evolving the Australian far-right Facebook group. Using our quantitative approach, we analyzed how problematic opinions emerge. The approach exhibits how problematic opinions appear over time and how they coincide.\n \nDemszky et al.\\cite{plrztnpolitk} highlights four linguistic dimensions of political polarization in social media, which includes; topic choice, framing, affect and apparent force. These features are quantified with existing lexical methods. Clustering of tweet embeddings is proposed to identify important topics for analysis in such events. The method is deployed on 4.4M tweets related to 21 mass shootings. Evidence proves the discussions on these events are highly polarized politically, and it is driven by the framing of biased differences rather than topic choice. The measures in this study provide connecting evidence that creates a big picture of the complex ideological division penetrating public life. The method also surpasses LDA-based approaches for creating common topics. \n \nWhile most typical use of social media is focused on disease outbreaks, protests, and elections, Khandpur et al. \\cite{cyberattack} explored social media to uncover ongoing cyber-attacks. The unsupervised approach detects cyber-attacks such as; breaches of private data, distributed denial of service (DDOS) attacks, and hijacking of accounts while using only a limited set of event trigger as a fixed input.\n \nCoordinated campaigns aim to manipulate and influence users on social media platforms. Pacheco et al. \\cite{coordcomp} work aim to unravel such campaigns using an unsupervised approach. The method builds a coordination network relying on random behavioural traces shared between accounts. A total of five case studies are presented in work, including U.S. elections, Hong Kong protests, the Syrian civil war, and cryptocurrency manipulation. Networks of coordinated Twitter accounts are discovered in all these cases by inspecting their identities, images, hashtag similarities, retweets, or temporal patterns. The authors propose using the presented approach for uncovering various types of coordinated information warfare scenarios.\n \nCoordinated campaigns can also influence people towards offline violence. Xian Ng et al. \\cite{coordinariots} investigates the case of capital riots. They introduce a general methodology to discover coordinated by analyzing messages of user parleys on Parler. The method creates a user-to-user coordination network graph prompted by a user-to-text graph and a similarity graph. The text-to-text graph is built on a textual similarity of posts shared on Parler. The study of three prominent user groups in the 6 January 2020 Capitol riots detected networks of coordinated user clusters that posted similar textual content in support of different disinformation narratives connected to the U.S. 2020 elections. \n \nWanzheng Zhu and Suma Bhat \\cite{drgEuphemisticPD} studies the specific case of the use of euphemisms by fringe groups and organizations that is expression substituted for one considered to be too harsh. The work claims to address the issue of Euphemistic Phrase detection without human effort for the first time. Firstly the phrase mining is done on raw text corpus to extract standard phrases; then, word embedding similarity is implemented to select candidates of euphemistic phrases. In the final phases, those candidates are ranked using a masked language model called SpanBERT.\n \nYang Yang et al. \\cite{hmntrfk} explore the use of  Network Structure Information (NSI) for detecting human trafficking on social media. They present a novel mathematical optimization framework that combines the network structure into content modelling to tackle the issue. The experimental results are proven effective for detecting information related to human trafficking.\n\n\\begin{table}[h!]\n  \\begin{center}\n    \\begin{tabular}{p{6cm}c}\n    \\toprule\n        \\textbf{Tweets} & \\textbf{Proposed dangerous event type }   \\\\\n         \\midrule\n            ``RT @KaitMarieox: This deranged leftist and LGBT activist named Keaton Hill assaulted and threatened to kill @FJtheDeuce, a black conservati\u2026'' & Action-based dangerous event  \\\\\n            ``RT @Lrihendry: When Trump is elected in 2020, I'm outta here. \nIt's a hate-filled sewer. \nIt is nearly impossible to watch the hateful at\u2026\" &Sentiment-based dangerous event  \\\\\n           ``Scientists predict a tsunami will hit Washington, DC on 1\/18\/2020\nWe Are Marching in DC\u2026 https:\/\/t.co\/3af4ZhyV3J\" & Scenario-based dangerous event \\\\\n            \\bottomrule\n    \\end{tabular} \\caption{Presumed types of dangerous events for tweets.}     \\label{tab:exde}\n  \\end{center}\n\\end{table}\n\nAuthors present Table \\ref{tab:exde} to clarify the intent of this work by providing an example of the collected tweets and their presumed techniques. Based on the existing methods for event detection, it gives a clear objective for using these methods for detecting dangerous events.\n\n\\subsection{Event Prediction}\n Event prediction is a complex issue that revolves around many dimensions. Various events are challenging to predict before they become apparent. For example, it is impossible to predict in case of natural disasters, and they can only be detected after the occurrence. Some events can be predicted while they are still in the evolving phase. Authors of \\cite{nourbakhsh2017breaking} identify events from local news sources before they may become breaking news globally. The use case of Covid-19 can be regarded as an example where it started locally and became a global issue later. \n \n A dataset is obtained from a recent Kaggle competition to explore the usability of a method for predicting disaster in tweets. The work in \\cite{twitterbertpred} tests the efficiency of BERT embedding, which is an advanced contextual embedding method that constructs different vectors for the same word in various contexts. The result shows that the deep learning model surpasses other typical existing machine learning methods for disaster prediction from tweets. \n \n Zhou et al. \\cite{covidfatlerate} proposed a novel framework called as Social Media enhAnced pandemic suRveillance Technique (SMART) to predict Covid-19 confirmed cases and fatalities. The approach consists of two parts where firstly, heterogeneous knowledge graphs are constructed based on the extracted events. Secondly, a module of time series prediction is constructed for short-and long-term forecasts of the confirmed cases and fatality rate at the state level in the United States and finally discovering risk factors for intervening COVID-19. The approach exhibits an improvement of 7.3\\% and 7.4\\%  compared to other state-of-the-art methods.\n \n Most of the other existing research targets particular scenarios of event prediction with limited scope. Keeping in mind the complexity of this problem, we only present a few related works, and the generalization is obscure. \n \n\\begin{landscape}\\centering\n\\vspace*{\\fill}\n\\begin{table}[htpb]\n\\begin{tabular}{{llllll}}\n\n & Event Type & Technique & Reference & Dataset & Year \\\\ \\cline{2-6} \nScenario-based & & & & & \\\\ \\hline\n & Natural Disasters & SVM\/SGD & \\cite{nourbakhsh2017breaking} & Twitter & 2017 \\\\\n & Earthquake & Classification(SVM)& \\cite{equake2010} & Twitter & 2010 \\\\\n & Crisis & CrisisBERT & \\cite{crisisbert} &Twitter (C6,C36) & 2021\\\\\n &Earthquake \\&Hurricane & Unsupervised & \\cite{hurrearth} &Twitter & 2019 \\\\\n \n &Fire and Haze Disaster & Classification (hotspots) & \\cite{firehaze} &NASA \\&Twitter & 2017 \\\\ \n & Emergency & Text-CNN, Linear SVC \\& Clustering & \\cite{emergemcyweibo} & Weibo & 2021 \\\\\n& ... & ... &... &... &...\\\\ \n \nSentiment-based & & && & \\\\ \\hline\n & Extreme Sentiments & Unsupervised learning & \\cite{extremsentilax} & misc. & 2020 \\\\\n & Covid19 Sentiments & word2vec & \\cite{covsent} & Twitter \\& Instagram & 2021 \\\\\n &Hate speech \\& offensive Language& BERT & \\cite{hateoffense} & HASOC(Twitter) &2021 \\\\ \n &Far-right Extremism & Classification & \\cite{farrightextreme} & Facebook, Twitter \\&Youtube & 2021 \\\\ \n & Political Polarization & Clustering & \\cite{plrztnpolitk}& Twitter &2019 \\\\ \n& ... & ... &... &... &...\\\\ \n \nAction-based & & & & & \\\\ \\hline\n& Cyber attack & Unsupervised & \\cite{cyberattack} & Twitter & 2017 \\\\\n& Coordinated campaigns & Unsupervised & \\cite{coordcomp} & misc. & 2021 \\\\ \n& Riots & Clustering & \\cite{coordinariots} & Parler & 2021 \\\\\n&Drugs Trafficking& SPANBert & \\cite{drgEuphemisticPD} &Text Corpus(subreddit) & 2021\\\\\n& Human Trafficking & Classification (NSI) & \\cite{hmntrfk}& Wiebo & 2018 \\\\\n& ... & ... &... &... &...\\\\ \n\\end{tabular}\\caption{Dangerous Events categorized under relevant types }\\label{detable}\n\\end{table} \n\\vfill\n\\end{landscape}\n \n\\section{Conclusion}\\label{sec13}\nIn this work, we laid the basis of the term ``Dangerous Events\" and explored different existing techniques and methods for detecting events on social media. Dangerous events contain a broad meaning, but we keep it essential to define the term better. We believe much more can be included in dangerous events, as we explored in the discussion section. Categorizing dangerous events into sub-categories can help specify the event and its features. The subcategories consist of scenario-based, sentiment-based and action-based dangerous events. The usefulness of social media these days provides a significant advantage in detecting such events initially. While in some cases, significant events also originate from social media and manifest in real life, such as; mass protests, communal violence and radicalization.  \n\nThe events on social media are mainly polarized. People use it to express their likes or dislike, which can also be classified as positive or negative. Not all extreme events are dangerous, but all dangerous events are extreme, as people can show their happiness using extreme emotions, which is an anomaly under normal circumstances. There is a common element in all dangerous events: we all want to avoid it because of the harm it brings. We believe there is an excellent scope for related work in future. As a proposal, we suggest a dataset containing all types of dangerous events. Secondly, different techniques can be applied to this dataset to further deepen the usefulness and evolve a technique that can be generalized for all kinds of such events. Considering the limitations of event detection, techniques covering only specific events, a joint base can help discover the universally applicable method. \n\n\\backmatter\n\n\n\\bmhead{Acknowledgments}\n\nThis work was supported by National Founding from the FCT- Fundac\u00e3o para a Ci\u00eancia e a Tecnologia, through the MOVES Project-PTDC\/EEI-AUT\/28918\/2017 and by operation Centro-01-0145-FEDER-000019-C4-Centro de Compet\u00eancias em Cloud Computing, co-financed by the European Regional Development Fund (ERDF) through the Programa Operacional Regional do Centro (Centro 2020), in the scope of the Sistema de Apoio \u00e0 Investiga\u00e7\u00e3o Cient\u00edfica e Tecnol\u00f3gica.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe zodiacal cloud is the name given to the population of small dust grains that permeate the Solar System's terrestrial planet region. The source of these grains has long been debated, with the asteroid belt, Kuiper belt, comets and interplanetary grains all being suggested as possible sources. Recent work by \\citet{2010ApJ...713..816N} and \\citet{2013MNRAS.429.2894R} shows that the majority of the dust (70-95\\%) comes from Jupiter Family Comets.\n\nAlthough often related to dust in the habitable zone, our zodiacal dust is distributed over a wide range of radii from the sun. It ranges from the Asteroid belt inwards, through the habitable zone, to the dust sublimation radius at $\\sim4$ Solar radii. The radial dust distribution globally follows an inward slowly increasing power-law that is locally modified by the interaction with the inner, rocky planets and the local production of dust through comet evaporation \\citep{1998ApJ...508...44K}. In the innermost regions it forms the Fraunhofer corona \\citep[F-corona,][]{1998EP&S...50..493K}.\n\nIn analogy to the zodiacal light, warm and hot dust around other stars is called exozodiacal dust (exozodi, emission: exozodiacal light). To observe this faint emission close to nearby main sequence stars, high angular resolution and contrast are needed. Thus, interferometric observations are the method of choice. In the mid-infrared the emission of habitable zone dust can be detected using nulling interferometry \\citep{2014ApJ...797..119M}. In the near-infrared hotter dust closer to the star can be detected using optical long baseline interferometry \\citep[e.g.,][]{2006A&A...452..237A}. The largest statistical survey for exozodis so far has been carried out in the near-infrared where a detection rate of ~15 - 20\\% has been found \\citep{2013A&A...555A.104A,2014A&A...570A.128E}.\n\nFor A stars, there is no apparent correlation between the presence of an exozodi and the age of the system and for FGK stars there is tentative evidence of a trend for older systems to be more likely to show evidence for an exozodi  \\citep{2014A&A...570A.128E}. This is particularly surprising as mid-infrared observations show a clear decline in presence of excess with age \\citep{2006ApJ...653..675S,2009ApJS..181..197C,2009ApJ...697.1578G} as is expected from a population evolving through collisions \\citep{2007ApJ...658..569W,2008ApJ...673.1123L}. If, as in the Solar System, comets are also responsible for the dust in exozodis then this appears to contradict with what would be expected, since the reservoir of planetesimals is not replenished, and the dust feeding rates should eventually diminish with time.\n\nThis has motivated several recent studies to investigate the origin of these exozodis, with particular focus on how they can be present in systems with ages of order of 100 Myr - 1 Gyr. A possible explanation is that the production of comets is delayed on large timescales (several 100 Myr).\nThis can be achieved if the system is in a phase similar to that of the Solar System Late Heavy Bombardment (LHB). In the Nice model of \\citet{2005Natur.435..466G}, the Solar system was originally more compact, with Neptune orbiting inside Uranus, and suffered major changes at $\\sim 800\\,$Myr. This lead Neptune to jump beyond Uranus, but also lead the four giant planets to adopt more eccentric orbits, thus enhancing close encounters with planetesimals and generating a short phase of intense production of active comets. However, whilst this mechanism appears to be a good candidate to explain the presence of exozodis in some old systems where there are other reasons to believe that a LHB-like event took place such as in the $\\eta\\,$Crv system \\citep{2009MNRAS.399..385B,2012ApJ...747...93L}, the detection rate of exozodis would be only $\\sim 0.1\\%$ if all were produced due to LHB-like events \\citep{2013MNRAS.433.2938B}.\n\nAnother possibility is that the levels of dust are maintained throughout the lifetime of these systems, as suggested by \\citet{2012A&A...548A.104B}, who investigated scattering of planetesimals from an outer cold belt to the inner parts of a system by a chain of planets.It was found that the necessary scattering rates could be obtained, although they required contrived multi-planet system architectures involving chains of tightly packed low-mass planets. The rates were found to be higher, or sustained on longer timescales if the outermost planet of the chain happens to migrate outwards into the belt while scattering planetesimals \\citep{2014MNRAS.441.2380B,2014MNRAS.442L..18R}. However, this requires additional conditions for migration to take place which depend highly on the mass of the planet and the disc surface density.\n\nIn \\citet{2015A&A...573A..87F}, a dynamical process is presented that shows that inner mean-motion resonances (MMR) with a planet on an orbit at least moderately eccentric ($\\mathrm{e_p}\\gtrsim 0.1$), is a valuable route to set planetesimals on highly eccentric orbits, potentially cometary-like, with periastrons small enough for bodies to sublimate. This mechanism combines several strengths : in contrast with scattering of the reservoir by a chain of planets, it involves only one single planet, and the generation of cometary-like orbits through this process can be delayed by timescales as large as several 100 Myr, as for a LHB-like event. In addition, it is expected to place bodies on cometary-like orbits continuously, potentially over large timescales. Therefore, this mechanism could provide a robust explanation for the presence of exozodis, especially in systems older than 100 Myr.\n\nThe general question of the ability of this mechanism to generate a flux of active comets compatible with the presence of an exozodi is addressed in this paper, with particular focus on old systems. We present this mechanism in more details in Sect.~\\ref{sec:analytical}, as well as the analytical background. We make predictions on the ability of a given MMR with a perturber of eccentricity 0.1 to scatter planetesimals on cometary-like orbits. These predictions are complemented by a numerical analysis, and we present a study case of a given MMR, namely the 5:2, in Sect.~\\ref{sec:results}. This will allow us to determine the efficiency of this mechanism in a quantitative manner (rates, timescales) and as a function of the characteristics of the planet (semi-major axis, mass). We compare achievable scattering rates with observations and show applications of this framework to the cases of a low mass disc around a Sun-like star and the Vega system in Sect.~\\ref{sec:applis}. Finally, in Sect.~\\ref{sec:conclusion}, we discuss the efficiency of other MMRs and the impact of the eccentricity of the planet, before drawing our conclusions.\n\n\n\\section{Analytical study}\\label{sec:analytical}\n\nIn this section, we will present in more detail the mechanism of \\citet{2015A&A...573A..87F}, and in particular we will show the analytical predictions that can be made on the ability of a given perturber and a given MMR to set planetesimals on cometary orbits. The question of whether this mechanism can induce sufficient feeding rates and the determination of its characteristic timescales (delays and duration) will be addressed using a numerical analysis in the next sections.\n\n\\subsection{Orbital evolution of planetesimals in MMRs with an outer eccentric planet}\n\nMMRs between a planetesimal and a planet, usually noted $\\mathrm{n:p}$, where $\\mathrm{n}$ and $\\mathrm{p}$ are integers, concern bodies with orbital periods achieving the $\\mathrm{p\/n}$ commensurability with that of the planet. Therefore, MMRs occur at specific locations relative to the orbit of the planetary perturber. The integer $\\mathrm{q=|n-p|}$ is called the order of the resonance. Resonances with $\\mathrm{n>p}$ correspond to \\emph{inner} resonances, that is, planetesimals orbiting inside the orbit of the planet, while $\\mathrm{n<p}$ denotes \\emph{outer} resonances.\n\nThe process we focus on here implies planetesimals initially on almost circular orbits and protected from close-encounters with the planet, which they are usually assumed to be at the end of the protoplanetary phase because of the eccentricity damping action of the gas. If these planetesimals are in inner MMR with an eccentric planet as shown in Fig.~\\ref{fig:MMR_schema}, they may then be driven to more eccentric orbits.\n\n\\begin{figure*}\n\\centering\n\\makebox[\\textwidth]{\\includegraphics[scale=0.5]{MMR_schema.jpg}}\n\\caption[]{Schematic edge-on view of the systems we focus on in this study. According to Eqs.~(\\ref{eq:MMRloc}) and ~(\\ref{eq:chaosin2}), both the location of the MMRs and the inner edge of the chaotic zone are proportional to the semi-major axis of the planet $\\mathrm{a_p}$. The size of the chaotic zone and the relative distance between its inner edge and the location of the MMRs, depend on the mass of the planet, which was set to $\\mathrm{m_p}=0.5\\,\\mathrm{M_{Jup}}$, and on its eccentricity, which was set to $\\mathrm{e_p}=0.1$.}\n\\label{fig:MMR_schema}\n\\end{figure*}\n\nWhen a planetesimal is trapped in MMR with a planet that is on a circular orbit, the evolution of its eccentricity is coupled to that of the semi-major axis. Since this semi-major axis undergoes small amplitude librations ($\\la 0.1$AU) around the exact resonance location, its eccentricity undergoes small amplitude variations as well. However, as soon as the eccentricity of the planet is larger than $\\sim 0.05$, the amplitude of the variations of the eccentricity can be increased up to large values \\citep[see][for details]{1995Icar..115...60M,1996Icar..120..358B}.\n\nThis can be understood from analytical considerations, and in particular by using phase-space diagrams.\nThese consist of lines of constant energy, and are obtained using the Hamiltonian of the system which has been reduced to a single degree of freedom beforehand. The resulting Hamiltonian thus depends only on $\\mathrm{(\\nu,e)}$, where $\\nu$ is the longitude of periastron of the planetesimal with respect to that of the planet, and $\\mathrm{e}$ is the eccentricity of the planetesimal. This means that the precession cycle of a planetesimal is coupled with its eccentricity evolution. In order to easily visualize the co-evolution of these two quantities, the Hamiltonian can then be evaluated for a grid of $(\\nu,e)$, and level curves can be drawn \\citep[for more details on the theoretical background, see ][]{1993CeMDA..57...99M,1995Icar..114...33M,1996Icar..120..358B,2000Icar..143..170B,2015A&A...573A..87F}.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.8\\columnwidth]{MMR52.jpg}\n\\caption[]{Theoretical phase space diagram of the 5:2 MMR with a planet of eccentricity $\\mathrm{e_p=0.1}$. This diagram is made of iso-Hamiltonian curves, so that the behaviour of the orbit of a planetesimal in 5:2 MMR with a planet that has an eccentricity of 0.1 can be evaluated by following the trajectories in the $\\mathrm{(\\nu,e)}$ phase-space, where $\\nu$ is the longitude of periastron of the planetesimal with respect to that of the planet, and $\\mathrm{e}$ is the eccentricity of the planetesimal.}\n\\label{fig:phase_space52}\n\\end{figure}\n\nIn Fig.~\\ref{fig:phase_space52}, we display the trajectories of a planetesimal in the $\\mathrm{(\\nu,e)}$ phase-space, when trapped in 5:2 MMR with a planet of eccentricity $\\mathrm{e_p}=0.1$. As can be read on the Y-axis, the eccentricity of the planetesimal can be increased from an initially quasi circular orbit up to values $\\sim 0.85$. This occurs when the orbit of the planetesimal is anti-aligned with that of the planet, that is, when there is an angle of $\\nu=180^{\\circ}$ between the periastron of the planetesimal and that of the planet (X-axis).\nSince the semi-major axis of the planetesimal only undergoes small amplitude librations, it is roughly constant. Therefore, when the eccentricity of the planetesimal increases, its periastron diminishes, while its apastron increases.\nNote that these theoretical paths in the $(\\mathrm{\\nu,e})$ phase-space are derived making the assumption that the amplitude of libration of the resonance is zero, so that these paths are ideal paths compared to the real ones that particles will adopt in our numerical simulations, and which will oscillate around the ideal path, as shown in Fig. 13 of \\citet{2000Icar..143..170B}.\n\nIn this paper we are interested in the rate of dust input into an exozodi. We are working on the assumption that comets provide this dust in a similar way to how comets in the Solar System provide dust to the zodiacal cloud \\citep[see ][]{2010ApJ...713..816N}. It is thought that the main mass loss for comets is their spontaneous disruption rather than the continual loss of dust through activity \\citep[e.g., ][]{1980A&A....85..191W,1994Icar..108..265C,2004come.book..301B,2005ASSL..328.....F,2006mspc.book.....J}. Nonetheless, the most likely causes of disruption are related to the physical changes a comet undergoes whilst it is active \\citep{1980A&A....85..191W,2004come.book..301B} and so we will focus on determining the number of planetesimals that become active. A comet becomes active when it comes close enough to the star for water ice to sublimate, which means that its periastron has to decrease to less than $\\mathrm{q_{sub}}$. This distance varies with the luminosity of the star. For Solar-type stars, for example, comets are expected to undergo sublimation at separations smaller than 3 AU \\citep[see e.g., ][]{2016arXiv160403790M}. If the eccentricity increase is sufficient, there will be two possible mechanisms for a planetesimal to develop onto an orbit that results in it becoming an active comet.\n\nFirstly, a MMR can potentially drive the periastron of a planetesimal to values smaller than $\\mathrm{q_{sub}}$, which is a direct route to set planetesimals on to cometary orbits. Secondly, a MMR can also potentially drive planetesimals' apastrons to values such that their orbit crosses the chaotic zone of the planet. The chaotic zone is the region around the planet where MMRs overlap. Interactions occurring there between a planet and a planetesimal can lead to major changes in the orbit of the planetesimal, that is, it can be scattered out of MMR on to a potentially comet-like orbit. This is an indirect mechanism in two steps, with a first phase of eccentricity increase while in MMR, followed by a scattering event.\n\nIn either case, planetesimals set on cometary orbits will endure cometary evaporation during phases of close approach to their host star, deposit dust, maybe even disrupt, and contribute to an exozodi. We describe hereafter the direct and indirect placement of planetesimals on to cometary orbits.\n\n\\subsection{Direct placement on cometary orbits}\n\nUsing Kepler's third law, the theoretical semi-major axis $\\mathrm{a_{MMR}}$ of a given $\\mathrm{n:p}$ MMR is linked to the planet semi-major axis by :\n\n\\begin{equation}\\label{eq:MMRloc}\n\\mathrm{a_{MMR}}= \\mathrm{a_p} \\mathrm{\\left(\\frac{p}{n} \\right)^{2\/3}}\\qquad.\n\\end{equation}\n\nFor a given MMR, one can estimate from the phase space diagram the maximum eccentricity $\\mathrm{e_{max}}$ that a planetesimal can acquire when starting from a low eccentricity orbit $(\\mathrm{e} \\lesssim 0.05)$. In Fig~\\ref{fig:phase_space1}, we compare phase-space diagrams of different MMRs, namely those shown on Fig.~\\ref{fig:MMR_schema}. One can see that the MMRs which induce the largest eccentricities are the 4:1, 3:1, 7:3 and 5:2 MMRs.\nOne can then retrieve the minimum periastron $\\mathrm{q_{min}=a_{MMR}(1-e_{max})}$ that can be achieved.\nThe 4:1 MMR is the only one that is able to increase the eccentricity of a planetesimal up to values $\\sim 1$. This means that this resonance can produce cometary orbits directly no matter what the semi-major axis of the planet is, without the need for an additional scattering event, as is necessary for the mechanism we consider in the rest of this paper. The other MMRs can do this too, but for a very limited range of planet semi-major axes. For a Solar-type star, where $\\mathrm{q_{sub}}=3\\,$AU, $\\mathrm{q_{min}}<\\mathrm{q_{sub}}$ when $\\mathrm{a_p} \\lesssim 15\\,$AU for the 3:1 and 7:3 MMRs, and $\\mathrm{a_p} \\lesssim 35\\,$AU for the 5:2 MMR. The 2:1 MMR does not trigger larger eccentricities $(\\sim 0.1)$, hence it would directly induce the production of active comets for semi-major axes smaller than $\\sim 5\\,$AU.\n\nIn the case of direct cometary production, all the planetesimals in MMR that can be driven on to cometary orbits will become comets during their first precession cycle. Therefore, one will expect a sharp peaked profile of the time-dependent cometary event rate, and an activity starting very quickly after the planet starts acting upon the planetesimals. \\citet{1996Icar..120..358B} found that the timescale for a planetesimal to reach its maximum eccentricity was of the order of $10^3-10^4$ planetary orbital periods, and depends also on the MMR itself. This translates into timescales of the order of 1 Myr at most for planets of semi-major axis smaller than 10-25 AU.\nIn this case, the timescales on which production of active comets will be sustained may be incompatible with the presence of an exozodi in old systems. \nHowever, this timescale is expected as well to increase with smaller planet-star mass ratios, and may also reach large values. Therefore, we will determine numerically in Sect.~\\ref{sec:results} if these values are large enough for direct placement on cometary orbits to produce active comets on timescales of several 100 Myr.\n\n\\subsection{Indirect mechanism: additional scattering event and large timescales}\\label{sec:direct}\n\nIn cases where $\\mathrm{q_{min}}$ remains greater than $\\mathrm{q_{sub}}$, a planetesimal can still be placed on to an active comet orbit if the apastron increases to the point where it crosses the chaotic zone of the planet.\nIf the planetesimal and the planet actually encounter in this orbital configuration, the planetesimal will be scattered by the planet, potentially on to a cometary-like orbit. \nWhen in MMR, a planetesimal will require time to travel through its phase-space and to reach a position where its orbit crosses the chaotic zone of the planet. The planetesimal will not necessarily encounter the planet at this point and it may take several precession cycles before a close encounter with the planet occurs. The number of cycles that is necessary will depend on the initial position of the planetesimal in the phase-space. In a realistic situation, this mechanism applies to a collection of planetesimals in MMR, which span a whole range of initial conditions. Therefore, at each precession cycle, different planetesimals will be brought within reach of the planet. It is thus expected that once the planetesimals which encounter the planet during the first precession cycle are scattered on to cometary orbits, the process generates a continuous flux of comets. The timescale for a precession cycle to complete is expected to depend on the semi-major axis of the planet, and also on the mass ratio between the planet and the star. This timescale can reach several 100 Myrs when the planet is Saturn- Neptune-sized and at large separation from the star \\citep[$\\sim 100\\,$AU][]{2015A&A...573A..87F}. Consequently, a first feature of this mechanism is that scattering events and cometary events can start occurring very late in the history of a system. \nBut more importantly, since the production of cometary-like orbits is expected to be continuous on at least several precession timescales, it could be sustained on very large, potentially Gyr timescales, in contrast to a short period of intense scattering on cometary orbits as in LHB-like events.\nIn addition, the cometary orbits, once produced, are expected to have a potentially significant dynamical lifetime against ejection \\citep[several 0.1-10 Myr][]{2015A&A...573A..87F}, and the planetesimals are not lost onto the star but continuously deposit dust at each passage. However, their lifetime against evaporation may differ from the dynamical lifetime of the orbit. These aspects will be discussed in Sect.~\\ref{sec:applis}. \n\n\\begin{figure*}\n\\centering\n\\makebox[\\textwidth]{\\includegraphics[width=0.4\\textwidth,height=0.4\\textwidth]{MMR21.jpg}\n\\includegraphics[width=0.4\\textwidth,height=0.4\\textwidth]{MMR31.jpg}\n\\includegraphics[width=0.4\\textwidth,height=0.4\\textwidth]{MMR41.jpg}}\n\\makebox[\\textwidth]{\n\\includegraphics[width=0.4\\textwidth,height=0.4\\textwidth]{MMR52.jpg}\n\\includegraphics[width=0.4\\textwidth,height=0.4\\textwidth]{MMR73.jpg}}\n\\caption[]{Theoretical phase-space diagrams (see caption to Fig.~\\ref{fig:phase_space52} of diverse MMRs with a planet of eccentricity $\\mathrm{e_p}=0.1$.}\n\\label{fig:phase_space1}\n\\end{figure*} \n\nThis indirect mechanism that we wish to study here applies to planets at large separations from their host star (several tens of AU).\nThere are usually two possible explanations for the presence of a planet at large distance from its host star, that is, with semi-major axis $\\mathrm{a_p}$ of several tens of AU, or even greater than a hundred AU. These are outward migration and planet-planet scattering. However, migrating planets tend to have nearly circular orbits \\citep{1997Icar..126..261W,2003ApJ...588..494M,2008ApJ...673..487I,2009ApJ...705L.148C}. \nTherefore, planet-planet scattering is in general favoured over the migration scenario, because it could explain the large orbital distance as well as the eccentricity of the perturber \\citep[see for instance][]{1996Sci...274..954R,1997ApJ...477..781L,2008ApJ...686..621F,2008ApJ...686..580C,2008ApJ...686..603J}.\n\nOne can expect a major instability such as one triggering planet-planet scattering to severely damage discs. If the planet-planet scattering is too violent then it would likely scatter all of the material from the disc and wipe it out. However, dynamical simulations of planetary systems do show a range of outcomes including many cases with moderate level of scattering that only partially depletes the disc \\citep{2011A&A...530A..62R}. We will therefore place our work under the assumption that disc material remains available after the instability for our mechanism to take place.\nNote as well that the Solar System is a system where scattering has likely taken place and depleted the comet reservoir \\citep{2005Natur.435..466G,2009MNRAS.399..385B} and we will show in Section~\\ref{sec:comp_obs} that such a faint disc could still provide enough material for a detectable exozodi.\n\n\n\\subsection{Indirect mechanism: analytical predictions}\n\nThe ability of a given planet to scatter planetesimals originating from a given MMR, potentially on a cometary-like orbit, can be predicted by determining whether the apastron of planetesimals in MMR can be increased up to the point where it crosses the chaotic zone of the planet.\n\nUsing the definition $\\mathrm{Q=a_{MMR}(1+e)}$ of the apastron of a planetesimal and combining with $\\mathrm{e_{max}}$ as defined above and Eq.~\\ref{eq:MMRloc}, one can obtain the maximum apastron that the orbit of the planetesimal can reach :\n\n\\begin{equation}\\label{eq:Qmax}\n\\mathrm{Q_\\mathrm{max}= a_\\mathrm{p} \\left(\\frac{p}{n} \\right)^{2\/3} (1+e_{max}) }\\qquad.\n\\end{equation}\n\nOn the other hand, following \\citet{2014MNRAS.443.2541P}, the inner chaotic zone of the planet starts at a semi-major axis $\\mathrm{a_{chaos,in}}=\\mathrm{q_p-\\Delta a_{chaos,in}}$, where, in the case of an eccentric planet, $\\mathrm{\\Delta a_{chaos,in}}$ corresponds to several times the Hill radius at periastron of the eccentric planet, $\\mathrm{R_{H,q_p}}$, which \\citet{2014MNRAS.443.2541P} find to be :\n\n\\begin{equation}\\label{eq:rhill_peri}\n\\mathrm{R_{\\mathrm{H,q}} \\approx a_{\\mathrm{p}}(1-e_{\\mathrm{p}})\\left[ \\frac{m_{\\mathrm{p}}}{(3+e_{\\mathrm{p}})M_{\\star}} \\right]^{\\frac{1}{3}}} \\qquad,\n\\end{equation}\n\nand thus,\n\n\\begin{equation}\\label{eq:chaosin1}\n\\mathrm{a_{chaos,in}=q_p-k\\times R_{\\mathrm{H,q}}} \\qquad,\n\\end{equation}\n\nwhere the factor $\\mathrm{k}$ has to be determined numerically. \\citet{2014MNRAS.443.2541P} focused on the size of a planet's outer chaotic zone, and found that $\\mathrm{k\\simeq 5}$ when setting $\\mathrm{a_{chaos,out}=Q_p+k\\times R_{\\mathrm{H,Q}}}$, where $\\mathrm{R_{\\mathrm{H,Q}}}$ is the Hill radius at apastron, that shall be used in the case of the outer chaotic zone. They did not carry out the study for the inner chaotic zone. However, since we simply want to make qualitative predictions that shall be checked later numerically, we will use $\\mathrm{k=5}$ in our calculations. We obtain :\n\n\\begin{equation}\\label{eq:chaosin2}\n\\mathrm{a_{chaos,in} \\approx a_{\\mathrm{p}}(1-e_{\\mathrm{p}})\\left[1-k\\left( \\frac{m_{\\mathrm{p}}}{(3+e_{\\mathrm{p}})M_{\\star}} \\right)^{\\frac{1}{3}}\\right]} \\qquad,\n\\end{equation}\n\nwhere $\\mathrm{M_{\\star}}$ is the mass of the central star.\n\nThe condition that must be verified for a given MMR to lead planetesimals to cross the chaotic zone of a given planet, that is $\\mathrm{Q_{max}\\geq a_{chaos,in}}$, finally reads:\n\n\\begin{equation}\\label{eq:ability1}\n\\mathrm{\\left(\\frac{p}{n} \\right)^{2\/3} (1+e_{max}) \\geq (1-e_{\\mathrm{p}})\\left[1-k\\left( \\frac{m_{\\mathrm{p}}}{(3+e_{\\mathrm{p}})M_{\\star}} \\right)^{\\frac{1}{3}}\\right]}\\qquad.\n\\end{equation}\n\nThis condition depends on the MMR itself, on the maximum eccentricity this MMR can induce on a planetesimal, and on the mass ratio between the planet and the star. \n\nNote that it does not depend on the semi-major axis of the planet. This was expected, since according to Eqs.~(\\ref{eq:MMRloc}) and ~(\\ref{eq:chaosin2}), all the quantities involved here, i.e. the location of the MMR and the inner edge of the chaotic zone, are proportional to the semi-major axis of the planet $\\mathrm{a_p}$. The size of the chaotic zone and the relative distance between a resonance location and the inner edge of the chaotic zone, depend on the mass and eccentricity of the planet. Thus if scattering events are expected for a given combination of planetary mass and eccentricity, they will be expected for any planetary semi-major axis.\n\nThe condition stated in Eq.~\\ref{eq:ability1} can be rewritten to find $\\mathrm{e_{crit}}$, the minimum eccentricity a planetesimal should acquire when in a given MMR to cross the chaotic zone of the planet, as a function of the planet mass and eccentricity :\n\n\\begin{equation}\\label{eq:ability2}\n\\mathrm{e \\geq e_{crit} = \\left( \\frac{n}{p} \\right)^{2\/3}(1-e_{\\mathrm{p}})  \\left(  1- k\\left[ \\frac{m_{\\mathrm{p}}}{(3+e_{\\mathrm{p}})M_{\\star}} \\right]^{\\frac{1}{3}} \\right) - 1 }\\qquad.\n\\end{equation}\n\n\\begin{figure*}\n\\centering\n\\makebox[\\textwidth]{\\includegraphics[width=0.5\\textwidth,height=0.5\\textwidth]{emin31_21_41.pdf}\n\\includegraphics[width=0.5\\textwidth,height=0.5\\textwidth]{emin52_73.pdf}}\n\\caption[]{The solid lines show the critical eccentricity $\\mathrm{e_{crit}}$ above which a planetesimal should be driven in order to cross the chaotic zone of a planet (solid lines), as a function of the mass ratio between the planet and the star, and for the 4:1, 3:1, 5:2, 7:3, and 2:1 MMRs. These results are independent from the semi-major axis of the planet, according to Eq.~(\\ref{eq:ability2}), and hold for a planetary orbital eccentricity $\\mathrm{e_p}=0.1$. The axis showing the mass of the planet is obtained by assuming $\\mathrm{M_{\\star}=1\\,M_{\\odot}}$. The horizontal dashed lines show the maximum eccentricity $\\mathrm{e_{max}}$ that a given MMR can induce on a planetesimal that has an initial eccentricity $\\lesssim 0.05$.}\n\\label{fig:e_min}\n\\end{figure*}\n\nAs displayed in Fig.~\\ref{fig:e_min}, this quantity can be plotted as a function of $\\mu$, or $\\mathrm{m_p}$ assuming one solar mass star, and a planetary eccentricity of $\\mathrm{e_p}=0.1$. If we compare this minimum eccentricity with the range of eccentricities an MMR can induce on a planetesimal, we notice that the 4:1, 5:2, and 7:3 MMRs are expected to be efficient at scattering planetesimals for a large range of planetary masses. The 3:1 MMR should be effective for a smaller range of planetary masses. \n\nSurprisingly, the 2:1 MMR can be expected to set planetesimals on cometary-like orbits as well, although its phase-space diagram shows that the increase in eccentricity this MMR can induce on a planetesimal is very small. This is due to the proximity of this resonance with the chaotic zone, as seen in Fig.~\\ref{fig:MMR_schema}. This case illustrates how the analysis proposed here complements the reading of the phase-space diagram.\n\nNote that a way to retrieve directly the mass regime at which an MMR is supposed to be effective is to consider that $\\mathrm{e_{max}}$ is independent of the planet mass, and can be approximated and fixed for a given $\\mathrm{n:p}$ MMR thanks to the corresponding phase-space diagram. One can then derive a minimum planet-star mass ratio $\\mathrm{\\mu_{min}}$ above which planetesimals trapped in this MMR will be expected to be able to cross the chaotic zone of the planet. Rewriting Eq.~\\ref{eq:ability1} in that sense gives :\n\n\\begin{equation}\\label{eq:mass_min}\n\\mathrm{\\mu \\geq \\mu_{min}=(3+e_{\\mathrm{p}}) \\left[\\frac{1}{k}\\left( 1- \\left(\\frac{p}{n} \\right)^{2\/3} \\frac{1+e_{max}}{1-e_p} \\right) \\right]^3} \\qquad.\n\\end{equation}\n\nAn upper limit can be set on the mass of the planet by considering the mass for which the chaotic zone of the planet will overlap with the resonance location, that is $\\mathrm{a_{chaos,in}<a_{MMR}}$. Using Eqs.~\\ref{eq:MMRloc} and ~\\ref{eq:chaosin2}, we find that the maximum planet-star mass ratio $\\mathrm{\\mu_{p,max}}$ allowed for a planet is :\n\n\\begin{equation}\\label{eq:mass_max}\n\\mathrm{\\mu \\leq \\mu_{max}=(3+e_{\\mathrm{p}}) \\left[\\frac{1}{k}\\left( 1- \\left(\\frac{p}{n} \\right)^{2\/3} \\frac{1}{1-e_p} \\right) \\right]^3} \\qquad.\n\\end{equation}\n\nIf the planet exceeds this mass, it will directly and immediately scatter planetesimals on cometary orbits. This would result in an intense and short lived comet scattering, rather than the one necessary to explain the observations of exozodis around old stars. Since the 2:1 MMR is the closer resonance to the chaotic zone, it will have the smallest maximum mass of $0.7\\,\\mathrm{M_{Jup}}$ around a Solar-mass star, while the 7:3, 5:2 and 3:1 are expected to lead to direct scattering for planets with masses 1.3, 1.6, and $2.6\\,\\mathrm{M_{Jup}}$, respectively, and with orbital eccentricity $\\mathrm{e_p}=0.1$.\n\nFinally, note that we considered a planetesimal starting to be acted upon when having an almost circular orbit, whereas this is not necessarily the case in reality. Indeed, we assumed that the system has suffered from an instability in order for the perturber at work to have been set onto an eccentric orbit. Therefore, the material remaining in the reservoir belt can be expected to exhibit a certain level of excitation. This means that the planetesimals' initial eccentricities in our mechanism are expected to be actually larger than the value of $0.05$ used throughout this paper. This is not expected to inhibit the mechanism that produces active comets, since, as one can see on phase space diagrams, the greater the initial eccentricity of the particles, the greater the eccentricities it will have access to. However, it could also be expected that the timescale required to reach the necessary high eccentricities and suffer a close encounter with the planet is reduced, hence leading to a more rapid production of comets.\n\n\\section{Numerical study}\\label{sec:results}\nStudying the behaviour of a swarm of planetesimals by N-body simulations is an excellent way to complement our analytical study. It will allow us to explore statistics and timescales of the mechanism that sets planetesimals on cometary orbits.\nIn this section where we focus on the 5:2 MMR as a study case, we first describe the method we followed (code, initial conditions, parameters explored), before describing the results of our numerical simulations.\n\n\\subsection{Numerical initial conditions}\n \nOur initial sets of planetesimals all contain 5,000 massless test particles, for which eccentricity and inclination are randomly and uniformly distributed between 0-0.05 and $\\pm 3^{\\circ}$, respectively, which correspond to what can be expected from a cold debris disc, as constrained from observations of the Kuiper belt where $\\mathrm{e_{max}}\\sim 0.1$ and $\\mathrm{i_{max}}\\sim 3^{\\circ}$ \\citep{2011AJ....142..131P}.\nThe value of 0.05 chosen for the maximum initial eccentricity of our particles is arbitrary compared to the 0.1 limit of \\citet{2011AJ....142..131P}.\nThe remaining characteristic angles of the test particles, the longitude of ascending node $\\Omega$, the longitude of periastron $\\omega$ and the mean anomaly $\\mathrm{M}$, are all randomly and uniformly distributed between 0 and $2\\pi$, that is, the orbits exhibit no preferential direction. \nTheir semi-major axis distribution spans 0.5 AU, among which the particles semi-major axes are randomly and uniformly distributed. These intervals are centred on the exact mean-motion resonance location, as given by Eq.(\\ref{eq:MMRloc}). Their location will depend on the MMR and on the semi-major axis of the planet, therefore, the location of our initial ring of planetesimals will change from one simulation to another. As seen in the previous section, analytical studies indicate that the architecture of the system scales with $\\mathrm{a_p}$. Therefore, this parameter is expected to have no influence on the feasibility of the process.\n\nIn other words, it is only the mass of the planet that is expected to determine the feasibility, while the scattering rates, as well as delays and duration, are expected to depend on both the mass and the semi-major axis of the planet.\nLarger semi-major axes, that is, a larger scale of the architecture, translates into longer periods, and thus larger dynamical timescales. On the other hand, dynamical timescales are expected to be larger at smaller $\\mathrm{m_p}$, since less massive planets are expected to make planetesimals travel slower across their phase-space. \nWe explore planetary masses $\\mathrm{m_p}=0.1-0.25-0.5-0.75-1\\,\\mathrm{M_{Jup}}$ and semi-major axes $\\mathrm{a_p}=25-50-75-100\\,$AU. The influence of the mass of the central star will be discussed in Sect.~\\ref{sec:applis} as well as its luminosity.\nIn all cases, the central star is $\\mathrm{M}_{\\star}=1\\,\\mathrm{M}_{\\odot}$, and we only consider the case where there is no mutual inclination between the ring and the orbit of the planet, that is, the orbital inclination of the planet is always zero. It is certainly not realistic regarding our assumptions on the origin of the eccentric perturber, because planets are expected to suffer changes in both eccentricity and inclination when being scattered.\n\nDetermining the effect of mutual inclination would imply carrying out a 3D analysis for the 5:2 MMR, as was done for the 4:1 MMR by \\citet{2000Icar..143..170B} and \\citet{2007A&A...466..201B}. This is beyond the scope of this paper, however, note that \\citet{2000Icar..143..170B} showed that mutual inclination of $(\\lesssim 10^{\\circ})$  between the orbit of an eccentric planet and a reservoir in 4:1 MMR does not inhibit the generation of active comets via direct placement\\\\ \\citep{2000Icar..143..170B}, as their results in the coplanar case and the case with a mutual inclination were almost identical.\n\nWe integrate the evolution of this system over 1 Gyr, using the symplectic N-body code SWIFT-RMVS \\citep{1994Icar..108...18L}. We used a typical time-step of $\\sim 1\/20\\,$ of the smallest orbital period, which ensures a conservation of energy with a typical error of $\\sim 10^{-6}$ over the whole integration, since the symplectic nature of the code ensures there is no long-term drift of energy error. Particles were removed when approaching the star at a distance of less than $\\sim 1 \\mathrm{R}_{\\odot}$. We record snapshots of our simulations every 1 Myr and for each snapshot, we examine the particles which have orbits with $\\mathrm{q<q_{sub}}$ and calculate how many pericentre passages they have per year. We refer to the total number of pericentre passages per year of all active comets as the cometary events per year.\n\n\\subsection{Results}\n\nA large proportion of our simulations gave rise to what was expected, that is a delayed production of active comets, and then a continuous production at an approximately constant rate, until the system reaches 1 Gyr. This continuous production of cometary orbits was expected, because planetesimals may take several precession cycles and cross the chaotic zone of the planet several times before being scattered on to a comet-like orbit. Also, as mentioned in the previous section, the delay gives us an estimate of the precession timescale. Therefore, it was also expected that in the case of large ($\\sim 100\\,$Myr) delays, the production of active comets would be maintained for at least a few precession cycles and therefore until 1 Gyr. However, in some cases the delay is very short, that is inferior to 50 Myr, and yet the activity is still maintained up to 1 Gyr. Consequently, the number of cycles necessary for all the planetesimals in MMR to encounter the planet is of the order of several tens.\n\nIn Table~\\ref{tab:simus}, we summarise the results of the delay in the appearance of active comets. These delays range from several 10 Myr to several 100 Myr. As expected, this timescale increases with decreasing mass of the planet (see Fig.~\\ref{fig:example_rate}). It also increases for increasing semi-major axes, however, there are exceptions to this rule, for which very few comets were produced and\/or the timescales do not match the expected trend. These exceptions occur for planets with masses $\\leq 0.5\\,(\\mathrm{M_{Jup}})$, and at 75 AU, which appears to contradict the expectation that the semi-major axis has no impact on the feasibility of the process. Although we do not understand yet what is the cause of these irregularities, studying the problem will probably require an extensively detailed analysis of the orbital evolution of the test-particles. This will be the subject of future work and we will focus here on the most common behaviour.\n\nThe cases where a planet is at 25 AU appear to be limiting cases. In these cases the production of active comets is almost immediate with delays of several Myr only. This was expected because at this semi-major axis, as we mentioned in the previous section, the resonance will sustain direct production of cometary orbits, without the need for an additional scattering event. We also expected a profile with a sharp peak of intense production of active comets, which we indeed observe for planets with masses $\\mathrm{m_p}\\geq 0.5\\,(\\mathrm{M_{Jup}})$ (see Fig.~\\ref{fig:direct}). As expected, the peak rate is at least one or two orders of magnitude larger than when the planet is at larger distances than 25 AU. Note however, that although the activity is not maintained until 1 Gyr, it was maintained for several 100 Myr, which may be sufficient to produce an exozodi in systems younger than this. Interestingly, at the same distance, planets with masses $\\mathrm{m_p}\\leq 0.5\\,(\\mathrm{M_{Jup}})$ can cause the direct production of active comets to be sustained for Gyr timescales and with rates comparable or greater than those obtained via the indirect mechanism (see Fig.~\\ref{fig:direct}).\n\nIn the remaining cases, the number of cometary events per year spans two orders of magnitude with a tendency for this rate to be smaller for planets on very distant orbits than for closer planets. This tendency can be easily explained, as comets that originate at larger distances have larger semi-major axes, and longer periods, which naturally diminishes the number of cometary events per year.\nThe dynamical lifetime of a comet-like orbit is in general inferior to 1 Myr (in $\\sim 75\\%$ of orbits), with a tendency for smaller mass planets to give the largest survival timescales.\n\nHowever, the values numerically found for the rates and survival lifetime must be taken with caution, because they are not necessarily realistic. We used 5,000 test particles in our simulations while real belts contain much more bodies. Besides, the dynamical lifetime of a cometary orbit may differ substantially from the actual lifetime of the comet against evaporation. \n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.55\\textwidth]{comet_events_C1e_C1c_C5e.jpg}\n\\caption[]{Evolution of the number of cometary events per year as a function of time, when the cometary event is generated via the 5:2 MMR with a planet of eccentricity $\\mathrm{e_p=0.1}$ with planet of mass $0.1\\,\\mathrm{M_{Jup}}$ at 100 AU \\emph{(solid line)}, a planet of mass $0.1\\,\\mathrm{M_{Jup}}$ at 50 AU \\emph{(dashed line)}, and a planet of mass $1\\,\\mathrm{M_{Jup}}$ at 100 AU \\emph{(dotted line)}. The delay increases with increasing semi-major axis and decreasing mass, as expected. The production of active comets can be sustained up to at least 1 Gyr and probably beyond, even when the delay is short. Note that the rates presented here are raw rates from our numerical simulations that shall be translated into more realistic values in Sect.~\\ref{sec:applis}, and that the curve is time-averaged over 10 Myr intervals to reduce some of the stochasticity that results from the resolution of the simulations.}\n\\label{fig:example_rate}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.55\\textwidth]{comet_events_C1b_C3b_C5b.jpg}\n\\caption[]{Evolution of the number of cometary events per year as a function of time, when the cometary event is generated via the 5:2 MMR with a planet of eccentricity $\\mathrm{e_p=0.1}$ at 25 AU, and with mass $0.1\\,\\mathrm{M_{Jup}}$ (\\emph{dotted line}), $0.5\\,\\mathrm{M_{Jup}}$ (\\emph{dashed line}), and $1\\,\\mathrm{M_{Jup}}$ (\\emph{solid line}). Here the placement of planetesimals on cometary orbits is produced directly by the MMR without the need for an additional scattering event. Therefore, as expected, planetesimals are set on cometary orbits in only few Myr, and the activity profile is more intense. If the planet has a mass of the order of that of Jupiter, the production of cometary orbits is sustained on timescales shorter than in the case the production is indirectly induced via scattering events. Note that it can nevertheless be maintained on timescales of the order of several 100 Myr. In the case of less massive planets, direct production can be sustained up to at least 1 Gyr and probably beyond.}\n\\label{fig:direct}\n\\end{figure}\n\n\\begin{table*}\n \\caption{Properties of the simulations as a function $\\mathrm{a_p}$, and $\\mathrm{m_p}$:  peak cometary event rate $\\mathrm{N_{max}}$, delay before active comets are seen $\\mathrm{t_{del}}$ in Myr, duration of the production of active comets $\\mathrm{t_{prod}}$ in Myr, or minimum duration if the production is still ongoing at the end of our simulations, and FWHM of the peak in Myr, or minimum FWHM in case the time for the profile to reach again half of the peak value exceeds the duration of our simulations. Note that in some cases, it is questionable whether the maximum rate is reached during the simulation. There were no active comets produced for a planet of 0.1 $\\mathrm{M_{Jup}}$ at 75 AU.}\n\\label{tab:simus}\n\n\\begin{tabular*}{\\textwidth}{@{\\extracolsep{\\fill}}c|cccccc}\n\\toprule\n  \\multirow{2}{*}{$\\mathrm{a_p}$ (AU)}   & & \\multicolumn{5}{c}{$\\mathrm{m_p}\\,(\\mathrm{M_{Jup}})$} \\\\[5pt]\n  & & 0.1 & 0.25 & 0.5 & 0.75 & 1 \\\\[5pt]\n\\midrule\n    & $\\mathrm{N_{max}}$ & $\\sim 3.3\\times 10^{-1}$ & $\\sim 6.2\\times 10^{-1}$ & $\\sim 8.5\\times 10^{-1}$ & $\\sim 7.9\\times 10^{-1}$ &  $\\sim 5.1\\times 10^{-1}$ \\\\[2pt]\n  25  & $\\mathrm{t_{del}}$ & $ \\sim$ 5   & $\\sim$ 5   & $\\sim$ 5   & $\\sim$ 5  & $\\sim$ 5\\\\[2pt]\n    & $\\mathrm{t_{prod}}$&$\\gtrsim$ 995 & $\\gtrsim$ 995 & $\\gtrsim$ 995 & $\\gtrsim$ 995 &  $\\sim$ 600 \\\\[2pt]     \n    & FWHM & $\\gtrsim$ 900 & $\\sim$ 900 & $\\sim$ 250 & $\\sim$ 100 &  $\\sim$ 50 \\\\[2pt]   \n\\midrule\n    & $\\mathrm{N_{max}}$&$ \\sim 7.2\\times 10^{-2}$ & $\\sim 8.6\\times 10^{-2}$ & $\\sim 8.0\\times 10^{-2}$ & $\\sim 8.9\\times 10^{-2}$ &  $\\sim 9.6\\times 10^{-2}$ \\\\[2pt]\n50  & $\\mathrm{t_{del}}$&$ \\sim$ 250 & $\\sim$ 60  & $\\sim$ 40  & $\\sim$ 30 & $\\sim$ 20    \\\\[2pt]\n    & $\\mathrm{t_{prod}}$&$ \\gtrsim$ 750 & $\\gtrsim$ 940 & $\\gtrsim$ 960 & $\\gtrsim$ 970 &  $\\gtrsim$ 980 \\\\[2pt]     \n    & FWHM & $\\gtrsim$ 500 & $\\gtrsim$ 700 & $\\sim$ 900 & $\\sim$ 700 &  $\\sim$ 300 \\\\[2pt]  \n\\midrule\n    &  $\\mathrm{N_{max}}$& &$  \\sim 3.5\\times 10^{-3}$ & $\\sim 3.3\\times 10^{-2}$ & $\\sim 4.1\\times 10^{-2}$ &  $\\sim 2.6\\times 10^{-2}$ \\\\[2pt]  \n75  &   $\\mathrm{t_{del}}$& &$  \\sim$ 600 & $\\sim$ 200 & $\\sim$ 60 & $\\sim$ 30   \\\\[2pt]\n    &   $\\mathrm{t_{prod}}$& &$ \\gtrsim$ 400 & $\\gtrsim$ 800 & $\\gtrsim$ 940 &  $\\gtrsim$ 970 \\\\[2pt]     \n    &  FWHM & & $\\gtrsim$ 300 & $\\gtrsim$ 400 & $\\sim$ 800 &  $\\sim$ 600 \\\\[2pt]  \n\\midrule\n    & $\\mathrm{N_{max}} $&$\\sim 1.4\\times 10^{-2}$ & $\\sim 2.1\\times 10^{-2}$ & $\\sim 3.6\\times 10^{-2}$ & $\\sim 3.7\\times 10^{-2}$ &  $\\sim 2.5\\times 10^{-2}$ \\\\[2pt]\n100 & $\\mathrm{t_{del}} $&$\\sim$ 500 & $\\sim$ 200 & $\\sim$ 100 & $\\sim$ 70 & $\\sim$ 40  \\\\[2pt]\n    & $\\mathrm{t_{prod}}$&$\\gtrsim$ 500 & $\\gtrsim$ 800 & $\\gtrsim$ 900 & $\\gtrsim$ 930 &  $\\gtrsim$ 960 \\\\[2pt]     \n    & FWHM & $\\gtrsim$ 200 & $\\gtrsim$ 500 & $\\gtrsim$ 700 & $\\gtrsim$ 750 &  $\\sim$ 700 \\\\[2pt]  \n\\bottomrule     \n\\end{tabular*}\n\n\\end{table*}\n\n\\section{Applications}\\label{sec:applis}\n\nIn this section, we apply our findings to specific cases and show how the raw rates from our simulations can be rescaled and interpreted for different situations. This will allow us to discuss the influence of the mass of the initial reservoir and of the star's luminosity. In particular, we will investigate the case of a low mass disc around a Solar-type star and demonstrate that even a low mass disc is a reservoir that can provide enough active comets to sustain the amount of dust necessary to produce the detected exozodis. The second application demonstrates how high the comet scattering can be when considering a much larger disc around an A-type star, namely Vega.\n\\subsection{Rescaling raw rates}\n\nFirstly, the lifetime of a comet against evaporation $\\mathrm{t_{evap}}$ may differ from the survival dynamical timescale $\\mathrm{t_{dyn}}$ of its orbit. If comets evaporate faster than their dynamical timescale, the raw rates will give an overestimation and should therefore be corrected by a factor $\\mathrm{t_{evap}\/t_{dyn}}$. \nThe dynamical timescale can be retrieved from our simulations, while $\\mathrm{t_{evap}}$ can be determined using the recent cometary evaporation models of \\citet{2016arXiv160403790M}.\n\nSecondly, our simulations contain $\\mathrm{N_{init}}=5,000$ test particles distributed over a radial range of 0.5 AU, which is extremely small compared to what would be expected in real debris discs.\nFor instance, in the Kuiper belt interior to 50 AU, there is an estimate of $5\\times 10^9$ comets \\citep{1995AJ....110.3073D}. The principal part of the Kuiper belt extends between the 2:3 and 1:2 MMR with Neptune, that is from 39.5 to 48 AU, which gives $\\sim 10\\,$AU in radial extension.\nTherefore, if our simulations were to reproduce realistically a Kuiper-belt analogue, these would have to contain $\\mathrm{N_{real}}=2.5 \\times 10^8$ test-particles.\nIt implies that the raw production rates obtained from our simulations have to be rescaled in order to obtain realistic values. As was expected, and as checked numerically by running additional simulations with a planet at 100 AU and with mass $0.25\\,\\mathrm{M_{Jup}}$, and with 1,000, 2500, and 10,000 test-particles, the rate scales linearly with the initial number of particles in the belt.\nConsequently, realistic rates are obtained by correcting the raw rates by a factor $\\mathrm{\\frac{N_{real}}{N_{init}}}$, and which value is $5\\times 10^4$ in the case the reservoir is a Kuiper belt analogue.\n\\subsection{Interpretation and comparison with observations}\n\nComparing our results with observations of hot exozodiacal dust around other Sun-like stars will rely on the assumption that we are looking at exact analogues to our own Solar System zodiacal cloud, and considering bodies sublimating below a 3 AU distance from their host star. \nThis approach is limited since the detected exozodis can be made of dust very close to the star (and therefore hot). Indeed, if we want to test in a direct way whether our mechanism can reproduce the amount and incidence rate of this dust, we must measure how much dust can be brought close enough to the star and 3 AU for a main sequence star is not necessarily close enough.\nHowever, the need for this assumption arises from the fact that it is difficult to derive the dust mass present in the detected exozodis. Indeed, it requires well constrained dust properties which are not available. On the other hand, the near-IR emission of our own Zodiacal cloud is poorly constrained, and strongly relies on models. Therefore, without this assumption, we would not be able to make any comparison, knowing mass feeding rates of our Zodiacal cloud, but very poorly that of exozodis, or knowning the near-IR emission of exozodis, but very poorly that of the Zodiacal cloud.\nUnder this assumption, we first estimate the amount of dust that is present in these systems. So far, only a few systems around A type stars have been modeled extensively and dust masses have been derived \\citep{2006A&A...452..237A,2011A&A...534A...5D,2013A&A...555A.146L}. The amount of hot dust in these systems is found to be about 100 to 1000 times higher than in our Solar system, where the uncertainties stem from both the modeling uncertainties and the poorly constrained amount of hot dust in our own Solar system \\citep{1998EP&S...50..493K,2002Icar..158..360H}. \nWe use this number to estimate the amount of dust needed to produce the same ($\\sim$1\\%) near-infrared dust-to-star flux ratio also observed around Sun-like stars \\citep{2013A&A...555A.104A,2014A&A...570A.128E}. We assume that the dust emission in all systems is dominated by thermal emission at the dust sublimation temperature. This is justified, because the interferometric constraints require the dust to be either extremely hot or the emission to be dominated by scattered light \\citep{2012A&A...546L...9D,2014A&A...570A.128E}. The latter has been largely ruled out by polarimetric non-detections of the excesses \\citep{2016arXiv160408286M} and the confirmation that the majority of the excesses are stable over time scales of years \\citep{2016arXiv160805731E}. \nAt a given (sublimation) temperature the total dust emission scales with the emitting surface and thus with the amount of dust present if we assume similar dust grain sizes and properties. Given the same flux ratio observed for A type and Sun-like stars, the total dust emission (and thus the dust mass) must scale with the star's emission. A Sun-like star is about an order of magnitude fainter in H band than an A-type star. Under the above assumptions the amount of dust around a Sun-like star required to produce the 1\\% flux ratio is about an order of magnitude lower than that required around an A-type star. We thus estimate that the amount of hot dust detected in the near-infrared around Sun-like stars is about one to two orders of magnitude higher than that in our Solar system.\nThen, under the assumption that the exozodis observed around Solar-type stars are exact analogs to our Zodiacal cloud, we can estimate the amount of dust released by comets from our simulations by using cometary evaporation models and estimates of the mass feeding rate of the Zodiacal cloud. This amount of dust can then be compared to that estimated from observations (10-100 zodis).\n\\subsection{Solar-type stars with a Kuiper belt analogue}\\label{sec:comp_obs}\nHere we take a look at the case of a Solar type star with a low mass debris disc and consider whether there is enough of a planetesimal reservoir to provide the necessary comets to explain the presence of exozodis around mature stars. To do this, we consider the star to have a disc analogous to the Kuiper belt both in terms of mass and location, but with a planet exterior to it.\n\\citet{2016arXiv160403790M} found that km-sized bodies survive for 480-4800 orbits around Solar-type stars. From our simulations, $\\mathrm{t_{evap}}\\sim 0.1-1\\,$Myr is the typical amount of time this number of orbits translates into. We retrieved $\\mathrm{t_{dyn}}$ by considering our simulation with a planet of mass $1\\,\\mathrm{M_{Jup}}$ at 75 AU, and running an additional 10 Myr integration, taking snapshots every 10,000 yrs. We find that the dynamical lifetime of the orbits is typically 0.1-2 Myr (see Fig.~\\ref{fig:orbit_life}), which is consistent with the lifetime of comets against evaporation around Solar-type stars. Consequently, the correction factor is close to, and taken to be unity.\nIn our simulations, the rates range from $\\sim 0.01$ to $\\sim 0.05$ cometary events per year. Applying a correction factor $5\\times 10^4$ to the raw rates, the realistic rates produced from a Kuiper-belt analogue would be $\\sim 500-2500$ cometary events per year.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.55\\textwidth]{orbit_life_C4dresol.jpg}\n\\caption[]{Histogram of the dynamical survival lifetime of cometary orbits, when the planet involved has eccentricity $\\mathrm{e_p=0.1}$, mass $0.75,\\mathrm{M_{Jup}}$, and is at 75 AU, that is, for a reservoir at the scale of the Solar System Kuiper belt. The majority of the orbits have survival timescales of less than 2 Myr, and of more than several 100,000 yrs, with a mean survival of $\\sim 1\\,$Myr.}\n\\label{fig:orbit_life}\n\\end{figure}\n\nWe wish to compare these rates of $\\sim 500-2500$ cometary events per year to the number of active comets that are required in the Solar system to sustain the Zodiacal dust. The rate at which the inner parts of the Solar system are replenished in dust has been estimated to be $10^3\\,\\mathrm{kg.s^{-1}}$ \\citep{1983A&A...118..345L,2010ApJ...713..816N}, that is $\\sim 5\\times 10^{-15}\\,\\mathrm{M_{\\oplus}.yr^{-1}}$.\n\nEvaporation and sublimation of a typical comet (2.5 km in radius) releases $\\sim 3.6\\times 10^{-15}\\,\\mathrm{M_{\\oplus}.yr^{-1}}$ on average between 2 and 3 AU \\citep{2016arXiv160403790M}, so that the rates of $\\sim 500-2500$ cometary events per year from our simulations would translate into mass feeding rates of $\\sim 10^{-12}-10^{-11}\\,\\mathrm{M_{\\oplus}.yr^{-1}}$.\nIf we take for example the case shown in Fig.~\\ref{fig:example_rate2}, that is for a planet of mass $0.75\\,\\mathrm{M_{Jup}}$ at 75 AU, the 5:2 MMR would be located at 40.7 AU, which corresponds to the location of the Solar System's Kuiper-belt. In this case, the expected rate is $\\sim 1000\\,$cometary events per year, that is a few $10^{-12}\\,\\mathrm{M_{\\oplus}.yr^{-1}}$, which is three orders of magnitude greater than the rate at which the Zodiacal cloud is replenished. This means that a planet with an eccentricity of 0.1 exterior to an analogue of the Kuiper-belt in terms of content and scale could produce an exozodi comparable to those that are detected.\n\nNote that according to the results of our simulations, these rates could be also achieved on Gyr timescales thanks to low mass planets $(\\lesssim 0.5\\,\\mathrm{M_{Jup}})$ at distances such that they trigger direct cometary orbits production ($\\lesssim 25\\,$AU).\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.55\\textwidth]{comet_events_C4dKuiper.jpg}\n\\caption[]{Evolution of the number of cometary events per year as a function of time, when the cometary event is generated via the 5:2 MMR with a planet with orbital eccentricity $\\mathrm{e_p=0.1}$, mass $0.75\\,\\mathrm{M_{Jup}}$ and semi-major axis 75 AU. In this case, the reservoir is at the location of the Kuiper Belt.}\n\\label{fig:example_rate2}\n\\end{figure} \n \nThese high rates raise the question whether even the massive, hot exozodiacal dust systems detected in the near-infrared \\citep{2013A&A...555A.104A,2014A&A...570A.128E} can be produced by our proposed scenario. \\citet{2014A&A...570A.128E} found no strong correlation between the presence of hot dust and that of a Kuiper-belt like cold debris disk. Our Kuiper belt would have remained undetected around other stars in available observations due to its low mass \\citep[e.g., with Herschel, ][]{2010A&A...520A..32V}. Thus, a scenario that can produce sufficient amounts of hot dust from such a belt could explain this lack of strong correlation.\n\n\\subsection{The Vega System}\n\nExtreme feeding rates are required around A-type stars because dust grains have a much shorter lifetime around these stars than around Solar-type stars. A famous example of an A-type star surrounded by an exozodi is Vega. With a luminosity of 40 times that of the sun \\citep{2010ApJ...708...71Y}, dust grains have a very short lifetime in its vicinity, of the order of 1 yr due to the short collision times and high radiation pressure \\citep{2006A&A...452..237A,2011A&A...534A...5D,2016arXiv160403790M}, whereas grains have lifetimes of $2-6\\times 10^{5}$ yrs in the Solar System's Zodiacal cloud due to PR drag \\citep{2002ApJ...578.1009F,2003Icar..164..384R}. The quantity of dust deposited by comets must be sufficient to replenish the exozodi on timescales of the order of the grains lifetime. Consequently, the number of comets necessary around Vega can be expected to be more important than it would be to produce a comparable exozodi around a Solar-type star.\n\nNotably, Vega possesses a Kuiper belt analogue which peaks at $\\sim 100\\,$AU \\citep{2005ApJ...628..487S,2010A&A...518L.130S}. Therefore, we ran a simulation specifically dedicated to Vega. According to \\citet{2010ApJ...708...71Y}, Vega has a mass of $\\mathrm{M_{\\star}=2\\,M_{\\odot}}$. The ring is centred on 100 AU, where it is in 5:2 MMR with an 0.1 eccentric planet, thus located at 184.2 AU. These parameters are summarized in Table \\ref{tab:simus3}. We set the mass of the planet to $1\\,\\mathrm{M_{Jup}}$, and otherwise used the exact same conditions as for the rest of our simulations.\nRecent cometary evaporation modelling find that the rate required around Vega spans values of 1-2500 cometary events per month, with comets sublimating below 1 AU, that is, where the dust has been observed around Vega \\citep{2016arXiv160403790M}. Therefore, we extract from our simulation cometary orbits with periastron smaller than 1 AU. The best-fit mass to the cold belt of Vega is $46.7\\mathrm{M_{\\oplus}}$ \\citep{2010ApJ...708.1728M}, while estimates for the Kuiper-belt give a mass of $0.1\\mathrm{M_{\\oplus}}$ \\citep{2001AJ....122.1051G}, that is 500 times less. Therefore, we use a first correction factor 500 times larger than the one used for the Kuiper belt. Around extremely luminous stars (several tens of $\\mathrm{L_{\\odot}}$), \\citet{2016arXiv160403790M} found that the number of orbits it takes for a comet to evaporate is 4-40. The cometary orbits from our simulations achieve this number of orbits in several 1,000-10,000 years. The comets evaporate on timescales $\\mathrm{t_{evap}}$ two orders of magnitude smaller than the typical survival timescale $\\mathrm{t_{dyn}}$ of an orbit and a second correction factor of 1\/100 is applied to the raw rates. We find that the rate achievable would be $\\sim 500$ events per year (see left panel of Fig.~\\ref{fig:Vega_rate}). Therefore, the mechanism we present here could sustain several tens of cometary events per month, which is compatible with the rates derived by \\citet{2016arXiv160403790M}.\nNote that the approach described above supposes that the dust is delivered where the exozodi is observed, which is not necessarily the case. Due to its sensitivity to stellar radiation effects, the dust deposited by comet sublimation is expected to spiral inwards towards the star due to Poynting-Robertson drag effects. On the other hand, the water ice sublimation radius is expected to become larger with increasing stellar luminosity. In the previous sections, cometary orbits had been defined as orbits with periastron smaller than 3 AU, which is relevant around Solar-type stars, however, in the case of an A-type star such as Vega, this sublimation radius is expected to be of the order of $\\sim 18$ AU \\citep{2016arXiv160403790M}. As can be seen in Fig.~\\ref{fig:Vega_rate}, where bodies on orbits with periastrons between 1 and 18 AU are included, the rate of cometary events is one order of magnitude greater. Although PR drag can only bring a limited amount of dust inwards because of competitive collisional processes \\citep{2006A&A...452..237A}, the rates extracted from our simulations by considering on site sublimation and disregarding PR drag actually underestimate the real feeding rates.\n\nIf this mechanism were at work in the Vega system, the presence of the eccentric planet would have set an imprint on the debris belt, by secularly forcing the eccentricity of the main non-resonant part of the belt to higher values $\\mathrm{e_f}$. Applying Eq.(2) of \\citet{2014A&A...563A..72F}:\n\n\\begin{equation}\\label{eq:forced_ecc}\n\\mathrm{e_f \\simeq \\frac{5}{4} \\frac{a}{a_p} \\frac{e_p}{1-e_p^2}} \\qquad,\n\\end{equation}\n\nthis would result into the shaping of the belt into a ring with an eccentricity of 0.07, with a center of symmetry that is offset by $\\sim 7\\,$AU from the star. Such a small eccentricity and offset would not be detectable in the available data. Although it is one of the most famous debris disks and the first one discovered, high angular resolution observations are scarce for this disk. The cool dust was found to be distributed in a radially very broad, face-on disk in mid- to far-infrared observations obtained with Spitzer \\citep{2005ApJ...628..487S} and Herschel \\citep{2010A&A...518L.130S}. An inner hole is visible in the Herschel\/PACS data in the star-subtracted images only \\citep{2010A&A...518L.130S,2013ApJ...763..118S}, where the exact stellar location is uncertain. The disk's measured extent seems to be sensitivity limited with its outer edge remaining undetected. In such a situation, measuring an offset of the disk from the central star or detecting a pericenter glow \\citep{1999ApJ...527..918W} is not possible. Scattered light or well resolved millimeter interferometric observations are difficult for such a broad disk due to its expected low surface brightness. Clumpy structures in (sub-)millimeter observations of the disk have been reported and attributed to the interaction with a companion \\citep{1998Natur.392..788H,2001ApJ...560L.181K,2002ApJ...569L.115W}, but could not be confirmed by more sensitive observations \\citep{2011A&A...531L...2P}.Finally, limits from direct imaging observations with Spitzer rule out the presence of planets with mass greater than 2$\\,\\mathrm{M_{Jup}}$ and semi-major axis inferior to 150 AU \\citep{2015A&A...574A.120J}.\nIn summary, the available observations cannot rule out the presence of the planet required in our scenario to explain the production of hot dust in this system. Considering the extent of the debris disc surrounding Vega \\citep[several hundreds of AU, ][]{2005ApJ...628..487S}, our scenario implies that the perturbing planet resides within the disc, and at a large separation. This does not seem unrealistic, since planets residing within or even outside a Kuiper belt analogue have been suggested, for instance, as an explanation for a gap in the disc around HD 107146 \\citep{2015ApJ...798..124R}, or for a sharp outer edge in the Fomalhaut system \\citep{2012ApJ...750L..21B}.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.55\\textwidth]{comet_events_Vega.jpg}\n\\caption[]{Evolution of the number of cometary events per year as a function of time, when the cometary event is generated via the 5:2 MMR with a planet of eccentricity $\\mathrm{e_p=0.1}$ and semi-major axis 184.2 AU. In this configuration, the 5:2 MMR is located at 100 AU, that is where Vega's cold belt is located. This belt is 500 times more massive than the Kuiper belt so the rate was scaled to what would be expected from such a belt. In addition, the raw rate was also scaled down by a factor 100 because comets are expected to survive on timescales 100 times shorter than their actual lifetime against evaporation around an A-type star. The cometary orbits are defined as orbits with periastron lower than 1 AU on the \\emph{left panel}, and 18 AU on the \\emph{right panel}.}\n\\label{fig:Vega_rate}\n\\end{figure}\n\n\\begin{table}\n\\caption{Parameters $\\mathrm{m_p}$ and $\\mathrm{a_p}$, along with the MMR explored and its corresponding location as a function of $\\mathrm{a_p}$, for additional simulations. The test-particles are distributed over 0.5 AU centred on the theoretical MMR location computed with Eq.~\\ref{eq:MMRloc}, and follow the initial distribution described in Sect.~\\ref{sec:results}.}\n\\label{tab:simus3}\n\n\\begin{tabular*}{\\columnwidth}{@{\\extracolsep{\\fill}}llllll}\n\\hline\\hline\\noalign{\\smallskip}\nCase &  $\\mathrm{m_p}\\,(\\mathrm{M_{Jup}})$ & $\\mathrm{a_p}$ (AU) & $\\mathrm{e_p}$ & MMR & $\\mathrm{a_{MMR}}$ (AU) \\\\\n\\hline\\noalign{\\smallskip}\nVega & 1 & 184.2 & 0.1 & 5:2 & 100\\\\\n2:1 MMR & 0.5 & 100 & 0.1 & 2:1 & 63.0\\\\\n7:3 MMR & 0.5 & 100 & 0.1 & 7:3 & 56.8\\\\\n3:1 MMR & 0.5 & 100 & 0.1 & 3:1 & 48.1\\\\\n$\\mathrm{e_p=0.2}$ & 1 & 100 & 0.2 & 5:2 & 54.3\\\\\n\\noalign{\\smallskip}\n\\end{tabular*}\n\n\n\\end{table}\n\n\\section{Discussion and Conclusion}\\label{sec:conclusion}\n\nWe found that the 5:2 MMR with a moderately eccentric planet could sustain the observed exozodis on Gyr timescales around Solar and A-type stars. The reservoir belts required are realistic, and could have remained undetected. We took here the example case of the 5:2 MMR, but this process is also expected to occur if the reservoir overlaps with the 7:3, 3:1, and 2:1 MMR. We discuss hereafter the impact of these resonances on the process. We also discuss the impact of the planet eccentricity on the process before drawing our conclusions.\n\n\\subsection{Impact of the nature of the MMR}\n\nThe rates of cometary events are expected to vary from one MMR to another. Therefore, we use the example case of a planet with an eccentricity of 0.1 at 100 AU, and run additional simulations for the 7:3, 3:1 and 2:1 MMR for comparison with the 5:2. We do not study the 4:1 MMR here, because, as mentioned previously, it would lead to direct placement on cometary orbits without the need for an additional scattering event, no matter the value of the planet semi-major axis. Since it was analytically predicted that the 2:1 MMR would start overlapping with the chaotic zone of the planet for masses greater than $0.7\\,\\mathrm{M_{Jup}}$, we chose to set the mass of the planet to $0.5\\,\\mathrm{M_{Jup}}$. The parameters for these simulations are summarized in Table~\\ref{tab:simus3}. \n\nIn Fig.~\\ref{fig:impact_MMR}, we display the results for the 3:1, 5:2 and 2:1 MMRs, with rates scaled for a reservoir of the mass of the Kuiper Belt as described in Sect.~\\ref{sec:applis}. \nThe production of active comets generated via the 2:1 MMR shows the same profile as in the case of the 5:2 MMR: it is delayed ($\\sim 200\\,$Myr), and sustained until 1 Gyr. However, compared with the 5:2 MMR, the rates achieved are several times greater. This means that the exozodis formed via this MMR would be several times more massive than those obtained via the 5:2 MMR.\n\nIn the case the planetesimals originate from the 3:1 MMR, the placement on cometary orbits starts almost immediately and the rates are one order of magnitude greater than in the case they originate from the 5:2 or 2:1 MMR. A closer examination of the semi-major axes of the comets formed reveals they are still in MMR while reaching periastrons smaller than 3 AU, which means they have reached eccentricities greater than $\\sim 0.95$. This is much larger than the theoretically achievable eccentricity. Indeed, as can be seen on the phase-space diagram, the highest eccentricity expected in this MMR for a planetesimal starting on a low eccentricity orbit (<0.05) is $\\sim 0.5-0.6$. Consequently, these results suggest that possible weak planet-particles interactions, e.g., when the planetesimal does not come close enough to the planet to be scattered but suffers nevertheless an impulsion, lead the planetesimals to drift progressively in the phase-space towards higher eccentricities instead of being scattered out of the MMR onto cometary orbits. The 7:3 MMR produced no comets, whereas it was expected to be a channel for cometary production according to analytical predictions in Sect.~\\ref{sec:analytical}. We ran an additional simulation for a 0.75 $\\mathrm{M_{Jup}}$ planet at 75 AU and found the same type of results. A more detailed examination at high resolution of the orbital evolution of test-particles will be needed to understand the behaviour of planetesimals in these two MMRs.\n\nFinally, it is expected that reservoir belts span radial extents of the order of several AU or tens of AU. As shown in Fig. 2 of \\citep{1996Icar..120..358B}, the radial width of a resonance behaves as a function of the reduced mass of the planet $\\mu$ (where $\\mathrm{m_p=\\mu (M_{\\star}+m_p)}$, which obviously reduces to the mass ratio $\\mathrm{m_p\/M_{\\star}}$ when $\\mathrm{m_p << M_{\\star}}$). For the cases studied in this paper, this mass ratio is of the order of $\\sim 0.001$ or smaller, which corresponds to a radial width of the order of 0.1 AU or smaller. \n\nTherefore, it could be expected that a realistic reservoir overlaps with several MMRs at the same time. Combining the results found for the case of a 0.5 $\\mathrm{M_{Jup}}$ planet orbiting at 100 AU, the total rate of cometary events achievable per year would reach values of more than 20,000 when starting from a low mass disc, thus reinforcing the idea that those discs are already sufficiently massive to produce the observed exozodis.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.55\\textwidth]{comet_events_A3e_C3e_D3e.jpg}\n\\caption[]{Evolution of the number of cometary events per year as a function of time, when the production of active comets is generated via the 2:1 \\emph{(solid line)}, 5:2 \\emph{(dashed line)}, and 3:1 MMRs \\emph{(dotted line)}, for a planet at 100 AU, of mass $\\mathrm{m_p}=0.5\\,\\mathrm{M_{Jup}}$, and eccentricity $\\mathrm{e_p=0.1}$. The rates are given for a reservoir as massive as the Kuiper Belt around a Solar-type star.}\n\\label{fig:impact_MMR}\n\\end{figure}\n\n\\subsection{Planet eccentricity regime}\n\nWe already discussed and explained in Sect.~\\ref{sec:analytical} why the process would not work if the planet does not have a sufficient eccentricity. The eccentricity we explored so far, that is $\\mathrm{e_p=0.1}$, is considered to be the minimum for the process to take place, however, we did not discuss how this process would be impacted by larger eccentricities.\n\nFirstly, varying the eccentricity of the planet will modify the topology of the phase space, and the planetesimals will be driven to higher eccentricities. In Fig.~\\ref{fig:impact_ecc2}, we display the phase space diagram of the 5:2 MMR for planets of eccentricity 0.2 and 0.3. When the planet has an eccentricity of 0.3, it drives directly part of the planetesimals to orbital eccentricities $\\sim 1$ and thus, direct comet production is expected in this case, and depending on the mass of the planet, in combination with either comets originating from an additional scattering event or direct scattering.\n\nIn addition, when the eccentricity of the planet increases, its periastron diminishes. For a given eccentricity\n$\\mathrm{e_{p,max}}$, the inner chaotic zone of the planet will start to overlap with the reservoir belt itself, that is for $\\mathrm{a_{chaos,in}}\\geq \\mathrm{a_{MMR}}$. Thus, using Eq.~\\ref{eq:MMRloc} and Eq.~\\ref{eq:chaosin2}, $\\mathrm{e_{p,max}}$ satisfies the equation:\n\n\\begin{equation}\\label{eq:ecc_cond}\n(1-e_{\\mathrm{p,max}}) \\left[ 1-k\\left( \\frac{m_{\\mathrm{p}}}{(3+e_{\\mathrm{p,max}})M_{\\star}} \\right)^{\\frac{1}{3}}\\right] - \\mathrm{\\left(\\frac{p}{n} \\right)^{2\/3}} = 0 \\qquad.\n\\end{equation}\n\nIf the planet has an orbital eccentricity that exceeds $\\mathrm{e_{p,max}}$, it will directly scatter bodies from the reservoir. It is expected to lead to a short and intense period  of cometary activity.\nThis condition translates into a maximum eccentricity allowed for the process to take place. For the MMR studied here, $\\mathrm{e_{p,max}}$ varies between $\\sim 0.2$ and $\\sim 0.4$ for planets of mass 0.1-1$\\mathrm{M_{Jup}}$. Since lower mass planets have smaller chaotic zones, this maximum eccentricity will be larger for lower mass planets. Note that this condition does not depend on the scale of the system. \n\nUsing the example where the planet has a mass 1 $\\mathrm{M_{Jup}}$ and semi-major axis 100 AU, we investigate the effect of the planet eccentricity on the placement of planetesimals on cometary orbits induced via the 5:2 MMR. \nUsing Eq.~\\ref{eq:ecc_cond}, the eccentricity for which the 5:2 MMR should start inducing direct comet production is $\\sim 0.2$. Therefore, we ran the corresponding simulation, and display in Fig.~\\ref{fig:impact_ecc1} the time profile of the cometary production. As expected, it is immediate and intense, with rates of several thousands of cometary events per year. It then decays rapidly towards a much smaller rate, maintained until 1 Gyr. This is in accordance with what is expected if a planet directly scatters planetesimals.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.55\\textwidth]{comet_events_C5e_e02.jpg}\n\\caption[]{Evolution of the number of cometary events per year as a function of time, when the production of active comets is generated via the 5:2 MMR for a planet at 100 AU, of mass $\\mathrm{m_p}=1\\,\\mathrm{M_{Jup}}$, and eccentricity $\\mathrm{e_p=0.2}$. The rates are given for a reservoir as massive as the Kuiper Belt around a Solar-type star.}\n\\label{fig:impact_ecc1}\n\\end{figure}\n\n\\begin{figure*}\n\\centering\n\\makebox[\\textwidth]{\n\\includegraphics[width=0.5\\textwidth]{MMR52_e02.jpg}\n\\includegraphics[width=0.5\\textwidth]{MMR52_e03.jpg}\n}\n\\caption[]{Theoretical phase space diagrams (see caption to Fig.~\\protect\\ref{fig:phase_space52} for more details) of the 5:2 MMR with a planet of eccentricity $\\mathrm{e_p=0.2}$ \\emph{(left)} and $\\mathrm{e_p=0.3}$ \\emph{(right)}.}\n\\label{fig:impact_ecc2}\n\\end{figure*}\n\n\\subsection{Conclusions} \n\nFor many reasons, the dynamical mechanism presented here, that is, resonant interactions between planetesimals and an outer eccentric planet followed by scattering on to cometary orbits, is a viable route to produce bright analogues to the Zodiacal Cloud, even in old systems.\nFirstly, it can induce production of active comets with potentially significant delays (several 100 Myr). More importantly, it can sustain this comet scattering on Gyr timescales, and at rates compatible with observations when starting from realistic reservoir belts.\n\nWe find that when applied around Solar-type stars, our proposed mechanism can sustain a comet production compatible with the formation of the exozodis which are detected, even with low mass reservoirs comparable to the Kuiper Belt. The rate of cometary events that have to be induced around more luminous A-type stars can be sustained as well, and applying our findings to the system of Vega, we suggest that the presence of a planet $\\sim 1\\,\\mathrm{M_{Jup}}$ at large distance ($\\sim 200\\,$AU) and with eccentricity of 0.1 interacting with the system's cold debris disc via MMR could explain the levels of exozodiacal dust detected around the host star.\n\nThe mass of the perturbing planet drives the feasibility of the process, while all important quantities at play scale with the semi-major axis of the planet. In other words, if the process occurs for a given planetary mass, it will occur at any scale. Note that at small scales, however, the MMR can induce placement on cometary orbits without the need for an additional scattering event. This is expected to occur with the 5:2 MMR if the planet has a semi-major axis inferior to $\\sim 35\\,$AU, and inferior to $\\sim 15\\,$AU for the 7:3 and 3:1 MMRs.\nThe process is also expected to occur for a large range of planetary masses (typically $0.1-1\\,\\mathrm{M_{Jup}}$) and semi-major axes $(\\mathrm{a_p \\gtrsim 10-25\\,AU})$. The eccentricity regime, $\\mathrm{e_p\\simeq 0.1-0.4}$, is more restricted.\n\nOur results show that the mechanism is efficient through the 5:2 and 2:1 MMRs. Although predicted to be able to sustain the mechanism, the 7:3 MMR was found to be inefficient, which may be due to its higher order. The 3:1 seems to be particularly efficient, although in this case, comets seem to have formed directly while in MMR thanks to a drift of their eccentricities towards large values. These cases shall be explored in more details in future work, along with the few unexpected irregularities noted in the behaviour of the process with the semi-major axis of the planet. A more extended parametric exploration as well as closer examination of the orbital evolution of test-particles at high temporal resolution should help us explain these cases. \n\nInner MMR with eccentric planets is a starting point from which planetesimals can be set on cometary orbits in various ways depending on the planetary mass, eccentricity and semi-major axis regime, as well as on the MMR itself. Planets on short semi-major axes tend to promote direct placement on cometary orbits, for which, interestingly, cases were found for low mass planets ($\\lesssim 0.5\\,\\mathrm{M_{Jup}}$) where the production of cometary orbits could be sustained up to 1 Gyr. \nOn the other hand, planets at larger separations tend to place planetesimals on cometary orbits indirectly, with a resonant evolution followed by a scattering event. Large planetary eccentricities tend to trigger placement on cometary orbits via direct scattering. Finally, MMR may promote or deny access to cometary orbits via phase space drifts effects.\n\nSome constraints can be drawn on the range of parameters for which each process will take place and how they sometimes superimpose each other. However, they remain crude and several aspects of these processes are for now poorly known and characterised. These aspects shall be the object of future work (Faramaz et al., in prep), performing  extended parametric explorations. \nIn particular, it would be useful to be able to predict observational signatures of each of these processes. For instance, with direct placement on cometary orbits while in MMR, planetesimals reach their maximum eccentricity for a specific orbit orientation, so that cometary orbits would have preferential orientations. With the indirect mechanism, additional secular evolution of the planetesimal with the eccentric planet once scattered out of MMR can lead to apsidal alignment with the planet and thus again a preferential orientation of the cometary orbits \\citep{2015A&A...573A..87F}. Such preferential alignment of cometary orbits has already been suggested around Beta Pictoris, for which exocomets have been suggested as an explanation for time-variable absorption features in the spectrum of the star, and the lack of variation of the radial velocity has been interpreted as a tendency for cometary orbits to be apsidally aligned \\citep{1990A&A...236..202B}. \n\nOur results suggest that with the mechanism we focused on here, exozodis we detect may originate from as yet undetected cold Kuiper belt analogues, interacting with distant exterior Jupiter-like or smaller planets. This suggests that such systems are likely ideal targets for direct imaging observations. We also note that, whilst we refer in this paper to Kuiper belt analogues, the cometary reservoirs do not necessarily have to be at the outer edge of a planetary system and much of what we discuss here could potentially apply to an asteroid belt analogue. Although we did not explore semi-major axes small enough for the reservoir to be considered an asteroid belt analogue, results for the smallest spatial scale investigated in this study suggest this scenario is worth exploring in more detail, that is, the exozodis that have been detected might instead originate from an MMR with eccentric planets exterior to asteroid belt analogues. \n\n\n\\section*{Acknowledgements}\n\nThe authors wish to thank the anonymous referee for helping us clarify this paper.\nVF and MB acknowledge support from FONDECYT Postdoctoral Fellowships, projects nos 3150106 and 3140479.\nVF, MB, and JC acknowledge support from the Millenium Nucleus RC130007 (Chilean Ministry of Economy) and Millennium Nucleus \"Protoplanetary Disks\". MB acknowledges support from the Deutsche Forschungsgemeinschaft (DFG) through project Kr 2164\/15-1. JC acknowledges support from CONICYT-Chile through FONDECYT (1141175) and Basal (PFB0609) grants.\nCS acknowledges funding by the VRI Summer Research programme for undergraduates of PUC.\nThe simulations were performed on the Geryon computers at the Center for Astro-Engineering UC (BASAL PFB-06, QUIMAL 130008, Fondequip AIC-57).\n\n\n\n\n\\bibliographystyle{mnras}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Introduction}\n\n\nThe relevant place of real and complex Hilbert spaces in so many areas of mathematics has made the question of its generalization to spaces over different scalar fields a most interesting one. The decision of which features of the classical definition should be maintained, and which will be allowed to change, has been central. A well developed theory has as its center normed spaces over valued fields, and crucial questions are related to areas of Functional Analysis.  A comprehensive study can be found in van Rooij's  classical book (see \\cite{Rooij}). A different approach started with the work of Keller, Gross and K\\\"unzi. They were interested in spaces with a bilinear form that induced a norm, and in which the Projection Theorem was valid. They termed them orthomodular spaces (see \\cite{G-Ku} and \\cite{Keller}).\\\\\nIn both cases fields have a non-archimedean valuation and the norm is also non-archimedean. But in the first one the value group is a subset of $\\R$ whereas in the second it is the union of the infinite  set of its convex subgroups. They are said to be valuations of rank one or of infinite rank respectively.\\\\\nAs the theory of generalized Hilbert spaces (that is, every closed  subspace has an orthocomplement) over arbitrary fields developed, it became clear  that in many interesting cases the set of vector norms was not related directly to the valued field or to the value group. Thus a far more general concept was introduced as a ``home'' for the set of norms. It was called a $G$-module, and the general setup was structured in \\cite{morado} and further developed in \\cite{base}.\\\\\nA new question was posed by Schikhof and Olivos: How do the features of the $G$-module intertwine with those of the vector spaces? To be precise, suppose  that the value group $G$ is just a cyclic group, but place no conditions a priori  on the $G$-module except that it has a convex base as in \\cite{base}. What can be said of NHS spaces?\nThis is the central theme of this paper. Finally we could like to recognize the debt that this paper has towards the research of W. Schikhof. He started the study of $G$-modules and the way their specific characteristics influenced the properties of non-archimedean vector spaces. The present work follows a line originated by him and the authors during his stays in Temuco, Chile.\\\\\n\nWe start with the definitions that shall be needed in what follows, the prominent one being that of a $G$-module with a convex base. Then we study the relevant theorems, first  for Banach spaces and then for NHS, in this new setup. We give full proofs, even in the cases when they are quite similar to classical ones. The paper ends with a new characterization of NHS that is both simple and useful.\n\n\n\\section{Basic concepts and notation}\n\n\n\\begin{definition} Let $\\langle G,\\, \\cdot\\rangle $ be an ordered group.\nA linearly ordered set $(X, \\leq)$ with no smallest element is called a {\\bf $G$-module} if $G$ acts on $X$ by the map $(g,\\, x)\\mapsto gx$ and for any $g,\\, g_1,\\, g_2\\in G,\\ x,\\, x_1,\\, x_2\\in X$ we have\\\\\n(i) $g_1\\leq g_2 \\Rightarrow g_1x\\leq g_2x$,\\\\\n(ii) $x_1\\leq x_2 \\Rightarrow gx_1\\leq gx_2$,\\\\\n(iii) The orbit $Gx$ is coinitial in $X$\n\\end{definition}\n\n\\begin{remark}\nIt is easily proven that $Gx$ is also cofinal in $X$ and that $X$ has no largest element.\\\\\nUsually we adjoin a smallest element to $X$, that will be denoted by 0.\\end{remark}\n\n\\begin{definition} Let $X$ and $Y$ be two $G$-modules, a map $\\varphi: X\\to Y$ is called a $G$-module map if $\\varphi$ is increasing and $\\varphi(gx)=g\\varphi(x)$ for all $g\\in G,\\  x\\in X$.\\end{definition}\n\nWe introduce now a central concept, the {\\bf convex base} of a $G$-module $X$. \\\\\nA subset $B$ of $X$ is called a {\\bf generating set} if $GB=X$, {\\bf independent} if for $b_1\\neq b_2\\in B$ we have $Gb_1\\cap Gb_2=\\varnothing$, and a {\\bf base} of $X$ if $B$ is a set that is both generating and independent. Clearly  $B$ is a base of $X$ if and only if $B$ contains one and only one element of any orbit $Gx$. Therefore two bases of a $G$-module $X$ have the same cardinality.\n\nThe next definition is not surprising,\n\\begin{definition} A convex base of a $G$-module $X$ is a base $B$ of $X$ that is convex as a subset of $X$.\\end{definition}\n\n\n\\begin{remark} $G$-modules with a convex base were studied in \\cite{base} for $G$ any abelian, multiplicatively written, totally ordered group. A strong characterization of them was given in Theorem 2.9. Here we restate this result for our case.\\end{remark}\n\n\n \n\\begin{theorem} Let $G$ be a cyclic group and $X$ a $G$-module. The following are equivalent.\\\\\n(i) $X$ has a convex base.\\\\\n(ii) As a $G$-module, $X$ is isomorphic to $B\\times G$ for some non empty chain $B$ (where the action of $G$ on $B\\times G$ is defined by $g'(b,\\, g)=(b,\\ g'g)$).\\end{theorem}\n\n\\begin{remark} By this theorem, given any cyclic group $G$ and any cardinal $\\alpha>\\aleph_0$ we can obtain a $G$-module $X$ with convex base, such that the cardinality of $X$ is $\\alpha$.\\end{remark}\n\nWe give now two examples of $G$-modules with a convex base.\\\\\n  \nLet $(K,\\, |\\ |)$ be a valued field, with value group $G$, a cyclic subgroup of $(\\R^+, \\cdot)$.\n\n\\begin{example}\\label{ex1} Define $X_1$ as $B_1\\times G$, with $B_1:=(0,\\, 1]\\subset\\R^+$ (Notice that $B_1$ is not a well ordered set, since it is not isomorphic to an ordinal).\\end{example}\n\\begin{example}\\label{ex2} Now let $X_2 := B_2\\times G$ where $B_2$ is the ordinal $\\omega_1$. (Notice that $X_2$ cannot be immersed in $\\R^+$, since $\\omega_1$ has no cofinal sequence).\\end{example}\n\n\n\n\n\\section{Banach spaces over a discretely valued field with norms on a $G$-module with convex base}\n\nWe shall study NHS in the case the field $K$ has a valuation of rank one.\\\\\n\nTo prove the following lemma we will use the ``main tool'' of \\cite{two}. \\\\\nLet $E=(E,\\,\\|\\ \\|)$ be an $X$-normed space and $x_0\\in X$. The $G$-module map $\\varphi:X\\to G^\\#$ defined by $\\varphi(x)=\\sup_{G^\\#}\\{g\\in G: gx_0\\leq x\\}$ induces a new norm in $E$ with values in $G^\\#$. It turned out that these two norms are equivalent. The space $E$ provided with the new norm is called $E_\\varphi$.\n\n\\begin{lemma}\\label{discrete} If $E$ is a NHS over a field $K$ whose valuation has rank one then the value group of $K$ is cyclic.\\end{lemma}\n\n\\begin{proof}\nIf the value group $G\\subseteq (0, \\infty)$ then so is its Dedekind completion.\\\\\nBy \\cite{two}  Theorem 2.7 if $E$ is a NHS then so is $E_\\varphi$ and by \\cite{Rooij} Theorem 5.16, if each closed subspace of $E_\\varphi$ has an orthocomplement then the valuation is discrete.\\end{proof}\n\nThis has a strong consequence.\n\n\n\n\\begin{theorem}\\label{Ban1.1} Let $G=\\langle g_0\\rangle$ be a cyclic group. Each $G$-module $X$ has a convex base. In fact, $A:=[a,\\, g_0a)$ $($as well as $(a,\\, g_0a] )$ is a convex base of $X$  for any $a\\in X$.\\end{theorem}\n\\begin{proof} It is proved in \\cite{base}, Lemma 4.10  that for any $a\\in X$ the set $A$ is a convex base of the submodule $GA$. Thus, we only need to show that $G[a,\\, g_0a)=X$. So, let $x\\in X$. As $\\{g_0^nx\\}_{n\\in\\Z}$ is cofinal and coinitial there exists $m\\in\\Z$ such that $g_0^mx<a\\leq g_0^{m+1}x$. Then $g_0^{m+1}x<g_0a$ so $g_0^{m+1}x\\in [a,\\, g_0a)$ i.e. $x\\in G[a,\\, g_0a)$.\n\n\\end{proof}\n\nThus in the spaces described by van Rooij (\\cite{Rooij}, theorems 5.13 and 5.16) the set of norms has, as a $G$-module, always a convex base.\\\\\n\nBut the standard construction given below shows that there are a host of possibilities for new examples of spaces in which the set of norms is a $G$-module with a convex base.\n\n\\begin{example}\nLet $B$ a chain, $E$ be the space of linear combinations of the set $V:=\\{v_b: b\\in B\\}$ over a valued field $K$ with value group $G=\\langle g_0\\rangle$, a cyclic group. We define for each $b\\in B$ the norm $\\|v_b\\|=(b, 1)$ and for any $v:=\\sum\\limits_{b}k_b v_b \\in E$, $\\|v\\|=\\max\\{\\|k_bv_b\\|\\}$. Therefore $\\|E\\setminus\\{0\\}\\|=B\\times G$.\n\\end{example}\n\n\n\nFrom now on $G=\\langle g_0\\rangle$ is a cyclic group, $X$ is a $G$-module and  $E$ is an $X$-normed Banach space (that is, complete in the topology of the $X$-valued norm function).\\\\\nNotice that by Theorem \\ref{Ban1.1}, the $G$-module $X$ has a convex base.\n\n\n\n\\begin{lemma}\\label{Ban2.1} Let $E$ be an $X$-normed Banach space over $K$. Then the valuation topology on $K$ and the norm topology on $E$ are (ultra)metrizable.\\end{lemma}\n\\begin{proof} Fixing an $a\\in E$, $a\\neq 0$ we see that $\\{\\lambda\\in K: |\\lambda|<g_0^n\\}_{n\\in\\Z}$ and $\\{ v\\in E:\\|v\\|<g_0^n\\|a\\|\\}_{n\\in\\Z}$ form neighbourhood bases of 0 in $K$ and $E$, respectively.\\end{proof}\n\n\n\\begin{theorem}\\label{Ban2.11} Let $E$ have an orthogonal base $\\{e_1,\\, e_2,\\, \\ldots\\}$. Then there is an orthogonal base $\\{f_1,\\, f_2,\\, \\ldots\\}$ such that $\\|f_1\\|>\\|f_2\\|>\\cdots$ and $f_n\\to 0$.\\end{theorem}\n\n\\begin{proof} Let $s_1,\\, s_2,\\, \\ldots \\in X$ such that $s_1>s_2>\\cdots$ and $s_n\\to 0$. Put $f_1:=e_1$. There is an $n_1\\in\\Z$ such that $\\|g_0^{n_1}e_2\\|<\\min\\{\\|e_1\\|,\\, s_1\\}$. Put $f_2:=g_0^{n_1}e_2$, and so on. Inductively we arrive at $f_1,\\, f_2,\\, \\ldots$, multiples of $e_1,\\, e_2,\\,\\ldots$ respectively (hence orthogonal) such that $\\|f_n\\|<s_n$ for each $n\\in\\N$. Hence $f_n\\to 0$.\\end{proof}\n\n\nWe now prove the following  theorems on the structure of Banach spaces. It appears in a more general setup in \\cite{morado}, but we include it with its short proof for convenience.\n\n\n\\begin{theorem}\\label{Ban2.5} A Banach space of countable type has an orthogonal base.\\end{theorem}\n\n\\begin{proof} We may assume $E$ is an infinite dimensional Banach space of countable type. Then there exists a linearly independent set $\\{v_1,\\, v_2,\\,\\ldots\\}$ whose linear hull is dense. Using spherical completeness of $D_n:=[v_1,\\, v_2,\\,\\ldots,\\, v_n]$ we construct inductively $e_1,\\, e_2,\\,\\ldots\\in E$ such that $[e_1,\\, e_2,\\,\\ldots,\\, e_n]=D_n$ and $e_{n+1}\\perp D_n$ for each $n$. Then clearly $e_1,\\, e_2,\\,\\ldots$ is an orthogonal base for $E_n$.\n\\end{proof}\n\\bigskip\n{\\bf We consider now $c_0(I)$ and define the expression ``to contain $c_0$''.}\\\\\n\nThe wording of the following definitions, given in \\cite{Rooij}, have been slightly modified in order to include our present cases.\n\n\n\n\\begin{definition} Let $I$ be an index set, the space $c_0(I)$ is the space of all $(\\lambda_i)_{i\\in I}\\in \\prod\\limits_IK$ for which\nthe set $ \\{i\\in I: |\\lambda_i|\\geq g\\}$  is finite for all $g\\in G$.\\\\\nIt is a Banach space with respect to the $G$-norm $(\\lambda_i)_{i\\in I}\\mapsto \\max_i|\\lambda_i|$. The unit vectors of $c_0(I)$ form an orthogonal base of $c_0(I)$. As usual we will write $c_0$ for $c_0(\\N)$.\n\\end{definition}\nWe have\n\n\n\\begin{definition} A $K$-Banach space $E$ is said  {\\bf to contain $c_0$ } if there exist $a\\in X$ and an orthogonal sequence $e_1,\\, e_2,\\,\\ldots$ in $E$ such that $\\|e_n\\|=a$ for each $n$.\n\\end{definition}\n\n\\begin{theorem}\\label{Ban2.14} Let $E$ be a Banach space, let $X:=\\|E\\setminus\\{0\\}\\|$ with convex base $B$. The following are equivalent.\n\n($\\alpha$) $E$ does not contain $c_0$.\n\n($\\beta$) For all orthogonal systems $\\{e_i:i\\in I\\}$ we have that the set $\\{i\\in I: \\|e_i\\|\\in Gb\\}$ is finite for each $b\\in B$.\n\n\n($\\gamma$) There is a maximal orthogonal system  $\\{e_i:i\\in I\\}$ such that the set $\\{i\\in I:\\|e_i\\|\\in Gb\\} $ is finite for each $b\\in B$.\\end{theorem}\n\n\\begin{proof} \n($\\alpha\\Rightarrow \\beta$) We may assume $I$ infinite. If $\\{i\\in I: \\|e_i\\|\\in Gb\\}$ is infinite for some $b\\in B$ we can select $\\lambda_i\\in K^*$, ($i\\in I$) and $i_1,\\, i_2,\\, \\ldots\\in I$ (distinct) such that $\\|\\lambda_{i_n}e_{i_n}\\|=b$ for each $n$, conflicting with ($\\alpha$).\n\n\n($ \\beta\\Rightarrow\\gamma$) is obvious. By Zorn the existence of a maximal orthogonal system is assured.\n\n\n($\\gamma \\Rightarrow\\alpha$) If $\\{f_i:i\\in I\\}$ is a second maximal orthogonal system then by \\cite{morado} 2.4.12, $\\{i\\in I: \\|f_i\\|\\in Gb\\}$ is finite for each $b\\in B$. Now if $E$ has orthogonal vectors $a_1,\\, a_2,\\, \\dots$ of the same length we can extend $\\{ a_1,\\, a_2,\\, \\ldots\\}$ to a maximal orthogonal system. This one violates ($\\gamma$).\n\n\\end{proof}\n\n\n\n\n\\section{Norm Hilbert spaces over $G$-modules with a convex base}\n\n\\begin{definition} An $X$-normed Banach space is a Norm Hilbert space (NHS) iff every closed subspace $D$ of $E$ is orthocomplemented. \\end{definition}\nThis section is the center of our research. We start with several characterization of Norm Hilbert spaces very much in the spirit of van Rooij's classical book  ( \\cite{Rooij} Theorems 5.13 and 5.16). But in his work the set of norms is a subset of $(0, \\infty)$ and in our case some assertions do not necessarily hold. Thus we prefer to give detailed proofs of the equivalences in theorem \\ref{Ban2.3}.\n\n\n\n\\begin{remark}\\label{NHS3.1} Recall (\\cite{morado} Proposition 4.1.2) that $E$ is a NHS iff each maximal orthogonal system in $E$ is an orthogonal base, iff for each closed subspace $D$ the set $\\{\\|v-d\\|: d\\in D\\}$ has a minimum in $X\\, \\cup\\{0\\}$.  \\end{remark}\n\n\\begin{theorem}\\label{Ban2.3} Let $K$ be a valued field, $G$ a cyclic group, value group of $K$, $X$ a $G$-module with convex base and $E$ a Banach space over $K$. Then the following are equivalent.\n\n\n\\begin{itemize}\n\\item[($\\alpha$)] $E$ is a NHS.\n\\item[($\\beta$)] If $e_1,\\, e_2,\\, \\ldots\\in E$ are orthogonal and $\\|e_1\\|>\\|e_2\\|>\\cdots$ then $e_n\\rightarrow 0$.\n\\item[($\\gamma$)] If $v_1,\\, v_2,\\, \\ldots\\in E$ and $\\|v_1\\|> \\|v_2\\|>\\cdots$ then $v_n\\rightarrow 0$.\n\\item[($\\delta$)] Each closed subspace of countable type is a NHS.\n\\item[($\\epsilon$)] Each closed hyperplane is orthocomplemented.\n\\item[($\\iota$)] $E$ has an orthogonal base and is spherically complete.\n\\end{itemize} \\end{theorem}\n\n\\begin{proof}\n\n($\\gamma$)$\\Rightarrow$($\\alpha$):  It suffices to prove (\\ref{NHS3.1}) that each maximal orthogonal system $\\{e_i:i\\in I\\}$ is an orthogonal base. So let $D:=\\overline{[e_i: i\\in I]}$; we must prove $D=E$. Suppose $v\\in E\\setminus D$, we derive a contradiction. Consider $V:=\\{\\|v-d\\|:d\\in D\\}$. If $\\|v-d_0\\|=\\min V$ then $v-d_0\\perp D$, a contradiction. So $\\min V$ does not exist and we can therefore find $d_1,\\, d_2,\\,\\ldots\\in D$ such that $\\|v-d_1\\|>\\|v-d_2\\|>\\cdots$. By assumption $\\|v-d_n\\|\\rightarrow 0$. But then $v\\in\\overline{D}=D$ and we have our contradiction.\n\n\n($\\alpha$)$\\Rightarrow$($\\beta$): Suppose $e_1,\\, e_2,\\, \\ldots\\in E$ orthogonal, $\\|e_1\\|>\\|e_2\\|>\\cdots >s$ for some $s\\in X$. We derive a contradiction. Consider $D:=\\overline{[e_i: i\\in I]}$ and $\\phi\\in D'$ given by $\\phi\\left(\\sum \\xi_ie_i\\right)=\\sum \\xi_i $, ($\\xi_i\\rightarrow 0$).\\\\\nNow $D$ is a NHS, so $\\ker\\phi$ has an orthocomplement $Ka$ in $D$. Without lost of generality $\\phi(a)=1$. Let $a:=\\sum\\limits_{n_1}^\\infty\\lambda_ne_n$. We have $$1=|\\phi(a)|=\\left|\\sum\\limits_{n=1}^\\infty\\lambda_n\\right|\\leq\\max\\limits_n|\\lambda_n|$$ We see that there exists $i\\in\\N$ with $|\\lambda_i|\\geq 1$, $\\|a\\|=\\max\\|\\lambda_ne_n\\|\\geq \\|\\lambda_ie_i\\|\\geq \\|e_i\\|>\\|e_{i+1}\\|$.\\\\\nBut $\\phi(a-e_{i+1})=0$, so $a-e_{i+1}\\in \\ker\\phi$, so $a\\perp a-e_{i+1}$, and so $|e_{i+1}|=\\max\\{ \\|a\\|,\\,\\|a-e_{i+1}\\|\\}\\geq a$. Contradiction.\n\nTo complete the link we shall prove, by contradiction, ($\\beta$)$\\Rightarrow$($\\gamma$): Suppose we have a decreasing sequence  $\\|v_1\\|> \\|v_2\\|>\\cdots>s$ ($v_1,\\, v_2,\\, \\ldots \\in E,\\, s\\in X$). \\\\\nLet $B$ be a convex base of $X$, thus $X=\\bigcup g_0^nB$. For any fixed $m\\in\\Z$ we observe that $I_m:=\\{ n\\in\\Z: \\|v_n\\|\\in g_0^mB\\}$ is finite (If $I_m$ is infinite then $\\{\\|v_n\\|:n\\in I_m\\}$ would consist of elements that are not equivalent mod $G$.Thus they would be orthogonal, but this is forbidden by assumption). But now, $s\\in g_0^mB$ for some $m$. We see that $\\|v_i\\|>g_0^{m-1}B$ for all $i$. Also $v_1\\in g_0^rB$ for some $r$. Hence the whole sequence $v_1,\\, v_2,\\, \\ldots$ is contained in $g_0^rB,\\, g_0^{r+1}B,\\, \\ldots,\\, g_0^{m-1}B$. This implies finiteness of $v_1,\\, v_2,\\, \\ldots$, a contradiction.\n\n\nThis completes the proof of the equivalence.\n\nClearly we have ($\\alpha$)$\\Rightarrow$($\\delta$). Now ($\\delta$)$\\Rightarrow$($\\gamma$) is easy by observing that $\\overline{[v_1,\\, v_2,\\,\\ldots]}$ is of countable type. ($\\alpha$)$\\Rightarrow$($\\epsilon$) is trivial. \n\nTo prove ($\\epsilon$)$\\Rightarrow$($\\alpha$), let $(e_i)_{i\\in I}$ be a maximal orthogonal system. It suffices to prove (\\ref{NHS3.1}) that $D:= \\overline{[e_i: i\\in I]}=E$.  Suppose not. Then take an $a\\in E\\setminus D$ and consider the map $\\lambda a+d\\mapsto \\lambda$, ($\\lambda\\in K, \\, d\\in D$) which is in $(Ka+D)'$. By the Hahn- Banach Theorem (\\cite{morado}), $f$ extends to a $g\\in E'$. Then $H:= {\\rm Ker}\\, g$ is a closed hyperplane as $a\\in H$. By assumption there is a $z\\in E\\setminus H$ with $z\\perp H$. But then, since $D\\subseteq H$, also $z\\perp D$ which conflicts with the maximality of $\\{e_i: i\\in I\\}$.\n\nWe now prove  ($\\alpha$)$\\Rightarrow$($\\iota$).  Clearly $E$ has an orthogonal base. To prove spherical completeness  let $\\{B_i\\}_{i\\in I}$ be a nest of balls in $E$ where $I$ is linearly ordered and $i<j \\Rightarrow B_i\\supseteq B_j$. By the remark following Definition 1.2.1 in \\cite{morado}, we may suppose that $B_i$ has the form $\\{v\\in E:\\|v-a_i\\|\\leq r_i\\}$, where we may suppose that $r_i<r_j$ if $i>j$.\n\nTo prove $\\bigcap B_i\\neq \\varnothing$ we may suppose that $I$ has no smallest element. Then there are $i_1,\\, i_2,\\,\\ldots\\in I$ which $r_{i_1}> r_{i_2}>\\cdots$. \\\\\nBy ($\\gamma$) we must have $r_{i_n}\\rightarrow 0$ showing that $\\bigcap B_i=\\bigcap\\limits_{n\\in\\N}B_{i_n}$ is a singleton set by ordinary completeness of $E$.\n\n\nFinally, we prove ($\\iota$)$\\Rightarrow$($\\alpha$): Let $\\{f_i\\}_{i\\in I}$ be a maximal orthogonal set in $E$. We prove that $D:=\\overline{[f_i: i\\in I]}=E$. \\\\\nNow $E$ has an orthogonal base (which has the cardinality of $I$), say $\\{e_i\\}_{i\\in I}$.\\\\\nAs $\\{e_i\\}_{i\\in I}$ is also a maximal orthogonal system we have that $D$ is isometrically isomorphic to $\\overline{[e_i: i\\in I]}=E$. Thus $D$ is spherically complete, so for each $v\\in E\\setminus D$, $\\min\\{\\|v-d\\|: d\\in D\\}$ exists, so that $v-d\\perp D$. This conflicts the maximality, so we have $D=E$.\n\nThis completes the proof of Theorem \\ref{Ban2.3}.\n\\end{proof}\n\n\n\\begin{corollary}\\label{Ban2.4} For each index set $I$ the space $c_0(I)$ is a NHS.\\end{corollary}\n\n\\begin{proof} We have $\\|c_0(I)\\|=G\\,\\cup\\, \\{0\\}$. Clearly every strictly decreasing sequence on $G$ tends to 0, so $c_0(I)$ satisfies ($\\gamma$) of \\ref{Ban2.3}, and we are done.\n\\end{proof}\n\nWe come back now to $c_0(I)$ and its link with isometries.\n\n\n\n\\begin{remark} As it has been said before, we find in \\cite{morado} Lemma 4.3.4, that a NHS over a field with an infinite rank valuation does not contain $c_0$.\\end{remark}\n\n\n\\begin{definition} $E$ is {\\bf rigid} if every linear isometry $ T: E\\to E$ is surjective.\\end{definition}\nThe subject was studied in \\cite{linear} for the case of NHS over fields with a valuation of (countable) infinite rank. Let $E$ be such a space, then $E$ is rigid, in fact no proper subspace can be isometrically isomorphic to the whole space and any isometry of a closed subspace into itself can be extended to an isometry of $E$ onto itself. Clearly this sharply contrast the case of the classical space $c_0$.\n\n\n\nThe following theorems show that the condition ``$E$ does not contain $c_0$'' will need additional hypotheses in order to ensure that $E$ is rigid.\n\n\\begin{theorem}\\label{Ban2.7} If $E$ is a spherically complete space that does not contain $c_0$ then $E$ is rigid.\\end{theorem}\n\\begin{proof} Let $T:E\\to E$ be a linear isometry, $TE\\neq E$; we derive a contradiction. Now $TE$ is spherically complete, so it has an orthocomplement in $E$, in particular, there exists a nonzero $v\\in E$, $v\\perp TE$. Inductively we find easily that $\\{v,\\, Tv,\\, T^2v,\\ldots \\} $ is orthogonal and that $\\|T^nv\\|=\\|v\\|$ for each $n$, so $E$ contains $c_0$, a contradiction.\n\n\\end{proof}\nWe obtain\n\\begin{corollary}\\label{Ban2.8} $E$ is a NHS and does not contain $c_0$  if and only if  $E$ is rigid  and has an orthogonal base.\\end{corollary}\n\nNot surprisingly, rigidity is a condition that forbids $E$ to contain $c_0$.\n\n\\begin{theorem}\\label{Ban2.9} A rigid space does not contain $c_0$.\\end{theorem}\n\\begin{proof} Suppose $E$ is rigid and contains $c_0$, that is there are $a_1,\\, a_2,\\, \\ldots$, orthogonal, $\\|a_n\\|=s$ for all $n$. Then $\\overline{[a_1,\\, a_2,\\, \\ldots]}$ is a NHS, it is spherically complete so it has an orthocomplement $D$.\\\\\nThen define $T$ in  $\\overline{[a_1,\\, a_2,\\, \\ldots]}$ by $Ta_n=a_{n+1}$ and $T(d)=d$ for all $d\\in D$. This gives us a nonsurjective linear isometry, a contradiction.\n\\end{proof}\n\n\n\\begin{theorem}\\label{Ban2.10} Let $E$ have an orthogonal base $\\{e_1,\\, e_2,\\, \\ldots\\}$ for which $n\\neq m$ implies $\\|e_n\\|\\notin G\\|e_m\\|$. Then $E$ does not contain $c_0$.\\end{theorem}\n\\begin{proof} Suppose we have an orthogonal set $\\{a_1,\\, a_2,\\,\\ldots\\}$ and $s\\in X$ such that $\\|a_n\\|=s$ for all $n$; we derive a contradiction. For each $n\\in\\N$ we have an expansion $$a_n=\\sum_{i=1}^\\infty \\lambda_j^ne_j\\hspace{1cm} (\\lambda_j^n\\in K)$$\nThere is a unique $j_0$ with $s\\in G\\|e_{j_0}\\|$. Thus $$s=\\|a_n\\|=\\max\\limits_j\\|\\lambda_j^ne_j\\|=\\|\\lambda_{j_0}^ne_{j_0}\\|$$ Clearly we have for each $n$, $\\|a_n-\\lambda_{j_0}^ne_{j_0}\\|<\\|a_n\\|$. Therefore, by the Perturbation Lemma, the sequence $n\\mapsto \\lambda_{j_0}^ne_{j_0}$ must be orthogonal, an absurdity.\\end{proof}\n\n\n\n \n\\begin{theorem}\\label{Ban2.16} Let $E$ have an orthogonal base and suppose there exists a sequence  $\\{v_n\\}$ in $E$ with $\\|v_1\\|>\\|v_2\\|>\\cdots$, $\\|v_n\\|\\nrightarrow 0$. Then $E$ is not rigid.\n\\end{theorem}\n\\begin{proof} By  Theorem \\ref{Ban2.3} $E$ is not a NHS, so by Corollary \\ref{Ban2.8} $E$ cannot be rigid.\n\n\\end{proof}\n\nNow it is easy to present an example of a space that does not contain $c_0$ but is not rigid. Let $B=\\{ b_1,\\, b_2,\\,\\ldots\\}\\subseteq\\R^+$ a denumerable chain where $b_1>b_2>\\cdots$. Let $X:= B\\times G$ ($G$ a cyclic group) thus $X$ has a convex base $B\\times\\{1_G\\}$. Let $E$ have an orthogonal base $\\{ e_1,\\, e_2,\\, \\ldots\\} $ with $\\|e_n\\|=(b_n, \\, 1_G)$ for each $n$. Clearly by \\ref{Ban2.10} $E$ does not contain $c_0$. By \\ref{Ban2.16} $E$ is not rigid.\\\\\n\nThe next section contains the surprising main result of this paper.\n\n\n\n\\section{A new characterization of NHS}\nWe recall the standard definition: A linear ordering $\\leq$ of a set $S$ is a well-ordering if every non-empty subset of $S$ has a least element.\n\\begin{theorem}\\label{Ban2.12} Let $K$ be a valued field and $G=\\langle g_0\\rangle$ its cyclic value group. $E$ is a $K$-Banach space and $X:=\\|E\\setminus\\{0\\}\\|$, the set of norms, is a $G$-module with convex base $B$.\\\\\nThen $E$ is a NHS if and only if $B$ is well ordered.\\end{theorem}\n\n\n\\begin{proof} Let $E$ be a NHS, let $b_1>b_2>\\cdots$ where $b_n\\in B$; we derive a contradiction. There are $v_1,\\, v_2,\\,\\ldots\\in E$ with $\\|v_n\\|=b_n$ for each $n$. Then $\\|v_1\\|>\\|v_2\\|>\\cdots$. But by \\ref{Ban2.3}, $\\|v_n\\|\\to 0$ an impossibility as for all $n$, $b_n>g_0^{-1}B$ .\n\nConversely. let $B$ be well-ordered, let $v_1,\\, v_2,\\, \\ldots \\in E$. $\\|v_1\\|>\\|v_2\\|>\\cdots$. We have $X=\\bigcup\\limits_{n\\in\\Z}g_0^nB$, where $$\\cdots< g_0^{-1}B<B<g_0B<g_0^2B<\\cdots$$ Since $\\{n\\in \\N:\\|v_n\\|\\in g_0^mB\\}$ is finite for each $m\\in \\Z$ (as $g_0^mB$ is well-ordered) we must have, for each $m\\in\\Z$, that $\\|v_n\\|<g_0^mB$ eventually, i.e. $\\|v_n\\|\\to 0$ so $E$ is a NHS by Theorem \\ref{Ban2.3}.\\end{proof}\n\n\n\n\\begin{remark} The previous theorem is a strong characterization of NHS over $G$-modules $X$ with a convex base and $G$ a cyclic group. If we come back to examples \\ref{ex1} and \\ref{ex2} it is now clear that an $X_1$-Banach space can never be a NHS, while an $X_2$-Banach space is necessarily a NHS.\\\\\n\nIn addition, as the value group $G=\\langle g_0\\rangle$ is a cyclic group, we have by Theorem 2.1 that  for any $a \\in X$ the interval $[a,g_0a)$ is a convex base of $X$. With this in mind we show in the next Corollary how easy it is to decide if the space $E$ is a NHS.\n\n \\end{remark}\n\n\n\\begin{corollary} Let $K$, $G$, $X$, $E$ be as in Theorem \\ref{Ban2.12}. Let $a\\in X$, then $E$ is a NHS if and only if the interval $[a,\\, g_0a)$ is well ordered.\\end{corollary}\n\n\n\n\n\n\n\\section*{References}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction} \n\n\nA cornerstone of fractal fluids is \nthe non-integer dimension \\cite{Fractal1,Fractal2,CM}. \nThe mass of fractal fluid  \nsatisfies a power law relation $M \\sim R^{D}$, \nwhere $M$ is the mass of the ball region with radius $R$,\nand $D$ is the mass dimension \\cite{TarasovSpringer}. \nFractal fluid can be described by four different approaches:\n(1) Using the methods of \"Analysis on fractals\" \n\\cite{Kugami,Strichartz-1,Strichartz-2,Harrison,Kumagai,DGV} \nit is possible to describe fractal media;\n(2) An application of fractional-differential continuum models \nsuggested in \\cite{CCC2001,CCC2002}, and then developed in \n\\cite{CCC2004a,CCC2004c,CCSPZ2009,CCC2009,Yang2013a,Yang2013b},\nwhere so-called local fractional derivatives \\cite{Yang2013c} \nare used; \n(3) Applying fractional-integral continuum models suggested in \n\\cite{PLA2005-1,AP2005-2,IJMPB2005-2,MPLB2005-1,TarasovSpringer} \n(see also \\cite{MOS-3}-\\cite{MOS-5} and \\cite{Bal-1}-\\cite{Bal-4}), where\nintegrations of non-integer orders and \na notion of density of states \\cite{TarasovSpringer} are used;  \n(4) Fractal media can be described by using\nthe theory of integration and differentiation for\na non-integer dimensional space \\cite{Collins,Stillinger,PS2004}\n(see also \\cite{CNSNS2015,JMP2014}.\n\n\nLet us note that main difference of \nthe continuum models with non-integer dimensional spaces\nform the fractional continuum models suggested in \n\\cite{PLA2005-1,AP2005-2,IJMPB2005-2,MPLB2005-1,TarasovSpringer}\nmay be reduced to the following.\n(a) Arbitrariness in the choice of the numerical factor\nin the density of states is fixed by the equation\nof the volume of non-integer dimensional ball region.\n(b) In the fractional continuum models suggested \nin \\cite{PLA2005-1,AP2005-2,TarasovSpringer}, \nthe differentiations are integer orders whereas\nthe integrations are non-integer orders.\nIn the continuum models with non-integer dimensional spaces\nthe integrations and differentiations are defined \nfor the spaces with non-integer dimensions.\n\n\nIn this paper, we consider approach based on\nthe non-integer dimensional space.\nThe power law $M \\sim R^{D}$ can be naturally \nderived by using the integrations \nin non-integer dimensional space \\cite{Collins},\nwhere the mass dimension of fractal fluid\nis connected with the dimension of this space.\nA vector calculus for non-integer dimensional space \nproposed in this paper\nallows us to use continuum models with\nnon-integer dimensional spaces to describe\nfor fractal fluids.\nThis is due to the fact that\nalthough the non-integer dimension  \ndoes not reflect completely the geometric\nproperties of the fractal media, it nevertheless permits\na number of important conclusions about the behavior\nof fractal structures.\nTherefore continuum models with non-integer dimensional spaces\ncan be successfully used to describe fractal fluids.\n\nIntegration over non-integer dimensional spaces \nare actively used in the theory of critical phenomena and \nphase transitions in statistical physics \n\\cite{WilsonFisher,WK1974}, and \nin the dimensional regularization of\nultraviolet divergences \nin quantum field theory \\cite{HV1972,Leibbrandt,Collins}. \nThe axioms for integrations in non-integer dimensional space are proposed in \\cite{Wilson,Stillinger} and this type of integration is considered in the book by Collins \\cite{Collins}\nfor rotationally covariant functions. \nIn the paper \\cite{Stillinger} a mathematical basis of integration on non-integer dimensional space is given. \nStillinger \\cite{Stillinger} suggested a generalization of \nthe Laplace operator for non-integer dimensional spaces also. \nUsing a product measure approach,  \nthe Stillinger's methods \\cite{Stillinger}\nhas been generalized by Palmer and Stavrinou \\cite{PS2004} \nfor multiple variables case\nwith different degrees of confinement in orthogonal directions. \nThe scalar Laplace operators suggested \nby Stillinger in \\cite{Stillinger} \nand Palmer, Stavrinou in \\cite{PS2004}\nfor non-integer dimensional spaces, \nhave successfully been used for \neffective descriptions in physics and mechanics. \nThe Stillinger's form of Laplacian \nfor the Schr\\\"odinger equation in non-integer dimensional space \nis used by He \\cite{XFHe1,XFHe2,XFHe3} to describe\na measure of the anisotropy and \nconfinement by the effective non-integer dimensions. \nQuantum mechanical models with non-integer (fractional) \ndimensional space have been discussed in \n\\cite{Stillinger,PS2004,Thilagam1997b,MA2001a,MA2001b,QM1,QM5}\nand \\cite{Muslih2010,MA2012,QM8,QM9}.\nRecent progress in non-integer dimensional space approach\nalso includes description of\nthe fractional diffusion processes in\nnon-integer dimensional space in \\cite{LSTLRL}, and\nthe electromagnetic fields in non-integer dimensional space\nin \\cite{MB2007,BGG2010,MSBR2010}\nand \\cite{ZMN2010,ZMN2011a,ZMN2011b,ZMN2011c,ZMN}. \n\n\nUnfortunately, \\cite{Stillinger,PS2004} proposed \nonly the second order differential operators \nfor scalar fields in the form of the scalar Laplacian \nin the non-integer dimensional space. \nA generalization of the vector Laplacian \\cite{VLap}\nfor the non-integer dimensional space is not suggested\nin \\cite{Stillinger,PS2004}. \nThe first order operators such as gradient, \ndivergence, curl operators  \nare not considered in \\cite{Stillinger,PS2004} also.\nIn the work \\cite{ZMN} \nthe gradient, divergence, and curl operators\nare suggested only as approximations of the square of \nthe Laplace operator.\nConsideration only the scalar Laplacian \nin the non-integer dimensional space approach\ngreatly restricts us in application of continuum models\nwith non-integer dimensional spaces for fractal fluids and material.\nFor example, we cannot use the Stillinger's form of Laplacian\nfor vector field ${\\bf v}({\\bf r},t)$ in hydrodynamics of fractal fluids, in fractal theory of elasticity and thermoelasticity, \nin electromagnetic theory of fractal media\nto describe processes\nin the framework non-integer dimensional space approach.\n\nIn this paper, we propose to use a vector calculus\nfor non-integer dimensional space, and we define\nthe first and second orders differential vector\noperations such as gradient, divergence, \nthe scalar and vector Laplace operators \nfor non-integer dimensional space.\nIn order to derive the vector differential operators \nin non-integer dimensional space\nwe use the method of analytic continuation in dimension.\nFor simplification we consider rotationally covariant \nscalar and vector functions that are independent of angles.\nIt allows us to reduce differential equations \nin non-integer dimensional space to\nordinary differential equations with respect to $r$. \nThe proposed operators allow us to describe fractal media\nto describe processes in the framework of continuum models\nwith non-integer dimensional spaces. \nIn this paper we describe a Poiseuille flow of \nan incompressible viscous fractal fluid in the pipe. \nA generalization of the Navier-Stokes equation \nfor non-integer dimensional space to describe \nfor fractal fluid are suggested. \nA solution of this equation for steady flow of \nfractal fluid in a pipe and corresponding \nfractal fluid discharge are derived.\n\n\n\n\\section{Fractal fluids} \n\n\nA basic characteristic of fractal fluids\nis the non-integer dimensions such as mass \nor \"particle\" dimensions \\cite{TarasovSpringer}. \nFor fractal fluids \nthe number of particles $N_D(W)$ or mass $M_D(W)$ \nin any region $W \\subset \\mathbb{R}^3$ of this fluid \nincrease more slowly \nthan the 3-dimensional volume $V_3(W)$ of this region.\nFor the ball region $W$ with radius $R$ in \nan isotropic fractal fluid, \nthis property can be described by the relation\nbetween the number of particles $N_D(W)$\nin the region $W$ of fractal fluid, and \nthe radius $R$ in the form\n\\be \\label{NDW} \nN_D(W) = N_0 (R\/ R_0)^D , \\quad  R\/R_0 \\gg 1 , \n\\ee\nwhere $R_0$ is the characteristic size of fractal fluid\nsuch as a minimal scale of self-similarity \nof a considered fractal fluid. \nThe number $D$ is called the \"particle\" dimension. \nIt is a measure of how the fluid particles\nfill the space.\nThe parameter $D$ does not depend on the shape \nof the region $W$.\nTherefore fractal fluids can be considered \nas fluid with non-integer \"particle\" or mass dimension. \n\nIf the fractal fluid consists of particles \nwith identical masses $m_0$, then relation (\\ref{NDW}) gives\n\\be \\label{MDW} \nM_D(W) = M_0 (R\/ R_0)^D , \\quad  R\/R_0 \\gg 1 , \n\\ee\nwhere $M_0=m_0 \\, N_0$. \nIn this case, the mass dimension coincides\nwith the \"particle\" dimension.\n\nAs the basic mathematical tool for\ncontinuum models of fractal fluids,\nwe propose to use the integration and differentiation\nin non-integer dimensional spaces.\nIn Section 7, we will show that \nthe power-law relation (\\ref{MDW}) for \nan isotropic fractal fluid  \ncan be naturally derived by using the integration over \nnon-integer dimensional space, where\nthe space dimension is equal to\nthe mass dimension of fractal fluid. \n\n\nIn order to describe fractal fluid by \ncontinuum models with non-integer dimensional spaces,\nwe use the concepts of \ndensity of states $c_3(D,{\\bf r})$\nthat describes how closely packed permitted \nplaces (states) in the space $\\mathbb{R}^3$, \nwhere the fractal fluid is distributed.\nThe expression $dV_D({\\bf r})=c_3(D,{\\bf r})dV_3$ \nis equal to the number of permitted places (states) \nbetween $V_3$ and $V_3 +dV_3$ in $\\mathbb{R}^3$. \nThe notation $d^D {\\bf r}$ \nalso will be used instead of $ d V_D ({\\bf r})$.\nNote that density of states and \ndistribution function are different concepts, \nand it is impossible to describe \nall properties of fractal fluids\nby the distribution function only.\n\nFor fractal fluids, we can use the equation\n\\be \ndN_D(W)= n({\\bf r}) \\, d V_D ({\\bf r}) , \n\\ee\nwhere $n({\\bf r})$ is a concentration of particles\nthat describes a distribution of number of particles\non a set of permitted places (possible states).\nThe density of states is chosen such that\n$d V_D ({\\bf r}) = c_3(D,{\\bf r}) \\, dV_3$\ndescribes the number of permitted states in $dV_3$.\n\nThe form of the function $c_3(D,{\\bf r})$ \nis defined by symmetries of \nconsidered problem and properties of \nthe described fractal fluid.\nA general property of density of states\nfor fractal fluids is a power-law type \nof these functions that reflects a scaling property \n(fractality) of the fractal fluid. \nTo simplify our consideration in this paper \nwe will consider only isotropic fractal fluids \nwith density of states that is independent of angles.\nIn this case, the form of density of states is\ndefined such that $dV_D$ is an elementary volume\nof the non-integer dimensional space.\n\nIn the continuum models of fractal fluids,\nwe should work with the dimensionless variables\n$x\/R_0 \\to x$, $y\/R_0 \\to x$, $z\/R_0 \\to x$, \n${\\bf r}\/R_0 \\to {\\bf r}$,\nin order to physical quantities of \nfractal fluids have correct physical dimensions. \n\n\n\n\n\n\n\\section{Vector differential operators \nin non-integer dimensional space}\n\nTo derive equations for vector differential operators \nin non-integer dimensional space,\nwe use equations for the differential operators \nin the spherical (and cylindrical) coordinates in $\\mathbb{R}^n$ \nfor arbitrary $n$ to highlight the explicit \nrelations with dimension $n$.\nThen the vector differential operators  \nfor non-integer dimension $D$ \ncan be defined by continuation in \ndimension from integer $n$ to non-integer $D$. \nTo simplify we will consider only scalar \nfields $\\varphi$ and vector fields ${\\bf v}$ \nthat are independent of angles \n\\[ \\varphi({\\bf r}) =\\varphi(r) ,\n\\quad {\\bf v}({\\bf r})={\\bf v} (r) = v_r\\, {\\bf e}_r , \\]\nwhere $r=|{\\bf r}|$ is the radial distance,\n${\\bf e}_r={\\bf r} \/r$ is the local orthogonal unit \nvector in the directions of increasing $r$,  \nand $v_r=v_r(r)$ is the radial component of ${\\bf v}$.\nWe will work with rotationally covariant functions only. \nThis simplification is analogous to the simplification\nfor definition of integration over non-integer dimensional space\ndescribed in Section 4 of the book \\cite{Collins}.\n\n\n\\subsection{Vector differential operators \nfor spherical and cylindrical cases}\n\n\n\nUsing the continuation from integer $n$ \nto arbitrary non-integer $D$,  \nwe can get explicit definitions of  differential operators\nfor non-integer dimensional space in the following forms.\nNote that the same expressions can be obtained by using \nthe integration in non-integer dimensional space\nand the correspondent Gauss's theorem \\cite{CNSNS2015}\n\nLet us define the differential vector operations\nsuch as  gradient, divergence, the scalar and \nvector Laplacian for non-integer dimensional space. \nFor simplifications, we assume that\nthe vector field ${\\bf v}={\\bf v}({\\bf r})$ \nbe radially directed and the scalar and vector \nfields $\\varphi({\\bf r})$, ${\\bf v}({\\bf r})$ \nare not dependent on the angles.\n\nThe divergence in non-integer dimensional space \nfor the vector field ${\\bf v}={\\bf v}(r)$ is\n\\be \\label{Div-D}\n\\operatorname{Div}^{D}_{r} {\\bf v} = \n\\frac{\\partial v_r}{\\partial r} + \\frac{D-1}{r} \\, v_r.\n\\ee\n\nThe gradient in non-integer dimensional space \nfor the scalar field $\\varphi=\\varphi (r)$ is\n\\be \\label{Grad-D}\n\\operatorname{Grad}^{D}_r \\varphi = \\frac{\\partial \\varphi}{\\partial r} \\, {\\bf e}_r .\n\\ee\n\nThe scalar Laplacian in non-integer dimensional space \nfor the scalar field $\\varphi=\\varphi (r)$ is\n\\be \\label{S-Delta-D}\n^S\\Delta^{D}_r \\varphi= \n\\operatorname{Div}^{D}_r \\operatorname{Grad}^{D}_{r} \\varphi =\n\\frac{\\partial^2 \\varphi}{\\partial r^2} + \\frac{D-1}{r} \\, \n\\frac{\\partial \\varphi}{\\partial r} .\n\\ee\n\nThe vector Laplacian in non-integer dimensional space \nfor the vector field ${\\bf v}=v(r) \\, {\\bf e}_r$ is\n\\be \\label{V-Delta-D}\n^V\\Delta^{D}_r {\\bf v} = \n\\operatorname{Grad}^{D}_r \\operatorname{Div}^{D}_{r} {\\bf v} =\n\\Bigl(\n\\frac{\\partial^2 v_r}{\\partial r^2} + \\frac{D-1}{r} \\, \n\\frac{\\partial v_r}{\\partial r}  -  \\frac{D-1}{r^2} \\, v_r\n\\Bigr) \\, {\\bf e}_r.\n\\ee\n\nIf $D=n$, equations (\\ref{Div-D}-\\ref{V-Delta-D})\ngive the well-known formulas for\ninteger dimensional space $\\mathbb{R}^n$.\n\n\nLet us consider a case of axial symmetry \nof the fluid, where the fields \n$\\varphi(r)$ and ${\\bf v}(r)=v_r(r) \\, {\\bf e}_r$\nare also axially symmetric.\nWe will direct the $Z$-axis along the axis of symmetry. \nTherefore we use a cylindrical coordinate system.\n\nThe divergence in non-integer dimensional space \nfor the vector field ${\\bf v}={\\bf v}(r)$ is\n\\be \\label{Div-DC}\n\\operatorname{Div}^{D}_{r} {\\bf v} = \n\\frac{\\partial v_r}{\\partial r} + \\frac{D-2}{r} \\, v_r.\n\\ee\n\nThe gradient in non-integer dimensional space \nfor the scalar field $\\varphi=\\varphi (r)$ is\n\\be \\label{Grad-DC}\n\\operatorname{Grad}^{D}_r \\varphi = \n\\frac{\\partial \\varphi}{\\partial r} \\, {\\bf e}_r .\n\\ee\n\nThe scalar Laplacian in non-integer dimensional space \nfor the scalar field $\\varphi=\\varphi (r)$ is\n\\be \\label{S-Delta-DC}\n^S\\Delta^{D}_r \\varphi = \n\\frac{\\partial^2 \\varphi}{\\partial r^2} + \\frac{D-2}{r} \\, \n\\frac{\\partial \\varphi}{\\partial r} .\n\\ee\n\nThe vector Laplacian in non-integer dimensional space \nfor the vector field ${\\bf v}=v(r) \\, {\\bf e}_r$ is\n\\be \\label{V-Delta-DC}\n^V\\Delta^{D}_r {\\bf v} = \n\\Bigl(\n\\frac{\\partial^2 v_r}{\\partial r^2} + \\frac{D-2}{r} \\, \n\\frac{\\partial v_r}{\\partial r}  -  \\frac{D-2}{r^2} \\, v_r\n\\Bigr) \\, {\\bf e}_r .\n\\ee\n\n\n\nEquations (\\ref{Div-DC}-\\ref{V-Delta-DC}) can be\neasy generalized for the case $\\varphi=\\varphi(r,z)$ and \n${\\bf v}(r,z)=v_r(r,z) \\, {\\bf e}_r+ v_r(r,z) \\, {\\bf e}_z$.\nIn this case the curl operator for ${\\bf v}(r,z)$\nis different from zero, and\n\\be \\label{Curl-DC}\n\\operatorname{Curl}^{D}_r {\\bf v} = \n\\left( \\frac{\\partial v_r}{\\partial z} -\n\\frac{\\partial v_z}{\\partial r} \\right) \\, {\\bf e}_{\\theta} .\n\\ee\n\n\nFor $D=3$ equations (\\ref{Div-D}) - (\\ref{V-Delta-DC}) and \n(\\ref{Curl-DC}) give the well-known\nexpressions for the gradient, divergence, curl operator,\nscalar and vector Laplacian operators\n\nThe suggested operators for $0<D<3$ allow us to reduce \n$D$-dimensional vector differentiations \n(\\ref{Div-D}) - (\\ref{V-Delta-D}) and \n(\\ref{Div-DC}) - (\\ref{V-Delta-DC})\nto derivatives with respect to $r=|{\\bf r}|$. \nIt allows us to reduce partial differential equations for\nfields in non-integer dimensional space to\nordinary differential equations with respect to $r$.\n\n\n\\subsection{Stillinger's Laplacian for \nnon-integer dimensional space}\n\n\nFor a function $\\varphi=\\varphi(r,\\theta)$ of radial distance $r$ \nand related angle $\\theta$ measured relative \nto an axis passing through the origin, \nthe scalar Laplacian in a non-integer dimensional space\nproposed by Stillinger \\cite{Stillinger} is\n\\be \\label{NI-1}\n^{St}\\Delta^{D} = \\frac{1}{r^{D-1}} \\frac{\\partial}{\\partial r} \\left( r^{D-1} \\,\\frac{\\partial}{\\partial r} \\right) +\n\\frac{1}{r^2 \\, \\sin^{D-2} \\theta} \n\\frac{\\partial}{\\partial \\theta} \n\\left(  \\sin^{D-2} \\theta \n\\frac{\\partial}{\\partial \\theta} \\right) ,\n\\ee\nwhere $D$ is the dimension of space ($0<D <3$),\nand the variables $r \\ge 0$, $0\\le \\theta \\le \\pi$.\nNote that\n$(\\, ^{St}\\Delta^{D} )^2 \\ne \\, ^{St}\\Delta^{2D}$.\nIf the function depends on radial distance $r$ \nonly ($\\varphi=\\varphi(r)$), then\n\\be \\label{NI-R}\n^{St}\\Delta^{D} \\varphi (r) = \\frac{1}{r^{D-1}} \\frac{\\partial}{\\partial r} \\left( r^{D-1} \\,\\frac{\\partial \\varphi (r)}{\\partial r} \\right) =\n\\frac{\\partial^2 \\varphi (r)}{\\partial r^2} +\n\\frac{D-1}{r} \\, \\frac{\\partial \\varphi (r)}{\\partial r} .\n\\ee\nIt is easy to see that \nthe Stillinger's form of Laplacian $\\, ^{St}\\Delta^{D}$\nfor radial scalar functions $\\varphi ({\\bf r})=\\varphi (r)$\ncoincides with the scalar Laplacian \n$\\, ^S\\Delta^{D}_r$ defined by (\\ref{S-Delta-D}), i.e.,\n\\be \\label{NI-R-2}\n^{St}\\Delta^{D} \\varphi (r) = \\, ^S\\Delta^{D} \\varphi (r) .\n\\ee\nThe Stillinger's Laplacian can be applied \nfor scalar fields only. It cannot be used \nto describe vector fields ${\\bf v}=v_r(r) \\, {\\bf e}_r$ \nbecause this Laplacian for $D=3$ is not equal to \nthe usual vector Laplacian for $\\mathbb{R}^3$,\n\\be\n^{St}\\Delta^{3} {\\bf v}(r) \\ne \\, \\Delta {\\bf v}(r) =\n\\Bigl( \n\\frac{\\partial^2 v_r}{\\partial r^2} + \\frac{2}{r} \\, \n\\frac{\\partial v_r}{\\partial r} - \\frac{2}{r^2} \\, v_r\n\\Bigr) \\, {\\bf e}_r  .\n\\ee\nThe gradient, divergence, curl operator and vector Laplacian are not considered by Stillinger in paper \\cite{Stillinger}.\n\n\n\n\n\\subsection{Differential operators for $d \\ne D-1$}\n\nLet us consider a ball region $B_D$ in the fractal fluids\nwith the boundary $S_d= \\partial B_D$ with dimensions\n\\be\n\\operatorname{dim} (B_D)=D, \\quad \n\\operatorname{dim} (S_d) = d .\n\\ee \nEquations (\\ref{Div-D}) - (\\ref{V-Delta-DC}) \ndefine differential operators for spaces\nwith non-integer dimension $D$ and \nboundary dimensions $d=D-1$.\nIn general, the dimension $D$ of the region of a fractal fluid \nand the dimension $d$ of boundary of this region\nare not related by the equation $d = D-1$, i.e.,\n\\be\n\\operatorname{dim} (\\partial B_D) \\ne D-1 . \n\\ee\n\nUsing the integration in non-integer dimensional space\nand the correspondent Gauss's theorem, we can define \nthe divergence for the case $d \\ne D-1$ by the equation\n\\be \\label{Div-Dd}\n\\operatorname{Div}^{D,d}_{r} {\\bf v} =\\frac{\\pi^{(d+1-D)\/2} \\, \\Gamma (D\/2) }{\\Gamma ((d+1)\/2)}\n\\left( \\frac{1}{r^{D-1-d}}\n\\frac{\\partial v_r(r)}{\\partial r} + \\frac{d}{r^{D-d}} \\, v_r(r) \n\\right) .\n\\ee\nFor $d=D-1$, we get (\\ref{Div-D}).\nWe can define the parameter\n\\be \\label{ar}\n\\alpha_r = D-d ,\n\\ee\nthat can be interpreted as a dimension of fractal fluid  \nalong the radial direction ${\\bf e}_r$. Using (\\ref{ar}), \nequation (\\ref{Div-Dd}) can be rewritten in the form\n\\be \\label{Div-Dd2}\n\\operatorname{Div}^{D,d}_{r} {\\bf v} = \n\\pi^{(1-\\alpha_r)\/2} \\, \n\\frac{\\Gamma ((d+\\alpha_r)\/2) }{ \\Gamma ((d+1)\/2)}\n\\left( \\frac{1}{r^{\\alpha_r-1}}\n\\frac{\\partial v_r(r)}{\\partial r} + \n\\frac{d}{r^{\\alpha_r}} \\, v_r(r) \\right) .\n\\ee\nThis is divergence operator for non-integer boundary dimension\n$d \\ne D-1$.\nFor $\\alpha_r=1$, we have $d=D-1$, and then equations \n(\\ref{Div-Dd}), (\\ref{Div-Dd2}) give (\\ref{Div-D}).\n\nThe gradient for the scalar field $\\varphi({\\bf r})= \\varphi(r)$ \nand the radial dimension $\\alpha_r \\ne 1$ is defined by\n\\be \\label{Grad-Dd}\n\\operatorname{Grad}^{D,d}_{r} \\varphi = \n\\frac{\\Gamma (\\alpha_r\/2)}{ \\pi^{\\alpha_r\/2} \\, r^{\\alpha_r-1}} \\,\n\\frac{\\partial \\varphi(r)}{\\partial r} \\, {\\bf e}_r .\n\\ee\nFor $\\alpha_r=1$, equation (\\ref{Grad-Dd}) gives (\\ref{Grad-D}).\n\nUsing the operators (\\ref{Grad-Dd}) and (\\ref{Div-Dd})\nfor the fields $\\varphi=\\varphi (r)$ and \n${\\bf v}=v(r) \\, {\\bf e}_r$,\nwe can get the scalar and vector Laplace operators \nfor the case $d \\ne D-1$ by\n\\be\n^S\\Delta^{D,d}_r \\varphi= \n\\operatorname{Div}^{D,d}_r \n\\operatorname{Grad}^{D,d}_{r} \\varphi , \\quad\n^V\\Delta^{D,d}_r {\\bf v} = \n\\operatorname{Grad}^{D,d}_r \n\\operatorname{Div}^{D,d}_{r} {\\bf v} .\n\\ee\n\nThe scalar Laplacian for $d \\ne D-1$ \nfor the field $\\varphi=\\varphi (r)$ is\n\\be \\label{S-Delta-Dd}\n^S\\Delta^{D,d}_r \\varphi= A(d,\\alpha_r)\n\\left(  \\frac{1}{r^{2 \\alpha_r-2}} \\, \n\\frac{\\partial^2 \\varphi}{\\partial r^2} + \n\\frac{d+1-\\alpha_r}{r^{2\\alpha_r-1}} \\, \n\\frac{\\partial \\varphi}{\\partial r} \\right) ,\n\\ee\nwhere\n\\be \\label{Ada}\nA(d,\\alpha_r) = \n\\frac{\\Gamma ((d+\\alpha_r)\/2) \\, \\Gamma (\\alpha_r\/2)}{ \n\\pi^{\\alpha_r-1\/2} \\, \\Gamma ((d+1)\/2)} .\n\\ee\n\nThe vector Laplacian for $d \\ne D-1$  \nfor the field ${\\bf v}=v(r) \\, {\\bf e}_r$ is\n\\be \\label{V-Delta-Dd}\n^V\\Delta^{D,d}_r {\\bf v} = A(d,\\alpha_r) \n\\left( \\frac{1}{r^{2 \\alpha_r-2}} \\, \n\\frac{\\partial^2 v_r}{\\partial r^2} \n+ \\frac{d+1-\\alpha_r}{r^{2\\alpha_r-1}} \\, \n\\frac{\\partial  v_r}{\\partial r} \n- \\frac{d \\, \\alpha_r}{r^{2\\alpha_r}} \\,  v_r\n\\right) \\, {\\bf e}_r.\n\\ee\n\nThe vector differential operators (\\ref{Grad-Dd}), \n(\\ref{Div-Dd}), (\\ref{S-Delta-Dd}) and (\\ref{V-Delta-Dd})\nallow us to describe complex fractal fluids with \nthe boundary dimensions $d \\ne D-1$\nby continuum models with non-integer dimensional spaces.\n\n\n\n\\section{Navier-Stokes equations in non-integer \ndimensional space for fractal fluid} \n\n\n\nA motion of an incompressible viscous fractal fluid\nin the framework of continuum model with \nnon-integer dimensional space is described by the equations\n\\be \\label{nse-1}\n\\operatorname{Div}^D_r {\\bf v} =0 ,\n\\ee\n\\be \\label{NSE}\n\\frac{d {\\bf v}}{dt} = {\\bf f}\n- \\frac{1}{\\rho} \\operatorname{Grad}^D_r p + \n\\nu \\, ^V\\Delta^D_r {\\bf v} , \n\\ee\nwhere ${\\bf f}$ is the vector field of mass forces,\n$\\nu$ is the kinematic viscosity is the ratio of the dynamic viscosity $\\mu$ to the density of the fluid $\\rho$, and\n$d\/dt$ is the material derivative\n\\be \\label{nse-2}\n\\frac{d {\\bf v}}{dt} = \\frac{\\partial {\\bf v}}{\\partial t} +\n\\Bigl({\\bf v},\\operatorname{Grad}^D_r {\\bf v}\\Bigr) .\n\\ee\nIn equations (\\ref{nse-1})-(\\ref{nse-2}) \nthe gradient $\\operatorname{Grad}^{D}_r$, \nthe divergence $\\operatorname{Div}^{D}_{r}$,\nand the vector Laplacian $^V\\Delta^{D}_r$\nare defined by equations\n(\\ref{Grad-D}), (\\ref{Div-D}), (\\ref{V-Delta-D})\nfor spherical symmetry.\nFor cylindrical symmetry,\nthese operators are defined by equations\n(\\ref{Grad-DC}), (\\ref{Div-DC}), (\\ref{V-Delta-DC}).\n\nIf the dimension $D$ of the region of a fractal fluid\nand the dimension $d$ of boundary of this region\nare not related by the relation $d = D-1$, i.e.,\n$\\alpha_r = D-d \\ne 1$, then we should use the equations\n\\be \\label{nse-1d}\n\\operatorname{Div}^{D,d}_r {\\bf v} =0 ,\n\\ee\n\\be \\label{NSEd}\n\\frac{d {\\bf v}}{dt} = {\\bf f}\n- \\frac{1}{\\rho} \\operatorname{Grad}^{D,d}_r p + \n\\nu \\, ^V\\Delta^{D,d}_r {\\bf v} , \n\\ee\n\\be \\label{nse-2d}\n\\frac{d {\\bf v}}{dt} = \\frac{\\partial {\\bf v}}{\\partial t} +\n\\Bigl({\\bf v},\\operatorname{Grad}^{D,d}_r {\\bf v}\\Bigr) ,\n\\ee\nwhere the gradient $\\operatorname{Grad}^{D,d}_r$, \nthe divergence $\\operatorname{Div}^{D,d}_{r}$,\nand the vector Laplacian $^V\\Delta^{D,d}_r$\nare defined by equations\n(\\ref{Grad-Dd}), (\\ref{Div-Dd2}), (\\ref{V-Delta-Dd}).\n\nEquations (\\ref{NSE}) and (\\ref{NSEd}) \ncan be called the Navier-Stokes equations\nfor non-integer dimensional space.\n\n\nIt is convenient to work in the dimensionless space variables\n$x\/R_0 \\to x$, $y\/R_0 \\to x$, $z\/R_0 \\to x$, $r\/R_0 \\to r$,\nthat yields dimensionless integration and dimensionless \ndifferentiation in non-integer dimensional space.\nHere $R_0$ is the characteristic size of a \nfractal fluid, which is always finite. \nFor example, $R_0$ can be the minimal scale of self-similarity \nof a considered fractal fluid. \nThen the density is properly scaled such that\nthe mass $Q$ of fractal fluid and the fields ${\\bf v}$, $p$, $f$\nhave correct physical dimensions. \n\n\n\nThe Navier-Stokes equations (\\ref{NSE}) and (\\ref{NSEd}) \ndescribe dynamics of fractal fluids in the framework of\ncontinuum models with non-integer dimensional spaces.\nThese equations allow us to describe the isotropic fractal fluid \nonly when the presence of spherical or cylindrical symmetry.\n\nEquations (\\ref{NSE}) and (\\ref{NSEd}) can be used, when\nthe fields $p$, ${\\bf v}$, ${\\bf v}$ have the form\n$p  =p(r)$ and ${\\bf v}=v_r(r) \\, {\\bf e}_r$,\n${\\bf f})=f_r(r) \\, {\\bf e}_r$\ndoes not depend on the angles.\n\nAs an example of application of the \nNavier-Stokes equations (\\ref{NSE}) and (\\ref{NSEd}),\nwe consider a steady flow of fractal fluid in a pipe, \nand fractal fluid discharge in the next sections.\nIn the next sections, we derive the Poiseuille equation \nfor fractal fluids from the Navier-Stokes equations (\\ref{NSE}) and (\\ref{NSEd}). \nEquations can be used for any other problems of\nhydrostatics and hydrodynamics of fractal fluids\nwithin the case of spherical and cylindrical symmetries\n\nTo consider anisotropic fractal fluids,\nand problems without spherical and cylindrical symmetries,\nwe cannot use the Navier-Stokes equations \n(\\ref{NSE}) and (\\ref{NSEd}).\nIn this case, we should apply a product measure approach\n\\cite{JMP2014}.\nThe product measure approach to describe fractal\nproperties of space-time has been considered in \\cite{Svozil}.\nThe product measure approach for the fractional spaces\nhas been suggested in \\cite{PRE2005,JPCS2005},\nwhere fractional phase space is considered with its \ninterpretation as a non-integer (fractional) dimensional space.\nFor non-integer dimensional spaces \nthe product measure approach is suggested in \\cite{PS2004},\nwhere each orthogonal coordinates has own dimension. \nThe product measure approach for \nthe fractional-integral continuum models\nhas been considered in \\cite{MOS-4,MOS-4b,MOS-4c,MOS-5}.\n\n\n\n\\section{Steady flow of fractal fluid in a pipe} \n\nIn this section, we derive the Poiseuille equation \nfor fractal fluids from the Navier-Stokes equations (\\ref{NSE}). \nLet us consider a simple problem of motion \nof an incompressible viscous fractal fluid.\nUsing the continuum models with non-integer dimensional space,\nwe describe a steady flow of fractal fluid \nin a pipe with circular cross-section. \nWe take the axis of the pipe as the $X$-axis. \nThe velocity for laminar of fractal fluid\nis along the $X$-axis at all points, \nand is a function of $r$ only\n\\be \n{\\bf v}={\\bf v}(r)=v_x(r) \\, {\\bf e}_x .\n\\ee\nThe equation of continuity is satisfied identically. \nThe components of the Navier-Stokes equation \nfor $Y$-axis and $Z$-axis \ngive that the pressure is constant over \nthe cross-section of the pipe. \n\nWe shall solve the equation for a pipe \nwith circular cross-section. \nTaking the origin at the center of the circle \nwe can use cylindrical symmetry\n${\\bf v}={\\bf v}(r)=v_x(r) \\, {\\bf e}_x$. \nUsing the Navier-Stokes equations (\\ref{NSE}), we have \n\\be \\label{NSE-X}\n^S\\Delta^D_r v_x(r)= \\frac{1}{ \\mu} \\,\\frac{dp}{dx} ,\n\\ee\nwhere $\\mu= \\rho \\, \\nu$, and $dp\/dx$ is a constant. \nThe pressure gradient $dp\/dx$ may be written $ - \\Delta p \/l$, \nwhere $\\Delta p$ is the pressure difference between \nthe ends of the pipe and $l$ is its length. \n\n\nUsing the scalar Laplacian \nfor non-integer dimensional space\nthe Navier-Stokes equation (\\ref{NSE-X}) takes the form\n\\be \\label{Cyl-1}\n\\frac{\\partial^2 v_x(r)}{\\partial r^2} + \\frac{D-2}{r} \\, \n\\frac{\\partial v_x}{\\partial r} - \\frac{1}{ \\mu} \\,\\frac{dp}{dx} = 0 .\n\\ee\nFor $1<D<3$ and $0<D<1$, \nthe general solution of (\\ref{Cyl-1}) is\n\\be \\label{vxD}\nv_x(r) =C_1 \\, r^{3-D} + C_2 + \\frac{1}{2 \\, (D-1) \\, \\mu} \n\\frac{dp}{dx} \\, r^2 \\quad (0<D<3, \\quad D \\ne 1).\n\\ee\nFor $D=3$, we have\n\\be\nv_x(r) =C_1 \\, \\ln(r) + C_2 + \\frac{1}{4 \\, \\mu} \n\\frac{dp}{dx} \\, r^2 .\n\\ee\nFor $D=1$, we get the general solution\n\\be\nv_x(r) =C_1 \\, r^2 + C_2 + \\frac{1}{4 \\, \\mu} \n\\frac{dp}{dx} \\, r^2 \\, \\left(2 \\, \\ln(r) - 1 \\right) .\n\\ee\nIt should be noted that dimensions $D=1$ of the fractal fluid \ndo not correspond to the distribution of particles \nalong the line. \nThe fractal media with $D=1$\ndescribe a distribution of fluid particles \nin 3-dimensional space \nsuch that the mass dimension of the distribution \nis equal to $D=1$.\n\n\nLet us determine a flow of fractal fluid in a pipe \nof annular cross-section with \nthe internal radius $R_1$ and external radius $R_2$. \nThe constants $C_1$ and $C_2$ \nin the general solution (\\ref{vxD}) \nare determined from the boundary conditions \n\\be \\label{vR1vR2}\nv_x(R_1) = v_x(R_2) = 0 .\n\\ee\nUsing (\\ref{vxD}), these conditions have the form\n\\be\nC_1 \\, R_1^{3-D} + C_2 + \\frac{1}{2 \\, (D-1) \\, \\mu} \n\\frac{dp}{dx} \\, R_1^2 =0 ,\n\\ee\n\\be\nC_1 \\, R_2^{3-D} + C_2 + \\frac{1}{2 \\, (D-1) \\, \\mu} \n\\frac{dp}{dx} \\, R_2^2 =0 .\n\\ee\nThen the constants are\n\\be \\label{C1}\nC_1 = \\frac{1}{2 \\, (D-1) \\, \\mu} \\frac{dp}{dx} \\, \n\\frac{R_2^2- R_1^2}{R_1^{3-D}-R_2^{3-D}} ,\n\\ee\n\\be \\label{C2}\nC_2 = \\frac{1}{2 \\, (D-1) \\, \\mu} \\frac{dp}{dx} \n\\, \\frac{ R_1^{3-D}\\, R_2^2- R_2^{3-D} \\, R_1^2}{R_1^{3-D}-R_2^{3-D}} .\n\\ee\nSubstitution of (\\ref{C1}) and (\\ref{C2}) into (\\ref{vxD}) gives\n\\be \\label{vxr}\nv_x(r) = \\frac{1}{2 \\, (D-1) \\, \\mu} \\, \\frac{dp}{dx}\n\\left( \\frac{ R_2^2- R_1^2}{R_1^{3-D}-R_2^{3-D}} \\, r^{3-D} + \n\\frac{R_1^{3-D}\\, R_2^2- R_2^{3-D} \\, R_1^2}{R_1^{3-D}-R_2^{3-D}} +  r^2 \\right) ,\n\\ee\nwhere $0<D<1$ and $1<D <3$.\n\nUsing the variables\n\\[ x=\\frac{R_2}{R_1} , \\quad  y=\\frac{r}{R_1} , \\]\nequation (\\ref{vxr}) can be represented in the form\n\\be \\label{vxr2}\nv(x,y) = \\frac{R^2_1}{2 \\, (D-1) \\, \\mu} \\, \\frac{dp}{dx}\n\\left( - \\frac{1-x^2}{1-x^{3-D}} \\, y^{3-D} + \n\\frac{x^2 - x^{3-D}}{1-x^{3-D}} +  y^2 \\right) ,\n\\ee\nTo demonstrate some properties of the velocity\n$v_x(r)$ defined by (\\ref{vxr}), \nwe can visualize the function (\\ref{vxr2}), \nfor $x \\in [1;100]$, $y \\in [1;100]$ \nand different values of dimensions \n$D=2.9$, $D=2.7$, $D=2.0$, $D=1.1$.\nThe plots of function (\\ref{vxr2}) are presented by Figures 1-4, \nwhere $\\mu=1$, $R_1=1$ and ${dp}\/{dx}=-1$.\n\n\n\n\n\n\n\\begin{figure}[H]\n\\begin{minipage}[h]{0.47\\linewidth}\n\\resizebox{11cm}{!}{\\includegraphics[angle=-90]{101.eps}} \n\\end{minipage}\n\\caption{Plot of the velocity function $z=v(x,y)$ \ndefined by (\\ref{vxr2}) for the ranges $x \\in [1;100]$,  \n$y \\in [1;100]$, and $D=2.9$.} \n\\label{Plot1}\n\\end{figure}\n\n\n\\begin{figure}[H]\n\\begin{minipage}[h]{0.47\\linewidth}\n\\resizebox{11cm}{!}{\\includegraphics[angle=-90]{102.eps}} \n\\end{minipage}\n\\caption{Plot of the velocity function $z=v(x,y)$ \ndefined by (\\ref{vxr2}) for the ranges $x \\in [1;100]$,  \n$y \\in [1;100]$, and $D=2.7$.} \n\\label{Plot2}\n\\end{figure}\n\n\n\\begin{figure}[H]\n\\begin{minipage}[h]{0.47\\linewidth}\n\\resizebox{11cm}{!}{\\includegraphics[angle=-90]{103.eps}} \n\\end{minipage}\n\\caption{Plot of the velocity function $z=v(x,y)$ \ndefined by (\\ref{vxr2}) for the ranges $x \\in [1;100]$,  \n$y \\in [1;100]$, and $D=2.0$.} \n\\label{Plot3}\n\\end{figure}\n\n\n\\begin{figure}[H]\n\\begin{minipage}[h]{0.47\\linewidth}\n\\resizebox{11cm}{!}{\\includegraphics[angle=-90]{104.eps}} \n\\end{minipage}\n\\caption{Plot of the velocity function $z=v(x,y)$ \ndefined by (\\ref{vxr2}) for the ranges $x \\in [1;100]$,  \n$y \\in [1;100]$, and $D=1.1$.} \n\\label{Plot4}\n\\end{figure}\n\n\n\n\nThe flow in a pipe of annular cross-section  \nwith the radius $R$, i.e. $R_1=0$ and $R_2=R$, we have \n\\be \\label{PE-FF}\nv_x(r) = - \\frac{1}{2 \\, (D-1) \\, \\mu} \\frac{dp}{dx} \\, R^2\n\\left( \\left(\\frac{r}{R}\\right)^{3-D} \n- \\left(\\frac{r}{R}\\right)^2 \\right) \\quad\n(0<D \\le 3, \\quad D \\ne 1).\n\\ee\nEquation (\\ref{PE-FF}) can be called \nthe Poiseuille equation for flow of fractal fluid. \nFor the case of non-fractal fluid ($D=3$), \nequation (\\ref{PE-FF}) gives the well-known Poiseuille equation\n\\be \\label{PE-FF-D3}\nv_x(r) = - \\frac{1}{4 \\, \\mu} \\frac{dp}{dx} \\, R^2\n\\left( 1 - \\left(\\frac{r}{R}\\right)^2 \\right) .\n\\ee\nThus the velocity distribution \nacross the pipe is parabolic for the non-fractal fluids. \nFor the fractal fluids, we have \nnon-integer power-law (\\ref{PE-FF}). \n\nNote that suggested Poiseuille equation \nfor fractal fluids, which are derived \nfrom the proposed Navier-Stokes equation \nfor non-integer dimensional space \ncan be used only to describe \nflow of fractal fluid in pipes.\nTo describe flow of fractal fluid between parallel planes, we should define new differential operators \nand the Navier-Stokes equations (\\ref{NSE}) and (\\ref{NSEd}), \nshould be modified by using the product measure approach\n\\cite{Svozil,PS2004,PRE2005,JPCS2005}.\n\n\n\n\\section{Fractal fluid with $\\alpha_r \\ne 1$}\n\nLet us derive the Poiseuille equation for fractal fluids \nfrom the Navier-Stokes equations (\\ref{NSEd}). \nThe Navier-Stokes equations for \nfractal fluid with $\\alpha_r=D-d \\ne 1$ has the form\n\\be \\label{Eq-Dda}\nA(d_x,\\alpha_r) \\left(  \\frac{1}{r^{2 \\alpha_r-2}} \\, \n\\frac{\\partial^2 v_x(r)}{\\partial r^2} + \n\\frac{d_x +1-\\alpha_r}{r^{2\\alpha_r-1}} \\, \n\\frac{\\partial v_x(r)}{\\partial r} \\right) -\n\\frac{1}{ \\mu} \\,\\frac{dp}{dx} = 0 ,\n\\ee\nwhere $A(d_x,\\alpha_r)$ is defined by (\\ref{Ada}),\n$d_x=d- \\alpha_x$, and\n$\\alpha_x$ is dimension along the $X$-axis.\nUsing $v_x(r)$ as an effective scalar field\n$\\varphi_{eff}(r)=v_x(r)$, we can apply equations\n(\\ref{Grad-Dd}), (\\ref{Div-Dd2}) and (\\ref{S-Delta-Dd})\nwhere $D \\to D_x=D- \\alpha_x$ and $d \\to d_x=d- \\alpha_x$\nto get (\\ref{Eq-Dda}).\nEquation (\\ref{Eq-Dda}) with $\\alpha_r=\\alpha_x=1$ \ngives (\\ref{Cyl-1}).\n\nFor $1<D<3$ and $0<D<1$, \nthe general solution of (\\ref{Eq-Dda}) is\n\\be \\label{vxDda}\nv_x(r) =C_1 \\, r^{\\alpha_r-d_x} + C_2 + \n\\frac{1}{2 \\, (d_x+ \\alpha_r) \\, \\alpha_r \\, A(d_x,\\alpha_r) \\, \\mu} \n\\frac{dp}{dx} \\, r^{2 \\alpha_r} \n\\quad (0<D<3, \\quad D \\ne 1) .\n\\ee\nFor $\\alpha_r=\\alpha_x=1$ equation (\\ref{vxDda}) gives (\\ref{vxD}).\nThe constants $C_1$ and $C_2$ \nin the general solution (\\ref{vxDda}) \nare determined by the boundary conditions \n\\be \\label{vR1vR2b}\nv_x(R_1) = v_x(R_2) = 0 .\n\\ee\nThese conditions give the equations\n\\be \nC_1 \\, R_1^{\\alpha_r-d_x} + C_2 + \n\\frac{1}{2 \\, (d_x+ \\alpha_r) \\, \\alpha_r \\, A(d_x,\\alpha_r) \\, \\mu} \n\\frac{dp}{dx} \\, R_1^{2 \\alpha_r} =0 ,\n\\ee\n\\be \nC_1 \\, R_2^{\\alpha_r-d_x} + C_2 + \n\\frac{1}{2 \\, (d_x+ \\alpha_r) \\, \\alpha_r \\, A(d_x,\\alpha_r) \\, \\mu} \n\\frac{dp}{dx} \\, R_2^{2 \\alpha_r} =0 .\n\\ee\nThen the coefficients are\n\\be \\label{C1da}\nC_1 = - \\frac{1}{2 \\, (d_x+ \\alpha_r) \\, \n\\alpha_r \\, A(d_x,\\alpha_r) \\, \\mu} \\frac{dp}{dx} \n\\frac{ R_1^{2\\alpha_r} - R_2^{2\\alpha_r} }{R_1^{\\alpha_r-d_x} - R_2^{\\alpha_r-d_x}} ,\n\\ee\n\\be \\label{C2da}\nC_3 = - \\frac{1}{2 \\, (d_x+ \\alpha_r) \\, \n\\alpha_r \\, A(d_x,\\alpha_r) \\, \\mu} \\frac{dp}{dx} \n\\frac{ R_2^{2\\alpha_r} R_1^{\\alpha_r-d_x} - R_1^{2\\alpha_r} R_2^{\\alpha_r-d_x} }{ R_1^{\\alpha_r-d_x} - R_2^{\\alpha_r-d_x}} ,\n\\ee\nSubstitution of (\\ref{C1da}) and (\\ref{C2da}) \ninto (\\ref{vxDda}) gives\n\\[\nv_x(r) = - \\frac{1}{2 \\, (d_x+ \\alpha_r) \\, \n\\alpha_r \\, A(d_x,\\alpha_r) \\, \\mu} \\frac{dp}{dx} \\Bigl(\n\\frac{ R_1^{2\\alpha_r} - R_2^{2\\alpha_r} }{R_1^{\\alpha_r-d_x} - R_2^{\\alpha_r-d_x}} \\, r^{\\alpha_r-d_x} + \\]\n\\be \\label{vxDda2}\n+ \\frac{ R_2^{2\\alpha_r} R_1^{\\alpha_r-d_x} - R_1^{2\\alpha_r} R_2^{\\alpha_r-d_x} }{ R_1^{\\alpha_r-d_x} - R_2^{\\alpha_r-d_x}} \n- r^{2 \\alpha_r} \\Bigr) \n\\ee\nfor $0<D<3$, where $D \\ne 1$.\n\nIf $R_1=0$ and $R_2=R$, then equation (\\ref{vxDda2}) has the form\n\\be \\label{vxDda3}\nv_x(r) = - \\frac{1}{2 \\, (d_x+ \\alpha_r) \\, \n\\alpha_r \\, A(d_x,\\alpha_r) \\, \\mu} \\frac{dp}{dx} \\,\nR^{2\\alpha_r}  \\, \\left(\n\\left(\\frac{r}{R}\\right)^{\\alpha_r-d_x} -\n\\left( \\frac{r}{R}\\right)^{2 \\alpha_r} \\right) .\n\\ee\nFor $\\alpha_r=\\alpha_x=1$ \nequation (\\ref{vxDda3}) gives (\\ref{PE-FF}).\nThe cases $\\alpha_r <1$ and\/or $\\alpha_x <1$ \ncorresponds to fractal fluid.\n\nWe can assume that $\\alpha_x >1$ can be used \nto describe fractal turbulent flow in pipe.\nThis assumption is based on the fact\nthat trajectories of the fluid particles \nare fractal curve, then $\\alpha_x>1$\n(for example, the Koch curve with \n$\\alpha_x=\\ln(4) \/ \\ln(3) \\approx 1.262$).\n\n\n\n\n\\section{Fractal fluid discharge}\n\n\nIn general, fractal fluids cannot be considered \nas a fluid on fractal.\nReal fractal fluids have a characteristic smallest length scale\nsuch as the radius, $R_0$, of a particle \n(for example, an atom or molecule).\nIn real fluids the fractal structure cannot be observed on all\nscales but only those for which $R >R_0$, \nwhere $R_0$ is the characteristic scale of the particles.\nThe concept of non-integer mass dimension of fractal fluid\nis based on the idea of how the mass \nof a fluid region scales with the region size,\nif we assume unchanged density.\nFor many cases, we can write the asymptotic form for the relation between\nthe mass $M_D(W)$ of a ball region $W$ of fluid,\nand the radius $R$ containing this mass as follows:\n\\be \\label{1-1-MR} \nM_D(W)=M_0 \\left( \\frac{R}{R_0} \\right)^D  \\ee\nfor $R\/R_0 \\gg 1$.\nThe constant $M_0$ depends on how the spheres of radius $R_0$ \nare packed.\nThe parameter $D$, which is interpreted as a dimension, \ndoes not depend on the shape of the region $W$,\nor on whether the packing of spheres of radius $R_0$ is \nclose packing,\na random packing or a porous packing with a uniform distribution of holes.\nThe non-integer mass dimension $D$ of fractal fluid \nis a measure of how the fluid fills \nthe integer $n$-dimensional Euclidean space it occupies.\nNote that the fact that a fluid is random or contains cavities\ndoes not necessarily imply that the fluid is fractal.\n\n\nUsing the non-integer dimensional space approach,\nwe can calculate the mass of fractal homogeneous fluids. \nScaling law (\\ref{1-1-MR}) is obtained naturally \nin the framework of this approach.\nWe can use the integration in a non-integer dimensional space \n\\cite{Stillinger} that is described by the equation\n\\be \\label{NI-2}\n\\int_{R^D} d^D {\\bf r} \\, \\varphi ({\\bf r}) =\n\\frac{2 \\pi^{(D-1)\/2}}{\\Gamma((D-1)\/2)}\n\\int^{\\infty}_0 dr \\, r^{D-1} \\,\n\\int^{\\pi}_0 d \\theta \\, \\sin^{D-2}\\theta \\, \\varphi (r,\\theta) ,\n\\ee\nwhere $d^D {\\bf r}$ represent the volume element\nin the non-integer dimensional space.\nUsing (\\ref{NI-2}) with $\\varphi (r,\\theta)=1$, and\n\\be \\int^{\\pi}_0 d \\theta \\, sin^{D-2}\\theta = \n\\frac{\\pi^{1\/2}\\, \\Gamma (D\/2-1)}{\\Gamma(D\/2)} , \\ee\nwe get the volume of $D$-dimensional ball with radius $R$ in the form\n\\be\nV_D = \\frac{\\pi^{D\/2}}{\\Gamma(D\/2+1)} \\, R^{D} .\n\\ee\nThe mass of fluid in $W$ is described by the integral \n\\be \\label{1-MW3} \nM_D(W) = \\int_{W} \\rho({\\bf r}) \\, d^D {\\bf r} ,\n\\ee\nwhere ${\\bf r}$ is dimensionless vector variable. \nFor a ball with radius $R$ and constant density \n$\\rho({\\bf r})=\\rho =\\operatorname{const}$, we get \n\\be \\label{M-D}\nM_D(W) = \\rho \\, V_D =\n\\frac{\\pi^{D\/2} \\, \\rho}{\\Gamma(D\/2+1)} \\, R^{D} .\n\\ee\nThis equation define the mass of the \nfractal homogeneous ball region of fluid with volume $V_D$. \nFor $D=3$, equation (\\ref{M-D}) gives the well-known \nequation for mass of non-fractal ball region\n$M_3 =(4 \\rho \\pi \/3) \\, R^3$ because \n$\\Gamma(3\/2) =\\sqrt{\\pi}\/2$ and $\\Gamma(z+1)=z \\, \\Gamma(z)$. \n\n\nLet us determine the mass $Q$ of fluid passing per unit time \nthrough any cross-section of the pipe (called the discharge). \nNot all pipe volume is occupied by fractal fluid.\nThere are areas unoccupied by particles of fractal fluid.\nIn continuum model of fractal fluids, \nwe take into account this fact by using the integration \nin space with non-integer dimension\n\\be \\label{Q-Def}\nQ= \\rho \\, \\frac{2 \\, \\pi^{d\/2}}{\\Gamma(d\/2)} \n\\int^R_0 v_x(r) \\, r^{d-1} \\, dr ,\n\\ee\nwhere $d=D-1$ is non-integer dimension of \nthe cross-section, $\\rho$ is a constant density, \nand $v_x(r)$ is defined by equation (\\ref{PE-FF}).\nSubstitution of (\\ref{PE-FF}) into (\\ref{Q-Def}) gives\n\\be \\label{QR}\nQ= - \\frac{\\rho \\, \\pi^{(D-1)\/2}}{2 \\, (D+1) \\, \\Gamma((D-1)\/2)\\, \\mu} \n\\frac{dp}{dx} \\, R^{D+1} \\quad (0<D \\le 3 \\quad D \\ne 1).\n\\ee\nNote that $\\rho$ has physical dimension of mass \n(for example, kilogram).\nThe mass of fractal fluid is thus proportional \nto $(D+1)$-power of the radius of the pipe.\n\nFor $D=3$, equation (\\ref{QR}) gives the well-known equation\n\\be \\label{QRD3}\nQ= - \\frac{\\rho \\, \\pi}{8\\, \\mu} \\frac{dp}{dx} \\, R^{4} .\n\\ee\nThe mass of non-fractal fluid is proportional to \nthe fourth power of the radius of the pipe. \nThe dependence of $Q$ on $dp\/dx$ and $R$ given by formula \n(\\ref{QRD3}) was established empirically by G. Hagen in 1839 and J. L. M. Poiseuille in 1840, and theoretically justified by G. G. Stokes in 1845.\n\nEquation (\\ref{QR}) can be rewritten in the form\n\\be\nQ= - \\frac{\\rho \\, \\pi}{8\\, \\mu_{eff}} \\frac{dp}{dx} \\, R^{D+1} \n\\ee\nwith the effective dynamic viscosity \n\\be \\label{meff}\n\\mu_{eff} = \n\\frac{D+1}{4} \\, \\pi^{(3-D)\/2} \\ \\Gamma((D-1)\/2) \\ \\mu .\n\\ee\nThe mass (discharge) of fractal fluid is proportional to \nthe non-integer power $(D+1)<4$ of the radius of the pipe,\nand the dynamic viscosity is effectively changed. \n\nFor $D=3$, equation (\\ref{meff}) gives $\\mu_{eff}=\\mu$.\nFor $1<D<3$, we have $\\mu_{eff}>\\mu$.\nIf $0<D<1$, then $\\mu_{eff} <0$.\nThe effective dynamic viscosity of \nfractal fluid with $D \\in (1;3)$ \nincreases with increasing a deviation of the dimension $D$ \nfrom three. \nWe can see an interesting effect of \na negative effective dynamic viscosity for fractal fluid \nwith dimension $D \\in (0;1)$. \nThe strong fractality of the fluid, \nwhich is caused by small dimension, \nleads to an increased fluid flow \ncompared with conventional medium.\nThis is probably due to an increase in freedom of particles motion for the fractal fluid similar to 3D Cantor dust.\n\nThe discharge function $Q=Q(R,D)$ defined by (\\ref{QR}) \nfor the different values of dimensions $0 < D < 3$ \nand the range $R\\in[0,1]$ are present on Figures 5-8, \nwhere $\\rho=1$, $\\mu=1$, and ${dp}\/{dx}=-1$.\n\n\n\n\n\n\\begin{figure}[H]\n\\begin{minipage}[h]{0.47\\linewidth}\n\\resizebox{11cm}{!}{\\includegraphics[angle=-90]{201.eps}} \n\\end{minipage}\n\\caption{Plot of the discharge function $Q=Q(R,D)$ defined by (\\ref{QR}) for the range $R\\in[0,1]$ and $D\\in[2,3]$.} \n\\label{Plot5}\n\\end{figure}\n\n\n\\begin{figure}[H]\n\\begin{minipage}[h]{0.47\\linewidth}\n\\resizebox{11cm}{!}{\\includegraphics[angle=-90]{202.eps}} \n\\end{minipage}\n\\caption{Plot of the discharge function $Q=Q(R,D)$ defined by (\\ref{QR}) for the range $R\\in[0,1]$ and $D\\in[1,2]$.} \n\\label{Plot6}\n\\end{figure}\n\n\n\\begin{figure}[H]\n\\begin{minipage}[h]{0.47\\linewidth}\n\\resizebox{11cm}{!}{\\includegraphics[angle=-90]{203.eps}} \n\\end{minipage}\n\\caption{Plot of the discharge function $Q=Q(R,D)$ defined by (\\ref{QR}) for the range $R\\in[0,1]$ and $D\\in[0.5,1.5]$.} \n\\label{Plot7}\n\\end{figure}\n\n\n\\begin{figure}[H]\n\\begin{minipage}[h]{0.47\\linewidth}\n\\resizebox{11cm}{!}{\\includegraphics[angle=-90]{204.eps}} \n\\end{minipage}\n\\caption{Plot of the discharge function $Q=Q(R,D)$ defined by (\\ref{QR}) for the range $R\\in[0,1]$ and $D\\in[0,3]$.} \n\\label{Plot8}\n\\end{figure}\n\n\n\n\n\n\\section{Conclusion}\n\nIn this paper, we propose a generalization of the Navier-Stokes \nequations to describe fractal fluids in the framework of\ncontinuum models with non-integer dimensional spaces. \nThese equations contain generalized \ndifferential vector operators for\nnon-integer dimensional space \\cite{CNSNS2015,JMP2014}.\nAs an example of application of the suggested\nNavier-Stokes equations for fractal fluids, \nwe consider a Poiseuille flow of \nan incompressible viscous fractal fluid in the pipe. \nThe solution for steady flow of \nfractal fluid in a pipe and corresponding \nfractal fluid discharge have been derived.\n\nIn this paper fractal fluid is described as a continuum \nin non-integer dimensional space. \nWe assume that suggested continuum models with non-integer\ndimensional spaces and the correspondent\nNavier-Stokes equations for fractal fluids\nmay be important for fractal theory of different type of media.\n\n\nAs the main object for application of the proposed \ncontinuum models is a two-component medium, \nwhere distribution of one component  (gas, liquid, solid) \ninto another component (fluid, gas or empty space) \ncan be characterized by non-integer mass or \n\"particle\" dimension.\nThis non-integer dimensional component can be considered \nas a fractal fluid. \nOne of the possible experimental methods \nfor determining the presence of \nfractal properties of the two-component medium \nmay be to use labels with radioactive isotopes \nfor particles of component that is assumed a fractal.\n\n\nA basic idealized model of fractal fluid \nis a liquid distributed in empty space $\\mathbb{R}^3$\nwith non-integer mass dimension $D<3$.\nIn some sense the fractal fluid is considered \nas a liquid analog of fractal porous solid material.\nFractal fluid can also be viewed as a two-phase medium \nconsisting of a liquid and a discharged gas instead \nof empty space,\nwhere the liquid is characterized by fractal mass dimension.\n\n\nAs an object of study, we also can consider \nan emulsion, when both the dispersed and the continuous phase \nare liquids, and the dispersed phase is fractally distributed \nin continuous phase.\nAn emulsion that is a mixture of two immiscible liquids,\none of which (the dispersed phase) is fractally dispersed \nin the other (the continuous phase). \nIn this case the dispersed phase can be described \nas a fractal fluid by suggested continuum models \nwith non-integer dimensional space.\nThe proposed models can be used for\na solution that is a homogeneous mixture \ncomposed of only one (liquid) phase,\nwhere one phase has a fractal dimension. \nWe can consider a fractal distribution of a solute \ndissolved in a non-fractal solvent,\nthen the solute is considered as a fractal fluid.\nThe solvent that is fractally homogeneously mixed with solute \ncan be considered as a fractal homogeneous fluid.\nThe homogeneity property of the fractal fluid means \nthat two regions $W_1$ and $W_2$ \nwith the equal volumes $V_n(W_1)=V_n(W_2)$ \nhave equal number of particles $N_D(W_1)=N_D(W_2)$. \nIn other words the fractal fluid is called homogeneous \nif the power law $N_D(W) \\sim R^{D}$ \n(or $M_D(W) \\sim R^{D}$) does not depend on \nthe translation of the region $W$. \n\n\nWe can consider a fractal distribution of small solid particles \nin the suspension. In this case, we have \nan internal phase (solid) that is fractally distributed through \nthe external phase (fluid) by mechanical agitation. \n\nAn object of investigations can be a liquid mixed \nwith a solid particles, \nwhere the distribution of these particles in space\ncan be characterized by non-integer mass dimensions,\nwhich can be caused by a power law distribution \nof particles by size or mass.\n\n\nAs a complex medium which may exhibit \nfractal properties can be considered the blood\nthat is composed of\nproteins, glucose, mineral ions, hormones, \ncarbon dioxide, blood cells \nand other particles suspended in water. \nWe assume that the blood as a multi-phase medium\ncan have attributes of a fractal distribution for \nsome blood components including bacteria, \nviruses and medicinal substances getting into the blood.\n\n\nWe assume that the suggested approach to describe \nfractal fluids by continuum models with non-integer\ndimensional spaces may be important for\nfractal theory of blood flow in cardiovascular system, \ndynamics of fractal media in hydrologic modeling \n\\cite{FLU-1,FLU-2,FLU-3,FLU-4} and \nit allows to develop the fractal dynamics \nof multi-phase media \\cite{Soo,Nigmatulin}.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\nPortraits make up a large percentage of the photos on the web nowadays.  ``Selfies'' have become a phenomenon, and recent studies~\\cite{facesinstagram} show that images with faces are more popular (+38\\% ``likes\" on Instagram) than other pictures in online social networks. Portraits are also used in web user profiles, in news articles, to represent celebrities and public figures, and they are an essential part of all kinds of IDs.\n\nGiven the huge volume of digital portraits, their broad usage, and their importance for people identification, surfacing the best digital portraits in terms of  photographic quality is of crucial importance. A system able to automatically score the aesthetic value of portraits could be used to select good images for a variety of applications such as  journalism, photo sharing websites, web search, PhotoBoosts, and many others.\n\nShooting photos of people is not a trivial task: human faces convey emotions, stories, lifestyles, and a good photographer needs to be able to capture their essence and personality. As a matter of fact, portrait photography is a stand-alone branch of photography literature, with  its own rules and compositional techniques, and tons of dedicated books  \\cite{child2008studio, hurter2007portrait, weiser1999phototherapy}. Systems that automatically rate the quality of digital portraits should be therefore specifically designed for face photos, unlike traditional visual aesthetics works \\cite{datta, ke2006design}, based on general photographic rules .\n\nIn spite of its importance, there has been little work in the research community to specifically address computational aesthetics of portraits.  Preliminary works ~\\cite{khan2012evaluating, li2010aesthetic} leave out many of the aspects that are specific to portraits (e.g., illumination, landmark representation, affective properties, etc.), and have experimented only with small datasets (less than 500 images). \n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=\\linewidth, natwidth=1362, natheight=921]{fig\/Dataset.pdf}\n\\caption{(a) Distribution of images per demographic category and aesthetic scores (b) Characteristics extracted from the portrayed subjects}\n\\label{Facefeatures}\n\\vspace{-0.5cm}\n\\end{figure}\nIn this paper, we try to fill this void and introduce a new framework to automatically evaluate portrait aesthetics\\footnote{The aim of this work is to estimate the photographic quality of the \\textit{representation} of the person, independent from the beauty of the subject represented. }. To do so, we design visual features to describe image quality and portrait-specific properties and present a large-scale analysis of a data set of  over 10,000 portraits.\nIn addition, we build predictive models that are able to determine the aesthetic score of digital portraits.\nMoreover, with such large scale study, we provide an analysis of what makes a portrait beautiful from a computational perspective. To our knowledge, this represents the first attempt in literature to understand the relevancy of features for portrait aesthetics. \n\nOur main contributions can be summarized as follows:\n\\\\\\textbf{(1) Dataset:} we build a large dataset of portraits annotated with physical characteristics (determined using facial analysis) by sampling the AVA~\\cite{murray2012ava} images.\n\\\\\\textbf{(2) Features:} we introduce new features to describe portrait composition, quality, illumination, memorability, emotions, and originality.\n\\\\\\textbf{(3) Feature analysis:} we perform analyses on a set of over 10,000 portraits and report observations. We find that race, gender, and age are largely uncorrelated with photographic beauty, but aesthetic score is related to sharpness of facial landmarks, image contrast, exposure, homogeneity, illumination pattern, uniqueness, and originality.\n\\\\\\textbf{(4) Aesethetic Prediction:} we develop predictive models to classify portraits as aesthetically beautiful or not.\n\nIn Sec. \\ref{related}, we describe related work, and explain our portrait dataset in Sec. \\ref{datasets}. Sec. \\ref{visual_features} presents the visual features and analyze their relations with portrait beauty in Sec. \\ref{analysis}, then present our classification experiments in Sec. \\ref{results}. %\n\n\\section{Related Work}\\label{related}\nOur work relates to research that applies image analysis techniques to  detect the visual presence of non-semantic, fuzzy concepts such as memorability~\\cite{isola2011}, emotions~\\cite{emotions,joshi2011aesthetics}, interestingness~\\cite{redi2012interestingness, interesting, gygli2013interestingness}, privacy~\\cite{zerr2012privacy}, and beauty~\\cite{datta}.\n In particular, this paper follows previous work on computational aesthetics~\\cite{datta,murray2012ava}, that explores the discriminative ability of visual features to automatically assess the beauty of images and videos. Pioneers in this field are~Datta et al. and Ke et al., \\cite{datta, ke2006design}, who built an aesthetic classification framework for images based on features inspired by photographic theory. In subsequent years, such works were improved by designing more discriminative features~\\cite{nishiyama2011aesthetic, interesting}, proving the effectiveness of generic features~\\cite{marchesotti2011assessing, murray2012ava}  and building more effective learning frameworks~\\cite{wu2011learning}. Similar frameworks were applied to automatic image composition and enhancement by Bhattacharya et al.~\\cite{bhattacharya2010framework}\n\nWhile these existing computational aesthetic works build general frameworks for photographs of any semantic category, we focus on a specific type of images, namely portraits, whose compositional  and aesthetic criteria constitute a separate subject of study in the photographic literature~\\cite{hurter2007portrait, weiser1999phototherapy, child2008studio}, and therefore need a separate computational framework for aesthetic assessment.\nThis aspect is also proven by our experiments: we show that our portrait-specific aesthetic framework performs much better than a general classifier  for portrait aesthetic assessment.\n\nA few works~\\cite{murray2012ava, luo2011content, interesting} perform topic-based aesthetic classification. They build category-specific subsets of images by sampling aesthetic databases according to given image tags (``city\", ``nature\", but also  ``humans\" or ``portraits\"), and then use general compositional features to build topic-specific models.  The framework in this paper differs from those works for two reasons. (1) \\textit{We build a rich large-scale portrait aesthetic database}. A dataset based on tag-based sampling as in \\cite{murray2012ava, luo2011content, interesting}, could ignore many face images without tags while including images with noisy tags (as shown in Section \\ref{datasets}).  In this paper, we adopt a \\textit{content-aware sampling strategy} based on detailed face analysis. We reduce a large scale aesthetic dataset \\cite{murray2012ava} to a subset of more than $10000$ face images annotated with information about the portrayed subject, useful for both analysis and feature extraction. (2) \\textit{We build portrait-specific aesthetic visual features}. The works in~\\cite{murray2012ava, luo2011content, interesting} use traditional aesthetic  features designed for a general case, and apply them to the topic-specific contexts. In our work, we design face-specific aesthetic features inspired by photographic literature, together with non-face features that describe crucial aspects of photographic portraiture, such as illumination, sharpness, manipulation detection, image quality, emotion and memorability. Moreover, we show their combined effectiveness for aesthetic assessment of face photographs compared to  traditional aesthetic features. \n\nThere are a few recent works that attempt to design portrait specific datasets and features. For example, Li et al.\\cite{li2010aesthetic} use  face expression, face pose and face position features to estimate the aesthetic value of the images in a dataset of $500$ face images annotated by micro-workers. This work was improved by the work in~\\cite{xuefeature}, that uses hand-crafted features together with low-level generic features, and by Khan et al.~\\cite{khan2012evaluating} using spatial composition rules specifically tailored for portrait photography, together with specific background contrast features and face brightness and size features. These works represent a first attempt towards portrait aesthetic classification. However, one major weak point of such works is that they rely on small datasets (<500 images), thus making the results less generalizable for large datasets like the one we consider. Moreover,  despite their focus on face analysis, the features proposed by those works miss many important aspects of portrait photography such as illumination, demographics, face landmark properties, affective dimension, semantics and post-processing. In our work, we use features that are able to capture these aspects, and prove their effectiveness  by showing that they outperform the features in \\cite{khan2012evaluating} when used in an aesthetic classification framework on the dataset used by Khan et al.~\\cite{khan2012evaluating}. Moreover, in this paper, we perform for the first time a deep analysis of the importance of each feature and each group of features for face photo aesthetics, giving interesting and probably unexpected insights about what makes a portrait beautiful.\n\n\\section{Large Scale Portrait Dataset}\\label{datasets}\nIn order to create a large scale corpus of face images annotated with beauty scores, we resort to the largest aesthetic database available in the literature, i.e. the AVA dataset \\cite{murray2012ava}, created from the photo challenge website dpchallenge.com, that contains more than $250,000$ images annotated with an aesthetic score, a challenge title, and semantic textual tags.\n\nAVA is a unique, rich dataset for visual aesthetics, and therefore a reliable source of data for our purposes.\nHowever, AVA images contain very diverse subjects other than faces.  Moreover, for analysis and classification purposes, we want to collect not only a reliable subset of portrait images, but also some rich information about the portrayed subject and its representation. With this in mind, we design a \\textit{content-aware sampling strategy} on the AVA dataset, based on both metadata-based filtering and face analysis:\n\\\\(1) \\textbf{Enhanched metadata-based filtering.} First, we select  from the AVA database not only the images tagged as ``Portrait'' but  also  all the images whose challenge title contains the words 'Portrait', 'Portraiture' or 'Portraits'. (e.g. \\textit{Portrait Of The Elderly}). A total of 21,719 images are collected at this stage.\n\\\\(2) \\textbf{Face detection-based filtering.}  We use  Face+\n~\\cite{face++} to filter the images collected after metadata-based filtering. We obtain  a subset of 10,141 images for which Face++  detected the presence of one or more faces (in case of multiple faces, we retain the information about the largest one only).\n\\\\(3) \\textbf{Subject properties.} We compute though Face++ basic information about the subject, such as position, orientation, demographics (race, gender, age), coordinates of facial landmarks (eyes, nose and mouth in relative coordinates), presence of smile, presence of glasses, etc. (for a complete list of features see Table \\ref{tab:features}). \n\nFor each of the resulting images, we assign the average aesthetic score (in a 1-10 range) according to the votes provided by the AVA dataset. Figure \\ref{Facefeatures} shows the composition of our dataset, highlighting the distribution, based on gender and other properties estimated by the Face++ detector. About 53\\% of the subjects are classified as female, and 1\/3 of the image corpus shows subjects between 14 and 26 years of age (Fig. \\ref{Facefeatures} (a)). Similar to the AVA dataset, the vast majority of the aesthetic scores lies between 4 and 6, with a peak around the mean, which stands at 5.5.\n\n\\section{Features for Portrait Aesthetic Assessment}\\label{visual_features}\n\\input{tables\/features}\nVisually stunning portrait photographs are often  the result of an artistic process that might not strictly follow  general rules of composition, or fulfill basic quality \trequirements. However, photographic portraiture literature\n \\cite{child2008studio, hurter2007portrait, weiser1999phototherapy} suggests that following some specific photographic principles can help making digital portraits more attractive, ensuring visual appeal and expressiveness. Among the various tips for good portraiture available in literature, we identified 5 main photographic dimensions, namely:\n\\\\\\textbf{Compositional Rules:} arrangement of lines, objects, lights and color, widely used in visual aesthetic literature \\cite{datta, luo2011content}.\n\\\\\\textbf{Scene Semantics:} where has the photo been shot? and which objects co-exist with the subject in the scene?\n\\\\\\textbf{Portrait-Specific Features:}  information about the subject (aspect, soft biometrics, demographics) and its representation (sharpness, illumination, etc.)\n\\\\\\textbf{Basic Quality Metrics:} principles that ensure the correct perception of the signal, without distorting the scene represented.  Rarely used in computational aesthetics, they can be fundamental for high-quality portraiture \\cite{hurter2007portrait}.\n\\\\\\textbf{Fuzzy Properties:}  portrait photographic beauty is related to non-objective properties such as emotions or uniqueness, which are unquantifiable with low level features. \n\nIn this work, we design 5 groups of features that aim at describing various aspects of each of these dimensions using computer vision techniques. \n\n\\subsection{Compositional Rules} \\label{sec:comp}\nAs highlighted in many previous works \\cite{datta, pere, bhattacharya2010framework}, the visual attractiveness of a  picture is strongly influenced by the arrangement of objects in the image, their lighting, their colors,  their perceptibility.\n\nSimilar compositional rules apply to portraits photography. However, since portraits generally focus on a single subject whose essence needs to be captured in the shot, two compositional aspects need particular consideration: lighting and sharpness. The correct illumination of the scene and the detailed representation of the subject ensures both  perceptibility and expressiveness. Given these observations, we design a set of new features that capture essential properties of image lighting and sharpness, and collect a set of existing features for image composition analysis.\n\\vspace{0.5cm}\n\\\\\\emph{Lighting  Features} \\\\\nThe lighting setup is crucial to determine the essence of the portrait. In previous works \\cite{datta,khan2012evaluating, luo2011content}, scene lighting is described using features based on overall image brightness. However, as proved by our results, the raw brightness channel information might not be enough to capture portrait lighting patterns. \n\nWe therefore design a new lighting feature to expose \\textbf{Lighting Patterns} based on  an illumination compensation algorithm originally created for face recognition \\cite{gross2003image}.  Such method considers an image $I$  as a product $I=R(I)\\cdot L(I)$, where $R(I)$ is the  'reflectance' of the image and $L(I)$ is its  ``illuminance'\" i.e. the perceived lighting distribution.\n\nIn order to infer the lighting  pattern of an image, we proceed as follows.\nFor each image, we calculate $L(I)$ and create an illuminance vector $V(I)$ by averaging its  illuminance $L(I)$ over local windows (25x25 subdivision).  Applying k-means clustering on the illuminance vectors of a set of training images, we group the illuminance vectors into 5 Lighting Patterns representing the most common lighting setups in our dataset (See Fig.\\ref{fig:otherfea}).\nFor a new image $J$, we assign its corresponding lighting pattern by looking at the closest cluster to its illuminance vector $V(J)$, and retain the cluster number as the \\textit{Lighting Pattern Feature}.\n\\vspace{0.2cm}\n\\\\\\emph{ Sharpness  Features} \\\\\nThe recognizability and sharpness of the subject is a basic requirement for good portraiture. To analyze the amount of sharpness in the image, we design two new features:\n\\\\\\textbf{Overall Sharpness}: \nSubject movements or camera defocus can affect the overall image sharpness, introducing disturbing blur in particular image regions. We compute the sharpness of a picture by calculating the strength of the edges after applying horizontal and vertical Sobel masks on the image, according to the Tenengrad method (as explained in \\cite{ng2001practical}).\n\\\\\\textbf{Camera Shake}: sometimes camera movements can create an overall blurriness in the image. In order to estimate this particular type of blur, we compute  the ratio between the number of pixels detected to be affected by camera shake and the total number of pixels, according to the camera motion estimation algorithm of Chakrabarti et al. \\cite{chakrabarti2010analyzing}.\n\\vspace{0.2cm}\n\\\\\\emph{Traditional Compoisitional Features}\\\\\nWe collect here a set of features from state-of-the art works that model compositional photographic rules using a computational approach.\n\\\\\\textbf{Color Features.} In order to capture color patterns and their relation with portrait aesthetics, we compute the following features extracted from literature: \\textit{Color names} \\cite{emotions}, \\textit{Hue, Saturation, Brightness (HSV)} \\cite{emotions, datta} , the \\textit{Pleasure, Arousal, Dominance} metrics \\cite{emotions}, the \\textit{Itten Color Histograms} \\cite{emotions}, and the corresponding \\textit{Itten Color Contrasts}: \\cite{emotions} Moreover, we compute 2 contrast metrics: \n\\textit{Contrast (Michelson)} \\cite{Michel}, and a traditional \\textit{Contrast} measure computed as the ratio between the difference of max-min values of the Y channel and the Y average.\n\\\\\\textbf{Spatial Arrangement Features.} The distribution of textures, lines and object in the image space is an important cue for aesthetic and affective image analysis, as proved in \\cite{datta, emotions, pere, interesting}. \nTo analyze spatial layout of objects and shapes in the scene, we  compute first two symmetry descriptors, namely \\textit{ Symmetry (Edges)}\\cite{interesting}, and \\textit{Symmetry (HOG)}, for which we retain the difference between the HOG \\cite{dalal2005histograms} descriptors from left half of the image, and  from the flipped right half. Moreover, we compute 2 new features that describe shapes and their distribution, namely the \\textit{Number of Circles}, and the \\textit{Rule of Thirds}, that, unlike previous works \\cite{datta, bhattacharya2010framework}, determines the rule of thirds by computing the amount of spectral saliency \\cite{hou2007saliency} in the 9 quadrants resulting from a 3x3 division of the image. \n\\\\\\textbf{Texture Features.} Textural features can help analyzing the overall smoothness, order and entropy of the image. We analyze image homogeneity by computing the \\textit{GLCM properties} \\cite{emotions}, the \\textit{Image Order} \\cite{redi2012interestingness}, and the \\textit{Level of Detail} \\cite{emotions}.\n\\subsection{ Semantics and Scene Content}\nAs proved by various works in visual aesthetics \\cite{redi2012interestingness, luo2011content, pere}, the content of the scene and the types of objects placed in the picture substantially influence the aesthetic assessment of pictures. In particular, in the portraiture context, it is important to analyze the setting where the photo has been shot, i.e. objects, scenery and overall harmony of subject with the scene. In order to estimate these properties, we compute an adapted version of the \\textbf{Object bank features} \\cite{li2010object} that retains the maximum probability of a pixel in the image to be part of one of the 208 objects in the Object Bank.\n\\subsection{Basic Quality Metrics}\nIn general, visually appealing portraits are also high-qiuality photographs, i.e. images where the degradation due to image registration or post-processing is not highly perceivable.\nIn order to deeply analyze this dimension, we design some rules to determine the perceived image degradation by looking at simple image metrics, independent of the composition, the content, or its artistic value, namely:\n\\\\\\textbf{Noise}: we compute the amount of camera noise by applying an image denoising algorithm \\cite{buades2005non}, and then computing the distance between the denoised image and the original one. \n\\\\\\textbf{Contrast Quality}: well-contrasted images, i.e. images where the contrast level allows to distinguish the picture shapes without  introducing disturbing over-saturated regions, can be recognized by the uniform distribution of the intensities on the image histogram.  We therefore compute the quality of the contrast  by negating \\footnote{We take the negative of the distance in order to have higher values of this features for higher contrast quality} of the distance between the original image and its contrast-equalized version. \n\\\\\\textbf{Exposure Quality}: the luminance histogram of an overexposed image is skewed towards the right part, while for an underexposed image it is skewed towards the left side. In order to capture this behavior, we convert the image to the YCbCr space,  we compute the skewness of the Y channel histogram over 255 bins. When the skewness is close to zero, the exposure is correct, when below or above zero, the image is under or over exposed. We negate the absolute value of the skewness as exposure balance metric.\n\\\\\\textbf{JPEG Quality}: when too strong, JPEG compression can cause disturbing effects such as blockiness or block smoothness. We implement the objective quality measure for JPEG images proposed by \\cite{wang2002no} and retain the JPEG quality score output by the algorithm.\n\\\\\\textbf{Image Manipulations}: more and more, digital pictures are post-processed after the shooting using editing tools. In order to understand the amount of post-processing applied on the image, we design 2 new quality metrics, inspired by blind image forensics techniques. First, we design a feature to compute the amount of  \\textit{Splicing Manipulation}: we retain the output of an SVM classifier trained with Markov Features \\cite{pevny2010steganalysis} computed on a training set of images annotated as spliced\/not spliced from the CASIA dataset \\cite{casia}  (85\\% accuracy on this set). \nNext, we build a feature to compute the amount of \\textit{Median Filtering Manipulation}, using the algorithm of Yuan et al. \\cite{yuan2011blind}.\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=\\linewidth,natwidth=1745,natheight=336]{fig\/patterns.pdf}\n\\caption{Illuminance Distribution of the 5 Lightning Patterns}\n\\label{fig:otherfea}\n\\vspace{-0.5cm}\n\\end{figure}\n\\subsection{Portrait-Specific Features}\\label{facefeatures}\nIn photographic portraiture, lot of effort should be spent on understanding the subject and its correct representation. Photographic portrait theory \\cite{hurter2007portrait} particularly stresses the importance of the focus, sharpness, lighting and position of the face landmarks (eyes,nose,mouth). \n\nIn order to describe the properties of the subject and its representation, we retain as candidate features all the values extracted automatically by the Face++ api, and we build on top of such values a set of features to deeply describe the face and landmark properties. Overall, the set of Face\/Subject features is as follows:\n\\\\\\textbf{Face++ description:} \\textit{Face Position}, namely x, y relative coordinates, plus relative width and height , \\textit{Face Orientation}, i.e.,  yaw, pitch and roll angle of the head, \\textit{Demographics  } like Race (white, black, asian), Age (in years) and Gender, \\textit{Landmark Coordinates}, namely Right\/Left Eye, Nose and Mouth position in relative coordinates, \\textit{Subject Expression  }, that  estimates wether the subject is smiling or not, and \\textit{Other Face Properties } such as  presence of glasses (none, sunglasses, normal glasses). \\\\\n\\textbf{Landmark Sharpness} for each landmark, we simply compute its sharpness by averaging the gradient magnitude over the landmark region.\n\\\\ \\textbf{Landmark Staitstics:}  for each landmark, we extract its average Hue and Brightness\n\\\\ \\textbf{Face\/Background Contrasts:} similar to the background contrast feature in \\cite{khan2012evaluating}, we analyze here the compositional differences between face region and background region. However,  while Khan et al.  \\cite{khan2012evaluating} simply retain the ratio between face region brightness and image brightness, we perform here a deeper analysis. We consider face ($F$) and background ($B$) as two separate sub-images. We then  compute the \\textit{Lighting Contrast} as the ratio between the average Lightning (see Sec. \\ref{sec:comp}) of F and the average Lightning of B,  the F\/B \\textit{Sharpness Contrast} (Sharpness is computed computed as for the Landmark Properties), and, similarly, the \\textit{Brightness Contrast}.\n\\subsection{ Fuzzy Properties}\nSome artistic traits of photographs cannot be directly captured by low-level features:\nmany times, photographic beauty is related to feelings vehiculated by the image, which not even words can describe. In our work, we try to model some of those 'fuzzy' properties using a computational approach, by re-using existing work on image memorability, originality and affective analysis.\n\\\\\\textbf{Emotion}, is the emotion aroused by the image positive or negative?  We address this question by training an emotion classifier (SVM, 75\\% accuracy) with traditional Compositional Features, using as a groundtruth a mixture of 3 affective dataset \\cite{dan2011geneva, borth2013large, emotions}. We binarize the annotation in order to reflect the positive\/negative trait of the emotion shown. For each image , we retain the emotion score predicted by such classifier as the image emotion feature.\n\\\\\\textbf{Originality} of the image composition is computed by retaining the output of an originality classifier trained with Compositional Features and the Photo.net database from \\cite{datta} (Support Vector Regression (SVR), 4,7\\% MSE ). \n\\\\\\textbf{Memorability} of the image content. We compute this by retaining the output of a memorability classifier trained with the Saliency Moments Features\\cite{sm}, and the  memorability database of Isola et al.  \\cite{isola2011} (SVR, 2\\% MSE). \n\\\\\\textbf{Uniqueness}: as in \\cite{redi2012interestingness}, we estimate the photo uniqueness as the euclidean distance between the average spectrum of the images in a database and the spectrum of each image.\n\n\\section{What Makes  a Portrait Beautiful?}\\label{analysis}\nAmong all the features in Section \\ref{visual_features}, which of them is more discriminative to identify beautiful  portraits in a computational framework? In this Section we explore the relations between the visual features extracted and portrait aesthetic scores, by first analyzing the importance of each feature group described in Sec \\ref{visual_features}, and by then looking at the relevance of each  single feature within dimensions defined.\n\\subsection{Feature Groups for Portrait Aesthetics}\\label{multicorr}\nTo measure the significance of the five feature sets, we perform regression analysis using LASSO~\\cite{tibshirani1996regression}\nfor the different groups of features (i.e. \\textit{Compositional Features}). Once the regression parameter vector is\nlearned, we use compute the Spearman correlation between the predicted scores and the original aesthetic scores. This gives us a multidimensional correlation\nmetric that indicates the relevance of feature group\nfor portrait aesthetic assessment.  \nWe split the data into 5 random partitions, using one of the partitions as the test set\nand the rest as training, and\nlearn regression coefficients to predict the aesthetic scores on the test set using the different groups of features.\n\nAs shown in Fig. \\ref{fig:analysis}(b) all the groups of features  correlate positively with aesthetic scores. As expected, given the importance face of representation for portraiture, the \\textit{Portrait-Specific Features} correlate the most among all the  groups of features proposed, with a correlation of $0.330 \\pm 0.029$.  \nDespite its rich semantic analysis, and the proved effectiveness for scene\nanalysis \\cite{li2010object}, the ObjectBank \\textit{Semantic Features}, with its\n$~190$ feature detectors, are not as predictive, achieving a correlation score of\n$0.211 \\pm 0.022$ in contrast to compositional features which achieve a\ncorrelation score of $0.290 \\pm 0.029$. In comparison to these large\nfeature sets, smaller sets of features such as \\textit{Basic Quality} and \\textit{Fuzzy\nProperties} with $6$ and $4$ dimensions respectively achieve a much lower correlation\nscore for portrait aesthetics assessment as a whole, despite the importance of\nsingle features within the groups.\n\nIn order to calculate the combined predictive power of the whole set of features proposed, we perform similar regression analysis on all features together , i.e. without logical grouping, and look at the behavior of the algorithm  as more and more features are taken into account.\nFigure \\ref{fig:analysis}(d) shows a plot of the Spearman correlation of the feature set as a function of the number of features used and chosen by LASSO.   Using one single feature (\\textit{Right\\_Eye\\_Sharpness}), the Spearman correlation between predicted aesthetic scores and original aesthetic scores is $0.252 \\pm 0.018$. The best correlation score of $0.398 \\pm 0.027$ is obtained taking into account all $~300$ features. However, adding more than $60$\nfeatures shows diminishing returns. The correlation with $60$ features stand at\n$0.37$. The smallest mean square error achieved on the test set stands at\n$0.430 \\pm 0.008$. \n\nTable~\\ref{tab:featuresrank} reports the weight of the features ranked by when they are first picked by LASSO. Also\nreported are the feature category and weights. Notice how all the feature groups appear in the top-10 features, thus confirming the importance of each dimension we consider for portrait aesthetic evaluation, with a predominance of face features. We can  also spot some first insights about the importance of single features: crucial for aesthetic prediction are  landmark sharpness (\\textit{Right\\_Eye} and \\textit{Left\\_Eye}), the Exposure Quality, and the high discriminative ability of the Fuzzy Properties \\textit{Uniqueness}.\n\\begin{table}\n\\scriptsize\n\\centering\n\\begin{tabular}{|c|c|c|c|}\n\\hline\n\\textbf{Rank} & \\textbf{Feature Name} & \\textbf{Feature Group} &\\textbf{Weight} \\\\\\hline\n1 & Left\\_Eye\\_Sharpness & Portrait Features & \\bf{0.061894} \\\\\n2 & Right Eye\\_Sharpness & Portrait Features & \\bf{0.074302} \\\\\n3 & Exposure\\_Balance & Basic Quality & -0.031212 \\\\\n4 & Uniqueness & Fuzzy Properties & \\bf{0.14232} \\\\\n5 & Smiling & Portrait Features & -0.045702 \\\\\n6 & Cluster4\\_Lighnting & Compositional & 0.017803 \\\\\n7 & Fence & Semantics & -0.022525 \\\\\n8 & Hue\\_Inner\\_Quadrant & Compositional & -0.045009 \\\\\n9 & Nose\\_Hue & Portrait Features & -0.03898 \\\\\n10 & Flower & Semantics & 0.026438 \\\\\n\\hline\n\\end{tabular}\n\\caption{Feature ranks based on Lasso regression}\n\\label{tab:featuresrank}\n\\vspace{-1cm}\n\\end{table}\n\\subsection{Single Features for Portrait Aesthetics}\\label{singlecorr}\nTo analyze in a more detailed manner which features correlate most with beautiful portraits, we partition the dataset into 5  subsets, as in Sec. \\ref{multicorr} and  average the Spearman correlation coefficient $\\rho$  between the individual features values and the aesthetic scores of each partition. \n\nIn Figure \\ref{fig:analysis} (a), we report the $\\rho$ coefficients of the features that show higher correlation with portrait aesthetics. We can notice how face sharpness and lighting  are of crucial importance for portrait beauty, as suggested by the Lasso analysis of discriminative features, and by portrait aesthetic literature.  4 out of the top 5 positively-correlated features correspond to the landmark sharpness features. Also, the contrast in sharpness between face and background strongly correlates with portrait beauty ($\\rho=0.12$), as well as the \\textit{Overall\\_Sharpness} metric.   As hypothesized, lighting patterns are also fundamental for a good portrait. This is shown by the positive $\\rho$ of the face\/background lighting contrast feature. Moreover, our analysis shows that there is a relation between image beauty and illumination patterns ( e.g. Clusters 3 has positive $\\rho$, while Cluster 4 has negative  $\\rho$). Overall, our new lighting features show higher relation with beauty than basic brightness features ($\\rho=0.054$ for the \\textit{Average\\_V} features), confirming the need of more complex lighting features for portrait aesthetic evaluation. \nSimilarly, contrast in colors and in gray levels (\\textit{GLCM\\_Contrast} and \\textit{Contrast\\_Michelson}) also show positive correlation with aesthetic scores.\n\nMoreover, negative $\\rho$ values for \\textit{Noise}  and positive correlation with \\textit{GLCM\\_Energy} make us conclude that visually appealing portraits should have a homogeneous, smooth composition without disturbing distortions. We can also see that the amount of \\textit{Median\\_Filtering} is negatively correlated with beauty, showing that too intensive post-processing results in a decrease of the portrait appeal. Surprisingly, \\textit{Exposure\\_Quality} is negatively correlated with beauty, suggesting that playing with over\/under exposure results in more appealing pictures. Moreover, negative $\\rho$ for some Color Names indicates that beautiful portraits tend to have little regions colored with non-skin colors such as green, purple, magenta. We can also notice the good outcome of our attempt of modeling fuzzy properties, given that properties such as \\textit{Originality} and \\textit{Uniqueness} positively correlate with beauty.\n\nIt was very interesting to notice how physical\/demographic properties such as gender, eye color, glasses, age, and race show very low correlation with image beauty, suggesting that any subject, no matter his\/her traits, can be part of a stunning picture, if the photographer is able to grasp the subject's essence.\n\nBy correlating gender properties with other visual features, we could find some side curious insights about portraiture. For example, female pictures tend to be more memorable, as well as  brighter and post-processed, while male tend to be represented with darker colors, and smile less than females.\n\\begin{figure*}[t]\n\\centering\n\\includegraphics[width=\\linewidth,natwidth=1008,natheight=288 ]{fig\/Analysis.pdf}\n\\caption{Analysis of the most relevant features and components for portrait aesthetic prediction (a,b,d). Classification Performances (c).}\n\\label{fig:analysis}\n\\vspace{-0.5cm}\n\\end{figure*}\n\n\\section{Predicting Portrait Beauty}\\label{results}\nIn order to test the effectiveness of our proposed features, and verify the findings of our analysis (see Sec. \\ref{analysis}), we perform 2 different classification experiments. First, we perform a small-scale experiment on the dataset provided in \\cite{khan2012evaluating}, showing the performances of our method and comparing them with the face-specific framework proposed by \\cite{khan2012evaluating}. Then, we design a large-scale classification framework, by looking at the ability  of our features to discriminate between beautiful\/non-beautiful pictures, using the large-scale dataset we built in Sec. \\ref{datasets}. We compare the classification performances of a framework based on our different groups of features with the one of a generic aesthetic classifier, i.e. based on traditional compositional features and trained on images with diverse subjects.\n\n \\subsection{Small-Scale Experiment}\nThe work that more closely relates to ours is the portrait aesthetic framework from Khan et al. \\cite{khan2012evaluating}: they design face-specific features and computes their effectiveness on a publicly available small-scale dataset of 150 pictures. \n\\input{tables\/result_small}\nIn order to test the performances of our approach, we compute the visual features in Sec. \\ref{visual_features} on the dataset from Khan et al. \\cite{khan2012evaluating} and we prove their effectiveness by using the same experimental setup, i.e. binarization of scores based on median, 10-folds cross validation on an SVM classifier in Weka, and average accuracy as evaluation metric. For fair comparison, we first evaluate the classification performances on our portrait-specific features only (see Sec. \\ref{facefeatures}), reporting results with and without feature selection in Table \\ref{tab:small_experiments}. Our group of portrait features alone outperforms the system in \\cite{khan2012evaluating}. Moreover, when we use all the features proposed in this work for the same classification task, we reach even higher classification accuracy, observing a substantial improvement of the performances compared to our baseline (and similar works such as the one from Li et al. \\cite{li2010aesthetic}).\n\\subsection{Large-Scale Aesthetic Categorization}\nWe now test the proposed approach for aesthetic classification on a large-scale, using the dataset of Sec. \\ref{datasets}.\nTo classify the images as ``Beautiful\" and ``Non-beautiful\", we use the binaries the average AVA \nscores it by labeling as positive any image with a score\ngreater than the mean user score ($5.55$). Similar to \\cite{murray2012ava}, we learn a SVM classifier using the publicly available libSVM\npackage. For this, the dataset is randomly divided into 5 partitions, as in Sec. \\ref{analysis}, and a SVM\nclassifier is learned per partitions. We use RBF kernel where the $\\gamma$ parameter is set\nto $1\/n$ where $n$ is the number of features. The cost parameter $C$ is obtained\nusing 10-fold cross-validation. All features are standardized to be zero mean\nand unit variance. \n\nFig \\ref{fig:analysis} (c)  shows the average classification accuracy on the test set for each group of features. As we can see, our framework benefits from the combination of diverse features, since the best performance is given by all features combined with early fusion, ($64.24\\% \\pm 1.76$) . Moreover, as expected by our analysis, we confirm that the classifier based on our rich portrait features outperforms the  classifiers based on the other groups of features, suggesting that detailed information of face properties and landmarks is more discriminative for portrait classification than traditional compositional features.\n\nResults reported in \\cite{murray2012ava, interesting} proved that a classifier trained on non-specific images performs better than a portrait-specific framework.  To prove the importance of building a portrait-specific framework, we compare our results with a baseline classifier built with traditional compositional features only (as in Sec. \\ref{sec:comp}), and trained on the dataset used in \\cite{pere}, namely  a database of  images belonging to 7 different categories, including ``Portraiture\",``Flower\", etc. and annotated with the corresponding aesthetic scores from DPchallenge.com (same source as our dataset, same score range). Unlike the findings in \\cite{murray2012ava, interesting}, we confirm the hypothesis that portraits need a separate computational framework for aesthetic assessment, showing that all the classifiers based on our proposed features perform better than this baseline (with all features, the improvement is more than 16\\%).\n\nAs in ~\\cite{murray2012ava}, we also performed SVM classification by introducing a $\\delta$\nparameter to discard ambiguous images from the training set (keeping all the\nimages in the test set). The $\\delta$ parameter was ranged from $0.1$ to $1.0$,\nbut unlike~\\cite{murray2012ava} we did not experience any increase in the classification\naccuracy. However, the performance with the $\\delta = 0.5$ is similar to when\n$\\delta = 0.0$, implying that the ambiguous images do not help for the task of\nclassification and can be discarded to speed up the learning time.  \n\n\\section{Conclusions}\\label{conclusions}\nIn this paper, we presented a complete framework for large-scale portrait aesthetic assessment based on visual features. We procured a dataset of digital portraits annotated with aesthetic scores and other information regarding traits\/demographics of the subjects in the portraits. We designed a set of discriminative visual features based on portrait photography literature. We analyzed the importance of each feature for portrait beauty, showing that rich facial features play a significant role in guiding the portrait aesthetics, and that the perceived portrait beauty is largely independent of the demographic characteristics of the subject. Finally, we built a classifier that is able to successfully distinguish between beautiful and non-beautiful portraits.\n\nIn our future work, we plan to broaden our framework by extending our database to include portrait images 'in the wild', exploring portrait aesthetics with a more challenging context.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\n\n\n\\section{Introduction}\\label{sec1}\nQuantum teleportation is a process based on classical communication\nthat transmits the quantum information from a location to another\nwith the help of shared quantum entanglement between the sender and\nreceiver. This process is a technique for transporting the state of\nan atom or photon to the remote recipient even in the absence of\nquantum communication channels connecting the sender of the quantum\nstate (called  Alice) to the recipient (called Bob) \\cite{nel}. The\noriginal protocol of this process was firstly introduced by Bennett\n\\emph{et al.} using the Einstein-Podolsky-Rosen (EPR) state as the\nquantum channel \\cite{Bennett93}. Quantum teleportation using two\nqubit systems is discussed also in \\cite{chakra10,liang13}.\n\nBecause of the strong connection between quantum entanglement and\nquantum teleportation, the usage of multiparticle entangled quantum\nstates other than two-particle entangled states for quantum\nteleportation has been the subject of various investigations\n\\cite{wang07,paul11}. In particular, quantum teleportation with\nthree-qubit GHZ state and W state is studied in\nRefs.~\\cite{karl98,wang10,gorb03,joo03,agra06,Hu1}. The\npossibility of teleportation of an unknown qubit using four-particle\nGHZ state is discussed in Ref.~\\cite{pati00}. It is also shown that\nthe state in the form $|{\\psi}\\rangle = \\frac{1}{\\sqrt{2}} \\left(\n|00q_1\\rangle + |11q_2\\rangle \\right)$ allows perfect two-party\nteleportation in which $|q_1\\rangle$ and $|q_2\\rangle$ are arbitrary\nnormalized single qubit states \\cite{jung07}.\n\nIn quantum information theory and quantum computation, fidelity is a\nmeasure to quantify the closeness of two quantum states \\cite{nel} and\nis closely related to quantum entanglement \\cite{4}, quantum phase\ntransitions \\cite{6,7,8}, and quantum chaos \\cite{5}. Fidelity can\nbe also used to quantify how much quantum information is lost due to\nnoisy channel between initial and final states. This reduction of fidelity is usually due to\nthe interaction of quantum states with environment which results in\nimperfect teleportation. Thus, the coherence of the entangled state\nmay be lost and it becomes a mixed state. Some efforts have been\nperformed in this direction to realize effective factors which cause\nthis phenomenon \\cite{Bennett93,horodecki99,banas01,jung08-1}. For\ninstance, Bennett \\emph{et al.} showed that the fidelity of\nteleportation and the range of accurately teleported states reduce in\nthe less entangled quantum channels \\cite{Bennett93}.\n\nThe existence of noise is an unavoidable property in quantum\nteleportation process which results is decoherence of states and the\nreduction of fidelity \\cite{0h02,han08,bhak08}. In particular, Oh\n\\emph{et al.} using a pair of EPR states showed that the average\nfidelity and the range of teleported states depend on the type of\nthe noise that acts on the quantum channel and confirmed Bennett\n\\emph{et al.} results \\cite{0h02}. They solved analytically and\nnumerically the master equation with Lindblad structure and found\nthe fidelity as a function of decoherence time and angles of an\nunknown teleported state.\n\nNote that analytically solving the Lindblad equation in the presence of the noise is\nnot a trivial task in general. Indeed, for multiparticle systems one\nneeds to solve many coupled differential equations that involve\ntedious computation. For example, for three-particle GHZ state\n(3GHZ) the master equation reduces to 8 diagonal coupled\ndifferential equations and 28 off-diagonal coupled differential\nequations \\cite{jung08-2}. The situation is even worse for\nfour-particle GHZ state (4GHZ) that involves 16 diagonal coupled\ndifferential equations and 120 off-diagonal coupled differential\nequations.\n\nIn this paper, we analytically solve the master equation for\n$n$-particle GHZ state ($n\\in\\{4,5,6\\}$) through various noisy\nchannels. The number of coupled differential equations for each case\nis considerably reduced by using a proper ansatz for the density\nmatrix.  The ansatz is determined from the temporal evolution of the\ninitial state of the system. We obtain the fidelity of teleportation\nand the average fidelity of teleportation that depend on the type of\nthe noisy channel and compare the results with three-particle GHZ\nstate. The goal of this paper is to find out which state is better\n(loses less quantum information) in the teleportation process with\nnoisy channels. Therefore, although various noisy channels were\nstudied in Ref.~\\cite{0h02}, we discuss noisy channels which cause\nthe quantum channels to be mixed to compare $n$GHZ states in the\nprocess of teleportation.\n\nThe organization of this paper is as follows: Section \\ref{sec2} is\ndevoted to general framework used to evaluate the two-party quantum\nteleportation circuit. In Sec.~\\ref{sec3}, we analytically solve the\nLindblad equation where the quantum channel is a four-particle GHZ\nstate, i.e., $| \\mbox{4GHZ}\\rangle$. We transmit 4GHZ state through\nisotropic and Pauli noises and compute the fidelity of\nteleportation. Moreover, we compare the robustness of 4GHZ state\nwith 3GHZ state in the noisy channels. Solving the master equation\nfor 5GHZ and 6GHZ states when Lindblad operators are in $x$ and $z$\ndirections is the subject of Secs.~\\ref{sec4} and \\ref{sec5},\nrespectively. We present our conclusions in Sec.~\\ref{sec6}.\n\n\\begin{figure}[t]\n\\input{fig2.tex}\n\\caption{\\label{fig1}A circuit for quantum teleportation through\nnoisy channels with EPR state. The two top lines belong to Alice and\nthe bottom line to Bob. $M$ denotes measurement and the dotted box\nrepresents noisy channel. The Lindblad operator is turned on inside\nthe dotted box.}\n\\end{figure}\n\n\n\\section{GHZ state, fidelity, and Lindblad equation}\\label{sec2}\nFor $n$-particle system, an $n$GHZ state is a quantum state defined\nas follows\n\\begin{equation}\n|n\\mbox{GHZ}\\rangle=\\frac{1}{\\sqrt{2}} \\left( |0\\rangle^{\\otimes n}\n+ |1\\rangle^{\\otimes n} \\right),\n\\end{equation}\nwhere $n>2$. Note that, teleportation with $|\\mbox{EPR}\\rangle$ through\nnoisy channels is depicted in Fig.~\\ref{fig1} and it is discussed in\nRef.~\\cite{0h02}. Also, teleportation of 3GHZ state through various\nnoisy channels has been previously studied in Ref.~\\cite{jung08-2}.\nHere, we are interested to investigate the teleportation process for\n$n$GHZ state through noisy channels for $n\\in\\{4,5,6\\}$. For this\npurpose, we need to solve the master equation with Lindblad form\n\\cite{lind76}\n\\begin{equation}\\label{Lindblad}\n\\frac{\\partial \\rho}{\\partial t} = -\\frac{i}{\\hbar} [H_S, \\rho] +\n\\sum_{i, \\alpha} \\left(L_{i,\\alpha} \\rho L_{i,\\alpha}^{\\dagger} -\n\\frac{1}{2} \\left\\{ L_{i,\\alpha}^{\\dagger} L_{i,\\alpha}, \\rho\n\\right\\} \\right),\n\\end{equation}\nin which $L_{i,\\alpha} = \\sqrt{\\kappa_{i,\\alpha}}\n\\sigma^{(i)}_{\\alpha}$ denote Lindblad operators that describe\ndecoherence and act on the $i$th qubit. Also,\n$\\sigma^{(i)}_{\\alpha}$ are the Pauli spin matrices of the $i$th\nqubit with $\\alpha =\\{ x,y,z\\}$, $\\kappa_{i,\\alpha}$ is the\ndecoherence rate, and $H_S$ is the Hamiltonian of the system.\n\n\\begin{figure}\n\\input{fig1.tex}\n\\caption{\\label{fig2}\n   A circuit for quantum teleportation through noisy channels with 4GHZ state.\n   The four top lines belong to Alice and the bottom line to Bob.\n   $M$ denotes measurement and the dotted box represents noisy channel.\n   The Lindblad operator is turned on inside the dotted box.}\n\\end{figure}\n\nThe unknown state to be teleportated can be written as a Bloch vector on a\nBloch sphere\n\\begin{equation}\n\\label{unknown} |\\psi_{\\text{in}}\\rangle = \\cos\n\\left(\\frac{\\theta}{2}\\right) e^{i \\phi \/ 2} |0\\rangle + \\sin\n\\left(\\frac{\\theta}{2}\\right) e^{-i \\phi \/ 2} |1\\rangle,\n\\end{equation}\nwhere $\\theta$ and $\\phi$ denote the polar and azimuthal angles,\nrespectively. Fig.~\\ref{fig2} shows a quantum teleportation circuit\nthrough noisy channels with 4GHZ state in which the input state\ninvolves five qubits as the product state of\n$|\\psi_{\\text{in}}\\rangle $ and $|\\mbox{4GHZ}\\rangle$. The four top\nlines (qubits) belong to Alice and bottom one belongs to Bob. The\ndifference of this circuit with the teleportation circuit for EPR\nstate (Fig.~\\ref{fig1}) is the presence of two more controlled-NOT\n$\\left(\\mbox{CNOT} \\right)$ gates between $\\psi_{\\text{in}}$ and\n$\\mbox{4GHZ}$ states. After measurement of the top four qubits, Bob\ngets the teleported state $|\\psi_{\\text{out}}\\rangle$. It is\nconvenient to describe the teleportation in terms of the density\noperator\n\\begin{equation}\n\\label{out1} \\rho_{\\text{out}} = \\mbox{Tr}_{1,2,3,4} \\left[\nU_{\\mbox{\\tiny tel}} \\rho_{in} \\otimes \\varepsilon\n(\\rho_{4\\mbox{\\tiny GHZ}}) U_{\\mbox{\\tiny tel}}^{\\dagger} \\right],\n\\end{equation}\nwhere $\\rho_{\\text{in}} = |\\psi_{\\text{in}}\\rangle \\langle\n\\psi_{\\text{in}}|$ is density matrix of the unknown initial state\nand $\\varepsilon(\\rho_{4\\mbox{\\tiny GHZ}})$ is the density matrix\nafter transmission through noisy channel which is given by the\nLindblad equation. In fact, $\\varepsilon$ is a quantum operation\nthat maps $\\rho_{4\\mbox{\\tiny GHZ}}$ to $\\varepsilon\n(\\rho_{4\\mbox{\\tiny GHZ}})$ because of noisy channel and\n$\\rho_{4\\mbox{\\tiny GHZ}}=|\\mbox{4GHZ}\\rangle\\langle\\mbox{4GHZ}|$.\nMoreover, $U_{\\mbox{\\tiny tel}}$ is the unitary operator\ncorresponding to the quantum circuit and $\\mbox{Tr}_{1,2,3,4}$ is\npartial trace over first four qubits which belong to Alice.\n\nFidelity can be used as a tool to measure how much information is\nlost or preserved through noisy quantum channels in quantum\nteleportation process. It can be written as the overlap between the\ninput state $|\\psi_{in}\\rangle$ and the density operator for the\nteleported state $|\\rho_{\\text{out}} \\rangle$,\n\\begin{equation}\n\\label{fidelity} F = \\langle \\psi_{\\text{in}} | \\rho_{\\text{out}} |\n\\psi_{\\text{in}}\\rangle,\n\\end{equation}\nthat depends on an input state and the type of noise. For the\nperfect teleportation the fidelity is equal to unity. Also, $1-F$\nindicates how much information is lost through the teleportation\nprocess. For all possible unknown input states,\nthe average fidelity is given by\n\\begin{equation}\n\\overline{}\\label{average} \\overline{F} = \\frac{1}{4 \\pi}\n\\int_{0}^{2 \\pi} \\mathrm{d} \\phi \\int_{0}^{\\pi} \\mathrm{d} \\theta\n\\sin \\theta F(\\theta, \\phi).\n\\end{equation}\nSimilarly, we find the unitary operator, fidelity and average\nfidelity for 5GHZ and 6GHZ states in the following sections.\n\n\\section{Four-qubit GHZ state with noisy channels}\\label{sec3}\nIn this section, we analytically solve the Lindblad equation,\nEq.~(\\ref{Lindblad}), for 4GHZ state through various noisy channels.\nFirst, consider $(L_{2,x},L_{3,x},L_{4,x},L_{5,x})$ noise channel\nwith $\\kappa_{2,x}=\\kappa_{3,x}=\\kappa_{4,x}=\\kappa_{5,x}=\\kappa$\nthat acts on 4GHZ state. Also, here and throughout the paper we\nassume $H_S=0$.\n\nFor this case, the Lindblad equation involves 16 diagonal and 120\noff-diagonal coupled linear differential equations which make this\nequation difficult to be solved analytically. To overcome this problem,\nwe find the time evolution of the density matrix for infinitesimal\ntime interval $\\delta t$ using the Lindblad equation as\n\\begin{equation}\\label{del}\n\\rho(\\delta t)= \\rho(0) + \\left[ \\sum_{i, \\alpha} \\left(L_{i,\\alpha}\n\\rho(0) L_{i,\\alpha}^{\\dagger} \\right) - \\frac{1}{2} \\left\\{\nL_{i,\\alpha}^{\\dagger} L_{i,\\alpha}, \\rho(0) \\right\\} \\right]\\delta\nt,\n\\end{equation}\nwhere\n\\begin{equation}\\label{ro0}\n\\rho(0)=|\\mbox{4GHZ}\\rangle\\langle\\mbox{4GHZ}|=\\frac{1}{2}\\left[|0\\rangle^{\\otimes\n4} \\langle 0 |^{\\otimes 4} + |0\\rangle^{\\otimes 4}\\langle\n1|^{\\otimes 4}+|1\\rangle^{\\otimes 4} \\langle 0 |^{\\otimes 4} +\n|1\\rangle ^{\\otimes 4}\\langle 1|^{\\otimes 4}\\right].\n\\end{equation}\nSubstituting $\\rho(0)$ in Eq.~(\\ref{del}) results in\n\\begin{eqnarray}\n\\varepsilon(\\rho_{4\\mbox{\\tiny GHZ}})\\Big|_{t=\\delta t} =\n\\frac{1}{2}{\\left(\n\\begin{smallmatrix}\n1 -4 \\kappa \\delta t & 0 & 0 & 0 & 0 & 0 &0 & 0 & 0 & 0 & 0 & 0 & 0 & 1-4 \\kappa \\delta t   \\\\\n0 & \\kappa \\delta t & 0 & 0& 0 & 0 & 0& 0 & 0 & 0 & 0 & 0 &  \\kappa \\delta t & 0   \\\\\n0 & 0 & \\kappa \\delta t & 0& 0 & 0& 0 & 0 & 0 & 0 & 0 & \\kappa \\delta t & 0 & 0   \\\\\n0 & 0 & 0 & {\\footnotesize \\textcircled{\\tiny 1}} & 0 & 0 & 0& 0 & 0 & 0& {\\footnotesize \\textcircled{\\tiny 1}} & 0 & 0 & 0 \\\\\n0 & 0 & 0& 0 & \\kappa \\delta t  & 0& 0 & 0 & 0 &  \\kappa \\delta t & 0& 0 & 0 & 0   \\\\\n0 & 0 & 0& 0& 0 & {\\footnotesize \\textcircled{\\tiny 2}} & 0 & 0 & {\\footnotesize \\textcircled{\\tiny 2}} & 0& 0& 0 & 0 & 0 \\\\\n0 & 0 & 0& 0& 0 &0  & \\kappa \\delta t &  \\kappa \\delta t & 0 & 0 & 0& 0& 0 & 0  \\\\\n0 & 0 & 0& 0& 0 &0  & \\kappa \\delta t & \\kappa \\delta t & 0 & 0 & 0& 0& 0 & 0 \\\\\n0 & 0 & 0& 0& 0 & {\\footnotesize \\textcircled{\\tiny 2}}  & 0& 0 & {\\footnotesize \\textcircled{\\tiny 2}} & 0& 0& 0 & 0 & 0   \\\\\n0 & 0 & 0& 0 & \\kappa \\delta t  & 0& 0 & 0 & 0 &  \\kappa \\delta t & 0& 0 & 0 & 0   \\\\\n0 & 0 & 0 & {\\footnotesize \\textcircled{\\tiny 1}} & 0 & 0 & 0& 0 & 0 & 0& {\\footnotesize \\textcircled{\\tiny 1}} & 0 & 0 & 0   \\\\\n0 & 0 &  \\kappa \\delta t & 0& 0 & 0& 0 & 0 & 0& 0 & 0 &  \\kappa \\delta t & 0 & 0   \\\\\n0 &  \\kappa \\delta t & 0 & 0& 0 & 0 & 0& 0 & 0 & 0 & 0 & 0 & \\kappa \\delta t & 0   \\\\\n1 -4 \\kappa \\delta t & 0 & 0 & 0 & 0 & 0 & 0& 0 & 0 & 0 & 0 &\n0 & 0 & 1 -4 \\kappa \\delta t \\end{smallmatrix}\\right),}\n\\end{eqnarray}\nwhere \\textcircled{\\emph{n}} denotes $n$ diagonal zeros. Now, because of the form of the density matrix at $t=\\delta t$, we\nuse the following ansatz for the density matrix for all times\n\\begin{eqnarray}\\label{mat1}\n\\varepsilon(\\rho_{4\\mbox{\\tiny GHZ}}) = { \\left(\n\\begin{smallmatrix}\na & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & a   \\\\\n0 & b & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & b & 0   \\\\\n0 & 0 & b & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & b & 0 & 0   \\\\\n0 & 0 & 0 & c & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & c & 0 & 0 & 0   \\\\\n0 & 0 & 0 & 0 & b & 0 & 0 & 0 & 0 & 0 & 0 & b & 0 & 0 & 0 & 0   \\\\\n0 & 0 & 0 & 0 & 0 & c & 0 & 0 & 0 & 0 & c & 0 & 0 & 0 & 0 & 0   \\\\\n0 & 0 & 0 & 0 & 0 & 0 & c & 0 & 0 & c & 0 & 0 & 0 & 0 & 0 & 0   \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & b & b & 0 & 0 & 0 & 0 & 0 & 0 & 0   \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & b & b & 0 & 0 & 0 & 0 & 0 & 0 & 0   \\\\\n0 & 0 & 0 & 0 & 0 & 0 & c & 0 & 0 & c & 0 & 0 & 0 & 0 & 0 & 0   \\\\\n0 & 0 & 0 & 0 & 0 & c & 0 & 0 & 0 & 0 & c & 0 & 0 & 0 & 0 & 0   \\\\\n0 & 0 & 0 & 0 & b & 0 & 0 & 0 & 0 & 0 & 0 & b & 0 & 0 & 0 & 0   \\\\\n0 & 0 & 0 & c & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & c & 0 & 0 & 0   \\\\\n0 & 0 & b & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & b & 0 & 0   \\\\\n0 & b & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & b & 0   \\\\\na & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & a\n\\end{smallmatrix} \\right).}\n\\end{eqnarray}\nInserting this matrix in the Lindblad equation, Eq.~({\\ref{Lindblad}}),\ngives us a set of three coupled differential equations\n\\begin{eqnarray}\\label{diff1}\n\\left\\{\n\\begin{array}{l}\n\\dot{a}(t) = 4k\\Big(b(t)-a(t)\\Big),\\\\\n\\dot{b}(t) = k\\Big(a(t)-4b(t)+3c(t)\\Big),\\\\\n\\dot{c}(t)=4k\\Big(b(t)-c(t)\\Big),\n\\end{array}\\right.\n\\end{eqnarray}\nsubject to the initial conditions $a(0)=1\/2$ and $b(0)=c(0)=0$ (see\nEq.~(\\ref{ro0})). The solutions are readily given by\n\\begin{eqnarray}\n\\label{elements} \\left\\{\n\\begin{array}{l}\na(t) =\\frac{1}{16}\\left( 1 + 6 e^{-4 \\kappa t} +  e^{-8 \\kappa t}\\right),\\\\\nb(t) =\\frac{1}{16}\\left( 1 - e^{-8 \\kappa t}\\right),\\\\\nc(t) =\\frac{1}{16}\\left( 1 - 2 e^{-4 \\kappa t} + e^{-8 \\kappa\nt}\\right).\n\\end{array}\\right.\n\\end{eqnarray}\nIn fact, the infinitesimal temporal behavior of the density matrix\nhelped us to properly suggest the solution and consequently reduced\n136 coupled differential equations to three coupled differential\nequations which are readily solved. It is now easy to check that\n$\\varepsilon(\\rho_{4\\mbox{\\tiny GHZ}})$, Eq.~(\\ref{mat1}), exactly\nsatisfies the Lindblad equation, Eq.~({\\ref{Lindblad}}), and the validity\nof the ansatz is verfied.\n\nHaving $\\varepsilon(\\rho_{4\\mbox{\\tiny GHZ}})$ and $U_{\\mbox{\\tiny\ntel}}$ which can be read off from Fig.~\\ref{fig2}, it is\nstraightforward to compute $\\rho_{out}$. Thus, the fidelity reads\n\\begin{eqnarray}\n\\label{xfidelity} & &F(\\theta, \\phi) = \\frac{1}{2} \\left[(1 + \\sin^2\n\\theta \\cos^2 \\phi) + e^{-4 \\kappa t} (\\cos^2 \\theta + \\sin^2 \\theta\n\\sin^2 \\phi) \\right],\n\\end{eqnarray}\nand the average fidelity is given by\n\\begin{eqnarray}\n\\label{xfbar} \\overline{F} = \\frac{2}{3} + \\frac{1}{3} e^{-4 \\kappa\nt}.\n\\end{eqnarray}\n\nNow consider $(L_{2,y},L_{3,y},L_{4,y},L_{5,y})$ and assume\n$\\kappa_{2,y}=\\kappa_{3,y}=\\kappa_{4,y}=\\kappa_{5,y}=\\kappa$.\nSimilar to the previous case, using the infinitesimal time evolution\nof the density matrix\n\\begin{eqnarray}\n\\varepsilon(\\rho_{4\\mbox{\\tiny GHZ}})\\Big|_{t=\\delta t} =\n\\frac{1}{2}{\\left(\n\\begin{smallmatrix}\n1 -4 \\kappa \\delta t & 0 & 0 & 0 & 0& 0 & 0 & 0& 0 & 0 & 0 & 0 & 0 & 1-4 \\kappa \\delta t   \\\\\n0 & \\kappa \\delta t & 0 & 0 & 0& 0 & 0 & 0 & 0 & 0 & 0 & 0 &  -\\kappa \\delta t & 0   \\\\\n0 & 0 & \\kappa \\delta t & 0& 0 & 0& 0 & 0 & 0& 0 & 0 & -\\kappa \\delta t & 0 & 0   \\\\\n0 & 0 & 0 & {\\footnotesize \\textcircled{\\tiny 1}} & 0 & 0 & 0& 0 & 0 & 0& {\\footnotesize \\textcircled{\\tiny 1}} & 0 & 0 & 0 \\\\\n0 & 0 & 0& 0 & \\kappa \\delta t  & 0& 0 & 0 & 0&  -\\kappa \\delta t & 0& 0 & 0 & 0   \\\\\n0 & 0 & 0& 0& 0 & {\\footnotesize \\textcircled{\\tiny 2}} & 0 & 0 & {\\footnotesize \\textcircled{\\tiny 2}}& 0& 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0& 0& 0 &0  & \\kappa \\delta t &  -\\kappa \\delta t & 0 & 0& 0& 0 & 0 & 0   \\\\\n0 & 0 & 0& 0& 0 &0  & -\\kappa \\delta t & \\kappa \\delta t & 0 & 0& 0& 0 & 0 & 0   \\\\\n0 & 0 & 0& 0& 0 & {\\footnotesize \\textcircled{\\tiny 2}} & 0 & 0 & {\\footnotesize \\textcircled{\\tiny 2}} & 0& 0& 0 & 0 & 0 \\\\\n0 & 0 & 0& 0 & -\\kappa \\delta t  & 0& 0 & 0 & 0& \\kappa \\delta t & 0& 0 & 0 & 0   \\\\\n0 & 0 & 0 & {\\footnotesize \\textcircled{\\tiny 1}}& 0 & 0 & 0& 0 & 0 & 0& {\\footnotesize \\textcircled{\\tiny 1}} & 0 & 0 & 0 \\\\\n0 & 0 &  -\\kappa \\delta t & 0& 0 & 0& 0 & 0 & 0& 0 & 0 &  \\kappa \\delta t & 0 & 0 \\\\\n0 &  -\\kappa \\delta t & 0 & 0& 0 & 0 & 0& 0 & 0 & 0 & 0 & 0 & \\kappa \\delta t & 0 \\\\\n1 -4 \\kappa \\delta t & 0 & 0 & 0 & 0 & 0 & 0& 0 & 0 & 0 & 0\n&0 & 0 & 1 -4 \\kappa \\delta t\n\\end{smallmatrix}\\right),}\n\\end{eqnarray}\nwe take the following ansatz\n\\begin{eqnarray}\n \\varepsilon(\\rho_{4\\mbox{\\tiny GHZ}}) =\n \\left(\n\\begin{smallmatrix}\na & 0 & 0 & 0 & 0 & 0 & 0& 0 & 0 & 0& 0 & 0 & 0 & 0 & 0 & a   \\\\\n0 & b& 0 & 0& 0 & 0 & 0& 0 & 0 & 0 & 0 & 0 & 0 & 0 & -b & 0   \\\\\n0 & 0 & b & 0& 0 & 0& 0 & 0 & 0& 0 & 0 & 0 & 0 & -b & 0 & 0   \\\\\n0 & 0 & 0 & c & 0 & 0& 0 & 0 & 0& 0 & 0 & 0& c & 0 & 0 & 0   \\\\\n0 & 0 & 0& 0 & b  & 0& 0 & 0 & 0& 0 & 0 & -b & 0& 0 & 0 & 0   \\\\\n0 & 0 & 0& 0& 0 & c  & 0& 0 & 0 & 0 & c & 0& 0& 0 & 0 & 0   \\\\\n0 & 0 & 0& 0& 0&0 & c  & 0& 0 & c & 0& 0& 0 & 0& 0 & 0   \\\\\n0 & 0 & 0& 0& 0 &0  & 0& b & -b & 0 & 0 & 0& 0& 0 & 0 & 0   \\\\\n0 & 0 & 0& 0& 0 &0  & 0& -b & b & 0 & 0 & 0& 0& 0 & 0 & 0   \\\\\n0 & 0 & 0& 0& 0&0 & c  & 0& 0 & c & 0& 0& 0 & 0& 0 & 0   \\\\\n0 & 0 & 0& 0& 0 & c  & 0& 0 & 0 & 0 & c & 0& 0& 0 & 0 & 0   \\\\\n0 & 0 & 0& 0 &- b  & 0& 0 & 0 & 0& 0 & 0 & b & 0& 0 & 0 & 0   \\\\\n0 & 0 & 0 & c & 0 & 0& 0 & 0 & 0& 0 & 0 & 0& c & 0 & 0 & 0   \\\\\n0 & 0 &- b & 0& 0 & 0& 0 & 0 & 0& 0 & 0 & 0 & 0 & b & 0 & 0   \\\\\n0 & -b & 0 & 0& 0 & 0 & 0& 0 & 0 & 0 & 0 & 0 & 0 & 0 & b & 0   \\\\\na & 0 & 0 & 0 & 0 & 0 & 0& 0 & 0 & 0& 0 & 0 & 0 & 0 & 0 & a\n\\end{smallmatrix}\n\\right).\n\\end{eqnarray}\nInserting this matrix in the Lindblad equation, Eq.({\\ref{Lindblad}}),\ngives the previous set of coupled differential equations, Eq.~(\\ref{diff1}), and consequently the solutions agree with\nEq.~(\\ref{elements}). For this case the fidelity becomes\n\\begin{eqnarray}\n\\label{yfidelity} & &F(\\theta, \\phi) = \\frac{1}{2} \\left[1 +( \\sin^2\n\\theta \\sin^2 \\phi + \\cos^2 \\theta) e^{-4 \\kappa t} + \\sin^2 \\theta\n\\cos^2 \\phi e^{-8 \\kappa t} \\right],\n\\\\   \\nonumber\n\\end{eqnarray}\nand the average fidelity reads\n\\begin{eqnarray}\n\\label{yfbar} \\overline{F} = \\frac{1}{2} +\\frac{1}{3}  e^{-4 \\kappa\nt} +\\frac{1}{6} e^{-8 \\kappa t}.\n\\end{eqnarray}\n\n\nFor the third case consider $(L_{2,z},L_{3,z},L_{4,z},L_{5,z})$ and\nassume $\\kappa_{2,z}=\\kappa_{3,z}=\\kappa_{4,z}=\\kappa_{5,z}=\\kappa$.\nThe infinitesimal time evolution of the density matrix gives\n\\begin{eqnarray}\n\\varepsilon(\\rho_{4\\mbox{\\tiny GHZ}})\\Big|_{t=\\delta t} =\n\\frac{1}{2} \\left( |0\\rangle^{\\otimes 4} \\langle 0 |^{\\otimes 4} +\n|1\\rangle ^{\\otimes 4}\\langle 1| ^{\\otimes 4}\\right) +\n\\frac{1-8\\kappa \\delta t}{2} \\left(|0\\rangle ^{\\otimes 4} \\langle 1|\n^{\\otimes 4}+ |1\\rangle ^{\\otimes 4}\\langle 0 |^{\\otimes 4} \\right).\n\\end{eqnarray}\nSo the ansatz is\n\\begin{eqnarray}\n\\varepsilon(\\rho_{4\\mbox{\\tiny GHZ}}) = a \\left( |0\\rangle^{\\otimes\n4} \\langle 0 |^{\\otimes 4} + |1\\rangle ^{\\otimes 4}\\langle 1|\n^{\\otimes 4}\\right) + b\\left(|0\\rangle ^{\\otimes 4} \\langle 1|\n^{\\otimes 4}+ |1\\rangle ^{\\otimes 4}\\langle 0 |^{\\otimes 4} \\right).\n\\end{eqnarray}\nInserting this matrix in the Lindblad equation, Eq.~({\\ref{Lindblad}}),\nresults in\n\\begin{eqnarray}\\label{diff2}\n\\left\\{\n\\begin{array}{l}\n\\dot{a}(t) = 0,\\\\\n\\dot{b}(t) =-8k\\, b(t) ,\n\\end{array}\\right.\n\\end{eqnarray}\nsubject to the initial condition $a(0)=b(0)=1\/2$. The solution is\n\\begin{equation}\n\\label{zmatrix} \\varepsilon(\\rho_{4\\mbox{\\tiny GHZ}}) = \\frac{1}{2}\n\\left( |0\\rangle^{\\otimes 4} \\langle 0 |^{\\otimes 4} + |1\\rangle\n^{\\otimes 4}\\langle 1| ^{\\otimes 4}\\right) + \\frac{1}{2} e^{-8\n\\kappa t} \\left(|0\\rangle ^{\\otimes 4} \\langle 1| ^{\\otimes 4}+\n|1\\rangle ^{\\otimes 4}\\langle 0 |^{\\otimes 4} \\right).\n\\end{equation}\nAlso, the fidelity and its average read\n\\begin{eqnarray}\n\\label{z,f,fbar}\n\\begin{array}{l}\\displaystyle\nF(\\theta, \\phi) = 1 - \\frac{1}{2} \\left(1 - e^{-8 \\kappa t} \\right)\n\\sin^2 \\theta, \\\\\\\\ \\displaystyle\\overline{F} = \\frac{2}{3} +\n\\frac{1}{3} e^{-8 \\kappa t}.\n\\end{array}\n\\end{eqnarray}\n\nThe next noisy channel is the isotropic noisy channel. For this case, the\nmaster equation involves twelve Lindblad operators\n$(L_{2,\\alpha},L_{3,\\alpha},L_{4,\\alpha},L_{5,\\alpha})$ with\n$\\alpha\\in\\{x,y,z\\}$. At $t=\\delta t$ we have\n\\begin{eqnarray}\n\\varepsilon(\\rho_{4\\mbox{\\tiny GHZ}})\\Big|_{t=\\delta t} =\n\\frac{1}{2}{\\left(\n\\begin{smallmatrix}\n1 -8 \\kappa \\delta t & 0 & 0 & 0 & 0 & 0 & 0 & 0& 0 & 0 & 0 & 0 & 0 & 1-16 \\kappa \\delta t \\\\\n0 & 2\\kappa \\delta t & 0 & 0& 0 & 0 & 0& 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 2\\kappa \\delta t & 0& 0 & 0& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & {\\footnotesize \\textcircled{\\tiny 1}} & 0 & 0 & 0& 0 & 0 & 0& {\\footnotesize \\textcircled{\\tiny 1}} & 0& 0 & 0 \\\\\n0 & 0 & 0& 0 & 2\\kappa \\delta t  & 0& 0 & 0 & 0& 0 & 0 &  0 & 0& 0 \\\\\n0 & 0 & 0& 0& 0 &{\\footnotesize \\textcircled{\\tiny 2}} & 0& 0 & {\\footnotesize \\textcircled{\\tiny 2}} &0 & 0 & 0& 0& 0 \\\\\n0 & 0 & 0& 0& 0 &0 & 2\\kappa \\delta t & 0 & 0 & 0& 0& 0 & 0 & 0  \\\\\n0 & 0 & 0& 0& 0 &0 & 0 & 2\\kappa \\delta t & 0 & 0 & 0& 0& 0 & 0 \\\\\n0 & 0 & 0& 0& 0 & {\\footnotesize \\textcircled{\\tiny 2}} & 0 & 0 & {\\footnotesize \\textcircled{\\tiny 2}} & 0& 0& 0 & 0 & 0   \\\\\n0 & 0 & 0& 0 & 0  & 0& 0 & 0 & 0&  2\\kappa \\delta t & 0& 0 & 0 & 0   \\\\\n0 & 0 & 0 & {\\footnotesize \\textcircled{\\tiny 1}} & 0 & 0& 0 & 0 & 0& 0 & {\\footnotesize \\textcircled{\\tiny 1}} & 0 & 0 & 0   \\\\\n0 & 0 &  0 & 0& 0 & 0& 0 & 0 & 0& 0 & 0 &  2\\kappa \\delta t & 0 & 0   \\\\\n0 &  0 & 0 & 0& 0 & 0 & 0& 0 & 0 & 0 & 0 & 0 & 2\\kappa \\delta t & 0   \\\\\n1 -16 \\kappa \\delta t & 0 & 0 & 0 & 0& 0 & 0 & 0& 0 & 0 & 0\n&0 & 0 & 1 -8 \\kappa \\delta t\n\\end{smallmatrix}\\right).}\n\\end{eqnarray}\nSo we take the ansatz\n\\begin{eqnarray}\n\\varepsilon(\\rho_{4\\mbox{\\tiny GHZ}}) =\n \\left(\n\\begin{smallmatrix}\na & 0 & 0 & 0 & 0 & 0 & 0& 0 & 0 & 0& 0 & 0 & 0 & 0 & 0 & d   \\\\\n0 & b& 0 & 0& 0 & 0 & 0& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0   \\\\\n0 & 0 & b & 0& 0 & 0& 0 & 0 & 0& 0 & 0 & 0 & 0 & 0 & 0 & 0   \\\\\n0 & 0 & 0 & c & 0 & 0& 0 & 0 & 0& 0 & 0 & 0& 0 & 0 & 0 & 0   \\\\\n0 & 0 & 0& 0 & b  & 0& 0 & 0 & 0& 0 & 0 & 0 & 0& 0 & 0 & 0   \\\\\n0 & 0 & 0& 0& 0 & c  & 0& 0 & 0 & 0 & 0 & 0& 0& 0 & 0 & 0   \\\\\n0 & 0 & 0& 0& 0&0 & c  & 0& 0 & 0 & 0& 0& 0 & 0& 0 & 0   \\\\\n0 & 0 & 0& 0& 0 &0  & 0& b & 0 & 0 & 0 & 0& 0& 0 & 0 & 0   \\\\\n0 & 0 & 0& 0& 0 &0  & 0& 0 & b & 0 & 0 & 0& 0& 0 & 0 & 0   \\\\\n0 & 0 & 0& 0& 0&0 & 0  & 0& 0 & c & 0& 0& 0 & 0& 0 & 0   \\\\\n0 & 0 & 0& 0& 0 & 0  & 0& 0 & 0 & 0 & c & 0& 0& 0 & 0 & 0   \\\\\n0 & 0 & 0& 0 & 0 & 0& 0 & 0 & 0& 0 & 0 & b & 0& 0 & 0 & 0   \\\\\n0 & 0 & 0 & 0 & 0 & 0& 0 & 0 & 0& 0 & 0 & 0& c & 0 & 0 & 0   \\\\\n0 & 0 &0 & 0& 0 & 0& 0 & 0 & 0& 0 & 0 & 0 & 0 & b & 0 & 0   \\\\\n0 & 0 & 0 & 0& 0 & 0 & 0& 0 & 0 & 0 & 0 & 0 & 0 & 0 & b & 0   \\\\\nd & 0 & 0 & 0 & 0 & 0 & 0& 0 & 0 & 0& 0 & 0 & 0 & 0 & 0 & a\n\\end{smallmatrix}\n\\right).\n\\end{eqnarray}\nInserting this solution in the Lindblad equation, Eq.~({\\ref{Lindblad}}),\nwe find\n\\begin{eqnarray}\n\\left\\{\n\\begin{array}{l}\n\\dot{a}(t) = 8k\\Big(b(t)-a(t)\\Big),\\\\\n\\dot{b}(t) = 2k\\Big(a(t)-4b(t)+3c(t)\\Big),\\\\\n\\dot{c}(t)=8k\\Big(b(t)-c(t)\\Big),\\\\\n\\dot{d}(t)=-16k\\,d(t),\\\\\n\\end{array}\\right.\n\\end{eqnarray}\nsubject to the initial conditions $a(0)=d(0)=1\/2$ and $b(0)=c(0)=0$.\nThe solutions are\n\\begin{eqnarray}\\left\\{\n\\begin{array}{l}\na(t) =\\frac{1}{16}\\Big( 1 + 6 e^{-8 \\kappa t} + e^{-16 \\kappa t}\\Big),\\\\\nb(t) =\\frac{1}{16}\\Big( 1 - e^{-16 \\kappa t}\\Big),\\\\\nc(t) = \\frac{1}{16}\\Big(1 - 2 e^{-8 \\kappa t} +  e^{-16 \\kappa t}\\Big),\\\\\nd(t) = \\frac{1}{2} e^{-16 \\kappa t}.\n\\end{array}\\right.\n\\end{eqnarray}\nAlso the fidelity is\n\\begin{eqnarray}\n& &F(\\theta, \\phi) = \\frac{1}{2} \\left[ 1 + e^{-8 \\kappa t} \\cos^2\n\\theta + e^{-16 \\kappa t}  \\sin^2 \\theta \\right],\n\\end{eqnarray}\nand\n\\begin{eqnarray}\n\\label{dfbar} \\overline{F} = \\frac{1}{6} \\left( 3 +  e^{-8 \\kappa t}\n+ 2 e^{-16 \\kappa t} \\right).\n\\end{eqnarray}\n\n\nTo this end, we only considered the noisy channels with the same\naxis. Now, as a different-axis noisy channel, consider\n$(L_{2,x},L_{3,y},L_{4,z},L_{5,x})$ noise with\n$\\kappa_{2,x}=\\kappa_{3,y}=\\kappa_{4,z}=\\kappa_{5,x}=\\kappa$ that\nexhibits the effects of noises in different directions. After an\ninfinitesimal time interval and using the Lindblad equation, the\ndensity matrix  can be written as\n\\begin{eqnarray}\n\\varepsilon(\\rho_{4\\mbox{\\tiny GHZ}})\\Big|_{t=\\delta t} =\n\\frac{1}{2}{\\left(\n\\begin{smallmatrix}\n1 -6 \\kappa \\delta t & 0 & 0 & 0 & 0 & 0& 0 & 0 & 0 & 0 & 0 & 1-10 \\kappa \\delta t \\\\\n0 & \\kappa \\delta t & 0 & 0& 0 & 0 & 0 & 0 & 0 & 0 & \\kappa \\delta t & 0 \\\\\n0 & 0 & {\\footnotesize \\textcircled{\\tiny 2}} & 0& 0 & 0 & 0 & 0 & 0 & {\\footnotesize \\textcircled{\\tiny 2}} & 0 & 0 \\\\\n0 & 0 & 0 & \\kappa \\delta t & 0 & 0 &0 & 0 & -\\kappa \\delta t & 0& 0 & 0 \\\\\n0 & 0 & 0 & 0& {\\footnotesize \\textcircled{\\tiny 2}}& 0 & 0 & {\\footnotesize \\textcircled{\\tiny 2}}& 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0& 0 & 0& \\kappa \\delta t  & \\kappa \\delta t & 0 & 0 & 0& 0 & 0 \\\\\n0 & 0 & 0& 0 & 0& \\kappa \\delta t  & \\kappa \\delta t & 0 & 0 & 0& 0 & 0 \\\\\n0 & 0 & 0 & 0& {\\footnotesize \\textcircled{\\tiny 2}}& 0 & 0 & {\\footnotesize \\textcircled{\\tiny 2}} & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & -\\kappa \\delta t & 0 & 0& 0& 0 & \\kappa \\delta t & 0& 0 & 0 \\\\\n0 & 0 &{\\footnotesize \\textcircled{\\tiny 2}}& 0 & 0 & 0 & 0& 0 & 0 & {\\footnotesize \\textcircled{\\tiny 2}} & 0 & 0 \\\\\n0 & \\kappa \\delta t & 0 & 0& 0 & 0 & 0& 0 & 0 & 0 & \\kappa \\delta t & 0 \\\\\n1 -10 \\kappa \\delta t & 0 & 0 & 0 & 0& 0 & 0 & 0 & 0\n&0 & 0 & 1 -6 \\kappa \\delta t\n\\end{smallmatrix}\\right).}\n\\end{eqnarray}\nSo, the elements of the density matrix for all time can be read off\nas\n\\begin{eqnarray}\n\\varepsilon(\\rho_{4\\mbox{\\tiny GHZ}}) =\n \\left(\n\\begin{smallmatrix}\na & 0 & 0 & 0 & 0 & 0 & 0& 0 & 0 & 0& 0 & 0 & 0 & 0 & 0 & g   \\\\\n0 & b& 0 & 0& 0 & 0 & 0& 0 & 0 & 0 & 0 & 0 & 0 & 0 & h & 0   \\\\\n0 & 0 & c & 0& 0 & 0& 0 & 0 & 0& 0 & 0 & 0 & 0 & m & 0 & 0   \\\\\n0 & 0 & 0 & d & 0 & 0& 0 & 0 & 0& 0 & 0 & 0& k & 0 & 0 & 0   \\\\\n0 & 0 & 0& 0 & b  & 0& 0 & 0 & 0& 0 & 0 & n & 0& 0 & 0 & 0   \\\\\n0 & 0 & 0& 0& 0 & d  & 0& 0 & 0 & 0 & k & 0& 0& 0 & 0 & 0   \\\\\n0 & 0 & 0& 0& 0&0 & d  & 0& 0 & f & 0& 0& 0 & 0& 0 & 0   \\\\\n0 & 0 & 0& 0& 0 &0  & 0& b & h & 0 & 0 & 0& 0& 0 & 0 & 0   \\\\\n0 & 0 & 0& 0& 0 &0  & 0& h & b & 0 & 0 & 0& 0& 0 & 0 & 0   \\\\\n0 & 0 & 0& 0& 0&0 & f  & 0& 0 & d & 0& 0& 0 & 0& 0 & 0   \\\\\n0 & 0 & 0& 0& 0 & k  & 0& 0 & 0 & 0 & d & 0& 0& 0 & 0 & 0   \\\\\n0 & 0 & 0& 0 & n & 0& 0 & 0 & 0& 0 & 0 & b & 0& 0 & 0 & 0   \\\\\n0 & 0 & 0 & k & 0 & 0& 0 & 0 & 0& 0 & 0 & 0& d & 0 & 0 & 0   \\\\\n0 & 0 & m & 0& 0 & 0& 0 & 0 & 0& 0 & 0 & 0 & 0 & c & 0 & 0   \\\\\n0 & h & 0 & 0& 0 & 0 & 0& 0 & 0 & 0 & 0 & 0 & 0 & 0 & b & 0   \\\\\ng & 0 & 0 & 0 & 0 & 0 & 0& 0 & 0 & 0& 0 & 0 & 0 & 0 & 0 & a\n\\end{smallmatrix}\n\\right),\n\\end{eqnarray}\nwhich leads to two sets of four and six coupled differential\nequations, namely\n\\begin{eqnarray}\n\\left\\{\n\\begin{array}{l}\n\\dot{a}(t) = 3\\kappa\\Big(b(t)-a(t)\\Big),\\\\\n\\dot{b}(t) = \\kappa\\Big(a(t)-3b(t)+2d(t)\\Big),\\\\\n\\dot{c}(t) = 3\\kappa\\Big(d(t)-c(t)\\Big),\\\\\n\\dot{d}(t) = \\kappa\\Big(2b(t)-3d(t)+c(t)\\Big),\n\\end{array}\\right.\n\\end{eqnarray}\nand\n\\begin{eqnarray}\n\\left\\{\n\\begin{array}{l}\n\\dot{f}(t) = \\kappa\\Big(-5f(t)+2h(t)-m(t)\\Big),\\\\\n\\dot{g}(t) = \\kappa\\Big(-5g(t)+2h(t)-n(t)\\Big),\\\\\n\\dot{h}(t) = \\kappa\\Big(f(t)+g(t)-5h(t)-k(t)\\Big),\\\\\n\\dot{k}(t) = \\kappa\\Big(-h(t)-5k(t)+m(t)+n(t)\\Big),\\\\\n\\dot{m}(t)=\\kappa\\Big(-f(t)+2k(t)-5m(t)\\Big),\\\\\n\\dot{n}(t)=\\kappa\\Big(-g(t)+2k(t)-5n(t)\\Big),\n\\end{array}\\right.\n\\end{eqnarray}\nsubject to $a(0)=g(0)=1\/2$ and\n$b(0)=c(0)=d(0)=f(0)=h(0)=k(0)=m(0)=n(0)=0$. The solutions are\nreadily found\n\\begin{eqnarray}\\left\\{\n\\begin{array}{l}\na(t) =e^{2 \\kappa t}g(t)=\\frac{1}{16}\\Big( 1 + 3 e^{-2 \\kappa t} +3 e^{-4 \\kappa t} + e^{-6 \\kappa t}\\Big),\\\\\nb(t) =e^{2 \\kappa t}h(t)=-e^{2 \\kappa t}n(t)=\\frac{1}{16}\\Big( 1 + e^{-2 \\kappa t} - e^{-4 \\kappa t} - e^{-6 \\kappa t}\\Big),\\\\\nc(t) =-e^{2 \\kappa t}m(t)=\\frac{1}{16}\\Big( 1 - 3 e^{-2 \\kappa t} +3 e^{-4 \\kappa t} - e^{-6 \\kappa t}\\Big),\\\\\nd(t) =e^{2 \\kappa t}f(t)=-e^{2 \\kappa t}k(t)=\\frac{1}{16}\\Big( 1 -\ne^{-2 \\kappa t} - e^{-4 \\kappa t} + e^{-6 \\kappa t}\\Big).\n\\end{array}\\right.\n\\end{eqnarray}\nThus, the fidelity, $F(\\theta, \\phi)$, and its average, $\\overline{F}$, are given by\n\\begin{eqnarray}\n& &F(\\theta, \\phi) = \\frac{1}{2} \\left[ 1 + e^{-2 \\kappa t} \\cos^2\n\\theta + e^{-4 \\kappa t} \\sin^2\n\\theta \\cos^2 \\phi + e^{-6 \\kappa t}  \\sin^2 \\theta \\sin^2 \\phi \\right],\n\\end{eqnarray}\nand\n\\begin{eqnarray}\n\\label{mixfbar} \\overline{F} = \\frac{1}{6} \\left( 3 + e^{-2 \\kappa t}+ e^{-4 \\kappa t}\n+ e^{-6 \\kappa t} \\right).\n\\end{eqnarray}\n\n\\begin{table}\n\\caption{\\label{tab1}Summary of $F(\\theta, \\phi)$ and $\\overline{F}$\nthrough various noisy channels.}\n\\begin{ruledtabular}\n\\begin{tabular}{c|ccc}\n{} & Noise & 3GHZ & 4GHZ \\\\ \\hline\n{} & Pauli-X & $\\frac{1}{2} \\bigg[(1 + \\sin^2 \\theta \\cos^2 \\phi$) & $\\frac{1}{2} \\bigg[ (1 + \\sin^2 \\theta \\cos^2 \\phi$) \\\\\n{} & {} & $+ e^{-4 \\kappa t} (\\cos^2 \\theta + \\sin^2 \\theta \\sin^2 \\phi) \\bigg]$ & $+ e^{-4 \\kappa t} (\\cos^2 \\theta + \\sin^2 \\theta \\sin^2 \\phi) \\bigg]$ \\\\\n$F(\\theta,\\phi)$ & Pauli-Y &$\\frac{1}{2} \\bigg[1 + \\sin^2 \\theta \\sin^2 \\phi e^{-2 \\kappa t} +\\cos^2 \\theta e^{-4 \\kappa t}$   & $\\frac{1}{2} \\bigg[1 + (\\sin^2 \\theta \\sin^2 \\phi +\\cos^2 \\theta) e^{-4 \\kappa t}$    \\\\\n{} & {} & $+ \\sin^2 \\theta \\cos^2 \\phi e^{-6 \\kappa t} \\bigg]$ & $+\\sin^2 \\theta \\cos^2 \\phi e^{-8 \\kappa t} \\bigg]$\\\\\n{} & Pauli-Z & $1 - \\frac{1}{2} (1 -e^{-6 \\kappa t}) \\sin^2 \\theta$  &   $1 - \\frac{1}{2} (1 - e^{-8\\kappa t})\\sin^2 \\theta$\\\\\n{} & isotropic & $\\frac{1}{2} (1 + \\cos^2 \\theta e^{-8 \\kappa t} +\\sin^2 \\theta e^{-12 \\kappa t} )$ & $\\frac{1}{2} (1 + \\cos^2 \\theta e^{-8 \\kappa t} + \\sin^2 \\theta e^{-16 \\kappa t} )$\\\\\\hline\n{} & Pauli-X & $\\frac{2}{3} +\\frac{1}{3} e^{-4 \\kappa t}$ & $\\frac{2}{3} + \\frac{1}{3} e^{-4\\kappa t}$\\\\\n$\\overline{F}$ & Pauli-Y &$\\frac{1}{6} (3 + e^{-2 \\kappa t} + e^{-4 \\kappa t} + e^{-6 \\kappa t})$    & $\\frac{1}{6} (3 + 2 e^{-4 \\kappa t} + e^{-8 \\kappa t})$\\\\\n{} & Pauli-Z & $\\frac{2}{3} +\\frac{1}{3} e^{-6 \\kappa t}$ & $\\frac{2}{3} + \\frac{1}{3} e^{-8\\kappa t}$\\\\\n{} & isotropic & $\\frac{1}{6} (3 + e^{-8 \\kappa t} + 2 e^{-12 \\kappa t} )$ & $\\frac{1}{6} (3 + e^{-8 \\kappa t} + 2 e^{-16 \\kappa t})$\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\n\nIn Table \\ref{tab1}, a summary of fidelity and average fidelity for\n3GHZ \\cite{jung08-2} and 4GHZ states is reported and compared. Also,\ntheir average fidelity versus time is depicted in Fig.~\\ref{fig3}\nfor various noisy channels. Comparing 3GHZ and 4GHZ states shows\nthat for $(L_{2,x},L_{3,x},L_{4,x},L_{5,x})$ noise both states have\nthe same fidelity. This result also agrees with Bell state\n$|\\beta_{00}\\rangle=\\frac{1}{\\sqrt{2}}(|00\\rangle+|11\\rangle)$\n\\cite{0h02}. However, for other cases 3GHZ state is more robust,\ni.e., loses less quantum information in the quantum teleportation\nprocess with respect to 4GHZ state. Note that, for the isotropic\ncase, the fidelities are approximately equal. These results and\nthose obtained in Refs.~\\cite{jung08-2,0h02} show that increasing\nthe number of qubits can enhance the rate of information lost in\nquantum teleportation process. Moreover, using a proper ansatz for\nthe density matrix, we reduced the number of coupled differential\nequations from 136 to at most four coupled equations.\nFig.~\\ref{fig7} shows average fidelity for 4GHZ state through\nvarious noises. As it can be seen from the figure,\n$(L_{2,x},L_{3,x},L_{4,x},L_{5,x})$ noise does lose less quantum\ninformation with respect to others. The next noise with small\ninformation lost is $(L_{2,x},L_{3,y},L_{4,z},L_{5,x})$  for $\\kappa\nt<0.2$. However, for $\\kappa t>0.2$,\n$(L_{2,z},L_{3,z},L_{4,z},L_{5,z})$ noise represents a better\nbehavior. Moreover, the isotropic noise and the noise in $y$\ndirection always result in low fidelity quantum teleportation. In\nthe following sections, we exactly solve the Lindblad equation for\n5GHZ and 6GHZ states through two types of noisy channels.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=8cm]{fig4GHZ.eps}\n\\caption{\\label{fig7} The plot of time dependence of average\nfidelity through noisy channels for\n4GHZ state.}\n\\end{figure}\n\n\n\\begin{figure}\n\\centerline{\\begin{tabular}{ccc}\n\\includegraphics[width=8cm]{figF1.eps}\n &\\hspace{2.cm}&\n\\includegraphics[width=8cm]{figF2.eps}\\\\\n\\includegraphics[width=8cm]{figF3.eps}\n &\\hspace{2.cm}&\n\\includegraphics[width=8cm]{figF4.eps}\n\\end{tabular}}\n\\caption{\\label{fig3} The plot of time dependence of average\nfidelity for Pauli-X (left up),\nPauli-Y (right up),\nPauli-Z (left down), and isotropic\n(right down) noisy channels.}\n\\end{figure}\n\n\\section{Five-qubit GHZ state with noisy channels}\\label{sec4}\nIn this section, we teleport 5GHZ state through noisy channels as\ndepicted in Fig.~\\ref{fig4}. For this case the solution of the\nLindblad equation is a $32\\times32$ matrix that results in a set of\n32 diagonal and 496 off-diagonal coupled differential equations.\nHowever, we show that the number of required equations can be\nconsiderably reduced by choosing appropriate ansatz for the density\nmatrix.\n\n\\begin{figure}[t]\n\\input{fig3.tex}\n\\caption{\\label{fig4}\n   A circuit for quantum teleportation through noisy channels with 5GHZ state.\n   The five top lines belong to Alice and the bottom line to Bob.\n   $M$ denotes measurement and the dotted box represents noisy channel.\n   The Lindblad operator is turned on inside the dotted box.}\n\\end{figure}\n\nFirst, consider ($L_{2,x}$,$L_{3,x}$,$L_{4,x}$,$L_{5,x}$,$L_{6,x}$)\nnoise and assume\n$\\kappa_{2,x}=\\kappa_{3,x}=\\kappa_{4,x}=\\kappa_{5,x}=\\kappa_{6,x}=\\kappa$.\nThe infinitesimal time evolution of the density matrix now reads\n\n\\begin{eqnarray}\n\\hspace{-1cm} \\varepsilon(\\rho_{5\\mbox{\\tiny GHZ}})\\Big|_{t=\\delta\nt} = \\frac{1}{2}{ \\left(\n\\begin{smallmatrix}\n1- 5 \\kappa \\delta t & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1- 5 \\kappa \\delta t  \\\\\n0 & \\kappa \\delta t & 0 & 0 & 0 & 0 & 0 & 0& 0 & 0 & 0 & 0 & 0 & 0 & 0 &  0 & \\kappa \\delta t & 0   \\\\\n0 & 0 & \\kappa \\delta t & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\kappa \\delta t & 0 & 0   \\\\\n0 & 0 & 0 & {\\footnotesize \\textcircled{\\tiny 1}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\\footnotesize \\textcircled{\\tiny 1}} & 0 & 0 & 0   \\\\\n0 & 0 & 0& 0 & \\kappa \\delta t  & 0 & 0& 0 & 0 & 0 & 0 & 0 & 0 & \\kappa \\delta t & 0& 0 & 0 & 0   \\\\\n0 & 0 & 0& 0 &0 & {\\footnotesize \\textcircled{\\tiny 3}} & 0 & 0& 0 & 0 & 0 & 0 & 0 & {\\footnotesize \\textcircled{\\tiny 3}} & 0& 0 & 0 & 0   \\\\\n0 & 0 & 0& 0 &0 & 0 & \\kappa \\delta t  & 0 & 0 & 0 & 0 & \\kappa \\delta t & 0 & 0 & 0& 0 & 0 & 0   \\\\\n0 & 0 & 0& 0 &0 &0 & 0& {\\footnotesize \\textcircled{\\tiny 6}} & 0 & 0 & {\\footnotesize \\textcircled{\\tiny 6}} & 0 & 0 & 0 & 0& 0 & 0 & 0   \\\\\n0& 0 & 0 & 0& 0& 0 & 0 & 0 & \\kappa \\delta t & \\kappa \\delta t & 0 & 0 & 0& 0& 0 & 0 & 0 & 0 \\\\\n0& 0 & 0 & 0& 0& 0 & 0 & 0 & \\kappa \\delta t & \\kappa \\delta t & 0 & 0 & 0& 0& 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0& 0 &0 &0 & 0& {\\footnotesize \\textcircled{\\tiny 6}} & 0& 0 & {\\footnotesize \\textcircled{\\tiny 6}} & 0 & 0 & 0 & 0& 0 & 0 & 0   \\\\\n0 & 0 & 0& 0 &0 & 0 & \\kappa \\delta t  & 0 & 0 & 0 & 0 & \\kappa \\delta t & 0& 0 & 0& 0 & 0 & 0   \\\\\n0 & 0 & 0& 0 &0 & {\\footnotesize \\textcircled{\\tiny 3}} & 0 & 0 & 0 & 0 & 0 & 0 & {\\footnotesize \\textcircled{\\tiny 3}} &0 & 0& 0 & 0 & 0   \\\\\n0 & 0 & 0& 0 & \\kappa \\delta t & 0 & 0 & 0 & 0 & 0 & 0 &0 & 0 & \\kappa \\delta t & 0& 0 & 0 & 0   \\\\\n0 & 0 & 0 & {\\footnotesize \\textcircled{\\tiny 1}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\\footnotesize \\textcircled{\\tiny 1}} & 0 & 0 & 0   \\\\\n0 & 0 & \\kappa \\delta t & 0 & 0 & 0 & 0 & 0 & 0& 0& 0 & 0 & 0 & 0 & 0 & \\kappa \\delta t & 0 & 0   \\\\\n0 & \\kappa \\delta t & 0 & 0 & 0 & 0 & 0 & 0& 0 & 0 & 0& 0 & 0 & 0 & 0 & 0 & \\kappa \\delta t & 0   \\\\\n1- 5 \\kappa \\delta t & 0 & 0 & 0 & 0 & 0 &\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0& 0 & 0 & 0\n& 1- 5 \\kappa \\delta t\n\\end{smallmatrix} \\right).}\n\\hspace{0.5cm}\n\\end{eqnarray}\nSo we take the ansatz as\n\\begin{eqnarray}\n\\hspace{-1cm} \\varepsilon(\\rho_{5\\mbox{\\tiny GHZ}}) = \\left(\n\\begin{smallmatrix}\na & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & a \\\\\n0 & b & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &  0 & b & 0   \\\\\n0 & 0 & b & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & b & 0 & 0   \\\\\n0 & 0 & 0 & \\tiny\\textcircled{\\emph{c}}_{1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\tiny\\textcircled{\\emph{c}}_{1} & 0 & 0 & 0   \\\\\n0 & 0 & 0& 0 & b & 0 & 0& 0 & 0 & 0 & 0 &0 & 0 & b & 0& 0 & 0 & 0   \\\\\n0 & 0 & 0& 0 &0 & \\tiny\\textcircled{\\emph{c}}_{3} & 0 & 0 & 0 & 0 & 0 & 0 & \\tiny\\textcircled{\\emph{c}}_{3} &0 & 0& 0 & 0 & 0   \\\\\n0 & 0 & 0& 0 &0 & 0 & b  & 0 & 0 & 0 & 0 & b & 0 & 0 & 0& 0 & 0 & 0   \\\\\n0 & 0 & 0& 0 &0 &0 & 0& \\tiny\\textcircled{\\emph{c}}_{6} & 0 & 0 & \\tiny\\textcircled{\\emph{c}}_{6} &0 &  0 & 0 & 0& 0 & 0 & 0   \\\\\n0& 0 & 0 & 0& 0& 0 & 0 & 0& b & b & 0 & 0 & 0& 0& 0 & 0 & 0 & 0 \\\\\n0& 0 & 0 & 0& 0& 0 & 0 & 0 & b & b & 0 & 0 & 0& 0& 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0& 0 &0 &0 & 0& \\tiny\\textcircled{\\emph{c}}_{6} & 0 & 0 & \\tiny\\textcircled{\\emph{c}}_{6} &0 &  0 & 0 & 0& 0 & 0 & 0   \\\\\n0 & 0 & 0& 0 &0 & 0 & b & 0 & 0 & 0 & 0 & b & 0& 0 & 0& 0 & 0 & 0   \\\\\n0 & 0 & 0& 0 &0 & \\tiny\\textcircled{\\emph{c}}_{3} & 0 & 0 & 0 & 0 & 0 & 0 & \\tiny\\textcircled{\\emph{c}}_{3} &0 & 0& 0 & 0 & 0 \\\\\n0 & 0 & 0& 0 & b &0 & 0& 0 & 0 & 0 & 0 &0 & 0 & b & 0& 0 & 0 & 0   \\\\\n0 & 0 & 0 &\\tiny\\textcircled{\\emph{c}}_{1} & 0 & 0& 0 & 0 & 0& 0 & 0 & 0 & 0 & 0 & \\tiny\\textcircled{\\emph{c}}_{1} & 0 & 0 & 0   \\\\\n0 & 0 & b & 0 & 0 & 0 & 0& 0 & 0 & 0& 0 & 0 & 0 & 0 & 0 & b & 0 & 0   \\\\\n0 & b& 0 & 0 & 0& 0 & 0 & 0& 0 & 0 & 0& 0 & 0 & 0 & 0 &  0 & b & 0   \\\\\na & 0 & 0 & 0 & 0 & 0 & 0& 0 & 0 & 0& 0 & 0 & 0 & 0 & 0 & 0 & 0 & a\n\\end{smallmatrix} \\right).\n\\hspace{0.5cm}\n\\end{eqnarray}\nHere $\\textcircled{\\emph{c}}_{n}$ denotes\n$n$ diagonal $c$.\n\nNow inserting this matrix in Lindblad equation, Eq.~(\\ref{Lindblad}), four\ncoupled differential equations are obtained as follows\n\\begin{eqnarray}\n\\left\\{\n\\begin{array}{l}\n\\dot{a}(t) = 5k\\Big(b(t)-a(t)\\Big),\\\\\n\\dot{b}(t) = k\\Big(a(t)-5b(t)+4c(t)\\Big),\\\\\n\\dot{c}(t)=2k\\Big(b(t)-c(t)\\Big).\\\\\n\\end{array}\\right.\n\\end{eqnarray}\nSolving this set of equations with the initial conditions\n$a(0)=1\/2$, $b(0)=c(0)=0$, leads to the following solution\n\\begin{eqnarray}\n\\left\\{\n\\begin{array}{l}\na(t) =\\frac{1}{32}\\Big( 1 + 10 e^{-4 \\kappa t} + 5e^{-8 \\kappa t}\\Big),\\\\\nb(t) =\\frac{1}{32}\\Big( 1 + 2 e^{-4 \\kappa t}- 3e^{-8 \\kappa t}\\Big),\\\\\nc(t) = \\frac{1}{32}\\Big(1 - 2 e^{-4 \\kappa t} +  e^{-8 \\kappa\nt}\\Big).\n\\end{array}\\right.\n\\end{eqnarray}\nSubstituting $\\varepsilon(\\rho_{5\\mbox{\\tiny GHZ}})$ in\nEq.~(\\ref{out1}) and using Eqs.~(\\ref{fidelity}) and (\\ref{average})\nfidelity and its average are given by\n\\begin{eqnarray}\nF(\\theta, \\phi) &=& \\frac{1}{2} \\left[ 1 + \\sin^2 \\theta \\cos^2\n\\phi+ e^{-4 \\kappa t} (\\cos^2 \\theta +\n     \\sin^2 \\theta \\sin^2 \\phi )\\right],\\\\\n\\overline{F} &=& \\frac{1}{3} \\left( 2 + e^ {-4 \\kappa t} \\right).\n\\end{eqnarray}\n\nFor ($L_{2,z}$,$L_{3,z}$,$L_{4,z}$,$L_{5,z}$,$L_{6,z}$) noise with\n$\\kappa_{2,z}=\\kappa_{3,z}=\\kappa_{4,z}=\\kappa_{5,z}=\\kappa_{6,z}=\\kappa$,\nthe infinitesimal evolution matrix is\n\\begin{eqnarray}\n\\varepsilon(\\rho_{5\\mbox{\\tiny GHZ}})\\Big|_{t=\\delta t} =\n\\frac{1}{2} \\left( |0\\rangle^{\\otimes 5} \\langle 0 |^{\\otimes 5} +\n|1\\rangle ^{\\otimes 5}\\langle 1| ^{\\otimes 5}\\right) +\n\\frac{1-10\\kappa \\delta t}{2} \\left(|0\\rangle ^{\\otimes 5} \\langle\n1| ^{\\otimes 5}+ |1\\rangle ^{\\otimes 5}\\langle 0 |^{\\otimes 5}\n\\right).\n\\end{eqnarray}\nUsing the ansatz\n\\begin{eqnarray}\n \\varepsilon(\\rho_{5\\mbox{\\tiny GHZ}}) =\na \\left( |0\\rangle^{\\otimes 5} \\langle 0 |^{\\otimes 5} + |1\\rangle\n^{\\otimes 5}\\langle 1| ^{\\otimes 5}\\right) + b\\left(|0\\rangle\n^{\\otimes 5} \\langle 1| ^{\\otimes 5}+ |1\\rangle ^{\\otimes 5}\\langle\n0 |^{\\otimes 5} \\right),\n\\end{eqnarray}\nwe obtain two coupled equations\n\\begin{eqnarray}\n\\left\\{\n\\begin{array}{l}\n\\dot{a}(t) =0,\\\\\n\\dot{b}(t) = -10kb(t),\\\\\n\\end{array}\\right.\n\\end{eqnarray}\nsubject to $a(0)=1\/2$, $b(0)=0$. Therefore, the density matrix reads\n\\begin{equation}\n\\label{zmatrix5} \\varepsilon(\\rho_{5\\mbox{\\tiny GHZ}}) = \\frac{1}{2}\n\\left( |0\\rangle^{\\otimes 5} \\langle 0 |^{\\otimes 5} + |1\\rangle\n^{\\otimes 5}\\langle 1| ^{\\otimes 5}\\right) + \\frac{1}{2} e^{-10\n\\kappa t} \\left(|0\\rangle ^{\\otimes 5} \\langle 1| ^{\\otimes 5}+\n|1\\rangle ^{\\otimes 5}\\langle 0 |^{\\otimes 5} \\right),\n\\end{equation}\nand the fidelity and its average are given by\n\\begin{eqnarray}\nF(\\theta, \\phi) &=&  1 - \\frac{1}{2} \\left( 1 - e^{-10 \\kappa t}\\right) \\sin^2 \\theta ,\\\\\n\\overline{F} &=& \\frac{1}{3} \\left( 2 + e^ {-10 \\kappa t} \\right).\n\\end{eqnarray}\n\n\\section{Six-qubit GHZ state with noisy channels}\\label{sec5}\nA quantum circuit for  teleportation through noisy channels with\n6GHZ state is depicted in Fig.~\\ref{fig5}. In the dotted box the\nLindblad operators act on the $64\\times64$ density matrix that\ninvolves five Alice's qubits and one Bob's qubits. The Lindblad\nequation, Eq.~(\\ref{Lindblad}), leads to 64 diagonal and 2016 off-diagonal\nlinear coupled differential equations. However, similar to previous\nsections, we first study infinitesimal temporal behavior of the\ndensity matrix and use a proper ansatz to considerably reduce the\nnumber of required equations.\n\n\n\\begin{figure}[t]\n\\input{fig4.tex}\n\\caption{\\label{fig5}\n   A circuit for quantum teleportation through noisy channels with 6GHZ state.\n   The six top lines belong to Alice and the bottom line to Bob.\n   $M$ denotes measurement and the dotted box represents noisy channel.\n   The Lindblad operator is turned on inside the dotted box.}\n\\end{figure}\n\n\nFor ($L_{2,x}$,$L_{3,x}$,$L_{4,x}$,$L_{5,x}$,$L_{6,x},L_{7,x}$)\nnoise and\n$\\kappa_{2,x}=\\kappa_{3,x}=\\kappa_{4,x}=\\kappa_{5,x}=\\kappa_{6,x}=\\kappa_{7,x}=\\kappa$,\nthe Lindblad operators after an infinitesimal time transform the\ninput density matrix\n$\\rho(0)=|6\\mbox{GHZ}\\rangle\\langle6\\mbox{GHZ}|$ to\n\\begin{eqnarray}\n\\hspace{-1cm}\\varepsilon(\\rho_{6\\mbox{\\tiny GHZ}})\\Big|_{t=\\delta\nt}= \\frac {1}{2}\\left(\\mbox{\\footnotesize$\n\\begin{smallmatrix}\n     1 - 6 \\kappa \\delta t  & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &  1 - 6 \\kappa \\delta t \\\\\n    0 & \\kappa \\delta t & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\kappa \\delta t & 0 \\\\\n    0 & 0 & \\kappa \\delta t& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\kappa \\delta t & 0 & 0 \\\\\n    0 & 0 & 0 & {\\footnotesize \\textcircled{\\tiny 1}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\\footnotesize \\textcircled{\\tiny 1}} & 0 & 0 & 0 \\\\\n    0 & 0 & 0 & 0 & \\kappa \\delta t & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\kappa \\delta t & 0 & 0 & 0 & 0 \\\\\n    0 & 0 & 0 & 0 & 0 & {\\footnotesize \\textcircled{\\tiny 3}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\\footnotesize \\textcircled{\\tiny 3}} & 0 & 0 & 0 & 0 & 0 \\\\\n    0 & 0 & 0 & 0 & 0 & 0 & \\kappa \\delta t & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\kappa \\delta t & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n    0 & 0 & 0 & 0 & 0 & 0 & 0 & {\\footnotesize \\textcircled{\\tiny 7}} & 0 & 0 & 0 & 0 & 0 & 0 & {\\footnotesize \\textcircled{\\tiny 7}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n    0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\kappa \\delta t & 0 & 0 & 0 & 0 & \\kappa \\delta t & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n    0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\\footnotesize \\textcircled{\\tiny 14}} & 0 & 0 & {\\footnotesize \\textcircled{\\tiny 14}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n    0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\kappa \\delta t & \\kappa \\delta t & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n    0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\kappa \\delta t & \\kappa \\delta t & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n    0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\\footnotesize \\textcircled{\\tiny 14}} & 0 & 0 &{\\footnotesize \\textcircled{\\tiny 14}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n    0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\kappa \\delta t & 0 & 0 & 0 & 0 & \\kappa \\delta t & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n    0 & 0 & 0 & 0 & 0 & 0 & 0 & {\\footnotesize \\textcircled{\\tiny 7}} & 0 & 0 & 0 & 0 & 0 & 0 & {\\footnotesize \\textcircled{\\tiny 7}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n    0 & 0 & 0 & 0 & 0 & 0 & \\kappa \\delta t & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\kappa \\delta t & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n    0 & 0 & 0 & 0 & 0 & {\\footnotesize \\textcircled{\\tiny 3}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\\footnotesize \\textcircled{\\tiny 3}} & 0 & 0 & 0 & 0 & 0 \\\\\n    0 & 0 & 0 & 0 & \\kappa \\delta t & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\kappa \\delta t & 0 & 0 & 0 & 0 \\\\\n    0 & 0 & 0 & {\\footnotesize \\textcircled{\\tiny 1}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\\footnotesize \\textcircled{\\tiny 1}} & 0 & 0 & 0 \\\\\n    0 & 0 & \\kappa \\delta t& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\kappa \\delta t & 0 & 0 \\\\\n    0 & \\kappa \\delta t & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\kappa \\delta t & 0 \\\\\n     1 - 6 \\kappa \\delta t  & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &  1 - 6 \\kappa \\delta t\n\\end{smallmatrix}$}\\right).\n\\end{eqnarray}\nSo consider\nthe ansatz\n\\begin{eqnarray}\n\\hspace{-2.3cm}\\varepsilon(\\rho_{6\\mbox{\\tiny GHZ}})=\n\\left(\\mbox{\\footnotesize$\n\\begin{smallmatrix}\na & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & a \\\\\n0 & b & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & b & 0 \\\\\n0 & 0 & b & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & b & 0 & 0 \\\\\n0 & 0 & 0 & c & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & c & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & b & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & b & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & \\tiny\\textcircled{\\emph{c}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\tiny\\textcircled{\\emph{c}} & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & d & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & d & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & b & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & b & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\tiny\\textcircled{\\emph{c}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\tiny\\textcircled{\\emph{c}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & d & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & d & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & c & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & c & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\tiny\\textcircled{\\emph{d}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\tiny\\textcircled{\\emph{d}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & c & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & c & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & b & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & b & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\tiny\\textcircled{\\emph{c}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\tiny\\textcircled{\\emph{c}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & d & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & d & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & c & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & c & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\tiny\\textcircled{\\emph{d}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\tiny\\textcircled{\\emph{d}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\tiny\\textcircled{\\emph{c}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\tiny\\textcircled{\\emph{c}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\tiny\\textcircled{\\emph{d}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\tiny\\textcircled{\\emph{d}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & c & 0 & 0 & 0 & 0 & 0 & 0 & c & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & d & 0 & 0 & 0 & 0 & d & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\tiny\\textcircled{\\emph{c}} & 0 & 0 & \\tiny\\textcircled{\\emph{c}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & b & b & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & b & b & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\tiny\\textcircled{\\emph{c}} & 0 & 0 & \\tiny\\textcircled{\\emph{c}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & d & 0 & 0 & 0 & 0 & d & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & c & 0 & 0 & 0 & 0 & 0 & 0 & c & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\tiny\\textcircled{\\emph{d}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\tiny\\textcircled{\\emph{d}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\tiny\\textcircled{\\emph{c}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\tiny\\textcircled{\\emph{c}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\tiny\\textcircled{\\emph{d}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\tiny\\textcircled{\\emph{d}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & c & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & c & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & d & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & d & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\tiny\\textcircled{\\emph{c}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\tiny\\textcircled{\\emph{c}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & b & 0 & 0 & 0 & 0 & 0 & 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\\tiny\\textcircled{\\emph{c}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\tiny\\textcircled{\\emph{c}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & b & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & b & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & d & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & d & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & \\tiny\\textcircled{\\emph{c}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\tiny\\textcircled{\\emph{c}} & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & b & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & b & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & c & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & c & 0 & 0 & 0 \\\\\n0 & 0 & b & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & b & 0 & 0 \\\\\n0 & b & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & b & 0 \\\\\na & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & a \\\\\n\\end{smallmatrix}$}\\right),\n\\end{eqnarray}\nwhere $\\small\\textcircled{\\emph{c}}$ and $\\small\\textcircled{\\emph{d}}$ denote two diagonal $c$ and $d$, respectively. Substituting this matrix\ninto the Lindblad equation leads to four coupled equations\n\\begin{eqnarray}\n\\left\\{\n\\begin{array}{l}\n\\dot{a}(t) = 6k\\Big(b(t)-a(t)\\Big),\\\\\n\\dot{b}(t) = k\\Big(a(t)-6b(t)+5c(t)\\Big),\\\\\n\\dot{c}(t)=2k\\Big(b(t)-3c(t)+2d(t)\\Big),\\\\\n\\dot{d}(t)=-6k\\Big(c(t)-d(t)\\Big),\\\\\n\\end{array}\\right.\n\\end{eqnarray}\nsubject to $a(0)=1\/2$ and $b(0)=c(0)=d(0)=0$. Thus, the solutions\nread\n\\begin{eqnarray}\n\\left\\{\n\\begin{array}{l}\na(t) =\\frac{1}{64}\\left( 1 + 15 e^{-4 \\kappa t} + 15 e^{-8 \\kappa t} + e^{-12 \\kappa t}\\right),\\\\\nb(t) =\\frac{1}{64}\\left( 1 +5 e^{-4 \\kappa t} - 5 e^{-8 \\kappa t} - e^{-12 \\kappa t}\\right),\\\\\nc(t) =\\frac{1}{64}\\left( 1 -  e^{-4 \\kappa t} -  e^{-8 \\kappa t} +\ne^{-12 \\kappa\nt}\\right),\\\\\nd(t) =\\frac{1}{64}\\left( 1 - 3 e^{-4 \\kappa t} + 3 e^{-8 \\kappa t}-\ne^{-12 \\kappa t}\\right),\n\\end{array}\\right.\n\\end{eqnarray}\nand finally\n\\begin{eqnarray}\nF(\\theta, \\phi) &=&  \\frac{1}{2} \\left[ 1 + \\sin^2 \\theta \\cos^2\n\\phi+ e^{-4 \\kappa t} (\\cos^2 \\theta +\n     \\sin^2 \\theta \\sin^2 \\phi )\\right],\n   \\\\ \\overline{F} &=& \\frac{1}{3} \\left( 2 + e^ {-4 \\kappa t}\n\\right).\n\\end{eqnarray}\n\n\nFor the last case, we study\n($L_{2,z}$,$L_{3,z}$,$L_{4,z}$,$L_{5,z}$,$L_{6,z},L_{7,z}$) noise\nwith\n$\\kappa_{2,z}=\\kappa_{3,z}=\\kappa_{4,z}=\\kappa_{5,z}=\\kappa_{6,z}=\\kappa_{7,z}=\\kappa$.\nFor this case, the temporal evolution matrix is\n\\begin{eqnarray}\n\\varepsilon(\\rho_{6\\mbox{\\tiny GHZ}})\\Big|_{t=\\delta t} =\\frac{1}{2}\n\\left( |0\\rangle^{\\otimes 6} \\langle 0 |^{\\otimes 6} + |1\\rangle\n^{\\otimes 6}\\langle 1| ^{\\otimes 6}\\right) + \\frac{1-12\\kappa\\delta\nt}{2} \\left(|0\\rangle ^{\\otimes 6} \\langle 1| ^{\\otimes 6}+\n|1\\rangle ^{\\otimes 6}\\langle 0 |^{\\otimes 6} \\right),\n\\end{eqnarray}\nTherefore, using the ansatz\n\\begin{eqnarray}\n\\varepsilon(\\rho_{6\\mbox{\\tiny GHZ}}) = a \\left( |0\\rangle^{\\otimes\n6} \\langle 0 |^{\\otimes 6} + |1\\rangle ^{\\otimes 6}\\langle 1|\n^{\\otimes 6}\\right) + b\\left(|0\\rangle ^{\\otimes 6} \\langle 1|\n^{\\otimes 6}+ |1\\rangle ^{\\otimes 6}\\langle 0 |^{\\otimes 6} \\right),\n\\end{eqnarray}\nwe obtain two simple differential equations\n\\begin{eqnarray}\n\\left\\{\n\\begin{array}{l}\n\\dot{a}(t) =0,\\\\\n\\dot{b}(t) = -12kb(t),\\\\\n\\end{array}\\right.\n\\end{eqnarray}\nsubject to $a(0)=b(0)=1\/2$. So the solution is given by\n\\begin{equation}\n\\label{zmatrix6} \\varepsilon(\\rho_{6\\mbox{\\tiny GHZ}}) = \\frac{1}{2}\n\\left( |0\\rangle^{\\otimes 6} \\langle 0 |^{\\otimes 6} + |1\\rangle\n^{\\otimes 6}\\langle 1| ^{\\otimes 6}\\right) + \\frac{1}{2} e^{-12\n\\kappa t} \\left(|0\\rangle ^{\\otimes 6} \\langle 1| ^{\\otimes 6}+\n|1\\rangle ^{\\otimes 6}\\langle 0 |^{\\otimes 6} \\right),\n\\end{equation}\nand the fidelity and its average read\n\\begin{eqnarray}\nF(\\theta, \\phi) &=&  1 - \\frac{1}{2} \\left( 1 - e^{-12 \\kappa t}\\right) \\sin^2 \\theta ,\\\\\n\\overline{F} &=& \\frac{1}{3} \\left( 2 + e^ {-12 \\kappa t} \\right).\n\\end{eqnarray}\n\n\\section{Conclusions}\\label{sec6}\nIn this paper, we studied quantum teleportation through noisy\nchannels for $n$GHZ states, $n\\in\\{4,5,6\\}$), so that the noisy\nchannels lead to the quantum channels to be mixed states. We\nexactly solved the Lindblad equation and obtained corresponding\ndensity matrices after the transmission process. The Lindblad\noperators are responsible for the decoherence of quantum states and\nare defined to be proportional to the Pauli matrices. Solving the\nLindblad equation for $n>2$ is not a trivial task in general. For\ninstance, we need to solve 2080 coupled differential equations to\nfind the density matrix for 6GHZ state. We overcame this problem by\nstudying the temporal evolution of the input state and using a\nproper ansatz for the density matrix. Therefore, we reduced 2080\ncoupled equations to at most four coupled equations which are\nreadily solved. We found the fidelity and the average fidelity for\nvarious cases and showed that for the Lindblad operators\ncorresponding to $x$ direction the fidelity is the same for EPR and\n$n$GHZ states where $n\\in\\{3,4,5,6\\}$. However, 3GHZ state does lose\nless quantum information for other types of noisy channel. Note\nthat, In Ref.~\\cite{jung08-2} the authors only studied the same-axis\nnoisy channels and conjectured that ``average fidelity with\nsame-axis noisy channels are in general larger than average fidelity\nwith different-axis noisy channels''. However, we showed the failure\nof this conjecture for 4GHZ state which is apparent in\nFig.~\\ref{fig7}. In the appendix we showed this conjecture also\nfails for 3GHZ state (see Fig.~\\ref{fig6}). In fact, for\ndifferent-axes noises, the analytical solutions can be obtained in\nthe same way, but the number of coupled differential equations\nusually increases with respect to the same-axes noises.\n\n\\acknowledgments\nWe would like to thank Robabeh Rahimi for fruitful discussions and\nsuggestions and for a critical reading of the paper.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nSince the seminal work of \\citet{Coo95}, \\textit{statistical shape models} became an emerging tool to capture natural shape variability within a given class of objects.\nAs a result, a large number of shape models were developed during the last decades.\nThe probably most well-known models were built for the human face using textured 3D face scans.  \nIntroduced by \\citet{Bla99}, this class of statistical shape models is commonly known as \\textit{3D Morphable Models} (3DMMs) and includes models such as the \\textit{Basel Face Model} (BFM) and \\textit{Large Scale Facial Model} (LSFM) presented by  \\citet{Pay09} and \\citet{Boo16}, respectively.\nWell-studied applications of 3DMMs include face recognition, expression transfer between individuals, face animation, and 3D face reconstruction from a single 2D photograph \\citep{Egg20}.\nEspecially in the last decade, shape analysis also gained popularity in the field of computational anatomy, where statistical shape models are successfully used to model variations of anatomical objects such as bones and organs. \nLater, these models are utilized for a variety of medical applications including (but not limited to) image segmentation, surgical simulation, therapy planning, and motion analysis \\citep{Amb19}.\nDespite the popularity of statistical shape models in the aforementioned areas, to date and to the best of our knowledge, no publicly available 3D statistical shape model of the female breast exists.\n\nWith breast cancer being the most common malignant neoplasm among women \\citep{Sun21}, successful \\textit{breast reconstruction surgery} (BRS) is crucial for patients undergoing mastectomy.\nIn order to give patients a first impression about what their breast might look like after BRS, surgical outcomes are more and more often simulated using patient-specific 3D breast scans, acquired in a standing position (see Section \\ref{sec:related_work} for an overview). \nTypically, simulations are performed using physically motivated deformable models of the breast.\nWhile these models take into account material properties and physical effects such as gravity, they may \\textit{not always} produce realistic-looking shapes as no prior knowledge in form of example shapes is included \\citep{Roo06,Su11}.\nHence, simulated outcomes might be physically plausible, but definitely lack \\textit{statistical plausibility} in the sense that generated breast shapes are somehow \\textit{likely} or similar to those typically observed within a target population.\nAs the ultimate goal of BRS is an outcome looking as \\textit{natural} as possible, we believe that simulation of surgical outcomes must not only rely on physically based deformable models but also should take into account statistical effects.\nIndeed, the phenomenon that humans tend to compare themselves with others clearly underlines the importance of the fact that simulated breast shapes should look similar to the breasts within the target population.\nAs a first step towards combining physical \\textit{and} statistical plausibility of simulated breast surgery outcomes, we propose to use 3D statistical shape models built from natural-looking female breasts.\nIn addition, by introducing such models into the breast shape domain, many of the aforementioned applications from other domains could be transferred to the breast as we will exemplary show later in this article.\n\nTo this end, this paper introduces the first publicly available 3D statistical shape model of the female breast built from 110 breast scans.\nTogether with the model, we present a fully automated, pairwise registration pipeline especially tailored for 3D breast scans and its application in the context of statistical shape modeling.\nOur method is computationally efficient and requires only four landmarks to guide the registration process.\n\n\\subsection{Challenges}\nCompared to shape modeling of most parts of the human body, building a statistical shape model for the female breast imposes some new challenges as discussed in the following.\n\n\\textbf{Data acquisition.}\nFirstly, acquiring a sufficient amount of high-quality training data is challenging.\nUsually, 3D scanning and \\textit{manual} landmark detection is an uncomfortable situation for the participants, in which their upper body is required to be naked.\nMoreover, landmarks can be identified only through palpation and by using a regular tape measure, both requiring a physical examination in a clinical environment.\nIn addition, during the \\textit{whole} examination, a specified posture needs to be held fixed ensuring a similar pose across all subjects.\nThis can be very exhausting, especially if 3D breast scans are taken in a standing position in which both arms should ideally be held away from the body in order to capture the breast as isolated as possible.\nAs a result, 3D scanning protocols used in clinical practice are often designed to be carried out relatively fast, thus lacking necessary precision for pose standardization, see Figure \\ref{fig:database_samples}.\nNote that the problem of only \\textit{quasi-similar} postures was also recently observed by \\citet{Maz21}.\nAll in all, the aforementioned factors definitely hinder the implementation of large-scale, high-quality data surveys.\n\n\\begin{figure}[b]\n\\includegraphics{.\/images\/1_database_samples}\n\\caption{Three typical 3D breast scans sampled from our database.\nAlthough a common pose was declared during data acquisition, a lot of pose variations are still present (indicated through a skeleton drawn in red). \nThese mainly emerge from the arms and shoulders.}\n\\label{fig:database_samples}\n\\end{figure}\n\n\\begin{figure}[t]\n\\includegraphics[width=88.5mm]{.\/images\/2_anatomy}\n\\caption{A brief overview of common landmarks and key anatomical structures of the female breast and thorax. \nThe illustration was adapted from \\citet{Hwa15}.}\n\\label{fig:anatomy}\n\\end{figure}\n\n\\textbf{Correspondence estimation.}\nSecondly, establishing correspondence among 3D breast scans by means of surface registration (rigid and non-rigid) is difficult due to the lack of reliable landmarks.\nIn essence, only four valid landmarks can be used for non-rigid registration.\nThese include both nipples as well as both lower breast poles (throughout this work defined as \\textit{the lowest (most caudal) point of the breast}, see Figure \\ref{fig:anatomy}). \nAnatomical landmark points outside the breast region (such as the sternal notch or processus coracoideus) cannot be used for non-rigid registration purposes as they would recover undesired shape variations during statistical shape modeling.\nThis way, the processus coracoideus disqualifies as its position depends strongly on the position of the arms and shoulders.\nOn the other hand, the xiphoid, located at the center of the thorax, \\textit{could} technically be used for registration. \nHowever, it cannot be reliably and consistently located across a wide range of differently-shaped breasts because of its dorsal tilt, effectively hindering the identification of a unique point.\nAnother fundamental problem is the complete lack of robust landmarks on the lateral part of the breast. \nAlthough \\citet{Har20} defined a lateral breast pole as the orthogonal intersection between the anterior axillary line and a line passing through the nipple, this point cannot be used for non-rigid registration as its position is also affected by the pose of the arms and shoulders.\nFinally, already the initial, rigid alignment of 3D breast scans is challenging due to the lack of reliable landmarks that do not undergo large soft tissue deformations.\n\n\\textbf{Non-separability of breast and thorax.} \nLast but most importantly, the region of interest, i.e. the breast, obviously cannot be well separated from the rest of the thorax when considering 3D breast scans.\nThis is primarily because the chest wall separating the breast from the thorax cannot be captured using 3D surface scanning devices, but also due to the reason that no commonly accepted and \\textit{exact} definition of the breast contour exists, see e.g. \\citet{Lot21}. \nA statistical shape model built from 3D breast scans will therefore \\textit{necessarily} capture also shape variations of the thorax, even after pose standardization.\nIn particular, these include morphological shape variations of the underlying chest wall and upper abdomen, but also those emerging from the arms, armpits, and shoulders due to improper pose standardization (also seen from Figure \\ref{fig:database_samples}).\nIf not reduced to a minimum, this will cause the following unwanted effect.\nSince breast and thorax shapes are tightly correlated in the subspace spanned by the model, the range of representable breast shapes is limited.\nThe reason is that a considerably large part of the subspace accounts for the unwanted shape variations of the surrounding regions.\nHence, the breast area should be decoupled from the thorax as much as possible in order to build an expressive and well-performing statistical shape model.\n\n\\subsection{Contributions}\nAs key contribution, this work presents the \\textit{Regensburg Breast Shape Model} (RBSM) -- a 3D statistical shape model of the female breast.\nIn order to weaken the strong coupling between the breast and surrounding regions, we propose to minimize the \\textit{variance} outside the breast region as much as possible.\nTo achieve this goal, a novel concept called \\textit{breast probability masks} (BPMs) is introduced. \nA BPM assigns probabilities to each point of a 3D breast scan, telling how \\textit{likely} it is that a particular point belongs to the breast region. \nLater, during pairwise registration, we use the BPMs to align the template to the target as accurately as possible \\textit{inside} the breast region and only roughly outside. \nThis way, only the most prominent and global shape variations outside the breast region will be recovered, effectively reducing the unwanted variance in these areas to a minimum. Figure \\ref{fig:overview_approach} illustrates this idea.\n\nTo summarize, the contributions of this paper are three-fold:\n\\begin{itemize}\n\\item We introduce the \\textit{Regensburg Breast Shape Model} (RBSM) -- an open-access 3D statistical shape model of the female breast built from 110 breast scans. It is available at \\url{https:\/\/www.rbsm.re-mic.de\/}.\n\\item We propose a fully automated, pairwise registration pipeline used to establish correspondence among our 3D breast scans. It uses BPMs to decouple the breast from the surrounding regions as much as possible and requires only four landmarks to guide the registration process.\n\\item We present two exemplary applications demonstrating how the RBSM can be used for surgical outcome simulation and the prediction of a missing breast from the remaining one.\n\\end{itemize}\nThe remainder of this paper is organized as follows: Section \\ref{sec:related_work} briefly reviews some related work. \nSection \\ref{sec:methodology} describes the entire model building pipeline used to construct the RBSM. \nIn particular, it formally introduces the notion of a BPM, followed by a detailed description of the proposed registration pipeline.\nSection \\ref{sec:evaluation} presents an extensive evaluation of the RBSM in terms of the common metrics compactness, generalization, and specificity. \nUsing the RBSM, two exemplary applications are showcased in Section \\ref{sec:applications}.\nFinally, Section \\ref{sec:discussion} discusses the results whereas Section \\ref{sec:conclusion} concludes this article.\n\n\n\\section{Related work}\n\\label{sec:related_work}\nIn this section, we briefly summarize some related work concerning breast surgery simulation, statistical shape models of the female breast, and popular techniques for pairwise surface registration used within the context of statistical shape modeling in general.  \nNote that we have limited our review to the 3D case. \n\n\\textbf{Breast surgery simulation.}\nMost of the existing methods for pre-operative breast surgery simulation are designed to simulate alloplastic, implant-based breast augmentation procedures, either for aesthetic reasons or after mastectomy as part of BRS.\nTypically, those methods first generate a patient-specific, geometric representation of the breast (using tetrahedral meshes, for instance). \nAfterwards, the soft tissue deformation caused by implant insertion is simulated using a geometric and biomechanical model of the implant and breast, respectively.\n\n\\begin{figure}[t]\n\\includegraphics[]{.\/images\/3_overview_approach}\n\\caption{The proposed concept of BPMs (top row) allows to minimize unwanted shape variations of the thorax by registering a template surface as precisely as possible \\textit{inside} the breast region and only roughly outside. \nThe breast region is defined by BPMs in a probabilistic manner (top left, red areas correspond to a high probability of belonging to the breast region). \nThis simple yet effective strategy decouples the breast from the surrounding regions by reducing the variance outside the breast area. \nIn the last column, the per-vertex variance over the whole dataset is visualized on the resulting mean shape. \nThe regions showing the highest variance (red) are almost coincident with the breasts in our proposed BPM-based approach.\nContrary, without BPMs (bottom row), a lot of unwanted variance is present in the surrounding regions, especially around the arms, armpits, shoulders, and upper abdomen. \nThis implies a strong coupling between breast and thorax in the final statistical shape model.}\n\\label{fig:overview_approach}\n\\end{figure}\n\nAs such, \\citet{Roo06} used the tensor-mass model introduced by \\citet{Cot00}, a combination of classical finite element and mass-spring models, for implant-based breast augmentation planning.\n\\citet{Cie12} proposed a web-based tool for breast augmentation planning which requires only 2D photographs and anthropometric measurements as input and allows the user to choose from a variety of different implants. \nTheir method automatically reconstructs a 3D breast model and subsequently applies a tissue elastic model closely resembling the finite element model.\n\\citet{Geo14} utilizes patient-specific finite element models generated from 3D breast scans. \nCombined with a novel mechanism called \\textit{displacement template}, geometric implant models are no longer required, thus breaking up the coupling between implant and enclosing breast.\n\n\\begin{figure*}[t!]\n\\centering\n\\includegraphics{.\/images\/4_overview_pipeline}\n\\caption{An overview of the pipeline used to build the RBSM. \nWe first establish dense correspondence by means of pairwise surface registration. \nBased on BPMs, our method starts by rigidly aligning the template to the target. \nSubsequently, a non-rigid alignment is applied in a hierarchical, multi-resolution fashion. \nIn this step, BPMs are used to precisely recover only shape variations of the breast.\nFinally, we perform classical \\textit{Generalized Procrustes Analysis} (GPA) and \\textit{Principal Component Analysis} (PCA) to build the model.}\n\\label{fig:overview_pipeline}\n\\end{figure*}\n\nBesides the simulation of breast augmentation procedures using implants, some methods exist especially addressing the simulation of BRS \\textit{without} implant insertion, for example by means of autologous fat tissue.\nBased on Pascal's principle and volume conservation, \\citet{Cos01} developed a novel approach for real-time physically based simulation of deformable objects, called \\textit{Long Elements Method} (LEM).\nAn extension of LEM, known as the \\textit{Radial Elements Method} \\citep{Bal03}, was later used by \\citet{Bal06} for cosmetic and reconstructive breast surgery simulation.\n\\citet{Wil03} employ a finite element approach incorporating the Mooney-Rivlin hyperelastic material model for the realistic simulation of soft tissue to simulate \\textit{transverse rectus abdominis myocutaneous} flap breast reconstruction.\n\nFor most of the existing methods, however, various authors highlighted the disagreement between simulated and actual outcomes \\citep{Cru15,Roo06} or only partially satisfying results for certain types of breast shapes \\citep{Vor17}.\n\n\\textbf{Statistical shape models of the female breast.}\nLiterature about 3D statistical shape models of the female breast is sparse.\nIn the early work of \\citet{Seo07}, a 3D statistical shape model was built from 28 breast scans with the goal of analyzing breast volume and surface measurements.\nHowever, they assume symmetric breasts obtained by simply mirroring the right breast.\nTo date and to the best of our knowledge, this is the only work primarily addressing the construction of a statistical shape model from 3D breast scans, which is thus closest to our work.\nBesides, \\citet{Rui18} utilized a 3D statistical shape model built from 310 breast scans for the validation of a novel weighted regularized projection method used for 3D reconstruction.\nAs their focus did not lie on the construction of a well-performing statistical shape model of the breast, they did not provide detailed information about the registration method used to establish correspondence, the training data nor a comprehensive evaluation in terms of common metrics.\n\n\\textbf{Pairwise surface registration.}\nDuring the last few decades, countless algorithms were proposed tackling the problem of (pairwise) non-rigid surface registration.\nTo date, however, none of them were used for 3D breast scan registration.\n\nOne of the most widely used methods is the \\textit{Optimal Step Non-rigid Iterative Closest Point} (NICP) framework proposed by \\citet{Amb07} and based on the work of \\citet{All03}. \nNICP is the only method already used in the female breast shape domain to reconstruct a 3D breast model from a sequence of depth images \\citep{Lac19}.\nMoreover, NICP was employed to construct the BFM and LSFM.\nA second class of non-rigid registration methods is based on splines, such as \\textit{Thin Plate Splines} (TPS) or B-splines.\nPioneered by \\citet{Boo89}, TPS were utilized by \\citet{Pau02} to construct a statistical shape model of the human ear canal.\nB-splines are extensively used in the \\textit{free-form deformations} framework and, among others, used for the creation of shape models of the human heart \\citep{Ord07}.\nA third, recently introduced framework is based on \\textit{Gaussian Process Morphable Models} (GPMMs) introduced by \\citet{Lue18}.\nGPMMs are statistical shape models themselves generalizing classical point-based models as proposed by \\citet{Coo95}.\nBy means of GPMMs, expected deformations can be modeled using analytic covariance functions and later used as a \\textit{prior} for non-rigid surface registration, thus effectively reducing the search space. \nThis approach was successfully used by \\citet{Ger17} to build an improved version of the BFM.\nLastly, pairwise non-rigid surface registration algorithms are built upon the \\textit{as-rigid-as-possible} (ARAP) real-time mesh deformation framework proposed by \\citet{Sor07}.\nThis method was successfully transferred to the registration domain, yielding to similar methods constraining deformations to be \\textit{as conformal as possible} \\citep{Yos14} or \\textit{as similar as possible} \\citep{Jia17,Yam13}. \nRecently, a variant of ARAP was utilized by \\citet{Dai20} to built a shape model of the full human head. \n\n\\section{Methodology}\n\\label{sec:methodology}\nThis section describes the entire pipeline used to build the RBSM. \nAs outlined in Figure \\ref{fig:overview_pipeline}, we start by establishing correspondence among our training data.\nBased on a novel concept called \\textit{breast probability masks} (Section \\ref{subsec:bpms}), this is achieved by means of a fully automated, pairwise registration pipeline as proposed in Section \\ref{subsec:registration}.\nFinally, we follow the typical workflow used to build a point-based statistical shape model by applying a \\textit{Generalized Procrustes Analysis} and a \\textit{Principal Component Analysis} to the registered data set (briefly summarized in Section \\ref{subsec:model_building}).\n\nIn what follows, 3D breast scans are represented using triangular surface meshes.\nA triangle mesh $\\mathcal{M}=(V,E,\\mathcal{P})$ is fully specified by a set of $n$ vertices $V\\subset\\mathbb{N}$, edges $E\\subset V\\times V$, and an embedding $\\mathcal{P}=\\{\\mathbf{p}_1,\\mathbf{p}_2\\ldots,\\mathbf{p}_n\\}\\subset\\mathbb{R}^3$. \nSometimes, however, instead of arranging points $\\mathbf{p}_i$ into a set, it is more convenient to use a matrix notation $\\mathbf{P}=(\\mathbf{p}_1,\\mathbf{p}_2,\\ldots,\\mathbf{p}_n)^\\top\\in\\mathbb{R}^{n\\times 3}$.\nHence, we will denote a triangle mesh either as $\\mathcal{M}=(V,E,\\mathcal{P})$ or equivalently as $\\mathcal{M}=(V,E,\\mathbf{P})$.\n\n\\subsection{Breast probability masks}\n\\label{subsec:bpms}\nGiven a 3D breast scan represented as triangle mesh $\\mathcal{M}=(V,E,\\mathcal{P})$, we call\n\\begin{equation}\np_\\mathcal{M}:\\mathcal{P}\\longrightarrow(0,1]\n\\end{equation}\na \\textit{breast probability mask} (BPM). Technically, a BPM is a scalar field defined over $\\mathcal{M}$ assigning each point $\\mathbf{p}_i$ of a 3D breast scan a probability $p_\\mathcal{M}(\\mathbf{p}_i)$ telling how \\textit{likely} it is that $\\mathbf{p}_i$ belongs to the breast region.\n\n\\textbf{Concrete mapping.} As a concrete mapping for $p_\\mathcal{M}$ we propose to use a normalized sum of \\textit{elliptical basis functions} (EBFs), centered at the nipples.\nWe use EBFs instead of ordinary \\textit{radial basis functions} (RBFs) because we found that they better capture the natural teardrop shape of the breast (see Figure \\ref{fig:comparison_bpms} for a comparison between RBFs and EBFs).\nTechnically, EBFs are a generalization of RBFs using the Mahalanobis distance instead of an ordinary vector norm.\nFormally, an EBF $\\phi:[0,\\infty)\\longrightarrow\\mathbb{R}$ centered at a point $\\mathbf{c}\\in\\mathbb{R}^n$ is of the form $\\phi(\\mathbf{x})=\\phi\\left(d_M\\left(\\mathbf{x},\\mathbf{c}\\right)\\right)$. \nHere, $d_M$ is the Mahalanobis distance, defined as\n\\begin{equation}\nd_M(\\mathbf{x},\\mathbf{c})\\coloneqq\\sqrt{\\left(\\mathbf{x}-\\mathbf{c}\\right)^\\top\\mathbf{S}^{-1}\\left(\\mathbf{x}-\\mathbf{c}\\right)}\\,,\n\\end{equation}\nwhere $\\mathbf{S}\\in\\mathbb{R}^{n\\times n}$ is a symmetric positive definite matrix, also called \\textit{covariance matrix}.\nTo stress that the Mahalanobis distance depends on $\\mathbf{S}$, we write $d_M(\\mathbf{x},\\mathbf{c};\\mathbf{S})$ in the following.\n\nNow, in order to define a concrete BPM using EBFs, let $\\mathbf{p}_\\text{N}^\\tau\\in\\mathcal{P}$ denote the position of the left (L) and right (R) nipple, respectively, and $\\tau\\in\\{\\text{L, R}\\}$.\nWe first construct two individual probability masks for the left and the right breast, given as\n\\begin{equation}\np_\\mathcal{M}^\\tau(\\mathbf{p}_i)=\\phi\\left(d_M\\left(\\mathbf{p}_i,\\mathbf{p}_\\text{N}^\\tau;\\mathbf{S}_\\tau\\right)\\right).\n\\end{equation}\nHereby, we define $\\phi:[0,\\infty)\\longrightarrow(0,1]$ as \n\\begin{equation}\n\\phi(x)=\\exp\\left(-x^2\\right).\n\\end{equation}\nFinally, the BPM for a whole 3D breast scan is given as the normalized sum\n\\begin{equation}\np_\\mathcal{M}(\\mathbf{p}_i)=\\frac{1}{4}\\left(p_\\mathcal{M}^\\text{L}(\\mathbf{p}_i)+\\hat{p}_\\mathcal{M}^\\text{L}(\\mathbf{p}_i)+p_\\mathcal{M}^\\text{R}(\\mathbf{p}_i)+\\hat{p}_\\mathcal{M}^\\text{R}(\\mathbf{p}_i)\\right),\n\\end{equation}\nwhere \n\\begin{equation}\n\\hat{p}_\\mathcal{M}^\\tau(\\mathbf{p}_i)=\\phi\\left(d_M\\left(\\mathbf{p}_i,\\mathbf{\\hat{p}}_\\text{N}^\\tau;\\mathbf{\\hat{S}}_\\tau\\right)\\right)\n\\end{equation}\nare shifted BPMs of the left and right breast added to better mimic the teardrop shape, and $\\mathbf{\\hat{p}}_\\text{N}^\\tau=\\mathbf{p}_\\text{N}^\\tau+\\mathbf{t}_\\tau$ with translation vectors $\\mathbf{t}_\\tau\\in\\mathbb{R}^3$.\n\n\\begin{figure}[h!]\n\\includegraphics[]{.\/images\/5_comparison_bpms}\n\\caption{From left to right: comparison between RBF, EBF, and a sum of two EBFs, illustrated as contour plots. \nWhile a simple RBF or EBF is not able to accurately mimic the typical teardrop shape of the breast, a sum of two EBFs comes close.}\n\\label{fig:comparison_bpms}\n\\end{figure}\n\n\\textbf{Parameter selection.} In order to fully define a BPM, appropriate matrices $\\mathbf{S}_\\tau,\\mathbf{\\hat{S}}_\\tau\\in\\mathbb{R}^{3\\times 3}$ and translation vectors $\\mathbf{t}_\\tau\\in\\mathbb{R}^3$ need to be chosen first.\nAs such, a total of 30 values are required to be properly determined (six for each $\\mathbf{S}_\\tau$ and $\\mathbf{\\hat{S}}_\\tau$, and three for each $\\mathbf{t}_\\tau$). \nTo simplify that task, we assume diagonal covariance matrices and utilize previously marked landmarks on the 3D breast scans.\nSpecifically, denote the landmark points shown in Figure \\ref{fig:anatomy} as $\\mathbf{p}_\\text{SN},\\mathbf{p}_\\text{XI}\\in\\mathcal{P}$ for sternal notch and xiphoid, and $\\mathbf{p}_\\text{LaBP}^\\tau,\\mathbf{p}_\\text{LBP}^\\tau\\in\\mathcal{P}$ for left and right lateral and lower breast pole, respectively.\nWe then define\n\\begin{equation}\n\\begin{split}\n\\mathbf{S}_\\tau&=\n\\!\\begin{multlined}[t][60mm]\n\\frac{1}{2}\\diag\\Bigl(d_G\\left(\\mathbf{p}^\\tau_\\text{LaBP},\\mathbf{p}^\\tau_\\text{N}\\right)+d_G\\left(\\mathbf{p}^\\tau_\\text{N},\\mathbf{p}_\\text{XI}\\right),\\\\\nd_G\\left(\\mathbf{p}^\\tau_\\text{N},\\mathbf{p}^\\tau_\\text{LBP}\\right), d_G\\left(\\mathbf{p}^\\tau_\\text{LaBP},\\mathbf{p}^\\tau_\\text{N}\\right)\\Bigr)\\,,\n\\end{multlined}\\\\\n\\mathbf{\\hat{S}}_\\tau&=\n\\!\\begin{multlined}[t][60mm]\n\\frac{1}{2}\\diag\\Bigl(d_G\\left(\\mathbf{p}^\\tau_\\text{LaBP},\\mathbf{p}^\\tau_\\text{N}\\right)+d_G\\left(\\mathbf{p}^\\tau_\\text{N},\\mathbf{p}_\\text{XI}\\right),\\\\\nd_G\\left(\\mathbf{p}^\\tau_\\text{N},\\mathbf{p}_\\text{SN}\\right),d_G\\left(\\mathbf{p}^\\tau_\\text{LaBP},\\mathbf{p}^\\tau_\\text{N}\\right)\\Bigr)\\,,\n\\end{multlined}\\\\\n\\mathbf{t}_\\tau&=\\mathbf{p}^\\tau_\\text{N}+\\frac{1}{5}\\left(0,d_G\\left(\\mathbf{p}^\\tau_\\text{N},\\mathbf{p}_\\text{SN}\\right),0\\right),\n\\end{split}\n\\end{equation}\nwhere $d_G$ denotes the Geodesic distance between two points on the surface mesh.\nNote that $\\mathbf{S}_\\tau$ and $\\mathbf{\\hat{S}}_\\tau$ differ only in the second diagonal element.\n\n\\subsection{Registration of 3D breast scans}\n\\label{subsec:registration}\nFollowing Figure \\ref{fig:overview_pipeline}, the proposed pairwise registration pipeline is mainly composed of rigid alignment (Section \\ref{subsubsec:rigid}) and non-rigid alignment (Section \\ref{subsubsec:non-rigid}). \nFollowing common practice, the latter is carried out in a hierarchical, multi-resolution fashion (Section \\ref{subsubsec:multi_res_fitting}).\n\nBoth phases make extensive use of BPMs in order to align a template surface $\\mathcal{S}=(V,E,\\mathbf{P})$ to a target $\\mathcal{T}$ as accurately as possible \\textit{inside} the breast region and only roughly outside, effectively decoupling the breast from the rest of the thorax by reducing the \\textit{variance} outside the breast region to a minimum.\nThis is justified as the covariance $\\cov(x,y)$ becomes smaller if $\\var(x)$ or $\\var(y)$ is lowered, following from the well known fact that $\\left|\\cov(x,y)\\right|\\leq\\sqrt{\\var(x)}\\sqrt{\\var(y)}$ (which holds via the Cauchy\u2013Schwarz inequality).\n\nFinally, note that the target surface $\\mathcal{T}$ can be given in any representation that allows for closest point search.\nWe use a triangular surface mesh but write $\\mathcal{T}\\subset\\mathbb{R}^3$ for the sake of notational simplicity.\n\n\\subsubsection{Rigid alignment}\n\\label{subsubsec:rigid}\nThe overall goal of the rigid alignment is to move the template as close as possible to the \\textit{rigid part} of the target, which we define as \\textit{the thorax without the breast}.\nIn particular, we expect that the thoraxes of two subjects without the breast region can be sufficiently well aligned if we assume the breast to be the only part of the thorax that deforms non-rigidly.\nBased on this assumption, the absence of suitable landmarks, and due to the fact that our initial 3D breast scans are already reasonably well aligned (see Section \\ref{subsec:data}) we propose a modified version of the \\textit{ Iterative Closest Point} (ICP) algorithm, originally introduced by \\citet{Bes92}.\n\nEssentially, compared to the standard version of the ICP algorithm, our modified version differs in the following three aspects: (i) a scaling factor is added to the rigid transformation effectively allowing for Euclidean similarity transformations \\citep{Du07,Zin05}. \nSecondly, (ii) to ensure that only the rigid parts of the 3D breast scans are used for alignment, correspondences, where both points have a high probability belonging to the breast region, are discarded.\nThis is implemented by thresholding the template and target BPMs.\nFinally, (iii) rotations are restricted to the $x$-axis corresponding to the transversal plane.\nRotations around the $y$- and $z$-axis (sagittal and coronal plane), possibly introduced due to severe overweight in conjunction with an uneven distribution of abdominal fat could destroy the initial alignment and lead to misalignment.\nIn any case, asymmetries introduced due to the thorax should \\textit{not} affect the rigid alignment of the template.\n\n\\subsubsection{Non-rigid alignment}\n\\label{subsubsec:non-rigid}\nGiven the rigidly aligned template $\\mathcal{S}=(V,E,\\mathbf{P})$, the goal of the non-rigid alignment is to gradually deform $\\mathcal{S}$ into a new surface $\\mathcal{S}'=(V,E,\\mathbf{P}')$ with identical topology such that $\\mathcal{S}'$ is as close as possible to the target $\\mathcal{T}$ \\textit{inside} the breast region.\nFollowing various authors including \\citet{Jia17} and \\citet{Yam13}, we formulate our non-rigid registration problem using the following non-linear energy functional\n\\begin{equation}\nF\\left(\\mathbf{P}'\\right)=F_D\\left(\\mathbf{P}'\\right)+\\alpha F_R\\left(\\mathbf{P}'\\right)+\\beta F_L\\left(\\mathbf{P}'\\right),\n\\label{eq:non_rigid_cost}\n\\end{equation}\nwhere $F_D$ is a distance term used to penalize the point-to-point distance between the template and target surface, $F_R$ is a regularization term constraining deformations \\textit{as similar as possible}, and $F_L$ constitutes a landmark term ensuring certain points to be matched.\n$\\alpha,\\beta\\geq 0$ are weights controlling the individual contribution of each term to the cost function.\nMinimizing $F$ finally leads to the new points $\\mathbf{P}'$ of the deformed template surface $\\mathcal{S}'$, i.e.\n\\begin{equation}\n\\mathbf{P}'=\\argmin_{\\mathbf{P'}\\in\\mathbb{R}^{n\\times 3}}F(\\mathbf{P'}).\n\\label{eq:non_rigid_opt_problem}\n\\end{equation}\nAdapting the strategy proposed by \\citet{All03}, instead of computing (\\ref{eq:non_rigid_opt_problem}) only once, we minimize $F$ several times but each time lowering the regularization weight $\\alpha$ in (\\ref{eq:non_rigid_cost}).\nAs later demonstrated by \\citet{Amb07}, this strategy is able to recover the whole range of global and local non-rigid deformations efficiently.\nFollowing various authors \\citep{Jia17,Sor07}, the optimization problem in (\\ref{eq:non_rigid_opt_problem}) is solved using an alternating minimization (AM) approach as briefly summarized in \\ref{app:solve_am}.\n\n\\textbf{Distance term.}\nThe distance term $F_D$ is used to attract the template $\\mathcal{S}$ to the target $\\mathcal{T}$.\nAssuming fixed correspondences between both surfaces, i.e. $\\left\\{(\\mathbf{p}_1,\\mathbf{q}_1),(\\mathbf{p}_2,\\mathbf{q}_2),\\ldots,(\\mathbf{p}_n,\\mathbf{q}_n)\\right\\}$ with $\\mathbf{q}_i\\in\\mathcal{T}$ being the closest point to $\\mathbf{p}_i$, the distance term can be written as\n\\begin{equation}\nF_D(\\mathbf{P}')=\\frac{1}{2}\\left\\Vert\\mathbf{C}\\left(\\mathbf{P}'-\\mathbf{Q}\\right)\\right\\Vert^2_F,\n\\end{equation}\nwhere $\\mathbf{C}\\coloneqq\\text{diag}(c_1,c_2,\\ldots,c_n)$, $c_i\\geq 0$ for all $i\\in\\left\\{1,2,\\ldots,n\\right\\}$ are weights used to quantify the reliability of a match, and $\\mathbf{Q}\\coloneqq\\left(\\mathbf{q}_1,\\mathbf{q}_2,\\ldots,\\mathbf{q}_n\\right)^\\top\\in\\mathbb{R}^{n\\times 3}$.\nUsing the BPMs $p_\\mathcal{S}$ and $p_\\mathcal{T}$ of the template and target, we set\n\\begin{equation}\nc_i=\\frac{p_\\mathcal{S}(\\mathbf{p}_i)+p_\\mathcal{T}(\\mathbf{q}_i)}{2}.\n\\end{equation}\nThis way, correspondences $(\\mathbf{p}_i,\\mathbf{q}_i)$ mapping from one breast region to the other have a greater impact on the overall distance term as $c_i\\in(0,1]$ becomes large in this case.\nConversely, the influence tends to zero if $c_i\\rightarrow 0$, i.e. if both points are less likely to belong to the breast region.\nAs such, the deformation of points $\\mathbf{p}_i$ on the template with a small value for $c_i$ is mainly controlled by the regularization term, as previously described by \\citet{All03}.\n\n\\textbf{Regularization term.}\nThe regularization term $F_R$ should prevent the template surface from shearing and distortion while simultaneously ensuring structure preservation and smooth deformations.\nTo do so, we adapt the \\textit{consistent as-similar-as-possible} (CASAP) regularization technique in which deformations are constrained to be \\textit{locally} as similar as possible \\citep{Jia17,Yam13}.\nSpecifically, given a local neighborhood $E_i\\subset E$ around each point $\\mathbf{p}_i$, the template surface is only allowed to move in terms of an Euclidean similarity transformation\n\\begin{equation}\n\\mathbf{p}'_j-\\mathbf{p}'_k=s_i\\mathbf{R}_i\\left(\\mathbf{p}_j-\\mathbf{p}_k\\right)\\qquad\\forall(j,k)\\in E_i,\n\\label{eq:asap}\n\\end{equation}\nwhere $s_i>0$ is a scaling factor and $\\mathbf{R}_i\\in\\SO(3)$ a rotation matrix.\nFollowing \\citet{Cha10}, we define $E_i$ as \\textit{the set containing all (directed) edges of triangles incident to} $\\mathbf{p}_i$, also known as \\textit{spokes-and-rims}.\nFinally, our CASAP regularization term reads\n\\begin{multline}\nF_R(\\mathbf{P}')=\\frac{1}{2}\\sum_{i=1}^n w_i\\left[\\sum_{(j,k)\\in E_i}w_{jk}\\left\\Vert\\left(\\mathbf{p}'_j-\\mathbf{p}'_k\\right)-s_i\\mathbf{R}_i\\left(\\mathbf{p}_j-\\mathbf{p}_k\\right)\\right\\Vert^2_2+\\right.\\\\\n\\left.\\lambda\\sum_{l\\in N_i}w_{il}\\left\\Vert\\mathbf{R}_i-\\mathbf{R}_l\\right\\Vert^2_F\\right]\\,,\n\\label{eq:casap_regularization}\n\\end{multline}\nwhere weights $w_i>0$ are added to individually control the amount of regularization for each particular point.\nAs mentioned above, since the deformation of points $\\mathbf{p}_i$ with a small value for $c_i$ is mainly controlled by the regularization term, we define\n\\begin{equation}\nw_i=\\frac{1}{(h-1)c_i+1}\\qquad\\text{with}\\qquad\\frac{1}{h}\\leq w_i<1\n\\end{equation}\nfor all $i\\in\\{1,2,\\ldots,n\\}$ and $h\\in\\mathbb{N}^+$ (we used $h=2$ throughout this paper).\nAs seen, this strategy keeps points $\\mathbf{p}_i$ of the template relatively stiff if (i) $\\mathbf{p}_i$ has a low probability belonging to the breast region and (ii) if the corresponding point on the target is also not likely to be part of the breast region (because $w_i\\rightarrow 1$ if $c_i\\rightarrow 0$), thus effectively preventing the template from adapting too close to the target outside the breast region.\nLastly, $N_i\\subset V$ in (\\ref{eq:casap_regularization}) denotes the one-ring neighborhood of the $i$-th point and $w_{jk}\\in\\mathbb{R}$ are the popular cotangent weights, see e.g. \\citet{Bot10}.\n$\\lambda\\geq 0$ is usually set to $0.02A$, where $A\\geq 0$ is the total surface area of $\\mathcal{S}$ \\citep{Lev14}.\n\n\\textbf{Landmark term.}\nThe goal of the landmark term $F_L$ is to keep certain positions (i.e. landmarks) fix during the registration process.\nLet $I\\subset\\mathbb{N}$ be an index set containing the indices of the $m$ landmarks specified on the template surface $\\mathcal{S}$.\nDefine a matrix $\\mathbf{D}\\in\\mathbb{R}^{m\\times n}$ as\n\\begin{equation}\n\\mathbf{D}=(d_{ij}):=\n\\begin{cases}\n1, & \\text{if }j\\in I,\\\\\n0, & \\text{otherwise}\n\\end{cases}\n\\end{equation}\nfor $i=1,2,\\ldots,m$ and $j=1,2,\\ldots,n$.\nNext, denote the corresponding landmarks on the target surface by $\\{\\mathbf{q}_1,\\mathbf{q}_2,\\ldots,\\mathbf{q}_m\\}\\subset\\mathcal{T}$. \nThen, the landmark term is defined as\n\\begin{equation}\nF_L(\\mathbf{P}')=\\frac{1}{2}\\left\\Vert\\mathbf{DP}'-\\mathbf{Q}_L\\right\\Vert^2_F,\n\\end{equation}\nwhere $\\mathbf{Q}_L\\coloneqq\\left(\\mathbf{q}_1,\\mathbf{q}_2,\\ldots,\\mathbf{q}_m\\right)^\\top\\in\\mathbb{R}^{m\\times 3}$.\n\n\\subsubsection{Multi-resolution fitting strategy}\n\\label{subsubsec:multi_res_fitting}\nFollowing common practice, instead of applying the previously described non-rigid alignment only once, we employ a hierarchical, multi-resolution fitting strategy composed of initial fitting, coarse fitting, and fine fitting (see also Figure \\ref{fig:overview_pipeline}).\n\n\\textbf{Initial fitting.}\nHaving a low-resolution instance of the rigidly aligned template at hand, the goal of the initial fitting is to roughly adapt the coarse template to the key features (i.e. landmarks) of the target.\nTo do so, we strictly prioritize the landmark constraints and do not use BPMs in this phase.\n\n\\textbf{Coarse fitting.}\nIn this step, the initially fitted low-resolution template is gradually deformed towards the target.\n\n\\textbf{Upsampling.}\nNext, the deformation obtained from the previous step is applied to the original, full-resolution template. \nThis is achieved using a concept called \\textit{Embedded Deformation}, introduced by \\citet{Sum07}.\nIn essence, the deformation of the coarse template obtained from the previous step is transferred to the template by linearly interpolating the transformation at each point.\n\n\\textbf{Fine fitting.}\nLastly, the upsampled template is fitted to the target again to produce the final result. \n\n\\subsection{Model building}\n\\label{subsec:model_building}\nOnce the data set is brought into correspondence, we follow the typical workflow used to build a classical point-based statistical shape model as proposed by \\citet{Coo95}.\nFor notational simplicity, instead of stacking points $\\mathcal{P}$ of a triangular mesh $\\mathcal{M}=(V,E,\\mathcal{P})$ into a matrix $\\mathbf{P}\\in\\mathbb{R}^{n\\times 3}$ as before, we use a vectorized representation, denoted as $\\mathbf{x}=\\vecz(\\mathbf{P})\\in\\mathbb{R}^{3n}$ in the following.\n\nBriefly, given a set of $k$ breast scans $\\{\\mathbf{x}_1,\\mathbf{x}_2,\\ldots,\\mathbf{x}_k\\}\\subset\\mathbb{R}^{3n}$ in correspondence, we first perform a \\textit{Generalized Procrustes Analysis} (GPA) as introduced by \\citet{Gow75}.\nGPA iteratively aligns the objects to the arithmetic mean $\\mathbf{\\bar{x}}\\in\\mathbb{R}^{3n}$ (successively estimated from the data) by using an Euclidean similarity transformation, effectively transforming the objects into the shape space.\nSecondly, a \\textit{Principal Component Analysis} (PCA) is carried out on the Procrustes-aligned shapes.\nLet $\\{\\lambda_1,\\lambda_2,\\ldots,\\lambda_q\\}\\subset\\mathbb{R}^+$ be the $q<k$ non-zero eigenvalues (also called \\textit{Principal Components} (PCs) in this context) of the empirical covariance matrix calculated from the data and sorted in a descending order.\nDenote the corresponding eigenvectors as $\\{\\mathbf{u}_1,\\mathbf{u}_2,\\ldots,\\mathbf{u}_q\\}\\subset\\mathbb{R}^{3n}$.\nThen, a statistical shape model can be interpreted as a linear function $M:\\mathbb{R}^q\\longrightarrow\\mathbb{R}^{3n}$ defined as\n\\begin{equation}\nM(\\boldsymbol\\alpha)=\\mathbf{\\bar{x}}+\\boldsymbol\\alpha\\mathbf{U},\n\\end{equation}\nwhere $\\mathbf{U}\\coloneqq(\\mathbf{u}_1,\\mathbf{u}_2,\\ldots,\\mathbf{u}_q)^\\top\\in\\mathbb{R}^{q\\times 3n}$.\nTo ensure plausibility of the newly generated shapes, $\\alpha_i$ is usually restricted to $\\left|\\alpha_i\\right|\\leq 3\\sqrt{\\lambda_i}$ for all $i\\in\\{1,2,\\ldots,q\\}$.\nIf a (possibly unseen) shape $\\mathbf{x}'\\in\\mathbb{R}^{3n}$ is in correspondence with the model and properly aligned, it can be reconstructed from $M$ in a least-squares sense by using\n\\begin{equation}\n\\boldsymbol\\alpha^*=\\mathbf{U}^{-1}\\left(\\mathbf{x}' - \\mathbf{\\bar{x}}\\right)\n\\end{equation}\nas the model parameters, i.e. $\\mathbf{x}'\\approx M(\\boldsymbol\\alpha^*)$.\nThe number $q<k$ of retained PCs is chosen so that the model typically represents a fixed proportion of the total variance, e.g. 98\\%.\n\n\\section{Evaluation}\n\\label{sec:evaluation}\nBased on our 3D breast scan database which we present in Section \\ref{subsec:data}, several experiments were conducted in order to evaluate the proposed statistical shape model (Section \\ref{subsec:exp_res}).\n\n\\begin{table}[h]\n\\caption{An overview of the 3D breast scan database used to build the RBSM. It includes 110 textured 3D breast scans.}\n\\label{tab:overview_database} \n\\vspace{2mm}\n\\centering\n\\begin{tabular}{p{22.1mm} l l}\n\\hline\n& Mean ($\\pm$ SD) & Range [min -- max]\\\\\n\\hline\nAge [years] & 40.78 ($\\pm$ 14.30) & 18.00 -- 83.00\\\\\nBMI [kg\/m$^2$] & 23.66 ($\\pm$ 3.57) & 16.90 -- 38.27\\\\\n\\hline\n\\multicolumn{3}{l}{Breast volume [cc]} \\\\\n\\hspace{3mm} Left & 477.23 ($\\pm$ 242.94) & 70.60 -- 1,258.90\\\\\n\\hspace{3mm} Right & 481.45 ($\\pm$ 240.27) & 80.00 -- 1,609.30\\\\\n\\hspace{3mm} Difference & \\hspace{.95mm} 80.12 ($\\pm$ 81.34) & \\hspace{.95mm} 0.30 -- 367.40\\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\\subsection{Data}\n\\label{subsec:data}\nOur database consists of 110 textured 3D breast scans collected at our institution (St. Josef Hospital Regensburg) using the portable Vectra H2$^{\\text{TM}}$ scanning system (Canfield Scientific, New Jersey, USA).\nThe H2 system employs photogrammetry to reconstruct a 3D surface mesh from a series of 2D images with a resolution of 3.5 mm and a maximal capture volume of $70\\times 41\\times 40$ cm$^3$ (width $\\times$ height $\\times$ depth).\nIn our case, in total three photos were taken of each subject: one from a frontal view and two from a lateral view ($\\pm 45^\\circ$).\nNote that due to the standardized distance and angles from which the photos are taken, the reconstructed 3D breast scans are already reasonably well aligned and consistently oriented.\n\n\\begin{figure*}[t!]\n\\centering\n\\includegraphics{.\/images\/7_exp1_model}\n\\caption{From left to right: compactness, generalization, and specificity of statistical shape models built from 3D breast scans registered either with (i.e. the RBSM) or without using BPMs during registration.}\n\\label{fig:exp1_model}\n\\end{figure*}\n\nA previously developed standardized scanning protocol \\citep{Har20} was used to ensure a common pose and same imaging conditions for all participants.\nIn brief, all subjects were asked to stand in an upright posture and abduct both arms by an angle of $45^\\circ$ (a telescopic stick was used to support the subjects in holding their arms fixed at the specified angle).\nThis posture produces a natural-looking breast shape and allows to capture the whole breast as isolated as possible.\nMoreover, it is fast and easy to adopt for all participants regardless of age, body weight, or medical history.\n\nA compact overview of some key parameters in our database is given in Table \\ref{tab:overview_database}. \nIn addition, 62 out of 110 participants have at least one ptotic breast.\nOf these, 15 women show different ptosis grades on the left and right breast.\n72 out of 110 participants received some kind of breast surgery.\n\n\\subsection{Experiments and results}\n\\label{subsec:exp_res}\nFollowing common practice, we evaluate our statistical shape model using the well-known metrics compactness, generalization, and specificity as proposed by \\citet{Sty03}.\nRegistration results are assessed by the classical distance-based \\textit{mean squared error} (MSE) between deformed template and target.\nIn addition, the \\textit{angle distortion} between the deformed template $\\mathcal{S}'$ and the original template $\\mathcal{S}$ is determined in order to quantify the amount of shearing introduced due to non-rigid registration. \nThis is achieved by simply averaging the absolute deviation between the inner-triangle angles of $\\mathcal{S}$ and $\\mathcal{S}'$ over all triangles \\citep{Wan16}.\n\nFor all experiments, the same template $\\mathcal{S}$ was used during registration.\nIn particular, the 3D breast scan of a healthy, non-operated subject was chosen and subsequently mirrored along the sagittal plane to produce a perfectly symmetrical template. \nAfter isotropic remeshing and Laplacian smoothing, a regular surface mesh with an average edge length of 2.05 mm was obtained.\nIt consists of 30,924 vertices.\n\nThe complete registration pipeline was implemented in C++.\nStatistical shape modeling was performed with an open-source framework called Scalismo (\\url{https:\/\/scalismo.org\/}), implemented in Scala.\n\n\n\\subsubsection{On the effect of BPMs}\nThe first experiment should evaluate our novel BPM-based registration technique, its influence on the resulting statistical shape models, and, in particular, whether or not BPMs are able to weaken the strong coupling between the breast and surrounding regions.\nAs such, we compare two different models: the first one is constructed from 3D breast scans brought into correspondence using the proposed approach based on BPMs (i.e. the RBSM). \nThe second model is built from 3D breast scans registered without using BPMs.\n\n\\begin{figure}[h]\n\\includegraphics[]{.\/images\/6_exp1_registration}\n\\caption{MSE (left) and angle distortion (right), calculated from 3D breast scans registered with and without BPMs. }\n\\label{fig:exp1_registration}\n\\end{figure}\nFigure \\ref{fig:exp1_model} shows the evaluation metrics of the resulting statistical shape models. \nAs seen, although the model built without BPMs is more compact than with BPMs (i.e. the RBSM) when considering only the very first PCs, it is also the model that generalizes worse.\nSpecifically, when using all 109 available PCs a generalization error of 0.65 mm is reported.\nFor comparison, the RBSM shows a generalization error of only 0.17 mm when using the same number of PCs. \nFinally, the RBSM constantly achieves the lower specificity error of about 2.8 mm if more than 30 PCs are used.\n\nFigure \\ref{fig:exp1_registration} summarizes the registration results, quantified in terms of MSE and angle distortion.\nAs expected, registration without using BPMs clearly achieves the lower MSE of 1.05 mm as the goal here is to register the template to the target as close as possible.\nFor comparison, the MSE achieved when using BPMs is 2.98 mm.\nHowever, it is important to remember that the goal of the BPM-based registration is to align the template as precisely as possible to the target only \\textit{inside} the breast region.\nTherefore, the overall MSE between template and target might not reflect the actual registration quality well (see e.g. Figure \\ref{fig:overview_approach}).\nIn terms of angle distortion, only a small difference is noticeable between registration with and without BPMs, respectively.\nIn particular, registration without BPMs caused the angle distortion to increase by only 0.008 degrees on average.\n\n\\begin{figure}[h!]\n\\includegraphics{.\/images\/8_exp1_random_samples}\n\\caption{Comparison between five random samples from the RBSM (top row, generated from 3D breast scans registered using BPMs) and five random samples from a model constructed without using BPMs (bottom row). \nThe samples are colored according to the distance from the mean $\\mathbf{\\bar{x}}$. \nAs such, the red areas underwent a high deformation, whereas blue regions are rather stiff. For samples drawn from the RBSM, the regions showing the highest variation are almost always the breast region, indicating a solid decoupling between breast shape and thorax.}\n\\label{fig:exp1_random_samples}\n\\end{figure}\nIn order to further investigate to which extend BPM-based registration is able to weaken the strong coupling between breast region and thorax, Figure \\ref{fig:exp1_random_samples} shows some random samples from the RBSM and the model built from 3D breast scans registered without BPMs.\nThe samples are colored according to their distance to the mean shape, effectively providing a \\textit{measure of variation}.\nInterestingly, the areas of the highest variation in the samples drawn from the RBSM are almost always located at the breasts.\nThe surrounding regions are rather stiff, indicating low variance and a quite well decoupling between breast and thorax.\nHence, the RBSM is able to produce a variety of breast shapes without altering the whole thorax too much, also reflected in the principal modes of variation shown in Figure \\ref{fig:modes_rbsm}. \nContrary, the area of the highest variation in the samples generated from the model built without BPMs is oftentimes \\textit{not} the breast region.\n\n\n\n\n\\begin{figure*}[t!]\n\\includegraphics[width=\\textwidth]{.\/images\/9_modes_rbsm}\n\\caption{The first eight principal modes of variation from the RBSM, visualized by either adding (top row) or subtracting (bottom row) $3\\sqrt{\\lambda_i}\\mathbf{u}_i$ from the mean $\\mathbf{\\bar{x}}$, displayed on the left. \nTogether, they represent about 85\\% of the total variance present in the dataset.}\n\\label{fig:modes_rbsm}\n\\end{figure*}\n\n\n\n\n\\begin{figure*}[b!]\n\\centering\n\\includegraphics{.\/images\/10_exp2_model}\n\\caption{From left to right: compactness, generalization, and specificity of statistical shape models built from randomly sampled subsets of the training data.}\n\\label{fig:exp2_model}\n\\end{figure*}\n\n\\subsubsection{How much data is needed?}\nAn often arising question in the context of statistical shape modeling concerns the amount of training data needed to build a reasonably well-performing model.\nGenerally, this question very much depends on the amount of variability samples from a particular class of objects are expected to show.\nAs a rule of thumb, a good training set should always reflect the whole bandwidth of possible variations likely to occur within a target population.\nThe goal of this experiment is to test how much data is needed to build a well-performing statistical shape model of the female breast.\nTo do so, we randomly sample 30, 60, and 90 breast scans from our database and subsequently compare the resulting models with the model containing all 110 breast scans (i.e. the RBSM).\nTo avoid sampling bias, the whole procedure was repeated three times, and results were averaged.\n\n\n\n\nFigure \\ref{fig:exp2_model} shows the results in terms of compactness, generalization, and specificity.\nRegarding compactness, it can be clearly seen that the model built with 30 breast scans is the most compact one, followed by the models constructed from 60 and 90 breast scans and the RBSM.\nOn the other hand, the model built from only 30 breast scans shows the worst generalization ability (about 1.07 mm when using all 29 PCs available).\nThe model containing 90 breast scans and the RBSM achieved very similar results, both showing a generalization error of about 0.6 mm when considering the first 29 PCs.\nAs expected, whereas the RBSM is the most specific one, the model learned from only 30 breast scans clearly performs the worst. \n\n\\subsubsection{Building a statistical shape model from operated breasts}\nAs already mentioned earlier, a quite big proportion of our 3D breast scans (72 out of 110) result from subjects who already received some kind of breast surgery.\nIt is thus natural to assume that the RBSM might be biased towards operated breasts and is therefore not able to explain non-operated breasts well.\nThe goal of this experiment is thus to verify how well a model built from operated breasts is able to reconstruct 3D breast scans from participants that did not undergo breast surgery.\nIn particular, a statistical shape model is built from a random subset of 37 (out of 72) operated breasts.\nAfterwards, generalization ability is evaluated using the 38 non-operated breasts as test set.\nAgain, to avoid sampling bias, the sampling procedure was repeated three times, and results were averaged.\n37 out of the 38 non-operated breasts are the basis for a second model, calculating the generalization ability using leave-one-out cross-validation.\n\n\n\n\n\n\nThe results are shown in Figure \\ref{fig:exp3_model}.\nAs it can be seen, although our database is obviously biased towards operated breasts, a model built only from operated breasts \\textit{is} able to explain non-operated breasts quite well.\nSpecifically, starting with a generalization error of over 3.5 mm, the generalization ability increases constantly as the number of PCs increase.\nFinally, a generalization error of 0.77 mm is achieved when using 36 PCs.\nFor comparison, the model constructed from 37 non-operated breasts is able to reconstruct non-operated breasts only slightly better with an error of 0.71 mm when using the same number of PCs.\n\n\n\n\n\n\\begin{figure}[t]\n\\includegraphics{.\/images\/11_exp3_model}\n\\caption{Generalization ability of a statistical shape model built from 37 operated breasts, calculated using the 38 non-operated breasts as test set. \nFor comparison purposes, a second model was trained with the same number of non-operated breasts. \nIts generalization ability was evaluated on the 38 non-operated breasts using leave-one-out cross-validation.}\n\\label{fig:exp3_model}\n\\end{figure}\n\n\\subsubsection{Generalization ability using clinical parameters}\nOur final experiment evaluates the generalization ability of the RBSM using three common, \\textit{clinical} parameters typically obtained on the breast.\nThis way, we aim to show whether or not reconstructions obtained from the RBSM not only well explain unseen objects in general, but also preserve meaningful clinical parameters.\nTechnically, this is quite similar to the generalization ability as defined by \\citet{Sty03}.\nHowever, instead of calculating point-to-point distances between an unseen 3D breast scan and its corresponding reconstruction from the RBSM, we compare three anthropometric measurements obtained on each mesh.\nThe following three distances are measured:\n\\begin{itemize}\n\\item sternal notch to left nipple (SN-NL),\n\\item sternal notch to right nipple (SN-NR), and\n\\item left nipple to right nipple (NL-NR).\n\\end{itemize}\nAfterwards, the absolute difference between the ground-truth distance taken on the original 3D breast scan and the distance obtained from the reconstruction is calculated for each of the three anthropometric measurements.\nSimilar to ordinary generalization based on point-to-point distances, we employ leave-one-out cross-validation for all 110 breast scans to calculate our newly defined generalization ability.\n\n\n\n\n\\begin{figure}[h]\n\\includegraphics{.\/images\/12_exp4_model}\n\\caption{Generalization ability of the proposed RBSM, evaluated by comparing anthropometric measurements obtained on an original 3D breast scan and its corresponding reconstruction from the model. \nThe average generalization error of the three measurements considered is also shown.}\n\\label{fig:exp4_model}\n\\end{figure}\n\nThe results are shown in Figure \\ref{fig:exp4_model}.\nAs it can be seen, using less than 30 PCs leads to a high generalization error for all three anthropometric measurements, ranging between 15 mm and 45 mm. \nHowever, as the number of PCs increases, the generalization error drops significantly and remains low for the SN-NL and SN-NR distances.\nInterestingly, only the NL-NR distance increases again up to 20 mm.\nIn summary, all three anthropometric measurements are best preserved when using 109 PCs, showing an average generalization error of 1.82 mm.\nThe individual generalization errors when using 109 PCs are 0.66 mm (SN-NL), 3.24 mm (SN-NR), and 1.57 mm (NL-NR).\n\n\n\\section{Applications}\n\\label{sec:applications}\nThis section demonstrates two exemplary applications for the RBSM that may be used for breast surgery simulation. \n\n\\subsection{Breast shape editing}\nSo far, our statistical shape model provides a convenient way to generate new breast shapes by simply varying its parameter $\\boldsymbol\\alpha\\in\\mathbb{R}^q$ within a suitable range.\nHowever, a major drawback of this parametrization is the non-interpretability of $\\boldsymbol\\alpha$ in the sense that it does not correlate with any meaningful features of the breast.\nThis clearly hinders the generation of breast shapes with certain properties or the ability to alter an existing shape according to some clinical parameters, for instance.\nTo this end, the feature editing framework proposed by \\citet{Bla99} and extended by \\citet{All03} overcomes this drawback by linearly relating features with shape parameters.\nBriefly, being $l$ feature values $\\{f_{i1},f_{i2},\\ldots,f_{il}\\}\\subset\\mathbb{R}^l$ for each individual given and stacked into a feature vector $\\mathbf{f}_i\\coloneqq(f_{i1},f_{i2},\\ldots,f_{il},1)\\in\\mathbb{R}^{l+1}$, $i=1,2,\\ldots,k$, a matrix $\\mathbf{F}\\coloneqq\\left(\\mathbf{f}_1,\\mathbf{f}_2,\\ldots,\\mathbf{f}_k\\right)\\in\\mathbb{R}^{(l+1)\\times k}$ is defined. \nArranging the shape parameters of each individual in a matrix $\\mathbf{A}\\coloneqq\\left(\\boldsymbol\\alpha_1,\\boldsymbol\\alpha_2,\\ldots,\\boldsymbol\\alpha_k\\right)\\in\\mathbb{R}^{q\\times k}$ and following \\citet{All03}, $\\mathbf{F}$ and $\\mathbf{A}$ can be related by means of an unknown transformation matrix $\\mathbf{M}\\in\\mathbb{R}^{q\\times(l+1)}$ satisfying\n\\begin{equation}\n\\mathbf{MF}\\overset{!}{=}\\mathbf{A}\\quad\\Longrightarrow\\quad\\mathbf{M}=\\mathbf{F}^+\\mathbf{A},\n\\end{equation}\nwhere $\\mathbf{F}^+$ denotes the Moore\u2013Penrose inverse of $\\mathbf{F}$.\nUsing the matrix $\\mathbf{M}$, the shape of an individual can be edited by simply providing new feature values.\nIf $\\Delta\\mathbf{f}$ denotes the component-wise difference between a target feature vector and the actual feature vector of an individual, the new shape parameter $\\boldsymbol\\alpha'$ is obtained as $\\boldsymbol\\alpha'=\\boldsymbol\\alpha+\\mathbf{M}\\Delta\\mathbf{f}$.\nThe edited shape is given by $M(\\boldsymbol\\alpha')$.\n\n\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics{.\/images\/13_feature_editing}\n\\caption{Systematic breast shape editing for two examples (top row and bottom row) using the RBSM and three common anthropometric distances as features. \nNote that all measurements are given in centimeters.}\n\\label{fig:feature_editing}\n\\end{figure}\n\n\nFor our exemplary application, we use the clinical parameters SN-NL, SN-NR, and NL-NR as features and set $q=k-1$.\nTwo exemplary results are shown in Figure \\ref{fig:feature_editing}.\n\n\\subsection{Breast shape prediction}\nWhile our first application is designed to alter key clinical parameters of the breast, this section demonstrates how the RBSM can be used to predict a missing breast from the existing one by means of \\textit{posterior} shape models \\citep{Alb13}.\n\nGiven an unseen 3D breast scan in correspondence with the template and represented as $\\mathbf{x}\\in\\mathbb{R}^{3n}$, the area enclosing the missing breast is marked first.\nDenote the indices of marked points as $J\\subset\\mathbb{N}$.\nFollowing \\citet{Alb13}, the remaining $r\\coloneqq n-\\left|J\\right|$ points provide known observations which we use to compute a posterior mean $\\mathbf{\\bar{x}}_p\\in\\mathbb{R}^{3n}$. \nIt is the likeliest reconstruction of the missing breast.\nFormally, denote as $\\mathbf{\\bar{x}}_*\\in\\mathbb{R}^{3r}$ and $\\mathbf{U}_*\\in\\mathbb{R}^{q\\times 3r}$ the sub-vector and sub-matrix obtained by removing those entries from $\\mathbf{\\bar{x}}$ and $\\mathbf{U}$ corresponding to the selected points in $J$.\nSimilarly, $\\mathbf{x}_*\\in\\mathbb{R}^{3r}$ represents the target 3D breast scan after removing the marked points.\nThen, \n\\begin{equation}\n\\mathbf{\\bar{x}}_p=\\mathbf{\\bar{x}}+\\mathbf{U}^\\top\\left(\\mathbf{U}_*\\mathbf{U}_*^\\top+\\sigma^2\\mathbf{I}_q\\right)^{-1}\\mathbf{U}_*\\left(\\mathbf{x}_*-\\mathbf{\\bar{x}}_*\\right),\n\\end{equation}\nwhere $\\sigma^2\\geq 0$ is a small variance accounting for the deviation of $\\mathbf{x}_*$ from the model \\citep{Alb13} and $\\mathbf{I}_q\\in\\mathbb{R}^{q\\times q}$ is the identity matrix.\n\n\n\n\n\n\\begin{figure}[b]\n\\includegraphics[]{.\/images\/14_breast_prediction}\n\\caption{Predicting the right breast (marked as \\textit{missing} in red) from the left one for two examples (top row and bottom row) using the RBSM.}\n\\label{fig:breast_prediction}\n\\end{figure}\n\n\n\nTwo exemplary results are shown in Figure \\ref{fig:breast_prediction}.\nNote that, compared to simply mirroring the remaining breast, the prediction obtained from the RBSM is equipped with a statistical probability.\nHence, it may not be the most symmetrical result, but, more importantly, the likeliest and probably most natural one.\nThis is crucial especially in BRS, where the goal is to reconstruct a breast looking as natural as possible.\n\n\\section{Discussion}\n\\label{sec:discussion}\nThe following section discusses the most important insights gained from experimental evaluation and also summarizes some limitations of our method.\n\n\\subsection{Decoupling between breast and thorax}\nThe experimental evaluation showed that BPM-based registration is able to decouple the breast region from the thorax well.\nThe areas with the highest variation in random samples drawn from the RBSM are almost always coincident with the breast region whereas the thorax is kept relatively stiff.\nHence, a variety of different breast shapes can be generated independently from the surrounding areas or without altering the thorax too much, implying a quite well decoupling between breast and thorax.\nThis can be also verified from Figure \\ref{fig:overview_approach}, showing that the unwanted variance of the surrounding regions is properly reduced when using BPM-based registration.\nFurthermore, the fact that our feature editing application is able to change only the requested features without altering the shape of the thorax supports our observations.\n\nFinally, we also want to note that there might be an alternative strategy for breaking up the strong correlation between breast and surrounding regions. \nIn particular, the framework proposed by \\citet{Wil17} allows constructing statistical shape models with a locality assumption by manipulating the sample covariance matrix.\nDistant areas are decoupled by explicitly setting non-zero covariances between those points to zero.\nThis way, covariances between points in the breast region and points outside could be set to zero, effectively decoupling those regions.\nA similar but probably easier way to localize statistical shape models is by using GPMMs as described by \\citet{Lue18}.\nWe believe, however, that minimizing variances instead of covariances (between pairs of points) is a more intuitive way to weaken the coupling between breast and thorax. \nNote that there is a strong relation between both approaches since $\\left|\\cov(x,y)\\right|\\leq\\sqrt{\\var(x)}\\sqrt{\\var(y)}$.\n\n\\subsection{BPM-based registration}\nOur quantitative evaluation clearly indicates that BPM-based registration produces superior results than registration without BPMs, affecting both, registration \\textit{and} model quality.\nWhen using BPMs, only a little distortion was introduced. \nIn addition, although only four landmarks are provided, pretty well correspondences are established. \nWe observed only minor correspondence errors when randomly sampling from the RBSM. \n\nRegarding the EBF-based representation of the BPMs, we can conclude that EBFs are quite well suited to capture the natural teardrop shape of the breast.\nHowever, in some cases, we observed that the current BPMs are not always able to properly capture the \\textit{entire} breast region, especially if the breast does not follow a typical teardrop shape.\nThe reason is the restriction of the covariance matrices (occurring in the Mahalanobis distance) to be diagonal, pulling off a lot of flexibility for the BPM to capture unusual breast shapes.\nWe expect that, if the full covariance matrix would be specified, better BPMs can be constructed. \nHowever, this will clearly hinder the automatic parameter selection as 30 values need to be carefully determined for each 3D breast scan.\n\nLastly, it is important to note that our rigid alignment will completely fail to estimate the correct transformation if the initial alignment is bad.\nThis is a well-known behavior of the ICP algorithm as it does not optimize for a globally optimal transformation.\nSince our 3D breast scans are initially quite well aligned and consistently oriented due to the standardized data acquisition using photogrammetry, we did not run into this problem. \n\n\\subsection{3D breast scan database}\nAlthough our 3D breast scan database contains nearly twice as many operated than non-operated breasts (72 operated, 38 non-operated), the RBSM is likely to show only a minor bias towards operated breasts.\nThis is supported by the fact that a model built from a randomly chosen subset of 37 operated breasts is able to reconstruct non-operated breasts with an almost similar error than a model trained \\textit{and} tested on the same number of non-operated breasts. \nSpecifically, an absolute difference of 0.06 mm between both reconstructions was reported.\nIntuitively, this suggests that the operated breasts contained in our database look quite natural (or, at least, similar to non-operated ones), making it challenging to decide whether a particular patient received breast surgery or not.\nIt is, however, important to note that this experiment was conducted by considering only a subset of 37 operated breasts.\nThis is because otherwise, a model built from an equal amount of non-operated breasts could not be constructed for comparison purposes.\nOn the other hand, the RBSM contains 72 operated breasts.\nHowever, even if the absolute difference between a reconstruction obtained from a model trained on operated breasts and one that includes only non-operates breasts scales \\textit{exponentially} with the number of operated breasts considered, an absolute difference of 0.23 mm would be reported for the RBSM.\nIt is thus valid to assume that the bias introduced due to operated breasts will not hinder the RBSM from explaining non-operated breasts sufficiently well.\n\nRegarding the amount of data, our experiments confirmed what was expected: the more data, the better. \nThe RBSM (containing all 110 breast scans) clearly outperforms the other models trained only on a subset of the data.\nOur experiments do not allow a conclusion about whether or not 110 breast scans are already enough in order to represent the vast amount of possible breast shapes well.\nHowever, the difference between 90 and 110 breast scans in terms of generalization and specificity is quite small (see Figure \\ref{fig:exp2_model}) strengthening the impression of being near convergence.\n\nBesides, we believe that in order to provide better pose-standardized training data, a more advanced technical set-up is needed in which the movement of arms and shoulders is further constrained. \nThis, however, would lengthen the whole data acquisition which might then become unfeasible to be applied in a clinical environment.\n\n\\subsection{Landmarks guiding the registration process}\nUsing only four landmarks (both nipples as well as both lower breast poles) for non-rigid registration is quite uncommon.\nMost of the existing methods need a lot more points to guide the registration process reasonably well.\nFor instance, the registration of facial 3D scans for the construction of statistical shape models is usually guided by about 60 to 80 landmarks, see e.g. \\citet{Boo16} or \\citet{Ger17}.\nDue to human anatomy, collecting such a huge amount of landmarks on the female breast is challenging and a fundamental problem of 3D breast scan registration. \nExcept for the nipples (and partially the lower breast poles), there are no landmarks that can be \\textit{reliably} detected only through visual inspection and consistently across a wide range of differently-shaped breasts.\n\nBesides anatomical points, additional landmarks based on easy-detectable, non-bony structures may be used.\nThis way, the lower breast pole was simply defined as \\textit{the lowest (most caudal) point of the breast}.\nFrom a medical point of view, this is indeed a valid definition of a landmark that is also used in clinical practice.\nBased on our collected data, however, it turns out that this definition is not sufficiently precise to be used for registration purposes.\nThe actually determined lower breast poles were oftentimes \\textit{not} accurately located at the lowest point of the breast.\nAs a result, the initially detected points had to be manually refined in about 40\\% of the subjects.\n\nAnother fundamental issue is the non-existence of easy-detectable landmarks on the lateral part of the breast.\nThe absence of those points poses a serious problem in our 3D breast scan registration.\nIn particular, when not already quite well aligned initially, the template has no chance of being pulled laterally due to the missing guidance.\n\nTo conclude, detecting landmarks on 3D breast scans reliably is challenging.\nUltimately, since a physical examination is required to detect anatomical landmarks, automatic landmark detection algorithms based on 3D breast scans (only capturing the surface of the body) are almost impossible to develop.\nAdditionally, it is important to note that even in some participants clearly detectable landmark points would be present, one has to keep in mind that those landmarks always need to be consistently located across \\textit{all} subjects.\nBy means of the proposed registration pipeline, however, we demonstrated that 3D breast scan registration \\textit{is} generally possible by providing only four landmarks.\n\n\\section{Conclusion and outlook}\n\\label{sec:conclusion}\nThis paper proposed the RBSM -- the first publicly available 3D statistical shape model for the female breast, learned from 110 breast scans. \nOur extensive evaluation reveals a generalization ability of 0.17 mm and a specificity error of about 2.8 mm when using all 109 PCs available.\nTogether with the model, a fully automated, pairwise surface registration pipeline was presented which requires only four landmarks to guide the registration process.\nIn order to break up the strong coupling between breast and thorax and thus being able to capture the actual breast shape as isolated as possible, we proposed to minimize the unwanted variance of the surrounding regions by means of a novel concept called BPMs.\nDefined over a surface mesh and based on EBFs, BPMs allow for a probabilistic definition of the breast region and hence overcomes the difficulty in determining an exact delineation of the breast.\nLater, BPMs are incorporated into the registration pipeline in order to align the template as accurately as possible inside the breast region and only roughly outside.\nWith our experiments, we could show that this strategy effectively reduces the unwanted variance outside the breast region and hence, decouples the breast from the surrounding areas very well.\nAs a result, the RBSM is able to generate a variety of different breast shapes quite independently from the thorax.\n\nIn our future work, we ultimately plan to combine physically motivated deformable models and statistical shape models of the breast in order to enable more realistic and statistically plausible surgical outcome simulation for BRS. \nThe proposed RBSM is seen as a first step towards this goal.\n\nBesides the two exemplary applications shown in this paper, there is room for plenty of different applications, extensions, and improvements of the RBSM.\nIn particular, since our model is generative, it could be conveniently used for data augmentation in the machine learning domain.\nAdditionally, the RBSM may be used as a prior for surface registration algorithms, effectively reducing the search space of possible deformations.\nFollowing the ideas of \\citet{Boo16}, so-called \\textit{bespoke models} (statistical shape models trained from a dedicated subset of the training set) may be built, for example for different classes of BMIs, ptosis grades, or breast volumes.\nFurthermore, instead of one global model accounting for the whole breast, two individual models for the left and right breast could be built and subsequently combined.\nLastly, our rigid alignment needs to be improved to be able to deal with an arbitrary initial alignment.\nFor instance, more robust ICP variants such as Go-ICP \\citep{Yan16} may be used or advanced feature detectors (based on heat or wave kernel signatures \\citep{Aub11,Sun09} or multi-scale mean curvatures \\citep{Pan10}, for instance) combined with a robust outlier detection to find reliable correspondences for transformation calculation. \n\n\\section*{Declaration of Competing Interest}\nThe authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.\n\n\\section*{Acknowledgments}\nWe thank our colleagues from the University Center of Plastic, Aesthetic, Hand and Reconstructive Surgery at the University Hospital Regensburg and the St. Josef Hospital Regensburg for their help during data collection.\nThis research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.\n\n\\bibliographystyle{model2-names.bst}\\biboptions{authoryear}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nThe momentum-dependence of single-nucleon mean-field potentials is one of the fundamental features of finite-range nucleon-nucleon interactions in nuclear matter.\nA moving nucleon experiences different potentials from its surroundings compared to static ones, either under the mean fields or those considering correlations.\nMomentum-dependent nuclear potentials in fact play very important roles in heavy-ion collisions\\,\\cite{Gale1987,Bertsch1988,Pra88,Welke,Gale90,Aichelin1990,Pan,Zhang94,Hart,Pawel,Das2003,LiBA04}.\nThe mean-field potential $U_J$ (with the subscript $J$ denoting protons (p) or neutrons (n)) of a nucleon with momentum $\\v{k}$ at zero temperature is generally a function of neutron density $\\rho_{\\rm{n}}$, the proton density $\\rho_{\\rm{p}}$, i.e., $U_J=U_J(\\rho_{\\rm{n}},\\rho_{\\rm{p}},|\\v{k}|)$.\nIt can be written equivalently in the form of $U_J(\\rho,\\delta,|\\v{k}|)$ with $\\rho=\\rho_{\\rm{n}}+\\rho_{\\rm{p}}$ the total density and $\\delta=(\\rho_{\\rm{n}}-\\rho_{\\rm{p}})\/\\rho$ the isospin asymmetry of the nucleon system under consideration.\nThe equation of state (EOS) of cold asymmetric nucleonic matter (ANM) can be conveniently denoted by the nucleon specific energy $E(\\rho,\\delta)$. The latter could be obtained by certain integration of $U_J(\\rho_{\\rm{n}},\\rho_{\\rm{p}},|\\v{k}|)$ over $\\v{k}$, i.e., $\\sim\\int^{\\v{k}}U_J(\\rho,\\delta,|\\v{k}|)\\d\\v{k}+\\rm{``kinetic\\;term''}\\to E(\\rho,\\delta)$.\nEffects of the momentum $\\v{k}$ dependence of the single-nucleon potential on the EOS are implicit as the momentum is integrated out.\nConnecting the momentum-dependence and the correlations\/fluctuations of the interactions\/potentials with the EOS as well as understanding how they influence each other are fundamentally important and interesting for resolving many related issues in both nuclear physics and astrophysics.\n\n\nThe EOS $E(\\rho,\\delta)$ of ANM can conventionally be expanded around the symmetric nuclear matter (SNM) with $\\delta=0$, leading to the approximation,\n\\begin{equation}\nE(\\rho,\\delta)\\approx E_0(\\rho)+E_{\\rm{sym}}(\\rho)\\delta^2+E_{\\rm{sym},4}(\\rho)\\delta^4+\\mathcal{O}(\\delta^6),\n\\end{equation}\nhere $E_{\\rm{sym}}(\\rho)=2^{-1}\\partial^2E(\\rho,\\delta)\/\\partial\\delta^2|_{\\delta=0}$ is the conventional nuclear symmetry energy besides the EOS $E_0(\\rho)\\equiv E(\\rho,0)$ of SNM,  $E_{\\rm{sym},4}(\\rho)\\equiv 24^{-1}\\partial^4E(\\rho,\\delta)\/\\partial\\delta^4|_{\\delta=0}$ is the fourth-order symmetry energy, etc. Owing to its fundamental importance in nuclear\nphysics\\,\\cite{LiBA98,Dan02,Bar05,Ste05,Che07a,LCK08,Tsa12,Che14}\nand astrophysics\\,\\cite{Gle00,Lat04,Lat12,Lat14,Oze16,Oer17}, \nunderstanding nuclear symmetry energy $E_{\\rm{sym}}(\\rho)$,\nespecially its density dependence (both at sub- and supra-saturation densities), is currently one of the most\nimportant goals in contemporary nuclear astrophysics\\,\\cite{Hor14,EPJA}. \n\nThanks to the significant efforts made by many people in the nuclear astrophysics community during\nthe last two decades, much knowledge on the symmetry energy around the saturation density\n$\\rho_0\\approx0.16\\,\\rm{fm}^{-3}$ has been obtained. However, the density dependence of\n$E_{\\rm{sym}}(\\rho)$ especially at supra-saturation densities is still\npoorly known. In particular, several nuclear structure observables are known as sensitive probes of symmetry energy at sub-saturation\ndensities, e.g., the correlation between the slope parameter $L(\\rho)$ of the symmetry energy and the\nneutron skin thickness of heavy nuclei\\,\\cite{Cen09,Zha13} as well\nas the correlation between $L(\\rho)$ and the electric dipole $\\alpha_{\\rm{D}}$\npolarizability in $^{208}\\rm{Pb}$\\,\\cite{Maz13,Zha14,Zha15}; the\ncorrelation between the isobaric analog states (IAS) and the\nsymmetry energy\\,\\cite{Dan14}, to name a few. While analyses of neutron star observables (mostly the radii and tidal polarizability) especially since GW170817 \nhave provided strong constraints on the symmetry energy around $1\\rho_0\\mbox{$\\sim$}2\\rho_0$\\,\\cite{Ste10,LiBA21,Huth22}, the situation at higher densities is unclear\\,\\cite{ZL21,Fr22} from studying the mass and radius data of the currently known most massive neutron star \nPSR J0740+6620 from the NICER Collaboration\\,\\cite{Miller21,Riley21}. On the other hand, intermediate-relativistic energy heavy-ion collisions (HICs) have been playing a significant role in determining the symmetry energy from sub-saturation to supra-saturation densities, see, e.g., Refs.\\,\\cite{LCK08,Tsa12,Xia09}. Unfortunately, various analyses of heavy-ion reaction observables based on transport model simulations have not reached a firm conclusion regrading the high-density behavior of nuclear symmetry energy, \nfor a recent review, see e.g., Ref.\\,\\cite{EPJA-review}. It is mostly because of the still existing relatively large model dependences in simulating heavy-ion collisions especially above the pion production threshold\\,\\cite{Xu16,Zhang18,Ono19,Colonna21,TMEP}.  The single-nucleon potential especially its momentum dependence in dense neutron-rich matter is among the most uncertain but very critical physics inputs of transport models. It is known to have significant impact on observables of heavy-ion reactions\\,\\cite{LiBA04,LiChen05,Cozma17,WangLi,Cozma21}. \n\nThe symmetry energy can conventionally be decomposed into a kinetic and a potential part in the quasi-particle picture, i.e.,\n$E_{\\rm{sym}}(\\rho)=E_{\\rm{sym}}^{\\rm{kin}}(\\rho)+E_{\\rm{sym}}^{\\rm{pot}}(\\rho)$.\nEach individual term of this decomposition may have its own distinct and important physical ramifications on different problems. For instance, the critical\nformation density of the charged $\\Delta(1232)$ states in neutron\nstars depends on $E_{\\rm{sym}}^{\\rm{kin}}(\\rho)$ and\n$E_{\\rm{sym}}^{\\rm{pot}}(\\rho)$ separately\\,\\cite{Cai15a}; while on\nthe other hand, the potential part $E_{\\rm{sym}}^{\\rm{pot}}(\\rho)$\nis extremely relevant to certain dynamical processes and observables in\nHICs\\,\\cite{Li15,Li15a,Hen15b,Yon17}. Thus it is not only important\nto explore the density dependence of the total symmetry energy, but also its\nindividual terms and their physical origins. Since the total symmetry energy at the saturation density $\\rho_0$, as mentioned above, is relatively well constrained\\,\\cite{LiBA13}, the decomposition of symmetry energy has to satisfy an asymptotic sum rule at $\\rho_0$. In the traditional free Fermi gas (FFG) model of nuclear matter,\nwhere the single-nucleon momentum distribution function\n$n_{\\v{k}}^J$ takes the form of a step function, the kinetic\nsymmetry energy is simply\n$E_{\\rm{sym}}^{\\rm{kin}}(\\rho)=k_{\\rm{F}}^2\/6M$ with $k_{\\rm{F}}$ being\nthe nucleon Fermi momentum in SNM, and $M$ is the average static mass of nucleons. It will be used as a reference of kinetic symmetry energy in our following discussions.\nIn this work, we use the abbreviation ``FFG'' (``HMT'') to denote the calculation\/model in which the momentum distribution $n_{\\v{k}}^J$ is a step function (the function containing the HMT), even in cases where a nuclear potential is included.\n\nIn most calculations of cold nuclear matter EOSs especially within phenomenological models and\/or input EOSs as well as initializations of transport model simulations of nuclear reactions, often a step function is used for the $n_{\\v{k}}^J$ as if the system is a FFG at zero temperature. However, this assumption has to be modified if one considers the HMT induced by the SRC in nuclear matter. Moreover, this modification is known to have significant ramifications on the EOS especially the kinetic symmetry  energy\\,\\cite{CXu10,Lee11,Lee14,Wang12}. \nIn a more realistic picture of nuclear matter, the form of the\n$n_{\\v{k}}^J$ should be different from the step function, owing to, e.g., the (poorly known) spin-isospin dependence of the three-nucleon forces and\/or the isospin dependence of SRC induced by the tensor forces and\/or the hard repulsive core. \nIndeed, It has been known for a long time that the SRC leads to a high (low)\nmomentum tail (depletion) in the single-nucleon momentum\ndistribution above (below) the nucleon Fermi surface, in cold\nnucleonic matter\\,\\cite{Bethe,Ant88,Arr12,Cio15}. Significant\nefforts have been made in the past\nyears\\,\\cite{Wei15,Wei15a,Cruz2018,Weiss2019-1,Weiss2019-2,Weiss2021,Hen14,Hen15,Col15,Egi06,kk1,kk2,kk3,kk4,kk5,\nHen2017RMP,Duer2018,Schmookler2019,Schmidt2020,Li22} to constrain the\n(isospin-dependent) parameters characterizing the SRC-modified\n$n_{\\v{k}}^J$ in neutron-rich matter using both\nexperimental data and microscopic model calculations. For example,\nit has been found via analyzing electron-nucleus scattering data\nthat the percentage of nucleons in the HMT above the Fermi surface is as high as about 28\\% in SNM but\ndecreases gradually to about only 1\\% in a pure neutron matter\n(PNM)\\,\\cite{Hen14,Hen15}, indicating a totally different momentum\nbehavior of the $n_{\\v{k}}^J$ due to the strong isospin dependence of SRC. \n\nDespite of the long history and impressive progresses especially in recent years in studying various features and impacts of SRC\/HMT, many interesting issues regarding the size, shape, density and isospin dependence of SRC\/HMT remain to be clearly resolved. \nFor example, although various many-body theories\nhave consistently predicted effects of the SRC on the HMT\nqualitatively consistent with the experimental findings, the\npredicted size of the HMT still depends on the model and interaction\nused. For instance, the self-consistent Green's function (SCGF)\ntheory adopting the AV18 interaction predicts a 11\\%$\\sim$13\\% HMT for\nSNM at saturation density\\,\\cite{Rio09,Rio14}. Moreover, the latest\nBruckner-Hartree-Fock (BHF) calculations predict a HMT ranging from\nabout 10\\% via the next-next-next-leading order 450 (N$^3$LO450)\nto over 20\\% using several microscopic interactions\\,\\cite{ZHLi},\nthe latest VMC calculations for $^{12}$C gives a 21\\%\nHMT\\,\\cite{VMC,Wir16}, etc. While based on the observation that the SRC\nbetween a neutron-proton pair is about 18$\\sim$20 times that of two\nprotons, the HMT in PNM was estimated to be about\n1\\%$\\sim$2\\%\\,\\cite{Hen15b}. Furthermore, some recent calculations\nindicate a significantly higher HMT in PNM, e.g., the SCGF\npredicted a 4\\%$\\sim$5\\% HMT in PNM\\,\\cite{Rio09,Rio14}. \n\nThe uncertain properties of SRC\/HMT in cold dense neutron-rich matter may bring large uncertainties to both the kinetic and potential parts of the symmetry energy as well as the single-nucleon potentials. \nFor example, because of the momentum-squared weighting in\ncalculating the average nucleon kinetic energy, the strong isospin\ndependence of the HMT makes the kinetic symmetry energy dramatically\ndifferent from the FFG model\nprediction\\,\\cite{Cai16c,Hen15b,CXu11,CXu13,Vid11,Lov11,Car12,Rio14,Car14,Cai15}.\nSpecifically, the kinetic symmetry energy is significantly reduced even\nto negative values in some model studies. Roughly speaking, nuclear symmetry\nenergy is the energy difference between PNM and SNM within the parabolic\napproximation of the total ANM EOS. It is also known that the SRC is dominated by the isosinglet neutron-proton tensor interaction. It thus increases significantly the average kinetic energy per nucleon\nin SNM but has little effect on that in PNM, leading to a reduction\nof the kinetic symmetry energy. Consequently, the potential part of the $E_{\\rm{sym}}(\\rho)$, i.e., $E_{\\rm{sym}}^{\\rm{pot}}(\\rho)$, should be modified correspondingly to satisfy the sum rule of symmetry energy at $\\rho_0$. Moreover, the \nSRC\/HMT may affect directly the potential symmetry energy if nuclear interactions are momentum dependent. Of course, it is also expected to affect the momentum dependence of both isoscalar and isovector parts of the single-nucleon potential and the associated effective masses of nucleons in dense neutron-rich matter.\n\nInterestingly, SRC\/HMT effects on the ANM EOS and some properties of neutron stars have been studied recently within relativistic mean field (RMF) models by several groups\\,\\cite{Cai16b,Lou22a,Lu2022,Bur22,Hong22,Lou22,Lu21,Souza20}. \nMost of these studies were carried out using nonlinear RMF models where the interaction is momentum independent. Given the results of these studies and those mentioned earlier, there still exist many interesting\/important questions to be  investigated\/answered to deepen our understanding of the SRC physics and its ramifications, e.g.,\n\\begin{enumerate}\n\\item[(a)]How does the SRC\/HMT affect the single-nucleon potential in neutron-rich matter, especially the momentum dependence of its isoscalar and isovector parts, in comparisons with existing experimental observations of nucleon optical potentials and the associated nucleon effective masses at $\\rho_0$? Answers to this question are potentially important for modeling HICs involving high-energy rare isotope beams.\n\\item[(b)]How does the SRC\/HMT affect the ANM EOS (including both the SNM EOS and the symmetry energy) qualitatively and quantitatively, if a non-relativistic potential\/interaction is adopted? This question is relevant since many such kind of interactions and energy density functionals are widely used successfully in studying nuclear structures, calculating the EOS for astrophysical purposes and\/or simulating heavy-ion reactions.\n\\item[(c)] Due to the SRC\/HMT induced sizable enhancement of SNM pressure and the reduction of total symmetry energy in nonlinear RMF models, the maximum mass of cold neutron stars was found to increase, see, e.g., Ref.\\,\\cite{Cai16b}. One may then wonder whether this conclusion would still hold or not if a non-relativistic and\/or momentum-dependent potential is used.\n\\end{enumerate}\n\nIn a Gogny-like energy density functional (EDF) that has been widely used in simulating heavy-ion reactions, see, e.g., Refs.\\,\\cite{Gale1987,Bertsch1988,Pawel,Das2003,LCK08,LiBA04,LiChen05,Cozma17,Cozma21}, the momentum-dependent part of the single-nucleon potential has the form of \n$\\int\\d\\v{k}'f_{J'}(\\v{r},\\v{k}')\\Omega(\\v{k},\\v{k}')$ while the corresponding potential energy in the ANM EOS has the form of\n$\\int\\d\\v{k}\\d\\v{k}'f_J(\\v{r},\\v{k})f_{J'}(\\v{r},\\v{k}')\\Omega(\\v{k},\\v{k}')$ where the kernel $\\Omega(\\v{k},\\v{k}')=[1+(\\v{k}-\\v{k}')^2\/\\Lambda^2)]^{-1}$ is from the Yukawa-type finite-range two-nucleon interaction, $\\Lambda$ is the momentum-scale parameter while $f_J(\\v{r},\\v{k})$ and $f_{J'}(\\v{r},\\v{k}')$ are the momentum distribution functions of the two interacting nucleons $J$ and $J'$. Analytical integrations of the two integrals involved after incorporating the HMT in the $f_J(\\v{r},\\v{k})$ and $f_{J'}(\\v{r},\\v{k}')$ have been found formidable\\,\\cite{CLC18}, while they are necessary for fixing the model parameters using empirical properties of nuclear matter and nucleon optical potentials at $\\rho_0$.\n\nIn this work, using a parametrized $\\Omega(\\v{k},\\v{k}')$ as the surrogate of its original form we derive analytical expressions for all relevant terms and their characteristic parameters in the ANM EOS and single-nucleon potential. The surrogate approach adopted here not only grasps all major physical features of the momentum-dependence of the original Gogny-like potential but also enables analytical expressions for all the relevant physical quantities. There are different physical origins of the HMT in various many-body systems under different conditions\\,\\cite{Coleman15}. We notice that while during heavy-ion collisions, a HMT is automatically generated in the $f_J(\\v{r},\\v{k})$ and $f_{J'}(\\v{r},\\v{k}')$ due to finite temperatures. Nevertheless, most of the parameters of the input single-nucleon potentials in transport models have to be determined by the EOS properties of nuclear matter at zero temperature. Thus, the SRC\/HMT in cold nuclear matter is important for determining the EOS of hot matter formed during heavy-ion reactions. \nWhile the EOS of hot dense neutron-rich matter is a necessary input for simulating neutron star mergers and understanding post-merger spectrum of high-frequency gravitational waves.\nMoreover, the HMT at zero temperature is also useful for investigating properties of cold neutron stars, e.g., the proton fraction and the core-crust transition density\/pressure in neutron stars at $\\beta$-equilibrium.\n\nWith the motivations concerning the questions outlined above, the main contributions\/findings of this work can be summarized as follows:\n\\begin{enumerate}\n\\item[(a)]Qualitatively, the pressure $P_0$ of SNM is enhanced while the $E_{\\rm{sym}}(\\rho)$ is reduced (both at sub- and supra-saturation densities) considering the SRC-induced HMT, which are consistent with the predictions from the nonlinear RMF models\\,\\cite{Cai16b}.\nQuantitatively, however, the reduction of $E_{\\rm{sym}}(\\rho)$ is much stronger than that from the RMF models, while the enhancement of $P_0$ is weaker compared with the one predicted from the RMF models.\n\\item[(b)]Due to the much stronger reduction of $E_{\\rm{sym}}(\\rho)$ as pointed out in (a), the maximum mass of neutron stars is found to decrease correspondingly, compared with calculations using the FFG momentum distribution.\nIn this sense, it is not quite clear whether the SRC-induced HMT tends to help support heavy neutron stars (around $2M_{\\odot}$).\nMore studies\/possible mechanisms for stiffening high-density EOS within the Gogny-like EDFs are still needed.\n\\item[(c)]The surrogate Gogny-like EDF is shown to produce reasonable (kinetic) energy-dependence of the single-nucleon potential compared with known nucleon optical potentials at $\\rho_0$,\nthus it improves another aspect of the ANM EOS with the HMT, compared with the nonlinear RMF models\\,\\cite{Cai16b}.\n\\item[(d)]The isovector (symmetry) potential $U_{\\rm{sym}}(\\rho,|\\v{k}|)$ is largely enhanced as the $E_{\\rm{sym}}^{\\rm{pot}}(\\rho)$ is enhanced (while the $E_{\\rm{sym}}^{\\rm{kin}}(\\rho)$ is reduced).\nThis is a new feature originated from the SRC\/HMT within the Gogny-like EDF compared to the nonlinear RMF models where there is no momentum-dependence. It provides a novel connection linking the SRC-induced HMT with the dynamics of HICs involving neutron-rich nuclei.\n\\item[(e)]Parametrized forms of the EOS $E_0(\\rho)$ and single-nucleon potential $U_0(\\rho,\\v{k}$) in SNM as well as the symmetry energy $E_{\\rm{sym}}(\\rho)$ and isovector potential $U_{\\rm{sym}}(\\rho,|\\v{k}|)$ are provided to facilitate their future applications in both nuclear physics and astrophysics.\n\\end{enumerate}\n\nA brief report on some preliminary results of our study on some issues listed above can be found in a conference proceedings\\,\\cite{CLC18}. The rest of this paper is organized as follows: In section \\ref{sec2}, the single-nucleon momentum distribution function $n_{\\v{k}}^J$ as well as other relevant parameters in the calculations are given. The\nmomentum-dependent nucleon potential used in the present work will also be discussed in this section.\nIn section \\ref{sec4}, several analytical expressions for the characteristics of the NM EOS are given. \nSection \\ref{sec5} is devoted to numerical demonstrations of the formulas developed in section \\ref{sec4}. Section \\ref{sec6} gives the summary of the present work. The detailed\nderivations of most formulas are given in the appendix.\n\n\\setcounter{equation}{0}\n\\section{Gogny-like energy density functional encapsulating SRC\/HMT effects}\\label{sec2}\n\n\n\\subsection{Characteristics of the ANM EOS}\n\nWe first briefly recall definitions of\nseveral quantities characterizing the EOS of ANM.\nAround the saturation density $\\rho _{0}$, the $E_{0}(\\rho )$ can be\nexpanded up to the third-order in density as,\n\\begin{equation}\nE_{0}(\\rho )\\approx E_{0}(\\rho _{0})+\\frac{1}{2}K_0\\chi\n^{2}+\\frac{1}{6}J_0\\chi ^{3}, \\label{DenExp0}\n\\end{equation}%\nwhere $\\chi =(\\rho -\\rho _{0})\/3\\rho _{0} $ is a dimensionless\nvariable characterizing the deviations of the density from $\\rho _{0}$. The first term $E_{0}(\\rho _{0})$ on\nthe right-hand-side of Eq.\\,(\\ref{DenExp0}) is the binding energy per\nnucleon in SNM at $\\rho _{0}$ and,\n\\begin{align}\nK_{0} =&\\left. 9\\rho _{0}^{2}\\frac{\\text{d} ^{2}E_{0}(\\rho\n)}{\\text{d} \\rho ^{2}}\\right\\vert _{\\rho =\\rho\n_{0}},~~J_0=\\left.27\\rho_0^3\\frac{\\d^3E_0(\\rho)}{\\d\\rho^3}\\right|_{\\rho=\\rho_0},\n\\label{K0}\n\\end{align}\nare the incompressibility coefficient and the\nskewness\\,\\cite{Ste10,Cai14,Sel14} of the SNM, respectively.\n\n\nSimilarly, one can expand the $E_{\\mathrm{sym}}(\\rho )$ around the\nsaturation density as\n\\begin{equation}\nE_{\\text{sym}}(\\rho)\\approx\nE_{\\text{sym}}(\\rho_0)+L\\chi\n+\\frac{1}{2}K_{\\rm{sym}}\\chi^2+\\frac{1}{6}J_{\\rm{sym}}\\chi^3, \\label{EsymLKr}\n\\end{equation}\nwith the slope parameter $L$ defined as,\n\\begin{align}\nL\\equiv&\\left.3\\rho_0\\frac{\\text{d}E_{\\mathrm{sym}}(\\rho)}{\\text{d}\\rho}\\right|_{\\rho\n=\\rho_{0} }.\n\\end{align}\nThe curvature and the skewness\\,\\cite{ZL18,ZL19,ZL21,ZL22} of the symmetry energy are defined similarly as \n\\begin{align}\nK_{\\rm{sym}} =&\\left.9\\rho _{0}^{2}\\frac{\\text{d} ^{2}E_{\\rm{sym}}(\\rho\n)}{\\text{d} \\rho ^{2}}\\right|_{\\rho =\\rho\n_{0}},\\\\\nJ_{\\rm{sym}}=&\\left.27\\rho_0^3\\frac{\\d^3E_{\\rm{sym}}(\\rho)}{\\d\\rho^3}\\right|_{\\rho=\\rho_0}.\n\\end{align}\n\n\\subsection{Single-nucleon Momentum Distribution}\n\nWe describe the SRC-modified single-nucleon momentum\ndistribution function encapsulating a HMT used in the present work.\nMore details can be found in Ref.\\,\\cite{Cai15}. The single-nucleon\nmomentum distribution function in cold ANM takes the following\nform\\,\\cite{Cai15},\n\\begin{equation}\\label{MDGen}\nn^J_{\\v{k}}(\\rho,\\delta)=\\left\\{\\begin{array}{ll}\n\\Delta_J,~~&0<|\\v{k}|<k_{\\rm{F}}^J,\\\\\n\\displaystyle{C}_J\\left({k_{\\rm{F}}^{J}}\/{|\\v{k}|}\\right)^4,~~&k_{\\rm{F}}^J<|\\v{k}|<\\phi_Jk_{\\rm{F}}^J.\n\\end{array}\\right.\n\\end{equation}\nwhere $k_{\\rm{F}}^J=k_{\\rm{F}}(1+\\tau_3^J\\delta)^{1\/3}$ is the nucleon Fermi\nmomentum with $k_{\\rm{F}}=(3\\pi^2\\rho\/2)^{1\/3}$ and\n$\\tau_3^{\\rm{n}}=+1$, $\\tau_3^{\\rm{p}}=-1$.\nThe above form of nucleon momentum distribution function is\nconsistent with the well-known predictions of microscopic nuclear\nmany-body theories\\,\\cite{Bethe,Ant88,Arr12,Cio15} and the recent\nexperimental\nfindings\\,\\cite{Hen14,Hen15,Wei15,Wei15a,Col15}.\nVery recently, an analysis of the energy spectra of protons emitted in reactions of 47\\,MeV\/u  (several light to medium sized) projectiles on Sn and Au targets provides new evidence for the $1\/k^4$ shape of the HMTs in the intrinsic momenta spectra of the projectiles\\,\\cite{Hag21}. \n\n\\begin{figure}[h!]\n\\centering\n \n  \\includegraphics[width=7.cm]{Fig1.eps}\n  \\caption{Sketch of the $n_{\\v{k}}^J$ with a HMT. Taken from Refs.\\,\\cite{Cai15,Cai16b}.}\n  \\label{mom-dis}\n\\end{figure}\n\nIn (\\ref{MDGen}), the $\\Delta_J$ measures the depletion of the Fermi\nsphere at zero momentum with respect to the FFG model. The sketch of\n$n^J_{\\v{k}}(\\rho,\\delta)$ is shown in FIG.\\,\\ref{mom-dis}. The\nisospin structure of the parameters $C_J$ and $\\phi_J$\nis found to be $Y_J=Y_0(1+Y_1\\tau_3^J\\delta)$\\,\\cite{Cai15}. The\namplitude ${C}_J$ and high-momentum cutoff coefficient $\\phi_J$\ndetermine the fraction of nucleons in the HMT via\n\\begin{equation}\\label{xPNM}\nx_J^{\\rm{HMT}}=3C_{{J}}\\left(1-\\phi_J^{-1}\\right).\n\\end{equation}\nThe normalization condition between the density $\\rho_J$\nand the distribution $n_{\\v{k}}^J$, i.e.,  $\n[{2}\/{(2\\pi)^3}]\\int_0^{\\infty}n^J_{\\v{k}}(\\rho,\\delta)\\d\\v{k}=\\rho_J={(k_{\\rm{F}}^{J})^3}\/{3\\pi^2}\n$ requires that only two of the three parameters, i.e., ${C}_J$,\n$\\phi_J$ and $\\Delta_J$, are independent. Here we choose the first\ntwo as independent and determine the $\\Delta_J$ by\n\\begin{equation}\\label{DeltaJ}\n\\Delta_J=1-3{C}_J\\left(1-\\phi_J^{-1}\\right)=1-x_J^{\\rm{HMT}}.\n\\end{equation}\n\nThe ${C}\/{|\\v{k}|^4}$ shape of the HMT both for SNM and PNM is\nstrongly supported by recent studies both theoretically and\nexperimentally. Combining the results from analyzing cross sections\nof $\\rm{d}(\\rm{e}, \\rm{e}^{\\prime}\\rm{p})$ reactions\\,\\cite{Hen15}\nand medium-energy photonuclear absorptions\\,\\cite{Wei15}, the $C_0$\nwas found to be $C_0\\approx0.161\\pm0.015$ in the HMT-exp\nmodel\\,\\cite{Cai15}. With this $C_0$ and the value of\n$x^{\\rm{HMT}}_{\\rm{SNM}}\\approx28\\%\\pm4\\%$\\,\\cite{Hen14,Hen15,Hen15b}\nobtained from systematic analyses of inclusive (e,e$'$) reactions\nand data from exclusive two-nucleon knockout reactions, the HMT\ncutoff parameter in SNM is determined to be\n$\\phi_0=(1-x_{\\rm{SNM}}^{\\rm{HMT}}\/3{C}_0)^{-1}\\approx2.38\\pm0.56$\\,\\cite{Cai15}.\nFurthermore, the value of $C_{\\rm{n}}^{\\rm{PNM}}= C_0(1+C_1)$ was\nextracted by applying the adiabatic sweep theorem\\,\\cite{Tan08} to\nthe EOS of PNM constrained by predictions of microscopic nuclear\nmany-body theories\\,\\cite{Sch05,Epe09a,Tew13,mm,Gez13,Gez10} and the\nEOS of ultra-cold atoms under unitary\ncondition\\,\\cite{Tan08,Stew10,Kuh10}. And more specifically,\n$C_{\\rm{n}}^{\\rm{PNM}}\\approx0.12$ and $C_1\\approx-0.25\\pm0.07$ were\nobtained\\,\\cite{Cai15}. Moreover, by inserting the values of\n$x_{\\rm{PNM}}^{\\rm{HMT}}\\approx1.5\\%\\pm0.5\\%$\\,\\cite{Hen14,Hen15,Hen15b}\nextracted in the same way as the $x_{\\rm{SNM}}^{\\rm{HMT}}$ and\n$C_{\\rm{n}}^{\\rm{PNM}}$ into Eq.\\,({\\ref{xPNM}), the high momentum\ncutoff parameter for PNM was determined to be\n$\\phi_{\\rm{n}}^{\\rm{PNM}}\\equiv\n\\phi_0(1+\\phi_1)=(1-x_{\\rm{PNM}}^{\\rm{HMT}}\/3C_{\\rm{n}}^{\\rm{PNM}})^{-1}\\approx1.04\\pm0.02$\\,\\cite{Cai15},\nthus $\\phi_1\\approx-0.56\\pm0.10$\\,\\cite{Cai15} was obtained. \nSee Ref.\\,\\cite{Cai15} for more details on these discussions.\nOn the other hand, if we take $x_{\\rm{SNM}}^{\\rm{HMT}}\\approx12\\%$ and\n$x_{\\rm{PNM}}^{\\rm{HMT}}\\approx4\\%$ in the HMT-SCGF model, and using\na very similar scheme to determine the parameters in the HMT-exp\nmodel, we obtain $\\phi_0\\approx1.49, \\phi_1\\approx-0.25$,\n$C_0\\approx0.121$ and $C_1\\approx-0.01$\\,\\cite{Cai16c}.\nIn the following, the parameter sets adopting the experimental verification and the SCGF are denoted as HMT-exp and HMT-SCGF, respectively, and the corresponding calculations are made under the HMT-exp model or the HMT-SCGF model.\n\n\nBesides the effects of the SRC-induced HMT on the kinetic symmetry\nenergy, several related interesting problems are explored recently,\nincluding the enhancement of the isospin quartic term\\,\\cite{Cai15},\nthe predictions on the effective E-mass together with its isospin\nsplitting at the Fermi surface\\,\\cite{Cai16a,LiBA15,LiBA16}, the\ncorrelation between the neutron skin thickness and the corresponding\nkinetic energy density in the surface region of heavy\nnuclei\\,\\cite{Cai16c}, and the effects on the neutron star\nmass-radius relation\\,\\cite{Cai16b,Hen16}. \nRecently, the EOS of ANM considering the HMT in a general space of dimension $d$ is investigated\\,\\cite{CaiLi2022,CaiLi2022a}, see Ref.\\,\\cite{LiBA2018PPNP} for a recent review on more related issues\\,\\cite{Souza2020,YongGC2017PRC,YongGC2018PLB,\nWang2017PRC,Guo2021PRC,Yong2022,Yang2019,Bulgac2022,Bulgac2022a,Miller2019}.\n\n\n\n\\subsection{Surrogate Momentum-dependent Single-nucleon Potential in ANM}\n\nThe total EOS of ANM could be decomposed into its kinetic and potential parts as\n\\begin{equation}\\label{Gen-EOS}\nE(\\rho,\\delta)=E^{\\rm{kin}}(\\rho,\\delta)+E^{\\rm{pot}}(\\rho,\\delta)\n\\end{equation}\nwith the kinetic part given by\n\\begin{equation}\\label{Ekin}\nE^{\\rm{kin}}(\\rho,\\delta)=\\frac{1}{\\rho}\\frac{2}{(2\\pi)^3}\\sum_{J=\\rm{n,p}}\\int_0^{\\infty}\\frac{\\v{k}^2}{2M}n_{\\v{k}}^J(\\rho,\\delta)\\d\\v{k}\n\\end{equation}\nwhere $n_{\\v{k}}^J(\\rho,\\delta)$ is the single-nucleon momentum\ndistribution function, i.e., a step function in the FFG model or the one given in Eq.\\,(\\ref{MDGen}) in the HMT model. The potential part\nin the present work adopts the following form\\,\\cite{CLC18},\n\\begin{align}\\label{Epot}\nE^{\\rm{pot}}(\\rho,\\delta)=&\\sum_{J,J'}\\frac{C_{J,J'}}{\\rho\\rho_0}\\int\\d\\v{k}\\d\\v{k}'f_J(\\v{r},\\v{k})f_{J'}(\\v{r},\\v{k}')\\Omega(\\v{k},\\v{k}')\\notag\\\\\n&+\\frac{A_\\ell(\\rho_{\\rm{p}}^2+\\rho_{\\rm{n}}^2)}{2\\rho\\rho_0}\n+\\frac{A_{\\rm{u}}\\rho_{\\rm{p}}\\rho_{\\rm{n}}}{\\rho\\rho_0}\\notag\\\\\n&+\\frac{B}{\\sigma+1}\\left(\\frac{\\rho}{\\rho_0}\\right)^{\\sigma}\\left(1-x\\delta^2\\right),\n\\end{align}\nhere \\begin{equation}\nA_\\ell=A_\\ell^0+{2xB}\/({1+\\sigma}),~~A_{\\rm{u}}=A_{\\rm{u}}^0-{2xB}\/({1+\\sigma}).\n\\end{equation}\nBesides the first line of (\\ref{Epot}), the remaining terms are\ntotally the same as those in the original Momentum Dependent Interaction (MDI)\\,\\cite{Gale1987,Bertsch1988,Pawel,Das2003,LCK08,LiBA04,LiChen05,Cozma17,Cozma21,Che05,XuJ10,XuJ15}. \nMoreover, $A_\\ell,\nA_{\\rm{u}},B,\\sigma$ and $x$ are five phenomenological\nparameters characterizing the density dependence of the nucleon\npotential. In particular, if the replacements\n$t_3=16B\/(\\sigma+1)\\rho_0^{\\sigma}$ and $x_3=(3x-1)\/2$ are made,\nthen the parameters $B,\\sigma$ and $x$ are reduced to the ones widely used to describe the\neffective three-nucleon forces\\,\\cite{XuJ10}. Furthermore, the\nparameter $x$ only affects the slope parameter of the symmetry energy. It does not affect the magnitude of the symmetry energy at $\\rho_0$, providing a convenient way to investigate the density dependence of the symmetry energy\\,\\cite{Das2003,Che05}.\nFor the same type of nucleons, one has $C_{J,J}\\equiv C_\\ell$, while for the\nunlike ones,  then $C_{J,\\overline{J}}\\equiv C_{\\rm{u}}$,\nhere $\\overline{\\rm{n}}=\\rm{p}$ and $\\overline{\\rm{p}}=\\rm{n}$.\n\nThe single-nucleon potential corresponding to the EOS of (\\ref{Gen-EOS}) is given by\n\\begin{align}\\label{Gen-U}\nU_J(\\rho,\\delta,|\\v{k}|)=&\\frac{A_\\ell\\rho_J}{\\rho_0}+\\frac{A_{\\rm{u}}\\rho_{\\overline{J}}}{\\rho_0}\n+B\\left(\\frac{\\rho}{\\rho_0}\\right)^{\\sigma}\\left(1-x\\delta^2\\right)\\notag\\\\\n&-4x\\tau_3^J\\frac{B}{\\sigma+1}\\frac{\\rho^{\\sigma-1}}{\\rho_0^{\\sigma}}\\delta\\rho_{\\overline{J}}\n\\notag\\\\\n&+\\sum_{J'}\\frac{2C_{J,J'}}{\\rho_0}\\int\\d\\v{k}'f_{J'}(\\v{r},\\v{k}')\\Omega(\\v{k},\\v{k}').\n\\end{align}\n The momentum-dependence of the single-nucleon potential is controlled by\nthe last term, which is composed of two intrinsically different\nterms, i.e., $f_J(\\v{r},\\v{k})$ and $\\Omega(\\v{k},\\v{k}')$, both are\nfunctions of momentum, leading to a non-trivial momentum behavior of\nthe nucleon potential $U_J(\\rho,\\delta,|\\v{k}|)$ once the HMT in the $n_{\\v{k}}^J$ is\nconsidered.\n\nThe $f_J$ used in several integrations above is the phase space distribution function of nucleon $J$. It is related to the nucleon momentum distribution $n_{\\v{k}}^J(\\rho,\\delta)$ by\n\\begin{equation}\nf_J(\\v{r},\\v{k})=\\frac{2}{h^3}n_{\\v{k}}^J(\\rho,\\delta)=\\frac{1}{4\\pi^3}n_{\\v{k}}^J(\\rho,\\delta),~~\\hbar=1.\n\\end{equation}\nFor example, in the FFG model,\n$n_{\\v{k}}^J=\\Theta(k_{\\rm{F}}^J-|\\v{k}|)$ with $\\Theta$ the\nstandard step function, then\n$f_J(\\v{r},\\v{k})=(1\/4\\pi^3)\\Theta(k_{\\rm{F}}^J-|\\v{k}|)$. As mentioned briefly earlier, the $\\Omega(\\v{k},\\v{k}')$ in (\\ref{Epot}) is the function characterizing\nthe momentum dependence of the single-nucleon potential due to the finite-range of nuclear interactions.\nFor instance, for a Yukawa form of the two-nucleon interaction\n$\\exp[-\\mu|\\v{r}|]\/|\\v{r}|$ with $|\\v{r}|=|\\v{x}-\\v{x}'|$\nthe distance between two nucleons,  one obtains $\n\\Omega(\\v{k},\\v{k}')=[1+(\\v{k}-\\v{k}')^2\/\\Lambda^2)]^{-1}$, which\nis the familiar MDI interaction\\,\\cite{Gale1987,Bertsch1988,Das2003}. \nIn applying the above formalism to transport model simulations of nuclear reactions, the $f_J (\\v{r}, \\v{k})$ and $n_{\\v{k}}^J$ become time dependent and they are calculated self-consistently from solving\ndynamically the coupled Boltzmann-Uehling-Uhlenbeck (BUU) transport equations for quasi-nucleons\\,\\cite{Bertsch1988,Aichelin1990,Sto86}. \nSimilarly, in the original Gogny interaction\\,\\cite{GOG}, the finite-range two-nucleon interaction has the form of a Gaussian\nregulator $\\exp[-\\mu^2|\\v{r}|^2]$, and the corresponding $\\Omega$ takes the form of $\\exp[-\\mu^2|\\v{k}-\\v{k}'|^2\/4]$.\n\n\n\n\\begin{figure}[h!]\n\\centering\n \n  \\includegraphics[width=8.cm]{Fig2.eps}\n  \\caption{Sketch of the function $\\Omega(\\v{k},\\v{k}')$, where the blue line is only effectively applicable.}\n  \\label{fig_Omega}\n\\end{figure}\n\nIt is useful to emphasize here the following basic properties of $\\Omega(\\v{k},\\v{k}')$: \n\\begin{enumerate}\n\\item[(a)] It is symmetric between the momentum $\\v{k}$ and $\\v{k}'$, i.e.,\n$\\Omega(\\v{k},\\v{k}')=\\Omega(\\v{k}',\\v{k})$, reflecting the scalar\nproperty of the nucleon potential. Thus $\\Omega$ is a function of\n$\\v{k}+\\v{k}'$ and $\\v{k}\\cdot\\v{k}'$, and only the second term\ncouples the momenta of the two nucleons.\nIn general, one can show that it only depends on the magnitude of the difference $|\\v{k}-\\v{k}'|$.\nBoth the Yukawa potential and the Gogny force have such form.\n\n\\item[(b)] At zero momentum, $\\Omega(\\v{k},\\v{k}')$ is a constant (which can\nalways be normalized to unity), indicating\n$\\Omega(\\v{k},\\v{k}')\\approx1+\\rm{``corrections''}$ when\n$|\\v{k}|\\approx0,|\\v{k}'|\\approx0$. Furthermore, as the momentum\nscale increases, the momentum-dependent interaction between nucleons\ndecreases, even to become free\/negative, leading to that\n$\\Omega(|\\v{k}|\\to\\infty,|\\v{k}'|\\to\\infty)\\to0$ or approaches a\nsmall negative quantity $\\mathcal{N}$. This property is relevant to\nthe fact, e.g., that the single-nucleon potential in SNM at large energies\nsaturates. Consequently, the above ``corrections'' in\n$\\Omega(\\v{k},\\v{k}')$ beyond the leading contribution ``1'' at\nmoderate momenta should be negative.\nFor example, we have $\\Omega(\\v{k},\\v{k}')\\approx 1-|\\v{k}-\\v{k}'|^2\/\\Lambda^2$ for the Yukawa potential and \n$\\Omega(\\v{k},\\v{k}')\\approx1-\\mu^2|\\v{k}-\\v{k}'|^2\/4$ for the Gogny force.\nIn fact, both of them can be approximated as $\\Omega(\\v{k},\\v{k}')\\approx 1+c_1\\v{k}^2\/\\Lambda^2+c_2\\v{k}'^2\/\\Lambda^2+c_3\\v{k}\\cdot\\v{k}'\/\\Lambda^2$, where the coefficients $c_i$'s depend on the detailed form of the interactions.\nWe will use this perturbative expansion as a hint to construct a reasonable surrogate $\\Omega(\\v{k},\\v{k}')$, see (\\ref{MFunc}).\n\n\\item[(c)] Based on the last point, a perturbation-effective version of the\nfunction $\\Omega(\\v{k},\\v{k}')$ can be constructed generally as\n$\\Omega(\\v{k},\\v{k}')\\approx\n1+af(|\\v{k}-\\v{k}'|)+bg(|\\v{k}-\\v{k}'|)+\\cdots$, where the\nphenomenological functions $f,g,\\cdots$ are small at moderate\nmomentum scale. This effective $\\Omega(\\v{k},\\v{k}')$ can only be\nused up to a momentum scale around $\\Lambda$, see the sketch illustrating this feature in FIG.\\,\\ref{fig_Omega}, where the $\\Omega(\\v{k},\\v{k}')$ denoted by the\nblue line is perturbatively effective up to around a certain high momentum\ncutoff. Namely, the surrogate $\\Omega(\\v{k},\\v{k}')$ constructed in this manner grasps the main features of the original $\\Omega(\\v{k},\\v{k}')$ up to the high momentum cutoff.\n\\end{enumerate}\nConsidering the SRC-induced HMT in the single-nucleon momentum distribution and the kernel $\\Omega(\\v{k},\\v{k}')$ for the momentum dependent single-nucleon potential either with the Yukawa or Gogny force, their generally complicated form and nature hinder us from integrating out analytically the relevant integrations over momentum in the expressions of nucleon specific energy and single-nucleon potential. While these integrations are necessary for fixing all the EOS parameters using empirical properties of nuclear matter, symmetry energy, nucleon effective mass, and nucleon optical potentials at $\\rho_0$, etc. \n\nMaking use of the main features of $\\Omega$ outlined above, we adopt in this work the following parametrization as a surrogate of $\\Omega(\\v{k},\\v{k}')$ \n\\begin{equation}\\label{MFunc}\n\\Omega(\\v{k},\\v{k}')=1+{a}\\left[\\left(\\frac{\\v{k}\\cdot\\v{k}'}{\\Lambda^2}\\right)^2\\right]^{1\/4}\n+{b}\\left[\\left(\\frac{\\v{k}\\cdot\\v{k}'}{\\Lambda^2}\\right)^2\\right]^{1\/6},\n\\end{equation}\nwhere $a$ and $b$ are two phenomenological parameters, and\n$\\Lambda\\approx1\\,\\rm{GeV}$ is an effective momentum scale. The\nparameter $a$ should be negative according to the above discussion (see point (b)).\nAs pointed out in Ref.\\,\\cite{CLC18}, the advantages of using this surrogate is twofold: (1) the final 1\/2 and 1\/3 power of $\\v{k}\\cdot\\v{k}'\/\\Lambda^2$ is relevant for describing properly the kinetic energy dependence of nucleon optical potential, as we shall show in FIG.\\,\\ref{fig_ab_U0}; (2) it enables analytical expressions for both the EOS of ANM and $U_J(\\rho,\\delta, |\\v{k}|)$. \nMoreover,  the $\\Omega$ function is only perturbatively effective at momenta smaller than the momentum scale $\\Lambda$, indicating that the EDF thus constructed can only be used to a restricted range of momentum and\/or density.  For example,  the cutoff of the HMT about $2.4k_{\\rm{F}}\\approx631\\,\\rm{MeV}$ or the density $\\rho\\approx7\\rho_0\\mbox{$\\sim$}8\\rho_0$ corresponding to $k_{\\rm{F}}\\approx503\\,\\rm{MeV}\\mbox{$\\sim$}526\\,\\rm{MeV}$ should be safely smaller than the $\\Lambda$ parameter $\\approx1\\,\\rm{GeV}$.\nIn this sense, the parameterized $\\Omega$ (\\ref{MFunc}) is a reliable surrogate\/effective model\\,\\cite{SURR} for studying properties of neutron stars and intermediate-relativistic energy heavy-ion reactions where the maximum density reached is less than the above estimates.\n\n\\setcounter{equation}{0}\n\\section{Analytical expressions of the SNM EOS, nucleon effective mass, symmetry energy, isoscalar and isovector potentials}\\label{sec4}\nIn this section, we present all the analytical expressions of quantities relevant for describing the ANM EOS using the surrogate function (\\ref{MFunc}) for $\\Omega(\\v{k},\\v{k}')$. Detailed derivations are given in the APPENDIX \\ref{app1}. More specifically, in subsection \\ref{sb_EOS_SNM} we give the SNM EOS together with the isoscalar nucleon effective mass $M_0^{\\ast}(\\rho)$, the coefficient of incompressibility $K_0(\\rho)$ and the single-nucleon potential $U_0(\\rho)$ in SNM. In subsection \\ref{sb_Esym}, the expression for the symmetry energy is derived. For general expressions of the ANM EOS $E(\\rho,\\delta)$, see Eq.\\,(\\ref{HMT-Erd}) (together with Eqs.\\,(\\ref{HMT_Erd1}), (\\ref{HMT_Erd2}) and (\\ref{HMT_Erd3})) and Eq.\\,(\\ref{FFG-Erd}).\\\\\n\n\\subsection{EOS of SNM}\\label{sb_EOS_SNM}\n\n For the kinetic EOS of SNM, we\nhave\\,\\cite{Cai15}\n\\begin{equation}\\label{E0kin-HMT}\nE^{\\rm{kin}}_0(\\rho)=\\frac{3}{5}E_{\\rm{F}}(\\rho)\\left[\n1+{C}_0\\left(5\\phi_0+\\frac{3}{\\phi_0}-8\\right)\\right],\n\\end{equation}\nwhere $E_{\\rm{F}}=k_{\\rm{F}}^2\/2M$ is the Fermi kinetic energy. The\nhigh momentum cutoff $\\phi_0$ is generally larger than unity,\nthus $\\Upsilon_0=1+C_0(5\\phi_0+3\/\\phi_0-8)\\geq1$, indicating\nthat the $E^{\\rm{kin}}_0(\\rho)$ in the presence of HMT is enhanced\ncompared to the prediction of using the FFG nucleon momentum distribution\\,\\cite{Cai15}. This conclusion has\nimportant consequences on the total EOS of SNM once the potential\npart is taken into account, i.e., as the kinetic part is enhanced,\nthe potential part should be correspondingly reduced, leading to a\nreduction of the single-nucleon potential. Furthermore, the\ndensity-dependent terms in the EOS is the same as those in the MDI\nmodel\\,\\cite{Das2003,LCK08,Che05,XuJ10,XuJ15}. The total EOS of SNM in\nthe HMT model is thus,\n\\begin{widetext}\n\\begin{align}\\label{HMT-E0-mm}\nE_0(\\rho)=&\\frac{3k_{\\rm{F}}^2}{10M}\\left(5\\phi_0+\\frac{3}{\\phi_0}-8\\right)\n+\\frac{1}{4}A_{\\rm{tot}}\\left(\\frac{\\rho}{\\rho_0}\\right)+\\frac{B}{\\sigma+1}\\left(\\frac{\\rho}{\\rho_0}\\right)^{\\sigma}\\notag\\\\\n&+\\frac{1}{2}C_{\\rm{tot}}\\left(\\frac{\\rho}{\\rho_0}\\right)\n\\left[1+\\frac{24a}{49}\\left[\\Delta_0+7C_0\\left(1-\\frac{1}{\\sqrt{\\phi_0}}\\right)\\right]^2\\left(\\frac{k_{\\rm{F}}}{\\Lambda}\\right)+\n\\frac{243b}{400}\\left[\\Delta_0+5C_0\\left(1-\\frac{1}{\\phi_0^{2\/3}}\\right)\\right]^2\\left(\\frac{k_{\\rm{F}}}{\\Lambda}\\right)^{2\/3}\\right],\n\\end{align}\n\\end{widetext}\nsee also (\\ref{HMT-E0}), where\n$A_{\\rm{tot}}=A_\\ell+A_{\\rm{u}}=A_\\ell^0+A_{\\rm{u}}^0$ and\n$C_{\\rm{tot}}=C_\\ell+C_{\\rm{u}}$. The two factors $\\Pi_1$ and $\\Pi_2$ characterizing the HMT\nin Eq.\\,(\\ref{HMT-E0-mm}) are given by\n\\begin{align}\n\\Pi_1=&\\Delta_0+7C_0\\left(1-\\frac{1}{\\sqrt{\\phi_0}}\\right),\\label{def_Pi1}\\\\\n\\Pi_2=&\\Delta_0+5C_0\\left(1-\\frac{1}{\\phi_0^{2\/3}}\\right).\n\\end{align}\n\nIn the FFG model (as a reminder, the ``FFG'' here means that the $n_{\\v{k}}^J$ takes the step function while the potential part is also included in calculating the EOS and related quantities), the high momentum cutoff $\\phi_0$ becomes $\\phi_0=1$ and $\\phi_1=0$, we have $\n\\Delta_0=1-3C_0(1-\\phi_0^{-1})\\to1$,  and the two factors $\\Pi_1$ and $\\Pi_2$ become $\\Pi_1\\to1,\\Pi_2\\to1$. In the same limit, the Eq.\\,(\\ref{HMT-E0-mm}) is then reduced to\n\\begin{align}\\label{FFG-E0-mm}\nE_0(\\rho)=&\\frac{3k_{\\rm{F}}^2}{10M}+\\frac{1}{4}A_{\\rm{tot}}\\left(\\frac{\\rho}{\\rho_0}\\right)+\\frac{B}{\\sigma+1}\\left(\\frac{\\rho}{\\rho_0}\\right)^{\\sigma}\\notag\\\\\n&+\\frac{1}{2}C_{\\rm{tot}}\\left(\\frac{\\rho}{\\rho_0}\\right)\\left[1+\\frac{24a}{49}\\frac{k_{\\rm{F}}}{\\Lambda}\n+\\frac{243b}{400}\\left(\\frac{k_{\\rm{F}}}{\\Lambda}\\right)^{2\/3}\\right],\n\\end{align}\nwhich is derived (independently) in the FFG model, see derivations leading to Eq.\\,(\\ref{FFG-E0}).\n\nThe density structure of both the $E_0(\\rho)$ in the\nFFG and the HMT model is the same, i.e.,\n\\begin{equation}\\label{str}\nE_0(\\rho)=\\nu_1\\rho^{2\/3} +\\nu_2\\rho+\\nu_3\\rho^{\\sigma}+\n\\nu_4\\rho^{4\/3}+\\nu_5\\rho^{11\/9},\n\\end{equation}\nwith different coefficient $\\nu_j$'s in the FFG\/HMT model.\nThus, $E_0(\\rho)$ is a generalized polynomial of density $\\rho$ (or equivalently a polynomial of Fermi momentum $k_{\\rm{F}}$),\nindicating that it is only effectively applicable to low densities.\nConsequently, the density structure of the pressure of SNM and the incompressibility can be worked out as\n\\begin{align}\nP_0(\\rho)\/\\rho=&\\frac{2}{3}\\nu_1\\rho^{2\/3} +\\nu_2\\rho\n+\\sigma\\nu_3\\rho^{\\sigma}\\notag\\\\\n&+\\frac{4}{3}\\nu_4\\rho^{4\/3}\n+\\frac{11}{9}\\nu_5\\rho^{11\/9},\\\\\nK_0(\\rho)=&-2\\nu_1\\rho^{2\/3}+9\\sigma(\\sigma-1)\\nu_3\\rho^{\\sigma}\\notag\\\\\n&+4\\nu_4\\rho^{4\/3}+\\frac{22}{9}\\nu_5\\rho^{11\/9}.\n\\end{align}\nSpecifically, the expressions for the pressure $P_0(\\rho)$ and incompressibility $K_0(\\rho)$ in the FFG model are\ngiven in Eqs.\\,(\\ref{FFG-p0}) and (\\ref{FFG-K0}), while those in the\nHMT model are given in Eqs.\\,(\\ref{HMT-p0}) and (\\ref{HMT-K0}),\nrespectively.\n\nBesides the SNM EOS $E_0(\\rho)$ in the HMT model, the single-nucleon potential as a function of density $\\rho$ and momentum $\\v{k}$ in SNM could also be given,\n\\begin{align}\\label{HMT-U0-mm}\n&U_0(\\rho,|\\v{k}|)=\\frac{1}{2}A_{\\rm{tot}}\\left(\\frac{\\rho}{\\rho_0}\\right)+B\\left(\\frac{\\rho}{\\rho_0}\\right)^{\\sigma}\n+C_{\\rm{tot}}\\left(\\frac{\\rho}{\\rho_0}\\right)\\notag\\\\\n&\\times\\left[1+\\frac{4a}{7}\\Pi_1\\left(\\frac{|\\v{k}|k_{\\rm{F}}}{\\Lambda^2}\\right)^{1\/2}\n+\\frac{27b}{40}\\Pi_2\\left(\\frac{|\\v{k}|k_{\\rm{F}}}{\\Lambda^2}\\right)^{1\/3}\\right],\n\\end{align}\nsee Eq.\\,(\\ref{HMT-U0}) for a detailed derivation. Again the above expression can be reduced to the FFG one, see Eq.\\,(\\ref{FFG-U0}). The\nmomentum structure of the $U_0$ could be written explicitly as (due to the function $\\Omega$),\n\\begin{equation}\nU_0(\\rho,|\\v{k}|)=\\zeta_0(\\rho)+\\zeta_1(\\rho)|\\v{k}|^{1\/2}+\\zeta_2(\\rho)|\\v{k}|^{1\/3},\n\\end{equation}\nwith\n$\\zeta_0(\\rho)=(A_{\\rm{tot}}\/2+C_{\\rm{tot}})(\\rho\/\\rho_0)+B(\\rho\/\\rho_0)^{\\sigma}$. The latter is unchanged in the presence of HMT, while the form of $\\zeta_1(\\rho)$ and $\\zeta_2(\\rho)$ should change correspondingly, compare (\\ref{FFG-U0}) and (\\ref{HMT-U0}).\nThe concavity of the $U_0$ at the saturation density\\,\\cite{Ham90} indicates that $C_{\\rm{tot}}$ is negative according to $\n\\zeta_1(\\rho)=4C_{\\rm{tot}}(\\rho\/\\rho_0)a\\Pi_1k_{\\rm{F}}^{1\/2}\/7\\Lambda>0$,\nwhere $a$ is a negative parameter as expected by general discussions (given in the last section) and\n$\\Pi_1$ is positive (see the expression (\\ref{def_Pi1})). The momentum-dependence of the $U_0$ is characterized by introducing the nucleon effective k-mass\\,\\cite{LiBA2018PPNP,LiBA15,LiBA16}, i.e.,\n\\begin{equation}\nM_0^{\\ast}(\\rho,|\\v{k}|)\/M=\\left[1+\\frac{M}{|\\v{k}|}\\frac{\\partial\nU_0}{\\partial|\\v{k}|}\\right]^{-1}.\n\\end{equation}\nSpecifically, we obtain for the current EDF the following expression \n\\begin{align}\n\\frac{M_0^{\\ast}(\\rho,|\\v{k}|)}{M}=\\Bigg[&1+C_{\\rm{tot}}M\\left(\\frac{\\rho}{\\rho_0}\\right)\n\\Bigg[\\frac{2a\\Pi_1}{7}\\left(\\frac{k_{\\rm{F}}}{\\Lambda^2}\\right)^{1\/2}|\\v{k}|^{-3\/2}\\notag\\\\\n&+\\frac{9b\\Pi_2}{40}\\left(\\frac{k_{\\rm{F}}}{\\Lambda^2}\\right)^{1\/3}|\\v{k}|^{-5\/3}\\Bigg]\\Bigg]^{-1},\n\\end{align}\nwhich is further evaluated at the Fermi surface $|\\v{k}|=k_{\\rm{F}}$ to give the density dependence of the $M_0^{\\ast}(\\rho)\/M$ as,\n\\begin{align}\\label{FFG-M0-mm}\n\\frac{M_0^{\\ast}(\\rho)}{M}\n=\\Bigg[&1+C_{\\rm{tot}}M\\left(\\frac{\\rho}{\\rho_0}\\right)\n\\Bigg[\\frac{2a\\Pi_1}{7}\\frac{1}{k_{\\rm{F}}\\Lambda}\\notag\\\\\n&+\\frac{9b\\Pi_2}{40}\\left(\\frac{1}{\\Lambda^2k_{\\rm{F}}^4}\\right)^{1\/3}\\Bigg]\\Bigg]^{-1}.\n\\end{align}\nIn the high density limit, we have\n\\begin{equation}\\label{def_11}\n\\lim_{\\rm{large }\\rho}\\frac{M_0^{\\ast}(\\rho)}{M}\n\\approx\\left[1+\\frac{2M\\Pi_1aC_{\\rm{tot}}}{7k_{\\rm{F}}\\Lambda}\\left(\\frac{\\rho}{\\rho_0}\\right)+\\mathcal{O}\\left(k_{\\rm{F}}^{-1\/3}\\right)\\right]^{-1},\n\\end{equation}\nwhich approaches zero in the limit $\\rho\\to\\infty$. In the FFG\nmodel, the density dependence of the effective mass is given in Eq.\\,(\\ref{FFG-M0}), i.e., by taking $\\Pi_1=1$ in (\\ref{def_11}).\n\n\n\\subsection{Symmetry Energy}\\label{sb_Esym}\n\nWe now discuss the symmetry energy, its slope parameter, and the isovector (symmetry) potential $U_{\\rm{sym}}(\\rho,|\\v{k}|)$. As derived in\ndetailed in APPENDIX \\ref{app1}, the symmetry energy for the general HMT model is given by,\n\\begin{widetext}\n\\begin{align}\\label{HMT-Esym-mm}\nE_{\\rm{sym}}(\\rho)=&\\frac{k_{\\rm{F}}^2}{6M}\\left[1+C_0(1+3C_1)\\left(5\\phi_0+\\frac{3}{\\phi_0}-8\\right)\n+3C_0\\phi_1\\left(1+\\frac{5}{3}C_1\\right)\\left(5\\phi_0-\\frac{3}{\\phi_0}\\right)+\\frac{27C_0\\phi_1^2}{5\\phi_0}\\right]\n\\notag\\\\\n&+\\frac{1}{4}C_\\ell\\Delta_0^2\\left(\\frac{\\rho}{\\rho_0}\\right)\\left[\nY_{10}+Y_{11}a\\frac{k_{\\rm{F}}}{\\Lambda}+Y_{12}b\\left(\\frac{k_{\\rm{F}}}{\\Lambda}\\right)^{2\/3}\n\\right]\n+\\frac{1}{4}C_{\\rm{u}}\\Delta_0^2\\left(\\frac{\\rho}{\\rho_0}\\right)\n\\left[\nZ_{10}+Z_{11}a\\frac{k_{\\rm{F}}}{\\Lambda}+Z_{12}b\\left(\\frac{k_{\\rm{F}}}{\\Lambda}\\right)^{2\/3}\n\\right]\\notag\\\\\n&+\\frac{3}{2}C_\\ell\\Delta_0C_0\\left(\\frac{\\rho}{\\rho_0}\\right)\\left[\nY_{20}+Y_{21}a\\frac{k_{\\rm{F}}}{\\Lambda}+Y_{22}b\\left(\\frac{k_{\\rm{F}}}{\\Lambda}\\right)^{2\/3}\n\\right]\n+\\frac{3}{2}C_{\\rm{u}}\\Delta_0C_0\\left(\\frac{\\rho}{\\rho_0}\\right)\\left[\nZ_{20}+Z_{21}a\\frac{k_{\\rm{F}}}{\\Lambda}+Z_{22}b\\left(\\frac{k_{\\rm{F}}}{\\Lambda}\\right)^{2\/3}\n\\right]\\notag\\\\\n&+\\frac{9}{4}C_\\ell C_0^2\\left(\\frac{\\rho}{\\rho_0}\\right)\\left[\nY_{30}+Y_{31}a\\frac{k_{\\rm{F}}}{\\Lambda}+Y_{32}b\\left(\\frac{k_{\\rm{F}}}{\\Lambda}\\right)^{2\/3}\n\\right]+\\frac{9}{4} C_{\\rm{u}} C_0^2\\left(\\frac{\\rho}{\\rho_0}\\right)\\left[\nZ_{30}+Z_{31}a\\frac{k_{\\rm{F}}}{\\Lambda}+Z_{32}b\\left(\\frac{k_{\\rm{F}}}{\\Lambda}\\right)^{2\/3}\n\\right]\\notag\\\\\n&+\\frac{1}{4}A_{\\rm{d}}\\left(\\frac{\\rho}{\\rho_0}\\right)-\\frac{Bx}{\\sigma+1}\\left(\\frac{\\rho}{\\rho_0}\\right)^{\\sigma},\\end{align}\n\\end{widetext}\nwith the expressions for $Y_{ij}$ and $Z_{ij}$ given in Eqs.\\,(\\ref{def_Y1}), (\\ref{def_Y20}), (\\ref{def_Y21}), (\\ref{def_Y22}),\n(\\ref{def_Y30}), (\\ref{def_Y31}), (\\ref{def_Y32}),\n (\\ref{def_Z1}), (\\ref{def_Z20}), (\\ref{def_Z21}), \n(\\ref{def_Z22}), (\\ref{def_Z30}), (\\ref{def_Z31}), and (\\ref{def_Z32}), respectively.\nThe second line of Eq.\\,(\\ref{HMT-Esym-mm}) represents the\ncontribution from the depletion in the $n_{\\v{k}}^J$ characterized by $\\Delta_0^2$, while the fourth line has the HMT contribution indicated by the\n$C_0^2$. The third line of Eq.\\,(\\ref{HMT-Esym-mm}) is the mixing of\nthe depletion and the HMT characterized by $\\Delta_0C_0$. In the FFG\nmodel, only the first two lines of Eq.\\,(\\ref{HMT-Esym-mm}) survive.\nMoreover, $A_{\\rm{d}}$ and $C_{\\rm{d}}$ are the\ndifference between the like and unlike terms,\n\\begin{equation}\\label{ad}\nA_{\\rm{d}}=A_{\\rm{d}}^0+\\frac{4xB}{1+\\sigma},~~C_{\\rm{d}}=C_\\ell-C_{\\rm{u}},\n\\end{equation}\nwith $\nA_{\\rm{d}}^0=A_\\ell^0-A_{\\rm{u}}^0$. The density structure of\nthe symmetry energy is the same as that of the SNM EOS, see Eq.\\,(\\ref{str}), thus it is straightforward to obtain the slope\nparameter $L(\\rho)$, see Eq.\\,(\\ref{HMT-L}).\n\nSimilarly, the isovector (symmetry) potential $U_{\\rm{sym}}(\\rho,|\\v{k}|)$ is defined through the following Lane potential \\,\\cite{Lan62} \n\\begin{equation}\nU_J(\\rho,\\delta,|\\v{k}|)\\approx\nU_0(\\rho,|\\v{k}|)+U_{\\rm{sym}}(\\rho,|\\v{k}|)\\tau_3^J\\delta+\\mathcal{O}(\\delta^2).\n\\end{equation}\nSuch a decomposition of single-nucleon potential in ANM has been verified by various phenomenological model analyses of nucleon-nucleus scattering data and predictions of microscopic nuclear many-body theories, see, e.g., Refs. \\cite{mu04,Li04,Ron06,Dal1,zuo05,Beh11,LiX13,LiX15}. In the HMT model, it is given by\n\\begin{align}\\label{HMT-Usym-mm}\n&U_{\\rm{sym}}(\\rho,|\\v{k}|)=\\frac{1}{2}A_{\\rm{d}}\\left(\\frac{\\rho}{\\rho_0}\\right)-\\frac{2Bx}{\\sigma+1}\\left(\\frac{\\rho}{\\rho_0}\\right)^{\\sigma}\\notag\\\\\n&+\nC_{\\rm{d}}\\left(\\frac{\\rho}{\\rho_0}\\right)\\left[\\gamma_0+\\gamma_1a\\left(\\frac{|\\v{k}|k_{\\rm{F}}}{\\Lambda^2}\\right)^{1\/2}\n+\\gamma_2b\\left(\\frac{|\\v{k}|k_{\\rm{F}}}{\\Lambda^2}\\right)^{1\/3}\\right].\n\\end{align}\nThe expressions for $\\gamma_0,~\\gamma_1$ and $\\gamma_2$ are given in\nEqs.\\,(\\ref{def_gamma0}), (\\ref{def_gamma1}) and (\\ref{def_gamma2}), respectively.\nNotice that in the FFG model, besides $\n\\Delta_0\\to1$, we also have\n\\begin{align}\n\\Delta_0\\Delta_1=&-{3C_0[C_1(\\phi_0-1)+\\phi_1]}\/{\\phi_0}\\to0,\\\\\n\\Delta_0\\Delta_2=&-{3C_0\\phi_1(C_1-\\phi_1)}\/{\\phi_0}\\to0,\n\\end{align}\nsee (\\ref{Def_Delta012}), thus,\n\\begin{align}\n&Y_{10}\\to2,~~Y_{11}\\to{32}\/{21},~~Y_{12}\\to{33}\/{20},\\\\\n&Z_{10}\\to-2,~~Z_{11}\\to-{8}\/{7},~~Z_{12}\\to-{27}\/{20},\n\\end{align}\ntogether with $Y_{2j}\\to0,~Z_{2j}\\to0,~Y_{3j}\\to0,~Z_{3j}\\to0$ with $j=0,1,2$, see\nthe expressions (\\ref{def_Y20}), (\\ref{def_Y21}), (\\ref{def_Y22}),\n(\\ref{def_Z20}), (\\ref{def_Z21}), (\\ref{def_Z22}), (\\ref{def_Y30}),\n(\\ref{def_Y31}), (\\ref{def_Y32}), (\\ref{def_Z30}),\n(\\ref{def_Z31}) and (\\ref{def_Z32}) and the consequent result is the\nsymmetry energy in the FFG model, see Eq.\\,(\\ref{FFG-Esym}).\nMoreover, the expressions for the slope parameter $L(\\rho)$ and the symmetry potential\n$U_{\\rm{sym}}(\\rho,|\\v{k}|)$ in the FFG model are given in Eq.\\,(\\ref{FFG-L}) and Eq.\\,(\\ref{FFG-Usym}), respectively.\n\n\nConsidering the expression for the symmetry energy, the factor characterizing the effects of the HMT, i.e.,\n\\begin{align}\\label{def_Upsilon}\n&\\Upsilon_{\\rm{sym}}=1+C_0(1+3C_1)\\left(5\\phi_0+\\frac{3}{\\phi_0}-8\\right)\\notag\\\\\n&+3C_0\\phi_1\\left(1+\\frac{5}{3}C_1\\right)\\left(5\\phi_0-\\frac{3}{\\phi_0}\\right)\n+\\frac{27C_0\\phi_1^2}{5\\phi_0},\n\\end{align}\nis generally smaller than unity, leading to a reduction of the\nkinetic symmetry energy\\,\\cite{Cai15}. According to the single-nucleon potential decomposition of the symmetry\nenergy\\,\\cite{XuC10,XuC11,CheR12} based on the  Hugenholtz-Van Hove (HVH) theorem\\,\\cite{Hug58}, a contribution of\n$2^{-1}U_{\\rm{sym}}(\\rho,|\\v{k}|=k_{\\rm{F}})$ to the\nsymmetry energy is obtained. Thus a reduction of the kinetic symmetry energy\nwill induce an enhancement of the symmetry potential, i.e., the\nsymmetry energy encapsulating the SRC effects will intrinsically\nhave impacts on the symmetry potential.\nIn addition, by using the expression of $A_{\\rm{d}}$ in Eq.\\,({\\ref{ad}) the last two terms in Eq.\\,(\\ref{HMT-Esym-mm}) can be rewritten as\n\\begin{equation}\n\\frac{1}{4}A_{\\rm{d}}^0\\left(\\frac{\\rho}{\\rho_0}\\right)+\\frac{xB}{1+\\sigma}\\left(\\frac{\\rho}{\\rho_0}\\right)\n\\left[1-\\left(\\frac{\\rho}{\\rho_0}\\right)^{\\sigma-1}\\right],\n\\end{equation}\nindicating that the $x$ does not affect the symmetry energy at\n$\\rho_0$\\,\\cite{Che05,XuJ10,XuJ15}. On the other hand, the corresponding contribution to the slope parameter of symmetry energy\n$L(\\rho)$ is $\n3A_{\\rm{d}}^0\/4+3xB({1-\\sigma})\/({1+\\sigma})$ at $\\rho=\\rho_0$. Thus\na larger (positive) $x$ corresponds to a softer symmetry energy if\n$B(1-\\sigma)<0$, once the other phenomenological parameters are\nfixed\\,\\cite{Che05}, see TABLE \\ref{tab_para}.\n\n\n\n\n\n\\setcounter{equation}{0}\n\\section{Numerical Demonstrations}\\label{sec5}\n\n\n\\subsection{Schemes for Determining the Coupling Constants}\\label{sb_parameter}\n\n\nIn this subsection, we discuss the scheme to determine the model couplings, i.e., $A_{\\rm{tot}},~B,~C_{\\rm{tot}},~A_{\\rm{d}},~C_{\\rm{d}},~\\sigma,~a,~b,~x$ and $\\Lambda$.\nAs pointed out in Ref.\\,\\cite{CLC18} that the function $\\Omega(\\v{k},\\v{k}')$ is invariant under the transformations,\n\\begin{equation}\na\\to a\/\\xi^{3\/2},~~b\\to\\xi b,~~\\Lambda\\to\\Lambda\/\\xi^{3\/2},\n\\end{equation}\nwith $\\xi>0$ being any scaling factor, i.e., we have the freedom to first fix one of them without affecting the physical results. We set $b = 2$ in the following and then determine other parameters using known empirical constraints.\n\n\n\\begin{table*}[t!]\n\\caption{Coupling constants used in the two models (right side) and\nsome empirical properties of ANM used to fix\nthem (left side), $b=2$ and $\\Lambda=1.6\\,\\rm{GeV}$ are fixed,\n$K_0\\equiv K_0(\\rho_0),~M_0^{\\ast}\\equiv M_0^{\\ast}(\\rho_0),~L\\equiv\nL(\\rho_0)$.\nSee also Ref.\\,\\cite{CLC18}.\n}\\label{tab_para} {\\normalsize\\begin{tabular}{lr||cccc}\n\\hline\\hline Quantity& Value & Coupling & FFG&HMT-SCGF&HMT-exp\\\\\n\\hline $\\rho_0$ (fm$^{-3}$) & $0.16$ & $A_\\ell^0$ $(\\rm{MeV})$ &\n$-266.2934$&$520.3611$&$146.6085$\\\\\n\\hline $E_0(\\rho_0)$\n$(\\rm{MeV})$&$-16.0$&$A_{\\rm{u}}^0$ $(\\rm{MeV})$&$-86.8331$&$805.3082$&$1216.2500$\\\\\n\\hline $M_0^{\\ast}\/M$ &$0.58$&$B$ $(\\rm{MeV})$&$517.5297$&$-256.9850$&$-64.5669$\\\\\n\\hline $K_0$ ($\\rm{MeV}$) &$230.0$&$C_\\ell$ $(\\rm{MeV})$&$-155.6406$&$-154.7508$&$-37.3249$\\\\\n\\hline $U_0(\\rho_0,0)$ ($\\rm{MeV}$)\n&$-100.0$&$C_{\\rm{u}}$ $(\\rm{MeV})$&$-285.3256$&$-351.0989$&$-679.5379$\\\\\n\\hline $E_{\\rm{sym}}(\\rho_0)$ ($\\rm{MeV}$)&$31.6$&$\\sigma$&$1.0353$&$0.9273$&$0.6694$\\\\\n\\hline $L$ ($\\rm{MeV}$)&$58.9$&$a$&$-5.4511$&$-5.0144$&$-4.1835$\\\\\n\\hline $U_{\\rm{sym}}(\\rho_0,1\\,\\rm{GeV})$ ($\\rm{MeV}$)&$-20.0$&$x$&$0.6144$&$0.3774$&$-0.2355$\\\\\n\\hline\\hline\n\\end{tabular}}\n\\end{table*}\n\n\nIn our scheme, the values of $E_0(\\rho_0),~ \\rho_0, ~K_0\\equiv\nK_0(\\rho_0), ~M_0^{\\ast}(\\rho_0)$ and $U_0(\\rho_0,0\\,\\rm{MeV})$\nfor SNM, and those of $E_{\\rm{sym}}(\\rho_0), ~L\\equiv L(\\rho_0)$ and\n$U_{\\rm{sym}}(\\rho_0,1\\,\\rm{GeV})$ for the symmetry energy, its slope and the\nsymmetry potential are fixed at their currently known most probable empirical values. For example, the value of the incompressibility $K_0=230\\pm20\\,\\rm{MeV}$ was\ndetermined from analyzing nuclear giant resonances\n(GMR)\\,\\cite{You99,Shl06,Pie10,Che12,Col14,Stone2014PRC,Garg2018PPNP,LiBA2021PRC,XuJ2021PRC,ZhangZ2021CPC}, and the nucleon effective mass at $\\rho_0$ is fixed at $0.58M$\\,\\cite{CLC18,LiBA2018PPNP}.\nFor the $E_\\text{sym}(\\rho_0)$ and $L $, all existing constraints extracted\nso far from both terrestrial laboratory measurements and\nastrophysical observations are found to be essentially consistent\nwith the global averages of $E_{\\text{sym}}({\\rho _{0}}) \\approx\n31.6\\pm2.66$ MeV and $L \\approx 58.9 \\pm 16$ MeV\\,\\cite{LiBA13}, see also Ref.\\,\\cite{Chen17,LiBA21}. Consequently, we can fix for the SNM five phenomenological parameters, i.e.,\n$A_{\\rm{tot}},~B,~C_{\\rm{tot}},~\\sigma$ and $a$, while for the symmetry\nenergy and the symmetry potential, the remaining three parameters,\ni.e., $A_{\\rm{d}},~C_{\\rm{d}}$ and $x$ can be fixed. Equivalently,\nthe totally eight parameters, i.e.,\n$A_\\ell^0,~A_{\\rm{u}}^0,~B,~C_\\ell,~C_{\\rm{u}},~\\sigma,~a$ and $x$ can be\ndetermined.\n\nNext, we need to select the cutoff $\\Lambda$.\nAt first glance, the $\\Lambda$ could be determined by adopting another empirical constraint, e.g., the symmetry energy at two times the saturation density $E_{\\rm{sym}}(2\\rho_0)$\\,\\cite{LiBA21}.\nBut it does not work in this manner.\nIn particular, one can demonstrate that once we fix the above eight quantities, i.e., $E_0(\\rho_0),~\\rho_0,~K_0,~M_0^{\\ast}(\\rho_0),~U_0(\\rho_0,0\\,\\rm{MeV}),~E_{\\rm{sym}}(\\rho_0),~L$ and\n$U_{\\rm{sym}}(\\rho_0,1\\,\\rm{GeV})$, these quantities at other densities or other energy scales do not further depend on the cutoff $\\Lambda$, see APPENDIX \\ref{app2} for more discussions.\nThe role of the $\\Lambda$ is different from the other model parameters.\nHowever, the $\\Lambda$ parameter may affect quantities which are not used in the fixing scheme, e.g., $E_{\\rm{sym},4}(\\rho)$.\nIn the following we select the $\\Lambda$ parameter based on the empirical fact that the fourth-order symmetry energy at $\\rho_0$ in the FFG model is smaller than about 3\\,MeV\\,\\cite{Cai12,Sei14,Gon17,PuJ17,CWZC2022}.\n\nWe construct three models of EOS of ANM using\ndifferent HMT input parameters described in section \\ref{sec2},\ni.e., the FFG model as well as the HMT-SCGF and HMT-exp models. Again, we emphasize that the ``FFG'' or the ``HMT'' is only used to label the $n_{\\v{k}}^J(\\rho,\\delta)$ and the potential EOS is included in all of these models.\nSee TABLE \\ref{tab_para} for the values of the model parameters, here the cutoff $\\Lambda$ is set to be 1.6\\,GeV to fulfill the aforementioned constraint (specifically $1.40\\,\\rm{GeV}\\lesssim\\Lambda\\lesssim1.65\\,\\rm{GeV}$).\nIt is necessary to point out that since we fixed the parameters of\nthe HMT by using experimental data and\/or microscopic-theory calculations at the saturation density, the possible density dependence of those\nparameters, i.e., $C_0,~C_1,~\\phi_0$ and $\\phi_1$ is not\nexplored here (also see discussions given in Ref.\\,\\cite{Cai15}). The density dependence of the various terms\nin the kinetic EOS is thus only due to that of the nucleon Fermi momentum $k_{\\rm{F}}$.\nFrom the table, one can clearly find that the parameter $a$ is negative, i.e., the moving nucleons feel weaker interaction compared to the static ones,\nsee the discussions given after the formula (\\ref{MFunc}).\nThe EDF constructed is abbreviated as the abMDI\\,\\cite{CLC18}.\n\n\n\n\\begin{figure}[h!]\n\\centering\n \n  \\includegraphics[height=2.8cm]{CC-Lambda-a.eps}\\quad\n  \\includegraphics[height=2.8cm]{CC-Lambda-Ctot.eps}\\quad\n  \\includegraphics[height=2.8cm]{CC-Lambda-RA.eps}\n  \\caption{The $\\Lambda$-dependence of the coupling constants $a$ (left) and $C_{\\rm{tot}}$ (middle) and the ratio $C_{\\rm{tot}}\/a^2$ (right) in the three models.}\n  \\label{fig_CC-Lambda}\n\\end{figure}\n\nWe show in FIG.\\,\\ref{fig_CC-Lambda} the $\\Lambda$-dependence of the coupling constants $a$ (left panel) and $C_{\\rm{tot}}$ (middle panel) as well as the ratio $C_{\\rm{tot}}^2\/a$ (right panel) in the three models. The parameters $a$ and $C_{\\rm{tot}}$ evolve according to $a=a^0(\\Lambda\/\\Lambda_0)^{1\/3}$ and $C_{\\rm{tot}}=C_{\\rm{tot}}^0(\\Lambda\/\\Lambda_0)^{2\/3}$, respectively, see (\\ref{CC-aa}), and thus $C_{\\rm{tot}}\/a^2=C_{\\rm{tot}}^0\/(a^{0})^2=\\rm{const.}$, with the constant depending on the fitting scheme, i.e., it depends on the values of $\\rho_0,~E_0(\\rho_0),~K_0,~M_0^{\\ast}(\\rho_0)$ and $U_0(\\rho_0,0)$ via the specific model.\nSee the right panel of FIG.\\,\\ref{fig_CC-Lambda}, e.g., the ratio $C_{\\rm{tot}}\/a^2$ in the HMT-exp model is about \n$-40.95\\,\\rm{MeV}$ (see TABLE \\ref{tab_para}).\nOne can also show numerically that as $\\Lambda\\to0$, both $a$ and $C_{\\rm{tot}}$ approach zero, but their ratio keeps a constant.\nThe $\\Lambda$-dependence of other coupling constants ($A_{\\ell}^0,~A_{\\rm{u}}^0,~C_{\\ell},~C_{\\rm{u}}$) could be obtained similarly, and would not be analyzed here further (see APPENDIX B for more discussions).\n\n \n \n\\subsection{SNM EOS $E_0(\\rho)$ and Single-nucleon Potential $U_0(\\rho,|\\v{k}|)$}\n\nIn FIG.\\,\\ref{fig_ab_E0}, we show the SNM EOSs in the three models.\nIt is first interesting to see that the HMT models predict a\nslightly harder SNM EOS at supra-saturation densities than the\nFFG model, while by design they all have the same values of\n$M_0^{\\ast}$, $\\rho_0$, $E_0(\\rho_0)$ and $K_0$. Physically, this is\neasy to understand because of the large contribution to the kinetic EOS by the\nhigh momentum nucleons in the HMT models\\,\\cite{Cai16b}. For\nexample, the skewness of the SNM characterizing the high density\nbehavior of the EOS in the FFG model is found to be\n$J_0(\\rm{FFG})\\approx-381\\,\\rm{MeV}$, while that in the two HMT\nmodels is about $J_0(\\rm{HMT-SCGF})\\approx-376\\,\\rm{MeV}$ and\n$J_0(\\rm{HMT-exp})\\approx-329\\,\\rm{MeV}$, respectively. The small\nchange in the skewness from the FFG model to the HMT-SCGF model can\nbe traced back to the enhancement factor\n$\\Upsilon_0=1+C_0(5\\phi_0+3\/\\phi_0-8)$, which is unity in the FFG\nmodel and becomes about 1.18\/1.83 in the HMT-SCGF\/HMT-exp model. It\nis thus not surprising that the predictions on the pressure in SNM\nis also very similar, and can all pass through the constraints from analyzing \nthe collective flow date from relativistic heavy-ion collisions\\,\\cite{Dan02},\nsee FIG.\\,\\ref{fig_ab_Flow}.\nIt is necessary to point out that since the high momentum nucleon fraction in the HMT-SCGF is relatively lower than that in the HMT-exp model, the former is much closer to the FFG model (see the black dash line and the blue dash-dotted line shown in the inset of FIG.\\,\\ref{fig_ab_E0}).\n\n\\begin{figure}[h!]\n\\centering\n \n  \\includegraphics[width=6.5cm]{E0-EOS.eps}\n  \\caption{EOS of SNM in the FFG model and the HMT-SCGF as well as the HMT-exp models.}\n  \\label{fig_ab_E0}\n\\end{figure}\n\n\n\n\n\\begin{figure}[h!]\n\\centering\n \n  \\includegraphics[width=6.5cm]{P0Flow.eps}\n  \\caption{Comparison between the pressure $P_0$ of SNM with the experimental constraints from analyzing nuclear collective flows in heavy ion collisions\\,\\cite{Dan02}.}\n  \\label{fig_ab_Flow}\n\\end{figure}\n\n\nIn FIG.\\,\\ref{fig_ab_U0}, we show the single-nucleon potential in SNM\n$U_0$ as a function of kinetic energy $E_{\\rm{kin}}=[|\\v{k}|^2+M^2]^{1\/2}+U_0(\\rho_0,|\\v{k}|)-M$\nin the three models. As discussed earlier, the SRC-induced HMT (slightly) enhances\nthe kinetic EOS of SNM, and in order to maintain the total EOS of SNM\nat the saturation density to be consistent with empirical constraints the\npotential part should be correspondingly reduced, which is reflected\nin FIG.\\,\\ref{fig_ab_U0}. As the difference of the fraction of high\nmomentum nucleons in SNM and in PNM becomes larger, the kinetic EOS\nof SNM is much more enhanced, leading to a much softer potential\npart, i.e., the $U_0$ in the HMT-exp model\nis smaller than that in the HMT-SCGF model. Also shown in FIG.\\,\\ref{fig_ab_U0} are the predictions on the $U_0(\\rho_0,|\\v{k}|)$\nfrom other approaches, including the global relativistic fitting of\nelectron-scattering data up to about 1\\,GeV\\,\\cite{Ham90} (blue ``+''); the\npredictions by the neutron optical model up to about\n200\\,MeV\\,\\cite{LiX15} (dashed black band), and those from the chiral effective field\ntheories\\,\\cite{Hol16} (cyan band). It is obvious that the predictions on $U_0$\nboth in the HMT-SCGF and the HMT-exp models are consistent with these approaches, at kinetic energies $\\lesssim600\\,\\rm{MeV}$.\nIn this sense, the momentum-dependence in the function (\\ref{MFunc})\ngives a reasonable description of the kinetic-energy-dependence of the single-nucleon isoscalar\npotential. As the kinetic energy $E_{\\rm{kin}}$ becomes larger, the\ndeviation between the predictions on the $U_0$ in the three models and\nthe global relativistic fitting also becomes larger, indicating the breakdown of the perturbative construction for $\\Omega(\\v{k},\\v{k}')$.\n\n\n\\begin{figure}[h!]\n\\centering\n \n  \\includegraphics[width=6.5cm]{U0Hama.eps}\n  \\caption{single-nucleon potential in SNM as a function of kinetic energy in the FFG model and\n  HMT-SCGF\/HMT-exp model. Constraints from other approaches are shown for comparison.}\n  \\label{fig_ab_U0}\n\\end{figure}\n\n\n\n\\begin{figure}[h!]\n\\centering\n \n  \\includegraphics[width=6.5cm]{Mstar.eps}\n  \\caption{Nucleon effective k-mass as a function of density in the FFG and HMT-SCGF\/HMT-exp models.}\n  \\label{fig_ab_M0}\n\\end{figure}\n\n\n\nIn FIG.\\,\\ref{fig_ab_M0}, the density dependence of the effective k-mass is\nshown. It is straightforward to understand that the three models give very\nsimilar results. On one hand, three points of the effective mass are\nfixed, i.e., $M_0^{\\ast}(0)\/M=1$, $M_0^{\\ast}(\\rho_0)\/M=0.58$ and\n$M_0^{\\ast}(\\infty)\/M=0$ (see Eq.\\,(\\ref{def_11}) since both $a$ and $C_{\\rm{tot}}$ are negative, leading to $aC_{\\rm{tot}}>0$). Moreover, the\n$M_0^{\\ast}\/M$ monotonically decreases in the whole density range\nand is concave, e.g., for large density $\\rho$, one has from (\\ref{def_11}) that $\nM_0^{\\ast}\/M\\sim\\rho^{-2\/3}$. Meeting all of these common constraints the\n$M_0^{\\ast}(\\rho)\/M$ in the three models behaves thus very similarly (see similar discussions given in Ref.\\,\\cite{Cai16b} for the nonlinear RMF model counterpart).\n\n\n\n\n\\subsection{Reduction of the Kinetic Symmetry Energy $E_{\\textmd{sym}}^{\\textmd{kin}}(\\rho)$ and Enhancement of the Symmetry Potential $U_{\\textmd{sym}}(\\rho,|\\v{k}|)$}\n\n\nWe now discuss effects of the SRC\/HMT on the symmetry energy and isovector (symmetry) potential.\nIn FIG.\\,\\ref{fig_ab_Esym}, we show the density dependence of the nuclear\nsymmetry energy in the three models\\,\\cite{CLC18}.\nAlso shown include the constraints on the $E_{\\rm{sym}}(\\rho)$\naround the saturation density from analyses of heavy-ion collisions\\,\\cite{Tsa12} and isobaric analog state studies\\,\\cite{Dan14}. Although the symmetry energy from the three models can pass through these constraints, they have very different behaviors at super-saturation\ndensities\\,\\cite{CLC18}. Specifically, as discussed in the above sections, the\nreduction of the kinetic symmetry energy should be compensated by\nthe potential part to keep the total symmetry energy around $\\rho_0$ consistent with certain empirical constraints\\,\\cite{LiBA13,Chen17,LiBA21}.\nMore quantitatively, the factor $\\Upsilon_{\\rm{sym}}$ (defined in Eq.\\,(\\ref{def_Upsilon})) in the HMT-SCGF model is found to be about\n$\\Upsilon_{\\rm{sym}}\\approx0.71$, which is not far different from\nthat in the FFG model (which is unity), while the\n$\\Upsilon_{\\rm{sym}}$ factor in the HMT-exp model is about\n$\\Upsilon_{\\rm{sym}}\\approx-1.12$\\,\\cite{Cai15}, totally different from the FFG\nprediction. Thus it is not surprising that the effects of HMT in the\nHMT-exp model is much more apparent. \n\nIn addition, there are two critical densities $\\rho_1$ and $\\rho_2$ corresponding to the point above which the symmetry energy starts decreasing and its zero-point when the symmetry energy vanishes. They are defined as $L(\\rho_1)=0$ and $E_{\\rm{sym}}(\\rho_2)=0$, respectively. The latter case indicates the onset of the so-called isospin separation instability, namely it is energetically more favorable to split SNM into pure neutron matter and proton matter when the  symmetry energy becomes negative\\,\\cite{Kut93,Kut94,LiBA02}. Possible appearances and ramifications of such instability in heavy-ion reactions and neutron stars have been studied in several works, see., e.g., Refs.\\,\\cite{Kut93,Kut94,LiBA02,Szm06,wen,Kho96,Bas07,Ban00,Kubis1}.\nWithout considering the SRC-induced HMT, the FFG model predicts $\\rho_1^{\\rm{FFG}}\\approx4.4\\rho_0$ and $\\rho_2^{\\rm{FFG}}\\approx11.2\\rho_0$, while the HMT reduces the $\\rho_1$ to $3.5\\rho_0$ ($1.9\\rho_0$) and the $\\rho_2$ to $8.5\\rho_0$ ($4.1\\rho_0$) in the HMT-SCGF (HMT-exp) model.\nWe see that the experimental HMT constraints largely reduce the zero-point density $\\rho_2$, with an effect of about 63\\%.\nSince the $\\rho_2^{\\rm{HMT}\\mbox{-}\\rm{exp}}\\approx4.1\\rho_0$ is quite low considering neutron star issues, it is expected that the reduction of the symmetry energy at supra-saturation densities in the HMT-exp model may have sizable effects on properties of neutron stars, e.g., the mass-radius relation.\nOn the other hand, the reduction of the $E_{\\rm{sym}}(\\rho)$ at sub-saturation densities (see the inset of FIG.\\,\\ref{fig_ab_Esym}) may also induce changes on, e.g., the core-crust transition densities in neutron stars.\nThese issues are investigated in some details in subsection \\ref{sb_NS}.\n\nFurthermore, we also show in FIG.\\,\\ref{fig_ab_Esym} two constraints on the $E_{\\rm{sym}}(2\\rho_0)$. The pink solid circle at $E_{\\rm{sym}}(2\\rho_0)\\approx45\\pm3\\,\\rm{MeV}$ is from chiral perturbation theories with consistent nucleon-nucleon and three-nucleon interactions up to the fourth-order\\,\\cite{Dri20}. The grey diamond at $E_{\\rm{sym}}(2\\rho_0)\\approx51\\pm13\\,\\rm{MeV}$ is the fiducial value from surveying analyses of both relativistic heavy-ion reactions and neutron star properties since GW170817\\,\\cite{LiBA21}. It is clearly seen that the calculations with the SRC\/HMT-induced reduction of $E_{\\rm{sym}}(2\\rho_0)$ are consistent with these constraints\\,\\cite{LiBA21,Dri20,CL2021}.\n\n\\begin{figure}[h!]\n\\centering\n \n  \\includegraphics[width=6.5cm]{Esym.eps}\n  \\caption{Density dependence of the symmetry energy in the FFG model and HMT-SCGF\/HMT-exp model. Constraints from heavy-ion collisions\\,\\cite{Tsa12} and the isobaric analog state studies\\,\\cite{Dan14} are shown for comparison.\nSee also Ref.\\,\\cite{CLC18} and the text for details on the pink solid circle and the grey diamond.}\n  \\label{fig_ab_Esym}\n\\end{figure}\n\n\n\nThe reduction of the\nsymmetry energy found here is qualitatively consistent with the one from the nonlinear RMF\nmodels\\,\\cite{Cai16b}. \nHowever, they give quantitatively different results. \nIn particular, due to the specific structure of the nonlinear RMF model, the symmetry energy should never becomes negative.\nOn the other hand, the EOS in a non-relativistic EDF could be treated as an effective expansion in density. \nIn order to investigate the probable origin of this quantitative difference, it is enough for our purpose to assume that the symmetry energy takes the form $E_{\\rm{sym}}(\\rho)=E_{\\rm{sym}}^{\\rm{kin}}(\\rho_0)(\\rho\/\\rho_0)^{2\/3}+s_1(\\rho\/\\rho_0)^{\\alpha}+s_2(\\rho\/\\rho_0)^{\\beta}$, where $\\beta>\\alpha>2\/3$, and $E_{\\rm{sym}}^{\\rm{kin}}(\\rm{FFG})\\approx12.3\\,\\rm{MeV}$.\nThe coefficients $s_1$ and $s_2$ can be determined via the values of $E_{\\rm{sym}}(\\rho_0)$ and $L\\equiv L(\\rho_0)$. Specifically, the expression for $s_2$ is given by,\n\\begin{equation}\ns_2=\\frac{1}{\\beta-\\alpha}\\left(\\alpha-\\frac{2}{3}\\right)E_{\\rm{sym}}^{\\rm{kin}}(\\rho_0)+\\frac{L-3\\alpha E_{\\rm{sym}}(\\rho_0)}{\\beta-\\alpha}.\n\\end{equation}\nThe second term here is negative since $\\alpha>2\/3$ and thus $L-3\\alpha E_{\\rm{sym}}(\\rho_0)<0$ while $\\beta-\\alpha>0$. \nSimilarly, the coefficient of the first term of $E_{\\rm{sym}}^{\\rm{kin}}(\\rho_0)$ is positive (due to the same consideration), thus:\n\\begin{enumerate}\n\\item[(a)]One has $s_2^{\\rm{FFG}}>s_2^{\\rm{HMT}}$ as $E_{\\rm{sym}}^{\\rm{kin}}(\\rm{FFG})>E_{\\rm{sym}}^{\\rm{kin}}(\\rm{HMT})$. \n\\item[(b)]In addition, $s_2<0$ for the HMT-exp model, as the $E_{\\rm{sym}}^{\\rm{kin}}(\\rho_0)\\approx-13.8\\,\\rm{MeV}$ is negative. \n\\end{enumerate}\nThe point (a) explains why the $E_{\\rm{sym}}(\\rho)$ considering the HMT should be reduced while the point (b) indicates that in the HMT-exp model (with a negative $E_{\\rm{sym}}^{\\rm{kin}}(\\rho_0)$) the $E_{\\rm{sym}}(\\rho)$ must start decreasing above some critical density, and become negative at even higher densities. In our EDF, $\\beta=4\/3$ and one can find actually that the $s_2$'s in all three models are negative, see TABLE \\ref{tab_para1}.\nThis explains why the reduction of symmetry energy using the non-relativistic EDF should be much stronger than the one in the nonlinear RMF models.\n\n\nThe reduction of the symmetry energy at supra-saturation densities also generates corresponding reduction of its curvature coefficient $K_{\\rm{sym}}$. Quantitatively, its value changes from $-109$\\,MeV in the FFG model to about\n$-121\\,\\rm{MeV}$ ($-223$\\,MeV) in the HMT-SCGF (HMT-exp) model.\nWhile the $K_{\\rm{sym}}\\approx-223\\,\\rm{MeV}$ in the HMT-exp model is somewhat smaller than its current fiducial value of about $-107\\pm 88$ MeV (see FIG.\\,2 of Ref.\\,\\cite{LiBA21}), those from the FFG and the HMT-SCGF model are very consistent with the fiducial value. Thus, the general tendency is that the SRC\/HMT reduces the symmetry energy curvature coefficient $K_{\\rm{sym}}$. This may have some other consequences. For example, the pressure of neutron-rich nucleonic matter is given by\n\\begin{align}\nP\/3\\rho\\approx&L(\\rho)\\delta^2+3\\rho\\frac{\\d E_0(\\rho)}{\\d\\rho}\\notag\\\\\n\\approx&\nL\\delta^2\\left(1+3\\chi+\\frac{9}{2}\\chi^2\\right)\n+K_0\\chi(1+3\\chi)+\\frac{1}{2}\\chi^2J_0\\notag\\\\\n&+K_{\\rm{sym}}\\delta^2\\chi(1+3\\chi)\\left[1+\\frac{J_{\\rm{sym}}}{K_{\\rm{sym}}}\n\\frac{\\chi}{2(1+3\\chi)}\\right]\\notag\\\\\n\\approx&L\\delta^2\\left(1+3\\chi+\\frac{9}{2}\\chi^2\\right)\n+K_0\\chi(1+3\\chi)\\notag\\\\\n&+\\frac{1}{2}\\chi^2J_0+K_{\\rm{sym}}\\delta^2\\chi(1+3\\chi),\n\\end{align}\nwhere the skewness coefficient $J_{\\rm{sym}}$ is neglected in the last step.\nSince the $K_0$ and $L$ are fixed in our study,  and as discussed earlier the $J_0$ is slightly increased by the SRC\/HMT, the reduction of $K_{\\rm{sym}}$ may reduce the pressure $P$ at densities below about $3\\rho_0$ above which the $J_{\\rm{sym}}$ becomes important (the $\\chi$ is 2\/3 for $\\rho=3\\rho_0$ and thus $\\chi\/[2(1+3\\chi)]=1\/9$). Moreover,  the reduction of $K_{\\rm{sym}}$ may also modify the saturation line (on which the pressure is zero) of ANM. More quantitatively, the incompressibility coefficient of ANM along its saturation line is $K(\\delta)=K_0+K_{\\rm{sat},2}\\delta^2$\\,\\cite{LWChen2009} where the $K_{\\rm{sat},2}$ is\n \\begin{equation}\nK_{\\rm{sat,2}}=K_{\\rm{sym}}-6L-\\frac{J_0L}{K_0}.\n\\end{equation}\nIt changes from $-365\\,\\rm{MeV}$ in the FFG model to $-378\\,\\rm{MeV}$ ($-492\\,\\rm{MeV}$) in the HMT-SCGF (HMT-exp) model.  \nIt is seen that the maximum reduction is about 127\\,\\rm{MeV}.\n\n\\begin{table*}[t!]\n\\caption{Parameterizations of the EOS of SNM and the symmetry energy as\nwell as the nucleon potential. Momentum $|\\v{k}|$ is measured in MeV and\nthe density $\\rho$ in $\\rm{fm}^{-3}$,\n$E_0(\\rho),E_{\\rm{sym}}(\\rho),U_0(\\rho,|\\v{k}|)$ and\n$U_{\\rm{sym}}(\\rho,|\\v{k}|)$ are in MeV.}\\label{tab_para1}\n{\\normalsize\n\\begin{tabular}{c||c}\n\\hline\\hline Model for $n_{\\v{k}}^J$& Parametrization of the Physical Quantity\\\\\\hline \n\\hline FFG  &$x_{\\rm{SNM}}^{\\rm{HMT}}=0\\%,x_{\\rm{PNM}}^{\\rm{HMT}}=0\\%$\n\\\\\n\\hline $E_0(\\rho)$ &\n$E_0(\\rho)\\approx75.00\\rho^{2\/3}+1695.42\\rho^{1.04}-551.76\\rho-1378.02\\rho[1-0.81\\rho^{1\/3}+0.55\\rho^{2\/9}]$\\\\\n\\hline $U_0(\\rho,|\\v{k}|)$ &$U_0(\\rho,|\\v{k}|)\\approx3450.68\\rho^{1.04}-1103.52\\rho-2756.04\\rho[1-0.04|\\v{k}|^{1\/2}\\rho^{1\/6}+0.08|\\v{k}|^{1\/3}\\rho^{1\/9}]$\\\\\n\\hline $E_{\\rm{sym}}(\\rho)$ & $E_{\\rm{sym}}(\\rho)\\approx41.67\\rho^{2\/3}+1101.25\\rho-1041.63\\rho^{1.04}-229.34\\rho^{4\/3}+180.92\\rho^{11\/9}$\\\\\n\\hline $U_{\\rm{sym}}(\\rho,|\\v{k}|)$ &\n$U_{\\rm{sym}}(\\rho,|\\v{k}|)\\approx-2083.27\\rho^{1.04}+\\rho[2202.51-40.52|\\v{k}|^{1\/2}\\rho^{1\/6}+69.80|\\v{k}|^{1\/3}\\rho^{1\/9}]$\\\\\\hline\n\\hline HMT-SCGF  &$x_{\\rm{SNM}}^{\\rm{HMT}}=12\\%,x_{\\rm{PNM}}^{\\rm{HMT}}=4\\%$\n\\\\\n\\hline $E_0(\\rho)$   & $E_0(\\rho)\\approx88.28\\rho^{2\/3}-729.40\\rho^{0.93}+2071.36\\rho-1580.78\\rho[1-0.79\\rho^{1\/3}+0.57\\rho^{2\/9}]$\\\\\n\\hline $U_0(\\rho,|\\v{k}|)$  &$U_0(\\rho,|\\v{k}|)\\approx-1405.76\\rho^{0.93}+4142.72\\rho-3161.56\\rho[1-0.04|\\v{k}|^{1\/2}\\rho^{1\/6}+0.08|\\v{k}|^{1\/3}\\rho^{1\/9}]$\\\\\n\\hline $E_{\\rm{sym}}(\\rho)$ & $E_{\\rm{sym}}(\\rho)\\approx29.53\\rho^{2\/3}-146.18\\rho+275.30\\rho^{0.93}-467.07\\rho^{4\/3}+343.07\\rho^{11\/9}$\\\\\n\\hline $U_{\\rm{sym}}(\\rho,|\\v{k}|)$ &\n$U_{\\rm{sym}}(\\rho,|\\v{k}|)\\approx550.60\\rho^{0.93}+\\rho[-292.37-57.07|\\v{k}|^{1\/2}\\rho^{1\/6}+106.42|\\v{k}|^{1\/3}\\rho^{1\/9}]$\\\\\\hline\n\\hline HMT-exp  &$x_{\\rm{SNM}}^{\\rm{HMT}}=28\\%,x_{\\rm{PNM}}^{\\rm{HMT}}=1.5\\%$\n\\\\\n\\hline $E_0(\\rho)$  & $E_0(\\rho)\\approx137.31\\rho^{2\/3}-131.89\\rho^{0.67}+2129.47\\rho-2240.19\\rho[1-0.77\\rho^{1\/3}+0.63\\rho^{2\/9}]$\\\\\n\\hline $U_0(\\rho,|\\v{k}|)$ &$U_0(\\rho,|\\v{k}|)\\approx-220.18\\rho^{0.67}+4258.93\\rho-4480.39\\rho[1-0.04|\\v{k}|^{1\/2}\\rho^{1\/6}+0.08|\\v{k}|^{1\/3}\\rho^{1\/9}]$\\\\\n\\hline $E_{\\rm{sym}}(\\rho)$ & $E_{\\rm{sym}}(\\rho)\\approx-46.83\\rho^{2\/3}+392.52\\rho-31.06\\rho^{0.67}-1837.48\\rho^{4\/3}+1421.05\\rho^{11\/9}$\\\\\n\\hline $U_{\\rm{sym}}(\\rho,|\\v{k}|)$ &\n$U_{\\rm{sym}}(\\rho,|\\v{k}|)\\approx-62.11\\rho^{0.67}+\\rho[785.04-156.10|\\v{k}|^{1\/2}\\rho^{1\/6}+348.23|\\v{k}|^{1\/3}\\rho^{1\/9}]$\\\\\n\\hline\\hline\n\\end{tabular}}\n\\end{table*}\n\n\\begin{figure}[h!]\n\\centering\n \n  \\includegraphics[height=3.8cm]{Usym-1.eps}\n  \\includegraphics[height=3.8cm]{Usym-2.eps}\n  \\caption{Symmetry potential in the FFG model and the HMT-SCGF\/HMT-exp model at $\\rho=\\rho_0$ (left) and\n  at $\\rho=3\\rho_0$ (right).\nConstraints on the symmetry potential at $\\rho=\\rho_0$ from other approaches are shown for\ncomparisons.}\n  \\label{fig_ab_Usym}\n\\end{figure}\n\nIn FIG.\\,\\ref{fig_ab_Usym}, we show the momentum dependence of the\nsymmetry potential $U_{\\rm{sym}}(\\rho,|\\v{k}|)$ at $\\rho_0$ (left panel) and $3\\rho_0$ (right panel), respectively. The predictions on\n$U_{\\rm{sym}}(\\rho_0,|\\v{k}|)$ from other microscopic\napproaches\\,\\cite{LiX15,Hol16,Fri05} are also shown for comparisons. The empirical band is parametrized as $U_{\\rm{sym}}(E)\\approx\n(28\\pm6)\\,\\rm{MeV}-(0.15\\pm0.05)E$\\,\\cite{Hol16}. It can be found \nthat the uncertainties on the symmetry potential are larger than\nthose of the isoscalar potential $U_0$. This is mainly due to the poorly-known isospin dependence of nuclear interactions\\,\\cite{LCK08,LiBA2018PPNP}.\nWhile the prediction by the HMT-exp on the\n$U_{\\rm{sym}}(\\rho_0,|\\v{k}|)$ has certain deviation from the microscopic model predictions, the FFG and HMT-SCGF model\npredictions are quite consistent with the optical model fitting\\,\\cite{LiX15} as well as the chiral effective theory prediction\\,\\cite{Hol16}.\nSince there exists no direct constraint on the symmetry potential at, e.g., a large density $\\rho=3\\rho_0$, and it is beyond the current application domain of microscopic nuclear many-body theories, it will be interesting to test the predictions shown in the right window of  FIG.\\,\\ref{fig_ab_Usym}, thus the SRC\/HMT effects in dense neutron-rich matter, using nuclear reactions with high-energy radioactive beams. Of course, for this purpose it is first necessary to investigate experimental observables sensitive to the high-density symmetry potential. The results presented here especially the various analytical expressions and parameterizations will facilitate the explorations of this topic using transport models for nuclear reactions with radioactive beams. \nListed in TABLE \\ref{tab_para1} are parametrizations of the SNM EOS $E_0(\\rho)$, nucleon isoscalar potential $U_0(\\rho,|\\v{k}|)$, symmetry energy $E_{\\rm{sym}}(\\rho)$ and isovector potential $U_{\\rm{sym}}(\\rho,|\\v{k}|)$ corresponding to the three models (with their parameters given in TABLE \\ref{tab_para}), for future applications.\n\\begin{figure}[h!]\n\\centering\n \n  \\includegraphics[height=3.3cm]{Esym-xpara-1.eps}\n  \\includegraphics[height=3.3cm]{Esym-xpara-2.eps}\n  \\includegraphics[height=3.3cm]{Esym-xpara-3.eps}\n  \\caption{Symmetry energy with different effective three-nucleon force parameter $x$ in the three models indicated. }\n  \\label{fig_ab_Esym_x}\n\\end{figure}\n\nIn our calculations up to now, the effective three-nucleon force parameter $x$ is self-consistently determined by fixing the $L(\\rho_0)$, see TABLE \\ref{tab_para}.\nSince the $x$ parameter only affects the slope parameter of the symmetry energy instead of its magnitude at $\\rho_0$ itself, we can adjust $x$ to investigate how the density dependence of $E_{\\rm{sym}}(\\rho)$ changes when the SRC-induced HMT is taken into account. In FIG.\\,\\ref{fig_ab_Esym_x}, the symmetry energy obtained within the three\nmodels indicated by only changing the $x$ parameter (corresponding to changing only the $L$ coefficient) while fixing at the same time all the other seven parameters listed\nin TABLE \\ref{tab_para}, is shown. For the FFG model, the $x$-dependence of $E_{\\rm{sym}}(\\rho)$ is the same as in the conventional MDI\nmodel\\,\\cite{Das2003,Che05,LCK08,XuJ10,XuJ15}. In the presence of the SRC\/HMT, the large\nuncertainties of $E_{\\rm{sym}}(\\rho)$ at high densities can come from the poorly known physics of either the three-nucleon forces and\/or the SRC\/HMT.\n\n\n\\begin{figure}[h!]\n\\centering\n \n  \\includegraphics[height=3.7cm]{Esym-frac-1.eps}\\quad\n  \\includegraphics[height=3.7cm]{Esym-frac-2.eps}\n  \\caption{The symmetry energy at $2\\rho_0$ and $3\\rho_0$ with different fractions of high momentum nucleons in SNM and\/or in PNM, respectively. The black circles are for the default set on $x_{\\rm{SNM}}^{\\rm{HMT}}$ and $x_{\\rm{PNM}}^{\\rm{HMT}}$, i.e., $x_{\\rm{SNM}}^{\\rm{HMT}}=28\\%$ and $x_{\\rm{PNM}}^{\\rm{HMT}}=1.5\\%$.}\n  \\label{fig_ab_EsymFrac}\n\\end{figure}\n\nAs discussed in the INTRODUCTION, there are still uncertainties about the fraction of high momentum nucleons in either SNM and\/or PNM and thus its isospin dependence\\,\\cite{Hen14,Rio14}. Thus, it is interesting to study how the relevant quantities change when the HMT fractions in SNM and\/or PNM are modified.\nIn FIG.\\,\\ref{fig_ab_EsymFrac}, we show the symmetry energy at two\nreference densities, i.e., $\\rho_{\\rm{f}}=2\\rho_0$ and\n$\\rho_{\\rm{f}}=3\\rho_0$, by continuously changing the high\nmomentum nucleon fraction $x_{\\rm{SNM}}^{\\rm{HMT}}$ in SNM when fixing the $x_{\\rm{PNM}}^{\\rm{HMT}}$ in PNM at the\ndefault value of the HMT-exp model (left panel) (in PNM when fixing\nthat in SNM at the default value of the HMT-exp model (right panel)), respectively. \nThe values of $C_0$ and $C_{\\rm{n}}^{\\rm{PNM}}=C_0(1+C_1)$ are fixed at 0.161 and 0.12, respectively, while the values of  $\\phi_0$ and $\\phi_1$ are readjusted correspondingly according to the given $x_{\\rm{SNM}}^{\\rm{HMT}}$ and $x_{\\rm{PNM}}^{\\rm{HMT}}$ values. The black circles are for the default parameter set of $x_{\\rm{SNM}}^{\\rm{HMT}}$ and $x_{\\rm{PNM}}^{\\rm{HMT}}$, i.e., $x_{\\rm{SNM}}^{\\rm{HMT}}=28\\%$ and $x_{\\rm{PNM}}^{\\rm{HMT}}=1.5\\%$.\nAs the difference between the $x_{\\rm{SNM}}^{\\rm{HMT}}$ and\n$x_{\\rm{PNM}}^{\\rm{HMT}}$, i.e., $\\delta x=x_{\\rm{SNM}}^{\\rm{HMT}}-x_{\\rm{PNM}}^{\\rm{HMT}}$, increases, the reduction of the\n$E_{\\rm{sym}}(\\rho_{\\rm{f}})$ becomes much more apparent. While the\nsymmetry energy at the saturation density is fixed (at $31.6\\,\\rm{MeV}$),\nthese effects on $E_{\\rm{sym}}(2\\rho_0)$ are relatively minor compared with those at $\\rho=3\\rho_0$.\n\n\n\\begin{figure}[h!]\n\\centering\n \n  \\includegraphics[height=3.6cm]{Ksat2-frac-1.eps}\\quad\n  \\includegraphics[height=3.6cm]{Ksat2-frac-2.eps}\n  \\caption{The same as FIG.\\,\\ref{fig_ab_EsymFrac} but for the isospin incompressibility coefficient $K_{\\rm{sat},2}=K_{\\rm{sym}}-6L-J_0L\/K_0$. }\n  \\label{fig_Ksat2_frac}\n\\end{figure}\n\nSimilarly, we show in FIG.\\,\\ref{fig_Ksat2_frac} the dependence of the isospin incompressibility coefficient $K_{\\rm{sat},2}$ on the high momentum nucleon fraction.\nAs the $L$ and $K_0$ are fixed in our calculations, the dependence of the coefficient $K_{\\rm{sat},2}$ on $x_{\\rm{SNM}}^{\\rm{HMT}}$ and\/or\n$x_{\\rm{PNM}}^{\\rm{HMT}}$ mainly reflects the dependence of the curvature coefficient $K_{\\rm{sym}}$ and the skewness parameter $J_0$ on them.\nThe $K_{\\rm{sat},2}$ shown in FIG.\\,\\ref{fig_Ksat2_frac} is in good agreement with the estimate of $K_{\\rm{sat},2}\\approx-550\\pm100\\,\\rm{MeV}$ from analyzing different types\nof experimental data currently available\\,\\cite{Col14}, as shown using the cyan bands.\nFor example, if $x_{\\rm{PNM}}^{\\rm{HMT}}\\approx1.5\\%$ is fixed, then the constraint on $K_{\\rm{sat},2}$ leads to about $x_{\\rm{SNM}}^{\\rm{HMT}}\\gtrsim24\\%$. Similarly, if $x_{\\rm{SNM}}^{\\rm{HMT}}\\approx28\\%$ is fixed, then we can roughly find the constraint that $x_{\\rm{PNM}}^{\\rm{HMT}}\\lesssim12\\%$.\n\n\n\\subsection{SRC\/HMT Effects on the Proton Fraction $x_{\\textmd{p}}$, Core-crust Transition Density $\\rho_{\\textmd{t}}$ and pressure $P_{\\textmd{t}}$ as well as the Mass-radius Relation of Cold Neutron Stars at $\\beta$-equilibrium}\\label{sb_NS}\n\nAs an example of applying the Gogny-like EDFs encapsulating the SRC\/HMT in astrophysical studies, we investigate here the density profile of proton fraction $x_{\\textmd{p}}(\\rho)=[1-\\delta(\\rho)]\/2$, the core-crust transition density $\\rho_{\\rm{t}}$ and transition pressure $P_{\\rm{t}}$ as well as the mass-radius correlation in cold neutron stars at $\\beta$-equilibrium.\nThe transition density $\\rho_{\\rm{t}}$ is the nucleon number\ndensity that separates the liquid core from the inner crust in\nneutron stars. It plays an important role in determining many\nproperties of neutron\nstars\\,\\cite{Hor01,Pro06,Duc08a,Duc08b,XuJ09}.\nOne simple and widely used approach to determine the core-crust transition density $\\rho\n_{\\rm{t}}$ is the thermodynamical method. Normally, neutron star matter has to obey the intrinsic stability\ncondition\\,\\cite{Lat04,XuJ09,NBZ19,Kub07}\n\\begin{align}\n\\mathcal{U}_{\\rm{ther}}(\\rho)=&2\\rho \\frac{\\partial E(\\rho\n,x_{\\rm{p}})}{\\partial \\rho }+\\rho ^{2}\\frac{\\partial\n^{2}E(\\rho ,x_{\\rm{p}})}{\\partial \\rho ^{2}}\\notag\\\\\n& -\\left. \\left( \\frac{\\partial ^{2}E(\\rho ,x_{\\rm{p}})}{\\partial\n\\rho\n\\partial x_{\\rm{p}}}\\rho \\right) ^{2}\\right\/ \\frac{\\partial ^{2}E(\\rho ,x_{\\rm{p}})%\n}{\\partial x_{\\rm{p}}^{2}}\\notag\\\\\n=&\\rho^2\\left(\\frac{\\partial^2E(\\rho,x_{\\rm{p}})}{\\partial x_{\\rm{p}}^2}\\right)^{-1} \n\\left[\\frac{\\partial\\mu_{\\rm{n}}}{\\partial\\rho_{\\rm{n}}}\\frac{\\partial \\mu_{\\rm{p}}}{\\partial\\rho_{\\rm{p}}}-\\left(\\frac{\\partial\\mu_{\\rm{n}}}{\\partial\\rho_{\\rm{p}}}\\right)^2\\right]>0. \\label{Vther}\n\\end{align}\nIn the above, the density profile $x_{\\rm{p}}=x_{\\rm{p}}(\\rho)$ is determined by the $\\beta$-equilibrium and charge neutrality condition of neutron star matter consisting of neutrons, protons and electrons (the core-crust transition occurs around $\\rho_0\/3\\mbox{$\\sim$}\\rho_0\/2$ where muons generally do not emerge). It is uniquely determined by the density dependence of nuclear symmetry energy\\,\\cite{Lat04}.\nOnce the above stability condition is broken, the speed of sound in the uniform outer core becomes imaginary and small density oscillations will then grow exponentially, indicating the onset of a transition from the uniform liquid core to the crust.  \nIn (\\ref{Vther}), $E(\\rho ,x_{\\rm{p}})$ is the energy per nucleon in\nthe $\\beta $-equilibrated neutron star matter.  The corresponding pressure is obtained as $P=P_{\\rm{N}}+P_{\\rm{e}}$ with\n$P_{\\rm{N}}$ and $P_{\\rm{e}}$ denoting contributions from nucleons and electrons, respectively.\nThe electron pressure $P_{\\rm{e}}$ could be obtained through the FFG model\\,\\cite{XuJ09}, i.e., \n\\begin{equation}\\label{dd_e}\n\\varepsilon_{\\rm{e}}(\\rho,\\delta)=\\eta_{\\rm{e}}\\Phi_{\\rm{e}}(t_{\\rm{e}}),\n\\end{equation}\nhere \\begin{equation}\n\\eta_{\\rm{e}}={m_{\\rm{e}}}\/{8\\pi^2\\lambda_{\\rm{e}}^3},~~\n\\lambda_{\\rm{e}}={1}\/{m_{\\rm{e}}},~~t_{\\rm{e}}=\\lambda_{\\rm{e}}\\left(3\\pi^2\\rho_{\\rm{e}}\\right)^{1\/3}\n,\\end{equation} and,\n\\begin{equation}\\label{dd_e1}\n\\Phi_{\\rm{e}}(t_{\\rm{e}})=t_{\\rm{e}}\\left(1+2t_{\\rm{e}}^2\\right)\\sqrt{1+t_{\\rm{e}}^2}-\\ln\\left(t_{\\rm{e}}+\\sqrt{1+t_{\\rm{e}}^2}\\right)\n,\\end{equation} \nwhere $m_{\\rm{e}}$ and $\\rho_{\\rm{e}}$ are the electron mass and density respectively.\nConsequently, $P_{\\rm{e}}(\\rho,\\delta)=\\rho_{\\rm{e}}\\mu_{\\rm{e}}-\\varepsilon_{\\rm{e}}(\\rho,\\delta)\n$, with $\n\\mu_{\\rm{e}}=[{k_{\\rm{e}}^2+m_{\\rm{e}}^2}]^{1\/2}\\approx k_{\\rm{e}}$ and $k_{\\rm{e}}=(3\\pi^2\\rho_{\\rm{e}})^{1\/3}$.\n\nShown in FIG.\\,\\ref{fig_ab_rhot} are our results for the core-crust transition density $\\rho_{\\rm{t}}$ (upper panel) and pressure $P_{\\rm{t}}$ (lower panel) as functions of the slope parameter $L(\\rho_{\\rm{r}})$ at a sub-saturation density $\\rho_{\\rm{r}}=0.11\\,\\rm{fm}^{-3}$.}\nIt is clear that the SRC\/HMT affects significantly the\ntransition density $\\rho_{\\rm{t}}$, e.g., a reduction as high as about\n58\\% with the HMT-exp parameter set compared to the FFG model. More quantitatively, the crust-core transition density in the FFG model is found to be about\n$\\rho_{\\rm{t}}\\approx0.086\\,\\rm{fm}^{-3}$, while that in the HMT\nmodels is about $\\rho_{\\rm{t}}\\approx0.079\\,\\rm{fm}^{-3}$\n(HMT-SCGF) and $\\rho_{\\rm{t}}\\approx0.036\\,\\rm{fm}^{-3}$ (HMT-exp), respectively.\nCorrespondingly, the transition pressure in the FFG model and the\nHMT-SCGF (HMT-exp) model is about\n$0.28\\,\\rm{MeV}\/\\rm{fm}^{3}$ and $0.26\\,\\rm{MeV}\/\\rm{fm}^{3}$\n($0.17\\,\\rm{MeV}\/\\rm{fm}^{3}$), respectively. \n\nAs discussed in the\nabove sections, the symmetry energy $E_{\\rm{sym}}(\\rho)$ at sub-saturation densities is reduced by the SRC\/HMT while its magnitude and slope at $\\rho_0$ are fixed at their currently known empirical values. Thus, the $E_{\\rm{sym}}(\\rho)$ is effectively hardened within the sub-saturation density region (see the inset of FIG.\\,\\ref{fig_ab_Esym}), leading to an enhancement of the slope\nparameter $L(\\rho)$ for $\\rho\\lesssim\\rho_0$. It is known that a larger $L(\\rho_{\\rm{r}})$ with $\\rho_{\\rm{r}}\\lesssim\\rho_0$\\,\\cite{Zha13} leads to a\nsmaller core-crust transition density as studied carefully in, e.g.,\nRefs.\\,\\cite{XuJ09}. The observed SRC\/HMT effects on the crust-core transition density and pressure are qualitatively consistent with expectations based on previous studies. \n\\begin{figure}[h!]\n\\centering\n \n \\includegraphics[height=2.7cm]{rhot.eps}\\\\\n \\hspace{0.cm}\n \\includegraphics[height=2.7cm]{Pt.eps}\n  \\caption{Correlation between the core-crust transition density (upper) and the transition pressure (lower) in $\\beta$-stable neutron star matter\n  and the slope parameter $L(\\rho_{\\rm{r}})$ at $\\rho_{\\rm{r}}=0.11\\,\\rm{fm}^{-3}$.}\n  \\label{fig_ab_rhot}\n\\end{figure}\n\nAs we discussed earlier, the fundamentally important quantity that is growing from the FFG to HMT-SCGF then HMT-exp is the strength of the underlying isospin dependence of SRC\/HMT, namely, \nthe difference $\\delta x=x_{\\rm{HMT}}^{\\rm{SNM}}-x_{\\rm{HMT}}^{\\rm{PNM}}$ is increasing. Our results shown in FIG.\\,\\ref{fig_ab_rhot} indicates an anti-correlation between the $\\delta x$  and the transition density $\\rho_{\\rm{t}}$.\nIt provides an important connection to the study of the thickness of inner crusts\\,\\cite{XuJ09} as well as the related observational phenomena\\,\\cite{Gear11,Wen12,Newton12,Newton14,Hooker15,Zhou21}, such as the crustal oscillations and glitches as well as the $r$-mode stability window of rapidly rotating neutron stars. Our findings here may also have important implications on the heat transport process in neutron stars\\,\\cite{Cac08}.\n\nWithin the minimal model of neutron stars assuming no hadron-quark phase transition as well as baryon resonance and hyperon production, the compositions of neutron stars are \ndetermined by the requirement of charge neutrality and $\\beta$-equilibrium conditions. From the nucleon specific energy $E(\\rho,\\delta)$ we can calculate numerically the proton fraction $x_{\\rm{p}}(\\rho)$ for $\\beta$-stable matter and examine its dependence on the strength of SRC\/HMT. More specifically, for neutrino free $ \\beta $-stable matter, the chemical\nequilibrium for the reactions $\n\\rm{n}\\rightarrow\\rm{p}+\\rm{e}^{-}+\\overline{\\nu}_{\\rm{e}}$ and\n$\\rm{p}+\\rm{e}^{-}\\rightarrow \\rm{n}+\\nu _{\\rm{e}}$ requires $ \\mu\n_{\\rm{e}}=\\mu_{\\rm{n}}-\\mu_{\\rm{p}}\\approx\n4E_{\\rm{sym}}(\\rho)\\delta+8E_{\\rm{sym,4}}(\\rho)\\delta^3+12E_{\\rm{sym},6}(\\rho)\\delta^5+\\cdots$, here $E_{\\rm{sym},6}(\\rho)\\equiv 120^{-1}\\partial^6E(\\rho,\\delta)\/\\partial\\delta^6|_{\\delta=0}$ is the sixth-order symmetry energy.\nFor relativistically degenerate electrons, we have $ \\mu _{\\rm{e}}=[\nm_{\\rm{e}}^{2}+(3\\pi ^{2}\\rho x_{\\rm{e}})^{2\/3}] ^{1\/2}\\approx\n( 3\\pi ^{2}\\rho x_{\\rm{e}}) ^{1\/3} $, where\n$m_{\\rm{e}}\\approx0.511\\,$MeV is the electron mass, and\n$x_{\\rm{p}}=x_{\\rm{e}}$ because of charge neutrality. Above a certain density where $\\mu _{\\rm{e}}$ exceeds the muon\nmass $m_{\\mu }\\approx105.7\\,$MeV, the reactions\n$\\rm{e}^{-}\\rightarrow \\mu ^{-}+\\nu\n_{\\rm{e}}+\\overline{\\nu}_{\\mu }$, $\\rm{p}+\\mu ^{-}\\rightarrow \\rm{n}+\\nu _{\\mu }$ and $%\n\\rm{n}\\rightarrow\\rm{p}+\\mu ^{-}+\\overline{\\nu}_{\\mu }$ are\nenergetically allowed. Then, both electrons and muons are present\nin $\\beta $-stable matter. This alters the $\\beta$-stability condition\nto $\\mu_{\\rm{e}}=\\mu _{\\mu }=[ m_{\\mu }^{2}+(3\\pi ^{2}\\rho x_{\\mu\n})^{2\/3}] ^{1\/2}$ with $x_{\\rm{p}}=x_{\\rm{e}}+x_{\\mu }$. \nGenerally, when the nucleon momentum distribution function $n_{\\v{k}}^J(\\rho,\\delta)$ has a low- (high-) momentum depletion (tail), the nucleon chemical potential is not given by \n$k_{\\rm{F}}^{\\rm{n\/p},2}\/2M+U_{\\rm{n\/p}}(\\rho,\\delta,k_{\\rm{F}}^{\\rm{n\/p}})$, i.e., it could not be obtained by simply letting the momentum $|\\v{k}|$ to be the Fermi momentum in the single-nucleon potential. \nIn fact, determining the relation between the nucleon chemical potential and the single-nucleon potential for a general $n_{\\v{k}}^J(\\rho,\\delta)$ is known as a fundamental problem in nuclear many-body theories\\,\\cite{AGD1960}.\nWe thus have to calculate numerically the chemical potential $\\mu_J$ for a nucleon $J$ from the energy density $\\varepsilon$ of neutron star matter via\n$\n  \\mu_J=\\partial\\varepsilon(\\rho,\\delta)\/\\partial\\rho_J.\n$\n\n\\begin{figure}[h!]\n\\centering\n \n \\includegraphics[width=4.cm]{abMDI-xp.eps}\\quad\n \\includegraphics[width=4.05cm]{abMDI-sound.eps}\n  \\caption{The proton fraction $x_{\\rm{p}}$ (left) and the square of the speed of sound $s^2$ (right) in neutron star matter within the FFG and HMT-exp model, respectively.\n  The black circle and the cyan star correspond to the central densities of the neutron star in the two models.}\n  \\label{fig_abMDI-xp}\n\\end{figure}\n\nShown in the left panel of FIG.\\,\\ref{fig_abMDI-xp} is the proton fraction $x_{\\rm{p}}$ in neutron star matter as a function of density within the FFG and HMT-exp model, respectively.\nSince the HMT-SCGF result is similar to the one using the FFG model, we thus in the following focus on the comparison between the predictions using the FFG and the HMT-exp models.\nObviously, the most important observation is that due to the reduction of the high-density symmetry energy in the HMT-exp model the proton fraction $x_{\\rm{p}}$ in this model starts to decrease quickly above some critical density. \nIt thus prompts the role played by the symmetry energy in the core of neutron stars as $\\delta$ in the HMT-exp model is much closer to 1 than that in the FFG model.\nTherefore, it may have important ramifications on properties of neutron stars. For example, it is well known that the  $x_{\\rm{p}}$ determines the cooling mechanisms of protoneutron stars and the associated neutrino emissions\\,\\citep{LPPH}. \nMore quantitatively, the critical proton fraction $x^{\\rm{DU}}_{\\rm{p}}$ enabling the direct URCA process (DU) for fast cooling is given by\n$x^{\\rm{DU}}_{\\rm{p}}=1\/[1+(1+x_{\\rm{e}}^{1\/3})^3]$\nwith the $x_{\\rm{e}}\\equiv \\rho_{\\rm{e}}\/\\rho_{\\rm{p}}$ between 1 and 0.5 leading to $x^{\\rm{DU}}_{\\rm{p}}$ between 11.1\\% to 14.8\\%\\,\\citep{Kla06}, see the red dashed lines in the left panel of FIG.\\,\\ref{fig_abMDI-xp}.\nThe black circle and the cyan star in the left panel of FIG.\\,\\ref{fig_abMDI-xp} correspond to the central densities of neutron stars in the two models studied here. It is seen that around these central densities the direct URCA is allowed in the FFG model but forbidden in the HMT-exp model. \n\nThe speed of sound squared $s^2=\\d P\/\\d\\varepsilon$ is a measure of the stiffness of nuclear EOS. Shown in the right panel of FIG.\\,\\ref{fig_abMDI-xp} are the predicted $s^2$ by the two models considered. It is seen that the EOS in the HMT-exp model is much softer than the FFG model prediction at densities above about $2.5\\rho_0$. Since the ANM EOS $E(\\rho,\\delta)\\approx E_0(\\rho)+E_{\\rm{sym}}(\\rho)\\delta^2+\\cdots$, the reduced symmetry energy but increased $\\delta^2$ at high densities together may effectively soften the EOS. However, its net effects depend on how strongly the SNM EOS $E_0(\\rho)$ is stiffened by the SRC\/HMT. The results shown here indicate clearly that the SRC\/HMT effect on the symmetry energy is winning against that on the SNM EOS. \nThis result is expected to have some dynamical effects on properties of neutron stars. Since the symmetry energy and SNM EOS in different density regions have different effects on properties of neutron stars, the SRC\/HMT effects through these two terms of the ANM EOS have to be studied quantitatively as we shall do next. \n\nThe mass-radius correlation of neutron stars is obtained from integrating the Tolman-Oppenheimer-Volkoff (TOV) equations\\,\\cite{Misner1973}\n\\begin{align}\n\\frac{\\d P(r)}{\\d r}=&-\\frac{[\\varepsilon(r)+P(r)][M(r)+4\\pi r^3P(r)]}{r[r-2M(r)]},\\\\\n\\frac{\\d M(r)}{\\d r}=&4\\pi r^2\\varepsilon(r),\n\\end{align}\nwhere $r$ is the radial distance from the center of the star, and\n$M(r)$ is the mass enclosed within $r$ (adopting natural unit in which $c=G=1$).\nThe pressure of $\\beta$-stable and charge neutral npe$\\mu$ matter is given by $\nP(\\rho,\\delta)=P_{\\rm{N}}(\\rho,\\delta)\n+P_{\\rm{e}}(\\rho,\\delta)+P_{\\mu}(\\rho,\\delta)$,\nwhere $P_{\\rm{N}}(\\rho,\\delta) $ is the nucleon pressure. \nThe lepton pressure is further given as $\nP_\\ell(\\rho,\\delta)=\\rho_\\ell\\mu_\\ell-\\varepsilon_\\ell(\\rho,\\delta)\n$, with $\n\\mu_\\ell=[{k_\\ell^2+m_\\ell^2}]^{1\/2}$ and $k_\\ell=(3\\pi^2\\rho_\\ell)^{1\/3}$ for $\\ell=\\rm{e},\\mu$, here $\\varepsilon_{\\ell}(\\rho,\\delta)=\\eta_{\\ell}\\Phi_{\\ell}(t_{\\ell})$ (see (\\ref{dd_e1}) for the definition of $\\Phi_{\\ell}$).\nFor the whole system, we have the self-consistency relation between the total pressure $P$ and the total energy density $\\varepsilon$, i.e., \n$\nP=\\rho^2{\\d(\\varepsilon\/\\rho)}\/{\\d\\rho}$.\nThe inner crust of neutron stars with densities ranging between $\\rho_{\\text{out}}=2.46\\times\n10^{-4}\\,\\rm{fm}^{-3}$ corresponding to the neutron dripline and the\ncore-crust transition density $\\rho _{\\text{t}}$ is the region where\nsome complex and exotic structures\\,---\\,the ``nuclear pasta'' may exist.  Because of our very poor knowledge\nabout this region we adopt the polytropic EOSs expressed in\nterms of the pressure $P$ as a function of the total energy density\n$\\varepsilon$ as\n$P=c+d\\varepsilon^{4\/3}$\\,\\cite{XuJ09,Hor03}. The constants $c$ and\n$d$ are determined by the quantities at $\\rho\n_{\\text{t}}$ and $\\rho _{\\text{out}}$\\,\\cite{XuJ09}, i.e.,\n\\begin{equation}\nc=\\frac{P_{\\rm{out}}\\varepsilon_{\\rm{t}}^{4\/3}-P_{\\rm{t}}\\varepsilon_{\\rm{out}}^{4\/3}}{\\varepsilon_{\\rm{t}}^{4\/3}-\\varepsilon_{\\rm{out}}^{4\/3}},\n~~\nd=\\frac{P_{\\rm{t}}-P_{\\rm{out}}}{\\varepsilon_{\\rm{t}}^{4\/3}-\\varepsilon_{\\rm{out}}^{4\/3}}\n.\\end{equation}\nFor the outer\ncrust\\,\\cite{BPS71,Iida1997}, we use the conventional Baym-Pethick-Sutherland (BPS) EOS for the region with $6.93\\times 10^{-13}\\,\\rm{fm}^{-3}\\lesssim\\rho \\lesssim\\rho _{\\text{out}}$ and the\nFeynman-Metropolis-Teller (FMT) EOS for $4.73\\times 10^{-15}\\,\\rm{fm}^{-3}\\lesssim\\rho \\lesssim6.93\\times\n10^{-13}\\,\\rm{fm}^{-3}$, respectively.\nThe total neutron star matter EOSs in the two models are shown in FIG.\\,\\ref{fig_abMDI-beta}.\n\n\\begin{figure}[h!]\n\\centering\n \n \\includegraphics[width=6.5cm]{abMDI-beta.eps}\n  \\caption{The EOS of neutron star matter in the FFG model and the HMT-exp model. See the text for details.}\n  \\label{fig_abMDI-beta}\n\\end{figure}\n\nThe resulting mass-radius correlation of neutron stars is shown in FIG.\\,\\ref{fig_abMDI-MR} for the FFG and the HMT-exp model, respectively. Firstly, we notice that both models give a radius of about 12\\,km for canonical neutron stars of mass 1.4$M_{\\odot}$ consistent with many analyses of the GW170817 observation, see, e.g., Refs.\\,\\cite{EPJA-review,Baiotti,Capano20,David,Kat20,AngLi} for reviews. \nThis is mostly by design as our EDF parameters are fixed such that all the EOS characteristics at $\\rho_0$ are consistent with existing constrains from both astrophysical observations and terrestrial experiments. \nSecondly, it is interesting to see that the two models indeed predict significantly different maximum masses and the corresponding radii. More quantitatively, the maximum mass of neutron stars in the FFG model is about $M_{\\rm{NS}}^{\\max}\\approx1.98M_{\\odot}$, with the corresponding radius being about 9.9\\,km. While the $M_{\\rm{NS}}^{\\max}$ in the HMT-exp model is about $1.72M_{\\odot}$ with a radius of about 10.3\\,km. \nThe reduction of the maximum mass $M_{\\rm{NS}}^{\\max}$ due to the SRC\/HMT is thus about 13\\%. \nThe point corresponding to the maximum mass on the $P(\\varepsilon)$ plane is also shown in FIG.\\,\\ref{fig_abMDI-beta}, by the black circle (cyan star) for the FFG (HMT-exp) model.\nThirdly, since the HMT largely affects the transition density $\\rho_{\\rm{t}}$ (see FIG.\\,\\ref{fig_ab_rhot}), the radii of low-mass neutron stars in the FFG model and the HMT-exp model show obvious differences.\n\nThe central density $\\rho_{\\rm{cen}}$ corresponding to $M_{\\rm{NS}}^{\\max}$ is about $\\rho_{\\rm{cen}}^{\\rm{FFG}}\\approx7.3\\rho_0$ ($\\rho_{\\rm{cen}}^{\\rm{HMT}\\mbox{-}\\rm{exp}}\\approx7.9\\rho_0$) for the FFG (HMT-exp) model (see the points shown in FIG.\\,\\ref{fig_abMDI-xp}).\nAlthough the EDF constructed is non-relativistic in nature, it could be effectively applied to neutron stars and fulfill the principle of causality (set by the speed of sound $s^2=\\d P\/\\d\\varepsilon=1$).\nIn particular, we find that the square of the speed of sound in the FFG model at $\\rho_{\\rm{cen}}^{\\rm{FFG}}$ is about $s^2_{\\rm{FFG}}\\approx0.84$, while that for the HMT-exp model is about $s_{\\rm{HMT}\\mbox{-}\\rm{exp}}^2\\approx0.35$, see the right panel of FIG.\\,\\ref{fig_abMDI-xp} for the density dependence of the $s^2$ in the two model. Thus, due to the softening of the symmetry energy above a critical density and the non-relativistic nature of the Gogny-like EDF, the high-density $s^2$ in the HMT-exp model is \nmuch smaller than that in the FFG model. In fact, it may not exceed 1 at even larger densities when the FFG becomes acausal (the density violating the principle of causality in the FFG model is about $9.8\\rho_0$).\n\n\\begin{figure}[h!]\n\\centering\n \n \\includegraphics[width=6.5cm]{abMDI-MR.eps}\n  \\caption{The mass-radius relation for neutron stars in the FFG and the HMT-exp models.}\n  \\label{fig_abMDI-MR}\n\\end{figure}\n\nTo this end, it is interesting to note that some earlier studies on the mass-radius relation of neutron stars adopting the nonlinear RMF models found that the SRC\/HMT generally increases the maximum mass $M_{\\rm{NS}}^{\\max}$\\,\\cite{Cai16b,Lou22a,Lu2022,Hong22,Lou22,Lu21,Souza20}. In these models the SRC\/HMT induced increase in SNM pressure dominates over the reduction of symmetry energy. \nOne main reason is that in the nonlinear RMF models the symmetry energy itself plays a minor role in determining the maximum mass $M_{\\rm{NS}}^{\\max}$ as pointed out already a long time ago\\,\\cite{Ser79}. Moreover, the softening of the $E_{\\rm{sym}}(\\rho)$ in the RMF models is weaker than that in the current (non-relativistic) Gogny-like EDF. Of course, this then raises an interesting question: how can this kind of EDFs encapsulating the SRC\/HMT effects support the currently observed most massive neutron star PSR J0740+6620 which has a mass of $2.08^{+0.07}_{-0.07}M_\\odot$\\,\\cite{Fon21}? In fact, even without considering the SRC\/HMT effects, it has been a challenging problem to find mechanisms to support heavy neutron stars (with masses around and above $2M_{\\odot}$) within non-relativistic energy density functionals, see, e.g., Refs.\\,\\cite{Rios-G,Zha16,ZhouY2019,Gon18,Gon19,Lou20}. Our results presented above indicate that this problem may become more challenging and certainly deserves further investigations.\n\n\\setcounter{equation}{0}\n\\section{Summary and Outlook}\\label{sec6}\n\nIn summary, we investigated SRC\/HMT effects on the EOS and single-nucleon potential in cold neutron-rich matter within the Gogny-like EDF. The SRC effects are incorporated into the EDF by using a single-nucleon\nmomentum distribution function $n_{\\v{k}}^J(\\rho,\\delta)$ that has an isospin-dependent high-momentum (low-momentum) tail (depletion) with its parameters determined by the available SRC experiments. We introduced a parametrization as a surrogate for the momentum-dependent kernel of the Gogny-like EDF to facilitate derivations of analytical expressions for all relevant physical quantities describing the EOS and single-nucleon potential in ANM. The surrogate is shown to be effective for problems with momentum (density) scale smaller than about 1 GeV\/c (several times the saturation density). The resulting expressions for all terms in the ANM EOS and single-nucleon potential will facilitate further explorations of nuclear interactions in dense neutron-rich matter as well as their impacts on properties of neutron stars and heavy-ion reactions induced by high-energy radioactive beams. \nIn particular, \n\\begin{enumerate}\n\\item[(a)]We have derived the analytical expressions for the EOS of SNM $E_0(\\rho)$,  the corresponding pressure $P_0(\\rho)$,  the incompressibility coefficient $K_0(\\rho)$, the nucleon effective k-mass $M_0^{\\ast}(\\rho)$, the isoscalar potential $U_0(\\rho,|\\v{k}|)$, the symmetry energy $E_{\\rm{sym}}(\\rho)$ together with its slope parameter $L(\\rho)$, and the isovector (symmetry) potential $U_{\\rm{sym}}(\\rho,|\\v{k}|)$.\nNumerical constructions based on these analytical expressions are given through three models for the nucleon momentum distribution, namely the FFG model without the SRC-induced HMT in the $n_{\\v{k}}^J(\\rho,\\delta)$, the\nHMT-SCGF\/HMT-exp model with different properties of the HMT, to fulfill certain empirical constraints on the EOS of ANM.  All three models contain proper potential parts.\n\\item[(b)] While the same set of available empirical constraints on the ANM EOS at $\\rho_0$ are all maintained in the three models, the EOS $E_0(\\rho)$ of SNM at high densities is made slightly harder once the SRC\neffects are considered.  In addition, the symmetry energy in the presence of HMT becomes softer at large densities even to start decreasing above a critical density. Although qualitatively consistent with the nonlinear RMF model predictions, the reduction of the high-density symmetry energy $E_{\\rm{sym}}(\\rho)$ is much stronger while the enhancement of the SNM pressure $P_0$ is much weaker in the Gogny-like EDF with SRC\/HMT. Consequently, as a result of the competition between these two effects, the maximum mass of a cold neutron star is found to be reduced compared with the prediction of the Gogny-like EDF without considering the SRC\/HMT.\n\\item[(c)] The single-nucleon potential $U_0(\\rho,|\\v{k}|)$ in SNM at $\\rho_0$ in all three models is consistent with the isoscalar nucleon optical potential from earlier analyses of various experimental data. The nucleon isovector (symmetry) potential $U_{\\rm{sym}}(\\rho,|\\v{k}|)$ in ANM is largely enhanced due to the corresponding enhancement of the potential symmetry energy $E_{\\rm{sym}}^{\\rm{pot}}(\\rho)$.\nThese are the direct consequences of the momentum-dependence of the nucleon potential in the Gogny-like EDF due to the finite ranges of nuclear interactions.  Such momentum-dependence is absent in the nonlinear RMF approach.\n\\end{enumerate}\n\nThe formulas, table of coupling constants, parameterizations and quantitative demonstrations of the relevant physical quantities given in this work provide a useful and convenient starting point to further investigate the nature of strong interactions at short distances and novel effects of SRC\/HMT on properties of dense neutron-rich matter. In particular, the generally density- and momentum-dependent single-nucleon potential in neutron-rich matter is expected to play a significantly role in understanding the dynamics and observables of heavy-ion reactions induced by high-energy radioactive beams, the cooling mechanisms of protoneutron stars as well as properties of cold neutron stars at $\\beta$-equilibrium and their mergers. For instance, with the coupling parameters given in Table   \\ref{tab_para}, one can use the single-nucleon potential of Eq. (\\ref{Gen-U}) in simulating heavy-ion reaction with the $f_J(\\v{r},\\v{k})$ self-consistently generated by the BUU transport model. After comparing the predicted observables with experimental data, one can then make better informed predictions for the EOS of hot, dense and neutron-rich matter. The latter is a necessary input for simulating mergers of two neutron stars based on principles of general relativity \\cite{Ra1,Ra2}. In particular, it is important for understanding the high-frequency spectrum of gravitational waves from post-mergers that carry critical information about the nature of possible phase transitions and the separation boundary\/gap between neutron stars and blackholes. Interestingly, it has already been found that the density dependence of nuclear symmetry energy play a significant role in answering many intriguing questions in simulating  neutron star mergers \\cite{Most21}. \n\nAs discussed earlier, our approach has some caveats and some challenging issues remain to be resolved. Fortunately, multimessenger nuclear astrophysics with high-precision X-ray and gravitational wave detectors as well as advanced rare isotope beam facilities are expected to provide us with various kinds of new and more accurate observational data. Combined analyses of these data using the same Gogny-like EDF with SRC\/HMT may help us address some of the remaining issues. For example, the strong enhancement of the symmetry potential $U_{\\rm{sym}}(\\rho,|\\v{k}|)$ due to the SRC\/HMT especially at supra-saturation densities, e.g., $\\rho\\approx2\\rho_0\\mbox{$\\sim$}3\\rho_0$,  is expected to affect significantly the dynamics and some observables of nuclear reactions with high-energy radioactive beams. While the corresponding reduction of the total symmetry energy reduces significantly the maximum mass of neutron stars that the resulting EOS can support. A joint analysis of these effects using a unified Gogny-like EDF with SRC\/HMT is expected to be fruitful. Besides the traditionally used forward predictions with the different model parameter sets, the expected new data and existing ones together will enable us to get new insights from inferences of relevant model parameters using machine leaning techniques, such as the Bayesian model selection \\cite{MAL}. These studies are on the top of our working agenda. \n\n\\section*{Acknowledgement} We would like to thank Lie-Wen Chen, Xiao-Tao He, Che Ming Ko and Xiao-Hua Li for useful discussions over the years on some of the issues studied in this work.\nThis work is supported in part by the U.S. Department of Energy, Office of Science,\nunder Award No. DE-SC0013702, the CUSTIPEN (China-\nU.S. Theory Institute for Physics with Exotic Nuclei) under\nUS Department of Energy Grant No. DE-SC0009971.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nInflation theory is proposed to solve fundamental problems of standard cosmology\\cite{Guth:1980zm, Linde1, Albrecht}; it also explains the origin of the primordial fluctuations. Although observational data support the inflation theory in general, there is no fundamental theory that can describe the nature of this theory. In the simplest model, the inflaton field, which is responsible for inflating the universe, rolls down in an almost flat potential (slow-roll regime).\nObservational data indicate that in single field models, the monomial potentials are disfavored, including the famous potential $m^2\\phi^2$. Other models with more intricate potentials, especially with exponential tails, provide good fits to data. Brane inflation is another model that is consistent with the Planck data\\cite{Planck 2018a, Planck2018b}.\nOne example of brane inflation is $D3-\\bar{D}3$ which is a well-motivated scenario\\cite{brane1,brane2,brane3,brane4}. In this scenario, due to an attractive force, the $\\bar{D}3$-brane is sitting down at the bottom of a warped throat, while the $D3$ -brane is relatively mobile. When the $D3$-brane and the $\\bar{D}3$-brane collide and annihilate, inflation ends. When the branes start close to or inside the throat, we can approximate the potential with a simple expression\\cite{shandera06}.\n\nIn general, inflation can be derived from non-canonical fields. This kind of model is investigated in k-inflation\\cite{mukhanov1,mukhanov2} and general multi-field inflation context\\cite{langlois2008,koyoma2008}. Dirac-Born-Infeld(DBI) model of inflation\\cite{Alishahiha}, which is first considered in the context of brane-inflation, is one of these models. Under some constraints, the DBI model is consistent with observational data \\cite{Planck2019,gomes2018}.\n\nOn the other hand, in recent years, $F(R)$ theories have attracted attention. These theories, which are an extension of Hilbert-Einstein's action, are another approach to explain the acceleration periods of our universe. Maybe the simplest and most famous model of $F(R)$ theory is $R+cR^2$. Being within the Planck $68\\%$ confidence level constraints arouse enthusiasm for this model\\cite{Planck2018b}.\nThis model is proposed many years ago by Starobinsky \\cite{Staro1, Stao2} as a model for inflation. It is usual to write the action in the Jordan frame. If one transforms to the Einstein frame, the action is equivalent to a scalar field plus Hilbert-Einstein action \\cite{witt}. This dual scalar field, which is called scalaron, can take the role of inflaton with exponential potential.\n\nInspired by string theory and high-energy physics, there is motivation to have more than one field. A multi-field model has more phenomenology than a single field model. Many works consider extra fields in $F(R)$ theory\\cite{Bamba:2015uxa, Myrzakulov2015,Canko_2020,Gomes_2017,Mori17, Elizalde_2019, Antoniadis19}. The simplest extension of the Starobinsky model is considering extra-canonical scalar fields in $R+cR^2$ gravity. It is shown that these models, with minimal and non-minimal couplings, are robust models\\cite{Bruck2015, Starobinsky-like two-field inflation}.\nIt is also possible to add fields with more phenomenology such as the Higgs field or fields inspired by super-gravity and other fundamental theories\\cite{Ketov2020-2,Higgs_dependent_scalaron}. Recently Starobinsky inflation is explored within the context of supergravity \\cite{Ketov2020-3,Ketov2020-1,Aldabergenovetal2020}. In\\cite{dbisupergravity,Ketov2018-1,Ketov2018-3,Ketov-supergravity}\n DBI field and D-brane models are investigated in the extension of supergravity to Starobinsky inflation.\n\nTherefore, the natural question that arises is whether adding a new field with a nontrivial kinetic term to the $R+cR^2$ model would make a robust model.\nThis work aims to investigate the existence of a DBI action in the context of $R+cR^2$ gravity. From another point of view, we would like to study the brane inflation in the context of Starobinsky gravity. Apart from the theoretical origin, the square root feature of DBI action makes several novelties. In principle, every $F(R)$ theory can be reformulated  as a scalar-tensor theory. Therefore the DBI field in the $R+cR^2$ gravity is equivalent to the DBI field, supplemented by another scalar field. In our case, the  scalar field is scalaron. When the scalaron traps at its minimum, the DBI field governs inflation. Especially when we have heavy scalaron, only the DBI field impact on cosmological perturbation, so we only consider the DBI field perturbations in the observational parameters such as spectral index.\nAs mentioned before, we transform into the Einstein frame and redefine the fields. This redefinition causes the DBI field to couple with the dual field. This coupling modifies the dynamics of the DBI field and hence affects the cosmological parameters.\n\nThis paper is organized as follows: in section (\\ref{Setup}), the setup of the model is described. In section (\\ref{DBI}), the background solution is considered. The field perturbations are investigated in section (\\ref{perturbation}). We also do some numerical analysis in (\\ref{numerical analysis}). We summarized our results in section (\\ref{conclusion}).\n\\section{The Setup }\\label{Setup}\nIn principle, a generic $F(R)$ model is given by the below action;\n\\ba\nS=\\frac{1}{2\\kappa^2}\\int d^4x\\sqrt{-g}f\\left(R\\right)\n\\ea\nThis model is connected to the scalar-tensor theory via Legendre transformation as,\n \\ba\n S=\\frac{1}{2\\kappa^2}\\int d^4x\\sqrt{-g}\\left(f\\left(\\phi\\right)+f'\\left(\\phi\\right)\\left(R-\\phi\\right)\\right).\n \\ea\nWhere we defined $\\Omega^2\\equiv f'\\left(\\phi\\right)$ and $\\phi$  is a real scalar field. With this definition, we rewrite the above action as,\n\\ba\nS=\\int d^4x\\sqrt{-g}\\left(\\frac{1}{2\\kappa^2}\\Omega^2R-V\\left(\\phi\\right)\\right),\n\\ea\nwith $V\\left(\\phi\\right)\\equiv\\frac{1}{2}\\left(\\phi f'\\left(\\phi\\right)-f\\left(\\phi\\right)\\right)$.\nThe stability of classical and quantum gravity requires having $f'\\left(R\\right)>0$ and $f''\\left(R\\right)>0$ where $'$ denotes derivative with respect to $R$.\n\n It is also possible to add a matter sector. We are interested in matter with the non-canonic kinetic term; the general action can be written as below,\n\\ba\nS=\\int d^4x\\sqrt{-g}\\left(\\frac{1}{2\\kappa^2}\\Omega^2R-V\\left(\\phi\\right)\\right)+\\int d^4x\\sqrt{-g}P\\left(\\mathbb{X}_\\chi,\\chi\\right)\n\\ea\nwhere $\\chi$ is the non-canonic field and $\\mathbb{X}_\\chi=-\\frac{1}{2}g_{\\mu\\nu}\\partial_\\mu\\chi\\partial_\\nu\\chi$ is its kinetic term.  $P$ denotes the Lagrangian density of the matter field; it is a function of both $\\mathbb{X}$ and  $\\chi$.\n\nIt is feasible  to go to  Einstein-frame under a conformal transformation $\\tilde{g}_{\\mu\\nu}=\\Omega^2g_{\\mu\\nu}$, we define a new field as $\\Omega^2=f'\\left(\\phi\\right)=e^{2\\alpha\\psi}$.\nFirst, we consider the gravitational part of the action,\n\\ba\\nonumber\nS'_G&=&\\int d^4x\\sqrt{-\\tilde{g}}\\left(\\frac{\\tilde{R}}{2\\kappa^2}-\\frac{1}{2}\\tilde{g}^{\\mu\\nu}\\tilde{\\partial}_\\mu\\psi\\tilde{\\partial}_\\nu\\psi-\n\\tilde{V}\\left(\\psi\\right)\\right),\n\\ea\nwhere $\\alpha=\\frac{\\kappa}{\\sqrt{6}}$ with  $\\kappa^2=8\\pi G=M^{-2}_{pl}$. $M_{pl}$ is the reduced Planck mass. Under transformation to the Einstein frame, the matter part transforms as below,\n\\ba\nS'_M=\\int d^4x \\sqrt{-\\tilde{g}}e^{-4\\alpha\\psi}P\\left(\\mathbb{\\tilde{X}}_\\chi,\\chi\\right)\n\\ea\nwhere $\\mathbb{\\tilde{X}}_\\chi=-\\frac{1}{2}\\tilde{g}_{\\mu\\nu}\\tilde{\\partial}_\\mu\\chi\\tilde{\\partial}_\\nu\\chi$; $\\tilde{\\partial}$ indicates the derivative with respect to $\\tilde{g}_{\\mu\\nu}$.\nIn the Einstein frame, there are two fields; the first one is scalaron, denoted by $\\psi$. It comes from the correction of Einstein's gravity. The second one is  $\\chi$, the matter field. Both of these fields influence inflation. The conformal transformation causes  $\\chi$ to be coupled with $\\psi$. In fact, F(R) theories are equivalent to scalar-tensor models\\cite{Ketov2020-2,Maeda_book}; this equivalence permits us to apply the same  formalism to F(R) models.\nIn the next section, we focus on $R+cR^2$  and a special kind of non-canonic field i.e. Dirac-Born-Infeld(DBI) field.\n\n\\section{DBI field Dynamics in Starobinsky Model}\\label{DBI}\nIn this section, we consider Starobinsky action. As mentioned before, the Starobinsky model is a robust model\\cite{Robustness-Starobinsky}.\n We choose a DBI field as the non-canonic field,\n\\ba\\label{action1}\nS&=&\\frac{1}{2\\kappa^2}\\int d^4x\\sqrt{-g}(R+\\mu R^2)\\\\\\nonumber\n&+&\\int d^4x\\sqrt{-g}[\\frac{1}{f(\\chi)}\\left(1-\\sqrt{1+f(\\chi)g^{\\mu\\nu}\\partial_\\mu\\chi\\partial_\\nu\\chi}\\right)-U\\left(\\chi\\right)],\n\\ea\nwhere $\\kappa^2=8\\pi G=M^{-2}_{pl}$.\nIn the following, we will work in natural units in which $\\kappa^2=1$. The coupling parameter $\\mu$, with units $[mass]^{-2}$, is assumed to satisfy the condition $\\mu \\gg \\kappa^2$. In the following, we set $\\mu\\sim10^{9}M^{-2}_{pl}$, as it is fixed by the observed CMB amplitude. In the DBI part, $f\\left(\\chi\\right)\\approx\\frac{\\lambda}{\\chi^4}$ is the warp factor of DBI field and $U(\\chi)$ is its potential. Originally, this model proposed in the context of $D3-\\bar{D}3$ brane-inflation in a warped throat. We assume D3-brane starts inside the throat, so the effective potential takes the simple form as\\cite{shandera06},\n\\ba\\label{chi-potential}\nU\\left(\\chi\\right)&=&\\frac{1}{2}m^2\\chi^2+V_0\\left(1-\\frac{vV_0}{4\\pi^2}\\frac{1}{\\chi^4}\\right)\n\\ea\n $V_0$ is the effective cosmological constant ; it depends on the warp factor of the $\\bar{D}3$ branes position.  The constant v  depends on the properties of the warped throat,  we choose v = 27\/16 ( see \\cite{shandera06} and references therein).\n\n Therefore, The total action in Einstein-frame is given by,\n\\ba\\label{full action}\nS'&=&\\int d^4x\\sqrt{-\\tilde{g}}\\left(\\frac{\\tilde{R}}{2\\kappa^2}-\\frac{1}{2}\\tilde{g}^{\\mu\\nu}\\tilde{\\partial}_\\mu\\psi\\tilde{\\partial}_\\nu\\psi-\ne^{-4\\alpha\\psi}\\frac{\\left(e^{2\\alpha\\psi}-1\\right)^2}{8\\kappa^2\\mu}\\right)\\\\\\nonumber\n&+&\\int d^4x\\sqrt{-\\tilde{g}}e^{-4\\alpha\\psi}\\left(\\frac{1}{f\\left(\\chi\\right)}\n\\left(1-\\sqrt{1+f\\left(\\chi\\right)e^{2\\alpha\\psi}\\tilde{g}^{\\mu\\nu}\\tilde{\\partial}_\\mu\\chi\\tilde{\\partial}_\\nu\\chi}\\right)-U\\left(\\chi\\right)\\right).\n\\ea\nFrom the above equation, we can read the potential of $\\psi$ as\n\\ba\\nonumber\nw\\left(\\psi\\right)\\equiv \\frac{1}{8\\kappa^2\\mu}e^{-4\\alpha\\psi}\\left(e^{2\\alpha\\psi}-1\\right)^2.\n\\ea\n The mass of scalaron is also defined as $m^2_\\psi=\\frac{1}{6\\mu}$. We assume the metric of space-time is flat FRW, $ds^2=-dt^2+a^2\\left(t\\right)d\\vec{x}^2$;\n then the equations of motion for $\\psi$ and $\\chi$  are as follows,\n\\ba\\label{eqm1}\n\\ddot{\\psi}+3H\\dot{\\psi}+w_{,\\psi}=-\\alpha e^{-4\\alpha\\psi}T^b_{DBI},\n\\ea\n\\ba\\label{eqm2}\n\\ddot{\\chi}&+&3H\\gamma^{-2}\\dot{\\chi}+e^{-2\\alpha\\psi}\\frac{f_{,\\chi}}{2f^2}\\left(1+2\\gamma^{-3}-3\\gamma^{-2}\\right)+e^{-2\\alpha\\psi}\\gamma^{-3}U_{,\\chi}\\\\\\nonumber\n&=&\\alpha\\dot{\\psi}\\dot{\\chi}\\left(3\\gamma^{-2}-1\\right).\n\\ea\nWhere, $\\gamma=1\/\\sqrt{1-e^{2\\alpha\\psi}f\\dot{\\chi}^2}$, is the modified boost factor of DBI field. The presence of $e^{2\\alpha\\psi}$ under the square root affects the dynamics of $\\chi$. $T^b_{DBI}\\equiv [f^{-1}\\left(\\chi\\right)\\left(4-\\gamma-3\\gamma^{-1}\\right)-4U\\left(\\chi\\right)]$ is the trace of the energy-momentum tensor of the DBI part. $()_{,\\psi}$ and $()_{,\\chi}$ denote derivative with respect to the fields $\\psi$ and $\\chi$, respectively.\n  Einstein's field equations in flat FRW background are given as below,\n\\ba\n3H^2&=&\\frac{1}{2}\\dot{\\psi}^2+e^{-4\\alpha\\psi}\\frac{\\left(e^{2\\alpha\\psi}-1\\right)^2}{8\\kappa^2\\mu}\n+\\rho_{DBI}\\label{H}\\\\\n-2\\dot{H}&=&\\dot{\\psi}^2+e^{-2\\alpha\\psi}\\dot{\\chi}^2\\gamma,\\label{dotH}\n\\ea\nwhere $\\rho_{DBI}=e^{-4\\alpha\\psi}[f^{-1}\\left(\\gamma-1\\right)+U\\left(\\chi\\right)]$.\n\n  We solve the equations of motion, (\\ref{eqm1}) and (\\ref{eqm2}) together with  Einstein's field equations, (\\ref{H}) and (\\ref{dotH}) to arrive at the evolution of the fields which are plotted in FIG.\\ref{fig1}.\n\\begin{figure*}[t!]\n \\begin{subfigure}[t]{0.5\\textwidth}\n  \\includegraphics[height=3cm]{beta=1.pdf}\n   \\caption{mass ratio=1}\n \\end{subfigure}\n \\begin{subfigure}[t]{0.5\\textwidth}\n    \\includegraphics[height=3cm]{beta=10.pdf}\n      \\caption{mass ratio=10}\n \\end{subfigure}\n \\begin{subfigure}[t]{0.5\\textwidth}\n  \\includegraphics[height=3cm]{beta=50.pdf}\n   \\caption{mass ratio=50}\n \\end{subfigure}\n \\begin{subfigure}[t]{0.5\\textwidth}\n  \\includegraphics[height=3cm]{beta=100.pdf}\n   \\caption{mass ratio=100}\n \\end{subfigure}\n\\caption{The blue(thick) and red(dashed) curves depict the evolution of $\\psi$ and  $\\chi$, respectively. The free parameters are chosen as $\\lambda=2\\times10^{12}$ and $V_0=10^{-12}$.  The horizontal axis, $N_e$, is the number of e-folds. }\n\\label{fig1}\n\\end{figure*}\n To satisfy the constraint on the maximum length of the throat\\cite{Mcallister}, we choose the initial value of $\\chi$ equals 1.5  (which is less than the initial value of $\\psi$). At the end of inflation $\\chi$ decreases to a small value( from brane-inflation viewpoints, branes and anti-branes annihilate near the bottom of the throat). We define the mass ratio parameter, $\\beta$, as $\\beta=\\frac{m_\\psi}{m_\\chi}$. These figures show that when $\\beta$ becomes much larger than one, the scalaron traps at its minimum and the energy density of the DBI field overcome the energy density of scalaron; thus DBI field governs the dynamics. The effect of scalaron is hidden in the boost factor; $\\psi$ provides enough e-folds and keep the boost factor around 1, which allows us to use slow-roll approximation and also assume that the  DBI field is potential-dominated.\n\n\\subsection{Background}\\label{background}\nThe scalaron rolls down in the effective potential to go to its minimum, where it is trapped. The effective potential depends on both fields. It is written as follows,\n \\ba\n U_{eff}=\\frac{1}{8\\kappa^2\\mu}e^{-4\\alpha\\psi}{\\left(e^{2\\alpha\\psi}-1\\right)^2}-\\frac{1}{4}e^{-4\\alpha\\psi}T^b_{DBI}.\n \\ea\nThe extremum  value at $\\psi_{min}$  satisfies ,\n \\ba\\nonumber\n [\\frac{\\alpha}{2\\kappa^2\\mu}e^{-4\\alpha\\psi}\\left(e^{2\\alpha\\psi}-1\\right)+\\alpha e^{-4\\alpha\\psi}T_{DBI}^b]\\mid_{\\psi_{min}}=0,\n \\ea\n solving the above equation gives,\n \\ba\\label{psi min}\n \\psi_{min}=\\frac{1}{2\\alpha}\\ln{\\left(1-2\\kappa^2\\mu T_{DBI}^b\\right)},\n \\ea\n\n the condition for having a minimum ($d^2U_{eff}\/d\\psi^2\\mid_{\\psi_{min}}>0$) is always satisfied because we have,\n  \\ba\n \\frac{d^2U_{eff}}{d\\psi^2}\\mid_{\\psi_{min}}=\\frac{\\alpha^2}{\\kappa^2\\mu}e^{-2\\alpha\\psi_{min}}>0.\n  \\ea\n We assume that the fields are potential dominated  i.e.  $T^b_{DBI}\\simeq-4U\\left(\\chi\\right)$ and $e^{2\\alpha\\psi_{min}}\\approx1+8\\kappa^2\\mu U\\left(\\chi\\right)$ , the Friedmann equations can be approximated as,\n  \\ba\\label{approx}\n   3H^2&\\simeq&\\frac{1}{8k^2\\mu}e^{-4\\alpha\\psi}[(e^{2\\alpha\\psi}-1)^2+8k^2\\mu U(\\chi)]\\mid_{\\psi_{min}},\\label{H min}\\\\\\nonumber\n       &\\simeq&e^{-2\\alpha\\psi}U(\\chi)\\\\\\nonumber\n   -2\\dot H&\\simeq&e^{-2\\alpha\\psi}\\gamma\\dot{\\chi}^2.\\label{dot H min}\n  \\ea\n From now on, we dropped the index min. As mentioned before, when $\\psi$ is trapped at its minimum, the dynamics is controlled by $\\chi$. Comparing with usual DBI in the general relativity context shows that the effect of $\\psi$  or equivalently $R^2$  term appears in $e^{-2\\alpha\\psi}$ factor.\n  To arrive at the above equations, we assumed $\\dot\\psi^2\\ll e^{-2\\alpha\\psi}\\gamma\\dot\\chi^2$ in (\\ref{dotH}). This assumption is equivalent to,\n  \\ba\\gamma\\gg\\frac{4\\kappa^2}{9\\alpha^2\\beta^2}\\frac{\\chi^2}{1+\\frac{2\\kappa}{3\\beta}\\chi^2}.\\ea\n This condition is satisfied when $\\beta\\gg1$,  i.e. when there is a heavy scalaron. In the following, we assume this condition is satisfied.\n  Differentiating (\\ref{psi min}) with respect to time gives the change of the minimum of $\\psi$, as $\\chi$ evolves,\n  \\ba\\label{psi min diff}\n  \\dot{\\psi}=\\frac{4\\kappa^2\\mu}{\\alpha}e^{-2\\alpha\\psi}U_{,\\chi}\\dot{\\chi}.\n  \\ea\n   The coupling between the two fields causes Hubble friction term and potential terms  dominate in the DBI equation of motion (\\ref{eqm2}), then we have\n  \\ba\\label{app  dot chi}\n  \\dot\\chi\\simeq-e^{-2\\alpha\\psi}\\frac{U_{,\\chi}}{3H\\gamma}\n  \\ea\n  We  also define the slow-roll parameters as usual,\n\\ba\n\\epsilon\\equiv-\\frac{\\dot{H}}{H^2}=\\frac{3}{2}\\frac{\\dot{\\psi}^2+\\gamma e^{-2\\alpha\\psi}\\dot{\\chi}^2}{\\rho_{DBI}+\\frac{1}{2}\\dot{\\psi}^2+w\\left(\\psi\\right)},\n\\ea\nUsing (\\ref{approx})we arrive at,\n  \\ba\\label{epsilon}\n  \\epsilon&\\approx&\\frac{3}{2}\\frac{\\gamma\\dot{\\chi}^2}{U\\left(\\chi\\right)}\\\\\\nonumber\n  &\\approx&\\frac{1}{2}e^{-2\\alpha\\psi}(\\frac{U_{,\\chi}}{U})^2.\n  \\ea\nSimilar to previous results the only difference with usual DBI model is $e^{-2\\alpha\\psi}$ factor. As usual we have $\\ddot{a}\/a=H^2\\left(1-\\epsilon\\right)$.\nThe inflation ends when $\\epsilon$ gets larger than 1. In our numerical analysis, we get around 55 e-folds.\n  Differentiate (\\ref{epsilon}) with respect to time, we arrive at the rate of change of this slow-roll parameter,\n  \\ba\\label{dotepsilon}\n  \\frac{\\dot\\epsilon}{2H\\epsilon}\\simeq\\frac{1}{2}s-\\delta+2\\left(1+12\\kappa^2\\mu U\\left(\\chi\\right)\\right)\\epsilon,\n  \\ea\n  with\n  \\ba\\label{slow param}\n  s=-\\frac{\\dot\\gamma}{H\\gamma}\\hspace{1.5cm}and\\hspace{1.5cm}\\delta=\\frac{1}{\\gamma}\\frac{U_{,\\chi\\chi}}{U}.\n  \\ea\n   where \"s\" measures the rate of change of the sound speed and $\\delta$ is equivalent to $\\eta$ parameter. Note that both of these parameters has implicit dependence on $\\psi$ through $e^{-2\\alpha\\psi}$ factor in $\\gamma$ .\n   From a mathematical point of view, our model is equivalent to a scalar-tensor theory\\cite{ Bruck2010, Bruck2011}. But the physics behind these models is different. In our case, the canonical scalar field originated from higher-order gravity and quantum corrections rather than put inside the theory by hand.\n\nIt is worth mentioning that there are other models which have interesting motivations for deriving inflation, for example in [44], it is shown explicitly that the quantum potential plays the role of the cosmological constant and also produces the exponential expansion.\n\n\\section{Perturbations evolution and cosmological parameters}\\label{perturbation}\nFirst, we consider the evolution of linear perturbation of this model. We perturb the action (\\ref{full action}) in a standard way by decomposition of the fields $\\psi$ and $\\chi$ into a homogeneous and perturbed part,\n\\ba\n \\psi\\left(t,\\mathbf{x}\\right)=\\psi\\left(t\\right)+\\delta\\psi\\left(t,\\mathbf{x}\\right)\\hspace{2cm} \\chi\\left(t,\\mathbf{x}\\right)=\\chi\\left(t\\right)+\\delta\\chi\\left(t,\\mathbf{x}\\right).\n\\ea\nThe field perturbations are of linear order. We shall work in Fourier space in which the spatial derivative,$\\partial$, can be replaced by $-ik$. Assume that the anisotropic stress is absent, in longitudinal gauge, the scalar perturbation of the flat FRW metric is expressed as below,\n\\ba\n ds^2=-\\left(1+2\\Phi\\right)dt^2+a^2\\left(t\\right)\\left(1-2\\Phi\\right)\\delta_{ij}dx^{i}dx^{j}.\n\\ea\n\nThe equations of field perturbation are as follows\n\\ba\\label{perturbeqm1}\\nonumber\n&\\ddot{\\delta\\psi}&+3H\\dot{\\delta\\psi}-4\\dot{\\Phi}\\dot{\\psi}+\\alpha\\dot{\\delta\\chi}\\dot{\\chi}e^{-2\\alpha\\psi}\n(3\\gamma-\\gamma^3)\\\\\\nonumber\n&+&\\delta\\psi(\\frac{k^2}{a^2}+w_{,\\psi\\psi}-4\\alpha^2e^{-4\\alpha\\psi}[f^{-1}(4-3\\gamma^{-1}-\\gamma)-4U(\\chi)]+\n\\alpha^2e^{-2\\alpha\\psi}(3\\gamma-\\gamma^3)\\dot{\\chi}^2)\\\\\\nonumber\n&+&\\delta\\chi(-4\\alpha e^{-4\\alpha\\psi}U_{,\\chi}-\\alpha e^{-4\\alpha\\psi}\\frac{f_{,\\chi}}{2f^2}(8-3\\gamma^{-1}-6\\gamma+\\gamma^3))\\\\\\nonumber\n&+&2\\Phi(w_{,\\psi}+\\alpha e^{-4\\alpha\\psi}[f^{-1}(4-3\\gamma^{-1}-\\gamma)-4U(\\chi)]+\\alpha e^{-2\\alpha\\psi}(3\\gamma-\\gamma^3)\\dot{\\chi}^2)=0,\\ea\nand\n\\ba\\label{perturbeqm2}\\nonumber\n&\\ddot{\\delta\\chi}&+(3H+3\\frac{\\dot\\gamma}{\\gamma}-2\\alpha\\dot{\\psi})\\dot{\\delta\\chi}-\\dot{\\Phi}\\dot{\\chi}(1+3\\gamma^{-2})\n+\\alpha\\dot{\\delta\\psi}\\dot{\\chi}(1-3\\gamma^{-2})\\\\\\nonumber\n&+&\\delta\\chi\\{\\gamma^{-2}\\frac{k^2}{a^2}+\\gamma^{-3}U_{,\\chi\\chi}e^{-2\\alpha\\psi}+\\frac{f_{,\\chi}}{f}\\frac{\\dot{\\chi}\\dot{\\gamma}}{\\gamma}-\n\\frac{1}{2}U_{,\\chi}f_{,\\chi}\\gamma^{-1}\\dot{\\chi}^2\\\\\\nonumber\n&+&\\frac{1}{2}e^{-2\\alpha\\psi}(1-\\gamma^{-1})^2\\gamma^{-2}\n[\\gamma(\\frac{f_{,\\chi}}{f^2})_{,\\chi}+(\\frac{f_{,\\chi}}{f})_{,\\chi}\\frac{1}{f}(1+\\gamma^{-1})\\gamma^2]\\}\\\\\\nonumber\n&-&\\alpha\\delta\\psi(\\gamma^{-1}(1+\\gamma^{-2})U_{,\\chi}e^{-2\\alpha\\psi}+\\frac{f_{,\\chi}}{f^2}\\gamma^{-1}(1-\\gamma^{-1})^2e^{2\\alpha\\psi}\n-2\\dot{\\chi}\\frac{\\dot{\\gamma}}{\\gamma})\\\\\\nonumber\n&+&\\Phi(e^{-2\\alpha\\psi}\\gamma^{-1}(1+\\gamma^{-2})U_{,\\chi}\n-2\\dot{\\chi}\\frac{\\dot{\\gamma}}{\\gamma}+\\frac{f_{,\\chi}}{f^2}e^{-2\\alpha\\psi}\\gamma^{-1}(1-\\gamma^{-1})^2)=0.\n\\ea\nIt is convenient to introduce gauge-invariant quantity, so-called Sasaki-Mokhanuv variables\\cite{Bruck2010,Lalak2007},\n\\ba\n Q_\\psi\\equiv\\delta\\psi+\\frac{\\dot{\\psi}}{H}\\hspace{2cm} Q_\\chi\\equiv\\delta\\chi+\\frac{\\dot{\\chi}}{H},\n\\ea\nwhich are the scalar field perturbations in the flat gauge. In terms of these new variables the equations form a closed system,\n\\ba\n\\ddot{Q_\\psi}&+&3H\\dot{Q_\\psi}+B_\\psi\\dot{Q_\\chi}+\\left(\\frac{k^2}{a^2}+C_{\\psi\\psi}\\right)Q_\\psi+C_{\\psi\\chi}Q_\\chi=0,\\\\\n\\ddot{Q_\\chi}&+&\\left(3H-2\\alpha\\dot{\\psi}+3\\frac{\\dot{\\gamma}}{\\gamma}\\right)\\dot{Q_\\chi}\\\\\\nonumber\n&+&B_\\chi\\dot{Q_\\psi}\\left(\\frac{k^2}{a^2\\gamma^2}+C_{\\chi\\chi}\\right)Q_\\chi+C_{\\chi\\psi}Q_\\psi=0.\n\\ea\n with the coefficients as\n\\ba\\nonumber\nB_\\chi&=& -\\alpha(\\frac{3}{\\gamma^2}-1)\\dot{\\chi}-\\frac{\\dot{\\psi}\\dot{\\chi}}{2H}(1 - \\frac{1}{\\gamma^2}),\\\\\\nonumber\nB_\\psi&=&-e^{-2\\alpha\\psi}\\gamma^3B_\\chi,\\\\\\nonumber\nC_{\\psi\\psi}&=&-\\alpha\\frac{\\dot{\\psi}}{H} e^{-4\\alpha\\psi}f^{-1}(\\frac{3}{\\gamma} + 1)(1 -\\gamma)^3 -\\alpha^2 e^{-4\\alpha\\psi}f^{-1}(16 - 8\\gamma - \\frac{9}{\\gamma}+\\gamma^3) + 3\\dot{\\psi}^2 \\\\\\nonumber\n&-&\\gamma^3(1 + \\frac{1}{\\gamma^2}) e^{-2\\alpha\\psi}\\frac{\\dot{\\psi}^2\\dot{\\chi}^2}{4H^2}-\\frac{\\dot{\\psi^4}}{2H^2}+\\alpha e^{-4\\alpha\\psi}\\frac{2\\dot{\\psi}}{H}(\\frac{1}{2\\kappa\\mu}(1-e^{2\\alpha\\psi})-4U(\\chi))\\\\\\nonumber&+&\\alpha^2e^{-4\\alpha\\psi}\n\\left(\\frac{1}{\\kappa^2\\mu}(2-e^{2\\alpha\\psi})+16U(\\chi)\\right),\\\\\\nonumber\nC_{\\psi\\chi}&=&\\frac{e^{-4\\alpha\\psi}\\dot{\\psi}}{4H}\\frac{f_{,\\chi}}{f^2}\\frac{1}{\\gamma}(1 -\\gamma)^2(\\gamma^2 + 2\\gamma- 1) + 3\\gamma e^{-2\\alpha\\psi}\\dot{\\psi}\\dot{\\chi}-\\gamma^4(1 + \\frac{1}{\\gamma^2})e^{-4\\alpha\\psi}\\frac{\\dot{\\psi}\\dot{\\chi}^3}{4H^2}\\\\\\nonumber\n&+&\\frac{1}{2}\\alpha e^{-4\\alpha\\psi}f^{-1}(\\frac{3}{\\gamma}+ 1)(1 - \\gamma)^3\\left(\\frac{f_{,\\chi}}{f}-\\frac{e^{-2\\alpha\\psi}\\dot{\\chi}\\gamma}{H}\\right)-\\gamma e^{-2\\alpha\\psi}\\frac{\\dot{\\psi}^3\\dot{\\chi}}{2H^2}+e^{-4\\alpha\\psi}\\frac{\\dot{\\psi}}{H}U(\\chi)_{,\\chi}\\\\\\nonumber\n&+&\\alpha\\gamma e^{-6\\alpha\\psi}\\frac{\\dot{\\chi}}{H} (\\frac{1}{2\\kappa^2\\mu}(1-e^{2\\alpha\\psi}-1)-4U(\\chi))-4\\alpha e^{-4\\alpha\\psi}U_{,\\chi},\\\\\\nonumber\nC_{\\chi\\chi}&=&e^{-4\\alpha\\psi}\\frac{\\dot{\\chi}}{H}\\frac{f_{,\\chi}}{f^2}(1-\\frac{1}{\\gamma})^2-(\\frac{f_{,\\chi}}{f}+\n\\frac{e^{-2\\alpha\\psi}\\dot{\\chi}\\gamma}{H})\\frac{\\dot{\\gamma}}{\\gamma}\\dot{\\chi}\\\\\\nonumber\n&-&\\frac{1}{2\\gamma}f_{,\\chi}\\dot{\\chi}^2U_{,\\chi}+\\frac{1}{2}e^{-2\\alpha\\psi}\n(1-\\frac{1}{\\gamma})^2[\\frac{1}{\\gamma}(\\frac{f_{,\\chi}}{f^2})_{,\\chi}+(1+\\frac{1}{\\gamma})f^{-1}(\\frac{f_{,\\chi}}{f}t)_{,\\chi}]\\\\\\nonumber\n&+&\\frac{3}{2}e^{-2\\alpha\\psi}\\dot{\\chi}^2\\gamma(1+\\frac{1}{\\gamma^2})-e^{-4\\alpha\\psi}\\gamma^2\\frac{\\dot{\\chi}^4}{2H^2}-\ne^{-2\\alpha\\psi}\\gamma(1+\\frac{1}{\\gamma^2})\\frac{\\dot{\\chi}^2\\dot{\\psi}^2}{4H^2}\\\\\\nonumber\n&+&e^{-4\\alpha\\psi}\\frac{\\dot{\\chi}}{H}(1+\\frac{1}{\\gamma^2})U_{,\\chi}+\\frac{1}{\\gamma^3}e^{-2\\alpha\\psi}U_{,\\chi\\chi},\\\\\\nonumber\n\\ea\n\\ba\\nonumber\nC_{\\chi\\psi}&=&(-2e^{-2\\alpha}+\\frac{\\dot{\\psi}}{H})(\\frac{1}{2}e^{-2\\alpha\\psi}\\frac{f_{,\\chi}}{f^2\\gamma}(1-\\frac{1}{\\gamma})^2\n-\\frac{\\dot\\gamma}{\\gamma}\\dot\\chi)+2\\alpha\\frac{e^{-4\\alpha\\psi}\\dot{\\chi}}{H}f^{-1}(1-\\frac{1}{\\gamma})^2\n\\\\\\nonumber\n&-&\\gamma\\frac{e^{-2\\alpha\\psi}\\dot\\psi\\dot\\chi^3}{2H^2}+\\frac{1}{2}(1+\\frac{1}{\\gamma^2})\\\\\\nonumber\n&&\\left(3\\dot\\psi\\dot\\chi-\\frac{\\dot\\phi^3\\dot\\chi}{2H^2}-\\frac{2\\alpha}{\\gamma}e^{-2\\alpha\\psi} U_{,\\chi}+\\frac{\\dot\\psi e^{-2\\alpha\\psi}U_{,\\chi}}{\\gamma H}+\\alpha e^{-4\\alpha\\psi}(\\frac{1}{2k^2\\mu}(e^{2\\alpha\\psi}-1)-4U(\\chi))\\frac{\\dot\\chi}{H}\\right).\\nonumber\n\\ea\n\nSimilar to single-field perturbation analysis in canonical  and DBI models, we introduce two auxiliary fields as,\n\\ba\nu_\\psi=a Q_\\psi,\\hspace{2cm}u_\\chi=ae^{-\\alpha\\psi}c_s^{-3\/2} Q_\\chi.\n\\ea\nThe equations of motion in terms of conformal time can be rewritten in a more symmetric form,\n\\ba\nu''_\\psi&-&Bu'_\\chi+[k^2+a^2C_{\\psi\\psi}-\\frac{r''_\\psi}{r_\\psi}]u_\\psi+[\\frac{r_\\psi}{r_\\chi}a^2C_{\\psi\\chi}+B\\frac{r'_\\chi}{r_\\chi}]u_\\chi=0\\label{upsi}\\\\\nu''_\\chi&+&Bu'_\\psi+[k^2c_s^2+a^2C_{\\chi\\chi}-\\frac{r''_\\chi}{r_\\chi}]u_\\chi+[\\frac{r_\\chi}{r_\\psi}a^2C_{\\chi\\psi}-B\\frac{r'_\\psi}{r_\\psi}]u_\\psi=0\\label{uchi}\n\\ea\n where $()'$ denotes the derivative with respect to conformal time and we define $c_s=1\/\\gamma$, $r_\\chi=ae^{-\\alpha\\psi}\\gamma^{3\/2}$, $r_\\psi=a$, and $B=r_\\chi B_\\chi$.\nThe co-moving curvature perturbation,\ncan be express in terms of gauge invariant variables $Q_\\psi$ and $Q_\\chi$  in a simple form\\cite{Bruck2010},\n\\ba \\label{R}\\mathcal{R}=\\frac{H}{-2\\dot{H}}[\\dot\\psi Q_\\psi+e^{-2\\alpha\\psi}\\gamma\\dot\\chi Q_\\chi].\\ea\n\n\\subsubsection*{The evolution of perturbations for a trapped scalaron:}\nThe contribution of $Q_\\psi$ in curvature perturbation can be ignored when the scalaron,  $\\psi$,  traps in the minimum of the effective potential. In this case, It is possible to treat  the system of equations as a single field DBI model with modified boost factor. Numerical analysis supports this approximation\\footnote{In our numerical code we got some help from numerical code mTransport\\cite{numeric}}(Fig(\\ref{fig2})).\nAfter that, the dynamics is governed by the DBI field.\n\n\\begin{figure}[H]\n\\includegraphics[keepaspectratio=true,scale=0.6]{curvatureperturbationn3.pdf}\n\\caption{We depict  the contribution of the scalaron (blue thick curve) and DBI field (red  dashed curve) in curvature perturbation (\\ref{R}),  after $\\psi$ trapped at the minimum of the effective potential. The  horizontal axis is time. }\n\\label{fig2}\n\\end{figure}\n\n Therefore, the perturbation equations (\\ref{upsi} and \\ref{uchi}) are estimated as follows,\n\\ba\nu''_\\chi+[k^2c_s^2+a^2C_{\\chi\\chi}-\\frac{r''_\\chi}{r_\\chi}]u_\\chi\\simeq0.\n\\ea\n  insertion of (\\ref{app  dot chi}) into (\\ref{slow param}), gives $C_{\\chi\\chi}$ and the derivative of  $r_\\chi$  in terms of slow-roll parameters (up to the first order) as\n \\ba\\label{c chi chi}\na^2 C_{\\chi\\chi}&\\simeq&3\\mathcal{H}^2[\\delta-s-2\\epsilon+8\\kappa^2\\mu U\\epsilon\\left(1-\\gamma^{-2}\\right)],\n \\ea\n and\n \\ba\\label{r''}\n \\frac{r''_\\chi}{r_\\chi}&\\simeq&\\mathcal{H}^2\\left(2-\\frac{2}{9}s-\\left(1-24\\kappa^2\\mu U\\right)\\epsilon\\right)\n \\ea\nwhere $\\mathcal{H}=a'\/a$ and $\\mathcal{H}'=\\mathcal{H}^2\\left(1-\\epsilon\\right)$. The background variable z is defined as usual,\n\\ba z\\equiv\\frac{a\\gamma\\sqrt{\\rho+p}}{H}=a\\gamma\\sqrt{2\\epsilon}\\ea\nwhere we used the fact $-2\\dot H=\\rho+p$. Combination of (\\ref{c chi chi}) and (\\ref{r''}) gives,\n\\ba\na^2C_{\\chi\\chi}-\\frac{r''_\\chi}{r_\\chi}\\simeq-\\frac{z''}{z}+24\\kappa^2\\mu U\\epsilon\\mathcal{H}^2\\left(1+c_s^2\\right).\\ea\n The first term is almost the same as single field k-inflation\\cite{mukhanov1},  in which $\\frac{u_\\chi}{z}$ is constant for small k; the second term is a small correction of order $\\epsilon$ which is proportional to $\\left(1+c_s^2\\right)$. At lowest order, we ignore the second term;\n \\ba\\label{u} u''_\\chi+\\left(k^2c_s^2-\\mathcal{H}^2[2-\\frac{3}{2}s-3\\delta+\\left(5+72\\kappa^2\\mu U\\right)\\epsilon]\\right)u_\\chi\\simeq0.\\ea\n\n Ignoring the perturbation of $\\psi$ in the co-moving curvature perturbation (\\ref{R}) gives,\n \\ba \\mathcal{R}\\simeq\\frac{e^{-\\alpha\\psi}\\gamma^{1\/2}}{\\sqrt{2\\epsilon}}Q_\\chi=\\frac{u_\\chi}{z}.\\ea\n The power spectrum is as\n \\ba\\label{powersp}\\mathcal{P}_{\\mathcal{R}}\\simeq\\frac{k^3}{2\\pi^2}|\\frac{u_\\chi}{z}|^2.\\ea\n\n For solving eq.(\\ref{u}) we follow the approach in \\cite{Bruck2011}, define a new time variable as,\n \\ba\\label{y}y\\equiv\\frac{c_s k}{a H}=\\frac{c_s k}{\\mathcal{H}}.\\ea\n With this definition at sound horizon crossing,  we have  $y=1$.\n  The derivatives of $u_\\chi$ can be expressed in terms of slow-roll parameters,\n \\ba\\nonumber u'_\\chi=-c_s k\\left(1-\\epsilon-s\\right)\\frac{du_\\chi}{d\\tau},\\ea\n  and\n  \\ba\\nonumber u''_\\chi=\\mathcal{H}^2[\\left(1-\\epsilon-s\\right)^2y^2\\frac{du_\\chi^2}{dy^2}-s\\left(1-\\epsilon-s\\right)y\\frac{du_\\chi}{dy}],\\ea\n  where we have used  $\\mathcal{H}'=\\mathcal{H}^2\\left(1-\\epsilon\\right)$. Substituting in (\\ref{u}) gives,\n  \\ba\\label{uy}\\nonumber\n   y^2\\frac{d^2u_\\chi}{dy^2}+(1-2p) y\\frac{du_\\chi}{dy}+\\left(l^2y^2+p^2-\\nu^2\\right)u_\\chi=0,\n   \\ea\n  with\n  \\ba\n  p&=&\\frac{1}{2}(1+s),\\\\\n  l&=&(1-\\epsilon-s)^{-1},\\\\\n  \\nu&=&\\frac{3}{2}+s-\\delta+3\\epsilon(1+8\\kappa^2\\mu U.)\n  \\ea\n\nThe solution of (\\ref{uy}) is of the form $u_\\chi=y^pJ_\\nu(ly)$. $J_\\nu$  denotes Bessel function of order $\\nu$. Instead of Bessel functions, we write the solution in terms of Hankel functions, which are more appropriate for our purpose. In the short wavelength limit , ($y\\gg1$), the solution is given by positive frequency mode,\n$\\frac{1}{\\sqrt{2c_sk}}e^{-ic_sk\\tau}$, where $\\tau$ is conformal time.\nOnly $H^{(1)}_\\nu( ly)$ can satisfy this initial condition; therefore, the solution is\n\\ba u_\\chi( y)=\\frac{1}{2}\\sqrt{\\frac{\\pi}{c_sk}}\\sqrt{\\frac{y}{1-\\epsilon-s}}H^{(1)}_\\nu(\\frac{y}{1-\\epsilon-s}).\\ea\nIn the long-wavelength limit ($y\\ll1$) we have,\n$H^{(1)}_\\nu(ly)\\sim\\sqrt{\\frac{2}{\\pi}}e^{-i\\pi\/2}2^{\\nu-\\frac{3}{2}}\\frac{\\Gamma(\\nu)}{\\Gamma(3\/2)}y^{-\\nu}$; so,  the solution is\n\\ba |u_\\chi|\\sim2^{\\nu-\\frac{3}{2}}\\frac{\\Gamma(\\nu)}{\\Gamma(3\/2)}(1-\\epsilon-s)^{\\nu-\\frac{1}{2}}\\frac{y^{\\nu-\\frac{1}{2}}}{\\sqrt{2c_sk}}.\\ea\nReplacing in (\\ref{powersp}) we arrive at\n\\ba\\mathcal{P}_{\\mathcal{R}}^{1\/2}\\simeq(\\frac{\\mathcal{V}(\\nu)}{\\pi})\\frac{H}{\\sqrt{c_s\\epsilon}}y^{\\frac{3}{2}-\\nu},\\ea\nwith $\\mathcal{V}\\equiv2^{\\nu-3}(1-\\epsilon-s)^{\\nu-\\frac{1}{2}}\\Gamma(\\nu)\/\\Gamma(\\frac{3}{2})$.\n\n It can be shown that $\\frac{d}{dy}(\\frac{H}{\\sqrt{c_s\\epsilon}}y^{\\frac{3}{2}-\\nu})\\simeq0,$, which insures us that the power spectrum is independent of y and can be evaluated at any preferred y value\\cite{Bruck2011,generalizations of slow-roll,y1,y2}, hence the sound crossing formalism is applicable.\n\nUsing this gives the spectral index (up to first order in slow-roll parameters) as,\n\\ba\nn_s-1&=&3-2\\nu\\\\\\nonumber\n     &=&-2s+2\\delta-6\\epsilon(1+8\\kappa^2\\mu U).\n\\ea\nReplacing the slow-roll parameters we arrive  at\n\\ba\nn_s-1&=&2\\frac{\\dot\\gamma}{H\\gamma}-8\\frac{1}{\\gamma\\chi^2}.\n\\ea\nAt the end of inflation, only the DBI field drives inflation, so we ignore the perturbation of scalaron. It is reasonable to assume that the results obtained in the DBI inflation are applicable to this model; for example, the tensor-to-scalar ratio must be $r\\simeq16\\epsilon c_s$.\nThe non-Gaussianity is also given by $f^{DBI}_{NL}\\simeq-0.3\\left(c_s^{-2}-1\\right)$. To be more precise, one can apply the result of \\cite{langlois2008} and \\cite{koyoma2008}to this model and obtain the third-order action.\nThe effect of $R^2$ gravity on the DBI field keeps the sound speed close to one (see Fig.(\\ref{fig6})) i.e keeps $f^{DBI}_{NL}$ very small.\n\n\\begin{figure}[H]\n\\includegraphics[keepaspectratio=true,scale=0.5]{sound-speed.pdf}\n\\caption{The sound speed versus  the number of e-folds is shown. Parameters value are chosen as $\\lambda=2\\times10^{12}$, $V_0=10^{-12}$,  and $\\beta=55$.}\n\\label{fig6}\n\\end{figure}\n\n\n\\subsection{Numerical Analysis}\\label{numerical analysis}\nIn this section, we check the compatibility of our model with Planck 2018 data.\nOur analysis shows that the amount of inflation depends on the $\\psi$ initial value. We pick the initial value of $\\psi$ so that to obtain enough e-folds.\nMotivated by brane inflation, we choose the initial value of $\\chi$ around 1\\cite{Mcallister}(through this work we choose 1.5). In the DBI part, we have three undetermined parameters,the mass of $\\chi$ (m), $\\lambda$, and $V_0$. We investigate the effect of varying these parameters in this section.\nThere is also another parameter in the $R^2$ part of the action, which is denoted by $\\mu$. As mentioned before, we select $\\mu\\sim10^9$. Since the mass square of the scalaron is proportional to the inverse of $\\mu$, we have $m_\\psi\\sim1.3\\times10^{-5}$ (in natural units).\n\nWe change the mass ratio parameter, $\\beta$, to obtain the spectral index and the tensor-to-scalar ratio. As previously stated, in the DBI part of the action, there are also two other parameters, $\\lambda$ and $V_0$.\nWe check the different values of these parameters. First, we inspect the different values of $V_0$; we depict the tensor-to-scalar ratio versus the spectral index in figure(\\ref{fig2a}).\nBy increasing the mass ratio, the spectral index also increases; but the tensor-to-scalar ratio remains almost constant.\n The tensor-to-scalar ratio value is very small(in comparison with Planck upper limit 0.064). This result is similar to ordinary Starobinsky inflation\\cite{Planck2018b}. To be more clear we plot spectral index (Fig(\\ref{fig2b}))and the tensor-to-scalar ratio (Fig(\\ref{fig2c}))with respect to mass ratio ($\\beta$).\n\n \\begin{figure}[H]\n  \\includegraphics[keepaspectratio=true]{Plotnsrv0.pdf}\n   \\caption{The tensor to scalar ratio versus the spectral index is depicted, we choose $\\lambda=2\\times10^{12}$. The colored regions are  $68\\%$ and $95\\%$  confidence level  of TT,TE,EE+lowE+lensing Planck2018 data.}\n   \\label{fig2a}\n  \\end{figure}\n \\begin{figure}[H]\n  \\includegraphics[keepaspectratio=true]{Plotvaryingv0ns.pdf}\n   \\caption{ We plot $n_s$ versus beta (the mass ratio), the narrow gray band shows the Planck limit. We choose $\\lambda=2\\times10^{12}$.}\n   \\label{fig2b}\n \\end{figure}\n \\begin{figure}[H]\n  \\includegraphics[keepaspectratio=true]{Plotvaryingv0r.pdf}\n   \\caption{The tensor to scalar ratio is shown versus beta (the mass ratio). We choose $\\lambda=2\\times10^{12}$.\n   Our results are much smaller than the Planck limit $r<0.064$}\n   \\label{fig2c}\n \\end{figure}\n\n From the above figures, it is obvious that very small and very large values of $V_0 $ are not compatible with observations. For a small value of $V_0$, the DBI potential is almost $\\frac{1}{2}m^2\\phi^2$. So we can conclude that this famous potential is not compatible with the Planck data, even for the DBI field. On the opposite side, for large values of $V_0$, the DBI potential is almost constant.\n  It seems that to get good results, we need both parts of the potential. Therefore, we choose an intermediate value (in the other parts of this work we choose $V_0=10^{-12}$).\n\n \\begin{figure}[H]\n  \\includegraphics[keepaspectratio=true]{Plotnsrl.pdf}\n   \\caption{The tensor to scalar ratio versus the spectral index is depicted, we choose $V_0=10^{-12}$. The colored regions are  $68\\%$ and $ 95\\%$ confidence level of TT,TE,EE+lowE+lensing Planck2018 data.}\n    \\label{fig3a}\n  \\end{figure}\n \\begin{figure}[H]\n  \\includegraphics[keepaspectratio=true]{Plotvaryinglns.pdf}\n   \\caption{ We plot $n_s$ versus beta (the mass ratio), we choose $V_0=10^{-12}$. The narrow gray band shows the Planck limit.}\n    \\label{fig3b}\n \\end{figure}\n \\begin{figure}[H]\n  \\includegraphics[keepaspectratio=true]{Plotvaryinglr.pdf}\n   \\caption{The tensor to scalar ratio is shown versus beta (the mass ratio), we choose $V_0=10^{-12}$. These plots are for different value of  constant part of DBI potential. Our results are much smaller than the Planck limit $r<0.064$. }\n    \\label{fig3c}\n \\end{figure}\nTo find out the effect of the other parameter, $\\lambda$ we again plot r with respect $n_s$ by varying the mass ratio($\\beta$) for different value of $\\lambda$ (Fig(\\ref{fig3a})). We also plot r (Fig(\\ref{fig3b})) and $n_s$  (Fig(\\ref{fig3c})) with respect to $\\beta$ separately.\nThese figures indicate that, only intermediate values,  around $10^{12}$ to $10^{13}$ gives  compatible results, therefore for a closer look, we plot r (Fig(\\ref{fig4})) and $n_s$ (Fig(\\ref{fig5})) for $\\lambda$ in this range.\n\n \\begin{figure}[H]\n\\includegraphics[keepaspectratio=true]{Plotns.pdf}\n\\caption{ We plot the spectral index by varying the $\\lambda$ parameter. The gray area is allowed value by Planck2018. As before $V_0=10^{-12}$ }\n\\label{fig4}\n\\end{figure}\n\\begin{figure}[H]\n\\includegraphics[keepaspectratio=true]{Plotr.pdf}\n\\caption{ We plot the tensor to scalar ratio by varying the $\\lambda$ parameter. The gray area is allowed value by Planck2018. As before $V_0=10^{-12}$ }\n\\label{fig5}\n\\end{figure}\n\n\nOur analysis shows that it is possible to get the spectral index  and the tensor to scalar ratio in the  Planck range for the appropriate choice of parameters and initial conditions.\n\nIn addition, from figures \\ref{fig2c}, \\ref{fig3c} and \\ref{fig5}\n one can conclude that in this model the scalar-tensor-ratio is very small, $r<0.01$ regardless of the spectral index. This property is different from other models, which consider extra fields in the context of F(R) gravity.\n\n\n\nOur numerical analysis shows that regardless of the mass ratio, the sound speed is near one (see FIG\\ref{fig6}. As before we set the initial values of $\\chi=1.5$ and $\\psi=5.3$. According to our analysis, the main results are not sensitive to initial conditions.\n\n\n\n\n\n\n\n\\section{Conclusion}\\label{conclusion}\nWe studied the effect of the existence of a DBI field in the Starobinsky inflation, i.e. $R+cR^2$ gravity.\nIn this model, there are two fields: the DBI field and the scalaron. Therefore, our model is within the context of a general multi-field model.\nWe have shown that when the mass of the scalaron is much greater than the mass of the DBI field, the DBI field drives the inflation. In this case, the scalaron is trapped at its minimum. Before the trapping of $\\psi$, the DBI field is almost constant. From the brane inflation point of view, it means that the branes move very slowly. After $\\psi$ traps at its minimum, the DBI field begins to decrease i.e. the branes get closer together. Although the DBI field drives inflation, the boost factor and other quantities have implicit dependence on $\\psi$. In this model, the boost factor is smaller than the single DBI model due to the existence of $e^{2\\alpha\\psi}$ in the square root. Hence in Starobinsky gravity the level of non-Gaussianity of DBI model decreases.\nIt is possible to ignore the fluctuation of scalaron when it is trapped at the minimum, so only the DBI field contributes to curvature perturbation, spectral index, tensor to scalar ratio, and other quantities that are related to the field perturbations.\nBefore trapping, the scalaron contribution to the energy density is much greater than the contribution of the DBI field. Therefore, the Hubble parameter, and consequently, the maximum number of e-folds, have a strong dependence on the scalaron. But due to the heaviness of scalaron, its perturbations are suppressed, and only the perturbations of the DBI field contribute to the curvature perturbation. This issue has been checked numerically.\n\n The main result of all these works is reducing the boost factor of the DBI field. Therefore the amount of non-Gaussianity is also decreased, which is compatible with observational data. But as stated before, our numerical results show that the spectral index, which is caused by simple potential $\\frac{1}{2}m^2\\phi^2$, is not compatible with Planck2018 data. To overcome this problem, we considered a well-motivated potential for the DBI part rather than a simple square potential.\n\n From mathematical point of view, this model is equivalent to a scalar-tensor model. In \\cite{Bruck2010} and \\cite{Bruck2011} DBI field in scalar-tensor theories are investigated. Our mathematical analysis is very similar to them. Our results are also\ncompatible with their results.Even though mathematics is the same, the physics of these models are different. In our case, the canonical field originates from quantum corrections, which are included in the $R^2$ term. We consider the brane inflation in Starobinsky gravity and investigate the effect of the existence of the DBI field in this theory.\n We have shown that with appropriate initial conditions, we get 50-60 e-folds at the end of inflation. As previously mentioned, this model is compatible with the Planck constraints on the spectral index and the tensor to scalar ratio. (see figure  (\\ref{fig3a})).\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nDirect production of new particles (with special emphasis on Higgs\nand supersymmetric partners) and possible indirect effects due to \ndeviations from the predictions of the Standard Model (hereafter denoted as\nSM), in particular in the presence of anomalous triple gauge \ncouplings, will be soon thoroughly searched for at LEP2 in the \nforthcoming few years. In both cases, typical and sometimes spectacular \n experimental signatures would exist, allowing to draw unambiguous \nconclusions if a certain type of signal were discovered.\n \n\tAt LEP2, one extra Z (to be called $ Z'$ from now on) would not \nbe directly produced, owing to the already existing mass limits from \nTevatron.Its indirect effects on several observables might be, \nthough, sizeable, since it would enter the theoretical expressions \nat tree level.\n In this sense, $ Z'$ effects at LEP2 would be of similar \ntype to those coming from anomalous triple gauge couplings\n (hereafter denoted as AGC), although the responsible mechanism \nwould be of totally different physical origin.\nThis peculiar feature of a $ Z'$ at LEP2 has substantially oriented \nthe line of research of our working group. In fact, we have tried in \nthis report to answer two complementary questions. \n\nThe first one was \nthe question \"which information on a  $Z'$ can one derive if no indirect\nsignal of\nany type is seen at LEP2? \". To answer this point leads to the derivation, to a\ncertain  conventionally chosen confidence limit, of (negative) bounds on the \n$Z'$ mass $M_{Z'}$. This has been done for a \nnumber of \"canonical\" $Z'$ models, and the resulting bounds (that are \ntypically in the TeV range) will be shown in section 2 together with\nthose for more general $Z'$ 's that might not be detected at an hadronic\ncollider. \n\nThe second question is: \"if a signal of indirect type were seen at LEP2,\n would it be possible to decide whether it may come from a $Z'$ or,\ntypically,  from a model with anomalous triple gauge couplings?\". The\nanswer to  this question , which is essentially provided from measurements\nin the\n final leptonic channel, is given in section 3. In section 4, the role \nof the final WW channel, that might a priori not be negligible for a \n$Z'$ of most general type, has also been investigated and shown to be \nirrelevant at LEP2. Section 5 is devoted to a short final discussion, that \nwill conclude our work. \n   \n\n\\section{Derivation of bounds.}\nTheoretical motivations for the existence of a $Z'$ have already been\ngiven by several authors, and excellent reviews are available \n\\cite{review}, where the most studied models are listed and \nsummarized. In the following, we will limit ourselves in defining as \n\"canonical\" cases those of a $Z'$ from  \neither $E_6$ \\cite{E6} or Left-Right symmetric models \\cite{LR} type. \n For sake of completeness we shall also \nconsider the often quoted case of a \"Sequential\" Standard Model $Z'$ \n\\cite{ALT}  \n(hereafter denoted as SSM), whose couplings to \nfermions are the same of those of the SM $Z^0$. For these models, \nderivations of bounds for $Z'$ parameters ($Z- Z'$ mixing angle and \n$M_{Z'}$) have been obtained from present data\\cite{present},\\cite{mix1}\nand calculations of discovery limits for $M_{Z'}$  performed for future \ncolliders\\cite{future},\\cite{mix2}, including also a discussion of \n$Z'$ model \nidentification. Therefore, the first question that we shall answer in our \nreport will be that of how do the LEP2 indirect mass limits compare \nto the direct ones achievable at Tevatron now and in a not too far future \n(i.e. assuming an integrated luminosity of $1 fb^{-1}$). In fact a motivation \nof our work was also that of  deriving limits on a $Z'$ whose couplings to \nfermions are completely  free, including cases that would not be detectable\nby any\npresent or future hadronic collider (for example for negligibly small $Z'$ \n quark couplings). For this purpose, the final leptonic channel at \nLEP2 provides all the necessary experimental information, and we shall \nconsequently begin our analysis with the detailed examination of the \nrole of this channel.\n\nTo fix our normalization and conventions, we shall write the expression\n of the invariant amplitude for the process $e^+e^- \\rightarrow l^+l^-$ \n(where l is a generic charged lepton) in the Born approximation and in\n the presence of a $Z'$. Denoting $q^2$ as the squared center of mass \nenergy this amplitude reads in our notations:\n\n\\begin{equation}\nA_{ll'}^{(0)}(q^2) = A_{ll'}^{(0) \\gamma, Z }(q^2) + A_{ll'}^{(0) Z'}\n(q^2)\n\\end{equation}\nwhere:\n\\begin{equation} \n A_{l{l'}}^{(0) \\gamma}(q^2) = i \\frac{e_0^2}{q^2} \n{\\bar v}_l \\gamma_{\\mu} u_l {\\bar u}_{l'} \\gamma^{\\mu} v_{l'}\n\\end{equation}\n\n\\begin{equation} \n A_{l{l'}}^{(0) Z}(q^2) = \\frac{i}{q^2-M^2_Z}  \\frac{g_0^2}{4c_0^2} \n{\\bar v}_l \\gamma_{\\mu} ( g_{Vl}^{(0)}-\\gamma^5 g_{Al}^{(0)} ) u_l \n{\\bar u}_{l'} \\gamma_{\\mu} (g_{V{l'}}^{(0)}-\\gamma^5 g_{A{l'}}^{(0)})\n v_{l'} \n\\end{equation}\nand(note the particular normalization):\n\\begin{equation} \n A_{ll'}^{(0) Z'}(q^2) = \\frac{i}{q^2-M^2_{Z'}}  \\frac{g_0^2}{4c_0^2} \n{\\bar v}_l \\gamma_{\\mu} ({g'}_{Vl}^{(0)}-\\gamma^5 {g'}_{Al}^{(0)}) u_l \n{\\bar u}_{l'} \\gamma_{\\mu} ({g'}_{Vl'}^{(0)}-\\gamma^5 {g'}_{A{l'}}^{(0)}) \n v_{l'} \n\\end{equation}\nIn the previous equations $e_0 = g_0 s_0$, $c_0^2 = 1-s_0^2$, \n$g_{Al}^{(0)}= I_{3L}=- \\frac{1}{2}$ and $g_{Vl}^{(0)}=-\n\\frac{1}{2}+2s_0^2$. Following the usual approach we shall treat the SM\nsector at one loop  and the $Z'$ contribution at tree level. The $Z'$ width\nwill be assumed  \"sufficiently\" small \nwith respect to $M_{Z'}$ to be safely neglected in the $Z'$ propagator.\n Moreover the $Z- Z'$ mixing angle will be ignored since the limits \nfor this quantity provided by LEP1 data from the final leptonic channel\n are enough constraining to rule out the possibility of any observable \neffect in the final leptonic channel at LEP2 (this has been shown in \na previous paper \\cite{mix1} for the \"canonical\" cases and for a \ngeneral $Z'$ case in a more recent preprint \\cite{mix2}).\nIf we stick ourselves to the final charged leptonic states, we must \ntherefore only deal with two \"effective\" parameters, that might be \nchosen as the adimensional quantities\n $g'_{Vl} \\sqrt{\\frac{q^2}{M^2_{Z'}-q^2}}$ and \n$g'_{Al} \\sqrt{\\frac{q^2}{M^2_{Z'}-q^2}}$ (this would be somehow \nreminiscent of notations that are common for models with AGC, with \n$M_{Z'}$ playing the role of the scale \nof new physics). In practice, for the specific purpose of the \nderivation of bounds, a convenient choice was that of the following \nrescaled couplings \\cite{rl1}:\n\\begin{equation}\nv_l^N = g'_{Vl} \\sqrt{\\frac{q^2}{M^2_{Z'}-q^2}}\n \\sqrt{\\frac{\\alpha}{16c_W^2s_W^4}}\n\\label{cs5} \n\\end{equation}\nand \n\\begin{equation}\na_l^N = -g'_{Al} \\sqrt{\\frac{q^2}{M^2_{Z'}-q^2}}\n\\sqrt{\\frac{\\alpha}{16c_W^2s_W^4}}\n\\label{cs6} \n\\end{equation}\nwith $\\alpha = \\frac{1}{137}$ and $s_W^2=1- c_W^2=0.231$.\n\nAs previously stressed our first task has been that of the derivation \nof constraints for the two previous rescaled couplings from the non \nobservation of effects in the leptonic channel. We considered as \nobservables at LEP2 the leptonic cross section \n${\\sigma}_l$ and the forward-backward asymmetry $A_{FB}^l$, obtained \nfrom measurements of $\\mu$ and  $\\tau$ final states, and also the final\n$\\tau$ polarization $A_\\tau$ (we have not yet included the electron channel \nbecause a full assessment of the corresponding experimental precision\n is more complicated at this stage). Three different energy-integrated \nluminosity configurations were considered, i.e. $ \\sqrt{q^2}=140$ GeV \nand $\\int L dt=5 pb^{-1}$, 175(500) and 192(300).\nTable 1 gives the SM predictions for the three leptonic quantities \ntogether with the expected experimental accuracies in the three cases. \n For each energy the first block of three lines contains convoluted \nquantities, whereas the second does not.\n\n\nA short technical discussion about the way we have calculated \n the effects of QED emission is now appropriate. In fact two main \napproaches exist that use either complete Feynmann diagrams \nevaluation to compute photonic emission from external legs \\cite{bb} or \nthe so called QED structure function formalism \\cite{sf}, based on \nthe analogy with QCD factorization and on the use of the Lipatov-Altarelli-\nParisi evolution equation \\cite{russ}. In the calculation of the \n limits on rescaled parameters performed in this section we have used \nthe code ZEFIT \\cite{ZEFIT} together with ZFITTER \\cite{ZFITTER}.\n These programs utilize the first approach \\cite{bb}.\nMore precisely ZFITTER has all SM corrections and all possibilities to \napply kinematical cuts.The code ZEFIT contains the additional $Z'$ \ncontributions including the full first order QED correction with soft \nphoton exponentiation. The first version of this combination, \napplied to LEP1 data,\nwhich  was restricted to definite $Z'$ models, has been adapted for the\nmodel independent analysis that we are now performing at LEP2. \n\nIn practice the \nlargest contribution is due to initial state radiation. The \ncorresponding expressions for the cross section and forward-backward \nasymmetry read:\n\\begin{equation}\n\\sigma_T = \\int^{\\Delta}_0 dk \\sigma_T^0(s') R_T^e(k)\n\\end{equation}\n\n\\begin{equation}\nA_{FB} = \\frac{1}{\\sigma_T} \\int^{\\Delta}_0 dk \\sigma_{FB}^0(s') \nR_{FB}^e(k)\n\\end{equation}\nThe reduced energy $s'$ reads $s'=q^2(1-k)$ and \n$\\Delta=\\frac{E_{\\gamma}}{E_{beam}}$. \nTo first order in $\\alpha$, improved by soft photon exponentiation, \nthe two functions  $R_T^e(k)$ and $R_{FB}^e(k)$ are given by the \nfollowing expressions:\n\\begin{equation}\nR_{T,FB}^e(k) = (1+S_e) \\beta_e k^{\\beta_e -1}+ H_{T,FB}^e (k)\n\\end{equation}\nwhere\n\\begin{equation}\n \\beta_e = 2 \\frac{\\alpha}{\\pi} e_e^2  \\ln( \\frac{q^2}{m_e^2}-1)\\ ,\n\\end{equation}\nthe soft radiaton part reads:\n\\begin{equation}\n S_e = \\frac{\\alpha}{\\pi} e_e^2 \\left( \\frac{\\pi^2}{3}-\\frac{1}{2}+\n\\frac{3}{2} \\ln( \\frac{q^2}{m_e^2}-1) \\right)\n\\end{equation}\nand the hard radiation parts:\n\\begin{equation}\n H_T(k) = \\frac{\\alpha}{\\pi} e_e^2 \\left\\{ \\frac{1+(1-k)^2}{k}\n \\ln( \\frac{q^2}{m_e^2}-1) \\right\\} -\\frac{\\beta_e}{k}\n\\end{equation}\n\n\\begin{equation}\n H_{FB}(k) = \\frac{\\alpha}{\\pi} e_e^2 \\left( \\frac{1+(1-k)^2}{k}\n\\frac{1-k}{(1-\\frac{k}{2})^2}\n ( \\ln \\frac{q^2}{m_e^2}-1-\\ln \\frac{1-k}{(1-\\frac{k}{2})^2}) \\right) \n-\\frac{\\beta_e}{k}\n\\end{equation}\nThe value of $\\Delta$ is chosen by requiring that the invariant mass \nof the final fermion pair $M_{ll}^2=(1-\\Delta) q^2$ is \"sufficiently\"\n greater than $M_Z$, to exclude the radiative return to the $Z$ peak.\n This has very important implications for searches of $Z'$ effects, \nsince it is well known that the radiative tail can enhance the SM cross \nsection by a factor $2-3$, then completely diluting the small $Z'$ \neffects, as fully discussed in a previous paper \\cite{djouadi} . \nResults shown in table 1 are obtained for an  invariant \nmass of the final fermion pair $M_{ll}$ greater than $120$ GeV. \n\nOne clearly sees from inspection of Table 1 that the most promising of\nthe three energy-luminosity combinations for what concerns the relative size\nof the error is $175$ GeV and $500 pb^{-1}$.\n\n>From now on  we shall therefore  \nconcentrate on this configuration and evaluate the bounds on $Z'$ \nrescaled parameters that would follow from the non observation of any \nvirtual effect. \nWith this purpose we have made full use of the code ZEFIT and chosen \n$\\Delta=.7$ ,although smaller values like for instance the one used \nin table 1 would lead to practically the same conclusions.\n \nTo obtain exclusion limits, we calculate the SM predictions of all \nobservables $O_i(SM)$ and compare them with the prediction \n $O_i(SM,v_i^N,a_i^N)$ from a theory including a $Z'$. In our fits we \nuse the errors $\\Delta O_i$ calculated using the same assumptions as \nin Table 1 and define: \n\n\\begin{equation}\n\\chi^2 =  \\sum_{O_i} \\left( \\frac{O_i(SM)-O_i(SM,Z')}{\\Delta O_i}\n \\right)^2.\n\\label{chi2}\n\\end{equation}\n${\\chi}^2 < \\chi^2_{min} + 5.99$ corresponds to $95\\%$ confidence \nlevel for one sided exclusion bounds for two parameters.\n\n\n\\newpage\n\n\n\\null\n\\vskip 3cm\n\n\n\\hskip -8mm\\begin{tabular}{|r|r|r|r|r|r|r|r|r|r|r|r|r|}\n\\hline\n&$E_{cm}$&Lumi&Acc&$\\sigma(pb)$&$\\Delta\\sigma_{stat}$&$\\Delta\n\\sigma_{syst}$&$\\Delta\\sigma$&Error&$A_{FB}$&$(\\Delta\nA_{FB})$&$A_{\\tau}$&$(\\Delta A_{\\tau})$\\\\\n\\hline\n$\\mu$&140.&5.&.90&6.43&1.20&.05&1.20&18.6\\%&.684&.136&&\\\\\n$\\ell$&140.&5.&.90&6.51&0.85&.04&0.85&13.1\\%&.684&.095&&\\\\\n$\\tau$&140.&5.&&&&&&&&&-.104&0.61\\\\\n\\hline\n$\\mu$&140.&5.&.90&6.98&1.25&.05&1.25&17.9\\%&.666&.133&&\\\\\n$\\ell$&140.&5.&.90&6.98&0.88&.04&0.88&12.6\\%&.666&.094&&\\\\\n$\\tau$&140.&5.&&&&&&&&&-.102&0.59\\\\\n\\hline\n$\\mu$&175.&500.&.90&4.13&0.10&.03&0.10&2.4\\%&.602&.019&&\\\\\n$\\ell$&175.&500.&.90&4.15&0.07&.03&0.07&1.7\\%&.602&.013&&\\\\\n$\\tau$&175.&500.&&&&&&&&&-.082&0.08\\\\\n\\hline\n$\\mu$&175.&500.&.90&4.01&0.09&.03&0.10&2.5\\%&.586&.01&&\\\\\n$\\ell$&175.&500.&.90&4.01&0.07&.02&0.07&1.8\\%&.586&.01&&\\\\\n$\\tau$&175.&500.&&&&&&&&&-.079&0.08\\\\\n\\hline\n$\\mu$&192.&300.&.90&3.47&0.11&.02&0.12&3.3\\%&.579&.02&&\\\\\n$\\ell$&192.&300.&.90&3.49&0.08&.02&0.08&2.4\\%&.579&.01&&\\\\\n$\\tau$&192.&300.&&&&&&&&&-.085&0.11\\\\\n\\hline\n$\\mu$&192.&300.&.90&3.28&0.11&.02&0.11&3.4\\%&.565&.02&&\\\\\n$\\ell$&192.&300.&.90&3.28&0.08&.02&0.08&2.5\\%&.565&.02&&\\\\\n$\\tau$&192.&300.&&&&&&&&&-.081&0.11\\\\\n\\hline\n\\end{tabular}\n\\begin{table}[hbtp]\n\\caption{SM predictions for leptonic observables including\nexperimental  accuracies.\nAs a simple simulation of the detector acceptance, \nwe impose that the angle between the outgoing leptons and the beam \naxis is larger than $20^{\\circ}$, leading to an acceptance of about $0.9$. \nThe first line gives the muon cross section and the forward-backward \nasymmetry and errors. The second line \ngives the averaged $\\mu$ and $\\tau$ cross section and asymmetry (and \nerrors) whereas the third line contains the $\\tau$ polarization(obtained \nby using only the $\\rho$ and  $\\pi$ channels and assuming an average \nsensitivity of $.5$). Concerning systematics we assumed $.5\\%$ relative \nerror for $\\mu$ and $\\tau$ selections and also for luminosity. For all \nasymmetries we did not consider any systematic error. \nAll quoted errors refer to a single LEP experiment. Taking into account \nthe type of systematic errors and the relative size of systematics vs \nstatistics, it is a good approximation to divide the error by 2 to \nestimate the combined error of the four experiments.}\n\\end{table}\n\n\n\\newpage\n\n\n\n\\begin{figure}\n\n\\hbox to\\hsize{\\epsfxsize=0.5\\hsize\n\\epsffile[0 0 340 340]{fi1lep2.ps}\n\\hfill\\epsfxsize=0.5\\hsize\n\\epsffile[0 0 340 340]{fi2lep2.ps}}\n\n\\noindent {\\bf Fig.~1} {\\eight The areas in the $(a^N_{\\ell},v^N_{\\ell})$\nplane excluded with\n$95\\%$ CL at LEP 2 by different observables i.e. $\\sigma_{\\ell}$\n(ellipse) and $A^l_{FB}$ (crossed lines). The remaining contours, that do\nnot improve the limits, would correspond to a measurement of $A_{\\tau}$\nwith an accuracy twice better than the realistic one quoted in Table~1.\\\\\n\\noindent{\\bf Fig.~2} The area in the $(a^N_l,v^N_l)$ plane excluded\nwith\n$95\\%$ confidence at LEP 2 by the combination of all the leptonic\nobservables.}\n\\end{figure}\n\n\n\n\n\n\n\n\n\n Figures 1 and 2 give our model independent constraints to\n the rescaled leptonic $Z'$ couplings including all radiative corrections\n, i.e. the QED radiation and the electroweak corrections, that have \na very small influence on the results. In figure 1 the constraint \nfrom each observable is shown separately. The combined\nexclusion region is depicted in the  next figure 2 and \na few comments on the previous figure are now appropriate. It \nrepresents in fact the most general type of constraints that can be \nderived on a $Z'$ from the absence of signals in the leptonic channel \nat LEP2, under the assumption that the $Z'$ couples to charged leptons \nin a universal way. In particular, from this figure one \nmight derive bounds on the parameters of  $Z'$ that were only coupled \nto leptons  and would therefore escape detection at any hadronic machine. \nFor such models, the limit on $M_{Z'}$ would be then derivable to a \nvery good(and conservative) approximation, for given $Z'$ couplings, by \nthe simple expression(derived from  eq.~(\\ref{cs5}) and eq.~(\\ref{cs6})):\n\\begin{equation}\n\\frac{M_{Z'}^2}{q^2}\\geq \\frac{1}{r^2}\n\\frac{{g'}_{Vl}^2+{g'}_{Al}^2}{\\frac{\\alpha}{16c_w^2s_w^4}}\n\\end{equation}\nwhere r is the distance from the origin in the ($v_l^N,a_l^N$) plane of \nthe intersection between the boundary region of figure 2 and the \nstraight line:  \n \\begin{equation}\n \\frac{g'_{Vl}}{g'_{Al}} = -  \\frac{v_l^N}{a_l^N} = k\n\\end{equation}\nwhere k is fixed by the considered model and r will vary between $.01$ and \n$.015$.\n(Numerically $\\frac{\\alpha}{16c_w^2s_w^4} \\sim 81$.)\n\n\n\n\n\n\\vskip 15mm\n\n\\hskip -10mm\\begin{tabular}{|r|r|r|r|r|r|r|r|r|r|r|r|r|}\n\\hline\n&$E_{cm}$&Lumi&Acc&$\\sigma(pb)$&$\\Delta\\sigma_{stat}$&$\\Delta\n\\sigma_{syst}$&$\\Delta\\sigma$&Error&$A_{FB}$&$(\\Delta\nA_{FB})$&$R$&$Error$\\\\\n\\hline\n$b$&140.&5.&0.25&10.46&2.89&.31&2.91&27.8\\%&.499&.379&&\\\\\n$h$&140.&5.&1.00&59.03&3.44&.66&3.50&5.9\\%&&&&\\\\\n$R_h$&140.&5.&&&&&&&&&9.181&14.3\\%\\\\\n$R_b$&140.&5.&&&&&&&&&0.177&28.4\\%\\\\\n\\hline\n$b$&140.&5.&0.25&10.62&2.92&.32&2.93&27.6\\%&.509&.373&&\\\\\n$h$&140.&5.&1.00&60.49&3.48&.68&3.54&5.9\\%&&&&\\\\\n$R_h$&140.&5.&&&&&&&&&8.666&13.9\\%\\\\\n$R_b$&140.&5.&&&&&&&&&0.176&28.2\\%\\\\\n\\hline\n$b$&175.&500.&0.25&5.15&0.20&.15&.26&5.0\\%&.543&.052&&\\\\\n$h$&175.&500.&1.00&31.24&0.25&.35&.43&1.4\\%&&&&\\\\\n$R_h$&175.&500.&&&&&&&&&7.572&2.1\\%\\\\\n$R_b$&175.&500.&&&&&&&&&0.165&5.1\\%\\\\\n\\hline\n$b$&175.&500.&.25&4.69&0.19&.14&0.24&5.1\\%&.562&.054&&\\\\\n$h$&175.&500.&1.00&28.90&0.24&.32&0.40&1.4\\%&&&&\\\\\n$R_h$&175.&500.&&&&&&&&&7.215&2.1\\%\\\\\n$R_b$&175.&500.&&&&&&&&&0.162&5.3\\%\\\\\n\\hline\n$b$&192.&300.&0.25&4.08&0.23&.12&.26&6.5\\%&.554&.079&&\\\\\n$h$&192.&300.&1.00&25.22&0.29&.28&.40&1.6\\%&&&&\\\\\n$R_h$&192.&300.&&&&&&&&&7.265&2.8\\%\\\\\n$R_b$&192.&300.&&&&&&&&&0.162&6.6\\%\\\\\n\\hline\n$b$&192.&300.&.25&3.62&0.22&.11&0.25&6.8\\%&.577&.078&&\\\\\n$h$&192.&300.&1.00&22.79&0.28&.25&0.38&1.6\\%&&&&\\\\\n$R_h$&192.&300.&&&&&&&&&6.942&2.9\\%\\\\\n$R_b$&192.&300.&&&&&&&&&0.159&6.9\\%\\\\\n\\hline\n\\end{tabular}\n\\begin{table}[hbtp]\n\\caption{SM predictions for hadronic observables including\nexperimental  accuracies.\nThe first line gives the b quark cross section and forward-backward \nasymmetry and the corresponding experimental errors. The second line \ngives the total hadronic cross section (and errors) whereas the \nthird line contains the ratio $R_h=\\frac{\\sigma_h}{\\sigma_l}$ \nand the fourth line the ratio $R_b=\\frac{\\sigma_b}{\\sigma_h}$\n. Concerning systematics we assumed $1\\%$ relative \nerror for hadron selection and $3\\%$ relative error for\n b quark selection.We did not consider\n any systematic error for all asymmetries. Concerning the b quark cross\nsection we assume a tagging efficiency of $25\\%$ (vertex tag) and \n $10\\%$ (lepton tag) for the asymmetry. The first block of \nnumbers (upper four lines) refer to convoluted quantities where proper\ncuts have been applied, whereas the lower four lines contain deconvoluted \nquantities.}\n\\end{table}\n\n\n\\newpage\n\n\nIn a less special situation, the $Z'$ couplings to quarks will \nnot be vanishing. In these cases, to derive meaningful bounds, the \nfull information coming from the final hadronic channel should be also  \nexploited. At LEP2, we assumed the availability of three different \nmeasurements, i.e. those of the total hadronic cross section \n${\\sigma}_h$ and those of the cross section and forward-backward \nasymmetry for $b \\bar b$ production,${\\sigma}_b$ and $A_{FB}^b$. In \ntable 2 we give the related expected experimental accuracies, for the \nthree energy-luminosity configurations already investigated for the \nfinal leptonic channel in Table 1, and under the same general \nassumptions listed in the discussion preceding the presentation of \nthis table.  \n\n>From the combination of the leptonic and hadronic channels, a fully \ngeneral investigation of the six rescaled $Z'$ couplings (there would \nbe four extra rescaled couplings for \"up\" and \"down\" type quarks) might be,\n in principle, carried through if at least four hadronic independent \nobservables were measured at LEP2. This could be obtained if one more \nhadronic asymmetry were measured. In practice, though, the utility of \nsuch an approach is somehow obscured by practical considerations( like \nthe realistic achievable experimental accuracy). For these reasons, we \nhave therefore decided to make full use of the hadronic observables to \nderive limits on $M_{Z'}$ only for a number of \"canonical\" models \nwhere the $Z'$ couplings to fermions are constrained.\nAs relevant examples to be investigated, we shall discuss $E_6$ models \n\\cite{E6} and Left-Right symmetric models \\cite{LR}, for which the \n$Z'$ current can be decomposed as:\n \\begin{equation}\n J_{Z'}^{\\mu} = J_{\\chi}^{\\mu} \\cos \\beta +  J_{\\psi}^{\\mu} \\sin \\beta \n\\end{equation}\n or\n \\begin{equation}\n J_{Z'}^{\\mu} = J_{LR}^{\\mu} \\alpha_{LR} - J_{B-L}^{\\mu} \\frac{1}\n{2 \\alpha_{LR}} \n\\end{equation}\n\n\n\\begin{table}[hbtp]\n\\begin{center}\n\nTable 3a\n\n\\medskip\n\n\\begin{tabular}{|c|c|c|}\n\\hline\nf&$\\frac{{g'}_{Vf}}{s_w}$&$\\frac{{g'}_{Af}}{s_w}$\\\\\n\\hline\n l &$\\frac{2}{\\sqrt 6} \\cos \\beta$&$\\frac{1}{\\sqrt 6} \\cos \\beta + \n\\frac{\\sqrt 10}{6} \\sin \\beta $\\\\\n\\hline\n u &0&$-\\frac{1}{\\sqrt 6} \\cos \\beta + \\frac{\\sqrt 10}{6} \\sin \\beta $\\\\\n\\hline\n d &$-\\frac{2}{\\sqrt 6} \\cos \\beta$&$\\frac{1}{\\sqrt 6} \\cos \\beta + \n\\frac{\\sqrt 10}{6} \\sin \\beta $\\\\\n\\hline\n\\end{tabular}\n\n\\vskip 4truemm\n\nTable 3b\n\n\\medskip\n\n\\begin{tabular}{|c|c|c|}\n\\hline\nf&$\\frac{{g'}_{Vf}}{s_w}$&$\\frac{{g'}_{Af}}{s_w}$\\\\\n\\hline\n l &$\\frac{1}{\\alpha_{LR}}- \\frac{\\alpha_{LR}}{2}$&$\\frac{\\alpha_{LR}}{2}$\\\\ \n\\hline\n u &$-\\frac{1}{3 \\alpha_{LR}} +\n\\frac{\\alpha_{LR}}{2}$&$-\\frac{\\alpha_{LR}}{2}$\\\\ \n\\hline\n d &$-\\frac{1}{3 \\alpha_{LR}} - \\frac{\\alpha_{LR}}{2}$&$\\frac{\\alpha_{LR}}{2}$\\\\\n\\hline\n\\end{tabular}\n\n\\caption{Couplings of ordinary fermions (f=l,u,d) to $Z'$ boson \na) from $E_6$ models as a function of the parameter $\\cos \\beta$\nb) from Left-Right models as a function of the parameter $\\alpha_{LR}$.}\n\\end{center}\n\\end{table}\n\n\n\n\\vspace{0.3cm}\n\n\n\nIn table 3 we have given the $Z'$ couplings to l, u and d fermions for\nthe two models. Some specific relevant cases in the $E_6$ sector are \nthe so called $\\chi$ model (corresponding to $\\cos \\beta =1$), $\\psi$ model\n($\\cos \\beta =0$) and  $\\eta$ model\n($\\arctan \\beta = - \\sqrt {\\frac{5}{3}}$). Special \ncases for Left-Right symmetric models are obtained for\n $\\alpha_{LR} = \\sqrt {\\frac{2}{3}}$ (this case reproduces the $\\chi$ \n model) and  $\\alpha_{LR} = \\sqrt 2$ (the so called manifestly L-R \nsymmetric model). Finally we also chose the $Z'$ of the Sequential \nStandard Model(which has the same fermionic couplings as those of the \nSM Z) as an additional benchmark.\n\nTable 4 shows the  CL bounds on  $M_{Z'}$ obtainable from the \nnon observation of any effect at LEP2 in the configuration: $175$ GeV, \n $500 pb^{-1}$. In fact one can easily show that for virtual \n$Z'$ searches this configuration is the best of the three that \nwe have considered for LEP2, since the simple scaling law for the achievable\n limit ${(M_{Z'})}_{max} \\sim {\\left( q^2 \\int L \\right)}^{\\frac{1}{4}}$ \n applies. The different lines show the influence of the \nhadronic observables. As one sees, this is indeed relevant for the \nSSM $Z'$. In the other cases it improves the bounds derived from purely\n leptonic observables by an amount of less than (typically) a relative \n$10\\%$.\n\n\\begin{table}[hbtp]\n\\begin{center}\n\\begin{tabular}{|l|ll|ll|ll|ll|l|}\n\\hline\n&$\\chi$&&$\\psi$&&$\\eta$&&LR&&SSM\\\\\n\\hline\n$\\sigma_{\\ell},A^{\\ell}_{FB}$&870&&640&&525&&838&&1238\\\\\n\\hline\n$\\sigma_{\\ell},A^{\\ell}_{FB},\\sigma_{had}$&900&&642&&550&&854&&1530\\\\\n\\hline\n$\\sigma_{\\ell},A^{\\ell}_{FB},R_b,A^{b}_{FB},\\sigma_{had}$&930&&666&&560&&880\n&&1580\\\\\n\\hline\n\\end{tabular}\n\n\\caption{Maximal $Z'$ masses $M_Z'$ excluded by leptonic and hadronic\nobservables.\n${\\chi}^2 < \\chi^2_{min} + 2.7$ ($95\\%$ CL, one sided limits).}\n\\end{center}\n\\end{table}\n\n\n\nThe more general analysis of the two models of extra gauge, that \ncorresponds to values of $\\cos \\beta$ ranging from $-1$ to $+1$ \n(positive  $\\sin \\beta$) and   $\\alpha_{LR}$ ranging from \n$\\sqrt {\\frac{2}{3}}$ to $\\sqrt 2$, has been summarized in figures 3 \nand 4 (full lines). One sees that the best $ M_{Z'}$ limits correspond \nto models where  $\\cos \\beta \\sim1$ and where $\\alpha_{LR}$ is at the \nboundary of its allowed interval, for which the bounds derivable at \nLEP2 would be about 1 TeV.\n\n\n\n\\newpage\n\n\\begin{figure}\n\n\\hbox to\\hsize{\\epsfxsize=0.5\\hsize\n\\epsffile[8.5 22.0 541.3 747.1]{fi3lep2.ps}\n\\hfill\\epsfxsize=0.5\\hsize\n\\epsffile[8.5 22.0 541.3 747.1]{fi4lep2.ps}}\n\n\n\\noindent {\\eight Maximal $Z'$ masses excluded at LEP2 (full\ncurve) and at  Tevatron with a luminosity of $1 fb^{-1}$ (dashed curve) or $20\npb^{-1}$ (dotted curve) for $E_6$ models (Fig.~3) and for Left-Right models\n(Fig.~4).}\n\n\\end{figure}\n\n\n\nThe values that we have derived should be compared with those \nalready available and with those reachable \n in a not too far future at Tevatron. To fix the scales for the \ncomparison, we have considered the limits that would correspond to an \nenergy of $1.8$ TeV with an integrated luminosity of  $1 fb^{-1}$ and \ndrawn on the same figures 3 and 4 (dashed lines) the expected Tevatron \nlimits, that would \"compete\" with the LEP2 results (the present Tevatron \nlimits for $20 pb^{-1}$  correspond to the dotted curves). The  \nlimits correspond to $95 \\%$ CL bounds on $M_{Z'}$ based on 10 events \nin the $e^+e^- + \\mu^+ \\mu^-$ channel, assuming that $Z'$ can only \ndecay in the three conventional fermion families. The values  that we \nplot are in agreement with those quoted in a recent report  \\cite{CVETIC}. \nOne sees from figures 3 and 4 that for the $E_6$ models the LEP2 \nlimits are in a sense complementary to those of Tevatron in the \nfuture configuration, providing better or worse indications depending \non which range is chosen for $\\cos \\beta$. LEP2 is better if \n$\\cos \\beta$ lies in the vicinity of $-1$ and $\\cos \\beta \\geq 0.4$. \nOn the contrary, for Left-Right symmetric models LEP2 appears to do \nmuch better, except for $\\alpha_{LR}$ ranging between $1$ and $1.2$ \nwhere Tevatron could provide limits a bit higher;\nconcerning the SSM $Z'$, LEP2 does systematically better since \nit can reach $1.5$ TeV whereas the future Tevatron limit \nis around $900$ GeV. Note that, should other exotic or supersymmetric\nchannel be open for $Z'$ decay, the Tevatron limits might decrease by \nas much as $30 \\%$, depending on the considered model \\cite{CVETIC}. \nWe conclude therefore that, until the Tevatron luminosity will reach \nvalues around $10 fb^{-1}$, the canonical LEP2 bounds will be, least \nto say, strongly competitive.\n \nOur determination of bounds is at this point finished for what concerns \nthe final fermionic channel. In the next section we shall try to derive \nsome model-independent criterion to identify $Z'$ signals at LEP2.\n  \n\\section{Search for signals: the leptonic channel.}\nIn this section, we shall assume that some virtual signal has been seen \nat LEP2 in the leptonic channel. In this case, we shall show that it would \nbe possible to conclude whether this signal is due to a $Z'$ or not.\n\n This can be easily understood if one \ncompares the $Z'$ effect to a description that includes the SM \neffects at one loop, and we shall briefly summarize the main points.\nFor what concerns the treatment of the SM sector, a prescription has \nbeen very recently given \\cite{CV2}, that corresponds to a \n\"Z-peak substracted\" representation of four fermion processes, in \nwhich a modified Born approximation and \"substracted\" one loop \ncorrections are used. These corrections, that are \"generalized\" \nself-energies, i.e. gauge-invariant combinations of self-energies, \nvertices and boxes, have been called in \\cite{CV2} (to which we refer \nfor notations and conventions) \n$\\widetilde {\\Delta} \\alpha (q^2)$, $R(q^2)$ and $V(q^2)$ respectively.\n As shown in ref \\cite{CV2}, \nthey turn out to be particularly useful whenever effects of new physics \nmust be calculated. In particular, the effect of a general $Z'$ would  \nappear in this approach as a particular modification of purely \"box\" \ntype to the SM values of $\\widetilde {\\Delta} \\alpha (q^2)$, $R(q^2)$ \nand $V(q^2)$ given by the following prescriptions:\n\\begin{equation}\n{\\widetilde {\\Delta}}{\\alpha} ^{(Z')} (q^2) = - \\frac{q^2}{M^2_{Z'}-q^2}\n\\frac{1}{4c_1^2s_1^2} g^2_{Vl} {(\\xi_{Vl} - \\xi_{Al})}^2\n\\label{rese1} \n\\end{equation}\n\n\\begin{equation}\nR^{(Z')}(q^2) = (\\frac{q^2-M^2_Z}{M^2_{Z'}-q^2}) \\xi_{Al}^2\n\\label{rese2}\n\\end{equation}\n\n\\begin{equation}\nV^{(Z')}(q^2) = -(\\frac{q^2-M^2_Z}{M^2_{Z'}-q^2}) \n\\frac{g_{Vl}}{2c_1s_1} \\xi_{Al} (\\xi_{Vl} - \\xi_{Al})\n\\label{rese3}\n\\end{equation}\nwhere we have used the definitions:\n\\begin{equation}\n \\xi_{Vl} = \\frac{g'_{Vl}}{g_{Vl}}\n\\label{res3}\n\\end{equation}\n\n\\begin{equation}\n \\xi_{Al} = \\frac{g'_{Al}}{g_{Al}}\n\\label{res4}\n\\end{equation}\n\nwith $g_{Vl}= \\frac{1}{2} (1-4s_1^2)$;$g_{Al}= -\\frac{1}{2}$ and \n$c_1^2s_1^2 = \\frac{\\pi \\alpha (0)}{\\sqrt 2 G_{\\mu} M^2_Z}$. \nTo understand the philosophy of our approach it is convenient to write \nthe expressions at one-loop of the three independent leptonic \nobservables that will be measured at LEP2, i.e. the leptonic cross section, \nthe forward-backward asymmetry and the final $\\tau$ polarization. \nLeaving aside specific QED corrections extensively discussed in the previous\nsection, these  expressions read:\n\n\\begin{eqnarray}  \\sigma_l(q^2)=&&\\sigma^{Born}_l(q^2)\\bigm\\{1+{2\\over\n\\kappa^2(q^2-M^2_Z)^2+q^4}[\\kappa^2(q^2-M^2_Z)^2\n\\tilde{\\Delta}\\alpha(q^2)\\nonumber\\\\\n&&-q^4(R(q^2)+{1\\over2}V(q^2))]\\bigm\\}\n\\label{rese4} \\end{eqnarray}\n\n\n\n\\begin{eqnarray}  A^l_{FB}(q^2)=&&A^{l,Born}_{FB}(q^2)\\bigm\\{1+\n{q^4-\\kappa^2(q^2-M^2_Z)^2\n\\over\\kappa^2(q^2-M^2_Z)^2+q^4}[\n\\tilde{\\Delta}\\alpha(q^2)+R(q^2)]\\nonumber\\\\\n&&+{q^4\\over\\kappa^2(q^2-M^2_Z)^2+q^4}V(q^2)]\\bigm\\}\n\\label{rese5} \\end{eqnarray}\n\n\n\\begin{eqnarray} A_{\\tau}(q^2)=&&A^{Born}_{\\tau}(q^2)\n\\bigm\\{ 1+[{\\kappa(q^2-M^2_Z)\n\\over\\kappa(q^2-M^2_Z)+q^2}-{2\\kappa^2(q^2-M^2_Z)^2\\over\n\\kappa^2(q^2-M^2_Z)^2+q^4}]\n[\\tilde{\\Delta}\\alpha(q^2)\\nonumber \\\\\n&&+R(q^2)]\n-{4c_1s_1\\over v_1}V(q^2) \\bigm\\}\n\\label{rese6} \\end{eqnarray}\n\n\n\nwhere $\\kappa$ is a numerical constant \n($\\kappa^2 =  {(\\frac{\\alpha}{3 \\Gamma_l M_Z})}^2 \\simeq 7$) and we \nrefer to \\cite{CV2} for a more detailed derivation of the previous \nformulae.\n \nA comparison of  eqs.~(\\ref{rese4}-\\ref{rese6}) with eqs.~(\\ref{rese1}\n-\\ref{rese3}) shows that, in the three leptonic observables, only two \neffective parameters, that could be taken for instance as \n$\\xi_{Vl} \\frac{M_Z} {\\sqrt{M^2_{Z'}-q^2}}$ and \n$(\\xi_{Vl} - \\xi_{Al})\\frac{M_Z} {\\sqrt{M^2_{Z'}-q^2}}$(to have \ndimensionless quantities, other similar definitions would do equally \n well), enter. This leads to the conclusion that it must be possible \nto find a relationship between the relative $Z'$ shifts \n$\\frac{\\delta \\sigma_l}{\\sigma_l}$, $\\frac{\\delta A^l_{FB}}{A^l_{FB}}$ \n and  $\\frac{\\delta A_{\\tau}}{A_{\\tau}}$ (defining, for each observable\n $O_i =O_i^{SM} + \\delta O_i^{Z'}$) that is completely independent of \nthe values of these effective parameters. This will correspond to a \nregion in the 3-d space of the previous shifts that will be fully \ncharacteristic of a model with the most general type of $Z'$ that we \nhave considered. We shall call this region \"Z' reservation\".\n\nTo draw this reservation would be rather easy if one relied on a \ncalculation in which the $Z'$ effects are treated in first \napproximation, i.e. only retaining the leading effects, and not taking \ninto account the QED radiation. After a rather straightforward \ncalculation one would then be led to the following approximate \nexpressions that we only give for indicative purposes:\n\\begin{equation}\n({\\frac{\\delta A_{\\tau}}{A_{\\tau}}})^2 \\simeq f_1 f_3\n\\frac{8 c_1^2 s_1^2}{v_1^2} \\frac{\\delta \\sigma_l}{\\sigma_l} \n\\left( \\frac{\\delta A^l_{FB}}{A^l_{FB}} +\n\\frac{1}{2} f_2 \\frac{\\delta \\sigma_l}{\\sigma_l} \\right)\n\\label{rese7}  \n\\end{equation}\n\nwhere the $f_i$'s are numerical constants, whose expressions can be found\nin~\\cite{CV2}.\n\nEq.~(\\ref{rese7}) is an approximate one. A more realistic description \ncan only be obtained if the potentially dangerous QED effects are \nfully accounted for.In order to accomplish this task, the QED structure \nfunction formalism\\cite{sf} has been employed as a reliable tool for \nthe treatment of large undetected initial-state photonic radiation. \nUsing the structure function method amounts to writing, in analogy with \nQCD factorization, the QED corrected cross section as a convolution \nof the form:\n\n\\begin{equation}\n\\sigma (q^2) = \\int dx_1 dx_2 D(x_1,q^2) D(x_2,q^2) \n\\sigma_0 ((1-x_1x_2)q^2) (1+ {\\delta}_{fs}) \\Theta (cuts)\n\\label{rese8}\n\\end{equation}\nwhere $\\sigma_0$ is the lowest order kernel cross section, taken at \nthe energy scale reduced by photon emission,$ D(x,q^2)$ is the electron\n(positron) structure function, ${\\delta}_{fs}$ is the correction factor \ntaking care of QED final-state radiation and $\\Theta (cuts)$ represents \nthe rejection algorithm to implement possible experimental cuts. Its \nexpression, obtained by solving the Lipatov-Altarelli-Parisi evolution equation\n in the non-singlet approximation, can be found in \\cite{russ} together \nwith a complete discussion of the method. \nIn order to proceed with the numerical simulation of the $Z'$ effects\nunder realistic experimental conditions, the master formula \neq.~(\\ref{rese8}) has  been implemented in a Monte Carlo event \ngenerator which has been first checked against currently used LEP1 \nsoftware \\cite{nicrosini}, found to be in very good agreement and then \nused to produce our numerical results. The $Z'$ contribution has been \nincluded in the kernel cross section $\\sigma_0$ computing now the \ns-channel Feynman diagrams associated to the production of a leptonic \npair in $e^+e^-$ annihilation mediated by the exchange of a photon, a SM \nZ and an additional $Z'$ boson. In the calculation, which has been \ncarried out within the helicity amplitude formalism for massless fermions and \n with the help of the program for algebraic manipulations SCHOONSCHIP \n\\cite{veltman}, the $Z'$ propagator has been included in the \nzero-width approximation. Moreover, the bulk of non QED corrections has \nbeen included in the form of the Improved Born Approximation, choosing \n $\\bar \\alpha (s), M_Z, G_F$ and $\\Gamma_Z$ as input parameters. The \nvalues used for the numerical simulation are \\cite{SMLEP}: \n$M_Z= 91.1887$ GeV, $\\Gamma_Z =2.4979$ GeV. The center of mass energy \nhas been fixed to  $\\sqrt q^2=175$ GeV and the cut $x_1x_2 > 0.35$ (that \nwould correspond to the choice $\\Delta = 0.65$ in the notations of the \nprevious section) has been imposed in order to remove the events due to \nZ radiative return and hence disentangle the interesting virtual $Z'$ \neffects. These have been investigated allowing the previously defined \nratios $\\xi_{Vl}$ and $\\xi_{Al}$ to vary within the ranges \n$-2 \\le  \\xi_{Al} \\le 2$ and $-10 \\le  \\xi_{Vl} \\le 10$. Higher values \nmight be also taken into account; the reason why we chose the previous \nranges was that, to our knowledge, they already include all the most \nknown models. \n\nThe results of our calculation are shown in figure 5 \\cite{MNPRV}. \nOne sees that \nthe characteristic features of a general $Z'$ effect are the fact that \nthe shifts in the leptonic cross section are essentially negative. This \ncan be qualitatively  predicted from the Born approximation formula   \neq.~(\\ref{rese4}) because the dominant photon exchange contribution to \n$\\sigma_l$ is clearly negative since \n${\\widetilde {\\Delta}}^{(Z')} \\alpha (q^2)$ is negative. Away from \n$\\xi_{Al} \\simeq 0$ the forward-backward asymmetry will be also negative, \nas easily inferred from eq.~(\\ref{rese5}).\n\n\n\n\\begin{figure}\n\\vskip -3.5cm\n\n\\hbox to\\hsize{\\epsfxsize=0.50\\hsize\n\\epsffile[0 0 612 792]{fi5lep2.ps}\n\\hfill\\epsfxsize=0.50\\hsize\n\\epsffile[0 0 612 792]{fi6lep2.ps}}\n\n\\vskip -1cm\n\n\\noindent {\\bf Fig.~5} {\\eight ${{\\delta A^\\tau} \\over {A^\\tau}} $ versus \n${{\\delta \\sigma^\\mu} \\over {\\sigma^\\mu}} $  and \n${{\\delta A^\\mu_{FB}} \\over {A^\\mu_{FB}}} $. \nThe central ''dead'' area\nwhere a signal would not be distinguishable corresponds to an assumed\n(relative) experimental error of 1.5\\% for $\\sigma_\\mu$ and to \n1\\% (absolute)\nerrors on the two asymmetries. The region that remains outside \nthe dead area\nrepresents the $Z'$ reservation at LEP2, to which the effect of \nthe most\ngeneral $Z'$ must belong.}\\\\\n\\noindent {\\bf Fig.~6} {\\eight The same as Fig.~5, comparing the realistic\nresults  obtained via\nMonte Carlo simulation with the approximate ones according to \nBorn approximation.}\n\n\\vskip -2cm\n{\n\\def\\epsfsize#1#2{0.41#1}\n\n\\hskip 1,8cm\\epsfbox{fi7lep2.ps}\n}\n\n\\vskip -2cm\n\n\\noindent {\\bf Fig.~7} {\\eight The region \ncorresponding to Anomalous Gauge Couplings according \nto a Born approximation.}\n\n\n\n\\end{figure}\n\n\n\\newpage\n\n\nOne might be interested in knowing how different the realistic figure \n5 is from the approximate Born one, corresponding to the simplest \nversion given in eq.~(\\ref{rese7}). This can be seen in figure 6 where \nwe have drawn the allowed regions, the points corresponding to the \nrealistic situation, already shown in figure 5. The region inside the \nparallelepiped , where a signal would not be detectable, corresponds to \nan assumed relative experimental error of $1.7\\%$ for $\\sigma_l$ and to \nan absolute error of $1\\%$ for $A^l_{FB}$. For the $\\tau$ asymmetry an \nabsolute error of $2\\%$ has been assumed, that is extremely optimistic. \nThe domain that remain \noutside this area represents the $Z'$ reservation at LEP2, to which the \neffect of the most general $Z'$ must belong. One sees that the simplest \nBorn calculation is, qualitatively, a reasonable approximation to a \nrealistic estimate, which could be very useful if one first wanted to \nlook for sizeable effects. \n\nThe next relevant question that should be now answered is whether the \ncorrespondence between $Z'$ and reservation is of the one to one type, \nwhich would lead to a unique identification of the effect. We have \ntried to answer this question for one specific and relevant case, that \nof virtual effects produced by anomalous gauge couplings. In particular,\nwe have considered the case of the most  general dimension 6 $SU(2) \n\\otimes U(1)$\ninvariant effective  lagrangian recently proposed \\cite{Hagi}. This has\nbeen fully \ndiscussed in a separate paper \\cite{cv3}, where the previously \nmentioned \"Z-peak substracted\" approach has been used. The resulting \nAGC reservation in the ($\\sigma_l$, $A^l_{FB}$, $A_{\\tau}$) has been \ncalculated for simplicity in the Born approximation, as suggested by \nthe previous remarks. This AGC area is plotted in figure 7. As one sees, the \ntwo domains do not overlap in the meaningful region. Although we cannot \nprove this\nproperty in general, we can at least conclude that, should  a clear virtual\neffect\nshow up at LEP2, it would be possible to decide  unambiguously to which\namong two\nwell known proposed models it does  belong.\n  \nThe results that we have shown so far have been obtained by exploiting the \ninformation provided by the final fermionic channel. We shall devote the next\nsection 4 to a brief  discussion of the WW channel at LEP2.\n\n\n\n\n\n\n\n\n\n\\section{Search for effects in the WW channel.}\nThe virtual effects of a $Z'$ in W pair production from $e^+e^-$ \nannihilation can be described, at tree level, by adding to the photon, \n SM Z and neutrino exchanges the diagram with an additional $Z'$ boson  \nexchange. The overall effect in the scattering amplitude reads:\n\n\\begin{equation}\nA_{lW}^{(0)}(q^2) = A_{lW}^{(0) (\\gamma, Z,\\nu)}(q^2) + A_{lW}^{(0) Z'}\n(q^2)\n\\label{WW1}\n\\end{equation}\nwhere we assume universal couplings.\nSeparate expressions can be easily derived for eq.~(\\ref{WW1}). We \nshall only give here the relevant $Z'$ contribution: \n\\begin{equation} \n A_{lW}^{(0) Z'}(q^2) = \\frac{i}{q^2-M^2_{Z'}}  \\frac{g_0}{2c_0} \n{\\bar v}_l \\gamma_{\\mu} ({g'}_{Vl}^{(0)}-\\gamma^5 {g'}_{Al}^{(0)}) u_l\ne_0 g_{ZWW} P^{\\alpha \\beta \\mu} \\epsilon^{\\star}_{\\alpha}(p_1)\n\\epsilon^{\\star}_{\\beta}(p_2)\n\\label{WW2}\n\\end{equation}\nwhere:\n\\begin{equation} \n P_{\\alpha \\beta \\mu} = g_{\\mu \\beta}(p_1 + 2 p_2)_{\\alpha} + \n g_{\\beta \\alpha}(p_1 - p_2)_{\\mu} - g_{\\mu \\alpha}(2p_1 + p_2)_{\\beta} \n\\end{equation} \nand $p_{1,2}$ are the four momenta of the outgoing W bosons. In the \nexpression eq.~(\\ref{WW2}) we have assumed that the $Z'WW$ vertex has \nthe usual Yang-Mills form. We do not consider here the possibility of \nanomalous magnetic or quadrupole type of couplings. \n An analysis with anomalous $ZWW$ and $Z'WW$ couplings is possible \nalong the lines of \\cite{CV4} but is beyond the scope of this report. \nOur analysis will be nevertheless rather general as the trilinear \n$Z'WW$ coupling will be treated as a free parameter, not necessarily \nproportional to the $Z-Z'$ mixing angle as for example it would appear \nin a \"conventional\" $E_6$ picture.\n\nFor the purposes of this working group, it will be particularly \nconvenient to describe the virtual $Z'$ effect as an \"effective\" \nmodification of $Z$ and $\\gamma$ couplings to fermions and W pairs. As \none can easily derive, this corresponds to the use of the following modified \ntrilinear couplings that fully \ndescribe the effect in the final  process  $e^+e^- \\rightarrow W^+W^-$:\n\\begin{equation}\ng_{\\gamma WW}^{\\star} = g_{\\gamma WW} +  g_{Z' WW}\n\\frac{q^2}{M^2_{Z'}-q^2} g_{Vl} (\\xi_{Vl} - \\xi_{Al})\n\\label{WW3}\n\\end{equation} \n\n\\begin{equation}\ng_{ZWW}^{\\star} = g_{ZWW} - g_{Z' WW} \n\\frac{q^2- M^2_Z}{M^2_{Z'}-q^2} \\xi_{Al}\n\\label{WW4}\n\\end{equation} \n\nIn the previous equations, the same definitions as in eq.~(\\ref{res3}) \nand in eq.~(\\ref{res4}) have been used.  In the following we shall use \nthe results obtained on $\\xi_{Vl}$ and $\\xi_{Al}$ in the previous section. \nOur normalisation for  trilinear couplings is such that: \n$ g_{\\gamma WW} = 1 $ and\n$g_{ZWW} = \\frac{c_0}{s_0}$.\n \nAdopting the notations that are available in the recent literature \n\\cite{KNEUR}, we find for the $Z'$ effect:\n\n\\begin{equation}\n {\\delta}_{\\gamma}^{(Z')} = g_{\\gamma WW}^{\\star} - 1 =\n   g_{Z' WW}\n\\frac{q^2}{M^2_{Z'}-q^2} g_{Vl} (\\xi_{Vl} - \\xi_{Al})\n\\label{WW5}\n\\end{equation} \n\n\n\\begin{equation}\n {\\delta}_Z^{(Z')} = g_{ZWW}^{\\star} - \\cot {\\Theta}_W =\n - g_{Z' WW} \n\\frac{q^2- M^2_Z}{M^2_{Z'}-q^2} \\xi_{Al}\n\\label{WW6}\n\\end{equation} \n\n>From eq.~(\\ref{WW5}) and eq.~(\\ref{WW6}) one can derive the following \nconstraint:\n\n\\begin{equation}\n \\frac{{\\delta}_{\\gamma}^{(Z')}}{{\\delta}_Z^{(Z')}} = \n \\frac{- q^2}{q^2-M^2_Z} \n\\left( \\frac{ \\xi_{Vl} - \\xi_{Al}}{\\xi_{Al}} \\right) g_{Vl}  \n\\label{WW7}\n\\end{equation} \nWe then notice that the virtual effect of a general $Z'$ in the WW \nchannel is, at first sight, quite similar to that of a possible model \nwith anomalous gauge couplings, that would also produce shifts \n${\\delta}_{\\gamma}$, ${\\delta}_Z$ both in the $\\gamma WW$ and in the \n$ZWW$ couplings. But the $Z'$ shifts satisfy in fact the constraint \ngiven in eq.~(\\ref{WW7}), that corresponds to a certain line in the \n(${\\delta}_{\\gamma}, {\\delta}_Z$)  plane whose angular coefficient is \nfixed by the model i.e. by the values of  $\\xi_{Al}$ and  \n$(\\xi_{Vl} - \\xi_{Al})$.\n \nWe shall now introduce the following ansatz concerning the theoretical \nexpression of $g_{Z' WW}$, that we shall write as: \n\n\\begin{equation}\n g_{Z' WW} = \\left( c \\frac{M^2_Z}{M^2_{Z'}} \\right) \\cot {\\Theta}_W\n\\label{WW8}\n\\end{equation}   \n\nThe constant c would be of the order of one for the \"conventional\" \nmodels where the $Z'$ couples to W only via the $Z-Z'$ mixing (\nessentially contained in the bracket of eq.~(\\ref{WW8})). But for a \ngeneral model, c could be larger, as one can see for some special cases \nof composite models\\cite{Y} or when the $Z'$ originates from a strong coupling \nregime\\cite{BESS}.\nIn fact, a stringent bound on c comes from the request that the partial \n$Z'$ width into WW has to be \"small\" compared to the $Z'$ mass. \nImposing the reasonable limit: \n\n\\begin{equation}\n \\Gamma_{Z'WW} \\leq \\frac{1}{10}{M_{Z'}}\n\\label{WW9}\n\\end{equation} \n\nleads to the condition: \n\n\\begin{equation}\n c \\leq 10\n\\label{WW10}\n\\end{equation}\nthat we consider a rather \"extreme\" choice.\n \nWe shall now discuss the observability limits on ${\\delta}_{\\gamma}$ \nand ${\\delta}_Z$. According to \\cite{KNEUR}, six equidistant bins in \nthe cosine of the production angle are chosen for the generation of \ndata, such that each bin contains a reasonable number of events (\n$\\ge 4$). A binned maximum likelihood method has been used. The result\nfor one parameter fit ${\\delta}_Z$ is:$-0.2 \\le  \\delta_Z \\le 0.25$ for \nthe configuration $\\sqrt q^2=175$ GeV and $\\int L dt=500 pb^{-1}$ and similarly \nfor ${\\delta}_{\\gamma}$. We have then considered a number of possible \nillustrative situations, as extensively discussed in \\cite{CRV} and \nfound that even in correspondance to the available present \n CDF limits and for the optimistic choice \n$c=10$, one would get an effect of about $1 \\%$, i.e. well below the \nexpected LEP2 observability limit.\n\nIn conclusion, a $Z'$ of even pathologically small mass, for extreme \nvalues of its assumed couplings, would be unable to produce observable \neffects in the WW channel at LEP2. Therefore in the derivation of \nbounds or searches for visible effects, the final fermionic channel \nprovides all the relevant information.   \n\n\\section{Concluding remarks.}\n\nWe have tried in this report to be as concise and essential as possible, \npartially owing to the lack of space. In this spirit, we feel that a proper \nconclusion to our work might be that of stressing that LEP2, under \nrealistic experimental conditions and in a rather near future, will be able to\n perform a clean and competitive, in some respects quite unique, search \nfor effects of a $Z'$ whose mass is not above the TeV boundary. For\n$M_{Z'}$ values  beyond this limit, only more energetic machines will be\nable to continue this task.\n \n\n\\section*{Acknowledgements.}\nThis work has been partially supported by EC contract CHRX-CT94-0579.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe Very Long Baseline Interferometry (VLBI) technique measures time delay - the difference between times of the signal arrival on two radio telescopes separated by a long baseline. All measurements are referred to the Solar system barycentre, which moves around the Sun with an orbital velocity about 30~km\/s. This makes the Earth a natural flying platform and VLBI a very effective tool to detect a tiny effect of special relativity. Each baseline of thousand kilometres length may serve as a ``flying rod\", which is traditionally used for theoretical calculation. Precision of each single group delay is about 10~mm and since many observations are collected over a long period of time (20~years or more) the estimate of the time dilation effect will be very accurate.\n\nGeodetic VLBI has been used to test general relativity theory either in the frame of the Parameterized Post-Newtonian (PPN) formalism or the Standard-Model Extension (SME) (e.g., \\citet{Robertson84}, \\citet{Shapiro04}, \\citet{Lambert09}, or \\citet{LePoncinL16}). However, it has not been considered for testing special relativity in spite of its interferometric nature directly linked to the Michelson-Morley and Kennedy-Thorndike interferometers yet. In this paper we show a possible application of the geodetic VLBI to this experimental work.\n\nThe conventional group delay $\\tau_{g}$ model to approximate the observed VLBI data is given\nby~\\citet[chap.~11]{iers10} as\n\\begin{equation}\\label{groupdelay_gcrs}\n\\tau_{g}=\\frac{-\\frac{(\\boldsymbol{b}\\cdot\\boldsymbol{s})}{\\textrm{c}}\\Big(1-\\frac{2GM}{\\textrm{c}^{2} R} -\\frac{|\\boldsymbol{V}|^{2}}{2\\textrm{c}^{2}}-\\frac{(\\boldsymbol{V}\\cdot\\boldsymbol{w}_{2})}{\\textrm{c}^2}\\Big) -\\frac{(\\boldsymbol{b}\\cdot\\boldsymbol{V})}{\\textrm{c}^{2}}\\Big(1+\\frac{(\\boldsymbol{s}\\cdot\\boldsymbol{V})}{2\\textrm{c}}\\Big)}\n    {1+\\frac{\\boldsymbol{s}\\cdot(\\boldsymbol{V}+\\boldsymbol{w}_{2})}{\\textrm{c}}}\n\\end{equation}\nwhere $\\boldsymbol{b}$ is the vector of baseline $\\boldsymbol{b} = \\boldsymbol{r_2}-\\boldsymbol{r_1}$, $\\boldsymbol{s}$ is the barycentric unit vector of radio source, $\\boldsymbol{V}$ is the barycentric velocity of the geocentre, $\\boldsymbol{w}_{2}$ is the geocentric velocity of the second station, \\textrm{c} is the speed of light, \\textrm{G} is the gravitational constant, $M$ is the mass of the Sun, and $R$ is the geocentric distance to the Sun.\\\\\nThe term $\\frac{2GM}{\\textrm{c}^{2} R}$ is related to the general relativity effect and we won't focus on it in this note. The impact of $\\boldsymbol{w}_{2}$ is small and may be ignored for the sake of simplicity. After these alterations, Equation~(\\ref{groupdelay_gcrs}) is given by\n\\begin{equation}\\label{groupdelay_gcrs_simple}\n\\tau_{g}=\\frac{-\\frac{(\\boldsymbol{b}\\cdot\\boldsymbol{s})}{\\textrm{c}}\\Big(1 -\\frac{|\\boldsymbol{V}|^{2}}{2\\textrm{c}^{2}}\\Big) -\\frac{1}{\\textrm{c}^{2}}(\\boldsymbol{b}\\cdot\\boldsymbol{V}) \\Big(1+\\frac{(\\boldsymbol{s}\\cdot\\boldsymbol{V})}{2\\textrm{c}}\\Big)}\n    {1+\\frac{1}{\\textrm{c}}(\\boldsymbol{s}\\cdot\\boldsymbol{V})} .\n\\end{equation}\nUsing the Taylor series expansion $(1+x)^{-1} = 1-x+x^2$\nfor $\\Big(1+\\frac{(\\boldsymbol{s}\\cdot\\boldsymbol{V})}{\\textrm{c}}\\Big)^{-1}$ and noting the $\\frac{|\\boldsymbol{V}|^2}{\\textrm{c}^2}$ terms only, Equation~(\\ref{groupdelay_gcrs_simple}) reduces to\n\\begin{equation}\\label{groupdelay_gcrs_simple_v2c3}\n\\begin{aligned}\n\\tau_{g}=&\\frac{(\\boldsymbol{b}\\cdot\\boldsymbol{s})}{\\textrm{c}}\n\\frac{|\\boldsymbol{V}|^{2}}{2\\textrm{c}^{2}} - \\frac{1}{\\textrm{c}^{2}} (\\boldsymbol{b}\\cdot\\boldsymbol{V})\n\\frac{(\\boldsymbol{s}\\cdot\\boldsymbol{V})}{2\\textrm{c}} -\\\\\n&-\\frac{(\\boldsymbol{b}\\cdot\\boldsymbol{s})(\\boldsymbol{s}\\cdot\\boldsymbol{V})^2 }{\\textrm{c}^3} +\n\\frac{1}{\\textrm{c}^{2}} (\\boldsymbol{b}\\cdot\\boldsymbol{V}) \\frac{(\\boldsymbol{s}\\cdot\\boldsymbol{V})}{\\textrm{c}} =\\\\\n=&\\frac{(\\boldsymbol{b}\\cdot\\boldsymbol{s})|\\boldsymbol{V}|^{2}}{2\\textrm{c}^{3}} -\n\\frac{(\\boldsymbol{b}\\cdot\\boldsymbol{s})(\\boldsymbol{s}\\cdot\\boldsymbol{V})^2}{2\\textrm{c}^{3}} \\\\\n&-\\frac{(\\boldsymbol{b}\\cdot\\boldsymbol{V})(\\boldsymbol{s}\\cdot\\boldsymbol{V})}{2\\textrm{c}^{3}} +\n\\frac{(\\boldsymbol{b}\\cdot\\boldsymbol{V})(\\boldsymbol{s}\\cdot\\boldsymbol{V})}{\\textrm{c}^{3}} -\n\\frac{(\\boldsymbol{b}\\cdot\\boldsymbol{s})(\\boldsymbol{s}\\cdot\\boldsymbol{V})^2}{2\\textrm{c}^{3}} = \\\\\n=&\\frac{(\\boldsymbol{b}\\cdot\\boldsymbol{s})\\Big(|\\boldsymbol{V}|^{2}-(\\boldsymbol{s}\\cdot\\boldsymbol{V})^2\\Big)}{2\\textrm{c}^{3}} +\n\\frac{(\\boldsymbol{s}\\cdot\\boldsymbol{V})\\Big((\\boldsymbol{b}\\cdot\\boldsymbol{V})-(\\boldsymbol{b}\\cdot\\boldsymbol{s})(\\boldsymbol{s}\\cdot\\boldsymbol{V})\\Big)}{2\\textrm{c}^{3}} .\n\\end{aligned}\n\\end{equation}\nIn Fig.~\\ref{fig_triangle} we introduce the following angles\n$|\\boldsymbol{V}| \\cos\\theta = (\\boldsymbol{s}\\cdot\\boldsymbol{V})$,\n$|\\boldsymbol{b}| \\cos\\varphi = (\\boldsymbol{b}\\cdot\\boldsymbol{s})$ and\n$|\\boldsymbol{b}||\\boldsymbol{V}| \\cos\\psi = (\\boldsymbol{b}\\cdot\\boldsymbol{V})$, and from the equation of spherical trigonometry we get $\\cos\\psi = \\cos\\theta \\cos\\varphi + \\sin\\theta \\sin\\varphi \\cos A$. After applying the substitution\n\\begin{equation}\\label{trigfunc1}\n\\begin{aligned}\n(\\boldsymbol{b}\\cdot\\boldsymbol{s})\\Big(|\\boldsymbol{V}|^{2}-(\\boldsymbol{s}\\cdot\\boldsymbol{V})^2\\Big) =\n&|\\boldsymbol{b}| \\cos\\varphi \\Big(|\\boldsymbol{V}|^2-|\\boldsymbol{V}|^2 \\cos^2\\theta\\Big) =\\\\\n=&|\\boldsymbol{b}| |\\boldsymbol{V}|^2 \\cos\\varphi \\sin^2\\theta\n\\end{aligned}\n\\end{equation}\nand\n\\begin{equation}\\label{trigfunc2}\n\\begin{aligned}\n&(\\boldsymbol{s}\\cdot\\boldsymbol{V})\\Big((\\boldsymbol{b}\\cdot\\boldsymbol{V})-(\\boldsymbol{b}\\cdot\\boldsymbol{s})(\\boldsymbol{s}\\cdot\\boldsymbol{V})\\Big) = \\\\\n&=|\\boldsymbol{V}| \\cos\\theta \\Big( |\\boldsymbol{b}||\\boldsymbol{V}| \\cos\\psi -  (|\\boldsymbol{b}| \\cos\\varphi)   (|\\boldsymbol{V}| \\cos\\theta)       \\Big) = \\\\\n&=|\\boldsymbol{b}||\\boldsymbol{V}|^2 \\cos\\theta \\Big(\\cos\\theta \\cos\\varphi  +  \\sin\\theta \\sin\\varphi \\cos A - \\cos\\theta \\cos\\varphi\\Big)\n\\end{aligned}\n\\end{equation}\nwe get the Equation~(\\ref{groupdelay_gcrs_simple_v2c3}) in the following form\n\\begin{equation}\\label{groupdelay_sphtrig}\n\\tau_{g}= \\frac{|\\boldsymbol{b}||\\boldsymbol{V}|^{2}}{2\\textrm{c}^3} \\cos\\varphi \\sin^2\\theta +\n             \\frac{|\\boldsymbol{b}||\\boldsymbol{V}|^{2}}{2\\textrm{c}^3} \\sin\\varphi \\sin\\theta\\cos\\theta\\cos A .\n\\end{equation}\n\\\\\n\nThe major term of the geometric delay is\n\\begin{equation}\\label{tau_geom}\n\\tau_{g}=-\\frac{(\\boldsymbol{b}\\cdot\\boldsymbol{s})}{\\textrm{c}} = -\\frac{|\\boldsymbol{b}|\\cos\\varphi}{\\textrm{c}} .\n\\end{equation}\nThe components of the baseline vector $\\boldsymbol{b}$ and the source position vector $\\boldsymbol{s}$ can be estimated from a large set of data within an adjustment. The observational delay from a correlator is approximated by the theoretical delay (Equation~(\\ref{groupdelay_gcrs})), and the difference between the observational and the theoretical delay is modelled as follows:\n\\begin{equation}\\label{dtau}\n\\tau_{obs} - \\tau_{calc} = \\frac{\\partial \\tau}{\\partial b} \\Delta b + \\frac{\\partial \\tau}{\\partial s} \\Delta s\n\\end{equation}\nor, by applying Equation~(\\ref{tau_geom}) one gets\n\\begin{equation}\\label{dtau1}\n\\tau_{obs} - \\tau_{calc} = -\\Delta b\\frac{1}{\\textrm{c}} \\cos\\varphi + \\Delta s\\frac{|\\boldsymbol{b}|}{\\textrm{c}}\\sin\\varphi\n\\end{equation}\nwhich means that the corrections to the baseline vector components are calculated with the partials proportional to $\\cos\\varphi$ and corrections to the source vector components need partials proportional to $\\sin\\varphi$.\nTherefore, the first part of Equation~(\\ref{groupdelay_sphtrig}) is a variation of the baseline vector (i.e., of the Earth scale) as it is proportional to the factor $(|\\boldsymbol{b}|\\cos{\\varphi})$, and the second part is a variation of the source positions $(|\\boldsymbol{b}|\\sin{\\varphi})$.\n\n\n\n\\begin{figure}[tbp]\n          \\includegraphics[trim=0 0 0 0,clip, width=\\hsize]{Fig1.png}\n  \\caption{Schematic view of the introduced angles with vectors placed in the geocentre.}\n     \\label{fig_triangle}\n\\end{figure}\n\n\n\n\n\\section{Lorenz transformation}\n\\label{sec:LorenzTrafo}\nThere are many approaches to generalise the standard Lorenz transformation. The first known is the Robertson-Mansouri-Sexl (RMS) formalism \\citep{Robertson49, Mansouri77a, Mansouri77b} that assumes a possibility of anisotropic speed of light, i.e., tantamount of a preferred reference frame existing. Due to some critical discussion \\citep{Rybicki16}, we develop a new functional form of three functions $A(V), B(V),$ and $D(V)$ instead of the original RMS formalism. \\\\\nLet's consider the Lorenz transformation in a general form between the preferred reference frame $S(\\boldsymbol{x},t)$ and another reference frame $S'(\\boldsymbol{x'},t')$ moving with a relative velocity $\\boldsymbol{V}$\n\\begin{equation}\\label{xt}\n\\begin{aligned}\n\\boldsymbol{x'} = & D\\boldsymbol{x}-(D-B)\\frac{(\\boldsymbol{V}\\boldsymbol{x})\\boldsymbol{V}}{|\\boldsymbol{V}|^2} - B\\boldsymbol{V}t \\\\\nt' = & A\\Bigg(t - \\frac{\\boldsymbol{V}}{\\textrm{c}^2}\\boldsymbol{x}\\Bigg) .\n\\end{aligned}\n\\end{equation}\nFor the sake of simplicity, we adopt the Einstein synchronisation here. In special relativity $A(V) = B(V) =\\\\ \\bigg(\\sqrt{1-\\frac{|\\boldsymbol{V}|^2}{\\textrm{c}^2}}\\bigg)^{-1} = \\gamma$ and $D(V) = 1$. This form of generalisation is similar to those proposed by \\citet{Will92} with some alterations.\\\\\nThe velocity transformation between frames $S'$ and $S$ may be derived from Equation~(\\ref{xt}) as\n$\\boldsymbol{v'_x} = \\frac{\\boldsymbol{dx'}}{dt'}$ and $\\boldsymbol{v_x} = \\frac{\\boldsymbol{dx}}{dt}$\n\\begin{equation}\\label{dxdt}\n\\begin{aligned}\n\\boldsymbol{dx'} = & D\\boldsymbol{dx}-(D-B)\\frac{(\\boldsymbol{V}\\boldsymbol{dx})\\boldsymbol{V}}{|\\boldsymbol{V}|^2} - B\\boldsymbol{V}dt \\\\\ndt' = & A\\Bigg(dt - \\frac{\\boldsymbol{V}}{\\textrm{c}^2}\\boldsymbol{dx}\\Bigg)\n\\end{aligned}\n\\end{equation}\n\\begin{equation}\\label{vx}\n\\begin{aligned}\n\\boldsymbol{v_x'} =& \\frac{\\boldsymbol{dx'}}{dt'} =\n\\frac{ D\\frac{\\boldsymbol{dx}}{dt}-(D-B)\\frac{ \\Big(\\boldsymbol{V}\\frac{\\boldsymbol{dx}}{dt} \\Big)\\boldsymbol{V}}{|\\boldsymbol{V}|^2} - B\\boldsymbol{V}}\n{A\\bigg(1 - \\frac{\\boldsymbol{V}\\frac{\\boldsymbol{dx}}{dt}}{\\textrm{c}^2}\\bigg)} \\\\\n =& \\frac{ D\\boldsymbol{v_x}-(D-B)\\frac{ (\\boldsymbol{V}\\boldsymbol{v_x} )\\boldsymbol{V}}{|\\boldsymbol{V}|^2} - B\\boldsymbol{V}}\n{A\\bigg(1 - \\frac{\\boldsymbol{V}\\boldsymbol{v_x}}{\\textrm{c}^2}\\bigg)} .\n\\end{aligned}\n\\end{equation}\nFor the propagation of light from an extragalactic radio source one has to assign the unit vector in the direction of the source apparent position (moving system $S'$) as $\\boldsymbol{s'} = -\\frac{\\boldsymbol{v'_x}}{\\textrm{c}}$, and in the preferred reference frame as $\\boldsymbol{s} = -\\frac{\\boldsymbol{v_x}}{\\textrm{c}}$. Therefore, the transformation between $\\boldsymbol{s'}$ and $\\boldsymbol{s}$ is given by\n\\begin{equation}\\label{s}\n\\boldsymbol{s'} = \\frac{ D\\boldsymbol{s}-(D-B)\\frac{(\\boldsymbol{V}\\boldsymbol{s})\\boldsymbol{V}}{|\\boldsymbol{V}|^2} +\n\\frac{B\\boldsymbol{V}}{\\textrm{c}}}\n{A\\Big(1 + \\frac{(\\boldsymbol{V}\\boldsymbol{s})}{\\textrm{c}}\\Big)} .\n\\end{equation}\nNow we apply the traditional expansion \\citep{Mansouri77a, Will92}:\n\\begin{equation}\\label{ABD}\n\\begin{aligned}\nA(V) &= 1+\\alpha \\frac{|\\boldsymbol{V}|^2}{\\textrm{c}^2} + ... \\\\\nB(V) &= 1+\\beta \\frac{|\\boldsymbol{V}|^2}{\\textrm{c}^2} + ... \\\\\nD(V) &= 1+\\delta \\frac{|\\boldsymbol{V}|^2}{\\textrm{c}^2} + ...\n\\end{aligned}\n\\end{equation}\nto the Equation~(\\ref{s}) which gives\n\\begin{equation}\\label{s_traditional}\n\\boldsymbol{s'} = \\frac{ \\Big(1+\\delta \\frac{|\\boldsymbol{V}|^2}{\\textrm{c}^2}\\Big)\\boldsymbol{s}+\\Big(\\beta-\\delta\\Big)\\frac{(\\boldsymbol{V}\\boldsymbol{s})\\boldsymbol{V}}{\\textrm{c}^2} +\n\\Big(1+\\beta \\frac{|\\boldsymbol{V}|^2}{\\textrm{c}^2}\\Big)\\frac{\\boldsymbol{V}}{\\textrm{c}}}\n{\\Big(1+\\alpha \\frac{|\\boldsymbol{V}|^2}{\\textrm{c}^2}\\Big)\\Big(1 + \\frac{(\\boldsymbol{V}\\boldsymbol{s})}{\\textrm{c}}\\Big)} .\n\\end{equation}\nWith the help of the Taylor series expansion $(1+x)^{-1} = 1-x+x^2$ and keeping the terms to $\\frac{\\boldsymbol{V}^2}{\\textrm{c}^2}$ only, we obtain from Equation~(\\ref{s_traditional})\n\\begin{equation}\\label{s_traditional_tillV2c2}\n\\boldsymbol{s'} = \\boldsymbol{s} +\n\\frac{\\boldsymbol{V}-\\boldsymbol{s}(\\boldsymbol{V}\\boldsymbol{s})}{\\textrm{c}} +\n\\frac{\\boldsymbol{s} \\Big( |\\boldsymbol{V}|^2 (\\delta - \\alpha) + (\\boldsymbol{V}\\boldsymbol{s})^2\\Big)}{\\textrm{c}^2}  +\n\\frac{(\\beta-\\delta-1)\\boldsymbol{V}(\\boldsymbol{V}\\boldsymbol{s})}{\\textrm{c}^2} .\n\\end{equation}\nThe second term $\\sim\\frac{\\boldsymbol{V}}{\\textrm{c}}$ represents the annual aberration. The third term (proportional to the vector $\\boldsymbol{s}$ only) does not affect the apparent position of the celestial objects. It could be ignored by the traditional observational astrometric techniques but it is essential for the geodetic VLBI. Finally, the last term in the Equation~(\\ref{s_traditional_tillV2c2}) describes the second order correction ($\\sim\\frac{|\\boldsymbol{V}|^2}{\\textrm{c}^2}$) in the radio source positions.\\\\\nNow we convert the Equation~(\\ref{s_traditional_tillV2c2}) to the geometric delay $\\tau$\n\\begin{equation}\\label{geomdel_s}\n\\begin{aligned}\n\\tau = &-\\frac{(\\boldsymbol{b}(\\boldsymbol{s'}-\\boldsymbol{s}))}{\\textrm{c}} =\n-\\frac{(\\boldsymbol{b}\\boldsymbol{V})-(\\boldsymbol{b}\\boldsymbol{s})(\\boldsymbol{V}\\boldsymbol{s})}{\\textrm{c}^2} \\\\ &-\\frac{(\\boldsymbol{b}\\boldsymbol{s})(|\\boldsymbol{V}|^2(\\delta-\\alpha)+(\\boldsymbol{V}\\boldsymbol{s})^2)}{\\textrm{c}^3}-\n\\frac{(\\beta-\\delta-1)(\\boldsymbol{b}\\boldsymbol{V})(\\boldsymbol{V}\\boldsymbol{s})}{\\textrm{c}^3} .\n\\end{aligned}\n\\end{equation}\nSince in special relativity $\\alpha = \\frac{1}{2}, \\beta = \\frac{1}{2}$ and $\\delta = 0$, the Equation~(\\ref{geomdel_s}) may be presented as\n\\begin{equation}\\label{geomdel_spec}\n\\begin{aligned}\n\\tau = &-\\frac{(\\boldsymbol{b}(\\boldsymbol{s'}-\\boldsymbol{s}))}{\\textrm{c}} =\n-\\frac{(\\boldsymbol{b}\\boldsymbol{V})-(\\boldsymbol{b}\\boldsymbol{s})(\\boldsymbol{V}\\boldsymbol{s})}{\\textrm{c}^2} \\\\ &-\\frac{(\\boldsymbol{b}\\boldsymbol{s})\\Big(-\\frac{1}{2}|\\boldsymbol{V}|^2+(\\boldsymbol{V}\\boldsymbol{s})^2\\Big)}{\\textrm{c}^3}+\n\\frac{(\\boldsymbol{b}\\boldsymbol{V})(\\boldsymbol{V}\\boldsymbol{s})}{2\\textrm{c}^3} \\\\\n=&-\\frac{(\\boldsymbol{b}\\boldsymbol{V})-(\\boldsymbol{b}\\boldsymbol{s})(\\boldsymbol{V}\\boldsymbol{s})}{\\textrm{c}^2} +\n\\frac{(\\boldsymbol{b}\\boldsymbol{s})\\Big(|\\boldsymbol{V}|^2-(\\boldsymbol{V}\\boldsymbol{s})^2\\Big)}{2\\textrm{c}^3}\\\\\n&+\\frac{(\\boldsymbol{b}\\boldsymbol{V})(\\boldsymbol{V}\\boldsymbol{s})}{2\\textrm{c}^3}\n-\\frac{(\\boldsymbol{b}\\boldsymbol{s})(\\boldsymbol{V}\\boldsymbol{s})^2}{2\\textrm{c}^3}\n\\end{aligned}\n\\end{equation}\nwhere the terms $\\sim\\frac{|\\boldsymbol{V}|^2}{\\textrm{c}^2}$ coincide to Equation~(\\ref{groupdelay_gcrs_simple_v2c3}).\n\nIt is essential to note that the sum of the two last terms in Equation~(\\ref{geomdel_s})\n\\begin{equation}\\label{geomdel_s_2terms}\n\\begin{aligned}\n\\Delta\\tau = &\\frac{(\\boldsymbol{b}\\boldsymbol{s})|\\boldsymbol{V}|^2(\\alpha-\\delta)}{\\textrm{c}^3}\n-\\frac{(\\boldsymbol{b}\\boldsymbol{s})(\\boldsymbol{V}\\boldsymbol{s})^2}{\\textrm{c}^3}-\n\\frac{(\\beta-\\delta-1)(\\boldsymbol{b}\\boldsymbol{V})(\\boldsymbol{V}\\boldsymbol{s})}{\\textrm{c}^3} \\\\\n= & \\frac{|\\boldsymbol{b}||\\boldsymbol{V}|^2(\\alpha-\\delta)}{\\textrm{c}^3}\\cos\\varphi -\n\\frac{|\\boldsymbol{b}||\\boldsymbol{V}|^2}{\\textrm{c}^3}\\cos\\varphi\\cos^2\\theta \\\\\n&-\\frac{|\\boldsymbol{b}||\\boldsymbol{V}|^2(\\beta-\\delta-1)}{\\textrm{c}^3} (\\cos\\varphi \\cos^2\\theta - \\sin\\theta \\cos\\theta \\sin\\varphi \\cos A)\\\\\n= & \\frac{|\\boldsymbol{b}||\\boldsymbol{V}|^2(\\alpha-\\delta)}{\\textrm{c}^3}\\cos\\varphi -\n\\frac{|\\boldsymbol{b}||\\boldsymbol{V}|^2 (\\beta -\\delta)}{\\textrm{c}^3}\\cos\\varphi\\cos^2\\theta \\\\\n& + \\frac{|\\boldsymbol{b}||\\boldsymbol{V}|^2(\\beta-\\delta-1)}{\\textrm{c}^3}  \\sin\\theta \\cos\\theta \\sin\\varphi \\cos A\\\\\n= & \\frac{|\\boldsymbol{b}||\\boldsymbol{V}|^2(\\alpha-\\beta)}{\\textrm{c}^3}\\cos\\varphi +\n\\frac{|\\boldsymbol{b}||\\boldsymbol{V}|^2 (\\beta -\\delta)}{\\textrm{c}^3}\\cos\\varphi\\sin^2\\theta \\\\\n& + \\frac{|\\boldsymbol{b}||\\boldsymbol{V}|^2(\\beta-\\delta-1)}{\\textrm{c}^3}  \\sin\\theta \\cos\\theta \\sin\\varphi \\cos A\n\\end{aligned}\n\\end{equation}\nmanifests itself as a combination of the Kennedy-Thorndike experiment testing the factor $(\\alpha-\\beta+1)$ and the Michelson-Morley experiment testing the factor $(\\beta+\\delta-\\frac{1}{2})$ \\citep{Michelson87, Kennedy32, Mansouri77b}. In special relativity, the factors equal to $(\\alpha-\\beta) = 0$ and $(\\beta-\\delta) = \\frac{1}{2}$, therefore only the two last terms are included in the Equation~(\\ref{groupdelay_gcrs}) describing the group delay. The difference $(\\beta-\\delta)$ is tested twice, once with the geodetic parameters (second term in the Equation~(\\ref{geomdel_s_2terms})), and another time with the apparent displacement of the extragalactic radio sources (third term in the Equation~(\\ref{geomdel_s_2terms})). Thus, the second order aberration effect in the source positions is an alteration of the Michelson-Morley experiment, which was already noted in~\\citet{Klioner12}.\\\\\nIf $(\\alpha-\\beta) \\neq 0$ in the Equation~(\\ref{geomdel_s_2terms}), the Earth scale factor estimated from geodetic VLBI data would differ from unity by the factor $\\frac{|\\boldsymbol{V}|^2 (\\alpha - \\beta)}{\\textrm{c}^2}$. In this context, the discrepancy between the VLBI and the SLR (Satellite Laser Ranging) scale factor ($1.37\\pm0.10$~ppb), reported by~\\citet{Altamimi16} for the International Terrestrial Reference Frame ITRF2014, may be interpreted as the violation of the local Lorenz invariance at the level of $(\\alpha-\\beta) \\approx 0.14\\pm0.01$. However, the modern laboratory Kennedy-Thorndike tests of the Lorenz invariance (e.g., \\citet{Herrmann09}) rule this interpretation out.\n\nAll three terms in the Equation~(\\ref{geomdel_s_2terms}) can be presented in the form of Equation~(\\ref{dtau1}), i.e., as a combination of estimated parameters and partial derivatives. Therefore, the corrections $\\Delta b$ and $\\Delta s$ are given by the following equations (note, that the angle $A$ shows the difference of directions from the source to vectors $\\boldsymbol{b}$ and $\\boldsymbol{V}$, and therefore the function $\\cos A$ is a part of the partial derivative):\n\\begin{equation}\\label{delta_bs}\n\\begin{aligned}\n\\Delta b &= -\\frac{|\\boldsymbol{b}||\\boldsymbol{V}|^2(\\alpha-\\beta)}{\\textrm{c}^2} -\n\\frac{|\\boldsymbol{b}||\\boldsymbol{V}|^2 (\\beta -\\delta)}{\\textrm{c}^2}\\sin^2\\theta \\\\\n\\Delta s &= \\frac{|\\boldsymbol{V}|^2(\\beta-\\delta-1)}{\\textrm{c}^2}  \\sin\\theta \\cos\\theta .\n\\end{aligned}\n\\end{equation}\nThe barycentric velocity $\\boldsymbol{V}$ is well-known from the high-precision Solar system ephemerids. Therefore, the Equation~(\\ref{delta_bs}) is a standard part of the delay model, i.e., Equation~(\\ref{groupdelay_gcrs}).\\\\\nLet's apply the Equation~(\\ref{delta_bs}) to a hypothetical preferred reference frame. In this preferred reference frame (e.g., Cosmic Microwave Background (CMB)) the direction of the velocity vector $\\boldsymbol{V}$ is constant. Therefore, if $(\\beta-\\delta) \\neq \\frac{1}{2}$ the shift in source positions depends on the angle $\\theta$ only, i.e., the systematic shift across the sky would show a quadrupole pattern.\n\nIf we define the vector $\\boldsymbol{s} = (\\cos \\hat{\\alpha} \\cos \\hat{\\delta}, \\sin \\hat{\\alpha} \\cos \\hat{\\delta},\\\\ \\sin \\hat{\\delta})$ where $\\hat{\\alpha}$ and $\\hat{\\delta}$ are the right ascension and declination of the source, respectively, and the vector $\\boldsymbol{V} = (V_x, V_y, V_z)$, we can further reformulate the Equation~(\\ref{delta_bs}) for $\\Delta s$ using the relationship $|\\boldsymbol{V}| \\cos\\theta = (\\boldsymbol{s}\\cdot\\boldsymbol{V})$, the spherical law of cosine for the angle $\\theta$, and the division of $\\sin \\theta$ in two equations which gives us the corresponding corrections to the source coordinates $\\Delta \\hat{\\alpha} \\cos \\hat{\\delta}$ and $\\Delta \\hat{\\delta}$ as\n\\begin{equation}\\label{RA}\n\\begin{aligned}\n\\Delta \\hat{\\alpha} \\cos \\hat{\\delta} = &\\frac{(\\beta-\\delta-1)}{\\textrm{c}^2} \\bigg( \\Big(V^2_y - V^2_x\\Big) \\cos \\hat{\\delta} \\sin \\hat{\\alpha} \\cos \\hat{\\alpha} \\\\\n&+ V_x V_y \\cos 2\\hat{\\alpha} \\cos \\hat{\\delta} - V_x V_z \\sin \\hat{\\alpha} \\sin \\hat{\\delta} \\\\\n&+ V_y V_z \\cos \\hat{\\alpha} \\sin \\hat{\\delta} \\bigg)\n\\end{aligned}\n\\end{equation}\nand\n\\begin{equation}\\label{De}\n\\begin{aligned}\n\\Delta \\hat{\\delta} = &\\frac{(\\beta-\\delta-1)}{\\textrm{c}^2} \\Bigg( -\\frac{1}{2} V^2_x \\sin 2\\hat{\\delta} \\cos^2 \\hat{\\alpha} -  \\frac{1}{2} V^2_y \\sin 2\\hat{\\delta} \\sin^2 \\hat{\\alpha} \\\\\n&+\\frac{1}{2} V^2_z \\sin 2\\hat{\\delta} - \\frac{1}{2} V_x V_y \\sin 2\\hat{\\delta} \\sin 2\\hat{\\alpha} \\\\\n&+ V_x V_z \\cos 2\\hat{\\delta} \\cos \\hat{\\alpha}\n+ V_y V_z \\cos 2\\hat{\\delta} \\sin \\hat{\\alpha}\n\\Bigg) .\n\\end{aligned}\n\\end{equation}\nA similar equation is developed for the scale factor from the $\\Delta b$ in the Equation~(\\ref{delta_bs}) using the formula $\\sin^2 \\theta = 1 - \\cos^2 \\theta$ in addition:\n\\begin{equation}\\label{b}\n\\begin{aligned}\n\\Delta b =& -\\frac{|\\boldsymbol{b}||\\boldsymbol{V}|^2(\\alpha-\\beta)}{\\textrm{c}^2} -\n\\frac{|\\boldsymbol{b}|(\\beta -\\delta)}{\\textrm{c}^2} \\\\\n&\\Bigg(-V^2_x \\cos^2 \\hat{\\alpha} \\cos^2 \\hat{\\delta} -V^2_y \\sin^2 \\hat{\\alpha} \\cos^2 \\hat{\\delta}  \\\\\n&-V^2_z \\sin^2 \\hat{\\delta} - V_x V_y \\cos^2 \\hat{\\delta} \\sin 2\\hat{\\alpha} \\\\\n&- \\frac{1}{2} V_x V_z \\cos \\hat{\\alpha} \\sin 2\\hat{\\delta}- \\frac{1}{2} V_y V_z \\sin \\hat{\\alpha} \\sin 2\\hat{\\delta}\n\\Bigg)\n\\end{aligned}\n\\end{equation}\nwhich means that the scale factor magnitude depends on the equatorial coordinates of the observed radio sources.\n\n\\section{Methodology comments}\nAdditional consideration of the special relativity in the geodetic VLBI may be developed in a form of an arbitrary synchronization of clock instead of the Einstein synchronization adopted in Equation~(\\ref{xt}) or equivalent to the introduction of the preferred reference frame as in~\\citet{Klioner12}.\n\nAny non-Einstein synchronisation comes down to replacement of the velocity $\\boldsymbol{V}$ in the second equation of the Equation~(\\ref{xt}) by a function $\\varepsilon(\\boldsymbol{V}) \\neq \\boldsymbol{V}$. In this sense, the geodetic velocity of the second station $\\boldsymbol{w}_{2}$, so far ignored throughout this paper, could be explicitly introduced to modify the equation for the time transformation in Equation~(\\ref{xt}) as follows:\n\\begin{equation}\\label{xt2}\nt' = A\\Bigg(t - \\frac{\\boldsymbol{V}+\\boldsymbol{w}_{2}}{\\textrm{c}^2}\\boldsymbol{x}\\Bigg) .\n\\end{equation}\nThis modification immediately results in the appearance of additional terms in equation of the relativistic time delay:\n\\begin{equation}\\label{d_tau_geom}\n\\Delta\\tau_{g}=-\\frac{(\\boldsymbol{b}\\cdot\\boldsymbol{s})}{\\textrm{c}} \\cdot \\frac{(\\boldsymbol{w}_{2}\\cdot\\boldsymbol{s})}{\\textrm{c}}\n\\end{equation}\nas a part of Equation~(\\ref{groupdelay_gcrs}) and in addition to the Equation~(\\ref{groupdelay_gcrs_simple}). This correction changes the VLBI scale factor only and it is not relevant to astrometric positions of reference radio sources, therefore, it is not part of the final equation by~\\citet{Klioner12}. Further analysis of this term is essential because it lies besides the traditional discussion on the clock synchronization (see, for example, discussion by~\\citet{Cole76}).\n\nTherefore, there is no need in the introduction of the preferred reference frame explicitly in the VLBI relativistic group delay equation. The barycentric and geocentric celestial reference systems (BCRS and GCRS) are fully sufficient to test the Lorentz invariance. The fact of the matter is that GCRS is moving with respect to BCRS along with the Earth. Therefore, direction of the relative velocity of GCRS with respect to BCRS changes as time progresses. This effectively introduces a set of different inertial reference frames along the Earth's orbit. So, introduction of BCRS and GCRS does not mean that we deal with only two frames. Instead of that an infinite number of the inertial frames along the Earth's orbit are introduced. Therefore, tracking the consistency of VLBI time-delay residuals over one orbital revolution allows to compare VLBI observations conducted at different time of year from different locations of GCRS moving in various directions and, thus, to test Lorentz invariance.\n\n\\section{Conclusions}\nWe can conclude that a variety of opportunities is allowed by the geodetic VLBI technique to test the Lorenz invariance in a frame of the kinematic RMS formalism. However, precision of the ground based VLBI measurements is not competitive to the laboratory experiments. While the geodetic VLBI is able to reach an accuracy of the estimation of the $\\alpha, \\beta$ and $\\delta$ parameters at the level of $\\sim10^{-2}$ using the barycentric velocity of the Earth in approximation \\citep{Smoot77}, the laboratory tests set bounds on the anisotropy of the speed of light to $\\sim10^{-12}$ with the Michelson-Morley experiments \\citep{Herrmann09} and to $\\sim10^{-8}$ with the Kennedy-Thorndike experiments \\citep{Tobar10} using the velocity of the Sun with respect to the CMB which is about $\\sim 370$~km\/s \\citep{Smoot77}. Theoretically, space VLBI observations within, e.g., the RadioAstron mission at baselines $\\sim50$ times longer than the Earth radius reduced to the CMB reference frame (as proposed by~\\citet{Klioner12}) may provide an accuracy of $\\sim10^{-6}$ for the $\\alpha, \\beta$ and $\\delta$ parameter combinations.\n\n\n\n\n\\begin{acknowledgements}\nThe authors thank the anonymous reviewers for their suggestions and comments which helped to improve the manuscript significantly. We acknowledge the IVS and all its components for providing VLBI data \\citep{Nothnagel15}. Hana Kr{\\'a}sn{\\'a} works within the Hertha Firnberg position T697-N29, funded by the Austrian Science Fund (FWF). This paper has been published with the permission of the Geoscience Australia CEO.\n\\end{acknowledgements}\n\n\\bibliographystyle{spbasic}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIn a series of papers in the 1980s, Coleman formulated a $p$-adic theory of line integration on curves and higher-dimensional varieties with good reduction at $p$\nand gave numerous applications in arithmetic geometry. This includes the computation of $p$-adic polylogarithms~\\cite{coleman:dilogarithms}, \ntorsion points on Jacobians of curves~\\cite{coleman:torsion}, rational points on certain curves with small Mordell-Weil rank~\\cite{coleman:chabauty}, \n$p$-adic heights on curves (joint with Gross)~\\cite{coleman-gross}, and $p$-adic regulators in $K$-theory (joint with de Shalit)~\\cite{coleman-deshalit}. \nIn \\cite{coleman-deshalit}, Coleman and de Shalit  also introduced a theory of {iterated} $p$-adic integration on curves, which plays an important role in Kim's \nnonabelian Chabauty program \\cite{kim:chabauty} to compute rational points on curves.\n\n\nBesser and de Jeu~\\cite{besser-dejeu} gave the first algorithm for explicit computation of these integrals---now known as \\emph{Coleman integrals}---in the case of iterated \nintegrals on $\\P^1\\setminus\\{0,1,\\infty\\}$. These integrals compute $p$-adic polylogarithms, which are conjecturally related to special values of $p$-adic $L$-functions. \nBalakrishnan, Bradshaw and Kedlaya~\\cite{bbk} gave an algorithm to compute single \nColeman integrals on odd degree models of hyperelliptic curves, which was further generalized to iterated Coleman integrals on arbitrary\nhyperelliptic curves in~\\cite{balakrishnan:iterated, balakrishnan:even}. These algorithms all rely on an algorithm for computing the action of Frobenius on \n$p$-adic cohomology to realize Dwork's principle of \\emph{analytic continuation along Frobenius}. In the case of odd degree hyperelliptic \ncurves, this is achieved by Kedlaya's algorithm~\\cite{kedlaya:mw}.\n\nBecause of all of the applications mentioned above, it is of interest to develop practical \nalgorithms to carry out Coleman integration on \\emph{any} smooth curve. \nIn the present work, we do this by building on work of \nTuitman~\\cite{tuitman:p1, tuitman:pc-general}, which generalizes Kedlaya's algorithm to this setting. We give algorithms to \ncompute single Coleman integrals on curves and develop the precision bounds necessary to obtain provably\ncorrect results. Moreover, we provide a complete implementation~\\cite{colemangit} of our algorithms in the computer algebra system \\texttt{Magma}~\\cite{magma} and present a selection of examples, computing torsion points on Jacobians and carrying out the Chabauty-Coleman method for finding rational points on curves.  The case of iterated Coleman integrals will be discussed in a subsequent paper. \n\nThe structure of the paper is as follows: \nFirst, in Section \\ref{sec:padiccoh} we recall what we need from the theory of $p$-adic cohomology and the algorithms from~\\cite{tuitman:p1,tuitman:pc-general}. \nIn Section~\\ref{sec:integrals}, we present our algorithms for Coleman integration. Next, in Section~\\ref{sec:precision}, we discuss the precision bounds necessary to obtain provably correct results. \nFinally, in Section~\\ref{sec:examples} we conclude with a collection of examples computed with our implementation~\\cite{colemangit}.\n\n\\section{$p$-adic cohomology}\\label{sec:padiccoh}\n\nLet $X$ be a nonsingular projective curve of genus~$g$ over $\\QQ$ given by a (possibly singular) \nplane model $Q(x,y)=0$ with $Q(x,y) \\in \\ZZ[x,y]$ a polynomial that is irreducible and monic in $y$. \nLet $d_x$ and $d_y$ denote the degrees of the morphisms $x$ and $y$, respectively, from $X$ to the projective line. Note\nthat this corresponds to the degrees of $Q$ in the variables $y$ and $x$, respectively.\n\n\\begin{defn} \\label{defn:Delta}\nLet $\\Delta(x) \\in \\ZZ[x]$ denote the discriminant of $Q$ with respect to the variable $y$.\nMoreover, define $r(x) \\in \\ZZ[x]$ to be the squarefree polynomial with the same zeros as $\\Delta(x)$,\nin other words, $r=\\Delta\/(\\gcd(\\Delta,\\frac{d\\Delta}{dx}))$.\n\\end{defn}\n\n\\begin{defn}\\label{defn:intbases} Let $W^0 \\in Gl_{d_x}(\\QQ[x,1\/r])$ and\n$W^{\\infty} \\in Gl_{d_x}(\\QQ[x,1\/x,1\/r])$ denote matrices such that, if we denote \n\\[ b^0_j = \\sum_{i=0}^{d_x-1} W^0_{i+1,j+1} y^i \\; \\; \\; \\; \\mbox{ and } \\; \\; \\; \\; b^{\\infty}_j = \\sum_{i=0}^{d_x-1} W^{\\infty}_{i+1,j+1} y^i \\] \nfor all $0 \\leq j \\leq d_x-1$, then\n\\begin{enumerate}\n\\item $[b^{0 \\;}_0,\\ldots,b^{0 \\;}_{d_x-1}]$ is an integral basis for $\\QQ(X)$ over $\\QQ[x]$,\n\\item $[b^{\\infty}_0,\\ldots,b^{\\infty}_{d_x-1}]$ is an integral basis for $\\QQ(X)$ over $\\QQ[1\/x]$,\n\\end{enumerate}\nwhere $\\QQ(X)$ denotes the function field of $X$. Moreover, let $W \\in Gl_{d_x}(\\QQ[x,1\/x])$ denote\nthe change of basis matrix $W=(W^0)^{-1} W^{\\infty}$.\n\\end{defn}\n\nThere are good algorithms available to compute such matrices, e.g. \\cite{hess,bauch}. \n\n\\begin{defn}\nWe say that the triple $(Q,W^0,W^{\\infty})$ has good reduction at a prime number~$p$ if the conditions\nfrom \\cite[Assumption 1]{tuitman:pc-general} are satisfied. These conditions imply that the curve $X$ has good reduction\nat~$p$ but are stronger. Note that any triple $(Q,W^0,W^{\\infty})$ has good reduction at all but finitely many\nprime numbers~$p$. \n\\end{defn}\n\nFrom now on we fix a prime~$p$ at which $(Q,W^0,W^{\\infty})$ has good reduction and we let $X^{an}$ denote \nthe rigid analytic space over  $\\QQ_p$ associated to $X$.\nThere is a specialization map from $X^{an}$ to its special fibre $\\overline{X}$. \nThe fibres of this map are called \\emph{residue disks}. \n\n\\begin{defn}\nWe say that a point of $X^{an}$ is\n\\emph{very infinite} if its $x$-coordinate is $\\infty$ and \\emph{very bad} if\nit is either very infinite or its $x$-coordinate is a zero of $r(x)$. From the fact\nthat $(Q,W^0,W^{\\infty})$ has good reduction at~$p$, it follows that a residue disk\ncontains at most one very bad point and that this point is defined over an unramified\nextension of $\\QQ_p$. For a very\nbad point $P$, we will denote the ramification index of the map $x$ by $e_P$. \n\\end{defn}\n\n\nWe let $U$ denote the complement of the very bad points in $X$. \n\n\\begin{defn}\nWe say that a residue\ndisk (as well as any point inside it) is \\emph{infinite} or \\emph{bad} if it contains \na very infinite or a very bad point, respectively. A point or residue disk is called \\emph{finite} if \nit is not infinite and \\emph{good} if it is not bad. Note that all infinite points and infinite \nresidue disks are bad.\n\\end{defn}\n\n\\begin{rem}\nIf a point is very bad, this can mean one of three things: \n\\begin{enumerate}\n\\item $x(P) = \\infty$,\n\\item the fiber of $X$ above $x(P)$ contains a ramification point,\n\\item the fiber of $X$ above $x(P)$ contains a point mapping to a singularity of the plane model $Q(x,y)=0$. \n\\end{enumerate}\n\\end{rem}\n\n\\begin{defn} We denote \n\\begin{align*}\nS &=\\QQ_p \\langle x, 1\/r \\rangle,         &S^{\\dag} &=\\QQ_p \\langle x, 1\/r \\rangle^{\\dag}, \\\\\nR &= \\QQ_p \\langle x, 1\/r, y \\rangle\/(Q), &R^{\\dag} &=\\QQ_p \\langle x, 1\/r, y \\rangle^{\\dag}\/(Q).\n\\end{align*}\nHere $\\langle \\rangle^\\dag$ denotes the ring of overconvergent functions obtained by weak completion of the corresponding\npolynomial ring. A Frobenius lift  $\\Frob_p:R^{\\dag} \\rightarrow R^{\\dag}$ is defined as a continuous ring homomorphism that \nreduces to the $p$-th power Frobenius map modulo $p$. \n\\end{defn}\n\n\\begin{thm} \\label{thm:froblift} There exists a Frobenius lift $\\Frob_p: R^{\\dag} \\rightarrow R^{\\dag}$ \nfor which $\\Frob_p(x)=x^p$. \n\\end{thm}\n\n\\begin{proof}\nSee \\cite[Thm. 2.6]{tuitman:pc-general}.\n\\end{proof}\n\n\\begin{defn}\nFor a point $P$ on a curve, we let $\\ord_P$ denote the corresponding discrete valuation\non the function field of the curve. In particular, $\\ord_0$ and $\\ord_{\\infty}$ are\nthe discrete valuations on the rational function field $\\QQ(x)$ corresponding to the points $0$ and \n$\\infty$ on $\\PP^1_{\\QQ}$. We extend these definitions to matrices by taking the minimum over \ntheir entries.\n\\end{defn}\n\nFrom the assumption that $(Q,W^0,W^{\\infty})$ has good reduction at~$p$, it follows that the rigid\ncohomology spaces $\\Hrig^1(U \\otimes \\QQ_p)$ and $\\Hrig^1(X \\otimes \\QQ_p)$ are isomorphic to their \nalgebraic de Rham counterparts~\\cite{baldachiar}. \n\n\\begin{defn}\nLet $[\\omega_1,\\ldots,\\omega_{2g}]$ be $p$-adically integral $1$-forms on $U$ such that\n\\begin{enumerate}\n\\item $\\omega_1,\\ldots,\\omega_{g \\;}$ form a basis for $H^0(X,\\Omega^1_X)$,\n\\item $\\omega_1,\\ldots,\\omega_{2g}$ form a basis for $\\Hrig^1(X \\otimes \\QQ_p)$,\n\\item $\\ord_P(\\omega_i) \\geq -1$ for all~$i$ at all finite very bad points $P$,\n\\item $\\ord_P(\\omega_i) \\geq -1+(\\ord_0(W)+1)e_P$ for all~$i$ at all very infinite points $P$.\n\\end{enumerate}\n\\end{defn}\nIn~\\cite{tuitman:p1,tuitman:pc-general} it is explained how such $1$-forms can be computed.\nActually, property (1) is omitted there, but the algorithm\ncan be easily adapted so that (1) is satisfied as well.\n\n\\begin{defn} \\label{defn:frobdecomp}\nThe $p$-th power Frobenius $\\Frob_p$ acts on $\\Hrig^1(X \\otimes \\QQ_p)$, so there exist a\nmatrix $\\Phi \\in M_{2g \\times 2g}(\\QQ_p)$ and functions $f_1,\\ldots,f_{2g} \\in R^{\\dag}$\nsuch that \n\\[\n\\Frob_p^*(\\omega_i) = df_i + \\sum_{j=1}^{2g} \\Phi_{ij} \\omega_j  \n\\]\nfor $i=1, \\ldots, 2g$.\n\\end{defn}\n\nLet us briefly recall from \\cite{tuitman:p1, tuitman:pc-general} how the matrix~$\\Phi$ and the functions~$f_1,\\ldots,f_{2g}$ are computed. \n\n\\begin{alg} \\mbox{ } \\label{alg:pcc}\n\\begin{enumerate} \n\\item \\textbf{Compute the Frobenius lift.} Determine $\\Frob_p$ as in Theorem~\\ref{thm:froblift}, i.e. set $\\Frob_p(x)=x^p$ and\ndetermine the elements $\\Frob_p(1\/r) \\in S^{\\dag}$ and $\\Frob_p(y) \\in R^{\\dag}$ by Hensel lifting.  \n\\item \\textbf{Finite pole order reduction.} For $i=1,\\ldots,2g$, find $f_{i,0} \\in R^{\\dag}$ \nsuch that\n\\[\n\\Frob_p^*(\\omega_i) = df_{i,0} + G_i \\left( \\frac{dx}{r(x)} \\right),\n\\]\nwhere $G_i \\in R$ only has poles at very infinite points. \n\\item \\textbf{Infinite pole order reduction.} For $i=1,\\ldots,2g$, find $f_{i,\\infty} \\in R$  such that\n\\[\n\\Frob_p^*(\\omega_i) = df_{i,0} + df_{i,\\infty} + H_i \\left(\\frac{dx}{r(x)}\\right),\n\\]\nwhere $H_i \\in R$ still only has poles at very infinite points $P$ and satisfies\n$$\\ord_{P}(H_i) \\geq (\\ord_0(W)-\\deg(r)+2)e_P$$ \nat all these points. \n\\item \\textbf{Final reduction.} For $i=1,\\ldots,2g$, find $f_{i,end} \\in R$ \nsuch that\n\\[\n\\Frob_p^*(\\omega_i) = df_{i,0} + df_{i,\\infty} + df_{i,end} + \\sum_{j=1}^{2g} \\Phi_{ij} \\omega_j, \n\\]\nwhere $\\Phi \\in M_{2g \\times 2g}(\\QQ_p)$ is the matrix of $\\Frob_p^*$ on $\\Hrig^1(U \\otimes \\QQ_p)$ with respect\nto the basis $[\\omega_1,\\ldots,\\omega_{2g}]$. \\\\\n\\end{enumerate}\n\\end{alg}\n\nThe matrix $\\Phi$ and the functions \n$f_i=f_{i,0}+f_{i,\\infty}+f_{i,end}$\nare exactly what we need from~\\cite{tuitman:p1,tuitman:pc-general} to compute Coleman integrals. \n\n\n\\section{Coleman integrals}\\label{sec:integrals}\n\nOur goal is to compute the Coleman integral \n$\\int_{P}^{Q} \\omega$ of a $1$-form $\\omega \\in \\Omega^1(U \\otimes \\QQ_p)$ of the second kind  \nbetween points $P,Q \\in X(\\QQ_p)$. \n\nThe Coleman integral satisfies several key properties, which we will use throughout our integration algorithms:\n\\begin{thm}[Coleman, Coleman--de Shalit] Let $\\eta, \\xi$ be $1$-forms on a wide open $V$ of $X^{an}$ and $P,Q,R \\in V$. The definite Coleman integral has the following properties:\n\\begin{enumerate}\\item Linearity: $\\int_P^Q (a\\eta + b\\xi) = a \\int_P^Q \\eta + b\\int_P^Q \\xi.$\n\\item Additivity in endpoints: $\\int_P^Q \\xi = \\int_P^R \\xi + \\int_R^Q \\xi.$\n\\item Change of variables: If $V' \\subset X'$ is a wide open subspace of a rigid analytic space $X'$ and $\\phi: V \\rightarrow V'$ a rigid analytic map then $\\int_P^Q \\phi^* \\xi = \\int_{\\phi(P)}^{\\phi(Q)} \\xi.$\n\\item Fundamental theorem of calculus: $\\int_P^Q df = f(Q) - f(P)$ for $f$ a rigid analytic function on $V$.\n\\end{enumerate}\n\\end{thm}\n\\begin{proof} \n\\cite{coleman:torsion} for $1$-forms of the second kind and~\\cite{coleman-deshalit} for general $1$-forms.\n\\end{proof}\n\nLet us first explain how we specify a point $P$. Note that giving $(x,y)$-coordinates might not be \nsufficient even for a finite very bad point, since there may be multiple points on \n$X$ lying above a singular point $(x,y)$ of the plane model defined by $Q$. However, a point $P$ is determined \nuniquely by the value of $x$ ($1\/x$ if $P$ is infinite) together with the values of the functions $b^0$ \n($b^{\\infty}$ if $P$ is infinite).  Note that all of these values are $p$-adically integral.\nIn our implementation, we therefore specify a point $P$ by storing three values $(\\texttt{P`x,P`b,P`inf})$:\n\\begin{enumerate}\n\\item \\texttt{P`x}: the $x$-coordinate of $P$ ($1\/x$ if $x$ is infinite),\n\\item \\texttt{P`b}: the values of the functions $b^0$ ($b^{\\infty}$ if $P$ is infinite),\n\\item \\texttt{P`inf}: true or false, depending on whether the point $P$ is infinite or not.\n\\end{enumerate}\n\nWe will often need power series expansions of functions in terms of a \\emph{local coordinate} (i.e., a uniformizing parameter) \n$t$ at $P$. This local coordinate should not just \nbe a local coordinate at~$P$ on $X \\otimes \\QQ_p$, but on the model $\\mathcal{X}$ over \n$\\ZZ_p$ obtained from the triple $(Q,W^0,W^{\\infty})$ as in~\\cite[Prop. 2.3]{tuitman:pc-general}. Then it follows \nthat the reduction modulo~$p$ of $t$ is a local coordinate at the reduction modulo~$p$ of $P$ and that the residue \ndisk at $P$ is given by $\\lvert t \\rvert<1$. In a bad residue disk, we will always expand functions at the very bad\npoint. Therefore, in the following proposition, we only consider points that are either good or very bad.\n\\begin{prop} \\label{prop:locparam} Let $P \\in X(\\QQ_p)$ be a point. Assume that $P$ is either good\nor very bad. As a local coordinate at $P$, we can take\n\\begin{equation*}\nt = \\begin{cases} x-x(P) &\\;\\textrm{if}\\; e_P=1\\; (\\textrm{or}\\; t=1\/x \\;\\textrm{if}\\; P \\;\\textrm{is infinite}),\\\\\n b^0_i-b^0_i(P) \\;\\textrm{for some}~i &\\;\\textrm{otherwise}\\; (\\textrm{with}\\; b^0 \\;\\textrm{replaced by}\\; b^{\\infty} \\;\\textrm{if}\\; P \\;\\textrm{is infinite}).\n\\end{cases}\n\\end{equation*}\n\\end{prop}\n\n\\begin{proof} \nBy definition $e_P=\\ord_P(x-x(P))$ (or $e_p=\\ord_P(1\/x$) if $P$ is infinite). So if $e_P=1$ then \n$t=(x-x(P))$ (or $t=1\/x$ if $P$ is infinite) is a local coordinate at $P$ on $X \\otimes \\QQ_p$.\nIf $e_P \\geq 2$, then at least one of the $b^0_i-b^0_i(P)$ (with $b^0$ replaced by $b^{\\infty}$ if \n$P$ is infinite) has to be a local coordinate at $P$ on  $X \\otimes \\QQ_p$, since otherwise there\nwould be no functions on $X \\otimes \\QQ_p$ of order~$1$ at~$P$. In both cases, since\n$(Q,W^0,W^{\\infty})$ has good reduction at~$p$, the divisor defined by $t$ on $\\mathcal{X}$ is\nsmooth over $\\ZZ_p$, so that $t$ is also a local coordinate at~$P$ in the stronger sense explained above.\n\\end{proof}\n\nAfter choosing a local coordinate~$t$ at~$P$, in our implementation we compute \\texttt{xt,bt} where\n\\begin{enumerate}\\item \\texttt{xt} is the power series expansion in $t$ of the function $x$ ($1\/x$ if $P$ is infinite),\n\\item \\texttt{bt} is the tuple of power series expansions in $t$ of the functions $b^0$ ($b^{\\infty}$ if $P$ is infinite).\n\\end{enumerate}\nNote that all of these power series have $p$-adically integral coefficients.\nFrom \\texttt{xt,bt} we will be able to determine the power series expansion in $t$ of any function which is regular at $P$.\n\nA $1$-form $\\omega \\in \\Omega^1(U \\otimes \\QQ_p)$ is of the form $f dx$ with $f \\in R$. We will usually represent it as follows:\n\\begin{eqnarray*} \\label{eqn:omega}\n\\omega = \\sum_{i=0}^{d_x-1} \\sum_{j \\in J} \\frac{f_{ij}(x)}{r(x)^j} b^0_i \\frac{dx}{r} \n\\end{eqnarray*}\nwith $f_{ij} \\in \\QQ_p[x]$ such that  $\\deg(f_{ij}) < \\deg(r(x))$ for all $i,j$, since $\\omega$ needs\nto be in this form to start the cohomological reduction procedures outlined in Section~\\ref{sec:padiccoh}. \\\\\n\nWe begin by describing the computation of \\emph{tiny} integrals. \n\n\\begin{defn}A tiny integral $\\int_P^Q \\omega$ is a Coleman integral with endpoints $P,Q \\in X(\\QQ_p)$ that lie in the same residue disk.\n\\end{defn}\n\n\\begin{alg}[Computing the tiny integral $\\int_P^Q \\omega$]\\label{alg:tiny} \\mbox{ }\n\\begin{enumerate} \n\\item If the residue disk of $P,Q$ is bad, then find the very bad point $P'\\in X(\\QQ_p)$, otherwise set $P'=P$.\n\\item Compute a local coordinate $t$ and the power series expansions \\emph{\\texttt{xt,bt}} at $P'$. \n\\item Integrate using $t$ as coordinate: \n$$\\int_P^Q \\omega = \\int_{t(P)}^{t(Q)} \\omega(t).$$ \nThe Laurent series expansion $\\omega(t)$ can be determined from \\emph{\\texttt{xt,bt}}. Note that $\\omega$ is of the second kind, \nso the coefficient of $t^{-1}dt$ is zero.\n\\end{enumerate} \n\\end{alg}\n\n\\begin{rem}\nThe calculation of tiny integrals does not require \ncomputing the action of Frobenius on the cohomology space $\\Hrig^1(X \\otimes \\QQ_p)$. \nThis can be a useful consistency check for the integration algorithms that follow, which do use the \ncomputation of the action of Frobenius. \n\\end{rem}\n\nWhen $P,Q \\in X(\\QQ_p)$ do not lie in the same residue disk, this approach breaks down since the Laurent series expansions\ndo not converge anymore. In this case we will compute the Coleman integrals $ \\int_{P}^Q \\omega_i$ for $i=1,\\ldots,2g$ \nby solving a linear system imposed by the $p$-th power Frobenius map $F_p$. We first assume that the functions $f_1,\\ldots,f_{2g}$ from \nSection~\\ref{sec:padiccoh} converge at $P,Q$. Note that \n$f_1,\\ldots,f_{2g}$ converge at all good points, but only at bad points that are not too close to the corresponding very bad point.\nThis will be made more precise in the next section.\n\n\\begin{alg}[Compute the $\\int_P^Q \\omega_i$ assuming $f_1,\\ldots,f_{2g}$ converge at $P,Q$]\\label{alg:colemangoodi} \\mbox{ } \n\\begin{enumerate}\n\\item Compute the action of Frobenius on $\\Hrig^1(X \\otimes \\QQ_p)$ using Algorithm \\ref{alg:pcc} and store $\\Phi$ and $f_1,\\ldots,f_{2g}$.\n\\item Determine the tiny integrals $\\int_P^{F_p(P)} \\omega_i$ and $\\int_{F_p(Q)}^Q \\omega_i$ for $i=1,\\ldots,2g$ using Algorithm \\ref{alg:tiny}.\n\\item Compute $f_i(P) - f_i(Q)$ for $i = 1, \\ldots, 2g$ and use the system of equations  \n\\begin{equation*}\n\\sum_{j=1}^{2g} (\\Phi-I)_{ij} \\left( \\int_P^Q \\omega_j \\right) = f_i(P)-f_i(Q) - \\int_P^{F_p(P)} \\omega_i - \\int_{F_p(Q)}^Q \\omega_i\n\\end{equation*} to solve for all $\\int_P^Q \\omega_i$, as in \\cite[Algorithm 11]{bbk}.\n\\end{enumerate} \n\\end{alg}\n\n\\begin{rem}\nNote that the matrix $\\Phi-I$ is invertible, since the eigenvalues of $\\Phi$ are algebraic numbers of complex absolute value $p^{1\/2}$. \n\\end{rem}\n\n\nWhen $P$ or $Q$ are bad points and $f_1,\\ldots,f_{2g}$ do not converge there, the idea is simply to find points \n$P',Q'$ in the residue disks of $P$ and $Q$ where these functions do converge, compute the integrals between \nthe new points, and correct for the difference with tiny integrals.\n\n\\begin{alg}[Computing the $\\int_P^Q \\omega_i$ in general]\\label{alg:colemani} \\mbox{ } \n\\begin{enumerate}\n\\item Determine $P',Q'$ in the residue disks of $P,Q$ at which all functions $f_1,\\ldots,f_{2g}$ converge.\n\\item Compute the tiny integrals $\\int_P^{P'} \\omega_i$ and $\\int_{Q'}^Q \\omega_i$ for $i=1,\\ldots,2g$ using Algorithm~\\ref{alg:tiny}.\n\\item Determine $\\int_{P'}^{Q'} \\omega_i$ for $i=1,\\ldots,2g$ using Algorithm~\\ref{alg:colemangoodi}.\n\\item Compute $$\\int_P^Q \\omega_i = \\int_P^{P'} \\omega_i + \\int_{P'}^{Q'} \\omega_i +\\int_{Q'}^Q \\omega_i.$$\n\\end{enumerate} \n\\end{alg}\n\nIn general we have to take the points $P',Q'$ to be defined over some (totally ramified) \nextension $K$ of $\\QQ_p$ to get far enough away from the very bad point \nin the bad residue disk. We will always take this extension to be of the form\n$\\QQ_p(p^{1\/e})$ for some positive integer~$e$. Note that Algorithms~\\ref{alg:tiny} and~\\ref{alg:colemangoodi} \ncan still be applied in this case and that we may take $P' \\in X(\\QQ_p)$ in Algorithm~\\ref{alg:tiny}. Since computing in \nextensions is more expensive, integrals involving bad points are usually the hardest to compute. \\\\\n\nFor more general $1$-forms of the second kind $\\omega \\in \\Omega^1(U \\otimes \\QQ_p)$, we can now compute the Coleman integrals $\\int_{P}^Q \\omega$ as follows\nfrom the output of Algorithms~\\ref{alg:colemangoodi} and~\\ref{alg:colemani}.\n\\begin{alg}[Computing $\\int_P^Q \\omega$] \\label{alg:colemangood} \\mbox{ }\n\\begin{enumerate}\n\\item Use Steps (2),(3) and (4) of Algorithm~\\ref{alg:pcc} to find $f \\in R$ and $c_i \\in \\QQ_p$ for $i=1,\\ldots,2g$ such that $$\\omega = df+ \\sum_{i=1}^{2g} c_i \\omega_i.$$ \n\\item Compute $f(Q)-f(P)$ and determine\n\\[\n\\int_{P}^{Q} \\omega = f(Q)-f(P)  + \\sum_{i=1}^{2g} c_i \\int_{P}^Q \\omega_i. \n\\]\n\\end{enumerate}\n\\end{alg}\n\n\\begin{rem}\nNote that we are only considering points $P,Q$ defined over $\\QQ_p(p^{1\/e})$ for some positive integer $e$. It is\npossible to extend our work to points defined over arbitrary finite extensions of $\\QQ_p$ as in~\\cite[Remark 12]{bbk}.\nHowever, we have not attempted to make this practical or implement it yet, so we leave it out of the discussion\nhere.\n\\end{rem}\n\n\n\\section{Precision bounds} \\label{sec:precision}\n\nSo far we have not paid any attention to the fact that we can only compute to finite $p$-adic and $t$-adic\nprecision in our algorithms. By \\emph{precision} we will always mean \\emph{absolute} $p$-adic precision, i.e., \nthe valuation of the error term. We extend the $p$-adic valuation and the notion of precision to all \nfinite extensions of $\\QQ_p$, where they will take non-integer values in general. \\\\ \n\nLet us start with tiny integrals.\n\n\\begin{prop} \\label{prop:prectiny} Let $e$ be a positive integer and $P,Q \\in X(\\QQ_p(p^{1\/e}))$ two points in \nthe same residue disk accurate to precision $N$. Let $t$ be a local coordinate (in the sense of \nProposition~\\ref{prop:locparam}) at the point $P'$ from Algorithm~\\ref{alg:tiny}. \nSuppose that $\\omega = g(t) dt$ is a differential of the second kind with \n$$g(t) = a_{-k} t^{-k} + a_{-k+1} t^{-k+1} + \\ldots  \\in \\ZZ_p[[t]][t^{-1}]$$\nfor some positive integer $k$. If $g$ is accurate to $p$-adic precision $N$ and truncated modulo $t^l$, then the tiny integral $\\int_P^Q \\omega$ \ncomputed as in Algorithm~\\ref{alg:tiny} is correct to precision $\\min\\{\\nu_1,\\nu_2,\\nu_3\\}$ where:\n\\begin{align*}\n\\nu_1 &= \\min_{i \\geq l}   \\{ (i+1)\/e-\\lfloor \\log_p(i+1) \\rfloor \\}, \\\\\n\\nu_2 &= \\min_{i \\leq l-1} \\{ N+i\/e- \\lfloor \\log_p(i+1) \\rfloor \\}, \\\\\n\\nu_3 &= N-k \\max \\{ \\ord_{p}(t(P)),\\ord_p(t(Q))\\}-\\lfloor \\log_p(k-1) \\rfloor. \n\\end{align*}\n\\end{prop}\n\\begin{proof} Recall from Algorithm~\\ref{alg:tiny} that\n\\[\n\\int_{P}^Q \\omega = \\int_{t(P)}^{t(Q)} \\omega(t) = \\sum_{i=-k}^{\\infty} \\frac{a_i}{i+1} \\left( t(Q)^{i+1} - t(P)^{i+1} \\right).\n\\]\nwhere $a_{-1}=0$ since $\\omega$ is of the second kind. Since $P,Q$ both lie in the residue disk given by $\\lvert t \\rvert <1$,\nwe have that $\\ord_p(t(P)),\\ord_p(t(Q)) \\geq 1\/e$.\n\nFirst, we bound the error introduced by omitting the terms with $i \\geq l$. Note that $\\ord_{p}(t(P)^{i+1}),\\ord_p(t(Q)^{i+1}) \\geq (i+1)\/e$ \nand $\\ord_{p}(i+1) \\leq \\lfloor \\log_p(i+1) \\rfloor$. Therefore, the valuation of this error is at least $\\nu_1$.\n\nNext, we consider the error coming from terms with $0 \\leq i \\leq l-1$. Since $t(P),t(Q)$ are accurate to precision $N$ \nand have valuation at least~$1\/e$, we have that $t(P)^{i+1},t(Q)^{i+1}$ are correct to precision $N+i\/e$. Therefore, the valuation of\nthis error is at least $\\nu_2$.\n\nFinally, we  bound the error coming from terms with $-k \\leq i \\leq -2$. This time $t(P)^{i+1},t(Q)^{i+1}$ are correct to precision at least\n$N+i\\ord_{p}(t(P)),N+i\\ord_{p}(t(Q))$, respectively (since the loss of precision of an inversion is $2$ times the valuation). Therefore, the valuation \nof the error is at least $\\nu_3$ this time.\n\\end{proof}\n\n\\begin{rem}\nSince we always have that $\\nu_2 \\leq N$, there is no point in increasing the $t$-adic precision $l$ further if \n$\\nu_1 \\geq N$ already. Therefore, in our implementation we take $l$ to be minimal such that $\\nu_1 \\geq N$. \n\\end{rem}\n\nTo compute integrals that are not tiny, in Algorithm~\\ref{alg:colemangoodi} we have to evaluate the functions \n$$f_i=f_{i,0}+f_{i,\\infty}+f_{i,end}$$ from Section~\\ref{sec:padiccoh} at the endpoints for $i=1,\\ldots,2g$. \nEvaluating an element of \n$R^{\\dag}$ at a bad point leads to problems with convergence and loss of precision. We first recall \nfrom~\\cite{tuitman:p1,tuitman:pc-general} what we know about the poles of the functions\n$f_{i,0},f_{i,\\infty},f_{i,end} \\in R^{\\dag}$.\n\nThe only poles of infinite order are those of the $f_{i,0}$ at the finite very bad points. \nIt follows from~\\cite[Prop. 2.12, Prop. 3.3, Prop. 3.7]{tuitman:pc-general} that\n\\begin{equation} \\label{eq:f0i} f_{i,0} = \\sum_{j=0}^{d_x-1} \\sum_{k=1}^{\\infty} \\frac{c_{ijk}(x)}{r(x)^k} b^0_j,\\end{equation}\nfor all $i$, where the $c_{ijk}$ are elements of $\\QQ_p[x]$ of degree smaller than $\\deg(r)$ that satisfy\n\\begin{equation} \\label{eq:cijk} \\ord_p(c_{ijk}) \\geq \\lfloor k\/p \\rfloor +1 - \\lfloor \\log_p(k e_0) \\rfloor\\end{equation}\nwith $e_0 = \\max\\{e_P \\colon P \\mbox{ finite very bad point}\\}$. \n\nSimilarly, it follows from~\\cite[Prop. 2.12, Prop. 3.4, Thm. 3.6]{tuitman:pc-general} that\n\\begin{equation} \\label{eq:finftyi}\nf_{i,\\infty} = \\sum_{j=0}^{d_x-1} \\sum_{k=0}^{\\kappa_1} c_{ijk} x^k b^0_j = \\sum_{j=0}^{d_x-1} \\sum_{k=\\kappa_2}^{\\kappa_3} d_{ijk} x^k b^{\\infty}_j\n\\end{equation}\nfor all $i$, where the $c_{ijk},d_{ijk}$ are elements of $\\QQ_p$ and \n$$\\kappa_3 \\leq -\\min\\{p (\\ord_0(W)+1) ,(\\ord_{\\infty}(W^{-1})+1)\\}.$$\nNote that this determines bounds on $\\kappa_1,\\kappa_2$ as well.\n\nFinally, it follows from~\\cite[Thm. 3.6]{tuitman:pc-general} that\n\\begin{equation} \\label{eq:fendi}\nf_{i,end} = \\sum_{j=0}^{d_x-1} \\sum_{k=0}^{\\lambda_1} c_{ijk} x^k b^0_j  = \\sum_{j=0}^{d_x-1} \\sum_{k=\\lambda_2}^{\\lambda_3} d_{ijk} x^k b^{\\infty}_j\n\\end{equation}\nfor all $i$, where the $c_{ijk}, d_{ijk}$ are elements of $\\QQ_p$ and \n\\[\n\\lambda_3 \\leq -(\\ord_0(W)+1). \n\\]\nNote that this determines bounds on $\\lambda_1,\\lambda_2$ as well.\n\n\\begin{prop}\nOn a finite bad residue disk, the functions $f_{i,0}$ converge outside of the \nclosed disk defined by $\\ord_p(r(x)) \\geq 1\/p$.\n\\end{prop}\n\n\\begin{proof}\nThis is clear from \\eqref{eq:f0i} and \\eqref{eq:cijk}. \n\\end{proof}\n\n\\begin{rem}\\label{rem:nearbdprec}\nLet $t$ denote a local coordinate at the very bad point of a finite bad residue disk.\nThen we have that $\\ord_p(r(x)) < 1\/p$ is equivalent to the condition $\\ord_p(t) < \\frac{1}{pe_P}$. \nConsequently, for the functions $f_{i,0}$ to converge at a point $P' \\in X(\\QQ_p(p^{1\/e}))$ \nin the residue disk of $P$, we need to take $e > pe_P$. \n\\end{rem}\n\nWhen $f_1,\\ldots,f_{2g}$ do converge at a point $P$, their computed values at this point will\nsuffer some loss of $p$-adic precision in general. In the next three propositions we quantify this precision loss for good, finite bad, and infinite points, respectively.\n\n\\begin{prop} \\label{prop:precgood}\nSuppose that the functions $f_{i,0},f_{i,\\infty},f_{i,end}$ are accurate to \nprecision~$N$. Moreover, let $e$ be a positive integer and let\n$P \\in X(\\QQ_p(p^{1\/e}))$ be a good point that is accurate to precision $N$. \nThen the computed values $f_i(P)$ are correct to precision $N$ as well.\n\\end{prop}\n\\begin{proof} \nNote that a good point is always finite. Since we have that $\\ord_p(x(P)) \\geq 0$ and $\\ord_p(r(x(P)))=0$,\nthere is no loss of precision in evaluating~\\eqref{eq:f0i} and the expressions in the middle\nof~\\eqref{eq:finftyi} and~\\eqref{eq:fendi}.\n\\end{proof}\n\n\\begin{prop} \\label{prop:precfinitebad}\nSuppose that the functions $f_{i,0},f_{i,\\infty},f_{i,end}$ are accurate to precision $N$. Moreover, \nlet $e$ be a positive integer and let $P \\in X(\\QQ_p(p^{1\/e}))$ be a finite bad point that is accurate to precision $N$. \nLet $\\epsilon=\\ord_p(r(Q))$ and suppose that $\\epsilon < 1\/p$. Define a function $\\pi$ on positive integers by\n\\[\n\\pi(k) =  \\max \\{N, \\lfloor k\/p \\rfloor +1 - \\lfloor \\log_p(k e_0) \\rfloor \\},\n\\]\nwhere $e_0 = \\max\\{e_P \\colon P \\mbox{ finite bad point }\\}$.\nThen the computed values $f_i(P)$ are correct to precision\n$$\\min_{k \\in \\NN} \\{ \\pi(k) - k \\epsilon \\}.$$\n\\end{prop}\n\\begin{proof} \nIn this case $\\ord_p(x(P)) \\geq 0$, but $\\ord_p(r(x(P))) = \\epsilon$ with $0 < \\epsilon < 1\/p$.\nClearly there is still no loss of precision in evaluating the expressions in the middle\nof~\\eqref{eq:finftyi} and~\\eqref{eq:fendi}. However for the $f_{i,0}$ there will be\nloss of precision. After dropping the terms with valuation greater than or equal to~$N$\nin~\\eqref{eq:f0i}, the coefficient $c_{ijk}$ will be correct to precision $\\pi(k)$ for all $k$. \nDividing by $r(x(P))^k$ leads to the loss of $k \\epsilon$ \ndigits of precision, so the terms corresponding to $k$ will be correct to precision $\\pi(k)-k\\epsilon$.\nTaking the minimum over $k$, we obtain the result. \n\\end{proof}\n\n\\begin{prop} \\label{prop:precinfinite}\nSuppose that the functions $f_{i,0},f_{i,\\infty},f_{i,end}$ are accurate to precision $N$. \nMoreover, let $e$ be a positive integer and let\n$P \\in X(\\QQ_p(p^{1\/e}))$ be an infinite point that is accurate to precision $N$.\nLet $\\epsilon=\\ord_p(1\/x(P))$. Then the computed values $f_i(P)$ are correct to precision\n\\[\nN + \\epsilon \\min \\{ \\ord_{\\infty}(W^{-1})+1), p (\\ord_0(W)+1) \\}. \n\\]\n\\end{prop}\n\\begin{proof} \nIn this case $\\ord_p(x(P)) = - \\epsilon < 0$ and $\\ord_p(r(x(P))) = \\deg(r) \\epsilon$. Let us first consider the $f_{i,0}$. \nDetermining the $b^0_j(P)$ from the $b^{\\infty}_j(P)$ in $\\eqref{eq:f0i}$ leads to a precision loss of $-\\ord_{\\infty}(W^{-1}) \\epsilon $, \nbut since \n$$\\ord_p \\left( \\frac{c_{ijk}(x(P))}{r(x(P))^k} \\right) \\geq \\epsilon $$\nfor all $k\\geq 1$, we recover precision $\\epsilon$ and the loss of precision will be at most $-(\\ord_{\\infty}(W^{-1})+1)\\epsilon$. \nEvaluating the expressions on the right of~\\eqref{eq:finftyi} and~\\eqref{eq:fendi} leads to precision loss at most \n$$-\\min\\{p (\\ord_0(W)+1) ,(\\ord_{\\infty}(W^{-1})+1)\\} \\epsilon$$\nand \n$$-(\\ord_0(W)+1) \\epsilon,$$ respectively. The result follows easily from this.\n\\end{proof}\n\nNow all that is left to analyze in Algorithm~\\ref{alg:colemangoodi} is the precision loss from solving the linear system, \ni.e. computing the matrix $(\\Phi-I)^{-1}$ and multiplying by it.\n\n\\begin{prop}\nSuppose that the matrix $\\Phi$ is $p$-adically integral and accurate to precision $N$. \nMoreover, let $e$ be a positive integer and let $P,Q \\in X(\\QQ_p(p^{1\/e}))$ be points accurate to precision $N$. \nSuppose that the right hand side of (3) in Algorithm~\\ref{alg:colemangoodi} is \naccurate to precision $N' \\leq N$ according to \nPropositions~\\ref{prop:prectiny},~\\ref{prop:precgood},~\\ref{prop:precfinitebad}, and~\\ref{prop:precinfinite}.\nThen the integrals $\\int_{P}^Q \\omega_i$ as computed in Algorithm~\\ref{alg:colemangoodi} are correct to precision\n\\[\nN' - \\ord_p(\\det(\\Phi-I)). \n\\]\n\\end{prop}\n\\begin{proof}\nThis follows since $(\\Phi-I)^{-1}$ has valuation at least $-\\ord_p(\\det(\\Phi-I))$ and is \ncorrect to precision $N-\\ord_p(\\det(\\Phi-I))$.\n\\end{proof}\n\n\\begin{rem}\nIf we do not assume that $\\Phi$ is $p$-adically integral, then we can show\nthat the integrals $\\int_{P}^Q \\omega_i$ as computed in Algorithm~\\ref{alg:colemangoodi} \nare correct to precision $$N' - \\ord_p(\\det(\\Phi-I))-\\delta$$ \nwith $\\delta$ defined as in~\\cite[Definition 4.4]{tuitman:pc-general}.\n\\end{rem}\n\n\\begin{rem}\nTo analyze the loss of precision in Algorithm~\\ref{alg:colemangood}, we proceed as follows. First, we use~\\cite[Prop. 3.7, Prop. 3.8]{tuitman:pc-general} \nto determine the precision to which $f$ and the $c_i$ are correct. Then we proceed as in Propositions~\\ref{prop:precgood},\n~\\ref{prop:precfinitebad}, and~\\ref{prop:precinfinite} to determine the precision of the computed values of $f(P)$, $f(Q)$ and $\\int_{P}^Q \\omega_i$\nfor $i=1,\\ldots,2g$. Finally, we determine the precision to which $\\int_{P}^Q \\omega$ is correct, taking into account the valuations of the $c_i$\nas well. \n\\end{rem}\n\n\n\\section{Examples}\\label{sec:examples}\n\n\\subsection{An example from the work of Bruin--Poonen--Stoll} \\mbox{ } \\\\\n\nLet $X\/\\QQ$ be the genus 3 curve given by the following plane model: $$Q(x,y) = y^3 + (-x^2 - 1)y^2 - x^3y + x^3 + 2x^2 + x = 0.$$  Bruin, Poonen, and \nStoll~\\cite[Prop. 12.17]{bruin-poonen-stoll} show that, under the assumption of the Generalized Riemann Hypothesis, the Jacobian of $X$ has Mordell-Weil \nrank~1. (Note that our working plane model is given by taking the equation in  \\cite[\\S 12.9.2]{bruin-poonen-stoll}, provided by D. Simon, \nand setting $x:=1, z:=x$.)\n\nWe have $W^0 = I$, which means that $b^0 = [1,y,y^2]$  is an integral basis \nfor the function field of $X$ over $\\QQ[x]$. Moreover, we have\n$$W^{\\infty}=\n\\begin{pmatrix}\n1 & 0 & 0 \\\\\n0 & 1\/x^2 & 0 \\\\\n0 & -1\/x, & 1\/x^3\n\\end{pmatrix}\n,$$\nso that $b^{\\infty}=[1,y\/x^2,-y\/x+y^2\/x^3]$ is an integral basis \nfor the function field of $X$ over $\\QQ[1\/x]$.\n\nWe consider the following points on $X$ : $P_1 = (0,0), P_2 = (0,1), P_3 = (-3,4), P_4 = (-1,0), P_5 = (-1,1)$, as well as three very infinite  points: \n$P_6$ with $b^{\\infty}$-values $[1,0,1]$ $P_7$ with $b^{\\infty}$-values $[1,1,1]$ and $P_8$ \nwith $b^{\\infty}$-values $[1,0,0]$.\n\nIn \\cite[Prop. 12.17]{bruin-poonen-stoll}, the authors compute $X(\\QQ)$ by using the fact that $[(P_3)-(P_2)]$ is of infinite order in $J(\\QQ)$ and running a 3-adic Chabauty-Coleman argument.  \nIn particular, by computing $3$-adic tiny integrals between $P_2$ and $P_3$, they produce a two-dimensional subspace of regular $1$-forms \nannihilating rational points on $X$ and use the Coleman integrals of these differentials to show that these eight points are all of the \nrational points on $X$. \n\nHere we show how to produce a basis for the two-dimensional space of annihilating $1$-forms without immediately appealing to tiny integrals. While it is desirable to use tiny integrals \nwhenever possible, some curves do not readily admit points of infinite order in $J(\\QQ)$ that are given as small integral combinations of known rational points that \nallow a tiny integral computation. Consequently, in such a scenario, some arithmetic in the Jacobian (working in the kernel of reduction) would be needed to reduce the necessary \nColeman integral computation to a tiny integral computation. The computation below shows how one might bypass \nthe Jacobian arithmetic by using Coleman integrals that are not necessarily tiny integrals. \n\nWe have $r =  x(x+1)(x^8 + 7x^7 + 21x^6 + 31x^5 + 3x^4 - 51x^3 - 69x^2 - 23x + 4)$. Taking $p = 3$ makes all eight points  various types of bad:\n\\begin{center}\n\\begin{tabular}{| l |  c  | c  |c | }\n\\hline\nPoint $P$ & $r(x(P))$ & Type of point  \\\\ \n\\hline\n$P_1 = (0,0)$  & 0      & finite very bad\\\\\n$P_2 = (0,1)$  & 0      & finite very bad\\\\\n$P_3 = (-3,4)$ & $-600$ & finite bad\\\\\n$P_4 = (-1,0)$ & 0      & finite very bad\\\\\n$P_5 = (-1,1)$ & 0      & finite very bad \\\\\n$1\/x(P_6) = 0, b^{\\infty} = [1,0,1] $ & $\\infty$ & very infinite\\\\\n$1\/x(P_7) = 0, b^{\\infty} = [1,1,1] $ & $\\infty$ & very infinite \\\\\n$1\/x(P_8) = 0,     [ 0, 0, 1 ],\n    [ -1, 0, 1 ],\n    [ 3, 0, 1 ],\n    [ 1, -2, 1 ],\n    [ 1, 1, 0 ] = [1,0,0] $ & $\\infty$ & very infinite \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\nWe compute the $3$-adic Coleman integrals on a basis of $\\Hrig^1(X \\otimes \\QQ_p)$, in particular, the regular $1$-forms \nare given by\n\\begin{align*}\n\\omega_1 &=  (b^0 \\cdot (-8x^8 - 8x^7 + 86x^6 + 192x^5 + 118x^4 + 12x,\\\\\n&\\qquad\\qquad -31x^7 - 98x^6 - 75x^5 + 70x^4 + 183x^3 + 234x^2 + 83x - 12, \\\\\n&\\qquad\\qquad 31x^5 + 60x^4 - 52x^3 - 246x^2 - 119x + 12))\\frac{dx}{r}, \\\\\n\\omega_2  &= (b^0 \\cdot (2x^8 - 4x^7 - 56x^6 - 120x^5 - 76x^4 + 6x^2, \\\\\n&\\qquad\\qquad13x^7 + 44x^6 + 45x^5 - 22x^4 - 81x^3 - 144x^2 - 77x + 12,\\\\\n&\\qquad\\qquad -13x^5 - 24x^4 +  28x^3 + 138x^2 + 77x - 12))\\frac{dx}{r},\\\\\n\\omega_3 &=  (b^0 \\cdot (4x^7 + 22x^6 + 44x^5 + 30x^4 + 4x^3, \\\\\n&\\qquad\\qquad  -3x^7 - 10x^6 - 11x^5 + 6x^4 + 19x^3 + 42x^2 + 27x - 4,\\\\\n&\\qquad\\qquad  3x^5 + 4x^4 - 12x^3 - 46x^2 - 27x + 4))\\frac{dx}{r},\\end{align*}\nproducing the following values:\n\\begin{align*}\n\\int_{P_1}^{P_2} \\omega_1 &= 2 \\cdot 3^{2} + 3^{3} + 2 \\cdot 3^{5} + 3^{6} + 2 \\cdot 3^{7} + 3^{8} + O(3^{9}), \\\\\n\\int_{P_1}^{P_2} \\omega_2 &= 3^{3} + 3^{4} + 2 \\cdot 3^{5} + 2 \\cdot 3^{6} + 3^{7} + O(3^{9}),\\\\\n\\int_{P_1}^{P_2} \\omega_3 &= 3 + 2 \\cdot 3^{2} + 3^{3} + 3^{4} + 3^{5} + O(3^{9}).\n\\end{align*}\n\nWe use the values of these three integrals (i.e., by computing the kernel of the associated $3 \\times 1$ matrix) to compute that the two differentials\n\\begin{align*}\n\\xi_1 &= (1 + O(3^9))\\omega_1 + O(3^9)\\omega_2 + (430\\cdot3 + O(3^9))\\omega_3 \\\\\n\\xi_2 &=  O(3^9)\\omega_1 + (1 + O(3^9))\\omega_2 + (569\\cdot3^2 + O(3^9))\\omega_3\n\\end{align*}\ngive a basis for the regular $1$-forms annihilating rational points. Indeed, we can numerically see that the values of the two integrals \n$\\int_{P_1}^P \\xi_1, \\int_{P_1}^P \\xi_2$ vanish for all $P = P_3, P_4, \\ldots, P_8$.  The code for this example along with the complete \ncomputation which proves that $X(\\QQ)=\\{P_1,P_2,P_3,P_4,P_5,P_6,P_7,P_8\\}$ can be found in the file~\\texttt{.\/examples\/bps.m} in~\\cite{colemangit}.\n\n\n\\subsection{The modular curve $X_0(44)$} \\mbox{ } \\\\\n\nWe consider the genus $4$ curve $X = X_0(44)$. We work with the plane model found by Yang \\cite{yang}: $$Q(x,y) =y^5+12x^2y^3-14x^2y^2+(13x^4+6x^2)y-(11x^6+6x^4+x^2) = 0.$$ \nWe have \n\\[W^0=\n\\begin{bmatrix}\n1 & 0  &  0 & 0  & 0 \\\\\n0 & 1 &  0 & 0 & 0 \\\\\n0 & 0 & 1 & 0 & 0 \\\\\n0 & 0 & 0 & \\frac{1}{x} & 0 \\\\\n\\frac{-10x^3}{x^4 + 6x^2 + 1} &  \\frac{-6x^3 - 13x}{x^4 + 6x^2 + 1} &  \\frac{x^3 + 12x}{x^4 + 6x^2 + 1} &  \\frac{-x}{x^4 + 6x^2 + 1} & \\frac{1}{x^5 + 6x^3 + x} \\end{bmatrix}.\\]\n\n\\noindent Indeed, this plane model is singular, as we see $W_0 \\neq I$.  We have $$r = x(x^4 + 6x^2 + 1)(45753125x^8 + 8440476x^6 + 1340814x^4 + 69756x^2 + 3125)$$\nand\n$$b^0 = \\left[1, y, y^2, \\frac{y^3}{x}, \\frac{-10x^4-(6x^4-13x^2)y + (x^4+12x^2)y^2 - x^2y^3+1}{x^5 + 6x^3 +x}\\right].$$\nWe have that $(Q,W^0, W^{\\infty})$ has good reduction at $p=7$.\nLet $P_1$ be the (good) point $(1,1)$. We consider the point $P_2$ which lies over the singularity $x=0,y=0$ of the plane model (which is of degree $5$ in $y$). \nAs a point on the smooth model, $P_2$ has ramification index equal to $5$, the $b^0$ which is a local coordinate at $P_2$ is $y^3\/x$, so $b^0 = (1,0,0,0,0)$.  \nComputing $7$-adic integrals gives \n$$\\left(\\int_{P_1}^{P_2} \\omega_1, \\int_{P_1}^{P_2} \\omega_2, \\int_{P_1}^{P_2} \\omega_3, \\int_{P_1}^{P_2} \\omega_4\\right) = (O(7^9),O(7^9),O(7^9),O(7^9)),$$ \n\n\\noindent which seems to suggest that $[(P_2)-(P_1)]$ is a torsion point in the Jacobian of $X$. A computation in Magma verifies that $15[(P_2)-(P_1)] = 0.$\nThe code for this example can be found in the file \\texttt{.\/examples\/X0\\_44.m} in \\cite{colemangit}.\n\n\\subsection{A superelliptic genus 4 curve} \\mbox{ } \\\\\n\nWe consider the superelliptic genus $4$ curve $X\/\\QQ$ given by the plane model $$Q(x,y) = y^3 - (x^5 - 2x^4 - 2x^3 - 2x^2 - 3x) = 0.$$ \nUsing the \\texttt{Magma} intrinsic \\texttt{RankBounds}, which is based on~\\cite{poonenschaefer} and implemented by Creutz, we find that the Mordell-Weil rank of its Jacobian is $1$. A \nsearch yields the rational points \n$$P_1=(1,-2), P_2=(0,0), P_3=(-1,0), P_4=(3,0), P_5=\\infty.$$\nWe have $b^0=[1,y,y^2]$ and $r=x^5 - 2x^4 - 2x^3 - 2x^2 - 3x$. A basis for the regular $1$-forms on $X$ is given by \n\\[\n\\omega_1 = \\frac{y dx}{r}, \\; \\; \\; \\omega_2 = \\frac{xy dx}{r}, \\;\\;\\; \\omega_3 = \\frac{x^2y dx}{r}, \\;\\;\\; \\omega_4=\\frac{y^2 dx}{r}. \n\\]\nNow we take $p=7$ and compute\n\\[\n\\int_{P_1}^{P_2} \\omega_1 =  12586493\\cdot7 + O(7^{10}).\n\\]\nSince this integral does not vanish, $[(P_2) - (P_1)]$ is non-torsion in the Jacobian. \n\nThe space of annihilating regular $1$-forms is $3$-dimensional, and a basis is given by\n\\begin{align*}\n\\xi_1 &= (1 + O(7^{10})) \\omega_1 + O(7^{10}) \\omega_2 + O(7^{10}) \\omega_3 - (139167240 + O(7^{10})) \\omega_4 \\\\\n\\xi_2 &= O(7^{10}) \\omega_1 + (1 + O(7^{10}))\\omega_2 + O(7^{10}) \\omega_3 + (93159229 + O(7^{10})) \\omega_4\\\\\n\\xi_3 &= O(7^{10}) \\omega_1 + O(7^{10}) \\omega_2 + (1 + O(7^{10})) \\omega_3 + (8834289 + O(7^{10})) \\omega_4.\n\\end{align*}\nIndeed, we can numerically see that the values of the $3$ integrals \n$\\int_{P_1}^P \\xi_1, \\int_{P_1}^P \\xi_2, \\int_{P_1}^P \\xi_3$ vanish for $P = P_3, P_4,P_5$.  \nThe code for this example along with the complete computation which proves that $X(\\QQ)=\\{P_1,P_2,P_3,P_4,P_5\\}$ can be found in the file \\texttt{.\/examples\/C35.m} in \\cite{colemangit}. \\\\\n\nFurther examples illustrating how to call and use the code are available in the file \\texttt{examples.pdf} in \\cite{colemangit}.\n\n\\section*{Acknowledgements}\nWe would like to thank Amnon Besser, Netan Dogra, Alan Lauder and Steffen M\\\"{u}ller for helpful discussions. Balakrishnan is supported in part by NSF grant DMS-1702196, the Clare Boothe Luce Professorship (Henry Luce Foundation), and Simons Foundation grant \\#550023.\nTuitman is a Postdoctoral Researcher of the Fund for Scientific Research FWO - Vlaanderen. \\\\\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:introduction}\nDirect Coupling Analysis (DCA) is a collective term for a number\nof related techniques to learn the parameters in Ising\/Potts models\nfrom data and to use these inferred parameters in biological data analysis~\\cite{Nguyen-2017a}.\nDCA has led to a breakthrough in identifying\nepistatically linked sites in proteins from protein sequence data~\\cite{Weigt-2009a,Morcos-2011a,Stein-2015a,Michel-2017a,Cocco-2018a},\nwhich in turn has been used to predict spatial contacts from the sequence data~\\cite{Ovchinnikov-2017a,Michel-2017b,Ovchinnikov-2018a}.\nDCA has also been used to identify \nnucleotide-nucleotide contacts of RNAs~\\cite{DeLeonardis-2015a},\nmultiple-scale protein-protein interactions~\\cite{Gueudre-2016a,Uguzzoni-2017a},\namino acid--nucleotide interaction in RNA-protein complexes~\\cite{Weinreb-2016a}\nand synergistic effects not necessarily related to spatial contacts~\\cite{Figliuzzi-2016a,Hopf-2017a,CouceE9026,Schubert325993}.\n\nSkwark \\textit{et al} applied a version of DCA to\nwhole-genome sequencing data of a population of \\textit{Streptoccoccus pneumoniae}~\\cite{Skwark-2017a},\nand were able to retrieve interactions\nbetween members of the Penicillin-Binding Protein (PBP) family\nof proteins as well as other predictions.\n\\textit{S. pneumoniae} (pneumococcus) is an important human pathogen \nwhere resistance to antibiotics in the $\\beta$-lactam family of compounds\nare associated to alterations in their target enzymes, which are the PBPs~\\cite{Hakenbeck2012}.\nFurther results were recently given in~\\cite{Puranen-2017a}\nshowing robustness by using sequencing data from a \npneumococcal population from another continent, \nand identifying a novel seasonal phenotype signal.\nThree of the authors of the current article additionally recently\nshowed that DCA analysis on the bacterial genome scale does not\nneed supercomputing resources, but can be carried out\nin a reasonable time (hours) on a standard desktop computer~\\cite{GaoZhouAurell2018}.    \n\nThese advances raise the question why DCA works at all,\nand if one can identify from the outset when that is the case.\nAs discussed by one of us ``max-entropy'' arguments sometimes\nevoked in the literature are not pertinent to this issue~\\cite{Aurell2016}.\nInstead, we will here argue that at least for genome-scale data\nthe answer lies in a very different direction.\nWe will show that the \\textit{Quasi-Linkage Equilibrium} (QLE) \nof Kimura~\\cite{Kimura1956,Kimura1964,Kimura1965},\nas extended by Neher and Shraiman to statistical genetics on the genome scale~\\cite{NeherShraiman2009,NeherShraiman2011},\nprovides a natural and rational basis for DCA. \nAccording to this theory a population evolving \nwith sufficient amount of exchange of genetic material\n(recombination, or any form of sex) will settle down\nto a dynamic equilibrium where the \ndistribution of genotypes is of the form assumed by DCA.\nIn the opposite case of little exchange of genetic material\n(little sex) the distribution over genotypes\nis different and dominated by clones, \nidentical or very similar individuals\ndescended from a common ancestors.\nIn such a setting DCA is not an appropriate approach, and is likely to yield nonsense results.\n\nWe will also discuss the inference task of DCA in the context of QLE\nas realistically applied to biological data.\nWe will first show that DCA can give a much more sparse\nrepresentation of the data than correlations (covariances).\nThis is in line with the intended meaning of the acronym DCA: \nthe parameters in a Potts or an Ising model can be considered\n``direct couplings'', and while these are typically reflected in correlations (covariances),\nthe latter also includes combined effects, or ``indirect couplings''.\nSecond, the authors of~\\cite{NeherShraiman2009,NeherShraiman2011}\nassumed that a genotype can be described by a Boolean vector\n\\textit{i.e.} a string of $0$'s and $1$'s. This is almost never\nthe case for population-wide whole-genome sequencing data\ndue to varying gene content, which have to be represented as gaps.\nWe have therefore generalized the theory\nto categorical data and to a model of bacterial recombination.\nThird, as surveyed in~\\cite{Nguyen-2017a}, DCA as a methodology has matured \nconsiderably\nover the last decade.\nFor the mathematical task of inferring parameters in a Potts or Ising model   \nfrom data which was generated from such a model, the \nSmall-Interaction Expansion (SIE) used in~\\cite{NeherShraiman2009,NeherShraiman2011}\nis inferior to many other inference methods.\nWe will show that it is also inferior when applied to\nreal data in the sense of yielding much less sparse\nresults, and would also have specific problems when applied to simulation data. \nA conclusion of this work is hence that when\nQLE is combined with DCA on the genome scale, it should be combined\nwith one of the modern and more powerful versions of DCA.\n\nThe paper is organized as follows.\nSections~\\ref{sec:QLE}-\\ref{sec:log-Prob}\nreformulate the theory of \\cite{NeherShraiman2011}\nin a way suitable to our presentation and for categorical data.\nSection~\\ref{sec:QLE} hence contains a non-mathematical\noverview, while Sections~\\ref{sec:Neher-Shraiman} and~\\ref{sec:modifications} \ncontain the specific changes needed for categorical data and our model\nof bacterial recombination.\nSection~\\ref{sec:log-Prob} formulates the dynamics of\nPotts model parameters in QLE phase, which is a central result of the theory.\nSection~\\ref{sec:DCA-sparse} presents results for \nreal sequence data and Section~\\ref{sec:QLE-genetic-drift} for simulation data.\nSection~\\ref{sec:discussion} contains discussion and outlook for future work.\nTechnical details such as \na derivation of SIE for categorical data,\nsequence and code availability are given in appendices.\n\n\n\\section{Statistical genetics and Quasi-Linkage Equilibrium}\n\\label{sec:QLE}\nWe will here first present key concepts and results in a mostly\nnon-mathematical manner.\nThe driving forces of evolution are assumed to be genetic drift, \nmutations, recombination, and fitness variations. The first refers to the element of chance; in a finite \npopulation it is not certain which genotypes will reproduce and leave\ndescendants in later generations. The three latter are deterministic,\ndescribing the expected success or failure of different\ngenotypes. Mutations are hence random genome changes described by mean rates. \n\nRecombination (or sex) is the mixing of genetic material between different\nindividuals.\nIn diploid organisms every individual inherits \nhalf of its genetic material\nfrom the mother, and half from the father. This material is further\nmixed up in the process called cross-over so that each chromosome\nof the child consists of segments alternately inherited from the mother and the father.\nBy sequencing the parents and children in a single family\nthe per generation mutation rate and number of cross-over segments in human \nhas been measured to be about $30$ and $100$~\\cite{Roach636},\nnumbers that are in line with previous estimates.\nBy this measure recombination is hence in human about three times faster than mutations.\nIn bacteria recombination happens by transformation, transduction, and conjugation.\nThe ratio of recombination to mutations \ndiffer greatly between different bacterial species \nand can also differ between different strains and different environments of the same species.  \nIn this work we use data from \\textit{S. pneumoniae}\nwhere this ratio has been estimated from less than one to over \nforty, but with an average close to nine~\\cite{Chaguza2016}.\nSimilarly to the analysis in~\\cite{NeherShraiman2011}\nwe will for the most part here assume that\nrecombination is a faster and stronger effect than mutations.\n\nFitness means in statistical genetics a propensity \nfor a given genotype to propagate its genomic material to the next generation.\nLike mutation and recombination fitness is hence here a rate, measured in units $\\left(\\hbox{time}\\right)^{-1}$.\nFitness variations refer to the variations of these rates.\nConsider then then the effects of recombination and fitness\non correlated variations in a population, ignoring mutations and\ngenetic drift. The correlation between alleles $\\alpha$ and\n$\\beta$ at loci $i$ and $j$ is $M_{ij}(\\alpha,\\beta)=f_{ij}(\\alpha,\\beta)\n- f_{i}(\\alpha)f_{j}(\\beta)$ where $f_{i}(\\alpha)$ is the frequency\nof allele $\\alpha$ at locus $i$ and similarly $f_{j}(\\beta)$,\nand where $f_{ij}(\\alpha,\\beta)$ is the frequency of simultaneously finding\nalleles $\\alpha$ and $\\beta$ at loci $i$ and $j$. \nIf there is recombination between $i$ and $j$\nbut \nare no fitness variations at all,\nthen it is trivial to see that $M_{ij}(\\alpha,\\beta)$ must decay to zero.\nThis state is called \\textit{Linkage Equilibrium} (LE).\n\nIf now instead fitness variations are small but non-zero,\nthen non-zero correlations\nmay persist.\nWe will assume that the fitness of genotype $\\mathbf{g}$ which carries allele $g_i$ on locus $i$ depends on single-locus\nvariations and pair-wise co-variations, that is\n\\begin{equation}\n  \\label{eq:fitness}\n  F(\\mathbf{g}) = F_0 + \\sum_i F_i(g_i) + \\sum_{ij}F_{ij}(g_{i},g_{j})\n\\end{equation}\nIf so, the first central result of statistical genetics is that when recombination is\nsufficiently strong, the distribution will have the form\n\\begin{equation}\n  \\label{eq:Potts}\nP(\\mathbf{g}) = \\frac{1}{Z}\\exp\\left(\\sum_i h_i(g_i) + \\sum_{ij} J_{ij}(g_i,g_j)\\right) \n\\end{equation}\nThe above distribution is also the Gibbs-Boltzmann distribution over variables $\\mathbf{g}$\nwith energy terms $h_i$ and $J_{ij}$, and where $Z$ (the partition function) is the normalization.\nThe second central result is that\n\\begin{equation}\n  \\label{eq:main-Neher-Shraiman-eq}\n  J_{ij}(\\alpha,\\beta) = \\frac{F_{ij}(\\alpha,\\beta)}{rc_{ij}}\n\\end{equation}\nwhere $r$ is an overall recombination rate and $c_{ij}$ is the probability that alleles at loci $i$ and $j$ are inherited from the\nsame parent.  \nFor the most part we will in the following assume that $c_{ij}$ equals $\\frac{1}{2}$, appropriate if\nrecombination is sufficiently strong and \nloci $i$ and $j$ are sufficiently far apart on the genome.\nNote that the right-hand\nside of \\eqref{eq:main-Neher-Shraiman-eq} is the ratio of two rates, and therefore dimension-less.\nFor the distribution \\eqref{eq:Potts} the parameters\n$J_{ij}(\\alpha,\\beta)$ carry the same information as the correlations $M_{ij}(\\alpha,\\beta)$ but in a de-convoluted or ``direct'' manner.\n\nInferring $J_{ij}(\\alpha,\\beta)$ from data is what \nthe methods known as DCA achieve~\\cite{Nguyen-2017a}.\nFrom \\eqref{eq:main-Neher-Shraiman-eq} this gives\nfitness parameters $F_{ij}(\\alpha,\\beta)$ up to a proportionality (the overall rate $r$),\nand for pairs of loci sufficiently far apart on the genome so that $c_{ij}$ is approximately constant.\nRecombination does not change single-locus frequencies; in a stationary state parameters $h_i(\\alpha)$ instead result from a dynamic equilibrium between\nfitness and mutations. In the absence of mutations QLE in an infinite population\nis in fact only a long-lived transient while the $h_i(\\alpha)$ change slowly in time\nas the population drifts towards fixation. \nIn a finite population both $h_i(\\alpha)$ and\n$J_{ij}(\\alpha,\\beta)$\nalso fluctuate in time, and the prediction \\eqref{eq:main-Neher-Shraiman-eq} does not apply directly.\nAll these aspects have to be taken into account when applying DCA techniques to \nanalyze a QLE phase.\n\nThe third central prediction of statistical genetics is that when fitness variations still have the form \\eqref{eq:fitness} but\nare not small compared to recombination, then the distributions\nwill not be of the form \\eqref{eq:Potts}. In that phase, in~\\cite{NeherShraiman2009,NeherShraiman2011}\ncalled \\textit{Clonal Competition} (CC), the distribution is instead better described\nas\n\\begin{equation}\n  \\label{eq:mixture-model}\n  P(\\mathbf{g}) = \\sum_c \\mu_c P_c(\\mathbf{g})\n\\end{equation}\nwhere the sum goes over clones, $\\mu_c$ is the weight of clone $c$, and $P_c(\\mathbf{g})$ is some distribution peaked\naround clone center $\\mathbf{g}_c$.\nStatistical genetics hence predicts a parameter-dependent transition between the two canonical distribution families\nin high-dimensional statistics, namely the exponential model (\\eqref{eq:Potts}) and the mixture model\n(\\eqref{eq:mixture-model}). \nA further difference between QLE and CC is that in QLE the joint distribution over more than\none genotype approximately factorizes, $P(\\mathbf{g}_1,\\ldots,\\mathbf{g}_N)\\approx P(\\mathbf{g}_1)\\cdots P(\\mathbf{g}_N)$.\nIn CC phase this is not so; genomes related by descent do not vary independently. \nA different analogy, discussed in~\\cite{NeherShraiman2009,NeherShraiman2011}\nis that of equilibrium states in disordered\nsystems~\\cite{MezardParisiVirasoro,MezardMontanari};\n\\eqref{eq:Potts} is like a high-temperature replica-symmetric para-magnetic phase, while \\eqref{eq:mixture-model} is like a low-temperature replica symmetry-breaking\nspin glass phase. \n\n\\section{Statistical genetics for categorical data}\n\\label{sec:Neher-Shraiman}\nIn this section we summarize statistical genetics as formulated in~\\cite{NeherShraiman2009,NeherShraiman2011}\nin a more technical manner, and generalize the theory to categorical data \\textit{i.e.} to\nwhen there can be more than two alleles per locus.\nLet there be $M_i$ alleles at locus $i$ and\nlet the allele be indicated by a variable $l_i$ that takes values $1,2,\\ldots,M_i$.\nThe frequency of allele $\\alpha$ at locus $i$ is\n\\begin{equation}\n  f_i(\\alpha) = \\left< \\mathbf{1}_{l_i,\\alpha} \\right>\n\\end{equation}\nThese quantities satisfy $\\sum_{\\alpha=1}^{M_i} f_i(\\alpha) = 1$.\nThe covariance matrix between loci $i$ and $j$ is\n\\begin{equation}\n    \\label{eq:c-ij}\n  M_{ij}(\\alpha,\\beta) = \\left< \\mathbf{1}_{l_i,\\alpha} \\mathbf{1}_{l_j,\\beta}\\right> - f_i(\\alpha) f_j(\\beta) \n\\end{equation}\nThese quantities satisfy $\\sum_{\\alpha=1}^{M_i}M_{ij} (\\alpha,\\beta) = \\sum_{\\beta=1}^{M_j}M_{ij}(\\alpha,\\beta) = 0$.\nThe variance matrix at one locus is\n\\begin{equation}\n  \\label{eq:c-ii}\n  M_{ii}(\\alpha,\\beta) = \\mathbf{1}_{\\alpha,\\beta} f_i(\\alpha)  - f_i(\\alpha) f_j(\\beta)  \n\\end{equation}\nand satisfies $\\sum_{\\alpha=1}^{M_i}M_{ii} (\\alpha,\\beta) = \\sum_{\\beta=1}^{M_j}M_{ii}(\\alpha,\\beta) = 0$.\n\nStatistical genetics are evolution equations for the distributions over genotypes\n\\begin{equation}\n  \\frac{d}{dt} P(\\mathbf{g}) =   \\frac{d}{dt}|_{mut} P(\\mathbf{g}) + \\frac{d}{dt}|_{fitness} P(\\mathbf{g}) + \\frac{d}{dt}|_{recomb} P(\\mathbf{g}) \n\\end{equation}\nwhere the three terms on the right-hand side represent the changes due to mutations, fitness variations and recombination.\nThe mechanisms of mutations and fitness are classical in population genetics, and known as Wright-Fisher models.\n\nSingle-locus mutations are hence modelled by matrices $\\mu^{(i)}_{\\alpha, \\beta}$\nwhich give the rate at which allele $\\alpha$ on locus $i$ changes to allele $\\beta$.\nLet $F^{(i)}_{\\alpha, \\beta}$ be the operator which if the allele at locus $i$ is $\\alpha$ changes\nit to $\\beta$, and otherwise does nothing. In the dynamic equation for probability\nmutations hence enter as\n\\begin{equation}\n  \\label{eq:evolution-mutations}\n  \\frac{d}{dt}|_{\\hbox{mut}} P \\left(\\mathbf{g} \\right)= \\sum_i \\sum_{\\alpha\\beta} \\mathbf{1}_{g_i,\\alpha}\\left(\n  \\mu^{(i)}_{\\beta,\\alpha} P \\left( F^{(i)}_{\\alpha,\\beta} \\mathbf{g}\\right) - \\mu^{(i)}_{\\alpha,\\beta}  P \\left(\\mathbf{g} \\right) \\right)\n\\end{equation}\nThis gives contributions to the dynamic equations for the frequencies and correlations as\n\\begin{eqnarray}\n  \\label{eq:f-i-mutations}\n  \\frac{d}{dt}|_{\\hbox{mut}} f_i(\\alpha) &=& \\sum_{\\gamma} \\mu^{(i)}_{\\gamma,\\alpha} f_i(\\gamma) - \\sum_{\\gamma} \\mu^{(i)}_{\\alpha,\\gamma} f_i(\\alpha) \\\\\n    \\label{eq:c-ij-mutations}\n    \\frac{d}{dt}|_{\\hbox{mut}} M_{ij}(\\alpha,\\beta) &=& \\sum_{\\gamma} \\mu^{(i)}_{\\gamma,\\alpha} M_{ij}(\\gamma,\\beta) - \\sum_{\\gamma} \\mu^{(i)}_{\\alpha,\\gamma} M_{ij}(\\alpha,\\beta) + \\nonumber \\\\\n                                                && \\sum_{\\delta} \\mu^{(j)}_{\\delta,\\beta} M_{ij}(\\alpha,\\delta) - \\sum_{\\delta} \\mu^{(j)}_{\\beta,\\delta} M_{ij}(\\alpha,\\beta)  \n\\end{eqnarray}\nIn the simulations reported below all transition rates $\\mu^{(i)}_{\\alpha, \\beta}$ are the same.\nAs discussed above it is often a reasonable assumption to take mutations a weaker effect than\nfitness variations and recombination.\n\nFitness variations, such as \\eqref{eq:fitness} above, act on the distributions over genotypes as\n\\begin{equation}\n  \\label{eq:evolution-fitness}\n  \\frac{d}{dt}|_{\\hbox{fitness}} P \\left(\\mathbf{g} \\right)= \\left(F(\\mathbf{g})-\\left<F\\right>\\right) P \\left(\\mathbf{g} \\right)\n\\end{equation}\nwhere $\\left<F\\right>=\\sum_{\\mathbf{g}}F(\\mathbf{g}) P \\left(\\mathbf{g}\\right)$ is the instantaneous average of fitness over the population.\n\nPotts models are defined as \n\\begin{equation}\n  P(\\mathbf{g}) = \\frac{1}{Z} \\exp\\left( \\sum_{i,\\alpha} h_{i}(\\alpha) \\mathbf{1}_{g_i,\\alpha} + \\sum_{i,j,\\alpha,\\beta} J_{ij}(\\alpha,\\beta) \\mathbf{1}_{g_i,\\alpha} \\mathbf{1}_{g_j,\\beta}\\right)\n\\end{equation}\nAs written the model over-parametrized since the same distribution is found by shifting all\n$h_{i}(\\alpha)$ by a constant $c_i$ or all $J_{i,j}(\\alpha,\\beta)$ by a vector $c_{ij}(\\beta)$, or a vector $c_{ij}(\\alpha)$.\nIn the DCA literature it is customary to go to the Ising gauge~\\cite{Weigt-2009a,Ekeberg-2014a} given by\n\\begin{equation}\n  \\sum_{\\alpha} h_{i}(\\alpha) = \\sum_{\\alpha} J_{i,j}(\\alpha,\\beta) = \\sum_{\\beta} J_{i,j}(\\alpha,\\beta) = 0\n\\end{equation}\n\n\\section{Bacterial recombination in statistical genetics}\n\\label{sec:modifications}\nRecombination (or sex) takes many different forms depending on if the organism is haploid or diploid\nand the type of recombination. The mechanism formulated in~\\cite{NeherShraiman2009,NeherShraiman2011}\nis specifically for sexual reproduction in haploid yeast, where two parents each produce a mating body (copy of parent genome),\nand these two mating bodies merge and produce one new genome while the other half of the genetic material of the two mating bodies is discarded.\nAs closer to our data we consider instead a form of bacterial recombination,\nfor which however the evolution essentially turns out to be the same,\nmodulo a \\textit{Stosszahlansatz}.\n\nRecombination is thus (we assume) distinguished by\ntwo genomes merging and forming two new genomes. \nIn an elementary step two genotypes are therefore lost (the parents) and\ntwo genotypes are gained (the offspring). Let $E_{\\mathbf{g}_1,\\mathbf{g}_2 \\to \\mathbf{g}_1',\\mathbf{g}_2'}$ be the event that\ntwo individuals with genotypes $\\mathbf{g}_1$ and $\\mathbf{g}_2$\nrecombine and give two individuals $\\mathbf{g}_1'$ and $\\mathbf{g}_2'$.\nTo describe the kinetics of the individual process we assume that recombination between the two parents happen with\nrate $r Q(\\mathbf{g}_1,\\mathbf{g}_2)$ where $r$ an overall rate of recombination\nand $Q(\\mathbf{g}_1,\\mathbf{g}_2)$ a relative rate.\nThe two new genotypes\n$\\mathbf{g}_1'$ and $\\mathbf{g}_2'$ are specified by an indicator variable $\\mathbf{\\xi}$:\n\\begin{eqnarray}\n  \\mathbf{g}_1':\\quad   g^{(1)'}_i &=&  \\xi_i g_i^{(1)}  + (1-\\xi_i) g_i^{(2)} \\\\\n  \\mathbf{g}_2':\\quad   g^{(2)'}_i &=& (1-\\xi_i) g_i^{(1)} + \\xi_i g_i^{(2)} \n\\end{eqnarray}\nand this outcome of the recombination happens with probability $C(\\mathbf{\\xi})$.\nThe total rate of the individual event is hence $r Q(\\mathbf{g}_1,\\mathbf{g}_2)C(\\mathbf{\\xi})$.\nThe change of the distribution over genotypes due to recombination is given by\n\\begin{eqnarray}\n  \\label{eq:Potts-recomb-bact}\n  \\frac{d}{dt}|_{rec} P(\\mathbf{g}) &=&  r \\sum_{\\mathbf{\\xi},\\mathbf{g}'}   C(\\mathbf{\\xi})\n  \\big[ Q(\\mathbf{g}_1,\\mathbf{g}_2) P_2(\\mathbf{g}_1,\\mathbf{g}_2) - \\nonumber \\\\\n         && Q(\\mathbf{g},\\mathbf{g}')    P_2(\\mathbf{g},\\mathbf{g}') \\big]\n\\end{eqnarray}\nIn practice it is hard to use \\eqref{eq:Potts-recomb-bact} without assuming that the pair probabilities factorize,\nas in gas collisions at low densities in statistical physics.\nWe assume for simplicity also\nthat $Q$ depends only on the overlap $q$ between\nthe two genotypes $\\mathbf{g}$ and $\\mathbf{g}'$:\n\\begin{equation}\n  q(\\mathbf{g},\\mathbf{g}') = \\frac{1}{L}\\sum_{i=1}^L \\mathbf{1}_{g_i,g_i'}\n\\end{equation}\nRecombination as modelled above does not change the overlap.\nThis can be seen as follows:: $q(\\mathbf{g}_1,\\mathbf{g}_2) = 1 - \\frac{1}{L}\\sum_{i=1}^L \\mathbf{1}_{l_i^{(1)},l_i^{(2)}}$\nand $\\mathbf{1}_{l_i^{(1)},l_i^{(2)}}=\\xi_i(1-\\xi_i)\\mathbf{1}_{l_i,l_i} + \\xi_i^2\\mathbf{1}_{l_i,l_i'} + (1-\\xi_i)^2\\mathbf{1}_{l_i',l_i} + (1-\\xi_i)\\xi_i \\mathbf{1}_{l_i',l_i'}$. As the indicator variable takes values zero and one \nthis gives $\\mathbf{1}_{l_i^{(1)},l_i^{(2)}}=\\mathbf{1}_{l_i,l_i'}$.\n   \nBy this invariance of overlaps \nand the factorization assumption the right-hand side of \\eqref{eq:Potts-recomb-bact} simplifies to:\n\\begin{eqnarray}\n  &&r \\sum_{\\mathbf{\\xi},\\mathbf{g}'} C(\\mathbf{\\xi}) P(\\mathbf{g}')   Q(\\mathbf{g},\\mathbf{g}') \\sum_{i,j,\\alpha,\\beta} J_{i,j}(\\alpha,\\beta) \\Big(\n  \\mathbf{1}_{g_i^{(1)},\\alpha} \\mathbf{1}_{g_j^{(1)},\\beta} + \\nonumber \\\\\n  &&\\qquad \\mathbf{1}_{g_i^{(2)},\\alpha} \\mathbf{1}_{g_j^{(2)},\\beta}- \\mathbf{1}_{g_i,\\alpha} \\mathbf{1}_{g_j,\\beta}-\\mathbf{1}_{g_i',\\alpha} \\mathbf{1}_{g_j',\\beta}\\Big) \\nonumber \\\\\n  \\label{eq:NS-B4b}\n  &=& \\sum_{i,j,\\alpha,\\beta} c_{ij} J_{i,j}(\\alpha,\\beta) \\Big(\n                                                        \\mathbf{1}_{g_i,\\alpha} \\hbox{E}_{Q}\\left[\\mathbf{1}_{g_j',\\beta} \\right]\n                                                       + \\hbox{E}_{Q}\\left[\\mathbf{1}_{g_i',\\alpha}\\right] \\mathbf{1}_{g_j,\\beta}\n                                                       \\nonumber \\\\\n &&\\qquad\\qquad                                        - \\left<Q\\right> \\mathbf{1}_{g_i,\\alpha}  \\mathbf{1}_{g_j,\\beta}\n                                                       - \\hbox{E}_{Q}\\left[\\mathbf{1}_{g_i',\\alpha} \\mathbf{1}_{g_j',\\beta} \\right] \\Big)\n\\end{eqnarray}\nwhere we have used the abbreviations\n\\begin{eqnarray}\nc_{ij} &=&\\sum_{\\mathbf{\\xi}} C(\\mathbf{\\xi})\\left(\\xi_i(1-\\xi_j)+(1-\\xi_i)\\xi_j\\right) \\\\\n\\left<Q\\right> &=& \\sum_{\\mathbf{g}'}Q(\\mathbf{g},\\mathbf{g}')P(\\mathbf{g}') \\\\\n\\hbox{E}_Q\\left[\\mathbf{1}_{g_i',\\alpha} \\right] &=& \\sum_{\\mathbf{g}'} \\mathbf{1}_{g_i',\\alpha} Q(\\mathbf{g},\\mathbf{g}')P(\\mathbf{g}') \\\\\n\\hbox{E}_Q\\left[\\mathbf{1}_{g_i',\\alpha}\\mathbf{1}_{g_j',\\beta} \\right] &=& \\sum_{\\mathbf{g}'} \\mathbf{1}_{g_i',\\alpha} \\mathbf{1}_{g_j',\\beta}  Q(\\mathbf{g},\\mathbf{g}')P(\\mathbf{g}')\n\\end{eqnarray}\nThe first of these is the probability that two loci are\ninherited from the same parent and does not (for this model)\ndepend on the genotype $\\mathbf{g}$.\nThe last three averages on the other hand depend on $\\mathbf{g}$. However, if\nthe function $Q$ is not too sharply focused the dependence can be taken\nweak. In particular, we assume that $\\left<Q\\right>$ is self-averaging, and essentially does not depend on $\\mathbf{g}$. \nIn spin glass physics language we hence assume that\n$\\left<Q\\right>$, $\\left<Q \\mathbf{1}_{l_i',\\alpha}\\right>$ and $\\left<Q\\mathbf{1}_{l_i',\\alpha}\\mathbf{1}_{l_j',\\beta}\\right>$\nare self-averaging in the ``paramagnetic'' phase where QLE is expected to hold.\n\n\n\\section{Evolution equation for log-probability in QLE}\n\\label{sec:log-Prob}\nIn QLE the evolution equation can conveniently be written for the logarithmic probability\n\\begin{equation}\n  \\label{eq:logPotts}\n  \\frac{d}{dt}\\log P(\\mathbf{g})= -\\frac{\\dot{Z}}{Z} + \\sum_{i,\\alpha}\\dot{h}_{i}(\\alpha)\n                                  \\mathbf{1}_{g_i,\\alpha} + \\sum_{i,j,\\alpha,\\beta}\\dot{J}_{i,j}(\\alpha,\\beta) \\mathbf{1}_{g_i,\\alpha} \\mathbf{1}_{g_j,\\beta}     \n\\end{equation}\nand the various terms identified. Fitness enters (\\ref{eq:logPotts}) as\n\\begin{equation}\n  \\frac{d}{dt}|_{fitness} h_{i}(\\alpha) = f_{i}(\\alpha) \\qquad\n  \\frac{d}{dt}|_{fitness} J_{i,j}(\\alpha,\\beta) = f_{i,j}(\\alpha,\\beta)              \n\\end{equation}\nand if there were higher-order terms in fitness (more than pair-wise dependencies)\nthey would enter higher than quadratic terms in the QLE distribution in the same way.\nIgnoring mutations and genetic drift we have for the pair-wise dependencies\n\\begin{equation}\n  \\label{eq:final-result}\n  \\dot{J}_{i,j}(\\alpha,\\beta) = f_{i,j}(\\alpha,\\beta) -r \\left<Q\\right> c_{ij}  J_{i,j}(\\alpha,\\beta)\n\\end{equation}\nwhere the contribution from recombination is simply read off from \\eqref{eq:NS-B4b}.\n\\eqref{eq:final-result} is a relaxation equation which pushes the Potts model parameter $J_{i,j}(\\alpha,\\beta)$\nto be the ratio of two rates, \\eqref{eq:main-Neher-Shraiman-eq} above.\nWhen the data is from one population in a stationary state the average relative rate $\\left<Q\\right>$\ncan be subsumed in the overall rate $r$.\nWhen gentoypes are Boolean vectors this gives the same result as Eq.~25 in~\\cite{NeherShraiman2011}.\n\nAs observed above recombination does not change single-locus frequencies,\nand without mutations the $f_{i}(\\alpha)$ will drift towards fixation (taking values $0$ or $1$).\nOnce the population has reached fixation at locus $i$ there can no longer\nbe any non-zero correlation of Potts parameter involving $i$, and in \nsuch a setting \nQLE is therefore only a long-lived quasi-stationary state (for the correlations, and for the $J_{i,j}(\\alpha,\\beta)$'s).\nNote that by \\eqref{eq:NS-B4b} recombination terms enter in the \nevolution equations of $h_{i}(\\alpha)$, in combination with the quantities $J_{i,j}(\\alpha,\\beta)$.\nThis is no contradiction, because   \nwhen correlations are non-zero there is not a one-to-one relation between\nsingle-locus frequencies ($f_{i}(\\alpha)$) and Potts model magnetization parameters ($h_{i}(\\alpha)$);\nrecombination can influence the latter but not the former.\n\nThe break-down of the relaxational equation \\eqref{eq:final-result} when the \nsingle-locus frequencies go to fixation can be understood as follows.\nIn such a setting the $J_{i,j}(\\alpha,\\beta)$'s would first remain of order unity, \nwhile the $h_{i}(\\alpha)$'s would tend to $\\pm\\infty$.\nWhen the $h_{i}(\\alpha)$'s become large enough that the minor alleles in a finite-$N$\npopulation are likely to be present only in a few copies, \na few random events can remove all of the remaining ones at once, \nwhich sets the correlation and the $J_{i,j}(\\alpha,\\beta)$ to zero in one go.\nAlternatively the argument can be made starting from Eqs.~37-29 in~\\cite{NeherShraiman2009},\nwhich are stochastic differential equations for the frequencies and pair-wise correlations\n($f_i(\\alpha)$ and $M_{ij}(\\alpha,\\beta)$ in our case).\nWhen translated  \nto equations for $h_{i}(\\alpha)$ and $J_{i,j}(\\alpha,\\beta)$ near fixation\nthe noise will be large, which destabilizes \\eqref{eq:final-result}.\n\n\n\\section{DCA for whole-genome sequencing data}\n\\label{sec:DCA-sparse}\nA set of whole-genome sequences of the human pathogen \\textit{S. pneumoniae} obtained in the Maela collaboration (Materials \\& Methods) \ncan be represented as about $3,000$ genotypes \nof about $100,000$ loci each. \nCorrelations and Potts model terms obtained from this data have qualitatively different distributions,\nas shown in Fig~\\ref{fig:distributions-accumulated} and Fig~\\ref{fig:distributions-genomic-distance}. \n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.95\\linewidth]{distribution-all-semilogy}\n\\caption{Cumulative distributions of (a) correlations; (b) pseudo-likelihood maximization (PLM); (c) small-interaction expansion (SIE); \nSemi-logarithmic scale visualizing the distributions of of approximately $10^{10}$ elements. \nThe scalar value associated to each pair of loci $i$ and $j$ is the Frobenius norm\nof the $3\\times 3$-correlation matrix (case a) or the Frobenius norm of the inferred \nPotts model $3\\times 3$-matrix element (cases b and c). \n}\n\\label{fig:distributions-accumulated}\n\\end{figure}\n\nThe number of correlations larger than a cut-off $c$ grows quickly when $c$\ndecreases below its maximum value,\nwhile the cumulative distribution of inferred Potts model couplings\nhave a much more pronounced tail.\nThis implies that the set of largest DCA couplings is better separated than the largest correlations from the unavoidable\nbackground due to under-sampling~\\cite{Xu-2017a,Xu2018}.\nCorrelations also generally have a more uniform distribution\nacross genomic distance, \nwhile the representation as a Potts model is more sparse (Fig~\\ref{fig:distributions-genomic-distance} (a) and (b)),.\nThe first-order perturbative version of DCA employed in~\\cite{NeherShraiman2009,NeherShraiman2011}\nhere called SIE\ngives in both instances results closer to correlations, \\textit{see} Fig~\\ref{fig:distributions-accumulated} (c) and Fig~\\ref{fig:distributions-genomic-distance} (c).\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.95\\linewidth]{statistics-distance-all-semilogx}\n\\caption{Distributions as functions of genomic distance: (a) correlations; \n(b) pseudo-likelihood maximization (PLM).\n(c) small-interaction expansion (SIE).\nSame data and same norms as in Fig.~\\protect\\ref{fig:distributions-accumulated}.\nFigures show averages in windows of genomic distance.\nBlue: maximum, red: top-$1\\%$, yellow: top-$5\\%$, violet: top-$10\\%$, green: mean, all in window.\nThe curves in (b) show a sharp initial decrease with genomic distance which generally much lower\nvalues beyond genomic distance $10^3$ where recombination can be expected to act effectively.\n}\n\\label{fig:distributions-genomic-distance}\n\\end{figure}\n\nFig~\\ref{fig:Maela-scatter-plot} shows pair-wise scatter-plots of correlations and DCA terms\nobtained by PLM and SIE.\nIn all three cases the scatter-plots are ``clouds of points'', indicating that DCA and correlations\nmeasure different properties of the data. \nThe scatter-plot of PLM vs correlations shows a weak trend,\nsuch that larger PLM scores are associated to larger correlations.\nSuch trends are absent in correlations-SIE and PLM-SIE, and\nSIE values are also numerically large.\nIn fact, the correlation matrix is under-sampled, hence smaller correlations reflect sampling noise \nand this is also an issue for SIE, as well as the sensitive dependence on almost fixed alleles in this procedure.\nPLM scores and correlations were compared graphically for this data in~\\cite{GaoZhouAurell2018}, \nwith a cut-off excluding short-range interactions.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.95\\linewidth]{maela3K-N_81506-B_3145-x_1--comp-PLM-SIE-cor-linear}\n\\caption{Pair-wise scatter plot of correlations, SIE and PLM: \n(a) PLM vs correlations; \n(b) SIE vs PLM;\n(c) SIE vs PLM.\nSame data and same norms as in \nFig~\\protect\\ref{fig:distributions-accumulated} and Fig~\\protect\\ref{fig:distributions-genomic-distance}. \nThe numerical scale in each direction depends on the \ndetails of the norms and inference procedure, \\textit{e.g}\n$PLM$ scores depend on $L_2$ regularization parameters.\nCorrelations and PLM scores are numerically similar while SIE scores are not, as discussed in text.\n}\n\\label{fig:Maela-scatter-plot}\n\\end{figure}\n\n\n\\section{The QLE phase is obscured by genetic drift}\n\\label{sec:QLE-genetic-drift}\nGenetic drift is the random changes from one generation to the next due to chance events.\nIn a finite population statistical genetics as described above only holds on the \naverage; when following one population in time fluctuations of order $N^{-\\frac{1}{2}}$ appear\nfor observables such as single-locus frequencies and pair-wise loci-loci correlations.\nFig.~\\ref{fig:time-development} \nreports simulations using the FFPopSim software showing that these fluctuations can in practice be quite large, even \nfor populations that are not small.\n\n\\begin{figure}\n\\includegraphics[width=.95\\linewidth]{magnetization-and-correlation-1-2-3-1.pdf}\n\\caption{\\label{fig:time-development}  \nTemporal behavior of (a) all magnetizations defined as $\\chi_i=f_i(2)-f_i(1)$\nand ((b) and (c)) two selected correlations defined as $\\chi_{ij}=f_{ij}(1,1)-f_{ij}(1,2)-f_{ij}(2,1)+f_{ij}(2,2)$ \nin a simulation of a population of Ising genomes (two alleles per locus). \nNumber of loci $L$ of the genotypes is 256, number of genotypes $N$ in the population is 50000,\nother simulation parameters as reported in Materials \\& Methods.\nData is taken every five generations, total simulation time is $2500$ generations.\n}\n\\end{figure}\n\nAccording to the theory developed \nin~\\cite{NeherShraiman2011} (Appendix C)\ndynamics of correlations is relaxational and the curves of correlations\nvs time hence should fluctuate around an equilibrium value, which \nis the one given in \\eqref{eq:main-Neher-Shraiman-eq} above.\nThe fluctuations in Fig.~\\ref{fig:time-development} are however large compared to the pair-wise fitness values,\nand DCA inference from instantaneous values of the ensemble correlations\nare not good predictors for pairwise fitness (data not shown).\nThe dynamics of frequencies is not relaxational, and one may hence observe\nlarge changes where the population at one locus changes from one allele to another.\nA further conclusion is that SIE should not here be an appropriate \ninference procedure also because fluctuations \nin the frequencies are large and have long time scales; flavors of DCA\nthat rely only on correlations should exhibit better performance.\n\n\\section{Discussion}\n\\label{sec:discussion}\nThe main question addressed in this work is if and when DCA can be expected to work\nfor genome-scale epistasis analysis. We have given an answer in the context\nof statistical genetics: for a population evolving under \nrecombination, mutations and fitness variations this is so when \nrecombination is sufficiently fast. The joint distribution of the population over genotypes\nthen approximately factorizes into a product of identical Potts distributions (\\ref{eq:Potts}). \nTreating a set of genomes as independent samples from such a distribution\nallows to infer fitness parameters ($F_{ij}$) from Potts model parameters ($J_{ij}$)\nby inverting \\eqref{eq:main-Neher-Shraiman-eq}, and this is essentially what using DCA on such data means.\nWe now discuss limits to the analysis and further directions. \n\nThe first limitation is that \nDCA cannot be expected to yield meaningful results when recombination is weak.\nOne example of such an effect was already given in~\\cite{Skwark-2017a} \nwhere also data from \\textit{Streptococcus pyogenes} was presented (Fig. 6 of~\\cite{Skwark-2017a}).\nAnother example was recently given on\n\\textit{Vibrio parahaemolyticus}, a human gastrointestinal pathogen\nin panmixia \\textit{i.e.} where all strains are able to recombine, but  \nhaving a very low overall recombination rate~\\cite{Falush2018}.\nThe problem of inferring fitness from data on populations that are in the CC phase \nappears to be both conceptually and practically important; \nwe hope to be able to return to such questions in future work.\n\nA second limitation concerns finite populations, particularly simulated data, where the population has to be \nof moderate size.\nAccording to the theory developed for Ising genomes in~\\cite{NeherShraiman2011} (Appendix~C)\nand qualitatively confirmed above in Fig.~\\ref{fig:time-development},  \nfrequencies and correlations follow stochastic differential equations \nwith noise strength scaling as $N^{-\\frac{1}{2}}$. \nIn principle Potts model parameters ($h_i(\\alpha)$ and $J_{ij}(\\alpha,\\beta)$)\nfor categorical data also follow stochastic differential equations, but of a more complicated form\ndue to the inverse Ising\/Potts relations.\nApplying the DCA procedure to finite-$N$ data thus requires \nparameter inference from a high-dimensional stochastic time series\nwith a complicated deterministic part. This may not be an easy task.\n\nA third limitation is the neglect of spatial and environmental separation. \nBacteria such as the human pathogen~\\textit{Helicobacter pylori} readily recombine if they meet, but can only do so when\ntheir human host populations overlap~\\cite{Thorell2017}. \nAllele frequencies may be different for different bacterial populations, reflecting\ndifferences in the host populations and environments. \nIf data from the different populations is pooled, this will be a confounding factor for some flavors of DCA \n\\textit{e.g.} for PLM and SIE, \nThese types of issues appear to merit further study.  \n\n\\section*{Acknowledgments}\nE.A. thanks Prof Boris Shraiman for enlightening discussions.\nThis research was supported by National Science Foundation of China (grant numbers \n11647601 and 11421063). The numerical computations were partly carried out \nat the HPC cluster of ITP-CAS.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction} \n\nLow-energy nuclear reaction dynamics has successfully been treated within the stationary-state multi-channel scattering theory including complex potentials \\cite{Feshbach,Taylor}. \nHowever, this \\emph{cannot} account for quantum decoherence \\cite{Zeh0,Zurek,Schlosshauer}, which is a key aspect of irreversibility in open dynamical systems \\cite{Breuer}, when unavoidably a limited number of degrees of freedom and reaction channels is used \\cite{Alexis0}. This issue can be tackled through the present time-dependent approach. The coupled-channels density-matrix ({\\sc ccdm}) technique was first introduced in studies of quantum molecular dynamics \\cite{Saalfrank}, and has recently been applied to investigate the coupling-assisted quantum tunneling in heavy-ion fusion \\cite{Alexis1,Alexis2}. Ref. \\cite{Alexis3} provides a didactic discussion on the {\\sc ccdm} approach. In contradistinction to this approach, most of the dynamical models \\cite{Alexis1} of dissipative nuclear collisions do \\emph{not} treat the relative motion of the nuclei quantum-mechanically and\/or use \\emph{incoherent} (statistically averaged) rather than \\emph{decoherent} (partially coherent) reaction channels.  \n\n\\begin{figure}\n\\includegraphics[width=0.50\\textwidth,angle=0]{Fig1.eps}\n\\caption{A low-energy nuclear collision represented by an open quantum system (relative motion + a few intrinsic, low-lying collective states). The bath of single-particle states surrounding a giant resonance state represents the environment. It gradually destroys the coherent quantum superposition of the reduced-system collective states, as the nuclei approach.}\n\\label{Fig1}\n\\end{figure}\n\nFigure \\ref{Fig1} illustrates the innovative ideas of the {\\sc ccdm} approach. The situation can be imagined as an orchestra (representing the reduced quantum system), where the director plays the role of the relative motion and the musicians correspond to a few intrinsic, low-lying collective states. Now imagine an airplane appearing overhead, representing the environment of innumerable nucleonic states. When the airplane is not present, the orchestra plays a marvelous music, all musicians are in sync, like in a coherent quantum superposition. But when the airplane approaches, the listener percieves two effects due to interference: (i) the music gets attenuated (dissipation) and, most importantly, (ii) the quality of the music changes, as the musicians play out of sync (decoherence).  \n \nDecoherence, which always accompanies dissipation in open quantum systems \\cite{Breuer}, means dynamical dislocalization of coherent quantum superpositions due to entanglement of the system with its environment \\cite{Zeh1}. Coherent quantum superpositions are the basis of the coherent coupled-channels approach to near-barrier reaction dynamics, which manifest themselves through experimental fusion-barrier distributions \\cite{Nanda0}. Dissipation of energy and angular momentum is revealed in heavy-ion deep-inelastic scattering that occurs at near-barrier energies as well \\cite{Hinde1}. Cross sections of deep-inelastic collisions also indicate loss of angular-momentum coherence \\cite{Simbel1,Rosa1}. While the coherent coupled-channels calculations \\cite{Balantekin1} are able to explain several collision observables, major problems are unresolved. Foremost is the inability to describe the elastic and quasi-elastic scattering and fusion processes simultaneously \\cite{Newton1} and the related, more recent failure to describe consistently below-barrier quantum tunneling and above-barrier fusion yields \\cite{Nanda1}. New, precise fusion measurements have inevitably led to phenomenological (sometimes contradictory) adjustments \\cite{Rae,Esbensen1,Hagino1} to stationary-state coupled-channels models to fit the experimental data, but without a physically consistent foundation. Complementary to fusion experiments, precision measurements of back-angle scattering energy spectra \\cite{Gasques1,Evers1,Lin1} and quasi-elastic barrier distributions \\cite{Piasecki1} clearly indicate that complex, dynamical processes (other than the low-lying collective excitation of the interacting nuclei \\cite{Yusa}) play an important role in low-energy heavy-ion collisions. \n\nThis paper suggests that quantum decoherence and dissipation should be simultaneously included in a consistent description of low-energy reaction dynamics, when a restricted set of (relevant) degrees of freedom is employed. A possible description is the {\\sc ccdm} approach whose formalism is described in Sec. II. For the sake of simplicity and a specific application, without loss of generality, the collision of an inert spherical $^{16}$O projectile on a deformed $^{154}$Sm target is studied. It includes the couplings between the relative orbital angular momentum $\\hat{L}$ of the reactants and the spin $\\hat{I}$ of the ground-state rotational-band levels of $^{154}$Sm. Formulae of asymptotic observables that can reveal effects of quantum decoherence are obtained. A method for calculating the energy-resolved observables is presented. In Sec. III, model calculations are carried out and discussed, while a summary is given in Sec. IV.  \n\n\\section{Formalism}\n\\subsection{Initial density matrix}\n\nThe origin of the reference frame is in the overall center of mass. The vector $\\vec{r}$ refers to the relative vector between the projectile and the target, while $\\hat{r}$ and $\\hat{\\xi}$ are respectively the orientation angles of $\\vec{r}$ and the target intrinsic symmetry axis, with respect to the laboratory fixed z-axis defined by the beam direction. \n\nInitially, the target is considered to be unpolarized at the ground-state of spin $I_0$, being the rotational state the $Y_{I_0 M_{I_0}}(\\hat{\\xi})$ spherical harmonic. It is coupled to the orbital motion described with $Y_{L M_L}(\\hat{r})$. The total \nangular-momentum state, $|I_0 L;JM \\rangle = \\sum_{M_L M_{I_0}} C_{LM_L I_0 M_{I_0}}^{JM}Y_{LM_L}(\\hat{r})Y_{I_0M_{I_0}}(\\hat{\\xi)}$, describes the angular variables, denoting $C$ the Clebsch-Gordan coefficients. The radial motion is described with a Gaussian $\\psi_{{k}_0}(r)$ wave-packet, centered at $r_0$ with the average radial wave-number $k_0$ and the spatial dispersion $\\sigma$:\n\\begin{equation}\n\\psi_{k_0}(r) = N \\exp \\, [-\\frac{(r-r_0)^2}{2\\sigma^2} ] \\, \n                              e^{ik_0 \\, r}, \n\\label{eq1}\n\\end{equation}\nwhere the constant $N$ is specified below. Thus, the initial state is $|\\chi \\rangle = \\, \\sum_{LJM}\\psi_{k_0}(r) \\, |I_0 L;JM \\rangle$, and the initial density operator is $\\hat{\\rho}_0=(2I_0+1)^{-1} \\, |\\chi \\rangle \\langle \\chi |$. The latter can be represented as \n\n\\begin{equation}\n\\hat{\\rho}_0 \\, = \\sum_{\\alpha,\\alpha ',rs} \\, |r ) |\\alpha \\rangle \\, \\rho_{\\alpha \\alpha '}^{rs}(t=0) \\,\n\\langle \\alpha '| ( s|,  \\label{eq2}\n\\end{equation}\nwhere $\\alpha \\equiv (IL;JM)$, $|\\alpha \\rangle$ and $|r )$ are the coupled angular-momentum basis and the discrete grid-basis describing the internuclear separations, respectively. The initial density matrix is: \n\n\\begin{eqnarray}\n\\rho_{\\alpha \\alpha '}^{rs}(t=0)&=& (2I_0 + 1)^{-1} N^2 \\exp \\, [-\\frac{(r-r_0)^2}{2\\sigma^2} ] \\, e^{ik_0 \\, r} \n \\nonumber \\\\ \n&& \\times \\exp \\, [-\\frac{(s-r_0)^2}{2\\sigma^2} ] e^{-ik_0 \\, s} \\,\\, \\delta_{II_0} \\,\\, \\delta_{I'I_0},\n\\label{eq3}\n\\end{eqnarray}\nwhere $N$ is determined from the normalization condition $\\sum_{r \\alpha} \\rho_{\\alpha \\alpha}^{rr} = 1$.\n\n\\subsection{Time evolution}\n\\subsubsection{Non-dissipative case}\n\nThe Liouville-von Neumann master equation dictates the time evolution of the density matrix operator $\\hat{\\rho}(t)$ with the initial value (\\ref{eq2}). It reads as\n\\begin{equation}\ni\\hbar \\, \\frac{\\partial \\hat{\\rho}}{\\partial t} = [\\hat{H}, \\hat{\\rho}],\n\\label{eq4}\n\\end{equation}\nwhere $\\hat{H}$ is the system (Hermitian) Hamiltonian specified below. Inserting the expansion (\\ref{eq2}) for $\\hat{\\rho}(t)$ in (\\ref{eq4}), and projecting onto the basis states, we get a system of coupled equations for the matrix elements $\\rho_{\\alpha \\alpha '}^{rs}(t)$:\n\\begin{equation}\ni\\hbar \\, \\dot{\\rho}_{\\alpha \\alpha '}^{rs} = \\sum_{t\\beta} \\, ( \\, H_{\\alpha \\beta}^{rt} \\, \n\\rho_{\\beta \\alpha '}^{ts} \\, - \\, \\rho_{\\alpha \\beta}^{rt} \\, H_{\\beta \\alpha '}^{ts} \\, ), \n\\label{eq5}\n\\end{equation}\nwith the initial values (\\ref{eq3}). The system Hamiltonian contains different pieces: (i) the radial kinetic energy operator $\\hat{T}$, (ii) the total bare (Coulomb+nuclear+centrifugal) nucleus-nucleus real potential $\\hat{U}$, (iii) the total (Coulomb+nuclear) real coupling potential $\\hat{V}$ between the relative motion and the intrinsic rotational states of the target, and (iv) the target intrinsic Hamiltonian $\\hat{H_0}$. In terms of these operators, the coupled equations (\\ref{eq5}) read as\n\\begin{eqnarray}\ni\\hbar \\, \\dot{\\rho}_{\\alpha \\alpha '}^{rs}&=&\\sum_{t}\\, (\\, T^{rt} \\, \n\\rho_{\\alpha \\alpha '}^{ts} \\, - \\, \\rho_{\\alpha \\alpha '}^{rt} \\, T^{ts}\\, ) \\nonumber \\\\\n&& + \\, [\\, U_{\\alpha} (r) \\, - \\, U_{\\alpha '} (s)\\, ]\\, \\rho_{\\alpha \\alpha '}^{rs} \\nonumber \\\\\n&& + \\, \\sum_{\\beta}\\, [\\, V_{\\alpha \\beta}(r) \\, \n\\rho_{\\beta \\alpha '}^{rs} \\, - \\, \\rho_{\\alpha \\beta}^{rs} \\, V_{\\beta \\alpha '} (s)\\, ] \\nonumber \\\\\n&& + \\, (\\, \\varepsilon_{\\alpha} \\, - \\, \\varepsilon_{\\alpha '} \\, )\\, \\rho_{\\alpha \\alpha '}^{rs} . \n\\label{eq6}\n\\end{eqnarray}\n\n\\subsubsection{Dissipative case}\n\nIn practice, however, a truncated model space of reaction channels (\\emph{reduced system}) is employed \\cite{Amos}. The impact of the excluded model space (\\emph{environment}) on the reaction dynamics is usually treated through complex potentials \\cite{Feshbach,Taylor,Amos}, making the system Hamiltonian non-Hermitian. This results in loss of probability and energy from the reduced system to the environment, \\emph{but} the quantum coherence in the basis states of the reduced system is preserved, as recently demonstrated in Ref. \\cite{Alexis0}. It is inconsistent with the irreversible dynamics of an open quantum system \\cite{Breuer}, as energy dissipation always goes hand in hand with quantum decoherence \\cite{Zeh0,Zurek,Schlosshauer}. \n\nIrreversibility can be consistently described by the Lindblad master equation \\cite{Lindblad1,Gorini,Sandulescu1,Ronnie1}. Here, quantum decoherence and dissipation are incorporated into the dynamics through a dissipative Liouvillian, \n$i \\hbar \\mathcal{L}_D\\, \\hat{\\rho}$, which is added to the r.h.s. of Eq. (\\ref{eq4}). \n$\\mathcal{L}_D\\, \\hat{\\rho}$ reads as \n\\begin{equation} \n\\mathcal{L}_D \\, \\hat{\\rho} \\, = \\, \\sum_\\nu \\bigl( \\hat{\\mathcal C}_\\nu \\, \\hat{\\rho} \\, \\hat{\\mathcal C}_\\nu^{\\dag} -\n\\frac{1}{2} \\bigl[\\hat{\\mathcal C}_\\nu^{\\dag} \\, \\hat{\\mathcal\nC}_\\nu ,\\hat{\\rho} \\bigl]_{+} \\bigl)\\, \n\\label{eq6a}\n\\end{equation}\nwhere $[\\ldots]$ and $[\\ldots]_{+}$ denote the commutator and anti-commutator, respectively. Each $\\hat{\\mathcal C}_\\nu$ is a Lindblad operator for a \ndissipative coupling, physically motivated according to the specific problem. \nIt is assumed that each dissipative coupling $\\nu \\equiv (\\alpha \\alpha')$ between \ngiven states $|\\alpha \\rangle$ and $|\\alpha' \\rangle$ has an\nassociated (local) radial-dependent decay rate $\\Gamma_{\\alpha \\alpha'}^{r r}$ \\cite{Saalfrank2}, i.e., the \\emph{spontaneous emission} Lindblad operator $\\hat{\\mathcal C}_{\\alpha \\alpha'}=\\sqrt{\\Gamma_{\\alpha \\alpha'}^{rr}} |\\alpha \\rangle \\langle \\alpha'|$.\n\nIn order to also describe decay to the Hilbert space of excluded, intrinsic degrees of freedom, environmental states \\cite{Irene,Bertlmann} are considered in addition to the basis $|\\alpha \\rangle$ of the reduced system. All these states are assumed to be orthonormal, resulting in an enlarged basis $|\\bar{\\alpha} \\rangle$. Using this and the discrete grid-basis, the matrix elements of (\\ref{eq6a}) read as \\cite{Erratum} \n\\begin{eqnarray}\n(\\mathcal{L}_D\\, \\hat{\\rho})_{\\bar{\\alpha} \\bar{\\alpha}'}^{r s} &=&\n\\delta_{\\bar{\\alpha} \\bar{\\alpha}'}\\, \\sum_{\\mu} \\sqrt{\\Gamma_{\\bar{\\alpha} \\mu}^{rr}} \\, \n\\rho_{\\mu \\mu}^{rs} \\, \\sqrt{\\Gamma_{\\bar{\\alpha} \\mu}^{ss}} \\, \\nonumber \\\\ \n&& - \\, \\frac{1}{2}\\, \\sum_{\\mu}\\, ( \\, \\Gamma_{\\mu \\bar{\\alpha}}^{rr} \\, + \\, \n\\Gamma_{\\mu \\bar{\\alpha}'}^{ss} \\,)\\, \n\\rho_{\\bar{\\alpha} \\bar{\\alpha}'}^{rs},\n\\label{eq6b}\n\\end{eqnarray}\nwhere $\\mu$ runs over all the $\\bar{\\alpha}$ states, and \n$\\Gamma_{\\bar{\\alpha} \\bar{\\alpha}}^{rr} = \\sum_{\\mu \\ne \\bar{\\alpha}} \\Gamma_{\\mu \\bar{\\alpha}}^{rr}$ \\cite{Saalfrank3}.\n\nWe now distinguish two kinds of density matrix elements $\\rho_{\\bar{\\alpha} \\bar{\\alpha}'}^{r s}$, namely one type associated with the reduced-system states $|\\alpha \\rangle$ only, and another type related to the environmental states \n$|\\bar{\\alpha} \\rangle, \\, \\bar{\\alpha} \\ne  \\alpha$. These obey the following equations of motion: \n\\begin{eqnarray}\ni\\hbar \\, \\dot{\\rho}_{\\alpha \\alpha '}^{rs}&=&\\sum_{t}\\, (\\, T^{rt} \\, \n\\rho_{\\alpha \\alpha '}^{ts} \\, - \\, \\rho_{\\alpha \\alpha '}^{rt} \\, T^{ts}\\, ) \\nonumber \\\\\n&& + \\, [\\, U_{\\alpha} (r) \\, - \\, U_{\\alpha '} (s)\\, ]\\, \\rho_{\\alpha \\alpha '}^{rs} \\nonumber \\\\\n&& + \\, \\sum_{\\beta}\\, [\\, V_{\\alpha \\beta}(r) \\, \n\\rho_{\\beta \\alpha '}^{rs} \\, - \\, \\rho_{\\alpha \\beta}^{rs} \\, V_{\\beta \\alpha '} (s)\\, ] \\nonumber \\\\\n&& + \\, (\\, \\varepsilon_{\\alpha} \\, - \\, \\varepsilon_{\\alpha '} \\, )\\, \\rho_{\\alpha \\alpha '}^{rs}  \\nonumber \\\\\n&& + \\, i\\hbar \\, \\, \\{ \\, \\delta_{\\alpha \\alpha'}\\, \\sum_{\\mu} \\sqrt{\\Gamma_{\\alpha \\mu}^{rr}} \\, \n\\rho_{\\mu \\mu}^{rs} \\, \\sqrt{\\Gamma_{\\alpha \\mu}^{ss}} \\, \\nonumber \\\\ \n&& - \\, \\frac{1}{2}\\, \\sum_{\\mu}\\, ( \\, \\Gamma_{\\mu \\alpha}^{rr} \\, + \\, \n\\Gamma_{\\mu \\alpha'}^{ss} \\,)\\, \\rho_{\\alpha \\alpha'}^{rs} \\, \\}, \n\\label{eq6c}\n\\end{eqnarray}\nfor matrix elements of the reduced-system states, whereas \n\\begin{eqnarray}\n\\dot{\\rho}_{\\bar{\\alpha} \\bar{\\alpha}'}^{rs}&=&\n\\delta_{\\bar{\\alpha} \\bar{\\alpha}'}\\, \\sum_{\\mu} \\sqrt{\\Gamma_{\\bar{\\alpha} \\mu}^{rr}} \\, \n\\rho_{\\mu \\mu}^{rs} \\, \\sqrt{\\Gamma_{\\bar{\\alpha} \\mu}^{ss}} \\, \\nonumber \\\\ \n&& - \\, \\frac{1}{2}\\, \\sum_{\\mu}\\, ( \\, \\Gamma_{\\mu \\bar{\\alpha}}^{rr} \\, + \\, \n\\Gamma_{\\mu \\bar{\\alpha}'}^{ss} \\,)\\, \\rho_{\\bar{\\alpha} \\bar{\\alpha}'}^{rs}, \n\\label{eq6d} \n \\end{eqnarray}\nfor matrix elements involving the environmental states, i.e., either $\\bar{\\alpha}$ or \n$\\bar{\\alpha}' \\ne \\alpha$, at least. It is worth emphasizing that the environmental states are not reaction channels, but auxiliary states \\cite{Irene} supplying a probability drain and rendering the reduced-system states decoherent. \n\nThe initial values for Eqs. (\\ref{eq6c}) are given by (\\ref{eq3}), while for Eqs. (\\ref{eq6d}) these are zero as the nuclei are initially (far apart) at the ground-states. Hence, the off-diagonal elements $\\rho_{\\bar{\\alpha} \\bar{\\alpha}'}^{rs}$ in (\\ref{eq6d}) remain zero. However, the diagonal terms $\\rho_{\\bar{\\alpha} \\bar{\\alpha}}^{rs}$ absorb probabilities only, provided there is no flux back from the environment to the reduced system. \n\nIn the calculations below, it will be considered that the off-diagonal elements of the decay-rate matrix $\\Gamma$ are \\emph{nonzero} only for transitions from the reduced system to the environment, i.e., $\\Gamma_{\\bar{\\alpha} \\alpha}^{rr}, \\, \\bar{\\alpha} \\ne \\alpha$. \n(Still, environment-induced transitions among the states of the reduced system may occur.) The decay rates are given by $\\Gamma_{\\bar{\\alpha} \\alpha}^{rr} = W_{\\alpha}(r) \/\\hbar$, where $W_{\\alpha}(r) > 0$ are decay functions. These are here treated as \\emph{empirical} functions, but it is hoped that a microscopic theory on damping of collective excited states \\cite{Matsuo,Guo,Ring}, including dynamical modifications of excited state properties due to the close proximity of other nucleus \\cite{McIntosh}, can provide them. \n\nVarious types of environments can coexist in a nuclear collision, which may be specific to particular degrees of freedom, such as isospin asymmetry or weak binding. Among these environments, which can be coupled to specific states or to all states of the reduced system, are (i) the high level-density of one- and multi-nucleonic excitations in different mass\/charge partitions (transfer), (ii) the continuum of non-resonant decay states of weakly-bound nuclei (breakup), and (iii) the innumerable nuclear molecular (compound nucleus) states (fusion). These can be treated separately, and their effects can be distinguished within the {\\sc ccdm} approach.\n\n\\subsection{Asymptotic observables}\n\nHaving the solution of Eqs. (\\ref{eq6c}), after a long period of time $t_f$, when the centroid of the recoiled body of the wave-packet is at a large internuclear distance and a quasi-stationary probability current establishes in all the $\\alpha$ channels, we then calculate asymptotic observables. For instance, these can be the angular distribution of the target excitations, and their integrated values. These are calculated as follows.\n\nWe now introduce the projector $\\hat{P}_{I M_I} = |I M_I \\rangle \\langle I M_I |$ associated with a specific state of the target. The new operator $\\hat{\\rho}(t_f)\\,\\hat{P}_{I M_I}$ describes the scatterred waves in this target state\n\\begin{eqnarray}\n\\hat{\\rho}(t_f)\\,\\hat{P}_{I M_I}&=&\\langle I M_I |\\hat{\\rho}(t_f)|I M_I \\rangle \n= \\nonumber \\\\\n&& \\sum_{p} \\, C_{Lm I M_I}^{JM} \\, Y_{Lm}(\\hat{r}) \\, \\, |r ) \\, \\,\n\\rho_{\\gamma \\lambda }^{rs}(t_f) \\nonumber \\\\ \n&& \\times \\, ( s| \\, \\, C_{L'm' I M_I}^{J'M'} \\, Y_{L'm'}^*(\\hat{s}),\n\\label{eq7}\n\\end{eqnarray}       \nwhere $p \\equiv (r,s,L,m,J,M,L',m',J',M')$, $\\gamma \\equiv (IL;JM)$ and \n$\\lambda \\equiv (IL';J'M')$. The reduced density matrix, $\\rho_{\\gamma \\lambda }^{rs}(t_f)$, is normalized with its trace, i.e., $\\sum_{r \\gamma} \\rho_{\\gamma \\gamma}^{rr}(t_f)$.  \n\nWe note that (\\ref{eq7}) is still an operator in $\\vec{r}$ and $\\vec{s}$. The radial projector $\\hat{P}_{r'} = |r') \\, ( r'|$ associated with a specific separation between the nuclei is then introduced. With this projector we now act on (\\ref{eq7}) and set \n$\\hat{r}=\\hat{s}=\\hat{r}'$. A new operator is thus obtained, whose partial trace (sum over all separations $r'$) provides the probability for producing the target in state ($I,M_I$) with the relative coordinate in the direction $\\hat{r}'$:\n\\begin{eqnarray}\n\\frac{d\\mathcal{W}}{d\\Omega}(I,M_I)&=&\\sum_{q} \\, C_{Lm I M_I}^{JM} \\, Y_{Lm}(\\hat{r}') \\, \n\\mathcal{S}_{\\gamma \\lambda } (t_f) \\nonumber \\\\\n&& \\times \\, C_{L'm' I M_I}^{J'M'} \\, Y_{L'm'}^*(\\hat{r}'),\n\\label{eq8}\n\\end{eqnarray}  \nwhere $q \\equiv (L,m,J,M,L',m',J',M')$ and $\\mathcal{S}_{\\gamma \\lambda }(t_f) = \\sum_{r'} \\rho_{\\gamma \\lambda }^{r'r'}(t_f)$. The latter contains information about the coherence of angular momenta.\n\nIntegrating (\\ref{eq8}) over all directions $\\hat{r}'$ of solid angles, and summing over all $M_I$, the total probability for producing the target in state $I$ (population) is obtained:\n\\begin{equation}\n\\mathcal{W}(I) = \\sum_{M_I} \\, \\sum_{LmJM} \\, (\\, C_{Lm I M_I}^{JM} \\,)^2 \\, \n\\mathcal{S}_{\\gamma \\gamma } (t_f).\n\\label{eq9}\n\\end{equation}\n\n\\subsection{Energy-resolved observables} \n\nThe observables (\\ref{eq8}) and (\\ref{eq9}) correspond to average values for the range of energies contained in the incident wave-packet (\\ref{eq1}). The energy-resolved scattering information can be obtained using a window operator \\cite{Schafer}. \nThe key idea is to calculate, for definite $\\gamma \\lambda$ indices, the energy spectrum \n$\\mathcal{P}(E_k)$ of the initial and final reduced density matrices. $E_k$ is the centroid of a \\emph{total energy bin} of width $2\\epsilon$. A matrix of reflection coefficients, \n$\\mathcal{R}_{\\gamma \\lambda } (E_k)$, is determined by the ratio\n\\begin{equation}\n\\mathcal{R}_{\\gamma \\lambda } (E_k)\\, = \\, \n\\frac{\\mathcal{P}_{\\gamma \\lambda}^{\\, final}(E_k)}\n{\\sum_{\\gamma} \\, \\mathcal{P}_{\\gamma \\gamma}^{\\, initial}(E_k)}, \n\\label{eq10a}\n\\end{equation}\nwhich replaces the matrix $\\mathcal{S}_{\\gamma \\lambda } (t_f)$ in (\\ref{eq8}) and (\\ref{eq9}). Expression (\\ref{eq10a}) generalizes the wave-packet formulation of the reflection coefficient \\cite{Tannor,Yabana}.  \n\nThe energy spectrum $\\mathcal{P}(E_k)\\,=\\,\\widetilde{\\textnormal{Tr}}$($\\hat{\\Delta}\\hat{\\rho}$), where $\\widetilde{\\textnormal{Tr}}$ denotes a partial trace involving the radial indices only, and $\\hat{\\Delta}$ is the window operator \\cite{Schafer}:\n\\begin{equation}\n\\hat{\\Delta}(E_k,n,\\epsilon)\\, \\equiv \\, \\frac{\\epsilon^{2^n}}  \n{[(\\hat{\\mathcal{H}}\\, - \\, E_k)^{2^n}\\, + \\, \\epsilon^{2^n}]}, \n\\label{eq10b}\n\\end{equation}      \nwhere $\\hat{\\mathcal{H}}$ is the system asymptotic Hamiltonian, and $n$ determines the shape of the window function. As $n$ is increased, this shape rapidly becomes rectangular with very little overlap between adjacent energy bins \\cite{Schafer}, remaining the bin width constant at $2\\epsilon$. The spectrum is constructed for a set of $E_k$ where $E_{k+1}=E_k + 2\\epsilon$. Thus, scattering information over a range of incident energies can be extracted from a time-dependent density matrix that has been calculated on a grid.\n\n\\subsubsection{Example}\n\nFigure \\ref{Fig2} shows for a single-channel {\\sc ccdm} calculation with \n$L=0$ \\cite{Alexis1}: (a) the energy spectrum ($n=4$, $2\\epsilon=1$ MeV) of the initial (solid line) and final (dashed line) density matrices, and (b) the final-to-initial ratio of the energy spectrum providing energy-resolved reflection coefficients (full squares). These very well agree with those (dotted line) of a time-independent Schr\\\"odinger equation with a short-range imaginary potential or an ingoing-wave boundary condition at small radii, as implemented in the {\\sc ccfull} code \\cite{CCFULL}.\n \n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.5\\textwidth,angle=0]{Fig2a.eps} \\\\\n\\includegraphics[width=0.5\\textwidth,angle=0]{Fig2b.eps}\n\\caption{(a) Energy spectrum of the initial (solid line) and final (dashed line) density matrices for the elastic scattering of $^{16}$O off the $^{154}$Sm target at the average total energy $E_0 = 60$ MeV. (b) Energy-resolved reflection coefficients provided by the final-to-initial ratio of the energy spectrum (full squares). These very well agree with those of a stationary Schr\\\"odinger equation (dotted line).}\n\\label{Fig2}\n\\end{center}\n\\end{figure}\n\nIn the {\\sc ccdm} calculation above, the grid ($r=0-100$ fm) was evenly spaced with $512$ radial points. The incident wave-packet was initially centered at $r_0 = 50$ fm, with width $\\sigma = 5$ fm, and was boosted toward the potential with the appropriate average kinetic energy for the total energy $E_0 = 60$ MeV. \n\nThe time evolution of the density matrix was carried out using a Faber propagator \\cite{Faber} and the Fourier method \\cite{Kosloff2} for the commutator between the kinetic energy and density operator. The time step for the density-matrix propagation was $\\Delta t = 10^{-22}$ s.\n\nThe form of the bare nuclear potential between the $^{16}$O and $^{154}$Sm nuclei is a Woods-Saxon potential with ($V_0, r_0,a_0) \\equiv$ ($-$165 MeV, 0.95 fm, 1.05 fm). The Coulomb potential is that for two point charges. These yield a $s$-wave Coulomb barrier of $V_B = 59.41$ MeV at the radius $R_B = 10.81$ fm. \n\nThe irreversible capture of the two nuclei in the nucleus-nucleus potential pocket inside the Coulomb barrier (fusion) is caused by an environmental coupling \\cite{Alexis1,Alexis2} with a decay function $W(r)$ taken as a Fermi function with depth $10$ MeV and diffuseness $0.1$ fm, located at the pocket radius of $5.77$ fm.\n\n\\section{Model calculations}\n\nIn the model calculations the $^{16}$O projectile was taken to be inert and the $^{154}$Sm target was allowed to be excited up to the $4^{+}$ state of the ground-state rotational band. The all order nuclear coupling of the ground-state $0^{+}$ to the states $2^{+}$ and $4^{+}$, with excitation energies $E_{2^+}=0.08$ MeV and $E_{4^+}=0.27$ MeV, has a macroscopic deformed Woods-Saxon form with the radius parameter of $1.06$ fm, and the deformation parameters of \n$\\beta_2=0.322$ and $\\beta_4=0.027$. The Coulomb coupling includes terms up to second order with respect to $\\beta_2$ and to the first order of $\\beta_4$. The total coupling-potential matrix in the coupled angular-momentum basis is calculated using the {\\sc ccfull} code \\cite{CCFULL} and the Kermode-Rowley matrix technique \\cite{KR}.\n\nWith such a coupling Hamiltonian, the time propagation on the grid employed in Fig. \\ref{Fig2} is very time consuming and memory demanding, beyond the present limit of our computational capability. That is why the present calculations aim at exploring \\emph{qualitative} effects only, for which a smaller grid ($r=0-40$ fm) with $64$ evenly spaced radial points suffices. The initial wave-packet is then centered at $r_0 = 25$ fm, with width $\\sigma = 3$ fm and the incident, average total energy $E_0 = 60$ MeV. Relative partial waves up to $20 \\hbar$ are included in the calculations.\n\nTwo types of calculations (without energy projection) are carried out including: (i) only the effects of the \\emph{fusion} environment highlighted above, and (ii) in addition the effects of a \\emph{surface} environment specified below. While the fusion environment is coupled to all the $^{154}$Sm states, the surface environment is considered to be coupled to the ground-state only. The latter can be associated with complex, multi-nucleon\/cluster transfers from the ground-state of the colliding nuclei to other mass (or charge) partitions. The corresponding decay function is taken as a Gaussian with width of $1$ fm, centered at the contact radius that is estimated as $1.2(16^{1\/3} + 154^{1\/3})$ fm. This function is physically motivated by the spatial localization of transfer processes in heavy-ion reactions \\cite{Satchler}. The measure of coherence \\cite{Alexis0,Klum} in the reduced system is the ratio Tr$(\\hat{\\rho}^2)\/[$Tr$(\\hat{\\rho})]^2$, whose time evolution is presented in Fig. \\ref{Fig3}. The fusion environment essentially preserves coherence (solid line), while the surface environment results in decoherence (dashed line). Comparing the calculation (ii) to (i), we learn how the surface environment-induced decoherence impacts on the asymptotic observables (\\ref{eq8}) and (\\ref{eq9}). \n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.5\\textwidth,angle=0]{Fig3.eps}\n\\caption{Time evolution of the measure of coherence in the reduced system. While the fusion environment essentially preserves coherence (solid line), it is destroyed by the surface environment (dashed line).}\n\\label{Fig3}\n\\end{center}\n\\end{figure}  \n\nFigure \\ref{Fig4} shows the angular distribution of the $^{154}$Sm excitations, which corresponds to expression (\\ref{eq8}) summed over all $M_I$. The solid and dashed lines are results of the calculations (i) and (ii), respectively. Clearly, the surface environment varies the quantum interference effects, destroying the coherence of relative partial waves and changing by a few degrees the minimum of the inelastic probability distributions [Figs. \\ref{Fig4}(b) and (c)]. It also affects significantly the asymptotic population of the $^{154}$Sm states and the fusion probability, as presented in Table \\ref{Table}. The surface environment-induced decoherence hinders the probability flow from the elastic to the inelastic and fusion channels.           \n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.5\\textwidth,angle=0]{Fig4a.eps} \\\\\n\\includegraphics[width=0.5\\textwidth,angle=0]{Fig4b.eps} \\\\\n\\includegraphics[width=0.5\\textwidth,angle=0]{Fig4c.eps}\n\\caption{Angular distribution of the $^{154}$Sm excitation probabilities, when an inert $^{16}$O projectile scatters off the $^{154}$Sm target at the average total energy $E_0 = 60$ MeV: (a) $0^{+}$, (b) $2^{+}$, and (c) $4^{+}$. The solid lines are outcomes including the effects of the fusion environment only, while the dashed lines include in addition the effects of the surface environment. The latter destroys the coherence of relative partial waves, changing by a few degrees the minimum of the inelastic distributions (small figures inserted).}\n\\label{Fig4}\n\\end{center}\n\\end{figure}\n\n\\begin{table} \\caption{Asymptotic population (\\ref{eq9}) of the $^{154}$Sm states and the fusion probability for calculations including (i) only the fusion environment (Environment 1), and (ii) in addition the surface environment (Environment 2). The latter hinders the probability flow from the elastic to the inelastic and fusion channels, due to decoherence.} \\label{Table}\n\\medskip\n\\begin{center}\n\\begin{tabular}{c|c|c}\n\\hline\\hline &\\multicolumn{1}{c}{Environment 1} &\n\\multicolumn{1}{|c}{Environments 1 and 2}\n\\\\\n\\hline\nStates &~~ Population~~&~~ Population ~\\\\\n\\hline\n$0^+$ &0.8376&0.8770\\\\\n$2^+$ &0.1206&0.0961\\\\\n$4^+$ &0.0418&0.0268\\\\\n\\hline\nFusion Probability &3.127E-3&9.106E-4\\\\\n\\hline\\hline \\end{tabular}\n\\end{center}\n\\end{table}\n\n\\section{Summary} \n\nThe innovative, coupled-channels density-matrix approach to low-energy reaction dynamics has been developed further, including full angular momentum couplings and a method for extracting energy-resolved scattering information from the time-dependent density matrix. These developments have enabled a first exploration of decoherence effects on asymptotic observables, such as the angular distribution of the target excitations, and their integrated values. These are significantly affected by decoherence induced by a surface environment (related to complex, multi-nucleon\/cluster transfers), which changes by a few degrees the minimum of the back-angle inelastic probability distributions and hinders the probability flow from the elastic to the inelastic and fusion channels. To test the present theory against experiments, both extensive calculations (with energy projection) and high-precision measurements of fusion and scattering observables (including those investigated here) are required. It is hoped that decoherence effects can help resolve major problems in low-energy nuclear reaction physics, such as the current inability to simultaneously describe fusion and scattering measurements of heavy ions within the standard, coupled-channels framework.          \n  \n\\begin{acknowledgments}\nThe author thanks Jeff Tostevin for discussions, and Ron Johnson for constructive comments on the importance of the density-matrix off-diagonal elements (with respect to the total angular momentum) for calculating the angular distributions. Useful discussions with participants in the ECT$^*$ Workshop on Decoherence in Quantum Dynamical Systems, Trento, April 26--30, 2010, are also acknowledged. The work was supported by the UK Science and Technology Facilities Council (STFC) Grant No. ST\/F012012\/1.\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec:int}\n\n\nRecommender systems that try to assess probabilities, e.g.~for estimating probabilities based on the context of a particular user, may be faced with ambiguous statistical evidence.\nFor example, consider the task:\n\\emph{Customer is known to belong to categories A and B, each of which is known to increase the probability of buying product C by $50\\%$, how do we estimate the probability that she will buy product C? Is it increased by $50\\%$, $100\\%$, or perhaps $125\\%$?}\n\n\nOf course we {\\em may\\\/} have enough data on $S_1$, $S_2$, and $S_3$ to make this assessment by computing the observed probability.\nBut for rare terms or products there may not be enough data to directly produce such an estimate --- perhaps we never directly observed a connection between $S_1$, $S_2$, and $S_3$.\nIn the extreme case, what can we do when there is {\\em no support\\\/} for estimating the probability by simply computing the observed frequency?\nMost likely, even the number of observations of proper subsets of $S_1$, $S_2$, and $S_3$ will then be small enough that there is nonnegligible uncertainty about the pairwise correlations.\n\n\nThe difficulty of estimating probabilities of events occurring clearly depends on the distribution of the input, and on how much information we have about this distribution.\nSo rather than a classical approach that considers worst-case data, we should\nconsider ideas from statistical analysis.\nThe Maximum Entropy (maxent) Model is a method of statistical inference that based on partial knowledge of a distribution provides a \\emph{maximum entropy estimate}.\nInformally, it provides a probability prediction based on the distribution that has ``the least bias possible'' based on the given observations.\nIn this paper we consider the use of maximum entropy estimates in information retrieval contexts where estimates are sought of the probability that an item or term is of interest to a user.\n\n\n\\subsection{Motivating examples}\n\n{\\bf Movie recommendation.}\nSuppose you know that a user loved ``The Rock'' and ``The Matrix''. What is the probability that he already saw and gave 5 stars to ``One Flew Over the Cuckoo's Nest''? We will try to answer this question using the smallest possible sample of the MovieLens data set, which is examined further in \\Cref{sec:exp}. The difficulty is that only about 1 in 1000 people has seen all three movies and given them 5 stars. This means that to get a statistically significant answer (in absence of other information) we need to ask a very large set of people. Obviously, the less mainstream films you consider, the bigger this problem will become. See \\Cref{fig:intro:ex2} for visualization of the setting.\n\nHowever, it is considerably easier to obtain information on pairs of movies. About $2.5\\%$ of people will have seen at least two of these movies and given them $5$ stars. This means that we can reliably estimate conditional probabilities based on significantly less data. Using the information that your user loved ``The Rock\" gives a probability of about $11\\%$ that he loves ``One Flew Over the Cuckoo's Nest\". However, if we instead use the information that he loved ``The Matrix'' we get an estimate of about $7\\%$. It seems that both these movies make it more likely that he will love ``One Flew Over the Cuckoo's Nest'', but how do we combine these pieces of information? It seems that anywhere in the range $11$-$18\\%$ is a reasonable guess.\n\nTo resolve this ambiguity we again use a maximum entropy estimate based on subset frequencies. This estimate takes the correlation of ``The Rock'' and ``The Matrix'' into account, and arrives at an estimate of $14\\%$ based on a data set of $673$ people in which nobody has given all three movies $5$ stars. When we consider a data set $100$ times larger it is possible to see how well this estimate fares: In the larger data set, $15\\%$ of those who gave 5 stars to ``The Rock'' and ``The Matrix'' also gave 5 stars to ``One Flew Over the Cuckoo's Nest''. We generally find that maximum entropy estimates are surprisingly accurate across a wide range of data sets from different areas. \n\\begin{figure}\n\n        \\centering\n        \\begin{subfigure}[b]{0.45\\textwidth}\n                \\centering\n                \\includegraphics[width=\\textwidth]{sampledist}\n                \\caption{Venn diagram of distribution based on the sample.}\n                \\label{fig:intro:sample}\n        \\end{subfigure}%\n        ~\n         \n        \\begin{subfigure}[b]{0.45\\textwidth}\n                \\centering\n                \\includegraphics[width=\\textwidth]{fulldist}\n                \\caption{Venn diagram of distribution for the whole data set.}\n                \\label{fig:intro:full}\n        \\end{subfigure}\n       \n        \\caption{Two distributions of movie watchers who love three selected movies. \\Cref{fig:intro:sample} shows the observed probabilities in a sample of 1\\% of the data set, from which we want to approximate $S_1 \\cap S_2 \\cap S_3$. This is consistent with $S_1$, $S_2$, and $S_3$ never occurring together. \\Cref{fig:intro:full} shows the probabilities in the whole data set, and indeed it is not the case that loving two of the movies precludes loving the third one. Our findings are that a maximum entropy estimate in such a case is well-concentrated as opposed to independence or extrapolation estimators. In this example an independence assumption yields an estimate of $36$ occurrences, our maxent estimate yields $82$, while the true number of occurrences is $90$.}\n        \\label{fig:intro:ex2}\n\\end{figure}\n\n\n\n{\\bf Query completion.}\nConsider the case where a search engine user types the words ``{\\tt jordan} {\\tt air}'' (followed by a space).\nWhat words should be suggested to complete the query?\n\nWe consider the simple method of ignoring the order of words and relying on association rules, in our case obtained from a set of 2.1 million queries from a major US search engine.\nSuppose we have two competing suggestions, ``{\\tt force}'' and ``{\\tt wholesale}'' (occurring in around 0.03\\% and 0.06\\% of queries, respectively).\nAround 9\\% of the queries that contain the word ``{\\tt air}'' also contain the word ``{\\tt force}''.\nOn the other hand, less than $0.00003\\%$ of past queries contain ``{\\tt jordan}'' {\\em and\\\/} ``{\\tt force}'', which means that the maximum entropy estimate for the probability of completing with ``{\\tt force}'' becomes less than 1\\%.\nFor comparison, both ``{\\tt air}'' and ``{\\tt jordan}'' significantly increase the probability that the word ``{\\tt wholesale}'' occurs, to around 0.3\\% and 1\\%, respectively.\n\\Cref{fig:intro:ex1} summarizes the association rules involving two words.\n\n\nIf the combination of words had been just slightly more rare, we might have had no past queries containing them.\nThus, for ``long tail'' queries we need to rely on other methods for estimating the likelihood of a particular completion.\n\nA maximum entropy estimate, or more precisely the approximation formula of~(\\ref{eq:simplest}), shows that the user has a $5\\%$ probability of completing with ``{\\tt wholesale}''.\nThis is consistent with the data, which contains 31 queries including $\\{${\\tt jordan}, {\\tt air}, {\\tt wholesale}$\\}$ out of a total of 575 queries containing ``{\\tt jordan}'' and ``{\\tt air}''.\n\\begin{figure}\n        \\centering\n        \\begin{tabular}{|c|l|}\n\t\t\t\\hline\n\t\t\t{\\bf Assoc.~rule} & {\\bf Confidence}\\\\\n\t\t\t\\hline\n\t\t\t{\\tt air => force} & 0.091\\\\\n\t\t\t{\\tt jordan => wholesale} & 0.010\\\\\n\t\t\t{\\tt air => wholesale} & 0.0030\\\\\n\t\t\t{\\tt jordan => force } & 0.00097\\\\\n\t\t\t\\hline\n        \\end{tabular}\n        \\caption{ Association rules for the words {\\tt force} and {\\tt wholesale} in the  query set  {\\tt jordan} and {\\tt air}, respectively. The probabilities are sorted in decreasing order.\n\t\tIn the whole data set, {\\tt wholesale} occurs in a fraction 0.00064 of queries, and {\\tt force} in a fraction 0.00028.\n\t\tBy the first association rule, presence of the word {\\tt air} increases the probability of the word {\\tt force} hundreds of times, to over 9\\%. However, a maximum entropy estimate correctly predicts less than 1\\% probability of seeing {\\tt force} when both {\\tt air} and {\\tt jordan} are present.\n\t\tOn the other hand, the presence of {\\tt air} and {\\tt jordan} yield a maximum entropy probability for {\\tt wholesale} of 3\\%, close to the observed frequency of 5\\%. }\n        \\label{fig:intro:ex1}\n\\end{figure}\n\n\n\n\nWe will argue by experimental evidence that in scenarios such as the one above, the maximum entropy estimate will\ngive a better prediction than both extrapolation and the independence model (see \\Cref{sec:ests} for definitions), while still being efficiently computable.\n\n\n\\subsection{Our results}\n\n{\\bf Problem definition.} We consider the problem of estimating probabilities of conjunctions of boolean random variables, where each such conjunction occurs a statistically insignificant number of times in a data set of samples from the joint distribution (e.g., given by a complete data set). For some big data set $\\mathcal{D}$ we consider a sample $D \\subset \\mathcal{D}$. Given such $D$ we wish to estimate event frequencies of $\\mathcal{D}$ also in the difficult cases where the events do not occur in $D$.\nIn particular we will focus on triples: Let $I$ be the set of possible items, $|I|=n$, and $D$ be an $m \\times n$ binary data set where each of the $m$ rows $D_i$ encodes a transaction $D_i \\subseteq I$. For all singleton- and pair-subsets of $I$ we assume that we know the number of transactions which they occur in, i.e., all singleton and pair frequencies are known. For each $X \\subset I$, $|X|=3$ where the frequency $\\theta_X$ in $D$ is~$0$ we then wish to estimate $\\theta_X$ in $\\mathcal{D}$.\n\n{\\bf Our contribution.} We consider triple frequency estimation based on the principle of maximum entropy.\nOur main theoretical result is that a maximum entropy estimate based on a sample, which implies that the frequencies used as input to the estimator will have some relative error $\\varepsilon$, will yield an estimate close to the true triple frequency under the maximum entropy assumption. We show this through a surprisingly simple estimator $\\tilde{p}$ that approximates the maximum entropy estimate well when triple frequencies are small.\n\\begin{theorem}\\label{thm:maxent:main}\nConsider boolean random variables $X,Y,W$ where $\\Pr(X)$, $\\Pr(Y)$, $\\Pr(W)$, $\\Pr(XY)$, $\\Pr(XW)$ and $\\Pr(YW)$ are given.\nAssume that the maximum entropy distribution consistent with these probabilities satisfies\n\\begin{align*}\n\\Pr(XYW)\\leq \\varepsilon \\min(& \\Pr(\\overline{X}YW),\\Pr(X\\overline{Y}W),\\Pr(XY\\overline{W}),\\Pr(\\overline{XY}W),\\\\\n& \\Pr(\\overline{X}Y\\overline{W}),\\Pr(X\\overline{YW}),\n\\Pr(\\overline{XYW})) \\enspace .\n\\end{align*}\nThen given probability estimates $p_{\\cdot\\cdot\\cdot}$ such that\n\\begin{align*}\n& \\tfrac{p_{\\overline{X}YW}}{\\Pr(\\overline{X}YW)},\n\\tfrac{p_{X\\overline{Y}W}}{\\Pr(X\\overline{Y}W)},\n\\tfrac{p_{XY\\overline{W}}}{\\Pr(XY\\overline{W})},\n\\tfrac{p_{\\overline{XY}W}}{\\Pr(\\overline{XY}W)},\n\\tfrac{p_{\\overline{X}Y\\overline{W}}}{\\Pr(\\overline{X}Y\\overline{W})},\n\\tfrac{p_{X\\overline{YW}}}{\\Pr(X\\overline{YW})},\n\\tfrac{p_{\\overline{XYW}}}{\\Pr(\\overline{XYW})}\\\\\n& \\in [(1-\\varepsilon),(1+\\varepsilon)],\n\\end{align*}\nit holds that\n\\begin{align*}\n\\tilde{p} &= \\frac{p_{XY\\overline{W}} p_{X\\overline{Y}W} p_{\\overline{X}YW} p_{\\overline{XYW}}}{ p_{X\\overline{YW}}  p_{\\overline{X}Y\\overline{W}}  p_{\\overline{XY}W} }\\\\\n&\\in \\left[ (1- \\mathcal{O}(\\varepsilon))\\Pr(XYW), (1+ \\mathcal{O}(\\varepsilon))\\Pr(XYW)  \\right]\n\\end{align*}\n\\end{theorem}\nIt follows from \\Cref{thm:maxent:main} that a) using sampled data to perform maximum entropy estimates of probabilities in the bigger data is theoretically well-founded b) there is a simple explicit estimator, $\\tilde{p}$, that approximates the maximum entropy estimate in the interesting case where the triple frequency is significantly smaller than the pair frequencies.\n\nIt is instructive to consider a less precise, but even simpler estimator for the case where, informally, there is no strong positive correlation among $X$, $Y$, and $W$, and $\\Pr(\\overline{XYW})$ is close to~1.\nThen $p_{XY\\overline{W}}\/p_{p_{X\\overline{YW}}} \\approx p_{Y|X}$, the observed probability of $Y$ given $X$, and similarly $p_{X\\overline{Y}W}\/p_{p_{\\overline{XY}W}} \\approx p_{X|W}$ and $p_{\\overline{X}YW}\/p_{p_{\\overline{X}Y\\overline{W}}} \\approx p_{W|Y}$, so we can approximate the triple frequency by:\n\\begin{align}\\label{eq:simplest}\n\tp^* = p_{Y|X} p_{X|W} p_{W|Y}\n\\end{align}\n\\noindent\nApplying (\\ref{eq:simplest}) to estimate $\\Pr(W|XY)$, we get the estimator \\[p^{\\#} = p_{Y|X} p_{X|W} p_{W|Y} \/ p_{XY} = p_{W|X} p_{W|Y} \/ p_W \\text{; }\\]that is, the factors by which conditioning on $X$ and $Y$ influence the probability of $W$ get multiplied.\n\n{\\bf Empirical study.} Our experimental evaluation on real data sets shows that maximum entropy estimates give meaningful, and often quite precise, frequency predictions also in cases where the independence estimate $\\theta^i_X$ and the extrapolation estimate $\\theta^e_X$ do not. The error of the estimator is well modeled by the assumption that transactions are independently sampled from a distribution having the estimated subset probabilities.\n\n\n{\\bf Overview.}\nIn \\Cref{sec:prel} we introduce basic terms and notation followed by a description in \\Cref{sec:ent} of how the maximum entropy estimate of an item set is computed. We then prove in \\Cref{sec:noise} that the maxent estimate is not too sensitive to error on the input distribution. For the experimental evaluation we first show in \\Cref{sec:exgen} that the maximum entropy estimate achieves better concentration in general and then in \\Cref{sec:exlow} we discuss results on item sets of low statistically insignificant support.\n\n\\subsection{Related work}\\label{sec:rel}\n\nThe principle of maximum entropy dates far back, but was introduced to information theory in a seminal work by E. T. Jaynes~\\cite{1957PhRv..106..620J}. It has since seen applications in a large number of areas.\n\nThe maximum entropy distribution of $n$ random variables is known to be computable in time exponential in $n$ using the well-known Iterative Scaling algorithm~\\cite{588021}. The running time is due to the fact that for $n$ variables, there are $2^n$ subsets of variables. In the general case, that is with no knowledge of the distribution, Tatti has proved\nthat querying the model is PP-hard \\cite{Tatti:2006:CCQ:1146068.1711152}, which is (believed to be) harder than NP.\n\n{\\em Association rule mining\\\/}~\\cite{Han:2005:DMC:1076797,Agrawal:1993:MAR:170036.170072} is a well-known and extensively studied problem, where a rule has the form $X \\implies Y$ with $X,Y$ being disjoint subsets of random variables. In transactional data sets association rule mining traditionally relies on finding \\emph{frequent itemsets}, i.e., for some set of items $I$ and a set of transactions $D$ over $I$ then one wishes to report back the sets $X \\subseteq I$ that are contained in more than $s$ transactions, for a fixed threshold $s$.\n\n\nThe maxent distribution has been used as a model to measure how significant an itemset is, in the framework of frequent itemset mining, e.g. \\cite{10.1109\/ICDM.2007.43,Mampaey:2011:TMI:2020408.2020499}. The general approach is to compute the maximum entropy distribution (via the Iterative Scaling algorithm) and then compute the Kullback-Leibler divergence with the empirical distribution, from which a $p$-value can be found that is used to rank the item sets. Our approach is that we observe some sample of the subsets of the set of interest and then use these subsets to efficiently compute the maxent estimate.\n\nFor the case where all frequencies of the strict subsets are known, the maximum entropy model has been used by R. Meo~\\cite{Meo:2000:TDV:363951.363956} by comparing the probability estimate under maximum entropy to the empirical probability in order to achieve a measure of ranking an itemset. One of the main open problems of \\cite{Meo:2000:TDV:363951.363956} was determining the existence of a closed form formula for a maxent estimate given the subset probabilities. This was partially resolved in \\cite{971763}, where the author provides a formula for the $1$-dimensional search space in which the maxent estimate lies, which is then traversed by binary search that is shown to converge to the maxent estimate.\n\n{\\bf Comparisons.} Our estimator takes as parameters all the single and pair-frequencies. The hypothesis present implicitly in our model is that data generally has weak third-order dependencies. This is also the reason Chow-Liu trees\\cite{1054142}, which model only first and second-order dependencies, are known to be a good approximations of many observed distributions. The maximum entropy estimate used can be seen as an application of \\cite{Meo:2000:TDV:363951.363956}, where singleton and pair frequencies are used to efficiently compute a maxent estimate in the interesting case where the estimand has no observed occurrences. One of the main open problems of \\cite{Meo:2000:TDV:363951.363956} is an explicit formula for the maxent estimate - \\Cref{thm:maxent:main} in this paper shows an explicit formula for an approximation of the maxent estimate under certain conditions.\nAs the $1D$ search of the maximum entropy estimate $\\theta^m_X$ is determined when having all subset frequencies of $X$, we can compute a good maximum entropy estimate using a small constant number of iterations of binary search as opposed to computing the full maximum entropy distribution. We give a proof of this that is similar to that of Meo~\\cite{971763}, with the distinction that where she shows that there exists constants such that the maxent estimate can be classified by setting particular equations to be equal to each other, in our proof the constants are explicitly stated.\n\n\\section{Frequency estimates of itemsets}\\label{sec:mod}\n\\subsection{Preliminaries}\\label{sec:prel}\nWe provide some definitions and notation that will be used throughout the paper.\n\nRemember that a boolean random variable $A$ is a variable with values in $\\{0,1\\}$. A binary data set $D$ of observations of boolean random variables is\nan $m \\times n$ matrix consisting of $m$ binary $n$-sized vectors, index $0 \\leq i \\leq n-1$ corresponding to the outcome of boolean random variable $A_i$.\nFor a particular subset of the boolean variables $S \\subseteq [n]$ a binary vector $\\omega$ \\emph{covers} $S$ iff $A_i = 1$ implies $\\omega_i$ for every $i \\in S$.\nThe \\emph{frequency} $\\theta_S$ of a set of boolean variables is the proportion of the $m$ row vectors in $D$ that covers $S$.\\\\\nA \\emph{distribution} $p$ over data $D$ is mapping \\[ p: \\{0,1\\}^n \\mapsto [0,1] \\text{ s.t.} \\sum_{\\omega \\in \\{0,1\\}^n} p(\\omega) = 1\\text{.} \\]\nFor a distribution $p$ and a vector of $1$s $v$ we denote by $p(S = v) = p(S = 1) = \\theta_S$ the probability $\\Pr(\\omega \\text{ covers } S)$.\n\nThe \\emph{empirical distribution} over data set $D$ is given as\n\\begin{equation}\\label{eq:emp}\n\tq_D(a_1 = v_1, \\ldots, a_n = v_n) =  \\frac{ \\left| \\{ t \\in D | t = v \\}\\right|}{m}\n\\end{equation}\nWe will denote by \\emph{empirical frequency} the frequency according to the empirical distribution\n\nLet a family of random variable sets $\\mathcal{F}$ be \\emph{satisfied} by a distribution $p: \\{0,1\\}^n \\mapsto [0,1]$\niff for every $S \\in \\mathcal{F}$ it holds that $p( \\omega \\textrm{ covers } S ) = \\theta_S$.\n\nFinally, we say that a set of binary variables $X$ is \\emph{downard closed} if for all strict subsets $S \\subset X$ we have $\\theta_S$, e.g., if we consider\nthe triple of items $s = \\{I_1, I_2, I_3\\}$ then $s$ is downward closed if we know the empirical singleton frequencies $\\theta_{I_1}, \\theta_{I_2}, \\theta_{I_3}$ and empirical pair frequencies $\\theta_{\\{I_1,I_2\\}}, \\theta_{\\{I_2,I_3\\}}, \\theta_{\\{I_1,I_3\\}}$.\n\nWe consider specifically the case where all itemsets of size $3$ (triples) are downward closed and the triples are the itemsets which we wish to estimate the frequency of.\n\n\n\\subsection{Estimation by extrapolation and independence assumption}\\label{sec:ests}\nWe briefly describe the two estimators used for comparison. \n\nLet $X = \\{I_1, I_2, I_3\\}\\subset I$, $|X|=3$, be the triple of interest from sampled data set $D \\subset \\mathcal{D}$. The independence model assumes the occurrences of random variables to not be correlated, thus we have:\n\\begin{equation*}\n\\theta^i_X = \\theta_{I_1} \\theta_{I_2} \\theta_{I_3}\n\\end{equation*}\nFor the extrapolation estimator, let $occ(X)$ be the number of occurrences of $X$ in $D$. To estimate the frequency $\\theta_X$ in $\\mathcal{D}$ we have:\n\\begin{equation*}\n\\theta^e_X = occ(X)\/|D|\n\\end{equation*}\nFollowing from Chernoff bounds on independent variables $\\theta^e_X$ is known to be a good estimate when $\\theta_X$ is significant in $D$.\n\n\\subsection{Maximum entropy of itemsets}\\label{sec:ent}\nWe are interested in estimating the frequency of a specific set of items $G \\subseteq I$. We will estimate such a frequency using\nthe maximum entropy distribution, which intuitively can be thought of as the most uniform distribution given some\nobserved frequencies. We will compute the needed entries of the maximum entropy distribution to be used to compare to the empirical distribution.\n\nFor a family of itemsets $\\mathcal{F}$ and a variable set $G$ let the projected family $\\mathcal{F}_G$ be defined as \\[ \\mathcal{F}_G = \\{ X \\in \\mathcal{F} \\vert X \\subset G, X \\neq \\emptyset\\}.\\]\nThen letting $\\mathcal{P}$ be the set of all possible probability distributions that satisfy $\\mathcal{F}_G$,\nthe entropy of a distribution $p$ is given as\n\\begin{equation}\n\tH(p) =\t- \\sum_{ \\omega \\in \\{0,1\\}^{|G|}} p(\\omega) \\log p(\\omega)\n\\end{equation}\nThe maximum entropy distribution $p*$ can be found by maximizing $H(p)$ over all $p \\in \\mathcal{P}$\n\\begin{equation}\n\tp^* = \\arg\\max_{p \\in \\mathcal{P}} H(p)\n\\end{equation}\nWe note that $|\\mathcal{F}_G| = \\mathcal{O}\\left( 2^{|G|} \\right)$ and if $\\mathcal{F}_G = \\emptyset$ then\n $p^*$ is the uniform distribution. The set $\\mathcal{P}$ contains the empirical distribution (\\Cref{eq:emp}) and hence is\nnon-empty by construction. We shall denote by \\emph{maximum entropy estimate}  $\\theta^m_X$ of an itemset $X$ the frequency of the itemset according to the maximum entropy distribution.\n\n\\subsubsection{Classifying and computing the maxent estimate}\\label{sec:maxentcomp} For completeness we will state the approach used to compute the maximum entropy estimate of itemset frequency. We note that a similar proof of how to find the maxent estimate appears in \\cite{971763}, however we give explicit constants in \\Cref{eq:tmax} whereas their proof shows existence of the constants.\n\n{\\bf High-level description}. The overview of the proof is that for any $z>1$ variables, when $\\Set{\\theta_X}{X \\subset G}$ is given for each variable then\nthe subspace of the joint probability space of $z$ random boolean variables is $1$-dimensional. It follows from this that the frequency estimate according to the maximum entropy distribution $p^*$ is located on a line segment. The estimate thus be computed by doing a simple binary search on this line segment and the main point of doing this is that we avoid computing the entire maximum entropy distribution.\n\nGiven $z$ random boolean variables and the marginal probabilities $\\Set{\\theta_X}{X \\subset G}$ then the joint probability space is given by $2^z$ linear equations, where\nthe rank of the corresponding matrix is $2^z-1$. Let the $z$ boolean random variables $A_1, \\ldots, A_z$ have probabilities $\\Pr[A_1], \\ldots, \\Pr[A_z]$.\nFor $x \\subseteq [z]$, $y = [z]\\setminus x$ then denote by $f^=_{x}$ the probability $\\Pr\\left[\\forall_{i \\in x} A_i = 1 \\wedge \\forall_{j \\in y} A_j = 0\\right]$.\nand by $\\theta_x$ the probability $\\Pr\\left[\\forall_{i \\in x} A_i = 1\\right]$.\n\n\n\\begin{lemma}\\label{lem:marg}For random $z$ boolean variables $A_1, \\ldots, A_z$ with feasible marginal probabilities $\\Set{\\theta_X}{X \\subset [z]}$ then the space of feasible distributions is determined entirely by $f^{=}_{[z]}$. More generally,\nfor $S \\subseteq [z]$ then $f^{=}_S$ follows from \\Cref{eq:pincl}.\n\\begin{equation}\\label{eq:pincl}\nf^{=}_S = \\sum_{S' \\supseteq S} (-1)^{\\left|S' \\setminus S\\right|} \\theta_{S'}\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nWe prove this by induction on $|S|$. For $|S|=i$ we have \\[ f^=_S = \\sum_{S' \\supseteq S} (-1)^{\\left| S' \\setminus S \\right|} \\theta_{S'} \\]\nFor the inductive step we assume \\Cref{eq:pincl} to hold for $|S| > i$.\nThen for $|S| = i$, \\Cref{eq:base} holds as the sum is over supersets of $S$.\nBy applying \\Cref{eq:pincl} to the second term of the right hand side of \\Cref{eq:base} we get\n\\Cref{eq:sub}.\n\\begin{align}\nf^{=}_S &= \\theta_S - \\sum_{S' \\supset S} f^=_{S'}\\label{eq:base}\\\\\nf^{=}_S &= \\theta_S - \\sum_{S' \\supset S} \\sum_{S'' \\supseteq S'} (-1)^{\\left|S'' \\setminus S'\\right|} \\theta_{S''}\\label{eq:sub}\n\\end{align}\nThe double sum of \\Cref{eq:sub} can be split into two as shown in \\Cref{eq:maxent:split}\n\\begin{equation}\nf^{=}_S = \\theta_S - \\left( \\sum_{S' \\supseteq S} \\sum_{S'' \\supseteq S'} (-1)^{\\left|S'' \\setminus S'\\right|} \\theta_{S''} - \\sum_{S' = S} \\sum_{S'' \\supseteq S'} (-1)^{\\left|S'' \\setminus S'\\right|} \\theta_{S''} \\right)   \\label{eq:maxent:split}\n\\end{equation}\nThe first double sum can be split into two parts \\[ \\sum_{S' \\supset S} \\sum_{S'' \\supset S'} (-1)^{\\left|S'' \\setminus S'\\right|} \\theta_{S''} + \\sum_{S' = S} \\sum_{S'' = S'} (-1)^{\\left|S'' \\setminus S'\\right|} \\theta_{S''}\\]\nthe first for which we will use \\Cref{fa:sum} below.\n\\begin{fact}\\label[fact]{fa:sum} For a set space $X$ and function $g: X \\to \\mathbb R$, for a double sum of sign-alternating supersets of any $x \\in X$ we have \\[ \\sum_{x' \\supset x} \\sum_{x'' \\supset x'} (-1)^{\\left|x'' \\setminus x'\\right|} g(x) = 0 \\] due to summands canceling out.\n\\end{fact}\nThen by application of \\Cref{fa:sum} to \\Cref{eq:maxent:split} we get \\Cref{eq:merge}, where the first double sum consists\nonly of the element $\\theta_S$, hence we now arrive at the induction basis in \\Cref{eq:basis}.\n\\begin{align}\nf^{=}_S &= \\theta_S - \\left( \\sum_{S' = S} \\sum_{S'' = S'} (-1)^{\\left|S'' \\setminus S'\\right|} \\theta_{S''} - \\sum_{S' = S} \\sum_{S'' \\supseteq S'} (-1)^{\\left|S'' \\setminus S'\\right|} \\theta_{S''} \\right)\\label{eq:merge}\\\\\nf^{=}_S &= \\theta_S - \\left( \\theta_S - \\sum_{S' = S} \\sum_{S'' \\supseteq S'} (-1)^{\\left|S'' \\setminus S'\\right|} \\theta_{S''} \\right) \\notag\\\\\nf^{=}_S &=  \\sum_{S'' \\supseteq S} (-1)^{\\left|S'' \\setminus S\\right|} \\theta_{S''} \\label{eq:basis}\n\\end{align}\n\\end{proof}\nIntuitively, \\Cref{lem:marg} states that the space of probability distributions that satisfy\nthe marginal probabilities can be traversed by varying $f^{=}_{[z]}$. We note that \\Cref{lem:marg} was used earlier by Calders \\& Goethals~\\cite[eq. (1)]{Calders:2007:NIM:1231311.1231319} and that we include the proof for sake of completeness.\n\nWe shall now argue that on this line through $2^z$ dimensional space, there is a unique point, i.e. a unique feasible distribution $p$,\nthat maximizes the entropy $H(p)$ and thus computing this point allows us to query the the maximum entropy distribution $p^*$ for a\n$z$-sized set of variables.\nGiven a feasible distribution $x$ consisting of entries $x_S \\geq 0$ for every $S \\subseteq [z]$,\nthen by \\Cref{lem:marg} the feasible distribution space $\\mathcal{P}_f$ can be traversed by \\Cref{eq:distspace}.\n\\begin{equation}\\label{eq:distspace}\n\\mathcal{P}_f = x + t \\cdot v, t \\in (l,r)\n\\end{equation}\nwhere $v$ is a $2^z$-sized $\\{-1,+1\\}$-vector with entries $v_S = (-1)^{z -|S|}$ for every $S \\subseteq [z]$ and the range\n$(l,r)$ is the range for which all coordinates in the vector $x + t \\cdot v$ are non-negative.\nConsider the partitioning of subsets $S\\subseteq [z]$ into\n$S_{\\textup{even}} = \\Set{S}{\\left( z - |S| \\right) \\text{is even}}$  and $S_{\\textup{odd}} = \\Set{S}{\\left( z - |S| \\right) \\text{is odd}}$.\nThen the $t$-value borders are given below.\n\\begin{align*}\nl &= \\max \\Set{x_S}{S \\in S_{\\textup{even}} }\\\\\nr &= \\min \\Set{x_S}{S \\in S_{\\textup{odd}}}\n\\end{align*}\nWe note that feasible solution $x$ exists by construction since we have observed a feasible distribution and that vector $v$ corresponds to\nthe null space of the $2^z$-row matrix denoting the linear equalities which the joint distribution adheres to.\n\nLocation of the unique point corresponding to querying the max entropy distribution $p^*$ is shown in \\Cref{lem:diff} below.\n\\begin{lemma}\\label{lem:diff}\nFor a feasible distribution space $\\mathcal{P}_f = x + t \\cdot v, t \\in (l,r)$ there exists a point $p_{max} \\in \\mathcal{P}_f$ s.t.\n$ \\forall p \\in\\mathcal{P}_f, p \\neq p_{max} \\implies H(p_{max}) \\geq H(p)$.\nIn particular, \\Cref{eq:diffres} below holds.\n\\begin{align}\n\\frac{\\d{}}{\\d{t}} H(x + t \\cdot v) &= \\frac{\\d{}}{\\d{t}} \\left( \\sum_{S \\subseteq [z]} ( x_S + (-1)^{z - |S|}t) \\log \\left( \\frac{1}{x_S + (-1)^{z-|S|} t}\\right) \\right) \\label{eq:diffbase}\\\\\n&= \\sum_{S \\subseteq [z]} (-1)^{z - |S|} \\log \\left(\\frac{1}{x_S + (-1)^{z-|S|} t} \\right)\\label{eq:diffres}\n\\end{align}\n\\end{lemma}\n\\begin{proof}\nWe wish to show the existence of the scalar $t_{max}$ that optimizes the entropy,\ni.e. $p_{max} = x + t_{max} \\cdot v$.\n\\Cref{eq:diffres} follows form standard deriviate rules, from which we arrive at\n\\[\\frac{\\d{}}{\\d{t}} H(x + t \\cdot v) = \\sum_{S \\subseteq [z]} (-1)^{z - |S|}\\left( \\log \\left(\\frac{1}{x_S + (-1)^{z-|S|} t} \\right) + \\frac{\\left( x_S + (-1)^{z-|S|} t \\right)^2}{\\left( x_S + (-1)^{z-|S|} t \\right)^2} \\right) \\]\nwhere the rightmost term will cancel out due to there being an equal number of odd- and even-sized subsets of $[z]$ for any integer $z$.\n\\end{proof}\n\\begin{corollary}\\label{cor:diff}\nThe value $t_f \\in (l,r)$ that maximizes $H(x + t \\cdot v)$ while $x + t \\cdot v$ is a feasible solution can be found as $t_f = \\textup{median} \\{l,r,t_{max} \\}$.\n\\end{corollary}\n\\begin{proof}\nFrom \\Cref{lem:diff} we have that $t_{max}$ is the $t$-value that maximizes the entropy function $H(x + t \\cdot v)$, which is at its unique maximum when \\Cref{eq:tmax} holds.\n\\begin{equation}\\label{eq:tmax}\n \\sum_{S \\in S_{\\textup{odd}}} \\log \\left(\\frac{1}{x_S - t} \\right) = \\sum_{S \\in S_{\\textup{even}}} \\log \\left(\\frac{1}{x_S + t} \\right)\n\\end{equation}\nLet $t_f = \\textup{median} \\{l,r,t_{max} \\}$. For the line segment spanned by end points $l$ and $r$, we have 3 cases: $t_{max} < l$ then $t_f = l$, $l \\leq t_{max} \\leq r$ then $t_f = t_{max}$ and $t_{max} > r$ then $t_f = r$.\nThe median of the values $l,r,t_{max}$ distinguishes these cases.\n\\end{proof}\n\n\n\n\n\n\nThe $t$ for which \\Cref{eq:tmax} holds is $t_{\\text{max}}$, that is, the equation holds strictly under maximum entropy. Since the solution space space of $(l,r)$ is always $1$-dimensional if all frequencies of the $2^z-1$ subsets are given, then we compute the value $t_{\\text{max}}$ by binary search in the space. We note that this search is an approximation, but the binary search converges to values arbitrarily close to $t_{\\text{max}}$ quickly in practice, e.g., $30$ iterations of binary search was used to produce the results in this paper. For an itemset $X$, the $t$-value we hold after $30$ such iterations in the search space is then the maximum entropy estimate $\\theta^m_X$.\n\nWe note that by using a constant number of iterations for each binary search the computation takes time linear in the number of itemsets for which we wish to compute the maximum entropy estimate. Up to a constant factor this is equivalent to the extrapolation and independence assumption estimators.\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsubsection{Maxent on noisy inputs}\\label{sec:noise}\nRecall that a view on data is that all data comes from some smaller sample $D$ sampled independently from a larger data set $\\mathcal{D}$. As we wish to use the data from $D$ to reason about triples from $\\mathcal{D}$ we will show that the error introduced by sampling does not hurt the maxent estimate too much, in particular we will show that a maxent estimate based on $D$ will not be far from a maxent estimate based on $\\mathcal{D}$.\nWe will show that the maximum entropy estimate on a triple computed on input with relative error $0 \\leq \\varepsilon < 0.5$ is only a factor $\\mathcal{O}(\\varepsilon)$ from the maximum entropy estimate computed using the true distribution from $\\mathcal{D}$ as input.\n\nFor a distribution $d$ over $3$ variables $X,Y,W$ we have \\[ d = (\\Pr(XYW), \\Pr(XY \\wedge \\overline{W}), \\ldots, \\Pr(\\overline{XYW}))\\text{,}\\] i.e., $|d| = 2^3$. Let $|d_i|$ be the number of non-negated literals in $d_i$. The entries of $d$ can (as in \\Cref{sec:maxentcomp}) be partitioned into two sets, $u = u_1, \\ldots, u_4$ and $ l = l_1, \\ldots, l_4$ where $d_i \\in u$ if $3-|d_i|$ is odd and $d_i \\in l$ otherwise.\n\nLet two polynomials $L_{\\varepsilon_1}(t)$ and $U_{\\varepsilon_1}(t)$ with error $0 \\leq |\\varepsilon_1| < 1$ be defined\n\\begin{align}\nL_{\\varepsilon_1}(t) & = (l_1 + \\varepsilon_1 + t)(l_2 + \\varepsilon_1 + t)(l_3 + \\varepsilon_1 + t)t \\label{eq:upol} \\\\\nU_{\\varepsilon_1}(t) & = (u_1 + \\varepsilon_1 - t)(u_2 + \\varepsilon_1 - t)(u_3 + \\varepsilon_1 - t)(u_4 + \\varepsilon_1 - t) \\label{eq:lpol}\n\\end{align}\nwhere $t = \\Pr(XYW)$ is the triple frequency. When we have\n\\begin{equation}\nL_0(t) = U_0(t)\n\\end{equation}\nthen $t$ is the triple frequency under maximum entropy. We seek to bound the error on the output caused by the input error $\\varepsilon_1$. Letting $t_0$ be the solution to $L_0(t) = U_0(t)$ and $t_{\\varepsilon_1}$ be the solution to $L_{\\varepsilon_1}(t) = U_{\\varepsilon_1}(t)$ we wish to bound the error on $t_{\\varepsilon_1}$ in terms $t_0$.\nWe will show the following lemma.\n\\begin{lemma}\\label{lem:bounds}\nLet two polynomials with additive error $t$ on the terms and where $0 < l_i, u_i, \\leq 1$ be defined as below.\n\\begin{align}\n\\tilde{L}(t) &= (l_1+t)(l_2+t)(l_3+t)\\\\\n\\tilde{U}(t) &= (u_1-t)(u_2-t)(u_3-t)(u_4-t)\n\\end{align}\nFor all $t \\in \\left[0; \\varepsilon \\min{u_1, \\ldots, u_4, l_1, \\ldots, l_3}\\right]$, where $ 0 < \\varepsilon \\leq 1$, \nthe following bounds on \\Cref{eq:upol,eq:lpol} hold \n\\begin{align}\n\\tilde{L}(0)t &\\leq L_0(t) \\leq (1+\\varepsilon)^3 \\tilde{L}(0)t \\label{eq:uin}\\\\\n(1-\\varepsilon)^4 U_0(0) &\\leq \\tilde{U}(t) \\leq U_0(0) \\label{eq:lin}\n\\end{align}\n\\end{lemma}\n\\begin{proof}\nThe first inequality of \\cref{eq:uin} follows trivially from $t > 0$ and from $\\tilde{L}$ being monotonically increasing in $t$. For the second inequality let $l_{\\min} = \\min{l_1,\\ldots,l_3}$ and $t \\leq \\varepsilon l_{\\min}$, $0 < \\varepsilon \\leq 1$. It follows that\n\\begin{align*}\nL_0( t) & \\leq (l_1+\\varepsilon l_{\\min})(l_2+ \\varepsilon l_{\\min})(l_3+ \\varepsilon l_{\\min})\\varepsilon l_{\\min}\\\\\n& \\leq (1+\\varepsilon)^3 l_1 l_2 l_3 t\\\\\n&= (1+\\varepsilon)^3 \\tilde{L}(0) t\n\\end{align*}\nThe first inequality of \\cref{eq:lin} follows analogously; let $u_{\\min} = \\min{u_1,\\ldots,u_4}$ and $t\\leq \\varepsilon u_{\\min}$, $0 < \\varepsilon \\leq 1$. We then have\n\\begin{align*}\n\\tilde{U}(t) &\\geq (u_1-\\varepsilon u_{\\min})(u_2-\\varepsilon u_{\\min})(u_3-\\varepsilon u_{\\min})(u_4-\\varepsilon u_{\\min})\\\\\n &\\geq (1-\\varepsilon)^4 U_0(0)\n\\end{align*}\nThe second inequality again follows from $t>0$ and $\\tilde{U}$ being monotonically decreasing in $t$.\n\\end{proof}\nIt follows that a simple approximate formula for maximum entropy estimates on triples exist.\n\\begin{lemma}\\label{lem:tint}\nFor a triple with distribution over entries $\\mathcal{D} = l_1, \\ldots, l_3, u_1, \\ldots, u_4$ let\n\\begin{equation*}\n\\tilde{t} = \\frac{u_1 u_2 u_3 u_4}{l_1 l_2 l_3}\n\\end{equation*}\nThe triple frequency under maximum entropy $t < \\varepsilon \\min{\\mathcal{d}}$ for $0 < \\varepsilon \\leq 0.5$ can be bounded in terms of $\\tilde{t}$\n\\begin{equation}\\label{eq:tbound}\nt \\in \\left[ (1-6.5\\varepsilon) \\tilde{t},  \\tilde{t}  \\right]\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nRecall that $t$ takes value under maximum entropy when $U_0(t)\/L_0(t) = 1$.\nBy \\Cref{lem:bounds} we have a lower bound on $U_0(t)\/L_0(t)$\n\\begin{align*}\n(1-\\varepsilon)^4 U_0(0) \/ (1+\\delta)^3 \\tilde{L}(0) &\\leq U(t)_0\/L_0(t)\\\\\n(1-\\varepsilon)^4 u_1 u_2 u_3 \/ (1+\\varepsilon)^3 l_1 l_2 l_3 &\\leq U_0(t)\/L_0(t) \\\\\n\\frac{(1-\\varepsilon)^4}{(1+\\varepsilon)^3} \\tilde{t} & \\leq U_0(t)\/L_0(t)\n\\end{align*}\nBy expansion and using $0 < \\varepsilon \\leq 0.5$ we get the needed lower bound of \\cref{eq:tbound}.\n\\begin{align*}\n1 - c*\\varepsilon \\geq \\frac{(1-\\varepsilon)^4}{(1+\\varepsilon)^3} \\\\\nc \\geq -\\varepsilon^3 + 5\\varepsilon^2 - 3\\varepsilon + 7\\\\\nc \\geq 6.5\n\\end{align*}\nEquivalently we have the needed upper bound\n\\begin{align*}\nU_0(t)\/L_0(t) &\\leq U_0(0) \/ \\tilde{L}(0)\\\\\nU_0(t)\/L_(t) &\\leq u_1 u_2 u_3 \/ l_1 l_2 l_3\\\\\nU_0(t)\/L_(t) &\\leq \\tilde{t}\n\\end{align*}\n\\end{proof}\nWe will now assume relative error on the distribution entries $l_i, u_i$. The following lemma holds analogously to \\Cref{lem:tint}.\n\\begin{lemma}\\label{lem:appr}\nLet distribution $d = (l_1, \\ldots l_3, u_1, \\ldots, u_4)$. Let an approximation of distribution $d$ be defined by $u'_i \\in [(1-\\varepsilon)u_i, (1+\\varepsilon)u_i]$ and  $l'_i \\in [(1-\\varepsilon)l_i, (1+\\varepsilon)l_i]$ for each $u_i, l_i \\in d$.\n\\begin{align*}\n\\tilde{t}_{r} &= \\frac{u'_1 u'_2 u'_3 u'_4}{l'_1 l'_2 l'_3}\\\\\n\\tilde{t} &= \\frac{u_1 u_2 u_3 u_4}{l_1 l_2 l_3}\n\\end{align*}\nIt holds that\n\\begin{equation}\\label{eq:tapprox}\n\\tilde{t}_{r} \\in \\left[ (1-6.5\\varepsilon') \\tilde{t},  \\tilde{t}  \\right]\n\\end{equation}\n\\end{lemma}\n\\begin{proof}(\\Cref{thm:maxent:main})\nThe theorem follows directly from \\Cref{lem:tint,lem:appr}.\n\\end{proof}\n\n\\section{Experimental Results}\\label{sec:exp}\nOur experiments were conducted on $5$ real datasets shown in \\Cref{tab:data}. For each dataset we prune the singletons with support below a specified threshold. We perform this pruning in order to construct datasets where intuitively the independence model has a chance to do well as it relies solely on the singletons, but also to keep the number of items down to a practical level, as the time complexity with $n$ items is $\\mathcal{O}(n^3)$ per experiment.\nAOL Queries \\footnote{\\url{http:\/\/www.gregsadetsky.com\/aol-data\/}} is a uniform sample of the infamous AOL seach terms dataset. Docwords is transactions of words occurring together in documents. From MovieLens \\footnote{\\url{http:\/\/www.grouplens.org\/node\/73}} we create a dataset where a transaction is the set of movies which a particular user rated 5\/5 stars. Retail \\footnote{\\url{http:\/\/fimi.ua.ac.be\/data\/}} is shopping baskets from an anonymous Belgian supermarket.\n\n{\\bf Overview of experiments.} We start by showing how the independence and maxent estimators perform on the full datasets, i.e., when there is significant support for the triple whose frequency is to be estimated as well as its subsets. The maxent estimator shows better concentration and less variance even for the low-support triples. We then perform experiments for all datasets where $1\\%$ of the transactions are sampled and we wish to estimate the frequency of triples in the whole data set. For every triple $X$ with $30 \\leq occ(X) \\leq 100$ we use the three estimators $\\theta^m_X$, $\\theta^i_X$ and $\\theta^e_X$. As $X$ occurs at most $100$ times in the full dataset, it occurs in expectation at most once in the sample.\n\nWe also study precision and recall for the problem of approximating the set of frequent triples based on the estimators. Again we consider two cases: 1) Estimates are based on the full dataset, where the set to approximate consists of the $10\\%$ highest frequencies among all triples, and 2) as in the first case, but with estimates based on a $1\\%$ sample. In the latter case, if the  threshold for being in top 10\\% is $\\Delta$, then we include in the estimate all triples that are estimated to have at least $0.9 \\Delta$ occurrences. The number 0.9 was experimentally found to yield a good precision\/recall tradeoff for the maximum entropy estimates.\n\nFinally, using again a $1\\%$ sample of the data, we compute the average ratio between the error made by our maximum entropy estimate and estimates made by independence and extrapolation, respectively.\n\nIn summary our experiments show:\n\\begin{enumerate}\n\\item In most cases, the maximum entropy estimator provides the best estimate for low-support triples.\n\\item Sampling with higher probability increases concentration greatly for $\\theta^m_X$ even when the sample is still of insufficient size for $\\theta^e_X$ to be useful.\n\\item In almost all cases studied, the precision and recall are strictly better for $\\theta^e_X$ (see \\cref{tab:prfull,tab:prsample}).\n\\item When predicting the occurences of triples in the full dataset using a $1\\%$ sample, we show $\\theta^m_X$  improves the absolute error by a factor $3$ -- $14$ compared to $\\theta^i_X$ and $\\theta^e_X$ (see \\Cref{tab:estratio}).\n\\end{enumerate}\n\\begin{table}[!t]\n\\centering\n\\begin{tabular}{|c|ccc|}\n\\hline\nName & Occ. Threshold& \\#Items & \\#Transactions\\\\\n\\hline\nAOL Queries & $500$ & $211$ & $144038$\\\\\nDocwords & $3000$ & $142$ & $49078$\\\\\nMovielens & $3000$ & $85$ & $67312$\\\\\nRetail & $500$ & $85$ & $88162$\\\\\n\\hline\n\\end{tabular}\n\\caption{Datasets used. All datasets are from real data and have previously been used for data mining purposes.}\\label{tab:data}\n\\end{table}\n\n\\begin{table}[t!]\n\\centering\n\\begin{tabular}{|l|llll|}\n\\hline\n & \\multicolumn{2}{ c }{Independence} & \\multicolumn{2}{ c| }{Maxent}\\\\\nDataset & Precision & Recall & Precision & Recall\\\\\n\\hline\nAOL Queries & $0.0$ & $0.0$ & $0.54$ & $1.0$\\\\\nDocwords & $0.93$ & $0.43$ & $0.92$ & $0.93$\\\\\nMovieLens & $1.00$ & $0.003$ & $0.80$ & $0.96$\\\\\nRetail & $1.00$ & $0.43$ & $0.99$ & $0.97$\\\\\n\\hline\n\\end{tabular}\n\\caption{Precision-recall for full datasets. Relevant triple threshold $\\Delta$ is such that relevant triples are among the $10\\%$ most frequent.}\\label{tab:prfull}\n\\end{table}\n\n\\begin{table}\n\\centering\n\\begin{tabular}{|l|llllll|}\n\\hline\n & \\multicolumn{2}{ c }{Ind.} & \\multicolumn{2}{ c }{Maxent} & \\multicolumn{2}{ c| }{Extrapol.} \\\\\nDataset & Prec & Rec & Prec & Rec & Prec & Rec\\\\\n\\hline\nAOL Queries & $1.0$ & $0.001$ & $0.23$ & $0.89$ & $0.31$ & $0.67$ \\\\\nDocwords & $0.88$ & $0.44$ & $0.56$ & $0.72$ & $0.53$ & $0.58$ \\\\\nMovieLens & $1.0$ & $0.009$ & $0.51$ & $0.90$ & $0.49$ & $0.85$ \\\\\nRetail & $0.93$ & $0.40$ & $0.51$ & $0.78$ & $0.49$ & $0.73$ \\\\\n\\hline\n\\end{tabular}\n\\caption{Precision-recall for sampled datasets.  Relevant triple threshold $\\Delta$ is such that relevant triples are among the $10\\%$ most frequent. Triples are reported when $occ \\geq 0.9\\Delta$ as this was observed to maximize precision\/recall for $\\theta_X^e$.}\\label{tab:prsample}\n\\end{table}\n\n\\begin{table}\n\\centering\n\\begin{tabular}{|l|ll|}\n\\hline\nDataset & Independence & Extrapolation \\\\\n\\hline\nAOL Queries & $7.91$ & $7.58$ \\\\\nDocwords & $6.31$ & $14.32$ \\\\\nMovieLens & $11.55$ & $7.22$ \\\\\nRetail & $3.22$ & $4.42$ \\\\\n\\hline\n\\end{tabular}\n\\caption{For $n$ estimates and two estimators $est_1$ and $est_2$ the table shows the normalized absolute error ratio: $\\frac{1}{n} \\sum_X \\left(|est_1(X) - occ(X)| \/ |est_2(X) - occ(X)|\\right)$, where $est_2$ is maximum entropy and $est_1$ is independence and extrapolation respectively for the two columns.}\n\\label{tab:estratio}\n\\end{table}\n\n\\subsection{Maxent vs. independence for full datasets}\\label{sec:exgen}\nFor all datasets we ran the estimators on every triple with occ $>30$ on the full (unsampled) dataset. The figures show the concentration of the estimates by for each triple plotting its observed value (Y-axis) and estimated value (X-axis). The plots are shown in \\Cref{fig:full:1,fig:aol:all}.\n\nFor the Docwords dataset both the maxent estimate (\\Cref{fig:doc:ent}) and the independence estimate (\\Cref{fig:doc:ind}) are concentrated around the $X=Y$, which denote the line of optimal estimations while for MovieLens we observe similar concentration for maxent (\\Cref{fig:mov:ent}) while the independence estimator underestimates slightly (\\Cref{fig:mov:ind}) and Retail behaves equivalently in this setting. For AOL Queries (\\Cref{fig:aol:all}) using independence estimates we observe similar great under estimation due to high positive correlation, while the maxent estimator overestimates slightly but is far more concentrated.\n\nOur precision and recall computations (\\Cref {tab:prfull}) is based on relevance threshold $\\Delta$ being such that relevant triples are among the top $10\\%$ most frequent and we report a triple if the estimate is at least $\\Delta$. Note that the high precision for the independence estimate is due to high underestimation - the estimators report too few triples as relevant, as the recall shows. An extreme case of this is the AOL dataset that reports zero triples. Maxent has similar precision but much higher recall for all datasets.\n\\begin{figure*}\n        \\centering\n        \\begin{subfigure}[b]{0.5\\textwidth}\n                \\centering\n                \\includegraphics[width=\\textwidth]{full_docwords_maxent}\n               \n                \\caption{Max. entropy estimates for all triples of occ $>30$ in Docwords.}\n                \\label{fig:doc:ent}\n        \\end{subfigure}%\n        ~\n         \n        \\begin{subfigure}[b]{0.5\\textwidth}\n                \\centering\n                \\includegraphics[width=\\textwidth]{full_docwords_ind}\n               \n                \\caption{Independence estimates for all triples of occ $>30$ in Docwords.}\n                \\label{fig:doc:ind}\n        \\end{subfigure}\n\n        \\begin{subfigure}[b]{0.5\\textwidth}\n                \\centering\n                \\includegraphics[width=\\textwidth]{full_movies_maxent}\n               \n                \\caption{Max. entropy estimates for all triples of occ $>30$ in Movielens.}\n                \\label{fig:mov:ent}\n        \\end{subfigure}%\n        ~\n         \n        \\begin{subfigure}[b]{0.5\\textwidth}\n                \\centering\n                \\includegraphics[width=\\textwidth]{full_movies_ind}\n               \n                \\caption{Independence estimates for all triples of occ $>30$ in Movielens.}\n                \\label{fig:mov:ind}\n        \\end{subfigure}\n        \\caption{Concentration plots for Docwords and Movielens datasets. Each point is a triple, the Y-value of a point is the empirical number of occurrences (occ) of the triple while the X-value is the estimated number of occurences, using either independence or maxent estimators. The red line is X=Y. We observe better concentration on the maxent estimates, in particular when the statistical significance is high.} \\label{fig:full:1}\n\\end{figure*}\n\n\n\\begin{figure*}\n        \\centering\n        \\begin{subfigure}[b]{0.5\\textwidth}\n                \\centering\n                \\includegraphics[width=\\textwidth]{full_aol_maxent}\n               \n                \\caption{Maxent estimates for all triples of occ $>30$ in AOL Queries.}\n                \\label{fig:aol:maxent}\n        \\end{subfigure}%\n        ~\n         \n        \\begin{subfigure}[b]{0.5\\textwidth}\n                \\centering\n                \\includegraphics[width=\\textwidth]{full_aol_ind}\n               \n                \\caption{Independence estimates for all triples of occ $>30$ in AOL Queries.}\n                \\label{fig:aol:ind}\n        \\end{subfigure}\n        \\caption{Concentration plots for AOL Queries. Each point is a triple, the Y-value of a point is the empirical number of occurrences (occ) of the triple while the X-value is the estimated number of occurences, using either independence or maxent estimators. The red line is X=Y. We observe better concentration for Maxent for all occurences. \\label{fig:aol:all}}\n\\end{figure*}\n\n\\subsection{Low-support itemsets}\\label{sec:exlow}\nOn the same datasets we now examine triples $X$ where $30 \\leq occ(X) \\leq 100$ and we sample independently at random every transaction with probability $1\/100$. We wish to estimate the $\\theta_X$ in the full dataset, but since we have $occ(X) \\leq 100$ then following from independent sampling the expected number of occurrences of $X$ in our sample is $\\leq 1$. We will perform estimates using the extrapolation estimate $\\theta^e_X$, the independence estimate $\\theta^i_X$ and the maxent estimate $\\theta^m_X$. We restrict ourselves to triples where all pairs occur in the sample.\n\nThe extrapolation estimator $\\theta^e_X$ performs similarly on all datasets. While $\\theta^e_X$ is an unbiased estimator of $\\theta_X$, the variance is large as a consequence of the sampling - by the mode of independent Bernoulli trials we have that the most likely outcome of a triple is $\\lceil \\mu \\rceil = 1$, with the outcome $0$ and $2$ slightly less probable. On the AOL, Retail and MovieLens datasets our experiments conclude similarly: $\\theta_X^e$ doesn't give meaningful estimates, e.g., zero for the unsampled triples and overestimates on the sampled triples, while $\\theta_X^i$ underestimates greatly and $\\theta_X^m$ is fairly concentrated. See \\Cref{fig:aol:ratio} for example of all three estimators on AOL. For Docwords in this setting, we get better concentration for $\\theta_X^i$ than $\\theta_X^m$ - this can be explained by pair frequencies being more vulnerable to noise introduced by sampling than single frequencies and $\\theta_X^i$ being well concentrated for Docwords (recall \\Cref{fig:doc:ind}). However, we observe that if we raise the sampling probability to $1\/20$ from $1\/100$ then $\\theta_X^m$ has better concentration. In summary, there is a sampling rate where $\\theta_X^e$ is very poorly concentrated where $\\theta_X^m$ outperforms $\\theta_X^i$ on all datasets.\n\nPrecision and recall values (\\Cref{tab:prsample}) were computed by setting the relevance threshold $\\Delta$ to be s.t. if $occ(X) \\geq \\Delta$ then triple $X$ is among the $10\\%$ most frequent triples in the entire dataset with $occ(X) \\geq 30$. Triples are reported if the occurrence estimate, which is based on the $1\/100$ independent sample of all transactions, is at least $0.9\\Delta$ as this value was observed experimentally to yield a good tradeoff. AOL and MovieLens using $\\theta_X^i$ has full precision due to reporting very few triples. Note that $\\theta_X^e$ has an advantage in terms of precision\/recall due to it mostly performing overestimates, i.e., if a triple occurs in the sample then it will likely be reported -- even so we see $\\theta_X^m$ being better than $\\theta_X^e$ except for AOL where only recall is better.\n\nIn the same setting, i.e., triples where $occ(X) \\geq 30$ and using an independent $1\\%$ sample we compute the normalized ratio of the absolute error between estimators $\\theta_X^i$, $\\theta_X^e$ and $\\theta_X^m$. That is, letting $est_m(X),est_i(X)$ denote estimates of triple $X$ by maxent and independence respectively, the normalized ratio is $\\frac{1}{n} \\sum_X \\frac{|est_i(X) - occ(X)|}{|est_m(X) - occ(X)|}$. This experiments show (see \\Cref{tab:estratio}) that for all datasets in this setting we would improve, on average, our absolute error by a factor of $3$ -- $14$ by switching from independence or extrapolation to maximum entropy.\n\n\n\n\n\n\n \\begin{figure}\n         \\centering\n          \\includegraphics[width=0.5\\textwidth]{low_aol_ratio_all}\n         \\caption{est\/obs distribution plots for AOL Queries sampled at $1\/100$. We observe that independence underestimates greatly, while maxent has better concentration than extrapolation, in particular for the low occurence triples.  \\label{fig:aol:ratio}}\n \\end{figure}\n\n\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe question of how to measure utility had already been raised since the time of Bentham (1789). This question took on a special place in economics as Walras (1874) incorporated utility into his model and this model spread as a general equilibrium theory. Because of the difficulty of the problem, economics at one time placed the greatest emphasis on analyses in which utility need not be measured. That is the so-called axiomatic approach. The axiomatic approach does not require any particular form of utility, but assumes only `axioms' that seem reasonable and deduces properties about the economy from them. The existence of equilibrium and the fundamental theorems of welfare economics are the typical results of this approach.\n\nHowever, Debreu (1974) showed the Sonnenschein-Mantel-Debreu theorem, which demonstrated the limitation of the axiomatic approach. According to this theorem, for any continuous function on the unit simplex, we can construct an economy such that the given function coincides with the excess demand function of this economy except for some neighborhood of the boundary. The only conditions imposed on this economy are the following: the economy is a pure exchange economy and all consumers have continuous, increasing, and strictly quasi-concave utility functions, which are all natural and frequently used in the axiomatic approach. This result was taken to mean that there is almost nothing that can be said about the economy from the usual axioms used in the axiomatic approach.\n\nAn alternative to the axiomatic approach, which has therefore come into use for economic analysis, is the method of estimating utility functions from data. However, an important problem arises here. That is, the only data available to us for estimating utility functions is data on the consumer's purchase behavior. In essence, therefore, we can estimate not the utility function, but only the demand function. Because of this problem, the estimation method currently used is basically a method of parameter estimation where the demand function and the utility function is exogenously assumed to correspond one-to-one through their parameters. This method is known as calibration.\n\nOur purpose is to construct a general theory that calculates the utility function from an arbitrary demand function without giving such a exogenous assumption. The history of such works is very old and was discussed by Antonelli (1886) long before the axiomatic approach reached its zenith. This work was succeeded by Pareto (1906), followed by Samuelson (1950) and then greatly improved by Hurwicz and Uzawa (1971) to its present form. This research area is known as integrability theory.\n\nIn this context, Hosoya (2017) gave a specific method for calculating the corresponding utility function from a given candidate of the demand function. This paper also discussed how to handle the case where the demand function contains errors. Let us elaborate on this issue. In order to obtain the demand function, we need to estimate it from the purchase behavior of the consumer, as discussed above. The problem addressed by Hosoya (2017) is the following. If the error in the demand function is small enough, is the error in the utility function calculated by our constructed method also small enough?\n\nThe problem is essentially that of continuity. In other words, it requires the property that when the demand function moves a little, the corresponding utility function also moves only a little. And the most important aspect of the continuity problem is the topology. Because Hosoya (2017) assumed that the demand function is assumed to be continuosly differentiable, the local $C^1$ topology was used for the space of demand functions. However, this result was problematic in many ways. The most significant problem is that there are few econometric results that discuss the space of the demand function with the local $C^1$ topology. This is because there is almost no currently known estimation method that allows the error to converge to $0$ in the local $C^1$ topology as the data size increases. Therefore, even if good results are obtained in this topology, the results cannot be used in econometrics.\n\nTherefore, a weaker topology is needed. A common use in econometric theory for this problem is the topology of compact convergence: that is, the topology whose convergence concept coincides with the uniform convergence on any compact set. However, using this topology means that the space of continuously differentiable functions is not complete. We therefore need to reproduce the results of Hosoya (2017) without assuming differentiability for the demand function. This is the research objective of this paper.\n\nHowever, if we weaken the assumption of differentiability to that of continuity, we are confronted with the problem found by Mas-Colell (1977). He showed the existence of a continuous demand function such that there are two corresponding continuous utility functions representing different preference relations. Applying this to our context means that in the space of continuous demand functions, even if the error of the demand function is $0$, the estimated error of the utility function may not be $0$. This is highly undesirable for us.\n\nThus, as a compromise, we assume that the demand function is locally Lipschitz. Any locally Lipschitz function is differentiable almost everywhere by Rademacher's theorem, and thus, it can be seen that for a locally Lipschitz candidate of the demand function, we can define the Slutsky matrix almost everywhere. In this paper, we first use this result to extend Theorem 1 of Hosoya (2017). Namely, when a candidate of the demand function satisfies Walras' law and locally Lipschitz, if its Slutsky matrix is symmetric and semi-negative definite almost everywhere, then the corresponding utility function can be constructed through the calculation of a differential equations (Theorem 1). We present an example of such a calculation where this method works effectively (Example 1).\n\nThis result has several important consequences. First, using this result, we can obtain two necessary and sufficient condition for a candidate of the demand function to actually be a demand function (Corollary 1). As was the case under continuous differentiability, symmetry and negative semi-definiteness of the Slutsky matrix ``almost everywhere'' are one of them. More important necessary and sufficient condition is another one, which is the existence of a unique global concave solution to some partial differential equation for an arbitrary initial value condition. Because this property is robust for limit manipulation, it provides an important stepping stone for the next arguments in this paper.\n\nMoreover, our method of constructing utility functions is upper semi-continuous if it is restricted on the range of the demand function, and we guarantee the uniqueness of the corresponding upper semi-continuous preference relation on the same space. For outside of the range of the demand function, Hosoya (2020) derives a construction method for an upper semi-continuous utility function and ensures the uniqueness of the corresponding upper semi-continuous preference relation when the range of the demand function is sufficiently wide. We slightly variate this construction method and prove that, again, if the range of the demand function is sufficiently wide, an upper semi-continuous utility function can be constructed outside the range of the demand function and the corresponding upper semicontinuous preference relation is unique (Corollary 2). Hence, Mas-Colell's (1977) counterexample vanishes in such a demand function.\n\nWith these results as a groundwork, we can finally discuss what we wanted to discuss, namely the continuity of the mapping from the demand function to the utility function. As already mentioned, the topology given to the space of demand functions is only the topology corresponding to uniform convergence on compact sets. On this space, even if a sequence of locally Lipschitz demand functions converges to some function (which is not necessarily demand function), it is not guaranteed that the limit is locally Lipschitz. However, if the limit happens to be locally Lipschitz, we can derive that the limit of a demand function is still a demand function. (Theorem 2)\n\nWith this in mind, we construct a certain space of functions. It is the space of demand functions such that they satisfy Walras' law and have a Lipschitz constant uniformly on any compact set. Using Theorem 2, we can prove the completeness of this space (Corollary 3). That is, on this space, if a sequence of demand functions is a Cauchy sequence, then it converges to a demand function.\n\nOn the other hand, however, for a unique derivation of the utility function, the range of the demand function must be sufficiently wide, as discussed in the explanation of the results of Corollary 2 above. We have found an example of a sequence of demand functions that satisfy all the assumptions of Corollary 3, all of which have a sufficiently wide range, yet in the limit the range is very small (Example 2). Therefore, another additional assumption is needed for the continuity result we wanted. Namely, the range of the function in the limit must also be sufficiently wide. In addition to this, the wanted continuity proposition can be obtained when all functions satisfy an axiom called the ``C axiom'', which is introduced by Hosoya (2017, 2020) (Theorem 3). That is, when a sequence of demand functions converges to a demand function with respect to the topology discussed above, then the corresponding sequence of utility functions also converges to the corresponding utility function uniformly on any compact set in the positive orthant.\n\nNote that this result does not hold at the corner. We can only derive more naive results at the corner (Corollary 4). Actually, we construct an example where the sequence of the value of the utility function do not converge to the value of the limit utility function at the corner (Example 3). This is an unfortunate result, but we have to accept it.\n\nFinally, by strengthening the C axiom, we can strengthen the result of Corollary 3 and Theorem 3. We construct a new space of demand functions such that the C axiom is uniformly imposed on the whole functions. Then, we can show that this space is complete, and furthermore, the mapping from this space to the corresponding utility function is also continuous (Corollary 5). This is our desired result.\n\nThe results in the second half of the paper are specifically constructed with a view to discussing the consistency of estimation methods. For an estimation method of the true value $x$, it is said to be consistent if the estimated value $x_N$ converges in probability to $x$ as the data size $N$ increases. To summarise our results, we obtain the following: if the estimation method for the demand function satisfies consistency, then the estimation method for the utility function that is constructed by the given estimation method for the demand function and the computational process in Theorem 1 also satisfies consistency. We believe that this presents a new way of estimating utility functions for econometric theory. In particular, this result can be applied to any estimation method, whether parametric or non-parametric.\n\nThe structure of this paper is as follows. First, Section 2 defines several terms in consumer theory that are necessary for reading this paper. Section 3 presents the main result, whereas Subsection 3.1 discusses the construction method of the utility function, and Subsection 3.2 discusses the completeness of the space and the continuity of the calculation results. Section 4 discusses the position of the present work in the context of related research and provides a list of open problems. Section 5 is the conclusion. As many of the theorems in this paper have long proofs, all proofs are placed in Section 6.\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Preliminary}\nFix $n\\ge 2$. Throughout this paper, we use the following notations: $\\mathbb{R}^n_+=\\{x\\in\\mathbb{R}^n|x_i\\ge 0\\mbox{ for all }i\\in \\{1,...,n\\}\\}$, and $\\mathbb{R}^n_{++}=\\{x\\in \\mathbb{R}^n|x_i>0\\mbox{ for all }i\\in \\{1,...,n\\}\\}$. The former set is called the \\textbf{nonnegative orthant} and the latter set is called the \\textbf{positive orthant}. We write $x\\ge y$ if $x-y\\in \\mathbb{R}^n_+$ and $x\\gg y$ if $x-y\\in \\mathbb{R}^n_{++}$.\n\nLet $\\Omega$ denote the consumption set. We assume that $\\Omega=\\mathbb{R}^n_+$ unless otherwise stated. A set $A\\subset \\Omega^2$ is called a {\\bf binary relation} on $\\Omega$.\n\nFor a binary relation $A\\subset \\Omega^2$, we say that it is\n\\begin{itemize}\n\\item {\\bf complete} if for every $(x,y)\\in \\Omega^2$, either $(x,y)\\in A$ or $(y,x)\\in A$,\n\n\\item {\\bf transitive} if $(x,y)\\in A$ and $(y,z)\\in A$ imply $(x,z)\\in A$,\n\n\\item {\\bf upper semi-continuous} if for every $x\\in \\Omega$, the set $U(x)=\\{y\\in \\Omega|(y,x)\\in A\\}$ is closed,\n\n\\item {\\bf continuous} if $A$ is closed in $\\Omega^2$,\n\n\\item {\\bf upper semi-continuous on $B$} if for every $x\\in B$, the set $U^B(x)=\\{y\\in B|(y,x)\\in A\\}$ is closed with respect to the relative topology of $B$, and\n\n\\item {\\bf continuous on $B$} if $A\\cap B^2$ is closed in $B^2$.\n\\end{itemize}\n\nA binary relation $\\succsim$ on $\\Omega$ is called a {\\bf weak order} if it is complete and transitive. For a weak order $\\succsim$, we write $x\\succsim y$ instead of $(x,y)\\in \\succsim$ and $x\\not\\succsim y$ instead of $(x,y)\\notin \\succsim$. Moreover, we write $x\\succ y$ if $x\\succsim y$ and $y\\not\\succsim x$, and $x\\sim y$ if $x\\succsim y$ and $y\\succsim x$.\n\nSuppose that $\\succsim$ is a weak order on $\\Omega$. If there exists a function $u:\\Omega\\to \\mathbb{R}$ such that\n\\[x\\succsim y\\Leftrightarrow u(x)\\ge u(y),\\]\nthen we say that $u$ {\\bf represents} $\\succsim$, or $u$ is a {\\bf utility function} of $\\succsim$.\n\nConsider a function $f:\\mathbb{R}^n_{++}\\times\\mathbb{R}_{++}\\to \\Omega$. The following is called the {\\bf budget inequality}.\n\\[p\\cdot f(p,m)\\le m.\\]\nIf the budget inequality holds for all $(p,m)\\in \\mathbb{R}^n_{++}\\times \\mathbb{R}_{++}$, then we call this function $f$ a {\\bf candidate of demand} (CoD). If, moreover,\n\\[p\\cdot f(p,m)=m\\]\nfor all $(p,m)\\in \\mathbb{R}^n_{++}\\times \\mathbb{R}_{++}$, then we say that this CoD $f$ satisfies {\\bf Walras' law}.\n\nLet $\\succsim$ be a weak order on $\\Omega$. For each $(p,m)\\in \\mathbb{R}^n_{++}\\times \\mathbb{R}_{++}$, we define\n\\[\\Delta(p,m)=\\{x\\in \\Omega|p\\cdot x\\in m\\},\\]\n\\[f^{\\succsim}(p,m)=\\{x\\in \\Delta(p,m)|x\\succsim y\\mbox{ for all }y\\in \\Delta(p,m)\\}.\\]\nWe call the set-valued function $f^{\\succsim}$ the {\\bf demand relation} of $\\succsim$, and if it is single-valued, then we call this function $f^{\\succsim}$ the {\\bf demand function} of $\\succsim$. If $u$ represents $\\succsim$, then $f^u$ denotes $f^{\\succsim}$. For a CoD $f$, if $f=f^{\\succsim}$, then we say that $f$ corresponds to $\\succsim$ and $\\succsim$ corresponds to $f$. Of course, if $f=f^u$, then we say that $f$ corresponds to $u$ and $u$ corresponds to $f$.\n\nLet $f$ be a CoD. We say that it is {\\bf income-Lipschitzian} if for every compact subset $C\\subset \\mathbb{R}^n_{++}\\times \\mathbb{R}_{++}$, there exists $L>0$ such that if $(p,m_1),(p,m_2)\\in C$, then\n\\[\\|f(p,m_1)-f(p,m_2)\\|\\le L|m_1-m_2|.\\]\nMoreover, we say that $f$ is {\\bf locally Lipschitz} if for every compact subset $C\\subset \\mathbb{R}^n_{++}\\times \\mathbb{R}_{++}$, there exists $L>0$ such that if $(p_1,m_1),(p_2,m_2)\\in C$, then\n\\[\\|f(p,m_1)-f(p,m_2)\\|\\le L\\|(p_1,m_1)-(p_2,m_2)\\|.\\]\nObviously, every locally Lipschitz CoD is income-Lipschitzian. It is well known that every continuously differentiable CoD is locally Lipschitz.\n\nSuppose that $f$ is a CoD and $f$ is differentiable at $(p,m)$. Define\n\\[s_{ij}(p,m)=\\frac{\\partial f_i}{\\partial p_j}(p,m)+\\frac{\\partial f_i}{\\partial m}(p,m)f_j(p,m),\\]\nand let $S_f(p,m)$ denote the $n\\times n$-matrix whose $(i,j)$-th component is $s_{ij}(p,m)$. This matrix-valued function $S_f(p,m)$ is called the {\\bf Slutsky matrix}. An alternative expression of this matrix is as follows:\n\\[S_f(p,m)=D_pf(p,m)+D_mf(p,m)f^T(p,m),\\]\nwhere $f^T(p,m)$ denotes the transpose of $f(p,m)$. Note that, if $f$ is locally Lipschitz, then by Rademacher's theorem, $f$ is differentiable almost everywhere, and thus the Slutsky matrix is defined almost everywhere. We say that $f$ satisfies (S) if $S_f(p,m)$ is symmetric almost everywhere, and (NSD) if $S_f(p,m)$ is negative semi-definite almost everywhere.\n\nFor a CoD $f$, we define $R(f)$ as the range of $f$. That is,\n\\[R(f)=\\{x\\in \\Omega|x=f(p,m)\\mbox{ for some }(p,m)\\in \\mathbb{R}^n_{++}\\times \\mathbb{R}_{++}\\}.\\]\n\nFinally, suppose that $f$ is a CoD such that $R(f)$ includes $\\mathbb{R}^n_{++}$. Define\n\\[G^f(x)=\\left\\{p\\in \\mathbb{R}^n_{++}\\left|\\sum_ip_i=1,\\ f(p,p\\cdot x)=x\\right.\\right\\}\\]\nfor each $x\\in \\mathbb{R}^n_{++}$. This multi-valued function is called the {\\bf inverse demand correspondence} of $f$. We say that $f$ satisfies the {\\bf C axiom} if $G^f$ is compact-valued, convex-valued, and upper hemi-continuous on $\\mathbb{R}^n_{++}$.\n\n\n\n\\section{Main Results}\n\\subsection{First Result: Constructing a Reverse Calculation Method}\nThe first result of this study is to construct a rigorous and effective method for calculating a utility function corresponds to the given CoD.\n\n\\vspace{12pt}\n\\noindent\n{\\bf Theorem 1}. Suppose that $f$ is a locally Lipschitz CoD that satisfies Walras' law, (S), and (NSD). Fix $\\bar{p}\\gg 0$, and define $u_{f,\\bar{p}}(x)$ as follows. First, if $x\\notin R(f)$, then define $u_{f,\\bar{p}}(x)=0$. Second, if $x=f(p,m)$ for some $(p,m)$, then consider the following differential equation\n\\begin{equation}\\label{eq3.1}\n\\dot{c}(t)=f((1-t)p+t\\bar{p},c(t))\\cdot (\\bar{p}-p),\\ c(0)=m,\n\\end{equation}\nand define $u_{f,\\bar{p}}(x)=c(1)$. Then, the following holds.\n\\begin{enumerate}[1.]\n\\item $u_{f,\\bar{p}}$ is well-defined,\\footnote{That is, $c(1)$ is always defined, and independent of the choice of $(p,m)$.} and $f=f^{u_{f,\\bar{p}}}$.\n\n\\item $u_{f,\\bar{p}}$ is upper semi-continuous on $R(f)$.\n\n\\item If $f=f^{\\succsim}$ for some weak order $\\succsim$ on $\\Omega$ that is upper semi-continuous on $R(f)$, then for every $x,y\\in R(f)$,\n\\[x\\succsim y\\Leftrightarrow u_{f,\\bar{p}}(x)\\ge u_{f,\\bar{p}}(y).\\]\n\\end{enumerate}\n\nAs a corollary, we obtain the following result.\n\n\\vspace{12pt}\n\\noindent\n{\\bf Corollary 1}. Suppose that $f$ is a locally Lipschitz CoD that satisfies Walras' law. Then, the following four statements are equivalent.\n\\begin{enumerate}[(i)]\n\\item $f=f^{\\succsim}$ for some weak order $\\succsim$ on $\\Omega$.\n\n\\item $f=f^{u_{f,\\bar{p}}}$, where $u_{f,\\bar{p}}$ is defined in Theorem 1.\n\n\\item $f$ satisfies (S) and (NSD).\n\n\\item For every $(p,m)\\in \\mathbb{R}^n_{++}\\times\\mathbb{R}_{++}$, the following partial differential equation\n\\begin{equation}\\label{eq3.2}\nDE(q)=f(q,E(q)),\\ E(p)=m,\n\\end{equation}\nhas a unique concave solution.\n\\end{enumerate}\n\n\\vspace{12pt}\nThere are a few remarks on Theorem 1 and Corollary 1. In Hosoya (2017), the same result as Theorem 1 was obtained for continuously differentiable CoDs. Hosoya (2018) showed that the same result holds for differentiable and locally Lipschitz CoDs. In these theorem, the Slutsky matrix is assumed to be symmetric and negative semi-definite at {\\bf every} $(p,m)$. In contrast, our Theorem 1 only requires the symmetry and negative semi-definiteness of the Slutsky matrix at {\\bf almost every} $(p,m)$. This change heavily increase the difficulty of the proof by the following two reasons. First, in (\\ref{eq3.1}), $f((1-t)p+t\\bar{p},c(t))$ appears. However, the set $A=\\{((1-t)p+t\\bar{p},c(t))|t\\in [0,1]\\}$ is a null-set with respect to the Lebesgue measure. Hence, the Slutsky matrix may be undefined in all points of $A$. This fact denies many techniques used in related research. Second, the classical techniques that derives such a result is constructed in Hurwicz and Uzawa (1971). However, in Hurwicz-Uzawa's proof, the weak axiom of revealed preference was first derived (lemma 5 in their paper), and then the main result was proved using the weak axiom. Because the claim of the weak axiom includes a strict inequality, this property vanishes by any limit manipulation. This indicates that the usual approximation approach cannot work well for the proof of Theorem 1.\n\nIn Hosoya (2021), similar result is obtained by using several techniques of partial differential equation. The proof of Hosoya (2021) solved the above difficulties by some purterbation techniques on partial differential equation, although this is difficult to understand. In contrast, we construct the proof of Theorem 1 based on the knowledge of ordinary differential equations. Note also that every continuously differentiable CoD is locally Lipschitz, and thus Theorem 1 is a pure extension of Hosoya's (2017) result.\n\nWe mention that (iv) of Corollary 1 is a new necessary and sufficient condition for a CoD to be a demand function of some weak order. It is well known that the strong axiom of revealed preference is necessary and sufficient for a CoD to be a demand function (Richter (1966), Mas-Colell et al. (1995)), and recently it is shown that (S) and (NSD) are necessary and sufficient for a continuously differentiable CoD with Walras' law to be a demand function (Hosoya (2017)). Our condition (iv) is a new alternative necessary and sufficient condition for a locally Lipschitz CoD with Walras' law to be a demand function. Later, we discuss that this condition is crucial for this paper.\n\nNote that, equation (\\ref{eq3.2}) is deeply related to the {\\bf expenditure function}. For given weak order $\\succsim$ on $\\Omega$ and $x\\in \\Omega$, define\n\\[E^x(p)=\\inf\\{p\\cdot y|y\\succsim x\\}.\\]\nThis function is called the expenditure function. Actually, $E^x$ coincides with the value function of the following minimization problem:\n\\begin{align*}\n\\min~~~~~&~p\\cdot y\\\\\n\\mbox{subject to. }&~y\\in \\Omega,\\\\\n&~y\\succsim x.\n\\end{align*}\nThis is traditionally called the {\\bf expenditure minimization problem} in consumer theory. It is well known that the expenditure function is concave and continuous. Moreover, if $f=f^{\\succsim}$ and $f$ is continuous, and if $x=f(p,m)$, then it satisfies (\\ref{eq3.2}). This result is usually called {\\bf Shephard's lemma}. See Lemma 1 of Hosoya (2020). Therefore, (i) implies (iv). It is obvious that (ii) implies (i), and it is easy to show that (iv) implies (iii). Finally, Theorem 1 claims that (iii) implies (ii). This is the background logic of Corollary 1.\n\nIf $f=f^{\\succsim}$ and $f$ is continuous and income-Lipschitzian, then we can easily show that $E^x(p)$ is the unique solution to (\\ref{eq3.2}), and $u_{f,\\bar{p}}(x)=E^x(\\bar{p})$ for all $x\\in R(f)$. If $R(f)$ includes $\\mathbb{R}^n_{++}$ and is open in $\\mathbb{R}^n_+$, then by applying a similar proof as Theorem 1 of Hosoya (2020), we obtain the following result.\n\n\n\\vspace{12pt}\n\\noindent\n{\\bf Corollary 2}. Suppose that $f$ is a locally Lipschitz CoD that satisfies Walras' law, (S), and (NSD). Moreover, suppose that $R(f)$ includes $\\mathbb{R}^n_{++}$ and is relatively open in $\\Omega$. Define\n\\begin{equation}\\label{eq3.3}\nv_{f,\\bar{p}}(x)=\\begin{cases}\nu_{f,\\bar{p}}(x) & \\mbox{if }x\\in \\mathbb{R}^n_{++},\\\\\n\\inf_{\\varepsilon>0}\\sup\\{u_{f,\\bar{p}}(y)|y\\in\\mathbb{R}^n_{++},\\ \\|y-x\\|<\\varepsilon\\} & \\mbox{if }x\\notin \\mathbb{R}^n_{++}.\n\\end{cases}\n\\end{equation}\nThen, $f=f^{v_{f,\\bar{p}}}$ and $v_{f,\\bar{p}}$ is upper semi-continuous. Moreover, the following holds.\n\\begin{enumerate}[1)]\n\\item The function $f$ satisfies the C axiom if and only if $v_{f,\\bar{p}}$ is continuous on $\\mathbb{R}^n_{++}$.\n\n\\item If $f=f^{\\succsim}$ for some upper semi-continuous weak order $\\succsim$, then $v_{f,\\bar{p}}$ represents $\\succsim$. In particular, such a $\\succsim$ must be unique.\n\n\\item $f=f^{\\succsim}$ for some continuous weak order $\\succsim$ if and only if $v_{f,\\bar{p}}$ is continuous.\n\\end{enumerate}\n\n\n\\vspace{12pt}\n\\noindent\n{\\bf Example 1}. Consider the utility function $u(x_1,x_2)=\\sqrt{x_1}+x_2$. The corresponding demand function is:\n\\[f(p,m)=\\begin{cases}\n\\left(\\frac{m}{p_1},0\\right) & \\mbox{if }p_2^2\\ge 4p_1m,\\\\\n\\left(\\frac{p_2^2}{4p_1^2},\\frac{4p_1m-p_2^2}{4p_1p_2}\\right) & \\mbox{otherwise.}\n\\end{cases}\\]\nThis function satisfies all requirements of Theorem 1 but is not continuously differentiable. Moreover, $R(f)=\\{(x_1,x_2)\\in \\mathbb{R}^2_+|x_1>0\\}$, and thus this also satisfies all requirements of Corollary 2. Set $\\bar{p}=(1,1)$, and choose any $x\\in R(f)$. Then, $x=f^u(p,m)$ for some $(p,m)\\in \\mathbb{R}^2_{++}\\times \\mathbb{R}_{++}$. If necessary, we can replace $p_2$ with $\\min\\{2\\sqrt{p_1m},p_2\\}$, and thus we can assume that $p_2^2\\le 4p_1m$. Moreover, again if necessary, we can replace $(p,m)$ with $\\frac{1}{p_2}(p,m)$, and thus we can assume that $p_2=1$. Let us try to solve (\\ref{eq3.1}) and to determine $u_{f,\\bar{p}}(x)$ and $v_{f,\\bar{p}}(x)$.\n\nFirst, define\n\\[f^1(q,w)=\\left(\\frac{w}{q_1},0\\right),\\ f^2(q,w)=\\left(\\frac{q_2^2}{4q_1^2},\\frac{4q_1w-q_2^2}{4q_1q_2}\\right),\\]\nand consider\n\\begin{equation}\\label{eq3.4}\n\\dot{c}_i(t)=f^i((1-t)p+t\\bar{p},c_i(t))\\cdot (\\bar{p}-p).\n\\end{equation}\nTo solve (\\ref{eq3.4}), we have that\n\\[c_1(t)=c_1(s)\\frac{p_1+t(1-p_1)}{p_1+s(1-p_1)},\\]\nand because $p_2=\\bar{p}_2=1$,\n\\[c_2(t)=c_2(s)-\\frac{1}{4}\\left[\\frac{1}{p_1+t(1-p_1)}-\\frac{1}{p_1+s(1-p_1)}\\right].\\]\nIn particular, if $s=0$ and $c_2(0)=m$, then\n\\[c_2(t)=m-\\frac{1}{4}\\left[\\frac{1}{p_1+t(1-p_1)}-\\frac{1}{p_1}\\right].\\]\n\nSecond, suppose that $4c_2(1)\\ge 1$, where $c_2(0)=m$. By our initial assumption, $4p_1c_2(0)\\ge 1$. Moreover, $(p_1+t(1-p_1))c_2(t)$ is monotone in $t$. Therefore, $c(t)=c_2(t)$ is a solution to (\\ref{eq3.1}) defined on $[0,1]$, and thus we obtain that\n\\[c(1)=c_2(1)=m+\\frac{1-p_1}{4p_1}.\\]\nIn this case, $4m+\\frac{1-p_1}{p_1}\\ge 1$, and thus we have that\n\\begin{equation}\\label{eq3.5}\n4p_1m+1\\ge 2p_1.\n\\end{equation}\nBecause\n\\[x_1=\\frac{1}{4p_1^2},\\ x_2=m-\\frac{1}{4p_1},\\]\nwe have that (\\ref{eq3.5}) is equivalent to\n\\[u(x)\\ge \\frac{1}{2}.\\]\nMoreover,\n\\[c(1)=\\sqrt{x_1}+x_2-\\frac{1}{4}.\\]\nTherefore, we obtain that\n\\[u_{f,\\bar{p}}(x)=u(x)-\\frac{1}{4},\\]\nif $u(x)\\ge \\frac{1}{2}$.\n\nThird, suppose that $4c_2(1)<1$, where $c_2(0)=m$. By the same argument as above, we have that this assumption is equivalent to\n\\[u(x)<\\frac{1}{2}.\\]\nNote that $1\\neq p_1$. If $1>p_1$, then $(p_1+t(1-p_1))c_2(t)$ is increasing, and thus we have that $4p_1m<1$, which contradicts our initial assumption. Thus, we have that $1<p_1$. We guess that $c(t)=c_2(t)$ on $[0,t^*]$, and $c(t)=c_1(t)$ on $[t^*,1]$, where $c(t^*)=c_1(t^*)=c_2(t^*)$ and $\\dot{c}_1(t^*)=\\dot{c}_2(t^*)$. Then,\n\\[\\frac{c(t^*)(1-p_1)}{p_1+t^*(1-p_1)}=\\dot{c}_1(t^*)=\\dot{c}_2(t^*)=\\frac{1-p_1}{4(p_1+t^*(1-p_1))^2},\\]\nand thus\n\\[c(t^*)=c_1(t^*)=c_2(t^*)=\\frac{1}{4(p_1+t^*(1-p_1))}.\\]\nThen, \n\\[c_2(t^*)=m-\\frac{1}{4}\\left[\\frac{1}{p_1+t^*(1-p_1)}-\\frac{1}{p_1}\\right]=\\frac{1}{4(p_1+t^*(1-p_1))},\\]\nand hence, we obtain\n\\[t^*=\\frac{p_1}{1-p_1}\\left[\\frac{1-4p_1m}{4p_1m+1}\\right].\\]\nBecause $1-p_1<0$ and $1\\le 4p_1m$, we have that $t^*\\ge 0$. Moreover, because $(p_1+t(1-p_1))c_2(t)$ is decreasing in $[0,1]$, $4c_2(1)<1$ and $4(p_1+t^*(1-p_1))c_2(t^*)=1$, we have that $t^*<1$. Therefore, $t^*\\in [0,1]$. Hence,\n\\[c(t)=\\begin{cases}\nm-\\frac{1}{4}\\left[\\frac{1}{p_1+t(1-p_1)}-\\frac{1}{p_1}\\right] & \\mbox{if }t\\le t^*,\\\\\n\\frac{(p_1+t(1-p_1))}{4(p_1+t^*(1-p_1))^2} & \\mbox{if }t\\ge t^*.\\\\\n\\end{cases}\\]\nIn particular,\n\\[c(1)=\\frac{1}{4(p_1+t^*(1-p_1))^2}=\\frac{16p_1^2m^2+8p_1m+1}{16p_1^2}.\\]\nBecause\n\\[x_1=\\frac{1}{4p_1^2},\\ x_2=m-\\frac{1}{4p_1},\\]\nwe have that\n\\[c(1)=(\\sqrt{x_1}+x_2)^2=(u(x))^2.\\]\nTherefore, if $u(x)<\\frac{1}{2}$, then\n\\[u_{f,\\bar{p}}(x)=(u(x))^2.\\]\nIn conclusion, we obtain that\n\\[u_{f,\\bar{p}}(x)=\\begin{cases}\nu(x)-\\frac{1}{4} & \\mbox{if }x_1>0,\\ u(x)\\ge \\frac{1}{2},\\\\\n(u(x))^2 & \\mbox{if }x_1>0,\\ u(x)<\\frac{1}{2},\\\\\n0 & \\mbox{if }x_1=0.\n\\end{cases}\\]\nOf course, in this case,\n\\[v_{f,\\bar{p}}(x)=\\begin{cases}\nu(x)-\\frac{1}{4} & \\mbox{if }u(x)\\ge \\frac{1}{2},\\\\\n(u(x))^2 & \\mbox{if }u(x)<\\frac{1}{2}.\n\\end{cases}\\]\nThis completes our calculation.\n\n\n\\subsection{Second Result: Continuity of Calculation}\nWe are now able to calculate a utility function $u_{f,\\bar{p}}$ from a CoD $f$. However, in real world, we can only obtain a finite data of $f$, and because $f$ includes infinite data, we cannot obtain $f$ rigorously. Hence, our CoD $f$ must be considered as an estimated value of the true demand function. In this view, we need a continuity result: that is, we wish that if $f'$ is near to $f$, then $u_{f',\\bar{p}}$ is also near to $u_{f,\\bar{p}}$. If this condition is violated, then we are confronted with a methodological difficulty: that is, ensuring ``consistency'' for the estimated utility function becomes hard.\n\nWe should explain this point in detail. Consider some estimation problem with the true value $x\\in X$. Suppose that there is a given estimation method, and for some data with size $N$, let $x_N$ be the estimated value of $x$. Then, $x_N$ is a random variable on $X$. This estimation method is said to {\\bf consistent} if $x_N$ converges to $x$ in probability as $N\\to \\infty$.\n\nSuppose that there is an estimation method for the true demand function $f$, and $f_N$ is an estimated value of $f$ for some data set with size $N$. Suppose also that this estimation method is consistent with respect to some topology on the space of demand functions. For each $f_N$, we can calculate the utility function $u_{f_N,\\bar{p}}$, and thus $u_{f_N,\\bar{p}}$ can be treated as an estimated value of the ``true utility function'' $u_{f,\\bar{p}}$. Our question is as follows: does $u_{f_N,\\bar{p}}$ converge to $u_{f,\\bar{p}}$? If not, our estimation method violates the consistency, and thus it is not useful.\n\nHence, the continuity of $u_{f,\\bar{p}}$ with respect to $f$ is very important. In this regard, we first show the following result.\n\n\n\\vspace{12pt}\n\\noindent\n{\\bf Theorem 2}. Suppose that $(f^k)$ is a sequence of locally Lipschitz demand functions that satisfy Walras' law, and for every compact set $C\\subset \\mathbb{R}^n_{++}\\times\\mathbb{R}_{++}$, $f^k$ converges to a CoD $f$ uniformly on $C$ as $k\\to \\infty$. If $f$ is locally Lipschitz, then $f$ is also a demand function.\n\n\n\\vspace{12pt}\nAs a corollary, we obtain an important result. Let $\\Delta_{\\nu}=[\\nu^{-1},\\nu]^{n+1}$. Define a metric\n\\[\\rho(f,f')=\\sum_{\\nu=1}^{\\infty}\\frac{1}{2^{\\nu}}\\arctan\\left(\\sup_{(p,m)\\in \\Delta_{\\nu}}\\|f(p,m)-f'(p,m)\\|\\right)\\]\nwhere $f,f'$ are CoDs. We can easily show that $\\rho$ is actually a metric in the space of CoDs, and a sequence $(f^k)$ converges to $f$ with respect to $\\rho$ if and only if for every compact set $C\\subset \\mathbb{R}^n_{++}\\times \\mathbb{R}_{++}$, $(f^k)$ converges to $f$ uniformly.\n\nSuppose that $L=(L_{\\nu})$ is a sequence of positive real numbers. Define $\\mathscr{F}_L$ as the set of demand functions such that $f\\in \\mathscr{F}_L$ if and only if $f$ satisfies Walras' law and\n\\[\\|f(p,m)-f(q,w)\\|\\le L_{\\nu}\\|(p,m)-(q,w)\\|\\]\nfor all $(p,m),(q,w)\\in \\Delta_{\\nu}$.\n\n\n\\vspace{12pt}\n\\noindent\n{\\bf Corollary 3}. The space $\\mathscr{F}_L$ is complete with respect to the metric $\\rho$.\n\n\\vspace{12pt}\nBy the way, for obtaining the uniqueness of the upper semi-continuous weak order corresponding to $f$, we need to use Corollary 2, and thus an additional assumption is needed. That is, $R(f)$ includes $\\mathbb{R}^n_{++}$ and is open in $\\mathbb{R}^n_+$. Suppose that $(f^k)$ is a sequence of $\\mathscr{F}_L$ that converges to $f$, and for every $k$, $f^k$ satisfies all requirements in Corollary 2. Does $f$ also satisfy requirements in Corollary 2? Unfortunately, the following example indicates that the answer is negative.\n\n\\vspace{12pt}\n\\noindent\n{\\bf Example 2}. Consider the class of CES utility functions:\n\\[u^{\\sigma}(x)=(x_1^{\\sigma}+x_2^{\\sigma})^{\\frac{1}{\\sigma}},\\]\nwhere $\\sigma<1$ and $\\sigma\\neq 0$. The corresponding demand function is\n\\[f_i^{\\sigma}(p,m)=\\frac{p_i^{\\frac{-1}{1-\\sigma}}m}{p_1^{\\frac{-\\sigma}{1-\\sigma}}+p_2^{\\frac{-\\sigma}{1-\\sigma}}}.\\]\nWe assume that $\\sigma<0$. To differentiate this function, we have that for $j\\neq i$ and $(p,m)\\in \\Delta_{\\nu}$,\n\\begin{align*}\n\\left|\\frac{\\partial f_i^{\\sigma}}{\\partial p_i}(p,m)\\right|=&~\\frac{[(1-\\sigma)p_i^{\\frac{-2}{1-\\sigma}}+p_i^{\\frac{-2+\\sigma}{1-\\sigma}}p_j^{\\frac{-\\sigma}{1-\\sigma}}]m}{(1-\\sigma)(p_1^{\\frac{-\\sigma}{1-\\sigma}}+p_2^{\\frac{-\\sigma}{1-\\sigma}})^2}\\le \\frac{\\nu^5}{2},\\\\\n\\left|\\frac{\\partial f_i^{\\sigma}}{\\partial p_j}(p,m)\\right|=&~\\frac{-\\sigma p_i^{\\frac{-1}{1-\\sigma}}p_j^{\\frac{-1}{1-\\sigma}}m}{(1-\\sigma)(p_1^{\\frac{-\\sigma}{1-\\sigma}}+p_2^{\\frac{-\\sigma}{1-\\sigma}})^2}\\le\\frac{\\nu^5}{4},\\\\\n\\left|\\frac{\\partial f_i^{\\sigma}}{\\partial m}(p,m)\\right|=&~\\frac{p_i^{\\frac{-1}{1-\\sigma}}}{p_1^{\\frac{-\\sigma}{1-\\sigma}}+p_2^{\\frac{-\\sigma}{1-\\sigma}}}\\le\\frac{\\nu^2}{2}.\n\\end{align*}\nand thus, if we define $L_{\\nu}=\\nu^5$, then $f_i^{\\sigma}\\in \\mathscr{F}_L$. Moreover, for every $\\sigma<0$, we have that $R(f^{\\sigma})=\\mathbb{R}^2_{++}$. However, it is well known that $f^{\\sigma}$ converges to a function $f$ with respect to the metric $\\rho$, where\n\\[f_1(p,m)=f_2(p,m)=\\frac{m}{p_1+p_2},\\]\nand $R(f)=\\{(c,c)|c>0\\}$, even though $R(f^{\\rho})=\\mathbb{R}^n_{++}$. This fact implies that the limit manipulation in $\\mathscr{F}_L$ may shrinks the range of the demand function.\n\n\n\\vspace{12pt}\nThe above example shows that for our purpose, an additional assumption is needed. One of the easiest way to solve this problem is assuming that the limit CoD $f$ satisfies all assumptions of Corollary 2. The result is as follows.\n\n\n\\vspace{12pt}\n\\noindent\n{\\bf Theorem 3}. Let $(f^k)$ be a sequence on $\\mathscr{F}_L$ that converges to $f$. Suppose that $R(f^k)$ includes $\\mathbb{R}^n_{++}$ and $f^k$ satisfies the C axiom for all $k$, and $f$ also satisfies these conditions. Then, for every compact set $D\\subset \\mathbb{R}^n_{++}$,\n\\[\\sup_{x\\in D}|u_{f^k,\\bar{p}}(x)-u_{f,\\bar{p}}(x)|\\to 0\\]\nas $k\\to \\infty$.\n\n\\vspace{12pt}\nThe definition of $v_{f,\\bar{p}}$ in Theorem 2 only depends on the values if $u_{f,\\bar{p}}$ on $\\mathbb{R}^n_{++}$. Hence, it seems to be shown that $v_{f^k,\\bar{p}}$ converges to $v_{f,\\bar{p}}$ pointwise, here $v_{f^k,\\bar{p}}$ is defined in Corollary 2. However, there are several technical difficulties, and thus we cannot obtain such a result. Instead, we can show the following result.\n\n\\vspace{12pt}\n\\noindent\n{\\bf Corollary 4}. In addition to the assumptions of Theorem 3, suppose that $R(f)$ and all $R(f^k)$ are relatively open in $\\Omega$. Then, $\\limsup_{k\\to \\infty}v_{f^k,\\bar{p}}(x)\\le v_{f,\\bar{p}}(x)$ for every $x\\in \\Omega$.\n\n\\vspace{12pt}\nBecause $v_{f^k,\\bar{p}}(x)\\ge 0$ for every $x\\in \\Omega$, if $v_{f,\\bar{p}}(x)=0$, then $v_{f^k,\\bar{p}}(x)$ converges to $v_{f,\\bar{p}}(x)$. However, in other case, whether $v_{f^k,\\bar{p}}(x)$ converges to $v_{f,\\bar{p}}(x)$ or not is unknown. Actually, we find the following example in which $\\lim_{k\\to \\infty}v_{f^k,\\bar{p}}(x)\\neq v_{f,\\bar{p}}(x)$.\n\n\\vspace{12pt}\n\\noindent\n{\\bf Example 3}. Suppose that $h:\\mathbb{R}_{++}\\to \\mathbb{R}_{++}$ is $C^{\\infty}$, nondecreasing and $\\lim_{c\\to 0}h(c)=0,\\lim_{c\\to \\infty}h(c)=\\infty$. Choose any $(x_1,x_2)\\in \\mathbb{R}^2_{++}$, and consider the following equation\n\\[(x_1^{\\frac{1}{1+\\frac{1}{c}}}+x_2^{\\frac{1}{1+\\frac{1}{c}}})^{1+\\frac{1}{c}}=h(c).\\]\nBy the same arguments as in the proof of Example 4 in Hosoya (2020), we can show that there exists a unique solution $c^*>0$ to the above equation, and if we define $u^h(x_1,x_2)=c^*$, $u^h$ is $C^{\\infty}$, monotone, and strictly quasi-concave. If $(x_1,x_2)\\in \\mathbb{R}^2_+\\setminus\\mathbb{R}^2_{++}$, then we define\n\\[u^h(x_1,x_2)=\\inf_{\\varepsilon>0}\\sup\\{u^h(x_1',x_2')|(x_1',x_2')\\in\\mathbb{R}^2_{++},\\ \\|(x_1,x_2)-(x_1',x_2')\\|<\\varepsilon\\}.\\]\nBy the same arguments as in the proof of Corollary 2, we can show that $u^h$ is upper semi-continuous on $\\mathbb{R}^2_+$. Moreover, again by the same arguments as in the proof of Example 4 in Hosoya (2020), we can show that $f^{u^h}$ is also $C^{\\infty}$, and $R(f^{u^h})=\\mathbb{R}^2_{++}$. Furthermore, if $h'\\to h$ in the sense of $C^2$, then we can easily show that $u^{h'}\\to u^h$ with respect to the local $C^2$ topology on $\\mathbb{R}^n_{++}$, and by Proposition 2.7.2 of Mas-Colell (1985), we have that $f^{h'}\\to f^h$ with respect to the metric $\\rho$.\n\nLet $h:\\mathbb{R}_+\\to \\mathbb{R}_+$ be a $C^{\\infty}$ nondecreasing function such that $h(0)=0, h(1)=h(2)=1$, $h'(c)>0$ if $c\\notin [1,2]$, and $\\lim_{c\\to \\infty}h(c)=+\\infty$. Also, let $\\eta:\\mathbb{R}_+\\to \\mathbb{R}_+$ be a $C^{\\infty}$ function such that $\\eta(c)\\equiv 0$ on $[0,1]$ and $[4,+\\infty[$, $\\eta(c)$ is increasing on $[1,2]$, constant on $[2,3]$, decreasing on $[3,4]$, and $\\max_{c\\in [1,4]}|\\eta'(c)|<\\min_{c\\in [3,4]}h'(c)$. Define $h^k(c)=h(c)+k^{-1}\\eta(c)$. Then, $h^k\\to h$ with respect to $C^2$ topology, and thus $f^{h^k}\\to f^h$ with respect to $\\rho$. On the other hand, choose $\\bar{p}=(1,1)$. Then, we can easily check that\n\\[v_{f^{h^k},\\bar{p}}(1,0)=\\frac{1}{2},\\]\nfor all $k$, and\n\\[v_{f^h,\\bar{p}}(1,0)=\\frac{1}{\\sqrt{2}},\\]\nwhich implies that $\\lim_{k\\to \\infty}v_{f^{h^k},\\bar{p}}(1,0)\\neq v_{f^h,\\bar{p}}(1,0)$.\n\n\\vspace{12pt}\nFinally, we can obtain another completeness result. Choose a sequence $M=(M_{\\nu})$ of positive real numbers and define $\\mathscr{F}_{L,M}$ as the set of all $f\\in \\mathscr{F}_L$ such that $R(f)$ includes $\\mathbb{R}^n_{++}$, $f$ satisfies the C axiom, and if $x\\in ]\\nu^{-1},\\nu[^n$, then for all $p\\in G^f(x)$ and $i\\in \\{1,...,n\\}$, $p_i\\ge M_{\\nu}$.\n\n\\vspace{12pt}\n\\noindent\n{\\bf Corollary 5}. Suppose that $(f^k)$ is a sequence on $\\mathscr{F}_{L,M}$ that converges to $f$ with respect to the metric $\\rho$. Then, $f\\in \\mathscr{F}_{L,M}$, and for every compact set $D\\subset \\mathbb{R}^n_{++}$,\n\\[\\sup_{x\\in D}|u_{f^k,\\bar{p}}(x)-u_{f,\\bar{p}}(x)|\\to 0\\]\nas $k\\to \\infty$.\n\n\n\\vspace{12pt}\nNote that, in our Theorem 3, the C axiom is required. Corollary 2 stated that if $R(f)$ is relatively open in $\\Omega$, this axiom is equivalent to the continuity of $v_{f,\\bar{p}}$ on $\\mathbb{R}^n_{++}$. Actually, the requirement that $R(f)$ is relatively open in $\\Omega$ does not used in the proof of this fact, and thus we can show by the same proof as this theorem that under assumptions of our Theorem 3, $u_{f^k,\\bar{p}}$ and $u_{f,\\bar{p}}$ are continuous on $\\mathbb{R}^n_{++}$. Actually, we use continuity of $u_{f,\\bar{p}}$ on $\\mathbb{R}^n_{++}$ in the proof of Theorem 3: see Lemma 4. However, $v_{f,\\bar{p}}$ is not necessarily continuous on $\\Omega$ itself even if $R(f)$ is open: see Example 4 of Hosoya (2020).\n\nIf $f$ is a continuously differentiable demand function such that the rank of $S_f(p,m)$ is always $n-1$, then we can show that the inverse demand correspondence $G^f(x)$ is a single-valued continuously differentiable function, and thus the C axiom is automatically satisfied. For a proof, see Proposition 1 of Hosoya (2013). In this regard, if $(f^k)$ is a sequence on $\\mathscr{F}_L$ such that every $f^k$ is continuously differentiable and the rank of $S_{f^k}(p,m)$ is always $n-1$, then there exists a sequence $M=(M_{\\nu})$ such that every $f^k$ is in $\\mathscr{F}_{L,M}$ if and only if for every compact set $C\\subset \\mathbb{R}^n_{++}$, $\\inf_k\\min_i\\min_{x\\in C}G_i^{f^k}(x)>0$. This is another sufficient condition for the limit function $f$ of $(f^k)$ to satisfy all requirements of Theorem 3.\n\n\n\\section{Discussion}\n\\subsection{Comparizon with Related Literature}\nThe history of integrability theory begins with Antonelli (1886). This theory aims to calculate a utility function from the consumer's purchase behavior. Hurwicz (1971) classified this theory into two categories. One derives the inverse demand function from the purchase behavior and then calculates a utility function by finding a function that satisfies Lagrange's first-order conditions for the inverse demand function, which is called the indirect approach. On the other hand, the other derives the demand function from the purchase behavior, solves Shepherd's lemma as a partial differential equation to calculate the expenditure function, and then calculates the utility function from this expenditure function, which is called the direct approach. Antonelli (1886) used the indirect approach and most of the classical results including Samuelson (1950), Katzner (1970), and Debreu (1972), also used the indirect approach. On the other hand, Hurwicz and Uzawa (1971) are a classical result for the direct approach. A recent paper, Hosoya (2013), is an example of the indirect approach, while Hosoya (2017) is an example of the direct approach. Theorem 1 in this paper is categorized as a direct approach.\n\nTo understand the position of this paper in integrability theory with direct approach, let us look at the classical result due to Hurwicz-Uzawa. They showed that if a CoD is differentiable and local Lipschitz, satisfies Walras' law, (S), (NSD), and a condition called the ``strong income-Lipschitzian'' requirement, then this CoD is a demand function. Although they do not specify in their theorem how to derive the utility function, their utility function appears in the proof essentially coincides with our $u_{f,\\bar{p}}$. After this paper, several studies tried to remove the ``strong income-Lipschitzian'' requirement, and Hosoya (2017, 2018) finally succeeded in doing so.\n\nLet us explain the logic that allowed us to eliminate the strong income-Lipschitzian requirement. In the proof of Hurwicz-Uzawa's Theorem 2, this condition is only used for deriving the existence of the solution to the partial differential equation (\\ref{eq3.2}). In fact, it is well known that the necessary and sufficient condition for the existence of local solutions to (\\ref{eq3.2}) is (S), which was proved in Theorem 10.9.4 of Dieudonne (1969). Hurwicz-Uzawa constructed a similar proof as theorem of Nikliborc (1929) to verify the existence of a global solution to (\\ref{eq3.2}), in which the strong income-Lipschitzian condition and (S) were used. Then, from it and (NSD), they proved the claim that is the same as Step 4 in the proof of our Theorem 1. Once the existence of global solutions to (\\ref{eq3.2}) and the statement in Step 4 have been shown, we do not need to use any differentiability of $f$ from that point onwards for proving this theorem. In contrast, Hosoya (2017) brought (NSD) into the proof of the existence of the global solution to (\\ref{eq3.2}) and verified that the existence of global solutions can be shown without the strong income-Lipschitzian condition. This is why we can eliminate the strong income-Lipschitzian condition from this theorem. Hosoya (2017) treated continuously differentiable CoDs, and Hosoya (2018) extended this result to differentiable and locally Lipschitz CoDs.\n\nOn the other hand, this paper removes even differentiability and assumes only local Lipschitz conditions for CoDs. This poses two difficulties in the proof. First, as already mentioned, the Lebesgue measure of the trajectory of $((1-t)p+t\\bar{p},c(t))$ in (\\ref{eq3.1}) is $0$. By Rademacher's theorem, any locally Lipschitz function is differentiable almost everywhere. However, the demand candidate $f$ may not be differentiable at any point on the above trajectory because this trajectory is a null set. And even if it is differentiable, the Slutsky matrix may not have good properties. This means that most of the tools used in integrability theory up to now cannot be used as they are. This problem is solved for perturbing the solution to the differential equation by the income. See Lemma 1 in the proof. Another difficulty arises in Step 4 of the proof of Theorem 1. This Step claims that $p\\cdot y>m$. The problem is that this inequality is strong. The proof using the past techniques is no longer possible, because the strong inequality is replaced by a weak inequality by the limit manipulation. This can be solved by rigorous evaluation fot the inequality of the perturbed trajectory, but this evaluation is not so straightforward. See the proof in Step 4.\n\nCorollary 1 is derived from Theorem 1. This is one of the newest results in integrability theory. As implied by Houthakker (1950) and shown by Uzawa (1960) and Richter (1966), a necessary and sufficient condition for a CoD to be a demand function is the strong axiom of revealed preference. For this result, there is no topological condition imposed on the CoD. On the other hand, if a CoD satisfies Walras' law and is continuously differentiable, then a necessary and sufficient conditions for it to be a demand function are (S) and (NSD). This is shown in Hosoya (2017). Corollary 1 shows that the same result holds even if the CoD is not differentiable but only locally Lipschitz. The most important thing about Corollary 1, however, is that it shows another necessary and sufficient condition for a CoD to be a demand function. That is the existence of a concave global solution to the partial differential equation (\\ref{eq3.2}).\n\nTo illustrate the importance of this result, we begin by recalling the strong axiom of revealed preference. A CoD $f$ satisfies the strong axiom of revealed preference if and only if for every finite sequence $x^1,...,x^k$ such that $x^i=f(p^i,m^i)$ and $p^i\\cdot x^{i+1}\\le m^i$, $p^k\\cdot x^1>m^k$. The problem is that there is a strong inequality in this last claim. Even if $f^k$ satisfies this condition, this strong inequality changes a weak inequality in the limit $f$, and thus whether $f$ is actually a demand function becomes unknown. This implies that the strong axiom of revealed preference cannot be used for proving results such as Theorem 2.\n\nA similar problem arises when we discuss this problem using conditions (S) and (NSD). Even if $f^k$ converges to $f$ with respect to the metric $\\rho$, it is uncertain that the derivative converges. Therefore, even if $f^k$ satisfies (S) and (NSD), $f$ may violate (S) or (NSD), which fails to prove that $f$ is a demand function. If we change the metric and use a stronger topology, we can show that (S) and (NSD) hold in the limit $f$. In this case, however, it becomes difficult to find results on econometric theory corresponding to such a topology. Therefore, these conditions are also undesirable.\n\nIn contrast, condition (iv) of Corollary 1 fundamentally resolves this problem. Indeed, in the proof of Theorem 2, we confirm that $f$ satisfies condition (iv). This property (iv) is not broken by the convergence with respect to $\\rho$, which makes such a proof possible. We can say that all the rest of our results also depend on this property (iv).\n\nTheorem 3 requires the C axiom. This axiom was first discovered by Hosoya (2017). This axiom did not appear in Hosoya (2015), who showed a result similar to Theorem 3 in integrability theory with the indirect approach. Since $G^f(x)$ is assumed to be a single-valued, continuously differentiable function in the indirect approach, this axiom automatically holds. This is the reason why the C axiom disappears in Hosoya (2015). The C axiom is known to be equivalent to another axiom called the NLL axiom. In Theorem 2 of Hosoya (2020), it was shown that for an income-Lipschitzian demand function $f$ that satisfies Walras' law, $f=f^u$ for some function $u:\\Omega\\to \\mathbb{R}$ such that $u$ is continuous on $\\mathbb{R}^n_{++}$ if and only if $f$ satisfies the C axiom. In this paper, this result is required for proving Theorem 3. See Lemma 4.\n\n\n\\subsection{Several Open Problems}\nIn this paper, we have attempted to produce the best results as far as possible. However, there remained several problems that we just could not solve. Here we describe a few of them, which we consider important.\n\nFirst, in Corollary 1, we proved the equivalence of (i) and (iv) by assuming that $f$ is locally Lipschitz. Can this equivalence also be proved when $f$ is continuous and income-Lipschitzian? If it could be said, then the income-Lipschitzian requirement is sufficient for the proof of Theorem 2, so the result of Theorem 2 can be discussed on a wider space than the space of local Lipschitz CoDs. This would also strengthen Corollary 3 and Theorem 3.\n\nSecond, the question remains as to whether Corollary 4 can be strengthened. We have only shown that $\\limsup_{k\\to \\infty}v_{f^k,\\bar{p}}(x)\\le v_{f,\\bar{p}}(x)$, and find an example such that the inequality becomes strong. However, maybe this inequality could be turned into an equality with some weak additional assumption. In particular, the equality may be guaranteed when $x$ is an element of $R(f)$. If we could prove this, our result would be much better.\n\nThe third problem is related to the second problem. In Example 2, we saw an example where $R(f)$ shrinks at the limit. However, we do not yet know of any examples where $R(f)$ expands by limit operations. What we want to know is that whether $R(f)\\subset \\limsup_kR(f^k)$ when $R(f)$ is an open set and $f^k\\to f$. If this can be said, then the difficulty in solving the second problem will be reduced.\n\nThe fourth problem concerns whether the condition that $R(f)$ is an open set is necessary in the first place. In the proof of Corollary 2, we show that $v_{f,\\bar{p}}$ coincides with the utility function defined in Hosoya (2020). It is necessary for some technical reasons that $R(f)$ is an open set in order to guarantee that $f=f^{v_{f,\\bar{p}}}$. However, there is no known counterexample of $f$ such that $R(f)$ is not an open set and $f\\neq f^{v_{f,\\bar{p}}}$. Maybe $f=f^{v_{f,\\bar{p}}}$ can be said even if $R(f)$ is not an open set.\n\nFinally, there remains the task of identifying the conditions for $v_{f,\\bar{p}}$ to be continuous. Condition a. of Theorem 6 in Hurwicz and Uzawa (1971) is a frequently used condition in this context. This condition states that if $p\\ge 0, p\\neq 0$ and $p_i=0$ for some $i$, then for any convergent sequence $(p^k)$ to $p$ on $\\mathbb{R}^n_{++}$ and any $(q,w)\\in \\mathbb{R}^n_{++}$, $f(p^k,v_{f,p^k}(f(q,w)))$ is unbounded. However, when discussing this condition, Hurwicz-Uzawa restricts the domain of the utility function to $R(f)$. Therefore, whether the continuity of $v_{f,\\bar{p}}$ outside $R(f)$ can be guaranteed by this condition is one of the open questions.\n\n\n\n\n\n\n\\section{Conclusion}\nIn this study, we obtained a calculate procedure for a utility function from a given locally Lipschitz CoD that satisfies Walras' law. Using this procedure, we found two necessary and sufficient conditions for a locally Lipschitz CoD that satisfies Walras' law to be a demand function. Moreover, under the assumption that the range of this CoD includes positive orthant and is open in the consumption space, we obtained the uniqueness result for corresponding upper semi-continuous weak order to this CoD, and derived an upper semi-continuous utility function that represents this weak order.\n\nUsing these results, we proved some kind of completeness for the space of demand functions. That is, we showed that for every sequence of demand functions that is locally Lipschitz and satisfies Walras' law, if it converges to some function with respect to topology of compact convergence, then the limit function is also a demand function. From this result, we showed that the space of demand function that has uniform Lipschiz constant on any compact set is complete under this topology. \n\nFurthermore, we showed that if every function has sufficiently wide range and satisfies the C axiom, then our derived utility function is continuous on the demand function. Using this result, we showed that the space of demand functions that has uniform local Lipschitz constant and uniformly satisfies the C axiom is complete, and the mapping from the space of demand functions into the space of utility function is continuous.\n\nWe also provided three examples. The first exhibits how our calculating procedure of utility function works well. The second example shows that the range of the CoD may shrink by limit manipulation. The third example indicates that our continuity result never holds in the corner of the consumption space. We think that all examples are meaningful in this context.\n\nAlthough there are many open problems concerning our results, we believe that the results in this paper is sufficiently good and worthwhile for applied economic research. In particular, we think that our results are a foundation to apply integrability theory into econometric theory.\n\n\n\n\\section{Proofs}\n\\subsection{Mathematical Knowledge on Lipschitz Analysis and Differential Equations}\nWe must use the technique of the Lipschitz analysis in the proof of our theorems repeatedly. However, the Lipschitz property of the solution function (defined later) for ODEs are not so known, and thus we provide knowledge on these facts in this subsection.\n\nFirst, recall the definition of the locally Lipschitz function. Let $f:U\\to \\mathbb{R}^N$ be some function, where $U\\subset \\mathbb{R}^M$ is open. This function is called {\\bf locally Lipschitz} if for every compact set $C\\subset U$, there exists $L>0$ such that for every $x,y\\in C$,\n\\[\\|f(x)-f(y)\\|\\le L\\|x-y\\|.\\]\nBecause the following property is important, we present a proof in this subsection.\n\n\\vspace{12pt}\n\\noindent\n{\\bf Fact 1}. Let $f:U\\to \\mathbb{R}^N$, where $U\\subset \\mathbb{R}^M$ is open. Then, $f$ is locally Lipschitz if and only if for every $x\\in U$, there exists $r>0$ and $L>0$ such that if $y,z\\in U,\\ \\|y-x\\|\\le r,$ and $\\|z-x\\|<r$, then\n\\[\\|f(y)-f(z)\\|\\le L\\|y-z\\|.\\]\n\n\\vspace{12pt}\n\\noindent\n{\\bf Proof of Fact 1}. Suppose that $f$ is locally Lipschitz. For each $x\\in U$, there exists $r>0$ such that $\\bar{B}_r(x)\\equiv \\{y\\in \\mathbb{R}^M|\\|y-x\\|\\le r\\}\\subset U$, and $\\bar{B}_r(x)$ is compact. This implies that there exists $L>0$ such that if $y,z\\in \\bar{B}_r(x)$, then\n\\[\\|f(y)-f(z)\\|\\le L\\|y-z\\|.\\]\nTo prove the converse relationship, we use the contraposition method. Suppose that $f$ is not locally Lipschitz. Then, there exists a compact set $C$ and sequences $(x^k),(y^k)$ on $C$ such that for all $k$,\n\\[\\|f(x^k)-f(y^k)\\|>k\\|x^k-y^k\\|.\\]\nBecause $C$ is compact, we can assume that $x^k\\to x^*,\\ y^k\\to y^*$ as $k\\to \\infty$. Then,\n\\[\\|f(x^*)-f(y^*)\\|\\ge k\\|x^*-y^*\\|\\]\nfor every $k$, which implies that $x^*=y^*$. Choose any $r>0$ and $L>0$. Then, there exists $k>L$ such that $x^k,y^k\\in \\bar{B}_r(x^*)$, and thus the latter claim of this fact is violated. This completes the proof. $\\blacksquare$\n\n\n\n\\vspace{12pt}\nBy the above fact, we have that every continuously differentiable function is locally Lipschitz. Of course, the converse is not true: consider $f(x)=|x|$.\n\nThe next fact is known as {\\bf Rademacher's theorem}. Because the proof of this fact is long, we omit the proof.\\footnote{See, for example, Heinonen (2004).}\n\n\n\\vspace{12pt}\n\\noindent\n{\\bf Fact 2}. Suppose that $f:U\\to \\mathbb{R}^N$, where $U\\subset \\mathbb{R}^M$ is open. If $f$ is locally Lipschitz, then it is differentiable almost everywhere.\n\n\\vspace{12pt}\nNext, we explain some notion on ordinary differential equations (ODEs). First, consider the following ODE:\n\\begin{equation}\\label{eq6.00}\n\\dot{x}(t)=g(t,x(t)),\\ x(t_0)=x^*,\n\\end{equation}\nwhere $g:U\\to \\mathbb{R}^N$ and $U\\subset \\mathbb{R}\\times\\mathbb{R}^N$ is open. We call a subset $I$ of $\\mathbb{R}$ an {\\bf interval} if it is a convex set containing at least two points. We say that a function $x:I\\to \\mathbb{R}^N$ is a {\\bf solution} to (\\ref{eq6.00}) if and only if 1) $I$ is an interval containing $t_0$, 2) $x(t_0)=x^*$, 3) $x$ is absolutely continuous on every compact subset $C\\subset I$, and 4) $\\dot{x}(t)=g(t,x(t),y)$ for almost every $t\\in I$. Let $x:I\\to \\mathbb{R}^N$ and $y:J\\to \\mathbb{R}^N$ be two solutions. Then, we call $x$ an {\\bf extension} of $y$ if $J\\subset I$ and $y(t)=x(t)$ for all $t\\in J$. A solution $x:I\\to \\mathbb{R}^N$ is called a {\\bf nonextendable solution} if there is no extension except $x$ itself. The next fact is well known, and thus we omit the proof.\\footnote{See, for example, Theorem 1.1 and Theorem 3.1 of chapter 2 in Hartman (1982).}\n\n\\vspace{12pt}\n\\noindent\n{\\bf Fact 3}. Suppose that $g$ is locally Lipschitz. Then, for every interval $I$ including $t_0$, there exists at most one solution to (\\ref{eq6.00}) defined on $I$, and if there exists a solution, it is continuously differentiable. In particular, there exists a unique nonextendable solution $x:I\\to \\mathbb{R}^N$ to (\\ref{eq6.00}), where $I$ is open and $x(t)$ is continuously differentiable. Moreover, for every compact set $C\\subset U$, there exists $t_1,t_2\\in I$ such that if $t\\in I$ and either $t<t_1$ or $t_2<t$, then $(t,x(t))\\notin C$.\n\n\\vspace{12pt}\nNext, consider the following parametrized ODE:\n\\begin{equation}\\label{eq6.01}\n\\dot{x}(t)=h(t,x(t),y),\\ x(t_0)=z,\n\\end{equation}\nwhere $h:U\\to \\mathbb{R}^N$ and $U\\subset \\mathbb{R}\\times \\mathbb{R}^N\\times \\mathbb{R}^M$ is open. We assume that $h$ is locally Lipschitz. Fix $(y,z)$ such that $(t_0,z,y)\\in U$. Then, (\\ref{eq6.01}) can be seen as (\\ref{eq6.00}), where $g(t,x)=h(t,x,y)$ and $x^*=z$. Hence, we can define a nonextendable solution $x^{y,z}:I\\to \\mathbb{R}^N$ according to Fact 3. We write $x(t;y,z)=x^{y,z}(t)$, and call this function $x:(t,y,z)\\mapsto x(t;y,z)$ the {\\bf solution function} of (\\ref{eq6.01}). The following fact is needed but not so known, and thus we prove this fact in this paper.\n\n\\vspace{12pt}\n\\noindent\n{\\bf Fact 4}. Under the assumption that $h$ is locally Lipschitz, the domain of the solution function is open, and the solution function is locally Lipschitz.\n\n\n\\vspace{12pt}\n\\noindent\n{\\bf Proof of Fact 4}. First, we introduce a lemma.\n\n\\vspace{12pt}\n\\noindent\n{\\bf Lemma 1} (Gronwall's inequality). Suppose that $g:[t_0,T]\\to \\mathbb{R}$ is continuous, and\n\\[0\\le g(t)\\le \\int_{t_0}^t[Ag(s)+B]ds\\]\nfor every $t\\in [t_0,T]$, where $A>0$ and $B\\in \\mathbb{R}$. Then,\n\\[g(t)\\le \\frac{B}{A}(e^{A(t-t_0)}-1).\\]\n\n\\vspace{12pt}\n\\noindent\n{\\bf Proof}. Omitted. See, for example, Problem 5.2.7 of Karatzas and Shreve (1991) or Lemma 8 of Hosoya (2017).\n\n\\vspace{12pt}\nLet $V\\subset \\mathbb{R}\\times\\mathbb{R}^M\\times \\mathbb{R}^N$ be the domain of the solution function $x(t;y,z)$. Choose any $(t^*,y,z)\\in V$. By Fact 3, there exists an open interval $I$ such that $t\\mapsto x(t;y,z)$ is a nonextendable solution, and $t^*\\in I$. Choose $t_1,t_2\\in I$ such that $t_1<\\min\\{t^*,t_0\\}\\le \\max\\{t^*,t_0\\}<t_2$. Consider the following differential equation:\n\\begin{equation}\\label{eq6.02}\n\\dot{x}(t)=h(t,x(t)+z'-z,y'),\\ x(t_0)=z.\n\\end{equation}\nIf $z'=z$ and $y'=y$, then $x^*:t\\mapsto x(t;y,z)$ is a solution to (\\ref{eq6.02}). Choose $a,b>0$ so small and define\n\\[\\Pi(a,b)=\\{(t,x,y',z')|t\\in [t_1,t_2],\\ \\|x-x^*(t)\\|\\le a,\\ \\|y'-y\\|\\le b,\\ \\|z'-z\\|\\le b\\},\\]\n\\[\\Pi'(a,b)=\\{(t,x+z'-z,y')|(t,x,y',z')\\in \\Pi(a,b)\\}.\\]\nBy definition, $\\Pi(a,b)$ and $\\Pi'(a,b)$ are compact. We assume that $a,b$ is sufficiently small that $\\Pi'(a,b)\\subset U$. Because $h$ is locally Lipschitz, there exists $L>0$ such that for every $(t_1',z_1',y_1'), (t_2',z_2',y_2')\\in \\Pi'(a,b)$,\n\\[\\|h(t_1',z_1',y_1')-h(t_2',z_2',y_2')\\|\\le L[|t_1'-t_2'|+\\|z_1'-z_2'\\|+\\|y_1'-y_2'\\|].\\]\nSuppose that $\\|y'-y\\|\\le b,\\ \\|z'-z\\|\\le b$, and define $t^+_2(d',c^*)$ as the supremum of the set of all $t\\in ]t_0,t_2]$ such that there exists a solution $\\tilde{x}:[t_0,t]\\to \\mathbb{R}^N$ to (\\ref{eq6.02}) and $(s,\\tilde{x}(s),y',z')\\in \\Pi(a,b)$ for all $s\\in [t_0,t]$. By Fact 3, we have that $t^+_2(y',z')>t_0$. If $t\\in [t^*,t^+_2(y',z')[$, then\n\\begin{align*}\n\\|\\tilde{x}(t)-x^*(t)\\|\\le&~\\int_{t_0}^t\\|h(s,\\tilde{x}(s)+z'-z,y')-h(s,x^*(s),y)\\|ds\\\\\n\\le&~\\int_{t_0}^t\\|h(s,\\tilde{x}(s)+z'-z,y')-h(s,x^*(s)+z'-z,y')\\|ds\\\\\n &~+\\int_{t_0}^t\\|h(s,x^*(s)+z'-z,y')-h(s,x^*(s),y)\\|ds\\\\\n\\le&~\\int_{t^*}^t[L\\|\\tilde{x}(s)-x^*(s)\\|+L(\\|y'-y\\|+\\|z'-z\\|)]ds.\n\\end{align*}\nTherefore, by Lemma 1,\n\\begin{equation}\\label{eq6.03}\n\\|\\tilde{x}(t)-x^*(t)\\|\\le (\\|y'-y\\|+\\|z'-z\\|)(e^{L(t-t_0)}-1).\n\\end{equation}\nChoose $b'\\in ]0,b[$ so small that\n\\begin{equation}\\label{beta}\nb'(e^{L(t_2-t_0)}-1)<a.\n\\end{equation}\nWe show that if $\\|y'-y\\|+\\|z'-z\\|\\le b'$, then $t^+_2(y',z')=t_2$. Suppose not. If $\\tilde{x}$ is defined at $t^+_2(y',z')$, then $(t^+_2(y',z'),\\tilde{x}(t^+_2(y',z')),y',z')\\in \\Pi(a,b)$, and thus $\\|\\tilde{x}(t^+_2(y',z'))-x^*(t^+_2(y',z'))\\|<a$, which contradicts the definition of $t^+_1(y',z')$. Therefore, $\\tilde{x}$ cannot be defined at $t^+_2(y',z')$. By Fact 3, there exists $t_3<t^+_2(y',z')$ such that if $t_3<t<t^+_2(y',z')$, then $(t,\\tilde{x}(t)+z'-z,y')\\notin \\Pi'(a,b)$, which contradicts (\\ref{eq6.03}). Hence, if $b'$ satisfies (\\ref{beta}) and $\\|y'-y\\|+\\|z'-z\\|\\le b'$, then $\\tilde{x}(t)$ is defined on $[t_0,t_2]$ and if $t\\in [t_0,t_2]$, then\n\\[\\|\\tilde{x}(t)-x^*(t)\\|\\le (\\|y'-y\\|+\\|z'-z\\|)(e^{L(t_2-t_1)}-1).\\]\nBy a symmetric argument, we have that if $b'\\in ]0,b[$ is so small that\n\\begin{equation}\\label{beta2}\nb'(e^{L(t_0-t_1)}-1)<a,\n\\end{equation}\nand $\\|y'-y\\|+\\|z'-z\\|\\le b'$, then (\\ref{eq6.02}) has a solution $\\tilde{x}$ defined on $[t_1,t_0]$, and if $t\\in [t_1,t_0]$,\n\\[\\|\\tilde{x}(t)-x(t)\\|\\le (\\|y'-y\\|+\\|z'-z\\|)(e^{L(t_2-t_1)}-1).\\]\nClearly,\n\\[x(t;y',z')=\\tilde{x}(t)+z'-z,\\]\nand thus the domain $V$ of the solution function $x$ includes \n\\[[t_1,t_2]\\times \\{(y',z')|\\|y'-y\\|+\\|z'-z\\|\\le b'\\},\\]\nwhich is a neighborhood of $(t^*,y,z)$. Moreover,\n\\[\\max_{t\\in [t_0,t_1]}\\|x(t;y',z')-x(t;y,z)\\|\\le (\\|y'-y\\|+\\|z'-z\\|)(e^{L(t_2-t_1)}-1),\\]\nand thus by Fact 1, we have that the solution function is locally Lipschitz. This completes the proof. $\\blacksquare$\n\n\n\\vspace{12pt}\nFinally, we mention the formula of the solution to linear differential equations. Consider the following ODE:\n\\[\\dot{x}(t)=a(t)x(t),\\]\nwhere $I$ is an interval including $t_0$ and $a:I\\to \\mathbb{R}$ is a bounded measurable function on $I$. Then, the formula of the solution to the above equation is the following:\n\\[x(t)=x(t_0)e^{\\int_{t_0}^ta(s)ds}.\\]\nFor the proof, use Theorem 1 of section 0.4 of Ioffe and Tikhomirov (1979).\n\n\n\n\n\n\\subsection{Proof of Theorem 1}\nFirst, consider the following parametrized ODE:\n\\begin{equation}\\label{eq6.1}\n\\dot{c}(t)=f((1-t)p+tq,c(t))\\cdot (q-p),\\ c(0)=w,\n\\end{equation}\nand let $c(t;p,q,w)$ denote the solution function of (\\ref{eq6.1}). We introduce two lemmas.\n\n\\vspace{12pt}\n\\noindent\n{\\bf Lemma 2}. Let $U=\\mathbb{R}^n_{++}\\times\\mathbb{R}_{++}$. Choose any $(p,m)\\in U$. Suppose that $W\\subset U$ and the Lebesgue measure of $U\\setminus W$ is zero. Moreover, suppose that $q\\in \\mathbb{R}^n_{++}$ and there exists $i^*\\in \\{1,...,n\\}$ such that $q_{i^*}\\neq p_{i^*}$, and the domain of the solution function $c(t;p,q,w)$ of the ODE $(\\ref{eq6.1})$ includes $[0,t^*]\\times \\bar{P}_1\\times \\bar{P}_2$ for $t^*>0$, where $P_1$ is a bounded open neighborhood of $q$ and $P_2$ is a bounded open neighborhood of $m$, and $\\bar{P}_j$ denotes the closure of $P_j$. For every $(t,\\tilde{r},w)\\subset \\mathbb{R}^{n+1}$ such that $t\\in [0,t^*]$, $r\\in P_1$ for\n\\[r_i=\\begin{cases}\n\\tilde{r}_i & \\mbox{if }i<i^*,\\\\\nq_i & \\mbox{if }i=i^*,\\\\\n\\tilde{r}_{i-1} & \\mbox{if }i>i^*,\n\\end{cases}\\]\n and $w\\in P_2$, define\n\\[\\xi(t,\\tilde{r},w)=((1-t)p+tr,c(t;p,r,w)).\\]\nThen, the Lebesgue measure of $\\xi^{-1}(U\\setminus W)$ is also zero.\n\n\\vspace{12pt}\n\\noindent\n{\\bf Proof}. Without loss of generality, we assume that $i^*=n$. Throughout the proof of Lemma 2, we use the following notation. If $r\\in \\mathbb{R}^n$, then $\\tilde{r}=(r_1,...,r_{n-1})\\in \\mathbb{R}^{n-1}$. Conversely, if $\\tilde{r}\\in \\mathbb{R}^{n-1}$, then $r=(r_1,...,r_{n-1},q_n)$.\n\nLet $\\tilde{P}_1=\\{\\tilde{r}|r\\in P_1\\}$. We show that $\\xi$ is one-to-one on the set $]0,t^*]\\times \\tilde{P}_1\\times P_2$. Suppose that $t_1\\neq 0\\neq t_2$ and $\\xi(t_1,\\tilde{r}_1,w_1)=\\xi(t_2,\\tilde{r}_2,w_2)=(z,c)$. Because $z_n=(1-t_1)p_n+t_1q_n=(1-t_2)p_n+t_2q_n$ and $p_n\\neq q_n$, we have $t_1=t_2$. Because $z_i=(1-t_1)p_i+t_1r_{1i}=(1-t_1)p_i+t_1r_{2i}$ and $t_1\\neq 0$, we have that $r_{1i}=r_{2i}$, and thus $\\tilde{r}_1=\\tilde{r}_2$. Therefore, it suffices to show that $c(t;p,r,w)$ is increasing in $w$. Suppose that $w_1<w_2$ and $c(t;p,r,w_1)\\ge c(t;p,r,w_2)$. Because $c(0;p,r,w_1)=w_1<w_2=c(0;p,r,w_2)$, by the intermediate value theorem, there exists $s\\in [0,t]$ such that $c(s;p,r,w_1)=c(s;p,r,w_2)$. Then, by Fact 3, we have $w_1=c(0;p,r,w_1)=c(0;p,r,w_2)=w_2$, which is a contradiction.\n\nNext, define\n\\[V^{\\ell}=\\xi([\\ell^{-1}t^*,t^*[\\times \\tilde{P}_1\\times P_2).\\]\nWe show that $\\xi^{-1}$ is Lipschitz on $V^{\\ell}$. Define\n\\[t(v)=\\frac{v_n-p_n}{q_n-p_n},\\]\n\\[\\tilde{r}(v)=\\frac{1}{t(v)}[(t(v)-1)\\tilde{p}+\\tilde{v}].\\]\nSuppose that $(v_1,c_1), (v_2,c_2)\\in V^{\\ell}$ and $(v_j,c_j)=\\xi(t_j,\\tilde{r}_j,w_j)$. Then, we have $t_j=t(v_j)$ and $\\tilde{r}_j=\\tilde{r}(v_j)$. Clearly, the functions $t(v)$ and $\\tilde{r}(v)$ are Lipschitz on $\\xi([\\ell^{-1}t^*,t^*]\\times \\tilde{P}_1\\times P_2)$, and hence they are Lipschitz on $V^{\\ell}$. Next, consider the following ODE:\n\\[\\dot{d}(s)=f((1-(s+t-t_2))p+(s+t-t_2)r(v),d(s))\\cdot (r(v)-p),\\ d(t_2)=c.\\]\nLet $d(s;t,v,c)$ be the solution function of the above ODE. Define $\\hat{P}_1$ as the closure of $\\tilde{P}_1$. If $(v,c)=\\xi(t,\\tilde{r},w)$ for some $(t,\\tilde{r},w)\\in [{\\ell}^{-1}t^*,t^*]\\times \\hat{P}_1\\times \\bar{P}_2$, then $d(s;t,v,c)=c(s+t-t_2;p,r,w)$. Moreover, the set\n\\[\\{(t,v,c)|t\\in [\\ell^{-1}t^*,t^*],\\ (v,c)=\\xi(t,\\tilde{r},w)\\mbox{ for some }(\\tilde{r},w)\\in \\hat{P}_1\\times \\bar{P}_2\\}\\]\nis compact, and by Fact 4, $(t,v,c)\\mapsto d(t_2-t;t,v,c)$ is Lipschitz on this set. Therefore,\n\\begin{align*}\n|w_1-w_2|=&~|d(t_2-t_1;t_1,v_1,c_1)-d(t_2-t_2;t_2,v_2,c_2)|\\\\\n\\le&~L[|t_1-t_2|+\\|(v_1,c_1)-(v_2,c_2)\\|]\\\\\n=&~L[|t(v_1)-t(v_2)|+\\|(v_1,c_1)-(v_2,c_2)\\|]\\\\\n\\le&~L(M+1)\\|(v_1,c_1)-(v_2,c_2)\\|,\n\\end{align*}\nwhere $L,M>0$ are some constants, and therefore our claim is correct.\n\nNow, recall that the Lebesgue measure of $U\\setminus W$ is zero. Because $\\xi^{-1}$ is Lipschitz on $V^{\\ell}$, we have that the Lebesgue measure of\n\\[\\xi^{-1}(V^{\\ell}\\cap (U\\setminus W))\\]\nis zero. Therefore, the Lebesgue measure of\n\\[\\cup_{\\ell}\\xi^{-1}(V^{\\ell}\\cap (U\\setminus W))\\]\nis also zero. Clearly, the Lebesgue measure of\n\\[\\xi^{-1}(U\\setminus W)\\setminus \\left(\\cup_{\\ell}\\xi^{-1}(V^{\\ell}\\cap (U\\setminus W))\\right)\\]\nis zero, because this set is included in $\\{(t,\\tilde{r},w)|t\\in \\{0,t^*\\}\\}$. This completes the proof of Lemma 2. $\\blacksquare$\n\n\\vspace{12pt}\n\\noindent\n{\\bf Lemma 3}. Choose any $(p,m)\\in \\mathbb{R}^n_{++}$. Then, there exists a solution $E:\\mathbb{R}^n_{++}\\to \\mathbb{R}_{++}$ of the following partial differential equation\n\\begin{equation}\\label{eq6.1.1}\nDE(q)=f(q,E(q)),\\ E(p)=m,\n\\end{equation}\nif and only if the domain of the solution function of (\\ref{eq6.1}) includes $[0,1]\\times \\{p\\}\\times \\mathbb{R}^n_{++}\\times \\{m\\}$. Moreover, in this case\n\\[E(q)=c(1;p,q,m).\\]\n\n\\vspace{12pt}\n\\noindent\n{\\bf Proof}. Suppose that a solution $E:\\mathbb{R}^n_{++}\\to \\mathbb{R}_{++}$ to (\\ref{eq6.1.1}) exists. Choose any $q\\in \\mathbb{R}^n_{++}$. Let $d(t)=E((1-t)p+tq)$. Then, $d(0)=E(p)=m$ and\n\\[\\dot{d}(t)=f((1-t)p+tq,d(t))\\cdot (q-p),\\]\nand by the uniqueness of the solution to ODE (Fact 3), we have that $d(t)\\equiv c(t;p,q,m)$. Hence, the domain of the solution function $c$ includes $[0,1]\\times \\{p\\}\\times \\mathbb{R}^n_{++}\\times \\{m\\}$, and moreover, $E(q)=d(1)=c(1;p,q,m)$.\n\nWe show the converse is also true. Suppose that the domain of the solution function $c$ is defined on $[0,1]\\times \\{p\\}\\times \\mathbb{R}^n_{++}\\times \\{m\\}$. Define $E(q)=c(1;p,q,m)$. We show that $E(q)$ is a solution to (\\ref{eq6.1.1}).\n\nFirst, let $\\Delta^*$ be the set of all $(q,w)$ such that $f$ is differentiable and $S_f$ is symmetric and negative semi-definite at $((1-t)p+tq,c(t;p,q,w))$ for almost every $t\\in [0,1]$, and the mapping $r\\mapsto c(t;p,r,w)$ is differentiable at $r=q$ for almost every $t\\in [0,1]$. Suppose that $(q,w)\\in \\Delta^*$ and let $e_i$ denote the $i$-th unit vector. Then, for each $i\\in \\{1,...,n\\}$,\\footnote{In this step, we frequently abbreviate several variables for simplicity.}\n\\begin{align*}\n&~\\lim_{h\\to 0}\\frac{c(t;p,q+he_i,w)-c(t;p,q,w)}{h}\\\\\n=&~\\lim_{h\\to 0}\\frac{1}{h}\\times\\\\\n&~\\left[\\int_0^tf((1-s)p+s(q+he_i),c(s;p,q+he_i,w))\\cdot (q+he_i-p)ds\\right.\\\\\n&~-\\left.\\int_0^tf((1-s)p+sq,c(s;p,q,w))\\cdot (q-p)ds\\right]\\\\\n=&~\\int_0^t\\left[f_i+\\sum_{j=1}^n\\left[s\\frac{\\partial f_j}{\\partial p_i}+\\frac{\\partial f_j}{\\partial m}\\frac{\\partial c}{\\partial q_i}\\right](q_j-p_j)\\right]ds,\n\\end{align*}\nby the dominated convergence theorem, and thus $\\frac{\\partial c}{\\partial q_i}(t;p,q,w)$ is defined for all $t\\in [0,1]$ and absolutely continuous in $t$. Define the following absolutely continuous function\n\\[\\varphi(t)=\\frac{\\partial c}{\\partial q_i}(t;p,q,w)-tf_i((1-t)p+tq,c(t;p,q,w)).\\]\nBy the above evaluation and the symmetry of the Slutsky matrix, we have that\n\\begin{align*}\n\\dot{\\varphi}(t)=&~f_i+\\sum_{j=1}^n\\left[t\\frac{\\partial f_j}{\\partial p_i}+\\frac{\\partial f_j}{\\partial m}\\frac{\\partial c}{\\partial q_i}\\right](q_j-p_j)-f_i-t\\sum_{j=1}^n\\left[\\frac{\\partial f_i}{\\partial p_j}+\\frac{\\partial f_i}{\\partial m}f_j\\right](q_j-p_j)\\\\\n=&~t\\sum_{j=1}^n\\left[\\frac{\\partial f_j}{\\partial p_i}-\\frac{\\partial f_i}{\\partial p_j}-\\frac{\\partial f_i}{\\partial m}f_j\\right](q_j-p_j)+\\frac{\\partial c}{\\partial q_i}\\sum_{j=1}^n\\frac{\\partial f_j}{\\partial m}(q_j-p_j)\\\\\n=&~\\varphi(t)\\sum_{j=1}^n\\frac{\\partial f_j}{\\partial m}(q_j-p_j)\\\\\n\\equiv&~a(t)\\varphi(t),\n\\end{align*}\nwhere $a(t)$ be some bounded measurable function. By the formula of the solution to linear ODEs, we have that\n\\[\\varphi(t)=\\varphi(0)e^{\\int_0^ta(s)ds}.\\]\nHowever, we can easily check that $\\varphi(0)=0$, and thus $\\varphi(t)\\equiv 0$. In particular, $\\varphi(1)=0$ and thus\n\\[\\frac{\\partial c}{\\partial q_i}(1;p,q,w)=f_i(q,c(1;p,q,w)).\\]\n\nSecond, suppose that $q_n\\neq p_n$ and $i\\in \\{1,...,n-1\\}$. By Lemma 2 and Fubini's theorem, there exists $\\delta>0$ and a sequence $(q^k,w^k)$ on $\\Delta^*$ such that $q^k\\to q, w^k\\to m$ as $k\\to \\infty$, and for every $k$, $i\\in \\{1,...,n-1\\}$ and almost every $h\\in ]-\\delta,\\delta[$, $(q^k+he_i,w^k)\\in \\Delta^*$. Then, for every $h\\in ]-\\delta,\\delta[$,\n\\[c(1;p,q^k+he_i,w^k)-c(1;p,q^k,w^k)=\\int_0^hf_i(q^k+se_i,c(1;p,q^k+se_i,w^k))ds.\\]\nBy the dominated convergence theorem, we have that\n\\[E(q+he_i)-E(q)=\\int_0^hf_i(q+se_i,E(q+se_i))ds,\\]\nwhich implies that\n\\[\\frac{\\partial E}{\\partial q_i}(q)=f_i(q,E(q)).\\]\n\nThird, suppose that $q_n=p_n$ and $i\\in \\{1,...,n-1\\}$. Let $e=(1,1,...,1)$ and define $q^k=q+k^{-1}e$. Then, $q_n^k\\neq p_n$, and thus, for every $h\\in ]-q_i,q_i[$,\n\\[E(q^k+he_i)-E(q^k)=\\int_0^hf_i(q^k+se_i,E(q^k+se_i))ds,\\]\nand by the dominated convergence theorem,\n\\[E(q+he_i)-E(q)=\\int_0^hf_i(q+se_i,E(q+se_i))ds,\\]\nwhich implies that\n\\[\\frac{\\partial E}{\\partial q_i}(q)=f_i(q,g(q)).\\]\nTo summarize the above result, we obtain the following: for every $q\\in \\mathbb{R}^n_{++}$ and $i\\in \\{1,...,n-1\\}$,\n\\begin{equation}\\label{eq6.3}\n\\frac{\\partial E}{\\partial q_i}(q)=f_i(q,E(q)).\n\\end{equation}\nReplacing the role of $n$ to that of $1$ and repeating the above arguments, we can show that (\\ref{eq6.3}) holds for $i=n$, and thus $DE(r)=f(r,E(r))$. This completes the proof. $\\blacksquare$\n\n\n\\vspace{12pt}\nWe now complete the preparation to prove Theorem 1. We separate the proof of Theorem 1 into ten steps.\n\n\n\\vspace{12pt}\n\\noindent\n{\\bf Step 1}. Suppose that $t^*>0$ and the domain of the solution function $c(t;p,q,w)$ of (\\ref{eq6.1}) includes $[0,t^*]\\times \\{(p,q,m)\\}$. Define $p(t)=(1-t)p+tq$ and $x(t)=f(p(t),c(t;p,q,m))$. Then, $p\\cdot x(t^*)\\ge m$ and $p(t^*)\\cdot x(0)\\ge c(t^*;p,q,m)$.\n\n\\vspace{12pt}\n\\noindent\n{\\bf Proof of Step 1}. We show only the former claim, because the latter claim can be shown symmetrically. Define $p(t,r)=(1-t)p+tr$, and let $\\Delta(t^*)$ be the set of all $(r,w)$ such that the domain of $t\\mapsto c(t;p,r,w)$ includes $[0,t^*]$, and for almost every $t\\in [0,t^*]$, $f$ is differentiable and $S_f$ is symmetric and negative semi-definite at $(p(t,r),c(t;p,r,w))$. By Lemma 2 and the Fubini's theorem, there exists a sequence $(q^k,w^k)\\in \\Delta(t^*)$ that converges to $(q,m)$ as $k\\to \\infty$. Define $d(t)=p\\cdot x(t)$ and $d^k(t)=p\\cdot f(p(t,q^k),c(t;p,q^k,w^k))$. Then, $d^k$ is absolutely continuous, and for almost all $t\\in [0,t^*]$, \n\\[\\dot{d}^k(t)=p^TS_f(p(t,q^k),c(t;p,q^k,w^k))(q^k-p).\\]\nOn the other hand, to differentiate both side of the Walras' law, we obtain that\n\\[(p(t,q^k))^TS_f(p(t,q^k),c(t;p,q^k,w^k))(q^k-p)=0.\\]\nSubtracting the latter from the former, we have that\n\\[\\dot{d}^k(t)=-t(q^k-p)^TS_f(p(t,q^k),c(t;p,q^k,w^k))(q^k-p)\\ge 0,\\]\nand thus $d^k(t)$ is nondecreasing. Because $d^k(t)\\to d(t)$ for every $t$, we have that $d(t)$ is also nondecreasing, and thus\n\\[p\\cdot x(t^*)=d(t^*)\\ge d(0)=m,\\]\nwhich completes the proof of Step 1. $\\blacksquare$\n\n\n\\vspace{12pt}\n\\noindent\n{\\bf Step 2}. The domain of the solution function $c(t;p,q,w)$ includes $[0,1]\\times \\mathbb{R}^n_{++}\\times \\mathbb{R}^n_{++}\\times \\mathbb{R}_{++}$.\n\n\\vspace{12pt}\n\\noindent\n{\\bf Proof of Step 2}. Suppose not. Then, there exists $p,q\\in \\mathbb{R}^n_{++}$ and $m\\in\\mathbb{R}_{++}$ such that $c(t;p,q,m)$ is defined only on $[0,t^*[$, where $t^*\\le 1$. Let $p(t)=(1-t)p+tq$ and $x=f(p,m)$. By Fact 3, we have that the trajectory of the function $(p(t),c(t;p,q,m))$ is excluded from any compact set in $\\mathbb{R}^n_{++}\\times\\mathbb{R}_{++}$ as $t\\to t^*$, and thus either $\\limsup_{t\\to t^*}c(t;p,q,m)=+\\infty$ or $\\liminf_{t\\to t^*}c(t;p,q,m)=0$. By Step 1, $\\max\\{p\\cdot x,q\\cdot x\\}\\ge p(t)\\cdot x\\ge c(t;p,q,m)$ for every $t\\in [0,t^*[$, and thus there exists an increasing sequence $(t^k)$ such that $t^k\\to t^*$ and $c(t^k;p,q,m)\\to 0$ as $k\\to \\infty$. Define $x^k=f(p(t^k),c(t^k;p,q,m))$. Because $p\\cdot x^k\\ge m=p\\cdot x$ and $p(t^k)\\cdot x\\ge c(t^k;p,q,m)=p(t^k)\\cdot x^k$, we have that $q\\cdot x\\ge q\\cdot x^k$. Hence, $x^k$ is a sequence in the following compact set\n\\[\\{y\\in \\Omega|q\\cdot y\\le q\\cdot x\\}.\\]\nTherefore, without loss of generality, we can assume that $x^k\\to x^*$ as $k\\to \\infty$. Because $p\\cdot x^*\\ge m$, we have that $x^*\\neq 0$. However,\n\\[0<p(t^*)\\cdot x^*=\\lim_{k\\to \\infty}p(t^k)\\cdot x^k=\\lim_{k\\to \\infty}c(t^k;p,q,m)=0,\\]\nwhich is a contradiction. This completes the proof of Step 2. $\\blacksquare$\n\n\n\\vspace{12pt}\n\\noindent\n{\\bf Step 3}. For all $t\\in [0,1]$, $c(1-t;p,q,m)=c(t;q,p,c(1;p,q,m))$. Moreover, if $m>m'$, then $c(t;p,q,m)>c(t;p,q,m')$ for every $t\\in [0,1]$.\n\n\\vspace{12pt}\n\\noindent\n{\\bf Proof of Step 3}. First,\n\\[\\frac{d}{dt}c(1-t;p,q,m)=f((1-t)q+tp,c(1-t;p,q,m))\\cdot (p-q),\\]\nand thus by the uniqueness of the solution to ODE (Fact 3), we have that\n\\[c(1-t;p,q,m)=c(t;q,p,c(1;p,q,m)).\\]\nNext, suppose that $c(t;p,q,m)\\le c(t;p,q,m')$ for every $t\\in [0,1]$. By the intermediate value theorem, there exists $s\\in [0,1]$ such that $c(s;p,q,m)=c(s;p,q,m')$. Again by the uniqueness of the solution to ODE, we have that\n\\[m=c(0;p,q,m)=c(0;p,q,m')=m',\\]\nwhich is a contradiction. This completes the proof of Step 3. $\\blacksquare$\n\n\n\\vspace{12pt}\n\\noindent\n{\\bf Step 4}. Suppose that $x\\neq y,\\ x=f(p,m),\\ y=f(q,w)$ and $w\\ge c(1;p,q,m)$. Then, $p\\cdot y>m$.\n\n\\vspace{12pt}\n\\noindent\n{\\bf Proof}. First, define $m^*=c(1;q,p,w)$. By Step 3, we have that $c(t;q,p,w)=c(1-t;p,q,m^*)$, and thus $m^*\\ge m$. Moreover, $w>c(1;p,q,m)$ if and only if $m^*>m$.\n\nDefine $p(t)=(1-t)p+tq$ and $d(t)=p\\cdot f(p(t),c(t;p,q,m^*))$. We have already shown in the proof of Step 1 that $d(t)$ is nondecreasing. Therefore, if $m^*>m$, then\n\\[p\\cdot y=p\\cdot f(q,w)=d(1)\\ge d(0)=m^*>m,\\]\nas desired. Thus, hereafter we assume that $m^*=m$. In this regard, we have that $w=c(1;p,q,m)$, and $c(1-t;q,p,w)=c(t;p,q,m)$.\n\nDefine $\\Delta^*$ as the set of all $(r,c)$ such that $f$ is differentiable and $S_f$ is symmetric and negative semi-definite at $((1-t)p+tr,c(t;p,r,c))$ for almost all $t\\in [0,1]$. By Lemma 2 and Fubini's theorem, we can show that there exists a sequence $(q^k,w^k)$ on $\\Delta^*$ such that $(q^k,w^k)\\to (q,w)$ as $k\\to \\infty$. Let $2\\varepsilon=\\|x-y\\|$, and define $p^k(t)=(1-t)p+tq^k$ and $x^k(t)=f(p^k(t),c(t;p,q^k,w^k))$. Then, $x^k(1)\\to y$ and $x^k(0)\\to x$ as $k\\to \\infty$, and thus without loss of generality, we can assume that $\\|x^k(1)-x^k(0)\\|\\ge \\varepsilon$ for all $k$. By assumption, $x^k(t)$ is a Lipschitz function defined on $[0,1]$. If $f$ is differentiable at $(p^k(t),c(t;p,q^k,w^k))$, then define $S_t^k=S_f(p^k(t),c(t;p,q^k,w^k))$. By our assumption, $S_t^k$ can be defined and is symmetric and negative semi-definite for almost all $t\\in [0,1]$. Because $S_t^k$ is symmetric and negative semi-definite, there exists a positive semi-definite matrix $A_t^k$ such that $S_t^k=-(A_t^k)^2$. Moreover, the operator norm $\\|A_t^k\\|$ is equal to $\\sqrt{\\|S_t^k\\|}$.\\footnote{Because $S_t^k$ is symmetric, there exists an orthogonal transform $P$ such that\n\\[S_t^k=P^T\\begin{pmatrix}\n\\lambda_1 & 0 & ... & 0\\\\\n0 & \\lambda_2 & ... & 0\\\\\n\\vdots & \\vdots & \\ddots & \\vdots\\\\\n0 & 0 & ... & \\lambda_n\n\\end{pmatrix}P,\\]\nwhere $\\lambda_i$ is some eigenvalue of $S_t^k$. Because $S_t^k$ is negative semi-definite, $\\lambda_i\\le 0$ for every $i$. Hence, if we define\n\\[A_t^k=P^T\\begin{pmatrix}\n\\sqrt{-\\lambda_1} & 0 & ... & 0\\\\\n0 & \\sqrt{-\\lambda_2} & ... & 0\\\\\n\\vdots & \\vdots & \\ddots & \\vdots\\\\\n0 & 0 & ... & \\sqrt{-\\lambda_n}\n\\end{pmatrix}P,\\]\nthen $S_t^k=-(A_t^k)^2$. Moreover, because the operator norm $\\|S_t^k\\|$ (resp. $\\|A_t^k\\|$) coincides with $\\max_i|\\lambda_i|$, (resp. $\\max_i\\sqrt{|\\lambda_i|}$,) all our claims are correct.} Because $f$ is locally Lipschitz, there exists $L>0$ such that $\\|S_t^k\\|\\le L$ for all $k$ and almost all $t\\in [0,1]$. Define $d^k(t)=p\\cdot x^k(t)$, and choose $\\delta>0$ such that $\\varepsilon^2>2L^2\\delta\\|q^k-p\\|^2$ for every sufficiently large $k$. By the same arguments as in the proof of Step 1, we can show that\n\\[\\dot{d}^k(t)=-t(q^k-p)^TS_t^k(q^k-p)=t\\|A_t^k(q^k-p)\\|^2,\\]\n\\[\\dot{x}^k(t)=S_t^k(q^k-p)\\]\nfor almost all $t\\in [0,1]$. Therefore,\n\\begin{align*}\n\\varepsilon^2\\le \\|x^k(1)-x^k(0)\\|^2=&~\\left\\|\\int_0^1\\dot{x}^k(t)dt\\right\\|^2\\le \\int_0^1\\|\\dot{x}^k(t)\\|^2dt=\\int_0^1\\|S_t^k(q^k-p)\\|^2dt\\\\\n\\le&~\\int_0^1\\|A_t^k\\|^2\\|A_t^k(q^k-p)\\|^2dt\\le L\\int_0^1\\|A_t^k(q^k-p)\\|^2dt\\\\\n=&~L\\int_0^{\\delta}\\|A_t^k(q^k-p)\\|^2dt+L\\int_{\\delta}^1\\frac{1}{t}\\dot{d}^k(t)dt\\\\\n\\le&~L^2\\delta\\|q^k-p\\|^2+\\frac{L}{\\delta}(d^k(1)-d^k(\\delta)),\n\\end{align*}\nand thus,\n\\[\\frac{\\delta \\varepsilon^2}{2L}\\le d^k(1)-d^k(\\delta).\\]\nLetting $k\\to \\infty$, we have that\n\\[p\\cdot y=d(1)\\ge d(\\delta)+\\frac{\\delta \\varepsilon^2}{2L}>d(0)=m,\\]\nas desired. This completes the proof of Step 4. $\\blacksquare$\n\n\\vspace{12pt}\n\\noindent\n{\\bf Step 5}. If $x\\neq y,\\ x=f(p,m),\\ y=f(q,w)$ and $p\\cdot y\\le m$, then $q\\cdot x>w$.\\footnote{In other words, $f$ satisfies the {\\bf weak axiom of revealed preference}.}\n\n\\vspace{12pt}\n\\noindent\n{\\bf Proof of Step 5}. By the contrapositive of Step 4, we have that $c(1;p,q,m)>w$. By Step 3, $m>c(1;q,p,w)$. By Step 4, we obtain that $q\\cdot x>w$, as desired. This completes the proof of Step 5. $\\blacksquare$\n\n\n\\vspace{12pt}\n\\noindent\n{\\bf Step 6}. For every $q\\in \\mathbb{R}^n_{++}$, $c(1;q,\\bar{p},c(1;p,q,m))=c(1;p,\\bar{p},m)$.\n\n\\vspace{12pt}\n\\noindent\n{\\bf Proof of Step 6}. Define $p(t)=(1-t)p+tq$, $m^*=c(1;p,\\bar{p},m)$ and $w(t)=c(1;\\bar{p},p(t),m^*)$. By Step 3, $c(1-t;p,\\bar{p},m)=c(t;\\bar{p},p,m^*)$, and thus $w(0)=c(0;p,\\bar{p},m)=m$. Moreover, by Lemma 3 and Step 2, we have that $E(r)=c(1;\\bar{p},r,m^*)$ satisfies the following differential equation:\n\\[DE(r)=f(r,E(r)).\\]\nTherefore,\n\\begin{align*}\n\\dot{w}(t)=&~\\frac{d}{dt}c(1;\\bar{p},(1-t)p+tq,m^*)=\\frac{d}{dt}E((1-t)p+tq)\\\\\n=&~f((1-t)p+tq,E((1-t)p+tq))\\cdot (q-p)\\\\\n=&~f((1-t)p+tq,w(t))\\cdot (q-p).\n\\end{align*}\nBy the uniqueness of the solution to ODE, we have that\n\\[w(t)=c(t;p,q,m)\\]\nfor all $t\\in [0,1]$. Now, define $m^+=c(1;p,q,m)$. Then,\n\\[c(1;\\bar{p},q,m^*)=w(1)=m^+.\\]\nBy Step 3, we have $c(1-t;\\bar{p},q,m^*)=c(t;q,\\bar{p},m^+)$, and thus\n\\[m^*=c(1;q,\\bar{p},m^+),\\]\nas desired. This completes the proof of Step 6. $\\blacksquare$\n\n\n\\vspace{12pt}\n\\noindent\n{\\bf Step 7}. Suppose that $x=f(p,m)=f(q,w)$. Then, $c(1;p,\\bar{p},m)=c(1;q,\\bar{p},w)$.\n\n\\vspace{12pt}\n\\noindent\n{\\bf Proof of Step 7}. Let $p(t)=(1-t)p+tq$ and $m(t)=(1-t)m+tw$. Suppose that $f(p(t),m(t))=y\\neq x$ for some $t\\in [0,1]$. By Walras' law, $p(t)\\cdot y=m(t)=p(t)\\cdot x$, and thus $p\\cdot y>m$ and $q\\cdot y>w$ by Step 5. However, this implies that $p(t)\\cdot y>m(t)$, which is a contradiction. Therefore, $f(p(t),m(t))\\equiv x$, and thus\n\\[\\dot{m}(t)=w-m=x\\cdot (q-p)=f(p(t),m(t))\\cdot (q-p).\\]\nBy the uniqueness of the solution to ODE,\n\\[m(t)=c(t;p,q,m),\\]\nand thus $w=c(1;p,q,m)$. Therefore, by Step 6, we have that $c(1;p,\\bar{p},m)=c(1;q,\\bar{p},w)$. This completes the proof of Step 7. $\\blacksquare$\n\n\\vspace{12pt}\nBy Steps 2 and 7, we can define $u_{f,\\bar{p}}(x)$ for all $x\\in \\Omega$, and our definition of $u_{f,\\bar{p}}(x)$ is independent of the choice of $(p,m)$.\n\n\\vspace{12pt}\n\\noindent\n{\\bf Step 8}. $f=f^{u_{f,\\bar{p}}}$.\n\n\\vspace{12pt}\n\\noindent\n{\\bf Proof of Step 8}. Suppose that $x=f(p,m), y\\neq x$ and $p\\cdot y\\le m$. If $y\\notin R(f)$, then $u_{f,\\bar{p}}(y)=0<u_{f,\\bar{p}}(x)$. If $y=f(q,w)$ for some $(q,w)$, then by the contrapositive of Step 4, we have that $c(1;p,q,m)>w$. By Step 3,\n\\[c(t;q,\\bar{p},c(1;p,q,m))>c(t;q,\\bar{p},w)\\]\nfor every $t\\in [0,1]$. By Step 6,\n\\[u_{f,\\bar{p}}(x)=c(1;p,\\bar{p},m)=c(1;q,\\bar{p},c(1;p,q,m))>c(1;q,\\bar{p},w)=u_{f,\\bar{p}}(y).\\]\nTherefore, $x=f^{u_{f,\\bar{p}}}(p,m)$, as desired. This completes the proof of Step 8. $\\blacksquare$\n\n\n\\vspace{12pt}\n\\noindent\n{\\bf Step 9}. $u_{f,\\bar{p}}$ is upper semi-continuous on $R(f)$.\n\n\\vspace{12pt}\n\\noindent\n{\\bf Proof of Step 9}. Suppose that $x=f(p,m)$ and $u_{f,\\bar{p}}(x)<a$. By Fact 4, the solution function $c$ is continuous, and thus there exists $\\varepsilon>0$ such that $c(1;p,\\bar{p},m+\\varepsilon)<a$. Define $y=f(p,m+\\varepsilon)$. Then, the set\n\\[U=\\{z\\in R(f)|p\\cdot z<m+\\varepsilon\\}\\]\nis a neighborhood of $x$ in the relative topology of $R(f)$, and for every $z\\in U$, $u_{f,\\bar{p}}(z)<u_{f,\\bar{p}}(y)<a$. This completes the proof of Step 9. $\\blacksquare$\n\n\\vspace{12pt}\n\\noindent\n{\\bf Step 10}. Suppose that $f=f^{\\succsim}$ for some weak order $\\succsim$, and $\\succsim$ is upper semi-continuous on $R(f)$. Then, for every $x,y\\in R(f)$,\n\\[x\\succsim y\\Leftrightarrow u_{f,\\bar{p}}(x)\\ge u_{f,\\bar{p}}(y).\\]\n\n\\vspace{12pt}\n\\noindent\n{\\bf Proof of Step 10}. First, choose any $x\\in R(f)$ and suppose that $x=f(p,m)$. Define $z=f(\\bar{p},u_{f,\\bar{p}}(x))$, and let\n\\[E^x(q)=\\inf\\{q\\cdot w|w\\succsim x\\}.\\]\nBy Lemma 1 of Hosoya (2020), we have that\n\\[DE^x(q)=f(q,E^x(q)),\\ E^x(p)=m.\\]\nDefine $d(t)=E^x((1-t)p+t\\bar{p})$. Then,\n\\[\\dot{d}(t)=f((1-t)p+t\\bar{p},d(t))\\cdot (\\bar{p}-p),\\]\nwhich implies that $d(t)=c(t;p,\\bar{p},m)$ for every $t\\in [0,1]$. In particular,\n\\[E^x(\\bar{p})=d(1)=c(1;p,\\bar{p},m)=u_{f,\\bar{p}}(x).\\]\nNow, choose any $\\varepsilon>0$. Then, there exists $w\\in \\Omega$ such that $p\\cdot w<E^x(\\bar{p})+\\varepsilon$ and $w\\succsim x$. Define $z^{\\varepsilon}=f(\\bar{p},E^x(\\bar{p})+\\varepsilon)$. Then, $z^{\\varepsilon}\\succsim w$, and thus $z^{\\varepsilon}\\succsim x$. Letting $\\varepsilon\\to 0$, by the upper semi-continuity of $\\succsim$, we obtain that $z\\succsim x$.\n\nNext, define\n\\[E^z(q)=\\inf\\{q\\cdot w|w\\succsim z\\}.\\]\nBy the same arguments as above, we can show that $E^z(p)=m$, and thus $x\\succsim z$. Hence, $x\\sim f(\\bar{p},u_{f,\\bar{p}}(x))$ for all $x\\in R(f)$.\n\nNow, choose any $x,y\\in R(f)$. Then,\n\\[x\\succsim y\\Leftrightarrow f(\\bar{p},u_{f,\\bar{p}}(x))\\succsim f(\\bar{p},u_{f,\\bar{p}}(y))\\Leftrightarrow u_{f,\\bar{p}}(x)\\ge u_{f,\\bar{p}}(y),\\]\nas desired. This completes the proof of Step 10.\n\n\\vspace{12pt}\nSteps 8-10 indicates that all our claims in Theorem 1 is correct. This completes the proof. $\\blacksquare$\n\n\n\\subsection{Proof of Corollary 1}\nIt is obvious that (ii) implies (i).\n\nSuppose that (i) holds, and choose any $(p,m)\\in\\mathbb{R}^n_{++}\\times\\mathbb{R}_{++}$. Let $x=f(p,m)$ and define\n\\[E^x(q)=\\inf\\{q\\cdot y|y\\succsim x\\}.\\]\nBy Lemma 1 of Hosoya (2020), this $E^x$ is a concave solution to (\\ref{eq3.2}). Suppose that $E$ is another soution of (\\ref{eq3.2}). For each $q\\in \\mathbb{R}^n_{++}$, define $c_1(t)=E^x((1-t)p+tq)$ and $c_2(t)=E((1-t)p+tq)$. Then,\n\\[\\dot{c}_i(t)=f((1-t)p+tq,c_i(t))\\cdot (q-p),\\ c_i(0)=m,\\]\nand thus, by the uniqueness of the solution to ODE (Fact 3), we have that $c_1\\equiv c_2$. In particular, $E^x(q)=c_1(1)=c_2(1)=E(q)$, and thus $E=E^x$. Therefore, the solution is unique, and (iv) holds.\n\nSuppose that (iv) holds. Choose any $(p,m)\\in \\mathbb{R}^n_{++}\\times \\mathbb{R}_{++}$ such that $f$ is differentiable at $(p,m)$. Let $E$ be a concave solution to (\\ref{eq3.2}). By easy calculation, we obtain that\n\\[D^2E(p)=S_f(p,m).\\]\nBecause $E$ is concave, the left-hand side of the above equation is negative semi-definite. Moreover, by extended Young's theorem,\\footnote{See Lemma 3.2 of Hosoya (2021).} the left-hand side of the above equation is symmetric. Therefore, $f$ satisfies (S) and (NSD), and (iii) holds.\n\nFinally, our Theorem 1 says that (iii) implies (ii). This completes the proof. $\\blacksquare$\n\n\n\n\\subsection{Proof of Corollary 2}\nDefine\n\\[w_{f,\\bar{p}}(x)=\\begin{cases}\nu_{f,\\bar{p}}(x) & \\mbox{if }x\\in R(f),\\\\\n\\inf_{\\varepsilon>0}\\sup\\{u_{f,\\bar{p}}(y)|y\\in R(f),\\ \\|y-x\\|<\\varepsilon\\} & \\mbox{if }x\\notin R(f).\n\\end{cases}\\]\nTheorems 1-2 of Hosoya (2020) stated the following facts: 1) $f=f^{w_{f,\\bar{p}}}$, 2) $w_{f,\\bar{p}}$ is upper semi-continuous, 3) $w_{f,\\bar{p}}$ is continuous on $\\mathbb{R}^n_{++}$ if and only if $f$ satisfies the C axiom, and 4) if $\\succsim$ is an upper semi-continuous weak order on $\\Omega$ such that $f=f^{\\succsim}$, then for each $x,y\\in \\Omega$,\n\\[x\\succsim y\\Leftrightarrow w_{f,\\bar{p}}(x)\\ge w_{f,\\bar{p}}(y).\\]\nWe first show that $w_{f,\\bar{p}}(x)=v_{f,\\bar{p}}(x)$ for all $x\\in \\Omega$.\n\nIf $x\\in \\mathbb{R}^n_{++}$, then $w_{f,\\bar{p}}(x)=u_{f,\\bar{p}}(x)=v_{f,\\bar{p}}(x)$.\n\nSuppose that $x\\notin R(f)$. Choose any $\\varepsilon>0$, $\\|y-x\\|<\\varepsilon$ and $y\\in R(f)$. Then, there exists $z\\in \\mathbb{R}^n_{++}$ such that $z\\gg y$ and $\\|z-x\\|<\\varepsilon$. If $z=f(p,m)$, then $p\\cdot y<m$, and thus $u_{f,\\bar{p}}(z)>u_{f,\\bar{p}}(y)$. This indicates that\n\\[\\sup\\{u_{f,\\bar{p}}(y)|y\\in R(f),\\ \\|y-x\\|<\\varepsilon\\}=\\sup\\{u_{f,\\bar{p}}(y)|y\\in\\mathbb{R}^n_{++},\\ \\|y-x\\|<\\varepsilon\\},\\]\nand thus, $v_{f,\\bar{p}}(x)=w_{f,\\bar{p}}(x)$.\n\nSuppose that $x\\in R(f)\\setminus \\mathbb{R}^n_{++}$. Let $e=(1,1,...,1)$ and define $x^k=x+k^{-1}e$. Then, $x^k\\in \\mathbb{R}^n_{++}$. It is easy to show that\n\\[\\lim_{k\\to \\infty}u_{f,\\bar{p}}(x^k)=v_{f,\\bar{p}}(x).\\]\nBecause $u_{f,\\bar{p}}$ is upper semi-continuous on $R(f)$, we have that\n\\[\\lim_{k\\to \\infty}u_{f,\\bar{p}}(x^k)\\le u_{f,\\bar{p}}(x).\\]\nOn the other hand, if $x^k=f(p^k,m^k)$, then $p^k\\cdot x<m^k$, and thus $u_{f,\\bar{p}}(x^k)>u_{f,\\bar{p}}(x)$. Therefore, we have that\n\\[\\lim_{k\\to \\infty}u_{f,\\bar{p}}(x^k)\\ge u_{f,\\bar{p}}(x).\\]\nTo combine these inequalities, we have that\n\\[v_{f,\\bar{p}}(x)=\\lim_{k\\to \\infty}u_{f,\\bar{p}}(x^k)=w_{f,\\bar{p}}(x),\\]\nas desired. Hence, $v_{f,\\bar{p}}(x)=w_{f,\\bar{p}}(x)$.\n\nThe rest of the claim of this corollary is 3). If $v_{f,\\bar{p}}$ is continuous, then $f=f^{\\succsim}$ for a continuous weak order $\\succsim$ defined as\n\\[x\\succsim y\\Leftrightarrow v_{f,\\bar{p}}(x)\\ge v_{f,\\bar{p}}(y).\\]\nLet us show the converse. Suppose that there exists a continuous weak order $\\succsim$ on $\\Omega$ such that $f=f^{\\succsim}$. Debreu (1954) showed that there exists a continuous function $u:\\Omega\\to \\mathbb{R}$ such that $u$ represents $\\succsim$. By the above arguments, $v_{f,\\bar{p}}$ also represents $\\succsim$, and thus $v_{f,\\bar{p}}$ and $u$ have the same order. On the other hand, in the proof of Theorem 1 of Hosoya (2020), it was shown that if $v_{f,\\bar{p}}(x)>0$, then\n\\[v_{f,\\bar{p}}(x)=v_{f,\\bar{p}}(f(\\bar{p},v_{f,\\bar{p}}(x)))\\]\nfor all $x\\in \\Omega$, and $v_{f,\\bar{p}}(0)=0$. Therefore, if $v_{f,\\bar{p}}(x)>0$, then\n\\[u(x)=u(f(\\bar{p},v_{f,\\bar{p}}(x))),\\]\nand if $v_{f,\\bar{p}}(x)=0$, then\n\\[u(x)=u(0).\\]\nIf $v_{f,\\bar{p}}(x)>0$, then for every sufficiently small $\\varepsilon>0$, there exists $\\delta>0$ such that if $y\\in \\Omega$ and $\\|y-x\\|<\\delta$, then\n\\[u(f(\\bar{p},v_{f,\\bar{p}}(x)-\\varepsilon))<u(y)<u(f(\\bar{p},v_{f,\\bar{p}}(x)+\\varepsilon)),\\]\nwhich implies that $|v_{f,\\bar{p}}(y)-v_{f,\\bar{p}}(x)|<\\varepsilon$. If $v_{f,\\bar{p}}(x)=0$, then for every $\\varepsilon>0$, there exists $\\delta>0$ such that if $y\\in \\Omega$ and $\\|y-x\\|<\\delta$, then\n\\[u(0)\\le u(y)<u(f(\\bar{p},v_{f,\\bar{p}}(x)+\\varepsilon)),\\]\nwhich implies that $0\\le v_{f,\\bar{p}}(y)<\\varepsilon$. Therefore, $v_{f,\\bar{p}}$ is continuous. This completes the proof. $\\blacksquare$\n\n\n\n\n\n\n\n\\subsection{Proof of Theorem 2}\nDefine\n\\[I(t,c,g,p,q)=g((1-t)p+tq,c)\\cdot (q-p),\\]\nwhere $g:\\mathbb{R}^n_{++}\\times \\mathbb{R}_{++}\\to \\mathbb{R}^n$ is locally Lipschitz. Consider the following ODE:\n\\begin{equation}\\label{eq6.4}\n\\dot{c}(t)=I(t,c(t),g,p,q),\\ c(0)=m,\n\\end{equation}\nand let $c(t;g,p,q,m)$ be the solution function. If the domain of $c$ includes $[0,1]\\times\\{f\\}\\times \\mathbb{R}^n_{++}\\times \\mathbb{R}^n_{++}\\times \\mathbb{R}_{++}$, then by Corollary 1, we have that $f$ is a demand function. Hence, it suffices to show that for every $(p,m)\\in\\mathbb{R}^n_{++}\\times\\mathbb{R}_{++}$ and $q\\in \\mathbb{R}^n_{++}$, the domain of the mapping $t\\mapsto c(t;f,p,q,m)$ includes $[0,1]$.\n\nChoose any $(p,m)\\in \\mathbb{R}^n_{++}\\times \\mathbb{R}_{++}$ and $q\\in \\mathbb{R}^n_{++}$, and define $p(t)=(1-t)p+tq$. For every continuous function $g:\\mathbb{R}^n_{++}\\times\\mathbb{R}_{++}\\to \\mathbb{R}^n$, define\n\\[H_K(g)=\\sup_{(r,c)\\in [K^{-1},K]^{n+1}}\\|g(r,c)\\|.\\]\nHereafter, we abbreviate $c(t;f,p,q,m)$ as $c(t)$ and $c(t;f^k,p,q,m)$ as $c^k(t)$. Note that $c^k(t)$ is defined on $[0,1]$ for all $k$ according to Corollary 1.\n\nBecause $f^k$ is a demand function, there exists a weak order $\\succsim_k$ on $\\Omega$ such that $f^k=f^{\\succsim_k}$. Define\n\\[E^k(r)=\\inf\\{r\\cdot y|y\\succsim_kf^k(p,m)\\}.\\]\nThen, by Shephard's lemma (Lemma 1 of Hosoya (2020)), we have that $c^k(t)=E^k(p(t))$. Define\n\\[\\phi(r,w)=\\max\\{p\\cdot y|y\\in \\Omega,\\ r\\cdot y=w\\}.\\]\nThen, $\\phi(r,0)=0$, and by Berge's maximum theorem, $\\phi$ is continuous. Therefore, there exists $w_0>0$ such that if $p(t)\\cdot y=w_0$ for some $t\\in [0,1]$, then $p\\cdot y<m$, and thus $y\\not\\succsim^kf^k(p,m)$ for all $k$. By definition, $E^k(p(t))\\ge w_0$, and thus $c^k(t)\\ge w_0$ for all $k$ and $t\\in [0,1]$. On the other hand,\n\\begin{align*}\nc^k(t)=&~E^k(p(t))\\le p(t)\\cdot f^k(p,m)\\\\\n\\le&~(1-t)m+t\\max\\{q\\cdot x|p\\cdot x=m\\}\\\\\n\\le&~\\max\\{(1-s)m+s\\max\\{q\\cdot x|p\\cdot x=m\\}|s\\in [0,1]\\}\\equiv w_1\n\\end{align*}\nfor all $k$ and $t\\in [0,1]$. Choose $K\\in\\mathbb{R}$ such that $p,q\\in [K^{-1},K]^n$ and $w_0,w_1\\in ]K^{-1},K[$. Because $f$ is locally Lipschitz, there exists $L>0$ such that if $(p',m'), (q',w')\\in [K^{-1},K]^{n+1}$, then\n\\[\\|f(p',m')-f(q',w')\\|\\le L\\|(p',m')-(q',w')\\|.\\]\nLet $I^*$ be the set of all $t\\in [0,1]$ such that $c(s)$ is defined and $c(s)\\in [K^{-1},K]$ for all $s\\in [0,t]$, and suppose that $I^*=[0,t^*[$. For any $t\\in [0,t^*[$,\n\\begin{align*}\n|c^k(t)-c(t)|\\le&~\\int_0^t\\|f^k(p(s),c^k(s))-f(p(s),c(s))\\|\\|q-p\\|ds\\\\\n\\le&~\\int_0^t[\\|f^k(p(s),c^k(s))-f(p(s),c^k(s))\\|\\\\\n&~+\\|f(p(s),c^k(s))-f(p(s),c(s))\\|]\\|q-p\\|ds\\\\\n\\le&~\\int_0^t[L\\|q-p\\||c^k(s)-c(s)|+H_K(f^k-f)\\|q-p\\|]ds,\n\\end{align*}\nand thus, by Lemma 1,\n\\begin{equation}\\label{eq6.5}\n|c^k(t)-c(t)|\\le \\frac{H_K(f^k-f)}{L}(e^{L\\|q-p\\|}-1).\n\\end{equation}\nThis indicates that $c^k(t)\\to c(t)$ as $k\\to \\infty$, and thus $c(t)\\in [w_0,w_1]$ for all $t\\in [0,t^*[$. Because $c(t)$ is a nonextendable solution to (\\ref{eq6.4}), by Fact 3, we have that $c(t)$ is defined at $t^*$. By continuity of $c(t)$, $c(t^*)\\in [w_0,w_1]\\subset [K^{-1},K]$, which is a contradiction. Therefore, $I^*=[0,t^*]$. If $t^*<1$, then $c(t^*)\\in ]K^{-1},K[$, and thus there exists $t>t^*$ such that $t\\in I^*$, which is a contradiction. Thus, $t^*=1$ and $I^*=[0,1]$, which implies that $c(t)$ is defined on $[0,1]$. This completes the proof. $\\blacksquare$\n\n\n\n\\subsection{Proof of Corollary 3}\nSuppose that $(f^k)$ is a Cauchy sequence in $\\mathscr{F}_L$. It is easy to show that $f^k\\to f$ for some function $f:\\mathbb{R}^n_{++}\\times \\mathbb{R}_{++}\\to \\Omega$, and for every compact set $C\\subset \\mathbb{R}^n_{++}\\times\\mathbb{R}_{++}$,\n\\[\\lim_{k\\to \\infty}\\sup_{(p,m)\\in C}\\|f^k(p,m)-f(p,m)\\|=0.\\]\nClearly $f$ satisfies Walras' law. Because $\\Delta_{\\nu}$ is compact, we have that $(f^k)$ converges to $f$ uniformly on $\\Delta_{\\nu}$. Moreover, if $(p,m),(q,w)\\in \\Delta_{\\nu}$, then\n\\[\\|f(p,m)-f(q,w)\\|=\\lim_{k\\to \\infty}\\|f^k(p,m)-f^k(q,w)\\|\\le L_{\\nu}\\|(p,m)-(q,w)\\|,\\]\nwhich implies that $f\\in \\mathscr{F}_L$. This completes the proof. $\\blacksquare$\n\n\n\n\\subsection{Proof of Theorem 3}\nFirst, we show the following lemma.\n\n\\vspace{12pt}\n\\noindent\n{\\bf Lemma 4}. Suppose that $f$ is a locally Lipschitz demand function that satisfies Walras' law, and $R(f)$ includes $\\mathbb{R}^n_{++}$. If $f$ satisfies the C axiom, then $u_{f,\\bar{p}}$ is continuous on $\\mathbb{R}^n_{++}$.\\footnote{Note that, in this lemma, $R(f)$ need not be relatively open in $\\Omega$, and thus Corollary 2 cannot be directly applied.}\n\n\\vspace{12pt}\n\\noindent\n{\\bf Proof}. Recall the differential equation (\\ref{eq3.1}):\n\\[\\dot{c}(t)=f((1-t)p+t\\bar{p},c(t))\\cdot (\\bar{p}-p),\\ c(0)=m.\\]\nLet $c(t;p,m)$ be the solution function. We have that $u_{f,\\bar{p}}(x)=c(1;p,m)$ if $x=f(p,m)$. Choose any sequence $(x^k)$ in $\\mathbb{R}^n_{++}$ such that $x^k\\to x\\in \\mathbb{R}^n_{++}$ as $k\\to \\infty$, and suppose that $u_{f,\\bar{p}}(x^k)\\not\\to u_{f,\\bar{p}}(x)$. Taking a subsequence, we can assume that there exists $\\varepsilon>0$ such that $|u_{f,\\bar{p}}(x^k)-u_{f,\\bar{p}}(x)|>\\varepsilon$ for every $k$. Choose any $p^k\\in G^f(x^k)$. Again, taking a subsequence, we can assume that $p^k\\to p^*\\in G^f(x)$. Then, \n\\[u_{f,\\bar{p}}(x^k)=c(1;p^k,p^k\\cdot x^k)\\to c(1;p^*,p^*\\cdot x)=u_{f,\\bar{p}}(x),\\]\nwhich is a contradiction. Therefore, $u_{f,\\bar{p}}$ is continuous on $\\mathbb{R}^n_{++}$. This completes the proof. $\\blacksquare$\n\n\n\\vspace{12pt}\nChoose any compact set $D\\subset \\mathbb{R}^n_{++}$. Let $x\\in D$. We first show that there exists an open neighborhood $U$ of $x$ and $\\varepsilon>0$ such that if $p\\in G^{f^k}(y)$ for some $y\\in U$ and $k$, then $p_i\\ge \\varepsilon$ for all $i\\in \\{1,...,n\\}$.\n\nSuppose not. Then, there exists $p^{\\ell}\\in\\mathbb{R}^n_{++}$ and $z^{\\ell}=f^{k(\\ell)}(p^{\\ell},p^{\\ell}\\cdot z^{\\ell})$ such that $\\sum_jp^{\\ell}_j=1,\\ z^{\\ell}\\to x$ and $\\min_jp^{\\ell}_j\\to 0$ as $\\ell\\to \\infty$. First, suppose that $k(\\ell)=k$ for infinitely many $\\ell$. Taking a subsequence, we can assume that $k(\\ell)=k$ for any $\\ell$. By the C axiom, the inverse demand correspondence $G^{f^k}$ is compact-valued and upper hemi-continuous. Moreover, $p^{\\ell}\\in G^{f^k}(z^{\\ell})$ for all $\\ell$ and $z^{\\ell}\\to x$ as $\\ell\\to \\infty$. Therefore, if we choose\n\\[\\varepsilon=\\frac{1}{2}\\min\\{\\min_ip_i|p\\in G^{f^k}(x)\\},\\]\nthen $\\varepsilon>0$ and $\\min_jp^{\\ell}_j\\ge \\varepsilon$ for sufficiently large $\\ell$, which is a contradiction. Hence, we can assume that $k(\\ell)$ is increasing. Taking a subsequence, we can assume that $p^{\\ell}\\to p^*\\in\\mathbb{R}^n_+$, where $\\sum_jp^*_j=1$ and $\\min_ip^*_i=0$. Choose $i,j\\in \\{1,...,n\\}$ such that $p^*_i>0$ and $p^*_j=0$.\n\nLet $e_j$ denote the $j$-th unit vector. Choose a small $\\delta>0$, and set $y_j=x_j+2$, $y_i(\\delta)=x_i-\\delta$, and $y_m=x_m$ for every $m\\in \\{1,...,n\\}\\setminus\\{i,j\\}$. Let $y(\\delta)=(y_1,...,y_i(\\delta),...,y_n)$. Because $G^f$ is upper semi-continuous, there exists a sequence $(\\delta_{\\nu})$ of positive real numbers such that $\\delta_{\\nu}\\to 0$ as $\\nu\\to 0$ and there exists $p^{\\nu}\\in G^f(y(\\delta_{\\nu}))$ such that $p^{\\nu}\\to p^+\\in G^f(y(0))$ as $\\nu\\to \\infty$. Because $p^+\\cdot y(0)>p^+\\cdot (x+e_j)$, we have that $p^{\\nu}\\cdot y(\\delta_{\\nu})>p^{\\nu}\\cdot (x+e_j)$ for sufficiently large $\\nu$. Choose such a $\\nu$, and define $q=p^{\\nu}$ and $y=y(\\delta_{\\nu})$. Then, $q\\in G^f(y)$ and thus $y=f(q,q\\cdot y)$. Define $y^k=f^k(q,q\\cdot y)$. Then, $y^k\\to y$ as $k\\to \\infty$. By assumption, $q\\cdot z^{\\ell}+q_j<q\\cdot y$ for sufficiently large $\\ell$, and thus we have that $q\\cdot z^{\\ell}\\le q\\cdot y^{k(\\ell)}$ for sufficiently large $\\ell$. However,\n\\[\\lim_{\\ell\\to \\infty}p^{\\ell}\\cdot z^{\\ell}=p^*\\cdot x>p^*\\cdot y=\\lim_{\\ell\\to \\infty}p^{\\ell}\\cdot y^{k(\\ell)},\\]\nand thus $p^{\\ell}\\cdot z^{\\ell}\\ge p^{\\ell}\\cdot y^{k(\\ell)}$ if $\\ell$ is sufficiently large, which contradicts the weak axiom of $f^{k(\\ell)}$. Therefore, our initial claim is correct. Define $U_x$ as such a neighborhood that corresponds with $x\\in D$. Then $(U_x)$ is an open covering of $D$, and thus there exists $\\varepsilon^*>0$ such that if $\\sum_jp_j=1$ and $f^k(p,p\\cdot x)=x$ for some $x\\in D$ and $k$, then $p_i\\ge \\varepsilon^*$ for all $i\\in \\{1,...,n\\}$.\n\nLet\n\\[C=\\{p\\in\\mathbb{R}^n_{++}|p\\in G^f(x)\\mbox{ for some }x\\in D\\},\\]\n\\[C_k=\\{p\\in\\mathbb{R}^n_{++}|p\\in G^{f^k}(x)\\mbox{ for some }x\\in D\\}.\\]\nBy the compact-valuedness and upper hemi-continuity of the inverse demand functions, we have that $C,C_k$ are compact. Because of our previous arguments, there exists a compact set $K\\subset \\mathbb{R}^n_{++}$ that includes $C$ and all $C_k$. Define $m_1=\\min\\{p\\cdot x|p\\in K, x\\in D\\}>0$ and $m_2=\\max\\{p\\cdot x|p\\in K,x\\in D\\}>0$.\n\nIt suffices to show that $\\sup_{x\\in D}|u_{f^k,\\bar{p}}(x)-u_{f,\\bar{p}}(x)|\\to 0$ as $k\\to \\infty$. Suppose not. Then, there exist $\\varepsilon>0$ and a sequence $(x^{\\ell})$ in $D$ such that $|u_{f^{k(\\ell)},\\bar{p}}(x^{\\ell})-u_{f,\\bar{p}}(x^{\\ell})|\\ge \\varepsilon$ for all $\\ell$, where $k(\\ell)$ is increasing in $\\ell$. Because $D$ is compact, we can assume that $x^{\\ell}\\to x^*\\in D$ as $\\ell\\to \\infty$. Suppose that $x^{\\ell}=f^{k(\\ell)}(p^{\\ell},m^{\\ell})$, where $p^{\\ell}\\in C_{k(\\ell)}$ and $m^{\\ell}=p^{\\ell}\\cdot x^{\\ell}$. Taking a subsequence, we can assume that $p^{\\ell}\\to p^*\\in K$. Define $m^*=p^*\\cdot x^*$. Then, $(p^{\\ell},m^{\\ell}), (p^*,m^*)\\in K\\times [m_1,m_2]$, and thus\n\\begin{align*}\n\\|f^{k(\\ell)}(p^{\\ell},m^{\\ell})-f(p^*,m^*)\\|\\le&~\\|f^{k(\\ell)}(p^{\\ell},m^{\\ell})-f(p^{\\ell},m^{\\ell})\\|\\\\\n&~+\\|f(p^{\\ell},m^{\\ell})-f(p^*,m^*)\\|\\to 0,\n\\end{align*}\nas $\\ell\\to \\infty$. This implies that $f(p^*,m^*)=x^*$.\n\nNow, consider the following differential equation:\n\\begin{equation}\\label{eq6.6}\n\\dot{c}(t)=I(t,c(t),g,p,m),\\ c(0)=m^*,\n\\end{equation}\nwhere $g:\\mathbb{R}^n_{++}\\times \\mathbb{R}_{++}\\to \\Omega$ is locally Lipschitz and $I(t,c,g,p,m)=g((1-t)p+t\\bar{p},c+m-m^*)\\cdot (\\bar{p}-p)$. Let $c(t;g,p,m)$ be the solution function of (\\ref{eq6.6}). We abbreviate $c(t;f,p^*,m^*)$ as $c^*(t)$ and $c(t;f^{k(\\ell)},p^{\\ell},m^{\\ell})$ as $c^{\\ell}(t)$. By Theorem 1, the domain of $c^*(t)$ and $c_{\\ell}(t)$ includes $[0,1]$, $u_{f,\\bar{p}}(x^*)=c^*(1)$, and $u_{f^{k(\\ell)},\\bar{p}}(x^{\\ell})=c^{\\ell}(1)+m^{\\ell}-m^*$.\n\nChoose $a>0$ and $b>0$ so small, and define\n\\[\\Pi=\\{(c,p,m)|\\exists t\\in [0,1]\\mbox{ s.t. }|c^*(t)-c|\\le a,\\ \\|p-p^*\\|+|m-m^*|\\le b\\},\\]\nand $\\hat{\\Pi}$ as the set of all $g\\in \\mathscr{F}_L$ such that $\\|g-f\\|\\le b$, where\n\\[\\|h\\|=\\sup_{(t,c,p,m)\\in [0,1]\\times \\Pi}\\|h((1-t)p+t\\bar{p},c+m-m^*)\\|.\\]\nDefine $\\tilde{\\Pi}=\\Pi\\times \\hat{\\Pi}$. We assume that $a,b$ are so small that 1) $\\Pi$ is a compact set in $\\mathbb{R}_{++}\\times \\mathbb{R}^n_{++}\\times \\mathbb{R}_{++}$, 2) $\\min\\{c+m-m^*|(c,p,m)\\in \\Pi\\}>0$, 3) there exists $L>0$ such that if $(t.c,p,m,g), (t,c',p,m,g)\\in [0,1]\\times \\tilde{\\Pi}$, then \n\\[\\|g((1-t)p+t\\bar{p},c+m-m^*)-g((1-t)p+t\\bar{p},c'+m-m^*)\\|\\|\\bar{p}-p\\|\\le L|c-c'|,\\]\nand 4) there exists $B>0$ such that if $(t,c,p,m,g)\\in [0,1]\\times \\tilde{\\Pi}$, then\n\\[|I(t,c,g,p,m)-I(t,c,f,p^*,m^*)|\\le B[\\|p-p^*\\|+|m-m^*|+\\|g-f\\|].\\]\nNow, for any sufficiently large $\\ell$, $|m^{\\ell}-m^*|\\le a$, $\\|p^{\\ell}-p^*\\|+|m^{\\ell}-m^*|\\le b$ and $\\|f^{k(\\ell)}-f\\|\\le b$. For such an $\\ell$, define $t_{\\ell}=\\sup\\{t\\in [0,1]|\\forall s\\in [0,t], (c_{\\ell}(s),p^{\\ell},m^{\\ell})\\in \\Pi\\}$. Because $c_{\\ell}(0)=m^*=c^*(0)$, we have that $t_{\\ell}$ is well-defined and positive. If $t\\in [0,t_{\\ell}]$, then\n\\begin{align*}\n&~|c_{\\ell}(t)-c^*(t)|\\\\\n\\le &~\\int_0^t|I(\\tau,c_{\\ell}(\\tau),f^{k(\\ell)},p^{\\ell},m^{\\ell})-I(\\tau,c^*(\\tau),f,p^*,m^*)|d\\tau\\\\\n\\le &~\\int_0^t|I(\\tau,c_{\\ell}(\\tau),f^{k(\\ell)},p^{\\ell},m^{\\ell})-I(\\tau,c^*(\\tau),f^{k(\\ell)},p^{\\ell},m^{\\ell})|d\\tau\\\\\n&+\\int_0^t|I(\\tau,c^*(\\tau),f^{k(\\ell)},p^{\\ell},m^{\\ell})-I(\\tau,c^*(\\tau),f,p^*,m^*)|d\\tau\\\\\n\\le&~\\int_0^t[L|c_{\\ell}(\\tau)-c^*(\\tau)|+B(\\|p^{\\ell}-p^*\\|+|m^{\\ell}-m^*|+\\|f^{k(\\ell)}-f\\|_{C^1})]d\\tau.\n\\end{align*}\nBy Lemma 1, we have for any $t\\in [0,t_{\\ell}]$,\n\\begin{align*}\n|c_{\\ell}(t)-c^*(t)|\\le&~\\frac{B(\\|p^{\\ell}-p^*\\|+|m^{\\ell}-m^*|+\\|f^{k(\\ell)}-f\\|)}{L}(e^L-1)\\\\\n\\equiv&~C(\\|p^{\\ell}-p^*\\|+|m^{\\ell}-m^*|+\\|f^{k(\\ell)}-f\\|)\n\\end{align*}\nfor some $C>0$. Now, choose any $b'\\in ]0,b]$ with $Cb'<a$. If $\\ell$ is sufficiently large, then $\\|p^{\\ell}-p^*\\|+|m^{\\ell}-m^*|+\\|f^{k(\\ell)}-f\\|_{C^1}\\le b'$. For such $\\ell$, we have that $t_{\\ell}=1$: if not, then $a\\le |c_{\\ell}(t_{\\ell})-c^*(t_{\\ell})|\\le Cb'<a$, which is a contradiction. Therefore,\n\\[|c_{\\ell}(1)-c^*(1)|\\le C(\\|p^{\\ell}-p^*\\|+|m^{\\ell}-m^*|+\\|f^{k(\\ell)}-f\\|_{C^1}),\\]\nand thus if $\\ell$ is sufficiently large, then\n\\[|u_{f^{k(\\ell)},\\bar{p}}(x^{\\ell})-u_{f,\\bar{p}}(x^*)|\\le |c_{\\ell}(1)-c^*(1)|+|m^{\\ell}-m^*|<\\frac{\\varepsilon}{2}.\\]\nBy Lemma 4, $u_{f,\\bar{p}}$ is continuous at $x^*$, and thus $|u_{f,\\bar{p}}(x^{\\ell})-u_{f,\\bar{p}}(x^*)|<\\frac{\\varepsilon}{2}$ if $\\ell$ is sufficiently large. Then,\n\\begin{align*}\n\\varepsilon \\le&~|u_{f^{k(\\ell)},\\bar{p}}(x^{\\ell})-u_{f,\\bar{p}}(x^{\\ell})|\\\\\n\\le&~|u_{f^{k(\\ell)},\\bar{p}}(x^{\\ell})-u_{f,\\bar{p}}(x^*)|+|u_{f,\\bar{p}}(x^*)-u_{f,\\bar{p}}(x^{\\ell})|\\\\\n<&~\\varepsilon,\n\\end{align*}\nwhich is a contradiction. This completes the proof. $\\blacksquare$\n\n\n\n\\subsection{Proof of Corollary 4}\nBy Theorem 3, it suffices to show that $\\limsup_{k\\to \\infty}v_{f^k,\\bar{p}}(x)\\le v_{f,\\bar{p}}(x)$ for all $x\\in \\Omega\\setminus \\mathbb{R}^n_{++}$. Choose any $x\\in \\Omega\\setminus \\mathbb{R}^n_{++}$. Let $e=(1,1,...,1)$ and define $x^{\\ell}=x+\\ell^{-1}e$. By the same argument as in the proof of Corollary 2, we can show that $u_{f,\\bar{p}}(x^{\\ell})>v_{f,\\bar{p}}(x)$ and $\\lim_{\\ell\\to \\infty}u_{f,\\bar{p}}(x^{\\ell})=v_{f,\\bar{p}}(x)$. The same fact is true for $v_{f^k,\\bar{p}}$. Now, suppose that $\\limsup_{k\\to \\infty}v_{f^k,\\bar{p}}(x)=a>v_{f,\\bar{p}}(x)$. Because $v_{f^k,\\bar{p}}(x)<u_{f^k,\\bar{p}}(x^1)$ and $u_{f^k,\\bar{p}}(x^1)$ converges to $u_{f,\\bar{p}}(x)$ as $k\\to \\infty$, we have that the sequence $(v_{f^k,\\bar{p}}(x))$ is bounded, and thus $a<+\\infty$. Taking a subsequence, we can assume that $\\lim_{k\\to \\infty}v_{f^k,\\bar{p}}(x)=a$. Let $3\\varepsilon=a-v_{f,\\bar{p}}(x)$. Choose $\\ell$ so large that $u_{f,\\bar{p}}(x^{\\ell})<v_{f,\\bar{p}}(x)+\\varepsilon$. By Theorem 3, for sufficiently large $k$, $v_{f^k,\\bar{p}}(x)\\le u_{f^k,\\bar{p}}(x^{\\ell})<u_{f,\\bar{p}}(x^{\\ell})+\\varepsilon<v_{f,\\bar{p}}(x)+2\\varepsilon$, which is a contradiction. This completes the proof. $\\blacksquare$\n\n\n\\subsection{Proof of Corollary 5}\nFirst, choose any $x\\in \\mathbb{R}^n_{++}$. Then, there exists $\\nu$ such that $x\\in ]\\nu^{-1},\\nu[^n$. Choose $p^k\\in G^{f^k}(x)$ for each $k$. Because $(p^k)$ is a sequence of the compact set\n\\[P_{\\nu}=\\{p\\in \\mathbb{R}^n_{++}|\\sum_ip_i=1,\\ \\min_ip_i\\ge M_{\\nu}\\},\\]\nthere exists a subsequence $p^{\\ell(k)}$ such that $\\lim_{k\\to \\infty}p^{\\ell(k)}=p^*\\in P_{\\nu}$. Because $f^{\\ell(k)}$ converges to $f$ uniformly on any compact set,\n\\[\\lim_{k\\to \\infty}f^{\\ell(k)}(p^{\\ell(k)},p^{\\ell(k)}\\cdot x)=f(p^*,p^*\\cdot x),\\]\nand thus $p^*\\in G^f(x)$. This implies that $R(f)$ includes $\\mathbb{R}^n_{++}$.\n\nSecond, choose any $x\\in ]\\nu^{-1},\\nu[^n$, and suppose that there exists $p\\in G^f(x)$ such that $\\min_ip_i<M_{\\nu}$. Let $x^k=f^k(p,p\\cdot x)$. Then, $x^k\\to x$ as $k\\to \\infty$, and thus $x^k\\in ]\\nu^{-1},\\nu[^n$ for sufficiently large $k$. Because $p\\in G^{f^k}(x^k)$, we have that $f^k\\notin \\mathscr{F}_{L,M}$, which is a contradiction. Therefore, if $p\\in G^f(x)$, then $\\min_ip_i\\ge M_{\\nu}$. Because $G^f(x)$ is obviously closed, this implies that $G^f$ is compact-valued.\n\nThird, it is easy to show that for any demand function $f'$ and $x\\in R(f')$, $G^{f'}(x)$ is convex. Therefore, $G^f$ is convex-valued.\n\nLast, suppose that $G^f$ is not upper semi-continuous at $x$. Then, there exists an open neighborhood $U$ of $G^f(x)$, sequences $(x^{\\ell})$ and $(p^{\\ell})$ such that $x^{\\ell}\\to x$ as $\\ell\\to \\infty$ and $p^{\\ell}\\in G^f(x^{\\ell})\\setminus U$ for all $\\ell$. Choose $\\nu$ such that $x^{\\ell}\\in ]\\nu^{-1},\\nu[^n$ for all $\\ell$. Then, $(p^{\\ell})$ is a sequence in the compact set $P_{\\nu}$, and thus by taking a subsequence, we can assume that $p^{\\ell}\\to p^*\\in P_{\\nu}$ as $\\ell\\to \\infty$. Because $f$ is continuous, we have that $p^*\\in G^f(x)\\subset U$, which is a contradiction. Therefore, $f$ satisfies the C axiom. The rest of the claim of Corollary 5 follows from Theorem 3. This completes the proof. $\\blacksquare$\n\n\n\\section*{Acknowledgement}\nThis work was supported by JSPS KAKENHI Grant Number JP21K01403.\n\n\n\\section*{Reference}\n\\begin{description}\n\\item{[1]} Antonelli, G. B. (1886) Sulla Teoria Matematica dell' Economia Politica. Tipografia del Folchetto, Pisa. Translated by Chipman, J. S. and Kirman, A. P. (1971) On the Mathematical Theory of Political Economy. In: Chipman, J. S., Hurwicz, L., Richter, M. K., Sonnenschein, H. F. (Eds.) Preferences, Utility and Demand. pp.114-148, Harcourt Brace Jovanovich, Inc., New York. pp.333-364.\n\n\\item{[2]} Bentham, J. (1789) An Introduction to the Principles of Morals and Legislation. Reprinted by Batoche Books, Kitchener. \n\n\\item{[3]} Debreu, G. (1954) Representation of a Preference Ordering by a Numerical Function. In: Thrall, R. M., Coombs, C. H., Davis, R. L. (Eds.) Decision Processes. Wiley, New York, pp.159-165.\n\n\\item{[4]} Debreu, G. (1972) Smooth Preferences. Econometrica 40, 603-615.\n\n\\item{[5]} Debreu, G. (1974) Excess Demand Functions. J. Math. Econ. 1, 15-21.\n\n\\item{[6]} Dieudonne, J. (1969) Foundations of Modern Analysis. Academic Press, London.\n\n\\item{[7]} Hartman, P. (1997) Ordinary Differential Equations. 2nd ed. Birkh\\\"auser Verlag AG, Boston.\n\n\\item{[8]} Heinonen, J. (2004) Lectures on Lipschitz Analysis. Lectures at the 14th Jyv\\\"askyl\\\"a Summer School.\n\n\\item{[9]} Hosoya, Y. (2013) Measuring Utility from Demand. J. Math. Econ. 49, 82-96.\n\n\\item{[10]} Hosoya, Y. (2015) A Theory for Estimating Consumer's Preference from Demand. Adv. Math. Econ. 18, 33-55.\n\n\\item{[11]} Hosoya, Y. (2017) The Relationship between Revealed Preference and the Slutsky Matrix. J. Math. Econ. 70, 127-146.\n\n\\item{[12]} Hosoya, Y. (2018) First-Order Partial Differential Equations and Consumer Theory. Discret. Contin. Dyn. Syst. - Ser. S 11, 1143-1167.\n\n\\item{[13]} Hosoya, Y. (2020) Recoverability Revisited. J. Math. Econ. 90, 31-41.\n\n\\item{[14]} Hosoya, Y. (2021) Consumer Optimization and a First-Order PDE with a Non-Smooth System. Oper. Res. Forum 2-66.\n\n\\item{[15]} Houthakker, H. S. (1950) Revealed Preference and the Utility Function. Economica 17, pp.159-174.\n\n\\item{[16]} Hurwicz, L. (1971) On the Problem of Integrability of Demand Functions. In: Chipman, J. S., Hurwicz, L., Richter, M. K., Sonnenschein, H. F. (Eds.) Preferences, Utility and Demand. Harcourt Brace Jovanovich, Inc., New York, 174-214.\n\n\\item{[17]} Hurwicz, L. and Uzawa, H. (1971) On the Integrability of Demand Functions. In: Chipman, J. S., Hurwicz, L., Richter, M. K., Sonnenschein, H. F. (Eds.) Preferences, Utility and Demand. Harcourt Brace Jovanovich, Inc., New York, pp.114-148, \n\n\\item{[18]} Ioffe, A. D. and Tikhomirov, V. M. (1979) Theory of Extremal Problem. North Holland, Amsterdam.\n\n\\item{[19]} Karatzas, I and Shreve, S. E. (1998) Brownian Motion and Stochastic Calculus: 2nd Edition. Springer, New York.\n\n\\item{[20]} Katzner, D. W. (1970) Static Demand Theory. Macmillan, London.\n\n\\item{[21]} Mas-Colell, A. (1977) The Recoverability of Consumers' Preferences from Market Demand Behavior. Econometrica 45, 1409-1430.\n\n\\item{[22]} Mas-Colell, A. (1985) The Theory of Genaral Economic Equilibrium: a Differentiable Approach. Cambridge Universtiy Press, Cambridge.\n\n\\item{[23]} Mas-Colell, A., Whinston, M. D., and Green, J. (1995) Microeconomic Theory. Oxford University Press, Oxford.\n\n\\item{[24]} Nikliborc, W. (1929) Sur les \\'equations lin\\'eaires aux diff\\'erentielles totales. Studia Mathematica 1, 41-49.\n\n\\item{[25]} Pareto, V. (1906) Manuale di Economia Politica con una Introduzione alla Scienza Sociale. Societa Editrice Libraria, Milano.\n\n\\item{[26]} Richter, M. K. (1966) Revealed Preference Theory. Econometrica 34, 635-645.\n\n\\item{[27]} Samuelson, P. A. (1950) The Problem of Integrability in Utility Theory. Economica 17, 355-385.\n\n\\item{[28]} Uzawa, H. (1960) Preferences and Rational Choice in the Theory of Consumption. In: Arrow, K. J., Karlin, S., Suppes, P. (Eds.) Mathematical Models in the Social Science, 1959: Proceedings of the First Stanford Symposium. Stanford University Press, Stanford, pp.129-149. Reprinted in: Chipman, J. S., Hurwicz, L., Richter, M. K., Sonnenschein, H. F. (Eds.) Preferences, Utility, and Demand. Harcourt Brace Jovanovich, Inc., New York. (1971).\n\n\\item{[29]} Walras, L. (1874) El\\'ements d' Economie Politique Pure, ou Theorie de la Richesse Sociale. Lausanne.\n\n\\end{description}\n\n\n\n\n\n\n\n\n\n\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec_introduction}\nThe Cell Formation Problem as a part of Group Technology (GT) was introduced by \\citet{5} and \\citet{4}.\nIn the most general formulation it is designed to reduce production costs by grouping machines and parts into manufacturing cells (production shops).\nThe goal of such kind of grouping is to set up manufacturing process in a way that maximizes loading of machines within the cells and minimizes movement of parts from one cell to another.\nIn classical formulation the problem is defined by a binary matrix $A$ with $m$ rows representing machines and $p$ columns representing parts. In this machine-part matrix $a_{ij} = 1$ if part $j$ is processed on machine $i$.\nThe objective is to form production cells, which consist  of machines and parts together, optimizing some production metrics such as machine loading and intercell movement.\n\nAs an example of input data we will consider the instance of \\citet{1} shown in Table~\\ref{table1}. This instance consists of 5 machines and 7 parts.\nThe ones in a machine-part matrix are called \\textit{operations}.\nIn Table~\\ref{table2} a solution with 2 manufacturing cells is presented. The first manufacturing cell contains machines $m_{1}$, $m_{4}$ with parts $p_{1}$, $p_{7}$ and the second manufacturing cell contains machines $m_{2}$,$m_{3}$,$m_{5}$ with parts $p_{2}$,$p_{3}$,$p_{4}$,$p_{5}$,$p_{6}$. Some parts have to be moved from one cell to another for processing (e.g. part $p_{6}$ needs to be processed on machine $m_{1}$, so it should be transported from its cell 2 to cell 1).\nThe operations lying outside cells are called \\textit{exceptional elements} or \\textit{exceptions}. There can be also non-operation elements inside cells ($a_{ij}$ = 0). These elements reduce machine load and are called \\textit{voids}.\nSo the goal is to minimize the number of exceptions and the number of voids at the same time.\n\n\n\\begin{table}[H]\n\\centering\n        \\begin{tabular}{ c|ccccccc|}\n                  \\multicolumn{1}{c}{~} & \\multicolumn{1}{c}{$p_{1}$ } & \\multicolumn{1}{c}{$p_{2}$ } & \\multicolumn{1}{c}{$p_{3}$ } & \\multicolumn{1}{c}{$p_{4}$ } & \\multicolumn{1}{c}{$p_{5}$ } & \\multicolumn{1}{c}{$p_{6}$ } & \\multicolumn{1}{c}{$p_{7}$} \\\\ \\hhline{~-------}\n                  $m_{1}$ & \\hspace{0.5em}1\\hspace{0.5em}  & \\hspace{0.5em}0\\hspace{0.5em}  &\\hspace{0.5em} 0\\hspace{0.5em}  &\\hspace{0.5em}0\\hspace{0.5em}  &\\hspace{0.5em}1\\hspace{0.5em} &\\hspace{0.5em}1\\hspace{0.5em} &\\hspace{0.5em}1\\hspace{0.5em} \\\\ \n                  $m_{2}$ &0 &1 &1 &1 &1 &0 &0 \\\\ \n                  $m_{3}$ &0 &0 &1 &1 &1 &1 &0 \\\\ \n                  $m_{4}$ &1 &1 &1 &1 &0 &0 &0 \\\\ \n                  $m_{5}$ &0 &1 &0 &1 &1 &1 &0 \\\\ \\hhline{~-------}\n        \\end{tabular}\n        \\caption{Machine-part 5 $\\times$ 7 matrix from \\citet{1}}\n        \\label{table1}\n\\end{table}\n\n\\begin{table}[H]\n\\centering\n        \\begin{tabular}{ c|ccccccc|}\n                  \\multicolumn{1}{c}{~} & \\multicolumn{1}{c}{$p_{7}$ } & \\multicolumn{1}{c}{$p_{1}$ } & \\multicolumn{1}{c}{$p_{6}$ } & \\multicolumn{1}{c}{$p_{5}$ } & \\multicolumn{1}{c}{$p_{4}$ } & \\multicolumn{1}{c}{$p_{3}$ } & \\multicolumn{1}{c}{$p_{2}$} \\\\ \\hhline{~-------}\n                  $m_{1}$ &\\hspace{0.5em}\\cellcolor{yellow}1\\hspace{0.5em}  & \\cellcolor{yellow}\\hspace{0.5em}1\\hspace{0.5em}  &\\hspace{0.5em}1\\hspace{0.5em}  &\\hspace{0.5em}1\\hspace{0.5em}  &\\hspace{0.5em}0\\hspace{0.5em} &\\hspace{0.5em}0\\hspace{0.5em} &\\hspace{0.5em}0\\hspace{0.5em} \\\\ \n                  $m_{4}$ &\\cellcolor{yellow}0 &\\cellcolor{yellow}1 &0 &0 &1 &1 &1 \\\\ \n                  $m_{2}$ &0 &0 &\\cellcolor{yellow}0 &\\cellcolor{yellow}1 &\\cellcolor{yellow}1 &\\cellcolor{yellow}1 &\\cellcolor{yellow}1 \\\\ \n                  $m_{3}$ &0 &0 &\\cellcolor{yellow}1 &\\cellcolor{yellow}1 &\\cellcolor{yellow}1 &\\cellcolor{yellow}1 &\\cellcolor{yellow}0 \\\\ \n                  $m_{5}$ &0 &0 &\\cellcolor{yellow}1 &\\cellcolor{yellow}1 &\\cellcolor{yellow}1 &\\cellcolor{yellow}0 &\\cellcolor{yellow}1 \\\\ \\hhline{~-------}\n        \\end{tabular}\n        \\caption{Solution with 2 production cells}\n        \\label{table2}\n\\end{table}\n\n\n\\newpage\n\\subsection {Related work}\nMany different approaches are proposed for solving the cell formation problem. The majority of them provide heuristic solutions and only a few exact methods have been suggested.\n\n\\citet{2} provided two MINpCUT exact models based on the well-known k-cut graph partition problem. The objective function considered in this research is minimization of the exceptional elements number for a fixed number of cells. Unfortunately this objective function does not address the load inside cells.\n\n\\citet{3} presented a mixed-integer linear programming model which maximizes the most popular objective for the cell formation problem - the grouping efficacy, introduced by ~\\citet{6}. These authors suggested to apply Dinkelbach algorithm since the grouping efficacy is a fractional objective function. Their model allows solving the cell formation problem only if the number of production cells is predefined. Thus the suggested approach cannot guarantee global optimality of the obtained solutions with respect to a variable number of production cells. In many cases the computational times for this model are quite long or memory limitations are exceeded and the optimal solutions cannot be found.\n\n\\citet{7} introduced two approaches for solving the cell formation problem with the grouping efficacy objective. The first is a mixed-integer linear programming model which is based on a general two-mode clustering formulation with some simplifying assumptions (e.g. the numbers of clusters by rows and columns are equal). This model looks interesting, but requires too much time to be solved for many even medium-sized instances. The second approach is a branch-and-bound algorithm combined with a relocation heuristic to obtain an initial solution. The branch and bound approach is able to solve about two times more problem instances and the computational times are greatly improved as well. Generally it runs fine on well-structured (with grouping efficacy value $>$ 0.65 - 0.7) medium-sized problems.\nTwo major assumptions are made for both of these approaches: singletons are permitted (manufacturing cells containing only one machine or one part) that is quite a common practice;  residual cells are permitted (cells containing only machines without parts, or only parts without machines). Also the number of production cells is predefined for the both approaches, but for some test instances several values for the number of cells are considered in computational experiments.\n\nAnother model is provided in our earlier paper \\citep{8}. There we present a mixed-integer linear programming formulation for the cell formation problem with a variable number of production cells. It deals well with small-sized instances,  but nevertheless the number of variables and constraints is huge -  O($m^2p$). This does not allow obtaining solutions even for some moderate-sized test instances and in some cases this model runs too slowly. \n\nSome authors used biclustering approaches to solve the cell formation problem.\n\\citet{64} applied simultaneous clustering for both dimensions (machines and parts) and minimized the number of voids plus the number of exceptional elements. \\citet{62} reduced the cell formation problem to another biclustering problem - bicluster graph editing problem and suggested an exact method and a linear programming model which provides good computational results for the grouping efficacy objective.\n\n\\subsection {Contributions of this research}\n\nIn this paper we develop a fast compact model providing optimal solutions for the cell formation problem with a variable number of manufacturing cells and the grouping efficacy objective. Unlike the majority of linear programming models our model does not contain a direct assignment of machines or parts to cells. We use  machine-machine and part-machine assignments instead of the widely used machine-part-cell assignment. This leads to a compact and elegant formulation considering only constraints which ensure a block-diagonal structure of solutions. It allows us to drastically reduce the number of variables and constraints in our programming model and obtain globally optimal solutions even for some large-sized problem instances.\n\nComputational experiments show that our model outperforms all known exact methods. We have solved 63 of 67 problem instances to the global optimum with respect to a variable number of production cells. We have also found several new solutions unknown before.\n\nWe would like to highlight that many researchers in the field use the 35 GT instances dataset provided by \\citet{9}. These instances are taken from different cell formation research papers (references to the original sources are shown in Table \\ref{testset_a_problems}). Some problem instances in this 35 GT dataset have errors and differ from the ones presented in the original papers. Many researchers including \\citet{3} and \\citet{62} have performed their computational experiments using these data from \\citet{9}. We have reviewed all the original sources, comparing and forming the corrected version of this popular dataset. We have also collected many other problem instances less popular and formed a new dataset. All data can be downloaded from website \\href{http:\/\/opt-hub.com\/problems\/cfp}{opt-hub.com} or \\href{https:\/\/researchgate.net\/publication\/316648108_Test_instances_and_solutions_for_the_cell_formation_problem}{researchgate.net} (full urls can be found in references).\n \n\nThe paper is organized as follows. In Section \\ref{sec_formulation} we provide the formulation of the cell formation problem.\nThen in Section \\ref{sec_model} we present our new mixed-integer linear programming model.\nSections \\ref{sec_datasets} contains the information about datasetes we used for our experiments and  the computational results and comparisons to other exact approaches are given in Section \\ref{sec_results}.\nThe conclusion is provided in Section \\ref{sec_conclusion}.\n\\label {subsec_contrib}\n\n\\section{Problem formulation}\n\\label{sec_formulation}\nCellular manufacturing systems apply are aimed to process similar parts within the same production cell, balance machines workload and minimize parts movement from one cell to another during the production process. The most popular objective for the cell formation problem is the grouping efficacy introduced by \\citet{6}:\n\n\\[\n\\tau =\\frac{n_1^{in} }{n_1 +n_0^{in}\n}\n\\] \nwhere\n\n$n_1$ -- the total number of operations (ones) in the machine-part matrix,\n\n$n_1^{in}$ -- the number of operations performed inside cells,\n\n$n_0^{in}$ -- the number of voids (zeros inside cells). \\newline\n\nIn comparisson to the other objectives the grouping efficacy function  addresses the best block-diagonal structure of the cell formation problem solutions \\citep{63}.\n\n\nIn the literature several constraints related to the minimal size of a cell could be found. The following are the three most popular consideratons:\n\n\\begin{itemize}\n  \\item allowing residual cells (cells containing only machines or parts)\n  \\item allowing singletons (cells with one machine and several parts or vice versa) and prohibiting residual cells \n  \\item allowing only cells with at least 2 machines and 2 parts \\newline\n\\end{itemize}\n\nThe most popular option is allowing singletons and prohibiting residual cells. In this section for the classical formulation we assume that singletons can appear in solutions and residual cells are prohibited. In our new model and in computational experiments we consider the first two options. \n\nA straightforward integer fractional programming (IFP) model for the cell formation problem with the grouping efficacy objective function  allowing singletons and prohibiting residual cells is given below.\nWe use the following notation: $m$ is the number of machines, $p$ is the number of parts, $a_{ij}$ equals to 1 if machine $i$ processes part $j$ and $c$ is the maximal possible number of production cells. Since each production cell has to contain at least one machine and at least one part we set $c  = \\min(m,p)$. \\newline \\\\\n\\text{(IFP model):}\\\\\n\nDecision variables: \\newline\n\\[\n x_{ik} = \n  \\begin{cases} \n   1,  & \\text{if machine $i$ belongs to cell $k$,} \\\\\n   0,        & \\text{otherwise }\n  \\end{cases}\n\\] \n\\[\n y_{jk} = \n  \\begin{cases} \n   1, & \\text{if part $j$ belongs to cell $k$, \\quad} \\\\\n   0,           & \\text{otherwise }\n  \\end{cases}\n\\]\\newline\n\\begin{equation}\n\\label{general_objective}\n  max  \\quad \\frac{\\sum_{i=1}^{m}\\sum_{j=1}^{p}\\sum_{k=1}^{c} a_{ij}x_{ik}y_{jk}}{\\sum_{i=1}^{m}\\sum_{j=1}^{p}a_{ij} +\\sum_{i=1}^{m}\\sum_{j=1}^{p}\\sum_{k=1}^{c} (1 - a_{ij})x_{ik}y_{jk}}\n\\end{equation} \n\\\\\n\n\\text{Subject to:}\n\\\\\n\\begin{equation}\n\\label{eq:machines_assigned}\n\\sum_{k=1}^{c} {x_{ik}}  = 1  \\quad  i = 1,...,m\n\\end{equation} \n\n\\begin{equation}\n\\label{eq:parts_assigned}\n\\sum_{k=1}^{c} {y_{jk}}  = 1  \\quad j = 1,...,p\n\\end{equation} \n\n\\begin{equation}\n\\label{eq:no_zerotones_1}\n\\sum_{i=1}^{m}x_{ik} \\leq m\\cdot\\sum_{j=1}^{p}y_{jk}  \\quad k = 1,...,c\n\\end{equation} \n\n\\begin{equation}\n\\label{eq:no_zerotones_2}\n\\sum_{j=1}^{p}y_{jk} \\leq p\\cdot\\sum_{i=1}^{m}x_{ik}  \\quad k = 1,...,c\n\\end{equation} \n\nObjective function \\eqref{general_objective} is the grouping efficacy measure where the numerator is the number of ones inside cells ($n_1^{in}$) and two sums in the denominator are the total number of ones ($n_1$) and the number of zeros inside cells ($n_0^{in}$) respectively.\nConstraints \\eqref{eq:machines_assigned} and \\eqref{eq:parts_assigned} require that each machine and each part is assigned to exactly one production cell.\nThe following two inequalities \\eqref{eq:no_zerotones_1} and  \\eqref{eq:no_zerotones_2} prohibit residual cells (without machines or  parts). The left part of \\eqref{eq:no_zerotones_1} is the total number of machines assigned to the particular cell (this sum is not greater than $m$) and the right part is the total number of parts assigned to that cell (multiplied by $m$). It means that if we have at least one machine assigned to some cell there should be at least one part assigned to this cell. This model allows us to have any number of cells in the optimal solution. For example if optimal solution has only two cells then variables \n$x_{ik}$ and $y_{jk}$ will be zero for all $k$ except only two values of $k$.\n\n\n\\section{MILP model}\n\\label{sec_model}\n\\subsection{Objective linearization}\nIn our paper ~\\citet{8} we have proposed a mixed-integer linear programming model for the cell formation problem which is very similar to the one described in the previous section. One of the most important points there was linearization of the grouping efficacy objective. Our previous idea was to linearize the grouping efficacy objective function by fixing the value of denominator $n_1 +n_0^{in}$ and considering a range of all possible numbers of voids $n_0^{in}$. The lower bound for $n_0^{in}$ equals to 0 because generally there can be a solution without any voids. The upper bound is computed using the following proposition.\n\n\\newtheorem*{thm*}{Proposition 1 \\citep{8}}\n\\begin{thm*}\nThe number of voids in the optimal solution satisfies the following inequality:\n\\[\nn_0^{in} \\leq \\left\\lfloor \\frac{1 - \\tau}{\\tau}n_{1} \\right\\rfloor\n\\]\nwhere $\\tau$ is the grouping efficacy value of any feasible solution.\n\\end{thm*}\n\nSo if we know a feasible solution we can limit the range of possible values for the number of voids. Unfortunately, the performance of this approach strongly depends on the feasible solution we use for \nobtaining our bounds. This way solving problem instances where grouping efficacy value is low takes too much computational resources (since the number of subtasks is too large) and sometimes we are unable to solve even medium-sized cell formation instances.\n\nIn this paper together with using our new mixed-integer linear model we use another way of linearization -- \\citet{65} algorithm.\nThe parametric approach introduced by W.Dinkelbach is one of the most general and popular strategies for fractional programming. It reduces the solution of a fractional programming problem to the solution of  a sequence of simpler problems.\nIf we consider a fractional programming model with the following objective function:\n\n\\begin{equation}\n\\label{fract-lin-objective}\nQ(x) = \\frac{P(x)}{D(x)},\n\\end{equation} \nthen Dinkelbach procedure is the following: \n\n\\begin{itemize}\n  \\item \\textbf{Step 1} take some feasible solution $x^{0}$, compute $\\lambda_{1} = \\frac{P(x^{0})}{D(x^{0})}$ and  let $k=1$\n  \\item \\textbf{Step 2} solve the original problem with objective function $Q(x)$ replaced with  $F(\\lambda_{k}) =  P(x) - \\lambda_{k}D(x) \\rightarrow max$\nand let $x^{k}$ be the optimal solution\n  \\item \\textbf{Step 3} If $F(\\lambda_{k})$ is equal to $0$ (or less than some predefined tolerance) then stop the procedure and return $x^{k}$ as the optimal solution. \\newline\n  Else $ k = k+1, \\lambda_{k} = \\frac{P(x^{k})}{D(x^{k})}$ and goto step 2.\n\\end{itemize}\n\n \\citet{3} have also used Dinkelbach approach for linearization of grouping efficacy measure. Although their computational times are quite high and the results are given only for the particular fixed number of production cells.\n\n\\subsection{Suggested two-index model}\n\\label{subsec_2ind}\nDue to a large number of variables and constraints in three-index model \\citep{8} CPLEX spends too much computational resources solving even small-sized instances (we have solved only 14 of 35 problem instances).\nHere we introduce a two-index mixed-integer linear programming model for the cell formation problem. The key idea of this model is removing machine-part-cell \nrelation as it has been done in many works before. Instead of mapping elements to cells we use a simple fact that machines within the same production cell have the same set of parts assigned to that cell. The two-index model combines well with the Dinkelbach algorithm and shows impressing performance even on large-sized problem instances.\n\nTwo-index model:\n\\[\n x_{ik} = \n  \\begin{cases} \n   1,  & \\text{if machines $i$ and $k$ are in the same cell,\\quad } \\\\\n   0,        & \\text{otherwise }\n  \\end{cases}\n\\] \n\\[\n y_{ij} = \n  \\begin{cases} \n   1, & \\text{if machine $i$ and part $j$ are in the same cell,} \\\\\n   0,           & \\text{otherwise }\n  \\end{cases}\n\\]\n\\\\\n\\begin{equation}\n\\label{2IND_obj}\nmax \\quad\\sum_{i=1}^{m} \\sum_{j=1}^p a_{ij}y_{ij} - \\lambda \\cdot(\\sum_{i=1}^{m} \\sum_{j=1}^p (1-a_{ij})y_{ij} + \\sum_{i=1}^{m} \\sum_{j=1}^p a_{ij})\n\\end{equation} \n\\text{Subject to:}\n\\\\\n\\begin{equation}\n\\label{2IND_rect1}\n2x_{ik} - y_{ij} - y_{kj} \\geq -1 \\quad  i,k = 1,\\dots,m  \\quad  j = 1,\\dots,p \n\\end{equation} \n\\begin{equation}\n\\label{2IND_rect2}\ny_{ij} - y_{kj} - x_{ik} \\geq -1 \\quad  i,k = 1,\\dots,m  \\quad  j = 1,\\dots,p  \n\\end{equation}\n\\begin{equation}\n\\label{2IND_rect3}\ny_{kj} - y_{ij} - x_{ij} \\geq -1 \\quad  i,k = 1,\\dots,m  \\quad  j = 1,\\dots,p  \n\\end{equation}\n\\begin{equation}\n\\label{2IND_machine_with_part}\n\\sum_{j=1}^{p}y_{ij} \\geq 1 \\quad  i = 1,...,m\n\\end{equation} \n\\begin{equation}\n\\label{2IND_part_with_machine}\n\\sum_{i=1}^{m}y_{ij} \\geq 1 \\quad  j = 1,...,p\n\\end{equation}\n\nTechnically matrix $[x_{ik}]$ here can be replaced by the one with part-part relations, however the number of machines in problem instances is usually lower than the number of parts (for large-sized instances the difference is significant). As a result we have $m^2$ variables from matrix $[x_{ik}]$ and $mp$ variables from matrix $[y_{ij}]$.\n\nObjective function \\eqref{2IND_obj} is the grouping efficacy measure linearized using Dinkelbach method. Constraints \\eqref{2IND_rect1}, \\eqref{2IND_rect2}, \\eqref{2IND_rect3} set relations between machines and parts to ensure the solution can be transformed into the block-diagonal form (which means its feasibility). The last two inequalities \\eqref{2IND_machine_with_part} and \\eqref{2IND_part_with_machine} are optional and prohibit residual cells. \n\nWe start with setting $\\lambda$ equal to the best known efficacy value for the considered cell formation problem instance. Then we sequentially solve several two-index problems according to the Dinkelbach algorithm and update $\\lambda$ value with the solutions found until our objective function is above 0.\n\\begin{table}[H]\n  \\centering\n  \\scriptsize\n  \\caption{Testset A - Instances}\n    \\resizebox{\\columnwidth}{!}{%\n    \\begin{tabular}{cccc}\n    \\toprule\n    \\multirow{2}[2]{*}{\\textbf{ID}} & \\multirow{2}[2]{*}{\\textbf{Source}} & \\multirow{2}[2]{*}{\\textbf{m}} & \\multirow{2}[2]{*}{\\textbf{p}} \\\\\n         &      &      &  \\\\\n   \\midrule\n    A1   & \\citet{13} - Figure 1a& 5    & 7 \\\\\n    A2   & \\citet{14} - Problem 2 & 5    & 7 \\\\\n    A3   & \\citet{15} & 5    & 18 \\\\\n    A4   & \\citet{16} & 6    & 8 \\\\\n    A5   & \\citet{17} & 7    & 11 \\\\\n    A6   & \\citet{18} - Example 1 & 7    & 11 \\\\\n    A7   & \\citet{19} & 8    & 12 \\\\\n    A8   & \\citet{20} & 8    & 20 \\\\\n    A9   & \\citet{21} & 8    & 20 \\\\\n    A10  & \\citet{22} & 10   & 10 \\\\\n    A11  & \\citet{24} & 15   & 10 \\\\\n    A12  & \\citet{25} & 14   & 24 \\\\\n    A13  &\\citet{26} & 14   & 24 \\\\\n    A14  & \\citet{27} & 16   & 24 \\\\\n    A15  & \\citet{10} & 16   & 30 \\\\\n    A16  &\\citet{28} & 16   & 43 \\\\\n    A17  & \\citet{29} & 18   & 24 \\\\\n    A18  & \\citet{23} & 20   & 20 \\\\\n    A19  & \\citet{30} & 23   & 20 \\\\\n    A20  & \\citet{29} & 20   & 35 \\\\\n    A21  & \\citet{31} & 20   & 35 \\\\\n    A22  & \\citet{32} - Dataset 1& 24   & 40 \\\\\n    A23  & \\citet{32} - Dataset 2& 24   & 40 \\\\\n    A24  & \\citet{32} - Dataset 3& 24   & 40 \\\\\n    A25  & \\citet{32} - Dataset 5& 24   & 40 \\\\\n    A26  & \\citet{32} - Dataset 6& 24   & 40 \\\\\n    A27  & \\citet{32} - Dataset 7& 24   & 40 \\\\\n    A28  & \\citet{27} & 27   & 27 \\\\\n    A29  & \\citet{29} & 28   & 46 \\\\\n    A30  &\\citet{33} & 30   & 41 \\\\\n    A31  & \\citet{26} - Figure 5 & 30   & 50 \\\\\n    A32  & \\citet{26} - Figure 6 & 30   & 50 \\\\\n    A33  &\\citet{34} & 30   & 90 \\\\\n    A34  & \\citet{27} & 37   & 53 \\\\\n    A35  & \\citet{35} & 40   & 100 \\\\\n    \\bottomrule\n    \\end{tabular}%\n    }\n  \\label{testset_a_problems}%\n\\end{table}%\n\\section{Test instances}\n\\label{sec_datasets}\n\nFor our computational experiments we have used two datasets, \\textit{Testset A} and \\textit{Testset B}.\n  \n\\textbf {Testset A - Classic}. The first dataset is a classical 35 GT problem set taken from \\citet{9}. It contains 35 test instances with\nsizes from $5 \\times 7$ up to $40 \\times 100$ (machines $\\times$ parts notation). This dataset is very popular among cell formation researchers and there are lots of computational results obtained by different methods (heuristics and metaheuristics generally). As we highlighted before some problem instances in this dataset have inconsistencies with the original papers they are published in. We have compared these instances to the original ones and corrected the dataset. \n\n\n\\textbf{Testset B - Extra}.\nAnother dataset named \\textit{Testset B} consists of other instances taken from different papers. We have looked through many papers on the cell formation problem and formed this new set. There are 32 test instances with sizes from $6 \\times 6$ to $50 \\times 150$. A couple of instances from this set have been adopted to the classical formulation of the cell formation problem.\n\nSince the number of machines determines the size of our model  (the number of decision variables and constraints) we consider 3 classes of problem instances.\n\\begin{itemize}\n  \\item small (less than 10 machines)\n  \\item medium (from 10 to 20 machines)\n  \\item large (20 machines or greater)\n\\end{itemize}\n\nFor our data we can conclude that Testset A has 2 times more large instances, but  less medium and small instances (see Table~\\ref{instances_sizes}).\n\n\n\n\\begin{table}[htbp]\n  \\centering\n  \\normalsize\n  \\caption{Testsets instances size}\n    \\begin{tabular}{cccc}\n    \\toprule\n         & small & medium & large \\\\\n    \\midrule\n    Testset A & 9    & 8    & 18 \\\\\n    Testset B & 11   & 13   & 8 \\\\\n    \\bottomrule\n    \\end{tabular}%\n  \\label{instances_sizes}%\n\\end{table}%\n\n\n\\section{Computational results}\n\\label{sec_results}\nFor our computational experiments we consider two most popular versions of cell size constraints:\n\\begin {enumerate}\n\\item Residual cells are prohibited, singletons are allowed (each cell has at least 1 machine and 1 part)\n\\item Residual cells are allowed (cells with only machines or only parts can appear in the final solution)\n\\end {enumerate}\n\n\nSeveral state-of-art exact approaches have been chosen for comparisons. \nAs a platform for our computations we have used  a system with Intel Xeon processor running at 3.4 GHz with 16GB RAM and CPLEX 12.4.0 installed.\nDue to high-quality initial solutions the Dinkelbach algorithm makes only one or, in rare cases, two iterations.\n\\begin{table}[H]\n  \\scriptsize\n  \\caption{Testset B - Instances}\n    \\resizebox{\\columnwidth}{!}{%\n    \\begin{tabular}{cccc}\n    \\toprule\n    \\multirow{2}[2]{*}{\\textbf{ID}} & \\multirow{2}[2]{*}{\\textbf{Source}} & \\multirow{2}[2]{*}{\\textbf{m}} & \\multirow{2}[2]{*}{\\textbf{p}} \\\\\n         &      &      &  \\\\\n   \\midrule\n    B1    & \\citet{39} & 6    & 6 \\\\\n    B2    &\\citet{40} & 6    & 7 \\\\\n    B3    & \\citet{41} & 6    & 11 \\\\\n    B4   & \\citet{24} & 7    & 5 \\\\\n    B5   & \\citet{17} & 7    & 8 \\\\\n    B6   & \\citet{42} & 7    & 8 \\\\\n    B7   & \\citet{44} & 7    & 10 \\\\\n    B8   & \\citet{45} & 8    & 8 \\\\\n    B9  & \\citet{46} & 8    & 10 \\\\\n    B10   &\\citet{47} & 8    & 10 \\\\\n    B11   & \\citet{30} & 9    & 15 \\\\\n    B12   & \\citet{49} & 10   & 8 \\\\\n    B13   & \\citet{50} & 10   & 12 \\\\\n    B14   &\\citet{43} & 10   & 38 \\\\\n    B15   & \\citet{44} & 11   & 10 \\\\\n    B16   & \\citet{51} & 11   & 22 \\\\\n    B17   & \\citet{53} & 12   & 19 \\\\\n    B18   & \\citet{37} & 14   & 14 \\\\\n    B19   & \\citet{24} - Fig.3a & 15   & 10 \\\\\n    B20   & \\citet{12} - Fig.6b & 15   & 15 \\\\\n    B21   & \\citet{12} - Fig.6c & 15   & 15 \\\\\n    B22   & \\citet{12} - Fig.6d & 15   & 15 \\\\\n    B23   & \\citet{54} & 17   & 20 \\\\\n    B24   & \\citet{55} & 18   & 24 \\\\\n    B25   & \\citet{58} & 20   & 10 \\\\\n    B26   & \\citet{59} & 20   & 51 \\\\\n    B27   & \\citet{44} & 26   & 28 \\\\\n    B28   & \\citet{12} - Fig.7 & 28   & 35 \\\\\n    B29   & \\citet{60} & 35   & 15 \\\\\n    B30   & \\citet{60} & 41   & 50 \\\\\n    B31   & \\citet{12} - Fig.12 & 46   & 105 \\\\\n    B32   & \\citet{61} & 50   & 150 \\\\\n    \\bottomrule\n    \\end{tabular}%\n    }\n  \\label{testset_b_problems}%\n\\end{table}%\n\n\\subsection{Testset A}\n\\subsubsection{Experiments}\nThe instances from Table~\\ref{testset_a_problems} have been studied widely in the literature.\nWe report results separately for the formulation where minimal cell size is $1 \\times 1$ (Table~\\ref{testset_a_single} and Figure~\\ref{testset_a_single_running_times}) and the formulation with residual cells allowed (Table~\\ref{testset_a_residual}  and Figure~\\ref{testset_a_residual_running_times}).\nIn the first case we have selected two approaches for the results comparison:\n\\begin{enumerate}\n  \\item The  MILP model by \\citet{3}\n  \\item  The MILP model by \\citet{8}\n\\end{enumerate}\n\\citet{3} considered a simplified formulation of the cell formation problem solving every problem instance only for one fixed number of production cells. \nThese authors have performed computational experiments on an AMD processor 2.2 GHz with 4GB RAM.\nFor Testset A we use the best efficacy results from the literature as initial values for $\\lambda$ parameter.\n\nIn case of unrestricted cell sizes (residual cells are allowed) we have compared our results to the following approaches:\n\\begin{enumerate}\n  \\item The branch-and-bound algorithm by \\citet{7}\n  \\item The ILP model by \\citet{62}\n  \\item The iterative exact method by \\citet{62}\n\\end{enumerate}\n\n\n\\citet{7} considers several values for the number of cells for some problem instances, so in this case we compare our computational time with these times summed up for every test instance. \nAs hardware platforms \\citet{7} reports 3.4 GHz Intel Core i7-2600 with 8GB RAM and \\citet{62} the same 3.4 GHz Intel Core i7-2600  with 32 GB RAM.\n\nSince \\citet{3} and \\citet{7} do not consider all possible numbers of production cells we show the number of cells in brackets for these approaches.\n\n\\subsubsection{Results}\nThe results for Testset A are summarized in Table~\\ref{testset_a_single} and Table~\\ref{testset_a_residual}. For each algorithm we report  the grouping efficacy value and the running time in seconds. Since our testset is larger than the one used by \\citet{7} the missing results are marked as ''-''. For some problems exact solutions have not been obtained because CPLEX runs too long or takes too much memory. These instances are marked as \"*\". \n\\begin{figure}[H]\n    \\centering\n    \\includegraphics[scale=0.35]{single_merged_3.png}\n    \\caption{Testset A - No residual cells. Running times comparison.}\n    \\label{testset_a_single_running_times}\n\\end{figure}\nTable~\\ref{testset_a_single} shows the results for the case where we prohibit cells without machines or parts. Our previous model from \\citet{8} also considers a variable number of production cells, but due to its complexity and not very effective linearization technique it is able to solve only 14 test problems of 35. The model suggested by \\citet{3} solved 27 problem instances but only for the one fixed number of production cells for each problem instance. Our new model provides global optimal solutions (with respect to any possible number of cells) for 31 of 35 problem instances.  For problem instance A33 we have found a new solution with grouping efficacy 0.48 unknown before.\n\nFor 17 instances: A14-A21, A23-A26, A28, A30, A31, A34 and A35 we are the first to prove the global optimality of the best known solutions with respect to a variable number of production cells.\n\n\\begin{figure}[H]\n    \\centering\n    \\includegraphics[scale=0.35]{residual_merged_3.png}\n    \\caption{Testset A - Allowed residual cells. Running times comparison.}\n    \\label{testset_a_residual_running_times}\n\\end{figure}\n\nRunning times bar chars for Table~\\ref{testset_a_single} are presented in Figure~\\ref{testset_a_single_running_times}. Here we have used logarithmic scale with base 10 for Y axis (running time). Our new model shows really good performance, it works from 7 to 43383 times faster than the model from \\citet{3} and from 11 to 1833 times faster than the model from \\citet{8}.  We must underline that although we use a better hardware platform than \\citet{3}, our problem formulation is more complicated than a formulation with a fixed number of cells.\n\nThe results for the formulation with no constraints on cell sizes are summarized in Table~\\ref{testset_a_residual}. The model suggested by \\citet{62} solved 27 problem instances to the global optimum. Our approach has obtained exact solutions for 32 of 35 test instances. In addition for problem instances A18, A33 and A34 we have found new solutions unknown before.\n\nRunning times bar charts for Table~\\ref{testset_a_residual} are presented in Figure~\\ref{testset_a_residual_running_times}. Here we have chosen the ILP model from \\citet{62} for comparison since it has a better performance than the exact iterative method of the same authors. In Figure~\\ref{testset_a_residual_running_times} we have also used logarithmic scale with base 10 for the first and second plots (instances with running times less than 60 seconds and less than 5000 seconds). For the last plot (instances with running times less than 1500000 seconds) we have used logarithmic scale with base 100. We can see that  the two-index model runs up to 1 million times faster than the branch-and-bound algorithm by \\citet{7} and up to 161 times faster than the ILP  model by \\citet{62}.\n\n\n\n\\subsubsection{Inconsistencies}\nThe classical dataset of 35 GT problems from \\citet{9} have been used for many years by the cell formation researchers for computational experiments and results comparison.\nUnfortunately, the dataset contains several inconsistencies with the original sources: papers from \\citet{13} to \\citet{35} (see Table~\\ref{testset_a_problems}). Many researchers have used corrupted instances and sometimes add some new inconsistencies. Therefore obtaining results for these problems and comparing it to results of other approaches becomes a really difficult task. One of the goals of this paper is to provide correct data for the cell formation researchers. In this paper we mark usage of inconsistent data with superscript \\textsuperscript{E}. \n\nWe have not been able to obtain results reported in \\citet{3} for problem instances A15 and A31 using both possible data sets - dataset from  \\citet{9} and our corrected version. Probably some other data have been used.\n\n\\begin{table}[h]\n \\centering\n \\captionsetup{font=small}\n \\setlength{\\tabcolsep}{4pt}\n  \\small\n  \\caption{Computational experiments on the data provided by \\citet{9} }\n    \\begin{tabular}{|c|rr|cc|}\n    \\toprule\n   \n    \\multicolumn{1}{|c|}{\\multirow{2}{*}{\\textbf{\\#}}}  &  \\multicolumn{2}{c|}{\\textbf{Time, sec}} & \\multicolumn{2}{c|}{\\textbf{Efficacy}} \\\\\n   \n    \\multicolumn{1}{|c|}{} &   \\multicolumn{1}{c|}{Pinheiro et al.} &  \\multicolumn{1}{c|}{two-index} &  \\multicolumn{1}{c|}{Pinheiro et al.} &  \\multicolumn{1}{c|}{two-index}\\\\\n  \n    \\multicolumn{1}{|c|}{} &  \\multicolumn{1}{c|}{(2016)} &   \\multicolumn{1}{c|}{} & \\multicolumn{1}{c|}{(2016)} &   \\multicolumn{1}{c|}{}\\\\\n\n    \\midrule\n     A1 & 0.01  &  0.01  &  0.7500 & 0.7500 \\\\\n     A7 & 0.03   &  0.01  & 0.6944 & 0.6944  \\\\\n    A14 & 144.91  &  4.99  &  0.5333  &  0.5333 \\\\\n    A15 &  0.54  &  0.17  &  0.6992 & 0.6992 \\\\\n    A17 & 42.32  &  3.51  & 0.5773  & 0.5773 \\\\\n    A20 & 14.55 &  0.11  & 0.7938  & 0.7938\\\\\n    A21 & 305.48  & 15.08 & 0.5879 & 0.5879  \\\\\n    A25 & 48743.90 & 678.53 &  0.5329  &   0.5329 \\\\\n    A30 & 41.53 & 8.58  &  0.6304 &  0.6304 \\\\\n   \\bottomrule\n    \\end{tabular}  \n  \\label{table_inconsistent_results}%\n\\end{table}\n\nSeveral instances provided by \\citet{9}, which are different from its original sources  (papers from \\citet{13} to \\citet{35}, see Table~\\ref{testset_a_problems}),  have been also used by \\citet{62}.\nThese instances are A1, A7, A14, A15, A17, A20, A21, A25 and A30. For a fair comparison we have also run our model using the same input data (see Table ~\\ref{table_inconsistent_results}).\nOur experiments have confirmed all the results obtained by \\citet{62}. Also we can conclude that the running times of our model have not changed much on these input data.\n\n\n\n\\subsection{Testet B results}\n\nSince the test instances from Table~\\ref{testset_b_problems} are less popular in research papers our goal is just to obtain optimal solutions for this set.\nWe have used our multi-start local search heuristic \\citep{11} to get good solutions which are then passed as initial values for $\\lambda$ parameter (we pick the best solution found by the heuristic within 30 seconds). \n\nThe results for Testset B are shown in Table~\\ref{testset_b_results}. Here we have found optimal solutions for 31 of 32 test problems.\nAnother result is an excellent performance of our multi-start local search heuristic algorithm: only one of 32 instances solved by the heuristic differs from the exact solution (instance B18). Due to the high computational complexity we are unable to solve the largest problem in the set -- problem B32 (50 $\\times$ 150).\n\n\\section{Conclusion}\n\\label{sec_conclusion}\nThe cell formation problem is a well known combinatorial optimization problem with a  high computational complexity. \nA very few authors have suggested exact approaches for the most popular problem formulation with the grouping efficacy objective function. The majority of these works assume that the number of production cells is predefined. In this paper we suggest a new compact and effective integer linear programming model for the cell formation problem with a variable number of production cells. The model is based on the machine-machine and part-machine relations instead of the widely used machine-part-cell relation. It allows us to drastically reduce the number of variables and constraints in the resulting integer linear program.\nComputational experiments show that our new model outperforms the state-of-art exact methods. We have solved 63 of 67 problem instances to the global optimum with respect to a variable number of production cells. We have also found several new solutions unknown before.\nUnfortunately many problem instances from the cell formation datasets have inconsistencies with the original papers. This makes it really difficult to perform computational experiments and compare results to other approaches in the field. We have extracted and checked over 67 problem instances. All these data are available for downloading from website \n \\href{http:\/\/opt-hub.com\/problems\/cfp}{opt-hub.com} or \\href{https:\/\/www.researchgate.net\/publication\/316648108_Test_instances_and_solutions_for_the_cell_formation_problem}{researchgate.net} and we hope it will help the researchers in this area. \n The suggested model can be also used for solving biclustering problems and this is one of the directions of our future work.\n\n\n\\section{Acknowledgments}\nThis work was conducted at National Research University Higher School of Economics, Laboratory of\nAlgorithms and Technologies for Network Analysis and supported by RSF grant 14-41-00039.\\newline\n\\label{sec_ack}\n\n\n \n\n\n\\begin{table*}[t]\n \\centering\n \\captionsetup{font=small}\n \\setlength{\\tabcolsep}{4pt}\n  \\normalsize\n  \\caption{Testset A - Computational results (residual cells are prohibited, singletons are allowed) }\n    \\begin{tabular}{|c|rrr|lrr|}\n    \\toprule\n   \n    \\multicolumn{1}{|c|}{\\multirow{4}{*}{\\textbf{\\#}}}  &  \\multicolumn{3}{c|}{\\textbf{Time, sec}} & \\multicolumn{3}{c|}{\\textbf{Efficacy}} \\\\\n   \n    \\multicolumn{1}{|c|}{} &  \\multicolumn{1}{c|}{Elbenani \\& }& \\multicolumn{1}{c|}{Bychkov}    & \\multicolumn{1}{c|}{}    &\\multicolumn{1}{c|}{Elbenani \\&} & \\multicolumn{1}{c|}{Bychkov} & \\multicolumn{1}{c|}{} \\\\\n  \n    \\multicolumn{1}{|c|}{} &  \\multicolumn{1}{c|}{Ferland (2012)}&  \\multicolumn{1}{c|}{et al.} & \\multicolumn{1}{c|}{two-index} & \\multicolumn{1}{c|}{Ferland (2012)} & \\multicolumn{1}{c|}{et al.} &\\multicolumn{1}{c|}{ two-index } \\\\\n   \n    \\multicolumn{1}{|c|}{} &  \\multicolumn{1}{c|}{\\textbf{}}&  \\multicolumn{1}{c|}{(2014)}  & \\multicolumn{1}{c|}{model} & \\multicolumn{1}{c|}{(cells)} & \\multicolumn{1}{c|}{(2014)}  & \\multicolumn{1}{c|}{model} \\\\\n    \\midrule\n     A1 & 2.3 &0.63  &  0.00  & \\qquad0.8235(2) & 0.8235 & 0.8235\\\\\n     A2 & 1.6 &2.29  &  0.00   &\\qquad0.6957(2) & 0.6957 & 0.6957  \\\\\n     A3 & 3.1 &5.69  &  0.00  & \\qquad0.7959(2) & 0.7959 &  0.7959   \\\\\n     A4 & 2.0   &1.86  &  0.09   & \\qquad0.7692(2) & 0.7692  &  0.7692  \\\\\n     A5 & 30.6 &9.14  &  0.17   & \\qquad0.6087(5) & 0.6087 & 0.6087 \\\\\n     A6 & 4.3  &5.15   &  0.01  & \\qquad0.7083(4) & 0.7083 & 0.7083 \\\\\n     A7 & 9.6  &13.37   &  0.02  & \\qquad0.6944(4) & 0.6944 & 0.6944 \\\\\n     A8 & 3.1  &18.33   &  0.01  & \\qquad0.8525(3) & 0.8525 & 0.8525\\\\\n     A9 & 3.5 &208.36  &  0.45   &\\qquad0.5872(2) & 0.5872 & 0.5872\\\\\n    A10 & 1.1 & 6.25   &  0.00  & \\qquad0.7500(5) & 0.7500  & 0.7500\\\\\n    A11 & 1.6 & 2.93  &  0.02  & \\qquad0.9200(3) & 0.9200  & 0.9200 \\\\\n    A12 & 2188.7 & 259.19 & 0.19  & \\qquad0.7206(7) & 0.7206  &  0.7206 \\\\\n    A13 & 593.2 & 179.21 & 0.23   & \\qquad0.7183(7) & 0.7183 &  0.7183 \\\\\n    A14 & 15130.5 & *  &  4.24  & \\qquad0.5326(8) & *  &  \\textbf{0.5326}\\\\\n    A15 & 252.5 & *  &  0.25  & \\qquad0.6953(6)\\textsuperscript{E} & * & \\textbf{0.6899}\\\\\n    A16 & 183232.5 & *  &  4.80  & \\qquad0.5753(8) & *  & \\textbf{0.5753}\\\\\n    A17 & 2345.6 & *  &  3.82  & \\qquad0.5773(9) & *  & \\textbf{0.5773}\\\\\n    A18 & * & *      & 32243.10 & \\qquad* & * & \\textbf{0.4345}  \\\\\n    A19 & 131357.5 & *  & 245.59  & \\qquad0.5081(7) & *  & \\textbf{0.5081}\\\\\n    A20 & 31.1 & *   &  0.22  & \\qquad0.7791(5) & *  &  \\textbf{0.7791} \\\\\n    A21 & 14583.6 & *    &  24.34 & \\qquad0.5798(5) & * &  \\textbf{0.5798}  \\\\\n    A22 & 11.3 &1.64  & 0.14  & \\qquad1.0000(7) & 1.0000     & 1.0000 \\\\\n    A23 & 230.7 & *   &  0.12   & \\qquad0.8511(7) & *  & \\textbf{0.8511} \\\\\n    A24 & 1101.1 & *   &  0.16  & \\qquad0.7351(7) & *  & \\textbf{0.7351} \\\\\n    A25 & * & *       &  1026.96 & \\qquad* & *  &   \\textbf{0.5329} \\\\\n    A26 & * & *          & 178182.24  & \\qquad* & *  & \\textbf{0.4895} \\\\\n    A27 & * & *           & *  & \\qquad* & * & * \\\\\n    A28 & 958714.1 & *    &  1964.00 & \\qquad0.5482(5) & * &  \\textbf{0.5482}\\\\\n    A29 & * & *      & *  & \\qquad* & *  & * \\\\\n    A30 & 378300.0 & *     &  8.72  &\\qquad0.6331(14) & *&  \\textbf{0.6331} \\\\\n    A31 & * & *     & 136.00 & \\qquad0.6012(13)\\textsuperscript{E} & * & \\textbf{0.5977} \\\\\n    A32 & * & *     & * & \\qquad* & *  & *  \\\\\n    A33 & * & *     & * & \\qquad*  & * & \\underline{0.4800} \\\\\n    A34 & 268007.6 & *   & 16323.71 & \\qquad0.6064(3) & *   & \\textbf{0.6064}\\\\\n    A35 & 7365.3 & *  &  1.34 & \\qquad0.8403(10) &  * & \\textbf{0.8403} \\\\\n    \\bottomrule\n    \\end{tabular}\n  \\label{testset_a_single}%\n\\end{table*}\n\n\\begin{table*}[t]\n \\centering\n \\captionsetup{font=small}\n \\setlength{\\tabcolsep}{4pt}\n  \\normalsize\n  \\caption{Testset A - Computational results (residual cells are allowed) }\n    \\begin{tabular}{|c|rrrr|llr|}\n    \\toprule\n   \n    \\multicolumn{1}{|c|}{\\multirow{4}{*}{\\textbf{\\#}}}  &  \\multicolumn{4}{c|}{\\textbf{Time, sec}} & \\multicolumn{3}{c|}{\\textbf{Efficacy}} \\\\\n   \n    \\multicolumn{1}{|c|}{} &  \\multicolumn{1}{c|}{}& \\multicolumn{1}{c|}{Pinheiro et al.} & \\multicolumn{1}{c|}{Pinheiro et al.}   & \\multicolumn{1}{c|}{}    &\\multicolumn{1}{c|}{Brusco} & \\multicolumn{1}{c|}{Pinheiro} & \\multicolumn{1}{c|}{} \\\\\n  \n    \\multicolumn{1}{|c|}{} &  \\multicolumn{1}{c|}{Brusco}&  \\multicolumn{1}{c|}{(2016)} &  \\multicolumn{1}{c|}{(2016)} & \\multicolumn{1}{c|}{two-index} & \\multicolumn{1}{c|}{(2015)} & \\multicolumn{1}{c|}{et al.} &\\multicolumn{1}{c|}{two-index} \\\\\n   \n    \\multicolumn{1}{|c|}{} &  \\multicolumn{1}{c|}{(2015)}&  \\multicolumn{1}{c|}{IM} &  \\multicolumn{1}{c|}{ILP} & \\multicolumn{1}{c|}{} & \\multicolumn{1}{c|}{(cells)} & \\multicolumn{1}{c|}{(2016)}  & \\multicolumn{1}{c|}{\\textbf{}} \\\\\n    \\midrule\n     A1 & 0.01 & 0.16 & 0.01  &  0.01  & \\qquad0.8235(2,3,4) & 0.7500\\textsuperscript{E} & 0.8235 \\\\\n     A2 & 0.01 & 0.07 & 0.01  &  0.01   & \\qquad0.6957(2,3,4) & 0.6956 & 0.6957  \\\\\n     A3 & 0.02 & 0.09 & 0.03  &  0.01  & \\qquad0.8085(2,3,4) & 0.8085 &  0.8085   \\\\\n     A4 & 0.01 & 0.02  & 0.01  &  0.01   & \\qquad0.7916(2,3,4) & 0.7917  &  0.7917  \\\\\n     A5 & 0.6  & 0,29 & 0.06  &  0.17   & \\qquad0.6087(2,3,4,5,6) & 0.6087 & 0.6087\\\\\n     A6 & 0.04  & 0.14 & 0.01   &  0.01  & \\qquad0.7083(2,3,4,5) & 0.7083 &  0.7083  \\\\\n     A7 & 0.08  & 0.18 & 0.03   &  0.01  & \\qquad0.6944(2,3,4,5) & 0.6944\\textsuperscript{E} & 0.6944  \\\\\n     A8 & 0.01  & 2.06 & 0.04   & 0.01  & \\qquad0.8525(2,3,4) & 0.8525 & 0.8525 \\\\\n     A9 & 35.86 & 81.46 & 4.94  &  0.45   & \\qquad0.5872(2,3,4) & 0.5872 & 0.5872\\\\\n    A10 & 0.06 & 0.03 &  0.01  &  0.01  & \\qquad0.7500(2,3,4,5,6) & 0.7500  & 0.7500 \\\\\n    A11 & 0.01 & 0.01 & 0.02  &  0.02  & \\qquad0.9200(2,3,4) & 0.9200  & 0.9200 \\\\\n    A12 & 633.91 & 0.49 & 0.09 & 0.03  & \\qquad0.7424(6,7,8) & 0.7424  & 0.7424 \\\\\n    A13 & 2631.76 & 0.49 & 0.11 & 0.03   & \\qquad0.7285(6,7,8) & 0.7286 &  0.7286 \\\\\n    A14 & 24716.34 & 600.98 & 144.91  &  4.88  & \\qquad0.5385(8) & 0.5333\\textsuperscript{E}  &  0.5385 \\\\\n    A15 & 1279.93 & 7.24 & 0.54  &  0.16  & \\qquad0.6992(5,6,7) & 0.6992\\textsuperscript{E} & 0.6992 \\\\\n    A16 & - & 1156.23  & 125.62 &  4.24  & \\qquad- & 0.5804  & 0.5804  \\\\\n    A17 & 20840.55 & 87.13 & 42.32  &  3.84  & \\qquad0.5773(9) & 0.5773\\textsuperscript{E}  & 0.5773 \\\\\n    A18 & - & *   & *   & 52810.10 & \\qquad- & * & \\textbf{0.4397}  \\\\\n    A19 & 1375608.66 & 23928.70& 1771.99  & 249.52  & \\qquad0.5081(7) & 0.5081  & 0.5081 \\\\\n    A20 & 4830.00 & 1.78  & 14.55 &  0.09  & \\qquad0.7888(5,6,7) & 0.7938\\textsuperscript{E}  & 0.7888\\\\\n    A21 & - & 2145.24  & 305.48  & 22.60 & \\qquad- &0.5879\\textsuperscript{E} & 0.5860  \\\\\n    A22 & 0.01 & 0.02 & 0.15  &  0.14  & \\qquad1.0000(7) & 1.0000     & 1.0000\\\\\n    A23 & 42.29& 10.08 & 0.44   &  0.14   & \\qquad0.8511(7) & 0.8511  &  0.8511  \\\\\n    A24 & 208158.02 & 17.46 & 0.78   &  0.20 & \\qquad0.7351(7) & 0.7351  & 0.7351 \\\\\n    A25 & - & 371233.00  &  48743.90 & 759.70 & \\qquad- & 0.5329\\textsuperscript{E}  &   0.5329 \\\\\n    A26 & - & *  & * & 134418.65 & \\qquad- & *  & \\textbf{0.4895} \\\\\n    A27 & - & *  & * & * & \\qquad- & * & * \\\\\n    A28 & - & *  &  * & 46361.97 & \\qquad- & * & \\textbf{0.5482} \\\\\n    A29 & - & *  & * & * & \\qquad- & *  & * \\\\\n    A30 & - & 183.71  &  41.53 & 8.00  & \\qquad- & 0.6304\\textsuperscript{E} &  0.6331 \\\\\n    A31 & - & 13807.50  & 2622.06 & 64.82 & \\qquad- & 0.5977 & 0.5977  \\\\\n    A32 & - & *  & * & 234055.90 & \\qquad- & *  & \\textbf{0.5084} \\\\\n    A33 & - & *  & * & * & \\qquad-  & * & \\underline{0.4829} \\\\\n    A34 & - & *  & * & 14212.57 & \\qquad- & *   & \\textbf{0.6131} \\\\\n    A35 & - & 325.53  &  18.22 & 1.61 & \\qquad- &  0.8403 & 0.8403 \\\\\n    \\bottomrule\n    \\end{tabular}\n  \\label{testset_a_residual}%\n\\end{table*}\n\n\\begin{table*}[t]\n  \\centering\n  \\normalsize\n \\setlength{\\tabcolsep}{6pt}\n \n  \\caption{Testset B - Computational results}\n    \\begin{tabular}{|crrrrr|}\n    \\toprule\n    \\multicolumn{1}{|c}{\\multirow{4}[4]{*}{\\textbf{\\#}}} & \\multicolumn{2}{c}{\\textbf{Time}} & \\multicolumn{3}{|c|}{\\textbf{Efficacy}}  \\\\\n    \\multicolumn{1}{|c}{} & \\multicolumn{1}{|c|}{two-index}  & \\multicolumn{1}{c}{two-index} & \\multicolumn{1}{|c|}{Heuristic}  & \\multicolumn{1}{c|}{two-index}  & \\multicolumn{1}{c|}{two-index} \\\\\n    \\multicolumn{1}{|c}{} & \\multicolumn{1}{|c|}{(no residual} & \\multicolumn{1}{c}{(allow } & \\multicolumn{1}{|c|}{bound} &  \\multicolumn{1}{c|}{(no residual} & \\multicolumn{1}{c|}{(allow } \\\\\n \\multicolumn{1}{|c}{}  & \\multicolumn{1}{|c|}{cells)}& \\multicolumn{1}{c}{residual cells)} & \\multicolumn{1}{|c|}{} & \\multicolumn{1}{c|}{cells)}& \\multicolumn{1}{c|}{residual cells)}\\\\\n   \\midrule\n  B1  & 0.01 & 0.01  &0.8095  & 0.8095 & 0.8095\\\\\n  B2  & 0.01 & 0.01  &0.7222  &0.7222& 0.7222\\\\\n  B3  & 0.25 & 0.03  &0.6071  & 0.6071 & 0.6071\\\\\n  B4 & 0.01 & 0.01  &0.8889  & 0.8889 & 0.8889\\\\\n  B5 & 0.01 & 0.01 &0.7500  & 0.7500 & 0.7500 \\\\\n  B6 & 0.01 & 0.01  &0.7391  & 0.7391 & 0.7391 \\\\\n  B7 & 0.01 & 0.01  &0.8148  & 0.8148 & 0.8148 \\\\\n  B8 & 0.01  & 0.01  & 0.7222 & 0.7222 & 0.7222 \\\\\n  B9 & 0.01 & 0.01  & 0.7576  & 0.7576& 0.7576  \\\\\n  B10 & 0.01 & 0.01  & 0.9000 &0.9000 & 0.9000 \\\\\n  B11 &  0.01 & 0.02  & 0.7273 & 0.7273 & 0.7297 \\\\\n  B12 & 0.01 & 0.01  & 0.8276  &0.8276 & 0.8276 \\\\\n  B13 & 0.36 &0.80  & 0.5962  & 0.5962 & 0.6042 \\\\\n  B14 & 0.25  &0.30  & 0.6405  & 0.6405& 0.6405 \\\\\n  B15 & 0.01  &0.01  & 0.8333  & 0.8333& 0.8333 \\\\\n  B16 & 0.16  &0.06  & 0.7391  & 0.7391  & 0.7444\\\\\n  B17 & 0.98 & 0.26  & 0.6552  & 0.6552  & 0.6842\\\\\n  B18  & 1.82 & 1.65  &0.6027 & 0.6129 & 0.6129\\\\\n  B19 & 0.03 & 0.06  & 0.8000  & 0.8000  & 0.8113\\\\\n  B20 & 0.05 & 0.03  &0.8710  & 0.8710  & 0.8710\\\\\n  B21 & 0.03 & 0.04  &0.8333  & 0.8333  & 0.8333\\\\\n  B22 & 0.05 & 0.01  &0.7258  & 0.7258 & 0.7258 \\\\\n  B23 & 0.05  & 0.06  &0.8111  & 0.8111  & 0.8111\\\\\n  B24 & 4.79 & 7.80  & 0.5673  & 0.5673   & 0.5728\\\\\n  B25 & 0.20 & 0.10  &0.7600  & 0.7600  & 0.8000\\\\\n  B26 & 13.81 & 25.75  &0.6068  & 0.6068 & 0.6078 \\\\\n  B27 & 0.25 & 0.28  &0.7248  & 0.7248  & 0.7248\\\\\n  B28 & 0.83 & 1.04  &0.6729  & 0.6729  & 0.6729\\\\\n  B29 & 33.82 & 51.76  &0.5730  & 0.5730  &0.5745\\\\\n  B30 & 4.76 & 8.67  &0.7308  & 0.7308  & 0.7325\\\\\n  B31 & 19.69 & 17.50  &0.6799  & 0.6799  & 0.6799\\\\\n  B32 & * & *  &0.6193  & *   & * \\\\ \n    \\bottomrule\n    \\end{tabular}%\n  \\label{testset_b_results}%\n\\end{table*}%\n\n\n\\newpage\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nLongstanding interest in hybrid mesons continues because they offer a window into the unknown dynamics of nonperturbative gluonic degrees of freedom in Quantum Chromodynamics (QCD). In particular, robust experimental programs are underway by the GlueX collaboration at Jefferson Lab\\cite{Dobbs} and by the PANDA collaboration at FAIR\\cite{Lutz:2009ff}. The spectrum, radiative transitions, and strong decay widths are of interest to the experimental community as these are crucial to informing experimental design and the interpretation of experimental results. A simple model of hybrid structure is developed in the following and is applied to the strong decays of charmonium hybrids, which is a focal point for the PANDA collaboration.\n\n\nA variety of hybrid models have been developed since the 1970s; these chiefly fall into two camps: constituent gluon models and flux tube models. The former include bag models\\cite{bags} or simple constituent glue models\\cite{HM}, while the latter include string models\\cite{string} or more elaborate flux tube models\\cite{flux}. Most of this work predates the substantial progress that has been made in understanding hybrid properties by the lattice gauge community.\nThis effort had revealed flaws in the early models, most of which are severe enough to invalidate the models\n(see Ref. \\cite{Meyer:2015eta} for a recent review of the experimental and theoretical status of hybrids). Given this situation, and the impending data from GlueX and PANDA, it is incumbent to revisit hybrid structure modelling. Of course, it is preferable that new models incorporate the features of gluodynamics that the past 20 years of lattice gauge computations have revealed. Among these are the spectrum of adiabatic gluonic excitations\\cite{Juge:1997nc,Juge:1999ie}, the gluelump (bound states of gluons and a static adjoint colour source) spectrum\\cite{gluelump}, and properties of charmonium hybrids\\cite{Liu:2012ze,Cheung:2016bym,Knechtli:2019bqx}. Of particular interest is the confirmation that the heavy quark multiplet structure anticipated in Ref. \\cite{Juge:1999ie} is reflected in the charmonium spectrum\\cite{Liu:2012ze}. It is interesting, and very suggestive, that this multiplet structure can be reproduced by degrees of freedom consisting of a quark, an antiquark, and an axial gluon with quantum numbers $J^{PC} = 1^{+-}$\\cite{Guo:2008yz,Dudek:2011bn}.  \n\nIndeed, work has started along these lines, starting with an examination of the adiabatic gluon potentials in a constituent gluon model based on the Hamiltonian of QCD\\cite{Swanson:1998kx}. This work obtained decent agreement with lattice but noted that the level ordering was incorrect. Subsequently it was realized that gluonic three-body interactions can correct the level ordering problem\\cite{Szczepaniak:2006nx}, and this was used to construct models for gluelumps\\cite{Guo:2007sm} and heavy quark hybrids\\cite{Guo:2008yz}. Here we shall follow a similar approach with an eye to developing the simplest model possible that captures the necessary features revealed by the lattice. This will be described in Section \\ref{sec:model}. The resulting hybrid wavefunctions will then form the starting point for the computation of strong decays of charmonium hybrids.\n\nIt is useful to note that advances in the application of effective field theory to hybrids have also been occurring. This approach has led to a more thorough understanding of systematics in heavy quark hybrids, including subtleties concerning angular momentum and the structure of subleading spin-dependent operators\\cite{Berwein:2015vca, Brambilla:2018pyn}. Although these results are interesting, and inform our model building, they are not of direct relevance to the present work because gluonic degrees of freedom are integrated out, precluding the application of the formalism to hadronic transitions.\n\nHybrid meson decay models also have a long history. The earliest work we are aware of is due to Tanimoto\\cite{Tanimoto} (see also the first of Ref.\\cite{HM}), who assumed a simple $P$-wave symmetric Gaussian wavefunction for all hybrids and decay via gluon dissociation to a quark-antiquark pair. In contrast, the model of Isgur and Paton assumed that gluon dynamics can be described by a nonrelativistic string\\cite{Isgur:1984bm} and that strong decays occur via a string breaking mechanism with vacuum quantum numbers\\cite{Isgur:1985vy}. A model for hybrid decay that employed the same philosophy but assumed vector quark pair creation was developed by Swanson and Szczepaniak\\cite{Swanson:1997wy} and applied in Ref. \\cite{Page:1998gz}. \n\nThe approach adopted here is predicated on the simple constituent gluon model we adopt and is guided by the QCD Hamiltonian. Namely, we shall assume that the gluon simply dissociates into a light quark pair via the perturbative quark-gluon coupling. Thus we resurrect the Tanimoto decay model, but with the addition of a reasonable model of hybrid structure and with the benefit of lattice predictions of the charmonium hybrid spectrum. The decay model will be described in Section \\ref{sec:decay} followed by conclusions in Section \\ref{sec:conc}.\n\n\n\n\n\n\\section{Constituent Gluon Model of Hybrid Mesons}\n\\label{sec:model}\n\n\nOur model for hybrid structure will assume constituent quarks and gluons are the dominant degrees of freedom and that these interact according to QCD. Interactions are made especially transparent by adopting the Hamiltonian of QCD in Coulomb gauge\\cite{Schwinger, Christ:1980ku}. Perhaps the strongest reason for adopting this approach is that all degrees of freedom are manifest (\\textit{i.e.}, there are no ghosts) and that an explicit instantaneous ``Coulomb\" interaction exists between valence quarks and gluons. The philosophy advocated here was developed extensively in Ref. \\cite{Szczepaniak:2001rg}, where a mean field model of the gluonic vacuum was used to derive an effective gluon dispersion and a nonperturbative expression for the Coulomb operator (to be explained further below) that recapitulates linear confinement as observed in lattice gauge computations.\n\n\nThe construction of a hybrid meson proceeds by assuming a dynamical constituent gluon (thus, the gluon remains transverse) coupled to quarks. This involves a Clebsch-Gordan coefficient that couples the canonical gluon spin projection $s$ to the gluon angular momentum $(\\ell_g,m_g)$ to total gluon spin, $j_g$. Converting to the gluon helicity basis and assuming  that  $\\ell_g = j_g$ reduces the product of two Wigner matrices to one and produces a factor of\n\n\\begin{equation}\n\\chi^{(-)}_{\\lambda,\\mu} \\equiv \\langle 1 \\lambda \\ell_g 0| \\ell_g \\mu\\rangle = \n\\begin{cases} 0, \\ell_g = 0 \\\\ \\frac{\\lambda}{\\sqrt{2}} \\delta_{\\lambda,\\mu}, \\ell_g \\geq 1 \\end{cases}.\n\\end{equation}\nThis represents a transverse electric (TE) gluon in our model and forms the explicit realisation of the axial constituent gluon. Alternatively, one may set $\\ell_g = j_g\\pm 1$ and obtain a transverse magnetic (TM) gluon with a Clebsch factor given by\n\n\\begin{equation}\n\\chi^{(+)}_{\\lambda,\\mu} = \\frac{1}{\\sqrt{2}} \\delta_{\\lambda,\\mu}.\n\\end{equation}\n\nThe end result for a generic hybrid meson creation operator is then \n\n\n\\begin{align}\n&|JM [LS \\ell j_g \\xi]\\rangle = \\frac{1}{2} T_{ij}^A\\int \\frac{d^3q}{(2\\pi)^3}\\, \\frac{d^3k}{(2\\pi)^3} \\,\n\\Psi_{j_g,\\ell}({\\bf k}, {\\bf q})\\, \\sqrt{\\frac{2 j_g +1}{4\\pi}} \\, D_{m_g\\mu}^{j_g*}(\\hat k) \\, \\chi^{(\\xi)}_{\\mu,\\lambda} \\nonumber \\\\\n& \\times \n\\langle \\frac{1}{2} m \\frac{1}{2} \\bar{m} | S M_{S} \\rangle \\,\n\\langle \\ell m_\\ell, j_g m_g| L M_L\\rangle \\, \n\\langle S M_S, L M_{L} | J M \\rangle \\,\nb_{{\\bf q}-\\frac{{\\bf k}}{2},i,m}^\\dagger \\,\n  d_{-{\\bf q}-\\frac{{\\bf k}}{2},j, \\bar{m}}^\\dagger \\,\na^\\dagger_{{\\bf k}, A, \\lambda} |0\\rangle,\n\\label{eq:PSI}\n\\end{align} \nThe momenta of the constituents are chosen as a convenient rescaling of Jacobi coordinates, \n$\\vec k = -2\/\\sqrt{6} \\vec p_\\lambda$ and $ \\vec q = \\vec p_\\rho\/\\sqrt{2}$.\nFinally, the hybrid state is an eigenstate of parity and charge conjugation with eigenvalues given by\n\n\\begin{equation}\nP=\\xi (-1)^{\\ell + j_g+1}\n\\end{equation}\nand\n\\begin{equation}\n  C= (-1)^{\\ell+S+1}.\n\\end{equation}\n\n\nAs mentioned above, we choose to model hybrid structure with a simplified version of the QCD Hamiltonian. The Hamiltonian in Coulomb gauge is written as\\cite{Schwinger, Christ:1980ku}\n\n\n\\begin{equation}\nH_{QCD} = \\int d^3x\\, \\left[ \\psi^\\dagger\\left( -i \\alpha\\cdot\\bdel\n + \\beta m\\right) \\psi + \n\\frac{1}{2}\\left( {\\cal J}^{-1\/2}\\bPi {\\cal J} \n\\cdot \\bPi {\\cal J}^{-1\/2} + {\\bf B}\\cdot{\\bf B}\\right) \n -g \\psi^\\dagger \\bm{\\alpha}\\cdot \\A \\psi \\right] + H_C\n\\label{eq:h}\n\\end{equation}\nwith\n\n\\begin{equation}\nH_C = \\frac{1}{2}\\int d^3x\\, d^3y\\, {\\cal J}^{-1\/2} \\rho^A({\\bf x}) \n {\\cal J}^{1\/2} \\hat K_{AB}({\\bf x},\\y;\\A) {\\cal J}^{1\/2} \\rho^B(\\y) {\\cal J}^{-1\/2}.\n\\label{eq:hc}\n\\end{equation}\nThe latter expression is the instantaneous interaction that emerges from the imposition of Gauss's law in Coulomb gauge. This involves the Faddeev-Popov determinant\n\n\\begin{equation}\n{\\cal J} \\equiv {\\rm det}(\\bnabla\\cdot {\\bf D}),\n\\end{equation}\nwhich is the remnant of the ghost in this gauge. Here $D$ represents the adjoint covariant derivative \n\n\n\\begin{equation}\n {\\bf D}^{AB} \\equiv \\delta^{AB} \\bnabla  - g f^{ABC}\\A^C.\n\\end{equation}\n\nThe colour charge density that appears in Eq. \\ref{eq:hc} is given by\n\n\\begin{equation}\n\\rho^A({\\bf x}) = \n f^{ABC} {\\bf A}^B({\\bf x}) \\cdot {\\bf \\Pi}^C({\\bf x}) + \\psi^{\\dag}({\\bf x})T^A\\psi({\\bf x}).\n\\label{eq:rho}\n\\end{equation}\n\n\nThe kernel of the Coulomb interaction can be formally written as\\cite{Christ:1980ku}\n\n\n\\begin{equation}\n\\hat K^{AB}({\\bf x},{\\bf y};\\A) \\equiv \\langle{\\bf x},A|\n \\frac{ g }{ \\bnabla\\cdot {\\bf D}}(-\\bnabla^2)\n \\frac{ g }{ \\bnabla\\cdot {\\bf D}}|{\\bf y},B\\rangle.\n\\label{eq:K}\n\\end{equation}\nFinally, $\\A$ is the vector potential and ${\\bm \\Pi}$ is the conjugate momentum given by the negative of the transverse chromoelectric field.\n\nEquations \\ref{eq:h} to \\ref{eq:K} are a full field-theoretic representation of QCD and are therefore difficult to solve. In particular, the Coulomb interaction involves infinitely many gluons that build $n$-body operators in Fock space. The two-body operator can be related to the Wilson loop linear confinement potential in a rather direct way\\cite{Szczepaniak:2001rg, Zwanziger:2002sh}. Here we accept the mapping to linear confinement as a phenomenologically useful device and simply set \n\n\\begin{equation}\n\\hat K^{AB}({\\bf r},0) \\to\nK^{AB}({\\bm r},0) = \\delta^{AB}\\left( \\frac{a_S}{r} - \\frac{3}{4} b r\\right).\n\\label{eq:V}\n\\end{equation}\nOf course, this reproduces the successes of the Cornell potential in nonrelativistic quark models. Higher terms in the $n$-body expansion of $\\hat K$ can be incorporated in the formalism as required. The effects of the Faddeev-Popov determinant are assumed to be largely confined to restricting the gauge field to the fundamental modular region\\cite{Greensite:2011zz}, and are therefore ignored in the following. The vector potential is approximated by its Abelian analogue when making field expansions:\n\n\\begin{equation}\n\\A^B({\\bf x}) = \\int \\frac{d^3k}{(2\\pi)^3}\\, \\frac{1}{\\sqrt{2\\omega(k)}}\\left( {\\bm a}^B({\\bf k}) + {\\bm a}^{B\\dagger}(-{\\bf k})\\right) {\\rm e}^{i {\\bf k} \\cdot {\\bf x}}.\n\\label{eq:A}\n\\end{equation}\nThe gluon dispersion is denoted as $\\omega$ in this expression and need not take on the perturbative form $\\omega = k$ because the gluon is a quasiparticle in this approach. In fact, explicit computation in a mean field vacuum model gives a dispersion that is well approximated by\\cite{Szczepaniak:2001rg}\n\n\\begin{equation}\n\\omega^2 = k^2 + m_g^2{\\rm e}^{-k\/b_g}\n\\label{eq:omega}\n\\end{equation}\nwhere the dynamical gluon mass is $m_g \\approx 600$ MeV and the parameter $b_g \\approx 6000$ MeV.\nFinally, the nonrelativistic limit is taken at all opportunities. \n\n\nThe resulting model can be thought of as a minimal extension of the nonrelativistic potential quark model with the addition of constituent gluon degrees of freedom and possible many-body couplings (for example, the tri-linear gluon coupling). It can also be extended to include relativistic effects in the inverse mass expansion that may be important for spin-dependent effects.\n\n\n\nOur immediate goal is to obtain a rough description of the lowest hybrid charmonium mesons. As mentioned above, these fall into multiplets that are associated with interpolating fields as shown in Table \\ref{tab:JPCMulti}. A model description of these multiplets is sought by employing a TE constituent gluon with $j_g=1$. This leaves the quark angular momentum ($\\ell$) and the total angular momentum ($L$) to be specified. Doing so gives columns three to six in the table. Finally combining these degrees of freedom with the total quark spin gives the quantum numbers shown in the last column (the entries in brackets correspond to $S=1$).\n\n\n\\begin{table}[h]\n\\caption{$J^{PC}$ Hybrid Multiplets. With generating operators and corresponding constituent gluon model quantum numbers.}\n\\begin{tabular}{c|c|cccc|l}\n\\hline\\hline\nmultiplet & operator & $\\xi$ & $j_g$ & $\\ell$ & $L$ & $J^{PC}$ \\\\\n\\hline\n$H_1$ & $\\psi^\\dagger \\vec B \\chi$ & -1 & 1 & 0 & 1 & $1^{--}$, $(0,1,2)^{-+}$ \\\\\n$H_2$ & $\\psi^\\dagger \\nabla \\times \\vec B \\chi$  & -1 & 1 & 1 & 1 & $1^{++}$, $(0,1,2)^{+-}$ \\\\\n$H_3$ &$\\psi^\\dagger \\nabla \\cdot \\vec B \\chi$ & -1 & 1 & 1 & 0 & $0^{++}$, $(1^{+-})$ \\\\\n$H_4$ & $\\psi^\\dagger [\\nabla \\vec B]_2 \\chi$ & -1 & 1 & 1 & 2 & $2^{++}$, $(1,2,3)^{+-}$ \\\\\n\\hline\\hline\n\\end{tabular}\n\\label{tab:JPCMulti}\n\\end{table}\n\n\nIn this preliminary calculation we choose a simple product Ansatz for the hybrid wavefunction, namely\n\n\\begin{equation}\n\\Psi_{j_g,\\ell}({\\bf k}, {\\bf q}) = \\varphi_{j_g}(k;\\beta)\\, \\varphi_{\\ell}(q; \\alpha) Y_{\\ell,m_\\ell}(\\hat q)\n\\label{eq:ansatz}\n\\end{equation}\nThis makes explicit the angular momentum dependence in the $q$ coordinate. The gluon angular momentum dependence is contained  in the Wigner rotation matrix in Eq. \\ref{eq:PSI}. In the following we shall employ simple harmonic oscillator (SHO) wavefunctions for the $\\varphi$ functions and determine the width parameters  $\\alpha$ and $\\beta$ variationally:\n\n\\begin{equation}\n\\frac{\\partial}{\\partial \\alpha(\\beta)} \\langle J'M'[L' S' \\ell' j_g' \\xi']|H_{QCD}|JM[L S \\ell j_g \\xi]\\rangle =0.\n\\label{eq:var}\n\\end{equation}\n\n\nThe computation amounts to evaluating the diagram illustrated in Fig. \\ref{fig:Hint}, where three possible two-body interactions are shown.   As mentioned in the Introduction, it has been found that three-body interactions are important for describing higher gluonic excitation surfaces. However, these are zero for the lowest lying surface\\cite{Szczepaniak:2006nx}, with which we deal exclusively.\n\n\\begin{figure}[ht]\n\\includegraphics[width=0.3\\linewidth]{Hinteraction.pdf}\n\\caption{Interactions Contributing the Hybrid Bound State.}\n\\label{fig:Hint}\n\\end{figure}\n\n\n\nModel parameters were fixed to those used previously in a potential quark model\\cite{Swanson:1992ec} along with the dispersion of Eq. \\ref{eq:omega}. Thus all parameters have been fixed by previous work. Since this preliminary work is not concerned with fine structure in the spectrum we restrict attention to the nonrelativistic limit and hence seek spin-averaged masses. Details of the calculation are presented in the Appendices; the results for  spin averaged $S$-wave and $P$-wave charmonia and the spin averaged hybrid masses are shown in Fig. \\ref{fig:Hmasses}. We remark that the method for spin-averaging hybrid multiplets is not obvious; here we simply take these values as the masses of the spin zero member of the multiplet. This procedure is justified in Ref. \\cite{Lebed:2017xih}. The figure shows two sets of lattice charmonia masses obtained with light quark masses that correspond to pion masses of 400 MeV (dark boxes) and 236 MeV (light boxes). Model results have been centered on the lattice $\\eta_c$--$J\/\\psi$ multiplet in the first column. The resulting spin-averaged $H_2$ and $H_3$ hybrid mass predictions are high by 100 -- 200 MeV. Somewhat surprisingly, given that no parameters were adjusted, the other mass predictions agree rather well with the more recent lattice field spectrum.  We find these initial results encouraging and look forward to performing more detailed computations. In the meantime, we adopt the hybrid wavefunction width parameters and move on to hybrid decays.\n\n\n\n\\begin{figure}[ht]\n\\includegraphics[width=0.75\\linewidth]{Hmasses.pdf}\n\\caption{Spin Averaged Meson and Hybrid Masses. Dark grey boxes: lattice results with $m_\\pi$ = 400 MeV\\cite{Liu:2012ze}. \nLight grey boxes: lattice results with $m_\\pi$ = 236 MeV\\cite{Cheung:2016bym}.  Solid box: model results.}\n\\label{fig:Hmasses}\n\\end{figure}\n\n\n\\section{Hybrid Decay Model}\n\\label{sec:decay}\n\nIt is feasible and perhaps obligatory to consider strong hybrid decays once a reasonably robust model of hybrid structure is in hand. The first issue is determining the decay operator. Possibilities include absorbing the valence gluon into a quark or the Coulomb operator and then pair producing via a valence-valence-transverse tri-gluon vertex. This is rather elaborate and is suppressed by two powers of a mass scale (that is roughly 1 GeV). We choose, instead, to adopt the simplest and leading operator, which is quark pair production via gluon dissociation (Fig. \\ref{fig:Hdecay}).\n\n\\begin{figure}[ht]\n\\includegraphics[width=0.3\\linewidth]{Hdecay.pdf}\n\\caption{Hybrid Decay Model.}\n\\label{fig:Hdecay}\n\\end{figure}\n\n\n\nThe calculation will be made using the hybrid wavefunctions found in the previous section. In a similar fashion, we also employ SHO wavefunctions for the final state mesons. Masses for the hybrids are taken from the lattice \\cite{Liu:2012ze,Cheung:2016bym} and measured masses are used for all mesons. Details are provided in the Appendix along with a description of the $P$-wave mixing angles used.\n\n\nThe decay model contains two selections rules: (i) spin zero hybrids do not decay to spin zero mesons, (ii) TE hybrids do not decay to mesons with identical spatial wavefunctions. The former follows from the ${}3S_1$ quantum numbers of the vertex, while the latter is proved in the Appendix. These selection rules are typical of hybrid decay models, and there is reason to believe that they are generally true\\cite{Page:1996rj}.\n\n\nThe partial widths obtained in the model generally follow the pattern of ``${}^3S_1$\" models, namely the leading partial wave is much larger than subleading ones. It is interesting that exceptions do exist; for example $H_2(1+-) \\to  D^*\\bar D$ has an S\/D ratio of 0.16. It is possible that this unusual circumstance can aid in the eventual identification of hybrid candidates. More generally, partial wave ratios tend to be closer to unity for decay models in which the vertex carries ${}^3P_0$ quantum numbers. In fact this can be used to distinguish strong decay models in the case of canonical meson decays\\cite{Ackleh:1996yt}.\n\n\n\n\nThe major open issue in the current work is the selection of the remaining parameter, the quark-gluon coupling  $g$. In contrast to all other model parameters, which have their values fixed by external data or by direct computation, the quark-gluon coupling is unconstrained. While it may be natural to assign\n\n\\begin{equation}\ng = \\sqrt{4\\pi a_S}\n\\end{equation}\nthis is not necessarily accurate. The parameter $a_S$ results from a fit to either the meson spectrum or lattice Wilson loop computations of the adiabatic interquark potential. It therefore parameterises an effective description of  nonperturbative gluodynamics. It is possible, for example, that $a_S$ is approximated by the string splitting\\cite{LW} \n\n\\begin{equation}\na_S \\approx  \\frac{\\pi}{12}\n\\end{equation}\nat intermediate scales (say, 1\/2 fm) and therefore has little direct contact with $g$.\nA third alternative is that one should use the QCD coupling at the scale of hybrid mesons, hence \n\n\\begin{equation}\ng \\approx g_{QCD}(4.5\\ \\textrm{GeV}) \\approx 1.6.\n\\end{equation} \nAlternatively, setting $g = g_{QCD}$ at the average gluon momentum gives $g \\approx 2.5$.\n\n\n\nThe natural way to resolve the problem is to set the coupling from a well-established hybrid decay width. Unfortunately, no lattice field computations of open flavour heavy hybrid decay have been made. \n\nTurning to experiment, the quantum number exotic state $R_{c0}(4240)$ is reported in the Review of Particle Properties\\cite{rpp}. The claimed quantum numbers are $J^{PC} = 0^{--}$, which are forbidden to fermion-antifermion systems. The state is seen to decay to $\\pi^-\\psi(2S)$ with a total width of $\\Gamma = 220$ MeV with large errors. Note that the decay mode implies that the $R_{c0}$ is also flavour-exotic. A $0^{--}$ hybrid can be accommodated within the current model, but is a TM hybrid, and therefore  heavier than the 4.0 -- 4.5 GeV masses seen in the lightest multiplets. The poor agreement with model expectations, the exotic flavour content, and the closed flavour decay mode make it impossible to use this state for setting the coupling. \n\n\n\nThe $\\psi(4260)$ (previously known as the $Y(4260)$) is a longstanding charmonium hybrid candidate. Its mass is reported as $4320(6)$ MeV and width as $55(19)$ MeV\\cite{rpp}. The state is seen to decay to $\\pi\\pi J\/\\psi$, either directly, or via resonances as $f_0 J\/\\psi$ or $Z_c(3900)\\pi$. Since the $Y$ does not fit with well-established quark model expectations\\cite{Barnes:2005pb} it has been suggested that it is a hybrid candidate, with a natural identification with $H_1(1^{--})$. The lattice predictions for the mass of this state are 4285(14) MeV at $m_\\pi=400$ MeV or 4411(17) MeV at $m_\\pi=236$ MeV. The former value looks promising, so it is unfortunate that decreasing the pion mass towards its physical value worsens the agreement. This appears to be a general feature of the computation -- with all higher mass mesons shifting upwards in mass by 100 -- 200 MeV (these mesons have an excitation energy of approximately 1400 MeV; those with an excitation energy of approximately 900 MeV shift upwards by tens of MeV). An additional possible problem is that the closed flavour decay $H_1(1^{--}) \\to \\pi\\pi J\/\\psi$ must be accompanied by a spin flip, which is generally suppressed. Countering this is the existence of anomalies in the hadronic transitions $\\Upsilon(nS) \\to \\pi\\pi \\Upsilon(n'S)$ also imply a failure of naive expectations with respect to ``spin flip\". This can happen if intermediate states such as $B^*\\bar B$ can propagate for a long time and effectively decorrelate the heavy quark spin. Indeed, the observation of the $Y$ in $Z_c(3900)\\pi$ indicates that such a mechanism could be relevant here. Thus, although it is tempting, it is not prudent to set the coupling by comparison with the $\\psi(4260)$.\n\n\n\nIn view of the lack of external information on the value of $g$, we proceed with the initial impulse and set $g = \\sqrt{4\\pi a_S} \\approx 2.73$ and bear in mind that the results may require substantial modification pending the arrival of new information. Of course this modification amounts to simply scaling the results, so at least it is not difficult to make.\n\n\nTable \\ref{tab:widths} presents partial decay widths for the complete set of low-lying hybrid mesons to a variety of channels. In this case we take the hybrid masses to be those obtained on the lattice with $m_\\pi = 400$ MeV. Although these are somewhat lighter than those obtained with more physical light quark masses, the calculation illustrates the effect of assumptions about the hybrid masses and might be relevant to nature if it is confirmed that the $\\psi(4260)$ does indeed contain substantial valence gluonic degrees of freedom.\n\nThe table reveals that members of the $H_1$ multiplet are  very narrow and should appear as sharp peaks in their final states. In contrast, all other hybrids have typical hadronic widths. Of course, this statement relies on the assumed value of the quark-gluon coupling constant.  \nIt is evident that S+P decay modes dominant total widths when these channels are open. This repeats a rule of thumb found in previous hybrid decay models, such as Ref. \\cite{Isgur:1985vy}.\n\n\n\\begin{table}[ht]\n\\caption{Hybrid Decay Widths (MeV), computed with the $m_\\pi= 400$ MeV Hybrid Spectrum. Suppressed channels are denoted as follows: $-$ = quantum number, \\o = threshold, 0 = selection rule, x = negligible.}\n\\begin{tabular}{l|ccccccccccccc|c}\n\\hline\\hline\nstate & $D^*D$ & $D_0D$ & $D_{1L}D$ & $D_{1H}D$ & $D_2D$ & $D_0D^*$ & $D_{1L}D^*$ & $D_{1H}D^*$ & $D_2D^*$ & $D_s D_s^*$ & $D_{s0} D_s$ & $D_{s0} D_s^*$ & $D_{s1L} D_s$ & width\\\\\n\\hline\n$1^{--}$ $(H_1)$ & x & - & \\o & \\o & \\o & \\o & \\o & \\o & \\o & \\o & - & \\o & \\o & x \\\\\n$0^{-+}$ $(H_1)$ & x & \\o & \\o & \\o & \\o & - & \\o & \\o & \\o & \\o & \\o & - & \\o & x\\\\\n$1^{-+}$ $(H_1)$ & x & - & \\o & \\o & \\o & \\o & \\o & \\o & \\o & 6 & - & \\o & \\o & x\\\\\n$2^{-+}$ $(H_1)$ & x &  x &  x & x & 6 &\\o & \\o &\\o &\\o & \\o &  \\o &  \\o & \\o & x \\\\\n\\hline\n$1^{++}$ $(H_2)$ & x  & - & 3 & 11 & x & 4 & \\o & \\o & \\o & \\o & - & \\o & \\o & 18\\\\\n$0^{+-}$ $(H_2)$ & x &  x & 21 & 3 & x  & 4 & \\o & \\o   & \\o  & x  & x& \\o & \\o & 28 \\\\\n$1^{+-}$ $(H_2)$ & x &  12 & 1  & 1 & x & \\o  & \\o  & \\o  &\\o  & 1  &  6  &\\o  &\\o & 21  \\\\\n$2^{+-}$ $(H_2)$ & x &  15 &  3 &  2 & 3 &  2  &  \\o &  \\o &  \\o &  x &  12 &  \\o  &   \\o & 37 \\\\\n\\hline\n$0^{++}$ $(H_3)$ & - & - & 3 & 13 & - & 10 & 2 & 2 & \\o & - & - & 3 & 1 & 44\\\\\n$1^{+-}$ $(H_3)$ & x &  4 &  6  &  2  & 4  & 6  &  3 &   2  &  x  & x &  4  &   2  &   1 & 34 \\\\\n\\hline\n$2^{++}$ $(H_4)$ &  x  &  x  &   0 &  17  & x  & 12  & 9 & 2 &  1 &  x &  x &  6 & 0 & 47\\\\\n$1^{+-}$ $(H_4)$ &  x & 24  & 8 & 4 & x  &  10  &   1  &  1  & x &  x & 28 & 6 &  3 & 85\\\\\n$2^{+-}$ $(H_4)$ & x &  x  & 26 &  6 & 1 &  20 &  5 &  5 & 2 &  x &  x &  12 &  13 & 90 \\\\\n$3^{+-}$ $(H_4)$& x &  x &   0  &   x   &   10  & x   & x & 26 & 7 & x &  x  &   x & 0 & 43\\\\\n\\hline\\hline\n\\end{tabular}\n\\label{tab:widths}\n\\end{table}\n\n\n\n\\begin{table}[ht]\n\\caption{Hybrid Decay Widths (MeV), computed with the $m_\\pi= 236$ MeV Hybrid Spectrum. Suppressed channels are denoted as follows: $-$ = quantum number, \\o = threshold, 0 = selection rule, x = negligible.}\n\\begin{tabular}{l|ccccccccccccc|c}\n\\hline\\hline\nstate & $D^*D$ & $D_0D$ & $D_{1L}D$ & $D_{1H}D$ & $D_2D$ & $D_0D^*$ & $D_{1L}D^*$ & $D_{1H}D^*$ & $D_2D^*$ & $D_s D_s^*$ & $D_{s0} D_s$ & $D_{s0} D_s^*$ & $D_{s1L} D_s$ & width\\\\\n\\hline\n$1^{--}$ $(H_1)$ & x & - & 8 & 33 & x & 25 & \\o & \\o & \\o & x & - & \\o & \\o & 66\\\\\n$0^{-+}$ $(H_1)$ & x & 76 & \\o & \\o & \\o & - & \\o & \\o & \\o & 34  & \\o & \\o & \\o & 76\\\\\n$1^{-+}$ $(H_1)$ & x & - & 28 & \\o & \\o & \\o & \\o & \\o & \\o & x  & - & \\o & \\o & 28\\\\\n$2^{-+}$ $(H_1)$ & x & x  &  x & 1 & 20 &  x & 16 & 24 &\\o & x & x & x & x & 60\\\\\n\\hline\n$1^{++}$ $(H_2)$ & x  & - & 5 & 18 & x & 11 & 5 & x & \\o & x & - & 4 & 2 & 45\\\\\n$0^{+-}$ $(H_2)$ & x &  x & 30 & 5 & x  & 17 & x & \\o   & \\o  & x  & x & x & 1 & 53  \\\\\n$1^{+-}$ $(H_2)$ & x & 16  & 4  & 3 & 5 & 4  & x  & \\o  & \\o  & x  & 16   & x  & x & 48  \\\\\n$2^{+-}$ $(H_2)$ & x & 16  & 4  & 3  & 6 & 7  & 14  & 5 & 1  & x & 20  & 4   & 2 & 82  \\\\\n\\hline\n$0^{++}$ $(H_3)$ & - & - & 4 & 15 & - & 15 & 7 & 21 & 14 & - & - & 14 & 6 & 96 \\\\\n$1^{+-}$ $(H_3)$ & x & 4  & 6   & 3   & 4  & 9  & 11  & 16  & 5   & x & 5   & 8  & 5 & 76   \\\\\n\\hline\n$2^{++}$ $(H_4)$ & x &  x  &   0 & 17   & x & 15  & 23   & 9 & 14  & x  &  x & 18 & 0 & 96\\\\\n$1^{+-}$ $(H_4)$ & x  & 17  & 7 & 4 & x  & 12   & 2  & 4  & 3 & x  & 31 & 17 & 8 & 105  \\\\\n$2^{+-}$ $(H_4)$ & x &  x  &  23  & 6  & 2 & 22  & 10  & 14  & 9 & x   &  x & 28 & 29 & 139  \\\\\n$3^{+-}$ $(H_4)$&  x   &  x    &   0  &  x    &  9 & x   & x & 40 & 13 & x & x   & x   & 0 & 62\\\\\n\\hline\\hline\n\\end{tabular}\n\\label{tab:widthsHeavy}\n\\end{table}\n\nTable \\ref{tab:widthsHeavy} presents predicted widths when the hybrid masses are fixed according to the lattice results with $m_\\pi=236$ MeV\\cite{Cheung:2016bym}. In this case more S+P channels open for the $H_1$ multiplet and these states attain typical charmonium widths. \nAs an illustration of an application of these predictions, we show total hybrid widths for $m_\\pi= 236$ MeV hybrid  masses in Table \\ref{tab:totals}. Widths have been normalized by identifying the $H_1(1^{--})$ as the $\\psi(4660)$. (It is worth stressing that this identification is not entirely satisfactory as corresponding lattice mass is 4411 MeV.) One observes that the predicted widths happen to be close to those obtained with our assumed coupling (Table \\ref{tab:widthsHeavy}).\n\n\n\n\\begin{table}[ht]\n\\caption{Total Hybrid Widths (MeV). Normalized by assuming $H_1(1^{--}) \\leftrightarrow \\psi(4660)$.}\n\\begin{tabular}{llll|llll|ll|llll}\n\\hline\\hline\n\\multicolumn{4}{c|}{$H_1$} & \\multicolumn{4}{c|}{$H_2$} & \\multicolumn{2}{c|}{$H_3$} & \\multicolumn{4}{c}{$H_4$} \\\\\n$1^{--}$ & $0^{-+}$ & $1^{-+}$ & $2^{-+}$ & $1^{++}$ & $0^{+-}$ & $1^{+-}$ & $2^{+-}$ & $0^{++}$ & $1^{+-}$ & $2^{++}$ & $1^{+-}$ & $2^{+-}$ & $3^{+-}$ \\\\\n\\hline\n70 & 83 & 30 & 66 & 50 & 57 & 53 & 90 & 103 & 81 & 103 & 113 & 154 & 68 \\\\\n\\hline\\hline\n\\end{tabular}\n\\label{tab:totals}\n\\end{table}\n\nIt is interesting to enquire into the feasibility of using decay characteristics as a signature of gluonic content. Table \\ref{tab:comparison} shows partial decay widths for the cryptoexotic $H_1(1^{--})$ at an assumed mass of 4111 MeV and the $4{}^3S_1$ charmonium vector at the same mass. The latter decay rates have been computed with the ``3P0\" model\\cite{Barnes:2005pb}. Although the predicted total widths are similar, it is evident that the decay patterns are quite different; especially in the S+S channels and in the reversed roles played by $D_{1L}D$\/$D_{1H}D$ and $D_2D$\/ $D_0D^*$ decays.\n\n\n\n\\begin{table}[ht]\n\\caption{Cryptoexotic $H_1(1^{--})$ and $\\psi(4{}^3S_1)$ Decay Modes (assuming masses of 4411 MeV).}\n\\begin{tabular}{l|cccccccccc|c}\n\\hline\\hline\nstate & $DD$ & $D^*D$ & $D^*D^*$  & $D_{1L}D$ & $D_{1H}D$ & $D_2D$ & $D_0D^*$ & $D_sD_s$ & $D_s D_s^*$ & $D_s^*D_s^*$ & width\\\\\n\\hline\n$H_1(1^{--})$ & 0 & 0.078 & 0 &  8 & 33 & 0.0035 & 25 & 0 & 0.2 & 0 & 66 \\\\\n$\\psi(4{}^3S_1)$ & 0.4 & 2.3 & 16 & 31 & 1 & 23 & 0 & 1.3 & 2.6 & 0.7 & 78 \\\\\n\\hline\\hline\n\\end{tabular}\n\\label{tab:comparison}\n\\end{table}\n\n\n\\section{Conclusions}\n\\label{sec:conc}\n\n\nWe have developed a simple model of hybrid structure that  leverages guidance from QCD in Coulomb gauge and from lattice field theory computations. In particular we assume that an axial constituent gluon provides a reasonable facsimile of low lying gluonic degrees of freedom. Certainly, this is in agreement with recent lattice computations of the charmonium spectrum. Furthermore, when the interactions of the constituent gluon are sufficiently tuned (in particular, by including three-body interactions), it is possible to maintain consistency with the gluelump spectrum and the gluonic adiabatic energy surfaces. Using these observations as a starting point has led to a reasonable spin-averaged spectrum shown in Fig. \\ref{fig:Hmasses} and lends confidence that a more detailed calculation should be able to provide an accurate representation of the lowest hybrid multiplets. It will be interesting to improve the spectrum computation and to include spin-splittings. Having a detailed structure model will also permit the computation of other observables of interest, such as radiative couplings.\n\n\nAdopting a constituent gluon model for hybrid structure leads naturally to the assumption that quark pair production from the valence gluon describes strong hybrid meson decay, resurrecting an old model of Tanimoto. The model incorporates several familiar selection rules: spin zero hybrids do not decay to spin zero mesons, TE hybrids do not decay to identical mesons or mesons with identical spatial wavefunctions, and S+P wave decay modes are preferred. We note that the structure of the decay vertex satisfies the general conditions of Ref. \\cite{Barnes:2007xu}, and therefore hybrid mesons will experience no relative mass shifts if spin-averaged hadron masses are assumed. \n\nWith respect to the present work, the major remaining issue is the determination of the quark-gluon coupling. The most likely way forward is a direct lattice computation of an open flavour decay. Alternatively, the dependence of hybrid masses on the light sea quark mass evident in Table \\ref{tab:spectrum} (the quoted lattice computations are unquenched) is an indication that relatively large couplings of these states to the continuum occur. The fact that the mass shifts are comparable to those of canonical charmonia is also an indication that the widths predicted here are of the correct magnitude.\n\n\n\n\\acknowledgments\n\nSwanson acknowledges support by the U.S. Department of Energy under contract DE-SC0019232. Garcia-Tecocoatzi acknowledges financial support from Consejo Nacional de Ciencia y Tecnolog\u00eda, Mexico.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nIt was  found \\cite{Mancini96} (see also \\cite{IbortPhysScr}) that the quantum states can\nbe described by fair probability distributions, which contain the same complete\ninformation of quantum states that is contained in density operator \\cite{Landau1927}\nand different its representations like Wigner function \\cite{Wigner32},\nHusimi Q-function \\cite{Husimi40} and diagonal representation P-function\nof Glauber \\cite{Glauber63} and Sudarshan \\cite{Sudarshan63}.\nSuch a probability distributions depend on homodyne variables and reference frames parameters\nand are called  optical tomograms \\cite{BerBer,VogRis} and symplectic tomograms\n\\cite{Mancini95}.\n\nIt is obvious, that not all of functions of these variables can be a tomogram\nof a physical state of quantum or classical system. In Ref. \\cite{OConnell}\ntwo sets of conditions, which are necessary and sufficient for a function\ndefined on phase space to be a Wigner function, were discussed.\nIn Ref. \\cite{Simoni2010} conditions for a symplectic tomogram-like function to be \na tomogram were formulated based on association of symplectic tomograms \nwith a unitary representation of the Weyl-Heisenberg group.\n\nThe evolution equation of symplectic tomograms of quantum systems was obtained in \\cite{ManciniFoundPhys97}.\nFor the optical tomograms such a dynamical equation was firstly  found in \\cite{Korarticle2}.\nWe also tried to obtain the tomographic evolution equations for relativistic\nsystems in Ref. \\cite{Korarticle1}. \n\nThe aim of our article is to explore the conditions, which must be satisfied by\noptical tomograms of both quantum and classical states. \nWe focus on the problem of the conditions for optical tomograms due to these\ntomograms are directly measured by homodyne detectors in quantum optical\nexperiments (see \\cite{Raymer93,LvovRayRevModPhys}).\n\nWe also study the conditions under which the tomographic evolution equations \nretain the normalization of their solutions.\nIt turns out that these conditions play a decisive role in numerical findings of the solutions\nof such an equations because these equations conserves normalizations only \nfor a narrow class of functions, which obey to the special relations,\nand limited accuracy of numerical calculations can lead to non-normalized solutions.\n\nThe paper is organized as follows. In Sec.\\ref{Section2} we review the definitions and main properties of optical\nand symplectic tomograms for both classical and quantum states.\nIn Sec.\\ref{Section3} conditions for  optical and symplectic tomogram-like functions  to be \ntomograms of a physical states are given.\nIn Sec.\\ref{Section4} conditions for optical tomogram-like functions to retain normalization\nduring evolution are obtained.\nIn Sec.\\ref{Section5} conditions for conservation of \nnormalization of the symplectic tomograms during evolution are found.\nIn Sec.\\ref{Section6} we show that Liuoville equation, dynamical equations for Wigner function and Husimi function\ndo not demand of any additional conditions to conserve\nnormalizations of their solutions.\nIn Sec.\\ref{Section7} conclusion and prospectives are presented.\n\n\n\\section{\\label{Section2}Definitions and properties of optical and symplectic tomograms}\n\nFor spinless $N$ dimensional quantum systems  with the density matrix $\\hat \\rho(t)$, which can be \ndependent on time,  the corresponding \noptical $w(\\mathbf X,\\bm\\theta,t)$ and symplectic $M(\\mathbf X,\\bm\\mu,\\bm\\nu,t)$ tomograms are defined as follows\n(see \\cite{Korarticle5,IbortPhysScr})\n\\be\t\t\\label{eq_43}\nw(\\mathbf X,\\bm\\theta,t)=\\mathrm{Tr}\\left\\{\n\\hat\\rho(t) \\prod_{\\sigma=1}^{N}\n\\delta \\left(X_\\sigma-\\hat q_\\sigma\\cos\\theta_\\sigma\n-\\hat p_\\sigma\\frac{\\sin\\theta_\\sigma}{m_\\sigma\\omega_\\sigma}\\right)\n\\right\\},\n\\ee\n\\be\t\t\\label{eq_44}\nM(\\mathbf X,\\bm\\mu,\\bm\\nu,t)=\\mathrm{Tr}\\left\\{\n\\hat\\rho(t) \\prod_{\\sigma=1}^{N}\n\\delta \\left(X_\\sigma-\\mu_\\sigma \\hat q_\\sigma-\\nu_\\sigma \\hat p_\\sigma\\right)\n\\right\\},\n\\ee\nwhere frequency $\\omega_{\\sigma}$ and mass $m_\\sigma$ dimensional constants \nfor $\\sigma-$th degree of freedom  are chosen from convenience considerations for particular Hamiltonian.\nTo simplify the formulas hereafter we choose the system of units\nsuch that $m_\\sigma=\\omega_\\sigma=\\hbar=1$.\nGiven the Wigner function of the state of quantum system or the classical\ndistribution function on phase space $W(\\mathbf q,\\mathbf p,t)$, for optical and symplectic tomograms\nwe can write\n\\be\t\t\\label{optfromWig}\nw(\\mathbf X,\\bm\\theta,t)=\n\\int~W(\\mathbf q,\\mathbf p,t)\\prod_{\\sigma=1}^{N}\n\\delta \\left(X_\\sigma-q_\\sigma\\cos\\theta_\\sigma\n-p_\\sigma\\sin\\theta_\\sigma\\right)\nd^Nq~d^Np,\n\\ee\n\\be\t\t\\label{sympfromWig}\nM(\\mathbf X,\\bm\\mu,\\bm\\nu,t)=\n\\int~W(\\mathbf q,\\mathbf p,t)\\prod_{\\sigma=1}^{N}\n\\delta (X_\\sigma-\\mu_\\sigma q_\\sigma-\\nu_\\sigma p_\\sigma)d^Nq~d^Np.\n\\ee\nWe suppose that  the normalization of the Wigner function is unity.\n\nThe inverse transforms of maps (\\ref{optfromWig}) and (\\ref{sympfromWig})\nare given by the Fourier integrals: \n\\be\t\t\\label{eq_52}\nW(\\mathbf q,\\mathbf p,t)=\\int\\limits_{0}^{\\pi}d^N\\theta\n\\int\\limits_{-\\infty}^{+\\infty}\\frac{d^N\\eta~d^NX}{(2\\pi)^{2N}}\nw(\\mathbf X,\\bm\\theta,t)\\prod_{\\sigma=1}^{N}\n|\\eta_\\sigma|\\exp\\left\\{i\\eta_\\sigma\\left(X_\\sigma\n-q_\\sigma\\cos\\theta_\\sigma-p_\\sigma\\sin\\theta_\\sigma\\right)\\right\\} ,\n\\ee\n\\be\t\t\\label{eq_53}\nW(\\mathbf q,\\mathbf p,t)=\\frac{1}{(2\\pi)^{2N}}\\int M(\\mathbf X,\\bm\\mu,\\bm\\nu,t)\n\\prod_{\\sigma=1}^{N}\\exp\\left\\{ i \\left(X_\\sigma-\\mu_\\sigma q_\\sigma\n-\\nu_\\sigma p_\\sigma\\right)\\right\\}d^NX~d^N\\mu~d^N\\nu.\n\\ee\n\nTomograms  are nonnegative and normalized by the conditions\n\\be\t\t\t\t\\label{eqnormOpt}\n\\int w(\\mathbf X,\\bm\\theta,t)d^NX=1,\n\\ee\n\\be\t\t\t\t\\label{eqnormSymp}\n\\int M(\\mathbf X,\\bm\\mu,\\bm\\nu,t)d^NX=1.\n\\ee\n\n\nSince the Dirac delta-function in Eq.(\\ref{eq_44}) is homogeneous function,\ni.e. $\\delta(\\lambda y)=|\\lambda|^{-1}\\delta(y)$, the symplectic tomogram is also the homogeneous function\n\\be                             \\label{eq_3}\nM(\\lambda_\\sigma X_\\sigma,\\lambda_\\sigma\\mu_\\sigma,\\lambda_\\sigma\\nu_\\sigma,t)=\nM(\\mathbf X,\\bm\\mu,\\bm\\nu,t)\\prod_{\\sigma=1}^N|\\lambda_\\sigma |^{-1}.\n\\ee\nThe optical tomogram is  even function. It means that the optical tomogram has the property\n\\be\t\t\\label{eq_14}\nw(-X_\\sigma,\\theta_\\sigma+\\pi,t)=w(\\mathbf X,\\bm\\theta,t).\n\\ee\nIt is obvious, that \n\\be\t\t\\label{OptSym}\nw(\\mathbf X,\\bm\\theta,t)=M(\\mathbf X,\\mu_\\sigma=\\cos\\theta_\\sigma,\\nu_\\sigma=\\sin\\theta_\\sigma,t).\n\\ee\nThe homogeneity condition (\\ref{eq_3}) provides the relation of the optical and symplectic tomograms opposite to (\\ref{OptSym}).\nOne can transform the integral (\\ref{sympfromWig}) using polar coordinates \n$\\mu_\\sigma=r_\\sigma\\cos\\theta_\\sigma$, $\\nu_\\sigma=r_\\sigma\\sin\\theta_\\sigma$. After some calculations\nfor $N-$dinentional case we can write\n\\be                             \\label{eq_6}\nM(\\mathbf X,\\bm\\mu,\\bm\\nu)=w\\left(\\frac{X_\\sigma\\mathrm{sgn}(\\nu_\\sigma)}{\\sqrt{\\mu_\\sigma^2+\\nu_\\sigma^2}},\n\\cot^{-1}\\frac{\\mu_\\sigma}{\\nu_\\sigma}\\right)\n\\prod_{\\sigma=1}^N\\frac{1}{\\sqrt{\\mu_\\sigma^2+\\nu_\\sigma^2}}.\n\\ee\nAlso the symplectic tomogram satisfy an extra homogeneous\ndifferential relation. Making differentiation of (\\ref{eq_3}) by $\\lambda_i$\nand substituting $\\lambda_i=1$ we obtain\n$$\nM+X_i\\partial_{X_i}M+\\mu_i \\partial_{\\mu_i}M\n+\\nu_i\\partial_{\\nu_i}M=0.\n$$\nIn addition, optical tomograms must satisfy the entropic uncertainty relation\n(see \\cite{MAMankoAIP2011}) (Hirshman criterion)\n\\be\t\t\t\\label{Hirschman}\n-\\int w(\\mathbf X,\\bm\\theta,t)\\ln w(\\mathbf X,\\bm\\theta,t)d^NX\n-\\int w(\\mathbf X,\\bm\\theta+\\bm{\\pi}\/2,t)\\ln w(\\mathbf X,\\bm\\theta+\\bm{\\pi}\/2,t)d^NX \\geq N\\ln(\\pi e).\n\\ee\nHere $S(\\bm\\theta,t)=-\\int w(\\mathbf X,\\bm\\theta,t)\\ln w(\\mathbf X,\\bm\\theta,t)d^NX$\nis the tomographic Shannon entropy associated with optical tomogram.\n\nIn Ref. \\cite{Simoni2010} it was mentioned that non-negativity, normalization and homogeneity\nproperties are not sufficient for the symplectic tomogram-like function $f(\\mathbf X,\\bm\\mu,\\bm\\nu)$\nto be a tomogram. Also non-negativity, normalization (\\ref{eqnormOpt}), parity (\\ref{eq_14}),\nand satisfaction of Hirshman criterion (\\ref{Hirschman}) are not sufficient for optical \ntomogram-like function $f(\\mathbf X,\\bm\\theta)$ to be a tomogram of any quantum or\nclassical state.\nFor example, consider\n\\be\t\t\\label{example1}\nf(X,\\theta)=\\frac{1}{\\sqrt\\pi}e^{-(X-\\cos^3\\theta)^2}.\n\\ee\nIn spite of this function is positive and satisfy  conditions (\\ref{eqnormOpt}),\n(\\ref{eq_14}), and (\\ref{Hirschman}), it is not an optical tomogram. More over, as it will\nbe seen from Section \\ref{Section4}, the normalization of this function will not be conserved \nduring time evolution.\n\n\\section{\\label{Section3}Conditions for tomogram-like functions  to be \ntomograms of physical states}\n\nIn Ref. \\cite{OConnell} necessary and sufficient conditions for a phase-space function \nto be a Wigner distribution of physical system are studied. Such function is a Wigner function\nif and only if it defines a kernel (density matrix) for a positive trace-class operator\nwith trace equal to one. The authors consider two sets of necessary and\nsufficient conditions and show that these sets are formally equivalent.\n\n\\textbf{The first} more familiar \\textbf{set} is that the function $W(\\mathbf q,\\mathbf p)$ must be\nsquare integrable, and for every Wigner function $W_\\psi(\\mathbf q,\\mathbf p)$\nof a pure state $\\psi$ the following inequality must be valid\n\\be\t\t\\label{conditionW1}\n\\int W(\\mathbf q,\\mathbf p)W_\\psi(\\mathbf q,\\mathbf p)d^Nq~d^Np\\geq 0,\n\\ee\nand the function $W(\\mathbf q,\\mathbf p)$ must be normalized to unity\nover all phase space,\n\\be\t\t\\label{normWig}\n\\int W(\\mathbf q,\\mathbf p)d^Nq~d^Np=1.\n\\ee\nOne may find the proof that these conditions are necessary and\nsufficient in \\cite{PhisRep1984}.\n\nLet us note that from the physical point of view the integral in inequality \n(\\ref{conditionW1}) is a transition probability from the state with Wigner function \n$W(\\mathbf q,\\mathbf p)$ to the state $\\psi$ and, obviously, it must be non-negative.\n\n\\textbf{The second set} of conditions is called in \\cite{OConnell} as KLM conditions,\nafter Kastler \\cite{Kastler}, Loupias, and Miracle-Sole \\cite{Loupias1,Loupias2},\nand applied to the symplectic Fourier transform of the \nphase space function $W(\\mathbf q,\\mathbf p)$\n\\be\t\t\\label{sympFour}\n\\widetilde W(\\mathbf{u},\\mathbf{v})=\\int W(\\mathbf{q},\\mathbf{p})\ne^{i(\\mathbf q\\mathbf v-\\mathbf u\\mathbf p)}d^Nq~d^Np.\n\\ee\nThe first KLM condition, which is equivalent to (\\ref{conditionW1}), is that the function \n$\\widetilde W(\\mathbf{u},\\mathbf{v})$ must be continuous and of $\\hbar-$positive type,\ni.e. for every choce of poins \n$(\\mathbf u_1,\\mathbf v_1)$, $(\\mathbf u_2,\\mathbf v_2)$, ..., $(\\mathbf u_n,\\mathbf v_n)$,\nthe $n\\times n$ matrix  with entries \n\\be\t\t\\label{matrixnonneg1}\nZ_{jk}=e^{i(\\mathbf u_j\\mathbf v_k-\\mathbf u_k\\mathbf v_j)\/2}\n\\widetilde W(\\mathbf u_j-\\mathbf u_k,\\mathbf v_j-\\mathbf v_k)\n\\ee\nis non-negative (in our system of units $\\hbar=1$).\nThe second KLM condition is equivalent to (\\ref{normWig})\n\\be\t\t\\label{normWig2}\n\\widetilde W(\\mathbf u=0,\\mathbf v=0)=1.\n\\ee\n\nUsing the formulas of maps of the Wigner function  to the tomogram and vice versa\none can find necessary and sufficient conditions for tomographic functions corresponding to (\\ref{conditionW1}).\nNormalized according to (\\ref{eqnormOpt}) and (\\ref{eqnormSymp}) functions $w(\\mathbf X,\\bm\\theta)$ and $M(\\mathbf X,\\bm\\mu,\\bm\\nu)$ \nobeying relations  (\\ref{eq_14}) or (\\ref{eq_3})  are tomograms of\nstates of quantum systems if and only if for any quantum pure state $\\psi$ with\nthe tomogram $w_\\psi(\\mathbf X,\\bm\\theta)$ (or $M_\\psi(\\mathbf X,\\bm\\mu,\\bm\\nu)$),\nthe following inequalities are valid\n\\be\t\t\t\\label{conditwpos}\n\\int w(\\mathbf X,\\bm\\theta)w_\\psi(\\mathbf X,\\bm\\theta)\ne^{i\\bm\\eta(\\mathbf X-\\mathbf X')}|\\eta_1||\\eta_2|\\dots|\\eta_n|\nd^nX~d^NX'~d^n\\eta~d^N\\theta\\geq 0, \n\\ee\n\\be\t\t\t\\label{conditMpos}\n\\int M(\\mathbf X,\\bm\\mu,\\bm\\nu)M_\\psi(\\mathbf X',\\bm\\mu,\\bm\\nu)\ne^{i(\\mathbf X-\\mathbf X')}\nd^nX~d^nX'~d^N\\mu~d^N\\nu\\geq 0. \n\\ee\nAlso with the map between functions $W(\\mathbf q,\\mathbf p,t)$ and $M(\\mathbf X,\\bm\\mu,\\bm\\nu,t)$\n the symplectic Fourier transform (\\ref{sympFour}) is expressed as follows\n\\be\t\t\\label{symFsym}\n\\widetilde W(\\mathbf{u},\\mathbf{v})=\\int M(\\mathbf X,\\mathbf{v},-\\mathbf{u})\ne^{i\\mathbf X}d^NX.\n\\ee\nSubstituting (\\ref{symFsym}) to the first KLM condition with the set of points\n$\\{(\\bm\\mu_j,\\bm\\nu_j):\\mathbf{v}_j=\\bm\\mu_j,\\mathbf{u}_j=-\\bm\\nu_j \\}$,\nwhich will also be arbitrary, we obtain the expression for non-negative matrix $Z_{jk}$\nfrom symplectic tomogram\n\\be\t\t\\label{matrixnonneg2}\nZ_{jk}=e^{i(\\bm\\mu_j\\bm\\nu_k-\\bm\\mu_k\\bm\\nu_j)\/2}\n\\int M(\\mathbf X,\\bm\\mu_j-\\bm\\mu_k,\\bm\\nu_j-\\bm\\nu_k)e^{i\\mathbf X}d^NX.\n\\ee\nWith the help of relation (\\ref{eq_6}) between $M(\\mathbf X,\\bm\\mu,\\bm\\nu)$ and\n$w(\\mathbf X,\\bm\\theta)$ we immediately obtain the expression for the matrix $Z_{jk}$\nin terms of the optical tomogram\n\\bea\t\t\nZ_{jk}&=&e^{i(\\bm\\mu_j\\bm\\nu_k-\\bm\\mu_k\\bm\\nu_j)\/2} \\nonumber \\\\[3mm]\n&&\\times\\int w\\left(X_\\sigma,\n\\cot^{-1}\\frac{\\mu_{\\sigma j}-\\mu_{\\sigma k}}{\\nu_{\\sigma j}-\\nu_{\\sigma k}}\\right)\n\\nonumber \\\\[3mm]\n&&\\times\\exp\\left[i\\sum_{\\sigma=1}^N X_\\sigma\\mathrm{sgn}(\\nu_{\\sigma j}-\\nu_{\\sigma k})\n\\sqrt{(\\mu_{\\sigma j}-\\mu_{\\sigma k})^2+(\\nu_{\\sigma j}-\\nu_{\\sigma k})^2}\\right]d^NX.\n\\label{matrixnonneg3}\n\\eea\nAs for condition (\\ref{normWig2}), it will be valid as the limit case \n\\be\t\t\\label{normWig3}\n\\widetilde W(0,0)=\\int M(\\mathbf X,0,0)e^{i\\mathbf X}d^NX=\n\\int w\\left(\\mathbf X ,\\cot^{-1}\\frac{0}{0}\\right)d^NX=1,\n\\ee\nbecause\nthe symplectic tomogram transforms to the delta-function\n$M(\\mathbf X,0,0)=\\delta(\\mathbf X)$, and the function $\\cot^{-1}(x)$ is confined at any argument.\nThus, instead of (\\ref{normWig3}) it is more preferably to use conditions (\\ref{eqnormOpt}) and (\\ref{eqnormSymp}),\nwhich must be valid for any $(\\bm\\mu,\\bm\\nu)\\neq0$, or phase vector $\\bm\\theta$ $\\{\\theta_j\\in[0,\\pi]\\}$.\n\nIn Ref. \\cite{Simoni2010} the same expression (\\ref{matrixnonneg2}) \nfor non-negative matrix $Z_{jk}$ was obtained from considerations of group theory\nand called as $\\omega-$positivity condition. It was shown \\cite{Simoni2009} that symplectic tomograms \nare associated with nontrivial unitary irreducible representations\n(we generalize here the results of \\cite{Simoni2010,Simoni2009} to $N-$dimensional case\nand to optical tomograms)\n\\be\t\t\\label{Weilrep}\n\\hat U_g(\\bm\\mu,\\bm\\nu,\\xi)=\\exp[i(\\bm\\mu\\hat{\\mathbf q}+\\bm\\nu\\hat{\\mathbf p}\\,)]e^{i\\xi}\n\\ee\nof the Weyl-Heisenberg group WH(2N). So, according to Naimark's theorem\n\\cite{Naimark} the function \n\\bea\t\t\n\\varphi(\\bm\\mu,\\bm\\nu,\\xi)&=&\\mathrm{Tr}\\left\\{\\hat\\rho\n\\exp\\left[i\\left(\\bm\\mu\\hat{\\mathbf q}+\\bm\\nu\\hat{\\mathbf p}\\,\\right)\\right]\\right\\}e^{i\\xi}\n=e^{i\\xi}\\int M(\\mathbf X,\\bm\\mu,\\bm\\nu)e^{i\\mathbf X}d^NX \\nonumber \\\\[3mm]\n&=&e^{i\\xi}\\int w\\left(\\mathbf X,\\cot^{-1}(\\mu_\\sigma\/\\nu_\\sigma)\\right) \n\\exp\\left[i\\sum_{\\sigma=1}^N X_\\sigma\\mathrm{sgn}(\\nu_{\\sigma})\n\\sqrt{\\mu_{\\sigma}^2+\\nu_{\\sigma}^2}\\right]d^NX\n\\label{funcongr}\n\\eea\nis a positive definite function on the Weyl-Heisenberg group, i.e.\nfor any $n-$tuple of group elements $(g_1,g_2,...,g_n)\\in \\mathrm{WH}(2N)$\nthe $n\\times n$ matrix with elements\n\\be\t\t\\label{matrixnonneg4}\nZ_{jk}=\\varphi(g_jg_k^{-1})\n\\ee\nis non-negative. Substitution of the expression of representation of the \ngroup element $g_jg_k^{-1}$\n\\be\t\t\\label{prod1}\n\\hat U_{g_j}(\\bm\\mu_j,\\bm\\nu_j,\\xi_j)\\hat U_{g_k}^{-1}(\\bm\\mu_k,\\bm\\nu_k,\\xi_k)=\n\\hat U_{g_jg_k^{-1}}\\Big[\\bm\\mu_j-\\bm\\mu_k,\\bm\\nu_j-\\bm\\nu_k,\\xi_j-\\xi_k+\n(\\bm\\mu_j\\bm\\nu_k-\\bm\\mu_k\\bm\\nu_j)\/2\\Big]\n\\ee\nto  definition (\\ref{funcongr}) of the function $\\varphi$ and omission of the \ninessential factors $e^{i\\xi_j}$ and $e^{i\\xi_k}$ lead us to expressions of the \nmatrix $Z_{jk}$ for symplectic (\\ref{matrixnonneg2}) and optical (\\ref{matrixnonneg3})\ntomograms.\n\nIn order to functions $w_\\mathrm{cl}(\\mathbf X,\\bm\\theta)$ or $M_\\mathrm{cl}(\\mathbf X,\\bm\\mu,\\bm\\nu)$\nbe tomograms of classical systems, in addition to their smoothness, normalization in accordance with \nconditions (\\ref{eqnormOpt}, \\ref{eqnormSymp}), parity (\\ref{eq_14}) or homogeneity (\\ref{eq_3}) respectively,\nnecessary and sufficient conditions is that they must define the function $W_\\mathrm{cl}(\\mathbf q,\\mathbf p)$ \nwith formula (\\ref{eq_52}) or (\\ref{eq_53}), which consist in a class of distribution functions,\ni.e. positive normalized functions.\n\nIn terms of group theory  the function $\\phi(\\bm\\mu,\\bm\\nu)$ \nwith the definition\n\\bea\t\t\n\\phi(\\bm\\mu,\\bm\\nu)&=&\\int W_\\mathrm{cl}(\\mathbf q,\\mathbf p)\n\\exp\\left[i\\left(\\bm\\mu\\mathbf{q}+\\bm\\nu\\mathbf{p}\\,\\right)\\right]d^Nq~d^Np\n=\\int M_\\mathrm{cl}(\\mathbf X,\\bm\\mu,\\bm\\nu)e^{i\\mathbf X}d^NX \\nonumber \\\\[3mm]\n&=&\\int w_\\mathrm{cl}\\left(\\mathbf X,\\cot^{-1}(\\mu_\\sigma\/\\nu_\\sigma)\\right) \n\\exp\\left[i\\sum_{\\sigma=1}^N X_\\sigma\\mathrm{sgn}(\\nu_{\\sigma})\n\\sqrt{\\mu_{\\sigma}^2+\\nu_{\\sigma}^2}\\right]d^NX\n\\label{funcongrcl}\n\\eea\non the translational group with group law \n\\be\t\t\t\\label{transgrlaw}\n(\\bm\\mu,\\bm\\nu)\\circ(\\bm\\mu\\,',\\bm\\nu\\,')=(\\bm\\mu+\\bm\\mu\\,',\\bm\\nu+\\bm\\nu\\,')\n\\ee \nmust be a positive definite function on this group (see \\cite{Simoni2010}), \ni.e. for every choice of points\n$(\\bm\\mu_1,\\bm\\nu_1)$, $(\\bm\\mu_2,\\bm\\nu_2)$, ..., $(\\bm\\mu_n,\\bm\\nu_n)$\nthe $n\\times n$ matrix  with entries $\\phi_{jk}=\\phi(\\bm\\mu_j-\\bm\\mu_k,\\bm\\nu_j-\\bm\\nu_k)$\nmust be non-negative.\nThen, by the Bochner theorem the function $\\phi(\\bm\\mu,\\bm\\nu)$ is the Fourier transform \nof the probability measure on the phase space.\n\nNote \\cite{Simoni2010}  that the function $\\phi(\\bm\\mu,\\bm\\nu)=e^{-i\\xi}\\varphi(\\bm\\mu,\\bm\\nu,\\xi)$\nmay be simultaneously of $\\hbar-$positive and positive type on the translational group\nwith group law (\\ref{transgrlaw}).\n In this case the corresponding tomogram-like function \n$w(\\mathbf X,\\bm\\theta)$ or $M(\\mathbf X,\\bm\\mu,\\bm\\nu)$ may be interpreted as a quantum as a classical tomogram.\nSuch an example is the tomogram of the ground state of the harmonic oscillator.\n\nLet's additionally note that conditions considered are always necessary but they are sufficient if and only if \nthe Radon integral  of tomogram-like function of type (\\ref{optfromWig})  \nor (\\ref{sympfromWig}) with $W(\\mathbf q,\\mathbf p,t)$\ndefined by (\\ref{eq_52}) or (\\ref{eq_53})  exist and equal to the function itself.\nAs an example, examine the function\n\\be\t\t\t\\label{exampfunc1}\nf_1(X,\\mu,\\nu)=\\frac{e^{1\/4}}{\\sqrt\\pi}e^{-X^2-\\mu^2\/4-\\nu^2\/4}.\n\\ee\nEvidently, that this function is neither quantum no classical tomogram.\nNevertheless, it is easy to see that both the first and the second KLM conditions for this\nfunction are satisfied:\n\\bdm\n\\widetilde f_1(\\mu,\\nu)=\\int f_1(X,\\mu,\\nu)e^{iX}dX=\ne^{-\\mu^2\/4-\\nu^2\/4},~~~~\\widetilde f_1(X,0,0)=1,\n\\edm\nand $\\widetilde f_1(\\mu,\\nu)$ is $\\hbar-$positive function.\nThe thing is that, if we apply transformation (\\ref{eq_53}) to this function, \nwe obtain the classical distribution $W_1(q,p)=\\pi^{-1\/2}e^{-q^2-p^2}$ \n(which in this special example  can also be interpreted as\na Wigner function of  the ground state of the harmonic oscillator),\nbut if we apply to $W_1(q,p)$ map (\\ref{sympfromWig}), which is inverse to (\\ref{eq_53}), we  obtain \nthe function that does not equal to $f_1(X,\\mu,\\nu)$.\nSo, Radon integral (\\ref{eq_53})  of the function $f_1(X,\\mu,\\nu)$ exist but does not \nequal to the function $f_1(X,\\mu,\\nu)$, and KLM conditions are not  sufficient \nfor the tomogram-like function $f_1(X,\\mu,\\nu)$ to be a tomogram of the state of a physical quantum \nor classical system.\n\n\\section{\\label{Section4}Conditions for conservation of normalization \\\\ \nof optical tomogram-like functions during evolution}\nFor Hamiltonians\n\\be                             \\label{r18_2}\n\\hat H=\\sum_\\sigma\\frac{\\hat p_\\sigma^2}{2m_\\sigma}+V({\\mathbf q},t)\n\\ee\nthe evolution equation \nof the optical tomogram  $w({\\mathbf X},{\\bm\\theta},t)$ \nfirstly was found in \\cite{Korarticle2}\n\\begin{eqnarray}                             \n&&\n\\frac{\\partial}{\\partial t}w({\\mathbf X},{\\bm\\theta},t)=\n\\sum_{\\sigma=1}^N\\omega_{0\\sigma}\n\\left[\\cos^2\\theta_\\sigma\\frac{\\partial}{\\partial\\theta_\\sigma}\n-\\frac{1}{2}\\sin2\\theta_\\sigma\\left\\{1+X_\\sigma\\frac{\\partial}{\\partial X_\\sigma}\\right\\}\n\\right]\nw({\\mathbf X},{\\bm\\theta},t)\\nonumber \\\\[3mm]\n&&+\n\\frac{2}{\\hbar}\\left[\\mathrm{Im}~V\\left\\{\n\\sin\\theta_\\sigma\\frac{\\partial}{\\partial\\theta_\\sigma}\n\\left[\\frac{\\partial}{\\partial X_\\sigma}\\right]^{-1}\n+X_\\sigma\\cos\\theta_\\sigma+i\\frac{\\hbar\\sin\\theta_\\sigma}\n{2m_\\sigma\\omega_{0\\sigma}}\n\\frac{\\partial}{\\partial X_\\sigma}\\right\\}\\right]\nw({\\mathbf X},{\\bm\\theta},t), \n\\label{r20_2}\n\\end{eqnarray}\nwhere we introduced the designation \\cite{Korarticle5}\n\\be\t\t\t\\label{invder}\n\\left[\\frac{\\partial}{\\partial X_\\sigma}\\right]^{-n}f(X_\\sigma)=\\frac{1}{(n-1)!}\n\\int(X_\\sigma-X_\\sigma')^{n-1}\\Theta(X_\\sigma-X_\\sigma')f(X_\\sigma')dX_\\sigma',\n\\ee\nwhere $\\Theta(X_\\sigma-X_\\sigma')$ is a Heaviside step function.\nOptical tomographic representation of the classical non-relativistic Liouville equation\n\\cite{Korarticle2} is the limit case of (\\ref{r20_2}) when $\\hbar\\to 0$\n\\begin{eqnarray}                             \n&&\\ds{\\frac{\\partial}{\\partial t}w_\\mathrm{cl}(\\mathbf X,\\bm\\theta,t)=\n\\sum_{\\sigma=1}^{N}\\omega_\\sigma \\left[\\cos^2\\theta_\\sigma\\frac{\\partial}{\\partial\\theta_\\sigma}\n-\\frac{1}{2}\\sin2\\theta_\\sigma\\left\\{1+X_\\sigma\\frac{\\partial}{\\partial X_\\sigma}\\right\\}\n\\right]w_\\mathrm{cl}(\\mathbf X,\\bm\\theta,t)} \\nonumber \\\\[3mm]\n&&\\ds{+\\left[\\sum_{\\sigma=1}^{n}\\frac{\\partial}{\\partial q_\\sigma}~V\\left\\{\nq_\\sigma\\rightarrow\\sin\\theta_\\sigma\\frac{\\partial}{\\partial\\theta_\\sigma}\n\\left[\\frac{\\partial}{\\partial X_\\sigma}\\right]^{-1}\n+X_\\sigma\\cos\\theta_\\sigma\\right\\}\n\\frac{\\sin\\theta_\\sigma}{m_\\sigma\\omega_\\sigma}\\frac{\\partial}{\\partial X_\\sigma}\\right]\nw_\\mathrm{cl}(\\mathbf X,\\bm\\theta,t).}\n\t\t\\label{eq_54}\n\\end{eqnarray}\nEvolution equation of the  tomogram for arbitrary spinless\nquantum Hamiltonian  was found in \\cite{Korarticle1}.\n\nEquations (\\ref{r20_2}), (\\ref{eq_54}) can be presented in the following compact form\n\\be \t\t\t\\label{eqcompFF}\n\\partial_t w({\\mathbf X},{\\bm\\theta},t)=\\hat F_\\mathrm{k}({\\mathbf X},{\\bm\\theta})~w({\\mathbf X},{\\bm\\theta},t)\n+\\hat F_\\mathrm{p}({\\mathbf X},{\\bm\\theta},t)~w({\\mathbf X},{\\bm\\theta},t),\n\\ee\nwhere $\\hat F_\\mathrm{k}({\\mathbf X},{\\bm\\theta})$ is a time independent operator\ncorresponding to the kinetic part of the Hamiltonian, and  \n$\\hat F_\\mathrm{p}({\\mathbf X},{\\bm\\theta},t)$ is an operator\ncorresponding to the potential energy of the system. \nGenerally speaking, $\\hat F_\\mathrm{p}({\\mathbf X},{\\bm\\theta},t)$ can be dependent on time\nif the potential $V(\\mathbf q,t)$ depend on time. More over, the operator \n$\\hat F_\\mathrm{k}$ for equations (\\ref{r20_2}) and (\\ref{eq_54}) is the same,\nbut the operators $\\hat F_\\mathrm{p}$ are different for the classical and quantum cases.\n\nIf we know the initial condition $w_0({\\mathbf X},{\\bm\\theta})$\nfor the equation with the form (\\ref{eqcompFF}),\nwe can write its formal general solution with the help of chronological exponential operator \n\\be\t\t\t\\label{solexp}\nw({\\mathbf X},{\\bm\\theta},t)=\\mathrm{T}\\exp\\left\\{\n\\int\\limits_0^t\\left[\\hat F_\\mathrm{k}({\\mathbf X},{\\bm\\theta})+ \n\\hat F_\\mathrm{p}({\\mathbf X},{\\bm\\theta},t)\\right]dt\\right\\}w_0({\\mathbf X},{\\bm\\theta}).\n\\ee \n\nWhich of the properties the tomograms must satisfy for conservation\nof the normalization  during evolution?\nLet's integrate left and right sides of equation (\\ref{eqcompFF})\nover the hyperspace $X^N$. If $w({\\mathbf X},{\\bm\\theta},t)$ is a continuous\nfunction normalized by unity\nand tends to zero faster then any finite negative power of $X_i$ when $X_i$\ntends to infinity, then the following relation must be valid\n\\be \t\t\t\\label{intzero}\n\\partial_t \\int w({\\mathbf X},{\\bm\\theta},t)dX^N=0=\n\\int\\hat F_\\mathrm{k}({\\mathbf X},{\\bm\\theta})~w({\\mathbf X},{\\bm\\theta},t)dX^N\n+\\int\\hat F_\\mathrm{p}({\\mathbf X},{\\bm\\theta},t)~w({\\mathbf X},{\\bm\\theta},t)dX^N.\n\\ee\nUsing integration by parts and taking into account condition (\\ref{eqnormOpt})\nit is easy to show that \n\\bdm\n\\int\\hat F_\\mathrm{k}({\\mathbf X},{\\bm\\theta})~w({\\mathbf X},{\\bm\\theta},t)d^NX=0.\n\\edm\nFor free motion $\\hat F_\\mathrm{p}\\equiv 0$ and equation (\\ref{intzero}) is valid\nwithout any additional properties of the tomogram. For linear potential\n$V(\\mathbf q)$ equation (\\ref{intzero}) is valid too.\nFor quadratic potential equation (\\ref{eqcompFF}) transforms to \n\\be\t\t\t\\label{eqossil}\n\\frac{\\partial}{\\partial t} w({\\mathbf X},{\\bm\\theta},t)=\\sum_{\\sigma=1}^{N} \\omega_\\sigma\n\\frac{\\partial}{\\partial\\theta_\\sigma}w({\\mathbf X},{\\bm\\theta},t),\n\\ee\nand we see that there is only normalization property (\\ref{eqnormOpt})\nmust be satisfied for tomograms to be normalized during evolution, i.e. for validity of equation (\\ref{intzero}).\n\nFurther, to simplify the formulas, we consider one-dimensional motion and choose a system \nof physical units in which $\\hbar=m=\\omega_0=1$. The generalization to multidimensional case is straightforward.\n\nAs we discuss the related motion, then without loss of physical \ngenerality, it suffices to consider the case when $V(q)$ is \na polynomial, because according to Chebyshev theorem, every continuous function $V(q)$ \non a finite interval with any accuracy can be approximated by a polynomial.\nEach term of the $n$-th power of $q$ of the polynomial potential will give \ncontribution to (\\ref{intzero}) as a sum of terms proportional to\n$\\cos^k\\theta\\sin^l\\theta,$ where $k+l=n$.\n\nSince equation (\\ref{intzero}) must be satisfied for any fixed \n$\\theta$ and $t$, then the sum of the coefficients of each term of the form \n$\\cos^k\\theta\\sin^l\\theta$ must be zero. Hence, we obtain \na system of equations for tomogram, which  must be satisfied \nfor any fixed $\\theta$ and $t$.\n\nSuppose that the potential $V(q)$ contains a term with $q^3$. \nThen, after some calculations, we can show that the quantum case \nequation (\\ref{intzero}) will have the form\n\\begin{eqnarray}                             \n0&=&-\\frac{\\sin^3\\theta}{4}\\int\\partial _Xw(X,\\theta,t)dX\n\\nonumber \\\\[3mm]\n&&+3\\sin\\theta\\cos^2\\theta\\left(2\\int Xw(X,\\theta,t)dX\n+\\int X^2\\partial _Xw(X,\\theta,t)dX\n\\right)\n\\nonumber \\\\[3mm]\n&&+6\\sin^2\\theta\\cos\\theta~\\partial_\\theta\\left(\n\\int \\partial^{-1}_Xw(X,\\theta,t)dX\n+\\int Xw(X,\\theta,t)dX\n\\right)\n\\nonumber \\\\[3mm]\n&&+3\\sin^3\\theta\\left(\n\\partial^2_\\theta \\int\\partial^{-1}_Xw(X,\\theta,t)dX\n-\\int Xw(X,\\theta,t)dX\n\\right).\n\\label{Vq3}\n\\end{eqnarray}\nThe same equation but without term in the first row is obtained for a classical motion. \nUsing integration by parts it is easy to show that \neach of the first three rows in equation (\\ref{Vq3}) is equal to zero, \nand fourth line leads us to the equation\n\\be\t\t\t\\label{ostatok}\n\\partial^2_\\theta \\int Xw(X,\\theta,t)dX\n+\\int Xw(X,\\theta,t)dX=0.\n\\ee\nIt is easy to see that this is a classical harmonic oscillator equation with variable $\\theta$, \nand it is performed only if\n\\be\t\t\t\\label{garmsol}\n\\int Xw(X,\\theta,t)dX=A(t)\\cos\\theta+B(t)\\sin\\theta.\n\\ee\nThus, we have the integral condition, which must be satisfied \nfor the optical tomogram both in the classical and in the quantum case \nin order to its normalization will be remained during evolution in the third degree polynomial potential.\nNote that from definition of tomogram (\\ref{eq_43}) \nwe can see that the left side of equality (\\ref{garmsol}) is a mean value of the\nhomodyne variable $X$,\n\\be\t\t\t\\label{garmsolQP}\n\\int Xw(X,\\theta,t)dX=\\langle\\hat q\\rangle\\cos\\theta\n+\\langle\\hat p\\rangle\\sin\\theta,\n\\ee\nwhere $\\langle\\hat q\\rangle$ and $\\langle\\hat p\\rangle$ are average quantum (classical) \nvalues of the position and momentum respectively.\nSo, we have $A(t)=\\langle q\\rangle$ and $B(t)=\\langle p\\rangle$.\n\nSuppose that the potential $V(q)$ contains a term with $q^4$. \nAfter similar calculations, taking into account (\\ref{eqnormOpt}) and (\\ref{ostatok})\nwe obtain the differential equation for the second moment \nvariable $X$, which is valid both in the quantum and in the classical case:\n\\be\t\t\t\\label{ostatok2}\n\\partial^3_\\theta \\int X^2w(X,\\theta,t)dX\n+4\\partial_\\theta \\int X^2w(X,\\theta,t)dX=0.\n\\ee\nThe solution of (\\ref{ostatok2}) contains three constants independent on $\\theta$, but dependent on time \n\\be\t\t\t\\label{garmsol2}\n\\int X^2w(X,\\theta,t)dX=A_1(t)\\cos2\\theta+B_1(t)\\sin2\\theta +C_1(t).\n\\ee\nFrom definition of tomogram  (\\ref{eq_43}) and the expression for the \nmean value of $X^2$ we can find the constants\n$A_1(t)$, $B_1(t)$, $C_1(t)$,\n\\be\t\t\t\\label{garmsolQP2}\n\\int X^2w(X,\\theta,t)dX=\\langle\\hat q^2\\rangle\\cos^2\\theta\n+\\langle\\hat p^2\\rangle\\sin^2\\theta\n+\\langle\\hat q\\hat p+\\hat p\\hat q\\rangle\\sin\\theta\\cos\\theta,\n\\ee\n\\bdm\nA_1(t)=\\frac{\\langle\\hat q^2\\rangle-\\langle\\hat p^2\\rangle}{2},~~~\nB_1(t)=\\frac{\\langle\\hat q\\hat p+\\hat p\\hat q\\rangle}{2},\n~~~ C_1(t)=\\frac{\\langle\\hat q^2\\rangle+\\langle\\hat p^2\\rangle}{2}.\n\\edm\nExpressing  condition (\\ref{garmsol2}) - (\\ref{garmsolQP2}) through the Wigner function we obtain\n\\bea\t\t\t\n\\int X^2w(X,\\theta,t)dX&=&\n\\left(\\int q^2W(q,p,t)dq~dp\\right)\\cos^2\\theta\n+\\left(\\int p^2W(q,p,t)dq~dp\\right)\\sin^2\\theta \\nonumber \\\\[3mm]\n&&+2\\left(\\int qp\\,W(q,p,t)dq~dp\\right)\\sin\\theta\\cos\\theta.\n\\label{garmsolQP2W}\n\\eea\nArguing similarly, after some calculations we can show that if $V(q)$ contains a term with $q^n$, \nthen all average values $\\langle X^m\\rangle$ of the variable $X$, where $m=1,~2,...,n-2$, \nmust satisfy one of the following differential equation:\n\\be\t\t\t\\label{Gendifeven}\n\\left\\{\\frac{\\partial}{\\partial\\theta}\\prod_{k=1}^{m\/2}\n\\left[\\frac{\\partial^2}{\\partial\\theta^2}+(2k)^2\\right]\\right\\}\n\\int X^{m}\\,w(X,\\theta,t)dX=0,\n\\ee\nif $m$ is even number, or \n\\be\t\t\t\\label{Gendifodd}\n\\left\\{\\prod_{k=0}^{(m-1)\/2}\n\\left[\\frac{\\partial^2}{\\partial\\theta^2}+(2k+1)^2\\right]\\right\\}\n\\int X^{m}\\,w(X,\\theta,t)dX=0,\n\\ee\nif $m$ is odd number.\n\nGeneral solutions of these equations are respectively equal:\n\\be\t\t\t\\label{Gensoleven}\n\\int X^{m}\\,w(X,\\theta,t)dX=A_0(t)+\\sum_{k=1}^{m\/2}[A_k(t)\\cos(2k\\theta)\n+B_k(t)\\sin(2k\\theta)],~~~~m~~\\mathrm{is~~even},\n\\ee\n\\be\t\t\t\\label{Gensolodd}\n\\int X^{m}\\,w(X,\\theta,t)dX=\\sum_{k=0}^{(m-1)\/2}\\{A_k(t)\\cos[(2k+1)\\theta]\n+B_k(t)\\sin[(2k+1)\\theta]\\},~~~~m~~\\mathrm{is~~odd},\n\\ee\nwhere $A_k(t)$, $B_k(t)$ are independent on $\\theta$, but dependent on time $t$.\n\nConsistent with (\\ref{Gensoleven}) or (\\ref{Gensolodd}), from formula (\\ref{optfromWig}) we have that\nall average values $\\langle X^m\\rangle$ \nare the sums of terms of the form $A_{k\\,m}(t)\\cos^k\\theta\\sin^{m-k}\\theta$, \nwhere $k=0,~1,...,m$ and $A_{k\\,m}(t)$ are the constants that depend only on time. \nIn other words the following conditions must be satisfied:\n\\be\t\t\t\\label{garmsolQPnW}\n\\int X^{m}\\,w(X,\\theta,t)dX=\n\\sum_{k=0}^{m}A_{km}(t)\n\\cos^k\\theta\\sin^{m-k}\\theta,~~~m=1,~2,...,~n-2,\n\\ee\n\\be\t\t\t\\label{coefAkm}\t\t\t\nA_{km}(t)=\nC_{m}^k\\int q^kp^{m-k}\\,\nW(q,p,t)dq~dp,\n\\ee\nwhere $C_{m}^k$ is a binomial coefficient.\n\nIf the potential $V(q)$ is an analytic function representing an endless \nseries of powers of $q$,\nthen the tomogram $w(X,\\theta,t)$ must satisfy an infinite number of conditions \nof the form (\\ref{garmsolQPnW}), where $n=\\infty$. \n\nObviously, if we consider the classical case, \nthen in formula (\\ref{coefAkm}) the Wigner function $W(q,p,t)$ must be replaced \nwith classical distribution function $W_\\mathrm{cl}(q,p,t)$.\n\nOne can show that to all these conditions be satisfied, the function\n$w(X,\\theta,t)$ must allow to be expressed as an expansion in Hermite polynomials of the following form:\n\\be\t\t\t\\label{razlozhH}\nw(X,\\theta,t)=\\sum_{n,\\,m}\\rho_{nm}(t)\n\\frac{e^{i\\theta(m-n)}e^{-X^2}}{\\sqrt{\\pi2^{n+m}n!m!}}\nH_n(X)H_m(X),\n\\ee\nwhere for the quantum case the matrix $\\rho_{nm}(t)$ is obviously a density matrix\nof the quantum state in Fock basis representation, and for the classical case the set of quantities\n$\\rho_{nm}^\\mathrm{cl}(t)$ can be expressed in terms of classical distribution function\n$W_\\mathrm{cl}(q,p,t)$ as follows:\n\\bea\t\t\t\\label{rhoclas}\n\\rho_{nm}^\\mathrm{cl}(t)&=&\\frac{1}{\\sqrt{\\pi2^{n+m}n!m!}}\n\\int W_\\mathrm{cl}\\left(\\frac{q+q'}{2},p,t\\right) \\nonumber\\\\[3mm]\n&&\\times\\exp\\left(ip\\,(q-q')\n-\\frac{q^2}{2}-\\frac{q'^2}{2}\\right)\nH_n(q)H_m(q')\\,dq~dq'~dp.\n\\eea\nSo that, for the normalization of the optical tomogram be preserved during evolution \nin accordance with the equation (\\ref{r20_2}) or (\\ref{eq_54}),\nit is necessary and sufficient that the tomogram must be a sum of the form\n(\\ref{razlozhH}), which can be written for $N$-dimensional system as:\n\\bea\t\t\t\\label{razlozhHN}\nw(\\mathbf X,\\bm\\theta,t)=\\sum_{{n_1...n_N}\\atop{m_1...m_N}}\n\\rho_{n_1...n_Nm_1...m_N}(t)\\frac{1}{\\sqrt{\\pi^N}}\n\\prod_\\sigma^N\n\\frac{e^{i\\theta_\\sigma(m_\\sigma-n_\\sigma)}e^{-X_\\sigma^2}}\n{\\sqrt{2^{n_\\sigma+m_\\sigma}n_\\sigma!m_\\sigma!}}\nH_{n_\\sigma}(X_\\sigma)H_{m_\\sigma}(X_\\sigma).\n\\eea\n\n\nNote that the property of conservation of normalization in itself, \nadditionally to non-negativity,  parity, and satisfaction of Hirschmann criterion,\nis also not sufficient for the function to be a tomogram. Consider the example\n\\be\t\t\t\\label{example3}\nw_1(X,\\theta)=\\frac{e^{-X^2}}{\\sqrt\\pi}\\left(X^4+\\frac{X^2}{4}+\\frac{1}{8}\\right).\n\\ee\nDespite the fact that the function $w_1(X,\\theta)$ has all properties listed above,\nit is not a tomogram of any quantum or classical state,\nbecause the ``probability\" of finding a quantum system in state $|0\\rangle\\langle0|$\nis negative (equal to $-1\/8$) or, if we apply to $w_1(X,\\theta)$ transformation (\\ref{eq_52}),\nwe obtain the function\n\\be\t\t\t\\label{example4}\nW_1(q,p)=\\frac{e^{-q^2-p^2}}{\\pi}\\left[(q^2+p^2)^2-\\frac{3}{4}(q^2+p^2)-\\frac{1}{4}\\right],\n\\ee \nwhich is not a classical distribution function.\n\n\\section{\\label{Section5}Conditions for conservation of normalization \\\\\nof symplectic tomogram-like functions during evolution}\nFor Hamiltonians  (\\ref{r18_2})\nthe evolution equation \nof the symplectic tomogram  $M({\\mathbf X},{\\bm\\mu,\\bm\\nu},t)$ \nof the quantum system has the form \\cite{ManciniFoundPhys97}\n\\begin{eqnarray}\n\\frac{\\partial}{\\partial t}M({\\mathbf X},{\\bm\\mu},{\\bm\\nu},t)&=&\n\\left[\\sum_{\\sigma=1}^N\\frac{\\mu_\\sigma}{m_\\sigma}\\frac{\\partial}{\\partial\\nu_\\sigma}\\right]\nM({\\mathbf X},{\\bm\\mu},{\\bm\\nu},t) \\nonumber \\\\[3mm]\n&+&\n\\frac{2}{\\hbar}\n\\left[\\mathrm{Im}~\nV\\left\\{-\\left[\\frac{\\partial}{\\partial X_\\sigma}\\right]^{-1}\n\\frac{\\partial}{\\partial\\mu_\\sigma}+\\frac{i\\nu_\\sigma\\hbar}{2}\n\\frac{\\partial}{\\partial X_\\sigma}\\right\\}\\right]\nM({\\mathbf X},{\\bm\\mu},{\\bm\\nu},t),\n\\label{eq_46}\n\\end{eqnarray}\nand for classical system in the potential field $V(\\mathbf q,t)$\nwe have \\cite{OlgaJRLR97}\n\\begin{eqnarray}\n\\frac{\\partial}{\\partial t}M_\\mathrm{cl}(\\mathbf X,\\bm\\mu,\\bm\\nu,t)&=&\n\\left[\\sum_{\\sigma=1}^N\\frac{\\mu_\\sigma}{m_\\sigma}\\frac{\\partial}{\\partial\\nu_\\sigma}\\right]\nM_\\mathrm{cl}(\\mathbf X,\\bm\\mu,\\bm\\nu,t) \\nonumber \\\\[3mm]\n&+&\n\\left[\\sum_{\\sigma=1}^{N}\\frac{\\partial}{\\partial q_\\sigma}\nV\\left\\{q_\\sigma\\rightarrow-\\left[\\frac{\\partial}{\\partial X_\\sigma}\\right]^{-1}\n\\frac{\\partial}{\\partial\\mu_\\sigma}\\right\\}\n\\nu_\\sigma\\frac{\\partial}{\\partial X_\\sigma}\\right]\nM_\\mathrm{cl}(\\mathbf X,\\bm\\mu,\\bm\\nu,t).\n\t\t\\label{eq_55}\n\\end{eqnarray}\n\nMaking the similar calculations as in the previous section for the term\nin the potential $V(q)$ with $q^3$ we can find the equation\nfor the first momentum of the variable $X$\n\\be \t\t\t\\label{condit1}\n\\partial^2_\\mu\\int XM(X,\\mu,\\nu,t)dX=0.\n\\ee\nThe solution of this differential equation has the linear form with\nrespect to the variable $\\mu$\n\\be \t\t\t\\label{solutionM1}\n\\int XM(X,\\mu,\\nu,t)dX=A_1(\\nu,t)+\\mu B_1(\\nu,t),\n\\ee\nand taking into account the homogeneity condition (\\ref{eq_3})\nwe find that $B_1(\\nu,t)=B_1(t)$ is independent on $\\nu$\nand $A_1(\\nu,t)=\\nu A'_1(t)$, where $A'_1(t)$ is independent on $\\nu$\n\\be \t\t\t\\label{solutionM1cor}\n\\int XM(X,\\mu,\\nu,t)dX=\\mu B_1(t)+\\nu A'_1(t),\n\\ee\nor from definition (\\ref{eq_44})\nthis relation can be written as\n\\be\t\t\t\\label{garmsolQPM}\n\\int XM(X,\\mu,\\nu,t)dX=\\langle\\hat q\\rangle\\mu\n+\\langle\\hat p\\rangle\\nu.\n\\ee\nAnalogously, for the term $q^4$ in the potential $V(q)$ under conditions (\\ref{eq_44}) and (\\ref{condit1})\nwe can find the condition\n\\be \t\t\t\\label{condit2}\n\\partial^3_\\mu\\int X^2M(X,\\mu,\\nu,t)dX=0,\n\\ee\nwhich is valid, because \n\\bea\t\t\t\n\\int X^2M(X,\\mu,\\nu,t)dX&=&\n\\left(\\int q^2W(q,p,t)dq~dp\\right)\\mu^2\n+\\left(\\int p^2W(q,p,t)dq~dp\\right)\\nu^2 \\nonumber \\\\[3mm]\n&&+2\\left(\\int qp\\,W(q,p,t)dq~dp\\right)\\mu\\nu.\n\\label{garmsolQPMW}\n\\eea\nFor the polynomial potential of the power of $q^n$ as in the cases of (\\ref{condit1}) and (\\ref{condit2})\nwe can write \n\\be \t\t\t\\label{conditn}\n\\partial^{m-1}_\\mu\\int X^{m-2}M(X,\\mu,\\nu,t)dX=0,~~~m=2,~3,...,~n,\n\\ee\nand as in the case of (\\ref{garmsolQPnW})\nwe have\n\\be\t\t\t\\label{garmsolQPnWM}\n\\int X^{m}\\,M(X,\\mu,\\nu,t)dX=\n\\sum_{k=0}^{m}A_{km}(t)\n\\mu^k\\nu^{m-k},~~~m=1,~2,...,~n-2,\n\\ee\nwhere $A_{km}(t)$ are defined by formula (\\ref{coefAkm}),\nand so, the symplectic tomogram must be represented in the form of the sum \\cite{OlgaJRLR97}\n\\bea\nM(X,\\mu,\\nu,t)&=&\\sum_{n,m}\\frac{\\rho_{nm}(t)}\n{\\sqrt{\\pi(\\mu^2+\\nu^2)2^{(n+m)}n!m!}}\n\\frac{(\\nu+i\\mu)^m(\\nu-i\\mu)^n}{(\\mu^2+\\nu^2)^{(n+m)\/2}}\n\\nonumber \\\\[3mm]\n&&\\times\\exp\\left(-\\frac{X^2}{\\mu^2+\\nu^2}\\right)\nH_n\\left(\\frac{X}{\\sqrt{\\mu^2+\\nu^2}}\\right)\nH_m\\left(\\frac{X}{\\sqrt{\\mu^2+\\nu^2}}\\right),\n\\label{razlMH}\n\\eea\nwhich is the necessary and sufficient condition for preservation\nof the normalization  during evolution of the symplectic tomogram.\nFor $N-$dimensional systems the latter formula has the form\n\\bea\t\t\t\nM(\\mathbf X,\\bm\\mu,\\bm\\nu,t)&=&\\sum_{{n_1...n_N}\\atop{m_1...m_N}}\n\\frac{\\rho_{n_1...n_N\\,m_1...m_N}(t)}{\\sqrt{\\pi^N}}\n\\prod_\\sigma^N\n\\frac{(\\mu_\\sigma+i\\nu_\\sigma)^{m_\\sigma}(\\mu_\\sigma-i\\nu_\\sigma)^{n_\\sigma}}\n{\\sqrt{2^{n_\\sigma+m_\\sigma}(\\mu_\\sigma^2+\\nu_\\sigma^2)^{n_\\sigma+m_\\sigma+1}\nn_\\sigma!m_\\sigma!}} \\nonumber \\\\[3mm]\n&&\\times\n\\exp\\left(-\\frac{X_\\sigma^2}{\\mu_\\sigma^2+\\nu_\\sigma^2}\\right)\nH_{n_\\sigma}\\left(\\frac{X_\\sigma}{\\sqrt{\\mu_\\sigma^2+\\nu_\\sigma^2}}\\right)\nH_{m_\\sigma}\\left(\\frac{X_\\sigma}{\\sqrt{\\mu_\\sigma^2+\\nu_\\sigma^2}}\\right).\n\\label{HermExpanN}\n\\eea\n\nAs in the case with the optical tomograms, conditions of conservation of normalization of type \n(\\ref{conditn}), or (\\ref{garmsolQPnWM}), or (\\ref{razlMH}) are not sufficient for the non-negative\ntomogram-like function $f(\\mathbf X,\\bm\\mu,\\bm\\nu)$ obeying to (\\ref{eqnormSymp}) and (\\ref{eq_3})\nto be a tomogram of any quantum or classical state.\nProving example it is easy to obtain from expression (\\ref{example3}) applying the formula\n(\\ref{eq_6}):\n\\be\t\t\t\\label{example5}\nM_1(X,\\mu,\\nu)=\\frac{1}{\\sqrt\\pi\\sqrt{\\mu^2+\\nu^2}}\n\\left(\\frac{X^4}{(\\mu^2+\\nu^2)^2}+\\frac{X^2}{4(\\mu^2+\\nu^2)}+\\frac{1}{8}\\right)\n\\exp\\left(-\\frac{X^2}{\\mu^2+\\nu^2}\\right).\n\\ee\nThe function $M_1(X,\\mu,\\nu)$ is non-negative, normalized, homogeneous, and it conserves\nthe normalization during time evolution, but it is not a tomogram.\n\n\n\n\\section{\\label{Section6}Conservation of normalizations of the Wigner function,\nclassical distribution function, and Husimi function\nduring evoluton}\n\nIn previous sections we have found that in order to as classical as quantum\ndynamical equations for optical and symplectic\ntomograms retain normalizations, the tomograms \nhave to satisfy a set of conditions.\nAnd only normalization of the tomogram is insufficient for \nthis condition remain valid during time evolution.\n\nOn the contrary, Moyal equation for the Wigner function\n\\cite{Moyal1949}\n\\be\t\t\t\\label{Moyal}\n\\frac{\\partial }{\\partial _t} W({\\bf q},{\\bf p},t)=\n-\\sum_{\\sigma=1}^{N} \\frac{p_\\sigma}{m_\\sigma}\n\\frac{\\partial}{\\partial_{q_\\sigma }}W({\\bf q},{\\bf p},t)\n-2\\left[\\mathrm{Im}~V\\left\\{\n{\\bf q}+\\frac{i}{2} \\frac{\\partial}{\\partial {\\bf p}}\n\\right\\}\\right]W({\\bf q},{\\bf p},t),\n\\ee\nLiouville equation for the distribution function\n\\be\t\t\t\\label{Vlasov}\n\\frac{\\partial }{\\partial _t} W_\\mathrm{cl}({\\bf q},{\\bf p},t)=\n-\\sum_{\\sigma=1}^{N} \\frac{p_\\sigma}{m_\\sigma}\n\\frac{\\partial}{\\partial_{q_\\sigma }}W_\\mathrm{cl}({\\bf q},{\\bf p},t)\n-\\left[\\frac{\\partial }{\\partial {\\bf q}}~V({\\bf q})\\right]\\frac{\\partial}{\\partial {\\bf p}}\nW_\\mathrm{cl}({\\bf q},{\\bf p},t),\n\\ee\nand  dynamical equation for the Husimi \nfunction \\cite{Mizrahi1986}\n\\bea\t\t\t\\label{HusimiEQ}\n\\frac{\\partial }{\\partial _t} Q({\\bf q},{\\bf p},t)&=&\n-\\sum_{\\sigma=1 }^{N} \\frac{1}{m_\\sigma}\\left[p_\\sigma\n\\frac{\\partial}{\\partial_{q_\\sigma }}\n+\\frac{1}{2}\\frac{\\partial}{\\partial_{p_\\sigma }}\n\\frac{\\partial}{\\partial_{q_\\sigma }}\\right]\nQ({\\bf q},{\\bf p},t) \\nonumber\\\\[3mm]\n&&-2\\left[\\mathrm{Im}~V\\left\\{\n{\\bf q}+\\frac{1}{2}\\frac{\\partial }{\\partial {\\bf q}}\n+\\frac{i}{2} \\frac{\\partial}{\\partial {\\bf p}}\n\\right\\}\\right]Q({\\bf q},{\\bf p},t),\n\\eea\nretain normalization for any normalized and quickly decaying functions at $q_i$, $p_j$ \ntending to infinity.\nAs in the previous parts let's approximate the potential $V(\\bf q)$ \nwith a polynomial and integrate both sides of the dynamical equations \n(\\ref{Moyal}) -- (\\ref{HusimiEQ}) over the hyperspace ${\\bf q}\\times{\\bf p}$. \nEasy to see that for any functions $W({\\bf q},{\\bf p},t)$, \n$W_\\mathrm{cl}({\\bf q},{\\bf p},t)$ and $Q({\\bf q},{\\bf p},t)$  from the class considered\nthe right sides of equations (\\ref{Moyal}) -- (\\ref{HusimiEQ})\nare becoming identically equal to zero.\n\nThus, Moyal equation, Liouville equation and dynamical equation for the Husimi function \nretain normalizations of their solutions without any additional conditions, \nlike (\\ref{Gendifeven}, \\ref{Gendifodd}) for optical\nor (\\ref{conditn})  for symplectic tomograms and that is a principal \ndifference of equations (\\ref{Moyal}) -- (\\ref{HusimiEQ}) from corresponding \nequations (\\ref{r20_2}, \\ref{eq_54}, \\ref{eq_46}, \\ref{eq_55}) for tomograms.\n\n\\section{\\label{Section7}Conclusion}\n\nIn summary we point out the main results of our paper.\nWe discussed the properties of optical and symplectic tomograms. \n\nIt was demonstrated that optical tomograms could be associated with a \nunitary representation of the Weyl-Heisenberg group as well as symplectic \ntomograms. This fact has allowed to formulate an autonomous criterion for optical\ntomogram-like function to be a tomogram of quantum or classical state\nbased on positivity properties of the function on the group\ndetermined with the help of the latter tomogram-like function.\n\nWe shown that for the arbitrary potential the tomograms must satisfy  a set \nof relations to be normalized during evolution.\nWe found that all moments of homodyne variable $X$ \nas for optical as for symplectic tomography\nmust satisfy a set of specific linear differential equations.\nThese equations were obtained explicitly and their general \nsolutions were presented.\n\nWe also illustrated that Moyal equation, Liouville equation, and dynamical equation\nfor the Husimi function retain the normalization for any normalized and decreasing\nrapidly at infinity initial conditions, unlike tomographic dynamical equations.\n\nThe relations allowing to conserve the normalization of the tomogram\nduring time evolution determine a narrow class of functions, which can \nevolve without dramatic growth maintaining the physical sense.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe Herschel Space Observatory was launched from Kourou in May 2009 aboard an Ariane 5 rocket. Two of three scientific instruments on the focal plane (PACS and SPIRE) are capable to observe the infrared sky with unprecedented angular\nresolution and sensitivity, providing photometric observations in 6\ndifferent bands \\citep[$70$$\\mu$m, $100$$\\mu$m, $160$$\\mu$m, $250$$\\mu$m,\n$350$$\\mu$m\\ and $500$$\\mu$m :][and reference therein]{Pilbratt10}.\n\nThe PACS photometer is composed of two bolometer arrays: a $64 \\times 32$ pixel matrix arranged from 8 monolithic subarrays of $16 \\times 16$\npixels each centered on the $70\\mu$m and $100\\mu$m wavelength (blue and green bands), and a $32 \\times 16$ pixel matrix organized in two subarrays for the band centered on $160\\mu$m (red band), see \\citet{Poglitsch10}.  \n\nSPIRE comprises a three band photometer, operating in spectral bands centered on $250\\mu$m, $350\\mu$m and $500\\mu$m. Every band uses a matrix of germanium bolometers (139, 88 and 43 respectively) coupled to hexagonally packed conical feed horns \\citep{Griffin10}.  \n\nIn order to handle science data provided by the Herschel instruments, including the data retrieval from the Herschel Science Archive, the data reduction through the standard pipelines and the scientific analysis, an official software environment called HIPE\n\\citep[Herschel Interactive Processing Environment,][]{HIPE} is available from ESA. \n\nThe raw data provided by the satellite are reduced in HIPE to generate scientific data (so-called Level 2) and\nintermediate products of the data reduction process (Level 0 and Level\n1 data).\n\nIn this paper, we describe the dedicated pipeline created to obtain maps for Hi-GAL\\ \\citep[Herschel Infrared Galactic Plane\nSurvey,][]{Molinari10_PASP}. Hi-GAL aims to homogeneously cover with observations in 5 contiguous IR bands between 70$\\mu$m and 500$\\mu$m  a 2 degrees wide stripe of galactic plane between $l=-70^{\\circ}$ and $l=70^{\\circ}$.\n\nThe Galactic plane shows emission varying from point-like sources to large-scale structures and with intensity varying over a wide dynamic range. In this work we show  that  the existing standard reduction strategy (which is based on HIPE version 4.4.0, released in November, 11th 2010) is not optimized to reduce Hi-GAL data and that a dedicated pipeline can enhance the quality of the Level 2 products.\n\n After Herschel successfully passed the Performance\nVerification Phase (PV Phase), two fields of the Hi-GAL\\ survey were acquired\nduring the Science Demonstration Phase (SDP): 2x2 square degree areas\nof the Galactic plane centered on 30$^{\\circ}$ of longitude\n(hereafter, $\\textit{l}=30^{\\circ}$) and on 59$^{\\circ}$ ($\\textit{l}=59^{\\circ}$).\n\nWe describe the data reduction tools used to obtain high quality maps from SDP data, with the aim to provide a reliable environment for the\nRoutine Phase (RP) data. The maps provided by our pipeline are successfully used for several works like, e.g. , \\citet{Molinari10_special}, \\citet{Martin10}, \\citet{Peretto10}.\n\nThe paper is organized as follows: in Section \\ref{sec:higal} we\ndescribe the acquisition strategy for Hi-GAL\\ data; in Section \\ref{sec:preprocessing} we\ndescribe the pre-processing steps of the data reduction pipeline, necessary to prepare data for the map making\nand the tools that we have developed to that\npurpose. In Section \\ref{sec:mapmaking} we describe the ROMAGAL map\nmaking algorithm used in order to estimate the maps. ROMAGAL is used\nin place of the MadMap code which is the map making algorithm offered \nin HIPE. The quality of the ROMAGAL maps for both PACS and SPIRE instruments, related to the SDP observations, will be analyzed in Section\n\\ref{sec:SDP_results}; in Section \\ref{sec:Conclusions} we draw our\nconclusions. \n\n\\section{The Hi-GAL\\ acquisition strategy}\\label{sec:higal}\n\nHi-GAL\\ data are acquired in PACS\/SPIRE Parallel mode\\footnote{http:\/\/herschel.esac.esa.int\/Docs\/PMODE\/html\/parallel\\_om.html}, in which the same sky\nregion is observed by moving the satellite at a constant speed of 60$\\arcsec$\/sec and\nacquiring images simultaneously in five photometric bands: $70$$\\mu$m\\ and $160$$\\mu$m\\ for PACS and $250$$\\mu$m, $350$$\\mu$m\\ and $500\\mu$m for SPIRE.\n\nThe whole data acquisition is subdivided in $2^\\circ\\times2^\\circ$ fields of sky\ncentered on the Galactic plane. Every\nHi-GAL\\ field is composed of the superposition of two orthogonal AOR (Astronomical Observation Requests).\nEach of them is based on a series of consecutive, parallel and partly overlapped scan legs covering $2^\\circ\\times2^\\circ$ square degrees. The scanning strategy adopted for Hi-GAL is fully described in \\citep{Molinari10_PASP}. The superposition is performed in order to obtain a suitable data redundancy and for better sampling the instrumental effect like the high-frequency detector response \\citep{Molinari10_PASP}.\n\nThe acquisition rate for the parallel\nmode is 40 Hz for PACS and 10 Hz for SPIRE, although the PACS\ndata are averaged  on-board for an effective rate of 5\\,Hz and 10\\,Hz\nfor the $70$$\\mu$m\\ and $160$$\\mu$m\\ array respectively. The implications of the PACS data compression are detailed in Section \\ref{sec:signal_striping}. \n\nAn example of the scanning strategy of the Hi-GAL\\ survey is shown\nin Figure \\ref{fig:coverage}. The coverage map of the PACS blue array is shown on the left panel and on the right panel we highlight the\nsuperposition of one scan leg to the following ones, by enlarging the\nbottom right corner of the left image. Two calibration blocks for each\nAOR, during which the instrument observes two internal calibration\nsources located on the focal plane, were provided during the SDP\nobservations. They appear as higher than mean coverage areas and are marked\nby black and green circles for the 2 AORs in Figure\n\\ref{fig:coverage}. Higher coverage zones are also clearly visible in\nthe slewing region at the end of each scan leg, where the telescope\ndecelerates and then accelerates before initiating the next scan leg. \n\n\\begin{figure}\n\\centering\n\\includegraphics[width=4cm]{figure1.eps}\n\\includegraphics[width=4cm]{figure2.eps}\n\\caption{Left: the coverage map of PACS blue array. Right: a zoom of the bottom right corner where is clear the\n  effect of the superposition from one scan-leg to the next. Black and\n  green circles highlight the calibration blocks that, during PV and SDP phase, were observed twice during which the internal calibration sources are observed.} \n\\label{fig:coverage}\n\\end{figure}\n\n\\section{Map making pre-processing}\\label{sec:preprocessing}\n\nThe aim of the pre-processing is to prepare Hi-GAL\\ data for\nmapmaking. \n\nWhile map making is performed\nusing a Fortran parallel code borrowed from cosmological observations\n(see Section \\ref{sec:mapmaking}), the preprocessing is done through a series of\nIDL and jython tools to be run on the data one after the other. After having\ntried to map Hi-GAL\\ data using the standard tools provided within HIPE,\nwe decided to develop our own routines that we tailored\nspecifically for the reduction of data affected by bright and irregular\nbackground, as in the Galactic plane. In fact, high-pass filtering\nused in HIPE to cure the long-time drift also removes a large part of\nthe diffuse Galactic emission. Furthermore, the standard deglitching\nembedded in HIPE , the MMT  \\citep[Multiresolution Median\n  Transform,][]{Starck95}, generates false detections in correspondence\nto the very bright sources when we apply this task to the PACS data. On the other hand, the deglitching procedure based on wavelet analysis used by SPIRE does not affect the data, given also the lower spatial resolution compared to the PACS one. We therefore use the HIPE task for SPIRE data only.\n\nHerschel data are stored in subsequent snapshots of the sky acquired\nby the entire detector array, called frames. In a first processing\nstep, the standard modules within HIPE are used to generate Level 1 data for both PACS (except the deglitching task) and SPIRE. Thus, the data are rearranged into one time series per detector pixel, called Time\nOrdered Data (TOD), in which the calibrated flux ( Jy\\,beam$^{-1}$ for SPIRE and Jy\\,sr$^{-1}$ for PACS) and the celestial coordinates are included. \nAt the end of this step, TODs are exported outside HIPE in fits format. In the subsequent processing steps, TODs are managed by a series of IDL tools, in order to produce final TODs free\nof systematic effects due to electronics and of glitches and corrupted chunks of data due to cosmic\nrays. To each TOD a flag file is attached to keep track\nof any flagging done during data reduction steps.\n\nPreprocessing includes identification of corrupted TODs\n(or part of them), drift removal and deglitching.  The following steps will be the Noise Constraint Realization (NCR) and ROMAGAL mapmaking; they will be both described in detail in the next Sections. The summary of the entire pipeline is shown in the diagram in Figure \\ref{fig:higalpipe}. \n\n\\begin{figure}\n\\centering\n\\includegraphics[width=4cm]{figure3.eps}\n\\caption{Schematic representation of the Hi-GAL\\ data reduction pipeline} \n\\label{fig:higalpipe}\n\\end{figure}\n\n\\subsection{Corrupted TODs}\n\nA TOD can be partially corrupted by the random hiting of\ncharged particles (cosmic rays) which produce strong and spiky signal\nvariations called glitches. Two\ndifferent effects on the TODs can be identified: the glitch corrupts a single or few consecutive samples, generating\nspiky changes along the TOD. This is the most common effect and in\nSection \\ref{sec:deglitching} we describe how to detect and mask the\ndata for PACS, as well as to mask any possible residual glitches for SPIRE. Very powerful glitches, on the other hand,\nmake the detector signal unstable for a considerable\ninterval of time \\citep[see, e.g.,][]{Billot10}. This effect depends on the glitch amplitude and on the\nbolometers time response. These events affect a much larger part of\nthe TOD that cannot be used for mapmaking.\n\nTheir impact results in a bias on the bolometer \ntimeline with a low-frequency drift which can involve a considerable\nnumber of samples, as shown in\nFigure~\\ref{fig:bad_bolometer}. \n\nIn that Figure, blue crosses represent the\nobserved timeline of one bolometer of the blue\narray. \n\nAutomatic identification of the (partially) corrupted TODs exploits\nthe first derivative of the TOD to detect extraordinary ``jumps'' in\nthe signal. In order to determine what portion of the TOD is to be\nflagged, the flagging tool exploits the fact that the detector pixels response that have\nbeen hit by a cosmic ray is mostly exponential. Data samples ranging from the\njump to the time sample at which an exponential fit reaches 98\\% of\nthe signal level before the event are identified as bad data and\nstored as such in the corresponding flag file. In case the exponential\nfit does not reach 98\\% of the starting value before the end of the\nTOD, then all data starting from the hit will be flagged as is the case\nin Figure~\\ref{fig:bad_bolometer}. This procedure is applied both in the cases of a changing in responsivity or a detector offset alteration. In the latter, we estimate the fit with an exponent equal to 0. The described procedure was adopted to process both PACS and SPIRE data. \n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=8cm]{figure4.eps}\n\\caption{Timeline of a PACS blue bolometer. The exponential decay illustrates the change in responsivity after frame 40000 due to the impact of a powerful glitch.}\n\\label{fig:bad_bolometer}\n\\end{figure}\n\n\\subsection{Drift removal}\n\nAfter having identified corrupted data we proceed to the elimination of changes in responsivity over time. The procedures are in principle\nidentical for PACS and SPIRE data, the only differences account for\ndifferent composition of the detector arrays and the data acquisition\nof the two instruments. \n\nThe signal in PACS TODs  that exit HIPE does not represent the true sky\nbut is dominated by the telescope background and the (unknown) zero level of the electronics. The\nelectronics further introduce significant pixel-to-pixel\noffsets. For each TOD, we mitigate the effect of pixel-to-pixel offset by calculating and then subtracting the median level for each pixel from each readout.  This ensures that all pixels have median level equal to 0.  The median value is preferred over mean because the median is a much better representation of the sky+telescope background flux, and is much less sensitive to signal from astrophysical sources.\n\nAlso, the subtraction of this offset from each TOD does not\nalter the signal in the final map, but it\nintroduces only a global offset constant over the entire area covered\nin the observation. However it should be kept in mind that bolometers\nare inherently differential detectors which bear no knowledge of an\nabsolute signal value; besides, any optimized map making methods like\nthe one we employ (see Section \\ref{sec:mapmaking}) produce maps with\nan unknown offset value which needs to be independently calibrated. So\nit is important to reduce all the bolometers to the same median value,\nregardless of its amount. The pixel-to-pixel median subtraction has the effect\nseen in Figure~\\ref{fig_drift_2}. Diffuse emission and compact sources\nare clearly visible in the frame. \n\n\\begin{figure}\n\\centering\n\\includegraphics[width=8cm]{figure5.eps}\n\\caption{\\textbf{Blue PACS frame after the median subtraction on each pixel. Diffuse emission and compact source are visible in the frame.}}\n\\label{fig_drift_2}\n\\end{figure}\n\n\n\nStill, when plotting a detector pixel timeline we see that the signal decreases\nsystematically from the start to the end of the observation.  This trend is likely due to a combination of small changes in the thermal bath of the array and to small drifts in the electronics. The former affects the entire array, while the latter affects subunits of the detector array (in fact, PACS blue is divided into 8 units, PACS red into 2, electronically independent units \\citep{Poglitsch08}). The drift is then a combination of these two effects: drifts of the entire array and drift of a single subunit. These effects are dominant with respect to the $1\/f$ noise pattern, which will be described in the next Sections. \n\n\\begin{figure}\n\\centering\n\\includegraphics[width=8cm]{figure6.eps}\n\\caption{Median behavior computed on the whole array for each frame. The\n  slow-drift behavior is due to the electronics and the thermal bath.} \n\\label{fig_drift_4}\n\\end{figure}\n\nIt is in principle not a trivial task to decide which drift has to be\nsubtracted first: the drift from the thermal bath (affecting the entire detector\narray) or the drift from the readout electronics (affecting sub-arrays\ndifferently)? Ideally\nboth should be subtracted, if only it were possible to separate each\ncomponent accurately, as the net effect  in the data is the sum\nof both.\n\nOur methodology for removing the correlated signal drifts (on both the bolometer module\/unit level and the array level) is based on tracing the low signal envelope of the unit or array median levels.  In Figure \\ref{fig_drift_4}, this envelope is the curve defined by the lowest signal values. and estimated as follows: \n\n\\begin{itemize}\n\\item[i] We compute the median value of the entire bolometer array\/unit for each bolometer readout.  Figure \\ref{fig_drift_4} shows one example for the entire array.  \n\\item[ii] The median values thus obtained are segmented and grouped by scan legs. Each scan leg is composed of $\\sim1000$ frames and we observed 54 scan leg for each $2^\\circ\\times2^\\circ$ Hi-GAL field.\n\\item[iii] For each scan leg we compute the minimum value of the array\/unit medians.\n\\item[iv] The resulting set of minimum median values for all scan legs are fit with a polynomial.\n\\end{itemize}\n\nThe median value for each array\/unit readout is chosen because it is closest to the actual sky+telescope background.  However, as clearly seen in Figure \\ref{fig_drift_4}, in the presence of strong astrophysical sources the median value is incorrect for our purposes.  The strong sources appear as signal spikes in Figure \\ref{fig_drift_4}.  Hence, we take the next step of finding the minimum value from the set of medians belonging to a single scan leg.  The idea is that at some point during the scan leg the median was indeed a true representation of the local sky+telescope and relatively free of source emission.  This step allows us to reject the sources at the expense of degrading our time-resolution to scan-leg duration ($\\sim240$ sec).  The polynomial fit allows us to estimate the drift behavior at the same time resolution as the PACS signal (5Hz and 10Hz for 70$\\mu$m and160$\\mu$m band respectively).  We further note that the correlated signal drift is relatively flat over a single scan leg; hence, the minimum value is not significantly affected by the presence of the monotonic signal drift itself in the first place.\n\nThe minimum median method discussed above removes background levels from spatial emission structures that are of the order or larger than the scan legs yet preserves the spatial structures smaller than scan legs.  In essence, information about the absolute calibration zero-point \\citep{Miville05} is lost but all spatial structures within our map boundaries are preserved.\n\nIn Figure \\ref{fig_drift_6} we reported the minimum median values of each subarray. The common downward trend is due to the (common) thermal bath drift, while the different readout electronics are responsible for the differences in the subarray slopes.\n\n\nWe therefore decide to subtract the subarray drifts in order to consider both the thermal bath and the readout electronics behaviors, but separately for each subunit.\n\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=8cm]{figure7.eps}\n\\caption{Interpolation of the minima of the\n  median evaluated on every scan leg. Each curve refers to a PACS subarray. The curves mimic the same behavior with slopes due to the different subarray electronics.} \n\\label{fig_drift_6}\n\\end{figure}\n\nOnce the individual subarray drift is removed, the remaining dispersion on the whole array is only a residual scatter due to the intrinsic efficiency of the removal tool, as shown in Figure \\ref{fig_drift_7}. \n\n\\begin{figure}\n\\centering\n\\includegraphics[width=8cm]{figure8.eps}\n\\caption{Minima of the median evaluated on the whole PACS blue array after the subarray drift\n  subtraction. The dispersion is due to the\n  intrinsic efficiency of the drifts subtraction tool. There is no\n  residual behavior and the scatter is one order of magnitude under\n  the value of the initial drift.} \n\\label{fig_drift_7}\n\\end{figure}\n\nSPIRE array detectors are not divided into subarrays, so every\nprocedure that has to be run 8 or 2 times for PACS data is only\nperformed once per SPIRE band. \nSPIRE uses blind bolometers (5 in total) as thermistors to\nevaluate the most relevant correlated noise component: the bath\ntemperature fluctuations. A standard pipeline module uses this\ninformation to perform an effective removal of the common drift\npresent along the scan observation. HIPE also corrects for the delay between\nthe detector data and the telescope position along the scan, using an\nelectrical low-pass filter response correction. But despite these\ninitial (and very effective) corrections, we apply the drift removal\ntool to SPIRE data in the same way as for PACS\ndata: we fit a polynomial to the minimum of the median of each scan\nleg (calculated over the entire detector array), that we then subtract\nto all TODs. Experience on the data shows that a residual long time\ndrift is often present in SPIRE data.  \n\nFinally, when removing drifts it is important to know how the\nobservational scans are oriented. In fact, as the Galactic plane is\nvery bright, scans across the plane will give rise to an increase of\nsignal, on top of the general drift. On the other hand, when the scan\nis almost parallel to the plane of the Galaxy, the signal can\ndominated by its bright emission, also on the evaluation of the minima of the median. \n\n\\textit{In this case, the curve to fit is estimated only in the scan legs where the median is not affected by the signal.}\n\n\n\nSince the procedure is not automatic, care has to used when choosing what\npolynomial to fit and subtract from the data, in order not to\neliminate genuine signal from the sky. From our experience the best choice is a first or a second degree polynomial, depending on the signal behavior observed. \n\nHigher polynomial degree can be necessary when part of the drift has to be extrapolated in order to avoid signal contamination.\n\n\nAn example for one subarray is shown in Figure~\\ref{fig:drift_galassia}. \n\n\\begin{figure}\n\\centering\n\\includegraphics[width=8cm]{figure9.eps}\n\\caption{Blue curves: interpolation of the minima of the median for one subarray\n  of PACS blue channel when the scan direction is almost parallel to the Galactic plane. Red line: the fit that we\n  choose to evaluate the drift without considering the minima affected\n  by the Galactic emission (central bump).} \n\\label{fig:drift_galassia}\n\\end{figure}\n\nEach scan leg is bigger then the overlapping region of the two scan directions for the scanning strategy adopted by Hi-GAL (see Figure \\ref{fig:coverage}). Since the Hi-GAL fields are squared regions, the slowly-traversed direction of the AOR within the overlapping region have a length comparable with the scan leg. Thus, we assume that even if there is a signal gradient along the slowly-traversed direction of the AOR, it is not filtered out by the array medians subtraction.\n\n\\subsection{Deglitching}\\label{sec:deglitching}\n\nTo remove outliers in the time series of the bolometers we exploit the spatial redundancy provided by the telescope movement which ensures that each sky pixel of the final map is observed with different bolometers. Outliers detection is done with the standard sigma-clipping algorithm: given a sample of $N$ values, first estimates for the mean and for the standard deviations are derived; then all the values that differ from the mean by more than $n$ standard deviations are considered outliers and removed. \n\nFor this algorithm the choice of $n$, the parameter that defines the threshold above which a value is considered an outlier, is usually arbitrary: a certain $n$ is chosen, very often equal to 3, without any statistical justification. Recently \\citet{Pezzuto11} has derived a formula that starting from the properties of the error function for a Gaussian distribution and exploiting the discreteness of the number of available measures, relates $n$ to the size of the sample. The formula is\n\n\\begin{equation}\nn=-0.569+\\sqrt{-0.072+4.99\\log(N)}\\label{nfromN}\n\\end{equation}\n\nAs a consequence, in the central region of the map, where the coverage (and so $n$) is high, the number of standard deviation is larger than in the outskirt of the map where the coverage is low. For instance, if a sky pixel has been observed with 40 bolometers, the above formula gives $n=2.25$; so, once we have estimated the mean $m$ and the standard deviations $\\sigma$, all the values $x_i$ such that ABS$(x_i-m)>2.25\\sigma$ are flagged as outliers. If a pixel has been observed with 20 bolometers the threshold lowers to 1.96$\\sigma$.\n\nThis procedure is automatically iterated until outliers are no longer found. However, the procedure converges within 1 iteration in $\\sim$98\\% of the cases in which we have applied the analysis.\n\nThe outliers detection is done in this way for both instruments, however for SPIRE, as explained before, we also make use of the standard deglitching algorithm (wavelet based) implemented in the official pipeline. But we found some weak glitches left in the SPIRE TODs so that we decided to run our deglitching algorithm also on SPIRE data.\n\nThe number of glitches found is on the average about 15\\%, a value which is likely larger than the real percentage. For PACS we are now working on a different way to associate each bolometer to the sky pixels, taking into account the finite size of the sky pixels. For the first test cases we run, the percentage of detected glitches is now around 5-6\\%.\n\n\\section{The ROMAGAL Map Making algorithm}\\label{sec:mapmaking}\n\nThe ROMAGAL algorithm is based on a Generalized Least Square (GLS)\napproach \\citep{Lupton93}. Since the TOD is a linear combination of\nsignal and noise, we can model our dataset $\\textbf{d}_{k}$ for each\ndetector $k$ as \\citep{Wright96}: \n\n\\begin{equation}\\label{eq:tod}\n\\textbf{d}_{k}=P\\textbf{m}+\\textbf{n}_{k}\n\\end{equation}\n\n\\noindent where $P$ is the pointing matrix, which associates to every\nsample of the timeline a direction in the sky, $\\textbf{m}$ is our map estimator of the\n``true\" sky and $\\textbf{n}_{k}$ is the noise\nvector.  \n\nThe observed sky, $P\\textbf{m}$, is the map estimator of the ``true sky\" convolved with the instrumental\ntransfer function and the optical beam. However, in case of circularly\nsymmetric beam profile, $\\mathbf{m}$ is a beam smeared, pixelised\nimage of the sky.\n\nIn this case the pointing matrix has only a non-zero\nentry per row corresponding to the sky pixel observed at a given\ntime. Since the beam profiles for PACS  \\citep{Poglitsch08} and SPIRE\n\\citep{Griffin09} are only weakly asymmetric we can put ourselves in\nthis simple case. Note that the transpose of the $P$ operator performs\na data binning (without averaging) into the sky pixels. \n\nEquation~\\ref{eq:tod} holds only if the noise vector of each detector $\\textbf{n}_{k}$ is\ncomposed of statistical random noise, with Gaussian distribution and\nnull average. All the relevant systematic effects (offset, glitches) have then to be removed with an accurate data preprocessing\nbefore map production, as explained in Section\n\\ref{sec:preprocessing}. \n\n\nThe formalism can be easily extended in the case of multidetector analysis. In this case the vector $\\textbf{d}$ contains the data relative to each detector. Rather, one has to take care to upgrade also the noise vector $\\textbf{n}$, accordingly with the correct noise value for each detector.\n\n\nThe GLS algorithm produces minimum noise variance sky maps. Noise\nproperties for each detector  have to be previously estimated and provided in input to\nthe algorithm as described in Section \\ref{sec:noise_filters}.  \n\nThe GLS estimate for the sky, $\\tilde\\textbf{m}$, is \\citep{Natoli01}\n\\begin{equation}\\label{eq:GLS}\n\\tilde\\textbf{m}=(P^{T}\\textbf{N}^{-1}P)^{-1}P^{T}\\textbf{N}^{-1}\\textbf{d}\n\\end{equation}\n\\noindent where $\\textbf{N}=\\langle \\textbf{nn}^{T}\\rangle$ is the\nnoise covariance matrix, which takes into account noise time\ncorrelation between different samples. Such correlation is\nparticularly high at low frequencies because of the $1\/f$ (or long\nmemory) noise. In case of uncorrelated noise (or white noise) the $\\textbf{N}$\nmatrix becomes diagonal and the problem is greatly simplified.  If we\nfurther assume stationary uncorrelated noise, Equation \\ref{eq:GLS} reduces\nto: \n\\begin{equation}\\label{eq:naive}\n\\tilde\\textbf{m}=(P^{T}P)^{-1}P^{T}\\textbf{d} .\n\\end{equation}\n$P^{T}P$ is the number of observations of a pixel of the map, so we\nare averaging the different TOD values into that pixels assigning the\nsame weight to each sample. We will refer to this map estimate as ``naive\" or ``binned\" in the following.  \n\nWhen non negligible noise correlation is present, as in the case of\nPACS \\citep{Poglitsch08} and SPIRE \\citep{Schulz08},  Equation~\\ref{eq:GLS} must be solved. This is a challenging\ncomputational task since it requires, in principle, the inversion of\nthe large (of the order of the number of pixels in the map) matrix\n$P^{T}\\textbf{N}^{-1}P$, which is the covariance matrix of the GLS\nestimator \\citep{Lupton93}. One key simplifying assumption is to\ninvoke that the noise is stationary. In this case, the $\\textbf{N}$ matrix has\na Toeplitz form which can be approximately treated as circulant,\nignoring boundary effects \\citep{Natoli01}. A circulant matrix is\ndiagonal in Fourier space and its inverse is also circulant, so the\nproduct between $\\textbf{N}^{-1}$ and a vector is a convolution\nbetween the same vector and a filter provided by any of the rows of\nthe matrix. In the following we will refer to any of these rows as a\nnoise filter. Its Fourier transform is the inverse of the noise frequency power\nspectrum.  \n\nConsidering the conditions listed above, the GLS map making algorithm\nperforms the following operations, starting with rewriting the Equation \\ref{eq:GLS} in the form \n\n\\begin{equation}\\label{eq:gls_rewrite}\n(P^{T}\\textbf{N}^{-1}P)\\textbf{m}_{0}-P^{T}\\textbf{N}^{-1}\\textbf{d}=\\textbf{r}\n\\end{equation}\n\nwhere $\\textbf{m}_{0}$ is the starting map used at the first iteration, generally the naive map. \n\nThe $P \\textbf{m}_{0}$ product projects the map onto a timeline.  Application of $\\textbf{N}^{-1}$: this is a convolution which can be performed in Fourier space. Application of $P^T$: this step projects the convolved timeline back into a map. \n\nThe second term performs the convolution with the filter (applying  $\\textbf{N}^{-1}$ to the data vector $\\textbf{d}$ in Fourier space) and then the projection of the convolved timeline into a map (applying $P^T$ to the product $\\textbf{N}^{-1}\\textbf{d}$). \n\nThen, we need to evaluate the residual \\textbf{r}. If the residual is higher than a fixed threshold, it is used to produce a new map, $\\textbf{m}_{1}$, as described in \\citet{CG}. This map will be considered instead of $\\textbf{m}_{0}$ for evaluating again the Equation \\ref{eq:gls_rewrite}, until convergence. This is achieved by running a Conjugate Gradient algorithm, an iterative\n  method useful to obtain a numerical solution of a system \\citep{CG},\n  until convergence is reached with the residual lower then the threshold.\n\nThe algorithm outlined is the same described in \\citep[``unroll, convolve and bin\":][]{Natoli01} and is implemented in the ROMAGAL\ncode. The next section explains the strategy employed to estimate the\nnoise filters used by ROMAGAL directly from in-flight data. \n\n\\subsection{Noise estimation}\\label{sec:noise_filters}\n\nIn order to estimate the noise filters for ROMAGAL we need to\ninvestigate the noise statistical properties in the timelines. Data\nare mostly affected by two kind of statistical noise: $1\/f$ noise due\nboth to the electronics and thermal background radiation from the telescope\nor the instruments, and photon noise (see \\citealt{Poglitsch08,\n  Schulz08}).  \n\nThe detector $1\/f$ noise arises in the electronic chain and it impacts\nparticularly regions with low signal-to-noise ratio (SNR), where only\ndiffuse emission is present. In those regions it can be of the same\norder of magnitude of the signal or even higher. In these cases GLS\ntreatment is particularly effective.  \n\nPhoton noise is due to statistical fluctuation in the photon\ncounts. This process follows poissonian statistic, so the SNR is\nproportional to the square root of counts. Since poisson distribution\ntends to be Gaussian for large numbers, we can approximate photon\nnoise as Gaussian on the map if the number of counts is large\nenough.\n \nSince bolometers are organized in matrix and sub-matrix, the signal of a bolometer can be correlated with the signal of another, generally adjacent, bolometer. These effects could be both\nstatistical and deterministic. We already described how to remove the deterministic common mode from\nTOD (like the thermal bath variations, see Section \\ref{sec:preprocessing}).\n\nOne possible source of statistical cross-correlated noise is the crosstalk between bolometers: the signal of\none pixel may contaminate the signal of its neighbors through the\ncapacitive inductive coupling, generating a common mode called\n``electrical crosstalk''. On the contrary, ``optical crosstalk'' is\ndue to diffraction or aberrations in the optical system which could\ndrive an astronomical source to fall on inappropriate detectors\n\\citep{Griffin09_pipeline}. \n\nWe then analyze the residual contribution of the statistical component\nof the correlated noise. We found that the residual correlated noise\nlevel in each pixel is negligible with respect to the intrinsic\ndetector noise level for both PACS and SPIRE instruments, as described\nin the following. \n\nIn principle, the noise\nproperties vary significantly across the array and we had to estimate\nthe noise power spectrum for each bolometer. \nTo do that we have processed ``blank\" sky mode (i.e. filled with\nnegligible contribution from sky signal) data acquired during the\nPV Phase. \n \nIn Figure \\ref{fig:PACS_blue_noise_spectrum} we show a typical noise spectrum\nestimated for a pixel of the $160\\mu$m  PACS band (black) and the\ncross-spectrum between two adjacent bolometers (red). The\ncross-spectrum evaluates the impact of the cross-correlated noise in\nthe frequency domain between two different bolometers. The level of\nthe cross-correlated noise is at least 4 order of magnitude below the\nauto-correlated noise power spectrum of each pixel. Note that this means we do not see any relevant cross-correlated noise, despite the fact that crosstalk can be present into the timeline. \n\nIn Figure \\ref{fig:SPIRE_psw_noise_spectrum} we\nshow noise power spectra of the 250$\\mu$m SPIRE band bolometers. \nAlso in this case the cross-spectrum is negligible. \n \nNoise spectra of both PACS and SPIRE display low-frequency noise\nexcess ($1\/f$). In case of SPIRE spectra (Figure\n\\ref{fig:SPIRE_psw_noise_spectrum}) a high\nfrequency rise is also evident, which is due to the deconvolution of the bolometric\nresponse function. PACS spectra do not show this behavior because the\nbolometer transfer function is not deconvoluted by the standard\npipeline. \n\n\\begin{figure}\n\\centering\n\\includegraphics[width=8cm] {figure10.ps\n\\caption{Black line: typical noise spectrum of a PACS $160$$\\mu$m\\ detector,\n  estimated on blank sky data. Red line: cross spectrum between two\n  detectors of the same subarray. The level of the cross-correlated noise is\n  significantly under the noise level of each single bolometer, so we\n  can reasonably neglect it.} \n\\label{fig:PACS_blue_noise_spectrum}\n\\end{figure}\n\\begin{figure}\n\\centering\n\\includegraphics[width=8cm] {figure11.ps\n\\caption{Same as Figure \\ref{fig:PACS_blue_noise_spectrum} for a SPIRE\n  $250$$\\mu$m\\ bolometer. For SPIRE also the noise level of cross-spectrum is\n  reasonably negligible with respect to the auto spectrum level.} \n\\label{fig:SPIRE_psw_noise_spectrum}\n\\end{figure}\n\n\\subsection{From ROMA to ROMAGAL}\\label{sec:roma_romagal}\n \nThe ROMAGAL GLS code has been optimized to recover the features in the\nHi-GAL\\ fields with high accuracy. \n \nHi-GAL observes the Galactic plane where the dynamic range of sources\nspans over several orders of magnitudes. This poses strong constraints\non the map making algorithm: both the weak diffuse emission and the\nbright compact sources in e.g., star forming regions have to be\nrecovered with high accuracy. \nThe signal often exhibits steep gradients that are hard to follow for\nthe GLS solver, which relies on the assumption that the sky signal\ndoes not vary significantly within a resolution element (see below\n\\ref{sec:signal_striping}). At the same time, several systematics\naffect the dataset. As explained above, many of them are cured at the\npreprocessing level. However, their removal generates a conspicuous\namount of transient flagging, that must be correctly handled by the\nGLS code. \n \nThe core of ROMAGAL is the same of the ROMA code \\citep{deGasperis05}\nwhere the input-output routines have been deeply modified to adapt to\nthe HIPE generated dataset. ROMAGAL inputs are the TOD generated by\nHIPE, pointing and transient flagging. These have the same format for\nboth PACS and SPIRE. ROMAGAL outputs are fits file containing the\noptimal map, arranged in standard gnomonic projection routines. \nThe code is written in FORTRAN 95 and relies on the MPI library for\nparallel calls. It runs on the Hi-GAL\\ dedicated machine, called BLADE,\na cluster of 104 processors at 2.5 GHz each and 208 Gb RAM total. Its\nnodes are interconnected with MPI-infiniBAND. The machine is located\nat IFSI-Rome. \n\nAs explained in the previous Section, the computation of Equation~\\ref{eq:GLS} is unfeasible due to the size of the map's covariance\nmatrix. However, we assume the noise of each Hi-GAL\\ field to be\nstationary to set up an FFT based solver built upon a conjugate\ngradient iterative algorithm (see Section \\ref{sec:mapmaking}). Such\na scheme can estimate the final maps with a precision of order of\n$\\epsilon=10^{-8}$ in $\\sim$ 150 iterations for Hi-GAL\\ . ROMAGAL\ncomputational time scales linearly with the size of the dataset and\nonly weakly with the number of pixels in the output maps. The scaling\nwith the number of processors is highly optimal in the range of cores\nrequired for the Hi-GAL\\ analysis ($< 50$). For the largest channels\n(PACS blue band), a final GLS map of a $2^\\circ \\times 2^\\circ$ field requires about 16 Gbytes of RAM and 1400 sec on 8\ncores.  Due to the high number of array pixels (2048), this channel is\nthe largest dataset to analyze as well as the most demanding in terms\nof computational resources. Further information on resource\nconsumptions can be found in Table \\ref{tab:computing_resources}. \n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|c|c|c|}\n\\hline  \\textbf{Band} & \\textbf{Total Time (sec)}  & \\textbf{RAM (Gb)}\\\\ \n\\hline  $70\\mu$m &  $\\sim$ 1400 & 16\\\\ \n\\hline  $160\\mu$m &  $\\sim$ 1000 & 8\\\\ \n\\hline  $250\\mu$m & $\\sim$  180 & 4\\\\ \n\\hline  $350\\mu$m &  $\\sim$ 130& 1\\\\ \n\\hline  $500\\mu$m &  $\\sim$ 100& 1\\\\ \n\\hline \n\\end{tabular} \n\\end{center}\n\\caption{Time and minimum RAM amount required from ROMAGAL for each\n  PACS and SPIRE band using 8 BLADE processors.} \n\\label{tab:computing_resources}\n\\end{table}\n\n\\subsection{Optimal treatment of transient flagging}\n\nAs mentioned above, the timelines are inspected for bad data samples\nthat must be excluded from map making as part of the preprocessing\npipeline. Bad data can arise due to a variety of reasons. They are\ngenerally caused by transient events, either unexpected (e.g.,\nglitches, anomalous hardware performance) or expected (e.g.,\ndetectors saturating because of a bright source, observation of a\ncalibrator).    \nOnce identified, a flag record is generated and stored for these\nanomalous events, so that their contribution can be safely excluded\nfrom map making. Flags in the TOD pose a potential problem for ROMAGAL\nbecause its solver is based on the FFT as discussed in the previous\nsection. The FFT requires the timeline to be continuous and uniformly\nsampled. Since noise in the PACS and SPIRE data is correlated, just\nexcising the flagged sampled to fictitiously create a continuous\ntimeline would interfere  with noise deconvolution, and is thus not a\nsafe option. Instead, we advocate using a suitable gap-filling\ntechnique. The rest of this section is mostly devoted to defining\nwhich, among the various options for gap filling, is best suited for the\nHi-GAL\\ maps.  \n\nIt is important to realize that the output map will depend on the\ncontent of the flagged sections of the TOD even if these values are\nnot associated to any (real) map pixel. This is due to the convolution\nperformed within the solver that ``drags'' out data from the flagged\nsection of the timeline, even if, when the data are summed back into a\nmap, the $P$ operator is applied only to the unflagged samples. Since\none is not in control of the content of the flagged section of the\ntimeline, a kind of gap-filling must be performed in any case. We have\ntested different recipes for gap-filling making extensive use of\nsimulations. We have treated separately the signal and noise\ncomponents of the timelines, running separately noise dominated and\nsignal dominated cases, because the behavior towards flags of the two\ncomponents is different, as it will be shown below. \n\nThe simplest form of gap filling is to remove the content of the flags\naltogether, replacing the flagged sections with a nominal constant\n(usually null) value. This works very well on a signal-only simulation\nof the Hi-GAL field. However, it fails dramatically when noise is\npresent, as evident from Figure  \n\\ref{fig:zeroflags_noiseonly_noCNR} (left panel), where a markedly\nstriped pattern in the reconstructed map is seen (in this simulation,\nthe Hi-GAL\\ noise has been amplified by a factor 100 to make its\npattern more evident). The reason for this failure is readily\nunderstood: The GLS map making employed requires noise stationarity\n(see Section  \\ref{sec:roma_romagal} above), which is obviously not\npreserved by zeroing the gaps. A less obvious effect is that even if\ngaps are filled with a noise realization with the correct statistical\nproperties, but unconstrained, the GLS map making is bound to fail as\nwell, as shown in the middle panel of Figure\n\\ref{fig:zeroflags_noiseonly_noCNR}. A noise realization is said to be\nconstrained when it respects certain boundary conditions\n\\citep{1991ApJ...380L...5H}, which in our case are represented by the\nnoise behavior outside the gap. Unconstrained noise inside the gap,\ndespite having the correct statistical properties, creates a border\ndiscontinuity that causes the GLS map maker to behave sub-optimally\n\\citep{2002PhRvD..65b2003S}. We have employed a code to create\nGaussian noise constrained (NCR) realizations, based on the algorithm\nset forth by \\citet{1991ApJ...380L...5H}. The code uses in input the\nnoise pattern and statistical properties, as measured from the\ntimelines. The results on noisy simulations are excellent, as set forth\nby the third (rightmost) panel in Figure\n\\ref{fig:zeroflags_noiseonly_noCNR}. Note, however, that Figure\n\\ref{fig:zeroflags_noiseonly_noCNR} refers to a noise dominated\nsimulation. We now turn to discuss the effect of a non negligible\nsignal contribution (a far more realistic case for Hi-GAL\\ ). \n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=10cm] {figure12.eps}\n\\caption{Shown are the results obtained in a noise\ndominated regime (normal Hi-GAL\\ noise is amplified by a factor\n100). Left panel is the map obtained replacing flagged data samples\nwith null values (clearly it does not work), middle panel is the map\nobtained replacing data samples with unconstrained noise realization\n(does not work either), right panel shows the map obtained using our\nNCR code (does work). \n\\label{fig:zeroflags_noiseonly_noCNR}}\n\\end{figure*}\n\nWe have verified that the presence of non negligible signal in the\ntimelines does not affect the results provided that the NCR is\nperformed using the underlying noise field as a baseline. Measuring\nthe latter in presence of signal is however impractical. It would be\nsignificantly simpler if the NCR could be run directly on the\ntimelines themselves, constraining thus the (fake) noise realization\nwithin the gap to the outside noise plus signal (true)\nvalues. Unfortunately, this poses a problem for Hi-GAL\\ data: large\nsignal jumps are present in the field and the resulting gap filling\nrealizations are affected in a non negligible manner by the boundary\nconditions, at least with the present version of ROMAGAL. This\nbehavior is different from what happens for experiments aimed at the\nCosmic Microwave Background (see e.g. \\citet{Masi06})\nwhere NCR codes are routinely run on the timelines as they are. In\norder to find a workaround that would spare us the inconvenience of\nestimating the underlying noise field to serve as a NCR input, we have\nmodified the flag treatment of the ROMAGAL code as explained in the\nfollowing. \n\nThe original version of ROMAGAL makes use of a single extra pixel\n(dubbed ``virtual pixel'') to serve as junk bin where the contents of\nthe gaps are sent when applying the $P^T$ operator within the ROMAGAL\nsolver. This approach, as stated above, works excellently in presence\nof both signal and noise, irrespective of their relative amplitude,\nprovided the NCR code assumes the underlying noise field as a baseline\nto perform the realization. In order to relax this assumption, we have\nmodified ROMAGAL to take into account not a single virtual pixel but\nan entire virtual map. In other words, we introduce a virtual\ncompanion for each pixel of the map, and use it as a junk bin to\ncollect the output from the gaps they correspond to. The hope is to\nredistribute the content of the flagged sections more evenly,\npreventing artifacts. This approach obtains satisfactory results when\nthe NCR code is run on the original (signal plus noise) timelines, as\nshown in Figure \\ref{fig:reldiff_realcnr_nvirtcnr_zeroflags_vs_input}.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=8cm] {figure13a.eps}\n\\includegraphics[width=8cm] {figure13b.eps}\n\\caption{Top row: For a signal dominated case, shown are the relative\n  differences between the input simulated map and the output obtained\n  with ROMAGAL (in the ``virtual map'' mode) with NCR performed on the\n  underlying noise (left), on the original signal plus noise TOD\n  (middle), and without NCR after replacing the gaps with null\n  values. The latter case is clearly striped, but no signal related\n  artifacts are present. Bottom row: For a noise dominated case, shown\n  again are the relative difference versus the input map obtained by\n  ROMAGAL with ``virtual map'',  assuming NCR on underlying noise only\n  (left), without NCR (middle) and with NCR on the signal plus noise\n  TOD (right). As expected, the left panel shows the best residuals,\n  but the right one appears as a good compromise (see also text). \n\\label{fig:reldiff_realcnr_nvirtcnr_zeroflags_vs_input}}\n\\end{figure}\nTo summarize our findings:\n\\begin{itemize}\n\n\\item Using a NCR realization code is {\\it always} necessary in order to avoid artifact striping effect into the map\n\n\\item If the NCR code is run on the underlying noise-only timelines\n  (which are cumbersome to estimate) we obtain the best quality output\n  map, with no signal related artifacts and a noise standard deviation\n  which is lower (by a factor $\\sim 2$) with respect to the case in\n  which no NCR is performed. \n\\item Running the NCR on the original timelines is possible with the\n  ``virtual map'' approach. No signal artifacts are detected in the\n  difference maps and the advantage in terms of noise standard\n  deviation with respect to the no NCR case is still present, and of\n  order 10\\% to 20\\% on average. \n\\end{itemize}\nWe have therefore chosen this  latter approach as the baseline for the Hi-GAL\\ pipeline. \n\n\\section{ROMAGAL maps}\\label{sec:SDP_results} \n\nIn this section we analyze the final map obtained running our dedicated pipeline. We analyze the Point Spread Function (PSF) for the five bands in order to fix the resolution of the ROMAGAL maps. We compare the final GLS map with the naive map and we\npoint out the differences and the capability to recover the diffuse\nemission from the GLS map. Finally we discuss about the noise\nresiduals on the maps. \n\n\\subsection{Point Spread Function and pixel size}\\label{sec:PSF}\n\nThe angular resolution (pixel size) of the final map is a free\nparameter. Its choice is a compromise between competing\nrequirements. A small pixel size assures a correct sampling of the PSF;\n\nindeed, assuming a Gaussian profile for the PSF (which is reasonable as discussed in the following), the Nyquist theorem imposes that to better sample a 2-d image we need\nto set a pixel size which is at most one third of its FWHM value.  \n\nOn the other hand, a too small pixel size can cause the loss of\nredundancy, useful to reduce the white noise level, and (even) some\nnon-observed pixels in the final map. \n\nThe diffraction limited beam of PACS at $70\\mu$m is $5.2\\arcsec$. Thus,\nwe should build the map with a pixel size of at least $1.8\\arcsec$. However, due to the limited bandwidth available for\ntransmission, especially in PACS\/SPIRE Parallel mode, PACS frames are coadded on-board the satellite before broadcasting. For\nthe $70$$\\mu$m\\ channel, a coaddition of 8 consecutive frames is applied by\non-board software.  Since the acquisition rate is 40Hz and the scanning\nspeed for Hi-GAL\\ is set to $60\\arcsec \/s$, two close frames are\nsnapshots of the sky acquired $1.5\\arcsec$ apart. Due to coaddition,\nthe satellite provides one averaged frame every $12\\arcsec$; in\nspatial coordinates, this is twice the beam width of the PACS blue channel. \nThe measured PSF then is not the diffraction limited beam but it\nresults in an elongation along the scan direction due to the\nconvolution of the coaddition with the beam. As shown in \\citet{Lutz09},\nthe observations of Vesta and $\\alpha$Tau with the blue channel\nevidenced a FWHM equal to $5.86\\arcsec \\times12.16\\arcsec$ as a result\nof a 2-d Gaussian fitting, elongated in the scan direction.  \n\nPACS $160\\mu$m is also affected by the averaging on-board, but only 4\n frames are coadded together. The nominal instrumental beam is\n $12.0\\arcsec$, while the measured is $11.64\\arcsec\\times15.65\\arcsec$\n \\citep{Lutz09}, elongated along the scan direction. However, in this\n case we can sample the beam without issues, and the effect of coaddition\n on the final map is negligible.\n \nFor the Hi-GAL\\ fields the scanning strategy consists of two orthogonal AORs, therefore the redundancy regularizes the\nPSF, resulting in approximately 2-d symmetric\nGaussian profile, as shown in Table  \\ref{tab:time_beam_pixel}.  \n\nAccording to the values reported in  Table  \\ref{tab:time_beam_pixel}, we observe\n quasi-symmetric beams with an averaged\nellipticity of less than $15\\%$ for both blue and red channel and the\naxis oriented randomly with respect to the scan direction.\n\nWe choose a pixel size of $3.2\\arcsec$ for the PACS $70\\mu$m\nband which samples the observed beam at Nyquist frequency. Below this threshold, the diffuse emission\nareas become too noisy due to the low SNR.\nSimilarly, we can choose a pixel size of\n $4.5\\arcsec$ for red band without loosing redundancy.  \n\nSPIRE does not suffer from on-board coadding and the detectors were\nbuilt to reach the band diffraction limit. In-flight data show\nthat the SPIRE beam is well approximated by a 2-d asymmetric Gaussian\ncurve with the major axis orientation independent of the\nscan direction, with an ellipcticity not bigger then $10\\%$ (see \\citet{Sibthorpe10}). We set the pixel size for\neach SPIRE band equal to one third of the nominal beam. \nIn Table \\ref{tab:time_beam_pixel} we also report the beam measured in\nthe SPIRE maps. \n\nThe average ellipticity we observe agrees with found by \\citet{Sibthorpe10}. On the contrary, while the FWHM for the two axis found by \\citet{Sibthorpe10} are in agreement with the nominal (within the error), our measured beam results in a FWHM larger then the nominal of $\\sim25\\%$. \n\n\\begin{table*}\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline  \\textbf{Band} & \\textbf{Nominal Beam (arcsec)}  & \\textbf{Measured Beam (arcsec)} & \\textbf{Ellipticity} & \\textbf{Pixel Size (arcsec)}\\\\ \n\\hline  $70\\mu$m & $5.2\\times 5.2$ & $\\sim 9.7\\times \\sim 10.7$ & 14.6\\% &  3.2 \\\\ \n\\hline  $160\\mu$m &  $12.0\\times 12.0$ & $\\sim 13.2\\times \\sim 13.9$ & 14.7\\%& 4.5\\\\ \n\\hline  $250\\mu$m &  $18\\times 18$ & $\\sim 22.8\\times \\sim 23.9$ & 8.3\\%& 6.0\\\\ \n\\hline  $350\\mu$m &  $24\\times 24$ & $\\sim 29.3\\times 31.3$ & 8.8\\%& 8.0\\\\ \n\\hline  $500\\mu$m &  $34.5\\times 34.5$ & $\\sim 41.1\\times \\sim 43.8$ & 9.7\\%& 11.5\\\\ \n\\hline \n\\end{tabular} \n\\end{center}\n\\caption{Nominal (2nd column), map-measured beam (two AOR, 3rd column) and ellipticity (4th column) of each band.} \n\\label{tab:time_beam_pixel}\n\\end{table*}\n\n\\subsection{Hi-GAL SDP results}\n\nThe quality of the outcome can be qualified by the comparison between the igls maps with the naive maps. In fact, the naive map is the simple averaging of signal recorded by every spatial pixel and it represents ``the least compromised view of the sky\". \n\nSince the TOD are created at the end of the preprocessing steps, when the data are a combination of only signal and $1\/f$ noise, we expect the $1\/f$ residuals in the naive map as well as a ``pure\" sky map produced by ROMAGAL. \n\nIn Figure \\ref{fig:PACS_l30_igls_vs_bin} a comparison between the\nnaive map and the ROMAGAL map of $\\textit{l}=30^{\\circ}$ field at $70\\mu$m is shown. The\nGLS code is capable to remove the $1\/f$ residuals without loosing any\nsignal both on bright sources and on diffuse emission. \n\n\\begin{figure}\n\\centering\n\\includegraphics[width=8cm]{figure17.eps}\n\\caption{Left: particular of the ROMAGAL map of the red PACS array,  $\\textit{l}=30^{\\circ}$ field. Right: particular of the naive map, same\n  band and field. The $1\/f$ noise is evident on the naive map, as well\n  as its minimization is evident on the GLS map without loosing the\n  signal of the diffuse component.} \n\\label{fig:PACS_l30_igls_vs_bin}\n\\end{figure}\n\n In particular, we choose three main proxies:\n\n\\begin{itemize}\n\\item   the difference between naive and igls should show only a pattern due to the $1\/f$ noise residuals in the binned map. The pattern of this low-frequency noise is recognizable as stripes superimposed on the sky signal in the naive map. The stripes are the consequence of the $1\/f$ noise due to the adopted scanning strategy.\\\\\nIn Figure \\ref{fig:diff_PACS_l30_red_igls_bin} we show the difference\nbetween igls map and naive map. The $1\/f$ noise is removed in the igls\nmap but not in the naive. The residual stripes due to the\nlow-frequency noise are clearly visible. \n\n\\begin{figure}\n\\centering\n\\includegraphics[width=5.25cm]{figure18.eps}\n\\caption{Particular of map difference between PACS $160\\mu$m igls and naive, same region of the previous Figure. The\n  stripes due to $1\/f$ removal made in the igls are evident.} \n\\label{fig:diff_PACS_l30_red_igls_bin}\n\\end{figure}\n\n\\item the source fluxes should be the same between the igls and naive maps.\\\\\n This item is quantified by the map difference, where the pattern is due only to the noise without any residual signal, except across the very brilliant source. This last effect is discussed on the next Section.\n\n\\item a statistical analysis of the background noise level should show a decrease of the rms value in the igls with respect to the naive map.\\\\\nIn Tables\n\\ref{tab:rms_PACS} and \\ref{tab:rms_SPIRE} we report the rms residuals of the PACS and SPIRE maps respectively, calculated\nin a diffuse emission area of each map. Since the flux between the maps is conserved, a decreasing of the rms noise level assures an increasing of S\/N ratio in the ROMAGAL maps.\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|}\n\n\\hline\n\\multicolumn{4}{|c|}{\\textbf{PACS $\\textit{l}=30^{\\circ}$ field}}\\\\\n\\hline \\textbf{Band}&  \\textbf{rms igls (MJy\/pixel)}& \\textbf{rms naive(MJy\/pixel)}& \\textbf{ratio}\\\\ \n\\hline  $70\\mu$m &  0.0085&  0.026& $\\sim$ 3.1 \\\\ \n\\hline  $160\\mu$m &  0.047&  0.102&  $\\sim$ 2.2\\\\ \n\\hline \n\\hline\n\\multicolumn{4}{|c|}{\\textbf{PACS $\\textit{l}=59^{\\circ}$ field}}\\\\\n\\hline \\textbf{Band}&  \\textbf{rms igls (MJy\/pixel)}& \\textbf{rms naive(MJy\/pixel)}& \\textbf{ratio}\\\\ \n\\hline  $70\\mu$m &  0.004545&  0.02208& $\\sim$ 4.9 \\\\ \n\\hline  $160\\mu$m &  0.01899&  0.03586&  $\\sim$ 1.9\\\\ \n\\hline \n\\end{tabular} \n\\end{center}\n\\caption{rms compared from GLS and naive map on both the SDP observations,\n  for PACS bands, measured on a background region of $50\\times$50\n  pixels. In the last column is reported the ratio between the naive\n  and GLS rms.} \n\\label{tab:rms_PACS}\n\\end{table}\n\nThe ratio between the naive and igls rms shows an improvement of a factor $\\sim 2-5$ in the PACS ROMAGAL maps, and a factor $\\sim 1-2$ in the SPIRE case. The difference is mostly due to an intrinsically different $1\/f$ noise level.\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|}\n\\hline\n\\multicolumn{4}{|c|}{\\textbf{SPIRE $\\textit{l}=30^{\\circ}$ field}}\\\\\n\\hline \\textbf{Band}&  \\textbf{rms igls (MJy\/beam)}& \\textbf{rms naive(MJy\/beam)}& \\textbf{ratio}\\\\ \n\\hline  $250\\mu$m &  0.1749&  0.2868& $\\sim$ 1.6 \\\\ \n\\hline  $350\\mu$m &  0.1569&  0.2302&  $\\sim$ 1.5\\\\ \n\\hline  $500\\mu$m &  0.2659&  0.4065&  $\\sim$ 1.5\\\\ \n\\hline \n\\hline\n\\multicolumn{4}{|c|}{\\textbf{SPIRE $\\textit{l}=59^{\\circ}$ field}}\\\\ \n\\hline \\textbf{Band}&  \\textbf{rms igls (MJy\/beam)}& \\textbf{rms naive(MJy\/beam)}& \\textbf{ratio}\\\\ \n\\hline  $250\\mu$m &  0.09857&  0.1123& $\\sim$ 1.1 \\\\ \n\\hline  $350\\mu$m &  0.0734&  0.08164&  $\\sim$ 1.1\\\\ \n\\hline  $500\\mu$m &  0.1073&  0.2101&  $\\sim$ 1.9\\\\ \n\\hline \n\\end{tabular} \n\\end{center}\n\\caption{rms compared from GLS and naive map on both the SDP observations,\n  for SPIRE bands, measured on a background region of $50\\times$50\n  pixels. In the last column is reported the ratio between the naive\n  and GLS rms. The $1\/f$ noise is less evident in the SPIRE bolometers\n  with respect to the PACS one, but its effect is still remarkable.}\n\\label{tab:rms_SPIRE}\n\\end{table}\n\n\\end{itemize}\n\n\\subsection{Consistency of data analysis assumptions}\\label{sec:signal_striping}\n\nOne of the assumptions of ROMAGAL, as well as of all Fourier based GLS map making codes, is that the underlying sky signal is constant within a chosen resolution element. If this is not the case, artifacts (stripes) will be generated in the final map, contributing to the so called pixelization noise \\citep{Poutanen06}. In the case of Hi-GAL the situation is complicated by several effects: \n\n\\begin{itemize}\n\n\\item On-board coaddition of samples: each PACS 70$\\mu$m (160  $\\mu$m) frame is the result of an on-board average of eight (four)\nconsecutive frames  reducing the effective sampling frequency of the instrument (see Section \\ref{sec:PSF}). Thus, sky signal is low-pass filtered by only partially effective data-windowing, rather then telescope response, leaving room for signal aliasing.\n\n\nThe map making code is quite sensitive to aliasing since it works in Fourier space. The situation is worsened by the large dynamic range of the Hi-GAL fields, especially when scanning across bright sources.\n\n\\item Time bolometer response induces signal distortions along the scan. While within HIPE the SPIRE detector response is deconvoluted from the data \\citep{Griffin09_pipeline}, the same is not true for PACS. Redundancy in each pixel is obtained by scans coming from different directions, thus the effect contributes further signal mismatch at the pixel level.\n\n\\item Pointing error: as analyzed in detail in \\citet{Herschel} the pointing performance of Herschel, which mean the capability of assign coordinates to each sample in a given reference frame, can be affected by several pointing error effects; the main contributor is due to the angular separation between the desired direction and the actual instantaneous direction, based on the position-dependent bias within the star-trackers. \n\nThe Herschel AOCS goal is that the mismatch between real coordinates and assigned\ncoordinates along the scan-leg is smaller than $1.2\\arcsec$ at 1 sigma \\citep{Herschel}. So that a 2$\\sigma$ event becomes significant compared to the PSF of the PACS blue band.\n\n\\end{itemize} \n\n\\begin{figure}\n\\centering\n\\includegraphics[width=6cm]{figure16.eps}\n\\caption{Zoom around a compact sources for the PACS blue optimal\n  map. The striping dragged along the scan directions are\n  remarkable.}  \n\\label{fig:crosses_blue}\n\\end{figure}\n\nAll of the above effects challenge the basic assumptions (no signal aliasing and no sub-pixel effect) under which ROMAGAL works.  Our simulations suggest that signal aliasing contribute significantly more than the other two. The net result on maps is the striping of bright sources in the GLS maps. An example is shown in Figure \\ref{fig:crosses_blue} for the PACS blue band.\n\nIt is important to notice that the stripes are present only around the point-like sources (where - of course - the signal aliasing is more evident), regardless of their magnitude. However, magnitude influences the amplitude of the stripes. For a source peak which is only 10 times higher that the rms background, the intensity of the stripes are within $1\\sigma$ from the background dispersion. For the more intense sources, the stripes magnitude in the pixels surrounding the PSF shape can be $100\\sigma$ times away from the background value.\n\nSince these stripes are produced within the GLS solver, which performs repetitive convolutions along the scan directions, but do not affect the naive map the obvious workaround is to implement dedicated map making which considers a different analysis around the bright sources.\n \n However, the detailed accounting of the above effects and the enhanced map making strategies to address them  will be the subject of a forthcoming paper \\cite{Piazzo11}.\n\n\\section{Summary and conclusions} \\label{sec:Conclusions}\n\nThis paper describes in detail all steps of the processing of Herschel data, from originally downloaded frames to high-quality maps, used in the framework of Hi-GAL survey. Hi-GAL data are taken in fast scan mode ($60\\arcsec$\/sec ) and simultaneously by PACS and SPIRE (Parallel mode). We test our pipeline reducing data from the Science Demonstration Phase and present results, taking as proxy for the quality of the final images their comparison with naive maps. \n\nWe divided data processing into two distinct phases: preprocessing and mapmaking. Pre-processing aims to accurately remove systematics and random effects in the data, in order to prepare them for the ROMAGAL map making algorithm, which implements the minimum variance GLS approach in order to minimize the noise into the data. It turns out that NCR is a fundamental step in the pre-processing because ROMAGAL, as an FFT mapmaking code needs continuous and regular data time series as input.\n\nNoise residuals in the diffuse emission of the two test fields (SDP Hi-GAL data, two 2$^{\\circ}\\times2^{\\circ}$ tiles centered on the Galactic plane at l = 30$^{\\circ}$ and l = 59$^{\\circ}$) show that we obtain optimal maps, getting rid of glitches and systematic drifts, as well as minimizing the 1\/ f and white noise component. The remaining effects, which do not affect the overall quality of the maps except across the bright source on the PACS 70$\\mu$m maps, are under investigation that will appear in a dedicated publication to be available shortly. \n\n\\section*{Acknowledgements}\nSpecial thanks to G\\\"oran Pilbratt for allowing the using of PV Phase\nblank data. \n\n\\bsp\n\n\\bibliographystyle{mn2e}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nElectromagnetic periodic structures have received a lot of attention due to their peculiar properties and their relatively easy integration in complex designs. Among the applications are the design of Frequency-Selective Surfaces (FSS), reflect- and transmit-arrays, metasurfaces, metamaterials, etc. The efficient study of these materials requires suitable numerical tools.\n\nStructures periodic in two directions can be simulated using the Method of Moments (MoM), a spectral method based on surface integral equations \\cite{ref1}. If the excitation is quasi-periodic, \\textit{i.e.} it shares the periodicity of the structure within a linear phase-shift, the response of the structure can be obtained from the study of a single unit cell. The periodicity is accounted for through the use of a modified Green's function that accounts for the periodicity of the structure. First, the impedance matrix of the structure is computed. Then, the resulting system of equation is solved to obtain the response of the structure to a given excitation. For unit cells of small or moderate size, most of the time is devoted to the computation of the periodic impedance matrix. Non-periodic sources can be handled using the Array Scanning Method (ASM), which consists of decomposing the non-periodic source into a superposition of quasi-periodic sources and solving each subproblem separately \\cite{ref6}.\n\nOne drawback of the periodic Method of Moments is that the response of the structure must be computed for each frequency and relative phase-shift between consecutive unit cells. This problem becomes critical when using the ASM, for which a large number of phase shifts may be required to get accuracte results \\cite{ref9}. One way to counter this drawback is to use interpolation techniques \\cite{ref12, ref10, ref11}. The value of the impedance matrix is computed for few reference frequencies and\/or phase shifts. Its value at different frequencies and\/or phase shifts is then interpolated. Brute-force polynomial interpolation is inefficient due to the rapidly oscillating behaviour of the Green's function and the singularity arising from plane-waves with grazing incidence. To improve the convergence of the method, it was proposed to use a Model-Based Parameter Estimation (MBPE) technique \\cite{ref12, ref11}. Based on the physics, a model is developed to describe the generic behaviour of the periodic impedance matrix entries. Then, the evolution of each entry of the impedance matrix is obtained by adjusting few coefficients involved in the model. In \\cite{ref10}, the periodic impedance matrix for normally incident plane-waves is divided into two contributions: the resonant modes of the substrate and the rest. Then, the relative contribution of the different terms is estimated to fit the actual value of the periodic impedance matrix at few sampling frequencies. In \\cite{ref11}, it is proposed to extend the technique to oblique incidence by removing an additional phase term. In this way, frequency interpolation of the periodic impedance matrix can be carried out for non-vanishing phase shifts between consecutive unit cells. However, these studies concentrate on the interpolation of the periodic impedance matrix vs. frequency. It seems that the interpolation with respect to the phase shift has received little attention so far.\n\nIn this paper, we propose a method to interpolate the periodic impedance matrix for different phase shifts between consecutive unit cells. First, the contribution of the leading Floquet modes is removed from the impedance matrix. Then, an additional phase term is extracted. It is shown that the resulting function is smooth with respect to the phase shift between consecutive unit cells. The contribution of the dominant Floquet modes is evaluated numerically instead of being fitted, so that very few sampling points are required for the interpolation. It is in contrast with MBPE techniques for which at least one sampling point per coefficient is required. It is also shown that removing the contribution of the main Floquet modes breaks the periodicity of the impedance matrix, so that one impedance matrix can be used to obtain several sampling points.\n\nThe paper is organized as follows. In Section II, the working principle of the periodic MoM is briefly described. Then, in Section III, the behaviour of the periodic impedance matrix is studied and the interpolation technique is described. Last, in Section IV, the interpolation technique is validated through few examples. \n\n\\section{The Periodic Method of Moments}\nSurface Integral Equation based numerical methods are relying on the surface equivalence principle. The response of a structure to incident fields is modeled using equivalent currents. The fields generated by these currents correspond to the fields scattered by the structure. The amplitude of the equivalent currents for a given excitation can be determined by enforcing boundary conditions across each interface separating different media. Using the Method of Moments, the unknown currents are discretized using a predefined set of basis functions (BF) and the boundary conditions are imposed on average along a predefined set of testing functions (TF) \\cite{ref8}. First, the impact of each BF on the boundary conditions along each TF is computed and stored in the so-called impedance matrix. Then, the amplitude of the currents is found by solving the resulting system of equations:\n\\newcommand{\\mat}[1]{\\underline{\\underline{#1}}}\n\\newcommand{\\vect}[1]{\\mathbf{#1}}\n\\begin{equation}\n\\mat{Z} \\, \\vect{x} = - \\vect{b},\n\\end{equation}\nwith $\\mat{Z}$ the impedance matrix, $\\vect{x}$ a vector containing the unknown coefficients used to expand the equivalent currents using the BF and $\\vect{b}$ the impact of the incident fields on the boundary conditions.\nOne natural set of boundary conditions that can be used is the continuity of the tangential electric and magnetic fields, leading to the Poggio-Miller-Chang-Harrington-Wu-Tsai (PMCHWT) formulation. Using this formulation, the impedance matrix corresponds to the sum of the fields generated by the BF along the TF through the media on both sides of the interface.\n\nThis method can be extended to periodic structures provided that the fields radiated by replicas of the BFs are included when computing the impedance matrix \\cite{ref1}. In that case, one obtains the following system of equations\n\\newcommand{\\mat{\\tilde{Z}}}{\\mat{\\tilde{Z}}}\n\\begin{equation}\n\\mat{\\tilde{Z}}(\\boldsymbol{\\varphi}) \\, \\vect{x} = -\\vect{b}(\\boldsymbol{\\varphi})\n\\end{equation}\nwith $\\mat{\\tilde{Z}}$ the periodic impedance matrix and $\\boldsymbol{\\varphi}$ the phase-shift between consecutive unit cells in the two directions of periodicity. \n\nOne way to compute the periodic impedance matrix is to first compute the the periodic Green's function and convolve it spatially with the BFs and integrate the result along the TF. In that case, several methods can be used to accelerate the computation of the periodic Green's function (see \\textit{e.g.} \\cite{ref2, ref3, ref4, ref5}). Another possibility is to compute directly the impedance matrix in the spectral domain. In that case, each entry of the impedance matrix can be expressed as an infinite series of spectral term, each term corresponding to one Floquet mode of the structure \\cite{ref14}. \n\n\n\\section{Interpolation scheme}\n\\newcommand{\\dir}[1]{\\hat{\\mathbf{#1}}}\nWe consider a homogeneous medium of permittivity and permeability $\\varepsilon$ and $\\mu$. In this medium, a quasi-periodic set of BF of period $d_x$ and $d_y$ and phase-shifts $\\varphi_x$ and $\\varphi_y$ in the $\\dir{x}$ and $\\dir{y}$ directions emit electric ($E$) and magnetic ($H$) fields, which are tested with the TFs. We define the direction $\\dir{z} = \\dir{x}\\times \\dir{y}$. We consider that the TF is located either above or below the BF, \\textit{i.e.} the sign of $\\dir{z} \\cdot (\\vect{r}' - \\vect{r})$ is identical for any pair of points $\\vect{r}'$ and $\\vect{r}$ on the BF and TF, respectively. This condition is always met in planar geometries, so that we will restrain ourselves to this particular case for the rest of the paper. Then, using the notations of \\cite{ref9}, the corresponding entry of the periodic impedance matrix reads \n\\newcommand{\\sumInf}[1]{\\sum_{#1=-\\infty}^\\infty}\n\\newcommand{\\tilde{f}_{B,e} (\\mathbf{k}_{pq})}{\\tilde{f}_{B,e} (\\mathbf{k}_{pq})}\n\\newcommand{\\tilde{f}_{B,m} (\\mathbf{k}_{pq})}{\\tilde{f}_{B,m} (\\mathbf{k}_{pq})}\n\\newcommand{\\tilde{f}_{T,e} (-\\mathbf{k}_{pq})}{\\tilde{f}_{T,e} (-\\mathbf{k}_{pq})}\n\\newcommand{\\tilde{f}_{T,m} (-\\mathbf{k}_{pq})}{\\tilde{f}_{T,m} (-\\mathbf{k}_{pq})}\n\\begin{align}\n\\tilde{Z}^{EJ}&(\\boldsymbol{\\varphi}) = \n\\dfrac{\\eta k}{2 d_x d_y}\n\\sumInf{p} \\sumInf{q} \\dfrac{1}{\\gamma_{pq}}\n\\label{eq:01}\n\\\\\n&  \\hspace{-1cm} \\times\n\\Big( \\tilde{f}_{B,e} (\\mathbf{k}_{pq}) \\tilde{f}_{T,e} (-\\mathbf{k}_{pq})  + \\tilde{f}_{B,m} (\\mathbf{k}_{pq}) \\tilde{f}_{T,m} (-\\mathbf{k}_{pq}) \\Big)\n\\nonumber\n\\\\\n\\tilde{Z}^{EM}&(\\boldsymbol{\\varphi}) = \n\\dfrac{ k}{2 d_x d_y}\n\\sumInf{p} \\sumInf{q}\n\\dfrac{1}{\\gamma_{pq}}\n\\label{eq:02} \n\\\\\n& \\hspace{-1cm} \\times\n\\Big( \\tilde{f}_{B,e} (\\mathbf{k}_{pq}) \\tilde{f}_{T,m} (-\\mathbf{k}_{pq}) - \\tilde{f}_{B,m} (\\mathbf{k}_{pq}) \\tilde{f}_{T,e} (-\\mathbf{k}_{pq}) \\Big) \\nonumber\n\\end{align}\nwith $\\boldsymbol{\\varphi} = (\\varphi_x, \\varphi_y)$, $\\eta = \\sqrt{\\mu\/\\varepsilon}$ the impedance of the medium through which the interactions are computed, $k = \\omega \\sqrt{\\varepsilon \\mu}$ the wavenumber of the medium, $\\omega$ the angular frequency, $\\mathbf{k}_{pq} = (k_{x,p}, k_{y,q}, \\pm \\gamma_{pq})$ the wave-vector associated to Floquet harmonic $(p,q)$, $\\tilde{f}_{B\/T,e\/m}$ the Fourier transform of the $\\dir{e}$ or $\\dir{m}$ component of the BF ($B$) and TF ($T$), with:\n\\begin{align}\n\\dir{e}(\\mathbf{k}_{pq}) \n&= \n\\dfrac{1}{k_{t,pq}} \\big(-k_{x,p}, k_{y,q}, 0 \\big) \n\\\\\n\\dir{m}(\\mathbf{k}_{pq}) \n&=\n- \\dfrac{\\gamma_{pq} }{k k_{t,pq}} \\bigg(\\pm k_{x,p}, \\pm k_{y,q}, - \\dfrac{k_{t,pq}^2}{\\gamma_{pq}} \\bigg)\n\\\\\n\\mathbf{k}_{t,pq} &= \\big(k_{x,p}, k_{y,q}, 0 \\big) \\\\\n\\gamma_{pq} &= \\sqrt{k^2-k_{t,pq}^2} \\\\\nk_{t,pq} &= \\sqrt{\\mathbf{k}_{t,pq} \\cdot \\mathbf{k}_{t,pq}} \\\\\nk_{x,p} &= \\dfrac{2 \\pi p + \\varphi_x}{d_x} \\label{eq:08}\\\\\nk_{y,q} &= \\dfrac{2 \\pi q + \\varphi_y}{d_y} \\label{eq:09}\n\\end{align}\nNote that where $\\pm$ is indicated, the $+$ sign is used if the TF is located above the BF, and the $-$ sign otherwise. Similarly, note that $b = \\sqrt{a}$ is defined such that its imaginary part is negative and, if $b$ is real, it is positive.\n\nFrom \\eqref{eq:01} and \\eqref{eq:02}, the dependence of the impedance matrix on $\\boldsymbol{\\varphi}$ can be analyzed. Looking at each term of the sum separately and factoring out the phase term due to the spatial shift between the BF and TF, one can highlight four different factors: \n\\newcommand{\\tilde{f}_{B,e\/m}^0 (\\mathbf{k}_{pq})}{\\tilde{f}_{B,e\/m}^0 (\\mathbf{k}_{pq})}\n\\newcommand{\\tilde{f}_{T,e\/m}^0 (-\\mathbf{k}_{pq})}{\\tilde{f}_{T,e\/m}^0 (-\\mathbf{k}_{pq})}\n\\newcommand{\\mathbf{r}_B^0}{\\mathbf{r}_B^0}\n\\newcommand{\\mathbf{r}_T^0}{\\mathbf{r}_T^0}\n\\newcommand{\\mathbf{k}_{pq}}{\\mathbf{k}_{pq}}\n\\begin{equation}\n\\label{eq:03}\n\\dfrac{1}{\\gamma_{pq}}\n\\end{equation}\n\\begin{equation}\n\\label{eq:04}\n\\tilde{f}_{B,e\/m}^0 (\\mathbf{k}_{pq}) \\tilde{f}_{T,e\/m}^0 (-\\mathbf{k}_{pq}) \n\\end{equation}\n\\begin{equation}\n\\label{eq:05}\n\\exp\\bigg(j \\mathbf{k}_{t,pq} \\cdot (\\mathbf{r}_B^0-\\mathbf{r}_T^0)\\bigg)\n\\end{equation}\n\\begin{equation}\n\\label{eq:06}\n\\exp\\bigg(\\pm j \\gamma_{pq} \\dir{z} \\cdot (\\mathbf{r}_B^0-\\mathbf{r}_T^0)\\bigg)\n\\end{equation}\nwith $\\tilde{f}_{B,e\/m}^0 (\\mathbf{k}_{pq})$ and $\\tilde{f}_{T,e\/m}^0 (-\\mathbf{k}_{pq})$ the Fourier transforms of the BF and TF when they are centered at the origin and $\\mathbf{r}_B^0$ and $\\mathbf{r}_T^0$ the position of the BF's and TF's actual centers with respect to the origin.\n\nWe now look at each factor separately.\n\\subsubsection*{Factor \\eqref{eq:03}}\nFor $k_{t,pq} \\simeq k$, this factor varies rapidly vs. $\\varphi_x$ and $\\varphi_y$ that may be hard to interpolate. However, for $k_{t,pq} \\gg k$, the variation is slow with respect to $\\boldsymbol{\\varphi}$ and thus the interpolation is expected to work well.\n\\subsubsection*{Factor \\eqref{eq:04}}\nThe BF and TF are generally well-behaving functions, so their Fourier transform does not exhibit any sharp feature. Moreover, the BF and TF being usually much smaller than the unit cell, their Fourier transform is expected to vary very slowly with $\\boldsymbol{\\varphi}$.\n\\subsubsection*{Factor \\eqref{eq:05}}\nThe variation of this factor with respect to $\\boldsymbol{\\varphi}$ does not depend on indices $(p,q)$ of the Floquet mode considered and can be factored out as \n\\begin{equation}\n\\label{eq:07}\n\\exp\\bigg(j \\Big(\\dfrac{\\varphi_x}{d_x} \\dir{x} + \\dfrac{\\varphi_y}{d_y} \\dir{y}\\Big) \\cdot (\\mathbf{r}_B^0-\\mathbf{r}_T^0)\\bigg).\n\\end{equation}\n\\subsubsection*{Factor \\eqref{eq:06}}\nFour different cases can be highlighted, depending on the distance $\\Delta z$ between the BF and the TF in the $\\dir{z}$ direction and depending on $k_{t,pq}$, the amplitude of the transverse wave-vector. On one hand, if $\\Delta z \\gg 0$, variations of the factor are rapid if $k_{t,pq} \\lesssim k_0$. However, if $k_{t,pq} \\gg k_0$, \\eqref{eq:06} becomes negligible and the contribution of the corresponding terms to series \\eqref{eq:01} and \\eqref{eq:02} is negligible. On the other hand, if $\\Delta z \\simeq 0$, variation with respect to $\\gamma_{pq}$ and thus $k_{t,pq}$ is slow. It should be noticed that for $k_{t,pq} = k$, despite the relatively slow variation, the function is not analytic. Thus, interpolation is not expected to provide accurate results when crossing this limit.\n\nSummarizing all these observations, the high-order $(p,q)$ terms, and thus their sum, are expected to be easy to interpolate. Thus the interpolation method is the following:\n\\begin{enumerate}\n\\item Compute the periodic impedance matrix for few phase shifts.\n\\item Remove the contribution of the dominant Floquet modes ($|p|, |q| \\leq N$).\n\\item Remove the phase term of \\eqref{eq:07}.\n\\item Interpolate the remaining function for the phase shift of interest using a polynomial interpolation technique.\n\\item Re-introduce the phase term of \\eqref{eq:07}.\n\\item Re-add the contribution of the dominant Floquet modes.\n\\end{enumerate}\n\nOne interesting feature of the technique is that $\\mat{\\tilde{Z}}$ is periodic with respect to $\\varphi_x$ and $\\varphi_y$, so that \n\\begin{equation}\n\\mat{\\tilde{Z}}(\\varphi_x, \\varphi_y) = \\mat{\\tilde{Z}}(\\varphi_x + 2\\pi, \\varphi_y) = \\mat{\\tilde{Z}}(\\varphi_x, \\varphi_y +2\\pi).\n\\end{equation}\nHowever, the Floquet modes that are removed do depend on the phase shift considered since, according to \\eqref{eq:08} and \\eqref{eq:09}, we have\n\\begin{equation}\nk_{x,p}(\\varphi_x +2\\pi) = k_{x,p+1}(\\varphi_x)\n\\end{equation}\n\\begin{equation}\nk_{y,q}(\\varphi_y + 2\\pi) = k_{y,q+1}(\\varphi_y)\n\\end{equation}\nIt means that from a single periodic impedance matrix, one can evaluate the value of the interpolant at several different locations. Let's take a simplified 1D example, as illustrated in Fig. \\ref{fig:02}. On the top graph, one can see the evolution of the Floquet modes of the series \\eqref{eq:01} and \\eqref{eq:02} with the phase-shift $\\varphi$.  Clearly, if $\\varphi$ is increased by $2\\pi$, the Floquet modes will be translated by one single period. Hence, when summing the contribution of the Floquet modes, one obtains the same value for $\\varphi$ and $\\varphi +2\\pi$. However, after removing the contribution of the $2N+1$ dominant Floquet modes, the remaining terms in series \\eqref{eq:01} and \\eqref{eq:02} will be different for different phase shifts $\\varphi$ (cf. bottom graphs). The resulting interpolant $Z_\\text{rem}(\\varphi)$ will be non-periodic. Thus, from a single periodic impedance matrix, by choosing properly the Floquet modes that are being removed, one can determine the value of the interpolant at several different locations.\n\n\\begin{figure}\n\\center\n\\includegraphics[width = 6 cm]{Illustration_Zper} \\\\\n\\includegraphics[width = 7cm, trim={1cm 0 1.2cm 0},clip]{Illustration_Zrem}\n\\caption{Top graph: Qualitative illustration of the evolution of the Floquet modes of \\eqref{eq:01} and \\eqref{eq:02} with respect to the phase shift of the periodic impedance matrix. Different colors correspond to different phase shifts. Bottom graphs: Floquet modes included in the interpolant $Z_\\text{rem}(\\varphi)$ for $\\varphi = -2\\pi, 0, 2\\pi$. The black dashed Floquet modes are not accounted for in order to smoothen the interpolant (Floquet modes extraction). Through the extraction of these modes, the periodicity of the series of \\eqref{eq:01} and \\eqref{eq:02} is broken.}\n\\label{fig:02}\n\\end{figure}\n\nFor example, considering the original 2D periodic scenario, the periodic impedance matrix $\\mat{\\tilde{Z}}(\\boldsymbol{\\varphi} = (0,0))$ can be used to estimate the value of the interpolant at points $(-2\\pi, -2\\pi)$, $(-2\\pi, 0)$, $(-2\\pi, 2\\pi)$, $(0, -2\\pi)$, $(0, 0)$, $(0, 2\\pi)$, $(2\\pi, -2\\pi)$, $(2\\pi, 0)$ and $(2\\pi, 2\\pi)$. For each of these sampling points, a different set of Floquet harmonics will be extracted from the periodic impedance matrix and thus the value of the interpolant will be different.\n\n\\section{Numerical results}\nIn order to illustrate the improvement of the interpolant with each successive transformation, we consider the interaction between two identical co-planar rooftop basis functions aligned in the $\\dir{x}$ direction.  Each half of the rooftop is corresponding to a square of size $\\lambda\/18$, $\\lambda$ being the wavelength. The periodicity in the $\\dir{x}$ and $\\dir{y}$ directions is $\\lambda\/1.8$. The TF corresponds to the replica of the BF translated by half a unit cell in the $\\dir{x}$ and $\\dir{y}$ directions. The evolution of the periodic interaction with phase shift is illustrated in the top graphs of Fig. \\ref{fig:01}. Graphs in the left-hand side illustrate the real part and graphs on the right-hand side illustrate the imaginary part of the periodic interaction. The middle graphs illustrate the value of the periodic impedance matrix after removing the contribution of the 9 dominant Floquet modes ($N=1$). The bottom graphs illustrate the value of the interpolant after removing both the contribution of the 9 dominant Floquet modes and the phase term of \\eqref{eq:07}. It can be seen that, at each step, the function becomes much smoother and easier to interpolate. Similar results are found for different pairs of BF and TF. It is worth noting that the value of the interpolant is illustrated for phase-shifts varying from $-2\\pi$ to $2\\pi$. While one is generally interested in estimating the periodic impedance matrix for phase shifts ranging from $-\\pi$ to $\\pi$, this extended zone can be used to add sampling points that may improve the polynomial interpolation. \n\n\\begin{figure}\n\\center\n\\includegraphics[width = 8cm]{fig1.pdf}\n\\caption{Evolution of the periodic interaction with the contribution of all the Floquet modes (top graphs), without the contribution of the 9 dominant Floquet modes (middle graphs) and after removing the remaining phase term (bottom graphs). Both the real (LHS) and imaginary (RHS) parts of the functions are displayed. The red square delineates the zone of phase-shifts going from $-\\pi$ to $\\pi$.}\n\\label{fig:01}\n\\end{figure}\n\nIn order to check the accuracy of the interpolation procedure, we meshed a whole plane of the unit cell using 200 rooftop TFs oriented along the $\\dir{x}$ and $\\dir{y}$ directions. The periodic interaction between a single BF and the 200 TF was computed for several different heights $\\Delta z$ between the BF and the TF. For the interpolation, we used four different periodic impedance matrices: $\\mat{\\tilde{Z}}(0,0)$, $\\mat{\\tilde{Z}}(0, \\pi)$, $\\mat{\\tilde{Z}}(\\pi,0)$ and $\\mat{\\tilde{Z}}(\\pi, \\pi)$. From these impedance matrices, the interpolant has been evaluated on 25 sampling points using the described procedure with $N=1$ (9 Floquet modes extracted), the coordinates of the sampling points corresponding to $\\varphi_x, \\varphi_y \\in \\{-2\\pi, -\\pi, 0, \\pi, 2\\pi\\}$. Then, from these sampling points, the interpolant was approximated using a polynomial of order 4. Due to the high number of sampling points, the coefficients of the polynomial were obtained by solving an overconstrained system of equations, the relative weight attributed to the most distant points being a hundred times smaller than the weight attributed to the nine central points. \n\nPeriodic impedance matrices were computed using the method of \\cite{ref15}. Since only one BF is used, the resulting impedance ``matrices\" correspond to vectors. The error is estimated as the norm of the difference between the interpolated impedance matrix and the computed one, relative to the norm of the computed one:\n\\begin{equation}\ne(\\boldsymbol{\\varphi}) = \\dfrac{\\|\\mat{\\tilde{Z}}_{interp}-\\mat{\\tilde{Z}}\\|_2}{\\|\\mat{\\tilde{Z}}\\|_2}\n\\end{equation}\nwith $\\mat{\\tilde{Z}}_{interp}$ the periodic impedance matrix obtained using the interpolation technique and $\\|\\cdot \\|$ the $L2$ norm. For any distance and phase-shift, the maximum error was found to be approximately $0.2\\%$.\n\nWe compared the times required to prepare the interpolation procedure and apply it. To do so, we considered the electric and magnetic fields generated by the plane on itself (200x400 interactions). In \\cite{ref15}, the periodic Green's function is first tabulated and then integrated over the BF and TF. For each phase-shift the tabulation of the periodic Green's function required 20\" and the computation of the impedance matrix required 6\". Thus computing the four periodic impedance matrices required 1'45\". Then, as explained earlier, the four periodic impedance matrices have been used to estimate the value of the interpolant at 25 different points. The estimation of the interpolant took about 10\". Last, to interpolate the value of the matrix for a given phase-shift, it took about 0.5\". The duration of the different steps is summarized in Table \\ref{tab:01}. It should be emphasized that the computation of the reference impedance matrices and the extraction of the dominant Floquet modes only needs to be performed once.\n\n\\begin{table}\n\\center\n\\caption{Computation times involved in the interpolation technique (in seconds).}\n\\begin{tabular}{|c|c|c|}\n\\hline\nComputation of four\n&\nExtraction of the dominant \n&\nInterpolation of\n\\\\\nperiodic impedance\n&\nFloquet modes and\n&\none matrix\n\\\\\nmatrices $\\mat{\\tilde{Z}}$\n&\nlinear phase shift\n&\n~\n\\\\\n\\hline\n105 & 10 & 0.5\n\\\\\n\\hline\n\\end{tabular}\n\\label{tab:01}\n\\end{table}\n\nLast, the interpolation technique has been used to compute the radiation pattern of the leaky-wave antenna proposed in \\cite{sup1} and used in \\cite{ref9} to validate the periodic MoM solver. The leaky wave antenna is made of a ground plane and a periodic array of patches located 19 mm above the ground plane. The patches are rectangular, with dimensions of 12.5 mm and 1 mm in the $\\dir{x}$ and $\\dir{y}$ directions. The periodicity of the patches is 13.5 mm in the $\\dir{x}$ direction and 3 mm in the $\\dir{y}$ direction. The structure is excited using a $\\dir{x}$-directed dipole located 9.5 mm below the center of one patch. The radiation pattern of the structure in the $E$ and $H$ planes($x-z$ and $y-z$ planes, respectively) is studied at 9.5 GHz.\n\nThe radiation pattern was computed twice, the results being displayed in Fig. \\ref{fig:03}. In both cases, the code of \\cite{ref9} was used to compute the periodic impedance matrices. The first radiation pattern was obtained by computing explicitly the periodic impedance matrix for each different phase shift. The second radiation pattern was obtained by computing the periodic impedance matrix for four different phase shifts and then interpolating the value of the periodic impedance matrix for the other phase shifts. The parameters used for the interpolation were the same as those used in the previous example. The radiation patterns obtained using both techniques and their difference are displayed in Fig. \\ref{fig:03}. It can be seen that both radiation patterns are in perfect agreement and that the relative error introduced by the interpolation technique is about 0.1\\%.\n\n\\begin{figure}\n\\center\n\\includegraphics[width = 7cm]{final_graph_1.pdf}\n\\caption{Radiation pattern of the leaky-wave antenna described in \\cite{sup1} and simulated in \\cite{ref9}. Blue and red curves correspond to the radiation pattern in the E and H-planes, respectively. Continuous lines: reference radiation patterns for which each periodic impedance matrix was computed explicitly. Dashed lines: radiation patterns obtained using the interpolation technique. Dotted dashed lines: error due to the interpolation technique.}\n\\label{fig:03}\n\\end{figure}\n\n\\section{Conclusion}\nTo conclude, we proposed a technique to efficiently interpolate the Method of Moments periodic impedance matrix vs. the phase shift between consecutive unit cells. To improve accuracy, the interpolant is smoothened by extracting the contribution of the dominant Floquet modes and a linear phase term. The efficiency of the method partly relies on the fact that a single periodic impedance matrix can be used to sample the interpolant at many different points. The accuracy of the method has been validated on numerical examples.\n\n\\section*{Acknowledgment}\nThis project has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sk\\l odowska-Curie grant agreement No. 842184.\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Acknowledgments}{#1}}\n\\newcommand{\\mykeywords}[1]{}\n\\bibliographystyle{alpha}\n\n\\fi\n\n\\newenvironment{proofidea}{\\noindent{\\textit{Proof idea.}}}{\\hfill$\\square$\\medskip}\n\n\\newcommand{\\bigdotcup}{\\ensuremath{\\mathaccent\\cdot{\\bigcup}}}\n\\newcommand{\\floor}[1]{\\left\\lfloor#1\\right\\rfloor}\n\\newcommand{\\card}[1]{\\lvert#1\\rvert}\n\\renewcommand{\\dim}{{n}}\n\n\\newcommand{\\RR}{\\ensuremath{\\mathbb{R}}}\n\\newcommand{\\ones}{\\ensuremath{\\mathbbm{1}}}\n\\newcommand{\\expectation}{\\operatorname{\\mathbb{E}}}\n\\newcommand{\\e}{\\expectation}\n\\newcommand{\\conv}{\\operatorname{conv}}\n\\newcommand{\\suchthat}{\\mathrel{:}}\n\\newcommand{\\norm}[1]{{\\lVert#1\\rVert}}\n\\newcommand{\\diag}{\\operatorname{diag}}\n\\newcommand{\\eps}{\\epsilon}\n\n\\newcommand{\\poly}{\\operatorname{poly}}\n\\newcommand{\\polylog}{\\operatorname{polylog}}\n\\newcommand{\\abs}[1]{\\lvert#1\\rvert}\n\\newcommand{\\vol}{\\operatorname{vol}}\n\\newcommand{\\expdist}[1]{\\mathrm{Exp({#1})}}\n\\newcommand{\\noname}[1]{}\n\n\\newcommand{\\dimension}{n}\n\n\\newtheorem{problem}{Problem}\n\n\\title{Efficient Learning of Simplices}\n\n\\date{}\n\n\\begin{document}\n\\maketitle\n\\begin{abstract}\nWe show an efficient algorithm for the following problem: Given uniformly random points from an arbitrary $\\dim$-dimen\\-sional simplex, estimate the simplex. The size of the sample and the number of arithmetic operations of our algorithm are polynomial in $\\dim$.\nThis answers a question of Frieze, Jerrum and Kannan~\\cite{FJK}.\nOur result can also be interpreted as efficiently learning the intersection of $\\dim+1$ half-spaces in $\\RR^\\dim$ in the model where the intersection is bounded and we are given polynomially many uniform samples from it.\nOur proof uses the local search technique from Independent Component Analysis (ICA), also used by \\cite{FJK}. Unlike these previous\nalgorithms, which were based on analyzing the fourth moment, ours is based on the third moment.\n\nWe also show a direct connection between the problem of learning a simplex and ICA: a simple randomized reduction to ICA from the problem of learning a simplex. The connection is based on a known representation of the uniform measure on a simplex. Similar representations lead to a reduction from the problem of learning an affine transformation of an $n$-dimensional $\\ell_p$ ball to ICA. \n\\end{abstract}\n\n\\mykeywords{independent component analysis, randomized reductions, learning convex bodies, method of moments}\n\n\\section{Introduction}\nWe are given uniformly random samples from an unknown convex body in $\\RR^\\dim$, how many samples are needed to approximately reconstruct\nthe body? It seems intuitively clear, at least for $\\dim = 2, 3$, that\nif we are given sufficiently many such samples then we can reconstruct (or learn) the body with very little error. For general $n$, it is known to require $2^{\\Omega(\\sqrt{\\dim})}$\nsamples~\\cite{GR09} (see also \\cite{KOR} for a similar lower bound in a different but related model of learning).\nThis is an information-theoretic lower bound and no computational considerations are involved. As mentioned\nin \\cite{GR09}, it turns out that if the body has few facets (e.g. polynomial in $\\dim$), then polynomial in $\\dim$ samples are\nsufficient for approximate reconstruction.\nThis is an information-theoretic upper bound and no efficient algorithms (i.e., with running time poly$(n)$) are known.\n(We remark that to our knowledge the same\nsituation holds for polytopes with poly$(n)$ vertices.)  In this paper we study the\nreconstruction problem for the special case when the input bodies are restricted to be (full-dimensional) simplices. We show that\nin this case one can in\nfact learn the body efficiently. More precisely, the algorithm  knows that the input body is a simplex but only up to an affine transformation, and the problem is to recover this affine transformation.\nThis answers a question of \\noname{Frieze~et~al.~}\\cite[Section 6]{FJK}.\n\n\nThe problem of learning a simplex is also closely related to the well-studied problem of learning intersections of half-spaces.\nSuppose that\nthe intersection of $\\dim+1$ half-spaces in $\\RR^\\dim$ is bounded, and we are given poly$(\\dim)$ uniformly random samples from it. Then our\nlearning simplices result directly implies that we can learn the $\\dim+1$ half-spaces. This also has the advantage of being a proper\nlearning algorithm, meaning that the output of the algorithm is a set of $\\dim+1$ half-spaces, unlike many of the previous algorithms.\n\n\n\n\n\\paragraph{Previous work.}\n\nPerhaps the first approach to learning simplices that comes to mind is to find a minimum volume simplex containing the samples. This can be shown\nto be a good approximation to the original simplex. (Such minimum volume estimators have been studied in machine learning literature, see e.g. \\cite{ScholkopfPSSW01} for the problem of estimating the support of a probability distribution. We are not aware of any\ntechnique that applies to our situation and provides theoretical guarantees.)\nHowever, the problem of finding a minimum volume simplex is in general\nNP-hard~\\cite{Packer}. This hardness is not directly applicable for our problem because our input is a random sample and not a general\npoint set. Nevertheless, we do not have an algorithm for directly finding a minimum volume simplex; instead we use ideas similar to those used in Independent Component Analysis (ICA).\nICA studies the following problem:\nGiven a sample from an affine transformation of a random vector with independently distributed coordinates, recover the affine transformation (up to some unavoidable ambiguities).\n\\noname{Frieze~et~al.~}\\cite{FJK} gave an efficient algorithm for this problem\n(with some restrictions on the allowed distributions, but also with some weaker requirements than full independence)\nalong with most of the details of a rigorous analysis (a complete analysis of a special case can be found in \\noname{Arora et al.~}\\cite{Arora2012}; see also \\noname{Vempala and Xiao~}\\cite{VempalaX11} for a generalization of ICA to subspaces along with a rigorous analysis). \nThe problem of learning parallelepipeds from uniformly random samples is a special case of this problem. \n\\cite{FJK} asked if one could learn other convex bodies, and in particular simplices, efficiently from uniformly random samples.\n\\noname{Nguyen and Regev~}\\cite{NguyenR09} gave a simpler and rigorous algorithm and analysis for the case of learning parallelepipeds with similarities to the popular FastICA algorithm of \\noname{Hyv\\\"{a}rinen~}\\cite{Hyvarinen99}.\nThe algorithm in \\cite{NguyenR09} is a first order algorithm unlike Frieze~et~al.'s second order algorithm.\n\nThe algorithms in both \\cite{FJK, NguyenR09} make use of the fourth moment function of the probability distribution. Briefly, the fourth moment in direction $u \\in \\RR^n$ is $\\mathbb{E}(u\\cdot X)^4$, where $X \\in \\RR^n$ is\nthe random variable distributed according to the input distribution. The moment function can be estimated from\nthe samples. The independent components of the distribution correspond to local maxima or minima of the\nmoment function, and can be approximately found by finding the local maxima\/minima of the moment function\nestimated from the sample.\n\n\n\nMore information on ICA including historical remarks can be found in \\cite{ICAbook, BlindSS}.\nIdeas similar to ICA have been used in\nstatistics in the context of projection pursuit since the mid-seventies.\nIt is not clear how to apply ICA to the simplex learning problem directly as there is no clear independence among the components. Let us\nnote that \\noname{Frieze et al.~}\\cite{FJK} allow certain kinds of dependencies among the components, however this does not appear to be useful\nfor learning simplices.\n\n\n\nLearning intersections of half-spaces is a well-studied problem in learning theory. The problem\nof PAC-learning intersections of even two half-spaces is open, and there is evidence that it is\nhard at least for sufficiently large number of half-spaces: E.g., \\noname{Klivans and Sherstov~}\\cite{KlivansS09} prove\nthat learning intersections of $n^\\epsilon$ half-spaces in $\\RR^\\dim$ (for constant $\\epsilon>0$) is\nhard under standard cryptographic assumptions (PAC-learning is possible, however, if one also\nhas access to a membership oracle in addition to random samples \\cite{KwekP98}). Because of this, much effort has been expended\non learning when the distribution of random samples is some simple distribution,\nsee e.g. \\cite{KlivansS07, Vempala10, VempalaFocs10} and references therein. This line of work makes substantial progress\ntowards the goal of learning intersections of $k$ half-spaces efficiently, however it falls short of\nbeing able to do this in time polynomial in $k$ and $n$; in particular, these algorithms do not seem to be\nable to learn simplices.  The distribution of samples in\nthese works is either the Gaussian distribution or the uniform distribution over a ball.\n\\noname{Frieze et al.~}\\cite{FJK} and \\noname{Goyal and Rademacher~}\\cite{GR09} consider the uniform distribution over\nthe intersection. Note that this requires that the intersection be bounded. Note also that one\nonly gets positive samples in this case unlike other work on learning intersections of\nhalf-spaces. The problem of learning convex bodies can also be thought of as learning a distribution or density estimation\nproblem for a special class of distributions.\n\n\n\\noname{Gravin et al.~}\\cite{Gravin12} show how to reconstruct a polytope with $N$ vertices in $\\RR^n$, given its first $O(nN)$ moments\nin $(n+1)$ random directions. In our setting, where we have access to only a polynomial number of random samples, it's not clear\nhow to compute moments of such high orders to the accuracy required for the algorithm of \\cite{Gravin12} even for simplices.\n\nA recent and parallel work of \\noname{Anandkumar et al.~}\\cite{AnandTensor} is closely related to ours. They show that tensor decomposition\nmethods can be applied to low-order moments of various latent variable models to estimate their parameters. The latent variable \nmodels considered by them include Gaussian mixture models, hidden Markov models and latent Dirichlet allocations. The tensor \nmethods used by them and the local optima technique we use seem closely related. One could view our work, as well as theirs, as \nshowing that the method of moments along with existing algorithmic techniques can be applied to certain unsupervised learning problems.\n\n\\paragraph{Our results}\n\nFor clarity of the presentation, we use the following machine model for the running time: a random access machine that allows the following exact arithmetic operations over real numbers in constant time: addition, subtraction, multiplication, division and square root.\n\nThe estimation error is measured using total variation distance, denoted $d_{TV}$ (see Section \\ref{sec:preliminaries}).\n\n\\begin{theorem} \\label{thm:main}\nThere is an algorithm (Algorithm~\\ref{alg:simplex} below) such that given access to random samples from a simplex\n$S_{INPUT} \\subseteq \\RR^\\dim $, with probability at least $1-\\delta$ over the sample and the randomness of the algorithm, it outputs $n+1$ vectors that are the vertices of a simplex $S$ so that $d_{TV}(S, S_{INPUT}) \\leq \\eps$.\nThe algorithm runs in time polynomial in $\\dim$, $1\/\\eps$ and $1\/\\delta$.\n\\end{theorem}\n\nAs mentioned earlier, our algorithm uses ideas from ICA. \nOur algorithm uses the third moment instead of the fourth moment used in certain versions of ICA. The third moment is not useful for learning symmetric bodies such as the cube as it is identically $0$. \nIt is however useful for learning a simplex where it provides useful information, and is easier to handle than the fourth moment.\nOne of the main contributions of our work is the understanding of the third moment of a simplex and the structure of local maxima. This is more involved than in previous work as the simplex has no obvious independence structure, and the moment polynomial one gets has no obvious structure unlike for ICA.\n\nThe probability of success of the algorithm can be ``boosted'' so that the dependence of the running time on $\\delta$ is only linear in $\\log(1\/\\delta)$ as follows: The following discussion uses the space of simplices with total variation distance as the underlying metric space. Let $\\eps$ be the target distance. Take an algorithm that succeeds with probability $5\/6$ and error parameter $\\eps'$ to be fixed later (such as Algorithm \\ref{alg:simplex} with $\\delta = 1\/6$). Run the algorithm $t = O(\\log 1\/\\delta)$ times to get $t$ simplices. By a Chernoff-type argument, at least $2t\/3$ simplices are within $\\eps'$ of the input simplex with probability at least $1- \\delta\/2$.\n\nBy sampling, we can estimate the distances between all pairs of simplices with additive error less than $\\eps'\/10$ in time polynomial\nin $t, 1\/\\epsilon'$ and $\\log{1\/\\delta}$ so that all estimates are correct with probability at least $1-\\delta\/2$.\nFor every output simplex, compute the number of output simplices within estimated distance $(2+1\/10)\\eps'$. With probability at least\n$1-\\delta$ both of the desirable events happen, and then necessarily there is at least one output simplex, call it $S$, that has $2t\/3$ output simplices within estimated distance $(2+1\/10)\\eps'$. Any such $S$ must be within $(3+2\/10) \\eps'$ of the input simplex. Thus, set $\\eps'=\\eps\/(3+2\/10)$.\n\nWhile our algorithm for learning simplices uses techniques for ICA, we have to do \nsubstantial work to make those techniques work for the simplex problem. \nWe also show a more direct connection between the problem of learning a simplex and ICA: a randomized reduction from the problem of learning a simplex to ICA. The connection is based on a known representation of the uniform measure on a simplex as a normalization of a vector having independent coordinates. Similar representations are known for the uniform measure in an $n$-dimensional $\\ell_p$ ball (denoted $\\ell_p^n$) \\cite{barthe2005probabilistic} and the \\emph{cone measure} on the boundary of an $\\ell_p^n$ ball \\cite{schechtman1990volume, rachev1991approximate, MR1396997} (see Section \\ref{sec:preliminaries}  for the definition of the cone measure). \nThese representations lead to a reduction from the problem of learning an affine transformation of an $\\ell_p^n$ ball to ICA. These reductions show connections between estimation problems with no obvious independence structure and ICA. \nThey also make possible the use of any off-the-shelf implementation of ICA. \nHowever, the results here do not supersede our result for learning simplices because to our knowledge no rigorous analysis is available for \nthe ICA problem when the distributions are the ones in the above reductions. \n\n\\paragraph{Idea of the algorithm.}\n\n\n The new idea for the algorithm is that after putting\nthe samples in a suitable position (see below), the third moment of the sample can be used to recover the simplex using a simple \nFastICA-like algorithm. We outline our algorithm next.\n\nAs any full-dimensional simplex can be mapped to any other full-dimensional simplex by an invertible affine transformation, it is enough to determine the translation and linear transformation that would take the given simplex to some canonical simplex.\nAs is well-known for ICA-like problems (see, e.g., \\cite{FJK}),\nthis transformation can be determined \\emph{up to a rotation} from the mean and the covariance matrix of the uniform distribution on the given simplex.\nThe mean and the covariance matrix can be estimated efficiently from a sample.\nA convenient choice of an $n$-dimensional simplex is the convex hull of the canonical vectors in $\\RR^{\\dim+1}$.\nWe denote this simplex $\\Delta_n$ and call it the \\emph{standard simplex}.\nSo, the algorithm begins by picking an arbitrary invertible affine transformation $T$ that maps $\\RR^\\dim$ onto the hyperplane $\\{x \\in \\RR^{\\dim+1} \\suchthat \\ones \\cdot x = 1 \\}$. We use a $T$ so that $T^{-1}(\\Delta_n)$ is an isotropic\\footnote{See Section \\ref{sec:preliminaries}.} simplex. In this case, the algorithm brings the sample set into isotropic position and embeds it in $\\RR^{\\dim+1}$ using $T$.\nAfter applying these transformations we may assume (at the cost of small errors in the final result) that our sample set is obtained by sampling from an unknown rotation of the standard simplex\nthat leaves the all-ones vector (denoted $\\ones$ from now on) invariant (thus this rotation keeps the center of mass of the standard simplex fixed), and the problem is to recover this rotation.\n\n\nTo find the rotation, the algorithm will find the vertices of the rotated simplex approximately. This can be done efficiently because of the following characterization of the vertices: Project the vertices of the simplex onto the hyperplane through the origin orthogonal to $\\ones$ and normalize the resulting vectors. Let $V$ denote this set of $n+1$ points. Consider the problem of maximizing the third moment of the uniform distribution in the simplex along unit vectors orthogonal to $\\ones$. Then $V$ is the complete set of local maxima and the complete set of global maxima (Theorem~\\ref{thm:maxima}). A fixed point-like iteration (inspired by the analysis of FastICA~\\cite{Hyvarinen99} and of gradient descent in \\cite{NguyenR09}) starting from a random point in the unit sphere finds a local maximum efficiently with high probability. By the analysis of the coupon collector's problem, $O(\\dim \\log \\dim)$ repetitions are highly likely to find all local maxima.\n\n\\paragraph{Idea of the analysis.}\n\nIn the analysis, we first argue that after putting the sample in isotropic position and mapping it through $T$, it is enough to analyze the algorithm in the case where the sample comes from a simplex $S$ that is close to a simplex $S'$ that is the result of applying a rotation leaving $\\ones$ invariant to the standard simplex. The closeness here depends on the accuracy of the sample covariance and mean as an estimate of the input simplex's covariance matrix and mean. A sample of size $O(n)$ guarantees\n(\\cite[Theorem 4.1]{MR2601042}, \\cite[Corollary 1.2]{1106.2775}) that the covariance and mean are close enough so that the uniform distributions on $S$ and $S'$ are close in total variation. We show that the subroutine that finds the vertices (Subroutine \\ref{alg:vertex}), succeeds with some probability when given a sample from $S'$. By definition of total variation distance, Subroutine \\ref{alg:vertex} succeeds with almost as large probability when given a sample from $S$ (an argument already used in \\cite{NguyenR09}). As an additional simplifying assumption, it is enough to analyze the algorithm (Algorithm \\ref{alg:simplex}) in the case where the input is isotropic, as the output distribution of the algorithm is equivariant with respect to affine invertible transformations as a function of the input distribution.\n\n\\iffalse\nWe may assume at the cost of small error in the final result that the sample set comes from\n\nAn interesting aspect of the algorithm is that it also uses vectors outside the plane $\\norm{x}_1=1$.\n\nThere are several kinds of approximations that need to be handled by a successful algorithm and its analysis:\n\n(isotropic, exact 3rd moment, real arithmetic) The algorithm can evaluate the gradient of the 3rd moment of the canonical simplex in some direction exactly.\n\n(isotropic, approximate 3rd moment, real arithmetic) The algorithm has access to samples from the canonical simplex. This allows the approximation of the gradient.\n\n(nearly isotropic, approximate 3rd moment, real arithmetic) The algorithm has access to samples from an affine perturbation of the canonical simplex.\n\n(nearly isotropic, approximate 3rd moment, limited precision) The algorithm has access to samples from an affine perturbation of the canonical simplex. The samples have limited numerical precision, arithmetic operations also have limited precision.\n\\fi\n\n\n\n\n\\paragraph{Organization of the paper.} \nStarting with some preliminaries in Sec.~\\ref{sec:preliminaries}, we state some\nresults on the third moment of simplices in Sec.~\\ref{sec:moment}. \nIn Sec.~\\ref{sec:standard} we give an algorithm that\nestimates individual vertices of simplices in a special position; using this algorithm as a subroutine in Sec.~\\ref{sec:general} we give\nthe algorithm for the general case. \nSec.~\\ref{sec:maxima} characterizes the set of local maxima of the third moment. \nSec.~\\ref{sec:prob} gives the probabilistic results underlying the reductions from learning simplices and $\\ell_p^n$ balls to ICA.\nSec.~\\ref{sec:reduction} explains those reductions.\n\n\n\n\\section{Preliminaries} \\label{sec:preliminaries}\nAn $\\dim$-simplex is the convex hull of $\\dim+1$ points in $\\RR^{\\dim}$ that do not lie on an $(\\dim-1)$-dimensional affine hyperplane.\nIt will be convenient to work with the standard $\\dim$-simplex\n$\\Delta^{\\dim}$ living in $\\RR^{\\dim+1}$ defined as the convex hull of the $\\dim+1$ canonical unit\nvectors $e_1, \\ldots, e_{\\dim+1}$; that is\n\\begin{align*}\n\\Delta^{\\dim} = \\{(x_0, \\ldots, x_{\\dim}) \\in \\RR^{\\dim+1} &\\suchthat x_0 + \\dotsb + x_{\\dim} = 1 \\optionalbreak \n\\text{ and } x_i \\geq 0 \\text{ for all } i\\}.\n\\end{align*}\nThe canonical simplex $\\Omega^\\dim$ living in $\\RR^\\dim$ is given by\n\\begin{align*}\n\\{(x_0, \\dotsc, x_{\\dim-1}) \\in \\RR^{\\dim} &\\suchthat x_0 + \\dotsb + x_{\\dim-1} \\leq 1 \\optionalbreak\n\\text{ and } x_i \\geq 0 \\text{ for all } i\\}.\n\\end{align*}\nNote that $\\Delta^\\dim$ is the facet of $\\Omega^{\\dim+1}$ opposite to the origin.\n\nLet $B_n$ denote the $n$-dimensional Euclidean ball.\n\n\nThe complete homogeneous symmetric polynomial of degree $d$ in variables $u_0, \\ldots, u_{\\dim}$, denoted $h_n(u_0, \\ldots, u_{\\dim})$,\nis the sum of all monomials of degree $d$ in the variables:\n\\begin{align*}\nh_d(u_0, \\ldots, u_\\dim)\n&= \\sum_{k_0 + \\dotsb + k_\\dim = d} u_0^{k_0} \\dotsm u_\\dim^{k_\\dim} \\optionalbreak\n= \\sum_{0 \\leq i_0 \\leq i_1 \\leq \\dotsb \\leq i_d \\leq \\dim} u_{i_0} u_{i_1} \\dotsm u_{i_d}.\n\\end{align*}\n\nAlso define the $d$-th power sum as\n$$\np_d(u_0, \\ldots, u_\\dim) = u_0^d + \\ldots + u_\\dim^d.\n$$\n\nFor a vector $u = (u_0, u_1, \\ldots, u_\\dim)$, we define\n\\[\nu^{(2)} = (u_0^2, u_1^2, \\ldots, u_\\dim^2).\n\\]\nVector $\\ones$ denotes the all ones vector (the dimension of the vector will be clear from the context).\n\nA random vector $X \\in \\RR^\\dim$ is \\emph{isotropic} if $\\e(X)=0$ and $\\e(X X^T) = I$. A compact set in $\\RR^\\dim$ is isotropic if a uniformly distributed random vector in it is isotropic. The inradius of an isotropic $n$-simplex is $\\sqrt{(n+2)\/n}$, the circumradius is $\\sqrt{n(n+2)}$.\n\nThe total variation distance between two probability measures is $d_{TV}(\\mu, \\nu) = \\sup_A \\abs{\\mu(A) - \\nu(A)}$ for measurable $A$. For two compact sets $K, L \\subseteq \\RR^\\dim$, we define the total variation distance $d_{TV}(K, L)$ as the total variation distance between the corresponding uniform distributions on each set. It can be expressed as\n\\[\n    d_{TV}(K,L) =\n    \\begin{cases}\n        \\frac{\\vol{K \\setminus L}}{\\vol K} & \\text{if $\\vol K \\geq \\vol L$}, \\\\\n        \\frac{\\vol{L \\setminus K}}{\\vol L} & \\text{if $\\vol L > \\vol K$.}\n    \\end{cases}\n\\]\nThis identity implies the following elementary estimate:\n\\begin{lemma}\\label{lem:bmtv}\nLet $K, L$ be two compact sets in $\\RR^\\dim$. Let $0 < \\alpha \\leq 1 \\leq \\beta$ such that\n$\n\\alpha K \\subseteq L \\subseteq \\beta K$.\nThen $d_{TV}(K,L) \\leq 2\\left(1-(\\alpha\/\\beta)^\\dim\\right)$.\n\\end{lemma}\n\\begin{proof}\nWe have $d_{TV}(\\alpha K, \\beta K) = 1-(\\alpha\/\\beta)^\\dim$. Triangle inequality implies the desired inequality.\n\\end{proof}\n\n\\begin{lemma}\\label{lem:coupons}\nConsider the coupon collector's problem with $n$ coupons where every coupon occurs with probability at least $\\alpha$. Let $\\delta >0$. Then with probability at least $1-\\delta$ all coupons are collected after $\\alpha^{-1} (\\log n + \\log 1\/\\delta)$ trials.\n\\end{lemma}\n\\begin{proof}\nThe probability that a particular coupon is not collected after that many trials is at most\n\\[\n(1-\\alpha)^{\\alpha^{-1} (\\log n + \\log 1\/\\delta)} \\leq e^{-\\log n - \\log 1\/\\delta} = \\delta\/n.\n\\]\nThe union bound over all coupons implies the claim.\n\\end{proof}\n\nFor a point $x \\in \\RR^{\\dimension}$, $\\norm{x}_p = (\\sum_{i = 1}^{\\dimension} |x_i|^p)^{1\/p}$ is the standard $\\ell_p$ norm.  The unit $\\ell_p^n$ ball is defined by \n\\[\nB_p^{\\dimension} = \\{x \\in \\RR^{\\dimension}: \\norm{x}_p \\leq 1 \\}.\n\\]\n\nThe Gamma distribution is denoted as $\\mathrm{Gamma}(\\alpha, \\beta)$ and has density $f(x; \\alpha, \\beta) = \\frac{\\beta^\\alpha}{\\Gamma(\\alpha)} x^{\\alpha - 1} e^{- \\beta x } 1_{x \\geq 0}$, \nwith shape parameters $\\alpha, \\beta >0$.\n$\\mathrm{Gamma}(1, \\lambda)$ is the exponential distribution, denoted $\\expdist{\\lambda}$. \nThe Gamma distribution also satisfies the following additivity property: If $X$ is distributed as $\\mathrm{Gamma}(\\alpha, \\beta)$ and $Y$ is distributed as $\\mathrm{Gamma}(\\alpha', \\beta)$, then $X+Y$ is distributed as $\\mathrm{Gamma}(\\alpha + \\alpha', \\beta)$.\n\nThe cone measure on the surface $\\partial K$ of centrally symmetric convex body $K$ in $\\RR^{\\dimension}$ \\cite{barthe2005probabilistic, schechtman1990volume, rachev1991approximate, MR1396997} is \ndefined by\n\\[\n\\mu_{K}(A) = \\frac{\\vol(ta; a \\in A, 0 \\leq t \\leq 1)}{\\vol(K)}.\n\\] \nIt is easy to see that $\\mu_{B_p^{\\dimension}}$ is uniform on $\\partial B_p^{\\dimension}$ for $p \\in \\{1, 2, \\infty\\}$.\n\nFrom \\cite{schechtman1990volume} and \\cite{rachev1991approximate} we have the following representation of the cone measure on $\\partial B_p^n$:\n\\begin{theorem}\\label{thm:cone}\nLet $G_1, G_2, \\dotsc, G_n$ be iid random variables with density proportional to $\\exp(-\\abs{t}^p)$. Then the random vector $X = G\/\\norm{G}_p$\nis independent of $\\norm{G}_p$. Moreover, $X$ is distributed according to $\\mu_{B_p^{\\dimension}}$.\n\\end{theorem}\n\nFrom \\cite{barthe2005probabilistic}, we also have the following variation, a representation of the uniform distribution in $B_p^n$:\n\\begin{theorem} \\label{theorem:fullnaor} \nLet $G = (G_1, \\dotsc, G_{\\dimension})$ be iid random variables with density proportional to $\\exp(-|t|^p)$. Let $Z$ be a random variable distributed as $\\expdist{1}$, independent of $G$. Then the random vector\n\\[\nV = \\frac{G}{\\bigl( \\sum_{i=1}^{\\dimension} \\abs{G_i}^p + Z\\bigr)^{1\/p}}\n\\]\nis uniformly distributed in $B_p^n$.\n \\end{theorem} \n\n See e.g. \\cite[Section 20]{MR1324786} for the change of variable formula in probability.\n\n\\section{Computing the moments of a simplex}\\label{sec:moment}\nThe $k$-th moment $m_k(u)$ over $\\Delta^\\dim$ is the function\n\\[\nu \\mapsto \\mathbb{E}_{X \\in \\Delta_\\dim}((u\\cdot X)^k).\n\\]\nIn this section we present a formula for the moment over $\\Delta^\\dim$.\nSimilar more general formulas appear in \\cite{LasserreA}.\nWe will use the following result from \\cite{GrundmannM78} for $\\alpha_i\\geq 0$:\n\\[\n\\int_{\\Omega^{\\dim+1}} x_0^{\\alpha_0} \\dotsm x_\\dim^{\\alpha_\\dim} dx = \\frac{\\alpha_0! \\dotsm \\alpha_\\dim!}{(\\dim+1+ \\sum_i \\alpha_i)!}.\n\\]\n\nFrom the above we can easily derive a formula for integration over $\\Delta^\\dim$:\n$$\n\\int_{\\Delta^\\dim} x_0^{\\alpha_0} \\dotsm x_\\dim^{\\alpha_\\dim} dx = \\sqrt{n+1} \\cdot \\frac{\\alpha_0! \\dotsm \\alpha_\\dim!}{(\\dim + \\sum_i \\alpha_i)!}.\n$$\n\nNow\n\\begin{align*}\n\\int_{\\Delta^\\dim} &(x_0 u_0 + \\ldots + x_\\dim u_\\dim)^k dx \\\\\n&= \\sum_{k_0 + \\dotsb + k_\\dim = k} \\binom{k}{k_0!, \\ldots, k_\\dim!} u_0^{k_0} \\ldots u_\\dim^{k_\\dim} \\int_{\\Delta^\\dim} x_0^{k_0} \\ldots x_\\dim^{k_\\dim} dx \\\\\n&= \\sum_{k_0 + \\dotsb + k_\\dim = k} \\binom{k}{k_0!, \\ldots, k_\\dim!} u_0^{k_0} u_0^{k_0} \\ldots u_\\dim^{k_\\dim}\n\\frac{ \\sqrt{\\dim+1} \\cdot k_0! \\ldots k_\\dim!}{(\\dim + \\sum_i k_i)!} \\\\\n&= \\frac{k! \\sqrt{\\dim+1}}{(\\dim+k)!}  \\sum_{k_0 + \\dotsb + k_\\dim = k} u_0^{k_0} \\ldots u_\\dim^{k_\\dim} \\\\\n&= \\frac{k! \\sqrt{\\dim+1}}{(\\dim+k)!} \\; h_k(u).\n\\end{align*}\n\nThe variant of Newton's identities for the complete homogeneous symmetric polynomial gives the following relations which\ncan also be verified easily by direct computation:\n$$ 3 h_3(u) = h_2(u) p_1(u) + h_1(u) p_2(u) + p_3(u),$$\n$$ 2 h_2(u) = h_1(u) p_1(u) + p_2(u) = p_1(u)^2 + p_2(u).$$\n\nDivide the above integral by the volume of the standard simplex $|\\Delta_n|=\\sqrt{\\dim+1}\/{\\dim!}$\nto get the moment:\n\\begin{eqnarray*}\nm_3(u) & = & \\frac{3! \\sqrt{\\dim+1}}{(\\dim+3)!} h_3(u) \/ |\\Delta_n| \\\\\n       & = & \\frac{2 (h_2(u) p_1(u) + h_1(u) p_2(u) + p_3(u))}{(\\dim+1)(\\dim+2)(\\dim+3)}  \\\\\n       & = & \\frac{(p_1(u)^3 + 3 p_1(u) p_2(u) + 2 p_3(u))}{(\\dim+1)(\\dim+2)(\\dim+3)} .\n\\end{eqnarray*}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Subroutine for finding the vertices of a rotated standard simplex} \\label{sec:standard}\nIn this section we solve the following simpler problem: Suppose we have $\\text{poly}(\\dim)$ samples from a rotated copy\n$S$ of the standard simplex, where the rotation is such that it leaves $\\ones$ invariant. The problem\nis to approximately estimate the vertices of the rotated simplex from the samples.\n\nWe will analyze our algorithm in the coordinate system in which the input simplex is the standard simplex. This is only\nfor convenience in the analysis and the algorithm itself does not know this coordinate system.\n\n\nAs we noted in the introduction, our algorithm is inspired by the algorithm of \\noname{Nguyen and Regev~}\\cite{NguyenR09} for\nthe related problem of learning hypercubes and also by the FastICA algorithm in \\cite{Hyvarinen99}. New ideas are\nneeded for our algorithm for learning simplices; in particular, our update rule is different. With the right update\nrule in hand the analysis turns out to be quite similar to the one in \\cite{NguyenR09}.\n\n\nWe want to find local maxima of the sample third moment. A natural approach to do this would be to use gradient\ndescent or Newton's method (this was done in \\cite{FJK}). Our algorithm, which only uses first order information, can\nbe thought of as a fixed point algorithm leading to a particularly simple analysis and fast convergence. Before stating\nour algorithm we describe the update rule we use.\n\nWe will use the abbreviation $C_\\dim = (\\dim+1)(\\dim+2)(\\dim+3)\/6$. Then, from the expression for $m_3(u)$ we get\n\\[\n\\nabla m_3(u) =  \\frac{1}{6C_\\dim} \\left(3 p_1(u)^2 \\ones + 3 p_2(u) \\ones + 6 p_1(u) u + 6 u^{(2)}\\right).\n\\]\nSolving for $u^{(2)}$ we get\n\\begin{align}\nu^{(2)} &= C_\\dim \\nabla m_3(u) - \\frac{1}{2} p_1(u)^2 \\ones - \\frac{1}{2} p_2(u) \\ones -  p_1(u) u \\nonumber \\\\\n       &=  C_\\dim  \\nabla m_3(u) - \\frac{1}{2} (u \\cdot \\ones)^2 \\ones - \\frac{1}{2} (u \\cdot u)^2 \\ones - (u \\cdot \\ones) u.\\label{equ:squaring}\n\\end{align}\n\nWhile the above expressions are in the coordinate system where the input simplex is the canonical simplex, the\nimportant point is that all terms in\nthe last expression can be computed in any coordinate system that is obtained by a rotation leaving $\\ones$ invariant.\nThus, we can compute $u^{(2)}$ as well independently of what coordinate system we are working in. This immediately\ngives us the algorithm below.\nWe denote by $\\hat{m}_3(u)$ the sample third moment, i.e., $\\hat{m}_3(u)=\\frac{1}{t}\\sum_{i=1}^{t}(u \\cdot r_i)^3$ for $t$ samples.\nThis is a polynomial in $u$, and the gradient is computed in the obvious way. Moreover, the gradient of the sample moment is clearly an unbiased estimator of the gradient of the moment; a bound on the deviation is given in the analysis (Lemma \\ref{lem:gradient}).\nFor each evaluation of the gradient of the sample moment, we use a fresh sample.\n\nIt may seem a bit alarming that the fixed point-like iteration is squaring the coordinates of $u$, leading to an extremely fast growth (see Equation \\ref{equ:squaring} and Subroutine 1). But, as in other algorithms having quadratic convergence like certain versions of Newton's method, the convergence is very fast and the number of iterations is small. We show below that it is $O(\\log(n\/\\delta))$, leading to a growth of $u$ that is polynomial in $n$ and $1\/\\delta$. The boosting argument described in the introduction makes the final overall dependence in $\\delta$ to be only linear in\n$\\log (1\/\\delta)$.\n\nWe state the following subroutine for $\\mathbb{R}^n$ instead of $\\mathbb{R}^{n+1}$ (thus it is learning a rotated copy of\n$\\Delta^{n-1}$ instead of $\\Delta^n$). This is for notational convenience so that we work with $n$ instead of $n+1$.\n\n\\begin{subroutine}\n\\caption{Find one vertex of a rotation of the standard simplex $\\Delta^{n-1}$ via a fixed point iteration-like algorithm}\\label{alg:vertex}\n\\begin{algorithmic}\n\\State Input: Samples from a rotated copy of the $n$-dimensional standard simplex (for a rotation that leaves $\\ones$ invariant).\n\\State Output: An approximation to a uniformly random vertex of the input simplex.\n\\vspace{.2em}\n\\hrule\n\\vspace{.2em}\n\\State  Pick $u(1) \\in S^{\\dim-1}$, uniformly at random.\n\\myFor{$i=1$ to $r$}\n\\begin{align*}\nu(i+1) := &C_{\\dim-1} \\nabla \\hat{m}_3(u(i)) - \\frac{1}{2} (u(i) \\cdot \\ones)^2 \\ones \\optionalbreak\n- \\frac{1}{2} (u(i) \\cdot u(i))^2 \\ones - (u(i) \\cdot \\ones) u(i).\n\\end{align*}\n\\State Normalize $u(i+1)$ by dividing by $\\norm{u(i+1)}_2$.\n\\myEndFor\n\\State Output $u(r+1)$.\n\\end{algorithmic}\n\\end{subroutine}\n\n\\begin{lemma}\\label{lem:gradient}\nLet $c>0$ be a constant, $\\dim > 20$, and $0<\\delta<1$. Suppose that Subroutine 1 uses a sample of size\n$t = 2^{17}\\dim^{2c+22}(\\frac{1}{\\delta})^2\\ln\\frac{2\\dim^5 r}{\\delta}$ for each evaluation of the gradient and runs for\n$r =\\log \\frac{4(c+3) n^2\\ln{n}}{\\delta}$\niterations. Then with probability at least $1-\\delta$\nSubroutine 1 outputs a vector within distance $1\/n^c$ from a vertex\nof the input simplex.  With respect of the process of picking a sample and running the algorithm, each vertex is equally likely to be the nearest.\n\n\n\\end{lemma}\n\n\nNote that if we condition on the sample, different vertices are not equally likely over the randomness of the algorithm. That is, if we try to find all vertices running the algorithm multiple times on a fixed sample, different vertices will be found with different likelihoods.\n\\begin{proof}\nOur analysis has the same outline as that of \\noname{Nguyen and Regev~}\\cite{NguyenR09}.\nThis is because the iteration that we get is the same as that of \\cite{NguyenR09} except that cubing is replaced by\nsquaring (see below); however some details in our proof are different.\nIn the proof below, several of the inequalities are quite loose and are so chosen to make the computations\nsimpler.\n\nWe first prove the lemma assuming that the gradient computations are exact and then show how to handle samples.\nWe will carry out the analysis in the coordinate system where the given simplex is the standard simplex. This is\nonly for the purpose of the analysis, and this coordinate system is not known to the algorithm.\nClearly, $u(i+1) =  (u(i)^2_1, \\ldots, u(i)_{\\dim}^2)$.\nIt follows that, $$u(i+1) = (u(1)_1^{2^i}, \\ldots, u(1)_{\\dim}^{2^i}).$$\nNow, since we choose $u(1)$ randomly, with probability at least $(1-(n^2-n)\\delta')$\n one of the\ncoordinates of $u(1)$ is greater than all the other coordinates in absolute value by a factor of at least $(1+ \\delta')$,\nwhere $0 < \\delta' < 1$.\n(A similar argument is made in \\cite{NguyenR09} with different parameters.\n We briefly indicate the proof for our case:\nThe probability that the event in question does not happen is less than the probability that there are two coordinates $u(1)_a$\nand $u(1)_b$ such that their absolute values are within factor $1+\\delta'$, i.e.\n$1\/(1+\\delta') \\leq |u(1)_a|\/|u(1)_b| < 1+\\delta'$. The probability that for given $a, b$ this event happens can be seen as the\nGaussian area of the four sectors (corresponding to the four choices of signs of $u(1)_a, u(1)_b$) in the plane each with angle\nless than $2\\delta'$. By symmetry, the Gaussian volume of these sectors is $2\\delta'\/(\\pi\/2) < 2\\delta'$.\nThe probability that\nsuch a pair $(a,b)$ exists is less than $2 \\binom{n}{2} \\delta'$.)\nAssuming this happens, then after $r$ iterations, the ratio between the largest coordinate (in absolute value) and the absolute value of any other coordinate is at least\n$(1+\\delta')^{2^r}$.\nThus, one of the coordinates is very\nclose to $1$ and others are very close to $0$, and so $u(r+1)$ is very close to a vertex of the input simplex.\n\nNow we drop the assumption that the gradient is known exactly. For each evaluation of the gradient we use\na fresh subset of samples of $t$ points. Here $t$ is chosen so that each evaluation of the gradient is within $\\ell_2$-distance\n$1\/\\dim^{c_1}$ from its true value with probability at least $1-\\delta''$, where $c_1$ will be set at the end of the proof.\nAn application of the Chernoff bound yields that we can take\n$t = 200\\dim^{2c_1+4}\\ln\\frac{2\\dim^3}{\\delta''}$; we omit the details. Thus all the $r$ evaluations of the gradient are\nwithin distance $1\/\\dim^{c_1}$ from their true values with probability at least $1-r\\delta''$.\n\n\n\nWe assumed that our starting vector $u(1)$ has a coordinate greater than every other coordinate by a factor\nof\n$(1+\\delta')$ in absolute value; let us assume without loss of generality that this is the first coordinate.\nHence $|u(1)_1| \\geq 1\/\\sqrt{n}$.\nWhen expressing $u^{(2)}$ in terms of the gradient, the gradient gets multiplied by $C_{n-1} < n^3$ (we are assuming $n>20$),\nkeeping this in mind and letting $c_2 = c_1-3$ we get for $j \\neq 1$\n\\begin{align*}\n\\frac{|u(i+1)_1|}{|u(i+1)_j|} \\geq \\frac{u(i)_1^2-1\/n^{c_2}}{u(i)_j^2+1\/n^{c_2}} \\geq \\frac{u(i)_1^2 (1-n^{-(c_2-1)})}{u(i)_j^2+1\/n^{c_2}}.\n\\end{align*}\n\n\nIf $u(i)_j^2 > 1\/\\dim^{c_2-c_3}$, where $1\\leq c_3 \\leq c_2 -2$ will be determined later, then we get\n\\begin{align} \\label{eq:if}\n|u(i+1)_1|\/|u(i+1)_j|\n&> \\frac{1-1\/n^{c_2-1}}{1+1\/n^{c_3}} \\cdot \\left(\\frac{u(i)_1}{u(i)_j}\\right)^2 \\nonumber\\\\\n&>  (1-1\/n^{c_3})^2 \\left(\\frac{u(i)_1}{u(i)_j}\\right)^2.\n\\end{align}\n\nElse,\n\\begin{align*}\n|u(i+1)_1|\/|u(i+1)_j|\n&> \\frac{1\/n-1\/n^{c_2}}{1\/n^{c_2-c_3}+1\/n^{c_2}} \\\\\n&> \\left(1-\\frac{1}{n^{c_3}}\\right)^2 \\cdot n^{c_2-c_3 -1} \\\\\n&> \\frac{1}{2} n^{c_2-c_3-1},\n\\end{align*}\nwhere we used $c_3 \\geq 1$ and $n>20$ in the last inequality.\n\n\n\nWe choose $c_3$ so that\n\\begin{align}\\label{eq:c3}\n\\left(1-\\frac{1}{n^{c_3}}\\right)^2(1+\\delta') > (1+\\delta'\/2).\n\\end{align}\nFor this, $\\delta' \\geq 32\/n^{c_3}$ or equivalently $c_3 \\geq (\\ln{(32\/\\delta')})\/\\ln{n}$ suffices.\n\nFor $c_3$ satisfying \\eqref{eq:c3} we have $(1-\\frac{1}{n^{c_3}})^2(1+\\delta')^2 > (1+\\delta')$.\nIt then follows from \\eqref{eq:if} that\nthe first coordinate continues to remain the largest in absolute value by a factor of at least $(1+\\delta')$ after each iteration.\nAlso, once we have $|u(i)_1|\/|u(i)_j| > \\frac{1}{2} n^{c_2-c_3-1}$, we\nhave  $|u(i')_1|\/|u(i')_j| > \\frac{1}{2} n^{c_2-c_3-1}$ for all $i'>i$.\n\n\\eqref{eq:if} gives that after $r$ iterations we have\n\\begin{align*}\n\\frac{|u(r+1)_1|}{|u(r+1)_j|}\n&>  (1-1\/n^{c_3})^{2+2^2+\\ldots+2^r} \\left(\\frac{u(1)_1}{u(1)_j}\\right)^{2^r} \\\\\n&\\geq (1-1\/n^{c_3})^{2^{r+1}-2} (1+\\delta')^{2^r}.\n\\end{align*}\n\nNow if $r$ is such that $(1-1\/n^{c_3})^{2^{r+1}-2} (1+\\delta')^{2^r} > \\frac{1}{2} n^{c_2-c_3-1}$, we will be guaranteed that\n$|u(r+1)_1|\/|u(r+1)_j| > \\frac{1}{2} n^{c_2-c_3-1}$. This condition is satisfied if we have\n$(1-1\/n^{c_3})^{2^{r+1}} (1+\\delta')^{2^r} > \\frac{1}{2} n^{c_2-c_3-1}$, or equivalently\n$((1-1\/n^{c_3})^2 (1+\\delta'))^{2^r} \\geq \\frac{1}{2} n^{c_2-c_3-1}$.\nNow using \\eqref{eq:c3} it suffices to choose $r$ so that\n$(1+\\delta'\/2)^{2^r} \\geq \\frac{1}{2} n^{c_2-c_3-1}$. Thus we can take $r = \\log(4(c_2-c_3)(\\ln{n})\/\\delta')$.\n\n\n\nHence we get $|u(r+1)_1|\/|u(r+1)_j| > \\frac{1}{2} n^{c_2-c_3-1}$.\nIt follows that for\n$u(r+1)$, the $\\ell_2$-distance from the vertex $(1, 0, \\ldots, 0)$ is at most $8\/n^{c_2-c_3-2} < 1\/n^{c_2-c_3-3}$ for $\\dim > 20$; we omit\neasy details.\n\n\n\nNow we set our parameters: $c_3 = 1+(\\ln(32\/\\delta')\/\\ln{n})$ and $c_2 - c_3 - 3 = c$ and\n$c_1 = c_2 + 3 = 7 + c + \\ln(32\/\\delta')\/\\ln{n}$ satisfies all the constraints we imposed on $c_1, c_2, c_3$.\nChoosing $\\delta'' = \\delta'\/r$, we get that the procedure succeeds with probability at least\n$1-(\\dim^2-\\dim)\\delta' - r\\delta'' > 1-\\dim^2\\delta'$. Now setting $\\delta'=\\delta\/n^2$ gives the overall probability of error $\\delta$,\nand the number of samples and iterations as claimed in the lemma.\n\\end{proof}\n\n\n\n\\section{Learning simplices} \\label{sec:general}\n\nIn this section we give our algorithm for learning general simplices, which uses the subroutine from the previous section.\nThe learning algorithm uses an affine map $T:\\RR^{\\dim} \\to \\RR^{\\dim+1}$ that maps some isotropic simplex to the standard simplex.\nWe describe now a way of constructing such a map: Let $A$ be a matrix having as columns an orthonormal basis of $\\ones^\\perp$ in $\\RR^{\\dim+1}$.\nTo compute one such $A$, one can start with the $(n+1)$-by-$(n+1)$ matrix $B$ that has ones in the diagonal and first column, everything else is zero.\nLet $QR=B$ be a QR-decomposition of $B$. By definition we have that the first column of $Q$ is parallel to $\\ones$ and the rest of the columns span $\\ones^\\perp$.\nGiven this, let $A$ be the matrix formed by all columns of $Q$ except the first.\nWe have that the set $\\{A^T e_i\\}$ is the set of vertices of a regular $n$-simplex.\nEach vertex is at distance\n\\[\n\\sqrt{\\left(1-\\frac{1}{n+1}\\right)^2 + \\frac{n}{(n+1)^2}} = \\sqrt{\\frac{n}{n+1}}\n\\]\nfrom the origin, while an isotropic simplex has vertices at distance $\\sqrt{n(n+2)}$ from the origin. So an affine transformation that maps an isotropic simplex in $\\RR^\\dim$ to the standard simplex in $\\RR^{\\dim+1}$ is $T(x) = \\frac{1}{\\sqrt{(n+1)(n+2)}} A x + \\frac{1}{n+1} \\ones_{\\dim+1}$.\n\n\n\\begin{algorithm}\n\\caption{Learning a simplex.}\\label{alg:simplex}\n\\begin{algorithmic}\n\\State Input: Error parameter $\\eps>0$. Probability of failure parameter $\\delta>0$. Oracle access to random points from some $n$-dimensional simplex $S_{INPUT}$.\n\n\\State Output: $V =\\{v(1), \\dotsc, v(\\dim+1)\\} \\subseteq \\RR^{\\dim}$ (approximations to the vertices of the simplex).\n\\vspace{.2em}\n\\hrule\n\\vspace{.2em}\n\\State Estimate the mean and covariance using $t_1=\\poly(n,1\/\\eps, 1\/\\delta)$ samples $p(1), \\dotsc, p(t_1)$:\n\\[\n\\mu = \\frac{1}{t_1} \\sum_i p(i),\n\\]\n\\[\n\\Sigma = \\frac{1}{t_1} \\sum_i (p(i)-\\mu) (p(i)-\\mu)^T.\n\\]\n\\State Compute a matrix $B$ so that $\\Sigma = BB^T$ (say, Cholesky decomposition).\n\n\\State Let $U = \\emptyset$.\n\\myFor{$i=1$ to $m$ (with $m = \\poly (n, \\log 1\/\\delta))$}\n\n\\State Get $t_3=\\poly(n, 1\/\\eps, \\log 1\/\\delta)$ samples $r(1),\\dotsc r(t_3)$ and use $\\mu, B$ to map them to samples $s(i)$ from a nearly-isotropic simplex: $s(i) = B^{-1}(r(i)-\\mu)$.\n\n\\State Embed the resulting samples in $\\RR^{\\dim+1}$ as a sample from an approximately rotated standard simplex:\nLet $l(i) = T(s(i))$.\n\n\\State Invoke Subroutine \\ref{alg:vertex} with sample $l(1), \\dotsc, l(t_3)$ to get $u \\in \\RR^{\\dim+1}$.\n\\State Let $\\tilde u$ be the nearest point to $u$ in the affine hyperplane $\\{x \\suchthat x \\cdot \\ones = 1\\}$. If $\\tilde u$ is not within $1\/\\sqrt{2}$ of a point in $U$, add $\\tilde u$ to $U$. (Here $1\/\\sqrt{2}$ is half of the edge length of the standard simplex.)\n\\myEndFor\n\\State Let\n\\begin{align*}\nV &= B T^{-1}(U) + \\mu \\optionalbreak\n= \\sqrt{(n+1)(n+2)} B A^T\\left(U-\\frac{1}{n+1}\\ones \\right) + \\mu.\n\\end{align*}\n\\end{algorithmic}\n\\end{algorithm}\n\nTo simplify the analysis, we pick a new sample $r(1), \\dotsc, r(t_3)$ to find every vertex, as this makes every vertex equally likely to be found when given a sample from an isotropic simplex. (The core of the analysis is done for an isotropic simplex; this is enough as the algorithm's first step is to find an affine transformation that puts the input simplex in approximately isotropic position. The fact that this approximation is close in total variation distance implies that it is enough to analyze the algorithm for the case of exact isotropic position, the analysis carries over to the approximate case with a small loss in the probability of success. See the proof below for the details.) A practical implementation may prefer to select one such sample outside of the for loop, and find all the vertices with just that sample---an analysis of this version would involve bounding the probability that each vertex is found (given the sample, over the choice of the starting point of gradient descent) and a variation of the coupon collector's problem with coupons that are not equally likely.\n\n\\iffalse\n\\begin{theorem}\nAlgorithm \\ref{alg:simplex}, with probability at least $1-\\delta$ over the sample and the randomness of the algorithm, outputs $n+1$ vectors that are the vertices of a simplex $S$ so that $d_{TV}(S, S_{INPUT}) \\leq \\eps$. The algorithm runs in time polynomial in $\\dim$, $1\/\\eps$ and $1\/\\delta$.\n\\end{theorem}\n\\fi\n\n\\begin{proof}[\\myproof of Theorem~\\ref{thm:main}]\nAs a function of the input simplex, the distribution of the output of the algorithm is equivariant under invertible affine transformations.\nNamely, if we apply an affine transformation to the input simplex, the distribution of the output is equally transformed.\\footnote{To see this: the equivariance of the algorithm as a map between distributions is implied by the equivariance of the algorithm on any given input sample. Now, given the input sample, if we apply an affine transformation to it, this transformation is undone except possibly for a rotation by the step $s(i) = B^{-1}(r(i)-\\mu)$. A rotation may remain because of the ambiguity in the characterization of $B$. But the steps of the algorithm that follow the definition of $s(i)$ are equivariant under rotation, and the ambiguous rotation will be removed at the end when $B$ is applied again in the last step.}\nThe notion of error, total variation distance, is also invariant under invertible affine transformations.\nTherefore, it is enough to analyze the algorithm when the input simplex is in isotropic position.\nIn this case $\\norm{p(i)} \\leq n+1$ (see Section \\ref{sec:preliminaries}) and we can set $t_1 \\leq \\poly(n,1\/\\eps', \\log(1\/\\delta))$ so that $\\norm{\\mu} \\leq \\eps'$ with probability at least $1-\\delta\/10$ (by an easy application of Chernoff's bound), for some $\\eps'$ to be fixed later.\nSimilarly, using results from\n\\cite[Theorem 4.1]{MR2601042}, a choice of $t_1 \\leq n {\\eps'}^{-2} \\polylog(1\/\\eps') \\polylog(1\/\\delta)$ implies that the empirical second moment matrix \\[\\bar \\Sigma = \\frac{1}{t_1} \\sum_i p(i) p(i)^T\\] satisfies $\\norm{\\bar \\Sigma - I} \\leq \\eps'$ with probability at least $1-\\delta\/10$. We have $\\Sigma = \\bar \\Sigma - \\mu \\mu^T$ and this implies $\\norm{\\Sigma - I} \\leq \\norm{\\bar \\Sigma - I} + \\norm{\\mu \\mu^T} \\leq 2\\eps'$.\nNow, $s(1), \\dotsc, s(t_3)$ is an iid sample from a simplex $S'=B^{-1}(S_{INPUT}-\\mu)$. Simplex $S'$ is close in total variation distance to some isotropic simplex\\footnote{The isotropic simplex $S_{ISO}$ will typically be far from the (isotropic) input simplex, because of the ambiguity up to orthogonal transformations in the characterization of $B$.} $S_{ISO}$. More precisely, Lemma~\\ref{lem:tv} below shows that\n\\begin{equation}\\label{equ:tv}\nd_{TV}(S', S_{ISO}) \\leq 12 \\dim \\eps',\n\\end{equation}\nwith probability at least $1-\\delta\/5$.\n\nAssume for a moment that $s(1), \\dotsc, s(t_3)$ are from $S_{ISO}$. The analysis of Subroutine \\ref{alg:vertex} (fixed point-like iteration) given in Lemma~\\ref{lem:gradient} would guarantee the following: Successive invocations to Subroutine~\\ref{alg:vertex} find approximations to vertices of $T(S_{ISO})$ within Euclidean distance $\\eps''$ for some $\\eps''$ to be determined later and $t_3 = \\poly(\\dim,1\/\\eps'', \\log 1\/\\delta)$. We ask for each invocation to succeed with probability at least $1-\\delta\/(20m)$ with $m = n (\\log n + \\log 20\/\\delta)$. Note that each vertex is equally likely to be found. The choice of $m$ is so that, if all $m$ invocations succeed (which happens with probability at least $1-\\delta\/20$), then the analysis of the coupon collector's problem, Lemma~\\ref{lem:coupons}, implies that we fail to find a vertex with probability at most $\\delta\/20$. Overall, we find all vertices with probability at least $1-\\delta\/10$.\n\nBut in reality samples $s(1), \\dotsc, s(t_3)$ are from $S'$, which is only \\emph{close} to $S_{ISO}$. The estimate from \\eqref{equ:tv} with appropriate $\\eps' = \\poly(1\/n, \\eps'', \\delta)$ gives\n\\[\nd_{TV}(S', S_{ISO}) \\leq \\frac{\\delta}{10} \\frac{1}{ t_3 m},\n\\]\nwhich implies that the total variation distance between the joint distribution of all $t_3 m$ samples used in the loop and the joint distribution of actual samples from the isotropic simplex $S_{ISO}$ is at most $\\delta\/10$, and this implies that the loop finds approximations to all vertices of $T(S_{ISO})$ when given samples from $S'$ with probability at least $1-\\delta\/5$. The points in $U$ are still within Euclidean distance $\n\\eps''$ of corresponding vertices of $T(S_{ISO})$.\n\nTo conclude, we turn our estimate of distances between estimated and true vertices into a total variation estimate, and map it back to the input simplex. Let $S''=\\conv T^{-1} U$. As $T$ maps an isotropic simplex to a standard simplex, we have that $\\sqrt{(n+1)(n+2)} T$ is an isometry, and therefore the vertices of $S''$ are within distance $\\eps''\/\\sqrt{(n+1)(n+2)}$ of the corresponding vertices of $S_{ISO}$. Thus, the corresponding support functions are uniformly within \\[\\eps''' = \\eps''\/\\sqrt{(n+1)(n+2)}\\] of each other on the unit sphere. This and the fact that $S_{ISO} \\supseteq B_n$ imply\n\\[\n(1 - \\eps''') S_{ISO}\\subseteq S'' \\subseteq (1 + \\eps''')S_{ISO}.\n\\]\nThus, by Lemma \\ref{lem:bmtv}, $d_{TV}(S'', S_{ISO}) \\leq\n1 - (\\frac{1-\\eps'''}{1+\\eps'''})^\\dim \\leq 1-(1-\\eps''')^{2n}\\leq 2n \\eps''' \\leq 2\\eps''$\nand this implies that the total variation distance between the uniform distributions on $\\conv V$ and the input simplex is at most $2 \\eps''$. Over all random choices, this happens with probability at least $1-2\\delta\/5$. We set $\\eps'' = \\eps\/2$.\n\\end{proof}\n\n\\begin{lemma}\\label{lem:tv}\nLet $S_{INPUT}$ be an $\\dim$-dimensional isotropic simplex. Let $\\Sigma$ be an $\\dim$-by-$\\dim$ positive definite matrix such that $\\norm{\\Sigma - I} \\leq \\eps < 1\/2$. Let $\\mu$ be an $n$-dimensional vector such that $\\norm{\\mu} \\leq \\eps$. Let $B$ be an $n$-by-$n$ matrix such that $\\Sigma = BB^T$. Let $S$ be the simplex $B^{-1}(S_{INPUT}-\\mu)$. Then there exists an isotropic simplex $S_{ISO}$ such that $d_{TV}(S, S_{ISO}) \\leq 6 \\dim \\eps$.\n\\end{lemma}\n\\begin{proof}\nWe use an argument along the lines of the orthogonal Procrustes problem (nearest orthogonal matrix to $B^{-1}$, already in \\cite[Proof of Theorem 4]{NguyenR09}): Let $U D V^T$ be the singular value decomposition of $B^{-1}$. Let $R = U V^T$ be an orthogonal matrix (that approximates $B^{-1}$). Let $S_{ISO} = R S_{INPUT}$.\n\nWe have $S = UDV^T (S_{INPUT} - \\mu)$. Let $\\sigma_{min}$, $\\sigma_{max}$ be the minimum and maximum singular values of $D$, respectively. This implies:\n\\begin{align}\n\\sigma_{min} UV^T (S_{INPUT} - \\mu) &\\subseteq S \\subseteq \\sigma_{max}UV^T (S_{INPUT} - \\mu), \\nonumber \\\\\n\\sigma_{min} (S_{ISO} - R\\mu)  &\\subseteq S \\subseteq \\sigma_{max}( S_{ISO} - R\\mu).\\label{equ:isotropy}\n\\end{align}\nAs $S_{ISO} \\supseteq B_n$, $\\norm{\\mu} \\leq 1$, $R$ is orthogonal and $S_{ISO}$ is convex, we have\n\\[\nS_{ISO}-R\\mu \\supseteq (1-\\norm{\\mu}) S_{ISO}.\n\\]\nAlso,\n\\begin{align*}\nS_{ISO} - R\\mu &\\subseteq S_{ISO} + \\norm{\\mu} B_n \\\\\n&\\subseteq S_{ISO} (1+ \\norm{\\mu}).\n\\end{align*}\nThis in \\eqref{equ:isotropy} gives\n\\[\n\\sigma_{min} (1-\\norm{\\mu}) S_{ISO} \\subseteq S \\subseteq \\sigma_{max}(1+\\norm{\\mu}) S_{ISO}.\n\\]\nThis and Lemma \\ref{lem:bmtv} imply\n\\[\nd_{TV}(S, S_{ISO}) \\leq 2\\left(1- \\left(\\frac{\\sigma_{min} (1-\\norm{\\mu}) }{ \\sigma_{max}(1+\\norm{\\mu})}\\right)^\\dim\\right).\n\\]\nThe estimate on $\\Sigma$ gives $\\sigma_{min} \\geq \\sqrt{1-\\eps}$, $\\sigma_{max} \\leq \\sqrt{1+\\eps}$.  Thus\n\\begin{align*}\nd_{TV}(S, S_{ISO}) &\\leq 2\\left(1- \\left(\\frac{1-\\eps}{1+\\eps} \\right)^{3\\dim\/2}\\right) \\\\\n& \\leq 2\\left(1- \\left(1-\\eps\\right)^{3\\dim}\\right) \\\\\n& \\leq 6\\dim \\eps.\n\\end{align*}\n\\end{proof}\n\n\\section{The local and global maxima of the 3rd moment of the standard simplex and the isotropic simplex} \\label{sec:maxima}\n\n\nIn this section we study the structure of the set of local maxima of the third moment as a function of the direction (which happens to be essentially $u \\mapsto \\sum u_i^3$ as discussed in Section \\ref{sec:moment}). This is not necessary for our algorithmic result, however it gives insight into the geometry of the third moment (the location of local maxima\/minima and stationary points) and suggests that more direct optimization algorithms like gradient descent and Newton's method will also work, although we will not prove that.\n\n\\begin{theorem}\\label{thm:maxima}\nLet $K \\subseteq \\RR^\\dim$ be an isotropic simplex. Let $X$ be random in $K$. Let $V = \\{x_i\\}_{i=1}^{\\dim+1} \\subseteq \\RR^\\dim$ be the set of normalized vertices of $K$. Then $V$ is a complete set of local maxima and a complete set of global maxima of $F:S^{\\dim-1} \\to \\RR$ given by $F(u) = \\e ((u \\cdot X)^3)$.\n\\end{theorem}\n\\noindent\\emph{Proof idea:} Embed the simplex in $\\RR^{\\dim+1}$. Show that the third moment is proportional to the complete homogeneous symmetric polynomial of degree 3, which for the relevant directions is proportional to the sum of cubes. To conclude, use first and second order optimality conditions to characterize the set of local maxima.\n\\begin{proof}\nConsider the standard simplex\n\\[\n\\Delta^n = \\conv \\{e_1, \\dotsc, e_{\\dim+1} \\} \\subseteq \\RR^{\\dim+1}\n \\]\nand identify it with $V$ via a linear map $A :\\RR^{\\dim+1} \\to \\RR^{\\dim}$ so that $A(\\Delta^n) = V$. Let $Y$ be random in $\\Delta^n$. Consider $G: S^{\\dim} \\to \\RR$ given by $G(v) = m_3(v) = \\e ((v \\cdot Y)^3)$. Let $U = \\{ v \\in \\RR^{\\dim+1} \\suchthat v \\cdot \\ones =0, \\norm{v}=1 \\}$ be the equivalent feasible set for the embedded problem.\nWe have $G(v) = cF(A v)$ for any $v \\in U$ and some constant $c > 0$ independent of $v$.\nTo get the theorem, it is enough to show that the local maxima of $G$ in $U$ are precisely the normalized versions of the projections of the canonical vectors onto the hyperplane orthogonal to $\\ones = (1, \\dotsc, 1)$. According to Section \\ref{sec:moment}, for $v \\in U$ we have\n\\[\nG(v) \\propto p_3(v).\n\\]\nUsing a more convenient but equivalent constant, we want to enumerate the local maxima of the problem\n\\begin{equation}\\label{equ:opt}\n\\begin{aligned}\n\\max \\frac{1}{3} p_3(v)& \\\\\n\\text{s.t.} \\quad v \\cdot v &= 1 \\\\\nv \\cdot \\ones &= 0 \\\\\nv &\\in \\RR^{\\dim+1}.\n\\end{aligned}\n\\end{equation}\nThe Lagrangian function is\n\\[\nL(v, \\lambda_1, \\lambda_2) = \\frac{1}{3} \\sum_i v_i^3 - \\lambda_1 \\sum_i v_i - \\lambda_2\\frac{1}{2} \\left(\\biggl(\\sum_i v_i^2 \\biggr) -1 \\right).\n\\]\nThe first order condition is $\\nabla_v L = 0$, that is,\n\\begin{equation}\\label{equ:foc}\nv_i^2 = \\lambda_1 + \\lambda_2 v_i \\quad \\text{for $i= 1, \\dotsc, \\dim+1$.}\n\\end{equation}\nConsider this system of equations on $v$ for any fixed $\\lambda_1$, $\\lambda_2$. Let $f(x) = x^2$, $g(x)= \\lambda_1 + \\lambda_2 x$. The first order condition says $f(v_i) = g(v_i)$, where $f$ is convex and $g$ is affine. That is, the $v_i$s can take at most two different values. As our optimization problem \\eqref{equ:opt} is symmetric under permutation of the coordinates, we conclude that, after putting the coordinates of a point $v$ in non-increasing order, if $v$ is a local maximum of \\eqref{equ:opt}, then $v$ must be of the form\n\\[\nv = (a, \\dotsc, a, b, \\dotsc, b),\n\\]\nwhere $a>0>b$ and there are exactly $\\alpha$ $a$s and $\\beta$ $b$s, for $\\alpha, \\beta \\in \\{1, \\dotsc, \\dim\\}$.\n\nWe will now study the second order necessary condition (SONC) to eliminate from the list of candidates all vectors with $\\alpha >1$. It is easy to see that the surviving vectors are exactly the promised scaled projections of the canonical vectors. This vectors must all be local and global maxima: At least one of them must be a global maximum as we are maximizing a continuous function over a compact set and all of them have the same objective value so all of them are local and global maxima.\n\nThe SONC at $v$ asks for the Hessian of the Lagrangian to be negative semidefinite when restricted to the tangent space to the constraint set at $v$ \\cite[Section 11.5]{MR2423726}.\nWe compute the Hessian (recall that $v^{(2)}$ is the vector of the squared coordinates of $v$):\n\\[\n\\nabla_v L = v^{(2)} - \\lambda_1 \\ones - \\lambda_2 v\n\\]\n\\[\n\\nabla^2_v L = 2 \\diag(v) - \\lambda_2 I\n\\]\nwhere $\\diag(v)$ is the $(\\dim + 1)$-by-$(\\dim+1)$ matrix having the entries of $v$ in the diagonal and 0 elsewhere.\n\nA vector in the tangent space is any $z \\in \\RR^{\\dim+1}$ such that $z \\cdot \\ones = 0$, $v \\cdot z = 0$, and definiteness of the Hessian is determined by the sign of $z^T \\nabla^2_v L z$ for any such $z$, where\n\\[\nz^T \\nabla^2_v L z = \\sum_{i=1}^{\\dim+1} z_i^2 (2 v_i - \\lambda_2).\n\\]\nSuppose $v$ is a critical point with $\\alpha \\geq 2$. To see that such a $v$ cannot be a local maximum, it is enough to show $2a > \\lambda_2$, as in that case we can take $z = (1, -1, 0,\\dotsc, 0)$ to make the second derivative of $L$ positive in the direction $z$.\n\nIn terms of $\\alpha, \\beta, a, b$, the constraints of \\eqref{equ:opt} are $\\alpha a + \\beta b = 0$, $\\alpha a^2 + \\beta b^2 = 1$, and this implies $a = \\sqrt{\\frac{\\beta}{\\alpha (\\dim+1) }}$, $b = - \\sqrt{\\frac{\\alpha}{\\beta (\\dim+1) }}$. The inner product between the first order condition \\eqref{equ:foc} and $v$ implies $\\lambda_2 = \\sum v_i^3 = \\alpha a^3 + \\beta b^3$. It is convenient to consider the change of variable $\\gamma = \\alpha\/(\\dim+1)$, as now candidate critical points are parameterized by certain discrete values of $\\gamma$ in $(0,1)$. This gives $\\beta = (1-\\gamma)(\\dim+1)$, $ a = \\sqrt{(1-\\gamma)\/(\\gamma(\\dim+1))}$ and\n\\begin{align*}\n\\lambda_2 &= (\\dim+1)\\biggl[\\gamma \\left(\\frac{1-\\gamma}{\\gamma (\\dim+1)}\\right)^{3\/2} \\\\\n&\\qquad - (1-\\gamma)\\left(\\frac{\\gamma}{(1-\\gamma) (\\dim+1)}\\right)^{3\/2}\\biggr] \\\\\n&= \\frac{1}{\\sqrt{(\\dim+1) \\gamma (1-\\gamma)}} \\left[(1-\\gamma)^2 - \\gamma^2\\right] \\\\\n&= \\frac{1}{\\sqrt{(\\dim+1) \\gamma (1-\\gamma)}} [1- 2\\gamma].\n\\end{align*}\nThis implies\n\\begin{align*}\n2 a - \\lambda_2 &= \\frac{1}{\\sqrt{(\\dim+1) \\gamma (1-\\gamma)}} [2(1-\\gamma) -1 + 2\\gamma] \\\\\n    &= \\frac{1}{\\sqrt{(\\dim+1) \\gamma (1-\\gamma)}}.\n\\end{align*}\nIn $(0,1)$, the function given by $\\gamma \\mapsto 2a-\\lambda_2 = \\frac{1}{\\sqrt{(\\dim+1) \\gamma (1-\\gamma)}}$ is convex and symmetric around $1\/2$, where it attains its global minimum value, $2\/ \\sqrt{\\dim+1}$, which is positive.\n\\end{proof}\n\n\\section{Probabilistic Results} \\label{sec:prob} \n\nIn this section we show the probabilistic results underlying the reductions from learning simplices and $\\ell_p^n$ balls to ICA. The results are Theorems \\ref{thm:simplexscaling} and \\ref{thm:lpscaling}. They each show a simple non-linear rescaling of the respective uniform distributions that gives a distribution with independent components (Definition \\ref{def:ic}).\n\nTheorem \\ref{thm:simplexscaling} below gives us, in a sense, a ``reversal'' of the representation of the cone measure on $\\partial B_p^n$, seen in Theorem \\ref{thm:cone}. Given any random point in the standard simplex, we can apply a simple non-linear scaling and recover a distribution with independent components. \n\n\\begin{definition}\\label{def:ic}\nWe say that a random vector $X$ has \\emph{independent components} if it is an affine transformation of a random vector having independent coordinates.\n\\end{definition}\n\n\\begin{theorem}\\label{thm:simplexscaling}\nLet $X$ be a uniformly random vector in the $(\\dimension-1)$-dimensional standard simplex $\\Delta_{n-1}$. Let $T$ be a random scalar distributed as $\\mathrm{Gamma}(n, 1)$. Then the coordinates of $T X$ are iid as $\\expdist{1}$.\n\nMoreover, if $A:\\mathbb{R}^{\\dimension} \\rightarrow \\mathbb{R}^{\\dimension}$ is an invertible linear transformation, then the random vector $TA(X)$ has independent components.\n\\end{theorem}\n\\begin{proof}\nIn the case where $p = 1$, Theorem \\ref{thm:cone} restricted to the positive orthant implies that for random vector $G = (G_1, \\dotsc, G_{\\dimension})$, if each $G_i$ is an iid exponential random variable $\\expdist{1}$, then $( G\/\\norm{G}_1, \\norm{G}_1)$ has the same (joint) distribution as $(X,T)$. Given the measurable function $f(x,t) = xt$, $f(X,T)$ has the same distribution as $f( G\/\\norm{G}_1, \\norm{G}_1)$. That is, $XT$ and $G$ have the same distribution\\footnote{See \\cite[Theorem 1.1]{MR1456629} for a similar argument in this context.}.\n\nFor the second part, we know $T A(X) = A(TX)$ by linearity. By the previous argument the coordinates of $TX$ are independent. This implies that $A(TX)$ has independent components.\n\\end{proof}\n\n\n\n\nThe next lemma complements the main result in \\cite{barthe2005probabilistic}, Theorem 1 (Theorem \\ref{theorem:fullnaor} elsewhere here). They show a representation of the uniform distribution in $B_p^n$, but they do not state the independence that we need for our reduction to ICA.\n\\begin{lemma} \\label{lemma:independence}\nLet $p \\in [1, \\infty)$. Let $G=(G_1, \\dotsc, G_n)$ be iid random variables with density proportional to $\\exp(-\\abs{t}^p)$. Let $W$ be a nonnegative random variable with distribution $\\expdist{1}$ and independent of $G$. Then the random vector\n\\[\n\\frac{G}{(\\norm{G}_p^p + W)^{1\/p}}\n\\]\nis independent of $(\\norm{G}_p^p + W)^{1\/p}$.\n\\end{lemma}\n\n\\begin{proofidea}\nWe aim to compute the join density, showing that it is a product of individual densities. To avoid complication, we raise everything to the $p$th power, which eliminates extensive use of the chain rule involved in the change of variables. \n\\end{proofidea}\n\n\\begin{proof}\nIt is enough to show the claim conditioning on the orthant in which $G$ falls, and by symmetry it is enough to prove it for the positive orthant. Let random variable $H = (G_1^p, G_2^p, \\dotsc, G_n^p)$.\nSince raising (strictly) positive numbers to the $p$th power is injective, it suffices to show that the random vector \n\\[\nX = \\frac{H}{\\sum_{i=1}^n{H}_i + W}\n\\]\nis independent of the random vector $Y = \\sum_{i=1}^n{H}_i + W$. \n \nFirst, let $U$ be the interior of the support of $(X,Y)$, that is $U = \\{ x \\in \\RR^{n} : x_i > 0, \\sum_i x_i < 1 \\}\\times \\{y \\in \\RR :y>0\\}$  and consider $h: U \\rightarrow \\RR^{\\dimension}$ and $w: U \\rightarrow \\RR$ where\n\\[\nh(x,y) = x y\n\\] and \n\\[\nw(x,y) = y - \\sum_{i=1}^n h(x,y)_i = y - \\sum_{i=1}^n x_i \\cdot y = y\\left(1 - \\sum_{i=1}^n x_i\\right).\n\\]\nThe random vector $(H,W)$ has a density $f_{H,W}$ supported on $V  = \\operatorname{int} \\RR^{n+1}_+$ and \n$$(x,y) \\mapsto (h(x,y), w(x,y))$$\nis one-to-one from $U$ onto $V$. \nLet $J(x,y)$ be the determinant of its Jacobian. This Jacobian is\n\\[\n\\begin{pmatrix}\ny & 0 & \\cdots & 0 & x_1 \\\\\n0 & y & \\cdots & 0 & x_2 \\\\\n\\vdots & & & & \\vdots \\\\\n0 & 0 & \\cdots & y & x_n \\\\\n-y & -y & \\cdots & -y & 1 - \\sum_{i=1}^n x_i \\\\\n\\end{pmatrix}\n\\]\nwhich, by adding each of the first $n$ rows to the last row, reduces to\n\\[\n\\begin{pmatrix}\ny & 0 & \\cdots & 0 & x_1 \\\\\n0 & y & \\cdots & 0 & x_2 \\\\\n\\vdots & & & & \\vdots \\\\\n0 & 0 & \\cdots & y & x_n \\\\\n0 & 0 & \\cdots & 0 & 1 \\\\\n\\end{pmatrix},\n\\]\nthe determinant of which is trivially $J(x,y) = y^n$. \n\n\nWe have that $J(x,y)$ is nonzero in $U$. Thus, $(X,Y)$ has density $f_{X,Y}$ supported on $U$ given by\n\\begin{align*}\nf_{X,Y}(x,y) &= f_{H,W}\\bigl(h(x,y),w(x,y)\\bigr) \\cdot \\abs{J(x)}.\n\\end{align*}\n\n\n\nIt is easy to see\\footnote{See for example \\cite[proof of Theorem 3]{barthe2005probabilistic}.} that each $H_i = G_i^p$ has density $\\mathrm{Gamma}(1\/p, 1)$ and thus $\\sum_{i=1}^n H_i$ has density $\\mathrm{Gamma}(n\/p,1)$ by the additivity of the Gamma distribution.  We then compute the joint density\n\\begin{align*}\nf_{X,Y}(x,y) &= f_{H,W}\\bigl(h(x,y),w(x,y)\\bigr) \\cdot y^n \\\\\n&= f_{H,W}\\Bigl(xy, y(1 - \\sum\\limits_{i=1}^n x_i)\\Bigr) \\cdot y^n.\n\\end{align*}\nSince $W$ is independent of $H$,\n\\begin{align*}\nf_{X,Y}(x,y) &=  f_{W}\\bigg(y\\Big(1 - \\sum\\limits_{i=1}^n x_i\\Big)\\bigg) \\cdot y^n \\prod_{i=1}^n f_{H_i}(x_iy) \n\\end{align*}\nwhere\n\\begin{align*}\n\\prod_{i=1}^{\\dimension}\\Big(f_{H_i}(x_iy)\\Big) \\cdot f_{W}\\bigg(y\\Big(1 - \\sum\\limits_{i=1}^n x_i\\Big)\\bigg) \\cdot y^n \n&\\propto \\prod_{i=1}^n \\left[e^{-x_iy}(x_iy)^{\\frac{1}{p} - 1}\\right] \\exp\\left(-y(1-\\sum\\limits_{i=1}^n x_i)\\right) y^n \\\\\n&\\propto \\biggl(\\prod_{i=1}^n x_i^{\\frac{1}{p} - 1}\\biggr) y^{n\/p}.\n\\end{align*}\nThe result follows.\n\\end{proof}\n\nWith this in mind, we show now our analog of Theorem \\ref{thm:simplexscaling} for $B_p^{\\dimension}$.\n\n\\begin{theorem}\\label{thm:lpscaling\nLet $X$ be a uniformly random vector in $B_p^{\\dimension}$. Let $T$ be a random scalar distributed as $\\mathrm{Gamma}((n\/p)+1,1)$. Then the coordinates of $T^{1\/p} X$ are iid, each with density proportional to $\\exp(-\\abs{t}^p)$.\nMoreover, if $A: \\RR^{\\dimension} \\rightarrow \\RR^{\\dimension}$ is an invertible linear transformation, then the random vector given by $T^{1\/p}A(X)$ has independent components.\n\\end{theorem}\n\n\\begin{proof}\nLet $G = (G_1, \\dotsc, G_{\\dimension})$ where each $G_i$ is  iid as $\\mathrm{Gamma}(1\/p, 1)$. Also, let $W$ be an independent random variable distributed as $\\expdist{1}$. Let $S = \\bigl(\\sum_{i=1}^n |G_i|^p + W \\bigr)^{1\/p}$. \n\nBy Lemma \\ref{lemma:independence} and Theorem \\ref{theorem:fullnaor} we know $(G\/S,S)$ has the same joint distribution as $(X,T^{1\/p})$. Then for the measurable function $f(x,t) = xt$, we immediately have $f(X,T^{1\/p})$ has the same distribution as $f(G\/S, S)$ and thus $XT^{1\/p}$ has the same distribution as $G$. \n\nFor the second part, since $T$ is a scalar, we have $T^{1\/p}A(X) = A(T^{1\/p}X)$. By the previous argument we have that the coordinates of $T^{1\/p}X$ are independent. Thus, $A(T^{1\/p}X)$ has independent components. \n\\end{proof}\n\n\nThis result shows that one can obtain a vector with independent components from a sample in a linearly transformed $\\ell_p$ ball.\nIn Section \\ref{sec:reduction} we show that they are related in such as way that one can recover the linear transformation from the independent components via ICA.\n\n\n\\section{Learning problems that reduce to ICA}\\label{sec:reduction}\nIndependent component analysis is a certain computational problem and an associated family of algorithms. Suppose that $X$ is a random $\\dimension$-dimensional vector whose coordinates are independently distributed. The coordinates' distributions are unknown and not necessarily identical. The ICA problem can be stated as follows: given samples from an affine transformation $Y=AX+b$ of $X$, estimate $A$ and $b$ (up to a certain intrinsic indeterminacy: permutation and scaling of the columns of $A$). We will state more precisely below what is expected of a an ICA algorithm.\n\n\nWe show randomized reductions from the following two natural statistical estimation problems to ICA:\n\n\\begin{problem}[simplex] \nGiven uniformly random points from an $n$-dimensional simplex, estimate the simplex.\n\\end{problem} \n\nThis is the same problem of learning a simplex as in the rest of the paper, we just restate it here for clarity.\n\nTo simplify the presentation for the second problem, we ignore the estimation of the mean of an affinely transformed distribution. That is, we assume that the $\\ell_p^n$ ball to be learned has only been \\emph{linearly} transformed.\n\n\\begin{problem}[linearly transformed $\\ell_p^n$ balls]\nGiven uniformly random points from a linear transformation of the  $\\ell_p^n$-ball, estimate the linear transformation.\n\\end{problem}\n\nThese problems do not have an obvious independence structure. Nevertheless, known representations of the uniform measure in an $\\ell_p^n$ ball and the \\emph{cone measure} (defined in Section \\ref{sec:preliminaries}) on the surface of an $\\ell_p^n$ ball can be slightly extended to map a sample from those distributions into a sample with independent components by a non-linear scaling step.\nThe use of a non-linear scaling step to turn a distribution into one having independent components has been done before \\cite{MR2756189, DBLP:conf\/nips\/SinzB08}, but there it is applied \\emph{after} finding a transformation that makes the distribution axis-aligned. This alignment is attempted using ICA (or variations of PCA) on the original distribution \\cite{MR2756189, DBLP:conf\/nips\/SinzB08}, without independent components, and therefore the use of ICA is somewhat heuristic. One of the contributions of our reduction is that the rescaling we apply is ``blind'', namely, it can be applied to the original distribution. In fact, the distribution does not even need to be isotropic (``whitened''). The distribution resulting from the reduction has independent components and therefore the use of ICA on it is well justified.\n\nThe reductions are given in Algorithms \\ref{alg:simplexreduction} and \\ref{alg:lpreduction}. To state the reductions, we denote by $ICA(s(1), s(2), \\ldots)$ the invocation of an ICA routine. \nIt takes samples $s(1), s(2), \\ldots$ of a random vector $Y=AX + \\mu$, where the coordinates of $X$ are independent, and returns an approximation to a square matrix $M$ such that $M(Y-\\e(Y))$ is isotropic and has independent coordinates. \nThe theory of ICA \\cite[Theorem 11]{comon1994independent} implies that if $X$ is isotropic and at most one coordinate is distributed as a Gaussian, then such an $M$ exists and it satisfies $M A = DP$, where $P$ is a permutation matrix and $D$ is a diagonal matrix with diagonal entries in $\\{-1, 1\\}$. We thus need the following definition to state our reduction: Let $c_{p,n} = (\\e_{X \\in B_p^n}(X_1^2))^{1\/2}$. That is, the uniform distribution in $B_p^n\/c_{p,n}$ is isotropic.\n\nAs we do not state a full analysis of any particular ICA routine, we do not state explicit approximation guarantees.\n\n\\begin{algorithm}\n\\caption{Reduction from Problem 1 to ICA}\\label{alg:simplexreduction}\n\n\\begin{algorithmic}\n\\State Input: A uniformly random sample $p(1), \\dotsc, p(t)$ from an $n$-dimensional simplex $S$.\n\\State Output: Vectors $\\tilde v(1), \\dotsc, \\tilde v(n+1)$ such that their convex hull is close to $S$.\n\\vspace{.2em}\n\\hrule\n\\vspace{.2em}\n\\State Embed the sample in $\\RR^{n+1}$: Let $p'(i) = (p(i), 1)$.\n\n\\State For every $i = 1, \\dotsc, t$, generate a random scalar $T(i)$ distributed as $\\mathrm{Gamma}(n+1, 1)$. Let $q(i) = p'(i) T(i)$.\n\\State Invoke $ICA(q(1), \\dotsc, q(t))$ to obtain a approximately separating matrix $\\tilde M$.\n\\State Compute the inverse of $\\tilde M$ and multiply every column by the sign of its last entry to get a matrix $\\tilde A$.\n\\State Remove the last row of $\\tilde A$ and return the columns of the resulting matrix as $\\tilde v(1), \\dotsc, \\tilde v(n+1)$.\n\\end{algorithmic}\n\\end{algorithm}\nAlgorithm \\ref{alg:simplexreduction} works as follows: Let $X$ be an $(n+1)$-dimensional random vector with iid coordinates distributed as $\\expdist{1}$. Let $V$ be the matrix having columns $(v(i),1)$ for $i=1, \\dotsc, n+1$. Let $Y$ be random according to the distribution that results from scaling in the algorithm. Theorem \\ref{thm:simplexscaling} implies that $Y$ and $VX$ have the same distribution. Also, $X-\\ones$ is isotropic and $Y$ and $V(X-\\ones) + V \\ones$ have the same distribution. Thus, the discussion about ICA earlier in this section gives that the only separating matrices $M$ are such that $M V = D P$ where $P$ is a permutation matrix and $D$ is a diagonal matrix with diagonal entries in $\\{-1, 1\\}$. That is, $V P^T = M^{-1} D$. As the last row of $V$ is all ones, the sign change step in Algorithm \\ref{alg:simplexreduction} undoes the effect of $D$ and recovers the correct orientation.\n \n\\begin{algorithm}\n\\caption{Reduction from Problem 2 to ICA}\\label{alg:lpreduction}\n\\begin{algorithmic}\n\\State Input: A uniformly random sample $p(1), \\dotsc, p(t)$ from $A(B_p^n)$ for a known parameter $p \\in [1, \\infty)$, where $A:\\RR^n \\to \\RR^n$ is an unknown invertible linear transformation.\n\\State Output: A matrix $\\tilde A$ such that $\\tilde A B_p^n$ is close to $A(B_p^n)$.\n\\vspace{.2em}\n\\hrule\n\\vspace{.2em}\n\n\\State For every $i = 1, \\dotsc, t$, generate a random scalar $T(i)$ distributed as $\\mathrm{Gamma}((n\/p)+1,1)$. Let $q(i) = p(i)  T(i)^{1\/p}$.\n\\State Invoke $ICA(q(1), \\dotsc, q(t))$ to obtain an approximately separating matrix $\\tilde M$. \n\\State Output $\\tilde A = c_{p,n}^{-1} \\tilde M^{-1}$.\n\\end{algorithmic}\n\\end{algorithm}\n\nSimilarly, Algorithm \\ref{alg:lpreduction} works as follows: Let $X$ be a random vector with iid coordinates, each with density proportional to $\\exp(-\\abs{t}^p)$. Let $Y$ be random according to the distribution that results from scaling in the algorithm. Theorem \\ref{thm:lpscaling} implies that $Y$ and $AX$ have the same distribution. Also, $X\/c_{p,n}$ is isotropic and we have $Y$ and $Ac_{p,n}(X\/c_{p,n})$ have the same distribution. Thus, the discussion about ICA earlier in this section gives that the only separating matrices $M$ are such that $M A c_{p,n} = D P$ where $P$ is a permutation matrix and $D$ is a diagonal matrix with diagonal entries in $\\{-1, 1\\}$. That is, $A P^{T} D^{-1} = c_{p,n}^{-1} M^{-1}$. The fact that $B_p^n$ is symmetric with respect to coordinate permutations and sign changes implies that $A P^{T} D^{-1} B_p^n = A B_p^n$ and is the same as $c_{p,n}^{-1} M^{-1}$. When $p \\neq 2$, the assumptions in the discussion above about ICA are satisfied and Algorithm \\ref{alg:lpreduction} is correct. When $p = 2$, the distribution of the scaled sample is Gaussian and this introduces ambiguity with respect to rotations in the definition of $M$, but this ambiguity is no problem as it is counteracted by the fact that the $l_2$ ball is symmetric with respect to rotations.\n\n\n\n\n\\section{Conclusion}\nWe showed, in two different ways, that the problem of learning simplices can be solved efficiently using techniques for ICA. We also showed that when the sample is one that may not satisfy the requirement of independent components, we can efficiently obtain from it a sample that guarantees this property and from which the original distribution can be estimated. \nMany questions remain: \nCan we do this for other polytopes? Can we do this when the points come from the Gaussian distribution with labels instead\nof the uniform distribution in the polytope? In particular, does any one of the two techniques that we used \nin this paper for learning simplices extend to learning polytopes or to \nlatent variable models?\n\n\\acks{We thank Santosh Vempala for telling us about the polytope learning problem, the approach of using higher-order moments and for helpful discussions.\nWe also thank Keith Ball, Alexander Barvinok, Franck Barthe, Mikhail Belkin, Adam Kalai, Assaf Naor, Aaditya Ramdas, Roman Vershynin and James Voss for helpful discussions.}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\subsection{\\refname}%\n      \\@mkboth{\\MakeUppercase\\refname}{\\MakeUppercase\\refname}%\n      \\list{\\@biblabel{\\@arabic\\c@enumiv}}%\n           {\\settowidth\\labelwidth{\\@biblabel{#1}}%\n            \\leftmargin\\labelwidth\n            \\advance\\leftmargin\\labelsep\n            \\@openbib@code\n            \\usecounter{enumiv}%\n            \\let\\p@enumiv\\@empty\n            \\renewcommand\\theenumiv{\\@arabic\\c@enumiv}}%\n      \\sloppy\n      \\clubpenalty4000\n      \\@clubpenalty \\clubpenalty\n      \\widowpenalty4000%\n      \\sfcode`\\.\\@m}\n     {\\def\\@noitemerr\n       {\\@latex@warning{Empty `thebibliography' environment}}%\n      \\endlist}\n\n\\usepackage{hyperref}\n\\hypersetup{\npdftitle={\\@title},\npdfauthor={\\@author},\nbreaklinks=true,\ncolorlinks=true,\n linkcolor=blue,\n citecolor=darkred,\n urlcolor=darkblue,\n}\n\n\\newenvironment{sectionauthors}{}{\n\\author{Jean-Guillaume Dumas\\footnote{Universit\\'e de Grenoble;\n    Laboratoire Jean Kuntzmann, (umr CNRS 5224, Grenoble INP, INRIA,\n    UJF, UPMF);\n    \\href{mailto:Jean-Guillaume.Dumas@imag.fr}{Jean-Guillaume.Dumas@imag.fr};\n    51, rue des Math\\'ematiques, \n    BP 53X, F-38041 Grenoble, France.\n  } \\and Cl{\\'e}ment Pernet\\footnote{INRIA, Universit\\'e de Grenoble;\n    Laboratoire LIG (umr CNRS 5217, Grenoble INP, INRIA, UJF, UPMF);\n    \\href{mailto:Clement.Pernet@imag.fr}{Clement.Pernet@imag.fr};\n    ENSIMAG Antenne de Montbonnot, \n    51, avenue Jean Kuntzmann, \n    F-38330 Montbonnot Saint-Martin, France.\n  }\n}\n\\maketitle\n\\setcounter{tocdepth}{5}\n\\@starttoc{toc}\n\\newpage\n}\n\\renewenvironment{table}[1][empty]\n               {\\@float{table}[#1]\\begin{center}}\n               {\\end{center}\\end@float}\n\\makeatother\n\n\\usepackage[english]{babel}\n\\usepackage{babelbib}\n\n\\textwidth 365pt\n\\oddsidemargin 52pt\n\\evensidemargin 52pt\n\n\\begin{document} \n\\input{sec134.tex}\n\n\\subsection{Acknowledgment}\nWe thank an anonymous referee for numerous helpful suggestions that\nconsiderably improved the paper.\n\n\\bibliographystyle{babplain}\n\n\\section{\\titlename}\\label{sec134}\n\n\n\\begin{sectionauthors}\n\\sectionauthor{Jean-Guillaume Dumas}{Universit{\\'e} de Grenoble}\n\\sectionauthor{Cl{\\'e}ment Pernet}{Universit{\\'e} de Grenoble}\n\\end{sectionauthors}\n\nWe present here algorithms for efficient computation of\nlinear algebra problems over finite fields. \nImplementations\\footnote{\\url{http:\/\/magma.maths.usyd.edu.au},\n  \\url{http:\/\/www.maplesoft.com}, \\url{http:\/\/sagemath.org},\n  \\url{http:\/\/www.shoup.net\/ntl},\n  \\url{http:\/\/www.flintlib.org}, \n  \\url{http:\/\/www.cs.uwaterloo.ca\/~astorjoh\/iml.html},\n  \\url{http:\/\/m4ri.sagemath.org},\n  \\url{http:\/\/linalg.org}}\nof the proposed algorithms are available through the {\\sc Magma}, {\\sc\n  Maple} (within the \\verb!LinearAlgebra[Modular]! subpackage) and\n{\\sc Sage} systems; some parts can also be found within the C\/C++ libraries\nNTL,\nFLINT, IML, M4RI and the special purpose {\\sc LinBox} template library\nfor exact, high-performance linear algebra computation with dense,\nsparse, and structured matrices over the integers and over finite\nfields \\cite{DumGauGieGioHovKalSauTurVil02}.  \n\n\\subsection{Dense matrix multiplication}\\label{ssec:blas}\n\n\\begin{definition} \\label{def:mm}\nFor $A \\in {\\mathbb F}_q^{m\\times k}$ and $B \\in {\\mathbb F}_q^{k\\times n}$ with elements $A_{i,j}$ and $B_{i,j}$,\nthe matrix $C=A\\times B$ has $C_{i,j} = \\sum_{l=1}^{k} A_{i,l}\nB_{l,j}$. We denote by $\\MatrixMul(m,k,n)$ a time complexity bound on the\nnumber of field operations necessary to compute $C$.\n\\end{definition}\nClassical triple loop implementation of matrix multiplication makes\n$\\MatrixMul(m,k,n)\\leq2mkn$. The best published estimates to date\ngives $\\MatrixMul(n,n,n) \\leq \\BigO{n^\\omega}$ with $\\omega \\approx\n2.3755$ \\cite{MR1056627}, though improvements to $2.3737$ and $2.3727$\nare now claimed \\cite{Sto10,Vas11}. For very rectangular matrices one\nalso have astonishing results like $\\MatrixMul(n,n,n^\\alpha) \\leq\n\\BigO(n^{2+\\epsilon})$ for a constant $\\alpha > 0.294$ and any\n$\\epsilon>0$ \\cite{MR1449760}.\nNowadays practical implementations mostly use Strassen-Winograd's\nalgorithm, see section \\ref{ssec:wino}, with an intermediate complexity and\n$\\omega \\approx 2.8074$.\n\n\\subsubsection{Tiny finite fields}\\label{sssec:tiny}\nThe practical efficiency of matrix multiplication depends highly on\nthe representation of field elements. \nWe thus present three kinds of compact representations for elements of a\nfinite field with\nvery small cardinality: bitpacking (for ${\\mathbb F}_2$), bit-slicing (for say\n${\\mathbb F}_3, {\\mathbb F}_5, {\\mathbb F}_7, {\\mathbb F}_{2^3}$, or ${\\mathbb F}_{3^2}$) and Kronecker\nsubstitution\\index{Kronecker!substitution}. These representations are\ndesigned to allow\nefficient linear algebra operations, including matrix multiplication.\n\\index{Four Russians (Method!of)}\n\\index{bit-packing}\n\n\n\\begin{algorithm}{[Greasing]}\\label{alg:Greasing}\\index{Greasing}\nOver ${\\mathbb F}_2$, the method of \\textit{the four Russians}~\\cite{MR0269546}, also called\n\\textit{Greasing} can be used as follows:\n\\begin{itemize}\n\\item A 64 bit machine word can be used to represent a row vector of dimension~64.\n\\item Matrix multiplication of a $m\\times k$ matrix  $A$ by a $k\\times n$\nmatrix $B$ can be done by first storing all $2^k$ $k$-dimensional linear\ncombinations of rows of $B$ in a table. Then the i-th row of the\nproduct is copied from the row of the table indexed by the i-th row of $A$.\n\\item By ordering indices of the table according to a binary Gray Code, each\nrow of the table can be deduced from the previous one, using only one row\naddition. This brings the bit operation count to build the table from $k2^kn$ to $2^kn$.\n\\item Choosing $k=\\log_2n$ in the above method implies $\\MatrixMul(n)=\\BigO{n^3\/\\log n}$\nover~${\\mathbb F}_2$.\n\\end{itemize}\n\\end{algorithm}\n\n\\index{bit-slicing}\n\\begin{definition}\\cite{BooBra09}\nBitslicing consists in representing an $n$-dimensional vector of $k$-bit sized\ncoefficients using $k$ binary vectors of dimension $n$. In particular, one can\nuse boolean word instruction to perform arithmetic on 64 dimensional vectors.\n\\begin{itemize}\n\\item Over ${\\mathbb F}_3$, the binary representation $0 \\equiv [0,0], 1\\equiv\n[1,0], -1 \\equiv [11]$ allows to add and subtract two elements in 6 boolean operations:\n$$\\begin{array}{ll}\n\\text{Add}([x_0,x_1],[y_0,y_1]) : & s \\leftarrow x_0\\oplus y_1 , t \\leftarrow x_1 \\oplus y_0\\\\\n      & \\text{Return} (s\\wedge t, (s\\oplus x_1) \\vee (t\\oplus y_1))\\\\\n\\text{Sub}([x_0,x_1],[y_0,y_1]) : & t\\leftarrow x_0\\oplus y_0 \\\\\n                                 & \\text{Return}(t\\vee (x_1\\oplus y_1), (t\\oplus\n                                 y_1)\\wedge(y_0\\oplus x_1))\n\\end{array}\n$$\n\\item Over ${\\mathbb F}_5$ (resp. ${\\mathbb F}_7$), a redundant representation $x=x_0+2x_1+4x_2 \\equiv\n[x_0,x_1,_2]$ allows to add two elements in 20 (resp. 17) boolean operations,\nnegate in 3 (resp. 6) boolean operations and double in 0 (resp. 5) boolean operations.\n\\end{itemize}\n\\end{definition}\n\n\\begin{table}[htbp]\n\\begin{tabular}{l|ccc}\n         & ${\\mathbb F}_3$ & ${\\mathbb F}_5$ & $F_7$\\\\\nAddition & 6 & 20 & 17\\\\\nNegation & 1 & 5 &  3\\\\\nDouble   &   & 5 &  0\\\\\n\\end{tabular}\n\\caption{boolean operation counts for basic arithmetic using bit slicing}\n\n\\end{table}\n\n\n\n\\index{bit-packing}\n\\begin{definition}\nBitpacking consists in representing a vector of field elements as an integer\nfitting in a single machine word using a $2^k$-adic\nrepresentation: $$(x_0,\\dots,x_{n-1})\\in {\\mathbb F}_q^n \\equiv X=x_0+2^kx_1+\\dots\n+(2^k)^{n-1}x_{n-1} \\in {\\mathbb Z}_{2^{64}}$$ \nElements of extension fields are viewed as\npolynomials and stored as the evaluation of this polynomial at the\ncharacteristic of the field. The latter evaluation is called {\\em\n  Kronecker substitution}\\index{Kronecker!substitution}.\n\\end{definition}\nWe first need a way to simultaneously reduce coefficients modulo the\ncharacteristic, see \\cite{MR2500374}.\n\\begin{algorithm}{[REDQ: Q-adic REDuction]}\\label{alg:REDQ}\\index{REDQ}\n\\begin{algorithmic}[1]\n\\REQUIRE Three integers $p$, $q$ and $\\tilde{r}  = \\sum_{i=0}^d \\widetilde{\\mu_i} q^i \\in {\\mathbb Z}$.\n\\ENSURE $\\rho \\in {\\mathbb Z}$, with $\\rho = \\sum_{i=0}^d \\mu_i q^i$\nwhere $\\mu_i = \\widetilde{\\mu_i} \\bmod p$.\n\\UNDERL{REDQ COMPRESSION\\index{REDQ!Compression}}\n\\STATE $s = \\left\\lfloor \\frac{\\tilde{r}}{p} \\right\\rfloor$;\n\\FOR{$i=0$ to $d$}\n\\STATE $u_i = \\left\\lfloor \\frac{\\tilde{r}}{q^i} \\right\\rfloor -  p \\left\\lfloor \\frac{s}{q^i} \\right\\rfloor$;\n\\ENDFOR\n\\UNDERL{REDQ CORRECTION\\index{REDQ!Correction}} \\hfill\\COMMENT{only when $p\\nmid q$, otherwise $\\mu_i=u_i$ is correct}\n\\STATE $\\mu_{d}=u_{d}$;\n\\FOR{$i=0$ to $d-1$}\n\\STATE $\\mu_i = u_i-qu_{i+1} \\bmod p$;\n\\ENDFOR\n\\\\[5pt]\\STATE Return $\\rho = \\sum_{i=0}^d \\mu_i q^i$;\n\\end{algorithmic}\n\\end{algorithm}\n\nOnce we can pack and simultaneously reduce coefficients of finite\nfield in a single machine word, the obtained parallelism can be used\nfor matrix multiplication. Depending on the respective sizes of the\nmatrix in the multiplication one can pack only the left operand or\nonly the right one or both \\cite{DumFouSal11}. We give here only a generic\nalgorithm for packed matrices, which use multiplication of a right\npacked matrix by a non packed left matrix.\n\\begin{algorithm}{[Right packed matrix\n  multiplication\\index{packed!matrix multiplication}]}\\label{alg:rightcomp}\n\\begin{algorithmic}[1]\n\\REQUIRE A prime $p$ and $A_c \\in {\\mathbb F}_p^{m\\times k}$ and $B_c \\in {\\mathbb F}_p^{k\\times n}$,\nstored with several field elements per machine word.\n\\ENSURE $C_c=A_c\\times B_c\\in {\\mathbb F}_p^{m\\times n}$\n\\STATE $A=\\operatorname{Uncompress}(A_c)$; \\hfill\\COMMENT{extract the\n  coefficients}\n\\STATE $C_c=A\\times B_c$; \\hfill\\COMMENT{Using e.g., algorithm\n  \\ref{alg:fgemm}}\n\\STATE Return $\\operatorname{REDQ}(C_c)$;\n\\end{algorithmic}\n\\end{algorithm}\n\nThen, over extensions, fast floating point operations can be used on\nthe Kronecker substitution\\index{Kronecker!substitution} of the\nelements. \nIndeed, it is very often desirable to use floating point\narithmetic, {\\em exactly}.\nFor instance floating point\nroutines can more easily use large hardware registers, they can more easily\noptimize the memory hierarchy usage \\cite{MR2501869,WhaPetDon01} and\nportable implementations are more widely available. \nWe present next the dot product and the matrix\nmultiplication is then straightforward\n\\cite{MR2035233,MR2500374,DumFouSal11}. \n\\begin{algorithm}{[Compressed Dot product over extension fields]\\label{alg:FGDP}}\n\\begin{algorithmic}[1]\n\\REQUIRE A field ${\\mathbb F}_{p^k}$ with elements represented as exponents of\na generator of the field;\n\\REQUIRE two vectors $v_1$ and $v_2$ of elements of ${\\mathbb F}_{p^k}$;\n\\REQUIRE a sufficiently large integer $q$.\n\\ENSURE $R \\in {\\mathbb F}_{p^k}$, with $R = v_1^T\\cdot v_2$.\n\\newline{\\COMMENT{Tabulated conversion: uses tables from exponent to floating point evaluation}}\n\\STATE Set $\\widetilde{v_1}$ and $\\widetilde{v_2}$ to the floating\npoint Kronecker substitution\\index{Kronecker!substitution} of the elements of $v_1$ and $v_2$.\n\\STATE Compute $\\tilde{r} = \\widetilde{v_1}^T \\cdot\\widetilde{v_2}$;\n\\hfill\\COMMENT{The floating point computation}\n\\STATE $r = REDQ\\_COMPRESSION(\\tilde{r},p,q)$;\\index{REDQ!Compression}\n\\hfill\\COMMENT{Computing a radix\n    decomposition}\n\\newline{\\COMMENT{Variant of REDQ\\_CORRECTION\\index{REDQ!Correction}:\n    $\\mu_i = \\widetilde{\\mu_i} \\bmod p$ for\n$\\tilde{r} = \\sum_{i=0}^{2k-2} \\widetilde{\\mu_i} q^i$}}\n\\STATE Set $L = representation( \\sum_{i=0}^{k-2} \\mu_i X^i )$;\n\\STATE Set $H = representation( X^{k-1} \\times \\sum_{i=k-1}^{2k-2} \\mu_i X^{i-k+1} )$;\n\\STATE Return $R = H + L \\in {\\mathbb F}_{p^k}$;\n\\hfill\\COMMENT{Reduction in the field}\n\\end{algorithmic}\n\\end{algorithm}\n\n\n\\subsubsection{Word size prime fields}\n\\label{sec:fflas}\n\\index{BLAS}\\index{FFLAS}\nOver word-size prime fields one can also use the reduction to floating\npoint routines of algorithm \\ref{alg:FGDP}.\nThe main point is to be able to perform efficiently the\nmatrix multiplication of blocks of the initial matrices without\nmodular reduction. Thus delaying the reduction as much as possible,\ndepending on the algorithm and internal representations, in\norder to amortize its cost. We present next such a delaying with the\nclassical matrix multiplication algorithm and a centered\nrepresentation \\cite{MR2738206}.\n\n\\begin{algorithm}{[{\\texttt fgemm}: Finite Field GEneric Matrix Multiplication]\\label{alg:fgemm}}\n\\begin{algorithmic}[1]\n\\REQUIRE An odd prime $p$ of size smaller than the floating point mantissa $\\beta$\nand $F_p$ elements stored by values between\n$\\frac{1-p}{2}$ and \n$\\frac{p-1}{2}$\n\\REQUIRE $A \\in {\\mathbb F}_p^{m\\times k}$ and $B \\in {\\mathbb F}_p^{k\\times n}$\n\\ENSURE $C=A\\times B\\in {\\mathbb F}_p^{m\\times n}$\n\\IF {$n(p-1)^2<2^{\\beta+1}$}\n\\STATE Convert $A$ and $B$ to floating point matrices $A_f$ and $B_f$;\n\\STATE Use floating point routines to compute $C_f=A_f \\times B_f$;\n\\STATE $C = C_f \\mod p$;\n\\ELSE\n\\STATE Cut $A$ and $B$ into smaller blocks;\n\\STATE Call the algorithm recursively for the block multiplications;\n\\STATE Perform the block additions modulo $p$;\n\\ENDIF\n\\end{algorithmic}\n\\end{algorithm}\n\n\\subsubsection{Large finite fields}\\label{sssec:large}\nIf the field is too large for the strategy \\ref{alg:fgemm} over\nmachine words, then two main approaches would have to be considered:\n\\begin{itemize}\n\\item Use extended arithmetic, either arbitrary of fixed\n  precision, if the characteristic is large, and a polynomial\n  representation for extension fields. \n  The difficulty here is to preserve an optimized memory\n  management and to have an almost linear time extended precision polynomial\n  arithmetic.\n\\item Use a residue number system and an evaluation\/interpolation\n  scheme: one can use algorithm \\ref{alg:fgemm} for each prime in the\n  RNS and each evaluation point. For ${\\mathbb F}_{p^k}$, the number of needed\n  primes is roughly $2\\log_{2^\\beta}(p)$ and the number of evaluations\n  points is $2k-1$.\n\\end{itemize}\n\n\\subsubsection{Large matrices: subcubic time complexity}\\label{ssec:wino}\n\nWith matrices of large dimension, sub-cubic time complexity algorithms, such as\nStrassen-Winograd's~\\cite{MR0297115} can be used to decrease the number of\noperations. Algorithm~\\ref{alg:strwin} describes how to compute one recursive\nlevel of the algorithm, using seven recursive calls and 15 block additions.\n\n\\begin{algorithm}{[Strassen-Winograd]}\n\\label{alg:strwin}\n$$A=\\begin{bmatrix}\n  A_{11} & A_{12}\\\\\n  A_{21} & A_{22}\\\\\n\\end{bmatrix};\n B=\n\\begin{bmatrix}\n  B_{11} & B_{12}\\\\\n  B_{21} & B_{22}\\\\\n\\end{bmatrix};\nC=\\begin{bmatrix}\n  C_{11} & C_{12}\\\\\n  C_{21} & C_{22}\\\\\n\\end{bmatrix};$$\n\n$$\n\\begin{array}{lllll}\n S_1  \\leftarrow A_{21} + A_{22};  &  T_1 \\leftarrow B_{12} - B_{11}; & P_1 \\leftarrow A_{11} \\times B_{11}; & P_2 \\leftarrow A_{12} \\times B_{21};\\\\\n S_2  \\leftarrow S_1 - A_{11};   &  T_2 \\leftarrow B_{22} - T_1;   & P_3  \\leftarrow S_4 \\times B_{22};  &P_4  \\leftarrow A_{22} \\times T_4;\\\\\n S_3  \\leftarrow A_{11} - A_{21};  &  T_3 \\leftarrow B_{22} - B_{12}; & P_5 \\leftarrow S_1 \\times T_1; & P_6\\leftarrow S_2 \\times T_2; \\\\\n S_4  \\leftarrow A_{12} - S_2;\t& T_4 \\leftarrow T_2 - B_{21};   &  P_7 \\leftarrow S_3 \\times T_3; \\\\\n\\end{array}$$\n$$\\begin{array}{llll}\nC_{11}  \\leftarrow P_1 + P_2; & U_2  \\leftarrow P_1 + P_6; & U_3  \\leftarrow U_2 +\nP_7; & U_4  \\leftarrow U_2 + P_5;\\\\\nC_{12} \\leftarrow U_4 + P_3; & C_{21} \\leftarrow U_3 - P_4;& C_{22} \\leftarrow U_3 + P_5;  \\\\\n\\end{array}$$\n\\end{algorithm}\n\n\nIn practice, one uses a threshold in the matrix dimension to switch to a\nbase case algorithm, that can be any of the one previously described.\nFollowing section~\\ref{sec:fflas}, one can again delay the\nmodular reductions, \nbut the intermediate computations of Strassen-Winograd's algorithm impose a\ntighter bound:\n\n\\begin{theorem}\\cite{MR2738206}\nLet $A \\in {\\mathbb Z}^{m \\times k}$, $B \\in {\\mathbb Z}^{k \\times n}$  $C \\in {\\mathbb Z}^{m\n  \\times n}$ and $\\beta \\in {\\mathbb Z}$ with\n\n$a_{i,j},b_{i,j},c_{i,j},\\beta \\in \\{0\\dots p-1\\}$. \nThen \nevery intermediate value $z$ involved in the computation of  $A \\times B +\n\\beta C$ with $l$ ($l\\geq 1$) recursive levels of algorithm~\\ref{alg:strwin} satisfy: \n\\[ \\left|z \\right| \\leq \\left(\\frac{1+3^l}{2}\\right)^2 \\left\\lfloor{\n    \\frac{k}{2^l}}\\right\\rfloor (p-1)^2 \\]\n\nMoreover, this bound is tight.\n\\end{theorem}\nFor instance, on a single Xeon 2.8GHz core with gcc-4.6.3, Strassen-Winograd's\nvariant implemented with LinBox-1.2.1 and GotoBLAS2-1.13 \ncan be 37\\% faster for the multiplication of $10\\,000\\times 10\\,000$\nmatrices over ${\\mathbb F}_{2^{19}-1}$, in less than $1'49\"$.\n\\subsection{Dense Gaussian elimination and echelon\n  forms}\\label{ssec:echelon}\nIn this section, we present algorithms computing the determinant and\ninverse of square matrices; the rank, rank profile, nullspace, and\nsystem solving for arbitrary shape and rank matrices. All these\nproblems are solved a la Gaussian elimination, but recursively in\norder to effectively incorporate matrix multiplication. The latter is\ndenoted generically \\texttt{gemm} and, depending on the underlying\nfield, can be implemented using any of the techniques of sections\n\\ref{sssec:tiny}, \\ref{sec:fflas} or \\ref{sssec:large}.\n\n A special care\nis given to the asymptotic time complexities: the exponent is reduced to that of matrix\nmultiplication using block recursive algorithms, and the constants are also\ncarefully compared. Meanwhile, this approach is also effective for\nimplementations: grouping arithmetic operations into matrix-matrix products allow\nto better optimize cache accesses.\n\n\\subsubsection{Building blocks}\nAlgorithms~\\ref{alg:trsmrec}, \\ref{alg:trmm}, \\ref{alg:trtri} and~\\ref{alg:trtrm} show how to reduce the\ncomputation of triangular matrix systems, triangular matrix multiplications, and\ntriangular matrix inversions to matrix-matrix multiplication. Note that they do\nnot require any temporary storage other than the input and output arguments.\n\n\\begin{algorithm}{[\\texttt{trsm}: Triangular System Solve with Matrix\n    right hand side]\\label{alg:trsmrec}}\n\\begin{algorithmic}[1]\n\\begin{minipage}{0.58\\columnwidth}\n\\REQUIRE{$A \\in {\\mathbb F}_q^{m \\times m}$ non-singular upper triangular, $B \\in {\\mathbb F}_q^{m \\times n}$}\n\\ENSURE{$X \\in {\\mathbb F}_q^{m \\times n}$ s.t. $AX= B$}\n\\IFTHEN{ m=1 }{{\\algorithmicreturn} $X= A_{1,1}^{-1} \\times B$}\n  \\STATE  $X_2=${\\tt trsm($A_3,B_2$)};\n  \\STATE $B_1= B_1 - A_2X_2$; \\COMMENT{using\n    \\hypnamref[alg:fgemm]{gemm}, e.g., via alg. \\ref{alg:fgemm}}\n  \\STATE $X_1=${\\tt trsm($A_1,B_1$)}; \n\\RETURN $X=\n\\begin{bmatrix}\nX_1 \\\\ X_2\\end{bmatrix};\n$\n\\end{minipage}\\hfill\n\\begin{minipage}{0.36\\columnwidth}\nUsing the conformal block \\mbox{decomposition}:\\\\ \n$\n  \\begin{bmatrix}\n  A_1&A_2\\\\&A_3\n  \\end{bmatrix}\n  \\begin{bmatrix}\n  X_1\\\\X_2\n  \\end{bmatrix}\n  =\n  \\begin{bmatrix}\n  B_1\\\\B_1\n  \\end{bmatrix}\n  $\n\\end{minipage}\n\\end{algorithmic}\n\\end{algorithm}\n\\index{TRSM}\n\n\\begin{algorithm}{[\\texttt{trmm}: Triangular Matrix Multiplication]\\label{alg:trmm}}\n\\begin{algorithmic}[1]\n\\begin{minipage}{0.58\\columnwidth}\n\\REQUIRE{ $A \\in {\\mathbb F}_q^{m \\times m}$ upper triangular, $B \\in {\\mathbb F}_q^{m \\times n}$}\n\\ENSURE{$C \\in {\\mathbb F}_q^{m \\times n}$ s.t. $AB= C$}\n\\IFTHEN{ m=1 }{{\\algorithmicreturn} $C= A_{1,1} \\times B$}\n  \\STATE $C_1=${\\tt trmm($A_1,B_1$)}; \n  \\STATE $C_1= C_1 + A_2B_2$; \\COMMENT{using \\hypnamref[alg:fgemm]{gemm}}\n  \\STATE $C_2=${\\tt trmm($A_3,B_2$)}; \n\\RETURN $C=\n\\begin{bmatrix}\nC_1 \\\\ C_2\\end{bmatrix};\n$\n\\end{minipage}\\hfill\n\\begin{minipage}{0.36\\columnwidth}\nUsing the conformal block \\mbox{decomposition}:\\\\ \n$\n  \\begin{bmatrix}\n  A_1&A_2\\\\&A_3\n  \\end{bmatrix}\n  \\begin{bmatrix}\n  B_1\\\\B_2\n  \\end{bmatrix}\n  =\n  \\begin{bmatrix}\n  C_1\\\\C_2\n  \\end{bmatrix}\n  $\n\\end{minipage}\n\\end{algorithmic}\n\\end{algorithm}\n\n\\begin{algorithm}{[\\texttt{trtri}: Triangular Matrix Inversion]\\label{alg:trtri}}\n\\begin{algorithmic}[1]\n\\begin{minipage}{0.58\\columnwidth}\n\\REQUIRE{ $A \\in {\\mathbb F}_q^{n \\times n}$ upper triangular and non-singular}\n\\ENSURE{$ C=A^{-1}$}\n\\IFTHEN{ m=1 }{{\\algorithmicreturn} $ C= A_{1,1}^{-1}$}\n  \\STATE $C_1=A_1^{-1}$; \\COMMENT{using \\hypnamref[alg:trtri]{trtri} recursively}\n  \\STATE $C_3=A_3^{-1}$; \\COMMENT{using \\hypnamref[alg:trtri]{trtri} recursively}\n  \\STATE $C_2=A_2C_3$; \\COMMENT{using  \\hypnamref[alg:trmm]{trmm} }\n  \\STATE $C_2=-C_1C_2$; \\COMMENT{using \\hypnamref[alg:trmm]{trmm} }\n\\RETURN $C=\n\\begin{bmatrix}\nC_1 & C_2\\\\&C_3\\end{bmatrix};\n$\n\\end{minipage}\\hfill\n\\begin{minipage}{0.36\\columnwidth}\nUsing the conformal block \\mbox{decomposition}:\\\\\n$\n  \\begin{bmatrix}\n  A_1&A_2\\\\&A_3\n  \\end{bmatrix},\n  \\begin{bmatrix}\n  C_1&C_2\\\\&C_3\n  \\end{bmatrix}\n  $\n\\end{minipage}\n\\end{algorithmic}\n\\end{algorithm}\n\n\n\\begin{algorithm}{[\\texttt{trtrm}: Upper-Lower Triangular Matrix Multiplication]\\label{alg:trtrm}}\n\\begin{algorithmic}[1]\n\\begin{minipage}{0.58\\columnwidth}\n\\REQUIRE{ $L \\in {\\mathbb F}_q^{n \\times n}$ lower triangular}\n\\REQUIRE{ $U \\in {\\mathbb F}_q^{n \\times n}$ upper triangular}\n\\ENSURE{$ A=UL$}\n\\IFTHEN{ m=1 }{{\\algorithmicreturn} $ A= U_{1,1}L_{1,1}$}\n  \\STATE $A_1=U_1L_1$; \\COMMENT{using \\hypnamref[alg:trtrm]{trtrm} recursively}\n  \\STATE $A_1=A_1+U_2L_2$; \\COMMENT{using \\hypnamref[alg:fgemm]{gemm}}\n  \\STATE $A_2=U_2L_3$; \\COMMENT{using \\hypnamref[alg:trmm]{trmm} }\n  \\STATE $A_3=U_3L_2$; \\COMMENT{using \\hypnamref[alg:trmm]{trmm} }\n  \\STATE $A_4=U_3L_3$; \\COMMENT{using \\hypnamref[alg:trtrm]{trtrm} recursively}\n\\RETURN $A=\n\\begin{bmatrix}\nA_1 & A_2\\\\A_3&A_4\\end{bmatrix};\n$\n\\end{minipage}\\hfill\n\\begin{minipage}{0.36\\columnwidth}\nUsing the conformal block \\mbox{decomposition}:\\\\ \n$\\begin{bmatrix}\n  L_1\\\\L_2&L_3\n  \\end{bmatrix},\\begin{bmatrix}\n  U_1&U_2\\\\&U_3\n  \\end{bmatrix},\\begin{bmatrix}\n  A_{1}&A_{2}\\\\A_{3}&A_{4}\n  \\end{bmatrix}$\n\\end{minipage}\n\\end{algorithmic}\n\\end{algorithm}\n\n\n\n\\subsubsection{PLE decomposition}\\index{PLE decomposition}\n\nDense Gaussian elimination over finite fields can be reduced to matrix\nmultiplication, using the usual techniques for the LU decomposition of\nnumerical linear algebra~\\cite{MR0331751}. \nHowever, in applications over a finite field, the input matrix often\nhas non-generic rank profile and special care needs to be taken about\nlinear dependencies and rank deficiencies.\nThe PLE decomposition is thus a generalization of the PLU\ndecomposition for matrices with any rank\\index{Rank} profile. \n\\begin{definition}\nA matrix is in row-echelon form if all its zero rows occupy the last row\npositions and the leading coefficient of any non-zero row except the first one is strictly to the right of the\nleading coefficient of the previous row.\nMoreover, it is said to be in reduced row-echelon form, if all coefficients\nabove a leading coefficient are zeros.\n\\end{definition}\n\n\\begin{definition}\nFor any matrix $A\\in F_q^{m\\times n}$ of rank\\index{Rank} $r$, there is a PLE decomposition $A=PLE$\nwhere $P$ is a permutation matrix, $L$ is a $m\\times r$ lower triangular matrix\nand $E$ is a $r\\times n$ matrix in row-echelon form, with unit leading coefficients.\n\\end{definition}\n\nAlgorithm~\\ref{alg:pledec} shows how to compute such a decomposition by a block\nrecursive algorithm, thus reducing the complexity to that of matrix\nmultiplication.\n\n\\begin{algorithm}{[PLE decomposition]\\label{alg:pledec}}\n\\begin{algorithmic}[1]\n\\REQUIRE{ $A \\in {\\mathbb F}_q^{m \\times n}$}\n\\ENSURE{$(P,L,E)$ a PLE decomposition of $A$}\n\\IF{$n=1$}\n  \\IFTHEN{$A=0_{m\\times 1}$}{{\\algorithmicreturn} $(I_m,I_0, A)$;}\n     \\STATE Let $j$ be the column index of the first non zero entry of $A$\n     and $P=T_{1,j}$ the transposition between indices $1$ and $j$;\n     \\RETURN $(P,PA, [1])$;\n\\ELSE \n\\begin{minipage}{\\columnwidth}\n\\begin{minipage}{0.58\\columnwidth}\n   \\STATE $(P_1,L_1,E_1) = \\texttt{PLE}(A_1)$; \\COMMENT{recursively}\n   \\STATE $A_2 = P_1A_2$;\n   \\STATE $A_3 = L_{1,1}^{-1}A_3$; \\COMMENT{using \\hypnamref[alg:trsmrec]{trsm}}\n   \\STATE $A_4 = A_4 - L_{1,2}A_3$; \\COMMENT{using \\hypnamref[alg:fgemm]{gemm}}\n   \\STATE $(P_2,L_2,E_2) = \\texttt{PLE}(A_4)$; \\COMMENT{recursively}\n\\end{minipage}\n\\begin{minipage}{0.36\\columnwidth}\nSplit $A$ columnwise in halves: \n$A=\\begin{bmatrix}A_1&A_2\\end{bmatrix}$\\\\\nSplit $A_2=\\begin{bmatrix}A_{3}\\\\A_4\\end{bmatrix}$,\n$L_1=\\begin{bmatrix}L_{1,1}\\\\L_{1,2}\\end{bmatrix}$ \nwhere $A_3$ and $L_{1,1}$ have $r_1$ rows.\n\\end{minipage}\n\\end{minipage}\n   \\RETURN ($\n   P_1   \\begin{bmatrix}   I_{r_1}\\\\&P_2  \\end{bmatrix}, \n   \\begin{bmatrix}\n   L_{1,1}\\\\P_2L_{1,2}&L_2\n   \\end{bmatrix}, \n   \\begin{bmatrix}\n   E_1&A_3\\\\&E_2\n   \\end{bmatrix}\n$);\n\\ENDIF\n\\end{algorithmic}\n\n\\end{algorithm}\n\n\\subsubsection{Echelon forms}\n\nThe row-echelon and reduced row-echelon forms can be obtained from the PLE\ndecomposition, using additional operations: \\texttt{trsm}, \\texttt{trtri}\nand \\texttt{trtrm}, as shown in algorithm~\\ref{alg:rowech} and~\\ref{alg:redrowech}.\n\n\\begin{algorithm}{[\\texttt{RowEchelon}\\label{alg:rowech}]}\n\\begin{algorithmic}[1]\n\\label{alg:echelon}\n\\REQUIRE{ $A \\in {\\mathbb F}_q^{m \\times n}$ }\n\\ENSURE{$(X,E)$ such that $XA=E$, $X$ is non-singular and  $E$ is in row-echelon\nform}\n\\STATE $(P,L,E) = \\texttt{PLE}(A)$; \\\\\n\\begin{minipage}{\\columnwidth}\n\\begin{minipage}{0.58\\columnwidth}\n\\STATE $X_1=L_1^{-1}$; \\COMMENT{using \\hypnamref[alg:trtri]{trtri}}\n\\STATE $X_2=-L_2X_1$; \\COMMENT{using \\hypnamref[alg:trmm]{trmm}}\n\\end{minipage}\\hfill\n\\begin{minipage}{0.36\\columnwidth}\nSplit $L = \n\\begin{bmatrix}\nL_1\\\\L_2\n\\end{bmatrix}$, $L_1: r\\times r$.\n\\end{minipage}\n\\end{minipage}\n\\RETURN  $\\left(X = \n\\begin{bmatrix}\nX_1\\\\X_2&I_{m-r}\\end{bmatrix} P^T, E\\right);$\n\\end{algorithmic}\n\\end{algorithm}\n\n\n\\begin{algorithm}{[\\texttt{ReducedRowEchelon}\\label{alg:redrowech}]}\n\\begin{algorithmic}[1]\n\\label{alg:redechelon}\n\\REQUIRE{ $A \\in {\\mathbb F}_q^{m \\times n}$ }\n\\ENSURE{$(Y,R)$ such that $YA=R$, $Y$ is non-singular and  $R$ is in reduced row-echelon\nform}\n\\STATE $(X,E) = \\texttt{RowEchelon}(A)$;\n\\STATE Let $Q$ be the permutation matrix that brings the leading row\ncoefficients of E to the diagonal;\n\\STATE Set $EQ=\n\\begin{bmatrix}\nU_1&U_2\\end{bmatrix}$; \\COMMENT{where $U_1$ is  $r\\times r$ upper triangular}\n\\STATE $Y_1=U_1^{-1}$; \\COMMENT{using \\hypnamref[alg:trtri]{trtri}}\n\\STATE $Y_1=Y_1X_1$; \\COMMENT{using  \\hypnamref[alg:trtrm]{trtrm}}\n\\STATE $R = \n\\begin{bmatrix}\nI_r&U_1^{-1}U_2\n\\end{bmatrix} Q^T$; \\COMMENT{using \\hypnamref[alg:trsmrec]{trsm}}\n\\RETURN $\\left(Y=\n\\begin{bmatrix}\nY_1\\\\U_2&I_{n-r}\n\\end{bmatrix} P^T, R\\right) ;$\n\\end{algorithmic}\n\\end{algorithm}\n\nFigure~\\ref{fig:reductions} shows the various steps between the classical\nGaussian elimination (LU decomposition), the computation of the echelon form and\nof the reduced echelon form, together with the various problems that each of\nthem solve. Table~\\ref{tab:gaussconstants} shows the leading constant $K_\\omega$ in the\nasymptotic time complexity of these algorithms, assuming that two $n\\times n $\nmatrices can be multiplied in $C_\\omega n^\\omega + o(n^\\omega)$.\n\\begin{figure}\n\\begin{center}\n\\includereduc\n\\end{center}\n\\caption{Reductions from PLE decomposition to Reduced echelon form}\\label{fig:reductions}\n\\end{figure}\n\\begin{table}\n    \\begin{tabular}{llll}\n    Algorithm  & Constant $K_\\omega$ & $K_3$ & $K_{\\log_27}$\\\\\n        \\hline\n    \\texttt{gemm}  & $C_\\omega$& 2 & 6 \\\\\n    \\texttt{trsm} & $\\frac{C_\\omega}{2^{\\omega-1}-2}$ & $1$  & $4$\\\\\n    \\texttt{trtri}& \n    $\\frac{C_\\omega}{(2^{\\omega-1}-2)(2^{\\omega-1}-1)}$&$\\frac{1}{3}\\approx 0.33$&$\\frac{8}{5}=1.6$\\\\\n      \\texttt{trtrm}, \\texttt{PLE}\n    &\n    $\\frac{C_\\omega}{2^{\\omega-1}-2}-\\frac{C_\\omega}{2^{\\omega}-2}$ &\n    $\\frac{2}{3}\\approx 0.66$ & $\\frac{14}{5} =2.8$ \\\\\n    \\texttt{Echelon}&  $\\frac{C_\\omega}{2^{\\omega-2}-1}-\\frac{3C_\\omega}{2^{\\omega}-2}$ & 1 &\n      $\\frac{22}{5}\\approx 4.4$\\\\\n    \\texttt{RedEchelon} &$\\frac{C_\\omega(2^{\\omega-1}+2)}{(2^{\\omega-1}-2)(2^{\\omega-1}-1)}$& 2 &\n    $\\frac{44}{5}= 8.8$\\\\\n \n\n\\end{tabular}\n\\caption{Complexity of elimination algorithms}\\label{tab:gaussconstants}\n\\end{table}\n\n\\begin{remark}\nNote that, if the rank $r$ is very small compared to the dimensions $m\\times n$ of\nthe matrix, a system $A x = b$ can be solved in time bounded by\n$\\BigO{(m+n)r^2}$ \\cite[Theorem~1]{MR1805128}. \n\\end{remark}\n\n\\subsection{Minimal and characteristic polynomial of a dense matrix}\n\\index{Characteristic polynomial}\n\\index{Minimal polynomial}\n\n\\begin{definition}\n\\begin{enumerate}\n\\item A {\\em Las-Vegas algorithm} is a randomized algorithm which is always\n  correct. Its expected running is time is always finite.\n\\item A {\\em Monte-Carlo algorithm} is a randomized algorithm which is correct with a\n   certain probability. Its running time is deterministic.\n\\end{enumerate}\n\\end{definition}\n\nThe computation of the minimal and characteristic polynomials is closely related\nto that of the Frobenius normal form.\n\\begin{definition} Any matrix $A \\in {\\mathbb F}_q^{n\\times n}$ is similar to a\n  unique block diagonal matrix $F=P^{-1} A P =\n  diag(C_{f_1},\\ldots,C_{f_t})$ where the blocks $C_{f_i}$ are\n  companion matrices of the polynomials $f_i$, which satisfy\n  $f_{i+1}|f_i$. The $f_i$ are the {\\em invariant factors} of\n  $A$ and $F$ is the {\\em Frobenius normal form} of $A$. \n\\end{definition}\nMost algorithms computing the minimal and characteristic polynomial or the\nFrobenius normal form rely on Krylov basis computations.\n\\begin{definition}\n\\begin{enumerate}\n\\item The Krylov matrix of order $d$ for a vector $v$ w.r.t a matrix $A$ is the\n  matrix \n$K_{A,v,d}=\n\\begin{bmatrix}\n v&Av&\\dots&A^{d-1}v\n \\end{bmatrix} \\in {\\mathbb F}_q^{n\\times d}$.\n\\item The minimal polynomial $P_\\text{min}^{A,v}$ of $A$ and $v$\n is the least degree monic polynomial $P$ such that $P(A)v=0$.\n\\end{enumerate}\n\\end{definition}\n\n\\begin{theorem}\n\\begin{enumerate}\n\\item $AK_{A,v,d}=K_{A,v,d}C_{P_\\text{min}^{A,v}}$, where\n  $d=\\deg(P_\\text{min}^{A,v})$.\n\\item For lineraly independent vectors $(v_1,\\dots,v_k)$, if   $K=\\begin{bmatrix}K_{A,v_1,d_1}&\\dots& K_{A,v_k,d_k}\\end{bmatrix}$ is non\n  singular. Then $AK=\nK\\begin{bmatrix}\nC_{P_\\text{min}^{A,v_1}}&B_{1,2}&\\dots&B_{1,k}\\\\\nB_{2,1}&C_{P_\\text{min}^{A,v_1}}&\\dots&B_{2,k}\\\\\n\\vdots&\\vdots&\\ddots&\\vdots\\\\\nB_{k,1}&B_{k,2}&&C_{P_\\text{min}^{A,v_k}}\n\\end{bmatrix}$, where the blocks $B_{i,j}$ are zero except on the last column.\n\\item For linearly independent vectors $(v_1,\\dots,v_k)$, let $(d_1,\\dots\n  d_k)$ be the lexicographically largest sequence of degrees such that\n  $K=\\begin{bmatrix}K_{A,v_1,d_1}&\\dots& K_{A,v_k,d_k}\\end{bmatrix}$ is non-singular. Then \n\\begin{equation}\nK^{-1}AK=\n\\begin{bmatrix}\nC_{P_\\text{min}^{A,v_1}}&B_{1,2}&\\dots&B_{1,k}\\\\\n&C_{P_\\text{min}^{A,v_1}}&\\dots&B_{2,k}\\\\\n&&\\ddots&\\vdots\\\\\n&&&C_{P_\\text{min}^{A,v_k}}\n\\end{bmatrix}=H\n\\label{eq:hessenberg}\n\\end{equation}\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{remark}\n\\begin{enumerate}\n\\item Some choice of vectors $v_1,\\dots, v_k$ lead to a matrix $H$ block\n  diagonal: this is the Frobenius normal form~\\cite{MR1657129}\n\\item The matrix obtained at equation (\\ref{eq:hessenberg}) is called a Hessenberg\n  form. It suffices to compute  the characteristic polynomial from its diagonal blocks.\n\\end{enumerate}\n\\end{remark}\n\n\\begin{theorem}\n The Frobenius normal form can be computed:\n\\begin{enumerate}\n\\item\\label{alg:charp:det} by a deterministic algorithm~\\cite{StoVil00} in $6n^3+\\BigO{n^2\\log^2n}$ field\n  operations,(only $(2+\\frac{2}{3})n^3+\\BigO{n^2}$ for the characteristic polynomial~\\cite{MR2280540})\n\\item by a deterministic algorithm~\\cite{MR1948725} in $\\BigO{n^\\omega\\log n \\log\\log\n  n}$, together with a transformation matrix, (only $\\BigO{n^\\omega\\log n}$ for the characteristic polynomial~\\cite{MR796306} )\n\\item by a Las-Vegas algorithm~\\cite{MR1834824} in $\\BigO{n^\\omega\\log n}$\n  field operations for any field, together with a transformation matrix\n\\item \\label{alg:charp:arith} by a Las-Vegas algorithm~\\cite{MR2402276} in $\\BigO{n^\\omega}$ for\n  $q>2n^2$, without transformation matrix.\n\\end{enumerate} \n\n The minimal\\index{Minimal polynomial} and characteristic\\index{Characteristic polynomial} polynomials,  obtained\n as the first invariant factor and the product of all invariant factors, can be\n computed with the same complexities.\n\\end{theorem}\n\n\\begin{remark}\nThese algorithms are all based Krylov bases. Algorithm~(\\ref{alg:charp:det}.)\niteratively compute the Krylov iterates one after the other. Their cubic time\ncomplexity with a small leading constant makes them comparable to Gaussian\nelimination.\nA fast exponentiation scheme by Keller-Gehrig~\\cite{MR796306} achieves a\nsub-cubic time complexity for the characteristic polynomial, off by a logarithmic factor of n from the matrix\nmultiplication. \nThe choice for the appropriate vectors that will generate the Frobenius normal\nform can be done either probabilistically (Las-Vegas) or deterministically with an\n$\\log\\log n$ factor.\nAlgorithm~(\\ref{alg:charp:arith}.) uses a\ndifferent iteration where the size of the Krylov increases according to an\narithmetic progression rather than geometric (as all others) and the transformation matrix is not\ncomputed. This allows it to match to the complexity of matrix\nmultiplication. This reduction is practical and is implemented as in {\\sc\n  LinBox}.\n\\end{remark}\n\n\\begin{remark}\\label{rem:extensions}\nThese probabilistic algorithms depend on the ability to sample uniformly from a\nlarge set of coefficients from the field.\nOver small fields, it is always possible to embed the problem into an\nextension field, in order to make the random sampling set\nsufficiently large. In the worst case, this could add a\n$\\BigO{\\log(n)}$ factor to the arithmetic cost and prevent most of the\nbit-packing techniques.\nInstead, the effort of~\\cite{MR1834824} is to handle cleanly the small finite\nfield case.\n\\end{remark}\n\n\n\n\\subsection{Blackbox iterative methods}\\label{ssec:blackbox}\nWe consider now the case where the input matrix is sparse, i.e., has\nmany zero elements, or has a structure which enables fast\nmatrix-vector products. Gaussian elimination would fill-in the sparse\nmatrix or modify the interesting structure. Therefore one can use\niterative methods instead which only use matrix-vector iterations\n({\\em blackbox methods} \\cite{MR1056629}). \nThere are two major differences with numerical iterative \nroutines: over finite fields there exists isotropic vectors and there is no\nnotion of convergence, hence the iteration must proceed until exactness of the result~\\cite{Lam96}.\nProbabilistic early termination can nonetheless be applied when the\ndegree of the minimal polynomial is smaller than the dimension of the\nmatrix~\\cite{MR1687279,DumVil02,Ebe03}. More generally the probabilistic nature\nof the algorithms presented in this section is subtle: e.g., the computation of\nthe minimal polynomial is Monte-Carlo, but that of system solving, using the\nminimal polynomial, is Las-Vegas (by checking consistency of the produced\nsolution with the system). Making some of the Monte-Carlo solutions Las-Vegas is\na key open-problem in this area.\n \n\\subsubsection{Minimal\\index{Minimal polynomial} Polynomial and the Wiedemann algorithm\n }\\index{Wiedemann}\nThe first iterative algorithm and its analysis are due to D. Wiedemann\n\\cite{MR831560}. The algorithm computes the minimal\\index{Minimal polynomial}\npolynomial in the Monte-Carlo probabilistic fashion.\n\\begin{definition} For a linearly recurring sequence $S=(S_i)$, its\n  minimal\\index{Minimal polynomial} polynomial is denoted by $\\Pi_S$.\n\\begin{itemize}\n\\item The minimal\\index{Minimal polynomial} polynomial of a matrix is denoted $\\Pi_A = \\Pi_{(A^i)}$.\n\\item For a matrix $A$ and a vector $b$, we note\n  $\\Pi_{A,b}=\\Pi_{(A^i\\cdot b)}$.\n\\item With another vector $u$, we note\n  $\\Pi_{u,A,b}=\\Pi_{(u^T\\cdot A^i \\cdot b)}$.\n\\end{itemize}\n\\end{definition}\n\n\\begin{algorithm}{[Wiedemann minimal\\index{Minimal polynomial} polynomial]\\label{alg:wiedemann}}\n\\begin{algorithmic}[1]\n\\REQUIRE $A \\in {\\mathbb F}_q^{n\\times n}$, $u, b \\in {\\mathbb F}_q^n$.\n\\ENSURE $\\Pi_{u,A,b}$.\n\\STATE Compute $S=(u^T A^i b)$ for $i \\leq 2n$;\n\\STATE Use the Berlekamp-Massey algorithm to compute the minimal\\index{Minimal polynomial}\npolynomial of the scalar sequence $S$;\n\\end{algorithmic}\n\\end{algorithm}\n\n\\begin{definition}\\seedefTotient\nWe extend Euler's totient function by \n$\\Phi_{q,k}(f)=\\prod (1-q^{-kd_i}),$ where ${d_i}$ are the degrees of the distinct monic\nirreducible factors of the polynomial~$f$.\n\\index{function!Euler's $\\Phi$}\n\\end{definition}\n\n\\begin{theorem} For vectors $u_1, \\ldots, u_j$ selected uniformly at random,\n  the probability that $\\operatorname{lcm}(\\Pi_{u_j,A,b})=\\Pi_{A,b}$ is at\n  least $\\Phi_{q,k}(\\Pi_{A,b})$.\n\\end{theorem}\n\n\\begin{theorem} For vectors $b_1, \\ldots, b_k$ selected\n  uniformly at random, the probability that\n  $\\operatorname{lcm}(\\Pi_{A,b_i})=\\Pi_A$ is at least\n  $\\Phi_{q,k}(\\Pi_A)$.\n\\end{theorem}\n\\subsubsection{Rank\\index{Rank}, Determinant and\n  Characteristic\\index{Characteristic polynomials} Polynomial}\nIt is possible to compute the rank\\index{Rank}, determinant, and\ncharacteristic\\index{Characteristic polynomials} polynomial of a\nmatrix from its minimal\\index{Minimal polynomial} polynomial. All\nthese reductions require to precondition the matrix so that the\nminimal\\index{Minimal polynomial} polynomial of the obtained matrix\nwill reveal the information sought, while keeping a low cost for the\nmatrix-vector product\n\\cite{MR1809985,MR1229306,DumVil02,Tur02,MR1849765,Vil03,MR1878939}. \n\\begin{theorem}\\cite{MR1809985}\\label{thm:EK} \n  Let S be a finite subset of a field ${\\mathbb F}$ that does not include~$0$. \n  Let $A \\in {\\mathbb F}^{m \\times n}$ having rank\\index{Rank} $r$. \n  Let $D_1 \\in S^{n \\times n}$ and $D_2 \\in S^{m \\times m}$ be two\n  random diagonal matrices then \n  $degree(minpoly( D_1 \\times A^t \\times D_2 \\times A \\times D_1 ))=r$, \n  with probability at least $1 - \\frac{11.n^2-n}{2|S|}$.\n\\end{theorem}\n\n\\begin{theorem}\\cite{Tur02}\\label{thm:det} \n  Let S be a finite subset of a field ${\\mathbb F}$ that does not include~$0$. \n  Let $U\\in S^{n \\times n}$ be a unit upper bi-diagonal matrix where\n  the second diagonal elements $u_1,\\ldots,u_{n-1}$ are randomly\n  selected in $S$.\n  For $A \\in {\\mathbb F}^{n \\times n}$, the term of degree $0$ of the\n  minimal\\index{Minimal polynomial} polynomial of $UA$ is the\n  determinant of $A$ with probability at least $1-\\frac{n^2-n}{2|S|}$.\n\\end{theorem}\n\\begin{remark}\nIf $A$ is known to be non-singular the algorithm can be repeated with\ndifferent matrices $U$ until the obtained minimal polynomial is of\ndegree $n$. Then it is the characteristic polynomial of $UA$ and the\ndeterminant is certified. \nAlternatively if the matrix is singular then $X$ divides the minimal\npolynomial. As Wiedemann's algorithm always returns a factor of the\ntrue minimal polynomial, and $U$  is invertible,\nthe algorithm can be repeated on $UA$ until either the obtained\npolynomial is of degree $n$ or it is divisible by $X$.\nOverall\nthe determinant has a Las-Vegas blackbox solution.\n\\end{remark}\n\n\\begin{theorem}\\cite{MR1849765,Vil03}\\label{thm:rankupdate}\n  Let S be a finite subset of a field ${\\mathbb F}$\n  that does not include~$0$ and $A\\in {\\mathbb F}^{n\\times n}$ with $s_1,\n  \\ldots, s_t$ as invariant factors. Let $U \\in S^{n\\times k}$ and $V\n  \\in S^{k\\times n}$ be randomly chosen rank\\index{Rank} $k$ matrices\n  in ${\\mathbb F}$. Then $\\operatorname{gcd}(\\Pi_A, \\Pi_{A+UV})=s_{k+1}$ with\n  probability at least $1 -\\frac{nk + n + 1}{|S|}$.\n\\end{theorem}\n\\begin{remark}\nUsing the divisibility of the invariant factors and the fact that\ntheir product is of degree $n$, one can see that the number of degree\nchanges between successive invariant factors is of order\n$\\BigO{\\sqrt{n}}$~\\cite{MR1849765}. Thus by a binary search over\nsuccessive applications of theorem \\ref{thm:rankupdate} one can\nrecover all of the invariant factors and thus the\ncharacteristic\\index{Characteristic polynomials} polynomial of the\nmatrix in a Monte-Carlo fashion. \n\\end{remark}\n\\subsubsection{System solving and the Lanczos algorithm}\\index{Lanczos}\nFor the solution of a linear system $Ax=b$, one could compute the\nminimal\\index{Minimal polynomial} polynomial $\\Pi_{A,b}$ and then derive a solution of the\nsystem as a linear combination of the $A^ib$. \nThe following Lanczos approach is more efficient for system solving as\nit avoids recomputing (or storing) the latter vectors\n\\cite{MR1809985,MR1805174}.\n\\begin{algorithm}{[Lanczos system solving]\\label{alg:lanczos}}\n\\begin{algorithmic}[1]\n\\REQUIRE $A \\in {\\mathbb F}^{m\\times n}$, $b \\in {\\mathbb F}^m$.\n\\ENSURE $x\\in {\\mathbb F}^n$ such that $Ax=b$ or {\\em failure}.\n\\STATE\\label{lin:datdad} Let $\\tilde{A}=D_1 A^T D_2 A D_1$ and $\\tilde{b} = D_1 A^T D_2\nb+\\tilde{A} v$ with $D_1$ and $D_2$ random diagonal matrices and $v$ a\nrandom vector;\n\\STATE $w_0=\\tilde{b}$; $v_1=\\tilde{A}w_0$; $t_0=v_1^T w_0$;\n$\\gamma=\\tilde{b}^tw_0 t_0^{-1}$; $x_0 = \\gamma w_0$;\n\\REPEAT\n\\STATE $\\alpha=v^T_{i+1}v_{i+1} t_i^{-1}$; $\\beta=v^T_{i+1}v_{i}\nt_{i-1}^{-1}$; $w_{i+1} = v_{i+1} -\\alpha w_i -\\beta w_{i-1}$;\n\\STATE $v_{i+2}= \\tilde{A} w_{i+1}$; $t_{i+1}=w_{i+1}^T v_{i+2}$;\n\\STATE $\\gamma=\\tilde{b}^tw_{i+1}t_{i+1}^{-1}$; $x_{i+1}=x_i + \\gamma w_{i+1}$;\n\\UNTIL{$w_{i+1}=0$ or $t_{i+1}=0$;}\n\\STATE Return $x=D_1(x_{i+1}-v)$;\n\\end{algorithmic}\n\\end{algorithm}\n\nThe probability of success of algorithm \\ref{alg:lanczos} follows also\ntheorem \\ref{thm:EK}. \n\\begin{remark}\nOver small fields, if the rank of the matrix is known, the\ndiagonal matrices of line \\ref{lin:datdad} can be replaced by sparse\npreconditioners with $\\BigO{n\\log(n)}$ non-zero coefficients to avoid the\nneed of field extensions\\cite[corollary 7.3]{MR1878939}. \n\\end{remark}\n\\begin{remark}\nIf the system with $A$ and $b$ is known to have a solution then the\nalgorithm can be turned Las-Vegas by checking that the output $x$\nindeed satisfies $Ax=b$. In general, we do not know if this algorithm\nreturns failure because of bad random choices or because the system is\ninconsistent. However, Giesbrecht, Lobo and Saunders have shown that\nwhen the system is inconsistent, it is possible to produce a\ncertificate vector $u$ such that $u^T A=0$ together with \n$u^T b \\neq 0$ within the same complexity \n\\cite[Theorem 2.4]{MR1805174}. Overall, system solving can be\nperformed by blackbox algorithms in a Las-Vegas fashion.\n\\end{remark}\n\\subsection{Sparse and structured methods}\\index{Sparse matrix}\n\\index{Structured matrix}\\index{Gaussian elimination}\\index{Fill-in}\nAnother approach to sparse linear system is to use Gaussian\nelimination with pivoting, taking into account the zero coefficients. \nThis algorithm modifies the structure of the matrix and might suffer\nfrom fill-in. Consequently the available memory is usually the\nbottleneck. From a triangularization one can naturally derive the\nrank\\index{Rank}, determinant, system solving and nullspace. \nComparisons with the blackbox approaches above can be found e.g., in\n\\cite{DumVil02}.\n\\subsubsection{Reordering}\\index{Reordering}\n\\begin{algorithm}{[Gaussian elimination with linear pivoting]\\label{alg:reord}}\n\\begin{algorithmic}[1]\n\\REQUIRE a matrix $A \\in {\\mathbb F}^{m \\times n}$;\n\\ENSURE An upper triangular matrix $U$ such that there exists a\nunitary lower-triangular matrix $L$ and permutations matrices $P$ and\n$Q$ over ${\\mathbb F}$, with $A=P\\cdot L \\cdot U \\cdot Q$;\n\\FORALL{elimination steps}\n\\STATE Choose as pivot row the sparsest remaining row;\n\\STATE In this row choose the non zero pivot with lowest number of non\nzero elements in its column;\n\\STATE Eliminate using this pivot;\n\\ENDFOR\n\\end{algorithmic}\n\\end{algorithm}\n\n\\begin{remark} \nYannakakis showed that finding the minimal fill-in (or\nequivalently the best pivots) during\nGaussian elimination is an NP-complete task\n\\cite{MR604513}. \nIn numerical algorithms, heuristics have been developed and  \ncomprise minimal degree ordering, cost functions or nested dissection\n(see\ne.g.,\n\\cite{MR1135327,MR2124398,MR1642639}).\nThese heuristics for reducing fill-in in the numerical setting, often\nassume symmetric and invertible matrices, and do not take into account\nthat new zeros may be produced by elimination operations ($a_{ij} =\na_{ij} + \\delta_i * a_{kj}$), as is the case with matrices over finite\nfields. \\cite{DumVil02} thus proposed the heuristic \\ref{alg:reord} to\ntake those new zeros into account, using a local optimization of a\ncost function at each elimination step.\n\\end{remark}\n\n\\subsubsection{Structured matrices and displacement\n  rank\\index{Displacement rank}}\nOriginating from the seminal paper \\cite{MR537629} most of the\nalgorithms dealing with structured matrices use the displacement rank\\index{Displacement rank}\napproach \\cite{MR1843842}.\n\\begin{definition}\nFor $A\\in {\\mathbb F}^{m\\times m}$ and $B\\in{\\mathbb F}^{n\\times n}$, the {\\em Sylvester\n  (resp. Stein)\nlinear displacement operator $\\bigtriangledown_{A,B}$\n(resp. $\\bigtriangleup_{A,B}$)} satisfy for $M\\in{\\mathbb F}^{m\\times n}$:\n\\begin{align*}\n\\bigtriangledown_{A,B}(M) = AM - MB\\\\\n\\bigtriangleup_{A,B}(M) = M - AMB\n\\end{align*}\nA pair of matrices $(Y,Z)\\in{\\mathbb F}^{m\\times \\alpha} \\times {\\mathbb F}^{n\\times\n  \\alpha}$ is a {\\em $A,B$-Sylvester-generator\\index{Sylvester\n    generator} of length $\\alpha$} (resp. Stein\\index{Stein\n  generator}) for $M$ if $\\bigtriangledown_{A,B}(M) = Y Z^T$\n(resp. $\\bigtriangleup_{A,B}(M) = Y Z^T$). \n\\end{definition}\nThe main idea behind algorithms for structured matrices is to use such\ngenerators as a compact data structure, in cases where the\ndisplacement has low rank\\index{Rank}. \n\nUsual choices of matrices $A$ and $B$ are diagonal matrices and\ncyclic down shift matrices:\n\\begin{definition}\n$\\mathbb{D}_x, x\\in{\\mathbb F}^n$ is the diagonal matrix whose $(i,i)$\n  entry is $x_i$.\\\\\n$\\mathbb{Z}_{n,\\varphi},\\varphi\\in{\\mathbb F}$ is the $n\\times n$ unit\n  circulant matrix having $\\varphi$ at position $(1,n)$, ones in the\n  subdiagonal $(i+1,i)$ and zeros elsewhere.\n\\end{definition}\n\n\\begin{table}[htbp]\n\\begin{tabular}{|c|c|c|c|c|}\n\\multicolumn{2}{|c|}{operator matrices}& class of structured & rank\\index{Rank} of\n& number of flops \\\\\nA & B & matrices $M$ & $\\bigtriangledown_{A,B}(M)$ & for computing $M\n\\cdot v$\\\\\n\\hline\n$\\mathbb{Z}_{n,1}$& $\\mathbb{Z}_{n,0}$& Toeplitz\\index{Toeplitz matrix} and its inverse& $\\leq\n2$& $\\BigO{(m+n)\\log(m+n)}$\\\\\n$\\mathbb{Z}_{n,1}$& $\\mathbb{Z}_{n,0}^T$& Hankel\\index{Hankel matrix} and its inverse& $\\leq 2$& $\\BigO{(m+n)\\log(m+n)}$\\\\\n$\\mathbb{Z}_{n,0}+\\mathbb{Z}_{n,0}^T$&$\\mathbb{Z}_{n,0}+\\mathbb{Z}_{n,0}^T$ & Toeplitz\\index{Toeplitz matrix}+Hankel\\index{Hankel matrix}& $\\leq 4$& $\\BigO{(m+n)\\log(m+n)}$\\\\\n$\\mathbb{D}_{x}$& $\\mathbb{Z}_{n,0}$& Vandermonde\\index{Vandermonde matrix}& $\\leq 1$& $\\BigO{(m+n)\\log^2(m+n)}$\\\\\n$\\mathbb{Z}_{n,0}$& $\\mathbb{D}_{x}$ & inverse of Vandermonde\\index{Vandermonde matrix}& $\\leq 1$& $\\BigO{(m+n)\\log^2(m+n)}$\\\\\n$\\mathbb{Z}_{n,0}^T$& $\\mathbb{D}_{x}$ & transposed of Vandermonde\\index{Vandermonde matrix}& $\\leq 1$& $\\BigO{(m+n)\\log^2(m+n)}$\\\\\n$\\mathbb{D}_{x}$& $\\mathbb{D}_{y}$ & Cauchy\\index{Cauchy matrix} and its inverse& $\\leq 1$& $\\BigO{(m+n)\\log^2(m+n)}$\\\\\n\\end{tabular}\n\\caption{Complexity of the matrix-vector product for some structured matrices}\\label{tab:struct}\n\\end{table}\nAs computing matrix vector products with such structured matrices have\nclose algorithmic correlation to computations with polynomials and\nrational functions, these matrices can be multiplied by vectors fast,\nin nearly linear time as shown on table \\ref{tab:struct}. \nTherefore the algorithms of section \\ref{ssec:blackbox} can naturally\nbe applied to structured matrices, to yield almost $\\BigO{n^2}$ time\nlinear algebra.\n\nNow, if the displacement rank\\index{Displacement rank} is small there exists \nalgorithms quasi linear in $n$, the dimension of the matrices, which\nover finite fields are essentially variations or extensions of the\nMorf\/Bitmead-Anderson divide-and-conquer \\cite{Mor80,MR591427} or\nCardinal's \\cite{MR1696212} approaches.\nThe method is based on dividing the original problem repeatedly into\ntwo subproblems with one leading principal submatrix and the related\nSchur complement. This leads to $\\BigO{\\alpha^2n^{1+o(1)}}$ system solvers,\nwhich complexity bound have recently been reduced to\n$\\BigO{\\alpha^{\\omega-1}n^{1+o(1)}}$ \\cite{MR2463004,JeaMou10}.\nWe few exceptions, all algorithms thus need matrices in generic rank\nprofile. Over finite fields this can be achieved using Kaltofen and\nSaunders unit upper triangular Toeplitz preconditioners\n\\cite{MR1229306}  and by controlling the displacement rank growth and\nnon-singularity issues \\cite{Kal94}. \n\n\\subsection{Hybrid methods}\n\\subsubsection{Hybrid sparse-dense methods}\nOverall, as long as the matrix fits into memory, Gaussian elimination\nmethods are usually faster than iterative methods, over finite fields\n\\cite{DumVil02}.\nThere are then heuristics trying to take the best of both\nstrategies. Among those we briefly mention the most widely used:\n\\begin{itemize}\n\\item Perform the Gaussian elimination with reordering\n  \\ref{alg:reord} until the matrix is almost filled-up. If the\n  remaining non-eliminated part would fit as a dense matrix, switch to\n  the dense methods of section \\ref{ssec:echelon}.\n\\item Maintain two sets of rows (or columns), sparse and\n  dense. Favor elimination on the sparse set. This is\n  particularly adapted to index calculus \\cite{LamOdl91}. \n\\item Perform a preliminary reordering in order to cut the matrix into\n  four quadrants, the upper left one being triangular. This, together\n  with the above strategies has proven effective on matrices which are\n  already quasi-triangular, e.g., Gr{\\\"o}bner bases\n  computations in finite fields \\cite{FauLac10}.\n\\item If the rank\\index{Rank} is very small compared to the dimension of the\n  matrix, one can use left and right highly rectangular projections to\n  manipulate smaller structures \\cite{MR2402272}.\n\\item The arithmetic cost and thus timing predictions are easier on\n  iterative methods than on elimination methods. On the other hand the\n  number of non-zero elements at a given point of the elimination is\n  usually increasing during an elimination, thus providing a lower\n  bound on the remaining time to triangularize. Thus a heuristic is to\n  perform one matrix-vector product with the original matrix and then\n  eliminate using Gaussian elimination. If at one point the lower\n  bound for elimination time surpasses to predicted iterative one or\n  if the the algorithm runs out of memory, stop the elimination and\n  switch to the iterative methods \\cite{DurSauWan03}.\n\\end{itemize}\n\n\\subsubsection{Block-iterative methods}\n\nIterative methods based on one-dimensional projections, such as Wiedmann and\nLanczos algorithm can be generalized with block projections.\nVia efficient preconditioning \\cite{MR1878939} these extensions to the\nscalar iterative methods can present enhanced properties:\n\\begin{itemize}\n\\item Usage of dense sub-blocks, after multiplications of blocks of\n  vectors with the sparse matrix or the blackboxes, allows for a\n  better locality and optimization of memory accesses, via the\n  application of the methods of section~\\ref{ssec:blas}.\n\\item Applying the matrix to several vectors simultaneously introduces more\n  parallelism \\cite{MR1236735,MR1192970,MR1687279}.\n\\item Also, their probability of success augments with the size of the\n  considered blocks, especially over small fields \\cite{MR1270621,Vil97}.\n\\end{itemize}\n\n\\begin{definition}\nLet $X\\in {\\mathbb F}_q^{k\\times n}$, $Y\\in {\\mathbb F}_q^{n\\times k}$ and $H_i =XA^iY$ for\n$i=0\\dots n\/k$.\nThe matrix minimal\\index{Minimal polynomial} polynomial of the sequence $H_i$ is the matrix polynomial\n$F_{X,A,Y} \\in {\\mathbb F}_q[X]^{k\\times k}$ of least degree, with its leading degree\nmatrix column-reduced, that annihilates the sequence $(H_i)$.\n\\end{definition}\n\n\\begin{theorem}\nThe degree $d$ matrix minimal\\index{Minimal polynomial} polynomial of a block sequence $(H_i) \\in (F_q^{k\\times\nk})^{\\mathbb Z}$ can be computed in $\\BigO{k^3d^2}$ using block versions of Hermite-Pade\napproximation and extended Euclidean algorithm~\\cite{MR1779720} or Berlkamp-Massey\nalgorithm~\\cite{MR1192970,MR1270621,Vil97}. Further improvement\nby~\\cite{MR1779720,MR2049765,MR2035204,MR2120701}  bring this complexity down\nto $\\BigO{k^\\omega d}^{1+o(1)}$, using a matrix extended Euclidean algorithm.\n\\end{theorem}\n\n\\begin{algorithm}{[Nullspace vector]\\label{alg:nullspace}}\n\\begin{algorithmic}\n\\REQUIRE{ $A\\in {\\mathbb F}_q^{n\\times n}$}\n\\ENSURE {$\\omega \\in {\\mathbb F}_q^n$ a vector in the nullspace of $A$}\n\\STATE Pick $X\\in {\\mathbb F}_q^{k\\times n},Y\\in {\\mathbb F}_q^{n\\times k}$ uniformly at random;\n\\STATE Compute the sequence $H_i=XA^iY$;\n\\STATE Compute $F_{X,A,Y}$ the matrix minimal\\index{Minimal polynomial} polynomial;\n\\STATE Let $f=f_rx^r+\\dots+f_dx^d$ be a column of $F_{X,A,Y}$;\n\\STATE Return $\\omega=Yf_r +AYf_{r+1}+ \\dots +A^{d-r}Yf_{d}$;\n\\end{algorithmic}\n\\end{algorithm}\n\n\\begin{remark}\nThese block-Krylov techniques are used to achieve the best known time\ncomplexities for several computations with black-box matrices over a finite\nfield or the ring of integers: computing the\ndeterminant, the characteristic\\index{Characteristic polynomials} polynomial~\\cite{MR2120701} and the  solution of a linear system\nof equations~\\cite{MR2396196}.\n\\end{remark}\n\n\\AllRefCited{MR2124398,MR0269546,MR1779720,MR591427,BooBra09,MR2463004,MR0331751,MR1696212,MR1878939,MR1236735,MR1192970,MR1449760,MR1056627,MR2500374,DumFouSal11,DumGauGieGioHovKalSauTurVil02,MR2035233,MR2738206,MR2280540,DumVil02,DurSauWan03,MR1834824,Ebe03,MR2396196,MR1809985,FauLac10,MR1657129,MR1805174,MR2035204,MR2501869,MR1642639,JeaMou10,MR537629,MR1687279,Kal94,MR1270621,MR1229306,MR1056629,MR2120701,MR796306,LamOdl91,Lam96,MR2402272,Mor80,MR1805128,MR1843842,MR2402276,MR1948725,StoVil00,Sto10,MR2049765,Tur02,Vas11,Vil97,MR1849765,Vil03,WhaPetDon01,MR831560,MR0297115,MR604513,MR1135327}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\n\n\n\nNext generation cellular networks \\cite{QWu2017}\\cite{MAgiwal2016} are envisioned to have some unique beneficial characteristics, among which ultra-dense deployment \\cite{XGe2016} of network nodes is one of the most crucial paradigms. Ideally, ultra-dense deployment of nodes enhances the coverage and capacity of the system \\cite{LZhou2014}. However, there are still some problems with this concept if the deployment and system parameters are not properly optimized. Since the number of transmitters in such networks is very large, this causes huge interference. Consequently, energy-efficient transmission is one of the important research endeavors in this arena. In order to obtain fruitful outcome in this context, designing an effective and efficient radio access technology \\cite{TEdler2014} is one of the possible solutions. Non-orthogonal multiple access (NOMA) is considered as the most promising radio access technique for next generation wireless systems \\cite{ZDing2014}\\cite{KHIGUCHI2015}\\cite{VWong2017}. Through experimentation and theoretical analysis \\cite{Higuchi2013}\\cite{YEndo2012}\\cite{JUmehara2012}\\cite{NOtao2012}, it is proved that the NOMA technique is able to provide enhanced performance in different sectors of wireless communication systems comparing with other orthogonal multiple access (OMA) techniques. Conceptually, power-domain NOMA \\cite{SMRIslam2017} allows multiple users to occupy the same resource. This leads to additional interference for NOMA-equipped networks and their neighboring networks. Consequently, existing resource management techniques \\cite{DFeng2013} in conventional networks, especially the energy-efficient ones, need to be revisited due to the incorporation of additional interference this new technology brings. \n\nAlthough the NOMA technique allows multiple users to be superimposed on the same frequency channel, due to the usage of practical, finite-length and possibly non-Gaussian codes, it is not an optimal design to assign large number users to the same channel. Consequently, dedicated spectrum of a system needs to be subdivided into multiple subchannels in order to support increased number of users~\\cite{LDai2015}. At the same time, how to allocate these subchannels among the users in a multiplexed manner given the maximum allowable number of users that can utilize a subchannel simultaneously, is an important problem. While targeting on the maximization of the overall throughput, some research has been conducted on the downlink and uplink subchannel and power allocation in such NOMA systems. Based on some assumption of having constant power on the subchannels, existing works typically provide some heuristic solutions for the subchannel-user mapping problem. Once the subchannel-user mapping is done, in order to enhance the performance of the system further, different existing works have provided different schemes for the power allocation. For example, in~\\cite{YSaito2013, ABenjebbour2013}, the authors use the fractional transmit power allocation technique among users and apply the equal power allocation technique across subchannels.~\\cite{MRHojeij2015} uses the water filling-based approach for the power allocation, and in \\cite{PParida2014}, the authors use difference of convex (DC) programming-based approach~\\cite{NVucic2010} for the power allocation at both the user and subchannel levels. For uplink systems, there are some works as well, such as~\\cite{KKumaran2003, MAlImari2015, MMollanoori2014}. In~\\cite{MAlImari2015}, the authors used the iterative water filling idea~\\cite{WeiYu2004} for both the subchannel-user mapping problem and their granular power allocation. On the other hand, in~\\cite{MMollanoori2014}, the authors assumed that the resources are time slots instead of frequency channels. Although the work in~\\cite{MAlImari2015} can ensure that users can be assigned to multiple subchannels, each user is assigned to at most one resource block via the scheme in~\\cite{MMollanoori2014}. While the base station (BS) of all these works is assumed to be equipped with the half-duplex functionality, the authors in \\cite{YSun2017} have proposed a joint subchannel and power allocation scheme for such a system based on the assumption that the BS is equipped with the full-duplex functionality to serve uplink and downlink users simultaneously. \n\n\\begin{comment}\nAlthough NOMA technique allows multiple users to be superimposed on the same frequency channel, due to the error propagation in the successive interference cancellation (SIC) mechanism~\\cite{NIMiridakis2013}, it is not an optimal assign to assign large number users on the same channel. Consequently, dedicated spectrum of a system needs to be subdivided into multiple subchannels in order to support increased number of users. At the same time, how to allocate these subchannels among users in a multiplexed manner given the allowable maximum number of users that can use a subchannel simultaneously, is an important problem. While targeting on maximizing overall throughput of the system, some research has been conducted on the downlink and uplink subchannel and power allocation in such NOMA systems. Based on some assumption of having constant power on the subchannels, typically, existing works provide some heuristics for subchannel-user mapping task. Once the subchannel-user mapping is done, different existing works provide different schemes for power allocation. For example, for downlink systems, in \\cite{YSaito2013, ABenjebbour2013}, the authors use fractional transmit power allocation technique among users and equal power allocation across subchannels.~\\cite{MRHojeij2015} uses water filling based approach for power allocation, and in \\cite{PParida2014}, the authors use difference of convex (DC) programming-based approach for the power allocation in both user and subchannel levels. For uplink systems, there are some works as well, such as~\\cite{KKumaran2003, MMollanoori2014}. \n\nIn~\\cite{KKumaran2003}, the authors used iterative water filling idea~\\cite{WeiYu2004} for both subchannel and user mapping and their granular power allocation. On the other hand, in~\\cite{MMollanoori2014}, the authors assumed that the resource is time slot instead of frequency channel. Although the work in~\\cite{KKumaran2003} can ensure that users can obtain multiple subchannels, each user obtains at most one time slot via the scheme in~\\cite{MMollanoori2014}. \n\\end{comment}\n\nWith the increasing desire to have green communications in the recent years, reducing energy consumption has become the prime concern for researchers, and the fifth generation (5G) systems have also targeted energy efficiency as one of the major milestones to achieve~\\cite{AAbrol2016}. Although the minimization of used power reported in \\cite{LeiLei2016} could be one of the objectives of energy-efficient transmissions, the resultant solution of such a type of formulation is not spectrum-efficient. In order to achieve spectrum efficiency with the minimal power, the ideal objective of an energy-efficient transmission is to maximize the achievable bits of a channel under the unit Joule of power consumption~\\cite{OJumira2012}. For a NOMA-equipped single cell, which has two users, an energy-efficient power allocation scheme is studied in~\\cite{SHan2014}. In this work, the authors found the relationship between energy efficiency and spectrum efficiency under the total system power constraint. At the same time, energy efficiency is also studied for NOMA-equipped MIMO systems in~\\cite{QSun2015}. The closest to ours and the most recent energy-efficient resource allocation scheme appeared in~\\cite{FFang2016}. The system model that this paper considered is similar to ours, and our target is to enhance the performance of the proposed scheme in this paper further. This work has proposed an energy-efficient downlink subchannel and power allocation scheme for a cellular network under total power constraint at the BS with a number of drawbacks. For example, through their scheme, each user cannot be assigned to multiple subchannels, the concept of which fails to exploit the multi-user diversity of wireless systems~\\cite{DavidTse2005}. Moreover, each subchannel can be assigned to at most two users, which also conflicts with the NOMA concept. Furthermore, they have used the DC programming-based approach in order to allocate power across all the subchannels and users of the system. Actually, through this power allocation, the entire power at the BS is used up, and hence the system fails to reach the optimal energy-efficient state. On the other hand, they also have utilized the DC programming-based approach in order to calculate the optimal instantaneous rate of a subchannel, which is an essential intermediate step of their subchannel-user mapping algorithm. However, the DC programming-based approach is appropriate only in this case because of provisioning at most two users to a subchannel. If the allowable number of users per subchannel is increased, the DC programming-based approach is not appropriate to solve this problem any more as the problem can no longer be expressed as a difference of convex functions under this realistic consideration.\n\n\n\n\n\nThe contribution of this paper is the design of an energy-efficient downlink subchannel and power allocation scheme in a cellular network with enhanced performance compared to the work in~\\cite{FFang2016}. Since this problem considers subchannels assignment which are associated with discrete variables in the formulated problem, the problem is NP-hard in general. Moreover, even if the subchannel assignment information is known, the power allocation problem is non-convex~\\cite{DPBertsekas1999}. As a result, joint subchannel assignment and power allocation of this problem can be considered as an mixed integer non-linear programming (MINLP) problem. Considering the importance of the overall energy efficiency for such a system from the perspective of green communications, since the existing solution of~\\cite{FFang2016} is approximate, the investigation of the better ones is required. Consequently, we find that decomposing the problem into a subchannel allocation problem followed by a power loading problem and then solving each problem individually while considering inter-dependency, is an elegant approach to solve this problem. In the first step, under the assumption that total power of the BS is subdivided equally among all the subchannels, we solve the subchannel-user mapping problem via a many-to-many matching model~\\cite{KHamidouche2014, ARoth1984}. However, unlike the scheme in~\\cite{FFang2016}, our proposed subchannel-user mapping scheme can utilize the multi-user diversity efficiently. On the other hand, one of the assumptions of the subchannel-users mapping algorithm in~\\cite{FFang2016} is, at most $2$ users can be assigned to a subchannel. Because of this assumption, the power allocation problem of a subchannel is amenable to the DC programming-based approach. If a subchannel is assigned to more than $2$ users, the power allocation problem is no longer amenable to the DC programming-based approach. In contrast, our proposed scheme is general, and the number of users assigned to a subchannel is arbitrary. Our discovery to solve this problem via the geometric programming (GP)-based~\\cite{Boyd2007, MChiang2005} approach is one of the specific contributions of this paper. Then, in the second step, once the subchannel-user mapping information are known, we adopt the GP-based approach in order to allocate power across all subchannel-user slots upon approximating the energy efficiency of system by a ratio of two posynomials. The DC programming-based approach in~\\cite{FFang2016} allocates power across the subchannels while ignoring the detailed granularity of the user power level. Our proposed power loading scheme using the GP-based approach allocates power to each user's data stream on each of its allocated subchannels. Compared to the most relevant existing work~\\cite{FFang2016}, the major contributions of the paper are summarized as follows. \n\n\n\n\n\\begin{itemize}\n\n\\item For the subchannel-user mapping task, unlike the one in~\\cite{FFang2016}, we have adopted a many-to-many matching model that exploits the multi-user diversity of wireless systems efficiently. Through this scheme, not only each subchannel can be used to serve multiple users, but also each user can utilize multiple subchannels. However, assigning more and more users to a subchannel cannot necessarily enhance the energy efficiency of the system further. Our proposed algorithm can smartly adapt user assignment to each subchannel given the information of some maximal allowable number of users to each subchannel.\n\n\\item While solving the subchannel-user mapping problem, it is required to determine the optimal instantaneous sum-rate of a subchannel in the intermediate decision making steps. When a subchannel can serve at most $2$ users, the approach in~\\cite{FFang2016} is appropriate to determine the instantaneous sum-rate of a subchannel. However, in practice, having more than $2$ users on a subchannel can enhance the system performance. On the other hand, when a subchannel is assigned to more than $2$ users, the DC programming-based approach is no longer a suitable method to determine the optimal instantaneous sum-rate of that subchannel. This is because the objective function cannot be expressed as a difference of convex functions in this case. However, this problem is amenable to GP via the single condensation heuristic method after approximating the objective function of the problem by a ratio of two posynomials. Consequently, upon the approximation, we have adopted the single condensation method to determine the optimal instantaneous sum-rate of each subchannel while making the decision about each subchannel-user assignment.\n\n\\item Once the subchannel-user mapping information are known from the first step, the DC programming-based approach in~\\cite{FFang2016} can allocate power optimally across the subchannels. However, this power allocation scheme ignores the detailed granularity of each subchannel-user slot, and the entire power of the BS is used up. While targeting on the optimality and considering the detailed granularity of each subchannel-user slot, we approximate the original problem via replacing the log function by the first term of its approximated series~\\cite{RNave} to facilitate the problem solving via the GP-based approach. Upon approximating the objective function of the problem via a ratio of two posynomials, we adopt the GP-based single condensation heuristic to solve the approximated problem. The proposed scheme allocates power to each user's data stream on each of its allocated subchannels. As a result, the total required power, to reach the optimal overall energy-efficient state, is much less than the total power of the BS. From the perspective of green communications, this phenomenon of our power loading scheme is a big advantage over the DC programming-based approach. Although the proposed GP-based approach provides fine-grained power allocation across each subchannel and user with higher energy efficiency, this approach is computationally intensive with the growing number of users and subchannels due to the implementation limitation of the off-the-shelf GP solvers~\\cite{cvx}. Therefore, based on the insights of the optimal solution, we also have proposed a computationally-efficient suboptimal solution for the fine-grained energy-efficient power loading problem, which has a polynomial time complexity.\n\n\\item Extensive simulation has been conducted in order to verify the effectiveness of our proposed resource allocation scheme. The results demonstrate that our scheme always outperforms the scheme in~\\cite{FFang2016} under various realistic scenarios.\n\n\\end{itemize}   \n\n\n\nThe rest of the paper is organized as follows. Along with the background information and the description of the system, in Section~\\ref{sec:sysmodel}, we formulate our energy-efficient resource allocation problem. The detailed solution approach is provided in Section~\\ref{sec:sol}. Followed by the simulation methodology, we evaluate the performance of our proposed resource allocation schemes in Section~\\ref{sec:eval}. Finally, Section~\\ref{sec:concl} concludes the paper with some implication.    \n\n\n\\section{System Model and Problem Formulation}\n\\label{sec:sysmodel}\n\nIn this paper, we consider a downlink scenario of a cellular network, which has one BS. Time is divided into frames, and the entire pre-assigned spectrum for the system is divided into $N$ subchannels in each frame. The resultant subchannels are the elements of a set, denoted by \\textbf{N}. There are $M$ users in the system, and the corresponding set holding these users is denoted by \\textbf{M}. Using the subchannels in set \\textbf{N} as the transmission media, the BS transmits data to the users in set \\textbf{M}. Both the BS and the users in the system are equipped with NOMA technologies. The BS transmits its data to a set of users using the superposition coding (SC) technique over a set of subchannels. Whereas, the receivers (i.e., the users) apply the SIC technique on each subchannel to decode the superimposed signals for extracting their own individual signal. However, before the downlink transmission operation, it is required to schedule subchannels and power across the users optimally so that the overall energy efficiency of the system is maximized. We assume that the scheduling scheme in the system is centralized, and the BS is appointed to conduct the entire scheduling operation. To develop this scheduling scheme, since the entire channel state information (CSI) of the system is required, the BS is aware of all these information. At the beginning of each time frame, all users send their CSI to the BS via some reliable control channels. The CSI of the subchannels in set \\textbf{N} follows the block fading model~\\cite{DavidTse2005}, where the CSI of each subchannel is constant for a time slot and varies in an i.i.d. manner from time slot to time slot. The BS has the maximal power constraint, denoted by $p^{\\mbox{max}}$. \n\n\n\nWe assume that the BS assigns $M_n$ users to the $n$th subchannel, and the corresponding set holding these users is denoted by $\\textbf{M}_n$. Given this, let denote the symbol transmitted by the BS on subchannel $n$ as $x_n$, and it is given by $\\sum_{m \\in \\textbf{M}_n}\\sqrt{p_m^n}{s_m}$. Here, $s_m$ is the modulated symbol of the $m$th user on subchannel $n$ and $p_m^n$ is the power level assigned to user $m$ on subchannel $n$. Consequently, the received signal of user $m$ on subchannel $n$ can be represented as\n\n\n\n\\[\ny_m^n = \\sqrt{p_m^n}h_m^ns_m + \\displaystyle\\sum_{i=1, i \\neq m}\\sqrt{p_i^n}h_m^ns_i + z_n,\n\\]\n\n\n\n\\noindent \nwhere $h_m^n$ is the channel gain of user $m$ on the $n$th subchannel. $z_n$ is the noise power over subchannel $n$, which follows Additive White Gaussian Noise (AWGN)~\\cite{KMcClaning2000} distribution with mean zero and variance ${\\sigma}_n^2$, i.e., $z_n \\approx {\\cal{CN}}(0, {\\sigma}_n^2)$. The noise power of subchannel $n$ is statistically same for all users. In NOMA systems, each subchannel is shared by multiple users. Consequently, each user on subchannel $n$ receives its signal as well as the the interference signals from the other users that share the same subchannel. Let us denote the set holding the users that cause interference to user $m$ on subchannel $n$ as $\\textbf{M}_n^m$. Given the information of set $\\textbf{M}_n^m$, the received signal-to-interference-ratio (SINR) of the $m$th user on subchannel $n$ is given by\n\n\n\\begin{equation}\n\\mbox{SINR}_m^n = \\frac{p_m^n|h_m^n|^2}{{\\sigma}_n^2 + \\displaystyle\\sum_{i \\in \\textbf{M}_n^m}p_i^n|h_m^n|^2} = \\frac{p_m^n{g_m^n}}{{\\sigma}_n^2 + \\displaystyle\\sum_{i \\in \\textbf{M}_n^m}p_i^ng_m^n},\n\\end{equation}\n\n\\noindent\nwhere ${\\sigma}_n^2 = E[|z_n|^2]$ is the noise power on subchannel $n$, and $g_m^n = |h_m^n|^2$ represents the power gain of the $m$th user on subchannel $n$. \n\n\nSince the NOMA users are equipped with the SIC technique, the way a NOMA user retrieves information depends on the order of the allocated power level among all the NOMA users in the corresponding subchannel. In other words, a particular NOMA user can decode the information of other users that are assigned to larger power level compared to itself. On the other hand, this user can decode its own information considering the power level of other remaining users (that are assigned to lower power level compared to itself) as noise. Therefore, if the order of the power level assigned to the NOMA users are known, the decoding order is straightforward for a particular NOMA user and vice versa. It is clear that the optimal decoding order of the NOMA users and their power assignment are coupled to each other and is not a straightforward problem. However, in order to reduced the complexity of the proposed solution scheme in this paper, we intend to fix the order of the power level among the NOMA users while solving the entire energy efficiency problem. Ideally, in order to maximize the energy efficiency of the system, the optimal power allocation among the NOMA users of a subchannel should follow the water filling-based approach. On the other hand, when the fairness of the system is considered, the order of the ideal power allocation should be in-between the water filling-based approach and the opposite of the water filling one. If we would fix the order of the power level among the NOMA users according to the water filling norm, it is very likely that an ideal power loading scheme\\footnote{The proposed power loading scheme in this paper is particularly optimal for the applications that work in the low SNR regime.} assigns very low power to the users with worse channel condition. In this case, those users (with worse channel condition) may not be able to decode their information or this is not a fair attitude to those users. Therefore, while considering the fairness of the system, we have adopted the opposite norm of the water filling technique so that the users with worse channel have fair provision. In other words, in order to reduce the overall computational complexity of the solution schemes and considering the fairness of the system, similar to~\\cite{FFang2016}, the users with larger gain are assigned to lower power level compared to the users with lower gain. As a result, following the norm of the SIC technique, the set of users that cause interference to user $m$ on subchannel $n$ is given by $\\textbf{M}_n^m = \\{m'|_{g_{m'}^n > g_m^n}, m' \\in \\textbf{M}_n\\}$. Consequently, the sum-rate of subchannel $n$, denoted by $R_n(\\textbf{M}_n, \\{p_m^n\\})$, can be obtained by the Shannon's capacity formula~\\cite{HMichiel2001}.\n\n\\[\nR_n(\\textbf{M}_n, \\{p_m^n\\}) = \\displaystyle\\sum_{m \\in \\textbf{M}_n}\\mbox{log}_2\\left(1 + \\mbox{SINR}_m^n\\right).\n\\]\n\n\n\nIf the circuit power consumption on subchannel $n$ is denoted by $p_c$, the energy efficiency of subchannel $n$ is given by\n\n\\[\nE_n = \\frac{R_n(\\textbf{M}_n, \\{p_m^n\\})}{p_c + \\sum_{m \\in \\textbf{M}_n}p_m^n}.\n\\]     \n\n\n\n\nIn this work, given the power constraint of the BS, our objective is to allocate the subchannels in set \\textbf{N} across all the users in set \\textbf{M} so that the aggregate energy efficiency of all subchannels, $\\sum_{n \\in \\textbf{N}}E_n$, is maximized. Clearly, this is an optimization problem. To formulate this problem, we define binary variables ${\\beta}_m^n, m \\in \\textbf{M}_n, n \\in \\textbf{N}$. ${\\beta}_m^n = 1$ implies that subchannel $n$ is allocated to user $m$, and ${\\beta}_m^n = 0$ means the other case. Ideally, more and more users are assigned to a subchannel, the better the system performance. However, due to the increasing level of interference with more and more assigned users and the decoding complication of the SIC technique, not necessarily more users assigned to a subchannel will enhance the system energy efficiency. While giving weight to this observation and insight, we assume that maximum $K$ users can be assigned to a subchannel\\footnote{Setting $K$ to $M$, the multi-user diversity of a subchannel can be fully exploited. Due to the limited power level and the superposition-coded interference level caused by other users, an ideal resource allocation scheme should smartly select the users that are beneficial for that subchannel in terms of energy efficiency}. Therefore, if the achieved instantaneous rate of user $m$ is $R_m$ and its minimum required rate is $R_m^{\\mbox{min}}$ (to satisfy its quality-of-service requirement), the energy-efficient downlink subchannel and power allocation problem in this context can be formulated as (\\ref{eq:opt-prob1})-(\\ref{eq:opt-prob5}). Note that the aggregate power level of subchannel $n$ is denoted by $p_n$, and $\\sum_{m \\in \\textbf{M}_n}p_m^n = p_n$.\n\n\n\n\\begin{eqnarray}\n\\label{eq:opt-prob1} &\\displaystyle\\max_{\\{p_m^n, {\\beta}_m^n\\}}\\displaystyle\\sum_{n \\in \\textbf{N}}\\frac{1}{p_c + \\displaystyle\\sum_{m \\in \\textbf{M}_n}p_m^n}\\displaystyle\\sum_{m \\in \\textbf{M}_n}{\\beta}_m^n\\mbox{log}\\left(1 + \\mbox{SINR}_m^n\\right), \\\\\n& \\nonumber \\mbox{subject to} \\\\\n\\label{eq:opt-prob2} & \\displaystyle\\sum_{m \\in \\textbf{M}}{\\beta}_m^n \\le K,~ \\forall{n \\in \\textbf{N}},~~~\\displaystyle\\sum_{n \\in \\textbf{N}}{\\beta}_m^n \\le N,~ \\forall{m \\in \\textbf{M}}, \\\\\n\\label{eq:opt-prob3} & {\\beta}_m^n \\in \\{0, 1\\},~ \\forall{m \\in \\textbf{M}},~ \\forall{n \\in \\textbf{N}}, \\\\\n\\label{eq:opt-prob4} & \\displaystyle\\sum_{n \\in \\textbf{N}}\\displaystyle\\sum_{m \\in \\textbf{M}_n}{\\beta}_m^np_m^n \\le p^{\\mbox{max}},~ p_m^n \\ge 0, \\\\\n\\label{eq:opt-prob5}& R_m  \\ge R_m^{\\mbox{min}}, \\forall{m \\in \\textbf{M}}.\n\\end{eqnarray}\n\n\nIn the above formulation, there are two types of variables, i.e., $\\{{\\beta}_m^n\\}$ and $\\{p_m^n\\}$. $\\{{\\beta}_m^n\\}$ are the set of discrete variables, and the problem is NP-hard because of these variables. The NP-hardness property of this problem can be proved by mapping it to the classical relaxed bin-packing problem~\\cite{SMartello1990}. Let us assume that the subchannels in set \\textbf{N} are the possible bins, the users in set \\textbf{M} are items and the capacity of each bin is $K$. Now, if we assume that an item can reside into multiple bins and if we transform the objective of the problem from minimizing the number of used bins to maximizing the system energy efficiency, reduction of our problem to the relaxed bin packing problem is completed. Thus, the NP-hardness property of our problem is proved. On the other hand, even if the information of set $\\{{\\beta}_m^n\\}$ are known, it is straightforward to prove that the problem is non-convex with respect to (w.r.t.) $\\{p_m^n\\}$. Moreover, in order to reduce the complexity of the solution method proposed in the following section, the ordering of the power level and gain of the users in set $\\textbf{M}_n$ are considered to be fixed and given as follows. For any $2$ users $m$ and $m'$ in set $\\textbf{M}_n$, if they follow $g_m^n > g_{m'}^n$, they must hold $p_m^n < p_{m'}^n$.  \n\n\n\n\n\\section{Solution Approach}\n\\label{sec:sol}\n\n\nIn this section, we explore the solution approach of the energy-efficient downlink resource allocation problem of a NOMA system described in (\\ref{eq:opt-prob1})-(\\ref{eq:opt-prob5}). Apparently, due to the discrete nature of subchannel assignment (i.e., variables $\\{{\\beta}_m^n\\}$) and the continuous nature of power assignment (i.e., variables $\\{p_m^n\\}$), this is an MINLP problem. Furthermore, the assignment issue of users on the same subchannel brings further complication in the solution strategy. Therefore, we realize that decomposing the problem into a subchannel allocation subproblem followed by a power loading subproblem and then solving each subproblem based on the insights of the optimal solution, is a good strategy to solve the joint problem. Consequently, based on the assumption that the maximal power of the BS ($p^{\\mbox{max}}$) is equally subdivided among all the subchannels, we solve the subchannel-user mapping subproblem using a many-to-many matching model. Compared to the one-to-many matching model adopted in~\\cite{FFang2016}, we believe that a many-to-many one can capture the structure of the problem well. This is because one user can be assigned to multiple subchannels in order to exploit the multi-user diversity of wireless systems, and one subchannel can be assigned to multiple users to take the advantages of the NOMA technology. While solving the subchannel-user mapping subproblem, given the power constraint of each subchannel, it is required to assign power optimally among the allocated users so that the overall rate of that subchannel is maximized. For this purpose, we have noticed that if the value of $K$ is larger than $2$, the resultant optimization problem cannot be written as the difference of convex functions, and hence the DC programming-based approach is not appropriate to solve this problem. However, we found that GP is a well-fit technique to solve this problem after approximating the objective function of the original problem by a ratio of two posynomials. For the solution of the second part of the problem, we assume that the subchannel-user mapping information are available. Given this information,~\\cite{FFang2016} has utilized the DC programming-based approach to allocate power across the subchannels while ignoring the detailed granularity of the user power level. On the attempt of finding an elegant solution for the power loading subproblem, we find that the GP-based approach is a good one as well upon approximating the problem by a ratio of two posynomials. \n\n\\subsection{Subchannel and User Mapping Scheme}\nIntuitively, assignment of many subchannels to a user and allocating multiple users to a subchannel (to follow the guidelines of the NOMA technique) is expected to enhance the overall energy efficiency of the system. However, due to the usage of practical, finite-length and possibly non-Gaussian codes, we assume that maximum $K$ users can be assigned to a subchannel. Given the total power constraint of the BS, this problem is NP-hard. The nature of the problem implies that a many-to-many matching model~\\cite{KHamidouche2014, ARoth1984} is appropriate to capture the aforementioned behavior. Given that maximum $K$ users can be multiplexed to a subchannel, $M$ users in set \\textbf{M} and $N$ subchannels in set \\textbf{N} are two sets of players of this many-to-many matching model. Note that each user $m$ can have infinite ($N$ in practice) subchannels if possible. In this case, since the BS has the maximal power constraint $p^{max}$, this should be subdivided equally among all the subchannels. \n\n\\noindent\n\\textbf{Definition 1:} A many-to-many matching model $\\mu$ is a mapping from set \\textbf{M} to set \\textbf{N} such that every $m \\in \\textbf{M}$ and $n \\in \\textbf{N}$ satisfy the following properties:\n\n\\begin{itemize}\n\n\\item ${\\mu}(m) \\subseteq \\textbf{N}$ and ${\\mu}(n) \\subseteq \\textbf{M}$\n\n\\item $|{\\mu}(m)| \\le \\infty, \\forall{m \\in \\textbf{M}}$\n\n\\item $|{\\mu}(n)| \\le K, \\forall{n \\in \\textbf{N}}$\n\n\\item $n \\in {\\mu}(m)$ if and only if $m \\in {\\mu}(n)$\n\n\\end{itemize}\n\n\\noindent\nwhere ${\\mu}(m)$ is the set of partners for user $m$ and ${\\mu}(n)$ is the set of partners for subchannel $n$ under the matching model $\\mu$. The definition states that each user in set \\textbf{M} is matched to a subset of subchannels in set \\textbf{N}, and vice versa. However, before accomplishing these assignment operations, each user needs to have a preference list based on some criteria. The criterion of constructing the preference list for the users is based on their gain from the subchannels, which is similar to that in~\\cite{FFang2016}. The preference of each subchannel $n$ is based on the overall benefit (i.e., energy efficiency) of the system. For example, if user $m$ chooses subchannel $n$, this subchannel only accepts this user if and only the energy efficiency of the system is enhanced by this allocation. \n\n\n\nTo solve our subchannel-user mapping problem, we are interested to look at a stable solution, in which there are no players that are not matched to one another but they all prefer to be partners. Since the subchannel players give preference to the overall energy efficiency of the system while choosing partners from set \\textbf{M}, a stable solution is considered to be an elegant solution of this problem\\footnote{At this point, none of the players can enhance their performance further by choosing alternative partners, and hence this point is considered as a stable point.}. In the many-to-many matching models~\\cite{ARoth1984}, many stability concepts can be considered based on the number of players that can improve their utility by forming new partners among one another. However, due to the large number of players ($\\textbf{M} \\cup \\textbf{N}$) in our problem, identifying optimal subset of partners for a player is computationally intractable. Consequently, we choose to solve the matching problem by identifying the partners one by one from the opposite set. This way of choosing the partners in the matching relation brings pair-wise stability. In \\textit{Definition 1} and \\textit{Definition 2}, we highlight some properties of a pair-wise stable matching relation. For the sake of these definitions, we define some notations. For example, $\\textbf{C}_m(\\textbf{N})$ denotes the choice set of user $m$, which basically follows $\\textbf{C}_m(\\textbf{N}) \\subseteq \\textbf{N}$. Moreover, ${\\textbf{M}'}_n {\\succ}_m {\\textbf{M}''}_n$ implies that subchannel $n$ prefers the users in set  ${\\textbf{M}'}_n$ over that in set ${\\textbf{M}''}_n$.\n\n\\noindent\n\\textbf{Definition 1:} Consider that pair $(m,n)$ is not the element of matching model $\\mu$, that is $m \\not\\in {\\mu}(n)$ and $n \\not\\in {\\mu}(m)$. Now consider another pair $(\\phi, \\varphi)$, that satisfies $\\phi \\in \\textbf{C}_m({\\mu}(n) \\cup \\{n\\})$ and $\\varphi \\in \\textbf{C}_n({\\mu}(n) \\cup \\{m\\})$. For the matching relation $\\mu$ to be pairwise stable, it is not possible that both $\\phi {\\succ}_m {\\mu}(m)$ and $\\varphi {\\succ}_n {\\mu}(n)$ are satisfied.\n\n\n\\noindent\n\\textbf{Definition 2:} Let ${\\textbf{M}}_n$ (${\\textbf{M}}_n \\in \\textbf{M}$) be the set of matched users of subchannel $n$ under the matching relation $\\mu$, and $|{\\textbf{M}}_n| = K$. Consider $m$ is one of the matched users in set ${\\textbf{M}}_n$. Now, another user $m' \\in \\textbf{M}$ has come to be matched with subchannel $n$. Note that $m' \\not\\in {\\textbf{M}}_n$. However, if user $m$ is replaced by user $m'$, the performance of subchannel $n$ is enhanced. Therefore, in order to have stable matching, ${\\textbf{M}}_n = ({\\textbf{M}}_n \/\\ m) \\cup m'$ should hold. This phenomenon of stable matching is called substitutability. \n\n\n\n\nWhile satisfying the properties of a stable many-to-many matching relation (e.g, substitutability~\\cite{ARoth1984}), we have proposed an algorithm in \\textit{Algorithm~\\ref{alg:sc-usr-map}}. In order to bring the stability in this matching relation or enhance the overall energy efficiency of the system, we have adopted a few heuristics or strategies. The description of the algorithm is as follows. In order to realize the outcome of each step of the algorithm, we introduce one more type of set variable: ${\\bm{\\Omega}}_m$ that holds the allocated subchannels for user $m$. First, ${\\bm{\\Omega}}_m, m \\in \\textbf{M}$ and $\\textbf{M}_n, n \\in \\textbf{N}$ are initialized with $\\emptyset$. These sets are gradually filled up as we go through the iterations in between step $3$ and step $26$. At the initialization phase, each user $m \\in \\textbf{M}$ also constructs its subchannel preference list based on the descending order of their gain. Then, inside the outer-most loop (between step $3$ and step $26$), if no assignment is possible, the algorithm terminates\\footnote{At this point, it is assumed that the system has reached a stable situation or the improvement of energy efficiency is no longer possible.}. Inside the inner loop (between step $4$ and step $25$), each user $m$ chooses its most preferred unallocated (and not rejected already) subchannel $n$. At this point, two conditions are possible. The first condition is that the number of allocated users to subchannel $n$ can be less than $K$ (maximum allowable number of users per subchannel), and the second condition is the other case. If the first condition is true, we can apply the addition strategy for this subchannel-user assignment: user $m$ can be added to this subchannel and this strategy is added to set \\textbf{S} (which was initialized before sorting out the possible strategies for use $m$). Whereas, for the second condition, only substitution strategy (already allocated user $m'$ can be replaced by new user $m$) is possible. After filling the strategy set \\textbf{S}, no matter the number of allocated users to subchannel $n$ is less than or equal to $K$, the elements of \\textbf{S} are filtered based on the previous sum-rate of subchannel $n$ before this new possible assignment. The filtered strategy set is \\textbf{CS} in \\textit{Algorithm~\\ref{alg:sc-usr-map}}. Finally, $s^{\\mbox{Best}}$ strategy is chosen based on the sum-rate (${\\Gamma}_{s}^n$ in \\textit{Algorithm~\\ref{alg:sc-usr-map}}) each strategy achieves. If $s^{\\mbox{Best}}$ is empty, the innermost loop continues, and the next user is chosen from set \\textbf{M} for building its possible strategy set. For the other case, the corresponding strategy is executed. As a result, set ${\\bm{\\Omega}}_m$, set ${\\bm{\\Omega}}_{m'}$ (only for the substitution strategy) and set $\\textbf{M}_n$ are updated. By analyzing the algorithm, we conclude \\textit{Proposition 1}, \\textit{Proposition 2} and \\textit{Theorem 1}. \n\n\n\n\\begin{algorithm}\n\\caption{The energy-efficient subchannel-user mapping algorithm via a many-to-many matching model.}\n\\label{alg:sc-usr-map}\n\\begin{algorithmic}[1]\n\\STATE ${\\bm{\\Omega}}_m \\leftarrow \\emptyset, \\forall{m \\in \\textbf{M}}$; $\\textbf{M}_n \\leftarrow \\emptyset, \\forall{n \\in \\textbf{N}}$.\n\\STATE Each user $m \\in \\textbf{M}$ produces its preference list in the descending order of its gain on subchannel $n \\in \\textbf{N}$\n\n\\REPEAT\n\\FOR{$m \\in \\textbf{M}$}\n\n\\STATE $n$ $\\leftarrow$ The best unallocated (and not rejected previously) subchannel of user $m$. \n\\STATE ${\\Gamma}_{\\mbox{prev}}^n \\leftarrow$ The aggregate rate of subchannel $n$.\n\\STATE $\\textbf{S} \\leftarrow \\emptyset$, $\\textbf{CS} \\leftarrow \\emptyset$, and $\\textbf{TRate} \\leftarrow \\emptyset$. \n\\IF{The number of assigned users on subchannel $n$ is less than $K$}\n\\STATE Construct strategy $s$ by adding user $m$ to set $\\textbf{M}_n$, and add $s$ to set \\textbf{S}.\n\\ELSIF{The number of assigned users on subchannel $n$ is equal to $K$}\n\\STATE Construct each strategy $s$ by replacing each user $m' \\in \\textbf{M}_n$ by user $m$, and add $s$ to set \\textbf{S}.\n\\ENDIF\n\\FOR{$s \\in \\textbf{S}$}\n\\STATE Due to strategy $s$, determine the power level of user $m$ in $\\textbf{M}_n$ using (18)~\\cite{FFang2016} or the approach in Section~\\ref{sssec:GP-indiv}, and ${\\Gamma}_{s}^n \\leftarrow$ The aggregate rate of subchannel $n$ due to strategy $s$.\n\\STATE Add ${\\Gamma}_{s}^n$ to set \\textbf{TRate} and add strategy $s$ to \\textbf{CS} if and only if ${\\Gamma}_{s}^n - {\\Gamma}_{\\mbox{prev}}^n > 0$.\n\\ENDFOR\n\n\\STATE $s^{\\mbox{Best}} \\leftarrow \\argmax_{s \\in \\textbf{CS}}\\textbf{TRate}(s)$.\n\\IF{$s^{\\mbox{Best}}$ is the substitution strategy}\n\\STATE ${\\bm{\\Omega}}_m \\leftarrow {\\bm{\\Omega}}_m \\cup \\{n\\}$, ${\\bm{\\Omega}}_{m'} \\leftarrow {\\bm{\\Omega}}_{m'} \/\\ \\{n\\}$, and $\\textbf{M}_n \\leftarrow (\\textbf{M}_n \/\\ \\{m'\\}) \\cup \\{m\\}$. \\COMMENT{$m'$ is the to-be-replaced user and $m$ is the new user on subchannel $n$}\n\\STATE Update the preference list of user $m$ and user $m'$.\n\\ELSIF{$s^{\\mbox{Best}}$ is the addition strategy}\n\\STATE ${\\bm{\\Omega}}_m \\leftarrow {\\bm{\\Omega}}_m \\cup \\{n\\}, \\textbf{M}_n \\leftarrow \\textbf{M}_n \\cup \\{m\\}$.\n\\STATE Update the preference list of user $m$.\n\\ENDIF\n\\ENDFOR\n\\UNTIL{The overall performance of the system cannot be enhanced}\n\\end{algorithmic}\n\\end{algorithm}\n\n\n\n\n\n\n\n\n\n\n\n\n\\noindent\n\\textbf{Proposition 1:} Given a certain value of $K$, allocating more and more users to a subchannel cannot enhance the energy efficiency further.\n\n\n\\noindent\n\\textbf{Proof:} See \\textit{Appendix A}.\n\n\n\n\\noindent\n\\textbf{Proposition 2:} Once a user is rejected while to be assigned with a subchannel through the addition or the substitution strategy, that rejection is final. More precisely, even if that user wants to be assigned to that particular subchannel later, this assignment no longer enhances the system energy efficiency.\n\n\n\n\\noindent\n\\textbf{Proof:} See \\textit{Appendix B}.\n\n\n\n\\noindent\n\\textbf{Theorem 1:} The subchannel-user mapping algorithm (i.e., \\textit{Algorithm 1}) is guaranteed to converge to a pair-wise stable matching relation.\n\n\\noindent\n\\textbf{Proof:} See \\textit{Appendix C}.\n\n\\noindent\n\\textbf{\\textit{Computational Complexity of \\textit{Algorithm~\\ref{alg:sc-usr-map}}:}} As discussed and proved in \\textit{Proposition 2}, if user $m$ is rejected by subchannel $n$ once, that user does not propose subchannel $n$ to be matched with anymore. This is because the energy efficiency of this subchannel $n$ is not enhanced in any way if previously rejected user is given preference in the following iterations. The enhancement of the energy efficiency for each subchannel implies the enhancement of the overall energy efficiency in the system. The termination of the outer-most loop (between step $3$ and step $26$) depends on the improvement of the overall energy efficiency. To be precise, if user $m$ in the inner loop proposes a subchannel $n$ and the corresponding $s^{\\mbox{Best}}$ results in empty value, the next user in set \\textbf{M} builds its possible strategy set. If none of the users in set \\textbf{M} can be matched with any of the subchannel in set \\textbf{N}, the outer-most loop terminates. Therefore, intuitively, each user $m \\in \\textbf{M}$ approaches at most $N$ subchannels to be matched with. Consequently, the outer-most loop runs at most $MN$ times. On the other hand, the complexity of finding the best strategy for each user inside the inner loop is $\\mbox{O}(K)$, and the corresponding reason is as follows. If the number of users assigned to subchannel $n$ is less than $K$, only one strategy (the addition strategy) is possible. If the number of users assigned to subchannel $n$ is exactly $K$, then $K$ strategies are possible due to the $K$ possible replacement policies. Since the value of $K$ is usually not large in practice, we consider the complexity of finding the best strategy as constant. The remaining operations inside the inner-most loop happen in constant time, and so we can ignore the complexity of these operations as well. Consequently, the overall complexity of the algorithm is $\\mbox{O}(MN)$.\n\n\n \n\\subsection{Power Allocation Scheme}\n\\label{ssec:GP}\n\nFrom \\textit{Algorithm~\\ref{alg:sc-usr-map}}, we know the subchannel-user mapping information, i.e., $\\textbf{M}_n, n \\in \\textbf{N}$ and ${\\bm{\\Omega}}_m, m \\in \\textbf{M}$. This information is derived based on the assumption that the maximal power level $p^{\\mbox{max}}$ of the BS is equally subdivided among all subchannels, i.e., $p_n = p^{\\mbox{max}}\/N, n \\in \\textbf{N}$. However, in (\\ref{eq:opt-prob1}), we see that the instantaneous rate of user $m$ assigned to subchannel $n$ is the increasing function of $p_m^n$ and the decreasing function of the interference power level caused by other users, i.e., $p_i^n, i \\in \\textbf{M}_n^m$. Consequently, even if the information about $\\textbf{M}_n, n \\in \\textbf{N}$ and ${\\bm{\\Omega}}_m, m \\in \\textbf{M}$ are known, the power allocation across all subchannel-user slots, i.e., $p_m^n, m \\in \\textbf{M}, n \\in \\textbf{N}$, is a non-convex optimization problem. In the literature, the transformation of such a type of problem to convex form is not reported yet, and hence the dual-based method~\\cite{CWang2016} or the bisection search~\\cite{QSun22015} cannot be adopted to solve this problem. Therefore, our next objective is to design a method that allocates power across all the subchannel-user slots in such a manner that near-optimality is achieved. We have adopted the GP-based optimization technique to solve this power allocation problem, the description of which is provided in the following discussions.\n\n\n\\begin{figure}\n  \\begin{center}\n    \\includegraphics[width=0.5\\columnwidth]{log_expansion2.eps}\n    \\caption{A sample verification of natural log series expansion~\\cite{RNave}.}\n    \\label{fig:log-expansion}\n  \\end{center}\n\\end{figure}\n\n\nA GP~\\cite{Boyd2007, MChiang2005} is a type of mathematical optimization problem with the characteristics that the objective and constraint functions must be monomial(s) or posynomial(s). Two problems that we have discussed in the following have a certain form, such as $\\frac{G_n(q_1,\\cdots,q_T)}{G_d(q_1,\\cdots,q_T)}$. Here, both functions $G_n(q_1,\\cdots,q_T)$ and $G_d(q_1,\\cdots,q_T)$ are posynomials; $q_1,\\cdots,q_T$ are the variables in the problem; and $T$ is the number of variables. This type of function is still not amenable to GP. To make this problem amenable to GP, there are some heuristics, such as the single condensation method and the double condensation method~\\cite{Boyd2007, MChiang2005}. We plan to apply the single condensation method in this context, which requires the denominator (i.e., $G_d(q_1,\\cdots,q_T)$) to be approximated via some monomial ${\\lambda}\\prod_{i=1}^T{q_i}^{a_i}$. Given the values of $q_1,\\cdots,q_T$, the unknown variables of this monomial can be approximated by $a_i = \\frac{q_i}{G_d(.)}\\frac{{\\partial}G_d(.)}{{\\partial}q_i}$ and $\\lambda = \\frac{G_d(.)}{\\prod_{i=1}^T{q_i}^{a_i}}$. As a result, the approximated objective function becomes the ratio of a posynomial and a monomial, which is a posynomial. Consequently, we require an iterative process in order to solve this problem step by step. The steps of this iterative process are provided in Section III-A of~\\cite{RRuby20152}.\n\n\n\n\n\\begin{comment}\n\n\\begin{enumerate}\n\\item Set the initial values for variables $q_1,\\cdots,q_T$.\n\\item Determine the values of $a_1,\\cdots,a_T$, and $\\lambda$.\n\\item Solve the problem associated with resultant objective function, i.e., the ratio of the posynomial and the monomial, via GP.\n\\item If the absolute difference between the current and previous values of the variables is less than or equal to some small number, i.e., $\\epsilon$, go to step $2$, and if otherwise, stop the iterative process. \n\\end{enumerate}\n\nThe final values of the variables obtained in the last iteration of the iterative process is the solution of our defined optimization problem.\n\\end{comment}\n\n\n\n\n\\subsubsection{Joint Power Allocation among all Subchannels}\n\\label{sssec:GP-joint}\n\nOnce the subchannel-user assignment information, i.e., $\\textbf{M}_n, n \\in \\textbf{M}_n$, in order to obtain the energy-efficient power level of each  subchannel-user tuple given the power constraint at the BS, we need to solve an optimization problem, which is as follows.\n\n\n\n\n\n\\begin{eqnarray}\n\\label{eq:GP-joint}\n\\nonumber & \\displaystyle\\max_{\\{p_m^n\\}}\\displaystyle\\sum_{n \\in \\textbf{N}}\\frac{1}{p_c+\\displaystyle\\sum_{m \\in \\textbf{M}_n}p_m^n}\\displaystyle\\sum_{m \\in \\textbf{M}_n}\\mbox{log}_2\\left(1 + \\frac{g_m^np_m^n}{{\\sigma}_n^2 + \\displaystyle\\sum_{i\\in\\textbf{M}_n^m}p_i^ng_m^n}\\right),  \\\\\n\\nonumber & \\mbox{s.t.}~\\displaystyle\\sum_{n \\in \\textbf{N}}\\displaystyle\\sum_{m \\in \\textbf{M}_n}p_m^n \\le p^{max}, \\\\\n& \\displaystyle\\sum_{m\\in \\textbf{M}_n}p_m^n \\ge p_n^{\\mbox{min}},~\\forall{n \\in \\textbf{N}}.\n\\end{eqnarray}\n\n\n\\begin{equation}\n\\mbox{log}_2\\left(1 + \\frac{g_m^np_m^n}{{\\sigma}_n^2 + \\displaystyle\\sum_{i\\in\\textbf{M}_n^m}p_i^ng_m^n}\\right) = \\frac{\\frac{g_m^np_m^n}{{\\sigma}_n^2 + \\displaystyle\\sum_{i\\in\\textbf{M}_n^m}p_i^ng_m^n}}{2 + \\frac{g_m^np_m^n}{{\\sigma}_n^2 + \\displaystyle\\sum_{i\\in\\textbf{M}_n^m}p_i^ng_m^n}}\n\\end{equation}\n\n\n\n\n\nThe relations in~(\\ref{eq:opt-prob5}) basically are associated with $M$ equations. The number of variables (i.e., $\\{p_m^n\\}_{m \\in \\textbf{M}_n, n \\in \\textbf{N}}$) in all these equations is $\\sum_{n \\in \\textbf{N}}|\\textbf{M}_n|$. However, using (18)~\\cite{FFang2016}, the number of variables in all these equations can be reduced to $N$. Therefore, for $M \\ge N$, intuitively, by solving all these equations jointly, the relations in the last constraint of~(\\ref{eq:GP-joint}) are achievable. For the other case (i.e., $M < N$), the last constraint of~(\\ref{eq:GP-joint}) can be approximated as well based on some simple assumption. Moreover, on subchannel $n$, the set of users, $\\textbf{M}_n^m$, that cause interference to user $m$ is determined based on the decoding order described in Section~\\ref{sec:sysmodel}. Clearly, the objective function here is not amenable to GP (not even any GP-based heuristic method) because of the energy efficiency factor $\\frac{1}{p_c+\\sum_{m \\in \\textbf{M}_n}p_m^n}$. Therefore, we have decided to expand the log series. Since the SINR of user $m$ on subchannel $n$ is larger than zero, we can expand the log function according to $\\mbox{ln}(1+x) = 2\\left[\\frac{x}{2+x} + \\frac{1}{3}(\\frac{x}{2+x})^3+\\cdots\\right]$~\\cite{RNave}. The accuracy of the log expansion is verified in Fig.~\\ref{fig:log-expansion}. For the sake of simplicity, using the first term of the expanded log function, we transform the original objective function to a form, which becomes the ratio of two posynomials. As shown in Fig.~\\ref{fig:log-expansion}, both the original log function and the first term of the expanded series are increasing with the increasing value of $x$. Moreover, the difference between the original log function and the first term of the log series increases with the increasing value of $x$. However, this difference is negligible when the value of $x$ is $\\le 3$. If we look at the objective function in ~(\\ref{eq:GP-joint}), $x$ is mapped to the SINR term of each user on a particular subchannel. The SINR of a user on a subchannel is a function of its and other users' gain and assigned power level. If the values of gain and maximal power level of the system are such that the SINR of a user on a subchannel does not exceed $5$ dB (the absolute value of which is around 3), the resultant approximated problem using the first term of the log series does not deviate much from the original problem. On the other hand, the search range of the SINR for a user over a subchannel may exceed $5$ dB at higher gains and a larger power level of the system. However, the first term of the log series dominates the entire log function. Moreover, since the nature of the log function as well as the first term of the log series are increasing, the sum of the log function as well as the sum of the first term (of the log series) are increasing. Therefore, when the search range of a SINR exceeds $5$ dB, although the approximated function is deviated from the original function, the resultant solution of the approximated problem should not be very bad. The accuracy of the resultant solution can be improved by approximating the problem using the first two terms of the log series instead of only the first one. Since the problem is a maximization problem, we apply the inverse function\\footnote{The inverse function of $\\frac{A}{B}$ is $\\frac{B}{A}$.} to transform it to the minimization one. Consequently, the problem becomes solvable via GP, as described in Section~\\ref{ssec:GP}. The outcome of the proposed fine-grained power allocation scheme strengthens the well-established truth~\\cite{SHan2014} that the two objectives of maximizing the energy efficiency and maximizing the spectrum efficiency cannot be achieved simultaneously at the same operating point.\n\n\n\n\n\n\n\n\\begin{comment}\n\n\\noindent\n\\textbf{Proposition 3:} If the overall energy efficiency achieved by a certain power allocation scheme is better than that of some benchmark scheme, it does not necessarily implies that the overall throughput achieved by that scheme will be better compared to the benchmark scheme.\n\n\\noindent\n\\textbf{Proof:} See \\textit{Appendix D}.\n\\end{comment}\n\n\n\n\\subsubsection{Power Allocation of a Subchannel}\n\\label{sssec:GP-indiv}\n\n\nFor subchannel $n$, once $\\textbf{M}_n$ and $p_n$ are known, we would like to determine $\\{p_m^n\\}_{m \\in {\\textbf{M}}_n}$. If $K=2$,~\\cite{FFang2016} has proved that this problem can be solved using the DC programming-based approach. However, if $K > 2$, this problem can no longer be solvable via this approach. To prove this statement, let us assume $K=3$, and $g_1^n > g_2^n > g_3^n$ holds for subchannel $n$. Under this circumstances, the instantaneous rate of the $3$rd user on subchannel $n$ can be given by $\\mbox{log}_2(1 + g_3^np_3^n\/(1+g_3^np_1^n+g_3^np_2^n))$. The interference term in the instantaneous rate of the $3$rd user is a barrier to express the problem in terms of a difference of convex functions. If the value of $K$ is even larger, the nonconforming nature of the problem to the DC programming-based approach becomes even more severe. In general, the energy-efficient power allocation problem of subchannel $n$ can be written as follows.\n\n\n\n\\begin{eqnarray}\n\\label{eq:GP-indiv1}&  \\displaystyle\\max_{\\{p_m^n\\}}\\frac{1}{p_c+\\displaystyle\\sum_{m \\in \\textbf{M}_n}p_m^n}\\displaystyle\\sum_{m \\in \\textbf{M}_n}\\mbox{log}_2\\left(1 + \\frac{g_m^np_m^n}{{\\sigma}_n^2 + \\sum_{i\\in\\textbf{M}_n^m}p_i^ng_m^n}\\right), \\\\\n\\label{eq:GP-indiv2}&\\mbox{s.t.}~\\displaystyle\\sum_{m \\in \\textbf{M}_n}p_m^n \\le p_n.\n\\end{eqnarray}\n\nExpanding the log function and then taking the first term of the series, the objective function in (\\ref{eq:GP-indiv1})-(\\ref{eq:GP-indiv2}) becomes a ratio of two posynomials. Upon applying the inverse function, the objective function is transformed to a form (i.e., $\\frac{G_n(.)}{G_d(.)}$) as described in Section~\\ref{ssec:GP}. \n\n\n\\subsubsection{Joint Computationally-Efficient Suboptimal Power Allocation among all Subchannels}\n\\label{sssec:sopt-joint}\n\n\n\n\n\nThe solution of the fine-grained power allocation problem in Section~\\ref{sssec:GP-joint} is computationally intensive due to the cumbersome iterative process of the single condensation method as well as the implementation complication of the off-the-shelf GP solvers~\\cite{cvx}. The complexity of this iterative process is proportional to the number of degrees of freedom in the system. If the number of users and subchannels in the system is moderately large, the solution of the problem cannot be obtained in a reasonable amount of time. Consequently, in this subsection, we have proposed another suboptimal solution for the fine-grained power allocation across the subchannels and users with much less computational overhead even with the off-the-shelf GP solvers~\\cite{cvx}. The mechanism of this suboptimal solution is developed based on the insights of the optimal solution, the description of which is given as follows. The idea of this suboptimal solution is to obtain the energy-efficient solution of the subchannels one by one. For this, using (18)~\\cite{FFang2016}, we plan to approximate the problem in (\\ref{eq:GP-indiv1})-(\\ref{eq:GP-indiv2}) to a one-variable optimization problem no matter the value of $|\\textbf{M}_n|$ is. If the energy-efficient power level of subchannel $n$ is $p_n$, using (18)~\\cite{FFang2016}, we have $p_m^n = {\\gamma}_m^np_n,~m \\in \\textbf{M}_n$, where $\\sum_{m\\in\\textbf{M}_n}{\\gamma}_m^n = 1$. Then, we can transform the problem in (\\ref{eq:GP-indiv1})-(\\ref{eq:GP-indiv2}) to either the problem in (\\ref{eq:sopt-DC1})-(\\ref{eq:sopt-DC2}) or that in (\\ref{eq:sopt-GP1})-(\\ref{eq:sopt-GP2}). Since we do not know the allocated power level of subchannel $n$ in advance, we assume that the maximal power level at the BS, $p^{\\mbox{max}}$, is the upper limit to be allocated at the beginning. It is proved in~\\cite{FFang2016} that $\\mbox{log}_2\\left(1 + p_nA_m(\\{{\\gamma}_m^n\\},\\{g_m^n\\},{\\sigma}^2) \\right)\/(p_c+\\sum_{m \\in \\textbf{M}_n}{\\gamma}_m^np_n)$ is a concave function, and hence the DC-programming-based approach can be used to solve the problem in (\\ref{eq:sopt-DC1})-(\\ref{eq:sopt-DC2}). On the other hand, in (\\ref{eq:sopt-GP1})-(\\ref{eq:sopt-GP2}), both the $C(\\{{\\gamma}_m^n\\},\\{g_m^n\\},p_n,{\\sigma}^2)$ and $D(\\{{\\gamma}_m^n\\},\\{g_m^n\\},p_n,{\\sigma}^2)$ are the polynomial functions of variable $p_n$. Using the first term of the log series, this problem can be transformed to a form which is the ratio of two posynomials as well. Consequently, we can adopt the GP-based single condensation method to obtain an elegant energy efficiency of subchannel $n$. From our experiments, the latter solution approach provides better solution for this problem, and so we adopt it as a part of our proposed suboptimal energy-efficient power allocation scheme. The number of iterations required for the single condensation method is constant as the number of optimization variable is one in this case no matter the value of $|\\textbf{M}_n|$ for subchannel $n$ is. \n\n\\begin{eqnarray}\n\\label{eq:sopt-DC1}&  \\displaystyle\\max_{p_n}\\frac{1}{p_c+\\displaystyle\\sum_{m \\in \\textbf{M}_n}{\\gamma}_m^np_n}\\displaystyle\\sum_{m\\in\\textbf{M}_n}\\frac{\\mbox{log}_2\\left(1 + p_nA_m(\\{{\\gamma}_m^n\\},\\{g_m^n\\},{\\sigma}^2) \\right)}{\\mbox{log}_2\\left(1 + p_nB_m(\\{{\\gamma}_m^n\\},\\{g_m^n\\},{\\sigma}^2)\\right)}, \\\\\n\\label{eq:sopt-DC2}&\\mbox{s.t.}~\\displaystyle\\sum_{m \\in \\textbf{M}_n}{\\gamma}_m^np_n \\le p^{\\mbox{max}}.\n\\end{eqnarray}\n\n\\begin{eqnarray}\n\\label{eq:sopt-GP1} &  \\displaystyle\\max_{p_n}\\frac{1}{p_c+\\displaystyle\\sum_{m \\in \\textbf{M}_n}{\\gamma}_m^np_n}\\mbox{log}_2\\left(1 + \\frac{C(\\{{\\gamma}_m^n\\},\\{g_m^n\\},p_n,{\\sigma}^2)}{D(\\{{\\gamma}_m^n\\},\\{g_m^n\\},p_n,{\\sigma}^2)}\\right), \\\\\n\\label{eq:sopt-GP2}&\\mbox{s.t.}~\\displaystyle\\sum_{m \\in \\textbf{M}_n}{\\gamma}_m^np_n \\le p^{\\mbox{max}}.\n\\end{eqnarray}\n\n\n\n\n\\begin{comment}\n\nconvex nature of the energy efficiency problem of a subchannel. Given the power constraint of $p^{\\mbox{max}}$, the energy-efficient power allocation of subchannel $n$ can be obtained by solving the optimization problem in~(\\ref{eq:sopt-indiv}). \n\\begin{eqnarray}\n\\label{eq:sopt-indiv}\n\\nonumber &  \\displaystyle\\max_{\\{p_m^n\\}}\\displaystyle\\sum_{m \\in \\textbf{M}_n}\\mbox{log}_2\\left(1 + \\frac{g_m^np_m^n}{{\\sigma}_n^2 + \\sum_{i\\in\\textbf{M}_n^m}p_i^ng_m^n}\\right), \\\\\n&\\displaystyle\\sum_{m \\in \\textbf{M}_n}p_m^n \\le p^{\\mbox{max}}.\n\\end{eqnarray}\nIt is proved in~\\cite{FFang2016} that the problem\nin~(\\ref{eq:sopt-indiv}) is convex, and hence it is straightforward to find the solution of this problem.\n\n\\end{comment}\n\nAfter solving the problem in (\\ref{eq:sopt-GP1})-(\\ref{eq:sopt-GP2}) for all the subchannels, the resultant total power $\\sum_{n \\in \\textbf{N}}\\sum_{m \\in {\\textbf{M}}_n}{\\gamma}_m^np_n$ can be $>$ or $\\le$ $p^{\\mbox{max}}$. For the latter case, the resultant total power, $\\{{\\gamma}_m^np_n\\}_{n \\in {\\textbf{M}}_n, n \\in \\textbf{N}}$, can be considered as an elegant solution. On the other hand, for the former case, we need to revisit the solution of the optimization problem further. In order to develop a good solution for this case, we need to know the energy efficiency rate per unit of power level for all subchannels. Once these information are known, we can develop a suboptimal scheme, that assigns unit power to the subchannels one by one until the total allocated power to all the subchannels results in $p^{\\mbox{max}}$. In order to know which subchannel has higher rate of energy efficiency, we construct an energy efficiency rate matrix \\textbf{EEM}, the dimension of which is $(\\lfloor{p^{\\mbox{max}}\/{\\Delta}}\\rfloor+1) \\times N$. The $1$st row of this \\textbf{EEM} matrix contains the energy efficiency of the subchannels given that each subchannel is assigned with $\\Delta$ level of power. Each element of the $2$nd row contains the subtracted energy\nefficiency when the corresponding subchannel is allocated with\n$2\\Delta$ and $\\Delta$ level of power. Finally, each element of the last row contains the subtracted energy efficiency of the\ncorresponding subchannel when it is assigned to $\\lfloor{p^{\\mbox{max}}\/{\\Delta}}\\rfloor \\Delta$ and $p^{\\mbox{max}}$ level of power. The energy efficiency of each subchannel is computed using the formulation in (\\ref{eq:sopt-GP1})-(\\ref{eq:sopt-GP2}) given the maximal power constraint designated for each element in matrix \\textbf{EEM}. Based on the information of this matrix, power is assigned to each subchannel one by one in an iterative manner. For example, in one iteration, $(a, n),~a \\in [1,\n\\lfloor{p^{\\mbox{max}}\/{\\Delta}}\\rfloor+1]$ tuple in \\textbf{EEM} matrix has been selected since it has the highest value. In this case, subchannel $n$ is assigned to $a\\Delta$ level of power. This process continues until the allocated power level to all subchannels, $\\sum_{n \\in \\textbf{N}}p_n$, is equal to $p^{\\mbox{max}}$. The detailed  steps of the entire iterative process are provided in \\textit{Algorithm \\ref{alg:sopt-pwr-alloc}}.\n\n\\begin{algorithm}[h!]\n\\caption{The computationally-efficient fine-grained energy-efficient power allocation scheme.}\n\\label{alg:sopt-pwr-alloc}\n\\begin{algorithmic}[1]\n\\FOR{$n \\in \\textbf{N}$}\n\\STATE Solve the problem in~(\\ref{eq:sopt-GP1})-(\\ref{eq:sopt-GP2}) for subchannel $n$.\n\\ENDFOR\n\n\\IF{$\\sum_{n \\in \\textbf{N}}\\sum_{m \\in \\textbf{M}_n}{\\beta}_m^np_n > p^{\\mbox{max}}$}\n\\STATE Construct matrix \\textbf{EEM}.\n\\REPEAT\n\\STATE Select an un-marked tuple $(a, n)$ from matrix \\textbf{EEM}, which\nhas the highest non-zero value.\n\\IF{$a \\in [1, \\lfloor{p^{\\mbox{max}}\/{\\Delta}}\\rfloor]$}\n\\STATE Assign $a\\Delta$ level of power to subchannel $n$.\n\\ELSE\n\\STATE Assign $p^{\\mbox{max}}$ level of power to subchannel $n$.\n\\ENDIF\n\\STATE Mark tuple $(a,n)$ in matrix \\textbf{EEM}.\n\\UNTIL{$\\sum_{n \\in \\textbf{N}}\\sum_{m \\in {\\textbf{M}}_n}{\\beta}_m^np_n  = p^{\\mbox{max}}$}\n\\ENDIF\n\\end{algorithmic}\n\\end{algorithm}\n\n\\noindent\n\\textbf{Computational complexity of \\textit{Algorithm\n\\ref{alg:sopt-pwr-alloc}}:} Ideally, the GP-based single condensation method is supposed to adopt a conventional optimization solver, such as the interior-point method to solve the transformed convex problem in each iteration. Since the number of iterations is constant for the single condensation method to solve the problem in (\\ref{eq:sopt-GP1})-(\\ref{eq:sopt-GP2}), its computational complexity is equivalent to the complexity of the adopted convex optimization solver. In~\\cite{YNesterov1994}, it is shown that the interior-point method can obtain the $\\epsilon$-optimal solution in polynomial time, and hence we can say that the complexity of the single condensation method to solve the problem in (\\ref{eq:sopt-GP1})-(\\ref{eq:sopt-GP2}) has a polynomial time complexity. Moreover, although the off-the-shelf GP solvers do not solve a GP problem in a native manner, since the number of variable(s) of the problem in (\\ref{eq:sopt-GP1})-(\\ref{eq:sopt-GP2}) is one, the computational complexity of the resultant solution is constant and quick unlike that in Section~\\ref{sssec:GP-joint}. The steps in between $1$ and  $3$, the problem in~(\\ref{eq:sopt-GP1})-(\\ref{eq:sopt-GP2}) needs to be solved $N$ times. If the resultant total power does not satisfy the power constraint, we need to construct matrix \\textbf{EEM}. Except the $1$st row, for each tuple of matrix\n\\textbf{EEM}, the problem in (\\ref{eq:sopt-GP1})-(\\ref{eq:sopt-GP2}) needs to be solved twice while replacing $p^{\\mbox{max}}$ by different values. Therefore, the process of constructing matrix \\textbf{EEM} takes a polynomial time multiplied with $(\\lfloor{p^{\\mbox{max}}\/{\\Delta}}\\rfloor+1) \\times N$. Since each tuple is marked once it is selected for the assignment of power, the steps in between $6$ and $14$ run at most $(\\lfloor{p^{\\mbox{max}}\/{\\Delta}}\\rfloor+1) \\times N$ times. As a result, the complexity of constructing matrix \\textbf{EEM} is dominated by the rest of the other steps in \\textit{Algorithm\n\\ref{alg:sopt-pwr-alloc}}.\n\n\n\n\n\n\n\n\n \\section{Performance Evaluation}\n \\label{sec:eval}\n\nIn this section, via simulation, we evaluate the performance of our proposed energy-efficient downlink resource allocation schemes. Followed by the methodology, we exhibit the detailed outcome of the simulation in order to verify the effectiveness of the proposed schemes.\n \n \\subsection{Simulation Setup}\n \nThe cellular network, that we consider, is isolated from the neighboring networks. It has a circular-like shape and suitable for an office environment. We place the BS at the center of the network, and the users are uniformly spread surrounding the BS within $500$ m distance. Typically, the minimum relative distance between two emploees in an office environment could be $40$ m, and hence we set the minimum distance between two users as $40$ m. Although we set a minimum relative distance between two users, the performance of each NOMA user is independent of this distance~\\cite{MBasit2018}. We also set the minimum distance from each user to the BS as $50$ m. As mentioned previously, continuous time is divided into frames. Each time frame is equivalent to $1$ s. During each time frame, the designated spectrum is equally subdivided among $20$ subchannels, and these subchannels are available to be allocated among $M$ users in the system. Each subchannel is assumed to have $200$ KHz bandwidth. According to~\\cite{XQiu1999}, the theoretical limit of the channel capacity is given by $\\frac{-1.5}{\\mbox{ln}(5P_b)}$, where $P_b$ denotes the Bit Error Rate (BER). The BER for each subchannel is configured as $10^{-6}$. We consider that the shadow and Rayleigh are two main fading components of a wireless channel between the BS and a user in the system. The shadow fading follows the log-normal distribution with variance $3.76$.\n\nIn order to calculate the quantity of the shadow fading over a subchannel, we assume the reference distance as $1$ km and the SNR for this reference distance is $28$ dB. The Rayleigh fading for all users over all subchannels follows the Rayleigh distribution with $0$ mean and $4.3$ variance. Using all these parameters, the gain of each subchannel for a user towards the BS is computed following ($22$) in~\\cite{RRuby2015}. We employ the SIC technique in~\\cite{SVanka2012, NIMiridakis2013} for a user in order to decode the SC-coded signal of each subchannel transmitted by the BS. \n\n\n \n\nIn addition to implementing our proposed resource allocation schemes, we have implemented the scheme in~\\cite{FFang2016}. In the following, this scheme is referred as ``Scheme in~\\cite{FFang2016}''. In order to show that the many-to-many matching model applied in the subchannel-user mapping algorithm outperforms the one-to-many matching model, we have one version (i.e., Proposed Scheme-1) which uses \\textit{Algorithm~\\ref{alg:sc-usr-map}} to solve the subchannel-user mapping problem and the DC programming-based approach for the final power allocation across the subchannels and users. Note that in this version, for step $14$ of \\textit{Algorithm~\\ref{alg:sc-usr-map}}, ($18$)~\\cite{FFang2016} is used. On the other hand, in order to show the superiority of our fine-grained power allocation scheme via the GP technique over the DC programming-based power allocation approach, we have another version (i.e., Proposed Scheme-$2$). This version adopts the subchannel-user mapping algorithm in~\\cite{FFang2016} for the first step, and adopts the scheme in Section~\\ref{sssec:GP-joint} in order to have power allocation across each allocated subchannel-user slot. We have three more versions of our proposed schemes, namely Scheme-$3$, Scheme-$4$ and Scheme-$5$, in order to exhibit a tradeoff between optimality and computational complexity. Both Scheme-$3$ and Scheme-$4$ adopt the GP technique in Section~\\ref{sssec:GP-joint} for the final power allocation. However, Scheme-$3$ uses ($18$)~\\cite{FFang2016} for step $14$ in \\textit{Algorithm~\\ref{alg:sc-usr-map}}, and Scheme-$4$ uses the GP technique in Section~\\ref{sssec:GP-indiv}. Consequently, step $14$ in \\textit{Algorithm~\\ref{alg:sc-usr-map}} happens in constant time for Scheme-$3$ version. On the other hand, this operation of Scheme-$4$ has polynomial time complexity w.r.t. $\\textbf{M}_n$. Since the maximum value of $\\textbf{M}_n$ is $K$ (e.g., $4$ in our simulation), this complexity of Scheme-$4$ can be considered negligible, but provides better solution compared to Scheme-$3$. Moreover, our Scheme-$5$ version uses the technique in Section~\\ref{sssec:GP-indiv} for step $14$ of the subchannel-user mapping algorithm, and uses \\textit{Algorithm 2} for the final fine-grained power allocation across the subchannels and users. The complexity of the suboptimal power allocation algorithm (i.e., \\textit{Algorithm 2}) is elaborately explained in Section~\\ref{sssec:sopt-joint}. Apparently, both \\textit{Algorithm 2} and the fine-grained power allocation scheme in Section~\\ref{sssec:GP-joint} have polynomial time complexity. However, since the complexity of the GP technique is computationally intensive with the large number of optimization variables (e.g., $\\sum_{n \\in \\textbf{N}}|\\textbf{M}_n|$), the proposed Scheme-$3$ or Scheme-$4$ may run very slowly with the growing number of users and subchannels in the system. On the other hand, although step $2$ of \\textit{Algorithm 2} is associated with solving the problem in~(\\ref{eq:sopt-GP1})-(\\ref{eq:sopt-GP2}) via the GP technique, the corresponding problem has only one optimization variable which results in constant time complexity for this step. Other steps of \\textit{Algorithm 2} have constant time complexity, and hence the complexity of Scheme-$5$ can be considered negligible compared to Scheme-$3$ or Scheme-$4$. However, because of the intense search of the fine-grained power allocation scheme in Section~\\ref{sssec:GP-joint}, Scheme-$3$ and Scheme-$4$ (especially Scheme-$4$) achieve much better performance compared to Scheme-$5$, which has been verified in the subsequent discussions. Finally, in order to demonstrate the global optimal performance, we apply the brute-force search on both the subchannel-user mapping and power allocation problems for a system with $10$ and $20$ users. In the following subsection, for each data point, we conduct the simulation over $10000$ times.\n\n \n \\subsection{Simulation Results}\n \n\n\\begin{figure}\n  \\begin{center}\n    \\includegraphics[width=0.5\\columnwidth]{syseff_usrs.eps}\n    \\caption{Comparison of overall energy efficiency with the increasing number of users.}\n    \\label{fig:syseff-usrs}\n  \\end{center}\n\\end{figure}\n\n\n\n\n\n\n\n\n\nFor $K = 4$, $p^{\\mbox{max}} = 23~\\mbox{dBW}$ and $p_c = 1.75~\\mbox{dBW}$, Fig.~\\ref{fig:syseff-usrs} presents the increasing energy efficiency with the increasing number of users. Given the constant power level at the BS, the more the users, the higher the sum-rate, and hence the larger the efficiency is. This trend is similar for all the schemes no matter it is ours or the scheme in~\\cite{FFang2016}. Via the many-to-many matching model, if a user has better gain over many subchannels compared to other users, that user can be assigned to as many subchannels as possible if such assignments increase the system energy efficiency. On the other hand, via the one-to-many matching model, it is possible that a user with worse gain is forced to be assigned to a subchannel, despite such assignment is not necessarily beneficial for the system energy efficiency. This is because each user can be assigned to at most one subchannel via this matching model. As a result, both intuitively and empirically, we see the evidence of enhanced performance for our Scheme-$1$ compared to that in~\\cite{FFang2016}, even when we use (18)~\\cite{FFang2016} for step $14$ in \\textit{Algorithm~\\ref{alg:sc-usr-map}} and the DC programming-based approach for the final power allocation. To provide further evidence, for $M = 40$, we plot Fig.~\\ref{fig:usrs-vs-sc} and Fig.~\\ref{fig:scs-vs-usr}, which are the outcome of the subchannel-user mapping algorithms. In Fig.~\\ref{fig:scs-vs-usr}, we compare the number of allocated subchannels to each individual user (the users are sorted in the ascending order of their distance from the BS) between the one-to-many and our model. From our observation, we see that via the many-to-many matching model, a user can be assigned to multiple subchannels, and this number depends on the subchannels over which that particular user can achieve enhanced energy efficiency compared to other users. Our channel model is such that the quantity of the shadow fading is dominated by that of the Rayleigh fading. Therefore, a user that is closer to the BS is likely to have better gain over more subchannels compared to a farther user. Therefore, in Fig.~\\ref{fig:scs-vs-usr}, we see that a user that is the closest to the BS is assigned to the largest number of subchannels. On the other hand, based on the dynamics of users' gain, in order to enhance the energy efficiency further, some subchannels accommodate more users compared to other subchannels.\n\n\n\n\n\\begin{comment}\n\n\\begin{figure}\n  \\begin{center}\n    \\includegraphics[width=0.5\\columnwidth]{usrs_vs_sc.eps}\n    \\caption{Comparison of allocated users to each individual subchannel when $M=40$.}\n    \\label{fig:usrs-vs-sc}\n  \\end{center}\n\\end{figure}\n\n\\begin{figure}\n  \\begin{center}\n    \\includegraphics[width=0.5\\columnwidth]{scs_vs_usr.eps}\n    \\caption{Comparison of allocated subchannels to each individual user when $M=40$.}\n    \\label{fig:scs-vs-usr}\n  \\end{center}\n\\end{figure}\n\\end{comment}\n\n\n\n\\begin{figure}[h!]%\n    \\centering\n    \\subfloat[Comparison of allocated users to each individual subchannel.\\label{fig:usrs-vs-sc}]{{\\includegraphics[scale=0.6]{usrs_vs_sc.eps} }}%\n   ~\n    \\subfloat[Comparison of allocated subchannels to each individual user.\\label{fig:scs-vs-usr}]{{\\includegraphics[scale=0.6]{scs_vs_usr.eps} }}%\n    \\caption{Detailed subchannel-user assignment study when $M=40$.}%\n   \n\\end{figure}\n\n\n\n\n\n\nIn Fig.~\\ref{fig:usrs-vs-sc}, we compare the number of assigned users to each individual subchannel between two subchanel-user mapping algorithms. In this figure, although $K=4$, not necessarily all subchannels have $4$ allocated users. In general, it is seen that if the difference of gain between any two assigned users over any subchannel is larger compared to other subchannels, that subchannel achieves better sum-rate as well as better energy efficiency. The numerical degree of users' gain over a subchannel also plays a crucial role in the decision whether additional user will be allocated to that subchannel or not. Because of these user dynamics, some subchannels cannot be assigned to many users as the allocation of more and more users may decrease the sum-rate as well as its energy efficiency, which is proved in \\textit{Proposition 1}. Due to the structure of the formulation, unlike the DC programming-based approach, the GP technique can provide fine-grained energy-efficient power allocation across all subchannel-user tuples. Hence, the energy efficiency is much better for this case (i.e., Scheme-$3$) even if we use (18)~\\cite{FFang2016} for step $14$ of \\textit{Algorithm~\\ref{alg:sc-usr-map}}. If we use the GP technique instead of (18)~\\cite{FFang2016} for step $14$ in \\textit{Algorithm~\\ref{alg:sc-usr-map}} (i.e., Scheme-$4$), we obtain better organization in the subchannel-user map due to the better power allocation among the users of each subchannel. Therefore, this scheme has the best performance compared to all the others. The elegance of the fine-grained GP technique is further evident from the enhanced performance of Scheme-$2$ over the scheme in \\cite{FFang2016}. In Scheme-$2$, we take the same subchannel-user mapping algorithm (via the one-to-many matching model) as that in \\cite{FFang2016}. On the other hand, we argued previously that the fine-grained power allocation using the GP technique is computationally quite intensive. Consequently, we developed \\textit{Algorithm~\\ref{alg:sopt-pwr-alloc}} in order to have fine-grained energy-efficient power allocation across all the allocated subchannel-user tuples in a low complexity manner. Because of adopting the insights of the optimal solution, the resource allocation scheme via this algorithm (i.e., Scheme-$5$) has very close performance to that of the Scheme-$3$ and Scheme-$4$ versions, and obviously outperforms the scheme with the DC programming-based approach.\n\n\n\n\n\n\\begin{figure}\n  \\begin{center}\n    \\includegraphics[width=0.5\\columnwidth]{syspwr_usrs.eps}\n    \\caption{Comparison of total power consumption with the increasing number of users.}\n    \\label{fig:syspwr-usrs}\n  \\end{center}\n\\end{figure}\n\n\n\n\n\nGiven $p^{\\mbox{max}} = 23~\\mbox{dBW}$ and $p_c = 1.75~\\mbox{dBW}$, Fig.~\\ref{fig:syspwr-usrs} presents total power consumption with the increasing number of users. This is natural that the more the users in the system, the diversity of the users that spread among all the allocated subchannels (i.e., multi-user diversity) increases. Therefore, with the increasing number of users, it is more likely that each subchannel is assigned to at least one user with good channel. On the other hand, from our observation, it is seen that if a subchannel has at least one user with good channel, it requires less power for that subchannel to reach the optimal energy-efficient state. Consequently, a lower power level should be required to reach the optimal energy-efficient state for a system with more users compared to that with less users. This observation and insights hold for both the GP technique and \\textit{Algorithm~\\ref{alg:sopt-pwr-alloc}}. However, since the mechanism of \\textit{Algorithm~\\ref{alg:sopt-pwr-alloc}} is somewhat suboptimal (although developed based on the insights of the optimal solution) compared to that in Section~\\ref{sssec:GP-joint}, total consumed power in this case is slightly larger compared to the other one. The findings of this figure are quite interesting in a sense that the schemes with the fine-grained power allocation via the GP technique and \\textit{Algorithm~\\ref{alg:sopt-pwr-alloc}} use much less power compared to that with the DC programming-based approach. If we compare Fig.~\\ref{fig:syseff-usrs} with this figure, it becomes even more interesting as our schemes achieve much better energy efficiency using much less power compared to that with the DC programming-based approach. On the other hand, the schemes with the DC programming-based approach use full power of the BS, but incur much less energy efficiency. From the perspective of green communications, the results presented in this figure verify our original motivation towards pursuing this work.\n\n\n\\begin{figure}\n  \\begin{center}\n    \\includegraphics[width=0.5\\columnwidth]{systhr_usrs.eps}\n    \\caption{Comparison of energy-efficient total throughput with the increasing number of users.}\n    \\label{fig:systhr-usrs}\n  \\end{center}\n\\end{figure}\n\n\nSimilar to our work, the conventional definition of energy efficiency is achievable blocks of bits from a channel under the usage of unit power level. The Shannon's information capacity theorem has already established that two objectives, i.e., minimizing the consumed energy and maximizing the spectral efficiency are not achievable simultaneously at the same operating point. Consequently, under the consideration of fixed circuit power, there always exist two separate optimal points in the energy efficiency versus spectrum efficiency curve.  In Fig.~\\ref{fig:systhr-usrs}, we compare the energy-efficient total throughput acheievd by all aforementioned schemes. The more the users, the better the utilization of limited resources because of the enhanced multi-user diversity. Hence, we see the increasing trend in the energy-efficient total throughput, achieved by all the schemes, with the increasing number of users. However, due to the fact in the Shannon's information capacity theorem, since the scheme in~\\cite{FFang2016} does not achieve the optimal energy-efficient state, it is possible that the total energy-efficient throughput achieved by this scheme is larger than our schemes. This is what observed in this figure.\n\n\n\n\n\n\n\n\\begin{comment}\n\\begin{figure}\n  \\begin{center}\n    \\includegraphics[width=0.5\\columnwidth]{syseff_pwr.eps}\n    \\caption{Comparison of overall energy efficiency with the increasing power at the BS.}\n    \\label{fig:syseff-pwr}\n  \\end{center}\n\\end{figure}\n\n\\begin{figure}\n  \\begin{center}\n    \\includegraphics[width=0.5\\columnwidth]{syspwr_pwr.eps}\n    \\caption{Comparison of total power consumption with the increasing power at the BS.}\n    \\label{fig:syspwr-pwr}\n  \\end{center}\n\\end{figure}\n\\end{comment}\n\n\\begin{figure}[h!]%\n    \\centering\n    \\subfloat[Overall energy efficiency.\\label{fig:syseff-pwr}]{{\\includegraphics[scale=0.6]{syseff_pwr.eps} }}%\n   ~\n    \\subfloat[Total power consumption.\\label{fig:syspwr-pwr}]{{\\includegraphics[scale=0.6]{syspwr_pwr.eps} }}%\n    \\caption{Comparison of overall energy efficiency and total power consumption with the increasing power at the BS.}%\n   \n\\end{figure}\n\n\nIn Fig.~\\ref{fig:syseff-pwr}, given $M = 40$ and $p_c = 1.75~\\mbox{dBW}$, we show the energy efficiency with the increasing power at the BS. As we saw in the previous results that the scheme in~\\cite{FFang2016} fails to exploit the multi-user diversity of wireless systems as well as uses the DC programming-based approach to obtain the coarse-grained power allocation, this scheme has the lowest energy efficiency no matter the power constraint of the BS is. For this case, we see that with the increasing power level, the trend of energy efficiency is decreasing. This is because there is a tradeoff between the transmission capacity and the energy-efficient power consumption. Whereas, for our case, since the GP technique provides a fine-grained elegant power allocation, the overall energy efficiency is much better compared to the benchmark scheme. Moreover, since the GP technique provides the unique solution while consuming much less power, no matter we increase the power level of the BS, the energy efficiency remains same at the unique point. Similar trend is observed in the case of our suboptimal \\textit{Algorithm~\\ref{alg:sopt-pwr-alloc}} although the energy efficiency achieved by this scheme is slightly lower compared to the fine-grained GP-based power allocation scheme. In order to show the power consumption for this case, we plot Fig.~\\ref{fig:syspwr-pwr}. Since the optimal energy-efficient state of the system is unique and our proposed GP technique is able to search this state to some extent, we see the constant level of used power no matter how much power the BS has. In the similar manner, given the subchannel-user mapping matrix, via our suboptimal \\textit{Algorithm~\\ref{alg:sopt-pwr-alloc}}, we obtain an elegant energy-efficient point for each individual subchannel. Consequently, although this suboptimal scheme does not achieve as good solution as by that in Section~\\ref{sssec:GP-joint}, the suboptimal unique energy efficiency is still achieved at the unique power level. Whereas, the schemes, which use the DC programming-based approach, consume full power of the BS. Therefore, in this case, we see the increasing power consumption with the increasing total power at the BS.\n\n\n\\begin{comment}\n\\begin{figure}\n  \\begin{center}\n    \\includegraphics[width=0.5\\columnwidth]{figures\/syseff_pc.eps}\n    \\caption{Comparison of overall energy efficiency with the increasing $p_c$.}\n    \\label{fig:syseff-pc}\n  \\end{center}\n\\end{figure}\n\n\\begin{figure}\n  \\begin{center}\n    \\includegraphics[width=0.5\\columnwidth]{figures\/syspwr_pc.eps}\n    \\caption{Comparison of total power consumption with the increasing $p_c$.}\n    \\label{fig:syspwr-pc}\n  \\end{center}\n\\end{figure}\n\\end{comment}\n\n\\begin{figure}[h!]%\n    \\centering\n    \\subfloat[Overall energy efficiency.\\label{fig:syseff-pc}]{{\\includegraphics[scale=0.6]{syseff_pc.eps} }}%\n   ~\n    \\subfloat[Total power consumption.\\label{fig:syspwr-pc}]{{\\includegraphics[scale=0.6]{syspwr_pc.eps} }}%\n    \\caption{Comparison of overall energy efficiency and total power consumption with the increasing $p_c$.}%\n   \n\\end{figure}\n\n\n\nIn Fig.~\\ref{fig:syseff-pc}, given $p^{\\mbox{max}} = 23~\\mbox{dBW}$ and $M = 40$, we show the decreasing overall energy efficiency with the increasing circuit power consumption. This is a natural trend achieved by all the schemes. Since circuit power is not used in enhancing system throughput but is added as consumed power, the overall energy efficiency is decreasing with the increasing circuit power. However, our subchannel-user mapping scheme can better exploit the multi-user diversity of wireless systems and we use the GP technique for the final fine-grained energy-efficient power allocation, our schemes, even via our suboptimal \\textit{Algorithm~\\ref{alg:sopt-pwr-alloc}}, always outperform the scheme in~\\cite{FFang2016}. In order to show the total power consumption in this case, we plot Fig.~\\ref{fig:syspwr-pc} with the increasing $p_c$. Increasing $p_c$ means, a subchannel requires higher power to reach its energy-efficient state. Consequently, the optimal energy efficiency is achieved at larger power level with the increasing $p_c$. For the similar reason, via our suboptimal \\textit{Algorithm~\\ref{alg:sopt-pwr-alloc}}, the same trend is observed with the increasing $p_c$. On the other hand, since the DC programming-based approach uses full power of the BS no matter the value of $p_c$ is, the total power consumption is constant (i.e., $p^{\\mbox{max}}$) via the scheme in~\\cite{FFang2016} and our Scheme-$1$ version.    \n\n\\section{Conclusion}\n\\label{sec:concl}\n\nIn this paper, we proposed an energy-efficient downlink subchannel and power allocation scheme for NOMA systems with enhanced performance compared to the most relevant existing work in the literature. Due to the discrete nature of subchannel assignment and the characteristics of the NOMA technique, this is an MINLP problem. Therefore, similar to an existing work, we solved the problem via decomposing it into a subchannel allocation subproblem followed by a power loading subproblem. However, unlike the existing work, via a many-to-many matching model, we better exploited the multi-user diversity of wireless systems in the solution of the subchannel-user mapping subproblem. In the second step, unlike the DC programming-based approach, via the GP technique, we were able to allocate the power level across the allocated subchannel-user slots in a fine-grained manner such that better energy efficiency is achieved compared to the benchmark scheme. Since the fine-grained power allocation via the GP technique is computationally intensive using the off-the-shelf GP solvers, we also proposed a suboptimal fine-grained power allocation algorithm with much lower computational complexity. Under various realistic scenarios, extensive simulation had been conducted to verify that our scheme (even via our computationally-efficient suboptimal power allocation algorithm) can outperform the existing scheme while consuming much less power in the system. \n\n\nBesides achieving an elegant energy-efficient state via the better resource allocation schemes, the implication of this work is extended to a certain extent. Via our schemes (even the suboptimal one), since better energy efficiency is achieved at a lower power level, the interference effect to the neighboring networks is expected to be mitigated. At the same time, unused power in the system can be used for other purposes.\n\n\\begin{appendices}\n\n\\section{Proof of Proposition 1}\n\n\\begin{figure}\n  \\begin{center}\n    \\includegraphics[width=0.5\\columnwidth]{prop1_proof.eps}\n    \\caption{A sample example that $[R_n(\\{1,2,3\\})-R_n(\\{1,2\\})]$ can be both positive and negative.}\n    \\label{fig:prop1-proof}\n  \\end{center}\n\\end{figure}\n\nConsider a subchannel $n$, and it has two allocated users, indexed by $1$ and $2$. Moreover, we set $K = 4$, and there is one assumption, i.e., $R_n(\\{1, 2\\}) > R_n(\\{1\\})$. It implies that the sum-rate of user $1$ and user $2$ is larger than that of only user $1$. Now, user $3$ has come to be assigned with subchannel $n$. There are two possible conditions for this assignment, which are $R_n(\\{1,2,3\\}) > R_n(\\{1,2\\})$ and $R_n(\\{1,2,3\\}) < R_n(\\{1,2\\})$. According to step $15$ of \\textit{Algorithm~\\ref{alg:sc-usr-map}}, we only consider this user to construct an addition strategy if and only if the former case is true. In practice, both the conditions for any subchannel $n$ can happen. This statement can be proved from the result of Fig.~\\ref{fig:prop1-proof}. In this figure, we plot $R_n(\\{1,2,3\\}) - R_n(\\{1,2\\})$ w.r.t. $g_3^n$. It is obvious that $R_n(\\{1,2,3\\}) - R_n(\\{1,2\\})$ can be both positive and negative. The energy efficiency of subchannel $n$ is enhanced if and only if the aforementioned value is positive, and user $3$ is not added to subchannel $n$ for the negative case. This completes the proof. Note that $g_1^n = 0.4141,~g_2^n = 6.2512$ and $p_n = 50$ in Fig.~\\ref{fig:prop1-proof}. \n\n\\section{Proof of Proposition 2}\n\n\\begin{figure}\n  \\begin{center}\n    \\includegraphics[width=0.5\\columnwidth]{prop2_proof.eps}\n    \\caption{A sample example that $R_n(\\{1,2\\})$ is increasing w.r.t. $g_2^n$.}\n    \\label{fig:prop2-proof}\n  \\end{center}\n\\end{figure}\n\nWithout loss of generality, let us assume $K=2$. Consider that subchannel $n$ has already $2$ users, indexed by $1$ and $2$, and $g_1^n > g_2^n$ holds. At this point, user $3$ has come to be assigned with subchannel $n$ with $g_3^n > g_2^n~\\mbox{and}~g_1^n > g_3^n$. Since the nature of function $R_n(\\{1,2\\})$ is increasing w.r.t. $g_2^n$ (as shown in Fig.~\\ref{fig:prop2-proof}), we can conclude $R_n(\\{1,3\\}) > R_n(\\{1,2\\})$. Moreover, using (18)~\\cite{FFang2016}\nand then simplifying, $R_n(\\{1,3\\})$ and $R_n(\\{3,2\\})$ are given by\n\n\\begin{eqnarray}\n\\label{prop2:f}  & R_n(\\{1,3\\}) = \\\\\n& \\nonumber \\mbox{log}_2\\left((1+\\frac{p_n(g_1^n)^{1+\\alpha}}{(g_1^n)^{\\alpha}+(g_3^n)^{\\alpha}})(1\n+ \\frac{p_n(g_1^n)^{1+\\alpha}}{(g_1^n)^{\\alpha}+(g_3^n)^{\\alpha}+p_ng_3^n(g_1^n)^{\\alpha}})\\right),\n\\\\\n\\label{prop2:l} & R_n(\\{3,2\\}) = \\\\\n& \\nonumber \\mbox{log}_2\\left((1+\\frac{p_n(g_3^n)^{1+\\alpha}}{(g_2^n)^{\\alpha}+(g_3^n)^{\\alpha}})(1\n+ \\frac{p_n(g_2^n)^{1+\\alpha}}{(g_2^n)^{\\alpha}+(g_3^n)^{\\alpha}+p_ng_2^n(g_3^n)^{\\alpha}})\\right),~\\mbox{respectively}.\n\\end{eqnarray}\n\nSince $g_1^n > g_3^n > g_2^n$, from (\\ref{prop2:f}) and\n(\\ref{prop2:l}), for any $\\alpha < 0$, it is straightforward to say $R_n(\\{1,3\\}) > R_n(\\{3,2\\})$. Now, if another user $4$ comes, two possible cases are possible. The first case is, user $4$ will replace further user $3$ if both $g_4^n > g_3^n~\\mbox{and}~g_1^n > g_4^n$ hold. In order to prove this proposition, we need to show that rejected user $2$ cannot match with subchannel $n$ further as the assignment of user $2$ to subchannel $n$ cannot enhance the energy efficiency any more. For this case, since $g_3^n > g_2^n$, via substituting user $3$ by user $4$ in (\\ref{prop2:f}) and (\\ref{prop2:l}), $R_n(\\{1,4\\}) > R_n(\\{2,4\\})$ can be\nproved. For the sake of soundness, the second possible case is, user $4$ approaches subchannel $n$ with $g_4^n < g_3^n$. In this case, user $3$ is not substituted by user $4$ as $R_n(\\{1,4\\})$ is an increasing function of $g_4^n$ according to Fig.~\\ref{fig:prop2-proof}. Now, our objective is to prove $R_n(\\{1,3\\}) > R_n(\\{3,4\\}) > R_n(\\{2,4\\})$. While substituting user $2$ by user $4$ in (\\ref{prop2:f}) and (\\ref{prop2:l}),\n$R_n(\\{1,3\\}) > R_n(\\{3,4\\})$ is straightforward as $g_1^n > g_3^n > g_4^n$ is true.  In the similar manner,  $R_n(\\{3,4\\}) > R_n(\\{2,4\\})$ can be proved as well due to the $g_3^n > g_4^n$ and $g_3^n > g_2^n$ relations. Thus, the proof of this proposition is completed.\n\n\n\n\n\\section{Proof of Theorem 1}\n\nLet us prove this theorem by contradiction. Consider a matching relation $\\mu$ and a user-subchannel pair $(m, n)$, which satisfy $m \\not\\in {\\mu}(n)$ and $n \\not\\in {\\mu}(m)$. Although this pair is not matched, both user $m$ and subchannel $n$ prefer each other over the remaining other subchannels and users, respectively. Now, according to the steps of \\textit{Algorithm \\ref{alg:sc-usr-map}}, two possible cases can happen. The first case is, $|\\bm{\\Omega}_m| < K$, and user $m$ proposes subchannel $n$ to be matched with (step $5$ in \\textit{Algorithm \\ref{alg:sc-usr-map}}). Over receiving the proposal, according to \\textit{Proposition 1}, if the energy efficiency of subchannel $n$ is enhanced by adding user $m$, subchannel $n$ is paired with user $m$, otherwise not. From the latter case, it can be concluded that although user $m$ prefers subchannel $n$ over the remaining other subchannels, subchannel $n$ does not prefer user $m$ over the remaining other users. The second case is, subchhanel $n$ is already matched with user $m$ through the addition strategy. However, later, in some iteration, user $m'$ proposes subchannel $n$ to be matched with, and subchannel $n$ prefers user $m'$ over user $m$ because of the enhanced energy efficiency. Consequently, user $m$ will be replaced by user $m'$ (step $19$ - step $20$) in ${\\mu}(n)$. From this discussion, it can be concluded that if both the user $m$ and subchannel $n$ prefer each other over the remaining other players, there is no way that a matching relation will not be established between them. Consequently, the initial statement of this proof is shown to be false. This concludes the proof of this theorem. \n\n\\begin{comment}\n\n\\section{Proof of Proposition 3}\n\n\\begin{figure}\n  \\begin{center}\n    \\includegraphics[width=1.0\\columnwidth]{figures\/eeth_figure.eps}\n    \\caption{Comparison of total energy efficiency (EE) and total throughput (TH) for the same increasing level of $p_1$ and $p_2$.}\n    \\label{fig:EE-vs-TH}\n  \\end{center}\n\\end{figure}\n\n\n\nThis statement can be proved by contradiction. We\nassume that the two objectives of maximizing energy efficiency and maximizing spectrum efficiency can be achieved simultaneously. Without loss of generality, let assume $K = 2$, $\\textbf{N} = \\{1,2\\}$, $\\textbf{M} = \\{1,2,3,4\\}$, $\\textbf{M}_1 = \\{1,2\\}$,\n$\\textbf{M}_2 = \\{3,4\\}$. For the given value of $p_n$, each user in subchannel $n$ obtains power level based on (18)~\\cite{FFang2016}. Fig.~\\ref{fig:EE-vs-TH} plots the overall throughput and energy efficiency for different values of $p_1$ and $p_2$. From the figure, it is obvious that there are certain values $p_1$ and $p_2$ for which the energy efficiency is better than the overall throughput, and there are other values of those parameters for which the overall throughput is better. More specifically, when $p_1 = p_2 = 4.9$, the overall throughput is better than the energy efficiency (right subfigure), and when $p_1 = 3.5, p_2 = 4.1$, the energy efficiency is\nbetter than the overall throughput (left subfigure). Therefore, the optimal\nenergy-efficient power allocation is associated with the latter case, and the throughput-optimal power allocation belongs to the former case. This completes the proof.\n\n\\end{comment}\n\n\\end{appendices}\n\n\n\n\\section*{References}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe puzzle of dark matter remains an outstanding problem of particle physics. One of the more attractive approaches to this problem exploits the fact that weakly interacting massive particles (WIMPs) in thermal equilibrium produce the relic dark matter abundance in the right ballpark. Particles of this type appear in many extensions of the Standard Model. In this work, we explore the possibility that the Standard Model is connected to the dark sector through the Higgs portal~\\cite{Silveira:1985rk},\n\\begin{equation}\nV_{\\rm portal}= \\lambda_{h \\phi} H^\\dagger H \\phi^\\dagger \\phi \\;,\n\\label{eq1}\n\\end{equation}\nwhere $H$ is the Higgs field and $\\phi$ is the ``hidden Higgs'', that is, the field responsible for breaking the gauge symmetry of the hidden sector. We assume the dark sector to be endowed with U(1) or SU(N) gauge symmetry. In that case, the massive gauge fields can play the role of WIMP--type dark matter quite naturally.\nIndeed, they are weakly coupled to the Standard Model and, owing to inherent discrete symmetries, can be stable.\n\nThe U(1) case was considered in Ref.~\\cite{Lebedev:2011iq} where the stabilising $Z_2$ symmetry was found to be related to charge conjugation.\nThe SU(2) example was worked out in Ref.~\\cite{Hambye:2008bq} and the DM stability was attributed to a custodial SO(3). In this work, we extend these ideas to larger SU(N) Lie groups and uncover the nature of the underlying discrete symmetries. We focus on weakly coupled theories, although\nconfining hidden sectors represent a viable alternative~\\cite{Hambye:2009fg,Boddy:2014yra}.\nFinally, we perform a comprehensive study of DM phenomenology in all of these cases, which includes direct and indirect DM detection as well as an analysis of the DM relic abundance. Our main conclusion is that massive hidden gauge fields serve as viable and attractive DM candidates.\n\n\n\\section{Massive U(1) and SU(2) gauge fields as vector DM}\nIn this section, we review the cases of massive U(1) and SU(2) gauge fields as vector DM and identify the underlying symmetries leading to stability of vector dark matter.\nIn what follows we assume that the hidden sector consists only of gauge fields and of scalar multiplets which are necessary to make these gauge fields massive.\n\n\\subsection{Hidden U(1)}\nAn Abelian gauge sector provides the simplest example of the vector DM model endowed\nwith a natural $Z_2$ symmetry~\\cite{Lebedev:2011iq}. In this case, the $Z_2$ corresponds to the charge conjugation symmetry.\n \n Consider a U(1) gauge theory with a single charged scalar $\\phi$,\n \\begin{equation}\n {\\cal L_{\\rm hidden}}= -{1\\over 4} F_{\\mu\\nu} F^{ \\mu\\nu} + (D_\\mu \\phi)^\\dagger D^\\mu \\phi -V(\\phi) \\;,\n \\end{equation}\n where $F_{\\mu\\nu}$ is the field strength tensor of the gauge field $A_\\mu$ and \n $V(\\phi)$ is the scalar potential. For easier comparison to the non--Abelian case, we take the charge of $\\phi $ to be +1\/2. Suppose at the minimum of the scalar potential\n $\\phi$ develops a VEV, $\\langle \\phi \\rangle= 1\/\\sqrt{2}~ \\tilde v$ . The imaginary part of $\\phi$ gets eaten by the gauge field which\n now acquires the mass $m_A= \\tilde g \\tilde v \/2$, where $\\tilde g$ is the gauge coupling.\n The real part of $\\phi$ remains as a degree of freedom. Denoting it by $\\rho$ and normalising it canonically, $\\phi=1\/\\sqrt{2}~ (\\rho +\\tilde v)$ , we get the following gauge--scalar interactions:\n \\begin{eqnarray}\n&& \\Delta {\\cal L}_{\\rm s-g}= {\\tilde g^2\\over 4} \\tilde v \\rho \\; A_\\mu A^{ \\mu} +\n {\\tilde g^2\\over 8} \\rho^2 \\; A_\\mu A^{\\mu} \\;. \n \\end{eqnarray}\nThe system possesses the $Z_2$ symmetry \n\\begin{equation}\nA_\\mu \\rightarrow - A_\\mu ~,\n\\end{equation} \nwhich is the usual charge conjugation symmetry. In terms of the original scalar field, this symmetry acts as $\\phi \\rightarrow \\phi^*$ and $A_\\mu \\rightarrow - A_\\mu$, which is preserved by both the Lagrangian and the vacuum. The $Z_2$ makes the massive gauge field stable.\nNote that this symmetry applies to a sequestered U(1) which has no tree\nlevel mixing with the hypercharge, in which case no mixing is generated radiatively either.\n\nInteractions with the visible sector proceed through the Higgs portal coupling\n\\begin{equation}\n {\\cal L_{\\rm portal}}= - \\lambda_{h \\phi} \\vert H \\vert^2 \\vert \\phi \\vert^2 \\;,\n \\end{equation}\nwhich also leads to the Higgs mixing with $\\rho$. \nIn the unitary gauge, the Higgs field is given by $H^T= (0, v+h)\/\\sqrt{2}$.\nThe fields $\\rho$ and $h$ are then to be expressed in terms of the mass eigenstates $h_{1,2}$ as follows: \n\\begin{eqnarray}\n&& \\rho= - h_1 \\; \\sin\\theta + h_2 \\; \\cos\\theta \\;, \\nonumber\\\\\n&& h = h_1 \\; \\cos\\theta + h_2 \\; \\sin\\theta \\;,\n \\end{eqnarray}\nwhere the mixing angle $\\theta$ is constrained by various experiments, most notably, by LEP and LHC. \nThe upper bound on $\\sin \\theta$ depends on the mass of the heavier state $h_2$ and is around 0.3 for $m_{h_2}$ of the order of a TeV, see e.g.~figure 3 of Ref.~\\cite{Falkowski:2015iwa} for details.\nThe lighter state $h_1$ is identified with the 125 GeV Higgs, while the mass of the second state can vary in a wide range.\n\nAs the Higgs portal necessarily preserves the $Z_2$ symmetry, $A_\\mu$ \nis a viable DM candidate.\nAll the relevant scattering processes proceed through an exchange of $h_1$ and $h_2$, which include DM annihilation into the SM particles as well as DM scattering on nucleons.\n \n \n\\subsection{Hidden SU(2)}\nThe U(1) considerations can easily be extended to SU(2), albeit with a modification of the stabilising symmetry.\nConsider an SU(2) gauge theory with one doublet $\\phi$,\n \\begin{equation}\n {\\cal L_{\\rm hidden}}= -{1\\over 4} F_{\\mu\\nu}^a F^{a \\; \\mu\\nu} + (D_\\mu \\phi)^\\dagger D^\\mu \\phi -V(\\phi) \\;,\n \\end{equation}\nwhere $a=1,2,3$.\nThe potential is assumed to have a minimum at a nonzero VEV of $\\phi$. In the unitary gauge, $\\phi$ takes the form \n\\begin{equation}\n\\phi = {1\\over \\sqrt{2}}\\; \\left( \n\\begin{matrix}\n 0 \\\\\n \\rho +\\tilde v\n\\end{matrix}\n\\right)~,\n\\end{equation}\nwith $\\rho$ being a real field and $\\tilde v$ being the VEV. Denoting the gauge coupling by $\\tilde g$,\nthis leads to the gauge boson mass $m_A= \\tilde g \\tilde v\/2$. The scalar--gauge and gauge--gauge field interactions are given by\n\\begin{eqnarray}\n&& \\Delta {\\cal L}_{\\rm s-g}= {\\tilde g^2\\over 4} \\tilde v \\rho \\; A_\\mu^a A^{ a\\; \\mu} +\n {\\tilde g^2\\over 8} \\rho^2 \\; A_\\mu^a A^{ a\\; \\mu} \\;, \\nonumber \\\\\n&& \\Delta {\\cal L}_{\\rm g-g} = - \\tilde g \\epsilon^{abc} (\\partial_\\mu A_\\nu^a) A^{\\mu \\;b}\nA^{\\nu \\;c} -{\\tilde g^2\\over 4} \\left( (A_\\mu^a A^{\\mu \\;a})^2 -\n A_\\mu^a A_\\nu^a \\; A^{\\mu \\;b} A^{\\nu \\;b} \\right) ~. \n\\end{eqnarray} \nAlthough the triple gauge vertex breaks the parity of the previous section,\nit follows that the system possesses a $Z_2 \\times Z_2'$ symmetry,\n\\begin{eqnarray}\n&& Z_2: ~~ A^1_\\mu \\rightarrow - A^1_\\mu ~~,~~ A^2_\\mu \\rightarrow - A^2_\\mu \\;, \\nonumber \\\\\n&& Z_2':~~ A^1_\\mu \\rightarrow - A^1_\\mu ~~,~~ A^3_\\mu \\rightarrow - A^3_\\mu \\;.\n\\end{eqnarray}\nAs a result, all three $A^a_\\mu$ fields are stable and can play the role of dark matter.\nWhile the above symmetry is \nsufficient to ensure stability of DM, it actually generalises in this simple case to a custodial \nSO(3)~\\cite{Hambye:2008bq}. As we will see, for larger SU(N) groups, it is the discrete\nsymmetry that plays a crucial role. The first $Z_2$ is associated with a gauge transformation, while the $Z_2'$ generalises the charge conjugation symmetry, i.e.~it\ncorresponds to complex conjugation of the group elements. \n\n\nAs before, the dark sector couples to the visible one through the Higgs portal,\n$\\lambda_{h \\phi} \\vert H \\vert^2 \\vert \\phi \\vert^2 $. The discussion of the Higgs mixing with $\\rho$ of the previous section applies here as well.\nClearly, the $Z_2 \\times Z_2'$ symmetry is preserved by the Higgs portal and the hidden gauge fields couple to the visible sector only in pairs. \n\n \n\\section{Extension to SU(3)}\nWhile the SU(2) case is straightforward, larger SU(N) groups exhibit more complicated breaking patterns. In a phenomenologically viable set--up, the symmetry must be broken completely to avoid the existence of massless fields (barring confinement). One possibility \nwould be to break SU(3) by two scalar multiplets in the fundamental representation, i.e.~triplets. One may also explore other options involving SU(3) tensors. As we detail below,\nour conclusion is that a single irreducible representation with two indices cannot break SU(3) completely, which leaves the two triplet option as the minimal one. \n\n\n\\subsection{Breaking SU(3) by tensor fields}\nThe lowest order SU(3) tensor whose generic VEV can break SU(3) completely is the symmetric tensor $\\phi_{ij}$, that is {\\bf 6} of SU(3). \nGauge transformations act on $\\phi_{ij}$ as\n\\begin{equation}\n\\phi \\rightarrow U \\phi \\, U^T \\;.\n\\end{equation}\nBy virtue of Takagi's matrix decomposition, this allows one to bring $\\phi_{ij}$\nto the diagonal form, \n\\begin{equation}\n\\phi = \\left( \n\\begin{matrix}\n \\phi_1 & 0 & 0 \\\\\n 0 & \\phi_2 & 0 \\\\\n 0 & 0 & \\phi_3\n\\end{matrix}\n\\right)~,\n\\end{equation}\nwhere $\\phi_1, \\phi_2, \\phi_3$ are real up to an overall complex phase.\nIf the VEVs of $\\phi_1, \\phi_2 $ and $\\phi_3$ are all different, SU(3) is broken completely. However,\nwhen some of them coincide, the residual gauge group is at least SO(2).\n\nIn order to determine what VEVs are possible, let us write down the most general gauge invariant potential for $\\phi_{ij}$.\nIt is easy to convince oneself that the potential has the form\n\\begin{equation}\nV= m^2 \\; {\\rm Tr} \\phi^\\dagger \\phi + \\lambda_1 {\\rm Tr} (\\phi^\\dagger \\phi)^2\n+ \\lambda_2 \\left( {\\rm Tr} \\phi^\\dagger \\phi \\right)^2 +\n \\left( \\mu \\; {\\rm Det} \\phi + {\\rm h.c.} \\right) \\;,\n\\end{equation}\nwhere $m^2$ can be negative.\nThe minima of this potential determine the $\\phi_i$ VEVs.\nActing upon $V$ with the operator $\\phi_i \\partial\/\\partial \\phi_i$ (no summation over $i$), one finds that all nonzero $ \\langle \\phi_i \\rangle $ satisfy the same equation.\nThis implies that $ \\langle \\phi_i \\rangle $ are either degenerate or zero. \nFor the case $\\mu=0$,\nsuch behaviour has been noticed in~\\cite{Li:1973mq,Jetzer:1983ij}.\nWe find (analytically and numerically) that it persists in the case of nonzero $\\mu$ as well.\nAs a result, the residual gauge symmetry is at least SO(2) and the corresponding gauge bosons remain massless.\\footnote{ Ref.~\\cite{D'Eramo:2012rr} has considered $\\langle \\phi \\rangle$ proportional to the unit matrix, which entails unbroken SO(3). The corresponding gauge bosons may be confined in glueballs.}\nThus, the model with a single symmetric tensor is unrealistic. \n\nOne may also consider the possibility of breaking SU(3) by an antisymmetric tensor.\nBy virtue of Youla's decomposition, it can be gauge--transformed to the block--diagonal form \n\\begin{equation}\n\\phi = \\left( \n\\begin{matrix}\n 0 & \\phi_1 & 0 \\\\\n -\\phi_1 & 0 & 0 \\\\\n 0 & 0 & 0\n\\end{matrix}\n\\right)~.\n\\end{equation} \nTherefore, the residual gauge symmetry is at least Sp(1)=SU(2), which can also be understood via the equivalence between the antisymmetric tensor and the (anti)funda\\-mental representation. Again we find that the model is unrealistic.\n\nSimilarly, complete SU(3) breaking cannot be achieved by a VEV of an adjoint scalar.\nIn this case, the group rank is preserved and thus there are massless gauge fields.\n\nThe next simplest option is to combine the symmetric and antisymmetric tensors. This system would have 9 complex degrees of freedom, while two SU(3) triplets only have 6.\nThus SU(3) breaking with two triplets is minimal and sufficient for our purposes. \n\n\n\\subsection{Breaking SU(3) by triplets and $Z_2 \\times Z_2'$ }\nMisaligned VEVs of two triplets break SU(3) completely. This breaking pattern can be understood in stages: the first triplet VEV reduces the symmetry to SU(2), while the second breaks the remaining SU(2). VEVs misaligned in SU(3) space represent a generic situation, that is, they result from the minimisation of a general scalar potential consistent with SU(3) symmetry. Therefore, such a breaking pattern is phenomenologically viable. \n\nBefore going into details, let us identify the Lie group discrete symmetries which eventually lead to DM stability. One way to find them is to analyse the SU(3) structure constants (using the usual Gell-Mann basis),\n\\begin{eqnarray}\n&& f^{123}=1 \\;, \\nonumber\\\\\n&& f^{147}=-f^{156}= f^{246}=f^{257}=f^{345}=-f^{367}={1\\over 2} \\;, \\nonumber\\\\\n&& f^{458}=f^{678}= {\\sqrt{3}\\over 2} \\;.\n\\end{eqnarray}\nIdentifying the transformation properties of the generators with those of the gauge fields, we define the ``parity'' transformation as\n\\begin{equation}\nA^a_\\mu \\rightarrow \\eta (a) A^a_\\mu \\;.\n\\end{equation}\nIt is easy to see that the structure constants are invariant if the parities are\n\\begin{eqnarray}\n Z_2: ~~&&\\eta (a) = -1 ~~{\\rm for }~~a=1,2,4,5 ~, \\nonumber \\\\\n &&\\eta (a) = +1 ~~{\\rm for }~~a=3,6,7,8 ~,\n\\end{eqnarray}\nand also\n\\begin{eqnarray}\n Z_2': ~~&&\\eta (a) = -1 ~~{\\rm for }~~a=1,3,4,6,8 ~, \\nonumber \\\\\n &&\\eta (a) = +1 ~~{\\rm for }~~a=2,5,7 ~.\n\\end{eqnarray}\nOne may notice that the first $Z_2$ acts on the off--diagonal generators with nonzero entries in the first row, while the second reflects the real SU(3) generators.\nIn Section \\ref{SUN}, we will show that these symmetries generalise to arbitrary SU(N) and \nthat the first $Z_2$ is a gauge transformation, whereas the second corresponds to an outer automorphism of the group, i.e.~complex conjugation of the group elements.\n\nThese symmetries are inherited by the Yang--Mills Lagrangian. If {\\it CP} is conserved, they are also preserved in the matter sector leading to stable dark matter. Below we study the relevant interactions in detail.\n \n\n\\subsection{Explicit example}\nConsider an SM extension by two complex fields $\\phi_1$ and $\\phi_2$ transforming as triplets of hidden gauge SU(3).\nThe Lagrangian of the model is\n\\begin{equation}\n{\\cal L}_{\\rm SM} + {\\cal L}_{\\rm portal} + {\\cal L}_{\\rm hidden} \\;,\n\\end{equation}\nwhere \n\\begin{subequations}\n\\bal\n-{\\cal L}_{\\rm SM} &\\supset V_{\\rm SM} =\\frac{\\lambda_{H}}{2} |H|^4+m_{H}^2 |H|^2 \\;,\n \\\\\n-{\\cal L}_{\\rm portal} &= V_{\\rm portal}= \\lambda_{H11} \\, |H|^2 |\\phi_1|^2 + \\lambda_{H22} \\, |H|^2 | \\phi_2|^2 - ( \\lambda_{H12} \\, |H|^2 \\phi_1^\\dagger \\phi_2 +{\\textrm{h.c.}})\\;,\n \\\\\n{\\cal L}_{\\rm hidden} &= - \\frac12 \\textrm{tr} \\{G_{\\mu \\nu} G^{\\mu \\nu}\\} + |D_\\mu \\phi_1|^2 + |D_\\mu \\phi_2|^2 -V_{\\rm hidden} \\,.\n\\eal\n\\end{subequations}\nHere, $G_{\\mu \\nu}=\\partial_\\mu A_\\nu - \\partial_\\nu A_\\mu + i \\tilde g [A_\\mu,A_\\nu]$ is the field strength tensor of the SU(3) gauge fields $A_\\mu^a$, $D_\\mu \\phi_i = \\partial_\\mu \\phi_i + i \\tilde g A_{\\mu} \\phi_i$ is the covariant derivative of $\\phi_i$, $H$ is the Higgs doublet, and the most general renormalisable hidden sector scalar potential is given by \n\\bal\nV_{\\rm hidden}(\\phi_1,\\phi_2) &=\nm_{11}^2 |\\phi_1|^2\n+ m_{22}^2 |\\phi_2|^2\n- ( m_{12}^2 \\phi_1^\\dagger \\phi_2 + {\\textrm{h.c.}} )\n\\nonumber \\\\ \n& \n+ \\frac{\\lambda_1}{2} |\\phi_1|^4\n+ \\frac{\\lambda_2}{2} |\\phi_2|^4\n+ \\lambda_3 |\\phi_1|^2 |\\phi_2|^2\n+ \\lambda_4 | \\phi_1^\\dagger\\phi_2 |^2\n\\nonumber \\\\ \n& \n+ \\left[\n\\frac{ \\lambda_5}{2} ( \\phi_1^\\dagger\\phi_2 )^2\n+ \\lambda_6 |\\phi_1|^2\n( \\phi_1^\\dagger\\phi_2)\n+ \\lambda_7 |\\phi_2|^2\n( \\phi_1^\\dagger\\phi_2 )\n+ {\\textrm{h.c.}} \\right] \\,.\n\\eal\nUsing SU(3) gauge freedom, 5 real degrees of freedom of $\\phi_1$ and 3 real degrees of freedom of $\\phi_2$ can be removed. Therefore, \nin the unitary gauge $\\phi_1$, $\\phi_2$ read\n\\begin{equation} \\label{unitarygauge}\n\\phi_1={1\\over \\sqrt{2}} \\,\n\\left( \\begin{array}{c}\n0\\\\0\\\\v_1+\\varphi_1\n\\end{array} \\right) \\,,\n\\quad \n\\phi_2= {1 \\over \\sqrt{2}}\\,\n\\left( \\begin{array}{c}\n0\\\\v_2+\\varphi_2\\\\(v_3+\\varphi_3) + i (v_4+\\varphi_4)\n\\end{array} \\right) ~,\n\\end{equation}\nwhere the $v_i$ are real VEVs and $\\varphi_{1-4} $ are real scalar fields.\nAnalogously, we express the Higgs field in the unitary gauge as $H^T= (0,v+h)\/\\sqrt{2}$.\n\n\nIn what follows, we make two assumptions which are crucial for stability of vector dark matter:\n\\begin{itemize}\n\\item\nthe scalar potential is {\\it CP} invariant \n\\item\nthe VEVs of $\\phi_1$, $\\phi_2$ are real.\n\\end{itemize}\nThe first condition implies that the scalar couplings are real, while the second assumes\nthat no spontaneous {\\it CP} violation occurs ($v_4=0$). As a result, {\\it CP}--even and {\\it CP}--odd fields do not mix.\n\nIn this case, the $Z_2 \\times Z_2'$ symmetry extends to the Higgs sector as well. \nUnder the first $Z_2$ all $\\varphi_i$ are even, while the second $Z_2'$ \n reflects $\\varphi_4$ and leaves the other fields intact. As we detail in Section \\ref{SUN},\nthis assignment follows from the explicit form of the first $Z_2$ as a gauge transformation and the fact that the second $Z_2'$ acts as complex conjugation. In any case, these are explicit symmetries of the Lagrangian and the vacuum. \nThe full list of the parities is presented in Table~\\ref{parities}.\nClearly, the lightest states with non--trivial parities cannot decay to the Standard Model particles.\n \\begin{table}[h]\n \\begin{center}\n \\begin{tabular}{|c|c|}\n\\hline\n fields & $Z_2 \\times Z_2'$\n \\\\ \\hline \\hline \n$h, \\varphi^1, \\varphi^2, \\varphi^3, A_\\mu^7$ & $(+,+)$\n\\\\ \\hline \n$A_\\mu^2,A_\\mu^5$& $(-,+)$\n\\\\ \\hline \n$A_\\mu^1,A_\\mu^4$& $(-,-)$\n\\\\ \\hline \n$\\varphi^4,A_\\mu^3,A_\\mu^6, A_\\mu^8$& $(+,-)$\n\\\\ \\hline\n\\end{tabular}\n\\end{center}\n\\caption{\\label{parities} $Z_2 \\times Z_2'$ parities of the scalars and dark gauge bosons.\n}\n \\end{table}\n \n \nWe now discuss the Lagrangian in more detail, starting with the covariant derivates of $\\phi_1$ and $\\phi_2$,\n\\begin{equation} \\label{dphi}\n|D_\\mu \\phi_i|^2 = |\\partial_\\mu \\phi_i|^2 + i \\tilde g A_\\mu^a \\left((\\partial^\\mu \\phi_i)^\\dagger T^a \\phi_i - {\\textrm{h.c.}} \\right)\n+\\tilde g^2 A_\\mu^a A^{\\mu b} \\, \\phi_i^\\dagger T^a T^b \\phi_i \\,.\n\\end{equation}\nInserting the parametrization~(\\ref{unitarygauge}) with $v_4=0$,\nwe get the kinetic terms for the scalars\n\\begin{equation} \n|\\partial_\\mu \\phi_1|^2+ |\\partial_\\mu \\phi_2|^2 \n=\n\\frac12\n\\sum_{i=1}^4\n(\\partial_\\mu \\varphi_i)^2 \\,,\n\\end{equation}\nthe mass terms for the gauge fields, the mixing terms \nas well as the gauge--scalar interactions.\nLet us first discuss the terms quadratic in the fields.\nThe third term on the r.h.s. of Eq.~(\\ref{dphi}) contains the mass terms for the gauge bosons, \n\\begin{equation}\n{\\cal L} \\supset\n\\frac12 \\left( \\vec{A}_{(1,4)}^T {\\cal M}_{(1,4)} \\vec{A}_{(1,4)}\n+ \\vec{A}_{(2,5)}^T {\\cal M}_{(2,5)} \\vec{A}_{(2,5)}\n+ \\vec{A}_{(3,6,8)}^T {\\cal M}_{(3,6,8)} \\vec{A}_{(3,6,8)}\n+ {\\cal M}_{(7)} A_\\mu^7 A^{\\mu 7} \\right),\n\\end{equation}\nwhere we have used the shorthand notation \n$\\vec{A}_{(1,4)}^T \\equiv (A_\\mu^1,A_\\mu^4)$ and similarly for the other gauge bosons.\nThe mass--squared matrices are\n\\bal \\label{Amasses}\n{\\cal M}_{(1,4)}&={\\cal M}_{(2,5)}\n= \\frac{\\tilde g^2}{4}\n\\left(\n\\begin{array}{cc}\nv_2^2&v_2 v_3\n\\\\\nv_2 v_3&v_1^2+v_3^2\n\\end{array}\n\\right)~,\n\\nonumber \\\\\n{\\cal M}_{(3,6,8)}\n&= \\frac{\\tilde g^2}{4}\n\\left(\n\\begin{array}{ccc}\nv_2^2&-v_2 v_3&-v_2^2\/\\sqrt{3}\n\\\\\n-v_2 v_3&v_1^2+v_2^2+v_3^2&-v_2 v_3\/\\sqrt{3}\n\\\\\n-v_2^2\/\\sqrt{3}&-v_2 v_3\/\\sqrt{3}&(4 v_1^2+v_2^2+4 v_3^2)\/3\n\\end{array}\n\\right)~,\n\\nonumber \\\\\n{\\cal M}_{(7)}\n&= \\frac{\\tilde g^2}{4} ( v_1^2+v_2^2+v_3^2 ) \\,.\n\\eal\nThe second term on the r.h.s. of Eq.~(\\ref{dphi}) contains the $A_\\mu \\partial^\\mu \\varphi$ mixing terms\n\\begin{equation}\n{\\cal L} \\supset \\frac{\\tilde g v_2}{2} A^{\\mu 6} \\partial_\\mu \\varphi_4\n-\\frac{\\tilde g v_3}{\\sqrt{3}} A^{\\mu 8} \\partial_\\mu \\varphi_4\n+ \\frac{\\tilde g v_3}{2} A^{\\mu 7} \\partial_\\mu \\varphi_2\n- \\frac{\\tilde g v_2}{2} A^{\\mu 7} \\partial_\\mu \\varphi_3 \\,.\n\\end{equation}\nIn general, terms of the type $\\kappa_{ai} A_\\mu^a \\partial^\\mu \\varphi^i$ can be removed by the field redefinition \n\\begin{equation} \\label{removemix}\n\\tilde A_\\mu^a = A_\\mu^a + \\partial_\\mu Y^a \\;,\n\\quad \\textrm{where} \\quad\nY^a\\equiv ({\\cal M}^{-1})_{ab}\\, \\kappa_{bi}\\,\\varphi^i\\,.\n\\end{equation}\nWe then have\n\\bal\n&- \\frac12 \\textrm{tr} \\{G_{\\mu \\nu} G^{\\mu \\nu}\\} + \\frac12 (\\partial_\\mu \\varphi^i)^2 + \\frac12 {\\cal M}_{ab} A_\\mu^a A^{\\mu b} + \\kappa_{ai} A_\\mu^a \\partial^\\mu \\varphi^i =\n\\nonumber\n\\\\\n& - \\frac12 \\textrm{tr} \\{\\tilde G_{\\mu \\nu} \\tilde G^{\\mu \\nu}\\} + \\frac12 \\gamma_{ij} (\\partial_\\mu \\varphi^i) (\\partial^\\mu \\varphi^j) + \\frac12 {\\cal M}_{ab} \\tilde A_\\mu^a \\tilde A^{\\mu b} \\,,\n\\eal\nwhere\n\\begin{subequations}\n\\bal\n\\tilde G_{\\mu \\nu} &= \\partial_\\mu \\tilde A_\\nu - \\partial_\\nu \\tilde A_\\mu + i \\tilde g [\\tilde A_\\mu - \\partial_\\mu Y,\\tilde A_\\nu - \\partial_\\nu Y] \\, ,\n\\\\\n\\gamma_{ij} &=\\delta_{ij} - \\kappa^T_{ia}{\\cal M}^{-1}_{ab}\\kappa_{bj} \\, ,\n\\eal\n\\end{subequations}\nand we have defined $Y=Y^a T^a$.\nThe kinetic terms for the vector fields are still canonically normalised, unlike those for the scalar fields.\nThe latter can be normalised canonically by a further field redefinition\n\\begin{equation} \\label{normalizescalars}\n\\tilde \\varphi^i = \\omega_{ik} \\varphi^k \\;, \\quad \\textrm{where} \\quad (\\omega^T \\omega)_{ij}= \\gamma_{ij} \\,.\n\\end{equation}\nNote that the term $- \\frac12 \\textrm{tr} \\{\\tilde G_{\\mu \\nu} \\tilde G^{\\mu \\nu}\\} $ includes additional couplings of the gauge fields to the scalars.\nThe resulting vertices are obtained by \n replacing $A_\\mu^a$ in the triple and quartic gauge boson terms by $-\\partial_\\mu Y^a$\n such that interactions of the type \n $(\\tilde A)^3 \\partial Y, (\\tilde A)^2 (\\partial Y)^2,$ etc. arise.\n The analysis of the general case is very cumbersome, so further we will focus on the simple case of $v_3 = 0$ which retains all the relevant physics.\n In that case, the couplings involving $\\partial Y$ play no role in the DM phenomenology.\n \n\n\\subsection{Detailed study for $v_{3,4} = 0$}\nAs long as $v_1$ and $v_2$ are nonzero, SU(3) is broken completely. Hence it suffices\nto consider the case $v_3 = 0$, which simplifies the analysis.\nThen the only mixing term among the gauge bosons is $ A^3_\\mu A^{\\mu 8}$ and the only\ngauge--scalar mixing terms are \n$ A^6_\\mu \\partial^\\mu \\varphi_4$, $ A^7_\\mu \\partial^\\mu \\varphi_3$.\n\nIn what follows, we restrict ourselves to the case $v_3 = 0$.\n\n\n\\subsubsection{Masses}\n\n\n\\paragraph{Gauge boson masses}\n\nThe gauge boson mass eigenstates are\n\\bal\n\\left( \\begin{array}{c}\nA'^3_\\mu\\\\\nA'^8_\\mu\n\\end{array} \\right)\n=\n \\left( \\begin{array}{c}\n\\cos \\alpha \\, A^3_\\mu + \\sin \\alpha \\, A^8_\\mu\\\\\n\\cos \\alpha \\, A^8_\\mu - \\sin \\alpha \\, A^3_\\mu\n\\end{array} \\right)\\,,\n\\quad\n\\textrm{where}\n\\quad\n\\tan2 \\alpha = \\frac{\\sqrt{3} v_2^2}{2 v_1^2-v_2^2} \\;,\n\\eal\nand the masses are\\footnote{Note that\n$\\frac{\\tan \\alpha}{\\sqrt{3}}=\\frac{v_2^2}{4 v_1^2} + \\ensuremath{\\mathcal O}\\left(\\frac{v_2^6}{v_1^6}\\right)$,\n$\\cos \\alpha = 1- \\frac{3 v_2^4}{32 v_1^4} + \\ensuremath{\\mathcal O}\\left(\\frac{v_2^8}{v_1^8}\\right)$\nand\n$\\sin \\alpha = \\frac{\\sqrt{3} v_2^2}{4 v_1^2} + \\ensuremath{\\mathcal O}\\left(\\frac{v_2^6}{v_1^6}\\right)$.\n}\n\\begin{equation}\nm^2_{A'^3}=\\frac{\\tilde g^2 v_2^2}{4} \\Big(1-\\frac{\\tan \\alpha}{\\sqrt{3}}\\Big) \\,,\n\\qquad\nm^2_{A'^8}=\\frac{\\tilde g^2 v_1^2}{3} \\frac{1}{1-\\frac{\\tan \\alpha}{\\sqrt{3}}} \\,,\n\\label{A3prime-mass}\n\\end{equation}\nwhile the other gauge boson masses can be read off directly from Eq.~(\\ref{Amasses}) for $v_3=0$:\n\\begin{equation}\nm^2_{A^1}=m^2_{A^2}=\\frac{\\tilde g^2 }{4}v_2^2\\;,\n\\quad\nm^2_{A^4}=m^2_{A^5}=\\frac{\\tilde g^2 }{4}v_1^2\\;,\n\\quad\nm^2_{A^6}=m^2_{A^7}=\\frac{\\tilde g^2 }{4}(v_1^2+v_2^2) \\;.\n\\end{equation}\nFor $v_2< v_1$, the light fields are $A_\\mu^{1,2}$ and $A_\\mu^{\\prime 3}$, with the latter being the lightest.\nIt is instructive\nto consider the case $v_2^2 \\ll v_1^2$, so that $\\tan \\alpha $ is small and positive.\nThen $A'^3_\\mu$ is slightly lighter than $A^1_\\mu$ and $A^2_\\mu$ by a factor of $(1-\\frac{\\tan \\alpha}{\\sqrt{3}})^{1\/2}$, while the other five dark gauge bosons are all much heavier, by a factor of order $v_1\/v_2$. The mass degeneracy between $A^1_\\mu$ and $A^2_\\mu$ persists at loop level by symmetry arguments (see Section~\\ref{SUN}).\n\nOne can easily verify that the heavier states $A^{4-7}_\\mu$ and $A^{\\prime 8}_\\mu$ \nall decay into the light states and the SM particles. The decay proceeds via emission\n(off--shell or on--shell) of the {\\it CP} even scalars which couple to the SM Higgs\n and thus to all other SM fields.\n\nThe 3 lightest states all have different $Z_2 \\times Z_2^\\prime$ parities such that they cannot decay into each other by emitting SM particles. The only scalar with negative parity is $\\varphi_4$, however it is generally heavy (see below) and does not contribute to the above decay. Hence, $A_\\mu^{1,2}$ and $A_\\mu^{\\prime 3}$ are stable.\\footnote{The parities allow for a decay of one DM component into two others, however this is forbidden kinematically. }\n\n\n\\paragraph{Gauge boson - scalar mixing}\n\nAccording to Eq.~(\\ref{removemix}), the $A_\\mu \\partial^\\mu \\varphi$ mixing terms are removed by the redefinition (which does not affect the gauge boson masses)\n\\bal\n\\tilde A^6_\\mu&=A^6_\\mu+ \\partial_\\mu Y^6 \\;, \\quad \\textrm{where}\\quad Y^6=\\frac2{\\tilde g} \\frac{v_2}{v_1^2+v_2^2} \\varphi^4\\;,\n\\nonumber\n\\\\\n\\tilde A^7_\\mu&=A^7_\\mu+ \\partial_\\mu Y^7 \\;, \\quad \\textrm{where}\\quad Y^7=-\\frac2{\\tilde g} \\frac{v_2}{v_1^2+v_2^2} \\varphi^3\\,,\n\\eal\nwhile, according to Eq.~(\\ref{normalizescalars}), the canonically normalised scalars are\n\\begin{equation}\n\\tilde \\varphi^3 = \\frac{v_1}{\\sqrt{v_1^2+v_2^2}} \\varphi^3 \\,,\n\\quad\n\\tilde \\varphi^4 = \\frac{v_1}{\\sqrt{v_1^2+v_2^2}} \\varphi^4 \\,.\n\\end{equation}\n\n\n\\paragraph{Scalar masses}\nThe general scalar potential has many parameters. To make our discussion more transparent, let us assume the symmetry \n $\\phi_2 \\to - \\phi_2$ which does not affect the essence of our considerations and \n requires $m_{12}^2=\\lambda_{H12}=\\lambda_6=\\lambda_7=0$.\nIn this case, it turns out that the potential has no local minima with all of $v,v_1,v_2, v_3$ being nonzero such that setting $v_3=0$ is actually required.\nIt should be noted that in practice we are considering the limit in which \nthe above quantities are very small but nonzero such that the decay channels for heavy particles, e.g. $\\tilde\\varphi_3$, into the SM fields are open. \n\n\nThe VEVs $v,v_1,v_2$ can be expressed in a compact form using the matrix\n\\bal\n\\mathbf \\Lambda \\equiv\n\\left( \\begin{array}{ccc} \n\\lambda_H & \\lambda_{H11} & \\lambda_{H22} \\\\\n\\lambda_{H11} & \\lambda_1 & \\lambda_3 \\\\\n\\lambda_{H22} & \\lambda_3 & \\lambda_2 \n\\end{array} \\right) \n\\eal\nas well as the matrices $\\mathbf \\Lambda_{ij}$ defined as $(-1)^{i+j}$ times the matrix obtained by deleting the $i$-th row and $j$-th column of $\\mathbf \\Lambda$, i.e.~$\\det \\mathbf \\Lambda_{ij}$ is the $(i,j)$-cofactor of $\\mathbf \\Lambda$.\nOne finds\n\\bal\nv^2&= - 2 ( m_H^2 \\det{\\mathbf \\Lambda_{11}} +m_{11}^2 \\det{\\mathbf \\Lambda_{21}}+m_{22}^2 \\det{\\mathbf \\Lambda_{31}})\/\\det{\\mathbf \\Lambda} \\,, \\nonumber\n\\\\\nv_1^2&= -2 ( m_H^2 \\det{\\mathbf \\Lambda_{12}} +m_{11}^2 \\det{\\mathbf \\Lambda_{22}}+m_{22}^2 \\det{\\mathbf \\Lambda_{32}})\/\\det{\\mathbf \\Lambda} \\,, \\nonumber\n\\\\\nv_2^2&= -2 ( m_H^2 \\det{\\mathbf \\Lambda_{13}} +m_{11}^2 \\det{\\mathbf \\Lambda_{23}}+m_{22}^2 \\det{\\mathbf \\Lambda_{33}})\/\\det{\\mathbf \\Lambda} \\,.\n\\eal\nThe mass terms for the scalars are\n\\begin{equation}\n- {\\cal L} \\supset \\frac12 \\Phi^T \\mathbf{m}^2 \\Phi + \\frac14 (\\lambda_4+\\lambda_5) (v_1^2+v_2^2) \\, \\tilde \\varphi_3^2 + \\frac14 (\\lambda_4-\\lambda_5) (v_1^2+v_2^2) \\, \\tilde \\varphi_4^2 \\;,\n\\label{phi4-mass}\n\\end{equation}\nwhere $\\Phi^T\\equiv (h,\\varphi_1,\\varphi_2)$ and\n\\bal\n\\mathbf{m}^2 =\n\\left( \\begin{array}{ccc} \n\\lambda_H v^2 & \\lambda_{H11} v v_1 & \\lambda_{H22} v v_2 \\\\\n\\lambda_{H11} v v_1 & \\lambda_1 v_1^2& \\lambda_3 v_1 v_2\\\\\n\\lambda_{H22} v v_2& \\lambda_3 v_1 v_2 & \\lambda_2 v_2^2\\\\\n\\end{array} \\right) \\,.\n\\eal\nNote that $\\tilde \\varphi_3$ and $\\tilde \\varphi_4$ are generally heavier than $A^1,A^2,A'^3$ since their masses involve $v_1$\n(unless $\\lambda_{4,5}$ are very small). \n\nThe matrix $\\mathbf{m}^2$ is positive definite if and only if $\\det \\mathbf \\Lambda>0$, $\\det \\mathbf \\Lambda_{33}\\equiv \\lambda_H \\lambda_1- \\lambda_{H11}^2>0$ and $\\lambda_H>0$.\nIt can be diagonalised by an orthogonal transformation\n$O^T \\mathbf{m}^2 O = \\textrm{diag}(m_{H_1}^2,m_{H_2}^2,m_{H_3}^2) ,$\nwhere\n\\bal\nO&= \\left( \\begin{array}{ccc}\nc_{12} c_{13}& s_{12} & c_{12}s_{13}\\\\\n-c_{13} c_{23} s_{12} - s_{13} s_{23} & c_{12} c_{23} &-c_{23} s_{12} s_{13}+ c_{13} s_{23} \\\\\n-c_{23} s_{13} + c_{13} s_{12} s_{23} &-c_{12} s_{23} &c_{13} c_{23} + s_{12} s_{13} s_{23}\n\\end{array} \\right) \\;\n\\eal\nand we have used the abbreviation $s_{ij} \\equiv \\sin \\theta_{ij}, c_{ij} \\equiv \\cos \\theta_{ij}$. Instead of providing the most general formulae, let us focus on a simplified case.\n Suppose that the $(12)$ and $(23)$ entries of $\\mathbf{m}^2 $ are much smaller than the other matrix elements, in other words, that $\\lambda_{H11}$\nand $\\lambda_3$ are very small. In this case, the Higgs mixing with\n$\\varphi_2$ is the dominant one. The reason behind this choice is that \nthe DM constituents $A^1,A^2$ and $A'^3$ all have a significant coupling to $\\varphi_2$, see Table~\\ref{AAscalar}.\nGiven that the mixing between the Higgs and $\\varphi_2$ is substantial,\nthis facilitates DM annihilation into the SM fields. \nClearly, similar considerations also apply to the general case with all the angles $\\theta_{ij}$ being significant.\n\n\nAssuming small $\\mathbf{m}^2_{12},\\mathbf{m}^2_{23}$, one finds\n\\begin{subequations}\n\\bal\n& \\theta_{12} \\approx p_{32}\\, \\mathbf{m}^2_{12} + q \\, \\mathbf{m}^2_{23} \\;,\n\\\\\n& \\theta_{23} \\approx p_{21}\\, \\mathbf{m}^2_{23} + q \\, \\mathbf{m}^2_{12} \\;,\n\\\\\n& \\tan 2 \\theta_{13} \\approx \\frac{2 \\lambda_{H22} v v_2}{\\lambda_2 v_2^2-\\lambda_H v^2} \\,,\n\\eal\n\\end{subequations}\nwhere $p_{ij}=(\\mathbf{m}^2_{ii}-\\mathbf{m}^2_{jj})\/s$, $q=\\mathbf{m}^2_{13}\/s$ and $s=(\\mathbf{m}^2_{13})^2+(\\mathbf{m}^2_{11}-\\mathbf{m}^2_{22})(\\mathbf{m}^2_{22}-\\mathbf{m}^2_{33})$.\nThe mass eigenstates are\n\\bal\n\\left( \\begin{array}{c}\nh_1\\\\\n{\\cal H}\\\\\nh_2\n\\end{array} \\right)\n\\equiv O^T \\Phi\n\\approx\n \\left( \\begin{array}{c}\nc_{13} h - s_{13} \\varphi_2 \\\\\n\\varphi_1 \\\\\nc_{13} \\varphi_2 + s_{13} h\n\\end{array} \\right)\n- \\theta_{12}\n \\left( \\begin{array}{c}\n c_{13} \\varphi_1 \\\\\n-h \\\\\ns_{13} \\varphi_1\n\\end{array} \\right)\n- \\theta_{23}\n \\left( \\begin{array}{c}\n s_{13} \\varphi_1 \\\\\n \\varphi_2\\\\\n-c_{13} \\varphi_1\n\\end{array} \\right)\n\\,. \\label{rotation-matrix}\n\\eal\nand the mass--squared eigenvalues are \n\\begin{subequations}\n\\bal\n& m^2_{h_1,h_2} \\approx \\frac12 ( \\lambda_2 v_2^2+\\lambda_H v^2) \\mp \\frac{ \\lambda_2 v_2^2-\\lambda_H v^2}{2 \\cos 2 \\theta_{13}}\n\\\\\n& m^2_{\\cal H} \\approx \\lambda_1 v_1^2 \\,.\n\\eal\n\\end{subequations}\nThe eigenstates $h_1,h_2$ are typically the lighter ones, while ${\\cal H}$\nis heavy (unless $\\lambda_1$ is very small). In our analysis of DM phenomenology, we retain only the former states.\nIn summary, the relevant light fields are the DM components $A^1,A^2$ and $A'^3$ as well as the mediators $h_1,h_2$ which link the dark sector to the SM fermions and gauge bosons.\n \n\n\\subsubsection{Couplings}\n\nThe full list of the couplings is not necessary for our DM studies.\nThe important couplings are those with two gauge bosons and one or two scalars at the vertex. In terms of the variables $\\tilde A_\\mu^a $\nand $\\tilde \\varphi^i$, most of these are listed in Table~\\ref{AAscalar} \nand Table~\\ref{AAscalarscalar}. For our applications, the couplings of $h_1$ and $h_2$ are obtained from these tables using the relation\n (\\ref{rotation-matrix}), in which one may neglect $\\theta_{12}$\n and $\\theta_{23}$.\n\nWe focus on the case $v_1 \\gg v_2$ so that\n DM consists of $A^1,A^2$ and $A'^3$. Other fields with non--trivial parities\n decay into these states and the SM particles. For instance, \n the processes $\\tilde \\varphi_4 \\rightarrow A^1 A^2 \\, +$~SM and \n $\\tilde \\varphi_4 \\rightarrow A^{3\\prime}\\, +$~SM are allowed. \n (When $v_1$ and $v_2$ are close, the composition of DM depends on the mass splittings.)\n DM annihilation and scattering proceeds through an exchange of $h_1$\n and $h_2$. Therefore, only the vertices involving these fields \n play a significant role.\n\n\n\\section{Generalisation to arbitrary SU(N)} \\label{SUN}\nSU(N) is broken completely by generic VEVs of $N-1$ fields $\\phi_i$ in the fundamental representation.\nThe $\\phi_i$'s can be gauge--transformed to the form\n\\begin{equation}\n\\phi_1= \\left( \\begin{matrix} \n   0\\\\ \n   0 \\\\\n   ...\\\\\n   0 \\\\\n   \\rho_1 \n  \\end{matrix}\n\\right) ~~,~~\n\\phi_2= \\left( \\begin{matrix} \n   0\\\\ \n   0 \\\\\n   ...\\\\\n   \\rho_2^{(1)} \\\\\n   \\rho_2^{(2)} e^{i \\xi_2}\n   \\end{matrix} \n\\right) ~~,~~...~~,~~\n\\phi_{N-1}= \\left( \\begin{matrix}\n    0\\\\ \n   \\rho_{N-1}^{(1)} \\\\\n   ...\\\\\n   \\rho_{N-1}^{(N-2)} e^{i \\xi_{N-1}^{(N-3)}}\\\\ \n   \\rho_{N-1}^{(N-1)} e^{i \\xi_{N-1}^{(N-2)}}\n   \\end{matrix} \n\\right) ~~.\n\\end{equation}\nHere the radial fields $\\rho_i^{(j)}$ and the phases $\\xi_i^{(j)}$ are real. We label the scalars such that \nthe lightest gauge fields are associated with the SU(2) subgroup which gets broken\nat the last stage by a VEV of $\\rho_{N-1}^{(1)}$. We assume that the VEVs as well as the couplings in the scalar potential are all real so that {\\it CP} is preserved in the hidden sector.\n\nThe generalisation of the $Z_2 \\times Z_2'$ parity to SU(N) is as follows. \nThe transformation properties of the gauge fields are identified with those of the corresponding SU(N) generators. \nThe basis of the $N(N-1)$ off--diagonal generators $T^{ab}, \\tilde T^{ab}$ can be chosen as \n\\begin{eqnarray}\n&& ( T^{ab} )_{ij}= \\delta_{ia} \\delta_{jb} + \\delta_{ib} \\delta_{ja} \\;, \\nonumber\\\\\n&& ( \\tilde T^{ab})_{ij}= -i \\delta_{ia} \\delta_{jb} + i \\delta_{ib} \\delta_{ja} \\;,\n\\end{eqnarray}\nwhere $a=1,..,N-1$ and $b=2,..,N$.\nWith the Cartan generators denoted by $H^\\alpha$, the $Z_2$ associated with \ncomplex conjugation of the group elements acts as\n\\begin{eqnarray}\n&& T^{ab} \\rightarrow -T^{ab} ~~,~~ \\tilde T^{ab} \\rightarrow \\tilde T^{ab}\n~~,~~ H^\\alpha \\rightarrow - H^\\alpha \\;. \\label{Z21N}\n\\end{eqnarray}\nThis is a well known outer automorphism of SU(N) which entails the corresponding symmetry of the Yang--Mills Lagrangian.\n\nAnother $Z_2$ can be defined by reflecting the off--diagonal generators containing nonzero elements in the first row:\n\\begin{eqnarray}\n&& T^{1a} \\rightarrow -T^{1a} ~~,~~ \\tilde T^{1a} \\rightarrow - \\tilde T^{1a}\n~~, \\nonumber\\\\\n&& T^{bc} \\rightarrow T^{bc} ~~~~~,~~ \\tilde T^{bc} \\rightarrow \\tilde T^{bc}\n~~ ~~~(b,c \\geq 2), \\nonumber\\\\\n&& H^\\alpha \\rightarrow H^\\alpha \\;. \\label{Z22N}\n\\end{eqnarray}\nIt is easy to show that this $Z_2$ is an inner automorphism. Indeed, it corresponds to the group transformation with\n\\begin{equation}\nU= e^{ { i \\pi \\over N}} \\; {\\rm diag}(-1,1,...1) \\;. \\label{U}\n\\end{equation}\nThe gauge fields $A^{1-3}_\\mu$ associated with the \nupper left SU(2) block \n\\begin{equation}\nT^{12},~~\\tilde T^{12},~~ H^1= {\\rm diag}(1,-1,0,...,0)\n\\end{equation}\ntransform under these parities the same way as did the SU(2) gauge fields\nunder $Z_2 \\times Z_2'$ of the previous section. The above gauge transformation of course leaves the Yang--Mills Lagrangian invariant.\n\nThese symmetries are preserved by gauge interactions with scalars in our set--up. The $Z_2$\nassociated with complex conjugation acts on scalars by reflecting the complex phases,\nwhich therefore correspond to odd fields under the transformation~(\\ref{Z21N}). This symmetry is guaranteed by {\\it CP} invariance of the Lagrangian and preserved by the vacuum (assuming no spontaneous {\\it CP} violation).\nThe second $Z_2$ is a gauge transformation. On vectors of the form \n$(0,a_1,...,a_{N-1})$, it acts as multiplication by an overall \nconstant phase which cancels in all the Lagrangian terms. It is therefore a valid symmetry in the broken phase as well. \n\nAs long as the $\\phi_i$ have a zero first component,\nthe interaction vertices contain an even number of $T^{1a}$ and $\\tilde T^{1a}$.\n The gauge fields associated with $a>2$ are heavier than those corresponding to $a=2$. By virtue of \n the vertices involving $T^{12} T^{1k}$ $(k>2)$ and the matter fields, they decay \n to the lighter fields such that only the final SU(2) block remains stable. \n (Similar considerations also apply to other heavy gauge fields.)\nThen DM is composed mostly of the aforementioned $A^{1-3}_\\mu$ whose stability is enforced by $Z_2 \\times Z_2'$.\\footnote{As before, we take the ``phase'' fields to be heavy since they get\ntheir masses from large VEVs $\\langle \\phi_i \\rangle$, unlike $A^{1-3}_\\mu$. In the presence of more than $N-1$ fundamentals, this logic no longer applies and the light phase fields can constitute DM.} \n \nAs in the SU(3) case, DM consists of three components, two of which are degenerate in mass,\n\\begin{equation}\nm_{ A^{1\\prime }} =m_{ A^{2\\prime }} \\not= m_{ A^{3\\prime }} \\;,\n\\end{equation}\nwhere $ A^{1\\prime -3 \\prime}_\\mu$ are the mass eigenstates consisting mostly of $ A^{1-3 }_\\mu$\nwith some admixture of other gauge fields (see the SU(3) example). The degeneracy persists at loop level. This can be seen as follows. The SU(N) Lie algebra possesses a discrete symmetry which interchanges the real and imaginary generators with nonzero entries in the first row,\n\\begin{equation}\nT^{1a} \\rightarrow \\tilde T^{1a} ~~,~~ \\tilde T^{1a} \\rightarrow - T^{1a}~~,\n\\label{A1-A2}\n\\end{equation}\nwhile all the other generators remain intact.\nThis is achieved by the group transformation\n\\begin{equation}\nU'= e^{ { -i \\pi \\over 2 N}} \\; {\\rm diag}(i,1,...1) \\;,\n\\end{equation}\nwhich can be recognised as the square root of $U$ in (\\ref{U}).\nThis gauge transformation acts on $\\phi_i$ with a zero first entry as an overall\nconstant phase multiplication. Since such a phase cancels in all of the Lagrangian terms,\n(\\ref{A1-A2}) remains a valid symmetry even in the broken phase.\n\n Consider now the mass matrix for the gauge fields associated with $T^{1a}$ and $\\tilde T^{1a}$. By virtue of (\\ref{Z22N}), only fields corresponding to $T^{1a}$ and $\\tilde T^{1a}$\n can mix, while (\\ref{Z21N}) forbids a mixing between the tilded and untilded fields.\n The resulting mass matrix for $T^{1a}$ is then identical to that of $\\tilde T^{1a}$\n according to (\\ref{A1-A2}). Analogous considerations apply to the kinetic terms.\n Hence the lightest eigenstates have the same mass. \n \n The resulting DM phenomenology is analogous to that for the SU(3) case. \n \n\n\\section{Dark matter phenomenology }\n\n In what follows, we consider direct and indirect detection as well as relic abundance constraints on vector DM. \n In the U(1) and SU(2) cases, all the scattering processes are mediated by $h_1$ and $h_2$. For SU(3) and larger groups, further states can contribute.\nHowever, we make the simplifying assumption that the Higgs mixing with those states is small and\/or such states are heavy. In that case, it suffices to consider an exchange of $h_1$ and $h_2$ only. We note that earlier phenomenological analyses of vector dark matter \nhave appeared in various contexts~\\cite{Kanemura:2010sh}-\\cite{Duch:2014xda}.\nThe current collider searches for dark matter, e.g. in the form of a monojet plus missing energy, do not set further useful constraints on the model, see e.g. Ref.~\\cite{Kim:2015hda}.\n\n\n\\subsection{U(1) dark matter}\nLet us start with the U(1) case. The relevant terms in the Lagrangian are\n\\bal\n{\\cal L} &\\supset {1\\over 2} m_A^2 A_\\mu A^\\mu + {\\tilde g \\, m_A \\over 2} \n\\left(-h_1 \\sin\\theta + h_2 \\cos\\theta \\right) A_\\mu A^\\mu \\nonumber \\\\\n&+\n{\\tilde g^2 \\over 8 } \\left( h_1^2 \\sin^2 \\theta -2 h_1 h_2 \\sin\\theta \\;\\cos\\theta +\nh_2^2 \\cos^2 \\theta \\right) A_\\mu A^\\mu \\;.\n\\label{DMinteractions}\n\\eal\nThe couplings of $h_1$ and $h_2$ to the SM fields are those of the SM Higgs up\nto the suppression factors of $\\cos\\theta$ and $\\sin\\theta$, respectively.\nA phenomenological analysis of this model in the decoupling limit $m_{h_2} \\gg m_{h_1}$,\n$\\sin\\theta \\rightarrow 0$ can be found in~\\cite{Kanemura:2010sh} and \n\\cite{Lebedev:2011iq,Djouadi:2011aa}. \nRelated studies have also appeared in~\\cite{Farzan:2012hh,Baek:2012se}. \n\n\nThe DM scattering on nucleons proceeds through the $t$--channel exchange of $h_{1,2}$\n and leads to the following spin--independent cross\nsection (see e.g.~\\cite{Lebedev:2011iq}),\n\\begin{equation}\n\\sigma^{\\rm SI}_{A-N}= {g^2 \\tilde g^2 \\over 16 \\pi} \\; {m_N^4 f_N^2 \\over m_W^2} \\;\n{ (m_{h_2}^2 - m_{h_1}^2 )^2 \\sin^2 \\theta \\; \\cos^2 \\theta \\over m_{h_1}^4 m_{h_2}^4} \\;,\n\\end{equation}\nwhere $m_N$ is the nucleon mass and $f_N \\simeq 0.3$ parametrizes the Higgs--nucleon coupling. One should keep in mind that there is significant uncertainty in $f_N$ and here\nwe use the value somewhat smaller than the one assumed in~\\cite{Lebedev:2011iq}. \nAs expected, $h_1$ and $h_2$ contribute with opposite signs. Since \n$m_{h_2}^2 \\gg m_{h_1}^2$ in realistic cases, the cancellation is not very significant.\n %\n\\begin{figure}[h] \n\\centering{\n\\includegraphics[scale=0.283]{DMDMtoH.pdf}\n\\includegraphics[scale=0.283]{DMDMtoHtoHH.pdf}\n\\includegraphics[scale=0.283]{DMDMtoHH_Tchannel.pdf}\n\\includegraphics[scale=0.283]{DMDMtoHH.pdf}\n }\n\\caption{ \\label{diagrams1}\nLeading diagrams for vector DM annihilation.\n}\n\\end{figure}\n\n\nThe calculation of the DM annihilation cross section is more involved due to a few contributing diagrams (Fig.~\\ref{diagrams1}). To compute the DM relic abundance, we use the software package micrOMEGAs 4.1.8~\\cite{Belanger:2014vza}. It is important to note that the leading $s$--channel annihilation amplitude\ndue to the $h_{1,2}$--exchange is proportional to the factor\n \\begin{equation}\n {\\cal A}^{\\rm annih} \\propto \\sin\\theta \\;\\cos\\theta \\; \\left( {1\\over s-m_{h_1}^2 } -\n {1\\over s-m_{h_2}^2 } \\right) \\;,\n\\end{equation}\nwhere the $s$--parameter can be approximated by $s \\simeq 4 m_A^2$. Therefore,\nthe amplitude is highly suppressed at $m_A \\gg m_{h_2}$. \nAlthough other annihilation channels remain available, this makes DM annihilation\ninefficient for heavy masses and the corresponding parameter space is challenged by the direct detection constraint. \n\nClearly, the $s$--channel annihilation becomes very efficient around the resonances,\n$m_A \\simeq m_{h_1}\/2$ and $m_A \\simeq m_{h_2}\/2$. In this case, a very small gauge coupling is sufficient to obtain the right relic abundance. \nThe first resonance is quite narrow due to the small width of the SM Higgs, whereas \nthe second resonance is broad since many decay channels are available to $h_2$. For \n$m_{h_2} > 2 m_{h_1}$, the decay $h_2 \\rightarrow h_1 h_1$ becomes important. Its significance depends on $\\lambda_{h \\phi}$ of Eq.~(\\ref{eq1}) with BR${_{h_2 \\rightarrow h_1 h_1}}$ increasing for larger $\\lambda_{h \\phi}$ (see the explicit formulae in~\\cite{Falkowski:2015iwa}). The resonance is widened further by the thermal averaging over the DM momentum.\n\nThe most important features of DM annihilation are associated with the $s$--channel.\nOther channels play a less significant role. Similar considerations apply to the indirect detection constraint due to the gamma ray emission in the process of DM annihilation (FERMI). \n\nThe last constraint we impose is of theoretical nature. We require that the theory be perturbative at the TeV scale. One way to enforce it is to demand perturbative unitarity at tree level, for instance, in the process $h_i h_i \\rightarrow h_j h_j$. The resulting constraint was estimated in~\\cite{Chen:2014ask} to be \n\\begin{equation}\n\\lambda_i < {\\cal O} (4\\pi\/3) \\;,\n\\label{unitarity}\n\\end{equation} \nwhere $\\lambda_i$ are the scalar quartic couplings.\nWe define the quartic couplings involving $\\phi$ by \n$\\Delta V_{\\rm quart} = \\lambda_{h \\phi} \\vert H \\vert^2 \\vert \\phi \\vert^2 +\n{1\\over 2} \\lambda_{\\phi} \\vert \\phi \\vert^4 $. They can be expressed in terms of \nthe masses, the gauge coupling and the mixing angle as (see e.g.~\\cite{Falkowski:2015iwa}) \n\\begin{eqnarray}\n&& \\lambda_{h \\phi}= \\tilde g ~\\sin 2 \\theta~ { m_{h_2}^2-m_{h_1}^2 \\over 4 v m_A } \\;,\n\\nonumber \\\\ \n&& \\lambda_{\\phi}= \\frac{ 4 \\, \\lambda_{h \\phi}^2 }{\\sin^2 2 \\theta } \\frac{v^2}{m_{h_2}^2-m_{h_1}^2} \\left(\\frac{m_{h_2}^2}{m_{h_2}^2-m_{h_1}^2} -\\sin^2 \\theta \\right) \\;,\n\\end{eqnarray} \n where we have used $\\tilde v =2 m_A\/ \\tilde g $. \nThis implies that both $\\lambda_{h \\phi}$ and $\\lambda_\\phi$ become large for heavy $h_2$ \nor for light dark matter. As a result, Eq.~(\\ref{unitarity}) imposes an important constraint \non our model. We further require the standard perturbativity constraint \n\\begin{equation}\n{\\tilde g^2 \\over 4 \\pi} < 1 ~,\n\\end{equation} \nwhich we find less significant for our purposes. \n\\begin{figure}[h] \n\\centering{\n\\includegraphics[scale=0.43]{U1ST03MHp280DM1TeV.pdf}\n\\includegraphics[scale=0.43]{U1ST03MHp1000DM2TeV.pdf}\n\\includegraphics[scale=0.43]{U1ST02MHp3000DM3TeV.pdf}\n\\includegraphics[scale=0.43]{SU2ST03MHp280DM1TeV.pdf}\n\\includegraphics[scale=0.43]{SU2ST03MHp1000DM2TeV.pdf}\n\\includegraphics[scale=0.43]{SU2ST02MHp3000DM3TeV.pdf}\n}\n\\caption{ \\label{plots}\nConstraints on the gauge coupling $\\tilde g$ vs DM mass $m_A$ for U(1) and SU(2).\nThe area between the red lines is favoured by the DM relic abundance, while the regions above \nthe dashed blue, dotted black and green lines are ruled out by direct detection, indirect detection and perturbativity of $\\lambda_i$, respectively.\n{\\it Upper row:} U(1) dark matter with $\\sin\\theta=0.3, m_{h_2}= 280$ GeV (left),\n $\\sin\\theta=0.3, m_{h_2}= 1$ TeV (center), $\\sin\\theta=0.2, m_{h_2}= 3$ TeV (right).\n {\\it Lower row:} SU(2) dark matter with $\\sin\\theta=0.3, m_{h_2}= 280$ GeV (left),\n $\\sin\\theta=0.3, m_{h_2}= 1$ TeV (center), $\\sin\\theta=0.2, m_{h_2}= 3$ TeV (right). \n}\n\\end{figure}\n\nOur results for U(1) DM are presented in Fig.~\\ref{plots}, upper row. \nWe include the constraints from PLANCK~\\cite{Ade:2015xua} (relic abundance), LUX~\\cite{Akerib:2013tjd} (direct detection),\nFERMI~\\cite{Ackermann:2015zua} (indirect detection) and perturbativity of $\\lambda_i$.\nThe area between the red lines is consistent with the thermal relic DM abundance measured by PLANCK. The LUX data provide the strongest constraint on the allowed parameter space. \nThe FERMI bound is typically relevant for light DM, while the perturbativity bound becomes\nimportant for heavy $h_2$. \nIn each panel, the mixing angle is chosen such that, for a given $m_{h_2}$,\nit is consistent with the LHC and EW precision data~\\cite{Falkowski:2015iwa}. Heavier\n$h_2$ imply smaller $\\sin\\theta$, so we take $\\sin\\theta=0.3$ for the left and center panels,\nand $\\sin\\theta=0.2$ for the right panel.\n\n\nThe two dips are associated with the resonant annihilation through $h_1$ and $h_2$. The second\nresonance gets broader with increasing $h_2$ due to the increase in $\\lambda_{h \\phi}$ and\navailability of the decay $h_2 \\rightarrow h_1 h_1$. The area around this resonance constitutes the largest parameter space consistent with all of the constraints. For $m_{h_2}= 280$ GeV,\nthe allowed DM mass range is about 100 GeV; for $m_{h_2}= 1$ TeV, it widens to 1 TeV, and\nfor $m_{h_2}= 3$ TeV, it reaches more than 3 TeV. The resonance is broadened by the thermal averaging\nover the DM momentum, so even though it appears very broad for $m_{h_2}= 3$ TeV,\nit is still consistent with perturbativity. \n\nThe dip associated with the resonant annihilation through $h_1$ is quite narrow and does not open up further significant areas of parameter space. \nOther features of the PLANCK curve are local peaks corresponding to the kinematic opening of additional annihilation channels. For instance, the peak at $m_A \\sim 80$ GeV is associated with the $W^+ W^-$ final state. There are further visible peaks at $(m_{h_2} + m_{h_1})\/2$ and $m_{h_2}$. \n\n\nAway from the resonances, there appears to be a further allowed region at $m_A > m_{h_2}$.\nSince the $h_2$ production is not suppressed by $\\sin\\theta$, \nin this case the $t$--channel annihilation $AA \\rightarrow h_2 h_2$\nand the quartic interactions dominate.\nThe Planck--allowed strip is dangerously close to the LUX bound, so the conclusion\ndepends strongly on the nucleon--Higgs coupling $f_N \\simeq 0.3$, which suffers from substantial uncertainties.\n\n\n\\subsection{SU(2) dark matter}\nAside from the gauge self--interactions, the Lagrangian (\\ref{DMinteractions}) applies\nto the SU(2) case as well, up to the summation over the 3 species, $A_\\mu A^\\mu \n\\rightarrow A_\\mu^a A^{a \\mu }$. The main change compared to the U(1) case is that \nthe annihilation cross section decreases since only the species with the same group index \ncan annihilate through the Higgs--like states. In order to keep the same relic abundance, one needs to increase the gauge coupling. Since the $s$--channel annihilation through $h_{1,2}$\noften dominates, this amounts approximately to\n\\begin{equation}\n\\tilde g \\rightarrow \\sqrt{3} \\tilde g \\;,\n\\end{equation}\nwhereas if \n the $t$--channel and the quartic interactions dominate, the rescaling factor is closer\n to $3^{1\/4}$. The direct detection constraint remains the same as in the U(1) case since particles with different group indices scatter the same way on nucleons.\nThis decreases somewhat the allowed parameter space compared to the Abelian case (Fig.~\\ref{plots}).\n\nNon--Abelian DM features a semi--annihilation channel $AA \\rightarrow A h_{1,2}$\n(Fig.~\\ref{diagrams2}).\nIn some regimes, for example at large $\\tilde g$ and small $\\sin\\theta$, it can even dominate~\\cite{D'Eramo:2012rr} (see also~\\cite{Arina:2009uq,Khoze:2014xha}). \nUsing the analytical results of~\\cite{D'Eramo:2012rr},\nwe find that DM semi--annihilation is insignificant in the relevant parameter regions.\nFor example, at $m_A > m_{h_2}$ the $\\sin\\theta$--unsuppressed and potentially important channel $AA \\rightarrow A h_2$ opens up, yet it is dominated by $AA \\rightarrow h_2 h_2$.\nAlso, around the resonances the gauge coupling is rather small which diminishes the relative importance of semi--annihilation. \n\n\\begin{figure}[h] \n\\centering{\n\\includegraphics[scale=0.32]{DMDMtoDMH_Schannel.pdf} \\hspace{0.8cm}\n\\includegraphics[scale=0.32]{DMDMtoDMH_Tchannel.pdf}\n }\n\\caption{ \\label{diagrams2}\nSemiannihilation of vector DM.\n}\n\\end{figure}\n\nNo firm conclusion can be reached as to whether the region $m_A > m_{h_2}$ is allowed.\nAs stated above, the uncertainties in $f_N$ play a critical role due to the proximity of the Planck band and the LUX bound.\n\n\n\\subsection{SU(3) dark matter}\nIn the SU(3) case, DM is composed again of 3 species with two of them being degenerate \n($A^1_\\mu, A^2_\\mu$) in mass and the third one being lighter ($A^{3\\prime}_\\mu$). This is a result of the mixing between the gauge bosons corresponding to the Cartan generators of SU(3). Therefore, while the couplings of $A^1_\\mu, A^2_\\mu$ to the Higgs like scalars remain the same\nas in the SU(2) case, the coupling of $A^{3\\prime}_\\mu$ changes. In terms of\n$m_A \\equiv m_{A^{1,2}}$, the relevant Lagrangian reads\n\\bal\n{\\cal L} & \\supset \\frac12 m_{A^{}}^2 \\Big(\\sum_{a=1,2} A^a_\\mu A^{a\\mu}+\\Big(1-{\\tan\\alpha \\over \\sqrt{3}}\\Big) A'^3_\\mu A'^{3\\mu}\\Big)\n\\\\\n&+ \\frac{ \\tilde g \\, m_{A^{}}}2 (- h_1 \\sin\\theta + h_2 \\cos\\theta) \\Big(\\sum_{a=1,2} A^a_\\mu A^{a\\mu}+\\Big(\\cos\\alpha-{\\sin\\alpha\\over \\sqrt{3}}\\Big)^2 A'^3_\\mu A'^{3\\mu}\\Big)\n\\nonumber\n\\\\\n&+ \\frac{ \\tilde g^2}8 (h_1^2 \\sin^2\\theta -2 h_1 h_2 \\sin\\theta \\cos\\theta + h_2^2 \\cos^2 \\theta) \\Big( \\! \\sum_{a=1,2} \\!\\! A^a_\\mu A^{a\\mu} \\! + \\! \\Big(\\cos \\alpha-{\\sin \\alpha \\over\\sqrt{3}}\\Big)^2 A'^3_\\mu A'^{3\\mu}\\Big) .\n\\nonumber\n\\eal\nThe mass of the lighter DM component is reduced by the factor \n$( 1- {1\\over \\sqrt{3}} \\tan\\alpha )^{1\/2} $ compared to that of \n$A^1_\\mu$ and $A^2_\\mu$, while the gauge--scalar couplings decrease by a factor\n$( \\cos\\alpha- {1\\over \\sqrt{3}} \\sin\\alpha )^{ 2}$.\nTherefore, the lighter state has a smaller annihilation cross section. \nNote that \n$\\sin \\alpha \\simeq \\frac{\\sqrt{3} v_2^2}{4 v_1^2}$ and even a factor \nof two difference in the triplet VEVs leads to a rather small $\\alpha \\sim 10^{-1}$.\nIn that case, there is no tangible difference between the SU(2) and the SU(3) analyses.\n %\n\\begin{figure}[t] \n\\centering{\n\\includegraphics[scale=0.6]{SU3.pdf}\n}\n\\caption{ \\label{plot2}\n Illustration of the main features of SU(3) vs SU(2) DM in the $extreme$ case. Here $v_1\/v_2 = 1.1$ and $\\sin\\theta=0.3, m_{h_2}= 280$ GeV. In the SU(3) case, $M_{\\rm DM}$ stands for the mass of the (dominant) lighter component $A_\\mu^{3\\prime}$.\n See Fig.~\\ref{plots} for further details. \n \\label{su3plot}\n}\n\\end{figure}\n\nTo understand where differences can appear, it is instructive to consider the limit $v_1 \\simeq v_2$. Although in this case there are further relatively light states\nthat can mediate DM annihilation (e.g. $A_\\mu^{8\\prime}$), let us consider the simplified example in which only the same states are allowed to contribute in the SU(2) and SU(3) \nset--ups. The main features (Fig.~\\ref{su3plot}) are that the gauge coupling must be larger in the SU(3) case\nin order to allow for efficient annihilation of $A^{3\\prime}_\\mu$ and that the resonant\ndips are slightly shifted due to a different freeze--out temperature (see e.g.~\\cite{Griest:1990kh}). The DM density today is dominated by the lighter component. Since it couples to nucleons weaker than $A^{1,2}_\\mu$ do, the direct detection constraint relaxes. Understanding further features of the model\nwould require precise knowledge of the spectrum and the couplings, which we\nrelegate to future work~\\cite{nextpaper}.\n\n\nFinally, one should keep in mind that \n there exists the coupling $A^{3\\prime}_\\mu \\; \\tilde\\varphi_3 \\;\\tilde\\varphi_4$,\nwhere $A^{3\\prime}$ and $\\tilde\\varphi_4$ have the same parities.\nIf $\\tilde\\varphi_4$\nwere light, that is, $\\lambda_4-\\lambda_5 \\ll 1$ and\/or $v_1 \\sim v_2$ (see Eq.~(\\ref{phi4-mass})), \nthe decay $A^{3\\prime}_\\mu \\rightarrow \\tilde\\varphi_4 \\, +$~SM would occur. In that case, DM would consist of both the vector and scalar components.\nWe defer a detailed study of this scenario to future work~\\cite{nextpaper}.\n\n\n\\section{Summary and conclusions}\nIn this paper, we have considered the possibility that the hidden sector enjoys SU(N) gauge symmetry and couples to the Standard Model through the Higgs portal. We find that when endowed with a ``minimal'' matter content, such hidden sectors lead naturally to stable vector dark matter. The underlying Lie group symmetries which stabilise DM \nare associated with complex conjugation of the group elements and discrete gauge transformations.\n\nWe require complete breaking of hidden SU(N) by scalar multiplets to avoid massless states (barring confinement in some cases). That can be done in a minimal fashion by introducing $N-1$ scalar multiplets in the fundamental representation, which develop generic VEVs.\nIf the scalar sector preserves {\\it CP}, the above--mentioned discrete symmetries of the Lie group \ngeneralise to full--fledged symmetries of the model and lead to stable gauge fields.\nWhen sufficiently light, they constitute all of dark matter. \nIn this case, DM consists of 3 components \nassociated with an SU(2) subgroup which hosts the lightest gauge fields $A_\\mu^{1\\prime}$, $A_\\mu^{2\\prime}$ and $A_\\mu^{3\\prime}$. Two of them ($A_\\mu^{1\\prime}, A_\\mu^{2\\prime}$) are\nalways degenerate in mass, while for $N=2$ all 3 components have the same mass.\n\nWe have performed phenomenological analyses of U(1), SU(2) and SU(3) gauge field dark matter. We find that there are vast regions of parameter space where all of the relevant constraints are satisfied. In many of these regions, DM annihilation is facilitated by \nthe broad resonances associated with the Higgs--like scalars.\nWe also find that the SU(3) case appears very similar to that of SU(2), \nunless the scalar VEVs breaking SU(3) are close in magnitude.\n\n\n\\vspace{10pt}\n{\\bf Acknowledgements.} \n The authors are indebted to Sasha Pukhov and Genevieve Belanger for their\nhelp with the new version of micrOMEGAs 4.1.8. C.G. and O.L. acknowledge support\nfrom the Academy of Finland, project ``The Higgs boson and the Cosmos''.\n\nThis work was also supported by the Spanish MICINN's Consolider-Ingenio 2010\nProgramme under grant Multi-Dark {\\bf CSD2009-00064}, the contract\n{\\bf FPA2010-17747} and the France-US PICS no. 06482. \nY.M. acknowledges\npartial support from the European Union FP7 ITN INVISIBLES (Marie Curie\nActions, PITN- GA-2011- 289442) and the ERC advanced grants\n Higgs@LHC and MassTeV. This\nresearch was also supported in part by the Research Executive Agency\n(REA) of the European Union under the Grant Agreement PITN-GA2012-316704\n(``HiggsTools\"). \n\nThe authors would like to thank the Instituto de Fisica Teorica (IFT\nUAM-CSIC) in Madrid for its support via the Centro de Excelencia Severo\nOchoa Program under Grant SEV-2012-0249, during the Program\n``Identification of Dark Matter with a Cross-Disciplinary Approach'' where\nsome of the ideas presented in this paper were developed.\n\n\n\\begin{appendix}\n\\section*{Appendix}\n\n\n\\section{Hidden SU(3) vector--scalar couplings for $v_3=0$}\nThe tables below provide a list of most important gauge--scalar couplings. These\nare relevant to DM phenomenology as well as to understanding the decay channels of the heavier gauge fields.\n\n %\n \\begin{table}[h]\n \\begin{center}\n \\begin{tabular}{|c|c||c|c||c|}\n\\hline\na&b&i&j&coeff. of $\\tilde A_\\mu^a \\tilde A^{\\mu b} \\tilde \\varphi^i \\tilde \\varphi^j$\n\\\\\n\\hline \\hline\n$4$&$4$&$1$&$1$& \\multirow{4}{*}{$\\tilde g^2\/8$}\n\\\\\n$5$&$5$&$1$&$1$&\n\\\\\n$6$&$6$&$1$&$1$&\n\\\\\n$7$&$7$&$1$&$1$&\n\\\\\n\\hline\n$8$&$8$&$1$&$1$& $\\tilde g^2\/6$\n\\\\\n\\hline \\hline\n$1$&$1$&$2$&$2$&\\multirow{5}{*}{$\\tilde g^2\/8$}\n\\\\\n$2$&$2$&$2$&$2$&\n\\\\\n$3$&$3$&$2$&$2$& \n\\\\\n$6$&$6$&$2$&$2$& \n\\\\\n$7$&$7$&$2$&$2$& \n\\\\\n\\hline\n$8$&$8$&$2$&$2$&$\\tilde g^2\/24$\n\\\\\n\\hline\n$3$&$8$&$2$&$2$&$- \\frac{1}{4 \\sqrt{3}}\\tilde g^2$\n\\\\\n\\hline \n\\end{tabular}\n\\quad\n \\begin{tabular}{|c|c||c|c||c|}\n\\hline\na&b&i&j&coeff. of $\\tilde A_\\mu^a \\tilde A^{\\mu b} \\tilde \\varphi^i \\tilde \\varphi^j$\n\\\\\n\\hline \\hline\n$4$&$4$&$3$&$3$& \\multirow{4}{*}{$\\frac 18 \\tilde g^2 \\frac{v_1^2+v_2^2}{v_1^2}$}\n\\\\\n$5$&$5$&$3$&$3$&\n\\\\\n$6$&$6$&$3$&$3$&\n\\\\\n$7$&$7$&$3$&$3$&\n\\\\\n\\hline\n$8$&$8$&$3$&$3$& $\\frac16 \\tilde g^2 \\frac{v_1^2+v_2^2}{v_1^2}$\n\\\\\n\\hline \\hline\n$4$&$4$&$4$&$4$& \\multirow{4}{*}{$\\frac18 \\tilde g^2 \\frac{v_1^2+v_2^2}{v_1^2}$}\n\\\\\n$5$&$5$&$4$&$4$&\n\\\\\n$6$&$6$&$4$&$4$&\n\\\\\n$7$&$7$&$4$&$4$&\n\\\\\n\\hline\n$8$&$8$&$4$&$4$& $\\frac16 \\tilde g^2 \\frac{v_1^2+v_2^2}{v_1^2}$\n\\\\\n\\hline \\hline\n$1$&$4$&$2$&$3$& \\multirow{2}{*}{$\\frac14 \\tilde g^2 \\frac{\\sqrt{v_1^2+v_2^2}}{v_1}$}\n\\\\\n$2$&$5$&$2$&$3$&\n\\\\\n\\hline\n$3$&$6$&$2$&$3$&$-\\frac14 \\tilde g^2 \\frac{\\sqrt{v_1^2+v_2^2}}{v_1}$\n\\\\\n\\hline\n$6$&$8$&$2$&$3$& $- \\frac{1}{4 \\sqrt{3}}\\tilde g^2 \\frac{\\sqrt{v_1^2+v_2^2}}{v_1}$\n\\\\\n\\hline \\hline\n$1$&$5$&$2$&$4$& $\\frac14 \\tilde g^2 \\frac{\\sqrt{v_1^2+v_2^2}}{v_1}$\n\\\\\n\\hline\n$2$&$4$&$2$&$4$&\\multirow{2}{*}{$-\\frac14 \\tilde g^2 \\frac{\\sqrt{v_1^2+v_2^2}}{v_1}$}\n\\\\\n$3$&$7$&$2$&$4$&\n\\\\\n\\hline\n$7$&$8$&$2$&$4$& $- \\frac{1}{4 \\sqrt{3}}\\tilde g^2 \\frac{\\sqrt{v_1^2+v_2^2}}{v_1}$\n\\\\\n\\hline \n\\end{tabular}\n\\end{center}\n\\caption{\\label{AAscalarscalar} Non--derivative couplings $\\tilde A_\\mu^a \\tilde A^{\\mu b} \\tilde \\varphi^i \\tilde \\varphi^j$.\n}\n \\end{table}\n\n \\begin{table}[h]\n \\begin{center}\n \\begin{tabular}{|c|c||c||c|}\n\\hline\na&b&i&coeff. of $\\tilde A_\\mu^a \\tilde A^{\\mu b} \\tilde \\varphi^i$\n\\\\\n\\hline \\hline\n$4$&$4$&$1$& \\multirow{4}{*}{$\\frac{1}{4}\\tilde g^2 v_1$}\n\\\\\n$5$&$5$&$1$& \n\\\\\n$6$&$6$&$1$& \n\\\\\n$7$&$7$&$1$& \n\\\\\n\\hline\n$8$&$8$&$1$& $\\frac{1}{3}\\tilde g^2 v_1$\n\\\\\n\\hline \\hline\n$1$&$1$&$2$& \\multirow{5}{*}{$\\frac{1}{4}\\tilde g^2 v_2 $}\n\\\\\n$2$&$2$&$2$& \n\\\\\n$3$&$3$&$2$& \n\\\\\n$6$&$6$&$2$& \n\\\\\n$7$&$7$&$2$& \n\\\\\n\\hline\n$8$&$8$&$2$& $\\frac{1}{12}\\tilde g^2 v_2 $\n\\\\\n\\hline\n$3$&$8$&$2$& $- \\frac{1}{2 \\sqrt{3}}\\tilde g^2 v_2 $\n\\\\\n\\hline \n\\end{tabular}\n\\quad\n \\begin{tabular}{|c|c||c||c|}\n\\hline\na&b&i&coeff. of $\\tilde A_\\mu^a \\tilde A^{\\mu b} \\tilde \\varphi^i$\n\\\\\n\\hline \\hline\n$1$&$4$&$3$& \\multirow{2}{*}{$\\frac{1}{4}\\tilde g^2 \\frac{v_2 \\sqrt{v_1^2+v_2^2} }{v_1} $}\n\\\\\n$2$&$5$&$3$& \n\\\\\n\\hline\n$3$&$6$&$3$& $-\\frac{1}{4}\\tilde g^2 \\frac{v_2 \\sqrt{v_1^2+v_2^2} }{v_1} $\n\\\\\n\\hline\n$6$&$8$&$3$& $- \\frac{1}{4 \\sqrt{3}}\\tilde g^2 \\frac{v_2 \\sqrt{v_1^2+v_2^2} }{v_1} $\n\\\\\n\\hline \n\\hline\n$1$&$5$&$4$& $\\frac{1}{4}\\tilde g^2 \\frac{v_2 \\sqrt{v_1^2+v_2^2} }{v_1} $\n\\\\\n\\hline\n$2$&$4$&$4$& \\multirow{2}{*}{$-\\frac{1}{4}\\tilde g^2 \\frac{v_2 \\sqrt{v_1^2+v_2^2} }{v_1} $}\n\\\\\n$3$&$7$&$4$& \n\\\\\n\\hline\n$7$&$8$&$4$& $- \\frac{1}{4 \\sqrt{3}}\\tilde g^2 \\frac{v_2 \\sqrt{v_1^2+v_2^2} }{v_1} $\n\\\\\n\\hline \n\\end{tabular}\n\\end{center}\n\\caption{\\label{AAscalar} Non--derivative couplings $\\tilde A_\\mu^a \\tilde A^{\\mu b} \\tilde \\varphi^i$. \n}\n \\end{table}\n %\n\n\n\n\\end{appendix} \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nLet $\\mathbb{R}$  be the real field. A tensor can be regarded as a high-order generalization of a matrix, which takes a form\n\\begin{equation*}\n\\mathcal{A}=(a_{i_1i_2\\dots i_m}),\\quad a_{i_1i_2\\dots i_m}\\in\\mathbb{R},\\quad 1\\leq i_1,i_2,\\ldots,i_m\\leq n.\n\\end{equation*}\nSuch a multi-array $\\mathcal{A}$ is said to be an $m$-order $n$-dimensional real tensor. We denote the set of all $m$-order $n$-dimensional real tensor\nby $\\mathbb{R}^{[m,n]}$.\nLet $x\\in\\mathbb{R}^{n}$, the $n$ dimensional vector $\\mathcal{A}x^{m-1}$ is defined as \\cite{Qi1}:\n\\begin{equation}\\label{eq2}\n (\\mathcal{A}x^{m-1})_{i}=\\sum\\limits_{i_2,\\ldots,i_m}^{n}a_{ii_2,\\ldots,i_m}x_{i_2}\\ldots x_{i_m},\\; i=1,2,\\ldots,n,\n\\end{equation}\nwhere $x_i$ denotes the $i$th component of $x$.\n\nFor any $q\\in\\mathbb{R}^{n} $, we consider the tensor complementarity problem, a special class of nonlinear complementarity problems, denoted by TCP$(\\mathcal{A},q)$: finding $x\\in\\mathbb{R}^{n}$ such that\n \\[x\\geq 0,\\quad\\mathcal{A}x^{m-1}+q\\geq0,\\quad x^{\\top}(\\mathcal{A}x^{m-1}+q)=0. \\]\nThis is a generalization of the linear complementarity problem. So far many\nresearchers have paid attention to this topic \\cite{Bai2016Global,Che2016Positive,Ding2015P,Du2018,Gowda2016Z,Huang2017Formulating,Luo2017The,Song2015Properties,Song2016Tensor,Xie2017An} because of its applications such\nas DNA micro-arrays, communication and $n$-person non-cooperative game \\cite{Huang2017Formulating, Luo2017The}. In \\cite{Song2015Properties}, Song and Qi showed that TCP$(\\mathcal{A}, q)$  has a solution if and only if $\\mathcal{A}$ is nonnegative with its diagonal entries being positive. Song and Qi \\cite{Song2016Tensor} discussed the solution of TCP$(\\mathcal{A}, q)$, when $\\mathcal{A}$ is strictly semi-positive.\n Che, Qi and Wei \\cite{Che2016Positive} discussed the existence and uniqueness of solution\nof TCP$(\\mathcal{A}, q)$ with some special tensors. Luo, Qi and Xiu \\cite{Luo2017The} obtained the sparsest solutions to TCP$(\\mathcal{A}, q)$ with a $\\mathcal{Z}$-tensor. Song and Yu \\cite{Song2016Properties} obtained global\nupper bounds of the solution of the TCP$(\\mathcal{A}, q)$ with a strictly semi-positive\ntensor. Gowda, Luo, Qi and Xiu \\cite{Gowda2016Z} studied the various equivalent conditions for the existence of solution to TCP$(\\mathcal{A}, q)$ with a $\\mathcal{Z}$-tensor.\nDing, Luo and Qi \\cite{Ding2015P} showed the properties of TCP$(\\mathcal{A}, q)$ with a $P$-tensor.\nBai, Huang and Wang \\cite{Bai2016Global} considered the global uniqueness and solvability\nfor TCP$(\\mathcal{A}, q)$ with a strong $P$-tensor. Wang, Huang and Bai \\cite{Wang2016Exceptionally} gave the solvability of TCP$(\\mathcal{A}, q)$ with exceptionally regular tensors.\n\nNumerical algorithms for solving tensor complementarity problems have been\nproposed recently. Xie, Li and Xu \\cite{Xie2017An} presented numerical methods for finding the least solution\nto the TCP$(\\mathcal{A}, q)$ with a $\\mathcal{Z}$-tensor. Liu, Li and Vong \\cite{Liu2017Tensor} proposed the modulus equation for TCP$(\\mathcal{A}, q)$ and based on this equation, they developed the corresponding nonsmooth Newton's method for solving TCP$(\\mathcal{A}, q)$.  Huang and Qi \\cite{Huang2017Formulating} proposed a smoothing type algorithm. Han \\cite{Han2018} introduced a Kojima-Megiddo-Mizuno type continuation method for solving TCP$(\\mathcal{A}, q)$.\nDu et al. \\cite{Du2018} showed that the tensor absolute equation is equivalent\nto a generalized tensor complementarity problem and proposed an inexact Levenberg-Marquardt method for solving the tensor absolute equation. Also, Du and Zhang \\cite{Du2018a} gave a mixed integer programming model to solve the TCP$(\\mathcal{A}, q)$.\n\n\nThe introduction of dynamic models in optimization started in 1980s\n\\cite{chua1984nonlinear,hopfield1985neural}. Since then, significant research results have been achieved for various optimization problems,\nsuch as linear programming \\cite{zak1995solving}, quadratic programming \\cite{bouzerdoum1993neural}, linear complementarity problems \\cite{LIZHI19999}, and nonlinear programming \\cite{rodriguezvazquez1990nonlinear}. The essence of dynamic approach for optimization is\nto establish an energy function (nonnegative).\n The dynamic system is normally in the form of first-order ordinary differential\nequations. It is expected that for an initial state, the dynamic system will approach its static\nstate (or equilibrium point) which corresponds the solution of the underlying optimization problem.\nAn important requirement is that the energy function decreases monotonically as the dynamic system\napproaches an equilibrium point.\n\n\nThe gradient dynamical system (GDS) has now been regarded as a powerful alternative for online computation \\cite{Zhang2006A},\nlinear complementarity problems \\cite{LIZHI19999} and nonlinear complementarity problems \\cite{LIAO2001},  in view of its high speed processing nature and its convenience of hardware implementation in practical applications \\cite{Feng2006Gradient,Ramezani2013Nonlinear}. To effectively solve the linear complementarity problem and nonlinear complementarity problems, linear gradient dynamical system (LGDS) is thus obtained \\cite{LIZHI19999,LIAO2001}. Note\nthat the LGDS with application to online linear (nonlinear) complementarity problems solving have been investigated by the previous work \\cite{LIZHI19999,LIAO2001}.\nThe existence and the convergence of the trajectory of the dynamical system are addressed in detail in \\cite{LIZHI19999,LIAO2001}. In addition, Liao, Qi and Qi \\cite{LIAO2001} also explore the stability properties, such as the stability in the sense of Lyapunov, the asymptotic stability and the exponential stability, for the dynamical system model.\nHowever, to the best of our knowledge, there exists few research results on solving the TCP$(\\mathcal{A}, q)$ via the nonlinear gradient dynamical system (NGDS). Motivated by this\nreason, we thus design, propose and investigate different NGDS for solving TCP$(\\mathcal{A}, q)$ by defining error-monitoring functions. It is theoretically proved that defined  NGDS converge to the theoretical solution.\nThrough illustrative computer-simulation examples, the efficacy and the superiority of the proposed dynamical system model for online computation of the TCP$(\\mathcal{A}, q)$ with $\\mathcal{A}x^{m-1}+q$ is a $P$-function is well-verified.\n\nThe main contributions of the paper are listed as follows.\n\n(1) One type of NGDS for solving the TCP$(\\mathcal{A}, q)$ with $\\mathcal{A}x^{m-1}+q$ for any $q\\in \\mathbb{R}^{n}$ is a $P$-function are presented;\n\n(2) Theoretical analysis shows the convergence of the presented gradient neural networks to the theoretical solution of the TCP$(\\mathcal{A}, q)$ with $\\mathcal{A}x^{m-1}+q$ for any $q\\in \\mathbb{R}^{n}$ is a $P$-function;\n\n(3) Computer simulation results via illustrative examples are presented, compared and\ndiscussed. Generated numerical results comparatively substantiate that the NGDS with nonlinear activation function are much more efficient in solving the TCP$(\\mathcal{A}, q)$ with $\\mathcal{A}x^{m-1}+q$ for any $q\\in \\mathbb{R}^{n}$ is a $P$-function, as compared to the NGDS with linear activation function proposed in this paper.\n\nThis paper is organized as follows. In Section \\ref{sec2}, we recall some preliminary definitions and\nresults. Dynamical system models with different nonlinear activation functions for online solution of the TCP$(\\mathcal{A}, q)$ with $\\mathcal{A}x^{m-1}+q$  for any $q\\in \\mathbb{R}^{n}$ is a $P$-function are presented in Section \\ref{sec3}. Convergence properties of the presented dynamical system models will be discussed in Section \\ref{sec4}.\nIllustrative numerical examples are presented in Section \\ref{Examples}.\n\n\n\\section{Preliminaries}\\label{sec2}\nHan \\cite{HAN201749} gave a method to partially symmetrize tensor $\\mathcal{A}=(a_{i_1i_2\\ldots i_m})\\in \\mathbb{R}^{[m,n]}$ with respect to the indices $i_2\\ldots i_m$, which will be used in sequel. In detail, the partially symmetrized tensor $\\mathcal{\\widehat{A}}=(\\widehat{a}_{i_1i_2\\ldots i_m})$ as follows\n\\[\\widehat{a}_{i_1i_2\\ldots i_m}=\\frac{1}{(m-1)!}\\sum\\limits_{\\pi}a_{i_1\\pi(i_2\\ldots i_m)},\\]\nwhere the sum is over all the permutations $\\pi(i_2\\ldots i_m)$. For any $\\mathcal{A}\\in \\mathbb{R}^{[m,n]}$, we can get a partially symmetrized tensor $\\mathcal{\\widehat{A}}\\in \\mathbb{R}^{[m,n]}$ such that $\\mathcal{A}x^{m-1}=\\mathcal{\\widehat{A}}x^{m-1}$, by an averaging procedure.\n\n\n\\subsection{Function tensors and matrices}\nWe first recall the definitions of $P$-matrix, $P$-tensor and $P$-function as follows.\n\\begin{lemma}{\\bf (\\cite{Berman1994Plemmons})}\\label{pmat}\nLet $A\\in \\mathbb{R}^{n\\times n}$, then\n $A$ is called a $P$-matrix if all the principal minors of $A$ are positive.\n\n\\end{lemma}\n\n\n\nThroughout the paper, we assume that $F:\\mathbb{R}^n\\rightarrow \\mathbb{R}^n$ is a continuously differentiable function.\n\\begin{definition}{\\bf (\\cite{Facchinei2003Finite})}\\label{pfunction}\nA function $F:K\\subseteq \\mathbb{R}^n\\rightarrow \\mathbb{R}^n$ is called a\n $P$-function on $K$ if for all $x,y\\in \\mathbb{R}^n$ with $x\\neq y$, it holds that\n\\begin{equation}\\label{peq1}\n \\max_{i}(x_i-y_i)[F_{i}(x)-F_{i}(y)]>0.\n\\end{equation}\n\\end{definition}\n\\begin{lemma}{\\bf (\\cite{Facchinei2003Finite})}\\label{plema}\n$F$:\\;$\\Omega\\supset K\\subseteq \\mathbb{R}^n\\rightarrow \\mathbb{R}^n$ be continuously differentiable on the\nopen set $\\Omega$ containing the set $K$. $F$ is a $P$-function on $K$ if and only if Jacobian matrix $F'(x)$ is a $P$-matrix for all $x\\in K$.\n\n\n\\end{lemma}\n\\begin{definition}{\\bf (\\cite{Bai2016Global,Liu2017Tensor})}\\label{ptensor}\nLet $\\mathcal{A}\\in\\mathbb{R}^{[m,n]}$, then $\\mathcal{A}$ is called\\\\\n{\\rm (i)} a $P$-tensor, if for each nonzero $x\\in \\mathbb{R}^n$ there exists index $i$ such that\n\\begin{equation}\\label{peq2}\n x_i(\\mathcal{A}x^{m-1})_i> 0;\n\\end{equation}\n{\\rm (ii)} a strong $P$-tensor, if and only if $\\mathcal{A}x^{m-1}$ is a $P$-function;\\\\\n{\\rm (iii)} a strong strictly semi-positive tensor if $\\mathcal{A}x^{m-1}+q$ is a $P$-function in $\\mathbb{R}_{+}^{n}$ for any $q\\in \\mathbb{R}^n$, where $\\mathbb{R}_{+}^{n}$ denotes the set of all the nonnegative vectors.\n\\end{definition}\nFrom the definitions of the $P$-tensor, the strong $P$-tensor and the strong strictly semi-positive tensor, it is easy to see that every strong $P$-tensor must be a $P$-tensor, a strong strictly semi-positive tensor is a $P$-tensor. In additional, we can obtain if for any $q\\in \\mathbb{R}^n$, $\\mathcal{A}x^{m-1}+q$ is a $P$-function, then tensor $\\mathcal{A}$ is a $P$-tensor, furthermore, we have Jacobian matrix $(\\mathcal{A}x^{m-1})'$ is a $P$-matrix.\n\nBai, Huang, and Wang \\cite{Bai2016Global} proved that a TCP possesses the\nglobal uniqueness and solvability property if the tensor is a strong $P$-tensor.  Liu {\\it et al.} \\cite{Liu2017Tensor} showed that a TCP possesses the\nglobal uniqueness and solvability property if the tensor is strong strictly semi-positive tensor.  We summarize their results in the following theorem.\n\\begin{lemma}{\\bf (\\cite{Bai2016Global,Liu2017Tensor})}\\label{pexist} Let $\\mathcal{A}\\in \\mathbb{R}^{[m,n]}$,\\\\\n{\\rm (i)} if $\\mathcal{A}$ is a $P$-tensor, then, for any $q\\in \\mathbb{R}^{n}$, the solution set of $TCP(\\mathcal{A},q)$ is nonempty and compact;\\\\\n{\\rm (ii)} if $\\mathcal{A}$ is a strong $P$-tensor, then $TCP(\\mathcal{A},q)$ has the global uniqueness and solvability property;\\\\\n{\\rm (iii)} if $\\mathcal{A}$ is a strong strictly semi-positive tensor, then $TCP(\\mathcal{A},q)$ has the global uniqueness and solvability property.\n\\end{lemma}\n\\subsection{The Fischer-Burmeister NCP function}\nBelow, we introduce the classical nonlinear complementarity problem (NCP). It will be\nshown that the tensor complementarity problem TCP$(\\mathcal{A},q)$ is a\nspecial kind of nonlinear complementarity problem.\n\\begin{definition}{\\bf (\\cite{Facchinei2003Finite})}\\label{ncp}\nGiven a mapping $F$ : $\\mathbb{R}^n\\rightarrow \\mathbb{R}^n$, the nonlinear complementarity problem,\ndenoted by NCP(F), is to find a vector $x\\in \\mathbb{R}^n $ satisfying\n\\begin{equation}\\label{eqncp}\n  x\\geq 0,\\quad F(x)\\geq 0,\\quad x^{\\top}F(x)=0.\n\\end{equation}\n\\end{definition}\nNote that if $F(x)=\\mathcal{A}x^{m-1}+q$, then NCP reduces to TCP$(\\mathcal{A},q)$.\n\nMany solution methods developed for NCP or related problems are based on\nreformulating them as a system of equations using so-called NCP-functions. Here,\na function $\\phi:\\;\\mathbb{R}^2\\rightarrow \\mathbb{R}$ is called an NCP-function if\n\\begin{equation}\\label{ncpfun}\n \\phi(a,b)=0\\Leftrightarrow ab=0,\\quad a\\geq 0, \\quad b\\geq 0.\n\\end{equation}\nHere, we use the following Fischer-Burmeister NCP-function \\cite{ Fischer1992A},\n\\[\\phi(a,b)=\\sqrt{a^2+b^2}-a-b,\\]\nwhich are widely used in nonlinear complementarity problems.\n\n\\begin{lemma}\\label{lefb}\nThe square of $\\phi(a,b)$ is continuously differentiable; $\\phi(a,b)$ is twice continuously differentiable everywhere except at the origin; but it is strongly semismooth at the origin.\n\\end{lemma}\nBy a simple calculation procedure, we have\n\\[\\partial \\phi(a,b)=\\left\\{\\begin{array}{lc}\n                             \\left(\\frac{a}{\\sqrt{a^2+b^2}}-1,\\frac{b}{\\sqrt{a^2+b^2}}-1\\right), & a^2+b^2\\neq 0, \\\\\n                             (\\alpha-1,\\beta-1)\\;\\text{with}\\; \\alpha^2+\\beta^2\\leq 1, & a^2+b^2=0.\n                           \\end{array}\\right.\n\\]\nFrom Lemma \\ref{lefb}, we have $\\partial \\phi(a,b)$ is a singleton except at the region.\n\\subsection{Activation function}\\label{subsec2.3}\n\nThe matrix-valued activation function $\\mathcal{F}(E)$, $E=(e_{ij})$, is defined as $(f(e_{ij}))$, $i,j=1,2,\\ldots,n$, where $f(\\cdot)$ is a scalar-valued monotonically-increasing odd function. The following real-valued linear and nonlinear odd and monotonically increasing functions $f(\\cdot)$ are widely used.\\\\\n\nLinear function\n  $$f_{\\rm{lin}}(x)=x;$$\n\nBipolar-sigmoid function\n$$\n f_{\\rm{bs}} (x,q)=\\frac{1+\\exp(-q)}{1-\\exp(-q)}\\frac{1-\\exp(-qx)}{1+\\exp(-qx)},\\;q>2;\n$$\n\nPower-sigmoid function\n$$\n f_{\\rm{ps}}(x,p,q)=\\left\\{\\begin{array}{cl}\n                         x^p, & \\mathrm{if }\\ |x|\\geq1 \\\\\n                         \\frac{1+\\exp(-q)}{1-\\exp(-q)}\\frac{1-\\exp(-qx)}{1+\\exp(-qx)},    &  \\mathrm{otherwise}\n                       \\end{array},\\right.\\;q\\geq2,\\;p\\geq3;\n$$\n\nSmooth power-sigmoid function\n$$\nf_{\\rm{sps}}(x,p,q)=\\frac{1}{2}x^{p}+\\frac{1+\\exp(-q)}{1-\\exp(-q)}\\frac{1-\\exp(-qx)}{1+\\exp(-qx)},\\;p\\geq3,\\;q>2.\n$$\n\nIn general, any monotonically increasing odd activation function $f(\\cdot)$  can be used for\nthe construction of the dynamical system.\nAs it was shown in \\cite{Li2013Accelerating,Zhang2009Global}, the\nconvergence rate can be thoroughly improved by an appropriate activation function. So far,\nthe influence of various nonlinear activation functions was investigated for different dynamical system models. We\ninvestigate this scenario on several dynamical system models which are introduced in this paper.\n\n\n\\section{Nonlinear dynamical system methods}\\label{sec3}\nIn this section, the error monitoring function is designed for deriving a gradient dynamical system (GDS). Specifically, by defining different EFs, different GDSs can be obtained for online solution of the  TCP$(\\mathcal{A},q)$. We construct nonlinear gradient dynamical system models, called NGDS, and consider their convergence.\n\nSince  TCP$(\\mathcal{A},q)$ can be equivalently reformulated as finding a solution of the following equation:\n\\begin{equation}\\label{ERRf}\n \\Phi(x)=\\left(\n           \\begin{array}{c}\n             \\phi(x_1,(\\mathcal{A}x^{m-1}+q)_1) \\\\\n                 \\phi(x_2,(\\mathcal{A}x^{m-1}+q)_2) \\\\\n             \\vdots \\\\\n            \\phi(x_n,(\\mathcal{A}x^{m-1}+q)_n) \\\\\n           \\end{array}\n         \\right)=0.\n\\end{equation}\nWe note that $\\Phi(x)$ is locally Lipschitz continuous everywhere, so that Clarke's \\cite{Clarke1983Optimization} generalized Jacobian\n$\\partial\\Phi(x)$ is well defined at any point.\n\nThus, we can define the error monitoring function as\n\\[\\varepsilon(t)=\\varepsilon(x(t))=\\frac{1}{2}\\|\\Phi(x(t))\\|_{2}^2.\\]\nThe function $\\varepsilon(x(t))$ is continuously differentiable \\cite{facchinei1997a}, which follows from the semi-smoothness of $\\Phi(x)$.\n\n\nIn order to force $\\varepsilon(t)$ to converge to zero, the\nnegative of the gradient (i.e., -$\\partial(\\varepsilon(t))\/\\partial x$) is used as the\ndescent direction, which leads to the so-called GDS design formula in the form of a first-order differential\nequation:\n\\begin{equation}\\label{GNNF}\n\\frac{\\mathrm{d}x}{\\mathrm{d}t}=-\\gamma\\frac{\\partial(\\varepsilon(t))}{\\partial x}=-\\gamma V^{\\top}\\Phi(x),\n\\end{equation}\nwhere  $V\\in\\partial\\Phi(x)$ and $\\gamma$ is a positive scaling constant. Note that $\\gamma$ corresponds to the reciprocal of a capacitance parameter, of\nwhich the value should be set as large as the hardware\nwould permit, or appropriately large for modeling and experimental purposes. The dynamic equation (\\ref{GNNF}) will be simply termed the LGDS model.\n\n\n\nFollowing the principle of nonlinear activation in the LGDS model defined in Section \\ref{subsec2.3}, the conventional LGDS model (\\ref{GNNF}) can be improved\ninto the following nonlinear GDS model by exploiting a nonlinear activation function array\n$\\mathcal{F}(\\cdot)$:\n\\begin{equation}\\label{GNN-I2}\n\\frac{\\mathrm{d}x}{\\mathrm{d}t}=-\\gamma V^{\\top}\\mathcal{F}(\\Phi(x)),\n\\end{equation}\nwhere $\\mathcal{F}(\\cdot)$ denotes a matrix-valued activation function. The dynamic equation (\\ref{GNN-I2}) will be simply termed the NGDS model.\n\nWe give the precise definition of $V$ which is necessary for the implementation of our model.\n\\begin{lemma}{\\bf(\\cite{deluca1996a})}\nAny $V$ with the following structure is an element of $\\partial\\Phi(x)$\n\\[V=D_{a}(x)+(m-1)D_{b}(x)\\mathcal{\\widehat{A}}x^{m-2},\\]\nwhere $D_{a}(x)=$diag$(a_1(x),\\ldots,a_n(x))$, $D_{b}(x)=$diag$(b_1(x),\\ldots,b_n(x))$ are diagonal matrices whose $i$th diagonal elements are given by\n\\[a_i(x)=\\frac{x_i}{\\sqrt{x_i^2+(\\mathcal{A}x^{m-1}+q)_i^2}}-1,\\quad b_i(x)=\\frac{(\\mathcal{A}x^{m-1}+q)_i}{\\sqrt{x_i^2+(\\mathcal{A}x^{m-1}+q)_i^2}}-1\\]\nif $(x_i,(\\mathcal{A}x^{m-1}+q)_i)\\neq (0,0)$, and by\n\\[a_i(x)=\\alpha_i-1,\\quad b_i(x)=\\beta_i-1\\]\nfor every $(\\alpha_i,\\beta_i)\\in \\mathbb{R}^2$ such that $\\alpha_i^2+\\beta_i^2\\leq 1$ if $(x_i,(\\mathcal{A}x^{m-1}+q)_i)= (0,0)$.\n\\end{lemma}\n\n\n\\section{Stability analysis}\\label{sec4}\nIn this section, we address the stability issues on the dynamical system (\\ref{GNN-I2}) to the solution of the  TCP$(\\mathcal{A},q)$.\n\nFor given $x_{*}\\in \\mathbb{R}^{n}$ with $x_{*}$ is a solution of the  TCP$(\\mathcal{A},q)$, define $\\delta=\\min\\|x_{*}-u\\|_2$ for any $u\\in \\mathbb{R}^{n}$. Thus, for any $0<\\widehat{\\delta}\\leq\\delta$, we define a neighbourhood of $x_{*}$ as\n\\begin{equation}\\label{BQOP:equation16}\n\\mathbb{B}(x_{*};\\widehat{\\delta}):=\\{x\\mid\\|x-x_{*}\\|_2\\leq\\widehat{\\delta}\\},\n\\end{equation}\nwhere $x\\in\\mathbb{R}^n$.\n\nNow we recall some stability results from \\cite{Zabczyk2015Mathematical} on the following differential equation:\n\\begin{equation}\\label{eqnnew}\n \\frac{\\mathrm{d}x(t)}{\\mathrm{d}t}=f(x(t)), \\quad x(t_0)\\in\\mathbb{R}^n.\n\\end{equation}\n\nThe following classical results on the existence and uniqueness of the solution to (\\ref{eqnnew}) hold.\n\\begin{definition}{\\bf(\\cite{Zabczyk2015Mathematical})} \\label{def4.1}\nLet $x(t)$ be a solution of (\\ref{eqnnew}). An isolated equilibrium point $x^*$ is Lyapunov stable if for any $x(t_0)$ and any scalar $\\epsilon>0$ there exists a $\\hat{\\delta}>0$\nso that if $x(t_0)\\in \\mathbb{B}(x_{*},\\widehat{\\delta})$ then $\\|x(t)-x_*\\|_2<\\epsilon$ for $t\\geq t_0$.\n\\end{definition}\n\\begin{definition}{\\bf(\\cite{Zabczyk2015Mathematical})} \\label{def4.2}\n An isolated equilibrium point $x_*$ is said to be asymptotic stable\nif in addition to being a Lyapunov stable it has the property that\n$x(t)\\rightarrow x_*$ as $t\\rightarrow +\\infty $, if $x(t_0)\\in \\mathbb{B}(x_{*},\\hat{\\delta})$.\n\\end{definition}\n\nThen we focus on a particular case where the equilibrium point is isolated. Let $S$ denote the solution set of the  TCP$(\\mathcal{A},q)$ and $x\\in S$  implies $\\Phi(x)=0$. Togather with   $\\mathcal{F}(\\cdot)$ is odd and monotonically increasing function, we have $\\mathcal{F}(\\Phi(x))=0$, consequently, $\\frac{\\mathrm{d}x}{\\mathrm{d}t}=0$. Hence, we have the following result.\n\\begin{theorem}\\label{theq}\nEvery solution to the TCP$(\\mathcal{A},q)$ is an equilibrium point of the dynamical system (\\ref{GNN-I2}).\nConversely; if $x\\in \\mathbb{R}^n$ is an equilibrium of (\\ref{GNN-I2}) and for any $q\\in \\mathbb{R}^n$, $\\mathcal{A}x^{m-1}+q$ is a $P$-function, then $x\\in S$.\n\\end{theorem}\n\\begin{proof}\nWe only need to address the second part of the Theorem. Since  $\\mathcal{A}x^{m-1}+q$ is a continuously differentiable $P $-function, it's Jacobian matrix is a $P $-matrix for all $x\\in \\mathbb{R}^{n}$. Analogy to the proof of  \\cite[Corollary  4.4]{deluca1996a}, we have  a stationary point of (\\ref{GNN-I2}) is a solution to the TCP$(\\mathcal{A},q)$.\nWe complete our proof.\n\\end{proof}\n\nNote that if $\\mathcal{A}$ is a strong $P$-tensor (strong strictly semi-positive tensor), then $TCP(\\mathcal{A},q)$ has the global uniqueness and solvability property. Together with a strong $P$-tensor (strong strictly semi-positive tensor) is a $P$-tensor, we obtain the following results.\n\\begin{theorem}\\label{strongp}\nLet $\\mathcal{A}$ be a strong $P$-tensor (strong strictly semi-positive tensor) and $x$ be a solution of the TCP$(\\mathcal{A},q)$, then $x$ is an unique equilibrium point of the dynamical system (\\ref{GNN-I2}).\n\\end{theorem}\n\\begin{proof}\nSince $\\mathcal{A}$ is a strong $P$-tensor (strong strictly semi-positive tensor), then $TCP(\\mathcal{A},q)$ has the global uniqueness and solvability property. That is if $x$ is a solution of the TCP$(\\mathcal{A},q)$, then $x$ is a unique solution of the TCP$(\\mathcal{A},q)$. From the Theorem \\ref{theq}, we get $x$ is an unique equilibrium point of the neural network (\\ref{GNN-I2}). The proof is thus completed.\n\\end{proof}\nWe have the following result of the convergence of the NGDS model.\n\\begin{theorem} \\label{theorem4.1}\nGiven $\\widehat{\\delta}>0$ and $x_{*}$ be a solution of the TCP$(\\mathcal{A},q)$, if nonzero vector $x(0)\\in \\mathbb{B}(x_{*};\\widehat{\\delta})$ and  $\\mathcal{A}x(t)^{m-1}+q$ is a $P$-function for any $q\\in \\mathbb{R}^n$, then the state $x(t)$ of the {\\em NGDS} model (\\ref{GNN-I2}), starting from the initial state $x(0)\\in \\mathbb{B}(x_{*};\\widehat{\\delta})$, converges to the solution $x_*\\in \\mathbb{R}^{n}$ of the  TCP$(\\mathcal{A},q)$.\n\\end{theorem}\n\\begin{proof} We construct the following Lyapunov function:\n\\begin{equation}\\label{lyap}\n  L(t)=\\varepsilon(x(t))=\\frac{1}{2}\\|\\Phi(x(t))\\|_{2}^2\\geq 0,\n\\end{equation}\nwith its time derivative being\n\\begin{equation}\\label{eq4.2}\n\\begin{array}{ccl}\n \\frac{\\mathrm{d}L(t)}{\\mathrm{d}t}&= &\\frac{1}{2}\\frac{\\mathrm{d}}{dt} \\mathrm{Tr}\\left(\\Phi(x(t))^{\\top}\\Phi(x(t))\\right)\\\\\n &= &\\left(\\Phi(x(t))^{\\top}\\frac{\\mathrm{d}\\Phi(x(t)))}{\\mathrm{d}t}\\right)\\\\\n  &=&\\left(\\Phi(x(t))^{\\top}V\\frac{\\mathrm{d}x}{\\mathrm{d}t}\\right)\\\\\n    &=&-\\gamma\\left(\\Phi(x(t))^{\\top}VV^{\\top}\\mathcal{F}(\\Phi(x(t)))\\right)\\\\\n &=&-\\gamma\\mathrm{Tr}\\left(VV^{\\top}\\Phi(x(t))^{\\top}\\mathcal{F}(\\Phi(x(t)))\\right).\\\\\n \\end{array}\n \\end{equation}\nSince $V$ is a $P$-matrix, which follows from $\\mathcal{A}x^{m-1}+q$ is a $P$-function for any $q\\in \\mathbb{R}^n$, then, $VV^{\\top}$ is a symmetric positive definite matrix, and then, we have\n\n\\[\\begin{array}{cll}\n     \\lambda_{\\min}\\mathrm{Tr}\\left(\\Phi(x(t))^{\\top}\\mathcal{F}(\\Phi(x(t)))\\right) & \\leq& \\mathrm{Tr}\\left(VV^{\\top}\\Phi(x(t))^{\\top}\\mathcal{F}(\\Phi(x(t)))\\right) \\\\\n     & \\leq & \\lambda_{\\max}\\mathrm{Tr}\\left(\\Phi(x(t))^{\\top}\\mathcal{F}(\\Phi(x(t)))\\right),\n  \\end{array}\n\\]\nwhere $ \\lambda_{\\min}$ and $ \\lambda_{\\max}$ denote the smallest eigenvalue and the largest eigenvalue of matrix $VV^{\\top}$, respectively.  $\\mathrm{Tr}(A)$ is the trace of the matrix $A$.\n\nSince the scalar-valued function $f(\\cdot)$ is an odd and monotonically increasing function, it immediately follows $f(-x)=-f(x)$ and\n\\[f(x)\\left\\{\\begin{array}{cl}\n               >0 ,& \\mathrm{if}\\; x>0, \\\\\n               =0, & \\mathrm{if} \\;x=0, \\\\\n               <0 ,& \\mathrm{if} \\;x<0,\n             \\end{array}\n\\right. \\]\nwhich implies\n\\[xf(x)\\left\\{\\begin{array}{cl}\n               >0, & \\mathrm{if}\\; x\\neq 0, \\\\\n               =0,& \\mathrm{if} \\;x=0.\n             \\end{array}\n\\right. \\]\nIt follows that\n\\[\\mathrm{Tr}\\left(\\Phi(x(t))^{\\top}\\mathcal{F}(\\Phi(x(t)))\\right)\\left\\{\\begin{array}{cl}\n               >0, & \\mathrm{if}\\; \\Phi(x(t))\\neq 0, \\\\\n               =0, & \\mathrm{if} \\;\\Phi(x(t))=0.\n             \\end{array}\n\\right.\\]\nDue to the fact that the design parameter satisfies $\\alpha>0$, in view of (\\ref{eq4.2}), it follows that\n\\[\\frac{\\mathrm{d}L_1(t)}{\\mathrm{d}t}\\left\\{\\begin{array}{cl}\n              <0, & \\mathrm{if}\\; \\Phi(x(t))\\neq 0, \\\\\n               =0, & \\mathrm{if} \\;\\Phi(x(t))=0.\n              \n             \\end{array}\n\\right.\\]\n\nBy the Lyapunov theory,  $\\Phi(x(t))$ can converge to zero; or, equivalently speaking, state $x(t)$ of NGDS model (\\ref{GNN-I2}) is asymptotic stable at one of solutions $x_{*}$ with $\\Phi(x_{*})=0$ starting from an initial state $x(0)\\in \\mathbb{B}(x_{*};\\widehat{\\delta})$. The proof is thus complete.\n\\end{proof}\n\nWe summarize the convergence result of the NGDS  method (\\ref{GNN-I2}) when $\\mathcal{A}$\nis strong $P$-tensor (strong strictly semi-positive tensor) in the following theorem.\n\\begin{theorem} \\label{theorem4.2}\nSuppose that $\\mathcal{A}\\in \\mathbb{R}^{[m,n]}$ is a strong $P$-tensor (strong strictly semi-positive tensor) and $q\\in \\mathbb{R}^n$. The state $x(t)$\nof the {\\em NGDS} model (\\ref{GNN-I2}), starting from an arbitrary initial state $x(0)\\in \\mathbb{R}^{n}$, converges to the unique solution $x_*\\in \\mathbb{R}^{n}$ of the TCP$(\\mathcal{A},q)$.\n\\end{theorem}\n\\begin{proof}\nSince $\\mathcal{A}$ is a strong $P$-tensor (strong strictly semi-positive tensor), from Theorem \\ref{strongp}, we know $TCP(\\mathcal{A},q)$, for given $q\\in \\mathbb{R}^{n}$, has the global uniqueness and solvability property. Together with the proof of Theorem \\ref{theorem4.1}, we can show that the state $x(t)$\nof the { NGDS} model (\\ref{GNN-I2}), starting from an arbitrary initial state $x(0)\\in \\mathbb{R}^{n}$, converges to the unique solution $x_*\\in \\mathbb{R}^{n}$ of the TCP$(\\mathcal{A},q)$. We complete our proof.\n\\end{proof}\n\\section{Numerical examples}\\label{Examples}\nIn this section, some computer-simulation examples are\ndemonstrated to verify the efficacy and the superiority of\nthe proposed neural network models. We apply the NGDS to the TCP$(\\mathcal{A},q)$.\n\n\nAll computations are carried out in Matlab Version 2014a, which has a unit roundoff $ 2^{-53}\\approx 1.1\\times 10^{-16}$, on a laptop with Intel Core(TM) i5-4200M CPU (2.50GHz) and 7.89GB RAM.\n\n\n\\begin{exmple} \\label{eg1}\nConsider the tensor $\\mathcal{A}\\in \\mathbb{R}^{[4,2]}$, from \\cite{Bai2016Global,Liu2017Tensor}, defined by:\n\\[a_{1111}=1,\\;a_{1222}=-1,\\;a_{1122}=1,\\;a_{2222}=1,\\;a_{2111}=-1,\\;a_{2211}=1,\\]\nand $a_{i_1i_2i_3i_4}=0$ otherwise. This tensor is a $P$-tensor, but not a strong $P$-tensor \\cite{Bai2016Global}. However, it is a strong strictly semi-positive tensor \\cite{Liu2017Tensor}.\n\\end{exmple}\nIn our tests, we take different vectors $q\\in \\mathbb{R}^2$ and initial vector as $x_0={\\rm{rand}}(2,1)$. Trajectories of state variables corresponding to the LGDS with $\\gamma=10$ are shown in Figure \\ref{figure 1} (a), (c), (e) and Figure \\ref{figure 2} (a) and (c), respectively.\nResidual errors\n\\begin{equation}\\label{eq6.2}\n \\mathrm{Res}=\\|\\Phi(x)\\|_2,\n\\end{equation}\n\\begin{figure}[H]\n\\centering\n\\subfigure[$\\gamma=10$, $q=(-5,-3)^{\\top}$.]{\\includegraphics[width=3in, height=1.8in]{ex1q1state.jpg}}\n\\subfigure[$\\gamma=1\\times 10^6$, $q=(-5,-3)^{\\top}$.]{\\includegraphics[width=3in, height=1.8in]{ex1q1error.jpg}}\\\\\n\\subfigure[$\\gamma=10$, $q=(-5,3)^{\\top}$.]{\\includegraphics[width=3in, height=1.8in]{ex1q2state.jpg}}\n\\subfigure[$\\gamma=1\\times 10^6$, $q=(-5,3)^{\\top}$.]{\\includegraphics[width=3in, height=1.8in]{ex1q2error.jpg}}\\\\\n\\subfigure[$\\gamma=10$, $q=(5,3)^{\\top}$.]{\\includegraphics[width=3in, height=1.8in]{ex1q3state.jpg}}\n\\subfigure[$\\gamma=1\\times 10^6$, $q=(5,3)^{\\top}$.]{\\includegraphics[width=3in, height=1.8in]{ex1q3error.jpg}}\n\\caption{Trajectories of state variables and residual errors of the NGDSs for Example \\ref{eg1}.}\\label{figure 1}\n\\end{figure}\n\n\nderived by employing the NGDS with $\\gamma=1\\times 10^6$  shown in Figures \\ref{figure 1} (b), (d), (f) and Figure \\ref{figure 2} (b) and (d), respectively, where  blue stars indicate that $f(\\cdot)$ is the linear function, red stars  correspond to the bipolar-sigmoid function $f_{bs}(x,5)$,\npink stars denote that power-sigmoid function $f_{ps}(x,3,5)$  and green stars are generated using $f_{sps}(x,3,7)$ as the smooth power-sigmoid function.\n\\begin{figure}[!h]\n\\centering\n\\subfigure[$\\gamma=10$, $q=(2,-3)^{\\top}$.]{\\includegraphics[width=3in, height=2.2in]{ex1q4state.jpg}}\n\\subfigure[$\\gamma=1\\times 10^6$, $q=(2,-3)^{\\top}$.]{\\includegraphics[width=3in, height=2.2in]{ex1q4error.jpg}}\\\\\n\\subfigure[$\\gamma=10$, $q=(0,-5)^{\\top}$.]{\\includegraphics[width=3in, height=2.2in]{ex1q5state.jpg}}\n\\subfigure[$\\gamma=1\\times 10^6$, $q=(0,-5)^{\\top}$.]{\\includegraphics[width=3in, height=2.2in]{ex1q5error.jpg}}\\\\\n\\caption{Trajectories of state variables and residual errors of the NGDSs for Example \\ref{eg1}.}\\label{figure 2}\n\\end{figure}\n\\begin{exmple} \\label{eg2}\nLet $\\mathcal{A}\\in \\mathbb{R}^{[5,3]}$ be defined by $a_{kkkkk}=k$ for $k=1,2,3$ and\n $a_{i_1i_2i_3i_4i_5}=0$ otherwise. This tensor is both a strong $P$-tensor \\cite{Bai2016Global} and a strong strictly semi-positive \\cite{Han2018}.\n\\end{exmple}\nFor different vectors $q\\in \\mathbb{R}^3$, we take  initial vector as $x_0={\\rm{rand}}(3,1)$. Trajectories of state variables corresponding to the LGDS with different $\\gamma$ are shown in Figure \\ref{figure 3} (a), (c), (e) and Figure \\ref{figure 4} (a), (c) and (e), respectively.\nResidual errors (\\ref{eq6.2})\nderived by employing the NGDS with $\\gamma=1\\times 10^6$  shown in Figures \\ref{figure 3} (b), (d), (f) and Figure \\ref{figure 4} (b), (d) and (f), respectively.\n\n\\begin{figure}[H]\n\\centering\n\\subfigure[$\\gamma=1000$, $q=(1,2,3)^{\\top}$.]{\\includegraphics[width=3in, height=2.2in]{ex2q1state.jpg}}\n\\subfigure[$\\gamma=1\\times 10^6$, $q=(1,2,3)^{\\top}$.]{\\includegraphics[width=3in, height=2.2in]{ex2q1error.jpg}}\\\\\n\\subfigure[$\\gamma=1000$, $q=(1,-2,3)^{\\top}$.]{\\includegraphics[width=3in, height=2.2in]{ex2q2state.jpg}}\n\\subfigure[$\\gamma=1\\times 10^6$, $q=(1,-2,3)^{\\top}$.]{\\includegraphics[width=3in, height=2.2in]{ex2q2error.jpg}}\\\\\n\\subfigure[$\\gamma=100$, $q=(-3,-2,-3)^{\\top}$.]{\\includegraphics[width=3in, height=2.2in]{ex2q3state.jpg}}\n\\subfigure[$\\gamma=1\\times 10^6$, $q=(-3,-2,-3)^{\\top}$.]{\\includegraphics[width=3in, height=2.2in]{ex2q3error.jpg}}\n\\caption{Trajectories of state variables and residual errors of the NGDSs for Example \\ref{eg2}.}\\label{figure 3}\n\\end{figure}\n\\begin{figure}[H]\n\\centering\n\\subfigure[$\\gamma=100$, $q=(3,3,3)^{\\top}$.]{\\includegraphics[width=3in, height=2.2in]{ex2q4state.jpg}}\n\\subfigure[$\\gamma=1\\times 10^6$, $q=(3,3,3)^{\\top}$.]{\\includegraphics[width=3in, height=2.2in]{ex2q4error.jpg}}\\\\\n\\subfigure[$\\gamma=100$, $q=(-3,-1,-2)^{\\top}$.]{\\includegraphics[width=3in, height=2.2in]{ex2q5state.jpg}}\n\\subfigure[$\\gamma=1\\times 10^6$, $q=(-3,-1,-2)^{\\top}$.]{\\includegraphics[width=3in, height=2.2in]{ex2q5error.jpg}}\\\\\n\\subfigure[$\\gamma=100$, $q=(-1,-1,-2)^{\\top}$.]{\\includegraphics[width=3in, height=2.2in]{ex2q6state.jpg}}\n\\subfigure[$\\gamma=1\\times 10^6$, $q=(-1,-1,-2)^{\\top}$.]{\\includegraphics[width=3in, height=2.2in]{ex2q6error.jpg}}\n\\caption{Trajectories of state variables and residual errors of the NGDSs for Example \\ref{eg2}.}\\label{figure 4}\n\\end{figure}\n\nWhere  blue stars indicate that $f(\\cdot)$ is the linear function, red stars  correspond to the bipolar-sigmoid function $f_{bs}(x,7)$,\npink stars denote that power-sigmoid function $f_{ps}(x,5,7)$  and green stars are generated using $f_{sps}(x,5,9)$ as the smooth power-sigmoid function, respectively.\n\n\n\n\n\\begin{exmple} \\label{eg3}\nLet $\\mathcal{A}\\in \\mathbb{R}^{[4,2]}$ be defined by\n\\[a_{1111}=1,\\;a_{1112}=-2,\\;a_{1122}=1,\\;a_{2222}=1,\\]\nand all other $a_{i_1i_2i_3i_4}=0$. This tensor is  a $P$-tensor \\cite{Bai2016Global}.\n\\end{exmple}\nTaking $q=(0,-1)^{\\top}$, it is easy to see that $x_{*}=(0,1)^{\\top}$ and $x_{*}=(1,1)^{\\top}$ are the solutions to the TCP$(\\mathcal{A},q)$.\n\nWe choose  initial vector as $x_0=(0.1,0.5)^{\\top}$ for $x_{*}=(0,1)^{\\top}$ and $x_0=(1.5,1.1)^{\\top}$ for $x_{*}=(1,1)^{\\top}$. Trajectories of state variables corresponding to the LGDS with  $\\gamma=100$ and  $\\gamma=10000$ are shown in Figure \\ref{figure 5} (a) and Figure \\ref{figure 6} (a), respectively.\n\n\n\nResidual errors (\\ref{eq6.2}) derived by employing the NGDS with $\\gamma=1\\times 10^6$  shown in Figures \\ref{figure 5} (b) and Figures \\ref{figure 6} (b), where  blue stars indicate that $f(\\cdot)$ is the linear function, red stars  correspond to the bipolar-sigmoid function $f_{bs}(x,7)$,\npink stars denote that power-sigmoid function $f_{ps}(x,5,9)$  and green stars are generated using $f_{sps}(x,7,11)$ as the smooth power-sigmoid function, respectively.\n\n\\begin{figure}[H]\n\\centering\n\\subfigure[$\\gamma=100$, $x_{*}=(0,1)^{\\top}$.]{\\includegraphics[width=3in, height=2.2in]{ex3q2state1.jpg}}\n\\subfigure[$\\gamma=1\\times 10^6$, $x_{*}=(0,1)^{\\top}$.]{\\includegraphics[width=3in, height=2.2in]{ex3q1error.jpg}}\n\\caption{Trajectories of state variables and residual errors of the NGDSs for Example \\ref{eg3}.}\\label{figure 5}\n\\end{figure}\n\n\\begin{figure}[h]\n\\centering\n\\subfigure[$\\gamma=10000$, $x_{*}=(1,1)^{\\top}$.]{\\includegraphics[width=3in, height=2.2in]{ex3q2state2.jpg}}\n\\subfigure[$\\gamma=1\\times 10^6$, $x_{*}=(1,1)^{\\top}$.]{\\includegraphics[width=3in, height=2.2in]{ex3q2error2.jpg}}\n\\caption{Trajectories of state variables and residual errors of the NGDSs for Example \\ref{eg3}.}\\label{figure 6}\n\\end{figure}\n\nFrom the computer simulation results derived in the illustrative examples, the following conclusions can be highlighted:\n\n1. The NGDS models with different types of activation functions  presented in section 3, exactly and efficiently solve the TCP$({\\mathcal A},q)$ with $P$-tensors, strong $P$-tensor and strong strictly semi-positive tensor.\n\n2. NGDS models could achieve different performances if different activation function arrays are used. In general, the convergence\nperformance of nonlinear activation functions is superior to that of the\nlinear activation function.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\t\\label{intro}\n\t\t\nThe Moran construction is a typical way to generate self-similar fractals, and has been studied extensively in the literature (e.g. \\cite{Moran},  \\cite{Beardon}, \\cite{hutch}, \\cite{Mandelbrot},\\cite{Li}, \\cite{Su}, \\cite{inhom}, and references therein).  In this paper, we extend ideas from iterated function systems (IFS) and Moran constructions by describing a new process that allows for the functions to be updated at every iteration while still maintaining the computational simplicity of an IFS.  This process provides more variance in the limit sets (such as non-self-similarity) using an analogous approach to an IFS procedure.  We also give estimates of the Hausdorff dimension of the limit sets created from such a process, and provide concrete examples.\n\t\t\nThe classic construction of Moran sets was introduced in \\cite{Moran}.  We reproduce the definition here with a more current interpretation to introduce notations.\n\t\n\tLet $\\{n_k\\}_{k\\geq1}$ be a sequence of positive integers for $k \\geq 1$.  For any $k \\in {\\mathbb{N}}$, define \n\t\\begin{equation}\n\t\\label{Dk}\n\tD_{k} = \\{ (i_1, i_2, \\cdots, i_k) : 1 \\leq i_j \\leq n_k, 1 \\leq j \\leq k\\} \\text{ and } D = \\bigcup_{k \\geq 0} D_{k}. \n\t\\end{equation} \n\tWe define $D_0 = \\emptyset$.  Let $\\sigma = (\\sigma_1, \\cdots, \\sigma_k) \\in D_k$ and $\\tau = (\\tau_1, \\cdots, \\tau_m) \\in D_m$, then denote $\\sigma * \\tau = (\\sigma_1, \\cdots, \\sigma_k, \\tau_1, \\cdots, \\tau_m)$.\n\tUsing this notation, we may express\n\t\\begin{equation}\n\t\\label{dksigma}\n\tD_{k} =\\{ \\sigma * j | \\sigma \\in D_{k-1}, 1 \\leq j \\leq n_k\\}\n\t\\end{equation} to emphasize the process of moving between generations.\n\t\n\tSuppose $\\mathcal J := \\{J_{\\sigma} : \\sigma \\in D\\}$ is a collection of subsets of ${\\mathbb{R}}^N$.  \n\tSet \n\t\\begin{equation}\n\t\\label{eqn: Limit_set}\n\tE_{k} = \\bigcup_{\\sigma \\in D_{k}} J_{\\sigma}, \\text{ and } F = \\bigcap_{k \\geq 0} E_{k}.\n\t\\end{equation}\n\tWe call $F$ the limit set associated with the collection $\\mathcal{J}$.\n\t\n\t\n\t\\begin{definition}[\\cite{Su}]\n\t\tSuppose that $J \\subset {\\mathbb{R}}^N$ is a compact set with nonempty interior.  Let $\\{n_k\\}_{k \\geq 1}$ be a sequence of positive integers, and $\\{ \\Phi_k\\}_{k \\geq 1}$ be a sequence of positive real vectors with \n\t\t\\begin{equation}\n\t\t\\Phi_k = (c_{k,1}, c_{k,2}, \\dots, c_{k,n_k}), \\sum_{1 \\leq j \\leq n_k} c_{k,j} \\leq 1, k \\in {\\mathbb{N}}.\n\t\t\\end{equation} \n\t\tSuppose that ${\\mathcal F} := \\{J_\\sigma : \\sigma \\in D\\}$ is a collection of subsets of ${\\mathbb{R}}^N,$ where $D$ is given in (\\ref{Dk}).  We say that the collection ${\\mathcal F}$ fulfills the Moran Structure provided it satisfies the following Moran Structure Conditions (MSC):\n\t\t\\begin{enumerate}\n\t\t\t\\item[MSC(1)] $J_{\\emptyset} = J.$\n\t\t\t\\item[MSC(2)] For any $\\sigma \\in D$, $J_\\sigma$ is geometrically similar to $J$.  That is, there exists a similarity $S_\\sigma: {\\mathbb{R}}^N \\to {\\mathbb{R}}^N$ such that $J_\\sigma = S_\\sigma(J)$. \n\t\t\t\\item[MSC(3)] For any $k \\geq 0$ and $\\sigma \\in D_k$, $J_{\\sigma \\star 1}, \\dots , J_{\\sigma \\star n_k}$ are subsets of $J_\\sigma$, and $int(J_{\\sigma \\star i}) \\cap int(J_{\\sigma \\star j}) = \\emptyset$ for $i \\neq j$.\n\t\t\t\\item[MSC(4)] For any $k \\geq 1$ and $\\sigma \\in D_{k-1}, 1 \\leq j \\leq n_k,$ \n\t\t\t\\begin{equation}\n\t\t\t\\label{ckj}\n\t\t\t\\dfrac{diam(J_{\\sigma \\star j})}{diam(J_{\\sigma})} = c_{k,j}.\t\n\t\t\t\\end{equation}\n\t\t\t\n\t\t\\end{enumerate}\n\t\t\n\t\tFor the collection ${\\mathcal F}$ fulfilling the MSC, the limit set $F$ given in (\\ref{eqn: Limit_set}) is a nonempty compact set. This limit set $F$ is called the Moran set associate with the collection ${\\mathcal F}$.  This Moran set is self-similar, and has been studied extensively by many authors with various approaches (e.g. \\cite{Moran}, \\cite{hutch}, \\cite{Falconer}, \\cite{Beardon}, \\cite{peres}).\n\t\\end{definition}\n\t\n   \tSeveral approaches have been used to relax MSC in order to create more general limit sets.  The dimension (e.g. Hausdorff, Box, Packing, \\dots) of these sets has been a fruitful area of study.  For example, in \\cite{mcmullen}, MSC(2) has been expanded to affine maps.  In this setting, however, calculations of the dimension of some limit sets can become particularly difficult.  One could also study the limit sets generated by infinitely many similarities, as in \\cite{mauldin}. In \\cite{Li}, the authors  removed MSC(2), but required $\\overbar{int(J_\\sigma)} = J_\\sigma$ in their construction, and studied the dimension of the resulting fractals.  In \\cite{inhom}, Holland and Zhang studied a construction that replaced similarity maps in MSC(2) with a more general class of functions that are not necessarily contractions.  In \\cite{PW}, Pesin and Weiss removed the requirement for similarities from MSC(2), but also relaxed MSC(3) from non-intersecting basic sets to non-intersecting balls contained in the basic sets.  In particular they pursued sufficient conditions for which the Box-counting and Hausdorff dimensions coincide.  For more examples of modifications to the Moran set definition, see \\cite{Su} and the references therein.\n\t\n\tA special case of Moran sets can be constructed from an iterated function system (IFS).  An iterated function system $\\{S_1, S_2, \\cdots, S_m\\}$ is a finite family of similarities for a fixed natural number $m\\geq2$ (see \\cite{Falconer} for more details and applications).  In MSC(2), define $n_k=m$ and set $S_{\\sigma} = S_{i_k} \\circ S_{i_{k-1}} \\circ \\cdots \\circ S_{i_1}$ for $\\sigma = (i_1, i_2, \\cdots, i_k) \\in D$.  Then the resulting Moran set is self-similar and agrees with the attractor of the IFS $\\{S_1, S_2, \\cdots, S_m\\}$.  The dimension of the limit set can be quickly calculated from the Moran-Hutchinson formula in \\cite{hutch}.  Using iterated function systems is a popular way to construct fractals, and has been used to great effect (e.g. \\cite{barnsley}, \\cite{Falconer}, \\cite{hutch}, \\cite{BV} ).   \n\t\n\tA natural question arises: Can we construct more general fractals (e.g. non-self-similar Moran type sets) using an analogous approach while preserving the computational simplicity of the IFS?  In this paper, we present a method to do so.\n\t\n\tWe first make the following observations.  Note that in the above construction,\n\t\\begin{equation}\n\t\\label{Jsigmai}\n\tJ_{\\sigma * i} = S_i (J_\\sigma), \\text{ for all } i=1, ... m, \\text{ and } \\sigma \\in D.\n\t\\end{equation}\n\t\n\tSuppose that there is a tuning parameter in the expression of the function $S_i$ (e.g. the coefficients $a_i, b_i$ in a linear function $S_i(x) = a_i x+ b_i$).  One can tune the values of the parameter to get a comparable function.  When $J_{\\sigma}$ is given, applying the comparable function to $J_{\\sigma}$, as in equation (\\ref{Jsigmai}), will not significantly change the computational complexity of constructing $J_{\\sigma * i}$. The advantage of doing this at each iteration is that we introduce some variance into the limit set.  Another observation is about which space the functions are defined. In classical IFS constructions, the functions are usually defined on all of the ambient space ${\\mathbb{R}}^N$ (as in \\cite{inhom}, the functions are $C^{1+\\alpha}$ diffeomorphisms on ${\\mathbb{R}}^N$).  For our construction, we wish to relax the condition MSC(2) as well.  Instead of restricting our attention to functions of higher regularity defined on the whole ambient space ${\\mathbb{R}}^N$, we use maps from a collection of subsets to itself.  \n\n\t  This article is organized as follows.  In section 2 we find bounds for the Hausdorff dimension of the limit sets in a general metric space setting of a collection of bounded sets, not necessarily satisfying the MSC conditions. Then in section 3 we formulate the general setup for the construction of Moran-type limit sets using the ideas from a modified IFS procedure, as discussed in the previous paragraph.  In our construction we relax MSC(2) so that the limit set is not necessarily self-similar.  More importantly, we drop MSC(4) from the construction process so that there are no limitations on the ratios of the diameters of the sets.  Specifically, the ratio $\\displaystyle \\dfrac{diam(J_{\\sigma * j})}{diam(J_\\sigma)}$ in (\\ref{ckj}) is not limited to depend on just $k$ and $j$, but varies with $\\sigma$.  This change allows us to produce a mosaic of possible fractals.  An important observation is that the computational complexity of generating these fractals is the same as using an analogous, standard IFS.  In section 4 we give estimates of the Hausdorff dimension of the limit sets created from the general construction.  Finally, in section 5 we apply the results to specific examples, including modifications of the Cantor set, the Sierpinski triangle, and the Menger sponge.\n\t  \n\t\\section{Hausdorff Dimension of the Limit Sets}\n\tIn this section we investigate the Hausdorff dimension $\\dim_H(F)$ of the fractals $F$ defined in (\\ref{eqn: Limit_set}), which is not necessarily satisfying MSC conditions.  To start, we determine an upper bound for the dimension of the limit set $F$ by considering the step-wise relative ratios between the diameters of sets.\n\t\n\t\\begin{proposition}\\label{thm: upper}\n\t\tSuppose $\\mathcal J := \\{J_{\\sigma} : \\sigma \\in D\\}$ is a collection of bounded subsets of a metric space $(X,d)$, and $s>0$. \n\t\tLet\t$E_{k} = \\bigcup_{\\sigma \\in D_{k}} J_{\\sigma}, \\text{ and } F = \\bigcap_{k \\geq 0} E_{k}$ be defined as in (\\ref{eqn: Limit_set}).  If there exists a sequence of positive numbers $\\{c_k\\}_{k=1}^{\\infty}$ such that\n\t\t\\[\\liminf_{k\\rightarrow \\infty} \\prod_{i=1}^k c_i =0\\]\n\t\tand\n\t\t\\begin{equation}\n\t\t\t\\label{eqn: c}\n\t\t\t\\sum_{j=1}^{n_k} \\left (\\textsl{diam}(J_{\\sigma*j})\\right)^s \\leq c_k \\left( \\textsl{diam}(J_\\sigma)\\right)^s,\n\t\t\\end{equation}\n\t\tfor all $\\sigma\\in D_{k-1}$ and all $k=1,2,\\cdots$, then $dim_H(F) \\le s$.\n\t\\end{proposition}\n\t\n\t\\begin{proof}\n\t\tWe prove by using mathematical induction that for $k=1,2,\\cdots,$\n\t\t\\begin{equation}\n\t\t\t\\label{general_form}\n\t\t\t\\sum_{\\sigma \\in D_k}(\\textsl{diam}(J_{\\sigma}))^s \\le \\left(\\prod_{i=1}^k c_i \\right) (diam(J_\\emptyset))^s.\n\t\t\\end{equation}\n\t\tWhen $k=1$, (\\ref{general_form}) follows from (\\ref{eqn: c}). Now assume (\\ref{general_form}) is true for some $k\\ge 1$.  Then by (\\ref{dksigma}), (\\ref{eqn: c}), and (\\ref{general_form}),\n\t\t\n\t\t\\begin{eqnarray*}\n\t\t\t\\sum_{\\sigma\\in D_{k+1}} (\\textsl{diam}(J_{\\sigma}))^s &=&\\sum_{\\sigma\\in D_k} \\left(\\sum_{j=1}^{n_{k+1}} (\\textsl{diam}(J_{\\sigma*j}))^s\\right)\\\\\n\t\t\t &\\le& c_{k+1} \\sum_{\\sigma\\in D_k}(\\textsl{diam}(J_{\\sigma}))^s \\le \\left(\\prod_{i=1}^{k+1} c_i \\right)(\\textsl{diam}(J_\\emptyset))^s\n\t\t\\end{eqnarray*}\n\t\t\n\t\tas desired. By the induction principle, (\\ref{general_form}) holds for all $k=1,2,\\cdots.$\n\t\t\n\t\tFor each $k$, set\n\t\t\\[\\delta_k=\\max\\{\\textsl{diam}(J_\\sigma): \\sigma\\in D_k\\}>0.\\]\n\t\tThen, by (\\ref{general_form}),  $\\delta_k\\le  \\left(\\prod_{i=1}^k c_i \\right)^{1\/s}\\textsl{diam}(J_\\emptyset)$.  Moreover, by (\\ref{general_form})\n\t\t\\[\\mathcal{H}_{\\delta_k}^{s}(F) \\le \\mathcal{H}_{\\delta_k}^{s}(E_k)\\le \\sum_{\\sigma\\in D_k} \\alpha(s)\\left(\\frac{\\textsl{diam}(J_\\sigma)}{2}\\right)^s \\le \\left(\\prod_{i=1}^k c_i \\right)\\alpha(s)\\left(\\frac{\\textsl{diam}(J_\\emptyset)}{2}\\right)^s.\\]\n\t\tSince $\\liminf_{k\\rightarrow \\infty} \\prod_{i=1}^k c_i =0$, there exists a sequence $\\{k_t\\}_{t=1}^\\infty$ such that\\\\\n\t\t$\\lim_{t\\rightarrow \\infty} \\prod_{i=1}^{k_t} c_i =0$. Thus, $\\delta_{k_t}\\rightarrow 0$ as $t\\rightarrow \\infty$, and\n\t\t$\\mathcal{H}^{s}(F) =\\lim_{t\\rightarrow \\infty}\\mathcal{H}_{\\delta_{k_t}}^{s}(F)=0 $, and hence $\\dim_H(F) \\le s$.\n\t\\end{proof}\n\t\n\tConversely, a lower bound on the Hausdorff dimension of the limit set $F$ can also be obtained as follows. \n\t\n\t\\begin{proposition}\\label{thm:lower}\n\t\tSuppose $\\mathcal J := \\{J_{\\sigma} : \\sigma \\in D\\}$ is a collection of compact subsets of Euclidean space ${\\mathbb{R}}^N$, and let $F$ be the limit set of $\\mathcal{J}$ as given in (\\ref{eqn: Limit_set}).  If for some $s>0$,\n\t\t\\begin{equation}\n\t\t\t\\label{lower_eqn}\\sum_{j=1}^{n_k} \\textsl{diam}(J_{\\sigma*j})^s\n\t\t\t\\geq  \\textsl{diam}(J_\\sigma)^s\n\t\t\\end{equation}\n\t\tfor all $\\sigma\\in D_{k-1}$ and all $k=1,2,\\cdots$, then $dim_H(F)\\ge s$.\n\t\\end{proposition}\n\t\n\t\\begin{proof}\n\t\t\n\t\tWe first show that under condition (\\ref{lower_eqn}), there exists a probability measure $\\mu$ on ${\\mathbb{R}}^N$ concentrated on $F$ such that for each Borel subset $B$ of ${\\mathbb{R}}^N$, \n\t\t\\begin{equation}\\label{mu_c} \\mu(B)\\le \\left(\\frac{\\textsl{diam}(B)}{\\textsl{diam}(J_\\emptyset)}\\right)^s.\\end{equation}\n\t\t\n\t\tLet $\\mu(J_\\emptyset)=1$, and for each $\\sigma \\in D_k$ for $k>0$ and $i=1, \\cdots, n_k$, we inductively set\n\t\t\\[\\mu(J_{\\sigma *i}) =\\frac{\\textsl{diam}(J_{\\sigma*i})^s}{\\sum_{j=1}^{n_k} \\textsl{diam}(J_{\\sigma *j})^s} \\mu(J_{\\sigma}).\\]\n\t\tThen by Proposition 1.7 in \\cite{Falconer}, $\\mu$ can be uniquely extended to a probability measure on ${\\mathbb{R}}^N$, concentrated on $F$.\n\t\tFor any Borel set $B$, the value\n\t\t\\[ \\mu(B) = \\inf \\left \\{ \\sum_{i=1}^{\\infty} \\mu(J_{\\sigma_i}) : B \\cap F \\subset \\bigcup_{i=1}^{\\infty} J_{\\sigma_i} \\text{ and } J_{\\sigma_i} \\in \\mathcal J \\right \\}. \\]\n\t\tThus, to prove (\\ref{mu_c}) for each Borel set $B$, it is sufficient to prove (\\ref{mu_c}) for $J_{\\sigma}$, $ \\forall \\sigma \\in D$.  We proceed by using induction on $k$ when $\\sigma \\in D_k$. It is clear for $k=0$.  Now assume that (\\ref{mu_c}) holds for each $\\sigma \\in D_k$ for some $k$.  Then by induction assumption and (\\ref{lower_eqn}), for each $i=1,\\cdots, n_{k+1}$,\n\t\t\\begin{align*}\n\t\t\t\\mu(J_{\\sigma*i}) &=\\frac{\\textsl{diam}(J_{\\sigma*i})^s}{\\sum_{j=1}^{n_k} \\textsl{diam}(J_{\\sigma*j})^s} \\mu(J_{\\sigma}) \\\\\n\t\t\t&\\leq \\frac{\\textsl{diam}(J_{\\sigma*i})^s}{\\sum_{j=1}^{n_k} \\textsl{diam}(J_{\\sigma*j})^s} \\left(\\frac{\\textsl{diam}(J_{\\sigma})}{\\textsl{diam}(J_{\\emptyset})}\\right)^s\\\\\n\t\t\t&\\leq  \\left(\\frac{\\textsl{diam}(J_{\\sigma*i})}{\\textsl{diam}(J_\\emptyset)}\\right)^s.\n\t\t\\end{align*}\n\t\tThis proves inequality (\\ref{mu_c}).\n\t\t\n\t\t\n\t\tNow, for any $\\delta>0$, let $\\{B_i\\}$ be any collection of closed balls with $\\textsl{diam}(B_i)\\le \\delta$ and $F\\subseteq \\cup_i B_i$. Then, by (\\ref{mu_c}),\n\t\t\\[\\sum_i  \\alpha(s)\\left(\\frac{\\textsl{diam}(B_i)}{2}\\right)^s \n\t\t\\ge \\alpha(s)\\left(\\frac{\\textsl{diam}(J_\\emptyset)}{2}\\right)^s \\sum_i  \\mu(B_i)\\ge c \\mu \\left(\\bigcup_i B_i \\right)\\ge c\\mu(F) = c,\\]\n\t\twhere $c=\\alpha(s)\\left(\\frac{\\textsl{diam}(J_\\emptyset)}{2}\\right)^s$.\n\t\tThus, $\\mathcal{H}^{s}(F) =\\lim_{\\delta \\rightarrow 0}\\mathcal{H}_{\\delta}^{s}(F)\\ge c>0 $, and hence $\\dim_H(F) \\ge s$.\n\t\\end{proof}\n\t\n\t\\section{General Setup of ${\\mathcal F}$-Limit sets}\n\t\\label{general setup}\n\t\n\tWe now formalize the ideas from section 1 to give a description of the construction of such fractals.  We concentrate on the maps in order to take advantage of the computational nature of an IFS, but allow for the maps to be updated and changed at each iteration.\n\t\n\tIn this section let $\\mathcal{X}$ be a collection of nonempty compact subsets of a metric space.\n\t\n\t\\begin{definition}\n\t  A mapping $f: \\mathcal{X} \\to \\mathcal{X}$ is called a \\textit{compression} on $\\mathcal{X}$ if $f(E) \\subseteq E$ for each $E \\in \\mathcal{X}$. \n\t\\end{definition}\n\nFor each natural number $m$, let $$\\mathcal{C}_m(\\mathcal{X}) = \\{ (f^{(1)}, f^{(2)}, \\dots, f^{(m)}): f_i \\text{ is a compression on } \\mathcal{X}, i=1, \\dots, m\\}.$$\n\n\t\\begin{definition}\n\t\tLet $\\mathcal{M}$ be a nonempty set.  A mapping \t\n\t\t\\begin{eqnarray}\n\t\t&\\mathcal{F}: &\\mathcal{M} \\to \\mathcal{C}_m(\\mathcal{X})  \\\\\n\t\t&&k \\to f_k = (f_k^{(1)}, f_k^{(2)}, \\cdots , f_k^{(m)}). \\label{f_k}\n\t\t\\end{eqnarray} is called a marking of $\\mathcal{C}_m(\\mathcal{X})$ by $\\mathcal{M}$.\n\tEach element $ k \\in \\mathcal{M}$ is called the marker of $f_{k}$.\n\t\t\n\t\\end{definition}\n\nGiven a marking ${\\mathcal F}$ and an initial set $E_0 \\in \\mathcal{X}$, we will construct a generalized Moran set from any sequence of markers in $\\mathcal{M}$.  Note that any sequence $\\{k_{\\ell}\\}_{\\ell = 0}^{\\infty}$ in $\\mathcal{M}$ can be represented as a mapping from the ordered set $D$ to $\\mathcal{M}$.\n\n\n\\begin{definition}\n\t\\label{3.0.3}\n\tLet $\\mathcal{F}$ be a marking of $\\mathcal{C}_m(\\mathcal{X})$ by $\\mathcal{M}$, let $E_0$ be any element in $\\mathcal{X}$, and $D$ be as in (\\ref{Dk}).  Suppose $\\vec{k} : D \\to \\mathcal{M}$ is a map sending $\\sigma$ to $k_{\\sigma}$. \n\tFor each $\\sigma \\in D$ and $1 \\leq j \\leq m$, we recursively define $J_{\\emptyset} = E_0$ and\n\t\\begin{equation}  \n\t\\label{Jsigj}\n\tJ_{\\sigma * j} = f_{\\k_{ \\sigma}}^{(j)} (J_{\\sigma}), \n\t\\end{equation}\n\t\n\twhere $f_{k_{\\sigma}}$ is given by $\\mathcal{F}$ as in (\\ref{f_k}).\n\t\n\tThe limit set $\\displaystyle F = \\bigcap_{k\\geq 1} \\bigcup_{\\sigma \\in D_k} J_{\\sigma}$ associated with $\\mathcal{J}(\\vec{k})=\\{J_\\sigma: \\sigma \\in D\\}$ is called the $\\mathcal{F}$-limit set generated by $\\vec{k}$ with the initial set $ E_0$.  \n\t\n\t\n\\end{definition}\n\nWe now make two observations relating the concepts of an ${\\mathcal F}$-limit set with the attractor of an IFS.\t\n\n\t\tWe first observe that the attractor of an IFS $\\{ S_1, S_2, \\dots, S_m\\}$ on a closed subset $\\Delta$ of ${\\mathbb{R}}^N$ can be viewed as an ${\\mathcal F}$-limit set as follows.\n\t\t\n\t\tLet $\\mathcal{X} = \\{ E : E \\text{ is a non-empty compact subset of } \\Delta,  S_i(E) \\subseteq E, \\text{ for all } i\\}$.  Since each $S_i$ is a contraction on $\\Delta$, the set $E_r := \\Delta \\cap \\overbar{B(0,r)}$ is a non-empty compact subset of $\\Delta$, and $S_i(E_r) \\subseteq E_r$ for each $i$ when $r$ is sufficiently large.  In other words, $E_r \\in \\mathcal{X}$ for sufficiently large $r$.  Also, each contraction map $S_i$ acting on $\\Delta$ naturally determines a map $f^{(i)} :\\mathcal{X} \\to \\mathcal{X}$ given by \n\t\t\\begin{equation}f^{(i)}(E) = S_i(E) := \\left \\{ S_i(x) | x \\in E \\subseteq \\Delta \\right \\} \\label{fie}\\end{equation} \n\t\t\\noindent for each $E \\in \\mathcal{X}$.  Since $f^{(i)}(E) = S_i(E) \\subseteq E$, $f^{(i)}$ is a compression for each $i$.  Set $$ f=(f^{(1)}, f^{(2)}, \\dots , f^{(m)} ).$$  For any non-empty set $\\mathcal{M}$, define the marking ${\\mathcal F}$ of $\\mathcal{C}_{m}(\\mathcal{X})$ to be the constant function ${\\mathcal F}(k)=f$ for all $k \\in \\mathcal{M}$.  \n\t\t Thus, for each $\\sigma \\in D_k$ and $i=1, \\dots ,m$, we have that $J_{\\sigma * i} = S_i(J_{\\sigma})$ from (\\ref{Jsigj}). As a result, for any map $\\vec{k}: D \\to \\mathcal{M}$, the collection $\\mathcal{J}(\\vec{k}) = \\{ J_{\\sigma} : \\sigma \\in D\\}$ is independent of the choice of $\\vec{k}$.  Thus, the associated $\\mathcal{F}$-limit set $\\displaystyle F = \\bigcap_{k\\geq 1} \\bigcup_{\\sigma \\in D_k} J_{\\sigma}$  agrees with the attractor of the given IFS $\\{ S_1, S_2, \\dots, S_m\\}$.\n\n\t\n\tConversely, let $\\mathcal{F}$ be a marking of $\\mathcal{C}_m(\\mathcal{X})$ by $\\mathcal{M}$ where $\\mathcal{X}$ is a collection of non-empty compact subsets of $\\Delta$.  Suppose there is a mapping $\\vec{k} : D \\to \\mathcal{M}$ such that the sequence $\\{ f_{k_{\\sigma}}\\}_{\\sigma \\in D}$ is constant in $\\mathcal{C}_{m}(\\mathcal{X})$ (i.e. there exists an $f \\in \\mathcal{C}_{m}(\\mathcal{X})$ such that $f_{k_{\\sigma}} =f$ for all $\\sigma \\in D$)  and for each $i=1,2,\\dots,m$, there exists a contraction $S_i$ on $\\Delta$ such that equation (\\ref{fie}) holds for each $E \\in \\mathcal{X}$. Then the $\\mathcal{F}$-limit set $F$ generated by $\\vec{k}$ is the attractor of the IFS $\\{S_1, S_2, \\dots, S_m \\}$.\n\t\t\nTherefore, choosing $\\vec{k}: D \\to \\mathcal{M}$ to be a constant map will result in a limit set $F$ that is the attractor of an IFS.\n\tIn the above sense, our approach is a generalization of the standard IFS construction.\n\t\n\t\tAn important observation is that replacing $\\{k_{\\sigma}\\}_{\\sigma \\in D}$ by another sequence  $\\{\\tilde{k}_{\\sigma}\\}_{\\sigma \\in D}$ in (\\ref{Jsigj}) will not change the computational complexity of the construction of $\\mathcal{J}(\\vec{k})$.  Thus, generating the limit set $F$ will have a similar computational complexity as generating the attractor of a comparable IFS.\n\t\n\t In the following section we will compute the Hausdorff dimension of the constructed ${\\mathcal F}$-limit sets.  In section 5 we will provide examples along with their dimensions.\n\t\n\t\\section{Hausdorff dimensions of ${\\mathcal F}$-Limit sets}\n\t\\label{dimensions}\n\t\n\tAs indicated in Propositions \\ref{thm: upper} and \\ref{thm:lower} in section 2, the relative ratio between the diameters of the sets plays an important role in the calculation of the dimension of the limit set.  Therefore, we introduce the following definition.\n\t\n\t\\begin{definition}\n\t\tFor any compression $g: \\mathcal{X}\\to \\mathcal{X}$, define\n\t\t\\begin{equation}\n\t\t\tU(g)=\\sup_{E\\in \\mathcal{X}} \\frac{\\textsl{diam}(g(E))}{\\textsl{diam}(E)}, \\text{ and } L(g)=\\inf_{E\\in \\mathcal{X}} \\frac{\\textsl{diam}(g(E))}{\\textsl{diam}(E)}.\n\t\t\\end{equation}\n\t\\end{definition}\n\tNote that, for each $E\\in \\mathcal{X}$,\n\t\\begin{equation}\n\t\\label{LgUg}\n\tL(g)\\cdot \\textsl{diam}(E)\\le \\textsl{diam}(g(E)) \\le U(g) \\cdot \\textsl{diam}(E).\n\t\\end{equation}\n\t\n\tFor any $\\mathbf{k}\\in \\mathcal{M}$ and $f_{\\mathbf{k}} = (f_{\\k}^{(1)}, f_{\\k}^{(2)}, \\cdots , f_{\\k}^{(m)}) \\in \\mathcal{C}_m(\\mathcal{X})$, define\n\t\\[\\mathbf{U_k}=\\left( U(f_{\\mathbf{k}}^{(1)}), \\cdots,  U(f_{\\mathbf{k}}^{(m)})\\right) \\in \\mathbb{R}^m,\\]\n\tand \n\t\\[ \\mathbf{L_k}=\\left( L(f_{\\mathbf{k}}^{(1)}), \\cdots,  L(f_{\\mathbf{k}}^{(m)})\\right) \\in \\mathbb{R}^m.\\]\n\t\n\tAlso, for each $x=(x_1, \\cdots, x_m)\\in \\mathbb{R}^m$ and $s>0$, denote\n\t\\[||x||_s=\\left(\\sum_{i=1}^m |x_i|^s\\right)^{\\frac{1}{s}}.\\]\n\t\n\tThese notations, Proposition \\ref{thm: upper} and Proposition \\ref{thm:lower} motivate our main theorem.\n\t\\begin{theorem} \\label{thm:ratio_bounds}\n\t\tLet $F$ be the ${\\mathcal F}$-limit set generated by a sequence $\\{ \\k_{\\sigma} \\}_{\\sigma \\in D}$ with initial set $J_{\\emptyset}$, and $s> 0$. \n\t\t\\begin{enumerate}\n\t\t\t\\item[(a)] If \n\t\t\t\\[\\inf_{\\sigma \\in D} \\{||\\mathbf{L}_{\\k_\\sigma}||_{s}\\}\\ge 1,\\]\n\t\t\tthen $\\dim_H(F) \\ge s$.\n\t\t\t\\item[(b)] If\n\t\t\t\\[\\sup_{\\sigma \\in D}\\{||\\mathbf{U}_{\\k_\\sigma}||_{s}\\}<1,\\]\n\t\t\tthen $\\dim_H(F)\\le s$.\n\t\t\\end{enumerate}\n\t\\end{theorem}\n\t\n\t\\begin{proof}\n\t\t(a)  By \\ref{Jsigj} and \\ref{LgUg}, for all $\\sigma \\in D$, \n\t\t\\begin{eqnarray*}\n\t\t\t& &\\sum_{j=1}^{m} \\textsl{diam}(J_{\\sigma * j})^s =\\sum_{j=1}^{m} \\textsl{diam}\\left(f_{\\k_{\\sigma}}^{(j)}(J_{\\sigma})\\right)^s \n\t\t\t\\ge  \\sum_{j=1}^{m}\\left(L(f_{\\k_\\sigma}^{(j)})\\right)^s \\textsl{diam}(J_{\\sigma})^s \\ge \\textsl{diam}(J_{\\sigma})^s.\n\t\t\\end{eqnarray*}\n\t\tThus, by Proposition \\ref{thm:lower}, $\\dim_H(F)\\ge s$.\n\t\t\n\t\t(b) Similarly, for all $\\sigma \\in D$, \n\t\t\\[\\sum_{j=1}^{m} \\textsl{diam}(J_{\\sigma*j})^s \\le \\sum_{j=1}^{m}\\left(U(f_{\\k_\\sigma}^{(j)})\\right)^s \\textsl{diam}(J_{\\sigma})^s \\le c \\cdot \\textsl{diam}(J_{\\sigma})^s,\\]\n\t\n\t\twhere\n\t\t\\[c:=\\sup_\\sigma\\{(||\\mathbf{U}_{\\k_\\sigma}||_{s})^s\\}<1.\\]\n\t\tBy Proposition \\ref{thm: upper}, $\\dim_H(F)\\le s$.\n\t\\end{proof}\n\t\n\t\n\t\n\t\n\t\\begin{remark}\n\t\t\n\t\tFor practical reasons, we find that it is more convenient to represent the mapping $\\vec{k} : D \\to \\mathcal{M}$ by a sequence $\\{k_{\\ell}\\}_{\\ell=0}^{\\infty} \\subseteq \\mathcal{M}$.  For each $\\sigma=(i_1, i_2, \\dots, i_k) \\in D_k$, let \n\t\t\\begin{equation}\\ell(\\sigma) = \\sum_{p=0}^{k-1} m^p i_{k-p}\\end{equation}\n\t\tbe the ordering of $\\sigma$ in the ordered set $D$.  Using this notation, we can rewrite Definition \\ref{3.0.3} as follows. \n\t\t\n\t\t\\noindent \\textbf{Definition \\ref{3.0.3}'.}\n\t\tLet ${\\mathcal F}$ be a marking of $\\mathcal{C}_m(\\mathcal{X})$ by $\\mathcal{M}$, let $\\{\\k_{\\ell}\\}_{\\ell=0}^{\\infty} $ be a sequence in $\\mathcal{M}$, and $E_0 \\in \\mathcal{X}$ be a starting set.  For each $\\ell =0,1, 2,\\cdots$ and $j=1,2,\\cdots, m$, we iteratively denote the set\n\t\t\\[E_{m\\ell+j}=f_{\\k_\\ell}^{(j)}(E_\\ell) \\in \\mathcal{X},\\]\n\t\twhere $f_{\\k_{\\ell}}$ is given by $\\mathcal{F}$ as in (\\ref{f_k}).\n\t\t\n\t\tLet $\\mathcal{G}_m(0)=0$ and for $n \\ge 1$,\n\t\t\\begin{equation} \\label{eqn:p_m}\n\t\t\\mathcal{G}_m(n)=m+m^2+\\cdots+m^n=\\frac{m^{n+1}-m}{m-1}\n\t\t\\end{equation}  \n\t\tdenote the number of sets in the $n^{th}$ generation, i.e. the cardinality of $D_n$.\n\t\t\n\t\tThe limit set \\begin{equation} \n\t\t F=\\bigcap_{n=1}^{\\infty} \\bigcup_{\\ell=\\mathcal{G}_m(n-1)+1}^{\\mathcal{G}_m(n)} E_{\\ell} \\label{F_limit_set} \\end{equation}\n\t\tis called the ${\\mathcal F}$-limit set generated by the triple $({\\mathcal F}, \\{\\k_{\\ell} \\}_{\\ell =0}^{\\infty} , E_0)$. \n\t\t\n\t\tIn the following, we will use the notation from Definition 3.0.3' to describe the construction of the ${\\mathcal F}$-limit sets.  Clearly, using this notation, Theorem \\ref{thm:ratio_bounds} simply says that if\t$\\displaystyle \\inf_{\\ell} \\{||\\mathbf{L}_{\\k_{\\ell}}||_{s}\\}\\ge 1,$ then $\\dim_H(F) \\ge s$, and if $ \\displaystyle \\sup_{\\ell}\\{||\\mathbf{U}_{\\k_\\ell}||_{s}\\}<1,$\tthen $\\dim_H(F)\\le s$.\n\t\n\t\\end{remark}\n\n\n\tWhen both $\\{||\\mathbf{L}_{\\k_\\ell} ||_s\\}_{\\ell=0}^{\\infty}$ and $\\{||\\mathbf{U}_{\\k_\\ell} ||_s\\}_{\\ell =0}^{\\infty}$ are convergent sequences, the following corollary enables us to quickly estimate the dimension of $F$.\n\n\n\n\t\\begin{corollary}\\label{cor:limit_sup_inf}\n\t\tLet $F$ be the limit set generated by the triple $({\\mathcal F}, \\{ \\k_{\\ell} \\}_{\\ell =0}^{\\infty} , E_0)$. Then,\n\t\t\n\t\t\\begin{equation}\\label{eqn: s_star}\n\t\t\ts_* \\le \\dim_H(F)\\le s^*,\n\t\t\\end{equation}\n\t\twhere \n\t\t\\[s_*=\\sup \\left\\{s: \\liminf_{\\ell \\rightarrow \\infty}\\{||\\mathbf{L}_{\\k_\\ell} ||_s\\}>1 \\right\\}\\text{, and } \n\t\ts^*=\\inf \\left \\{s: \\limsup_{\\ell \\rightarrow \\infty}\\{||\\mathbf{U}_{\\k_\\ell}||_s\\}<1 \\right\\} .\\]\n\t\\end{corollary}\n\t\\begin{proof}\n\t\tFor any $0< s< s_*$, by the definition of $s_*$,\n\t\t\\[\\liminf_{\\ell \\rightarrow \\infty}\\{||\\mathbf{L}_{\\k_\\ell} ||_s\\}>1.\\]\n\t\tThus, when $\\ell_*\\in \\mathbb{N}$ is large enough,\n\t\t\\[\\inf_{\\ell\\ge \\ell_*}\\{||\\mathbf{L}_{\\k_\\ell}||_s \\} \\ge 1, \\qquad \\text{i.e. } \\inf_{\\ell\\ge 0}\\{||\\mathbf{L}_{\\k_{\\ell_*+\\ell}}||_s\\} \\ge 1.\\]\n\t\tSince $F\\cap E_{\\ell_*}$ is the set generated by\n\t\tthe triple $({\\mathcal F}, \\{\\k_{\\ell_*+\\ell} \\}_{\\ell =0}^{\\infty} , E_{\\ell_*})$,  by Theorem \\ref{thm:ratio_bounds}, it follows that $\\dim_H(F\\cap E_{\\ell_*}) \\ge s$ for any $\\ell_*$ large enough. This implies that $\\dim_H(F) \\ge s$ for any $s<s_*$ and hence $\\dim_H(F) \\ge s_*$.\n\t\t\n\t\tSimilarly, we also have $\\dim_H(F)\\le s^*$.\n\t\\end{proof}\n\t\n\tIn the following corollaries, we will see that bounds of the dimension of $F$ can also be obtained from corresponding bounds on $\\mathbf{L}_{\\k_\\ell}$ and $\\mathbf{U}_{\\k_\\ell}$. \n\t\n\t\\textbf{Notation.} For any two points $x=(x_1,\\cdots, x_m)$ and $y=(y_1,\\cdots, y_m)$ in $\\mathbb{R}^m$, we say $x \\le y$ if $x_i \\le y_i$ for each $i=1,\\cdots, m$.\n\t\n\t\\begin{corollary} \\label{cor: bound_on_L_U}\n\t\tLet $\\mathbf{t}=(t_1,\\cdots, t_m) $  and $\\mathbf{r}=(r_1,\\cdots, r_m) $ be two points in $(0,1)^m \\subset \\mathbb{R}^m$. Let $s_*$ and $s^*$ be the solutions to $||\\mathbf{t}||_{s_*}=1$, and $||\\mathbf{r}||_{s^*}=1$ respectively,  i.e. \n\t\t\\[t_1^{s_*}+t_2^{s_*}+\\cdots+t_m^{s_*}=1,  \\text{ and } r_1^{s^*}+r_2^{s^*}+\\cdots+r_m^{s^*}=1.\\]\n\t\t\\begin{enumerate} \n\t\t\t\\item[(a)] If $\\mathbf{L}_{\\k_\\ell}\\ge \\mathbf{t}$ for all $\\ell$, then $\\dim_H(F)\\ge s_*$.\n\t\t\t\n\t\t\t\\item[(b)]If $\\mathbf{U}_{\\k_\\ell} \\le \\mathbf{r}$ for all $\\ell$, then\n\t\t\t$\\dim_H(F) \\le s^*$. \n\t\t\t\n\t\t\t\\item[(c)] If $ \\mathbf{L}_{\\k_\\ell}=\\mathbf{r} =\\mathbf{U}_{\\k_\\ell}$ for all $\\ell$, then $\\dim_H(F)=s^*$.\n\t\t\\end{enumerate}\n\t\\end{corollary}\n\t\\begin{proof}\n\t\t(a) Let $0< s<s_*$. Then,\n\t\t\\[\\inf_\\ell\\{||\\mathbf{L}_{\\k_\\ell}||_{s}\\}\\ge ||\\mathbf{t}||_s \\ge ||\\mathbf{t}||_{s_*}= 1.\\]\n\t\tThus, by Theorem \\ref{thm:ratio_bounds}, $\\dim_H(F)\\ge s$ for any $s<s_*$, and hence $\\dim_H(F)\\ge s_*$.\n\t\t\n\t\t(b) Similarly,  let $0<s^*<s$. Then,\n\t\t\\[\\sup_\\ell\\{||\\mathbf{U}_{\\k_\\ell}||_{s}\\}\\le ||\\mathbf{r}||_s < ||\\mathbf{r}||_{s^*}= 1.\\]\n\t\tThus, by Theorem \\ref{thm:ratio_bounds},  $\\dim_H(F)\\le s$ for any $s>s^*$, and hence $\\dim_H(F)\\le s^*$.\n\t\t\n\t\t(c) follows from (a) and (b).\n\t\\end{proof}\n\t\n\tA special case of Corollary \\ref{cor: bound_on_L_U} gives the following explicit formulas for the bounds on the dimension of $F$.\n\t\\begin{corollary}\\label{cor:uniform_bounds}\n\t\tLet $F$ be the limit set generated by the triple $({\\mathcal F}, \\{ \\k_{\\ell} \\}_{\\ell =0}^{\\infty} , E_0)$.  Let\n\t\t\\[\\mathbf{t}=(t, \\cdots, t) \\text{ and } \\mathbf{r}=(r, \\cdots, r),\\]\n\t\tfor some $0<t, r<1$. \n\t\t\\begin{enumerate}\n\t\t\t\\item[(a)] If $\\mathbf{L}_{\\k_\\ell}\\ge \\mathbf{t}$ for all $\\ell$, then $\\dim_H(F)\\ge \\frac{\\log{m}}{-\\log t}$.\n\t\t\t\n\t\t\t\\item[(b)]If $\\mathbf{U}_{\\k_\\ell} \\le \\mathbf{r}$ for all $\\ell$, then\n\t\t\t$\\dim_H(F) \\le \\frac{\\log{m}}{-\\log r}$. \n\t\t\t\n\t\t\t\\item[(c)] If $ \\mathbf{L}_{\\k_\\ell}=\\mathbf{r} =\\mathbf{U}_{\\k_\\ell}$ for all $\\ell$, then $\\dim_H(F)=\\frac{\\log{m}}{-\\log r}$.\n\t\t\\end{enumerate}\n\t\\end{corollary}\n\t\n\tOther types of bounds on $\\mathbf{L}_{\\k_\\ell}$ and $\\mathbf{U}_{\\k_\\ell}$ can also be used to provide bounds on $\\dim_H(F)$, as indicated by the following result.\n\t\\begin{corollary}\\label{cor: mean}\n\t\tLet $F$ be the limit set generated by the triple $({\\mathcal F}, \\{\\k_{\\ell} \\}_{\\ell =0}^{\\infty} , E_0)$. \n\t\t\\begin{enumerate}\n\t\t\t\\item[(a)] If \n\t\t\t\\[w:=\\inf_{\\ell}\\{||\\mathbf{L}_{\\k_\\ell}||_1\\} \\ge 1,\\]\n\t\t\tthen $\\dim_H(F) \\ge \\frac{\\log(m)}{\\log(m)-\\log(w)}$.\n\t\t\t\\item[(b)]\n\t\t\tIf \n\t\t\t\\[u:=\\sup_{\\ell}\\{||\\mathbf{U}_{\\k_\\ell}||_1\\} < 1,\\]\n\t\t\tthen $\\dim_H(F) \\le \\frac{\\log(m)}{\\log(m)-\\log(u)}$.\n\t\t\\end{enumerate}\n\t\\end{corollary}\n\t\\begin{proof}\n\t\t(a). In this case, for $s = \\frac{\\log(m)}{\\log(m)-\\log(w)} \\ge 1$, we have\n\t\t\\[\\frac{\\sum_{j=1}^{m} \\left(L\\left(f_{\\k_\\ell}^{(j)}\\right) \\right)^{s}}{m}\\ge \\left(\\frac{\\sum_{j=1}^{m}L\\left(f_{\\k_\\ell}^{(j)}\\right)}{m}\\right)^s \\ge \\left(\\frac{w}{m}\\right)^s\\]\n\t\tfor each $\\ell$. Thus,\n\t\t\\[\\inf_\\ell\\{||\\mathbf{L}_{\\k_\\ell}||_s\\}\\ge m^{\\frac{1}{s}}\\frac{w}{m} = 1,\\]\n\t\tthen by Theorem \\ref{thm:ratio_bounds}, $\\dim_H(F) \\ge s$.\n\t\t\n\t\t(b).  In this case, for any $1 \\ge s > \\frac{\\log(m)}{\\log(m)-\\log(u)}$, we have\n\t\t\\[\\frac{\\sum_{j=1}^{m}\\left(U\\left(f_{\\k_\\ell}^{(j)}\\right)\\right)^{s}}{m}\\le \\left(\\frac{\\sum_{j=1}^{m}U\\left(f_{\\k_\\ell}^{(j)}\\right)}{m}\\right)^s \\le \\left(\\frac{u}{m}\\right)^s\\]\n\t\tfor each $\\ell$. Thus,\n\t\t\\[\\sup_\\ell\\{||\\mathbf{U}_{\\k_\\ell}||_s\\}\\le m^{\\frac{1}{s}}\\frac{u}{m} <1.\\]\n\t\tBy Theorem \\ref{thm:ratio_bounds}, $\\dim_H(F) \\le s$. Hence, $\\dim_H(F) \\le \\frac{\\log(m)}{\\log(m)-\\log(u)}$.\n\t\\end{proof}\n\t\n\tNote that this corollary generally provides better bounds on $\\dim_H(F)$ than those obtained from directly applying Theorem \\ref{thm:ratio_bounds}.\n\t\n\t\n\t\n\t\\section{Examples of ${\\mathcal F}$-Limit sets}\n\t\n\tIn this section we describe the construction of both classical fractals and generalized Moran sets in the language of Section \\ref{general setup}, and calculate the dimension using the results from Section \\ref{dimensions}.  \n\t\n\t\\begin{subsection}{Cantor-Like Sets}\n\t\n\tWe first consider Cantor-like sets.  Let $$\\mathcal{X} = \\{ [a,b] : a,b \\in {\\mathbb{R}}\\}$$ be the collection of closed intervals, $m=2$, and let $\\mathcal{M} = [0,1]^2 \\subseteq {\\mathbb{R}}$. For each $\\mathbf{k}=(k^{(1)},k^{(2)}) \\in \\mathcal{M}$, we consider the following two maps, \n\t\\begin{eqnarray*}\n\t\tf^{(1)}_\\mathbf{k} : & \\mathcal{X} &\\to \\mathcal{X} \\\\\n\t\t&[a,b]   &\\mapsto  [a, k^{(1)}(b-a)+a]\n\t\\end{eqnarray*}\n\t\\begin{eqnarray*}\n\t\tf^{(2)}_\\mathbf{k} : &\\mathcal{X} &\\to \\mathcal{X} \\\\\n\t\t& [a,b]   &\\mapsto [k^{(2)}(a-b)+b, b].\n\t\\end{eqnarray*}\n\n\n\n\tNote that both $f^{(1)}_\\k$ and $f^{(2)}_{\\k}$ are compression maps for any $\\k \\in \\mathcal{M}$.  Thus, this defines a marking\n\t\t\\begin{eqnarray*}\n\t\t\t{\\mathcal F} : &\\mathcal{M} &\\to \\mathcal{C}_{2}(\\mathcal{X}) \\\\\n\t\t&\\k   &\\mapsto  f_{\\k}= (f^{(1)}_{\\k}, f^{(2)}_{\\k}).\n\t\\end{eqnarray*}\n\t\n\t\n\tHere, for each $\\mathbf{k}=(k^{(1)}, k^{(2)}) \\in \\mathcal{M}$, one can clearly see that \\[\\textsl{diam} \\left(f^{(i)}_\\mathbf{k}([a,b])\\right)=k^{(i)} \\cdot \\textsl{diam}([a,b]).\\]  Thus, $L \\left(f^{(i)}_\\mathbf{k} \\right)=k^{(i)}=U\\left(f^{(i)}_\\mathbf{k}\\right)$, and hence \\begin{equation}\\label{cantor_Lk} \\mathbf{L_k=k=U_k}.\\end{equation}\n\t\n\n  \n  Let $E_0 = [0,1] \\in \\mathcal{X}$ be fixed. For any sequence $\\{\\k_{\\ell}\\}_{\\ell=0}^{\\infty} \\in \\mathcal{M}$, we define the following:\n\t\n\t\\begin{eqnarray*}\n\t\t&E^{(0)} &= E_0\\\\\n\t\t&E^{(1)} &= f^{(1)}_{\\mathbf{k}_0}  \\left( E_0 \\right) \\cup  f^{(2)}_{\\mathbf{k}_0}  \\left( E_0 \\right) =: E_{1} \\cup E_{2}\\\\\n\t\t&E^{(2)} &= f^{(1)}_{\\mathbf{k}_1}  \\left( E_{1} \\right) \\cup  f^{(2)}_{\\mathbf{k}_1}  \\left( E_{1} \\right) \\cup  f^{(1)}_{\\mathbf{k}_2}  \\left( E_{2} \\right) \\cup  f^{(2)}_{\\mathbf{k}_2}  \\left( E_{2} \\right) \\\\\n\t\t& &:=\\qquad E_{3} \\qquad  \\cup \\quad  E_{4} \\quad \\cup \\quad  E_{5} \\quad  \\cup \\quad  E_{6}\\\\\n\t\t&  \\vdots \\\\\n\t\t&E^{(n)}&= \\bigcup_{i=2^{n-1}-1}^{2^n-2} \\left(f_{\\k_i}^{(1)}(E_i) \\cup f_{\\k_i}^{(2)}(E_i) \\right):=\\bigcup_{i=2^{n-1}-1}^{2^n-2} \\left(E_{2i+1} \\cup E_{2i+2} \\right)=\\bigcup_{ \\ell= 2^n-1}^{2(2^n-1)} E_{\\ell}.\n\t\\end{eqnarray*}\n\n\tNote that when $\\mathbf{k}_{\\ell}=(\\frac{1}{3}, \\frac{1}{3})$ for all $\\ell$, $E^{(n)} $ is the $n^{th}$-generation of the Cantor set ${\\mathcal C}$ and $\\displaystyle F =\\lim_{n \\to \\infty} E^{(n)} = \\bigcap_{n} E^{(n)} ={\\mathcal C} $. \n\t\n\t\n\t\t Observe that the process of constructing the sequence  $\\{E^{(n)}\\}_{n=0}^{\\infty}$ is independent of the values of $\\{\\k_{\\ell}\\}_{\\ell=0}^{\\infty}$. To allow for more general outcomes, we can update the linear functions $f^{(1)}_\\k$ and $f^{(2)}_\\k$ simply by changing the value of $\\k$ at each stage of the construction, which does not change the computational complexity of the process.  Using this idea, we now construct some examples of Cantor-like sets by choosing suitable sequences $\\{\\k_{\\ell}\\}_{\\ell=0}^{\\infty}$.\n\t\n\t\\begin{figure}[h]\n\t\t\\centering\n\t\t\\includegraphics[width=4in]{compare3.png} \n\t\t\\caption{Comparison of classical Cantor set (blue) and new Cantor-like set (red) }\n\t\t\\label{comparecant}\n\t\\end{figure}\n\t\n\t\\begin{example}\\label{example: 1}\n\t\tLet $\\mathbf{k}_{\\ell} = \\left(\\frac{\\ell+1}{4\\ell+6}, \\frac{2\\ell +5}{8\\ell+16}\\right)$ for $\\ell \\geq 0$, and let $F$ be the ${\\mathcal F}$-limit set generated by the triple $({\\mathcal F}, \\{\\k_{\\ell}\\}_{\\ell=0}^{\\infty}, E_0)$.  In Figure \\ref{comparecant} we plot the usual Cantor set ${\\mathcal C}$ (in blue) below the set $F$ (in red) to illustrate the comparison. We can see that the set $F$ has the same basic shape as the Cantor set ${\\mathcal C}$, but is no longer strictly self-similar.\n\t\t\n\t\tIn order to compute the Hausdorff dimension of the new Cantor-like set $F$, we apply Corollary \\ref{cor:limit_sup_inf}. Note that by \\ref{cantor_Lk},\n\t\t\\[\\lim_{\\ell \\rightarrow \\infty}||\\mathbf{L}_{\\mathbf{k}_\\ell}||_s=\\lim_{\\ell \\rightarrow \\infty}||\\mathbf{k}_\\ell ||_s=\\frac{2^{\\frac{1}{s}}}{4}.\\]\n\t\tSo,  \n\t\t\\[\n\t\ts_* =\\sup_s\\{\\liminf_{\\ell \\rightarrow \\infty}||\\mathbf{L}_{\\mathbf{k}_\\ell}||_s>1\\} = \\sup_s \\left\\{\\frac{2^{\\frac{1}{s}}}{4}>1\\right\\}=\\frac{1}{2}.\n\t\t\\]\n\t\tSimilarly, we also have $s^*=\\frac{1}{2}$. By (\\ref{eqn: s_star}), $\\dim_H(F)=\\frac{1}{2}$.\n\t\t\n\t\\end{example}\n\t\n\tIn the next example, we will construct a random Cantor-like set as follows.\n\t\n\t\\begin{example} \\label{example: 2}\n\t For each $\\ell\\ge 0$, we take $\\mathbf{k}_{\\ell}=\\left(q_\\ell, \\frac{1}{2}-q_\\ell \\right)$ where $q_\\ell$ is a random number between $\\frac{1}{8}$ and $\\frac{3}{8}$. Let $F$ be the corresponding ${\\mathcal F}$-limit set generated by the triple $({\\mathcal F}, \\{\\k_{\\ell}\\}_{\\ell=0}^{\\infty}, E_0)$.  We plot $F$ in Figure \\ref{cantorrand}.  In this example, the total length of the $n^{th}$ generation $E^{(n)}$ is chosen to be $(\\frac{1}{2})^n$, while the scaling factors of the left subintervals at each stage are randomly chosen.\n\t\t\n\t\t\\begin{figure}[h]\n\t\t\t\\centering\n\t\t\t\\includegraphics[width=3.1in]{example2_new.png} \n\t\t\t\\caption{A randomly generated Cantor-like set}\n\t\t\t\\label{cantorrand}\n\t\t\\end{figure}\n\t\t\n\t\tWe now estimate the dimension of $F$. By (\\ref{cantor_Lk}),\n\t\t\\[\\left(\\frac{1}{8},\\frac{1}{8}\\right)\\le \\mathbf{L}_{\\mathbf{k}_\\ell} =\\mathbf{k}_\\ell=\\mathbf{U}_{\\mathbf{k}_\\ell} \\le \\left(\\frac{3}{8},\\frac{3}{8}\\right).\\]\n\t\tBy Corollary \\ref{cor:uniform_bounds}, \n\t\t\\[\\frac{\\log(2)}{-\\log(1\/8)}\\le \\dim_H(F) \\le \\frac{\\log(2)}{-\\log(3\/8)}.\\]\n\t\tThat is,\n\t\t\\[\\frac{1}{3}\\le \\dim_H(F) \\le \\frac{\\log(2)}{\\log(8\/3)}\\approx 0.7067.\\]\n\t\t\n\t\t\n\t\\end{example}\n\t\n\t\n\t\\begin{example} \\label{example: 3}\n\tIn this example, we create a sequence $\\{\\k_{\\ell}\\}_{\\ell=0}^{\\infty}$ that results in a limit set with a given measure, e.g. 1\/3.   Of course, the classic example of such a limiting set is the fat Cantor set.  For a different approach, let $\\sum_{n=0}^{\\infty} a_n$ be any convergent series of positive terms with limit $L$.  We consider a sequence $\\{\\k_{\\ell}\\}_{\\ell=0}^{\\infty}$ defined in the following way.  \n\t\t\n\t\tLet $n\\geq1$ be the generation of the construction and for each $\\ell$ with $2^{n-1}-1 \\leq \\ell \\leq 2^{n}-2$, define $\\mathbf{k}_\\ell=(b_n, b_n)$ where $$\\displaystyle b_1:= \\dfrac{\\frac{3}{2}L-a_0}{2\\left(\\frac{3}{2}L\\right)} \\quad \\text{ and } \\quad b_n := \\dfrac{\\frac{3}{2}L-\\sum_{i=0}^{n-1} a_i}{2\\left(\\frac{3}{2}L-\\sum_{i=0}^{n-2}a_i\\right)} \\text{ for } n\\ge 2.$$ \n\t\tWith this sequence $\\{\\k_{\\ell}\\}_{\\ell =0}^{\\infty}$, one can find that the length of each interval in the $n^{th}$ generation is $$b_1 b_2 \\cdots b_n=\\dfrac{\\frac{3}{2}L-\\sum_{i=0}^{n-1} a_i}{2^n \\cdot \\frac{3}{2}L}.$$  Thus, the total length of the $n^{th}$ generation is \\[ \\dfrac{\\frac{3}{2}L-\\sum_{i=0}^{n-1} a_i}{\\frac{3}{2}L}=1-\\frac{2}{3L}\\sum_{i=0}^{n-1} a_i \\] which converges to 1\/3 as desired.  \n\t\t\n\t\tAs an example, we take the convergent series $\\displaystyle \\sum_{n=0}^{\\infty} \\dfrac{1}{n!} = e$ and use it to create the ${\\mathcal F}$-limit set $F$ with measure 1\/3.  The first few generations are shown in Figure \\ref{example3cant}.\n\t\t\\begin{figure}[h]\n\t\t\t\\centering\n\t\t\t\\includegraphics[width=3.1in]{example3_1.png} \n\t\t\t\\caption{Fractal of measure $\\frac{1}{3}$ created by using $\\sum_{n=0}^{\\infty} \\frac{1}{n!} = e$  }\n\t\t\t\\label{example3cant}\n\t\t\\end{figure}\n\t\\end{example}\n\t\\end{subsection}\n\t\n\t\\begin{subsection}{Sierpinski Triangle}\n\t\tThe Sierpinski triangle is another well known fractal.\n\t\t\n\t\tFollowing the general setup in Section \\ref{general setup}, we take \\begin{equation} \\mathcal{X} = \\{ (A,B,C) | A,B,C \\in {\\mathbb{R}}^2\\} \\label{xdefinesierp} \\end{equation} representing the collection of all triangles $\\Delta ABC$ in ${\\mathbb{R}}^2$, $m=3$, and $\\mathcal{M} = [0,1]^6 \\subseteq {\\mathbb{R}}^6$.  For each $\\mathbf{k}=\\left(k^{(1)}, k^{(2)}, k^{(3)}, k^{(4)}, k^{(5)}, k^{(6)}\\right) \\in \\mathcal{M}$ and $i=1,2,3$ we can define affine transformations $f_{\\k}^{(i)} : \\mathcal{X} \\to \\mathcal{X}$ as \n\t\t\\begin{eqnarray*}\n\t\t\tf^{(1)}_{\\k} (A,B,C) &= &(A,A+k^{(1)}(B-A), A+k^{(2)}(C-A))\\\\\n\t\t\tf^{(2)}_{\\k} (A,B,C) &= &(B+k^{(4)}(A-B),B, B+k^{(3)}(C-B))\\\\\n\t\t\tf^{(3)}_{\\k} (A,B,C) &= &(C+k^{(5)}(A-C),C+k^{(6)}(B-C), C)\n\t\t\\end{eqnarray*}\n\t\tfor every $(A,B,C) \\in \\mathcal{X}$.\n\t\t\n\n\t\tNote that each $f^{(i)}_\\k$ is a compression map for $i=1,2,3$ and  any $\\k \\in \\mathcal{M}$.  Thus, this defines a marking\n\t\t\\begin{eqnarray*}\n\t\t\t{\\mathcal F} : &\\mathcal{M} &\\to \\mathcal{C}_{3}(\\mathcal{X}) \\\\\n\t\t\t&\\k   &\\mapsto  f_{\\k}= (f^{(1)}_{\\k}, f^{(2)}_{\\k}, f^{(3)}_{\\k}).\n\t\t\\end{eqnarray*}\n\t\n\t \n\t\n\t\n\tOf course, to prevent overlaps we can require that $k^{(1)} + k^{(4)} \\leq 1, k^{(2)} +k^{(5)} \\leq 1, k^{(3)}+k^{(6)} \\leq 1$. When each of the inequalities are strict, the images of $f^{(i)}_{\\k}$ are three disconnected triangles, as illustrated in Figure (\\ref{disconnect}). When all equalities hold, the images are connected, as illustrated in Figure (\\ref{connect}). \n\t\n\t\t\t\\begin{figure}[h]\n\t\t\t\\centering\n\t\t\t\\begin{subfigure}[b]{0.3\\textwidth}\n\t\t\t\t\\includegraphics[width=2in]{sierpinski_mod.png}\n\t\t\t\t\\caption{}\n\t\t\t\t\\label{disconnect}\n\t\t\t\\end{subfigure}\n\t\t\t~\n\t\t\n\t\t\t\\qquad \\qquad\n\t\t\t\\begin{subfigure}[b]{0.3\\textwidth}\n\t\t\t\t\\includegraphics[width=2in]{sierpinski_mod_2.png}\n\t\t\t\t\\caption{}\n\t\t\t\t\\label{connect}\n\t\t\t\\end{subfigure}\n\t\t\t\\caption{First generation of disconnected and connected triangles}\\label{gen1}\n\t\t\\end{figure}\n\t\t\n\tIn the case of the connected sets, the values of $\\mathbf{k}=\\left(k^{(1)}, k^{(2)}, k^{(3)}, k^{(4)}, k^{(5)}, k^{(6)}\\right) $ are determined by $k^{(1)}, k^{(2)}, k^{(3)}$ since $k^{(4)} = 1-k^{(1)}$, $k^{(5)}=1-k^{(2)}$, $k^{(6)}=1-k^{(3)}$.  In this sense, we may also view $\\k=\\left(k^{(1)}, k^{(2)}, k^{(3)}\\right)$ as a vector in $[0,1]^3 \\subseteq {\\mathbb{R}}^3$.  \n\t\n\t\n\t\n\t\tTo create the normal Sierpinski triangle, we choose\n\t\\begin{equation}\n\tE_0 = \\begin{bmatrix}\n\t-1\/2 & 1\/2 & 0\\\\\n\t0&0&\\sqrt{3}\/2\n\t\\end{bmatrix},\n\t\\label{e0sierp}\n\t\\end{equation}\n\tthe equilateral triangle of unit side length, and $\\k_{\\ell} \\in \\mathcal{M}$ to be the constant sequence $\\k_{\\ell} =\\k=(1\/2, 1\/2, 1\/2, 1\/2, 1\/2, 1\/2)$ so that each iteration maps a triangle to three triangles of half the side length with the desired translation.  In this case the ${\\mathcal F}$-limit set generated by $({\\mathcal F}, \\{\\k_{\\ell}\\}_{\\ell=0}^{\\infty}, E_0)$ corresponds to the standard Sierpinski Triangle.  \n\t\n\tTo generate Sierpinski-like fractals, we now adjust the values of the marking parameters $\\{\\k_{\\ell}\\}_{\\ell=0}^{\\infty}$.  For each $\\mathbf{k}=(k^{(1)}, k^{(2)},\\cdots, k^{(6)})\\in \\mathcal{M}$ and $1\\le i \\le 3$, \n\t\\begin{eqnarray*}\n\t\tU\\left(f_{\\mathbf{k}}^{(i)}\\right)=\\sup_{(A,B,C)\\in \\mathcal{X}}\\frac{diam \\left(f_{\\mathbf{k}}^{(i)}(A,B,C)\\right)}{diam\\left((A,B,C)\\right)} =\\max\\left \\{k^{(2i-1)}, k^{(2i)} \\right\\},\n\t\\end{eqnarray*}\n\tand\n\t\\begin{eqnarray*}\n\t\tL\\left(f_{\\mathbf{k}}^{(i)} \\right)=\\inf_{(A,B,C)\\in \\mathcal{X}}\\frac{diam \\left(f_{\\mathbf{k}}^{(i)}(A,B,C)\\right)}{diam\\left((A,B,C)\\right)} =\\min \\left \\{k^{(2i-1)}, k^{(2i)}\\right \\}.\n\t\\end{eqnarray*}\n\t\nWhen $\\k$ is bounded, i.e. if $\\lambda \\le k^{(j)} \\le \\Lambda<1$ for all $j=1,\\cdots ,6$, then\n\\[\\mathbf{U_k} \\le \\mathbf{r}:=(r,\\cdots,r) \\text{ and } \\mathbf{L_k}\\ge \\mathbf{s}:=(s,\\cdots,s),\\]\nwhere $r=\\max\\{1-\\lambda, \\Lambda\\}$ and $s=\\min\\{1-\\lambda, \\Lambda\\}$.\n\t\t\n\t\tFollowing our general process, we construct some random Sierpinski-like sets by introducing randomness into the choice of the sequence $\\{\\k_{\\ell}\\}_{\\ell=0}^{\\infty}$.\n\t\t\\begin{example}\\label{example: 4}\n\t\t\tLet $\\{\\k_{\\ell}\\}_{\\ell=0}^{\\infty} =\\left \\{\\left (k_\\ell^{(1)}, k_\\ell^{(2)}, k_\\ell^{(3)} \\right )\\right \\}_{\\ell=0}^{\\infty}$ be a sequence in $[0,1]^3$ with each $k_{\\ell}^{(i)}$ a random number between given numbers $\\lambda$ and $\\Lambda$ for each $i=1,2,3$.  Let $F$ be the ${\\mathcal F}$-limit set generated by $({\\mathcal F}, \\{\\k_{\\ell}\\}_{\\ell=0}^{\\infty}, E_0)$.  Then the $6^{th}$ generation of the construction results in images like Figure \\ref{rand1}. Here, in Figure \\ref{rand1a}, $\\lambda=\\frac{1}{4}$ and $\\Lambda=\\frac{3}{4}$;  while in Figure \\ref{rand1b}, $\\lambda=0.45$ and $\\Lambda=0.55$. Note that the sets are no longer self-similar.\n\t\t\\end{example}\n\t\t\\begin{figure}[h]\n\t\t\t\\centering\n\t\t\t\\begin{subfigure}[b]{0.42\\textwidth}\n\t\t\t\t\\includegraphics[width=2.5in]{sierp_rand_4.png}\n\t\t\t\t\\caption{Each $ k_{\\ell}^{(i)}$ is random in $[\\frac{1}{4}, \\frac{3}{4}]$.}\n\t\t\t\t\\label{rand1a}\n\t\t\t\\end{subfigure}\n\t\t\t~\n\t\t\t\n\t\t\n\t\t\t  \\qquad \\qquad\n\t\t\t\\begin{subfigure}[b]{0.42\\textwidth}\n\t\t\t\t\\includegraphics[width=2.5in]{2dim_example1.png}\n\t\t\t\t\\caption{Each $ k_{\\ell}^{(i)}$ is random in $[0.45, 0.55]$.}\n\t\t\t\t\\label{rand1b}\n\t\t\t\\end{subfigure}\n\t\t\t\\caption{Generation 6 of Random Sierpinski triangle}\\label{rand1}\n\t\t\\end{figure}\n\n\t\t\n\t\tIn Figure \\ref{rand1b}, we pick $\\lambda=0.45$ and $\\Lambda=0.55$. By Corollary \\ref{cor:uniform_bounds},\n\t\t\\[\\frac{\\log(m)}{-\\log(s)}\\le \\dim_H(F) \\le \\frac{\\log(m)}{-\\log(r)},\\]\n\t\twhere $m=3$, $r=0.55$ and $s=0.45$. That is,\n\t\t\\[1.3758 \\le \\dim_H(F)\\le 1.8377.\\]\n\t\t\n\t\t\n\t\t\\begin{example}\\label{example: 5}\n\t\t\t\n\t\t\tAs in Example \\ref{example: 4}, but replacing $E_0$ with  $\\tilde{E_0} = \\begin{bmatrix} 0 & 1 & 0\\\\0 & 0&1\\end{bmatrix}$, the $7^{th}$ generation of the construction results in an image like Figure \\ref{rand2}, when $\\lambda=\\frac{1}{4}$ and $\\Lambda=\\frac{3}{4}$.\n\t\t\n\t\t\\end{example}\n\t\t\t\t\t\\begin{figure}[h]\n\t\t\t\\centering\n\t\t\n\t\t\t\\includegraphics[width=3in]{sierp_right_3.png} \n\t\t\t\\caption{Generation 7 of a Random Sierpinski triangle}\n\t\t\t\\label{rand2}\n\t\t\\end{figure}\t\n\n\t\t\\begin{example}\\label{example: 5new}  \n\t\t\tFor each $\\ell=0,1, \\cdots$, let $\\mathbf{k}_\\ell=\\left(k_{\\ell}^{(1)}, k_{\\ell}^{(2)}, \\cdots, k_{\\ell}^{(6)}\\right)$ where\n\t\t\t\\begin{eqnarray*}\n\t\t\t\tk_{\\ell}^{(1)} &=&\\frac{1}{2}+\\frac{a_\\ell}{\\sqrt{\\ell+1}},\\quad  k_{\\ell}^{(2)}=1-k_{\\ell}^{(1)},\\\\\n\t\t\t\tk_{\\ell}^{(3)} &=& \\frac{1}{2}+\\frac{b_\\ell}{\\sqrt{\\ell+1}}, \\quad k_{\\ell}^{(4)}=1-k_{\\ell}^{(3)},\\\\\n\t\t\t\tk_{\\ell}^{(5)} &=& \\frac{1}{2}+\\frac{c_\\ell}{\\ell+1}, \\quad k_{\\ell}^{(6)}=1-k_{\\ell}^{(5)}.\n\t\t\t\\end{eqnarray*}\n\t\t\tfor random numbers $a_\\ell, b_\\ell, c_\\ell \\in [-\\frac{1}{3}, \\frac{1}{3}]$.  Let $F$ be the ${\\mathcal F}$-limit set $F$ generated by $({\\mathcal F}, \\{\\k_{\\ell}\\}_{\\ell=0}^{\\infty}, E_0)$.\n\t\t\tThen the seventh generation of the construction of $F$ results in an image like Figure \\ref{rand3}. \n\t\t\t\n\t\t\t\t\t\\begin{figure}[h]\n\t\t\t\\centering\n\t\t\n\t\t\t\\includegraphics[width=3in]{sierp_controlled.png} \n\t\t\t\\caption{Generation 6 of a Sierpinski-type triangle with controlled dimension}\n\t\t\t\\label{rand3}\n\t\t\\end{figure}\n\t\t\n\t\t\tIn this case, we can calculate the exact value of the Hausdorff dimension of $F$.  Indeed, by Corollary \\ref{cor:limit_sup_inf},\n\t\t\t\\[\\lim_{\\ell \\rightarrow \\infty} (||\\mathbf{U}_{\\mathbf{k}_\\ell}||_s)^s=\\frac{3}{2^s}=\\lim_{\\ell \\rightarrow \\infty} (||\\mathbf{L}_{\\mathbf{k}_\\ell}||_s)^s.\\]\n\t\t\tThus, $\\dim_H(F)=\\frac{\\log(3)}{\\log(2)}$.\n\t\t\\end{example}\n\n\t\\end{subsection}\n\t\n\t\\begin{subsection}{Menger Sponge}\n\t\t\n\tLet \\begin{equation} \\mathcal{X} = \\left \\{ (O,A,B,C) | O,A,B,C \\in {\\mathbb{R}}^3 \\right \\} \\label{xdefinemeng} \\end{equation} representing the collection of all rectangular prisms $(OABC)$ in ${\\mathbb{R}}^3$, $m=20,$ and\n\t\t$$\\mathcal{M} = \\left \\{ \\left(k^{(1)}, k^{(2)}, k^{(3)}, k^{(4)}, k^{(5)}, k^{(6)}\\right) \\in [0,1]^6 :  k^{(1)} \\leq k^{(2)},  k^{(3)} \\leq k^{(4)}, k^{(5)} \\leq k^{(6)} \\right \\}.$$\n\t  For each $\\k \\in \\mathcal{M}$ and $i =1, 2, \\dots , 20$, we can define affine transformations $f^{(i)}_{\\k} : \\mathcal{X} \\to \\mathcal{X}$ as follows.  \n\t\t\n\t\tFor any $\\k=(k^{(1)}, k^{(2)}, k^{(3)}, k^{(4)}, k^{(5)}, k^{(6)}) \\in \\mathcal{M}$, define  \n\t\t$$ T = \\begin{bmatrix} 0& k^{(1)}& k^{(2)}& 1 \\end{bmatrix}, \\quad R= \\begin{bmatrix} 0& k^{(3)}& k^{(4)}& 1 \\end{bmatrix}, \\quad S= \\begin{bmatrix} 0& k^{(5)}& k^{(6)}& 1 \\end{bmatrix}.$$\n\t\tLet\n\t\t\\begin{eqnarray*}\n\t\t\tI&=& \\{ (a,b,c) | 1 \\leq a,b,c\\leq3 \\text{ with } a,b,c \\in {\\mathbb{Z}}, \\text{and no two of $a,b,c$ equal to 2} \\}.\n\t\t\\end{eqnarray*}\n\t\tFor each $(a,b,c) \\in I$ and $\\k \\in \\mathcal{M}$, define \n\t\t\n\t\t\\begin{equation*}\n\t\t\tM_{\\k}(a,b,c)=\\begin{bmatrix} \n\t\t\t\t1-( T(a) + R(b) +S(c)) & T(a) & R(b) & S(c)\\\\\n\t\t\t\t1-( T(a+1) + R(b) +S(c)) & T(a+1) & R(b) & S(c)\\\\\n\t\t\t\t1-( T(a) + R(b+1) +S(c)) & T(a) & R(b+1) & S(c)\\\\\n\t\t\t\t1-( T(a) + R(b) +S(c+1)) & T(a) & R(b) & S(c+1)\\\\\n\t\t\t\\end{bmatrix}.\n\t\t\\end{equation*}\n\t\t\n\t\tNote that the set $I$ contains 20 elements, so we can express it as $$I = \\{ (a_i, b_i, c_i) | 1 \\leq i \\leq 20\\}.$$\n\t\t\n\t\tFor each $\\k \\in \\mathcal{M}$ and $1 \\leq i \\leq 20$, we consider the affine transformation $f^{(i)}_{\\k} : \\mathcal{X} \\to \\mathcal{X}$ given by\n\t\t\\begin{equation} \n\t\t\tf^{(i)}_{\\k} (O,A,B,C)  = M_{\\k} (a_i,b_i,c_i)  \\begin{bmatrix} O\\\\A\\\\B\\{\\mathcal C}\\end{bmatrix} \n\t\t\t\\label{fdefinemeng_f} \n\t\t\\end{equation}\n\t\t\n\t\tfor every $(O, A, B, C) \\in \\mathcal{X}$.  \n\t\t\n\t\tNote that for $i=1, \\dots, 20$ and $\\k \\in \\mathcal{M}$, $f^{(i)}_{\\k}$ is a compression.   Thus, we can define a marking ${\\mathcal F}$  by\n\t\t\t\\begin{eqnarray*}\n\t\t\t{\\mathcal F} : &\\mathcal{M} &\\to \\mathcal{C}_{20}(\\mathcal{X}) \\\\\n\t\t\t&\\k   &\\mapsto  f_{\\k}= (f^{(1)}_{\\k}, \\dots , f^{(20)}_{\\k}).\n\t\t\\end{eqnarray*}\n\t\n\t\tUsing this, for any starting rectangular prism $E_0 = (O,A,B,C) \\in \\mathcal{X}$, we can generate a sequence of sets that follows a similar construction to the Menger Sponge.\n\t\t\n\t\t\\begin{example}\\label{example: 7}\n\t\t\t\\label{standardmeng}\n\t\t  Let\n\t\t\t\n\t\t\t\\begin{equation}\n\t\t\t\tE_0 = \\begin{bmatrix}\n\t\t\t\t\t0 & 1 & 0 & 0\\\\\n\t\t\t\t\t0&0&1&0\\\\\n\t\t\t\t\t0&0&0&1\n\t\t\t\t\\end{bmatrix}\n\t\t\t\t\\label{e0meng}\n\t\t\t\\end{equation}\n\t\t\tbe the cube of unit side length and choose $\\k_{\\ell} \\in \\mathcal{M}$ to be the constant sequence $\\k_{\\ell} =\\k=(1\/3, 2\/3, 1\/3, 2\/3, 1\/3, 2\/3)$.  Then the ${\\mathcal F}$-limit set $F$ generated by the triple $({\\mathcal F}, \\{\\k_{\\ell}\\}_{\\ell =0}^{\\infty}, E_0 )$ is the classical Menger sponge.  \\end{example}\n\t\t\n\t\tNow we consider variations of Menger Sponge. For each $\\mathbf{k}=(k^{(1)}, k^{(2)}, \\cdots, k^{(6)})\\in \\mathcal{M}$ and $1\\le i \\le 20$, \n\t\t\\begin{eqnarray*}\n\t\t\tU\\left(f_{\\mathbf{k}}^{(i)}\\right)&=&\\sup_{(O,A,B,C)\\in \\mathcal{X}}\\frac{diam \\left(f_{\\mathbf{k}}^{(i)}(O,A,B,C)\\right)}{diam\\left((O,A,B,C)\\right)} \\\\\n\t\t\t&=&\\sup_{(O,A,B,C)\\in \\mathcal{X}}\\frac{diam \\left(M_{\\mathbf{k}}(a_i, b_i, c_i)[O,A,B,C]'\\right)}{diam\\left((O,A,B,C)\\right)} \\\\\n\t\t\t&=&\\max\\{T(a_{i+1})-T(a_i), R(b_{i+1})-R(b_i), S(c_{i+1})-S(c_i)\\}.\n\t\t\\end{eqnarray*}\n\t\tSimilarly,\n\t\t\\begin{eqnarray*}\n\t\t\tL\\left(f_{\\mathbf{k}}^{(i)}\\right)=\\min\\{T(a_{i+1})-T(a_i), R(b_{i+1})-R(b_i), S(c_{i+1})-S(c_i)\\}.\n\t\t\\end{eqnarray*}\n\t\tWhen $k^{(2j)}=1-k^{(2j-1)}$ for each $j=1, 2, 3$, it is easy to check that\n\t\t\\begin{eqnarray*}\n\t\t\t\\sum_{i=1}^{20}U(f_{\\mathbf{k}}^{(i)})^s&=&\\sum_{i=1}^{20}\\max\\{T(a_{i+1})-T(a_i), R(b_{i+1})-R(b_i), S(c_{i+1})-S(c_i)\\}^s\\\\\n\t\t\t&=&8\\max\\{k^{(1)}, k^{(3)}, k^{(5)}\\}^s+4\\max\\{1-2k^{(1)}, k^{(3)}, k^{(5)}\\}^s  \\\\\n\t\t\t& & +4\\max\\{k^{(1)}, 1-2k^{(3)}, k^{(5)}\\}^s +4\\max\\{k^{(1)}, k^{(3)}, 1-2k^{(5)}\\}^s.\n\t\t\\end{eqnarray*}\n\t\t\n\t\t\n\t\t\\begin{example}\\label{example: 8}\n\t\t Let $$\\tilde{E_0} = \\begin{bmatrix} 0& 3 & 0 & 0\\\\ 0 &0&1&0\\\\0&0&0&2\\end{bmatrix}.$$ Let $\\left(k^{(1)}, k^{(2)}, k^{(3)}, k^{(4)}, k^{(5)}, k^{(6)}\\right) \\in \\mathcal{M}$ where each $k^{(i)}$ is a random number in $[0,1]$, but still satisfying the condition $k^{(1)} \\leq k^{(2)}, k^{(3)} \\leq k^{(4)}, k^{(5)} \\leq k^{(6)}$.  Then the first generation $E^{(1)}$ of the construction results in a set like Figure \\ref{meng1}.  \n\t\t\t\\begin{figure}[h]\n\t\t\t\t\\centering\n\t\t\t\t\\includegraphics[width=2.5in]{gen_1_6.png}\n\t\t\t\t\\caption{First generation of a randomly generated Menger sponge}\n\t\t\t\t\\label{meng1}\n\t\t\t\\end{figure}\n\t\t\\end{example}\n\t\t\n\t\t\\begin{example}\\label{example: 9}\n\t\t\t  Let $\\k_{\\ell} =\\left(k_\\ell^{(1)}, k_\\ell^{(2)}, k_\\ell^{(3)}, k_\\ell^{(4)}, k_\\ell^{(5)}, k_\\ell^{(6)}\\right) \\in \\mathcal{M}$ with each $k_\\ell^{(2j-1)}$ a random number between given parameters $\\lambda$ and $\\Lambda$ and $k_\\ell^{(2j)}=1-k_\\ell^{(2j-1)}$ for each $j=1,2,3$.  Let $F$ be the ${\\mathcal F}$-limit set generated by $({\\mathcal F}, \\{\\k_{\\ell}\\}_{\\ell =0}^{\\infty}, E_0)$. Then the third iteration of the construction of $F$ results in images like Figure \\ref{meng3}. Here, in Figure \\ref{meng3a} the parameters $\\lambda=0$ and $\\Lambda=\\frac{1}{2}$, while in Figure \\ref{meng3b} the parameters $\\lambda=0.32$ and $\\Lambda=0.35$.\n\t\t\t\n\t\t\t\n\t\t\t\\begin{figure}[h]\n\t\t\t\t\\centering\n\t\t\t\t\\begin{subfigure}[b]{0.45\\textwidth}\n\t\t\t\t\t\\includegraphics[width=3.6in]{random_3rd_generation.png}\n\t\t\t\t\t\\caption{$\\lambda=0, \\quad \\Lambda= \\frac{1}{2}$}\n\t\t\t\t\t\\label{meng3a}\n\t\t\t\t\\end{subfigure}\n\t\t\t\n\t\t\t\t~\n\t\t\t\n\t\t\t\t  \\qquad \\qquad\n\t\t\t\t\\begin{subfigure}[b]{0.45\\textwidth}\n\t\t\t\t\t\\includegraphics[width=2.5in]{3dim_example4.png}\n\t\t\t\t\t\\caption{$\\lambda=0.32, \\Lambda= 0.35$}\n\t\t\t\t\t\\label{meng3b}\n\t\t\t\t\\end{subfigure}\n\t\t\t\t\\caption{Generation 3 of random Menger sponge}\\label{meng3}\n\t\t\t\\end{figure}\n\t\t\\end{example}\n\t\t\n\t\tWe now calculate the dimension of the limit fractal $F$ illustrated by Figure \\ref{meng3b} in Example \\ref{example: 9}.\n\t\tNote that in general, when $\\lambda \\le k^{(2j-1)} \\le \\Lambda$ for each $j=1, 2, 3$, it follows that\n\t\t\\[\n\t\t(||\\mathbf{U_k}||_s)^s=\\sum_{i=1}^{20}U\\left(f_{\\mathbf{k}}^{(i)}\\right)^s \\le 8\\Lambda^s +12\\max\\{1-2\\lambda, \\Lambda\\}^s.\n\t\t\\]\n\t\tSimilarly,\n\t\t\\begin{eqnarray*}\n\t\t\t(||\\mathbf{L_k}||_s)^s \\ge 8\\lambda^s +12\\min\\{1-2\\Lambda, \\lambda\\}^s. \n\t\t\\end{eqnarray*}\n\t\tIn particular, when $\\lambda=0.32$ and $\\Lambda=0.35$,  for any $s>2.901$,\n\t\t\\begin{eqnarray*}\n\t\t\t(||\\mathbf{U_k}||_s)^s &\\le & 8\\Lambda^s+12\\max\\{1-2\\lambda, \\Lambda\\}^s \\le 8*0.35^s+12*0.36^s \\\\\n\t\t\t&<& 8*0.35^{2.901}+12*0.36^{2.901}\\approx 1.000.\n\t\t\\end{eqnarray*}\n\t\tBy Theorem \\ref{thm:ratio_bounds}, $\\dim_H(F)\\le 2.901$. Similarly, for any $s \\le 2.546$,\n\t\t\\begin{eqnarray*}\n\t\t\t(||\\mathbf{L_k}||_s)^s\n\t\t\t&\\ge & 8\\lambda^s+12\\min\\{1-2\\Lambda, \\lambda\\}^s \\\\\n\t\t\t&\\ge & 8*0.32^s+12*0.3^s \\ge 8*0.32^{2.546}+12*0.3^{2.546}\\approx 1.000.\n\t\t\\end{eqnarray*}\n\t\tBy Theorem \\ref{thm:ratio_bounds} again, $\\dim_H(F)\\ge 2.546$. As a result,\n\t\t\\[2.546 \\le \\dim_H(F)\\le 2.901.\\]\n\t\t\n\t\t\n\t\t\\begin{example}\n\t\t\tFor each $\\ell \\ge 0$, let $\\mathbf{k}_\\ell=\\left(k_{\\ell}^{(1)}, k_{\\ell}^{(2)}, \\cdots, k_{\\ell}^{(6)}\\right)$ where\n\t\t\t\\begin{eqnarray*}\n\t\t\t\tk_{\\ell}^{(1)} &=&\\frac{1}{3}+\\frac{(-1)^\\ell}{12(\\ell+1)},\\quad  k_{\\ell}^{(2)}=1-k_{\\ell}^{(1)},\\\\\n\t\t\t\tk_{\\ell}^{(3)} &=& \\frac{1}{3}-\\frac{(-1)^\\ell}{6(\\ell+1)}, \\quad k_{\\ell}^{(4)}=1-k_{\\ell}^{(3)},\\\\\n\t\t\t\tk_{\\ell}^{(5)} &=& \\frac{1}{3}+\\frac{(-1)^\\ell}{18(\\ell+1)}, \\quad k_{\\ell}^{(6)}=1-k_{\\ell}^{(5)}.\n\t\t\t\\end{eqnarray*}\n\t\tLet $F$ be the ${\\mathcal F}$-limit set generated by $({\\mathcal F}, \\{\\k_{\\ell}\\}_{\\ell =0}^{\\infty}, E_0)$.  Then the third generation of the construction of $F$ leads to an image like Figure \\ref{meng4}.\n\t\t\t\n\t\t\t\\begin{figure}[h] \n\t\t\t\t\\centering\n\t\t\t\t\\includegraphics[width=3.5in]{3dim_example3.png} \n\t\t\t\t\\caption{Generation 3 of random Menger sponge with controlled dimension}\n\t\t\t\t\\label{meng4}\n\t\t\t\\end{figure}\n\t\t\\end{example}\n\t\t\n\t\tIn this case, we can still calculate the exact Hasudorff dimension of $F$. By direct computation, \n\t\t\\[\\lim_{\\ell \\rightarrow \\infty} (||\\mathbf{U}_{\\mathbf{k}_\\ell}||_s)^s=\\frac{20}{3^s}=\\lim_{\\ell \\rightarrow \\infty} (||\\mathbf{L}_{\\mathbf{k}_\\ell}||_s)^s.\\]\n\t\tThus, by Corollary \\ref{cor:limit_sup_inf}, $\\dim_H(F)=\\frac{\\log(20)}{\\log(3)} \\approx 2.7268$.\n\t\\end{subsection}\n\t\n\t","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nReaction-diffusion equations of the form\n\\begin{equation}\\label{eqn-classical}\nu_{t}=u_{xx}+f(t,x,u),\\quad (t,x)\\in\\mathbb{R}\\times\\mathbb{R}\n\\end{equation}\nare widely used to model diffusive systems in applied sciences.\nThe nonlinearity $f$ arising from many diffusive systems in biology or physics possesses two zeros representing two states, say $0$ and $1$, that is, $f(t,x,0)=f(t,x,1)=0$ for all $(t,x)\\in\\mathbb{R}\\times\\mathbb{R}$.\nSince the pioneering works of Fish (see \\cite{Fish37}) and Kolmogorov, Petrowsky and Piscunov (see \\cite{KPP37}) on the traveling waves of \\eqref{eqn-classical} connecting the two constant states, i.e., $u\\equiv 0$ and $u\\equiv 1$, in the case $f(t,x,u)=u(1-u)$, a vast amount of literature has been carried out to the understanding of front-like solutions connecting $u\\equiv 1$ and $u\\equiv 0$ in such an equation and its generalized forms. We refer to \\cite{ArWe75,ArWe78,FiMc77,FiMc80,Kam76,Uch78,Xi00} and references therein for works in homogeneous media, i.e., $f(t,x,u)=f(u)$. Recently, there is a lot of progress concerning \\eqref{eqn-classical} in heterogeneous media. We refer to \\cite{BeHa02,DiHaZh14,MeRoSi10,MNRR09,Na14,NoRy09,NRRZ12,TaZhZl14,We02,Zl12} and references therein for works in space heterogeneous media, i.e., $f(t,x,u)=f(x,u)$, and to \\cite{AlBaCh99,NaRo12,Sh99-1,Sh99-2,Sh06,Sh11,ShSh14,ShSh14-1} and references therein for works in time heterogeneous media, i.e., $f(t,x,u)=f(t,x)$. There are also some works in space-time heterogeneous media (see e.g. \\cite{KoSh14,LiZh1,LiZh2,Na09,Na10, Sh04,Sh11-1,We02}), but it remains widely open.\n\n\nWhen using equation \\eqref{eqn-classical} to model a diffusive system in applied sciences, it is implicitly assumed that the internal\ninteraction range of organisms in the system is infinitesimal  and that the internal dispersal can be described by random walk.\n However, in practice,  a diffusive system may exhibit long range internal interaction. Equation \\eqref{eqn-classical} is then no long suitable to model such a system. More precisely, the random dispersal operator $\\partial_{xx}$ is no long suitable. As a substitute, the nonlocal dispersal operator is introduced (see e.g. \\cite{Fi03,GHHMV05} for some background) and we are now concerned with the following integral equation\n\\begin{equation}\\label{main-eqn}\nu_{t}=J\\ast u-u+f(t,x,u),\\quad (t,x)\\in\\mathbb{R}\\times\\mathbb{R},\n\\end{equation}\nwhere $J$ is a convolution kernel and $[J\\ast u](x)=\\int_{\\mathbb{R}}J(x-y)u(y)dy=\\int_{\\mathbb{R}}J(y)u(x-y)dy$.\nThere is also a great amount of research toward  the understanding of front-like solutions of \\eqref{main-eqn} connecting $u\\equiv 0$ and $u\\equiv 1$. See \\cite{BaCh06, BaCh02, BaFiReWa97,CaCh04,Ch97,Cov-thesis,CoDu05,CoDu07,Sc80} and references therein for the study in the  homogeneous case $f(t,x,u)\\equiv f(u)$.  See \\cite{BaCh99, FCh} for the study in the case that $f(t,x,u)\\equiv f(t,u)$ is periodic or almost periodic in $t$. In \\cite{CDM13,ShZh10,ShZh12-1,ShZh12-2}, the authors investigated \\eqref{main-eqn} in space periodic monostable media, i.e., $f(t,x,u)=f(x,u)$ is of monostable type and periodic in $x$, and proved the existence of spreading speeds and periodic traveling waves. In \\cite{RaShZh}, the authors studied the existence of spreading speeds and traveling waves of \\eqref{main-eqn} in space-time periodic monostable media. Very recently, both Berestycki, Coville and Vo (see \\cite{BCV14}), and Lim and Zlato\\v{s} (see \\cite{LiZl14}) investigated \\eqref{main-eqn} in space heterogeneous monostable media. While Berestycki, Coville and Vo studied principal eigenvalue, positive solution and long-time behavior of solutions, Lim and Zlato\\v{s} proved the existence of transition fronts in the sense of Berestycki-Hamel (see \\cite{BeHa07,BeHa12}). In \\cite{ShSh14-2,ShSh14-3}, the authors studied \\eqref{main-eqn} in the time heterogeneous media of ignition type, and prove the existence, regularity and stability of transition fronts.\n\n\nHowever, comparing to the classical random dispersal case, i.e., \\eqref{eqn-classical}, results concerning front propagation for \\eqref{main-eqn} are still very limited. One of the difficulties arising in the study of front propagation dynamics of \\eqref{main-eqn} is that solutions of \\eqref{main-eqn} do not become smoother as time $t$ elapses due to the fact that the semigroup generated by the nonlocal dispersal operator $u\\mapsto J\\ast u-u$ has no regularizing effect. The objective of the present paper is to investigate the space regularity of transition fronts of \\eqref{main-eqn} with various nonlinearities, including the monostable nonlinearity, the ignition nonlinearity, and the bistable nonlinearity. The results to be developed in this paper have important applications in the study of stability, uniqueness, asymptotic, etc. of transition fronts of \\eqref{main-eqn}.\n\nTo state the main results of this paper, we first introduce two standard hypotheses.\n\n\n\\begin{itemize}\n\\item[\\bf{(H1)}] $J:\\mathbb{R}\\to\\mathbb{R}$ satisfies $J\\not\\equiv0$, $J\\in C^{1}$, $J(x)=J(-x)\\geq0$ for $x\\in\\mathbb{R}$, $\\int_{\\mathbb{R}}J(x)dx=1$ and\n\\begin{equation*}\n\\int_{\\mathbb{R}}J(x)e^{\\gamma x}dx<\\infty,\\quad \\int_{\\mathbb{R}}|J'(x)|e^{\\gamma x}dx<\\infty,\\quad\\forall\\gamma\\in\\mathbb{R}.\n\\end{equation*}\n\\end{itemize}\n\n\\begin{itemize}\n\\item[\\bf{(H2)}] There exist $C^{2}$ functions $f_{B}:[0,1]\\to\\mathbb{R}$ and  $f_{M}:[0,1]\\to\\mathbb{R}$ such that\n\\begin{equation*}\nf_{B}(u)\\leq f(t,x,u)\\leq f_{M}(u),\\quad (t,x,u)\\in\\mathbb{R}\\times\\mathbb{R}\\times[0,1];\n\\end{equation*}\nmoreover, the following conditions hold:\n\\begin{itemize}\n\\item $f:\\mathbb{R}\\times\\mathbb{R}\\times[0,1]\\to\\mathbb{R}$ is continuous and continuously differentiable in $x$ and $u$, and satisfies\n\\begin{equation*}\n\\sup_{(t,x,u)\\in\\mathbb{R}\\times\\mathbb{R}\\times[0,1]}|f_{x}(t,x,u)|<\\infty\\quad\\text{and}\\quad\\sup_{(t,x,u)\\in\\mathbb{R}\\times\\mathbb{R}\\times[0,1]}|f_{u}(t,x,u)|<\\infty;\n\\end{equation*}\n\n\\item there exist $\\theta_{1}\\in(0,1)$ such that $f_{u}(t,x,u)\\leq0$ for all $(t,x,u)\\in\\mathbb{R}\\times\\mathbb{R}\\times[\\theta_{1},1]$;\n\n\\item $f_{B}$ is of standard bistable type, that is, $f_{B}(0)=f_{B}(\\theta)=f_{B}(1)=0$ for some $\\theta\\in(0,1)$, $f_{B}(u)<0$ for $u\\in(0,\\theta)$, $f_{B}(u)>0$ for $u\\in(\\theta,1)$ and $\\int_{0}^{1}f_{B}(u)du>0$; moreover, $u_{t}=J\\ast u-u+f_{B}(u)$ admits a traveling wave $\\phi_{B}(x-c_{B}t)$ with $\\phi_{B}(-\\infty)=1$, $\\phi_{B}(\\infty)=0$ and $c_{B}\\neq0$;\n\n\\item  $f_{M}$ is of standard monostable type, that is, $f_{M}(0)=f_{M}(1)=0$ and $f_{M}(u)>0$ for $u\\in(0,1)$.\n\\end{itemize}\n\\end{itemize}\n\n\nWe remark that $c_{B}$ must be positive. Observe that nonlinear functions satisfying (H2) include the monostable nonlinearity, the ignition nonlinearity, and the bistable nonlinearity satisfying the conditions below.\n\n\\begin{itemize}\n\\item[\\rm(i)] Monostable or Fisher-KPP nonlinearity.\n\n\\item[] Standard monostable nonlinearity $f(\\cdot)$: $f(0)=f(1)=0$, $f(u)>0$ for $u\\in(0,1)$, and $\\frac{f(u)}{u}$ is decreasing in $u$.\n\n\\item[] General monostable nonlinearity $f(\\cdot,\\cdot,\\cdot)$: $f_{\\min}(u)\\le f(t,x,u)\\le f_{\\max}(u)$ for $(t,x)\\in\\mathbb{R}\\times\\mathbb{R}$ and $u\\in[0,1]$, where $f_{\\min}(\\cdot)$ and $f_{\\max}(u)$ are two standard monostable nonlinearities, and $\\frac{f(t,x,u)}{u}$ decreasing in $u$ for any $(t,x)\\in\\mathbb{R}\\times\\mathbb{R}$.\n\n\\item[{\\rm (ii)}] Ignition nonlinearity.\n\n\\item[] Standard ignition nonlinearity $f(\\cdot)$: $f(u)=0$ for $u\\in [0,\\theta]\\cup\\{1\\}$,\n$f(u)>0$ for $u\\in(\\theta,1)$, and $f_u(1)<0$, where $\\theta\\in(0,1)$ is referred to as the {\\it ignition temperature}.\n\n\\item[] General ignition nonlinearity $f(\\cdot,\\cdot,\\cdot)$: $f_{\\min}(u)\\le f(t,x,u)\\le f_{\\max}(u)$ for $(t,x)\\in\\mathbb{R}\\times\\mathbb{R}$ and $u\\in[0,1]$,\nwhere $f_{\\min}(\\cdot)$ and $f_{\\max}(\\cdot)$ are two standard ignition nonlinearities.\n\n\\item[{\\rm (iii)}] Bistable nonlinearity.\n\n\\item[] Standard bistable nonlinearity $f(\\cdot)$: $f(0)=f(\\theta)=f(1)=0$, $f(u)<0$ for $u\\in(0,\\theta)$, $f(u)>0$ for $u\\in(\\theta,1)$, and $f_u(0)<0$, $f_u(1)<0$, $f_u(\\theta)>0$,\nwhere $\\theta\\in(0,1)$.\n\n\\item[] General bistable nonlinearity $f(\\cdot,\\cdot,\\cdot)$: $f_{\\min}(u)\\le f(t,x,u)\\le f_{\\max}(u)$ for $(t,x)\\in\\mathbb{R}\\times\\mathbb{R}$ and $u\\in [0,1]$, where $f_{\\min}(\\cdot)$ and $f_{\\max}(\\cdot)$ are two standard bistable nonlinearities with $\\int_{0}^{1}f_{\\min}(u)du>0$ and $u_{t}=J\\ast u-u+f_{\\min}(u)$ having traveling waves with nonzero speed.\n \\end{itemize}\n\n\nWe remark that (H2) can also be applied to a general bistable nonlinearity $f(t,x,u)$ with $\\int_0^1 f_{\\max}(u)du<0$ and $u_{t}=J\\ast u-u+f_{\\max}(u)$ having traveling waves with nonzero speed. In fact, let $v(t,x)=1-u(t,x)$. Then, $v(t,x)$ satisfies\n$$\nv_t=J\\ast v-v+\\tilde f(t,x,v),\\quad (t,x)\\in\\mathbb{R}\\times\\mathbb{R},\n$$\nwhere $\\tilde f(t,x,v)=-f(t,x,1-v)$. Hence\n$$\n\\tilde f_{\\min}(v)\\le \\tilde f(t,x,v)\\le \\tilde f_{\\max}(v),\\quad (t,x,v)\\in\\mathbb{R}\\times\\mathbb{R}\\times[0,1]\n$$\nwhere $\\tilde f_{\\min}(v)=-f_{\\max}(1-v)$ and $\\tilde f_{\\max}(v)=-f_{\\min}(1-v)$.\nClearly, $\\tilde f_{\\min}(\\cdot)$ and $\\tilde f_{\\max}(\\cdot)$ are two standard bistable nonlinearities and\n$\\int_0^1 \\tilde f_{\\min}(v)dv>0$ and $u_{t}=J\\ast u-u+\\tilde{f}_{\\min}(u)$ admits traveling waves with nonzero speed.\n\nWe also remark that, besides monostable, ignition, and bistable nonlinearities, nonlinear functions satisfying (H2) include those having more than one zeros between $0$ and $1$.\n\n\nNext, we recall the definition of transition fronts of \\eqref{main-eqn} connecting $u\\equiv 0$ and $u\\equiv 1$.\n\n\\begin{defn}\\label{transition-front-defn}\nSuppose that $f(t,x,0)=f(t,x,1)=0$ for all $(t,x)\\in\\mathbb{R}\\times\\mathbb{R}$. A global-in-time solution $u(t,x)$ of \\eqref{main-eqn} is called a (right-moving) {\\rm transition front} (connecting $0$ and $1$) in the sense of Berestycki-Hamel (see \\cite{BeHa07,BeHa12}, also see \\cite{Sh99-1,Sh99-2}) if $u(t,x)\\in(0,1)$ for all $(t,x)\\in\\mathbb{R}\\times\\mathbb{R}$ and there exists a function $X:\\mathbb{R}\\to\\mathbb{R}$, called {\\rm interface location function}, such that\n\\begin{equation*}\n\\lim_{x\\to-\\infty}u(t,x+X(t))=1\\,\\,\\text{and}\\,\\,\\lim_{x\\to\\infty}u(t,x+X(t))=0\\,\\,\\text{uniformly in}\\,\\,t\\in\\mathbb{R}.\n\\end{equation*}\n\\end{defn}\n\nThe notion of a transition front is a proper generalization of a traveling wave in homogeneous media and a periodic (or pulsating) traveling wave in periodic media. The interface location function $X(t)$ tells the position of the transition front $u(t,x)$ as time $t$ elapses, while the uniform-in-$t$ limits (the essential property in the definition) shows the \\textit{bounded interface width}, that is,\n\\begin{equation*}\n\\forall\\,\\,0<\\epsilon_{1}\\leq\\epsilon_{2}<1,\\quad\\sup_{t\\in\\mathbb{R}}{\\rm diam}\\{x\\in\\mathbb{R}|\\epsilon_{1}\\leq u(t,x)\\leq\\epsilon_{2}\\}<\\infty.\n\\end{equation*}\nNotice, if $\\xi(t)$ is a bounded function, then $X(t)+\\xi(t)$ is also an interface location function. Thus, interface location function is not unique. But, it is easy to check that if $Y(t)$ is another interface location function, then $X(t)-Y(t)$ is a bounded function. Hence, interface location functions are unique up to addition by bounded functions.\n\nWe see that neither the definition nor the equation \\eqref{main-eqn} guarantees any space regularity of transition fronts. In fact, there is even no guarantee that a transition front $u(t,x)$ of \\eqref{main-eqn} is continuous in $x$ (we refer to \\cite{BaFiReWa97} for the existence of discontinuous traveling waves in the bistable case). This lack of space regularity indeed causes a lot of troubles in studying transition fronts because $\\rm(i)$ space regularity of approximating solutions is required to ensure the convergence to transition fronts; $\\rm(ii)$ space regularity of transition fronts lays the foundation for applying various techniques for reaction-diffusion equations to nonlocal equations, and hence, for further studying various qualitative properties such as stability and uniqueness. Hence, it is very important to study the space regularity of special solutions.\n\nIn the present paper, we intend to establish some general criteria concerning the space regularity of transition fronts of \\eqref{main-eqn}. More precisely, we want to know whether a transition front $u(t,x)$ of \\eqref{main-eqn} is continuously differentiable in $x$. To state our first result, we further introduce the following hypothesis.\n\n\\begin{itemize}\n\\item[\\bf{(H3)}] There exist $\\theta_{0}\\in(0,\\theta_{1})$ and $\\kappa_{0}>0$ such that\n\\begin{equation*}\nf_{u}(t,x,u)\\leq1-\\kappa_{0},\\quad (t,x,u)\\in\\mathbb{R}\\times\\mathbb{R}\\times[0,\\theta_{0}].\n\\end{equation*}\n\\end{itemize}\n\nWe prove\n\n\\begin{thm}\\label{thm-regularity-of-tf}\nSuppose (H1)-(H3). Let $u(t,x)$ be an arbitrary transition front of \\eqref{main-eqn}. Then, $u(t,x)$ is regular in space in the following sense:  for any $t\\in\\mathbb{R}$, $u(t,x)$ is continuously differentiable in $x$ and satisfies $\\sup_{(t,x)\\in\\mathbb{R}\\times\\mathbb{R}}|u_{x}(t,x)|<\\infty$.\n\\end{thm}\n\nWe remark that Theorem \\ref{thm-regularity-of-tf} can be proven essentially due to the assumptions $\\int_{0}^{1}f_{B}(u)du>0$ and the existence of traveling waves of \n\\begin{equation}\\label{bistable-tw-eqn-000000}\nu_{t}=J\\ast u-u+f_{B}(u)\n\\end{equation}\nwith nonzero speed in (H2), which corresponds to the unbalanced case in the bistable case. If we drop these assumptions, then, in the bistable case, there is no hope that Theorem \\ref{thm-regularity-of-tf} is true without additional assumptions on $f(t,x,u)$, since discontinuous traveling waves of \\eqref{bistable-tw-eqn-000000} with zero speed were constructed in \\cite{BaFiReWa97}, where necessary conditions for the existence of discontinuous traveling waves are also given. Thus, our results are kind of sharp and are compatible with the results for traveling waves in the monostable case, the ignition case and the unbalanced bistable case. It is worthwhile to point out that in the case \\eqref{bistable-tw-eqn-000000} admits discontinuous traveling waves, it may also admits non-monotone waves (see \\cite[Theorem 5.4]{Ch97}).\n\nAlthough the assumptions on $f_{B}$ are automatically true in the monostable case and the ignition case, it does cause some restrictions in the bistable case, and hence, there remains an interesting problem, that is, whether transition fronts in the bistable case are regular in space when \\eqref{bistable-tw-eqn-000000} admits only smooth stationary waves, i.e., smooth traveling waves with speed zero. \n\nThe proof of Theorem \\ref{thm-regularity-of-tf} is based on the rightward propagation estimate of transition fronts and the analysis of the  growth and the  decay of $\\frac{u(t,x+\\eta)-u(t,x)}{\\eta}$. The rightward propagation estimate reads as\n\\begin{equation}\\label{right-propagation-intro}\nX(t)-X(t_{0})\\geq c_{1}(t-t_{0}-T_{1}),\\quad t\\geq t_{0}\n\\end{equation}\nfor some $c_{1}>0$ and $T_{1}>0$, which is established in Theorem \\ref{lem-propagation-estimate} (to show \\eqref{right-propagation-intro},  we need $\\int_{0}^{1}f_{B}(u)du>0$ and the existence of traveling waves of \\eqref{bistable-tw-eqn-000000} with nonzero speed). To control the behavior of $\\frac{u(t,x+\\eta)-u(t,x)}{\\eta}$ when $x$ is close to $\\infty$, we need (H3). A key ingredient in proving Theorem \\ref{thm-regularity-of-tf}, i.e., controlling the term $\\frac{u(t,x+\\eta)-u(t,x)}{\\eta}$, is the observation that for fixed $x$, the term $\\frac{u(t,x+\\eta)-u(t,x)}{\\eta}$ can only grow for a period of time that is independent of $x$. Moreover, since we directly study transition fronts, which may not come from approximating solutions, we are lack of a priori information, which immediately causes the possible blow-up behavior of $\\frac{u(t_{0},x+\\eta)-u(t_{0},x)}{\\eta}$ as $\\eta\\to0$ at the initial time $t_{0}$. To overcome this technical difficulty, we utilize the fact that transition fronts are global-in-time, which means we can take $t_{0}$ to approach $-\\infty$ along subsequences.\n\nClearly, (H3) rules out many monostable nonlinearities, and it does not cover all Fisher-KPP nonlinearities. Our next result is trying to cover the monostable nonlinearities at the cost of putting some restrictions on transition fronts. We prove\n\n\\begin{thm}\\label{thm-regularity-of-tf-ii}\nSuppose (H1) and (H2). Suppose, in addition, that\n\\begin{equation*}\n\\sup_{(t,x,u)\\in\\mathbb{R}\\times\\mathbb{R}\\times[0,1]}|f_{xu}(t,x,u)|<\\infty\\quad\\text{and}\\quad\\sup_{(t,x,u)\\in\\mathbb{R}\\times\\mathbb{R}\\times[0,1]}|f_{uu}(t,x,u)|<\\infty.\n\\end{equation*}\nLet $u(t,x)$ be an arbitrary transition front of \\eqref{main-eqn} satisfying\n\\begin{equation}\\label{harnack-type-intro}\nu(t,x)\\leq Ce^{r|x-y|}u(t,y),\\quad (t,x,y)\\in\\mathbb{R}\\times\\mathbb{R}\\times\\mathbb{R}\n\\end{equation}\nfor some $C>0$ and $r>0$. Then, $u(t,x)$ is regular in space in the following sense:  for any $t\\in\\mathbb{R}$, $u(t,x)$ is continuously differentiable in $x$ and satisfies $\\sup_{(t,x)\\in\\mathbb{R}\\times\\mathbb{R}}\\frac{|u_{x}(t,x)|}{u(t,x)}<\\infty$.\n\\end{thm}\n\nThe key assumption in Theorem \\ref{thm-regularity-of-tf-ii} is the Harnack-type inequality \\eqref{harnack-type-intro}, which is the case for some transition fronts in the Fisher-KPP case (see \\cite{LiZl14,ShSh14-kpp}). The importance of \\eqref{harnack-type-intro} lies in the fact that it allows the comparison of $J\\ast u$ and $J'\\ast u$ with $u$. More precisely, by \\eqref{harnack-type-intro}, we find $\\frac{1}{C}\\int_{\\mathbb{R}}J(x)e^{-r|x|}dx\\leq\\frac{J\\ast u}{u}\\leq C\\int_{\\mathbb{R}}J(x)e^{r|x|}dx$ and similarly for $J'\\ast u$, which plays crucial roles in controlling $\\frac{u(t,x+\\eta)-u(t,x)}{\\eta u(t,x)}$.\n\nThe next result gives space regularity of transition fronts under the exact decay assumption.\n\n\\begin{thm}\\label{thm-regularity-of-tf-iii}\nSuppose (H1) and (H2). Suppose, in addition, that\n\\begin{equation*}\n\\sup_{(t,x,u)\\in\\mathbb{R}\\times\\mathbb{R}\\times[0,1]}|f_{xu}(t,x,u)|<\\infty\\quad\\text{and}\\quad\\sup_{(t,x,u)\\in\\mathbb{R}\\times\\mathbb{R}\\times[0,1]}|f_{uu}(t,x,u)|<\\infty.\n\\end{equation*}\nLet $u(t,x)$ be an arbitrary transition front of \\eqref{main-eqn} with interface location function $X(t)$. There exists $r_{0}>0$ such that if \n\\begin{equation}\\label{condition-exact-decay-intro}\n\\lim_{x\\to\\infty}\\frac{u(t,x+X(t))}{e^{-rx}}=1\\quad\\text{uniformly in}\\quad t\\in\\mathbb{R}\n\\end{equation}\nfor some  $r\\in(0,r_{0}]$,\nthen, $u(t,x)$ is regular in space in the following sense:  for any $t\\in\\mathbb{R}$, $u(t,x)$ is continuously differentiable in $x$ and satisfies $\\sup_{(t,x)\\in\\mathbb{R}\\times\\mathbb{R}}\\frac{|u_{x}(t,x)|}{u(t,x)}<\\infty$.\n\\end{thm}\n\nWe remark that the exact decay assumption \\eqref{condition-exact-decay-intro} is the case for some transition fronts in the Fisher-KPP case (see \\cite{LiZl14,ShSh14-kpp}). The importance of \\eqref{condition-exact-decay-intro} lies in the fact that it allows the comparison of $J\\ast u$ and $J'\\ast u$ with $u$ near $x=\\infty$. Note that if the limit in \\eqref{condition-exact-decay-intro} is some other positive number instead of $1$, then we only need to shift the exponential function on the bottom correspondingly. The number $r_{0}$ corresponds to the possible decay rate of Fisher-KPP transition fronts, and thus, it would be interesting to determine the optimal $r_{0}$.\n\n\nThe study of space regularity of transition fronts of \\eqref{main-eqn} was initiated in \\cite{ShSh14-3}, where the space regularity of well-constructed transition fronts in time heterogeneous media of ignition type is obtained. Those well-constructed transition fronts are uniformly Lipschitz continuous in space and their interface location functions can be chosen to be continuously differentiable with uniformly positive first order derivatives. These properties, which may not be true for an arbitrary transition front, play important roles in the study of space regularity of well-constructed transition fronts in \\cite{ShSh14-3}. Our results, Theorem \\ref{thm-regularity-of-tf}, Theorem \\ref{thm-regularity-of-tf-ii} and Theorem \\ref{thm-regularity-of-tf-iii}, generalize the work in \\cite{ShSh14-3} to arbitrary transition fronts (with some additional assumptions in Theorem \\ref{thm-regularity-of-tf-ii} and Theorem \\ref{thm-regularity-of-tf-iii}) of \\eqref{main-eqn} with a large class of nonlinearities. As already shown in \\cite{ShSh14-3}, space regularity of transition fronts is of great significance in the study of stability, which together with uniqueness and asymptotic speeds, will be studied elsewhere.\n\n\nFinally, we study the existence of continuously differentiable and increasing interface location functions. As mentioned before, if $X(t)$ is an interface location function of a transition front $u(t,x)$ of \\eqref{main-eqn}, then for any bounded function $\\xi(t)$, $X(t)+\\xi(t)$ is also an interface location function of $u(t,x)$. Hence, interface location functions of a transition front are not unique and may not be continuous. But, near each interface location function, we are able to find a continuously differentiable and increasing function, as the new interface location function. This is given by the following\n\n\\begin{thm}\\label{thm-modified-interface-location}\nSuppose (H1) and (H2). Let $u(t,x)$ be an arbitrary transition front of \\eqref{main-eqn} with interface location function $X(t)$ satisfying\n\\begin{equation}\\label{upper-avg-speed}\nX(t)-X(t_{0})\\leq c_{2}(t-t_{0}+T_{2}),\\quad t\\geq t_{0}\n\\end{equation}\nfor some $c_{2}>0$ and $T_{2}>0$. Then, there are constants $0<\\tilde{c}_{\\min}\\leq\\tilde{c}_{\\max}<\\infty$ and a continuously differentiable function $\\tilde{X}(t)$ satisfying\n\\begin{equation*}\n\\tilde{c}_{\\min}\\leq\\dot{\\tilde{X}}(t)\\leq \\tilde{c}_{\\max},\\quad t\\in\\mathbb{R}\n\\end{equation*}\nsuch that\n\\begin{equation*}\n\\sup_{t\\in\\mathbb{R}}|\\tilde{X}(t)-X(t)|<\\infty.\n\\end{equation*}\nIn particular, $\\tilde{X}(t)$ is also an interface location function of $u(t,x)$.\n\\end{thm}\n\nWe see that (H1) and (H2) do not ensure the space regularity of  transition fronts. It will be clear later that the proof of Theorem \\ref{thm-modified-interface-location} does not need the space regularity of transition fronts.\n\nWe refer to $\\tilde{X}(t)$ in Theorem \\ref{thm-modified-interface-location} as the modified interface location function, which gives a better characterization of the propagation of transition fronts and has been verified to be of great technical importance in studying the stability of transition fronts in time heterogeneous media (see e.g. \\cite{ShSh14-1,ShSh14-2,ShSh14-3}). Moreover, for reaction-diffusion equations in space heterogeneous media, the rightmost interface location at some constant value continuously moves to the right (see e.g. \\cite{MeRoSi10,MNRR09,NoRy09,Zl13}). But for nonlocal equations in space heterogeneous media, the rightmost interface location at some constant value jumps in general due to the nonlocality, which makes the modified interface location function more important.\n\nThe condition \\eqref{upper-avg-speed} is a technical assumption saying a transition front moves to the right at most at linear speed in the average sense, which together with \\eqref{right-propagation-intro} allow us to find the modified interface location function. Arguing as in the proof of Theorem \\ref{lem-propagation-estimate}, we readily check that the condition \\eqref{upper-avg-speed} is always true in the bistable case. This condition can also be verified in other cases including the monostable case and the ignition case as in the following two corollaries.\n\n\\begin{cor}\\label{cor-modified-interface}\nSuppose (H1) and (H2). Let $u(t,x)$ be an arbitrary transition front of \\eqref{main-eqn}. If there exists $\\tilde{\\theta}\\in(0,\\theta)$ such that\n\\begin{equation*}\nf(t,x,u)\\leq0,\\quad(t,x,u)\\in\\mathbb{R}\\times\\mathbb{R}\\times[0,\\tilde{\\theta}],\n\\end{equation*}\nthen \\eqref{upper-avg-speed} is true. In particular, the conclusions of Theorem \\ref{thm-modified-interface-location} hold.\n\\end{cor}\n\nClearly, Corollary \\ref{cor-modified-interface} covers, in particular, all bistable and ignition nonlinearities, but rules out all monostable nonlinearities, which are covered by the next result.\n\n\\begin{cor}\\label{cor-modified-interface-ii}\nSuppose (H1) and (H2). Suppose, in addition, that $J$ is compactly supported. Let $u(t,x)$ be an arbitrary transition front of \\eqref{main-eqn} with interface location function $X(t)$. If there exist $r>0$ and $h>0$ such that\n\\begin{equation*}\nu(t,x+X(t))\\leq e^{-r(x-h)},\\quad (t,x)\\in\\mathbb{R}\\times\\mathbb{R},\n\\end{equation*}\nthen \\eqref{upper-avg-speed} is true. In particular, the conclusions of Theorem \\ref{thm-modified-interface-location} hold.\n\\end{cor}\n\nWe remark that in the monostable case, uniform exponential decay as $x\\to\\infty$ may be necessary for transition fronts to travel at linear speeds, since slower decay near $x=\\infty$ may cause super-linear propagation (see \\cite{HaRo10}). It is worthwhile to point out that in the bistable case, a discontinuous traveling wave may not converge to zero exponentially as $x\\to\\infty$ (see \\cite[Theorem 5.1]{BaChm99}). Here, we need $J$ to be compactly supported, since we will use results obtained in \\cite{CaCh04} and \\cite{ShZh12-2} in the proof, which were proven when $J$ is compactly supported. It should be pointed out that the arguments and results in\n\\cite{ShZh12-2} can be extended to dispersal kernels $J$ which are not compactly supported, but satisfy (H1) (see \\cite{AZh}).\n\n\nAs a direct application of the results, in particular, Theorem \\ref{thm-regularity-of-tf} and Corollary \\ref{cor-modified-interface}, obtained in the present paper, we study in \\cite{ShSh14-bistable} the equation \\eqref{main-eqn} in time heterogeneous media of general bistable type, that is, $f(t,x,u)=f(t,u)$ satisfies\n\\begin{equation*}\nf_{\\min}(u)\\leq f(t,u)\\leq f_{\\max}(u),\\quad (t,u)\\in\\mathbb{R}\\times[0,1],\n\\end{equation*}\nwhere the $C^{2}$ functions $f_{\\min}$ and $f_{\\max}$ are standard bistable nonlinearities on $[0,1]$ with the unbalanced condition $\\int_{0}^{1}f_{\\min}(u)du>0$. Provided that there is a space nonincreasing transition front, we show by means of Theorem \\ref{thm-regularity-of-tf} and Corollary \\ref{cor-modified-interface} that (i) all transition fronts are asymptotically stable and enjoy decaying estimates; (ii) transition fronts are unique up to space shifts; (iii) all transition fronts become periodic traveling waves in the periodic media and have asymptotic speeds in the uniquely ergodic media. The assumption on the existence of a space nonincreasing transition front can be verified if, for example, $f(t,u)$ is of standard bistable type in $u$ for each $t$ and their middle zeros are the same.\n\n\nThe rest of the paper is organized as follows. In Section \\ref{sec-prop-estimate}, we establish the rightward propagation estimate of transition fronts. We will see, in particular, that any transition fronts moves front left infinity to right infinity as time goes from $-\\infty$ to $\\infty$. In Section \\ref{subsec-unconditional-regularity}, we prove Theorem \\ref{thm-regularity-of-tf}. In Section \\ref{sec-proof-2}, we prove Theorem \\ref{thm-regularity-of-tf-ii}. In Section \\ref{sec-proof-iafjiaf}, we prove Theorem \\ref{thm-regularity-of-tf-iii}. In Section \\ref{sec-proof-thm-cor}, we prove Theorem \\ref{thm-modified-interface-location}, Corollary \\ref{cor-modified-interface} and Corollary \\ref{cor-modified-interface-ii}.  We end up the present paper with an appendix, Appendix \\ref{app-ig-tw}, on ignition traveling waves.\n\n\n\n\n\n\\section{Rightward propagation estimates}\\label{sec-prop-estimate}\n\nIn this section, we study the rightward propagation estimates of transition fronts. Throughout this section, we assume (H1) and (H2).\n\nIn what follows in this section, $u(t,x)$ will be an arbitrary transition front of \\eqref{main-eqn} with interface location function $X(t)$. For $\\lambda\\in(0,1)$, we define $X_{\\lambda}^{-}(t)$ and $X_{\\lambda}^{+}(t)$ by setting\n\\begin{equation}\\label{defn-interface-locations}\n\\begin{split}\nX_{\\lambda}^{-}(t)&=\\sup\\{x\\in\\mathbb{R}|u(t,y)>\\lambda,\\,\\,\\forall y\\leq x\\},\\\\\n X_{\\lambda}^{+}(t)&=\\inf\\{x\\in\\mathbb{R}|u(t,y)<\\lambda,\\,\\,\\forall y\\geq x\\}.\n\\end{split}\n\\end{equation}\nNote that if $u(t,x)$ is continuous in $x$, then $X_{\\lambda}^{-}(t)$ and $X_{\\lambda}^{+}(t)$ are nothing but the leftmost and rightmost interface locations at $\\lambda$. Trivially, $X_{\\lambda}^{-}(t)\\leq X_{\\lambda}^{+}(t)$ and $X_{\\lambda}^{\\pm}(t)$ are decreasing in $\\lambda$. Due to the possible discontinuity of $u(t,x)$ in $x$, it may happen that $u(t,X_{\\lambda}^{-}(t))<\\lambda$ or $u(t,X_{\\lambda}^{+}(t))>\\lambda$.\n\nFrom the definition of transition fronts, we have the following simple lemma.\n\n\\begin{lem}\\label{lem-bounded-interface-width}\nThe following statements hold:\n\\begin{itemize}\n\\item[\\rm(i)] for any $0<\\lambda_{1}\\leq\\lambda_{2}<1$, there holds $\\sup_{t\\in\\mathbb{R}}[X^{+}_{\\lambda_{1}}(t)-X^{-}_{\\lambda_{2}}(t)]<\\infty$;\n\n\\item[\\rm(ii)] for any $\\lambda\\in(0,1)$, there hold $\\sup_{t\\in\\mathbb{R}}|X(t)-X^{\\pm}_{\\lambda}(t)|<\\infty$.\n\\end{itemize}\n\\end{lem}\n\\begin{proof}\n$\\rm(i)$ By the uniform-in-$t$ limits $\\lim_{x\\to-\\infty}u(t,x+X(t))=1$ and $\\lim_{x\\to\\infty}u(t,x+X(t))=0$, there exist $x_{1}$ and $x_{2}$ such that $u(t,x+X(t))>\\lambda_{2}$ for all $x\\leq x_{2}$ and $t\\in\\mathbb{R}$, and $u(t,x+X(t))<\\lambda_{1}$ for all $x\\geq x_{1}$ and $t\\in\\mathbb{R}$. It then follows from the definition of $X_{\\lambda_{2}}^{-}(t)$ and $X_{\\lambda_{1}}^{+}(t)$ that $x_{2}+X(t)\\leq X_{\\lambda_{2}}^{-}(t)$ and $x_{1}+X(t)\\geq X_{\\lambda_{1}}^{+}(t)$ for all $t\\in\\mathbb{R}$. The result then follows.\n\n$\\rm(ii)$ Let $\\lambda_{1}=\\lambda=\\lambda_{2}$ in the proof of $\\rm(i)$, we have  $x_{2}+X(t)\\leq X_{\\lambda}^{-}(t)$ and $x_{1}+X(t)\\geq X_{\\lambda}^{+}(t)$ for all $t\\in\\mathbb{R}$. In particular,\n\\begin{equation*}\nx_{2}+X(t)\\leq X_{\\lambda}^{-}(t)\\leq X_{\\lambda}^{+}(t)\\leq x_{1}+X(t),\\quad t\\in\\mathbb{R}.\n\\end{equation*}\nThis completes the proof.\n\\end{proof}\n\nThe next result gives the rightward propagation estimate of $u(t,x)$ in terms of $X(t)$.\n\n\\begin{thm}\\label{lem-propagation-estimate}\nThere exist $c_{1}>0$ and $T_{1}>0$ such that\n\\begin{equation*}\nX(t)-X(t_{0})\\geq c_{1}(t-t_{0}-T_{1}),\\quad t\\geq t_{0}.\n\\end{equation*}\n\\end{thm}\n\\begin{proof}\nFix some $\\lambda\\in(\\theta,1)$. We  write $X^{-}(t)=X^{-}_{\\lambda}(t)$. Since $\\sup_{t\\in\\mathbb{R}}|X(t)-X^{-}(t)|<\\infty$ by Lemma \\ref{lem-bounded-interface-width}, it suffices to show\n\\begin{equation}\\label{propagation-estimate-1}\nX^{-}(t)-X^{-}(t_{0})\\geq c(t-t_{0}-T),\\quad t\\geq t_{0}\n\\end{equation}\nfor some $c>0$ and $T>0$.\n\nRecall $f_{B}$ is as in (H2). Let $(c_{B},\\phi_{B})$ with $c_{B}>0$ be the unique solution of\n\\begin{equation*}\n\\begin{cases}\nJ\\ast\\phi-\\phi+c\\phi_{x}+f_{B}(\\phi)=0,\\\\\n\\phi_{x}<0,\\,\\, \\phi(0)=\\theta,\\,\\, \\phi(-\\infty)=1\\,\\,\\text{and}\\quad\\phi(\\infty)=0\n\\end{cases}\n\\end{equation*}\n(see \\cite{BaFiReWa97} for the existence and uniqueness of $(c_B,\\phi_B)$). That is, $c_{B}$ is the unique speed and $\\phi_{B}$ is the normalized profile of traveling waves of\n\\begin{equation}\\label{eqn-bistable-homo-B}\nu_{t}=J\\ast u-u+f_{B}(u).\n\\end{equation}\n\nLet $u_{0}:\\mathbb{R}\\to[0,1]$ be a uniformly continuous and nonincreasing function satisfying\n\\begin{equation}\\label{some-special-initial-data-132184}\nu_{0}(x)=\\begin{cases}\n\\lambda,\\quad& x\\leq x_{0},\\\\\n0,\\quad & x\\geq0,\n\\end{cases}\n\\end{equation}\nwhere $x_{0}<0$ is fixed. By the definition of $X^{-}(t)$, we see that for any $t_{0}\\in\\mathbb{R}$, there holds $u(t_{0},x+X^{-}(t_{0}))\\geq u_{0}(x)$ for all $x\\in\\mathbb{R}$, and then, by $f(t,x,u)\\geq f_{B}(u)$ and the comparison principle, we find\n\\begin{equation}\\label{result-after-cp-42492}\nu(t,x+X^{-}(t_{0}))\\geq u_{B}(t-t_{0},x;u_{0}),\\quad x\\in\\mathbb{R},\\,\\,t\\geq t_{0},\n\\end{equation}\nwhere $u_{B}(t,x;u_{0})$ is the unique solution to \\eqref{eqn-bistable-homo-B} with $u_{B}(0,\\cdot;u_{0})=u_{0}$. By the choice of $u_{0}$ and the stability of bistable traveling waves (see e.g. \\cite{BaFiReWa97}), there are constants $x_{B}=x_{B}(\\lambda)\\in\\mathbb{R}$, $q_{B}=q_{B}(\\lambda)>0$ and $\\omega_{B}>0$ such that\n\\begin{equation*}\nu_{B}(t-t_{0},x;u_{0})\\geq\\phi_{B}(x-x_{B}-c_{B}(t-t_{0}))-q_{B}e^{-\\omega_{B}(t-t_{0})}, \\quad x\\in\\mathbb{R},\\,\\,t\\geq t_{0}.\n\\end{equation*}\nHence,\n\\begin{equation*}\nu(t,x+X^{-}(t_{0}))\\geq \\phi_{B}(x-x_{B}-c_{B}(t-t_{0}))-q_{B}e^{-\\omega_{B}(t-t_{0})}, \\quad x\\in\\mathbb{R},\\,\\,t\\geq t_{0}.\n\\end{equation*}\nLet $T_{0}=T_{0}(\\lambda)>0$ be such that $q_{B}e^{-\\omega_{B}T_{0}}=\\frac{1-\\lambda}{2}$ (making $q_{B}$ larger so that $q_{B}>\\frac{1-\\lambda}{2}$ if necessary) and denote by $\\xi_{B}(\\frac{1+\\lambda}{2})$ the unique point such that $\\phi_{B}(\\xi_{B}(\\frac{1+\\lambda}{2}))=\\frac{1+\\lambda}{2}$. Setting $x_{*}=x_{B}+c_{B}(t-t_{0})+\\xi_{B}(\\frac{1+\\lambda}{2})$, the monotonicity of $\\phi_{B}$ implies that for all $t\\geq t_{0}+T_{0}$ and $x\\leq x^{*}-1$\n\\begin{equation*}\n\\begin{split}\nu(t,x+X^{-}(t_{0}))&\\geq\\phi_{B}(x_{*}-1-x_{B}-c_{B}(t-t_{0}))-q_{B}e^{-\\omega_{B}T_{0}}\\\\\n&>\\phi_{B}(x_{*}-x_{B}-c_{B}(t-t_{0}))-q_{B}e^{-\\omega_{B}T_{0}}\\\\\n&=\\phi_{B}(\\xi_{B}(\\frac{1+\\lambda}{2}))-q_{B}e^{-\\omega_{B}T_{0}}=\\lambda.\n\\end{split}\n\\end{equation*}\nThis says that $x_{*}-1+X^{-}(t_{0})\\leq X^{-}(t)$ for all $t\\geq t_{0}+T_{0}$, that is,\n\\begin{equation}\\label{estimate-long-time}\nX^{-}(t)-X^{-}(t_{0})\\geq x_{B}-1+c_{B}(t-t_{0})+\\xi_{B}(\\frac{1+\\lambda}{2}),\\quad t\\geq t_{0}+T_{0}.\n\\end{equation}\n\nWe now estimate $X^{-}(t)-X^{-}(t_{0})$ for $t\\in[t_{0},t_{0}+T_{0}]$. We claim that there exists $z=z(T_{0})<0$ such that\n\\begin{equation}\\label{estimate-finite-time}\nX^{-}(t)-X^{-}(t_{0})\\geq z,\\quad t\\in[t_{0},t_{0}+T_{0}].\n\\end{equation}\nLet $u_{B}(t,x;u_{0})$ and $u_{B}(t;\\lambda):=u_{B}(t,x;\\lambda)$ be solutions of \\eqref{eqn-bistable-homo-B} with $u_{B}(0,x;u_{0})=u_{0}(x)$ and $u_{B}(0;\\lambda)=u_{B}(0,x;\\lambda)\\equiv\\lambda$, respectively. By the comparison principle, we have $u_{B}(t,x;u_{0})<u_{B}(t;\\lambda)$ for all $x\\in\\mathbb{R}$ and $t>0$, and $u_{B}(t,x;u_{0})$ is strictly decreasing in $x$ for $t>0$.\n\nWe see that for any $t>0$, $u_{B}(t,-\\infty;u_{0})=u_{B}(t;\\lambda)$. This is because that $\\frac{d}{dt}u_{B}(t,-\\infty;u_{0})=f_{B}(u_{B}(t,-\\infty;u_{0}))$ for $t>0$ and $u_{B}(0,-\\infty;u_{0})=\\lambda$. Since $\\lambda\\in(\\theta,1)$, as a solution of the ODE $u_{t}=f_{B}(u)$, $u_{B}(t;\\lambda)$ is strictly increasing in $t$, which implies that $u_{B}(t,-\\infty;u_{0})=u_{B}(t;\\lambda)>\\lambda$ for $t>0$. As a result, for any $t>0$ there exists a unique $\\xi_{B}(t)\\in\\mathbb{R}$ such that $u_{B}(t,\\xi_{B}(t);u_{0})=\\lambda$. Moreover, $\\xi_{B}(t)$ is continuous in $t$.\n\nSetting $x_{**}=\\xi_{B}(t-t_{0})$, we find $u(t,x+X^{-}(t_{0}))>\\lambda$ for all $x\\leq x_{**}$ by the monotonicity of $u_{B}(t,x;u_{0})$ in $x$, which  together with \\eqref{result-after-cp-42492}\nimplies that\n\\begin{equation*}\nX^{-}(t)\\geq x_{**}+X^{-}(t_{0})=\\xi_{B}(t-t_{0})+X^{-}(t_{0}),\\quad t>t_{0}.\n\\end{equation*}\nThus, \\eqref{estimate-finite-time} follows if $\\inf_{t\\in(t_{0},t_{0}+T_{0}]}\\xi_{B}(t-t_{0})>-\\infty$, that is,\n\\begin{equation}\\label{not-to-negative-finity}\n\\inf_{t\\in(0,T_{0}]}\\xi_{B}(t)>-\\infty.\n\\end{equation}\n\nWe now show \\eqref{not-to-negative-finity}. Since $u_{0}(x)=\\lambda$ for $x\\leq x_{0}$, continuity with respect to the initial data\n (in sup norm) implies that for any $\\epsilon>0$ there exists $\\delta>0$ such that\n\\begin{equation*}\nu_{B}(t;\\lambda)-\\lambda\\leq\\epsilon\\quad\\text{and}\\quad\\sup_{x\\leq x_{0}}[u_{B}(t;\\lambda)-u_{B}(t,x;u_{0})]=u_{B}(t;\\lambda)-u_{B}(t,x_{0};u_{0})\\leq\\epsilon\n\\end{equation*}\nfor all $t\\in[0,\\delta]$, where the equality is due to monotonicity. By (H1), $J$ concentrates near $0$ and decays very fast as $x\\to\\pm\\infty$. Thus, we can choose $x_{1}=x_{1}(\\epsilon)<<x_{0}$ such that\n\\begin{equation*}\n\\int_{-\\infty}^{x_{0}}J(x-y)dy\\geq1-\\epsilon,\\quad x\\le x_1.\n\\end{equation*}\nNow, for any $x\\leq x_{1}$ and $t\\in(0,\\delta]$, we have\n\\begin{equation*}\n\\begin{split}\n\\frac{d}{dt}u_{B}(t,x;u_{0})&=\\int_{\\mathbb{R}}J(x-y)u_{B}(t,y;u_{0})dy-u_{B}(t,x;u_{0})+f_{B}(u_{B}(t,x;u_{0}))\\\\\n&\\geq\\int_{-\\infty}^{x_{0}}J(x-y)u_{B}(t,y;u_{0})dy-u_{B}(t,x;u_{0})+f_{B}(u_{B}(t,x;u_{0}))\\\\\n&\\geq(1-\\epsilon)\\inf_{x\\leq x_{0}}u_{B}(t,x;u_{0})-u_{B}(t;\\lambda)+f_{B}(u_{B}(t,x;u_{0}))\\\\\n&=-(1-\\epsilon)\\sup_{x\\leq x_{0}}[u_{B}(t;\\lambda)-u_{B}(t,x;u_{0})]-\\epsilon u_{B}(t;\\lambda)+f_{B}(u_{B}(t,x;u_{0}))\\\\\n&\\geq-\\epsilon(1-\\epsilon)-\\epsilon(\\lambda+\\epsilon)+f_{B}(u_{B}(t,x;u_{0}))>0\n\\end{split}\n\\end{equation*}\nif we choose $\\epsilon>0$ sufficiently small, since then $f_{B}(u_{B}(t,x;u_{0}))$ is close to $f_{B}(\\lambda)$, which is positive. This simply means that $u_{B}(t,x;u_{0})>\\lambda$ for all $x\\leq x_{1}$ and $t\\in(0,\\delta]$, which implies that $\\xi_{B}(t)>x_{1}$ for $t\\in(0,\\delta]$. The continuity of $\\xi_{B}$ then leads to \\eqref{not-to-negative-finity}. This proves \\eqref{estimate-finite-time}. \\eqref{propagation-estimate-1} then follows from \\eqref{estimate-long-time} and \\eqref{estimate-finite-time}. This completes the proof.\n\\end{proof}\n\nAs a simple consequence of Theorem \\ref{lem-propagation-estimate}, we have\n\n\\begin{cor}\\label{cor-propagation-estimate}\nThere holds $X(t)\\to\\pm\\infty$ as $t\\to\\pm\\infty$. In particular,  $u(t,x)\\to1$ as $t\\to\\infty$ and $u(t,x)\\to0$ as $t\\to-\\infty$ locally uniformly in $x$.\n\\end{cor}\n\\begin{proof}\nWe have from Lemma \\ref{lem-propagation-estimate} that\n\\begin{equation*}\nX(t)-X(t_{0})\\geq c_{1}(t-t_{0}-T_{1}) ,\\quad t\\geq t_{0}.\n\\end{equation*}\nSetting $t\\to\\infty$ in the above estimate, we find $X(t)\\to\\infty$ as $t\\to\\infty$. Setting $t_{0}\\to-\\infty$, we find $X(t_{0})\\to-\\infty$ as $t_{0}\\to-\\infty$.\n\\end{proof}\n\nThis corollary shows that any transition front travels from the left infinity to the right infinity. Thus, steady-state-like transition fronts, blocking the propagations of solutions, do not exist.\n\n\n\\section{Proof of Theorem \\ref{thm-regularity-of-tf}}\\label{subsec-unconditional-regularity}\n\nWe prove Theorem \\ref{thm-regularity-of-tf} in this section. Throughout this section, we assume that (H1)-(H3) hold and $u(t,x)$ is an arbitrary transition front of \\eqref{main-eqn} with interface location function $X(t)$.\n\nTo prove Theorem \\ref{thm-regularity-of-tf}, we first do some preparations and prove several lemmas. Fix some $0<\\delta_{0}\\ll1$. For $(t,x)\\in\\mathbb{R}\\times\\mathbb{R}$ and $\\eta\\in\\mathbb{R}$ with $0<|\\eta|\\leq\\delta_{0}$, we set\n\\begin{equation*}\nv^{\\eta}(t,x)=\\frac{u(t,x+\\eta)-u(t,x)}{\\eta}.\n\\end{equation*}\nIt is easy to see that $v^{\\eta}(t,x)$ satisfies\n\\begin{equation}\\label{lipschitz-eqn-987654}\nv^{\\eta}_{t}(t,x)=b^{\\eta}(t,x)-v^{\\eta}(t,x)+a^{\\eta}(t,x)v^{\\eta}(t,x)+\\tilde{a}^{\\eta}(t,x),\n\\end{equation}\nwhere\n\\begin{equation*}\n\\begin{split}\na^{\\eta}(t,x)&=\\frac{f(t,x,u(t,x+\\eta))-f(t,x,u(t,x))}{u(t,x+\\eta)-u(t,x)},\\\\\n\\tilde{a}^{\\eta}(t,x)&=\\frac{f(t,x+\\eta,u(t,x+\\eta))-f(t,x,u(t,x+\\eta))}{\\eta},\\\\\nb^{\\eta}(t,x)&=\\int_{\\mathbb{R}}J(x-y)v^{\\eta}(t,y)dy=\\int_{\\mathbb{R}}\\frac{J(x-y+\\eta)-J(x-y)}{\\eta}u(t,y)dy.\n\\end{split}\n\\end{equation*}\nHence, for any fixed $x$, treating \\eqref{lipschitz-eqn-987654} as an ODE in the variable $t$, we find from the variation of constants formula that for any $t\\geq t_{0}$\n\\begin{equation}\\label{diff-integral-sol}\n\\begin{split}\nv^{\\eta}(t,x)&=v^{\\eta}(t_{0},x)e^{-\\int_{t_{0}}^{t}(1-a^{\\eta}(s,x))ds}+\\int_{t_{0}}^{t}b^{\\eta}(\\tau,x)e^{-\\int_{\\tau}^{t}(1-a^{\\eta}(s,x))ds}d\\tau\\\\\n&\\quad\\quad+\\int_{t_{0}}^{t}\\tilde{a}^{\\eta}(\\tau,x)e^{-\\int_{\\tau}^{t}(1-a^{\\eta}(s,x))ds}d\\tau.\n\\end{split}\n\\end{equation}\n\n\nMoreover, we set\n\\begin{equation*}\nL_{0}=1+\\delta_{0}+\\sup_{t\\in\\mathbb{R}}|X(t)-X_{\\theta_{0}}^{+}(t)|\\quad\\text{and}\\quad L_{1}=1+\\delta_{0}+\\sup_{t\\in\\mathbb{R}}|X(t)-X_{\\theta_{1}}^{-}(t)|,\n\\end{equation*}\nwhere $\\theta_{1}$ and $\\theta_{0}$ are as in (H2) and (H3), respectively. By Lemma \\ref{lem-bounded-interface-width}, $L_{0}<\\infty$ and $L_{1}<\\infty$. We also set\n\\begin{equation*}\n\\begin{split}\nI_{l}(t)&=(-\\infty,X(t)-L_{1}),\\\\\nI_{m}(t)&=[X(t)-L_{1},X(t)+L_{0}],\\\\\n I_{r}(t)&=(X(t)+L_{0},\\infty)\n\\end{split}\n\\end{equation*}\nfor $t\\in\\mathbb{R}$. Clearly, $I_{l}(t)$, $I_{m}(t)$ and $I_{r}(t)$ are disjoint and $I_{l}(t)\\cup I_{m}(t)\\cup I_{r}(t)=\\mathbb{R}$. Since $X(t)$ may jump, so do $I_{l}(t)$, $I_{m}(t)$ and $I_{r}(t)$.\n\nSince $X(t)\\to\\pm\\infty$ as $t\\to\\pm\\infty$ by Corollary \\ref{cor-propagation-estimate}, for any fixed $x\\in\\mathbb{R}$, there hold $x\\in I_{r}(t)$ for all $t\\ll-1$ and $x\\in I_{l}(t)$ for all $t\\gg1$. Thus, for any fixed $x\\in\\mathbb{R}$,\n\\begin{equation*}\n\\begin{split}\nt_{\\rm first}(x)&=\\sup\\big\\{\\tilde{t}\\in\\mathbb{R}\\big|x\\in I_{r}(t)\\,\\,\\text{for all}\\,\\,t\\leq \\tilde{t}\\big\\},\\\\\nt_{\\rm last}(x)&=\\inf\\big\\{\\tilde{t}\\in\\mathbb{R}\\big|x\\in I_{l}(t)\\,\\,\\text{for all}\\,\\,t\\geq \\tilde{t}\\big\\}\n\\end{split}\n\\end{equation*}\nare well-defined. We see that if $X(t)$ is continuous, then $t_{\\rm first}(x)$ and $t_{\\rm last}(x)$ are the first time and the last time that $x$ is in $I_{m}(t)$. Clearly, $-\\infty<t_{\\rm first}(x)\\leq t_{\\rm last}(x)<\\infty$ (notice, $t_{\\rm first}(x)=t_{\\rm last}(x)$ may happen since $X(t)$ may jump). Although the functions $x\\mapsto t_{\\rm first}(x)$ and $x\\mapsto t_{\\rm last}(x)$ are unbounded, their difference is a bounded function as given by\n\n\\begin{lem}\nThere holds\n\\begin{equation}\\label{lipschitz-growth-time}\nT:=\\sup_{x\\in\\mathbb{R}}[t_{\\rm last}(x)-t_{\\rm first}(x)]<\\infty.\n\\end{equation}\n\\end{lem}\n\\begin{proof}\nTo see this, we suppose $t_{\\rm first}(x)<t_{\\rm last}(x)$. Due to possible jumps, we consider two cases.\n\nIf $x\\notin I_{r}(t_{\\rm first}(x))$, then $x\\in I_{l}(t_{\\rm first}(x))\\cup I_{m}(t_{\\rm first}(x))$, that is, $x\\leq X(t_{\\rm first}(x))+L_{0}$. Thus, for all $t\\geq t_{\\rm first}(x)+T_{1}+\\frac{L_{0}+L_{1}+1}{c_{1}}$, we see from Theorem \\ref{lem-propagation-estimate} that\n\\begin{equation*}\nx\\leq X(t_{\\rm first}(x))+L_{0}\\leq X(t)-c_{1}(t-t_{\\rm first}(x)-T_{1})+L_{0}\\leq X(t)-L_{1}-1.\n\\end{equation*}\nThis, implies that $x\\in I_{l}(t)$ for all $t\\geq t_{\\rm first}(x)+T_{1}+\\frac{L_{0}+L_{1}+1}{c_{1}}$, and hence, by definition\n\\begin{equation}\\label{growth-time-1}\nt_{\\rm last}(x)\\leq t_{\\rm first}(x)+T_{1}+\\frac{L_{0}+L_{1}+1}{c_{1}}.\n\\end{equation}\n\nIf $x\\in I_{r}(t_{\\rm first}(x))$, then we can find a sequence $\\{t_{n}\\}$ satisfying $t_{n}>t_{\\rm first}(x)$, $t_{n}\\to t_{\\rm first}(x)$ as $n\\to\\infty$ and $x\\notin I_{r}(t_{n}(x))$. Then, similar arguments as in the case $x\\notin I_{r}(t_{\\rm first}(x))$ lead to $t_{\\rm last}(x)\\leq t_{n}+T_{1}+\\frac{L_{0}+L_{1}+1}{c_{1}}$. Passing to the limit $n\\to\\infty$, we find \\eqref{growth-time-1} again. Hence, we have shown \\eqref{lipschitz-growth-time}.\n\\end{proof}\n\nTo control $v^{\\eta}(t,x)$, it is crucial to control $1-a^{\\eta}(t,x)$, which is achieved by means of $t_{\\rm first}(x)$ and $t_{\\rm last}(x)$ and is given in the following\n\n\\begin{lem}\nFor any $(t,x)\\in\\mathbb{R}\\times\\mathbb{R}$ and $0<|\\eta|\\leq\\delta_{0}$, there holds\n\\begin{equation}\\label{lipschitz-coefficient-1}\na^{\\eta}(t,x)\\leq\\begin{cases}\n1-\\kappa_{0},&\\quad t< t_{\\rm first}(x),\\\\\n0,&\\quad t> t_{\\rm last}(x).\n\\end{cases}\n\\end{equation}\n\\end{lem}\n\\begin{proof}\nBy the definition of $t_{\\rm first}(x)$ and $t_{\\rm last}(x)$, we see\n\\begin{equation}\\label{result-from-defn}\nx\\in\\begin{cases}\nI_{r}(t),&\\quad t< t_{\\rm first}(x),\\\\\nI_{l}(t),&\\quad t> t_{\\rm last}(x).\n\\end{cases}\n\\end{equation}\nBy (H2), (H3) and the choices of $L_{0}$ and $L_{1}$, we find that for any $(t,x)\\in\\mathbb{R}\\times\\mathbb{R}$ and $0<|\\eta|\\leq\\delta_{0}$\n\\begin{equation*}\na^{\\eta}(t,x)\\leq\\begin{cases}\n1-\\kappa_{0},&\\quad x\\in I_{r}(t),\\\\\n0,&\\quad x\\in I_{l}(t).\n\\end{cases}\n\\end{equation*}\nThe lemma then follows from \\eqref{result-from-defn}.\n\\end{proof}\n\nThe above lemma says that for any fixed $x$, the solution $v^{\\eta}(t,x)$ of the ODE \\eqref{lipschitz-eqn-987654} can only grow for a period of time that is not longer than $T$.\n\nIn the next lemma, we show that $u(t,x)$ is continuous in space.\n\n\\begin{lem}\\label{lem-tf-continuity}\nFor any $t\\in\\mathbb{R}$, $u(t,x)$ is continuous in $x$. Moreover, there holds\n\\begin{equation*}\n\\sup_{t\\in\\mathbb{R}}\\sup_{x\\neq y}\\bigg|\\frac{u(t,x)-u(t,y)}{x-y}\\bigg|<\\infty.\n\\end{equation*}\n\\end{lem}\n\\begin{proof}\nSince $0<u(t,x)<1$, we trivially have\n\\begin{equation*}\n\\sup_{t\\in\\mathbb{R}}\\sup_{|x-y|\\geq\\delta_{0}}\\bigg|\\frac{u(t,x)-u(t,y)}{x-y}\\bigg|<\\infty.\n\\end{equation*}\nThus, we only need to show\n\\begin{equation}\\label{local-lipschitz-uniform-190}\n\\sup_{t\\in\\mathbb{R}}\\sup_{0<|x-y|\\leq\\delta_{0}}\\bigg|\\frac{u(t,x)-u(t,y)}{x-y}\\bigg|<\\infty.\n\\end{equation}\nTo do so, we consider \\eqref{diff-integral-sol} for some fixed $x\\in\\mathbb{R}$. We are going to take $t_{0}\\to-\\infty$ along some subsequence, and so $t_{0}\\ll t_{\\rm first}(x)$. For $t$, there are  three cases: $t< t_{\\rm first}(x)$, $t\\in[t_{\\rm first}(x),t_{\\rm last}(x)]$ and $t> t_{\\rm last}(x)$. Here, we only consider the case $t> t_{\\rm last}(x)$; other two cases can be treated similarly and are simpler.\n\nWe first note\n\\begin{equation*}\nM_{0}:=\\sup_{(t,x)\\in\\mathbb{R}\\times\\mathbb{R}}\\sup_{0<|\\eta|\\leq\\delta_{0}}\\Big[|b^{\\eta}(t,x)|+|\\tilde{a}^{\\eta}(t,x)|\\Big]<\\infty\n\\end{equation*}\nand the following uniform-in-$\\eta$ estimates hold:\n\\begin{equation}\\label{lipschitz-estimates}\n\\begin{split}\ne^{-\\int_{r}^{t_{\\rm first}(x)}(1-a^{\\eta}(s,x))ds}&\\leq e^{-\\kappa_{0}(t_{\\rm first}(x)-r)},\\quad r\\leq t_{\\rm first}(x),\\\\\ne^{-\\int_{r}^{t_{\\rm last}(x)}(1-a^{\\eta}(s,x))ds}&\\leq e^{C_{a}T},\\quad r\\in[t_{\\rm first}(x),t_{\\rm last}(x)],\\\\\ne^{-\\int_{r}^{t}(1-a^{\\eta}(s,x))ds}&\\leq e^{-(t-r)},\\quad r\\in[t_{\\rm last}(x),t],\n\\end{split}\n\\end{equation}\nwhere $C_{a}:=\\sup_{(t,x)\\in\\mathbb{R}\\times\\mathbb{R}}\\sup_{0<|\\eta|\\leq\\delta_{0}}|1-a^{\\eta}(t,x)|<\\infty$ by (H2). They are simple consequences of \\eqref{lipschitz-growth-time} and \\eqref{lipschitz-coefficient-1}.\n\nFor the second and third terms on the right-hand side of \\eqref{diff-integral-sol}, we have\n\\begin{equation*}\n\\begin{split}\n&\\frac{1}{M_{0}}\\bigg|\\int_{t_{0}}^{t}\\big[b^{\\eta}(\\tau,x)+\\tilde{a}^{\\eta}(\\tau,x)\\big]e^{-\\int_{\\tau}^{t}(1-a^{\\eta}(s,x))ds}d\\tau\\bigg|\\leq \\int_{t_{0}}^{t}e^{-\\int_{\\tau}^{t}(1-a^{\\eta}(s,x))ds}d\\tau\\\\\n&\\quad\\quad=\\int_{t_{0}}^{t_{\\rm first}(x)}e^{-\\int_{\\tau}^{t}(1-a^{\\eta}(s,x))ds}d\\tau+\\int_{t_{\\rm first}(x)}^{t_{\\rm last}(x)}e^{-\\int_{\\tau}^{t}(1-a^{\\eta}(s,x))ds}d\\tau+\\int_{t_{\\rm last}(x)}^{t}e^{-\\int_{\\tau}^{t}(1-a^{\\eta}(s,x))ds}d\\tau.\n\\end{split}\n\\end{equation*}\nFor the three terms on the right-hand side of the above estimate, we use \\eqref{lipschitz-estimates} to deduce\n\\begin{equation*}\n\\begin{split}\n&\\int_{t_{0}}^{t_{\\rm first}(x)}e^{-\\int_{\\tau}^{t}(1-a^{\\eta}(s,x))ds}d\\tau\\\\\n&\\quad\\quad=\\int_{t_{0}}^{t_{\\rm first}(x)}e^{-\\int_{\\tau}^{t_{\\rm first}(x)}(1-a^{\\eta}(s,x))ds}e^{-\\int_{t_{\\rm first}(x)}^{t_{\\rm last}(x)}(1-a^{\\eta}(s,x))ds}e^{-\\int_{t_{\\rm last}(x)}^{t}(1-a^{\\eta}(s,x))ds}d\\tau\\\\\n&\\quad\\quad\\leq\\int_{t_{0}}^{t_{\\rm first}(x)}e^{-\\kappa_{0}(t_{\\rm first}(x)-\\tau)}e^{C_{a}T}e^{-(t-t_{\\rm last}(x))}d\\tau\\leq\\frac{e^{C_{a}T}}{\\kappa_{0}},\n\\end{split}\n\\end{equation*}\n\n\\begin{equation*}\n\\begin{split}\n\\int_{t_{\\rm first}(x)}^{t_{\\rm last}(x)}e^{-\\int_{\\tau}^{t}(1-a^{\\eta}(s,x))ds}d\\tau&=\\int_{t_{\\rm first}(x)}^{t_{\\rm last}(x)}e^{-\\int_{\\tau}^{t_{\\rm last}(x)}(1-a^{\\eta}(s,x))ds}e^{-\\int_{t_{\\rm last}(x)}^{t}(1-a^{\\eta}(s,x))ds}d\\tau\\\\\n&\\leq\\int_{t_{\\rm first}(x)}^{t_{\\rm last}(x)}e^{C_{a}T}e^{-(t-t_{\\rm last}(x))}d\\tau\\leq Te^{C_{a}T}\n\\end{split}\n\\end{equation*}\nand\n\\begin{equation*}\n\\begin{split}\n\\int_{t_{\\rm last}(x)}^{t}e^{-\\int_{\\tau}^{t}(1-a^{\\eta}(s,x))ds}d\\tau\\leq\\int_{t_{\\rm last}(x)}^{t}e^{-(t-\\tau)}d\\tau\\leq1.\n\\end{split}\n\\end{equation*}\nHence, we have shown\n\\begin{equation}\\label{first-estimate-190}\n\\bigg|\\int_{t_{0}}^{t}\\big[b^{\\eta}(\\tau,x)+\\tilde{a}^{\\eta}(\\tau,x)\\big]e^{-\\int_{\\tau}^{t}(1-a^{\\eta}(s,x))ds}d\\tau\\bigg|\\leq M_{0}\\bigg(\\frac{e^{C_{a}T}}{\\kappa_{0}}+Te^{C_{a}T}+1\\bigg).\n\\end{equation}\nNotice the above estimate is universal.\n\nFor the first term on the right hand side of \\eqref{diff-integral-sol}, we choose $t_{0}$ such that $t_{\\rm first}(x)-t_{0}=\\frac{1}{|\\eta|}$ and claim that\n\\begin{equation}\\label{diff-claim-2}\nv^{\\eta}(t_{0},x)e^{-\\int_{t_{0}}^{t}(1-a^{\\eta}(s,x))ds}\\to0\\quad\\text{as}\\quad\\eta\\to0.\n\\end{equation}\nIn fact, from $|v^{\\eta}(t_{0},x)|\\leq\\frac{1}{|\\eta|}$ and \\eqref{lipschitz-estimates}, we see\n\\begin{equation*}\n\\begin{split}\n\\bigg|v^{\\eta}(t_{0},x)e^{-\\int_{t_{0}}^{t}(1-a^{\\eta}(s,x))ds}\\bigg|&\\leq\\frac{1}{|\\eta|}e^{-[\\int_{t_{0}}^{t_{\\rm first}(x)}+\\int_{t_{\\rm first}(x)}^{t_{\\rm last}(x)}+\\int_{t_{\\rm last}(x)}^{t}](1-a^{\\eta}(s,x))ds}\\\\\n&\\leq\\frac{1}{|\\eta|}e^{-\\kappa_{0}(t_{\\rm first}(x)-t_{0})}e^{C_{a}T}e^{-(t-t_{\\rm last}(x))}\\\\\n&\\leq\\frac{1}{|\\eta|}e^{-\\frac{\\kappa_{0}}{|\\eta|}}e^{C_{a}T}\\to0\\quad\\text{as}\\quad\\eta\\to0.\n\\end{split}\n\\end{equation*}\n\nConsequently, choosing $t_{0}$ such that $t_{\\rm first}(x)-t_{0}=\\frac{1}{|\\eta|}$, we conclude from \\eqref{diff-integral-sol}, \\eqref{first-estimate-190} and \\eqref{diff-claim-2} that\n\\begin{equation*}\n\\sup_{(t,x)\\in\\mathbb{R}\\times\\mathbb{R}}\\sup_{0<|\\eta|\\leq\\delta_{0}}|v^{\\eta}(t,x)|\\leq\\tilde{C}+M_{0}\\bigg(\\frac{e^{C_{a}T}}{\\kappa_{0}}+Te^{C_{a}T}+1\\bigg),\n\\end{equation*}\nwhere $\\tilde{C}=\\tilde{C}(\\kappa_{0},\\delta_{0},T)>0$. This proves \\eqref{local-lipschitz-uniform-190}, and hence, the lemma follows.\n\\end{proof}\n\n\nWe see that for any $(t,x)\\in\\mathbb{R}\\times\\mathbb{R}$, as $\\eta\\to0$, we have\n\\begin{equation}\\label{diff-limits-1}\n\\begin{split}\n&a^{\\eta}(t,x)\\to f_{u}(t,x,u(t,x)),\\\\\n&\\tilde{a}^{\\eta}(t,x)\\to f_{x}(t,x,u(t,x)),\\\\\n&b^{\\eta}(t,x)\\to\\int_{\\mathbb{R}}J'(x-y)u(t,y)dy.\n\\end{split}\n\\end{equation}\nNotice the first two convergence in \\eqref{diff-limits-1} need the continuity of $u(t,x)$ in $x$. By \\eqref{diff-limits-1}, \\eqref{lipschitz-coefficient-1} holds with $a^{\\eta}(t,x)$ replaced by $f_{u}(t,x,u(t,x))$, that is,\n\\begin{equation}\\label{lipschitz-coefficient-2}\nf_{u}(t,x,u(t,x))\\leq\\begin{cases}\n1-\\kappa_{0},&\\quad t< t_{\\rm first}(x),\\\\\n0,&\\quad t> t_{\\rm last}(x).\n\\end{cases}\n\\end{equation}\n\n\nNow, we prove Theorem \\ref{thm-regularity-of-tf}.\n\n\\begin{proof}[Proof of Theorem \\ref{thm-regularity-of-tf}]\n\nLet us consider \\eqref{diff-integral-sol}. We are going to prove the existence of the limit $\\lim_{\\eta\\to0}v^{\\eta}(t,x)$. As in the proof of Lemma \\ref{lem-tf-continuity}, we assume $t_{0}\\ll t_{\\rm first}(x)$ and $t>t_{\\rm last}(x)$ in the rest of the proof. We treat three terms on the right-hand side of \\eqref{diff-integral-sol} separately.\n\n\nFor the second term on the right-hand side of \\eqref{diff-integral-sol}, we claim\n\\begin{equation}\\label{diff-claim-1}\n\\begin{split}\n&\\int_{t_{0}}^{t}b^{\\eta}(\\tau,x)e^{-\\int_{\\tau}^{t}(1-a^{\\eta}(s,x))ds}d\\tau\\to\\int_{t_{0}}^{t}\\bigg(\\int_{\\mathbb{R}}J'(x-y)u(\\tau,y)dy\\bigg)e^{-\\int_{\\tau}^{t}(1-f_{u}(s,x,u(s,x)))ds}d\\tau\\\\\n&\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad \\text{as}\\,\\, \\eta\\to0\\,\\,\\text{ uniformly in}\\,\\, t_{0}\\ll t_{\\rm first}(x).\n\\end{split}\n\\end{equation}\nTo see this, we notice\n\\begin{equation*}\n\\begin{split}\n&\\bigg|\\int_{t_{0}}^{t}b^{\\eta}(\\tau,x)e^{-\\int_{\\tau}^{t}(1-a^{\\eta}(s,x))ds}d\\tau-\\int_{t_{0}}^{t}\\bigg(\\int_{\\mathbb{R}}J'(x-y)u(\\tau,y)dy\\bigg)e^{-\\int_{\\tau}^{t}(1-f_{u}(s,x,u(s,x)))ds}d\\tau\\bigg|\\\\\n&\\quad\\quad\\leq\\int_{-\\infty}^{t}\\bigg|b^{\\eta}(\\tau,x)e^{-\\int_{\\tau}^{t}(1-a^{\\eta}(s,x))ds}-\\bigg(\\int_{\\mathbb{R}}J'(x-y)u(\\tau,y)dy\\bigg)e^{-\\int_{\\tau}^{t}(1-f_{u}(s,x,u(s,x)))ds}\\bigg|d\\tau.\n\\end{split}\n\\end{equation*}\nBy \\eqref{diff-limits-1}, the integrand converges to $0$ as $\\eta\\to0$ pointwise. Thus, by dominated convergence theorem, we only need to make sure that the integrand is controlled by some integrable function that is independent of $\\eta$. Writing\n\\begin{equation*}\nb^{0}(\\tau,x)=\\int_{\\mathbb{R}}J'(x-y)u(\\tau,y)dy\\quad\\text{and} \\quad a^{0}(s,x)=f_{u}(s,x,u(s,x)),\n\\end{equation*}\nwe only need to make sure that the function\n\\begin{equation*}\n\\tau\\mapsto\\sup_{0\\leq|\\eta|\\leq\\delta_{0}}\\bigg|b^{\\eta}(\\tau,x)e^{-\\int_{\\tau}^{t}(1-a^{\\eta}(s,x))ds}\\bigg|\n\\end{equation*}\nis integrable over $(-\\infty,t]$.\n\nSetting $M:=\\sup_{(t,x)\\in\\mathbb{R}\\times\\mathbb{R}}\\sup_{0\\leq|\\eta|\\leq\\delta_{0}}|b^{\\eta}(t,x)|<\\infty$, we have\n\\begin{equation*}\n\\sup_{0\\leq|\\eta|\\leq\\delta_{0}}\\bigg|b^{\\eta}(\\tau,x)e^{-\\int_{\\tau}^{t}(1-a^{\\eta}(s,x))ds}\\bigg|\\leq M\\sup_{0\\leq|\\eta|\\leq\\delta_{0}}e^{-\\int_{\\tau}^{t}(1-a^{\\eta}(s,x))ds}.\n\\end{equation*}\nTo bound the last integral uniformly in $0\\leq|\\eta|\\leq\\delta_{0}$, according to \\eqref{lipschitz-estimates} and \\eqref{lipschitz-coefficient-2}, we consider three cases:\n\n\\paragraph{\\textbf{Case i. $\\tau< t_{\\rm first}(x)$}} In this case,\n\\begin{equation*}\n\\begin{split}\n\\sup_{0\\leq|\\eta|\\leq\\delta_{0}}e^{-\\int_{\\tau}^{t}(1-a^{\\eta}(s,x))ds}&=\\sup_{0\\leq|\\eta|\\leq\\delta_{0}}e^{-[\\int_{\\tau}^{t_{\\rm first}(x)}+\\int_{t_{\\rm first}(x)}^{t_{\\rm last}(x)}+\\int_{t_{\\rm last}(x)}^{t}](1-a^{\\eta}(s,x))ds}\\\\\n&\\leq e^{-\\kappa_{0}(t_{\\rm first}(x)-\\tau)}e^{C_{a}T}e^{-(t-t_{\\rm last}(x))};\n\\end{split}\n\\end{equation*}\n\n\\paragraph{\\textbf{Case ii. $\\tau\\in[t_{\\rm first}(x),t_{\\rm last}(x)]$}} In this case,\n\\begin{equation*}\n\\sup_{0\\leq|\\eta|\\leq\\delta_{0}}e^{-\\int_{\\tau}^{t}(1-a^{\\eta}(s,x))ds}=\\sup_{0\\leq|\\eta|\\leq\\delta_{0}}e^{-[\\int_{\\tau}^{t_{\\rm last}(x)}+\\int_{t_{\\rm last}(x)}^{t}](1-a^{\\eta}(s,x))ds}\\leq e^{C_{a}T}e^{-(t-t_{\\rm last}(x))};\n\\end{equation*}\n\n\\paragraph{\\textbf{Case iii. $\\tau\\in(t_{\\rm last}(x),t]$}} In this case, $\\sup_{0\\leq|\\eta|\\leq\\delta_{0}}e^{-\\int_{\\tau}^{t}(1-a^{\\eta}(s,x))ds}\\leq e^{-(t-\\tau)}$.\n\nThus, setting\n\\begin{equation*}\nh(\\tau)=\\begin{cases}\ne^{-\\kappa_{0}(t_{\\rm first}(x)-\\tau)}e^{C_{a}T}e^{-(t-t_{\\rm last}(x))},&\\quad\\tau< t_{\\rm first}(x)\\\\\ne^{C_{a}T}e^{-(t-t_{\\rm last}(x))},&\\quad\\tau\\in[t_{\\rm first}(x),t_{\\rm last}(x)]\\\\\ne^{-(t-\\tau)},&\\quad\\tau\\in(t_{\\rm last}(x),t],\n\\end{cases}\n\\end{equation*}\nwe find for any $\\tau\\in(-\\infty,t]$\n\\begin{equation*}\n\\begin{split}\n&\\sup_{0<|\\eta|\\leq\\delta_{0}}\\bigg|b^{\\eta}(\\tau,x)e^{-\\int_{\\tau}^{t}(1-a^{\\eta}(s,x))ds}-\\bigg(\\int_{\\mathbb{R}}J'(x-y)u(\\tau,y)dy\\bigg)e^{-\\int_{\\tau}^{t}(1-f_{u}(s,x,u(s,x)))ds}\\bigg|\\\\\n&\\quad\\quad\\leq2\\sup_{0\\leq|\\eta|\\leq\\delta_{0}}\\bigg|b^{\\eta}(\\tau,x)e^{-\\int_{\\tau}^{t}(1-a^{\\eta}(s,x))ds}\\bigg|\\leq 2h(\\tau).\n\\end{split}\n\\end{equation*}\nTo show \\eqref{diff-claim-1}, it remains to show $\\int_{-\\infty}^{t}h(\\tau)d\\tau<\\infty$. But, we readily compute\n\\begin{equation}\\label{diff-uniform-bound}\n\\begin{split}\n\\int_{-\\infty}^{t}h(\\tau)d\\tau&=\\int_{-\\infty}^{t_{\\rm first}(x)}h(\\tau)d\\tau+\\int_{t_{\\rm first}(x)}^{t_{\\rm last}(x)}h(\\tau)d\\tau+\\int_{t_{\\rm last}(x)}^{t}h(\\tau)d\\tau\\\\\n&\\leq\\int_{-\\infty}^{t_{\\rm first}(x)}e^{-\\kappa_{0}(t_{\\rm first}(x)-\\tau)}e^{C_{a}T}e^{-(t-t_{\\rm last}(x))}d\\tau\\\\\n&\\quad\\quad+\\int_{t_{\\rm first}(x)}^{t_{\\rm last}(x)}e^{C_{a}T}e^{-(t-t_{\\rm last}(x))}d\\tau+\\int_{t_{\\rm last}(x)}^{t}e^{-(t-\\tau)}d\\tau\\\\\n&\\leq\\frac{e^{C_{a}T}}{\\kappa_{0}}+Te^{C_{a}T}+1.\n\\end{split}\n\\end{equation}\nThus, we have shown \\eqref{diff-claim-1}. Note that the last bound is uniform in $(t,x)\\in\\mathbb{R}\\times\\mathbb{R}$.\n\nFor the third term on the right hand side of \\eqref{diff-integral-sol}, we claim\n\\begin{equation}\\label{diff-claim-3}\n\\begin{split}\n&\\int_{t_{0}}^{t}\\tilde{a}^{\\eta}(\\tau,x)e^{-\\int_{\\tau}^{t}(1-a^{\\eta}(s,x))ds}d\\tau\\to\\int_{t_{0}}^{t}f_{x}(t,x,u(t,x))e^{-\\int_{\\tau}^{t}(1-f_{u}(s,x,u(s,x)))ds}d\\tau\\\\\n&\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad \\text{as}\\,\\, \\eta\\to0\\,\\,\\text{ uniformly in}\\,\\, t_{0}\\ll t_{\\rm first}(x).\n\\end{split}\n\\end{equation}\nThe proof of \\eqref{diff-claim-3} is similar to that of \\eqref{diff-claim-1}. So, we omit it. Notice\n\\begin{equation}\\label{diff-uniform-bound-1}\n\\begin{split}\n&\\int_{-\\infty}^{t}\\bigg|f_{x}(t,x,u(t,x))e^{-\\int_{\\tau}^{t}(1-f_{u}(s,x,u(s,x)))ds}\\bigg|d\\tau\\\\\n&\\quad\\quad\\leq\\bigg[\\sup_{(t,x,u)\\in\\mathbb{R}\\times\\mathbb{R}\\times[0,1]}|f_{x}(t,x,u)|\\bigg]\\int_{-\\infty}^{t}h(\\tau)d\\tau\\\\\n&\\quad\\quad\\leq\\bigg[\\sup_{(t,x,u)\\in\\mathbb{R}\\times\\mathbb{R}\\times[0,1]}|f_{x}(t,x,u)|\\bigg]\\bigg(\\frac{e^{C_{a}T}}{\\kappa_{0}}+Te^{C_{a}T}+1\\bigg).\n\\end{split}\n\\end{equation}\n\nFor the first term on the right hand side of \\eqref{diff-integral-sol}, we have \\eqref{diff-claim-2}, that is,\n\\begin{equation}\\label{diff-claim-4}\n\\text{with}\\quad t_{\\rm first}(x)-t_{0}=\\frac{1}{|\\eta|},\\quad v^{\\eta}(t_{0},x)e^{-\\int_{t_{0}}^{t}(1-a^{\\eta}(s,x))ds}\\to0\\quad\\text{as}\\quad\\eta\\to0.\n\\end{equation}\n\n\nHence, choosing $t_{0}$ such that $t_{\\rm first}(x)-t_{0}=\\frac{1}{|\\eta|}$ and passing to the limit $\\eta\\to0$ in \\eqref{diff-integral-sol}, we conclude from \\eqref{diff-claim-1}, \\eqref{diff-claim-3} and \\eqref{diff-claim-4} that\n\\begin{equation}\\label{a-formula-for-ux}\n\\begin{split}\nu_{x}(t,x)&=\\lim_{\\eta\\to0}v^{\\eta}(t,x)\\\\\n&=\\int_{-\\infty}^{t}\\bigg[\\int_{\\mathbb{R}}J'(x-y)u(\\tau,y)dy+f_{x}(\\tau,x,u(\\tau,x))\\bigg]e^{-\\int_{\\tau}^{t}(1-f_{u}(s,x,u(s,x)))ds}d\\tau.\n\\end{split}\n\\end{equation}\nFrom which, we see that $u_{x}(t,x)$ is continuous in $(t,x)\\in\\mathbb{R}\\times\\mathbb{R}$. Moreover, by \\eqref{diff-uniform-bound}, \\eqref{diff-uniform-bound-1} and \\eqref{a-formula-for-ux}, we have $\\sup_{(t,x)\\in\\mathbb{R}\\times\\mathbb{R}}|u_{x}(t,x)|<\\infty$. This completes the proof.\n\\end{proof}\n\n\n\\begin{rem}\\label{rem-a-formula-for-ux}\nFrom \\eqref{a-formula-for-ux}, \\eqref{diff-uniform-bound} and \\eqref{diff-uniform-bound-1}, we see\n\\begin{equation*}\n\\sup_{(t,x)\\in\\mathbb{R}\\times\\mathbb{R}}|u_{x}(t,x)|\\leq\\bigg[\\|J'\\|_{L^{1}(\\mathbb{R})}+\\sup_{(t,x,u)\\in\\mathbb{R}\\times\\mathbb{R}\\times[0,1]}|f_{x}(t,x,u)|\\bigg]\\bigg(\\frac{e^{C_{a}T}}{\\kappa_{0}}+Te^{C_{a}T}+1\\bigg),\n\\end{equation*}\nwhere $C_{a}$ depends only on $\\sup_{(t,x,u)\\in\\mathbb{R}\\times\\mathbb{R}\\times[0,1]}|f_{u}(t,x,u)|$ and $T$ is controlled by \\eqref{growth-time-1}, and hence, $T$ depends only on $f_{B}$ and the shape of $u(t,x)$.\n\\end{rem}\n\n\n\\section{Proof of Theorem \\ref{thm-regularity-of-tf-ii}}\\label{sec-proof-2}\n\nThis whole section is devoted to the proof of Theorem \\ref{thm-regularity-of-tf-ii}. Let $u(t,x)$ be a transition front as in the statement of Theorem \\ref{thm-regularity-of-tf-ii}. Hence, there exist $C>0$ and $r>0$ such that\n\\begin{equation}\\label{harnack-type-inequality}\nu(t,x)\\leq Ce^{r|x-y|}u(t,y),\\quad (t,x,y)\\in\\mathbb{R}\\times\\mathbb{R}\\times\\mathbb{R}.\n\\end{equation}\n\nLet $X(t)$ and $X^{\\pm}_{\\lambda}(t)$ be interface location functions of $u(t,x)$, where $X^{\\pm}_{\\lambda}(t)$ are given in \\eqref{defn-interface-locations}. As in \\cite{LiZl14}, for $(t,x)\\in\\mathbb{R}\\times\\mathbb{R}$ and $\\eta\\in\\mathbb{R}$ with $0<|\\eta|\\leq\\delta_{0}\\ll1$, we consider\n\\begin{equation*}\nv^{\\eta}(t,x)=\\frac{u(t,x+\\eta)-u(t,x)}{\\eta}\\quad\\text{and}\\quad w^{\\eta}(t,x)=\\frac{v^{\\eta}(t,x)}{u(t,x)}.\n\\end{equation*}\nWe readily check $w^{\\eta}(t,x)$ satisfies\n\\begin{equation}\\label{eqn-9182}\nw^{\\eta}_{t}=a_{1}^{\\eta}+a_{2}^{\\eta}w^{\\eta},\n\\end{equation}\nwhere $a_{1}^{\\eta}=\\frac{J\\ast v^{\\eta}}{u}+\\frac{\\tilde{a}^{\\eta}}{u}$ and $a_{2}^{\\eta}=-\\frac{J\\ast u}{u}+a^{\\eta}-\\frac{f}{u}$ with\n\\begin{equation}\\label{1-term}\n\\tilde{a}^{\\eta}=\\frac{f(t,x+\\eta,u(t,x+\\eta))-f(t,x,u(t,x+\\eta))}{\\eta}\n\\end{equation}\nand\n\\begin{equation}\\label{2-term}\na^{\\eta}=\\frac{f(t,x,u(t,x+\\eta))-f(t,x,u(t,x))}{u(t,x+\\eta)-u(t,x)}.\n\\end{equation}\n\nTo bound the solution of \\eqref{eqn-9182}, we first analyze $a_{1}^{\\eta}$ and $a_{2}^{\\eta}$. For $a^{\\eta}_{1}$, we first see\n\\begin{equation*}\n\\bigg|\\frac{J\\ast v^{\\eta}}{u}\\bigg|=\\frac{1}{u(t,x)}\\bigg|\\int_{\\mathbb{R}}\\frac{J(x-y+\\eta)-J(x-y)}{\\eta}u(t,y)dy\\bigg|\\leq C\\int_{\\mathbb{R}}\\bigg|\\frac{J(x+\\eta)-J(x)}{\\eta}\\bigg|e^{r|x|}dx,\n\\end{equation*}\nwhere we used \\eqref{harnack-type-inequality}. Next, setting $C_{1}:=\\sup_{(t,x,u)\\in\\mathbb{R}\\times\\mathbb{R}\\times[0,1]}|f_{xu}(t,x,u)|$, we have\n\\begin{equation*}\n\\begin{split}\n\\bigg|\\frac{\\tilde{a}^{\\eta}}{u}\\bigg|\\leq\\frac{|f_{x}(t,x+\\eta_{*},u(t,x+\\eta))|}{u(t,x)}\\leq C_{1}\\frac{u(t,x+\\eta)}{u(t,x)}\\leq C_{1}Ce^{r|\\eta|},\n\\end{split}\n\\end{equation*}\nwhere we used Taylor expansion, the fact $f_{x}(t,x,0)=0$ and \\eqref{harnack-type-inequality}. Hence,\n\\begin{equation}\\label{a-1}\nC_{2}:=\\sup_{(t,x)\\in\\mathbb{R}\\times\\mathbb{R}}\\sup_{0<|\\eta|\\leq\\delta_{0}}|a_{1}^{\\eta}|<\\infty.\n\\end{equation}\n\nFor $a_{2}^{\\eta}$, we first see from \\eqref{harnack-type-inequality} that\n\\begin{equation*}\n\\frac{1}{C}\\int_{\\mathbb{R}}J(x)e^{-r|x|}dx\\leq\\frac{J\\ast u}{u}\\leq C\\int_{\\mathbb{R}}J(x)e^{r|x|}dx,\n\\end{equation*}\nand thus, setting $C_{3}:=\\frac{1}{C}\\int_{\\mathbb{R}}J(x)e^{-r|x|}dx$ and $C_{4}:=C\\int_{\\mathbb{R}}J(x)e^{r|x|}dx$, we find\n\\begin{equation*}\n-C_{4}\\leq-\\frac{J\\ast u}{u}\\leq-C_{3}.\n\\end{equation*}\nTo control the term $a^{\\eta}-\\frac{f}{u}$ in $a_{2}^{\\eta}$, we set\n\\begin{equation*}\n\\tilde{\\theta}_{0}:=\\min\\bigg\\{\\frac{\\theta_{1}}{2},\\frac{C_{3}}{2}\\Big[\\sup_{(t,x,u)\\in\\mathbb{R}\\times\\mathbb{R}\\times[0,1]}|f(t,x,u)|\\Big]^{-1}\\bigg\\},\n\\end{equation*}\nand define\n\\begin{equation*}\nL_{0}=1+\\delta_{0}+\\sup_{t\\in\\mathbb{R}}|X(t)-X^{+}_{\\tilde{\\theta}_{0}}(t)|\\quad\\text{ and}\\quad L_{1}=1+\\delta_{0}+\\sup_{t\\in\\mathbb{R}}|X(t)-X^{-}_{\\theta_{1}}(t)|.\n\\end{equation*}\nAs in the proof of Theorem \\ref{thm-regularity-of-tf}, we set\n\\begin{equation*}\n\\begin{split}\nI_{l}(t)&=(-\\infty,X(t)-L_{1}),\\\\\nI_{m}(t)&=[X(t)-L_{1},X(t)+L_{0}],\\\\\n I_{r}(t)&=(X(t)+L_{0},\\infty)\n\\end{split}\n\\end{equation*}\nfor $t\\in\\mathbb{R}$, and for any fixed $x\\in\\mathbb{R}$, define\n\\begin{equation*}\n\\begin{split}\nt_{\\rm first}(x)&=\\sup\\{\\tilde{t}\\in\\mathbb{R}|x\\in I_{r}(t)\\,\\,\\text{for all}\\,\\,t\\leq \\tilde{t}\\},\\\\\nt_{\\rm last}(x)&=\\inf\\{\\tilde{t}\\in\\mathbb{R}|x\\in I_{l}(t)\\,\\,\\text{for all}\\,\\,t\\geq \\tilde{t}\\}.\n\\end{split}\n\\end{equation*}\nThen, there hold $T:=\\sup_{x\\in\\mathbb{R}}[t_{\\rm last}(x)-t_{\\rm first}(x)]<\\infty$ and for all $x\\in\\mathbb{R}$\n\\begin{equation*}\\label{result-from-defn-2}\nx\\in\\begin{cases}\nI_{r}(t),&\\quad t< t_{\\rm first}(x),\\\\\nI_{l}(t),&\\quad t> t_{\\rm last}(x).\n\\end{cases}\n\\end{equation*}\n\nNow, for $0<|\\eta|\\leq\\delta_{0}$, we have\n\\begin{itemize}\n\\item if $t< t_{\\rm first}(x)$, then $x\\in I_{r}(t)$, in particular, $x\\geq X^{+}_{\\tilde{\\theta}_{0}}(t)+\\delta_{0}$, and hence, $u(t,x)\\leq\\tilde{\\theta}_{0}$ and $u(t,x+\\eta)\\leq\\tilde{\\theta}_{0}$; it then follows from Taylor expansion that\n\\begin{equation*}\n\\bigg|a^{\\eta}-\\frac{f}{u}\\bigg|=|f_{u}(t,x,u_{*})-f_{u}(t,x,u_{**})|\\leq|u_{*}-u_{**}|\\sup_{(t,x,u)\\in\\mathbb{R}\\times\\mathbb{R}\\times[0,1]}|f_{uu}(t,x,u)|\\leq\\frac{C_{3}}{2},\n\\end{equation*}\nwhere $u_{*}$ is between $u(t,x)$ and $u(t,x+\\eta)$, and $u_{**}$ is between $0$ and $u(t,x)$, and hence, both $u_{*}$ and $u_{**}$ are between $0$ and $\\tilde{\\theta}_{0}$, so $|u_{*}-u_{**}|\\leq\\tilde{\\theta}_{0}$;\n\n\\item if $t> t_{\\rm last}(x)$, then $x\\in I_{l}(x)$, in particular, $x\\leq X^{-}_{\\theta_{1}}(t)-\\delta$, and hence, $u(t,x)\\geq\\theta_{1}$ and $u(t,x+\\eta)\\geq\\theta_{1}$; it then follows from (H2) that $a^{\\eta}\\leq0$, which leads to $a^{\\eta}-\\frac{f}{u}\\leq0$;\n\n\\item if $t\\in[t_{\\rm first}(x),t_{\\rm last}(x)]$, then $|a^{\\eta}-\\frac{f}{u}|\\leq2\\sup_{(t,x,u)\\in\\mathbb{R}\\times\\mathbb{R}\\times[0,1]}|f_{u}(t,x,u)|$.\n\\end{itemize}\nTherefore, we have the following for $a_{2}^{\\eta}$: for $0<|\\eta|\\leq\\delta_{0}$ and $x\\in\\mathbb{R}$\n\\begin{equation}\\label{a-2}\na_{2}^{\\eta}\\leq\\begin{cases}\n-\\frac{C_{3}}{2},\\quad& t\\leq t_{\\rm first}(x),\\\\\nC_{5},\\quad & t_{\\rm first}(x)\\leq t\\leq t_{\\rm last}(x),\\\\\n-C_{3},\\quad & t\\geq t_{\\rm last}(x),\n\\end{cases}\n\\end{equation}\nwhere $C_{5}:=C_{4}+2\\sup_{(t,x,u)\\in\\mathbb{R}\\times\\mathbb{R}\\times[0,1]}|f_{u}(t,x,u)|$.\n\nWith the help of \\eqref{a-1} and \\eqref{a-2}, we are now able to bound the solution of \\eqref{eqn-9182}. Notice, the solution of \\eqref{eqn-9182} can be written as\n\\begin{equation}\\label{eqn-integral-9812}\nw^{\\eta}(t,x)=e^{\\int_{t_{0}}^{t}a_{2}^{\\eta}(s,x)ds}w^{\\eta}(t_{0},x)+\\int_{t_{0}}^{t}e^{\\int_{\\tau}^{t}a_{2}^{\\eta}(s,x)ds}a_{1}^{\\eta}(\\tau,x)d\\tau,\\quad x\\in\\mathbb{R},\\quad t\\geq t_{0}.\n\\end{equation}\n\nUsing \\eqref{a-1}, \\eqref{a-2} and \\eqref{eqn-integral-9812}, we first argue as in the proof of Lemma \\ref{lem-tf-continuity} to conclude that\n\\begin{equation*}\n\\sup_{(t,x)\\in\\mathbb{R}\\times\\mathbb{R}}\\sup_{0<|\\eta|\\leq\\delta_{0}}\\bigg|\\frac{u(t,x+\\eta)-u(t,x)}{\\eta u(t,x)}\\bigg|=\\sup_{(t,x)\\in\\mathbb{R}\\times\\mathbb{R}}\\sup_{0<|\\eta|\\leq\\delta_{0}}|w^{\\eta}(t,x)|<\\infty,\n\\end{equation*}\nwhich in particular implies that $u(t,x)$ is continuous in $x$, since $u(t,x)\\in(0,1)$.\n\n\nNow, we set $a^{0}_{1}=\\frac{J'\\ast u}{u}+f_{x}$ and $a_{2}^{0}=-\\frac{J\\ast u}{u}+f_{u}-\\frac{f}{u}$. Using the continuity of $u(t,x)$ in $x$, we have the pointwise limits $a^{\\eta}_{1}\\to a^{0}_{1}$\nand $a^{\\eta}_{2}\\to a^{0}_{2}$ as $\\eta\\to0$. Then, using \\eqref{a-1} and \\eqref{a-2}, we can show that as \\eqref{diff-claim-1},\n\\begin{equation*}\n\\int_{t_{0}}^{t}e^{\\int_{\\tau}^{t}a_{2}^{\\eta}(s,x)ds}a_{1}^{\\eta}(\\tau,x)d\\tau\\to\\int_{t_{0}}^{t}e^{\\int_{\\tau}^{t}a_{2}^{0}(s,x)ds}a_{1}^{0}(\\tau,x)d\\tau\n\\end{equation*}\nuniformly in $t_{0}\\ll t_{\\rm first}(x)$ as $\\eta\\to0$, and as \\eqref{diff-claim-2},\n\\begin{equation*}\n\\text{with}\\,\\,t_{\\rm first}(x)-t_{0}=\\frac{1}{|\\eta|},\\quad e^{\\int_{t_{0}}^{t}a_{2}^{\\eta}(s,x)ds}w^{\\eta}(t_{0},x)\\to0\\quad\\text{as}\\quad\\eta\\to0.\n\\end{equation*}\nHence, setting $t_{\\rm first}(x)-t_{0}=\\frac{1}{|\\eta|}$ in \\eqref{eqn-integral-9812} and passing to the limit $\\eta\\to0$, we find\n\\begin{equation*}\n\\frac{u_{x}(t,x)}{u(t,x)}=\\lim_{\\eta\\to0}w^{\\eta}(t,x)=\\int_{-\\infty}^{t}e^{\\int_{\\tau}^{t}a_{2}^{0}(s,x)ds}a_{1}^{0}(\\tau,x)d\\tau,\\quad (t,x)\\in\\mathbb{R}\\times\\mathbb{R}.\n\\end{equation*}\nThis completes the proof.\n\n\n\n\\section{Proof of Theorem \\ref{thm-regularity-of-tf-iii}}\\label{sec-proof-iafjiaf}\n\nWe prove Theorem \\ref{thm-regularity-of-tf-iii} in this section. Throughout this section, we assume (H1) and (H2). To prove Theorem \\ref{thm-regularity-of-tf-iii}, we need the following three lemmas. For $r>0$, let\n\\begin{equation*}\n\\Gamma_{r}(x)=\\min\\{1,e^{-rx}\\}=\\begin{cases}\n1,\\quad&x\\leq0,\\\\\ne^{-rx},\\quad&x\\geq0.\n\\end{cases}\n\\end{equation*}\n\n\\begin{lem}\\label{lem-090-1}\nThere exist two continuous functions $M:(0,\\infty)\\to(0,\\infty)$ and $\\gamma:(0,\\infty)\\to(0,\\infty)$ with $\\gamma(r)\\to0$ such that\n\\begin{equation*}\n\\bigg|\\frac{[J\\ast\\Gamma_{r}](x)}{\\Gamma_{r}(x)}-1\\bigg|\\leq\\gamma(r),\\quad x\\geq M(r)\n\\end{equation*}\nfor all $r>0$.\n\\end{lem}\n\\begin{proof}\nFix $r>0$ and write $\\Gamma=\\Gamma_{r}$. We see\n\\begin{equation*}\n\\begin{split}\n\\frac{[J\\ast\\Gamma](x)}{\\Gamma(x)}-1&=\\int_{-\\infty}^{0}J(x-y)\\frac{\\Gamma(y)}{\\Gamma(x)}dy+\\int_{0}^{\\infty}J(x-y)\\frac{\\Gamma(y)}{\\Gamma(x)}dy-1\\\\\n&=\\int_{-\\infty}^{0}J(x-y)\\frac{\\Gamma(y)}{\\Gamma(x)}dy+\\int_{0}^{\\infty}J(x-y)e^{r(x-y)}dy-1\\\\\n&=\\int_{-\\infty}^{0}J(x-y)\\frac{\\Gamma(y)}{\\Gamma(x)}dy+\\bigg[\\int_{\\mathbb{R}}J(y)e^{ry}dy-1\\bigg]-\\int_{x}^{\\infty}J(y)e^{ry}dy.\n\\end{split}\n\\end{equation*}\nDue to the decay of $J$ at $\\pm\\infty$, it is not hard to see that $\\lim_{x\\to\\infty}\\int_{-\\infty}^{0}J(x-y)\\frac{\\Gamma(y)}{\\Gamma(x)}dy=0$. In fact, for all large $x$,\n\\begin{equation*}\n\\int_{-\\infty}^{0}J(x-y)\\frac{\\Gamma(y)}{\\Gamma(x)}dy=\\int_{-\\infty}^{0}J(x-y)e^{rx}dy\\leq\\int_{-\\infty}^{0}J(x-y)e^{r(x-y)}dy\\to0\\,\\,\\text{as}\\,\\,x\\to\\infty.\n\\end{equation*}\nSince $\\int_{\\mathbb{R}}J(y)dy=1$, we find $\\lim_{r\\to0}\\int_{\\mathbb{R}}J(y)e^{ry}dy=1$ by dominated convergence theorem. Clearly, $\\lim_{x\\to\\infty}\\int_{x}^{\\infty}J(y)e^{ry}dy=0$. The lemma then follows.\n\\end{proof}\n\n\\begin{lem}\\label{lem-090-2}\nThere exists $r_{0}>0$ such that if $r\\in(0,r_{0}]$ is such that\n\\begin{equation}\\label{exact-decay-condition}\n\\lim_{x\\to\\infty}\\frac{u(t,x+X(t))}{e^{-rx}}=1\\quad\\text{uniformly in}\\quad t\\in\\mathbb{R},\n\\end{equation}\nthen, there exists $M(r)>0$ such that\n\\begin{equation}\\label{result-1}\n\\frac{1}{2}\\leq\\frac{[J\\ast u(t,\\cdot+X(t))](x)}{u(t,x+X(t))}\\leq\\frac{3}{2},\\quad x\\geq M(r)\\quad \\text{and}\\quad t\\in\\mathbb{R},\n\\end{equation}\nand\n\\begin{equation}\\label{result-2}\n\\frac{|[J'\\ast u(t,\\cdot+X(t))](x)|}{u(t,x+X(t))}\\leq2,\\quad x\\geq M(r)\\quad \\text{and}\\quad t\\in\\mathbb{R}.\n\\end{equation}\n\\end{lem}\n\\begin{proof}\nLet $u=u(t,x+X(t))$. For $r\\in(0,r_{0}]$, where $r_{0}$ is to be chosen, we let $\\Gamma=\\Gamma_{r}$. Write\n\\begin{equation*}\n\\begin{split}\n\\frac{J\\ast u}{u}-1&=\\frac{J\\ast\\Gamma}{\\Gamma}\\bigg[\\frac{J\\ast(u-\\Gamma)}{J\\ast\\Gamma}+1\\bigg]\\frac{\\Gamma}{u}-1\\\\\n&=\\bigg[\\frac{J\\ast\\Gamma}{\\Gamma}-1\\bigg]\\bigg[\\frac{J\\ast(u-\\Gamma)}{J\\ast\\Gamma}+1\\bigg]\\frac{\\Gamma}{u}+\\frac{J\\ast(u-\\Gamma)}{J\\ast\\Gamma}\\frac{\\Gamma}{u}+\\bigg[\\frac{\\Gamma}{u}-1\\bigg].\n\\end{split}\n\\end{equation*}\nBy Lemma \\ref{lem-090-1} and \\eqref{exact-decay-condition}, we only need to treat the term $\\frac{J\\ast(u-\\Gamma)}{J\\ast\\Gamma}$ for large $x$.\n\nBy \\eqref{exact-decay-condition}, for any $\\epsilon>0$, there exists $M(\\epsilon,r)>0$ such that\n\\begin{equation*}\n|u(t,x+X(t))-e^{-rx}|\\leq\\epsilon e^{-rx},\\quad x\\geq M(\\epsilon,r)\\quad \\text{and}\\quad t\\in\\mathbb{R}.\n\\end{equation*}\nThen, for $\\epsilon>0$, we have\n\\begin{equation*}\n\\begin{split}\n\\bigg|\\frac{J\\ast(u-\\Gamma)}{J\\ast\\Gamma}\\bigg|&=\\frac{\\Gamma}{J\\ast\\Gamma}\\bigg|\\frac{J\\ast(u-\\Gamma)}{\\Gamma}\\bigg|\\\\\n&\\leq\\frac{\\Gamma}{J\\ast\\Gamma}\\bigg[\\int_{-\\infty}^{M(\\epsilon,r)}J(x-y)\\frac{|u(t,y+X(t))-\\Gamma(y)|}{\\Gamma(x)}dy\\\\\n&\\quad\\quad\\quad\\quad\\quad+\\int_{M(\\epsilon,r)}^{\\infty}J(x-y)\\frac{|u(t,y+X(t))-e^{-ry}|}{e^{-rx}}dy\\bigg]\\\\\n&\\leq\\frac{\\Gamma}{J\\ast\\Gamma}\\bigg[\\int_{-\\infty}^{M(\\epsilon,r)}J(x-y)\\frac{|u(t,y+X(t))-\\Gamma(y)|}{\\Gamma(x)}dy+\\epsilon\\int_{M(\\epsilon,r)}^{\\infty}J(x-y)e^{r(x-y)}dy\\bigg]\\\\\n&\\leq\\frac{\\Gamma}{J\\ast\\Gamma}\\bigg[\\int_{-\\infty}^{M(\\epsilon,r)}J(x-y)\\frac{|u(t,y+X(t))-\\Gamma(y)|}{\\Gamma(x)}dy+\\epsilon\\int_{\\mathbb{R}}J(y)e^{ry}dy\\bigg]\n\\end{split}\n\\end{equation*}\nDue to the decay of $J$ at $\\pm\\infty$, we have\n\\begin{equation*}\n\\lim_{x\\to\\infty}\\int_{-\\infty}^{M(\\epsilon,r)}J(x-y)\\frac{|u(t,y+X(t))-\\Gamma(y)|}{\\Gamma(x)}dy=0\\quad\\text{uniformly in}\\quad t\\in\\mathbb{R}.\n\\end{equation*}\nIt then follows from Lemma \\ref{lem-090-1} that for any $\\epsilon>0$, there exists $\\tilde{M}(\\epsilon,r)>0$ such that\n\\begin{equation*}\n\\bigg|\\frac{J\\ast(u-\\Gamma)}{J\\ast\\Gamma}\\bigg|\\leq\\tilde{\\gamma}(r)\\bigg[\\frac{\\epsilon}{2}+\\epsilon\\int_{\\mathbb{R}}J(y)e^{ry}dy\\bigg],\\quad x\\geq\\tilde{M}(\\epsilon,r)\\quad\\text{and}\\quad t\\in\\mathbb{R},\n\\end{equation*}\nwhere $\\tilde{\\gamma}(r)\\to0$ as $r\\to0$. The result \\eqref{result-1} follows by first fixing $\\epsilon$, say $\\epsilon=\\frac{1}{100}$, and then choosing $r_{0}$ small so that if $r\\in(0,r_{0}]$, then $|\\frac{J\\ast u}{u}-1|\\leq\\frac{1}{2}$ for all large $x$ depending on $r$. Note that the term $\\frac{J\\ast\\Gamma}{\\Gamma}-1$ and Lemma \\ref{lem-090-1} restrict $r_{0}$. Similar arguments lead to \\eqref{result-2}, since\n\\begin{equation*}\n\\bigg|\\frac{J'\\ast u}{u}\\bigg|\\leq\\frac{|J'|\\ast\\Gamma}{\\Gamma}\\bigg[\\frac{|J'|\\ast(u-\\Gamma)}{|J'|\\ast\\Gamma}+1\\bigg]\\frac{\\Gamma}{u}.\n\\end{equation*}\nThis completes the proof.\n\\end{proof}\n\n\\begin{lem}\\label{lem-away-from-0}\nLet $u(t,x)$ be an arbitrary transition front of \\eqref{main-eqn} with interface location function $X(t)$. Then, there holds\n\\begin{equation*}\n\\forall L>0,\\quad \\inf_{t\\in\\mathbb{R}}\\inf_{x\\leq L+X(t)}u(t,x)>0.\n\\end{equation*}\n\\end{lem}\n\\begin{proof}\nFix $L>0$ and $x_{*}>0$. Let $\\lambda_{1}\\in(\\frac{1}{2},1)$ and $\\lambda_{2}\\in(0,\\frac{1}{10})$. We define $u_{0}^{B}:\\mathbb{R}\\to[0,1]$ and $u_{0}^{M}:\\mathbb{R}\\to[0,1]$ by setting\n\\begin{equation*}\nu_{0}^{B}(x)=\\begin{cases}\n\\lambda_{1},\\quad &x\\leq-x_{*},\\\\\n-\\frac{\\lambda_{1}}{x_{*}}x,\\quad& x\\in[-x_{*},0],\\\\\n0,\\quad &x\\geq0\n\\end{cases}\n\\quad\\text{and}\\quad\nu_{0}^{M}(x)=\\begin{cases}\n1,\\quad &x\\leq0,\\\\\n\\frac{\\lambda_{2}-1}{x_{*}}x+1,\\quad&x\\in[0,x_{*}],\\\\\n\\lambda_{2},\\quad &x\\geq x_{*}.\n\\end{cases}\n\\end{equation*}\nClearly, $u_{0}^{B}(\\cdot-X^{-}_{\\lambda_{1}}(t))\\leq u(t,\\cdot)\\leq u_{0}^{M}(\\cdot-X^{+}_{\\lambda_{2}}(t))$ for all $t\\in\\mathbb{R}$. Now, denote by $u_{B}(t,x;u_{0}^{B})$ and $u_{M}(t,x;u_{0}^{M})$ the solutions of $u_{t}=J\\ast u-u+f_{B}(u)$ and $u_{t}=J\\ast u-u+f_{M}(u)$, respectively, with initial data $u_{B}(0,\\cdot;u_{0}^{B})=u_{0}^{B}$ and $u_{M}(0,\\cdot;u_{0}^{M})=u_{0}^{M}$. It then follows from comparison principle and homogeneity that\n\\begin{equation}\\label{some-result-8472828373}\nu_{B}(T,x-X^{-}_{\\lambda_{1}}(t-T);u_{0}^{B})\\leq u(t,x)\\leq u_{M}(T,x-X^{+}_{\\lambda_{2}}(t-T);u_{0}^{M}),\\quad (t,x)\\in\\mathbb{R}\\times\\mathbb{R}\n\\end{equation}\nfor all $T\\geq0$. Now, we consider a small $T$ and let $\\lambda>0$ be small. Let $\\xi_{\\lambda}^{B}(T)$ be such that $u_{B}(T,\\xi_{\\lambda}^{B}(T);u_{0}^{B})=\\lambda$. We then see from the first inequality in \\eqref{some-result-8472828373} that if $x\\leq\\xi^{B}_{\\lambda}(T)+X^{-}_{\\lambda_{1}}(t-T)-1$, then the monotonicity of $u_{B}(t,x;u_{0}^{B})$ in $x$ yields\n\\begin{equation*}\nu(t,x)\\geq u_{B}(T,\\xi_{\\lambda}^{B}(T)-1;u_{0}^{B})>u_{B}(T,\\xi_{\\lambda}^{B}(T);u_{0}^{B})=\\lambda,\n\\end{equation*}\nwhich then leads to\n\\begin{equation*}\nX^{-}_{\\lambda}(t)\\geq\\xi^{B}_{\\lambda}(T)+X^{-}_{\\lambda_{1}}(t-T)-1,\\quad t\\in\\mathbb{R}.\n\\end{equation*}\nNote that if we can find some $C>0$ such that\n\\begin{equation}\\label{a-key-step-9828318317}\nX^{-}_{\\lambda_{1}}(t-T)\\geq X(t)-C,\\quad t\\in\\mathbb{R},\n\\end{equation}\nthen we can make $\\lambda$ closer to $0$ such that $\\xi^{B}_{\\lambda}(T)$ is so large that $X_{\\lambda}^{-}(t)\\geq L+X(t)+1$ for all $t\\in\\mathbb{R}$, which then leads to\n\\begin{equation*}\n\\inf_{t\\in\\mathbb{R}}\\inf_{x\\leq L+X(t)}u(t,x)\\geq \\inf_{t\\in\\mathbb{R}}\\inf_{x\\leq X_{\\lambda}^{-}(t)-1}u(t,x)>\\lambda>0.\n\\end{equation*}\nHence, to finish the proof, we only need to show \\eqref{a-key-step-9828318317}.\n\nWe now show \\eqref{a-key-step-9828318317}. Let us look at the interface locations for $u(t,x)$ and $u_{M}(t,x;u_{0}^{M})$ at $\\frac{1}{2}$. From the second inequality in \\eqref{some-result-8472828373}, we see\n\\begin{equation}\\label{some-inequalities-198314717419741}\n X^{+}_{\\frac{1}{2}}(t)\\leq \\xi^{M}_{\\frac{1}{2}}(T)+X^{+}_{\\lambda_{2}}(t-T)+1,\\quad t\\in\\mathbb{R},\n\\end{equation}\nwhere $\\xi^{M}_{\\frac{1}{2}}(T)$ is such that $u(T,\\xi^{M}_{\\frac{1}{2}}(T);u_{0}^{M})=\\frac{1}{2}$. Notice choosing $T$ or $\\lambda_{2}$ smaller, we can guarantee that $\\xi^{M}_{\\frac{1}{2}}(T)$ is well-defined. We then deduce from \\eqref{some-inequalities-198314717419741} that\n\\begin{equation*}\n\\begin{split}\nX_{\\lambda_{1}}^{-}(t-T)&\\geq X_{\\lambda_{2}}^{+}(t-T)-C_{1}\\\\\n&\\geq X^{+}_{\\frac{1}{2}}(t)-\\xi_{\\frac{1}{2}}^{M}(T)-1-C_{1}\\\\\n&\\geq X(t)-C_{2}-\\xi_{\\frac{1}{2}}^{M}(T)-1-C_{1}\n\\end{split}\n\\end{equation*}\nfor all $t\\in\\mathbb{R}$, where $C_{1}=\\sup_{t\\in\\mathbb{R}}|X_{\\lambda_{1}}^{-}(t)-X_{\\lambda_{2}}^{+}(t)|$ and $C_{2}=\\sup_{t\\in\\mathbb{R}}|X_{\\frac{1}{2}}^{+}(t)-X(t)|$. Setting $C=C_{2}+\\xi_{\\frac{1}{2}}^{M}(T)+1+C_{1}$, we find \\eqref{a-key-step-9828318317}, and hence, the lemma follows.\n\\end{proof}\n\n\nWe are ready to prove Theorem \\ref{thm-regularity-of-tf-iii}.\n\n\\begin{proof}[Proof of Theorem \\ref{thm-regularity-of-tf-iii}]\nLet $r_{0}>0$ be as in Lemma \\ref{lem-090-2} and fix $r\\in(0,r_{0}]$. Let $u(t,x)$ be an arbitrary transition front of \\eqref{main-eqn} satisfying\n\\begin{equation}\\label{exact-decay-condition-134141}\n\\lim_{x\\to\\infty}\\frac{u(t,x+X(t))}{e^{-rx}}=1\\quad\\text{uniformly in}\\quad t\\in\\mathbb{R}.\n\\end{equation}\n\nTo prove the theorem, we proceed as in the proof of Theorem \\ref{thm-regularity-of-tf-ii}. Thus, we only need to bound\n$a_{1}^{\\eta}=\\frac{J\\ast v^{\\eta}}{u}+\\frac{\\tilde{a}^{\\eta}}{u}$\nas in \\eqref{a-1} and estimate $a_{2}^{\\eta}=-\\frac{J\\ast u}{u}+a^{\\eta}-\\frac{f}{u}$ as in \\eqref{a-2}, where $\\tilde{a}^{\\eta}$ and $a^{\\eta}$ are given in \\eqref{1-term} and \\eqref{2-term}, respectively.\n\nFor $a_{1}^{\\eta}$, we have\n\\begin{equation*}\n|a_{1}^{\\eta}|\\leq\\bigg|\\frac{J\\ast v^{\\eta}}{u}\\bigg|+\\bigg|\\frac{\\tilde{a}^{\\eta}}{u}\\bigg|\\leq\\frac{1}{u(t,x)}\\int_{\\mathbb{R}}\\frac{|J(x-y+\\eta)-J(x-y)|}{\\eta}u(t,y)dy+C_{1}\\frac{u(t,x+\\eta)}{u(t,x)},\n\\end{equation*}\nwhere $C_{1}=\\sup_{(t,x,u)\\in\\mathbb{R}\\times\\mathbb{R}\\times[0,1]}|f_{xu}(t,x,u)|$. For a sufficiently large $M_{1}>0$, we see from \\eqref{result-2} and \\eqref{exact-decay-condition-134141} that\n\\begin{equation*}\n\\sup_{t\\in\\mathbb{R}}\\sup_{x\\geq M_{1}+X(t)}\\sup_{0<|\\eta|\\leq\\delta_{0}}|a_{1}^{\\eta}|<\\infty.\n\\end{equation*}\nSince $\\inf_{t\\in\\mathbb{R}}\\inf_{x\\leq M_{1}+X(t)}u(t,x)>0$ by Lemma \\ref{lem-away-from-0}, we have\n\\begin{equation*}\n\\sup_{t\\in\\mathbb{R}}\\sup_{x\\leq M_{1}+X(t)}\\sup_{0<|\\eta|\\leq\\delta_{0}}|a_{1}^{\\eta}|<\\infty.\n\\end{equation*}\nHence, we obtain\n\\begin{equation*}\n\\sup_{(t,x)\\in\\mathbb{R}\\times\\mathbb{R}}\\sup_{0<|\\eta|\\leq\\delta_{0}}|a_{1}^{\\eta}|<\\infty.\n\\end{equation*}\n\nFor $a^{\\eta}_{2}$, we first see from \\eqref{result-1} that we can find a sufficiently large $M_{2}>0$ such that\n\\begin{equation*}\n\\frac{1}{2}\\leq\\frac{J\\ast u}{u}\\leq\\frac{3}{2},\\quad x\\geq M_{2}+X(t)\\quad\\text{and}\\quad t\\in\\mathbb{R}.\n\\end{equation*}\nSince $\\inf_{t\\in\\mathbb{R}}\\inf_{x\\leq M_{2}+X(t)}u(t,x)>0$ by Lemma \\ref{lem-away-from-0}, there exist $C_{1}>0$ and $C_{2}>0$ such that\n\\begin{equation*}\nC_{1}\\leq\\frac{J\\ast u}{u}\\leq C_{2},\\quad x\\leq M_{2}+X(t)\\quad\\text{and}\\quad t\\in\\mathbb{R}.\n\\end{equation*}\nThen, setting $C_{3}=\\min\\{\\frac{1}{2},C_{1}\\}$ and $C_{4}=\\max\\{\\frac{3}{2},C_{2}\\}$, we have\n\\begin{equation*}\n-C_{4}\\leq-\\frac{J\\ast u}{u}\\leq-C_{3},\\quad(t,x)\\in\\mathbb{R}\\times\\mathbb{R}.\n\\end{equation*}\nWe then follow the arguments as in the proof of Theorem \\ref{thm-regularity-of-tf-ii} to conclude an estimate for $a_{2}^{\\eta}$ as in \\eqref{a-2}.\n\nThe rest of the proof can be done along the same line as in the proof of Theorem \\ref{thm-regularity-of-tf-ii} and then we complete the proof.\n\\end{proof}\n\n\n\\section{Proof of Theorem \\ref{thm-modified-interface-location} and Corollaries \\ref{cor-modified-interface}-\\ref{cor-modified-interface-ii}}\\label{sec-proof-thm-cor}\n\nIn this section, we prove Theorem \\ref{thm-modified-interface-location},  Corollary \\ref{cor-modified-interface} and Corollary \\ref{cor-modified-interface-ii}. We first prove Theorem \\ref{thm-modified-interface-location}.\n\n\\begin{proof}[Proof of Theorem \\ref{thm-modified-interface-location}]\nLet $u(t,x)$ be an arbitrary transition front of \\eqref{main-eqn} with interface location function $X(t)$ as in the statement of Theorem \\ref{thm-modified-interface-location}. Then, by Theorem \\ref{lem-propagation-estimate} and the assumption, we have\n\\begin{equation}\\label{lower-upper-estimate}\nc_{1}(t-t_{0}-T_{1})\\leq X(t)-X(t_{0})\\leq c_{2}(t-t_{0}+T_{2}),\\quad t\\geq t_{0}.\n\\end{equation}\nWe modify $X(t)$ within two steps by means of \\eqref{lower-upper-estimate}. The first step gives a continuous modification. The second step gives the continuously differentiable modification as in the statement of the theorem. We remark that two inequalities in \\eqref{lower-upper-estimate} play different roles in the following arguments. While the first inequality in \\eqref{lower-upper-estimate} pushes $X(t)$ move to the right, the second inequality in \\eqref{lower-upper-estimate} controls the possible jumps of $X(t)$.\n\n\\paragraph{\\bf{Step 1.}} We show there is a continuous function $\\tilde{X}:\\mathbb{R}\\to\\mathbb{R}$ such that $\\sup_{t\\in\\mathbb{R}}|\\tilde{X}(t)-X(t)|<\\infty$. Fix some $T>0$. At $t=0$, let\n\\begin{equation*}\nZ^{+}(t;0)=X(0)+c_{2}(T+T_{2})+\\frac{c_{1}}{2}t,\\quad t\\geq0\n\\end{equation*}\nBy the second inequality in \\eqref{lower-upper-estimate}, $X(t)<Z^{+}(t;0)$ for all $[0,T]$. By the first inequality in \\eqref{lower-upper-estimate}, we have $X(t)>Z^{+}(t;0)$ for all large $t$. Define $T_{1}^{+}=\\inf\\{t\\geq0|X(t)\\geq Z^{+}(t;0)\\}$. By \\eqref{lower-upper-estimate}, it is easy to see that $T_{1}^{+}\\in[T,\\frac{c_{2}(T+T_{2})+c_{1}T_{1}}{c_{1}\/2}]$. At the moment $T_{1}^{+}$, $X(t)$ may jump, but, due to the second inequality in \\eqref{lower-upper-estimate}, the jump is at most $c_{2}T_{2}$. Thus, we obtain\n\\begin{equation*}\n\\begin{split}\nX(t)&<Z^{+}(t;0)\\,\\,\\text{for}\\,\\,t\\in[0,T_{1}^{+}),\\\\\nX(T_{1}^{+})&\\in[Z^{+}(T_{1}^{+};0)-c_{2}T_{2},Z^{+}(T_{1}^{+};0)+c_{2}T_{2}).\n\\end{split}\n\\end{equation*}\n\n\nNext, at $t=T_{1}^{+}$, let\n\\begin{equation*}\nZ^{+}(t;T_{1}^{+})=X(T_{1}^{+})+c_{2}(T+T_{2})+\\frac{c_{1}}{2}(t-T_{1}^{+}),\\quad t\\geq T_{1}^{+}.\n\\end{equation*}\nThen, $T_{2}^{+}=\\inf\\{t\\geq T_{1}^{+}|X(t)\\geq Z^{+}(t;T_{1}^{+})\\}$ is well-defined, and $T_{2}^{+}-T_{1}^{+}\\in[T,\\frac{c_{2}(T+T_{2})+c_{1}T_{1}}{c_{1}\/2}]$. Moreover, there hold\n\\begin{equation*}\n\\begin{split}\nX(t)&<Z^{+}(t;T_{1}^{+})\\,\\,\\text{for}\\,\\,t\\in[T_{1}^{+},T_{2}^{+}),\\\\ X(T_{2}^{+})&\\in[Z^{+}(T_{2}^{+};T_{1}^{+})-c_{2}T_{2},Z^{+}(T_{2}^{+};T_{1}^{+})+c_{2}T_{2}).\n\\end{split}\n\\end{equation*}\n\nRepeating the above arguments, we obtain the following, there is a sequence of times $\\{T_{n-1}^{+}\\}_{n\\geq1}$ satisfying $T_{0}^{+}=0$, $T_{n}^{+}-T_{n-1}^{+}\\in[T,\\frac{c_{2}(T+T_{2})+c_{1}T_{1}}{c_{1}\/2}]$  and\n\\begin{equation}\\label{an-estimate-0198932r592}\n\\begin{split}\nX(t)&<Z^{+}(t;T_{n-1}^{+})\\,\\,\\text{for}\\,\\,t\\in[T_{n-1}^{+},T_{n}^{+}),\\\\\nX(T_{n}^{+})&\\in[Z^{+}(T_{n}^{+};T_{n-1}^{+}),Z^{+}(T_{n}^{+};T_{n-1}^{+})+c_{2}T_{2}),\n\\end{split}\n\\end{equation}\nfor all $n\\geq1$, where $Z^{+}(t;T_{n-1}^{+})=X(T_{n-1}^{+})+c_{2}(T+T_{2})+\\frac{c_{1}}{2}(t-T_{n-1}^{+})$.\n\nWe define $Z^{+}:[0,\\infty)\\to\\mathbb{R}$ by setting\n\\begin{equation*}\nZ^{+}(t)=Z^{+}(t;T_{n-1}^{+}),\\quad t\\in[T^{+}_{n-1},T^{+}_{n}),\\quad n\\geq1\n\\end{equation*}\nSince $\\sup_{n\\geq1}[T^{+}_{n-1},T^{+}_{n})=[0,\\infty)$, $Z^{+}(t)$ is well-defined. It follows from \\eqref{an-estimate-0198932r592} that $X(t)<Z^{+}(t)$ for all $t\\geq0$. Moreover, for $t\\in[T^{+}_{n-1},T^{+}_{n})$,\n\\begin{equation*}\n\\begin{split}\nZ^{+}(t)-X(t)&\\leq X(T_{n-1}^{+})+c_{2}(T+T_{2})+\\frac{c_{1}}{2}(t-T_{n-1}^{+})-[X(T_{n-1}^{+})+c_{1}(t-T_{n-1}^{+}-T_{1})]\\\\\n&\\leq c_{2}(T+T_{2})-\\frac{c_{1}}{2}(t-T_{n-1}^{+})+c_{1}T_{1}\\leq c_{2}(T+T_{2})+c_{1}T_{1}.\n\\end{split}\n\\end{equation*}\nHence, $0<Z^{+}(t)-X(t)\\leq c_{2}(T+T_{2})+c_{1}T_{1}$ for all $t\\in[0,\\infty)$. Modifying $Z^{+}(t)$ near $T_{n-1}^{+}$ for $n\\geq1$, we find a continuous function $\\tilde{Z}^{+}:[0,\\infty)\\to\\mathbb{R}$ such that $\\sup_{t\\in[0,\\infty)}|\\tilde{Z}^{+}(t)-X(t)|<\\infty$.\n\nClearly, we can mimic the above arguments to find a continuous function $\\tilde{Z}^{-}:(-\\infty,0]\\to\\mathbb{R}$ such that $\\sup_{t\\in(-\\infty,0]}|\\tilde{Z}^{-}(t)-X(t)|<\\infty$. Combining $\\tilde{Z}^{\\pm}(t)$ and modifying near $0$, we find a continuous function $\\tilde{X}:\\mathbb{R}\\to\\mathbb{R}$ such that $\\sup_{t\\in\\mathbb{R}}|\\tilde{X}(t)-X(t)|<\\infty$.\n\n\\paragraph{\\bf{Step 2.}} By Step 1, we assume, without loss of generality, that $X(t)$ is continuous. We proceed as in Step 1.\n\nFix any $t_{0}\\in\\mathbb{R}$ and consider it as an initial moment. At the initial moment $t_{0}$, we define $Z(t;t_{0})=X(t_{0})+C_{0}+\\frac{c_{1}}{2}(t-t_{0})$ for $ t\\geq t_{0}$, where $C_{0}>0$ is so large that $C_{0}>c_{2}T_{2}$. Clearly, $X(t_{0})<Z(t_{0};t_{0})$. By the first inequality in \\eqref{lower-upper-estimate} and continuity, $X(t)$ will hit $Z(t;t_{0})$ sometime after $t_{0}$. Let $T_{1}(t_{0})$ be the first time that $X(t)$ hits $Z(t;t_{0})$, that is, $T_{1}(t_{0})=\\min\\big\\{t\\geq t_{0}\\big|X(t)=Z(t;t_{0})\\big\\}$. It follows that $X(t)<Z(t;t_{0})$ for $t\\in[t_{0},T_{1}(t_{0}))$ and $X(T_1(t_{0}))=Z(T_{1}(t_{0});t_{0})$. Moreover, $T_{1}(t_{0})-t_{0}\\in\\Big[\\frac{C_{0}-c_{2}T_{2}}{c_{2}-c_{1}\/2},\\frac{C_{0}+c_{1}T_{1}}{c_{1}\/2}\\Big]$, which is a simple result of \\eqref{lower-upper-estimate}.\n\nNow, at the moment $T_{1}(t_{0})$, we define $Z(t;T_{1}(t_{0}))=X(T_{1}(t_{0}))+C_{0}+\\frac{c_{1}}{2}(t-T_{1}(t_{0}))$ for $t\\geq T_{1}(t_{0})$. Similarly, $X(T_{1}(t_{0}))<Z(T_{1}(t_{0});T_{1}(t_{0}))$ and $X(t)$ will hit $Z(t;T_{1}(t_{0}))$ sometime after $T_{1}(t_{0})$. Denote by $T_{2}(t_{0})$ the first time that $X(t)$ hits $Z(t;T_{1}(t_{0}))$. Then, $X(t)<Z(t;T_{1}(t_{0}))\\,\\,\\text{for}\\,\\,t\\in[T_{1}(t_{0}),T_{2}(t_{0}))$ and $X(T_{2}(t_{0}))=Z(T_{2}(t_{0});T_{1}(t_{0}))$, and $T_{2}(t_{0})-T_{1}(t_{0})\\in\\Big[\\frac{C_{0}-c_{2}T_{2}}{c_{2}-c_{1}\/2},\\frac{C_{0}+c_{1}T_{1}}{c_{1}\/2}\\Big]$.\n\n\nRepeating the above arguments, we obtain the following: there is a sequence of times $\\{T_{n-1}(t_{0})\\}_{n\\in\\mathbb{N}}$ satisfying $T_{0}(t_{0})=t_{0}$ and\n\\begin{equation}\\label{uniform-time-interval}\nT_{n}(t_{0})-T_{n-1}(t_{0})\\in\\bigg[\\frac{C_{0}-c_{2}T_{2}}{c_{2}-c_{1}\/2},\\frac{C_{0}+c_{1}T_{1}}{c_{1}\/2}\\bigg],\\quad\\forall\\,\\,n\\in\\mathbb{N},\n\\end{equation}\nand  for any $n\\in\\mathbb{N}$\n\\begin{equation*\n\\begin{split}\nX(t)<Z(t;T_{n-1}(t_{0}))\\,\\,\\text{for}\\,\\,t\\in[T_{n-1}(t_{0}),T_{n}(t_{0}))\\quad\\text{and}\\quad X(T_{n}(t_{0}))=Z(T_{n}(t_{0});T_{n-1}(t_{0})),\n\\end{split}\n\\end{equation*}\nwhere $Z(t;T_{n-1}(t_{0}))=X(T_{n-1}(t_{0}))+C_{0}+\\frac{c_{1}}{2}(t-T_{n-1}(t_{0}))$. Moreover, for any $n\\in\\mathbb{N}$ and $t\\in[T_{n-1}(t_{0}),T_{n}(t_{0}))$, we conclude from \\eqref{lower-upper-estimate} that\n\\begin{equation*}\n\\begin{split}\n&Z(t;T_{n-1}(t_{0}))-X(t)\\\\\n&\\quad\\quad\\leq X(T_{n-1}(t_{0}))+C_{0}+\\frac{c_{1}}{2}(t-T_{n-1}(t_{0}))-\\big[X(T_{n-1}(t_{0}))+c_{1}(t-T_{n-1}(t_{0})-T_{1})\\big]\\\\\n&\\quad\\quad=C_{0}+c_{1}T_{1}-\\frac{c_{1}}{2}(t-T_{n-1}(t_{0}))\\leq C_{0}+c_{1}T_{1}.\n\\end{split}\n\\end{equation*}\n\nNext, define $\\tilde{Z}(\\cdot;t_{0}):[t_{0},\\infty)\\rightarrow\\mathbb{R}$ by setting\n\\begin{equation}\\label{definition-new-fun}\n\\tilde{Z}(t;t_{0})=Z(t;T_{n-1}(t_{0}))\\quad\\text{for}\\,\\, t\\in[T_{n-1}(t_{0}),T_{n}(t_{0})),\\,\\,n\\in\\mathbb{N}.\n\\end{equation}\nSince $[t_{0},\\infty)=\\cup_{n\\in\\mathbb{N}}[T_{n-1}(t_{0}),T_{n}(t_{0}))$ by \\eqref{uniform-time-interval}, $\\tilde{Z}(t;t_{0})$ is well-defined for all $t\\geq t_{0}$. Notice $\\tilde{Z}(t;t_{0})$ is strictly increasing and is linear on $[T_{n-1}(t_{0}),T_{n}(t_{0}))$ with slope $\\frac{c_{1}}{2}$ for each $n\\in\\mathbb{N}$, and satisfies\n\\begin{equation*}\n0\\leq\\tilde{Z}(t;t_{0})-X(t)\\leq C_{0}+c_{1}T_{1},\\quad t\\geq t_{0}.\n\\end{equation*}\n\nDue to \\eqref{uniform-time-interval}, we can modify $\\tilde{Z}(t;t_{0})$ near each $T_{n}(t_{0})$ for $n\\in\\mathbb{N}$ as follows. Fix some $\\delta_{*}\\in\\Big(0,\\frac{1}{2}\\frac{C_{0}-c_{2}T_{2}}{c_{2}-c_{1}\/2}\\Big)$. We modify $\\tilde{Z}(t;s)$ by redefining it on the intervals $(T_{n}(t_{0})-\\delta_{*},T_{n}(t_{0}))$, $n\\in\\mathbb{N}$ as follows: define\n\\begin{equation*}\nX(t;t_{0})=\\begin{cases}\n\\tilde{Z}(t;t_{0}),\\quad t\\in[t_{0},\\infty)\\backslash\\cup_{n\\in\\mathbb{N}}(T_{n}(t_{0})-\\delta_{*},T_{n}(t_{0})),\\\\\nX(T_{n}(t_{0}))+\\delta(t-T_{n}(t_{0})),\\quad t\\in(T_{n}(t_{0})-\\delta_{*},T_{n}(t_{0})),\\,\\,n\\in\\mathbb{N},\n\\end{cases}\n\\end{equation*}\nwhere $\\delta:[-\\delta_{*},0]\\rightarrow[-\\frac{1}{2}c_{1}\\delta_{*},C_{0}]$ is twice continuously differentiable and satisfies\n\\begin{equation*}\n\\begin{split}\n&\\delta(-\\delta_{*})=-\\frac{c_{1}}{2}\\delta_{*},\\quad\\delta(0)=C_{0},\\\\\n&\\dot{\\delta}(-\\delta_{*})=\\frac{c_{1}}{2}=\\dot{\\delta}(0),\\quad\\dot{\\delta}(t)\\geq\\frac{c_{1}}{2}\\,\\,\\text{for}\\,\\,t\\in(-\\delta_{*},0)\\quad\\text{and}\\\\\n&\\ddot{\\delta}(-\\delta_{*})=0=\\ddot{\\delta}(0).\n\\end{split}\n\\end{equation*}\nThe existence of such a function $\\delta(t)$ is clear. Moreover, there exist $c_{\\max}=c_{\\max}(\\delta_{*})>0$ and $\\tilde{c}_{\\max}=\\tilde{c}_{\\max}(\\delta_{*})>0$ such that $\\dot{\\delta}(t)\\leq c_{\\max}$ and $|\\ddot{\\delta}(t)|\\leq\\tilde{c}_{\\max}$ for $t\\in(-\\delta_{*},0)$. Notice the above modification is independent of $n\\in\\mathbb{N}$ and $t_{0}$. Hence, $X(t;t_{0})$ satisfies the following uniform in $t_{0}$ properties:\n\\begin{itemize}\n\\item $0\\leq X(t;t_{0})-X(t)\\leq d_{\\max}$ for some $d_{\\max}>0$,\n\\item $\\frac{c_{1}}{2}\\leq\\dot{X}(t;t_{0})\\leq c_{\\max}$,\n\\item $|\\ddot{X}(t;t_{0})|\\leq\\tilde{c}_{\\max}$.\n\\end{itemize}\nSince $X(t)$ is continuous, so locally bounded, we may apply Arzel\\`{a}-Ascoli theorem to conclude the existence of some continuously differentiable function $\\tilde{X}:\\mathbb{R}\\to\\mathbb{R}$ such that $X(t;t_{0})\\to \\tilde{X}(t)$ and $\\dot{X}(t;t_{0})\\to\\dot{\\tilde{X}}(t)$ locally uniformly in $t$ as $t_{0}\\to-\\infty$ along some subsequence. It's easy to see that $\\tilde{X}(t)$ satisfies all the properties as in the statement of the theorem.\n\\end{proof}\n\nNext, we prove Corollary \\ref{cor-modified-interface}. Recall that for a given transition front $u(t,x)$ of \\eqref{main-eqn}, $X^{\\pm}_{\\lambda}(t)$ are defined in \\eqref{defn-interface-locations}.\n\n\\begin{proof}[Proof of Corollary \\ref{cor-modified-interface}]\nWe modify the proof of Theorem \\ref{lem-propagation-estimate}. Let $u(t,x)$ be an arbitrary transition front of \\eqref{main-eqn} with interface location function $X(t)$. Since $f(t,x,u)\\leq0$ for $(t,x,u)\\in\\mathbb{R}\\times\\mathbb{R}\\times[0,\\tilde{\\theta}]$, we can find a function $f_{I}(u)$ such that $f(t,x,u)\\leq f_{I}(u)$ for $(t,x,u)\\in\\mathbb{R}\\times\\mathbb{R}\\times[0,1]$, where $f_{I}:[0,1]\\to\\mathbb{R}$ is $C^{2}$ and is of standard ignition type, that is, there exists $\\theta_{I}\\in(0,1)$ such that\n\\begin{equation*}\nf_{I}(u)=0,\\,\\,u\\in[0,\\theta_{I}]\\cup\\{1\\},\\quad f_{I}(u)>0,\\,\\,u\\in(\\theta_{I},1)\\quad\\text{and}\\quad f_{I}'(1)<0.\n\\end{equation*}\n\n\n\n\nFix some $\\lambda\\in(\\theta,1)$. We  write $X^{+}(t)=X^{+}_{\\lambda}(t)$. Since $\\sup_{t\\in\\mathbb{R}}|X(t)-X^{+}(t)|<\\infty$ by Lemma \\ref{lem-bounded-interface-width}, it suffices to show\n\\begin{equation}\\label{propagation-estimate-1-ig}\nX^{+}(t)-X^{+}(t_{0})\\leq c(t-t_{0}+T),\\quad t\\geq t_{0}\n\\end{equation}\nfor some $c>0$ and $T>0$.\n\nTo do so, we fix some $\\tilde{\\theta}_{I}\\in(0,\\theta_{I})$. Let $(c_{I},\\phi_{I})$ with $c_{I}>0$ be the unique solution of\n\\begin{equation*}\n\\begin{cases}\nJ\\ast\\phi-\\phi+c\\phi_{x}+f_{I}(\\phi)=0,\\\\\n\\phi_{x}<0,\\,\\,\\phi(0)=\\theta_{I},\\,\\,\\phi(-\\infty)=1\\,\\,\\text{and}\\,\\,\\phi(\\infty)=\\tilde{\\theta}_{I}.\n\\end{cases}\n\\end{equation*}\nNote that $\\phi_{I}$ connects $\\tilde{\\theta}_{I}$ and $1$ instead of $0$ and $1$ (see Appendix \\ref{app-ig-tw} for more properties about $\\phi_{I}$; in Appendix \\ref{app-ig-tw}, we consider traveling waves connecting $0$ and $1$, but by simple shift, all results there apply here).\n\nLet $u_{0}:\\mathbb{R}\\to[0,1]$ be a uniformly continuous and nonincreasing function satisfying $u_{0}(x)=1$ for $x\\leq 0$ and $u_{0}(x)=\\tilde{\\theta}_{I}$ for $x\\geq x_{0}$, where $x_{0}>0$ is fixed. Clearly, $u(t_{0},\\cdot+X^{+}_{\\tilde{\\theta}_{I}}(t_{0}))\\leq u_{0}$. Applying comparison principle and Lemma \\ref{lem-ignition-property}, we find\n\\begin{equation*}\nu(t,x+X^{+}_{\\tilde{\\theta}_{I}}(t_{0}))\\leq u_{I}(t-t_{0},x;u_{0})\\leq\\phi_{I}(x-c_{I}(t-t_{0})-\\xi_{I})+\\epsilon_{I}e^{-\\omega_{I}(t-t_{0})},\\quad x\\in\\mathbb{R},\\quad t\\geq t_{0}.\n\\end{equation*}\nLet $\\xi_{I}(\\frac{\\lambda}{2})$ be the unique point such that $\\phi_{I}(\\xi_{I}(\\frac{\\lambda}{2}))=\\frac{\\lambda}{2}$ and $T>0$ be such that $\\epsilon_{I}e^{-\\omega_{I}T}=\\frac{\\lambda}{2}$ (we may make $\\epsilon_{I}>\\frac{\\lambda}{2}$ if necessary). Setting $x_{*}=c_{I}(t-t_{0})+\\xi_{I}+\\xi_{I}(\\frac{\\lambda}{2})$, we conclude from the monotonicity of $\\phi_{I}$ that for $x\\geq x_{*}+1$ and $t\\geq t_{0}+T$,\n\\begin{equation*}\n\\begin{split}\nu(t,x+X^{+}_{\\tilde{\\theta}_{I}}(t_{0}))&\\leq\\phi_{I}(x_{*}+1-c_{I}(t-t_{0})-\\xi_{I})+\\epsilon_{I}e^{-\\omega_{I}T}\\\\\n&<\\phi_{I}(x_{*}-c_{I}(t-t_{0})-\\xi_{I})+\\epsilon_{I}e^{-\\omega_{I}T}=\\lambda.\n\\end{split}\n\\end{equation*}\nIt then follows from the definition of $X^{+}(t)$ that\n\\begin{equation*}\nX^{+}(t)\\leq x_{*}+1+X^{+}_{\\tilde{\\theta}_{I}}(t_{0})=c_{I}(t-t_{0})+\\xi_{I}+\\xi_{I}(\\frac{\\lambda}{2})+1+X^{+}_{\\tilde{\\theta}_{I}}(t_{0}),\\quad t\\geq t_{0}+T.\n\\end{equation*}\nSetting $C_{*}:=\\sup_{t_{0}\\in\\mathbb{R}}|X^{+}(t_{0})-X^{+}_{\\tilde{\\theta}_{I}}(t_{0})|<\\infty$ due to Lemma \\ref{lem-bounded-interface-width}, we conclude\n\\begin{equation*}\nX^{+}(t)-X^{+}(t_{0})\\leq c_{I}(t-t_{0})+\\xi_{I}+\\xi_{I}(\\frac{\\lambda}{2})+1+C_{*},\\quad t\\geq t_{0}+T.\n\\end{equation*}\n\nIt remains to show that\n\\begin{equation}\\label{what-to-show}\nX^{+}(t)-X^{+}(t_{0})\\leq\\xi_{*},\\quad t\\in[t_{0},t_{0}+T]\n\\end{equation}\nfor some $\\xi_{*}>0$ independent of $t_{0}$. To do so, let $\\tilde{u}_{0}$ be the $u_{0}$ in the proof of Theorem \\ref{lem-propagation-estimate}. Then, we have $\\tilde{u}_{0}(\\cdot-X^{-}(t_{0}))\\leq u(t_{0},\\cdot)\\leq u_{0}(\\cdot-X^{+}_{\\tilde{\\theta}_{I}}(t_{0}))$, where $X^{-}(t)=X^{-}_{\\lambda}(t)$. Since $f_{B}\\leq f\\leq f_{I}$, we apply comparison principle to conclude that\n\\begin{equation*}\nu_{B}(t-t_{0},x-X^{-}(t_{0});\\tilde{u}_{0})\\leq u(t,x)\\leq u_{I}(t-t_{0},x-X^{+}_{\\tilde{\\theta}_{I}}(t_{0});u_{0}),\\quad x\\in\\mathbb{R},\\quad t\\geq t_{0}.\n\\end{equation*}\nWe then conclude \\eqref{what-to-show} from the continuity of $u_{B}(t-t_{0},x-X^{-}(t_{0})$ and $u_{I}(t-t_{0},x-X^{+}_{\\tilde{\\theta}_{I}}(t_{0});u_{0})$, and the fact $\\sup_{t_{0}\\in\\mathbb{R}}|X^{-}(t_{0})-X^{+}_{\\tilde{\\theta}_{I}}(t_{0})|<\\infty$ due to Lemma \\ref{lem-bounded-interface-width}. This completes the proof.\n\\end{proof}\n\nFinally, we prove Corollary \\ref{cor-modified-interface-ii}.\n\n\\begin{proof}[Proof of Corollary \\ref{cor-modified-interface-ii}]\nNote first we can find a $C^{2}$ Fisher-KPP nonlinearity $f_{\\rm KPP}:[0,1]\\to\\mathbb{R}$ such that $f(t,x,u)\\leq f_{\\rm KPP}(u)$ for all $(t,x,u)\\in\\mathbb{R}\\times\\mathbb{R}\\times[0,1]$. Let $u(t,x)$ be the transition front as in the statement of Corollary \\ref{cor-modified-interface-ii}, that is, there exist $r>0$ and $h>0$ such that\n\\begin{equation*}\nu(t_{0},x+X(t_{0}))\\leq e^{-r(x-h)},\\quad (t_{0},x)\\in\\mathbb{R}\\times\\mathbb{R},\n\\end{equation*}\n\nFixed $\\lambda\\in(0,1)$. Setting $h_{0}:=h+\\sup_{t_{0}\\in\\mathbb{R}}|X(t_{0})-X^{+}_{\\lambda}(t_{0})|<\\infty$, we find\n\\begin{equation*}\nu(t_{0},x+X^{+}_{\\lambda}(t_{0})+h_{0})\\leq e^{-rx},\\quad (t_{0},x)\\in\\mathbb{R}\\times\\mathbb{R}.\n\\end{equation*}\nThen, we can find some uniformly continuous function $u_{0}:\\mathbb{R}\\to[0,1]$ satisfying\n\\begin{equation*}\n\\lim_{x\\to-\\infty}u_{0}(x)=1\\quad\\text{and}\\quad\\lim_{x\\to\\infty}\\frac{u_{0}(x)}{e^{-rx}}=1\n\\end{equation*}\nsuch that\n\\begin{equation*}\nu(t_{0},x+X^{+}_{\\lambda}(t_{0})+h_{0})\\leq u_{0}(x),\\quad (t_{0},x)\\in\\mathbb{R}\\times\\mathbb{R}.\n\\end{equation*}\nNote that we may assume, without loss of generality, that $r$ is so small that it is the decay rate of some traveling wave of $\\phi_{r}(x-c_{r}t)$ (satisfying $\\phi_{r}(-\\infty)=1$ and $\\phi_{r}(\\infty)=0$) with speed $c_{r}=\\frac{\\int_{\\mathbb{R}}J(y)e^{ry}dy-1+f_{\\rm KPP}'(0)}{r}>0$ of\n\\begin{equation}\\label{eqn-kpp-homo}\nu_{t}=J\\ast u-u+f_{\\rm KPP}(u),\n\\end{equation}\nthat is, $\\lim_{x\\to\\infty}\\frac{\\phi_{r}(x)}{e^{-rx}}=1$ (see \\cite{CaCh04} and \\cite{ShZh12-2}). In particular, we have\n\\begin{equation}\\label{exact-decay-rate-2015201}\n\\lim_{x\\to\\infty}\\frac{u_{0}(x)}{\\phi_{r}(x)}=1.\n\\end{equation}\nMoreover, there holds\n\\begin{equation}\\label{exact-decay-rate-20152015}\n\\lim_{x\\to\\infty}\\frac{\\phi_{r}'(x)}{\\phi_{r}(x)}=-r.\n\\end{equation}\nTo see this, we notice $\\frac{J\\ast\\phi_{r}}{\\phi_{r}}-1+c_{r}\\frac{\\phi_{r}'}{\\phi_{r}}+\\frac{f_{\\rm KPP}(\\phi_{r})}{\\phi_{r}}=0$. Clearly, $\\lim_{x\\to\\infty}\\frac{f_{\\rm KPP}(\\phi_{r}(x))}{\\phi_{r}(x)}=f_{\\rm KPP}'(0)$. For $\\frac{J\\ast\\phi_{r}}{\\phi_{r}}$, we have\n\\begin{equation*}\n\\frac{[J\\ast\\phi_{r}](x)}{\\phi_{r}(x)}=\\frac{e^{-rx}}{\\phi_{r}(x)}\\int_{\\mathbb{R}}J(y)e^{ry}\\frac{\\phi_{r}(x-y)}{e^{-r(x-y)}}dy\\to J(y)e^{ry}dy\\quad\\text{as}\\quad x\\to\\infty\n\\end{equation*}\nby \\eqref{exact-decay-rate-2015201} and dominated convergence theorem. From which, we conclude \\eqref{exact-decay-rate-20152015}.\n\nThen, arguing as in the proof of Corollary \\ref{cor-modified-interface}, we conclude the result from the stability of $\\phi_{r}(x-c_{r}t)$, that is,\n\\begin{equation}\\label{stability-kpp}\n\\lim_{t\\to\\infty}\\bigg|\\frac{u_{\\rm KPP}(t,x;u_{0})}{\\phi_{r}(x-c_{r}t)}-1\\bigg|=0,\n\\end{equation}\nwhere $u_{\\rm KPP}(t,x;u_{0})$ is the solution of \\eqref{eqn-kpp-homo} with initial data $u_{\\rm KPP}(0,\\cdot;u_{0})=u_{0}$. We remark that \\eqref{stability-kpp} follows from \\cite[Theorem 2.6]{ShZh12-2}. Also, by means of \\eqref{exact-decay-rate-2015201} and \\eqref{exact-decay-rate-20152015}, it can be proven as that of \\cite[Theorem 1.3]{ShSh14-kpp}.\n\\end{proof}\n\n\n\n\\section*{Acknowledgements} \n\nThe authors would like to thank the referee for carefully reading the manuscript, pointing out some problems that we were not aware of, and drawing our attention to the reference \\cite{BaChm99}.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section*{Introduction}\nThis paper is a contribution to the representation theory of\nHopf-Galois extensions, as originated by Schneider in\n\\cite{Schneider1}. More specifically, we consider the following\nquestions. Let $H$ be a Hopf algebra, and $A$, $B$ right\n$H$-comodule algebras. Moreover, assume that $A$ and $B$ are\nright faithfully flat $H$-Galois extensions.\n\\begin{enumerate}\n\\item If $A$ and $B$ are Morita equivalent, does it follow that\n$A^{{\\rm co}H}$ and $B^{{\\rm co}H}$ are also Morita equivalent?\n\\item Conversely,\nif $A^{{\\rm co}H}$ and $B^{{\\rm co}H}$ are Morita equivalent, when does it follow\nthat $A$ and $B$ are Morita equivalent?\n\\end{enumerate}\nThese questions have been considered in \\cite{Marcus} in the context of\nstrongly group graded algebras, the motivation coming from problems raised in\nthe modular representation theory of finite groups. The results of the present\npaper generalize the results of \\cite[Sections 2 and 3]{Marcus}.\\\\\nGiven a right $H$-comodule algebra $A$, and a left $H$-comodule\nalgebra $B$, we consider $(A\\otimes B,H)$-Hopf modules. These are at\nthe same time left $A\\otimes B$-modules and right $H$-comodules, with\na suitable compatibility condition. There are various ways to look\nat these Hopf modules: they are Doi-Hopf modules (see \\cite{Doi})\nover a certain Doi-Hopf datum (with two possible descriptions of\nthe underlying module coalgebra), and they can also be viewed as\ncomodules over a coring (see \\seref{3}). The main result of\n\\seref{2}, and also the main tool used during the rest of the\npaper, is a structure Theorem for $(A\\otimes B,H)$-Hopf modules,\nstating that the category of $(A\\otimes B,H)$-Hopf modules is\nequivalent to the category of left modules over the cotensor\nproduct $A\\square_H B$, under the condition that $A$ is a\nfaithfully\nflat $H$-Galois extension.\\\\\nThe results from \\seref{2} can be applied to relative Hopf bimodules: let $A$ and\n$B$ be right $H$-comodule algebras, and consider $(A,B)$-bimodules with a\nright $H$-coaction, satisfying a certain compatibility condition. The category\nof relative Hopf bimodules is then isomorphic to the category of\n$(A\\otimes B^{\\rm op},H)$-Hopf modules. In \\seref{4}, we state the Structure Theorem\nfor relative Hopf bimodules, and  we investigate the compatibility of\nthe category equivalence with the Hom and tensor functors.\\\\\nIn \\seref{5}, we apply our results to discuss the two problems\nstated above. We introduce the notion of $H$-Morita contexts, and\nwe show that if two right faithfully flat $H$-Galois extensions are\nconnected by a (strict) $H$-Morita context, then the algebras of\ncoinvariants are also connected by a (strict) Morita context. Our\nmain result is the following converse result: if the algebras of\ncoinvariants are Morita equivalent, in such a way that the bimodule\nstructure on one of the connecting modules can be extended to a\nleft-action by the cotensor product $A\\square_H B^{\\rm op}$, then\n$A$ and $B$ are $H$-Morita\nequivalent.\\\\\nIn \\seref{6}, we show that the Morita equivalence coming from a strict $H$-Morita\ncontext between two faithfully flat $H$-Galois extensions respects the Miyashita-Ulbrich\naction.\nIn \\seref{7}, we investigate the behavior of $H$-Morita equivalences with respect to Hopf subalgebras. \\\\\nThe category of relative Hopf modules and $A$-linear (not necessarily $H$-colinear)\nmodules is an $H$-colinear category. If two right $H$-comodule algebras are\n$H$-Morita equivalent, then the induced equivalence between their categories of\nrelative Hopf modules is $H$-colinear. In \\seref{8}, we study the converse property:\nwhen does every $H$-colinear equivalence between two categories of relative Hopf\nmodules come from a strict $H$-Morita context. This leads to a generalization of\nthe Eilenberg-Watts Theorem (\\prref{8.5}). The main result is \\coref{8.7}, stating that\nevery $H$-colinear equivalence comes from a strict $H$-Morita context if the \nHopf algebra $H$ is projective, and the $H$-comodule algebras\n$A$ and $B$ are $H$-Galois extensions of their subalgebras of coinvariants.\\\\\nFor basic results on Hopf algebras, we refer the reader to\n\\cite{DNR} or \\cite{Mont}. For a concise treatment of corings and their applications,\nwe refer to \\cite{BrzezinskiWisbauer}.\n\n\\section{Preliminary results}\\selabel{1}\nThroughout this paper $H$ is a Hopf algebra over a commutative\nring $k$, with bijective antipode $S$. We use the Sweedler\nnotation for the comultiplication on $H$: $\\Delta(h)= h_{(1)}\\otimes\nh_{(2)}$. $\\mathcal{M}^H$ (respectively ${}^H\\mathcal{M}$) is the category of\nright (respectively left) $H$-comodules. For a right $H$-coaction\n$\\rho$ (respectively a left $H$-coaction $\\lambda$) on a\n$k$-module $M$, we denote\n$$\\rho(m)=m_{[0]}\\otimes m_{[1]}\\quad{\\rm and}\\quad\\lambda(m)=m_{[-1]}\\otimes m_{[0]}.$$\nThe submodule of coinvariants $M^{{\\rm co}H}$ of a right\n(respectively left) $H$-co\\-mo\\-du\\-le $M$ consists of the elements\n$m\\in M$ satisfying $\\rho(m)=m\\otimes 1$ (respectively\n$\\lambda(m)=1\\otimes m$).\\\\\nLet $A$ be a right $H$-comodule algebra. ${}_A\\mathcal{M}^H$ and $\\mathcal{M}^H_A$ are\nthe categories of left and right relative Hopf modules. We have two pairs of\nadjoint functors $(F_1=A\\otimes_{A^{{\\rm co}H}}-,\\ G_1=(-)^{{\\rm co}H})$\nand $(F_2=-\\otimes_{A^{{\\rm co}H}}A,\\ G_2=(-)^{{\\rm co}H})$ between the categories\n${}_{A^{{\\rm co}H}}\\mathcal{M}$ and ${}_A\\mathcal{M}^H$, and between\n$\\mathcal{M}_{A^{{\\rm co}H}}$ and $\\mathcal{M}^H_A$. The unit and counit of the adjunction\n$(F_1,G_1)$ are given by the formulas\n$$\\eta_{1,N}:\\ N\\to (A\\otimes_{A^{{\\rm co}H}} N)^{{\\rm co}H},\\quad\\eta_{1,N}(n)=1\\otimes n;$$\n$$\\varepsilon_{1,M}:\\ A\\otimes_{A^{{\\rm co}H}} M^{{\\rm co}H}\\to M,\\quad\\varepsilon_{1,M}(a\\otimes m)=am.$$\nThe formulas for the unit and counit of $(F_2,G_2)$ are similar. Consider the canonical maps\n$${\\rm can}:\\ A\\otimes_{A^{{\\rm co}H}} A\\to A\\otimes H,~~{\\rm can}(a\\otimes b)=ab_{[0]}\\otimes b_{[1]};$$\n$${\\rm can}':\\ A\\otimes_{A^{{\\rm co}H}} A\\to A\\otimes H,~~{\\rm can}(a\\otimes b)=a_{[0]}b\\otimes a_{[1]}.$$\nIt is well-known (see for example \\cite{KT}) that ${\\rm can}$ is an isomorphism if and only if\n${\\rm can}'$ is an isomorphism.\n\n\\begin{theorem}\\thlabel{1.1}Let $A$ be a right $H$-comodule algebra. \nConsider the following statements:\n\\begin{enumerate}\n\\item $(F_2,G_2)$ is a pair of inverse equivalences;\n\\item $(F_2,G_2)$ is a pair of inverse equivalences and $A\\in {}_{A^{{\\rm co}H}}\\mathcal{M}$ is flat;\n\\item ${\\rm can}$ is an isomorphism and $A\\in {}_{A^{{\\rm co}H}}\\mathcal{M}$ is faithfully flat;\n\\item $(F_1,G_1)$ is a pair of inverse equivalences;\n\\item $(F_1,G_1)$ is a pair of inverse equivalences and $A\\in \\mathcal{M}_{A^{{\\rm co}H}}$ is flat;\n\\item ${\\rm can}'$ is an isomorphism and $A\\in \\mathcal{M}_{A^{{\\rm co}H}}$ is faithfully flat.\n\\end{enumerate}\nWe have the following implications:\n$$(3)\\Longleftrightarrow (2)\\Longrightarrow (1)~~;~~(6)\\Longleftrightarrow (5)\\Longrightarrow (4).$$\nIf $H$ is flat as a $k$-module, then $(1)\\Longleftrightarrow (2)$ and $(4)\\Longleftrightarrow (5)$.\\\\\nIf $k$ is field then the six conditions are equivalent.\n\\end{theorem}\n\nIf the first three conditions of \\thref{1.1} hold, then we call $A$\na left faithfully flat $H$-Galois extension; if the three other conditions hold, then we call\n$A$ a right faithfully flat $H$-Galois extension.\n\n\\begin{proof}\nThe equivalence of (2) and (3) is well-known. It is essentially \\cite[Theorem 3.7]{Schneider},\nwhich is an improvement of \\cite[Theorem 2.11]{DT}.\nFor the equivalence of (5) and (6), we observe that $A$ is a left $H^{\\rm cop}$-comodule,\nso, by the left handed version of the equivalence $(3)\\Longleftrightarrow (2)$,\n(6) is equivalent to flatness of $A\\in \\mathcal{M}_{A^{{\\rm co}H}}$ and equivalence between\nthe categories ${}_{A^{{\\rm co}H}}\\mathcal{M}$ and ${}_A^{H^{\\rm cop}}\\mathcal{M}\\cong {}_A\\mathcal{M}^H$.\\\\\nThe implications $(2)\\Longrightarrow (1)$ and $(5)\\Longrightarrow (4)$ are trivial.\\\\\nIf $H$ is flat as a $k$-module, then $\\mathcal{M}_A^H$ is an abelian category and the\nforgetful functor $\\mathcal{M}_A^H\\to \\mathcal{M}_A$ is exact. \nIf $F_2$ is an equivalence, then the functor $-\\otimes_{A^{{\\rm co}H}}:\\\n\\mathcal{M}_{A^{{\\rm co}H}}\\to \\mathcal{M}_A$ is exact since it is the composition of the forgetful\nfunctor and the equivalence $F_2$. This shows that $A$ is flat as a left\n$A^{{\\rm co}H}$-module, and the implication $(1)\\Longrightarrow (2)$ follows.\n$(4)\\Longrightarrow (5)$ can be proved in a similar way.\\\\\nIf $k$ is a field, then the equivalence of the six statements in the Theorem follows\nfrom \\cite[Theorem I]{Schneider0}.\n\\end{proof}\n\nLet $M$ be a right $H$-comodule, and $N$ a left $H$-comodule. The cotensor\nproduct $M\\square_H N$ is the $k$-module\n$$M\\square_H N= \\{\\sum_i m_i\\otimes n_i\\in M\\otimes N~|~\\sum_i \\rho(m_i)\\otimes n_i=\n\\sum_i m_i\\otimes \\lambda(n_i)\\}.$$\nIf $H$ is cocommutative, then $M\\square_H N$ is also a right (or left) $H$-comodule.\n\n\\begin{proposition}\\prlabel{1.2}\nLet $R$ be a $k$-algebra, and assume that $P\\in \\mathcal{M}_R$ is flat.\nTake $M\\in {}_R\\mathcal{M}^H$\nand $N\\in {}^H\\mathcal{M}$, and assume that we have a right $H$-coaction on $M$ that\nis left $R$-linear. Then the map\n$$P\\otimes_R (M\\square_H N)\\to (P\\otimes_R M)\\square_H N,~~\np\\otimes (\\sum_i m_i\\otimes n_i)\\mapsto \\sum_i (p\\otimes m_i)\\otimes n_i$$\nis bijective.\n\\end{proposition}\n\n\\section{A structure theorem for $(A\\otimes B,H)$-Hopf modules}\\selabel{2}\nUnder our assumption on $H$, $H\\otimes H^{\\rm cop}$ is also a Hopf\nalgebra, and $H$ is a left $H\\otimes H^{\\rm cop}$-module coalgebra;\nthe left $H\\otimes H^{\\rm cop}$-action is given by\n$$(k\\otimes l)\\cdot h=khS(l),$$\nfor all $h,k,l\\in H$.\\\\\n\nWe present an alternative description of $H$ as a left $H\\otimes H^{\\rm cop}$-module coalgebra.\n$H\\otimes H^{\\rm cop}\\in {}_{H\\otimes H^{\\rm cop}}\\mathcal{M}_H$, with right\n$H$-action induced by the comultiplication on $H$, and $k\\in {}_H\\mathcal{M}$ via $\\varepsilon$,\nso we have the left $H\\otimes H^{\\rm cop}$-module $(H\\otimes H^{\\rm cop})\\otimes_H k$.\n$(H\\otimes H^{\\rm cop})\\otimes_H k$ is a coalgebra with comultiplication and counit given by\n$$\\Delta((h\\otimes h')\\otimes_H 1)=(h_{(1)}\\otimes h'_{(2)})\\otimes_H 1\\otimes (h_{(2)}\\otimes h'_{(1)})\\otimes_H 1;$$\n$$\\varepsilon((h\\otimes h')\\otimes_H 1)=\\varepsilon(hh').$$\nIt is easy to show that $(H\\otimes H^{\\rm cop})\\otimes_H k$ is an $H\\otimes H^{\\rm cop}$-module coalgebra.\n\n\\begin{proposition}\\prlabel{2.0}\n$(H\\otimes H^{\\rm cop})\\otimes_H k$ and $H$ are isomorphic as\n$H\\otimes H^{\\rm cop}$-module coalgebras.\n\\end{proposition}\n\n\\begin{proof}\nDefine\n$$f:\\ (H\\otimes H^{\\rm cop})\\otimes_H k\\to H,~~f((h\\otimes h')\\otimes_H 1)=hS(h');$$\n$$g:\\ H\\to (H\\otimes H^{\\rm cop})\\otimes_H k,~~g(h)=(h\\otimes 1)\\otimes_H 1.$$\n$f$ is well-defined since for all $h,h',l\\in H$\n$$f((h\\otimes h')l\\otimes_H 1)=hl_{(1)}S(h'l_{(2)})=hS(h')\\varepsilon(l)=\nf((h\\otimes h')\\otimes_H \\varepsilon(l)).$$\n$f$ is $H\\otimes H^{\\rm cop}$-linear since  for all $h,h',k,k'\\in H$\n$$f((kh\\otimes k'h')\\otimes_H 1)=khS(k'h')=(k\\otimes k')\\cdot (hS(k'))=(k\\otimes k')f((h\\otimes h')\\otimes_H 1).$$\n$f$ is a coalgebra map since for all $h,h'\\in H$\n$$((f\\otimes f)\\circ\\Delta)((h\\otimes h')\\otimes_H 1)=h_{(1)}S(h'_{(2)})\\otimes h_{(2)}S(h'_{(1)})=\n\\Delta(hS(h')),$$\nand\n$$(\\varepsilon\\circ f)((h\\otimes h')\\otimes_H 1)=\\varepsilon(hS(h'))=\\varepsilon(hh').$$\nIt is obvious that $f\\circ g=H$. Finally  for all $h,k\\in H$\n\\begin{eqnarray*}\n&&\\hspace*{-2cm}\n(g\\circ f)((h\\otimes k)\\otimes_H 1)= g(hS(k))\\\\\n&=&(hS(k)\\otimes 1)\\otimes_H 1\n= (hS(k_{(1)})\\otimes 1)\\otimes_H\\varepsilon(k_{(2)})\\\\\n&=&\n(hS(k_{(1)})k_{(2)}\\otimes k_{(3)})\\otimes_H 1=(h\\otimes k)\\otimes_H 1.\n\\end{eqnarray*}\n\\end{proof}\n\nLet $A$ be a right $H$-comodule algebra, and $B$ a left\n$H$-comodule algebra. Then $A\\otimes B$ is a right $H\\otimes H^{\\rm\ncop}$-comodule algebra, with coaction\n$$\\rho(a\\otimes b)=a_{[0]}\\otimes b_{[0]}\\otimes a_{[1]}\\otimes b_{[-1]}.$$\nThen $(H\\otimes H^{\\rm cop}, A\\otimes B,H)$ is a left-right Doi-Hopf datum (see\n\\cite{CMZ} or \\cite{Doi} for details), and we can consider the category\n${}_{A\\otimes B}\\mathcal{M}(H\\otimes H^{\\rm cop})^H$ of Doi-Hopf modules. The objects of this\ncategory are $k$-modules $M$ with a left $A\\otimes B$-action and a right $H$-coaction such\nthat\n$$\\rho((a\\otimes b)m)=(a_{[0]}\\otimes b_{[0]})m_{[0]} \\otimes a_{[1]}m_{[1]}S(b_{[-1]}),$$\nfor all $a\\in A$, $b\\in B$ and $m\\in M$.\nThe objects of ${}_{A\\otimes B}\\mathcal{M}(H\\otimes H^{\\rm cop})^H$ are called $(A\\otimes B,H)$-Hopf modules.\nIt is well-known and easily verified that  $A\\otimes B\\in {}_{A\\otimes B}\\mathcal{M}(H\\otimes H^{\\rm cop})^H$,\nwith coaction defined by\n$$\\rho(a\\otimes b)= a_{[0]}\\otimes b_{[0]}\\otimes a_{[1]}S(b_{[-1]}).$$\n\n\\begin{lemma}\\lelabel{2.1}\nWith notation as above, we have that $(A\\otimes B)^{{\\rm co}H}=A\\square_H B$.\n\\end{lemma}\n\n\\begin{proof}\nTake $x=\\sum_i a_i\\otimes b_i \\in (A\\otimes B)^{{\\rm co}H}$. Then\n$$\\sum_i a_i\\otimes b_i\\otimes 1= \\sum_i a_{i[0]}\\otimes b_{i[0]}\\otimes a_{i[1]}S(b_{i[-1]}).$$\nApply $\\lambda$ to the second tensor factor. Then switch the second and fourth\ntensor factor, and multiply the third and fourth tensor factor. It follows that\n$$\\sum_i a_i\\otimes b_{i[0]}\\otimes b_{i[-1]}=\\sum_i a_{i[0]}\\otimes b_{i[0]}\\otimes a_{i[1]}S(b_{i[-2]})b_{i[-1]}\n=\\sum_i a_{i[0]}\\otimes b_i \\otimes a_{i[1]},$$ and then $x\\in A\\square_H\nB$. The converse inclusion is proved in a similar way.\n\\end{proof}\n\nRecall (see for example \\cite{Doi}) that we have a pair of adjoint functors $(F,G)$:\n$$F:\\ {}_{A\\square_H B}\\mathcal{M}\\to {}_{A\\otimes B}\\mathcal{M}(H\\otimes H^{\\rm cop})^H,~~\nF(N)=(A\\otimes B)\\otimes_{A\\square_H B} N;$$\n$$G:\\  {}_{A\\otimes B}\\mathcal{M}(H\\otimes H^{\\rm cop})^H\\to {}_{A\\square_H B}\\mathcal{M},~~\nG(M)=M^{{\\rm co}H}.$$\nThe unit and counit of the adjunction are the following:\n$$\\eta_N:\\ N\\to \\bigl((A\\otimes B)\\otimes_{A\\square_H B} N\\bigr)^{{\\rm co}H},~~\n\\eta_N(n)=1_A\\otimes 1_B\\otimes n;$$\n$$\\varepsilon_M:\\ (A\\otimes B)\\otimes_{A\\square_H B} M^{{\\rm co}H}\\to M,~~\n\\varepsilon_M(a\\otimes b\\otimes m)=(a\\otimes b)m.$$\n\n\\begin{proposition}\\prlabel{2.2}\nAssume that $H$ is flat as a $k$-algebra. Let $A$ be a right\n$H$-comodule algebra, and $B$ a left $H$-comodule algebra. We have\na right $H$-colinear map\n$$f:\\ A\\otimes_{A^{{\\rm co}H}}(A\\square_HB)=F_1(A\\square_H B)\\to A\\otimes B,~~\nf(a\\otimes(\\sum_i a_i\\otimes b_i))=\\sum_i aa_i\\otimes b_i.$$\nIf $A$ is a right faithfully flat $H$-Galois extension, then $f$ is an isomorphism.\n\\end{proposition}\n\n\\begin{proof}\n$f$ is right $H$-colinear since\n\\begin{eqnarray*}\n&&\\hspace*{-2cm}\n\\rho\\bigl(f(a\\otimes(\\sum_i a_i\\otimes b_i))\\bigr)=\n\\sum_i a_{[0]}a_{i[0]}\\otimes b_{i[0]}\\otimes a_{[1]}a_{i[1]}S(b_{i[-1]})\\\\\n&=& \\sum_i a_{[0]}a_{i}\\otimes b_{i[0]}\\otimes a_{[1]}b_{i[-2]}S(b_{i[-1]})\\\\\n&=&\\sum_i a_{[0]}a_{i}\\otimes b_i\\otimes a_{[1]}\\\\\n&=& (f\\otimes H)(\\rho(a\\otimes(\\sum_i a_i\\otimes b_i))).\n\\end{eqnarray*}\nOn $A\\otimes_{A^{{\\rm co}H}}A$ and $A\\otimes H$, we consider the following right $H$-coactions:\n$$\\rho(a\\otimes b)=a\\otimes b_{[0]}\\otimes b_{[1]}~~;~~\\rho(a\\otimes h)=a\\otimes h_{(1)}\\otimes h_{(2)}.$$\nThen ${\\rm can}:\\ A\\otimes_{A^{{\\rm co}H}}A\\to A\\otimes H$ is right $H$-colinear, so we can\nconsider the map ${\\rm can}\\square_H B: (A\\otimes_{A^{{\\rm co}H}}A)\\square_H B\\to (A\\otimes H)\n\\square_H B$. If $A$ is a right faithfully flat $H$-Galois extension, then ${\\rm can}\\square_H B$\nis bijective, and applying \\prref{1.2}, we see that $f$ is the composition of the\nfollowing isomorphisms:\n\\begin{eqnarray*}\n&&\\hspace*{-2cm}\nA\\otimes_{A^{{\\rm co}H}}(A\\square_HB)\\cong (A\\otimes_{A^{{\\rm co}H}}A)\\square_HB\\\\\n&\\cong&(A\\otimes H)\\square_H B\\cong A\\otimes (H\\square_H B)\\cong A\\otimes B.\n\\end{eqnarray*}\n\\end{proof}\n\nThe following structure theorem is the main result of this\nsection.\n\n\\begin{theorem}\\thlabel{2.3} Let $A$ be a right $H$-comodule algebra, and $B$ a left\n$H$-comodule algebra. If $A$ is a right faithfully flat $H$-Galois\nextension, then $(F,G)$ is a pair of inverse equivalences between\nthe categories ${}_{A\\square_H B}\\mathcal{M}$ and ${}_{A\\otimes B}\\mathcal{M}(H\\otimes\nH^{\\rm cop})^H$.\n\\end{theorem}\n\n\\begin{proof}\nTake $N\\in {}_{A\\square_H B}\\mathcal{M}$. We have a well-defined algebra map\n$A^{{\\rm co}H}\\to A\\square_H B$, sending $a$ to $a\\otimes 1_B$, and $N$ is a\nleft $A^{{\\rm co}H}$-module, by restriction of scalars. Consider the isomorphism\n\\begin{eqnarray*}\n&&\\hspace*{-1cm}\n\\alpha_N=f\\otimes_{A\\square_H B} N:\\\nF_1(N)=A\\otimes_{A^{{\\rm co}H}} N\\cong A\\otimes_{A^{{\\rm co}H}} (A\\square_H B)\n\\otimes_{A\\square_H B} N\\\\\n&\\to& F(N)= (A\\otimes B)\\otimes_{A\\square_H B} N.\n\\end{eqnarray*}\nIt is easy to see that $\\alpha_N(a\\otimes n)=(a\\otimes 1)\\otimes_{A\\square_H B} n$, and $\\alpha_N$\nis right $H$-colinear since\n$$(\\alpha_N\\otimes H)((a_{[0]}\\otimes n)\\otimes a_{[1]})=((a_{[0]}\\otimes 1)\\otimes_{A\\square_H B}n)\\otimes a_{[1]}\n=\\rho((a\\otimes 1)\\otimes n).$$\nIt follows that $\\alpha_N$ restricts to  an isomorphism\n$$\\alpha_N^{{\\rm co}H}:\\ (A\\otimes_{A^{{\\rm co}H}} N)^{{\\rm co}H}\\to\n((A\\otimes B)\\otimes_{A\\square_H B} N)^{{\\rm co}H}.$$\nIt is then easily seen that\n$$\\eta_N=\\alpha_N^{{\\rm co}H}\\circ \\eta_{1,N}.$$\n$\\eta_{1,N}$ is an isomorphism by \\thref{1.1}, and it follows that $\\eta_N$ is\nan isomorphism.\\\\\nTake $M\\in {}_{A\\otimes B}\\mathcal{M}(H\\otimes H^{\\rm cop})^H$. Then $M$ is a left $A$-module,\nby restriction of scalars, and a relative Hopf module since\n$$\\rho(am)=\\rho((a\\otimes 1)m)=(a_{[0]}\\otimes 1)m_{[0]}\\otimes a_{[1]}m_{[1]}S(1)=\na_{[0]}m_{[0]}\\otimes a_{[1]}m_{[1]}.$$\nIt is then easy to see that\n$$\\varepsilon_M\\circ \\alpha_{M^{{\\rm co}H}}=\\varepsilon_{1,M}.$$\nIt follows from \\thref{1.1} that $\\varepsilon_{1,M}$ is an isomorphism, and this\nimplies that $\\varepsilon_M$ is an isomorphism.\n\\end{proof}\n\n\\section{Connection to comodules over corings}\\selabel{3}\nLet $A$ be a ring. Recall that an $A$-coring $\\mathcal{C}$ is a comonoid in the monoidal\ncategory ${}_A\\mathcal{M}_A$. For a detailed discussion of the theory of corings and\ncomodules, we refer to \\cite{BrzezinskiWisbauer}. One of the results is that we\ncan associate a coring to a Doi-Hopf datum, and that the category of Doi-Hopf modules\nis isomorphic to the category of comodules over this coring.\\\\\nLet us describe the $A\\otimes B$-coring $\\mathcal{C}$ associated to the\nleft-right Doi-Hopf datum $(H\\otimes H^{\\rm cop}, A\\otimes B, H)$ that we\nhave discussed in the previous section. We have that $\\mathcal{C}=H\\otimes\nA\\otimes B$, with left and right $A\\otimes B$-action given by\n$$(a'\\otimes b')(h\\otimes a\\otimes b)(a''\\otimes b'')=\na'_{[1]}hS(b'_{[-1]})\\otimes a'_{[0]}aa''\\otimes b'_{[0]}bb''.$$\nThe comultiplication and counit are given by the formulas\n$$\\Delta(h\\otimes a\\otimes b)=(h_{(2)}\\otimes 1_A\\otimes 1_B)\\otimes_{A\\otimes B} (h_{(1)}\\otimes a\\otimes b);$$\n$$\\varepsilon(h\\otimes a\\otimes b)=\\varepsilon(h)a\\otimes b.$$\nThe category ${}^\\mathcal{C}\\mathcal{M}$ of left $\\mathcal{C}$-comodules is isomorphic to\n${}_{A\\otimes B}\\mathcal{M}(H\\otimes H^{\\rm cop})^H$.\\\\\nA Galois theory for corings can be developed (see\n\\cite{Brzezinski02,Caenepeel03}). Let $x$ be a grouplike element\nof a coring $\\mathcal{C}$, and let $$A^{{\\rm co}\\mathcal{C}}= \\{a\\in\nA~|~ax=xa\\}.$$ Then we have an adjoint pair of functors between\n${}_{A^{{\\rm co}\\mathcal{C}}}\\mathcal{M}$ and ${}^\\mathcal{C}\\mathcal{M}$. If this adjoint pair is\na pair of inverse equivalences, then the map\n$${\\rm can}:\\ A\\otimes_{A^{{\\rm co}\\mathcal{C}}}A\\to \\mathcal{C},\\quad{\\rm can}(a\\otimes b)=axb$$\nis an isomorphism of corings (see \\cite[Proposition\n3.1]{Caenepeel03}). If, in addition, $A$ is flat as a right\n$A^{{\\rm co}\\mathcal{C}}$-module, then it also follows that $A$ is\nfaithfully flat as a right $A^{{\\rm co}\\mathcal{C}}$-module (see\n\\cite[Proposition 3.8, $2)\\Rightarrow 1)$]{Caenepeel03}). We will apply\nthis to the coring $\\mathcal{C}=H\\otimes A\\otimes B$. $1_H\\otimes 1_A\\otimes 1_B$ is a grouplike element of $H\\otimes A\\otimes B$, and\nthe associated pair of adjoint functors is precisely $(F,G)$. It\ncan be easily verified that the corresponding canonical map is\nprecisely the map\n\\begin{eqnarray}\n&&\\hspace*{-2cm}\n{\\rm can}:\\ (A\\otimes B)\\otimes_{A\\square_H B}(A\\otimes B)\\to H\\otimes A\\otimes B ,\\nonumber\\\\\n&&{\\rm can}((a\\otimes b)\\otimes (a'\\otimes b'))=a_{[1]}S(b_{[-1]})\\otimes a_{[0]}a'\\otimes b_{[0]}b'\\eqlabel{coring1}\n\\end{eqnarray}\n\n\\begin{proposition}\\prlabel{coring}\nLet $A$ be a right $H$-comodule algebra, and $B$ a left\n$H$-comodule algebra. Assume that $A$ is a right faithfully flat\n$H$-Galois extension. Then ${\\rm can}$\nis an isomorphism. Furthermore, $A\\otimes B$ is faithfully flat as a right\n$A\\square_H B$-module.\n\\end{proposition}\n\n\\begin{proof}\nIt follows from \\thref{2.3} that $(F,G)$ is a pair of inverse equivalences, hence\n${\\rm can}$ is an isomorphism.\\\\\nWe will now show that $A\\otimes B$ is flat as a right\n$A\\square_H B$-module.\nAssume that $N\\to N'$ is a monomorphism of left $A\\square_H B$-modules.\nUsing \\prref{2.2} and the fact that $A$ is flat as a right $A^{{\\rm co}H}$-module,\nwe find that\n$$(A\\otimes B)_{A\\square_HB} N\\cong A\\otimes _{A^{{\\rm co}H}} N\n\\to A\\otimes _{A^{{\\rm co}H}}N'\\cong (A\\otimes B)_{A\\square_HB} N'$$\nis injective. As explained above, it then follows from  \\cite[Proposition 3.8]{Caenepeel03} and \\leref{2.1}\nthat $A\\otimes B$ is faithfully flat as a right\n$A\\square_H B$-module.\n\\end{proof}\n\n\\section{Application to Hopf bimodules}\\selabel{4}\nNow let $A$ and $B$ be right $H$-comodule algebras.\nA two-sided\nrelative Hopf module is a $k$-module with a left $A$-action, a right $B$-action,\nand a right $H$-coaction, such that\n$$\\rho(amb)= a_{[0]}m_{[0]}b_{[0]}\\otimes a_{[1]}m_{[1]}b_{[1]},$$\nfor all $a\\in A$, $b\\in B$ and $m\\in M$.\n${}_A\\mathcal{M}_B^H$ is the category of two-sided relative Hopf modules with\n$k$-module maps that are $A$-linear, $B$-linear and $H$-colinear.\\\\\n\n $B^{{\\rm op}}$ is a\nleft $H$-comodule algebra, with left coaction $\\lambda$ given by\n$\\lambda(b)=S^{-1}(b_{[1]})\\otimes b_{[0]}$.\nWe can then apply the above results to $A$ and $B^{\\rm op}$. In particular,\n$A\\otimes B^{\\rm op}$ is a right $H\\otimes H^{\\rm cop}$-comodule algebra.\n\n\\begin{lemma}\\lelabel{2.4}\nLet $A$ and $B$ be right $H$-comodule algebras. Then the\nDoi-Hopf modules category\n${}_{A\\otimes B^{\\rm op}}\\mathcal{M}(H\\otimes H^{\\rm cop})^H$ is isomorphic to the\ncategory of two-sided relative Hopf modules ${}_A\\mathcal{M}_B^H$.\n\\end{lemma}\n\n\\begin{proof}\nIt is well-known that ${}_{A\\otimes B^{\\rm op}}\\mathcal{M}$ is isomorphic to\nthe category of bimodules ${}_A\\mathcal{M}_B$. The isomorphism respects the compatibility of\nthe action and coaction.\n\\end{proof}\n\n$A\\otimes B^{\\rm op}$ is a two-sided Hopf module, with coaction\n$\\rho(a\\otimes b)=a_{[0]}\\otimes b_{[0]}\\otimes a_{[1]}b_{[1]}$.\nFurthermore\n$(A\\otimes B^{\\rm op})^{{\\rm co}H}=A\\square_H B^{{\\rm op}}$.\nApplying \\thref{2.3}, we obtain the following Structure Theorem for\ntwo-sided Hopf modules.\n\n\\begin{theorem}\\thlabel{2.4a}\nLet $H$ be a Hopf algebra over the commutative ring $k$, with bijective antipode, and consider two right\n$H$-comodule algebras $A$ and $B$.\nWe have a pair of adjoint functors $(F=A\\otimes B^{\\rm op}\\otimes_{A\\square_H B^{\\rm op}}-,\\\nG=(-)^{{\\rm co}H})$ between the categories ${}_{A\\square_H B^{\\rm op}}\\mathcal{M}$\nand ${}_A\\mathcal{M}_B^H$. If $A$ is a right faithfully\nflat $H$-Galois extension, then $(F,G)$ is a pair of inverse equivalences.\n\\end{theorem}\n\n\\begin{remark} \\label{r:equiv} Assume that $A$  (resp. $B$) is a right\n(resp. left) faithfully\nflat $H$-Galois extension. The proof of \\thref{2.3} shows that\nvia appropriate transport of structure, the functors\n$$(A\\otimes B^{\\rm op})\\otimes_{A\\square B^{\\rm op}}-,\\ \\\nA\\otimes_{A^{{\\rm co}H}}-,\\ \\ -\\otimes_{B^{{\\rm\nco}H}}B:{}_{A\\square B^{\\rm op}}\\mathcal{M}\n\\to{}_A\\mathcal{M}^H_B$$ are naturally isomorphic equivalences of\ncategories. It follows immediately that we may define the functors\n$$-\\otimes_{A^{{\\rm co}H}}-: \\ _{B\\square A^{\\rm op}}\\mathcal{M}\\times \\\n_A\\mathcal{M}_C\\to{}_B\\mathcal{M}_C,$$\n$$-\\otimes_{A^{{\\rm co}H}}-: \\ _B\\mathcal{M}_A\\times{}_{A\\square\nC^{\\rm op}}\\mathcal{M}\\to \\ _B\\mathcal{M}_C.$$\n\\end{remark}\n\n\\begin{proposition}\\prlabel{3.1}\nLet $A,B,C$ be right $H$-comodule algebras. If $M\\in {}_A\\mathcal{M}_B^H$\nand $N\\in {}_B\\mathcal{M}_C^H$, then $M\\otimes_B N\\in  {}_A\\mathcal{M}_C^H$. If $A$ and $B$ are right\nfaithfully flat $H$-Galois extensions, then the map\n$$f:\\ M^{{\\rm co}H}\\otimes_{B^{{\\rm co}H}} N^{{\\rm co}H}\\to (M\\otimes_B N)^{{\\rm co}H},~~\nf(m\\otimes n)=m\\otimes n,$$\nis an isomorphism. Consequently $M^{{\\rm co}H}\\otimes_{B^{{\\rm co}H}} N^{{\\rm co}H}$\nis a left $A\\square_H C^{\\rm op}$-module.\n\\end{proposition}\n\n\\begin{proof}\nIt is clear that $M\\otimes_B N$ is an $(A,C)$-bimodule. A right $H$-coaction on\n$M\\otimes_B N$ is defined as follows:\n$$\\rho(m\\otimes_B n)=m_{[0]}\\otimes_B n_{[0]}\\otimes m_{[1]}n_{[1]}.$$\nIt is easy to show that $\\rho$ is well-defined, and that this coaction makes\n$M\\otimes_B N\\in {}_A\\mathcal{M}_C^H$.\\\\\nBy restriction of scalars, $M\\in {}_A\\mathcal{M}^H$ and $N\\in {}_B\\mathcal{M}^H$.\nIt follows from \\thref{1.1} that\n$$\\varepsilon_{1,M}:\\ A\\otimes_{A^{{\\rm co}H}} M^{{\\rm co}H}\\to M~~{\\rm and}~~\n\\varepsilon_{1,N}:\\ B\\otimes_{B^{{\\rm co}H}} N^{{\\rm co}H}\\to N$$\nare isomorphisms. Let $g$ be the composition of the maps\n$$\\varepsilon_{1,M}\\otimes_{B^{{\\rm co}H}} N^{{\\rm co}H}:\\\nA\\otimes_{A^{{\\rm co}H}}M^{{\\rm co}H}\\otimes_{B^{{\\rm co}H}}N^{{\\rm co}H}\n\\to M\\otimes_{B^{{\\rm co}H}}N^{{\\rm co}H} $$\nand\n$$M\\otimes_B\\varepsilon_{1,N}:\\ M\\otimes_{B^{{\\rm co}H}}N^{{\\rm co}H}\\cong M\\otimes_BB\\otimes_{B^{{\\rm co}H}}N^{{\\rm co}H}\n\\to M\\otimes_BN.$$\n$g$ is bijective, and is given by the formula\n$$g(a\\otimes m\\otimes n)=am\\otimes_B n,$$\nfor $a\\in A$, $m\\in M^{{\\rm co}H}$ and $n\\in N^{{\\rm co}H}$. It is clear that $g$ is\nleft $H$-linear. $g$ is also right $H$-colinear, since\n$g(a_{[0]}\\otimes m\\otimes n)\\otimes a_{[1]}=a_{[0]} m\\otimes n\\otimes a_{[1]}=\\rho(am\\otimes_B n)$, and it follows\nthat $g$ is an isomorphism in $ {}_A\\mathcal{M}^H$, and, by \\thref{1.1} that \n$$g^{{\\rm co}H}:\\ \\bigl(A\\otimes_{A^{{\\rm co}H}}M^{{\\rm co}H}\\otimes_{B^{{\\rm co}H}}N^{{\\rm co}H}\\bigr)^{{\\rm co}H}\n\\to (M\\otimes_BN)^{{\\rm co}H}$$\nis an isomorphism. The map $f$ is an isomorphism since it is the composition of $g^{{\\rm co}H}$ and the\nisomorphism\n$$\\eta_{1,M^{{\\rm co}H}\\otimes_{B^{{\\rm co}H}}N^{{\\rm co}H}}:\\\nM^{{\\rm co}H}\\otimes_{B^{{\\rm co}H}}N^{{\\rm co}H}\\to \\bigl(A\\otimes_{A^{{\\rm co}H}}M^{{\\rm co}H}\\otimes_{B^{{\\rm co}H}}N^{{\\rm co}H}\\bigr)^{{\\rm co}H}.$$\nFinally, the left $A\\square_HC^{\\rm op}$-action on $(M\\otimes_BN)^{{\\rm co}H}$ can be transported\nusing $f$ to $M^{{\\rm co}H}\\otimes_{B^{{\\rm co}H}}N^{{\\rm co}H}$.\n\\end{proof}\n\nIn the sequel, we will use the adjoint pair of functors $(F,G)$, with unit $\\eta$ and counit\n$\\varepsilon$ introduced after \\leref{2.1}, in the cases where the algebras involved are\nrespectively $A$ and $B^{\\rm op}$, $A$ and $C^{\\rm op}$ and $B$ and $C^{\\rm op}$.\nIf $A$ and $B$ are right faithfully flat $H$-Galois extensions, then these three adjunctions\nare pairs of inverse equivalences, by \\thref{2.4a}. We will use the same notation $(F,G)$\nand $(\\eta,\\varepsilon)$ for the three adjunctions, no confusion will arise from this.\\\\\nTake $M_1\\in {}_{A\\square_H B^{\\rm op}}\\mathcal{M}$ and $N_1\\in {}_{B\\square_H C^{\\rm op}}\\mathcal{M}$,\nand denote\n$$M= (A\\otimes B^{{\\rm op}})\\otimes_{A\\square_H B^{\\rm op}}M_1\\in {}_A\\mathcal{M}_B^H;$$\n$$N= (B\\otimes C^{{\\rm op}})\\otimes_{B\\square_H C^{\\rm op}}N_1\\in {}_B\\mathcal{M}_C^H.$$\nUsing \\thref{2.4a} and \\prref{3.1}, we find isomorphisms\n$$M_1\\otimes_{B^{{\\rm co}H}}N_1\\cong M^{{\\rm co}H}\\otimes_{B^{{\\rm co}H}} N^{{\\rm co}H}\n\\cong (M\\otimes_B N)^{{\\rm co}H}\\in {}_{A\\square_H C^{\\rm op}}\\mathcal{M}.$$\nTransporting structure, we find that $M_1\\otimes_{B^{{\\rm co}H}}N_1\\in {}_{A\\square_H C^{\\rm op}}\\mathcal{M}$,\nand we have a functor\n$$-\\otimes_{B^{{\\rm co}H}}-:\\ {}_{A\\square_H B^{\\rm op}}\\mathcal{M}\\times {}_{B\\square_H C^{\\rm op}}\\mathcal{M}\n\\to {}_{A\\square_H C^{\\rm op}}\\mathcal{M}.$$\n\n\\begin{corollary}\\colabel{3.2}\nLet $A,B,C$ be right $H$-comodule algebras, and\nassume that $A$ and $B$ are right faithfully flat $H$-Galois extensions. Take\n$M_1\\in {}_{A\\square_H B^{\\rm op}}\\mathcal{M}$ and $N_1\\in {}_{B\\square_H C^{\\rm op}}\\mathcal{M}$.\nWith notation as above, we have that\n$M_1\\otimes_{B^{{\\rm co}H}}N_1\\in {}_{A\\square_H C^{\\rm op}}\\mathcal{M}$,\nand we have an isomorphism \n$$h:\\  (A\\otimes C^{{\\rm op}})\\otimes_{A\\square_H C^{\\rm op}}\n(M_1\\otimes_{B^{{\\rm co}H}}N_1)\\to M\\otimes_B N$$\nin ${}_A\\mathcal{M}_C^H$. This isomorphism is natural in $M_1$ and $N_1$.\n\\end{corollary}\n\nFor later use, we observe that the naturality of $h$ means the following. Let\n$\\mu_1:\\ M_1\\to M'_1$ and $\\nu_1:\\ N_1\\to N'_1$ be morphisms in respectively\n${}_{A\\square_H B^{\\rm op}}\\mathcal{M}$ and ${}_{B\\square_H C^{\\rm op}}\\mathcal{M}$, and let\n$\\mu=F(\\mu_1)$, $\\nu=F(\\nu_1)$. Then $\\mu_1\\otimes_{B^{{\\rm co}H}}\\nu_1$ is a morphism\nin ${}_{A\\square_H C^{\\rm op}}\\mathcal{M}$, and the following diagram commutes\n\\begin{equation}\\eqlabel{3.2.2}\n\\xymatrix{\nF(M_1\\otimes_{B^{{\\rm co}H}}N_1)\n\\ar[rr]^{F (\\mu_1\\otimes \\nu_1)}\\ar[d]^{h}&&\nF(M'_1\\otimes_{B^{{\\rm co}H}}N'_1)\\ar[d]^{h}\\\\\nM\\otimes_BN\\ar[rr]^{\\mu\\otimes \\nu}&& M'\\otimes_BN'}\n\\end{equation}\n\nFrom now on, let $H$ be a projective Hopf algebra (this condition is always fulfilled\nif $k$ is a field); let $A$ be a right $H$-comodule algebra,\nand $M,N\\in {}_A\\mathcal{M}^H$. Then the map\n$$\\nu:\\ {}_A{\\rm Hom}(M,N)\\otimes H\\to {}_A{\\rm Hom}(M,N\\otimes H),~~\\nu(f\\otimes h)(m)=f(m)\\otimes h$$\nis injective (see for example \\cite[Prop. II.4.2, p. AII.75]{Bourbaki}).\nA direct computation shows that the map \n$\\tilde{\\rho}:\\ {}_A{\\rm Hom}(M,N)\\to {}_A{\\rm Hom}(M,N\\otimes H)$ defined by\n$$\\tilde{\\rho}(f)(m)=f(m_{[0]})_{[0]}\\otimes S^{-1}(m_{[1]})f(m_{[0]})_{[1]}$$\nis left $A$-linear. Let ${}_A{\\rm HOM}(M,N)$ be the $k$-submodule of\n${}_A{\\rm Hom}(M,N)$ consisting of the maps $f$ for which\n$\\tilde{\\rho}(f)$\nfactorizes through ${}_A{\\rm Hom}(M,N)$, or, equivalently, for which there\nexists $f_{[0]}\\otimes f_{[1]}\\in {}_A{\\rm Hom}(M,N)\\otimes H$ such that\n\\begin{equation}\\eqlabel{3.2.1}\nf_{[0]}(m)\\otimes f_{[1]}=f(m_{[0]})_{[0]}\\otimes S^{-1}(m_{[1]})f(m_{[0]})_{[1]},\n\\end{equation}\nfor all $m\\in M$. It follows from the injectivity of $\\nu$ that $f_{[0]}\\otimes f_{[1]}$\nis unique if it exists. ${}_A{\\rm HOM}(M,N)$ is called the rational part of\n${}_A{\\rm Hom}(M,N)$. If $H$ is finitely generated and projective, then\n$\\nu$ is bijective, and ${}_A{\\rm HOM}(M,N)={}_A{\\rm Hom}(M,N)$. We have a map\n$$\\rho=\\nu^{-1}\\circ \\tilde{\\rho}:\\ {}_A{\\rm HOM}(M,N)\\to {}_A{\\rm Hom}(M,N)\\otimes H,~~\n\\rho(f)=f_{[0]}\\otimes f_{[1]}.$$\n\n\\begin{proposition}\\prlabel{3.2x}\nLet $H$ be a projective Hopf algebra, $A$ a right $H$-comodule algebra,\nand $M,N\\in {}_A\\mathcal{M}^H$. Then $({}_A{\\rm HOM}(M,N),\\rho)$ is a right\n$H$-comodule.\n\\end{proposition}\n\n\\begin{proof}\n$N\\otimes H\\in {}_A\\mathcal{M}^H$ under the diagonal coaction.\nWe know that five of the six faces of the following diagram, namely all faces except the\ntop one, commute.\n$$\\xymatrix{\n\\scriptstyle{}_A{\\rm HOM}(M,N)\\ar[rr]^{\\rho}\\ar[dd]^{\\subset}\\ar[dr]^{\\rho}&&\\scriptstyle{}_A{\\rm Hom}(M,N)\\otimes H\\ar'[d][dd]^{\\nu}\n\\ar[dr]^{\\tilde{\\rho}\\otimes H}&\\\\\n&\\scriptstyle{}_A{\\rm Hom}(M,N)\\otimes H\\ar[dd]^(.7){\\nu}\\ar[rr]^(.35){(\\nu\\circ H)\\circ (id\\otimes \\Delta)}\n&&\\scriptstyle{}_A{\\rm Hom}(M,N\\otimes H)\\otimes H\\ar[dd]^{\\nu}\\\\\n\\scriptstyle{}_A{\\rm Hom}(M,N)\\ar'[r][rr]^(.3){\\tilde{\\rho}}\\ar[dr]^{\\tilde{\\rho}}&&\\scriptstyle{}_A{\\rm Hom}(M,N\\otimes H)\\ar[dr]^{\\tilde{\\rho}}&\\\\\n&\\scriptstyle{}_A{\\rm Hom}(M,N)\\otimes H)\\ar[rr]^{{}_A{\\rm Hom}(M,N\\otimes \\Delta)}\n&&\\scriptstyle{}_A{\\rm Hom}(M,N\\otimes H\\otimes H)}$$\nThis implies that the top face also commutes; this means that, for all\n$f\\in {}_A{\\rm HOM}(M,N)$,\n$$(\\nu\\otimes H)(f_{[0]}\\otimes \\Delta(f_{[1]}))=\\tilde{\\rho}(f_{[0]})\\otimes f_{[1]},$$\nand therefore $f_{[0]}\\otimes f_{[1]}\\in {}_A{\\rm HOM}(M,N)\\otimes H$. We then also have that\n$$(\\nu\\otimes H)(f_{[0]}\\otimes \\Delta(f_{[1]}))=(\\nu\\otimes H)(\\rho(f_{[0]})\\otimes f_{[1]}),$$\nand, since $\\nu\\otimes H$ is injective,\n$$f_{[0]}\\otimes \\Delta(f_{[1]})=\\rho(f_{[0]})\\otimes f_{[1]}.$$\nWe therefore have shown that $\\rho:\\ {}_A{\\rm HOM}(M,N)\\to {}_A{\\rm HOM}(M,N)\\otimes H$\nis a coassociative map. Finally, it follows immediately from \\equref{3.2.1}\nthat $\\varepsilon(f_{[1]})f_{[0]}=f$, for all $f\\in  {}_A{\\rm HOM}(M,N)$.\n\\end{proof}\n\nAn alternative description of ${}_A{\\rm HOM}(M,N)$ is the following:\n${}_A{\\rm Hom}(M,N)$ is a left $H^*$-module,\nwith action (see \\cite[6.5]{DNR} in the case where $k$ is a field):\n$$(h^*\\cdot f)(m)=\\langle h^*, S^{-1}(m_{[1]})f(m_{[0]})_{[1]}\\rangle f(m_{[0]})_{[0]}.$$\n${}_A{\\rm HOM}(M,N)$ is then the subspace of ${}_A{\\rm Hom}(M,N)$\nconsisting of left $A$-linear $f:\\ M\\to N$ for which there exists\na (unique) $f_{[0]}\\otimes f_{[1]}\\in {}_A{\\rm Hom}(M,N)\\otimes H$ such that\n$$(h^*\\cdot f)(m)=\\langle h^*,f_{[1]}\\rangle f_{[0]}(m).$$\n\n\\begin{proposition}\\prlabel{3.2c}\nLet $A$ be a right $H$-comodule algebra, with $H$ a projective Hopf algebra,\nand $M,N\\in {}_A\\mathcal{M}^H$. If $M$ is finitely generated projective as a left $A$-module, then\n${}_A{\\rm HOM}(M,N)$ coincides with ${}_A{\\rm Hom}(M,N)$. For $f\\in {}_A{\\rm HOM}(M,N)$,\nwe have\n\\begin{equation}\\eqlabel{3.2c.1}\n\\rho(f)=\\sum_i m_i^*\\cdot f(m_{i[0]})_{[0]}\\otimes S^{-1}(m_{i[1]})f(m_{i[0]})_{[1]},\n\\end{equation}\nwhere $\\sum_i m_i^*\\otimes_A m$ is a finite dual basis of $M\\in {}_A\\mathcal{M}$.\n\\end{proposition}\n\n\\begin{proof}\nWe used the following notation: for $m^*\\in {}_A{\\rm Hom}(M,A)$, and $n\\in N$, $m^*\\cdot n\n\\in {}_A{\\rm Hom}(M,N)$ is defined by\n$$(m^*\\cdot n)(m)=m^*(m)n.$$\nFor every $m\\in M$, we have that $m=\\sum_i m_i^*(m)m_i$, hence\n\\begin{equation}\\eqlabel{3.2c.2}\n\\rho(m)=\\sum_i m_i^*(m)_{[0]}m_{i[0]}\\otimes m_i^*(m)_{[1]}m_{i[1]}.\n\\end{equation}\nWe then compute that\n\\begin{eqnarray*}\n&&\\hspace*{-1cm}\nf(m_{[0]})_{[0]}\\otimes S^{-1}(m_{[1]})f(m_{[0]})_{[1]}\\\\\n&\\equal{\\equref{3.2c.2}}&\n\\sum_i f(m_i^*(m)_{[0]}m_{i[0]})_{[0]} \\otimes S^{-1}(m_i^*(m)_{[1]}m_{i[1]})f(m_i^*(m)_{[0]}m_{i[0]})_{[1]}\\\\\n&=&\n\\sum_i m_i^*(m)_{[0]}f(m_{i[0]})_{[0]} \\otimes S^{-1}(m_{i[1]})S^{-1} (m_i^*(m)_{[2]})\nm_i^*(m)_{[1]}f(m_{i[0]})_{[1]}\\\\\n&=& \\sum_i m_i^*(m)f(m_{i[0]})_{[0]}\\otimes S^{-1}(m_{i[1]})\n(m_{i[0]})_{[1]}\\\\\n&=& \\sum_i m_i^*\\cdot f(m_{i[0]})_{[0]}(m)\\otimes S^{-1}(m_{i[1]})f(m_{i[0]})_{[1]},\n\\end{eqnarray*}\nand \\equref{3.2c.1} follows from \\equref{3.2.1}.\n\\end{proof}\n\n\\begin{proposition}\\prlabel{3.3}\nLet $H$ be a projective Hopf algebra, and $A,B,C$ right $H$-comodule\nalgebras. If $M\\in {}_A\\mathcal{M}_B^H$ and $N\\in {}_A\\mathcal{M}_C^H$, then\n$${}_A{\\rm HOM}(M,N)\\in {}_B\\mathcal{M}_C^H.$$\nWe have a map\n$$\\beta:\\ {}_A{\\rm HOM}(M,N)^{{\\rm co}H}\\to {}_{A^{{\\rm co}H}}{\\rm Hom}(M^{{\\rm co}H},N^{{\\rm co}H}).$$\nIf $A$ is a right faithfully flat $H$-Galois extension, then $\\beta$ is an isomorphism of left $B\\square C^{\\rm op}$-modules.\n\\end{proposition}\n\n\\begin{proof}\nWe consider the following $(B,C)$-bimodule structure on ${}_A{\\rm Hom}(M,N)$:\n$$(b\\cdot f\\cdot c)(m)=f(mb)c.$$\nIt is clear that $b\\cdot f\\cdot c$ is then left $A$-linear.\nTake $f\\in {}_A{\\rm HOM}(M,N)$; in order to show that $b\\cdot f\\cdot c\\in {}_A{\\rm HOM}(M,N)$,\nit suffices to show that $b_{[0]}\\cdot f_{[0]}\\cdot c_{[0]}\\otimes b_{[1]}f_{[1]}c_{[1]}$\nsatisfies \\equref{3.2.1}. This can be seen as follows: for all $m\\in M$, we have\n\\begin{eqnarray*}\n&&\\hspace*{-2cm}\n(b_{[0]}\\cdot f_{[0]}\\cdot c_{[0]})(m)\\otimes b_{[1]}f_{[1]}c_{[1]}\\\\\n&=& f_{[0]}(mb_{[0]})c_{[0]}\\otimes b_{[1]}f_{[1]}c_{[1]}\\\\\n&=& f\\bigl((mb_{[0]})_{[0]}\\bigr)_{[0]}c_{[0]}\\otimes b_{[1]}S^{-1}\\bigl((mb_{[0]})_{[1]}\\bigr)\n f\\bigl((mb_{[0]})_{[0]}\\bigr)_{[1]}c_{[1]}\\\\\n&=& f(m_{[0]}b_{[0]})_{[0]}c_{[0]}\\otimes b_{[2]}S^{-1}(b_{[1]})S^{-1}(m_{[1]})\nf(m_{[0]}b_{[0]})_{[1]}c_{[1]}\\\\\n&=& f(m_{[0]}b)_{[0]}c_{[0]}\\otimes S^{-1}(m_{[1]})f(m_{[0]}b)_{[1]} c_{[1]}\\\\\n&=& \\bigl(f(m_{[0]}b)c\\bigr)_{[0]}\\otimes S^{-1}(m_{[1]})  \\bigl(f(m_{[0]}b)c\\bigr)_{[1]}\\\\\n&=& (b\\cdot f\\cdot c)(m_{[0]})_{[0]}\\otimes S^{-1}(m_{[1]}) (b\\cdot f\\cdot c)(m_{[0]})_{[1]},\n\\end{eqnarray*}\nas needed. This shows also that $\\rho(b\\cdot f\\cdot c)=\nb_{[0]}\\cdot f_{[0]}\\cdot c_{[0]}\\otimes b_{[1]}f_{[1]}c_{[1]}$, hence that\n${}_A{\\rm HOM}(M,N)\\in {}_B\\mathcal{M}_C^H$.\\\\\nNow take $ f\\in {}_A{\\rm HOM}(M,N)^{{\\rm co}H}$. Then $\\rho(f)=f_{[0]}\\otimes f_{[1]}$, so\n$$f(m)\\otimes 1=f(m_{[0]})_{[0]}\\otimes S^{-1}(m_{[1]})f(m_{[0]})_{[1]},$$\nfor all $m\\in M$. If $m\\in M^{{\\rm co}H}$, then it follows that $f(m)\\otimes 1=\\rho(f(m))$,\nso $f(m)\\in N^{{\\rm co}H}$. Thus $f$ restricts to a map $\\beta(f)=f^{{\\rm co}H}:\\\nM^{{\\rm co}H}\\to N^{{\\rm co}H}$. Using the fact that $f$ is left $A$-linear, we see\nthat the diagram\n$$\\diagram\nA\\otimes_{A^{{\\rm co}H}} M^{{\\rm co}H}\\rrto^{A\\otimes f^{{\\rm\nco}H}}\\dto_{\\varepsilon_{1,M}} &&\nA\\otimes_{A^{{\\rm co}H}} N^{{\\rm co}H}\\dto^{\\varepsilon_{1,N}}\\\\\nM\\rrto^{f}&& N\n\\enddiagram$$\ncommutes. If $A$ is right faithfully flat $H$-Galois, then we define the\ninverse of $\\beta$ as follows:\n$$\\beta^{-1}(g)=\\varepsilon_{1,N}\\circ (A\\otimes g)\\circ \\varepsilon_{1,M}^{-1}.$$\n\\end{proof}\n\nCombining \\prref{3.3} with \\thref{2.4a}, we obtain the following\nresult.\n\n\\begin{corollary}\\colabel{3.4}\nLet $A,B,C$ be right $H$-comodule algebras, and assume that $A$ and $B$ are right faithfully flat\n$H$-Galois extensions.\nLet $M_1\\in {}_{A\\square B^{\\rm op}}\\mathcal{M}$ and $N_1\\in {}_{A\\square C^{\\rm op}}\\mathcal{M}$,\nand consider\n$$M= (A\\otimes B^{\\rm op})\\otimes_{A\\square B^{\\rm op}}M_1\\in {}_A\\mathcal{M}_B^H,$$\n$$N= (A\\otimes C^{\\rm op})\\otimes_{A\\square C^{\\rm op}}N_1\\in {}_A\\mathcal{M}_C^H.$$\nThen\n$${}_{A^{{\\rm co}H}}{\\rm Hom}(M_1,N_1)\n\\cong {}_A{\\rm HOM}(M,N)^{{\\rm co}H}\\in {}_{B\\square C^{\\rm op}}\n\\mathcal{M}$$\nand\n$${}_A{\\rm HOM}(M,N)\\cong (B\\otimes C^{\\rm op})\\otimes_{B\\square C^{\\rm op}}\n{}_{A^{{\\rm co}H}}{\\rm Hom}(M_1,N_1).$$\n\\end{corollary}\n\n\\begin{proposition}\\prlabel{3.5}\nLet $A,B,C$ be right $H$-comodule algebras, and consider $M\\in {}_A\\mathcal{M}_B^H$,\n$N\\in {}_A\\mathcal{M}_C^H$. Then the evaluation map\n$$\\varphi:\\ M\\otimes_B {}_A{\\rm HOM}(M,N)\\to N,~~\\varphi(m\\otimes_B f)=f(m)$$\nis a morphism in ${}_A\\mathcal{M}_C^H$.\\\\\nIf $A$ and $B$ are right faithfully flat $H$-Galois extensions, then the evaluation map\n$$M^{{\\rm co}H}\\otimes_{B^{{\\rm co}H}} {}_{A^{{\\rm co}H}}{\\rm Hom}(M^{{\\rm co}H},N^{{\\rm co}H})\n\\to N^{{\\rm co}H}$$\nis left $A\\square_H C^{\\rm op}$-linear.\n\\end{proposition}\n\n\\begin{proof}\nWe first show that $\\varphi$ is right $H$-colinear.\n\\begin{eqnarray*}\n&&\\hspace*{-2cm}\n(\\varphi\\otimes H)(\\rho(m\\otimes f))=(\\varphi\\otimes H)(m_{[0]}\\otimes_B f_{[0]}\\otimes m_{[1]}f_{[1]}) \\\\\n&=& f_{[0]}(m_{[0]})\\otimes m_{[1]}f_{[1]}\\\\\n&\\equal{\\equref{3.2.1}}&f(m_{[0]})_{[0]}\\otimes m_{[2]}S^{-1}(m_{[1]})f(m_{[0]})_{[1]}\\\\\n&=& \\rho(f(m))=\\rho(\\varphi(m\\otimes_B f)).\n\\end{eqnarray*}\n$\\varphi$ is left $A$-linear and right $C$-linear since\n$$\\varphi(am\\otimes f\\cdot c)=(f\\cdot c)(am)=f(am)c=af(m)c=a\\varphi(m\\otimes_B f)c.$$\nThe composition\n\\begin{eqnarray*}\n&&\\hspace*{-8mm} M^{{\\rm co}H}\\otimes_{B^{{\\rm co}H}} {}_{A^{{\\rm\nco}H}}{\\rm Hom}(M^{{\\rm co}H},N^{{\\rm co}H}) \n\\stackrel{M^{{\\rm\nco}H}\\otimes\\beta}{\\longrightarrow}\nM^{{\\rm co}H}\\otimes_{B^{{\\rm co}H}} {}_A{\\rm HOM}(M,N)^{{\\rm co}H}\\\\\n&&\\hspace*{1cm}\n\\stackrel{f}{\\longrightarrow} (M\\otimes_B {}_A{\\rm HOM}(M,N))^{{\\rm co}H}\n\\stackrel{\\varphi^{{\\rm co}H}}{\\longrightarrow} N^{{\\rm co}H}\n\\end{eqnarray*}\nis the required evaluation map. If $A$ is right faithfully flat $H$-Galois, then\n$\\beta$ is an isomorphism of $B\\square_H C^{\\rm op}$-modules, by \\prref{3.3},\nand then $M^{{\\rm co}H}\\otimes\\beta$ is an isomorphism of $A\\square_H C^{\\rm op}$-modules,\nby \\coref{3.2}.\nIf $B$ is right faithfully flat $H$-Galois, then\n$f$ is an isomorphism of $A\\square_H C^{\\rm op}$-modules, by \\prref{3.1}.\n$\\varphi$ is a morphism in ${}_A\\mathcal{M}_C^H$, hence\n$\\varphi^{{\\rm co}H}$ is left $A\\square_H C^{\\rm op}$-linear, since $(-)^{{\\rm co}H}$\nis a functor from ${}_A\\mathcal{M}_C^H$ to ${}_{A\\square_H C^{\\rm op}}\\mathcal{M}$.\n\\end{proof}\n\n\\begin{proposition}\\prlabel{3.6}\nLet $A$ be a right $H$-comodule algebra, and $M\\in {}_A\\mathcal{M}^H$. Then ${}_A{\\rm END}(M)^{\\rm op}$\nis a right $H$-comodule algebra.\n\\end{proposition}\n\n\\begin{proof}\nApplying \\prref{3.3} (with $M=N$, $B=C=k$), we see that ${}_A{\\rm END}(M)$ is a right $H$-comodule.\nWe have to show the compatibility relation\n\\begin{equation}\\eqlabel{3.6.1}\n\\rho(g\\circ f)=g_{[0]}\\circ f_{[0]}\\otimes f_{[1]}g_{[1]},\n\\end{equation}\nfor all $f,g\\in {}_A{\\rm END}(M)$. To this end, it suffices to show that the right hand side of \\equref{3.6.1}\nsatisfies \\equref{3.2.1}. Indeed, for all $m\\in M$, we have\n\\begin{eqnarray*}\n&&\\hspace*{-13mm}\n(g_{[0]}\\circ f_{[0]})(m)\\otimes f_{[1]}g_{[1]}\\\\\n&\\equal{\\equref{3.2.1}}&\ng_{[0]}\\bigl(f(m_{[0]})_{[0]}\\bigr)\\otimes S^{-1}(m_{[1]})f(m_{[0]})_{[1]}g_{[1]}\\\\\n&\\equal{\\equref{3.2.1}}&\ng\\bigl(f(m_{[0]})_{[0]}\\bigr)_{[0]}\\otimes S^{-1}(m_{[1]})f(m_{[0]})_{[2]}\nS^{-1}\\bigl(f(m_{[0]})_{[1]}\\bigr)g\\bigl(f(m_{[0]})_{[0]}\\bigr)_{[1]}\\\\\n&=&g\\bigl(f(m_{[0]})\\bigr)_{[0]}\\otimes S^{-1}(m_{[1]})g\\bigl(f(m_{[0]})\\bigr)_{[1]}\\\\\n&=&(g\\circ f)(m_{[0]})_{[0]}\\otimes S^{-1}(m_{[1]}) (g\\circ f)(m_{[0]})_{[1]}.\n\\end{eqnarray*}\n\\end{proof}\n\n\\begin{proposition}\\prlabel{3.7}\nLet $A,B$ be right $H$-comodule algebras, and consider\n$M\\in {}_A\\mathcal{M}_B^H$. Then the map\n$$\\psi:\\ B\\to {}_A{\\rm END}(M),~~\\psi(b)(m)=mb$$\nis a morphism in ${}_B\\mathcal{M}_B^H$.\\\\\nIf $A$ is a right faithfully flat $H$-Galois extension, then the map\n$$\\psi^{{\\rm co}H}:\\ B^{{\\rm co}H}\\to {}_A{\\rm END}(M)^{{\\rm co}H}\\cong {}_{A^{{\\rm co}H}}\n{\\rm End}(M^{{\\rm co}H})$$\nis left $B\\square_H B^{\\rm op}$-linear.\n\\end{proposition}\n\n\\begin{proof}\nWe first show that $\\psi$ is right $H$-colinear and well-defined. Indeed,\n$$\\psi(b)_{[0]}\\otimes \\psi(b)_{[1]}=\\psi(b_{[0]})\\otimes b_{[1]},$$\nsince\n\\begin{eqnarray*}\n&&\\hspace*{-2cm}\n\\psi(b)(m_{[0]})_{[0]}\\otimes S^{-1}(m_{[1]}) \\psi(b)(m_{[0]})_{[1]}\\\\\n&=& (m_{[0]}b)_{[0]} \\otimes S^{-1}(m_{[1]}) (m_{[0]}b)_{[1]}\\\\\n&=& m_{[0]}b_{[0]}\\otimes S^{-1}(m_{[2]})m_{[1]}b_{[1]}\\\\\n&=& mb_{[0]}\\otimes b_{[1]}=\\psi(b_{[0]})(m)\\otimes b_{[1]}.\n\\end{eqnarray*}\n$\\psi$ is left and right $B$-linear since\n$$\\psi(b'bb'')(m)=mb'bb''=((b'\\cdot \\psi\\cdot b'')(b))(m),$$ for all $b,b',b''\\in B$\nand $m\\in M$. The second statement then follows immediately from \\coref{3.4}.\n\\end{proof}\n\n\\begin{remark}\\relabel{3.7b}\nThe map $\\psi$ in \\prref{3.7} is also a morphism of right $H$-comodule algebras\nbetween $B$ and ${}_A{\\rm END}(M)^{\\rm op}$.\n\\end{remark}\n\n\\begin{proposition}\\prlabel{3.8}\nLet $A,B,C$ be right $H$-comodule algebras, and consider\n$M\\in {}_A\\mathcal{M}_B^H$, $N\\in {}_A\\mathcal{M}_C^H$. Then the map\n$$\\mu:\\ {}_A{\\rm HOM}(M,A)\\otimes_A N\\to {}_A{\\rm HOM}(M,N),~~\n\\mu(f\\otimes n)(m)=f(m)n$$\nis a morphism in ${}_B\\mathcal{M}_C^H$. If $A$ is a right faithfully flat $H$-Galois extension, then\nthe map\n\\begin{eqnarray*}\n&&\\hspace*{-1cm}\n\\mu^{{\\rm co}H}:\\ {}_{A^{{\\rm co}H}}{\\rm Hom}(M^{{\\rm co}H},A^{{\\rm co}H})\\otimes_{A^{{\\rm co}H}}\n{N^{{\\rm co}H}}\n\\cong ({}_A{\\rm HOM}(M,A)\\otimes_A N)^{{\\rm co}H}\\\\\n&&\\hspace*{1cm}\\to\n{}_{A^{{\\rm co}H}}{\\rm Hom}(M^{{\\rm co}H},N^{{\\rm co}H})\\cong {}_A{\\rm HOM}(M,N)^{{\\rm co}H}\n\\end{eqnarray*}\nis left $B\\square_H C^{\\rm op}$-linear.\n\\end{proposition}\n\n\\begin{proof}\nIn order to prove that $\\mu$ is right $H$-colinear, we have to show that\n$$\\rho(\\mu(f\\otimes n))=\\mu(f_{[0]}\\otimes n_{[0]})\\otimes f_{[1]}n_{[1]}.$$\nIt suffices to compute that\n\\begin{eqnarray*}\n&&\\hspace*{-2cm}\n\\mu(f_{[0]}\\otimes n_{[0]})(m)\\otimes f_{[1]}n_{[1]}\n\\equal{\\equref{3.2.1}} f(m_{[0]})_{[0]}n_{[0]}\\otimes S^{-1}(m_{[1]})f(m_{[0]})_{[1]}n_{[1]}\\\\\n&=& (f(m_{[0]})n)_{[0]}\\otimes S^{-1}(m_{[1]})(f(m_{[0]})n)_{[1]}\\\\\n&=& (\\mu(f\\otimes n)(m_{[0]}))_{[0]}\\otimes S^{-1}(m_{[1]})(\\mu(f\\otimes n)(m_{[0]}))_{[1]}.\n\\end{eqnarray*}\n\nFinally, $\\mu$ is left $B$-linear and right $C$-linear, since\n$$(\\mu(b\\cdot f\\otimes nc))(m)=f(mb)nc=\\mu(f\\otimes n)(mb)c\n=(b\\cdot \\mu(f\\otimes n)\\cdot c)(m).$$\n\\end{proof}\n\n\n\\section{Morita equivalences}\\selabel{5}\nIn this section, we study Morita equivalences induced by two-sided relative\nHopf modules.\n\n\\begin{definition}\\delabel{5.1}\nLet $A$ and $B$ be right $H$-comodule algebras. An $H$-Morita context\nconnecting $A$ and $B$ is a Morita context $(A,B,M,N,\\alpha,\\beta)$ such that\n$M\\in {}_A\\mathcal{M}_B^H$, $N\\in {}_B\\mathcal{M}_A^H$,  $\\alpha:\\ M\\otimes_B N\\to A$\nis a morphism in ${}_A\\mathcal{M}_A^H$ and $\\beta:\\ N\\otimes_A M\\to B$ is a morphism in\n${}_B\\mathcal{M}_B^H$.\n\\end{definition}\n\nA morphism between two $H$-Morita contexts $(A,B,M,N,\\alpha,\\beta)$\nand $(A',B',$ $M',N',\\alpha',\\beta')$ is defined in the obvious way: it consists of a \nfourtuple $(\\kappa,\\lambda,$ $\\mu,\\nu)$, where $\\kappa:\\ A\\to A'$ and $\\lambda:\\\nB\\to B'$ are $H$-comodule algebra maps, $\\mu:\\ M\\to M'$ is a morphism in\n${}_A\\mathcal{M}_B^H$ and $\\nu:\\ N\\to N'$ is a morphism in ${}_B\\mathcal{M}_A^H$\nsuch that $\\kappa\\circ\\alpha=\\alpha'\\circ (\\mu\\otimes\\nu)$ and $\\lambda\\circ\\beta\n=\\beta'\\circ (\\nu\\otimes\\mu)$. $\\dul{\\rm Morita}^H(A,B)$ will be the subcategory of the category of\n$H$-Morita contexts, consisting of $H$-Morita contexts connecting $A$ and $B$,\nand morphisms with the identity of $A$ and $B$ as the underlying algebra maps.\n\n\\begin{proposition}\\prlabel{5.2}\nLet $(A,B,M,N,\\alpha,\\beta)$ be a strict $H$-Morita context. Then\nwe have a pair of inverse equivalences $(M\\otimes_B-, N\\otimes_A-)$\nbetween the categories ${}_A\\mathcal{M}^H$ and ${}_B\\mathcal{M}^H$.\n\\end{proposition}\n\n\\begin{proof}\nLet $P\\in {}_B\\mathcal{M}^H$. Then $M\\otimes_B P\\in {}_A\\mathcal{M}^H$, with right $H$-action\n$$\\rho(m\\otimes_B p)=m_{[0]}\\otimes_B p_{[0]}\\otimes m_{[1]}p_{[1]}.$$ The rest of the proof is\nstraightforward.\n\\end{proof}\n\nWe will now give an $H$-comodule version of \\cite[Prop. 4.2.1]{Bass}.\n\n\\begin{example}\\exlabel{5.3}\nLet $A$ be a right $H$-comodule algebra, and $M\\in {}_A\\mathcal{M}^H$. Then\n$B={}_A{\\rm END}(M)^{\\rm op}$ is also a right $H$-module algebra, by \\prref{3.6}.\nThen $M\\in {}_A\\mathcal{M}^H_B$, with right $B$-action given by $m\\cdot f=f(m)$, for all\n$f\\in B$ and $m\\in M$. Indeed, $(m\\cdot f)\\cdot g=m\\cdot (g\\circ f)$, and\n\\begin{eqnarray*}\n&&\\hspace*{-2cm}\nm_{[0]}\\cdot f_{[0]}\\otimes m_{[1]}f_{[1]}= f_{[0]}(m_{[0]})\\otimes m_{[1]}f_{[1]}\\\\\n&\\equal{\\equref{3.2.1}}& f(m_{[0]})_{[0]}\\otimes m_{[2]}S^{-1}(m_{[1]})f(m_{[0]})_{[1]}\\\\\n&=&\\rho(f(m))=\\rho(m\\cdot f).\n\\end{eqnarray*}\nIt follows from \\prref{3.3} that $N={}_A{\\rm HOM}(M,A)\\in {}_B\\mathcal{M}_A^H$, and from\n\\prref{3.5} that the map\n$$\\alpha:\\ M\\otimes_BN\\to A,~~\\alpha(m\\otimes n)=n(a)$$\nis a morphism in ${}_A\\mathcal{M}_A^H$. It follows from \\prref{3.8} that the map\n$$\\beta:\\ N\\otimes_A M\\to {}_A{\\rm END}(M),~~\\beta(n\\otimes m)(x)=n(x)m$$\nis a morphism in  ${}_B\\mathcal{M}_B^H$. Straightforward computations then show that\n$(A,B,M,N,\\alpha,\\beta)$ is an $H$-Morita context. We call it the $H$-Morita context\nassociated to $M\\in {}_A\\mathcal{M}^H$.\n\\end{example}\n\n\\begin{proposition}\\prlabel{5.4}\nThe $H$-Morita context associated to $M\\in {}_A\\mathcal{M}^H$ is strict if and only if $M$ is\na progenerator as a left $A$-module.\n\\end{proposition}\n\n\\begin{proof}\nIf the Morita context is strict, then $M$ is a left $A$-progenerator by \\cite[Theorem III.3.5]{Bass}.\nConversely, if $M$ is a left $A$-progenerator, then $M\\in {}_A\\mathcal{M}$ is finitely generated and\nprojective, hence ${}_A{\\rm Hom}(M,X)={}_A{\\rm HOM}(M,X)$, for all $X\\in {}_A\\mathcal{M}^H$. If we forget\nthe $H$-comodule structure in the $H$-Morita context, then we obtain the Morita context\nassociated to $M\\in {}_A\\mathcal{M}$, as in \\cite[Prop. II.4.1]{Bass}. By \\cite[Prop. II.4.4]{Bass}, this\nMorita context is strict.\n\\end{proof}\n\n\\begin{proposition}\\prlabel{5.5}\nLet $(A,B,M,N,\\alpha,\\beta)$ be a strict $H$-Morita context. Then $M$ is a left $A$-progenerator,\nand the $H$-Morita context is isomorphic to the $H$-Morita context associated to\n$M\\in {}_A\\mathcal{M}^H$.\n\\end{proposition}\n\n\\begin{proof}\n$M$ is a left $A$-progenerator by \\cite[Theorem III.3.5]{Bass}. Then ${}_A{\\rm End}(M)={}_A{\\rm END}(M)$,\nand by \\cite[Theorem II.3.4]{Bass}, $\\psi:\\ B\\to {}_A{\\rm END}(M)^{\\rm op}$, $\\psi(b)(m)=mb$\nis an isomorphism of $k$-algebras. It is an isomorphism of $H$-comodule algebras, by\n\\reref{3.7b}. It follows from \\cite[Theorem 3.4]{Bass} that\n$$\\varphi:\\ N\\to {}_A{\\rm HOM}(M,A)={}_A{\\rm Hom}(M,A),~~\\varphi(n)(m)=\\alpha(m\\otimes n)$$\nis an isomorphism of $(B,A)$-bimodules. We verify that $\\varphi$ is $H$-colinear.\nFor every $n\\in N$, we have to show that\n\\begin{equation}\\eqlabel{5.5.1}\n\\varphi(n_{[0]})\\otimes n_{[1]}=\\varphi(n)_{[0]}\\otimes \\varphi(n)_{[1]}.\n\\end{equation}\nUsing the right $H$-colinearity of $\\alpha$, we find\n\\begin{eqnarray*}\n&&\\hspace*{-15mm}\n\\alpha(m_{[0]}\\otimes_B n)_{[0]}\\otimes S^{-1}(m_{[1]})\\alpha(m_{[0]}\\otimes_B n)_{[1]}\\\\\n&=& \\alpha(m_{[0]}\\otimes_B n_{[0]})\\otimes S^{-1}(m_{[2]})m_{[1]}n_{[1]}\n= \\alpha(m\\otimes_B n_{[0]})\\otimes n_{[1]},\n\\end{eqnarray*}\nand \\equref{5.5.1} follows from \\equref{3.2.1}. From classical Morita theory (see \\cite{Bass}), we\nknow that $(A,\\psi,M,\\varphi)$ is an isomorphism of Morita contexts; since $\\psi$ and $\\varphi$\nare $H$-colinear, it follows that is an isomorphism of $H$-Morita contexts.\n\\end{proof}\n\n\\begin{definition}\\delabel{5.6}\nAssume that $A$ and $B$ are right faithfully flat $H$-Galois extensions of $A^{{\\rm co}H}$ and $B^{{\\rm co}H}$.\nA $\\square_H$-Morita context between $A^{{\\rm co}H}$ and $B^{{\\rm co}H}$ is a Morita context\n$(A^{{\\rm co}H},B^{{\\rm co}H},M_1,N_1,\\alpha_1,\\beta_1)$ such that $M_1$ (resp. $N_1$) is a left\n$A\\square_HB^{\\rm op}$-module (resp. $B\\square_HA^{\\rm op}$-module) and\n\\begin{itemize}\n\\item $\\alpha_1:\\ M_1\\otimes_{B^{{\\rm co}H}} N_1\\to A^{{\\rm co}H}$ is left $A\\square_HA^{\\rm op}$-linear,\n\\item $\\beta_1:\\ N_1\\otimes_{A^{{\\rm co}H}} M_1\\to B^{{\\rm co}H}$ is left $B\\square_HB^{\\rm op}$-linear.\n\\end{itemize}\n\\end{definition}\n\nA morphism between two $\\square_H$-Morita contexts connecting $A^{{\\rm co}H}$ and $B^{{\\rm co}H}$,\nis a morphism between Morita contexts of the form $(A^{{\\rm co}H},B^{{\\rm co}H},\\mu_1,\\nu_1)$,\nwhere $\\mu_1$ is left $A\\square_HB^{\\rm op}$-linear and $\\nu_1$ is left $B\\square_HA^{\\rm op}$-linear.\nThe category of $\\square_H$-Morita contexts connecting $A^{{\\rm co}H}$ and $B^{{\\rm co}H}$\nwill be denoted by $\\dul{\\rm Morita}^{\\square_H}(A^{{\\rm co}H},B^{{\\rm co}H})$.\n\n\\begin{theorem}\\thlabel{5.7}\nLet $A$ and $B$ be right faithfully flat $H$-Galois extensions of $A^{{\\rm co}H}$ and $B^{{\\rm co}H}$.\nThen the categories $\\dul{\\rm Morita}^H(A,B)$ and $\\dul{\\rm Morita}^{\\square_H}(A^{{\\rm co}H},$ $B^{{\\rm co}H})$\nare equivalent. The equivalence functors send strict contexts to strict contexts.\n\\end{theorem}\n\n\\begin{proof}\nLet $(A,B,M,N,\\alpha,\\beta)$ be an $H$-Morita context. It follows from\n\\thref{2.4a} that $M^{{\\rm co}H}\n\\in{}_{A\\square_H B^{\\rm op}}\\mathcal{M}$, and $N^{{\\rm co}H}\n\\in {}_{B\\square_H A^{\\rm op}}\\mathcal{M}$.\nIt follows from \\prref{3.1} that we have a left $A\\square_H A^{\\rm op}$-linear map\n$$\\alpha_1=\\alpha^{{\\rm co}H}\\circ f:\\ M^{{\\rm co}H}\\otimes_{B^{{\\rm co}H}} N^{{\\rm co}H}\n\\to (M\\otimes _B N)^{{\\rm co}H}\\to A^{{\\rm co}H},$$\nand\na left $B\\square_H B^{\\rm op}$-linear isomorphism\n$$\\beta_1=\\beta^{{\\rm co}H}\\circ f:\\ N^{{\\rm co}H}\\otimes_{A^{{\\rm co}H}} M^{{\\rm co}H}\n\\to (N\\otimes _A M)^{{\\rm co}H}\\to B^{{\\rm co}H}.$$\nFrom the description of $f$ in \\prref{3.1}, it follows that we have a commutative diagram\nof isomorphisms\n$$\\diagram\nM^{{\\rm co}H}\\otimes_{B^{{\\rm co}H}}N^{{\\rm co}H}\\otimes_{A^{{\\rm co}H}}M^{{\\rm co}H}\n\\rto\\dto & (M\\otimes_BN)^{{\\rm co}H}\\otimes_{A^{{\\rm co}H}}M^{{\\rm co}H}\\dto\\\\\nM^{{\\rm co}H}\\otimes_{B^{{\\rm co}H}}(N\\otimes_A M)^{{\\rm co}H}\\rto &\n(M\\otimes_BN\\otimes_A M)^{{\\rm co}H}\n\\enddiagram$$\nNow $\\alpha\\otimes_A M= M\\otimes_B \\beta$ implies $(\\alpha\\otimes_A M)^{{\\rm\nco}H}= (M\\otimes_B \\beta)^{{\\rm co}H}$, and it follows that\n$$\\alpha_1\\otimes_{A^{{\\rm co}H}} M^{{\\rm co}H}=M^{{\\rm\nco}H}\\otimes_{B^{{\\rm co}H}}\\beta.$$ In a similar way, we have that\n$$\\beta_1\\otimes_{B^{{\\rm co}H}} N^{{\\rm co}H}=N^{{\\rm\nco}H}\\otimes_{A^{{\\rm co}H}}\\alpha$$ and it follows that $(A^{{\\rm\nco}H}, B^{{\\rm co}H}, M^{{\\rm co}H},N^{{\\rm\nco}H},\\alpha_1,\\beta_1)$ is a Morita context. If\n$(A,B,M,N,\\alpha,\\beta)$ is strict, then $(A^{{\\rm co}H}, B^{{\\rm\nco}H}, M^{{\\rm co}H},$ $N^{{\\rm co}H},\\alpha_1,\\beta_1)$ is also\nstrict.\\\\\nConversely, let $(A^{{\\rm co}H},B^{{\\rm co}H},M_1,N_1,\\alpha_1,\\beta_1)$\nbe a $\\square_H$-Morita context. Then\n$M=F(M_1)=(A\\otimes B^{\\rm op})\\otimes_{A\\square_H B^{\\rm op}}M_1\\in {}_A\\mathcal{M}_B^H$ and\n$N=F(N_1)=(B\\otimes A^{\\rm op})\\otimes_{B\\square_H A^{\\rm op}}N_1\\in {}_B\\mathcal{M}_A^H$.\nAlso observe that $A\\cong F(A^{{\\rm co}H})= (A\\otimes A^{\\rm op})\\otimes_{A\\square_H A^{\\rm op}}A^{{\\rm co}H}$ and\n$B\\cong F(B^{{\\rm co}H})=(B\\otimes B^{\\rm op})\\otimes_{B\\square_H B^{\\rm op}}B^{{\\rm co}H}$. We define\n$\\alpha:\\ M\\otimes_B N\\to A$ and $\\beta:\\ N\\otimes_A M\\to N$ by the commutativity of the following\ntwo diagrams, where the isomorphisms $h$ are defined as in \\coref{3.2}.\n\\begin{equation}\\eqlabel{5.7.1}\n\\xymatrix{\n~~F(M_1\\otimes_{B^{{\\rm co}H}}N_1)~~\n\\ar[r]^-{F (\\alpha_1)}\\ar[d]^{h}&\n~~F(A^{{\\rm co}H})~~\\ar[d]^{\\cong}\\\\\nM\\otimes_BN\\ar[r]^-{\\alpha}& A}\n\\end{equation}\n\\begin{equation}\\eqlabel{5.7.2}\n\\xymatrix{\n~~F(N_1\\otimes_{B^{{\\rm co}H}}M_1)~~\n\\ar[r]^-{F (\\beta_1)}\\ar[d]^{h}&\n~~F(B^{{\\rm co}H})~~\\ar[d]^{\\cong}\\\\\nN\\otimes_AM\\ar[r]^-{\\beta}& B}\n\\end{equation}\nIt is clear that $\\alpha\\in {}_A\\mathcal{M}_B^H$ and $\\beta\\in {}_B\\mathcal{M}_A^H$. We claim that\n$(A,B,M,N,\\alpha,\\beta)$ is an $H$-Morita context. To this end, consider the following\ndiagram\n$$\\xymatrix{\nM\\otimes_B N\\otimes_A M \\ar[rr]^-{M\\otimes\\beta}\\ar[d]^{h^{-1}}&&\nM\\otimes_BB\\ar[d]^{h^{-1}}\\\\\nF(M_1\\otimes_{B^{{\\rm co}H}}N_1\\otimes_{A^{{\\rm co}H}}M_1)\\hbox{\\hspace*{5mm}}\n\\ar[rr]^-{F(M_1\\otimes \\beta_1)}\\ar[d]^{=}&&\n\\hbox{\\hspace*{5mm}}F(M_1\\otimes_{B^{{\\rm co}H}}B^{{\\rm co}H})\\ar[d]^{\\cong}\\\\\nF(M_1\\otimes_{B^{{\\rm co}H}}N_1\\otimes_{A^{{\\rm co}H}}M_1)\\hbox{\\hspace*{5mm}}\n\\ar[rr]^-{F(\\alpha_1\\otimes N_1)}\\ar[d]^{h}&&\n\\hbox{\\hspace*{5mm}}F(A^{{\\rm co}H}\\otimes_A^{{\\rm co}H}N_1)\\ar[d]^{h}\\\\\nM\\otimes_B N\\otimes_A M\\ar[rr]^-{\\alpha\\otimes_A M}&&A\\otimes_A M}$$\nThe top square and the bottom square commute by the definition of $\\alpha$ and $\\beta$,\nand because of the naturality of $h$ (see \\equref{3.2.2}). The square in the middle\ncommutes because $(A^{{\\rm co}H},B^{{\\rm co}H},M_1,N_1,\\alpha_1,\\beta_1)$\nis a Morita context. So the whole diagram commutes. The composition of the left\nvertical morphisms is the identity of $M\\otimes_BN\\otimes_A M$, and the composition of the\nright vertical morphisms is the natural isomorphism $M\\otimes_B B\\cong A\\otimes_AM$. So\nit follows that the diagram\n$$\\xymatrix{\nM\\otimes_B N\\otimes_A M \\ar[rr]^-{M\\otimes\\beta}\\ar[d]^{\\alpha\\otimes M}&&M\\otimes_BB\\ar[d]^{\\cong}\\\\\nA\\otimes_AM\\ar[rr]^-{\\cong}&&M}$$\ncommutes. The commutativity of the second diagram in the definition of a Morita context\nis proved in a similar way.\n\\end{proof}\n\nRecall that $M\\in {}_A\\mathcal{M}$ is a progenerator if and only if $A$ and $M$ are mutually\ndirect summands of finite direct sums of copies of the other. Now let $M\\in {}_A\\mathcal{M}^H$\nIf this property holds in the category ${}_A\\mathcal{M}^H$, then we call $M$ an $H$-progenerator.\n\n\\begin{corollary}\\colabel{5.8}\nAssume that $A$ and $B$ are right faithfully flat $H$-Galois extensions. If $(A,B,M,N,\\alpha,\\beta)$\nis a strict $H$-Morita context, then $M$ is an $H$-progenerator.\n\\end{corollary}\n\n\\begin{proof}\nLet $(A^{{\\rm co}H},B^{{\\rm co}H},M_1,N_1,\\alpha_1,\\beta_1)$ be the corresponding strict\n$\\square_H$-Morita context, as in \\thref{5.7}. It follows from classical Morita theory that\n$M_1$ is a left $A^{{\\rm co}H}$-progenerator. The claim then follows from the equivalence\n$A\\otimes_{A^{{\\rm co}H}}-$ between the categories ${}_{A^{{\\rm co}H}}\\mathcal{M}$ and ${}_A\\mathcal{M}^H$.\n\\end{proof}\n\n\\begin{theorem}\\thlabel{5.9}\nAssume that $A$ and $B$ are right faithfully flat $H$-Galois extensions, and let\n$(A^{{\\rm co}H},B^{{\\rm co}H},M_1,N_1,\\alpha_1,\\beta_1)$ be a strict Morita context.\nIf $M_1$ has a left $A\\square_H B^{\\rm op}$-module structure, then there is a unique\nleft $B\\square_H A^{\\rm op}$-module structure on $N_1$ such that \n$(A^{{\\rm co}H},B^{{\\rm co}H},M_1,N_1,\\alpha_1,\\beta_1)$ is a strict\n$\\square_H$-Morita context.\n\\end{theorem}\n\n\\begin{proof}\nWe know that $M=A\\otimes_{A^{{\\rm co}H}}M_1\\in {}_A\\mathcal{M}_B^H$.\nWe have seen in \\prref{3.7} that we have a morphism $\\psi:\\ B\\to {}_A{\\rm END}(M)$\nin ${}_B\\mathcal{M}_B^H$, and a left $B\\square_HB$-linear map\n$\\psi^{{\\rm co}H}:\\ B^{{\\rm co}H}\\to {}_A{\\rm END}(M)^{{\\rm co}H}\\cong {}_{A^{{\\rm co}H}}\n{\\rm End}(M^{{\\rm co}H})$, see also \\coref{3.4}. $\\psi^{{\\rm co}H}$ is an isomorphism, because\nthe Morita context is strict. Since $B$ is right faithfully flat $H$-Galois, it follows that\n$\\psi$ is an isomorphism in ${}_B\\mathcal{M}_B^H$. Since $M_1$ is a progenerator as\na left $A^{{\\rm co}H}$-module, $M$ is a progenerator as a left $A$-module.\nLet $N={}_A{\\rm HOM}(M,A)$. Then $N^{{\\rm co}H}\\cong {}_{A^{{\\rm co}H}}{\\rm Hom}(M_1,A^{{\\rm co}H})$\nas left $B\\square_HA^{\\rm op}$-modules\n(see \\coref{3.4}); ${}_{A^{{\\rm co}H}}{\\rm Hom}(M_1,A^{{\\rm co}H})$ and $N_1$\nare canonically isomorphic as $(B^{{\\rm co}H}\\hbox{-}A^{{\\rm co}H})$-bimodules, since the Morita context is strict. Using this isomorphism, the left $B\\square_HA^{\\rm op}$-module structure\ncan be transported to $N_1$. The $H$-Morita context $(A,B,M,N)$ associated to $M$\nis strict by \\prref{5.4}. The corresponding $\\square_H$-Morita context from \\thref{5.7}\nis canonically isomorphic to $(A^{{\\rm co}H},B^{{\\rm co}H},M_1,N_1,\\alpha_1,\\beta_1)$.\nThis proves the claim.\n\\end{proof}\n\nWe end this Section with the following result.\n\n\\begin{theorem}\\thlabel{5.10}\nLet $A$ be a (right) faithfully flat Galois extension of $A^{{\\rm co}H}$.\nAssume that $M\\in {}_A\\mathcal{M}^H$ is a progenerator as a left $A$-module. Then $B={}_A{\\rm END}(M)^{\\rm op}$\nis a (right) faithfully flat $H$-Galois extension of $B^{{\\rm co}H}$ if and only if $M$ is an $H$-progenerator.\n\\end{theorem} \n\n\\begin{proof}\nThe $H$-Morita context $(A,B,M,N={}_A{\\rm HOM}(M,A),\\alpha,\\beta)$\nfrom \\exref{5.3} is strict by \\prref{5.4}.\\\\\nIf $B$ is a faithfully flat $H$-Galois extension, then $M$ is an $H$-progenerator\nby \\coref{5.8}.\\\\\nConversely, let $M$ be an $H$-progenerator. $M\\in {}_A\\mathcal{M}_B^H$ (see \\exref{5.3}),\nhence $M_1=M^{{\\rm co}H}\\in {}_{A^{{\\rm co}H}}\\mathcal{M}_{B^{{\\rm co}H}}$.\nFrom the fact that the categories ${}_A\\mathcal{M}^H$ and ${}_{A^{{\\rm co}H}}\\mathcal{M}$\nare equivalent, it follows that $M_1$ is a left $A^{{\\rm co}H}$-progenerator. From\n\\prref{3.3}, we know that $B^{{\\rm co}H}\\cong {}_{A^{{\\rm co}H}}{\\rm End}(M_1)^{\\rm op}$\nand that $N^{{\\rm co}H}\\cong {}_{A^{{\\rm co}H}}{\\rm Hom}(M_1, B^{{\\rm co}H})$. The Morita context\n$$(A^{{\\rm co}H},B^{{\\rm co}H}\\cong {}_{A^{{\\rm co}H}}{\\rm End}(M_1)^{\\rm op},\nM_1,{}_{A^{{\\rm co}H}}{\\rm Hom}(M_1, B^{{\\rm co}H}),\\alpha_1,\\beta_1)$$\nassociated to $M_1\\in {}_{A^{{\\rm co}H}}\\mathcal{M}$ is strict, so $M_1\\otimes_{B^{{\\rm co}H}}-\n:\\ {}_{B^{{\\rm co}H}}\\mathcal{M}\\to  {}_{A^{{\\rm co}H}}\\mathcal{M}$ is a category equivalence.\n$A\\otimes_{A^{{\\rm co}H}}-:\\ {}_{A^{{\\rm co}H}}\\mathcal{M}\\to {}_A\\mathcal{M}^H$ is an equivalence\nsince $A$ is a right faithfully flat $H$-Galois extension, and $M\\otimes_B-:\\ {}_B\\mathcal{M}^H\\to\n{}_A\\mathcal{M}^H$ is also an equivalence (see \\prref{5.2}). Using the fact that\n$A\\otimes _{A^{{\\rm co}H}}M_1\\cong M$ ($A$ is a right  faithfully flat Galois extension), we\nsee easily that the following diagram of functors commutes;\n$$\\xymatrix{\n{}_{B^{{\\rm co}H}}\\mathcal{M}\\ar[rr]^{M_1\\otimes_{B^{{\\rm co}H}}-}\\ar[d]_{B\\otimes_{B^{{\\rm co}H}}-}&&\n{}_{B^{{\\rm co}H}}\\mathcal{M}\\ar[d]^{A\\otimes_{A^{{\\rm co}H}}-}\\\\\n{}_B\\mathcal{M}^H\\ar[rr]^{M\\otimes_B-}&& {}_B\\mathcal{M}^H}$$\nThree of the four functors in the diagram are equivalences, hence the fourth one,\n$B\\otimes_{B^{{\\rm co}H}}-$ is also an equivalence (see the observations following\n\\coref{7.3}). $M_1$, $A$ and $M$ are right faithfully flat over $B^{{\\rm co}H}$, $A^{{\\rm co}H}$ and $B$ respectively, hence it follows that $B$ is right faithfully flat over $B^{{\\rm co}H}$. Thus condition\n(5) of \\thref{1.1} is fulfilled, and it follows that $B$ is a right faithfully flat $H$-Galois extension.\n\\end{proof}\n\n\\section{Application to the Miyashita-Ulbrich action}\\selabel{6}\nLet $A$ be a right faithfully flat right $H$-Galois extension, and consider the map\n$$\\gamma_A={\\rm can}^{-1}\\circ(\\eta_A\\otimes H):\\ H\\to A\\otimes_{A^{{\\rm co}H}}A.$$\nFollowing \\cite{Schauenburg}, we use the notation\n$$\\gamma_A(h)=\\sum_i l_i(h)\\otimes_{A^{{\\rm co}H}} r_i(h).$$\n$\\gamma_A(h)$ is then characterized by the property\n$$\\sum_i l_i(h)r_i(h)_{[0]}\\otimes r_i(h)_{[1]}=1\\otimes h.$$\nThe following properties are then easy to prove (see \\cite[3.4]{Schneider1}):\nfor all $h,h'\\in H$ and $a\\in A$, we have\n\\begin{eqnarray}\n&&\\gamma_A(h)\\in (A\\otimes_{A^{{\\rm co}H}}A)^{A^{{\\rm co}H}};\\eqlabel{6.1.1}\\\\\n&&\\gamma_A(h_{(1)})\\otimes h_{(2)}=\n\\sum_i l_i(h)\\otimes_{A^{{\\rm co}H}} r_i(h)_{[0]}\\otimes r_i(h)_{[1]};\\eqlabel{6.1.2}\\\\\n&&\\gamma_A(h_{(2)})\\otimes S(h_{(1)})=\n\\sum_i l_i(h)_{[0]}\\otimes_{A^{{\\rm co}H}} r_i(h)\\otimes l_i(h)_{[1]};\\eqlabel{6.1.3}\\\\\n&&\\sum_i l_i(h)r_i(h)=\\varepsilon(h)1_A;\\eqlabel{6.1.4}\\\\\n&&\\sum_i a_{[0]}l_i(a_{[1]})\\otimes r_i(a_{[1]})=1\\otimes a;\\eqlabel{6.1.5}\\\\\n&&\\gamma(hh')=\\sum_{i,j} l_i(h')l_j(h)\\otimes_{A^{{\\rm co}H}}\nr_j(h)r_j(h').\\eqlabel{6.1.6}\n\\end{eqnarray}\nCombining \\equref{6.1.2} and \\equref{6.1.3}, we find\n\\begin{eqnarray}\n&&\\hspace*{-2cm}\n\\sum_i l_i(h)_{[0]}\\otimes_{A^{{\\rm co}H}} r_i(h)_{[0]}\\otimes l_i(h)_{[1]}\\otimes r_i(h)_{[1]}\\nonumber\\\\\n&\\equal{\\equref{6.1.3}}&\n\\sum_i l_i(h_{(2)})\\otimes_{A^{{\\rm co}H}} r_i(h_{(2)})_{[0]}\n\\otimes S(h_{(1)}) \\otimes r_i(h_{(2)})_{[1]}\\nonumber\\\\\n&\\equal{\\equref{6.1.2}}& \\sum_i l_i(h_{(2)})\\otimes_{A^{{\\rm co}H}}\nr_i(h_{(2)})\\otimes S(h_{(1)}) \\otimes h_{(3)}. \\eqlabel{6.1.7}\n\\end{eqnarray}\n\nLet $M$ be an $(A,A)$-bimodule. On $M^{A^{{\\rm co}H}}$, we can define a right $H$-action\ncalled the Miyashita-Ulbrich action. This was introduced in \\cite{DT}, and we follow\nhere the description given in \\cite{Schauenburg}. It is given by the formula\n$$m\\hbox{$\\leftharpoonup$} h= \\sum_i l_i(h)mr_i(h).$$\nIt follows from \\equref{6.1.1} and \\equref{6.1.6} that we have a well-defined right\n$H$-action. In particular, for $X,Y\\in \\mathcal{M}_A$, ${\\rm Hom}(X,Y)\\in {}_A\\mathcal{M}_A$,\nwith left and right $A$-action given by\n$$(a\\cdot f\\cdot a')(x)=f(xa)a'.$$\nIt is easy to see that $${\\rm Hom}(X,Y)^{A^{{\\rm co}H}}={\\rm Hom}_{A^{{\\rm\nco}H}}(X,Y),$$ and the Miyashita-Ulbrich action is then given by\n(see \\cite[Cor. 3.5]{Schneider1})\n$$(f \\hbox{$\\leftharpoonup$} h)(x)=\\sum_i f(xl_i(h))r_i(h).$$\n\n\\begin{lemma}\\lelabel{6.1}\nLet $A$ and $B$ be right faithfully flat right $H$-Galois extensions. For all $b\\in B$,\nwe have that\n$$x:=\\gamma(S^{-1}(b_{[1]}))\\otimes b_{[0]}\\in A\\otimes_{A^{{\\rm co}H}}\n(A\\square_H B^{\\rm op}).$$\n\\end{lemma}\n\n\\begin{proof} We have\n\\begin{eqnarray*}\n&&\\hspace*{-2cm}\n\\sum_i l_i(S^{-1}(b_{[1]})) \\otimes_{A^{{\\rm co}H}} r_i(S^{-1}(b_{[1]}))_{[0]}\n\\otimes r_i(S^{-1}(b_{[1]}))_{[1]}\\otimes b_{[0]}\\\\\n&\\equal{\\equref{6.1.2}}& \\gamma(S^{-1}(b_{[1]}))\\otimes S^{-1}(b_{[1]})\\otimes b_{[0]}.\n\\end{eqnarray*}\nhence $x\\in (A\\otimes_{A^{{\\rm co}H}} A)\\square_H B^{\\rm op}\\cong\nA\\otimes_{A^{{\\rm co}H}} (A\\square_H B^{\\rm op})$.\n\\end{proof}\n\nNow we assume that $(A,B,M,N,\\alpha,\\beta)$ is a strict $H$-Morita context connecting\nthe right faithfully flat $H$-Galois extensions $A$ and $B$. For $X\\in \\mathcal{M}_A$, we\nhave the isomorphism\n$$\\diagram \\varphi:\\ X\\otimes_{A^{{\\rm co}H}} M^{{\\rm co}H}\\cong X\\otimes_A A\n\\otimes_{A^{{\\rm co}H}} M^{{\\rm co}H}\\rrto^{\\hspace{3cm}X\\otimes_A\n\\varepsilon_{1,M}} && X\\otimes_A M, \\enddiagram$$ given by\n$$\\varphi(x\\otimes_{A^{{\\rm co}H}}m)=x\\otimes_A m.$$ We have that $X\\otimes_A M\\in \\mathcal{M}_B$,\nand its right $B$-action can be transported to $X\\otimes_{A^{{\\rm\nco}H}} M^{{\\rm co}H}$. We compute this action in our next Lemma.\n\n\\begin{lemma}\\lelabel{6.2}\nThe transported right $B$-action on $X\\otimes_{A^{{\\rm co}H}} M^{{\\rm co}H}$\nis given by the formula\n\\begin{equation}\\eqlabel{6.2.1}\n(x\\otimes_{A^{{\\rm co}H}} m)\\cdot b= \\sum_i xl_i(S^{-1}(b_{[1]}))\\otimes\n_{A^{{\\rm co}H}} (r_i(S^{-1}(b_{[1]}))\\otimes b_{[0]})m.\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\nObserve first that the action \\equref{6.2.1} is well-defined, since\n$M^{{\\rm co}H}\\in {}_{A\\square B^{\\rm op}}\\mathcal{M}$, and by \\leref{6.1}.\nFor the sake of simplicity, we introduce the following notation: for\n$\\sum_i a_i\\otimes b_i \\in A\\square_H B^{\\rm op}$ and $m\\in M^{{\\rm co}H}$, we write\n$$(\\sum_i a_i\\otimes b_i)\\cdot m=\\sum_i a_imb_i.$$\nWe have to show that $\\varphi$ is right $H$-linear. Indeed,\n\\begin{eqnarray*}\n&&\\hspace*{-2cm}\n\\varphi((x\\otimes_{A^{{\\rm co}H}} m)\\cdot b)=\n\\sum_i xl_i(S^{-1}(b_{[1]}))\\otimes _{A} r_i(S^{-1}(b_{[1]}))mb_{[0]}\\\\\n&=&\n\\sum_i xl_i(S^{-1}(b_{[1]})) r_i(S^{-1}(b_{[1]}))\\otimes _{A} mb_{[0]}\\\\\n&\\equal{\\equref{6.1.4}}&\n\\sum_i x\\varepsilon(S^{-1}(b_{[1]}))\\otimes _{A} mb_{[0]}=x\\otimes_A mb.\n\\end{eqnarray*}\n\\end{proof}\n\nConsider the setting of \\thref{5.10}: $(A,B,M,N,\\alpha,\\beta)$ is a strict\n$H$-Morita context connecting the right faithfully flat $H$-Galois extensions $A$ and $B$,\nand $(A^{{\\rm co}H},B^{{\\rm co}H},M^{{\\rm co}H},N^{{\\rm co}H},\\alpha_1,\\beta_1)$\nis the corresponding Morita context connecting $A^{{\\rm co}H}$ and $B^{{\\rm co}H}$.\nFor $X,Y\\in \\mathcal{M}_A$, we have an isomorphism\n\\begin{equation}\\eqlabel{6.3.1}\n\\phi:\\ {\\rm Hom}_{A^{{\\rm co}H}}(X,Y)\\to {\\rm Hom}_{B^{{\\rm co}H}}(X\\otimes_{A^{{\\rm co}H}}\nM^{{\\rm co}H}, Y\\otimes_{A^{{\\rm co}H}} M^{{\\rm co}H}),\n\\end{equation}\ngiven by $\\phi(f)= f\\otimes_{A^{{\\rm co}H}} M^{{\\rm co}H}$. It follows\nfrom \\leref{6.2} that ${\\rm Hom}(X\\otimes_{A^{{\\rm co}H}} M^{{\\rm co}H},\nY\\otimes_{A^{{\\rm co}H}} M^{{\\rm co}H})$ is a $(B,B)$-bimodule, and we\ncan consider the Mi\\-ya\\-shi\\-ta-Ulbrich action on ${\\rm Hom}_{B^{{\\rm\nco}H}}(X\\otimes_{A^{{\\rm co}H}} M^{{\\rm co}H}, Y\\otimes_{A^{{\\rm co}H}}\nM^{{\\rm co}H})$.\n\n\\begin{proposition}\\prlabel{6.3}\nWith notation as above, the map $\\phi$ from {\\rm \\equref{6.3.1}} preserves the\nMiyashita-Ulbrich action.\n\\end{proposition}\n\n\\begin{proof}\nWe will use the notation\n$$\\gamma_B(h)=\\sum_j k_j(h)\\otimes_{B^{{\\rm co}H}} q_j(h)\\in B\\otimes_{B^{{\\rm co}H}} B.$$\nWe have to show that\n$$\\phi(f)\\hbox{$\\leftharpoonup$} h=\\phi(f\\hbox{$\\leftharpoonup$} h),$$\nfor all right $A^{{\\rm co}H}$-linear $f:\\ X\\to Y$ and $h\\in H$. For $x\\in X$ and\n$m\\in M^{{\\rm co}H}$, we compute\n\\begin{eqnarray*}\n&&\\hspace*{-1cm}\n(\\phi(f)\\hbox{$\\leftharpoonup$} h)(x\\otimes_{A^{{\\rm co}H}} m)\n= \\sum_j \\phi(f)\\bigl( (x\\otimes_{A^{{\\rm co}H}} m)k_j(h)\\bigr) q_j(h)\\\\\n&=&\n\\sum_{i,j}\\Bigl[f\\bigl[ xl_i(S^{-1}(k_j(h)_{[1]}))\\bigr]\n\\otimes_{A^{{\\rm co}H}}\nr_i(S^{-1}(k_j(h)_{[1]}))m k_j(h)_{[0]}\\Bigr] q_j(h)\\\\\n&=&\n\\sum_{i,j,p}f\\bigl[ xl_i(S^{-1}(k_j(h)_{[1]}))\\bigr] l_p(S^{-1}(q_j(h)_{[1]}))\\\\\n&&\\hspace*{1cm}\\otimes_{A^{{\\rm co}H}}\nr_p(S^{-1}(q_j(h)_{[1]}))r_i(S^{-1}(k_j(h)_{[1]}))m k_j(h)_{[0]}q_j(h)_{[0]}\\\\\n&\\equal{\\equref{6.1.7}}&\n\\sum_{i,j,p}f\\bigl[ xl_i(S^{-1}(S(h_{(1)})))\\bigr] l_p(S^{-1}(h_{(3)}))\\\\\n&&\\hspace*{1cm}\\otimes_{A^{{\\rm co}H}}\nr_p(S^{-1}(h_{(3)}))r_i(S^{-1}(S(h_{(1)})))m k_j(h_{(2)})q_j(h_{(2)})\\\\\n&\\equal{\\equref{6.1.4}}&\n\\sum_{i,p} f(xl_i(h_{(1)}))l_p(S^{-1}(h_{(2)}))\n\\otimes_{A^{{\\rm co}H}}r_p(S^{-1}(h_{(2)}))r_i(h_{(1)})m1_B\\\\\n&\\equal{(*)}&\n\\sum_{i,p} f(xl_i(h_{(1)}))l_p(S^{-1}(h_{(2)}))\nr_p(S^{-1}(h_{(2)}))r_i(h_{(1)})\\otimes_{A^{{\\rm co}H}} m\\\\\n&\\equal{\\equref{6.1.4}}&\n\\sum_i f(xl_i(h))r_i(h) \\otimes_{A^{{\\rm co}H}} m\\\\\n&=& (f\\hbox{$\\leftharpoonup$} h)(x) \\otimes_{A^{{\\rm co}H}} m= (\\phi(f\\hbox{$\\leftharpoonup$} h))(x\\otimes_{A^{{\\rm co}H}} m).\n\\end{eqnarray*}\nThe equality $(*)$ can be justified as follows. From \\leref{6.1}, we deduce that, for\nall $i$:\n$$\n\\sum_i l_i(S^{-1}(k_j(h)_{[1]})) \\otimes_{A^{{\\rm co}H}} 1_A\n\\otimes_{A^{{\\rm co}H}} r_i(S^{-1}(k_j(h)_{[1]}))\\otimes k_j(h)_{[0]}$$\nand\n$$\n\\sum_p 1_A  \\otimes_{A^{{\\rm co}H}}l_p(S^{-1}(q_j(h)_{[1]}))\n\\otimes_{A^{{\\rm co}H}} r_p(S^{-1}(q_j(h)_{[1]}))\\otimes q_j(h)_{[0]}$$\nlie in $(A\\otimes_{A^{{\\rm co}H}}A)\\otimes_{A^{{\\rm co}H}}\n(A\\square_HB^{\\rm op})$. Consequently $(A\\otimes_{A^{{\\rm\nco}H}}A)\\otimes_{A^{{\\rm co}H}} (A\\square_HB^{\\rm op})$ also contains\n\\begin{eqnarray*}\n&&\\hspace*{-1cm}\n\\sum_{i,p} l_i(S^{-1}(k_j(h)_{[1]})) \\otimes_{A^{{\\rm co}H}}\nl_p(S^{-1}(q_j(h)_{[1]}))\\\\\n&&r_p(S^{-1}(q_j(h)_{[1]}))r_i(S^{-1}(k_j(h)_{[1]}))\\otimes k_j(h)_{[0]}q_j(h)_{[0]}\n\\end{eqnarray*}\n\\begin{eqnarray*}\n&\\equal{\\eqref{eq:6.1.4},\\eqref{eq:6.1.7}}&\n\\sum_{i,p} l_i(h_{(1)}) \\otimes_{A^{{\\rm co}H}} l_p(S^{-1}(h_{(2)}))\\otimes_{A^{{\\rm co}H}}\nr_p(S^{-1}(h_{(2)}))r_i(h_{(1)}) \\otimes 1_B\\\\\n&=:& Z\\otimes 1_B.\n\\end{eqnarray*}\nThis means that\n$$(A\\otimes_{A^{{\\rm co}H}} A\\otimes_{A^{{\\rm co}H}}\\rho_A)(Z)\\otimes 1_B=\nZ\\otimes 1_H\\otimes 1_B,$$ hence\n$$Z\\in (A\\otimes_{A^{{\\rm co}H}} A\\otimes_{A^{{\\rm co}H}} A)^{{\\rm co}H}\\cong\nA\\otimes_{A^{{\\rm co}H}} A\\otimes_{A^{{\\rm co}H}}A^{{\\rm co}H},\n$$\nsince $A\/A^{{\\rm co}H}$ is faihfully flat.\n\\end{proof}\n\n\\section{Hopf subalgebras}\\selabel{7}\nThroughout this Section, $H$ is a Hopf algebra with bijective antipode over a field $k$,\nand $K$ is a Hopf subalgebra of $H$. We assume that the antipode of $K$ is bijective,\nand that $H$ is (right) faithfully flat as a left $K$-module. Let $K^+={\\rm Ker}\\,(\\varepsilon_K)$.\nIt is well-known, and easy to prove (see \\cite[Sec. 1]{Ulbrich}) that\n$$\\overline{H}=H\/HK^+\\cong H\\otimes_K k$$\nis a left $H$-module coalgebra, with operations\n$$h\\cdot \\overline{l}=\\overline{hl},~~\\Delta_{\\overline{H}}(\\overline{h})=\\overline{h}_{(1)}\\otimes \\overline{h}_{(2)},~~\n\\varepsilon_{\\overline{H}}(\\overline{h})=\\varepsilon(h).$$\nThe class in $\\overline{H}$ represented by $h\\in H$ is denoted by $\\overline{h}$.\n$\\overline{1}$ is a grouplike element of $\\overline{H}$, and we consider coinvariants with\nrespect to this element. A right $H$-comodule $M$ is also a right $\\overline{H}$-comodule,\nby corestriction of coscalars:\n$$\\rho_{\\overline{H}}(m)=m_{[0]}\\otimes \\overline{m}_{[1]}.$$\nThe $\\overline{H}$-coinvariants of $M\\in \\mathcal{M}^H$ are then\n\\begin{eqnarray*}\nM^{{\\rm co}\\overline{H}}&=&\\{m\\in M~|~m_{[0]}\\otimes \\overline{m}_{[1]}=m\\otimes \\overline{1}\\}\\\\\n&=& \\{m\\in M~|~\\rho(m)\\in M\\otimes K\\}\\cong M\\square_H K.\n\\end{eqnarray*}\nIf $A$ is a right $H$-comodule algebra, then $A^{{\\rm co}\\overline{H}}$\nis a right $K$-comodule algebra, and $(A^{{\\rm co}\\overline{H}})^{{\\rm co}K}\n=A^{{\\rm co}H}$.\n\n\\begin{proposition}\\prlabel{7.1} {\\bf (\\cite[Remark 1.8]{Schneider1})}\nLet $H$, $K$ and $A$ be as above, and assume that $A$ is a faithfully\nflat $H$-Galois extension. Then $A$ is right faithfully flat as a right $A^{{\\rm co}\\overline{H}}$-module,\nand\n$${\\rm can}:\\ A\\otimes_{A^{{\\rm co}\\overline{H}}} A\\to A\\otimes \\overline{H},~~\n{\\rm can} (a\\otimes b)=ab_{[0]}\\otimes \\overline{b}_{[1]}$$\nis bijective. The functors $(A\\otimes_{A^{{\\rm co}\\overline{H}}}-,(-)^{{\\rm co}\\overline{H}})$\nform a pair of inverse equivalences between the categories\n${}_{A^{{\\rm co}\\overline{H}}}\\mathcal{M}$ and ${}_A\\mathcal{M}(H)^{\\overline{H}}$.\n\\end{proposition}\n\nWe also have an adjoint pair of functors $(F_4=A\\otimes_{A^{{\\rm co}\\overline{H}}}-,\nG_4=(-)^{{\\rm co}\\overline{H}}\\cong -\\square_H K)$ between the categories\n ${}_{A^{{\\rm co}\\overline{H}}}\\mathcal{M}^K$ and ${}_A\\mathcal{M}^H$. This can be seen directly,\n but it is also a consequence of a more general result: we apply\n \\cite[Theorem 1.3]{CaenepeelR} to the inclusion morphism between the\n Doi-Hopf data $(K, A^{{\\rm co}\\overline{H}}, K)$ and $(H,A,H)$.\\\\\n Take $N\\in {}_{A^{{\\rm co}\\overline{H}}}\\mathcal{M}^K$. Forgetting the $K$-coaction, we find\n that $N\\in  {}_{A^{{\\rm co}\\overline{H}}}\\mathcal{M}$. Then it is easy to see that the counit map\n $\\eta_N:\\ N\\to (A\\otimes_{A^{{\\rm co}\\overline{H}}}N^{{\\rm co}\\overline{H}}$ is a morphism in\n ${}_{A^{{\\rm co}\\overline{H}}}\\mathcal{M}^K$, and coincides with the counit map from the\n adjunction $(F_4,G_4)$. Since $\\eta_N$ is an isomorphism, the unit maps\n of the adjunction $(F_4,G_4)$ are isomorphisms. In the same way, we can conclude\n that the counit maps are isomorphisms, and we conclude\n \n \\begin{corollary}\\colabel{7.3}\n Let $H$, $K$ and $A$ be as above, and assume that $A$ is a faithfully\nflat $H$-Galois extension.\n Then the adjoint pair of functors\n $(F_4=A\\otimes_{A^{{\\rm co}\\overline{H}}}-,\nG_4=(-)^{{\\rm co}\\overline{H}}\\cong -\\square_H K)$ establishes a pair of inverse\nequivalences between the categories ${}_{A^{{\\rm co}\\overline{H}}}\\mathcal{M}^K$ and ${}_A\\mathcal{M}^H$.\n\\end{corollary}\n\nBefore stating our next corollary, we recall some elementary facts from category theory.\nIf $(F_1,G_1)$ and $(F_2,G_2)$ are pairs of adjoint functors, respectively\nbetween categories $\\mathcal{C}$ and $\\mathcal{D}$, and between $\\mathcal{D}$ and $\\mathcal{E}$,\nthen $(F=F_2\\circ F_1,G=G_1\\circ G_2)$ is a pair of adjoint functors between\n$\\mathcal{C}$ and $\\mathcal{E}$. If two of these three pairs are inverse equivalences, then the\nthird one is also a pair of inverse equivalences. This follows from the following\nformulas, which give the relations between the units and counits of the three\nadjunctions: for all $C\\in \\mathcal{C}$ and $D\\in \\mathcal{D}$, we have\n$$\\eta_C=G_1(\\eta_{2,F_1(C)})\\circ \\eta_{1,C}~~;~~\n\\varepsilon_E= \\eta_{2,E}\\circ F_2(\\varepsilon_{1,G_2(E)}).$$\n\nThis can be applied to the following situation. Assume that we are in the setting of\n\\prref{7.1} and \\coref{7.3}. We have adjunctions\n\\begin{itemize}\n\\item $(F_1=A\\otimes_{A^{{\\rm co}H}}-,\\ G_1=(-)^{{\\rm co}H})$ between\n${}_{A^{{\\rm co}H}}\\mathcal{M}$ and ${}_A\\mathcal{M}^H$; \\item $(F_3=A^{{\\rm\nco}\\overline{H}}\\otimes_{A^{{\\rm co}H}}-,\\ G_3=(-)^{{\\rm co}K})$ between\n${}_{A^{{\\rm co}H}}\\mathcal{M}$ and ${}_{A^{{\\rm co}\\overline{H}}}\\mathcal{M}^K$;\n\\item $(F_4=A\\otimes_{A^{{\\rm co}\\overline{H}}}-,\nG_4=(-)^{{\\rm co}\\overline{H}}\\cong -\\square_H K)$ between the categories ${}_{A^{{\\rm co}\\overline{H}}}\\mathcal{M}^K$ and ${}_A\\mathcal{M}^H$.\n\\end{itemize}\nIt is clear that $F_1=F_4\\circ F_3$ and $G_1=G_3\\circ G_4$.\n$(F_1,G_1)$ and $(F_4,G_4)$ are pairs of inverse equivalences, by \\thref{1.1}\nand \\coref{7.3}. Hence $(F_3,G_3)$ is also a pair of inverse equivalences, and using\n\\thref{1.1}, we obtain the following result.\n\n\\begin{corollary}\\colabel{7.2}\nLet $H$, $K$ and $A$ be as above, and assume that $A$ is a faithfully\nflat $H$-Galois extension. Then $A^{{\\rm co}\\overline{H}}$ is a right faithfully flat\n$K$-Galois extension.\n\\end{corollary}\n\n\\begin{theorem}\\thlabel{7.4}\nLet $H$ and $K$ be as before: $K\\subset H$ are Hopf algebras with invertible\nantipode over a field $k$, and $H$ is faithfully flat as a left $K$-module.\nLet $A$ and $B$ be (right) faithfully flat right $H$-Galois extensions, connected by\na strict $H$-Morita context $(A, B, M,N, \\alpha,\\beta)$.\n\\begin{enumerate}\n\\item $A^{{\\rm co}\\overline{H}}$ and $B^{{\\rm co}\\overline{H}}$ are connected\nby a strict $K$-Morita context, with connecting modules $M^{{\\rm\nco}\\overline{H}}$ and $N^{{\\rm co}\\overline{H}}$; \\item we have a pair of\ninverse equivalences $(M\\otimes_B-, N\\otimes_A-)$ between the categories\n${}_B\\mathcal{M}(H)^{\\overline{H}}$ and ${}_A\\mathcal{M}(H)^{\\overline{H}}$; \\item the\nfollowing diagram of categories and functors  commutes to within\nnatural equivalences of functors:\n$$\\diagram {}_A\\mathcal{M}(H)^{{\\bar H}}\\rrto<3pt>^{N\\otimes_A-}\\ddto<3pt>^{(-){{^{\\operatorname{co}{\\bar H}}}}}\n&& _B\\mathcal{M}(H)^{{\\bar H}}\\llto<3pt>^{M\\otimes_B-} \\ddto<3pt>^{(-){{^{\\operatorname{co}{\\bar H}}}}} \\\\ \\\\\n_{A{{^{\\operatorname{co}{\\bar\nH}}}}}\\mathcal{M}\\rrto<3pt>^{N{{^{\\operatorname{co}{\\bar\nH}}}}\\otimes_{A{{^{\\operatorname{co}{\\bar\nH}}}}}-}\\uuto<3pt>^{A\\otimes_{A{{^{\\operatorname{co}{\\bar H}}}}}-}\n&& _{B{{^{\\operatorname{co}{\\bar\nH}}}}}\\mathcal{M}\\llto<3pt>^{M{{^{\\operatorname{co}{\\bar\nH}}}}\\otimes_{B{{^{\\operatorname{co}{\\bar\nH}}}}}-}\\uuto<3pt>^{B\\otimes_{B{{^{\\operatorname{co}{\\bar\nH}}}}}-}\\enddiagram$$\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{proof} \n1) We have the following commutative diagram of inclusions\n$$\\diagram\nA^{{\\rm co}\\overline{H}}\\square_K (B^{\\rm op})^{{\\rm co}\\overline{H}}\\rrto^{\\subset}\n\\dto_{\\subset}&&A^{{\\rm co}\\overline{H}}\\otimes (B^{\\rm op})^{{\\rm co}\\overline{H}}\n\\dto^{\\subset}\\\\\nA\\square_H B^{\\rm op}\\rrto^{\\subset} &&A\\otimes B^{\\rm op}\n\\enddiagram$$\nBy \\thref{5.7}, we have\na strict $\\square_H$-Morita context $(A^{{\\rm co}H},B^{{\\rm co}H},M^{{\\rm co}H},$ $N^{{\\rm co}H},\n\\alpha_1,\\beta_1)$. By restriction of scalars, A is a left $A^{{\\rm co}\\overline{H}}\\square_K (B^{\\rm op})^{{\\rm co}\\overline{H}}$-module. Then we can apply Theorems \\ref{th:5.7} and \\ref{th:5.9},\nwith $H$ replaced by $K$, and taking into account that $A^{{\\rm co}\\overline{H}}$\nand $B^{{\\rm co}\\overline{H}}$ are right faithfully flat $K$-Galois extensions, by \\coref{7.2}.\nWe find that $A^{{\\rm co}\\overline{H}}$ and $B^{{\\rm co}\\overline{H}}$ are connected by\na strict $K$-Morita context. The first connecting module is\n\\begin{eqnarray*}\n&&\\hspace*{-2cm}\nA^{{\\rm co}\\overline{H}}\\otimes_{A^{{\\rm co}H}} M^{{\\rm co}H}\n=F_3G_1(M)\\cong G_4F_4F_3G_1(M)\\\\\n&\\cong& G_4F_1G_1(M)\\cong G_4(M)=M^{{\\rm co}\\overline{H}}.\n\\end{eqnarray*}\nIn a similar way, we find that the second connecting module is $N^{{\\rm co}\\overline{H}}$.\\\\\n2) The proof is an easy adaption of the proof of \\prref{5.2}.\\\\\n3) $B^{{\\rm co}\\overline{H}}$ is a right $K$-comodule algebra, and, by corestriction of coscalars,\na right $H$-comodule algebra, so we can consider the categories\n${}_{A^{{\\rm co}\\overline{H}}}\\mathcal{M}^K_{B^{{\\rm co}\\overline{H}}}$ and ${}_A\\mathcal{M}^H_{B^{{\\rm co}\\overline{H}}}$.\nIt is then easy to see that the inverse equivalent functors of \\coref{7.3}\nalso define a pair of inverse equivalences between these two categories of relative Hopf\nbimodules. Now $M\\in {}_A\\mathcal{M}^H_{B^{{\\rm co}\\overline{H}}}$, so\n$M\\cong A\\otimes_{A^{{\\rm co}\\overline{H}}} M^{{\\rm co}\\overline{H}}$ as right $B^{{\\rm co}\\overline{H}}$-modules.\nIt follows that we have, for all $P\\in {}_{B^{{\\rm co}\\overline{H}}}\\mathcal{M}$,\n$$A\\otimes_{A^{{\\rm co}\\overline{H}}} M^{{\\rm co}\\overline{H}}\\otimes_{B^{{\\rm co}\\overline{H}}}  P\n\\cong M\\otimes_{B^{{\\rm co}\\overline{H}}}  P\\cong M\\otimes_BB\\otimes_{B^{{\\rm co}\\overline{H}}}  P.$$\nIn a similar way, we can show that\n$$B\\otimes_{B^{{\\rm co}\\overline{H}}} N^{{\\rm co}\\overline{H}}\\otimes_{A^{{\\rm co}\\overline{H}}}  Q\\cong\nN\\otimes_AA\\otimes_{A^{{\\rm co}\\overline{H}}} Q,$$\nfor all $Q {}_{A^{{\\rm co}\\overline{H}}}\\mathcal{M}$. Finally, take $U\\in {}_A\\mathcal{M}(H)^{\\overline{H}}$. Then\n\\begin{eqnarray*}\n&&\\hspace*{-2cm}\n(N\\otimes_A U)^{{\\rm co}\\overline{H}}\\cong (N\\otimes_AA\\otimes_{A^{{\\rm co}\\overline{H}}} U^{{\\rm co}\\overline{H}})^{{\\rm co}\\overline{H}}\\\\\n&\\cong& (B\\otimes_{B^{{\\rm co}\\overline{H}}}N^{{\\rm co}\\overline{H}}\\otimes_{A^{{\\rm co}\\overline{H}}} U^{{\\rm co}\\overline{H}})^{{\\rm co}\\overline{H}}\\cong N^{{\\rm co}\\overline{H}}\\otimes_{A^{{\\rm co}\\overline{H}}} U^{{\\rm co}\\overline{H}},\n\\end{eqnarray*}\nand, in a similar way, for $V\\in {}_B\\mathcal{M}(H)^{\\overline{H}}$,\n$$(M\\otimes_B V)^{{\\rm co}\\overline{H}}\\cong\nM^{{\\rm co}\\overline{H}}\\otimes_{B^{{\\rm co}\\overline{H}}} V^{{\\rm co}\\overline{H}}.$$\n\\end{proof}\n\nFinally recall that if the algebras $A$ and $B$ are Morita\nequivalent, then there is a Morita equivalence between $A\\otimes\nA{{^{\\operatorname{op}}}}$ and $B\\otimes\nB{{^{\\operatorname{op}}}}$ sending $A$ to $B$. In particular, this\nimplies that the centers of $A$ and $B$ are isomorphic. In our\ncontext this generalizes as follows.\n\n\\begin{corollary}\\colabel{7.5}\nAssume that $(A,B,M,N,\\alpha,\\beta)$ is a strict $H$-Morita context.\n\\begin{enumerate}\n\\item Let  $K$ and $L$ be  Hopf subalgebras of $H$ with\nbijective antipodes, and assume that $H\\otimes H$ is faithfully\nflat as a right $K\\otimes L$-module. Then the categories\n$_{A{{^{\\operatorname{co}H\/HK^+}}}}\\mathcal{M}_{A{{^{\\operatorname{co}H\/HL^+}}}}$\nand\n$_{B{{^{\\operatorname{co}H\/HK^+}}}}\\mathcal{M}_{B{{^{\\operatorname{co}H\/HL^+}}}}$\nare equivalent.\n\\item There is an isomorphism\n$$C_A(A{{^{\\operatorname{co}H}}})\\cong\nC_B(B{{^{\\operatorname{co}H}}})$$ of left $H$-module right\n$H$-comodule algebras, where  $C_A(A{{^{\\operatorname{co}H}}})$ denotes the centralizer in $A$ of $A{{^{\\operatorname{co}H}}}$.\n\\end{enumerate}\n\\end{corollary}\n\n\\begin{proof}\n1) The objects $M\\otimes N\\in \\ _{A\\otimes A{{^{\\operatorname{op}}}}}\\mathcal{M}^{H\\otimes H}_{B\\otimes\nB{{^{\\operatorname{op}}}}}$ and $N\\otimes M\\in \\ _{B\\otimes\nB{{^{\\operatorname{op}}}}}\\mathcal{M}^{H\\otimes H}_{A\\otimes\nA{{^{\\operatorname{op}}}}}$ induce a Morita equivalence between\n$A\\otimes A{{^{\\operatorname{op}}}}$ and $B\\otimes\nB{{^{\\operatorname{op}}}}$. Now the assertion follows from\n\\thref{7.4}, where we replace $H$ by $H\\otimes H$,\n$K$ by $K\\otimes L$, $A$ by $A\\otimes A{{^{\\operatorname{op}}}}$\nand $B$ by $B\\otimes B{{^{\\operatorname{op}}}}$.\\\\\n2) Note that $$C_A(A{{^{\\operatorname{co}H}}})\\cong\n{\\rm End}_{A{{^{\\operatorname{co}H}}}\\otimes\nA{{^{\\operatorname{op}}}}}(A)$$ as $H$-module $H$-comodule\nalgebras. Since  under the equivalence of 1) (where we take $K=k$\nand $L=H$), $A$ corresponds to $B$, the statement follows from\n\\prref{6.3}.\n\\end{proof}\n\n\\section{$H$-colinear equivalences}\\selabel{8}\nLet $H$ be a projective Hopf algebra, and $A$ a right $H$-comodule algebra.\nLet ${}_A\\underline{\\mathcal{M}}^H$ be the category with relative Hopf modules as modules;\nthe set of morphisms between two objects $M$ and $N$ is ${}_A{\\rm HOM}(M,N)$.\n${}_A\\underline{\\mathcal{M}}^H$ is a right $H$-colinear category in the following sense:\n${}_A{\\rm HOM}(M,N)$ is a right $H$-comodule (see \\prref{3.2x}); the map\n$$\\varphi:\\ M\\otimes {}_A{\\rm HOM}(M,N)\\to N,~~\\varphi(m\\otimes f)=f(m)$$\nis right $H$-colinear (take $B=C=k$ in \\prref{3.5}); if $N$ is a third object in\n${}_A\\underline{\\mathcal{M}}^H$, then the composition\n$$\\psi:\\ {}_A{\\rm HOM}(L,M)\\otimes {}_A{\\rm HOM}(M,N)\\to {}_A{\\rm HOM}(L,N),~~\n\\psi(f\\otimes g)=g\\circ f$$\nis right $H$-colinear. The following result is then obvious.\n\n\\begin{proposition}\\prlabel{8.1}\nLet $H$ be a projective Hopf algebra.\nLet $(A,B,M,N,\\alpha,\\beta)$ be a strict $H$-Morita context connecting the\nright $H$-comodule algebras $A$ and $B$. Then the functors $M\\otimes_B-$\nand $N\\otimes_A-$ induce a pair of inverse right $H$-colinear equivalences\nbetween  ${}_B\\underline{\\mathcal{M}}^H$ and ${}_A\\underline{\\mathcal{M}}^H$.\n\\end{proposition}\n\nThe functors $F=M\\otimes_B-$ and $G=N\\otimes_A-$ are right $H$-colinear in the following\nsense: for $V,W\\in {}_B\\mathcal{M}^H$, the map\n$$F:\\ {}_B{\\rm HOM}(V,W)\\to {}_A{\\rm HOM}(M\\otimes_B V,M\\otimes_BW),~~F(f)=M\\otimes_B f$$\nis right $H$-colinear.\\\\\nIn this Section, we investigate when the converse of \\prref{8.1} holds: suppose that we have\na pair of inverse right $H$-colinear equivalences\nbetween  ${}_B\\underline{\\mathcal{M}}^H$ and ${}_A\\underline{\\mathcal{M}}^H$. Is this equivalence induced by\na strict $H$-Morita context? \nTo this end, we will give an $H$-colinear version of the Eilenberg-Watts Theorem.\n\n\\begin{proposition}\\prlabel{8.4}\nLet $A$ and $B$ be $H$-comodule algebras, and $T:\\ {}_{A}\\underline{\\mathcal{M}}^H\\to  {}_{B}\\underline{\\mathcal{M}}^H$\nan $H$-colinear functor. Then $N=T(A)\\in {}_A\\mathcal{M}_B^H$,\nand we have a natural transformation $\\psi:\\ F=N\\otimes_A-\\to T$, such that\n$\\psi_A:\\ N\\otimes_AA\\to T(A)=N$ is the natural isomorphism.\n\\end{proposition}\n\n\\begin{proof}\nIn the sequel, $V$ and $W$ will be objects in  ${}_{A}\\underline{\\mathcal{M}}^H$.\nThe fact that $T$ is right $H$-colinear means that\n\\begin{equation}\\eqlabel{8.4.a}\nT(f_{[0]})\\otimes f_{[1]}=\\rho(T(f)),\n\\end{equation}\nfor $f\\in {}A{\\rm HOM}(V,W)$. We claim that the map\n$$\\varphi_V:\\ V\\to {}_A{\\rm HOM}(A,V),~~\\varphi_V(v)(a)=av$$\nIs well-defined and right $H$-colinear. To this end, it suffices to show that\n\\begin{equation}\\eqlabel{8.4.b}\n\\varphi(v)_{[0]}\\otimes \\varphi(v)_{[1]}=\\varphi(v_{[0]})\\otimes v_{[1]},\n\\end{equation}\nfor all $v\\in V$. For all $a\\in A$, we have\n\\begin{eqnarray*}\n&&\\hspace*{-2cm}\n\\bigl(\\varphi_V(v)(a_{[0]})\\bigr)_{[0]}\\otimes S^{-1}(a_{[1]})\\bigl(\\varphi_V(v)(a_{[0]})\\bigr)_{[1]}\\\\\n&=& a_{[0]}v_{[0]}\\otimes S^{-1}(a_{[2]})a_{[1]}v_{[1]}\\\\\n&=& av_{[0]}\\otimes v_{[1]}=\\varphi_V(v_{[0]})(a)\\otimes v_{[1]},\n\\end{eqnarray*}\nand \\equref{8.4.b} follows using \\equref{3.2.1}.\n$\\varphi_V$ satisfies the following property:\n\\begin{equation}\\eqlabel{8.4.1}\n\\varphi_V(av)=\\varphi_V(v)\\circ \\varphi_A(a),\n\\end{equation}\nfor all $a\\in A$ and $v\\in V$. Indeed,\n$$\\varphi_V(av)(c)=cav=\\varphi_V(v)(ca)=(\\varphi_V(v)\\circ \\varphi_A(a))(c).$$\nOn $N=T(A)\\in {}_{B}\\mathcal{M}^H$, we define a right $A$-action as follows:\n$$na=T(\\varphi_A(a))(n),$$\nfor all $a\\in A$ and $n\\in N$. This makes $N$ an object of ${}_B\\mathcal{M}_A^H$,\nsince\n\\begin{eqnarray*}\n&&\\hspace*{-2cm}\nn(ac)= T(\\varphi_A(ac))(n)\\equal{\\equref{8.4.1}} T(\\varphi_A(a)\\circ \\varphi_A(c))(n)\\\\\n&=& \\bigl( T(\\varphi_A(a))\\circ T(\\varphi_A(c))\\bigr)(n)=(na)c;\\\\\n&&\\hspace*{-2cm}\n(bn)a=T(\\varphi_A(a))(bn)=bT(\\varphi_A(a))(n)=b(na);\\\\\n&&\\hspace*{-2cm}\nn_{[0]}a_{[0]}\\otimes n_{[1]}a_{[1]}= T(\\varphi(a_{[0]}))(n_{[0]}) \\otimes n_{[1]}a_{[1]}\\\\\n&\\equal{\\equref{8.4.b}}& T(\\varphi(a)_{[0]})(n_{[0]}) \\otimes n_{[1]}\\varphi(a)_{[1]}\\\\\n&\\equal{\\equref{8.4.a}}& T(\\varphi(a))_{[0]}(n_{[0]}) \\otimes n_{[1]}T(\\varphi(a))_{[1]}\\\\\n&\\equal{\\equref{3.2.1}}& T(\\varphi(a))(n_{[0]})_{[0]}\\otimes n_{[2]}S^{-1}(n_{[1]})\nT(\\varphi(a))(n_{[0]})_{[1]}\\\\\n&=& T(\\varphi(a))(n)_{[0]}\\otimes T(\\varphi(a))(n)_{[1]}=\\rho(na),\n\\end{eqnarray*}\nfor all $a,c\\in A$, $b\\in B$ and $n\\in N$.\\\\\n\nFor every $v\\in V$, $\\varphi_V(v):\\ A\\to V$ is left $A$-linear, hence $T(\\varphi_V(v)):\\\nT(A)=N\\to T(V)$ is left $B$-linear. By (\\ref{eq:8.4.a},\\ref{eq:8.4.b}), we also have that\n\\begin{equation}\\eqlabel{8.4.2}\nT(\\varphi_V(v_{[0]}))\\otimes v_{[1]}=\\rho(T(\\varphi_V(v))).\n\\end{equation}\nNow we define\n$$\\psi_V:\\ N\\otimes_A V\\to T(V),~~\\psi_V(n\\otimes_A v)= T(\\varphi_V(v))(n).$$\n$\\psi_V$ is well-defined since\n\\begin{eqnarray*}\n&&\\hspace*{-2cm}\n\\psi_V(n\\otimes_A av)=T(\\varphi_V(av))(n)\\equal{\\equref{8.4.1}}\n(T(\\varphi_V(v))\\circ T(\\varphi_V(a)))(n)\\\\\n&=& T(\\varphi_V(v))(na)=\n\\psi_V(n\\otimes_A v).\n\\end{eqnarray*}\n$\\psi_V$ is right $H$-colinear, since\n\\begin{eqnarray*}\n&&\\hspace*{-2cm}\n\\psi_V(n_{[0]}\\otimes_A v_{[0]})\\otimes n_{[1]}v_{[1]}= T(\\varphi_V(v_{[0]}))(n_{[0]})\\otimes n_{[1]}v_{[1]}\\\\\n&\\equal{\\equref{8.4.2}}& T(\\varphi_V(v))_{[0]}(n_{[0]})\\otimes n_{[1]}T(\\varphi_V(v))_{[1]}\\\\\n&\\equal{\\equref{3.2.1}}&\\bigl(T(\\varphi_V(v))(n_{[0]})\\bigr)_{[0]}\n\\otimes n_{[2]}S^{-1}(n_{[1]}) \\bigl(T(\\varphi_V(v))(n_{[0]})\\bigr)_{[1]}\\\\\n&=& \\rho\\bigl(T(\\varphi_V(v))(n)\\bigr)=\\rho(\\psi_V(n\\otimes_A v)).\n\\end{eqnarray*}\n$\\psi_V$ is left $B$-linear, since\n$$\\psi_V(bn\\otimes_A v)=T(\\varphi_V(v))(bm)=b(T(\\varphi_V(v))(m))=b\\psi_V(n\\otimes_A v).$$\nIn order to show that $\\psi$ is a natural transformation, we first observe the following\nproperty. For $f:\\ V\\to W$ in ${}_{A}\\mathcal{M}^H$, $v\\in V$ and $a\\in A$, we have\n$$\\varphi_W(f(v))(a)=af(v)=f(av)=f(\\varphi_V(v)(a),$$\nso $\\varphi_W(f(v))=f\\circ \\varphi_V(v)$. We can now show that the diagram\n$$\\diagram\nN\\otimes_AV\\rrto^{\\psi_V}\\dto_{N\\otimes_A f}&&T(V)\\dto^{T(f)}\\\\\nN\\otimes_AW\\rrto^{\\psi_W}&&T(W)\n\\enddiagram$$\ncommutes:\n\\begin{eqnarray*}\n&&\\hspace*{-2cm}\n(T(f)\\circ \\psi_V)(n\\otimes_A v)=(T(f)\\circ T(\\varphi_V(v)))(n)\\\\\n&=& T(f\\circ\\varphi_V(v))(n)=T(\\varphi_W(f(v)))(n)=\\psi_W(n\\otimes_A f(v))).\n\\end{eqnarray*}\nIt follows that $\\psi$ is a natural transformation. Finally, it is easy to compute that the map\n$\\psi_A:\\ N\\otimes_A \\to T(A)=A$ is given by\n$\\psi_A(n\\otimes a)=T(\\varphi_A(a))(n)=na$,\nas needed.\n\\end{proof}\n\nWe are now ready to prove the following generalization of the Eilenberg-Watts\nTheorem (cf. \\cite[II.2.3]{Bass}).\n\n\\begin{proposition}\\prlabel{8.5}\nWith notation and assumptions as in \\prref{8.4}, assume that $A$ is a generator\nof ${}_{A}\\mathcal{M}^H$, and that $T$, viewed as a functor ${}_{A}\\mathcal{M}^H\\to {}_{B}\\mathcal{M}^H$,\npreserves cokernels and arbitrary coproducts. Then the natural transformation\n$\\psi:\\ F=N\\otimes_A-\\to T$ from \\prref{8.4} is a natural isomorphism.\n\\end{proposition}\n\n\\begin{proof}\nLet $I$ be an index set, and $A^{(I)}$ the coproduct of copies of $A$ indexed by $I$.\nFor $i\\in I$, let $r_i:\\ A\\to A^{(I)}$ be the natural inclusion. Since $\\psi$ is a natural\ntransformation, we have a commutative diagram\n$$\\diagram\nF(A)\\rrto^{\\psi_A}\\dto_{F(r_i)}&&T(A)\\dto^{T(r_i)}\\\\\nF(A^{(I)})\\rrto^{\\psi_{A^{(I)}}}&&T(A^{(I)})\n\\enddiagram$$\nLet $n_i:\\ T(A)\\to T(A)^{(I)}$ be the natural inclusion. Then the diagram\n$$\\diagram\nF(A)^{(I)}\\rrto^{\\oplus_{i\\in I}(n_i\\circ\\psi_A)}\\dto_{\\oplus_{i\\in I}F(r_i)}&&T(A)^{(I)}\\dto^{\\oplus_{i\\in I}T(r_i)}\\\\\nF(A^{(I)})\\rrto^{\\psi_{A^{(I)}}}&&T(A^{(I)})\n\\enddiagram$$\nalso commutes. The vertical maps in the diagram are isomorphisms, since $F$ and $T$\ncommute with direct sums. We have seen in \\prref{8.4} that the top horizontal map is an isomorphim,\nso it follows that $\\psi_{A^{(I)}}$ is an isomorphism.\\\\\nNow take an arbitrary $V\\in {}_{A}\\mathcal{M}^H$. Since $A$ is a generator of ${}_{A}\\mathcal{M}^H$,\nwe have an exact sequence\n$$A^{(J)}\\mapright{\\pi} A^{(I)}\\mapright{\\varphi} V\\to 0$$\nin ${}_{A}\\mathcal{M}^H$. Since $\\psi$ is a natural transformation, and $F$ and $G$ preserve\ncokernels, we have the following commutative diagram with exact rows in ${}_{B}\\mathcal{M}^H$:\n$$\\diagram\nF(A^{(J)})\\dto^{\\psi_{A^{(J)}}}\\rrto^{F(\\pi)}&&\nF(A^{(I)})\\dto^{\\psi_{A^{(I)}}}\\rrto^{F(\\varphi)}&&\nF(V)\\dto^{\\psi_V}\\rrto^{}&&0\\\\\nT(A^{(J)})\\rrto^{T(\\pi)}&&T(A^{(I)})\\rrto^{T(\\varphi)}&&\nT(V)\\rrto^{}&&0\n\\enddiagram$$\nWe know from above that $\\psi_{A^{(J)}}$ and $\\psi_{A^{(I)}}$ are isomorphisms, and it\nfollows from the 5 lemma that $\\psi_V$ is also an isomorphism.\n\\end{proof}\n\n\\begin{theorem}\\thlabel{8.6}\nLet $A$ and $B$ be $H$-module algebras, and suppose that they generate the categories\n${}_{A}\\mathcal{M}^H$ and ${}_{B}\\mathcal{M}^H$. If $(T,U)$ is a pair of $H$-linear inverse equivalences between\nthe categories ${}_{A}\\underline{\\mathcal{M}}^H$ and ${}_{B}\\underline{\\mathcal{M}}^H$, then there exists a strict $H$-Morita \ncontext $(A,B,M,N,\\alpha,\\beta)$ such that $T\\cong N\\otimes_A-$ and $U\\cong M\\otimes_B-$.\n\\end{theorem}\n\n\\begin{proof}\nSince $(T,U)$ is also a pair of inverse equivalences between ${}_{A}\\mathcal{M}^H$ and ${}_{B}\\mathcal{M}^H$,\n$T$ and $U$ preserve coproducts and cokernels. Applying \\prref{8.5}, we find\n$M\\in {}_A\\mathcal{M}_B^H$ and $N\\in {}_B\\mathcal{M}_A^H$\nsuch that $T\\cong N\\otimes_A-$ and $U\\cong M\\otimes_B-$.\\\\\n$(T,U)$ is a pair of adjoint functors, and the unit $\\eta$ and the counit $\\varepsilon$ are\nnatural isomorphisms. We define $\\alpha=\\eta^{-1}_A:\\ M\\otimes_B N\\to A$. Then $\\alpha\\in\n{}_{A}\\mathcal{M}^H$. Let us show that $\\alpha$ is also right $A$-linear. For every $c\\in A$,\nthe map $f_c:\\ A\\to A$, $f_c(a)=ac$ is left $A$-linear. Since $\\eta$ is a natural transformation,\nthe diagram\n$$\\xymatrix{\nA\\ar[rr]^-{\\eta_A}\\ar[d]_{f_c}&&M\\otimes_BN\\otimes_AA\\ar[d]^{M\\otimes_BN\\otimes_Af_c}\\\\\nA\\ar[rr]^-{\\eta_A}&&M\\otimes_BN\\otimes_AA\n}$$\ncommutes. Evaluating the diagram at $1_A$, we find that $\\eta_A(ac)=\\eta_A(a)c$.\\\\\nWe define $\\beta=\\varepsilon_B:\\ N\\otimes_A M\\to B$. Applying the above argument to the\nadjunction $(U,T)$ with unit $\\varepsilon^{-1}$ and counit $\\eta^{-1}$, we find that\n$\\varepsilon_B$ is right $B$-linear.\\\\\nTake $W\\in {}_{B}\\mathcal{M}^H$. For every $w\\in W$, we consider the left $B$-linear map\n$g_w:\\ B\\to W$, $g_w(b)=bw$. Since $\\varepsilon$ is a natural transformation, the\ndiagram\n$$\\xymatrix{\nN\\otimes_A M\\otimes_B B\\ar[rr]^-{\\varepsilon_B}\\ar[d]_{N\\otimes_A M\\otimes_B g_w}&&B\\ar[d]^{g_w}\\\\\nN\\otimes_A M\\otimes_B W\\ar[rr]^-{\\varepsilon_W}&&W\n}$$\ncommutes. Evaluating the diagram at $n\\otimes_A m\\otimes_B 1$, we see that\n$\\varepsilon_W=\\varepsilon_B\\otimes_B W$.\\\\\nFrom the properties of adjoint functors, we know that\n$\\varepsilon_{T(V)}\\circ T(\\eta_V)=T(V)$, for all $V\\in {}_{A}\\mathcal{M}^H$ . Taking\n$V=A$ in this formula, we see that the diagram\n$$\\xymatrix{\nN\\otimes_A\\ar[rr]^-{N\\otimes_A\\eta_A}\\ar[d]_{\\cong}&& N\\otimes_A M\\otimes_B N\\ar[d]^{\\varepsilon_N=\\varepsilon_B\\otimes_BN}\\\\\nN\\ar[rr]^-{\\cong}&&B\\otimes_B N\n}$$\ncommutes. This diagram is one of the two diagrams in the definition of a Morita context.\nThe commutativity of the other diagram follows in a similar way.\n\\end{proof}\n\n\\begin{corollary}\\colabel{8.7}\nLet $H$ be a projective Hopf algebra, and assume that the right $H$-comodule\nalgebras $A$ and $B$ are $H$-Galois extensions of $A^{{\\rm co}H}$ and $B^{{\\rm co}H}$,\nrespectively. If $(T,U)$ is a pair of $H$-colinear inverse equivalences between\nthe categories ${}_{A}\\underline{\\mathcal{M}}^H$ and ${}_{B}\\underline{\\mathcal{M}}^H$, then there exists a strict $H$-Morita context $(A,B,M,N,\\alpha,\\beta)$ such that $T\\cong N\\otimes_A-$ and $U\\cong M\\otimes_B-$.\n\\end{corollary}\n\n\\begin{proof}\nIt is well-known that $A^{{\\rm co}H}$ is a generator of ${}_{A^{{\\rm co}H}}\\mathcal{M}$; since $(F_1,G_1)$\nis a pair of inverse equivalences (see \\thref{1.1}), $F_1(A^{{\\rm co}H})=A$ is a generator of\n${}_A\\mathcal{M}^H$. In a similar way, $B$ is a generator of  ${}_B\\mathcal{M}^H$, and we can apply\n\\thref{8.6}.\n\\end{proof}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\n\nIt is well  know that warm inflation, as opposed to the\nconventional cool inflation, presents the attractive feature that\nit avoids the reheating period \\cite{warm}. In these kind of\nmodels dissipative effects are important during the inflationary\nperiod, so that radiation production occurs concurrently together\nwith the inflationary expansion. If the radiation field is in a\nhighly excited state during inflation, and this has a strong\ndamping effect on the inflaton dynamics, then, it is found a\nstrong regimen of  warm inflation. Also, the dissipating effect\narises from a friction term which describes the processes of the\nscalar field dissipating into a thermal bath via its interaction\nwith other fields. Warm inflation shows how thermal fluctuations\nduring inflation may play a dominant role in producing the initial\nfluctuations  necessary for Large-Scale Structure (LSS) formation.\nIn these kind of models the density fluctuations arise from\nthermal rather than quantum fluctuations \\cite{62526}. These\nfluctuations have their origin in the hot radiation and influence\nthe inflaton through a friction term in the equation of motion of\nthe inflaton scalar field \\cite{1126}. Among the most attractive\nfeatures of these models, warm inflation end  at the epoch when\nthe universe stops inflating and \"smoothly\" enters in a radiation\ndominated Big-Bang phase\\cite{warm}. The matter components of the\nuniverse are created by the decay of either the remaining\ninflationary field or the dominant radiation field\n\\cite{taylorberera}.\n\n\nOn the other hand, the generalized Chaplygin gas    has been\nproposed as an  alternative model for  describing the present\naccelerating of the universe. The generalized Chaplygin gas is\ndescribed by an exotic equation of state of the form \\cite{Bento}\n\\begin{equation}\n p_{ch} = - \\frac{A}{\\rho_{ch}^\\beta},\\label{1}\n\\end{equation}\nwhere $\\rho_{ch}$ and $p_{ch}$ are the energy density and pressure\nof the generalized Chaplygin gas, respectively. $\\beta$ is a\nconstant that lies in  the range $0 <\\beta\\leq 1$, and $A$ is a\npositive constant. The original  Chaplygin gas corresponds to the\ncase $\\beta = 1$ \\cite{2}. The above equation of state leads to a\ndensity evolution in the form \\cite{Bento}\n\n\\begin{equation}\n \\rho_{ch}=\\left[A+\\frac{B}{a^{3(1+\\beta)}}\\right]^{\\frac{1}{1+\\beta}},\n \\label{2}\n\\end{equation}\n where $a$ is the\nscale factor  and $B$ is a positive integration constant.\n\nThe Chaplygin gas emerges as a effective fluid of a generalized\nd-brane in a (d+1, 1) space time, where the   action can be\nwritten as a generalized Born-Infeld action \\cite{Bento}. These\nmodels have been extensively studied in the literature\n\\cite{other}. The model parameters were constrained using current\ncosmological observations, such as, CMB \\cite{CMB} and  supernova\nof type Ia (SNIa) \\cite{SIa}.\n\n\n\nIn the  model of  Chaplygin inspired inflation usually the scalar\nfield, which drives inflation, is the standard inflaton field,\nwhere the energy density given by Eq.(\\ref{2}), can be extrapolate\nfor  obtaining a successful inflationary period\\cite{Ic}.\nRecently, tachyon-Chaplygin inflationary universe model was\nstudied in Ref.\\cite{SR}, and the dynamics of the early universe\nand the initial conditions for inflation in a model with radiation\nand a Chaplygin gas was considered  in Ref.\\cite{Monerat:2007ud}.\nThe main goal of the present work is to investigate the possible\nrealization of a warm-Chaplygin inspired inflationary model, where\nthe universe is filled with a self-interacting scalar field and a\nradiation field. We use astronomical data for constraining the\nparameters appearing in this model. Specifically, the   parameters\nare constrained from the WMAP observations\\cite{WMAP3,WMAP}.\n\n\n\n\n\n\n\nThe outline of the paper is a follows. The next section presents a\nshort review of  the modified  Friedmann equation and the\nwarm-Chaplygin Inflationary phase. Section \\ref{sectpert} deals\nwith the scalar and tensor perturbations.  In Section\n\\ref{exemple} it is presented a chaotic potential in the high\ndissipation approximation. Here, we give  explicit expressions for\nscalar power spectrum and tensor-scalar ratio for our models.\nFinally, sect.\\ref{conclu} summarizes our findings. We chose units\nso that $c=\\hbar=1$.\n\n\n\n\n\\section{The modified Friedmann equation and the Warm-Chaplygin Inflationary phase.\\label{secti} }\n\nWe start by writing down  the modified Friedmann equation, by\nusing the FRW metric. In particular, we assume that the\ngravitational dynamics  give rise to a modified Friedmann equation\nof the form\n\\begin{equation}\nH^2=\\kappa\\,[\\rho_{ch}+\\rho_\\gamma]=\n\\kappa\\left(\\left[A+\\rho_\\phi^{(1+\\beta)}\\right]^{\\frac{1}{1+\\beta}}\\;\\;+\\rho_\\gamma\\right),\n\\label{HC}\n\\end{equation}\nwhere $\\kappa=8\\pi G\/3=8\\pi\/3m_p^2$ (here $m_p$ represents the\nPlanck mass), $\\rho_\\phi=\\dot{\\phi}^2\/2+V(\\phi)$,\n  $V(\\phi)=V$ is\nthe scalar   potential and  $\\rho_\\gamma$ represents  the\nradiation energy density.  The modification is realized from an\nextrapolation of Eq.(\\ref{2}), so that\n\\begin{equation}\n \\rho_{ch}=\\left[A+\\rho_m^{(1+\\beta)}\\right]^{\\frac{1}{1+\\beta}}\\rightarrow\n \\;\\left[A+\\rho_\\phi^{(1+\\beta)}\\right]^{\\frac{1}{1+\\beta}},\n \\label{extr}\n\\end{equation}\nwhere $\\rho_m$ corresponds to the matter energy density \\cite{Ic}.\nThe generalized Chaplygin gas model may be viewed as a\nmodification of gravity, as described in Ref.\\cite{Ber}, and for\nchaotic inflation, in Ref.\\cite{Ic}. Different modifications of\ngravity have been proposed in the last few years, and there has\nbeen a lot of interest in the construction of early universe\nscenarios in higher-dimensional models motivated by\nstring\/M-theory \\cite{Ran}. It is  well-known that these\nmodifications can lead to important effects in the early universe.\nIn the following  we will take $\\beta=1$ for simplicity, which\ncorresponds to the usual Chaplygin gas.\n\n\n\n\n\n\n\n\n\n The dynamics of the\ncosmological model in the warm-Chaplygin inflationary scenario is\ndescribed by the equations\n \\begin{equation} \\ddot{\\phi}+\\,3H \\;\n\\dot{\\phi}+V_{,\\,\\phi}=-\\Gamma\\;\\;\\dot{\\phi}, \\label{key_01}\n \\end{equation}\nand \\begin{equation} \\dot{\\rho}_\\gamma+4H\\rho_\\gamma=\\Gamma\\dot{\\phi}^2\n.\\label{3}\\end{equation} Here $\\Gamma$ is the dissipation coefficient and it\nis responsible of the decay of the scalar field into radiation\nduring the inflationary era. $\\Gamma$ can be assumed to be a\nconstant or a function of the scalar field $\\phi$, or the\ntemperature $T$, or both \\cite{warm}. Here, we will take $\\Gamma$\nto be a function of $\\phi$ only.  In the near future we hope to\nstudy more realistic models in which $\\Gamma$ not only depends on\n$\\phi$ but also on $T$, expression which could be derived from\nfirst principles via Quantum Field Theory \\cite{Moss,Bastero}. On\nthe other hand, $\\Gamma$ must satisfy  $\\Gamma=f(\\phi)>0$ by the\nSecond Law of Thermodynamics. Dots mean derivatives with respect\nto time, and\n $V_{, \\,\\phi}=\\partial V(\\phi)\/\\partial\\phi$.\n\nDuring the inflationary epoch the energy density associated to the\nscalar field is of the order of the potential, i.e. $\\rho_\\phi\\sim\nV$, and dominates over the energy density associated to the\nradiation field, i.e. $\\rho_\\phi>\\rho_\\gamma$.  Assuming the set\nof slow-roll conditions, i.e. $\\dot{\\phi}^2 \\ll V(\\phi)$, and\n$\\ddot{\\phi}\\ll (3H+\\Gamma)\\dot{\\phi}$ \\cite{warm}, the Friedmann\nequation (\\ref{HC})  reduces  to\n\\begin{eqnarray}\nH^2\\approx\\kappa\\,\\sqrt{A+\\rho_\\phi^2}\\approx\\kappa\\,\\sqrt{(A+V^2)},\\label{inf2}\n\\end{eqnarray}\nand  Eq. (\\ref{key_01}) becomes\n\\begin{equation}\n3H\\left[\\,1+r \\;\\right ] \\dot{\\phi}\\approx-V_{,\\,\\phi},\n\\label{inf3}\n\\end{equation}\nwhere $r$ is the rate defined as\n\\begin{equation}\n r=\\frac{\\Gamma}{3H }.\\label{rG}\n\\end{equation}\nFor the high (weak) dissipation  regimen, we have $r\\gg 1$ ($r<\n1$).\n\n\nWe also consider that  during  warm inflation the radiation\nproduction is quasi-stable, i.e. $\\dot{\\rho}_\\gamma\\ll 4\nH\\rho_\\gamma$ and $ \\dot{\\rho}_\\gamma\\ll\\Gamma\\dot{\\phi}^2$.  From\nEq.(\\ref{3}) we obtained that the energy density of the radiation\nfield becomes\n \\begin{equation}\n\\rho_\\gamma=\\frac{\\Gamma\\dot{\\phi}^2}{4H},\\label{rh}\n\\end{equation}\nwhich  could be written as $\\rho_\\gamma= \\sigma T_r^4$, where\n$\\sigma$ is the Stefan-Boltzmann constant and $T_r$ is the\ntemperature of the thermal bath. By using Eqs.(\\ref{inf3}),\n(\\ref{rG}) and (\\ref{rh}) we get\n\\begin{equation}\n\\rho_\\gamma=\\sigma\\,T_r^4=\\frac{r}{12\\,\\kappa\\,(1+r)^2}\\,\\frac{V_{,\\,\\phi}^2}{\\sqrt{A+V^2}}\n.\\label{rh-1}\n\\end{equation}\nIntroducing the dimensionless slow-roll parameter $\\varepsilon$,\nwe write\n\\begin{equation}\n\\varepsilon\\equiv-\\frac{\\dot{H}}{H^2}\\simeq\\frac{1}{6\\kappa\\,(1+r)}\\,\n\\frac{V\\,V_{,\\,\\phi}^2}{(A+V^2)^{3\/2}},\\label{ep}\n\\end{equation}\nand the second slow-roll parameter  $\\eta$\n\\begin{equation}\n\\eta\\equiv-\\frac{\\ddot{H}}{H\n\\dot{H}}\\simeq\\,\\frac{1}{3\\kappa\\,(1+r)\\,\\sqrt{A+V^2}}\\,\\left[V_{,\\,\\phi\\phi}+\\frac{V_{,\\,\\phi}^2}{V}\n-\\frac{3}{2}\\,\\frac{V\\,V_{,\\,\\phi}^2}{(A+V^2)}\\right],\\label{eta}\n\\end{equation}\nwhere $V_{,\\,\\phi\\phi}=\\partial^2V\/\\partial\\phi^2$.\n\n\nNote that for $r=0$ (or $\\Gamma=0$), the parameters $\\varepsilon$\nand $\\eta$ given by Eqs.(\\ref{ep}) and (\\ref{eta})  reduced  to\n typical expression corresponding to cool inflation where a the Chaplygin gas\nis considered\\cite{Ic}.\n\nIt is possible to find a relation between the energy densities\n$\\rho_\\gamma$ and $\\rho_\\phi$  by using Eqs.(\\ref{rh-1}) and\n(\\ref{ep}), so that\n\\begin{equation}\n\\rho_\\gamma=\\frac{r}{2(1+r)}\\,\\frac{(A+\\rho_\\phi^2)}{\\rho_\\phi}\\,\\,\\varepsilon.\\label{c}\n\\end{equation}\n\n\nWarm inflation  takes place when the parameter $\\varepsilon$\nsatisfying  the inequality $\\varepsilon<1.$ This condition is\nanalogue to the requirement that  $\\ddot{a}> 0$. The condition\ngiven above is rewritten in terms of the energy densities\n$\\rho_\\phi$ and $\\rho_\\gamma$, so that\n\\begin{equation}\n\\frac{(A+\\rho_\\phi^2)}{\\rho_\\phi}>\n\\frac{2(1+r)}{r}\\;\\rho_\\gamma.\\label{cond}\n\\end{equation}\n\nAlso, inflation ends when the universe heats up at a time when\n$\\varepsilon\\simeq 1$, which implies\n\\begin{equation}\n\\frac{(A+\\rho_\\phi^2)}{\\rho_\\phi}\\simeq\n\\frac{2(1+r)}{r}\\;\\rho_\\gamma,\n\\end{equation}\nand the number of e-folds at the end of inflation is given by\n\\begin{equation}\nN=-3\\;\\kappa\\,\\int_{\\phi_{*}}^{\\phi_f}\\frac{\\sqrt{A+V^2}}{V_{,\\,\\phi}}(1+r)\nd\\phi\\,'.\\label{N}\n\\end{equation}\n\nIn the following, the subscripts  $*$ and $f$ are used to denote\nto the epoch when the cosmological scales exit the horizon and the\nend of  inflation, respectively.\n\n\n\\section{Perturbations\\label{sectpert}}\n\n\nIn this section we will study the scalar and tensor perturbations\nfor our model. Note that in the  case of scalar perturbations the\nscalar and the radiation fields are interacting. Therefore,\nisocurvature (or entropy) perturbations are generated besides of\nthe adiabatic ones. This occurs because warm inflation can be\nconsidered as an inflationary model with two basics fields\n\\cite{Jora1,Jora}. In this context dissipative effects  can\nproduce a variety of spectral, ranging between red and blue\n\\cite{62526,Jora}, and thus producing the running blue to red\nspectral suggested by WMAP observations\\cite{WMAP}.\n\nAs argued in Refs.\\cite{warm,Liddle}, the density perturbation\ncould be written as\n$\\delta_H=\\frac{2}{5}\\frac{H}{\\dot{\\phi}}\\,\\delta\\phi$. From\nEqs.(\\ref{inf3}) and (\\ref{rG}), this expression  becomes\n\\begin{equation}\n\\delta_H^2=\\frac{36\\,\\kappa^2}{25}\\left[\\frac{(A+V^2)\\,(1+r)^2}{V_{\\,,\\,\\phi}^2}\\right]\n\\,\\delta\\phi^2.\\label{331}\n\\end{equation}\n\n\nIn the case of high dissipation, the dissipation coefficient\n$\\Gamma$ is much greater that the  rate expansion $H$ , i.e.\n$r=\\Gamma\/3H\\gg 1$ and following  Taylor and Berera\\cite{Bere2},\nwe can write\n\\begin{equation}\n(\\delta\\phi)^2\\simeq\\,\\frac{k_F\\,T_r\\,}{2\\,\\pi^2},\\label{del}\n\\end{equation}\nwhere  the wave-number $k_F$ is defined by $k_F=\\sqrt{\\Gamma\nH\/V}=H\\,\\sqrt{3 r}\\geq H$, and corresponds to the freeze-out scale\nat which dissipation damps out to the thermally excited\nfluctuations. The freeze-out wave-number $k_F$ is defined at the\npoint where the inequality $V_{,\\,\\phi\\,\\phi}< \\Gamma H$, is\nsatisfied \\cite{Bere2}.\n\nFrom Eqs. (\\ref{331}) and (\\ref{del}) it follows that\n\\begin{equation}\n\\delta^2_H\\approx\\;\\frac{18\\,\\sqrt{3}}{25\\,\\pi^2}\\,\\frac{T_r}{V_{\\,,\\,\\phi}^2}\\,\\,\n(\\kappa\\,r\\,\\sqrt{A+V^2}\\,)^{5\/2}\\, .\\label{dd}\n\\end{equation}\n\n\nThe scalar spectral index $n_s$ is given by $ n_s -1 =\\frac{d\n\\ln\\,\\delta^2_H}{d \\ln k}$,  where the interval in wave number is\nrelated to the number of e-folds by the relation $d \\ln k(\\phi)=-d\nN(\\phi)$. From Eq.(\\ref{dd}), we get\n\\begin{equation}\nn_s  \\approx\\,\n1\\,-\\,\\left[5\\widetilde{\\varepsilon}-2\\widetilde{\\eta}-\\zeta\n\\right],\\label{ns1}\n\\end{equation}\nwhere, the  parameters $\\widetilde{\\varepsilon}$,\n$\\widetilde{\\eta}$ and $\\zeta$, (for $r\\gg 1$) are given by\n\\begin{equation}\n\\widetilde{\\varepsilon}\\approx\\frac{1}{6\\kappa\\,r}\\,\n\\frac{V\\,V_{,\\,\\phi}^2}{(A+V^2)^{3\/2}},\\,\\;\\;\\;\\;\\;\n\\,\\,\\widetilde{\\eta}\\approx\\,\\frac{1}{3\\kappa\\,r\\,\\sqrt{A+V^2}}\\,\\left[V_{,\\,\\phi\\phi}+\\frac{V_{,\\,\\phi}^2}{V}\n-\\frac{3}{2}\\,\\frac{V\\,V_{,\\,\\phi}^2}{(A+V^2)}\\right],\n\\end{equation}\nand\n\\begin{equation}\n\\zeta\\approx\\frac{V_{,\\,\\phi}^2}{\\kappa\\,r\\,\\sqrt{A+V^2}}\\,\\left(\\frac{V}{(A+V^2)}-\\frac{2}{3V}\\right)+\n\\frac{V_{,\\,\\phi}}\n{3\\kappa\\sqrt{A+V^2}}\\,\\frac{r_{,\\,\\phi}}{r^2},\n\\end{equation}\nrespectively.\n\nOne of the interesting features of the observations from WMAP is\nthat it hints at a significant running in the scalar spectral\nindex $dn_s\/d\\ln k=\\alpha_s$ \\cite{WMAP}.  From Eq.(\\ref{ns1}) we\nobtain that the running of the scalar spectral index becomes\n\n\\begin{equation}\n\\alpha_s\\approx\\frac{2\\,\\widetilde{\\varepsilon}\\,(A+V^2)}{V\\,V_{,\\,\\phi}}\\,\n[\\,5\\,\\widetilde{\\varepsilon}_{,\\phi}-2\\,\\widetilde{\\eta}_{,\\phi}-\\zeta_{,\\phi}\\,]\\label{dnsdk}.\n\\end{equation}\nIn models with only scalar fluctuations the marginalized value for\nthe derivative of the spectral index is approximately $-0.03$ from\nWMAP five year data\\cite{WMAP}.\n\n\n\n\n\nAs it was mentioned in Ref.\\cite{Bha} the generation of tensor\nperturbations during inflation would  produce stimulated emission\nin the thermal background of gravitational wave. This process\nchanges the power spectrum of the tensor modes by an extra\ntemperature dependently  factor given by $\\coth(k\/2T)$. The\ncorresponding spectrum  becomes\n\\begin{equation}\nA^2_g=\\frac{16\\pi}{m_p^2}\\left(\\frac{H}{2\\pi}\\right)^2\\,\\coth\\left[\\frac{k}{2T}\\right]\n\\simeq\\frac{4\\,\\kappa}{\\pi\\,m_p^2}\\,(A+V^2)^{1\/2}\\coth\\left[\\frac{k}{2T}\\right],\\label{ag}\n\\end{equation}\nwhere the spectral index $n_g$, results to be given by $\nn_g=\\frac{d}{d\\,\\ln k}\\,\\ln\\left[\n\\frac{A^2_g}{\\coth[k\/2T]}\\right]=-2\\,\\varepsilon$.  Here, we have\nused that  $A^2_g\\propto\\,k^{n_g}\\,\\coth[k\/2T]$ as described in\nRef.\\cite{Bha}.\n\n\n\n\nFor $r\\gg1$ and from expressions (\\ref{dd}) and (\\ref{ag}) we may\nwrite  the tensor-scalar ratio as\n\\begin{equation}\nR(k)=\\left.\\left(\\frac{A^2_g}{P_{\\cal R}}\\right)\\right|_{\\,k_*}\n\\simeq\\left.\\frac{m_p}{3\\,\\sqrt{8\\,\\pi}}\n\\left[\\frac{V_{\\,,\\,\\phi}^2}{T_r\\,\\,r^{5\/2}\\,(A+V^2)^{3\/4}}\\,\\coth\\left(\\frac{k}{2T}\\right)\\right]\\right|_{\\,k=k_*}.\n\\label{Rk}\\end{equation} Here, $\\delta_H\\equiv\\,2\\,P_{\\cal\nR}^{1\/2}\/5$ and  $k_*$  is referred to $k=Ha$, the value when the\nuniverse scale  crosses the Hubble horizon  during inflation.\n\nCombining  WMAP observations \\cite{WMAP3,WMAP} with the Sloan\nDigital Sky Survey (SDSS) large scale structure surveys\n\\cite{Teg}, it is found an upper bound for $R$ given by\n$R(k_*\\simeq$ 0.002 Mpc$^{-1}$)$ <0.28\\, (95\\% CL)$, where\n$k_*\\simeq$0.002 Mpc$^{-1}$ corresponds to $l=\\tau_0 k\\simeq 30$,\nwith the distance to the decoupling surface $\\tau_0$= 14,400 Mpc.\nThe SDSS  measures galaxy distributions at red-shifts $a\\sim 0.1$\nand probes $k$ in the range 0.016 $h$ Mpc$^{-1}$$<k<$0.011 $h$\nMpc$^{-1}$. The recent WMAP observations results give the values\nfor the scalar curvature spectrum $P_{\\cal\nR}(k_*)\\equiv\\,25\\delta_H^2(k_*)\/4\\simeq 2.3\\times\\,10^{-9}$ and\nthe scalar-tensor ratio $R(k_*)\\simeq0.055<0.2$. We will make use\nof these values  to set constrains on the parameters for our\nmodel.\n\n\n\\section{Chaotic potential in the high dissipation approach \\label{exemple}}\nLet us consider  an inflaton scalar field $\\phi$  with a chaotic\npotential. We write for the chaotic potential as $V=m^2\\phi^2\/2$,\nwhere $m$ is the mass of the scalar field. An estimation of this\nparameter is given for cool-Chaplygin inflation  in Ref.\\cite{Ic}\nand for warm inflation in Ref.\\cite{Bere2}. In the following, we\ndevelop models for constant and variable dissipation coefficient\n$\\Gamma$, and  we will restrict ourselves to the high dissipation\nregimen, i.e. $r\\gg 1$.\n\n\\subsection{ $\\Gamma =const.=\\Gamma_0$\ncase.\\label{GammaConst}}\n\n\nBy using the chaotic potential, we find that from Eq.(\\ref{inf3})\n\\begin{equation}\n\\dot{\\phi}=-\\frac{m^2\\,\\phi}{\\Gamma_0}\\Longrightarrow\\,\\phi(t)=\\phi_0\\,e^{-m^2\\,t\/\\Gamma_0},\n\\end{equation}\nand  during the inflationary scenario the scalar field decays due\nto dissipation into the radiation field. The Hubble parameter is\ngiven by\n\\begin{equation}\nH(t)=\\sqrt{\\kappa}\\,\\left(A+\\frac{m^4}{4}\\,\\phi_0^4\\,e^{-2m^2\\,t\/\\Gamma_0}\\right)^{1\/4}.\n\\end{equation}\nNote that in the limit $A\\longrightarrow 0$ the Hubble parameter\ncoincide with Ref.\\cite{Bere2}. The dissipation parameter $r$ in\nthis case is\n$$\nr(t)=\\frac{\\Gamma_0}{3\\,\\kappa^{1\/2}}\\,\\left(A+\\frac{m^4}{4}\\,\\phi_0^4\\,e^{-2m^2\\,t\/\\Gamma_0}\\right)^{-1\/4},\n$$\nwhere we observe that it increases when the cosmological time\nincreases.\n\nBy using Eq.(\\ref{c}), we can relate the energy density of the\nradiation field to the energy density of the inflaton field to\n\\begin{equation}\n\\rho_\\gamma=\\sqrt{\\frac{6}{\\pi}}\\;\\left(\\frac{m^2\\,m_p}\n{8\\,\\Gamma_0}\\right)\\,\\frac{\\rho_\\phi}{(A+\\rho_\\phi^2)^{1\/4}}.\\label{9}\n\\end{equation}\nNote again that in the limit $A\\longrightarrow 0$, Eq.(\\ref{9})\ncoincides with that  corresponding to the case where the Chaplygin\ngas is absent\\cite{Bere2}, i.e.\n$\\rho_\\gamma\\propto\\rho_\\phi^{1\/2}$.\n\n\nFrom Eq.(\\ref{dd}),  we obtain that the scalar power spectrum\nbecomes\n\\begin{equation}\nP_{\\cal R}(k)\\approx\\left.\n\\;\\frac{9\\;\\sqrt{3}}{4\\,\\pi^2}\\,\\frac{T_r}{m^2\\,V}\\,\n\\;\\left[\\frac{\\kappa^{1\/2}\\,\\Gamma_0\\,(A+V^2)^{1\/4}}{3}\\right]^{5\/2}\\right|_{\\,k=k_*},\\label{ppp}\n\\end{equation}\nand from Eq.(\\ref{Rk}) the tensor-scalar ratio is given by\n\\begin{equation}\nR(k)\\approx\\;\\left.\\frac{m_p}{3\\,\\sqrt{2\\pi}}\\;\n\\left(\\frac{3\\,\\kappa^{1\/2}}{\\Gamma_0}\\right)^{5\/2}\\;\\left[\\frac{m^2\\,V}{T_r\\,(A+V^2)^{1\/8}}\\,\n\\;\n\\;\\coth\\left(\\frac{k}{2T}\\right)\\right]\\right|_{\\,k=k_*}.\\label{rrrr}\n\\end{equation}\n\nBy using the WMAP observations where $P_{\\cal R}(k_*)\\simeq\n2.3\\times 10^{-9}$, $R(k_*)=0.055$, and choosing the parameter\n $T\\simeq\\,T_r$, we obtained from Eqs.(\\ref{ppp}) and\n(\\ref{rrrr}) that\n\\begin{equation}\nV_*\\simeq\\frac{4\\times10^{-5}\\,T_r}{m^2}\\,\\left[\\frac{m_p^2\\,\\Gamma_0^{2}}{\\coth(k_*\/2T_r)}\\right]^{5\/4},\\label{V}\n\\end{equation}\nand\n\\begin{equation}\nA=\\frac{10^{-21}\\,m_p^8}{\\coth^2(k_*\/2T_r)\\,}\\,-V_*^2.\\label{A}\n\\end{equation}\nFrom Eqs.(\\ref{V}) and (\\ref{A}), and since $A>0$, we obtained a\nlower limit for $m$ given by\n\\begin{equation}\nm^2>10^{6}\\,T_r\\;\\left[\\frac{\\Gamma_0^5}{m_p^3\\,\\sqrt{\\coth[k_*\/2T_r]}}\\right]^{1\/2},\n\\end{equation}\nand also we find an upper limit for the chaotic potential  when\nthe universe scale crosses the Hubble horizon  during inflation,\ngiven by\n\\begin{equation}\nV_*<10^{-11}\\,\\frac{m_p^4}{\\coth[k_*\/2T_r]}.\n\\end{equation}\nNow we consider the special case in which we fixe\n$T\\simeq\\,T_r\\simeq 0.24 \\times 10^{16}$ GeV and\n$\\Gamma_0\\simeq0.5\\times 10^{13}$ GeV. In this special case we\nobtained that the lower limit for the square mass of the scalar\nfield, is given by  $m^2>4\\times10^{-15}\\,m_p^2$, and for the\nchaotic potential we obtained an upper limit given by\n$V_*<10^{-14}m_p^4$.\n\n\n\\begin{figure}[th]\n\\includegraphics[width=5.0in,angle=0,clip=true]{f1.eps}\n\\caption{In this figure we plot the  parameter $(m\/m_p)^2$ as a\nfunction of the dissipation coefficient $\\Gamma_0\/m_p$. Here, we\nhave taken the values $T\\simeq\\,T_r\\simeq 0.24 \\times 10^{16}$ GeV\nand $k_*=0.002$ Mpc$^{-1}$.\n \\label{fig11}}\n\\end{figure}\n\n\n\n  In Fig.(\\ref{fig11})  we have plotted the parameter\n$(m\/m_p)^2$ versus the dissipation coefficient $\\Gamma_0\/m_p$. In\ndoing this, we have used Eqs. (\\ref{ns1}), (\\ref{V}) and\n(\\ref{A}). Here, we have taken the values $T\\simeq\\,T_r\\simeq 0.24\n\\times 10^{16}$ GeV and $k_*=0.002$ Mpc$^{-1}$.\n\n\n\n\\subsection{ $\\Gamma \\propto \\,\\phi^n$\ncase.\\label{Gammavariable}}\n\nIn this case we consider a power-law dissipation coefficient given\nby\n\\begin{equation}\n\\Gamma(\\phi)=\\alpha_n\\,\\phi^n,\n\\end{equation}\nwhere $n$ is a positive integer number and $\\alpha_n$ is a\npositive constant. The case $n=1$ was studied in Ref.\\cite{62526},\nthe case $n=2$ was developed in Ref. \\cite{Jora1}, and the general\ncase for any $n$ in Ref.\\cite{Lee:1999iv}.\n\n\nFrom Eq.(\\ref{inf3}) we find that\n\\begin{equation}\n\\phi(t)=\\left[\\phi_i^n-\\frac{n\\,m^2}{\\alpha_n}\\,t\\right]^{1\/n},\n\\end{equation}\nwhere $\\phi(t=0)=\\phi_i$. The dissipation parameter, $r$, as\nfunction of the cosmological time becomes\n\\begin{equation}\nr(t)=\\frac{\\alpha_n}{3\\,\\kappa^{1\/2}}\\,\\left[\\phi_i^n-\\frac{n\\,m^2}{\\alpha_n}\\,t\\right]\\,\n\\left[A+\\frac{m^4}{4}\\,\\left(\\phi_i^n-\\frac{n\\,m^2}{\\alpha_n}\\,t\\right)^{4\/n}\\right]^{-1\/4},\n\\end{equation}\nand in the high dissipation approach it is necessary to satisfy\nthat\n$$\n\\frac{\\alpha_n}{3\\,\\kappa^{1\/2}}\\,\\left[\\phi_i^n-\\frac{n\\,m^2}{\\alpha_n}\\,t\\right]\\gg\\,\\,\n\\left[A+\\frac{m^4}{4}\\,\\left(\\phi_i^n-\\frac{n\\,m^2}{\\alpha_n}\\,t\\right)^{4\/n}\\right]^{1\/4}.\n$$\n\nUsing Eq.(\\ref{c}), we can relate the energy density of the\nradiation field to the energy density of the inflaton field as\n\\begin{equation}\n\\rho_\\gamma=\\sqrt{\\frac{6}{\\pi}}\\;\\left(\\frac{m^{2+n}\\,m_p}\n{8\\,2^{n\/2}\\,\\alpha_n}\\right)\\,\\frac{\\rho_\\phi^{1-n\/2}}{(A+\\rho_\\phi^2)^{1\/4}}.\\label{91}\n\\end{equation}\n\n\n\n\n\nFrom Eq.(\\ref{dd}),  we obtain that the scalar power spectrum\nbecomes\n\\begin{equation}\nP_{\\cal R}(k)\\approx\\left.\n\\;\\frac{9\\;\\sqrt{3}}{4\\,\\pi^2}\\,\\frac{T_r\\,V^{\\frac{5n-4}{4}}}{m^{\\frac{5n+4}{2}}}\\,\n\\;\\left[\\frac{2^{n\/2}\\,\\kappa^{1\/2}\\,\\alpha_n\\,\\,(A+V^2)^{1\/4}}{3}\\right]^{5\/2}\\right|_{\\,k=k_*},\\label{p2}\n\\end{equation}\nand from Eq.(\\ref{Rk}) the tensor-scalar ratio becomes given by\n\\begin{equation}\nR(k)\\approx\\;\\left.\\frac{m_p}{3\\,\\sqrt{8\\pi}}\\;\n\\left(\\frac{3\\,\\kappa^{1\/2}}{\\alpha_n}\\right)^{5\/2}\\;\\left[\\frac{\nm^{\\frac{4+5n}{2}}\\,\\,(2\\,V)^{\\frac{4-5n}{4}}}{T_r\\,(A+V^2)^{1\/8}}\\,\n\\;\n\\;\\coth\\left(\\frac{k}{2T}\\right)\\right]\\right|_{\\,k=k_*}.\\label{r2}\n\\end{equation}\n\nBy using the WMAP observations where $P_{\\cal R}(k_*)\\simeq\n2.3\\times 10^{-9}$,  $R(k_*)=0.055$, and choosing the parameter\n $T\\simeq\\,T_r$, we obtained from Eqs.(\\ref{p2}) and\n(\\ref{r2}) that\n\\begin{equation}\nm\\approx\n\\left[\\frac{3.7\\times10^{-4}}{a^{1\/4}\\,b^{5\/4}\\,V_*^{\\frac{4-5n}{4}}}\\right]^{\\frac{2}{4+5n}},\\label{m2}\n\\end{equation}\nand\n\\begin{equation}\nA\\approx\\,5.9\\times 10^{-11}\\,\\left[\\frac{8.1\\times\n10^{-10}}{a^2\\,b^2}-1.7\\times 10^{10}\\,V_*^2\\right]\\label{A2},\n\\end{equation}\nwhere $a$ and $b$ are given by\n$$\na=\\frac{2^{\\frac{5n-8}{4}}\\,T_r\\,\\kappa^{5\/4}\\,\\alpha_n^{5\/2}}{\\pi^2},\\;\\;\\;\\mbox{and}\\;\\;\\;\\;\nb=\\frac{m_p\\,3^{3\/2}}{\\sqrt{\\pi}\\,\\,2^{\\frac{2+5n}{4}}}\\;\\frac{\\kappa^{5\/4}}{\\alpha_n^{5\/2}\\;T_r}\\,\n\\coth\\left(\\frac{k_*}{2\\,T_r}\\right),\n$$\nrespectively.\n\nUsing that $A>0$, then we find from Eqs. (\\ref{m2}) and (\\ref{A2})\na lower  limit for the mass of the inflaton $m$, independent of\nthe value $V_*$, and it  becomes\n\n\\begin{equation}\nm\\,>\\,\\left[\\frac{a^{3-5n}\\;10^{\\frac{44-95n}{2}}}{b^{1+5n}}\\right]^{\\frac{1}{2(4+5n)}}\\label{mass}.\n\\end{equation}\nFor instance, for $n=1$ we obtained from Eq.(\\ref{mass}), that\n$m>10^{-1}\\,[T_r^2\\,\\alpha_{n=1}^5\/\\kappa^5]^{1\/9}\\,[m_p\\,\\coth(k_*\/2T_r)]^{-1\/3}$$\\sim\n7\\times10^{-7}m_p$. Here, we have used that $T_r\\simeq\n10^{-3}m_p$, $k_*=0.002$Mpc$^{-1}$ and $\\alpha_{n=1}\\simeq\n10^{-5}$ (see Ref.\\cite{62526}).  Note that this lower limit for\n$m$ decreases when  the parameter $\\alpha_{n=1}$ decreases. For\n$n>1$ we noted that a similar situation occurs for the lower limit\nof $m$, i.e., this limit decreases when  the parameter\n$\\alpha_{n}$ decreases. In Fig.\\ref{rons} we have plotted  the\nrunning spectral index $\\alpha_s$ versus the scalar spectrum index\n$n_s$, for a dissipation coefficient\n$\\Gamma(\\phi)=\\alpha_{n=1}\\,\\phi$.\n In doing this, we have taken two different values for the\nparameter $\\alpha_{n=1}$ and choosing the parameters $T\\simeq\nT_r\\simeq0.24\\times 10^{16}$GeV and $k_*=0.002$Mpc$^{-1}$. Note\nthat for $\\alpha_{n=1}\\sim 10^{-5}-10^{-7}$ and for a given\n$n_s\\sim 0.95$ the values of $\\alpha_s$ becomes in agrement with\nthat registered by the WMAP observations. From Fig. \\ref{rons2} we\nnote that the parameter $A\\sim 10^{-25} m_p^8$  becomes smaller by\none order of magnitude when it is compared with the case of\nChaplygin cool-inflation  \\cite{Ic}, and smaller by two order of\nmagnitude when it is compared with the case of tachyon-Chaplygin\ninflationary universe model \\cite{SR}.\n\n\nAnother  interesting situation corresponds to the case when\n$\\Gamma=\\alpha_{n=2}\\,\\phi^2$. Following a  similar approach to\nthose of the previous cases,  we obtained that $\\alpha_{n=2}\\sim\n10^{-13}\/m_p-10^{-5}\/m_p$ and $A\\sim10^{-26}m_p^8$ in order to be\nin  agrement with  WMAP five year data\\cite{WMAP}. Note that this\nvalue of $A$ is similar to those found for the case $n=1$. Here,\nagain we have taken the parameters $T\\simeq T_r\\simeq0.24\\times\n10^{16}$GeV, $k_*=0.002$Mpc$^{-1}$ and $n_s\\simeq 0.95$.\n\n\nIn Fig.\\ref{fig4} we show the plot of the adimensional parameter\n$\\alpha_n\/m_p^{1-n}$ in terms of the parameters $n$ on the one\nhand, and $m\/m_p$ in addition. In doing this, we have used Eqs.\n(\\ref{ns1}), (\\ref{dnsdk}), (\\ref{m2}) and (\\ref{A2}). Here, we\nhave used the WMAP five year data, and  we have taken three\ndifferent values for the parameters $n$ and $m\/m_p$. Also, we have\nchosen $T\\simeq T_r\\simeq0.24\\times 10^{16}$GeV and\n$k_*=0.002$Mpc$^{-1}$, as above.\n\n\n\n\\begin{figure}[th]\n\\includegraphics[width=4.0in,angle=0,clip=true]{f2.eps}\n\\caption{Evolution of the  running  scalar spectral index\n$\\alpha_s$ versus the scalar spectrum index $n_s$ in the model\n$\\Gamma\\propto\\,\\phi$, for two different values of the parameter\n$\\alpha_{n=1}$.\n \\label{rons}}\n\\end{figure}\n\n\\begin{figure}[th]\n\\includegraphics[width=4.0in,angle=0,clip=true]{f3.eps}\n\\caption{The parameter $A$ versus the scalar spectrum index $n_s$\nin the model $\\Gamma\\propto\\,\\phi$, for two different values of\nthe parameter $\\alpha_{n=1}$.\n \\label{rons2}}\n\\end{figure}\n\n\n\n\n\n\\begin{figure}[th]\n\\includegraphics[width=7.8in,angle=0,clip=true]{f4.eps}\n\\caption{The left panel shows the parameter $\\alpha_n\/m_p^{1-n}$\nas a function of the parameter $n$. The right panel shows the\nparameter $\\alpha_n\/m_p^{1-n}$ versus the mass of the inflaton\nfield $m\/m_p$. In both cases, we have taken the parameters\n$T\\simeq T_r\\simeq0.24\\times 10^{16}$GeV and\n$k_*=0.002$Mpc$^{-1}$.\n \\label{fig4}}\n\\end{figure}\n\n\n\n\n\n\n\\section{Conclusions \\label{conclu}}\n\nIn this paper we have investigated the  warm-Chaplygin\ninflationary scenario. In the slow-roll approximation we have\nfound a general relationship between the radiation and scalar\nfield energy densities. This has led us to a general criterium for\n warm-Chaplygin inflation  to occur.\n\n\n\n\nOur  specific models are described by  a chaotic potential and we\nhave consider  different  cases for   the dissipation coefficient,\n$\\Gamma$. In the first case, we took $\\Gamma=constant=\\Gamma_0 $.\nHere, we have\n found  solutions for the inflaton field and the  Hubble parameter. The\n relation between  the radiation field and the inflaton field\n energy densities   presents the same\n characteristic to that corresponding to the   warm inflation case, except that\n it  depends on the extra parameter  $A$. For the case in which the\n dissipation coefficient $\\Gamma$\nis taken to be a function of the scalar field, i.e. $\\Gamma\n\\propto \\phi^n$, it was possible to describe an  appropriate warm\ninflationary universe model for $n=1$ and $n=2$. In these cases,\nwe have obtained  explicit expressions for the corresponding\nscalar spectrum  and the running of the scalar spectrum indices.\n\nFinally, by using the WMAP five year data\\cite{WMAP} and for the\nspecial case where $T\\simeq T_r\\simeq0.24\\times 10^{16}$GeV,\n$k_*=0.002$Mpc$^{-1}$ and  $n_s\\simeq 0.95$,  we have found new\nconstraints on the parameters $\\alpha_n$, $A$ and $m$.\n\n\n\n\n\n\n\n\n\n\\begin{acknowledgments}\n S.d.C. was\nsupported by COMISION NACIONAL DE CIENCIAS Y TECNOLOGIA through\nFONDECYT grant N$^0$ 1070306. Also, from UCV-DGIP N$^0$ (2008).\nR.H. was supported by the ``Programa Bicentenario de Ciencia y\nTecnolog\\'{\\i}a\" through the Grant ``Inserci\\'on de Investigadores\nPostdoctorales en la Academia\" \\mbox {N$^0$ PSD\/06}.\n\\end{acknowledgments}\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec:intro}\nThe standard model of particle physics (SM) is the most complete and successful explanation of the fundamental interactions. Despite its awesome phenomenological achievements, this model is not able to explain all the observed phenomena. Among them we can mention, for example, the presence of Dark Matter (DM) in the universe. Although  we know very well the amount of DM  in the Universe (in terms of relic density) \\cite{PLANCK}, we know nothing about it's nature.\\newline\nAnother problem is the neutrino mass generation mechanism. Due to the apparent absence of right-handed neutrinos, it is impossible to generate neutrino masses through Yukawa couplings as the other fermions do. Moreover, even if right handed neutrinos exist, the Yukawa coupling needed to explain the neutrino masses would be so small that the standard mechanism for mass generation seems to be unnatural for the neutrinos. In this context, it would be of great interest if these two apparently independent problems (dark matter and neutrino mass generation) could be connected. Part of the aim of this work is to propose a mechanism for studying one realization of this intriguing possibility.\n\nAmong the variety of proposed ideas that may describe the physics of dark matter, the minimal  dark matter program  \\cite{minimaldm} appears as specially appealing. Initially, it was developed for scalar and fermion fields. However, it was later shown that the mechanism worked very well also for vector fields. In reference \\cite{triplet_dm}, for instance, it was shown that the inclusion of a new massive field in the adjoint representation of $SU(2)_L$ provide a viable dark matter candidate. Additionally, the case of a massive vector field in the fundamental representation of $SU(2)_L$ was also successfully studied in reference \\cite{vector_dm}. Interestingly enough, this last model can be naturally extended by the inclusion of a sterile left-handed heavy neutral lepton (HNL) \\cite{masses_and_mixings}. In this version, the model is capable of solving both problems: dark matter and neutrino mass generation mechanism \\cite{vector_dm,masses_and_mixings,dong2021}. Indeed, this model has been shown to be able of tackling also the muon $(g-2)$ problem \\cite{dong2021}.   \n\n\nIn this work, we perform a phenomenological study for the production of these new particles ({\\it i.e.} the new massive vector boson and the left-handed HNL)  in the Large Hadron Collider (LHC). We use existing experimental data to constrain the model parameters, as a function of the sensitivity of current experiments. Additionally, we make predictions based on the kinematical features of the production process, in order to facilitate the search of these new particles in future measurements.\n\nThe paper is organized as follows: In Section \\ref{sec:theo} we describe the main features of the proposed model, while in Sections \\ref{sec:prod} and \\ref{cross} we discuss the main production mechanism of these new particles at the LHC and compute the cross section of the process, respectively. In Section \\ref{sec:uplims} we define an upper bound in the parameter space considering experimental constraints and Section \\ref{sec:kin} presents the kinematical analysis performed to find distinctive signals coming from the model. In Section \\ref{sec:lowlims} we obtain an estimation of lower limits for the parameter space coming from dark matter measurements. Finally, our conclusions are presented in Section \\ref{sec:conc}, while the Appendix is devoted to  general considerations for the parametrization of the ATLAS detector efficiency considered for our results.\n\n\n\n\\section{Theoretical background}\\label{sec:theo}\nLet us start by introducing the following set of vectors:\n\\begin{equation}\n    V_{\\mu}=\\begin{pmatrix}V^{+}_\\mu\\\\V^0_\\mu\\end{pmatrix}=\\begin{pmatrix}V^{+}_\\mu\\\\\\frac{1}{\\sqrt{2}}(V^1_\\mu+iV^{2}_\\mu)\\end{pmatrix},\n\\end{equation}\nwhich transforms as $(1,2,1\/2)$ under the SM group $SU(3)_c\\times SU(2)_L\\times U(1)_Y$. The dynamics of this new vector doublet is described by the following lagrangian:\n\\begin{equation}\n\\begin{split}\n    \\mathcal{L}=\\mathcal{L}_{SM}&-\\frac{1}{2}(D_\\mu V_\\nu- D_\\nu V_\\mu)^\\dagger (D^\\mu V^\\nu -D^\\nu V^\\mu)\n    +i\\frac{g'\\kappa_1}{2}V_\\mu^\\dagger B^{\\mu\\nu}V_\\nu+ig\\kappa_2 V_\\mu^\\dagger W^{\\mu\\nu}V_\\nu \\\\\n    &+M_V^2 V_\\mu^\\dagger V^\\mu-\\alpha_2(V_\\mu^\\dagger V^\\mu)(V_\\nu^\\dagger V^\\nu)-\\alpha_3(V_\\mu^\\dagger V^\\nu)(V_\\nu^\\dagger V^\\mu)\\\\&-\\lambda_2(\\Phi^\\dagger \\Phi)(V_\\mu^\\dagger V^\\mu)-\\lambda_3(\\Phi^\\dagger V_\\mu)(V^{\\mu\\dagger}\\Phi)-\\frac{\\lambda_4}{2}[(\\Phi^\\dagger V_\\mu)(\\Phi^\\dagger V^\\mu)+(V^{\\mu\\dagger}\\Phi)(V_\\mu^\\dagger \\Phi)],\n    \\end{split}\n\\end{equation}\nwhere $D_\\mu$ stands for the covariant derivative, $W^{\\mu\\nu}$ and $B^{\\mu\\nu}$ are the field strengths of $SU(2)_L$ and $U(1)_Y$, respectively, and $\\Phi$ is the SM Higgs doublet. This lagrangian was labeled by the authors as the Vector Doublet Dark Matter Model (VDDMM), since \nthe neutral component of the vector has been proven to account for DM (see reference \\cite{vector_dm}). \nIt's worth to mention the presence of non minimal gauge interactions. The coupling constants associated to these interactions ($\\kappa_1$ and $\\kappa_2$) are free parameters. In reference \\cite{vector_dm} the authors set their values as $\\kappa_1=\\kappa_2=1$, in order to avoid interactions between the photons and the neutral component of the vector doublet.\nThis restriction can be full filled with a more general condition: $\\kappa_1=\\kappa_2=\\kappa$. However, since the phenomenology of the model has been studied with $\\kappa=1$, we are going to keep this choice, despite the fact that a different choice of this value can produce an interesting phenomenology. Additionally, the lagrangian presents an accidental $Z_{2}$ symmetry, making the lightest neutral vector as a viable DM candidate.\n\\newline\n\n\nIn order to perform calculations, it's useful to define some quantities after the symmetry breaking:\n\\begin{equation}\n    \\begin{split}\n        &M_{V^\\pm}^2=\\frac{1}{2}(2M_V^2-v^2\\lambda_2)\\\\\n        &M_{V^1}^2=\\frac{1}{2}(2M_V^2-v^2[\\lambda_2+\\lambda_3+\\lambda_4])\\\\\n        &M_{V^2}^2=\\frac{1}{2}(2M_V^2-v^2[\\lambda_2+\\lambda_3-\\lambda_4])\\\\\n        &\\lambda_L=\\lambda_2+\\lambda_3+\\lambda_4.\n    \\end{split}\n\\end{equation}\nThe latter is the coupling that describes the interaction between the Higgs boson and $V^1$. Moreover, in this work, we are particularly interested on the production of these vector states, therefore, we don't need to pay attention on the quartic interactions, having in consequence  only 4 relevant parameters. For convenience, we perform a change of basis for them, and write the lagrangian in terms of the previously presented parameters:\n\\begin{equation}\n\\begin{split}\n    &\\lambda_2=\\lambda_L+2\\frac{M_{V^1}^2-M_{V^\\pm}^2}{v^2}\\\\\n    &\\lambda_3=\\frac{2M_{V^\\pm}^2-M_{V^1}^2-M_{V^2}^2}{v^2}\\\\\n    &\\lambda_4=\\frac{M_{V^2}^2-M_{V^1}^2}{v^2}\\\\\n    &M_V^2=M_{V^1}^2+\\frac{v^2}{2}\\lambda_L.\n    \\end{split}\n\\end{equation}\nHenceforth, we are going to consider the special case of $M_{V^1}= M_{V^2}\\equiv M_{V^0}$. This special case is motivated by the results obtained in reference \\cite{vector_dm}. In addition, we define the quantity $\\Delta M= M_{V^+}-M_{V^0}$.\\newline\nDue to the vector doublet quantum numbers and Lorentz invariance, it is not possible to link directly the new vector field to the SM fermions. However, if we introduce an exotic left-handed neutrino which is singlet of the SM, we can write the following interaction term:\n\\begin{equation}\n    \\mathcal{L}_{VNL}=-\\sum_{k=\\{e,\\mu,\\tau\\}}\\beta_k \\bar{L}_k \\gamma^\\mu \\tilde{V_\\mu} N_L  +\\text{h.c.} \\ ,\n    \\label{LVNL}\n\\end{equation}\nwith\n\\begin{equation}\n    \\tilde{V_\\mu}=i\\sigma_2 V_\\mu^*=\\begin{pmatrix}\\frac{1}{\\sqrt{2}}(V^1_\\mu-iV^{2}_\\mu)\\\\-V^{-}_\\mu\\end{pmatrix} \\ .\n\\end{equation}\nIn Eq.~\\ref{LVNL}, $N_L$ stand for a left-handed Majorana fermion with definite mass $M_N$, which acts as a portal between the vector doublet and SM leptons. If we define $N_L$ as $Z_2$ odd, the accidental symmetry is respected, and this new fermion can also be a DM candidate, depending if it's lighter than the vector candidate. This kind of interactions have been also studied with two \\cite{masses_and_mixings} and three \\cite{dong2021} HNLs. For simplicity, we will work with just one HNL. In order to keep track of the model variants, we will refer to the model treated in this work as VDM1N. \n\n\\section{Production mechanism at the LHC}\\label{sec:prod}\n\nAs we stated above, we are interested in the study of the VDM1N model in the context of LHC, particularly at the ATLAS detector. The signal to be searched is composed by two HNL in the final states, which can be used as smoking gun of this new physics.\nConsequently, we will focus on the following process $pp\\rightarrow j j N_L N_L l^+ l^-$.\n\nThe main contribution to this process comes from Vector Boson Fusion (VBF), as can be seen in Figure \\ref{vbf}. \n\\begin{figure}[!h]\n    \\centering\n\\includegraphics[scale=0.3]{figures\/vbf.png}\n\\caption{Leading order process for the production of Left-handed HNLs.}\n    \\label{vbf}\n\\end{figure}\n We used FeynRules  \\cite{fr1,fr2,ufo} to obtain the Feynman rules for the new sector, and Madgraph  \\cite{mg5} to compute the cross section for the process at $\\sqrt{s}=13$ [TeV], considering all the possible diagrams.\n\nWe have performed an analysis considering different values of $M_{V^+}$, restricting ourselves to the degenerate case where $\\Delta M=0$. As a first step, we considered only electrons in the final state (see Fig \\ref{vbf}). The study was carried out using the parameters presented in Table \\ref{simulation_params}, and from now on we will refer to these simulations as the signal sample.\n\\begin{table}[!h]\n    \\centering\n    \\begin{tabular}{|c|c|}\n    \\hline\n       variable  & value  \\\\\n       \\hline\n        $M_{V^+}$ & $200-300[$GeV$]$ with steps of $5[$GeV$]$\\\\\n        $M_N$ &$50 [$GeV$]$\\\\\n        $|\\beta_e|^2$& 1\\\\\n        $\\lambda_L$& 5\\\\\n        \\hline\n    \\end{tabular}\n    \\caption{Parameters used for the simulations.}\n    \\label{simulation_params}\n\\end{table}\n\\newline\n Additionally, we have simulated the SM background at $\\sqrt{s}=13$ [TeV]  (the background sample from now on), by computing the process $p p  \\rightarrow  j j e^+  e^- \\bar{\\nu}\\nu$ (where the neutrinos can be of any flavour) which is dominated by weak diboson production. \n\n\n\\section{Cross section of the process}\\label{cross}\nFor each signal sample, we have obtained the production cross section. Then, we considered the cuts shown in Table \\ref{cut_def} which are the same as the ones used by ATLAS for  neutralinos search \\cite{susylims}, and applied them to the generated events. The ratio of events matching the selection criteria to the total simulated events defines the acceptance $\\mathcal{A}$. With this information we computed the effective cross section as:\n\n\\begin{table}\n    \\centering\n    \\begin{tabular}{|c|c|c|}\n    \\hline\n     object    & definition &condition \\\\\n     \\hline\n$p_T(l+)$ & Transverse momentum of the positively charged lepton & $\\geq 25$ [GeV]\\\\\n    $p_T(l-)$    & Transverse momentum of the negatively charged lepton & $\\geq 25$[GeV]\\\\\n          $M_{ee}$ & Invariant mass of the SFOS lepton pair & $\\geq 12$[GeV]\\\\\n          $p_T^{miss}$ & Transverse component of the missing momentum vector &$>200$[GeV]\\\\\n          $\\Delta \\phi(jet_{1},p_T^{miss})$ &Azimuthal separation between $p_T^{miss}$ and the first jet & $\\geq 0.4$\\\\\n            $\\Delta \\phi(jet_{2},p_T^{miss})$ &Azimuthal separation between $p_T^{miss}$ and the second jet & $\\geq 0.4$\\\\\n          \\hline\n    \\end{tabular}\n    \\caption{Event selection criteria. These cuts were taken from reference \\cite{susylims}}\n    \\label{cut_def}\n\\end{table}\n \n \\begin{equation}\n     \\sigma_{eff}=\\epsilon\\mathcal{A} \\sigma,\n \\end{equation}\n where $\\epsilon$ parametrizes the detector efficiency.\n According to reference \\cite{lep_effs}, the electron reconstruction for high values of the $p_T$ is near to one, so we can approximate $\\epsilon\\approx 1$.\n After computing the acceptance for each simulated sample, we performed a fit to obtain the functional form of the effective cross section on $M_{V^+}$. The results for the cross sections can be seen in Figure \\ref{deltam0sim}. Notice that these cross sections where computed using $\\beta_e=1$.\n\\begin{figure}[!h]\n    \\centering\n    \\includegraphics[width=0.7\\textwidth]{figures\/deltam0sim.png}\n    \\caption{Results for the cross sections of the process before and after the cuts using $\\beta_e=1$. The solid black line represents the fitting function over the scaled cross sections.}\n    \\label{deltam0sim}\n\\end{figure}\nConsequently, the simulated results can be factorized by the coupling $|\\beta_e|^2$, obtaining:\n\\begin{equation}\n    \\sigma_{eff}^{ee}=|\\beta_e|^4 f(M_{V_{+}}) \\label{Eq:Sigma_Eff_ee},\n\\end{equation}\nwhere $f$ is the fitting function of the cross section.\nAt this point it is relatively easy to include processes that include muons (instead of electrons) in the final step. First we observe that in our kinematical regime, the muon mass is negligible. Additionally, the topology of the processes involving muons are exactly the same to those involving electrons. Consequently, we can write a generalized version of Eq. \\eqref{Eq:Sigma_Eff_ee}, in which the total cross section for the process considering a muon or electron lepton pair becomes:\n\\begin{equation}\\label{eqcross}\n    \\sigma_{eff}^{LL}=(|\\beta_e|^4 +|\\beta_\\mu|^4) f(M_{V_{+}}).\n\\end{equation}\nThis expression allows to define bounds on the parameter space, considering the current searches on new physics.\nNotice that we didn't include a $\\tau$ lepton pair in the final state because it's reconstruction is more difficult due to the presence of hadronic decay modes. However, the experimental reference have setted limits over muons and electrons in the final state \\cite{susylims}, which is suitable for our study. \n\n\n\\section{Setting the experimental limits}\\label{sec:uplims}\nThe background cross section reported in reference \\cite{susylims}  has a value of $\\sigma_{bkg}=102$[fb] for a luminosity of $14.7$[fb $^{-1}$]. Since the reference does not report a significant deviation from the Standard Model, we can define a threshold using the significance of a measurement. Consequently, we used the standard definition of significance:\n\n\\begin{equation}\\label{sig}\n    Z=\\frac{s}{\\sqrt{s+b}},\n\\end{equation}\nwhere $s$ and $b$ are the number of signal and background events, respectively. Moreover, the number of of events can be written in terms of the process cross section $\\sigma$ and the collider luminosity $\\mathcal{L}$: \n\\begin{equation}\\label{n=sigmaL}\n    n=\\sigma\\mathcal{L}.\n\\end{equation}\nWe can rewrite eq. \\eqref{sig} using eq. \\eqref{n=sigmaL}, and use this result to constrain $\\sigma_{eff}^{LL}$:\n\\begin{equation}\n    \\frac{\\sigma_{eff}^{LL}\\sqrt{\\mathcal{L}}}{\\sqrt{\\sigma_{bkg}+\\sigma_{eff}^{LL}}}\\leq 1\\implies \\sigma_{eff}^{LL}\\leq 2.67[\\text{fb}],\n\\end{equation}\nwith this bound we can use eq.~\\eqref{eqcross} to constrain the parameter space, as can be seen in Figure \\ref{paramlims}. Furthermore, it is possible to make predictions for the expected luminosity for a significant measurement of the process. The predictions show that is not possible to surpass the discovery threshold within the range of the HL-LHC, but significances of $Z=3\\sigma$ can be reached in the sensitivity range of current and future Collider Experiments.\n\\begin{figure}[!h]\n\\centering\n    \\begin{subfigure}{0.4\\textwidth}\n    \\includegraphics[width=\\textwidth]{figures\/crosslims.png}\n    \\caption{Available parameter space after applying the collider constraints.}\n\n\\end{subfigure}\n\\hspace{0.6cm}\n\\begin{subfigure}{0.4\\textwidth}\n    \\centering\n    \\includegraphics[width=\\textwidth]{figures\/lumilims.png}\n    \\caption{Predicted significance for projections of the Luminosity.}\n\\end{subfigure}\n\\caption{Collider limits and significance projections.}\n\\label{paramlims}\n\\end{figure}\n\n\\section{Kinematical Analysis}\\label{sec:kin}\nAfter completing the simulation, we studied the distribution of the relevant kinematical observables, which are defined and described in Table \\ref{kinobjects}.\n\n\\begin{table}[!h]\n    \\centering\n    \\begin{tabular}{|c|c|}\n    \\hline\n       object  & definition  \\\\\n       \\hline \n       $\\cos\\theta$  & angular distribution between the SFOS lepton pair\\\\\n        $H_T$ & Scalar sum of the $p_T$ of all outgoing quarks.\\\\\n       $M_{ee}$ &invariant mass of the SFOS lepton pair\\\\\n       $p_T^{miss}$ & $p_T$ of the left handed HNL pair\\\\\n       \\hline\n    \\end{tabular}\n    \\caption{Relevant kinematical observables  used in our study. Notice that $H_T$ was simulated only at parton level.}\n    \\label{kinobjects}\n\\end{table}\nFor each simulated sample, we obtained the distribution of these variables. In order to compare the results with the current experimental searches, we applied the same cuts defined in Section \\ref{cross}.\nThe resulting distributions are shown in Figure\n \\ref{merged}. We can see that all the signal samples have similar distributions, independently of the value of $M_{V^+}$. In addition, it can be seen that angular distribution and  dilepton invariant mass behave differently than the SM, therefore, they can be used as a smoking gun of this model. \n\\begin{figure}[!h]\n\n    \\begin{subfigure}{0.4\\textwidth}\n    \\includegraphics[width=\\textwidth]{figures\/srlow\/angdist.png}\n    \\caption{Angular distribution}\n\n\\end{subfigure}\n\\begin{subfigure}{0.4\\textwidth}\n    \\centering\n    \\includegraphics[width=\\textwidth]{figures\/srlow\/leppt.png}\n    \\caption{Parton level $H_T$}\n\\end{subfigure}\n\\begin{subfigure}{0.4\\textwidth}\n    \\centering\n    \\includegraphics[width=\\textwidth]{figures\/srlow\/lepm.png}\n\\caption{Di-electron invariant mass.}\n    \\end{subfigure}\n    \\begin{subfigure}{0.4\\textwidth}\n    \\centering\n    \\includegraphics[width=\\textwidth]{figures\/srlow\/misspt.png}\n\\caption{$p_T$ of the invisible particles.}\n\\end{subfigure}\n\n        \n\\caption{Kinematical Distributions for some representative simulations. The ATLAS cuts are applied in the samples.}\n\\label{merged}\n\\end{figure}\n\n Since in all cases the signal and the background distributions have some level of overlap, it's convenient to search for cuts that optimize the signal over the background. We find that our goal is achieved supplementing the ATLAS cuts with $M_{ee}>260\\text{[GeV]}$. \nThe implementation of the optimized cut increases the significance, making wider the parameter space that can be probed (see Figure \\ref{projections}).\n\n\n\n\\begin{figure}[!h]\n    \\begin{subfigure}{0.45\\textwidth}\n    \\includegraphics[width=\\textwidth]{figures\/significances\/lumi100.png}\n    \\caption{$\\mathcal{L}=100[\\text{fb}^{-1}]$}\n\\end{subfigure}\n    \\begin{subfigure}{0.45\\textwidth}\n    \\includegraphics[width=\\textwidth]{figures\/significances\/lumi300.png}\n    \\caption{$\\mathcal{L}=300[\\text{fb}^{-1}]$}\n\\end{subfigure}\n    \\begin{subfigure}{0.45\\textwidth}\n    \\includegraphics[width=\\textwidth]{figures\/significances\/lumi500.png}\n    \\caption{$\\mathcal{L}=500[\\text{fb}^{-1}]$}\n\\end{subfigure}\n    \\begin{subfigure}{0.45\\textwidth}\n    \\includegraphics[width=\\textwidth]{figures\/significances\/lumi700.png}\n    \\caption{$\\mathcal{L}=700[\\text{fb}^{-1}]$}\n\\end{subfigure}\n\n\n\n\\caption{Significance projections for different values of the luminosity.}\n\\label{projections}\n\\end{figure}\n\nA promising scenario is provided by the future HL-LHC. At this facility, the expected luminosity (${\\cal{L}}=300 [\\text{fb}^{-1}]$) will make possible to test a significant part of the parameter space, as can be seen in Figure \\ref{projections}.\n\n\\section{Lower limits}\\label{sec:lowlims}\n\nDepending on the kinematical regime of the model, some constraints on the couplings of the HNL can be obtained from astrophysical observations. For instance, in the case where the HNL is a DM candidate, we can use the DM abundance measured by Planck \\cite{PLANCK} as input to set limits.\n\nAccording to reference \\cite{dong2021} the annihilation cross section for the left handed HNL has the following form: \n\\begin{equation}\n    \\langle \\sigma v\\rangle=\\sum_{k,k'=\\{e,\\mu,\\tau\\}}|\\beta_k^*\\beta_{k'}|^2\\frac{M_N^2}{8\\pi}\\left(1+\\frac{8T_f}{M_N}\\right)\\left(\\frac{1}{M_{V^+}^4}+\\frac{4}{(M_{V^1}^2+M_{V^2}^2)^2}\\right).\n\\end{equation}\n\nIn order to avoid overabundance, the thermally averaged cross section must satisfy the following lower bound (for instance, see references \\cite{langacker,pdg2022,Profumo}):\n\\begin{equation}\n    \\langle \\sigma v\\rangle \\geq 3\\times10^{-9}[\\text{GeV}^{-2}],\n\\end{equation}\n\nwhich provides a restriction for the parameter space as seen in Figure \\ref{dmlims}.\n\\begin{figure}[!h]\n    \\centering\n    \\includegraphics[width=0.6\\textwidth]{figures\/dmlims.png}\n    \\caption{Annihilation cross section for the HNL in the region of interest. The white region is excluded by overabundance.}\n    \\label{dmlims}\n\\end{figure}\nThe collider limits presented in this work can be combined with the DM limits, as is shown in Figure \\ref{comblims}. \nThe lower limit in the coupling parameter space depends on the value of $|\\beta_\\tau|^2$, which is not constrained by the analysis presented in this work.\n\\begin{figure}[!h]\n    \\centering\n    \\includegraphics[width=0.6\\textwidth]{figures\/comblims.png}\n    \\caption{Limits for the ($|\\beta_e|^2$,$|\\beta_\\mu|^2$)plane, for $M_{V^+}=250$[GeV]}\n    \\label{comblims}\n\\end{figure}\n\n\\newpage\n\\section{Conclusions}\\label{sec:conc}\nIn this work we have studied a simplified extension of the SM based on a Massive Vector Doublet and a Left-Handed HNL. The results have shown that the model produces a characteristic signal composed by a SFOS lepton pair, 2 jets and missing energy. The angular distribution of the lepton pair is the more distinctive observable. We have performed an analysis to set limits on the parameter space using experimental cuts applied by current searches of new physics, finding that the surviving parameter space can't produce a significant signal in the upcoming years. In contrast, we have proposed a new set of cuts which improve the search of this kind of heavy neutral leptons. With the aforementioned cuts, we expect that this region of the parameter space can be probed in the HL-LHC.\n\\newline\nOn the other hand, we have constrained the parameter space, based on the dark matter relic abundance. The lower bound depends strongly on the exotic neutrino coupling to the three families of leptons, however, we restricted our analysis only to the coupling to electrons and muons. One possible way to constrain all flavors is to consider  Lepton Flavor Violation (LFV) processes. This type of processes can be suited in the model at 1 loop, but the calculation of these loops is not trivial, due to the non minimal gauge couplings. Therefore, a rigorous study of the radiative processes will be carried out in a future work.\n\n\n\\section*{Appendix A: Detector efficiency}\\label{app1}\nAccording to reference \\cite{lep_effs}, the detector efficiencty for electron reconstruction increases with the electron $p_T$ reaching values close to $1$ for values $p_T>50$[GeV]. The event selection removes all events with $p_T(l\\pm)<25$[GeV]. The number of events with $25<p_T(l\\pm)<50$[GeV] is very small, as can be seen in Figure \\ref{leps_pt}. Therefore, we can approximate $\\epsilon\\approx 1$.\n\n\\begin{figure}[!h]\n    \\begin{subfigure}{0.45\\textwidth}\n    \\includegraphics[width=\\textwidth]{figures\/pt1.png}\n    \\caption{Negatively charged lepton $p_T$.}\n\\end{subfigure}\n    \\begin{subfigure}{0.45\\textwidth}\n    \\includegraphics[width=\\textwidth]{figures\/pt2.png}\n    \\caption{Positively charged lepton $p_T$.}\n\\end{subfigure}\n\\caption{$p_T$ distributions for the outgoing simulated leptons}\n\\label{leps_pt}\n\\end{figure}\n\n\\section*{Aknowledgements}\nThis work was funded by ANID - Millennium Program - ICN2019\\_044. Also, we would like to thank to the DGIIP-UTFSM for founding during the development of this work.\n\n\\bibliographystyle{utphys}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nExtracellular electric potentials (EEP) recorded from the scalp by means of electroencephalography (EEG) and, invasively via local field potentials (LFP) and electrocorticography (ECoG),  reflect the average spatiotemporal interactions between neuronal sub populations and therefore, constitute a measure scale of neural constructs indicative of more overt types of behavioral phenomenon. The importance of these emergent population level field potentials has been reflected at both the research and clinical levels where EEPs have enabled interpretation of complex behavioral mechanisms such as Parkinson's disease \\cite{b1}, sleep and memory consolidation \\cite{b2}, and spatial navigation \\cite{b3}, among few. \n\nField potentials are highly non - stationary signals, exhibiting continuous fluctuations between unpredictable chaotic stages and predictable oscillatory stages \\cite{b4}. Transient ordered patterns in the oscillatory stages, also known as neuromodulations, are a direct consequence of synchronized synaptic interactions in the neuronal assemblies at the level of single neurons \\cite{b5}. These neuromodulations are immediately evident as waxing and waning oscillations in the raw as well as filtered EEP traces. The highly complex chaotic stages then correspond to spontaneous background activity which are known to be characterized by a $1\/f$ power spectrum. Walter J. Freeman further confirmed this two - component hypothesis, when he experimentally showed that neuronal networks at rest produce featureless activity with their amplitudes conforming to a Gaussian distribution; deviations from Normality were then observed during active stages in the form of stable neuromodulations \\cite{b6}. \n\nGiven the consequence of field potentials in understanding brain dynamics and their usefulness to the fields of engineering and medicine, the problem of knowledge extraction from field potentials has been widely addressed in literature \\cite{b7}. A solution was found in the `frequency content' of EEPs that appropriately characterizes the neuronal oscillations; ergo, making time - frequency analysis an integral part of brain studies. The task of spectral quantification of EEPs is notably demanding pertaining to the complex dynamics of non - stationary EEPs where it is required that an apt quantification of time variation also accounts for relevant frequency components.   \n\nSpectral analysis techniques are one of the most heavily exploited methods for feature extraction from field potentials. Although these methods seek to identify neuronal oscillations, they determine the global distribution of signal power across the frequency spectrum. Time - frequency decomposition models such as short - time Fourier Transforms (STFT), wavelets, etc build on piece-wise stationarity of signals while applying Fourier Transform techniques on short windowed segments of the signals in order to construct the signal power spectrum. However, window based methods are restricted by the infamous time - frequency trade - off which lower bounds the product of time and frequency resolutions \\cite{b8}, where an appropriate representation of time variation compromises relevant frequency variation and vice - versa. \n\nIn this paper, we present a model based local power estimation measure, we call MPP spectrogram, to capture finer time variations of neuromodulatory power using precise marked features of the oscillatory events. The markers are obtained from methods elucidated in our previous work \\cite{b9, b10, b11}. For the current study, our main objective was to relate the local power estimation achieved with our methods to the conventional power estimated by power spectrum density (PSD) methods. To achieve this, firstly, we demonstrate that power in neuromodulations, as obtained from the marked point process (MPP), are highly correlated with the PSD in the corresponding band. For this, we employ correntropy based measures for better quantification of the inter-dependency between the two measures. This is important because when applying our high resolution methodology for a given band, we do not estimate the PSD; while we still compare our methods with the vast literature that uses PSD quantification. Finally, we go on to show the ability of MPP spectrogram as a measure that goes beyond the pervasive power spectral measures where it offers not just the global power distribution but also enables access to time information at a resolution as high as the sampling frequency. We present our results as tested on two datasets in the $\\gamma$ (80 - 150 Hz) and $\\beta$ (15 - 30 Hz) frequency ranges, respectively: 1) LFPs recorded from 3 rats performing a two - alternative, instructed delay choice task across different days (sessions), 2) EEG recorded from 20 subjects across 6 channels in the visual cortex while performing a conditioned stimulus task. \n\n\\section{Methods}\n\n\\subsection{Transient Model for Neuromodulatory Events}\n\nDeriving motivation from the two - component hypothesis and building upon the concepts of conventional cortical rhythms, we define single channel bandpassed EEP traces,  $\\tilde{y}(t)$, as a linear combination of background noise, $n_{0}(t)$ and neuromodulations, $y(t)$, as shown in (\\ref{eq1}). Further, using shot - noise model representation \\cite{b12}, the neuromodulations are re-constructed as convolutions between weighted, shifted Kronecker delta functions ($\\boldsymbol{\\delta}$(t)) and corresponding impulse response of temporal filters representing `typical' neuromodulations obtained from the data, and summarized as the dictionary atoms in $D = \\{ d_{\\omega} \\}_{\\omega = 1}^{K}$ according to (\\ref{eq2}). Here, $a_{i}^{\\omega} $ and $\\tau_{i}^{\\omega} $ are the projection coefficient and time of occurrence, respectively of the $i^{th}$ neuromodulatory event described by the dictionary atom, $d_{i,\\omega}$. \n\n\\begin{eqnarray}\n\\tilde{y}(t) = n_{0}(t) + y(t) = n_{0}(t)  + \\displaystyle \\sum_{i=1}^{N} y_i(t),  \\label{eq1}\\\\\ny_i(t) = \\displaystyle \\sum_{m = -\\infty}^{\\infty} a_{i}^{\\omega} \\boldsymbol{\\delta}(t - \\tau_{i}^{\\omega} - m)d_{i,\\omega}(m). \\label{eq2} \n\\end{eqnarray}\n\nHowever, to realize such a representation, it is required that the neuromodulations are isolated from the background activity. We achieve this, in our first phase, we call `denoising' where a correntropy based similarity vector robustly separates the two components leveraging on the dissimilarities in their intrinsic distributions and the inherent predominance of background activity in EEP traces \\cite{b11, b15}. A threshold, $\\kappa$, defined as the minimum norm of the putative events is calculated which delineates the search space for neuromodulations in the subsequent phases. \n\nThe final phase follows an unsupervised learning framework paralleling a K - means clustering algorithm to learn the dictionary, $D$, representative of `typical' neuromodulations from the EEP trace. Alternations between updation of the dictionary atoms and sparse decomposition, then, constitute the model training. The dictionary atoms are updated using single value decomposition where a correntropy based cost function is optimized with the purpose of repressing outliers to avoid bias in the estimation of principal components. The latter step of sparse decomposition follows a traditional matching pursuit technique where the search space is outlined by the threshold, $\\kappa$, obtained from the denoising phase. \n\nThe model training is terminated when convergence is achieved in terms of an upper bound on the Frobenius difference norm between dictionaries estimated in successive iterations, or additionally, if a certain number of alternating optimization steps have been completed. The greedy approach of the methods demands that in order to avoid a local minima, the training be initialized with different initial seeds of the dictionary; wherefore, the final step in training includes determining the optimal dictionary with the maximum value of mutual coherence. Altogether, the model is able to learn representations of neuromodulations by adapting a data - driven approach while only depending on two hyperparameters: 1) maximum duration of neuromodulation, $M$ and 2) maximum number of dictionary atoms, $K$. A flowchart of the involved methods in the training phase is included in Fig. \\ref{Model Structure} and for a more elaborate explanation of the model, we direct the reader to \\cite{b9}. \n\n\n\\begin{figure}[!h]\n\\centering\n\\includegraphics[scale = 0.56]{Model_Power4.jpg}\n\\caption{Generative model. Training (top): After bandpass filtering the input EEP trace, the denoising phase exploits a correntropy - based similarity measure to calculate the threshold, $\\kappa$. Following this, the learning framework estimates the bases vectors, D, via alternations between dictionary update and event decomposition phases. Testing (bottom): Input EEP trace is bandpassed and denoised to remove background noise. A final decomposition using the dictionary summarizes the features of each neuromodulation as a marked point process.}\n\\label{Model Structure}\n\\end{figure}\n\n\\subsection{Neuromodulatory Features}\nTo this end, we have designed a dictionary prototypical of neuromodulations from a single channel EEP trace. Model testing then involves a simple convolution with the signal to obtain the markers of each neuromodulation, in terms of its time of occurence ($\\tau_{k}$), maximum amplitude ($\\alpha_{k}$) and duration ($\\delta_{k}$) constructing the MPP, via the intensity function of the timings and the joint probability distribution of the neuromodulatory features, duration and amplitude (Fig.\\ref{Model Structure}, bottom panel). Given complete access to each neuromodulatory event at time points, $\\tau_{k}: k = 1,2, ... N$, we extend our feature space by defining local power in each neuromodulation, $P_\\delta (\\tau_{k})$ according to (\\ref{eq3}).  A major advantage of $P_\\delta (\\tau_{k})$ is its ability to clearly isolate neuromodulatory power from power due to background noise. This property is emphasized in Fig. \\ref{Spectrogram} where power spectrum obscures the boundary between background activity and neuromodulations due to their dependence on fixed windows, while $P_\\delta (\\tau_{k})$ is able to maintain distinct boundaries between the two EEP components.   \n\n\\begin{eqnarray}\nP_\\delta(\\tau_k) =\\frac{1}{\\delta_{k}} \\displaystyle \\sum_{n = \\frac{-\\delta_k}{2}}^{\\frac{\\delta_k}{2}-1} \\tilde{y}^{2}(\\tau_{k} + n).  \\label{eq3}\n\\end{eqnarray}\n\n\\begin{eqnarray}\n\\lambda_\\alpha(t) = \\displaystyle \\sum_{n = -\\infty}^{\\infty}     \\left[ \\displaystyle \\sum_{k = 1}^{N}  P_{\\delta}(\\tau_k) \\boldsymbol{\\delta}(n - \\tau_k) \\right] \\kappa_\\sigma(t - n)  \\label{eq4}\n\\end{eqnarray}\n\n\\begin{eqnarray}\n\\rho_\\delta = \\frac{\\displaystyle \\sum_{k = 1}^{N}\\delta_{k}}{L}   \\label{eq5}\n\\end{eqnarray}\n\nFurther, we define MPP spectrogram ($\\lambda_\\alpha$) as the power weighted intensity function to capture the local variations in power due to $N$ detected neuromodulations from the EEP trace according to (\\ref{eq4})  where $\\kappa_\\sigma$ is a Guassian kernel of bandwidth $\\sigma$ and $\\boldsymbol{\\delta}(t)$ is the Kronecker delta function. It is important to note that, unlike power spectrum, $\\lambda_\\alpha$ is able to retain complete time information in the frequency range of interest while consistently limiting leakage from power due to background noise as it builds only on $P_\\delta (\\tau_{k})$.  \n\nIn addition to power measures, we also define a density measure, we call phasic event density ($\\rho_\\delta$), as the relative proportion of samples in the EEP trace of length $L$ that corresponds to neuromodulations. Phasic event density can be thought of as a measure that draws a parallel with the $l_0$ pseudonorm applied to the concept of sparse decomposition.  \n\n\\subsection{Correntropy Coefficient}\n\nThroughout the paper, we inspect inter-dependencies between random variables via a correntropy measure, termed as correntropy coefficient, $\\eta$ \\cite{b15}. Specifically designed as a generalized correlation coefficient, the estimate builds on cross - correntropy as shown in (\\ref{eq7}) for non - degenerate random variables $X  =\\{ {x_i}\\}_1^N$  and $Y = \\{{y_i}\\}_1^N$ where $\\kappa_\\sigma$ is a Gaussian kernel of bandwidth $\\sigma$. \nThe ``centered\" cross - correntropy (\\ref{eq6}) estimate is analogous to the conventional cross - covariance measure of the correlation coefficient where it is defined as the difference between the joint and the product of marginal expectations of $\\kappa(X,Y)$; the centering is pertinent to an explicit extraction of the mean value. Normalizing the centered cross correntropy with respect to their (centered) autocorrentropies, then, completely defines the correntropy coefficient (\\ref{eq7}).     \n\n\n\\begin{eqnarray}\n\\hat{u}_\\sigma(X,Y) = \\frac{1}{N}\\displaystyle \\sum_{i = 1}^{N} \\kappa_\\sigma(x_i - y_i)  - \\frac{1}{N^2}\\displaystyle \\sum_{i = 1} ^{N} \\sum_{j = 1}^{N} \\kappa_\\sigma(x_i - y_j)  \\label{eq6}\n\\end{eqnarray}\n\n\\begin{eqnarray}\n\\eta = \\frac{\\hat{u}_\\sigma(X,Y)}{\\sqrt{\\hat{u}_\\sigma(X,X) \\hat{u}_\\sigma(Y,Y)}}  \\label{eq7}\n\\end{eqnarray}\n\nThe rationale for implementing a correntropy - based measure, as opposed to correlation, lies in its ability to reject outliers by controlling the kernel bandwidth, $\\sigma$ \\cite{b15}. In this scenario, outliers are associated with local artifacts in the data or the poor estimation of events by the MPP algorithm, that are although rare, distort the correlation fits too much. Finally, relation between the random variables under study was also further scrutinized via best fit lines obtained from a maximum correntropy criterion (MCC) based regression to conclude the analysis of any inter-dependency \\cite{b15}. \n\n\\section{Experimental Setting}\nWe tested our power measures on both LFP and EEG data to validate our two premises: 1) neuromodulations, as detected by the model, maximally contribute to signal power and 2) MPP spectrogram is a finer measure of power spectrum which also reflects the global distribution of signal power. This section presents the details of the two datasets analyzed.   \n\n\\subsection{Dataset-1}\n\nLocal field potentials were recorded from the dorsal prelimbic cortex (dPrL) across 3 rats performing a two-alternative, instructed delay forced choice task as per studies reported in \\cite{b13}.  A microwire array was implanted in layer V of the prefrontal cortex in the right hemisphere such that electrodes 16 electrodes rested in the dPrL. The entire data was downsampled to 500 Hz for analysis. Further, all procedures involving animals were reviewed and approved by the Institutional Animal Care and Use Committee (IACUC).\n\nThe experiment was setup in an acoustically isolated operant conditioning chamber comprising of three nose poke holes: a fixation hole in the center was flanked by target holes on either side. Trials were self - initiated by the subject by positioning it's snout in the fixation hole. Before being presented with an instruction cue, the subject was required to place its nose in the fixation hole for 1s. The instruction cue was a variable pitch single tone delivered at 60 dB whose pitch determined the target hole; a low pitch tone corresponded to the right target hole, while a high pitch tone cued the left target hole. The instruction cue initiated a delay period, the length of which was chosen pseudo - randomly from a uniform distribution, $\\textbf{U}(1.5, 2)$. A Go cue, consisting of a white noise auditory stimulus delivered at 60 dB, then indicated the end of the delay period following which the subject was required to place its nose in the appropriate target hole. Visits to the instructed targets were rewarded immediately by delivering a 45mg food pellet (Bio-Serv, NJ), while incorrect visits were unrewarded. Fig.\\ref{Trial Structure}\\textbf{A} presents the trial structure for the experiment. A total of 8 sessions, that is, 1 session by subject 1, 3 sessions by subject 2 and 4 sessions by subject 3 were analyzed. \n\n\n\\subsection{Dataset-2}\n\nEEG data was recorded from 20 subjects while performing a task where an unconditioned stimulus was randomly interleaved with a conditioned stimulus in each trial. A total of 129 channels were recorded, of which 6 channels in the visual cortex were chosen for the current analysis (shaded channels, Fig. \\ref{Trial Structure}\\textbf{B}). Before trial initiation, the subject was required to look at a fixation cross that was presented at the center of the screen and lasted $\\sim3.6$ s. Each trial was initiated by presenting the conditioned stimuli (CS), a Gabor patch with a 1.5 degree left tilt and a Michelson contrast of 0.63, at the center of the screen. The CS was displayed for a total of 2.5s throughout the trial. The unconditioned stimulus (UCS), randomly paired with the CS in $\\sim50\\%$ of the trials, was a 96 dB sound delivered via 2 computer speakers positioned behind the participant. The UCS lasted for 1s and was set about 1.5s after trial initiation.\n\nEpochs analyzed in the current study were 5.1s in duration which included a 3.6s long display of a fixation cross in the center of the screen prior to trial initiation and 1.5s after CS onset as shown in Fig. \\ref{Trial Structure}\\textbf{B}. All data was sampled at 500 Hz and averaged to a single reference. The study was approved by the local Institutional Review Board. \n\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[scale = 0.12]{Trial}\n\\caption{Time course of the tasks. \\textbf{A} LFPs recorded from rat prefrontal cortex. Gradients arrows represent variable time periods. The cues last for 0.5s each. \\textbf{B} EEG data recorded from the human visual cortex. The epochs analyzed included the period [-3.6, 1.5]s. }\n\\label{Trial Structure}\n\\end{figure}\n\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[scale = 0.23]{Spectrogram}\n\\caption{Power represented via $P_\\delta$ compared to the power spectrum. In each sub-figure, top panel presents the spectrogram of the recorded field potenials, middle panel corresponds to power in neuromodulations evaluated via $P_\\delta$ in the frequency band under study and the bottom panel plots the raw field potentials throughout the trial. \\textbf{A} Dataset 1: $\\gamma$ (80 - 150 Hz); Session 1, channel 2, Trial 12. \\textbf{B} Dataset 2: $\\beta$ (15 - 30 Hz); Subject 1, channel 2, Trial 24.}\n\\label{Spectrogram}\n\\end{figure}\n\n\\section{Results}\nUsing the generative model for neuromodulatory events, we learnt the markers of $\\gamma$ neuromodulations from the LFPs of the rat subjects and of $\\beta$ neuromodulations from EEG data recorded from human subjects. The bandpassed traces in each frequency band were obtained using FIR filters of quality factor, Q $\\sim1.5$ (center frequencies - $\\gamma$: 115 Hz, $\\beta$: 22.5 Hz; order - $\\gamma$: 43; $\\beta$: 54). After thorough visual analysis of the EEP traces and taking into account the number of cycles of oscillations, maximum duration of $\\gamma$ and $\\beta$ neuromodulations, $M$, was set at 100 ms ($\\sim$10 cycles)  and 300 ms ($\\sim$7 cycles), respectively. Further, following similar analysis as in \\cite{b9}, we upper bounded the number of dictionary atoms, $K$, to $30$ for LFPs and $60$ for the EEG data. Correntropy measures implemented throughout the model used the kernel bandwidths as determined by the Silverman's rule \\cite{b14} for an appropriate projection into the joint space of the random variables involved. MPP spectrogram measure was obtained from the MPP representation using Gaussian kernels of bandwidth 80 and 100 ms for $\\gamma$ and $\\beta$ frequency bands, respectively. Finally, PSD measures throughout the analysis were calculated using STFT-Welch method with Gaussian window lengths set equal to 0.4 and 0.7s for LFPs ($\\gamma$ frequencies) and EEG data ($\\beta$ frequencies), respectively with 50$\\%$ overlap between windows.  \n\nWe began our analysis via simply juxtaposing spectrograms of the EEP traces (STFT, overlap - 70 $\\%$,  window lengths - $\\gamma$: 0.2 s, $\\beta$: 0.3 s) with their corresponding detections of neuromodulations represented as their power, $P_\\delta(\\tau_{k})$. Exemplary plots are shown in Fig.\\ref{Spectrogram} where it is evident that neuromodulations are detected by the MPP model in that portion of the signal where the PSD estimate (in the specified frequency band) is large; importantly, our method is able to precisely pin point the time of the event that is contributing to this increase in the PSD. Such an apt representation of neuromodulations by $P_\\delta(\\tau_{k})$  further motivated our analysis. Again, it is important to note the distinct boundaries evident between the neuromodulations and background activity as captured by $P_\\delta(\\tau_{k})$, which are absent in the spectrogram of the signal.  \n\nIn order for neuromodulations to be representative of signal power, it would be required that the two measures share a positive correlation. This premise was tested for recordings from both data sets by evaluating the correntropy coefficients between the normalized PSD (nPSD), as in (\\ref{eq8}) and total power in neuromodulations, $\\Sigma P_\\delta$,  in the specified frequency band for each trial. Such normalization of the PSD was required in order to account for the difference in the units of the two measures: that is, while PSD is calculated over a window of fixed length,  $P_\\delta$ calculates power only over the neuromodulatory event. Consequently, nPSD was normalized to represent power as captured by spectral analysis in the same duration as $P_\\delta$ with the help of phasic event density, $\\rho_\\delta$, as in (\\ref{eq8}) where $N_w$ is the number of windows used to calculate PSD.  \n\n\\begin{eqnarray}\nnPSD = PSD*N_{w}*\\rho_\\delta. \\label{eq8}\n\\end{eqnarray}\n\nResults from across 3 sessions of LFP recordings corresponding to each rat and across 4 subjects' EEG data are presented in Fig.\\ref{PINvsnPSD}\\textbf{A} and \\textbf{B}, respectively for a single channel. Moreover, best fit lines determined using   MCC further emphasize the interrelation between the two measures. It is worth noting that, in the plots, most of the detected neuromodulations exist along a line in the 2D space, but there are a few detections that are outliers. These outliers will bias the regression line and provide bogus correlation values. By contrast, correntopy is capable of aligning the regression line with the inherent trend, corresponding to a better estimation of the correlation between the two methods than linear regression does. Further, although the slope of the best fit line is very close to 1, in most cases, the slope was found to be slightly lower than that of the identity line. We attribute this to two plausible scenarios: 1) higher nPSD corresponding to power in background activity or 2) missed detections of neuromodulatory events. Scenario-1 reveals scope for analyses of brain processes that contribute to power in background activity; such analyses would merit future work.  \n\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[scale = 0.125]{PINvsnPSD}\n\\caption{Total power in neuromodulations plotted against nPSD in the corresponding frequency range for each trial. Each black marker corresponds to a trial and $\\eta$ is the correntropy coefficient. \\textbf{A} Dataset 1: $\\gamma$ (80 - 150 Hz); (L - R) Subjects 1 - 3; Sessions - 1, 3, 2, respectively;  Channel 7,  \\textbf{B} Dataset 2: $\\beta$ (15 - 30 Hz); (L - R) Subjects 1, 4, 7 and 17; Channel 73.}\n\\label{PINvsnPSD}\n\\end{figure}\n\nAdditionally, these scatter plots not just serve as a validation for our methods, but also as a device to detect noisy recordings; for any large deviations from the best fit line produced by an appropriately tuned generative model would most likely correspond to artifact noise or bad channels in the recordings. This was observed in an LFP recording from dataset-1 where noise in the signal contributed to a more dispersed scatter plot (Fig. \\ref{badCorr}). \n\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[scale = 0.125]{BadCorr}\n\\caption{Noisy signal detection. In the plot on the \\textit{right}, $\\Sigma P_\\delta$ plotted against nPSD obtained from $\\gamma$ neuromodulations for all trials in LFP recordings of a single channel from dataset-1 show large deviations from best fit line. Five outliers are mapped to their corresponding trial recordings on the \\textit{left} plot depict artifact noise. Subject 2; Session 2; Channel 2.}\n\\label{badCorr}\n\\end{figure}\n\n\nMean correntropy coefficient ($\\eta$) values across all channels for each LFP session and for each subject's EEG are summarized in Fig. \\ref{All_corrs}, respectively, along with their standard deviations and trial details. High dependencies between power in neuromodulations and PSD confirms a high correlation between the measures. Moreover, normalization of PSD with phasic event density shows that neuromodulations maximally contribute to signal power. These results ratify our argument by bringing out a two - fold conclusion: higher number of neuromodulations not just imply a higher signal power but also that the signal power is most dominant in these neuromodulations. Finally, for the analysis of $\\eta$, only 17 out of the total 20 subjects from dataset-2 were considered, as 3 subjects performed less than 30 trials and therefore, their results would not be representative of the inter - dependency between the variables under study.    \n\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[scale = 0.11]{All_corrs}\n\\caption{Summary of analysis on inter-dependency between neuromodulations and signal power. Primary histograms summarize the mean correntropy coefficients between $\\Sigma P_\\delta$ and nPSD calculated across all channels for each session\/subject analyzed. (Top inset) Histogram of standard deviations of coefficients. (Bottom inset) Histogram of number of trials performed in each session\/by each subject. \\textbf{A} Dataset 1: $\\gamma$ (80 - 150 Hz); \\textbf{B} Dataset 2: $\\beta$ (15 - 30 Hz). }\n\\label{All_corrs}\n\\end{figure}\n\n\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[scale = 0.21]{GlobalVSLocal}\n\\caption{Comparing power variations over time as captured by PSD and MPP spectrogram. In each subfigure, top panel correspond to the trial - averaged PSD and the bottom panels are trial - averaged $\\lambda_\\alpha$. Values of PSD and $\\lambda_\\alpha$ were normalized for visual purposes. \\textbf{A} Dataset 1: $\\gamma$ (80 - 150 Hz); (L - R) Subjects 1 - 3; Sessions - 1, 3, 2, respectively; \\textbf{B} Dataset 2: $\\beta$ (15 - 30 Hz); (L - R) Subjects 1, 4, 7 and 17}\n\\label{AM power}\n\\end{figure}\n\nHaving validated that neuromodulation power are a proxy for signal power as estimated via PSD in that frequency band, we were interested in assessing the ability of MPP spectrogram, which builds on power in these neuromodulations, to capture local variations in signal power. Such a representation would allow for greater access to time information at resolutions unachievable by other power spectral measures. For this, we evaluated the power spectrum of the signal using STFT and averaged the obtained spectrogram across the frequency band under study. These plots were then compared with those obtained using MPP spectrogram. Trial - averaged power spectrum plots juxtaposed with trial - averaged MPP spectrogram plots are presented in Fig. \\ref{AM power}\\textbf{A, B} corresponding to $\\gamma$ and $\\beta$ neuromodulations, respectively. Two observations are immediate from the figures that highlight the advantages of MPP spectrogram: 1) the general similarity in the variations of power over the trial period between the power spectrum and MPP spectrogram reflects the capability of MPP spectrogram to maintain the global power distribution of the signal, and 2) the finer details of variations that are only captured in the plots of MPP spectrogram. Therefore, MPP spectrogram is a high time - resolution power measure founded on concepts of neuromodulations that defines a finer, and more accurate representation of local as well as global signal power. Finally, it is worth noting that our methods do not lose any frequency resolution and are in fact a cumulative representation of \\textit{all} frequencies in the specified bandwidth, i.e., there is no `sampling' of frequencies, unlike PSD methods. \n\n\\section{Conclusion}\n\nThrough this paper, we introduce MPP spectrogram as a high - time resolution power measure obtained as a by - product of the neuromodulatory event detection model. The model's advantage lies in its ability to represent an EEP trace as a marked point process (MPP) by characterizing each neuromodulation in terms of its features - amplitude, duration and time of occurrence. We exploit this explicit access to neuromodulatory properties to expand the feature space to include power in neuromodulation ($P_\\delta(\\tau_k)$) as a clear - cut marker that distinguishes the power in neuromodulations from background power. Leveraging on the properties of $P_\\delta(\\tau_k)$, we construct MPP spectrogram to capture local variations in neuromodulatory power over time. Further, in order to validate the aptness of MPP spectrogram, we demonstrate firstly the dominance of neuromodulatory power in signal power using correntropy based techniques. We test our hypotheses on LFPs recorded from a rat prefrontal cortex and human EEG data recorded from the visual cortex. We show explicit variations in neuromodulatory power captured via MPP spectrogram as opposed to PSD, which, although closely resembles the MPP feature modulations, lacks the high time - resolution afforded by it. \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\label{sec:intro}\n\nAssembling workflow applications by putting together standalone binaries has become a popular approach to build complex scientific applications (e.g., modFTDock \\cite{Modftdock:12}, Montage\\cite{Montage:03} or BLAST \\cite{blast}). The processes spawned from these binaries communicate via temporary files stored on a shared storage system. In this setup, the workflow runtime engines are basically schedulers that build and manage a task-dependency graph based on the tasks' input\/output files (e.g., SWIFT \\cite{Swift:11}, Pegasus \\cite{Pegasus:05}).\n\nTo avoid accessing the platform's backend storage system (e.g., NFS or GPFS), the shared storage system can be co-deployed on the nodes allocated to the application. Aggregating node-local resources to provide such a shared \\textit{intermediate storage system} \\cite{versatile,Wozniak:09} offers a number of advantages: higher performance - as applications benefit from a wider I\/O channel obtained by striping data across several nodes; higher efficiency \u2013 as it improves resource utilization; incremental scalability \u2013 as it is possible to increase system capacity in small increments. This scenario also opens the opportunity for optimizing the intermediate storage system for the target workflow application: a storage system used by a single workflow, and co-deployed on the application nodes, can be configured specifically for the I\/O patterns generated by the workflow (e.g., configure striping and replication to eliminate hot spots, use a data placement policy to maximize data access locality) \\cite{Mosa-ccgrid:12}. \n\nThese benefits, however, come at a price: configuring the intermediate storage system becomes increasingly complex for multiple reasons. First, the optimization techniques commonly used in distributed environments expose trade-offs that rarely exist in centralized solutions \\cite{thereska:informed:06,strunk-utility}. Second, each workflow application obtains peak performance at a different configuration point, a consequence of different I\/O patterns \\cite{ursaminor}. Third, depending on the context, there are multiple metrics of interest to optimize: e.g., response time, throughput, energy, and, increasingly common in cloud computing environments, the cost of resources. \n\nTo illustrate these points, consider the following optimization techniques and their trade-offs. Data striping may accelerate data access, yet it decreases reliability and requires more resources. Increasing the number of storage nodes can avoid access bottlenecks, yet it may expose scalability bottlenecks. Higher redundancy levels (through replication or erasure codes) may accelerate data access and increase reliability, yet they also require more resources and complex consistency protocols. Finally, different data placement and\/or caching policies are beneficial to different workloads.\n\nPast work and our own experience support these assertions. They show that different applications achieve their best performance when using different storage configurations \\cite{versatile,ursaminor}. Also, different applications even benefit from different combinations of storage optimization techniques \\cite{Mosa-ccgrid:12,erss2011}, and that the choice of the optimal configuration point is not intuitive (Figure \\ref{fig:montage}). \n\nIn this scenario, the role of the application administrator\/user becomes non-trivial: in addition to being in charge with running the workflow application, she also becomes responsible for choosing the appropriate intermediate storage system configuration to maximize performance. Providing support for this activity is the focus of our project \\cite{grid2010}.\n\n\n\n\n\\begin{figure}[htb]\n\\centering\n\\includegraphics[width=0.5\\linewidth]{montage-large} \n\\caption{\\textit{Different storage system configurations deliver different performance and the choice of the optimal configuration point is not intuitive.} To demonstrate that this holds for workflow applications, we have executed the Montage workflow \\cite{Montage:03} on top of MosaStore \\cite{versatile} storage system with different storage configurations deployed over the Grid'5000 platform. We executed the same workload for several values of one configuration knob: \\textit{stripe width}. As we increase stripe width, we also increase the number of storage nodes to support it. For a low stripe width, performance is low due to congestion at the storage nodes and, for higher values, performance decreases due to connection handling and metadata access overheads. A user seeking the optimal point would pick a stripe width value close to five, a non-obvious choice. Other case studies can be found in \\cite{Mosa-ccgrid:12}. }\n\n\\label{fig:montage}\n\\end{figure}\n\\textbf{The Problem.} Configuring the intermediate-storage system to achieve high performance (e.g., in terms of application turnaround time, storage footprint, energy consumption, or financial cost), involves choosing a set of storage system configuration parameters (such as stripe width, data placement policy, and replication level). \n\n\n\\textit{Manually} fine-tuning the storage system configuration parameters is undesirable for multiple reasons. First, the user lacks a deep understanding of how different configuration choices affect the system's performance. Second, the workload may vary: new application versions may make one-time tuning obsolete. Finally, performance tuning is time-consuming due to the large configuration space to consider.\n\n\\textit{This paper presents our progress to date on designing and evaluating a performance prediction mechanism for object-based storage systems}. Given a storage system configuration, an application I\/O profile, and a characterization of the deployment platform based on a simple system identification process (e.g., storage nodes service time, network characteristics), the mechanism predicts the total application turnaround. \n\nThis approach can support four important tasks: First, \\textit{autotuning}, a software tool that relies on the proposed mechanism can enable efficiently configuring the storage system \\cite{grid2010,thereska:informed:06,strunk-utility}, through exploring the configuration space without actually running the application. Second, \\textit{resource provisioning}, it can inform the decision of how many resources to allocate to the application to achieve a certain performance level. Third, \\textit{new technology evaluation}, it can estimate the application performance on hardware that has not yet been procured (e.g., to help answer the question: \\textit{what} would be the performance improvement \\textit{if} we used SSDs?). Finally, the mechanism can help explore the impact of configuration choices in situations where direct measurement is difficult or requires specialized infrastructure (e.g., energy consumption). \n\nThis paper focuses on predicting the application turnaround time, but we note that the model and approach presented apply readily to other optimization metrics.\n\n\\textbf{The contributions of this paper} lay over multiple axes: \n\n\\begin{itemize}\n\n\\item Synthesizes the key requirements for a prediction mechanism that will be useful in practice (\\S\\ref{sec:requirements}).\n\n\\item Proposes a simple queue-based model for distributed, object-based storage systems (\\S\\ref{sec:model}). This model, at the base of the prediction mechanism, is: generic (i.e., covers the traditional design for object-based datastores), and homogeneous (all nodes are based on the same model). \n\n\\item Proposes a system identification procedure to seed the model that is simple, lightweight, effective, and does not require system changes to collect monitoring information (\\S\\ref{sec:seeding}).\n\n\\item Presents preliminary experience with using this prediction mechanism (\\S\\ref{sec:evaluation}) in differentiating between system configurations and identifying the best configuration for a certain workload. Also, the paper discusses our experience (\\S\\ref{sec:discussion}) to date with using the prediction mechanism beyond our original design goal: we have used the performance prediction to better understand and debug a distributed storage system that our group develops.\n\\end{itemize}\n\\section{The Design of a Performance Estimation Mechanism}\n\n\\label{sec:solution}\n\nBuilding an accurate and scalable distributed system performance predictor is challenging, which resulted in some simulation-based solutions instead of pure analytical models. At the one end of the design spectrum current practice (e.g., NS2 simulator \\cite{ns2}) suggests that while simulating a system at low granularity (e.g., packet level simulation in NS2) can provide high accuracy, the complexity of the simulation model and the number of events generated make simulating large-scale systems infeasible. At the other end, coarse grained simulations (e.g., \\cite{SimGrid,PeerSim}) can scale often at the cost of lower accuracy\n\nSimilarly, the proposed mechanism relies on a simulation-based approach. We target a scalable and adequately accurate performance prediction. While we simulate the storage system at chunk granularity to increase the accuracy, we exploit the following two observations to reduce simulation complexity and increase its scalability: First, as the goal of the simulation is to support configuration choice for a specific workload, achieving perfect accuracy is less critical (\\S\\ref{sec:evaluation}). Second, the focus is on the workload characteristics of workflow applications: relatively large files, single-write many reads and specific data access patterns. These observations enable us to reduce the simulation complexity by not simulating in detail some of the control paths that do not significantly impact accuracy (e.g., the chunk transfer time is dominated by the time to send the data, consequently not accounting the time of the acknowledgment messages or the metadata message transfer will not tangibly impact accuracy).\n\n These design decisions allowed us to simulate storage operations of runs of an entire application on a cluster in an order of magnitude faster than the actual run (\\S\\ref{sec:evaluation}).\n\nThe proposed mechanism uses a queue-based storage system model for the system components' operations and their interactions. The model requires three inputs from the user: the storage system configuration, a workload description, and the performance characteristics of storage system components (i.e., system identification). The simulator instantiates the storage system model with the specific component characteristics and configuration, and simulates the application run as described by the workload description.\n\nThis section discusses the requirements for a practical performance prediction mechanism (\\S\\ref{sec:requirements}) and presents the key aspects of the object-based storage system architecture modeled (\\S\\ref{sec:mosa}). Then, it focuses on the proposed solution: it presents the model (\\S\\ref{sec:model}), its implementation (\\S\\ref{sec:implementation}), the system identification process to seed the model (\\S\\ref{sec:seeding}), and an overview of the workload description (\\S\\ref{sec:workload}). \n\\subsection{Solution Requirements}\n\n\\label{sec:requirements}\n\nA practical performance prediction mechanism should meet the following, partially conflicting, requirements:\n\n\\begin{itemize}\n\n\\item \\textbf{Accuracy.} The mechanism should provide adequate accuracy. Of course better accuracy is desirable; however, in the face of practical limitations to achieve perfect accuracy, we note that there are decreasing incremental gains for improved accuracy in practical settings. For example, to support decisions about configuration choices a predictor only needs to correctly estimate their relative performance or trends of changing a configuration parameter. Even more, if two configurations offer near performance, their relative predicted performance is less important as long as the prediction mechanism places their performance as similar.\n\n\\item \\textbf{Scalability and Response Time.} The predictor should enable the quick exploration of the configuration space. To this end, the mechanism should offer performance predictions quickly and scale across at least two dimensions: (i) it should scale with the system size and be able to model large systems; and (ii) it should scale with the I\/O intensity and be able to model I\/O intensive applications. \n\n\\item \\textbf{Usability and Generality.} The predictor should not impose a burdensome effort to be used. Specifically, the bootstrapping\/seeding process should be simple and it should not require storage system redesign (or a particular initial design) to collect performance measurements. Additionally, ideally the prediction mechanism should model a generic object-based distributed storage design and using it should not require in-depth knowledge of storage system protocols and architecture. \n\n\\item \\textbf{Ability to explore \\textit{``what-if''} scenarios.} A prediction mechanism should be able to support exploring hypothetical scenarios, such as scenarios that assume new\/different hardware configurations (e.g., usage of SSDs). We note that there are two main categories for the models at the foundation of all performance prediction mechanisms: explanatory and agnostic models. The explanatory models try to mimic the key components of the system and their interaction at various levels of accuracy and granularity. The agnostic model aims to predict the output metric of interest being completely oblivious of system internals (e.g., a neural network based models would fit in this category). Supporting \\textit{``what-if''} scenarios exploration requires an approach based on an explanatory model.\n\n\\end{itemize}\n\\subsection{Object-based Storage System Design}\n\n\\label{sec:mosa}\n\nWe focus on a widely-adopted object-based storage system architecture (such as that adopted by GoogleFS \u200e\\cite{ghemawat2003google}, PVFS \\cite{pvfs}, MosaStore \\cite{versatile}, and UrsaMinor \\cite{ursaminor}). This architecture includes three main components: a centralized metadata manager, storage nodes, and a client-side system access interface (SAI). The manager maintains the stored files' metadata and system state. To speed up data storage and retrieval, the architecture employs striping \u200e\\cite{zebra95}: files are split into chunks stored across several storage nodes. Client SAIs implement data access protocols after they interact with the manager that stores data placement information. \n\n\\textit{Data placement}. The default data placement generally adopted is round-robin: when a new file is created on a stripe of \\textit{n} nodes the file's chunks are placed in a round-robin fashion across these nodes. Additionally, data placement policies that optimize for a specific application access patterns have seen higher adoption \\cite{Mosa-ccgrid:12,pnfs,Zhang:2012:DAD:2388996.2389112}. For instance, the following data placement policies are used to optimize for the workflow applications' data access patterns: local, co-locate and broadcast (detailed in \\S\\ref{sec:evaluation}).\n\n\\textit{Replication}. Replication is often used to increase reliability or to improve access performance. Data is replicated when new data is stored in the system, consequently, while a higher replication level will reduce contention on the node storing a popular file, it will increase the file write time and the storage space consumption.\n\nWe explore the accuracy of the prediction mechanism assuming that the stripe width, replication level, and data placement policy are configurable as suggested in \\cite{Mosa-ccgrid:12,ursaminor,versatile}. Our approach can be extended to support other configuration parameters.\n\\subsection{System Model}\n\n\\label{sec:model}\n\nThe process of developing the prediction mechanism involves several trade-offs between simplicity and accuracy. The solution space is bounded by the requirements listed in \\ref{sec:requirements}: we avoid changes in the storage system itself, and make the model and the system identification as simple as possible. \n\nWe use a queue-based model. All machines are modeled similarly, regardless of their specific role in the system (Figure \\ref{fig:model}): each machine hosts a network component and can host one or more system components (each modeled as a service with its own queue).\n\nA system service and its queue represent a specific functionality in the system. That is, the \\textit{manager} component is responsible for storing files' and storage nodes' metadata. The \\textit{storage} component is responsible for storing and replicating data chunks. Finally, the \\textit{client} component receives the read and write operations from the application, implements, at the high-level the storage system protocol by sending control or data requests to other services, and once a storage operation is terminated it communicates again with the application. Each of these components is modeled as service that takes requests from its queue (fed by the network service or by the application for the client service) and sends responses back through the network service (or directly to the application, again for the client). \n\nThe network service and its in- and out- queues model the network-related activity of a host. Key here is to model network-related contention while avoiding modeling the details of the transport protocol (e.g., dealing with packet loss, connection establishment and teardown details). The requests in the out-queue of a network component are broken in smaller pieces that represent network frames and sent to the in-queue of the destination host. Once the network service processes all the frames of a given request in the in-queue, it assembles the request and places it in the queue of the destination service.\n\n\n\n\\begin{figure}[htb]\n\\centering\n\\includegraphics[width=0.75\\linewidth]{storage-model} \n\\caption{Queue-based model of a distributed storage system. Each component (manager, client component, and storage component) has a single system service that processes requests from its queue. Additionally, each host has a network service with an in- and out- queue. The network core connects and routes the messages between the different components in the system and can model network latency and contention at the aggregate network fabric level. Solid lines show the flow going out from a storage system component while dashed lines show the in-flow path.}\n\\label{fig:model}\n\\end{figure}\n\nThe system services can be collocated on the same host (e.g., the client and storage services running on the same host). In this situation, requests between collocated services also go through the network, but have a faster service time than remote requests - representing a loopback data transfer (\\S\\ref{sec:seeding}).\n\nSpace limitations prevent us from presenting the full details of the model. As a rule, we uniformly model the data path at chunk-level granularity, and the control paths at a coarser granularity: modeling only one control message to initiate a specific storage function while an implementation may have multiple rounds of control message exchanges. We used our own experience with designing and implementing an object-based storage system \\cite{mosastore-website}.\n\n\\subsection{Model Implementation}\n\n\\label{sec:implementation}\n\nWe have implemented the above model as a discrete-event simulator in Java. The simulator receives as inputs: a summarized description of the application workload (described in \\S\\ref{sec:workload}) and a description of the deployed system which has two parts. The first part describes the system-wide configuration parameters (currently, replication level, stripe-width, chunk size, and data-placement system-wide) and the details of the system: number of hosts, number of storage nodes and clients, whether storage and clients nodes are collocated on the same hosts. The second part characterizes the performance of system services: service times for network, client, storage, and manager services (the process of identifying these values is described in \\S\\ref{sec:seeding}). \n\nOnce the simulator instantiates the storage system, it starts the application driver that processes the application workload. The driver reads the description of the application workload, creates the corresponding events (e.g., read from file $x$ at offset $y$, $z$ bytes) and places them in the client service queue. File-specific configuration (as proposed by \\cite{ursaminor,Mosa-ccgrid:12}) is described as part of the application workload description since it depends on the file that application reads\/writes.\n\nAs in a real system, the manager component maintains the metadata of the system (i.e., implements data placement policies by returning free chunks when requested by write operations, and keeps track of file to chunk mapping and chunk placement). To make the process clearer, consider the following example for a write operation where a client module processes a file write event. First, the client contacts the manager asking for free space. The manager replies specifying a set of free chunks and their storage services to be used during this write. Then, the client requests each storage service to store chunks in a round-robin fashion. After processing a request to store a chunk, a storage service replies to the client acknowledging the operation success. After sending all the chunks, the client sends to the manager the chunk-map (where each chunk is stored). Then, once the manager acknowledges, the client returns success to the application driver. In total the write operation generates two requests to the manager and one request per chunk to the storage nodes. \n\nNote that the specific set of storage services returned by the manager in the beginning of a write operation depends on the data placement policy used. A typical set is composed of stripe-width of storage services, but it can be composed of just one, e.g., when a local data placement is desired. \n\nTo model per-file optimizations, the client can overwrite system-wide configurations by requesting the manager to provide support for a specific data placement scheme. For example, the client may require that a file is stored locally, that is, on a storage service that is located on the same host. In this case, the manager attempts to allocate space on that specific storage service for that write operation (as opposed to striping the data across multiple storage services). The file-specific data placement policy is part of the workload description. \n\nAll communication among the system services uses the network. Each network request has its destination address, which is used by the simulator to determine which network queue should receive a packet. \n\nCurrently, the simulator reports the time spent, data transferred and storage used per each read or write. Additionally, one may request to collect aggregated information for specific points of the simulation.\n\\subsection{System Identification}\n\n\\label{sec:seeding}\n\nTo instantiate the storage system model, one needs specify the number of storage and client components in the system, and define the service times for the network ($\\mu^{net}$) and the system components (storage - $\\mu^{sm}$, manager - $\\mu^{ma}$, and client - $\\mu^{cli}$). \n\nThe system identification process is automated with a script as follows. First, to measure the service time per chunk\/request $T^{net}$), a script runs a network throughput measurement utility tool (e.g., \\textsf{iperf} \\cite{iperf}, ), to measure the throughput of both: remote and local (loopback) data transfers. Second, this script measures the time to read\/write a number of files to identify client and storage service time per data chunk. To this end, the system identification script deploys one client, one storage node and the manager on different machines, and writes\/reads a number of files. For each file read\/write the benchmark records the total operation time. At the end of its execution, the script computes the average read\/write time $T^{tot}$. The number of files read\/wrote is set to achieve 95\\% confidence intervals with $\\pm5\\%$ accuracy according to the procedure described in \\cite{jain91}.\n\nThe operation total time ($T^{tot}$) includes the client side processing time ($T^{cli}$), the storage node processing time ($T^{sm}$), the total time related to the manager operations ($T^{man}$) , and the network transfer time ($T^{net}$). The network service time for the network ($\\mu^{net}$) is based on a simple analytical model based on network throughput and proportional to the amount of data to be transferred in a packet.\n\nTo isolate just $T^{cli} + T^{man}$, the script runs a set of reads and writes of 0-size. This forces a request to go through the manager, but it does not touch the storage module. Since decomposing $T^{cli}$ and $T^{man}$ is not possible without probes in the storage system code, we opted to associate the $T^{cli} = 0$ and associate the whole cost of 0-size operations to the manager. While iperf can estimate $T^{net}$, and the script can infer $T^{cli} + T^{man}$, and therefore $T^{sm} = T^{tot} - T^{net} - T^{man}$. To obtain the service time per chunk, the times are normalized by chunk size. Therefore, $\\mu^{sm} = \\frac{T^{sm}}{chunkSize}$.\n\\subsection{Workload Description}\n\n\\label{sec:workload}\n\nThe simulator takes as an input a description of the workload to be simulated. The workload description contains two pieces of information: per client I\/O operations trace (i.e., open, read, write, close calls with the call details: timestamp, operation type, size, offset, and client id), and a files' dependency graph (capturing the operation dependency). The client traces can be obtained by running and profiling the application. The storage system logs often already provide these traces. Generating the file dependency graph can either be provided by the workflow scheduler (e.g., Swift \\cite{Swift:11}), by an expert user or automatically extracted from log files. Automating the extraction of the file dependency graph and client traces from storage system logs is an ongoing effort at our research group and is out of the scope of this paper.\n\\section{Evaluation}\n\n\\label{sec:evaluation}\n\nThis section aims to evaluate the mechanism's prediction accuracy and, more importantly, to demonstrate through a set of experiments the mechanism's ability to support correctly identifying the best configuration for a specific application pattern. To this end, we use a set of micro- and synthetic benchmarks, and a real application.. The microbenchmark-based evaluation compares the runtime of a simple workload of a single operation (read or write of a single file) to the predicted runtime provided by the simulator under a spectrum of configurations. The evaluation demonstrates that the mechanism can accurately predict in these simple scenarios the performance of the operations in all possible system configurations (varying stripe width and replication level). The synthetic benchmarks are designed to mimic real workflow application access patterns \\cite{Mosa-ccgrid:12}. The goal is to evaluate the mechanism's ability to predict performance under more complex system interactions that resemble the application ecosystem, we target and the storage system optimizations used in this context. Finally, we use BLAST \\cite{blast} as an example of a real application.\n\n\\textbf{Storage system.} The storage system used is MosaStore. The storage nodes are backed up by RAMDisk. We choose to experiment with RAMDisks as they are frequently used to support workflow applications: it offers higher performance and are the only option in some supercomputers that do not have spinning disks (e.g., IBM BG\/P machines). We briefly discuss a disk based evaluation in \\S \\ref{sec:discussion}.\n\n\\textbf{Deployment platform.} We use a testbed of 20 machines each with an Intel Xeon E5345 4-core, 2.33-GHz CPU, 4GB RAM, and 1-Gbps NIC. One machine runs the MosaStore manager while the other 19 machines each run both a storage node and a client access module.\n\n\\textbf{System identification.} The simulator is seeded according to the procedure described in \\S\\ref{sec:seeding}.\n\\subsection{Microbenchmarks}\n\n\\label{subsec:microbenchmarks}\nWe first focus on simple experiments composed of a set of read or write operations of 100MB files, triggered from one client. We used this experiment to guide the modeling\/implementation effort. To explore different configuration choices, we used: \\textit{stripe width} and \\textit{replication level} (with values from 1 to 5 for each), producing a total of 25 configuration scenarios. Figure \\ref{fig:relativerror} shows the relative error of the predicted runtime for the write microbenchmark (similar results were obtained for reads). For most of the cases, predicted performance is within 10\\% of the actual. \n\n\\begin{figure}[htb]\n\\centering\n\\includegraphics[width=0.4\\linewidth]{relative-error-writes} \n\\caption{Relative error for predicted performance. Values greater than 0 indicate that actual times are longer than predicted. We present averages over 30 runs. For most scenarios error within 5\\%, for all scenarios it is within 20\\% (Update Figure)}\n\\label{fig:relativerror}\n\\end{figure}\n\n\\subsection{Synthetic Benchmarks for Workflow Patterns}\n\\label{sec:eval:synt}\n\nThe microbenchmarks provide important data to understand the basic system operation and the impact of changes in the main knobs. They do not, however, capture the behavior of the system with multiple clients, the interaction among multiple applications, or the impact of data-placement policies designed to support workflow applications. To evaluate the accuracy of the prediction mechanism in more scenarios, we use synthetic benchmarks that mimic common data access patterns that exist in workflow applications \\cite{Mosa-ccgrid:12}. Specifically, this section focuses on pipeline, reduce, and broadcast patterns (Figure \\ref{fig:patterns}). These are among the patterns uncovered by studying over 20 scientific workflow applications by Wozniak et al. \\cite{Wozniak:09}, Shibata et al. \\cite{Shibata:10}, and Bharathi, et al. \\cite{Bharathi:08}). \n\n\\textbf{Experimental setup}. We use the MosaStore setup described above. We use the \\textit{DSS} label (from generic Distributed Storage System) for experiments where we use MosaStore default configuration: client and storage modules run on all machines, client stripes data over all 19 machines, and there is no data-access pattern optimization enabled. We use the \\textit{WASS} label (Workflow Aware Storage System) when the system configuration is optimized for a specific access pattern (including data placement, stripe width or replication) \\cite{Mosa-ccgrid:12}. All WASS experiments assume data location aware scheduling: for a given compute task, if all input file chunks exist on a single storage node, the task is scheduled on that node to increase access locality.\n\nThe goal of showing results for two different configurations choices is two-fold: (i) demonstrate the accuracy of the predictions for two different scenarios, and (ii), most important, show that the predictions correctly indicates which configuration is the best. To understand the impact of the data size, for each benchmark, we use (where possible) two workloads labeled as the \\textit{medium} and, a 10x larger, \\textit{large} workload. We omit results for a \\textit{small} workload, which is 10x smaller than medium, because it exhibits a high variability and similar performance between different configurations \\cite{Mosa-ccgrid:12}. \n\nFor actual performance, the figures show the average turnaround time and standard deviation for 15 trials. It is enough to guarantee a 9\n\n\n\n\\begin{figure}[htb]\n\\centering\n\\includegraphics[width=0.6\\linewidth]{patterns} \n\\caption{Pipeline, Reduce, and Broadcast benchmarks. Nodes represent workflow stages and arrows represent data transfers through files. The file sizes represent the \\textit{medium} workload. Files in the \\textit{large} workload are 10x larger.}\n\\label{fig:patterns}\n\\end{figure}\n\n\n\\textbf{Pipeline benchmark}. A set of compute tasks are chained in a sequence such that the output of one task is the input of the next task in the chain (Figure \\ref{fig:patterns}). A pipeline-optimized storage system will store the intermediate pipeline files on the storage node co-located with the application. Later, the workflow scheduler places the task that consumes the file on the same node, increasing data access locality. Here, 19 application pipelines run in parallel and go through three processing stages that read input from the intermediate storage and write the output to the intermediate storage. (We present only results for the medium workload as the large workload does not fit in the RAMdisk of the machines of our testbed). \n\n\\emph{Evaluation results.} Figure \\ref{fig:pipeline-medium-accuracy} shows the evaluation results. The simulator produces estimates equivalent to actual results for the optimized configuration (WASS). For no optimization (DSS), the prediction is 16\\% smaller (standard deviation considered). Note that for a case with default data placement policy, all clients stripe (write) data to all machines in the system; similarly, all machines read from all others. This creates, a complex interaction among all components in the system and some retries due to connection timeouts caused by network congestion which, we believe, is the source of the prediction inaccuracy. \n\n\\begin{figure}[htb]\n\\centering\n\\includegraphics[width=0.4\\linewidth]{actual_predicted_medium_pipeline_ram} \n\\caption{Actual and predicted performance for the pipeline benchmark and medium workload. Error bars show the standard deviation.}\n\\label{fig:pipeline-medium-accuracy}\n\\end{figure}\n\n\\textbf{Reduce or Gather benchmark}. A single compute task uses input files produced by multiple tasks. Real-world situations that generate this pattern include a task that checks the results of other tasks executed in parallel for a convergence criterion, or a task that calculates summary statistics from the output of many tasks. A possible data placement optimization is the use of collocation i.e., placing all these input files on one node and expose their location, which will later be used by the scheduler to run the reduce task on that machine. In the benchmark, 19 processes run in parallel on different nodes, consume an input file, and produce an intermediate file. In the next stage of the workflow, a single process reads all intermediate files and produces the reduce-file. Data sizes are indicated in Figure \\ref{fig:patterns}. In this scenario, for WASS configuration, the collocation optimization is enabled for the files used in the reduce stage, for the remaining files the locality optimization is enabled.\n\n\\emph{Evaluation results.} Similar to the pipeline benchmark, predictions for the reduce benchmark are within 20\\% of the actual performance (Figure \\ref{fig:reduce-ram}) and, more importantly, they capture the relative improvements that pattern-specific data placement policies policy can bring. \nWe note that Figure \\ref{fig:reduce-large-accuracy} captures the behavior of a heterogeneous scenario: We used a faster machine with a larger RAMDisk to run the reduce stage since the RAMDisk of the typical machine in the testbed is too small. Despite heterogeneity, the predictor captures the system performance with accuracy similar to a homogeneous system.\n\nWhen the collocation and locality optimizations are not enabled, the challenge of capturing exactly the system behavior is similar to the pipeline case: capture the complex interactions among all machines in the system. When the specific data placement is enabled though, the challenge is different: there is a high contention created by having several clients writing to the same storage machine (the one that performs the reduce phase). Figure \\ref{fig:reduce-large-stage} shows the results per-stage for the two stages of the large workload.\n\n\\begin{figure}\n\\begin{center}\n\\mbox{\n\\subfigure[Medium]\n{\\includegraphics[width=0.4\\linewidth]{actual_predicted_medium_reduce_ram}\n\\label{fig:reduce-medium-accuracy}}\n\\subfigure[Large]\n{\\includegraphics[width=0.4\\linewidth]{actual_predicted_large_reduce_ram}\n\\label{fig:reduce-large-accuracy}}\n\\subfigure[Large per Stage for WASS]\n{\\includegraphics[width=0.2\\linewidth]{actual_predicted_stages_large_reduce_ram}\n\\label{fig:reduce-large-stage}}\n\n}\n\\caption{Actual and predicted performance for the reduce benchmark for the medium, large workloads, and per stage for large workload. Error bars show the standard deviation.}\n\\label{fig:reduce-ram}\n\\end{center}\n\\end{figure}\n\n\n\\textbf{Broadcast benchmark}. A single task produces a file that is consumed by multiple tasks. In this benchmark, 19 processes running in parallel on different machines consume a file that is created in earlier stage by one task. A possible optimization for this pattern is to create replicas of the file that will be consumed by several different tasks. Data sizes are indicated in Figure \\ref{fig:patterns}.\n\n\\emph{Evaluation results.} Figure \\ref{fig:broadcast-medium-accuracy} shows the results for broadcast pattern with medium workload with the WASS system configured with 1, 2, or 4 replicas (the large workload shows a similar trend and we omit it here). For this benchmark all predictions matched the actual results: predictions were inside the interval of mean of actual $\\pm$ standard deviation, just 1-2\\% difference from the mean. This experiment highlights an interesting case for the predictor. According to the structure of the pattern and the results reported in \\cite{Mosa-ccgrid:12} (admittedly on a slightly different setup), creating replicas would improve the performance of the broadcast pattern. The results, however, show that creating replicas does not really help here. This happens because striping to many machines already avoids the contention of a single node holding the file. So, although creating replicas can alleviate the number of accesses to a given machine (since chunks are read in sequence), this gain is not paid off by the overhead of creating a replica. More importantly in our context, this is another situation where the predictor captures the impact of different configurations, showing, in this case that they are equivalent and the user can stick with one replica and save storage space.\n\n\\begin{figure}[htb]\n\\centering\n\\includegraphics[width=0.4\\linewidth]{actual_predicted_medium_broadcast_ram} \n\\caption{Actual and predicted performance for broadcast benchmark and medium workload. The experiment uses the WASS system while varying the replication level. Error bars show the standard deviation.}\n\\label{fig:broadcast-medium-accuracy}\n\\end{figure}\n\n\\textit{\\textbf{Synthetic Benchmark Evaluation summary}.} The evaluation with the microbenchmark and synthetic benchmarks demonstrates that a simulator can rely on proposed prediction approach to simulate complex system interactions generated by an application run on a complete cluster. Our approach leads to errors of 6\\% in average, lower than 9\\% in 90\\% of the studied scenarios and within 20\\% in the worst case. More importantly, the mechanism correctly differentiates between the different configurations and could support choosing the best configuration for each scenario in our evaluation. Finally, simulating the system takes roughly 10x less time and uses just one machine (200x fewer resources) than the actual execution of the benchmarks.\n\n\\subsection{Real Application}\n\\label{sec:eval:blast}\nThis section present the evaluation for prediction accuracy with an entire real application (BLAST). BLAST \\cite{blast} is a DNA search tool for finding similarities between DNA sequences. Each node in the cluster receives a set of DNA sequences as input (2 DNA queries in our evaluation with total size of 5.6KB) and each node searches, independently, the same database (size of 1.7GB in our evaluation) for matches with the input sequence, and writes the result (size of 82KB) to the storage system. A possible optimization for BLAST is to replicate the database file (broadcast pattern). We used replication levels of 1, 2, 4, 8, and 16. Finally, we note that, although BLAST is less I\/O intensive than the synthetic benchmarks, it generates an intensive I\/O workload: 550k I\/O operations for this experiment.\n\n\\emph{Evaluation results.} Figure \\ref{fig:blast} compares the actual and predicted performance for the BLAST workflow while varying the replication level. The results show a maximum prediction error of 10\\% of the actual performance. This highlights the ability of our approach to accurately predict performance of a complex application workload compromising thousands of IO operations and to help choosing a configuration.\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=0.5\\linewidth]{blast_actual_predicted_blast}\n\\caption{Average of the actual and predicted performance for the BLAST application.\n}\n\\label{fig:blast}\n\\end{figure}\n\\section{Related Work}\n\n\\label{sec:related}\n\nThis section describes briefly past work on storage system performance prediction and how it differs from our work.\n\nPast work used model-driven analysis to estimate performance of storage systems. For instance, Ergastulum \\cite{ergastulum} targets centralized storage solution based on one enclosure to recommend an initial configuration of the system, and Hippodrome \\cite{Hippodrome} relies on Ergastulum to improve the configuration based on online monitoring of the workload. The proposed predictor also support similar task. However, we consider a distributed system rather centralized storage, which brings more complex interaction among the components of the system and more configuration options.\n\nSimilar to this work, Thereska et al. \\cite{thereska:informed:06} proposed a predictor mechanism for a distributed storage system. The main difference to our work is the fact that their model is much more detailed (e.g., it model CPU and network adapter). To provide such information, they propose Stardust \\cite{stardust} a detailed monitoring information system that required changes to the storage system and kernel modules to add monitoring points. This approach enabled their predictor to achieve prediction within 20\\% of the actual predictions depending on the workload. Our approach have achieved similar accuracy on our target workload. \n\nAn important difference to past work on storage systems performance is our focus on a whole workflow application and the potential interaction among the workflow's phases instead of the average performance for a batch of operations or predicting performance of the system from the perspective of just one client.\n \nRecently, Z. Zhang et al. has proposed an approach to determine the storage bottleneck for a given many-tasks application (a class of workflow) based on a set of benchmarks and executions of the the application. The approach we propose enable a richer exploration of the system by a lower cost since the predictor is able to estimate performance of a scenario that add or reduce resources and change the configuration without requiring new runs of the application of the benchmarks.\n\\section{Discussion and Summary}\n\n\\label{sec:discussion}\n\nThis paper makes the case for a prediction mechanism to support the task of configuring an intermediate storage system. We focus on predicting the performance of workflow applications when running on top of an intermediate object-based storage system. Specifically, we propose a solution based on a queue-based model with a number of attractive properties: a generic and uniform system model; supported by a simple system identification process that does not require specialized probes or system changes to perform the initial benchmarking; with a low runtime to obtain predictions; and, finally, with adequate accuracy for the cases we study. \n\nWe highlight that this is an ongoing work. The discussion below aims to clarify our understanding of the limitations of our work and the lessons we have learned during this exercise so far, and to sketch our future work. \n\n\\vspace{+6pt}\n\n\\textbf{What are the main sources of inaccuracies?}\nCurrently, there are sources of inaccuracies at multiple levels: First, the model does not capture all the details of the storage system (e.g., support services like garbage collection or storage node heartbeats; the control paths are simplified to match what we believe generic object-based storage would do - while we know that a FUSE-based implementation would need more complex control paths; we model all control messages as having the same size) and the environment (e.g., contention at the network fabric level or scheduling). Second, we constrain and simplify system identification even further at the cost of additional accuracy loss. Third, we do not model the infrastructure in detail (e.g., we do not model the network protocols or the spinning disks). Finally, so far the application driver uses an idealized image of the workflow application (e.g., all pipelines are launched in the simulation exactly at the same time while in the experiments on real hardware coordination overheads make them slightly staggered). We believe the latter one is the main reason of current inaccuracies in the system and should be address by a richer workload description.\n\n\\vspace{+6pt}\n\n\\textbf{What is the accuracy when the intermediate storage is deployed on spinning disks?}\nSo far, we have focused on predicting performance when intermediate storage is deployed over RAMDisks for two reasons: This is a common setup on large systems (as some, do not even have spinning disks) and the individual performance of RAMdisks is simpler to predict (the service time for spinning disks is history dependent due to cache behavior and position of disk head). The storage service we use does not model history-dependent behavior, thus we expect lower accuracy predictions when the system is deployed over spinning disks. (This can be fixed, by using a more sophisticated model of the storage service and it is part of our future work). \n\n\n\\begin{figure}\n\\begin{center}\n\\mbox{\n{\\includegraphics[width=0.25\\linewidth]{actual_predicted_medium_hdd_reduce}\n\\label{fig:reduce-medium-accuracy-hdd}}\n{\\includegraphics[width=0.25\\linewidth]{actual_predicted_large_hdd_reduce}\n\\label{fig:reduce-large-accuracy-hdd}}\n}\n\\caption{Actual and predicted performance for the reduce benchmark on HDD - medium(left) and large (right) workload. The results for WASS is this experiment are not directly comparable to the corresponding ones for spinning disks as we use a slower node for data co-placement, keeping the system homogeneous.}\n\\label{fig:reduce-hdd}\n\\end{center}\n\\end{figure}\n\n\n\\vspace{+6pt}\n\n\\textbf{Can the performance prediction mechanism support the \\textit{development process} of a storage system? Do you have specific experience with using the mechanism in this context? }\nOne of the lessons we have learned so far is the utility of the mechanism to support development of the storage system itself. Back of the envelope calculations are a common mechanism to evaluate expected performance bounds for a given system. The predictor takes this a step further and is useful in complex scenarios where back of the envelope estimates were intractable. Not only developers can use it to evaluate the potential gains of implementing a new complex optimizations or to study the impact of faster network and nodes, but also the mechanism can be used as a baseline to detect performance anomalies. \n\nMore concretely, we have encountered a number of situations where the predicted and actual performance differed significantly. In some cases these highlighted simplifications in the model or in our simulator. But, more importantly, there were cases that highlighted complex performance-related anomalies that were fixed in the storage system such as: non-trivial implementation problems (e.g., limited randomness in the data placement decisions that created an artificial bottleneck, or unreasonable locking overheads at the manager). Similarly, the prediction mechanism helped us revisit assumptions about the middleware stack the storage system is implemented over (e.g., we have discovered the significant impact of the TCP connection initiation timeout of 3s in some scenarios); and highlighted shortcomings of the seeding process or incorrect assumptions about the deployment platform (e.g., we were ignoring platform heterogeneity). \n\n\\vspace{+6pt}\n\n\\textbf{How to decide when to stop increasing the level of detail in the model and the complexity of system identification?}\nWe aim to model only the key interactions between system components. Modeling all system subcomponents and all their interactions in detail would be too complex. Such complexity could improve prediction accuracy, but would have significant drawbacks: significantly more complex model (as complex as the actual storage system and the underlying environment (e.g., network protocols, operating system buffers, scheduling), complex seeding process, lower scalability, and loss of the model generality. Further, the improvement in accuracy may not add much value (e.g., when the prediction mechanism is used to decide between system configurations). \nWe followed a top-down approach: we started from a simple model and added more components or interactions' details until the accuracy of the all predictions was within 20\\% of actual performance (and the median error was within 5\\%) for the set of microbenchmarks we present in (\\S\\ref{subsec:microbenchmarks}). \n\n\\vspace{+6pt}\n\n\\textbf {What is next?}\nOur future work goes in several directions: (i) address the prediction shortcomings highlighted in \\S\\ref{sec:evaluation} -- besides the approaches already described, we also plan to enrich our system identification with data mining information \\cite{iron-models}; (ii) fully automate the process to extract the behavior of the application based on log information and pass to the simulator (currently the logs already provides us the information needed, but the process is not fully automated), (iii) explore a richer space of configuration knobs; and (iv) evaluate the system using additional synthetic benchmarks and real applications.\n\n\\section*{Acknowledgements}\n\nWe would like to thank Hao Yang for his collaboration in MosaStore development; Elizeu Santos-Neto and Abdullah Gharaibeh for their collaboration in keeping the quality of the code via code reviews.\n\n\\bibliographystyle{splncs}\n\n\\section{Introduction}\n\\label{sec:intro}\n\nAssembling workflow applications by putting together standalone binaries has become a popular approach to support large-scale science (e.g., modFTDock \\cite{Modftdock:12}, Montage\\cite{Montage:03} or BLAST \\cite{blast}). The processes spawned from these binaries communicate via temporary files stored on a shared storage system. In this setup, the workflow runtime engines are basically schedulers that build and manage a task-dependency graph based on the tasks' input\/output files (e.g., SWIFT \\cite{Swift:11}, Pegasus \\cite{Pegasus:05}).\n\nTo avoid accessing the platform's backend storage system (e.g., NFS or GPFS or Amazon S3), recent proposals  (e.g., \\cite{versatile,Wozniak:09}) advocate using some of the nodes allocated to the application to deploy a \\textit{intermediate storage system}. That is, aggregating (some of) the resources of an application allocation to provide a shared temporary storage system dedicated to (and co-deployed with) the application. \n\n\nAggregating node-local resources to provide the intermediate storage offers a number of advantages: higher performance - as applications benefit from a wider I\/O channel obtained by striping data across several nodes; higher efficiency \u2013 as it improves resource utilization; incremental scalability \u2013 as it is possible to increase system capacity in small increments. This scenario also opens the opportunity for optimizing the intermediate storage system for the target workflow application: a storage system used by a single workflow, and co-deployed on the application nodes, can be configured specifically for the I\/O patterns generated by the workflow (e.g., configure striping and replication to eliminate hot spots, use a data placement policy to maximize data access locality) \\cite{Mosa-ccgrid:12}. \n\nThese benefits, however, come at a price: configuring the intermediate storage system becomes increasingly complex for multiple reasons. First, the optimization techniques commonly used in distributed environments expose trade-offs that rarely exist in centralized solutions \\cite{thereska:informed:06,strunk-utility}. Second, each application may obtain peak performance at a different configuration point - not linear with the amount of resources of values for configuration parameters, a consequence of different I\/O patterns \\cite{ursaminor, cirne2002, strunk-utility, thereska:informed:06}. Third, depending on the context, there are multiple metrics of interest to optimize: e.g., response time, throughput, energy, and, increasingly common in cloud computing environments, the cost of resources. \n\nFurther complicating this scenario, the user faces resource allocation decisions that often entail trade-offs between cost and application turn-around time \\cite{cirne2002,herodotou2011,ubcwork2012}. Typical allocation choices involve deciding the number of nodes for the application in batch-computing environments (e.g., clusters) and specifying the nodes' 'type' in cloud-computing environments (i.e., per-node configuration in terms of compute, memory, storage and network capabilities).\n\nTo illustrate these points, consider the following optimization techniques and their trade-offs. Data striping may accelerate data access, yet it decreases reliability and requires more resources. Increasing the number of storage nodes can avoid data access bottlenecks, yet it may expose metadata management scalability bottlenecks. Higher redundancy levels (through replication or erasure codes) may accelerate data access and increase reliability, yet they also require more resources and complex consistency protocols. Finally, different data placement and\/or caching policies are beneficial to different workloads.\n\nPast work and our own experience support these assertions. They show that different applications achieve their best performance when using different storage configurations \\cite{versatile,ursaminor}. Also, different applications even benefit from different combinations of storage optimization techniques \\cite{Mosa-ccgrid:12,erss2011}, and that the choice of the optimal configuration point is not intuitive (Figure \\ref{fig:montage}). \n\nIn this scenario, the role of the application administrator\/user becomes non-trivial: in addition to being in charge with running the workflow application, the user has to choose how to partition the allocation between nodes to execute the application and nodes to run the intermediate storage system. Additionally, for each provisioning configuration (i.e., number of allocated nodes running the storage system), there is a wide spectrum of storage system configurations (e.g., replication level, chunk size, cache and data placement policies, or stripe width) that deliver different storage system performance \\cite{ursaminor, Mosa-ccgrid:12}. Consequently, provisioning the system entails searching a complex multi-dimensional configuration space to determine the user's ideal cost\/performance balance point. Providing support for this activity is the focus of our project \\cite{grid2010}.\n\n\n\n\n\\begin{figure}[htb]\n\\centering\n\\includegraphics[width=0.75\\linewidth]{montage-large} \n\\caption{\\textit{Different storage system configurations deliver different performance and the choice of the optimal configuration point is not intuitive.} To demonstrate that this holds for workflow applications, we have executed the Montage workflow \\cite{Montage:03} on top of MosaStore \\cite{versatile} storage system with different storage configurations deployed over the Grid'5000 platform. We executed the same workload for several values of one configuration knob: \\textit{stripe width}. As we increase stripe width, we also increase the number of storage nodes to support it. For a low stripe width, performance is low due to congestion at the storage nodes and, for higher values, performance decreases due to connection handling and metadata access overheads. A user seeking the optimal point would pick a stripe width value close to five, a non-obvious choice. Other case studies can be found in \\cite{Mosa-ccgrid:12}. }\n\n\\label{fig:montage}\n\\end{figure}\n\nThis work proposes an approach for exploring this space and helping the user select an allocation, a partitioning, and an intermediate storage system configuration that meets well the user's desired cost\/performance balance point.\n\n\\textbf{The contributions of this paper} lay over multiple axes: \n\n\\begin{itemize}\n\n\\item Highlights the system provisioning, allocation and configuration problem, presenting it as a multi-objective optimization problem, and presents an approach for exploring this complex decision space.\n\n\\item Synthesizes the key requirements for a prediction mechanism that will be useful in practice (\\S\\ref{sec:requirements}).\n\n\\item Proposes a simple queue-based model for distributed, object-based storage systems (\\S\\ref{sec:model}). This model, at the base of the prediction mechanism, is: generic (i.e., covers the traditional design for object-based datastores), and homogeneous (all nodes are based on the same model). \n\n\\item Proposes a system identification procedure to seed the model that is simple, lightweight, effective, and does not require system changes to collect monitoring information (\\S\\ref{sec:seeding}).\n\n\\item Presents experience with using this prediction mechanism when evaluated independently with synthetic benchmarks (\\S\\ref{sec:evaluation}) and in the context of making configuration choices for two different scenarios with a real application. These scenarios give a glimpse of how the application execution time can vary depending on the choices made and indicates that the predictor will be able to guide the search for the desired balance cost\/time-to-solution.  Our experience shows that the predictor is lightweight (200x to 2000x less resource intensive than running the application itself) and accurate (8\n\n\\item  Finally, this paper discusses our experience (\\S\\ref{sec:discussion}) to date with using the prediction mechanism beyond our original design goal: we have used the performance prediction to better understand and debug a distributed storage system that our group develops.\n\n\\end{itemize}\n\n\\textbf{The Problem.} Configuring the intermediate-storage system to achieve high performance (e.g., in terms of application turnaround time, storage footprint, energy consumption, or financial cost), involves choosing a set of storage system configuration parameters (such as stripe width, data placement policy, and replication level). \n\nAdditionally, for many-task applications, the user is concerned about resource allocation. Thus, the decision space we investigate revolves around three axes: provisioning - total number of nodes, deciding on node type(s) (or node specification); partitioning \u2013 splitting these nodes between the application and the intermediate storage system; and configuration choice \u2013 for each subsystem choosing an optimal configuration, in this case choosing replication level, stripe width, chunk size, cache\/prefetching and data placement policies for the intermediate storage system.\n\nIn this complex space, generally the user's goal is to optimize a multi-objective problem, in at least two dimensions: maximize performance (e.g., reduce application execution time) while minimizing cost (e.g., reduce the total CPU hours or dollar amount spent). More concretely the user is often interested in answering specific questions: How should the configuration of the storage system be? What is the allocation that can achieve the lowest total cost? How should I partition the allocation among application and storage nodes to achieve the highest performance? What is the allocation that is most cost efficient (i.e., has lowest cost per unit of performance)?\n\n\\textit{Manually} fine-tuning the storage system configuration parameters and allocation descisions is undesirable for multiple reasons. First, the user lacks a deep understanding of how different configuration choices affect the system's performance. Second, the workload may vary: new application versions may make one-time tuning obsolete. Finally, performance tuning is time-consuming due to the large configuration space to consider.\n\n\\textit{This works presents our progress to date on designing and evaluating a performance prediction mechanism for object-based storage systems in the context of workflow applications}. Given a storage system configuration, an application I\/O profile, and a characterization of the deployment platform based on a simple system identification process (e.g., storage nodes service time, network characteristics), the mechanism predicts the total application turnaround. \n\n This approach can support four important tasks: First, \\textit{autotuning}, a software tool that relies on the proposed mechanism can enable efficiently configuring the storage system \\cite{grid2010,thereska:informed:06,strunk-utility}, through exploring the configuration space without actually running the application. Second, \\textit{resource provisioning}, it can inform the decision of (i) the number of nodes to be allocated, (ii) how to partition allocated nodes between application and storage nodes to achieve a certain performance level. Third, \\textit{new technology evaluation}, it can estimate the application performance on hardware that has not yet been procured (e.g., to help answer the question: \\textit{what} would be the performance improvement \\textit{if} we used SSDs?). Finally, the mechanism can help explore the impact of configuration choices in situations where direct measurement is difficult or requires specialized infrastructure (e.g., energy consumption). \n\nThis paper focuses on predicting the application turnaround time, but we note that the model and approach presented apply readily to other optimization metrics.\n\\section{The Design of a Performance Estimation Mechanism}\n\\label{sec:solution}\n\nThis section discusses the requirements for a practical performance prediction mechanism (\\S\\ref{sec:requirements}) and presents the key aspects of the object-based storage system architecture modeled (\\S\\ref{sec:mosa}). Then, it focuses on the proposed solution: it presents the model (\\S\\ref{sec:model}), its implementation (\\S\\ref{sec:implementation}), the system identification process to seed the model (\\S\\ref{sec:seeding}), and an overview of the workload description (\\S\\ref{sec:workload}). \n\nMaking accurate performance predictions for distributed systems is a challenge. Since in most cases purely analytical models can not provide adequate accuracy, simulation is the commonly adopted solution. At the one end of the design spectrum, current practice (e.g., NS2 simulator \\cite{ns2}) suggests that while simulating a system at low granularity (e.g., packet level simulation in NS2) can provide high accuracy, the complexity of the model and the number of events generated make accurately simulating large-scale systems infeasible. At the other end, coarse grained simulations (e.g., \\cite{SimGrid,PeerSim}) scale at the cost of lower accuracy\n\nTwo key observations to reduce simulation complexity and increase its scalability: First, as the goal of the simulation is to support configuration choice for a specific workload, achieving perfect accuracy is less critical (\\S\\ref{sec:evaluation}). Second, we take advantage of the fact that we know the workload characteristics generated by workflow applications: relatively large files, single-write-many-reads, and specific data access patterns. These observations enable us to reduce the simulation complexity by not simulating in detail some of the control paths that do not significantly impact accuracy (e.g., the chunk transfer time is dominated by the time to send the data, consequently not accounting the time of the acknowledgment messages or the metadata message transfer will not tangibly impact accuracy).\n\\subsection{Solution Requirements}\n\\label{sec:requirements}\n\nA practical performance prediction mechanism should meet the following, partially conflicting, requirements that bound the solution space:\n\n\\begin{itemize}\n\n\\item \\textbf{Accuracy.} The mechanism should provide adequate accuracy. Of course better accuracy is desirable; however, in the face of practical limitations to achieve perfect accuracy, we note that there are decreasing incremental gains for improved accuracy in practical settings. For example, to support decisions about configuration choices a predictor only needs to correctly estimate their relative performance or trends of changing a configuration parameter. Even more, if two configurations offer near performance, their relative predicted performance is less important as long as the prediction mechanism places their performance as similar.\n\n\\item \\textbf{Scalability and Response Time.} The predictor should enable the quick exploration of the configuration space. To this end, the mechanism should offer performance predictions quickly and scale across at least two dimensions: (i) it should scale with the system size and be able to model large systems; and (ii) it should scale with the I\/O intensity and be able to model I\/O intensive applications. \n\n\\item \\textbf{Usability and Generality.} The predictor should not impose a burdensome effort to be used. Specifically, the bootstrapping\/seeding process should be simple and it should not require storage system redesign (or a particular initial design) to collect performance measurements. Additionally, ideally the prediction mechanism should model a generic object-based distributed storage design and using it should not require in-depth knowledge of storage system protocols and architecture. \n\n\\item \\textbf{Ability to explore \\textit{``what-if''} scenarios.} A prediction mechanism should be able to support exploring hypothetical scenarios, such as scenarios that assume new\/different hardware configurations (e.g., usage of SSDs). We note that there are two main categories for the models at the foundation of all performance prediction mechanisms: explanatory and agnostic models. The explanatory models try to mimic the key components of the system and their interaction at various levels of accuracy and granularity. The agnostic model aims to predict the output metric of interest being completely oblivious of system internals (e.g., a neural network based models would fit in this category). Supporting \\textit{``what-if''} scenarios exploration requires an approach based on an explanatory model.\n\n\\end{itemize}\n\\subsection{Object-based Storage System Design}\n\\label{sec:mosa}\n\nWe focus on a widely-adopted object-based storage system architecture (such as that adopted by GoogleFS \u200e\\cite{ghemawat2003google}, PVFS \\cite{pvfs}, MosaStore \\cite{versatile}, and UrsaMinor \\cite{ursaminor}). This architecture includes three main components: a centralized metadata manager, storage nodes, and a client-side system access interface (SAI). The manager maintains the stored files' metadata and system state. To speed up data storage and retrieval, the architecture employs striping \u200e\\cite{zebra95}: files are split into chunks stored across several storage nodes. Client SAIs implement data access protocols after they interact with the manager that stores data placement information. \n\n\\textit{Data placement}. The default data placement generally adopted is round-robin: when a new file is created on a stripe of \\textit{n} nodes the file's chunks are placed in a round-robin fashion across these nodes. Additionally, data placement policies that optimize for a specific application access patterns have seen higher adoption \\cite{Mosa-ccgrid:12,pnfs,Zhang:2012:DAD:2388996.2389112}. For instance, the following data placement policies are used to optimize for the workflow applications' data access patterns: local, co-locate and broadcast (detailed in \\S\\ref{sec:evaluation}).\n\n\\textit{Replication}. Replication is often used to increase reliability or to improve access performance. Data is replicated when new data is stored in the system, consequently, while a higher replication level will reduce contention on the node storing a popular file, it will increase the file write time and the storage space consumption.\n\nWe explore the accuracy of the prediction mechanism assuming that the stripe width, replication level, and data placement policy are configurable as suggested in \\cite{Mosa-ccgrid:12,ursaminor,versatile}. Our approach can be extended to support other configuration parameters.\n\\subsection{System Model}\n\\label{sec:model}\n\nOur solution uses a queue-based storage system model for the system components' operations and their interactions. The model requires three inputs from the user: the storage system configuration, a workload description, and the performance characteristics of storage system components (i.e., system identification). The simulator instantiates the storage system model with the specific component characteristics and configuration, and simulates the application run as described by the workload description.\n\nAll participating machines are modeled similarly, regardless of their specific role (Figure \\ref{fig:model}): each machine hosts a network component and can host one or more system components (each modeled as a service with its own queue).\n\nA system component and its queue represent a specific functionality: that is, the \\textit{manager} component is responsible for storing files' and storage nodes' metadata. The \\textit{storage} component is responsible for storing and replicating data chunks. Finally, the \\textit{client} component receives the read and write operations from the application, implements, at the high-level the storage system protocol by sending control or data requests to other services, and once a storage operation is terminated it communicates again with the application. Each of these components is modeled as service that takes requests from its queue (fed by the network service or by the application for the client service) and sends responses back through the network service (or directly to the application, again for the client). \n\nThe network component and its in- and out- queues model the network-related activity of a host. Key here is to model network-related contention while avoiding modeling the details of the transport protocol (e.g., dealing with packet loss, connection establishment and teardown details). The requests in the out-queue of a network component are broken in smaller pieces that represent network frames and sent to the in-queue of the destination host. Once the network service processes all the frames of a given request in the in-queue, it assembles the request and places it in the queue of the destination service.\n\n\n\n\\begin{figure}[htb]\n\\centering\n\\includegraphics[width=0.75\\linewidth]{storage-model} \n\\caption{Queue-based model of a distributed storage system. Each component (manager, client component, and storage component) has a single system service that processes requests from its queue. Additionally, each host has a network component with an in- and out- queue. The network core connects and routes the messages between the different components in the system and can model network latency and contention at the aggregate network fabric level. Solid lines show the flow going out from a storage system component while dashed lines show the in-flow path.}\n\\label{fig:model}\n\\end{figure}\n\nThe system components can be collocated on the same host (e.g., the client and storage components running on the same host). In this situation, requests between collocated services also go through the network, but have a faster service time than remote requests - representing a loopback data transfer (\\S\\ref{sec:seeding}).\n\nSpace limitations prevent us from presenting the full details of the model. As a rule, we accurately model the data paths at chunk-level granularity, and the control paths at a coarser granularity: modeling only one control message to initiate a specific storage function while an implementation may have multiple rounds of control message exchanges. We used our own experience with designing and implementing an object-based storage system \\cite{mosastore-website}.\n\\subsection{Model Implementation}\n\\label{sec:implementation}\n\nWe have implemented the above model as a discrete-event simulator in Java. The simulator receives as inputs: a summarized description of the application workload (described in \\S\\ref{sec:workload}) and a description of the deployed system which has two parts. The first part describes the system-wide configuration parameters (currently, replication level, stripe-width, chunk size, and data-placement system-wide) and the details of the system: number of hosts, number of storage nodes and clients, whether storage and clients nodes are collocated on the same hosts. The second part characterizes the performance of system components: service times for network, client, storage, and manager (the process of identifying these values is described in \\S\\ref{sec:seeding}). \n\nOnce the simulator instantiates the storage system, it starts the application driver that processes the application workload. The driver reads the description of the application workload, creates the corresponding events (e.g., read from file $x$ at offset $y$, $z$ bytes) and places them in the client service queue. File-specific configuration (as proposed by \\cite{ursaminor,Mosa-ccgrid:12}) is described as part of the application workload description since it depends on the file that application reads\/writes.\n\nAs in a real system, the manager component maintains the metadata of the system (i.e., implements data placement policies by returning free chunks when requested by write operations, and keeps track of file to chunk mapping and chunk placement). To make the process clearer, consider the following example for a write operation where a client module processes a file write event. First, the client contacts the manager asking for free space. The manager replies specifying a set of free chunks and their storage services to be used during this write. Then, the client requests each storage service to store chunks in a round-robin fashion. After processing a request to store a chunk, a storage service replies to the client acknowledging the operation success. After sending all the chunks, the client sends to the manager the chunk-map (where each chunk is stored). Then, once the manager acknowledges, the client returns success to the application driver. In total the write operation generates two requests to the manager and one request per chunk to the storage nodes. \n\nNote that the specific set of storage components returned by the manager in the beginning of a write operation depends on the data placement policy used. A typical set is composed of stripe-width of storage services, but it can be composed of just one, e.g., when a local data placement is desired. \n\nTo model per-file optimizations, the client can overwrite system-wide configurations by requesting the manager to provide support for a specific data placement scheme. For example, the client may require that a file is stored locally, that is, on a storage service that is located on the same host. In this case, the manager attempts to allocate space on that specific storage service for that write operation (as opposed to striping the data across multiple storage services). The file-specific data placement policy is part of the workload description. \n\nAll communication among the system services uses the network. Each network request has its destination address, which is used by the simulator to determine which network queue should receive a packet. \n\nCurrently, the simulator reports the time spent, data transferred and storage used per each read or write. Additionally, one may request to collect aggregated information for specific points of the simulation.\n\\subsection{System Identification}\n\\label{sec:seeding}\n\nTo instantiate the storage system model, one needs specify the number of storage and client components in the system, and define the service times for the network ($\\mu^{net}$) and the system components (storage - $\\mu^{sm}$, manager - $\\mu^{ma}$, and client - $\\mu^{cli}$). \n\nThe system identification process is automated with a script as follows. First, to measure the service time per chunk\/request $T^{net}$), a script runs a network throughput measurement utility tool (e.g., \\textsf{iperf} \\cite{iperf}, ), to measure the throughput of both: remote and local (loopback) data transfers. Second, this script measures the time to read\/write a number of files to identify client and storage service time per data chunk. To this end, the system identification script deploys one client, one storage node and the manager on different machines, and writes\/reads a number of files. For each file read\/write the benchmark records the total operation time. At the end of its execution, the script computes the average read\/write time $T^{tot}$. The number of files read\/wrote is set to achieve 95\\% confidence intervals with $\\pm5\\%$ accuracy according to the procedure described in \\cite{jain91}.\n\nThe operation total time ($T^{tot}$) includes the client side processing time ($T^{cli}$), the storage node processing time ($T^{sm}$), the total time related to the manager operations ($T^{man}$) , and the network transfer time ($T^{net}$). The network service time for the network ($\\mu^{net}$) is based on a simple analytical model based on network throughput and proportional to the amount of data to be transferred in a packet.\n\nTo isolate just $T^{cli} + T^{man}$, the script runs a set of reads and writes of 0-size. This forces a request to go through the manager, but it does not touch the storage module. Since decomposing $T^{cli}$ and $T^{man}$ is not possible without probes in the storage system code, we opted to associate the $T^{cli} = 0$ and associate the whole cost of 0-size operations to the manager. While iperf can estimate $T^{net}$, and the script can infer $T^{cli} + T^{man}$, and therefore $T^{sm} = T^{tot} - T^{net} - T^{man}$. To obtain the service time per chunk, the times are normalized by chunk size. Therefore, $\\mu^{sm} = \\frac{T^{sm}}{chunkSize}$.\n\\subsection{Workload Description}\n\\label{sec:workload}\n\nThe simulator takes as an input a description of the workload to be simulated. The workload description contains two pieces of information: per client I\/O operations trace (i.e., open, read, write, close calls with the call details: timestamp, operation type, size, offset, and client id), and a files' dependency graph (capturing the operation dependency). The client traces can be obtained by running and profiling the application. The storage system logs often already provide these traces. Generating the file dependency graph can either be provided by the workflow scheduler (e.g., Swift \\cite{Swift:11}), by an expert user or automatically extracted from log files. Automating the extraction of the file dependency graph and client traces from storage system logs is an ongoing effort at our research group and is out of the scope of this paper.\n\\section{Evaluation}\n\\label{sec:evaluation}\n\nThis section aims to evaluate the mechanism's prediction accuracy and, more importantly, to demonstrate through a set of experiments the mechanism's ability to support correctly identifying quise-optimal configuration for a specific application. To this end, we use a set of synthetic benchmarks and a real application. \n\nThe synthetic benchmarks are designed to mimic real workflow application access patterns \\cite{Mosa-ccgrid:12} of workflow applications. The goal is to evaluate the mechanism's ability to predict time consumed under more complex system interactions that resemble the application ecosystem, we target and the storage system optimizations used in this context. \n\nFinally, to understand how use the prediction mechanisms can be in a real set-up, we use BLAST \\cite{blast} as an example of a real scientific application. The goal is to evaluate how the ability to predict time-to-solution can support user's decisions when the user has to decide about the storage configuration and allocation.\n\n\\textbf{Storage system.} The storage system used is MosaStore. The storage nodes are backed up by RAMDisk. We choose to experiment with RAMDisks as they are frequently used to support workflow applications: it offers higher performance and are the only option in some supercomputers that do not have spinning disks (e.g., IBM BG\/P machines). We briefly discuss a disk based evaluation in \\S \\ref{sec:discussion}.\n\n\\textbf{Deployment platform.} We use a testbed of 20 machines each with an Intel Xeon E5345 4-core, 2.33-GHz CPU, 4GB RAM, and 1-Gbps NIC. One machine runs the MosaStore manager while the other 19 machines each run both a storage node and a client access module.\n\n\\textbf{System identification.} The simulator is seeded according to the procedure described in \\S\\ref{sec:seeding}.\n\n\\subsection{Synthetic Benchmarks for Workflow Patterns}\n\\label{sec:eval:synt}\n\nThis section evaluates the accuracy of the prediction mechanism capture the behavior of the system with multiple clients, the interaction among multiple applications, or the impact of data-placement policies designed to support workflow applications \\cite{Mosa-ccgrid:12,TR-Mosa}. We use synthetic benchmarks that mimic common data access patterns existent in workflow applications. Specifically, this section focuses on pipeline, reduce, and broadcast patterns (Figure \\ref{fig:patterns}). These are among the most used patterns uncovered by studying over 20 scientific workflow applications by Wozniak et al. \\cite{Wozniak:09}, Shibata et al. \\cite{Shibata:10}, and Bharathi et al. \\cite{Bharathi:08}). \n\nAdditionally, these synthetic benchmarks are designed to explore the limitations of the predictor and be a worst case in terms of accuracy as they are composed exclusively of I\/O operations, which leads to contention in the real storage system that are well-known for being difficult to model accurately \\ref{REF}.\n\n\\textbf{Experimental setup}. We use the MosaStore setup described above. We use the \\textit{DSS} label (from generic Distributed Storage System) for experiments where we use MosaStore default configuration: client and storage modules run on all machines, client stripes data over all 19 machines, and there is no data-access pattern optimization enabled. We use the \\textit{WASS} label (Workflow Aware Storage System) when the system configuration is optimized for a specific access pattern (including data placement, stripe width or replication) \\cite{Mosa-ccgrid:12}. All WASS experiments assume data location aware scheduling: for a given compute task, if all input file chunks exist on a single storage node, the task is scheduled on that node to increase access locality. The storage nodes are backed up by RAMDisk as they are frequently used to support workflow applications: they offer higher performance, is key part of the a intermediate storage system  to speed-up workflow application, and are the only option in some HPC centers that do not have spinning disks (e.g., IBM BG\/P machines). \n\nThe goal of showing results for two different configurations choices is two-fold: (i) demonstrate the accuracy of the predictions for two different scenarios, and (ii), most important, show that the predictions correctly indicates which configuration is the best. To understand the impact of the data size, for each benchmark, we use (where possible) two workloads labeled as the \\textit{medium} and, a 10x larger, \\textit{large} workload. We omit results for a \\textit{small} workload, which is 10x smaller than medium, because it already exhibits a similar performance between different configurations \\cite{Mosa-ccgrid:12}. \n\nFor actual performance, the figures show the average turnaround time and standard deviation for 15 trials. It is enough to guarantee a 9\n\n\n\n\\begin{figure}[htb]\n\\centering\n\\includegraphics[width=0.6\\linewidth]{patterns} \n\\caption{Pipeline, Reduce, and Broadcast benchmarks. Nodes represent workflow stages and arrows represent data transfers through files. The file sizes represent the \\textit{medium} workload. Files in the \\textit{large} workload are 10x larger.}\n\\label{fig:patterns}\n\\end{figure}\n\n\\textbf{Pipeline benchmark}. A set of compute tasks are chained in a sequence such that the output of one task is the input of the next task in the chain (Figure \\ref{fig:patterns}). A pipeline-optimized storage system will store the intermediate pipeline files on the storage node co-located with the application. Later, the workflow scheduler places the task that consumes the file on the same node, increasing data access locality. Here, 19 application pipelines run in parallel and go through three processing stages that read input from the intermediate storage and write the output to the intermediate storage. (We present only results for the medium workload as the large workload does not fit in the RAMdisk of the machines of our testbed). \n\n\\emph{Evaluation results.} Figure \\ref{fig:pipeline-medium-accuracy} shows the evaluation results. The simulator produces estimates equivalent to actual results for the optimized configuration (WASS). For no optimization (DSS), the prediction is 16\\% smaller (standard deviation considered). Note that for a case with default data placement policy, all clients stripe (write) data to all machines in the system; similarly, all machines read from all others. This creates, a complex interaction among all components in the system and some retries due to connection timeouts caused by network congestion which, we believe, is the source of the prediction inaccuracy. \n\n\\begin{figure}[htb]\n\\centering\n\\includegraphics[width=0.5\\linewidth]{actual_predicted_medium_pipeline_ram} \n\\caption{Actual and predicted performance for the pipeline benchmark and medium workload. Error bars show the standard deviation.}\n\\label{fig:pipeline-medium-accuracy}\n\\end{figure}\n\n\\textbf{Reduce or Gather benchmark}. A single compute task uses input files produced by multiple tasks. Real-world situations that generate this pattern include a task that checks the results of other tasks executed in parallel for a convergence criterion, or a task that calculates summary statistics from the output of many tasks. A possible data placement optimization is the use of collocation i.e., placing all these input files on one node and expose their location, which will later be used by the scheduler to run the reduce task on that machine. In the benchmark, 19 processes run in parallel on different nodes, consume an input file, and produce an intermediate file. In the next stage of the workflow, a single process reads all intermediate files and produces the reduce-file. Data sizes are indicated in Figure \\ref{fig:patterns}. In this scenario, for WASS configuration, the collocation optimization is enabled for the files used in the reduce stage, for the remaining files the locality optimization is enabled.\n\n\\emph{Evaluation results.} Similar to the pipeline benchmark, predictions for the reduce benchmark are within 20\\% of the actual performance (Figure \\ref{fig:reduce-ram}) and, more importantly, they capture the relative improvements that pattern-specific data placement policies policy can bring. \nWe note that Figure \\ref{fig:reduce-large-accuracy} captures the behavior of a heterogeneous scenario: We used a faster machine with a larger RAMDisk to run the reduce stage since the RAMDisk of the typical machine in the testbed is too small. Despite heterogeneity, the predictor captures the system performance with accuracy similar to a homogeneous system.\n\nWhen the collocation and locality optimizations are not enabled, the challenge of capturing exactly the system behavior is similar to the pipeline case: capture the complex interactions among all machines in the system. When the specific data placement is enabled though, the challenge is different: there is a high contention created by having several clients writing to the same storage machine (the one that performs the reduce phase). Figure \\ref{fig:reduce-large-stage} shows the results per-stage for the two stages of the large workload.\n\n\\begin{figure}\n\\begin{center}\n\\mbox{\n\\subfigure[Medium]\n{\\includegraphics[width=0.4\\linewidth]{actual_predicted_medium_reduce_ram}\n\\label{fig:reduce-medium-accuracy}}\n\\subfigure[Large]\n{\\includegraphics[width=0.4\\linewidth]{actual_predicted_large_reduce_ram}\n\\label{fig:reduce-large-accuracy}}\n\\subfigure[Large per Stage for WASS]\n{\\includegraphics[width=0.2\\linewidth]{actual_predicted_stages_large_reduce_ram}\n\\label{fig:reduce-large-stage}}\n}\n\\caption{Actual and predicted performance for the reduce benchmark for the medium, large workloads, and per stage for large workload. Error bars show the standard deviation.}\n\\label{fig:reduce-ram}\n\\end{center}\n\\end{figure}\n\n\n\\textbf{Broadcast benchmark}. A single task produces a file that is consumed by multiple tasks. In this benchmark, 19 processes running in parallel on different machines consume a file that is created in earlier stage by one task. A possible optimization for this pattern is to create replicas of the file that will be consumed by several different tasks. Data sizes are indicated in Figure \\ref{fig:patterns}.\n\n\\emph{Evaluation results.} Figure \\ref{fig:broadcast-medium-accuracy} shows the results for broadcast pattern with medium workload with the WASS system configured with 1, 2, or 4 replicas (the large workload shows a similar trend and we omit it here). For this benchmark all predictions matched the actual results: predictions were inside the interval of mean of actual $\\pm$ standard deviation, just 1-2\\% difference from the mean. This experiment highlights an interesting case for the predictor. According to the structure of the pattern and the results reported in \\cite{Mosa-ccgrid:12} (admittedly on a slightly different setup), creating replicas would improve the performance of the broadcast pattern. The results, however, show that creating replicas does not really help here. This happens because striping to many machines already avoids the contention of a single node holding the file. So, although creating replicas can alleviate the number of accesses to a given machine (since chunks are read in sequence), this gain is not paid off by the overhead of creating a replica. More importantly in our context, this is another situation where the predictor captures the impact of different configurations, showing, in this case that they are equivalent and the user can stick with one replica and save storage space.\n\n\\begin{figure}[htb]\n\\centering\n\\includegraphics[width=0.4\\linewidth]{actual_predicted_medium_broadcast_ram} \n\\caption{Actual and predicted performance for broadcast benchmark and medium workload. The experiment uses the WASS system while varying the replication level. Error bars show the standard deviation.}\n\\label{fig:broadcast-medium-accuracy}\n\\end{figure}\n\n\\textit{\\textbf{Synthetic Benchmarks: Summary}.} The evaluation so far demonstrates good accuracy: our approach leads to errors of 6\\% on average, lower than 9\\% in 90\\% of the studied scenarios, and within 20\\% in the worst case. More importantly, the mechanism correctly differentiates between the different configurations and could support choosing the best configuration for each scenario in our evaluation. \n\n\\subsection{Supporting decisions for a real application}\n\nSection \\ref{sec:eval:synt} presents the evaluation for the predictor's ability to accurately predict \nthe turnaround time of synthetic benchmarks that are impacted by different optimizations enabled in the storage system. This section targets a more complex scenario where the user has to deal with a real application, allocation decisions, and the storage system configuration. To this end, the evaluation shows the predictor's ability to correctly guide the user or a search algorithm to the desired configuration, specifically focusing on two provisioning scenarios:\n\n\\begin{itemize}\n\\item{Scenario I} assumes that the user has full access to a fixed-size cluster, a common set-up in several university research labs. The question we answer is how should the system be partitioned between application and storage nodes and what will be the intermediate storage system configuration for best application performance? \n\\item{Scenario II} explores the provisioning problem with cost constraints (e.g., in HPC centers with limited user budget or cloud environments). The question to answer is: for a fixed application workload; what is the cost\/turnaround time trade-off space among the deployment options?\n\\end{itemize}\n \n\nWorkload. We explore these two scenarios with a real workflow application: BLAST \\cite{blast} a DNA search tool for finding similarities between DNA sequences. Figure 4 shows the BLAST workflow. Each node receives a set of DNA sequences as input (a file for each node) and all nodes search the same database file. The workload includes 200 search queries using the RefSeq database (total size of 1.67 GB). The input files and the intermediary files are stored in the intermediate storage system.\n\n\n\\begin{figure}[htb]\n\\centering\n\\includegraphics[width=0.7\\linewidth]{Blast_CAC_13} \n\\caption{The BLAST database (1.67GB) is used by all nodes that search in parallel different queries. We assume the database is already loaded in intermediate storage.}\n\\label{fig:cac_3}\n\\end{figure}\n\n\\textbf{Testbed.} Among the 20 machines of the testbed; one node coordinates BLAST tasks' execution and runs the storage system manager. The remaining nodes can either execute tasks from the workload or act as storage nodes. \n\n\\textbf{Experimental methodology.} The plots in this paper section reports the average of at least 20 runs.  We have obtained 9\n\n\\subsubsection{Scenario I: Configuring a fixed size cluster.} This scenario addresses the following question: Given a fixed size cluster, how should the nodes be partitioned between the application and the intermediate storage, and what should be the intermediate storage system configuration to yield highest application performance?\n\nFigure \\ref{fig:cac_2} shows the application execution time with different partitioning and storage system configurations. As a proof of concept and to limit the number of possible configurations, we experiment with varying only one system configuration knob: chunk size. We chose chunk size because it has the highest impact on the BLAST workflow, and to present results for a know not covered with synthetic benchmarks.\n\nFigure \\ref{fig:cac_2} shows: First, the difference between the different configurations is significant (up to 10x difference between the best and the worst configuration given the same chunk size. Second, the results show that the system achieves the fastest processing time with a partitioning of 14 application nodes and 5 storage nodes, and chunk size of 256KB (4x smaller than the default size) a non-obvious configuration beforehand. Finally, the experiment shows that the predictor accurately captures the system performance given changes in the allocated partition, and on the storage system configuration. Therefore, it shows that the predictor can correctly lead the search algorithm to the desired configuration.\n\n\\begin{figure*}[htb]\n\\centering\n\\includegraphics[width=0.9\\linewidth]{apptime_19nodes_lines_cut}\n\\caption{Application runtime (log-scale) for a fixed-size cluster of 20 nodes. X-axis represents number of nodes allocated for the application\/storage. The three plots represent runtime for different storage configurations (chunk sizes). }\n\\label{fig:cac_2}\n\\end{figure*}\n\n\\begin{figure*}[htb]\n\\centering\n\\includegraphics[width=0.9\\linewidth]{apptime_19nodes_lines_cut}\n\\caption{Allocation cost (total CPU time on the left Y axis) and application time (right Y axis) for fixed size clusters of 11, 17, and 20 nodes while varying the chunk size. X-axis represents number of nodes allocated to the application\/storage. The figures show the actual (lines) and the predicted (arrows) cost\/performance. Note: Log scale on Y-axis.}\n\\label{fig:cac_4}\n\\end{figure*}\n\n\n\\subsubsection{Scenario II: Provisioning in an elastic and metered environment.}\n\nThis scenario assumes an environment where users are charged (as the cumulative CPU-hours used) and where they have a more complex tradeoff between cost and time-to-solution to make. We aim to inform the user's provisioning decisions by revealing the details of this trade-off. Specifically, this scenario helps the user to answer the following question: What is the allocation size, and how should it be portioned and configured to best fit my requirements?\n\nFigure \\ref{fig:cac_4} shows the application execution time and allocation cost measured in total allocation time (number of nodes x allocation time) with different cluster sizes, different partitioning, and different chunk size. The figure shows that an allocation of 11 nodes, with partitioning of 8 application, 2 storage nodes, and chunk size of 256KB offers the lowest cost. However, the user can analyze Figure \\ref{fig:cac_4}  to verify that an option with an allocation of 20 nodes actually offers almost 2x higher performance at a similar cost (less than \n\n\\subsection{Time to search the space}. Finally, an important point to evaluate is time to predict the performance of a given scenario. In our preliminary implementation, predicting simulations take roughly 10x to 100x less time than actual execution and uses only one machine. Since the application uses up to 20 nodes for actual runs, this means the simulator consumes approximately 200x to 2000x fewer resources than actual executions, enabling a deeper exploration of the search space.\n\\section{Related Work}\n\\label{sec:related}\n\nThis section describes briefly past work on storage system performance prediction and how it differs from our work.\n\nPast work used model-driven analysis to estimate performance of storage systems. For instance, Ergastulum \\cite{ergastulum} targets centralized storage solution based on one enclosure to recommend an initial configuration of the system, and Hippodrome \\cite{hippodrome} relies on Ergastulum to improve the configuration based on online monitoring of the workload. By considering a distributed system, our solution handles more complex interaction among the system components and more configuration options.\n\nSimilar to this work, Thereska et al. \\cite{thereska:informed:06} proposed an predictor mechanism for a distributed storage system with a detailed model. To provide such information, they propose Stardust \\cite{stardust} a detailed monitoring information system that required changes to the storage system and kernel modules to add monitoring points. This approach enabled their predictor to achieve prediction within 18\\% of the actual predictions depending on the workload. Our approach have achieved similar accuracy on our target workload, however with a lightweight approach to seed the model.\n\nAn important difference to past work on storage systems performance is our focus on a whole workflow application and the potential interaction among the workflow's phases instead of the average performance for a batch (e.g., \\cite{ergastulum,hippodrome}) of operations, and of predicting performance of the system from the perspective of just one client \\cite{thereska:informed:06}. Additionally, our work targets the partitioning problem of splitting the nodes among application and intermediate storage.\n \nRecently, Elastisizer \\cite{elastizer} targeted a similar problem automating allocation choices for an entire application. This work, however, does not address the aspects of workflow applications or storage system configuration since it focuses on a different class of application: Map Reduce jobs.\n\nRecently, Z. Zhang et al. has proposed an approach to determine the storage bottleneck for a given many-tasks application (a class of workflow) based on a set of benchmarks and executions of the the application. The approach we propose enable a richer exploration of the system by a lower cost since the predictor is able to estimate performance of a scenario that add or reduce resources and change the configuration without requiring new runs of the application of the benchmarks.\n\\section{Summary and Discussion}\n\\label{sec:discussion}\n\nThis paper makes the case for a prediction mechanism to support automating provisioning choices for workflow applications. We focus on predicting the performance of workflow applications when running on top of an intermediate object-based storage system. We propose a solution based on a queue-based model with a number of attractive properties: a generic and uniform system model; supported by a simple system identification process that does not require specialized probes or system changes to perform the initial benchmarking; with a low runtime to obtain predictions; and, finally, with adequate accuracy for the cases we study. \n\nWe intend to expand this work in multiple directions: (i) explore a richer space of configuration parameters, (ii) evaluate the system using additional benchmarks and applications, (iii) enable different optimization functions \\cite{strunk-utility, wilkes2009}, and including adding energy models \\cite{erss2011}, and (iv) explore different optimization solvers to search the configuration space. \n\nThe discussion below aims to clarify our understanding of the limitations of our work and the lessons we have learned during this exercise so far.\n\n\\textbf{What are the main sources of inaccuracies?}\nCurrently, there are sources of inaccuracies at multiple levels: First, the model does not capture all the details of the storage system (e.g., support services like garbage collection or storage node heartbeats; the control paths are simplified to match what we believe generic object-based storage would do - while we know that a FUSE-based implementation would need more complex control paths; we model all control messages as having the same size) and the environment (e.g., contention at the network fabric level or scheduling). Second, we constrain and simplify system identification even further at the cost of additional accuracy loss. Third, we do not model the infrastructure in detail (e.g., we do not model the network protocols or the spinning disks). Finally, so far the application driver uses an idealized image of the workflow application (e.g., all pipelines are launched in the simulation exactly at the same time while in the experiments on real hardware coordination overheads make them slightly staggered). We believe the latter one is the main reason of current inaccuracies in the system and should be address by a richer workload description.\n\n\\textbf{What is the accuracy when the intermediate storage is deployed on spinning disks?}\nSo far, we have focused on predicting performance when intermediate storage is deployed over RAMDisks for two reasons: This is a common setup on large systems (as some, do not even have spinning disks) and the individual performance of RAMdisks is simpler to predict (the service time for spinning disks is history dependent due to cache behavior and position of disk head). The storage service we use does not model history-dependent behavior, thus we expect lower accuracy predictions when the system is deployed over spinning disks. (This can be fixed, by using a more sophisticated model of the storage service and it is part of our future work). \n\nWe have evaluated, however, how the current (unchanged) model performs when using spinning disks for the simple synthetic benchmarks used in the evaluation section. Figure \\ref{fig:reduce-hdd} shows the results for the reduce pattern when using the medium and large workloads. The key observation here is that, although prediction accuracy is lower, predictions are good enough to make the correct choice between DSS and WASS, that is, the choice on whether to use the data co-placement optimization for each of the workloads (note that this optimization is beneficial in one case and it detrimental in the other one). Pipeline benchmark on HDDs shows similar results to RAMdisks (Figure \\ref{fig:pipeline-medium-accuracy}).\n\n\\begin{figure}\n\\begin{center}\n\\mbox{\n{\\includegraphics[width=0.25\\linewidth]{actual_predicted_medium_hdd_reduce}\n\\label{fig:reduce-medium-accuracy-hdd}}\n{\\includegraphics[width=0.25\\linewidth]{actual_predicted_large_hdd_reduce}\n\\label{fig:reduce-large-accuracy-hdd}}\n}\n\\caption{Actual and predicted performance for the reduce benchmark on HDD - medium(left) and large (right) workload. The results for WASS is this experiment are not directly comparable to the corresponding ones for spinning disks as we use a slower node for data co-placement, keeping the system homogeneous.}\n\\label{fig:reduce-hdd}\n\\end{center}\n\\end{figure}\n\n\n\\textbf{Can the performance prediction mechanism support the \\textit{development process} of a storage system? Do you have specific experience with using the mechanism in this context? }\nOne of the lessons we have learned so far is the utility of the mechanism to support development of the storage system itself. Back of the envelope calculations are a common mechanism to evaluate expected performance bounds for a given system. The predictor takes this a step further and is useful in complex scenarios where back of the envelope estimates were intractable. Not only developers can use it to evaluate the potential gains of implementing a new complex optimizations or to study the impact of faster network and nodes, but also the mechanism can be used as a baseline to detect performance anomalies. \n\nMore concretely, we have encountered a number of situations where the predicted and actual performance differed significantly. In some cases these highlighted simplifications in the model or in our simulator. But, more importantly, there were cases that highlighted complex performance-related anomalies that were fixed in the storage system such as: non-trivial implementation problems (e.g., limited randomness in the data placement decisions that created an artificial bottleneck, or unreasonable locking overheads at the manager). Similarly, the prediction mechanism helped us revisit assumptions about the middleware stack the storage system is implemented over (e.g., we have discovered the significant impact of the TCP connection initiation timeout of 3s in some scenarios); and highlighted shortcomings of the seeding process or incorrect assumptions about the deployment platform (e.g., we were ignoring platform heterogeneity). \n\n\n\\textbf{How to decide when to stop increasing the level of detail in the model and the complexity of system identification?}\nWe aim to model only the key interactions between system components. Modeling all system subcomponents and all their interactions in detail would be too complex. Such complexity could improve prediction accuracy, but would have significant drawbacks: significantly more complex model (as complex as the actual storage system and the underlying environment (e.g., network protocols, operating system buffers, scheduling), complex seeding process, lower scalability, and loss of the model generality. Further, the improvement in accuracy may not add much value (e.g., when the prediction mechanism is used to decide between system configurations). \nWe followed a top-down approach: we started from a simple model and added more components or interactions' details until the accuracy of the all predictions was within 10\\% of actual performance (and the median error was within 5\\%) for the set of microbenchmarks. \n\\section*{Acknowledgements}\nWe would like to thank Emalayan for his collaboration in MosaStore development; Elizeu Santos-Neto and Abdullah Gharaibeh for their collaboration in keeping the quality of the code via code reviews.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"# HARVEST THE WIND\n\n_America's Journey to Jobs,  \nEnergy Independence,  \nand Climate Stability_\n\nPHILIP WARBURG\n\nBEACON PRESS, BOSTON\nCONTENTS\n\nIntroduction\n\nCHAPTER ONE _Cloud County Revival_\n\nCHAPTER TWO _Early Adopters_\n\nCHAPTER THREE _Rust Belt Renewables_\n\nCHAPTER FOUR _The Chinese Are Coming_\n\nCHAPTER FIVE _Working the Wind_\n\nCHAPTER SIX _The Path to Cleaner Energy_\n\nCHAPTER SEVEN _Birds and Bats_\n\nCHAPTER EIGHT _The Neighbors_\n\nCHAPTER NINE _Greening the Grid_\n\nEPILOGUE\n\nTables\n\nAcknowledgments\n\nNotes\n\nSelected Bibliography\n\nIndex\nIntroduction\n\nTHIS BOOK IS BORN OF HOPE and frustration. I was an idealistic teenager when millions of Americans gathered for the first Earth Day in April 1970. At public rallies, on college campuses, and in high schools like mine, people demanded an honest reckoning with how we were destroying our environment. It was a time of tumult, a time of questioning.\n\nThen came the Arab oil embargo in 1973, which created panic at the gas pumps and set Washington abuzz with debate about the need to wean America off foreign oil. After college, I joined the energy staff of Senator Charles \"Chuck\" Percy of Illinois, a former corporate CEO and a Republican who took a pragmatic, bipartisan approach to promoting energy conservation and fossil fuel alternatives. President Jimmy Carter was a willing ally, laying out an energy program aimed at providing 20 percent of America's energy needs from solar, wind, and other renewable resources by the year 2000. As far-fetched as that goal seemed at the time, Congress got behind a number of measures to support investments in renewable energy projects, including wind farms.\n\nMy own, as well as the nation's, hopes for a clean energy future were dashed in 1981, when Ronald Reagan took office and immediately set about dismantling the Carter blueprint. Federal funding for renewable energy was slashed as the new president gave preferential treatment to coal, gas, oil, and nuclear development. Wind energy had begun to take hold in California, but new projects came to a near-halt by the mid-1980s\u2014about the time that Reagan symbolized his disdain for renewable energy by having Carter-era solar panels ripped off the White House roof.\n\nThe global momentum behind wind power shifted to Europe during the 1990s, with Denmark, Germany, and Spain leading the way. That decade was also a period when climate change moved out of scholarly publications and into the public arena. It was fast becoming clear that curbing our dependence on fossil fuels was an environmental and humanitarian imperative, not just a matter of national security.\n\nAfter working on environmental and human rights issues in the Middle East for many years, I returned to my native New England in 2003 to serve as president of the Conservation Law Foundation, an environmental advocacy group concerned with energy policy and climate change. From that new vantage point, I was struck by how little progress America had made toward Jimmy Carter's long-forgotten renewable energy target. Wind energy that year accounted for a tenth of a percent of America's power generation, and solar energy even less. Though George W. Bush had supported wind energy as the governor of his home state of Texas, as president he sent an unambiguous signal to the nation and the world: America was unwilling to lead, or even be part of, a serious global effort to curb greenhouse gas emissions.\n\nStymied by inaction in Washington, my colleagues and I pushed for saner energy policies and practices in New England. While we helped launch the nation's first multistate effort to curb carbon emissions via a \"cap-and-trade\" regime called the Regional Greenhouse Gas Initiative, we had a much tougher time moving renewable energy forward on a scale that would make a difference. In advocating for America's first offshore wind farm, Cape Wind, to be built near the Massachusetts coast, we found ourselves at loggerheads with an extremely well-funded opposition group that was determined to block any encroachment on their backers' ocean views. What made matters worse, Senator Ted Kennedy\u2014long a champion of renewable energy\u2014had come out against Cape Wind because it was to be sited about five miles from his family's Hyannis Port vacation compound. If the likes of Senator Kennedy were so unwilling to look beyond their own backyard interests, I wondered how and where America could make wind energy happen.\n\nIn 2009, I set out to explore this question. My journey began on the Kansas prairie, in half-forgotten Cloud County, where old-timers have long watched their children and grandchildren leave family farms and ranches in search of more lucrative and perhaps less arduous work elsewhere. A large commercial wind farm went into operation there in December 2008, its giant turbines generating enough electricity for 55,000 Kansas and Missouri households. To understand what wind power has brought to this struggling corner of rural America, I spent time getting to know the landowners who have rented out sections of their fields and pastures, the educators who have reshaped the local community college as a training center for wind energy technicians, and the developers who worked with community partners to move the Meridian Way Wind Farm from concept to completion.\n\nI also traveled to Rust Belt towns of the Midwest, where I learned about the thousands of new jobs that the wind industry has created for young and old alike, in retooled factories that once made car parts, printing presses, and construction cranes. And at wind farm sites across the heartland, truckers and construction crews expressed their excitement at being part of an emerging industry with a promising future, and newly trained technicians described the rigors of working atop 300-foot towers in hundred-degree heat and subzero cold.\n\nWind is not just a new presence on the American working landscape. It is also an expanding force in the global marketplace. To gain a better grasp of wind's broader market dynamics, I traveled to Denmark, which is proud to be producing one fifth of its power from wind. This small Scandinavian nation is also home to the world's top-ranked turbine supplier, Vestas Wind Systems, which made the machines that now spin out Cloud County's wind power. My next stop was China, vast and ambitious, outstripping all other nations in its wind energy use. Not surprisingly, the Asian giant is also emerging as a major rival to Europe and the United States in its wind energy manufacturing.\n\nToday, America gets about 3 percent of its electricity from wind\u2014hardly enough to get excited about. Yet, in raw terms, our wind energy resource\u2014like our solar potential\u2014vastly outstrips our current and future power needs. The U.S. Department of Energy conservatively predicts that, by 2030, wind energy could provide 20 percent of America's electricity. Other experts see wind generating more than half our nation's electricity by mid-century, so long as we invest in new transmission lines that will open up some of the remote areas where our wind resources are greatest, and so long as we develop a \"smart grid\" that can match the variable flow of wind-generated power to consumers' needs.\n\nLike any other energy technology, wind power has its downsides. Birds and bats may be killed by turbines, and the habitats of closely watched species like the sage grouse, the prairie chicken, and the Indiana bat may be at risk in some areas. Wind farms also can pose problems for people living nearby. Some object to the visual presence of towering turbines; others complain about the noise. At places as far-flung as Vermont's Taconic Mountains and the Flint Hills of Kansas, the tensions surrounding wind energy have caused bitter and lasting divisions among neighbors and former friends. Wind power's future success depends on our addressing these issues responsibly.\n\nAs I made my way across vast stretches of a country that I previously only knew through books and the media, I often thought of the advice that Franklin D. Roosevelt gave to a headstrong Wall Street banker who had just resigned from his Brain Trust, disillusioned with the New Deal's spending and monetary policies. \"Please get yourself an obviously secondhand Ford car,\" the president counseled. \"[P]ut on your oldest clothes and start west for the Pacific Coast, undertaking beforehand not to speak on the entire trip with any banker or business executive (except gas station owners), and to put up at no hotel where you have to pay more than $1.50 a night.\" The banker\u2014my father\u2014apparently felt no need to follow FDR's advice. I think he missed out on a lot.\n\nAlthough the motels where I stayed charged considerably more than $1.50 a night, the cross section of Americans whom I met\u2014farmers, ranchers, factory workers, shop owners, truckers, crane operators, and more\u2014deepened my appreciation and respect for the people who inhabit so much of this land. I also came to understand much more fully the ways that America's push toward renewable energy has changed people's lives for the better. Tens of thousands of Americans have found new jobs in the factory and field. Thousands of others have seen their flagging farm incomes boosted by generous, multi-year lease payments from wind developers.\n\nYet these people see more in our nation's wind energy investment than a pay stub or a rent check. The devastating consequences of climate change may still seem remote, but the people involved in our wind energy sector are fiercely proud of their role in advancing the nation's quest for energy independence. Too many of their family members, friends, and neighbors have been sent overseas to fight wars that they see, at least in part, as a struggle for control over waning energy resources. Too many loved ones have come back from those wars defeated in spirit, physically maimed, or in body bags. Too much taxpayer money has been spent fighting distant wars with no clear purpose\u2014funds that would have been far better spent creating jobs and new industries here at home.\n\nInvesting in wind energy will not bring an end to Middle East strife, but it _can_ begin to wean our nation off the fossil and nuclear fuels that we have come to associate far too closely with American prosperity. Now is the time to reshape our energy economy. With wind, we can tap an inexhaustible domestic energy resource while showing that America finally is willing to join\u2014and even lead\u2014the battle for climate stability.\n\nTurbine basics: The Vestas V90 3-megawatt turbines shown here are part of the Meridian Way Wind Farm in Cloud County, Kansas. At a well-sited wind farm, 1 megawatt of installed capacity produces enough electricity for roughly 270 average U.S. households. Graphic by Jason Fairchild of the Truesdale Group\/Recycled Paper Printing Inc.; photo by Philip Warburg.\n\n## CHAPTER ONE\n\n## Cloud County Revival\n\nWHEN THE U.S. POSTAL SERVICE commemorated a century and a half of Kansas statehood, those familiar with the state found it easy to fathom the stamp's chosen graphic: an array of windmills against a backdrop of broad blue sky. Even the state's name is drawn from the wind, or more precisely from an Indian tribe that honored the winds sweeping up into the region from the Gulf of Mexico. They were called _Kansa_ , or \"people of the South Wind.\"\n\nThe wind is at times a frighteningly destructive force in Kansas life. Killer tornadoes have leveled entire towns and have made tornado chasing a precarious summer sport for daredevils seeking grim moments of media attention. Yet it also offers a vast and largely untapped source of clean, renewable energy. According to a recent government survey, Kansas could install sufficient wind power to supply almost 90 percent of America's total present-day power consumption. Wind energy investments in the Sunflower State today have barely scratched the surface of that vast potential. About 7 percent of the state's electricity came from wind in 2010, with coal providing most of the remainder.\n\nThe story of putting the wind to good use could be told with slight variations throughout the Midwest and Great Plains states. As far back as the 1850s, Kansans recognized its value as an economic resource. Water-pumping windmills sprouted alongside farmhouses and in cattle pastures, greatly easing the burdens of frontier homesteaders. Initially made of wood, these early machines were easily repaired by self-sufficient farmers and ranchers. By the turn of the twentieth century, steel began to edge out older wooden designs and the maintenance burden gradually shifted to full-time itinerant wind millers. Some wind machines bore whimsical names like Albion, Climax, Eclipse, and Eureka; others had mechanical monikers like the Axtell Ever-Oiled and the I. X. L. Steel. Manufacturers were scattered across the Midwest and Great Plains states, and dozens based their operations in Kansas.\n\nWater-pumping windmills were widely used in Kansas until the late 1940s, when the electric grid finally reached rural sections of the state. Even then, many farmers continued to rely on wind to draw water up out of the ground; their windmills were already in place and, with minor tinkering, offered a free source of power. The remains of these machines still dot the Kansas prairie today. I encountered many of them in my travels across the state: spindly steel-frame obelisks, some standing bladeless, others with a few ragged slices of metal groaning and rattling in the breeze.\n\nA second wave of windmills, or \"wind chargers,\" brought a thin stream of electric power to rural Kansas in the 1920s. Hooked up to a car battery, these crude machines with their two-blade wooden rotors generated just enough juice to let farmers run a radio\u2014a much-valued novelty at the time\u2014or a few light bulbs. Older Kansans recall how these devices brought the wider world into their lives by opening up the airwaves. Larger wind chargers were also marketed at the time, one of them luring housewives with the promise that with homegrown electricity running their appliances, ironing would become a pleasure and cleaning a joy, but their price\u2014in the hundreds of dollars\u2014was beyond the reach of most farm families. In any case, wired-in electricity to most parts of the state made household-scale wind power obsolete by the late 1940s.\n\nOver the next half-century, cheap electricity from large conventional power plants kept renewable energy at bay in Kansas, as through much of the nation. Kansas largely missed out on the turbulent years of commercial wind energy experimentation that swept through California in the 1970s and early 1980s. Only after the turn of the millennium did wind energy find new life in Kansas. Spurred on by a property tax exemption adopted in 1999 and further bolstered by federal tax credits for renewable energy-based power, smart developers grasped the wind's enormous untapped potential and began to exploit its commercial possibilities.\n\nThe state's first large commercial wind facility, the Gray County Wind Energy Center, went on-line in 2001. Its \"installed capacity\"\u2014the maximum amount of power it can generate at any moment\u2014is 112 megawatts, or 112 million watts. No power plant operates at full capacity on a continuous basis, however. This is especially true for wind farms, which depend on a power source that can vary from hour to hour, day to day, and season to season. The Gray County Wind Energy Center operates, on average, at about 41 percent of its installed capacity\u2014higher than most other wind farms. At that rate, it churns out enough electricity to meet the needs of more than 32,000 Kansas households.\n\nThe Gray County project may have been the first big wind farm in the Sunflower State, but others soon followed. One of them was the Meridian Way Wind Farm, sited on a gently rolling stretch of prairie in Cloud County, just about a hundred miles north of Wichita on Interstate 81. The sixty-seven giant turbines at Meridian Way were dramatic newcomers to a predominantly horizontal landscape. Today they are a transformative force in a community that has been in steady decline ever since mechanization overran the local farm economy decades ago.\n\nKurt Kocher is the fourth generation in his family to raise cattle and till the soil on the family homestead, about nine miles south of Cloud County's sleepy administrative seat, Concordia. He fears he may be the last. \"There are simpler ways of making a living than farming,\" he says, pointing to the trailer where his son Kenton\u2014a high school junior\u2014is sweeping out crop waste. \"He thinks it's forced servitude because it's spring break this week!\"\n\nWhether Kenton will carry on the family farming tradition is an open question. \"He doesn't know what he wants to do,\" his father grumbles as he adjusts the faded red baseball cap that never leaves his close-cropped, graying head. He knows his son will have plenty of company if he ends up leaving the farm. Over the past half-century, young people have streamed away from Cloud County and other rural communities like it. Some search for steadier work, others want less physically demanding jobs, while still others are fleeing the isolation and tedium of life on the farm. It's not that farming is on the wane. Overall acreage under the plough across Kansas has actually held steady for decades. Yet today, fewer hands are needed to prepare the fields and harvest the wheat, soybeans, sorghum, and sunflowers that have long been the area's staples.\n\nFarms may have gotten bigger, with fewer people tending them, but not much else has changed in Cloud County for a very long time. Not much, that is, until the U.S. subsidiary of a Portuguese electric utility targeted the county for wind development. Nine of Meridian Way's wind turbines now soar above the Kochers' cattle pastures and grain fields, bringing in rent for the placement of each turbine, additional rent for the land where a transformer station now stands, payments for access roads and power-line easements, and a royalty on every kilowatt generated by the project. Kurt didn't talk numbers, but I could surmise from what I knew about landowner compensation elsewhere that the Kochers' wind-generated income must be closing in on six digits.\n\nThe income stream from wind may or may not help keep Kenton Kocher and his younger brother, Tyler, on the family farm, but it certainly brightens Kurt's outlook on the future. Crops can fail, cattle prices can plummet, but the wind blows steadily across the Kansas prairie, and Meridian Way's developers have promised to tap it for at least a quarter-century to come.\n\nKurt is, above all else, a practical man. You can see it in the soil stains that are ground deeply into his jeans. And you can hear it in the unsentimental way he talks about his prized herd of 150 Red and Black Angus cattle. \"These won't be hamburger,\" he tells me as he chases four young calves out of a trailer, setting them out to pasture for the summer. By early fall, they will be sent to a commercial feedlot, and by early winter, the more \"efficient\" among them will be turned into prime cuts of beef little more than a year after their birth. \"They're not an endangered species,\" he assures me.\n\nKurt regards Meridian Way's wind turbines in the same unadorned, pragmatic way he thinks about his Angus cattle and his fleet of farm machinery. A slight smile emerges from beneath his graying goatee as we look out over a large pasture where a half-dozen turbines face due south into the wind. \"For our business, we like to know which way the wind is blowing if we're applying weed killer or something along those lines,\" he says. He thinks of the wind turbines as very large weathervanes. Each turbine takes up no more than half an acre of ground, so he is free to continue tending his fields and grazing his cattle below them.\n\nHelen Kocher, like her son, is little bothered by the turbines. Her eyesight may be dimmed by cataracts, but her hearing remains sharp enough to register the swishing sound of the turning blades, which she says is especially noticeable in the evening, when all else is quiet. It reminds her of waves brushing up against a ship\u2014an unfamiliar sound on the Kansas prairie, but one she has encountered on her travels over the years. \"I don't know if it's the speed of the wind or the direction, but you get some pretty big waves sometimes!\" Usually, though, she finds the sound just blends in with the sounds of working life on a bustling prairie farm. The steady flow of funds from the wind farm's developers makes it easier for the Kochers to put up with a few minor annoyances here and there.\n\nRay Mason is another landowner who has made an easy adjustment to wind energy's arrival in Cloud County. Now in his early sixties, Ray works part time managing the American Legion men's club in Concordia. It was about ten o'clock on a Thursday morning when I arrived at the Legion's club hall, a building clad with sky-blue aluminum siding just off West Sixth Street, the spine of Concordia's four-block downtown business district. Toward the back of a dimly lit barroom, acrid from years of accumulated tobacco smoke, a few older men in checkered shirts and blue jeans sat playing cards at a small table. They looked up briefly as I entered. Ray, who stood beside them, gave me a warm smile before offering me his firm farmer's handshake. He then led me around the bar to a game room, slightly better lit, where we pulled two chairs up to an empty poker table.\n\nBorn in Concordia, Ray grew up on a farm just east of town and worked as a farmer and rancher until a few years ago, when he decided he needed a rest. He first encountered Meridian Way's developers back in 2003, and it quickly became clear that about five hundred acres of the family's upland pasture were good prospects for wind turbines. When he was offered several dollars an acre for a five-year option to develop wind on the land, the whole project seemed so far-fetched that Ray doubted it would ever happen. He spoke with a few neighbors, though, and together, they agreed to get on board. Then they waited.\n\nA few years into the option period, Ray began to get itchy. Next time the \"wind boys\" came to town, he laid the bait. \"I've been waiting on you guys for so long,\" he told them. \"Hell, I'm just gonna sell that damn ground down there. I'm not gonna mess around any longer.\"\n\nHis ploy worked. \"One younger guy got out his map, and I showed him where I was. He folded the map and looked at me, and he said, 'Don't you sell that land to anybody unless you sell it to me!' So I knew right then that we was gonna get at least one turbine.\"\n\nIn fact, Ray ended up with five. \"I was probably as shocked as anybody,\" he told me. Of the five turbines slated for his property, two were on 120 acres of marginal grazing land that his father bought for only $25 an acre back in 1957. \"It was a rough old farm, you know, with rocks kinda close to the top of the ground and short grass, over-grazed.\" Ray shook his head, and his eyes welled with tears as he thought of how pleased his father would be if he knew that rugged patch of land would one day bring in $20,000 a year, most of it from wind. \"He wouldn't believe it,\" Ray sighed.\n\nCombining his per-turbine earnings with other payments for access roads, Ray's annual cash flow from the wind farm must be close to $50,000, a welcome subsidy that will eventually benefit his children and\u2014if the wind farm stays in service long enough\u2014his grandchildren. Ray's family is a strong advertisement for community continuity, defying the trends driving younger people out of Cloud County and into areas with richer job prospects. All of his children live in Concordia, where his son works in a local metal shop and one of his daughters is a registered nurse.\n\nWhile he can't force his children or grandchildren to work the land, Ray is doing what he can to keep them tied to the family homestead, as landlords if not as farmers. \"I seen my folks work for that sucker pretty hard. I worked pretty hard too, so I don't want someone to just piss it away.\" He recently created a trust to maintain his family's ties to the land, at least for another generation or two. \"I got it set up so they can never mortgage the land or sell it,\" he says. \"So it'll pass on to my kids, and then to my grandkids. Then I quit.\"\n\nMeanwhile, Ray is happy to have the extra income that the wind farm is sending his way. \"Two months from now, I'll be sixty-two, and it fits in good to my retirement, y'know.\"\n\nThe man responsible for lining up the land deals for Meridian Way was Jim Roberts. An Oklahoma native, Jim was first dispatched to Cloud County in the spring of 2003 by Selim Zilkha and his son Michael, co-owners of Zilkha Renewable Energy in Houston. The Zilkhas had already proven themselves as highly successful energy entrepreneurs. When they sold their oil and gas holdings in the Gulf of Mexico for a billion dollars in 1998, the acreage they held reportedly surpassed all other prospectors operating on the Gulf's continental shelf. Their smart use of 3-D seismic data had yielded a drilling strike rate that was double the industry average. And now the Zilkhas were looking to get into wind.\n\nBefore joining up with the Zilkhas in 2001, Jim had worked for many years in the oil and gas industry. There he honed the diverse and subtle deal-making skills that are essential to negotiating leases for energy projects. His wife and family live in Edmond, Oklahoma, but Jim spends much of his time on the road. As soon as the land needs for one project are nailed down, he's on to the next, living out of motel rooms and trailers and eating most of his meals far from the family dinner table. He loves his job, though, and is happy to be promoting an alternative to the fossil fuels that he helped suck out of the ground for so many years. \"I believe in the need for clean, renewable energy,\" he told me as he began to recount his role in moving the Meridian Way Wind Farm from dream to reality. \"I'm very proud about my work in this industry.\"\n\nThe first wind project that Jim lined up for Zilkha Renewable Energy was the Blue Canyon project, now the largest in his home state. The company then sent him to scope out prospects for a wind farm in the Flint Hills of Kansas. To Jim and other wind developers, the Flint Hills were alluring. Located in the eastern part of the state, they are relatively close to Kansas City and other population centers. Just as important, winds in the Flint Hills are the strongest in the state. But Jim and his Zilkha colleagues were in for a beating. An alliance of ranchers and conservationists soon made it very clear that wind turbines were a bad match for one of the state's rare remaining expanses of untrammeled tallgrass prairie. Audubon of Kansas stood firmly against any wind farms in the area, and it was joined by state leaders of another leading conservation organization, the Nature Conservancy.\n\nJim's normally soft Oklahoma drawl took on a sharper edge when speaking about Flint Hills anti-wind organizers, whom he pegged as NIMBY, or \"not-in-my-backyard\" activists. One of those opponents was a professor of English at nearby Emporia State University, Jim Hoy. Professor Hoy is an expert in Great Plains folklore, with a special passion for the history of Flint Hills cowboys. When one wind supporter likened the proposed wind turbines to ballerinas, Hoy derided the idea of \"scores of 350-foot-tall ballerinas desecrating the native prairie grass that has covered the Flint Hills since the last Ice Age.\" In January 2004, as public pressures mounted to keep wind farms out of the area, Governor Kathleen Sebelius created a Wind and Prairie Task Force to try to resolve the controversy. While other wind developers put their project plans in neutral and waited for the task force's recommendations, the Zilkha team decided to look elsewhere for a community more welcoming to the company's ambitions. It didn't take long to find friendlier terrain about a hundred miles northwest of the Flint Hills.\n\nCloud County seemed like a great fit from the start. First, there were the southerly winds that the Kansa tribe had immortalized. Jim Roberts knew those winds would get an extra boost from the updraft off a long, south-facing ridgeline. He also knew that the area's higher elevation, about 1,600 feet above sea level, would give the winds yet a further push. Even before he set up equipment to do more precise measurements, it was clear that the area's climate and topography were in near-perfect alignment.\n\nThe land was favorable, too. Well over a century of continuous farming had transformed virtually all of Cloud County into a working landscape. Where the hills were too rugged for row crops, multiple generations of cattle had been set to graze. Greater prairie chickens\u2014grassland birds whose habitat has been decimated throughout most of the Midwest\u2014were still sighted occasionally, but Jim felt confident that measures to protect the breeding and roosting grounds for these birds could be developed.\n\nEvery bit as encouraging as the land and the winds were the people. The locals struck Jim as \"salt of the earth\"\u2014plainspoken and straightforward, with a no-nonsense agrarian pragmatism and a hands-off attitude toward government regulation. Unlike many other Kansas counties, Cloud County has no rural zoning, giving landowners a relatively free hand in deciding what to allow on their property. \"From the time we got there, we knew we were going to be accepted,\" he recalls.\n\nDespite his initial enthusiasm, Jim proceeded with the canniness and caution that are the stock of the land man's trade. \"Stealth\" is the word he uses to characterize his first steps in scoping out a possible wind farm site in the county. Having just retreated from the Flint Hills melee, he wanted to be sure he had core supporters among local landowners before going public with the project. He was also wary of competitors, one of whom had already put up a meteorological tower to collect more precise wind-quality data. \"We're not going to go in there with flags flying and whistles and alarms going off because it's a very competitive industry,\" he acknowledges.\n\nDescribing his next steps, Jim reveals no sources and names no names. \"We first made contact with a landowner,\" he tells me. \"We had strategically selected this person because of the land mass that this gentleman had, and we did some research and found he was kind of a ringleader.\" From that initial contact, the circles of consultation grew\u2014quietly and selectively. The landowner whom Jim had targeted for his opening gambit soon brought in just the catalyst Jim needed for the next stage: Kirk Lowell, a farmer-turned-business-booster who heads up CloudCorp, the county's economic development corporation. Kirk had already identified wind as a major economic growth prospect for the county, so he was more than willing to help out. A wind energy steering committee checked out Zilkha's credentials at Kirk's behest. When word came back that the company was legit, Jim knew it was time to reach out to other landowners.\n\nJim called a meeting for invited landowners only. He laid out a preliminary proposal for developing a wind farm on agricultural land about eight miles south of Concordia. In the first stage, he explained, participating property holders would be paid to give Zilkha a multi-year exclusive option to explore wind energy prospects on their land. If all went well, smaller parcels would later be leased for wind turbines, access roads, underground cables, and other infrastructure associated with the farm.\n\nJim teamed up with another Zilkha land agent. Working with a rapidly growing circle of landowners, the two of them assembled a huge swath of land for possible development: 22,000 acres in all. In six weeks' time, they had seventy landowners on board.\n\nThe Zilkha crew then geared up for the next stage of development: working their way through a dizzying array of studies and permit applications. Meteorological towers were erected to get a more accurate read on local wind patterns and climate conditions. The Federal Aviation Administration needed to sign off on the project, ensuring that the turbines would not endanger planes taking off and landing at Concordia's airstrip. The Department of Defense had to be satisfied that its radar installations in the area would continue to function unimpeded. The Federal Communications Commission was involved in protecting local radio airwaves.\n\nOn top of those issues, there was a lot of work to be done to protect the local environment. A joint project was launched with state conservation groups to set aside land for the mating and roosting of greater prairie chickens; protocols were prepared to protect the area's dung beetle population; and measures were developed to preserve low-lying wetlands in compliance with state and federal law.\n\nJim knew it was time to name the baby. A four-lane divided highway may seem an improbable inspiration for the naming of a forward-looking renewable energy project, but Interstate Highway 81 was exactly that. The highway runs right through the project area; its precursor was a dirt road that tracked just west of the Sixth Principal Meridian, the north-south map line used by government surveyors to lay out U.S. territories under the Kansas-Nebraska Act of 1854. That old thoroughfare was a lifeline connecting Cloud County farmers and ranchers to southern markets in Salina, Wichita, and beyond\u2014much the same role that Highway 81 serves today. People called it Meridian Road.\n\n\"Meridian Way\" was the name Jim and his team gave to the wind farm. In crafting the project's logo, artists were asked to link the excitement and promise of twenty-first-century technology to the aura of this historic road. They came up with a simple green graphic showing a highway with a dotted median converging on two tall wind turbines in the distance.\n\nWith the project named and the permitting process well under way, most of the pieces were falling into place. Before the wind farm could be built, though, two major challenges remained: finding a guaranteed buyer for the wind farm's electricity and raising the capital to build it.\n\nThe day was fast approaching when Zilkha would have to put real money on the table to buy and install the turbines and other equipment needed to turn nature's gift into a marketable commodity. In the old days, farmers might spend a few dozen to a few hundred dollars to buy a windmill. Today's giant wind turbines are in a different league entirely. Meridian Way ended up costing the developers $340 million\u2014an average of $1.7 million per installed megawatt for the 201-megawatt wind farm.\n\nFacing that anticipated financial burden while carrying the costs of several other wind farms, the Zilkhas decided they needed deeper pockets to carry the company's mission forward. In March 2005, they sold a controlling interest in Zilkha Renewable Energy to the then-robust global investment banking house, Goldman Sachs, which changed the company's name to Horizon Wind Energy. Only two years later, Goldman flipped its investment, selling the company\u2014valued at $2.2 billion\u2014to Energias de Portugal (EDP), the leading electric utility in Portugal and a huge player in the Iberian peninsula's natural gas sector. EDP, in turn, assigned Horizon to its renewable energy division, headquartered in Madrid.\n\nWith the substantial backing of Goldman Sachs and then of EDP Renewables, Horizon was able to take the next crucial step in advancing the Meridian Way project: finding a long-term buyer for the wind farm's electricity. Converting wind energy to electricity was one thing\u2014that was Horizon's job. Carrying the electricity to the households and businesses that needed it was a whole different story. For that, Horizon needed buy-in from a utility, ideally one with transmission lines already in the area. Lining up a power purchase agreement is the final threshold that shifts a wind farm's developers from paper-pushing to earth-moving.\n\nThe quest for a power purchaser led to a surprise development. In the fall of 2007, just as Empire District Electric, a Missouri-based utility, declared its willingness to buy 105 megawatts of Meridian Way's power, a second prospective energy buyer emerged: Westar, the largest electricity distributor in Kansas. Jim Roberts jumped at the challenge, shifting into super-high speed to assemble a second land package that could double Meridian Way's output.\n\nOnce again, Jim needed a ringleader who could help him work with landowners to put together another large, contiguous land parcel. Kurt Kocher was his man. Kurt commanded the respect among his peers that was vital to Jim's success. \"He kind of became the trumpeter of our praises,\" Jim recalls. \"He farms and ranches responsibly. He's a hard worker. He's got a good operation. He's well-known. . . . You like to get those people in your project up front.\" Jim pauses. \"I'm kinda giving away a little of our strategy here, but it's a no-brainer for any wind company. You're going to try to get your best landowners first.\"\n\nWithin just a few months, Jim's alliances with Kurt and others landed Horizon just what it needed to bring Westar into the deal. Wind development rights on another 12,000 acres were now in the company's hands.\n\nBy December 2007, Horizon had all the important preliminaries in place: the wind, the permits, more than a hundred local landowners on board, and commitments from not one but two power purchasers. The company now geared up for the final phase of project development: building the wind farm itself. Horizon wanted the wind farm up and running by the end of 2008, which was essential if it was to tap a 30 percent federal production tax credit that was due to expire on December 31 of that year.\n\nEnter Carole Engelder, a highly skilled, no-nonsense project manager who exemplifies many of the workers who have recently joined this frontier industry. A former employee of Amoco, BP, and a chemical startup, Carole came with a quarter-century of experience overseeing engineering, logistics, and procurement for major projects in the more traditional oil and chemical sectors. She had no wind industry background before joining Horizon in 2007, and she was unmoved by any high-minded idealism about freeing America from the stranglehold of foreign oil or the ravages of climate change. In brisk conversation, she sums up the attraction that wind held for her: \"It's fast-paced, it's cutting-edge, it's new in this country. . . . The fact that it's wind, I have to admit, doesn't have anything to do with it.\" What drew Carole to wind was the adrenaline rush of moving a new industry forward.\n\nMoving this particular project forward meant getting sixty-seven wind turbines in place and hooked up to the grid by Christmas 2008. Given the sheer size of these behemoths, this was no small feat. Horizon chose to use three-megawatt turbines manufactured by Vestas Wind Systems, a Danish company that is well established as a world leader in wind energy technology. The Vestas V90 has a three-bladed rotor whose diameter is the length of a football field. Each tower, transported in four segments, is nearly 260 feet tall and weighs 344,000 pounds. It is topped by a hefty mechanical chamber, or nacelle, weighing 154,000 pounds and containing the turbine's gearbox, generator, transformer, and control systems. With blades spinning, the V90 reaches 410 feet in the air\u2014just about the height of a forty-story building.\n\nFitting together the many pieces of the Meridian Way puzzle in just ten months required what Carole describes as \"choreography.\" Like a stage manager scrambling before the curtain goes up, she and on-site project manager Alvin Cargill had to make many big decisions and a myriad of smaller ones. They needed to recruit a full complement of contractors and subcontractors, some three hundred workers in all, each with their specialized trade, all of them demanding careful sitewide coordination. They needed precise locations for every tower, transformer, and power distribution line. They needed a long list of crucial components delivered to each turbine site. Access roads were a challenge in themselves. Some had to be planned and surfaced from scratch. Many of Cloud County's existing roads had to be widened and reinforced with concrete to withstand the weight of the massive, specially engineered semitrailers with their giant cargo of towers, gearbox assemblies, blades, and transformers.\n\nThe work required not only precision but flexibility. If a weather report showed an approaching thunderstorm, the dozen people poised to lift a multi-ton turbine tower segment into place would have to set down tools and stand clear of their 300-foot-high crane. With no grounding, this machine is its own lightning rod. \"Perfect planning isn't about planning the perfect project,\" Carole tells me. \"It's about perfect contingency planning.\"\n\nWith so much happening in a few short months, disruptions to the daily routines of local farmers and ranchers were inevitable. Skilled at orchestrating the logistics of a complicated and fast-paced construction job, Carole was less patient in dealing with disgruntled neighbors. \"Gnats,\" she calls them.\n\n\"N-A-T-S?\" I echo, thinking this is a wind industry acronym.\n\n\" _G\u2013n\u2013a\u2013t\u2013s_ ,\" Carole spells out. \"Annoyances. Alvin takes care of them so I don't have to hear about it.\"\n\nOne of the \"gnats\" Jim Roberts and Alvin Cargill had to contend with was Bonnie Sporer, a landowner who signed up to allow wind towers on her property. Bonnie, a seasoned rancher in her eighties, is a canny businesswoman. She's a Daughter of the American Revolution, proud of it, and equally proud to be the descendant of a Civil War doctor who brought his family to Cloud County in a covered wagon shortly after the Union victory. Farming and crop management have run in Bonnie's family for generations. Her family has farmed rich bottomland in the Republican River valley, just north of Concordia, for as long as anyone can remember. She grew up on that farm and continued to manage it as an adult, together with other family members. She also spent many years working for the U.S. Department of Agriculture, running the government's crop insurance program in Cloud County and points west\u2014the largest crop insurance program in Kansas, she boasts. She served, too, as the first woman on the county's Soil Conservation Board, and she made sure I knew it.\n\nBonnie began her relationship with the wind farm developers by driving a hard bargain with Jim Roberts and his land team back in 2003. \"They asked if I was interested in leasing some land,\" she says. \"I made them take all of it.\" That amounted to 1,800 acres of ranchland, for which the company paid her several dollars per acre for the option to tap the available wind. Once the wind farm's layout was finalized, she says she received close to $80,000 for an easement to accommodate underground electric cables, plus payments of over $16,000 a year for two wind turbines sited on her property. On top of that, she is paid rent for access roads to the turbines that have been erected on her land.\n\nJim won't discuss the specifics of Horizon's payments to Meridian Way landowners, though he allows that Bonnie can sometimes be a little loose with the details. \"One minute you'll be having a conversation with her, telling her that it looks like she'll have five wind turbines on her property. (He stresses that this is a hypothetical number.) A half-hour later, someone calls and says, 'We didn't know Bonnie was getting ten turbines!' \"\n\nI first encountered Bonnie when she strode into Meridian Way's field office on a warm afternoon in May 2009, on my second visit to Cloud County. She wore rubber boots up to the knee and left a boisterous border collie barking outside the door. Bursting with energy, she reeled off a long list of complaints with the way the construction process had unfolded. In one instance, a construction worker left one of her cattle gates open, she said, and her entire herd had wandered out before anyone noticed. Then those out-of-town greenhorns went racing around in their pickup trucks, trying to round up the creatures as they wandered off in all directions. \"That never works!\" she exclaimed, shaking her arms in the air for effect. \"It only panics the animals.\" One of the cows\u2014Number 41, she specifically recalls\u2014was never found.\n\nBonnie recalls sharing dreams with other family members that their unyielding ancestral terrain might surprise them in the future. \"We always laughed that someday we'd get lucky and hit oil or something.\" When Meridian Way's turbines started turning in December 2008, just ten short months after the first foundations were poured, Bonnie knew that, for all the aggravations, this was as close to a gusher as she was likely to see.\n\nLandowners, without a doubt, are the primary beneficiaries of Meridian Way, but there are other gains that reach the broader Cloud County community. During those intensive months of building the farm, about half of the 300 onsite workers were locals. If the farm doubles in size as planned, there will be similar opportunities for local workers, including truck drivers, metal welders, electricians, crane operators, and road construction crews. With all those workers hungry by noontime, local eateries and grocery stores can count on a substantial boost. In 2008, retail sales in Concordia\u2014and therefore local sales tax revenues\u2014jumped a full 24 percent.\n\nMeridian Way has also spawned a cutting-edge training program for wind energy technicians at Cloud County Community College. Bruce Graham, the program's creator and main driving force, has suddenly brought this little-known, two-year college into the limelight as one of the nation's leading pathways to a burgeoning, twenty-first-century industry. Raised on a Cloud County dairy farm, Bruce\u2014a former high school science teacher\u2014finds himself scrambling to keep pace with a program that, in just a few years, has grown from only four registered students to more than a hundred job-hungry enrollees. Until the recession hit, he had a hard time keeping students enrolled long enough to finish their two-year training before they got snapped up by wind developers. Even after the economic downturn cut the rate of new wind development by nearly half in 2010, Bruce still was able to place most of his students in entry-level jobs in the industry.\n\nAlong with teaching classes and hustling to raise funds for his program, Bruce finds himself in constant demand as a public lecturer. On an evening when I visited the program's makeshift classrooms in rented space at a Concordia shopping mall, he entertained a room full of aging, tattooed visitors from the Kansas-Nebraska Radio Club. A busload of Nebraska farmers had visited the school the previous week, wanting to know how they could bring wind energy technology and jobs to their own communities.\n\nBruce's wife, Michelle, is Meridian Way's administrative coordinator. Along with managing contracts and dealing with daily operational issues, she\u2014like her husband\u2014hosts a steady stream of curious wind farm visitors: church groups, schoolchildren, and more than the occasional journalist. Up on all the details, she fields questions deftly and conveys a contagious enthusiasm for the project.\n\nLooking at what Meridian Way means to the Cloud County economy, Kirk Lowell is every bit as bullish about wind as the Grahams. I met the Cloud County economic development director at Cloud-Corp's modest storefront headquarters, just a few doors away from Ray Mason's American Legion Post. As he ushered me into his office, it became obvious immediately that I was in the presence of a true believer. His polo shirt sported the Meridian Way logo. On his desk was a scale model of the Vestas V90, just like the turbines now operating at Meridian Way. And on his wall was a caricature of Grant Wood's _American Gothic_ farm couple, minus the pitchfork, with two tall wind turbines towering in the background.\n\nKirk knew that I was visiting from the East, and he wanted to make sure I understood the roll-up-your-sleeves spirit that pervades Cloud County. \"We have a very common-sense kind of people here. We know that we have to do something as a nation about our energy situation.\" He looked me straight in the eye. \"You can't just talk about it. It bothers people in the Midwest when you say, 'We need to do this, this, and this. _I'm_ not gonna do it, but we expect _you_ to do it.' \"\n\nThough he was too polite to mention it, I suspected that Kirk was alluding to the raging controversy back East over the long-stymied offshore Cape Wind project on Nantucket Sound. What happens to all those enlightened East Coast idealists, he must have been wondering, when tough choices have to be made about where to build new wind farms? His message was clear: \"Somebody has to step up to the plate and host these things.\"\n\nKirk's commitment to wind is rooted less in concern about climate change than in a weariness with wars fought over oil, thousands of miles away from the orderly calm of his home territory. His logic is elementary: \"Either we are going to have to put wind turbines _on_ our Kansas prairie or we're going to continue to put our fine young men and women _under_ it.\"\n\nCloud County is populated by more than the usual complement of hardened realists, but the county has its share of optimists as well. Wind energy alone may not be enough to reinvigorate Cloud County and places like it across rural America. That will take the concerted efforts of many people pulling together, reshaping and rebuilding their communities. There is no doubt, though, that Meridian Way has helped give this small slice of the American heartland a new pride of place and a belief in better things to come.\n\n## CHAPTER TWO\n\n## Early Adopters\n\nIT WAS HARDLY A SURPRISE that the giant wind machines at Meridian Way came from Denmark. Vestas Wind Systems has long been a wind energy technology leader, with a supply chain reaching back to America's first big wave of wind power development in the 1980s. Denmark has also distinguished itself as a global pioneer in generating its own electricity from wind. A fifth of the country's power today comes from wind, and plans are afoot to raise that contribution to as much as 80 percent of the country's power supply\u2014all part of a broader government plan to free Denmark from fossil fuels and reduce its greenhouse gas emissions by 80 to 95 percent by 2050.\n\nI traveled to Denmark to get a close look at wind turbine production. From Copenhagen I drove to the village of Lem, set on a flat stretch of farmland near the western rim of the Jutland peninsula. This is where Vestas got its start and where some of the company's manufacturing still takes place.\n\nLone Mortensen, director of people and culture in the Vestas blades division, met me at the entrance to a neat campus of white factory buildings. Appearing wholesome and informal in her slacks and Argyll sweater, Lone (pronounced \"LO-neh\") had recently returned from a two-year stint in Portland, Oregon, where she helped ramp up the company's North American communications. Despite both countries' refusal to sign on to any greenhouse gas reductions, the United States today ranks second only to China in both its annual installation of wind turbines and its cumulative wind power capacity (see tables 1 and 2). Responding to this market demand, Vestas has invested heavily in building new factories and assembly plants in those two countries. The massive scale of turbine equipment makes it important to keep transportation distances to a minimum. In addition to reducing costs, closer proximity cuts down on the greenhouse gas emissions associated with freight haulage\u2014something this environmentally attuned company monitors closely.\n\nLone's English is excellent, but she chooses her words slowly and carefully. She is wary of outsiders wanting to know details about the company's closely guarded manufacturing processes.\n\nAs we set out on our tour, Lone tells me about the company's founder, a blacksmith named H. S. Hansen who began making farm equipment in the village in 1898. Those early days are commemorated by a quaint outdoor cluster of iron figures depicting workers in the old Lem Forge, just a few blocks from the village center, with its handful of small shops sprinkled among modest red-brick homes.\n\nBy the end of World War II, H. S. Hansen's son Peder had taken over the business, naming it VEstjyskST\u00e5lteknik A\/S (fortuitously shortened to Vestas) and shifting the company's focus to the manufacture of farm trailers, ship engine intercoolers, and hydraulic cranes. In 1978, just when I migrated from college to Capitol Hill to promote renewable energy, Vestas built its first experimental wind turbines. After trying a few designs, the company's engineers decided on a three-blade turbine that went into production the following year. Eight years later, the company made wind energy systems its single focus, and it soon grew to be the world's top-ranked turbine manufacturer.\n\nIn 2004, Vestas acquired its prime Danish competitor, NEG Micon, and the consolidated company supplied 34 percent of the 8 gigawatts (8 billion watts) of new wind power capacity installed globally that year. Over the next six years, worldwide demand for wind turbines skyrocketed, reaching 40 gigawatts by 2010. Vestas remained number one in the trade, but its market share dropped to about 12 percent of all new installations. Still, the company brought in $9.2 billion in revenues that year and added nearly 3,000 people to its workforce\u2014healthy signs of strength coming out of the global recession.\n\nLone informs me that Vestas operates factories for blades, towers, and other wind turbine components at more than two dozen locations across Europe, Asia, and North America. Her job is to handle staff communications for the company's eight blade plants worldwide, fostering a sense of staff camaraderie across cultures and time zones from Colorado to China. Now and then, the company hosts planetary parties using satellite-beamed video-conference technology to connect workers celebrating at different plant locations around the world.\n\nRock music blares in the background as we step out onto the factory floor, our feet in company-issued red plastic clogs, our heads crowned by clean white hard hats, and our eyes shielded by safety glasses. This vast room, as brightly lit as the music is loud, is shaped like an enormous shoebox, nearly as long as a football field and about a third as wide. Lone first points to a machine that looks like an industrial-scale rotisserie. Workers clad in white protective clothing carefully feed long sheets of epoxy-saturated fiberglass around this squared-off steel mandrel, gradually expanding its girth. Layer upon layer of fiberglass is added, reinforced by wavy black veins of carbon fiber. Once heated, dried, and hardened, the resulting obelisk\u2014more than a hundred feet long\u2014will become the structural mainstay of a blade that is expected to withstand the forces of nature for two decades or more. In the trade, this structural spine is called the _spar_.\n\nLone directs me to the opposite end of the production room, where another team of workers is painting a protective sealant onto the inside surfaces of a giant double-mold that looks like an opened, elongated clamshell. Once the sealant has dried, epoxy-soaked fiberglass will be layered on each side of the mold, forming the shape of the blade's two halves. The bivalve will then be closed around the spar, an epoxy compound will be injected, and the parts will be fused into a completed blade.\n\nThe making of wind blades is a curious mix of high-tech engineering and traditional craftsmanship, demanding the sophistication and subtlety of aeronautic design along with the patient attention to detail found among builders of high-end yachts. When I ask how long it takes to produce a single blade, Lone is visibly uncomfortable. \"That's the type of information our competitors want to know,\" she explains. It's clear, though, that progress is measured in hours, even days. Producing 6,000 to 10,000 blades annually at plants around the world, the company's simultaneous demand for prodigious output and painstaking precision is daunting.\n\nA high-pitched, scratchy whine greets us on entering the factory's finishing department, where a team of masked workers uses handheld sanders to massage the creamy outer surfaces of each giant blade. Meticulously they shape the razorlike edges that will slice through the wind as efficiently and quietly as advanced aerodynamics allow. After all the surfaces are smoothed and painted, a tractor pulls each completed blade, mounted on a rubber-wheeled dolly, out into the factory yard. There a forklift gingerly stacks the blades, three and four high, in neatly ordered frames like bottled wine shelved in a vintner's cellar, waiting their turn to be shipped to customers.\n\nThough Danish innovators built a few dozen power-generating wind turbines in the early twentieth century, the seeds of Denmark's modern wind industry were planted by the 1973 oil crisis. Denmark at the time was more than 90 percent dependent on oil for its energy needs, and virtually all that oil was imported. To punish Western Europe and the United States for their support of Israel during the Yom Kippur War, Arab oil producers radically reduced the flow of oil to those countries, causing long gas station lines and tripling the price of oil. Denmark's political leaders recognized the need for decisive action, but initially the government focused more on fuel-switching than on energy efficiency or renewable energy development. The nation's first comprehensive energy plan, released in 1976, emphasized coal, natural gas, and nuclear power as the primary pathways to reduced dependence on foreign oil. In the years that followed, North Sea explorations yielded substantial supplies of oil and natural gas\u2014enough to make Denmark a substantial net exporter of both resources. No nuclear plants were built, however, and in 1985, responding to broad popular opposition, the Danish parliament officially banned their construction.\n\nThe same was not true for coal, which quickly gained a foothold as the primary fuel for the country's power plants. Like pre-embargo oil, nearly all of Denmark's coal was imported, but it came from a more reliable and diverse array of nations, including South Africa, Russia, Colombia, Australia, and the United States. Denmark's heavy reliance on coal continues to this day, despite an official ban on its use in new power plants as of 1997 and a commitment to phase out all coal-generated electricity by 2030. The pervasive reuse of waste heat from power production has substantially improved the overall efficiency of coal use in Denmark, with 60 percent of homes now kept warm during the long Scandinavian winters by a vast network of steam pipes running from the country's power plants. Nevertheless, the energy sector remains the main contributor to Denmark's greenhouse gas emissions, and within that sector, coal is the overwhelming culprit.\n\nWind grew more gradually to its current 20 percent share of Denmark's power supply. The same year that the government released its 1976 energy plan, Danish academicians produced their own, emphasizing the need for a major shift toward renewable energy. Two years later, a test lab and certification facility for wind turbines was established at Ris\u00f8 National Laboratory, and the year after that, the Danish parliament adopted a new law that gave a big boost to wind turbine ownership: wind power producers of any scale would be entitled to an up-front government grant amounting to 30 percent of the purchase price of the turbines, for all models approved by the Ris\u00f8 laboratory.\n\nMaking it easier to buy turbines was important, but two other steps were needed for wind energy to take hold. First, turbine owners had to be assured that the power generated by their wind machines would find its way onto Denmark's electric grid. With the government eager to encourage new wind projects, state-owned transmission companies agreed to cover 35 percent of the costs of connecting windmills to the grid. Second, turbine owners needed to be able to sell the power generated by their windmills at a reasonable profit. Here too, the major state-owned utilities settled on a formula that guaranteed wind power producers substantially more than utilities were paying for conventional coal-fired power.\n\nWith these incentives in place, tens of thousands of Danish land owners decided to enter the wind business. Some planted single, grid-connected turbines on their properties. Others joined local cooperatives, or _vindm\u00f8llelaugene_ , that pitched in together to build small arrays of turbines. Local ownership of wind power was seen as a virtue, by the landowners themselves as well as their political supporters in Copenhagen. When nonlocal speculators rushed in to take advantage of the public subsidies, the government restricted its grants to people living within 10 kilometers (6.2 miles) of their windmills or in the same municipality.\n\nDenmark's utility leaders soon became nervous as they watched windmills sprout across the countryside. To protect their large, central-station power plants, they successfully lobbied for the removal of the investment grants to wind developers. In 1989, the grants were abolished, but wind investors continued to enjoy guaranteed rates for the electricity they generated. They also benefited from a refund of the tax on carbon emissions that the Danish government charged to conventional electricity providers. With post-tax rates of return ranging from 5 to 22 percent per year, wind energy remained an attractive investment, and the number of investors continued to grow. According to one account, 50,000 Danes held ownership stakes in windmills in 1991\u2014nearly 10 percent of the country's population. By 1999, 150,000 Danes were reported to be full or partial owners of wind turbines, and during that period, more than 80 percent of the country's wind turbine capacity was locally owned.\n\nAppealing though it was for Danish landowners to invest in wind power, government planners quickly recognized the importance of building an export market for the country's emerging wind turbine industry. In a country with a smaller population than Massachusetts and little more than twice its physical size, it was clear that wind turbine producers would need to reach buyers outside Denmark's borders if they were to thrive. In 1982, the Ministry of Industry commissioned a report that looked at America as the next Danish wind energy frontier. The report triggered an immediate response among the country's leading turbine companies. According to one historian, sales representatives from Vestas and three other Danish turbine manufacturers were on a plane to the United States within a few days of the report's release, eager to test out their prospects.\n\nCalifornia was the prime target for Danish wind companies. A volatile combination of idealistic fervor and entrepreneurial bravado had spawned a new generation of California wind developers who were eager to cash in on some remarkably enticing state and federal investment incentives. As in Denmark, the 1973 oil embargo had sent shockwaves across America. Drivers long accustomed to cheap and plentiful gas suddenly found themselves lining up for hours outside filling stations. \"Rationing,\" a term that hadn't crossed most Americans' lips since World War II, reentered the nation's vocabulary. And America's political leadership awakened to the folly of depending on imported oil to meet the nation's energy needs.\n\nRichard Nixon, beleaguered by ongoing investigations into the Watergate scandal, grabbed hold of the energy crisis as a defining moment in his waning political career. On January 23, 1974, he delivered a Special Message to the Congress on the Energy Crisis, in which he outlined a lengthy slate of measures to bolster U.S. energy self-reliance. Along with creating a provisional framework for gas rationing, he called for expanded surface mining of U.S. coal, new mineral leases on federal lands, expanded offshore oil and gas production, and the accelerated licensing and construction of nuclear power plants. Research and development into \"new\" technologies was another feature of his program, though \"solar electric power\" was the only renewable technology mentioned, as a prospect \"for the far term,\" at the end of a long list of more immediate priorities focusing on fossil and nuclear fuels.\n\nThe historic nature of Nixon's energy agenda was made perfectly clear in his State of the Union address a week later: \"In all of the 186 State of the Union messages delivered from this place,\" he said to the packed House chamber, \"this is the first in which the one priority, the first priority, is energy.\" Under the rubric \"Project Independence,\" he set a high bar for immediate progress in weaning America off of foreign oil: \"Let this be our national goal,\" he said, barely glancing up from his text, with signature sweat beading on his chin and upper lip. \"At the end of this decade, in the year 1980, the United States will not be dependent on any other country for the energy we need to provide our jobs, to heat our homes, and to keep our transportation moving.\"\n\nThat goal was far from met. In 1973, 28 percent of our oil came from abroad: 6.3 million barrels a day out of a total U.S. daily consumption of 17.3 million barrels. Today we use substantially more oil than we did at the time of the embargo (19.1 million barrels a day in 2010), and we import about _60_ percent of it.\n\nWhile the 1973 embargo and its aftershocks failed to lessen America's oil appetite over the long run, it did open the door to a new level of public and private commitment to renewable energy. In the early days of Jimmy Carter's one-term presidency, energy independence emerged as the dominant theme. Just two weeks after taking office in the winter of 1977, President Carter appeared before the U.S. public in a televised address that came to be known as the \"sweater speech.\" Bundled in a beige cardigan and sitting by a glowing White House fireplace, the president called for a major commitment to energy conservation. In terms more befitting a pastor than a U.S. president, he decried Americans' wasteful ways and exhorted the public to \"make modest sacrifices\" and \"live thriftily.\" He also signaled the need for new energy technologies like solar power, but that message was largely lost on American TV viewers, who came away feeling stunned by the president's call for a new age of austerity in which thermostats would be kept at a cool 65 degrees during the day and an even chillier 55 degrees at night.\n\nIn his second televised energy address in April 1977, Carter struck a more conventional pose behind his desk in the Oval Office, but again he adopted a sermonizing tone. \"Tonight,\" he opened, \"I want to have an unpleasant talk with you about a problem unprecedented in our history.\" He went on to discuss the need to \"balance our demand for energy with our rapidly shrinking resources,\" and he cautioned that his proposed policies would \"test the character of the American people\" and be \"the 'moral equivalent of war,' except that we will be uniting our efforts to build and not destroy.\"\n\nJimmy Carter may not have won the hearts of the American public with his grim admonitions, but his administration's policies created a wholly new federal climate for advancing renewable energy. For Carter, as for Denmark's political leadership of the 1970s, coal was a big part of the solution to foreign oil dependence; he saw U.S. coal as \"our most abundant energy resource\" and advocated for its expanded use, along with the carefully supervised use of nuclear power. At the same time, he saw a real role for alternative energy resources including the sun, in which he invested major symbolic value. In 1977, he called for solar energy to be used in more than 2.5 million American homes, and two years later he led by example, installing solar water-heating panels on the White House roof.\n\nA further step that Carter took to advance solar energy was the creation of the Solar Energy Research Institute, a new national laboratory with Denis Hayes at its helm. Charismatic and articulate, Hayes had been the driving force behind the first Earth Day in 1970, and had more recently orchestrated the presidentially proclaimed Sun Day in May 1978. Hayes brought a broad definition of solar energy to the new national laboratory, including \"anything that uses sunlight within a few decades of the time it arrived at the surface of the planet.\" In his view, this included wind because of the sun's crucial role in creating the thermal gradients that draw air at different speeds and in different directions across the planet. As a practical matter, though, the research supported by his institute focused primarily on a narrower realm of solar thermal and solar electric technology.\n\nWhile solar energy received greater fanfare, Carter's policies brought enormous benefits to the wind industry as well. Most consequential was the Public Utility Regulatory Policies Act of 1978 (PURPA), which gave small-scale power producers a way to compete in a market that had, for decades, been monopolized by large public utilities. Wind farms and other renewable energy-based power plants were PURPA's primary targets, along with small \"cogeneration\" facilities that made use of the waste heat created from power production. Public utilities were obligated to buy the electricity generated by independent generators at a price that was \"just and reasonable\" to consumers and, at the same time, did not discriminate against this new breed of power providers.\n\nPURPA predictably triggered a backlash by a number of public utilities that were reluctant to open up their markets to competition. Some states also resented the federal government's encroachment onto turf traditionally dominated by their own public utility commissions. Two court challenges to PURPA's reach into state-held territory went all the way to the U.S. Supreme Court, but the Court ruled in PURPA's favor in both cases. This cleared the way for a new generation of daring and often idealistic energy entrepreneurs.\n\nJust as PURPA guaranteed wind developers a market for their electricity, a powerful blend of tax incentives drew investors into the field. Under the Energy Tax Act of 1978, wind energy projects enjoyed a double benefit: they could claim a business investment tax credit of ten percent, and they were on a short list of special energy projects that qualified for a second ten-percent tax credit for capital outlays. Then, in 1980, the Crude Oil Windfall Profits Tax Act increased the energy portion of the tax credit to 15 percent for solar, wind, and geothermal investments, bringing the total federal credit available to wind farmers to 25 percent. What's more, commercial wind farms\u2014like many other energy companies\u2014benefited from federal tax provisions that let them depreciate their equipment on an accelerated basis.\n\nIf Jimmy Carter was the stern preacher waging a righteous war for American energy independence, California governor Jerry Brown was the smart and savvy iconoclast, surrounding himself with creative people who were willing to challenge conventional thinking on everything from religion and social relationships to the environment. Thirty-six years old when he became governor for the first time in 1975, Brown later recalled his early days in office: \"This was the mid-1970s. It was the time of the Whole Earth Catalogue. I was dealing with people like Stewart Brand, Wendell Berry, Amory Lovins, Herman Kahn, and Dick Baker from the Zen Center. I mean, it was a hotbed of ideas. And there was a sense that we were on the threshold of a new politics. We were building something new. It was very exciting.\"\n\nAn eclectic and contradictory mix of community empowerment and space-age zeal inspired Jerry Brown's politics, leading a famously caustic newspaper columnist, Mike Royko, to brand him \"Governor Moonbeam.\" On one hand, Brown proposed that California launch its own communications satellite; on the other, he was an ardent proponent of backyard composters and rooftop solar panels. In the latter realm, he took particular inspiration from Amory Lovins, author of a widely read manifesto mapping out the social, economic, and political reasons to abandon our reliance on fossil fuels and centralized power generation.\n\nIn fact, Lovins's proposed paradigm for autonomous, community-based energy systems was much more radical than anything that Brown was able to advance as governor. Lovins was unequivocal in his condemnation of the status quo: \"Siting big energy systems pits central authority against local autonomy in an increasingly divisive and wasteful form of centrifugal politics,\" he wrote in his 1977 book _Soft Energy Paths_. \"In an electrical world, your lifeline comes not from an understandable neighborhood technology run by people you know who are at your own social level, but rather from an alien, remote, and perhaps humiliatingly uncontrollable technology run by a faraway, bureaucratized, technical elite who have probably never heard of you.\"\n\nSteps were certainly taken under Brown's leadership to help Californians recapture at least some of the power that they had drawn from distant, centralized generating plants for decades. A year after taking office, the governor turned to another energy mentor, Wilson Clark, and asked him to oversee the creation of California's Office of Appropriate Technology. Under Clark's guidance, this new agency sponsored demonstration projects and disseminated information on small-scale energy systems, along with other residential and community-based projects. The Brown administration also introduced hugely enticing tax incentives for residential renewable energy investments. Californians could claim 50 percent of the price of qualifying equipment as a personal income tax credit, up to a total of $3,000 per household.\n\nWhile thousands of homeowners used the residential tax credits to purchase rooftop solar water-heating panels, the state adopted a model for wind energy that strayed far from Lovins's idealized vision of grid-liberated local self-reliance. With new wind farms selling their power to big utilities like Southern California Edison and Pacific Gas & Electric, the state's renewable energy marketplace would soon be drawn back into the \"centrifugal politics\" of big energy systems that Lovins railed so passionately against.\n\nVarious state agencies under Jerry Brown were instrumental in helping this process along. Surveys conducted by the California Energy Commission identified sites for 13,000 megawatts of wind capacity in the state; half of these were viewed as ripe for near-term commercial development. Particularly promising were a number of the mountain passes where cool air blowing in from the Pacific accelerated as it flowed into warmer inland regions. The Altamont Pass, east of the Bay Area, was one such area. Two others were the Tehachapi Pass, southeast of Bakersfield, and San Gorgonio Pass, near Palm Springs, both drawing swift and sustained air currents from the west into the Mojave Desert.\n\nAlong with charting the winds, California offered a 25 percent tax credit to commercial wind investors, and it gave tax-free bonding authority of up to $10 million to \"alternative energy\" projects. California Democratic congressman Pete Stark minced no words in decrying these subsidies. \"These aren't wind farms,\" he grumbled. \"They're tax farms.\" Added to the federal tax incentives, the state enticements were enough to create what many have likened to the Gold Rush that had swept the state more than a century earlier. Paul Gipe, an early publicist for the wind industry, estimated that California's wind developers raised $2 billion from as many as 50,000 individual investors during the boom years of the early 1980s.\n\nAs wind investment capital flooded the California market, a scrappy collection of small, undercapitalized companies hustled to get their largely untested wares into the field. In 1981, wind developers at Tehachapi Pass installed 150 windmills; the following year they installed another 1,200 turbines; and in 1984, they topped out at 4,732 new units. By 1985, a grand total of 12,553 windmills were in place at California's three biggest wind energy complexes, and by 1987, more than 17,000 windmills had been installed statewide. The turbines came from more than a dozen manufacturers and ranged in size from a few tens of kilowatts up to 330 kilowatts of installed capacity per machine.\n\nWhile federal and state subsidies supercharged the wind farm investment climate, the government did little to make sure that turbines entering the market could actually perform as intended. There was no official rating system for turbines, and federal research-and-development (R&D) funding for wind technology singled out megawatt-plus prototypes\u2014equivalent in scale to today's commercial turbines but several times larger than the machines that were entering the market in the late 1970s and early 1980s. Much of the federal research funding went through NASA, which regarded wind energy as a promising new focus for a flagging aerospace industry and awarded plum contracts to corporate giants such as Boeing, McDonnell Douglas, Grumman Aerospace, Westinghouse, and General Electric (GE). According to one estimate, $350 million\u2014almost three-quarters of total U.S. R&D spending on wind between 1974 and 1992\u2014went to research on 1-, 2-, and 3-megawatt turbines, none of which made it to market.\n\nFledgling U.S. wind manufacturers learned their hard lessons in the field, experimenting wildly with a dizzying array of designs. Some had airplane-style propellers mounted on a horizontal shaft; others rotated around a vertical axis, their slender, convex metal blades resembling giant eggbeaters. One brand of turbine failed after only a few days of operation; it had been advertised as lasting a minimum of ten years. On other machines, gearboxes and generators failed, rotors spun out of control, blades cracked and collapsed, and towers buckled. In their rush to build windmills that could produce power on a commercial scale, some manufacturers simply relied on scaled-up versions of small residential wind generators; these often-flimsy machines were simply no match for the turbulent winds of California's mountain passes. As Dutch technology analyst Rinie van Est commented about this first generation of U.S. wind turbine manufacturers, \"they wanted to start dancing before they had learned to walk.\"\n\nIf U.S. turbines were inspired by the aerospace industry and its lofty ambitions, Danish manufacturers were tied to a tradition of rugged and enduring craftsmanship, firmly grounded in traditional agriculture and industry. American wind developers were ready to recognize the difference, with one of them pointedly observing that rockets and missiles, which played out their useful lives in no more than an hour, were a poor precedent for wind generators that needed to run reliably for years, if not decades. Just as Vestas could draw on decades of experience manufacturing farm trailers and cranes, another leading Danish company, Bonus, built irrigation systems, and a third, Nordtank, produced oil and water tanks.\n\nThe pragmatic spirit of these companies was reinforced by the Ris\u00f8 National Laboratory. Rather than focusing on speculative research into multi-megawatt prototypes, the lab devoted itself to testing and certifying much smaller market-ready designs. Its efforts converged with leading Danish manufacturers in favoring a single design that had demonstrated its reliability: a three-bladed rotor mounted on the upwind side of a supporting tower, with twin generators and a \"yawing\" system that mechanically adjusted the rotor to keep its blades facing into the wind. Experience gained operating turbines in Denmark strengthened the reputation of companies like Vestas as they pursued U.S. buyers. According to the Danish Wind Industry Association, some Vestas models had logged as many as 15,000 hours of operating time before being sold across the ocean. This compared very favorably to U.S. models, many of which found their way to California's wind farms after no more than 1,500 hours of operation.\n\nWith American turbines failing at an alarming rate in the early 1980s, the promise of greater reliability generated an American boom for Danish wind technology. In 1983, Danish turbines accounted for 11 percent of new installations in California; the following year they occupied 33 percent of the market, and by 1987, they had reached a staggering 90 percent of all new units. But even the relatively sturdy Danish machines had problems standing up to California's swift, blustery winds and dusty desert conditions, which placed much greater strain on turbines that had operated relatively well in Denmark's milder, steadier winds. Danish manufacturers were unfortunately slow to respond to these issues, in part due to a lack of timely information. Most California wind developers serviced their own equipment rather than relying on the original manufacturers, so company engineers back in Denmark had a hard time getting reliable feedback on issues as they arose.\n\nWhile the 1980s were turbulent times for Danish as well as U.S. wind turbine models, policy changes at the state and federal level played an even bigger role in breaking the frenetic pace of California's wind farm construction in the latter part of the decade. Since 1978, wind developers had benefited from a federal law that made it all but impossible to build new power plants primarily fueled by natural gas or petroleum. The Powerplant and Industrial Fuel Use Act of 1978 barred the use of these fuels for most new electric generation, with the goal of stimulating an increased reliance on coal, nuclear, and, to a lesser extent, renewables\u2014energy sources that were seen as less vulnerable to the fuel shortages that plagued U.S. industry and American consumers in the 1970s. By 1986, expanded domestic gas exploration, together with reduced demand, caused a market surplus and a drop in gas prices that led Congress to lift restrictions on the fuel's use for power production. Opening up the power market to natural gas made it much harder for wind and other renewable energy technologies to compete.\n\nThe other big problem facing wind developers and turbine manufacturers alike was the demise of both the federal and California tax credits that had been crucial to attracting wind farm investors. In the fall of 1985, Congress, with the full backing of the White House, refused to extend the federal investment tax credits for wind and certain other renewable technologies beyond December 31 of that year. The following spring, President Ronald Reagan ordered the removal from the White House roof of the solar panels that Jimmy Carter had in stalled seven years earlier. One of these panels is now on exhibit at the Smithsonian, a bittersweet symbol of two presidents' very different visions of America's energy needs.\n\nRonald Reagan, even before taking office in January 1981, declared open war on Jimmy Carter's energy policies, which he linked to our \"disintegrating economy\" and decried as \"based on the sharing of scarcity.\" Digging out more coal, drilling for more oil and gas, and building new nuclear plants became the cornerstones of Reagan's energy vision; advancing renewable energy did not really fit with this plan. Once elected, he canceled the federal tax credits for renewable energy projects and slashed federal funding for renewable energy R&D. From a high of $718.5 million during Carter's last year in office, research dollars for renewable energy dropped to $110.8 million by the end of Reagan's two-term presidency. Wind's share of the R&D budget dropped from $77.5 million to $8.7 million during the same period. The political momentum behind wind in Washington was gone.\n\nIn Sacramento, Republican governor George Deukmejian took office in 1983 with as much esteem for Jerry Brown's energy policies as Reagan had for Carter's. Stories about wind investment deals providing \"welfare for the wealthy\" and news footage showing broken-down wind turbines littering the California hills couldn't have helped. But even without the damning media reports, a statewide budget crisis gave Deukmejian the political cover he needed. In 1986, he rolled back the investment tax credit for commercial wind projects from 25 to 15 percent. The following year, he eliminated it entirely.\n\nEven after the plug was pulled on the state and federal tax credits, wind farm developers continued to install new turbines, thanks largely to contracts that had already been signed for the long-term supply of wind power to California's two big utilities, Pacific Gas & Electric and Southern California Edison. As these contractual obligations were met, however, new installations began a steep and steady decline. In 1985, nearly 400 megawatts of new wind power came on-line. In 1986, that number dropped to 275 megawatts; from there it plunged to 154 megawatts in 1987. New installations hit an all-time low in 1992, with new turbines adding up to just 19 megawatts of new capacity.\n\nAs new contracts for wind farms dried up, American and Danish turbine manufacturers alike fell into a deep slump. Their overreliance on a single market was taking its toll. Vestas had increased its workforce from 200 to 870 employees in 1982, largely to fulfill orders from a single large California wind developer. When the tax credits evaporated and petroleum prices plummeted in 1986, it was left with a huge inventory of turbines that it couldn't sell. That year Vestas and a Danish blade manufacturer declared bankruptcy. The next year Nordtank followed suit, and in 1988, four other Danish wind energy companies declared their insolvency. In the aftermath, a few firms, like Vestas, trimmed their staffs, reorganized, and survived; others disappeared.\n\nThe collapse of the California market could have driven Vestas out of the wind business. Instead, the company brought in new leadership and devoted itself to building new customer bases across Europe, Asia, the Pacific Rim, and other parts of the United States. In 1988, it won a bid to build six wind farms in India, financed by the Danish International Development Agency. A year later, it opened a subsidiary in Germany, and in the early 1990s, it expanded into new markets in Sweden, the United Kingdom, Australia, and New Zealand. In 1994, a joint venture with a local turbine producer, Gamesa, brought Vestas to Spain.\n\nGermany and Spain proved to be great strategic bets for Vestas and other wind energy entrepreneurs in the 1990s. In Germany, a law enacted in 1991 obligated utilities to pay solar and wind generators a fixed \"feed-in\" tariff amounting to 90 percent of the average consumer price for power. Along with enjoying a guaranteed price for their electricity, wind developers benefited from low-interest loans as well as planning guidelines that made it easier to site new projects. These incentives led to a dramatic surge in German wind power, from only 31 megawatts in 1990 to more than 6,000 megawatts of installed capacity a decade later\u2014nearly a 19,000 percent increase! This made Germany number one in wind power generation worldwide.\n\nSpain experienced similarly dramatic growth during the 1990s. From only 4 megawatts installed as of 1990, the country boosted its wind power capacity to 2,836 megawatts by the year 2000, placing it second only to Germany. As in Germany, this dramatic growth was triggered by a tariff that guaranteed Spanish wind producers a highly favorable rate for their electricity. Producers were paid the market price for power plus a premium that, in 2000, amounted to three euro cents per kilowatt hour.\n\nDuring the 1990s, Denmark continued to ramp up its own use of wind, stimulated by a succession of pro-wind government plans as well as a feed-in tariff that guaranteed wind producers an above-market rate for their electricity. This was also the decade when Denmark opened up a vast new horizon for renewable energy development: offshore wind farms. In 1991 and 1995, two pilot projects tested out the technology, with results so encouraging that the government, in its 1996 energy plan, set a target of building 4,000 megawatts of offshore wind power by 2030. By the year 2000, Danish wind power\u2014still mainly on land\u2014totaled 2,291 megawatts, ranking it fourth in the world, just behind the United States, which reached 2,539 megawatts of installed capacity that year.\n\nThe last leg of my Vestas tour in Denmark took me to the company's global headquarters, an attractive modern building a short distance from the city of Aarhus. There I met Peter Wenzel Kruse, the company's communications chief. Short in stature with a wiry build, he spoke in staccato phrases about his company's global positioning. He sees Denmark as a showcase for wind power, and in that spirit he worked hard to land Vestas a high-profile spot at the Copenhagen climate summit in December 2009. Visiting VIPs and reporters who attended the event could hardly miss the prominently labeled Vestas turbine that towered above the meeting's venue, a low-slung conference center on the outskirts of the capital city. Kruse said the exhibited turbine was \"a good gimmick,\" but he quickly added: \"We have no business in Denmark.\"\n\nDenmark may be ahead of all other nations in the _percentage_ of its power generated by wind. But in the _amount_ of electricity from wind, it barely earns a top-ten global ranking, lagging behind six other European nations plus India, the United States, and China (see tables 2 and 3). Even with the Danish government's plan to provide most of the country's power from wind, Kruse stresses that the local market for turbines will be minuscule relative to overall global demand. Denmark, after all, is a country with 5.5 million people in a world whose numbers already exceed 7 billion.\n\nAs one sure sign of the company's shift to global production and marketing, Vestas shut down four of its ten factories in Denmark in 2010, and laid off 2,000 Danish workers, bringing the company's total domestic labor force down to 5,600. The Lem blades plant survived, but layoffs hit that factory, too. In October of that year, Vestas CEO Ditlev Engel announced that the company would expand its U.S. workforce from 2,300 to 4,000 during 2011, filling slots in freshly built blade, tower, and nacelle factories in Colorado. By mid-2011, the company's American workforce had reached 3,100 and was expected to continue growing.\n\nIn all, wind turbine production has created about 20,000 new American factory jobs, and this number is expected to grow as turbine companies and their suppliers gear up to meet America's appetite for wind power. Wind power in America may have hit the doldrums following the California feeding frenzy of the 1980s, but its recent reemergence as a world wind energy leader has attracted vigorous competition among eager technology suppliers. Vestas is far from alone among foreign corporations that are seeking out U.S. buyers for their wind machines. Some of these businesses\u2014like Spain's Gamesa and India's Suzlon\u2014are hardly known outside the wind trade. Others are familiar global conglomerates like Siemens and Mitsubishi (see table 4). Whatever their corporate roots, most turbines sold in America are assembled on U.S. soil, with more than half of their components and subcomponents domestically made. Exploring this growing U.S. manufacturing base would be the next stop in my travels.\n\n## CHAPTER THREE\n\n## Rust Belt Renewables\n\nYOU DON'T HAVE TO BE an industrial giant to be a player in the American wind business. American Superconductor, a Massachusetts firm, markets and licenses state-of-the-art turbine designs to global manufacturers. In the Great Lakes region, about seventy firms are involved in manufacturing turbine components and subcomponents. One of them, just outside Cleveland, is a family-owned bolt-forging company called Cardinal Fastener. When Barack Obama made his pre-inaugural whistle-stop tour from Chicago to the nation's capital in January 2009, he made a point of visiting the Cardinal Fastener forge, where he talked up wind energy's importance in a spirited meeting with plant workers. In a video clip documenting the event, two muscle-bound workers firmly grip a yard-long piece of gunmetal-gray hardware. \"This is a 2\u00bc-inch heavy hex bolt,\" one of them says as he peers at the camera through his protective eyewear. \"Now it's called the 'Obama Bolt.' \"\n\nGeneral Electric supplies nearly half of America's wind turbines, but most of the equipment in its machines comes from smaller manufacturers scattered across the country and around the globe. Though GE has no real competition among U.S. turbine producers, one much smaller company, Clipper Wind, has worked valiantly to establish itself in the market. Clipper's road has been a rocky one, fraught with technology flaws and financial distress, but it has brought new jobs and new hope to hundreds of workers at its turbine assembly plant in Cedar Rapids, Iowa.\n\nSmall and sparsely populated, Iowa has captured a surprising number of new wind energy manufacturing jobs. Ranking thirtieth in the nation in population, the Hawkeye State has the fourth-largest workforce commitment to wind manufacturing, outflanked only by Texas, Illinois, and Colorado. When fully staffed, nine wind-dedicated factories scattered across cities and towns in southeastern Iowa will employ 2,300 workers. Some of these jobs are in foreign-owned workplaces like the blade factory that Siemens built in Fort Madison and the turbine assembly plant that Acciona, a Spanish company, now operates in West Branch. Others are in American-owned factories like the Trinity Structural Towers plant in Newton, Iowa, a major supplier of turbine towers to GE. Across a stretch of cornfields from the tower plant is a blade factory run by TPI Composites, a Rhode Island\u2013based firm. These two operations have filled at least part of the huge employment gap created when Maytag shut down its headquarters and appliance assembly plant in Newton in 2007. And then there are the many Iowa firms\u2014more than 200 of them\u2014that are part of a supply chain providing the 8,000 components and subcomponents that make up the modern-day wind turbine.\n\nClipper began assembling wind turbines in Cedar Rapids in 2006, and it has since brought approximately 350 jobs to a city that has seen one Rust Belt factory after another close down or move out of state. The company is a relatively small player, supplying 6 percent of the U.S. market in 2009 and only 1.4 percent in 2010, but it is emblematic of the hundreds of firms that are now contributing to the U.S. wind industry.\n\nWhen I traveled to Cedar Rapids in February 2010, Iowa's premier industrial city still looked badly beaten by the flood that had engulfed it two years earlier. The Cedar River's roiling waters had driven more than 8,000 people from their homes and had forced hundreds of city businesses to close, some forever. On my visit, I found entire streets of modest clapboard homes nailed shut, empty of life. If it weren't for the snow on the unshoveled sidewalks and pitched rooftops, this could have been post-Katrina New Orleans. A few doors down from the boarded-up Paramount Theatre, I read a spray-painted message on an abandoned store window: \"Bent, not broken.\"\n\nSited on the southern edge of the city, Clipper narrowly escaped the river's wrath. Veteran machinist Mark Meader told me the factory survived because it sits \"on the highlands,\" actually only a few feet above the waters that flooded out Clipper's nearest downhill neighbor, Casey's General Store. Just as Clipper managed to dodge the flood, Mark and his coworkers Mick Boots and Dave Wheatley have been spared the worst impacts of heavy industry's flight from Cedar Rapids. All three men, in their early sixties, were hired by Clipper within the past few years. They consider themselves lucky to have found jobs in their fields.\n\nWind energy may be a rapidly evolving twenty-first-century technology, but it has strong roots in heavy industry, in the kinds of trades that Mark, Mick, and Dave have plied for decades. All three worked many years at Goss Graphic Systems, manufacturing newspaper printing presses until that factory closed in 1999. Mark worked in the machine shop, Mick in assembly, and Dave in engineering. \"As a group, we were pretty much used to dealing with heavier, bigger, bulky items,\" Dave recalls. \"It's nice to see this kind of industry come back into the area.\"\n\nBob Loyd is the person responsible for bringing his former coworkers at Goss over to Clipper. He ran the assembly operations at Goss, and he's now the plant manager at Clipper. \"When we started off this plant, I literally picked up the phonebook and started calling people I knew,\" Bob recalls. \"I knew good mechanics and electricians, guys who know gears. If you think about a printing press, it's a big gearbox with hundreds of gears.\" He realized that wind turbine machinery isn't so different.\n\nAs he set about staffing the Clipper factory, Bob knew what he was looking for. He wanted a crew of seasoned industrial hands who could train a younger cohort coming from their first jobs or from courses in electronics, automobile mechanics, and engineering at Kirkwood Community College, just a mile or two up the road. Then, after five to ten years, the older guys could retire, having secured a next-generation workforce for this struggling Rust Belt city.\n\nBorn and raised in Wisconsin, Bob comes from a long line of mechanical and civil engineers. His great-grandfather had a bridge-building business; both grandfathers were engineers; and his father was the lead engineer at a company that built large diesel engines for U.S. Navy ships. College studies brought Bob to Iowa, where he started out at Iowa State and then transferred to the University of Iowa, earning an MBA and two engineering degrees. Despite his university credentials, he describes himself proudly as \"a wrench turner.\"\n\n\"We all have dirt under our fingernails,\" Bob tells me as he leans back from the piles of paper on his desk. His broad hands rest firmly on the molded plastic arms of his chair, and he smiles. \"I work on tractors, motorcycles, cars.\" His spartan office is just footsteps away from the vast factory floor, where workers assemble gearboxes, burnish cast-iron turbine hubs, and mount all manner of electronic equipment into the nacelles of Clipper's 2.5-megawatt Liberty turbine. The company is now developing larger machines for onshore and offshore wind farms, but the Liberty is Clipper's flagship.\n\nThe Liberty's Cedar Rapids birthplace, at 4601 Bowling Street SW, is a mammoth steel-frame shed whose history tracks the course of Iowa's rocky romance with heavy industry. It also holds important memories for Bob Loyd. The building opened in the mid-1960s, when Link-Belt Construction Equipment Company began manufacturing cranes and excavators at the site. Bob joined the company fresh out of college in 1973. By the mid-1980s, Link-Belt would be gone, bought out and moved south by a Japanese conglomerate, Sumitomo Heavy Industries. Bob later found himself back in the space as manager of the Goss printing press assembly plant. That company, too, did not last long in Cedar Rapids. Succumbing to fierce price wars with foreign competitors, Goss laid off hundreds of workers and eventually filed for bankruptcy.\n\nNext in was Maytag, which used the building as a warehouse for refrigerators. By 2006, Whirlpool had acquired the fading Maytag brand and soon announced that it would be shutting down the company's headquarters and most of its assembly operations in Iowa. About 1,800 local Maytag employees, from office workers to factory laborers, lost their jobs. Although some limited Whirlpool manufacturing remained in Iowa, there was no longer a need for warehousing at the Bowling Street facility.\n\nWhen Bob was hired by Clipper, he welcomed the chance to breathe some new life into the battle-scarred factory space that he knew so well, and he was even more excited to help a few hundred people find jobs once again in heavy industry in Iowa. With all the local factory closures, he knew plenty of people who had been forced out of the field. Dave Wheatley was one of them. A skilled mechanical engineer, Dave drove a FedEx truck, worked as a taxi driver, and built windows after the Goss debacle. Mark Meader found a job as a carpenter. Others migrated to food processing plants around town. Many of these dedicated industrial hands, Bob knew, would jump at the chance to return to building some real machines.\n\nFor Bob, Clipper's allure went further. He had worked for Rockwell and Raytheon, so he knew that \"heavy industry\" often involves a heavy dose of defense contracting. \"I wouldn't have been excited if we were making armaments for the military,\" he acknowledges. But when he talks about wind turbines and renewable energy, his voice becomes animated and his face brightens. \"This is a product the country needs . . . and it's a great thing for the state. It helps the farmers, and it helps kids get good jobs.\"\n\nNow nearing sixty, Bob has poured heart and soul into bringing youth and a hopeful future to the Clipper factory on Bowling Street. With obvious pride, he introduces me to four of the company's younger crew members. After seating us around a white laminate lunch table in the staff lounge, just off the factory supply room, he goes back out onto the factory floor, leaving us to talk. It's 3:30 in the afternoon and the room echoes with the pounding of the punch clock a few feet away. First-shift workers, mostly in their twenties and thirties, are streaming out of the building, and the second shift is coming in.\n\nMatt Lalley is one of Bob's next-generation hires. Boyish-looking and a little shy, he tells me about his work as second-in-command in the electrical department, where he troubleshoots issues that come up in the assembly of generators and turbine control systems. His work as a car mechanic helped prepare him for the job, and what he didn't learn in the car repair shop, he has picked up on the factory floor.\n\nJoel Peyton, twenty-seven, talks excitedly about his work as lead technician in the Remote Monitoring and Diagnostic Center, which tracks the performance of hundreds of Clipper turbines at wind farms stretching across the continent. He came to Clipper fresh out of college, where he studied management after training to be an electrician. Every day, he commutes forty miles each way from a farm where he and his father grow corn and beans. \"It's our hobby,\" he says of the farming, adding that his father's real job is in Clipper's gearbox assembly division. A concrete worker for many years, he followed his son to the company just a few months after Joel came on board in October 2006.\n\nMatt chimes in that _his_ dad works alongside Joel's dad in gearbox assembly. Then Mary Tiedeman tells me that she has an uncle and a cousin who work in the plant. Mary, also very young, is one of the few women I have seen. Trained in child care and family services, she works as a receptionist.\n\n\"For the record, my dad _doesn't_ work here!\" the fourth in our gathering pipes up. Everyone around the table breaks into laughter. Tyler Glass, a twenty-one-year-old fix-it guy and 3-D computer designer, has just finished his training in mechanical engineering at Kirkwood Community College. Cherubic and animated, Tyler verges on euphoria when he talks about his job. \"I love it\u2014I love coming to work every day. It might sound cheesy, but I wouldn't say I'm here just to get a paycheck,\" he says. \"I truly do believe in what we are doing.\"\n\nBob leads me through the gearbox assembly bay to a production zone where workers are busy installing electronic, mechanical, and hydraulic controls into Liberty nacelles. His people skills are obvious. Everyone gets a jovial, first-name greeting, followed by a question that shows Bob's awareness of exactly what that person is contributing to the Clipper effort.\n\nThe Clipper day begins at 7 a.m. with a shiftwide calisthenics class. In addition to staying fit, Bob wants people to feel part of the team. Some days he leads the exercises, but often he recruits others. \"We always pick on somebody,\" he explains in a gentle tone that makes it clear this is an act of inclusion, not punishment.\n\nReflecting more broadly on workplace morale, Bob describes a quick-turnaround electronic suggestion box that invites workers to air their concerns. He calls it \"VOICE,\" which is much easier to swallow than \"Valued Operational Improvement for Clipper Excellence.\" Each month, the worker with the best idea gets a gift certificate.\n\nOperational and design improvements are the name of the game for a small turbine company that is fighting its way into a market dominated by global giants like GE and Vestas. Technology glitches can be devastatingly costly, as the Clipper management team has learned in its bumpy start-up years. Joel Peyton is on the front lines in spotting problems as they pop up on the big-screen computer monitors that line the walls of the Remote Diagnostic and Monitoring Center, a crowded cluster of small rooms just around the corner from Clipper's staff lounge. We enter the center through a door prominently marked \"Tornado Safe.\" In the stormy Midwest, I imagine this is no joke.\n\nFor every Clipper turbine across the continent, dozens of parameters are transmitted on a real-time basis to the center's computer data bank. Joel leans down, types in a few codes, and opens a window that displays data about the Fowler Ridge Wind Farm in Benton County, Indiana. Ambient and equipment temperature, wind and rotor speed, generator revolutions per minute (RPM), power output, and much more are monitored in real time for each of the forty Clipper turbines operating at this site.\n\nA lot can go wrong when accelerating the rotational speed of a Liberty turbine from the top hub speed, 15.5 RPM, to the maximum generator speed of 1,133 RPM. Lubricating oil needs to be warm enough to flow to the spinning gears\u2014a challenge in the super-cold weather that strikes many of America's premier wind sites. Pre-heating the oil is essential. \"We don't want to push very thick oil through small holes,\" Joel explains. Without proper lubrication, bearings and gearboxes can overheat, causing a red icon to flash on the computer monitor and triggering an immediate shutdown.\n\nWorker safety is another factor carefully watched by Clipper's thirteen-member remote monitoring and diagnostic team. Joel switches to a screen showing a satellite map of the continental United States and beyond. Two circles surround every Clipper-equipped wind farm, a red one at a thirty-mile radius and a yellow one at fifty miles. Looking at the Bahamas, I point to brightly colored splotches on the screen, which Joel explains are lightning storms. \"In the summertime, we can get pretty busy with lightning alerts,\" he says. \"When technicians are busy up-tower, we want to give them a pre-warning so nobody gets hurt.\"\n\nThe Remote Monitoring and Diagnostic Center can make system-wide changes, where needed, to improve turbine operations. \"We can adjust parameters here on 400 turbines and make sure it's done correctly,\" Joel explains. \"Less human error, I guess you could say.\"\n\nEven with this fine-tuning in the field, quality control poses a particular challenge for Clipper, as it does for several other turbine companies that purchase most of the components and subcomponents in their machines from outside vendors rather than manufacturing them in-house. Unlike Vestas, which produces much of its own hardware, almost everything that goes into a Clipper turbine is brought in from outside. Its generators come from Mexico. Castings for gears and hubs are supplied by vendors in the United States, Germany, Brazil, and Spain. Main shafts connecting turbine rotors to generators are forged in the United States and Slovenia. Blades are made by the same Brazilian firm that produces blades for GE's turbines. And towers come from as close as Chattanooga, Tennessee, and as far away as China. In all, Clipper relies on 120 outside suppliers for its turbine components and subcomponents.\n\nWith parts coming from so many sources, maintaining uniformly high production standards demands constant vigilance. The steel used in gears, shafts, and bearings must be extremely durable to withstand the rigors of operating a machine that is in nearly perpetual motion, out in the elements, from one season to the next, year after year. Blades must be strong enough structurally and yet sufficiently flexible to accommodate widely varying wind conditions, and their surfaces have to withstand every kind of weather, from sub-zero cold to searing heat. And all moving parts must conform to very precise dimensional tolerances, ensuring their own internal integrity as well as their compatibility with components coming from other vendors.\n\nTo manage the global supply chain, Clipper has a full crew of quality-control technicians who inspect vendor factories across North and South America, Europe, and Asia. Quality-control experts are also on-site in Cedar Rapids, monitoring equipment as it arrives at the Bowling Street plant and testing key components on carefully calibrated machines. On our walk through the plant, Bob Loyd shows me a sound-insulated chamber, at the end of one long assembly bay, where every gearbox is run at up to 130 percent of maximum load under a variety of simulated wind conditions over a four-hour period. He likens this regimen to Detroit's factory testing of automobile drivetrains: both verify good mechanical performance but do not really mimic the stresses placed on equipment over many years of real-world operations.\n\nFor all its vigilance, Clipper encountered some serious defects in its Liberty turbine, starting in 2007. That year, the company reported a \"supplier quality deficiency\" that compromised the Liberty's drivetrain and gearboxes. Later the company divulged that more than a third of the turbines produced in 2006 and 2007 required drivetrain repairs. Defective blades added to Clipper's woes, requiring the blades on about 260 rotors to be reinforced. Most of these rotors had already been delivered to widely scattered wind farms, making repairs much more difficult and expensive. The combined price tag for repairs topped $107 million by the end of 2007.\n\nCracked blades and other defects continued to plague the company in the years that followed. Repair costs escalated to $222 million in 2008, and it took until early 2010 to repair all the damaged blades.\n\nClipper did what it could to minimize the costs of repairing its turbines in the field. Initially, to take down or put up a blade, Clipper repair crews would use a huge crane that had to be hauled to each site on fifteen to eighteen semitrailers. This cost the company a profit-killing $200,000 per job, or twice the price of the blade itself. But then the company's engineers came up with a new way to swap blades, using a technique that required only a few specialized tools and a cherry picker like the ones that phone company line crews use for repairs. The cost: $15,000 plus a few hours' labor. Similar savings were achieved by using the Clipper's built-in hoist, rather than a rented crane, to switch out faulty gearbox components.\n\nDespite its attempts to streamline repairs, the burden of so much unscheduled maintenance took a severe toll on Clipper's balance sheet. Then the financial crisis hit, causing wind farm developers to slow their implementation of existing projects and freeze the development of many future ones. By September 2009, Clipper's cash reserves had dropped to a precarious $40 million. The company was in need of a bailout, and United Technologies Corporation came to the rescue.\n\nUnder the deal with UTC, Clipper initially surrendered 49.5 percent ownership to the Connecticut engineering giant in exchange for a pledge of a quarter-of-a-billion dollars. Ultimately, as Clipper's financial condition worsened, UTC ended up buying the company's remaining shares for $112 million in the fall of 2010.\n\nIn taking on this relatively small, ailing enterprise, UTC's leadership saw a bigger strategic opportunity. Ari Bousbib, executive vice president and president of commercial companies at UTC, looked enthusiastically at the global market trends for the $50 billion wind industry. He commented to the press: \"Other than maybe elevators in China, I don't know many industries that have grown at 25 percent a year.\"\n\nWhile acknowledging that his company urgently needed an outside infusion of capital, Clipper's chief commercial officer Bob Gates sees the UTC buy-in as much more than a financial bailout. He talks about the knowledge transfer the merger brings, given UTC's long history of manufacturing precision-engineered products ranging from Otis elevators and Sikorsky helicopters to Pratt & Whitney jet engines and industrial turbines. \"You know, the elevators and escalators that Otis makes are all electronically controlled, and the elevator always stops at the right floor,\" he comments wryly. Gates is confident that Clipper will benefit from a higher level of quality control on the equipment it installs in its wind turbines.\n\nJust as Clipper is a story of Rust Belt renewal and pioneering innovation, it is a cautionary tale with some important lessons to companies that are looking to come in on the ground floor of an exciting but demanding new industry. Perhaps most obvious is the need for extreme vigilance before dispatching turbine technology of today's mammoth scale to far-flung wind farms. When blades get to be 150 feet long and are bolted onto hubs that are more than 250 feet in the air, companies better make sure they're sending out equipment that their customers can count on. The same is true for high-stress components like gearboxes and generators. Neither Clipper's quality-control teams nor the company's in-house simulators were successful at intercepting serious problems that later rang up such huge replacement costs in the field.\n\nAnother lesson relates to scale\u2014not of the turbines themselves, but of the companies manufacturing them. Competing in the global marketplace for wind technology demands a level of financial commitment and an ability to absorb setbacks that are far beyond the means of relatively small, modestly capitalized companies like Clipper. Remember the Zilkhas' decision to bring in Goldman Sachs when Meridian Way and other wind farm developments needed a big-time infusion of capital; as savvy energy entrepreneurs, they knew when it was time to reach beyond their own resources. Clipper seemed to wait until disaster was at the door before it turned to UTC for a bailout.\n\nWhile a company of Clipper's size simply may not have been equal to the challenges of fielding today's turbine technology prior to the UTC buyout, there _is_ a place for smaller companies in the wind energy supply chain. With thousands of components and subcomponents going into the typical commercial turbine, the opportunities abound for specialized manufacturers to enter the industry. From manufacturers of bolts and flanges to the makers of hydraulic pumps and electric motors, companies of many shapes and sizes have already joined the American wind industry. These businesses stand to flourish as turbine companies at home and abroad\u2014hopefully including a revitalized Clipper\u2014expand their production to meet the growing demand for wind power in the coming years.\n\nOne of the conditions that will help the U.S. wind industry build and maintain momentum is a stable federal policy climate that puts renewable energy on an equal footing with traditional fossil fuel and nuclear technologies. Unfortunately, federal policies have been erratic and unreliable, inviting a degree of caution among energy investors, technology manufacturers, and wind developers that has impeded their full commitment to the enterprise.\n\nCongress first approved a production tax credit for wind energy as part of the Energy Policy Act of 1992, with Senator Chuck Grassley of Iowa, fittingly, as its prime sponsor. Providing an income tax credit of 1.5 cents per kilowatt hour of electricity produced by wind, this law ushered in a wholly new approach to federal intervention in support of renewable energy. Unlike the investment tax credits that gave commercial wind its start in the 1980s, the production tax credit\u2014or \"PTC\" as it's called in the trade\u2014rewards the actual generation of power rather than the mere building of new power-generating capacity. It gives wind farm owners and operators an incentive to operate their facilities as efficiently and productively as possible, minimizing downtime for repairs and maximizing the power produced by every operating turbine.\n\nThough a valuable stimulant in concept, the PTC's actual impact has been compromised by the ebb and flow of legislative support over the years. Congress first let the PTC expire in June 1999, when its initial authorization under the 1992 energy law ran out. It was reinstated six months later, but for only two years. Then, at the end of 2001, it lapsed again for a number of months, and once more at the end of 2003 when a comprehensive energy bill authorizing its extension failed to pass Congress. When the PTC resumed in late 2004, its approved duration was little more than a year. And so the pattern continued up until the American Recovery and Reinvestment Act of 2009 extended the tax credit for a somewhat less truncated three-year period, through the end of 2012.\n\nBy the time this Obama-era legislation became law, inflation indexing had raised the PTC for wind-generated power to 2.1 cents per kilowatt hour. What's more, wind developers could choose between claiming the per-kilowatt-hour credit and taking a straight-up 30 percent investment tax credit on qualifying equipment and expenditures. As a third option, they could convert the investment tax credit to an outright grant amounting to 30 percent of the value of their installations. While some congressional lawmakers have attacked the degree to which these stimulus dollars have benefited foreign wind developers and manufacturers, American wind industry leaders have credited the stimulus package with preventing a U.S. jobs meltdown. According to the American Wind Energy Association, the wind industry lost about 10,000 jobs when new turbine orders lagged and new wind farm construction slowed in 2010 to about half the pace of the previous year. This brought the total wind industry workforce down from a pre-recession peak of 85,000. But with wind development regaining momentum in 2011, wind industry leaders are hopeful that jobs will regain and eventually outstrip their prior numbers.\n\nEven with the infusion of stimulus funds, wind farm developers and equipment manufacturers remain wary of the future. Clipper's Bob Gates points to his own experience negotiating with vendors. \"When you go to a foundry in Ohio and you say, 'Build a new plant to make more castings for wind turbine gearboxes,' the casting company says, 'What are you talking about? The PTC is going to expire in a year or two! I need ten or fifteen years to recover the cost of a plant.' \"\n\n\"The PTC was meant to level the playing field economically, which it does,\" Gates continues. \"What it doesn't do is level the playing field in the time dimension.\" He points to the uninterrupted subsidies enjoyed over decades by the oil, gas, and coal industries and wonders aloud why wind\u2014so much more benign\u2014has been supported so much more sporadically. As one sign of our nation's misaligned priorities, he laments our failure to internalize the huge environmental costs of coal: \"You burn it\u2014it's gone. You have the emissions, you have the CO2, the energy source is gone, and for that, you get an incentive.\" I find myself mentally adding to the list: the blackened lungs of miners, the ravaged mountainscapes and polluted streams of Appalachia, the endless procession of railroad cars carrying coal from the surfacemined moonscape of eastern Wyoming to power plants across the country.\n\nAs for nuclear, Gates points to the indemnity that Congress granted to power plant operators back at the dawn of the civilian nuclear industry in 1957. Under the continuously reauthorized Price Anderson Act, even the most catastrophic meltdown would expose the plant operator to less than half a billion dollars in liability. This level of federal protection for a civilian industry is without parallel in the United States, and its significance becomes very real when one contemplates the tens of billions of dollars of damage to property and people's lives caused by the recent catastrophe at Japan's Fukushima Daiichi nuclear plant.\n\n\"If a private company had to buy private insurance on a nuclear power plant, you wouldn't have any nuclear power plants,\" Gates flatly asserts. To that vast subsidy, we must add the billions of dollars the federal government has invested in trying to come up with a safe, long-term method for disposing of nuclear waste. Although no solution is in sight, civilian nuclear plants in America\u2014104 in all\u2014add to the nuclear waste burden every day. \"How much does nuclear really cost?\" He poses the question in exasperation, clearly not expecting an answer.\n\nBuilding a U.S. wind manufacturing base extends far beyond the dozen or so companies that actually assemble turbines. The Timken Company, for example, is a successful industrial innovator that has long served the automotive and aeronautics industries. Today it is making a strategic bet on wind. The company's politically conservative leadership defies any stereotypes about the wind industry being led by progressively minded twenty-first-century entrepreneurs who are eager to help America move away from carbon-based energy technology and are gung-ho about the battle against global warming.\n\nHeadquartered in Canton, Ohio, Timken is a world leader in the manufacture of precision bearings, ensuring the smooth operation of everything from helicopter rotors and jet engines to car transmissions and off-road construction equipment. With factories operating in twenty-seven countries, its 2010 sales totaled $4.1 billion.\n\nLorrie Crum, manager of Timken's global media relations and strategic communications, is passionate about the company's role in improving wind turbine performance. \"Wind energy has been identified as the company's most promising market,\" she says, pointing to the $200 million that Timken has already sunk into pursuing its potential in this field. The fit is perfect, she says, for a company that prides itself on building for endurance\u2014something of a mismatch with the automotive sector, where Timken's products substantially outlast the machinery that hosts them. \"We were able to make the million-mile axle, but you never capture their full value,\" she says. All of those million-mile axles end up in the landfill, junked along with the rusting carcasses of cars that seldom travel more than a few hundred thousand miles.\n\nWind turbines, by contrast, need to operate day-in, day-out, season-to-season, year after year, over an expected lifespan of two decades or more. \"This is extreme engineering,\" Crum explains. Timken specializes in making bearings for the wind industry\u2014meticulously crafted clusters of hardened steel rollers that hold key turbine components in place while allowing them to rotate with minimum friction. Ever since the company's founder, Henry Timken, began exploring ways to make sturdier wheel assemblies for horse-drawn carriages in the late nineteenth century, the study of friction, or \"tribology,\" has been a driving force behind Timken's technology development.\n\nOne of the \"mission-critical\" bearings that Timken provides to wind manufacturers is a tapered bearing that cradles the main shaft of the turbine, a multi-ton steel rod that carries the slow-spinning motion of the rotor\u2014usually in the range of 12 to 20 RPM\u2014to a gearbox that speeds up to match the demands of the turbine's electrical generator. This bearing has to hold the main shaft firmly in place through all kinds of weather, allowing it to absorb the relentless jostling and wildly varying velocities of the winds in locations that are specially chosen because of their high prevailing wind speeds.\n\nThe damaged main shaft bearing from a large commercial wind turbine arrives just as we are touring the factory floor at the Timken Technology Center. Timken didn't make the original bearing. Rather, the Technology Center's engineers are being called upon to diagnose why it failed and to replace it with a more durable, custom-designed Timken bearing.\n\nI watch as a diesel-powered mobile crane gingerly carries the main shaft assembly in through an open bay. Suspended by two triangulated wires from the crane's extended arm is a shiny silver-colored shaft, about ten feet long. At one end is the steel casing, about three feet in diameter, which contains the ailing main shaft bearing. Jim Charmley, who heads up the Technology Center's staff of 400 technicians, is on hand with me to witness this arrival. \"We did _not_ stage this for you!\" he insists.\n\nThe main shaft bearing is designed to last the full twenty-year predicted lifetime of a wind turbine, but Charmley tells me that this one made it through only about two years. When I ask if many other main shaft bearings have failed on this particular turbine model, the MIT-trained mechanical engineer is circumspect. \"I can only say that if [the manufacturer] had worked with us from the beginning, you might not see this here today.\" With that, he introduces me to Gary, a colleague whom he describes as one of the world's experts on nanocrystalline deposits\u2014specialized metals like tungsten that can be applied to bearing surfaces to minimize friction and enhance durability. Gary, he says, will be incorporating nanotechnology into the bearing's redesign.\n\nCharmley points to his own staff's dramatically shifted focus as a clear sign of Timken's commitment to wind. In 2005, there might have been three technicians working on wind applications at the Timken Technology Center. Five years later, he estimates that eighty to a hundred specialists, including many of the fifty PhDs working at the site, have been redeployed to develop new Timken products and services for the wind industry. He shouts out these impressive numbers as we walk through a cavernous room filled with forty high-speed electric motors that are testing out newly designed bearings. Salt and dirt are sprayed onto some bearings to test the durability of the rubber seals surrounding them; other bearings are jostled and shaken to test their ability to withstand tough conditions in the field. All these simulations are run on an accelerated schedule so that Timken can keep a leg up on the competition. Established European competitors like SKF and FAG are busily developing similar products, and Asian companies are entering the game as well.\n\nIn addition to securing a substantial share of the U.S. wind market, Timken has its eye on a growing customer base overseas. Just a few yards from the defective main shaft bearing, laid out on a platform, is a huge metal ring, larger in diameter than the men standing by it are tall. Tightly spaced cylindrical steel rollers are sandwiched between its inner and outer rims, like tapered teeth in the maw of a large beast. In the coming days, this supersized bearing is to be loaded into the belly of a 747 headed for Beijing. From there, it will make its way by truck to a wind installation in Hunan province, where it will be tested as a prototype on a turbine manufactured by XEMC Windpower. This Chinese company is competing for a share of the market for \"direct-drive\" turbines\u2014machines that generate electricity directly from the rotor, rather than relying on a main-shaft-and-gearbox configuration to feed a high-speed generator. Large, built-to-last bearings of this sort are at the upper end of Timken's price scale, easily exceeding $100,000 and sometimes reaching $200,000 for a bearing equipped with computerized sensors that provide automatic, temperature-controlled lubrication of all moving parts.\n\nThe prototype bearing that we are admiring was produced at a Timken plant in South Carolina, but Jim Charmley and his colleagues have no intention of flying or shipping multiple bearings of this size halfway around the world. If the prototype works to XEMC's satisfaction, production will shift to one of the half-dozen factories that Timken operates in the People's Republic. Already the company's Chinese manufacturing plants supply bearings, lubrication systems, and other related products and services to a number of Chinese heavy industries, and now it is looking to make wind a major part of its China business. Aside from the Chinese market's unparalleled growth potential, the starkly efficient way China goes about building its in dustrial and transportation infrastructure makes it an appealing home for Timken manufacturing.\n\nTimken's international marketing team is more than willing to do business in a nation run by central planners, where the state plays a dominant role in governing and, in some cases, owning the companies involved in turbine manufacturing and wind farm development. But here in America, Timken's leaders take a very different attitude toward government intervention in the industrial sector. Ward J. \"Tim\" Timken Jr., board chair since 2005, is a case in point.\n\nSpeaking at Ohio's Ashland University in October 2009, Tim Timken leveled a broadside attack on the Obama administration's economic recovery plan as a threat to American free enterprise. \"Not since the Great Depression has government interfered so dramatically and so decisively in the economic life of our nation,\" he declared, warning that the administration's emergency stimulus expenditures gravely undermined any remaining public confidence in the private sector. \"Nothing could be worse for our economy and our nation,\" he asserted. He has been equally vehement in his assault on legislative proposals that would cap U.S. carbon emissions and require power plants and other major industries to pay for any carbon dioxide (CO2) emissions above their allotted amounts. He told the Ashland University gathering that this sort of \"cap-and-trade\" regime would \"drive up the price of energy, deter American job creation, and send our jobs overseas.\"\n\nTimken's leaders have reason to be worried about the financial implications of any regime that attaches a price to carbon emissions. The company's bearings are manufactured from recycled steel, in electric arc furnaces that process the steel-content equivalent of 100,000 cars per month. Timken's Ohio facilities alone consume up to $50 million a year in electricity, mostly generated by coal. Communications manager Lorrie Crum contends that \"we couldn't possibly get enough sun or wind here to power our operations without relying heavily on conventional sources.\"\n\nCrum may be right that, today, her company is more or less stuck depending on coal to fuel its steel plants and other factories. Yet she discounts too readily the longer-term potential for electricity generated by wind and other renewable resources. While Timken's home state of Ohio has installed only a few megawatts of wind capacity to date, plans for building 1,000 megawatts of offshore wind on Lake Erie by 2020 have been announced. To the east, Pennsylvania has 748 megawatts of wind up and running, and New York developers have installed nearly 1,300 megawatts at their wind farms. To the west, wind power in Indiana has already topped 1,000 megawatts, Illinois has more than 1,500 megawatts of installed wind, and major offshore wind projects are now on the drawing boards for Lake Michigan. Reaching farther into the Great Plains, improved high-voltage transmission could open up truly vast resources in the heartland's wind belt, extending from North Dakota and Minnesota down through Iowa and Nebraska to Kansas, Oklahoma, and Texas. With the right policy priorities and market incentives, Timken and other Rust Belt companies could end up substantially curbing, if not eliminating, their reliance on conventional fuels for most of their power needs.\n\nWherever they lie on the political spectrum, and whatever their attitudes about climate change and the role of government in shifting us away from carbon-based fuels, U.S. manufacturers are finally embracing the new opportunities that wind energy offers to their own business interests and the American economy. The growing pains of smaller companies like Clipper should warn us that we are still in the early stages of developing an entirely new technology base for our nation's energy future. As in any pioneering technology field, the design challenges and operational surprises may stymie us at times, just as the financial setbacks may be harrowing. For decades, the federal government has generously nurtured our fossil fuel and nuclear industries. It behooves us to do the same for wind and other renewable energy technologies that can help create a less precarious platform for future development. With a serious commitment to cleaner energy choices, we can promote a vibrant, sustainable U.S. economy without perpetuating our reliance on foreign energy sources, without exposing ourselves to nuclear energy's hazards, and without mortgaging the global climate.\n\nThe challenges facing U.S. wind technology companies are complicated by the fact that they are not alone in looking for ways to consolidate their positioning in the clean energy marketplace. High-caliber equipment coming out of Denmark, Germany, and Spain has long competed for U.S. sales, but with European manufacturers shifting much of their production to America, the net gains to U.S. workers and the U.S. economy are substantial. Much harder to gauge is the role that Chinese wind technology will play in the years ahead. As in so many other sectors, China is a hugely ambitious, rapidly growing force in the global wind trade, benefiting from cheap labor costs and brashly protectionist policies at home as it begins to market its turbines abroad. Although Chinese companies are only making their first forays onto U.S. soil, American wind manufacturers have reason to be nervous.\n\n## CHAPTER FOUR\n\n## The Chinese Are Coming\n\nLEGEND HAS IT THAT Napoleon once pointed to a map of China and cautioned: \"Let China sleep, for when she wakes, she will shake the world.\" I felt the prophetic impact of those words as I looked around me at the thousands who had gathered for China Wind Power, a conference and trade show in Beijing promoting China's rapidly expanding wind energy industry.\n\nAs American companies look for ways to expand their niche in a growing global wind market, China is both an enormously alluring export target and an increasingly formidable competitor. Though a latecomer to the field, the People's Republic is now the world's biggest user of wind energy, each year installing more new wind power than any other nation. Predictably, General Electric, Timken, and other American companies that already have a strong presence in China are revamping their local plants to capture a corner of the Chinese wind market, while other U.S. firms are seeking out Chinese buyers for a dizzying array of wind energy equipment made in the United States. Breaking into the Chinese market is no small feat, however. Protectionist policies give Chinese companies an enormous advantage over foreign competitors\u2014even those that are employing Chinese workers at local factories.\n\nIn addition to being the world's biggest consumer of wind energy technology, China is quickly rising to the top ranks of wind equipment manufacturers\u2014largely by virtue of the enormous scale of China's internal market for wind. Four of its companies are now on the \"top ten\" list of global wind turbine producers, and seven of the top fifteen manufacturers worldwide are Chinese. Perhaps most worrisome to American and European turbine companies, China is selling its machines at cut-rate prices that may soon make them daunting competitors outside its borders. Already a few leading Chinese companies have opened sales offices in the United States, and one turbine manufacturer is building its own commercial-scale wind farm in Illinois, primarily to prove the worthiness of its turbines to American wind developers.\n\nSteve Sawyer, secretary-general of the Global Wind Energy Council and former head of Greenpeace, invoked Paul Revere's memory when he spoke about China's expanding presence in this hotly competitive field. \"Ever since China entered the wind market, we've been hoping, anticipating, fearing that 'the Chinese are coming,' \" he said at China Wind Power's opening session. Other speakers made more pointed remarks about China's protectionist rules and the special subsidies favoring domestic over foreign manufacturers. Li Junfeng, secretary-general of the Chinese Renewable Energy Industries Association, tried using humor to divert attention from the substance of these claims. \"Wind in China is in its puberty,\" he observed with a laugh. \"No one talks about what you do when you're in your fifties or sixties, but at sixteen, everyone's watching.\"\n\nChina's wind energy leaders may function within an industry that continues to be dominated by communist, state-run enterprise, but their entrepreneurial zeal poses a very real challenge to rivals who were raised in the cradles of European and American free enterprise. In wind energy, as in so many other arenas, China is fast becoming a nimble, dynamic and\u2014to many\u2014unnerving force in the global marketplace.\n\nA week before the China Wind Power gathering, I traveled to Tianjin, home to the world's biggest concentration of wind energy manufacturing facilities. Nearly 25,000 people work in the Tianjin wind trade, employed by several of the top global and Chinese wind technology companies. Vestas has the largest presence among turbine producers, initially opening a blade plant there in 2006 and, more recently, adding factories for nacelles, hubs, and generators. I was in Tianjin to attend the opening of the company's two latest facilities at a vast industrial campus that has become the most massive Vestas factory complex anywhere.\n\nAn eye-stinging haze enshrouded this city of 12 million as our sleek white bus moved through a dense fabric of elevated truck routes, high-rise apartment blocks, and vast, low-slung factory buildings. This was the China I had long heard about\u2014endless urban sprawl in a fog of filthy air that blocked out the sun and erased all hints of sky. The side panels of our bus, with their photo images of wind turbines against an azure sky, stood in sunny contrast to these bleak surroundings.\n\nAt the Vestas manufacturing complex, I joined hundreds of Chinese workers who were filing into a cavernous steel-frame shed to attend the factory-opening ceremony. Dressed in polo shirts with the Vestas logo, they were followed by European company officials in Western business attire who took their places alongside local dignitaries on a makeshift stage. Arrayed behind them was a blue-and-green electronic banner, in English and Chinese, proclaiming the company's mission in China: \"Building a World Class Production Base in the World's Fastest-Growing Wind Energy Market.\"\n\nLars Andersen, president of Vestas-China, came to the podium. Reading from an English-language text, he rolled out his company's local credentials. Vestas, he reported, was the first company to install grid-connected turbines in China in 1986. Its manufacturing complex in Tianjin has 1.4 million square feet of factory space and cost the equivalent of $363 million to build. Bounding to the mike, the next speaker was the Tianjin industrial zone's vice chairman, Ni Xiangyu, his exuberance palpable as he addressed members of the audience in their native tongue. \"This could be the best money-making machine in the world, using the best brains in the world!\" he exclaimed.\n\nNi's brazen boosterism and Tianjin's embrace of wind are local reflections of China's new dominance in the wind energy field. In 2003, the People's Republic brought 100 megawatts of new capacity on-line, little more than 1 percent of the 8 gigawatts installed globally that year. By 2009, it had outstripped all other countries in new installations, capturing 36 percent of the world market. In 2010, it took close to half of the market: its 18.9 gigawatts (18,900 megawatts) of new capacity gave it 49.5 percent of all newly installed wind power worldwide. Vestas had good reason to be locating its largest production facilities in the world in Tianjin.\n\nChina still lags far behind the United States and other Western nations in its per-capita wealth, but it is working hard to narrow that gap. At least in its larger cities, the quest for comfort, mobility, and personal expression through fashion has redefined the national ethos. Urban boulevards formerly filled with bicycle commuters are now gridlocked with Volkswagens, Audis, BMWs, Volvos, and Buicks, along with several Korean and domestic car brands. Glitzy shopping malls flaunt luxury designer clothing and accessories. Soaring office and residential towers line the streets, and beneath them run mile after mile of elegant and immaculate subways, with new lines opening every year.\n\nWhile much of rural China still may be quite poor, the Communist Party and its planners are expanding the economy at a feverish pace. Throughout the first decade of the millennium, China's gross domestic product (GDP) grew at an average of more than 9 percent annually, a rate that is likely to be matched, or nearly so, in the years ahead. In fueling that growth, China has overtaken the United States in its hunger for electric power. By 2020, the country is expected to consume twice the electricity that it uses today.\n\nSo what will be wind energy's role in fueling this phenomenal growth? A joint team of researchers from Harvard and Beijing's Tsinghua University has concluded that China's onshore wind resources could supply more than seven times the country's current electricity needs. Yet China is not Denmark, and its central planning body, the National Development and Reform Commission (NDRC), is not prepared to envision a future where wind turbines provide the world's most populous nation with the lion's share of its power. China's planners are spreading their bets across a broad array of power plants reliant on coal, nuclear, hydroelectricity, and natural gas. Of these, coal has been and will remain the primary fuel. A total of 80 percent of China's electricity today is produced from coal, and with several dozen new plants coming on-line every year, the country's coal-based power production will increase dramatically in the years ahead. Already consuming more than 3 billion tons of coal annually, China accounts for nearly 40 percent of global coal use. By 2015, its demand for coal is expected to top 4 billion tons per year.\n\nThe impacts of relying so heavily on the world's dirtiest fuel are legion. Mining disasters in China kill thousands each year, and that toll will inevitably rise as more and more coal is dug from the earth. Much of the air pollution that hangs over whole regions of China comes from coal burning, and that too will increase despite lip service paid to new \"clean coal\" technology. And burning all that coal will only push China further into the lead as the world's biggest contributor to global warming.\n\nIn addition to the new coal plants, China now has 27 nuclear reactors under construction\u2014three times as many as its closest rival, Russia, and nearly half of all nuclear plants being built worldwide. China also dwarfs all other nations in the 50 additional reactors now being planned and the 110 that have been proposed. As if Three Mile Island and Chernobyl didn't give us sufficient warning about the perils of nuclear power in the 1970s and 1980s, we now have the recent disaster at the Fukushima Daiichi reactor complex in Japan to refresh our recollections. It remains unclear whether China will heed this warning. Before peppering the countryside with nuclear reactors, China's central planners would be wise to take more careful stock of the hazards associated with this technology, particularly in light of the catastrophic earthquakes that have recently shaken parts of that nation.\n\nChinese energy planners are also moving ahead with a commitment to ramp up the country's use of hydropower, ostensibly greener than coal and nuclear in its reliance on a renewable energy resource. However, as the world's largest hydroelectric project moves toward completion at the Three Gorges Dam, along the Yangtze River, the human and ecological effects of giant dams are hard to ignore. More than 1 million people have already been forced from their homes by this project, and by the time it is complete, more than 5 million residents could be displaced.\n\nWith so many other energy projects under way or planned, wind energy remains a relatively small part of China's energy portfolio. Maximizing economic growth is clearly the order of the day\u2014far more important to China's political leadership than protecting the country's environment or reining in global warming. Yet China's dramatic progress in increasing its use of wind power reveals what its planners can do when they decide that a given investment is in the state's interest.\n\nChina's use of wind energy remained minimal throughout the 1980s and 1990s. Despite the government's issuance of a strategic plan calling for 1,000 megawatts of wind power by the year 2000, only 200 megawatts of new wind power had been installed nationwide by 1999\u2014roughly equal to the generating capacity of a single large wind farm in America like Cloud County's Meridian Way.\n\nBeginning in 2002, however, a number of reforms helped bring wind closer to the mainstream. The government that year invited competitive bids for the development of selected wind farm sites, and in just four years, China's installed wind power quintupled. Then, in 2005, the country's first Renewable Energy Law called for mid- to long-term national targets for renewable energy use and for plans at all levels of government to achieve those targets.\n\nThis development gave Ren Dongming and his colleagues the mandate they needed to expand wind energy's contribution to China's energy mix. Ren is deputy director of the Center for Renewable Energy Development, a branch of the NDRC that is charged with planning for China's wind energy future. On meeting him at the government-run Guohong Hotel, a few blocks from his Beijing office, I was surprised and relieved. Dressed in blue jeans, a denim shirt, and a tweed jacket, Ren peered at me through silver wire-rimmed glasses, looking to me more like a Berkeley professor than a Chinese government bureaucrat. As we talked over sodas in the hotel's noisy restaurant, his level-headed analysis cast light as well as hope on China's growing commitment to wind.\n\nRen spoke of China bringing 150 gigawatts of wind power online by the year 2020, or about 8 percent of the nation's projected power-generating capacity. To make this happen, he and his team have mapped out a series of wind farm megabases\u2014vast wind-rich zones where new projects are to be commissioned on a scale unprecedented in China or anywhere else in the world. In eastern Inner Mongolia, one zone has been slated for 30 gigawatts of wind power\u2014substantially more than China's overall wind power capacity at the time that the megabase program was officially unveiled in June 2009. Western Inner Mongolia has been targeted for 20 gigawatts. The Jiquan Wind Farm Base in Gansu province, first of the megaprojects to begin construction in 2009, includes thirty wind farms adding up to 12.7 gigawatts. This project alone, according to official projections, will displace nearly 10 million tons of coal combustion annually. In all, seven megabases have been designated in six Chinese provinces, adding up to 120 gigawatts of new wind-generation capacity at a cost of roughly $140 billion.\n\nWith other, smaller wind projects scattered across China and new offshore installations in various stages of planning, Ren Dongming's 150-gigawatt goal seems well within reach. Already the country has dipped its toes in the water with the 102-megawatt Donghai Bridge wind farm just off Shanghai's shores, and the NDRC maintains that offshore wind arrays could provide close to 500 gigawatts of new wind power capacity. That's in addition to the vastly greater wind resources available on land.\n\nBig state enterprises are the predicted winners in the bids for megabase wind projects, partially because they can readily tap the material and financial resources to deliver huge new increments of wind power. At the same time, private investors are finding their way into China's growing wind market. Typically, they are emerging as owners and operators of smaller wind farms that can be approved by local authorities without any national government review. Wind farms with fewer than 50 megawatts of installed capacity fall under local government jurisdiction, free from the NDRC's more cumbersome permit application process.\n\nUPC Renewables is the Chinese subsidiary of a private partnership founded in 1994 by a group of international investors, some of whom were pioneers in the California wind boom of the 1980s. After developing wind and solar projects in Europe, the United States, North Africa, and the Philippines, the firm's managers started looking for a way to break into China's wind energy market. They hired Guo Zheng, an electrical engineer trained at North China Electric Power University with two decades of experience in the oil and hydroelectric sectors.\n\nLike most Chinese nationals who are active in international business, Guo Zheng has adopted a Western name for his dealings with foreigners. He introduces himself as Wilson Guo when I visit UPC's China headquarters, on the seventeenth floor of a downtown Beijing office tower. I ask him to describe how he lines up new projects and am struck by the contrast with Jim Roberts's work as Horizon's land agent in Cloud County. He has no direct dealings with farmers and ranchers; Guo's only business is with local government officials. \"Farmers have the right to live on the land,\" he explains, \"but the land is owned by the government.\"\n\nWhen the Maoist collective farms were broken up in the 1980s, individual farmers received plots of land that they could either farm for themselves or lease to others. Actual ownership of the land remained with the collectives, however. Today, when a wind developer leases land from a collective, all members of the collective receive a share of the funds. \"The government has a standard formula for land compensation so you know what it will cost you from the beginning,\" Guo says. He compares this favorably to countries like the United States, where wind farms have to be assembled through individually negotiated contracts with multiple landowners.\n\nOthers express qualms about Chinese wind farm siting politics. Under-the-table payments to local officials are all too often part of the cost of doing business in rural China, as several close observers told me. Moreover, there is no real opportunity for members of the public to voice their concerns about badly sited projects. Charlie McElwee, a Shanghai-based American attorney and an adviser to wind energy developers, tells me that he hasn't heard of a single wind farm in China being stopped or altered because of concerns about migratory birds, human health, noise, or aesthetics. \"Tame\" is the word McElwee uses to describe China's environmental movement, where truly independent, citizen-based organizations simply don't exist. He can recall a few recent local demonstrations protesting major industrial projects\u2014a waste incinerator in one locality, proposed petrochemical plants in a few others\u2014but government security forces broke them up quickly. Beyond the local level, he says that the environmental movement generally consists of groups that are under the close supervision of Chinese governing authorities. He calls these groups \"GONGOs,\" or government-organized non-governmental organizations, to distinguish them from the NGOs, or non-governmental organizations, that operate with a high degree of independence in less-authoritarian societies.\n\nForeign investors like UPC are by no means alone in acquiring a stake in China's local and provincial wind farm development. Tianrun New Energy Investment Company, Ltd., is the Beijing-based development arm of Xinjiang Goldwind Science & Technology. Goldwind was an early pioneer in China's wind industry, producing its first turbines in 1998 under license from a German manufacturer. This practice of piggybacking on foreign turbine designs became the industry norm among Chinese manufacturers, allowing them to enter the market quickly, though not necessarily with the most updated technology. By 2007, Goldwind had become the country's market leader in turbine production, and it began investing in wind farms through Tianrun. Within two years, it had gained full ownership of eight wind farms and held a controlling stake in six others.\n\nUnlike UPC, which is a privately held foreign enterprise, Gold wind is a hybrid typical of companies operating on the seam between China's old-line communist state-owned enterprises and the new generation of private companies that are finding their place in the country's emerging market economy. The company, on the one hand, is majority-owned by private investors; its shares have been publicly traded on the Shenzhen Stock Exchange since 2007. On the other hand, several of Goldwind's biggest investors are state-owned enterprises; Tianrun officials estimate that 30 to 40 percent of the company's assets remain in state hands.\n\nWith multi-gigawatt megabases already under development and many companies like UPC and Tianrun moving forward with smaller projects, China's new wind capacity is growing at a breathtaking speed, accelerated by the ease of lining up land parcels on communal land, the very limited focus on environmental concerns, and the absence of any real public input into the siting and permitting processes. The contrast with America's methodical and highly participatory approach to project development couldn't be starker. Not surprisingly, transmission companies are having a hard time keeping pace with so much new, dispersed power production. In recent years, as much as a third of the country's installed wind power has not made its way onto the grid and either stands idle or remains landlocked in local areas that often can't make good use of the available power. The government has committed to build a vast new network of transmission lines, but with overall Chinese electricity use expected to nearly double by 2020, grid operators are in an ongoing race to keep up.\n\nEnsuring a steady supply of reasonably priced turbines to new Chinese wind farms is another challenge the government has addressed, often in ways that have aroused the ire of foreign turbine manufacturers. In the early years of China's wind power development, foreign companies supplied the lion's share of the technology. Today China's domestic brands dominate the market.\n\nHighly protectionist regulations have given Chinese manufacturers a decisive edge. Starting in 2003, local manufacture was made a key criterion in the review of bids for government-sponsored wind farm concessions. At first the minimum \"local content requirement\" was set at 50 percent of a turbine's value, but a year later it was raised to 70 percent. Soon this protectionist regime extended beyond government projects to include all turbines installed in the country.\n\nForeign manufacturers that relied heavily on imported components had a tough time meeting the local content hurdle. In 2004, international brands commanded three-quarters of turbine sales. Within five years the proportions had more than reversed, and by 2009, Chinese manufacturers had captured 86 percent of new installations. By 2010, the only non-Chinese manufacturers to make it into the top-ten ranking were Vestas, with less than 5 percent of the market, and Spain's Gamesa, which accounted for 3 percent of new wind capacity. General Electric, the sole U.S. player of any size in China, had a market share that barely exceeded 1 percent.\n\nFor years the United States had complained about Chinese anti\u2013free trade measures, but in 2009 the Obama administration singled out the local content requirements on wind turbines. While U.S. companies like Timken could sell bearings made at their Chinese factories to local turbine producers, dozens of other U.S. companies, producing their wares on U.S. soil using American workers, were effectively blocked from participating in the Chinese wind energy boom. China, under pressure, signaled its readiness to lift the local content requirements, but a year later the U.S. Trade Representative petitioned the World Trade Organization (WTO) for a special consultation on a broader range of policies that were seen as giving Chinese wind turbine producers an unfair edge over foreign competitors. Not only were the Chinese still making it hard for U.S. goods to enter the Chinese market, but leading Chinese turbine manufacturers were preparing to export their own technology to America. This, no doubt, raised the anxiety levels of U.S. officials, as well as the United Steelworkers, who instigated the WTO petition.\n\nChunhua Li, director of international business at Goldwind, made it clear to me that America is a prime target for Goldwind sales, when we met at the company's research and development center in an industrial park just south of Beijing. His words reflected cautious determination: \"We want to be sure that our turbines supplied to overseas developers make our international market successful. Now we feel we are ready.\" He proudly described the \"green-friendliness\" of Goldwind's direct-drive turbines\u2014simpler to build and easier to maintain than conventional gearbox-dependent machines. He also pointed to Goldwind's newly introduced 2.5- and 3-megawatt models as proof that the company can match the scale of equipment now being sold to the U.S. and European markets.\n\nAfter our meeting, I was taken past the company's soccer field and climbing wall to one of Goldwind's turbine assembly plants. Li Fan, a youthful-looking production manager, greeted me at the entrance. Eager to practice his English, he took me on a tour of the production room where 1.5- and 2.5-megawatt turbines were being assembled. Giant rotors and stators, the revolving and stationary parts of direct-drive turbine generators, were laid out on heavy wooden blocks in one quadrant of the factory. In another area, workers were carefully installing electronic controls inside turbine nacelles, and in a far corner of the factory, assembled generators were being tested on a full-scale turbine simulator.\n\nLi Fan had good reason to brush up on his English. In only a few days, he would fly to America to oversee the erection of three Goldwind turbines at a demonstration site near Pipestone, Minnesota. By selecting a site exposed to some of the harshest weather conditions in the lower forty-eight states, Chunhua Li and his international marketing team hope to persuade American wind developers that Goldwind's turbines can outperform better-known Western brands. As a further step toward making its technology known, Goldwind is now building a full-scale commercial wind farm in Lee County, Illinois. The company's U.S. affiliate, Goldwind USA, has already secured a twenty-year commitment from Commonwealth Edison to buy all the power generated by this 109-megawatt farm.\n\nGoldwind's strategists are competing fiercely with other Chinese companies that now have set their sights on the global market. The most formidable among them is Sinovel. Barely noticed in 2004, when it was launched as a subsidiary of the state-owned steel giant Dalian Heavy Industry Group, Sinovel entered the market two years later with a 1.5-megawatt turbine produced under license from a small German company, Fuhrl\u00e4nder. By 2008, it had eclipsed Goldwind to become the leading turbine supplier to the Chinese market, and by 2010, it held 23 percent of total domestic installations, compared to Goldwind's 20 percent share. Next in line were three other Chinese manufacturers; then came Vestas in sixth position, holding less than 5 percent of the Chinese market.\n\nSinovel's publicists are brash and uncompromising in their reach for global supremacy. Through a \"Three-Three-Five-One\" strategy announced in 2009, the company unveiled its ambition: within three years, to become one of the world's top three turbine manufacturers; and within five years, to be number one in the world. The company exceeded its first milestone in only a year, edging out General Electric to become the number-two global manufacturer in 2010. With 11 percent of all new installations worldwide, Sinovel ran just a single percentage point behind Vestas. General Electric and Goldwind came in third and fourth, each with about 10 percent of the market.\n\nExports remain a minor part of China's turbine manufacturing success story, yet the country's domestic market is big enough to create a hugely robust wind industry. In 2010, with nearly half of the world's new wind power installations taking place on the home front, China's turbine producers have been able to project a dominant presence on the global stage, even though their physical reach has largely remained within a single country's borders.\n\nChinese entrepreneurs are bullish about exploring new frontiers abroad, but it remains unclear whether their turbines will cut significantly into the European and U.S. markets where familiar, well-proven brands now have the edge. Industry analyst Matthew Kaplan describes Chinese turbines as based on borrowed \"tier three\" designs that are typically several years out of date, causing them to be less productive than the models that \"tier one\" Western manufacturers are placing on the market. In determining just how much of a bargain they are getting from a price discount on Chinese turbines, wind developers will need to weigh very carefully the full life-cycle performance of the equipment they are purchasing. Developers building wind farms in America will also have to reckon with the higher transportation costs of importing equipment from halfway around the world, and with the difficulties they may face raising capital for new wind farms using lower-quality turbines. Finally, the price advantage of Chinese hardware is being weakened by a drop in the price of American-manufactured turbines, caused by surplus capacity among existing suppliers to the U.S. market.\n\nRegardless of how effective Chinese companies turn out to be in breaking into new wind energy markets, China's planners and policymakers have cleared a startlingly expeditious path to developing their own country's wind energy potential. With more wind power installed in China than in any other nation today, the National Development and Reform Commission projects that the country's wind installations could reach 1,000 gigawatts by 2050. That's five times current global wind-generating capacity (see table 5) and would be enough, the NDRC estimates, to provide about 17 percent of China's total power needs that year. If the recent past is a reliable predictor, the Asian giant could end up far exceeding this goal well before 2050. It certainly has the wind resources to do so.\n\nChina's ability to match concrete programs to bold ambitions is practically unparalleled, but these transformative powers come at a high price. Private property rights scarcely exist. Government policies and plans are announced, not debated. And very few dare to protest government decisions on the streets or in the courtroom. _New York Times_ columnist Thomas Friedman has acknowledged his own ambivalence about China's staggering ability to get things done in _Hot, Flat, and Crowded_ , his manifesto for a newly defined green revolution. Decrying the political gridlock that prevents America from making essential, fundamental shifts in its infrastructure and economy, he writes: \"If only America could be China for a day\u2014just one day. _Just one day_!\" During that day, he dreams of our government being able to cut through the \"legacy industries, . . . the pleading special interests, . . . the bureaucratic obstacles, [and] . . . the worries of a voter backlash\" that prevent us from reshaping the way we use our resources and treat our environment. But then, once that day is done, he wants a vibrant and vigilant public interest sector to step back in to make sure government agencies and private businesses fulfill their responsibilities under the new order.\n\nFriedman's hybrid of strategically focused authoritarianism and participatory democracy may be a pipe dream, but his invocation of China is intended as both a lament and a challenge to Americans. He wants us to reckon honestly with just how hard it will be for our nation to make transformative decisions that will truly serve the longterm welfare of our nation and planet. At the same time, he is daring us to summon the political and civic courage to bring about those very changes.\n\nOn the long flight back from Beijing, I prepared myself for the next stage of my research. I would be going into the field to meet some of the thousands of people now working to build America's wind farms and keep them running for decades to come.\n\n## CHAPTER FIVE\n\n## Working the Wind\n\nBILL STOVALL LOOKS LIKE the kind of guy you'd expect to see blazing through town on a Harley chopper. His head wrap bears a grisly skull-and-crossbones design, and a tangled thicket of white hair bursts from its fringes. Opaque sunshades, lambchop sideburns, and a walrus mustache frame a full, reddened face. His forearms are a splotchy brown mosaic, pummeled by years of exposure to the sun.\n\nIn fact, Bill's motorcycle sits in the garage at his home in Gatesville, just west of Waco, Texas. \"I ain't rode it in a year and a half!\" he tells me with a low, rumbling laugh. Bill is a long-haul trucker, one of hundreds now delivering turbine components to wind farms across America. He works for Lone Star Transportation, a company based out of Fort Worth whose trucks feature telescoping trailers for transporting blades, low-riders that allow giant tower segments to slip under highway overpasses, and multi-axle decks that can carry nacelles, each weighing a hundred thousand pounds or more.\n\nBill and I meet at a railroad yard in Reynolds, a small town in northwest Indiana. This is the depot where blades, tower segments, and other turbine components are being held until they're needed at the construction site for the Meadow Lake Wind Farm, a 200-mega-watt power plant that may eventually be expanded to five times that size. Bill's trailer is loaded with a Vestas V82 blade, 130 feet long, that stretches his truck-trailer combo to 168 feet\u2014well over twice as long as a conventional semi-trailer. He pulls to the edge of the access road, steps down from the cab of his bright-red Peterbilt truck, and walks to the spot where a heavy steel rod, the \"kingpin,\" holds the trailer in place. Turning a crank, he lowers two steel legs so that he can yank the kingpin and \"dolly down\" the trailer, leaving it standing overnight while he drives off to get a decent night's sleep at a nearby motel. Tomorrow morning before seven o'clock, he will come back to pick up the blade and deliver it to the Meadow Lake Wind Farm, twenty miles away in a flat stretch of farm country just north of Lafayette.\n\nNow in his late fifties, Bill has been a long-haul trucker for years. He used to deliver fresh meats and produce, but for the past few years he has worked in the wind industry. The hours are long, and the time away from home is endless. It's late July, and he tells me it's been sixteen weeks since he's had a single day at home with his wife. Before that, he was on the road for nine weeks. \"So I've actually been gone twenty-five weeks with one day at home,\" he says. And last year was worse. \"I drove from May to December and never got home.\" His wife visited him twice on the road. \"That didn't hurt,\" he says with a smile that vanishes as quickly as it appears. His kids are grown, but he has five grandkids and one great-grandchild; rarely does he see them.\n\nWhat keeps Bill going, along with the wages, is pride. When he was delivering meats and produce, he saw himself as performing an important, if little-recognized, service. \"You go into the stores now and everyone wants fresh lettuce and eggs and fresh meat, and there's a whole lotta old boys out on the road that go to a lot of trouble on a regular basis to make sure that stuff is there every day.\" There's dignity, not bitterness, in his tone.\n\nWind energy brings Bill equal pride. \"When I go by these wind farms, I think, 'I did my part of that. I can _see_ that.' \" He's boned up on the basics of how turbines work, the power they produce, the fossil fuels they displace. \"I found out [one turbine] puts out electricity for a thousand homes, pays for itself in five years, and lasts fifty\u2014not a bad deal!\" His reasoning is plain: \"We don't buy the air. Somebody might start chargin' for it, but for now, it sounds better 'n coal.\"\n\nThe next morning, I set out from my motel at 6:30. My goal is to catch the nine-vehicle convoy that Bill will be joining: three trucks carrying one blade each, plus six escort vehicles\u2014one in front of each blade truck, one behind. Blades travel in matched sets from factory to wind farm, their weights carefully measured to make sure that every turbine has a well-balanced rotor.\n\nThe fog is so dense I can barely see fifty feet ahead of me as I make my way south on I-65. Eventually I spot blinking orange roof lights on a green Ford pickup truck in the right-hand lane. Ahead of it, I make out the contours of a giant white blade. Moving into the left-hand lane, I edge by one trio of trucks\u2014escort, blade, escort. Then I pass a second trio, then a third. The caravan is cruising at a surprisingly swift clip given the poor visibility: about fifty miles per hour, just slightly slower than the general traffic.\n\nBill calls the forward escort for his blade truck his \"front door\" and the rear one his \"back door.\" The back door has the tougher job: its driver has to steer the rig's rear axle remotely from a few dozen feet behind the truck, using a handheld electronic device that looks like a simplified TV control, all the while keeping his own vehicle on the road. It makes texting while driving seem easy.\n\nThe advantage of having a double-escort becomes obvious when we pull off I-65 and negotiate a tight left turn onto an overpass heading east on State Road 18. Before the blade truck enters the turn, the front-door driver hops out of his minivan and uproots the stop sign at the end of the exit ramp. Otherwise the blade's tip would level it on its wide swing around the ninety-degree turn. The back-door driver then navigates the turn, keeping the blade truck's rear axle heading straight while the cab veers off to the left. Once the last blade has made its way around the turn, the trailing escort replants the stop sign.\n\nA few miles down the road, the three blade trucks followed by their rear escorts peel off the pavement to the right, edging slowly onto a semicircular dirt path that allows them to make another ninety-degree left-hand turn. Completing the arc, the trucks head off into a fogbound sea of shoulder-high green corn. The front escorts, unneeded at this stage, stand idle in a grassy holding area while the trucks unload their wares at the foot of a nearby turbine tower.\n\nA friend who has dug into National Archives photos from the Cold War tells me these wide-swinging turnoffs remind him of the aerial surveillance shots that tipped off U.S. intelligence to Cuba's installation of missiles in the early 1960s. There were no giant wind turbines then, so what else would Fidel Castro have been moving around the countryside?\n\nFor the blades and other turbine components that have made their way to the Meadow Lake Wind Farm, the 25-mile ride from the Reynolds railroad depot is the final short leg of a very long journey. Some of the blades have come by truck from a Vestas factory in Windsor, Colorado, 1,000 miles due west. Others were shipped out of Aarhus, Denmark, destined for the Port of Burns Harbor, Indiana, on Lake Michigan's southern shore. The journey across the Atlantic, down the St. Lawrence Seaway, and through four Great Lakes covers 4,500 nautical miles and takes about fourteen days. Although each load is different, the ship that brought in Bill Stovall's cargo carried 94 blades, 30 power generators, and 30 hubs.\n\nArduous though shipping from Europe may be, the hands-down long-distance medalists at Meadow Lake are the towers. Purchased from a supplier in Vietnam, they were shipped in sections across the Pacific to the Port of Vancouver, Washington, on the Columbia River just north of Portland, Oregon. That leg of the trip racked up well over 6,000 nautical miles. The next leg took them on flatbed railcars cross country to Indiana\u2014about 2,200 miles. At the Reynolds rail yard, I watched as twinned pairs of tall cranes gingerly lifted these multi-ton tubes off their flatbeds and aligned them in neat rows down the full length of the yard. Hundreds of tower sections rested on heavy-gauge steel cradles, each stamped in bold black lettering: \"Made in Vietnam.\"\n\nWith increasing numbers of companies\u2014foreign and domestic\u2014opening up manufacturing and assembly plants on American soil, trips carrying turbines from factory to farm may soon be shorter. That comes as good news to equipment manufacturers and wind farm developers alike, as transportation accounts for about 10 percent of the cost of building a U.S. wind farm. But for now, wind energy is big business for shipping companies and the ports they use. Ports up and down the West Coast, from Longview, Washington, to San Diego, handle massive shipments coming from Asia. Along the Gulf Coast, five Texas ports\u2014Beaumont, Corpus Christi, Freeport, Galveston, and Houston\u2014serve as wind technology gateways. And in the Upper Midwest, the Great Lakes ports of Burns Harbor and Duluth enjoy an increasingly brisk wind trade. Ports along the eastern seaboard are not yet major players in the wind industry, but that may change if offshore wind farms are built in the Atlantic, as is expected in the coming years.\n\nRailways are also busy moving turbine components from ports and factories to their ultimate points of use. The Union Pacific is already moving thousands of carloads of wind turbine components from ports and assembly plants to wind farms annually, and it is prepared to handle up to 6,000 railcars loaded with wind equipment at its logistics center in Manly, Iowa.\n\nThough rail haulage is gaining momentum, trucks will remain at the center of U.S. wind technology transport. Needed, in any case, for carrying turbine components their final miles to construction sites amidst cornfields and cattle pastures, they are less cumbersome than rail for most trips shorter than several hundred miles. Shifting heavy loads from one transportation mode to another simply isn't worth the cost, time, and effort if the distances aren't truly great. Even when land travel is long, wind developers often opt for trucks, especially if they are carrying big-diameter tower segments that can't make it through railroad tunnels or super-long blades that require the use of two specially fitted railroad cars per unit. Looking at land traffic coming out of the Port of Longview, the proportions are telling: 90 percent of the turbine components leave by truck.\n\nAll of this gives plenty of work to people like Bill Stovall, who are now busy, year in, year out, moving wind energy equipment from ports, railroad depots, and factories to wind farms across the country. Beyond the haulers, thousands more have found work at wind farm construction sites, readying access roads, pouring turbine foundations, erecting towers, and installing electrical systems. And then there are the thousands of people who will operate and maintain the wind farms for decades to come.\n\nI arrive at Meadow Lake, Bill Stovall's destination, to find half the wind farm's 121 turbines already standing tall. Meadow Lake's developer, Horizon Wind Energy (the same company that built Meridian Way) has dispatched two familiar veterans to supervise construction: Carole Engelder and Alvin Cargill.\n\nCarole flew in last night, held meetings all morning, and slipped away at lunchtime without my catching so much as a glimpse of her. We had planned to meet here, but I think she has come to regard me as one of the \"gnats\" she complained about back in Kansas, getting in the way of expeditious project development.\n\nAlvin, Horizon's on-site manager, is more courteous. He takes a few minutes to speak with me in the company's field office, a minimally furnished trailer that sits in a row of identical white mobile offices housing the administrative staffs of the half-dozen firms most directly involved in building the wind farm. The trailers are lined up like dominoes at one end of the laydown yard, a graded dirt expanse where concrete is mixed, gravel and other construction materials are stockpiled, and dozens of construction crew pickups are parked alongside fuel trucks and road graders. Hanging from a sturdy chain-link fence surrounding the yard is a banner displaying the general contractor's slogan: \"WORK SAFE! Your family needs you.\" These words remind me of a visit I paid to another construction site, at the Grand Ridge Wind Farm just outside Marseilles (pronounced mar-SALES), in Illinois. There I stood in spring rain alongside Adam Hartman, the fresh-scrubbed site manager for Invenergy, a Chicago-based wind developer. Almost directly above our heads, a 42-ton tower segment was dangling from a 300-foot crane. Adam assured me that serious accidents are rare at wind farm construction sites, but he indulged in a bit of gallows humor as he touched the plastic rim of his hard hat. \"We call these brain buckets,\" he said, explaining with a smile that they're particularly useful in scooping up cranial matter in the event of a major mishap.\n\nAbout 350 workers are involved in building the Meadow Lake Wind Farm. Alvin breaks that number down for me. Bowen, the general contractor, has just over a hundred people on site doing \"civil\" work. Along with grading, widening, and reinforcing the access roads that are spread across tens of thousands of acres, they have poured all the foundations for the turbines. An electrical subcontractor, Hinkels and McCoy, employs about the same number, digging tens of miles of trenches and laying electric cables that will eventually carry power from turbines to the grid. Once the civil and electrical infrastructure is in place, Barnhart Crane & Rigging comes in with about ninety workers to do the heavy lifting\u2014stacking and securing the tubular segments of each steel tower, topping off each tower with a nacelle, and hoisting three-bladed rotors into position. Vestas is also on site with a few dozen technicians who make sure the turbines' mechanical, electrical, and hydraulic systems are working properly.\n\nPouring foundations not only demands a sizeable workforce; it also consumes formidable quantities of concrete and steel. At Meridian Way, I recall being told that each foundation contains 525 cubic yards of concrete and 40 tons of steel rebars, woven and then poured in the shape of a giant inverted mushroom. Emerging from each concrete base are the threaded tips of 144 eleven-foot-long bolts, arrayed in a dense double-ring that will be used to secure the lower tower section. Between these enormous bolts and the hundreds of shorter ones that hold upper sections of the turbine together, the American Wind Energy Association (AWEA) estimates that in 2008 alone, wind developers used 2.4 million oversized bolts to erect about 5,000 turbines. No wonder Cardinal Fastener, Ohio's custom-bolt manufacturer and home of the \"Obama Bolt,\" sees wind energy as such a promising new market!\n\nAlvin arranges for me to spend the morning with Steve Maples, site manager for Barnhart Crane & Rigging. In the Barnhart trailer, Steve introduces me briefly to Clint Newbold, the company's on-site quality assurance manager. A former Marine, Clint grew up on a ranch in South Dakota and was working for a county road department when Barnhart hired him. He tells me the company recruited heavily from his corner of South Dakota. When I asked why, he belts out his answer: \"Because we're good! We're hard workers. We like to get the job done. We're no-bullshit individuals.\" With eight people assigned to his staff, Clint keeps close tabs on every Barnhart work crew operating at Meadow Lake, making sure they adhere to a rigorous set of performance guidelines.\n\nSteve and I then hop into his pickup and set out through croplands on a matrix of ramrod-straight, unpaved county roads. Filling our field of vision are dozens of turbines, all planted in unwavering lines running along an east-west axis. Some are fully erected; others are white steel stumps waiting for upper tower sections, nacelles, and rotors to be lifted into place. Beginning our drive at row D, we head south at a good clip, leaving a wall of light-brown dust in our wake. Fifteen minutes later we arrive at row K; this first phase of the wind farm continues through row Q. I can only imagine how vast the fully built wind farm will be.\n\nPhase I of Meadow Lake will deliver 200 megawatts of wind power to the grid\u2014enough to meet the power needs of about 55,000 Indiana households. Horizon's land man, Martin Culik, tells me that the project could ultimately reach 1,000 megawatts, supplying 275,000 homes with clean energy. At that scale, it would be the largest wind farm in the state by a wide margin, stretching across 130,000 acres, covering a quarter of White County and spilling over into Benton County, to the west.  Flat as far as the eye can see, virtually all of this land today is planted in corn and soy. In the near future, it will be corn, soy, and windmills.\n\nAs he turns off the county road onto a dirt path through the fields, Steve tells me that Barnhart lured him out of retirement back in 2005. He will only volunteer that he's \"well past fifty,\" but my guess is he passed that milestone many years ago. Barnhart first hired Steve to oversee construction at a wind farm in upstate New York; he had previously managed projects for a mechanical and industrial contractor in Tennessee. He's happy to be working again, and he's pretty sure his wife is happy to have him out of the house. \"It gives her a break from me,\" he jokes, but then his tone turns earnest: \"You spend all your life workin' and all of a sudden you're not, and all your friends are still workin'. You need somethin' to do.\"\n\nSupervising Barnhart construction crews more than answers that need. Cranes are costly to rent, and keeping workers fed and housed runs up a hefty tab as well. All of that makes six-day workweeks routine, sometimes including double shifts and occasionally even spilling over onto Sundays. \"Some days I wonder what I'm doin\",' Steve admits, but he quickly adds: \"Other days I know, after a few weeks, I'd be extremely bored [sitting at home]. . . . All those years dealin' with plant managers and workers. I need that camaraderie in my life.\"\n\nThe close company Steve enjoys is palpable as we sit in his truck and watch one of his Barnhart crews \"fly a rotor,\" or lift it into place, connecting it to the turbine's main shaft atop 255 feet of steel tubing. It has just started raining. Sheets of water are pouring off the corrugated metal roof of a nearby barn, but the wind is calm and there's no lightning in the area, so the work goes on.\n\nA dozen men team up to fly the rotor, which has been preassembled on the ground by bolting three blades to a hub. (Barnhart has no women on its field crew at Meadow Lake, though Steve explains that's only because the Ironworkers Local hasn't sent any along.) Four men are on the ground, and four more are atop the tower. It takes a crew of two\u2014a Barnhart operator and a union apprentice called an \"oiler\"\u2014to manage the Manitowoc \"triple 9\" crane, mounted on tank tracks with a 300-foot, elbow-jointed boom. A single operator runs a smaller wheel-mounted crane, used to stabilize the rotor as it's lifted. And then there's the quality assurance supervisor from Clint Newbold's shop. \"It's A-A-A-L-L-L about teamwork,\" Steve yells to me above the crackling of his two-way radio and the revving of crane engines. \"Nobody's gonna like everybody, but you work as a team. You're only as good as the people you're surrounded by.\"\n\nThis seems particularly true when you're raising a 46-ton rotor that's broader in diameter than a Boeing 747 is long. A lot can go wrong if any member of the team isn't properly trained or is less than 100 percent alert. Most vulnerable, perhaps, are the crewmembers high in the tower who must guide the rotor, foot by foot and then inch by inch, as it approaches its docking position at the outer tip of the main shaft, protruding just slightly from the front of the nacelle.\n\nThe Manitowoc's engine howls as its cable becomes taut, tugging on the upper rim of the rotor's hub. Two blades rise, forming a perfect V high in the air. The third blade points straight downward, the smaller crane holding its tip just a few feet off the ground. A few dozen yards away, two men stand in a cleared stretch of cornfield, each grasping a rope looped around the tip of an elevated blade. With these slender tools, they hold the rotor steady as it rises into the air slowly, majestically. This is the largest kite I've ever seen flown.\n\nTwelve long minutes into this operation, the rotor hovers with its hub slightly higher than the nacelle. The rain has let up, and Rusty Fitz gets ready to guide the rotor through its final stages of flight. He is visible high above us, ant-sized, peering over the front edge of the nacelle like a sailor looking out from a ship's prow. He begins issuing radio instructions in a Louisiana drawl: \"Luffin' up, brother. Luffin' up.\" Rusty wants the crane to lift its upper boom slightly, creating a little more space between rotor and mount. Thirty slow seconds pass as the crane operator gently shifts the rotor's position.\n\nRusty then calls out to one of the ropers in the cornfield clearing. \"Comin' in on my left tag,\" he says, looking for a bit less slack in the line. Another twenty seconds pass. \"Alright on my left tag, comin' in on my right.\" The other roper complies. Then to the crane operator: \"Gimme a little cable down. I want 'bout eight, nine feet of cable down.\" The crane lets out some cable and the rotor descends. After a \"skootch\" to the right and another to the left to finalize the alignment, the first bolts are fastened between the main shaft and the rotor.\n\nStepping out onto the hub, Rusty disconnects the crane's cable. Even with a safety harness, this is no job for the acrophobe. As the freed cable rises, he sounds euphoric: \"Alright there, Big Daddy, all clear. FINE JOB! FINE JOB!\"\n\nWith the rotor flown, Steve takes me back to the laydown yard in his pickup. Next in line at the turbine site, Hinkels and McCoy will come in and hook up all the electrical systems. Then the Barnhart quality assurance team will make sure everything's ready for a walk-through by the owners. In a few months, he expects all 121 turbines to begin sending wind-generated power to the grid.\n\nSteve doesn't sound overly sentimental when he talks about his \"five aggravatin' grandchildren.\" He admits, though, that he's proud to be playing a role in opening America's horizons to wind, and he hopes that someday his grandkids will say, \"Grandpa had somethin' to do with it.\"\n\nErecting turbines has its challenges and risks; so does maintaining them. Those looking for a cushy desk job need not apply for any of the thousands of operations and maintenance\u2014\"O&M\"\u2014jobs that are opening up at America's wind farms. I learned this the hard way during an earlier visit to Cloud County.\n\nTo experience what wind technicians undertake a few times daily, I felt it was important that I try climbing a turbine\u2014straight up a series of narrow aluminum ladders inside the tower to the hatch on the underside of a Vestas V90 nacelle, 262 feet in the air. My folly should have been obvious from the start: I am mildly claustrophobic, vaguely uncomfortable with heights, not particularly robust, and, to echo Steve Maples, \"well past fifty.\"\n\nBefore setting out on the climb, I was given basic instruction by Meridian Way's operations manager, a two-tour Iraqi war veteran named Justin Van Beusekom, burly, sun-baked, his lower lip bulging with chewing tobacco. First, Justin had me review a multipage release form outlining the hazards of working up-tower, from hypothermia in the winter to dehydration and heat exhaustion in the summer. Then he fitted me with a hard hat, protective glasses, gloves for a better grip, and a harness heavily laden with clips and straps. I was already wearing the mandatory hard-toed work boots, purchased the previous day at Walmart.\n\nAs we drove to the turbine, Justin distracted me by talking about two of the local wildlife dangers listed on the release form: rattlesnakes and the small but surprisingly venomous brown recluse spider. With a slight flair for exaggeration, he warned that the spider's bite can cause a finger to \"swell up and explode.\" As for rattlers, I needed no convincing: the previous day, I had seen one slither across a packed-dirt farm road just a few feet in front of my car. In truth, though, my mind was on neither spiders nor snakes; I was girding my nerves for the climb.\n\nAt the site, Justin keyed in a command to pause the turbine's operation; no one goes up-tower when the rotor is turning. We entered the tower through a rounded, shiplike steel door, a few steps off the ground. Inside the air was dank with the smell of grease and plastics. Following Justin's instruction, I latched my harness, with its fallarresting clamp, to a cable running up the center of a short stretch of ladder to a grated steel platform just overhead. Getting to that first platform was easy. Then came the next span\u2014about fifty feet straight up into near-total darkness. And beyond that, I knew, were two more spans covering the remaining distance to the top\u2014nearly 200 feet into a cavity that would be shrinking from 12 feet across to only 7.5 feet in diameter at the top. I also knew that magnets were all that held these slender vertical spans to the tower's interior wall\u2014a means of preserving the maximal strength of this giant steel tube.\n\nWith every step I took, the temperature seemed to rise and the air became heavier, more stagnant. For several hours the sun had beaten down on the tower's exposed steel, warming it through and through. I tried not to look up; I tried not to look down. My arms strained, my thighs ached, my heart was pounding. Panic overcame pride; I knew I couldn't make it.\n\nI half-expected a soldier's scolding when I called down to Justin to say I'd had enough. Instead, he calmly coached me through the mechanics of the descent. As I exited the tower through its narrow hatch, I felt a huge rush of relief that left little room for humiliation.\n\nBack in Horizon's field office, I untangled myself from the web of harness gear and shared with Justin what my two teenaged daughters had said when I told them about my planned climb. \"Are you _crazy_?\" one exclaimed. The other insisted: \"You've _got_ to do it if you're writing a book about wind.\" Justin smiled and responded: \"They were both right!\"\n\nMy awe for the rigors of O&M work only deepened on my visit to the Grand Ridge Wind Farm outside Marseilles, Illinois. Leo Jessen is wind developer Invenergy's point man at this 99-megawatt project, supervising a crew of six O&M technicians. A clean-shaven head and fashionable earring somewhat mask his years, but at forty-seven, Leo is older than the other members of his team. Though still relatively quick at climbing turbines, he pithily observes, \"There's old climbers and bold climbers, but no old bold climbers.\" It's all about pacing, he claims. \"You might be in the greatest shape in the world; you might be able to run up the ladder; but you've got to understand that, when you get to the top, you've got to do the work.\" A young buck who races up the ladder in five minutes and then collapses in exhaustion for forty-five is less valuable, he says, than someone who takes twenty minutes to make the climb but then gets right to work.\n\nLadders on the GE turbines at Grand Ridge are equipped with counterweights that can reduce technicians' effective climbing weight by as much as 50 percent. Leo stresses, though, that technicians typically carry at least forty extra pounds in tools and safety gear, so getting to the top of a tower still demands real effort.\n\nWeather extremes, up-tower as well as en route, impose further physical stress on O&M technicians. Mechanical heat quickly dissipates when turbines are shut down, leaving workers aloft in winter temperatures that can easily drop below zero. In summertime, the temperature inside a nacelle\u2014ventilated but not air-conditioned\u2014often tops a hundred degrees. \"If you're a TV dinner, this is the ultimate job for you,\" Leo says. \"You'll freeze in the winter and cook in the summer.\"\n\nWhat do O&M technicians cope with, aside from the weather? First of all, turbines\u2014like any machine\u2014require scheduled maintenance. Gearboxes need periodic oil changes, using specially equipped pump trucks to siphon off the old oil and pump in the new. Grease must be applied to the gears that control the pitch of turbine blades and the mechanism that changes the \"yaw,\" or compass orientation, of the rotor, both of which are constantly adjusting to optimize rotor speed under different wind conditions. Also needing periodic checking and replacement are brake pads, used along with blade-feathering to hold the turbine still during maintenance and, in a pinch, to help keep the rotor from spinning out of control. Then there are the gearbox failures, generator misalignments, electrical defects, hydraulic control issues, and other non-routine problems that have to be handled as quickly as possible to minimize the time and cost of taking turbines out of production.\n\nLeo's goal is to keep the Grand Ridge turbines operating as close to 100 percent of the time as is physically possible. In its first half-year of operations, the turbines performed at 98 to 99 percent availability\u2014the term used to reflect the percentage of time that a turbine is fully operational, or _available_ to capture the wind.  Leo attributes this very high availability to the fact that the equipment is still new, but it also reflects the responsiveness and training of his O&M crewmembers.\n\nDemand for well-trained O&M technicians, on-site managers, and administrative support staff at American wind farms has been growing, but the rate of that growth has been directly affected by fluctuations in the U.S. economy. In 2009, new wind installations totaled nearly 10,000 megawatts. Although the economy had already slumped, turbine orders placed before the recession allowed the industry to maintain a brisk rate of development that year. To service the turbines that went on-line in that year alone, AWEA estimates that 1,000 new O&M jobs were created. The recession took its toll on wind development the following year, however, cutting newly installed wind capacity in half and substantially reducing demand for new O&M hires. Wind turbine technology has also become more reliable with every passing year, making it possible to spread O&M crews more sparingly across ever-larger turbine arrays.\n\nEven with these market fluctuations and technology developments, the need for trained technicians has been big enough to nurture a new generation of specialized training programs at more than two dozen centers of learning across the country. Cloud County Community College's story is emblematic.\n\nWhen Bruce Graham gave up his high school teaching career to launch Cloud County's wind program in 2007, he had to work hard at building faculty support. With only four students in his program, he had an uphill burden of persuasion. Before long, though, the school's top administrators came to embrace Bruce's effort as their poster child. Students were coming from several states across the lower Midwest, and money was flowing in\u2014about $1.8 million in federal grants and nearly $1 million in state funding. \"Wind has become our icon,\" the college's vice president for academic affairs proclaimed in March 2010. Today, Cloud County's Wind Energy Technology Department has more than a hundred students enrolled in its two-year program, taking courses that run the gamut from mechanics, hydraulics, and electronics to turbine siting, worker safety, and data acquisition. With an associate's degree in applied science, they emerge primed to enter a job market where they find themselves competing with a growing cohort of well-trained technicians, many of them coming out of similar programs at places like Iowa Lakes Community College.\n\nYou can tell that the wind industry has penetrated the national ethos when a community college's training program for wind technicians is featured on a prime-time TV ad. That is exactly what happened when a commercial for Duracell batteries showed young guys in hard hats climbing a turbine and inspecting circuits, with the voice-over saying: \"At Iowa Lakes Community College, the students learn to keep America's wind turbines going. And to keep them safe, the only battery they trust in their high-voltage meters are Duracell rechargeables.\"\n\nThe Wind Energy and Turbine Technology Program at Iowa Lakes is housed in an attractive modern building on the college's campus in Estherville, a small village in the northwest corner of the state. This is a far cry from Cloud County's temporary rented quarters, shoehorned between the Dollar General store and the Dragon House Chinese restaurant in a Concordia strip mall. Ahmad Hemami, an instructor in the fundamentals of electricity, guides me through a half-dozen well-appointed teaching labs and a nearly equal number of classrooms. In the labs, students gather around electronic control boards, computer monitors, and wind-measuring anemometers, looking up only briefly as we walk through.\n\nAhmad then takes me to see a Vestas V90 nacelle, neatly mounted as a floating classroom in a courtyard just off the main corridor. I climb a single short ladder leading up to the suspended capsule and finally get to eyeball what I would have seen if I'd completed my climb at Meridian Way. The nacelle's cramped metallic interior feels like a cross between a submarine engine room and a space shuttle cabin. To make room for students, there is no main shaft running the length of the chamber and other large components, such as the gearbox and generator, have also been removed. Even so, the nacelle is a crowded maze of circuit boxes, electric motors, hydraulic hoses, and cooling fans to keep workers and machinery within a bearable temperature range. It's not hard to imagine how tightly packed a nacelle must be when filled with all the machinery that's needed to turn motion into power.\n\nIowa Lakes may be in a remote corner of a rural state, but from a wind technology perspective, it's in an ideal spot. Aside from having its own fully functioning wind turbine just a half-mile from campus, the college sits in a region that is replete with commercial-scale wind farms. \"They're right smack-dab in northwest Iowa, where we need all the technicians,\" says Clipper's Bob Loyd, who sits on the Iowa Lakes board of advisers and is an enthusiastic booster of the program. The college also enjoys easy access to technology leaders at larger educational institutions like the University of Iowa and Iowa State. Barry Butler, dean of the University of Iowa's College of Engineering, is unstinting in his praise for the program and is in awe of the program's physical plant: \"The laboratories they have are just enviable, even by our standards,\" he says. \"I'd love to have some of the equipment they have.\"\n\nAmong the students at Iowa Lakes are seasoned tradespeople like Joe Brightwell and Richard Dunham. Joe was a unionized electrician in Montague, Michigan, until his work dried up with the collapse of the housing industry. Richard was a general contractor in Atlanta. When the recession devastated his business, Richard was already in his early fifties, yet he coolly set about searching for a new career. As a first step, he decided that he wanted to work in a field that would draw on his university training long ago in mechanical engineering. Next, he surveyed different energy technologies, ruling out several of them as less-than-promising employment prospects. \"I wasn't convinced that nuclear power was going to come back, and there's already a lot of people in coal,\" he reasons. \"So I looked at the alternative energies. Of course, wind seemed to be the most viable, and I knew it was something I would be interested in because there's a lot of mechanics as well as electrical work.\"\n\nThough sold on wind, Richard is far from a true believer when it comes to climate change. \"People are being persuaded falsely that there is a traumatic experience that's going to happen in the next fifty to a hundred years, like glacier melting and so forth,\" he says with obvious derision. On the other hand, he disdains coal company profiteering, loathes American reliance on Middle East oil, and believes that tapping the wind is an important way to secure American energy independence.\n\nWhile students like Joe and Richard have already logged in decades in the technical trades, most Iowa Lakes trainees are young, barely out of high school. Paul Johnson is one of them, coming to Iowa Lakes after a short stint at a small college in North Dakota. The valedictorian at his hometown high school in northern Minnesota, Paul has a little trouble squaring his current studies with the expectations built up over his high school years. \"In high school, these technical programs were kind of frowned upon,\" he acknowledges. \"It was expected you could do better.\" He tells me that he hasn't managed to get any of his high school classmates excited about wind power, yet he speaks with animation about the industry's value in creating well-paying rural jobs. Growing up on a small farm with two parents who needed outside jobs to make ends meet, he is painfully aware of the strains facing family farming and has a sober respect for the difficulties in finding new local jobs to fill the vacuum.\n\nWind developers make a point of hiring local labor wherever possible. They know that when county commissions and economic development agencies weigh the pros and cons of inviting a new wind farm into their communities, the promise of new jobs will be a major consideration. Construction crews like the one led by Steve Maples at Meadow Lake have large contingents of workers hired through local unions. O&M teams like Leo Jessen's at Grand Ridge also draw from neighboring areas; Leo himself grew up and now lives in the town of Ottawa, Illinois, a dozen miles from the wind farm field office.\n\nWhile many of the workers on wind farm construction crews and O&M teams are local, few of them are women. Project development teams often include women in supervisory and administrative roles; Carole Engelder is one example. But in the field, they are a rarity. At Meadow Lake, there were none in Steve Maples's ninety-man-strong construction crew. At Grand Ridge, I observed another construction team that included just one woman slogging through puddles on a rainy spring day, brushing and hosing mud off tower sections that had been lying in the fields. Meanwhile, the men busied themselves with their big machines, hoisting the freshly cleaned steel tubes into place.\n\nOn the operations side, Michelle Graham is an interesting and unusual hybrid. On one hand, she juggles a range of roles in the Meridian Way field office. She prepares daily reports on turbine availability, wind conditions, and needed repairs; she leads informational tours of the wind farm; and she maintains constant ties with the wind farm's participants and abutters, making sure they aren't caught off guard when an O&M crew goes out to repair a blade or replace a gearbox.\n\nAt the same time, Michelle has worked hard at defining a useful role for herself up-tower. To do this, she first had to get over her aversion to heights, so she began practice-climbing up a twenty-foot ladder that her husband Bruce\u2014a wind instructor at home as well as at Cloud County Community College\u2014had lashed to the side of their barn. \"I strapped on his climbing harness and climbed up and down that ladder fourteen times,\" she told me. Afterward, she collapsed on the couch, her body exhausted and her hands blistered. That was in July 2009. By December, she had reached the top of a Vestas V90, where she spent several hours learning how to audit the services performed by O&M technicians\u2014making sure bolts were tight, oil and grease levels were sufficient, and conditions were clean. Today, with her fears behind her, Michelle views up-tower quality assurance auditing as part of her on-the-job repertoire.\n\nRare among workers in the field, women are nearly absent from the student rolls at technical training programs. At Cloud County Community College, only four out of a hundred students in the 2010\u201311 academic year were women. The previous year, Iowa Lakes had just five women enrolled in its program.\n\nOne of them was Loma Roggenkamp, a native Pennsylvanian who came to the wind energy field after trying her hand at marine biology in Maine. \"I thought the ocean was our final frontier, and we were going to find things to save the world out there.\" Unable to land a job that drew on her studies, she worked in a graphics firm for several years. Then one day her mother spotted a wind technology ad on the Internet, and Loma immediately went online to explore possible points of entry. Training would be needed, she knew, and Iowa Lakes kept coming up as a great place to gear up for a career change. Soon she enrolled in the college with her family's blessing.\n\nImmediately on graduating from Iowa Lakes in June 2010, Loma was snapped up by Siemens Energy to work as an O&M technician. First she was dispatched to Houston for three weeks of training; then she was sent to her \"home farm,\" a 200-megawatt facility in Glenrock, Wyoming, 5,000 feet above sea level. At Top of the World, as the wind farm is fittingly called, Siemens operates forty-four 2.3-megawatt machines\u2014just over half the farm's installed capacity.\n\nIn a phone call at the end of her first day up-tower, Loma's exhilaration is palpable. An athlete, she relishes the climb\u2014nearly 300 feet straight up. \"I can climb every day, and I don't have to pay for a gym membership!\" she quips. And then there's the view. The Siemens nacelle opens like a clamshell, giving workers a stunning vista onto the surrounding countryside. \"I can troubleshoot a turbine and look at nature at the same time,\" she says.\n\n\"I'm making a difference and I'm needed in this industry,\" Loma tells me with confidence and pride. That difference isn't just advancing a technology she believes in; it's helping women find safe and welcoming pathways into the wind energy workforce. Not surprisingly, much of the safety training for O&M technicians is focused on men; she wants to help develop training guidance for women. \"Men aren't the only ones with appendages that are vulnerable,\" she says. Along with opting for the greater comfort of a safety harness configuration shaped like an \"X\" rather than an \"H\" in the chest area, women need to be cautioned not to wear flammable synthetic undergarments that can melt and adhere to the skin in case of fire.\n\nLoma is now working with a nonprofit group, Women of Wind Energy, to develop these training materials. She has found an enthusiastic ally and supporter in Kristen Graf, the organization's executive director. Kristen attributes the dearth of women in technical jobs partially to the stereotype that these sorts of positions are traditionally the domain of men. Women don't necessarily hear about wind energy training and hiring opportunities, so opening up the flow of information about schools and jobs is one needed step. Even for those women who stand up to the stereotypes, surmount the often-subtle biases, and get hired, Kristen says that working conditions built on a male culture can sometimes be unwelcoming and isolating. \"Some women are good at playing the role of being 'one of the guys,' \" she observes, \"but that shouldn't be a requirement if you have all the skills needed for a particular job.\" Her goal, and that of her organization, is \"to be more proactive in creating a working environment that fits everyone.\"\n\nAt least as troubling to Kristen is the underrepresentation of women in higher-level corporate positions. Women of Wind Energy, which has thirty state chapters and a few additional branches in Canada, provides fellowships for women to attend the AWEA annual meeting, a multiday conference and trade fair that draws tens of thousands of participants each year. Newcomers to the job market get easy access to wind energy employers at these gatherings. Kristen describes Denise Bode, CEO of AWEA, as \"a real champion\" in raising the profile of women in the industry. Even with those efforts, there remains a very large gender gap in wind energy management\u2014a gap that is reflected by the composition of the AWEA board: out of twenty-eight members serving as of April 2011, only five were women.\n\nJeanna Walters, a Cloud County graduate who has gone on to pursue a bachelor's degree in environmental science at Kansas State, sees the challenge facing women in wind as beginning with the school system. \"We really need to push the science and math on girls in high school,\" she insists. Even if women don't want to end up working in construction or climbing turbines, she is sure that opportunities abound for women in engineering, land acquisition, development oversight, financing, operations management, and marketing. \"This is definitely an industry that girls can be involved in\u2014and should be.\"\n\nIt's hard to predict how many U.S. jobs a thriving wind energy sector might provide in future years, but that hasn't kept the U.S. Department of Energy (DOE) from trying. In a report called _Wind Power in America's Future: 20% Wind Energy by 2030_ , it looked at the prospects for generating electricity from wind. If wind were to supply 20 percent of America's power by 2030, the DOE research team estimated that annual installations of new wind-generating capacity would have to reach 16 gigawatts by 2018\u2014an increase, but certainly not an impossible leap from an installation level that came very close to 10 gigawatts in 2009. Under this scenario, total installed wind power nationwide would reach 300 gigawatts by 2030\u2014sufficient to generate a fifth of the 5.8 billion megawatt-hours projected to be the overall U.S. demand for electricity by that date. This is _not_ a formula that optimistically flatlines U.S. electricity use between now and the target date; to the contrary, a 39 percent increase above total consumption in 2005 is built into the calculation. If we actually became a nation that valued energy conservation more than we do today, 300 gigawatts of installed wind power could end up providing well over 20 percent of the nation's power needs by 2030.\n\nUnder the _20% Wind Energy by 2030_ scenario, manufacturing jobs directly related to producing wind turbine components and subcomponents would top 30,000 by 2021, peaking at 32,835 in 2028. While factory work would somewhat slacken thereafter, ongoing expansion in generating capacity\u2014both onshore and offshore\u2014and the need to re-power aging wind plants would guarantee a continued high level of employment in the manufacturing sector. In construction, jobs would average over 70,000 a year from 2019 through 2030. And in wind farm operations, total jobs would reach 76,667 by 2030\u2014about 28,000 in on-site O&M and another 48,000 in utility services and subcontractors. Adding them all up, DOE foresees about 180,000 jobs directly linked to wind energy as the 2030 target date approaches.\n\nBeyond all the \"direct\" jobs in the wind energy economy, DOE also explores the \"indirect\" employment benefits of growing this sector. These jobs include the producers and suppliers of steel, fiberglass, and other materials that are used to build wind turbines; the companies that produce the parts that go into turbine components and subcomponents; and the providers of banking, accounting, legal, and other services to wind turbine manufacturers and wind farm contractors. These indirect jobs are expected to number about 100,000 annually in the years leading up to the 2030 target date.\n\nFinally, DOE draws an even wider circle around the \"induced\" jobs resulting from consumer spending by people directly and indirectly employed in the wind energy sector. A Clipper factory worker buys a new pair of jeans in a local store; an O&M technician takes his family out to dinner; a crane operator stays at a local motel. The DOE team attributes another 200,000 jobs per year to these induced economic activities.\n\nFolding induced jobs into the assessment of wind energy benefits may go farther down the speculative road than some are ready to travel. But even setting that outer circle of employment impacts aside, we are looking at a roster that rises to more than a quarter-million direct and indirect jobs if we pursue DOE's _20% by 2030_ goal. A technology commitment that advances America's energy independence and reduces our nation's carbon footprint while creating hundreds of thousands of new, skill-based jobs\u2014isn't this a path worth taking?\n\n## CHAPTER SIX\n\n## The Path to Cleaner Energy\n\nBEYOND BRINGING JOBS TO tens and eventually hundreds of thousands of Americans, can we expect wind energy to make a real difference in the way we fuel our economy? How much can we count on wind energy to rein in the pollution that is compromising our health and warming the Earth's atmosphere? And what will it cost us to make wind a major part of our energy diet?\n\nLooking at wind energy's contribution today, it may seem premature\u2014even presumptuous\u2014to think of the technology as a gamechanger. During 2010, wind supplied 2.9 percent of America's power needs. Coal, during the same period, delivered 45 percent of our electricity, and natural gas generated another 24 percent, followed closely by nuclear power, with a 20 percent share of overall output. Another 6 percent of our electricity came from conventional hydroelectric dams, while solar energy\u2014photovoltaics and thermal systems combined\u2014barely registered a blip on the screen at 0.03 percent.\n\nFrom the broader vantage point of energy use across all sectors, wind's current status looks humbler still. Electricity generation amounts to a little less than 40 percent of total U.S. energy consumption, with most of the rest relying on the direct burning of fossil fuels to run our vehicles, stoke our industries, and heat our buildings. So that means wind meets slightly more than 1 percent of our overall energy needs.\n\nWind energy's role in weaning America off fossil fuels may be modest today, but its untapped potential is vast. In February 2010, the government-run National Renewable Energy Laboratory (NREL) released the results of a mapping effort that gauges the full magnitude of land-based wind energy as a resource that can serve America's power needs (see table 1). First NREL identified \"windy land areas,\" which it defined as areas where the average winds are strong enough, at 80 meters (262 feet) above ground, to allow turbines to produce electricity at a minimum of 30 percent of their full installed or \"nameplate\" capacity, averaged annually. (A 30 percent \"capacity factor\" is considered moderately robust.) Areas with annual wind speeds averaging at least 14.6 miles per hour were regarded as meeting this threshold.\n\nFrom this gross measure, NREL then subtracted territory that it deemed unlikely to be developed for wind because of conflicting uses or characteristics\u2014urban areas, parks, designated wilderness, and areas with water features that could hinder wind development. In my home state of Massachusetts, NREL rated less than 1 percent of the state as sufficiently windy, and it disqualified 88 percent of that small area because of conflicts. At the other end of the spectrum is Kansas, where almost 90 percent of the land meets the threshold for windiness, and only 10 percent of that windy area has been sidelined by NREL because of conflicts.\n\nI remember loving those spin art kits that were toy store staples in the 1960s. After dribbling paints from ketchup-like dispensers onto a rectangular sheet of cardboard, I'd flip the switch of a battery-operated spinning wheel. Within seconds, a swirling, sometimes lurid maelstrom of colors emerged. On first glance, NREL's digitized wind map of the United States reminds me of those creations. In the Southeast, spring green splashes across the Atlantic coastal states, reaching as far inland as Tennessee, Alabama, and eastern Mississippi. This color connotes average wind speeds of less than 11.2 miles per hour\u2014well below what's needed to make wind a strong competitor with other power-generating fuels. Moving up into the Northeast, green yields to splotches of yellow and shades of brown, suggesting wind speeds edging up to 15 miles per hour. The same dappling of forest colors spreads from the Pacific Coast through Idaho, Utah, Nevada, and New Mexico. Then the real excitement begins. From the Rockies, across the Great Plains, stretching from eastern Montana and the Dakotas down to Texas, deep veins run from rust to violet to purple, plunging occasionally to midnight blue, revealing vast areas where average wind speeds range from 15 to 20 miles per hour. These expanses are the stuff of wind developers' dreams.\n\nAdding up all the qualifying windy areas, NREL projects that America could install up to 11,000 gigawatts of land-based wind power, yielding roughly 38.5 million gigawatt hours of electricity per year. _That's nine times our total electricity production today_!\n\nOffshore wind is another huge energy resource, virtually untapped in America, though a growing contributor to European power generation. NREL has concluded that U.S. ocean waters within fifty nautical miles of land, plus U.S. portions of the Great Lakes, could yield 4,000 gigawatts of wind-generating capacity, bringing our combined onshore and offshore wind energy potential to 15,000 gigawatts. _That's nearly fifty times the amount of wind energy that the Department of Energy says we will need to harness to supply 20 percent of our nation's power needs by 2030_.\n\nA whole host of environmental, aesthetic, and logistical concerns can make many a wind-rich site a poor prospect for building a wind farm. From my own advocacy for onshore and offshore wind energy in New England, I know how huge a leap it can be from identifying a resource to seeing wind blades spinning on the horizon. Whether the concerns are about harm to wildlife, noise, inadequate access to transmission, or simply the interrupted view, the realm of sites acceptable to policymakers and the public is considerably smaller than the universe of wind-worthy areas. We may not want to develop every promising wind site in the nation, but it's clear that wind energy could become a mainstay of our energy economy using just a small fraction of the available resource.\n\nWe may have more wind on hand than we can possibly use, but what will we gain by tapping this resource? One clear \"win\" is the opportunity to reduce America's contribution to global greenhouse gas emissions.\n\nThe carbon emissions gap between wind power and conventional fossil fuels is nothing short of stupendous. The Swedish utility Vattenfall, Europe's fifth-largest electricity producer, has prepared a life-cycle environmental assessment of a range of generating sources. Typical of cradle-to-grave environmental assessments, this evaluation looks at the materials used and the pollutants generated at all stages of a technology's implementation. For coal, this includes building the power plant; mining, purifying, transporting, and storing the coal; burning it to produce electricity; disposing of coal ash; and decommissioning the plant at the end of its productive lifespan. For natural gas, the study's scope is similar, beginning with drilling and running all the way through power plant decommissioning. For wind, the coverage extends from manufacturing and operating the turbines to end-of-life dismantling. Taking all these factors into account, Vattenfall reports that wind, averaged over its full life cycle, produces about 10 grams of carbon dioxide (CO2) per kilowatt hour of electricity generated, while coal releases 600 to 700 grams per kilowatt hour\u2014that's sixty to seventy times as much CO2 per unit of power. Gas, depending on the technology used, ranges from 400 to 1,300 grams of CO2 per kilowatt hour\u201440 to 130 times as much CO2 as wind.\n\nHere in the United States, NREL conducted its own life-cycle assessment of coal-based power production in 1999. It found that the average U.S. coal plant emitted 1,022 grams of CO2 per kilowatt hour, and newer coal-burning facilities released 941 grams per kilowatt hour. In more recent years, the performance of our coal plants hasn't improved very much. In 2008, the average coal-burning facility put out about 995 grams of CO2 per kilowatt hour. Newer, more efficient plants performed only slightly better, with the hundred best-performing units averaging about 865 grams per kilowatt hour.\n\nVestas has been an industry leader in preparing and publishing life-cycle analyses of its wind turbines, following guidelines established by the International Organization for Standardization (ISO). Looking at the V90 3-megawatt machine that stumped my climbing efforts at Meridian Way, the company estimates that each turbine will generate about 158,000 megawatt hours of electricity over the course of its twenty-year lifetime. By using wind rather than coal to generate this power, Vestas expects that a single turbine will spare the environment about 130,000 tons of CO2 pollution over the course of its useful life.\n\nAnother tool for evaluating different power-generating technologies is called an \"energy balance.\" In an energy balance, the energy consumed by a given technology is compared to the energy it produces over its full operational lifetime. Vestas analyzed the V90 in this manner. First it looked at the extraction and production of raw material inputs like steel. Next it evaluated the energy used in manufacturing, installing, operating, and ultimately decommissioning the turbine. (The steel used in the tower and other major components can be recycled once a turbine is taken down, by the way.) Balancing all those inputs against the turbine's estimated lifetime energy output, it found that each V90 produces enough electricity to cover its full life-cycle energy inputs within the first 6.6 months of operation. From that point onward, all of the turbine's generated power is essentially carbon-free.\n\nFor coal, the story is very different. For every moment a coal plant operates, from its first day in service to its last, huge quantities of energy are consumed and correspondingly huge amounts of carbon are released into the air. Giant excavators keep grinding away at coal seams, diesel-powered trains continue to pull endless chains of coal cars across hundreds of miles of open country, and power plant boilers burn the fuel in nearly unfathomable quantities right up until the day the plant shuts down. The energy balance for coal and every other fossil fuel begins in a deficit that only deepens with every new ton of fuel consumed.\n\nThe Department of Energy has looked at the CO2 emissions that we could avoid by generating 20 percent of our power from wind. It estimates that this switch to wind power would cut our yearly CO2 emissions by 825 million metric tons, or about 26 percent of all projected CO2 emissions from the power sector in 2030.\n\nIn addition to burning finite energy resources throughout their years of operation, coal and other fossil fuel\u2013based power plants spew massive amounts of dangerous pollutants into the atmosphere. Sulfur dioxide emissions are about 48 times higher and nitrogen oxide emissions are 42 times higher for coal plants than for wind farms, averaged over the full life cycle of each facility. (For wind farms, air emissions\u2014like the energy inputs already discussed\u2014are primarily associated with the fuels burned in producing, erecting, and dismantling the turbines, not with day-to-day operations.) Methane, an ozone precursor and a greenhouse gas that\u2014gram for gram\u2014is much more potent than CO2, is produced at levels 320 times higher for coal than for wind. Modern, combined-cycle gas facilities fare better, although the gap is still huge: sulfur dioxide levels are 4.6 times larger, nitrogen oxides are 11.4 times greater, and methane emissions are 124 times as high as wind. Adding this heavy dose of pollutants to the CO2 loadings caused by coal and gas facilities, it's clear that our ongoing reliance on fossil fuels comes at a very high environmental price.\n\nThe environmental drawbacks to fossil fuels, grave though they are, pale by comparison to the \"perils of the peaceful atom.\" In one of the most unsettling paradoxes of modern times, mounting concern about global warming has brought nuclear power to the fore as a \"carbonneutral\" technology that is ostensibly friendlier to the global climate than coal or other fossil fuels. A few veteran voices in the environmental community, like Stewart Brand, founder of _Whole Earth Catalog_ and an early mentor of California governor Jerry Brown, have even joined this chorus.\n\nWhen I was in graduate school in the early 1980s, I had a professor who was fond of explaining statistical probability with the reminder: \"Rare events _do_ happen.\" Those words haunted me at the time. Only a few years had passed since a combination of faulty design and human error had led to the partial meltdown of a nuclear reactor at Three Mile Island, near Harrisburg, Pennsylvania. Although no public health disaster ensued, the accident caused the American public to doubt the blithe assurances of nuclear industry leaders about the safety of their technology. Then Chernobyl melted down in 1986, and the nuclear nightmare that had been averted at Three Mile Island became a terrifying reality. Tens of thousands were exposed to dangerous levels of radiation, thyroid cancer began appearing at an alarming rate, and about 9 percent of the Ukraine and 23 percent of Belarus remain contaminated today.\n\nThe disaster at Japan's Fukushima Daiichi nuclear complex came as the next tragic reminder of nuclear power's menace. As I learned about the many tens of thousands of people who had to flee their homes and the hundreds of emergency workers who subjected themselves to extremely hazardous radiation levels while struggling to contain the damage, my thoughts kept circling back to my professor's warning about rare events. Should nuclear safety experts really have to plan for a double blow as improbable as an earthquake of practically unprecedented magnitude and a tidal wave of epic proportions hitting the same reactor complex in rapid succession? And yet this is exactly what happened within a few decades of the plant's construction. It's too soon to know the full impacts of this catastrophe, but the folly of relying on nuclear power to meet the twenty-first century's energy needs couldn't be clearer.\n\nAs horrific as accidental reactor meltdowns can be, nuclear power's hazards to humanity and the environment extend beyond the realm of the unintended. In the post-9\/11 era, we already know the havoc that can be created by commercial jetliners commandeered as guided missiles. In fact, targeting a nuclear reactor was an option considered by Al Qaeda in the period leading up to the World Trade Center and Pentagon attacks. While nuclear industry spokespeople have claimed that a plane's likelihood of breaking through a reactor's concrete containment and exposing the reactor core is low, the National Academy of Sciences has warned that an aerial attack on a power plant's spent fuel storage pool could drain the pool and trigger a high-temperature fire, releasing potentially deadly quantities of radioactivity into the environment.\n\nShort of a direct attack placing nearby populations at risk, it would be foolhardy to ignore the hazards of further proliferating nuclear fissile materials in an unstable world where rogue regimes and terrorist groups are hustling to develop their own nuclear stockpiles. Those who admire France's commitment to nuclear power often gloss over the huge risks involved in that country's reliance on fuel reprocessing to keep its reactors going. Separated plutonium\u2014a key product of reprocessing\u2014is a prime feedstock for nuclear weapons, much more valuable to potential bomb makers than decaying fuel rods. This danger has led Peter Bradford, former member of the U.S. Nuclear Regulatory Commission, to warn: \"In a world in which reprocessing becomes an integral part of the fuel cycle, the International Atomic Energy Agency's proliferation safeguard system would face challenges that it is not capable of dealing with because separated plutonium is so much easier and quicker to transfer into a weapon than . . . fuel rods that are in a reactor pool.\"\n\nThen there is the nuclear waste. Once the fuel rods reach the end of their useful lives, they are removed from the reactor core and are placed in temporary storage, usually at the reactor site but sometimes off-site, waiting for a safe method of long-term disposal to be discovered. In nuclear power parlance, \"long-term\" means tens and even hundreds of thousands of years for certain radioactive isotopes that are waste products of the fuel cycle. Eventually, when a reactor is decommissioned, much of the plant's hardware must also be sequestered until its radioactivity subsides. Because of the costs and dangers involved in dismantling them, only three of the twenty-four permanently closed U.S. power reactors have been fully decontaminated; nearly half are in a holding pattern that the Nuclear Regulatory Commission calls SAFSTOR, deferring decontamination to a later date.\n\nEver since the early 1980s, debate among experts and politicians has raged about the viability of a deep underground high-level nuclear waste repository at Yucca Mountain in Nevada. By the time President Obama delivered on a campaign promise to halt further work on the Yucca project in 2009, the government had already spent $7.7 billion at the site. Over its full operating lifetime, the facility would have cost taxpayers and electricity ratepayers close to $100 billion.\n\nWith Yucca on hold indefinitely, the search for a viable long-term storage solution for nuclear waste continues. Meanwhile, every operating reactor continues to produce about 20 metric tons of used nuclear fuel per year, filling up temporary storage facilities way beyond their planned capacities. Much of the waste is stored in pools just like the ones that erupted into steaming, highly radioactive cauldrons at Fukushima Daiichi. \"Hubris\" doesn't begin to plumb the depths of our imprudence in perpetuating and possibly expanding our reliance on an ultra-hazardous technology that we have no idea how to manage into the indefinite future.\n\nEven the carbon neutrality claim about nuclear power is notably weak. Like fossil-fuel power plants, nuclear reactors demand an ongoing supply of fresh fuel, in the form of highly enriched uranium. Each stage in the nuclear fuel cycle consumes energy\u2014from the equipment used to mine uranium, to the power demanded by the various stages of concentration, enrichment, and fuel-rod fabrication, to the transportation of nuclear materials from one stage to the next. And what about the massive amounts of carbon emitted by the huge mobilizations involved in containing and cleaning up Three Mile Island, Chernobyl, and Fukushima?\n\nBeyond wind energy's clear merits as a safe, low-carbon power producer, it offers some dramatic new ways to lessen our dependence on foreign energy resources. The opportunities begin with oil.\n\nWind energy opponents often argue that harvesting the wind will do little to reduce America's reliance on Middle East oil. They rightly point out that a very small fraction of America's electricity comes from oil-fired power plants, most of which were phased down, if not out, in the 1970s, when skyrocketing oil prices in the post\u2013oil embargo years triggered a massive switch to domestic coal. Evidence of our success in cutting back on the use of oil for electricity production is unequivocal: oil supplies less than 1 percent of our present-day power needs, way down from 16 percent in 1973. If only we had been remotely as effective in trimming our overall appetite for oil during this period! Total U.S. oil consumption is nearly 2 million barrels a day higher today than it was in 1973.\n\nWind power's advantage may not be in displacing oil-based power generation, but it could make a different kind of dent in our oil dependence by changing the way we move people and goods around the country. Transportation ranks second only to power generation as a U.S. energy user, accounting for 27 percent of all energy we consume. And of the energy we use for transportation, almost all of it\u201497 percent\u2014comes from oil. Anything we can do to trim our oil use in this sector will go a long way toward curbing America's greenhouse gas emissions. At the same time, the security gains could be enormous. The turmoil now sweeping North Africa and the Middle East only magnifies the madness of our reliance on foreign nations for so much of our oil supplies. Wouldn't it be a relief if our transportation lifeline led to wind farms in Kansas and Wyoming rather than oil fields in Iraq, Libya, Saudi Arabia, and the United Arab Emirates? By matching up large networks of electric cars with a robust, wind-supplied grid, this transformation is within our reach.\n\nLate at night, with dinner already prepared, the dishes washed, and televisions and computers shut off, the wind still blows. At such times, turbines can be taken off-line to avoid overloading the grid with excess electrons, but this wastes a fuel source that is essentially free. The capital costs of building a wind farm are considerable, as they are for conventional coal, gas, or nuclear power plants, but what distinguishes wind from those other power sources is the absence of any ongoing fuel cost. Increasingly, energy planners and innovators have been coming up with new ways to make fuller use of wind-generated power at night and other times when routine demand for electricity is low. Creating centrally managed networks of plug-in electric vehicles is one such solution that Denmark is now pioneering.\n\nKnud Pedersen is vice president for research and development at Denmark's utility conglomerate Danish Oil and Natural Gas (DONG). As its full name implies, DONG was once a company that focused largely on fossil fuel production and distribution. Today it is a leader in tapping wind and other forms of renewable energy. When we met in his glass-walled office overlooking one of Copenhagen's busier traffic arteries, Pedersen spoke excitedly about the steps DONG is taking to reduce its carbon footprint. The largely state-held company owns over half of Denmark's conventional power-generating capacity, including several of the country's biggest coal-fired power plants. It also operates distribution lines serving nearly a third of the nation's electricity customers. \"[E]ighty-five percent of our production capacity is fossil-based or CO2-emitting,\" Pedersen told me. \"Our goal is to turn that upside down by 2040.\" DONG is committed to cutting its CO2 emissions per kilowatt hour in half by 2020, as a first step toward having 85 percent of its electricity come from wind, biomass-derived fuels, and other renewable resources twenty years later. This would place DONG substantially ahead of the European Union's new mandate calling for an 80 percent reduction in regionwide greenhouse gas emissions by 2050.\n\nAlready DONG has invested heavily in wind power at home and abroad. With well over 1,000 megawatts of installed wind capacity, the company distributes some of its wind-generated power in Denmark and markets the rest to other countries in northern Europe. Its plan is to own at least 3,000 megawatts of wind-based generation by 2020. The more wind DONG acquires, the bigger will be the challenge of matching its customers' consumption patterns to the available supply of wind power, which varies with the strength of the winds.\n\nThis is where electric cars enter the picture. Together with Israeli innovator Shai Agassi and his Silicon Valley firm, Better Place, DONG is preparing a network of tens of thousands of electric vehicles whose owners will plug them in when they arrive home, leaving it to DONG to decide when it would be cheapest to charge them up for the next day's driving. Pedersen assured me that participants in this program will not be subjected to hardship conditions. \"You should see the car as your normal car, with the same convenience apart from no noise\u2014very speedy acceleration and very comfortable,\" he said, opening a brochure that features two of the models that will be used. One is a small SUV, the other is an attractive, mid-sized sedan; both will be manufactured under agreement with Renault-Nissan.\n\nIn its early implementation, Better Place is looking to develop electric car networks in relatively confined geographical areas. This is important because, in addition to off-peak charging, the Better Place system requires battery swap stations to be built along highways so that cars traveling longer distances won't be caught without a charge. Small countries like Denmark and Israel are ideal, as travel distances tend to be relatively short. For the same reason, Better Place is focusing much of its initial attention in the United States on Hawaii, although the company is also working with state and local officials in California to create an electric car network in the Bay Area.\n\nTo avoid the cost and inconvenience of battery replacement stations, a number of U.S. researchers are thinking about building plug-in networks based on hybrid-electric cars and trucks instead of the all-electric vehicles proposed by Better Place. Under this approach, utilities would have the same leeway to select low-demand times for recharging batteries, but the use of hybrid engines would lift the limitation on driving distances.\n\nIt's no flight of fancy to imagine half of America's cars and small trucks running on electricity in a few decades' time. Just as we geared up to produce ever-greater quantities of internal combustion vehicles through most of the twentieth century, we could shift that vast and now faltering productive capacity toward a challenge truly worthy of our engineering creativity and competitive zeal. What better opportunity is there for the U.S. auto industry to recover from decades of short-sighted and financially calamitous decision making than by retooling to become a world leader in the manufacture of electric vehicles? If we don't seize this opportunity, it's clear that our European and Asian competitors will.\n\nLooking to the future, America's vulnerability to foreign energy supplies may not be limited to oil. More than half of American households today are heated with natural gas, and more than a fifth of our electricity is gas-generated. While most of that gas still comes from within our borders, known and proven U.S supplies are limited, amounting to about 273 trillion cubic feet\u2014little more than eleven times the amount of gas that we now consume in a single year. American gas industry proponents are quick to point out that additional \"unproved\" but \"technically recoverable\" reserves may be as much as nine times that amount. They are less ready to own up to what is involved in accessing those reserves.\n\nUnconventional recovery methods have substantially expanded U.S. gas supplies in recent years. Shale formations in Appalachia, Texas, and the Rockies are prime sites for drilling operations that provide about a fifth of our natural gas supply\u2014up from only 1 percent at the turn of the millennium. These operations use high-pressure water, converted chemically to a viscous gel, to fracture rock formations that are often a mile or more beneath the surface. Effective in releasing trapped gas, hydraulic fracturing ('fracking\") is a huge drain on local water supplies, often using up to 3 million gallons of water per job. It also poses a pollution hazard whose dimensions are largely obscured by the companies' refusal to divulge the chemicals in their fracking compounds. Along with the fracking chemicals, leachates from shale rock enter the water stream, often polluting it with salt brines as well as radionuclides and heavy metals like arsenic and barium. Companies at times re-inject these fluids into shallow wells, but public concerns about drinking water contamination have brought this practice into serious question. The alternatives\u2014discharging into surface waters or evaporation ponds\u2014have their own environmental downsides.\n\nCoal-bed methane is another unconventional source of natural gas, produced most intensively in New Mexico, the Rocky Mountain states, and Appalachia. Coal seams tend to be closer to the surface than shale, but they have to be dewatered before methane can be extracted, yielding large volumes of water that is often highly saline. Discharging this water into surface streams, a common practice, alters natural habitats and also can endanger drinking water supplies.\n\nThrough deeper drilling, \"re-fracking\" shale gas wells that have already been tapped, and other technology advances, we may be able to access greater amounts of gas than are within our reach today. Yet gas industry insiders are already beginning to panic that their recent projections about future shale gas yields have relied on unduly optimistic assumptions about cost, well productivity, and technical feasibility. This may shorten today's romance with cheap domestic gas, shifting American utilities and other consumers toward foreign gas suppliers long before our own reserves run out. And where are the world's biggest gas resources? In Russia, whose known reserves outstrip America's by a factor of six; in Iran and Qatar, each of which has more than three times the known reserves we have here in America; in Saudi Arabia, whose known reserves roughly match our own; and in places like Algeria, Iraq, Libya, and Venezuela, where sentiments toward our country range from volatile to overtly hostile.\n\nFor those who doubt that we could sink into dependence on yet another imported fossil fuel, we need only look back at the decades that have passed since Richard Nixon and Jimmy Carter called upon lawmakers and the American public to rally behind an effort to free ourselves from the stranglehold of the Organization of Petroleum Exporting Countries (OPEC) in the 1970s. As we sat in those long fuel lines and watched gasoline prices skyrocket in the months following the 1973 oil embargo, who would have believed that we would allow our reliance on foreign oil to rise from 28 percent then to 60 percent today?\n\nWind energy won't eliminate our appetite for natural gas, just as it won't free us from our hunger for imported oil. It can, however, help keep our natural gas consumption within bounds. The Department of Energy predicts that we can reduce our reliance on natural gas\u2013fired power plants by half if we boost our wind energy production to 20 percent by 2030. This, DOE says, would reduce overall U.S. demand for natural gas by 11 percent.\n\nWe could curb our natural gas use further if Americans who now heat their homes with natural gas converted to electric heat pumps or other electric systems reliant on renewable energy\u2013based power. The American Gas Association reports that more than 64 million U.S. homes are heated with gas, an increase of 27 million customers since 1970. Many of these gas users had previously relied on oil to heat their homes; their switch to gas was largely in response to the rising price of home heating oil during the years following the OPEC oil embargo. Imagine what would happen to our natural gas demand if, over the _next_ forty years, we saw an equivalent shift in customers from natural gas to electric heat substantially generated by wind? And what if the 8 million households that still heat with oil switched over to renewable energy\u2013based electricity? The reduced fuel demand and lowered greenhouse gas emissions could be enormous.\n\nThere is no question that shifting tens of millions of cars and home-heating systems to electricity will demand a much more robust U.S. electric sector than we have today. For this to help rather than further harm the environment, we will need to ensure that our expanded generation is based on renewable energy. Meeting this demand with new coal-fired or nuclear power plants would only replace the petroleum used in cars and the gas used for home heating with other fuel choices that pose their own very grave environmental and security hazards. We will also have to build a much more robust and sophisticated transmission network that is capable of carrying large quantities of wind-generated power from remote locations, both onshore and offshore, to major population centers. Jon Wellinghoff, chairman of the Federal Energy Regulatory Commission (FERC), sees both of these challenges as well within our reach. He envisions a \"smart grid\" with the capacity to manage a U.S. electric system that is 50 percent or more reliant on wind.\n\nEven short of fulfilling the FERC chairman's long-term vision, we will need to increase our investment in wind substantially if we are to meet the Department of Energy's goal of drawing 20 percent of our power from wind by 2030. On average, we must install 16 gigawatts of new wind power capacity every year between now and 2030 to meet the 20 percent target. While this is an ambitious goal, it would certainly be within reach if we adopted federal pricing policies that accounted for the full costs of our fuel choices: the global warming effects and air pollution damage to people, property, and natural resources caused by carbon-based fuels; the practically incalculable hazards of nuclear power; and even the costs of waging military campaigns to protect our energy interests abroad. No such pricing is on the immediate political horizon, but lesser steps are certainly within reach. The extension of federal production and investment tax credits for renewable energy projects is one such measure. Adoption of a federal renewable electricity standard is another, requiring all states to ensure that the utilities under their jurisdiction draw a substantial portion of their power from wind and other forms of renewable energy.\n\nHappily, we needn't simply wait for the federal government to take action; more than thirty states have already adopted their own renewable electricity standards. Illinois, for example, has mandated that wind must provide three-quarters of its 25 percent renewable electricity quota for the year 2025, with three-quarters of that power mandated to come from wind. In Kansas, utilities must acquire 20 percent of their electricity from renewable energy sources by 2020. The nation's most ambitious renewable electricity standard, in California, requires utilities to draw 33 percent of their power from renewable sources by 2020. That's in addition to the abundant power that already comes from hydroelectricity\u2014about 15 percent of the state's overall power supply. Qualifying renewable resources\u2014mainly wind, geothermal, solar, and biomass\u2014supplied 18 percent of the power used by California's three big utilities in 2010, falling just short of the state's previous renewable electricity mandate, which set a 20 percent renewables target for that year.\n\nIn addition, eight states have passed laws requiring utilities to offer customers a green power option\u2014the choice of purchasing some or all of their power from renewable sources. And in forty-two states, net-metering laws obligate utilities to buy excess power generated by customers who have installed wind turbines, solar panels, or other renewable energy\u2013based power-generating systems. These measures, together with a range of state renewable energy grant and loan programs, have tilled fertile ground for wind energy development across much of the nation.\n\nWind energy's price relative to other power-generating technologies is critical to determining how big a role it will play in meeting America's electricity needs. The Lawrence Berkeley National Laboratory has gathered wholesale price data on 180 wind projects installed between 1998 and 2009, representing 38 percent of all installed wind power during that period. Through most of those years, wind power was at the low end of wholesale electricity prices, making it highly competitive with electricity coming from other energy plants. However, this relatively favorable relationship eroded in 2009, when a precipitous drop in natural gas prices\u2014largely the result of expanded shale gas development\u2014and a reduction in power demand as a result of the economic recession caused wholesale power prices to decline. By 2010, new wind projects commanded an average wholesale price of roughly $70 per megawatt hour, at a time when the wholesale power market averaged between $30 and $50 per megawatt hour.\n\nDespite this drop in gas prices, utilities have an ongoing stake in buying into the wind market, partially as a long-term hedge against the price instability of fossil fuels. Natural gas may be cheap today, but its price history has been a rollercoaster ride, and those swings very likely will continue in the future. Coal prices, too, may be hard to predict with any certainty if mine safety and environmental protection measures are made more stringent in the years ahead. Utilities also value wind as a way to diversify their portfolios in anticipation of eventual federal rules that make fossil fuels more costly or require a reduction in power plant carbon emissions.\n\nUtilities have a further\u2014and very important\u2014reason to invest in wind: they need it to meet their obligations under renewable electricity standards in the states that have adopted them. Take California as an example. Between 2002 and 2010, wind delivered about 83 percent of the power that utilities bought to meet the state's 20-percent-by-2010 renewable energy mandate. Looking ahead, California utilities will need substantial new increments of wind, generated in-state or purchased from its neighbors, to achieve the state's new commitment to deliver 33 percent of its electricity from renewable sources.\n\nWhile these and other considerations, such as customer preference and broader public opinion, may cause utilities to favor wind over other energy sources, pricing wind power competitively remains crucial. One major way to keep prices within bounds is to contain the costs of building and operating wind farms. The financial advisory firm Lazard Ltd. has examined the twenty-year life-cycle costs of different types of power plants, \"levelizing\" them by looking at the same parameters for each energy type, including up-front capital costs, the fixed and variable costs associated with operations and maintenance, and fuel costs. In 2010, it found the cost of new wind farms to range from $65 to $110 per megawatt hour, making them competitive with new coal plants ($69 to $152 per megawatt hour) as well as new combined cycle gas plants ($69 to $96 per megawatt hour). Solar was much more expensive, with photovoltaics costing $134 to $192 per megawatt hour and solar thermal plants (generating steam from concentrated solar heat) costing $119 to $194 per megawatt hour. Nuclear came in at $77 to $114 per megawatt hour, though experts assume that, post-Fukushima, its cost will rise substantially given the increased safety measures that will be applied to any new plant proposal.\n\nThese numbers highlight the importance of the federal production tax credit for wind. Without it, wind energy's cost as calculated by Lazard would rise by $20 per megawatt hour, making it much less competitive with coal and gas.\n\nFar and away the biggest factor affecting wind energy's cost is the price paid for turbines. Up-front capital outlays for equipment, construction, and financing account for 76 percent of all life-cycle costs associated with wind energy, and the turbine amounts to 75 percent of that initial investment. There's good news in this regard, stemming largely from the growing competition among turbine manufacturers discussed in earlier chapters. From a peak in 2007\u20132008, just when Horizon was building the Meridian Way Wind Farm, turbine prices have been steadily declining. By late 2010, the average contract price for turbines had dropped nearly 20 percent, and prices have continued their decline more recently.\n\nAlong with spending less on turbines, enhancing their performance is another way to get more from a given wind energy investment. Taller turbines and longer blades are among the changes that have had the greatest impact. Taller turbines generally produce more power because wind speeds at higher altitudes tend to be stronger, more constant, and less disrupted by near-ground turbulence, or wind shear, caused by landscape features like trees, buildings, and variations in the natural topography. This height advantage is magnified by the \"cube rule\" of wind power, establishing that the power available from the wind is the cube of its velocity. Given this relationship, it's clear that small increments of wind speed\u2014easily achieved by raising tower heights\u2014can yield big productivity gains. Increasing the average wind speed from 10 to 12 miles per hour boosts the wind power reaching a turbine by 73 percent; raising it to 14 miles per hour augments the available power by 175 percent.\n\nLonger blades further enhance the power-generating potential of turbines. The power available to a turbine is directly related to the \"swept area\" of the rotor, determined by the familiar formula for calculating the area of a circle: A = \u03c0r2. Applying this formula, we can see that small increases in blade length (that is, the radius) will translate into relatively large increases in a rotor's swept area, allowing much greater amounts of energy to be converted into electric power. A 50-meter blade has a swept area more than half again as large as a 40-meter blade. The biggest rotor that Vestas makes today\u2014120 meters in diameter\u2014has a swept area 36 times as large as the 10-meter rotors that it was importing to California in the mid-1980s. The resulting power gains are impressive.\n\nThe first few decades of the wind energy era have seen dramatic jumps in tower height, from latticework structures a few dozen feet high in the 1980s to today's sleek tubular towers, with hub heights approaching and sometimes exceeding 300 feet. We have also seen turbine blades grow from slender slivers that you could load into the back of a pickup truck to multi-ton behemoths stretching half the length of a football field. From 1998\u201399 to 2009, the average hub height rose from 185 to 258 feet\u2014an increase of 39 percent. The average rotor diameter expanded during the same period by 69 percent, from 159 feet to 268 feet. These growth trends appear to be slowing for land-based turbines, but the current race to build higher-output offshore turbines is spurring the development of bigger-diameter rotors at sea.\n\nOn many levels, wind power is superior to the fossil and nuclear technologies that we rely upon so heavily today. It taps a resource that is abundant beyond our wildest dreams. It can be harvested without stripping the Earth of its mineral resources. Its daily operations neither pollute the local environment nor burden the atmosphere with global-warming gases. And it will never force massive numbers of people to flee their homes or live in fear of nuclear devastation.\n\nCalling wind power cleaner than other energy technologies is not to say that it is in perfect harmony with the environment. Any energy technology will have some adverse impacts when applied on a scale large enough to provide a significant percentage of America's power needs. Opponents of wind energy tend to highlight those negative impacts without placing them in the broader context of the much more grievous damage caused by our current energy uses.\n\nA vibrant, if at times contentious, dialogue is helping to focus expertise, as well as public attention, on wind energy's environmental costs and benefits. Acknowledging the ways that wind farms can harm the environment is one cornerstone of a successful wind energy enterprise; managing those impacts responsibly is another. Wind developers are now working with a battery of technical experts and civic leaders to get both of those cornerstones in place as they gear up to expand wind's share of America's electricity supply.\n\n## CHAPTER SEVEN\n\n## Birds and Bats\n\nMANY ASSOCIATE ALTAMONT PASS with the mayhem that ran through a rock music festival at the local speedway in 1969, with 300,000 in attendance and Hells Angels hired as security guards. Unfortunately for wind developers, Altamont has also earned infamy as the place where closely packed clusters of wind turbines have slaughtered tens of thousands of birds, including the federally protected golden eagle.\n\nOver the years, multiple studies have documented the alarming rate of bird mortality at Altamont. In 2004, the Tucson-based Center for Biological Diversity filed suit against Altamont's multiple owners, alleging that 17,000 to 26,000 raptors, or birds of prey, had been killed since the wind energy complex began operating in the 1980s. The primary culprits identified in the suit were the \"small, inefficient, obsolete, first-generation wind turbine generators\"\u2014with more than 5,000 of them producing only 584 megawatts of rated power. That same year, the California Energy Commission reported the results of a multiyear survey of bird deaths at Altamont. Out of 1,766 to 4,721 birds killed each year, 75 to 116 were golden eagles. The report's authors suggested a number of mitigation actions, but their primary recommendation was to replace the wind farm's low-output turbines with a much smaller fleet of megawatt-plus machines.\n\nIn more recent years, the Center for Biological Diversity, five Bay Area Audubon chapters, and other citizen groups have continued to press for tougher bird protection measures, at times working in concert, at times bitterly disagreeing over the adequacy of proposed steps. Finally, in December 2010, the California Attorney General's Office presided over a settlement that committed the wind farm's largest stakeholder, NextEra Energy Resources, to replacing 2,400 aging Altamont turbines with 80 to 120 new ones of equivalent overall capacity by 2015. This effort is likely to be a boon rather than a burden to NextEra, as most of the targeted turbines are already in very poor shape and may be barely functional by the replacement deadline. An official of a smaller Altamont wind company commented at the time of the settlement: \"The wind smiths out there hold the turbines together with duct tape and wire.\"\n\nThe killing of birds remains a real concern at wind farms built in recent years, but local citizens and conservation groups have learned to be much more proactive in monitoring wind farm plans as they are being developed, raising objections to ill-chosen sites and intervening early enough in the game to bring about necessary changes. For some years now, Angelo Capparella, an ornithologist by training, has been closely tracking wind development plans in the heavily farmed counties surrounding Bloomington, Illinois, where he teaches systemic and evolutionary biology at Illinois State University. In 2006, he reviewed the draft environmental impact statement for the 100-turbine White Oak Wind Energy Center, a dozen miles northwest of the city\u2014all part of the due diligence on emerging wind projects that he performs for the local Audubon Society chapter. Studying the document, he noticed some glaring gaps. First, he claimed, it failed to give proper attention to adjacent nature reserves. Second, he maintained that it underestimated the number of raptors, including bald eagles, that migrate through the Mackinaw River valley, where the project was to be built. He was further disturbed that no one from the Chicago-based wind development company, Invenergy, had contacted him or, to his knowledge, anyone else at Audubon, Illinois State University, or the Mackinaw Ecosystem Partnership to gather data on local wildlife issues.\n\nAppearing before a county zoning board, Capparella flagged the study's shortcomings, urging the county officials to demand that the wind developer prepare a more thorough analysis. He also invited Invenergy's project manager to his office at Illinois State and showed him the site-specific databases that should have been included in the study. \"Your consultant is just looking at easy ways to find data,\" Capparella chided. He went on to warn the developer that Audubon would not be able to support the wind farm unless significant changes were made. Invenergy immediately responded, adjusting the project's footprint to include a one-mile setback from the Mackinaw River and a half-mile setback from a reservoir surrounded by protected land that provides valuable bird habitat.\n\nThough the outcome of his intervention was positive, Capparella remains wary. Wind projects are moving forward so quickly that there often isn't time to gather reliable data that can help developers really understand how their projects may affect surrounding wildlife. \"It is clear, to me at least, the farms are going up faster than the science,\" he told me as we sat in a motel lounge on the outskirts of Bloomington. I couldn't help feeling that the motel's bland d\u00e9cor matched the monotony of the surrounding countryside, utterly transformed by decades of industrialized farming.\n\nCapparella corroborated my impression that little native biodiversity is left in this vast stretch of central Illinois farm country. Born and raised in North Carolina, he has an outsider's candor when he talks about what's left of the open prairie. He chuckles as he recalls the outrage expressed by some local citizens at zoning hearings on proposed wind farms. \"You're ruining my view of the pristine prairie!\" they exclaim. His view is decidedly less romantic. \"This is all corn and soybean fields; this is _not_ pristine prairie! Perhaps people growing up here think this is natural, but it's really quite unnatural. . . . I just want to protect the remnants that are left.\"\n\nChanges in turbine technology and wind farm design have substantially reduced raptor kills at modern wind farms like White Oak. At Altamont, the turbines that have posed such a hazard to raptors stand low, with blades circling down to within a few dozen feet of the ground. California's early wind developers packed lots of these machines into limited space, thinking they would be able to harvest more energy from the lands they leased. The combined result was too little room for raptors to maneuver as they pursued rodents and other prey on the ground below. At more modern wind farms, developers have learned to space their machines much more generously to minimize interference with the air currents reaching each turbine. As a rule of thumb, the National Renewable Energy Laboratory (NREL) has recommended that the distance between turbines should be five to ten times the rotor's diameter. That can translate into gaps between turbines of a third of a mile or more for many of today's larger machines.\n\nWhile this broader spacing is driven by the quest for higher power output, a lucky by-product is the greater freedom of flight that the new design gives raptors and other birds within a wind farm area. The higher ground clearance of larger turbines yields a further margin of safety to raptors swooping down to hunt rodents; the lower reach of blades on today's turbines is well over 100 feet off the ground. And finally, the larger size and slower rotation of modern turbines make them easier for birds to spot and avoid colliding with. Instead of spinning at 30 to 40 revolutions per minute like the older machines at Altamont, large commercial turbines today rotate at about half that speed.\n\nTaller turbines with their longer blades are not an unqualified boon to birdlife, however. Biologist Paul Kerlinger has used radar to track bird migration patterns for more than a quarter-century. He reports that most migrating birds travel between 300 and 2,000 feet above the ground. With the blade tips of today's turbines extending 400 feet into the air and higher, migrating birds in certain areas may be increasingly at risk, especially when flying at night or in low-visibility daytime conditions. Some experts also believe that the pulsing safety lights required by the Federal Aviation Administration may attract migrating birds, particularly on nights with fog or low cloud cover. \"These birds orient by the stars, and we hypothesize that they must think the lights are some remnant stars on overcast nights,\" says Angelo Capparella. Birds tend to fly in circles around these lights in an apparent attempt to orient themselves, he adds. This hugely increases their risk of crashing into swishing blades.\n\nOne bird whose fate is being watched particularly closely is the nearly extinct whooping crane. Only a few hundred of these birds remain in the wild, and their migratory path cuts right through the heart of U.S. wind farm country\u2014the Dakotas, Colorado, Nebraska, Kansas, Oklahoma, and Texas. While whooping cranes generally migrate at 1,000 to 6,000 feet, they fly much closer to the ground when looking for roosting or foraging areas. To avert possible harm, ecologist Karl Kosciuch has recommended that 10-mile buffers be created around wetlands that are known to be critical stopover points. No whooping crane deaths have been linked to wind farms, and scientists like Kosciuch are committed to maintaining this perfect record. On the other hand, forty-four whooping cranes have died after colliding with power lines since 1956. As new transmission lines are built to serve an expanded wind power network in the coming years, this hazard may increase.\n\nTo ward off bird collisions with turbines, early warning and emergency shutdown systems have been installed at a few wind farms in the United States and abroad. Spanish wind developer Iberdrola uses radar to detect incoming flocks of migratory birds at its giant 404-megawatt Pe\u00f1ascal wind farm in southern Texas. A similar system is being used by AES Geo Energy, a subsidiary of the Virginia-based power conglomerate AES Corporation, at its St. Nikola Kavarna wind farm on Bulgaria's Black Sea coast. Because St. Nikola is sited along a major migratory flyway, the wind farm's environmental management plan requires an ornithologist to be on-site during migration periods, monitoring radar, overseeing fixed-point observers, and coordinating with a mobile tracker to determine when the wind farm's turbines might be placing birds at risk. To make sure wildlife protection is given priority, the decision to shut down turbines rests with the ornithologist, not the wind farm operator. In the fall of 2010, this system proved its worth when strong westerly winds blew more than 30,000 white storks off the sea and toward the wind farm. To protect the flock, ornithologist Pavel Zehtindjiev ordered the turbines to be halted thirty-seven times over a two-day period. Within just five seconds of the \"stop\" command, the turbines came to a standstill, and the birds passed safely through the wind farm area. Not a single stork was harmed.\n\nEven with improved technology and siting, bird deaths remain a reality at wind farms in the United States and around the globe. Estimates vary, but overall mortality at U.S. wind farms outside California seems to be around two birds per megawatt per year. At that rate, we might end up killing well over half a million birds per year as we approach the Department of Energy's target of installing 300,000 megawatts of wind capacity by 2030\u2014the amount needed to generate about a fifth of our nation's power. While most of the victims are likely to be common songbirds like horned larks, vesper sparrows, and bobolinks, that's still a disturbing number.\n\nBald and golden eagles, long the focus of controversy at Altamont, have recently become flash points in the dynamics between government regulators and wind developers at the national level. Acting under the Bald and Golden Eagle Protection Act's obligation to prevent the unauthorized \"taking\" of these iconic birds, the U.S. Fish and Wildlife Service issued a guidance document in February 2011 that outlined a series of steps that wind developers should follow to avoid harming eagle populations. Wind developers vehemently objected to these provisions, claiming that they present enormously cumbersome hurdles that will grind wind farm planning to a slow crawl across the West and, to a lesser degree, elsewhere in the nation. As of this writing, the debate continues.\n\nWind industry leaders insist that we look at birds killed by turbines within the context of the much more massive toll on bird life that we seem to accept quite readily, all in the name of comfort, convenience, and affinity for felines. I think back to my meeting in Denmark with Vestas communications chief Peter Kruse. \"Your neighbor's cat is a far more efficient killing machine than my turbines,\" he bantered as we surveyed the issues that wind energy opponents are prone to raise.\n\nBehind this flip remark lies a body of research that, over decades, has tried to gauge the risks to bird life from a wide range of causes. Every year in America alone, collisions with buildings may kill anywhere from 97.6 to 976 million birds, and road strikes may cause an additional 80 million deaths. Up to 50 million birds die when they crash into communication towers, and 130 to 174 million birds are killed when they collide with high-tension, or \"bulk,\" transmission lines. Beyond those hazards, there are many other ways we end up killing birds and destroying their habitats through energy industries like oil and gas drilling and refining, mountaintop removal and strip mining for coal, and power plant operations. Farming wreaks further havoc on bird populations by radically altering natural landscapes and saturating field crops with pesticides. And then, of course, there are Kruse's cats. The Fish and Wildlife Service estimates that hundreds of millions of songbirds and other bird species are hunted down and killed by domestic and feral cats in the United States every year. Researchers at the University of Georgia have arrived at an even higher number: they say that \"free-ranging domestic cats,\" themselves numbering between 117 and 157 million, kill more than 1 billion birds annually in the United States.\n\nKilling birds is one concern; disrupting their habitats is another. In Wyoming, wind energy proponents have found themselves at the center of a superheated controversy over the greater sage grouse, a large, richly feathered fowl that has long hovered on the brink of being listed as an endangered species by the U.S. Interior Department. A century ago, the sage grouse numbered about 16 million, its habitat spreading across sagebrush country throughout most of the American West, extending north into Alberta and Saskatchewan. Today, its population has dwindled to a few hundred thousand, with much of its remaining habitat in large sections of Wyoming where oil, gas, and coal development have already encroached heavily.\n\nBy the time wind developers began looking seriously at major new wind farm sites in Wyoming, conservation groups in the West had already launched efforts to get the sage grouse listed under the Endangered Species Act. The Interior Department ruled against listing the species in 2005, arguing that the bird's population had increased or stabilized in several states since 1985. This led to a federal lawsuit that yielded a stern court ruling in 2007, criticizing the lack of scientific rigor underlying the department's decision and requiring it to reconsider the matter.\n\nWyoming politicians and energy industry leaders then grew alarmed. Recalling the federal listing of the northern spotted owl as a threatened species in 1990, they knew how profoundly that decision had affected logging in the Pacific Northwest. A federal listing of the sage grouse, they feared, would shut down much of Wyoming's extremely lucrative coal, gas, and oil industries. To avert a crisis, Governor Dave Freudenthal convened a multi-stakeholder Sage Grouse Summit in June 2007, where he stressed the urgent need for a state conservation plan strong enough to fend off the bird's federal listing.\n\nThe following year, Freudenthal signed an executive order creating a Greater Sage-Grouse Core Area Protection plan. This measure gave protected status to about a fifth of Wyoming's territory, thought to contain more than 82 percent of the state's sage grouse population. Within the designated \"core areas,\" existing energy facilities\u2014oil and gas wells, pipelines, and even coal mines\u2014were allowed to continue operating. New projects, however, bore the burden of proving that they would not diminish the sage grouse population.\n\nWind farms were particular targets of this requirement. Unfortunately, they also became a focus for the governor's open disdain. He accused wind developers of being driven by a \"gold rush\" mentality that placed \"[s]eemingly every acre . . . up for grabs in the interest of 'green, carbon-neutral technologies' no matter how truly 'brown' the effects are on the land.\" He also likened wind developers' zeal in siting new projects to \"taking a shortcut to work through a playground of school children and claiming 'green' as a defense because you were driving a Toyota Prius.\"\n\nIn a state whose landscape has been punctured by oil and gas wells, scarred by pipelines, and torn apart by the nation's largest open-pit coal mines, Freudenthal's anti-wind rhetoric was surreal. But with more than half of the state's revenues coming from energy-related taxes and royalties, it's hardly surprising that Wyoming's chief executive would go to great lengths to protect that flow of income. In a second order issued in 2010 and now in effect, the bias toward the state's traditional energy industries is even more unmistakable. Not only are existing energy operations with their drill sites, pumping stations, access roads, and vehicle traffic allowed to continue, but new oil and gas wells are permitted in core sage grouse areas at a density of one drill pad per square mile. Although it's hard to fathom why, mines are also allowed at the same average density. Wind development, by contrast, is \"not recommended.\"\n\nThe actual impact of wind farms on the sage grouse remains unclear. Experts suspect that the birds may stay away from wind turbines, fearing that raptors\u2014their natural predators\u2014are using them as perches. They also worry that power lines, guy wires on meteorological towers, and turning turbine blades may pose collision risks. Few studies have been conducted to probe these and other issues, lending credence to Angelo Capparella's observation that wind farm development is several steps ahead of the available science.\n\nWyoming's brinkmanship, going tough on wind but easy on mineral extraction, is destined to become harder to sustain as the sage grouse moves ever closer to earning federal endangered species status. In March 2010, Barack Obama's Interior secretary, Ken Salazar, distanced his agency from the George W. Bush administration's decision not to consider the bird endangered. On one hand, he announced that listing the sage grouse was \"warranted but precluded\" in light of other species more urgently needing federal protection. On the other hand, he designated the bird a \"candidate species\" whose status is to be reviewed annually, clearly signaling close ongoing scrutiny and the very real possibility of a future listing.\n\nErik Molvar, a wildlife biologist who directs the nonprofit Biodiversity Conservation Alliance in Laramie, objects strongly to wind's uneven treatment under Governor Freudenthal's sage grouse protection program. \"There's nothing fair about it,\" he says, pointing to the oil and gas industry's enormous influence on state and federal politics. \"The oil and gas industry should be held to the same bar as the wind energy industry,\" he says, \"but it should be that higher bar of being compatible with protecting public lands and wildlife.\"\n\nFortunately for Wyoming wind developers, possible sites for new wind farms abound outside the state's designated core sage grouse areas. Molvar makes this point as he leads me through _Doing It Smart from the Start_ , his organization's guide to responsible wind energy development. The report overlays maps of the state that show critical ranges and migration corridors for birds, bats, and multiple big-game species including antelope, elk, bighorn sheep, and mule deer. It also looks at historic trails and other visually sensitive areas. Putting it all together, the report arrives at a composite map showing where wind power should and should not be developed. The results, as Molvar describes them, are encouraging. \"There are 5 million acres of 'go-zones' in Wyoming, and of those 5 million acres, 4 million acres have high wind potential.\" That's far more land than the state would need to achieve its maximum build-out of wind turbines, he assures me.\n\nThe greater prairie chicken is another member of the grouse family that has entered the debate about wind farm development. Once there were three subspecies of this bird. One is now extinct; another survives only in small areas of southeastern Texas; and the third continues to populate sections of mixed and tallgrass prairie in Kansas and several other Midwestern states. While greater prairie chickens have decreased dramatically in overall numbers as cultivated farmlands have encroached on the tallgrass prairie, they are not listed as endangered and continue to be legally hunted. Environmental biologist Robert Robel refuses to give an estimate of how many of this species remain in Kansas, but he calls the population \"healthy\" and insists that hunting prairie chickens\u2014a popular pastime\u2014does not pose a threat to their survival. He points to intrusions that interfere with mating and nesting as having a much greater impact on the bird's ability to maintain stable numbers. That's where wind farms come into play.\n\nMale prairie chickens compete with one another for sexual dominance through a mating dance called \"booming,\" which involves lots of shuffling, strutting, neck-puffing, and cooing. They carry out this ritual on open ground where there are no nearby trees or other high structures that could host birds of prey. Like the scientists who are studying the sage grouse, Robel wonders whether wind turbines cause prairie chickens to abandon nearby mating grounds. When I spoke with the retired Kansas State University professor in January 2010, he was midway through supervising a study of greater prairie chicken activity at the Meridian Way Wind Farm, whose footprint includes about 13,000 acres of mixed-grass ranchland along with several thousand acres of cultivated crops. While no results about the effects of Meridian Way's turbines were available at the time, Robel pointed to his own earlier collaboration on a study that looked at how other built structures affected a somewhat smaller relative, the lesser prairie chicken. Researchers in that study found that nesting birds stayed away from buildings, gas compressor stations, and a coal-fired power plant, and they further observed that nesting was unlikely to happen within 1,300 feet of transmission lines or paved roads.\n\nWhen Horizon Wind Energy settled on Cloud County as the site for Meridian Way, its environmental team recognized that the wind farm might harm local grassland habitats. The company, therefore, took the unusual step of committing to buy up conservation easements on an expanse of grassland equivalent in scale. It turned to the Ranchland Trust of Kansas, an offshoot of the Kansas Livestock Association, as its primary partner in negotiating easements with landowners in the Smoky Hills, a few dozen miles south of Meridian Way. With an undisclosed sum of money at her disposal, the trust's Smoky Hills coordinator, Stephanie Manes, began having conversations with area landowners. In signing a conservation easement, landowners would receive a one-time payment in exchange for maintaining their properties as grassland in perpetuity. They could continue to graze cattle and live on the land, so long as they adhered to stipulated practices for grazing, fencing, and weed control. They could not, however, convert pastures to cropland or sell off any of the property for development. And, of course, they could not put up any wind turbines.\n\nThe agreement between Horizon and the Ranchland Trust called for 13,100 acres to be placed under conservation easement, out of a total of 20,000 acres that were to be restored as grassland bird habitat. Through smart leveraging, Manes quickly pushed beyond that original goal. Within nine months, she had lined up 25,000 acres of easements and expected to bring another 5,000 acres under conservation protection with her remaining funds.\n\nManes is careful not to take a categorical stand on wind farms. She makes it clear that the Ranchland Trust's umbrella organization, the Kansas Livestock Association, has remained neutral on the subject. \"Our membership is pretty much split evenly down the middle,\" she says of Kansas ranchers. She lets me know that she has her concerns, though. From her training in wildlife management and grassland ecology at Oklahoma State, she knows what a powerful deterrent tall trees can be to grassland bird mating and roosting on the open prairie. \"One tree per ten acres to a human is like, 'Oh, what a beautiful tree!' \" she explains. \"Well, that tree will take out the entire ten acres of habitat for grassland nesting birds.\" Will grassland birds be similarly deterred by wind energy arrays, or will they learn to see turbines as non-threatening? Manes hopes that the study at Meridian Way, supervised by Robel, will begin to answer that question. \"They just haven't been around long enough for us to know,\" she says of the wind turbines.\n\nThough uncertain about wind turbines' impact on grassland birds, Manes has no doubt about the harm caused by another human activity: the annual burning of the Kansas tallgrass prairie by ranchers. She points in particular to the Flint Hills, where land agent Jim Roberts ran into such fierce opposition when he was scouting for possible wind farm sites on behalf of Horizon's predecessor, Zilkha Renewable Energy. Every spring, ranchers wielding kerosene torches race through the Flint Hills on their all-terrain vehicles, setting about 1.7 million acres of grasslands aflame. Thinking of this conflagration in New England terms, I am stunned to realize that it covers an area more than twice the size of Rhode Island.\n\nTo some degree, periodic burning is essential to the prairie's survival, especially in regions like central and eastern Kansas, which experience more rainfall than the semiarid sagebrush prairie to the west. Without fires to wipe out seedlings, larger plants, including trees, would invade quickly, converting prairie to forest in a matter of years. Long before white settlers arrived, Plains Indians relied on fires\u2014some natural, some set\u2014to maintain grasslands for their bison herds. Scientists today estimate that grassland fires are needed every three to four years to keep the prairie intact.\n\nToday's Flint Hills ranchers have pushed far beyond this threshold. Trucking in young cattle from out of state for a few months of intensive grazing every spring, they know that their heifers will gain weight much more quickly when they graze on the fresh grass that comes up after a burn. Quick weight gain translates into bigger profits, especially for those who double-stock their ranchlands, running two cohorts of cattle through shorter cycles in a single grazing season.\n\nIt's no small irony that Flint Hills ranchers, adamantly opposed to wind turbines in their midst, resort so readily to annual burning\u2014a practice that Stephanie Manes calls an ecological nightmare. They don't seem troubled that their blazes create enough air pollution to trigger ozone alerts in Wichita, Topeka, and other urban areas. They also seem unbothered that the fires rob prairie chickens and other grassland birds of the thatch they need to build their nests and the ground cover they rely on to hide their young from birds of prey. John Briggs, who directs the Konza Prairie Biological Station just outside Manhattan, Kansas, quantifies how much camouflage is needed to protect nesting birds: \"If you throw a football into a field and it lands and you can't see it, that's enough cover for prairie chickens,\" he tells me. After the spring burn, a billiard ball wouldn't pass this test.\n\nProfessor Robel and colleagues from Kansas State have looked at the effects of springtime burning on grassland bird nesting in eastern Kansas. Their results were stark, if predictable. On unburned fields, they found 372 nests. On burned fields, they spotted only 27 of them.\n\nThe verdict is still out on whether, and to what degree, Meridian Way's wind turbines have disrupted prairie chicken habitat in Cloud County. Whatever the impacts turn out to be, they will surely pale when compared to the devastation wrought by other human intrusions. Horizon's chief environmental officer, Rene Braud, makes no attempt to disguise her frustration with the degree to which wind developers are being called to account for environmental harms to ecosystems long ravaged by other human activities. She speaks of the Kansas prairie in much the same way Wyoming wind advocates describe their own state's eroded sagebrush terrain: \"There's no question it's been decimated by agriculture, cattle, oil and gas, housing, and other forms of development,\" she says, noting how unfair she feels it is to single out the wind energy industry for heightened scrutiny. \"We're taking the hit for the last two centuries.\"\n\nFrom the early days of wind energy development, it was obvious that birds were being killed by turbines. The toll on bats was initially less apparent. Even today, experts are careful to point out that most of the data on bat mortality at wind farms has come from surveys that were primarily focused on birds.\n\nAvailable studies may still be limited, but they reveal a level of bat mortality that is of real and mounting concern. Studies conducted at wind farms in the West and Midwest point to annual bat kill rates ranging from 0.8 to 8.6 per megawatt\u2014more or less in line with reported death rates for birds. Along some forested ridgelines in the East, however, the losses have been much higher, reaching as high as 41 bats per megawatt at the Tennessee Valley Authority's Buffalo Mountain wind farm near Oak Ridge, Tennessee.\n\nWhat draws bats to turbines is uncertain. Some experts assume that they come to feed on insects congregating in the clearings surrounding turbines; a related hypothesis is that the insects themselves are lured by the heat given off by turbine machinery. The mechanical or aerodynamic sounds that turbines produce may also appeal to bats. And then there's the possibility that bats confuse turbines with tall trees suitable for roosting.\n\nWhatever the attraction, bats are killed by wind turbines in two primary ways, as evidenced by carcass surveys. Some collide with rotors; apparently the radar that normally guides bats' flight has trouble detecting blades in motion. Most deaths don't result from collisions, however. Roughly 90 percent of bats are killed by internal hemor-rhaging, or barotrauma, caused by the rapid changes in atmospheric pressure that occur within the vortices of spinning blades.\n\nWhile no recorded deaths of threatened or endangered bat species have been linked to turbine operations, the development of one wind energy project\u2014the Beech Ridge Wind Farm in Greenbrier County, West Virginia\u2014has been curtailed to prevent possible harm to the Indiana bat, listed as endangered since 1967. The Indiana bat is tiny, weighing about a quarter of an ounce and with a body 1.5 to 2 inches long. Its habitat extends across much of the East and Midwest, though its numbers have dropped by more than 50 percent since it was first listed as endangered. During the warmer months, Indiana bats live in wooded and semiwooded areas, roosting under tree bark and in dead trees. In winter, they hibernate in caves.\n\nBeech Ridge Energy LLC, a subsidiary of Invenergy, announced plans to build its wind farm along a 23-mile stretch of Appalachian ridgeline in November 2005. As part of the project preparations, a consultant examined caves and conducted mist-net surveys to check for the possible presence of Indiana bats in the area. The consultant's reports showed three caves currently used by hibernating Indiana bats between 5 and 10 miles from the project site, but none within 5 miles of the site. No Indiana bats were captured by the nylon-mesh mist nets, strung up like badminton nets at multiple locations. Based on these findings, Beech Ridge sought and received permission from the West Virginia Public Service Commission to build its wind farm: 124 turbines totaling 186 megawatts of installed capacity.\n\nUnhappy with the state agency's decision, local citizens joined forces with a national nonprofit, the Animal Welfare Institute, in a suit claiming that already-dwindling colonies of the Indiana bat would be further diminished by the wind farm in violation of the Endangered Species Act. Federal judge Roger W. Titus agreed, invoking Ben Franklin when he ruled that \"like death and taxes, there is a virtual certainty that Indiana bats will be harmed, wounded, or killed imminently\" by the wind farm. Looking closely at the work performed by Beech Ridge's consultant and hearing testimony from other bat experts, he chastised the consultant for failing to consider acoustic monitoring data that indicated the likely presence of Indiana bats in the project area, and he took issue with the consultant's assumption that bats hibernating in caves 5 or more miles from the site would be unlikely to approach the wind farm. Other experts testifying in the case maintained that migrating bats flying to or from caves some distance from the wind farm could easily find themselves crossing through turbine zones.\n\nThe judge's order, issued in December 2009, barred Beech Ridge Energy from adding to the forty turbines already built and confined the wind farm's operations to the winter months, when the bats would be in hibernation. In January 2010, the parties reached a settlement agreement\u2014authorized by the court\u2014that expanded the permissible number of turbines to sixty-seven. The agreement also modified the winter-only restriction, allowing the wind farm to operate during daylight hours at other times of the year. Meanwhile, Beech Ridge Energy has applied to the U.S. Fish and Wildlife Service for an \"incidental take\" permit under the Endangered Species Act. This permit, if granted, might allow Beech Ridge to operate during nighttime hours so long as it complies with the terms of a habitat conservation program designed to minimize \"incidental\" deaths and other injuries caused to the Indiana bat.\n\nApprehensions may be greatest regarding endangered species like the Indiana bat, but wind companies are generally concerned about the toll that turbines are taking on bats, and they are experimenting with ways to reduce that toll. Some are weighing whether to take turbines out of service during the two hours after sunset, when insects are most abundant and bats are most likely to be pursuing them. Others are testing whether bat deaths can be reduced by shutting down turbines during low-wind periods, when bats and insects are much more likely to be flying than in higher-wind conditions.\n\nDon Furman at Iberdrola Renewables introduced me to his company's collaboration with a nonprofit group, Bat Conservation International, at the Casselman Wind Power Project in western Pennsylvania. During periods of peak bat activity over a two-year period, Iberdrola operated its turbines at three different \"cut-in speeds\"\u2014the minimum wind velocities at which rotors are allowed to start turning. At the low end was the turbines' normal cut-in speed: 3.5 meters per second, or 7.8 miles per hour. In the mid-range, turbines were stalled until the wind speed reached 5 meters per second (11.2 miles per hour). At the upper end, the turbines were allowed to operate only if the wind was blowing at a minimum of 6.5 meters per second (14.6 miles per hour). The research team found that, by raising the cut-in speed, bat fatalities could be reduced by 44 to 93 percent. Iberdrola also found that these curtailments, implemented during the peak bat migration period from late summer to early fall, caused relatively minor losses in power production: 0.3 percent of total annual output when the cut-in speed was 5 meters per second, and 1 percent when it was 6.5 meters per second.\n\nIt is hard to arrive at a reliable forecast for overall bat deaths as we build more wind farms in the years ahead. Technology may change; wind farm operations may become more responsive to surrounding conditions; and wind farm siting may increasingly avoid bat population hot spots. Projecting from current mortality rates and assuming no technology or design improvements, one study estimates that the annual death toll in the Mid-Atlantic Highlands alone could range from 33,000 to 111,000 bats by 2020. Nationwide, the total would likely be much higher, although bats generally have been found to be less vulnerable to wind turbines sited in unforested areas of the West and Midwest.\n\nThere is no denying that wind energy takes a toll on birds and bats, but there is also no question that the continued burning of coal and other fossil fuels exacts a much greater environmental cost. Beyond the localized devastation that accompanies our extraction and use of fossil fuels, we need to look at the utter transformation of our global environment\u2014the destruction of entire ecosystems, the wholesale elimination of species, the reduced availability of freshwater resources, and the massive displacement of human populations\u2014that will result from our failure to tame the global warming juggernaut. Wind energy technologists certainly should commit themselves to coming up with better ways to detect and respond to the presence of birds and bats at wind farm sites, just as government regulators and concerned citizens should demand rigorous planning and vigilant operation of wind farms (as well as all other energy facilities) to minimize damage to wildlife. The U.S. Fish and Wildlife Service's eagle conservation guidelines may require further streamlining to ensure necessary protections in a manner that doesn't stymie responsible wind development, but its leadership in this area bodes well for a more consistent approach to wildlife protection at wind farm sites.\n\nManaging wind energy's wildlife impacts is essential. If we fail to address them in earnest, we risk derailing a technology that, wisely implemented, can set us on a new energy course that will vastly reduce our local and global environment's exposure to much more fundamental dangers.\n\n## CHAPTER EIGHT\n\n## The Neighbors\n\nI ONCE TOLD A NEWSPAPER REPORTER that I find wind turbines beautiful. She responded sharply that, to her, they're ugly. Other than the noise turbines make, nothing about wind farms creates greater public discord than their visual appearance. Embroiled in the battle over building an offshore wind farm in Nantucket Sound, Cape Wind developer Jim Gordon often reassured skeptics that the proposed turbines would appear no higher than a thumbnail held at full arm's length when viewed from the nearest landfall. Anti\u2013Cape Wind activists countered by decrying the industrialization of their ocean horizon.\n\nIn the 1980s, most Americans would have associated wind farms with the chaotic turbine arrays that crowded California's Altamont, San Gorgonio, and Tehachapi passes. In their rush to build, developers purchased a haphazard variety of turbines, some with two-bladed rotors, others with three, some mounted on spindly steel-truss towers, others capping smooth metal tubes. The visual effect was jarring and, to some, alien. \"Spielberg and Lucas could not have done better,\" urban and regional planner Sylvia White wrote to the _Los Angeles Times_. \"Once-friendly pastoral scenes now bristle with iron forests,\" she fumed. Veteran wind advocate Paul Gipe hyperbolically likened the altered landscape at one California site, with its crude access roads gouged into mile after mile of barren hillsides, to parts of Appalachia where coal mining via mountaintop removal has devastated the natural topography.\n\nToday's wind farms are generally much kinder to the eye and more respectful of the landscape. Unlike the early California arrays, almost all modern wind farms stick to a uniform turbine profile, even where more than one manufacturer's equipment is used. Wind farm layout also tends to be much more orderly, with turbines generously spaced and careful consideration given to the turbines' appearance as part of a broader visual experience. Even with these changes, the much larger size of today's turbines undeniably makes them visually dominant at close range and noticeable from afar in many settings. One of those settings is Nantucket Sound, where my home state's long-proclaimed renewable energy ambitions ran headlong into the aesthetic passions of well-heeled vacationers, including our state's most revered political hero.\n\nI will never forget the sunny morning in April 2006 when a few dozen of us gathered outside Boston's historic Faneuil Hall. Senator Ted Kennedy had come to Boston to witness Governor Mitt Romney's signing of the Massachusetts health-care reform bill, and we were there to greet him. As the senator stepped briskly out of his black SUV, he faced a chorus of chants and a sea of red-white-and-blue posters showing wind turbines rising above three bold letters: YES. He rushed into the building, looking sheepish and stunned.\n\nThe Conservation Law Foundation, which I headed up at the time, usually didn't participate in public demonstrations. We preferred more orderly professional settings where our attorneys could best apply their skills. However, Kennedy's backroom politicking to block the Cape Wind project had pushed us over the edge. We weren't willing to let him undermine this pioneering project.\n\nDuring the preceding months, Kennedy had recruited Senator Ted Stevens and Representative Don Young, both Alaska Republicans, to help him blow Cape Wind out of the water. Stevens served on a House-Senate conference committee charged with finalizing the Coast Guard's annual reauthorization bill; Young was the committee's chair. A veil of secrecy surrounded the committee's deliberations, but word filtered out about the machinations that were taking place behind closed doors. First, Representative Young introduced language barring offshore wind turbines within a mile-and-a-half of any navigational channel\u2014a much bigger safety buffer than the federal government had ever required for oil rigs and other ocean installations. The measure's unstated goal was to reduce the scale of the Cape Wind project to such an extent that it would no longer be financially viable. When the media exposed this ploy, public furor mounted and several legislators objected that the measure violated basic principles of good governance. Then Senator Stevens stepped in, substituting new language giving governors of adjacent states veto power over the wind farm. Even though all of Cape Wind's 130 turbines were to be sited in federal ocean waters outside any state's jurisdiction, this provision would have allowed Governor Romney\u2014an avowed opponent of the project\u2014to stop the development in its tracks.\n\nI personally was deeply upset with Senator Kennedy for orchestrating these maneuvers; they seemed woefully out of character for a politician who had, for so many years, been a steadfast champion of policies that would open up the U.S. marketplace to renewable energy. It felt like a self-serving betrayal, all to preserve an unobstructed ocean vista from the Kennedy vacation compound in Hyannis Port, more than five miles from the nearest proposed wind turbine.\n\nWanting to get a better idea of how a project like Cape Wind might look to Massachusetts vacationers, I made it a priority to visit one of the world's largest offshore wind facilities when I traveled to Denmark in October 2009. The R\u00f8dsand Offshore Wind Farm lies just off Denmark's southern coast, in the Baltic Sea. I set my sights on the village of Nysted, whose snug harbor is a major attraction to Baltic boaters. Parking near a row of well-maintained brick houses, I walked to the water's edge, where a few dozen sailboats and motor cruisers were tied up along spindly wooden piers, their prows packed together like cars in a busy parking lot. I strained to look beyond the harbor, but I found myself distracted by the motor cruisers' flying bridges and the sailboats' tall masts with their wire stays twitching in the breeze. In the distance, about six miles from the shore, I could vaguely pick out a hazy sequence of finely drawn vertical shapes in evenly spaced rows, barely rising above the horizon. So these were the industrial intruders that Ted Kennedy found so offensive?\n\nWhen I visited the R\u00f8dsand wind farm, its first phase had been in operation for six years, with 162 turbines capable of delivering 373 megawatts of power. The second phase was under construction. A little over half as large as the first phase, R\u00f8dsand II was expected to meet the power needs of up to 200,000 Danish homes. Bjarne Haxgart, project manager for E.ON Climate and Renewables, described some of the challenges involved in building the project. To protect porpoises from the harsh sounds made by pile-driving, his crew was using an acoustic alarm to drive them away from the construction zone. This evacuation was only temporary, though. Porpoises have been observed returning to offshore wind farm construction areas within hours after pile-driving operations cease, and Haxgart says they are commonly spotted in the waters surrounding R\u00f8dsand's turbines. Fish also flourish around the wind farm, Haxgart notes, with the turbines' large concrete foundations serving as artificial reefs.\n\nAs with land-based wind farms, offshore projects like R\u00f8dsand may create some level of disruption to bird life in the area. Long-tailed ducks\u2014a particular focus of concern among conservationists\u2014may move away from the immediate vicinity of the wind farm, but the environmental impact study commissioned by E.ON predicts few bird collisions with turbines and no significant impact on the international population of this species.\n\nTurning to the wind farm's visual impacts, Haxgart took me to a large plate-glass window at the R\u00f8dsand field office, just across a short stretch of salt marsh from the shore. He handed me a pair of binoculars and pointed to where crane-equipped barges were lowering premolded concrete turbine foundations into the sea. Even with this visual aid, the turbine emplacements were so distant that I found it hard to make out what was happening. Haxgart then explained that the wind farm's visual impacts are greatest at night, when the aviation safety lights are flashing. Initially the turbines were equipped with red lights, synchronized via satellite to trigger simultaneously. On occasions when they went out of synch, people would get upset. \"It looked like Las Vegas at nighttime,\" he acknowledged.\n\nWith permission from the Danish aviation authorities, E.ON has replaced the blinking red beacons with a white strobe and a fixed, low-intensity red light. Apparently this is less jarring to the wind farm's neighbors. Even so, the environmental impact study for R\u00f8dsand II describes the wind farm's expected nighttime appearance as \"comparable to the effects of a small town,\" particularly from the vantage point of its closest landfall, just over a mile from the nearest turbine.\n\nHearing about the lights at R\u00f8dsand brought me back to my first glimpse of the Grand Ridge Wind Farm in Illinois. It was a clear evening in early May as I approached Marseilles on Interstate 80, heading west from Chicago. A few miles before the exit, I looked to my left and there, hovering in the distant darkness, was an eerie constellation of red lights, all blinking in unison. I later checked the map: Grand Ridge's closest turbines were ten miles from the highway.\n\nThe next morning I drove through Marseilles, a small town with a few fading industries on the banks of the Illinois River, and climbed up a short wooded rise onto the ridge that gave the wind farm its name. Seeing the turbines in the bright morning light came as a relief. They were generously spaced across the gently undulating fields, punctuating rather than crowding the horizon. Their profiles blended gracefully with the farmhouses and barns, some of peeling clapboard, some with aluminum siding, almost all of them white like the turbines. The turbines' clean geometric lines handsomely offset the ever-present silos, perfectly cylindrical, their galvanized steel glinting in the sun. And they were far less unsightly than the rusted skeletons of water-pumping windmills that still stand in so many barnyards.\n\nThe Grand Ridge Wind Farm wasn't the first power producer to reach this stretch of central Illinois farm country. Just a few hundred yards from the nearest turbine was the LaSalle County Nuclear Station, its boxy, windowless brick shell and candy-striped smokestack jarring in their juxtaposition to the adjacent fields of corn stubble. The sluiceway that carried cooling water in a long steaming arc from the plant to a large artificial pond also struck me as an uneasy contrast with the surrounding farmlands. A sign welcoming visitors to the LaSalle Lake State Fish and Wildlife Area did little to conceal the true function of the pond, with its ruler-straight borders and power plant looming in the background.\n\nI met with several farm families living in the midst of Grand Ridge's sixty-six turbines and heard very different attitudes about the wind farm. Frank and Sarah Diss told me that the two turbines on their 169-acre farm are easy neighbors, neither noisy nor visually intrusive. Beyond that, they're thrilled to be part of the green power revolution. \"We got on the bandwagon a little bit,\" Frank told me. \"Wind power, green energy\u2014we have to have it.\" Pointing at a wind tower on the far side of his barn, about a quarter-mile away, he said: \"One of those will provide enough electricity for 400 homes, so we don't have to fight foreign oil.\"\n\nBob and Ruth Widman, who live a mile or two away from the Disses, are much less happy having turbines in their midst. They have lived in the same house since 1955, and their son continues to farm the property. \"It changes the whole rural area,\" Ruth delicately commented. The turbines distract and disorient her when she's driving, and she welcomes the visual relief when she makes the occasional trip away from her home turf. \"It seems so good to get out someplace where we can look and see farms,\" she sighed. Bob spoke about the turbines in cruder and angrier terms: \"They're like cow manure\u2014they're all over the place.\"\n\nIn my visits with the Disses, the Widmans, and others, I found it hard to separate their divergent perspectives from their very different financial stakes in the project. The Disses get $8,250 in annual lease payments for each of the two turbines on their property, plus a dollar per year for each linear foot of access roads crossing their land. I assume that their enthusiasm about wind energy is reinforced, at least to some degree, by those benefits. The Widmans initially considered joining the project, but Bob bargained for higher lease payments than the project's developer, Invenergy, was willing to offer. He told the company's land agent that he wasn't willing to take less than $10,000 per turbine per year, so they ended up outside the project, earning nothing while looking out at their neighbors' turbines. I'm not surprised that they now feel alienated.\n\nAt Grand Ridge, as at dozens of other locations across America, wind farms may not be universally embraced, but they are fast becoming part of the working landscape. They are practical implements, akin to tractors and harvesters, even if their physical presence is more dominant and dynamic. Where wind developers run into stiffer opposition is in areas where their projects are seen as diminishing highly valued natural or historic landscapes. The Flint Hills of Kansas are certainly one such place.\n\nRose Bacon grew up on a farm in Iowa. When she was about twelve, her father brought her along on a trip to the Flint Hills to buy cattle. It was love at first sight, she recalls. \"I told my dad on the way home, 'That's where I belong.' \" It took a few decades to realize her dream, but in 1991, she and her husband, Kent, bought a 520-acre ranch in the heart of the Flint Hills, just south of the village of Council Grove. Since then, they have grazed up to a thousand head of cattle on their rich grassland pastures each summer, trucking them back to their out-of-state owners or sending them on to a feedlot after they've had their fill of prairie grass. Kent, a Vietnam vet, has a specially contoured prosthetic leg that allows him to sit comfortably in the saddle as he works the herds on horseback.\n\nIt may be Rose's outsider perspective that has made her such a passionate protector of the Flint Hills. \"When you move to a new area, you perhaps don't take it for granted the way you might if you grew up in that particular place,\" she observes. Iowa once was prairie; now it is cultivated farmland. She knows how utterly a landscape can be changed by human industry, and in her view, wind energy would bring just that sort of unwelcome transformation to the Flint Hills. Once the tallgrass prairie extended across 170 million acres of the eastern Great Plains. About 4 percent remains today, with two-thirds of that in the Flint Hills, extending in a 50-mile-wide swath from just south of the Nebraska border all the way through Kansas and down into Oklahoma.\n\nI visited RK Cattle, Rose and Kent's ranch, in the midst of a late-July heat wave. After serving me a bacon-and-eggs breakfast, Rose asked if I would be more comfortable touring the ranch by horse or on an all-terrain vehicle. I opted for the horse but admitted to being a bit nervous; I had last been on horseback as a camper in the 1960s. After a brief riding lesson, we headed up into the hills. Rose pointed to the rich layer of grasses and wildflowers beneath our horses' hooves and rattled off names like bluestem, buffalo grass, and ironweed\u2014just a few of the plants that inhabit this landscape. The grasses were still green, but by autumn they would be tall and brown, kept alive by roots that grow several feet into the rocky soil. I gazed all around me. Nothing more prominent than the occasional barbed-wire fence broke the flow of the grassy uplands. The only trees in sight were far below us, running along the path of a narrow creek.\n\nDown by the creek, Rose showed me a small grotto of layered flintstone (\"chert\" is the more accurate term, she told me) where water trickled into a shallow pool no more than a dozen feet across. I called it a spring; she corrected me, telling me it's a waterfall. \"The scale here is not like the Rockies\u2014we don't have thousand-foot waterfalls,\" she said. Then she quoted a well-rehearsed adage about the Flint Hills: \"It doesn't take your breath away, but it gives you a chance to catch your breath.\"\n\nRose's voice stayed calm as she shifted to talking about wind farms, but her steely determination as an anti\u2013wind farm fighter came through in her choice of metaphor. \"I personally look on the wind complexes as a rape of the landscape, and I don't use that term lightly.\" She reminded me that, before coming to Kansas, she worked as an emergency-room nurse. \"You see those people come in. They are battered and shattered and torn up. There's no doubt in your mind what happened.\" Then, she continued, a year or more may pass before the trial. By that time, the obvious scars will have healed, though not the deeper physical and psychological wounds. \"And the defense will say, 'See? Basically there was no damage here. Probably it was invited.' \" She likened this to the scars in the landscape caused by wind farm access roads and concrete-slab turbine foundations. Wagon-wheel ruts from the Santa Fe Trail are 150 years old, but you can still find them. Likewise, she said, the subterranean remnants of turbine footings may prevent prairie tallgrasses from sending roots deep into the soil, stunting their growth. \"You're going to have these little rocky, bare, weedy spots where everything dies on days like this.\" These wounds, she warned, will last long after the turbines have come down and the viewscape has been restored.\n\nRose Bacon's romance with the Flint Hills landscape unnerved me. Her self-avowed quest to preserve its pure prairie essence left no room for accommodating certain human intrusions, yet she seemed to accept others readily. She likened wind farms to the most heinous of criminal insults, but she dignified the burning of more than a million-and-a-half acres of tallgrass prairie as an \"annual rite\" that ushers in the renewal of lush green grass and countless varieties of wildflowers. What about the prairie chickens and other grassland birds that the flames rob of their nesting grounds, and what about all that air pollution, sufficient to set off ozone alerts in cities dozens of miles away?\n\nI was also bothered by Rose's apparent need to invalidate just about every possible argument for wind energy. She dismissed Al Gore's \"Chicken Little theory\" about climate change as \"a load of bull\" and derided wind technology as inefficient and unreliable. She railed against the subsidies given to wind developers, ignoring our government's long history of providing much more massive support to the fossil fuel and nuclear industries. And she extolled the beautiful pictures she'd seen of reclaimed surface coal mines. Could she really imagine that those gaping canyons, blasted, eviscerated, and bulldozed, were less devastating to the environment than a few dozen buried turbine footings spread across several miles of open landscape?\n\nRon Klataske, who directs Audubon's Kansas chapter, sees wind energy in less black-and-white terms than Rose Bacon. He acknowledges that some wind farms in Kansas are well sited, pointing in particular to two completed projects in the heavily farmed southwestern corner of the state. He also recognizes that the Flint Hills are a tempting target for wind developers, given the stiff, steady winds that blow through the area and the proximity of the Flint Hills to cities in eastern Kansas and over the border in Missouri. Yet he shares Rose's antipathy to wind in the Flint Hills and has been fighting what his organization has called \"a potential tsunami of industrial-scale wind turbine complexes\" and \"a killing field for migrating birds.\" He hopes that, over time, the U.S. Fish and Wildlife Service will acquire a million acres of conservation easements in the Flint Hills. In the meantime, he has been encouraging the purchase of conservation easements by groups like the Ranchland Trust of Kansas and the Nature Conservancy.\n\nRon arranged for me to meet some of his allies in the Flint Hills campaign against wind. We shared beer and burgers at the Hitchin' Post in the village of Matfield Green, on the Flint Hills Scenic Byway. Then we drove east on a dusty gravel road, crossing a stretch of soybean fields before reaching the open prairie. Just as the sun was setting, we pulled off the road on the crest of a windswept hill. Broad stretches of undulating green surrounded us, softened by the fading light. Thinly scattered herds of cattle peppered the landscape\u2014Black Angus, mainly. A nighthawk hovered overhead, crickets raged, and a coyote howled in the distance.\n\nBill Browning, a physician who still lives on his family's multigenerational ranch, opened the conversation. \"The beauty of it, for me, is where the hills meet the sky, morning and evening, and the shadows come across the hills and make all the contours stand out. If you're going to put a string of 400-foot steel behemoths across the horizon, it's gone. The loneliness, the emptiness, the absence of people, the absence of intrusions of people\u2014all that would be lost.\"\n\nJacque Sundgren, a rancher along with her husband, Steve, once went door to door petitioning neighboring ranchers to say no to a proposed wind farm in their area. The project was never built. \"This right here is my life,\" she said. \"You put industry on it and it's gone.\" Steve explained the choice their family has made. \"We could make hundreds of thousands of dollars off of wind, but there are other things to pass on to your kids.\"\n\nIt's no small wonder that Pete Ferrell has incurred the wrath of many Flint Hills ranchers and conservationists. Their passions and his pragmatism are a poor match. Pete's great-grandfather began assembling property for the family's ranch in 1888, with profits he earned from a dry goods store in Wichita. The family holdings grew to 7,000 acres, but Pete inherited only a third of the land. To keep the ranch intact, he bought back some of the acreage from his aunts and uncles. This left him carrying a crushing debt. \"If you inherit land and it's debt-free, you can probably make a decent living, but paying interest and principal on land?\" The last time farmers and ranchers could break even on land that they had to purchase was in the 1920s, he tells me.\n\nAs he struggled to amass enough capital to purchase the remainder of the ranch, Pete eventually realized the toll this was taking on his health and well-being. \"I was looking at a situation where I was going to join the Old Dead Ranchers Club, which is a group of great people who work 24\/7, 365 days a year, until they fall over dead. 'Congratulations, you're a member,' \" he says with a sardonic grin. We are sitting in his adobe-style ranch office, just across a rutted driveway from the hollow where he lives in a simple, unpainted wood cabin with his two dogs. The divorce from his wife some years ago, I can guess, only added to his financial stresses.\n\nIt was in the late 1980s that he became interested in the holistic management principles that economist Stan Parsons and ecologist Allan Savory had begun advancing through their teachings. Attending one of their training sessions, Pete was asked to catalog the full array of values on his property. He started underground with the oil resources that had yielded a handsome income for his mother over many years. (Even today, a half-dozen of those wells remain visible on his land, although Pete says they're drying up.) Then, rising to the surface, he reflected on the soils, the grass, the sun, and the water that make it possible to graze his cattle. Finally he was asked to delete those activities that are unsustainable or are not ecologically sound. Wind was on the short list of what was left.\n\nHalf-obscuring the windows in Pete's office today are two NREL maps showing the distribution of wind resources in Kansas. Even without those maps, he knew that the winds in his section of the Flint Hills were exceptional. This was confirmed when, in 1994, a representative of Oxbow Power Corporation knocked on his door, asking if he'd be interested in leasing out his land for a wind farm.\n\nPete found it strange that Oxbow would be interested in wind. The Palm Beach-based company was owned by Bill Koch, part-owner of Wichita-based Koch Industries until he split acrimoniously from the family-run conglomerate in 1983 and formed his own array of businesses with a heavy focus on traditional energy sources: coal, natural gas, and coke, an oil-refining byproduct used to power cement kilns and other industrial facilities. Pete also wasn't convinced that wind would be right for his ranch. In an effort to persuade him, Oxbow flew him out to California to visit some wind farms. There and elsewhere, Pete spoke with a number of wind farm hosts to see how the turbines affected their agricultural operations. They told him: \"What wind turbines? We don't even think about them anymore. We still farm or ranch just like we always did.\"\n\nReassured, Pete signed a thirty-five-year lease with Oxbow, giving it the option to build a wind farm on his property. Meteorological towers then went up, but Oxbow came back to him in 1998 with discouraging news. \"You have one of the finest wind resources we've ever seen,\" he was told, \"but we're pulling out of here.\" Oxbow's agent referred to the project's unfavorable economics, but he also mentioned that a hostile political environment in Kansas contributed to the decision.\n\nIt wasn't clear to Pete whether these hostilities were at the state level, in the Flint Hills, or among the notoriously feuding Koch brothers. Whatever the immediate reasons for Oxbow's abandonment of Flint Hills wind development, it is a sad irony that Bill Koch went from wind energy prospecting in Kansas to bankrolling anti-wind advocacy in Massachusetts. An avid sailor with a vacation home on Cape Cod, Koch\u2014a conservative Republican\u2014found common cause with liberal icon Ted Kennedy in fighting Cape Wind. _Forbes_ magazine reported that, by the fall of 2006, Koch had donated $1.5 million to the nonprofit Alliance to Protect Nantucket Sound, orchestrator of the multiyear campaign against the offshore wind project. He was the organization's co-chair at the time. Through Oxbow, he contributed another $620,000 to lobbyists who worked the halls of Congress in 2006 and 2007 on the anti\u2013Cape Wind amendments to the Coast Guard authorization bill and other measures aimed at stopping the project. Bill Koch may have estranged himself from Koch Industries, but his railing against Cape Wind was right in step with the far-right political activism of his brothers Charles and David, who have contributed well over $50 million to studies disputing climate change science, to policy initiatives opposed to curbing U.S. greenhouse gas emissions, and to politicians willing to fight for oil and gas industry interests in Congress.\n\nWhile Koch was navigating a quick course away from Flint Hills wind, Pete Ferrell continued collecting wind data from the meteorological towers Oxbow left on his ranch, and he began shopping for another developer. He ended up with seven offers and chose a small company called Greenlight Energy Resources, out of Charlottesville, Virginia. He had met the company's CEO, Sandy Reisky, on a bus tour of the Gray County wind farm in southwest Kansas, and the two immediately hit it off, signing a deal in 2001.\n\nThen came the real shock for Pete. He knew that some of his neighbors had their misgivings about wind turbines in the Flint Hills, but he assumed they would come to see its benefits once they learned more about the technology, just as he had. Looking back on the lawsuits, the fiery public hearings, and the angry words from people he once considered friends, he now realizes how naive he had been. \"We had TV cameras in the courtrooms, we had people pounding podiums and shaking their fists,\" he recalls with obvious sadness. \"You think abortion is a hot issue, you just try to build a wind farm in the Flint Hills.\"\n\nThe Butler County Commission approved Greenlight Energy's permit application in 2003, but by then the outcry against wind energy had reached such a feverish pitch that Governor Kathleen Sebelius intervened. Mindful that permitting authority resided with counties rather than the state, she convened a citizen task force and instructed its members to recommend voluntary guidelines that counties might follow in balancing wind energy development with prairie preservation. Rose Bacon was appointed to this panel. \"I think I was chosen because they didn't know me,\" she told me. \"As a ranch wife, what possible harm could I do?\"\n\nThe governor had hoped that consensual guidelines would emerge, but that was not in the cards. Although the task force's scope was statewide, the real friction points were in the Flint Hills. One faction, which Rose considered pro-wind, called for a three-tier classification system that would divide native grasslands into no-development zones, areas with restricted development, and areas with few restrictions. These zones, advisory in nature, were to leave the ultimate decision making with municipal and county authorities. Rose Bacon's faction took a much harder line, calling for an outright ban to be imposed by the state on wind energy projects in all areas with largely intact prairie, including a seven-mile buffer zone around those areas. It also recommended that the state's property tax exemption on wind energy investments be abolished, and favored consideration of a new statewide wind-development impact fee.\n\nFaced with this stalemate and sensing the growing vehemence of anti-wind forces in the Flint Hills, Governor Sebelius came up with an artful compromise in November 2004. On one hand, she asked\u2014but did not order\u2014wind developers with project proposals in an area she called the \"Heart of the Flint Hills\" to freeze their projects so that counties could have the time to prepare their own guidelines for wind development. The designation covered about two thirds of the Flint Hills\u2014far less than Rose Bacon and her faction had sought\u2014but the governor's action validated the Flint Hills defenders' claim to a unique historic landscape.\n\nAt the same time, the governor\u2014a supporter of wind energy for Kansas\u2014encouraged wind developers to build in other parts of the state, just as the Zilkhas decided to do when opposition mounted against their proposed project in the Flint Hills. As Greenlight Energy's wind farm site lay just a few miles outside the Heart of the Flint Hills' southern boundary, Sandy Reisky and Pete Ferrell moved ahead with their project. To raise sufficient capital for the wind farm, Reisky sold the project to PPM Energy of Portland, Oregon, in December 2004, which later became a subsidiary of the wind development giant Iberdrola Renewables. Construction began early in 2005, and the 150-megawatt Elk River Wind Power Project went into operation in December of that year.\n\nSebelius never converted her Heart of the Flint Hills freeze to an outright ban on wind development in the area, but Rose Bacon and other ranchers formed a group, Protect the Flint Hills, to lobby for a stronger state policy. At the same time, they worked on fellow ranchers to dissuade them from signing any new deals with wind developers. Rose's advice was unequivocal. \"If you care about your land, you won't lease it to wind. It's that simple.\"\n\nIn May 2011, anti-wind activists in the Flint Hills achieved another victory when Governor Sam Brownback announced a Road Map for Wind Energy Policy that discouraged any new wind development in an area more than twice the size of Sebelius's Heart of the Flint Hills. Though still not a categorical prohibition, the policy made it clear that wind developers would have a hard time building new projects in the 11,000-square-mile area that he dubbed the \"Tallgrass Heartland.\" Rose was elated. \"Ten years of work have gone into this, and we finally have a definitive decision on the future of the Flint Hills,\" she wrote me. \"It's green and beautiful here, the cattle are out to grass, and today this is the BEST place in the world to be!\"\n\nThe Elk River wind farm, with half its turbines on Pete Ferrell's property, falls inside the newly expanded boundaries of the Tallgrass Heartland. While expanding that project is now out of the question, Brownback has offered assurances that existing wind energy production in the Flint Hills will not be shut down. The governor also made it clear that transmission lines will not be blocked from crossing the Flint Hills. This comes as relief to wind developers in central and western Kansas, as well as points farther west, who are concerned about getting their power to eastern markets.\n\nDespite all the turmoil, Pete's enthusiasm for wind energy has by no means abated. Along with stabilizing his ranch's finances, he firmly believes he is helping wean America off energy resources that, in the greater scheme of history, are moments away from running out. To make this point, he unfurls a few dozen feet of computer paper, starting on the floor of his office, continuing out into the foyer, and then reaching across the full length of a second room where his ranch manager keeps track of cattle shipments. This long paper trail, ragged from repeated use, likens the past 750 million years of history to a single calendar year. Pete takes me to the final few perforated sheets and points to December 11, the date when mammals show up. _Homo erectus_ appears the day after Christmas; _Homo sapiens_ on December 30. At three seconds to midnight on December 31, we begin to burn petroleum, and by three seconds into the New Year, Pete tells me it will all be gone. \"We have a whole civilization built on something that's gonna run out,\" he told a roomful of fellow Grinnell alumni when he lectured on this topic at a recent college reunion.\n\nRose takes a cynical view of Pete's efforts. \"People don't believe he's a wind conqueror; they believe it was for the money. . . . I feel bad that he had to make that choice because he's lost so much.\" I ask what he has lost, and she tells me: \"Friendships and respect.\"\n\nPete, for his part, is now working on a project that may bring as much as 800 megawatts of wind power to cultivated farmland in southwestern Kansas\u2014more than five times as much as Elk River Wind's installed capacity. Luckily for Pete and other Kansas wind developers, there are ample opportunities to tap the state's enormous wind potential outside the Flint Hills.\n\nGrandpa's Knob, a few miles west of Rutland, Vermont, has a prominent place in the history of wind-generated electricity. There, on a 2,000-foot summit, an MIT-trained geologist-turned-energy innovator named Palmer Cosslett Putnam built America's first large-scale power-generating wind turbine in 1941. Equipped with two 70-foot-long aluminum alloy blades mounted on a 110-foot tower, the turbine was designed to produce up to 1.25 megawatts of power\u2014much larger than the wind machines that were widely used in California forty years later and in the same league as today's commercial turbines. As protection against damage from rough weather, the blades could be pivoted so that their broad surfaces stood perpendicular to incoming winds. This safety measure, using a centrifugal or \"flyball\" governor, anticipated the more-sophisticated electronics of modern-day blade pitch controls.\n\nPutnam's giant wind machine had a short and sporadic set of runs. When the main bearing failed little more than a year after the turbine began operating, resource constraints imposed by the war effort kept it out of service for over two years. Then, only a month after it was finally repaired in March 1945, one of the giant blades broke off, shutting down the turbine once again. In the immediate aftermath of World War II, one local company laid out tentative plans to rehabilitate the machine for power generation, proposing to make it a tourist magnet with a scenic toll road, summit restaurant, tower-top observation deck, and possibly a ski development. None of that happened.\n\nI climbed Grandpa's Knob in May 2010. To get there, I trudged up the same two-mile dirt road, steep and meandering, that had been used to truck turbine parts up to the site sixty years earlier. At the summit, all that remained of Putnam's wind machine were a few crumbling concrete footings and a small commemorative plaque. The site, however, did offer a panorama of three nearby ridgelines where a Vermont-based developer, financially backed by the Italian energy conglomerate Enel, had been negotiating to build a new 85-megawatt wind farm.\n\nLeasing land for the wind farm turned out to be the easy part. The developer, Vermont Community Wind, had lined up more than 4,000 acres of woodlands managed by a large timber company, Wagner Forest Management. What proved much more difficult was getting residents of the affected communities to come to terms with a wind farm in their midst. The town of Ira was a case in point. With only 460 residents, the town stood to gain $10,000 to $11,000 per turbine per year in contributions from the wind developer. Given plans for multiple turbines within Ira's boundaries, total payments would have easily exceeded the town's $160,000 annual budget. Yet this promised financial boon wasn't enough to overcome local residents' concerns about planting turbines on the forested ridges above their homes. Vermont Community Wind had originally identified sixty potential sites for turbines in five area towns. By June 2009, public opposition had reduced that number to forty-five, and by January 2010, the company had brought the total down to thirty-four.\n\nEven at this smaller scale, the wind farm had many local residents up in arms, not only about viewshed impacts, but also because of the noise that they feared the turbines would bring into their lives. Mary Pernal lives in a modest house in the town of Poultney, 500 feet below one of the ridges that Vermont Community Wind had targeted for turbines. An assistant professor of English at Green Mountain College, she described her apprehensions: \"Most of us live in this neighborhood because it's a wild, serene, beautiful place, and the noises that we hear are pleasant noises: the sound of a creek gurgling by us, the sounds of birds, the sounds of various wildlife occasionally, the sounds of rain on the roof.\" She worried that the thirteen proposed turbines would create an amphitheater effect, filling her tranquil surroundings with unwelcome noise.\n\nPernal and her neighbors were not just idly imagining the noise that wind turbines might bring into their lives. They had read or heard accounts of other New England residents living near wind farms\u2014people like Phil Bloomstein, neighbor to the Beaver Ridge Wind Project in Freedom, Maine, who reported that noise produced by a GE 1.5-megawatt turbine 1,000 feet away from his home \"can turn from an almost tolerable drone to a pulsating nightmare so oppressive that any outdoor activity is challenging.\" Or Wendy Todd, who testified before the Maine Governor's Task Force on Wind Power that there's no escaping the noise and vibrations coming from the Mars Hill wind farm, 2,600 feet from her home in Aroostook County. Todd likened the noise from these turbines\u2014the same GE model used at Beaver Ridge\u2014to \"a fleet of planes that are approaching but never arrive,\" and described particularly bad periods when \"a repetitive, pulsating, thumping noise . . . can go on for hours or even days.\"\n\nClimbing Grandpa's Knob was an intriguing hike into wind energy history, but my primary purpose in visiting Rutland was to get a better sense of the mounting controversy about wind farm noise. In its Winter 2010 _Heart Health_ newsletter, the Rutland Regional Medical Center featured one of its senior cardiologists linking wind turbine noise to sleep deprivation, which he said can cause an elevated risk of hypertension, heart attacks, atrial fibrillation, and stroke. The article angered a number of readers, who objected to the physician's speculative leaps and protested the medical center's apparent taking of sides in the stormy local debate about wind energy. In a postcard sent to all its readers, the medical center apologized for the communication, declared itself to have no position on wind energy, and announced that it would be holding a public forum on the technology's health effects.\n\nIn late April 2010, just about a week before the forum was to be held, Vermont Community Wind announced that it was freezing its plans for the wind farm. Spokesman Jeff Wennberg was candid about the company's frustrations. \"Vermont seems to say all the right things in terms of incentives and general state policy,\" he told a Vermont Public Radio reporter, \"but when you actually get to proposing specific projects, it's very clear that the words and the reality of what is possible to do in Vermont [are] very, very different.\"\n\nEven with the wind farm on indefinite hold, close to a hundred people crowded into the basement lecture hall at Rutland Medical Center. The forum featured a debate between Dr. Robert McCunney, an internist specializing in occupational and environmental health at Massachusetts General Hospital, and Dr. Michael Nissenbaum, a radiologist at Northern Maine Medical Center who has become a self-taught expert on the noise impacts of wind energy. McCunney had recently co-authored an expert review of wind turbine sound and health effects for the American Wind Energy Association (AWEA) and its Canadian counterpart. Nissenbaum had coauthored a searing critique of that study for an anti-wind group called the Society for Wind Vigilance.\n\nMcCunney started out by describing the two major components of sound: frequency, measured in hertz, or cycles per second; and volume or loudness, measured in decibels. Wind turbines produce a variety of sounds: mid- to high-frequency aerodynamic sounds given off by rotating blades; lower-frequency sounds emitted by turbine machinery such as gears and generators; and ultra-low-frequency \"infrasound\" sometimes associated with turbine operations. High-frequency sounds, though audible at lower volumes than low-frequency sounds, attenuate more quickly over distance and are absorbed more readily by landscape features such as trees and leafy crops. Infrasound isn't even detectable to the human ear at low volumes, but it can be experienced as vibration and can create secondary vibrations or rattles in nearby structures.\n\nNissenbaum called the m\u00e9lange of sounds produced by wind turbines \"an acoustic pizza.\" Some mechanical sounds are constant; other sounds pulsate, like the swishing of blades as they cut through the air at tip speeds that can reach 200 miles per hour. Some are tonal; others are not. When multiple turbines are within range, as is often the case at commercial wind farms, they produce even more complex combinations of sound.\n\nThe two physicians agreed that, above certain thresholds, the sounds generated by wind farms become unwanted noise. Reactions to this noise vary, said McCunney. \"Some people may be annoyed being stuck in a traffic jam, or being stuck too long in a line at the post office, or waiting too long for [their] son to come home at two a.m. on a Sunday night,\" he observed. Likewise, McCunney said, some people are more bothered by wind turbine noise than others. He referred to a Dutch study's finding that a small percentage of people\u2014about 5 percent of a sample of 725 people living within 2.5 kilometers (4 miles) of a wind turbine\u2014reported being annoyed by wind turbine sounds in the 35- to 40-decibel range. Two separate Swedish studies found that, when turbine sounds reached 40 to 45 decibels, 18 percent of respondents registered annoyance.\n\nNissenbaum shared McCunney's view that people have very different thresholds for perceiving and tolerating noise. \"One person in seven in this room will sense noise at six decibels lower than the average person,\" he said. \"We're all built differently.\" Where he parted ways with McCunney was over the health impacts of wind farm\u2013generated noise. McCunney insisted that there have been no rigorous studies showing a risk of adverse health effects from wind turbine noise. Nissenbaum countered with findings of his own limited investigation at the Mars Hill wind farm. In his evaluation of 22 people living within 3,500 feet of the turbines, 18 reported new or worsened sleep deprivation, 17 said they experienced persistent anger, 9 said they newly suffered from chronic headaches, and 8 had new or worsened depression. People living three or more miles away reported dramatically lower incidences of all these symptoms.\n\nSystematic studies have yet to explore the health effects of wind turbine noise on broad populations, but anecdotal accounts link a range of ailments to wind turbine noise. The U.S.-based Industrial Wind Action Group, whose declared mission is to \"counteract the misleading information promulgated by the wind energy industry and various environmental groups,\" has published numerous testimonials by aggrieved wind farm neighbors on its website. In Australia, health complaints by citizens led the government's National Health and Medical Research Council to evaluate their grievances. Based on this review, it stated: \"While a range of effects such as annoyance, anxiety, hearing loss, and interference with sleep, speech and learning have been reported anecdotally, there is not published scientific evidence to support these adverse effects of wind turbines on health.\"\n\nPhysician Nina Pierpont has coined the term \"Wind Turbine Syndrome\" as a catchall category for the health effects that her own anecdotal research has associated with wind turbine noise. In her interviews with ten families living near commercial-scale wind turbines, she heard about symptoms including sleep disturbance, headache, tinnitus (ringing of the ears), dizziness, nausea, visual blurring, tachycardia, irritability, concentration and memory problems, and panic episodes. Many of these conditions, Pierpont hypothesizes, occur when low-frequency sounds upset the normal balancing or \"vestibular\" function of the inner ear, as well as the brain's ability to process balance-related neural signals. She further hypothesizes that turbine-generated vibrations interfere with the gravity receptors in our visceral organs. These disruptions, taken together, create what she calls \"Visceral Vibratory Vestibular Disturbance.\"\n\nPierpont's study has been roundly attacked for its methodological flaws, including the lack of a control group living away from wind turbines that could provide a baseline against which Pierpont's reported health effects could be compared. Geoff Leventhall, a British acoustical expert, has taken Pierpont to task for her tenuous assumptions linking turbine-generated infrasound to ailments previously associated with much higher noise exposures by workers in aerospace and other heavy industries.\n\nPierpont, in turn, has scolded critics like Leventhall for overstepping their professional bounds. \"Deciding whether people have significant symptoms is not within the expertise of engineers or specialists in acoustics,\" she caustically noted in one of her talks. \"[T]he hallmark of a good doctor is one who takes symptoms seriously and pursues them until they are understood (and ameliorated).\"\n\nWhile Pierpont's findings may be sketchy, enough people are upset by wind farm noise to create real problems for an industry that seeks to cover a lot of new territory, quite literally, in the coming years. Even if we set aside speculation about various diseases arising from turbine noise, an increasingly vocal cohort of angry, sleep-deprived people will not be good ambassadors for an emerging industry that needs to win the public's support. Along with better research into health effects, what's needed are measurable and enforceable limits on noise. Also needed are siting guidelines that ensure neighbors a sufficiently generous setback from turbines to protect them from unwanted noise and other annoyances such as the pulsating light, called \"shadow flicker,\" that rotating blades can cast across the landscape when the sun is at a low angle on the far side of a turbine.\n\nTo set noise limits for wind farms, a decision has to be made about what level of turbine-generated noise is acceptable. Acoustic experts often approach this challenge by comparing wind turbine noise to other, more familiar sounds affecting our daily lives. A dishwasher in the next room produces about 50 decibels of noise. A library interior or a suburban area outdoors at nighttime might register 40 decibels\u2014which is perceived to be half as loud as 50 decibels given the logarithmic scale of sound measurement. Sound at 30 decibels\u2014half as loud again\u2014would approximate a quiet bedroom at night or a quiet rural area with no wind, insects, or traffic.\n\nToday there are no federal noise limits for wind turbines, but the U.S. Environmental Protection Agency (EPA) generally recommends that outdoor noise levels should be no higher than 55 decibels during the day and 45 decibels at night. The World Health Organization (WHO)'s Guidelines for Community Noise, issued in 1999, also call for a 45-decibel nighttime limit, measured outside the home on the assumption that noises in a bedroom with the window slightly open would be 15 decibels below that level. In an updated report focusing on Europe, the WHO has recommended a more-protective 40-decibel outdoor nighttime limit. Preventing sleep disturbance is the primary goal of this guideline.\n\nIn the absence of enforceable federal requirements, some states have adopted their own general noise standards. Maine, for example, has set general noise limits that mirror the EPA's daytime and nighttime guidelines. Most states, though, have treated noise as a matter to be governed by counties and municipalities. In some local jurisdictions, new ordinances specifically regulate wind farm noise. Others have deferred to individual wind developers, letting them negotiate ad hoc arrangements with neighbors at their project sites.\n\nSetting numerical limits for wind farm noise is complicated by the difficulty in capturing the _quality_ of the noise generated by turbines, not just the _quantity_ of that noise in decibels averaged over a certain period. Some early-design turbines with rotors mounted downwind of towers were known to make a distinctive thumping noise when their blades passed by the tower. The switch to upwind rotors has reportedly reduced this problem, although even today wind farm neighbors like Phil Bloomstein and Wendy Todd are annoyed by the pulsating sound that sometimes comes from the upwind turbines operating near their Maine homes. This noise apparently results from sound pressure that builds up during the blade's downward sweep. If a three-bladed turbine is rotating at fifteen revolutions per minute, aerodynamic noise can be expected to spike about once every 1.3 seconds. At twenty revolutions per minute, it peaks every second. Noise from multiple turbines can complicate the sound rhythm further, causing wind critics like Mike Nissenbaum to bridle when turbine-generated noise is likened to the steadier sound of a dishwasher or a refrigerator humming in an adjacent room.\n\nNina Pierpont challenges current turbine noise measurement techniques on another ground, taking issue with the commonly used A-weighted decibel scale, which reflects the human ear's sensitivity to different sound frequencies by underweighting inaudible sounds. Her contention is that this scale fails to capture the low-frequency and ultra-low-frequency sounds that may be having the greatest adverse health impact on wind turbine neighbors.\n\nRequiring turbines to be set back a minimum distance from nearby homes is another way to protect wind farm neighbors from noise. The advantage to relying on setbacks is their simplicity: compliance is easily measured, and ongoing enforcement isn't needed once a wind farm has been built. A very real downside, though, is their inaccuracy: setbacks are at best a crude proxy for numerical noise limits. Several environmental factors influence how far and in what direction turbine noise travels. People living downwind of turbines are more exposed to noise than those living on the upwind side. During temperature inversions, sounds stay closer to the ground and are audible at greater distances. Hard surfaces such as frozen ground reflect rather than absorb sound, causing it to carry farther. And the presence or absence of foliage can play a big role in how quickly sounds attenuate as they pass across cultivated fields, pastures, and woodlands.\n\nCertain wind conditions can conspire with topography to increase wind noise. When the wind is blowing at blade height but ground-level air is still, turbine noise can be particularly noticeable. This is often the case at Mars Hill, Wendy Todd told the governor's task force. \"There are many times when winds are high on the ridgeline but are near calm at our homes.\" On these occasions, she said, \"It doesn't matter which room you go to, there is no escape from the noise.\" Pete Ferrell told me he isn't bothered by turbine noise on his Flint Hills ranch, but he does hear the turbines when the winds don't reach down into the wooded hollow where his rustic cabin nestles.\n\nThe National Research Council reports that a commercial-scale turbine produces 90 to 105 decibels, with the sound attenuating to 50 to 60 decibels at a distance of 40 meters (131 feet) and 35 to 45 decibels at 300 meters (984 feet). If these fairly crude estimates were right, a setback of 1,000 feet would provide reasonable noise protection to wind farm neighbors. However, a greater buffer is very likely needed to accommodate acoustical differences between turbines, daily and seasonal changes in weather, varying topography, and the heightened sensitivity of certain people to noise. Maine-based developer Rob Gardiner of Independence Wind believes that setbacks in the 2,500- to 3,000-foot range may be needed, especially in areas where turbines are sited on ridgelines. As for Maine's 45-decibel nighttime limit for turbines, he thinks it's sufficiently stringent. \"All the problem sites in Maine have reported actual sound levels above that limit,\" he observes.\n\nKeeping turbines a half-mile or more away from homes may be what's needed, but such a broad margin of protection will not be an easy sell to many wind developers. Even before newly elected Wisconsin governor Scott Walker created a public uproar with his assault on the collective bargaining rights of labor unions, he angered wind developers when, in January 2011, he called for increasing his state's mandatory turbine setback to 1,800 feet from an adjacent landowner's property line. Not only did wind proponents object to the length of the setback, but they rightly felt it was arbitrary to use the neighbor's property line, rather than distance from the nearest residence, as the delineation point if the primary goal was to protect people in their homes from noise. Michael Vickerman, executive director of RENEW Wisconsin, protested \"the folly of Governor Walker's job-killing proposal\" and warned that its adoption by the legislature would drive $1.8 billion in new wind power development out of state. Vickerman and others favored the less-forbidding standard already adopted by the Public Service Commission, requiring the homes of wind farm neighbors to be set back by the lesser of 1,250 feet or 3.1 times the turbine's maximum blade-tip height. In March 2010, the same legislature that approved the governor's union-busting agenda suspended the commission's wind siting rules, leaving wind developers in limbo and leading AWEA to describe Wisconsin as \"closed for business.\"\n\nDrs. Nissenbaum and Pierpont's recommendations are far more restrictive than Governor Walker's controversial proposal. They have called for 2-kilometer (1.24-mile) setbacks from turbines in ordinary terrain, going up to 2 miles in mountainous areas. While these two doctors' positions may lie beyond the outer fringes of what most regulators are demanding, their outreach to state legislatures and local planning bodies gives their message a prominence that the wind industry can ill afford to ignore.\n\nCommercial power production, by virtue of its scale, is an intrusive presence in our lives. Wind farms are hardly exempt from this liability, as I have learned in my travels over the past two years and from my earlier advocacy work in New England. While the industry will inevitably have its detractors, there certainly are ways to help wind farm developers reach a more harmonious coexistence with their neighbors. Decisions about siting wind farms in areas where feelings about natural, scenic, and historic values run high may be aided by more coherent siting guidelines that balance the protection of those values with the need to take much fuller advantage of our renewable energy resources. The Biodiversity Conservation Alliance, already discussed in chapter 7 as an outspoken player in Wyoming's sage grouse debate, has done a particularly impressive job weighing factors ranging from visual concerns to wildlife protection in its delineation of go and no-go areas for wind energy development.\n\nRegarding turbine-generated noise, a more proactive governmental role is needed to reduce an obvious source of annoyance and discomfort. Current levels of exposure to wind farm noise may upset only a minority of wind farm neighbors today, but those numbers will grow as wind farms proliferate, particularly as we approach the Department of Energy's goal of having 20 percent of our power come from wind by 2030. The federal government could lead the way by adopting uniform noise standards and delegating enforcement authority to the states\u2014just as it has done with many of our federal environmental, public health, and workplace safety laws. Short of setting national standards, the government could surely offer more coherent guidance to state and local agencies as they develop their own. The current regulatory vacuum is substantially responsible for the discontentment that is emerging among at least some people living near wind turbines. Whether disgruntled neighbors turn to the media or take their grievances to court, the wind industry is not being well served by the absence of an adequate, enforceable framework for developing and operating wind farms.\n\nAs with any new industry, knowledge about wind power's impacts is growing as the technology gains traction in the field. Wind energy's enormous promise as a cleaner energy resource will be realized more fully if new information about noise and other concerns is openly and rigorously evaluated and appropriate measures are taken to protect public health and well-being.\n\n## CHAPTER NINE\n\n## Greening the Grid\n\nWHEN MERIDIAN WAY'S ROTORS are turning, electrons race down insulated cables to the base of each tower. From there, a network of collector lines gathers all the power produced by the wind farm's turbines and carries it via miles of underground collector lines to one of two transformer stations at the wind farm site. These transformers then boost the power to 230 kilovolts, readying it for dispatch to the grid. All of this happens almost instantaneously.\n\nHorizon Wind is lucky to have two high-voltage transmission lines nearby. Running through the countryside on tall steel stanchions, these lines carry power to Empire District Electric and Westar, the two utilities that have bought nearly equal shares of Meridian Way's output. Brad Beecher, chief operating officer of Empire District Electric, leads me through what happens next. Knowing roughly how much power his company's 160,000 customers use, he has to be sure to feed an equivalent amount of electricity to the grid. \"We know how much water our customers are taking from the bathtub and we have to put that much water in the bathtub,\" he explains metaphorically. Empire District's electricity comes from a variety of sources\u2014mostly from coal and gas, although wind now supplies about 15 percent of its customers' needs. In addition to owning half of Meridian Way's output, the company has contracted for all the power coming from Elk River, the Flint Hills wind farm that includes Pete Ferrell's land.\n\nSome of the electrons generated at Meridian Way may actually reach the homes, businesses, and factories of Empire District's customers, but electrons are notoriously promiscuous. \"You can't tell which electrons flow to our customers' meters,\" Beecher tells me. Once on the grid, electrons intermingle, not just with other electrons generated by a single company but with all the electrons produced by all the power suppliers that are part of an area's wholesale power pool. These markets typically cover multistate areas and, in much of the country, are governed by entities known as regional transmission organizations or independent system operators. Empire District is part of the Southwest Power Pool, a consortium of power producers, transmission providers, and electricity distributors stretching across all or part of eight states. The Southwest Power Pool's origins go back to 1941, when a group of utilities marshaled all available electricity for wartime aluminum production in Arkansas.\n\nWith transmission lines readily accessible, the developers of Meridian Way and Elk River had a relatively easy time getting their power onto the grid. This is far from the case in many other wind-blown parts of the country. On the National Renewable Energy Laboratory (NREL)'s color-coded maps, America's richest wind resources run through stretches of the eastern Rockies and Great Plains where people are few and power demand is pretty stable. Tapping the winds that sweep through these areas is relatively easy. Finding a reliable market for the wind-generated electricity is the hard part. Because existing conventional power plants meet most local needs, wind developers must seek out more distant buyers, often in population centers many hundreds of miles away.\n\nTo get a firsthand look at some of the hurdles involved in moving wind-generated power from where it's produced to where it's most needed, I visited Wyoming, home to some of the nation's strongest and steadiest winds. Nowhere are those winds more robust than along a stretch of Interstate 25 that runs from Casper down to Cheyenne, in the southeastern quadrant of the state.\n\nBob Whitton stood with me on a wooden porch just off the kitchen of his modest wood-frame ranch house, about sixty miles north of Cheyenne and just a mile or so east of I-25. He pointed to the rolled bales of hay that block the lower half of the windows on the west side of his home. These makeshift barriers help deflect the winds that blow hard off the foothills of the Rockies before crossing the gently rolling grasslands where he raises a small herd of Black Angus cattle. He bought his ranch when he retired from the Air Force in 1994, after thirty years of piloting F4s, F5s, and F15s.\n\nEarly one December morning not long ago, Bob gazed out his kitchen window off to the west. There, strewn along I-25 near the Bordeaux junction, he could make out the long rectangular shapes of three overturned semi-trailers, flipped during the night by the howling winds. Later he drove out to the highway and found three more semis lying on their sides.\n\nBob's ranch is in an area that has earned NREL's top ranking, with average winds rated at over 10 meters per second, or 22.4 miles per hour. In the cold winter months, they often reach 30 to 40 miles per hour, posing a particular danger to newborn calves. \"The wind will kill a calf pretty fast; the cold, not quite so fast,\" he says. Shielding calves from the elements is one wintertime ordeal; keeping his herd fed is another, as hay rolled out for feed often gets carried away by the wind. \"Every time the wind blows, it costs me a lot of money.\"\n\nThese winds can be a formidable burden, but Bob knows better than most how to translate them into an economic opportunity. Along with running his ranch, he chairs a coalition of pro-wind landowner associations called the Renewable Energy Alliance of Landowners (REAL). This innovative group has jettisoned the traditional model, whereby a wind developer scouts around for promising sites and then approaches landowners one by one, retaining the upper hand in lining up agreements for the lease of their property.\n\nREAL's 300 members recognize that, along with owning lands that are good for cattle, they hold a valuable resource in the winds that rip through their section of the state. Instead of waiting for a company like Horizon or Invenergy to come to them, they have formed associations that actively seek out takers for the wind on their ranchlands. Some neighbors have created limited liability corporations; others work together informally. Some associations include as many as forty landowners; others involve just a few like-minded property owners. Together, REAL's members have 800,000 acres of southeastern Wyoming lands that they are ready to market for wind.\n\nBack in 2008, Bob and some neighbors formed their own local group, the Bordeaux Landowners Association, now one of REAL's member associations. Putting 15,000 acres on the table, they began reaching out to wind companies with a neat package that included wind data, topographical maps, photos, and other documentation. Their prospectus went out to about fifty developers, and several responded. Negotiations ensued, and in February 2010, they signed on with Pathfinder Renewable Energy LLC, a Wyoming company backed by Dallas-based Sammons Enterprises, a private holding company with close to $45 billion in assets.\n\nI met Pathfinder's land agent Vic Garber at a public board meeting of the Wyoming Infrastructure Authority, a quasi-governmental agency that helps expand Wyoming's electric transmission infrastructure through planning assistance, financing, and even co-ownership of new lines and related facilities. Vic described Pathfinder's total ambition for its wind farm: 2.1 gigawatts of installed capacity sited across roughly 100,000 acres of land on both sides of Interstate 25. Built at that scale, it will dwarf all other wind farms operating in America today.\n\nWithin Wyoming, Pathfinder may ultimately be surpassed by another mega-project sited about 100 miles to the west. There, on a sprawling ranch covering 500 square miles of rugged upland territory, a local subsidiary of the Denver-based Anschutz Corporation is moving forward with a project that may harness as much as three gigawatts of Wyoming wind by 2015. Philip F. Anschutz, the privately held company's owner, built his multibillion-dollar fortune on oil and gas exploration ventures inherited from his father. In the late 1980s, he began acquiring railroad interests and later diversified into telecommunications, digital video data management, and ownership of a nationwide network of movie theaters. Today he is part-owner of the Los Angeles Lakers and other sports teams. Wind is no impulsive dream for this hard-headed entrepreneur.\n\nRecognizing the market potential for Wyoming wind in faraway places, public utilities and independent \"merchant\" developers are racing to build long-range transmission lines. One of these projects, the Zephyr, has as its starting point the village of Chugwater, just a few miles south of Bob Whitton's ranch. From there, it will travel due west into Idaho and then turn south toward Las Vegas, in the power-hungry Sun Belt. If all goes according to plan, this 1,100-mile electron expressway will be fully operational by 2016. Pathfinder Renewable Energy has committed to take two-thirds of the Zephyr's capacity; two other wind developers\u2014Horizon and BP Wind Energy\u2014have contracted for the remainder. The total project cost is estimated to be $3 billion, or a little less than $3 million per mile.\n\nNevada and California are also prime markets for the Chokecherry-Sierra Madre wind complex. Anschutz is planning on building its own transmission line, the TransWest Express, picking up power from the company's 1,000 or so planned wind turbines in south-central Wyoming and carrying it directly to southern Nevada.\n\nSome of the power moved by these two lines might be consumed in Nevada, which has a renewable electricity standard that calls for 25 percent of its power to come from renewable sources by 2025. The prize destination for Wyoming wind energy, though, is California, trumping all of its neighbors in both the scale of its power market and the ambition of its renewable energy mandate. California will need to import substantial amounts of power from out of state if it is to meet its 33-percent-by-2020 renewable electricity standard. In-state wind and solar projects just aren't being developed quickly enough to meet this mandate.\n\nThe Zephyr and TransWest Express, relying on extra-high voltage current, will be much more energy-conserving than the lower-voltage lines that carry Meridian Way's power to market. If the Zephyr, a 500-kilovolt line, were to rely on AC current, it would lose about 1.3 percent of its power every 100 miles. If it increased its voltage to 765 kilovolts, it would shed as little as a half-percent of its power over the same distance. However, because both projects plan to rely on more highly efficient DC current, they may see as much as a 20 percent further decrease in their line losses.\n\nBuilding a major new power line is not simply a matter of choosing the right technology. The transmission developer must comply with an array of federal laws, including those that govern the use of federal lands and those that protect threatened and endangered species.\n\nIn Wyoming, as throughout much of the West, the federal government is a major stakeholder in the planning and siting of new transmission lines. One look at the pattern of land ownership in Wyoming makes it clear why. Nearly half the land in the state\u2014about 30 million acres\u2014is owned and controlled by the federal government. Some of this land is in sprawling, contiguous parcels, while the rest follows a quirky pattern known as the Wyoming Checkerboard\u2014a holdover from the 1860s, when Congress passed two successive acts aimed at opening up the West. All along a rail right-of-way running across the state, alternating square-mile sections of land were handed over to the Union Pacific Railroad, extending out twenty miles on each side of the tracks.\n\nGiven the role of transmission in creating new frontiers for economic development, it's no coincidence that the labels given to several proposed interstate power lines resonate with the region's rail lore. TransCanada's Zephyr line shares its name with one of Amtrak's most dramatically scenic routes, rolling from Chicago across the Great Plains and the Rockies to San Francisco. The monikers of other planned lines similarly evoke the spirit of America's railroad heritage: the TransWest Express and High Plains Express, both running out of Wyoming, and the Green Power Express, rooted in the Dakotas.\n\nThe U.S. Bureau of Land Management (BLM), within the Department of the Interior, oversees a rigorous, multiyear process of evaluating transmission projects that cross federal lands. In doing so, it has to reckon with a range of factors, reflecting what one BLM official calls its \"schizophrenic\" mandate. On one hand, the agency is charged by various federal laws with promoting the exploitation of commodity resources on federal lands\u2014underground minerals as well as surface assets such as timber and grasslands. On the other hand, another whole set of federal laws calls upon the BLM to conserve our natural resources for future use while protecting the wildlife and habitats on those lands.\n\nAmidst these mixed signals, the Energy Policy Act of 2005 pointed the BLM toward wind development with its call for at least 10 giga-watts of non-hydropower renewable energy projects to be built on public lands by 2015. Tom Lahti, who is the BLM's Renewable Energy Chief in its Wyoming office, decided that the BLM should not simply wait for good projects to present themselves. Instead, he initiated a study to identify the most promising areas for wind development on Wyoming's federal lands. Access to present or proposed transmission lines is one priority. Another concern is preserving wildlife\u2014making sure that key habitats for vulnerable species like sage grouse, eagles, and migrating bats are taken into account. Viewshed protection is a third consideration, giving special attention to the state's extensive network of National Historic Trails.\n\nOnce a number of wind development areas have been identified, Tom expects that the BLM will lease them out on a competitive basis, just as it does today when offering large parcels of federal land for oil and gas drilling. This would be fairer, he feels, than the BLM's current ad hoc approach, whereby parcels are leased out in response to wind developers' individual requests. Under the present system, Tom says, \"The only competition is . . . who gets there first.\"\n\nThe BLM and other federal agencies are also taking a proactive approach to charting out corridors that would allow a range of energy resources\u2014oil, gas, and perhaps someday hydrogen, as well as electricity\u2014to move more easily across public lands. In November 2008, the government released its proposed West-Wide Energy Corridor, making it clear to transmission developers that, while they are not strictly confined to this corridor, it will be easier to gain federal approval for lines that stay within its alignment.\n\nObtaining federal rights-of-way is only one piece of a complex set of negotiations that the developers of projects like the Zephyr and the TransWest Express must undertake. States vary widely in the degree of deference given to local zoning boards and county commissions. In Nevada, for example, counties and municipalities largely govern the siting of transmission lines, whereas in Wyoming, primary control over transmission line authorization rests with two state agencies, the Industrial Siting Council and the Public Service Commission.\n\nAs it can take years to arrange rights-of-way over hundreds of miles of public and private lands, wind companies have to time their planning and development efforts carefully. On one hand, they don't want to have wind turbines sitting idle for months\u2014or even years\u2014while a transmission line is moving slowly toward completion. (This has been the situation in China, where wind developers have installed massive new generating capacity in Inner Mongolia and other rural provinces years before the State Grid has built the necessary transmission infrastructure.) On the other hand, wind developers don't want to find themselves carrying the cost of reserved space on a transmission line that is ready for business long before their turbines are in the ground and the blades have started spinning.\n\nBeyond federal and state agency review, proponents of new transmission lines must deal with the concerns of citizen groups, including those who suspect that certain of these lines may be stalking horses for coal power interests. Howard Learner heads up the Environmental Law and Policy Center, a multistate advocacy group headquartered in Chicago. His suspicions center on the Green Power Express, a 3,000-mile network of 765-kilovolt AC power lines, heralded by its proponent, ITC Transmission, as a $10- to $12-billion project \"designed to efficiently move up to 12,000 megawatts of renewable energy in wind-rich areas to major Midwest load centers.\" Howard directed me to a map on the ITC website, which shows the project's starting point in North Dakota's Antelope Valley. \"The wind belt in North Dakota is not principally around Antelope Valley,\" he said flatly. \"You've got to scratch the surface a little bit to figure out who the transmission is going to serve.\"\n\nI took Howard's advice and started scratching. Via the online watchdog group SourceWatch, I had no difficulty finding the Antelope Valley Station, an 870-megawatt coal-burning giant right where Howard said it would be. I also found another five coal-fired power plants within a forty-mile radius of Antelope Valley, adding up to more than 4,000 megawatts of capacity. Might the Green Power Express end up carrying coal-generated electrons out of North Dakota, along with or instead of power from renewables? Nothing under federal law or policy would seem to bar this outcome. To the contrary, the Federal Energy Regulatory Commission (FERC)'s \"open access\" transmission policy, adopted in 1996, is designed explicitly to ensure open, non-discriminatory access to the nation's transmission system.\n\nHoward and his colleagues at the Environmental Law and Policy Center worry that lines like the Green Power Express could become part of a larger network of new transmission lines that deliver coal power, produced and sold relatively cheaply in the Midwest, to higher-priced electricity markets in the Northeast. The Union of Concerned Scientists, based in Cambridge, Massachusetts, shares this concern. In a 2008 study, the group looked at the market dynamics that could emerge as a result of the Regional Greenhouse Gas Initiative (RGGI), the Northeast's cap-and-trade program for carbon emissions. The study pointed to a critical flaw in the RGGI regime's focus on power plants in a single region. Conceptually, the auction-based trading of carbon allowances among northeastern utilities is intended to stimulate a switch to cleaner fuels and energy efficiency measures. In practice, the upward pressure that RGGI places on the price of coal-generated power could entice enterprising Midwest utilities to export their dirtier, cheaper power to the Northeast. Building a new cohort of power lines connecting the two regions would only make it easier for this to happen.\n\nSome East Coast politicians and energy planners express a very different aversion to new power lines coming out of the Midwest. They fear that stepped-up transmission would give wind power generated in the Midwest too ready a market in the East, where large-scale wind farms have been slow to emerge. Representative Ed Markey of Massachusetts, for decades a champion of renewable energy development, expressed this view when I visited his Capitol Hill office in April 2010. \"The consensus in New England is to capture this wind revolution and make it our own,\" he told me, earning nods of approval from the three members of his staff who huddled with us around a coffee table strewn with books about U.S. energy policy. At the time, Markey held leadership positions on two key congressional energy committees\u2014positions he lost when the Republicans gained a majority in the House in November 2010.\n\nMarkey's jealous guarding of East Coast wind was based on data he had seen that pointed to the region's surplus of untapped, primarily offshore wind. He said that 30 to 40 gigawatts of offshore wind was already economically developable. (That's about three to four times the total installed power-generating capacity in Massachusetts today.) Hundreds of additional gigawatts would be within reach once deep-water wind technology matures, he added. This bullish embrace of offshore wind struck me as slightly odd, given the equivocal position Markey had long taken on Cape Wind during most of its tormented journey through state and federal permitting. It was not until November 2009, three months after Ted Kennedy's death, that Markey called explicitly for federal approval of the project. Presumably he had refrained from endorsing Cape Wind out of deference to Kennedy's outright hostility to wind turbines in Nantucket Sound. Few members of the Massachusetts congressional delegation had been willing to part ways with the state's senior senator on this controversial issue.\n\nFERC chairman Jon Wellinghoff takes a less-parochial stance on the nexus between wind and wires. He believes that wind could provide upwards of 50 percent of our power needs nationwide, and he is convinced that our most promising wind resources can only be accessed if we make an all-out commitment to building new transmission lines. He punctuates his public presentations with slides showing the advanced visions for grid development that are fast emerging overseas. On one slide, 25,000 miles of high-voltage conduits interconnect a future Europe with renewable resources ranging from Norwegian hydro to vast solar energy farms stretching across North Africa. On another slide, he shows China's fast-emerging supergrid, with row upon row of 800-kilovolt lines bringing power from wind-rich areas in the country's western reaches to major cities in the East.\n\nI asked the FERC chairman how he regarded environmentalists' concerns about coal piggybacking on new American power lines ostensibly built for wind. His response was blunt, even irritated: \"I think it's an urban myth.\" Coal plants like those near Antelope Valley have all the transmission they need, he said, adding that new coal-fired facilities are unlikely to be built in the coming years, given uncertainties about the future price of carbon-based power production. As evidence of the slowdown in coal plant development, he pointed to the tilt toward wind in the commissioning of new generating capacity in 2009: \"We had almost i0,000 megawatts of wind put on the system last year. That was almost ten times the amount of coal.\"\n\nA stepped-up investment in coal-fired power generation may not be as remote a possibility today as it was when I met with Wellinghoff in the spring of 2010. Carbon taxation and market-based trading of carbon allowances were prime targets of the anti-incumbent fervor that brought a new wave of Tea Party-inspired Republicans to Congress just a few months later. The American Clean Energy and Security Act of 2009, cosponsored by Markey and California congressman Henry Waxman, was unfortunately a lightning rod for this hostility. The nationwide cap-and-trade program that it proposed for carbon emissions from major industries was roundly attacked as a punitive drag on an already flagging economy. When the 112th Congress convened, cap-and-trade was dead. Also abandoned was Waxman and Markey's proposal for a nationwide electricity standard that would set explicit targets for power generation from renewable sources. In its stead, discussions shifted to a much more amorphous \"clean energy standard\" that would include nuclear power, as well as advanced methods of burning coal. If this reformulation finds its way into law, there may be added cause for concern about the encroachment of non-renewable electrons onto new \"green\" transmission lines.\n\nQuestions may remain about competing uses of an expanded and modernized grid, yet it's clear that without a major investment in new transmission, much of our nation's wind energy potential will remain beyond reach. Recognizing this, the drafters of the 2009 stimulus package allocated $6 billion in loan guarantees for renewable energy generation and electric transmission projects and another $4.5 billion in matching grants to modernize the grid. In his Earth Day 2009 speech before wind tower factory workers in Newton, Iowa, President Obama was passionate in endorsing this investment. \"The nation that leads the world in creating new sources of clean energy will be the nation that leads the twenty-first-century global economy,\" he declared, adding that \"we also need a smarter, stronger electricity grid to carry that energy from one end of this country to the other.\"\n\nIn fact, federal efforts to create priority transmission corridors for new energy investments predated the Obama administration's recovery program by several years. As early as 2002, the Department of Energy began exploring ways for FERC to break transmission bottlenecks when state agencies and regional planning bodies fail to move priority grid expansion projects forward. The Energy Policy Act of 2005 then stretched FERC's role in transmission line siting, giving it authority to issue permits for new transmission investments in designated national priority corridors if state siting bodies \"withheld approval\" for more than a year.\n\nA firestorm ensued when FERC interpreted this law as giving it the power to step in where a state has _denied_ a siting permit, as opposed to simply not acting on it. The Virginia-based Piedmont Environmental Council, together with other citizen groups and state public service commissions, brought suit and won a federal circuit court ruling that preserved the prerogative of states to turn down new transmission lines. When the Supreme Court declined to review this ruling, it became clear that without new federal legislation, states would remain in firm control of transmission line siting.\n\nToday the debate continues to rage over who should be involved in siting decisions about new interstate transmission lines. Two leading industry groups, AWEA and the Solar Energy Industries Association, insist that FERC should have ultimate control over the siting of multistate transmission lines, similar to the authority it currently exercises over interstate natural gas pipelines. Don Furman, a veteran wind developer who recently chaired AWEA's board, candidly refers to state officials as a mismatch for decision making on transmission investments. Governors and public utility commissioners have told him, \"We're not getting paid to go out there and build major interregional transmission. It needs to be done, but you ought not ask us to do it, to approve it, because that's not our job.\"\n\nFurman likens the U.S. grid to a patchwork of country roads. What's needed, he says, is equivalent to our interstate highway system. Just as the interstate's planners and builders relied heavily on the federal government's strong guiding hand, he favors firm federal leadership in building a superhighway network for electric power.\n\nHoward Learner remains wary of giving FERC too free a hand in authorizing new transmission lines. Although his group is an active proponent of wind power, the Environmental Law and Policy Center has also worked to strengthen and enforce state laws protecting wetlands and other natural habitats. These laws, he fears, could be compromised if their protections are downplayed or preempted by a new federal siting regime.\n\nChris Miller of the Piedmont Environmental Council is even more adamant about keeping the federal government away from transmission-line siting. Having won the court battle that now limits FERC's power to override state siting decisions, he strongly rejects the view that the global climate crisis calls for decisions that could trump local planning priorities. \"I don't see climate change and the international consensus of scientists as being any more politically valid than the desire of people here to have control over their viewshed,\" he tells me. At some level, he acknowledges that climate change is a problem. To address it, he argues half-heartedly for energy efficiency and greater use of passive solar building design. He concedes that these steps will fall far short of getting us the cutbacks in U.S. carbon emissions that so much of the scientific community is calling for, but he resents the \"energy geeks\" who are pressing for deeper cuts in pursuit of what he calls \"an unratified policy of 80 percent greenhouse gas reduction.\"\n\nI met with Chris at the Piedmont Environmental Council's headquarters, in a beautifully restored clapboard house just off Main Street in historic Warrenton, Virginia. Driving into Warrenton just before our meeting, I grasped just how far area planners are willing to go to preserve at least the surface trappings of colonial authenticity. Tract houses spill across the rolling hills, their fake clapboard aluminum siding only slightly more convincing than their make-believe window mullions. Post-and-rail fences made of white molded plastic only add to the effect, bracketing the road, dipping down into ravines, and running across empty expanses of close-cropped grass. This is horse country minus the horses.\n\nAs I drove out of Warrenton past shopping malls and medical offices designed to look like eighteenth-century brick manor houses, I couldn't help thinking how tough a time transmission planners will have building a green-power superhighway worthy of the name. However the balance is struck between state and federal decision making on transmission-line siting, tensions between widely divergent values and worldviews will only grow as we move from the abstract notion of a twenty-first-century supergrid to the reality of laying new wires across the American landscape.\n\nTo supply 20 percent of our electricity from wind by 2030, the Department of Energy estimates that we will need to invest about $60 billion in expanding the American grid. That would amount to about $30 per household per year, or a monthly cost of $2.50 per household, between 2012 and 2030. While this may sound like a very modest price for a monumental resource shift in our electricity sector, the dissension over who should pay for new transmission lines has been almost as explosive as the jurisdictional debate over siting them. FERC has refrained from imposing a one-size-fits-all allocation formula on these new transmission investments, instead inviting multi-state transmission groups to develop schemes that fairly reflect the range of generation needs and ratepayer benefits in their service areas.\n\nOne regional transmission organization, the PJM Interconnection, coordinates the flow of power across all or part of thirteen Midwest and Mid-Atlantic states and the District of Columbia. More than 160 gigawatts of generating capacity and more than 50,000 miles of transmission lines are under PJM's control. When PJM proposed to spread the costs of major new transmission lines evenly among all utilities in its system, it raised the hackles of Midwest utilities that rely mainly on lower-voltage power lines to deliver electricity over relatively short distances. The power companies claimed that PJM's cost-spreading formula would force them to underwrite transmission lines built primarily for the benefit of electricity consumers in the eastern PJM states.\n\nThe Southwest Power Pool has just adopted a more nuanced scheme for spreading the costs of new high-voltage transmission across its eight-state region. This \"Highway-Byway\" formula assigns the full costs of \"Electricity Highways\"\u2014high-voltage, longer-distance lines\u2014to all system users. \"Electricity Byways\"\u2014mid-voltage lines generally covering less terrain\u2014allocate a third of costs on a systemwide basis, leaving two-thirds to be paid within the zone where the project is located. And all the costs of lower-voltage lines are borne locally.\n\nAlthough it's an unlikely stretch given the prevailing antitax sentiments in Congress, some have proposed a new federal transmission tariff to supplement regional cost recovery mechanisms like the Highway\/Byway formula. There is precedent for such a tax in the federal portion of the gasoline tax, which has been a financial mainstay of the interstate highway system since the 1950s and, to a lesser extent, a revenue source for federally supported mass transit projects. Susan Tierney, who headed up the Department of Energy's policy efforts during Bill Clinton's administration, symbolically invokes the federal highway system when she calls for this tariff to support a nationwide \"interstate electric highway system.\" She asserts: \"State and utility-service-territory boundaries have no more meaning for the transmission grid than they do for the transportation highway system, since electrons flow across boundaries according to the laws of physics rather than the laws of states.\" More practically, she justifies this investment because of the essential national purposes it would serve: the provision of a \"reliable, economic, secure supply\" of power and the shift toward a U.S. economy based on clean, renewable energy.\n\nHowever the costs of new transmission lines are shared, there is no escaping the fact that building new power conduits across large expanses of the United States will come at a considerable cost to the American public. Arriving at a fair way to socialize these costs will be crucial to gaining public acceptance of a new energy infrastructure that will give wind and other dispersed renewables their rightful place in our energy mix.\n\nBuilding a wind-friendly transmission system is not just about stringing up wires and finding ways to pay for them. Accommodating the wind's variability demands constant vigilance. When winds drop off unexpectedly, grid operators must act immediately to make sure that electricity customers don't suddenly find themselves without electric power. When wind-generated output rises above predicted levels, system operators must take steps to absorb that power without destabilizing the grid.\n\nHydropower has long been used to help overstressed power systems cope with periods of peak demand, particularly on hot summer days when air conditioning needs outstrip the capacity of baseload-serving fossil fuel and nuclear power plants. As wind energy has become a more significant power resource in recent years, hydro dams and pumped-storage reservoirs have increasingly been called into play to keep power systems in balance. Within minutes, a hydro facility can step up its power output by channeling more water through its turbines. To soak up excess power, it can activate pumps that will transport water to an elevated storage reservoir. Coal and nuclear plants, which are slow to heat up and cool down, simply cannot make the swift shifts in power often needed to work in tandem with variable winds.\n\nWhile hydropower and quick-starting, gas-fired turbines play an important role in keeping the grid balanced, today's transmission innovators are coming up with a variety of new technologies and management tools. Jon Wellinghoff brims with enthusiasm when he talks about innovations like smart-metered electric vehicles that allow a centralized, automated dispatcher to send incremental pulses of power to and from plugged-in car batteries to help maintain a balance between energy demand and available electric current. Most cars, he says, sit idle for twenty to twenty-two hours a day. Connected to the grid during those many hours, tens of millions of vehicles in an electric car fleet can serve as a vast balancing resource, easily accessed by grid operators. He and Knud Pedersen of Denmark's DONG Energy would have a lot to discuss as Denmark moves forward with its own electric vehicle network.\n\nWellinghoff also gets excited about buildings whose heating and cooling can be controlled remotely to ease power use in low-wind periods and increase consumption when the winds are strong. In the same animated way, he speaks of \"sentient\" appliances like refrigerators with multiple functions\u2014defrosting, ice making, moisture control, and ordinary cooling\u2014programmed independently to modulate power use. \"You don't care when your refrigerator compressor is on. You don't care when your defrost cycle is on,\" he explains. \"You only care that you have a cold beer, nothing defrosts, and you have ice when you need it.\"\n\nUtility-scale flywheels are now used in a few areas to even out bumps in power flow from second to second and minute to minute. This service is called \"frequency regulation.\" Wellinghoff, a longtime Nevadan, tells me that my home state of Massachusetts is at the forefront in applying this technology. In Tyngsboro, the Beacon Power Corporation has installed 3 megawatts of flywheel storage and is on its way to integrating multiple flywheel units into a 20-megawatt storage system. These devices store excess power by converting it into kinetic energy: a carbon-fiber cylinder, vacuum-sealed and levitated by magnetic bearings to minimize friction, rotates at speeds of up to 16,000 revolutions per minute. When power is needed, the flywheel's stored energy is tapped to drive a generator that feeds electrons back into the grid. Flywheels are sometimes called grid \"shock absorbers\" because of their nimbleness in taking on and discharging energy in rapid response to grid-balancing needs.\n\nFlywheels may be nimble, but Throop Wilder maintains that his company's batteries will do a better job matching the variability of wind power over multihour periods. Wilder is president of 24M, a Massachusetts-based startup located in a small suite of offices adjacent to the MIT campus in Cambridge. According to Wilder, the 24M battery, which uses a lithium ion\u2013based semisolid suspension, will bring down the cost of batteries and reduce the space they require. \"We will be able to deliver 1 megawatt of power over a four-hour period from a battery the size of a small walk-in closet,\" he claims. This compactness will allow battery storage to be sited in urban locations, right near consumers, rather than relying on remote multi-acre sites now used for pumped hydro storage reservoirs\u2014a current mainstay in balancing power from multiple generation sources.\n\nUnderground injection of compressed air is being explored as another means of energy storage that could help level out the fluctuations in power produced by wind farms. With this technology, surplus electricity is used to compress air, which is then pumped into a geological formation such as an abandoned mine, a salt dome, or a capped layer of subterranean sandstone. When power is needed, the air is brought back to the surface. As the air is reheated, its expansion creates sufficient pressure to turn a set of turbines. In Iowa, a consortium of municipalities spent several years studying a thousand-acre site where air would be pumped into a sandstone aquifer two thousand feet below ground. One of the state's top proponents of community-based wind power, appropriately enough named Tom Wind, was a lead consultant on the project. Though this site was ultimately determined a poor match for the planned 270-megawatt project, two other utility-scale compressed-air energy storage facilities are already operating\u2014in Alabama and Germany.\n\nAs we work to green the American grid, we need new planning tools, siting provisions, and financing strategies that will help us create a grid that measures up to the extraordinary wind resources across our continent and off our shores. Regional transmission organizations have a vital role to play in coordinating current grid operations, assessing future needs, and coming up with fair and equitable cost-sharing mechanisms for new projects. Ultimately, though, strategic investments in new grid infrastructure may require a higher degree of national direction to break through state and local logjams.\n\nBuilding a twenty-first-century grid invites ingenuity as well as collaboration. From Beacon Power's flywheels to 24M's batteries, smart innovators are already rising to the challenge. The groundwork for a new power infrastructure that makes wind a major player is now being laid. The next steps will require clear vision and careful planning, as well as political leaders who are committed to moving America to a new era of energy self-reliance and energy security.\n\n## Epilogue\n\nNOW MORE THAN EVER, the winds of change are blowing. In the Middle East and North Africa, the upending of long-entrenched authoritarian regimes has laid bare just how much the United States relies on relationships of convenience with faraway countries whose political and social fabrics we scarcely understand. The need to free ourselves from an overwhelming dependence on energy resources outside our borders and beyond our control has never been more apparent. The OPEC oil embargo of 1973 stirred a new awareness of the need to wean ourselves off foreign oil, but all too soon that clarion call was muffled by a more dominant message: petroleum was there for the taking, even if it threatened to draw us into one war after another in the Middle East.\n\nOn the home front, the opening of the western coal frontier and the exploitation of new natural gas reserves have given us the sense that we can fuel our electricity needs by burning through vast quantities of carbon, with little regard for the environmental consequences. Even though we are the world leader in per-capita greenhouse gas emissions, we have consistently balked at joining the Kyoto Protocol and other international efforts to avert global warming. The cost to our industries and consumers would be too high, our political and industrial leaders have insisted. Unfortunately, the same metric has not been applied to the fortune that we have spent over the past decade fighting ill-defined wars in the Middle East\u2014more than a trillion dollars and counting.\n\nThe Fukushima reactor disaster in March 2011 put us on renewed notice about the perils of nuclear energy just as the United States was considering a new wave of power plant construction. Instead of buffering Japan's bitter memories of Hiroshima and Nagasaki, the world's pursuit of civilian nuclear power has brought ever-greater numbers of nations to a roulette table where the stakes are impossibly high. We have begun to learn the hard way that rare events do happen\u2014at Three Mile Island, Chernobyl, Fukushima, and who knows where next.\n\nIf wind energy looked promising in the first decade of this new millennium, it looks essential as we begin making our way through the second. Wind is no panacea, but the problems it poses are within our ability to address responsibly, and the scale of its potential contribution to our energy economy is staggering. By the end of 2012, we should have upwards of 50 gigawatts of installed capacity at wind farms scattered across most of our fifty states. That will get us a sixth of the way to the Department of Energy's goal of supplying a fifth of our power from wind by 2030, and at least start us down the path to FERC chairman Jon Wellinghoff's longer-term vision of having wind provide half of America's electricity. Happily, we have barely scratched the surface of the 11,000 gigawatts of wind power available to us on land. Equally wide open are the prospects for developing our wind energy resources at sea. Cape Wind, likely to become the nation's first offshore wind farm, has survived the obstacles thrown in its way and should be operational in a few years' time. That project will provide less than a half-gigawatt of installed wind power, leaving about 4,000 gigawatts of offshore wind resources for future development\u2014far more electricity than America could conceivably need in the foreseeable future.\n\nEven with our superabundance of available wind, it would be naive to believe that wind energy alone could free America from its reliance on fossil and nuclear fuels. Wind's variability requires balancing our wind-generated power with other technologies that are not constrained by intermittency. Realistically, fossil fuels like coal and gas, despite their drawbacks, will need to play a role in finding that balance during the next few decades. At the same time, we must make an all-out effort to develop a range of renewable technologies that can complement land-based wind power while doing less damage to the environment than fossil or nuclear fuels. Tidal and geothermal power, deepwater wind, carefully selected crop-derived fuels that have a net energy benefit and don't compete with our food production, and bio-gas drawn from sewage treatment, landfills, and livestock waste are among the options. Solar energy, intermittent like wind but with differently timed peaks and troughs, can further contribute to a balanced power supply, especially if its price becomes more competitive with other generation sources. The way forward will involve an integration of multiple technologies, with full recognition of the trade-offs involved with each.\n\nMaking wind a major American electricity provider will also require us to build a green-power superhighway worthy of the name. A new level of collaboration across state boundaries will be needed to plan these routes, and new mechanisms will have to be established to ensure that national energy priorities are given appropriate weight in decisions about new transmission lines. Beyond making sure that wind-generated power can be tapped where it is strongest and delivered to areas of greatest demand, sophisticated \"smart grid\" management tools and new power storage technologies will have to be developed to translate the wind's natural variability into an assured, continuous flow of power that perfectly matches consumer demand. The options abound, from flywheels and pumped hydro storage reservoirs to plug-in electric cars and industrial-scale batteries. All present timely opportunities to a U.S. technology sector that is looking for ways to make American industry more responsive to twenty-first-century needs.\n\nTo harness the wind's potential, we will need to sustain a level of federal support that places wind energy at least on a par with other new power plant investments. The U.S. government's stimulus program, with its flexible menu of federal production and investment tax credits convertible to treasury grants, has helped even out the playing field for wind energy in recent years. These programs were critical to averting a devastating setback to wind manufacturers and wind farm developers when the recession hit the industry hardest in 2010.  Continuing a program of wind energy tax credits will be a vital counterpoint to the subsidies, overt and hidden, that our government has long bestowed upon the fossil fuel and nuclear power industries.\n\nJust as we dare not pin our hopes for wind on the elimination of government subsidies for fossil fuels and nuclear energy, we would be foolish to chart a course that depended on congressional passage of a federal tax on carbon emissions. More clearly within reach is a federal standard akin to the renewable electricity standards that have already been adopted by more than thirty states, requiring utilities to provide a minimum percentage of their power from renewable energy. Yet this approach, too, has proven vulnerable to lobbying attacks by the fossil fuel and nuclear industries. President Obama's commitment to wind, so persuasively articulated during his early days in office, yielded all too quickly to a more ambiguous embrace of \"clean energy\" that risks diverting us from renewable energy investments and sending us down a path toward \"clean coal\" and even \"carbon-free nuclear energy,\" overlooking the horrendous hazards to health, safety, and the environment posed by each.\n\nEven as our political leaders waffle on renewable energy's role in shaping our national energy portfolio, many of our states continue to press forward with policies that significantly advance wind and other renewable technology investments. With renewable electricity standards adopted by nearly two-thirds of our states, the momentum will continue to build. Texas, the nation's oil king, now generates nearly 10 percent of its electricity from wind. In Oregon, 15 percent of the power supply is already based on wind. California now draws nearly 20 percent of its electricity from renewable sources, and it is charting a course toward 33 percent reliance on wind and other renewables by 2020.\n\nLeading proponents of renewable energy development in those and other states notably cut across party lines. Not only did Texas governor George W. Bush preside over the adoption of the state's first renewable electricity standard, but both Jerry Brown, California's current Democratic governor, and his Republican predecessor, Arnold Schwarzenegger, have carried the renewable energy banner with pride and determination. In November 2010, when the Koch brothers and other fossil fuel industry leaders bankrolled a referendum campaign to neutralize the renewable electricity standard and other elements of California's greenhouse gas agenda, one of the state's most vigorous defenders was George Shultz, Ronald Reagan's secretary of state and Richard Nixon's first budget director. Recalling the American car industry's shortsighted folly in opposing tighter air emission standards back in 1990, this lifelong conservative Republican observed: \"There is a long history here of the pessimists underestimating what American ingenuity can do.\"\n\nThe opportunities for ingenuity in developing America's wind energy potential abound. In my travels researching this book, I have witnessed this promise in the determined efforts of turbine manufacturer Clipper Wind. I have seen Rust Belt companies like Timken seize upon wind energy as its next big strategic bet. I have admired the pluck of family-run businesses like Cardinal Fastener as they carve out new niche markets. And I have been awed by the creativity of inventors and innovators who are racing to create a smarter grid.\n\nBut wind energy isn't just about innovation. It's about sound investment and careful planning by a growing corps of U.S. entrepreneurs. It's about creating a new generation of jobs in a renewable energy economy, from assembling turbines on the factory line to building and operating them in the field. It's about making America more energy-independent in an increasingly chaotic world. And it's about bringing much-needed stability to our unsettled global climate. With strong collaboration between government and the private sector, wind can truly become the heartland's new harvest.\n\n## Tables\n\n**Table 1. U.S. Onshore Wind Power Capacity by State in Gigawatts (GW) and Gigawatt Hours (GWH)**\n\n---\n\n**STATE** | **WIND ENERGY POTENTIAL**1 | **CURRENT CAPACITY**2\n\n**CAPACITY  \n(GW)** | **RANK** | **OUTPUT\/YR  \n(GWH)** | **CAPACITY  \n(GW)** | **RANK**\n\nTexas | 1,901.5 | 1 | 6,527,850 | 10.085 | 1\n\nIowa | 570. 7 | 7 | 2,026,340 | 3.675 | 2\n\nCalifornia | 34.1 | 20 | 105, 646 | 3.177 | 3\n\nMinnesota | 489.3 | 12 | 1,679,480 | 2.192 | 4\n\nWashington | 18.5 | 22 | 55,550 | 2.104 | 5\n\nOregon | 27.1 | 21 | 80,855 | 2.104 | 6\n\nIllinois | 249.9 | 15 | 763,529 | 2.046 | 7\n\nOklahoma | 516.8 | 9 | 1,788,910 | 1.482 | 8\n\nNorth Dakota | 770.2 | 6 | 2,983,750 | 1.424 | 9\n\nWyoming | 552.1 | 8 | 1,944,340 | 1.412 | 10\n\nIndiana | 148.2 | 16 | 443,912 | 1.339 | 11\n\nColorado | 387.2 | 13 | 1,288,490 | 1.299 | 12\n\nNew York | 25.8 | 20 | 74,695 | 1.275 | 13\n\nKansas | 952.4 | 2 | 3,646,590 | 1.074 | 14\n\nPennsylvania | 3.3 | 29 | 9,673 | .748 | 15\n\nSouth Dakota | 882.4 | 5 | 3,411,690 | .709 | 16\n\nNew Mexico | 492.1 | 11 | 1,644,970 | .700 | 17\n\nWisconsin | 103.8 | 17 | 300,136 | .469 | 18\n\nMissouri | 274.4 | 14 | 810,619 | .457 | 19\n\nWest Virginia | 1.9 | 33 | 5,820 | .431 | 20\n\nMontana | 944.0 | 3 | 3,228,620 | .386 | 21\n\nIdaho | 18.1 | 23 | 52,118 | .353 | 22\n\nMaine | 11.3 | 25 | 33,779 | .266 | 23\n\nUtah | 13.1 | 24 | 37,104 | .223 | 24\n\nNebraska | 918.0 | 4 | 3,540,370 | .213 | 25\n\nMichigan | 59.0 | 18 | 169,221 | .164 | 26\n\nArizona | 10.9 | 26 | 30,616 | .128 | 27\n\nMaryland | 1.5 | 35 | 4,269 | .070 | 28\n\nHawaii | 3.3 | 29 | 12,363 | .064 | 29\n\nNew Hampshire | 2.1 | 34 | 6,706 | .026 | 313\n\nMassachusetts | 1.0 | 36 | 3,323 | .018 | 32\n\nOhio | 54.9 | 19 | 151,881 | .011 | 33\n\nAlaska | 494.7 | 10 | 1,620,792 | .010 | 34\n\nVermont | 2.9 | 31 | 9,163 | .006 | 364\n\nArkansas | 9.2 | 27 | 26,906 | 0 | \u2014\n\nNevada | 7.2 | 28 | 20,823 | 0 | \u2014\n\nVirginia | 1.8 | 32 | 5,395 | 0 | \u2014\n\nOther | 2.2 | \u2014 | 6,411 | .041 | \u2014\n\n**TOTAL** | 10,956.9 | \u2014 | 38,552,706 | 40.180 | \u2014\n\n1. NREL, AWS Truewind, _80-Meter Wind Maps and Wind Resource Potential,_ February 4, 2010, <http:\/\/www.windpoweringamerica.gov\/>. Potential capacity is based on land areas where winds at 80 meters+ above ground would allow turbines to operate at a 30%+ capacity factor. States with less than 1 gigawatt of potential capacity are listed as \"Other.\"\n\n2. _AWEA U.S. Wind Industry Annual Market Report_ \u2014Year Ending 2010, figure 16, <http:\/\/www.awea.org\/>.\n\n3. Tennessee is ranked 30 in 2010, with 29 megawatts of installed capacity. It is listed as \"Other\" because its wind energy potential is less than 1 gigawatt (see note 1).\n\n4. New Jersey is ranked 35 in 2010, with 8 megawatts of installed capacity. It is listed as \"Other\" because its onshore wind energy potential is less than 1 gigawatt. It should be noted, however, that New Jersey has substantial offshore wind energy potential.\n\n**Table 2. Top 10 Countries by Cumulative Installed Wind Power Capacity, December 31, 2010**1\n\n---\n\n**COUNTRY** | **MEGAWATTS** | **%**\n\nChina | 44,715 | 22.7\n\nUnited States | 40,180 | 20.4\n\nGermany | 27,214 | 13.8\n\nSpain | 20,676 | 10.5\n\nIndia | 13,065 | 6.6\n\nItaly | 5,797 | 2.9\n\nFrance | 5,660 | 2.9\n\nUnited Kingdom | 5,204 | 2.7\n\nCanada | 4,009 | 2.1\n\nDenmark | 3,752 | 1.9\n\nRest of the World | 26,546 | 13.5\n\n**TOP 10 TOTAL** | 170,272 | 86.5\n\n**WORLD TOTAL** | 196,818 | 100\n\n1. GWEC, _Global Wind Statistics 2010_ , February 2, 2011, modified by updated reportage on China provided by CWEA, _China Wind Power Installed Capacity Data_ , March 18, 2011.\n\n**Table 3. Top 10 Countries by Wind Power Capacity Installed during 2010**1\n\n---\n\n**COUNTRY** | **MEGAWATTS** | **%**\n\nChina | 18,928 | 49.5\n\nUnited States | 5,115 | 13.4\n\nIndia | 2,139 | 5.6\n\nSpain | 1,516 | 4.0\n\nGermany | 1,493 | 3.9\n\nFrance | 1,086 | 2.8\n\nUnited Kingdom | 962 | 2.5\n\nItaly | 948 | 2.5\n\nCanada | 690 | 1.8\n\nSweden | 603 | 1.6\n\nRest of the World | 4,750 | 12.4\n\n**TOP 10 TOTAL** | 33,480 | 87.4\n\n**WORLD TOTAL** | 38,230 | 100\n\n1. GWEC, _Global Wind Statistics 2010_ , February 2, 2011, modified by updated reportage on China provided by CWEA, _China Wind Power Installed Capacity Data_ , March 18, 2011.\n\n**Table 4. U.S. Wind Power Installations by Manufacturer**1\n\n---\n\n**MANUFACTURER** | **ANNUAL INSTALLATIONS (IN MEGAWATTS)** | **% M ARKET SHARE 2009\/2010**\n\n**2005** | **2006** | **2007** | **2008** | **2009** | **2010**\n\nGE Wind (US) | 1,433 | 1,146 | 2,342 | 3,585 | 3,995 | 2,543 | 40.0\/49.7\n\nSiemens (GER) | 0 | 573 | 863 | 791 | 1,162 | 828 | 11.6\/16.2\n\nGamesa (SP) | 50 | 74 | 494 | 616 | 600 | 562 | 6.0\/11.0\n\nMitsubishi (JP) | 190 | 128 | 356 | 516 | 814 | 350 | 8.1\/6.8\n\nSuzlon (IN) | 25 | 92 | 197 | 736 | 702 | 312 | 7.0\/6.1\n\nVestas (DK) | 700 | 439 | 948 | 1,120 | 1,490 | 221 | 11.2\/4.3\n\nAcciona (SP) | 0 | 0 | 00 | 410 | 204 | 99 | 2.0\/1.9\n\nClipper (US) | 3 | 0 | 48 | 470 | 605 | 70 | 6.1\/1.4\n\nREpower (GER) | 0 | 0 | 0 | 94 | 330 | 68 | 3.3\/1.3\n\nDeWind (GER) | NA | NA | NA | NA | NA | 20 | NA\/0.4\n\nOther | 2 | 2 | 3 | 12 | 94 | 42 | 0.9\/0.8\n\n**TOTAL** | **2,402** | **2,454** | **5,249** | **8,350** | **9,994** | **5,115** | **100.0**\n\n1. DOE, _2009 Wind Technologies Market Report; AWEA U.S. Wind Industry Annual Market Report\u2014Year Ending 2010._\n\n**Table 5. Global Installed Wind Capacity (in Megawatts), 1996\u20132010**1\n\n---\n\n**YEAR** | **ANNUAL** | **CUMULATIVE**\n\n1996 | 1,280 | 6,100\n\n1997 | 1,530 | 7,600\n\n1998 | 2,520 | 10,200\n\n1999 | 3,440 | 13,600\n\n2000 | 3,760 | 17,400\n\n2001 | 6,500 | 23,900\n\n2002 | 7,270 | 31,100\n\n2003 | 8,133 | 39,431\n\n2004 | 8,207 | 47,620\n\n2005 | 11,531 | 59,091\n\n2006 | 15,245 | 74,052\n\n2007 | 19,866 | 93,820\n\n2008 | 26,560 | 120,291\n\n2009 | 38,610 | 158,738\n\n2010 | 38,230 | 196,818\n\n1. GWEC, _Global Wind Statistics 2010_ , modified by CWEA, _China Wind Power Installed Capacity Data_ , March 18, 2011.\n\n## Acknowledgments\n\nPEOPLE SAY THAT IT TAKES A VILLAGE to raise a child. Writing a book is not altogether different. So many people have helped at every stage\u2014ranchers and farmers who warmly welcomed me onto their lands and into their homes, construction workers and factory laborers who let me take a close look at what they do, wind industry leaders and transmission experts who patiently tutored me in their trades, colleagues and friends in the environmental community who responded to my barrage of queries, a superb crew of research assistants, inspiring editorial guides, and a publishing team that has been masterful in every aspect of creating this book.\n\nMy agent, Colleen Mohyde at the Doe Coover Agency, reassuringly embraced my proposal and was wonderfully sure-footed in leading me to Beacon Press, a publisher that is dedicated to both the authors they take on and the ideas they are advancing. Alexis Rizzuto, my editor at Beacon, has been brilliant in keeping me focused on what readers really need to know about a subject that gets dangerously technical very fast. She and the book's entire Beacon team (Tom Hallock, Susan Lumenello, Reshma Melwani, Will Myers, and P. J. Tierney) are superb at what they do.\n\nThe pages of this book bear the imprints of several other creative forces. Larry Tye has been the best of role models, contagious in his love of book writing and a supportive friend throughout. Rosemary Ahern has been a gifted writing coach, helping a circumspect lawyer bring his own voice to the fore. My sister, Sally Bliumis-Dunn, brought a poet's eye and ear to her reading of the manuscript. Joel Segel aided my early search for a way to infuse a technical subject with life and humor.\n\nFrom his perspective as a wind developer and lifelong environmentalist, Rob Gardiner combed every page for balance and technical accuracy. Seasoned journalist Ira Chinoy was equally vigilant in making sure my writing was well sourced. (I hope he'll forgive the rumored Napoleon quote that opens chapter 4. I liked it too much to delete it.) Dana Peck, a savvy wind project manager and long-ago coworker on Capitol Hill, scrutinized my writing on transmission issues, as did Seth Kaplan, a recent colleague at the Conservation Law Foundation. Renewable energy analyst Jan Hamrin filled out my understanding of California's wind energy boom and bust of the 1980s. Eric Lantz, Michael Milligan, and Suzanne Tegen at the National Renewable Energy Laboratory helped me grasp the economics of wind, as did Andrew Mills and Ryan Wiser at Lawrence Berkeley National Laboratory and Matt Kaplan of IHS Emerging Energy Research.\n\nResearching this book took me to remote reaches of America and even more distant corners of the world. Horizon Wind Energy's Tanuj TJ Deora\u2014now director of the Colorado Governor's Energy Office\u2014accompanied my first visit to Meridian Way, where Michelle Graham educated me about the wind farm and was amazingly resourceful in introducing me to Cloud County landowners, educators, and civic leaders. Horizon CEO Gabriel Alonso and company staff across several states were instrumental in demonstrating what it takes to finance, build, and operate a wind farm.\n\nAmong other wind developers, Invenergy's Mark Leaman, together with Art Fletcher, arranged for me to spend a fascinating few days meeting company staff and neighbors at the Grand Ridge Wind Farm. Don Furman at Iberdrola Renewables was enlightening about everything from his company's wildlife protection innovations to the challenges of getting wind power onto the grid.\n\nIn the manufacturing realm, the Vestas team in Denmark and China showed me extraordinary generosity. Michael Holm lined up informative factory visits in Denmark and responded unfailingly to my multiple follow-up queries. I'm sure he's as relieved as I am that this book is finally going to press! In China, Andrew Hilton and Tu Trinh Thai made it possible for me to hear from Danish and Chinese managers about the challenges of fusing Western and Eastern workplace norms.\n\nMary Paul Jesperson and Pernille Florin Elbech at the Danish Embassy shaped my Denmark visit, together with Christian Wedekinck Olesen at the Climate Consortium. Valuable insights on wind energy's Danish ascent were offered by Parliament members Anne Grete Holmsgaard and Per \u00d8rum J\u00f8rgensen, Lars Aagaard at the Danish Energy Association, and Rune Birk Nielsen at the Danish Wind Industry Association. I also want to thank Steen Hartvig Jacobsen for sharing his deep historical knowledge about Danish energy policies.\n\nArmond Cohen at the Clean Air Task Force guided me wisely in preparing my trip to China, and the Natural Resources Defense Council's China staff provided useful background on Chinese wind manufacturing. Lin Wei and Ming Sung arranged informative meetings, capably interpreted by Houji Zhuzhu. Sebastian Meyer of Azure International shared keen observations and timely data on Chinese wind energy development. Along with all I learned from the entrepreneurs and analysts who appear or are cited in chapter 4, I am grateful for the insights offered by Alfred Zhao at the Chinese Wind Energy Association, Ryan Chen at the China New Energy Chamber of Commerce, Ellen Carberry at the China Greentech Initiative, and Jasmine Zhang at PriceWaterhouseCoopers.\n\nStateside, Bob Loyd at Clipper Wind was a blessing. He opened the door to some great discussions with his multigenerational crew at the Cedar Rapids assembly plant, and he was a rich resource on the city's long struggle to hold onto a viable industrial base. Bob Gates, in Clipper's California headquarters, brought decades of wind energy experience to his reflections on the company's technology innovations, its financial hurdles, and the damage caused by erratic federal policies.\n\nOther manufacturers who helped me understand the wind trade were Jim Charmley, Hans Landin, and Lorrie Paul Crum at Timken; Daniel McGahn and Jason Fredette at American Superconductor; Bob Paxton at Broadwind; John Grabner at Cardinal Fastener; and Dheeraj Choudhary at Parker Hannifin. Richard Stuebi at the Cleveland Foundation kindly steered me toward several of these fine people.\n\nGeneral Electric eluded my persistent efforts over several months to arrange factory visits and interviews with company officials. This was unfortunate, given GE's central role in supplying America's turbines. Nevertheless, Millissa Rocker did provide useful speeches and testimony by Vic Abate, vice president of GE Energy's renewables division.\n\nA successful wind industry doesn't just depend on the companies that produce the technology and put up the turbines. I was privileged to meet several economic development boosters who have catalyzed factories and wind farms in their communities: Mayor Charles \"Chaz\" Allen of Newton, Iowa; Joe Jongewaard and Beth Govoni at the Iowa Department of Economic Development; Dennis Jordan at the Cedar Rapids Chamber of Commerce; CloudCorp's Kirk Lowell in Concordia, Kansas; Connie Neininger at the White County Economic Development Organization in Indiana; and Matt Sorensen at the McLean County Commission in Illinois. I also was inspired by the educators who are racing to create a well-trained corps of wind technicians: Bruce Graham at Cloud County Community College, Ahmad Hemami at Iowa Lakes Community College, and P. Barry Butler at the University of Iowa's College of Engineering top the list.\n\nIn exploring the energy industry's wildlife impacts, I was aided by several key people beyond those mentioned in chapter 7. Michelle Carder and Matt Gasner of Western Ecosystems Technology took me into the field to see how they monitor bats at Indiana's Meadow Lake Wind Farm. Shannon Anderson at the Powder River Basin Resource Council and the Sierra Club's Brad Mohrmann filled me in on the ecological devastation wrought by Wyoming's coal, gas, oil, and uranium industries. John Emmerich, executive director of the Wyoming Game and Fish Department, described his agency's efforts to protect the sage grouse and other wildlife.\n\nLocal battles over wind farm siting took me to the Flint Hills of Kansas, the Northern Laramie Range in Wyoming, and Vermont's Taconic Mountains. The Flint Hills controversy is well covered in the text, but I am especially grateful to Kansas Audubon's Ron Klataske for introducing me to local ranchers and conservation experts. I also thank ranchers Rose Bacon and Pete Ferrell for sharing their very different perspectives on a hotly divisive issue. John Briggs of the Konza Prairie Biological Station, Rob Manes and Brian Obermeyer of the Nature Conservancy, and Chuck Rice of Kansas State University gave me valued tutorials on tallgrass prairie ecology. Broader insights on wind energy's importance to Kansas came from Scott Allegrucci of the Great Plains Alliance for Clean Energy; Steve Baccus and Mike Irvin at the Kansas Farm Bureau; Rod Bremby, former secretary of the Kansas Department of Health and Environment; and Nancy Jackson at the Climate and Energy Project.\n\nSecuring wind energy's place in a state as richly endowed with fossil and nuclear fuels as Wyoming is no small undertaking. In probing that state's policies, I benefited hugely from my conversations with state senate president Jim Anderson; Cheryl Riley and David Picard of the Wyoming Power Producers Coalition; and Aaron Clark, who advised former governor Dave Freudenthal on energy policy. Although the hot debate over wind farm siting in the Northern Laramies didn't find its way into this book, I learned much about wind energy politics from rancher Rick Grant, Ken Lay of the Northern Laramie Range Alliance, and Ed Werner, chair of the Converse County Commission until wind farm opponents voted him out of office. I also spent an eye-opening day with L. J. and Karen Turner, Campbell County ranchers whose cattle graze on the edge of America's largest open-pit coal mine. I wish there was enough room in this book to tell every amazing story I encountered.\n\nIn Vermont's Taconic Mountains, concerns about noise and viewshed protection have dominated the debate about wind. Ken Kaliski of the Resource Systems Group was hugely instructive on the science of acoustics, and Dr. Michael Nissenbaum was forthcoming in explaining his own research on noise issues. I am also indebted to Annette Smith of Vermonters for a Clean Environment for her tour of the contested Vermont Community Wind site west of Rutland.\n\nPower transmission is a hugely technical, bureaucratically intertwined realm that several people made it easier to fathom: Brad Beecher of Empire District Electric; Jay Caspary and Carl Monroe of the Southwest Power Pool; Craig Cox at the Interwest Energy Alliance; Loyd Drain at the Wyoming Infrastructure Authority; Steve Gaw of the Wind Coalition; Mark Fagan of the Harvard Kennedy School; Chris Miller and Bri West at the Piedmont Environmental Council; John Nielsen of Western Resources Advocates; Todd Parfitt and Tom Schroeder of Wyoming's Department of Environmental Quality; Chris Petrie at the Wyoming Public Service Commission; and David Smith at TransWest Express LLC. Jon Wellinghoff, chairman of the Federal Energy Regulatory Commission, brought his powerful vision to our meeting, and Julia Bovey proved herself a true friend in arranging that interview.\n\nThe obstacles to getting truly transformative energy policies in place came to light in discussions with U.S. representatives Ed Markey and Henry Waxman. These two mavericks have fought long and hard to give our global climate crisis the priority it deserves. Hopefully, their day will come. David Osterberg of the Iowa Public Policy Project was uplifting in describing what his own state has done to bring renewable energy into the mainstream.\n\nIn addition to holding the most informative conferences and workshops in the trade, the American Wind Energy Association has a bright staff of professionals who have been highly responsive to my research needs. My particular thanks go to Rob Gramlich, Kathy Belyeu, Hans Detweiler, Michael Goggin, and Elizabeth Salerno.\n\nEach of my trips yielded new questions and a new crop of interviews to be transcribed. I was lucky to have a spirited and utterly reliable group of research assistants at the ready. Kate Davies uncovered a trove of studies on bird and bat concerns, turned recordings of varying quality into precise transcripts, and offered astute insights on the evolving manuscript. My thanks also go to Kathryn Boucher, Christine Cho, Michael Dorsi, Ari Peskoe, Christina Putz, and Allie Rosene-Mirvis. Vida Margaitis and Diane Sredl, librarians at Harvard University, were ever-resourceful in tracking down valuable government documents. I am grateful as well to my mother, Joan Warburg, for her many phone calls and e-mails alerting me to wind energy coverage in the media.\n\nMy wife, Tamar, has been both a tireless editor and an endless source of support and encouragement. Reading numerous drafts of this book, she praised generously, but she also dared to deliver the difficult news of a chapter's need for fundamental reshaping. I'm sure that she and our daughters, Tali and Maya, have heard more about wind energy than they ever wanted to know.\n\nSpeaking of Tali and Maya, this book is dedicated to them. While plenty of work still lies ahead for my generation, theirs will have to carry on the tough work of steering our national and global energy choices onto a saner course. I only hope that we're all up to the challenge.\n\n## Notes\n\nINTRODUCTION\n\n1. Letter from Franklin D. Roosevelt to James P. Warburg, May 23, 1934, quoted in James P. Warburg, _The Long Road Home_ (New York: Doubleday & Co., 1964), 156\u201357.\n\nCHAPTER ONE: CLOUD COUNTY REVIVAL\n\n1. See table 1, estimating total yearly wind generation potential in Kansas at 3.6 million gigawatt hours. U.S. power generation in 2010 totaled 4.1 million gigawatt-hours. U.S. Energy Information Administration (EIA), table 1.1, \"Net Generation by Energy Source: Total (All Sectors),\" released March 11, 2011, <http:\/\/www.eia.gov\/>.\n\n2. American Wind Energy Association, _U.S. Wind Industry Annual Market Report_ , 12, fig. 12.\n\n3. T. Lindsay Baker, _Field Guide to American Windmills_ , 33\u201340, 339\u2013421.\n\n4. Kansas Statutes 79\u2013201 (effective January 1, 1999). The federal production tax credit, more fully described in chapter 3, \"Rust Belt Renewables,\" had been intermittently available since passage of the Energy Policy Act of 1992 (P.L. 102\u2013486).\n\n5. Duane Schrag, \"Wind Turbines Could Resume Spinning Soon,\" _Salina (KS)Journal_ , April 28, 2009, <http:\/\/www.saljournal.com\/>, citing EIA data.\n\n6. Kurt and Helen Kocher, interviews with author, March 19, 2009, and May 14, 2009.\n\n7. Ray Mason, interview with author, May 13, 2009.\n\n8. Monica Perin, \"Wind Energy Visionary Flies under the Radar,\" _Houston Business Journal_ , April 27, 2003, <http:\/\/www.bizjournals.com\/>.\n\n9. Jim Hoy, \"Environmental Price Tag Too High for Wind Farms,\" undated, Protect the Flint Hills, www.protecttheflinthills.org\/, accessed September 30, 2011.\n\n10. Tim Unruh, \"Company Finds One of the Answers to the Energy Crisis Is Blowing in the Wind,\" _Salina Journal_ , October 8, 2008, <http:\/\/www.salina.com\/>.\n\n11. Tom Fowler, \"Zilkha Wind Firm Purchased by Goldman Sachs,\" _Houston Chronicle_ , March 22, 2005, <http:\/\/www.chron.com\/>.\n\n12. Sergio Goncalves, \"EDP to Buy $2.2 bln U.S. Horizon Wind Energy,\" Reuters, March 27, 2007, <http:\/\/www.reuters.com>.\n\n13. Jim Roberts, senior project manager, Horizon Wind Energy, phone interviews with author, May 12, 2009, and June 17, 2009.\n\n14. See Vestas, _V90\u20133.0 MW: An Efficient Way to More Power_ , <http:\/\/www.autonavzduch.cz\/>, accessed April 4, 2011.\n\n15. Carole Engelder, director of construction management, Midwest Region, Horizon Wind Energy, phone interview with author, June 11, 2009.\n\n16. Roberts phone interview, May 12, 2009. Although he did not discuss per-turbine payments in our conversations, Roberts is cited in media reportage as giving $4,000 to $8,000 per turbine as the range of payments made to Meridian Way landowners. Tim Unruh, \"Company Finds One of the Answers to the Energy Crisis Is Blowing in the Wind,\" _Salina Journal_ , October 8, 2008, <http:\/\/www.salina.com\/>.\n\n17. Bonnie Sporer, Cloud County landowner, phone interview with author, May 18, 2009.\n\n18. Carole Engelder, e-mail to author, June 19, 2009. Roughly 47 percent of the workforce was hired locally.\n\n19. Kirk Lowell, executive director, Cloud County Development Corporation, interview with author, May 20, 2009.\n\n20. Bruce Graham, chairman, Wind Energy Department, Cloud County Community College, interview with author, March 19, 2009, and e-mail to author, March 4, 2011.\n\nCHAPTER TWO: EARLY ADOPTERS\n\n1. Danish Commission on Climate Change Policy, _Green Energy_ , 32.\n\n2. BTM Consult, _World Market Update 2004 (Forecast 2005\u20132009)_ , March 31, 2005, <http:\/\/www.btm.dk\/>.\n\n3. John Acher, \"UPDATE 2\u2014China Rivals Narrow Gap on Wind Leader Vestas,\" Reuters, March 15, 2011, <http:\/\/af.reuters.com\/>.\n\n4. Vestas, _Annual Report\u20142010_ , <http:\/\/www.vestas.com\/>. Conversion to U.S. dollars at exchange rate applicable on December 31, 2010.\n\n5. Lone Mortensen, director of people and culture, Blades Division, Vestas Wind Systems A\/S, Lem, Denmark, October 7, 2009.\n\n6. See Vestas annual reports, 2008\u20132010, <http:\/\/www.vestas.com\/>.\n\n7. Danish Energy Authority, \"Combined Heat and Power Production in Denmark,\" undated fact sheet, <http:\/\/www.ambottawa.um.dk\/>, accessed August 1, 2011.\n\n8. Buen, \"Danish and Norwegian Wind Industry,\" 3890.\n\n9. Wind power producers were paid 85 percent of the pretax price charged to consumers. Birger T. Madsen, \"Public Initiatives and Industrial Development after 1979,\" in Nissen, _Wind Power_ , 52, 54.\n\n10. Toke, \"Wind Power in UK and Denmark,\" 83 et seq.\n\n11. Van Est, _Winds of Change_ , 89. This rule was issued in 1985.\n\n12. Buen, \"Danish and Norwegian Wind Industry,\" 3890.\n\n13. Sijm, _Performance of Feed-in Tariffs_ , 11.\n\n14. Gipe, \"Wind Energy Comes of Age,\" 760.\n\n15. \"Denmark Struggles with Regulations\u2014So Far, So Bad,\" _Windpower Monthly_ , April 1, 2000, <http:\/\/www.windpowermonthly.com\/>. See also Szarka, \"Wind Power,\" 3041, 3046.\n\n16. Toke, \"Wind Power in UK and Denmark,\" 92.\n\n17. Van Est, _Winds of Change_ , 87.\n\n18. Richard M. Nixon, Special Message to the Congress on the Energy Crisis, January 23, 1974, <http:\/\/www.presidency.ucsb.edu\/>.\n\n19. Richard M. Nixon, State of the Union address, January 30, 1974, <http:\/\/abcnews.go.com\/>.\n\n20. U.S. Energy Information Administration (EIA), \"Petroleum & Other Liquids: U.S. Imports by Country of Origin,\" released July 27, 2010, <http:\/\/www.eia.doe.gov\/>.\n\n21. Jimmy Carter, Report to the American People on Energy, February 2, 1977 <http:\/\/millercenter.org\/>.\n\n22. Jimmy Carter, Address to the Nation on Energy, April 18, 1977, <http:\/\/millercenter.org\/>.\n\n23. Denis Hayes, phone interview with author, July 14, 2010. See also D. Hayes, _Rays of Hope_ , 174\u201380.\n\n24. PURPA, sec. 210. The Federal Energy Regulatory Commission (FERC) encouraged states to set rates at the full \"avoided cost\" of generating the same increment of power in the absence of the independent power sources. See Gipe, _Wind Energy Comes of Age_ , 31.\n\n25. See Thomas A. Starrs, \"Legislative Incentives and Energy Technologies,\" 129\u201334, describing FERC v. Mississippi, 456 U.S. 742 (1982), and American Paper Institute, Inc. v. American Electric Power Service Corp., 461 U.S. 402 (1983).\n\n26. Energy Tax Act of 1978, PL 95\u2013618, 92 STAT. 3195, Sec. 301(a). See also Minan, \"Encouraging Solar Energy Development,\" 31\u201334.\n\n27. Crude Oil Windfall Profits Tax of 1980, PL 96\u2013223, 94 STAT. 230, Sec. 221(a).\n\n28. Fred Branfman, \"Moving toward the Abyss,\" _Salon_ , April 18, 1996, <http:\/\/www.salon.com\/>.\n\n29. Lovins, _Soft Energy Paths_ , 55.\n\n30. See Clark, _Energy for Survival_.\n\n31. See U.S. House of Representatives, _Renewable Energy Incentives_ , 147 (testimony of Warren D. Noteware, California Energy Commission).\n\n32. Righter, _Wind Energy in America_ , 208.\n\n33. Ellen Paris, \"The Great Windmill Tax Dodge,\" _Forbes_ , March 12, 1984, 40.\n\n34. Gipe, _Wind Energy Comes of Age_ , 31.\n\n35. See Righter, _Wind Energy in America_ , 215; Starrs, \"Legislative Incentives and Energy Technologies,\" 109, table 1; Smith, \"Wind Farms of the Altamont Pass Area,\" 153, fig. 3.\n\n36. Righter, _Wind Energy in America_ , 158. For testimony by representatives of NASA and major aerospace firms, see U.S. House of Representatives, _Wind Energy Systems Act of 1980_.\n\n37. Gipe, _Wind Energy Comes of Age_ , 106.\n\n38. Ibid., 96.\n\n39. See Asmus, _Reaping the Wind_ , 87\u2013131.\n\n40. Van Est, _Winds of Change_ , 54.\n\n41. Righter, _Wind Energy in America_ , 183, quoting interview with Paul Gipe, November 11, 1992.\n\n42. Danish Wind Industry Association, _Denmark\u2014Wind Power Hub: Profile of the Danish Wind Industry_ , 2008, 13, <http:\/\/www.windpower.org\/>.\n\n43. Van Est, _Winds of Change_ , 59; Righter, _Wind Energy in America_ , 181.\n\n44. Powerplant and Industrial Fuel Use Act of 1978, PL 95\u2013620, 42 USC 8301.\n\n45. Powerplant and Industrial Fuel Use Act Amendments of 1987, PL 100\u201342, 42 USC 8312.\n\n46. See Gipe, \"Wind Energy Comes of Age,\" 738; Starrs, \"Legislative Incentives and Energy Technologies,\" 119.\n\n47. Ronald Reagan, \"Acceptance Speech at the 1980 Republican Convention,\" July 17, 1980, <http:\/\/www.nationalcenter.org\/>.\n\n48. Sissine, _Renewable Energy: A New National Commitment?_ , unnumbered table, \"Dept. of Energy (DOE) Renewable Energy R&D Funding: FY74 to FY92.\"\n\n49. Gipe, \"Wind Energy Comes of Age,\" 758, fig. 4.\n\n50. Almost 70 percent of the world's wind-generated electricity came from California's big-three wind farms in 1985. Smith, \"Wind Farms of the Altamont Pass Area,\" 146.\n\n51. Van Est, _Winds of Change_ , 92.\n\n52. See European Monitoring Centre on Change, _EMCC Case Studies: Energy Sector: Vestas, Denmark_ , European Foundation for the Improvement of Living and Working Conditions, 2008, 4, <http:\/\/www.eurofound.europa.eu\/>; and _Reference for Business, Encyclopedia for Business_ , 2d ed., \"Vestas Wind Systems A\/S,\" <http:\/\/www.referenceforbusiness.com\/>.\n\n53. See Sijm, _Performance of Feed-in Tariffs_ , 7\u20138 and table 3.2.\n\n54. Ibid., table 3.2; DOE, U.S. Installed Wind Capacity, 2000 Year End Wind Power Capacity (map), <http:\/\/www.windpoweringamerica.gov\/>.\n\n55. Peter Wenzel Kruse, senior vice president for group communication and investor relations, Vestas, interview with author, October 8, 2009.\n\n56. With 327 megawatts of new wind power capacity in 2010, Denmark ranked eleventh among European nations that year. European Wind Energy Association, _Wind in Power: 2010 European Statistics_ , February 2011, <http:\/\/www.ewea.org\/>.\n\n57. Ditlev Engel, press conference on Q3 accounts 2010, Radisson Blue Royal Hotel, Copenhagen, October 26, 2011. Additional information provided via e-mail to author by Michael Holm, public relations manager, Vestas, October 26, 2010, and June 10, 2011.\n\n58. See American Wind Energy Association, _U.S. Wind Industry Annual Market Report_ , 36.\n\nCHAPTER THREE: RUST BELT RENEWABLES\n\n1. \"President-Elect Obama Visits and Speaks at Cardinal Fastener, Bedford Heights, OH,\" January 16, 2009, YouTube, <http:\/\/www.youtube.com\/>.\n\n2. American Wind Energy Association (AWEA), _U.S. Wind Industry Annual Market Report_ , 37.\n\n3. Beth Govoni and Elizabeth Conley, Iowa Department of Economic Development, e-mail to author, August 4, 2010.\n\n4. \"Stimulus Money IS Creating US Jobs: American Wind Energy Association Response to American University Study\/ABC World News Story,\" _Electric Energy Online_ , February 12, 2010, <http:\/\/www.electricenergyonline.com>.\n\n5. Elizabeth Salerno and Jessica Isaacs, \"The Economic Reach of a Thriving Sector,\" _Windpower Monthly, Special Report\u2013United States: Wind Industry a Driver of Economic Recovery_ , May 2009, 7.\n\n6. Interviews with Bob Loyd, plant manager, and employees at Clipper Wind Turbine Works were conducted during the author's site visit to Cedar Rapids, IA, February 2, 2010.\n\n7. David Pitt, \"Historic Maytag Factory Shuts Its Doors,\" Associated Press, October 25, 2007, <http:\/\/www.manufacturing.net\/>.\n\n8. Robert Gates, chief commercial officer, Clipper Windpower, phone interview with author, April 13, 2010. See also Jesse Broehl, \"Domestic Supply Chain Must Grow,\" _Windpower Monthly, Special Report\u2013United States: Wind Industry a Driver of Economic Recovery_ , May 2009, 11.\n\n9. \"Drive Train Retrofit Means More Delays\u2014Further Bad News from Clipper on Liberty Component Problems,\" _Windpower Monthly_ , October 2007, 42; \"Series of Blade Repairs Under Way\u2014More Clipper Problems,\" _Windpower Monthly_ , February 2008, 39; \"Growing Revenue but Higher Losses,\" _Windpower Monthly_ , June 2009, 31.\n\n10. \"Growing Revenue but Higher Losses,\" _Windpower Monthly_ , June 2009, 31; \"Analysis\u2013Changing Blades without a Tall Crane,\" _Windpower Monthly_ , April 2010, 22.\n\n11. See Karl-Erik Stromsta, \"Pertz Quits Over Losses,\" _Recharge News_ , March 12, 2010, 10; Robin Pagnamenta, \"Clipper Has the Wind beneath Its Wings after US Funding Deal,\" _Times (London)_ , December 11, 2009, <http:\/\/business.timesonline.co.uk>; Ben Backwell, \"Clipper in a Cash Crisis,\" _Recharge_ , September 24, 2010, www.rechargenews.com\/; and Associated Press, \"United Technologies Optimistic about Wind Power,\" February 10, 2011, <http:\/\/www.bloomberg.com\/>.\n\n12. \"Clipper Teams Up to Forge New Beginning,\" _Windpower Monthly_ , May 2010, 77.\n\n13. Gates phone interview, April 13, 2010.\n\n14. The production tax credit, under section 1914 of the Energy Policy Act of 1992 (P.L. 102\u2013486), originally targeted wind and biomass derived from field crops or trees grown exclusively for power production. Other forms of renewable energy, including solar, geothermal, landfill gas, trash combustion, and certain kinds of small-scale hydropower, were added later. Sissine, _Renewable Energy: Tax Credit, Budget, and Electricity Production Issues_ , 4.\n\n15. The production tax credit for wind has since been raised to 2.2 cents per kilowatt hour. \"Federal Incentives\/Policies for Renewables & Efficiency: Renewable Electricity Production Tax Credit,\" Database of State Incentives for Renewables and Efficiency (DSIRE), rev. April 25, 2011, <http:\/\/www.dsireusa.org\/>.\n\n16. See H.R. 1, P.L. 111\u20135, Sec. 1101. The PTC was raised again in 2010 to 2.2 cents per kilowatt hour. For further analysis, see Viva Hammer, \"Special Report: Alternative Energy Gets a Second Chance,\" _Tax Notes_ , November 22, 2010, <http:\/\/vivahammertax.com\/>.\n\n17. See Dan Eggen, \"Four Democratic Senators Aim to Halt Stimulus Wind Project,\" _Washington Post_ , March 4, 2010.\n\n18. AWEA, _U.S. Wind Industry Annual Market Report_ , 36; _AWEA Year-End 2009 Market Report_ , 37.\n\n19. Under the Price-Anderson Act, each licensed reactor pays for $375 million in private insurance covering off-site liability. If an accident were to cause damage above that amount, all 104 licensed U.S. reactors would be assessed a pro rata share of the excess, up to a maximum of $111.9 million apiece. Beyond the $11.6 billion covered by this industrywide pool, state and local governments would have to petition Congress for supplemental disaster relief. See Nuclear Regulatory Commission, \"Fact Sheet on Nuclear Insurance and Disaster Relief Funds,\" June 2011, <http:\/\/www.nrc.gov\/>.\n\n20. For a general overview of this company, see Pruitt, _Timken_.\n\n21. Lorrie Paul Crum, communications department, and James E. Charmley, director of product technology, Timken, interviews with author, March 5, 2010.\n\n22. Ward J. \"Tim\" Timken Jr., \"Private Enterprise: A Cornerstone of American Democracy,\" Major Issues Lecture, Ashland University, Ashbrook Center for Public Affairs, October 16, 2009, <http:\/\/www.ashbrook.org\/>.\n\n23. H. Josef Hebert, \"Timken, Belden, and Their Workers Keep an Eye on Climate Bills,\" Associated Press, October 11, 2009, <http:\/\/www.cantonrep.com\/>.\n\n24. Crum, e-mail to author, December 10, 2009.\n\n25. Associated Press, \"GE Partners to Create Wind Farm in Lake Erie Near Cleveland; Aims for 1,000 MW Farm by 2020,\" May 24, 2010.\n\nCHAPTER FOUR:THE CHINESE ARE COMING\n\n1. The top fifteen global turbine manufacturers in 2010, by rank and market share, were Vestas (Denmark) #1\u201312%; Sinovel (China) #2\u201311%; GE (U.S.) #3\u201310%; Goldwind (China) #4\u201310%; Enercon (Germany) #5\u20137%; Gamesa (Spain) #6\u20137%; Dongfang (China) #8\u20137%; Suzlon (India) #9\u20136%; Siemens (Germany) #10\u20135%; United Power (China) #10\u20134%; Mingyang (China) #11\u20133%; REpower (Germany) #12\u20132%; Sewind (China) #13\u20132%; Nordex (Germany) #14\u20132%; XEMC (China) #15\u20131%. Reuters, \"UPDATE 2-China Rivals Narrow Gap on Wind Leader Vestas,\" March 15, 2011, <http:\/\/www.reuters.com\/>, citing data from MAKE Consulting.\n\n2. This conference, attended by the author, was held on October 21\u201323, 2009.\n\n3. This ceremony, which took place on October 15, 2009, featured remarks by Lars Anderson, president, Vestas-China; Ni Xiangyu, vice chairman, the Tianjin Economic-Technological Development Area; and others.\n\n4. Chinese Wind Energy Association (CWEA), _China Wind Power Installed Capacity Data (2010)_ , March 18, 2011 (in Chinese), table 2; Global Wind Energy Council, _Global Wind Statistics 2010_ , <http:\/\/www.gwec.net\/>.\n\n5. China's total electricity consumption is predicted to reach 8,200 terawatt-hours by 2020, with total installed capacity growing to 1.885 terawatts. \"China Electricity Consumption to Almost Double by 2020: China Electricity Council,\" Xinhua News Agency, December 21, 2010, <http:\/\/news.xinhuanet.com\/>.\n\n6. McElroy et al., \"Potential for Wind-Generated Electricity in China,\" 1378\u201380.\n\n7. See Keith Bradsher, \"China Outpaces U.S. in Cleaner Coal-Fired Plants,\" _New York Times_ , May 11, 2009.\n\n8. U.S. Energy Information Administration (EIA), \"Independent Statistics and Analysis,\" July 2009, <http:\/\/www.eia.doe.gov\/>.\n\n9. See \"China's Coal Demand Likely to Hit 3.8 bln t billion metric tons] in 2015,\" June 9, 2010, iStockAnalyst, [www.istockanalyst.com\/. Projected coal use of 3.8 billion metric tons converts to 4.18 billion \"short\" U.S. tons.\n\n10. The official death toll for Chinese coal miners was 3,215 in 2008 and 2,631 in 2009. \"China Mine Accidents Multiply\u201428 Dead, 192 Missing,\" Agence France-Presse, April 2, 2010, <http:\/\/www.chinamining.org\/>.\n\n11. World Nuclear Association, \"World Nuclear Power Reactors & Uranium Requirements,\" table, March 2, 2011, <http:\/\/www.world-nuclear.org\/>.\n\n12. See Antoaneta Bezlova, \"Three Gorges Dam May Displace Millions More,\" IPS-Inter Press Service, October 12, 2007, <http:\/\/ipsnews.net\/>; Jim Yardley, \"Chinese Dam Projects Criticized for Their Human Costs,\" _New York Times_ , November 19, 2007.\n\n13. Lema, \"Between Fragmented Authoritarianism and Policy Coordination,\" 3882.\n\n14. See Li, _Study on the Pricing Policy of Wind Power in China_ , 7.\n\n15. Renewable Energy Law of the People's Republic of China, approved by the Standing Committee of the National People's Congress, February 28, 2005, articles 6\u20139.\n\n16. Wu Qi, \"Landmark Project Heralds Grand Vision,\" _Windpower Monthly Special Report\u2013China: Market Ambition Ramps Up a Gear_ , October 2009, 20\u201321. See also Louis Schwartz, \"China's New Generation: Driving Domestic Development,\" March 10, 2009, RenewableEnergyWorld.com, <http:\/\/www.renewableenergyworld.com\/>.\n\n17. Yvonne Chan, \"China Unveils $140bn Plan to Build Seven Giant Wind Farms by 2020,\" _BusinessGreen_ , June 30, 2009, <http:\/\/www.businessgreen.com\/>. See also Rujun Shen and Tom Miles, \"China's Wind-Power Boom to Outpace Nuclear by 2020,\" Reuters, April 20, 2009, <http:\/\/uk.reuters.com\/>.\n\n18. Wu Qi, \"New Milestone for Domestic Leader,\" _Windpower Monthly_ , May 2009, 31.\n\n19. Wilson Guo, China country manager, UPC, interview with author, October 20, 2009.\n\n20. Charles R. McElwee II, Squire Sanders & Dempsey LLP, phone interview with author, November 23, 2009.\n\n21. See Lewis, _Comparison of Wind Power Industry Development Strategies_ , 15\u201317.\n\n22. Goldwind Science & Technology Company, _Company Overview and Highlights_ , September\u2013October 2009, 12, www.goldwindglobal.com.\n\n23. Chunhua Li, director, International Business Department, Goldwind, and Eva Xie, vice director, Investment Department, Tianrun, interview with author, October 18, 2009; Liang Xuan, International Business Department, Goldwind, e-mail to author, June 30, 2010.\n\n24. See Wu Qi, \"Grid Quotas Aim to Connect 9 GW of Stuck Chinese Wind,\" _Windpower Monthly_ , June 2010, 17; Li Jing, \"China Plans for Renewable Energy,\" _China Daily_ , August 25, 2009, <http:\/\/www.chinadaily.com.cn\/>.\n\n25. \"China Electricity Consumption to Almost Double by 2020,\" _China Daily_ , December 22, 2010.\n\n26. Lewis, _A Review of the Potential International Trade Implications_ , 3.\n\n27. Azure International, Beijing, data provided to author via e-mail, September 30, 2010.\n\n28. CWEA, _China Wind Power Installed Capacity Data (2010)_ , March 18, 2011 (in Chinese), table 2. Rankings are based on megawatts installed in 2010.\n\n29. See Office of the U.S. Trade Representative, \"United States Requests WTO Dispute Settlement Consultations on China's Subsidies for Wind Power Equipment Manufacturers,\" December 22, 2010, <http:\/\/www.ustr.gov\/>. See also Keith Bradsher, \"To Conquer Wind Power, China Writes the Rules,\" _New York Times_ , December 14, 2010.\n\n30. See Office of the U.S. Trade Representative, \"United States Requests WTO Dispute Settlement Consultations.\"\n\n31. Chunhua Li interview, October 15, 2009.\n\n32. Wu Qi, \"Goldwind Wins New US Orders,\" _Windpower Monthly_ , April 25, 2011.\n\n33. CWEA, _China Wind Power Installed Capacity Data (2010)_ , table 2.\n\n34. See Lyn Harrison and Eric Prideaux, \"Conference Report China: World Asks China to Drop Its Barriers,\" _Windpower Monthly_ , 57, 60.\n\n35. Matthew Kaplan, associate director, North American Wind Energy Markets, IHS Emerging Energy Research, interview with author, June 7 2011.\n\n36. Jim Bai and Chen Aizhu, \"China Wind Power Capacity Could Reach 1,000 GW by 2050,\" Reuters, October 19, 2011, <http:\/\/www.reuters.com\/>.\n\n37. Friedman, _Hot, Flat, and Crowded_ , 372\u201373.\n\nCHAPTER FIVE: WORKING THE WIND\n\n1. Ira Chinoy, associate professor, Philip Merrill College of Journalism, University of Maryland, e-mail to author, March 1, 2011.\n\n2. _Inside Indiana Business_ , \"More Wind Turbine Components Arrive at Port of Indiana,\" June 19, 2009, <http:\/\/www.insideindianabusiness.com\/>.\n\n3. Jesse Broehl, \"Transportation Is the Wind beneath U.S. Industry Wings,\" _Windpower Monthly Special Report: Freight on Board_ , July 2010, 5.\n\n4. \"Western States\u2014Four Ports Make US Top Ten List,\" _Windpower Monthly, Special Report: U.S. Investment and Development_ , September 2010, 23.\n\n5. \"Texas and Oklahoma Lead Quiet Region,\" _Windpower Monthly, Special Report: U.S. Investment and Development_ , September 2010, 16.\n\n6. \"Trains Deliver the Wind,\" _Breakbulk Industry News_ , September 22, 2009; \"Iowa Cornfield Sprouts Wind Towers,\" _Breakbulk Industry News_ , September 22, 2009, <http:\/\/www.breakbulk.com\/> (archival access limited to subscribers).\n\n7. Mark Anderson, \"Trucking Turbines Gets Sophisticated,\" _Windpower Monthly_ , April 2009, 92.\n\n8. Site visit to Grand Ridge Wind Farm, Ransom, IL; Adam Hartman, construction manager, Invenergy, interview with author, May 8, 2009.\n\n9. Alvin Cargill, Horizon Wind Energy, project manager for construction\u2014Meadow Lake Wind Farm, White and Benton counties, IN, interview with author, July 22, 2009.\n\n10. Bruce Graham, presentation on Meridian Way Wind Farm to Kansas\u2013Nebraska Radio Club, Cloud County Community College, observed by author, March 19, 2009.\n\n11. American Wind Energy Association, \"Supply Chain\u2014Anatomy of a Wind Turbine,\" <http:\/\/www.awea.org\/>.\n\n12. See \"President-Elect Obama Visits and Speaks at Cardinal Fastener, Bedford Heights, OH,\" January 16, 2009, YouTube, <http:\/\/www.youtube.com\/>.\n\n13. Clinton Newbold, quality assurance manager, Barnhart, at Meadow Lake Wind Farm, interview with author, July 23, 2009.\n\n14. Martin Culik, Horizon Wind Energy, project manager for development, Meadow Lake Wind Farm, interview with author, July 23, 2009. See also Horizon Wind Energy website, Meadow Lake Wind Farm, <http:\/\/www.horizonwind.com\/>.\n\n15. Connie Neininger, economic development director, White County, IN, interview with author, July 18, 2009.\n\n16. Steve Maples, Barnhart Crane & Rigging, site manager, Meadow Lake Wind Farm, interview with author, July 23, 2009. All subsequent quotes attributed to Steve Maples are from the same source.\n\n17. Justin Van Beusekom, assistant operations manager, Meridian Way Wind Farm, Horizon Wind Energy, May 15, 2009.\n\n18. Leo Jessen, lead technician, Grand Ridge Wind Farm, Invenergy, interview with author, May 10, 2009.\n\n19. Ibid.\n\n20. American Wind Energy Association, _AWEA Year-End 2009 Market Report_ , 37\u201338.\n\n21. See American Wind Energy Association, \"AWEA Seal of Approval Program,\" <http:\/\/archive.awea.org\/>, accessed April 11, 2011.\n\n22. Cloud County Community College Grants List\u2014Wind Energy Technology Training Program, provided to author by Bruce Graham, March 2010.\n\n23. Brad Lowell, \"CCCC Becoming Leader in Wind Energy Education,\" _Concordia (KS) Blade-Empire_ , March 10, 2010, quoting Kim Krull, vice president for academic affairs, Cloud County Community College.\n\n24. Acme Idea Company, New Duracell Smart Power Commercial, December 4, 2009, YouTube, <http:\/\/www.youtube.com\/>.\n\n25. P. Barry Butler, dean, College of Engineering, University of Iowa, interview with author, February 1, 2010.\n\n26. Interviews with students, Wind Energy and Turbine Technology Program, Iowa Lakes Community College, conducted by author, February 5, 2010.\n\n27. Michelle Graham, Horizon Wind Energy, operations administrator, Meridian Way Wind Farm, e-mail to author, May 16, July 9, December 17, 2009, and September 4, 2010.\n\n28. Bruce Graham, e-mail to author, March 4, 2011.\n\n29. Ahmad Hemami, instructor, Iowa Lakes Community College, interview with author, February 5, 2010.\n\n30. Loma Roggenkamp, O&M technician, Siemens Energy, phone interview with author, September 7, 2010; e-mail to author, July 27, 2011.\n\n31. Kristen Graf, executive director, Women of Wind Energy, phone interview with author, September 3, 2010.\n\n32. AWEA Board of Directors, <http:\/\/www.awea.org\/>, accessed April 11, 2011.\n\n33. Jeanna Walters, former student, Wind Energy Technology Training Program, Cloud County Community College, phone interview with author, May 13, 2009.\n\n34. U.S. Department of Energy, _Wind Power in America's Future_ , 12.\n\n35. Ibid., appendix C: \"Wind-Related Jobs and Economic Impact,\" 207\u201310. Data are drawn, in part, from the explanatory text accompanying fig. C-7, with specific numerical projections underlying fig. C-7 provided by Suzanne Tegen, PhD, senior energy analyst, Strategic Energy Analysis Center, National Renewable Energy Laboratory (NREL), e-mail to author, August 31, 2010. Indirect jobs are based on \"the increase in economic activity that occurs . . . when a contractor, vendor, or manufacturer receives payment for goods or services and in turn is able to pay others who support their business.\" Induced employment is caused by \"the changes in wealth that result from spending by people directly and indirectly employed by the project.\" Ibid., 201.\n\nCHAPTER SIX: THE PATH TO CLEANER ENERGY\n\n1. Ryan H. Wiser, Lawrence Berkeley National Laboratory, \"Tracking the US Wind Industry: Update on Cost, Performance, and Pricing Trends,\" AWEA Windpower 2011 Conference, May 23, 2011. In previously published government data, wind was credited with a more modest 2.3 percent of total U.S. electricity supply in 2010: 94,647 gigawatt hours out of 4,120,028 gigawatt hours. U.S. Energy Information Administration (EIA), _Electric Power Monthly_ , table 1.1, accessed July 2011, <http:\/\/www.eia.doe.gov\/>\n\n2. EIA, _Electric Power Monthly_ , table 1.1A. Solar-generated power totaled 1.3 gigawatt hours in 2011.\n\n3. By way of comparison, a coal plant's capacity factor is typically in the 70 to 90 percent range, and a nuclear plant can operate at over 90 percent of its stated capacity.\n\n4. National Renewable Energy Laboratory (NREL) and AWS Truewind, \"Estimates of Windy Land Area and Wind Energy Potential by State for Areas >=30% Capacity Factor at 80m,\" February 4, 2010, U.S. Department of Energy (DOE), Wind Powering America, <http:\/\/www.windpoweringamerica.gov\/>.\n\n5. See David Appleyard, \"Record Growth for EU Offshore Wind in 2010,\" _Renewable Energy World_ , February 15, 2011, <http:\/\/www.renewableenergyworld.com\/>. Europe added 883 megawatts in new offshore wind capacity in 2010, about 10 percent of total European wind installations that year.\n\n6. Schwartz, _Assessment of Offshore Wind Energy Resources_ , 3\u20134, table 1. This study understates the total offshore wind resource because it does not include data for Florida, Alabama, or Mississippi; wind maps for these states were unavailable. On the other hand, over 60 percent of the wind resources in NREL's offshore database are in waters deeper than 150 feet, where wind energy development is not yet technically feasible.\n\n7. Vattenfall, _Life-Cycle Assessment_ , 2.\n\n8. Spath, _Life Cycle Assessment_ , 10\u201313, 29, table 29.\n\n9. Estimates of coal power plant CO2 emissions were calculated by David Schoengold, senior consultant, MSB Energy Associates, based on an analysis of DOE data for 2008. David Schoengold, e-mail to author, October 5, 2010; Armond Cohen, executive director, Clean Air Task Force, e-mail to author, September 30, 2010.\n\n10. Vestas, _Lifecycle Assessment of a V90\u20133.0 MW Onshore Wind Turbine_ , 7.\n\n11. Vestas, _Lifecycle Assessment of Offshore and Onshore Sited Wind Power Plants_ , 35\u201340.\n\n12. DOE, _Wind Power in America's Future_ , 13\u201314.\n\n13. Percentages derived from European Wind Energy Association, _Wind Energy_ , 326, table V.1.2.\n\n14. One of the earlier critiques of civilian nuclear power was published under this title. See Richard Curtis and Elizabeth Hogan, _Perils of the Peaceful Atom: The Myth of Safe Nuclear Power Plants_ (New York: Doubleday, 1969).\n\n15. See, e.g., Brit Liggett and Jill Fehrenbacher, \"Stewart Brand Says Nuclear Power Could Save the World,\" video interview, Inhabitat.com, February 19, 2011, <http:\/\/inhabitat.com\/>.\n\n16. See Adriana Petryna, \"The Work of Illness: The Science and Politics of Chernobyl-Exposed Populations,\" _Osiris_ , 2d series, vol. 19, _Landscapes of Exposure: Knowledge and Illness in Modern Environments_ (Chicago: University of Chicago Press, 2004), 250\u201365.\n\n17. See Behrens, _Nuclear Power Plants_ , 4.\n\n18. National Academy of Sciences, \"Spent Fuel Stored in Pools at Some U.S. Nuclear Power Plants Potentially at Risk From Terrorist Attacks; Prompt Measures Needed to Reduce Vulnerabilities,\" press release, April 6, 2005. The press release and a link to the full public report are available at <http:\/\/www8.nationalacademies.org\/>.\n\n19. Wald, \"Resolved,\" 52.\n\n20. One of those isotopes, plutonium-239, takes 24,000 years to lose half its radioactivity. Plutonium-242 has a half-life of 376,000 years. See _Nuclear Regulatory Commission_ , \"Backgrounder on Radioactive Waste,\" April 2007, and \"Fact Sheet on Plutonium,\" October 2003, <http:\/\/www.nrc.gov\/>.\n\n21. See U.S. Nuclear Regulatory Commission, \"Fact Sheet on Decommissioning Nuclear Power Plants,\" April 2011, <http:\/\/www.nrc.gov\/>.\n\n22. U.S. Energy Information Administration (EIA), _Electric Power Monthly_ , table 1.1 accessed July 2011, reporting that petroleum liquids produced 23.4 gigawatt hours of electricity in 2010.\n\n23. In 1973, petroleum products supplied to the U.S. market totaled 17.3 million barrels per day. In 2010, total U.S. oil consumption was 19.1 million barrels per day. U.S. Energy Information Administration, \"Short-Term Energy Outlook,\" March 8, 2011, <http:\/\/www.eia.doe.gov\/>.\n\n24. See Pew Center on Global Climate Change, \"Transportation Overview: Transportation Emissions in the United States,\" <http:\/\/www.pewclimate.org\/>, accessed March 29, 2011.\n\n25. DONG Energy Annual Report, 2009, 44\u201345, <http:\/\/www.dongenergy.com\/>.\n\n26. Knud Pedersen, vice president responsible for group R&D and regulatory issues, DONG Energy, Copenhagen, interview with author, October 9, 2009. See also Nelson D. Schwartz, \"In Denmark, Ambitious Plan for Electric Cars,\" _New York Times_ , December 1, 2009; Clive Thompson, \"Batteries Not Included,\" _New York Times Magazine_ , April 16, 2009.\n\n27. EIA, \"U.S. Crude Oil, Natural Gas, and Natural Gas Liquids Reserves,\" table 9, released November 30, 2010, <http:\/\/www.eia.doe.gov\/>.\n\n28. In 2010, U.S. consumption of natural gas totaled 24.1 trillion cubic feet. EIA, \"Natural Gas Consumption by End Use,\" released March 29, 2011, <http:\/\/www.eia.doe.gov\/>.\n\n29. See, e.g., Natural Gas Supply Association, \"Resources,\" <http:\/\/www.naturalgas.org\/>, accessed April 7, 2011. Among the sources cited here is the EIA's _Annual Energy Outlook 2010_ , which estimates that the United States has 2,349 trillion cubic feet of \"unproved\" technically recoverable gas resources.\n\n30. See Sandra Steingraber, \"The Whole Fracking Enchilada,\" _Orion_ , September\/October 2010, <http:\/\/www.orionmagazine.org\/>; Ian Urbina, \"Regulation Lax as Gas Wells' Tainted Water Hits Rivers,\" _New York Times_ , February 27, 2011.\n\n31. See Ian Urbina, \"Insiders Sound an Alarm Amid a Natural Gas Rush,\" _New York Times_ , June 26, 2011. In the immediate wake of these revelations, congressional leaders called on several federal agencies to investigate the industry's possibly exaggerated claims of shale gas productivity and profitability. See Ian Urbina, \"Lawmakers Seek Inquiry of Natural Gas Industry,\" _New York Times_ , June 28, 2011.\n\n32. EIA, \"International Gas Reserves and Resources,\" updated March 3, 2009, <http:\/\/www.eia.doe.gov\/>.\n\n33. DOE, _Wind Power in America's Future_ , 154, appendix A.\n\n34. Jon Wellinghoff, chairman, Federal Energy Regulatory Commission (FERC), interview with author, Washington, DC, April 15, 2010. Wellinghoff's ideas for developing a smart grid substantially supported by renewable energy are explored further in chapter 9, \"Greening the Grid.\"\n\n35. Pew Center on Global Climate Change, \"Renewable & Alternative Portfolio Standards,\" <http:\/\/www.pewclimate.org\/>. This webpage, accessed on March 17, 2011, lists thirty-one states and the District of Columbia with renewable or alternative electricity standards.\n\n36. H.B. 2369, Kan. Reg. Sess. (2009); Kan. State. Ann. Sec. 79\u2013201 (Supp. 2008).\n\n37. Todd Wood, \"California Utilities (Just) Miss Renewable Energy Deadline,\" _Grist_ , March 8, 2011, <http:\/\/www.grist.org\/>.\n\n38. Yin, \"Do State Renewable Portfolio Standards Promote In-State Renewable Generation?\" 1144.\n\n39. While not a direct indicator of the cost of a particular energy technology to consumers, wholesale prices\u2014paid by utilities to wind power producers\u2014are generally seen as a more accurate reflection of wind energy's competitiveness than the retail price of power, which can vary widely depending on the generation sources in a utility's portfolio, local distribution costs, the corporate structure of the utility (public- or investor-owned, regulated or deregulated), and the rate-setting policies of different states.\n\n40. DOE, _2009 Wind Technologies_ , 36\u201342; Ryan H. Wiser, Lawrence Berkeley National Laboratory, \"Tracking the U.S. Wind Industry: Update on Cost, Performance, and Pricing Trends,\" PowerPoint presentation, AWEA Windpower Conference, Anaheim, CA, May 23, 2011, 13\u201314 (summarizing data forthcoming in the DOE's _2010 Wind Technologies Market Report)_.\n\n41. The monthly natural gas price for electric power reached $11.84 per thousand cubic feet in October 2005, dropped as low as $5.76 in 2006, and then spiked again in 2008, topping out at $12.41. In 2010, it dropped to $4.44 in October, but was back up to $5.66 by December. EIA, _Monthly U.S. Natural Gas Electric Power Price_ , <http:\/\/www.eia.gov\/>.\n\n42. Nancy Rader, California Wind Energy Association, \"Wind Energy Development in the Desert,\" PowerPoint presentation, AWEA Windpower Conference, Anaheim, CA, May 23, 2011.\n\n43. Lazard Ltd., _Levelized Cost of Energy Analysis\u2014Version 4.0, 2010_ , cited in National Conference of State Legislatures, _Meeting the Energy Challenges of the Future: A Guide for Policymakers_ , 2010, 3, fig. 2, <http:\/\/www.ncsl.org\/>.\n\n44. Roland Berger Strategy Consultants, \"Innovating China's Wind Energy Market via Engineering Excellence & Supply Chain Integration.\" PowerPoint presentation, China Wind Power Conference, Beijing, October 21, 2009.\n\n45. Bloomberg New Energy Finance, \"Wind Turbine Prices Fall to their Lowest in Recent Years,\" February 7, 2011, <http:\/\/bnef.com\/>.\n\n46. Ibid., 32.\n\n47. DOE, _2009 Wind Technologies Market Report_ , 27.\n\nCHAPTER SEVEN:BIRDS AND BATS\n\n1. Center for Biological Diversity v. FPL Group, Inc. et al., Complaint for Violations of California Business and Professions Code Section 17200 et seq., Case No. RG04183113, Superior Court of the State of California, Alameda County, November 1, 2004, paragraphs 46\u201359.\n\n2. BioResource Consultants, _Developing Methods to Reduce Bird Mortality_ , table\n\n3. When Audubon reached a legal settlement with Alameda County in 2007, agreeing on a 50 percent mortality reduction goal, the Center for Biological Diversity strongly objected to the agreement's failure to spell out enforceable means of achieving that goal. It registered its views in a letter addressed to the members of the Alameda County Board of Supervisors, January 10, 2007, <http:\/\/www.biologicaldiversity.org\/>.\n\n4. Elizabeth McCarthy, \"Legal Battle over Altamont Windmill Bird Deaths Ends,\" _California Current_ , December 10, 2010, <http:\/\/www.cacurrent.com\/>.\n\n5. Scott Richardson, \"Audubon Society Wants Invenergy Study Redone,\" _Wind Watch: Industrial Wind Energy News_ , October 25, 2006, <http:\/\/www.wind-watch.org\/>.\n\n6. Dr. Angelo Capparella, associate professor of zoology, Illinois State University, Bloomington, interview with author, May 10, 2009.\n\n7. See National Renewable Energy Laboratory, _Power Technologies Energy Data Book_ , Wind Farm Area Calculator, <http:\/\/www.nrel.gov\/>, accessed August 1, 2011.\n\n8. Paul Kerlinger, \"Prevention and Mitigation of Avian Impacts at Wind Power Facilities\" in Resolve, Inc., _Proceedings of the Wind Energy and Birds\/Bats Workshop_ , 84\u201385.\n\n9. See Erickson, _Summary and Comparison of Bird Mortality from Anthropogenic Causes_ , 1033\u201334. The Federal Aviation Administration calls for lighting, at a minimum, on towers that define the periphery of all wind farms with turbines taller than 200 feet. See _FAA Advisory Circular: Obstruction Marking and Lighting_ , February 12, 2007, chapter 13, <http:\/\/www.windaction.org\/>.\n\n10. Capparella interview, May 10, 2009.\n\n11. Karl Kosciuch, \"Evaluating Whooping Crane Stop-over Habitat at Potential Wind Power Sites to Understand and Minimize Risk,\" AWEA Windpower Conference, Chicago, May 7, 2009.\n\n12. Diane Bailey, \"Whooping Cranes Use Wind Corridor\u2014Environmental Interests Clash,\" _Windpower Monthly_ , June 1, 2009, 52.\n\n13. Don Furman, senior vice president for development, Iberdrola Renewables, phone interview with author, February 19, 2010.\n\n14. AES Geo Energy, _Saint Nikola Kavarna Wind Farm Environmental Management and Monitoring Plan;_ Darius Snieckus, \"Early-Warning System Ensures the Safety of Millions of Migratory Birds,\" _Recharge_ , October 8, 2010, <http:\/\/www.rechargenews.com\/>.\n\n15. Erickson, _A Summary and Comparison of Bird Mortality from Anthropogenic Causes_ , 1035\u201336.\n\n16. National Research Council, _Environmental Impacts of Wind-Energy Projects_ , 75. See also Wallace P. Erickson, \"Bird Fatality and Risk at New Generation Wind Projects\" in Resolve, Inc., _Proceedings of the Wind Energy and Birds\/Bats Workshop_ , 29\u201331.\n\n17. See U.S. Fish and Wildlife Service, _Draft Voluntary Land-Based Wind Energy Guidelines;_ and _Draft Eagle Conservation Plan Guidance_. See also _Comments of the American Wind Energy Association on Draft Eagle Conservation Plan Guidance_ , May 19, 2011, <http:\/\/www.awea.org\/>.\n\n18. Erickson, _Avian Collisions with Wind Turbines_ , 7\u201312.\n\n19. U.S. Fish and Wildlife Service, _Migratory Bird Mortality: Many Human-Caused Threats Afflict Our Bird Populations_ , January 2002, <http:\/\/www.fws.gov\/>.\n\n20. Nico Dauphine and Robert J. Cooper, \"Impacts of Free-Ranging Domestic Cats _(Felis Catus)_ on Birds in the United States: A Review of Recent Research with Conservation and Management Recommendations,\" _Proceedings of the Fourth International Partners in Flight Conference: Tundra to Tropics_ , October 2009, 205\u20139, <http:\/\/www.abcbirds.org\/>.\n\n21. The Fish and Wildlife Service avoids giving numerical estimates of past and current sage grouse populations. See U.S. Fish and Wildlife Service, \"Questions and Answers for the Greater Sage-Grouse Status Review,\" 8, website updated March 5, 2010, <http:\/\/www.fws.gov\/>. However, when the Interior Department designated the bird as a candidate species under the Endangered Species Act (see discussion later in this chapter), press accounts reported total numbers of sage grouse as ranging from 200,000 to 500,000 today, down from 16 million a century ago. See, e.g., John M. Broder, \"No Endangered Status for Plains Bird,\" _New York Times_ , March 5, 2010.\n\n22. Hadassah Reimer, \"A Small Bird with a Big Footprint,\" _Wyoming Lawyer_ 31, no. 1 (February 2008): 2, <http:\/\/www.wyomingbar.org\/>.\n\n23. State of Wyoming Executive Order 2008\u20132, Greater Sage-Grouse Area Protection, August 1, 2008, paragraphs 2\u20133, <http:\/\/www.fws.gov\/>. The governor described the purposes of this order in a letter to State Senator Jim Anderson, chair of the legislature's Task Force on Wind Energy, May 18, 2009, <http:\/\/www.windaction.org\/>.\n\n24. Freudenthal letter to Anderson, May 18, 2009.\n\n25. For the 2011\u201312 biennium, mineral severance taxes and federal mineral royalties are projected to account for 51.4 percent of Wyoming's total state revenues. Consensus Revenue Estimating Group, _Wyoming State Government Revenue Forecast, Fiscal Year 2011\u2013Fiscal Year 2016_ , October 2010, <http:\/\/eadiv.state.wy.us\/>.\n\n26. State of Wyoming Executive Order 2010\u20134, Greater Sage-Grouse Area Protection, August 18, 2010, Specific Stipulations, <http:\/\/psc.state.wy.us\/>.\n\n27. See Wyoming Game and Fish Commission, \"Wildlife Protection Recommendations for Wind Energy Development in Wyoming,\" April 23, 2010, 7\u201310, <http:\/\/gf.state.wy.us\/>.\n\n28. John M. Broder, \"No Endangered Status for Plains Bird,\" _New York Times_ , March 5, 2010.\n\n29. Erik Molvar, executive director, Biodiversity Conservation Alliance, Laramie, WY, phone interview with author, January 11, 2010. See also Biodiversity Conservation Alliance, _Wind Power in Wyoming_.\n\n30. Natural Resources Conservation Service and Wildlife Habitat Council, \"Greater Prairie-Chicken (Tympanuchus cupido),\" _Fish and Wildlife Management Leaflet No. 27_ , February 2006, 1\u20132, <http:\/\/policy.nrcs.usda.gov\/>.\n\n31. Robert J. Robel, professor of environmental biology (ret.), Kansas State University, phone interview with author, January 13, 2010.\n\n32. Pitman, \"Location and Success of Lesser Prairie-Chicken Nests,\" 1267.\n\n33. Stephanie Manes, Smoky Hills coordinator, Ranchland Trust of Kansas, phone interview with author, December 23, 2009.\n\n34. Brian Obermeyer, Flint Hills project director, Nature Conservancy, interview with author, July 21, 2010.\n\n35. Rick Plumlee, \"Regulations Few in State's Flint Hills Burning Plan,\" _Wichita Eagle_ , December 7, 2010, <http:\/\/www.kansas.com\/>.\n\n36. John M. Briggs, director, Konza Prairie Biological Station, Kansas State University, interview with author, July 21, 2010.\n\n37. Robel, \"Spring Burning.\"\n\n38. Rene Braud, director of permitting and environmental affairs, Horizon Wind Energy, phone interview with author, December 17, 2009.\n\n39. See Kunz, \"Ecological Impacts of Wind Energy Development on Bats,\" 316, 320\u201322.\n\n40. Kunz, \"Assessing Impacts of Wind-Energy Development on Nocturnally Active Birds and Bats,\" 2450. Nationwide, three species of migratory, tree-dwelling bats\u2014the eastern red bat, the hoary bat, and the silver-haired bat\u2014make up almost 75 percent of fatalities from wind turbines. Kunz, \"Ecological Impacts of Wind Energy Development on Bats,\" 316.\n\n41. National Research Council, _Environmental Impacts of Wind Energy Projects_ , 97\u201398.\n\n42. Kunz, \"Ecological Impacts of Wind Energy Development on Bats,\" 318.\n\n43. Arnett, \"Patterns of Bat Fatalities,\" 64.\n\n44. Judge Roger W. Titus, Memorandum Opinion, Animal Welfare Institute, et al., v. Beech Ridge Energy LLC, et al., Case No. RWT 09cv1519, U.S. District Court, District of Maryland, December 8, 2009, 67, <http:\/\/www.mdd.uscourts.gov\/>.\n\n45. Judge Roger W. Titus, Order, Animal Welfare Institute, et al., v. Beech Ridge Energy LLC, et al., Case No. RWT 09cv1519, U.S. District Court, District of Maryland, December 8, 2009, <http:\/\/www.awionline.org\/>.\n\n46. \"Settlement Reached at Beech Ridge Industrial Wind Installation,\" _Charleston (WV) Daily Mail_ , January 27, 2010, <http:\/\/alleghenytreasures.wordpress.com\/>.\n\n47. Don Furman, senior vice president for development, transmission, and policy, Iberdrola Renewables, and AWEA president (2009\u201310), phone interview with author, February 19, 2010. See also \"Iberdrola Renewables, BWEC Second Year of Ground-Breaking Bat Study Again Shows Large Reduction in Bat Mortality,\" press release, November 11, 2010, <http:\/\/www.iberdrolarenewables.us\/>.\n\n48. Edward B. Arnett, \"Reducing Bat Fatalities at Wind Energy Facilities by Changing Turbine Cut-In Speed,\" Bat Conservation International, PowerPoint presentation, March 25, 2009, slide 23, U.S. Fish and Wildlife Service website, <http:\/\/www.fws.gov\/>.\n\n49. Kunz, \"Ecological Impacts of Wind Energy Development on Bats,\" 316\u201319.\n\n50. See also U.S. Fish and Wildlife Service, _Draft Voluntary Land-Based Wind Energy Guidelines_ , February 2011, <http:\/\/www.fws.gov\/>.\n\nCHAPTER EIGHT: THE NEIGHBORS\n\n1. Sylvia White, \"Towers Multiply, and Environment is Gone with the Wind,\" _Los Angeles Times_ , November 26, 1984.\n\n2. Paul Gipe, \"Aesthetic Guidelines for a Wind Power Future,\" in Pasqualetti, _Wind Power in View_ , 195, photo caption referring to the Zond Victory Garden wind farm in Tehachapi.\n\n3. A revealing account of this controversy appears in Williams, _Cape Wind_ , 248\u201398. See also American Wind Energy Association et al., \"Diverse Coalition Announces Campaign to Urge Congress to Reject Backroom Attack on Offshore Wind Power,\" press release, April 20, 2006, <http:\/\/www.capewind.org\/>; Philip Warburg and Susan Reid, \"Wind Power with No Direction,\" _Boston Globe_ , February 27, 2006; Philip Warburg and Susan Reid, \"Cape Wind Myths and Facts,\" _Cape Cod Times_ , May 16, 2006, <http:\/\/www.capewind.org\/>.\n\n4. The rapid return of porpoises to wind farm construction areas was documented at Horns Rev, Denmark's other large offshore wind energy complex. Jakob Tougaard et al., _Harbour Porpoises on Horns Reef\u2014Effects of the Horns Reef Wind Farm_ , Annual Status Report 2004 to Elsam Engineering A\/S (Roskilde, Denmark: National Environmental Research Institute, July 2005), 7, <http:\/\/www.hornsrev.dk\/>.\n\n5. E.ON Sverige, _R\u00f8dsand 2 Offshore Wind Farm: Environmental Impact Assessment\u2014Summary of the EIA-Report_ , E.On Sverige AB, June 2007, 12, www.dhi.dk.\n\n6. Bjarne Haxgart, site manager, R\u00f8dsand 2 Offshore Wind Farm, E.ON Climate & Renewables, R\u00f8dbyhavn, Denmark, interview with author, October 5, 2009.\n\n7. E.ON Sverige, _R\u00f8dsand 2\u2014Summary of the EIA-Report_ , 5, 9.\n\n8. Frank and Sarah Diss, Robert and Ruth Widman, Ransom, IL, interviews with author, May 8, 2009.\n\n9. Invenergy, _Grand Ridge Easement Agreement Summary_ , provided to author by Andrew Downey, land agent for the Grand Ridge Wind Farm, May 8, 2009. These fees escalate by either the cost-of-living index or 2 percent annually, whichever is greater.\n\n10. Bob Widman interview, May 8, 2009.\n\n11. Rose Z. Bacon, coproprietor, RK Cattle, Council Grove, KS. This and subsequent quotes are from interviews with author, January 5, 2010, and July 20, 2010.\n\n12. Rose Z. Bacon, \"Statement in Support of the Flint Hills National Heritage Area,\" undated, provided to author by R. Bacon.\n\n13. Ron Klataske, executive director, Audubon of Kansas, phone interview, December 21, 2009. Quotations are from Audubon of Kansas, \"Audubon of Kansas is a Leader in Efforts to Protect the Flint Hills,\" _Prairie Wings_ , Fall\/Winter 2004, 4\u20137.\n\n14. Garland P. (Pete) Ferrell III, rancher, Beaumont, KS, and principal, Energy for Generations, LLC, Tulsa. This and subsequent quotes are from interviews with author, January 19 and July 19, 2010.\n\n15. Leslie Wayne, \"Brothers at Odds,\" _New York Times Magazine_ , December 7, 1986.\n\n16. Tim Doyle, \"Koch's New Fight,\" _Forbes_ , September 21, 2006, <http:\/\/www.forbes.com\/>.\n\n17. \"Cape Wind Foes Spent $2 Million on Lobbying,\" _National Journal_ , February 23, 2009, <http:\/\/undertheinfluence.nationaljournal.com\/>. The multiple steps taken by Koch to fund anti\u2013Cape Wind efforts are documented in Greenpeace, _Bill Koch: The Dirty Money Behind Cape Wind Opposition_ , July 22, 2010, <http:\/\/www.greenpeace.org\/>.\n\n18. Greenpeace, _Koch Industries: Secretly Funding the Climate Denial Machine_ (March 2010) and _Koch Industries: Still Fueling Climate Denial_ (2011 rev'd. ed.), <http:\/\/www.greenpeace.org\/>. See also Tom Hamburger et al., \"Koch Brothers Now at Heart of GOP Power,\" _Los Angeles Times_ , February 6, 2011.\n\n19. See Wind and Prairie Task Force, _Final Report_ , 10, fig. 1.\n\n20. See Ken Vandruff, \"Governor Considers Making Kansas Flint Hills into a No-Wind-Farm Zone,\" _Wichita Business Journal_ , December 5, 2004, <http:\/\/www.bizjournals.com\/>.\n\n21. See Office of the Governor, State of Kansas, \"Governor Announces Road Map for Wind Energy Policy,\" news release, May 6, 2011, https:\/\/governor.ks.gov\/; Associated Press, \"Brownback Strikes Deal with Wind Farms,\" _Topeka Capital-Journal_ , May 8, 2011, <http:\/\/cjonline.com\/>.\n\n22. Rose Z. Bacon, e-mail to author, May 6, 2011.\n\n23. Pete Ferrell, Grinnell Alumni Lecture, \"Dances with Hooves,\" June 4, 2010, <http:\/\/loggia.grinnell.edu\/>. Ferrell attributes this timeline to Wes Jackson, founder of the Land Institute in Salina, Kansas, and a longtime advocate for sustainable agricultural practices in the prairie.\n\n24. See Gipe, _Wind Power_ , 129\u201346.\n\n25. See Clark, _Energy for Survival_ , 541\u201345; Hills, _Power from Wind_ , 274\u201375; and Righter, _Wind Energy in America_ , 126\u201336.\n\n26. \"Mountain-Top Wind Turbine To Be Built by Rutland Corporation,\" _Rutland (VT) Herald_ , January 21, 1946.\n\n27. Nina Keck, \"Ira Struggles with Proposed Wind Farm,\" Vermont Public Radio, June 9, 2009.\n\n28. Vermont Community Wind press releases: \"Vermont Community Wind Farm LLC Revises Project Scope,\" June 18, 2009; \"Vermont Community Wind Farm Removes Potential Susie's Peak Turbines from Project Plans,\" January 15, 2010, <http:\/\/vtcomwind.com\/>.\n\n29. Mary Pernal, assistant professor of English, Green Mountain College, Poultney, VT, interview with author, May 6, 2010.\n\n30. Phil Bloomstein, \"Living Next to a Wind Turbine,\" July 1, 2009, personal account, on Industrial Wind Action Group website, <http:\/\/www.windaction.org\/>.\n\n31. Wendy Todd, testimony presented to Governor's Task Force on Wind Power Development, State of Maine, September 26, 2007, <http:\/\/www.maine.gov\/>. See also Resource Systems Engineering, _Sound Level Study: Ambient & Operations Sound Level Monitoring_, June 21, 2007, <http:\/\/www.marshillwind.com\/>.\n\n32. Stanley M. Shapiro, MD, \"Can Wind Turbines Cause Heart Disease?\" _Heart Health News_ , Rutland Regional Medical Center, Winter 2010; \"The Industrial Wind Health Impact Debate Rages on at the Rutland (Vermont) Herald,\" February 6, 2010, on Allegheny Treasures website, <http:\/\/alleghenytreasures.wordpress.com\/>.\n\n33. Vermont Public Radio News, \"Plan for Ira Wind Is Tabled,\" April 27 and April 28, 2010, <http:\/\/www.vpr.net\/>.\n\n34. Colby, _Wind Turbine Sound and Health Effects_.\n\n35. Brett Horner et al., _Wind Energy Industry Acknowledgment of Adverse Health Effects_. The debate between Drs. McCunney and Nissenbaum, held on May 6, 2010, can be viewed on YouTube, RRMC Wind Health Forum, <http:\/\/www.youtube.com\/>.\n\n36. See Colby, _Wind Turbine Sound and Health Effects_ , sec. 3.4.1.\n\n37. Among the group living three miles or more away from the wind farm, one person reported new or worsened chronic sleep deprivation, one reported new chronic headaches, and none reported persistent anger or new or worsened depression. See Michael A. Nissenbaum, MD, \"Wind Turbines, Health, Ridgelines, and Valleys,\" media release, May 7, 2010, Society for Wind Vigilance website, <http:\/\/www.windvigilance.com\/>. Mars Hill residents have filed a civil lawsuit against the wind farm's owner, First Wind, seeking compensation for emotional and physical distress, as well as for decreased property values. See Jen Lynds, \"Mars Hill Windmills Prompt Civil Lawsuit,\" _Bangor (ME) Daily News_ , August 12, 2009, <http:\/\/new.bangordailynews.com\/>.\n\n38. See Industrial Wind Action Group website, <http:\/\/www.windaction.org>.\n\n39. Government of Australia, National Health and Medical Research Council, \"Wind Turbines and Health,\" July 2010, <http:\/\/www.nhmrc.gov.au\/>.\n\n40. Pierpont, _Wind Turbine Syndrome_ , 26, 48\u2013103.\n\n41. Geoff Leventhall, \"Direct Testimony on Behalf of Wisconsin Electric Power Company,\" _Application for a Certificate of Public Convenience and Necessity to Construct and Place in Service a Wind Turbine Electric Generation Facility Known as Glacier Hills Wind Park in Columbia County, Wisconsin_ , Docket No. 6630-CE-302, October 20, 2009, R1.80-R1.87, <http:\/\/www.co.whatcom.wa.us\/>; and Geoff Leventhall, \"Vibroacoustic Disease (VAD) and Wind Turbines,\" presented as Exhibit 20 at the same proceeding, <http:\/\/psc.wi.gov\/>.\n\n42. Nina Pierpont, letter to Geoff Leventhall, January 14, 2007, _Wind Turbine Syndrome_ book website, <http:\/\/www.windturbinesyndrome.com\/>. In this letter, Pierpont quotes her own remarks at a previous public presentation.\n\n43. See Tech Environmental, \"Acoustic Study of Vestas V82 Wind Turbines\u2014Fairhaven, Massachusetts,\" 4, table 1; and Resource Systems Group, Inc. (RSG), _Noise Impact Study for Georgia Mountain Community Wind_ , 4, fig. 2.\n\n44. See RSG, _Noise Impact Study for Georgia Mountain Community Wind_ , 6\u20137.\n\n45. WHO, _Night Noise Guidelines for Europe_ , 2009, xiv\u2013xviii.\n\n46. Maine Dept. of Environmental Protection, Chapter 375, \"No Adverse Environmental Effect Standard of the Site Location Law,\" sec. 10\u2014Control of Noise, effective January 18, 2006, <http:\/\/www.maine.gov\/>.\n\n47. Here are a few examples: In Ashe County, North Carolina, wind turbines rated above 20 kilowatts cannot produce more than 5 decibels above the average noise level on adjacent properties and are barred from exceeding 45 decibels except on properties leased or owned by the wind company; Ashe County, North Carolina, _An Ordinance to Regulate Wind Energy Systems_ , art. 6, sec. 2.7 (2007). In Chippewa, Minnesota, there is a 50-decibel daytime and nighttime noise limit at adjacent residences, but tighter limits may be imposed on pulsating and tonal sounds; Chippewa, Minnesota, _Windpower Management_ , sec. 12.8 (2005). In Huron County, Michigan, turbine noise measured outside residences, schools, hospitals, churches, and public libraries cannot exceed 50 decibels or 5 decibels above ambient levels, whichever is greater, for more than 10 percent of any hour; Huron County, Michigan, _Zoning Ordinance_ , art. X, sec. 3E (2009).\n\n48. Colby, _Wind Turbine Sound and Health Effects_ , sec. 3.1.3.\n\n49. See Pierpont, _Wind Turbine Syndrome_ , 112\u201321; Colby, _Wind Turbine Sound and Health Effects_ , secs. 3\u20137.\n\n50. Wendy Todd, testimony presented to the Governor's Task Force on Wind Power Development, State of Maine, September 26, 2007, <http:\/\/www.maine.gov\/>.\n\n51. National Research Council, _Environmental Impacts of Wind-Energy Projects_ , 157\u201358.\n\n52. Robert Gardiner, cofounder and co-owner, Independence Wind, e-mail to author, February 13, 2011.\n\n53. State of Wisconsin, 2011\u20132012 Legislature, January 2011 Special Session, Assembly Bill 9, 6\u20137, <http:\/\/www.renewwisconsin.org\/>.\n\n54. RENEW Wisconsin, \"Walker Proposal Would Torpedo $1.8 Billion in New Wind Power Investments,\" January 14, 2011, <http:\/\/www.renewwisconsin.org\/>.\n\n55. Public Service Commission of Wisconsin, \"Modifications to the Wind Siting Rules Approved by the Public Service Commission on December 9, 2010,\" PSC Docket 1-AC-231, 3, <http:\/\/www.renewwisconsin.org\/>.\n\n56. Jonathan Tilley, \"Wisconsin Suspends Wind Siting Rules,\" _Windpower Monthly_ , March 2, 2011, <http:\/\/www.windpowermonthly.com\/>.\n\n57. See Affidavit of Michael A. Nissenbaum, MD, State of Maine Board of Environmental Protection, In Re: Record Hill Wind, LLC, September 17, 2009, <http:\/\/www.wind-watch.org\/>.\n\n58. See, e.g., State of New Jersey, 214th Legislature, Senate Bill 2374, introduced November 8, 2010, <http:\/\/www.njleg.state.nj.us\/>. A stated purpose of this bill, which would establish a 2,000-foot turbine setback from residences and residentially zoned property, is to protect New Jersey residents from \"the ill health effects associated with 'wind turbine syndrome,' \" whose symptoms it then enumerates. The bill was referred to the Senate Environment and Energy Committee for consideration.\n\nCHAPTER NINE: GREENING THE GRID\n\n1. Bob Whitton, rancher, Bordeaux Junction, WY, and chair, Renewable Energy Alliance of Landowners, interview with author, November 6, 2010.\n\n2. U.S. Department of Energy (DOE), \"Wind Powering America: Wyoming Wind Map and Resource Potential,\" <http:\/\/www.windpoweringamerica.gov\/>.\n\n3. The Wyoming Infrastructure Authority has $1 billion in bonding authority for new transmission infrastructure. See <http:\/\/wyia.org\/>. See also Wyoming Statutes 37\u20135-301 et seq.\n\n4. Victor E. Garber, president, VeJay Energy & Land, Inc., and land agent for Pathfinder Renewable Energy, LLC, e-mail to author, November 19, 2010.\n\n5. See <http:\/\/www.powercompanyofwyoming.com>. See also Associated Press, \"Anschutz Corp. Has Wind Farm in the Works,\" May 25, 2009, <http:\/\/billingsgazette.com\/>.\n\n6. Wyoming Infrastructure Authority, Zephyr Project (ZTP), 2011, <http:\/\/wyia.org\/>.\n\n7. American Electric Power, \"Transmission Facts,\" undated, 4, <http:\/\/www.aep.com\/>, accessed June 19, 2011.\n\n8. Electric Power Research Institute, \"EPRI Finds Direct Current Power Uses 15% Less Electricity Than Alternating Current System at Duke Energy Data Center,\" news release, November 16, 2010, <http:\/\/my.epri.com\/>. A higher estimate of 20 percent loss reduction was cited in the summary of the EPRI High Voltage Direct Current & Flexible Alternating Current Transmission Systems Conference, October 28, 2010, <http:\/\/www.cvent.com\/>.\n\n9. See Bureau of Land Management, table 1\u20133: \"Mineral and Surface Acres Administered by the Bureau of Land Management, Fiscal Year 2009,\" <http:\/\/www.blm.gov\/>.\n\n10. Pacific Railway Act of 1862 (12 Stat. 489); Pacific Railway Act of 1864 (13 Stat. 356).\n\n11. Walter E. George, national project manager, BLM, Wyoming State Office, Cheyenne, interview with author, November 8, 2010.\n\n12. U.S. Secretary of the Interior, Order No. 3283, Enhancing Renewable Energy Development on the Public Lands, January 16, 2009.\n\n13. Tom Lahti, renewable energy chief, BLM, Wyoming State Office, Cheyenne, interview with author, November 8, 2010.\n\n14. In 2010 alone, the BLM held four public auctions for oil and gas leases in Wyoming, covering more than 300,000 acres. See Bureau of Land Management, _Competitive Lease Sale Notices & Results_, <http:\/\/www.blm.gov\/>.\n\n15. Lahti interview, November 8, 2010.\n\n16. See Energy Policy Act of 2005, Public Law 109\u201358 (August 8, 2005), Sec. 368: Energy Right-of-Way Corridors on Federal Land.\n\n17. See U.S. Departments of Agriculture, Energy, Interior, and Defense, News Release: \"Agencies Publish Final Environmental Impact Statement on Energy Corridor Designation in the West,\" November 26, 2008, <http:\/\/corridoreis.anl.gov\/>. These corridors have been legally challenged on a number of grounds by environmental groups. See Kate Galbraith, \"Environmentalists Sue over Energy Transmission Across Federal Lands,\" _New York Times_ , July 8, 2009. Western Resource Advocates and fourteen other environmental and conservation groups registered their concerns about the corridors in _Group Comments on the Draft Programmatic Environmental Impact Statement for the Designation of West-Wide Corridors_ , February 14, 2008, <http:\/\/www.westernresourceadvocates.org\/>.\n\n18. Holtkamp, _Transmission Siting in the Western United States_ , 28. This report, in part, describes the state-local division of authority over transmission infrastructure siting in eleven western states.\n\n19. See Wu Qi, \"Projects Waiting as Grid Build Lags Behind,\" _Windpower Monthly, Special Report\u2014China: Market Ambition Ramps Up a Gear_ , October 2009, 8\u20139.\n\n20. See ITC website, <http:\/\/www.itctransco.com\/>.\n\n21. Howard Learner, executive director, Environmental Law and Policy Center, phone interview with author, March 18, 2010.\n\n22. These plants include Antelope Valley Station (870 megawatts), Coal Creek Station (1,210 megawatts), Coyote Station (450 megawatts), R.M. Heskett Station (115 megawatts), Leland Olds Station (656 megawatts), and Milton R. Young Station (734 megawatts). See SourceWatch website, <http:\/\/www.sourcewatch.org\/>.\n\n23. Federal Energy Regulatory Commission (FERC), Promoting Wholesale Competition Through Open Access Non-Discriminatory Transmission Services by Public Utilities; Recovery of Stranded Costs by Public Utilities and Transmitting Utilities, Order No. 888, Final Rule, April 24, 1996, 75 FERC 61,080, Docket Nos. RM95-8-000 and RM94-7-001, <http:\/\/www.ferc.gov\/>.\n\n24. John Rogers et al., _Importing Pollution: Coal's Threat to Climate Policy in the U.S. Northeast_ , Union of Concerned Scientists, December 2008, <http:\/\/www.ucsusa.org\/>.\n\n25. Edward J. Markey, U.S. rep. (D-MA), Washington, DC, interview with author, April 16, 2010.\n\n26. See Beth Daley, \"Markey Urges Federal Cape Wind Approval,\" _Boston Globe_ , November 9, 2009.\n\n27. See, e.g., Wellinghoff, \"Federal Transmission Initiatives for Renewables.\"\n\n28. Jon Wellinghoff, chairman, FERC, interview with author, Washington, DC, April 15, 2010.\n\n29. The Waxman-Markey bill would have required retail electric distributors to provide an increasing percentage of power from renewable sources and electricity-saving measures, reaching 20 percent of all power sold in 2020. See H.R.2454, American Clean Energy and Security Act of 2009, 111th Congress, approved by the House of Representatives, June 26, 2009, Sec. 101, <http:\/\/www.govtrack.us\/>. The bill's summary can be found at the same site.\n\n30. See Ballard Spahr Andrews & Ingersoll, LLP, \"ARRA Appropriations Provisions Table,\" <http:\/\/baltimore.uli.org\/>.\n\n31. Barack Obama, \"Remarks by the President on Clean Energy,\" Newton, IA, April 22, 2009, <http:\/\/www.whitehouse.gov\/>.\n\n32. DOE, National Transmission Grid Study, May 2002, 53\u201359, <http:\/\/www.ferc.gov\/>.\n\n33. See Energy Policy Act of 2005, Public Law 109\u201358 (August 8, 2005), 16 U.S. Code Annotated Sec. 824p(i)(1), 824p(a), and 824p(b).\n\n34. See Regulations for Filing Applications for Permits to Site Interstate Electric Transmission Facilities, 71 Fed. Reg. 69,440, 69,444 (December 1, 2006).\n\n35. Piedmont Environmental Council et al. v. FERC, 558 F.3d 304 (4th Cir. 2009), cert denied, 130 S.Ct 1138 (2010). An enlightening analysis of this case appears in Dorsi, \"Case Comment.\"\n\n36. American Wind Energy Association, _Green Power Superhighways_ , 20\u201321.\n\n37. Don Furman, senior vice president for development, transmission, and policy, Iberdrola Renewables, and AWEA president (2009\u201310), phone interview with author, February 19, 2010.\n\n38. Chris Miller, president, Piedmont Environmental Council, Warrenton, VA, interview with author, April 14, 2010.\n\n39. DOE, _Wind Power in America's Future_ , 98.\n\n40. See Illinois Commerce Commission et al. v. FERC et al., 576 F.3d 470, 476 (2009).\n\n41. See FERC, _Order Accepting Tariff Revisions_ , June 17, 2010, 131 FERC 61,252, Docket No. ER10-1069-000, <http:\/\/www.ferc.gov\/>. See also Southwest Power Pool, \"FERC Approves New Cost Sharing Method for Expanding SPP's Transmission Grid,\" June 17, 2010, <http:\/\/www.spp.org\/>.\n\n42. Tierney, _A 21st-Century \"Interstate Electric Highway System,\"_ 44\u201345.\n\n43. See the discussion of DONG Energy's emerging plug-in vehicle network in chapter 6, \"The Path to Cleaner Energy.\"\n\n44. Wellinghoff interview, April 15, 2010.\n\n45. See Beacon Power, \"Smart Energy 25 Flywheel,\" <http:\/\/www.beaconpower.com\/>. See also Business Wire, \"Beacon Power Integrates Additional Megawatt of Flywheel Energy Storage on New England Power Grid,\" December 17, 2009, <http:\/\/www.businesswire.com\/>.\n\n46. Throop Wilder, president, 24M Technologies, phone interview with author, February 13, 2011. See also Gregory T. Huang, \"A123 Spinoff, 24M Technologies, Raises $10M to Develop Energy Storage Systems for Utilities, Electric Vehicles,\" Xconomy-Boston, August 16, 2010, <http:\/\/www.xconomy.com\/>.\n\n47. Tom Wind, Wind Utility Consulting PC, Jamaica, IA, interview with author, February 4, 2010. See also, \"Wrong Geology Sinks Iowa's $400m Wind Storage Dream,\" _Recharge_ , July 29, 2011, <http:\/\/www.rechargenews.com\/>.\n\nEPILOGUE\n\n1. See National Priorities Project, \"Cost of War,\" <http:\/\/costofwar.com\/>, accessed June 20, 2011.\n\n2. See, e.g., U.S. Partnership for Renewable Energy Finance, \"Prospective 2010\u20132012 Tax Equity Market Observations,\" July 2010, <http:\/\/www.uspref.org\/>.\n\n3. George P. Shultz, \"Viewpoints: Clean Air Law Is Key to Our Future,\" _Sacramento Bee_ , September 12, 2010, <http:\/\/www.sacbee.com\/>. Proposition 23 sought to freeze implementation of California's Global Warming Solutions Act until the state's unemployment rate dropped to 5.5 percent or below for four consecutive quarters. The state's jobless rate at the time stood at over 12 percent.\n\n##  _Selected Bibliography_\n\nAcoustic Ecology Institute. _AEI Special Report: Wind Energy Noise Impacts_ , November 17, 2009. <http:\/\/www.acousticecology.org\/>.\n\nAES Geo Energy. _Saint Nikola Kavarna Wind Farm\u2014Environmental Management and Monitoring Plan (EMMP)_ , November 2008. <http:\/\/www.aesgeoenergy.com\/>.\n\nAmerican Wind Energy Association. _An Agenda for the New President and Congress_ , November 2008. <http:\/\/www.newwindagenda.org\/>.\n\n\u2014\u2014\u2014. _AWEA Year-End 2009 Market Report_ , January 2010. <http:\/\/www.awea.org\/>.\n\n\u2014\u2014\u2014. _AWEA Mid-Year 2010 Market Report_ , July 2010. <http:\/\/www.awea.org\/>.\n\n\u2014\u2014\u2014. _U.S. Wind Industry Year-End 2010 Market Report_ , January 2011. <http:\/\/www.awea.org\/>.\n\n\u2014\u2014\u2014. _U.S. Wind Industry Annual Market Report\u2014Year Ending 2010_. April 2011. <http:\/\/www.awea.org\/>.\n\nAmerican Wind Energy Association, Blue Green Alliance, United Steelworkers. _Winds of Change: A Manufacturing Blueprint for the Wind Industry_ , June 2010. <http:\/\/www.awea.org\/>.\n\nAmerican Wind Energy Association, Solar Energy Industries Association. _Green Power Superhighways: Building a Path to America's Clean Energy Future_ , February 2009. <http:\/\/www.awea.org\/>.\n\nAmerica's Energy Future. _Electricity from Renewable Energy Resources: Status, Prospects, and Impediments_. Washington, DC: National Academies Press, 2010.\n\nAnderson, R., J. Tom, N. Neumann, et al. _Avian Monitoring and Risk Assessment at the San Gorgonio Wind Resource Area_. National Renewable Energy Laboratory Subcontract Report NREL\/SR-500\u201338054, August 2005. <http:\/\/www.nrel.gov\/>.\n\nArnett, Edward B., W. Kent Brown, Wallace P. Erickson, et al. \"Patterns of Bat Fatalities at Wind Energy Facilities in North America.\" _Journal of Wildlife Management_ 72, no. 1 (2008): 61\u201378. <http:\/\/www.batsandwind.org\/>.\n\nAsmus, Peter. _Reaping the Wind: How Mechanical Wizards, Visionaries, and Profiteers Helped Shape Our Energy Future_. Washington, DC: Island Press, 2001.\n\nAtkinson, Rob, Michael Shellenberger, Ted Nordhaus, et al. _Rising Tigers, Sleeping Giant: Asian Nations Set to Dominate the Clean Energy Race by Out-investing the United States_. Breakthrough Institute and the Information Technology and Innovation Foundation, November 2009. <http:\/\/www.thebreakthrough.org\/>.\n\nAwerbuch, Shimon, Leonard S. Hyman, and Andrew Vesey. _Unlocking the Benefits of Restructuring: A Blueprint for Transmission_. Vienna, VA: Public Utilities Reports, Inc., 1999.\n\nBaker, T. Lindsay. _Blades in the Sky: Windmilling through the Eyes of B. H. \"Tex\" Burdick_. Lubbock: Texas Tech University Press, 1992.\n\n\u2014\u2014\u2014. _A Field Guide to American Windmills_. Norman: University of Oklahoma Press, 1985.\n\nBarclay, Robert M. R., E. F. Baerwald, and J. C. Gruver. \"Variation in Bat and Bird Fatalities at Wind Energy Facilities: Assessing the Effects of Rotor Size and Tower Height.\" _Canadian Journal of Zoology_ 85, no. 3 (March 2007): 381\u201388.\n\nBehrens, Carl, and Mark Holt. _Nuclear Power Plants: Vulnerability to Terrorist Attack_. Congressional Research Service, Library of Congress, February 2005. <http:\/\/www.globalsecurity.org\/>.\n\nBenedetti, Tara. \"Running Roughshod? Extending Federal Siting Authority over Interstate Electric Transmission Lines.\" _Harvard Law Journal on Legislation_ 47 (Winter 2010): 253.\n\nBennett, Aaron. _Integration of Variable Generation Task Force (IVGTF)_. North American Electric Reliability Corporation, March 17, 2009.\n\nBerry, David. \"Innovation and the Price of Wind Energy in the U.S.\" _Energy Policy_ 37 (2009): 4493\u201399.\n\nBiodiversity Conservation Alliance. _Wind Power in Wyoming: Doing It Smart from the Start_ , November 2008. <http:\/\/www.voiceforthewild.org\/>.\n\nBioResource Consultants. _Developing Methods to Reduce Bird Mortality in the Altamont Pass Wind Resource Area_. California Energy Commission Public Interest Energy Research Program, August 2004. <http:\/\/www.energy.ca.gov\/>.\n\nBird, Lori, Mark Bolinger, Troy Gagliano, et al. \"Policies and Market Factors Driving Wind Power Development in the United States.\" _Energy Policy_ 33 (2005): 1397\u20131407.\n\nBranco, Castelo, and M. Alves-Pereira, \"Vibroacoustic Disease.\" _Noise and Health_ 6 (2004): 3\u201320. <http:\/\/www.noiseandhealth.org\/>.\n\nBreukers, Sylvia, and Maarten Wolsink. \"Wind Power Implementation in Changing Institutional Landscapes: An International Comparison.\" _Energy Policy_ 35 (2007): 2737\u201350.\n\nBrown, Ashley C., and Jim Rossi. \"Siting Transmission Lines in a Changed Milieu: Evolving Notions of the 'Public Interest' in Balancing State and Regional Considerations.\" _University of Colorado Law Review_ 81 (Summer 2010): 705.\n\nBrown, Brit T., and Benjamin A. Escobar. \"Wind Power: Generating Electricity and Lawsuits.\" _Energy Law Journal_ 28, no. 2 (2007): 489\u2013515. <http:\/\/bmpllp.com\/>.\n\nBrown, Lester R. _Plan B 4.0: Mobilizing to Save Civilization_. New York: W. W. Norton, 2009.\n\nBuen, Jorund. \"Danish and Norwegian Wind Industry: The Relationship Between Policy Instruments, Innovation and Diffusion.\" _Energy Policy_ 34 (2006): 3887\u201397.\n\nButler, Lucy, and Karsten Neuhoff. _Comparison of Feed in Tariff, Quota and Auction Mechanisms to Support Wind Power Development_. Cambridge, UK: University of Cambridge Working Papers in Economics, 2004.\n\nCharles River Associates. _SPP WITF Wind Integration Study_. Prepared for Southwest Power Pool, January 4, 2010. <http:\/\/www.uwig.org\/>.\n\nChief Medical Officer of Health. _The Potential Health Impact of Wind Turbines_. Ontario: Queen's Printer for Ontario, May 2010. <http:\/\/www.health.gov.on.ca\/>.\n\nChiras, Dan. _Power from the Wind: Achieving Energy Independence_. Gabriola Island, Canada: New Society Publishers, 2009.\n\nClark, Wilson. _Energy for Survival: The Alternative to Extinction_. Garden City, NY: Anchor Books, 1974.\n\nClarke, Alexi. \"Wind Energy: Progress and Potential.\" _Energy Policy_ 19, no. 8 (1991): 742\u201355.\n\nColby, W. David, Robert Dobie, Geoff Leventhall, et al. _Wind Turbine Sound and Health Effects: An Expert Panel Review_. Prepared for the American Wind Energy Association and the Canadian Wind Energy Association, December 2009. <http:\/\/www.canwea.ca\/>.\n\nConrad, Rebecca, and Susan Hess. _Tallgrass Prairie National Preserve Legislative History, 1920\u20131996_. Prepared for National Park Service Midwest Support Office, 1998.\n\nCrachilov, Constantin, Randall S. Hancock, and Gary Sharkey. _China Greentech Report 2009_. Mango Strategy, 2009. <http:\/\/www.china-greentech.com\/>.\n\nDanish Commission on Climate Change Policy. _Green Energy\u2014The Road to a Danish Energy System Without Fossil Fuels_ , September 28, 2010. <http:\/\/www.klimakommissionen.dk\/>.\n\nDeBlieu, Jan. _Wind: How the Flow of Air Has Shaped Life, Myth, and the Land_. New York: Houghton Mifflin, 1998.\n\nDietz, Brian. \"Turbines vs. Tallgrass: Law, Policy, and a New Solution to Conflict over Wind Farms in the Kansas Flint Hills.\" _Kansas Law Review_ 54 (2006): 1131\u201363.\n\nDorsi, Michael S. \"Case Comment: _Piedmont Environmental Council v. FERC.\" Harvard Environmental Law Review_ 34 (2010): 593\u2013603. <http:\/\/www.law.harvard.edu\/>.\n\nEnerNex Corporation. _Eastern Wind Integration and Transmission Study_. Prepared for the National Renewable Energy Laboratory, January 2010. <http:\/\/www.nrel.gov\/>.\n\nEnvironmental Law Institute. _Estimating U.S. Government Subsidies to Energy Sources: 2002\u20132008_. Washington, DC: September 2009.\n\n\u2014\u2014\u2014. _State Enabling Legislation for Commercial-Scale Wind Power Siting and the Local Government Role_. Washington, DC: May 2011.\n\nErickson, Wallace P., Gregory D. Johnson, and David P. Young Jr. _A Summary and Comparison of Bird Mortality from Anthropogenic Causes with an Emphasis on Collisions_. USDA Forest Service Technical Report PSW-GTR-191, 2005. <http:\/\/www.fs.fed.us\/>.\n\nErickson, Wallace P., Gregory D. Johnson, David P. Young Jr., et al. _Synthesis and Comparison of Baseline Avian and Bat Use, Raptor Nesting and Mortality Information from Proposed and Existing Wind Developments_. Prepared for Bonneville Power Administration, December 2002. <http:\/\/www.bpa.gov\/>.\n\nErickson, Wallace P., Gregory D. Johnson, M. Dale Strickland, et al. _Avian Collisions with Wind Turbines: A Summary of Existing Studies and Comparisons to Other Sources of Avian Collision Mortality in the United States_. National Wind Coordinating Committee Resource Document, August 2001. <http:\/\/www.west-inc.com\/>.\n\nEtherington, John. _The Wind Farm Scam: An Ecologist's Evaluation_. London: Stacey International, 2009.\n\nEuropean Wind Energy Association. _Wind Energy\u2014The Facts_ , 2009. <http:\/\/www.wind-energy-the-facts.org>.\n\nFagan, Mark L. _Understanding the Patchwork Quilt of Electricity Restructuring in the United States_. Regulatory Policy Program Working Paper RPP-2006\u201304. Cambridge, MA: Mossavar-Rahmani Center for Business and Government, John F. Kennedy School of Government, Harvard University, 2006.\n\nFerrey, Steven. \"Nothing but Net: Renewable Energy and the Environment, _MidAmerican_ Legal Fictions, and Supremacy Doctrine.\" _Duke Environmental Law & Policy Forum_ 14, no. 1 (2003): 1.\n\n\u2014\u2014\u2014. \"Power Future.\" _Duke Environmental Law & Policy Forum_ 15, no. 26 (Spring 2005): 261\u201373.\n\n\u2014\u2014\u2014.\"Restructuring a Green Grid: Legal Challenges to Accommodate New Renewable Energy Infrastructure.\" _Environmental Law_ 39 (2009): 977\u20131014.\n\nFershee, Joshua P. \"269 Levels of Green: State and Regional Efforts, in Wyoming and Beyond, to Reduce Greenhouse Gas Emissions.\" _Wyoming Law Review_ 7 (2007): 269.\n\nFielder, J. K., T. H. Henry, R. D. Tankersley, and C. P. Nicholson. _Results of Bat and Bird Mortality Monitoring at the Expanded Buffalo Mountain Windfarm, 2005_. Tennessee Valley Authority, June 2007. <http:\/\/www.tva.gov\/>.\n\nFox-Penner, Peter. _Electric Utility Restructuring: A Guide to the Competitive Era_. Vienna, VA: Public Utility Reports, 1998.\n\nFriedman, Thomas L. _Hot, Flat, and Crowded: Why We Need a Green Revolution\u2014And How It Can Renew America_. New York: Farrar, Straus and Giroux, 2008.\n\nGE Energy. _Western Wind and Solar Integration Study_ , May 2010. <http:\/\/www.nrel.gov\/>.\n\nGipe, Paul. _Wind Energy Comes of Age_. New York: John Wiley & Sons, 1995.\n\n\u2014\u2014\u2014. \"Wind Energy Comes of Age: California and Denmark.\" _Energy Policy_ 19, no. 8 (1991): 756\u201367.\n\n\u2014\u2014\u2014. _Wind Power: Renewable Energy for Home, Farm, and Business_. White River Junction, VT: Chelsea Green, 2004.\n\nGlobal Insight. _U.S. Metro Economies: Current and Potential Green Jobs in the U.S. Economy_. Prepared for the United States Conference of Mayors and the Mayors Climate Protection Center, October 2008. <http:\/\/www.usmayors.org\/>.\n\nGlobal Wind Energy Council. _Global Wind Energy Outlook 2010_ , October 2010. <http:\/\/www.gwec.net\/>.\n\n\u2014\u2014\u2014. _Global Wind 2008 Report_ , February 2009. <http:\/\/www.gwec.net\/>.\n\n\u2014\u2014\u2014. _Global Wind 2009 Report_ , March 2010. www.gwec.net\/.\n\nGodby, Rob, and Roger Coupal. \"The Impact of Wind Development on Local Economies\u2013Preliminary Wage Findings.\" Testimony Before the Wyoming Wind Energy Task Force, August 26\u201327, 2009. <http:\/\/legisweb.state.wy.us\/>.\n\nHadley, Stanton W., and Alexandra A. Tsvetkova. \"Potential Impacts of Plug-in Hybrid Electric Vehicles on Regional Power Generation.\" _Electricity Journal_ 22, no. 10 (December 2009): 56\u201368.\n\nHagen, Christian A., Brent E. Jamison, Kenneth M. Giesen, et al. \"Guidelines for Managing Lesser Prairie-Chicken Populations and Their Habitats.\" _Wildlife Society Bulletin_ 32 (2004): 69\u201382.\n\nHagen, Christian A., Brett K. Sandercock, James C. Pitman, et al. \"Spatial Variation in Lesser Prairie-Chicken Demography: A Sensitivity Analysis of Population Dynamics and Management Alternatives.\" _Journal of Wildlife Management_ 73, no. 8 (2009): 1325\u201332.\n\nHan, Jingyi, Arthur P.J. Mol, Yonglong Lu, et al. \"Onshore Wind Power Development in China: Challenges behind a Successful Story.\" _Energy Policy_ 37 (2009): 2941\u201351.\n\nHarris, Coy F., ed. _Windmill Tales_. Lubbock: Texas Tech University Press, 2004.\n\nHayes, Denis. _Rays of Hope: The Transition to a Post-Petroleum World_. New York: W. W. Norton, 1977.\n\nHeaps, Richard W. _Regional Economic Impact Analysis for the Georgia Mountain Community Wind Project_. Northern Economic Consulting, March 2009. <http:\/\/psb.vermont.gov\/>.\n\nHendricks, Bracken. _Wired for Progress: Building a National Clean-Energy Smart Grid_. Washington, DC: Center for American Progress, February 2009. <http:\/\/www.americanprogress.org\/>.\n\nHills, Richard L. _Power from Wind: A History of Windmill Technology_. New York: Cambridge University Press, 1994.\n\nHledik, Ryan. \"How Green Is the Smart Grid?\" _Electricity Journal_ 22, no. 3 (April 2009): 29\u201341.\n\nHoen, Ben, Ryan Wiser, Peter Cappers, et al. _The Impact of Wind Power Projects on Residential Property Values in the United States: A Multi-Site Hedonic Analysis_. Prepared for the Office of Energy Efficiency and Renewable Energy (Wind & Hydropower Technologies Program), U.S. Department of Energy, December 2009.\n\nHoltkamp, James A., and Mark A. Davidson. _Transmission Siting in the Western United States: Overview and Recommendations Prepared as Information to the Western Interstate Energy Board_. Holland & Hart, LLP, August 2009. <http:\/\/www.hollandhart.com\/>.\n\nHorn, Jason W., Edward B. Arnett, and Thomas H. Kunz. \"Behavioral Responses of Bats to Operating Wind Turbines.\" _Journal of Wildlife Management_ 72 (2008): 123\u201332.\n\nHorner, Brett, Richard R. James, Roy D. Jeffery, et al. _Wind Energy Industry Acknowledgement of Adverse Health Effects: An Analysis of the American\/Canadian Wind Energy Association Sponsored \"Wind Turbine Sound and Health Effects, An Expert Panel Review, December 2009.\"_ Society for Wind Vigilance, January 2010. <http:\/\/www.wind-watch.org\/>.\n\nHowe, Robert W., William Evans, and Amy T. Wolf. _Effects of Wind Turbines on Birds and Bats in Northeastern Wisconsin_. Report submitted to Wisconsin Public Service Corporation and Madison Gas and Electric Company, November 2002. <http:\/\/www.batsandwind.org\/>.\n\nHoy, Jim. _Flint Hills Cowboys: Tales of the Tallgrass Prairie_. Lawrence: University Press of Kansas, 2006.\n\nHyman, Leonard S., Andrew S. Hyman, and Robert C. Hyman. _America's Electric Utilities: Past, Present, and Future_. 8th ed. Vienna, VA: Public Utilities Reports, 2005.\n\nInformation Office of the State Council of the People's Republic of China. _China's Energy Conditions and Policies_ , December 2007. <http:\/\/en.ndrc.gov.cn\/>.\n\nISO New England. _New England 2030 Power System Study_. Report to the New England Governors, February 2010. <http:\/\/www.iso-ne.com\/>.\n\nJain, Aaftab, Paul Kerlinger, Richard Curry, et al. _Annual Report for the Maple Ridge Wind Power Project, Postconstruction Bird and Bat Fatality Study\u20142006_. Draft report prepared for PPM Energy and Horizon Energy and the Technical Advisory Committee for the Maple Ridge Project, February 2007. <http:\/\/www.wind-watch.org\/>.\n\nJames, Christopher A. \"Testimony on Behalf of the Sierra Club before the State Corporation Commission of Virginia.\" PATH Allegheny Virginia Transmission Corporation: Application for Approval of Electric Facilities Under the Utility Facilities Act, Docket No. PUE-2009\u201300043, October 14, 2009. <http:\/\/ceds.org\/>.\n\nJohnson, Anna, and Staffan Jacobsson. _The Emergence of a Growth Industry\u2014A Comparative Analysis of the German, Dutch, and Swedish Wind Turbine Industries_. Draft paper to be presented at the Joseph A. Schumpeter Society conference, University of Manchester, UK, June\u2013July 2000. <http:\/\/www.cric.ac.uk\/>.\n\nJohnson, Gregory D., Matthew K. Perlik, Wallace P. Erickson, et al. \"Bat Activity, Composition, and Collision Mortality at a Large Wind Plant in Minnesota.\" _Wildlife Society Bulletin_ 32, no. 4 (2004): 1278\u201388. <http:\/\/www.windrush-energy.com\/>.\n\nJohnson, Gregory D., Wallace P. Erickson, M. Dale Strickland, et al. \"Mortality of Bats at a Large-Scale Wind Power Development at Buffalo Ridge, Minnesota.\" _American Midland Naturalist_ 150 (2003): 332\u201342.\n\nJoskow, Paul L. \"Lessons Learned from Electricity Market Liberalization.\" Special issue, _Energy Journal_ 29, no. 2 (2008): 9\u201342.\n\nKahn, Edward. \"The Production Tax Credit for Wind Turbine Powerplants Is an Ineffective Incentive.\" _Energy Policy_ 24, no. 5 (1996): 427\u201335.\n\nKammen, Daniel M., Kamal Kapadia, and Matthias Fripp. _Putting Renewables to Work: How Many Jobs Can the Clean Energy Industry Generate?_ Report of the Renewable and Appropriate Energy Laboratory, University of California, Berkeley, April 2004. <http:\/\/rael.berkeley.edu\/>.\n\nKansas Energy Council. _Wind Energy Siting Handbook: Guideline Options for Kansas Cities and Counties_ , April 2005. <http:\/\/kec.kansas.gov\/>.\n\nKempton, Willett, Jeremy Firestone, Jonathan Lilley, et al. \"The Offshore Wind Power Debate: Views from Cape Cod.\" _Coastal Management_ 33 (2005): 119\u201349.\n\nKerlinger, Paul. _An Assessment of the Impacts of Green Mountain Power Corporation's Wind Power Facility on Breeding and Migrating Birds in Searsburg, Vermont, July 1996\u2013July 1998_. National Renewable Energy Laboratory, Subcontract Report NREL\/SR-500\u201328591, March 2002. <http:\/\/www.nrel.gov\/>.\n\nKerns, Jessica, and Paul Kerlinger. _A Study of Bird and Bat Collision Fatalities at the Mountaineer Wind Energy Center, Tucker Count, West Virginia: Annual Report for 2003_. Prepared for FPL Energy and Mountaineer Wind Energy Center Technical Review Committee, February 2004. <http:\/\/www.wvhighlands.org\/>.\n\nKirby, Brendan J. _Frequency Regulation Basics and Trends_ , Oak Ridge National Laboratory, December 2004. <http:\/\/www.ornl.gov\/>.\n\nKlassen, Ger, Asami Miketa, Katarina Larsen, et al. \"The Impact of R&D on Innovation for Wind Energy in Denmark, Germany, and the United Kingdom.\" _Ecological Economics_ 54 (2005): 227\u201340.\n\nKlein, Arne, Anne Held, Mario Ragwitz, et al. _Evaluation of Different Feed-In Tariff Design Options\u2014Best Practice Paper for the International Feed-In Cooperation_. Karlsruhe, Germany: Fraunhofer Institute for Systems and Innovation Research, October 2008. <http:\/\/www.worldfuturecouncil.org\/>.\n\nKomanoff, Charles. \"Whither Wind: A Journey Through the Heated Debate over Wind Power.\" _Orion_ , September\/October 2008. <http:\/\/www.orionmagazine.org\/>.\n\nKrapels, Edward N. _Integrating 200,000 MWs of Renewable Energy into the US Power Grid: A Practical Proposal_. Wakefield, MA: Anbaric Transmission, February 2009. <http:\/\/www.maine.gov\/>.\n\nKunz, Thomas H., Edward B. Arnett, Brian M. Cooper, et al. \"Assessing Impact of Wind-Energy Development on Nocturnally Active Birds and Bats: A Guidance Document.\" _Journal of Wildlife Management_ 71, no. 8 (2007): 2449\u201386. <http:\/\/www.batsandwind.org\/>.\n\nKunz, Thomas H., Edward B. Arnett, Wallace P. Erickson, et al. \"Ecological Impacts of Wind Energy Development on Bats: Questions, Research Needs, and Hypotheses.\" _Frontiers in Ecology and the Environment_ 5, no. 6 (2007): 315\u201324. <http:\/\/www.migrate.ou.edu\/>.\n\nKuvlesky, William P., Jr., Leonard A. Brennan, Michael L. Morrison, et al. \"Wind Energy Development and Wildlife Conservation: Challenges and Opportunities.\" _Journal of Wildlife Management_ 71, no. 8 (2007): 2487\u201398.\n\nLandworks. _Aesthetic Assessment of the Proposed Georgia Mountain Community WindProject_. Prepared for Georgia Mountain Community Wind, LLC, March 2009. <http:\/\/psb.vermont.gov\/>.\n\nLarrabee, Aim\u00e9e, and John Altman. _Last Stand of the Tallgrass Prairie_. New York: Metro Books, 2001.\n\nLazzari, Salvatore, and Jane Gravelle. _Effective Tax Rates on Solar\/Wind and Synthetic Fuels as Compared to Conventional Energy Resources_. Congressional Research Service, Library of Congress, May 1984.\n\nLeddy, Krecia L., Kenneth F. Higgins, and David E. Naugle. \"Effects of Wind Turbines on Upland Nesting Birds in Conservation Reserve Program Grasslands.\" _Wilson Bulletin of Ornithology_ 111, no. 1 (March 1999): 100\u2013104. <http:\/\/elibrary.unm.edu\/>.\n\nLema, Adrian, and Kristian Ruby. \"Between Fragmented Authoritarianism and Policy Coordination: Creating a Chinese Market for Wind Energy.\" _Energy Policy_ 35 (2007): 3879\u201390.\n\nLeventhall, Geoff. Vibroacoustic Disease (VAD) and Wind Turbines Critique. Submission to the Wisconsin Public Service Commission on Behalf of Wisconsin Electric Power Company, October 20, 2009. Docket No. 6630-CE-302, PSC Ref. No. 121879, Exhibit 20.\n\nLewis, Joanna I. _A Comparison of Wind Power Industry Development Strategies in Spain, India, and China_. Prepared for the Center for Resource Solutions, San Francisco, July 19, 2007. <http:\/\/www.newenergyindia.org\/>.\n\n\u2014\u2014\u2014. _A Review of the Potential International Trade Implications of Key Wind Power Industry Policies in China_. Prepared for the Energy Foundation China Sustainable Energy Program, October 2007. <http:\/\/www.resource-solutions.org\/>.\n\nLewis, Joanna I., and Ryan H. Wiser. \"Fostering a Renewable Energy Technology Industry: An International Comparison of Wind Industry Policy Support Mechanisms.\" _Energy Policy_ 35 (2007): 1844\u201357. <http:\/\/eetd.lbl.gov\/>.\n\nLi Junfeng, Gao Hu, Shi Pengfei, et al. _China Wind Power Report 2007_. Beijing: China Environmental Science Press, 2007. <http:\/\/www.greenpeace.org\/>.\n\nLi Junfeng, Shi Jingli, Xie Hongwen, et al. _A Study on the Pricing Policy of Wind Power in China_. Greenpeace, Global Wind Energy Council, Chinese Renewable Energy Industries Association, October 26, 2006. <http:\/\/www.greenpeace.org\/>.\n\nLinden, Eugene. _The Winds of Change: Climate, Weather, and the Destruction of Civilizations_. New York: Simon & Schuster, 2006.\n\nLovins, Amory B. _Soft Energy Paths: Toward a Durable Peace_. Cambridge, MA: Ballinger, 1977.\n\nMahan, Simon, Isaac Pearlman, and Jacqueline Savitz. _Untapped Wealth: Offshore Wind Can Deliver Cleaner, More Affordable Energy and More Jobs Than Offshore Oil_. Oceana, September 2010. <http:\/\/na.oceana.org\/>.\n\nMainzer, Elliot. Statement Before the Committee on Energy and Natural Resources, United States Senate, Hearing on the Role of Grid-Scale Energy Storage in Meeting Our Energy and Climate Goals. December 10, 2009. Bonneville Power Administration website: <http:\/\/www.bpa.gov\/>.\n\nManon Kamp, Linda. \"Danish and Dutch Wind Energy Policy 1970\u20132000: Lessons for the Future.\" _International Journal of Environment and Sustainable Development_ 5, no. 2 (2006): 213\u201320.\n\nManwell, J. F., J. G. McGowan, and A. L. Rogers. _Wind Energy Explained: Theory, Design, and Application_. Chichester, UK: John Wiley & Sons , 2002.\n\nMassey, Garth. \"Critical Dimensions in Urban Life: Energy Extraction and Community Collapse in Wyoming.\" _Urban Life_ 9, no. 2 (July 1980): 187\u201399. <http:\/\/jce.sagepub.com\/>.\n\nMcElroy, Michael B., Xi Lu, Chris P. Nielsen, et al., \"Potential for Wind-Generated Electricity in China,\" _Science_ 325, no. 5946 (September 2009): 1378\u201380.\n\nMenz, Frederic C., and Stephan Vachon. \"The Effectiveness of Different Policy Regimes for Promoting Wind Power: Experiences from the States.\" _Energy Policy_ 34 (2006): 1786\u201396.\n\nMeyer, Niels I. \"European Schemes for Promoting Renewables in Liberalised Markets.\" _Energy Policy_ 31 (2003): 665\u201376.\n\nMeyer, Niels I., and Anne Louise Koefoed. \"Danish Energy Reform: Policy Implications for Renewables.\" _Energy Policy_ 31 (2003): 597\u2013607.\n\nMiller, Christopher G. Testimony Before the Subcommittee on Energy and the Environment of the Committee on Energy and Commerce, U.S. House of Representatives. Hearing on the Future of the Grid: Proposals for Reforming National Transmission Policy, June 12, 2009.\n\nMills, Andrew, Ryan Wiser, and Kevin Porter. _The Cost of Transmission for Wind Energy: A Review of Transmission Planning Studies_. Berkeley, CA: Lawrence Berkeley National Laboratory, February 2009.\n\nMinan, John H., and William H. Lawrence. \"Encouraging Solar Energy Development Through Federal and California Tax Incentives.\" _Hastings Law Journal_ 32 (1980\u201381): 1\u201358.\n\nMunksgaard, Jesper, and Anders Larsen. \"Socio-economic Assessment of Wind Power\u2014Lessons from Denmark.\" _Energy Policy_ 26, no. 2 (1998): 85\u201393.\n\nMunksgaard, Jesper, and Poul Erik Morthorst. \"Wind Power in the Danish Liberalised Power Market\u2014Policy Measures, Price Impact, and Investor Incentives.\" _Energy Policy_ 36 (2008): 3940\u201347.\n\nMusgrove, Peter. _Wind Power_. New York: Cambridge University Press, 2010.\n\nNational Research Council. _Environmental Impacts of Wind-Energy Projects_. Washington, DC: National Academies Press, 2007.\n\nNational Wind Coordinating Committee. _Permitting of Wind Energy Facilities: A Handbook_ , August 2002. <http:\/\/www.nationalwind.org\/>.\n\n\u2014\u2014\u2014. _Wind Power Facility Siting Case Studies: Community Response_ , June 2005. <http:\/\/www.windaction.org\/>.\n\nNewman, James, and Edward Zillioux. _Comparison of Reported Effects and Risks to Vertebrate Wildlife from Six Electricity Generation Types in the New York\/New England Region_. New York State Energy Research and Development Authority, March 2009. <http:\/\/www.nyserda.org\/>.\n\nNissen, Povl-Otto, Therese Quistgaard, Jyette Thorndahl, et al. _Wind Power\u2014The Danish Way: From Poul la Cour to Modern Wind Turbines_. Askov, Denmark: Poul la Cour Foundation, September 2009.\n\nNoor, John. \"Herding Cats: What to Do When States Get in the Way of National Energy Policy.\" _North Carolina Journal of Law and Technology_ 11 (Fall 2009): 145.\n\nOffice of the Director of Defense Research and Engineering. _Report to the Congressional Defense Committees: The Effect of Windmill Farms on Military Readiness 2006_. U.S. Department of Defense, 2006. <http:\/\/www.defense.gov\/>.\n\nPasqualetti, Martin J., Paul Gipe, and Robert W. Righter, eds. _Wind Power in View: Energy Landscapes in a Crowded World_. London: Academic Press, 2002.\n\nPattanariyankool, Sompop, and Lester B. Lave. \"Optimizing Transmission from Distant Wind Farms.\" _Energy Policy_ 38 (2010): 2806\u201315.\n\nPedersen, Eja, Frits van den Berg, Roel Bakker, et al. \"Response to Noise from Modern Wind Farms in the Netherlands.\" _Journal of the Acoustical Society of America_ 126, no. 2 (August 2009): 634\u201343. <http:\/\/umcg.wewi.eldoc.ub.rug.nl\/>.\n\nPedersen, Eja, and Kerstin Persson Waye. \"Wind Turbine Noise, Annoyance, and Self-Reported Health and Well-Being in Different Living Environments.\" _Occupational Environmental Medicine_ 64 (March 2007): 480\u201386. <http:\/\/ncbi.nlm.nih.gov\/>.\n\nPedersen, Eja, and Pernilla Larsman. \"The Impact of Visual Factors on Noise Annoyance Among People Living in the Vicinity of Wind Turbines.\" _Journal of Environmental Psychology_ 28 (2008): 379\u201389. <http:\/\/www.mfe.govt.nz\/>.\n\nPercival, S. M. _Assessment of the Effects of Offshore Wind Farms on Birds_. Prepared for DTI Consulting. ETSU W\/13\/00565\/REP, DTI\/Pub URN 01\/1434, 2001. <http:\/\/files.zite3.com\/>.\n\nPetersen, Ib Krag. _Bird Numbers and Distributions in the Horns Rev Offshore Wind Farm Area\u2014Annual Status Report 2004_. Denmark: National Environmental Research Institute, 2005. <http:\/\/www.hornsrev.dk\/>.\n\nPew Charitable Trusts. _Who's Winning the Clean Energy Race? Growth, Competition, and Opportunity in the World's Largest Economies_. Philadelphia: G-20 Clean Energy Factbook, 2010. <http:\/\/www.pewtrusts.org\/>.\n\nPhilipson, Lorrin, and H. Lee Willis. _Understanding Electric Utilities and De-Regulation_. 2d ed. London: Taylor & Francis, 2006.\n\nPierpont, Nina. _Wind Turbine Syndrome: A Report on a Natural Experiment_. Santa Fe: K-Selected Books, 2009.\n\nPitman, James C., Christian A. Hagen, Robert J. Robel, et al. \"Location and Success of Lesser Prairie-Chicken Nests in Relation to Vegetation and Human Disturbance.\" _Journal of Wildlife Management_ 69, no. 3 (2005): 1259\u201369.\n\nPruett, Christin L., Michael A. Pattern, and Donald H. Wolfe. \"It's Not Easy Being Green: Wind Energy and a Declining Grassland Bird.\" _BioScience_ 59, no. 3 (2009): 257\u201362. <http:\/\/www.bioone.org\/>.\n\nPruitt, Bettye H. _Timken: From Missouri to Mars\u2014A Century of Leadership in Manufacturing_. Boston: Harvard Business School Press, 1998.\n\nPublicover, David. _A Methodology for Assessing Conflicts Between Windpower Development and Other Land Uses_. Appalachian Mountain Club, AMC Technical Report 04\u20132, May 2004. <http:\/\/www.outdoors.org\/>.\n\nRedlinger, Robert Y., Per Dannemand Andersen, and Paul Erik Morthorst. _Wind Energy in the 21st Century: Economics, Policy, Technology, and the Changing Electricity Industry_. New York: Palgrave, 2002.\n\nReece, Erik. _Lost Mountain: A Year in the Vanishing Wilderness_. New York: Riverhead Books, 2006.\n\nResolve, Inc. Proceedings of the Wind Energy and Birds\/Bats Workshop: Understanding and Resolving Bird and Bat Impacts, Washington, DC, May 18\u201319, 2004. September 2004. <http:\/\/www.ewashtenaw.org\/>.\n\nResource Systems Group, Inc. _Noise Impact Study for Deerfield Wind, LLC, Scarsburg\/Readsboro, Vermont_. Prepared for PPM Energy, December 2006. <http:\/\/www.sheffieldwind.com\/>.\n\n\u2014\u2014\u2014. _Noise Impact Study for George Mountain Community Wind, George and Milton, Vermont_. February 2009. <http:\/\/www.georgiamountainwind.com\/>.\n\n\u2014\u2014\u2014. _Noise Primer_. March 2008. <http:\/\/www.state.vt.us\/>.\n\nRighter, Robert W. _Wind Energy in America: A History_. Norman: University of Oklahoma Press, 1996.\n\nRobel, Robert J., John A. Harrington Jr., Christian A. Hagen, et al. \"Effect of Energy Development and Human Activity on the Use of Sand Sagebrush Habitat by Lesser Prairie Chickens in Southwestern Kansas.\" In _Transactions of the 69th North American Wildlife and Natural Resources Conference_ , Washington, DC: Wildlife Management Institute, 2004.\n\nRobel, Robert J., John P. Hughes, Scott D. Hull, et al. \"Spring Burning: Resulting Avian Abundance and Nesting in Kansas CRP.\" _Journal of Range Management_ 51, no. 2 (March 1998): 132\u201338.\n\nRoberts, Mark. \"Evaluation of the Scientific Literature on the Health Effects Associated with Wind Turbines and Low Frequency Sound.\" Wood Dale, IL: Exponent, Inc., October 20, 2009. Prepared for Wisconsin Public Service Commission, Docket No. 6630-CE-302, PSC Ref. No. 121885, Exhibit 27. <http:\/\/www.maine.gov\/>.\n\nRosen, Daniel H., and Trevor Houser. _China Energy: A Guide for the Perplexed_. Joint Project of the Center for Strategic and International Studies and the Peterson Institute for International Economics, May 2007. <http:\/\/www.iie.com\/>.\n\nRossi, Jim. \"The Political Economy of Energy and Its Implications for Climate Change Legislation.\" _Tulane Law Review_ 84 (2009): 379.\n\n\u2014\u2014\u2014. \"The Trojan Horse of Electric Power Transmission Line Siting Authority.\" _Environmental Law_ 39 (2009): 1015\u201348.\n\nSchwartz, Marc, Donna Heimiller, Steve Haymes, et al. _Assessment of Offshore Wind Energy Resources for the United States_. National Renewable Energy Laboratory, Technical Report NREL\/TP-500\u201345889, June 2010. <http:\/\/www.nrel.gov\/>.\n\nSijm, J. P. M. _The Performance of Feed-in Tariffs to Promote Renewable Energy Electricity in European Countries_. Energy Center of the Netherlands, November 2002. <http:\/\/www.ecn.nl\/>.\n\nSingh, Virinder, Jeffrey Fehrs, and BBC Research and Consulting. _The Work That Goes into Renewable Energy_. Renewable Energy Policy Project, Research Report, November 2001. <http:\/\/www.repp.org\/>.\n\nSissine, Fred. _Renewable Energy: A New National Commitment?_ Congressional Research Service, Library of Congress, October 1993. <http:\/\/fpc.state.gov\/>.\n\n\u2014\u2014\u2014. _Renewable Energy: Tax Credit, Budget, and Electricity Production Issues_. Congressional Research Service, Library of Congress, January 2006.\n\nSissine, Fred, and Michele Passarelli. _Renewable Energy Technology: A Review of Legislation, Research, and Trade_. Congressional Research Service, Library of Congress, March 1987.\n\nSmith, D. R. \"The Wind Farms of the Altamont Pass Area.\" _Annual Review of Energy_ 12 (1987): 145\u201383.\n\nSolar Energy Research Institute. _A New Prosperity: Building a Sustainable Energy Future_. Andover, MA: Brick House Publishing, 1981.\n\nSpath, Pamela L., Margaret K. Mann, and Dawn R. Kerr, _Life Cycle Assessment of Coal-Fired Power Production_. National Renewable Energy Laboratory. NREL\/TP-570\u201325119, June 1999. <http:\/\/www.nrel.gov\/>.\n\nSrivastava, Anurag K., Bharath Annabathina, and Sukumar Kamalasadan. \"The Challenges and Policy Options for Integrating Plug-In Hybrid Electric Vehicles into the Electric Grid.\" _Electricity Journal_ 23, no. 3 (April 2010): 83\u201391.\n\nStahl, Brent, Lisa Chavarria, and Jeff D. Nydegger. \"Wind Energy Laws and Incentives: A Survey of Selected State Rules.\" _Washburn Law Journal_ 49 (Fall 2009): 99\u2013142. <http:\/\/www.washburnlaw.edu\/>.\n\nStarrs, Thomas A. \"Legislative Incentives and Energy Technologies: Government's Role in the Development of the California Wind Energy Industry.\" _Ecology Law Quarterly_ 15 (1988): 103\u201358.\n\nSterzinger, George. _Component Manufacturing: Kansas' Future in the Renewable Energy Industry_. Renewable Energy Policy Project, Technical Report, April 2008. <http:\/\/www.repp.org\/>.\n\nSvedarsky, W. Daniel, Ronald L. Westemeier, Robert J. Robel, et al. \"Status and Management of the Greater Prairie-Chicken _Tympanuchus cupido pinnatus_ in North America.\" _Wildlife Biology_ 6, no. 4 (2000): 277\u201384. <http:\/\/www.wildlifebiology.com\/>.\n\nSwanstrom, Debbie, and Meredith M. Jolivert. \"DOE Transmission Corridor Designations & FERC Backstop Siting Authority: Has the Energy Policy Act of 2005 Succeeded in Stimulating the Development of New Transmission Facilities?\" _Energy Law Journal_ 30 (2009): 415\u201366. <http:\/\/www.felj.org\/>.\n\nSwofford, Jeffrey, and Michael Slattery. \"Public Attitudes of Wind Energy in Texas: Local Communities in Close Proximity to Wind Farms and Their Effect on Decision Making. _Energy Policy_ 38 (2010): 2508\u201319.\n\nSzarka, Joseph. \"Wind Power, Policy Learning, and Paradigm Change.\" _Energy Policy_ 34 (2006): 3041\u201348.\n\nTech Environmental, Inc. _Acoustic Study of Vestas V82 Wind Turbines, Fairhaven, Massachusetts_. Prepared for the Town of Fairhaven, May 2007. <http:\/\/masstech.org\/>.\n\nTierney, Susan F. _A 21st-Century \"Interstate Electric Highway System\"\u2014Connecting Consumers and Domestic Clean Power Supplies_. Prepared for AEP Transmission, October 31, 2008. <http:\/\/www.puc.nh.gov\/>.\n\nToke, Dave. \"Wind Power in UK and Denmark: 'Can Rational Choice Help Explain Different Outcomes?' \" _Environmental Politics_ 11, no. 4 (Winter 2002): 83\u2013100.\n\nTomain, Joseph P. \" 'Steel in the Ground': Greening the Grid with the iUtility.\" _Environmental Law_ 39 (2009): 931\u201376. <http:\/\/www.elawreview.org\/>.\n\nU.S. Department of Energy. _\"Grid 2030\": A National Vision for Electricity's Second 100 Years_ , July 2003. <http:\/\/www.climatevision.gov\/>.\n\n\u2014\u2014\u2014. _2008 Wind Technologies Market Report_ , July 2009. <http:\/\/eetd.lbl.gov\/>.\n\n\u2014\u2014\u2014. _2009 Wind Technologies Market Report_ , August 2010. <http:\/\/www1.eere.energy.gov\/>.\n\n\u2014\u2014\u2014. _Wind Power in America's Future: 20% Wind Energy by 2030_. Mineola, NY: Dover Publications, 2010.\n\nU.S. Energy Information Administration. _Emissions of Greenhouse Gases in the United States\u20132009_ , March 2011. <http:\/\/www.eia.gov\/>.\n\nU.S. Fish and Wildlife Service. Briefing Paper: \"Prairie Grouse Leks and Wind Turbines: U.S. Fish and Wildlife Service Justification for a 5-Mile Buffer from Leks; Additional Grassland Songbird Recommendations,\" July 2004. <http:\/\/www.fws.gov\/>.\n\n\u2014\u2014\u2014. _Draft Eagle Conservation Plan Guidance_ , dated January 2011; released February 8, 2011. <http:\/\/www.fws.gov\/>.\n\n\u2014\u2014\u2014. _Draft Voluntary Land-Based Wind Energy Guidelines_ , released February 8, 2011. <http:\/\/www.fws.gov\/>.\n\n\u2014\u2014\u2014. _Interim Guidelines to Avoid and Minimize Wildlife Impacts from Wind Turbines_. Department of the Interior, May 2003. <http:\/\/www.fws.gov\/>.\n\nU.S. General Accounting Office. _Export Promotion: Federal Efforts to Increase Exports of Renewable Energy Technologies_ , December 1992. <http:\/\/www.legistorm.com\/>.\n\nU.S. House of Representatives. _Comprehensive National Energy Policy Act: Hearing Before the Committee on Ways and Means_. 102nd Cong., 2d sess., April 28, 1992.\n\n\u2014\u2014\u2014. _Electricity Competition: Hearings Before the Subcommittee on Energy and Power of the Committee on Commerce_. 105th Cong., 1st sess., October 21 and 22, 1997.\n\n\u2014\u2014\u2014. _H.R. 1216\u2014Renewable Energy and Energy Efficiency Technology Competitiveness Act of 1989: Hearing Before the Subcommittee on Energy Research and Development of the Committee on Science, Space, and Technology_. 101st Cong., 1st sess., May 23, 1989.\n\n\u2014\u2014\u2014. _Independent Power Producers: Hearing Before the Subcommittee on Energy Conservation and Power of the Committee on Energy and Commerce_. 99th Cong., 2d sess., June 11, 1986.\n\n\u2014\u2014\u2014. _PURPA: Renewable Energy Programs: Hearing Before the Subcommittee on Energy and Power of the Committee on Energy and Commerce_. 101st Cong., 2d sess., June 14, 1990.\n\n\u2014\u2014\u2014. _Renewable Energy Development: Hearings Before the Subcommittee on Energy Research and Development of the Committee on Science, Space, and Technology_. 100th Cong., 1st sess., July 8 and 9, 1987.\n\n\u2014\u2014\u2014. _Renewable Energy Incentives: Hearing Before the Subcommittee on Energy Conservation and Power of the Committee on Energy and Commerce_. 99th Cong., 1st sess., June 20, 1985.\n\n\u2014\u2014\u2014. _Renewable Energy Industries: Hearing Before the Subcommittee on Energy Conservation and Power of the Committee on Energy and Commerce_. 99th Cong., 2d sess., December 16, 1986.\n\n\u2014\u2014\u2014. _Renewable Energy in the Eighties: Needs for Further R &D: Hearings Before the Subcommittee on Energy Development and Applications of the Committee on Science and Technology_. 97th Cong., 2d sess., May 28 and July 28, 1982.\n\n\u2014\u2014\u2014. _Renewable Energy Technologies: Hearing Before the Subcommittee on Energy and Power of the Committee on Energy and Commerce_. 100th Cong., 2d sess., April 27, 1988.\n\n\u2014\u2014\u2014. _Unlocking America's Energy Resources: Next Generation: Hearing Before the Subcommittee on Energy and Air Quality of the Committee on Energy and Commerce_. 109th Cong., 2d sess., May 18, 2006.\n\n\u2014\u2014\u2014. _Wind Energy: Hearing Before the Subcommittee on Energy of the Committee on Science and Astronautics_. 93rd Cong., 2d sess., May 21, 1974.\n\n\u2014\u2014\u2014. _Wind Energy Systems Act of 1980: Hearings Before the Subcommittee on Energy Development and Applications of the Committee on Science and Technology_. 96th Cong., 1st sess., September 18, 24, 26, and October 17, 1979.\n\nU.S. Offshore Wind Collaborative. _U.S. Offshore Wind Energy: A Path Forward_. October 2009. <http:\/\/www.usowc.org\/>.\n\nU.S. Senate. _Clean Energy: From the Margins to the Mainstream: Hearing Before the Committee on Finance_. 110th Cong., 1st sess., March 29, 2007.\n\n\u2014\u2014\u2014. _Energy Efficiency and Renewable Energy Research, Development, and Demonstration: Hearing Before the Subcommittee on Energy Research and Development of the Committee on Energy and Natural Resources.101st_ Cong., 1st sess., June 15, 1989.\n\n\u2014\u2014\u2014. _Energy Tax Incentives: Hearing Before the Subcommittee on Energy and Agricultural Taxation of the Committee on Finance_. 102nd Cong., 1st sess., June 13 and 14, 1991.\n\n\u2014\u2014\u2014. _Implementation of the Public Utility Regulatory Policies Act of 1978: Hearings Beforethe Committee on Energy and Natural Resources_. 99th Cong., 2d sess., June 3 and 5, 1986.\n\n\u2014\u2014\u2014. _National Energy Policy Act of 1989 (Energy Efficiency and Renewable Energy): Hearing before the Committee on Energy and Natural Resources_. 101st Cong., 1st sess., March 14, 1989.\n\n\u2014\u2014\u2014. _Renewable Electricity: Hearing Before the Committee on Energy and Natural Resources_. 110th Cong., 2d sess., June 17, 2008.\n\n\u2014\u2014\u2014. _Solar Development Initiative Act of 1987 and the Renewable Energy and Energy Conservation Competitiveness Act of 1987: Hearing Before the Subcommittee on Energy Research and Development of the Committee on Energy and Natural Resources_. 100th Cong., 1st sess., August 6, 1987.\n\nVan Est, Rinie. _Winds of Change: A Comparative Study of the Politics of Wind Energy Innovation in California and Denmark_. Utrecht, Netherlands: International Books, 1999.\n\nVann, Adam. _Wind Energy: Offshore Permitting_. Congressional Research Service, Library of Congress, September 2009. <http:\/\/assets.opencrs.com\/>.\n\nVattenfall. _Life-Cycle Assessment: Vattenfall's Electricity in Sweden_ , January 2005. <http:\/\/www.vattenfall.com\/>.\n\nVestas. _An Environmentally Friendly Investment: Lifecycle Assessment of a V82\u20131.65 MW Onshore Wind Turbine_ , undated. <http:\/\/www.vestas.com\/>.\n\n\u2014\u2014\u2014. _An Environmentally Friendly Investment: Lifecycle Assessment of a V90\u20133.0 MW Offshore Wind Turbine_ , undated. <http:\/\/www.vestas.com\/>.\n\n\u2014\u2014\u2014. _An Environmentally Friendly Investment: Lifecycle Assessment of a V90\u20133.0 MW Onshore Wind Turbine_ , undated. <http:\/\/www.vestas.com\/>.\n\n\u2014\u2014\u2014. _Life Cycle Assessment of Offshore and Onshore Sited Wind Power Plants Based on Vestas V90\u20133.0 MW Turbines_ , June 2006. <http:\/\/www.vestas.com\/>.\n\nWald, Matthew, Garry Brown, Mike Morris, et al. \"Resolved: Using Nuclear and Coal Power in an Environmentally Friendly Manner is the Path Forward in Controlling Climate Change.\" _Environmental Forum_ 27, no. 1 (January\/February 2010): 46\u201353.\n\nWellinghoff, Jon. \"Federal Transmission Initiatives for Renewables: The View from FERC.\" Keynote address at the Renewable Energy World Conference, Austin, TX, February 23, 2010. <http:\/\/www.ferc.gov\/>.\n\n\u2014\u2014\u2014. \" Testimony Before the Energy and Environment Subcommittee of the Committee on Energy and Commerce, U.S. House of Representatives.\" Hearing on the Future of the Grid: Proposals for Reforming National Transmission Policy, June 12, 2009. <http:\/\/www.ferc.gov\/>.\n\nWestern Electricity Coordinating Council. _Overview of Policies and Procedures for Regional Planning, Project Review, Project Rating Review, and Progress Reports_ , April 2005. <http:\/\/www.wecc.biz\/>.\n\nWestern Renewable Energy Zones. _Western Renewable Energy Zones\u2014Phase 1 Report_. June 2009. <http:\/\/www.westgov.org\/>.\n\nWhite, Sarah, and Jason Walsh. _Greener Pathways: Jobs and Workforce Development in the Clean Energy Economy_. Center on Wisconsin Strategy, University of Wisconsin, Madison, Workforce Alliance, Apollo Alliance, 2008. <http:\/\/apolloalliance.org\/>.\n\nWild Earth Guardians. _Lesser Prairie-Chicken: A Decade in Purgatory_ , June 2008. <http:\/\/www.wildearthguardians.org\/>.\n\nWilderness Society. \"Attn: Docket Nos. 2007-OE-01 and 2007-OE-02, Re: Proposed Designation of National Interest Electric Transmission Corridors.\" Comments submitted to the Office of Electricity Delivery and Energy Reliability, July 6, 2007. <http:\/\/wilderness.org\/>.\n\nWilliams, Wendy, and Robert Whitcomb. _Cape Wind: Money, Celebrity, Class, Politics, and the Battle for Our Energy Future on Nantucket Sound_. New York: Public Affairs, 2007.\n\nWillrich, Mason. _Electricity Transmission Policy for America: Enabling a Smart Grid, End-to-End_. MIT Industrial Performance Center Energy Innovation Working Paper 09\u2013003, July 2009. <http:\/\/web.mit.edu\/ipc\/>.\n\nWind and Prairie Task Force. _Wind and Prairie Task Force Final Report_. Kansas Geological Survey Open-file Report 2004\u201329, June 7, 2004. <http:\/\/kec.kansas.gov\/>.\n\nWind Energy Task Force. _Final Report and Recommendations_. Wyoming Legislative Service Office, November 2009. <http:\/\/legisweb.state.wy.us\/>.\n\n_Windpower Monthly_. \"China: Market Ambition Ramps up a Gear.\" Special issue. October 2009.\n\n\u2014\u2014\u2014. \"Europe 2020: Achieving 12\u201315% Electricity from Wind Power.\" Special issue. March 2009.\n\n\u2014\u2014\u2014. \"Growing Strong: China's Wind Energy Expands at Home and Abroad.\" Special issue. October 2010.\n\n\u2014\u2014\u2014. \"Investing in the U.S.: A Regional Guide to America's Wind Energy Market.\" Special issue. September 2010.\n\nWind Turbine Guidelines Advisory Committee. _Preamble and Policy Recommendations_. U.S. Fish and Wildlife Service, October 2009. <http:\/\/www.fws.gov\/>.\n\n\u2014\u2014\u2014. _White Paper_. U.S. Fish and Wildlife Service, October 2008. <http:\/\/www.fws.gov\/>.\n\nWizelius, Tore. _Developing Wind Power Projects: Theory and Practice_. London: Earth-scan, 2007.\n\nWorld Health Organization Europe. _Night Noise Guidelines for Europe_ , 2009. <http:\/\/www.euro.who.int\/>.\n\nWyoming Game and Fish Department. _Wildlife Protection Recommendations for Wind Energy Development in Wyoming_ , November 17, 2010. <http:\/\/gf.state.wy.us\/>.\n\nWyoming Wind Collector and Transmission Task Force. _Report to the Legislative Task Force on Wind Energy, Transmission Sub-Committee_ , October 9, 2009.\n\nYin, Haitao, and Nicholas Powers. \"Do State Renewable Portfolio Standards Promote In-State Renewable Generation?\" _Energy Policy_ 38 (2010): 1140\u201349.\n\nZoellner, Tom. _Uranium: War, Energy, and the Rock That Shaped the World_. New York: Viking, 2009.\n\n##  _Index_\n\nPlease note that page numbers are not accurate for the e-book edition.\n\nAcciona, 40, 188\n\nAES Corporation, and AES Geo Energy, 120\n\nAgassi, Shai, 106\n\nAgriculture Department. _See_ U.S. Department of Agriculture\n\nair pollution: in China, 61, 63; from coal, 51\u201352, 63, 99\u2013101; from prairie burning in Kansas, 127\u201328, 142, 168; from wind compared to other power sources, 100\u2013101. _See also_ carbon emissions; greenhouse gas emissions\n\nAlliance to Protect Nantucket Sound. _See_ Cape Wind\n\nAltamont Pass, California, 30, 116\u201317, 118, 119, 134\n\nAmerican Clean Energy and Security Act (2009), 171\n\nAmerican Gas Association, 109\n\nAmerican Recovery and Reinvestment Act (2009), 50\n\nAmerican Superconductor, 39\n\nAmerican Wind Energy Association (AWEA), 51, 80, 93, 152, 159, 172\u201373\n\nAndersen, Lars, 61\n\nanemometer, 88\n\nAnimal Welfare Institute, 130\n\nAnschutz, Philip F., and Anschutz Corporation, 164\u201365\n\nAntelope Valley Station, North Dakota, 168\u201369, 171\n\nArab oil embargo, vii, 22, 25, 26, 109, 180\n\nAudubon: Bay Area, 116; Illinois, 117\u201318; Kansas, 8, 142, 193\n\navailability (wind turbines), 87\n\navian impacts of wind farms, bats: and causes of death, 129\u201330; Indiana bats, x, 130\u201331; and mitigation efforts, 131\u201332; and mortality rates, 129, 132\n\navian impacts of wind farms, birds: bald and golden eagles, 116, 117, 121, 133; and comparative causes of death, 121\u201322; and Endangered Species Act, 122, 130\u201331; long-tailed ducks, 137; migratory birds, 119; prairie chicken habitats, x, 9, 11, 125\u201329, 142; raptor deaths and mitigation efforts, 116\u201319; sage grouse habitats, x, 122\u201325, 159; and turbine technology changes, 118\u201319; white storks, 120; whooping cranes, 119\u201320\n\nAWEA. _See_ American Wind Energy Association (AWEA)\n\nBacon, Rose and Kent, 140\u201342, 147\u201349\n\nbalancing wind with other power sources, 105\u20136, 176\u201378, 181, 182\n\nBarnhart Crane & Rigging, 80\u201383\n\nbats. _See_ avian impacts of wind farms, bats\n\nBeacon Power Corporation, 177, 179\n\nbearings: magnetic, for flywheel power storage, 177; maintenance of, in wind turbines, 45, 46; Timken and, 52\u201357, 59, 69, 184\n\nBeaver Ridge Wind Project, Maine, 151\n\nBeecher, Brad, 161\n\nBeech Ridge Energy LLC, 130\u201331\n\nBeech Ridge Wind Farm, West Virginia, 130\u201331\n\nBetter Place, 106\u20137\n\nBiodiversity Conservation Alliance, 124\u201325, 159\n\nbiomass energy, 111\n\nbirds. _See_ avian impacts of wind farms, birds\n\nblades: composition of, 21\u201322, 31, 149; controlling pitch of, 86, 150; defects in and repair of, 31, 47\u201348, 91, 150; detection of, by bats, 129; length of, 14, 49, 74, 114; length of, related to power output, 114; lower and upper reach of, 118\u201319; manufacture of, 19\u201322, 40, 46, 60; shadow flicker created by, 155; sounds produced by, 153, 156\u201357; swept area of, 114; tip speed of, 153; transportation of, 74\u201378\n\nBLM. _See_ U.S. Bureau of Land Management (BLM)\n\nBloomstein, Phil, 151, 156\n\nBlue Canyon Wind Farm, Oklahoma, 8\n\nBode, Denise, 93\n\nBoeing, 31\n\nBonus company, 32\n\nBoots, Mick, 41\n\nBousbib, Ari, 48\n\nBP Wind Energy, 165\n\nBradford, Peter, 103\n\nBrand, Stewart, 101\n\nBraud, Rene, 128\u201329\n\nBriggs, John, 128\n\nBrightwell, Joe, 89\n\nBrown, Jerry, 28\u201330, 34, 101, 183\n\nBrownback, Sam, 148\n\nBrowning, Bill, 143\n\nBuffalo Mountain Wind Energy Center, Tennessee, 129\n\nBush, George W., viii, 124, 183\n\nButler, Barry, 89\n\nCalifornia: bird deaths from wind turbines in, 116\u201318, 121; Danish turbine exports to, 24\u201325, 32\u201335, 114; electric vehicle network planned in, 107; energy policy in, 28\u201335, 183\u201384; hydroelectricity in, 111; need for out-of-state renewable power in, 165; public utility wind power purchases in, 28, 30, 34; renewable electricity standards for, 111, 112, 183\u201384; solar energy in, 29, 30, 111, 165; tax credits for wind industry in, 30\u201331, 33\u201335; visual chaos of early wind farms in, 31, 34, 116, 134\u201335; wind resources and wind energy use in, vii, 2, 25\u201335, 183\u201384\n\nCalifornia Energy Commission, 30, 116\n\nCalifornia Office of Appropriate Technology, 29\n\ncapacity factor (wind power), 3, 97, 114. _See also_ installed capacity\n\ncap-and-trade, viii, 56, 169, 171\n\nCape Wind: Alliance to Protect Nantucket Sound and, 145\u201346; East Coast perspective on, 18; Ted Kennedy's opposition to, viii, 135\u201336, 145, 170; Bill Koch's opposition to, 145\u201346; Ed Markey and, 170; Ted Stevens and, 135\u201336; Don Young and, 135\u201336\n\nCapparella, Angelo, 117\u201319, 124\n\ncarbon dioxide (CO2) emissions. _See_ carbon emissions; greenhouse gas emissions\n\ncarbon emissions: cap-and-trade regimes for, viii, 56, 169, 171; CO2 gap between coal, gas, and wind power, 98\u2013101; proposed federal tax on, 171, 183; reduction of, competing with other values, 173; Timken opposition to U.S. regulation of, 52, 56\u201357. _See also_ greenhouse gas emissions\n\nCardinal Fastener, 39, 80, 184\n\nCargill, Alvin, 14\u201316, 79, 80\n\nCarter, Jimmy, vii, viii, 26\u201328, 34, 109\n\nCasselman Wind Power Project, Pennsylvania, 131\u201332\n\nCenter for Biological Diversity, 116\n\nCharmley, James E., 54\u201355\n\nChernobyl nuclear disaster, 63, 101\u20132, 104, 181\n\nChina: air pollution in, 61, 63; coal production and use in, 63, 65; economy of, 62, 64; electricity use in, 62, 68; energy planning and policy in, 62\u201366, 72; environmental movement in, 67; export of wind turbines from, 60, 68\u201372; foreign investment in wind farms in, 66\u201368; hydroelectricity in, 63\u201364; land acquisition for wind farms in, 65\u201368; manufacture of wind turbines in, 21, 55\u201356, 59\u201362, 68\u201372; market for U.S. wind energy equipment in, 55\u201356, 59; nuclear power in, 63; protectionist rules of, 59, 60, 68\u201369; state and private ownership of wind industry in, 66, 67\u201368; transmission needs in, 168, 170; Vestas factories in, 60\u201362, 69, 71; wind resources and wind energy use in, ix, 19, 59, 61\u201362, 64\u201365, 187; World Trade Organization and, 69\n\nChina Wind Power conference, 59, 60\n\nChinese Renewable Energy Industries Association, 60\n\nChokecherry\u2013Sierra Madre wind complex, 164\u201365\n\nChunhua Li, 70\n\nClark, Wilson, 29\n\nclimate change: China's contribution to, 63; consequences of, 132\u201333; fossil fuels and, viii; skepticism about, 90, 142, 146, 173; wind energy as response to, xi, 13, 18, 57, 184. _See also_ carbon emissions; greenhouse gas emissions\n\nClipper Wind: Cedar Rapids, Iowa, and, 39\u201343; employees and workplace morale at, 40\u201345; financial problems of, 39, 47\u201350, 57; lessons to be learned from, 49\u201352, 57\u201358; Liberty turbine manufactured by, 42, 44; market share of, 188; quality control issues at, 47\u201348; remote monitoring by, 44\u201346; supply chain for, 46\u201347; United Technologies Corporation bailout of, 48, 49; worker safety at, 45\u201346\n\nCloudCorp, 10, 17\u201318\n\nCloud County, Kansas. _See_ Kansas\n\nCloud County Community College, 17, 87\u201388, 91\u201392\n\ncoal: air pollution from, 51\u201352, 63, 99\u2013101; capping carbon emissions from, 56; in China, 63, 65; cost of coal plants, 113; in Denmark, 22\u201323; energy balance for, 100; failure to internalize environmental costs of, 51, 110; landscape impacts of, 52, 63, 123; mining hazards from, 51\u201352, 63; price uncertainty of, 112; U.S. policies favoring, 27, 33, 34, 51; U.S. reliance on, 56\u201357, 96; Wyoming politics regarding sage grouse and, 123\u201324\n\ncoal-bed methane, 108\n\nColorado: wind manufacturing in, 37, 40; wind resources and wind energy use in, 185\n\nCommonwealth Edison, 70\n\ncompressed-air energy storage. _See_ storage of electricity\n\nconservation easements, 126\u201327, 143\n\nConservation Law Foundation, viii, 135\n\nconstruction of wind farms, 13\u201316, 77, 78\u201384, 90\u201391, 94, 113\n\ncost: of coal, gas, nuclear, and solar energy vs. wind, 105, 112\u201313; of modernizing the national grid, 174\u201376; of wars, 180. _See also_ economics of wind power\n\nCrude Oil Windfall Profits Tax (1980), 28\n\nCrum, Lorrie, 53, 56\u201357\n\nCube rule of wind power, 114\n\nCulik, Martin, 81\n\ncut-in speed (wind turbines), 131\u201332\n\nDalian Heavy Industry Group, 71\n\nDanish International Development Agency, 35\n\nDanish Oil and Natural Gas. _See_ DONG Energy\n\nDanish Wind Industry Association, 32\n\nDefense Department. _See_ U.S. Department of Defense\n\nDenmark: Arab oil embargo and, 22\u201323; coal use in, 22\u201323; DONG (Danish Oil and Natural Gas) in, 105\u20136, 177; electric vehicles in, 105\u20137, 177; nuclear plants banned by, 22; offshore wind farms in, 36, 136\u201338; oil and natural gas in, 22, 23; policies promoting wind in, 23\u201325, 36; policies reducing greenhouse gas emissions in, 19, 23, 106\u20137; population of, 37; Ris\u00f8 National Laboratory in, 23, 32; wind energy use in, viii, ix, 19, 22\u201325, 36\u201337, 106, 136\u201338, 187; wind manufacturing in, 32\u201333, 35\u201337, 58, 188. _See also_ Vestas Wind Systems\n\nDeukmejian, George, 34\n\nDeWind, 188\n\ndirect-drive turbines, 55, 70. _See also_ geardriven turbines\n\nDiss, Frank and Sarah, 139\n\nDOE. _See_ U.S. Department of Energy (DOE)\n\n_Doing It Smart from the Start_ (Biodiversity Conservation Alliance), 124\u201325\n\nDONG Energy, 105\u20136, 177\n\nDonghai Bridge offshore wind farm, China, 65\n\nDunham, Richard, 89\u201390\n\neconomics of wind power: benefits to rural communities, 3\u201310, 15\u201318, 43, 44, 90\u201391; California tax credits, 30, 33; feed-in tariffs (Europe), 35\u201336; capital outlays for wind power, 113; cost of turbines, 11\u201312, 113; life-cycle costs of wind compared to other power sources, 113; payments to landowners, ix, xi, 3\u20137, 15\u201316, 139, 145; productivity gains through technology changes, 113\u201315; utility purchases of wind power, 11, 12\u201313, 30, 34, 70, 161\u201362; wholesale price of wind power, 112. _See also_ employment related to wind power; wind power incentives, federal; wind power incentives, state\n\nEDP Renewables, 12\n\nelectricity: China's use of, 62, 68; from coal, 23, 56\u201357, 96; Denmark's shift to wind for, viii, ix, 19, 22\u201325, 36\u201337, 106, 136\u201338, 187; global supply of, by wind, 187\u201388; from hydroelectric dams, 63\u201364, 96, 111, 176, 182; from natural gas, 96, 107; number of households supplied with, from U.S. wind farms, ix, xii, 3, 81; present and potential price of, 112\u201315; purchase of, from U.S. wind farms, 11\u201313, 27\u201328, 30, 34, 35\u201336, 70, 111, 112, 161\u201362, 165, 183\u201384; renewable standards for, by states, 111, 112, 165, 183; supplied by wind (U.S.), ix, x, 1, 2, 94, 96, 97\u201398, 110, 112, 181, 185\u201386; Timken Company's use of, 56\u201357\n\nelectric vehicles, 106\u20137, 177, 182\n\nElk River Wind Power Project, Kansas, 148, 149, 161, 162\n\nEmpire District Electric, 12, 161\u201362\n\nemployment related to wind power: in construction, 13\u201316, 77, 78\u201384, 90\u201391, 94; \"indirect\" and \"induced\" jobs, 16, 95; in manufacturing, 20, 35, 37, 39\u201340, 51, 54, 94; in operations and maintenance (O&M), 84\u201387, 90\u201391; total jobs (present and projected), 37, 51, 94\u201395; training programs for technicians, ix, 17, 41, 44, 87\u201394; women in, 13\u201315, 17, 79, 82, 91, 92\u201394\n\nEndangered Species Act. _See_ avian impacts of wind farms, birds\n\nEnel, 150\n\nEnergias de Portugal (EDP), 12\n\nenergy balance, 100\n\nEnergy Department. _See_ U.S. Department of Energy\n\nEnergy Policy Act (1992), 50\n\nEnergy Policy Act (2005), 167, 172\n\nEnergy Tax Act (1978), 28\n\nEngel, Ditlev, 37\n\nEngelder, Carole, 13\u201315, 79, 91\n\nenvironmental issues. _See_ air pollution; avian impacts of wind farms, bats; avian impacts of wind farms, birds; carbon emissions; climate change; greenhouse gas emissions; noise from wind farms; visual appearance of wind farms; wildlife impacts of wind farms\n\nEnvironmental Law and Policy Center, 168, 169, 173\n\nEnvironmental Protection Agency. _See_ U.S. Environmental Protection Agency (EPA)\n\nE.ON Climate and Renewables, 137\n\nEPA. _See_ U.S. Environmental Protection Agency (EPA)\n\nEuropean Union: greenhouse gas reduction goal, 106\n\nFAG, 55\n\nfarms and farming: in China, 66; economic strains of, ix, 3, 4, 6, 90, 143\u201344; exodus of young people from, ix, 3\u20134, 7; in Illinois, 118; in Iowa, 44, 140; in Kansas, ix, 3\u20135, 9, 12\u201313, 15\u201318, 140\u201349; roots of Vestas turbines in equipment for, 20, 32; water-pumping windmills and, 1\u20132; wildlife impacts of, 122; wind farms as income supplement to, ix, xi, 3\u20137, 15\u201316, 44, 139, 145, 148; and working landscape of farms and ranches compatible with wind, 5, 9, 18, 118, 138\u201340, 141, 162\u201364. _See also_ landowners\n\nFederal Aviation Administration, 10, 119\n\nFederal Communications Commission, 11\n\nfederal energy policy: of George W. Bush, viii, 94\u201395; of Jimmy Carter, vii, viii, 26\u201328, 34, 109; on coal, 27, 33, 34, 51; of Richard Nixon, 25\u201326, 109; of Barack Obama, 39, 50\u201351, 56, 103, 124, 172, 183; of Ronald Reagan, vii, 34. _See also_ coal; natural gas; nuclear energy; oil; solar energy; wind power incentives, federal\n\nFederal Energy Regulatory Commission (FERC), 110, 169\u201374, 181\n\nfeed-in tariffs (Europe), 35\u201336\n\nFERC. _See_ Federal Energy Regulatory Commission (FERC)\n\nFerrell, Garland P. \"Pete,\" III, 143\u201349, 157\u201358, 161\n\nFish and Wildlife Service. _See_ U.S. Fish and Wildlife Service\n\nFitz, Rusty, 83\n\nFlint Hills, Kansas, x, 8\u20139, 127, 140\u201349\n\nflywheel power storage. _See_ storage of electricity\n\nFowler Ridge Wind Farm, Indiana, 45\n\nFrance: nuclear fuel reprocessing in, 103; wind power use in, 187\n\nFreudenthal, Dave, 123, 124\n\nFriedman, Thomas, 72\u201373\n\nFuhrl\u00e4nder, 71\n\nFukushima Daiichi, Japan, reactor disaster, 52, 63, 102, 104, 180\u201381\n\nFurman, Don, 131\u201332, 172\u201373\n\nGamesa, 35, 37, 69, 188\n\nGarber, Victor E., 164\n\nGardiner, Robert, 158\n\nGates, Robert, 48, 51, 52\n\ngear-driven turbines: assembly of gearboxes in, 41\u201344, 46, 51; contrast with direct-drive turbines, 55, 70; defects in and repair of, 31, 47\u201348; monitoring and maintenance of, 45\u201347, 86\u201387, 91; quality control in manufacture of, 46\u201347, 49; rotor-to-generator speed ratio in, 45, 53, 55; supply chain for, 46, 51\n\nGeneral Electric (GE), 31, 39, 40, 45, 46, 59, 69, 71, 86, 151, 188\n\ngeothermal energy, 28, 111\n\nGermany: compressed-air energy storage in, 178; feed-in tariff and other policies favoring solar and wind in, 35\u201336; manufacture of wind turbines in, 35, 58, 188; wind energy use in, viii, 36, 187\n\nGE Wind. _See_ General Electric (GE)\n\nGipe, Paul, 30, 134\n\nGlass, Tyler, 44\n\nglobal warming. _See_ climate change\n\nGlobal Wind Energy Council, 60\n\nGoldman Sachs, 12, 49\n\nGoldwind Science & Technology. _See_ Xinjiang Goldwind Science & Technology\n\nGordon, Jim, 134\n\nGore, Al, 142\n\nGraf, Kristen, 93\n\nGraham, Bruce, 17, 87\u201388, 91\n\nGraham, Michelle, 17, 91\n\nGrandpa's Knob, Vermont, 149\u201351\n\nGrand Ridge Wind Farm, Illinois, 79, 85\u201387, 90\u201391, 138\u201340\n\nGrassley, Chuck, 50\n\nGray County Wind Energy Center, Kansas, 3, 146\n\nGreat Lakes: manufacturing region for wind, 39; shipment of wind equipment via, 77\u201378; wind resources of, 57, 98\n\ngreenhouse gas emissions: California's agenda on, 184; coal's contribution to Denmark's, 23; Denmark's commitment to reducing, 19, 105\u20136; European Union commitment to reducing, 106; nuclear power and, 101, 104; Regional Greenhouse Gas Initiative (U.S. Northeast), viii, 169; U.S. as world leader in per capita, 180; U.S. resistance to international curbs on, viii, 19, 146, 180. _See also_ carbon emissions; climate change\n\nGreenlight Energy Resources, 146, 147\n\nGreenpeace, 60\n\nGreen Power Express, 166, 168\u201369\n\ngreen power option. _See_ wind power incentives, state\n\nGrumman Aerospace, 31\n\nGuo, Wilson, 66\n\nGuo Zheng, 66\n\nHansen, H. S., 20\n\nHansen, Peder, 20\n\nHartman, Adam, 79\n\nHarvard University, 62\n\nHawaii, 107, 186\n\nHaxgart, Bjarne, 137\n\nHayes, Denis, 27\n\nHemami, Ahmad, 88\u201389\n\nHigh Plains Express, 166\n\nHinkels and McCoy, 79\u201380, 83\n\nHorizon Wind Energy, 12\u201316, 79, 81, 85, 113, 126, 161, 163, 165\n\n_Hot, Flat, and Crowded_ (Friedman), 72\u201373\n\nhouseholds supplied by wind farms (U.S.), ix, xii, 3, 81\n\nHoy, Jim, 8\n\nhubs: attachment of blades to, 49, 82; height of, 49, 114; manufacture of, 42, 46, 61; raising of, 82\u201383; rotational speed of, 45; transport of, 77\n\nhydropower: in China, 63\u201364; power storage using, 176, 182; in U.S., 96, 111\n\nIberdrola Renewables, 120, 131\u201332, 148\n\nIllinois: Chinese-owned wind farm in, 70; eagle flyway near White Oak Wind Energy Center, 117\u201318; employment in wind manufacturing in, 40; Grand Ridge Wind Farm in, 85\u201387, 90\u201391, 138\u201340; nuclear energy in, 138; renewable electricity standards in, 111; wind resources and wind energy use in, 57 Illinois State University, 117\u201318\n\nIndependence Wind, 158\n\nIndia: wind energy use in, 35, 187; wind turbine manufacturing in, 37\u201338, 188\n\nIndiana: Fowler Ridge Wind Farm in, 45; Meadow Lake Farm in, 74\u201384, 90; wind resources and wind energy use in, 57, 81\n\nIndiana bats. _See_ avian impacts of wind farms, bats\n\nIndustrial Wind Action Group, 154\n\ninfrasound. _See_ noise\n\ninstalled capacity (wind power): in China, 66; defined, 3; in Denmark, 36, 106, 136\u201337; in Germany, 36; global, 189; of top ten countries, 187\u201388; of states, 185\u201386; of turbines, 13, 31, 99; U.S. total, 36, 181, 187; of U.S. wind farms, xii, 3, 92, 130, 149, 164\n\nInterior Department. _See_ U.S. Interior Department\n\nInternational Atomic Energy Agency, 103\n\nInternational Organization for Standardization (ISO), 99\n\nInvenergy, 79, 85\u201386, 117\u201318, 130, 163\n\ninvestment tax credits. _See_ wind power incentives, federal; wind power incentives, state\n\nIowa: factory closures in, 40, 41, 42\u201343; flooding in, 40\u201341; Iowa Lakes Community College in, 88\u201390, 92; population of, 39; wind resources and wind energy use in, 185. _See also_ Clipper Wind\n\nIowa Lakes Community College, 88\u201390, 92\n\nISO. _See_ International Organization for Standardization (ISO)\n\nITC Transmission, 168\n\nJapan: Fukushima Daiichi nuclear disaster in, 52, 63, 102, 104, 180\u201381\n\nJessen, Leo, 85\u201387, 90\u201391\n\nJiquan Wind Farm Base, China, 65\n\njobs. _See_ employment related to wind power\n\nJohnson, Paul, 90\n\nKansas: Cloud County Community College in, 17, 87\u201388, 91\u201392; early uses of wind power in, 1\u20132; economic benefits of wind power for Cloud County, 16\u201318; Flint Hills wind controversy in, x, 8\u20139, 140\u201349; land deals for wind farms in, 5\u201316, 144\u201346; prairie chickens in, 125, 128, 142; ranching practices in, 127\u201328, 143\u201344; whooping cranes in, 120; wind energy policy in, 8, 111, 146\u201348; wind farms in, ix, 3\u201318, 64, 79, 80, 146\u201349, 161; wind resources and wind energy use in, 1, 9, 97, 144\u201345, 149, 185\n\nKansas Livestock Association, 126, 127\n\nKansas State University, 93, 126, 128\n\nKaplan, Matthew, 71\u201372\n\nKennedy, Edward M. \"Ted,\" viii, 135\u201336, 145, 170\n\nKerlinger, Paul, 119\n\nKlataske, Ron, 142\u201343\n\nKoch, Bill, 145\u201346, 183\u201384\n\nKoch, Charles and David, 146, 183\u201384\n\nKocher, Kurt, and family, 3\u20135, 12\u201313\n\nKoch Industries, 145\u201346\n\nKonza Prairie Biological Station, Kansas, 128\n\nKosciuch, Karl, 120\n\nKruse, Peter Wenzel, 36\u201337, 121, 122\n\nKyoto Protocol, 180\n\nLahti, Tom, 167\n\nLalley, Matt, 43, 44\n\nlandowners: income from wind farms, ix, xi, 3\u20137, 15\u201316, 139, 145, 148; negotiations with wind developers, 5\u201316, 144\u201346, 150, 163\u201364; perspectives on wind farm noise, 5, 151\u201359; opposition to wind farms among, 8\u20139, 140\u201349, 150\u201359\n\nland used by wind farms: acreage leased or optioned to wind developers, 6, 10, 12\u201313, 15, 145, 150, 163\u201364; footprint of wind farms, 81, 164; \"go-zones\" in Wyoming, 125; land needed per turbine, 5; setbacks from residential buildings, 157\u201359; spacing between turbines, 118\u201319\n\nLaSalle County Nuclear Station, 138\n\nLawrence Berkeley National Laboratory, 112\n\nlayout of wind farms. _See_ visual appearance of wind farms\n\nLazard Ltd., 113\n\nLearner, Howard, 168\u201369, 173\n\nLeventhall, Geoff, 155\n\nLiberty turbines. _See_ Clipper Wind\n\nLi Fan, 70\n\nlife cycle environmental assessment, wind and other power sources, 98\u2013101\n\nLi Junfeng, 60\n\nLone Star Transportation, 74\n\nLovins, Amory, 29, 30\n\nLowell, Kirk, 10, 17\u201318\n\nlow-frequency sound. _See_ noise\n\nLoyd, Bob, 41\u201345, 47, 89\n\nMackinaw Ecosystem Partnership, 117\n\nMaine: Governor's Task Force on Wind Power, 151, 157; wind farm noise in, 151, 154, 156\u201358; wind resources and wind energy use in, 186\n\nmaintenance of wind turbines, 84\u201387, 90\u201391\n\nManes, Stephanie, 126\u201328\n\nmanufacture of wind turbines: in China, 21, 55\u201356, 59\u201362, 68\u201373; in Denmark, ix, 13\u201314, 19\u201325, 32\u201333, 35\u201337, 58; employment in, 20, 35, 37, 39\u201340, 94; in Germany, 35, 58; in India, 38; leading manufacturers serving U.S. market, 188; quality control in, 31\u201333, 46\u201349, 52\u201355; smaller companies in, 49\u201350, 57; in Spain, 35, 37, 58, 69; in U.S., 31\u201332, 37\u201358, 188; worker safety and, 45\u201346; workplace morale and, 44\u201345. _See also_ manufacturers of wind turbine components; manufacturers of wind turbines\n\nmanufacturers of wind turbine components. _See_ American Superconductor; Cardinal Fastener; FAG; SKF; Timken; TPI Composites; Trinity Structural Towers\n\nmanufacturers of wind turbines. _See_ Acciona; Bonus; Clipper Wind; DeWind; Fuhrl\u00e4nder; Gamesa; General Electric (GE); Mitsubishi; NEG Micon; Nordtank; REpower; Siemens; Sinovel; Suzlon; United Technologies Corporation (UTC); Vestas Wind Systems; XEMC Wind-power; Xinjiang Goldwind Science & Technology\n\nMaples, Steve, 80\u201384, 90, 91\n\nMarkey, Edward J. \"Ed,\" 169\u201371\n\nMars Hill wind farm, Maine, 151, 154, 157\n\nMason, Ray, 5\u20137, 17\n\nMassachusetts: flywheel power storage in, 177; Ed Markey on wind, 169\u201371; wind resources and wind energy use in, 97, 186. _See also_ Cape Wind\n\nMcCunney, Robert, 152\u201354\n\nMcDonnell Douglas, 31\n\nMcElwee, Charles R., II, 67\n\nMeader, Mark, 40\u201341, 43\n\nMeadow Lake Farm, Indiana: construction of, 77, 78\u201384, 90, 91; delivery of turbine parts to, 74\u201378; generating capacity of, 81; land used by, 81\n\nMeridian Way Wind Farm, Kansas: conservation easements and, 126\u201327, 143; construction of, 13\u201316, 78\u201380; cost of, 11\u201312; economic benefits of, for community, 16\u201318; financial backing for, 12, 49; generating capacity of, ix, 64; income for landowners from, 4\u20137, 15; jobs created by, 16\u201317; land acquisition for, 9\u201316, 127; location of, 3; naming of, 11; prairie chickens at, 126\u201328; purchase of power from, 11, 12\u201313, 161\u201362; transmission lines from, 161\u201362; Vestas wind turbines at, xii, 3, 4\u20136, 14, 84\u201385, 99\u2013100\n\nmeteorological towers, 9, 10, 124, 145\u201346\n\nmethane emissions: wind compared to other power sources, 101\n\nMiddle East: Arab oil embargo, vii, 22, 25, 26, 109, 180; gas resources in, 109; oil from, 104, 105; political changes in, 180\n\nMiller, Chris, 173\u201374\n\nMitsubishi, 38, 188\n\nMolvar, Erik, 124\u201325\n\nMortensen, Lone, 19\u201322\n\nnacelle: described, 14, 42, 44; failed climb attempt, 84; installation of, 80, 81, 82, 83; manufacturing of, 37, 42, 44, 61, 70; transport of, 74; working conditions in, 86, 88\u201389, 92\n\nNASA, 31\n\nNational Academy of Sciences, 102\n\nNational Health and Medical Research Council, 154\n\nNational Historic Trails, 167\n\nNational Renewable Energy Laboratory (NREL), 96\u201399, 119, 144, 162, 163\n\nNational Research Council, 158\n\nnatural gas: changing federal policies regarding use of, 33\u201334; China and, 63; environmental impacts of, 107\u201310, 122; federal land leases for extraction of, 167; federal subsidies for, 51; FERC jurisdiction over natural gas pipelines, 172; levelized cost of combined-cycle gas plants, 113; life-cycle air emissions of electricity generated by, 99\u2013101; ongoing role for, 181; prospects for shifting to wind power from, 109\u201310; unconventional recovery methods for (hydraulic fracturing), 108\u20139; U.S. and foreign supplies of, 107\u20138; use of, for electricity and household heating, 96, 107; volatile price of, 112; Wyoming sage grouse politics and, 122\u201325\n\nNature Conservancy, 8, 143\n\nNEG Micon, 20\n\nNevada: proposed nuclear waste repository in, 103; transmission line siting in, 165, 167; wind resources and wind energy use in, 97, 186\n\nNewbold, Clint, 80, 82\n\nNew York: wind resources and wind energy use in, 57, 81, 185\n\nNew Zealand, 35\n\nNextEra Energy Resources, 117\n\nNissenbaum, Michael, 152, 153\u201354, 157, 159\n\nnitrogen oxide emissions: wind compared to other power sources, 101\n\nNi Xiangyu, 61\n\nNixon, Richard, 25\u201326, 109, 184\n\nnoise from wind farms: annoyance thresholds for, 153\u201354; distance from turbines and, 151\u201359; downwind and upwind rotors and, 156; health concerns about, 152\u201359; setbacks as safeguard against, 157\u201359; setting noise limits for, 155\u201360; sound frequencies and, 153\n\nNordtank company, 32, 35\n\nNorth China Electric Power University, 66\n\nNorth Dakota: Antelope Valley Station and coal use in, 168\u201369; coal piggybacking on Green Power Express in, 168\u201369; wind resources and wind energy use in, 57, 185\n\nNREL. _See_ National Renewable Energy Laboratory (NREL)\n\nnuclear energy: \"carbon neutrality\" and \"clean energy\" claims for, 101, 104, 171; Carter's policy on, 27, 33; in China, 63; cost of, 113; decommissioning of, 103; Denmark's ban on, 22; disasters involving, 52, 63, 101\u20132, 104, 115, 180\u201381; in France, 103; in Illinois, 138; liability limits and other federal support for, 52, 57; Nixon's policy on, 25; number of nuclear power plants in U.S., 52; poorly matched to wind power balancing needs, 176; Reagan's policy on, vii, 34; share of U.S. electricity production, 96; waste disposal for, 52, 103\u20134; weapons proliferation risk of, 102\u20133\n\nObama, Barack, 39, 50\u201351, 56, 69, 103, 124, 172, 183\n\n\"Obama Bolt,\" 39, 80\n\noffshore wind power: Cape Wind, viii, 18, 134\u201336, 145\u201346, 170, 181; China and, 65; Denmark and, 36, 136\u201338; Great Lakes and, 57; porpoises and, 137; U.S. potential for, 98; visual appearance of, 134\u201338\n\nOhio: wind equipment manufacturing in, 51, 52\u201357; wind resources and wind energy use in, 57, 186\n\noil: Arab embargo of, vii, 22, 25, 26, 109, 180; in Denmark, 22, 23; drilling for, on federal land, 167; electric vehicles to reduce reliance on, 104\u20137; lobbying by leaders of, to block climate change policies, 145\u201346; Nixon's policy on, 25\u201326, 109; price history of, 33, 35, 112; Reagan's policy on, 34; subsidies for, 51; U.S. reliance on foreign sources of, 18, 26, 104\u20137, 180; U.S. consumption of, 26, 96, 104, 105, 107\u201310; wars fought over, xi, 18, 110\u201311, 180; Wyoming politics regarding sage grouse protection and, 123\u201325\n\nOklahoma: Blue Canyon Wind Farm in, 8; whooping crane migration through, 120; wind resources and wind energy use in, 185\n\nOPEC, 109, 180. _See also_ Arab oil embargo\n\noperations and maintenance (O&M) jobs, 84\u201387, 90\u201391\n\nopposition to wind farms: in California, 116\u201317, 134; in Flint Hills, x, 8\u20139, 140\u201349; in Vermont, 150\u201359; in Wyoming, 122\u201325. _See also_ avian impacts of wind farms, bats; avian impacts of wind farms, birds; Cape Wind; noise from wind farms; visual appearance of wind farms\n\nOregon: wind resources and wind energy use in,183, 185\n\nOrganization of Petroleum Exporting Countries (OPEC), 109, 180. _See also_ Arab oil embargo\n\nOxbow Power Corporation, 145\u201346\n\nPacific Gas & Electric, 30, 34\n\nParsons, Stan, 144\n\nPathfinder Renewable Energy LLC, 164\u201365\n\nPederson, Knud, 105\u20136, 177\n\nPe\u00f1ascal Wind Farm, Texas, 120\n\nPennsylvania: bat protection efforts at Casselman Wind Power Project in, 131\u201332; wind resources and wind energy use in, 57, 185\n\nPercy, Charles \"Chuck,\" vii\n\nPernal, Mary, 151\n\nPeyton, Joel, 44, 45\u201346\n\nPiedmont Environmental Council, Virginia, 172, 173\u201374\n\nPierpont, Nina, 154\u201355, 157, 159\n\nPJM Interconnection, 174\u201375\n\nPowerplant and Industrial Fuel Use Act (1978), 33\n\nPPM Energy, 147\u201348\n\nprairie chickens. _See_ avian impacts of wind farms, birds\n\nPrice Anderson Act, 52\n\nproduction tax credit (PTC). _See_ wind power incentives, federal\n\nProtect the Flint Hills, 148\n\nPTC. _See_ production tax credit\n\nPublic Utility Regulatory Policies Act (PURPA), 27\u201328\n\nPutnam, Palmer Cosslett, 149\u201350\n\nranches and ranching. _See_ farms and farming\n\nRanchland Trust of Kansas, 126\u201327, 143\n\nReagan, Ronald, vii, 34, 184\n\nREAL. _See_ Renewable Energy Alliance of Landowners (REAL)\n\nRegional Greenhouse Gas Initiative (RGGI), viii, 169\n\nReisky, Sandy, 146\u201348\n\nRen Dongming, 64\u201365\n\nrenewable electricity standards: adopted by states, 111, 112, 165, 183; proposed federal, 111\n\nRenewable Energy Alliance of Landowners (REAL), 163\u201364\n\nRENEW Wisconsin, 158\n\nREpower, 188\n\nresearch and development (R&D): on power storage, 176\u201378; on wind turbines, 23, 31\u201335\n\nRevere, Paul, 60\n\nRGGI. _See_ Regional Greenhouse Gas Initiative (RGGI)\n\nRis\u00f8 National Laboratory, Denmark, 23, 32\n\nRobel, Robert, 125\u201328\n\nRoberts, Jim, 7\u201316, 66, 127\n\nR\u00f8dsand Offshore Wind Farm, Denmark, 136\u201338\n\nRoggenkamp, Loma, 92\u201393\n\nRomney, Mitt, 135, 136\n\nRoosevelt, Franklin D., x\n\nrotors: axis of, 31; cut-in speed of, 131\u201332; expanding diameter of, 114; monitoring speed of, 86; number of blades on, 2, 14, 20, 32, 80, 82\u201383, 134; rotational speed of, 45, 119, 131\u201332, swept area of, 114; tip speeds of, 153\n\nRoyko, Mike, 29\n\nRutland Regional Medical Center, 152\n\nsafety. _See_ worker safety\n\nsage grouse. _See_ avian impacts of wind farms, birds\n\nSalazar, Ken, 124\n\nSammons Enterprises, 164\n\nSan Gorgonio Pass, California, 30, 134\n\nSavory, Allan, 144\n\nSawyer, Steve, 60\n\nSchwarzenegger, Arnold, 183\n\nSebelius, Kathleen, 8, 146\u201348\n\nsetbacks for wind turbines, from residential buildings, 157\u201359\n\n\"shadow flicker\" from wind turbines, 155\n\nShultz, George, 184\n\nSiemens\/Siemens Energy, 38, 40, 92, 188\n\nSinovel, 71\n\nsiting of wind farms: in China, 66\u201367; in general, 159\u201360; in Illinois, 117\u201318, 138\u201340; Indiana, 75; in Kansas, 5, 8\u20139, 127, 140\u201349; in Vermont, 149\u201359; in Wyoming, 163\u201364. _See also_ avian impacts of wind farms, bats; avian impacts of wind farms, birds; Cape Wind; noise impacts of wind farms; visual appearance of wind farms; wildlife, impacts from wind farms\n\nSKF, 55\n\nsmart grid. _See_ transmission of electricity\n\n_Soft Energy Paths_ (Lovins), 29\n\nsolar energy: in California, 29, 30, 111, 165; Carter's policy on, 26, 27, 34; cost of, 113, 182; definition of, 27; in Germany, 35; Nixon's policy on, 25; solar thermal power, 113; statistics on, 96; tax credit for, 28; White House solar water-heating panels, viii, 27, 34\n\nSolar Energy Industries Association, 172\n\nSolar Energy Research Institute, 27\n\nSourceWatch, 168\n\nSouthern California Edison, 30, 34\n\nSouthwest Power Pool, 162, 175\n\nSpain: wind energy in, viii, 36, 187; wind industry in, 12, 35, 37, 40, 69, 120, 188\n\nSporer, Bonnie, 15\u201316\n\nStark, Pete, 30\n\nStevens, Ted, 135\u201336\n\nSt. Nikola Kavarna wind farm, Bulgaria, 120\n\nstorage of electricity: batteries and, 178\u201379, 182; compressed air and, 178; electric vehicles and, 106\u20137, 177, 182; flywheels and, 177\u201379, 182; hydro dams and, 176, 182; smart-metered buildings and appliances and, 177\n\nStovall, Bill, 74\u201378\n\nsulfur dioxide emissions, 100\u2013101\n\nSundgren, Jacque and Steve, 143\n\nSupreme Court, U.S. _See_ U.S. Supreme Court\n\nSuzlon, 188\n\nswept area. _See_ blades; rotors\n\nTehachapi Pass, California, 30, 31, 134\n\nTennessee Valley Authority, 129\n\nTexas: drilling for natural gas in, 108; employment in wind manufacturing in, 40; greater prairie chicken in, 125; whooping cranes in, 120; wind energy use in, 183; wind farms in, 120; wind resources and wind energy use in, 185\n\nThree Mile Island nuclear accident, 63, 101, 104, 181\n\nTianrun New Energy Investment Company, Ltd., 67, 68\n\nTiedeman, Mary, 44\n\nTierney, Susan, 175\u201376\n\nTimken, Henry, 53\n\nTimken, Ward J. \"Tim,\" Jr., 56\n\nTimken Company, 52\u201357, 59, 69, 184\n\ntip speed. _See_ blades\n\nTitus, Roger W., 130\u201331\n\nTodd, Wendy, 151, 156, 157\n\nTop of the World Windpower Project, Wyoming, 92\n\ntowers: buckling of, 31; climbing of, 84\u201386; erection of, 14, 78\u201382, 91; height and weight of, ix, 14, 79, 149; height of, related to output, 114; manufacture of, 20, 37, 40, 46, 172; recycling of, 100; transport of, 14, 74, 77\u201378; visual appearance of, 114, 134; working conditions in, 84, 86, 91\u201392\n\nTPI Composites, 40\n\ntraining programs for wind energy technicians, 17, 87\u201393\n\nTransCanada's Zephyr line, 166\n\ntransmission of electricity: to California, 165; in China, 168, 170; citizen groups and, 168\u201374; coal piggybacking and, 171; costs of, and cost-sharing formulas for new transmission lines, 174\u201376; East Coast and, 169\u201370; in Europe, 170; federal role in, 166\u201374; \"smart grid\" management tools for, x, 176\u201378, 182; in Kansas, 161\u201362; from Midwest, 166, 168\u201370, 174\u201375; modernization of, 171\u201379, 182; in Nevada, 165, 167; and siting of new interstate transmission lines, 161\u201374; West-Wide Energy Corridor, 167; for wind energy, 161\u201379; in Wyoming, 164\u201367. _See also_ storage of electricity\n\ntransportation: cost of, for turbine transport, 105; electric vehicles for wind power storage and use, 106\u20137, 177, 182; energy consumed in U.S. for, 105; trucking of turbine components, 14, 74\u201377; use of rail and ships for turbine transport, 77\u201378\n\nTransWest Express, 165, 166, 167\n\nTrinity Structural Towers, 40\n\nTsinghua University, 62\n\nturbines. _See_ wind turbines\n\n24M, 178, 179\n\nUnion of Concerned Scientists, 169\n\nUnion Pacific Railroad, 78, 166\n\nUnited Kingdom: wind resources and wind energy use in, 35, 187, 188\n\nUnited Steelworkers, 69\n\nUnited Technologies Corporation (UTC), 48\n\nUniversity of Georgia, 122\n\nUniversity of Iowa, 89\n\nUPC Renewables, 66, 67, 68\n\nU.S. Bureau of Land Management (BLM), 166\u201367\n\nUSDA. _See_ U.S. Department of Agriculture\n\nU.S. Department of Agriculture, 15\n\nU.S. Department of Defense, 10\u201311\n\nU.S. Department of Energy (DOE), x, 94\u201395, 100, 109, 110, 121, 160, 172, 175, 181\n\nU.S. Environmental Protection Agency (EPA), 156\n\nU.S. Fish and Wildlife Service, 121, 122, 131, 133\n\nU.S. Interior Department, 122, 124, 166\n\nU.S. Nuclear Regulatory Commission, 103\n\nU.S. Supreme Court, 28, 172\n\nutility purchases of wind power: Commonwealth Edison, in Illinois, 70; Empire District Electric and Westar, in Kansas, 11\u201313, 161\u201362; feed-in tariffs as catalysts to, in Europe, 35\u201336; Nevada and California utilities as prime markets for, 30, 34, 165; PURPA's creation of market for, 27\u201328; renewable electricity standards as catalysts to, 111, 112, 183\u201384\n\nVan Beusekom, Justin, 84\u201385\n\nVan Est, Rinie, 32\n\nvariability of wind, and need for balancing, 105\u20137, 176\u201378, 181, 182\n\nVattenfall, 99\n\nVermont: Grandpa's Knob wind turbine in, 149\u201351; opposition to wind farms in, 150\u201359; Taconic Mountains in, x; wind resources and wind energy use in, 186\n\nVermont Community Wind, 150\u201352\n\nVestas Wind Systems: blade manufacture in Lem, Denmark, 19\u201322; in China, 60\u201362, 69, 71; and climb attempt, 84; compared with Clipper Wind, 45, 46; and construction of Meadow Lake Wind Farm, Illinois, 74, 77, 80; diagram and dimensions of turbines made by, xii, 14, 74, 84\u201385; employment by, 20, 35, 37; factory locations of, 20\u201321, 35, 37, 60\u201362; global headquarters of, 36\u201337; history of, 20, 32\u201336; life-cycle analyses of wind turbines by, 99\u2013100; markets for, 24\u201325, 35\u201336, 188; revenues of, 20; as world's top-ranked turbine supplier, ix, 20\n\nVickerman, Michael, 158\n\nVirginia: as home of Piedmont Environmental Council, 172\u201374; wind resources and wind energy use in, 186\n\nVisceral Vibratory Vestibular Disturbance, 155\n\nvisual appearance of wind farms: aviation lights on turbines, 137\u201338; Cape Wind and, 134\u201336; Flint Hills objections to, 8, 141\u201343; Illinois farm families' divergent perspectives on, 138\u201340; increasing size of turbines, 113\u201315, 119, 135; Kurt Kocher and, 5; layout of turbines and, 15, 81, 116, 118\u201319, 134\u201335, 138; National Historic Trails and, 167; R\u00f8dsand Offshore Wind Farm (Denmark) and, 136\u201338; spacing between turbines, 118\u201319, 138; turbine design changes and, 31, 114, 134\u201335, 138; visual chaos at older California wind farms, 31, 34, 116, 134\u201335; water-pumping windmills, 1\u20132\n\nWalker, Scott, 158\u201359\n\nWalters, Jeanna, 93\u201394\n\nwater-pumping windmills, 1\u20132\n\nWaxman, Henry, 171\n\nWellinghoff, Jon, 110, 170\u201371, 177, 181\n\nWennberg, Jeff, 152\n\nWestar Energy, 12, 13, 161\n\nWestinghouse, 31\n\nWest Virginia: bat protection effort in, 130\u201331; wind resources and wind energy use in, 186\n\nWest-Wide Energy Corridor, 167\n\nWheatley, Dave, 41, 43\n\nWhite, Sylvia, 134\n\nWhite Oak Wind Energy Center, Illinois, 117\u201318\n\nwhite storks. _See_ avian impacts of wind farms, birds\n\nWhitton, Bob, 162\u201365\n\nWHO. _See_ World Health Organization (WHO)\n\n_Whole Earth Catalog_ , 101\n\nwhooping cranes. _See_ avian impacts of wind farms, birds\n\nWidman, Bob and Ruth, 139\n\nWilder, Throop, 178\n\nwildlife, impacts from wind farms: on big-game species, 125; offshore wind farms, 137; and pacing of wind farm development, 118. _See also_ avian impacts of wind farms, bats; avian impacts of wind farms, birds\n\nWind, Tom, 178\n\nWind and Prairie Task Force (Kansas), 8\n\nwind chargers, 2\n\nwind farm developers. _See_ Beech Ridge Energy LLP; BP Wind Energy; EDP Renewables; Enel; E.ON Climate and Renewables; Greenlight Energy Resources; Horizon Wind Energy; Iberdrola Renewables; Independence Wind; Invenergy; NextEra Energy Resources; Oxbow Power Corporation; Pathfinder Renewable Energy; PPM Energy; Sammons Enterprises; Tianrun New Energy Investment Company, Ltd.; Vermont Community Wind\n\nwind farms (non\u2013U.S.). _See_ Donghai Bridge offshore wind farm, China; Jiquan Wind Farm Base, China; R\u00f8dsand Offshore Wind Farm, Denmark; St. Nikola Kaverna Wind Farm, Bulgaria\n\nwind farms (U.S.). _See_ Altamont Pass, California; Beaver Ridge Wind Project, Maine; Beech Ridge Wind Farm, West Virginia; Blue Canyon Wind Farm, Oklahoma; Buffalo Mountain Wind Energy Center, Tennessee; Cape Wind, Massachusetts; Casselman Wind Power Project, Pennsylvania; Chokecherry\u2013 Sierra Madre wind complex, Wyoming; Elk River Wind Power Project, Kansas; Fowler Ridge Wind Farm, Indiana; Grandpa's Knob, Vermont; Grand Ridge Wind Farm, Illinois; Gray County Wind Energy Center, Kansas; Mars Hill wind farm, Maine; Meadow Lake Wind Farm, Indiana; Meridian Way Wind Farm, Kansas; Pe\u00f1ascal Wind Farm, Texas; San Gorgonio Pass, California; Tehachapi Pass, California; Top of the World Windpower Project, Wyoming; White Oak Wind Energy Center, Illinois\n\nwind power capacity: as affected by wind speed (\"cube rule\"), 114; in California, 30\u201333, 185; in China, 19, 59\u201373, 187; as a factor of blade length, 114; global installed wind power capacity (1996\u20132010), 189; in Kansas, 1\u201318, 97, 144\u201345, 149, 185; in Massachusetts, 97, 186; ranking of installed wind power capacity, by country, 187\u201388; ranking of U.S. onshore wind power capacity (present and potential), by state, 185\u201386; in U.S., 96\u201398, 125, 185\u201386; U.S. offshore wind power potential, 98, 170, 181; in Wyoming, 125, 162\u201365, 185\n\n_Wind Power in America's Future_ (DOE report), 94\n\nwind power incentives, federal: grid modernization grants, 172; investment tax credits, 28, 34\u201335, 51; loan guarantees, 172; Obama stimulus measures, 50\u201351, 56, 172; production tax credit, 2\u20133, 13, 50\u201351, 111, 113, 182\u201383; technical training grants, 88. _See also_ wind power incentives, state\n\nwind power incentives, state: grants and loans, 111; green power option, 111; investment tax credits and tax-free bonds in California, 30\u201331, 33\u201335; net-metering laws, 111; property tax exemption in Kansas, 2, 147; renewable electricity standards, 111, 112, 165, 183. _See also_ wind power incentives, federal\n\nwind speed: and bat fatalities, 131\u201332; definition of \"windy land areas,\" 97; impact on turbine capacity factor, 97, 114; impact on turbine durability, 53\u201354; turbine tower height and, 114; and U.S. wind power potential, 96\u201398, 185\u201386\n\nwind turbine components. _See_ blades; hubs; nacelles; rotors; towers\n\nwind turbines: availability of, 87; capacity factor of, 3, 97, 114; climbing of, 84\u201385, 91, 92; cost of, 11\u201312, 113; cut-in speed for, 131\u201332; design of, 31\u201332, 34, 39, 113\u201317, 149\u201350; diagram of, xii; dimensions and weight of, xii, 14, 49, 74, 75, 84\u201385, 92, 114\u201315, 119; direct-drive vs. gear-driven, 31, 41\u201351, 53, 55, 70, 86\u201387, 91; erection of, 13\u201314, 81\u201384, 90\u201391; foundations for, 80; installed capacity of, 13, 31, 99; land required for, 5, 118\u201319; maintenance of, 84\u201387, 90\u201391; noise from and noise limits for, x, 5, 149\u201360; research and development on, 23, 31\u201335; rotational speed of, 119; setbacks for, from neighboring homes, 157, 158\u201359; \"shadow flicker\" from, 155; transport of, 74\u201378; wildlife impacts from, x, 116\u201333, 159, 167. _See also_ manufacture of wind turbines; visual appearance of wind farms; wind turbine components\n\nWind Turbine Syndrome, 154\u201355\n\nWisconsin: turbine setback controversy in, 158\u201359; wind resources and wind energy use in,185\n\nwomen, employment of: at wind farms, 13\u201315, 17, 79, 82, 91\u201394; in technical training programs, 91\u201394; in top corporate positions, 93\n\nWomen of Wind Energy, 93\n\nWood, Grant, 18\n\nworker safety: in operations and maintenance of turbines, 83, 86, 88, 92\u201393; at turbine manufacturing plants, 45\u201346\n\nWorld Health Organization (WHO), 156\n\nWorld Trade Organization (WTO), 69\n\nWyoming: coal mining in, 122, 123; coalition of pro-wind landowner associations (REAL) in, 163\u201364; electric transmission lines originating in, 164\u201367; federal lands in, 166\u201367; National Historic Trails in, 167; oil and gas resources in, 123\u201324; ranching in, 162\u201363; sage grouse in, 122\u201325, 159; wildlife conservation in, 122\u201323, 125, 159, 167; wind farm siting in, 123\u201325, 167; wind resources and wind energy use in, 124\u201325, 162\u201365, 185\n\nWyoming Infrastructure Authority, 164\n\nXEMC Windpower, 55\n\nXinjiang Goldwind Science & Technology, 67\u201368, 70\u201371\n\nYoung, Don, 135\n\nYucca Mountain, Nevada, 103\n\nZehtindjiev, Pavel, 120\n\nZephyr transmission line, 164\u201365, 166, 167\n\nZilkha Renewable Energy, 7, 8\u201312, 49, 127, 147\n_To Tali and Maya  \nMay your generation continue the journey_\n\nBeacon Press  \n25 Beacon Street  \nBoston, Massachusetts 02108-2892  \nwww.beacon.org\n\nBeacon Press books  \nare published under the auspices of  \nthe Unitarian Universalist Association of Congregations.\n\n\u00a9 2012 by Philip Warburg  \nAll rights reserved  \nPrinted in the United States of America\n\n15 14 13 12 8 7 6 5 4 3 2 1\n\nThis book is printed on acid-free paper that meets the uncoated paper  \nANSI\/NISO specifications for permanence as revised in 1992.\n\nText design by Wilsted & Taylor Publishing Services\n\nLibrary of Congress Cataloging-in-Publication Data  \nWarburg, Philip.  \nHarvest the wind : America's journey to jobs, energy  \nindependence, and climate stability \/ Philip Warburg.  \np.cm.  \nIncludes bibliographical references and index.  \nISBN 978-0-8070-0107-3 (alk. paper)  \nE-ISBN 978-0-8070-0108-0  \n1. Wind power plants\u2014United States. 2. Wind power  \nindustry\u2014United States. I. Title.  \nTK1541.W36 2012\n\n333.9'20973\u2014dc23 2011039420\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\n\n\nProduced by David Reed\n\n\n\n\n\nTHE LAST STETSON\n\nBy John Fox Jr.\n\n\n\n\nI.\n\nA MIDSUMMER freshet was running over old Gabe Bunch's water-wheel into\nthe Cumberland. Inside the mill Steve Marcum lay in one dark corner with\na slouched hat over his face. The boy Isom was emptying a sack of corn\ninto the hopper. Old Gabe was speaking his mind.\n\nAlways the miller had been a man of peace; and there was one time when\nhe thought the old Stetson-Lewallen feud was done. That was when Rome\nStetson, the last but one of his name, and Jasper Lewallen, the last\nbut one of his, put their guns down and fought with bare fists on a high\nledge above old Gabe's mill one morning at daybreak. The man who was\nbeaten was to leave the mountains; the other was to stay at home and\nhave peace. Steve Marcum, a Stetson, heard the sworn terms and saw the\nfight. Jasper was fairly whipped; and when Rome let him up he proved\ntreacherous and ran for his gun. Rome ran too, but stumbled and fell.\nJasper whirled with his Winchester and was about to kill Rome where he\nlay, when a bullet came from somewhere and dropped him back to the ledge\nagain. Both Steve Marcum and Rome Stetson said they had not fired the\nshot; neither would say who had. Some thought one man was lying, some\nthought the other was, and Jasper's death lay between the two. State\ntroops came then, under the Governor's order, from the Blue Grass, and\nRome had to drift down the river one night in old Gabe's canoe and on\nOut of the mountains for good. Martha Lewallen, who, though Jasper's\nsister, and the last of the name, loved and believed Rome, went with\nhim. Marcums and Braytons who had taken sides in the fight hid in the\nbushes around Hazlan, or climbed over into Virginia. A railroad started\nup the Cumberland. \"Furriners came in to buy wild lands and get out\ntimber.\" Civilization began to press over the mountains and down on\nHazlan, as it had pressed in on Breathitt, the seat of another feud,\nin another county. In Breathitt the feud was long past, and with good\nreason old Gabe thought that it was done in Hazlan.\n\nBut that autumn a panic started over from England. It stopped the\nrailroad far down the Cumberland; it sent the \"furriners\" home, and\ndrove civilization back. Marcums and Braytons came in from hiding, and\ndrifted one by one to the old fighting-ground. In time they took up the\nold quarrel, and with Steve Marcum and Steve Brayton as leaders, the\nold Stetson-Lewallen feud went on, though but one soul was left in the\nmountains of either name. That was Isom, a pale little fellow whom Rome\nhad left in old Gabe's care; and he, though a Stetson and a half-brother\nto Rome, was not counted, because he was only a boy and a foundling, and\nbecause his ways were queer.\n\nThere was no open rupture, no organized division--that might happen no\nmore. The mischief was individual now, and ambushing was more common.\nCertain men were looking for each other, and it was a question of\n\"draw-in' quick 'n' shootin' quick\" when the two met by accident, or of\ngetting the advantage \"from the bresh.\"\n\nIn time Steve Marcum had come face to face with old Steve Brayton in\nHazlan, and the two Steves, as they were known, drew promptly. Marcum\nwas in the dust when the smoke cleared away; and now, after three months\nin bed, he was just out again. He had come down to the mill to see Isom.\nThis was the miller's first chance for remonstrance, and, as usual,\nhe began to lay it down that every man who had taken a human life must\nsooner or later pay for it with his own. It was an old story to Isom,\nand, with a shake of impatience, he turned out the door of the mill, and\nleft old Gabe droning on under his dusty hat to Steve, who, being heavy\nwith \"moonshine,\" dropped asleep.\n\nOutside the sun was warm, the flood was calling from the dam, and the\nboy's petulance was gone at once. For a moment he stood on the rude\nplatform watching the tide; then he let one bare foot into the water,\nand, with a shiver of delight, dropped from the boards. In a moment his\nclothes were on the ground behind a laurel thicket, and his slim white\nbody was flashing like a faun through the reeds and bushes up stream.\nA hundred yards away the creek made a great loop about a wet thicket\nof pine and rhododendron, and he turned across the bushy neck. Creeping\nthrough the gnarled bodies of rhododendron, he dropped suddenly behind\nthe pine, and lay flat in the black earth. Ten yards through the\ndusk before him was the half-bent figure of a man letting an old army\nhaversack slip from one shoulder; and Isom watched him hide it with\na rifle under a bush, and go noiselessly on towards the road. It was\nCrump, Eli Crump, who had been a spy for the Lewallens in the old feud\nand who was spying now for old Steve Brayton. It was the second time\nIsom had seen him lurking about, and the boy's impulse was to hurry back\nto the mill. But it was still peace, and without his gun Crump was not\ndangerous; so Isom rose and ran on, and, splashing into the angry little\nstream, shot away like a roll of birch bark through the tawny crest of\na big wave. He had done the feat a hundred times; he knew every rock and\neddy in flood-time, and he floated through them and slipped like an eel\ninto the mill-pond. Old Gabe was waiting for him.\n\n\"Whut ye mean, boy,\" he said, sharply, \"reskin' the fever an' ager this\nway? No wonder folks thinks ye air half crazy. Git inter them clothes\nnow 'n' come in hyeh. You'll ketch yer death o' cold swimmin' this way\natter a fresh.\"\n\nThe boy was shivering when he took his seat at the funnel, but he did\nnot mind that; some day he meant to swim over that dam. Steve still lay\nmotionless in the corner near him, and Isom lifted the slouched hat\nand began tickling his lips with a straw. Steve was beyond the point of\ntickling, and Isom dropped the hat back and turned to tell the miller\nwhat he had seen in the thicket. The dim interior darkened just then,\nand Crump stood in the door. Old Gabe stared hard at him without a word\nof welcome, but Crump shuffled to a chair unasked, and sat like a toad\nastride it, with his knees close up under his arms, and his wizened face\nin his hands.\n\nMeeting Isom's angry glance, he shifted his own uneasily.\n\n\"Seed the new preacher comm' 'long today?\" he asked. Drawing one dirty\nfinger across his forehead, \"Got a long scar 'cross hyeh.\"\n\nThe miller shook his head.\n\n\"Well, he's a-comm'. I've been waitin' fer him up the road, but I reckon\nI got to git 'cross the river purty soon now.\"\n\nCrump had been living over in Breathitt since the old feud. He had been\n\"convicted\" over there by Sherd Raines, a preacher from the Jellico\nHills, and he had grown pious. Indeed, he had been trailing after Raines\nfrom place to place, and he was following the circuit-rider now to the\nscene of his own deviltry--Hazlan.\n\n\"Reckon you folks don't know I got the cirkit-rider to come over hyeh,\ndo ye?\" he went on. \"Ef he can't preach! Well, I'd tell a man! He kin\njus' draw the heart out'n a holler log! He 'convicted' me fust night,\nover thar in Breathitt. He come up thar, ye know, to stop the feud, he\nsaid; 'n' thar was laughin' from one eendo' Breathitt to t'other; but\nthar was the whoppinest crowd thar I ever see when he did come. The\nmeetin'-house wasn't big enough to hold 'em, so he goes out on the\naidge o' town, n' climbs on to a stump. He hed a woman with him from the\nsettlemints--she's a-waitin' at Hazlan fer him now-'n' she had a cur'us\nlittle box, 'n' he put her 'n' the box on a big rock, 'n' started in a\ncallin' 'em his bretherin' 'n' sisteren, 'n' folks seed mighty soon\nthet he meant it, too. He's always mighty easylike, tell he gits to the\nblood-penalty.\"\n\nAt the word, Crump's listeners paid sudden heed. Old Gabe's knife\nstopped short in the heart of the stick he was whittling; the boy looked\nsharply up from the running meal into Crump's face and sat still.\n\nWell, he jes prayed to the Almighty as though he was a-talkin' to him\nface to face, 'n' then the woman put her hands on that box, 'n' the\nsweetes' sound anybody thar ever heerd come outen it. Then she got to\nsingin'. Hit wusn't nuthin' anybody thar'd ever heerd; but some o' the\nwomen folks was a snifflin' 'fore she got through. He pitched right into\nthe feud, as he calls hit, 'n' the sin o' sheddin' human blood, I tell\nye; 'n' 'twixt him and the soldiers I reckon thar won't be no more\nfightin' in Breathitt. He says, 'n' he always says it mighty loud\n--Crump raised his own voice--\"thet the man as kills his feller-critter\nhev some day got ter give up his own blood, sartin 'n' shore.\"\n\nIt was old Gabe's pet theory, and he was nodding approval. The boy's\nparted lips shook with a spasm of fear, and were as quickly shut tight\nwith suspicion. Steve raised his head as though he too had heard the\nvoice, and looked stupidly about him.\n\n\"I tol' him,\" Crump went on, \"thet things was already a-gettin' kind o'\nfrolicsome round hyeh agin; thet the Marcums 'n' Braytons was a-takin'\nup the ole war, 'n' would be a-plunkin' one 'nother every time they got\ntogether, 'n' a-gittin' the whole country in fear 'n' tremblin'--now\nthet Steve Marcum had come back.\"\n\nSteve began to scowl and a vixenish smile hovered at Isom's lips.\n\n\"He knows mighty well--fer I tol' him--thet thar hain't a wuss man in all\nthese mountains than thet very Steve--\" The name ended in a gasp, and the\nwizened gossip was caught by the throat and tossed, chair and all, into\na corner of the mill.\n\n\"None o' that, Steve!\" called the miller, sternly. \"Not hyeh. Don't hurt\nhim now!\"\n\nCrump's face stiffened with such terror that Steve broke into a laugh.\n\n\"Well, ye air a skeery critter!\" he said, contemptuously. \"I hain't\ngoin' to hurt him, Uncl' Gabe, but he must be a plumb idgit, a-talkin'\n'bout folks to thar face, 'n' him so puny an' spindlin'! You git!\"\n\nCrump picked himself up trembling--\"Don't ye ever let me see ye on this\nside o' the river agin, now \"--and shuffled out, giving Marcum one look\nof fear and unearthly hate.\n\n\"Convicted!\" snorted Steve. \"I heerd old Steve Brayton had hired him to\nwaylay me, 'n' I swar I believe hit's so.\"\n\n\"Well, he won't hev to give him more'n a chaw o' tobaccer now,\" said\nGabe. \"He'll come purty near doin' hit hisseif, I reckon, ef he gits\nthe chance.\"\n\n\"Well, he kin git the chance ef I gits my leetle account settled with ole\nSteve Brayton fust. 'Pears like that old hog ain't satisfied shootin'\nme hisself.\" Stretching his arms with a yawn, Steve winked at Isom and\nmoved to the door. The boy followed him outside.\n\n\"We're goin' fer ole Brayton about the dark o' the next moon, boy,\" he\nsaid. \"He's sort o' s'picious now, 'n' we'll give him a leetle time to\ngit tame. I'll have a bran'-new Winchester fer ye, Isom. Hit ull be like\nole times agin, when Rome was hyeh. Whut's the matter, boy?\" he asked,\nsuddenly. Isom looked unresponsive, listless.\n\n\"Air ye gittin' sick agin?\"\n\n\"Well, I hain't feelin' much peert, Steve.\"\n\n\"Take keer o' yourself, boy. Don't git sick now. We'll have to watch Eli\nCrump purty close. I don't know why I hain't killed thet spyin' skunk\nlong ago, 'ceptin' I never had a shore an' sartin reason fer doin 'it.\"\n\nIsom started to speak then and stopped. He would learn more first; and\nhe let Steve go on home unwarned.\n\nThe two kept silence after Marcum had gone. Isom turned away from old\nGabe, and stretched himself out on the platform. He looked troubled. The\nmiller, too, was worried.\n\n\"Jus' a hole in the groun',\" he said, half to himself; \"that's whut we're\nall comm' to! 'Pears like we mought help one 'nother to keep out'n hit,\n'stid o' holpin' 'em in.\"\n\nBrown shadows were interlacing out in the mill-pond, where old Gabe's\neyes were intent. A current of cool air had started down the creek to\nthe river. A katydid began to chant. Twilight was coming, and the miller\nrose.\n\n\"Hit's a comfort to know you won't be mixed up in all this devilment,\"\nhe said; and then, as though he had found more light in the gloom:\n\"Hit's a comfort to know the new rider air shorely a-preachin' the right\ndoctrine, 'n' I want ye to go hear him. Blood for blood-life fer a life!\nYour grandad shot ole Tom Lewallen in Hazlan. Ole Jack Lewallen shot him\nfrom the bresh. Tom Stetson killed ole Jack; ole Jass killed Tom, 'n' so\nhit comes down, fer back as I can ricollect. I hev nuver knowed hit to\nfail.\" The lad had risen on one elbow. His face was pale and uneasy, and\nhe averted it when the miller turned in the door.\n\n\"You'd better stay hyeh, son, 'n' finish up the grist. Hit won't take\nlong. Hev ye got victuals fer yer supper?\"\n\nIsom nodded, without looking around, and when old Gabe was gone he rose\nnervously and dropped helplessly back to the floor.\n\n\"'Pears like old Gabe knows I killed Jass,\" he breathed, sullenly.\n\"'Pears like all of 'em knows hit, 'n' air jus' a-tormentin' me.\"\n\nNobody dreamed that the boy and his old gun had ended that fight on the\ncliff; and without knowing it, old Gabe kept the lad in constant torture\nwith his talk of the blood-penalty. But Isom got used to it in time,\nfor he had shot to save his brother's life. Steve Marcum treated him\nthereafter as an equal. Steve's friends, too, changed in manner towards\nhim because Steve had. And now, just when he had reached the point of\nwondering whether, after all, there might not be one thing that old Gabe\ndid not know, Crump had come along with the miller's story, which he had\ngot from still another, a circuit-rider, who must know the truth. The\nfact gave him trouble.\n\n\"Mebbe hit's goin' to happen when I goes with Steve atter ole Brayton,\"\nhe mumbled, and he sat thinking the matter over, until a rattle and a\nwhir inside the mill told him that the hopper was empty. He arose to\nfill it, and coming out again, he heard hoof-beats on the dirt road. A\nstranger rode around the rhododendrons and shouted to him, asking the\ndistance to Hazlan. He took off his hat when Isom answered, to wipe the\ndust and perspiration from his face, and the boy saw a white scar across\nhis forehead. A little awestricken, the lad walked towards him.\n\n\"Air you the new rider whut's goin' to preach up to Hazlan?\" he asked.\n\nRaines smiled at the solemnity of the little fellow. \"Yes,\" he said,\nkindly. \"Won't you come up and hear me?\"\n\n\"Yes, sir,\" he said, and his lips parted as though he wanted to say\nsomething else, but Raines did not notice.\n\n\"I wished I had axed him,\" he said, watching the preacher ride away.\n\"Uncle Gabe knows might' nigh ever'thing, 'n' he says so. Crump said\nthe rider said so; but Crump might 'a' been lyin'. He 'most al'ays is. I\nwished I had axed him.\"\n\nMechanically the lad walked along the millrace, which was made of hewn\nboards and hollow logs. In every crevice grass hung in thick bunches to\nthe ground or tipped wiry blades over the running water. Tightening\na prop where some silvery jet was getting too large, he lifted the\ntail-gate a trifle and lay down again on the platform near the old\nwheel. Out in the mill-pond the water would break now and then into\nripples about some unwary moth, and the white belly of a fish would\nflash from the surface. It was the only sharp accent on the air. The\nchant of the katydids had become a chorus, and the hush of darkness\nwas settling over the steady flow of water and the low drone of the\nmillstones.\n\n\"I hain't afeerd,\" he kept saying to himself. \"I hain't afeerd o'\nnothin' nor nobody;\" but he lay brooding until his head throbbed, until\ndarkness filled the narrow gorge, and the strip of dark blue up through\nthe trees was pointed with faint stars. He was troubled when he rose,\nand climbed on Rome's horse and rode homeward--so troubled that he\nturned finally and started back in a gallop for Hazlan.\n\nIt was almost as Crump had said. There was no church in Hazlan, and, as\nin Breathitt, the people had to follow Raines outside the town, and he\npreached from the roadside. The rider's Master never had a tabernacle\nmore simple: overhead the stars and a low moon; close about, the trees\nstill and heavy with summer; a pine torch over his head like a yellow\nplume; two tallow dips hung to a beech on one side, and flicking to the\nother the shadows of the people who sat under them. A few Marcums and\nBraytons were there, one faction shadowed on Raines's right, one on\nhis left. Between them the rider stood straight, and prayed as though\ntalking with some one among the stars. Behind him the voice of the woman\nat her tiny organ rose among the leaves. And then he spoke as he had\nprayed; and from the first they listened like children, while in their\nown homely speech he went on to tell them, just as he would have told\nchildren, a story that some of them had never heard before. \"Forgive\nyour enemies as He had forgiven his,\" that was his plea. Marcums and\nBraytons began to press in from the darkness on each side, forgetting\neach other as the rest of the people forgot them. And when the story\nwas quite done, Raines stood a full minute without a word. No one was\nprepared for what followed. Abruptly his voice rose sternly--\"Thou\nshalt not kill\"; and then Satan took shape under the torch. The man was\ntransformed, swaying half crouched before them. The long black hair fell\nacross the white scar, and picture after picture leaped from his tongue\nwith such vividness that a low wail started through the audience, and\nwomen sobbed in their bonnets. It was penalty for bloodshed--not in this\nworld: penalty eternal in the next; and one slight figure under the dips\nstaggered suddenly aside into the darkness.\n\nIt was Isom; and no soul possessed of devils was ever more torn than\nhis, when he splashed through Troubled Fork and rode away that night.\nHalf a mile on he tried to keep his eyes on his horse's neck, anywhere\nexcept on one high gray rock to which they were raised against his\nwill--the peak under which he had killed young Jasper. There it was\nstaring into the moon, but watching him as he fled through the woods,\nshuddering at shadows, dodging branches that caught at him as he passed,\nand on in a run, until he drew rein and slipped from his saddle at the\nfriendly old mill. There was no terror for him there. There every bush\nwas a friend; every beech trunk a sentinel on guard for him in shining\narmor.\n\nIt was the old struggle that he was starting through that night--the old\nfight of humanity from savage to Christian; and the lad fought it until,\nwith the birth of his wavering soul, the premonitions of the first dawn\ncame on. The patches of moonlight shifted, paling. The beech columns\nmottled slowly with gray and brown. A ruddy streak was cleaving the east\nlike a slow sword of fire. The chill air began to pulse and the mists to\nstir. Moisture had gathered on the boy's sleeve. His horse was stamping\nuneasily, and the lad rose stiffly, his face gray but calm, and started\nhome. At old Gabe's gate he turned in his saddle to look where, under\nthe last sinking star, was once the home of his old enemies. Farther\ndown, under the crest, was old Steve Brayton, alive, and at that moment\nperhaps asleep.\n\n\"Forgive your enemies;\" that was the rider's plea. Forgive old Steve,\nwho had mocked him, and had driven Rome from the mountains; who had\nthreatened old Gabe's life, and had shot Steve Marcum almost to death!\nThe lad drew breath quickly, and standing in his stirrups, stretched out\nhis fist, and let it drop, slowly.\n\n\n\n\nII.\n\nOLD Gabe was just starting out when Isom' reached the cabin, and the old\nman thought the boy had been at the mill all night. Isom slept through\nthe day, and spoke hardly a word when the miller came home, though the\nlatter had much to say of Raines, the two Steves, and of the trouble\npossible. He gave some excuse for not going with old Gabe the next day,\nand instead went into the woods alone.\n\nLate in the middle of the afternoon he reached the mill. Old Gabe sat\nsmoking outside the door, and Isom stretched himself out on the platform\nclose to the water, shading his eyes from the rich sunlight with one\nragged sleeve.\n\n\"Uncl' Gabe,\" he said, suddenly, \"s'posin' Steve Brayton was to step\nout'n the bushes thar some mawnin' 'n' pull down his Winchester on ye,\nwould ye say, 'Lawd, fergive him, fer he don't know whut he do'?\"\n\nOld Gabe had told him once about a Stetson and a Lewallen who were heard\nhalf a mile away praying while they fought each other to death with\nWinchesters. \"There was no use prayin' an' shootin',\" the miller\ndeclared. There was but one way for them to escape damnation; that was\nto throw down their guns and make friends. But the miller had forgotten,\nand his mood that morning was whimsical.\n\n\"Well, I mought, Isom,\" he said, \"ef I didn't happen to have a gun\nhandy.\"\n\nThe humor was lost on Isom. His chin was moving up and down, and his\nface was serious. That was just it. He could forgive Jass--Jass was dead;\nhe could forgive Crump, if he caught him in no devilment; old Brayton\neven--after Steve's revenge was done. But now--The boy rose, shaking his\nhead.\n\n\"Uncl' Gabe,\" he said with sudden passion, \"whut ye reckon Rome's\na-doin'?\"\n\nThe miller looked a little petulant. \"Don't ye git tired axin' me thet\nquestion, Isom? Rome's a-scratchin' right peert fer a livin', I reckon,\nfer hisself 'n' Marthy. Yes, 'n' mebbe fer a young 'un too by this time.\nEf ye air honin' fer Rome, why don't ye rack out 'n' go to him? Lawd\nknows I'd hate ter see ye go, but I tol' Rome I'd let ye whenever ye got\nready, 'n' so I will.\"\n\nIsom had no answer, and old Gabe was puzzled. It was always this way.\nThe boy longed for Rome, the miller could see. He spoke of him sometimes\nwith tears, and sometimes he seemed to be on the point of going to him,\nbut he shrank inexplicably when the time for leaving came.\n\nIsom started into the mill now without a word, as usual. Old Gabe\nnoticed that his feet were unsteady, and with quick remorse began to\nquestion him.\n\n\"Kinder puny, hain't ye, Isom?\"\n\n\"Well, I hain't feelin' much peert.\"\n\n\"Hit was mighty keerless,\" old Gabe said, with kindly reproach,\n\"swimmin' the crick atter a fresh.\"\n\n\"Hit wasn't the swimmin',\" he protested, dropping weakly at the\nthreshold. \"Hit was settin' out 'n the woods. I was in Hazlan t'other\nnight, Und' Gabe, to hear the new rider.\"\n\nThe miller looked around with quick interest. \"I've been skeered afore\nby riders a-tellin' 'bout the torments o' hell, but I never heerd\nnothin' like his tellin' 'bout the Lord. He said the Lord was jes as\npore as anybody thar, and lived jes as rough; thet He made fences and\nbarns n' ox-yokes 'n' sech like, an' He couldn't write His own name when\nHe started out to save the worl'; an' when he come to the p'int whar His\nenemies tuk hol' of Him, the rider jes crossed his fingers up over his\nhead 'n' axed us if we didn't know how it hurt to run a splinter into\na feller's hand when he's loggin' or a thorn into yer foot when ye're\ngoin' barefooted.\"\n\n\"Hit jes made me sick, Uncl' Gabe, hearin' him tell how they stretched\nHim out on a cross o' wood, when He'd come down fer nothin' but to save\n'em, 'n' stuck a spear big as a co'n-knife into His side, 'n' give Him\nvinegar, 'n' let Him hang thar 'n' die, with His own mammy a-stand-in'\ndown on the groun' a-cryin' 'n' watchin' Him. Some folks thar never\nheerd sech afore. The women was a-rockin', 'n' ole Granny Day axed right\nout ef thet tuk place a long time ago; 'n' the rider said, 'Yes, a long\ntime ago, mos two thousand years.' Granny was a-cryin', Uncl' Gabe, 'n'\nshe said, sorter soft, 'Stranger, let's hope that hit hain't so'; 'n'\nthe rider says, But hit air so; n' He fergive em while they was doin'\nit.' Thet's whut got me, Uncl' Gabe, 'n' when the woman got to singin',\nsomethin' kinder broke loose hyeh\"--Isom passed his hand over his thin\nchest--\"'n' I couldn't git breath. I was mos' afeerd to ride home. I jes\nlayed at the mill studyin', till I thought my head would bust. I reckon\nhit was the spent a-work-in me. Looks like I was mos' convicted, Uncl'\nGabe.\" His voice trembled and he stopped. \"Crump was a-lyin',\" he cried,\nsuddenly. \"But hit's wuss, Und' Gabe; hit's wuss! You say a life fer a\nlife in this worl'; the rider says hit's in the next, 'n' I'm mis'ble,\nUncl' Gabe. Ef Rome--I wish Rome was hyeh,\" he cried, helplessly. \"I\ndon't know whut to do.\"\n\nThe miller rose and limped within the mill, and ran one hand through\nthe shifting corn. He stood in the doorway, looking long and perplexedly\ntowards Hazlan; he finally saw, he thought, just what the lad's trouble\nwas. He could give him some comfort, and he got his chair and dragged it\nout to the door across the platform, and sat down in silence.\n\n\"Isom,\" he said at last, \"the Spent air shorely a-workin' ye, 'n' I'm\nglad of it. But ye mus 'n t worry about the penalty a-fallin' on Rome.\nSteve Marcum killed Jass--he can't fool me--'n' I've told Steve he's got\nthet penalty to pay ef he gits up this trouble. I'm glad the Spent's\na-workin' ye, but ye mus'n' t worry 'bout Rome.\"\n\nIsom rose suddenly on one elbow, and with a moan lay back and crossed\nhis arms over his face.\n\nOld Gabe turned and left him.\n\n\"Git up, Isom.\" It was the miller's voice again, an hour later. \"You\nbetter go home now. Ride the hoss, boy,\" he and, kindly.\n\nIsom rose, and old Gabe helped him mount, and stood at the door. The\nhorse started, but the boy pulled him to a standstill again.\n\n\"I want to ax ye jes one thing more, Uncl' Gabe,\" he said, slowly.\n\"S'posin' Steve had a-killed Jass to keep him from killin' Rome, hev he\ngot to be damned fer it jes the same? Hev he got to give up eternal life\nanyways? Hain't thar no way out'n it--no way?\"\n\nThere was need for close distinction now and the miller was deliberate.\n\n\"Ef Steve shot Jass,\" he said, \"jes to save Rome's life--he had the right\nto shoot him. Thar hain't no doubt 'bout that. The law says so.\nBut\"--there was a judicial pause--\"I've heerd Steve say that he hated Jass\nwuss' n anybody on earth, 'cept old Brayton; 'n' ef he wus glad o' the\nchance o' killin' him, why--the Lord air merciful, Isom; the Bible air\ntrue, 'n' hit says an 'eye fer an eye, a tooth fer a tooth,' 'n' I never\nknowed hit to fail--but the Lord air merciful. Ef Steve would only jes\nrepent, 'n' ef, 'stid o' fightin' the Lord by takin' human life, he'd\nfight fer Him by savin' it, I reckon the Lord would fergive him. Fer\nef ye lose yer life fer Him, He do say you'll find it agin\nsomewhar--sometime.\"\n\nOld Gabe did not see the sullen despair that came into the boy's tense\nface. The subtlety of the answer had taken the old man back to the days\nwhen he was magistrate, and his eyes were half closed. Isom rode away\nwithout a word. From the dark of the mill old Gabe turned to look after\nhim again.\n\n\"I'm afeerd he's a-gittin' feverish agin. Hit looks like he's convicted;\nbut\"--he knew the wavering nature of the boy--\"I don't know--I don't\nknow.\"\n\nGoing home an hour later, the old man saw several mountaineers climbing\nthe path towards Steve Marcum's cabin; it meant the brewing of mischief;\nand when he stopped at his own gate, he saw at the bend of the road a\nfigure creep from the bushes on one side into the bushes on the other.\n\nIt looked like Crump.\n\n\n\n\nIII.\n\nIT was Crump, and fifty yards behind him was Isom, slipping through the\nbrush after him--Isom's evil spirit--old Gabe, Raines, \"conviction,\"\nblood-penalty, forgotten, all lost in the passion of a chase which has\nno parallel when the game is man.\n\nStraight up the ravine Crump went along a path which led to Steve\nMarcum's cabin. There was a clump of rhododendron at the head of the\nravine, and near Steve's cabin. About this hour Marcum would be chopping\nwood for supper, or sitting out in his porch in easy range from the\nthicket. Crump's plan was plain: he was about his revenge early, and\nIsom was exultant.\n\n\"Oh, no, Eli, you won't git Steve this time. Oh, naw!\"\n\nThe bushes were soon so thick that he could no longer follow Crump by\nsight, and every few yards he had to stop to listen, and then steal on\nlike a mountain-cat towards the leaves rustling ahead of him. Half-way\nup the ravine Crump turned to the right and stopped. Puzzled, Isom\npushed so close that the spy, standing irresolute on the edge of\nthe path, whirled around. The boy sank to his face, and in a moment\nfootsteps started and grew faint; Crump had darted across the path, and\nwas running through the undergrowth up the spur. Isom rose and hurried\nafter him; and when, panting hard, he reached the top, the spy's\nskulking figure was sliding from Steve's house and towards the Breathitt\nroad; and with a hot, puzzled face, the boy went down after it.\n\nOn a little knob just over a sudden turn in the road Crump stopped, and\nlooking sharply about him, laid his gun down. Just in front of him were\ntwo rocks, waist-high, with a crevice between them. Drawing a long knife\nfrom his pocket, he climbed upon them, and began to cut carefully away\nthe spreading top of a bush that grew on the other side. Isom crawled\ndown towards him like a lizard, from tree to tree. A moment later the\nspy was filling up the crevice with stones, and Isom knew what he was\nabout; he was making a \"blind\" to waylay Steve, who, the boy knew, was\ngoing to Breathitt by that road the next Sunday. How did Crump know\nthat--how did he know everything? The crevice filled, Crump cut branches\nand stuck them between the rocks. Then he pushed his rifle through the\ntwigs, and taking aim several times, withdrew it. When he turned away\nat last and started down to the road, he looked back once more, and Isom\nsaw him grinning. Almost chuckling in answer, the lad slipped around the\nknob to the road the other way, and Crump threw up his gun with a gasp\nof fright when a figure rose out of the dusk before him.\n\n\"Hol' on, Eli!\" said Isom, easily. \"Don't git skeered! Hit's nobody but\nme. Whar ye been?\"\n\nCrump laughed, so quick was he disarmed of suspicion. \"Jes up the river\na piece to see Aunt Sally Day. She's a fust cousin o' mine by marriage.\"\n\nJsom's right hand was slipping back as if to rest on his hip. \"D'you say\nyou'd been 'convicted,' Eli?\"\n\nCrump's answer was chantlike. \"Yes, Lawd reckon I have.\"\n\n\"Goin' to stop all o' yer lyin', air ye,\" Isom went on, in the same\ntone, and Crump twitched as though struck suddenly from behind, \"an'\nstealin' 'n' lay-wayin'?\"\n\n\"Look a-hyeh, boy--\" he began, roughly, and mumbling a threat, started\non.\n\n\"Uh, Eli!\" Even then the easy voice fooled him again, and he turned.\nIsom had a big revolver on a line with his breast. \"Drap yer gun!\" he\nsaid, tremulously.\n\nCrump tried to laugh, but his guilty face turned gray. \"Take keer, boy,\"\nhe gasped; \"yer gun's cocked. Take keer, I tell ye!\"\n\n\"Drap it, damn ye!\" Isom called in sudden fury, \"'n' git clean away from\nit!\" Crump backed, and Isom came forward and stood with one foot on the\nfallen Winchester.\n\n\"I seed ye, Eli. Been makin' a blind fer Steve, hev ye? Goin' to shoot\nhim in the back, too, air ye? You're ketched at last, Eli. You've done\na heap o' devilment. You're gittin' wuss all the time. You oughter be\ndead, 'n' now--\"\n\nCrump found voice in a cry of terror and a whine for mercy. The boy\nlooked at him, unable to speak his contempt.\n\n\"Git down thar!\" he said, finally; and Crump, knowing what was wanted,\nstretched himself in the road. Isom sat down on a stone, the big pistol\nacross one knee.\n\n\"Roll over!\" Crump rolled at full length.\n\n\"Git up!\" Isom laughed wickedly. \"Ye don't look purty, Eli.\" He lifted\nthe pistol and nipped a cake of dirt from the road between Crump's feet.\nWith another cry of fear, the spy began a vigorous dance.\n\n\"Hol' on, Eli; I don't want ye to dance. Ye belong to the chu'ch now,\n'n' I wouldn't have ye go agin yer religion fer nothin'. Stan' still!\"\nAnother bullet and another cut between Crump's feet. \"'Pears like ye\ndon't think I kin shoot straight. Eli,\" he went on, reloading the empty\nchambers, \"some folks think I'm a idgit, 'n' I know 'em. Do you think\nI'm a idgit, Eli?\"\n\n\"Actin' mighty nateral now.\" Isom was raising the pistol again. \"Oh,\nLawdy! Don't shoot, boy--don't shoot!\n\n\"Git down on yer knees! Now I want ye to beg fer mercy thet ye never\nshowed--thet ye wouldn't 'a' showed Steve... Purty good,\" he said,\nencouragingly.\n\n\"Mebbe ye kin pray a leetle, seem' ez ye air a chu'ch member. Pray fer\nyer enemies, Eli; Uncl' Gabe says ye must love yer enemies. I know how\nye loves me, 'n' I want yer to pray fer me. The Lawd mus' sot a powerful\nstore by a good citizen like you. Ax him to fergive me fer killin' ye.\"\n\n\"Have mercy, O Lawd,\" prayed Crump, to command--and the prayer was\nsubtle--\"on the murderer of this Thy servant. A life fer a life, Thou\nhev said, O Lawd. Fer killin' me he will foller me, 'n' ef Ye hev not\nmussy he is boun' fer the lowes' pit o' hell, O Lawd--\"\n\nIt was Isom's time to wince now, and Crump's pious groan was cut short.\n\n\"Shet up!\" cried the boy, sharply, and he sat a moment silent. \"You've\nbeen a-spyin' on us sence I was borned, Eli,\" he said, reflectively.\n\n\"I believe ye lay-wayed dad. Y'u spied on Rome. Y'u told the soldiers\nwhar he was a-hidin' Y'u tried to shoot him from the bresh. Y'u found\nout Steve was goin' to Breathitt on Sunday, 'n' you've jes made a blind\nto shoot him in the back. I reckon thar's no meanness ye hain't done.\nDad's al'ays said ye sot a snare fer a woman once--a woman! Y'u loaded a\nmusket with slugs, 'n' tied a string to the trigger, 'n' stretched hit\n'cross the path, 'n' y'u got up on a cliff 'n' whistled to make her slow\nup jes when she struck the string. I reckon thet's yer wust--but I don't\nknow.\"\n\nSeveral times Crump raised his hands in protest while his arraignment\nwas going on; several times he tried to speak, but his lips refused\nutterance. The boy's voice was getting thicker and thicker, and he was\nnervously working the cock of the big pistol up and down.\n\n\"Git up,\" he said; and Crump rose with a spring. The lad's tone meant\nrelease.\n\n\"You hain't wuth the risk. I hain't goin' ter kill ye. I jus' wanted\nter banter ye 'n' make ye beg. You're a good beggar, Eli, 'n' a powerful\nprayer. You'll be a shinin' light in the chu'ch, ef ye gits a chance ter\nshine long. Fer lemme tell ye, nobody ever ketched ye afore. But you're\nketched now, an' I'm goin' to tell Steve. He'll be a-watchin' fer ye,\n'n' so 'll I. I tell ye in time, ef ye ever come over hyeh agin as long\nas you live, you'll never git back alive. Turn roun'! Hev ye got any\nballs?\" he asked, feeling in Crump's pockets for cartridges. \"No;\nwell\"--he picked up the Winchester and pumped the magazine empty--\"I'll\nkeep these,\" he said, handing Crump the empty rifle. \"Now git away--an'\ngit away quick!\"\n\nCrump's slouching footsteps went out of hearing, and Isom sat where he\nwas. His elbows dropped to his knees. His face dropped slowly into his\nhands, and the nettles of remorse began to sting. He took the back of\none tremulous hand presently to wipe the perspiration from his forehead,\nand he found it burning. A sharp pain shot through his eyes. He knew\nwhat that meant, and feeling dizzy, he rose and started a little blindly\ntowards home.\n\nOld Gabe was waiting for him. He did not answer the old man's querulous\ninquiry, but stumbled towards a bed. An hour later, when the miller was\nrubbing his forehead, he opened his eyes, shut them, and began to talk.\n\n\"I reckon I hain't much better 'n Eli, Und' Gabe,\" he said, plaintively.\n\"I've been abusin' him down thar in the woods. I come might' nigh\nkillin' him onct.\" The old man stroked on, scarcely heeding the boy's\nwords, so much nonsense would he talk when ill.\n\n\"I've been lyin' to ye, Uncl' Gabe, 'n' a-deceivin' of ye right along.\nSteve's a-goin' atter ole Brayton--I'm goin' too--Steve didn't kill\nJass--hit wusn't Steve--hit wusn't Rome--hit was--\" The last word stopped\nbehind his shaking lips; he rose suddenly in bed, looked wildly into the\nmiller's startled face, and dropping with a sob to the bed, went sobbing\nto sleep.\n\nOld Gabe went back to his pipe, and while he smoked, his figure shrank\nslowly in his chair. He went to bed finally, but sleep would not come,\nand he rose again and built up the fire and sat by it, waiting for day.\nHis own doctrine, sternly taught for many a year, had come home to him;\nand the miller's face when he opened his door was gray as the breaking\nlight.\n\n\n\n\nIV.\n\nTHERE was little peace for old Gabe that day at the mill. And when he\nwent home at night he found cause for the thousand premonitions that had\nhaunted him. The lad was gone.\n\nA faint light in the east was heralding the moon when Isom reached Steve\nMarcum's gate. There were several horses hitched to the fence, several\ndim forms seated in the porch, and the lad halloed for Steve, whose\nshadow shot instantly from the door and came towards him.\n\n\"Glad ter see ye, Isom,\" he called, jubilantly. \"I was jus' about to sen'\nfer ye. How'd ye happen to come up?\"\n\nIsom answered in a low voice with the news of Crump's \"blind,\" and Steve\nlaughed and swore in the same breath.\n\n\"Come hyeh!\" he said, leading the way back; and at the porch he had Isom\ntell the story again.\n\n\"Whut d' I tell ye, boys?\" he asked, triumphantly. \"Don't believe ye\nmore 'n half believed me.\"\n\nThree more horsemen rode up to the gate and came into the light. Every\nman was armed, and at Isom's puzzled look, Steve caught the lad by the\narm and led him around the chimney-corner. He was in high spirits.\n\n\"'Pears like ole times, Isom. I'm a-goin' fer thet cussed ole Steve\nBrayton this very night. He's behind Crump. I s'picioned it afore; now\nI know it for sartain. He's a-goin' to give Eli a mule 'n' a Winchester\nfer killin' me. We're goin' to s'prise him to-night. He won't be\nlook-in' fer us--I've fixed that. I wus jus' about to sen' fer ye. I\nhain't fergot how ye kin handle a gun.\" Steve laughed significantly.\n\"Ye're a good frien' o' mine, 'n' I'm goin' to show ye thet I'm a frien'\no' yourn.\"\n\nIsom's paleness was unnoticed in the dark. The old throbbing began to\nbeat again at his temple; the old haze started from his eyes.\n\n\"Hyeh's yer gun, Isom,\" he heard Steve saying next. The fire was blazing\ninto his face. At the chimney-corner was the bent figure of old Daddy\nMarcum, and across his lap shone a Winchester. Steve was pointing at it,\nhis grim face radiant; the old man's toothless mouth was grinning, and\nhis sharp black eyes were snapping up at him.\n\n\"Hit's yourn, I tell ye,\" said Steve again. \"I aimed jes to lend it\nto ye, but ye've saved me frum gittin' killed, mebbe, 'n' hit's yourn\nnow--yourn, boy, fer keeps.\"\n\nSteve was holding the gun out to him now. The smooth cold touch of the\npolished barrel thrilled him. It made everything for an instant clear\nagain, and feeling weak, Isom sat down on the bed, gripping the treasure\nin both trembling hands. On one side of him some one was repeating\nSteve's plan of attack. Old Brayton's cabin was nearly opposite, but\nthey would go up the river, cross above the mill, and ride back. The\nnight was cloudy, but they would have the moonlight now and then for the\nclimb up the mountain. They would creep close, and when the moon was hid\nthey would run in and get old Brayton alive, if possible. Then--the rest\nwas with Steve.\n\nAcross the room he could hear Steve telling the three new-comers, with\nan occasional curse, about Crump's blind, and how he knew that old\nBrayton was hiring Crump.\n\n\"Old Steve's meaner 'n Eli,\" he said to himself, and a flame of the old\nhate surged up from the fire of temptation in his heart. Steve Marcum\nwas his best friend; Steve had shielded him. The boy had promised to\njoin him against old Brayton, and here was the Winchester, brand-new, to\nbind his word.\n\n\"Git ready, boys; git ready.\"\n\nIt was Steve's voice, and in Isom's ears the preacher's voice rang after\nit. Again that blinding mist before his eyes, and the boy brushed at\nit irritably. He could see the men buckling cartridge-belts, but he sat\nstill. Two or three men were going out. Daddy Marcum was leaning on a\nchair at the door, looking eagerly at each man as he passed.\n\n\"Hain't ye goin', Isom?\"\n\nSomebody was standing before him twirling a rifle on its butt, a boy\nnear Isom's age. The whirling gun made him dizzy.\n\n\"Stop it!\" he cried, angrily. Old Daddy Marcum was answering the boy's\nquestion from the door.\n\n\"Isom goin'?\" he piped, proudly. \"I reckon he air. Whar's yer belt, boy?\nGit ready. Git ready.\"\n\nIsom rose then--he could not answer sitting down--and caught at a bedpost\nwith one hand, while he fumbled at his throat with the other.\n\n\"I hain't goin'.\"\n\nSteve heard at the door, and whirled around. Daddy Marcum was tottering\nacross the floor, with one bony hand uplifted.\n\n\"You're a coward!\" The name stilled every sound. Isom, with eyes afire,\nsprang at the old man to strike, but somebody caught his arm and forced\nhim back to the bed.\n\n\"Shet up, dad,\" said Steve, angrily, looking sharply into Isom's face.\n\"Don't ye see the boy's sick? He needn't go ef he don't want to. Time to\nstart, boys.\"\n\nThe tramp of heavy boots started across the puncheon floor and porch\nagain. Isom could hear Steve's orders outside; the laughs and jeers and\ncurses of the men as they mounted their horses; he heard the cavalcade\npass through the gate, the old man's cackling good-by; then the horses'\nhoofs going down the mountain, and Daddy Marcum's hobbling step on the\nporch again. He was standing in the middle of the floor, full in the\nfirelight, when the old man reached the threshold--standing in a trance,\nwith a cartridge-belt in his hand.\n\n\"Good fer you, Isom--\"\n\nThe cry was apologetic, and stopped short.\n\n\"The critter's fersakcn,\" he quavered, and cowed by the boy's strange\nlook, the old man shrank away from him along the wall. But Isom seemed\nneither to see nor hear. He caught up his rifle, and, wavering an\ninstant, tossed it with the belt on the bed and ran out the door. The\nold man followed, dumb with amazement.\n\n\"Isom!\" he called, getting his wits and his tongue at last. \"Hyeh's yer\ngun! Come back, I tell ye! You've fergot yer gun! Isom! Isom!\"\n\nThe voice piped shrilly out into the darkness, and piped back without\nanswer.\n\nA steep path, dangerous even by day, ran snakelike from the cabin down\nto the water's edge. It was called Isom's path after that tragic night.\nNo mountaineer went down it thereafter without a firm faith that only by\nthe direct help of Heaven could the boy, in his flight down through the\ndark, have reached the river and the other side alive. The path dropped\nfrom ledge to ledge, and ran the brink of precipices and chasms. In a\ndozen places the boy crashed through the undergrowth from one slippery\nfold to the next below, catching at roots and stones, slipping past\ndeath a score of times, and dropping on till a flood of yellow light\nlashed the gloom before him. Just there the river was most narrow; the\nnose of a cliff swerved the current sharply across, and on the other\nside an eddy ran from it up stream. These earthly helps he had, and he\nneeded them.\n\nThere had been a rain-storm, and the waves swept him away like\nthistle-down, and beat back at him as he fought through them and stood\nchoked and panting on the other shore. He did not dare stop to rest.\nThe Marcums, too, had crossed the river up at the ford by this time, and\nwere galloping towards him; and Isom started on and up. When he reached\nthe first bench of the spur the moon was swinging over Thunderstruck\nKnob. The clouds broke as he climbed; strips of radiant sky showed\nbetween the rolling masses, and the mountain above was light and dark\nin quick succession. He had no breath when he reached the ledge that ran\nbelow old Steve's cabin, and flinging one arm above it, he fell through\nsheer exhaustion. The cabin was dark as the clump of firs behind it;\nthe inmates were unsuspecting; and Steve Marcum and his men were not\nfar below. A rumbling started under him, while he lay there and grew\nfaint--the rumble of a stone knocked from the path by a horse's hoof.\nIsom tried to halbo, but his voice stopped in a whisper, and he\npainfully drew himself upon the rock, upright under the bright moon.\nA quick oath of warning came then--it was Crump's shrill voice in the\nBrayton cabin--and Isom stumbled forward with both hands thrown up and\na gasping cry at his lips. One flash came through a port-hole of the\ncabin. A yell broke on the night--Crump's cry again--and the boy swayed\nacross the rock, and falling at the brink, dropped with a limp struggle\nout of sight.\n\n\n\n\nV.\n\nTHE news of Isom's fate reached the miller by way of Hazlan before the\nnext noon. Several men in the Brayton cabin had recognized the boy in\nthe moonlight. At daybreak they found bloodstains on the ledge and on a\nnarrow shelf a few feet farther down. Isom had slipped from one to the\nother, they said, and in his last struggle had rolled over into Dead\nCreek, and had been swept into the Cumberland.\n\nIt was Crump who had warned the Braytons. Nobody ever knew how he had\nlearned Steve Marcum's purpose. And old Brayton on his guard and in his\nown cabin was impregnable. So the Marcums, after a harmless fusillade,\nhad turned back cursing. Mocking shouts followed after them,\npistol-shots, even the scraping of a fiddle and shuffling on the ledge.\nBut they kept on, cursing across the river and back to Daddy Marcum,\nwho was standing in the porch, peering for them through the dawn, with a\nstory to tell about Isom.\n\n\"The critter was teched in the head,\" the old man said, and this was\nwhat the Braytons, too, believed. But Steve Marcum, going to search for\nIsom's body next day, gave old Gabe another theory. He told the miller\nhow Daddy Marcum had called Isom a coward, and Steve said the boy had\ngone ahead to prove he was no coward.\n\n\"He had mighty leetle call to prove it to me. Think o' his takin' ole\nBrayton all by hisself!\" he said, with a look at the yellow, heaving\nCumberland. \"'N', Lord! think o' his swimmin' that river in the dark!\"\n\nOld Gabe asked a question fiercely then and demanded the truth, and\nSteve told him about the hand-to-hand fight on the mountain-side, about\nyoung Jasper's treachery, and how the boy, who was watching the fight,\nfired just in time to save Rome. It made all plain at last--Rome's and\nSteve's denials, Isom's dinning on that one theme,' and why the boy\ncould not go to Rome and face Martha, with her own blood on his hands.\nIsom's true motive, too, was plain, and the miller told it brokenly to\nSteve, who rode away with a low whistle to tell it broadcast, and left\nthe old man rocking his body like a woman.\n\nAn hour later he rode back at a gallop to tell old Gabe to search the\nriver bank below the mill. He did not believe Isom dead. It was just his\nfeelin', he said, and one fact, that nobody else thought important--the\nBrayton canoe was gone.\n\n\"Ef he was jus' scamped by a ball,\" said Steve, \"you kin bet he tuk the\nboat, 'n' he's down thar in the bushes somewhar now waitin' fer dark.\"\n\nAnd about dusk, sure enough, old Gabe, wandering hopefully through the\nthicket below the mill, stumbled over the canoe stranded in the bushes.\nIn the new mud were the tracks of a boy's bare feet leading into the\nthicket, and the miller made straight for home. When he opened his door\nhe began to shake as if with palsy. A figure was seated on the hearth\nagainst the chimney, and the firelight was playing over the face and\nhair. The lips were parted, and the head hung limply to the breast. The\nclothes were torn to rags, and one shoulder was bare. Through the upper\nflesh of it and close to the neck was an ugly burrow clotted with blood.\nThe boy was asleep.\n\nThree nights later, in Hazlan, Sherd Raines told the people of Isom's\nflight down the mountain, across the river, and up the steep to save his\nlife by losing it. Before he was done, one gray-headed figure pressed\nfrom the darkness on one side and stood trembling under the dips. It was\nold Steve Brayton, who had fired from the cabin at Isom, and dropping\nhis Winchester, he stumbled forward with the butt of his pistol held out\nto Raines. A Marcum appeared on the other side with the muzzle of his\nWinchester down. Raines raised both hands then and imperiously called on\nevery man who had a weapon to come forward and give it up. Like children\nthey came, Marcums and Braytons, piling their arms on the rock before\nhim, shaking hands right and left, and sitting together on the mourner's\nbench.\n\nOld Brayton was humbled thereafter. He wanted to shake hands with Steve\nMarcum and make friends. But Steve grinned, and said, \"Not yit,\" and\nwent off into the bushes. A few days later he went to Hazlan of his own\naccord and gave up his gun to Raines. He wouldn't shake hands with old\nBrayton, he said, nor with any other man who would hire another man\nto do his \"killin';\" but he promised to fight no more, and he kept his\nword.\n\nA flood followed on New Year's day. Old Gabe's canoe--his second\ncanoe--was gone, and a Marcum and a Brayton worked side by side at the\nmill hollowing out another. The miller sat at the door whittling.\n\n\"'Pears like folks is havin' bad luck with thar dugouts.\" said Brayton.\n\"Some trifin' cuss took old Steve Brayton's jes to cross the river,\nwithout the grace to tie it to the bank, let 'lone takin' it back. I've\nheard ez how Aunt Sally Day's boy Ben, who was a-fishin' that evenin,\nsays ez how he seed Isom's harnt a-floatin' across the river in it,\nwithout techin' a paddle.\"\n\nThe Marcum laughed. \"Idgits is thick over hyeh,\" he said. \"Ben's\na-gittin' wuss sence Isom was killed. Yes, I recollect Gabe hyeh lost a\ncanoe jus' atter a flood more'n a year ago, when Rome Stetson 'n' Marthy\nLewallen went a-gallivantin' out' n the mountains together. Hyeh's\nanother flood, 'n' old Gabe's dugout gone agin.\" The miller raised\na covert glance of suspicion from under his hat, but the Marcum was\nlaughing. \"Ye oughter put a trace-chain on this un,\" he added. \"A rope\ngits rotten in the water, 'n' a tide is mighty apt to break it.\"\n\nOld Gabe said that \"mebbe that wus so,\" but he had no chain to waste; he\nreckoned a rope was strong enough, and he started home.\n\n\"Old Gabe don't seem to keer much now 'bout Isom,\" said the Brayton.\n\"Folks say he tuk on so awful at fust that hit looked like he wus goin'\ncrazy. He's gittin' downright peert again. Hello!\"\n\nBud Vickers was carrying a piece of news down to Hazlan, and he pulled\nup his horse to deliver it. Aunt Sally Day's dog had been seen playing\nin the Breathitt road with the frame of a human foot. Some boys had\nfound not far away, behind a withered \"blind,\" a heap of rags and bones.\nEli Crump had not been seen in Hazlan since the night of the Marcum\nraid.\n\n\"Well, ef hit was Eli,\" said the Brayton, waggishly, \"we're all goin' to\nbe saved. Eli's case 'll come fust, an' ef thar's only one Jedgment Day,\nthe Lord 'll nuver git to us.\"\n\nThe three chuckled, while old Gabe sat dreaming at his gate. The boy had\nlain quiet during the weeks of his getting well, absorbed in one aim--to\nkeep hidden until he was strong enough to get to Rome. On the last night\nthe miller had raised one of the old hearth-stones and had given him the\nhire of many years. At daybreak the lad drifted away. Now old Gabe was\nfollowing him down the river and on to the dim mountain line, where the\nboy's figure was plain for a moment against the sky, and then was lost.\n\nThe clouds in the west had turned gray and the crescent had broken\nthe gloom of the woods into shadows when the miller rose. One star was\ncoming over Black Mountain from the east. It was the Star of Bethlehem\nto old Gabe; and, starlike on both sides of the Cumberland, answering\nfires from cabin hearths were giving back its message at last.\n\n\"Thar hain't nothin' to hender Rome 'n' Marthy now. I nuver knowed\nanybody to stay 'way from these mount'ins ef he could git back; 'n' Isom\nsaid he'd fetch 'em. Thar hain't nothin' to hender--nothin' now.\"\n\nOn the stoop of the cabin the miller turned to look again, and then on\nthe last Stetson the door was closed.\n\nTHE END\n\n\n\n\n\nEnd of the Project Gutenberg EBook of The Last Stetson, by John Fox Jr.\n\n*** ","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \n# How to Create Chemistry with Anyone\n**Other Books by Leil Lowndes**\n\n_How to Talk to Anyone:  \n92 Little Tricks for Big Success in Relationships_\n\n_How to Instantly Connect with Anyone:  \n96 More Little Tricks for Big Success in Relationships_\n\n_How to Make Anyone Fall in Love with You_\n\n_Good-Bye to Shy: 85 ShyBusters that Work!_\n\n_Undercover Sex Signals_\n\n_How to Be a People Magnet:  \nFinding Friends\u2014and Lovers\u2014and Keeping Them for Life_\n\n# How to  \nCreate  \nChemistry  \nwith Anyone\n\n**75 Ways to Spark It Fast . . . and Make It Last**\n\nLeil Lowndes\n\nCopyright \u00a9 2013 by Leil Lowndes\n\nAll rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher. For information, address Da Capo Press, 44 Farnsworth Street, 3rd Floor, Boston, MA 02210.\n\nEditorial production by Lori Hobkirk at the Book Factory.  \nSet in 11.5 point Minion Pro by the Perseus Books Group.\n\nCataloging-in-Publication data for this book is available from the Library of Congress.  \nFirst Da Capo Press edition 2013  \nISBN 978-0-7382-1612-6 (e-book)\n\nPublished by Da Capo Press  \nA Member of the Perseus Books Group  \nwww.dacapopress.com\n\nDa Capo Press books are available at special discounts for bulk purchases in the U.S. by corporations, institutions, and other organizations. For more information, please contact the Special Markets Department at the Perseus Books Group, 2300 Chestnut Street, Suite 200, Philadelphia, PA 19103, or call (800) 810-4145, ext. 5000, or e-mail special.markets@perseusbooks.com.\n\n10 9 8 7 6 5 4 3 2 1\n\n## Contents\n\n**PROLOGUE**\n\n**What Just Happened to Me?**\n\n **_How Fast Does Chemistry Happen?_**\n\n **_Does Timing Count?_**\n\n **_What's Happening to My Brain?_**\n\n **_Who Discovered Why We Feel Chemistry?_**\n\n **_But I Thought Chemistry Is Either There . . . Or It Isn't_**\n\n**Chemistry Makes the World Go 'Round**\n\n **_Why Did It Happen?_**\n\n **_Are You the Master or the Slave of Love?_**\n\n**Why Are We So Naive About Love?**\n\n **_Skull and Crossbones_**\n\n **_Now the Good News_**\n\n**A Few Important Housekeeping Notes (Don't Skip These!)**\n\nCHAPTER 1\n\n**LOVE IN LIMBIC LAND**\n\n**Where Love Lives**\n\n **_Your Amygdala Nickname_ : Ms. Emotions**\n\n **_Your Hippocampus Nickname_ : Mr. Memory**\n\n **_Your Hypothalamus Nickname_ : Mr. Action**\n\n **_Your Caudate Nucleus and Environs Nickname_ : Pleasure Island**\n\n **_Your Prefrontal Cortex Nickname_ : The Professor**\n\n**The Six Sexy Chemicals of Love**\n\n **_Dopamine_**\n\n **_Serotonin_**\n\n **_Testosterone_**\n\n **_Estrogen_**\n\n **_Oxytocin_**\n\n **_Vasopressin_**\n\nCHAPTER 2\n\n**CHEMISTRY AT FIRST SIGHT**\n\n**\"All Men Are Animals\"**\n\n**\"All Women Are Gold Diggers\"**\n\n**Huntresses, Cut the Animals Some Slack**\n\n **_Sex Scrambles His Brain_**\n\n **_Blame Mom and Co._**\n\n**Hunters, Cut the Gold Diggers Some Slack**\n\n**What Sparks a Hunter's Chemistry\u2014and Why**\n\n **_Babies \"R\" Us_**\n\n **_Let's Break It Down to Body Parts. Guys Do._**\n\n**What Sparks a Huntress's Chemistry\u2014and Why**\n\n**Planet Earth's Colossal Unisex Chemistry Sparker**\n\n **_Cliff Notes_**\n\nCHAPTER 3\n\n**HOW TO SPARK CHEMISTRY TO ATTRACT**\n\n**Forget the Golden Rule When Trying to Spark Chemistry**\n\n**Fight, Flight, or Chemistry at First Sight?**\n\n **_Get Wise to the Geisha in Every Woman_**\n\n **_Huntresses, Fight Your Instincts_**\n\n **_Eye Contact is Not Unisex_**\n\n**Hunters' Fishing Trip**\n\n **_Your Actions Speak Louder Than Looks_**\n\n **_Smiling Is Not Just Smiling_**\n\n **_Does She Really Care What I Wear?_**\n\n **_Why Coordination Counts_**\n\n**Huntresses' Fishing Trip**\n\n **_What Went Wrong?_**\n\n **_How the \"Beauty Challenged\" Can Ace the \"Tens\"_**\n\n **_A Hunter's Brain Is Like a Light Switch with a Delayed Connection_**\n\n **_Hunting Gear for Trapping First-Class Quarry_**\n\n **_Forget Fashion, Think Flirtation_**\n\n**Sizzling Sparkers for Serious Huntresses**\n\n **_One Huntress's Triple-X Tip_**\n\n **_Erotic Truth Is Stranger than Fiction_**\n\nCHAPTER 4\n\n**HOW TO SPARK CYBER CHEMISTRY**\n\n**The Successful Online Hunter**\n\n **_Hunters' Clothes_**\n\n **_Hunters' Background_**\n\n**The Successful Online Huntress**\n\n **_Huntresses' Clothes_**\n\n **_Huntresses' Background_**\n\n **_Number of Photos and Poses_**\n\n**A Devious Digital Tip for Both Sexes**\n\n**Your Online Name**\n\n**Your First Message**\n\n **_Who Is Your Quarry_ Really _Interested In?_**\n\n **_Talk About What You Read Between the Lines_**\n\n **_A Beautiful Tip from an Unsightly Man_**\n\nCHAPTER 5\n\n**HOW TO SPARK CHEMISTRY IN YOUR FIRST CONVERSATION**\n\n**Chemistry Sparking \"Pickup Lines.\" (Hunters, Bite Your Tongue!)**\n\n**What Every Woman Needs to Know\u2014But Few Do**\n\n **_Guys Just Don't Get It!_**\n\n**The Case for Female Proceptivity**\n\n **_Huntresses' Pickup Lines (Try a Few\u2014He'll Love It!)_**\n\n **_But Won't He Think I'm Being Aggressive?_**\n\n **_The Physical Side of \"Hello\"_**\n\n**Spark Your Quarry During the First Conversation**\n\n **_The Rapid Transit System Called the Female Brain_**\n\n **_She Reads Between the Lines\u2014So Talk Between the Lines_**\n\n **_A Pivotal Question_**\n\n **_Today the Who-Impresses-Whom Rules Are Reversed_**\n\n**Huntress, Are You One in a Million, or One Among a Million?**\n\n **_Begin the Bait and Switch_**\n\n **_Double Name Talk_**\n\n **_So Far, So Good_**\n\nCHAPTER 6\n\n**HOW TO SPARK CHEMISTRY ON DATES**\n\n**Hunters, \"Sell\" Her on the Date Like a Pro**\n\n**Crack the Old Hard-to-Get Chestnut**\n\n**How to Get the Date without Asking**\n\n**The Most Chemistry-Sparking First Date**\n\n **_The First Half of the Date_**\n\n **_The Second Half of the Date_**\n\n **_Eating Sparkers_**\n\n**Play Games with Your Quarry**\n\n**Chemistry-Sparking Conversation on Every Date**\n\n **_He Says\/She Says, What?_**\n\n **_Hunters, Follow Your Quarry's Rhythm_**\n\n **_Huntresses, Don't Fill in the Blanks_**\n\n **_Give a \"You Fit My LoveMap\" Feeling_**\n\n **_So . . . How Should I Listen?_**\n\n**I Want a Man Who Makes Me Laugh**\n\n **_What Makes Huntresses Laugh_**\n\n **_What Makes Hunters Laugh_**\n\nCHAPTER 7\n\n**HOW TO SPARK CHEMISTRY FOR SEX**\n\n**Huntresses, How Soon Should You Have Sex?**\n\n**Why Sex Blows Your Mind**\n\n **_Huntresses, Sex Has an Even Bigger Effect on You!_**\n\n **_Hunters, Hints for You on First-Date Sex_**\n\n**Foreplay Begins Way Long Before the Bedroom**\n\n **_Set the Stage for Sex_**\n\n **_When It's Time to Get Naked_**\n\n **_Oral Sex (It's Not What You're Thinking)_**\n\n **_The Educational Value of Daytime Soaps_**\n\n **_\"Oh, Great, How Do I Find Out What She Really Wants?\"_**\n\n**Why She Needs All That Stuff First**\n\n **_Squares versus Swirls_**\n\n **_Women Don't Come with Pull-Down Menus and Online Help_**\n\n **_Who Stole the G-Spot?_**\n\n**Huntresses Are Naturals**\n\n **_Go Where Women Aren't Allowed_**\n\n **_Better Yet, Get It Straight from the Horse's Mouth_**\n\n**Why Won't He Cuddle After Sex?**\n\n**Do You Decide Who Turns You On\u2014or Does Mother Nature?**\n\n **_If the First Kiss Stinks, Forget About It_**\n\n **_Huntresses on the Pill, Read This Carefully_**\n\nCHAPTER 8\n\n**HOW TO SPARK CHEMISTRY FOR A RELATIONSHIP**\n\n**Mars and Venus Told You the _What_ , Now the _Why_**\n\n**The Ball of Yarn vs. the Stack of Boxes**\n\n **_His Box Brain_**\n\n **_Her Yarn-Ball Brain_**\n\n **_P.S. for Huntresses_**\n\n**What Are You Thinking About?**\n\n**How Do You Feel About That?**\n\n **_Why He's Tongue-Tied_**\n\n **_Learn a Very Different Body Language_**\n\n**Shall We Talk It Out, or Slug It Out?**\n\nCHAPTER 9\n\n**HOW TO SPARK CHEMISTRY FOR FALLING IN LOVE**\n\n**Is This Quarry a Keeper?**\n\n **_Why Is He Afraid to Commit?_**\n\n**I See Life That Way Too!**\n\n**How Do You Define \"Togetherness\"?**\n\n**\"In Sickness and in Health\"**\n\n**The Magnificent Twenty-First-Century Mantra\u2014Personal Growth**\n\n**\"Vet\" Your Quarry**\n\n**May I Have a Word with Young Lovers?**\n\n **_Don't Marry Until You're Myelinated (Parents, Don't Miss This!)_**\n\n **_Shall We Make a Federal Case Out of It?_**\n\nCHAPTER 10\n\n**HOW TO SPARK CHEMISTRY FOR A LIFETIME OF LOVE**\n\n**Making Beautiful Beginnings Last Forever**\n\n**The Two-Year Itch**\n\n**A Little Mouse in the Meadow Holds a Secret of Togetherness**\n\n**How to Spark Long-Lasting Chemistry Anywhere, Anytime**\n\n **_Time Out, Sports Fans_**\n\n **_What If I'm Ticked and Don't Feel Like Touching Right Now?_**\n\n**Couples Who Play Together Stay Together**\n\n**Laughing Is Just Another Way to Say \"I Love You\"**\n\n**What About Long-Term Sizzling Sex?**\n\n **_Here's the Good News\u2014No, the Great News!_**\n\n **_Why \"Quickies\" Count_**\n\n **_Turnabout Is Fair Foreplay_**\n\n**Everyone Has Something They Need to Hear**\n\n**Hunters, _Why_ Her Feeling Loved Creates Lasting Chemistry**\n\n**Huntresses, _Why_ His Feeling Needed Creates Lasting Chemistry**\n\n**\"To What Do You Credit Your Lifelong Love?\"**\n\n**What to Expect When You're Expecting Lasting Love**\n\n**_References_**\n\n**_About the Author_**\n\n## Prologue\n\nLove. The word is but a pale shadow of the ecstasy it brings us. We spend years, sometimes a lifetime longing for it. If we're lucky, we find it\u2014and pray it lasts. But if it doesn't, we seek it again and again. Face it\u2014we're hooked!\n\nCupid's quiver contains chemicals that bathe the brain and compel us to act in ways we can't fathom. When you have finished reading _How to Create Chemistry with Anyone_ , though, you will. You'll know about love in a way that very few people currently do. And to the extent it is possible, you will be able to Spark romantic Chemistry with the man or woman of your choice.\n\nYou will also discover how and why nature injects you with different chemicals at various phases in your relationship and throughout your life. Finally, you will learn how to work with the changing Chemistry and create bonding chemicals so you both can \"live happily ever after\"\u2014not just \"until divorce do us part.\" Your new knowledge will not banish Cupid to the greeting card aisle, but it will help keep him in your life forever.\n\n### What Just Happened to Me?\n\nYou've felt it. Your heart pounds like a jackhammer, your tongue feels like sandpaper, your palms turn into a waterfall, and your words mysteriously start missing syllables. \"Hello\" becomes the hardest word in the English language.\n\nBut as your vital signs return to normal and a semblance of rationality sets in, you anxiously ask yourself if this person you just met felt the same Chemistry for you. If not, unrequited potential lovers used to think nothing could be done except sulk, swear, cry, or kick the cat. They were wrong, as you'll soon learn.\n\nWhy does this particular person blow you away like a tornado through a trailer camp, whereas with others, it's ennui at first sight? Because, unbeknownst to you, buried deeply in your brain, you have the ability to size someone up instantly. This skill has strong evolutionary roots, dating from when an instantaneous \"fight or flight\" decision was a matter of life or death.\n\nThroughout the centuries our crystal-ball capabilities have evolved with us. Just as DNA experts can tell a lot about a person from a sliver of his toenail, human beings have developed an incredible subconscious capability to sense whether someone will be fun to be with, fulfill their needs, and match a million other qualities on a very subjective laundry list of longings.\n\nYou've been setting the stage for love all through your life, just waiting for the star of the show to walk on the set. Your experiences from birth to the second you picked up this book draw a chart of the type of person you would\u2014or even could\u2014fall in love with. This map\u2014your \"LoveMap\" as it is called\u2014bears the mark of your unique individuality and can be specific down to details of physiognomy, personality, intellect, ambition, sense of humor, and hundreds of etceteras. A very new field called \"interpersonal neurobiology\" also shows how your brain constantly rewires itself through your relationships. Even a brief dalliance with someone you'd never want to see again could tweak your LoveMap forever.\n\nBy far the largest part of this map was charted in childhood before you were age five. If you were fortunate, you were surrounded by people you loved and who loved you, primarily your relatives. Their proximity programmed a tendency in you to feel Chemistry with someone who looks like she could be from the same gene pool or the type of man who was around when you were a child. Have you ever noticed how many couples look alike?\n\nThe influences on this diagram of who you could love ranges from milliseconds ago to millions of years before. To complicate matters further, your memories get rewritten every time they are invoked. Is it any wonder that Chemistry is confusing?\n\n#### _How Fast Does Chemistry Happen?_\n\nHow much time does it take to feel Chemistry? Compare the speed to a big ruckus in movie theaters back in the 1950s concerning subliminal advertising. Clever Madison Avenue types flashed words like \"Hungry?\" \"Get popcorn!\" and product names like \"Lipton Tea\" on the screen for less than a thousandth of a second during a movie. The messages flickered too fast for the audience to read. In fact, moviegoers reported they hadn't seen anything. But theaters sold a lot more popcorn on those particular days!\n\nUnbeknownst to the audience, the \"unseen\" flashes on the screen not only made them hungry or thirsty; it also told them specifically what brands they wanted. Previously, moviegoers who used to order just tea now requested it by brand\u2014Lipton, of course. Is it any surprise that neuroimaging shows you feel that Spark in a fifth of a second?\n\n#### _Does Timing Count?_\n\nUnquestionably. It could have a lot to do with other events happening in life when you meet. Perhaps he is in love with someone else or maybe she just suffered a tragic loss. You might find yourself in conversation with a smart, sensitive, stunning, sensuous single someone who would be the perfect partner for you. This spectacular specimen of humanity blows you away instantly, but she is stifling a yawn. Or he is glancing over your shoulder for someone he'd rather talk to.\n\n\"What's wrong with me?\" you scream at yourself between frantic heartbeats. Nothing. It's simply that thousands of other current emotional, physical, personal, or professional issues could close off his or her receptivity to you. Another time, another place, you might meet, and the magnetic field would make the two of you want to fall into each other's arms and bond together forever. Not only that, but for women it depends on the time of the month you meet  (More about that later.) Yes, timing counts.\n\n#### _What's Happening to My Brain?_\n\nBefore you spotted this special person the one hundred billion nerve cells in your head were, relatively speaking, in what Cognitive Science calls \"resting potential.\" But of course, they weren't really resting. The comparatively lethargic neurons were tediously _neurotransmitting_ (we'll call it \"texting\") messages to each other about the weather, the boring party, the tasteless snacks, the whatever.\n\nThen _KA-POW_! When your eyes spot this special someone across the crowded room, bar, bus station, McDonald's, or wherever, it's neurons gone wild! They hysterically contact their colleague neurons who live in other neighborhoods of the brain across tiny rivers called _synapses_ to tell them the electrifying news. That's what neurologists call _action potential\u2014_ and what we call _Wow!_\n\nPractically all of the techniques in the first sections of this book tell you how to create a neuronal response in someone to get feelgood chemicals, primarily one called _dopamine_. gushing into a brain region that Cognitive Science calls your _pleasure_ or _reward center_. This response is so incredible that your target person gets a high, chemically similar to cocaine.\n\n#### _Who Discovered Why We Feel Chemistry?_\n\nAs you are reading this, neuroscientists are breaking new ground tracking the galaxy of nerve cells in your brain that are wired together with a million billion connections. Obviously, romance was not their motivation. These pioneering professionals' goal was far more significant, saving lives and preserving human physical and mental health. However, seekers of love benefit greatly by grasping their extraordinary contributions. We can now understand\u2014and, to a certain extent, manipulate\u2014the chemicals that marinate his brain or flow through her body when you meet. Let's call this person your \"Quarry.\" And you are the \"Hunter\" or \"Huntress\" of hearts.\n\nIt is not only the cognitive community we must thank for unraveling the mysteries of romantic Chemistry; let's also tip our hats to evolutionary biologists and psychologists who further explore why we make such sudden and sometimes puzzling choices in relationships. These fields hold the key to why our neurons go berserk spotting one person and, comparatively speaking, hardly lift their sleepy heads for another.\n\n#### _But I Thought Chemistry Is Either There . . . Or It Isn't_\n\nMost people would agree with you. There are, of course, certain elements that, no matter what you say or do, can't change a thing when it comes to creating Chemistry. For instance, you can't change your or your potential partner's face, body, genotype, phenotype, or DNA. Additionally, by the time you are ready for love and sex, scillions of unconscious associations to pain or pleasure are etched in both your brains: How his stepdad dealt with him. How kindergarten schoolmates treated her. Who he previously loved. Who she hated. The instant you come into sight, all of the \"whos,\" \"whats,\" and \"whys\" of your potential partner's previous relationships resurrect neuronal activity from the past.\n\n\"Whew, if it's so complicated,\" you're thinking, \"I can't possibly make my Quarry feel Chemistry for me.\" Sure, and people once thought, \"The world is flat,\" \"Heavier objects fall faster than light,\" and \"All planets revolve around the earth.\" Scientists have tossed those three myths into the trash like dirty Kleenex. And the first just landed on top of them. Chemistry responds to very specific stimuli, many of which you _can_ affect.\n\nI do want to make one thing perfectly clear before we start, however: You can never have 100 percent control over a person's chemical reaction to you due to your Quarry's previous experiences, brain structure, and other factors listed above. But that's only part of the story. Mother Nature plays an equally large if not even bigger role in romance. You're going to learn how to be her coconspirator at the beginning of a relationship to capture your Quarry. Then you'll learn how to break away from the common forces that can demolish Chemistry between couples and destroy lasting love.\n\nYou are fortunate to be the first generation to benefit from fresh insights of Cognitive Science and the relatively new field called Developmental Evolution. Understanding what's in your Quarry's brain and how it develops will permit you to be more than just a pawn in nature's game. To a certain extent you can create specific chemically based emotions in the man or woman of your choice. The slight peek into neuroscience and new discoveries in evolutionary psychology in this book will give you further understanding of why and how you can create Chemistry and turn it into lasting love.\n\n### Chemistry Makes the World Go 'Round\n\nThey say \"Love makes the world go 'round.\" But just as accurate is \"Chemistry makes the world go 'round.\" Why? Because feeling Chemistry with someone is the precursor to passion, which leads to love, which leads to commitment. And that can lead to contentment, children, companionship, and many of life's greatest joys.\n\nWhat do you think? Is feeling Chemistry and then falling in love a decision? A destiny? A choice? Some say you _decide_ to fall in love. Sure, that's as easy as _deciding_ to stop breathing or cherishing your children. Others say it's _destiny_. Well, move to an archipelago in Antarctica and wait for your _destined_ one to come along. Many people think it's a _choice_. Of course, just like _choosing_ not to eat, drink, or sleep.\n\nNeurologists have proven that it is something else entirely. It is a _condition_ that involves neurons, neurotransmitters, hormones, receptors, and circuits in your brain. They define \"being in love\" as\u2014get ready for a long sentence\u2014an \"elevated activity in the brain pathways which cause feelings of euphoria, strong motivation, and heightened energy which can induce sleeplessness, loss of appetite, and obsessive thinking about the beloved.\"\n\nThat doesn't sound like much fun! And how crass to reduce it all to a rush of chemicals gushing through three pounds of gray slimy sponge under your scalp. Nobody wants to hear that, least of all me.\n\nWhen humans first began to explore the sun, some people worried that God would be pushed aside. Likewise, some now fear discovering that love is simply a \"condition\" will make it less magic. Not at all! Understanding what love is just makes us a lot smarter in our choice of partner and teaches us how to keep that love alive.\n\nBesides, who says a condition has to be crass or that a motivation system can't feel like magic? We know what it is. It's _love_ , the greatest happiness known to humankind, and it has no parallel in human experience. Skeptics aside, love truly can last a lifetime and get better and better. But only if you recognize the powerful neurological, chemical, and evolutionary forces controlling it.\n\n#### _Why Did It Happen?_\n\nDoes it really matter? The two of you fell in love, and life will never be the same. You have found \"the one\"\u2014the one you always knew would come along. Early love is the most exhilarating, extraordinary, ecstatic, and unforgettable part of the relationship. Your brain is brimming over with the intoxicating chemicals we'll soon meet called _dopamine, serotonin_ , and incipient _oxytocin_ and _vasopressin_. Hormones are at their highest levels ever. _Testosterone_ and _estrogen_ can hardly contain themselves. You and your lover want to cling tightly together and never part. \"It's delightful, it's delicious, it's delectable, it's delirious,\" Mother Nature is urging. \"Let yourself go!\" This period is the best part\u2014at least until the beauty of long-lasting lifetime love sets in.\n\nSo you cast off in the Love Boat with carefree abandon. But is it true that the course of true love never did run smooth? Sometimes it seems that way. Often, when embarking on a love affair, you think you are in control. When you first step into the Love Boat, it's like a sailboat on a calm sea on a sunny morning. Everything is beautiful. All's right with the world. You are sailing to Happily-Ever-After Land. An occasional gust of wind sweeps across the sea, and you break into song as it tingles your skin. Then the wind picks up, and that makes it even more exciting. You laugh out loud as you adjust the sails. The boat is a bit rocky, but you have faith it's propelling you toward ecstasy.\n\nSuddenly the wind changes and you fear you may be going off course. What happened? What went wrong? The tempest makes you all the more desperate to get back on track. Your heart beats faster. But each time you think you've regained control, a bigger wind comes up.\n\nNow there's an uncomfortable chill in the air, and the waves are rocky. Threatening clouds race across the sky and it starts to rain. The stabilizer cracks, and you desperately struggle to keep the boat afloat in the downpour. Soon darkness surrounds you, and you are helplessly tossed around. You are caught in the perfect storm. Will the Love Boat reach \"happily ever after?\" Or will it sink? It's up to you and how you'll use your new wisdom.\n\n#### _Are You the Master or the Slave of Love?_\n\nIf only we could all be as wise in love as an Italian captain of a cruise ship, whom I'll introduce you to later, was in his profession. Captain Giorgio Accornero was once offered the command of a 150,000-ton tanker. He refused. The flabbergasted shipowner asked, \"But why? No captain has ever refused before. It is a great honor to be the master of such a large ship.\"\n\nCaptain Accornero replied, \"Because, sir, I want to control the ship. I don't want her to control me.\"\n\nThe problem is that most people let the Love Boat master them rather than the other way around. The only way to be in control of the relationship is to admit that, in addition to magic, love is indeed a condition, a blissful one that we want to stay afloat forever.\n\nJust as knowing how to create rain, feed animals, and nurture plants doesn't make the world less wondrous, likewise, wisdom about creating, feeding, and nurturing Chemistry doesn't make it less magnificent. In fact, knowledge about love gives you the ability to make love stronger and last longer.\n\n### Why Are We So Naive About Love?\n\nOther than those life-sustaining needs like air, water, and shelter, I can think of no other need except love that begins at birth and lasts until we die. Yet I am shocked\u2014no, stunned\u2014by how little people know about this exquisite emotion that we idolize in music, film, and poetry, not to mention personally throughout our lives. We live for it. Some have died for it. In the song, \"Oh Love,\" Brad Paisley and Carrie Underwood sang, \"You're the simple truth. And you're the biggest mystery.\" Up until recently, that it was. Happily, however, love is no longer a mystery, and we're learning more about it every day.\n\nFor centuries philosophers, psychiatrists, and way too many lay people scratched their heads, made assumptions, and spouted theories about this phenomenon. Linguists, every bit as befuddled as the rest of them, wound up giving it a then-meaningless, cop-out name: \"Chemistry.\" Little did they know how right they were!\n\nPerhaps the wordsmiths were inspired by Aldous Huxley, who, in the early 1900s, wrote, \"In one way or another, all our experiences are chemically conditioned.\" In his day, thanks to visual tools like the telescope, scientists could explore outer space. But the astronomically closer inner space, the brain, was beyond the reach of the best of them. How ironic. They had the equipment to view planets light years away but not the tools to travel a few centimeters in the other direction. Today, however, extraordinary instruments like _f_ MRI, CT, PET scanners, and a few other brain-imaging devices with mysterious acronyms permit cognitive neuroscientists to examine precisely what's going on in that wrinkly gray tofu between your ears.\n\nThe relatively new field of Cognitive Science\u2014defined as the interdisciplinary study of the mind embracing philosophy, psychology, neuroscience, linguistics, and anthropology\u2014is making phenomenal progress in putting pieces of the puzzle together. Although there are still unanswered questions\u2014as there always will be in life\u2014scientists have uncovered the influences that generate the electrochemical activity in your brain, making you feel that magical chemical Spark with someone. They have also discovered how identical chemicals can influence a male brain and a female brain and nervous system in very different ways. Thus, the ways to Spark romantic Chemistry in this book will be gender specific\u2014 _very_ gender specific!\n\nMany people stubbornly insist that igniting Chemistry and making it last are not possible. They are wrong. Making it last is indeed doable, but _only_ if you understand what kind of Chemistry we're talking about and work within the context of the following:\n\n1.Your neuroanatomy is constructed in a very different way from that of the other sex, almost as different as your bodies.\n\n2.Your brain is saturated by dissimilar drugs that dictate how you think, feel, and act as well as what you respond to.\n\n3.Distinct evolutionary influences are working on males and females.\n\n4.Mother Nature has carved out diverse roles for you.\n\n5.As a relationship matures, it is natural and unavoidable that different chemicals flood your brain and body, affecting your emotions and desires.\n\nI will give you techniques called _Chemistry Sparkers_ to generate these specific bonding chemicals that keep you and your partner together. And, to the extent it is possible, regenerate some of the earlier thrilling ones.\n\n#### _Skull and Crossbones_\n\nAlthough love is the most exquisite emotion known to man and woman since the beginning of time, a sinister face sometimes hides behind an alluring mask. We all know beauty can be deceiving. If you're unaware of what you're dealing with, love can destroy you. Some people feel a Spark with someone who treats them badly because it's a familiar dynamic from childhood. Other common fragilities and shared unhealthy pathology can create damaging needs. Sadly, some people replicate these detrimental and sometimes risky relationships over and over again.\n\nYou've seen the TV ads for antidepressants that show previously despondent people dancing with delight through meadows tossing daisies at each other. Unless you listen carefully, you don't hear the lightning-speed voiceover saying, \"Side effects may include nausea, vomiting, internal hemorrhaging, epileptic seizures, respiratory arrest, coma, loss of hair, impotency, and, in rare cases, death. Ask your doctor if [this medication] is right for you.\"\n\nLikewise, when feeling that Spark, everyone is deaf to that internal voice warning, \"Side effects may include hyperactivity, loss of appetite, trembling, obsessive thinking, compulsive acts, and symptoms associated with mental illness. Ask yourself if [this particular person] is right for you.\"\n\nSometimes, when writing this book, I closed my laptop and gazed out the window, questioning whether I was writing a tutorial on how to create a bomb. The word _Chemistry_ , in the sense we're using it, is indeed like a chemical weapon that, if you are not careful, can claim you as a victim. I hope you get more from this book than you thought you would. Like learning to tell which mushrooms are poisonous, I want you to realize when you should run from Chemistry with someone before it thrusts you into a rotten relationship.\n\n#### _Now the Good News_\n\nThe other valuable lesson this book teaches is essential to your long-term happiness. Mother Nature makes the type of Chemistry between lovers change over time. Even in the best of relationships, it will. Don't let the books that tell you you'll want to jump each other's bones until you are eighty fool you. You can, however, create bonding chemicals that make love stay longer and generate delicious tides of sex more often. Unbeknownst to most people, these chemicals that replace the early hot ones are far more precious and essential to a fulfilling life than the sizzling kind. They are definitely not household names like _testosterone_ and _estrogen_ , and they are relatively unknown by people outside science-related fields; however, they are the essence of long-lasting happy relationships.\n\nFor the moment please use your new knowledge on how to create the more sizzling hot Chemistry that we'll talk about in the first sections wisely. Despite the overwhelming lust that you are absolutely sure is true love, slow down. Love can be as fragile and fleeting as an ice formation on the petal of a rose, or it can be as strong and enduring as an oak in the Petrified Forest.\n\nUsually, if you are very young, it's more like the former. It has nothing to do with intelligence. You could be as smart as Albert Einstein, Steven Hawking, and Lisa Simpson all rolled into one, but no matter how brilliant you are, your neural connections are not fully developed until about age twenty-five. In other words, you won't be able make the wisest decision about a partner until then. However, the love and passion chemicals gushing through your brain can blind you to reality at any age. That's part of Mother Nature's plan, which we'll discuss shortly. I pray this book will help you avoid the pattern of perceived love with the wrong person, marriage, disillusionment, divorce, and the tragedy of children left behind.\n\n### A Few Important Housekeeping Notes (Don't Skip These!)\n\nStaring at the blank computer screen with my anxious fingers hovering above the keyboard, I had to make a few choices before tapping the first key. Here's what the Ouija Board decided.\n\n#### _Gender Grammar_\n\nAh, the scourge of the English language, gender-specific pronouns! There was a day when the masculine \"he,\" \"his,\" and \"him\" graced most prose when referring to a nonspecific gender. Thankfully we are now an equal-opportunity pronoun language.\n\nIn the history of the world, however, there has never been progress without problems. The phrases \"he or she\" and \"his or her\" are unbearably clunky. Writer's Commandment Number One is \"Thou shalt avoid clunkiness at all costs.\" So when a sentence is not gender specific, I will sometimes alternate \"he\" or \"she\" in the same paragraph. After much soul searching, I chose minor confusion over major clunkiness. I hope you won't find it too distracting.\n\nYou may also encounter the signs for male and female, \u2642 and \u2640 respectively. The techniques for Hunters are marked with \u2642, and Huntresses with \u2640. The Chemistry Sparkers for both sexes are indicated by \u2640\u2642. You will find an almost equal number of Sparkers for men and for women. However don't skip the Sparkers for the other sex. Not only will they give you tremendous gender insights, but you'll also understand why your own Chemistry gets Sparked and by whom.\n\n#### _Gender Generalizations_\n\nUnfortunately, most sociological studies make a gross gender generalization saying females are primarily searching for a mate, and males just want a fling. There are many millions of exceptions, but until proven otherwise, I'll follow their findings that this is primarily the case. Keep in mind, however, that most men do want to marry and settle down someday. And I've yet to meet a modern woman who hasn't had a quickie tryst\u2014or at least hasn't thought about it!\n\nDevelopmental evolution shows that nothing is written in stone, especially human relationships. Are we headed toward a world of mainly Mr. Moms and predominately working wives? Probably not in our lifetime. But who knows what the future holds?\n\n#### _Same-Sex Relationships_\n\nGay and lesbian Chemistry is every bit as powerful as opposite-sex relationships, and often their love can last even longer due to samesex couples' neuroanatomical and neurochemical similarities. In this book, however, I speak primarily about male-female relationships because the structural, chemical, and functional brain differences are germane to the findings. They cause many of the most common relationship problems.\n\n#### _Political Correctness_\n\nThis brings me to a related maintenance issue. Should I obey or break the unspoken rule about avoiding gender stereotypes such as women being talkative and men being terse, women being in touch with their feelings and men asking, \"What feelings?\" I didn't have to think twice about this one because it is all biologically brain based. Whenever political correctness was contrary to the truth, I chose the latter.\n\nGo ahead, PC police. Arrest me.\n\n#### _Places Where Singles Mingle_\n\nHappily, most meetings don't take place in a bar these days. Because many studies were conducted in pubs and at singles parties, however, I will sometimes use those locales as examples when speaking generically. Obviously, please dub in a venue more pertinent to your lifestyle, whether it's educational events, religious gatherings, concerts, or anywhere\u2014including in front of your computer! The number of relationships that start with a cursor click is increasing exponentially. I'll also give you some deliciously devious but not untruthful ways to Spark Chemistry online before the first flesh-and-blood meeting.\n\n#### _Big Sparks and Little Sparks_\n\nThere are two ways to Spark Chemistry. The most renowned is the instantaneous, split-second BLAM. Unfortunately, unless you fit in your Quarry's LoveMap\u2014or invoke a \"forgotten\" intense adult experience\u2014it's an almost-impossible task. So sometimes when I talk of \"Sparkers,\" I'm referring to smaller pops that, when added up with a flurry of others, make the big Spark.\n\n#### _How to Make Anyone Fall in Love with You_\n\nThose of you who have read my first book on love by the above title will find this one quite different in two important ways. In _How to Make Anyone Fall in Love with You_ , I shared eighty-five techniques to, as the title promised, make someone fall in love with you. Each \"Little Trick,\" as I called them, was based on the latest sociological studies at the time. _How to Create Chemistry with Anyone_ , however, deals with the most recent and current findings in the groundbreaking Cognitive Sciences and neuroimaging fields, which, when I wrote my previous book, were barely in their formative stages.\n\nThe best thing about my first book is that, as readers told me, the techniques worked. The worst thing is that some were a tad scheming. Although I never condoned lying, some methods involved presenting yourself in a way that magnified or minimized certain qualities depending on what your Quarry wanted. Now, with the new neuroscientific insights on love, however, retroactively I feel a bit guilty.\n\nMark Twain said that speaking with an uneasy conscience was like a having a hair in your mouth. Nowadays, when someone from the media asks me to tell their readers, listeners, or viewers some of the techniques from _How to Make Anyone Fall in Love with You_ , I feel a follicle on my tongue.\n\nSo let's make a deal. I will give you techniques to kick your Quarry's nervous and hormonal systems into action. But when speaking with your Quarry, keep that hair out of your mouth! Don't misrepresent your qualities or magnify your attributes. Simply highlight the truths about yourself that relate to the pertinent Chemistry Sparking technique we're discussing. I say this not just on moral principles. When you embark on a serious love affair, any fabrication or falsehood could ruin two lives, your Quarry's and yours. Maybe more\u2014your eventual kids'.\n\n#### \" _Trailer\" of What's Coming_\n\nChapter 1, _Love in Limbic Land_ , gives you a mini-tour of the territory (your Quarry's brain) and the tools (chemicals) you have to create Chemistry with him or her.\n\nChapter 2, _Chemistry at First Sight_ , unravels the mysteries of that instant Spark that blows you away.\n\nChapter 3, _How to Spark Chemistry to Attract_ , gives you twelve surprising techniques to ignite it. (Be prepared, some are extreme\u2014but proven to work.)\n\nChapter 4, _How to Spark Cyber Chemistry_ , gives you six unique ways to Spark your Quarry into clicking on your dating site picture, then responding to your profile.\n\nChapter 5, _How to Spark Chemistry in Your First Conversation_ , presents thirteen verbal Chemistry Sparkers to turn \"Hello\" into \"Let's make a date.\"\n\nChapter 6, _How to Spark Chemistry on Dates_ , furnishes thirteen techniques to keep things Sparking between you two every time you go out together.\n\nChapter 7, _How to Spark Chemistry for Sex_ , provides ten unusual Sparkers to light erotic explosions in your Quarry's brain and keep them detonating time after time.\n\nChapter 8, _How to Spark Chemistry for a Relationship_ , gives you eight techniques to turn a new relationship into a serious one\u2014a seriously wonderful one.\n\nChapter 9, _How to Spark Chemistry for Falling in Love_ provides five methods to make your Quarry want to take the tumble\u2014into a beautiful life with you.\n\nChapter 10, _How to Spark Chemistry for a Lifetime of Love_ , gives you eight methods to make beautiful beginnings last forever.\n\n#### _Sizzling Techniques Can Lead to Lasting Love_\n\nIt is said that if you marry for any reason other than love, you pay for it every day for the rest of your life. Let me expand that. If you commit for the all-too-common initial madness of what you might mistake as \"love,\" you also run that risk. Only if you marry with your new knowledge on what love is, and then nurture it, can \"happily ever after\" really happen.\n\nThe first half of this book gives you \"little tricks\" to Spark that initial erotic Chemistry. Some readers might call them \"games.\" But ethical games have rules. The rules of this game of not-so-trivial pursuit are no deception, no dishonesty, and definitely no doing anything that would harm or mislead your Quarry. I warn you, however, that the ploys will involve a little strategy on your part\u2014and also courage!\n\nWomen especially: you may be shocked to find the early flirtation techniques fripperous or outrageous. But absorb them because, as the content of the book deepens along with your relationship, you'll retroactively understand the wisdom behind them. You will learn how to turn sizzling sexy lure into significant long-lasting love.\n\n#### _Sources_\n\nPractically all the information in this book comes from original studies referenced in the back of the book. While proofreading the manuscript, however, I found that phrases like \"a study proved,\" \"the [name of journal] proved,\" and so forth were a snore. I fell asleep reading phrases like \"the highly respected [neurologist, psychologist, sociologist, or anthropologist] found that . . .\" So I decided simply to state the fact and put a reference number beside each. For more information on each subject, I invite you to visit the annotated original source in the back of the book.\n\nThere are hundreds of contributors to the neuroscience of love to whom I'm indebted, and you will find many of their names in the references. Let me take a second, however, to express my admiration and gratitude to three of them for their extra-special contributions to the material in this book. First to Dr. Helen Fisher, biological anthropologist at Rutgers who, in addition to original research, wrote several beautifully written books for the layman on the subject. I am also in awe of the fascinating studies by Arthur Aron, PhD (SUNY, Stony Brook,) and David Buss, PhD (University of Texas at Austin) concerning the cognitive and evolutionary aspects of relationships. And a personal note of gratitude to my friend and \"fact checker,\" Dr. William Hoffman. This esteemed neurosurgeon made sure I followed the convoluted paths of the human brain accurately and didn't drown in some of its synapses.\n\nA multitude of research substantiates each fact in this book. For conciseness, however, I cite only one or two landmark studies on each finding. In my previous book, _How to Make Anyone Fall in Love with You_ , I also referenced a great many studies. Except when absolutely necessary, I avoided citing those again.\n\n#### _The Only Fiction You'll Find_\n\nAnyone who has witnessed the blue of the sky on a clear day, the green of the grass in spring, and the brightness of the sun at midday realizes Mother Nature's breathless beauty. Anyone who has survived a tornado, a blinding snowstorm, or a tempest at sea is in awe of her power. And anyone who has fallen in love has unknowingly been her ecstatically willing pawn\u2014for better or for worse. Mother Nature's main task is to keep the earth flourishing by assuring that all fish in the sea, birds in the sky, plants of the land, and mammals on earth, including humans, go forth and multiply.\n\nThe personification of this powerful force is the only fiction. Other than that, the entire book consists of information (proven), stories (true), and humor (weak).\n\n#### _How Complicated Is It?_\n\nSome of you have already surmised that, considering the magnitude of the subject, _How to Create Chemistry_ is, to say the least, simplified. I do this because the riveting field of Cognitive Science, which covers so many disciplines, needs a little simplification!\n\nTo readers who are more knowledgeable about Cognitive Science, I beg your pardon in advance for the oversimplification. _How to Create Chemistry with Anyone_ is written in layman's language. Sometimes too layman. Enjoy!\n\n|   \n---|---\n\n## CHAPTER | _1_\n\n## Love in Limbic Land\n\nWhere do we feel love? If you asked most people, they would answer, \"In the heart, of course.\" Valentine's Day proves it, right? Lovers give and receive cards, pendants, and chocolates in the shape of hearts. We carve our names surrounded by a heart in tree trunks and tell each other, \"My heart belongs to you.\" Only a fanatic neuroscientist would insist on accuracy. He would send his ladylove a Valentine's Day card with the image of a squishy gray blob evocative of a rotting cauliflower, the brain, because that's where romance really resides.\n\n### Where Love Lives\n\nLove dwells where all your emotions do, in the region I'm calling \"Limbic Land.\" More correctly, Cognitive Science labels it your _limbic system_. It's the part of your brain that houses the laughs, the tears, the hopes, the disappointments, the anger, the ardor, the ups, the downs, the in-betweens, and the profounds. The seed of _all_ your emotions resides in your limbic system, and one of the most powerful is love. Someday it will be a household word. Instead of saying, \"Don't get so emotional about it,\" chastising family, friends, and lovers will say, \"Oh, don't get so limbic!\"\n\nLet me present the five main residents of Limbic Land in alphabetical order. The \"relationship chemicals,\" which I'll introduce afterward, affect regions of a \u2642 brain and \u2640 brain differently. Limbic Land is a very sexist place to live.\n\n#### _Your Amygdala Nickname_ : Ms. Emotions\n\nI hesitate to present this brain region first because the _amygdala_ is hands down the emotional leader of Limbic Land. In love she can sometimes be flaky and totally irrational. But after you've read this book you'll never again need to tear your hair out over your partner and scream, \"She's nuts!\" or \"He's an animal!\" You will understand _why_ the two sexes are so different.\n\nIt should come as no surprise that women have a larger, deeper limbic system than men do. But here's some happy news for Huntresses: A man's love is often stronger and can last even longer for reasons to be explained.\n\nA Huntress also has a more direct neurological connection to her _amygdala_ because her neurons are more tightly packed. In fact, they have an ongoing dialogue about her day-by-day, night-by-night relationship with her Quarry. Think of the female brain's neurons as a group of women walking along on a clear day in the bright sunshine, all talking to each other at the same time. They are so close to each other that, amazingly enough, they don't miss a word.\n\nConversely, a male thinks primarily with his gray matter, of which he has six and a half times more of than a female. Gray matter is very good stuff because of its great ability to focus. Picking up on subtleties or communicating emotions, however, is not gray matter's specialty.\n\nImagine a Hunter's grey matter neurons are like a scattered herd of guys plodding along a rocky path in the pouring rain and gusty wind. They're so far apart that they'd have shout to even hear each other, let alone make any sense out of what the other guys are saying. Is it any wonder that women are more in touch with their emotions?\n\n#### _Your Hippocampus Nickname_ : Mr. Memory\n\nThe _hippocampus_ is memory\u2014short term, long term, and, most of the time, subconscious. This other Limbic Land dweller also has a closer relationship with the _amygdala_ when it comes to emotional issues in the female brain. It drives a Hunter crazy when his wife or girlfriend remembers all the details of the inconsiderate thing he did, the insinuation he made, or the birthday he forgot. \"Why can't she just let it go?\" he agonizes. Sorry, guys, it's not her fault. The more closely knit team of her _hippocampus_ and _amygdala_ won't let her.\n\n#### _Your Hypothalamus Nickname_ : Mr. Action\n\nNow comes the part everyone in the brain wants to influence, your _hypothalamus_ , which shoots messages down the spine 24\/7, controlling your central nervous system. He's the strong man who makes you physically act and react. Blame this brain region when, upon feeling that Spark of Chemistry, your mouth goes dry, your heart skips a beat, and your flushed face resembles a radish.\n\nYour _hypothalamus_ deals with sex, big time, especially in guys. In fact, it's more than twice as large in the male brain, and has an excellent working relationship with _testosterone._ This brain region is in charge of the temporary growth of a significant male appendage at tender moments\u2014and _testosterone_ cheers him on. In addition to erections, the _hypothalamus_ also monitors entities (at certain times considered secondary by men) such as eating, sleeping, and breathing.\n\n#### _Your Caudate Nucleus and Environs Nickname_ : Pleasure Island\n\nBefore we get to the \"brainy\" part of your brain, let's get to the happiest. It's a region called your _caudate nucleus_. This feel-good area encompasses the _nucleus accumbens_ and the nearby _ventral tegmental area (VTA_ ). Combined, Cognitive Science calls it your _pleasure_ or _reward center_. Excitement, enjoyment, and exhilaration hang out there. After a few gulps of a certain love chemical that I'll introduce next, _Pleasure Island_ lights up like a Las Vegas slot machine. Life would be hardly worth living if it weren't for this happy neighborhood in your brain.\n\n#### _Your Prefrontal Cortex Nickname_ : The Professor\n\nLast but most assuredly not least is the wisest part of your brain, which we're nicknaming the _Professor_. The _prefrontal cortex_ does not live in la-la _Limbic Land_ with the rest of the cast _of Chemistry_. His throne is at the front of your brain. This sage assiduously reviews neurotransmissions (which we'll call \"texts\" or \"tweets\") from _Limbic Land_ and tries to determine who is bad for you, who is better for you, and who is best for you.\n\nUnfortunately, most new lovers don't listen to the _Professor_. When a couple is in the early hot love \"condition,\" Mother Nature is blasting their brains with so many intoxicating chemicals that the signals from the rational brain section are weaker. The _Professor_ tries to prevent lovers from making a fool of themselves and doing self-destructive things while smitten. But when in love, who listens?\n\nThat's the emotional housing complex of the brain. Bear with me while I now introduce the chemicals that live inside. They play the biggest roles in love, and knowing about them is necessary to understand how to create Chemistry with someone.\n\n### The Six Sexy Chemicals of Love\n\n#### _Dopamine_\n\nThe star of the early show in love is _dopamin_ e. He's on the scene every time you feel sudden Chemistry with someone, and his influence is mind-blowing. (He has helper chemicals like _norepinephrine_ and _phenylethylamine_ , but he's the leader when you're \"crazy in love.\") When you first meet _dopamine_ , he introduces himself as ecstasy, euphoria, exaltation, exhilaration, and intoxication. Sometimes he appears in a split second; at other times he creeps up slowly. As you're falling in love, he scoops you up and whisks you off to that most beautiful place you will ever go, _Pleasure Island_ , or the _caudate nucleus_ , located deep in your limbic system.\n\n_Dopamine_ loves action and jumps into the fray when you feel fear. That's why movie thrillers filled with bloody bodies and crashing cars give some people a kick. _Dopamine_ is also summoned by beautiful music, stirring sermons, and activities you enjoy. People do desperate things to make him stay because, if he disappears or starts slipping away, they can become despondent, even self-destructive.\n\nIn fact, he has such a powerful ecstasy influence that many of today's most credible anthropologists, neurologists, and psychiatrists say that love is simply _dopamine_ 's effect on the _pleasure center_ in your brain. Personally, I don't like to think of it that way, but the evidence makes a pretty good case.\n\nDid someone liken this chemical to cocaine? Yes, everybody with even a passing knowledge of Cognitive Science does.\n\n#### _Serotonin_\n\nWhen _serotonin_ is high in your system, you feel superb. This blissful chemical floods your brain when you exercise, genuinely laugh, sunbathe, go sailing, skiing, surfing, dancing, or do anything else you love. She is so central to your happiness that if she starts to shrink drastically, you become melancholy, morose, and miserable. Some sad souls turn to alcohol\u2014or worse\u2014just to bring her back for short visits.\n\nIdeally, _serotonin_ and _dopamine_ would work well together. But there is a problem. If _dopamine_ makes you act too frenetic about your relationship, _serotonin_ doesn't like that. She'll start shrinking away, leaving you with insomnia, eating disorders, negative thoughts, anxiety, and worse. When _serotonin_ is artificially introduced, such as in antidepressants, she can also mess with your ability to love. It's up to couples to keep _serotonin_ and _dopamine_ in balance working together, without medication.\n\n#### _Testosterone_\n\nNeed I even introduce this famous performer? It is the most revered and reviled romance chemical of all. Sometimes _testosterone_ gets a bad rap and people blame him for starting street brawls, civil wars, stabbing adversaries in the back, piloting corporate takeovers, driving too fast, robbing, raping, sleeping around, and channel surfing. Although some of these unsavory types are, indeed, high-T men, it's not all true. _Testosterone_ is a good guy who plays a big role in developing creativity, intellect, thought patterns, assertiveness, and drive. He also gives guys large muscles, increased bone mass, and, of course, colossal, sometimes uncontrollable, sexual desire.\n\n#### _Estrogen_\n\nEqually well known, _estrogen_ is a hot commodity for women. Those who want to look younger and more beautiful\u2014like 100 percent of the world's female population\u2014crave her effects. Lucky ladies blessed with a lot of this hungered-after hormone have clearer complexions, shinier hair, rosier cheeks, fuller lips, larger breasts, and they are generally healthier. High- _estrogen_ women also get pregnant more easily. Due to Mother Nature's influence, that is a (very) subconscious Chemistry Sparker for guys, as you will soon learn.\n\nThis leads us to the family-friendly chemicals.\n\n#### _Oxytocin_\n\n_Oxytocin_ is wonderful. Everybody loves her because she generates affectionate feelings, increases trust, and promotes long-lasting relationships. It would be an exaggeration to say that _oxytocin_ is more of a woman thing, and the next \"togetherness chemical,\" called _vasopressin_ , a guy thing. Male and female brains create both. However, mixed with _estrogen, oxytocin_ has a powerful bonding effect.\n\n_Oxytocin_ and her colleague chemical, _vasopressin_ , help keep lovers together and make them more devoted parents. Neurologists don't call them the \"bonding\" or \"cuddle chemicals\" for nothing, and we'll learn more about them later in the book.\n\n#### _Vasopressin_\n\n_Vasopressin_ 's effects are similar to _oxytocin_ 's, and the two often have the hyphenated name _oxytocin-vasopressin_. They are both family folks. The latter does a terrific job of making a man bond with his lifetime partner and become a more involved dad. Unfortunately, _vasopressin_ and _testosterone_ don't get along well together. The former can drive the latter down. Because he's not coming on to his wife as much after childbirth, a wife often mistakenly fears the new dad is not sexually attracted to her anymore. But, women, look on the bright side: It means he's bonding better with you and the baby. Besides, less _testosterone_ reduces his desire to seek sex elsewhere.\n\nGirl, I know what you're thinking, but don't hold your breath. It will be a while before some profit-hungry drug company gets FDA approval for a _vasopressin_ pill you can slip in your man's orange juice.\n\nNow, let's talk about your specific qualities, which will play a major role in Sparking your Quarry's Chemistry, and how to get Mother Nature to help you do the job. You may wonder why I'm breaking it down into body parts. It's because the information is crucial to the Sparking techniques.\n\n|   \n---|---\n\n## CHAPTER | _2_\n\n## Chemistry at First Sight\n\nAs you read in the Introduction, due to DNA, past experiences, and a myriad of other elements, you can never have 100 percent control over the Chemistry between you and someone. That's just part of the picture, though. Mother Nature plays a _huge_ role. In fact, she is your best friend in helping you Spark your Quarry. Animals follow Mother Nature's guidance instinctively. When another of their species excites them, they growl, grunt, cluck, crow, pant, or emit some other form of indigenous whooping it up. (Humans have to be a bit more subtle about it.)\n\nAnimals strive to entice the best mate according to the neurological, chemical, and evolutionary influences defining \"best\" in their species. Although you may not think of it that way, but you do too. Something else comes into the human equation, though. You have a bigger brain and are, unfortunately, restricted by certain social conventions. Because you are more complicated, the Sparking process is too.\n\n### \"All Men Are Animals\"\n\nHuntresses, have you or a friend ever complained, \"Guys are only interested in sex\"? If not, you've probably thought it. My girlfriend Brandy from Baton Rouge certainly does. Brandy is a beautiful woman, a serial dater, and a constant complainer about men. She comes to New York every year to run the marathon. Here is our annual ritual: The night before the big event, we go to the same restaurant for the same pasta with olive sauce, and I listen to her same gripes about the men in her life.\n\nLast November, twirling spaghetti around her fork and making direct eye contact with a pimento-stuffed olive, she griped, \"Guys are only interested in sex. They're all big penises with a man attached. From the first date, they're pressuring me. Why can't I find a man I can connect with? Someone I can talk to . . . a guy who talks to me, not my breasts? I want a guy to cuddle, not fall asleep right after sex. Is that too much to ask?\"\n\n\"Well, Brandy, that last part is.\" I started to tell her about the universal male condition called \"postcoital narcolepsy\" and the chemical reason a man can't help zonking out after ejaculation. But she cut me off. There is no stopping Brandy once she's on a roll.\n\nIn previous Novembers I just nodded while she went on and on. But this year, while writing _How to Create Chemistry_ , I could hold it in no longer. I interrupted her diatribe and said, \"Well, hell-OH, Brandy, what do you expect?\"\n\nShocked, she glared at me. \"What . . . what do you mean?\"\n\n\"Brandy, my dear friend, you can take a male out of the jungle, but you can't take the jungle completely out of the male. In fact you wouldn't want to. Usually a guy's initial motive for even asking you out is because he wants sex, the sooner the better. He doesn't want to spend all night touching your soul. He'd prefer to touch your breasts. Brandy, you have to give it time!\"\n\n\"Sure,\" she said rolling her eyes. I would have continued, but I knew she needed sleep to face the challenge she'd been training for all year. I hugged her and gave her my annual \"Tear up the road tomorrow, Tiger.\" Then I made a mental note to send her this book.\n\nSo was Brandy wrong about men? Sex is definitely not the only thing most men think about when asking you out\u2014just usually the first. In addition to some of the aforementioned elements, it is what makes a guy feel that Spark for a particular woman. Most men do want a permanent relationship eventually. But sexual attraction is crucial because it's the match that lights the fire. Deep conversations, communicating, connecting, and commitment come later.\n\n### \"All Women Are Gold Diggers\"\n\nIf you've read my previous books, you are familiar with Phil, my dear friend and platonic male roommate, or \"friend without benefits.\" Phil makes my personal research into the male psyche quite convenient. Besides, he's the best roommate a gal could ever have. Naturally, every time he comes home from a date, I grill him mercilessly about it.\n\nOne night Phil returned with that \"She was the date from hell\" expression on his face. His wobbly walk and martini breath told me he'd consumed more than his customary one drink. He obviously wasn't in the mood to submit his social report.\n\nAs I was attempting an \"I'm not the least bit curious\" expression, my mild-mannered roomie uncharacteristically grumbled under his breath, \"Freakin' gold digger!\" The usually fastidious Phil tossed his coat on the couch and staggered unsteadily toward the bathroom. Under the rumble of his electric toothbrush, I heard him bellowing something in French, which he doesn't speak. \"Le Bernardin, Le Bernardin, Le Bernardin!\"\n\n\"Did you say something, Phil?\" I called.\n\nHe opened the door and, Crest foaming out of the sides of his mouth, shouted, \"Le Bernardin! It's the most %@!& expensive restaurant in New York!\" Spitting out the foamy water, he continued, \"This afternoon, I texted her the name of the restaurant where I wanted her to meet me. But oh no, that wasn't good enough for Goldilocks! She texted back that she preferred a 'nice little restaurant' near where she worked. Oh, it was nice all right. The bill came to over two hundred dollars, and all evening she's asking me creepy questions. Oh, ho ho. I read between the lines with no problem. She was trying to figure out how much money I make. Damn gold digger!\" he bellowed as he slammed his bedroom door.\n\nOf course, money is not a woman's major criterion for a man. However, no matter how much she herself has, Mother Nature is whispering in her ear, \"Make sure he'll be a good provider for you and your eventual babies, dear.\" Even if she earns enough herself to put an entire orphanage through college, most women feel her man should earn more than she does.\n\n### Huntresses, Cut the Animals Some Slack\n\nSisters, his constant pressure for sex is not his fault. You wouldn't blame a guy for limping due to a birth defect. Likewise, you can't blame him for his greater sexual drive, which nature also programmed into him before he was born. It's an almost insurmountable challenge for a male to push sex out of his mind. Mother Nature (MN for short) injected men at least with ten times more testosterone than you. It's not an exact proportion, but think about it: If you were ten times more ravenous for sex, wouldn't you act a little more needy, greedy, and aggressive about it? Women can go for months, sometimes years, between relationships without sex. But for men, it's practically impossible.\n\nIn fact, a guy's testosterone level shoots up a full third even casually chatting with a female stranger  Just thinking of you being naked makes dopamine gush through his brain like a cocaine rush. Unfortunately, that feel-good chemical short-circuits messages en route to his prefrontal cortex, the thinking part of his brain. Consider the ever-lengthening list of male politicians, sports figures, and other celebrities who throw away valuable careers for a few rolls in the hay. Girl, you have the power to turn a dignified gentleman into a dyslexic octopus when he's around you.\n\n#### _Sex Scrambles His Brain_\n\nIn fact, guys don't just make dumb decisions about sex; they can make pretty dangerous ones. Researchers on the Florida State University campus gathered a group of gutsy girls and directed them to approach male students they didn't know and ask, \"Will you have sex with me tonight?\" Seventy-five percent of the guys agreed immediately. That shocked even the relatively unshockable me. More recently, in a follow-up study after AIDS awareness, 69 percent of the males agreed. That stunned me even more and made me very, very sad.\n\nNeedless to say, when the study, \"Gender Receptivity to Sexual Offers,\" was reversed, and men approached the female students, none of them agreed to sex, pre-or post-STD awareness.\n\n#### _Blame Mom and Co_.\n\nHere is yet another complication. The guy brain is spinning with mixed signals. He probably grew up with a mother who implied that anything sexual was naughty. Maybe she scolded him for looking at his sister when she was dressing. Or was furious when she caught him looking at \"nasty pictures\" in magazines. She told him to keep his hands off his tiny crotch in public. One of my seminar students told me that his mom once spanked him when she walked into his room and caught him masturbating. In short, little guys grew up thinking sex was dirty.\n\nThen suddenly, a whole new set of signals replaced it. His high school buddies bragged about their sexual conquests; many self-proclaimed studs exaggerate them to gain status in their friends' eyes. No guy wants to admit he's still a virgin, so he becomes obsessed with curing himself of that \"contemptible\" condition. The social pressure to score is tremendous.\n\nI learned that at my girlfriend Sheila's sixteenth birthday party. At one point while we were talking, she leaned toward me and whispered, \"What are those boys doing over there?\" Five or six of them were comparing something they'd taken out of their pockets.\n\n\"I think it's their combs,\" I guessed.\n\n\"They must be pretty old,\" she murmured, \"because it looks like some of them lost a lot of teeth.\" When one boy proudly held up a totally toothless comb, they all cheered. Sheila called across the room, \"Hey, guys, whatcha doing?\" That detonated an outburst of hilarity that would have merited the acronym ROTFL. They were literally rolling on the floor laughing.\n\nIt wasn't until a week later that Sheila and I managed to wrestle her brother to the floor and demand he tell us what was going on at the party. Gasping for breath, he was forced to rat on his buddies. \"The guys were comparing combs. Every missing tooth represented sex with a different girl.\"\n\nA male who sleeps around is enviably called a \"stud.\" A female with the same sleeping habits is disparagingly called a \"slut.\" Fair? No. Understandable? Yes, at least in terms of evolutionary psychology.\n\n### Hunters, Cut the Gold Diggers Some Slack\n\nIt is impossible for a female to close her ears to the cries of her ancient foremothers. Here is her recurring subconscious nightmare: After a stupendous night of sex with you, you hop onto your trusty steed and ride off into the sunset . . . alone. She is left pregnant. Your fifteen minutes or so of bliss could mean fifteen years or so of burden for her.\n\n\"But that's not going to happen,\" you protest. Of course it isn't. She knows that too. But speak to the sticky cognitive modules floating around in her brain. Consult with her Cro-Magnon grandmother. Subconsciously she does, and the response is always, \"It happened to me, dearie, it can happen to you.\" Her primeval grannies also remind her that she will suffer morning sickness, migraines, throwing up, leg cramps, and hip pains.\n\nGentlemen, a bad bout of morning sickness is the same sensation you'd feel after scarfing down seafood cocktail and a dozen raw eggs, followed by a Big Mac, a hot fudge sundae, and a six-pack of beer. And then deciding to go for a spin on the roller coaster.\n\nIf your girlfriend gets pregnant, she pays a big biological cost. You can't have another whole human being residing inside you without crowding your own body space and causing some huge hassles, like sharing calcium, iron, vitamins, and other elements crucial to health and beauty.\n\nThen there's the weight gain, the dumpy clothes, and no sex. If you don't believe it, put a cement block inside a beach ball and strap it to your stomach. Then try to have sex.\n\nAfter that is the excruciating pain of childbirth, followed by changing diapers, the terrible twos, the tantrum threes, the fiendish fours, the feisty fives, and so on. Because her plight won't permit her to work through the early childrearing process, can you see why your financial status is at least a factor? Primordial memories stick like glue, so is it any wonder that a woman doesn't want to jump into bed with you right away?\n\nYour biological cost as a male? Zip. You only spin off sperm to start the whole process, and that's a whole lot of fun. Gentlemen, I applaud you. It is truly charming, if not naive, that when you fall deeply in love, you become blind to some of the realities of life. Females are usually more sensible about partner choice. Brain-imaging studies of women in love show activity in many more areas than yours. However, when a guy has flipped, in a sense, he's _really_ flipped, because more of his critical thinking pathways close down. She's making her decision with more of her head, whereas yours are made with, well, more of your other body parts.\n\nBe thankful that most women are more realistic in relationships. If she were as resistant to the reality of resources as you are when you fall in love, there would be more miserable marriages of the all-too-common kind in which he is broke, she's freaking, and the kids are hungry. Bottom line, brothers: Respect your Quarry's realization, that, in the long run, you really can't live on love alone.\n\nHere's food for thought: Is her immediate consideration of your wealth any worse than you immediately considering how physically attractive she is\u2014and how soon she'll go to bed with you?\n\n### What Sparks a Hunter's Chemistry\u2014and Why?\n\nHuntresses, if any of you are still harboring any doubt that your looks are the first thing a man cares about, let me quote one landmark study that doesn't mince words: \"A woman's physical attractiveness is the cardinal component of women's mate value.\" \"Mate value\" is the crass word that researchers use for how desirable someone is in the \"meat market.\" Yep, Huntresses, your looks matter, big time, at least until he gets to know you and your amazing qualities.\n\nA man can be sexually turned on by a gorgeous woman he would never dream of having a relationship with\u2014a mindless model, a knockout with bust size 38 and IQ to match, or a hot lady of the night he would be ashamed of the next morning. Those lasses may be delicious company for one night, maybe two if they're lucky. But definitely not someone he'd want to raise a family with. When thinking of a date, many guys' fantasies of the future end at orgasm.\n\nThe specifics may seem obvious. However, in this section I want to quickly review the particulars of what excites males, because only when you understand _why_ your Quarry gets whacked over certain features can you confidently and courageously use the Chemistry Sparkers in the next section.\n\n#### _Babies \"R\" Us_\n\nGuys don't realize it themselves and would surely deny it, but the number one subconscious Chemistry Sparker boils down to one word: _fertility_. Procreation is the name of nature's game. It's the bottom line, the nitty gritty of what guys find hot. Science calls it like it is: \"Instinctively, the male of all species wants to get the 'biggest reproductive payoff' for his,\" ahem, \"investment.\" After all, Mother Nature figures, why should a guy squander his sperm in a woman who can't bear him a healthy baby with good genes and a strong immune system? That would be a humongous waste of time.\n\nGentlemen, even if you are planning a peaceful life without the smell of baby poop and getting barfed on, Mother Nature doesn't know that. She programs you to be attracted to a female who looks like she'd be a good baby factory.\n\nSisters, Big Mama's got it all rigged to help that along. At ovulating time she gives you a natural cosmetic makeover and even makes you more symmetrical. Your eyes get larger, your lips get fuller, and your cheeks become rosier during the big O days (ovulation, not orgasm). It's like female apes whose lower cheeks get redder when they're in estrus. In case any guy ape would miss it, she grows a humongous pink butt at target time. Similarly, your rosy facial cheeks subconsciously signal guys that it's the best time to \"make the shot.\"\n\nIncidentally, ladies, if the big tough High-T guys aren't usually your preference, be extra vigilant during those big \"O\" days. MN slips you a chemical cocktail that makes you more attracted to them when you're most apt get pregnant.\n\n#### _Let's Break It Down to Body Parts. Guys Do_.\n\nA Hunter doesn't look at you and specifically think, \"Wow, smooth skin, big breasts, narrow waist, long shiny hair, and youth.\" It's the whole package that hits his eyes. But to have more \"ammunition\" on your hunting expedition, I'll tell you why each element counts.\n\n_**Smooth Skin.**_ Skin is a measuring stick for your hormonal state, which is, incidentally, more detectable in the lighter skin of blondes. That's why it is said that \"gentlemen prefer blondes.\" Do you remember the classic Marilyn Monroe film by that name? If a biologist were naming the film, it would have a less catchy but more telling title like, \"Gentlemen prefer females with lighter hair and skin to more easily detect anemia, cyanosis, jaundice, and other skin diseases.\" But then the movie wouldn't have been a box-office blockbuster.\n\n_**Long Hair.**_ Even short, shiny hair signals health and estrogen and, thus, that you're more easily impregnable. But long, shiny hair shows you've been healthy and chock-full of that enviable hormone for a couple of years. Natch', that means you'd be easier to impregnate with a healthy baby.\n\nWhile we're discussing hair, here's a bit of trivia that has absolutely no scientific basis, simply something from a men's website survey. Rather than the bushy look downstairs, the majority of men prefer the more youthful, completely shaved or famous \"landing strip\" look of Brazilian fame, shaved on both sides evenly.\n\nI once told my hairdresser that I had a great business idea for her: She should open a chic pubic hair trimming salon. I've had a lot of bad business ideas.\n\n_**Hourglass Figure.**_ So what's the big deal about a shapely body, you ask? Mother Nature murmurs in his ear, \"Dude, she'll get pregnant faster and easier and is less apt to have a miscarriage. A male subliminally recognizes that Ms. Shapely's roomier hips would give his unborn progeny's head more room to expand. To add more fuel to his firing neurons, her bigger breasts signal about 37 percent more estrogen.\n\n_**Eyes?**_ Larger \"ovulating signaling\" eyes are a bigger draw. Concerning color, blue-eyed men are drawn to similarly light-eyed women because it's a good cuckoldry indicator that his little one didn't have a brown-eyed daddy.\n\n_**Hey, What About My Face?**_ I always thought a woman's face was the first thing a Hunter looked at, so I was shocked at the studies that report a male registers your body first. He probably doesn't even realize it himself, but nanoseconds beforehand he has subconsciously sensed your body to see if your face is \"worth checking out.\" Disbelieving, I decided to ask Giorgio, the ship's captain I mentioned earlier who subsequently became my guy. Giorgio is an extremely cultured gentleman, so I naturally assumed that he'd say \"her face.\"\n\n\"Giorgio, if you could choose between a woman with an averagely attractive body and a spectacular face, or a dynamite body and an average face, which would you choose?\" I couldn't have counted to one before his answer zapped back, \"The latter.\" I sucked in my stomach and slunk back to the studies.\n\nA few weeks later, while rearranging my library, I came upon my freshman yearbook. Flipping through it reminded me of something that happened the first night in my new dorm that should have given me a clue way back then. A senior was showing us the previous year's book and chatting about some of the girls we'd soon meet on campus.\n\n\"And here's Shelly,\" she said pointing to one headshot. Several of us had to stifle an insensitive smile. It is only a slight exaggeration to say her face could have been the prototype for the Muppet's Miss Piggy. Our senior informant went on, \"The guys are crazy about her.\" The rest of us looked at each other in shock.\n\nA few weeks later I met Shelly at the campus pool. Her face was, indeed, evocative of the Muppet celebrity, but certainly not her body. She had the breasts of Beyonc\u00e9, the butt of Jennifer Lopez, and the waist of a wasp. And to think I still wondered why guys were so attracted to her.\n\n_**Does My Age Matter That Much?**_ Despicable perverts aside, a young girl doesn't usually begin Sparking a Hunter's Chemistry until she begins to get curvy around the time her periods start. Even preteen guys fantasize about older girls when they're all alone in their bedrooms with their doors locked. Mother Nature figures, \"Why fritter away fantasies on girls who can't get pregnant?\"\n\nThis brings us to an unsettling but inevitable fact: We grow old. So do guys, but that's no problem for Big Mama in the sky. She figures men can still spread sperm in younger women for as long as they can get it up. Depressingly, after menopause Mother Nature makes a female's breasts and butt start to sag, nipples soften, waists thicken, lips thin, complexions roughen, cheeks pale, and hair lose its shine. Personally, I think MN gave us a raw deal. She's telling guys, in no uncertain terms, \"That old lady is no longer fertile. So screw her\u2014or, rather, don't.\"\n\n### What Sparks a Huntress's Chemistry\u2014and Why?\n\nHunters, whenever a clearly stunning female slinks into the room, male eyeballs start to whirl like a beanie cap. So you probably assume that a great-looking guy\u2014tall, big muscles, flat abs\u2014would spin females' heads. In another era you would have been right. A cave woman needed a mate with enormous muscles to bash a wild boar on the head and spacious shoulders to carry the bloody beast home. Tall was better too, because he could run faster if he missed his target. The cave lady wanted her man oozing testosterone out of every pore, signifying aggression, high sexual desire, and strong genes to bear beefier babies.\n\nIn those days selecting a sensitive, compassionate, loving male would have been a really lamebrain choice. Imagine a caveman's success at intellectually reasoning with the tiger poised to pounce on him. If the ferocious animal didn't concede, her little one would have no daddy. So Mother Nature poked the primitive pretty and said, \"Better to go with the powerful jerk, dear.\"\n\nBeing drawn to brawny guys with a touch of the brute lasted a surprisingly long time. A scant fifty years ago women swarmed to movie theaters to swoon over the Rock Hudson, Burt Lancaster, and Clark Gable types. Their hearts fluttered when Rhett Butler told Scarlett O'Hara, \"Frankly, my dear, I don't give a damn.\" Now women tell men who don't live up to their standards, \"Frankly, mister, I don't give a damn.\"\n\nOne thing will never change, however: survival of the fittest. Huntresses are programmed to prefer men who match Darwin's definition \"best adapted for survival in the immediate, local environment.\" Today, however, the \"immediate local environment\" is dramatically different. Now the crucial survival qualities are cooperation, compassion, curiosity, kindness, competitiveness, and an open mind.\n\nEvolutionary psychologists who recognize that there are physiological spurts would call these \"neo-Darwinian\" qualities. The field encompassing that, developmental evolution, recognizes that the social environment changes in spurts, not at an even pace like evolutionists used to think.\n\nIn the twenty-first century something new has been added to a woman's wish list. She wants intellectual growth and to reach her full potential as a person and a professional. A Huntress seeks a partner who will be supportive and help her accomplish that.\n\nHunters, even for ladies who are just looking for one-night nookie, your success and \"good dad\" indicators are what shoot brief spikes of voltage through your Quarry's brain. But don't fret if you aren't a candidate for a _Men's Health_ cover with bulging biceps and deltoids on steroids. This isn't to say that a woman doesn't like a really ripped guy\u2014but only if it's attached to the right brain and attitude.\n\nHunters, when is the last time you strutted into a bar and said to yourself, \"Dude, tonight I am going to look so cooperative, compassionate, curious, kind, competitive, and show how supportive I am of a woman's personal growth that chicks will be lining up to meet me?\" (Stay tuned, aspiring studs. You'll learn how to do that in the next section.) In short, a woman's looks are the first thing a man notices. A man's character and personality are number one on a woman's most wanted in a partner list.\n\nNow we come to the most alluring physical quality for both sexes. It's one we seldom think of.\n\n### Planet Earth's Colossal Unisex Chemistry Sparker\n\nHaving a symmetrical face and body is like winning the lottery in attraction for both Hunters and Huntresses the world over because it screams \"healthy genes.\" You and I could see a good gene sitting next to a bad one on a plate, and it wouldn't say anything to us. But dump a batch of good ones into one baby and you'll see a big difference. Each tiny toe on his left foot looks exactly like the same one on the right. His little ears are mirror images of each other. Yet few would be conscious of its draw. Have you ever heard a woman swoon, \"Honey, your symmetrical nose drives me wild\"? Or a guy brag, \"Dude, she's got the most balanced eyebrows I've ever seen\"?\n\nWhen you look at most faces, you think, \"Sure, that's symmetrical.\" But I'm talking precise symmetry to the millimeter that only a fortunate one-tenth of 1 percent of the population has. If it's off one iota, it makes a difference. And it's not just perfectly balanced _faces_. We're talking symmetrical legs, shoulders, hips, thighs, fingers, elbows, breasts\u2014and yes, even both sides of \"it.\" Mother Nature draws females to symmetrical types, even if it's for one-night stands\u2014especially when they're ovulating  After all, she figures, even that one shot could result in another healthy earthling.\n\nSymmetry is also a powerful beauty plus in reptiles, insects, birds, and mammals because good genes also do a better job of battling environmental pollutants and parasites. Any self-respecting female fruit fly wouldn't even consider a guy fruit fly with asymmetrical spiracles.\n\nSymmetrical guys, go ahead and gloat because studies show:\n\n1.you start having sex four years earlier than your more symmetry-challenged brothers;\n\n2.you get laid more often and have a greater number of partners throughout your lifespan;\n\n3.you smell better to women than your more uneven, asymmetrical competitors; and\n\n4.you will give your girlfriends more and better orgasms than the more lopsided chaps do.\n\nHuntresses, here is a warning for you. They may look great, but be careful about those super-symmetrical guys. Statistically, they are less apt to stay with one partner and have more extramarital affairs throughout their lives. Also be extra vigilant if you don't want to get pregnant, because your vagina sucks up more sperm from the better orgasms that symmetrical Hunters give you.\n\n#### _Cliff Notes_\n\nYou'll see the reasoning behind the Sparkers in the next section, but for now let's tally the list of what pokes Chemistry and gets the opposite sex's hormones hopping.\n\n_Hunters . ._.\n\nThe _what_ : Your signs of health, mental strength, compassion, wit, and assets or potential for same.\n\nThe _why_ : Healthy men produce babies with stronger immune systems. Compassionate men care for them. Mentally strong men solve problems for them. And men with money give both baby and mother a better lifestyle.\n\n_Huntresses . ._.\n\nThe _what_ : Your large breasts, full lips, smaller waist, shapely hips, rosy cheeks, clear eyes, unblemished skin, and shiny hair.\n\nThe why: All the above qualities reflect high estrogen, meaning he can impregnate you quicker for a healthier baby.\n\nIn short . . .\n\n_A man is attracted to a woman's ability to grow a baby inside her_.\n\n_A woman is attracted to a man's ability to grow a baby outside him_.\n\nSo now we know _what_ charges the electrical fields in your Quarry's brain and _why_. Let's move on to the specific Chemistry Sparkers and _how_ to set the traps for the Big Catch.\n\n|   \n---|---\n\n## CHAPTER | _3_\n\n## How to Spark Chemistry to Attract\n\nLet's say one Saturday you and a few friends decide to go fishing. You get some poles, pack a bucket of worms for bait, and head out to the local lake. Trout are leaping onto the hooks, and you all have a fabulous time. Everyone laughs and cheers as they make catch after catch. The guys take turns carrying the big basket of fish to a friend's country cabin. You open a case of beer, turn on some music, and cook a great trout dinner.\n\nIt was such a blast that you plan a repeat experience the next Saturday. With fishing rods over your shoulders and toting a bucket of minnows for bait this time, your group treks to the lake, humming all the way. Excitedly, you put your bait on the hook and cast off. Everyone waits for the fish to swarm toward them like the previous week.\n\nAn hour goes by. No bites. Hmm. Two hours pass. Still nothing. After three hours you all give up and get Chinese takeout on the way back to the cabin.\n\nWhat went wrong? Those of you who fish probably know the answer. You used the wrong bait. Trout don't eat minnows.\n\nCut to a party. Now you're fishing for a bigger catch, someone to date, have sex with, love, and perhaps build a life with. Lots of attractive people are swimming around the bar and everybody is hoping for the best catch. But only a lucky few lovers succeed. Why doesn't the majority? They use the wrong bait.\n\nHere's the right kind.\n\n### Forget the Golden Rule When Trying to Spark Chemistry\n\nYou've heard of the Golden Rule: _Do unto others as you would have them do unto you_. That's fine for most situations. But not for igniting that Spark at first sight. Forget the Golden Rule and use the Platinum Rule, which I'll explain shortly.\n\nWhen a Huntress first spots a man, she is instinctively aware of his character, confidence, kindness, and demeanor. She hopes to find intelligence and humor. If she likes him, she wants respect from the gentleman from the first \"hello.\" In the back of her mind she's wondering if a relationship might be in the future.\n\nWhen a Hunter first spots a woman, he judges her looks and receptivity to him. In the front of his mind he's wondering if sex might be in the future\u2014and how soon.\n\nUsing the Golden Rule\u2014treat others as you'd like to be treated\u2014to appeal to a Quarry is one of the most foolish and common mistakes that potential lovers make. Huntresses are especially guilty of this. Girlfriend, at this point a guy is not judging your character, kindness, intelligence, and how much of a lady you are. Later, for a serious relationship with you, your character and other fine qualities are all important to him. _But not at first_. Luring him with those aspects of your personality before the first \"hello\" is the wrong bait because most guys are not thinking beyond the bedroom.\n\nConversely, Hunters, she is. Your abs of steel and bulging biceps pale as bait compared to other qualities she craves in a man. Later, when the two of you are in a loving relationship, she'll long to lick your killer pecs and squeeze your muscles. _But not at first_. She's visualizing beyond the bedroom. In fact, unless you have the \"beyond\" qualities, there probably won't even be the bedroom. Sexiness is the wrong bait for a guy to cast because she's thinking possible relationship.\n\n  **Chemistry Sparker #1**\n\n**Use the Platinum Rule When Luring Quarry**\n\nWhen trying to entice your Quarry, do not \"Do unto others as you would have them do unto you.\" Instead, \"Do unto others as _they_ would like to be done unto.\" Hunters, displaying your good-guy qualities and demonstrating respect for the lady is the best bait. Huntresses, sexiness and showing that you're receptive to him are what reels him in.\n\n### Fight, Flight, or Chemistry at First Sight?\n\nI didn't name this segment \"Eye Contact\" lest you skip it. I understand. I probably would have too. Millions of magazines, newspapers, books, and blogs\u2014including mine\u2014have done the subject to death. That aside, I would be remiss not to address it first.\n\nContinuing unbroken eye contact can cause even a small Spark to catch fire. In studies, opposite-sex strangers directed to have extended eye contact reported neurochemical reactions similar to romantic sentiments. Additionally, eyes locked for a period of time can give the sense that you are already in love. The _Journal of Research in Personality_ reported that people who are deeply in love gaze at each other 30 to 40 percent more than other couples do, and they are slower to look away during an intrusion.\n\nThere's more. Maintaining enduring eye contact also gives you the cachet of being a wiser abstract thinker because confident and creative individuals integrate incoming data more easily than concrete thinkers. They can continue looking into someone's eyes even during the silences.\n\nUnfortunately, most writers make it sound like \"eye contact\" is simply staring into each other's eyeballs. No, plain old eye contact between a man and a woman is all too common to Spark your Quarry majorly on its own without a strong message emanating from your eyes. I'll give you a few ways to use those grenades over your nose to capture your Quarry. But, first, gentlemen . . .\n\n#### _Get Wise to the Geisha in Every Woman_\n\nUnfortunately, there is a lot of misunderstanding about what eye contact signifies. Here is an all-too-common scenario: Guy smiles at girl. Girl looks away. Guy thinks it's rejection. Girl is disappointed. End of story.\n\nMost Western males think a female glancing away means she's not interested. Wrong! Japanese men don't fall for it. They know that the lady's little smile and demure downward glance is part of the game, the world-famous geisha ploy. Her shyly (or slyly) lowering gaze is part of the female International Courtship Ritual.\n\nThis eye contact edict has been in effect since the beginning of time. If you've seen the infamous painting, _The Fall of Man: Adam and Eve_ , you'll observe that the first human Hunter is looking at his Quarry's eyes. But Eve is demurely gazing elsewhere, all the while enticing him with the forbidden apple.\n\nGentlemen, it's not _if_ she looks away. She will. (Well, unless she's read this book.) It's _how_ she does it that counts. When you reward her with a smile, your Quarry will either . . .\n\n1. _Modestly sweep her eyes down at the floor as though admiring the carpet_. Experienced and confident Hunters know this is standard operating procedure for a flirting female and is as good as an engraved invitation. Go introduce yourself.\n\n2. _Turn away with her eyes parallel, like inspecting the wall's paint job_. She's thinking, \"Hmm, I'll reserve my judgment on you until after our first chat. The probability is good if you pass the \"chat test.\" Go for it.\n\n3. _Look up and away like checking the ceiling for ugly cracks_. Here she might as well be rolling her eyes into the next state and your chances of victory are slim. But, hey, you never know. Make a move\u2014but only if you're good at accepting rejection.\n\nGentlemen, here is another clue. After she looks away, if she glances back at you again within the minute, you must approach or else she'll be sorely disappointed.\n\n#### _Huntresses, Fight Your Instincts_\n\nHuntresses, if you are one of the vast majority of women who demurely look away when a man smiles at you, it's not your fault. Nature programmed you that way a very long time ago. Cro-Magnon males loved the thrill of the chase, just like guys today. But his female Quarry's feigned modesty didn't bother him one bit because the outcome would be the same either way. The confident Cro-Magnon just kept marching toward her with his erect club. However, Mother Nature didn't factor on modern males becoming so insecure that they would interpret the ladies' looking down as rejection and give up.\n\nGirl, do not look away. Gaze into his eyes, smile, and put down your book, drink, or anything else that a panicky Hunter could mistake as a warrior's brass shield. When he's within easy conversational distance, be the first to say \"hi.\" You will hear his palpable sigh of relief.\n\n#### _Eye Contact Is Not Unisex_\n\nHunters and Huntresses must use their eyes in diverse and complex tactical ways to keep the Chemistry churning. The following Sparker Number Two is for Hunters, Three and Four are for Huntresses, and Five for both.\n\n##### **A PPRAISING EYES**\n\nFor an attractive woman, being ogled is as common as hearing her own breath. A brief meeting of the eyes gives her no neural jolt\u2014unless your eyes stay persistently on hers and say something special. Here's how. Upon spotting your Quarry, keep your eyes on her face, not with an all-too-common hungry smile but in a mode I call \"appraising eyes\" to give her a several-layer chemical rush.\n\nI didn't comprehend what was happening to me at the time, but I now know I was the happy victim of that emotional tidal wave. At a dinner party I smiled at a distinguished gentleman near the end of the table. Apparently not noticing, he continued conversing with the lady on his left. When he tenderly touched her hand, I felt a jab of jealousy, assuming they were a couple.\n\nThen, while drowning my disappointment in the bowl of soup, I felt his eyes on me. Looking up, I almost dropped my spoon. He was stroking his chin and staring at me through squinted eyes like a radiologist inspecting an x-ray. It lasted only a few seconds, but it gave me emotional vertigo. When a smile flooded his face, I felt like I'd been saved from the lions. I wouldn't have appreciated it had his attention been too quick. Metaphorically, if I'd been hooked up to an _f_ MRI, the activity in my brain would have blown the circuits. Hunters, if you make it seem like she's earned your approval, she'll crave it all the more. Incidentally, the man at the end of the table was the ship's captain I mentioned earlier who is now my partner.\n\n  **Chemistry Sparker #2**\n\n**Don't Just Look At but \"Appraise\" Your Quarry**\n\nWhen she notices your gaze, don't give her the panting puppy dog smile that a lot of misguided guys do. Take charge and inspect her face like a jeweler appraising a valuable gem. Appear to be judging her demeanor, her disposition, her character, her comportment, her sensitivity, her soul, her whatever. Anything but her looks. The lady wants to feel that it was more than her beauty that caught your eye. Only after she sees you looking at her, let your lips form a slow smile of considered approval.\n\nHuntresses, the next two are for you. In a sense the techniques are opposites.\n\n##### **S UBMISSIVE EYES**\n\nMales are instinctively super-sensitive to hostile aggression.. Even if you are only half his size, a Hunter could find your extremely intense eye contact threatening. When he glances your way, look straight into his eyes, but let yours express instant admiration and slight submissiveness.\n\nYou can give him an extra Spark with a ploy that has animal origins. When two foxes fight in the wilderness, the one that loses submits and bares his exposed neck to the knifelike teeth of the victor. This vulnerability is an exhilarating feeling for the latter. Give him the homosapien version of this \"I submit to you\" signal.\n\n  **Chemistry Sparker #3**\n\n**Tilt Your Head and Caress Your Neck During Eye Contact**\n\nHuntresses, keep your eyes glued to his, but tilt your head to soften the exchange. Meanwhile, gently stroke your neck. Primal instinct tells him that you are protecting it because if it came to bloodshed, you know he'd win. It makes him feel like the masterful Hunter he imagines himself to be.\n\n##### **D ARING EYES**\n\nBe prepared, girl. Some of the following Chemistry-Sparking techniques are over the top and for gutsy Huntresses only. You must have the feminine version of brass balls to carry them off\u2014and the skill to turn him around later during the first conversation. However, I want to make one thing perfectly clear: I am _not_ suggesting you have loose morals or be less than the lady you are. I am _not_ suggesting you jump into bed with him any time soon. But you must lure him sexually so he can discover the real you. Casting the correct bait, then reeling in your Quarry, require two very different skill sets.\n\n  **Chemistry Sparker #4**\n\n**Scan His Body Shamelessly with Your Eyes**\n\nAs your eyes meet, let yours travel south. Start at his shoulders, work your way down his chest, then linger a second on his belt. (Courageous Huntresses, go a bit farther.) Explore his body top to crotch like an airport security officer inspecting a passenger in a body scanner.\n\nWhen you've finished your scrutiny, swiftly look up into his eyes and reward him with your smiling seal of approval.\n\nNeedless to say, gentlemen, for the sake of your freedom, do not attempt this on her. You do not need a police report on your record. The next Sparker is for both sexes and makes you feel a connection at the first meeting of the eyes.\n\n##### **L OVING EYES**\n\nDo you remember we talked about how the people who surrounded an infant in the first five years form a great part of the person's LoveMap? As a child, your Quarry's parents' eyes expressed unconditional love\u2014watching her sleeping in the crib, taking her first step, wobbling on his first bicycle ride, putting a bandage on his scraped knee. When he was a baby, he felt the love in Mommy's eyes while holding him. She sensed the adoration in Daddy's when he made funny faces to entertain her.\n\nHave you heard people say, \"She has loving eyes,\" or \"I could see the love in his eyes?\" It's true. That deep emotion does show in a person's eyeballs because internal thoughts produce physiological responses that your observer is subliminally aware of. So physiologically, you can come a step nearer to fitting your Quarry's LoveMap with this self-talk technique. To those of you who think the following Sparker is too far-fetched or \"touchie-feelie,\" I invite you to view the part of any video in which lovers fall in love at first sight. Press the pause button at the second their eyes meet. Now study all four eyes looking at their soon-to-be beloved and you'll see their pupils are ever so slightly enlarged.\n\n  **Chemistry Sparker #5**\n\n**Give Your Quarry \"Family Eyes\"**\n\nWhile maintaining eye contact, repeat to yourself, \"I feel you're like my family and therefore I love you.\" That automatically enlarges your pupils and softens your eyes, making them even more attractive and inviting. It simulates the admiration, acceptance, and unconditional love that your Quarry basked in as a child.\n\nI said earlier that Hunters and Huntresses should use their eyes in different ways to attract Quarry. Consider that a gross understatement when it comes to using your body, personality, and clothes. I flipped a coin to see which sex's strategy we'd explore first. Heads, Hunters. Tails, Huntresses.\n\nHeads won.\n\n### Hunters' Fishing Trip\n\nPickup Artists, or PUAs, as men on the prowl proudly refer to themselves, dig sexy women. So it's understandable that many misguided men assume \"sexy\" is near the top of a woman's wish list. But when was the last time you think a woman said, \"Hey girlfriend, I sure would like to meet that hairy half-naked stud over there who looks like his jeans were put on with a spray can?\" (That only happens on some TV so-called reality shows.) Guys have a hard time getting it into their heads that the female brain is totally different from theirs. Keep in mind that her hormones intuitively respond to qualities like compassion, confidence, consideration, intelligence, and financial security.\n\n\"Okay,\" you are saying, \"if I hear you correctly, these are qualities I can prove to her when we start talking and dating. But what's that got to do with her feeling quick Chemistry with me? I can't just go up to a chick and say, 'Hi, I'm caring, kind, clever, loaded with capital, can coach you to achieve your full potential, and be a great dad. Can I buy you a drink?' I mean, she doesn't even know me yet.\"\n\nOh yes she does! Compared to you, a female is a clairvoyant with x-ray vision. She senses more about you in a split second than you can in an hour of gaping at her. There are three main reasons:\n\n1.A female's neurons are more densely crowded in certain layers of the cortex that are responsible for signals coming in and out of the brain, so they have enhanced communication with each other\u2014and with subtleties in the outside world.\n\n2.A woman also has ten times more \"white matter\" in her brain\u2014and that's where she does a lot of her thinking. The significance of that? Neurons in white matter have more connections between them and have something like a greased tube called a _myelin sheath_. This makes her interbrain signals even quicker. Compare it to the clarity of a land line versus a cell phone.\n\n3.The connection between a female's left brain (logical and analytical) and right brain (intuitive and subjective) is stronger. That means she can put you under a microscope to examine your every expression, every gesture, and every syllable for its significance. Even what it revealed about you when you asked the bartender for more peanuts.\n\nShe's probably given you the thumbs up or cast you to the lions before you take your first step toward her, so from her first glance, you must demonstrate that you are an evolved twenty-first-century-kinda guy.\n\n#### _Your Actions Speak Louder Than Looks_\n\nLet's say you're at a Looking-4-Luv singles bar, and of course, the women are furtively checking you out. (It's a fact. Women who are single and searching check out every man who enters the room.) Gentlemen, carefully visualize yourself doing the following and then we'll analyze the coolness of your strategy step by step.\n\nYou are well dressed, pause in the doorway, survey the situation, and pretend not to notice your Quarry. You enter at a smooth pace, and as you head toward the bar, a couple of guys give you a fist bump. Then you greet a buddy and slap him on the back. Asking the bartender for a beer, he smiles and chats with you while two women try unsuccessfully to catch your eye. Then an attractive girl you obviously know approaches. When you whisper something in her ear, she laughs and you brush a hair off her cheek. You then put an arm around her shoulder and the two of you continue talking with your friends.\n\nThe reason your moves are Sparking her pleasure center (or caudate nucleus) is because this brain region has a tendency to live somewhat in the future. It fantasizes kicks\u2013to-come more than it pictures the joys of the present. She's instinctively and subconsciously imagining you playing a future role in her life.\n\nGentlemen, one by one, let's go over the dozen super-cool moves you made.\n\n_1. You dressed well_.  \nThe lady likes that. She knows good clothes will travel further in the business world than ripped jeans do.\n\n_2. You paused and looked around with a friendly expression_. That makes you look kind. If you come across as tough, Mother Nature nudges her, \"Watch out, dear, he might be ruthless to you and the family someday!\"\n\n_3. You strolled in at a smooth pace_.  \nYour walk evidences a strong immune system. If you limp into the room like a centipede with ninety-six missing legs, your uneven gait could signal a weak one. Huntresses pick up on the scent of bad genes like the smell of a dead mouse under the refrigerator.\n\n_4. You didn't appear to notice her_.  \nThat was a very cool move. It made you look like you weren't on the prowl.\n\n_5. A couple of guys gave you the fist bump_.  \nA man needs supportive colleagues to get ahead so having male friends is a big plus to women.\n\n_6. You greeted a buddy and slapped him on the back_. Excellent. That displays friendly dominance. Think about it, an employee doesn't slap his employer's back. A slave doesn't slap his master's back. It's the other way around. In other words, the slapper is \"on top.\"\n\n_7. The bartender smiled and chatted with you_.  \nHis respect proves you have a good reputation in that establishment.\n\n_8. Two women looked at you flirtatiously_.  \nYou are more of a prize if other women want you.\n\n_9. An attractive girl approached you_.  \n\"Uh oh,\" you might think. \"My Quarry might assume that's my girlfriend, that I'm 'taken.'\" Not a problem, gentlemen. It's okay to give her a little scare because it triggers fearful electrochemical activity in her amygdala. In a perverse way, however, she gets a kick out of it, like a scary movie.\n\nBesides, if you're in a relationship with a woman, that means you are \"preapproved.\" It's like buying a string of pearls at a yard sale versus a reputable antique store. At the yard sale you're taking a chance. In the antique store they're probably the real McCoy.\n\n_10. You whispered something in the woman's ear and she laughed_. That evidenced your good sense of humor. In Chapter 6 you'll learn some surprising reasons why women like men who make them laugh.\n\n_11. You brushed a hair off her cheek_.  \nThat was a killer move. Mother Nature tells her \"Girl, this guy is the protective sort and will take care of you and the kids.\"\n\n_12. Then you put your arm around her_.\n\nThat is a male version of \"playing hard to get.\" The jealousy gene bites and kicks your Quarry's competitive juices into action. Here's one that's in _Ripley's Believe It or Not_ , or at least \"Leil's Believe It or Not.\" I didn't until I read the studies. Most women don't mind doing what sociologists call \"mate poaching.\" Researchers asked women at Oklahoma State if they would actively pursue a man with qualities matching her wish list. Fifty-nine percent said yes. Then they asked, \"What if the man was already in a committed relationship?\" This time 90 percent of the women said they would go after him\n\nLet's sum it up. You've given the lady's pleasure center hints of good things to come with your smell of success, good genes, sense of humor, respect from other men, and caring qualities. Then you gave her brain an electrical jolt of fear that you're taken. Now when you approach your Quarry, she will be fully primed.\n\n  **Chemistry Sparker #6**\n\n**Don't Act \"Hot.\" Act \"Cool Dad.\"**\n\nHunters, in preparing for your next pursuit, the key words are not \"sexy hot\" or even \"handsome.\" Think \"caring.\" Think \"coach.\" Think \"kind,\" think \"clever,\" think \"capital.\" Think\u2014are you ready?\u2014\"cool dad.\" I know it's impossible to replicate the previous scenario, but squeeze in as many elements as possible to give tiny Sparks to any woman watching you.\n\n#### _Smiling Is Not Just Smiling_\n\nWhen the usual horny Hunter smiles at a woman, his objective is as obvious as a tarantula on his lips. You must show that you are different. I discovered the perfect expression to accomplish that while watching a teenager at a playground with his little sister. He obviously adored her and seldom took his eyes off her. I'll never forget the warm, protective look on his face as he helped her climb the precarious jungle gym. I call it the \"Little Sister Smile.\" Use the following visualization technique.\n\n  **Chemistry Sparker #7**\n\n**Bestow the \"Little Sister Smile\" on Your Quarry**\n\nLet your mind play a trick on your lips. When you first spot her, visualize a vulnerable someone you love and want to protect at all costs as you would a little sister. Your caring smile instinctively warms her heart and sets you apart from \"all those other animals\" with their leering grins.\n\n#### _Does She Really Care What I Wear?_\n\nIn a word, \"Yes!\" To her, your clothes are a projection of who you are. Never lose sight of the fact that a woman reads something into everything. Unlike you, she doesn't have sex-ray vision to undress her Quarry mentally. No matter what you think, she is not salivating to see the bulging biceps and washboard abs beneath your threads. She loves a great body but prefers it covered until impending intimacy.\n\nIn an infamous research project called the \"Hamburger Study,\" women saw photos of guys ranging from Tom Cruise look-alikes down to pictures of men even the photographer would be afraid to be alone with in the studio. The women didn't know the photos were of the same guys, sometimes wearing suits, sometimes in Burger King uniforms. Many of the former were chosen as \"husband material,\" but none of the poor fast-food slingers. Yes, a male's hunting gear is important to make the kill. To a woman, a well-dressed male is a bigger turn-on than an almost-naked one.\n\nWhen I was cruise director on a ship, the waiters were all Italian and gorgeous. At the grand finale dinner on the last night, we had a surprise tradition. Suddenly all the lights would go out. Just as passengers were freaking over a power outage, the music blared out. A hundred drop-dead gorgeous, sizzling waiters marched out of the kitchen wearing only tiny Speedo bathing suits and carrying candle-decorated baked Alaska above their shoulders.\n\nThe women hooted. The girls screamed. The men closed their eyes. Everyone laughed hysterically. The women told me they thought it was hilarious\u2014but not a turn-on.\n\nConversely, Hunters, if you saw a parade of practically naked beauties carrying cakes with candles above their heads, your eyeballs would jump out and dangle by the optic nerve. Even if you were starving, you wouldn't even notice the cake.\n\n#### _Why Coordinated Clothes Counts_\n\nSometimes, when my roommate Phil dresses for a date, he asks me how he looks. I break it to him gently, but I don't dare let him go out in his brown Timberland boots with navy pants and an olive, short-sleeved shirt. (Gentlemen, even if it's hot as a jet's exhaust outside, do not sport short sleeves. It short-circuits your sex appeal. Go for long sleeves rolled up.)\n\nThink \"coordinated.\" No black slacks with brown shoes; no brown belt with black shoes; no black socks with olive pants. And, horror of horrors, no piece of hairy leg showing in between. That merits you a female's \"yuck\" every time.\n\nYou've heard rumors that women always notice your shoes. Well, kill the rumor and file it under fact. Some have an uncanny ability to spot a pair of Pradas attached to the bottom of a male sitting on the other side of a crowded party. Buy one quality pair of shoes for your hunting expeditions.\n\n\"Why does the female population care about what I wear?\" you ask. Because coordinated clothes demonstrate creativity, taste, and intelligence. Perhaps you're thinking, \"But the kind of women I want wouldn't care about such surface stuff.\" Don't be so sure! Even the waitress from the One Horse Coffee Shop, who has never been outside of the town it's named after, has an uncanny sense of the quality of your clothes.\n\n  **Chemistry Sparker #8**\n\n**Dress Like You're Auditioning to Be Her Husband**\n\nMake sure your threads are good quality and soft because females are more sensitive to touch than you. Think quality, not quantity. And neatness counts. Even though the other guys might think it's cool to go out dressed like an unmade bed, set yourself apart. You can be casual, of course, but be casual chic, not casual cheap.\n\nDon't sweat it if your budget is more beer than champagne. Run the numbers. A small closet of fine-quality clothes costs a fourth of all the junk you most likely have stashed in it now. Unlike women, you can wear the same outfit more often\u2014as long as you air it out every now and then.\n\nA final tidbit: Before the big date that you're hoping will end with her asking \"Your place or mine?\" you have a tough decision. Boxers or briefs? Brief briefs or full briefs? Plain boxers or bright boxers? Colored briefs or white briefs?\n\nThere are no referenced studies for the following, but the results of the \"Official Dr. Lowndes Seminar Participants Survey\" says women prefer good-quality, low-cut white or black briefs on a man. But not too brief.\n\n### Huntresses' Fishing Trip\n\nHuntresses, now it's your turn to embark on the Quarry safari at the same location, the Looking-4-Luv singles spot. Let's say, because you are drop-dead gorgeous, you know men will soon swoop down on you like seagulls fighting over a crumb. But naturally you don't want just any ol' male sidling up to you, saying, \"What's a pretty girl like you doing in a place like this?\" You deserve a gentleman. So to attract more refined Quarry, you take a ladylike position and cross your legs daintily. Oops, your skirt rides up. A few men glance your way and you shyly look down. Another nice-looking man smiles at you. You quickly pull your skirt down lest he think you're cheap. When you look up again, he's chatting up some overly made-up hussy who is way less good looking than you. What an animal!\n\nAnother cool guy glances your way. As you demurely rearrange your blouse, he goes back to talking to his buddy. Ten minutes go by, and a few other Hunters smile at you. You sweetly look away and await their approach.\n\nBut it never happens! Why didn't any of them come over to say, \"hi\"? Not one of them offered to buy you a drink. You're thinking, \"I could be lying on the floor gasping for breath, dying of thirst, and no one would care.\" You figure every man in the bar is either blind, gay, or with a jealous girlfriend who packs a pistol in her purse.\n\n#### _What Went Wrong?_\n\nNo one approached because, when you sat down, your body looked like it was wearing a \"Private Property\" or \"Keep Out\" sign. When one gentleman glanced your way, you went into the archaic geisha act. Another stole a peek at your knees and you hid them. Then you modestly rearranged your blouse so guys couldn't see your alluring neckline.\n\nDo you really think a guy will tell his buddy, \"Hey Bro, think I'll go make a move on that prim little lady over there who turned away and hid her knees from me?\" To any woman who wants to capture the \"grand slam,\" as animal hunters call the biggest and the best of each species, I'm saying \"Girl, you have to change your stalking style!\" All the studies prove it: Friendly slightly suggestive aggressiveness is the name of the game if you want to win.\n\n#### _How the \"Beauty Challenged\" Can Ace the \"Tens_ \"\n\nHuntresses, earlier we talked about the extraordinary importance of looks. So one would logically assume that most men, spotting both an attractive woman and a plain one at a party, would approach the former. This is not true!\n\nUsing the common one-to-ten beauty-scale measurement, researchers engaged courageous young women of average attractiveness to go to pubs where singles mingle and to flirt overtly with men they didn't know. During all the activity the eagle-eyed investigators feverishly took notes on cocktail napkins, tracking which women various men approached\u2014the gorgeous women or the average-looking ladies in the experiment. Their findings?\n\nIf you are friendly and obviously flirtatious, men are more apt to approach you than they are the most gorgeous women in the room. But don't just take my word for it. Enjoy all the delicious details in any of dozens of studies. They are a thrilling read for us single-digit gals.\n\n#### _A Hunter's Brain Is Like a Light Switch with a Delayed Connection_\n\nYou've felt that instant Spark just spotting a guy on the street. Sure, guys can feel that too, _if_ the lady fits his previously established LoveMap. Perhaps she has just the right face, precise body, and DNA to invoke those scillions of pleasurable subconscious associations. But if you're not that one in a million, all is not lost. You can still give his brain brief spikes of voltage that he'll swear was an \"instant\" Spark. Why? Because the male neural \"instant\" is longer than yours due to those slower neurotransmissions and lack of lubricated myelin sheaths in his gray matter. His relative sluggishness on interpersonal subtleties is excellent news when trying to Spark Chemistry because you have time to flip his switch before the male \"instant\" is up. You toggle it by doing something suggestive.\n\nSisters, I'm assuming your first interest in meeting your Quarry is not just sex but also a possible relationship. You want him to be interested in you as a person, not a sex object. But of course, his first interest _is_ sex! So why shouldn't _you_ do what you expect _him_ to do for you? The Platinum Rule, remember? Don't approach him with _your_ interests in mind. Approach him with _his_. In other words, lure your Quarry with hints of sex\u2014then capture him with your substantial qualities.\n\nFor us women, it's just the opposite. We like a guy and, when we recognize his fine qualities, we become all the more sexually interested in him. But for a man, the sexual excitement must be there first. We'll talk about that now\u2014then we'll get to turning him around. A little later I'll give you techniques to show him that you are most definitely not just a sex object and dozens of ways to win his respect and love.\n\n#### _Hunting Gear for Trapping First-Class Quarry_\n\nIn light of what I wrote in the \"body parts\" paragraphs in Chapter 2, you have every right to assume I'll suggest you dress seductively.\n\nAbsolutely not! My Granny was no fool. Her words are as true today as when I was a teen. Once, as I was proudly spinning around, showing off my new micro-miniskirt, she took my hand and cautioned, \"Leilie, there are two kinds of girls in this world. The marrying kind and 'the other kind.'\" I rolled my eyes.\n\nThis many years later I owe her an apology. All serious studies support her wisdom. Instead of Granny's very unacademic designation of \"the other kind,\" however, researchers call them females with \"short-term mating strategies\" or \"munificent\" women. Yes, I had to look that word up too. It means \"bountiful,\" \"willing to give gifts,\" \"generous.\" Seeing her in too much makeup and too few clothes, no one needs to ask what she's generous with.\n\nWe've all seen guys at parties or at the mall salivating over these short-term-strategy ladies. But not one of those panting males would invite a woman with Cleopatra makeup and cleavage to her navel to his company's Christmas party. So tone down the makeup and torch the scanty outfit that you store in a matchbox. That gets you a large _quantity_ of attention, but not _quality_ attention.\n\nI told Hunters about the quality and fashion of their clothes. What about yours? Do men make judgments about that? Absolutely not. The coordination, the cost, how it reflects your deep, inner qualities, how much money you make, and whether your outfit augurs well for your future together is lambda calculus to them. In research similar to the Hamburger Study, men viewed women wearing everything from designer garb to bag-lady clothes. Men always chose the good-looking females even if they were sporting Salvation Army rejects. It doesn't matter what you wear; it's the little tricks you'll soon learn about using your clothes that count.\n\n#### _Forget Fashion, Think Flirtation_\n\nNaturally, you should match the intensity of the following techniques to your personality, the venue, the class, and the presumed mindset of your Quarry. But do not match them to your \"comfort level.\" Unless you are a hooker on heroin, I guarantee your comfort level is far too low. You'll have plenty of time later to demonstrate you are not a woman of easy virtue or \"that kind of girl.\" If you want a long-term relationship with him, you must start doing that during the first conversation.\n\nA Hunter doesn't notice your clothing, but he certainly notices what he sees of your body. That's where \"adjustable clothing\" comes in. Dress conservatively but in clothes you can shift around to reveal a little more of your body to _only_ your target man. An example is a soft blouse with a scoop neck that you can tug revealingly to the side to show more skin. Alternatively, you can let your blouse \"accidentally\" slip off one shoulder when he looks your way. Or a full skirt you can slide up to show more leg. Incidentally, here's a model's trick to make your legs shapelier when you do. With your legs crossed, push the calf of your top calf out with your bottom knee. Try it now and you'll see what I mean.\n\nIf a man's testosterone shoots up by a third just talking to an attractive woman, can you imagine how it skyrockets when you gently tug at your attire to expose a tad more of yourself? Getting a glimpse of even a relatively \"innocent\" part of your body like a thigh, a naked shoulder, or even a bare foot when you dangle a shoe on your toes gives him a little Spark\u2014and they all add up to a big one.\n\n  **Chemistry Sparker #9**\n\n**Don't Wear Revealing Clothes. Wear \"Adjustable\" Ones**\n\nDon't reveal cleavage or too much leg that any guy in the vicinity could gape at. Instead, sport outwardly conservative clothing that you can shift and slide around to reveal only _what_ you want, _when_ you want, and _how_ you want to excite the _who_ you want. Make it obvious that your ministrips are solely for your Quarry's viewing pleasure alone.\n\nBy the way, here's a neuroanatomical heads up: Don't waste your artillery more than forty degrees to his right or left during any of these maneuvers. A Hunter has fewer receptor rods and cones in his retina, so his peripheral vision is narrower. Stand as close as you can to his direct line of vision whenever casting bait.\n\n### Sizzling Sparkers for Serious Huntresses\n\nI learned this scorching Sparker in college. Not in class but from a petite brunette who had just moved into my dorm. Shannon was sweet but shy\u2014the reserved, quiet type. So we were all shocked when Carson, the Big Catch on Campus, who had only dated hot women, fell for her.\n\nLate one Saturday night after she and I had both returned from dates, I walked into the dorm's community shower room. Shannon was just slipping out of her dress revealing\u2014I gasped\u2014a skimpy, black lace garter belt, stockings, and a see-through bra! Embarrassed, she threw a towel around herself and scurried back to her room.\n\nSitting on the shower bench struggling to remove my tight pantyhose, I instantly realized what Carson saw in her. She was the type of girl he'd be proud to introduce to his family. Yet underneath, Shannon obviously was sexually savvier than any of his previous hot dates.\n\nI thought that was pretty cool and made it a point to become her friend to see what else I could learn. On our way to lunch one time we passed a Victoria's Secret lingerie store. I asked if she wanted to stop in.\n\n\"Why, Leil?\" Then I teased her about her sizzling undergarments.\n\n\"Laugh all you like,\" she said. \"It works.\"\n\n\"I'm sure it does, Shannon. And you go strutting around the campus in hot sexy underwear for all the guys to enjoy, ha ha!\"\n\n\"No!\" Taking mock offense, she continued, \"I'm very particular who gets to see it. In fact, that's how I met Carson.\" She looked around and lowered her voice. \"I knew he always had lunch at the campus canteen, so one day I sat at another table directly across from him. Then, pretending I didn't notice, I let the top of my garter show. Naturally, he looked over, and when he did, I winked at him before hiding it. Of course he came over and said 'hi.'\"\n\n\"Shannon, you've got to be kidding!\"\n\n\"No, I am dead serious!\"\n\n\"Girlfriend, you are one sassy lady! And when we met, I thought you were so shy.\"\n\n\"Not in all ways,\" she said coyly.\n\nSisters, it all has to do with your Quarry getting a private peek at the sexy forbidden, especially if you're revealing it especially for him. In the late nineteenth century savvy women in floor-length skirts flashed ankle. You need to raise the bar on this antiquity to just above your knees. Many men have told me that they find \"the peek\" at something usually hidden more exciting than nudity. Purposefully revealing a touch of a black garter or the top of a red lace bra says, \"I may look conservative outside. But on the inside I'm bitchin' hot _just for you._ \"\n\n  **Chemistry Sparker #10**\n\n**Look Sedate on the Outside, but Reveal Something Sizzling Underneath**\n\nWhen you're around small game, turn your body away, pull down your skirt, and button up your blouse. But when you spot your grand slam Quarry, reveal something sexy you are wearing underneath like the top of a lacy bra. It gives him a chock-full-of-testosterone Spark.\n\nI highly suggest garters and stockings because a man seldom sees them in the twenty-first century except in hot magazines. The somewhat uncomfortable but very sexy contraptions have come a long way since Granny's day. They now spell s-e-x.\n\nTake another tip from Shannon and put a touch of humor in your microstrip show. Your smile can express, \"I know you know what I'm doing and I'm having fun. Hope you are too.\" Winks work wonderfully during your self-presentation. The beauty of this ocular flutter is your later claim, \"What wink? I had something in my eye.\" (And you did\u2014him.)\n\n#### _One Huntress's Triple-X Tip_\n\nI can't in all good conscience put my seal of approval on this next one, but I do have evidence of its efficacy. Again I'd like to repeat, in no way whatsoever should you come across as cheap. But you can conspire with Mother Nature to bag your Quarry quickly. A clever girl in one of my dating seminars came up with an ingenious little trick. During the coffee break several men were surrounding a particular young lady who was quite poised, well spoken, and dressed in a medium-length skirt and a silk turtleneck. There was nothing cheap about her. I remember being pleased at my male students' good taste.\n\nWhen the class reconvened I asked participants to place their nametags higher on their right shoulders so I could see them from the podium. At that point I noticed Brittany's breasts\u2014not large, but tipped with little BB-like protrusions.\n\nBefore the break I had asked the group to write attraction techniques anonymously on a card, which, with their permission, I would later read to the class. When I got to Brittany's card, my jaw dropped. I looked up and saw the demure little lady grinning as wide as a banana. She squeezed her lips together signaling, \"Shh, it's our secret.\" The class was understandably confused when I broke out laughing.\n\nGentlemen, please skip the following. It's just a fashion tip (pun intended) for the ladies. Here's what Brittany had written on her card:\n\n  **Chemistry Sparker #11**\n\n**\"Modify\" Your Bra**\n\n\"Leil, please don't use my name if you read this to the class, but here's a technique that works like magic. Whenever I'm going somewhere there might be interesting guys, I dress conservatively but I snip the tips off my bras so the shape of my nipples shows under my blouse.\"\n\nThe class had no idea why I was laughing as I tucked the card away.\n\nLike all erotica, Mother Nature is behind the nipple magnetism for men. During sex your nipples become firmer, and when you're nursing, the area expands. Big nipples also help your baby find lunch. So girl, if you weren't blessed with protruding nipples, take a tip from Brittany. Big Mama in the sky would even give you the shears!\n\n#### _Erotic Truth Is Stranger than Fiction_\n\nYou've probably never consciously thought about it\u2014nor has he\u2014but a male is aroused by the scent of female underarms, therefore, also the sight of a lady's pits. As a further draw, Big Mama even makes them even spicier when you're ovulating.\n\nNapoleon, in his famous love letters to Josephine, wrote of the \"intoxicating pleasures\" of being with her. In one, he beseeched, \"I am arriving in Paris in three days. Don't wash.\"\n\nThis one sounds too silly to merit a specific Sparker, so just consider it a hint. If you are wearing a sleeveless top, lift your arm and pretend to be arranging the back of your hair. He's too far away to actually sniff your pheromones, but it invokes sweet subliminal memories of other times his nose has nuzzled that usually unseen female territory.\n\n##### **T HE OVULATING WALK**\n\nIn the 1600s, a young orphan named Catherine commissioned a cobbler to craft a pair of shoes with a higher heel in the back. This had the effect of arching her back, thrusting out her buttocks and bosom, and putting a wiggle in her walk. (Not so incidentally, Catherine de' Medici managed to bag the future king of France at the age of fourteen.) Marilyn Monroe also took hip swinging seriously. Rumor has it that she shaved down one heel of all her stilettos to put a swing in her stride.\n\nHowever, we're talking about a different gait that is more subtly sexually appealing than a wiggling walk or swinging stride. The best way to describe it would be an \"undulating, very slow-motion gallop.\" The study, \"Differences in Gait Across the Menstrual Cycle and Their Attractiveness to Men,\" demonstrated that women unconsciously dress more provocatively, and walk in this more sensuous manner during her big \"O\" days. Mother Nature knows its power and programmed it into the walk of ovulating women to signal males that \"it's time for you to make your move on me.\"\n\nLadies, don't be scandalized by some of the previous techniques. In Chapter 5, I will give you proof that if you handle your Quarry strategically in the ways I suggest, the blend of his weaker memory for details plus his stronger male ego will make him think he initiated the entire encounter. In fact, it's proven that females initiate two-thirds of all marriages, be it a smile, a come-hither look, or the first hello.\n\nObviously you should tailor your tricks to your desired catch. Refined Quarry or someone residing on Easy Street might be turned off by techniques that would excite a guy living on the other side of the tracks, and vice versa. But even if your Quarry is sitting on the top of the invisible class totem pole, you can use potent soft-x techniques to Spark him.\n\n  **Chemistry Sparker #12**\n\n**Do the Undulating Ovulating Walk When He's Watching**\n\nThe next time you're ovulating (around midpoint between your period) take note of how your walk instinctively changes. Or view a video of a high-fashion model in skyscraper heels swiveling her skeleton butt on the catwalk. Then practice, practice, practice. Just don't walk that way at church or work. It's for \"special occasions\" only, like when your Quarry is watching you.\n\n|   \n---|---\n\n## CHAPTER | _4_\n\n## How to Spark Cyber Chemistry\n\nWhen first introduced, online dating was a hush-hush humiliation for the pioneering few who clandestinely indulged in it. During that primitive period of romance, I remember asking one happy couple where they'd met. After an anxious change of glances, the wife nervously sputtered, \"Well, it was, er, umm, at . . .\" Her quick-thinking husband squeezed her hand and deftly jumped in, \"Oh, I happened to see her picture and knew immediately I had to meet her.\" He didn't say where he'd seen it! Now, however, online dating is standard operating procedure for finding love and may soon be responsible for the majority of meetings. When today's babies reach teenhood, they might ask, \"What were singles' bars?\"\n\nIf the term _computer dating_ still holds cringe-worthy associations for you, tell your hippocampus to banish the bad connotations. Dating sites are not just populated by cave trolls. Millions of single, divorced, and widowed fabulous folks are sincere love seekers just like you, and it may be just a click away.\n\nBecause information on writing a profile is plastered all over the web, and I promised to give you information that is found in few, if any, other places, I'll concentrate primarily on your picture. That's where the Chemistry starts.\n\n### The Successful Online Hunter\n\nGentlemen, a Huntress studies your face on the screen through her crystal ball to unearth what type of guy you are. Because she'll read between the lines on your face, choose a photo that shows character, not just a handsome guy. If wisdom is her big turn-on, she'll search for intelligence in your eyes. If she values a sense of humor, she'll look for credible laugh lines, not just a grimace for the camera. In the \"Truth Is Stranger Than Fiction\" category, researchers found that women can determine whether a guy wants kids just by looking at a picture of his face\n\nA good haircut is a big draw. You don't need to go to a stylist who charges by the follicle, but forget your neighborhood barber who moonlights as a butcher. The _Journal of Social Behavior and Personality_ discovered that glasses can be an asset for men. When asked why, women responded, \"They make him look more intelligent.\" The female mind doesn't stop there. Glasses are also a \"fashion statement.\" Are your frames up to date, meaning are you an _au courant_ type of guy? Are they the right ones for your facial shape, meaning are you smart enough to make the right choices in life?\n\nAnd, of course, your skin must scream \"good genes.\" Gentlemen, here's a \"You've got to be kidding\" moment, but hang in there. To prove to her that you've got all-class genes, furtively slip into the cosmetic aisle of a drugstore to buy a gooey skin-colored product called \"concealer.\" Just before the photo, lock your bathroom door and smear concealer on any pimples, rashes, broken capillaries, canker sores, large pores, or any other blight on your otherwise perfect face. (Do you think there is a newscaster, actor, or rock star alive who doesn't wear a little guy-liner while facing a camera or crowd?)\n\nFour photos are optimal. More, you look vain. Less, the ladies can't see enough to make a considered choice. Final caution: Never post a self-taken shot of you in the bathroom mirror. That says, \"I don't have a friend in the world, not even one I can talk into taking my picture.\"\n\n#### _Hunters' Clothes_\n\nGentlemen, your garb represents a possible future lifestyle to your Quarry. Does she see herself on the arm of a business tycoon in a suit, a surgeon in scrubs, a jock in a football uniform, or a casual fun-loving guy in jeans? Give the lady fodder for her fantasies.\n\nIf you select the \"I'm a nice, regular guy\" image, which is probably the best choice, follow the guidelines in Chapter 3 on clothing. The whole \"ensemble\" should look casually coordinated, perhaps a bit costly. Depending on your age and lifestyle, gentlemen, one shot in a suit doesn't hurt. It makes you look more successful professionally.\n\nYes, gentlemen, the job in your profile makes big difference. To cover your bets, unless you have controlling interest in all creation, some men told me they had more success listing a generic category (i.e., \"business,\" \"the arts,\" etc.) than stating their specific job. Some of you really hot-looking dudes might be asking, \"What about one shirtless shot showing my rippling muscles and ripped abs?\" No, that is way beyond cheesy as a choice. Your biceps may show success at the gym, but she's more interested in your success in life. She figures it's better that her future kids have a bright daddy than a beefcake daddy.\n\n#### _Hunters' Background_\n\nYou may not be aware of it, but Huntresses also scrutinize every megapixel of what's behind you in your picture. Your surroundings can make her respond\/delete decision. What's in the background of your outdoor shot? A battered truck, a motorcycle, a Subaru, or a Mercedes? What about an indoor shot? For some picky Huntresses, the click\/no-click decision comes down to curtains, blinds, or stark-naked dirty windows. Whatever is in the background, she's pondering, \"Would I like to live there or ride in that?\"\n\nIt doesn't hurt to have another woman in a secondary shot, but only if she's stunning. A study called, \"The Effects of Having a Physically Attractive Partner on Person Perception\" confirmed that when a man has a dazzling woman in tow, the world considers him richer, more accomplished, and better looking. If you decide to go this route, make sure that _all_ of the woman shows, not part of her cropped out. I've seen some pretty lame photos of Hunters with a woman's hand on his arm, and the rest of her sliced off. Your Quarry subconsciously fears that the next woman he slices out of his life will be her. Alas, our gentle sex reads something into everything.\n\n  **Chemistry Sparker #13**\n\n**Photo Tips: Show Character in Your Face and Have an Appealing Background**\n\nMake sure the lighting shows character and depth in your face and that you're not \"just another good looking guy.\" Don't forget the touch of concealer to signal, \"I have good genes.\" Pay extra attention to the background to make her feel, \"I'd like to be there with you.\"\n\n### The Successful Online Huntress\n\nGirl, let's say a Hunter writes you a cool message. You write an even cooler one back. You text a bit and then plan to talk. So far, so good. Visions of romance and maybe happily-ever-after dance through your head.\n\nBut stop. None of these pleasures will be part of your future if he doesn't like your cyber image. It's no news to women that their picture is the primary\u2014probably only\u2014factor that encourages a Hunter to click on their profiles. My only ubiquitous recommendation is to make sure it is recent and resembles you somewhat. I mention that merely because a male student in one of my relationship seminars told me that while waiting in a restaurant to meet his online date, a woman vaguely resembling the lady's picture walked through the restaurant door. He assumed she had sent her mother to apologize for her daughter being unable to keep the date. The rest of the men laughed knowingly. Obviously they had suffered a similar experience.\n\nMy regular readers may remember that, for a short while, I ran a modeling agency, and my girls gave me two great photo tips. First, the world's most photographed women close their eyes for a few seconds between camera clicks to magnify their pupils, making them more attractive. Second, a professional model doesn't \"suck\" in her tummy before the camera clicks. She first expels all her breath to slenderize herself\u2014then \"pulls\" in her tummy and lifts her breasts to \"hourglass\" herself.\n\nHuntresses sometimes ask me, \"Should I hire a professional photographer?\" No. Pros are great for wedding albums and business annual reports, but not your online picture. If you're itching to pay someone to make you look even more beautiful on the site, engage a local makeup artist and say you want natural-looking \"makeup for the camera,\" which is totally different from what you put on your face for a date.\n\nA client once hired a French makeup artist named Simone to \"fix\" my face for the cover of one of my audio programs. After an hour of sitting in the chair under bright lights while she futzed with brushes and pencils, pots and goo, I was dying of curiosity. I asked her if I could look in the mirror. Simone shook her head. I thought she was kidding, so I surreptitiously snatched it off the counter. I gasped at the ghoulish face plastered with gobs of makeup staring back at me. Simone pulled the mirror out of my hand, muttering, \"Zat ees makeup for zee camera only. It ees completely different.\" How right she was! It was the best and most natural-looking photo ever taken of me.\n\n#### _Huntresses' Clothes_\n\nHunters are not going to analyze your garb so, unless it's too revealing or too prim, don't worry about it. However, take a tip from an unusual source, the _Journal of Personality and Social Psychology_. Researchers showed men pictures of women wearing clothing in a wide variety of colors. Then the questioning began:\n\n\"Sir, tell us which women you would like to kiss.\"\n\n\"Which would you like to have sex with?\"\n\n\"Which would you be willing to spend a lot of money on?\"\n\nIn all three cases the winning lady was usually wearing red. Why? This sizzling hot hue physically stimulates a faster heartbeat and heavier breathing. So post your picture as a lady in red, and then check your in-box. If you don't have any red clothes, a red backdrop serves the same purpose.\n\n#### _Huntresses' Background_\n\nUnless you're in a jail, a house of ill repute, or surrounded by women more beautiful than you, don't worry about the background. Of these three negative settings, the last is the worst. Male subjects viewing a series of women's photos invariably judged a lady less appealing if the previous female had been more attractive. Ditto if any other woman in the same shot was more alluring. In fact, unless you're with a somewhat beauty challenged acquaintance, leave other women out of your secondary pictures. Why subject yourself to comparisons?\n\n#### _Number of Photos and Poses_\n\nMales joke about being a \"breast man,\" \"butt man,\" or \"leg man,\" and you want to cover all of these\u2014shall we say, \"special interests.\" Post one shot tastefully revealing your legs from just smidgeon higher than the knees down. Make the second a modestly covered head to waist shot, torso in profile, and no cleavage. To please the third set of aficionados, have one full body \"rear\" shot but with your face turned around toward the camera. That plus a couple of face shots and you've got all bases covered.\n\n  **Chemistry Sparker #14**\n\n**Photo Tips: Wear Red; Apply Camera-Ready Makeup; and Have a Shot for Each \"Special Interest\"**\n\nJust before the camera clicks, briefly close your eyes to enlarge your pupils. Exhale all your breath and further pull in your tummy. Wear red for your primary photo and, in addition, casual but conservative full-body shots\u2014make sure a leg man, breast man, or butt man can find what he's looking for. Now, with expertly applied camera ready makeup\u2014lots of it to look like none\u2014you'll get some possibly life-changing photographs.\n\n### A Devious Digital Tip for Both Sexes\n\nAs you now know, throughout nature a symmetrical insect, fish, bird, reptile, amphibian, invertebrate, or humanoid is considered hotter by other insects, fish, birds, reptiles, amphibians, invertebrates, and humanoids.\n\nUnless you are one of those rare genetically blessed beasts (.001 percent of the population) whose face is utterly symmetrical, turn it ever so slightly away from the camera. That disguises any microscopic imbalance that your Quarry could mistake as you being less than perfectly symmetrical.\n\nBeing a believer in truth in advertising, I hesitate to share the following. But with a little computer expertise\u2014and a lot _chutzpah\u2014_ you can rival any competing symmetrical creature on the service. Here's how.\n\n  **Chemistry Sparker #15**\n\n**Enhance Your Looks with the \"Mirror Trick\"**\n\nTake a full-face picture of you facing directly into the camera. Then make a computer image of one side of your face, flip it over, and put it on the other side. You'll be dumbfounded how fantastic you look with perfect symmetry. Is this cheating? Well, only half!\n\n### Your Online Name\n\nIf a neuroscientist had been sitting next to Shakespeare at the auditions for _Romeo and Juliet_ and heard the lines \"What's in a name? That which we call a rose by any other name would smell as sweet,\" he'd shout, WRONG! And he'd be right. Would a bouquet of flatulence smell as sweet?\n\nThe millisecond your name pops up on that dating service screen, your Quarry's hippocampus runs it through its gargantuan memory bank to see if your name has a positive, negative, or somewhat neutral association.\n\nIf you think names don't affect attraction, consider this. Hunters, let's say a friend tells you can have a blind date with a Hortense or a Heather. Which would you chose? Ladies, sight unseen, you must pick between a Bosworth and a Brad. I'll place my bets on Heather and Brad.\n\nTo prove the \"what's in a name\" concept, researchers posted a sign and a half-dozen girls' pictures on a college bulletin board announcing a supposed upcoming pageant. Written under each picture was a female's false name. The girls were all equally attractive and the students had to vote for one of six contestants for beauty queen. They then repeated the same study at another college with the same pictures but the girls' names were changed under the photos. Here are the results averaged. The girls named Heather got 59 votes, the ones called Jennifer received 52, and Kathy 47. The Gertrudes and Harriets got only 14 votes, and the poor Ethels just 11. That's 159 votes for the girls with the predetermined attractive names and 39 for the others.\n\nHunters, it seems the ladies are even pickier when it comes to your name. They preferred one-syllable monikers with a hard-stressed front consonant. Names like Curt, Dirk, Grant, Kent, Nash, Pierce, and Troy give females a tiny electrochemical jolt. If you are a Michael, Christopher, Daniel, Joseph, or Ronald, go by Mike, Chris, Dan, Joe, or Ron.\n\nConversely, gentlemen, if you are thirty-five or older, take the opposite counsel. With age and sophistication a lady prefers a two-syllable name. It sounds classier.\n\n  **Chemistry Sparker #16**\n\n**Choose a Real, Proven-to-Please Pseudonym**\n\nDon't try to be clever or cute with a made-up moniker. Huntresses, select a bona-fide girl's name with as much thought as you'd choose your future daughter's. Your fave is taken? No problem. Just add some underscores or a symbol to it.\n\nDitto, Hunters. But avoid names like \"Manson\" or \"Bundy.\" You don't want your Quarry subconsciously connecting it with a serial killer's.\n\nYou like her picture? You're impressed by his profile? Now it's time to plan your strategy. But first I must extend my sympathies to the millions of misguided Huntresses who suffer an atavistic ailment spread by word of mouth. They are under the impression that the male of our species must first write the female. Absolutely untrue. Those were the dark ages of computer dating, and it's time for us all to see the light.\n\n### Your First Message\n\nEveryone who has ever sent a message to an attractive someone on a dating site has endured the agony of a condition I'm all too familiar with\u2014writer's block. I feel your pain all the more because it's ten times more challenging to write about yourself. After the \"Hi,\" \"Hello,\" \"Hey,\" \"Yo,\" or \"How's it going,\" your fingers freeze. What should I say to this Quarry whose picture and profile I like?\n\nTo excavate pearls of wisdom on the subject, I ran an online search about how to present yourself in your first message\u2014and came up with a mere 333 million hits. Most of the suggestions were the same: Be upbeat, check your spelling, demonstrate your sense of humor, mention any shared elements, tell her about your hobbies and interests, tell him about your taste in music, movies, or thirteenth-century madrigals.\n\n#### _Who Is Your Quarry_ Really _Interested In?_\n\nPonder this: Of course your Quarry is curious about you. But who is she more interested in? Right, herself. Who is he most going to enjoy hearing about? Himself, of course. So in your introductory message, write about your Quarry's favorite subject\u2014him or herself. In fact, avoid using the word \"I\" as much as possible and try to start as many sentences as you can with _you_ or _your_. When you sprinkle _you_ like salt and pepper throughout your message, your reader finds it an irresistible spice. Gentlemen, applaud her qualities. Ladies, compliment his demeanor. How many people do you think will click \"delete\" in the middle of reading about themselves?\n\nThere's a saying in sales, \"When you're tellin', you're not sellin',\" and it's true here too. You don't need to sell yourself further because your picture and profile do the basic job. Besides, the most enticing thing about you is what's in your Quarry's imagination and fantasies. That's what keeps computer-dating companies in business.\n\n#### _Talk About What You Read Between the Lines_\n\nEven more tantalizing than talking about your Quarry's concrete qualities is what you, and only you, read between the lines about their intangible magnificence. Go through the profile again with a fine-tooth comb and get a sense of the inner person, not the outward physical appearance that any old love seeker ogles. Tell her you sense her creativity, gentleness, kindness, and honesty. Tell him you feel his integrity, dependability, loyalty, or leadership qualities.\n\nThis is not just speculation. The _Journal of Social Issues_ proved the overwhelmingly greater response to that approach in a study called, \"Can You See the Real Me?\" about detecting your Quarry's \"true self\" on the Internet.\n\n  **Chemistry Sparker #17**\n\n**Forget the External. Write About Your Quarry's Internal Qualities**\n\nRead your Quarry's profile with an \"inner\" eye. Then tell her about, say, the openness and gentleness that shines through her writing. Comment on his self-assured style that has depth and his humor that came jumping off the screen at you.\n\nYour Quarry will admire your insight and good taste. It confirms what they've known all along\u2014they are special and wonderful. And they want to meet the person who recognizes that.\n\nBefore sending, check your message yet again. If you've written something that could be sent to any other woman or man on the site, you're missing the target. Start over.\n\n#### _A Beautiful Tip from an Unsightly Man_\n\nAlthough the next Sparker is for both Hunters and Huntresses, it works better on females. In the play _Cyrano de Bergerac_ the beautiful Roxanne fell in love, sight unseen, with a hideous-looking man who, beneath her balcony, described a kiss as \"A wish that longs to be confirmed, a rosy circle drawn around the verb 'to love.'\"\n\nEvery woman has an inner Roxanne. If you're a sucker for a pretty face, she's more of one for a well-turned phrase. Words work wonders and, although you don't want to wax as poetically as Cyrano, take a hint from the verbal heartthrob by spicing up your adjectives. All you need to do is use the thesaurus feature on your computer to impress.\n\n  **Chemistry Sparker #18**\n\n**Create Magic in Your Message with a Thesaurus**\n\nDon't use words that are as common as weeds and just as unwelcome. If you want to tell your Quarry that her profile was \"interesting,\" run a synonym search and then substitute a word like, _enchanting, engrossing, or intriguing_.\n\nHuntresses, tell him his profile _mesmerized, impressed or captivated_ you. Your less common words and expressions of sentiment demonstrate that you're a cut above the rest of the trite, online love seekers waiting to ride off into the sunset with \"the one.\"\n\nNow let's go from written Sparkers to spoken Sparkers.\n\n|   \n---|---\n\n## CHAPTER | _5_\n\n## How to Spark Chemistry in Your First Conversation\n\nAfter the looking and luring\u2014either in person or online\u2014is over, the real road to the bed, beyond, and maybe forever begins at \"Hi,\" \"Hello,\" \"How do you do,\" or \"Hey.\" What follows in the next few seconds determines whether there will even be a next few seconds, minutes, hours, days, weeks, or years. Hunters and Huntresses must use diverse strategies if they want the first conversation to culminate in making a date.\n\nHunters, as you know, the minute you open your mouth, your Quarry is evaluating your intelligence, character, kindness, creativity, sense of humor, and everything else on her want-in-a-partner list. Huntresses, he is eyeing your attractiveness and how receptive you are to him. Clearly, the first conversation is a bigger challenge for males, so let's start there. However, women, do not skip it! These insights into the male psyche are crucial for you to have a successful hunting expedition.\n\n### Chemistry Sparking \"Pickup Lines.\" (Hunters, Bite Your Tongue!)\n\nMy shelf holding books for men on meeting women sags from the weight. Practically all the books include a chapter on clever opening lines, but most would only work on an eighty-year-old nymphomaniac when she was crunked. They are penned by elastic-facts players who brag they could seduce any woman, anytime, anywhere. Gentlemen, you'd have better luck finding the Loch Ness monster than a one-size-fits-all opening line.\n\nEvidence aside, the question of \"What is a good one\" is such a pathetic perennial, an invasive weed that won't die, that I must address it here:\n\nA group of sociologists who wanted their names forever enshrined in professional journals eavesdropped on male opening gambits in bars, restaurants, parties, laundromats, and other incubators of \"intergender acquaintanceship,\" as they called them. The analysts divided the overheard lines into three categories: the _direct_ , the _innocuous_ , and the _cute flippant_. Here are the stats on which approaches scored with the ladies and which hit the skids.\n\nWomen found the third category of opening lines, \"cute flippant,\" abominable. The second category, \"innocuous\" (a casual pleasantry to elicit conversation), came in second. The most successful was the \"direct\" approach, with no pretense of the pickup being anything else. Researchers wrote, \"The approaches which rated the highest were direct approaches displaying positive character traits as well as cultural knowledge.\" The conversation opener was even more welcome when the Hunter, with a confident demeanor, voiced a word of embarrassment.\n\nIf you're still not comfortable with that, here is the world's second-best approach. According to the \"Leil Lowndes Unofficial Survey of Women Sixteen to Sixty,\" women preferred it nine times out of ten, and it has stood the test of time. I prefer it. My mother preferred it. And her mother before her preferred it. The trumpet blare please: \"Hello, my name is ________. And yours?\" Try it. She'll like it.\n\nUnless a beautiful woman sounds like she's hog calling, you won't take special note of her voice, but she will yours. It can be beautiful music or microphone feedback to her. Mother Nature programs females to respond viscerally to a deep male voice because it signals more testosterone, which, in turn, makes your sperm stronger, which, in turn, makes you a better begetter of babies.\n\n  **Chemistry Sparker #19**\n\n**Forget Macho. Use a Slightly Self-Effacing Direct Approach**\n\nApproach your Quarry with total confidence\u2014but verbalize a slight touch of self-effacement. Example: \"Hi, I feel a little embarrassed about this, but I'd really like to meet you. My name is . . .\" Presenting yourself that way displays self-assurance, intelligence, and no pretense. You don't believe it works? Check out the original study called, quite aptly, \"Preference for Opening Lines\" in the references.\n\n### What Every Woman Needs to Know\u2014But Few Do\n\nThe word \"rejection\" mainlines terror into the veins of the most confident of men. They can hardly say the \"R word.\" Guys wrack their brains for pick-up lines, read books on how to be a \"player,\" take online courses, and practice personality tricks to make you swoon. Some spend thousands to attend seminars. I've spoken at these conferences and am shocked how it petrifies a guy that you won't accept his overture. He has nightmares of someone shouting out, \"Hey, everybody, look at that sniveling little critter. He actually thinks he can pick up the likes of her.\" The crowd laughs uproariously, and stripped of his manhood, he crawls away.\n\n#### _Guys Just Don't Get It!_\n\nGirl, you must make your interest outrageously blatant. One of the country's most respected sex and relationship researchers determined that 97 percent of men just don't \"get it\" when you signal them. It's not his fault. Never forget: The male brain is not constructed to sense subtleties in nonverbal communication.\n\nSome years ago, before my cousin, Rory, married his lovely wife, Camilla, he was visiting me. A nearby pub is known as one of the best meat markets in New York City, so off we went. I noticed a beautiful redhead sitting at the bar who kept smiling at Rory. To me and every other woman in the room, it was as obvious as a fly in a sugar bowl. Rory talked to a few girls and, not finding anyone special, suggested we leave. Walking home, I commented, \"Gee Rory, it's really too bad you didn't see any girls you liked.\"\n\n\"Oh, I did,\" he said.\n\n\"Really, which ones?\"\n\n\"Did you happen to notice the girl at the end of the bar with the long red hair?\"\n\n\"Yes!\" I sputtered. \"The one who was giving you the clear come-on signals.\"\n\n\"Giving me the what?\" he gasped.\n\n\"The poor girl kept flirting her little red head off trying to get your attention.\"\n\n\"C'mon Leil, don't tease me like that.\"\n\n\"Rory, I'm serious. Her signals were as clear as crystal to me and every other woman in the bar. And obviously clear as mud to you.\"\n\nTo this day, he thinks I was kidding.\n\n### The Case for Female Proceptivity\n\nFor female what? The word _proceptive_ is sociological\/anthropological lingo for the female initiating the relationship. In order to produce successful products, a company requires both quantity and quality. On Planet Earth, males are responsible for quantity of output and females are in charge of quality. It's true all throughout nature. Female mammals in Mother Nature's magnificent skies, seas, and earth are extremely picky about their \"sperm donors.\" A ladybug, bee, bird, or fish spots her preferred target bug, bee, bird, or fish. Then, by tongue flicking, self-licking, or obscene sucking sounds, she lures her chosen Quarry. Chuckling inside, if animals could chuckle, she pretends to run away. He chases her, convinced that all those spicy signals mean sure sex.\n\nLadies, learn from the lower creatures. They are smarter than you in this sense because they openly practice female proceptivity. Your mantra should be \"If you want a top-rate mate, you must initiate.\" Females are the logical pursuers. Society, however, has repressed our natural proclivities. Just as many gays and lesbians once felt obligated to hide theirs, it's time for us all to come out of the closet!\n\n#### _Huntresses' Pickup Lines (Try a Few, He'll Love It!_ )\n\nLadies, I'm not even going to bother with the kid stuff suggestions. You'll get the same old chestnuts about being the first to make eye contact and to smile from plenty of other sources. And, yes, it's good advice. To obliterate any lingering doubt about its overwhelming power, the research project, \"Giving Men the Come-on: Effect of Eye Contact and Smiling in a Bar Environment\" proves it. It's elementary. It's effortless. And yes, it works to lure him over. But we're not just talking \"lure\" in this book; we're talking about creating that chemical Spark.\n\nGirl, I cautioned Hunters to forget they ever heard the words \"pickup line.\" Conversely, I implore you to learn a few. When the bars\/restaurants\/parties\/laundromats study was reversed, males responded overwhelmingly to women's opening lines. \"Reactions to Heterosexual Opening Gambits\" proved that the direct, innocuous\u2014and even cute-flippant ones\u2014were equally effective when coming from a female.\n\nYou'd want to wear a bag over your head while delivering any of the following pickup lines, and I am _not_ suggesting any of them verbatim. I present them merely to give you an idea of the corny phrases proven to produce the desired results. Here were the researchers' scripted lines for the ladies:\n\n\u2022They directed some women to approach men with a smile and say, \"When I first saw you, I thought about introducing you to my girlfriend, but I'm not that generous.\"\n\n\u2022Others followed their Quarry with a pen in their hands and asked, \"Did you drop this?\" They then laughed and confessed. \"That was a pretty lamebrain excuse for me to talk to you, wasn't it?\"\n\n\u2022A third group said, \"Can I say I met a hot guy tonight, or do I have to lie to my diary?\"\n\nDo these lines sound cheesy? Yes. Corny? Yes. Dumb? Yes. _D\u00e9class\u00e9?_ Yes. Do they work? A resounding _yes!_ I hear you thinking, \"Oh, I couldn't do that!\" But consider this: If you don't awaken his napping neurons with something shocking, you may never meet him. Which is worse?\n\n  **Chemistry Sparker #20**\n\n**Huntresses, Plan Some Opening Lines!**\n\nSisters, think like a guy and plan phrases to meet your next Potential Love Partner, whom we'll call your PLP. There is no need to be quite as outrageous as the ladies in the study were directed, but write your own opening lines. Be creative and courageous. Raise the bar on your comfort level, and be a _little_ outrageous.\n\n#### _But Won't He Think I'm Being Aggressive?_\n\nThe short answer is no. Here is the longer one: It's not just the senile, the sleepless, and alcoholics who suffer memory loss. The male ego has an uncanny ability to rewrite history and conveniently disremember who approached whom. Remember, his prefrontal cortex is not the best egg in his basket when saturated with dopamine and testosterone. For the rest of your future life together, he will brag to your mutual friends that he, the successful Hunter, made the first move. Why? Because, as I said before, thanks to the male's more sluggish gray-matter transmissions, his \"instant\" is slower. When your blatant come-on finally dawns on him, it could be the first moment he thinks he's noticed you.\n\nAt a neighborhood gathering I ran into a friend, Melissa Richards, who had been married to her husband, Randy, for twelve years. I asked her how they'd met. \"Leil, I shamelessly stalked him,\" she laughed, and then revealed the details. When they were students at Stanford, Melissa spotted him studying at another table in the library. She instantly felt the Spark and kept sneaking peeks at Randy. When he noticed, she smiled. He returned the smile perfunctorily but went back to looking at his books. This scenario repeated itself the next day. Obviously Randy didn't feel the Spark for Melissa.\n\nOn evening three, Melissa decided to take action. She hid behind one of the shelves in the library at the usual time Randy came to study. Pretending to just enter, she came over to him and lied that she'd been sitting at that table earlier in the afternoon and had lost a contact lens. Randy crawled around the floor helping her look for it. But to no avail, of course, because Melissa never wore contacts. He then went back to his books.\n\nAlmost ready to give up, she decided to try one more ploy. She got a cup of coffee at the campus caf\u00e9, sat at his table, and started sipping it while studying. She reached for another book and\u2014oops\u2014the coffee just happened to spill. Randy, the perfect gentleman, raced into the men's room for towels to swab it up. When he finished she touched his arm and jokingly purred, \"Oh, you are my hero.\"\n\nFor the first time, it seemed, Randy looked her directly in the eyes. It was Melissa's physical touch that lit the Spark. \"Uh, let me get you another coffee,\" he stammered.\n\n\"Oh, thanks. I'll join you.\" Melissa replied. They walked side by side into the caf\u00e9. If it had been a movie, the music would crescendo as \"The End\" comes up on the big screen.\n\nSeveral weeks after the neighborhood get-together they invited me to dinner, and this time I thought it would be fun to ask Randy how they met. Here's his story:\n\n\"It was at the university library, and the first time I saw Melissa I knew she was the woman for me. I didn't even know if she drank coffee, but I'm glad she did because I asked her if she'd like one with me. And the rest,\" he grinned, \"is history.\" Melissa smiled demurely at him and kicked me under the table.\n\nSo sisters, secretly revel in your prowess and smile shamelessly into the eyes of your new Quarry. Go after him with barefaced abandon. He'll never remember how it happened.\n\nIf, despite all the evidence to the contrary, you still hesitate to deliver \"a line,\" there is the substitute testosterone tweaker that Melissa used\u2014the power of touch.\n\n  **Chemistry Sparker #21**\n\n**Trick Him into Touching You**\n\nFind an excuse for bodily contact\u2014any excuse! For example, wear a bracelet or necklace and, upon spotting your Quarry, surreptitiously click open the clasp. Then ask him, \"Oops, excuse me. Could I trouble you to help me put it back on?\" Another ploy is to ask him what time it is. When he tells you, pretend to be surprised and jokingly grab his arm to see his watch, as though you need proof that he's telling the truth.\n\nEven the most innocent physical contact gets his hormones hopping, _especially_ if he's rescuing a fair lady from a distressing situation.\n\n#### _The Physical Side of \"Hello_ \"\n\nHuntresses, you're in luck if you've met your Quarry in a somewhat professional setting because you can employ the power of bodily touch naturally. It's called shaking hands. People think, \"I can't read your mind,\" but cognitive science might take issue with that. Your thoughts produce physiological responses that are manifested in your auto-nomic, or involuntary, nervous system. Your every thought has a subtle influence on your heart rate, blood pressure, respiratory rate, and sweat glands. Lie detectors are built on that basis, and science has succeeded in implanting a device in the brain, enabling quadriplegics to control robots with just their thoughts.\n\nYou can secretly employ that cognitive power. While holding his hand, transmit a subliminal message by lightly placing your pointing finger on his pulse. In a sense, it is touching his heart, because his pulse is a wave traveling directly from the heart. Then press your whole hand against his so tightly that you couldn't shove a marble between your palms. After pumping the traditional few seconds, when he starts to pull his hand away, squeeze it and pull it slightly closer to you. Then harness the power of your mind to Spark him.\n\n  **Chemistry Sparker #22**\n\n**Think Hot Thoughts While Shaking His Hand**\n\nWhile holding his hand, gaze into his eyes and silently say, \"You are really hot and I want to have sex with you.\" Your soundless monologue\u2014some would call it \"dirty thoughts\"\u2014camouflaged by your professional demeanor, gives your face just the right expression to ignite a tiny Spark. To a certain extent, mind reading is possible\u2014especially if your mind is saying something he wants to hear.\n\nLadies, all of the above little Sparkers have a cumulative effect. He'll never guess that what he felt was due to your protracted handshake, pressing his pulse, silent salacious messaging, or surprise touch. He'll just tell your friends, \"I felt an instant Spark when I met her.\"\n\n### Spark Your Quarry During the First Conversation\n\nGentlemen, as you now know, the topic you discuss is way less crucial than what the casual chat reveals about you. Gentlemen age fifty-plus, you may remember the old eight-track music tapes in the 1970s. Your Quarry's brain is capable of listening to all tracks simultaneously. Track one: your appearance. Track two: how you move. Track three: your clothes. Track four: your voice. Track five: your intelligence. Track six: your socioeconomic level. Track seven: your personality. And track eight: how you treat her. Remember, her mind isn't one-track like most guys'.\n\n#### _The Rapid Transit System Called the Female Brain_\n\nHere is an only slightly exaggerated peek at the simultaneous tracks in the female brain. Let's say you're casually chatting with her about your health club. She's musing, _Great, he's a healthy guy_. But if you mention the gym again later in the conversation, it's, _He's a health freak_. Mention your church. _Nice. He has integrity_. Talk about your place of worship again, and it's, _This guy's a religious fanatic_. You tell her something nice about a buddy. _Good, he has male friends_. But bring up the same buddy later in the conversation? _Maybe he's gay_.\n\nScience buffs, you know a strip of blue litmus paper dunked in a vat of acid turns pink. And if you dip the tip in a single drop of acid on a glass slide, it turns just as pink. Well, the lady's mind is like litmus paper and can't help but make inferences from every drop of your conversation.\n\n  **Chemistry Sparker #23**\n\n**Don't Get Labeled by Repetition or Excessive Emphasis**\n\nHunters, be as careful as a hemophiliac in a razor blade factory during the first conversation. Beware of talking too much about the same thing, and don't return to a previous point unless really relevant. You don't need an undeserved \"fanatic\" label slapped on your forehead.\n\n#### _She Reads Between the Lines\u2014So Talk Between the Lines_\n\nYour goal in this first conversation is to show that you are boyfriend, maybe even husband material. If the woman is really gorgeous, Hunters, you might overlook a little thing like her prison record. Women don't. She secretly dons a Sherlock Holmes cap, packs a magnifying glass in her purse, and slinks stealthily through everything you say for clues to your character.\n\n\"But how can I show my good qualities without bragging?\" you ask. Here are a few ideas. Her silent reactions are in italics.\n\n\u2022Ask her favorite restaurants. Then drop the names of some of yours\u2014first-class ones, of course. Between the lines she's reading, _He's got good taste_.\n\n\u2022Compliment something she's wearing, say, her bracelet. Ask where she bought it because you want to buy your little sister a present. _How nice, he's good to his family_. But, darn, you couldn't find anything in the Tiffany's catalogue. _Maybe he's rich_.\n\n\u2022Whatever subject she brings up, say \"Tell me more.\" _He's interested in my mind, not just my body_.\n\n\u2022If at all relevant to the subject, quote something that you read about in the _Wall Street Journal_. If she's more the literary type, try the _New York Review of Books_. It will also impress her if you quote one of your favorite books. (But not this one!)\n\n\u2022If you know anyone in her field, be sure to say, \"You should meet . . .\" Now you get an A-plus for wanting to further her personal growth.\n\n  **Chemistry Sparker #24**\n\n**Talk Between the Lines, Saying, \"I Am Husband Material\"**\n\nIt makes sense that, if she's listening between the lines, you should talk between the lines. Never plan an \"opening line,\" but do stockpile ways to allude to your good-guy characteristics. Be subtle, however, very subtle! Don't worry\u2014her x-ray antennae will pick up on them.\n\n#### _A Pivotal Question_\n\nAs sure as a goose goes barefoot, your Quarry will soon ask, \"So what do you do?\" It is not necessarily _what_ your job is but _how you respond_ that could be the life or death sentence on your potential relationship. Prepare an answer that impresses her without seeming to.\n\nFrom what you now know about how Big Mama programmed the lovely, you might assume her query is solely for determining how much you make. In another era your assumption might be right. But for the most evolutionarily developed women, your vocation signifies something more important than money. It is a key to your character. For a growing number of twenty-first-century women, it doesn't matter whether you are a bigwig who owns half of the entire computer industry or work in the mailroom at Macy's. Women prefer a man of character who can get what he wants in life. If you are CEO of a global firm but despise your profession, you come off as a loser. Conversely, even if you cut cadavers or breed slugs for a living, you're a winner if you like your job because you've achieved what you want.\n\nWhen she asks, \"What do you do?\" here is the perfect answer:\n\n  **Chemistry Sparker #25**\n\n**Give a Playful but Passionate Answer About Your Work**\n\nWhen she asks what you do, respond with a playful smile and these precise words: \"For pleasure or work?\" Follow that with, \"I'm just kidding, I love my job.\" Then enthusiastically tell her more.\n\nSentence number one displays your sense of humor. Sentence two shows you are a positive person. Sentence three says you are winner because you're not stuck doing a job you don't like.\n\nA note for Huntresses: I know I may be asking the impossible, but if you refrain from asking, \"What do you do?\" it demonstrates that you are different from nearly every other female on the planet. I know it's not easy, girl, but give it a shot.\n\nGentlemen, the following conversation tip is gleaned from a study in which men were directed to \"confess\" a little fault to a new Quarry, something like you always forget your keys or where you parked the car. Because a woman is so used to a Hunter trying to impress her, she finds it charming when you do the opposite.\n\n  **Chemistry Sparker #26**\n\n**Make a Tiny Personal Confession**\n\nTo increase the Chemistry between you, confess a little fault. She'll find your revelatory repartee rare, honest, and endearing. But keep it small. She doesn't want to hear about your bankruptcy declaration, bigamy charge, or bitter divorce.\n\n#### _Today the Who-Impresses-Whom Rules Are Reversed_\n\nBeware of trying to impress your Quarry overtly, especially in the money department. There's a good chance she has more money than you anyway. In fact, single women today earn more than single men. Put your effort into showing how much she impresses you. Here's a Sparker that shows you're interested in her job.\n\nContinue asking your Quarry questions about herself and adjust your antennae to tune in to her self-image. Is she proud of being extremely bright? Spiritual? What about an extensive knowledge in a particular field? \"How fascinating,\" you exclaim, \"that you've read the twelfth-century _Dialogues de Scaccario_ in the original Latin. I'd like to hear more. When would you be free for dinner with me so you can tell me all about it?\"\n\n  **Chemistry Sparker #27**\n\n**Be Fascinated by Her Business Card and Ask Questions**\n\nWhen she hands you her business card, don't just peek at it and shove it in your back pocket. Hold it with two hands and gaze at respectfully. Glance at it several times while she tells you about her job. Occasionally nod admiringly and ask more questions about her job.\n\nHere's another question that is perfect for showing nonsexual interest. Ask her, \"What is your average day like?\"\n\n  **Chemistry Sparker #28**\n\n**Show Growing Interest in Her Mind**\n\nMen who express their interest gradually are more successful with attractive women than those who show they are immediately impressed. Convince your Quarry you are far more interested in what's between her ears than what is between her thighs. Otherwise, she figures it's just her looks, and you become as important to her as a screen door on a submarine.\n\n### Huntress, Are You One in a Million, or One Among a Million?\n\nHuntresses, if you're a gorgeous ten, little short of picking your nose, putting him down, or\u2014for some\u2014lighting a cigarette will turn him off sexually, so your first words are not as crucial. However, males have a derogatory term for women who use lascivious lures and \"don't deliver\" (hint: initials \"C. T.\"). It's only fair to lay the groundwork during the first conversation that you are not going to hop into bed with him immediately. After you've lured him with sexual signals it's time to show him that you are serious-relationship material. You are one in a million, not one among a million.\n\n#### _Begin the Bait and Switch_\n\nDid I just use that awful phrase? Shame on me. But I'm a great believer in \"the end justifies the means\" as long as it's best for both parties, is not illegal, and doesn't hurt anyone. Your \"bait\" was tweaking his testosterone in anticipation of hot times to come with you. Your \"switch\" is flashing something more important\u2014your superior qualities. In the back of his mind (albeit way back at first) are the characteristics he wants in a future wife, like intelligence, compassion, integrity, and fidelity.\n\nThere are lots of ways to hint at these. You could bring up a current news story\u2014not one in which celebrities are sleeping together but perhaps something international (intelligence). If it's about a subjugated population, express your dismay (compassion). Tell him how much you respect a friend's honesty (integrity). If the subject of past relationships comes up, make sure you never hint at any extracurricular activities (fidelity).\n\n  **Chemistry Sparker #29**\n\n**Reveal Your Substantial Qualities Early in the Conversation**\n\nAs soon as your sexy lure is successful, start showing the characteristics that quality Quarry considers long-term potential must-haves. Let nothing slip that could destroy that image. Make a conscious effort to plant the seeds that signal, \"I am a special woman of superior quality.\"\n\nHuntresses, this does not mean telling him _too_ much about yourself, however. If you read the previous advice for males, telling them to reveal a few tidbits of personal information to his Quarry in the first conversation, ignore it. It doesn't work the other way around. Instead, make your male Quarry wait for a date or two to discover more about you. Keep personal, psychologically revealing stuff to a minimum.\n\n  **Chemistry Sparker #30**\n\n**Stick to a Nondisclosure Policy**\n\nNo matter how much you're sure you're destined to be together, do not make the mistake of spilling your life story. (At this point he's not interested anyway.) Also, try to use the words \"I feel\" or \"I felt\" as few times as possible. Unlike you, at first a male is not fascinated by your feelings about anything\u2014except him.\n\n#### _Double Name Talk_\n\nCreating Chemistry, as you now know, means inciting neurons in your Quarry's brain to zap messages in all directions like a spreading wildfire. A recent National Institute of Health study showed excitation in several brain regions when hearing one's own name. Here's a way to double them to ignite the flicker into a little Spark.\n\n  **Chemistry Sparker #31**\n\n**Nudge Your Quarry's Neurons with a Double Name Whammy**\n\nJust saying his name is old hat. Double that power. Somewhere, further into the conversation, say your Quarry's first _and_ last name to give it a twofold punch. \"Palmer Smith, you are so funny.\" Or his first name twice in response to something he said: \"Oh, Palmer, Palmer.\"\n\n#### _So Far, So Good_\n\nHunters, 1) you've met the lady respectfully, 2) shown husband-material qualities, 3) made a charming tiny personal confession, and 4) proved that she fascinates you as a person.\n\nHuntresses, you 1) either lured him with sex or picked him up, 2) were upbeat, 3) a bit of a mystery woman, and, finally, 4) began the bait and switch by showing wife-worthy substantial qualities.\n\nOne or both of you decide it's time for a date.\n\n|   \n---|---\n\n## CHAPTER | _6_\n\n## How to Spark Chemistry on Dates\n\nDating is the best of times and the worst of times. You feel like you are dancing on the clouds. Then he does something that hurts you to the core. She says something that means you might lose her. You are crushed, and serotonin drops like a cannonball in the lake. Confusion and chaos result.\n\nHow heartbreaking that dating has become a competitive sport\u2014or that it is any type of game at all. In an ideal world potential partners would see each other, smile, feel confident, and form a relationship without all the difficulty and drama. But from the instant human animals sniff potential romance, electrical signals in their brains start zapping around like pinballs and their heads spin. Moments later, it's \"Tally ho, let the games begin!\"\n\nEven if you try to avoid them, as most of us do, sooner or later you and your Quarry find yourselves guessing about each other's motives, gathering evidence, envisioning, estimating, considering, and finally deciding whether to stay together or split. If that doesn't involve difficulty and drama, I don't know what does! Meanwhile, Mother Nature is loving it. She's kicking you both in your privates, encouraging your anxiety because that makes you crave your Quarry all the more.\n\nBut back to the game's starting line. Whether you become hungrier for her or you feel you've had enough of him often happens on the first date. But first you've got to get that date. Here are some ways to ensure success.\n\n### Hunters, \"Sell\" Her on the Date Like a Pro\n\nWhatever else you think about a used car salesmen, he's no dummy in one respect. After his pitch and carefully planned close, he sticks a pen between the prospect's thumb and forefinger\u2014point facing down\u2014and he deftly slides the contract under the tip of the pen. Instead of questioning, \"Well, do you want to buy it or not?\" he asks the prospect, \"When do you want delivery?\" They call it the \"assumptive close.\" He takes for granted that of course the customer wants his product.\n\nTake a tip from the pitchman and do the same. The tired old \"How about Saturday night?\" riff could spell quick rejection. Don't ask your Quarry, _if_ she'd like to have dinner with you. Take it as a foregone conclusion that of course she would and just ask _when_.\n\nTry this: \"I want to check out the new El Romantico Restaurant. What night would you be free to come with me?\" That shows confidence. Worded this way, unless she's thinking, \"You dish pit, I would never sit across a table from you,\" she's at a loss for what to say. She can't tell you she's given up eating or booked up for the rest of her life, so her only refusal resort is a revelatory humming or hawing. Then at least you know the score and won't put yourself out on a limb for another rejection later.\n\nAnother twist is what sales professionals call the \"alternative close.\" The salesman once again \"assumes\" his prospect wants the car and asks, \"Will you be taking the black one or the green one?\" Here's how to employ that tactic. Tell your date you'd like to take her to either El Romantico or L'Eleganti Restaurant. Which would she prefer?\n\n  **Chemistry Sparker #32**\n\n**Use the \"Assumptive\" or \"Alternative\" Close**\n\nWhen offering your Quarry the pleasure of spending a few hours with you, expand the window of opportunity. Simply ask, \"when?\" as though you're sure there's not a snowball's chance in a sauna that she'd say no. Another ploy is to ask her which of two places she'd prefer.\n\n### Crack the Old Hard-to-Get Chestnut\n\nHuntresses, you are talking to Mr. Magnificent Specimen and your white matter is making so many connections to every word that passes his lips that you can hardly concentrate on the order of them. But finally you hear the sequence you've been salivating for, \"Would you like to go . . .\"\n\nYou suppress shouting a jubilant \"Yes!\" before he finishes the sentence. Let's say he invites you to an excellent film you've been meaning to see. What should you do? Choose one.\n\n1.Look like you're pondering the question.\n\n2.Tell him that you're terribly sorry, but you're tied up that evening but perhaps another time.\n\n3.Give him a big smile and tell him you'd love to see that film.\n\nAnswer: None of the above! Instead, act pleased and exclaim, \"[his name], I'd love to go out with YOU!\" This lets your Quarry know your interest lies in spending time with _him_ , not in whatever he's suggesting.\n\n  **Chemistry Sparker #33**\n\n**Don't Say Yes to the Date. Say Yes to _Him_**\n\nLet your Quarry know it's not the activity you're interested in, it's the man. Your surprise answer gives him an immediate pleasure Spark. Whatever activity he suggests, here are your scripted words. Say his name and add, \"I'd love to go out with you.\"\n\n\"But,\" you may be wondering, \"what happens if he doesn't ask me out?\" Not to worry. Read on.\n\n### How to Get the Date without Asking\n\nReluctantly, you must admit that there is the rare possibility that asking you out didn't even cross his mind. The following tactic is rapidly gaining bigender and generational respect everywhere.\n\nSimply deliver a big smile and one of the following:\n\n\"Trevor, we should get together some evening.\"\n\n\"Patrick, let's go party one night.\"\n\n\"Lance, I'd love to go out with you sometime.\"\n\nThis one, followed by a wink, is my favorite: \"The next time you feel like asking somebody for a date, think of me.\"\n\nNow, that's female proceptivity at its finest!\n\n  **Chemistry Sparker #34**\n\n**Say \"Yes\" Before He Asks**\n\nIf he's lily-livered about asking you for a date, a broad hint gives him the guts to ask. If he's dense, it plants the seeds. And if he didn't think to ask you out (foolish man!), you do the job for him. This way you haven't actually popped the question, but you've made it obvious that his invitation will definitely not be met with his greatest fear, rejection.\n\n### The Most Chemistry-Sparking First Date\n\nThe constant quandary: Where should we go on our first date? Hunters, what would you most enjoy? A football game? A skating rink? An action movie? Fast action gives guys a dopamine rush.\n\nHuntresses, what about you? A French restaurant? An Italian restaurant? A Chinese restaurant? Fine cuisine and bonding conversation give females a dopamine and oxytocin rush.\n\nHmm, what he'd like to do (an exciting activity) and what she'd like to do (relaxed dining) are very different destinations. The solution? The best first date is both: an activity followed by dinner.\n\n#### _The First Half of the Date_\n\nStart the evening off with something exciting. A thriller movie, a strenuous physical activity, or maybe even something a little scary. The study \"Evidence for Heightened Sexual Attraction Under Conditions of High Anxiety\" proved a strong link between love and fear. Lovers and wannabe lovers on screen, on stage, and in novels face frightening forces together. Prehistoric beasts, ruthless killers, intergalactic invaders, and a passel of other evil forces threaten to tear the couple asunder. Did it ever cross your mind that the lovers might not even be turned on by each other if it weren't for the immense adversity they had to tackle together? Shakespeare knew its power. What would be the big thrill for Romeo and Juliet if the Montagues didn't want to kill the Capulets?\n\nA stimulating activity Sparks the dopamine thrill, resulting in a phenomenon called \"excitation transfer\" or \"transference effect,\" in which your brain assumes the thrill came from being with that particular person, not necessarily the activity. A form of psychotherapy called neurolinguistic programming (NLP) calls this effect \"anchoring.\" When you feel excitement on a date, even though it's due to an outside force, you connect or \"anchor\" it to your Quarry. Just seeing her face again can bring on the exciting stimulation you felt. Hearing his voice reinvokes the neuronal animation you experienced on the first date.\n\n  **Chemistry Sparker #35**\n\n**Date Part One: Do Something Stirring**\n\nDo something that generates electrochemical activity in your Quarry's brain as the first part of the date. Perhaps a physical activity or sports events for Hunters or an emotionally stirring experience like a concert or heart-wrenching film for Huntresses. Whatever either of you feels on this first date will rub off on the other through excitation transfer.\n\n#### _The Second Half of the Date_\n\nHunters, you may have the best of intentions and think you're being gracious by asking your date where she'd like to dine. But you should choose. Otherwise you might come across as unknowledgeable or indecisive. Make a reservation and confirm it a few hours beforehand. You look bad if your table isn't ready just because some dude named Brad Pitt made a last-minute reservation.\n\nThe eatery need not be pricey, but it should reflect your personality. Do you want your Quarry to think of you as artistic? Take her to a restaurant where artists go. Want to come off as a successful businessman? Take her to a restaurant where successful business-people go. You want her to see you as a cool dude? Take her to a restaurant where cool dudes go. There is one exception. You'd like her to think you're a jock? Do _not_ take her to a sports bar\u2014unless that's where she's dying to go. In that case, she's pretty special and you might consider proposing on the spot.\n\nHuntresses, if he is charmingly naive enough to let you choose the dining venue, don't make the same mistake that Phil's date, Goldilocks, did in Chapter 2. Unless you're positive he's a high roller, play lowball. Choose a charming little eatery within his budget. A place with a nice ambiance and low lighting is a plus because both of you will seem more attractive.\n\nIn one study, \"Effects of Aesthetic Surroundings,\" people were shown pictures of opposite sex individuals in various venues, some superior, some shoddy. Researchers then asked, \"Pick out the best-looking people.\" The folks in rooms with beautiful chandeliers, grand pianos, fine art, and other opulent etceteras were almost invariably chosen over those in greasy spoons. Of course, the photos were of the same people in both settings.\n\n  **Chemistry Sparker #36**\n\n**Date Part Two: Do Something Relaxing (Preferably with Food)**\n\nAfter the emotionally or physically electrically charged activity, wind down the date with a quiet little dinner where you can chat and enhance the bonding Chemistry between you. This two-part date sequence should leave no uncomfortable silences because, of course, you'll discuss how much you both loved the activity.\n\n#### _Eating Sparkers_\n\nHunters, as a real guy-type guy, you probably equate etiquette with extending a pinkie when drinking from a cup and wonder why your Quarry cares about manners. It's because Mother Nature reminds the lovely that no man who picks his teeth with the edge of a sugar packet is going to be promoted to the top. When gravy dribbles from your open mouth, her dopamine level dips down like a thermometer dropped in the snow.\n\nHuntresses, you can make it a more exciting dining experience for him by a few gentle, subtle, but not too suggestive moves like sliding your glass up next to his or running your fingers seductively up and down the stem of your wine glass. Let his fantasies do the rest.\n\nOn this and the many subsequent dates, you are both contemplating whether you want to continue with this PLP (Potential Love Partner) or call it quits. Where you go and what you talk about on each date factors into whether there will be a next. Here are dating Sparkers to use on every date right on up to the altar or moving-in together day.\n\n### Play Games with Your Quarry\n\nNow we're talking different type of games\u2014sports, hobbies, interests. I'm sure you've heard that females bond by talking and males by doing things together. It's not a myth.\n\nHuntresses, think back to the early days of your relationship. Were you moved when you discovered he felt deeply about something you believed in? Perhaps it increased your sense of closeness to him when you learned of his shared passion for animal rights, respect for the elderly, or concern for the environment. Likewise, a Hunter's bonding sentiments bubble up when you tell him you too enjoy bowling, bungee jumping, watching boxing matches, or whatever his passion.\n\nGirl, if you truly do enjoy his sport, highlight the heck out of it and suggest you do it a lot. Just make sure you truly do like it or else you could sentence yourself to years in noisy bowling alleys, sweaty gyms, or on the wrong end of a scary elastic rope. My regular readers know I usually subscribe to the \"fake it 'til you make it\" philosophy of life. But when it comes to serious relationships, swear off pretending. Otherwise, when your ruse is revealed, it's over. The axiom here is \"Fake it and you'll break it.\"\n\nBethany, a good friend of mine since high school, met the man she wanted to set sail with on the Love Boat, but it sank because of her lie. One time I had talked Bethany into coming on a scuba dive with me. She did, but later, while tearing off her tank, told me she'd hated it. But she certainly didn't hate what happened at the beach bar afterward. Don, a fellow diver sitting with us, introduced us to his buddy, Baird. \"A big-time diver,\" as Don described him.\n\nBaird asked, \"How did you enjoy the dive, girls?\"\n\nBethany squealed and, smiling as wide a crescent moon, lied, \"Oh, I just loved it! I can't wait to do it again!\" Right on her clever cue, Baird asked her to join him for a dive the following week. She did, and the following week, and the following week\u2014and the following week. Baird adored the fact that she was becoming such an avid new diver. They started dating regularly.\n\nBethany was so busy with Baird that I didn't get a chance to see her very often. The next time I heard from her was six months later. She was in tears.\n\n\"What happened, Beth?\" Between sobs, she told me the story. It seemed Baird wanted to go diving every weekend. All he'd talk about at dinner was their last dive and his plans for the next. From what I could determine, Bethany's ruse about her passion for diving became a drag. Her laughter at Baird's diving jokes sounded less sincere. Her smile froze as she listened to his stories of great dives.\n\nFinally, one Thursday evening, Bethany told Baird that she wasn't going diving with him that weekend because she hadn't had her hair done in months and she needed a facial due to all the wind and sand. The next weekend she found another excuse.\n\nBethany didn't hear from him one week, and when she called, he said he'd begun diving with some other friends and had met someone new, \"a dedicated diver,\" he told her. She didn't have to finish the story. Baird loved diving and he also loved women. He wanted both, and Bethany only filled half that bill.\n\nActivities that you enjoy shoot dopamine levels up just like the chemical rush you felt on your first dates. If you truly love the activity, you continue to \"transfer\" those euphoric feelings to your PLP. If you quit, the dopamine quits along with it.\n\n  **Chemistry Sparker #37**\n\n**Play the Same Games**\n\nIf there is an activity that your Potential Love Partner enjoys that you like too, play it up big time. Although it's true for both of you, joint activity is a higher priority for Hunters. It's a guy's way of bonding and bringing you closer. But be real\u2014or be wretched.\n\n### Chemistry-Sparking Conversation on Every Date\n\nAs a shy teen, just talking with someone of the male gender made my heart beat like a repeater pistol and my face look like a sunburned lobster. Before a date I'd torture myself trying to think of topics to discuss that would make him like me more. I wish I'd known the following.\n\n#### _He Says\/She Says, What?_\n\nThe gender-preferred topics are pretty common knowledge these days. If you missed some of the popular books on the subject, here are the crib notes.\n\n\u2022Huntresses tend to talk about people. Men concentrate on things.\n\n\u2022Huntresses speculate on feelings. Hunters stick to facts.\n\n\u2022Huntresses fancy the abstract. Hunters favor the concrete.\n\n\u2022Huntresses share emotions. Hunters prefer the logical.\n\n\u2022Huntresses conjecture harmony with colleagues. Hunters consider competition and who's on top.\n\nOnce again, like everything we're talking about in this book, it all makes exquisite evolutionary sense. \"Fight or flight\" has been an instinctive male reaction ever since his gorilla ancestor strutted down Noah's gangplank. Once on dry land he had to fight for dominance over the other simians, which gives insight into a Hunter's conversation preferences. He likes to talk of his dominance over today's human baboons\u2014his achievements, adventures, concepts, politics, objects, and big toys.\n\nThe neurological female equivalent to \"fight or flight\" is \"tend and befriend.\" It's the instinctive female response ever since her gorilla ancestress followed him down the gangplank. Once on dry land, tending to her tiny ones and keeping them safe was a full-time job. That gives insight into a Huntress's conversational preferences. She enjoys exploring relationships, feelings, intuition, and perceptions of other people.\n\n  **Chemistry Sparker #38**\n\n**Speak in Your Quarry's Gender**\n\nHuntresses, catch yourself if you find yourself talking too much about people and your feelings. Ditto, Hunters, if you start expounding excessively on facts and competitive situations. In short: Huntresses, stick more to things, facts, and the concrete. Hunters, try exploring abstracts and both your feelings.\n\nOh, did I forget to mention that men like to talk about sports? Girl, it's great if you can too. But be careful. Clashing with him on the MLB's top pitching rotation last year could strike you out in the first inning of your relationship.\n\n#### _Hunters, Follow Your Quarry's Rhythm_\n\nEven if you had the above crib sheet on your lap, having great conversations on dates can be tough because you don't _think_ like the other sex. But you can learn to speak in the other's style. Here are a few hints.\n\nHunters, you may have noticed that females have a different conversational rhythm. After stating an opinion, they often throw the conversation back with a tag line like, \"What about you?\" or \"What do you think?\"\n\n  **Chemistry Sparker #39**\n\n**Play Conversational Volleyball**\n\nGentlemen, after you've stated an opinion, ask hers. Once she picks her dropped jaw up off the table, she'll adore answering\u2014and you for asking. Make sure she is talking at least half the time, maybe more. As long as _she's_ talking, she'll think _you're_ a fascinating conversationalist.\n\nDon't let there be too many long silences. Those make females uncomfortable and fear you're not communicating.\n\n#### _Huntresses, Don't Fill in the Blanks_\n\nHuntresses, this may come as a surprise, but men don't need\u2014or even want\u2014to be communicating every minute the two of you are together. The male mind is quite comfortable with compatible silence. The lack of talking is not because he doesn't feel close to you. It is not because he has nothing to say. It's just that many smart men don't feel they need to fill up the air with words.\n\nAs we've discussed, your Quarry's neurotransmissions don't zap from one thing to the next as quickly. After you've said something, a male likes time to digest it and collect his thoughts before speaking. If you jump in trying to cover the uncomfortable (to you) silence, it could be jarring and destroy the closeness he's feeling during your mutual quiet time.\n\n  **Chemistry Sparker #40**\n\n**Don't Feel Obligated to Fill the Silences**\n\nWhen a lull comes in the conversation, simply enjoy it with him. A self-assured man is not the least bit uncomfortable with compatible silence. In fact, he'll appreciate it because you're probably his first date who doesn't talk just to make conversation.\n\n#### _Give a \"You Fit My LoveMap\" Feeling_\n\nYou can take a big step toward enhancing your relationship with a simple verbal tweak. Every word and tone has an emotional impact. Hearing a familiar or unfamiliar song creates a very different neurochemical reaction. A dancer named Svetlana whom I met at a neuroscience lecture told me about a brain scan she had volunteered for. During one scan, the researchers' played Stravinsky's _Firebird_ , a very exhilarating piece of music. They then played a piece from Tchaikovsky's _Swan Lake_. The _fMRI_ discovered that during the much calmer composition, the song from _Swan Lake_ , her brain was on fire with electrical reactions, but not during the more stimulating _Firebird_. Why? Because Svetlana had once danced in Swan Lake and felt closer to that composition. Every note evoked a familiar emotional, therefore neural, reaction. On a lesser scale, a familiar or less familiar word creates diverse neurotransmissions in your Quarry's brain.\n\nLet's say one Saturday afternoon you get a hunger pang, so you say to your honey, \"Hey, let's go out and get a hoagie.\" Or would you say \"submarine,\" \"grinder,\" or \"hero?\" It's all the same chock-full-of-stuff sandwich on a foot-long roll, but it has different names in different parts of the country. Ditto the regional words like \"cellar\" or \"basement,\" \"pavement\" or \"sidewalk,\" and \"sofa\" or \"couch.\"\n\nIt's not just different parts of the country. Kids learn language by listening to their parents, relatives, friends, and other familiar folks around whom they feel comfortable. Phoneticists observe that even people in different schools, groups, and parts of a city use ever-so-slightly different words.\n\n  **Chemistry Sparker #41**\n\n**Use Your Quarry's Words**\n\nTo make your Quarry feel like you're on the same \"wavelength,\" use the same words for anything or anyone. She might say, \"my mother,\" \"my mom,\" \"my mama,\" or \"my mommy.\" He might refer to \"Dad,\" \"Pop,\" \"Poppa,\" or \"my old man.\" Yet they can all live in the same neighborhood. Saying your Quarry's words for common things Sparks a subliminal LoveMap sense of familiarity and similarity.\n\n#### _So . . . How Should I Listen?_\n\nI've never actually been asked that question, but I thought I'd answer it anyway. A film of a man listening would, compared to that of a woman, look like a still life. A female's listening demeanor is closer to those toy-dunking ducks that kids balance on the side of a water glass. A man listens silently, but a woman nods to a soundtrack of trifling vocal interjections like \"uh huh,\" \"oh,\" \"umm,\" and a selection of other supportive cooing sounds.\n\nSometimes when speaking to a male, I exasperatingly assume he's not hearing a word I say because he's as silent as a dead cell phone. I mentioned it to one guy, and he had the nerve to tell me my little \"interruptions\" were annoying. (Imagine that!)\n\nWell, each gender is an expert on his own listening preferences, so to Spark similarity, go with your Quarry's flow.\n\n  **Chemistry Sparker #42**\n\n**Hunters, Listen with Nods and Supportive Sounds**\n\nGentlemen, signal you are listening by sprinkling short murmurs of understanding between your Quarry's sentences. \"Um, hum\" is okay but could get repetitive. Practice vocalizing a few supportive comments concentrating on _emotion_ words like: \"I know how you must have _felt,_ \" \"I would have _sensed_ the same thing,\" and \"I _sympathize_ with you.\"\n\nHuntresses, the reverse . . .\n\n  **Chemistry Sparker #43**\n\n**Huntresses, Don't Interject Emotional Affirmations**\n\nShh, ladies. Squelch supportive cooing, which disturbs his concentration and also takes away from your seriousness. Give the guy time between sentences to think about his next. Only during obvious pauses should you interject a comment, perhaps a supportive statement. Try little kudos like, \"That was smart of you,\" \"Excellent,\" or \"You did the right thing.\"\n\n### I Want a Man Who Makes Me Laugh\n\nYou've heard women lovingly say about a man, \"He makes me laugh.\" But have you ever heard a man say, \"She makes me laugh?\" Not likely, unless he's putting her down. Look on any dating site and you'll see \"good sense of humor\" so often that it has merited an acronym, GSOH. The desirability of GSOH has been proven by herds of serious humor researchers (not an oxymoron). Hunters and Huntresses find very different things humorous.\n\nStanford researchers engaged ten males and ten females to lie down on a narrow examination table, individually, not as a group exercise. Each subject slid head first into a hole in a huge metal brain-scanning machine with a series of cartoons projected on the circular ceiling. The examiners then measured the subjects' internal giggle or guffaw by tracking what parts of their brains lit up\u2014and how bright\u2014at each cartoon.\n\n#### _What Makes Huntresses Laugh_\n\nThe females reacted more slowly. It wasn't that they were slower to \"get it.\" It's just that, after reading the caption, the women processed it through their stronger linguistic sense, then ran it by their prefrontal cortex to see if it really made sense. The more unexpected or incongruous the caption was, the funnier females found it.\n\nHuntresses don't enjoy canned humor. They prefer Hunters who can grasp the wit in an unexpected situation and play off it. It all goes back to the evolutionary female dictum to get the best partner. Any old guy can memorize jokes. But picking up on an immediate occurrence, seeing the humor in it, and expressing it shows flexibility, intelligence, and cognitive fitness.\n\n  **Chemistry Sparker #44**\n\n**Don't Tell Your Quarry Jokes**\n\nHunters, save them for your friends, your compadres, your amigos, and your old buddy-roos. They'll love 'em. But definitely use off-the-cuff unexpected humor with women. Look on the light side of life and see something immediate and funny. Your clever comment will demonstrate your obvious superiority over all those other dull-witted Hunters vying for the lady's favors.\n\n#### _What Makes Hunters Laugh_\n\nMales, in their typically more rational, linear way of thinking, found the cartoons funnier faster. Why? Well, because they were cartoons. They were _supposed_ to be funny. Guys love sitting around with a six-pack of beer telling jokes for competitive reasons too. May the best joke win! Some guys like sight gags. But women sense the truth: Even a gorilla laughs at another slipping on a banana peel.\n\nHunters and Huntresses, there are a multitude of ingenious ways to make your Quarry take the tumble on dates. I don't want to be repetitive, however, so for more tips, I'll refer you to my previous book, _How to Make Anyone Fall in Love with You_. There you'll find eighty-five ways to do that based on sociological studies and subtle persuasion techniques. Just promise me you'll play fair and won't misuse any of the \"little tricks\" therein!\n\n|   \n---|---\n\n## CHAPTER | _7_\n\n## How to Spark Chemistry for Sex\n\nEvery author comes to a stumbling block when writing a book. Mine wasn't \"writer's block.\" It was \"shall I put this chapter in front of that one or the reverse?\" block. When it came to Sex Sparkers and Love Sparkers, should I put sex before love, or love before sex?\n\nIf I wrote about sex before love, my mother would shake her fist and shout at me from heaven. In most of today's world, however, it is antiquated to say there has to be commitment before sex. Besides, for both Hunters and Huntresses the ideal Love Boat has the same final destination, lasting togetherness. Initial lust and falling in love are just two ports along the way. Depending on whether a Hunter or Huntress is at the helm, however, the boat takes a different route.\n\nA male wants sex first, then fondness, then love.\n\nA female wants fondness first, then love and sex together.\n\nHuntresses, being choosier, usually need to like a guy a lot before sharing a pillow. Hunters want undercover activities right away. Considering this disparity, I couldn't decide which section, Love Sparkers or Sex Sparkers, to put first. So I turned to my relationship seminar students to help with the decision.\n\nOne particular program I gave in Sioux City was perfect for the task. Not only was it in the politically early indicator state of Iowa, but there were also an almost equal number of male and female participants. I told the group about this upcoming book and begged their counsel.\n\nI asked the class, \"How many of you suggest I put the love section first?\" About half the hands went up. Ninety percent of those hands looked suspiciously female. Everyone looked around and laughed, seeing how gender specific the choice was.\n\nI continued, \"How many think the sex part should come first?\" Male hands shot up like fireworks, accompanied by a chorus of affirmatives. \"Yeah,\" \"Sure,\" \"Definitely.\"\n\nI got the point. We then dove into a discussion of lust and love. All of us wound up agreeing that the two are hard to separate, so we decided to toss a coin. Heads, Love. Tails, Sex. I flipped the coin high in the air and it came back . . . tails. The girls booed. The guys cheered.\n\nI never defile the sacredness of the coin toss, so I will talk about sex before falling in love.\n\n### Huntresses, How Soon Should You Have Sex?\n\nHere I go, sounding like Granny again, but having sex on one of the first few dates is not such a good idea for many reasons\u2014and as you'll see, it's not just the obvious ones like \"He'll think I'm cheap.\" The suggestion of holding off on sex for a while sounds retro to you, right? After all, this is the twenty-first century. We know the steps to (somewhat) prevent STDs and unwanted pregnancy. So what's wrong with it?\n\nGirl, when you and a guy hop into bed and go at it, his testosterone shoots up off the charts, and his pleasure center gets drunk with dopamine. Just as he's coming, the bonding chemical, oxytocin, goes up 500 percent. In fact, that's the moment a man might blurt out, \"I love you\" to a woman he doesn't actually love. But for those few seconds, he feels he really does.\n\nBut then, _poof_ , after ejaculation, testosterone plunges, along with the \"I love you\" chemical. His dopamine crashes after coming, and the blast knocks him out. Snoring soon follows. (I hear you, girlfriend. We'll talk about his devastating not-wanting-to-cuddle condition later.) Your man's postcoital flushed face on the pillow might mumble, \"Give me an hour or so, honey, and I'll be ready to go again.\" And he may well be able to do that. But of course, it won't be as exciting for him as the first time.\n\nBut here's what most Huntresses don't understand. _Unless he's already started to develop feelings for you_ , sex with you won't be as thrilling for him the next night\u2014or the next\u2014or the next\u2014or even a week later! You could be the most exciting woman he ever met. But that extraordinary dopamine surge, like anything else that feels good, becomes less sensitive to the identical stimulus the second time, third time, and so forth. If a Hunter is having a lot of sex with the same woman (unless he's already developed feelings for her), she has to keep raising the bar. And a Huntress can only go so far with new erotic contortions, creativity, and sex toys.\n\nOne particular experiment made this dismal fact exceedingly clear. Researchers put a particular species of male lab rat into a cage with a new female. The duo indulged in a round-the-clock, nonstop orgy until the poor male zonked out, drowning in sweat and gasping for breath. But no matter how much time passed, he wasn't in the least bit interested in going at it again with the same female.\n\nHowever, when the researchers introduced a new lady into the cage, the lucky rat became ravenous again and hopped all over her. The experiment was repeated, and sure enough, the little guy could always get it up for a new girl in the cage. But the previous night's one-night stand was ho-hum.\n\nTo quote one of the landmark studies, \"Sexually satiated males cease copulating after several ejaculations with the same female; and the presence of an unknown receptive female renews copulation including ejaculation.\" This became known throughout the science world as the \"Coolidge effect.\"\n\n\"Why Coolidge?\" you ask. Well, one fine day, while inspecting a government farm, the President and Mrs. Coolidge were on separate tours. When Mrs. Coolidge passed the chicken pens, she spotted a rooster having sex. She inquired if the rooster copulates more than once each day. \"Dozens of times,\" replied the guide.\n\n\"Please tell that to the president,\" Mrs. Coolidge requested.\n\nWhen the president later passed the pens and was told about the rooster, he asked, \"Same hen every time?\"\n\n\"Oh no, Mr. President, a different one each time.\"\n\nThe president nodded slowly, then said, \"Tell that to Mrs. Coolidge.\"\n\nLaboratory rodents have a similar limbic system to human rats (and human nice guys as well), so they provide some pretty trustworthy evidence. Having sex with a man _before_ he starts to have feelings for you is not going to whet his appetite as much as wither it. Of course, _after_ he's starting to fall in love with you, it's an entirely different story. He can't get enough of you and would rather be in bed with you than any other woman in the world.\n\n### Why Sex Blows Your Mind\n\nIt makes sense that sex packs a pretty powerful punch because orgasm involves more areas of the brain than any other activity, short of an epileptic seizure. Your amygdala is in an emotional frenzy and commands your hippocampus, _Mr. Memory_ , to forget everything and just enjoy it. Your hypothalamus, _Mr. Action_ , kicks your whole body into a frenetic carnal dance. Meanwhile the various brain precincts are hollering at mission control, your prefrontal cortex, to chill and not think about anything.\n\nAll the sex chemicals are going bonkers. Testosterone swirls around with estrogen. Dopamine and his helpers have a wild party on _Pleasure Island_ (your caudate nucleus), which flashes like a strobe. Oxytocin and vasopressin light the fuse and explode at orgasm. No wonder sex is a BIG deal and blows your mind!\n\n#### _Huntresses, Sex Has an Even Bigger Effect on You!_\n\nBe careful, girl. Although men are more ravenous for sex as soon as possible, it has a stronger and longer-lasting impact on you. The powerful physiological effect from all those chemicals you release can persist for weeks, even months. If you don't feel a particular man is totally right for you, be careful, because the dopamine and oxytocin blast of sex can make you think you're in love with him. One of the most respected sex researchers in the world tells women in her lectures, \"Don't have sex with any man you're not prepared to fall in love with.\" Oxytocin has the phenomenal effect of creating trust and flushing out bad memories, closing your eyes to the fact that he's a creep.\n\nThe chemical's loving power was made very evident in one particular study in which subjects, half of whom were given oxytocin and half a placebo, were asked to recall their early childhood experiences with their mothers. The oxytocin sniffers talked much more highly of Mom. They described her being warmer and more loving than the placebo sniffers. Oxytocin even suppresses memories of bad experiences.\n\nNow I'm the one to say, \"It happened to me, dearie, it can happen to you.\" I once felt that Spark of instant Chemistry for a suave Spaniard named Santos. He had an electrifying air of mystery about him, and I fell into his bed much too quickly. I believed Santos was a man of morality and high standards. He went to church regularly and spoke often of his spirituality. Many of his stories underscored his commitment to truth and his honesty. He seemed ideal. Sex was wonderful, and I was smitten.\n\nThen one or two trivial matters sent up a tiny red flag. Santos had his caller ID blocked for outgoing calls, which I've never liked on anyone's phone. That was minor, though. Whenever I called from my cell, he'd pick up with a quick, \"Hi Babe.\" But the few times I called him from another phone with a number unrecognizable to him, he didn't even say \"hello\" until I spoke first. Apparently Santos didn't want to identify himself until he knew who was on the other end of the line. I was so in love with him, though, I didn't give it a second thought.\n\nI closed my eyes to other small indications that he might not be the honorable man I thought him to be. Whenever he got in his car, he assiduously turned on the radar detector and even attached it to my car when we rode in it. I found that a little strange because he never sped. Why was he so concerned about being stopped by the police?\n\nOnce we were going to have lunch at a diner, and driving into the parking lot, he saw several police cars parked in front of it. He made an abrupt U-turn. When I asked him why, he said he thought it was unethical that police are sometimes given free food and didn't want to witness such a disgrace. I assumed that was because he was so highly principled. But then a few drops of doubt began to splash on the cement.\n\nHis sister came to New York for a visit, and during dinner she told me there was \"a period of several years when nobody knew where Santos was.\" He just shrugged and said he'd had a falling out with his dad and didn't let the family know his whereabouts. I didn't like that, and my blinders started to slip.\n\nThen came the incident that ripped them off entirely. Santos was coming to my apartment for dinner at seven. While cooking, I looked out my kitchen window and saw someone on the fire escape across the courtyard jimmying up a neighbor's window. I had just hung up from 911 when Santos arrived. When I told him the police were on the way, he went ballistic. My beloved Dr. Jekyll suddenly turned into a vitriolic Mr. Hyde. Shouting at me that he didn't like \"snooping cops,\" he stormed out the door, leaving me devastated.\n\nIn my wounded state I started piecing together some of Santos's actions that, at the time, I had thought nothing of. First his refusal to say \"hi\" on the phone until he knew who was calling. Then his obsessive use of the radar detector for police cars. Later the avoidance of the diner where law enforcement officers were dining. Why hadn't I seen the evidence mounting that he was hiding something from the police? I subsequently discovered he was, and it was pretty serious.\n\nI wish I'd known at the time what researchers in the psychology of love have discovered. To quote one of the leading researchers in love, \"Newly smitten lovers often idealize their partner, magnifying their virtues and explaining away their flaws.\" That's what happened to me. The chemicals saturating my brain had clouded clear vision.\n\nSometimes, when I'd hear about people who'd had a bitter divorce, I used to wonder how good people who used to love each other could possibly wind up with such rancor. Now I understand. They thought it was love, but it was merely being \"in love,\" which is an entirely different neurochemical state. The couples tragically didn't wait to see if it was real love, the kind that generates different chemicals\u2014the ones we'll talk about in Chapter 10\u2014which last a lot longer and bring continued happiness. Don't let sex or the craving of it with one particular person make you think you're in love. Because love, until it's stood the test of time, really is blind.\n\n  **Chemistry Sparker #45**\n\n**Nix Sex on the First Couple of Dates**\n\nHuntresses, wait until your Quarry has feelings for you before having sex or else he can lose interest. A lot of old wives' tales are false, but this one is a neurochemical and biological fact.\n\nSex too early screws up your life in other ways too\u2014and not in ways that have anything to do with pregnancy, morality, or physical health. The sex chemicals in your brain could make you inadvertently do or say things that might mess up a potentially great relationship. Worse, it could make you think you're in love with a louse like it did me.\n\n#### _Hunters, Hints for You on First-Date Sex_\n\nGentlemen, I know this is going to fall on deaf ears, but sex on the first couple of dates is not such a good idea for you either. The oxytocin that momentarily streams through your brain at orgasm could make you say something you don't mean, like the \"L\" word. But please, Hunters, bite your tongue\u2014which is kind of hard to do just before coming\u2014and don't say anything untrue that could later hurt her. The lingering chemical could also trigger her trust circuits so much that she might not realize it if you're trying to edge out of the relationship. Men in my seminars have told me about girlfriends who, when they tried to move on, just didn't \"get it.\" Putting it more selfishly, gentlemen, you don't want a stalker on your hands! Play fair.\n\nHere's another strike against first-date sex: After a big slug of the testosterone\/dopamine cocktail, your brain can't possibly see the more subtle aspects of a woman's personality clearly. Having your mind blown by hot sex before you've even gotten to know her could blind you to a pretty terrific lady who could someday mean a lot more to you than one hot night.\n\n(Hmm, if the word got out that I was suggesting not trying for sex on the first date, no males would buy the book.) Don't hate me for saying this guys, but think about not pressuring her for sex too soon. In the long run it's better for both of you.\n\nLet's move on to subsequent dates, when sex is most likely in the picture. You probably know women want longer foreplay, but you have absolutely no idea how long! You may think it all starts when the two of you hit the sack, the couch, the floor, the wherever. Wrong. For her, foreplay began many hours, days, or even weeks before.\n\n### Foreplay Begins Way Long Before the Bedroom\n\nDear Hunters, I hope you don't take offense at the following. When it comes to giving women incredible orgasms, however, some men need a few insights and, well, a map. It's not your fault. Males are not as intuitive due to the natural neuroanatomical reasons mentioned before. Additionally, a woman often hesitates to give her man verbal or geographical directions for fear of hurting his feelings.\n\nGentlemen, during sex the lady has an infallible gauge on your ecstasy. Your sexual \"meter\" can't lie. Determining her real erotic gratification is, however, a bigger challenge for a guy-type brain. Maybe you've been keeping track of what makes her moan the most or scream the loudest. Among all of earth's miraculous creatures, a human being is the only one capable of being a conscious performer in bed. Your Quarry probably suffers from an untreatable female condition: the desire to please and make others feel good about themselves\u2014a quality that someday you will deeply appreciate.\n\nWhen a woman is falling in love with you, she's every bit as ravenous for sex as you, many times even more so. She thinks about it all the time and walks around in a state of sexual frenzy. However, when the big moment comes, it takes longer for her to push everything out of her mind and warm up. Her feelings and responses\u2014past, present, and fantasies of the future\u2014play a major role in her enjoyment of sex.\n\nWhen you meet her at the door, she may have no idea whether she'll want sex with you at the end of the evening. She waits to see if it \"feels right\" at the time. Everything depends on her memories and moment-to-moment reactions to you\u2014some momentous, some minor. Minutiae like the following can make the difference between major lovemaking and a quick \"G'nite.\" How respectful were you to her when you picked her up this evening? Did you take a long cell phone call at the table? Were you rude to the waiter? Did you leave a decent tip?\n\n\"What's _that_ got to do with sex?\" you ask. A lot, because if her hippocampus, which has a close relationship with her emotional amygdala, harbors any negative memories of the evening, sex is no-go. Unlike you, she can't box up her feelings in one brain region with no leakage. There is permanent seepage between relationship issues and sex. Whether your date with her ends horizontally or vertically can depend on what you'd call trivia. From the moment you meet at the rendezvous point, you must make her feel special throughout the whole evening.\n\nWith that in mind, let's skip to the end of the date that you hope will culminate in a lot more than a goodnight kiss.\n\n#### _Set the Stage for Sex_\n\nNeither gender can enjoy sex if preoccupied by other matters. However, you gentlemen have the enviable capability of turning your amygdala off like a single lamp on the night table and getting right into action. Your Quarry, conversely, has a slow dimmer on hers. It's up to you to bring it from its usual overactive brilliance down to dark romantic lighting, psychologically and physically. The key word is s-l-o-w-l-y.\n\nAn uncomfortable setting can also nix or fix her desire for sex. If you even suspect she might grace your pad with her presence at the end of the evening, fling any previously worn clothes on the closet floor. Then close the door\u2014tight. A woman is much more sensitive to the sense of smell. Tell your grungy socks to scurry under the bed, and figure out how to turn on the vacuum cleaner your mother gave you two birthdays ago.\n\nFinally, do an atmosphere check. Is the room too stuffy? Is it too hot, too cold? Did you know most women can't fully enjoy sex if their feet are cold? Seriously. Will your Quarry feel the lights are too bright? There are reasons she might fear this that you could never fathom\u2014like will you see her cellulite or smeared makeup? She can't let herself go and enjoy sex if she's freaking about things like that. What about the music? Does she like jazz, classical, country, heavy metal, or Nordic folk? When in doubt, ask her.\n\n  **Chemistry Sparker #46**\n\n**Hunters, Set the Stage for Sex Ahead of Time**\n\nTo a Huntress, the setting can mean the difference between \"yes\" and \"no.\" Do some preplanning before you arrive at your place with your Quarry. Your best bet is dim lighting, pleasant-smelling air, and a soft, previously made bed. Check the pillows for any long hairs, and turn off the spotlight by your bed. She's at your pad, not the gynecologist's.\n\n#### _When It's Time to Get Naked_\n\nOnce again, the key word is s-l-o-w-l-y. Instant stark nudity is not a turn-on for your Quarry. Some plucky researchers hooked females up to a vaginal photoplethysmograph to measure their vasoengorgement. (Don't ask what that is. Look it up if you're really interested.) It turns out that they were no more excited by videos of naked men than they were by naked women. In fact, women preferred panoramas of the snowcapped Himalayas to nude males.\n\nFor several years, while doing corporate consulting in Bermuda, I dated a brilliant executive there named Niles whenever I was on the island. Even though I really liked him, I had never agreed to sex. But holding hands with him across the table at lunch one day, I realized the foolishness of my ways. \"Niles,\" I purred, \"I have something for you.\" Reaching into my purse, I took an extra key card to my hotel room and placed it on the table in front of him. He took both my hands and told me how happy he was.\n\nNiles said he'd get there early, before I got back from an evening seminar, and would order room service for us. In the taxi on the way back to the hotel that night, I wondered what wine he had ordered for our predinner toast. Riding up in the elevator, I pondered what aroma would be rising through the little hole in the plate cover when I walked into the room. I'd had a hard time concentrating on my seminar during the day because I couldn't keep my mind off how wonderful sex was going to be with him that night. Excitedly, I slid my card into the lock and entered. The lights were out, but the full moon shining through the lace drapery gave a glow to the room. How exquisitely romantic!\n\nAs my eyes got used to the light though, instead of a room service table set for two, there was a naked Niles stretched on the bed with his arms behind his head. \"You can see that I've been waiting for you,\" he said huskily, referring to his excitement meter pointing straight up. I apologized profusely and feigned a splitting headache. After I practically pushed him out the door, I went down to the hotel pub, had a big hamburger, and swore never to see him again.\n\nUntil recently I feared my aversion to his no-foreplay nudity might be abnormal and that I was the perverse one, not he. I was relieved while reading the plethora of studies confirming it was a normal female reaction. Niles wasn't abnormal either. It's just that he wasn't in the know about a woman needing more time to get in the mood.\n\n  **Chemistry Sparker #47**\n\n**Let Her Start the Strip**\n\nHunters, don't you start peeling. After an appropriate length of time, if she hasn't, start gently with a button on her blouse or perhaps your shirt. But keep your hands away from your belt until it's obvious the feeling is mutual. Grabbing for your buckle can make her grab for her coat.\n\n#### _Oral Sex (It's Not What You're Thinking_ )\n\nIf the lady has extended the invitation to join her for a nightcap, do not pounce on her the moment she sits down. Start with a few caring moves, perhaps touching her cheek or brushing the hair out of her face. Now it's time to get \"oral,\" as in \"talking.\" It's tough because a guy's lack of ability to process emotions and say just right thing at the right time was formed in his mother's womb. It's hard for a guy's brain to wrap itself around the fact that words mean so much to his Quarry, so let me suggest a few phrases. Note, I did not say lines or lies! After all, if you didn't feel one of the following sentiments, you wouldn't be sitting next to the lady.\n\n  **Chemistry Sparker #48**\n\n**Don't Get Physical, Get Verbal**\n\nWords work wonders on your Quarry's need to calm down before heating up. Gaze deeply into her eyes and talk softly to the lady. Let her know how lovely she looks, how fascinating you find her, how happy you are that you are together tonight. Verbal foreplay is sexier to her than passionate pawing.\n\n#### _The Educational Value of Daytime Soaps_\n\nLikewise, Hunters, you'll get priceless insights into your Quarry's fantasies by watching a daytime TV soap opera gushing with strong, sensitive, respectful\u2014yet sometimes recklessly passionate\u2014males. One guy friend told me he learned \"the killer kiss\" from the _Young and the Restless_. He said, \"The dude cupped the woman's face in his hands and gave her a light kiss on the nose. Smiling, he pulled back and gently outlined her lips with his finger. Then he told her how beautiful she was and enveloped her in his arms.\"\n\n_Sigh_.\n\nJust to make it more complicated, no two women are alike in precisely what they want once the action begins.\n\n#### \" _Oh, Great, How Do I Find Out What She Really Wants?_ \"\n\nGentlemen, go with your strong suit, your gray matter. Simply ask her. Choose a tranquil time, preferably one when sex is not in the immediate picture. Say something like, \"You look especially beautiful when you're so relaxed.\" Lead the conversation around to the question, \"What physically relaxes you the most?\" She might say something like a warm bath, a back rub, a foot massage. Whatever it is, tell _Mr. Memory_ to store it away and remind you when it becomes pertinent. Then, when the time comes try to arrange her favorite atmosphere.\n\nI fell prey, blissfully, once to what might have called a ploy. But looking back, I don't care. It was lovely. Josh and I had been dating for a while, and at one point, while sitting on the couch, he took my hand tenderly and said, \"Leil, please forgive me if this is an improper question. But have you ever considered having sex with me?\" I didn't tell him that I'd thought about little else since we met. I managed a demure smile and a shy, tentative \"yes.\"\n\nJosh squeezed my hand and continued. \"If I should be so lucky, how would it happen?\" I melted like an ice cube under a blow torch. He teasingly asked me if I'd like music, what kind of wine, lighting. It seemed so sweet and silly that I started giggling. But on our next date everything was in place. And because of that, so was I.\n\n  **Chemistry Sparker #49**\n\n**Ask Her to Describe Her Secret Garden**\n\nHunters, I hesitate recommending this approach unless you feel confident you can carry it off, but I do want to give testimony to its efficacy. Lovingly\u2014and with perhaps a touch of humor\u2014ask her what she thinks is the perfect atmosphere for \"making love.\" (Forget you ever heard the word \"sex.\") Then have it all set up the next time you see her.\n\n### Why She Needs All That Stuff First\n\nWhen I was a kid, Mama gave me amazing birthday parties. The best one of all was a breakfast birthday with waffles and four different kinds of syrups\u2014strawberry, blueberry, raspberry, and maple. All the kids, about ten of us, were wide-eyed when Mama triumphantly carried in an almost two-foot-high stack of waffles. The girls clapped and the boys grabbed at the treat.\n\nOne of the oldest boys assiduously poured strawberry syrup in one of the squares. He then proceeded to spill blueberry in the next, raspberry in another, maple in a fourth. All the other guys thought it was cool and followed suit. We girls, however, mixed them together and made our own creative blends of several syrups. We then passed our plates around the table to share our concoctions with our friends. The boys weren't interested and continued to concentrate on dispensing syrups into the different squares.\n\n#### _Squares versus Swirls_\n\nA grownup guy does that with his life. His job goes in one little waffle square, sports in another, and sex in yet another. No problem. But here's the rub: In order to enjoy sex, _both_ Hunters and Huntresses alike need to make it an all-consuming experience and think of nothing else. Gentlemen, if you couldn't stop thinking about your boss hollering at you this morning, the SOB who got the promotion instead of you, or the power outage during your big PowerPoint presentation, how could you get it up for sex? The difference is that when sex is in the picture, you can instantly chuck garbage like that out of your mind and jump into a different waffle square.\n\nYour Quarry can't. It's neurologically impossible. Only when she suppresses memories and extinguishes her other thoughts can she thoroughly enjoy sex. Just as the girls at my birthday party had no dividing lines between their syrups and swirled them all together, we grownup girls blend all the elements in our adult lives.\n\nHunters, your Quarry needs time to clear out all that crappy real-life stuff. Otherwise, the mean boss and the bitch who beat her out at work are right there in bed with the two of you. The power outage during her presentation darkens her desire. Because a woman is more relationship oriented, everything you said to her today, yesterday, and last week swirls into her desire to have sex with you\u2014or not.\n\nThink of a Huntress's brain like a big dish of pasta. It's all interconnected. If the sex noodle is at the bottom of the bowl, she can't pull it out without several other gummy noodles sticking to it. And if you try to separate them too fast, you'll have more success playing pickup sticks with cooked spaghetti strings. If you really want your Quarry to forget everything and get into sex, you must clear her mind with caring words and caresses all over her body.\n\n\"But I don't have time to do that every time!\" you protest. I understand, Hunters. But stay tuned. All will become clear.\n\n#### _Women Don't Come with Pull-Down Menus and Online Help_\n\nYour caring words create the bonding chemical oxytocin, and your caresses create even more of that precious element, especially when you kiss her breasts. I don't expect you to find anything titillating about reading the _Gynecologic and Obstetric Investigation_ journal, so I'll translate. Studies on female sexual arousal stress the importance of fondling her breasts. It's not just because it feels good; they're a regular oxytocin factory because her breasts are the big deal in baby feeding.\n\nWhen it comes to love, oxytocin is huge for a female. Mother Nature doles out a major dose to her during pregnancy and then a whopping three hundred times more of it at childbirth. That attachment chemical just keeps on coming and coming during nursing and beyond. As a result, through thick and thin, Mom will forever bond with the little person who started in her womb. The togetherness chemical also makes her want to bond with you and have even more sex together.\n\nFemale skin is more sensitive and sensual all over than a man's. Hunters, comparatively, your body is a single-destination trip. She, however, has multiple sexual layover spots along the way. A different part of her brain lights up with each one. You exploring a much vaster territory stimulates her _Pleasure Island_ , and playing around on each part takes time.\n\nRutgers researchers\u2014rather cheeky ones I must say\u2014put women into an _fMRI_ machine with nothing but a dildo for company. The ladies wound up using it and their hands on four parts of their bodies\u2014nipples, cervix, vagina, and clitoris. Sometimes all four at once. But the clitoris was always involved.\n\n#### _Who Stole the G-Spot?_\n\nUnfortunately, a woman often feels it's her fault if she can't orgasm with no male attention to her clitoris, condescendingly referred to by Freud and Co. as the \"little penis.\" I was once discussing this distressing situation with a few of my girlfriends, and each confessed that she never gave her boyfriend a thorough geography lesson lest he feel his technique was being questioned.\n\n\"And while we're at it,\" my best friend, Stella, piped up, \"I've never been able to find that darn G-spot that every woman in the world has except me.\" I explained that Dr. Ernst Gr\u00e4fenberg was the person, male of course, who pushed it to stardom in the 1950s, claiming women had a vaginal climax separate from a clitoral. (Apparently Mrs. Gr\u00e4fenberg never spoke up.) For decades afterward medical examiners conducting autopsies scratched their heads wondering why the G-spot mysteriously disappeared posthumously.\n\nThey never found it because it never existed as the orgasmic gizmo it was reputed to be. So if coroners burrowing around in there with expensive equipment couldn't find it, don't depend on it giving her ultimate pleasure without equally attending to her hottest spot of all, the \"C-spot\"\u2014the clitoris.\n\n  **Chemistry Sparker #50**\n\n**Master Geography 101 for Hunters**\n\nWomen joke about how territorially challenged some men are when it comes to the female body. She is probably lying if she claims she can climax from vaginal entry alone without any contact with her clitoris. Ask your girlfriend for a geography lesson to her particular body because each female is slightly different. Once she's gratefully given you a personal tour of her unique favorite spots, be sure to give each lots of lengthy tender, loving care.\n\nSisters, a quick note for you: A softer way to guide him through the southern regions is simply to say, \"I especially love it when you touch me here, or there\u2014or both at the same time.\" It sounds obvious, but you'd be surprised how many women shy away from doing this.\n\n### Huntresses Are Naturals\n\nHuntresses, you are more romantically intuitive than males are, and you're natural pleasers. Not only that but you'll find scads of information everywhere you turn. Unfortunately, one of the most popular sources of information is _Cosmopolitan_ magazine, published in twenty-eight countries in sixteen languages. An issue I came across gives instructions for running her tongue around the rim of her boyfriend's penis while stroking his balls with her left hand and insinuating her right index finger up his anus. I am not making this up!\n\nAm I putting this invaluable font of knowledge down? Not really. My only regret is that although _Cosmo's_ published demographics start at eighteen, young girls now give a subscription to their girlfriends for their sweet sixteen parties. Ouch. The implication that all girls are having sex with their beau du jour and the lack of warnings about the psychological and physiological perils of promiscuity are pretty scary. My feminist side also fears the geisha mentality it imparts. That aside (and it's a pretty big aside), it is indeed informative in the gymnastics and erotic toys departments.\n\nOne _Cosmo_ feature was especially enlightening. The title was \"Thirty Things to Do with a Naked Man,\" and the subtitle touted, \"Sex Tips to Tease, Squeeze, and Totally Please Your Guy.\" The editorial advised young ladies to employ blindfolds in bed, frozen grapes in her mouth, Velcro restraints, gladiator heels, Gummi bears, sushi rolls, small mirrors, and a beaded necklace as a penis wrap.\n\nOf course, accoutrements are not always necessary. It informs the young reader that she can \"intensify his orgasm by placing two fingers an inch behind his balls and feeling for a dent\" or \"invite him to finish on your breasts.\" Another suggestion is to \"challenge\" him to have sex in five spots in your apartment before sundown.\n\nI have no problem with any of the above. (However, your male Quarry might with that last one.) In fact, sex toys and erotic contortions are fabulous fun during the period when Mother Nature is still plying the new lovers with drugs. Sex is a mind-blowing turn-on for a Huntress, and she can't get enough of her guy. The Sparks on _Pleasure Island_ spread like a brush fire on a windy day just knowing the thrills she's giving her Quarry and how much he'll love her for creative and strenuous gymnastics. It's action for a cause.\n\nIn the natural course of a relationship, however, a Huntress would tire of doing all that work. Additionally, if she's pushed the relationship too fast, it might dawn on her Quarry that he could get like favors from a less long-term, more munificent lady. He's thinking, \"I could get these benefits without her always asking me where our relationship is going or freaking if I even look at another girl.\"\n\n#### _Go Where Women Aren't Allowed_\n\nLasting Chemistry and true love don't come from doggy-style, convoluted Indian\u2013style, or even Viennese oyster\u2013style sex. It's not finding that dent or luxuriating in his essences all over your breasts. Fun stuff aside\u2014or included\u2014in order to get some long-term neural activity going, you need to go beyond the beaded necklaces and blindfolds. You must delve into the most erotic part of the body: his brain, of course.\n\nLet's discuss what his hundred billion neurons are up to while having sex. Until he's fallen deeply in love with you, a Hunter usually suspends relationship thoughts during the act. In fact, if he feels the \"situation\" softening a bit, he consciously switches gears, invoking hot memories of a previous encounter, perhaps a woman he spotted on the street today, a threesome, foursome, or more-some. Or perhaps just visions of isolated breasts, genitals, and other favored female body parts dance through his head.\n\nWhat kind of fantasies does he have? Well, let's see. How many stars are in the sky? How many angels can dance on the head of a pin? How many pickled peppers did Peter Piper pick? Sexual fantasies are as varied as the males who have them.\n\nOut of perverse curiosity, one time I asked an ex-boyfriend, who later became a buddy, if I could scrounge through the dusty stack of old porn magazines and videos he confessed he'd hidden under the bed. In the box that he'd labeled \"extra blanket\" I unearthed a gold mine of information about what men really find hot, and it's not frozen grapes and Gummi bears. One magazine was simply a selection of stories about sexy showgirls, naughty nurses, and warrior vixens. If you know the types of things your Quarry is fantasizing about during sex and embrace it, you're caressing his hottest organ by far\u2014his brain.\n\nTo get an authentic education on the male sexual psyche, start perusing mild men's mags like _Penthouse_ and _Playboy_. Then if you can stomach it, graduate (or sink to) reading _Gallery_ and _Swank_. After hitting rock bottom with _Hustler_ , you'll either be an expert on men's fantasies or decide to become a lesbian.\n\n  **Chemistry Sparker #51**\n\n**Read What _He_ Reads for Hot Hints**\n\nWhy read about what turns your Quarry on in magazines written by females? Forget _Cosmo_. Read the real story in the rags written by and for men. I'd also suggest visiting a few online porn sites, but the viruses, spyware, tracking cookies, Trojans, and other social-media transmitted diseases could be injurious to your computer's health.\n\n#### _Better Yet, Get It Straight from the Horse's Mouth_\n\nHuntresses, in order to excavate sexual buried treasure he's never shared with anyone, find a way to bring up the subject of erotic fantasies. Perhaps with a wink, let him know you find the topic titillating. Maybe tell him you were reading an article or book on them. (But not this one!) Better yet, reveal one of your own fantasies\u2014being taken by a handsome stranger, doing it in the airplane lavatory\u2014the usual stuff. The hotter, the better.\n\nAfter your \"big confession,\" casually ask if he has any fantasies. That's a rhetorical question, because of course he does. If he says, \"All I think about is you,\" that's sweet, but he's lying. Keep probing. Secretly, he'll love it. Let him know _nothing_ would shock you.\n\nAt that point your Quarry may surreptitiously look around, lean forward, and most likely confess a mundane fantasy\u2014which he feels is monstrous. The most common male fantasies are having sex with two women, watching them get it on with each other, or a little dab of dominance. Your playful smile and hearing, \"ooh, that's exciting\" floods his pleasure center with a tsunami of dopamine. That plus the torrent of testosterone create the perfect storm.\n\n  **Chemistry Sparker #52**\n\n**Conduct an Erotic Interview**\n\nAsk your Quarry what he fantasizes about during sex. After he recovers from the delicious shock of your question, he will be thrilled because you're probably the first female requesting entry into that tangled exotic erotic universe where anything goes. The neural pathways connecting you to his _Pleasure Island_ will open up like a superhighway.\n\n\"Does that mean I have to perform with another woman or wear a corset and crack a whip?\" you ask. Of course not. Most women have had \"rape fantasies.\" But that's the last thing they'd really want!\n\nIf his fantasy happens to be something you would enjoy doing, however, like wearing hot lingerie or stockings to bed, go for it. Just be prepared that his fantasy could be _him_ wearing the stockings and hot lingerie to bed! Unless you find it distasteful, letting him do his sex dream is the hottest sex Sparker in existence. It's the ultimate in sexy for him because _he_ wrote the script. Just keep in mind that doing his fantasy is usually not a one-night performance. If you're serious about the chap, you better decide if you'd like a lifetime of it. His tying you to the bedpost or wearing your panties as foreplay could go from big turn-on to turn-off.\n\n### Why Won't He Cuddle After Sex?\n\nRock stars, cinema heart throbs, or the Quarry we just met often pop into a female's sexual fantasies. But when we're in bed with a particular man, our erotica generally centers on him.After the lovemaking, we rerun the tape of what he did, what he didn't do, and what we wish he'd done\u2014and end by wishing he'd be more affectionate after sex.\n\nMy heart goes out to the many millions\u2014no, the majority\u2014of women like my friend Brandy who still thinks that if a man doesn't cuddle after sex it means he doesn't care for her. I now hope to dispel that myth once and for all with chemical evidence that it's not true.\n\nI tried to tell Brandy that a guy is neurologically cuddle-challenged, but she wouldn't listen. I wanted to explain that the big blast of oxytocin released during orgasm affects the male brain like a sleeping pill. It's a universal male condition called \"postcoital narcolepsy.\" When mixed with other sex chemicals, they produce an essence similar to melatonin, which plays a big role in a human's body clock. It's a tall order for a man to stay awake after sex and has nothing at all to do with how much he cares for you.\n\nIn one of my seminars I was talking about how women adored hearing loving words after sex and quoted a few romance novels.\n\n\"My darling girl, I want to hold you forever.\"\n\n\"I am in heaven when I'm holding you.\"\n\n\"I could die of happiness in your arms like this.\"\n\nThen, just for fun, I asked the men in the class if they had any favorite postcoital phrases for their girlfriends to suggest to the other guys. Blank faces stared back at me from all over the room.\n\n\"Uh, well, what you usually say?\" I queried.\n\nAfter a brief pause one guy raised his hand and said, \"Zzzzzzzz.\" The guys hooted and slapped their knees. The girls grimaced and nodded all too knowingly.\n\n  **Chemistry Sparker #53**\n\n**Let Him Have His Instantaneous Postsex Snooze**\n\nThe second a Hunter comes, it's like he's popped an Ambien. When he zonks out it does not mean he wouldn't like to hold and kiss you. He biologically practically can't! Complaining, accusing, or insisting on snuggle time is not going to help your relationship. Just cuddle up against your chemically challenged animal and get forty winks yourself.\n\nThe more evolutionarily curious among you might well ask why nature makes a male suffer this condition. (Although it's usually the girlfriend or wife who suffers the most.) Big Mama in the sky's reason for knocking him out after sex is because he can't kick Charlie into action for a second round right away. But if he has a rejuvinating little snooze, he might get it up for another shot at a tiny new earth dweller.\n\n### Do You Decide Who Turns You On\u2014or Does Mother Nature?\n\nHere's another little known fact about sex. Sometimes your relationship was just not meant to be, and Mother Nature lets you know in no uncertain terms during the first kiss.\n\n_Hunters_ , let's say the first date with her went beautifully. She was more alluring, affectionate, and fun than you could ever have hoped for. As you rowed the rented rowboat, she admired your muscles. In fact, you both were enjoying yourselves so much that you brought the boat back late. But you handled it beautifully. After the owner tied up the boat, you crossed his wet palm with some significant green. He thanked you profusely and she looked up at you admiringly. It was the perfect date. She shared your interests, was impressed by your subtle but intelligent humor, and even ordered the same pizza at the pub. You think maybe she is \"the one.\" By her warmth you can tell she probably feels the same.\n\nBack on her doorstep, she leans seductively toward you. You know she is cranked for your kiss. As your lips touch, she parts hers, inviting your tongue to enter. You oblige.\n\nSuddenly she stiffens and pushes you away. With a weak excuse, she races into her house. You're shell-shocked. What happened? Everything seemed so perfect. \"What did I do wrong?\" you ask yourself.\n\n_Huntresses_ , your date rings the bell. You take one last look in the mirror, splash a dab of perfume behind each ear, run down the stairs, and throw open the door. Wow, he's hotter than you remember him when you met him last week! You know you look awesome too. You see intelligence, warmth, and humor in his eyes. The evening is magic. You talk 'til dawn. You think you've found \"the one\" and suspect he feels it too.\n\nLater, just before sunrise, in the car he starts nibbling on your neck and you tenderly guide his head down to your cleavage. You arch your back and throw your head back. God, he feels good.\n\nBut suddenly he sits up, gives you a weak smile, a peck on the cheek, and revs up the car. You are flabbergasted. What happened? Everything seemed so perfect. \"What did I do wrong?\" you ask yourself.\n\nNeither of you did anything wrong! But Mother Nature was not happy. And when Big Mama's not happy, nobody's happy. So why didn't she want the two of you to be a couple? Because she figured your eventual kid might not live up to her survival standards.\n\n\"What? How in the world did she deduce that?\" you ask.\n\n#### _If the First Kiss Stinks, Forget About It_\n\nEveryone has a unique cluster of chromosomes that control the ability to fight off infections and disease. The tiny parasites in every human body reproduce faster than rabbits, and the germy little bastards are just waiting to attack your immune system. If you copulate with someone with similar bacteria, the stinky little slime balls join forces and will be victorious over your eventual baby's immune system. It's the reason brothers and sisters and other close relatives shouldn't mate. Twenty years later, if the little tyke with the feeble immune system lives that long, he or she won't be able to produce healthy babies either. Mother Nature doesn't like that one bit. So she uses her Chemistry set to make the smell or saliva of your date distasteful to you if your microbes are too much alike.\n\nI was once explaining this phenomenon, called your Major Histocompatibility Complex (MHC), in a relationship seminar. Feeling insulted, one student shot up his hand and said, \"Hey, I ain't got no complexes.\" I assured him that MHC is not a complex. It's a universal biological state.\n\nI hate to reference the following tasteless, overly publicized study because it's such an old coin that the buffalo has worn off. But it's sort of unforgettable. Researchers convinced women to sniff a selection of guys' grungy T-shirts they'd worn for a week.. It turns out the women actually liked the underarm aroma of guys with a _dissimilar_ smell, or MHC. ('Fess up, sisters, haven't you ever slept with some article of your boyfriend's clothing when he was away?) But the T-shirt stink of guys who had a fragrance too similar to their own, like a family member's, grossed them out. That's why the smell of opposite sex siblings doesn't turn most people on sexually.\n\nFar sweeter than any perfume is the natural odor of a sexual partner who has a different odor from yours. I'm not talking about flatulence, halitosis, or foot odor. I'm talking the hairy sweaty body parts, the fragrance of which is revealed through apocrine glands. Well, let's call a spade a spade and say it: It's your groin and armpits. These pits routinely spit out a lot of important internal social information.\n\nA woman perceives odors better than a man does, and it's not just so she can tell when her baby's diaper is full. Because she's the real mate chooser, she needs to be extra skilled at literally sniffing guys out. But watch out\u2014she can also get a whiff of your stinky feet a couple yards away.\n\nGentlemen, your Quarry might appreciate your favorite over-the-counter man scent when you first meet. It shows you are clean and take care of your appearance. But between the sheets she responds better to the aroma of raw (clean) male. A Hunter is less sensitive to this aromatic matter on the first kiss because he's thinking of what's to come with the rest of her body. \"Why let a little unsavory scent get in the way of a good thing?\" he figures.\n\nWhen searching for a sexually compatible partner, French kissing is a great test. White-fronted parrots have an ever better one. After the birds open their beaks and touch tongues, the male spews his lunch onto the female's chest.\n\nYou really needed to know that, didn't you?\n\nHuntresses, if your Quarry doesn't pass the taste test, no matter how much you adore him, he's not the right partner for you. Hunters, her fragrance may not be a big problem now, but sex with her in the long run could be odious. This is one time both of you should say, \"Thanks, Mother Nature, for letting us know it wouldn't be a marriage made in heaven.\"\n\nI hope the fragrance industry doesn't employ a slew of litigious attorneys, because the following Chemistry hint could ruin a good ROI on their annual advertising millions.\n\n  **Chemistry Sparker #54**\n\n**Don't Wear the Smell-Good Stuff During Sex**\n\nHuntresses, males get high on your natural smell. To a male, an ovulating female is subconsciously one of the sweetest smells in the world. The scent drives him crazy. (Lap dancers report they earn more on the big \"O\" days.) So wash off the perfume before slipping into bed with him.\n\nGentlemen, she will be more turned on by burying her nose in your (somewhat recently washed) armpits than by a coating of your favorite aftershave. Let your natural aroma sexually stimulate her.\n\n#### _Huntresses on the Pill, Read This Carefully_\n\nThe infamous stink study had opposite results for women on birth control pills. They were more attracted to guys who smelled _similar_ to them. Why? It's because the pill simulates the scent of pregnancy. Here's how Mother Nature sees it: \"If you're already pregnant, Miss, it's meaningless for you to be with a guy who turns you on. It's better for you to be around people who smell like you, your blood relatives, to help you raise Junior.\n\n\"Besides, if you still want sex with the father of your child all the time, it could keep him from going out and impregnating another woman to give me another earthling.\" Mother Nature stops at nothing!\n\nBottom line, girl, do not select the lucky man to be your permanent partner and father your children while on the pill. You don't want to wake up next to him someday when you're off it and think, \"Ugh, this guy stinks!\" Wait to see who smells right when you're off the pregnancy-simulating tablet.\n\nLet's say that, with your Quarry, it's \"So far, so good.\" So very, _very_ good. Each date leaves you breathlessly awaiting the next. Sex is spectacular, and you want to stay together under the covers ordering room service for three days straight. Mother Nature is ecstatic because her chemicals are working. Early in the game dopamine is the captain deliriously dancing on the bridge of the Love Boat. Testosterone and estrogen are euphorically singing at the helm. It seems you are right on course toward Happily-Ever-After Land. But if you don't keep a careful eye on the compass, the confused trio can steer you into stormy waters. With just a little more knowledge, however, you can control the deeper love Chemicals and keep the Love Boat afloat forever.\n\n|   \n---|---\n\n## CHAPTER | _8_\n\n## How to Spark Chemistry for a Relationship\n\nYou've been dating for almost a year now. It's fabulous fun, and the sex is mind blowing. You feel miserable when you're apart, and it's magic when you're together. Maybe he's \"the one.\" Perhaps she's the woman you could love forever. You're starting to feel serious\u2014seriously wonderful!\n\nWell, most of the time. Huntresses, you worry because at times he closes off for no reason and won't even tell you what he's thinking. Hunters, you fell for her hard and fast and truly do love her\u2014but you begin to wonder because sometimes she gets overly emotional over \"absolutely nothing.\" He gets angry or she gets pissy, and you both begin to fear your potential permanent Love Partner might become just a brief affair.\n\nThe biggest reason dating couples break up is not big fights. It's not infidelity. It's not even mistrust. It's gradually growing disappointment and irritation with each other. That's a pity because you could be perfect for each other in the important things in life: your beliefs, values, goals, and interests. But too often one person starts to find the other aggravating and throws in the towel on a potentially near-perfect relationship.\n\nWhy is that so common? Because people don't understand that they're practically dealing with a chemically and neuroanatomically different species! Books have tried to teach us what to do and what to say as well as to speak the other's language. But that's not enough. You must learn how to _think_ in a different world\u2014the other sex's brain.\n\n### Mars and Venus Told You the _What_ , Now the _Why_\n\nMany if not most relationship problems have deep roots in differences between the sexes. But you've known about those for decades. About twenty years ago the author John Gray cleverly and charmingly introduced the world to the non-astrological sense of the planets, Mars and Venus. Back then, readers took copious notes on the challenges that Mr. Mars and Ms. Venus faced with each other:\n\nMars hides his feelings. | Venus wants to share them.\n\n---|---\n\nMars doesn't listen. | Venus talks more.\n\nMars quickly offers solutions. | Venus gives unsolicited advice.\n\nMars wants to feel needed. | Venus needs to feel loved.\n\nMars reacts with anger. | Venus reacts with emotion and tears.\n\nNothing has changed in the years since the book _Men Are from Mars, Women Are from Venus_ was published. Males still hide feelings, don't listen, quickly offer solutions, want to be needed, and react with anger. Females still share feelings, talk more, give unsolicited advice, need to feel cherished, and react with emotion and tears. It isn't going to change in our lifetime. We are born that way.\n\nFor those of you who find gender differences as fascinating as I do, here's more: In preschool a little guy played more competitive games, got angry quicker and more often, fought more, talked later and less, didn't enthusiastically welcome other kids into his games, used more play space, was less sensitive to his friends' feelings, was more interested in objects than people, averaged thirty-six seconds for family goodbyes, identified with the hero or robber in stories, and played with trucks and action toys. If someone did force him to play with dolls, he used them as dive bombers.\n\nConversely, female tots played fewer competitive games, got angry more slowly and less often, fought less, talked earlier and more, welcomed other kids into their games, were more sensitive to their friends' feelings, used less play space, were more interested in people than objects, averaged ninety-three seconds for family goodbyes, identified with the victim in stories, and played with dolls\u2014not as dive bombers.\n\nHave we changed that much? Neuroscientist Dr. Rhawn Joseph wrote,\n\n_Within the core of each of us is the child we once were. This child constitutes the foundation of what we have become, who we are, and what we will be._\n\nSo why, these many years after Mars and Venus enlightened our planet, do half of all marriages still end in divorce? John Gray did a great job telling us how to change our behavior, but unfortunately, at that time he didn't have access to the neurological research because it didn't yet exist.\n\nThere is a big difference between knowing something and _really_ knowing it. You know that a car stalls, but unless you understand the construction under the hood, you can't fix it. Unless you know what chemicals to put in the gas tank, it's not going to stay purring very long. Likewise, unless you know the neuroanatomy, neurochemistry, and neuropsychology behind why Mars and Venus are so different, you won't remember what to say when she accuses you of \"what\" or he does \"that.\" Memorizing phrases isn't enough, and it makes avoiding the same mistakes almost impossible.\n\nWhen you were a kid you knew you shouldn't lick a frozen flagpole because your mother told you not to. But you don't really \"get it\" until you know why: The moisture on your tongue freezes instantly, forming an ice bond between your porous, now very painful body part and the pole. Of course you'll have plenty of time to ponder thermal conductivity while you await the rescue squad. But by then it's too late.\n\nIn many ways, in addition to technologically, a kid is smarter than his elders. Once he's felt like a thousand tweezers are ripping off his taste buds, he'll never lick the pole again\u2014no matter how much his big brother double-dog dares him.\n\nAdults, however, don't learn from their lessons. They marry and, not grasping the immense biological\/neurological difference\u2014and evolutionary influence\u2014of their mate's brain, get stuck. The ripping away is excruciatingly painful. And then they go out and get stuck all over again. Once again, with a new partner, they vow to stay together\u2014\"for richer, for poorer, in sickness or in health, to love and to cherish until death do us part.\" For far too many couples, unfortunately, it's \"until misunderstanding, jealousy, or 'I just don't love you anymore' do us part.\"\n\nSome fortunate few have the ability to swim through the synapses, figuratively speaking, of their Love Partner's brain. The other 99 percent of us need a life preserver to avoid drowning in the foreign rivers.\n\n### The Ball of Yarn vs. the Stack of Boxes\n\nJust last week Giorgio and I were sitting on the couch looking at movie listings and had decided on the latest Tom Hanks film. As we were considering the earlier or later show, I asked, \"Giorgio, did you know that Tom Hanks is a grandfather?\"\n\nGiorgio looked at me quizzically and asked, \"What? What's that got to do with whether we go to the six or eight o'clock show?\"\n\n\"Nothing.\" I mumbled, \"I just thought I'd mention it.\"\n\nA year ago his response would have exasperated me. But this time, finishing up this book, I caught myself.\n\nI just kissed him on the cheek and said, \"Giorgio, you are being so . . . guy.\"\n\n\"Would you want me to be anything else?\" he chuckled.\n\nI picked his arm up off the back of the couch and put it around me. \"No,\" I purred.\n\nHe kissed my cheek, and we went back to looking at the listings.\n\nThen I asked, \"Giorgio, don't you think it's interesting that Hanks is still a big sex symbol and he's a grandfather? How many grandmothers do you think are sex symbols?\"\n\nAt that, he abruptly stood up to do something in the other room. Oops, I realized. Once again I'd slipped again into \"female brain\" type of thinking. If I'd been talking with a girlfriend, it would have been perfectly logical for us to go from the movie times . . . to Tom Hanks the actor . . . to Tom Hanks the grandfather . . . to a male sex symbol's age . . . to joking about sex symbol grandmothers. That could even lead to discussing how my grandmother used to cook with a pressure cooker. So you see, it makes perfect sense that we could be talking about movie times one minute and pressure cookers the next. Any woman would understand that. But not Giorgio, not your brother, not your father, not your Quarry, nor your male Love Partner.\n\n#### _His Box Brain_\n\nHuntresses, think of a big box with lots of little boxes inside it. That's your man's brain compared to yours. Those interior squares aren't stacked tightly together. In fact, the perimeter of one box doesn't even touch the next. Your man may not neatly fold all his socks and place them in his drawers, but he neatly tucks all his thoughts into separate boxes in his brain like the little boys at my birthday party did with the waffle syrups.\n\nIn order to think quickly about another subject that women call \"related,\" he must make the comparatively long voyage across his wide synaptic rivers between neurons. Then he needs time to choose which box the thought should jump into. So when you swim quickly and smoothly from one subject to the next, he thinks you're scattered. And you're infuriated that he can't just casually discuss anything with you.\n\nGirl, be grateful he can compartmentalize like this. Men have the ability to concentrate and focus intently on one problem at a time better because all his thoughts on that matter are squished into one little box. He wants to stay right there until he figures it out and resists if you try to drag him out of it. Giorgio's box brain was still debating the early versus the late show. And a \"totally irrational female\" was babbling about grandmothers not being sex symbols.\n\n  **Chemistry Sparker #55**\n\n**Tell Him If You're Switching Boxes**\n\nHuntresses, when you're talking with your PLP and you want to say something you think is related to the subject, he might not see the association. Remember, a male's neurons are more sparsely packed, and this slows down his neural connections. His six and a half times more unmyelinated gray matter further detracts from the speed. So explain the connection to him before saying anything he could possibly think is irrelevant. If he seems confused or exasperated, wind up your point quickly by saying, verbatim, \"The bottom line is . . .\"\n\n#### _Her Yarn-Ball Brain_\n\nConversely, Hunters, her brain is not a set of organized containers. Think of it as a ball of yarn. Any knitter knows that to form the yarn for storage, you start with a loop, and then wrap the next string around the center of that. As it begins forming a ball, you gradually rotate it until the pieces of yarn are in place. Each string goes in a slightly different direction, but they're much more interconnected than your boxes. Her more closely knit neurons means she can jump effortlessly from one subject to the next, and you might not understand the association. The sudden switch makes some men think it's \"screwy thinking.\"\n\nDon't let it drive you up the wall. Try to grasp the connection of the new topic to what you're discussing. If you can't, just have faith that, in her brain, it _is_ relevant. The lady's intricate maze of millions of closely connected neurons can tweet messages all over her brain a thousand times quicker than a lizard's tongue. Gentlemen, you'll find this an advantage for you in some cases!\n\n  **Chemistry Sparker #56**\n\n**Understand Her Ball of Yarn**\n\nHunters, your lady love has ten times more of that speedy white matter, so of course her thoughts are on fast forward. To your more logical mind, her comment sounds like it's coming out of the blue. The next time she confuses you with something that seems irrelevant, either try to figure out the connection or just smile, shrug, and think, \"That's my girl.\"\n\n#### _P.S. for Huntresses_\n\nYou might want to try compartmentalizing some of your thoughts in boxes too. It not only helps you get along with your Love Partner better, but it can also keep you from being ticked off at a lot of things. A couple months ago understanding the Stacked Boxes vs. Yarn Ball Concept helped my life in two small ways. It not only saved me from being ticked at Giorgio, but it also got me some lovely lingerie I wouldn't have had.\n\nHe had foolishly made the mistake of going shopping with me and was waiting in the safety area near the lingerie department where embarrassed, bored, impatient husbands hang out. As I laid a bra I wanted to buy on the counter, I told the saleswoman I liked it so much that I was going to quickly grab the panties to match. When I returned seconds later, three ladies were in line. I stood on the side, naturally expecting the saleswoman to take me next because I'd been there first. In spite of my strident throat clearing, she continued to ignore me and wait on the others. I was furious and went storming over to Giorgio with a disgusted, \"Let's go!\"\n\n\"Why?\" he asked.\n\n\"Because I don't want to buy anything from that bitch!\" I sputtered, \"I was in line and just went away a few seconds to get something else. In the meantime she starts waiting on all those people who hadn't even been in line, without even a smile or apology to me!\"\n\n\"What's that got to do with the underwear. Do you want to buy it?\" he asked.\n\n\"Yes!\" I almost shouted. \"But . . .\"\n\n\"So?\" he interrupted. \"The other customers were ready to buy before you were.\" I started to blow up at him but, thanks to my new insights, I understood. For him, the lingerie was in one mental box, and my fury at the salesperson in another. He saw the two facts as separate, whereas I saw them all intertwined.\n\nI harrumphed, went back, scooped up my purchases and joined the line, realizing he was right. Thanks, Giorgio.\n\n### What Are You Thinking About?\n\nHuntresses, put your left hand on the Bible and raise your right in the air. Now try to tell me you've never had this conversation with your Quarry.\n\n_You_ : What are you thinking?\n\n_Him_ : Nothing.\n\n_You_ : Oh, come on. Tell me.\n\n_Him_ : Really, nothing.\n\n_You_ : What do you mean? You've got to be thinking about something.\n\n_Him_ : No, really!\n\nHe's telling the truth, Huntresses! Not thinking about something is inconceivable to us. We can never quit because our brain neighborhoods are constantly communicating\u2014on a ground line, no static. His, however, have long periods of silence. When there's nothing specific in one of his boxes to solve, his entire stock of thoughts drains out. His boxes can be completely empty. Nothing in them. Nada. Zip. Zilch.\n\nAs a relationship ripens, I've heard many women complain, \"He and I have nothing to talk about anymore.\" But have you ever heard a male groan, \"My wife and I can no longer find anything to talk about?\" He doesn't care because, for him, silence can spell togetherness.\n\nIt is said that one evening the poet William Wordsworth visited his good friend Samuel Taylor Coleridge. The two gentlemen sat silently by the fireside, contentedly smoking their pipes and saying nary a word. As the hours wore on, Mr. Wordsworth stood up, shook his friend's hand, and said, \"It has indeed been a delightful evening, Mr. Coleridge.\"\n\n\"The pleasure is all mine,\" Coleridge replied. And they both meant it!\n\nCan you imagine two girlfriends spending an evening together like that? Males enjoy sitting silently together and just chilling. Giorgio is one of those \"strong silent types,\" and I used to rack my brain for things for us to chat about when we were together. I felt we had to keep talking in order to be communicating.\n\n  **Chemistry Sparker #57**\n\n**Don't Bust into His Boxes**\n\nDon't ask your PLP what he's thinking about. When a Hunter is stressed out, retreating and curling up in his cozy \"nothing box\" is a very comforting place for him to be, and he gets exasperated if you try to yank him out. No matter how much he adores you, he sometimes prefers chilling out alone to chatting. It's crashing his private party to ask, \"What are you thinking?\" When he emerges, he'll feel closer than if you'd invaded.\n\nMy new neurological insight has been a real relationship booster for us. Now Giorgio and I sometimes go through an entire dinner with little casual chatting. I can tell that he's much happier in one of his boxes, and I am learning to enjoy my own yarn ball of thoughts sitting silently with him.\n\n### How Do You Feel About That?\n\nThat's the sequel to the perpetual male query, \"What does a woman really want?\" The stack of books in my basement oozes with ridiculous advice on \"how to get a man to share his feelings.\" Here are some of the worst suggestions I've read. (To avoid embarrassing the authors, some with recognizable names, I won't reference them.) They recommend telling him:\n\n1)\"Honey, I know the male role requires that you be tough, objective, and unemotional. But you'll be even more of a man in my eyes if you can open up to me.\"\n\n2)\"It's okay to have feelings. It's not a sign of weakness.\"\n\n3)\"I don't suspect you're hiding anything from me.\"\n\n4)\"I've always shared my feelings with you. Now all I'm asking is the same.\"\n\n5)Share your feelings with him honestly first. Then tell him, \"See, it wasn't so hard. Now it's your turn.\"\n\nThe ubiquitous, all-time worst advice in those books was \"Be sure to look him directly in the eyes when probing his feelings.\" Males definitely do _not_ communicate best gazing into another set of human eyeballs. They'd prefer to be staring down the barrel of a semiautomatic twelve-gauge shotgun. In fact, the deepest guy communication is done sitting side by side in a foxhole, pointing their guns at the same enemy.\n\nHuntresses, what if he insisted that you do something you were physically incapable of doing\u2014like picking him up and hoisting him over your head? Do you think you'd be able to do that if he told you any of the following?\n\n1)\"Honey, I know the female role requires that you be soft, feminine, and not muscular. But you'll be more of a woman in my eyes if you hoist me up over your head.\"\n\n2)\"It's okay lifting me up. It's not a sign of being too strong.\"\n\n3)\"I don't suspect you're hiding your physical strength from me.\"\n\n4)\"I've always lifted you up. Now all I'm asking is the same.\"\n\n5)Hoist her up. Then say, \"See, it wasn't so hard. Now it's your turn.\"\n\n#### _Why He's Tongue-Tied_\n\nAs I explained in the Introduction, language is housed in the left hemisphere of the brain, and emotions are in the right. Both male and female brains are divided right smack down the middle by a hedge of twenty million nerve fibers called the _corpus callosum_. But that division has a vastly different neuron density for you and your Quarry.\n\nA Hunter's barrier between language and emotions is the Wall of China compared to a Huntress's picket fence. While speaking from his language hemisphere, he has a heck of a time tapping into his emotions on the other side of the great wall. A Huntress hops back and forth over the barrier like a jump rope.\n\nSo don't ask your PLP to expound on how he feels. It's not that he doesn't want to talk about his feelings. He can't and, because the average guy doesn't like to admit he's incapable of anything, he gets annoyed. So sisters, if you decide to stay heterosexual, don't expect much discussion of feelings with your partner. If, however, getting your man's feelings on something is of life-and-death importance to you, here's a tip on how to do it.\n\n  **Chemistry Sparker #58**\n\n**Give Your Questions a Corporate Spin**\n\nChoose a time when he's not expected to do any deep gazing into your eyes. Then substitute more corporate phrases like, \"What is your take on that?\" \"Let me run a few questions past you,\" or \"What's your appraisal of the situation?\" (Never ever use the word _feel._ ) And, heaven forbid, don't ever, ever preface it with \"We need to talk.\"\n\nHunters, it's the opposite for a woman. Asking her feelings is a mammoth togetherness booster.\n\n  **Chemistry Sparker #59**\n\n**Ask Her \"How Do You Feel About That?\"**\n\nHunters, the next time you and your Quarry are chatting about anything, anytime, anywhere, that anyone could possibly have any emotions about\u2014good or bad\u2014gaze into her eyes and soulfully ask, \"How do you _feel_ about that?\" After you pick her up off the floor and revive her with smelling salts, she'll swoon with feelings of closeness.\n\n#### _Learn a Very Different Body Language_\n\nGentlemen, as tough as it is to express your feelings, it's tougher for you to recognize hers. If you put observation of your Love Partner's teary eyes in one box, her quivering lips in another, and her hanging head in another, how could you possibly be expected to connect the dots and determine she's upset? Besides, you've been brought up to believe that letting your emotions show is a sign of weakness. So you're probably assuming, \"I'm sure she wouldn't want me to notice.\"\n\nI'd like to help you change your thinking. Consider the usefulness of reading emotions. I'll put it in guy terms. There you are at the poker table. The lights are low and the smoke is thick. You stare over the mountain of chips at the guy across from you. It's just you and him now. You've got a sweet straight in your hand, but the dude just keeps raising the stakes. His expression is placid and his eyes are drilling a hole into your face. What's going on? Is he bluffing?\n\nYou take a chance. You call and lay down your cards.\n\nYES! The pot is yours.\n\nHow did you outsmart your opponent? You had developed the talent to read micro-expressions. It wasn't obvious. Your adversary hadn't sucked in his breath or glanced away even for a flash. But when he raised his eyebrows and pulled them together for a fraction of a second, you sensed it was sheer fear. You smelled it as clear as cow dung.\n\nThe biggest winners in business and sports are expert readers of emotions. A world-class chess player senses how firmly his opponent holds his piece and how quickly he places it. Your enemy's face and body telegraph his strategy, training, determination\u2014but only if you know how to read him. So get out there, dude, and master the skill of reading micro-expressions. You just need a little practice. How?\n\nGrit your teeth and hang in there with me on this next suggestion. Step one: Download a daytime soap opera and watch it with the sound off. You'll see the whole gamut of emotions\u2014anger, acceptance, anticipation, aversion, sadness, surprise, and hundreds in between. See how many you can name in the silent show.\n\nStep two: Now turn the sound on and suffer through it again. On average, a Huntress would recognize eighteen out of twenty emotions correctly. A guy's doing well if he can identify five.\n\nNow watch it yet again with the sound off and practice, practice, practice. You'll get the hang of reading emotions soon.\n\nFormer Army psychologist Paul Ekman, one of _Time_ magazine's world's one hundred most influential people in 2009, trains police interrogators and authorities in the art and science of face reading. Now he's developed a program to detect whether a couple is headed for the divorce court from just fifteen seconds of watching their interaction. Their flashes of indifference to each other's subtle feelings play a big role. To keep the Sparks flying, try to adjust your antennae to pick up her subtle expressions.\n\n  **Chemistry Sparker #60**\n\n**Detect Her Micro-Expressions**\n\nHunters, being sensitive to your Quarry's feelings is a biggie in making your Chemistry last. Practice the talent, and when you detect she's upset, ask her what's wrong. Then _listen_ , nod. _Listen_ , sympathize. _Listen_ , comfort her. But do not offer solutions\u2014until she asks.\n\nGuys, there's another reason to acknowledge her emotions: If you don't, she'll get angry and start arguing with you. And due to her brain's more direct connection between emotions and language, you'll probably lose!\n\n### Shall We Talk It Out, or Slug It Out?\n\nI'll never forget a totally confounding (at the time) experience I had in Seattle. A client who had booked me for a seminar picked me up at the airport. On the way to the hotel, Bernard briefed me about the group, the room setup, the AV arrangements, and he gave me my handouts he'd printed. Suddenly a Honda in front of us came to a screeching halt, just missing what could have been a major fender bender. I turned toward Bernard to tell him how lucky it was that he saw the car and stopped, but he was already shaking a furious fist at the other car. The other driver bellowed, \"I had to stop, you idiot, or I would have hit the car in front of me.\"\n\n\"You could have gotten us killed you sonnuva-bitch,\" Bernard bellowed. That was met with a hand darting out of the Honda's window, middle finger raised in the air\u2014the inner-city sign of war. Figuring the finger wasn't Hulk Hogan's or Mike Tyson's, Bernard leapt out of the car. All I saw was a flurry of flushed faces and flailing arms. Finally he stormed back, but not before banging his fist on the finger giver's car. Why did he react so physically? Because a male's limbic system is more wired to the physical than the linguistic.\n\n\"Asshole!\" he bellowed. I thought, but of course didn't say, \"Asshole\" had a point. He did have to stop suddenly to avoid hitting the next car. Grumbling, Bernard got back into the car, where I was pretending to go over the handouts.\n\nI lost something in his car that day: my respect for Bernard. If he had been my boyfriend, brother, or husband, I would have been furious and spent the rest of the ride trying to explain to him why the other guy was right. A.H. did indeed need to stop suddenly. If I'd known then what I know now, however, I would have considered Bernard's blowup just a rude male burp. His emotions, fueled by testosterone, shot straight to his biceps, bypassing any rational thought.\n\nThe rest of the time in Seattle, Bernard was an ideal client. Unfortunately, due to his irrational blowup, I was unable to give him the respect he deserved and the gratitude I wish I'd expressed.\n\nHuntresses, your man didn't exactly come out of the womb baring his toothlessness or clenching his tiny fists. But the anger and aggression circuits were already formed by the time he let out his first yelp. When the doctor held him upside-down by his tiny pink feet and spanked his bottom, the newborn dude probably wanted to kill him. While kindergarten guys were slugging it out in the playground over a dispute, we were talking out our differences in the sandbox.\n\nAnd of course, evolution gets into the act. If, instead of facing the small-fingered Honda driver, a Cro-Magnon Bernard found himself staring into the eyes of a livid lioness, what was he supposed to do? Say \"Down puddy cat,\" or \"Let's talk about this\"? No, he pulls out his homemade ax or hurls his hunga munga. The womenfolk back in the cave are delighted. And he is delighted when they don't scold him for his angry reaction.\n\nThere's more. Don't forget his childhood, the acculturated \"show 'em how tough you are, kid\" influence. Let's put this into Cognitive Science language: \"Limbic (emotional) activation in the female brain is linked to verbal response areas,\" and \"Limbic activation in the male brain is linked to motor\/physical response areas.\" In a sense, his anger chemicals go straight to his fists, hers to her tongue. When he's furious, his amygdala says \"Pull the trigger.\" Hers says, \"Tell one of your girlfriends.\"\n\n  **Chemistry Sparker #61**\n\n**Don't Talk When He's Fuming**\n\nHuntresses, between his limbic system being wired to the physical rather than the linguistic, plus evolution, plus his upbringing, plus ten times more testosterone, what do you expect?\n\nIgnore and forgive your Quarry's outbursts. It's anatomy, evolution, and upbringing all rolled up into one \"anger ball,\" and it will pass.\n\nHunters, your advice is just the opposite. Ask her to tell you all about the problem, every minute detail. If you have the misfortune of being the one she's pissed at, just listen quietly and then repeat one or more of the following phrases:\n\n\"I'm sorry I [fill in what she's accusing of you of].\"\n\n\"You're absolutely right. I had no right to [fill in what she's accusing of you of].\"\n\n\"I can't believe I was so thoughtless to [fill in what she's accusing of you of].\"\n\n\"Can you ever forgive me for [fill in what she's accusing of you of]?\"\n\n\"I promise [fill in what she's accusing of you of] will never happen again.\"\n\nThen, as soon as she's calmed down, hug her and tell her you love her.\n\nSee how easy that was?\n\n  **Chemistry Sparker #62**\n\n**Grill Her for the Details**\n\nWhen your Quarry is angry, Hunters, don't turn away. Soften your body language, look her directly in the eyes, and ask all the ins and outs of what's bothering her. Her emotions are more tied to the linguistic, so unlike you, she'll love talking it out. Nod often and toss in an occasional \"I understand.\" Then use a selection of the above phrases liberally and finish her off with a hug. Don't fight fire with fire. Snuff it out with the bonding chemical, oxytocin.\n\nCongratulations to both of you. The two of you have done everything right so far. Huntresses, you Sparked your Quarry with sexual bait and reeled him in with your fine qualities. Hunters, you Sparked hers by showing that you're husband material right up front. You both collaborated with Mother Nature to enhance the passion. You used neuroanatomical and chemical insights into your Quarry's box- or ball of yarn\u2013type brain to navigate the Love Boat away from the jagged rocks. You're ecstatically happy and tell everyone you are \"in love.\" But now let's learn how to Spark the Chemistry to make it the real thing, the kind of true love that makes your Quarry crave lasting togetherness with you.\n\n|   \n---|---\n\n## CHAPTER | _9_\n\n## How to Spark Chemistry for Falling in Love\n\nYou're in love . . . life is superb . . . your heart explodes with joy and everybody notices. You're always laughing and smiling and sometimes you burst into song. The world is brighter and more beautiful than ever, even on rainy days. Things that used to bother you, like traffic, annoying colleagues, and toilet tissue rolling from underneath instead of over the top, just make you smile. Nothing matters except being with your loved one. This exhilarating and electrifying stage of love consumes you. Your _Pleasure Island_ is deluged by dopamine, and it is sheer ecstasy. Enjoy every euphoric moment of it!\n\nFortunately, you now realize that love, in the song and sonnet sense, is a chemical assault on your brain, albeit a fabulous one. This knowledge can help you make the best choice of your life\u2014or save yourself from making a grave mistake. Before taking the big step, I suggest you run a few final tests on your Potential Love Partner. Squint and search beyond the glittering stars in your eyes to ensure that your love will last and last and last. And prove to your Potential Love Partner that you have the lasting qualities too.\n\nClear vision is tough when dopamine is hacking the texts between your amygdala and your prefrontal cortex. It's not that the _Professor_ hasn't been trying to warn you if he happens to feel this particular partner isn't right for you. He's been frantically tweeting all the other brain regions that trouble could be lurking behind the altar. But Mother Nature shushes him by shooting you up with more dopamine to make you even crazier for your Quarry. But all this drama is exciting, like a frightening film, and you can mistake it for true love. Here's how to tell if it really is\u2014and to show your PLP that you have the character and the qualities that make you \"the one.\" To do this, employ what I call the \"Big Four.\" Each is based on the latest studies of what both sexes seek in a permanent partner.\n\n### Is This Quarry a Keeper?\n\nConcentrate on clearing the paths between the emotional and rational parts of your brain. I'm not just speaking figuratively. You now understand how the love druggies ambush the messages so you're literally \"crazy in love.\" If you still love, respect, and feel wonderful about him\u2014and yourself\u2014after about a year and a half, don't let this one get away. You've found a keeper! It's only when you feel \"sanely\" in love with her and not \"crazy\" in love that can you take back the controls of the Love Boat.\n\n#### _Why Is He Afraid to Commit?_\n\nHuntresses, how many times have you heard that? A hundred? Two hundred? But you may not have heard this answer. No offense insinuated, but it's probably because when a smart guy gets serious, he gets smarter\u2014 _almost_ as smart as you've been all along! I find it interesting that men fall in love more often than women. But then they're more apt to balk. Why? Because if the relationship gets serious, men too become sensitive, consciously or subconsciously, to substantial qualities in their Potential Love Partner. They just take longer to come around to it.\n\nDuring the electrifying initial dating stages he doesn't analyze stuff to death. A guy-type brain doesn't consciously conjecture, \"What is her relationship with her mother and what does it signify?\" He's not going to sit down with a beer and a buddy and ask, \"Hey, dude, how much do you think she's going to help me realize my aspirations?\" or \"Do you think she's still going to enjoy cave spelunking with me a year from now?\" However, when a male finds himself contemplating the long term, he gets a hazy sense of these things. That's why sometimes everything can be going great in your relationship. The two of you seem to be running full-speed ahead toward total togetherness, even the altar. Then, just before you get there, _screech!_ He slams on the brakes.\n\nMany women would say, \"He got cold feet,\" or \"He's commitment-phobic.\" It's more likely that it finally got through to his dopamine\/testosterone-marinated skull that he and his girlfriend were not a fit in some of the more important matters in life. To demonstrate that you are the one for him, you must find subtle ways to hint to your Quarry that you are a match for the \"Big Four.\"\n\n1)Do you and your Potential Love Partner have similar deep beliefs and values in life?\n\n2)Do your definitions of \"togetherness\" match?\n\n3)Can your Potential Love Partner be depended on if adversity strikes?\n\n4)Will you each encourage your partner's personal and professional growth?\n\nWhile manifesting (not misrepresenting!) certain attributes, also keep a checklist on your PLP. Without certain components and similarities between you, long-lasting Chemistry is almost impossible. Your differences will chomp away at your love, bite by bite, until your plate is empty\u2014and you're both starving for love with someone else.\n\nLet's talk about the Big Four in order.\n\n### I See Life That Way Too!\n\nWhich statement expresses what you want in a life partner?\n\nA. _I want someone different from me_. Someone exciting who has a fascinating lifestyle, exposes me to new ideas, gives me fresh insights, shares adventures, and helps me view the world in a different way.\n\nB. _I want someone similar to me_. Someone who is in accord with my deep beliefs and principles, makes me feel secure, enjoys similar activities, likes my lifestyle, and understands how I look at life.\n\nOf course, your answer is \"both.\" Everyone wants someone similar in some of life's most important ways and different in the fun frills. We crave excitement, ecstasy, thrills, and chills. We also long for serenity, commitment, contentment, and comfort.\n\nThink back to the days when you were in school. Remember how there were various cliques? The preppies? The tough gang? The brains? The athletes? The self-proclaimed cool kids? Chances are that there weren't too many cross-friendships because people feel comfortable with their own.\n\nWhile moving into a college dormitory, college students were interviewed about their beliefs, values, ethics, mores, and morals. Researchers hypothesized that, as the students got to know each other, the ones who looked at life in the same way would gravitate toward each other. Sure enough, they hardly missed a beat. Cliques of similar kids formed.\n\nYour prefrontal cortex, the wise brainy _Professor_ , knows that if a relationship is going to work, you need a mate who looks at life in the same way as you. Let's face it\u2014life is confusing and scary. TV, magazines, newspapers, millions of blogs, and social media\u2014and all other methods of communication now known or currently being developed\u2014can make your head spin. When you find another human being who has come to the same conclusions about life, you feel protected and out of harm's way. Your opinions, morals, beliefs, and values are vindicated by your Love Partner's agreement. This floods your brain with oxytocin, the attachment chemical.\n\nObviously your agreement is not necessary or even desired on everything. That would be boring. What matters is not the number of agreements on small things but rather harmony on the more weighty matters in life that causes \"coupling,\" as researchers call it.\n\nAgreement on certain subtleties in life is often more important to females. Hunters, if you've pretended to share her dedication to animal rights, care for the elderly, and protecting the environment, how do you think she'll feel when you kick her cat, ignore your grandmother, or refuse to recycle?\n\nMen don't need agreement as much as they need respect that their choices are good. I'll never forget an episode in the then-popular TV show, _House_. The characters were musing about how much they valued their parents' love. Dr. Gregory House groused, \"All I wanted was for my father to say I was _right._ \" It confused me at the time, but now, after gaining a better understanding of the neuropsychology of the male brain, it makes perfect sense.\n\nHuntresses, if he's a vegetarian or macrobiotic, it doesn't mean you have to slurp seawood soup with him every night. Just show your support for his preference. Tell him he's smart to make such a disciplined and wise choice.\n\n  **Chemistry Sparker #63**\n\n**Show You Share or Respect Your Quarry's Values**\n\nTo stoke the fire in your relationship, find ways to highlight your similarities whenever you can. Emphasize your admiration for your PLP's beliefs.\n\nHunters, when you feel the same way about things she deeply cares about, oxytocin floods her brain. Huntresses, confirming his commitments and supporting his principles has the same effect on him.\n\nBut express the similarity only if it's true! Otherwise you're steering the Love Boat in a deadly direction, and your relationship goes aground. If it is going to stay afloat beyond the crazy dopamine-drenched days of early love, similar principles and convictions are must haves.\n\nBut that's not the whole picture. Three other major pieces must fit in the successful relationship puzzle.\n\n### How Do You Define \"Togetherness\"?\n\nHow close do you want to be to your partner? Together every day, every night? Inseparable? Having the same friends and interests as he does? Doing the same activities with her? Or is \"togetherness\" cohabitation, companionship? Living under the same roof, in the same bed, but coming and going as you please? You spending time with your friends and he with his? You going wherever you like, pursuing your passions, and she hers? In short, how would you define \"togetherness?\"\n\nTogetherness can mean one or the other of the above\u2014or something in between. The definition must be the same for both of you, or else the sticky wicket can shatter the bonding Chemistry. Finding out your PLP's views can be a tricky task, especially for women. Asking outright could make him feel like he's being water-boarded, so listen between the lines. What does he say about his parent's relationship? His friends? Does he joke about a buddy \"getting hooked,\" \"biting the dust,\" \"putting on the ball and chain\"? Or does he use words more like his buddy \"is hearing the ol' wedding bells,\" \"decided to settle down,\" or \"found the right one\"?\n\nGuys, what about her? Does she ever talk about any of her past boyfriends \"suffocating her,\" \"being too possessive,\" or \"never letting her out of his sight\"? Conversely, perhaps she complains about a previous boyfriend \"not being with her enough\" or that he was \"always running around with his buddies.\" Does she bemoan the fact that her father was never home?\n\nWhat about family feelings? How often does he talk to his mom? His dad? Are they close? Does she spend holidays with her family? Does he speak with his siblings, aunts, uncles, cousins, and other various and sundry relatives? If he hasn't been a bonder in his previous life, do you think he'll have a frontal lobotomy and start loving family life now? Psych it out and compare it to your togetherness quotient.\n\nResearchers call how you \"score\" on the togetherness chart your \"comparison level,\" and it plays a big part in keeping the love Chemistry between you. \"I thought he would change\" are some of the most foolish words I've ever heard.\n\nMy cousin Rory and his wife, Camilla, whom I knew way before they got married, have two beautiful young daughters, but they're going through a tough time this year. Rory has always had vast interests and continually finds new hobbies. He had a meditation phase, a bird-watching phase, and a bowling phase. Now he's into calligraphy and going to comic book conventions.\n\nA few months ago Camilla and I curled up on the couch with our coffee cups and had a poignant talk. She was lamenting that Rory was always off with his friends. I squeezed her hand and reminded her, \"Rory has a lot of diverse interests. Before you were married, I remember you telling me how fascinating he was because of it.\"\n\n\"I know, Leil, but I thought that when the girls came along, he would settle down.\"\n\n\"Camilla, what do you most enjoy doing?\" I asked her.\n\n\"Just being with my girls. I mean that's really my favorite thing to do. It's just that . . . well, I feel Rory should be here more. The family should be doing things together.\"\n\nHere's the tragedy. This very weekend Rory and Camilla are planning to tell their girls they are splitting. My eyes water as I write this. If only Rory and Camilla had discovered they had totally different definitions of \"togetherness.\" For Camilla, it is constant closeness. For Rory, it is loving cohabitation. Who will suffer the most from their not scoping it out before? Their girls, of course.\n\nThe so-called logical advice here would be to \"sit down now and talk about how you both define togetherness.\" Sounds good, but unfortunately that's not always realistic. Most Hunters are biologically challenged in illusive relationship discussions like that. It's like asking a mermaid to do the splits. And a Huntress's crystal ball is too chemically clouded during early passionate love to think straight.\n\nThe solution? Make an appointment for your wise _Professor_ and emotional amygdala to have a little non-dopamine-hacked neurotransmitting about what a relationship means to you.\n\n  **Chemistry Sparker #64**\n\n**Emphasize Your Similar Concept of Togetherness**\n\nMake it a point to define clearly and consciously what \"togetherness\" means to you. Then, to the best of your ability, determine what it means to your Potential Love Partner. If, hopefully, you discover that you and your PLP's are the same, find ways to underscore it in casual conversation.\n\nAnd remember (especially Huntresses), if you're thinking, \"I can change him,\" forget about it. If you succeed, call the _Guinness Book of World Records_.\n\n### \"In Sickness and in Health\"\n\nTo me, those words are the most moving most part of a wedding ceremony because I've seen how illness and adversity affects couples. After deep love has had time to take root, tragedy can draw a couple even closer together. Whether they have taken the actual marriage vows or not, taking care of a long-term Love Partner can intensify love and make it more enduring.\n\nInfirmity and adversity have the power to overshadow other problems that start to tear a couple apart. In many cases, when there are catastrophic challenges, major arguments vanish and money problems pale. \"You left the cap off the toothpaste\" and \"Take out the garbage\" are no longer declarations of war. In fact, those minor foibles can invoke smiles reminiscent of early posthoneymoon discoveries.\n\nI've seen it happen twice, once due to severe injury in a small plane crash and the other Parkinson's disease. Both my friends' marriages were tottering, but when tragedy struck, the other partner went instantly from being a quarreler to a loving caregiver. Life was harder, but their love became stronger.\n\nA few months into writing this book I learned firsthand the bonding effect of illness. I was glued to the keyboard morning to night and had put off a lot of \"should do\" things like having a mammogram. Giorgio bugged me incessantly about it. So despite my grumbling, he insisted and dragged me off to the hospital.\n\nA week later the hospital \"invited\" me back for a repeat visit. When I heard my gynecologist's sympathetic \"Hello\" on the phone a few days later, I realized there'd be a slight change of schedule during the next few months.\n\nChemo for breast cancer sucked, but I discovered a new closeness with Giorgio. He postponed taking command of the ship he was booked for in Italy, becoming my hospital chauffeur and staying with me during every depressing chemo session. Giorgio spent the months puffing up my pillows, rubbing my nauseous tummy, kissing my yucky red face, caressing my bald head, and lying to me, saying I looked beautiful.\n\nNow I'm completely okay. No, I'm better than okay because it showed me how vital togetherness is. Whether official vows have been exchanged or not, unspoken ones can be just as powerful. After that experience I have a deeper understanding of the distinction between \"being in love\" and loving. I'm sure the many millions of couples who have had experiences like mine understand what I'm talking about.\n\nSo here's a question you should ask yourself when considering exchanging togetherness promises: \"If tragedy struck, would my Potential Love Partner be able to take on the role of caregiver?\" When you're flat on your back, it's no fun lying there alone. And equally important, \"Would I do the same?\"\n\nI pray you will never need it, but think about it now because this is the person you plan to spend the rest of your life with in whatever shape either of you is in. Find ways to let your partner know you'll be there, for better or for worse. Here's how.\n\n  **Chemistry Sparker #65**\n\n**Show Your Quarry That You're a Caregiver**\n\nBe extra-attentive and loving when your Quarry isn't feeling well. Giving her flowers when she's sick is as meaningful as on her birthday, maybe more so. Taking him chicken soup when he's flat on his back gives him nurse fantasies of the nonsexual kind\u2014and those last a lot longer.\n\nAnd of course, check out your Quarry for care-giving qualities. It's bad enough to be sick. But to have no one to hold your hand when you're sick _really_ sucks.\n\n### The Magnificent Twenty-First-Century Mantra\u2014Personal Growth\n\nI don't need to tell you the twenty-first-century world is practically a different planet. We are inspired by different ideas, connected by diverse technologies, admired for distinct accomplishments, and desired for divergent qualities\u2014some old, some new. How lovely that we live in a land where free expression and diversity are encouraged and where poverty and plagues don't prohibit it as they often did in the past. In many parts of the world personal growth would be considered a luxury just for the rich. For those of us in more fortunate nations it is ubiquitous and adds a thrilling dimension to our lives.\n\nMarriage is no longer the economic and social institution it used to be. The expectations and psychological needs of the partners bear little resemblance to even a hundred years ago. Nora in Ibsen's late-nineteenth-century play _A Doll's House_ would be ecstatic knowing that, within a scant fifty years, a group of women calling themselves feminists would lay the cornerstone for the temple of personal growth at which both evolved men and women worship today. Sociologists call it \"self-expansion.\" The more a mate encourages your goals, the happier and more committed your togetherness will be.\n\nIn days gone by, happily in the past, it was primarily the woman who was expected to serve her partner. A \"dutiful wife\" brought hubby his slippers, didn't complain if he worked late, and entertained his colleagues whenever requested. Now it's a two-way street, and women who work just as hard expect and deserve the same deference. Both Hunters and Huntresses have their antennae out for a Love Partner who will be there for them both personally and professionally.\n\n  **Chemistry Sparker #66**\n\n**Show Signs That You Can Help Your Quarry Grow**\n\nToday you have the luxury of adding personal growth to your wish list and expecting it. Hunters, whenever the subject of her personal and professional goals comes up, tell her how supportive you are. Huntresses, when he speaks of his aspirations, share his enthusiasm and enhance his confidence that he can achieve them.\n\nBefore making the Big Decision, however, make sure it works the other way too. Chemistry can die quickly if your own creativity and growth are stifled.\n\n### \"Vet\" Your Quarry\n\nAs my regular readers know, I have a huge loft in New York City that, without rent control and a roommate, I could never afford. When I first found it, I was in the habit of having short-term roommates, students or professionals on short assignments in the city, for which I advertised in a then-popular weekly paper.\n\nOne of my ads came out on Thursday. On Friday, Keith, a thirty-six-year-old cellist with the Boston Symphony who was recently separated from his wife, called. He came over on Saturday and moved in on Sunday.\n\nKeith was a fabulous roommate, not like the previous one who left the toilet seat up with a vengeance. The only annoyance was the ghastly ringtone on his cell. Several times an evening, Manowar's _Dark Avenger_ blasted out on his phone. He'd answer it with an anguished expression and disappear into his room.\n\nI once joked about his ringtone. He shook his head, \"I hate it too.\"\n\n\"Well, um, Keith why do you have it?\"\n\n\"My wife\u2014soon to be EX-wife\u2014insisted.\"\n\n\"Oh, she was into heavy metal?\" I asked.\n\n\"No!\" he blurted out. \"She always liked classical. Then one day, right out of the blue, she starts listening to Slayer and Spinal Tap and turns the volume up as high as it will go.\"\n\n\"You mean it was all of a sudden?\" I queried.\n\n\"Well, not really; bizarre things started happening last year.\"\n\n\"Keith, please don't answer if I'm invading your privacy, but may I ask what?\"\n\nHe stared at me for a moment, took a deep breath, and said, \"Okay, here goes. The first thing I noticed was she'd start talking about something, and I couldn't follow her train of thought. At first I thought it was just me until one time at a dinner party, for no reason, she starts talking about Obi-Wan Kenobi, the _Star Wars_ character. The party's host asked who Obi-Wan Kenobi was. My wife breaks out laughing hysterically, and won't stop. Everybody looked at each other, wondering what the heck was going on.\n\n\"After that, there was all kinds of other weird stuff.\"\n\n\"Like?\"\n\nHe put his elbows on the table and his head in his hands. \"Jeez, Leil, the whole gambit. First she'd garble about some new age mumbo jumbo. Then she'd babble about a fatal accident if we didn't believe in Biblical miracles. I mean, she took them literally.\" When he raised his head, I saw his eyes watering. Embarrassed, he stood up. \"I dunno. Maybe it runs in the family. Her mom was a bit of a kook too.\" Just then Manowar's \"Dark Avenger\" blasted out on his phone. He slammed on the silencer and disappeared into his room.\n\nKeith's story rang a bell. A few friends in the past had told me that they'd known someone who had \"suddenly gone sort of nuts.\" They often followed it by saying they had heard the person's mother or father was \"very eccentric\" too. Other times I'd hear, \"She was abused as a child,\" or \"He was raised by an alcoholic father.\"\n\nLovers seldom realize that their potential partner's childhood can have a direct impact on their future relationships and desire for a long-term one. During casual discussions with your PLP try to steer the subject around to his childhood. Did his parents love him? Was there any brutality in his family? Did his mother suffer severe stress while he was still in her womb? Even that could have a detrimental effect on his later psychological development. And yes, it's chemical. Growing up in a loving environment increases oxytocin in his body, meaning he'll be a better bonder.\n\nHunters, what about her? Did she have a stable upbringing? Did any of her close blood relatives suffer from mental illness? Was she a troubled child? As callous as it sounds, if you intend to raise a family with this lady, it's worth determining if there was any potentially transmittable mental illness or childhood challenge that could surface in adulthood. It's not likely but definitely worth keeping your eyes open for when you start thinking about being together for the rest of your life.\n\nAs we discussed, often an affliction can create closeness between loved ones. But tragically, there are some behavioral conditions due to childhood or inherited genetic abnormalities that don't. There's a very good chance it won't ever manifest itself, so don't freak out about this. But it pays to be extra vigilant when considering someone you'll be with for the rest of your life. Don't dismiss clues like Keith did.\n\nThis sounds like pretty depressing stuff, doesn't it? So I'm going to just drop it here and beg you to read a document you'll find on the web. Before saying \"I do\" or giving up your rent-controlled apartment, run a search for the \"Surgeon General's Report on the Risk Factors of Certain Mental Conditions, Chapter Three, Section Two.\" Don't skip the part on \"Family and Genetic Factors.\" The percentage of inheritable and upbringing-influenced conditions is staggering.\n\n  **Chemistry Sparker #67**\n\n**Probe Your Chosen PLP's Childhood (and Genes!)**\n\nThis is what the dating process is all about\u2014figuring whether your future togetherness will be happy or horrible. While having fun together and fantasizing about a beautiful life, don't forget to climb out on the branches of your PLP's family tree to look for nuts. (Sorry, bad pun.) Scamper through your loved one's childhood to sniff out possible future challenges. Also, avoid doing anything irrational that your PLP might mistake as having deeper roots.\n\nLook at it this way: You wouldn't buy an expensive purebred dog without seeing its papers and assuring yourself that it has no inheritable diseases. And you're going to live with your loved one a lot longer than your dog!\n\n### May I Have a Word with Young Lovers?\n\n\"Life without him would be a life without meaning.\"\n\n\"I'll die if she ever leaves me.\"\n\n\"I could never fall in love with anyone else.\"\n\n\"It is fate we should be together forever.\"\n\nHave you ever felt this way when you were very young? We all did once or twice. Some thrice. At age seventeen I thought I'd found the love of my life, the boy I just _knew_ was the one. He was older and what some would call a tough guy. But he could be oh-so tender! I was ecstatic when he started talking about a future with me. \"Our forever together time,\" he called it.\n\nButch and I had been dating for about three months, but my mother hadn't met him yet. Instead of coming to the door to pick me up, he would park in front of my house and honk the horn. It was to the tune of \"shave-and-a-haircut, two bits,\" the then-sweetest sound in the world to me. Upon hearing it I'd bounce out the door and breathlessly race to his car for a few hours of blissful togetherness and discussing \"our forever together time.\"\n\nOne day my mother asked me, \"Leilie, do you have a date with Butch tonight?\"\n\n\"Yes,\" I crooned.\n\n\"He sounds so nice from what you've told me about him, dear. Why don't you invite him in for a few minutes? I'd love to meet him.\"\n\n\"But Mama . . .\"\n\n\"Just for a few minutes, dear. I promise.\"\n\nAt the anxiously awaited shave-and-a-haircut-two-bits, I ran out and begged him to come in to say \"hi\" to her.\n\nAfter a short, noticeably uncomfortable conversation, my mother asked, \"Leilie, can you come into my room for a second?\" She closed her door and literally lay down across the doorway. She pointed a furious finger up at me and gave notice: \"You will go out with that boy over my dead body!\" She stormed back to the living room, locking her door behind her with me inside. I never saw Butch again.\n\nThen, just several months ago, I got a letter written in a typeface I call \"nut font\" from, I kid you not, a return address of the Maryland Department of Corrections. Apparently, Butch's brother had found me on the web for him. Looking back, I shudder to think of what a nightmare my life would have been if we'd gone through with our \"forever together time.\"\n\nSo, Dear Reader, if you haven't reached the big two-oh, don't even think about the \"M\" word! It's practically impossible to make the right choice at your age. It's not because you're not smart. You could be the brightest crayon in the Crayola factory, but the biological structure of your brain is not prepared to make one of the most important decisions of your life. You don't have to believe Mom or Dad\u2014let neuroscience tell you why.\n\n#### _Don't Marry Until You're Myelinated (Parents, Don't Miss This!_ )\n\nUntil I'm what? No, it wasn't a typo. As you now know, the neurons in your brain are constantly communicating by electrical impulses. The tubular-shaped insulation I mentioned earlier, called a myelin sheath, protects them from hit-and-miss thinking. This sheath, a greased tube of sorts around the axons of your neurons, only develops gradually. When you were born, the myelin in your system was hardly detectable. That's why an as-yet myelinated baby sleeps in a crib. She might sense that falling out of bed would be a big \"ouch,\" but the message emanating from her little prefrontal cortex is too slow. Her tiny little _Professor_ doesn't have time to get the message to her central nervous system to stop her from rolling toward the cliff. In your early twenties, when your neurons become more fully myelinated, those greased tubes transmit clearer messages to the rest of your brain faster. (Sorry, young lovers, the full myelination process isn't complete until about age twenty-five.)\n\nI know it feels so intense. Your world revolves around your romance. But if it saves even one of you from ruining your life, I'm going to break my own commitment to not quote tedious excerpts from studies. This is from \"Defining the Brain Systems of Lust, Romantic Attraction, and Attachment,\" published in the _Archives of Sexual Behavior_ by five of the leading researchers in the field.\n\n_We were able to demonstrate that adolescents in early-stage intense romantic love did not differ from patients during a hippomanic stage . . . intense romantic love in teenagers is a \"psycho pathologically prominent stage._ \"\n\nPathological means \"evidencing a mentally disturbed condition.\" Okay, throw this book out the window or delete it. But I speak the truth.\n\n#### _Shall We Make a Federal Case Out of It?_\n\nIn my grandmother's day sex before marriage was a sin. Even in my mother's day, living together before marriage was a couple's guilty secret. In today's rapidly evolved society, however, I think premarital cohabitation should be a must, even a constitutional amendment! Permanent vows should come only after you have passed the test of togetherness, at least a year and a half to be safe. And don't even think about bringing another human being into our already-overpopulated world before then!\n\nIncidentally, if I should mysteriously get struck by lightning before this book is published, you know it was Mother Nature taking careful aim.\n\n|   \n---|---\n\n## CHAPTER | _10_\n\n## How to Spark Chemistry for a Lifetime of Love\n\n_Love is the most beautiful of dreams and the worst of nightmares_.\n\n\u2014William Shakespeare\n\nAt age eighteen this playwright married a lady of twenty-six, fathered a baby six months later, indulged in out-of wedlock sex, left his wife three years later, went on to multiple affairs with younger females, and wrote 126 love poems to a male known only as \"Young Lord\" or \"Fair Youth.\" Obviously the bard knew a thing or two about the nightmares part!\n\nI far prefer this quotation written by a man who had once been a motorcycle messenger and a mechanic. The insightful author, Louis de Berni\u00e8res, acknowledged the madness of romance yet understood that only those who wait until the frenzy passes will discover the magic of true love.\n\n_Love is a temporary madness. It erupts like volcanoes and then subsides. And when it subsides you have to make a decision. You have to work out whether your roots have become so entwined together that it is inconceivable that you should ever part. Because this is what love is_.\n\n_Love is not breathlessness, it is not excitement, it is not the promulgation of promises of eternal passion. . . . That is just being \"in love\" which any fool can do. Love itself is what is left over when \"being in love\" has burned away, and this is both an art and a fortunate accident. Your mother and I had it, we had roots that grew towards each other underground, and when all the pretty blossoms had fallen from our branches we found that we were one tree and not two_.\n\nAt the beginning of your relationship, the delirium of discovery and fantasies of the future make the chemicals in your brain gush like a new oil well. You're singing in the rain, dancing between the stars, frolicking on cloud nine. The frenzy is so fantastic that reality doesn't get through to the brighter part of your brain up front. You and your beloved close your eyes to all the signs: \"Warning!\" \"Danger Ahead!\" \"Proceed with caution!\" Early love is definitely not a drug-free zone.\n\n### Making Beautiful Beginnings Last Forever\n\nHuntresses, after you're engaged, girlfriends flock around you, oohing and aahing over your ring. Caterers call to tempt you with delicious finger foods that your wedding guests will devour. Bakers entice you with pictures of multitiered white castles with a miniature of you and your lucky man on top. On the big day you will wear the most exquisite dress you'll ever own, and he will look so dashing in his tux.\n\nHunters, you may be less involved in the preparations for the big day, but you're just as excited. Your buddies tease and slap you on the back. But under their joking put-downs, you know you've got one up on them because you've found the one. And besides, the guys throw you one helluva good bachelor party.\n\nYour family and friends will give you ridiculously costly crystal, kitchen gadgets, and _tchotchkes_. Then you'll cast off on the real Love Boat, a cruise to the Caribbean to drink Mai Tais on the beach, dance 'til dawn, and indulge in lovemaking that would make rabbits seem celibate.\n\nBut the bridal industry has a dirty little secret that wedding magazines and vendors don't tell you. When you come home from the honeymoon, no one makes a fuss over you anymore. No bakers begging, no caterers calling, no friends congratulating. You become just normal folks. At that point you look at each other and think, \"Where have all the hours gone? Is it really over? Where's all the excitement?\"\n\nIs either of you disappointed in your mate or wish you hadn't taken the big step? Of course not! You still love each other as much as ever and know in your heart you made the right choice. So what's happening? Why is it no longer as exhilarating?\n\nJournalists write about a phenomenon called \"postwedding blues.\" Sociologists say \"postnuptial depression.\" But only Cognitive Science can tell you precisely why you feel this way. Up until the honeymoon cruise ship sailed home, dopamine was gushing like a volcano. Your systems were swimming with serotonin, and both your pleasure centers were as bright as flashing cameras taking beautiful pictures of what was to come.\n\n\"Okay, no big deal,\" you shrug. Life is good, your mate is great, and your love is strong. The small discoveries about your loved one give you a chuckle. The way he scrunches the toothpaste and leaves the cap off is so cute. You smile at the way she \"gently hints\" you should take out the garbage. Dopamine and serotonin start dancing again.\n\nHowever, this time it's more of a waltz than a samba. _Pleasure Island_ lights up again, but now it's more like a night-light than a spotlight. You settle into a routine. Up at seven, off to work at eight, back home by six, dinner by seven, a few hours in front of a screen, and to bed by eleven.\n\nSex is still good, but she's put away the beaded penis wrap and mirrors. She doesn't scream as loud, and he's down from an hour of foreplay to ten minutes. And then there's her occasional headache. You no longer crave his company every minute or thrill to her tiniest touch. His phone calls, which you once awaited with fervor, become a familiar ring. Face it, we get used to things. Even the best experiences, when repeated, become ho-hum.\n\n### The Two-Year Itch\n\nTime continues to march on. You've been together for two years now. You're still happy together and think the other is terrific. But where's the sizzle? Where's the spice?\n\nStudies say the fizzle-on-the-sizzle effect inevitably takes place between eighteen months to two years of constant togetherness. His touch no longer ignites Sparks. Her body no longer incites animal hunger. But here's the real tragedy: Many couples feel that love is cooling along with the passion.\n\nLast Fourth of July, I was part of the oohing and aahing crowd straining their necks staring up at the fabulous Macy's fireworks show. A family with three little tots stood next to me. Unfortunately, Mommy and Daddy had only one pair of shoulders each. I hoisted one of the little boys up as high as I could. To me, his joyful squealing was far more exciting than the artistic explosions overhead. After the last incredible blast that lit up the sky as bright as midday, I put him down. \"NO!\" he howled. \"It's not over. It can't be.\"\n\nAs his little arms stretched up for me to lift him again, I said, \"I'm afraid it is over. But wasn't it beautiful?\"\n\n\"No, it can't be over,\" he screamed again. \"There's more. There's got to be.\" When I kneeled down and shook my head, tears flooded his eyes in disbelief.\n\nSadly, that's the way many couples feel when the fireworks in a relationship are over. When the deliciously passionate drugs of phase one diminish or disappear, they face a dilemma. They look at each other through tears and ask themselves, \"Is love really over?\"\n\n\"No! No! No!\" I want to shout at these people who once passionately loved each other. Robert Browning grasped the truth when he wrote, \"The last of love for which the first was made. . . . The best is yet to come.\"\n\nAnd neuroscience agrees.\n\nYou may have been devastated when you discovered the Tooth Fairy and Santa Claus were myths. But because of what you're going to learn in the rest of this chapter, you don't need to fear that \"happily ever after\" will be next.\n\n### A Little Mouse in the Meadow Holds a Secret of Togetherness\n\nIf you've been buried in neuroscience night and day for the past few years as I have, you're probably sick to death of hearing about the prairie vole. I was too\u2014at first. Until I fell in love with the little creature. Everyone does, except farmers whose crops he chomps on. When you hear more about this cute critter's lifestyle, you'll love him too.\n\nMany people mistake this tiny animal that lives in the prairies and meadows for a mouse, but he's got stockier legs, a shorter tail, teensy eyes, and almost hidden ears. He is absolutely adorable and very admirable.\n\nThe loveliest thing about these little meadow mice is that when they fall in love and mate, they stay together forever. Upon meeting his future lifelong partner, the male has wild round-the-clock love-making with her for twenty-four hours. After that he's a goner. The couple sleeps cuddled together, grooms each other, and raises the kids together as loving parents.\n\nUnlike the nasty lab rat we discussed earlier, this faithful little meadow mouse won't even look at another female. The experiment proving this is amazing to watch. Researchers put the \"husband\" meadow mouse in a center cage sandwiched between two other cages. One adjoining cage contains his \"wife\" and on the opposite side \"the other woman.\" No matter how hot the other female may be\u2014for a meadow mouse, that is\u2014he won't even go sniff her out. He prefers the side of the cage closest to his \"wife.\"\n\nInfidelity is practically unheard of in his species. If his true love dies, it's a rare meadow mouse that will go on to \"marry\" again. The big question: Why is this faithful mouse so different from practically all other male mammals, birds, and fish in our world?\n\nYou may have guessed. It's Chemistry, of course. Just like humans falling in love and staying in love, his devotion comes from naturally produced substances. The first time the two furry little creatures have sex, a big dose of the bonding chemicals we've discussed, oxytocin and vasopressin, flood through their systems. In the prairie vole it sticks. And here's what's so amazing. Without these chemicals they would become sexually promiscuous rats. In fact, when lab scientists extracted the fidelity chemicals from the devoted little male, he started chasing tail just as much as other species. In addition, his true love's motherly instinct went right out the cage window.\n\nThere is another 99 percent genetically identical animal called the \"montane vole.\" (I'll give Cognitive Science students time to hiss.) He is a real bastard when it comes to family values. After having sex Pop runs off immediately to find a hotter new mouse. And it's not just Dad who drops the family. Mom also abandons her babies soon after birth. At least the kids don't have any deep psychological problems with it. They just shrug it off, grow up, and repeat the whole dysfunctional process. And to think that the only difference between the loving mice and the lousy mice is a few drops of oxytocin and vasopressin!\n\n\"Oh, no,\" you may be thinking. \"Are you trying to tell me all those beautiful heartfelt, lasting emotional bonds come down to chemicals?\" I suppose one could look at it that way (neuroscience does), but here's another. Like all emotions, sentiments of love, devotion, and commitment emanate from your brain. When your brain feels those sentiments, it manufactures chemicals\u2014chemicals that create and influence feelings. So it's the story of which came first, the chicken or the egg? Or in this case, the chemicals or the emotions? And does it really matter?\n\nThe good news is that there are methods to create the loving chemicals in your lasting Love Partner's brain. You don't need to drug her with pills, inhalers, or liquids. You don't need to slip secret substances into his beer. There are actual techniques, Chemistry Sparkers, to make the brain and body of your Love Partner naturally produce these exquisite chemicals.\n\nDo I sound like a hack hawking a hair-growing magic potion from a stagecoach bandstand?\n\nI would understand if you answered \"yes\" to my question. It does sound preposterous, doesn't it? But brain imaging doesn't lie.\n\n### How to Spark Long-Lasting Chemistry Anywhere, Anytime\n\nOne of the easiest methods of stimulating oxytocin is speaking the primal language\u2014touch. The human need for touch starts while enveloped in the close quarters of the womb for nine months. The fetus's more than five million skin sensory cells thrive on touch to prepare him for the large and intimidating planet he'll inhabit.\n\nThe need for human touch doesn't stop at birth. The moment you make your grand, squealing entrance into the world, you need it continually for healthy development. Something as simple as loving touch can make the difference between life and death for children in an orphanage, and its power remains throughout life until the very end. Nursing homes report its phenomenal effect on residents' health, happiness, and longevity. And we now know the difference that loving touching makes on the health, happiness, and longevity of a couple.\n\nAll kinds of touch puts oxytocin in people's tanks\u2014kissing, hugging, holding hands, an affectionate caress, even incidental toe touching the other's body while sleeping. Each touch, accidental or intended, fleeting or abiding, shoots the trust chemical into the touchee's brain and tweaks their memories for the better. Patients who were touched in casual conversation thought their doctor had stayed with them longer. Waitresses who touched got higher tips. Dentists who touched got more referrals. In one study, women entering a brain scanner were told they were going to be given a shock. But when their partners touched them, their fear circuits shut down.\n\nI experienced something similar. Last year, sliding into one of those scary tubes for an MRI, I was silently freaking out. My partner, Giorgio, who was allowed to stay in the room, sensed this. He softly caressed one of my bare feet sticking out of the machine, and I relaxed like a midday snooze. Almost.\n\n#### _Time Out, Sports Fans_\n\nIn case you think I'm getting too sentimental about touch, guys, you'll appreciate this. Teams who touch more, win more! Professors from the University of California, Berkeley filmed ninety NBA games. The basketball teams who touched each other the most won the most. The Boston Celtics and the Los Angeles Lakers were the touchiest. No names here, but teams who touched the least lost the most. Even casual guy touching like high fives, fist bumps, and back slaps boost buddyship.\n\nGentlemen, you may think you're doing all that nonsexual touching just for her benefit. Well, your chances of getting same-night nookie are better if you affectionately touch her during the day. Don't forget, her foreplay began hours, days, and weeks before. Light touch now can mean heavy sex later.\n\nConversely, Huntresses, you should go overboard with touch of the sexy kind. Sitting on the couch, initiate a snuggle, massage his knee, rub his chest, plant a kiss on his cheek. Use your imagination like you used to do in your Gummi bear days.\n\n#### _What If I'm Ticked and Don't Feel Like Touching Right Now?_\n\nDo it anyway! Even when your long-term Love Partner (LP) is grumpy, you can boost the bonding chemicals through something called \"cognitive consistency,\" a natural phenomenon that means your brain and body struggle to be in agreement. It's like when you're feeling lousy, you grumble. When you grumble, you feel lousy. Well, when you feel love, you touch. When you touch, you feel love. Imagine your brain and body holding this conversation.\n\n_Brain_ : Hey there, Body, whatcha doin'? I notice you're touching him a lot. What are you trying to tell me?\n\n_Body_ : Yeah, I noticed that too, Brain. I guess I'm telling you that that I love him.\n\n_Brain_ : Well, there certainly is a lot of physical evidence. I guess you're right. You really do love him.\n\nHunters, on the street, if she's miffed, take her hand. She may yank it away, but she'll warm up faster. Huntresses, if he's grouchy, slide closer to him on the couch. If he stiffens, ignore it. Your attempted snuggle has done its job.\n\n  **Chemistry Sparker #68**\n\n**Find Any Excuse to Touch Your LP**\n\nHunters, give her daily light kisses, nonsexual caresses, hugs, and cheek brushings. Huntresses, initiate hand holding, putting your head on his shoulder, and making loving moves. Light touching all night long is a veritable oxytocin factory. Let your foot rest against his leg. Leave your hand on her shoulder. And of course, the ultimate is \"spooning,\"' sleeping like little spoons in a drawer.\n\nYou've heard that people have to work to keep a relationship together. Snore. Somehow I just don't picture \"working\" at a relationship being much fun. Let's talk about playing together to stay together. That works wonders\u2014just like enjoying the same activities sparked more Chemistry in dating.\n\n### Couples Who Play Together Stay Together\n\nAt the beginning of your love the two of you have a blast. Maybe movies, museums, and the beach. Perhaps bicycling, boating, bowling, or beachcombing. You think this high-dopamine life is pretty cool. You want it forever, so you move in together or marry.\n\nBut guess what? Real life happens! You no longer anxiously await the next moments when you do fun things together. He's there all the time. You're no longer wooing her, so the great dates diminish. No more picnics or discos. No more fancy restaurants or renting row-boats. Life is not as exciting, therefore you feel your Love Partner isn't.\n\nThis is due to the transference effect we talked about in Chapter 6. You're no longer connecting the concept of fun to your LP. Huntresses, you now subconsciously associate him with boring evenings in front of the TV, the computer, or the latest techno gizmo. Hunters, perhaps you feel neglected because she spends all her time with the kids.\n\nWhy did all the bicycling, boating, bowling, skateboarding, and snorkeling end? Many answer, \"There's not enough time, and besides, it's too expensive.\" Well, you made time when you were dating. And little do couples know how much more it's going to cost them in the future by not continuing to enjoy these things after they're married. You'll spend a lot more on divorce or marriage counseling later than you will for babysitters and movie tickets now.\n\nYou needn't be concerned about the relationship just because the sex doesn't give you the roller coaster thrills it once did. That's Mother Nature's plan. But having good shared experiences can keep you on a longer-lasting fun merry-go-round. The more you do, the more dopamine speeds up the carousel and makes it a more exciting ride.\n\n  **Chemistry Sparker #69**\n\n**Reinstate the Fun \"Excitation Transfer\" to Your LP**\n\nAh, if only all big problems could have such a simple solution. Think of all the cool things you used to do, and make a mutual bucket list. Now go out and enjoy those thrilling activities together to raise both your dopamine levels. You'll subconsciously connect the excitement to being with her. You'll think he's the reason you're having such a good time.\n\n### Laughing Is Just Another Way to Say \"I Love You\"\n\nYou have probably heard about the Asian laughing clubs, the endorphin-exploding health benefits of laughter, and how kids laugh six times more than adults do. I'll spare you repetition. Nor will I quote one of the unfunniest books I've ever read, Freud's _Joke and Its Relation to the Unconscious_ , which tells you how laughter reinvokes the joys of childhood. I'll just give you the bottom line: Laughter is very good stuff indeed.\n\nPeople\u2014primarily of the male gender\u2014have asked me, \"But what if I'm not funny?\" It doesn't matter. You may think laughter is inextricably connected to humor. Not at all! It has more to do with social interaction than it does with anything funny. Those serious humor researchers I mentioned earlier recorded friends hooting, hollering, and having a great time together. They found that the majority of laughter had no relation whatsoever to anything funny. It erupted simply from the joy of being together. Laughing with\u2014not at\u2014each other expresses joy at being with your lasting Love Partner.\n\nLet me bring cognitive consistency and the transference effect back for a quick encore: When your body is laughing, your brain thinks you're happy and, of course, your mate thinks you're the reason for the joy.\n\n_Brain_ : Hey, Body, why are you laughing so much? You must be happy.\n\n_Body_ : Yeah, I noticed all that chuckling and stuff. I guess I am happy when I'm with my partner.\n\n  **Chemistry Sparker #70**\n\n**Laugh with Your Long-Term Love Partner**\n\nLaughter is contagious, a socially transmitted condition. When you laugh, your Love Partner laughs. When your Love Partner laughs, you laugh. It doesn't make any difference who started it or what it's about. Whoever is around at the time gets the credit. Find excuses to laugh, giggle, guffaw, and crack up together. It's like spraying both of you with a refreshing mist of dopamine and oxytocin.\n\n### What About Long-Term Sizzling Sex?\n\nThis is the question that always comes up. Millions of writers who are more interested in telling readers what they want to hear than revealing the truth say that you can keep the _same_ early kind of intense passion alive for decades. Many of them, both best-sellers and flops, promise to keep you humping until the day you meet your maker. I put those in the fiction pile in my basement.\n\nThe media don't know, repress, or conveniently ignore the fact that there is a chemical reason for this. A few years of sex with the same person just doesn't give the same kicks because, as we've learned, dopamine diminishes with any oft-repeated pleasurable experience.\n\nJoyously, there will always be interludes of that early sizzling sex in a marriage. Vacations and celebrations are great for that. So are those special moments when, for a multitude of reasons, you feel a tremendous rush of love. But are there ways to poke the embers in between to get them blazing constantly again?\n\nYes, but I don't think you'd want some of the proven \"gotta have your body\" hot-sex stimulators\u2014like constant periods of separation or frequent squabbles and then a dramatic makeup. Both are dopamine boosters, but dangerous ones. One unintentional derision during the dispute could be the final blow. It could hit him too hard, hurt her too much. Your emotional amygdala could turn the delicious fun-fighting-then-making-up kind of dopamine into the detesting kind. And during one too-long separation, your Love Partner might meet a tempting someone else.\n\n\"So without separations or slugging it out for excitement, can I ever put that magnificent mind-blowing kind of sex back in our relationship?\" As I said at the very beginning, you can create a more magnificent kind of passion\u2014which is really mind-blowing because only a minority of couples attain it. The most important thing to remember is: _Don't think just because sex is not as hot as it was at first that love is cooling_. It could be just the opposite!\n\n\"How could that be?\" you ask. Life is unfair sometimes. Oxytocin and vasopressin, the attachment chemicals that the brain produces as love grows, actually reduce a man's testosterone. Here's another of Big Mama's chemical tricks: She further diminishes it when his wife is pregnant. \"Why waste all that precious sperm in her when he's already got her knocked up?\" she asks. During pregnancy a wife often fears that her husband no longer finds her attractive because he doesn't want sex as much. She's wrong. She's just as beautiful as ever to him. It has little to do with the big tummy and a lot to do with his lower testosterone.\n\nThere are other reasons long-term sex doesn't give the same fireworks. Your caudate nucleus, or reward center, which has that nasty habit of living in the future, isn't looking forward to anything new with the same partner. A Hunter knows how his wife looks when she's hot. A Huntress knows how her husband sounds when he comes. No sexual surprises are around the corner, so the lights on _Pleasure Island_ dim a little.\n\nThen there is the simple aging process. Mother Nature doesn't give a flying toot about human humping later in the game. Because she's so obsessed with propagation, she has no vested interest in you staying together much beyond childbearing. In her opinion, \"Why even have sex after menopause?\"\n\nOkay, so that's the bad news.\n\n#### _Here's the Good News\u2014No, the Great News!_\n\nWith devotion, respect, support, and emotional intimacy, a different type of powerful desire for each other grows. Sure, he no longer ravishes her like a rabbit on Viagra, and her screams don't disturb the neighbors every night. But sex is more fulfilling because this time it grows out of deep oxytocin-filled love, not a raging testosterone rush. Your hot sex can develop into warm long-term lovemaking by using some of the other concoctions in MN's chemistry set\u2014the same ones she injects into new parents to bond them to the amazing baby they created together. By using the touch, laughter, and doing things together we discussed, you create the bonding chemical that encourages continued warm lovemaking. That kind is fulfilling not only physically but also emotionally.\n\nThroughout history, all around the world people have sought aphrodisiacs such as the highly touted Spanish Fly, the Jamaican cow cod soup, Taiwanese deer penis, Asian dried lizard, and even (yuck) sick sperm whale vomit. None are as effective, however, as the following. This aphrodisiac to extend sexual desire for each other is an unspoken trade deal based on the irrefutable time honored \"Give and ye shall receive\" doctrine.\n\nHunters, you're going to love the first. Huntresses, the second is for you.\n\n#### _Why \"Quickies\" Count_\n\nHuntresses, sweet lovemaking for us means luxuriating in deep kisses, slow caresses, loving words\u2014maybe even a foot massage if we \"get lucky\"\u2014then the hot sex. But for a male, as much as he loves you, his main interest is the big finale. Unfortunately, Mother Nature is on his side, because his caressing your body does nothing for her propagation goal. His orgasm does.\n\nLet's say you're now living together. It's a typical weeknight. You're both exhausted. The heartless alarm will blast you out of bed in six hours. But your testosterone-filled sweetie is all set to go and craves some high-speed mingling of limbs. For you to really enjoy it, however, you need some sensual stroking, licking, and loving. On average it takes eighteen minutes to bring us women to orgasm. He can do the job in eight to eighteen thrusts.\n\nHave mercy, Huntresses. It's a tall order for a tired male to slow down and start the quarter of an hour minimum process of warming you up by giving proper attention to the countless supersensitive areas all over your body. You're tired too, and because quickies aren't as fulfilling for you, perhaps you feign a headache. Now he feels unloved.\n\nThink about it, girl. It takes it more time for you to hem and haw about a headache than for him to be fully satisfied. Sure you might not be in the mood right now, but never forget that he interprets a quickie as your love for him.\n\nHere's the reason quickies can literally keep the Chemistry between you alive. With repetition, the bonding oxytocin and vasopressin that flood his brain at ejaculation have a cumulative effect on his feelings of togetherness with you. He associates, or in neurolinguistic programming terms, \"anchors\" his joy of sex with you.\n\nAn added benefit is oxytocin's talent for burying unpleasant memories. If he's flaming mad at you for any reason, his ejaculation acts as a fire extinguisher.\n\n  **Chemistry Sparker #71**\n\n**Give Him Lots of \"Quickie\" Bonding**\n\nGirl, look at it this way: Constant sex plays a bigger role for a male, and you wanting it with him demonstrates your love. The chemicals released in his brain accumulate over the long term. And just think: A quickie can take less than five minutes.\n\nSo you must decide if quickies are worth it. If you ask him to slow down every time, _he_ just may figure it's not worth it. Compromise.\n\nI was discussing this chapter with a friend of mine who has been married for eight years, and whenever I see her with her husband, it's obvious that he's still deeply in love with her. In one of those girltalk moments, she said quickies are part of their regular love life. She would never tell her husband, but she jokingly said, \"I think of it as 'taking one for the team.'\" In fact, she initiates the quickies every so often. It's part of keeping \"the team\" together.\n\nIncidentally, don't feel you need to compete with Meg Ryan's infamous fake deli orgasm in the movie _When Harry Met Sally_. In fact, don't fake it during a quickie because he may think his speedy style is all you want. Tell him you enjoy it too because you love him. But do tell him how you love it even more when it's long and slow. Save your \"Oh, oh, yes, yes, YESSSSS!\" for the incredible real ones he'll want to give you when there's more time.\n\n#### _Turnabout Is Fair Foreplay_\n\nHunters, now it's time for you to \"give one for the team.\" Whenever there's time, thrill her with the tender leisurely lovemaking she longs for\u2014and make time more often. She craves those as much as you crave quickies.\n\nGentlemen, to increase the sexual Chemistry between you and your Love Partner, I suggest you go back and reread the Sparkers in Chapter 7:\n\n\u2022Chemistry Sparker number 46: _Set the Stage for Sex_. Creating her desired erotic atmosphere is just as important now. What kind of music puts her in the mood? Does she prefer dim lighting? Does she like a back rub or a foot massage before the action begins?\n\n\u2022Chemistry Sparker 47: _Let Her Start the Strip_. You can start the strip now\u2014but of _her_ clothes, not yours. While tenderly unbuttoning her blouse, express how much you care for her. Kiss her shoulder as you gently slip the top of her nightgown to the side. No quickies now. This is her time. Think \"Slowies.\" Slow and sweet.\n\n\u2022Chemistry Sparker 48: _Don't Get Physical, Get Oral_. Your tender words mean more to her now than ever, and you can truthfully use the \"L\" word. Say it often\u2014and not just during sex.\n\n\u2022Chemistry Sparker 49: _Play in Her Secret Garden_. You no longer need to ask what she wants because you're already familiar with her erotic paradise. Visit it with her frequently.\n\n\u2022And, of course, give yourself a refresher on Sparker 50: _Geography 101 for Hunters_.\n\nEarlier I suggested that women read magazines that show the average man's attitude toward sex. Turnabout is only fair play here too. Pick up a copy of _Cosmopolitan_. One cursory page flipping suffices to see the setting for sex through your lady's eyes. You'll find beautiful bedding, an occasional candle, and the omnipresent admiring and loving expression on the face of the male she's having sex with.\n\n  **Chemistry Sparker #72**\n\n**Don't Think \"Sex.\" Think \"Seduction.\"**\n\nCourting doesn't end at the altar, and neither do seduction techniques. Now that you know all the hidden pathways in her secret sexual garden, explore each one unhurriedly. The chemicals created during foreplay have an especially endearing and enduring effect on a Huntress.\n\n### Everyone Has Something They Need to Hear\n\nEvery decade or so, a TV commercial comes along that makes you think, \"Wow, I gotta get one of those thingamajigs.\" This was not one of them, and I do not even recommend their product. But we can learn a lot from their advertising campaign.\n\nMadison Avenue types spend millions of dollars annually for focus groups to discover what consumers want to hear and then pay millions more for just the right actors to dramatize it. This particular commercial is on target and gives great insight into the sentiments that your long-term Love Partner hungers to hear from you.\n\nThe following individuals flashed on screen for three seconds and delivered one line each into the camera. Here are the numbered characters, and their one-liners are in the alphabetized list below it. Figure out what each person wanted to hear and fill in the letter next to the desired expressions.\n\n1)Frazzled woman in a suburban home, dishes piled in the background, trying to control her frisky infant\n\n2)Rushed thirtyish middle-management male straightening his tie getting ready for work\n\n3)Kindly old man sitting alone on a park bench\n\n4)Young woman looking into the camera flirtatiously\n\n5)Poignantly smiling bald woman who had lost her hair obviously due to chemotherapy\n\n6)Male teenager making a video of his face on his computer for his girlfriend\n\n7)Ten-year-old kid in a superman costume\n\nHere are their one-line scripts. See if you can match the people above to what they most wanted to hear.\n\n___a)\" _Tell me_ we'll stay together forever, in sickness and in health.\"\n\n___b)\" _Tell me_ we'll grow old together.\"\n\n___c)\" _Tell me_ I'm still beautiful.\"\n\n___d)\" _Tell me_ you need me.\"\n\n___e)\" _Tell me_ you miss me.\"\n\n___f)\" _Tell me_ that I'm your superhero.\"\n\n___g)\" _Tell me_ you love me.\"\n\nThe Answers: a) 5; b) 3; c) 1; d) 2; e) 6; f) 7; g) 4\n\n  **Chemistry Sparker #73**\n\n**Say What Your Long-Term Love Partner Needs to Hear**\n\nGrow an extra antenna to pick up precisely what your Love Partner needs from you emotionally. Then find a hundred ways to repeat this truth over and over. She will never tire of hearing it. He will feel closer to you every time you say it.\n\nIn this case, you needn't feel hesitant about making gender-stereotyped generalizations. Hunters and Huntresses need to hear different things to get long-lasting love chemicals flowing through their brains.\n\nI'm embarrassed to reveal that it was a Hallmark Card Valentine's Day commercial. (Yes, sometimes a flower grows in dung, and this was such a blossom in a stench-filled field of TV ads.) The final line speaks the truth: \"Everyone has something they need to hear.\"\n\nIt is often said, \"Life is not a Hallmark card.\" That's true, but in long-lasting love it's beautiful to make it sound like one.\n\n### Hunters, _Why_ Her Feeling Loved Creates Lasting Chemistry\n\nEvolutionary psychologists once tucked all human behavior into an envelope neatly labeled \"Evolution\" and sealed the flap. They figured that, although we now reside in cities and suburbs rather than jungles and woods, we still needed precisely the same ancient toolbox between the ears. Then neuroscience and developmental evolution discovered that our brains are changing more quickly than expected. However, especially in the framework of love relationships, it's pretty clear that ancient influences continue to have a strong foothold in our thinking.\n\n  **Chemistry Sparker #74**\n\n**Give Her a Daily Dose of What She _Really_ Needs**\n\nHunters, at least once a day\u2014minimum\u2014tell your long-term Love Partner that you love her. And no matter what happens, bite your tongue before saying anything that she could, in the furthest stretch of her imagination, possibly interpret as \"I don't love you anymore.\" If you do, her Cro-Magnon grandmother will tell her, \"Girl, you're as good as dead.\"\n\nFemales know that if a man loves a woman, he stays with her. And up until recently, Mom needed a Hunter for both of them to survive. So she had a vested interest in being loved by her man for a very long time. In her mind, love equals survival. Thus, she _needs_ a man's love.\n\nA Hunter, of course, wants to be loved too. But a male doesn't need it as much because ancient voices are not echoing that he can't survive unless a woman protects him.\n\nOnce I was being interviewed as the guest expert on a show with four couples going through the two-year itch. The host asked the guys, \"When was the last time you told your wife you loved her?\" One responded, \"I told her I loved her when I married her. If anything changes, I'll let her know.\" Every woman in the audience wanted to hurl rotten tomatoes at him.\n\n### Huntresses, _Why_ His Feeling Needed Creates Lasting Chemistry\n\nI decided to web search the question: \"Why doesn't he tell me he loves me?\" At this writing, the question got 110 million hits from women asking precisely that. Yet I didn't get one hit from a man complaining, \"Why doesn't she tell me she needs me?\" In fact, a search on \"Why doesn't she tell me she needs me\" first brought up a woman asking about her toddler, \"Why doesn't she tell me she needs me . . . to take her to the potty?\"\n\nA man craves being needed by you as much as you hunger for his love. Buried deep in his brain, his brawny hirsute heroic club-carrying ancestor is alive and well. Because the womenfolk's lives depended on his wonderfulness, he got used to being dauntless and depended on.\n\nSisters, if you have ever dreamt about being swept up by the handsome prince, why can't he dream of being Prince Valiant? When you can't fix the electrical wire in a lamp, he wants to come riding into the living room on his white steed, shouting, \"Fear not, sweet damsel! I will rescue you from this dilemma.\" So, ladies, let him. And bite your tongue until it's bloody before giving him advice on how to go about doing it.\n\n  **Chemistry Sparker #75**\n\n**Find Ways to Say \"I Need You\"**\n\nIt's not just your guy. Practically all males have a neurological need to provide the solution to problems, especially his woman's. I've read dozens of books like _1000 Ways to Say I Love You_. Nice stuff. But there aren't any on _1000 Ways to Say I Need You to Your Man_. In fact, not even _One Way to Say I Need You to Your Man_. Start noodling on some now and express them often.\n\n### \"To What Do You Credit Your Lifelong Love?\"\n\nLooking out my window some winters in New York City, I expect to see polar bears strolling down my street. That's when I try to escape to Sarasota, Florida, for a mini-vacation. Even more beautiful than the sun and the sea there are the elder couples I see, some bicycling or kayaking together. Others strolling hand in hand or helping a partner in a walker navigate a grocery store aisle. I befriended several of the aging couples and learned a lot about the exquisite beauty and contentment that comes with long-term togetherness.\n\nRecently I saw a TV talk show host interviewing half a dozen happy couples who had passed their fiftieth anniversary. Some had canes, a few were in wheelchairs, and most were holding hands. Their kisses were for the camera, but their sincere smiles were for their lifelong mates. Yes, a couple can stay in love forever.\n\nNaturally the host asked, \"To what do you credit your lifelong love?\"\n\nOne septuagenarian answered, \"Because I didn't marry a woman I could live with. I married the woman I couldn't live without.\"\n\nThere was a resounding \"aww\" from the studio audience. Another octogenarian lady replied, \"Because my husband often buys two roses, saying, 'One is for the woman I love. The other for my best friend.' He gives them both to me.\"\n\nA crescendo of longer \"awws\" surged from the audience.\n\nFinally a feeble old man with a wavering voice said, \"From the moment I saw her at age eighteen, I knew life without her wouldn't be worth living.\" He then took her hand and kissed it. The audience reaction was something approaching a simultaneous group orgasm that lasted about five minutes.\n\nIf one of those octogenarians knew as much about neuroscience as you now do, his answer to the question would have been less romantic but more scientifically accurate. His frail voice would have murmured, \"It's because we knew about the chemicals affecting our brains and how to create the good ones. This wisdom and nurturing the valuable chemicals has kept us happily together.\"\n\nThe audience reaction would have been stunned silence, thinking the poor old goat had gone off his rocker.\n\nAnd yet the Chemistry they created together was one of the reasons they were still on earth to talk about their love. Long-term Love Partners live longer lives than people who reside alone. Oxytocin and vasopressin gently swim through the rivers in your brain during long-term love, making you astronomically healthier. It's not like early stage love, when dopamine and serotonin levels spike up and down like an electroencephalogram of a grasshopper in a frying pan. The attachment chemicals you create don't have the terrifying \"Side effects may include\" warning that TV ads spit out at the speed of light about hyperactivity, loss of appetite, compulsive acts, and symptoms associated with mental illness. There is no need to \"Ask your doctor if long-term togetherness is right for you.\" It has been proven that it is.\n\n### What to Expect When You're Expecting Lasting Love\n\nDuring the early crazy-in-love, passionate, gotta-have-you-round-the-clock days and nights, the sizzling chemicals swirl. Dopamine shoots through your brains like meteors. When he caresses your cheek, dopamine and estrogen party. When she brushes against your body, dopamine and testosterone do a dizzying dance. Serotonin swims through the synapses, and your caudate nucleus flashes like a neon sign.\n\nAfter a few years of living together dopamine levels dip, but there are magnificent spikes, especially when you do thrilling things with each other and laugh together. When you make love, a myriad of both the hot and the bonding chemicals intermingle. Sometimes sex is fast when time is tight. Sometimes lingering and loving, like on weekends and vacations.\n\nWhen kids come, an avalanche of chemicals consumes you. Testosterone and estrogen rise and fall like a roller coaster. Huntresses, you swim in an ocean of oxytocin while bonding with your baby. Hunters, big waves of vasopressin overcome you, especially when holding your newborn infant. There are problems, of course, and the frenetic fluctuating chemicals can cause frenzy and doubt. That is the time to cling together and not make foolish mistakes just because Mother Nature is making you antsy to move on.\n\nWhen you have similar beliefs and definitions of togetherness, when you enjoy doing things together, when you can depend on the other in adversity and help each other reach life goals, a fortress of oxytocin and vasopressin approaches like a gentle mist and engulfs you in contentment.\n\nAs you grow even older, naturally estrogen and testosterone levels go down, but without gushing testosterone fighting the bonding chemicals, you become closer, and Dad becomes a stronger bonder. That's when you reach the state Berni\u00e8res spoke of when he wrote that \"your roots have become so entwined together that it is inconceivable that you should ever part.\"\n\nI wish you the lasting love you so richly deserve.\nStay in touch! The only thing I _don't_ like about being a writer is not having the pleasure of meeting my readers. I'd love to hear from you, and you can contact me through my website, www.greatcommunicating.com. Ask a question or just let me know your thoughts.\n\nWhile you're visiting the site, sign up for my free, very short monthly hint to make you an even better communicator in your professional life, your social life, and your love life.\n\n## References\n\n### Prologue\n\n. Robert Sapolsky, _Biology and Human Behavior: The Neurological Origins of Individuality_ , 2nd ed. (Chantilly, VA: The Teaching Company, 2005).\n\n. John Money, _Lovemaps: Clinical Concepts of Sexual\/Erotic Health and Pathology, Paraphilia, and Gender Transposition of Childhood, Adolescence, and Maturity_ (New York: Irvington, 1986).\n\n. D. J. Siegel, _The Developing Mind: Toward a Neurobiology of Interpersonal Experience_ (New York: Guilford, 1999).\n\n. Money, _Lovemaps_.\n\n. Tamas Bereczkei, Petra Gyuris, and Glenn E. Weisfeld, \"Sexual Imprinting in Human Mate Choice,\" _Proceedings of the Royal Society of Biological Sciences_ 271, no. 1544 (2004): 1129\u20131134.\n\n. Sapolsky, _Biology and Human Behavior_.\n\n. Ingrid R. Olson and Christy Marshuetz, \"Facial Attractiveness Is Appraised in a Glance,\" _Emotion_ 5, no. 4 (December 2005): 498\u2013502.\n\n. Steven W. Gangestad, Christine E. Garver-Apgar, Jeffry A. Simpson, and Alita J. Cousins, \"Changes in Women's Mate Preferences Across the Ovulatory Cycle,\" _Journal of Personality and Social Psychology_ 92, no. 1 (January 2007): 151\u2013163.\n\n. Andreas Bartels and Semir Zeki, \"The Neural Basis of Romantic Love,\" _NeuroReport_ 2, no. 17 (2000): 12\u201315.\n\n. Helen Fisher, _Why We Love: The Nature and Chemistry of Romantic Love_ (New York: Henry Holt and Company, 2004).\n\n. Arthur Aron, Helen Fisher, Debra J. Mashek, Greg Strong, and Haifang Li, and Lucy L. Brown, \"Reward, Motivation, and Emotion Systems Associated with Early Stage Romantic Love,\" _Journal of Neurophysiology_ 94, no. 1 (July 2005): 327\u2013337.\n\n. Paula Tucker and Arthur Aron, \"Passionate Love and Marital Satisfaction at Key Transition Points in the Family Life Cycle,\" _Journal of Social and Clinical Psychology_ 12, no. 2 (1993): 135\u2013147.\n\n. Aldous Huxley, _The Doors of Perception_ (London: Chatto and Windus, 1954).\n\n. Zeenat F. Zaidi, \"Gender Differences in Human Brain: A Review,\" _The Open Anatomy Journal_ 2, no. 1 (2010): 37\u201355.\n\n. Pierce J. Howard, _The Owner's Manual for the Brain: Everyday Applications from Mind-Brain Research_ , 3rd ed. (Austin, TX: Bard Press, 2006).\n\n. Douglas T. Kenrick, Edward K. Sadalla, Gary Groth, and Melanie R. Trost, \"Evolution, Traits, and the Stages of Human Courtship,\" _Journal of Personality_ 58, no. 1 (March 1990), 97\u2013116.\n\n. Steven W. Gangestad and Jeffry A. Simpson, \"The Evolution of Human Mating: Trade-Offs and Strategic Pluralism,\" _Behavioral and Brain Sciences_ , 23, no. 4 (2000) 573\u2013587.\n\n. Bianca P. Acevedo, Arthur Aron, Helen E. Fisher, and L. L. Brown, \"Neural Correlates of Long-Term Intense Romantic Love,\" _Social Cognitive and Affective Neuroscience_ 7, no. 2 (February 2012): 145\u2013159.\n\n. Fisher, _Why We Love_.\n\n. Sapolsky, _Biology and Human Behavior_.\n\n. Jeffry A. Simpson and Steven W. Gangestad, \"Individual Differences in Sociosexuality: Evidence for Convergent and Discriminant Validity,\" _Journal of Personality and Social Psychology_ 60, no. 6 (June 1991): 870\u2013883.\n\n. William G. Axinn and Arland Thornton, \"The Transformation in the Meaning of Marriage,\" in _The Ties That Bind: Perspectives on Marriage and Cohabitation_ , edited by Linda J. Waite, 147\u2013165 (New York: De Gruyter, 2000); Susan Sprecher, Amy Wenzel, and John H. Harvey, eds., _Handbook of Relationship Initiation_ (New York: Psychology Press, 2008).\n\n. Alan S.Gurman, ed., _Clinical Handbook of Couple Therapy_ , 4th ed. (New York: Guilford Press, 2008).\n\n### Chapter 1\n\n. Pierce J. Howard, _The Owner's Manual for the Brain: Everyday Applications for Mind-Brain Research_ , 3rd ed. (Austin, TX: Bard Press, 2006).\n\n. Robert J. Sternberg and Susan Grajek, \"The Nature of Love,\" _Journal of Personality and Social Psychology_ 47, no. (3) (1984): 12\u201329.\n\n. Anne Moir and David Jessel, _Brain Sex: The Real Difference Between Men and Women_ (New York: Dell, 1991).\n\n. Ibid.\n\n. Ibid.\n\n. Howard, _The Owner's Manual for the Brain_.\n\n. Robert Sapolsky, _Biology and Human Behavior: The Neurological Origins of Individuality_ , 2nd ed. (Chantilly, VA: The Teaching Company, 2005).\n\n. Sandra L. Murray and John G. Holmes, \"A Leap of Faith? Positive Illusions in Romantic Relationships,\" _Personality and Social Psychology Bulletin_ 23, no. 6 (1997): 586\u2013604.\n\n. Helen Fisher, _Why We Love: The Nature and Chemistry of Romantic Love_ (New York: Henry Holt and Company, 2004).\n\n. Helen Fisher and J. Anderson Thomson Jr., \"Lust, Romance, Attraction, Attachment: Do The Side-Effects Of Serotonin-Enhancing Antidepressants Jeopardize Romantic Love, Marriage and Fertility?\" in _Evolutionary Cognitive Neuroscience_ , edited by Steven M. Platek, Julian Paul Kennan, and Todd K. Shakleford, 245\u2013283 (Cambridge, MA: MIT Press, 2007).\n\n. Malcolm Caruthers, _The Testosterone Revolution: Rediscover Your Energy and Overcome the Symptoms of Male Menopause_ (London: Thorsons, 2001).\n\n. Thomas R. Insel, \"Oxytocin\u2014A Neuropeptide for Affiliation: Evidence from Behavioral, Receptor Autoradiographic, and Comparative Studies,\" _Psychoneuroendocrinology_ 17, no. 1 (1992): 3\u201335.\n\n. Ilanit Gordon, Orna Zagoory-Sharon, James F. Leckman, and Ruth Feldman, \"Prolactin, Oxytocin, and the Development of Paternal Behavior Across the First Six Months of Fatherhood,\" _Hormones and Behavior_ 58, no. 3 (August 2010): 513\u2013518.\n\n. Lee T. Gettler, Thomas W. McWade, C. W. Kuzawa, and A.B. Feranil, \"Longitudinal Evidence That Fatherhood Decreases Testosterone in Human Males,\" edited by A. E. Storey, Proceedings of the National Academy of Sciences of the United States of America, 108, no. 39 (September 27, 2011): 16194\u201316199, <http:\/\/www.pnas.org\/content\/early\/2011\/09\/02\/1105403108.full.pdf+html>.\n\n. Sandra J. Berg and Katherine E. Wynne-Edwards, \"Changes in Testosterone, Cortisol, and Estradiol Levels in Men Becoming Fathers,\" _Mayo Clinic Proceedings_ 76, no. 6 (2001): 582\u2013592.\n\n### Chapter 2\n\n. A. H. Veenema, and I. D. Neuman, \"Central Vasopressin and Oxytocin Release: Regulation of Complex Social Behaviors,\" _Progress in Brain Research_ 170 (2008): 261\u2013276.\n\n. C. L., Clark, P. R. Shaver, and M. F. Abrahams, \"Strategic Behaviors in Romantic Relationship Initiation,\" _Personality and Social Psychology Bulletin_ 25, no. 6 (1999): 709\u2013722.\n\n. William G. Axinn and Arland Thornton, \"The Transformation in the Meaning of Marriage\" in _The Ties That Bind: Perspectives on Marriage and Cohabitation_ , edited by Linda J. Waite, 147\u2013165 (New York: Aldine de Gruter, 2000).\n\n. Susan Sprecher, Amy Wenzel, and John H. Harvey, eds., _Handbook of Relationship Initiation_ (New York: Psychology Press, 2008).\n\n. Zeenat F. Zaidi, \"Gender Differences in Human Brain: A Review,\" _The Open Anatomy Journal_ 2, no. 1 (2010): 37\u201355.\n\n. Louann Brizendine, _The Male Brain_ (New York: Broadway Books, 2010).\n\n. Samuel Vaknim, _Malignant Self-Love: Narcissism Revisited_. (Prague: Narcissus, 2007).\n\n. Russell D. Clark and Elaine Hatfield, \"Gender Differences in Receptivity to Sexual Offers,\" _Journal of Psychology and Human Sexuality_ 2 (1989): 39\u201355.\n\n. Russell D. Clark, \"The Impact of AIDS on Gender Differences in Willingness to Engage in Casual Sex,\" _Journal of Applied Social Psychology_ 20, no. 9 (May 1990): 771\u2013782.\n\n. Charles W. Hobart, \"The Incidence of Romanticism During Courtship,\" _Social Forces_ 36, no 4 (1958): 362\u2013367, 364.\n\n. Louann Brizendine, _The Female Brain_ (New York: Broadway Books, 2006).\n\n. Ibid.\n\n. David M. Buss and Todd K. Shackelford, \"Attractive Women Want It All: Good Genes, Economic Investment, Parenting Proclivities, and Emotional Commitment,\" _Evolutionary Psychology_ 6, no. 1 (2008): 134\u2013246.\n\n. B. Pawlowski and L. G. Boothroyd, D. I. Perrett, S. Kluska, \"Is Female Attractiveness Related to Final Reproductive Success?\" _Collegium Antropologicum_ 32, no. 2 (June 2008): 457\u2013460.\n\n. Ibid.\n\n. D. Scutt and J. T., Manning, \"Symmetry and Ovulation in Women,\" _Human Reproduction_ 11, no. 11 (1996): 2477\u20132480.\n\n. John Tierney, \"The Threatening Scent of Fertile Women,\" _New York Times_ , February 21, 2011, <http:\/\/www.nytimes.com\/2011\/02\/22\/science\/22tier.html>.\n\n. Steven W. Gangestad, Christine E. Garver-Apgar, Jeffry A. Simpson, and Alita J. Cousins, \"Changes in Women's Mate Preferences Across the Ovu la tory Cycle,\" _Journal of Personality and Social Psychology_ 92, no. 1 (January 2007): 151\u2013163.\n\n. P. Sorokowski, \"Do Men Prefer Blondes? The Influence of Hair Color on the Perception of Age and Attractiveness of Women,\" _Studia Psychologiczne_ 44, no. 3. (2006): 77\u201378.\n\n. L. Van der Berghe and P. Frost, \"Skin Color Preference, Sexual Dimorphism and Sexual Selection: A Case of Gene Co-Evolution,\" _Ethnic and Racial Studies_ 9 (1986): 87\u2013113.\n\n. Verlin Hinsz, David Matz, and Rebecca Patience, \"Does Women's Hair Signal Reproductive Potential?\" _Journal of Experimental Social Psychology_ 37, no. e (2001): 166\u2013172.\n\n. Grazyna Jasienska, Anna Ziomkiewicz, Peter T. Ellison, Susan F. Lipson, and Inger Thune, \"Large Breasts and Narrow Waists Indicate High Reproductive Potential in Women,\" _Proceedings of the Royal Society of Biological Sciences_ 271, no. 1545 (2004) 1213\u20131217; Pierce J. Howard, _The Owner's Manual for the Brain: Everyday Applications from Mind-Brain Research_ , 3rd ed. (Austin, TX: Bard Press, 2006).\n\n. Bruno Laeng, Ronny Mathisen, and Jan-Are Johnsen, \"Why Do Blue-Eyed Men Prefer Women with the Same Eye Color?\" _Behavioral Ecology and Sociobiology_ 61, no. 3 (2007): 371\u2013384.\n\n. Sprecher, Wenzel, and Harvey, _Handbook of Relationship Initiation_.\n\n. Helen Fisher, _Why We Love: The Nature and Chemistry of Romantic Love_ (New York: Henry Holt and Company, 2004).\n\n. Charles Darwin, _The Descent of Man, and Selection in Relation to Sex_ (New York: Addison, 1871).\n\n. Robert Sapolsky, _Biology and Human Behavior: The Neurological Origins of Individuality_ , 2nd ed. (Chantilly, VA: The Teaching Company, 2005).\n\n. P. M. La Cerra, \"Evolved Mate Preferences in Women: Psychological Adaptations for Assessing a Man's Willingness to Invest in Offspring,\" (doctoral dissertation, University of California, Santa Barbara, 1994).\n\n. Elaine Hatfield, \"What Do Women and Men Want from Love and Sex?\" In _Changing Boundaries: Gender Roles and Sexual Behavior_ , edited by Elizabeth R. Allgeir and Naomi B. McCormick, 106\u2013134 (Palo Alto, CA: Mayfield, 1982).\n\n. David M. Buss and Michael Barnes, \"Preferences in Human Mate Selection,\" _Journal of Personality and Social Psychology_ 50 (1989): 559\u2013570.\n\n. Ibid.\n\n. Randy Thornhill, \"The Biology of Beauty,\" _Newsweek_ ,June 2, 1996, <http:\/\/www.thedailybeast.com\/newsweek\/1996\/06\/02\/the-biology-of-beauty.html>.\n\n. Steven W. Gangestad, and Randy Thornhill, \"Menstrual Cycle Variation in Women's Preferences for the Scent of Symmetrical Men.\" _Proceedings of the Royal Society of London_ 265, no. 1399 (1998): 927\u2013933.\n\n. Howard, _The Owner's Manual for the Brain_.\n\n. Ibid.\n\n. Ibid.\n\n. Ibid.\n\n. Ibid.\n\n. Ibid.\n\n### Chapter 3\n\n. Joan Kellerman, James Lewis, and James Laird, \"Looking and Loving: The Effects of Mutual Gaze on Feelings of Romantic Love,\" _Journal of Research in Personality_ 23, no. 2 (1989): 145\u20131361.\n\n. Mark Cook, \"Gaze and Mutual Gaze in Social Encounters,\" _American Scientist_ 65 (1977): 328\u201333.\n\n. Zick Rubin, \"Measurement of Romantic Love,\" _Journal of Personality and Social Psychology_ 16, no. 2 (1970): 265\u2013273.\n\n. Timothy Perper, _Sex Signals: The Biology of Love_ (Philadelphia: ISI Press, 1985).\n\n. Pierce J. Howard, _The Owner's Manual for the Brain: Everyday Applications from Mind-Brain Research_ , 3rd ed. (Austin, TX: Bard Press, 2006).\n\n. William G. Iacono, \"Forensic 'Lie Detection': Procedures without Scientific Basis,\" _Journal of Forensic Psychology Practice_ 1, no. 1 (2001): 75\u201386.\n\n. E. H. Hess, \"Attitude and Pupil Size\" _American Scientist_ 212(1965): 46\u201354.\n\n. David M. Buss and Todd K. Shackelford, \"Attractive Women Want It All: Good Genes, Economic Investment, Parenting Proclivities, and Emotional Commitment,\" _Evolutionary Psychology_ 6, no. 1 (2006): 134\u2013146.\n\n. Howard, _The Owner's Manual for the Brain_.\n\n. Zeenat F. Zaidi, \"Gender Differences in Human Brain: A Review,\" _The Open Anatomy Journal_ 2, no. 1 (2010): 37\u201355.\n\n. Howard, _The Owner's Manual for the Brain_.\n\n. Timothy Perper, _Sex Signals: The Biology of Love_ (Philadelphia: ISI Press, 1985).\n\n. Hugo D. Critchley, Christopher J. Mathias, and Raymond J. Dolan, \"Neural Activity in The Human Brain Relating to Uncertainty and Arousal During Anticipation,\" _Neuron_ 29, no. 2 (February 2001): 537\u2013545.\n\n. Mark Cook and Robert McHenry, _Sexual Attraction_ (New York: Pergamon Press, 1978).\n\n. D. P. Schmidt, and D. M. Buss, \"Human Mate Poaching: Tactics and Temptations for Infiltrating Existing Mateships,\" _Personality and Social Psychology_ 80, no. 6 (June 2001): 894\u2013917.\n\n. Eric R. Bressler and Sigal Balshine, \"The Influence of Humor on Desirability,\" _Evolution and Human Behavior_ 27, no. 1 (2006): 29\u201339.\n\n. Schmidt and Buss, \"Human Mate Poaching.\"\n\n. Ibid.\n\n. P. N. Hamid, \"Changes in Person Perception as a Function of Dress,\" _Perceptual Motor Skills_ 29, no1 (1969): 191\u2013194.\n\n. Meredith L. Chivers, Michael C. Seto, and Ray Blanchard, \"Gender and Sexual Orientation Differences in Sexual Response to Sexual Activities Versus Gender of Actors in Sexual Films,\" _Journal of Personality and Social Psychology_ 93, no. 6 (2007): 1108\u20131121.\n\n. J. M. Townsend and G. D. Levy, \"Effect of Potential Partner's Costume and Physical Attractiveness on Sexuality and Partner Selection,\" _Journal of Psychology_ 124, no. 4 (1990): 371\u2013389.\n\n. Chivers, Seto, and Blanchard, \"Gender and Sexual Orientation Differences in Sexual Response to Sexual Activities Versus Gender of Actors in Sexual Films.\"\n\n. Howard, _The Owner's Manual for the Brain_.\n\n. Susan Sprecher, Amy Wenzel, and John H. Harvey, eds., _Handbook of Relationship Initiation_ (New York: Psychology Press, 2008).\n\n. Todd K. Shackelford, David Schmitt, and David Buss, \"Universal Dimensions Of Human Mate Preferences,\" _Personality and Individual Differences_ 39 (2005): 447\u2013458.\n\n. John M. Townsend and Gary D. Levy, \"Effects of Potential Partner's Physical Attractiveness and Socioeconomic Status on Sexuality and Partner Selection,\" _Archives of Sexual Behavior_ 19, no. 2 (1990): 149\u2013164.\n\n. Howard, _The Owner's Manual for the Brain: Everyday Applications from Mind-Brain Research_ , 3rd ed. (Austin, TX: Bard Press 2006).\n\n. R. K. Winkelmann, \"The Erogenous Zones: Their Nerve Supply and Its Significance,\" _Mayo Clinic Proceedings_ 34, no. 2 (1959): 39\u201347.\n\n. R. Ecochard and A. Gougeon \"Side of Ovulation and Cycle Characteristics in Normally Fertile Women,\" _Human Reproduction_ 15, no. 4 (2000): 752\u2013755.\n\n. Karl Grammer, LeeAnn Renninger, and Bettina Fischer, \"Disco Clothing, Female Sexual Motivation, and Relationship Status: Is She Dressed to Impress?\" _Journal of Sex Research_ 41, no. 1 (2004): 66\u201374; Meghan Provost, Vernon Quinsey, and Nikolaus Troje, \"Differences in Gait Across the Menstrual Cycle and Their Attractiveness to Men,\" _Archives of Sexual Behavior_ 37, no. 4 (2008): 598\u2013604.\n\n. M. M. Moore, \"Nonverbal Courtship Patterns in Women: Context and Consequences,\" _Ethnology and Sociobiology_ 6 (1985): 237\u2013247.\n\n### Chapter 4\n\n. James R. Roney, Katherine N. Hanson, Kristina M. Durante, and Dario Maestripieri, \"Reading Men's Faces: Women's Mate Attractiveness Judgments Track Men's Testosterone and Interest in Infants,\" _Proceedings of the Royal Society of Biological Sciences_ 273, no. 1598 (September 2006): 2169\u20132175.\n\n. Julie K. Hasart, Kevin L. Hutchinson, \"The Effects of Eyeglasses on Perceptions of Interpersonal Attraction,\" _Journal of Social Behavior and Personality_ 8, no. 3 (1993): 521\u2013528.\n\n. G\u00fcnter J. Hitsch, Ali Horta\u00e7su, and Dan Ariely, \"What Makes You Click? Mate Preferences and Mating Outcomes in Online Dating,\" University of Chicago (January 2010), <http:\/\/home.uchicago.edu\/~ghitsch\/Hitsch-Research\/Guenter_Hitsch_files\/Mate-Preferences.pdf>.\n\n. A. H. Maslow, and N. L. Mintz, \"Effects of Aesthetic Surroundings,\" _Journal of Psychology_ 41 (1956): 247\u2013254.\n\n. Harold Sigall and David Landy, \"Radiating Beauty: The Effects of Having a Physically Attractive Partner on Person Perception,\" _Journal of Personality and Social Psychology_ 28, no. 2 (1973): 218\u2013224.\n\n. Hitsch, Horta\u00e7su, and Ariely, \"What Makes You Click?\"\n\n. Andrew J. Elliot and Daniela Niesta, \"Romantic Red: Red Enhances Men's Attraction to Women,\" _Journal of Personality and Social Psychology_ 95 (2008): 1150\u20131164.\n\n. Douglas T. Kendrick and Sara E. Gutierres, \"Contrast Effects and Judgments of Physical Attractiveness: When Beauty Becomes a Social Problem,\" _Journal of Personality and Social Psychology_ 38, no. 1 (1980): 131\u2013140.\n\n. S. Gary Garwood, Lewis Cox, Valerie Kaplan, Neal Wasserman, and Jefferson L. Sulzer, \"Beauty Is Only 'Name' Deep: The Effect of First-Name on Rating Physical Attraction,\" _Journal of Applied Social Psychology_ 10, no. 5 (1980): 431\u2013435.\n\n. Amy Perfors, \"What's in a Name? The Effect of Sounds Symbolism on Perception of Facial Attractiveness,\" _Journal of Applied Social Psychology_ 10, no. 5 (1980): 431\u2013435.\n\n. John A. Bargh, Katelyn Y. A. McKenna, and Grainne M. Fitzsimons, \"Can You See the Real Me? Activation and Expression of the 'True Self' on the Internet.\" _Journal of Social Issues_ 58, no. 1 (2002): 34\u201380.\n\n. Edmond Rostad, _Cyrano de Bergerac_ , Act 3.\n\n### Chapter 5\n\n. N. B. McCormick, T. Perper, and A. J. Jones, \"Bar Hopping as Science: Results and Methodological Issues Related to Naturalistic Observational Research in Bars,\" Paper presented at the Eastern Region Conference of the Society for the Scientific Study of Sex, Philadelphia, April 1983.\n\n. Chris L. Kleinke, Frederick B. Meeker, and Richard A. Staneski, \"Preference for Opening Lines: Comparing Ratings by Men and Women,\" _Sex Roles_ 15, no. 11\u201312 (1986): 585\u2013600.\n\n. C. L. Apicella, D. R. Feinbert, and F. W. Marlowe, \"Voice Pitch Predicts Reproductive Success in Male Hunter-Gatherers,\" _Biology Letters_ 3, no. 6 (2007): 682\u2013684.\n\n. Jacquie D. Vorauer, Jessica J. Cameron, John G. Holmes, and Deanna G. Pearce, \"Invisible Overtures: Fears of Rejection and the Signal Amplification Bias,\" _Journal of Personality and Social Psychology_ 84, no. 4 (April 2003): 793\u2013812.\n\n. Timothy Perper, _Sex Signals: The Biology of Love_ (Philadelphia: ISI Press, 1985).\n\n. Debra G. Walsh and Jay Hewitt, \"Giving Men the Come-On: Effect of Eye Contact and Smiling in a Bar Environment,\" _Perceptual and Motor Skills_ 61, no. 3, pt. 1 (1985): 873\u2013874.\n\n. Michael R. Cunningham, \"Reactions to Heterosexual Opening Gambits: Female Selectiveness and Male Responsiveness,\" _Personality and Social Psychology Bulletin_ 15, no.1 (March 1989): 27\u201341.\n\n. Susan Young, \"Brain Chip Helps Quadriplegics Move Robotic Arms with Their Thoughts,\" _Technology Review_ , May 16, 2002, <http:\/\/www.technologyreview.com\/news\/427939\/brain-chip-helps-quadriplegics-move-robotic-arms\/>.\n\n. Morgan Worthy, Albert L. Gary, and Gay M. Kahn, \"Self-Disclosure as an Exchange Process,\" _Journal of Personality and Social Psychology_ 13, no. 1 (1969): 59\u201363.\n\n. Jean-Phillippe Laurenceau, Lisa Feldman Barrett, and Paula R. Pietromonaco, \"Intimacy as an Interpersonal Process. The Importance of Self Disclosure, Partner Disclosure, and Perceived Partner Responsiveness in Interpersonal Exchanges,\" _Journal of Personality and Social Psychology_ 74, no. 5 (1989): 1239\u20131251.\n\n. Conor Dougherty, \"Young Women's Pay Exceeds Male Peers',\" _Wall Street Journal_ , September 1, 2010, <http:\/\/online.wsj.com\/article\/SB10001424052748704421104575463790770831192.html>.\n\n. Brenda Major, Patricia I. Carrington, and Peter J. D. Carnevale, \"Physical Attractiveness and Self Esteem: Attributions for Praise from an Other Sex Evaluator,\" _Personality and Social Psychology Bulletin_ 10, no. 1 (1984): 43\u201350.\n\n. David M. Buss and Michael Barnes, \"Preferences in Human Mate Selection,\" _Journal of Personality and Social Psychology_ 50 (1989): 559\u2013570.\n\n. David R. Shaffer and Linda J. Pegalis, \"Gender Role Orientation and Prospect of Future Interaction as Determinants of Self-Disclosure,\" _Personality and Social Psychology Bulletin_ 22, no. 5 (May 1996): 495\u2013506.\n\n. Dennis P. Carmody and Michael Lewis, \"Brain Activation When Hearing One's Own and Others' Names,\" _Brain Research_ 1116, no. 1 (October 20, 2006): 153\u2013158.\n\n### Chapter 6\n\n. D. G. Dutton, and A. P. Aron, \"Some Evidence for Heightened Sexual Attraction under Conditions of High Anxiety,\" _Journal of Personality and Social Psychology_ 30, no. 4 (1974): 510\u2013517.\n\n. Richard Driscoll, Keith E. Davis, and Milton E. Lipetz, \"Parental Interference and Romantic Love: The Romeo and Juliet Effect,\" _Journal of Personality and Social Psychology_ 24, no. 1 (1972): 1\u201310.\n\n. Dolf Zillmann, \"Transfer of Excitation in Emotional Behavior,\" in _Social Psychophysiology: A Sourcebook_ , edited by John T. Cacioppo and Richard E. Petty, 215\u2013240 (New York: Guilford Press, 1983).\n\n. Richard Bandler and John Grinder, _Frogs into Princes: Neuro Linguistic Programming_ (Moab, UT: Real People Press, 1979).\n\n. A. H. Maslow and N. L. Mintz, \"Effects of Aesthetic Surroundings,\" _Journal of Psychology_ 41 (1956): 247\u2013254.\n\n. Shelly E. Taylor, Laura Cousino Klien, Brian P. Lewis, Tara L. Gruenewald, Regan A. R. Gurung, and John A Updegraff, \"Biobehavioral Responses to Stress in Females: Tend-and-Befriend, Not Fight-or-Flight,\" _Psychological Review_ 107, no. 3 (2000): 411\u2013429.\n\n. Deborah Tannen, _You Just Don't Understand: Women and Men in Conversation_ (New York: William Morrow, 1990).\n\n. Audrey Nelson, \"The Strong Silent Type: The Male Advantage,\" _Psychology Today_ , April 23, 2011, <http:\/\/www.psychologytoday.com\/blog\/he-speaks-she-speaks\/201104\/the-strong-silent-type-the-male-advantage>.\n\n. Azim Eiman and Dean Mobbs, Booil Jo, Vinod Menon, Allan L. Reiss, \"Sex Differences in Brain Activation Elicited by Humor,\" _Proceedings of the National Academy of Sciences_ 102, no. 45 (November 8, 2005): 16496\u201316501.\n\n. Willow Lawson, \"Humor's Sexual Side,\" _Psychology Today_ , September 1, 2005, <http:\/\/www.psychologytoday.com\/articles\/200508\/humors-sexual-side>.\n\n. Eric R. Bressler and Sigal Balshine, \"The Influence of Humor on Desirability,\" _Evolution and Human Behavior_ 27, no. 1 (2006): 29\u201339.\n\n### Chapter 7\n\n. Daniel G. Amen, Daniel G., M.D., Sex on the Brain: 12 Lessons to Enhance Your Love Life. 2008. (New York: Crown Publishing, Harmony, 2008).\n\n. Ibid.\n\n. Jos\u00e9 L. Tlachi-L\u00f3pez, Jos\u00e9 R. Eguibar, Alonso Fern\u00e1ndez-Guasti, Rosa Ang\u00e9lica Lucio, \"Copulation and Ejaculation in Male Rats Under Sexual Satiety and the Coolidge Effect,\" _Physiology and Behavior_ 106, no. 5 (July 2012): 626\u2013630.\n\n. Ibid.\n\n. Gordon Bermant and Juan M. Davidson, _Biological Bases of Sexual Behavior_ (New York: Harper and Row, 1974).\n\n. Gary Stix, \"Only Epilepsy Brings More Activity to Women's Brains Than Does 'Self-Stimulation' to Orgasm,\" _Scientific American_ , November 15, 2011, <http:\/\/blogs.scientificamerican.com\/observations\/2011\/11\/15\/only-epilepsy-brings-more-activity-to-womens-brains-than-does-self-stimulation-to-orgasm\/>.\n\n. Helen Fisher, _Why We Love: The Nature and Chemistry of Romantic Love_ (New York: Henry Holt and Company, 2004).\n\n. Ibid.\n\n. M. Kosfeld, M. Heinrichs, P. J. Zak, U. Fischbacher, and others. Fehr, \"Oxytocin Increases Trust in Humans,\" _Nature_ 435, no. 7042 (June 2, 2005): 673\u2013676.\n\n. Ibid.\n\n. Pamela C. Regan and Ellen Berscheid, _Love and Lust: What We Know about Human Sexual Desire_ (Thousand Oaks, CA: Sage Publications, 1999).\n\n. Deborah Blum, _Sex on the Brain: The Biological Differences Between Men and Women_ (New York: Penguin Books, 1998).\n\n. Anne Moir and David Jessel, _Brain Sex: The Real Difference Between Men and Women_ (New York: Dell Publishing, 1991)\n\n. Louann Brizendine, _The Female Brain_ (New York: Broadway Books, 2006).\n\n. Pierce J. Howard, _The Owner's Manual for the Brain: Everyday Applications from Mind-Brain Research_ , 3rd ed. (Austin, TX: Bard Press, 2006).\n\n. R. L. Doty, S. Applebaum, H. Zusho and R. G. Settle, \"Sex Differences in Odor Identification Ability,\" _Neuropsychologia_ 23, no. 5 (1985): 667\u2013672.\n\n. G. Holstege, J. R. Georgiadis, A. M Paans, L. C. Meiners and F. H. van der Graaf, \"Brain Activation During Human Male Ejaculation,\" _Journal of Neuroscience_ 23, no. 27 (October 8, 2003): 9158\u20139193.\n\n. Meredith L. Chivers, Michael C. Seto, and Ray Blanchard, \"Gender and Sexual Orientation Differences in Sexual Response to Sexual Activities Versus Gender of Actors in Sexual Films,\" _Journal of Personality and Social Psychology_ 93, no. 6 (2007): 1108\u20131121.\n\n. B. A. Shaywitz, Sally E. Shaywitz, Kenneth R. Pugh, R. T. Constanble, and P. Skudlarski, \"Sex Differences in the Functional Organization of the Brain for Language,\" _Nature_ 373, no. 6515 (February 16, 1995): 607\u2013609.\n\n. Brizendine, _The Female Brain_.\n\n. Roy Levin, and Cindy Meston, \"Nipple\/Breast Stimulation and Sexual Arousal in Young Men and Women,\" _Journal of Sexual Medicine_ 3, no. 3 (2006): 450\u2013454.\n\n. Barry R. Komisaruk and Nan Wise, \"Women's Clitoris, Vagina, and Cervix Mapped on the Sensory Cortex: fMRI Evidence,\" _Journal of Sexual Medicine_ 8, no. 10 (2011): 2822\u20132830.\n\n. A. Kilchevsky, Y. Vardi, L. Lowenstein, and I. Gruenwald, \"Is the Female G-Spot Truly a Distinct Anatomic Entity?\" _Journal of Sex Medicine_ 9, no. 3 (2012): 719\u2013726.\n\n. \"30 Things To Do With a Naked Man,\" _Cosmopolitan_ , <http:\/\/www.cosmopolitan.com\/sex-love\/tips-moves\/how-to-turn-him-on?click=main_sr#slide-1>.\n\n. Bruce J. Ellis and Donald Symons, \"Sex Differences in Sexual Fantasy: An Evolutionary Psychological Approach,\" _The Journal of Sex Research_ 27, no. 4 (1990): 527\u2013555.\n\n. Harold Leitenberg and Kris Henning, \"Sexual Fantasy,\" _Psychological Bulletin_ 117, no. 3 (May 1995): 469\u2013496.\n\n. Ibid.\n\n. Ellis and Symons, \"Sex Differences in Sexual Fantasy.\"\n\n. Brizendine, _The Female Brain_.\n\n. V. Apanius, D. Penn, P. R. Slev, L. R. Ruff, and W. K. Potts, \"The Nature of Selection on the Major Histocompatibility Complex,\" _Critical Reviews in Immunology_ 17, no. 2 (1997): 179\u2013224.\n\n. M. Milinski, \"The Major Histocompatibility Complex: Sexual Selection and Mate Choice,\" _Annual Review of Ecology, Evolution, and Systematics_ 37 (2006): 159\u2013186.\n\n. Claus Wedekind, Thomas Seebeck, Florence Bettens, and Alexander J. Paepke, \"MHC-Dependent Mate Preferences in Humans,\" _Proceedings of the Royal Society of London: Biological Sciences_ 260, no. 1359 (June 22, 1995): 245\u2013249.\n\n. Helen E. Fisher, \"The Biology of Attraction,\" _Psychology Today_ , April 1, 1993, <http:\/\/www.psychologytoday.com\/articles\/199303\/the-biology-attraction>.\n\n. \"Bizarre Animal Mating Rituals,\" Virgin Media, <http:\/\/www.virgin media.com\/digital\/features\/bizarre-animal-mating-rituals.php?ssid=9>.\n\n. John Tierney, \"The Threatening Scent of Fertile Women,\" _New York Times_ , February 21, 2011, <http:\/\/www.nytimes.com\/2011\/02\/22\/science\/22tier.html>.\n\n. G. Miller, J. M. Tybur, B. D. Jordan, \"Ovulatory Cycle Effects on Tip Earnings by Lap Dancers: Economic Evidence for Human Estrus?\" _Evolution and Human Behavior_ 28, no. 6 (2007): 375\u2013381.\n\n. S. C. Roberts, L. M. Gosling, V. Carter, and M. Petrie, \"MHC-Correlated Odor Preferences in Humans and the Use of Oral Contraceptives,\" _Proceedings of the Royal Society of Biological Sciences_ 275, no. 1652 (December 7, 2008): 2715\u20132722.\n\n### Chapter 8\n\n. John Gray, _Men Are from Mars, Women Are from Venus: A Practical Guide for Improving Communication and Getting What You Want in Your Relationships_ (New York: Harper Collins, 1993).\n\n. Pierce J. Howard, _The Owner's Manual for the Brain: Everyday Applications from Mind-Brain Research_ , 3rd ed. (Austin, TX: Bard Press, 2006).\n\n. Ibid.\n\n. Joseph Rhawn, _NeuroTheology: Brain, Science, Spirituality, Religious Experience_ (San Jose, CA: University Press, 2003). Italics added.\n\n. Zeenat F. Zaidi, \"Gender Differences in Human Brain: A Review,\" _The Open Anatomy Journal_ 2 (2010): 37\u201355.\n\n. Deborah Tannen, _You Just Don't Understand: Women and Men in Conversation_ (New York: William Morrow, 1990).\n\n. Zaidi, \"Gender Differences in Human Brain.\"\n\n. Paul Ekman, _Emotions Revealed: Recognizing Faces and Feelings to Improve Communication and Emotional Life_ , 2nd ed. (New York: Henry Holt and Co., 2007).\n\n. Ibid.\n\n. Howard, _The Owner's Manual for the Brain_.\n\n. Louann Brizendine, _The Male Brain_ (New York: Broadway Books, 2010).\n\n. Howard, _The Owner's Manual for the Brain_.\n\n### Chapter 9\n\n. Susan S. Hendrick and Clyde Hendrick, \"Gender Differences and Similarities in Sex and Love,\" _Personal Relationships_ 2 (1995): 55\u201365.\n\n. Donn Byrne, \"Interpersonal Attraction and Attitude Similarity,\" _Journal of Abnormal Social Psychology_ 62 (May 1961): 713\u2013715.\n\n. Theodore M. Newcomb, _The Acquaintance Process_ (New York: Holt, Rinehart and Wilson, 1961).\n\n. Byrne, \"Interpersonal Attraction and Attitude Similarity.\n\n. John Mordechai Gottman and Nan Silver, _Why Marriages Succeed or Fail_ (New York: Simon and Schuster, 1994).\n\n. G. J. Fletcher, J. A. Simpson, G. Thomas, and L. Giles, \"Ideals in Intimate Relationships,\" _Journal of Personality and Social Psychology_ 76, no. 1 (1999): 72\u201389.\n\n. John W. Thibaut and Harold H. Kellel, _The Social Psychology of Groups_ (New York: Wiley, 1959).\n\n. Mario Mikulincer and Gail S. Goodman, eds., _Dynamics of Romantic Love: Attachment, Caregiving, and Sex_ (New York: Guilford Press, 2006).\n\n. A. Aron, E. N. Aron, and C. Norman, \"The Self Expansion Model of Motivation and Cognition in Close Relationships and Beyond,\" In _Blackwell Handbook in Social Psychology, vol. 2: Interpersonal Processes_ , edited by M. Clark and G. Fletcher, 478\u2013501 (Oxford: Blackwell, 2001).\n\n. Tara Parker-Pope, \"The Happy Marriage Is the 'Me' Marriage,\" _New York Times_ , December 31, 2010, <http:\/\/www.nytimes.com\/2011\/01\/02\/week inreview\/02parkerpope.html>.\n\n. Jay Belsky, Laurence Steinberg, and Patricia Draper, \"Childhood Experience, Interpersonal Development, and Reproductive Strategy: An Evolutionary Theory of Socialization,\" _Child Development_ 62, no. 4 (1991): 647\u2013670.\n\n. Ibid.\n\n. Pedro M. Pereyra, Weixian Zhang, and Matthias Schmidt, and Laurence E. Becker, \"Development of Myelinated and Unmyelinated Fibers of Human Vagus Nerve During the First Year of Life,\" _Journal of the Neurological Sciences_ 110, no. 1\u20132 (July 1992): 107\u2013113.\n\n. J. N. Giedd, \"Structural Magnetic Resonance Imaging of the Adolescent Brain,\" _Annals of the New York Academy of Sciences_ 1021 (June 2004): 77\u201385.\n\n. H. E. Fisher, A. Aron, D. Mashek, H. Li, and L. L. Brown, \"Defining the Brain Systems of Lust, Romantic Attraction, and Attachment,\" _Archives of Sexual Behavior_ 31, no. 5 (2002): 413\u2013419. Italics added.\n\n### Chapter 10\n\n. \"The Fair Youth of the Shakespeare Sonnets,\" The Monument: Shakespeare's Sonnets, <http:\/\/www.shakespearesmonument.com\/page10.html>.\n\n. Louis de Berni\u00e8res, _Corelli's Mandolin_ (New York: Vintage Books, 1994).\n\n. Sadie Leder, \"From Bride to Blues: Examining the Prevalence of Post-Nuptial Depression,\" Science of Relationships, <http:\/\/www.scienceofrelationships.com\/home\/2012\/7\/16\/from-bride-to-blues-examining-the-prevalence-of-post-nuptial.html>.\n\n. D. Marazziti and D. Canale, \"Hormonal Changes When Falling in Love,\" _Psychoneuroendocrinology_ 29, no. 7 (August 2004): 931\u2013936.\n\n. Robert Browning, \"Grow Old Along with Me.\"\n\n. Thomas T. Insel and C. Sue Carter, \"The Monogamous Brain,\" _Natural History_ 104, no. 8 (1995): 12\u201314; Mary M. Cho, A. Courtney DeVries, Jessie R. Williams, and C. Sue Carter, \"The Effects of Oxytocin and Vasopressin on Partner Preferences in Male and Female Prairie Voles,\" _Behavioral Neuroscience_ 113, no. 5 (1999): 1071\u20131079.\n\n. Ibid.\n\n. Ibid.\n\n. James T. Winslow, Nick Hastings, C. Sue Carter, Carroll R. Harbaugh, and Thomas R. Insel, \"A Role for Central Vasopressin in Pair Bonding in Monogamous Prairie Voles,\" _Letters to Nature_ , <http:\/\/research.yerkes.emory.edu\/Young\/Getz\/1993%20Winslow%20N.pdf>.\n\n. Thomas R. Insel, and Lawrence E. Shapiroirie, \"Oxytocin Receptor Distribution Reflects Social Organization in Monogamous and Polygamous Voles,\" _Proceedings of the National Academy of Sciences_ 89, no. 13 (1992): 5981\u20135985.\n\n. Phyllis K. Davis, _The Power of Touch: The Basis for Survival, Health, Intimacy, and Emotional Well-Being_ (Carlsbad, CA: Hay House, 1999).\n\n. Allan Pease and Barbara Pease, _The Definitive Book of Body Language_ (New York: Bantam Books, 2006); James. E. Sheridan, \"Marriage Advice: Touch Is a Key to a Truly Good Marriage,\" _News Sentinel_ , October 25, 2011, <http:\/\/www.news-sentinel.com\/apps\/pbcs.dll\/article?AID=\/20111025\/NEWS01\/310269994>.\n\n. Diane Ackerman, \"The Brain on Love,\" _New York Times_ , March 24, 2012, <http:\/\/opinionator.blogs.nytimes.com\/2012\/03\/24\/the-brain-on-love\/>.\n\n. Michael W. Kraus, Cassy Huang, and Dacher Keltner, \"Tactile Communication, Cooperation, and Performance: An Ethological Study of the NBA,\" <http:\/\/socrates.berkeley.edu\/~keltner\/publications\/kraus.huang.keltner.2010.pdf>.\n\n. G. Strong and A. Aron, \"The Effect of Shared Participation in Novel and Challenging Activities on Experienced Relationship Quality: Is It Mediated by High Positive Affect?\" In _Self and Relationships: Connecting Intrapersonal and Interpersonal Processes_ , edited by Kathleen D. Vohs and Eli J. Finkel, 342\u2013359 (New York: Guilford Press, 2006).\n\n. Robert R. Provine, \"Laughter,\" _American Scientist_ 84, no. 1 (January\u2013February 1996): 38\u201347.\n\n. Rod A. Martin, _The Psychology of Humor: An Integrative Approach_ (Waltham, MA: Elsevier Academic Press, 2006).\n\n. Elaine Walster and Ellen Berscheid, \"Adrenaline Makes the Heart Grow Fonder,\" _Psychology Today_ , June 1971, 47\u201362; David M. Buss, _The Dangerous Passion: Why Jealousy Is as Necessary as Love and Sex_ (New York: Free Press, 2000).\n\n. Lee T. Gettler, Thomas W. McDade, Alan B. Feranil and Christopher W. Kuzawa, \"Longitudinal Evidence That Fatherhood Decreases Testosterone in Human Males,\" P _roceedings of the National Academy of Sciences of the United States of America_ , <http:\/\/www.pnas.org\/content\/early\/2011\/09\/02\/1105403108.full.pdf+html>.\n\n. Sandra J. Berg and Katherine E. Wynne-Edwards, \"Changes in Testosterone, Cortisol, and Estradiol Levels in Men Becoming Fathers,\" _Mayo Clinic Proceedings_ 76, no. 6 (2001): 582\u2013592.\n\n. Helen Fisher, _Why We Love: The Nature and Chemistry of Romantic Love_ (New York: Henry Holt and Company, 2004).\n\n. G. S. Berns, S. M. McClure, G. Pagnoni, and P. R. Montague, \"Predictability Modulates Human Brain Response to Reward,\" _Journal of Neuroscience_ 21, no. 8 (April 2001): 2793\u20132798.\n\n. Bianca P. Acevedo and Arthur Aron, \"Does a Long-Term Relationship Kill Romantic Love?\" _Review of General Psychology_ 13, no. 1 (2009): 59\u201365.\n\n. Wibke Blaicher, Doris Gruber, Christian Bieglmayer, Alex M. Blaicher, and Wolfgang Knogler, \"The Role of Oxytocin in Relation to Female Sexual Arousal,\" _Gynecologic and Obstetric Investigation_ 47, no. 2 (1999): 125\u2013126.\n\n. \"Ambergris, aka Floating Gold,\" squidoo, <http:\/\/www.squidoo.com\/ambergrease>.\n\n. Louann Brizendine, _The Male Brain_ (New York: Broadway Books, 2010).\n\n. Simon LeVay, _The Sexual Brain_ (Cambridge, MA: MIT Press, 1994).\n\n. M. Kosfeld, M. Heinrichs, P. J. Zak, U. Fischbacher, and others. Fehr, \"Oxytocin Increases Trust in Humans,\" _Nature_ 435, no. 7042 (June 2, 2005): 673\u2013676.\n\n. Paula Tucker and Arthur Aron, \"Passionate Love and Marital Satisfaction at Key Transition Points in the Family Life Cycle,\" _Journal of Social and Clinical Psychology_ 12, no. 2 (1993): 135\u2013147.\n\n. Howard, _The Owner's Manual for the Brain_.\n\n. Fisher, _Why We Love_.\n\n## About the Author\n\nLeil Lowndes is an internationally recognized communications expert who specializes in the subconscious interactions and subliminal approaches that unknowingly take place in all interpersonal communications. She conducts seminars for major corporations, associations, universities, and the public.\n\nShe is the author of ten books, including the top-selling _How to Talk to Anyone, How to Instantly Connect with Anyone, How to Make Anyone Fall in Love with You, Undercover Sex Signals, How to be a People Magnet_ , and _Good-bye to Shy_.\n\nPrior to her work in communications, Ms. Lowndes was founder and director of _The Project_ , a New York City\u2013based not-for-profit organization, which conducted personal relationship research and counseling. She is a member of the American Association of Sex Educators, Counselors, and Therapists.\n\nBased in New York City, Ms. Lowndes has been the guest communications expert on hundreds of television and radio programs.\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\nThis is a work of fiction. Names, characters, places, and incidents either are the product of the author's imagination or are used fictitiously. Any resemblance to actual persons, living or dead, events, or locales is entirely coincidental.\n\nText copyright \u00a9 2018 by Wonderline Productions LLC\n\nCover art copyright \u00a9 2018 by Victor Tongdee\n\nAll rights reserved. Published in the United States by Delacorte Press, an imprint of Random House Children's Books, a division of Penguin Random House LLC, New York.\n\nDelacorte Press is a registered trademark and the colophon is a trademark of Penguin Random House LLC.\n\nVisit us on the Web\n\nEducators and librarians, for a variety of teaching tools, visit us at RHTeachersLibrarians.com\n\nLibrary of Congress Cataloging-in-Publication Data\n\nName: Pessl, Marisha, author.\n\nTitle: Neverworld wake \/ Marisha Pessl.\n\nDescription: First edition. | New York : Delacorte Press, [2018] | Summary: \"A group of teens who all attended the same elite prep school reunite a year after graduation. After a night on the town, the teens are faced with an impossible choice\u2014only one of them can live and the decision must be unanimous\" \u2014 Provided by publisher.\n\nIdentifiers: LCCN 2017038509 (print) | LCCN 2017052001 (ebook) | ISBN 978-0-399-55395-0 (ebook) | ISBN 978-0-399-55392-9 (trade hardcover) | ISBN 978-0-399-55393-6 (library binding) | ISBN 978-0-525-64445-3 (intl. tr. pbk.)\n\nSubjects: | CYAC: Death\u2014Fiction. | Friendship\u2014Fiction. | Secrets\u2014Fiction. | Science fiction.\n\nClassification: LCC PZ7.1.P4479 (ebook) | LCC PZ7.1.P4479 Nev 2018 (print) | DDC [Fic]\u2014dc23\n\nEbook ISBN 9780399553950\n\nRandom House Children's Books supports the First Amendment and celebrates the right to read.\n\nv5.2\n\nep\n\n# Contents\n\nCover\n\nTitle Page\n\nCopyright\n\nDedication\n\nEpigraph\n\nPart 1\n\nChapter 1\n\nChapter 2\n\nChapter 3\n\nChapter 4\n\nChapter 5\n\nChapter 6\n\nChapter 7\n\nChapter 8\n\nChapter 9\n\nPart 2\n\nChapter 10\n\nChapter 11\n\nChapter 12\n\nChapter 13\n\nChapter 14\n\nChapter 15\n\nChapter 16\n\nChapter 17\n\nChapter 18\n\nPart 3\n\nChapter 19\n\nChapter 20\n\nChapter 21\n\nChapter 22\n\nChapter 23\n\nChapter 24\n\nChapter 25\n\nChapter 26\n\nChapter 27\n\nAcknowledgments\n\nAbout the Author\nFor David\n> Sometimes there are no answers,\n> \n> Sometimes you find love,\n> \n> Sometimes the dark has teeth,\n> \n> Sometimes it hides doves.\n> \n> The one thing you can expect in life,\n> \n> As you step down its twisted road,\n> \n> Is that you will be speechless.\n> \n> Then? Ask someone who knows.\n> \n> \u2014J. C. Gossamer Madwick, \n> \n> The Dark House at Elsewhere Bend\n\nI hadn't spoken to Whitley Lansing\u2014or any of them\u2014in over a year.\n\nWhen her text arrived after my last final, it felt inevitable, like a comet tearing through the night sky, hinting of fate.\n\n> Too long. WTF. #notcool. Sorry. My Tourette's again. How was your freshman year? Amazing? Awful?\n> \n> Seriously. We miss you.\n> \n> Breaking the silence bc the gang is heading to Wincroft for my bday. The Linda will be in Mallorca & ESS Burt is getting married in St. Bart's for the 3rd time. (Vegan yogi.) So it's ours for the weekend. Like yesteryear.\n> \n> Can you come? What do you say Bumblebee?\n> \n> Carpe noctem.\n\nSeize the night.\n\nShe was the only girl I knew who surveyed everybody like a leather-clad Dior model and rattled off Latin like it was her native language.\n\n\"How was your exam?\" my mom asked when she picked me up.\n\n\"I confused Socrates with Plato and ran out of time during the essay,\" I said, pulling on my seat belt.\n\n\"I'm sure you did great.\" She smiled, a careful look. \"Anything else we need to do?\"\n\nI shook my head.\n\nMy dad and I had already cleared out my dorm room. I'd returned my textbooks to the student union to get the 30 percent off for next year. My roommate had been a girl from New Haven named Casey who'd gone home to see her boyfriend every weekend. I'd barely seen her since orientation.\n\nThe end of my freshman year at Emerson College had just come and gone with the indifferent silence usually reserved for a going-out-of-business sale at a mini-mall.\n\n\"Something dark's a-brewin',\" Jim would have told me.\n\n* * *\n\n\u2014\n\nI had no plans all summer, except to work alongside my parents at the Captain's Crow.\n\nThe Captain's Crow\u2014the Crow, it's called by locals\u2014is the seaside caf\u00e9 and ice cream parlor my family owns in Watch Hill, Rhode Island, the tiny coastal village where I grew up.\n\nWatch Hill, Rhode Island. Population: You Know Everyone.\n\nMy great-grandfather Burn Hartley opened the parlor in 1885, when Watch Hill was little more than a craggy hamlet where whaling captains came to shake off their sea legs and hold their children for the first time before taking off again for the Atlantic's Great Unknowns. Burn's framed pencil portrait hangs over the entrance, revealing him to have the mad glare of some dead genius writer, or a world explorer who never came home from the Arctic. The truth is, though, he could barely read, preferred familiar faces to strange ones and dry land to the sea. All he ever did was run our little dockside restaurant his whole life, and perfect the recipe for the best clam chowder in the world.\n\nAll summer I scooped ice cream for tan teenagers in flip-flops and pastel sweaters. They came and went in big skittish groups like schools of fish. I made cheeseburgers and tuna melts, coleslaw and milk shakes. I swept away sand dusting the black-and-white-checkered floor. I threw out napkins, ketchup packets, salt packets, over-21 wristbands, Del's Frozen Lemonade cups, deep-sea fishing party boat brochures. I put lost cell phones beside the register so they could be easily found when the panic-stricken owners came barging inside: \"I lost my...Oh...thank you, you're the best!\" I cleaned up the torn blue tickets from the 1893 saltwater carousel, located just a few doors down by the beach, which featured faded faceless mermaids to ride, not horses. Watch Hill's greatest claim to fame was that Eleanor Roosevelt had been photographed riding a redhead with a turquoise tail sidesaddle. (It was a town joke how put out she looked in the shot, how uncomfortable and buried alive under her plate-tectonic layers of ruffled skirt.)\n\nI cleaned the barbecue sauce off the garbage cans, the melted Wreck Rummage off the tables (Wreck Rummage was every kid's favorite ice cream flavor, a mash-up of cookie dough, walnuts, cake batter, and dark chocolate nuggets). I Cloroxed and Fantasticked and Mr. Cleaned the windows and counters and doorknobs. I dusted the brine off the mussels and the clams, polishing every one like a gemstone dealer obsessively inspecting emeralds. Most days I rose at five and went with my dad to pick out the day's seafood when the fishing boats came in, inspecting crab legs and fluke, oysters and bass, running my hands over their tapping legs and claws, barnacles and iridescent bellies. I composed song lyrics for a soundtrack to a made-up movie called _Lola Anderson's Highway Robbery,_ drawing words, rhymes, faces, and hands on napkins and take-out menus, tossing them in the trash before anyone saw them. I attended grief support group for adolescents at the North Stonington Community Center. There was only one other kid in attendance, a silent boy named Turks whose dad had died from ALS. After two meetings he never returned, leaving me alone with the counselor, a jittery woman named Deb who wore pantsuits and wielded a three-inch-thick book called _Grief Management for Young People._\n\n\" 'The purpose of this exercise is to construct a positive meaning around the lost relationship,' \" she read from chapter seven, handing me a Goodbye Letter worksheet. \" 'On this page, write a note to your lost loved one, detailing fond memories, hopes, and any final questions.' \"\n\nSlapping a chewed pen that read TABEEGO ISLAND RESORTS on my desk, she left. I could hear her on the phone out in the hall, arguing with someone named Barry, asking him why he didn't come home last night.\n\nI drew a screeching hawk on the Goodbye Letter, with lyrics to a made-up Japanese animated film about a forgotten thought called _Lost in a Head._\n\nThen I slipped out the fire exit and never went back.\n\nI taught Sleepy Sam (giant yawn of a teenager from England visiting his American dad) how to make clam cakes and the perfect grilled cheese. _Grill on medium, butter, four minutes a side, six slices of Vermont sharp cheddar, two of fontina._ For July Fourth, he invited me to a party at a friend of a friend's. To his shock, I actually showed. I stood by a floor lamp with a warm beer, listening to talk about guitar lessons and Zach Galifianakis, trying to find the right moment to escape.\n\n\" _That,_ by the way, is Bee,\" said Sleepy Sam. \"She does actually speak, I swear.\"\n\nI didn't mention Whitley's text to anyone, though it was always in the back of my mind.\n\n* * *\n\n\u2014\n\nIt was the brand-new way-too-extravagant dress I'd bought but never taken out of the bag. I just left it there in the back of my closet, folded in tissue paper with the receipt, the tags still on, with intention of returning it.\n\nYet there was still the remote possibility I'd find the courage to put it on.\n\nI knew the weekend of her birthday like I knew my own: August 30.\n\nIt was a Friday. The big event of the day had been the appearance of a stray dog wandering Main Street. It had no tags and the haunted look of a prisoner of war. He was gray, shaggy, and startled with every attempt to pet him. A honk sent him skidding into the garbage cans behind the Captain's Crow.\n\n\"See that yellow salt-bed mud on his back paws? That's from the west side of Nickybogg Creek,\" announced Officer Locke, thrilled to have a mystery on his hands, his first of the year.\n\nThat stray dog had been the talk all that day\u2014what to do with him, where he'd been\u2014and it was only much later that I found my mind going back to that dog drifting into town out of the blue. I wondered if he was some kind of sign, a warning that something terrible was coming, that I should not take the much-exalted and mysterious Road Less Traveled, but the one well trod, wide-open, and brightly lit, the road I knew.\n\nBy then it was too late. The sun had set. Sleepy Sam was gone. I'd overturned the caf\u00e9 chairs and put them on the tables. I'd hauled out the trash. And anyway, that flew in the face of human nature. No one ever heeded a warning sign when it came.\n\nMy mom and dad assumed I was joining them at the Dreamland Theater in Westerly for the screwball comedy classics marathon, like I did every Friday.\n\n\"Actually, I made plans tonight,\" I said.\n\nMy dad was thrilled. \"Really, Bumble? That's great.\"\n\n\"I'm driving up to Wincroft.\"\n\nThey fell silent. My mom had just flipped the Closed sign in the window, and she turned, wrapping her cardigan around herself, shivering even though it was seventy-five degrees out.\n\n\"How long have you known about this?\" she asked.\n\n\"Not long. I'll be careful. I'll be back by midnight. They're up there for Whitley's birthday. I think it'll be good for me to see them.\"\n\n\"That's a long way to drive in the dark,\" said my dad.\n\nMy mom looked like I'd been given a prognosis of six weeks left to live. Sometimes when she got really upset, she chewed an imaginary piece of gum. She was doing that now.\n\n\"Part of the grieving process is confronting the past,\" I said.\n\n\"That's not the point. I\u2014\"\n\n\"It's all right, Victoria.\" My dad put a hand on her shoulder.\n\n\"But Dr. Quentin said not to put yourself in stressful situations that\u2014\"\n\n\"We've established that Dr. Quentin is an idiot,\" I said.\n\n\"Dr. Quentin is indeed an idiot,\" said my dad with a regretful nod. \"The fact that his name is one-half of a state prison should have been a red flag.\"\n\n\"You know I don't like it when you two gang up on me,\" said my mom.\n\nAt that moment, someone\u2014some red-faced weekender in seersucker shorts who'd had too many stouts at O'Malligan's\u2014tried to open the door.\n\n\"We're closed,\" my mom snapped.\n\n* * *\n\n\u2014\n\nThat was how I came to be driving my dad's ancient green Dodge RAM with the emphysema muffler fifty miles up the Rhode Island coastline.\n\n_Wincroft._\n\nThe name sounded like something out of a windswept novel filled with ghosts and madmen. The mansion was a sprawling collection of red brick, turrets, gardens, and crow gargoyles, built in the 1930s by a Great White Hunter who'd supposedly called Hemingway and Lawrence of Arabia his friends. He had traveled the world killing beautiful creatures, and thus Wincroft, his seaside estate, had never been lived in more than a few weeks in sixty years. When Whitley's weird ex-second-stepdad, Burt\u2014commonly called E.S.S. Burt\u2014bought it in foreclosure in the 1980s, he gut renovated the interiors in an unfortunate style Whitley called \"if Madonna threw up all over Cyndi Lauper.\"\n\nStill, it wasn't unusual to open a chest of drawers in the attic, or a musty steamer trunk, and find photographs of strangers gripping rifles and wearing fox furs or some weird piece of taxidermy\u2014a ferret, red frog, or rodent of unknown species. This gave every visit to Wincroft the mysterious feel of being on an archaeological expedition, as if all around us, inside the floors, walls, and ceiling, some lost civilization was waiting to be unearthed.\n\n\"We are our junk,\" said Jim once, pulling a taxidermy lizard out of a shoe box.\n\nLeaving the interstate, the road to get there turned corkscrewed and dizzying, as if trying to shake you. The coast of Rhode Island\u2014not the infamously uptight Newport part, with the stiff cliffs and colossal mansions smugly staring down at the tiny sailboats salting the harbor, but the rest of it\u2014was rough and tumbledown, laid-back and sunburnt. It was an old homeless beachcomber in a washed-out T-shirt who couldn't remember where he'd slept the night before. The grasses were wiry and wasted, the roads salty and cracked, sprouting faded signs and faulty traffic lights. Bridges elbowed their way out of the marshes before collapsing, exhausted, on the other side of the road.\n\nI still had their phone numbers, but I didn't want to call. I didn't even know if they'd be there. All these months later their plans could have changed. Maybe I'd knock and Whitley wouldn't answer, but her ex-second-stepdad, Burt, would, E.S.S. Burt with his too-long, curly gray hair; Burt, who a million years ago had written an Oscar-nominated song for a tragic love story starring Ryan O'Neill. Or maybe they would all be there. Maybe I wanted to see the looks on their faces when they first saw me, looks they hadn't rehearsed.\n\nThen again, if they didn't know I was coming, I could still turn around. I could still go join my parents at the Dreamland for _His Girl Friday,_ afterward head to the Shakedown for crab cakes and oysters, saying hi to the owner, Artie, pretending I didn't hear him whisper to my dad when I went to the bathroom, \"Bee's really come around,\" like I was a wounded racehorse they'd decided not to euthanize. Not that it was Artie's fault. It was the natural reaction when people found out what had happened: my boyfriend, Jim, had died senior year.\n\nSudden Death of the Love of Your Life wasn't supposed to happen to you as a teenager. If it did, though, it was helpful if it was due to one of the Top Three Understandable Reasons for Dying as a Kid: A. Car accident. B. Cancer. C. Suicide. That way, after you selected the applicable choice, the nearest adult could promptly steer your attention to the range of movies (many starring Timothy Hutton) and self-help books to help you Deal.\n\nBut when your boyfriend's death remains unsolved, and you're left staring into a black hole of guilt and the unknown?\n\nThere's no movie or self-help book in the world to help you with that.\n\nExcept maybe _The Exorcist._\n\nIf I was a no-show tonight, my old friends would come and go from Wincroft, and that would be that. Not showing up would be the final push of that old toy sailboat from my childhood, the one shove that would really send it drifting out toward the middle of the lake, far from the shoreline, forever out of reach.\n\nThen I'd never find out what happened to Jim.\n\nI kept driving.\n\nThe twisting road seemed to urge me onward, yellowed beech trees streaking past; a bridge; the sudden, startling view of a harbor where tall white sailboats crowded like a herd of feasting unicorns before vanishing. I couldn't believe how easily I remembered the way: left at the Exxon, right on Elm, right at the stop sign where you diced with Death, run-down trailers with strung-up laundry and flat tires in the yard. Then the trees fell away in deference to the most beautiful kiss of sky and sea, always streaked orange and pink at dusk.\n\nAnd there it was. The wrought-iron gate emblazoned with the _W._\n\nIt was open. The lamps were lit.\n\nI made the turn and floored it, oak branches flying past like ribbons come loose from a ponytail, wind howling through the open windows. Another curve and I saw the mansion, the windows golden and alive, all hulking red brick and slate, crow gargoyles perched forever on the roof.\n\nAs I pulled up I almost laughed aloud at the four cars parked there, side by side. I didn't recognize any of them\u2014except for Martha's Honda Accord with the bumper sticker HONK FOR GENERAL RELATIVITY. If pressed I could, with little trouble, match the other cars with their respective owners.\n\nI had changed so much. From the look of these cars, they had not.\n\nI checked my appearance in the rearview mirror, feeling immediate horror: messy ponytail, chapped lips, shiny forehead. I looked like I'd just run a marathon and come in last. I blotted my face on the roll of paper towels my dad kept in the door, pinched my cheeks, tucked the loose strands of dark brown hair behind my ears. Then I was sprinting up the stone steps and rapping the brass lion knocker.\n\nNothing happened.\n\nI rang the doorbell, once, twice, three times, all in one crazy, deranged movement, because I knew if I hesitated at all I'd lose my nerve. I'd sink, like some lost boot caught inside a lobster trap, straight back to the bottom of the sea.\n\nThe door opened.\n\nKipling stood there. He was wearing a chin-length pink wig, blue polo shirt, Bermuda shorts, flip-flops. He was extremely tan and chewing a red drink stirrer, though it fell out of his mouth when he saw me.\n\n\"Good Lord, strike me down dead,\" he said in his cotton-plantation drawl.\n\nGrand entrances don't happen in real life. Not the way you want.\n\nWhat you want is something between a Colombian telenovela (screaming, faces agog, running mascara) and a Meryl Streep Oscar\u2122 Moment (crackling dialogue, hugs, the whole world coming together to sing in harmony).\n\nInstead, they're awkward.\n\nMy sudden appearance at Wincroft was a poorly aimed torpedo. I had misfired, and now I was drifting aimlessly, explosive, but without a target. Standing in the foyer under the chandelier in my jean cutoffs, sneakers, Wreck Rummage\u2013stained T-shirt, faced with their freshly showered, glam selves, I felt ridiculous. I shouldn't have come.\n\nThey were heading to a sold-out punk rock concert at the Able Seaman in Newport, the beachfront dive bar where we'd spent many a weekend senior year with fake IDs and weekend passes, so they were greeting me, but also getting ready to go. So there was an awful feeling of distraction and poorly dubbed conversation.\n\nFirst Kip hugged me. Then he surveyed me politely, as if he were on an art museum tour and I were the tiny, underwhelming painting some guide was blathering on and on about.\n\nWhitley came running over.\n\n\"Oh, my God, Beatrice.\" She air-kissed me. \"You actually came. Wow.\"\n\nShe was even more jarringly beautiful than I remembered: thigh-high stiletto jean boots, oversized sweatshirt with a sequin mouth on the front, black fringe cutoffs, perfume of gardenia and leather. I was at once hit with the magazine ad that was her presence and also finding it impossible to believe she used to be my best friend. Countless nights at Darrow-Harker School in Warwick, Rhode Island\u2014home of the Crusaders\u2014we sat up illegally after curfew, cheeks polka-dotted with zit cream, wool socks on our feet. I had told her things I hadn't told anyone. Now that seemed like an out-of-place scene cut from some other movie.\n\n\"How are you, Bee?\" she asked, squeezing my hands.\n\n\"Good.\"\n\n\"This is the _best_ surprise. I mean, I could\u2014I'm\u2014 Oh, _shoot._ The patio cushions need to be brought in. It's supposed to rain, right?\"\n\nAnd then she was racing away, long blond hair carouseling her back. \"Kip was right,\" she called out as she vanished into the kitchen. \"He said you'd show up out of the blue like some presumed-dead character in a movie starring, like, Jake Gyllenhaal, but we told him he was nuts. I thought you'd rather die than see any of us again. Now I owe him, like, fifty dollars\u2014\"\n\n\"One hundred dollars,\" interrupted Kip, holding up a finger. \"Do not try to renege. Ghosting on debts is one of your worst qualities, Lansing.\"\n\n\"What? Oh, wait. We have to give Gandalf his Prozac or he'll pee everywhere.\"\n\n\"Gandalf is depressed,\" Kip explained to me with a prim nod. \"He also suffers from multiple personalities. He's a Great Dane who thinks he's a lapdog.\"\n\n\"I know Gandalf,\" I reminded him weakly.\n\n_\"Beatrice.\"_\n\nCannon was jogging barefoot down the staircase, Puma sneakers in hand. At the bottom he stopped, surveying me with a warm smile.\n\n\"I can't believe it. Sister Bee in the flesh. How's God?\"\n\n\"Funny.\"\n\nHe looked different too. He was still sporting his signature gray hacker's hoodie, but it was no longer misshapen and dusted with orange Cheez-Its powder after wearing it two weeks straight in the arctic subterranean computer room at Darrow. It was cashmere. Cannon had become semifamous when, sophomore year, he discovered a bug in Apple's OS X operating system: when you accidentally tapped certain keys, your screen froze, and your desktop turned into the surreal winter scene of Apple's Blue Pond wallpaper. He christened the bug Cannon's Birdcage, and it landed him on the front page of a million Silicon Valley blogs. Last I'd heard, he was attending Stanford for computer science.\n\nHe jumped off the stairs and hugged me. He smelled like expensive wood flooring.\n\n\"How's college? How's your mom and dad? They still run that little ice cream parlor?\"\n\n\"Yes.\"\n\nHe stared at me, his expression intense and unreadable. \"I love that place.\"\n\n\"Hello, Bee,\" called a solemn voice.\n\nTurning, I saw Martha. She was blinking at me from behind her thick, mad-scientist glasses, which gave her the all-seeing, telephoto-lens stare for which she was famous. She'd given up her khakis and boxy Oxford shirts for ripped black jeans and an oversized T-shirt proclaiming something in German: TORSCHLUSSPANIK. She'd also dyed her thin brown hair neon blue.\n\n\"Hi,\" I said.\n\n\"It's absurd how you haven't changed,\" drawled Kip, his smile like a tiny button on formal living room upholstery. \"You freeze-dry yourself in some cryogenic experiment? 'Cause it isn't fair, child. I got crow's-feet and gout.\"\n\nWhitley was back, avoiding eye contact, grabbing her flesh-colored Chanel purse.\n\n\"You're coming with us, right?\"\n\nShe seemed less than thrilled by the idea, now shoving her manicured feet into Lanvin flats.\n\n\"Actually, I\u2014\"\n\n\"Of course you are,\" said Cannon, throwing his arm around my shoulder. \"I'll scalp you a ticket. Or I'll scalp someone for a ticket. Either way, we'll figure it out.\"\n\n_\"Laissez les bon temps roulez,\"_ said Kip, raising his glass.\n\nThere was a Texas-sized stretch of silence as we filed outside, the only sounds our footsteps on the pavement and the wind ransacking the trees. My heart was pounding, my face red. I wanted nothing more than to sprint to my pickup and take off down the drive at a hundred miles an hour, pretend none of this had happened.\n\n\"We taking two cars?\" asked Martha.\n\n\"We're five,\" said Whitley. \"We'll squeeze into mine.\"\n\n\"Promise you'll glance in that rearview mirror at least once, child?\" asked Kip.\n\n\"You're hilarious.\"\n\nWe piled into her hunter-green convertible Jaguar. Whitley, with a severe look\u2014which I remembered meant she felt nervous\u2014pressed a series of buttons on the console screen. The engine did an elegant throat clear, and the top half of the car began to peel away like a hatching egg. Then we were speeding down the drive, Whitley accelerating like a veteran NASCAR driver, swerving into the grass, mowing through rhododendrons. I was in the backseat between Kip and Martha, trying not to lean too hard on either of them.\n\nKip tossed his pink wig into the air.\n\n\"Ahhhhh!\" he screamed, head back, as the wig landed in the driveway behind us. \"After a long absence, the band is back together! Let's never break up again! Let's go on a world tour!\"\n\n_What about the lead singer?_ I couldn't help wondering as I looked up at him.\n\n_Aren't you forgetting Jim?_\n\n* * *\n\n\u2014\n\nThe opening band had already started when we arrived. There wasn't time to talk. There was only this anxious pushing through the packed crowd outside while Whitley approached the bouncer. Martha went in to secure the table, and Cannon went around asking guys with buzz cuts and Budweiser breath if they had an extra ticket, all of which left me crammed pointlessly against the side railing.\n\n\"You guys go in without me!\" I shouted at Kip, who'd materialized beside me.\n\n\"Hush.\" He linked his arm through mine. \"Now that we found you again, we're never going to let you go. I'm your barnacle, child. Deal with it.\"\n\nI laughed. It seemed like the start of the first true conversation that night.\n\nKipling and I had always been close. Tall and lanky, with rust-red hair and \"an ancient gentleman face\"\u2014as he described himself\u2014he was the most fun stuffed into a single person I'd ever met. He was eccentric and strange, like some half-broken talisman you'd find on a dusty shelf at the back of an antiques store, hinting at a harrowing history and good luck. He was gay, though claimed to be more interested in a story well told than in sex, and saw Darrow more as a country club than as any institution in which he was meant to learn something. A study date in the library with Kipling meant constant interruption for his anecdotes and observations about life, friends, and the host of colorful characters populating his tiny hometown of Moss Bluff, Louisiana\u2014like we weren't holed up in muggy cubicles stressed about SATs, but relaxing on a porch shooing flies. While he was as rich as the others (\"defunct department store money\"), he had had what he called a \"busted childhood,\" thanks to his scary mom, Momma Greer.\n\nLittle was _actually_ known about Momma Greer, apart from the details Kipling let slip like a handful of confetti he loved to toss into the air without warning. When he was a toddler she locked him alone for days in Room 2 of the Royal Sonata Motel (\"ground floor by the vendin' machines so she could sneak out without payin' \"), nothing to eat but a stash of Moon Pies, no company but Delta Burke selling bangles on QVC. Her negligence had led to a pit bull, chained up in a backyard, attacking Kipling when he was five, biting off three fingers on his left hand, and leaving him with a \"mini shark bite\" on his chin\u2014disfigurations he paraded like a Purple Heart.\n\n\"Just call me Phantom of the Opera,\" he'd say, gleefully fanning his severed hand in front of your face. When the court finally removed Kip from his mom's custody, sending him to live with an infirm aunt, he kept running away to try to get back to Momma Greer.\n\nLast I'd heard she was in a mental institution in Baton Rouge.\n\nI wanted to ask how his year had been, but at that moment, Whitley, in true Whitley fashion, came over and without a word grabbed my wrist, pulling me through the crowd. She'd come to some understanding with the doorman. He let me in without a ticket, stamping my hand, and then we were all at a reserved table in the front watching a girl with stringy hair pretend she was Kurt Cobain.\n\nIt was strange. The drummer looked like Jim. I wasn't sure anyone else noticed, but he looked like Jim's younger brother, all milk-chocolate eyes and bedhead, the rueful air of a banished prince. It was deafening inside, too loud to talk, so all of us just stared at the band, lost in the swamps of our thoughts.\n\nMaybe I was the only one lost. Maybe they'd all had amazing experiences in college, which had shrunk what had happened to us in high school, turned even Jim's death into a faded T-shirt washed ten thousand times.\n\nOnce upon a time at Darrow, they'd been my family. They were the first real friends I'd ever had, a collection of people so vibrant and loyal that, like some child born into a grand dynasty, I couldn't help but be awed at my luck. We'd been a club, a secret society all the other students at Darrow eyed with envy\u2014not that we even paid attention. Friendship, when it runs deep, blinds you to the outside world. It's your exclusive country with sealed borders, unfair distribution of green cards, rich culture no foreigner could understand. To be cut off from them, exiled by my own volition as I had been for the past year, felt cheap and unsettled, a temporary existence of suitcases, rented rooms, and roads I didn't know.\n\nJim's death had been the earthquake that swallowed cities. Although I had spent the past year certain my friends knew much more about it than they'd let on, I also knew with every passing day the truth was drifting farther out of reach. I'd checked Whitley's Snapchat and every now and then I saw the four of them together. They looked so happy, so nonchalant.\n\n_Like nothing had happened._\n\nYet now, I could see that the dynamic between them had changed.\n\nKip kept drumming his disfigured hand on the table. Whitley kept checking her phone. Martha seemed to be in an unusually bad mood, throwing back shots the bartender kept sending to our table\u2014something called the Sinking of the _General Grant,_ which tasted like crude oil. I caught her staring at me once, her expression faintly accusatory. I smiled back, but she turned away like one of those jungle plants that shrivel at the faintest touch, refusing to look at me again. Once, as Cannon leaned forward to whisper something to Whitley, he tucked her hair behind her ear, which made me wonder if they were back together. Then it seemed more habit than anything else.\n\nWhen the opening band finished, I wanted to disappear. I wanted to take a taxi back to Wincroft, climb into my dad's truck, drive off, and never look back. What had I expected\u2014for the truth to be right there, obvious as a giant weed growing among tulips, waiting for me to yank it out?\n\nBut I stayed. I stayed for the next band, the band after that. I drank the Moscow Mules Whitley put in front of me. I let Kipling pull me to my feet, and I danced the Charleston with him, and the fox-trot, letting him spin me into the beach bums, and the prepsters, and the Harley-heads under the shaking paper lanterns and posters of sunken ships.\n\n_Just a little while longer,_ I kept thinking, _and I'll bring up Jim._\n\nWhen the next band finished, Whitley wanted to go back to Wincroft, only no one could find Cannon. As it turned out, he was in the bar's back alley, helping a girl who'd had too much to drink and was passed out by the fire exit.\n\n\"Here comes Lancelot,\" said Whitley.\n\nPerched along the railing, we watched while Cannon tracked down\u2014with the efficiency of a lobbyist working Capitol Hill\u2014the girl's missing friends, purse, sandals, and iPhone. He even located her hair clip, which he used to gently pin back her hair so she'd stop throwing up on it, which led the girl's newly located, equally drunk friends to stare up at him in wonder.\n\n\"Are you human, dude?\"\n\n\"Do you have a girlfriend?\"\n\n\"Who _are_ you?\"\n\nCannon ran a hand through his hair. \"I'm Batman.\"\n\n\"Here we go again,\" sighed Whitley.\n\nCannon was not handsome. He was slight, with dirty blond hair and pale, out-of-focus features. But he had atomic intensity, which never failed to shock and awe when unleashed upon the world. Moving like a highly charged ion, capable as a machine gun, the first week of freshman year Cannon hacked Darrow's intranet to display its flaws (becoming the school's de facto tech guru). He revamped the decrepit sculpture garden and the wrestling gym. He was class president, and organized marches, marathons, and fund-raisers for endangered species and girls' rights. Cannon was the first to admit that his outgoing, sociable nature and activism was compensation for being a tongue-tied computer geek as a child, worshipping Spielberg movies, eighties pop songs by the Cure, and Ray Kurzweil, no friends to speak of but an imaginary fly named Pete who lived inside his computer. He was adopted, raised by a single mother, a judge in the superior court of California. And while at first glance having Whitley Morrow as his girlfriend\u2014besting Darrow's country club boys who were IIIs and had middle names like Chesterton\u2014seemed like a mistaken case of the princess accidentally ending up with the sidekick, the more you knew Cannon, the more you realized the role of prince was far too trivial for him. He was the king\u2014at least, that was what he was aiming for. He was the most silently ambitious person I'd ever met.\n\n\"Any more distressed damsels you need to save?\" Whitley asked as Cannon strode back over, having helped the girl and her stumbling friends into an Uber.\n\nHe held out his arms in mock triumph. \"The bartender looks like he's coming down with a head cold. But no. My work here is done.\"\n\n\"Thank the Lord, 'cause I need my beauty sleep,\" said Kip with a yawn.\n\nWe piled into the Jaguar.\n\nThe problem was, no matter how many times Whitley pressed the buttons on the console screen, the convertible top wouldn't go up. It wouldn't go up manually either.\n\nCannon volunteered to drive, but Whitley insisted. It began to pour, so hard there was more rain in the air than air. The thirty-five-minute ride home was this terrible ordeal, all of us in the backseat hunched together, drunk and freezing. At one point Martha threw up all over her feet, all of us shivering under E.S.S. Burt's creepy London Fog trench coat, which Whitley had found in the trunk. Whitley began to cry that she couldn't see the road. Tearing around a curve, we nearly collided with a tow truck.\n\nThe driver blared his horn. Whitley jerked the wheel, tires screeching. Everyone screamed as we barreled off the road, bouncing to a halt in a ditch, Kip hitting his head on the seat. Killing the engine, Whitley started to sob, screaming at Cannon that it was all his fault, that as always he'd needed to impress a bunch of girls just to massage his screaming insecurity for five minutes and now we'd almost died. She snatched his baseball cap off his head and threw it into the dark. Then she scrambled out, shouting that she was finding her own ride home, running into the woods. I sensed her tantrum had to do with the rain and almost ending up in a car accident\u2014but also with me, how I'd shown up out of the blue.\n\nCannon went after her. A few minutes later, he brought her back. She was crying and wearing his hoodie. He tucked her carefully, like some wild bird with a broken wing, into the front seat, whispering, \"It's gonna be all right, Shrieks.\"\n\nIt was Cannon who got us home.\n\n* * *\n\n\u2014\n\nAs the five of us went clambering into Wincroft, dripping wet and drunk, it felt normal for the first time. It felt like the old days. Thank goodness for the defunct top on that convertible. Our brush with death had thawed the ice. We were giddy, teeth chattering as we pulled off our wet clothes, leaving them in a soggy pile on the floor, which Gandalf kept circling while whining. Whitley disappeared upstairs. Martha was on her hands and knees in front of the fireplace, moaning, \"I can't feel my legs.\" Cannon went down to the wine cellar, returning with four bottles of Chivas Regal Royal Salute, and poured shots in pink champagne glasses. Whitley dumped a giant mound of white terry-cloth bathrobes on the couch like a pile of dead bodies.\n\n\"I've never been so scared in my whole life,\" she said, giggling.\n\nThat was when the doorbell rang.\n\nWe all sat up, staring at each other, bewildered. Mentally counting. We were all here.\n\n\"Someone call Ghostbusters?\" slurred Martha.\n\n\"I'll go,\" volunteered Cannon. A sloppy salute, and he disappeared into the foyer. None of us said a word, listening, the only sound the rain drumming on the roof.\n\nA minute later, he was back.\n\n\"It's some old geezer. He's two hundred years old.\"\n\n\"It's Alastair Totters,\" said Martha.\n\n\"Who?\" Cannon snapped.\n\n\"Time-traveling villain in _The Bend,_ \" mumbled Martha.\n\n\"No, no,\" whispered Kip, gleeful. \"It's the proverbial kook with Alzheimer's who wandered away from his nursing home during Elvis Social Hour. _Without_ his medication. They're always without their medications.\"\n\n\"I'll invite him in for a nightcap?\" asked Cannon, sighing, a mischievous wink.\n\n\"No,\" hissed Whitley. \"That's how horror movies start.\"\n\n\"Chapter three,\" Martha muttered.\n\n\"Hey,\" said Cannon, pointing at Wit. \"That's not very nice. _I'm_ inviting him in\u2014\"\n\n_\"NO!\"_\n\nThen we were all racing, giggling, tripping over each other as we bumbled to the foyer to see for ourselves, tying up our bathrobes, taking turns to check the peephole, bumping heads. I assumed Cannon was somehow playing a trick on us, that no one would actually be there.\n\nBut there he was. An old man.\n\nHe was tall, with thick silver hair. Though I couldn't make out his face in the shadows, I could see that he was dressed in a dark suit and tie. He leaned in, smiling, as if he could see me peering out.\n\nCannon opened the door with a bow.\n\n\"Good evening, sir. How may we help you?\"\n\nThe man didn't immediately speak. Something about the way he surveyed us\u2014methodically inspecting each of our faces\u2014made me think he knew us from somewhere.\n\n\"Good evening,\" he said. His voice was surprisingly rich. \"May I enter the premises?\"\n\nNo one answered, the question being too presumptuous and strange. I gathered he was not senile. His eyes\u2014deep green, gleaming in the porch light\u2014were lucid.\n\n\"Oh, you live next door,\" said Whitley, stepping beside Cannon. \"Because if this is about Burt's sailboat, the _Andiamo,_ being marooned in front of your dock, he told me to tell you he had problems with the anchor and he's working on getting a tow next week.\"\n\n\"I do not live next door.\"\n\nHe stared at us another beat, his face expectant.\n\n\"It's really best if I come inside to explain.\"\n\n\"Tell us what you want right there,\" said Cannon.\n\nThe man nodded, unsurprised. It was then that I noticed two bizarre things.\n\nOne: he looked like Darrow's musical director, Mr. Joshua. For a moment my drunken mind believed that it _was_ Mr. Joshua, that something terrible had happened to him in the year since I'd last seen him. He'd suffered some tragedy and aged twenty-five years, his hair going silver, his face growing tattered. But it wasn't Mr. Joshua. Mr. Joshua was slight and rosy, quick to laugh. This man was bony, with a hawkish face, one that would look at home on foreign currency or atop a monument in a town square. It was as if he were the identical twin brother of Mr. Joshua, as if they'd been separated at birth and had totally different life experiences, Mr. Joshua's nurturing and this man's harrowing, bringing him to look the way he did.\n\nTwo: there was no car in the driveway, so the question of how he'd come here without an umbrella yet remained perfectly dry hung in the air, vaguely alarming, like a faint odor of gas.\n\n\"You're all dead,\" he said.\n\n\"Oh, dear. You'll have to excuse me. That's not accurate.\"\n\nThe old man placed a hand over his eyes, shaking his head. \"I overshot it. Went for the dramatic, _Masterpiece Theater_ effect. I apologize. Let's try that again, shall we?\"\n\nHe cleared his throat, smiling.\n\n\"You're all nearly dead. Wedged between life and death. Time for you has become snagged on a splinter, forming a closed-circuited potentiality called a Neverworld Wake.\"\n\nQuite pleased now, he nodded and took a deep breath.\n\n\"This phenomenon is not specific to you. There are such moments occurring simultaneously in the past, present, and future all around the world and across the universe, known and unknown, crumpled and unfolded. Time does not travel in a straight line. It bends and barrels across tunnels and bridges. It speeds up. Slows down. It even derails. Well then. This hitch, as we might call it, is where each of you exists at the moment. And it is where, until further notice, you will remain.\"\n\nHe bowed like the longtime ringmaster of a down-at-heel traveling circus, with gracious ease and a hint of exhaustion.\n\n\"I am the Keeper,\" he said. \"I have no other name. The way I look, act, the tone of my voice, my walk, face, everything I say and think is the sum total of your five lives as they were lived. Think of an equation. This moment equals your souls plus the circumstances of reality. Another example? Imagine if each of your minds was placed inside a blender. That blender is turned on high. The resulting smoothie is this moment. If there were someone else with you? It would be a slightly _different_ moment. I'd be saying something else. I'd have different hair. Different hands. Different shoes. Docksiders rather than Steve Maddens.\n\n\"I digress. The circumstances of reality. You're doubtlessly wondering what I meant by that. _Well._ \"\n\nHe sniffed, smiling.\n\n\"Each of you is, at present, lying kinda sorta dead on the side of a coastal road. This is due to a recent head-on collision with one Mr. Howard Heyward, age fifty-eight, of two hundred eighty-one Admiral Road, South Kingstown, who was driving a Chevrolet Kodiak tow truck. Time is standing still. It has become trapped inside an eighth of a second like a luna moth inside a mason jar. There is a way out, of course. There is one means by which the moth can escape and time can fly irrevocably free. Each of you must vote during the last three minutes of every wake. You must choose the single person among you who will survive. This person will return to life. The remainder of you will move on to true death, a state permanent yet wholly unknown. The decision must be unanimous, save one dissenter. There can be only one who lives. There are no exceptions. Do you have questions?\"\n\nNo one said a word.\n\nAll I could think was that he was senile after all. He also seemed to have once been an actor, because he had intoned his speech like the baritone narrator of some old 1950s TV Western starring John Wayne, his voice lilting, old-fashioned, and grand. There was an effortlessness to his every word, as if he'd given this memorized speech dozens of times before.\n\nHe was waiting for one of us to say something.\n\nKipling started to clap. \"Bravo.\"\n\n\"Hold on,\" said Martha, scowling. \"Is he selling Bibles?\"\n\n\"What do you want?\" demanded Cannon.\n\nThe man shrugged. \"I am a simple resource. I desire no compensation, monetary or otherwise. Nonetheless, I wish for you to succeed.\"\n\n\"Succeed at what?\" asked Whitley.\n\n\"The vote.\"\n\n\"Listen,\" said Cannon. \"It's been a long night. Tell us what you want.\"\n\n\"It appears my delivery was a bit rushed for your comprehension. Would you like the news a different way? Dramatic reenactment? Flash cards? A second language? Italian tends to soften the blow of even the most ominous prognosis, which was why Dante used it for the _Inferno._ \" He cleared his throat. _\"Buonasera. Tra la vita e la morte, il tempo \u00e8 diventato congelato\u2014\"_\n\n\"That's enough,\" snapped Cannon. \"Get the hell off this porch.\"\n\nThe man was unfazed. He smiled, revealing small gray teeth.\n\n\"Very well. Good luck to you all. Godspeed.\"\n\nHe hopped nimbly down the steps, striding out to the driveway. Within seconds he was drenched and vanishing into the yard beyond the lights. We listened to his footsteps sloshing through the grass.\n\n\"My brain just exploded,\" said Martha.\n\n\"Worst door-to-door salesman _ever,_ \" said Kip, shaking his head. \"I think he learned his sales techniques from Monty Python. What did he call us?\"\n\n\"Dead,\" I whispered.\n\n\"Right. I've been called many things. Deadhead. Dead _beat._ Never just plain old dead. Has a sort of bleak ring to it.\"\n\n\"He's a Jesus freak,\" said Whitley, nibbling her fingernail. \"Right? In some cult? Should I call the police? There may be others out there. They might be waiting to break in here and slaughter us or something.\"\n\n\"He's harmless,\" mumbled Cannon. Yet he seemed unnerved. Scowling out at the empty driveway, he suddenly seized an umbrella and barreled outside just as another monstrous clap of thunder exploded and the rain fell harder. He stomped into the yard, looking around, disappearing in the same spot as the old man.\n\nWe waited in silence, apprehensive.\n\nA minute later, Cannon reappeared.\n\n\"Must have headed back to the road. No sign of him.\"\n\n\"Let's check the security cameras,\" said Whitley.\n\nThey headed downstairs to the surveillance room, and Kip and Martha\u2014muttering about needing \"a stiff drink before the ensuing elderly zombie apocalypse\"\u2014shuffled back into the living room.\n\nI remained where I was, staring outside.\n\nThere had been something legitimately upsetting about the old man. All the eloquence, the formal speech, the accent\u2014at once like a cable newscaster's and someone who'd spent a year abroad in England\u2014seemed only to conceal a deep calculation. As if what he had told us were only one small piece of a grand plan.\n\nI watched the woods, searching for movement, trying to steady my drunken head.\n\nSuddenly, music erupted from inside, overlaying the storm with a soundtrack, softening the night's edge. With a deep breath, I shut the door and bolted it. Whitley was right. He was probably just looking to recruit people for his church.\n\nStill, I walked past Kip and Martha, curled up stroking Gandalf on the couch, and took out my phone, stepping into the hall. My mom answered on the first ring.\n\n\"Bee? Is everything all right?\"\n\nI could tell from her anxious tone that she and my dad were both still awake, doubtlessly reading in bed: Dad, one of his thirty-pound presidential biographies, Mom _trying_ to read a thriller by James Patterson, though she'd probably been skimming the same paragraph four, five times before blurting \"I don't understand why she had to go see them. They still have some mysterious hold on her.\" Then Dad, with the patient, knowing stare over his glasses: \"If she wants to see them she can, Victoria. She's an adult. She's stronger than you give her credit for.\"\n\nI realized I had no idea why I was calling, except to hear her voice.\n\n\"It's too late to drive back, so I'm spending the night,\" I said.\n\n\"Well, your father needs you at the Crow for opening. Sleepy Sam called to say he's having a tooth pulled.\"\n\n\"I'll be there.\"\n\nShe lowered her voice: \"How's it going with them? Can you talk? You sound upset.\"\n\n\"Everything's fine. I love you.\"\n\n\"We love you too, Bumble. We're here if you need us.\"\n\nI hung up, just as Whitley and Cannon were returning from the surveillance room.\n\n\"No sign of him on the cameras,\" said Cannon.\n\n\"He's gone,\" she said.\n\n\"This night gets an A-plus in weird,\" slurred Martha.\n\n\"Wasn't it hilarious how he asked to be called the Keeper?\" said Kip, shaking his head. \"The man looked like more of an Eastern European Santa Claus.\"\n\nWhitley wrinkled her nose. \"That was my Internet password for everything for _years._ I'm not even kidding. The keeper one-two-three.\"\n\nIn the end, the consensus was he was Just One of Those Things, one of life's untied shoelaces. As the thunderstorm raged on, however, lightning cracking and thunder yowling, at one point a giant oak branch crashed onto the back deck, demolishing the entire railing.\n\nWe jumped, staring at each other, doubtlessly imagining the same thing: here it was, the beginning of the horror to which that funny old man had been the creepy prelude.\n\nOnly nothing happened.\n\nAnother hour passed. Whitley talked about being sexually harassed by her boss at the San Francisco law firm where she'd had an internship all summer. Cannon couldn't tell if he was in love with his girlfriend, an international fencing champion.\n\n\"Love is this elusive bird,\" he said. \"You're the lifelong bird-watcher, looking for this rare red-plumed quail people spend entire lives trying to see for three seconds in a cherry tree on a mountaintop in Japan.\"\n\n\"You're mistaking love for perfection,\" I said. \"Real love when it's there? It's just _there._ It's a metal folding chair.\"\n\nWhen no one said anything, I realized, embarrassed, I'd blurted this as a clumsy way to bring up Jim. And I was about to. Then Whitley got up to get more Royal Salute, and Kipling muttered that he hadn't been this wasted since he was nine, and the moment was gone.\n\n\"I'll tell you what love is,\" said Martha, gazing at the ceiling. \"It's the Heisenberg uncertainty principle. Once you think it's there and give voice to it? It's not there anymore. It's over here. Then way over _here._ Then here. You can't trap it or contain it no matter how hard you try.\"\n\nIt was the first time I'd ever heard Martha speak in such a way\u2014the first time for the others too, if their surprised glances were any indication. Being allergic to romance was her shtick. If ever you asked her whom she had a crush on, she'd blink at you like you had three heads: \"Why would I waste _time_ \u2014a highly precious, constantly diminishing resource\u2014on transitory neurological fluctuations of adrenaline, dopamine, and serotonin?\" When she saw couples holding hands in the halls, she gave them a cartoonishly wide berth.\n\n\"In case they're contagious,\" she said. And she wasn't joking.\n\nThe conversation meandered on as rain peppered the windows.\n\nAt one point Kip started calling me Sister Bee again, which made Cannon blurt that I was the one person at school no one, not a teacher or student, a parent, a maintenance worker, or even an _ant,_ could ever say anything bad about.\n\n\"And your nice isn't even irritating,\" said Cannon.\n\n\"Remember how in biology,\" said Kip, smirking, \"Bee didn't even tell Mr. Jetty that Chad Burman had just thrown up his entire lunch all over the back of her blouse? She just sat there heroically answering his question about osmosis and then excused herself.\"\n\n\"And the field trip to D.C. when Mr. Miller had to go home to his pregnant wife, and rather than summon another teacher from campus to chaperone, Ms. Guild just asked _Bee._ \"\n\nThey cackled with laughter.\n\n\"It wasn't that big a deal,\" I said.\n\nDuring this conversation, Whitley remained tellingly silent, a smug expression on her face as she stared at the floor, as if she begged to differ, as if she wanted to laugh.\n\n_When is it coming?_ I wondered with a shiver. _The conversation about Jim?_\n\nThe absent leader. The sixth member. The killed one.\n\nWeren't they dying to talk about him? Jim, whose shadow stretched behind him long and dark, as captivating dead as he was when he was alive. Jim the poet. Jim the prince.\n\nOf course they were thinking about him. How could they not?\n\nYet it seemed he was the locked shed on the forsaken property everyone was too scared to approach, much less peer inside all the filthy windows.\n\nNot long after, I passed out. When I woke up, peeling my cheek from the couch cushions, Whitley and Cannon were asleep under a blanket in front of the fireplace. Kip was snoring on the love seat. Only Martha was awake. She appeared to have sobered up and was sitting across the room in a club chair, reading with her chin in her hand.\n\n\"Hey,\" I croaked, rubbing my eyes. \"What time is it?\"\n\n\"Four-fifteen.\"\n\nIt was still dark out, and still raining.\n\n\"Can't sleep,\" Martha said with a wan smile. \"It's that old man. I feel like he's still out there.\"\n\nHer remark made me glance out the windows, shivering.\n\nWhitley had turned on every light, and I could see the giant fallen branch, the gardens and pool, the stone path leading down to the dock.\n\n\"I'm sure it's fine,\" I whispered.\n\nWe went on talking, though eventually the silences between our words stretched out farther and farther, like the distance between a final chain of tiny islands before the open sea.\n\nMartha and I had never been close, though by rights we should have been. As the only scholarship kids at Darrow, we were two rescue mutts of humble bloodlines and skittish temperaments thrown into a kennel of world-class purebreds.\n\nShe'd attended Darrow on a physics scholarship established by a genius alumnus who'd worked on the God Particle. She'd been the first winner in twenty-eight years. Valedictorian of our class, she went to MIT on a full ride for mathematical engineering.\n\nShe'd been raised in South Philadelphia by a single dad, and her family was even poorer than mine. I never met her dad, though Cannon once said he owned a gas station and went by the nickname Mickey Peanuts. Jim told me Martha had had a considerably older sister who'd died of a drug overdose, and that death was the reason Martha's mother left. But Martha never mentioned a sister, and any talk of her mom was in connection with a single trip to Alaska she'd taken when she was ten.\n\nI'd spent hours in her company, yet I couldn't tell you who or what Martha ever loved beyond this weird underground fantasy novel called _The Bend._ The book was why she wallpapered her dorm room with mysterious posters of steam trains and scoured Reddit forums for other megafans, known as Benders. She even dressed up\u2014with a surprising lack of embarrassment\u2014once in a top hat and spectacles, or a gray barrister's wig\u2014to celebrate the apparent birthdays of the characters. She always kept a copy of the book at the bottom of her backpack, pulling out the doorstop of a thousand torn-up pages\u2014crudely Xeroxed, bound with frayed twine\u2014at the start of class, reading, it seemed, to avoid talking to anyone.\n\nAt heart she was Jim's friend. They'd met when they were kids at some invitation-only camp for the gifted, housed in a nineteenth-century mansion in upstate New York called Da Vinci's Daughters and Sons. Jim was there because he'd composed an entire musical about Napoleon, which had been staged at his Manhattan private school and gotten him profiled in _New York_ magazine. Martha was there because she'd built a working airplane engine in her garage.\n\nIt was Jim who urged Martha to apply to Darrow, Jim who sought to have her around. Over the years she'd become an integral part of our group, giving every situation its deadpan punctuation or making some awkward reference to a chapter in _The Bend_ that no one understood. And yet I always suspected Jim had been her only true champion, that Cannon, Wit, and Kip accepted her the way one accepts a lifelong inconvenience, like asthma or a spouse's beloved cat. He never stopped insisting she was amazing, that one day when we were sixty we'd look back and think with disbelief, _I was friends with Martha Ziegler._\n\n\"Which will be like saying you were friends with Steve Hawking. That's how big she's gonna be.\"\n\nThe two of them had a shabby shorthand, laughing at things only they found funny, arms slung around each other's necks like old cardigans. While it didn't make me jealous per se, it could lead me to notice something Martha did\u2014heavy glance, weird remark\u2014that would set off alarm bells, and I'd entertain my long-standing suspicion that she was harboring a burning secret: she was in love with Jim. It was why she'd never liked me.\n\nI could only assume she'd been heartbroken by his death. In the aftermath\u2014the ten or so days before summer break\u2014she was glum and taciturn, scuttling out of Final Chapel ahead of the entire school like some startled attic bat. She was agitated. Dimly I recalled how she'd left school suddenly the day before I did, vanishing without saying goodbye. Whitley, ever attentive to the embarrassing things people wished to hide, couldn't stop saying, \"Something's up with Martha.\"\n\nNow here she was, staring at me with that stark telephoto stare I'd always found nerve-wracking. Whatever she had felt about Jim's death, whatever had been uprooted, was hidden now, like a pod of blue whales thundering through the depths of an ocean with a still surface.\n\nI realized that she'd just asked me a question.\n\n\"What?\"\n\n\"I was wondering if you still made those dream soundtracks.\"\n\nShe was referring to my hobby of creating albums for movies I made up. It was just something I did. I didn't know why. As a child I'd always been painfully shy, so terrified of speaking in class, my teachers often thought I had a stutter or a hearing problem. I began crafting pocket-sized books with lyrics and hand-drawn art for movies I wished existed, like the soundtrack to a hit teen vampire movie called _Blood Academy._ Or _Dove Nova,_ the biopic of a Swedish teen pop star who vanished into thin air, her disappearance forever unsolved. There was no point to these albums. I couldn't even explain why I made them, except that I liked to imagine they were artifacts of some other world that existed beyond the one we could see, a world where I wasn't timid, and unsaid words didn't collect in my mouth like marbles, and I was brave. They were my what-ifs, my _glass menagerie,_ as Jim said.\n\nOne night freshman year during a snowstorm, the whole school was in the auditorium for Holiday Dance when the power went out. I had accidentally ripped the back of my dress, so I left Jim to run back to my dorm to change. To my surprise, I encountered Martha in the dark of the common room, reading _Pride and Prejudice_ with a flashlight, so absorbed she hadn't realized one of the windows was wide open and snow was collecting in the corner three inches thick. We ended up hanging out for two hours, just the two of us. It was the only time we ever did. For some reason, probably in the hopes of making things less uncomfortable between us, I'd shown Martha my collection of dream albums. Ever since then, when we were alone, she tended to ask about them, like they were some one-size-fits-all subject she could rely on to get me to talk. It could be a little unnerving.\n\n\"No,\" I said, feigning a yawn. \"Not really. I think I'll go find a bed upstairs.\"\n\nShe nodded, her face solemn. \"Good night, Beatrice.\"\n\nI slipped out\u2014Martha returning to her book\u2014and trudged upstairs, finding my favorite guest room at the end of the hall. I pulled back the comforter and slung myself into bed.\n\nAny other night I would have been kept awake by the memories inside that room. I was curled up under the heavy covers, same as always. The only thing missing was Jim snuggled beside me, composing lyrics by the light of his cell phone.\n\nI set my alarm for six and closed my eyes. I'd sneak out before any of them were awake.\n\nAnd that, for better or worse, would close my final chapter on Wincroft.\n\nWhen I awoke it was light out.\n\nI was freezing and covered in sweat. No, not sweat, I realized after a moment, blinking. It was rain. I was soaked because I was sitting in the backseat of the Jaguar convertible, the top still down. It had been parked, seemingly by someone very drunk, in a flower bed in the front yard of Wincroft.\n\nIt was still pouring rain. Kip and Martha were beside me, wearing confused expressions.\n\n\"What are you doing?\" Kip asked me. He was soaking wet, his eyes bloodshot. A raindrop dangled off the end of his nose. \"Where are you taking us?\"\n\nI had no clue what he was talking about. I scrambled out of the car, raced across the driveway to the mansion, and threw open the front door. I nearly collided with Whitley. She was frozen in the foyer, wearing the same outfit she'd had on last night. She surveyed me with a look so stunned, I understood immediately that something terrible had happened.\n\n\"What? What is it?\"\n\nShe only stepped past me, staring out the door, speechless.\n\nI hurried past her into kitchen. Shivering, I took inventory of my body. I felt fine. My head was clear. Yet somehow I'd overslept. I wasn't going to make it to the Crow by opening. My parents would be scrambling to keep up with the morning crowd, then lunch, and my dad would be so strapped he'd forget to tell people about the specials, and my mom would use this as an excuse to say they didn't need specials anymore, they were too expensive, which was sometimes enough of a spark to make them start arguing, which they rarely did.\n\nCannon was standing at the kitchen island typing on his open laptop.\n\n\"See, look!\" he shouted over his shoulder, seemingly believing I was Whitley. \" _New York Times._ It's the exact same thing.\"\n\nI stepped beside him. He was amped, like he'd had about six cups of coffee.\n\n\"What is it?\"\n\n\"What is it?\" he mocked, turning to me. He grabbed my head, directing it at the screen.\n\n\" 'Senate Pushes for New Immigration Initiative,' \" I read.\n\n\"The _date,_ \" he snapped.\n\n\"Friday, August thirtieth. So?\"\n\n\"So? _So?_ It's yesterday.\"\n\nScowling, he was tapping the keyboard, loading CNN.\n\n\"CNN. _The Post. Time._ All of them say the same thing.\"\n\nHe shoved his iPhone into my hands. I blinked stupidly down at the date overlaying a photo of what had to be his fencing-champion girlfriend.\n\nHe was right. _August 30. 5:34 p.m._\n\nThere had to be an error with the International Date Line. Terrorists had hacked the network. As if reading my mind, he held up his wristwatch, the hour and minute hands set to 5:35, the date turned to 30.\n\n\"How could hackers get into my TAG Heuer?\"\n\nI could only stare.\n\nAt that moment, his phone rang. Someone named Alexandra. He snatched the phone.\n\n\"Alex. Hold on. Now, wait a\u2014wait a\u2014 Tell me what day and time it is. The date and time. I'll explain in a sec\u2014would you tell me the goddamn _date_? I'm not asking you to recite the Declaration of\u2014 WOULD YOU PLEASE JUST SHUT UP AND TELL ME\u2014\"\n\nWhatever Alex's confused response was, Cannon furiously hurled the phone at the sliding glass doors. He collapsed on the couch, staring wild-eyed at the floor. I hurried to my purse and dug out my phone, which was actually pretty strange because the last time I'd seen it, it'd been upstairs.\n\nMy phone read the same thing. _August 30._ With a shiver of panic, I dialed my mom.\n\n\"Hi, Bumble\u2014\"\n\n\"Mom. _Mom?_ Where are you?\"\n\n\"On our way to the Dreamland to see _His Girl Friday._ What's the matter?\"\n\n\"You didn't see the movie yesterday?\"\n\n\"Yesterday?\"\n\n\"Mom, what day is it?\"\n\n\"What? Why are you shouting?\"\n\n\"What's the date?\"\n\n\"It's\u2014it's Friday, August thirtieth.\"\n\n\"Are you positive?\"\n\n\"I'm looking at the dashboard right now.\"\n\n\"It's the thirtieth,\" I heard my dad chime in.\n\n\"Mom, I called you last night, remember?\"\n\n\"Last night? What?\"\n\n\"Last night I called, and said I was spending the night at Wincroft, and you asked me to be in for opening because Sleepy Sam was getting a tooth pulled.\"\n\n\"Sam is out tomorrow? He called _you_? Sam is out tomorrow,\" she told my dad.\n\n\"He called Bee, after we've made sure he has our number about nineteen times?\"\n\n\"Bee, what's going on up there? Is it awful? Why don't we come get you?\"\n\nI hung up, blood rushing in my ears.\n\nMy mom called back, but I was too shaken to answer.\n\nI sat on the couch, trying to calm down. This had to be some kind of lucid dream. I willed myself to wake up. _Wake up._ After a moment, I realized Kip and Martha had drifted inside. They were standing stiffly with stricken expressions, like they'd just woken up from sleepwalking. Whitley had stepped back into the kitchen, her every gesture slow, as if pretending to walk on the moon.\n\n\"Y'all?\" whispered Kip, his voice scarcely audible. \"Was there an earthquake? Or some end-of-days world event we're just finding out about?\"\n\nThat was when the doorbell rang.\n\nI didn't wait for the others. I jumped off the couch, sprinting past Kip and Martha, and yanked open the front door.\n\n\"Perhaps this time I'll be invited in for tea,\" said the old man.\n\n\"The first thing you must do is stay calm,\" said the Keeper. \"Panic will get you nowhere.\"\n\nHe was making tea.\n\nHe had asked for tea when he'd strolled inside, and as we were all too alarmed to react to what he was saying, he had, incredibly, started making it for himself. He filled the kettle, turned on the gas stove, and grabbed a mug from the cabinet, as if he had visited this house many times before.\n\n\"If it's any reassurance, remember one thing,\" he continued, his fingers nimbly straightening his dark blue silk tie. It caught the overhead light, and I saw it had a discernable pattern of stags identical to the stag presiding over the entrance to Darrow.\n\n\"Others have gone through the Neverworld before you. Many more will after. Hundreds of millions of others will expire never having had the opportunity that each of you has. So you must look at this as a gift. A chance to change history, for your choice of who will live will affect billions of moments barreling into the future for infinity. In other words, there is a precedent, and you aren't alone. You must rely on each other. Each of you is a key, the others your locks. This isn't a nightmare, and it isn't a dream. It's a crack you will continue to fall through until you vote. The sooner you accept where you are, the sooner you will all escape.\"\n\nThe old man here, _again,_ wearing the same dark suit, speaking in the same grand voice, was so incongruous and strange, none of us could really pay attention to anything he was saying. Whitley and Kip were standing by the kitchen island, staring openmouthed at him, as if he were a poltergeist. Martha was on the couch, stone-faced, her feet planted like she felt faint. I was doing my best to follow what he said, in case there was some clue that might reveal who he actually was. Yet all the while my mind was screaming, _It's a prank. It's a prank._ It had to be. Somebody\u2014international terrorists, hackers from Anonymous or some other group\u2014was playing a cruelly ingenious trick.\n\nI noticed Cannon had disappeared upstairs. Now he reappeared, hauling his duffel.\n\n\"I'm out,\" he announced.\n\n\"What?\" asked Whitley, alarmed. \"Where are you going?\"\n\n\"Airport.\"\n\n\"But it's yesterday,\" said Kip.\n\n\"No, it's not. Of course it's not. Yeah, we can't explain it, but there is an explanation. I'm sure the physics department at Harvard is working on this as we speak.\"\n\n\"I'm afraid the physics department at Harvard is ignorant of your plight,\" interjected the Keeper, wringing out the tea bag on a spoon. \"They've got their hands quite full trying to solve quantum gravity. Specifically, the vacuum catastrophe.\"\n\nCannon surveyed him coldly. \"I'm going home.\"\n\n\"To do what?\" asked Kip. \"Complain? 'Ma? Uh, today's kinda yesterday'?\"\n\nCannon shrugged. \"I'll be damned if I'm staying here with him.\"\n\nHe left. We listened to the front door slam. Then, suddenly, Whitley was scrambling after him. And Kipling. Martha too. They were all moving, running away as if they'd just learned the old man was wearing explosives. They grabbed car keys, handbags, sweatshirts, phones. I didn't want to be left alone with him, so I grabbed my bag and ran out into the downpour too. They were sprinting to their cars, engines roaring to life, windshield wipers flying. By the time I'd started the Dodge truck and reversed, all four cars were gone.\n\nThe Keeper had walked out onto the front steps. He took a sip of his tea.\n\nThe reality of the situation, that we were just leaving him there in the house, a complete stranger, was too wild to fathom.\n\n\"Don't worry!\" he shouted cheerfully at me over the rain. \"I promise not to steal the silver.\"\n\nI floored the gas. As I roared down the driveway, I had the acute feeling of being chased. Yet, rounding another bend, I saw no one behind me. When I took a final glance back at Wincroft, the red brick mansion sinking behind the hill, even the Keeper appeared to be gone.\n\n* * *\n\n\u2014\n\nIt began to get dark. The rain was relentless, the sky black and blue. As soon as I'd gone a few miles, peering in at every driver to make sure they were alive and not ghosts, aliens, or zombies (most doing double takes, wondering what my problem was), I began to relax. All the drivers looked human, alive, and ordinary, chewing gum, fiddling with the radio, utterly at ease with what day it was, what time it was.\n\nEverything was normal.\n\nI called my mom again.\n\n\"Bee?\"\n\n\"Where are you?\"\n\n\"In the movie. What's going on? You scared us, the way you sounded before\u2014\"\n\nI drove straight to the Dreamland in Westerly. My parents were waiting outside, ashen. I parked in the fire lane, leaving the engine running. I wrenched the door and ran, throwing my arms around them.\n\nThey were real. I wasn't dreaming. It was going to be all right.\n\nMy mom was distraught. \"You're never speaking to any of those people again\u2014\"\n\n\"Victoria,\" admonished my dad.\n\n\"What? _Look_ at her. She's completely undone. We're not going through this again. No. Those kids are rotten. Spoiled. They'll live their entire lives without ever turning around to see the mess they've made, Mommy and Daddy always running after them with a maid and a checkbook.\"\n\n\"They're just kids.\"\n\n\" _Just kids_ left our daughter barely able to eat or sleep for two months, if you remember.\"\n\n\"That was shock. And grief.\"\n\nI was crying, but of course they couldn't understand the real reason, that it was relief. The day that had already happened\u2014whatever it was\u2014hadn't been real.\n\n_This was real._\n\nI managed to calm my parents down, and we went to dinner at the Shakedown. We talked with Artie, who gave us free apple pie. We strolled along the boardwalk and talked about the umpteenth offer from developers who told my dad he had to sell the Captain's Crow so they could build condos. Though my parents were alarmed, not just by my abrupt appearance, but by the uncharacteristic gusto with which I was approaching spending an evening with them\u2014something I had done with relative apathy all summer\u2014they said nothing. They pretended they believed my excuse for abruptly leaving Wincroft: \"I had to get out of there. We've all outgrown each other, you know?\"\n\nThey also humored my manic need to keep the night going, to walk a little farther down the boardwalk, to stare in at every sailboat painting in every window of every art gallery, to walk out to the old swings on the beach where someone had spray-painted on the wall LIFE IS BUT A DREAM, thereby postponing the inevitability of driving home and going to bed.\n\nI was afraid to sleep, because the glaring fact that _I had already lived this day_ nagged like a bad pop song that wouldn't leave my head.\n\nWe got home just after midnight. Dad drove the Dodge RAM as I said I was too tired, though the real reason was I was scared to be alone in the car. We filed into the house, my dad yawning. My mom loaded the dishwasher.\n\n\"Will you stay with me until I fall asleep?\" I asked her.\n\n\"Of course.\" She smiled, though I could tell the question worried her. The last time I'd asked her to do that, it was just after Jim died.\n\nShe sat on my bed as we talked about changing the menu at the Crow, the community vote to tear down a drawbridge. I knew she wanted to ask me about them, my old friends, what had happened tonight, but thought better of it.\n\nAt one point she stood up to inspect the white daisy wallpaper of my room.\n\n\"I can't believe it. Your dad said he fixed this.\"\n\nShe scratched at a seam in the corner, tugging the edge. A large chunk immediately peeled from the wall.\n\n\"Are you _kidding_ me? There's actually mold here.\"\n\n\"It's a sign you and Dad should sell the Crow and retire to Florida.\"\n\nShe crossed her arms. \"Do I look like someone who wears a visor?\"\n\nI began to feel heavy sleep falling over me. She said something about my dad's bad back, how it was hurting him more than he let on. I held her hand as I passed out.\n\nMy mom's hand was real. What had come before was not. So a day had decided to repeat itself. _Is it really that big a deal?_\n\nWhat lies the mind will tell to keep you safe.\n\nThe mind does its best to lessen the impact of any catastrophe. It really tries its best. But then the distance between reality and woven fantasy becomes too great for even the mind to bear. All those words of calm and relief, the hope that everything will be all right in the end, can't help stretching and tearing and fading to nothing.\n\nThen you wake up screaming.\n\n* * *\n\n\u2014\n\nI woke up in the downpour in the backseat of the Jaguar, Martha and Kipling beside me again. When I sprinted away from them into the house, I was shaking so badly I had to sit on the couch, feet apart, hands on my knees, trying not to hyperventilate.\n\nI was here again. I was back at Wincroft. At least I was alive.\n\nBut was this life?\n\nGandalf was running in circles around the living room, barking.\n\n\"No. No. No!\" shouted Cannon.\n\nHe was at the kitchen island typing on his laptop again, though\u2014undoubtedly after seeing that the date was the same\u2014he slammed it closed and threw it across the room.\n\nI realized dazedly, glancing up, that Whitley was outside, in the throes of one of her rages. Completely soaked, she was pulling the white umbrellas out of the patio tables and launching them over the railing.\n\nHer temper had been legendary at Darrow.\n\n\"Psychotic fits,\" the cattier girls used to hiss.\n\nI'd always found it enviable\u2014that Whitley could be so beautiful and smart, and on top of that so unconcerned about causing a scene or curbing her biblical emotions. It seemed unfairly glamorous, like she was the untamable heroine of a Victorian novel. (Even the oft-gossiped-about phrase around school\u2014 _Lansing's temper_ \u2014sounded gorgeously bygone, like the name of an exotic illness with no cure.) To be so wild\u2014it was how I longed to be. Wit surged into battle. I froze. Whitley threw her head back and screamed. I was mute. Her rages were Olympian, five-star, multiplatinum. They came from some boiling place inside her not even she could explain. Face flushed, eyes flashing, she'd demolish her dorm room, rip pages from every textbook, punch walls, overturn tables, tell off a teacher with zero care for tact, mercy, or an aftermath. It always seemed to me in those moments that Whitley was witnessing some alternative world invisible to the rest of us, something ugly and so vast it couldn't be fit into the English language.\n\nHer rages got her sent to the infirmary. They would have gotten her kicked out if it weren't for her mom, the Linda, CEO of the pharmaceutical group Lansing Drugs, flying in from St. Louis in her fat mink to smooth everything over, which meant funding another wing for the library. It was the reason Whitley got special permission to leave school to go see a psychologist up in Newport. Whenever a fit happened, I'd always run to her side and hold on to her, like some astronaut trying to make sure my colleague didn't float out into space.\n\nNow, as I watched her seize a deck chair and throw it over the railing screaming, I could only observe her blankly, unable to move. I couldn't help her. I couldn't help myself.\n\nKipling and Martha had wandered in and were looking around the kitchen like people visiting their property after a tornado.\n\n\"We have to call someone,\" Kipling said, his voice shaking. \"The FBI. The CIA?\"\n\n\"And say what?\" asked Martha, turning to him. \"Time has become a broken record?\"\n\n\"There've got to be others going through this. It's a national emergency.\"\n\n\"I'm sure Anderson Cooper's all over it,\" muttered Cannon. He was on the floor, hands linked around his neck like he was in a bomb shelter. \" 'Today. A new kind of breaking news. Yesterday is today. _Again._ More on this story as it never develops. Tweet us your experiences with hashtag Groundhog Day is real.' \"\n\nKipling grabbed the remote and turned on the TV, flipping through channels, every one yelping something normal. _Coming up, we'll show you how to make a three-minute omelet. Keeps whites white and colors brand-new._\n\nThe doorbell rang.\n\nNo one moved.\n\nWithin seconds the Keeper had strolled inside, a sympathetic, even grandfatherly look on his face. There was something insidious about him now: same suit, same tie. I felt like I was going to be sick.\n\n\"This will be the worst of it,\" he said. \"It's the second wake that feels the most catastrophic.\"\n\n\"Tell us what to do,\" said Martha.\n\n\"I did. Take the vote.\"\n\n_Take the vote._ As if it were just a matter of making a left turn rather than a right.\n\nWhitley must have spotted the old man from outside, because suddenly she heaved the sliding door open and stood in the doorway, panting and scowling at him, gusts of rain blasting around her like a storm scene in an old movie. Before anyone could stop her, she was sprinting inside. She grabbed a Chinese vase off a table and slung it at the old man's head.\n\nHe crumpled to the floor. Cannon ran to Whitley, but she brutally elbowed him off, grabbing the Keeper by his necktie and forcing him into a chair. Then she was barreling into the kitchen, pulling open drawers, tossing pots, ladles, cooking spoons to the floor.\n\n\"The cycle of violence is actually a pointless denial of reality,\" said the Keeper, holding his head.\n\nWhitley was back in front of him with cooking twine, brutally tying up his wrists, brandishing a fourteen-inch carving knife inches from his jaw as she sliced the string. Crouching, teeth gritted, she moved to his ankles. The Keeper didn't protest, only watching her, bemused, like a father when his four-year-old decides to bury him alive at the beach.\n\nShe dragged a stool over and sat in front of him, brushing her hair out of her eyes.\n\n\"Start talking.\"\n\n\"About what?\" asked the Keeper.\n\nShe smacked him hard across the cheek.\n\n\"Whitley,\" reproached Cannon.\n\n\"Tell us who did this and how we get out of here.\"\n\nThe Keeper closed his eyes. \"I've already told you. The vote. As for _who_? There is an infinite number of possibilities. The universe, God, the Absolute, the Supreme Being, He Who Actually Is, Adonai, Ahura Mazda\u2014\"\n\nShe slapped him again.\n\n\"Wit,\" whispered Kipling. \"You think it's wise to go all Tarantino on this poor man?\"\n\n\"He's not poor. He's toying with us.\"\n\nShe slapped him again. The Keeper remained unperturbed, blood trickling from his nose. I started to cry. And yet I made no attempt to stop her. None of us did. We stood there, frozen, all doubtlessly wondering\u2014terrible as it was to admit\u2014if hurting the Keeper might reveal something, something that would end this. He'd confess it was an elaborate game; the curtain would fall, scenery crashing. We'd laugh. _How hilarious. You really had me going there._ I also couldn't help hoping that, as with so many nightmares I'd had as a child, if things became sufficiently strange, the dream would at last puncture and I'd wake up.\n\nWhitley hit him again.\n\n\"The final three minutes of every wake you will each vote for the single person among you who will survive\u2014\"\n\n\"Why only one?\" asked Martha sharply, stepping beside Whitley.\n\n\"I can't explain the whys and hows of the Neverworld. They were determined by you.\"\n\n\"But if time has stopped,\" asked Cannon, \"why can we return to our normal lives?\"\n\n\"Only for eleven point two hours. Six hundred and seventy-two minutes. The length of your wake. For Cannon and Whitley it's six hundred and seventy-five. At the end of that time, you will all wake up in the Neverworld again, as surely as Cinderella's stagecoach turns back into a pumpkin. Even though your accident produced a snag in the space-time fabric, a crinkle in the cloth, the present world hasn't disappeared. It remains alive all around you, a bullet left in the gun chamber.\"\n\n\"What is the significance of our arrival time in the wake?\" asked Martha.\n\n\"The beginning and end of a wake are based on an infinite number of factors, including violent impact, strength of connection, and random chance.\"\n\nWhitley, seemingly unable to hear another word, flung down the knife. She seized her phone off the kitchen island and had a curt, unintelligible conversation before hanging up, shoving her feet into her Converse sneakers.\n\n\"What are you doing now?\" asked Cannon.\n\n\"Driving to T. F. Green.\"\n\nIt was the airport for private jets outside Providence.\n\n\"I booked the jet to Hawaii. We're leaving in an hour. Let's go.\"\n\n\"Won't change a thing, I'm afraid,\" said the Keeper.\n\nShe glared at him. \"We will be in a plane thirty-six thousand feet over the Pacific Ocean at the end of the\u2014what did you call it, the _wake_? What's going to happen? We just vanish out of our seats like some Willy Wonka magic trick?\"\n\n\"You'll see,\" said the Keeper.\n\n* * *\n\n\u2014\n\nEveryone went with Whitley except me.\n\nI couldn't. I was too devastated, too scared to move so far away from my parents, to be trapped in a box in the sky with them.\n\n_Them._\n\nThey were _them_ to me too now. I wasn't one of them, not anymore. If this situation had made anything clear, it was that: that the very people I'd once loved and trusted most in the world had become total strangers.\n\nWhat had I done to deserve this? To end up in hell with _them_?\n\nI couldn't think about it. No, I couldn't let my mind move ahead. It had to stay on a tight leash tied to this moment. It was all I could handle.\n\nI watched them pile with varying degrees of conviction into Cannon's Jeep. It was obvious that they suspected Whitley's plan, an impetuous flight westward to a tropical island, was futile. Yet they went ahead. For a show of solidarity? Some last, vain hope that it might actually work, that the Linda's Gulfstream V tearing through the pink cotton-candy clouds with its beige calfskin seats and trays of fanned-out mango slices would be the loophole, the wormhole, the Get Out of Jail Free card to puncture this nightmare?\n\nI stumbled down the steps, barely aware of the rain drenching me as I climbed into the truck. As I backed out, I saw that the Keeper had managed to free himself from Whitley's knots. Once again he was at the entrance, his face bloody and red, Gandalf at his side, as if the dog had always belonged to him.\n\nThis time the old man didn't utter a word. He didn't have to. His smile at me as I drove past him said it all.\n\n_See you later._\n\nYou can't stay awake.\n\nWe tried that. No matter how many cups of coffee or how many cans of Red Bull or Monster energy drinks you down, no matter how many caffeine pills or how much ginseng you take, your body gets pulled into the heaviest hollow of sleep you've ever felt in your life. The next thing you know, you're right back where you started.\n\nBack at the wake.\n\nYou can't kill yourself either.\n\nKip tried that. He hanged himself with one of E.S.S. Burt's belts in an upstairs bedroom. I didn't see him. Martha told me. The next wake, as usual, he was right beside me in the backseat of the Jaguar, a healthy color, no black-and-blue marks around his neck, no swollen face.\n\nLike nothing had happened.\n\n\"We're immortal,\" said Cannon. \"We should take over the White House.\"\n\n\"In eleven point two hours?\" said Martha. \"That's not enough time to drive to Chicago, much less rule the free world.\"\n\nTell your parents. Call the police. Have the Keeper arrested. Call a shrink. Check into the psych ward of Butler Hospital and ask the attending physician to make tomorrow arrive, please. Confess to a priest. Tell a bus driver, a cabdriver, the tired waitress at the twenty-four-hour diner who's seen it all, the bent-over elderly woman in the frozen foods section of Price Rite buying a shocking number of pepperoni Hot Pockets, the man in the leather jacket browsing engagement rings at Kmart. Read the two hundred and fifty-two books in the science section of the Warwick public library, and a zillion textbooks on Google Books, trying to determine if ever in the history of the world some sage like Copernicus, Aristotle, Darwin, or Hawking has ever written or even hinted about such a thing as errors in time, cosmic waiting rooms, lethal lotteries in limbo, human terrariums in hell.\n\n\"What's the subject you're searching for again?\" the librarian asked me.\n\n\"It's called a Neverworld.\"\n\nShe typed on the keyboard, shaking her head.\n\n\"Nothing comes up in the Library of Congress.\"\n\nWe tried every one of these things in the beginning.\n\nEvery time, we woke up in the exact same place, exact same time. We were songs on repeat, flies in a mason jar, echoing screams in a canyon that could not fade.\n\nThe ongoing experience of Recurring goes against the very heart of being human, and it is\u2014I will tell you this without flinching\u2014unbearable. The mind rages trying to disprove it. When it can't, the brain breaks down with shocking ease. The psyche is fragile. It is a child's sand castle in an incoming tide. Never before had I understood how little control we had over our world, or really anything except our own actions, and now my little life didn't even belong to me. We were helpless passengers strapped inside a spaceship circling Mars. The sun, the sky, the stars\u2014how long did I stare at them, lying on the deck chair by the pool in the pouring rain, wishing I could just be them, a collection of gas and fire? I'd even take a beetle, a blade of grass, anything, so long as it was outside the Neverworld.\n\n\"Take the vote,\" urged the Keeper. \"Just take the vote.\"\n\nWe took the vote. Of course we did.\n\nWe voted for the first time early in our arrival in the Neverworld. Twilight Zone. Purgatory. Doomed-Fate- _Survivor_ -Homeroom- _Freaky-Friday_ Bullshit. We called it all kinds of names, as if insulting the unknown forces keeping us here would make them change their minds.\n\nWe assembled in Wincroft's library like colorful characters in the final pages of a murder mystery waiting for the genius detective to unmask the killer. We sat in club chairs. Whitley served champagne. We wrote the name of our chosen survivor on a scrap piece of paper, the Keeper collecting them.\n\n\"There is no consensus,\" he announced.\n\nThe second time we voted, we each gave a speech beforehand in an attempt to persuade the others why we deserved life and not the others. We were defense attorneys in a courtroom speaking to a jury of the prosecutors, a circular setup of justice that would never work. The speeches ranged from altruistic (Cannon) to woodenly scientific (Martha) to childish and tone-deaf (Whitley, revealing a charitable streak she'd never had before, announcing she'd supply the entire continent of Africa with clean water). Kipling, when he stood up to speak, fell over, he was so drunk.\n\n\"You should vote for me because you _shouldn't_ vote for me,\" he said. \"I'm a mediocre, fucked-up shithead.\"\n\nI spoke last.\n\nAll I said was that I was an ordinary girl destined for an ordinary life, but they could vote for me because I'd make it my aim to do small acts of kindness every day.\n\nAs I said it, I was acutely aware that I sounded as disingenuous and desperate as they all did. Even worse, none of them were listening. They watched me, sure, but their attention was buried under the weight of their fates, fastidiously, hungrily inspecting it like Gollum inspecting the Ring, wondering if the Neverworld was real.\n\nI couldn't blame them. I was a blubbering mess too. Rarely had I passed the eleven point two hours without bawling as I drove to Westerly to see my parents at the Dreamland, usually just observing them without their knowledge, because to actually spend time with them made me sob uncontrollably the next wake. I'd tried explaining to them what was happening.\n\n\"I've been in a car accident, and I might die, and this limbo is called the Neverworld Wake according to this weird old man who won't leave us alone.\"\n\nThey always listened. Yet I could see that the only real feeling they had was devastation, believing that Jim's death had messed me up even more than they'd realized and I needed twenty-four-hour psychiatric care. So I'd gotten in the habit of sitting in the theater, unseen, a few rows behind them, beside this massively fat guy in a Brooklyn Book Drop T-shirt. I always smiled at him, thinking: _Do you realize how lucky you are? You have a tomorrow._ I ate popcorn, watched _His Girl Friday,_ and snuck out before the lights came on.\n\nThe result of that vote was no different.\n\n\"There is no consensus,\" said the Keeper.\n\nWe all voted for ourselves. I couldn't foresee a time when we wouldn't. It was all we had to keep us going: the possibility, however remote, of getting out of here, of getting back to life.\n\nAnd all the while the Keeper watched us.\n\nHe was still there, appearing when least expected. Sometimes he came inside and made tea. Sometimes he worked on the Wincroft grounds as a gardener, wearing a black hooded slicker. In spite of the rain\u2014which would, during some wakes, turn impossibly to snow, temperature dropping, swirls of snowflakes spinning like miniature tornadoes through the air\u2014the Keeper trimmed vines, rosebushes, ivy, and privet, the wisteria and lilac knotting the trellis. He swept the stone paths and hoed flowerbeds. He stood atop a green ladder and cleared dead leaves from the gutters, wiped the glass panes clean on lanterns and lamps. He removed lichen from the wings of the crow gargoyles silently cawing.\n\nOther times he could be spotted from a distance, a faceless silhouetted trespasser hurrying across the lawn and into the woods, as if taking a shortcut through Wincroft on his way somewhere else, somewhere unknown.\n\n* * *\n\n\u2014\n\nI don't know how long we'd been in the Neverworld when we had the fight.\n\nTime was vague here. It miraged and optical-illusioned the more you tried to look back on it, or fit it into a traditional monthly calendar. On close inspection, the hours were real. But if we tried to add them up into some larger understanding of the passage of time\u2014how long we'd been here\u2014they evaporated and grew unclear.\n\nThe passing of four wakes felt the same as four hundred.\n\nThe more wakes that passed, the more terrified I became. I could feel the others growing listless and distant, as if disinterested in ever actually leaving.\n\n\"I vote for Kanye!\" shouted Cannon, raising his glass. \"Kanye is my choice for who lives.\"\n\n\"There is no consensus,\" announced the Keeper.\n\nWhitley began to drink all day. So did Cannon and Kipling. Then all three started helping themselves to the pills E.S.S. Burt kept in his master suite, hundreds of orange bottles of uppers and downers lining the medicine cabinets like candy in a sweet shop. It wasn't uncommon for them all to be either manically hyper or unresponsive and lethargic. Kipling paced outside, having conversations with the rain, wearing nothing but that pink wig and a green silk peacock-patterned bathrobe belonging to one of Burt's girlfriends.\n\nOnce, while gathering everyone for the vote, I couldn't find him. Searching the mansion, I finally spotted him floating in the pool on a swan raft in the torrential rain.\n\n\"Kipling!\" I grabbed the leaf net and used it to haul him to the side.\n\nHe could barely open his eyes. \"Hello? You there, God? It's me. Judy.\"\n\n\"Kipling. Can you hear me?\"\n\n\"I'd like to order room service, please. I'd like the spaghetti Bolognese.\"\n\nHe rolled off the raft into the pool, sinking. I pulled off my shoes and raincoat and dove in after him, finding him drifting motionless along the bottom. Madly I kicked him back to the surface.\n\n\"Kipling! Can you hear me?\"\n\n\" 'It's the final countdown,' \" he sang, his eyes slits.\n\nI was the lone nurse working in a madhouse.\n\nWhile Martha had remained sane, she had also decided to remove herself, washing her hands of the situation, it seemed, ducking out without word at the beginning of every wake. She spent the day outside. A few times at dusk I caught sight of her wandering the woods fringing the far lawns, hauling her black bag, studying the treetops with a pair of binoculars like some professional bird-watcher, or an environmentalist recording evidence of acid rain. She'd fumble in her bag, which looked so heavy I wondered if inside was a copy of the same underground book, _The Bend,_ she'd lugged around Darrow. Instead, she'd remove a thin black notebook and scribble in it for a minute before trudging on. Once, I ran after her.\n\n\"Martha!\"\n\nShe kept walking, pretending she hadn't heard me.\n\n_\"Martha! Wait!\"_\n\nShe stopped and turned. I could see she didn't want to be bothered\u2014certainly not by me.\n\n\"I'm worried about them,\" I said.\n\nShe nodded. \"So?\"\n\n_So?_ I could only stare at her, rain coursing down my head and arms. Hadn't she witnessed what was going on? Didn't she care?\n\n\"They're going crazy. They're not taking it seriously anymore. I don't know what to do.\"\n\nShe shrugged. \"It's all part of the acceptance.\"\n\n\"What are you talking about?\"\n\n\"When criminals are sentenced to life in prison, there's a ninety-four-percent chance of mental collapse within the first year.\" She shrugged. \"Just leave them alone.\"\n\n\"No way. We have to stick together.\"\n\nTo my shock, with another awkward shrug, she began to walk away.\n\n\"Where are you going?\" I shouted.\n\nShe didn't answer.\n\n\"I need your help! _Please!_ Don't you want to get out of here?\"\n\nShe held up a hand\u2014a mild gesture of acknowledgment to a child having a tantrum\u2014and kept walking.\n\n* * *\n\n\u2014\n\nWe were shipwreck survivors in a raging sea. Now they were forcing me to let go of their hands so they could sink into the waves and drown.\n\nI was going to be stuck here forever.\n\nHere, in the Neverworld, where I'd never grow old.\n\nNever have a family.\n\nNever fall in love.\n\nI was an immortal vampire without any perks. No bewitching beauty, no golden eyes or shimmering skin, no ability to run three hundred miles an hour and flip cars over.\n\nI was a ghost with no haunt. I couldn't turn TVs to static or swivel porcelain doll heads 360 degrees, causing normal humans to have nervous breakdowns. I couldn't make toddlers stand in zombie trances in living rooms, captured in shaky found footage in the dead of night.\n\nI was a ticking clock in a timeless world.\n\nWithout time, nothing had meaning. Never before had I understood how crucial the passage of time was to caring about something. It gave it an expiry date, a wick, a rush, a burn. Without it, everything sat in place, dumbly waiting.\n\nIn my darkest moments I thought of Jim.\n\nI'd come to Wincroft to find out what happened to him. Now even that question, the one I'd spent the past year turning over and over in my mind, shriveled and flattened in the face of the Neverworld, like a little worm on the driveway in the beating sun.\n\n* * *\n\n\u2014\n\nThe night of the fight, I'd just returned from the Dreamland. Letting myself into Wincroft, I heard screaming coming from upstairs. I sprinted up the staircase, realizing they'd locked themselves in E.S.S. Burt's bathroom in the master suite.\n\nI knocked. \"Is everything okay?\"\n\nThere was no answer but snickering.\n\n\"It's almost time for the vote.\"\n\nThis was met with more laughter.\n\n\"Hello?\"\n\nThe door was flung open. Whitley stood there wearing an oversized red-sequined evening gown. Her eyes were bloodshot and smudged with eyeliner. Kipling was draped like an exhausted panther over the edge of the tub. Cannon was sitting on the counter, bandana tied marine-style around his forehead. It was obvious from their flushed faces\u2014and the array of empty Dom P\u00e9rignon bottles scattered across the tiles\u2014that they were wasted.\n\n\"Sister Bee, charmed to see you,\" Whitley said primly. \"We won't be joining you. _Ever._ \"\n\n\"What?\"\n\n\"We aren't voting. We're staying at Wincroft until the end of time. So there.\"\n\nShe rolled her eyes at the look on my face.\n\n\"Oh, God, Bee. Stop mothering everyone. Your good-girl nun act is never getting you chosen. In fact, it'll be over my maggot-infested body that I ever allow some Mother Teresa type to triumph on to life. No way. It goes against my very life philosophy that one must get filthy to live. You must get down in the dirt or you've done nothing.\"\n\n\"I'm not Mother Teresa. I'm not a nun. I'm not even that good.\"\n\nShe waved her hand as if shooing a fly and turned, idly surveying her reflection in the mirror.\n\n\"It's not about the vote,\" I went on. \"It's about staying together. We could lose ourselves forever in this place. Remember what Jim used to say about friendship? About _us_? What we have is a loyalty that can see us through anything.\"\n\nWhitley bit her bottom lip, trying not to laugh.\n\n\"You still love him. _Wow._ He was the only person you ever saw in a room. And it's still true, even though he's dead. By the way. Did you ever wonder why he chose you? Out of all the girls at school?\"\n\nShe rubbed some lipstick off her chin. I braced myself, because I knew what was coming. Her tantrums always began this way: she made some grand opening statement like a veteran prosecutor holding a jury rapt, the perfect set of words to slice her target in two.\n\n\"He chose you because a plain setting makes the diamond sparkle brighter.\"\n\nI said nothing, willfully reminding myself to ignore whatever Wit said when she was angry. Yet I felt my face flush, a nervous voice in my head chattering _It's not true._\n\n\"I disagree,\" said Cannon, frowning. \"The problem always was that _you_ loved Jim.\"\n\n\"He's right,\" muttered Kipling. \"It was obvious, child. Like a wart on a lifeguard's big toe at a public pool.\"\n\n\"Oh, please.\" Whitley glared at him. \"You were obsessed with him. Admit it. Don't think we didn't see you ogling him, your Southern accent going all syrupy around him, like you thought you could seduce him with some third-rate community-theater impression of Truman Capote. And _you._ \" She turned to Cannon. \"You were happy when he died.\"\n\n\"I was gutted,\" he answered in a clipped tone.\n\n\"Gutted with _glee,_ maybe.\"\n\nCannon glared at her, his face implacable. \"You hate the Linda? Well, too bad. You're her to a tee. All that's missing are the face-lifts, the cankles, and the army of men who have fled you like a storm warning for a Category Five hurricane. But don't worry, angel. That will come in time.\"\n\n\"There is no time,\" noted Kipling, holding up a finger, half asleep. \"Not anymore.\"\n\nWhitley stared at Cannon, mouth open, shoulders trembling.\n\n\"Cannon didn't mean that,\" I whispered, touching her arm.\n\nShe threw off my hand, seizing a bottle from the floor. Cannon ducked as it exploded against the mirror behind his head.\n\n\"You're all monsters! _Get out of my house!_ \"\n\nShe elbowed me out of the way as she fled. Seconds later, she reappeared at the end of the hall brandishing a shotgun, aiming for my head. I took off down the staircase as a shot blasted the ceiling, chandelier swinging, bits of plaster and molding crashing to the ground.\n\n_\"Get out! Termites! Leeches! Rats!\"_\n\nMore shots rang out as I reached the front door and pulled it open, colliding with Martha.\n\nShe was wearing a green poncho, soaking wet from the rain.\n\n\"Beatrice? What's the matter?\"\n\n_\"Worms! Maggots! Those disgusting fish at the bottom of the ocean with switchblade teeth! GET OUT! ALL OF YOU!\"_\n\nI didn't answer. I sprinted outside to my truck and took off, blasting across flowerbeds, mud puddles, broken branches, swerving back onto the driveway as I tried to catch my breath.\n\n* * *\n\n\u2014\n\nI had to get away from them. I had to clear my head.\n\nEverything they said, I kept reminding myself, was just the Neverworld talking. Being stuck here, day after day, made you think and feel the darkest things, as if daring the universe, God, whatever was out there, to prove that they weren't true.\n\n_A plain setting makes the diamond sparkle brighter. You loved Jim. You were happy when he died._\n\nI didn't want to think about it. I drove straight to the Captain's Crow, letting myself in with the spare key my dad kept stashed behind the outside wall thermometer. I'd make a grilled cheese, eat some Wreck Rummage, and fall asleep. I'd figure out what to do tomorrow, yesterday, today, whatever it was.\n\nThe moment I entered the restaurant, however, slipping through the tiger-stripe shadows, I realized something was very wrong.\n\nThe caf\u00e9 chairs, normally overturned on the tables, had been tossed all over the floor. The glass on the display of ice cream was cracked. Within the smells of toast and sunscreen was something else\u2014something rancid. I'd just slipped into the kitchen, wondering if Sleepy Sam had forgotten to take out the trash, when my sneaker kicked shards of glass. Bending down, I saw I'd stepped on my great-grandfather Burn's pencil portrait. It had moved from its usual place over the door. Somehow it had ended up by the stove, facedown, the frame broken.\n\n_There was a robbery._ That was my first thought.\n\nThen I felt the wake descending, the blackest of sleeps pulling over me like a coffin lid, and I realized something else was going on, something strange.\n\nI heard a faint tapping. Looking up, I screamed. In the window overlooking the alley by the sink, a face stared in at me.\n\nThe Keeper.\n\nHis gaze was neither hostile nor friendly, only stark, his jaw slashed by shadow. I realized that he was cutting away the ivy and vines of honeysuckle that had overtaken the wall, which my mom had never gotten around to pruning.\n\nWhen I stumbled outside to confront him, he was striding down the alleyway.\n\n\"Hey!\" I shouted after him. \"What do you want?\"\n\nHe ignored me, splashing through puddles, the clippings in a bag tossed over his shoulder, rounding the corner.\n\n\"Leave me alone!\"\n\nIt was then that it occurred to me what he was.\n\nThe Keeper was a reminder.\n\nThe vote. The vote. _The vote._\n\nAfter the fight, they went their separate ways. The moment they sprang back to the wake, Kip, Martha, Cannon, and Wit dispersed like seeds off a dead dandelion. They left without a word, sometimes without even looking at each other.\n\nI let them go. I had no choice.\n\nWas it depression? Probably. Fury over their fate? That too. Or maybe they just wanted to see what it felt like to climb beyond the Danger signs and Keep Out barricades, the barbed wire protecting the edges of the lookout atop the skyscraper, and jump.\n\nWhat happened to us didn't matter. Peril didn't exist. If the Neverworld Wake had one asset, it was that we could remain forever young, like the Alphaville song. We could live and die and live again, without consequence.\n\nKipling began hitchhiking.\n\nThe moment he appeared in the back of the Jaguar, he took off down the drive. After he did this countless times, his expression an enigmatic mixture of resolve and expectation\u2014as if he were actually looking forward to something\u2014I followed him. I tailed him out to the main road, where, just before the stone bridge, he began walking backward, sticking out his thumb.\n\nIt was always the sixth car that stopped for him. A brown Pontiac with a dented fender.\n\nI watched him disappear into that Pontiac so many times, I just had to know what was so captivating that he couldn't miss out on it, not even for one wake. So I caught up to him.\n\n\"Where are you going?\" I asked him.\n\nHe turned, startled to see me, then annoyed. _\"What?\"_\n\n\"Who picks you up in the Pontiac?\"\n\nHe kept walking. \"It doesn't matter.\"\n\n\"Where do you go?\"\n\n\"Leave me alone, Bee.\"\n\n\"Just tell me.\"\n\n\"No.\"\n\n\"Why not?\"\n\n\"None a your goddamn business.\"\n\n\"Then I'm coming with you.\"\n\n_\"No.\"_\n\nHe was furious. He actually looked like he was considering hitting me, or tying me to a tree so he could get away.\n\n\"Tell me and I'll go back,\" I said.\n\nHe scowled, wiping the streaming rain off his face. \"Her name's Shirley.\"\n\n\"And?\"\n\n\"And she takes me with her to her chemotherapy treatment in Providence. Then we go back to her crappy apartment by a Stop and Shop and watch _Night of the Living Dead._ I cook her shrimp jambalaya and make a tuna salad for her cat named Canary. She thinks I'm a runaway from Mississippi. Sometimes my name is James. Sometimes it's Jesus. She undresses in front of me and asks me to touch her. She's religious. Thinks I'm some kind of savior from a different planet because I know so much about her. We talk all night. Now would you please go find your own disturbin' experience to get lost in? This one is mine.\"\n\nAt that moment, the brown Pontiac rounded the bend. Probably because I was there, or because Kip had a fake smile on his face, quite different from his usual laid-back, lounging-porch-cat demeanor, the car slowed for a second\u2014revealing a plain-faced woman, brown hair, white T-shirt, radio blaring the Cure's \"Close to Me\"\u2014then accelerated away.\n\nKip ran after her, waving. \"Hold on! Wait for me! _Shirley!_ \"\n\nThe car tore around the bend, vanishing.\n\n\"Look what you did!\" he wailed.\n\n\"I'm sorry.\"\n\nShaking his head, he took off over the bridge. He tried flagging down the next car, a red pickup truck, then a van, but no one stopped.\n\n\"Leave me alone!\" he shouted as he took off jogging down the road.\n\nI let him go. I understood. He looked forward to Shirley because for some reason she made him forget he was in the Neverworld. It was probably only for a minute. But that was a priceless minute in a century of worthless ones.\n\n* * *\n\n\u2014\n\nAfter I'd learned where Kip went, I followed the others.\n\nI had to. If I had any hope of ever making it out of here alive, I had to make sure I didn't lose them completely, that they didn't fall into some psychological rabbit hole from which they'd never be able to emerge.\n\nI also needed a mission. I couldn't sit through _His Girl Friday_ one more time. I couldn't watch my mom tell my dad with only a look that she didn't like the seats he'd chosen because they were too near the screen. Then, two seconds later: the bearded homeless man dropped the can of Old Milwaukee on the floor, muttering, \"Shit, man,\" and the old woman behind him left to go report him as the man in the Brooklyn Book Drop T-shirt stuffed a handful of popcorn into his mouth (dropping three kernels in his lap). This symphony of normality played the same way every time. I knew every word, stutter, quip, throat clear, sniff, cough, scratch, and burp, like the stage manager who'd watched the same performance a million times from the wings.\n\nThen there was the fact that my parents seemed so happy together it made me feel even more alone.\n\nI followed Whitley and Cannon next.\n\nThey snapped back to the wake three minutes before the rest of us.\n\nWhen I sprinted into Wincroft, they were already gone. They left no note. The only evidence was red brake lights retreating down the drive. Yet their cars remained in front of the house. This meant that they left in some other car, and together, which suggested that whatever wounds their words had left from the fight, they'd already healed, like the skin of superheroes.\n\nThat didn't surprise me. They never stayed angry at each other for long.\n\nChecking E.S.S. Burt's classic car garage, I noticed wet tire marks on the floor. I went into his office, read through his insurance forms, and was able to figure out that the missing vehicle was a maroon 1982 Rolls-Royce Silver Spur.\n\nFor the next few wakes, I tried to catch up to them.\n\nIt seemed impossible. They hadn't driven to the highway or any of the obvious coastal roads, so where did they disappear to, and so swiftly? Only countless wakes later, when I turned down an unmarked, narrow dirt drive, did I see the black wood sign painted in elegant Victorian script.\n\nDAVY JONES'S LOCKER. Another mile and there was a second sign: MEMBERS ONLY.\n\nI pulled into the parking lot. Davy Jones's appeared to be some kind of exclusive marina crowded with yachts. There was a white clubhouse and an outdoor tiki bar. Tanned crewmen in blue polo shirts strode purposefully along the docks, wielding umbrellas and iPads.\n\nParked directly in front of me was one maroon 1982 Rolls-Royce Silver Spur.\n\nAlmost immediately I spotted Whitley and Cannon.\n\nThey were speaking to a group of retirees, three couples in their sixties or seventies, the women with short dyed hair and lean bodies like little bits of punctuation. The men were fat and bald. They were laughing. In fact, Wit and Cannon were laughing _so much_ as I slipped out of my truck, keeping the umbrella low so they wouldn't spot me, I couldn't help gaping, incredulous at their all-too-convincing impression of being totally normal\u2014like two people with tomorrows.\n\nThey seemed to be waiting for something.\n\nApparently it was an invitation to board the super-yacht the _Last Hurrah,_ docked beside them. Because not a minute later, they were stepping with phony wonder up the teak steps, past the helicopter landing pad, and vanishing inside.\n\nBewildered, I strolled up to the boat. The uniformed crew were preparing for departure.\n\n\"Where you headed?\"\n\n\"Bermuda.\"\n\nMinutes later, the yacht cast off. That night, like all other recent nights, Wit and Cannon never returned to Wincroft to vote. By the next wake they were already gone.\n\nSo what were they up to? And why did the question fill me with such dread?\n\n* * *\n\n\u2014\n\nI had thirty-three minutes.\n\nThere were forty-seven minutes between the time I woke up in the Jaguar and the time the _Last Hurrah_ cast off for Bermuda. By minute thirty-three it was too late. There were too many crew members buzzing around not to be spotted. I was caught a million times.\n\n\"Excuse me? _Who_ are you?\"\n\n\"Hey!\"\n\n\"You're not authorized to be here.\"\n\n\"Is this the _Dream Weaver_?\"\n\n\"Is this _Cleopatra III_?\"\n\n\"I'm looking for Captain Martin. I'm his niece.\"\n\nI'd leave, stuttering apologies, ignoring the looks of suspicion as I snuck back to my truck. I'd watch as Wit and Cannon boarded that same yacht and took off into the open sea.\n\nMy only hope lay in immediately, the instant I woke, grabbing Cannon's car keys and sprinting to his Mercedes\u2014twice as fast as my truck\u2014taking a shortcut along a dirt service road, and barreling ninety miles an hour through marshes and sand into the Davy Jones's Locker marina.\n\nI'd park behind a tree and speed-walk to the small cruiser beside _Last Hurrah,_ where, pretending to be boarding that boat, I'd wait for the teenage deckhand to check his cell phone, at which point I had twenty seconds to dash up the steps and duck into the first door I came to. It led into an ornate game room with a jukebox and pinball machines. I then had fifteen seconds to slip up three flights to the staterooms and vanish into the bedroom at the end of the hall.\n\nIt overlooked the marina. It was there that, by cracking the window, I was able to eavesdrop on the outrageous scene\u2014or, rather, con job. Whitley and Cannon, posing as newly married college sweethearts from Columbus, Ohio, had just been informed of a critical problem with their rented yacht, thereby leaving their honeymoon in tatters. Loudly they lamented their plight, which happened to be overheard by the owner of _Last Hurrah,_ Ted Daisy of Cincinnati, who invited the poor young couple aboard.\n\n\"Why don't you spend the week with us? Plenty of room here for everybody.\"\n\n\"That's very kind, sir,\" said Cannon. \"But we couldn't.\"\n\n\"Nonsense. The downside is you'll spend your honeymoon with a bunch of old geezers. But we promise to stay out of your way. You'll have a chef, an activities director, and a range of toys at your disposal.\"\n\n\"What do you say, sugar?\" Cannon asked Whitley.\n\nShe nibbled a fingernail. \"I'm not sure, honeybun.\"\n\nI marveled at the way they had their act down, like a couple of seasoned Broadway tap dancers. How many wakes had it taken them to figure out the perfect formula for eliciting the invitation to board the yacht? Ten? Ten thousand?\n\n\"You kids are coming with us. I insist on it. Ted Daisy. This is my wife, Patty.\"\n\n\"Artwell Calvin the third,\" said Cannon.\n\n\"Anastasia Calvin,\" said Whitley, shaking her head. \"I really don't know what I did in a previous life to deserve such kindness. I think I'm going to cry.\"\n\n* * *\n\n\u2014\n\nWhat had I expected aboard the _Last Hurrah_? A relaxed vacation cruise? A beautiful, distracting dream where Whitley and Cannon could forget the Neverworld?\n\nThat wasn't it. Not at all.\n\nI should have known. Their relationship at Darrow had always been incendiary. They had sex in closets and classrooms, on rooftops, in the woods, on the balcony of the chapel, never once getting caught. They stalked hallways with their arms around each other like boa constrictors, students and teachers alike eyeing them nervously, though no one complained. They were in the top five of our class, after all. Whitley talked about their love as an insatiable need. I saw it as a lethal bullet speeding toward a target. Whether that target was one of them or some unsuspecting third party, I had no idea. They fought, made up, hated each other, couldn't live without the other for even one second.\n\nThey called each other Sid and Nancy. They stole things for fun. Anything on campus, no matter how big or small, could be targeted, like Mrs. Ferguson's AP Physics exams; a $12,000 seascape from an art gallery; Rector Trask's XXL tartan vest, which he notoriously donned for Darrow's Holiday Feast; even a John Deere excavator from the library construction site. They'd help themselves to whatever it was, resulting in a weeklong uproar of faculty announcements and threats of expulsion, a few unsuspecting students being summoned into a dean's office to detail what they knew\u2014until, with equal quiet and swiftness, the object reappeared. Their knack for burglary was not due to the usual reasons for acting out, like anger or some perverse craving for attention. It was a simple love for the art of deceit\u2014being a step ahead of everyone\u2014not to mention their ongoing need to outdo each other.\n\nEveryone whispered they'd be legendary if they stayed together. I secretly thought their connection was _too_ close, like twins. Cannon didn't have Whitley's temper, but he had her intensity and knack for manipulation, dropping a word here, an inference there, that would be the gram of uranium to turn a benign situation nuclear. They broke up couples, made teachers cry. When they finally called it quits senior year, their breakup was eerily silent, a biological weapon that had abruptly dispelled with hardly any smoke, defying all scientific explanation.\n\n\"Everyone knows adolescent love has a short shelf life,\" Whitley explained with shrug.\n\nNow it was clear that Whitley and Cannon boarded the _Last Hurrah_ for no reason other than that they'd decided that boat was their mad, twisted playground to tear into, as if they were two wild monkeys locked in a cage.\n\nIt was their padded cell. The soundproof room where they could scream their heads off.\n\nThe first night, I watched Wit get so drunk she vomited all over the dinner table on the platters of lobster and sirloin steak.\n\n\"Whoops,\" she said, wiping her mouth.\n\nThe second night, she danced provocatively with Ted Daisy. When his wife, Patty, saw what was happening, she called out in a drunken voice, \"Ted! Ted?\" like their fifty-year marriage had suddenly turned into a phone call with poor reception.\n\nOn another occasion, Cannon and Wit stripped down to their underwear and, climbing up onto the ship's railing, screamed, _\"Carpe noctem!\"_ Holding hands, they jumped, falling the fifty feet into the sea. Alarms sounded. Women screamed. Engines gasped to a halt. The crew members swarmed, shouting orders, two diving in with life vests.\n\n\"Find them!\" shouted Ted Daisy, desperately peering over the railing. He looked like he was having a heart attack. \"I'll be damned if I'm going to jail because of those wackos!\"\n\n\"We're going to lose everything,\" wailed Patty.\n\n\"We should have tied them up the moment we realized something was mentally off with them. We should have called the coast guard.\"\n\n\"It's _your_ fault!\" screamed Patty, her stiff blond hair standing up like pieces of potato chips. \"You invited them aboard because you wanted to impress that little blond piece of ass. You thought you had a chance with her. Ha! Hope you're happy now!\"\n\nHysteria. Panic. Fury. Despair. Fear. Alarm.\n\nIt all happened aboard the _Last Hurrah_ on a day that would not stop happening.\n\nI watched from back rooms, spare bedrooms, an electrical supply closet. I put on the extra crew uniform I'd found, and no one looked at me twice. I kept waiting for the right moment to appear, to try to talk down Cannon and Wit, bring them back from the razor's edge. I couldn't find it. I knew them too well. When they were like this, there was no stopping them.\n\nI remained where I was, peering out at the nightmarish scene through a crack in the door, terrified, sick, sometimes crying, wondering when it would stop.\n\nThen one night Cannon smashed a decanter over Ted Daisy's head. Ted shoved him into a display case stacked with crystal goblets. They began wrestling, overturning coffee tables and the dining room table. Then Cannon was sitting on the man's chest, strangling him.\n\nI'd had enough. I ran out of the closet and knelt beside Cannon, trying to pry off his hands. The old man was spitting and blubbering.\n\n\"Stop it!\" I cried.\n\nIt took Cannon another minute to let go. I tried CPR, compressing the man's chest, counting the way my dad had taught me. I checked his pulse. He was alive, but barely.\n\n\"You have to stop,\" I whispered.\n\nCannon surveyed me like I was a distant relative whose name he couldn't recall.\n\n\"You're making it worse. Because _we_ remember. These people don't. But you do. And the destruction will eat away at you.\"\n\n\"Oh, shut up, Bee,\" said Wit.\n\nShe'd risen from the sofa, where she'd been passed out cold.\n\n\"What are you even doing here? Spying? When will you realize we want nothing to do with you? We're not your friends anymore. You blew that when you went MIA after Jim. You think you can just ditch your friends like that and get away with it?\"\n\nShe shuffled toward me, her eyes red and threatening. I turned and ran, barging past the other guests, who'd been woken by the noise and were now, in their white terry-cloth robes and matching slippers, gaping in shock at the scene. I ran to the third-floor deck and spent the rest of the wake in one of the rescue boats, sobbing, hoping no one would find me.\n\nI never returned to the _Last Hurrah._\n\nCannon and Wit went the very next wake. Wit left me a message scrawled in high-drama red lipstick across the kitchen counter.\n\n> STAY AWAY.\n\nThe threat was unnecessary. I could never go back there.\n\nWould repetition eventually render even the _Last Hurrah_ boring, whereupon they'd return to Wincroft? Would one of them decide they wanted to live, to escape the Neverworld, to vote? Or would they simply move on to devouring something or someone else? The Neverworld held an infinite number of playgrounds, so it was possible, horrifying as it was to consider, that I'd never see them again.\n\nI couldn't think about that. Not yet.\n\nInstead, I turned my attention to Martha.\n\nIt was funny how I'd almost forgotten her. And I suspected it was just what she wanted.\n\nMartha no longer spent the day hiking Wincroft. Now she would hurriedly enter the mansion, retrieve a raincoat, and drive off, never returning.\n\n\"Where do you go?\" I asked her.\n\nShe whipped around in surprise. She hadn't seen me sitting in the rocking chair on the porch. Recovering, she pulled out her car keys, then opened an umbrella.\n\n\"I visit this silly Baptist church buffet up in Newport.\" She shrugged, her face turning red. \"I've turned it into my personal biosphere. Like I find a guy and see if I can get him to say 'I love you' by the end of the night. Or I approach a woman and see if I can get her to leave her husband. I'm trying to prove a theory about human nature. That anyone is capable of anything at any time, given a certain set of conditions.\"\n\nShe was lying. I could tell.\n\n\"Can I come with you?\" I asked.\n\n\"I prefer to be alone, actually.\"\n\n\"What about when you were hiking around with binoculars? What were you doing?\"\n\n\"Bird-watching.\"\n\nShe was lying about that too. She seemed fully aware I didn't believe her, yet she stared back at me, undaunted.\n\n\"Aren't you worried?\" I asked, trying to ignore the anger in the pit of my stomach. \"Upset? Scared? We've lost them all now.\"\n\nShe smiled thinly. \"I suggest you resolve yourself to your fate, Bee.\"\n\nAnd with that she turned and hurried down the steps to her car.\n\n* * *\n\n\u2014\n\nThe next wake, I headed straight to my truck. While Martha was inside getting the raincoat, I hid in a driveway down the street, and when she pulled out, I followed her.\n\nUnfortunately, my confrontation appeared to have tipped her off, because as soon as I pulled up behind her on the interstate, though I was three cars back, she took the first exit and drove in meandering circles around deserted office parks before pulling into Birchwood Plaza. She spent the next four hours wandering Urban Outfitters and Barnes & Noble and eating a calzone in the food court.\n\nShe knew I was there, watching her. Yet she was unconcerned. The next wake, she did the same thing at a different mall. The third time, another.\n\nThere was no way Martha spent her Neverworld wandering malls. She was doing that because she knew I was following her. She seemed to be banking on my eventually growing bored and moving on.\n\nSo I did. I stopped. Instead, I devoted the next few wakes\u2014or was it a few thousand?\u2014to figuring out how to follow her unseen.\n\nAnd so began my illustrious career in grand theft auto.\n\n* * *\n\n\u2014\n\nI was a panicky and apologetic thief.\n\nHundreds of times I was caught red-handed.\n\n\"Who the hell are you and what are you doing in my garage?\"\n\n\" _Hi._ Sorry.\"\n\nThankfully\u2014probably because there was something intrinsically sad about me, which could only be blamed on the Neverworld\u2014everyone let me off the hook.\n\nThe only car I could steal without getting caught was a rusted white van emblazoned with the words MCKENDRICK PEST CONTROL.\n\nIt belonged to the McKendricks, a hyper militia-family of seven living in a modest ranch house four doors down from Wincroft. All seven McKendricks were always home, so to get my hands on the keys was the closing act of Cirque du Soleil.\n\nIt took me forever to get it right.\n\nOne: hide in rhododendrons outside the kitchen, waiting for Bud McKendrick to wander into the living room for his Camel Lights. Two: dart into the kitchen pantry, trying not to trip on the bags of Healthy Weight cat food or the Macaroni and Cheese Storage Bucket with Gamma Lid from Target. Three: wait for Pete McKendrick to grab a Kit-Kat and head to the basement to watch _The Adventures of Jimmy Neutron,_ Bud to go upstairs for a nap, and Gerry and Paul, the twins, to go play soccer in the front yard in the rain. Four: slip into the den, scaring Tupac, the cat, who jumps six feet into the air and climbs the curtains. Five: snatch the car keys off the table and duck behind the sofa as Laurel McKendrick takes forty dollars from her husband's wallet. \"Heading to the store!\" Six: run back into kitchen and try to avoid four-year-old Kendall McKendrick.\n\n\"Who are you?\" she asked me, eyes wide in surprise.\n\nThere was no avoiding Kendall. No matter what, she always caught me.\n\nIt was the most incendiary moment of all: finding the perfect recipe of words that would stop her from wailing like a smoke alarm. I had tried everything. Nothing worked.\n\n\"I'm an angel.\"\n\n\"I work for the tooth fairy.\"\n\n\"I'm the Elf on the Shelf, and I need to borrow your daddy's truck.\"\n\nHow many times had I expertly trapezed my way through the McKendricks', only to crash to the ground, thanks to Kendall yelling her head off, prompting every McKendrick to descend on me.\n\n\"Dad! _Dad!_ \"\n\nI'd run for my life as the McKendricks\u2014all with variations on the same bulldog marine face\u2014swarmed their front yard.\n\n\"Stop! Burglar!\" they shouted through the rain.\n\n\"Dad, you're letting her get away!\"\n\nGood old Bud McKendrick never called the police. Probably because after five kids, it took more than some teenage housebreaker to rattle him. Frowning quizzically after me from the porch, more than a little blas\u00e9, he always let me go.\n\nFinally there came the wake when I told Kendall the truth.\n\n\"My name is Beatrice Hartley. I'm trapped between life and death in a place called a Neverworld. I'm trying to make it out of here, and to make a long story short, I need you to be quiet and go watch cartoons with your brothers. _Now._ \"\n\nShe nodded mutely and padded downstairs.\n\nLight-headed with amazement, I snatched Bud's Rams baseball cap off a chair, grabbed his Oakley sunglasses, unlatched the door to the detached garage, and ran out. I pulled on Bud's coveralls, hat, sunglasses, climbed behind the wheel of the van. Starting the engine, I was just wondering how in the world I was going to drive past the twins playing soccer, when Paul punted the ball into a neighbor's yard. I inched down the drive, turned right, pulled into another drive a few houses down, my heart hammering.\n\nA minute later Martha drove past me.\n\nI followed her Honda Accord all the way to Providence, to Brown University, to the third floor of a redbrick building on Thayer Street, to a corner office.\n\nARNOLD BELORODA, PH.D. read the brass plaque on the door.\n\nI watched Martha knock. A male voice answered \"Yes?\" and she entered. I heard her say hi as the door closed, and though I slipped closer in the crowded hallway, straining to hear the muffled voices inside, I couldn't make out any more.\n\nI Googled the name. Arnold Winwood Beloroda. He was an award-winning psychiatrist and professor emeritus specializing in group dynamic theory. He taught a host of classes at Brown. Making Ethical Decisions: The Good, Bad, and the Ugly. The Psychology of Manipulation and Consent. The Fantasy of Free Will. A senior seminar, Laboratory for Experiments in Social Persuasion. He had published thirteen nonfiction books, winning a slew of awards for one from the nineties, _Heroes and Villains._ According to the _Wall Street Journal,_ it was about \"the master-slave dynamics of concentration camps\" and other situations in which \"a large populace allows themselves to be controlled by a select few.\"\n\nI scanned Beloroda's articles in the _Harvard Review,_ the _Economist,_ and _Scientific American._ What was so compelling about him? What was so critical that Martha had gone to such lengths to hide him?\n\nThen it hit me. It felt like a pair of hands had begun to squeeze my neck.\n\nWhile the rest of us had been wasting time warring against the reality of our circumstances, Martha had been using the Neverworld's infinity to study.\n\nBeloroda had been teaching her how to manipulate the group so we would choose her.\n\nShe was figuring out how to win.\n\nI tailed Martha over and over again. Every time, she drove to Brown's Cognitive, Linguistic, and Psychological Sciences building. Every time, she visited Beloroda. They remained holed up in his office for three, four hours. Clearly she'd figured out a way to hook him, captivate him with some high-level question about group dynamics or a detail mined from his own papers that served as the magic key to Open Sesame the close connection, the meeting of like minds. When they finally emerged, Beloroda\u2014an elfin man with a turned-up nose and an overmanicured inky beard like a Rorschach test\u2014was beaming at Martha (now hauling a pile of textbooks he'd given her, as well as a legal pad covered with notes), bewitched by the sudden appearance of such an engaging new student.\n\nSharing an umbrella, they always strolled outside, deep in conversation, and chatted for another twenty minutes on the sidewalk. Once I crept behind them, hiding in an alcove where a few students were smoking under the awning.\n\n\"You're absolutely correct,\" said Beloroda. \"But here I would cite the philosophy of M. Scott Peck. In all groups there are four stages. Pseudocommunity. Chaos. Emptiness. And true community.\"\n\n\"Could you tell me more about the Milgram experiment?\"\n\n\"Ah. The blind obedience to authority figures.\" Beloroda chuckled. \"There's nothing I'd like more, but I'm afraid I'm due to join my wife at a party. How about we resume this conversation tomorrow after my Group Cohesion lecture?\"\n\nHe was unlocking his car, climbing in.\n\n\"It was a delight to meet you, Miss Peters. Until tomorrow?\"\n\nHe drove off. Martha stared after him, her affable smile abruptly falling from her face as she pulled up her hood and took off. She sat for the next few hours in a window booth at Greek Taverna, poring over the books, taking notes. When the diner closed, she moved to her Honda and read in there, seat reclined, overhead light on.\n\nThe longer I watched from the darkness of the park across the street, the more I felt a choking anxiousness and fear, as if the Neverworld were closing in on me.\n\nMartha was brilliant. Martha understood. She was light-years ahead of the rest of us. She had summarily accepted the crushing reality of the Neverworld, and rather than fighting it, she had dedicated her time to figuring out how to master it.\n\nI wanted to live, didn't I? I wanted to be chosen. Yet, staring at the pale light inside Martha's car, fighting back tears, I sensed I was too late, that I'd already lost.\n\nMy gaze suddenly fell on a dark figure pushing a wheelbarrow toward me down the path through the park. It was heaped with black compost.\n\nI should have been used to the Keeper's presence by now. I should have ignored how no matter where I went, however near or far, when I least expected it, he would come to me like a terrifying thought, the Neverworld's omnipresent alarm, its memento, its tolling bell.\n\nThe vote. The vote. _The vote._\n\nThe temperature had dropped. The rain was turning to snow again.\n\nI sprinted to the McKendrick van, climbed in, and took off, swerving into the road so wildly I almost hit a streetlamp. The Keeper paused to watch me go, a shovel balanced on his shoulder.\n\nI caught a glimpse of his face through the swirling snowflakes, the chilling smile.\n\nI couldn't imagine what Martha was planning. Whatever it was, I suspected it'd be so well considered and masterful, none of us would ever see her coming.\n\nHow right I was.\n\nHow did I pass the next few wakes?\n\nWas it months? Or was it years?\n\nI was the only one left. Wincroft was my castle to rule, my tiny home planet. The solitude was infinite. Gandalf was there, but he backed away and barked whenever I tried to pet him, as if aware I wasn't quite real. I wandered the creaking hallways and musty rooms, had conversations with stuffed deer and grizzly bears. I read every book in E.S.S. Burt's library, sprawled across daybeds, love seats, and carpets; dining room tables, window seats, and grand pianos. I watched every show on every cable channel at every time. I ate chocolate. I played Scrabble by myself, and chess by myself, and sang pop songs. I drew everything I could think of\u2014eyes, faces, landscapes, shadows. I made a dream soundtrack, song lyrics to a fake four-hour movie about the end of the world called _Ned Gromby's Last Day Alive Ever,_ scribbling the mad rhymes about life and death, war and peace, all over the wallpaper and floors and ceilings of Wincroft. Wincroft was my bridge underpass spangled with my graffiti. I squeezed my eyes closed to beat back the silence, and sifted through memories of my old life as if inside them I'd find a key to a door that would lead me somewhere.\n\nI visited the elderly. They were my favorite. Because they were locked inside their own Neverworlds too, impenetrable rooms of repetition and loneliness. I made a habit of ringing their doorbells with an excuse about selling early Christmas calendars for my church. I ate their fruitcake and petted their old dogs with bad breath before they scampered away with twitching backs. I sipped the weird tea and watched TV, inhaled the curdled house odors the owner was oblivious to. Most of all, I listened to the stories. I untangled the gnarled pileups of anecdotes and convoluted tales of dead husbands, failing health, childhoods of taffeta and milk that cost ten cents.\n\nI figured if I remained in the Neverworld, alone, until the end of time, I would be like an ancient traveler wandering the side of the road with a calloused heart and hands, weighted with the world's tales and secrets.\n\nAt least then, if nothing else, I would be wise.\n\nIt was inevitable that I'd be sitting there, listening to the story about the broken engagement, the dead child, the cat, when suddenly I'd see the decay. It always came out of nowhere and made me jump. Every windowpane in every single window around me would be silently cracking. Or a family photo would suddenly drop down the wall with a thump, revealing a garish rectangle of wallpaper that hadn't seen daylight in forty years.\n\n\" _What_ in the name of Jesus is going on...?\"\n\nIn Mrs. Kahn's case, it began with a faint popping noise.\n\n\"Damn raccoon's got in again,\" she muttered, tightening her robe. When she started shrieking in the den, I ran to her, astonished, to find her prized collection of snow globes\u2014gifts from Paul, a lost suitor\u2014detonating like grenades, water and snow and glass, plastic Santas, Eiffel Towers, St. Peter's Basilicas, exploding around the room.\n\nMrs. Kahn shielded her face. \"It's the Day of Judgment!\"\n\nOf course, I'd noticed the deterioration before, back at my house with my mom. Again that night at the Crow. I didn't know why, or what it meant, but whenever I was away from Wincroft, the world began to decay and disintegrate around me.\n\nIt always made me scared. I ran away, muttering some excuse and that I'd be back tomorrow, leaving Mrs. Kahn, Mr. Appleton, Mrs. Janowitz, Miss Bellossi, bent over, disconcerted, as they inspected the rot, the mold, the cracks traveling like lengthening skeletal fingers along the windows. I'd sprint back to Wincroft to search the gardens and grounds for the Keeper. I wanted to confront him, demand to know what was happening.\n\nYet, bafflingly, whenever I willed him to appear, he stayed away.\n\nThe corrosion appeared to be getting stronger. What did it mean? Was the Neverworld going to swallow itself like a black hole? Were we running out of time to vote? Was it all because of what happened to Jim?\n\nThe answer jolted me like an electric shock.\n\nJim. It had to be because of Jim.\n\n* * *\n\n\u2014\n\nThen came the day Wit didn't leave.\n\nI discovered her upstairs, buried under an avalanche of duvet, her face swollen with tears as she watched _Heathers_ on her laptop. I stared at her, dumbfounded. I felt like some shipwreck survivor finding another person who'd washed up alive on my island.\n\nShe glared at me. \"Leave me alone, Bee.\"\n\nI was worried I'd frighten her away, so I did just what she said. I made her tea, left the mug on her bedside table, and ducked out.\n\nThe next wake, to my relief, she was there again, watching _The Breakfast Club;_ the wake after that, _Goonies._ I always left her tea. Then, one wake, as I did, she threw off the comforter and surveyed me with a sad smile.\n\n\"Want to watch _Ferris Bueller_ with me?\"\n\nCannon reappeared a few wakes later. Mayhem, as it turned out, wasn't as much fun without an accomplice. He was as exhausted as Whitley, holed up in the library with his laptop in DOS mode, typing some mysterious hacker's command as the screen belched code. I printed out an obscure article written by a Stanford doctorate student about the future of Internet security and left it next to his laptop for him. The next wake, it was an essay about Steven Spielberg and brain cloning written by a freshman scientist at Harvard, then a blog posting by some genius sixteen-year-old Cambridge student about the future of robotics.\n\n\"How are you finding these articles?\" Cannon asked once before I darted out. \"I mean, they're so obscure.\"\n\n_With all my free time in the Neverworld, I've read the entire Internet. Twice._\n\n\"I just stumbled upon them.\"\n\nHe smiled. \"They're really cool. Thanks, Bee.\"\n\nShortly after that, Kip stopped going hitchhiking. The moment he strolled into Wincroft, I couldn't help it. I threw my arms around him, hugging him.\n\n\"Sister Bee, you're breakin' my neck. I'm not Elvis back from the dead, child.\"\n\nHe pulled away, said nothing more, headed upstairs. Yet I could tell from his faint smile that he was happy to see me. That night I made him Boudreaux's Stomp Shrimp Gumbo, the recipe served in his favorite hole-in-the-wall caf\u00e9 in Moss Bluff. I left him a bowl in the bedroom where he was holed up watching _Hoarding: Buried Alive_ on TLC.\n\n\"How'd you get Auntie Mo's secret recipe?\" he blurted, incredulous.\n\n_I had a million wakes to make her believe I was her long-lost niece._\n\n\"Just whipped it up,\" I said with a shrug.\n\nSo there they were, three wild animals I was doing my best to cajole into remaining at the zoo in captivity, rather than roaming the wild.\n\nThen, one night, as the four of us sat reading in the library, I realized from the way Kip kept glancing curiously at the clock on the mantel that he was waiting for something. Martha must have said something to him about a group meeting, because when she appeared a few minutes after midnight, entering without a word, hauling her heavy black bag and taking a seat on the couch, he didn't look the least bit surprised.\n\n\"It's time,\" Martha announced.\n\nWhitley and Cannon surveyed her in shock.\n\n\"Nice to see you too,\" said Cannon.\n\nMartha gave him an official smile, clasping her hands like a judge.\n\n\"We've come back to where we started,\" she said. \"It's as if the Neverworld's walls are slippery and slanted, always sending us back to where we began. I suspect, like me, you were each pursued by the Keeper, often when you least expected it?\"\n\nI nodded. So did the others.\n\n\"He's our caretaker. He tends us, keeping us alive and thriving, making sure we have the sustenance we need but also keeping us in check. This means he's capable of anything, being at once a guide and a taskmaster, a custodian and a thorn. Maybe he leaves you alone, or offers you a sprinkling of advice. Or else he hounds you, reminding you of the one thing you wish to forget. He will become anything to make you grow in a certain direction. Most of all, he is the chairman of a grand design we can't see.\"\n\nNo one said a word, all of us listening in wonder, in shock. The way Martha sat there\u2014squared shoulders, steady stare. She was no longer the mute nerd who blurted unfunny comments at weird moments, the girl more comfortable buried in the pages of an underground fantasy novel than living in the real world. This was a new Martha, one who had studied with Beloroda. She was a confident presence now. I had no idea where she was going with this speech, but she'd given it considerable thought, her every word as carefully selected as stones in an ornate necklace, each one meticulously polished and gleaming.\n\n\"The Neverworld is real,\" she said. \"To understand and conquer it, we must first understand and conquer each other. I've thought it over. We must set aside the question of who should live. We're not prepared for that. Not yet. Because there's another mystery we have to solve. It's dogged each of us in different ways since it happened.\"\n\n\"What are you talking about?\" asked Cannon, frowning.\n\n\"Jim.\"\n\nHis name was like a gleaming sword pitching through the air, landing hard at our feet.\n\n\"It was suicide,\" whispered Whitley.\n\nMartha stared at her, stony. \"You don't _actually_ believe that.\"\n\nWit seemed too uncomfortable to answer.\n\n\"I've been studying the Neverworld,\" Martha went on. \"This place, among many things, makes us the most powerful detectives in the world. We can go back to the scene of the crime an infinite number of times. We can interview bystanders. Witnesses. The police. Every teacher, janitor, and student. We can polish our questions, manipulate, intimidate, blackmail. There are no penalties and no rules. We can find out what happened to Jim once and for all.\"\n\nMartha's dark eyes found mine as she said this, sending a shiver through me.\n\n\"But the case was unsolvable,\" said Cannon.\n\n\"Yeah,\" said Kipling in a low voice. \"The cops didn't get very far.\"\n\n\"They were pressured by the school board to wrap it up quickly. The sooner everyone believed suicide, the sooner Darrow could repaint the bloody walls. That's what our parents wanted. They wanted to sweep the scandal under the rug, for everyone to chalk the whole thing up to another doomed dream boy. The Legend of Jim Mason would be just another ghost story echoing through the halls.\"\n\n\"So we're Sherlocks for the foreseeable future,\" said Whitley.\n\nKipling raised an eyebrow. \"I've always had a thing for herringbone and bloodhounds.\"\n\n\"I'm in,\" said Cannon.\n\n\"Me too,\" whispered Whitley.\n\n\"Beatrice?\" asked Martha.\n\nThey all looked at me. I stared back, my heart pounding.\n\nIt was happening, after all this time: We were freeing the lion. Dredging the _Titanic_ up from the bottom of the sea. We were unburying the man who'd been sealed inside the walls that night we went searching for the cask of the Amontillado.\n\nWe were going to find out what happened to Jim.\n\n_My Jim._\n\nThe cat-and-mouse game had begun.\n\nThe strange circumstances of Jim Livingston Mason's death had always seemed unreal to me, even though I experienced them firsthand.\n\nAs I thought back on it now, holed up in that library with my four former best friends, returning to each detail felt like trying to recall the rules of an imaginary game I'd played as a child.\n\nSenior year, spring semester before finals week, my boyfriend, Jim, went missing.\n\nTwo days later, he was found dead, floating in the lake at Vulcan Quarry.\n\nHe was my first love, though those words don't begin to describe what he actually was. Moon. Voice in my head. Blood. Even though everyone and their grandmother will tell you young love never lasts, that its burn is much more fragile than it ever appears to the naked eye, I swore what Jim and I had was different.\n\nHe was beautiful in the unlikely way of some eighteenth-century hero galloping across moors on horseback: six foot three, honey-brown stare, uncombed black hair, cockeyed smile. But there was something else too. He was alive. If life force is a river's current, Jim's was so strong it could take off your fingers. He charged through an ordinary Monday as if he had been tasked with imparting a crucial secret about existence before Tuesday. He was a goofball, grandmaster of the Catchy Tune, the Double Entendre, the Shock Romantic Gesture, like giving me a vintage diamond Cartier pin in the shape of a bumblebee after he'd known me just a week. He wrote me a theme song called \"The Queen's Neck.\" The worst thing about Jim was that his intensity attracted everyone. He was the light on a porch at night. Men and women, young and old, swirled around him, as if mistaking the attention of Jim Mason for a miracle dip in Lourdes. I couldn't fault them. He made them feel important and less alone.\n\nHe called me Amish, and Cahoots, and Hedy Lamarr. He said I had some quality of the past that he could never put his finger on, that I was meant for some long-forgotten, more innocent time.\n\n\"You're a Dusky Flying Fox,\" he told me.\n\n\"A _what_?\"\n\n\"An extinct species of mammal known only by a single specimen. You were spotted once in 1874 on Percy Island off the coast of Queensland, Australia. No other examples of you were ever found. Yet here you are again, tucked away in an antiquated, not especially impressive boarding school in the wilds of Rhode Island. And no one knows about you but me.\"\n\nHe was analytical, agonizing, easily wounded, unable to let much go. The summer before senior year, he and a childhood friend were drinking and driving a speedboat off Long Island when they collided with a sandbar, hitting a fisherman's skiff. The fisherman and Jim's friend were fine, thank goodness, but Jim suffered a skull fracture and ended up unconscious for two days. As a result of his injury, he wrote six songs, four poems, and a rap song called \"Bang-Up\" about the incident. He vowed to give up alcohol. Once a month after the accident, he wrote letters to the fisherman, as if confessing to a priest.\n\nThat was just how Jim was. He saturated. He overflowed. He drowned.\n\n\"You have to design your life like it's a fresh America,\" he used to say, pulling his guitar onto his lap, his calloused fingertips dancing along the strings. \"An unseen brave new world sits before you. Every. Single. Day. What are you going to do about it?\"\n\nNow Whitley, Cannon, Kipling, and Martha were watching me, uneasy. We'd never done this before. We'd never talked together about Jim's death. This had had to do with timing as much as the devastation of it. When every fact had been released by police and the administration had made their statement, finals week was finished. In a state of shock, unable to leave my bed, barely able to speak, I allowed my stricken parents to whisk me away from the treacherous kingdom of Darrow, back to the calming shelter of Watch Hill. It was days before I could stop sobbing, months before I felt anything remotely resembling fine.\n\n\"The body shuts down when it's too sad,\" said my dad.\n\n* * *\n\n\u2014\n\n\"Where do we begin?\" I asked now.\n\n\"Excellent question,\" said Martha. She looked at me, her dark eyes glinting behind her glasses. \"What do _you_ think happened to Jim? I always wanted to ask you.\"\n\nThere it was, the question I asked myself every day. So much so, it had turned me into a secret freak of nature, like a man who wanders around for years with a bullet lodged in his brain, normal on the outside, a gruesome marvel on the inside.\n\nI was dying to spill my theories, what I knew but they didn't know I knew. It had been my whole reason for coming to Wincroft. But in this dizzying life-and-death dynamic in which we found ourselves, sharing them wouldn't necessarily be a wise idea. Not if I wanted to live. Martha asking this so pointedly sent a fresh wave of chills up my spine.\n\n\"I don't know,\" I said.\n\n\"It was an accident,\" interjected Kipling. \"Had to have been, right? Say Jim was out at the quarry. Maybe he decided to get wasted. Sure, he'd sworn off booze after that boat crash, but maybe he was depressed. He was stressed about his musical. Didn't think he could pull it off. Maybe he slipped. The swim team kept those Pabst Blue Ribbons stashed all over. So maybe he was wanderin' the tall grass, which on a windy night could be like gettin' caught in a car wash's Deluxe Wax Special, and, I don't know, he stepped too close to the edge and tripped?\"\n\n\"When did Jim Mason ever trip?\" asked Cannon.\n\nKipling shrugged, tipping back his head to squint at the ceiling.\n\n\"Freak possible,\" he said in a low voice. \"That's what Momma Greer calls it when worst-case scenarios on steroids actually happen. She says all the big mysteries of history, like Marilyn's death, JFK, the Black Dahlia, the Lost Colony? They all came down to the freak possible.\" He nodded as if trying to convince himself, giving a lazy wave of his hand. \"It's wild flukes. One-in-a-billion chances. Wrong places at the wrong time with a serious helping of bad luck. It's some crazy, gnarled tangle of destiny that can never be undone by any outside detective 'cause it'd sound too damn absurd.\" He looked at me, his face solemn. \"The freak possible's what happened to Jim. I'd bet my life on it.\"\n\n\"Yeah,\" said Whitley, shrugging. \"I mean, none of us knew he was heading to the quarry that night.\"\n\nEveryone nodded, glancing tentatively at the others.\n\nI suspected at least one of them was lying. I certainly was.\n\nAfter all, on the night of Jim's death, none of them had been where they'd claimed.\n\nI knew this because I'd gone looking for each of them, one by one.\n\nI'd found nothing and no one.\n\n* * *\n\n\u2014\n\nVulcan Quarry\u2014or Vulcanation, as Darrow's students called it\u2014was the abandoned quarry a mile from the center of campus.\n\nIf Darrow had one enduring legend, it was that quarry. Given its tantalizing proximity to school\u2014the seventeen-acre property bordered Darrow's southeast woods\u2014it was the off-limits no-man's-land kids whispered about and obsessed over, a far-off world to visit for pranks, hazing, hookups, and all other adolescent rites of passage, you name it.\n\nRumors about the quarry\u2014how to find it, what happened to students who went there (most of whom were long gone from Darrow, so events could never be verified)\u2014were part of the weekly goings-on at Darrow and served as a foundation to its lore. The quarry was as tightly woven into the fabric of the school as its official song, \"Oh, Lord, Unbind My Heart\"; its motto, \"Truth, Compassion, Enterprise\"; and even Marksman Library, the Gothic fortress of weather-beaten gray stones that stared out like a menacing stepfather from every brochure.\n\nAfter World War I, Vulcan Sandberg Corporation created the quarry for mining granite. By the 1950s, they were bankrupt, the quarry forsaken. In the ensuing years, the crater filled with water, creating a lake two hundred feet deep. The grounds overgrew, with grass that reached your neck. The Foreman's Lookout\u2014a wooden box like a pioneer-era saloon hoisted fifty feet into the sky, accessible only by scaling a narrow ladder\u2014began to lean northward. Then there was the quarry itself, a hole in the earth the size of a small town. It sat there, gaping and ominous, impossible to look away from. It seemed to reveal some terrifying truth about the world the grown-ups wanted to keep hidden from us.\n\nDarrow's football team used the quarry for Streak Night, the annual tradition of new recruits racing naked to the quarry and back. The crew team went swimming in the lake before state championships for good luck. Couples went there to lose their virginity, daredevils to brood. It was whispered that Vulcan Sandberg was actually a government cover-up, that the quarry had actually been the landing spot for an alien spaceship.\n\nFor Darrow's administration, Vulcan Quarry was a lawsuit waiting to happen, the enchanted wood they wanted to clear-cut to put an end to the dark fairy tales wafting off it like some toxic mist. There was always some board member protesting, collecting signatures to declare it a safety hazard, lobbying state representatives for it to be turned into a cultural center, a YMCA, a housing complex. In the meantime, it required new fencing and a twenty-four-hour police patrol. The town of Warwick\u2014partly out of resentment over being told what to do by uppity out-of-towners, partly out of ineptitude\u2014dragged their feet doing anything about it, though, and as long as I attended Darrow, the fencing around the quarry\u2014rusted, riddled with holes, its faded signs halfheartedly declaring KEEP OUT\u2014remained little more than a suggestion at best.\n\nAfter Jim was found dead, however, he became the poster boy for the board's cause. Last I'd heard, the quarry was going to be turned into a reservoir and there was brand-new, state-of-the-art fencing around it.\n\nNot that that would keep Darrow's students out.\n\nIf the administration knew the lengths to which the student body went to sneak out at night, to the quarry and everywhere else\u2014dorm rooms, basement gymnasiums, boiler rooms\u2014they wouldn't have believed it. There was a secret forum\u2014AlbanzHax.biz\u2014where students past and present anonymously revealed how to get in and out of every dorm without being caught.\n\n_All dark clothing. Porch ledge. Sneak past the window where Mr. Robertson is zonked out with an issue of_ Poets & Writers _over his chest. Get past him ur golden._\n\nThe six of us snuck out to Vulcan Quarry all the time. We were already in the habit of stealing away to each other's rooms after curfew, clambering across ledges and landings to hash over boys, teachers, sharing a cigarette in the dark before hightailing it home, stealing back into bed. Sophomore year, Cannon found the crude map and pointers for the quarry etched into the tiles of the forsaken gym in the old athletic center. At midnight we escaped our dorms, meeting at the entry to the Philosopher's Walk. Barely able to suppress our laughter, we took off running down the tangle of dirt paths to get there.\n\nThose were the best nights of my life.\n\nI couldn't say why, exactly, this was so\u2014only that I knew that as an old woman, when I thought back to my youth, I'd remember these nights, sitting with these five people along the harrowing window ledge of the Foreman's Lookout, gazing into that clear blue lake hundreds of feet below.\n\nOur friendship was born there. There we were bound together. Something about seeing each other against that spare, alien backdrop of rock, water, and sky\u2014not to mention the prohibited, dangerous thing we were doing\u2014it X-rayed us, revealed the unspoken questions we each were asking. You could feel life burning us, our scars as real as the wind whipping our faces. We knew that nothing would ever be the same, that youth was here and nearly gone already, that love was fragile and death was real.\n\n* * *\n\n\u2014\n\n\"What about the White Rabbit?\" asked Martha now. \"It never sat right with me. It was just too easy. The White Rabbit suddenly revealed to be Jim the _exact_ moment he turns up dead?\" She shook her head. \"It went against everything I knew about him.\"\n\n\"You think it was a cover-up?\" asked Whitley. \"Some grand conspiracy concocted by the administration and Jim was the fall guy?\"\n\n\"I don't know.\"\n\n\"You're right,\" I said to Martha. \"There's no way he was the White Rabbit.\"\n\n\"How do you know?\" Cannon asked me.\n\n_I just know._\n\n* * *\n\n\u2014\n\nThe White Rabbit.\n\nIt was what everyone called the drug dealer at Darrow, someone who circulated the student body, invisible and invasive as a virus.\n\nFor most of my time at the school he was a bogeyman. No one had actually ever seen him\u2014no one who would admit to it, at least. He did his deals in creative scavenger-hunt dead drops all around Darrow, like behind the frame of _Landscape #14_ in the art gallery, or inside the ripped seat cushion of seat 104, row E, Orchestra Hall.\n\nBy the time I was a senior, the name had garnered such cult status, whenever anything weird happened, it was said to be the work of the White Rabbit. Even teachers knew the name. They'd doubtlessly held emergency meetings about him, trying to determine whether he was real or it was just kids dreaming up some Keyser S\u00f6ze.\n\nThe biggest scandal concerned a freshman named Veronica Beers. She took some pills and went out of her mind during Winter Dance, fell down a flight of stairs, and got taken to the ER. She admitted the White Rabbit had sold her the pills. Tracing the phone number led to only a defunct prepaid phone.\n\nWas he a lone wolf or a gang of hoodlums? A student or someone from the outside?\n\nWhen the police found Jim, he'd been dead for two days. The cause of death was asphyxiation due to drowning, but he also had signs of a concussion and leg and spinal fractures, which the coroner believed were sustained when he hit the water.\n\nPolice searched Jim's room at Packer Hall, and they found hidden inside his Gibson guitar stashes of pot, Adderall, Ritalin, and cocaine. They concluded that Jim had been the infamous supplier, his death most likely suicide, though foul play in conjunction with some local criminal couldn't be ruled out.\n\nThe revelation spread like wildfire.\n\nFirst Jim Mason's disappearance and death, then the shocking reveal of his secret life. It was the perfect one-two punch to leave us all breathless at the end of a teen slasher flick.\n\n_Of course the White Rabbit was Jim,_ everyone whispered. _Totally._\n\nIt's always the one we worship the most we know the least.\n\nHe was, after all, Darrow's rock star, its heartthrob-musical-genius-Shakespeare, the boy who made spontaneous rapping, poetry, and wearing tweed caps cool (all small miracles unto themselves)\u2014the kid everyone loved, longed for, yet simultaneously wished dead.\n\nHe had _it._ An energy force field.\n\nHe was the giant lit-up window with no curtains at night. You couldn't help stopping to look closer on your silent walk past him.\n\nI never believed it.\n\nThere was no way Jim was the White Rabbit. Someone had set him up, I was sure. He never touched drugs or alcohol after his speedboat accident. And he wouldn't have sold it. He was a rescuer of broken-winged birds and lunchtime social outcasts. Nor did he need the money. His dad, Edgar Mason, was the inventor of the Van Gogh sneaker and the Poe hoodie, the man behind Starving Artist, a global leisurewear company he'd started in the back of his Jeep at nineteen. Jim's family was worth five billion, according to _Forbes._\n\nI'd spent the past year doubting my belief in Jim's innocence. I road-tested my theory obsessively, kicking the tires, trying to make the doors fall off. I wondered about all the occasions when Jim said he couldn't meet me in the canteen during Evening Spells, how he was on a tear and had to stay sequestered in his dorm room writing. I wondered if he'd been lying to me every time he said he was at work on _Nowhere Man,_ his musical about John Lennon.\n\nMy heart insisted no. He couldn't have been. He'd lied to me about other things.\n\nNot about that.\n\n* * *\n\n\u2014\n\n\"I know where to start,\" I announced.\n\nThe others had lapsed into thoughtful silence. Now they looked up at me, apprehensive.\n\n\"Vida Joshua.\"\n\n\"Kitten?\" yelped Kipling in surprise.\n\n\"She knows something about Jim's death. I'm positive.\"\n\n\"How do you know?\" asked Whitley sharply.\n\n\"Remember how Jim was acting that final week?\"\n\nKipling arched an eyebrow. \"Like I remember a hot summer with a water shortage, backed-up sewage, and zero air-conditionin'.\"\n\n\"He wasn't himself,\" I went on. \"He was moody. A short fuse.\"\n\n\"All because of his musical,\" said Whitley.\n\n\"Oh, _Lord Almighty,_ his musical,\" drawled Kipling, grimacing. \"It was eatin' him alive.\"\n\n\"He was stressed about his musical, definitely,\" I said. \"But there was something else going on too. Something I found out about.\"\n\nThey were watching me, rapt, waiting for me to go on.\n\n\"I'm pretty sure he was hooking up with Vida Joshua.\"\n\nNo one said a word. They just stared at me in shock.\n\n\"The day before he disappeared was the first night of Spring Vespers, remember?\"\n\nThey nodded.\n\nSpring Vespers\u2014it was a two-night performance of skits, speeches, and original songs commemorating the end of the year and preceding finals week.\n\nAround five, Jim texted me. He said he had a fever and chills, and was heading to the infirmary. I was shocked. After all, a medley of original songs from _Nowhere Man_ was being performed that night. It was the cornerstone of Spring Vespers, the first time anyone had heard it, so for Jim to abandon his own production on the eve of its debut was very strange. Even if he was nervous about it, he'd never quit. Later that night, after eight, I was running late for Vespers, having stayed longer than I realized in the library. I veered behind the cafeteria on my way to the auditorium. It was the shortcut Jim and I sometimes took. That was when I saw him. Sitting by the loading dock. _Not_ sick. At all. He was fine. Just sitting there in a black T-shirt and jeans, as if waiting for someone. Alone. I stood behind a tree and texted him.\n\n_How's the infirmary?_\n\n_One hundred and two fever,_ he wrote back, plus a sick emoji. I _watched_ him write this, completely nonchalant.\n\n_I'll come visit you,_ I wrote.\n\n_No. No. Don't. I'm going to sleep._\n\nI couldn't believe it. I was about to confront him right then and there. Only that was when a car slinked up. Slow. No headlights. Taking care not to be noticed. Jim hopped off the ledge and climbed right in. Vida Joshua was driving.\n\n\"You saw her?\" asked Cannon.\n\nI nodded. \"It was Mr. Joshua's beat-up red Nissan. The one he kept behind the music school with the keys in the ignition and the For Sale sign in the back window. The one the administration was always asking him to get towed.\"\n\n\"Poor Mr. Joshua,\" said Whitley. \"If his head wasn't attached, he'd lose it.\"\n\n\"Did you ever confront Jim, child?\" Kipling asked.\n\nI nodded. \"The next day. He didn't admit anything. But he was furious.\"\n\n\"Furious at you?\" asked Martha, squinting skeptically.\n\nI nodded.\n\n_Leave me alone, Beatrice. Stop spying on me. What are you, my father?_\n\nJim's reaction had scared me. I'd never seen him like that before: trembling hands, tears in his eyes, anger like a sudden venom in his veins, making him scowl and spit and contort his face so he was unrecognizable. He'd been on edge for weeks, a mood I'd attributed to the pressure of getting his musical ready for Spring Vespers and recording the producer's demo Mr. Joshua had set up. _I want it to be glorious, Bee. I'm going for glory._ He had turned inside out with anxiety, self-doubt, despair. _The notes have lost their velvet,_ he whined. What had once sounded like a haunting theme song had suddenly become shrill to him. His lyrics were clich\u00e9d. No amount of insistence on my part that they were good could convince him otherwise. Our relationship had become brittle, a series of botched conversations about stanzas and syncopated rhythms, finding a better rhyme for _dissipate._\n\n_Figure eight? Exonerate?_ I'd try.\n\n_Just forget it,_ Jim would snap.\n\nThat afternoon, I'd given Jim every chance to explain what I'd seen, tell me the innocent reason he'd climbed into Very Flexible Vida's Nissan that night and lied to me about it.\n\nBut he didn't.\n\n_You want to break up with me over this?_ he screamed. _Good. I've had enough of your insecurities and childishness and your totally annoying inability to see the bad in people. Sometimes there's evil in the world, okay? Sometimes the sickness is right in front of you._\n\nHis words had made me turn and sprint down the hill, hot tears blinding me. When I stopped and looked back, I saw in surprise that Jim wasn't following me as I'd expected. Instead, he was striding across the hill, a dark and consumed expression on his face, out of sight.\n\nLike he was over me. Like we really were done.\n\nThat was the last time I ever spoke to Jim.\n\nTwo days later he was dead.\n\n\"The day after Rector Trask announced that Jim had been found dead,\" I went on, \"it was all over the newspapers. That same day, Vida disappeared.\"\n\n\"What do you mean?\" asked Cannon.\n\n\"She immediately left town, remember? They made the announcement.\"\n\n\"That's _right,_ \" said Whitley slowly, wrinkling her nose. \"At Final Assembly. 'And in further news, Miss Joshua is taking a job in stem cell research at the University of Chicago.' \"\n\nKipling nodded, dubious. \"It was like hearin' a chimpanzee got employed at the State Department.\"\n\n\"You thought she was fleeing the scene of the crime?\" Martha asked me.\n\n\"The timing was strange,\" I said with a nod. \"Like she was afraid of something. Anyway, I checked Facebook, and she's working as an assistant chef at Angelo's Italian Palace, living at home again. I always wished I'd had the guts to confront her. Now I do.\"\n\nI took a deep breath and stood up.\n\n\"Who's coming with me?\"\n\nOne by one, with uneasy expressions, they raised their hands.\n\nVida Loretta Joshua was seven years older than all of us.\n\nShe was Mr. Joshua's only child. She'd graduated from Darrow and gone off to college in North Carolina, only she'd had some kind of mental breakdown\u2014the exact nature of which remained vague\u2014dropped out, and moved home.\n\nWhen you visited the Joshuas' modest Tudor cottage on Darrow's campus, it was like visiting two ordinary people housing a pet leopard. This was because: (1) Vida Joshua was stunningly beautiful, with black hair, far-apart blue eyes, alien cheekbones, a face so symmetrical and arresting when she finally looked at you (which she only did after a prolonged delay) that it was like finding a wildcat lazily regarding you from a mountaintop as you squinted through binoculars; and (2) Mr. and Mrs. Joshua seemed to be afraid of their daughter. They addressed her in soft tones. They tiptoed around (with no sudden movements) the spot where she could be found sunning herself on the living room couch with unwashed hair and baggy sweats, eating a bag of kale chips, watching _Real Housewives of Atlanta._ They seemed too scared to arrange her reentry to the wild (college) or get her into a rehabilitation sanctuary (therapy). So they just left her alone, bored and depressed, or whatever Vida was.\n\nNo one was really sure.\n\nSophomore year, Mr. Joshua twisted someone's arm to get her a job at Darrow. Vida started appearing in admissions, yawning as she shuffled unconvincingly between the copy machine and a computer; later she turned up in the Spanish department; then as assistant coach for JV field hockey, though when that didn't work out (apparently few were comfortable working alongside a big cat), they stuck her in the remote outpost of the art gallery. Most of the time she left the front desk unattended, and could be found outside in the back by the dumpsters, chatting with a random student, always a boy. Rumors swirled that her nickname at Darrow had been Very Flexible Vida, that she'd had sex with the entire wrestling team, that she'd fallen in love with a professor in college and stalked his wife, which had resulted in a restraining order that led to Vida's mysterious breakdown.\n\nThat was all I knew about her before I saw her with Jim. Once I saw them leave campus together, though, Mr. Joshua's nickname for his daughter, Kitten, seemed so fitting, because what had once been a cute little fluffball had suddenly morphed into a dangerous predator with a diet of horse carcasses and the capacity to kill without warning.\n\nI hadn't even known she and Jim were friends. He'd never mentioned her. She had surveyed Jim from her spot on the living room couch with only marginally less indifference than she'd regarded me and everyone else. After he was found dead, though, she suddenly vanished from her job at the art gallery. Her ergonomic swivel chair, her mug of pens, the gallery printouts of price sheets and artists' statements of purpose mixed in with a welcome pack for a gym membership at the Jam, and Thai take-out menus\u2014all sat there like nagging questions in the days following his death.\n\nI found myself calling into question my every private moment with Jim, as if I were a miser locking myself in my room to stare down at the stacks of cash I'd stashed under my mattress, counting the money for the millionth time to make sure it was all there, checking that it wasn't counterfeit. In the intervening year, I kept tabs on Vida. I stared at her Instagram photos of camping trips in Wisconsin with some friend named Jenni who wore Bermudas; her loud move into a new apartment in Wicker Park ( _Anyone know a good mover in the Chicago area????_ ), followed by a mute return home not three months later; her registration in a fashion design course; her interest in reflexology and a heavy-metal band called Eisenhower. I pored over these artifacts, looking for clues to her relationship with Jim. Occasionally I found them. She posted lyrics to a song Jim wrote\u2014\"Carpe\"\u2014to accompany a blurry photo she'd taken of a frog. _Once we had days when we ripped up the skies. Swore it was real, no deception, no lies._ On the one-year anniversary of his death, she posted a message on his Facebook page, which had become a living memorial. _Miss you Mason._\n\nIn some of my darker moments, I considered sending her a series of anonymous messages, a sort of _I Know What You Did Last Year,_ to see if I could smoke her out, get her to reveal what she knew, what she'd done.\n\nI never did.\n\nNever would I have believed Jim capable of betraying me by fooling around with Vida. Then again, we'd never had sex. We came close. At the last minute I always said no. Jim would roll onto his back, prop up his head, stare at the ceiling.\n\n\"What are you afraid of?\" he'd ask with genuine curiosity.\n\nI deferred with various versions of \"I'm not ready,\" never having the guts to tell him the truth: that I was scared of losing the last piece of dry land I was standing on. I loved Jim, but our relationship could feel like a blackout sometimes. I'd get swept up in him, then days, weeks later suddenly look around, unnerved, wondering where I was, what time it was.\n\n\"You want to wait for our wedding night? Fine,\" he'd tease.\n\n_No wonder he never broke up with me,_ I thought later. _He wasn't missing out on anything at all._\n\nHe had Kitten.\n\n* * *\n\n\u2014\n\nNone of us had been back to Darrow, not even inside the Neverworld.\n\nArriving felt the way it always did, as if we were traveling back in time to a lost past of driving goggles, candlestick telephones, and people wearing tweed sprinkling their sentences with _grand_ and calling good times a _gas._ Darrow had always been willfully old-fashioned, a quality the school went out of its way to proudly maintain, as if the place were not a school, but a sanctuary for some endangered bird. Classrooms had the same wooden tables as fifty years before, the chapel the same pews. Most of the teachers looked like walking daguerreotypes, with stiff necks and expressions suggesting great depressions.\n\nI peered out the window, trying to ignore the nervousness in the pit of my stomach. Throughout the past year I'd wondered what I'd say to Vida if I ever had the chance to confront her, but now every scenario I had considered sounded pathetic and insecure. _What were you doing with Jim that night? Why did you suddenly vanish when he died? Did you love him? Did he love you?_\n\nAhead I could see the old white wood sign swinging in the torrential rain.\n\nDARROW-HARKER SCHOOL. The bronze stag stood beside it. As we tore past I caught a fleeting glimpse of the antlers and eyes, FOUNDED IN 1887 flashing in the bloodred taillights before both sign and stage were swallowed by the dark.\n\n\"Don't worry, Bee,\" whispered Whitley, leaning her head on my shoulder. \"I'll take care of everything.\"\n\n\"Every time you say that, child, someone loses an eye,\" said Kipling from the front seat.\n\nShe smiled primly. \"I cannot vouch for the continued existence of anyone who messes with my best friends. They hurt Bee? They have to deal with unleashing the vengeful forces of the known universe.\"\n\n\"Well, Zeus, I'd pipe down if I were you,\" muttered Cannon, slowing the car.\n\nDarrow's security gatehouse was ahead.\n\n\"What are we going to say?\" asked Martha.\n\n\"Oh, the usual. We're former students. Kinda sorta dead? Stuck in a cosmic catacomb?\"\n\n\"That sounds _so tedious,_ \" whispered Whitley with a giggle, squeezing my hand.\n\nCannon pulled to a stop in front of the gate, unrolling the window. We watched in uneasy silence as Moses\u2014Darrow's notorious security guard\u2014took his time zipping his jacket, fixing his shirt collar, and opening a golf umbrella before ambling out. Grumpy, bent over like a question mark, he was whispered to have arrived on campus the same year the school was founded. He was a die-hard Christian, shoehorning God into most conversations, and a recovering alcoholic. Every Wednesday at midnight he secretly abandoned his post to attend an Alcoholics Anonymous meeting in the gym at St. Peter's, which meant there was a reliable two-hour window when you could stroll brazenly past the gatehouse, absconding from campus without getting caught, so long as you made it back before he returned.\n\n\"Evening,\" Moses shouted officially through the rain. \"How may I help you?\"\n\n\"You don't recognize us?\" asked Cannon.\n\nMoses peered closer, his bushy white eyebrows bunching together in surprise. \"Well, I'll be. Cannon Beecham. Kipling St. John. Whitley. Beatrice. And little Martha. What on earth are you kids doing here on a soggy night like this?\"\n\n\"We were in the neighborhood and wanted to take a quick drive around,\" said Cannon. \"We won't be long.\"\n\nMoses scowled in apparent consternation and checked his watch. When he glanced back at Cannon, he seemed uneasy.\n\n\"A quick look,\" he said, pointing at Cannon. \"But no mischief, you understand me?\"\n\nCannon nodded, waving as he rolled up the window, and we took off down the road.\n\n\"No mischief you'll remember tomorrow, old friend,\" he muttered.\n\n* * *\n\n\u2014\n\n\"My goodness. This _is_ a surprise.\"\n\nStanding in the doorway, Mr. Joshua looked exactly the same. He was still trim, with sparkling blue eyes, rosy cheeks, and a flagpole posture, plus a penchant for sweater vests.\n\n\"To what do I owe such a treat? Come in. Out of this tempest.\"\n\nHe beamed with genuine warmth, causing me to feel a pang of guilt as the five of us filed inside, dripping wet.\n\n\"We're here to visit Vida,\" said Whitley, smiling. \"Is dear Kitten at home?\"\n\nWe'd already seen her car in the driveway, the red Nissan, the many lit-up windows, so we pretty much knew the answer to that question.\n\nMr. Joshua blinked, puzzled.\n\n\" _Vida?_ Certainly. We're\u2014uh\u2014just having dinner. Come in. Come in. Please.\"\n\nWe moved after him through a quiet living room into the dining room, where we found Vida and Mrs. Joshua. Mrs. Joshua, wearing a yellow apron, was forking corn on the cob onto the three plates as Vida, seated idly at the head of the table, scrolled through her cell.\n\nIt seemed captivity had taken a toll on her, because she looked less intimidating than I remembered. She was stockier, with thinner, rattier hair. Though the five of us wordlessly assembled around her chair, she was totally oblivious, glancing up in apparent disinterest before returning to her phone. She was used to her father's students visits at all hours for guitar lessons and rehearsals.\n\n\"Peggy? Kitten? These are friends of Jim Mason's. You remember, Jim, my student? The, uh, wunderkind? One of the very best young lyricists I've ever come across.\" Mr. Joshua held up a finger, a soft smile. \"He was going to go far. His musical about Lennon was one of the most gorgeous\u2014 A veritable tapestry of music and words...\" He seemed to forget himself for a moment, blurting this with unabashed sadness. His face reddened. \" _Well._ What brings you to our neck of the woods?\"\n\nIt had never occurred to me to consider how Mr. Joshua had taken Jim's death\u2014not until now, standing in the shabby taupe modesty of his house, the deafening rain pounding the roof, the faint smell of mothballs, acoustic guitars mounted on the walls hinting at some unplayed song. It had been Mr. Joshua, after all, who'd been Jim's biggest champion, coaching him about out-of-town tryouts and a Broadway run. It was Mr. Joshua who had taken Jim's dozens of demos, recorded on Logic Pro, and transcribed them into sheet music, Mr. Joshua who had pushed him to dream up cleverer lyrics, sharper characters, _more variety for the ear,_ analyzing with him the ingenious phrasings and renegade words of Stephen Sondheim and Lin-Manuel Miranda and Tennessee Williams. It was Mr. Joshua who had arranged for a major New York producer to listen to Jim's demo of songs from _Nowhere Man._ The producer had loved what he'd heard, and a meeting, a lunch in New York, was being set up around the time Jim had died.\n\nI found myself wondering if Mr. Joshua had been in love with Jim. Or was it something else? Had he seen himself attached to Jim's rising star\u2014Jim, his one-way bus ticket out of town; Jim, his partner, pet student, meal ticket\u2014all those hopes and prospects null and void now that Jim was dead?\n\n\"Vida definitely remembers Jim,\" said Whitley, grinning. She tilted her head, arching an eyebrow. \"Kitten, dear? Now would be the optimal time to ask your parents to leave the room, unless you want them privy to all the gory details.\" She plopped easily into a chair at the head of the table, hitching one leg over the armrest, and helped herself to a green bean. \"We want to know everything,\" she went on, nibbling the end. \"Who started it? Who ended it? Where did the two of you sneak off to off campus? And why did you leave town the nanosecond Jim turned up dead?\"\n\nVida only stared, her mouth open, incredulous.\n\n\"Please don't insult our intelligence denyin' it, child,\" drawled Kipling, waving his hand in the air. \"For one? Beatrice doesn't lie. She's the kindest, most honest person you'll ever meet. Her middle name's Good Witch a the North. And two, we don't have a lot of patience. We've been livin' this same day, known as a wake, over and over again? And it's made us all a little tense.\"\n\n\"A little impossible to deal with,\" added Cannon.\n\n\"Excuse me?\" asked Mrs. Joshua. \"What on earth is happening here? Paul? _Paul!_ \"\n\nMr. Joshua, standing beside her, seemed unable to move or speak, sort of like a wren hovering nervously around a park bench where a few breadcrumbs have just fallen.\n\n\"You people are nuts,\" said Vida in a hoarse voice. \"I have no idea what you're talking about.\"\n\nSmiling, Whitley grabbed a plate of food and launched it into the air like a Frisbee. It sailed across the room, chicken, corn, rice flying, crashing against the windows.\n\nFor a moment, everyone was too stunned to move. Then Mrs. Joshua dashed to the side table and grabbed a phone. She dialed 911.\n\n\"I'd like to report a home invasion.\"\n\n\"Tell her to hang up,\" Cannon told Vida, scratching his nose.\n\n\"We need the police. _Now._ There are children\u2014teenagers\u2014trespassing in our house\u2014\"\n\n\"Tell her to hang up if you don't like orange jumpsuits,\" said Whitley.\n\nVida glanced at her, startled.\n\n\"Or public showers,\" said Cannon, plopping easily into a chair.\n\n\"They're terrorizing us. My husband's former students. Please come at once.\"\n\n\"Hang up the phone, Mom,\" said Vida.\n\n\"One of my pet peeves is when girls don't support other girls,\" said Whitley. \"When they just help themselves to someone's boyfriend like he's some free smoked gouda sample speared with a toothpick at Whole Foods. It's _so_ unforgivable. And out-of-date.\"\n\n\"The Darrow School\u2014\"\n\n\"I'd call off your pit bull of a mom,\" said Cannon.\n\n\"Mom,\" said Vida sharply.\n\nMrs. Joshua didn't hear her. \"Five forty-five Entrance Drive. Please hurry.\"\n\nVida leapt to her feet. She ran to her mother, shoving the woman aside as she wrenched the phone away and threw it across the room. It hit a small painting of a fox playing a violin, which immediately fell off the wall, revealing a bright rectangle of wallpaper spangled with black mold.\n\nEveryone fell silent in total bewilderment.\n\nVida stood there gaping at us, wild-eyed, trying to catch her breath.\n\n\"What did you do this time?\" Mr. Joshua asked her.\n\n* * *\n\n\u2014\n\n\"I don't know how many times I have to say it,\" growled Vida. \"We were _friends._ That's all. All he did was ask me for a ride. He wanted my help getting off campus. He hid under a blanket in the backseat of my car as I drove past Moses. And that was it, okay? I really don't know what the big deal is. You people are seriously insane.\"\n\n\"We don't believe you,\" said Cannon.\n\n\"That's your problem.\"\n\n\"Where did you take Jim?\" asked Martha.\n\n\"I already _said._ \"\n\n\"Tell us again,\" said Whitley.\n\n\"I don't know. Some shopping center?\"\n\n\"In Newport?\"\n\n\"Yeah.\"\n\n\"Do you remember where it was? Or what it was called?\" asked Martha.\n\n\"No.\"\n\n\"What _can_ you tell us?\" asked Cannon.\n\nVida shrugged. \"It was some dingy section of town. Dollar stores. A pet store. The parking lot had some man in a chicken costume handing out heart balloons.\"\n\n\"And why did Jim want to go there?\" asked Martha.\n\n\"Maybe he wanted to eat fried chicken and buy a pet iguana? I have no fucking clue.\"\n\n\"You must have drawn _some_ conclusion,\" said Wit.\n\nVida shrugged, irritated. \"I thought maybe he was trying to score some weed. There were these dime baggers loitering around the parking lot.\"\n\n\"What time was this?\"\n\n\"Eight? Nine at night?\" Vida sighed. \"I offered to stay, give him a ride back to school, but he said he'd make his own way. And that was _it,_ all right? I don't see what the big deal is, and I had nothing to do with his death. I mean, _please._ \"\n\nFor the past twenty minutes, contemptuous and huffy, Vida had been relating the same story over and over again: Jim had only asked her for a ride that night. That was all. There was nothing more to it. They had not been hooking up. They'd been accidental friends. She had not wavered in this explanation. And though I was inclined to believe her, listening to her was still like a knife through my heart, because even if it was true that there was nothing romantic about their relationship, it still meant Jim hadn't chosen to confide in me, that whatever he had been up to, whatever had upset him, he'd chosen to deal with it behind my back. And if he had lied to me about that, I couldn't help wondering what else he'd lied about.\n\n\"I find it pretty far-fetched that he'd ask you for a ride if you weren't more than friends,\" said Whitley.\n\nVida glared at her. \"Like I said. We _talked._ Occasionally. He visited me at the art gallery and gave me advice sometimes. Jim was a card-carrying genius. He understood stuff. I talked to him about my issues, you know, and he gave me better advice in ten minutes than six years of Dr. Milton Yeskowitz with the goatee, the too-long thumbnails, and the bookshelf full of seventies self-help manuals called _Learning to Love Yourself._ \"\n\nShe said this with a scathing glance at her father, who stared back blankly. Both Mr. and Mrs. Joshua had been listening to their daughter as if she were speaking a strange dialect in which only every third word was comprehensible.\n\n\"Jim reminded me again and again that my life was alive and I had to tame it. That was why I decided to move to Chicago. I told him I had the opportunity to intern at the lab, and he said I had to go. I had to seize the day. Even if I was afraid. Jim said when there's a break in the path in front of you and you're freaked out, you take a running leap and trust that you'll reach the other side. He inspired me. And I helped him, you know. He was really stressed about his musical. He wanted it to be great. He wanted it to be up there with _Oklahoma!_ and _Rent._ He aspired to greatness. More than anything, he wanted to go down in history. And he was going to. He showed me his notebook, and he'd written the most _insane_ rhymes. He was a genius.\" She shook her head. \"It's horrible what happened to him. But that's life, right? All the amazing people die too soon.\"\n\n\"When was the last time you saw him?\" I asked.\n\n\"Don't you people _listen_? I _told_ you. When I dropped him off at that mall.\"\n\n\"You didn't talk to him the night he died? He didn't tell you to meet him at Vulcan Quarry?\"\n\nShe scowled. \"What are you _talking_ about? I haven't been out to that quarry in forever.\" She sniffed, shaking her head. \"Soon as I heard what had happened, though, that he was not only dead but had been a major drug dealer or whatever? I went straight to the police and told them they were completely insane.\" She rolled her eyes. \"Whatever drugs they found in his dorm? They definitely weren't Jim's.\"\n\n\"How do you know?\" I asked.\n\n\"He told me who they belonged to.\"\n\n\"Excuse me?\" asked Martha sharply, with a surprised glance at me. \"What?\"\n\n\"Oh, yeah.\" Vida nodded. \"A couple of weeks before he died. It was all he could talk about. How he'd found out one of his best friends had been selling serious drugs to students for years.\"\n\n\"Did he say who it was?\" asked Kipling.\n\n\"He didn't have to. As he was telling me, the friend called. 'Speak of the devil,' he said. I saw the name right there on his phone.\" Vida wrinkled her nose. \"It was kind of weird. It was just one word.\"\n\n\"What was the word?\" asked Cannon.\n\n\"Shrieks.\"\n\nInstantly, a chorus of voices began to scream in my head: _No. No. Impossible._\n\nNo one moved.\n\nWhitley was gazing at the carpet, a blank look on her face.\n\n\"I gave the name to the police,\" Vida went on. \"Told them everything. I said this person had something to do with Jim's murder. I was sure. Knowing their secret was about to come out? They had to get rid of him. Shut him up, you know?\" She widened her eyes. \"But the police didn't care. Or they'd eaten too many Krispy Kremes to peel their asses off their swivel chairs and _do_ something, because I never heard anything about it. I wasn't surprised. No one does anything anymore. The only one who did, who went above and beyond for everyone, was Jim. And now he's gone.\"\n\nShe fell silent. There was a moment of uneasy stillness.\n\nThen Whitley leapt to her feet and ran out of the house.\n\nWe took off after her.\n\nAt the front door, as Cannon and the others barged into a downpour, I paused, taking a final glance back at the Joshuas, who were regarding each other sullenly, like three strangers held in a jail cell before charges were filed. Then I turned and sprinted out.\n\nWhitley was already disappearing down Entrance Drive.\n\n\"Whitley!\" shouted Kip.\n\n\"Come on!\" yelled Cannon. \"Where are you going?\"\n\nShe ignored them, veering off the road and vanishing over the hill. When I caught up, they were far below, Whitley a dark figure flying past the tennis courts and soccer fields, Martha, Cannon, and Kip fanning out behind her.\n\n\"Hold on!\" screamed Martha.\n\n\"Let's talk about this!\"\n\n\"Whitley Lansing! Stop!\"\n\nI raced after them as fast as I could, faint police sirens erupting somewhere behind me. _Whitley? The White Rabbit?_ How was it possible?\n\nThe rain was torrential now. Like ammunition blasting from the sky, it riddled my head and arms. It was hard to see where I was going. Tree branches cracked and thrashed overhead, thunder rumbling. I slipped and tripped my way to the bottom of the hill, where there was a swamp of thick, tarlike mud. My sneakers sank inches into the ground. I could see Wit and the others farther ahead, rounding the front of the girls' dormitories: Slate Hall, Stonington Manor, the Gothic arches hulking and dark, misshapen shadows stretching long and fingerlike under the yellowed lamps. I veered through the garden behind Morley House and nearly ran over Martha. Apparently she'd slipped and fallen facedown in the mud.\n\n\"Are you okay?\" I shouted.\n\n\"Go,\" she gasped, waving me on.\n\nI kept running. As I swerved around the front of the aquatic center, a hulking glass-and-slate building, I saw that one of the glass doors had been smashed with a brick. I opened the door and scrambled inside, disembodied shouts and footsteps echoing through the darkness in front of me. I hurried through the dark lobby, past the many display cases of trophies and first-place ribbons, black-and-white photographs of the swim team. I sprinted down the checkered corridor, my muddy sneakers slipping and sliding on the linoleum, and thrust open the double doors to the Olympic-sized pool.\n\nKipling and Cannon were inside. Whitley had dived into the water, and they were tracking her dark figure along the edge as she glided into the deep end.\n\nAfter a minute, she surfaced, panting.\n\n\"What are you doing?\" said Cannon. \"We just want to talk to you.\"\n\n\"You can't outrun us, child,\" said Kipling.\n\nGlaring at them, she only sank back underwater.\n\n\"She's going for the door again,\" said Cannon, running toward me. Sure enough, Whitley leapt up the ladder, shoving me aside so hard I tripped against a chair as she heaved the doors open, only to come face to face with Martha, who was covered head to toe in mud. Startled, Wit tried pushing past, but Martha was gripping one of the swimming trophies from the cases. She wheeled back and hit Whitley in the side of the head with it. Yowling in pain, Whitley fell to the ground.\n\n\"Behold the White Rabbit,\" said Martha, panting.\n\nShe slammed the doors and wedged the trophy between the handles to lock them.\n\n\"So it was you!\" shouted Cannon, staring down at Whitley. \"All along. How could you never say anything? How could you deceive me, day after day after\u2014 Unbelievable.\"\n\n\"I didn't mean to do it more than a few times,\" she muttered. She rolled upright, rubbing the side of her head. \"My number started getting passed around, and the myth of the White Rabbit was born. It was impossible to stop.\"\n\n\"How could you?\" I whispered in a low voice.\n\nWhitley glared at me. \"Yes, Bee, we all know that you'd never do something like that in a million years. That you're the good one. With a moral compass perfectly set toward sainthood. The rest of us aren't so lucky.\" She sniffed, staring gloomily at the ground.\n\nWe said nothing, reflections of the blued water of the pool trembling across our faces.\n\nI couldn't believe how she'd lied. For years. I'd never suspected her. Neither had Cannon, given the enraged look on his face.\n\nYet it made a sort of twisted sense, considering that Whitley's mother, the Linda, ruled a pharmaceutical empire. It wasn't rare to hear Whitley talk about her mother's business acumen with awe\u2014how she, armed with her mink coat, high school dropout's education, and Missouri-farm-girl common sense, could command New York City boardrooms and shareholder meetings, put macho bankers in their place with one of her perfectly timed ten-cent-gumball put-downs. _If stupid could fly, you'd be an Airbus A Three-Fifty._ The truth was\u2014and it used to make me cry thinking about it, though I was always careful never to say anything to Wit\u2014the Linda didn't love her daughter, not the way Whitley needed. Ever since Wit was a baby, she'd been shuffled between nannies and nurses and au pairs, summer camps and boarding schools and educational groups, like some lost suitcase. I somehow understood. Whitley had become the White Rabbit to prove to herself, or maybe even to her mother, that she was worthy.\n\nI hadn't forgotten Vida's comment that unmasking the White Rabbit had possibly played a role in Jim's death. _I said this person had something to do with Jim's murder. Knowing their secret was about to come out? They had to get rid of him._ The comment nagged at me, glimmering with the unmistakable sheen of truth, though I didn't want to admit it.\n\n\"How did Jim find out about you?\" I asked her.\n\nWit glanced up at me, sullen. \"He caught me.\"\n\n\"When?\"\n\n\"A week before finals. I always snuck out at three in the morning to do a drop. He was coming back from Vulcan Quarry and saw me entering the observatory. He followed me into one of the domes, watched what I was doing. And he went nuts on me. Made it into a bigger deal than it was. I mean, we were about to graduate. The White Rabbit was done. Jim started screaming about civic responsibility. Doing the right thing. He insisted that I confess to the administration.\"\n\n\"So you decided to frame him,\" said Martha. \"You put your drugs in his guitar so he'd take the fall for you.\"\n\n\"No.\" Whitley adamantly shook her head. \"That was an accident. I always kept my supply buried behind the old maintenance shed at the edge of Drury Field. You know that place everyone says is haunted? Well, it's not. It's just cruddy, with a lot of old athletic signs and banners. One day, I saw this general contractor inspecting it. I found out they were going to demolish the thing and build a greenhouse. That night at supper I excused myself and dug up my stash. I meant to keep it in my room, only they were painting the hallways that night. The place was crawling with maintenance. I was desperate. This huge stash in my bag? If I was stopped...\" She shuddered. \"I ducked next door into Packer, to Jim's room. I knew he kept his key under the carpet. I shoved the stash in his guitar. I meant to go back and move it the next day. _Obviously._ But that was when they announced Jim was missing. By the time I had a chance to go back, the cops had already searched his room and found everything.\"\n\nMartha stared at her. \"Jim was reported missing Thursday morning. The police found him dead late Friday night. So when exactly did you hide the drugs in his guitar?\"\n\nWhitley glanced up, uneasy. \"Maybe Wednesday?\"\n\n\"Stop lying,\" said Martha.\n\n\"I'm not.\"\n\n\"Your story would have worked, if not for one little problem.\"\n\n\"What?\"\n\n\"I had Jim's guitar.\"\n\nWe stared at her.\n\n\"He lent it to me so I could practice my song for Spring Vespers. I had it in my room for two weeks. They made the announcement Jim was missing Thursday morning. Late Thursday night, I returned the guitar to his room. That means only one thing. You knew Jim was missing when you put the drugs in his guitar.\" Martha looked at Wit, her face implacable. \"You saw his disappearance as an opportunity to get out of the mess you'd made. You set him up.\"\n\nWhitley glared at her.\n\n\"The night he died?\" I asked, my voice breaking. \"Were you with him at the quarry?\"\n\n_\"No.\"_ Whitley shook her head. \"I swear. I swear, Bee. I loved him as much as anybody.\" As she said this, she began to cry. \"Okay, fine. I set him up. I put the stash in his room. I was scared. I thought he'd already gone to the administration and told them I was the White Rabbit. I did it to save myself. It was awful. But I had nothing to do with his death, I swear to God.\" She stared at me, her eyes red. \"You have to believe me.\"\n\nAbruptly, there was loud pounding on the door.\n\n_\"Cannon Beecham, you in there?\"_\n\nIt was Moses.\n\n\"Please,\" Wit whimpered. \"I didn't know anything about Jim's death. I _still_ don't, I swear on\u2014\"\n\n\"Open up now. I got police with me.\"\n\n\"This is Warwick Police! You're trespassing on private grounds.\"\n\nThe door rattled but didn't budge.\n\n\"Let's get out of here,\" said Martha.\n\nAs the pounding continued, we sprinted to the opposite side of the pool, Whitley struggling to her feet and heading after us. There were bleachers. Girls' and boys' locker rooms.\n\nNo way out.\n\nI was about to suggest we surrender and spend the rest of the wake at the police station, when Cannon grabbed the lifeguard chair and, spinning, launched it at the wall of windows. The glass only cracked.\n\n\"Open this door!\"\n\nCannon picked up the chair again, hitting the wall a second time. I could see officers swarming outside. They were wielding flood flashlights. Another smash of the chair. Suddenly the glass shattered all at once. The five of us took off running, whooping and shouting at the top of our lungs, out into the night, exploding past the officers, their flashlights blinding us.\n\n\"Police! We command you to freeze!\"\n\n\"Warriors!\" whooped Kipling.\n\n\"Go to hell and back again!\"\n\nSomewhere behind me, Kip was howling. Whitley too. Cannon was singing. I ran blindly, weird, strung-out laughter hiccupping out of me as I willed the dark to swallow me. I could see one of the officers drawing his gun. I half expected him to shoot me out of sheer terror, thinking this was the beginning of some zombie apocalypse. I headed for the darkest part of the field, forcing my legs to go faster and faster, lungs tightening in pain, rain pummeling me. When I glanced back, I could see flashing red and blue police lights, figures swarming the aquatic center.\n\nNo one had followed me. I was alone.\n\nI slowed to a jog, then a walk, rain needling my face. I realized I'd reached the edge of the woods. I crossed onto a hiking trail and headed down it. Soon my mind quieted and I was aware only of my footsteps and the mud. It was everywhere, gelatinous and black as tar.\n\n_Jim._\n\nIt was so dark, I could almost feel him here, strolling beside me.\n\nI wanted so badly to scream at him, to demand the truth. _Why so many secrets?_ Jim was the painting I'd always thought was a one-of-a-kind masterpiece. As it turned out, there were countless versions of the same work floating around, smaller watercolors and rudimentary pencil sketches, cheap poster reprints selling for ninety-nine cents in an airport gift shop. True, none of them had the beauty and detail of my painting, but they still depicted the same scene and thus rendered it a little less special. Vida had her version. Mr. Joshua his. Whitley hers. Even Martha\u2014there had been something disconcerting about the way she'd announced it: _I had Jim's guitar._ Her feelings for Jim seemed to rise to the surface strong and strange, barely controlled, before sinking back into the depths.\n\nThe rain fell harder, thunder growling.\n\nI walked on. Now and then, with a wave of revulsion, I swore I caught a glimpse of the Keeper, dressed in his gardener's slicker, hiking between the trees, but every time I stopped, my heart pounding, staring into the woods to be sure, there was no one.\n\nSoon I could feel the leaden pull of the Neverworld taking hold, the now-familiar tingling grip starting at my feet, crawling up my shins.\n\nThe path had taken me to a clearing where there was strong wind and giant fallen trees all over the ground. Squinting, I saw a hulking oak teetering a few yards in front of me. Suddenly it fell with a deafening crack, the entire forest echoing with the sound.\n\nI froze in alarm.\n\nAnother crack rang out right beside me. Turning, I realized it was another oak coming loose. I tried to move out of the way, only to realize my feet were stuck in the cementlike mud. I managed to wrench them free, blindly throwing myself forward as the tree thundered to the ground, missing me by inches, branches shaking and snapping, whipping my head.\n\n_What was happening?_\n\nI lifted my head and crawled away, tried to take another step but fell facedown in the mud.\n\nThe Neverworld blackout was descending. The end of the eleven point two hours had come. I managed to roll onto my back, gasping as I blinked up at the sky, the rain. It felt like being buried alive under the weight of a million poured coins, my body sinking deeper and deeper into mud. Soon I would feel my limbs breaking apart and dissolving.\n\nAnother tree began to tear loose a few feet away.\n\nMy final thought was panic: panic that I was going to die here with no solution to the mystery. The confessions of Vida Joshua and Whitley had solved nothing. Would I ever know what really had happened to Jim? How could I win the vote and get back to life?\n\nAt that moment, I realized a dark figure was standing over me with a cruel expression.\n\nThe Keeper.\n\n\"In the dark there grows a tree. A castle tower shelters thee. When will I stop, when will I see? There is no poison but for me.\"\n\nI screamed as the oak tree fell on top of me and everything went black.\n\nI woke up, gasping, in the backseat of the Jaguar, Martha and Kip beside me.\n\nMy heart was still pounding from the massive tree collapsing on me, the swamplike mud, the sudden appearance of the Keeper, the insidious rhyme he'd recited.\n\nI felt nauseous, but there was no time to think. Martha and Kipling were scrambling out of the car, running toward the house. I took off after them. Like me, they seemed worried that Whitley wouldn't be there anymore; that, humiliated by the revelation that she had long been the White Rabbit, maybe even scared that we'd hold her accountable for Jim's death, she'd fled.\n\nInstead, we found her sitting with Cannon in the kitchen. I could see from their mutually subdued demeanors that they'd been having an intense conversation, perhaps even arguing.\n\nWhitley looked red-faced and sullen. There was a hint of relief on her face.\n\n\"Whitley has something to tell you,\" announced Cannon.\n\nShe glanced up with a feeble smile. \"I'm sorry. I'm sorry for what I did to Jim. To the other students. For lying to all of you. It was disgusting. And stupid. It could have ruined my life. It almost did. But I swear with every fiber of my being I had nothing to do with Jim's death. I understand you might not believe me now. But we will find out what happened to him that night, and you'll know I'm innocent.\"\n\nKipling and Martha, studying her, seemed to accept this. They nodded. I nodded too.\n\nAnd yet, considering she'd lied to us for so long, I had to remind myself Wit was still capable of lying. Though I couldn't imagine her plotting to harm Jim, I also couldn't ignore what she was capable of in a rage, or the fact that she now had a motive. If Whitley had believed that Jim was going to expose her secret\u2014that the administration and, most seriously of all, the Linda, were going to find out the terrible thing she'd been doing\u2014it wasn't outrageous to think she would have done anything to prevent that from happening.\n\nAbruptly, I was aware of everyone staring at me.\n\n\"What?\" I blurted.\n\n\"We were wondering what you thought of Vida's confession,\" said Martha.\n\nI shrugged. \"I think I believe her.\"\n\n\"Me too,\" said Kipling with a wry smile. \"A girl like Vida can't lie very well. How many times did she call Jim a genius? I'm surprised she didn't suggest he'd risen from the dead, like Jesus.\" He looked at me. \"You know I loved Jim to pieces. But the way he collected admirers could get a little old. His ego, bless his soul, could be an insatiable baby.\"\n\n\"He didn't do it on purpose,\" I said. \"People were drawn to him.\"\n\n\" 'Didn't do it on purpose,' \" said Kipling. \" _Sure._ That's like me holdin' up an iron rod in a football field during a storm and sayin' it's not my fault I got struck by lightning.\"\n\n\"She was especially impressed by Jim's lyrics for his musical,\" said Martha.\n\n\"Right. The sudden blast of brilliance. That _was_ somethin'. 'Member how he had nothin' written for weeks 'cept a few bad rhymes straight out of MC Hammer? He kept complainin' that he was finished, dried up\u2014torturin' us all, pretty much. Then, out of the blue, a masterpiece.\" Kipling waved his hand in the air, a drowsy gesture. \"Strands of lyrics like pearls. One after the other. All about the immense pain of being young and alive.\"\n\n\"Those lyrics were amazing,\" said Whitley.\n\n\"The performance at Spring Vespers was a hit,\" said Cannon thoughtfully, interlacing his fingers. \"The New York producer Mr. Joshua arranged loved the demo. Jim's destiny was teed up, on the brink, like he always wanted. So what happened?\"\n\n\"Life,\" said Kipling dryly.\n\n\"Or,\" said Martha, \"it had something to do with that ride from Vida Joshua.\" She gnawed a thumbnail. \"I'm wondering if we can track down that shopping center.\"\n\n\"All Vida gave us to go on was a pet store and a fast-food restaurant,\" said Wit.\n\n\"When was the last time you talked to Jim?\" Martha asked me suddenly, squinting.\n\n\"Tuesday afternoon.\" I cleared my throat. \"The second night of Vespers. I confronted him about lying to me about the infirmary.\"\n\n\"So you didn't speak to Jim at all the next day? The day he died?\"\n\nI shook my head.\n\nNo one said anything, all of them doubtlessly thinking how tragic it must have been for me, for that argument about Vida Joshua to be our last conversation.\n\nThe truth was, Wednesday I'd exiled myself to Marksman Library, hiding out in the fourth-floor attic stacks in the History of South America section, which was seldom visited by students. It reeked of mildew and served as a shadowy breeding ground for a range of freakishly large moths. All day I sat hunched over my European history and English literature textbooks in front of the lone window with the dirty glass, Beats headphones blasting the soundtrack to _Suicide Squad_ in my ears, forcing myself to focus on the French Revolution and World War II and _For Whom the Bell Tolls._ I kept my cell phone off all day because I didn't want to deal with Jim. The only time I exposed myself to the rest of campus was during my four-minute walk between the library and my room in Creston Hall around eleven o'clock.\n\nI waited until I'd put on my pajamas and climbed into bed at midnight before turning on my cell, whereupon I was hit by the torrent of texts. A few were from Kipling, Cannon, and Wit. Twenty-seven others were from Jim. They'd started at eight that morning, messages ranging from _?_ to _come on_ to _why are u bein like this_ to desperate voice mails, his mood ranging from teasing to despair to anger, all of which sounded crazy and heartbreaking the more I replayed them.\n\n> Call me.\n> \n> Call me Bumblebee.\n> \n> We need to talk.\n> \n> If you have any love left in your heart, call me.\n> \n> Why are you doing this?\n> \n> I need you. You know how I need you to survive.\n> \n> I hate you. I hate you so much. Because I love you.\n> \n> Don't do this.\n> \n> I'm going to the quarry. Meet me.\n\nThat was the last text I ever got from him. Received at 11:29 p.m.\n\nI deleted it. I deleted all of them.\n\nWhen they found Jim dead, two days later, I expected the police to ask me about his texts. I'd tell them I'd remained in my room all night.\n\nBut they never did ask. No one ever even questioned me.\n\nI was tempted to tell them that I knew Jim was going to Vulcan Quarry. But what if they didn't believe I'd stayed in my dorm all night?\n\nI'd be damned to the Neverworld forever. I'd have no chance\u2014 _none_ \u2014of ever making it out of here alive.\n\n\"What if we went to the police now?\" whispered Cannon.\n\n\"What?\" asked Whitley.\n\n\"What if we went to speak to the Warwick police about Jim? We could get our hands on his case file. They have to have pulled his cell phone records. We get our hands on those reports. We'll know a lot about his final days\u2014where he went and who he was with.\"\n\n\"The police never came up with anything substantial to make them think Jim's death was anything but suicide,\" said Martha.\n\n\"Unless the school forced them to cover up what they found,\" said Whitley.\n\n\"Or his family,\" added Kipling. \"If Edgar Mason thought somethin' damnin' was about to come out about his beloved dead son? The apple of his eye? He'd do anythin' to stop it from gettin' out. Remember the safe house?\"\n\nWe said nothing, all of us thinking back.\n\nChristmas break, senior year, Jim invited us to his family home in Water Mill, and we were shocked by the extreme security measures his family had adopted as totally routine.\n\nEdgar Mason had always been paranoid. Hoover, Jim called his dad, a not-exactly-joking reference to J. Edgar Hoover, the fanatical wiretapping founder of the FBI. For years, Edgar Mason had employed a private security firm called Torchlight to safeguard his family, which meant for the entirely of Jim's life, two armed ex\u2013Navy SEALs silently tailed him and every other member of his family when they left the house.\n\nThe Christmas visit revealed a new level of Edgar's obsession. Every inch of the Masons' many houses around the world was being recorded in HD, the feeds playing in a basement control room called the Eye. There was a cybersecurity team on staff in Washington, D.C., who monitored the family's servers twenty-four hours a day.\n\nThen there was the safe house.\n\n\"For home invasions, terrorist attacks, and Zero Days,\" Jim said, pointing out the black bunker peering out, crocodile-like, over the hill on the edge of the property. \"It has power generators, independent water supplies, a secure phone line that can call the director of Homeland Security in three seconds. When the end of the world happens, let's meet here.\"\n\nThe smile fell from his face, the lonely implications of such a structure hardly lost on him. He seemed reluctant to say more. After all, his dad's obsession with safety had everything to do with him. Edgar Mason had always been careful, but it was apparently Jim's boating accident the summer before senior year that had triggered this new level of mania.\n\n\"I say we go over there,\" said Cannon. \"Ask around. See what the police know.\"\n\n\"Or are trying to forget,\" said Whitley.\n\n\"Bee?\" prompted Martha.\n\nEveryone turned, waiting for me to weigh in.\n\nI stared back.\n\nTaking a look inside Jim's police file could mean his final texts to me would come to light. I'd have a lot to explain. But what else was in that file?\n\nThe decision was really no decision at all.\n\n\"Can I help you?\" asked the police officer. POLK read his name tag.\n\n\"We'd like to speak to Detective Calhoun,\" Whitley said sweetly.\n\nCalhoun had been the lead investigator on Jim's case. He'd given the few public statements and briefings. We had decided it was easy enough to start with him, with his thick gray beard and rodent eyes, his wan blinks at the news cameras as a gaudy rash of embarrassment seeped across his neck. One sensed he wanted nothing more than to get away from the glare of such a high-profile case and go back to working petty crimes like public bench vandalism.\n\n\"What do you want with Calhoun?\" demanded another officer, now approaching. His name tag read MCANDRESS.\n\n\"We wanted to ask him about a case he worked on,\" said Martha.\n\n\"Which one?\"\n\n\"The death of Jim Mason,\" I said.\n\n\"That case is closed,\" said a third officer.\n\nAfter ten more minutes of hostile questioning\u2014they seemed wary of outsiders\u2014we made it to Calhoun's office, finding the man in question marooned behind a desk piled high with papers, like a giant bullfrog hiding in a bog.\n\nI wasn't sure what I expected\u2014maybe that movie scene where the grizzled old detective, asked about the cold case still haunting him after All These Years, begins to talk and talk.\n\nInstead, Detective Calhoun was a concrete wall.\n\n\"Mason case was solved. Suicide,\" he belched.\n\n\"What made you rule suicide?\" Martha asked, frowning. \"Usually with suicides there's a ritual or preparation before the act. A note left behind. Glasses removed, as well as shoes and socks. Was there any sign of that with Jim?\"\n\nCalhoun flashed a grin that was really more of a sneer.\n\n\"That case is closed.\"\n\n* * *\n\n\u2014\n\n_That case is closed._\n\nThe Warwick police station, located just off the highway, was a quaint banana-yellow bungalow with white shutters and a sign on the bulletin board that read LIFE IS BETTER WITH COFFEE. The place seemed ill-suited for solving crimes, better for selling homemade muffins.\n\nLittle did we know how terrifying it would be\u2014that our time with the Warwick police would be pandemonium.\n\nThere was no other way to put it.\n\nWe tried different tactics: warm, curt, nervous, sexy (Wit, wearing a low-cut red dress, perched on a desk). We tried surprising them. We tried a brazen arrival at Calhoun's private residence after his wife went to bed and Calhoun stayed up late drinking Harpoon IPA, eating gummy worms, and watching _Better Call Saul._ No matter what we said, and where, how, or what time we said it, Calhoun refused to tell us anything about Jim's case.\n\n\"Can't help you.\"\n\n\"You Nancy Drews get out of here.\"\n\n\"How dare you accost me at my home!\"\n\n\"You kids get out of here, or I'll make sure every one of you spends the rest of your life flipping byproduct burgers at the local Mickey D's, because a fryolator and an automatic mixer to make a few watery milk shakes are all you'll be fit for after I get through with you. UNDERSTAND ME?\"\n\nWe decided to give up on Calhoun and bribe the office manager, Frederica.\n\nWe waited for her to leave the station in workout clothes, watched her getting drenched in the downpour as her umbrella went inside out. As she fumbled for the keys to her Kia, Cannon ran to her, armed with a golf umbrella and a grin.\n\nWe watched from the Mercedes as he made his pitch: ten thousand dollars cash to go back inside and steal the Jim Mason file. Frederica blushed, nodded, and power-walked back inside.\n\nCannon turned, grinning, giving us the thumbs-up. Then Frederica emerged with the entire Warwick police force in tow, eight officers hightailing it for Cannon.\n\n\"Get on your knees! Police!\"\n\nWith a yelp, Whitley backed the car out, tearing through the lot to pick up Cannon, who, sprinting for his life, hurled himself into the backseat. We barreled through the grass, over a curb, and through a red light into a twelve-lane intersection, tires squealing, and nearly collided with a cement truck.\n\n\"Move!\" Cannon screamed. _\"Move!\"_\n\n* * *\n\n\u2014\n\nWe spent the next few wakes at Roscoe gun range learning how to shoot as we formulated our new plan. We would raid the Warwick police. We paid the owner of Last Resort Twenty-Four-Hour Pawn Shop fifty dollars for the name of a guy who sold guns without a license out of the back of his RV, called Big Bobby. Out of Big Bobby, we bought three guns: two Ruger LC9 strikers, and a Heckler & Koch HK45.\n\nOur siege would take place at eleven-fifteen, when we believed all officers, apart from Polk and McAndress, had gone home for the night. We'd take them by surprise and lock them in the supply closet. Then we'd have the station to ourselves and we could find Jim's file.\n\nThe first time we attempted it, Officer McAndress\u2014displaying moves apparently learned from a fruitful career moonlighting in mixed martial arts\u2014struck Kip with an uppercut to the face, elbowed Wit in the ribs, then, spinning, back-kicked Cannon's stomach, sending him gasping to the floor. Meanwhile, Officer Polk had me pinned to the ground, his left foot gouging my back as he zip-tied Martha to a desk chair.\n\n\"The Warwick police aren't _actually_ police,\" said Kipling with unabashed wonder the next wake. \"They're the spawn of Satan.\"\n\nHe wasn't kidding.\n\nHow many times at midnight, at one, two, three o'clock in the morning, did the five of us storm that station? Was it a hundred times? Was it ten thousand?\n\nHow many different entry SWAT formations did we attempt, after finding on the Internet the _Ground Reconnaissance Operations Handbook,_ a training manual for marines?\n\nSingle file, double file, double file advancement with variation, through front doors, back doors, barred windows, fire escapes, drainpipes, adjacent-building rooftops. We downloaded something called _The Criminal's Guide to Revelry_ off the Deepnet, a hand-typed, poorly photocopied manual written by Anonymous Doe that detailed strategies for nullifying people who were aggressive, hysterical, or empowered by Superman fantasies, during any raid, heist, or robbery. How many terrifying masks did we try (clown, pig, _Clockwork Orange_ droogs)? How many bullets did we shoot into the ceiling? How many threats and warnings did we scream?\n\nIt should have been easy to get our hands on one case file.\n\nAs the wakes went on, we transformed from disorganized teenagers acting out a makeshift version of _Mission: Impossible_ into a real five-person platoon. We were able to advance without noise, reading each other's thoughts and movements with nothing but a look.\n\nWake after wake, we were thwarted.\n\nThis was due not to Officers Polk and McAndress, but to their wild card, Officer Victoria Channing, assigned to Traffic Safety. She was always in the women's bathroom, and she advanced on us out of nowhere, displaying an eagerness to kill that was psychotic.\n\n\"Take that, you little shitheads!\" she screamed.\n\nShe was terrifying, even if we were immortal.\n\nThough Cannon and Whitley had learned to nullify Polk and McAndress, Channing eluded them every time. She was able to slip like vapor through the back stairwell, the front stairwell, a panel in the ceiling, firing her Glock without warning into Cannon's chest.\n\nOr Kip's stomach.\n\nOr Whitley's forehead.\n\nThe time she shot Martha in the temple\u2014as if blithely turning the knob on a gumball machine\u2014I froze, stunned, staring down at her and the rest of my friends lying on the police station carpet, blood gurgling out of their foreheads and necks like water trickling from a hose.\n\n\"Hands up or you'll be joining them in hell!\" screeched Officer Channing, aiming at me.\n\n* * *\n\n\u2014\n\n\"Death feels like floating in a warm bath,\" said Martha, the next wake.\n\nI was always the only one left alive. This was because I wasn't a natural warrior. I tended to freeze when I needed to act. As a result, Cannon had decided my job was to locate Jim's case file.\n\n_Find Jim's file._ It was all I had to do.\n\nThe closed homicide cases were kept in the basement. It was an unnerving, neglected fish tank of a room: humming green fluorescent light, smells of mildew, pipes yowling with steam. Row after row of metal shelves extended, dreamlike, in every direction, floor to ceiling, packed with cardboard boxes. Each box was scribbled with a victim's name.\n\nAppleton, Janice\n\nAvery, Jennifer\n\nAzella, Robert P.\n\nAt every break-in, no matter the hell being unleashed upstairs, I headed straight to the back stairwell and raced to the basement. I'd fling open the wooden doors marked STORAGE and sprint into the labyrinth of shelves, madly looking for the _M_ s, my footsteps squeaking on the orange linoleum. Every time, Channing caught me and I spent the rest of the wake sitting in a jail cell hearing her tell the other officers a phony story about having to kill everyone else in self-defense.\n\nAnd yet, each wake, while my friends were dealing with the mayhem upstairs, I was dealing with my own torment in that droning green basement, a trial that had nothing to do with Officer Channing.\n\nIt began when I accidentally kicked a box off a bottom shelf.\n\nHendrews, Holly\n\nThe box tipped over, sending a plastic bag marked _Evidence_ spinning across the floor. Inside, there was a blood-encrusted Christmas scarf decorated with snowmen and reindeer, but what struck me\u2014I stopped dead, blinking in alarm\u2014was that the bag was spangled with black mildew.\n\nThe box was also leaking.\n\nI kicked it closer, peering inside. An oil-like liquid had pooled in the corners, as if one of the evidence bags had leaked. Glancing up, I saw, stunned, that it wasn't just this box. There were others. At least four or five boxes had that same black liquid seeping through the bottom.\n\nThen there were the shelves.\n\nThey were hulking and metal. Yet sometimes as I raced past them, madly searching for MASON, JIM, the slightest brush of my shoulder would send the giant shelf toppling over as if it were nothing but cardboard. It would land with a deafening clang, sending the one beside it over, the one beside that too, until all the shelves in the basement were falling around me like massive dominos, hundreds of boxes thundering to the ground. All I could do was scramble out of the way, press my back against the nearest wall, pray I didn't get hit until it was over.\n\nAfterward I'd try to sift through the rubble for Jim's box before Officer Channing caught me red-handed, as she always did.\n\nI never mentioned to the others what was happening. I was too scared.\n\n\"How's it going in the basement?\" Cannon asked me. \"Are you close to finding Jim's file? I heard a lot of banging downstairs this last time.\"\n\n\"There's a lot to sift through,\" I said. \"I'm close.\"\n\nThe key, I'd found, was to sprint through the shelves as fast as I could, allowing them to fall after me as I kept running and running toward the row of _M_ s at the very back of the basement. I had just perfected the optimal path through the maze when, once, barreling too fast, I missed the correct row and was forced to backtrack. I was careful not to graze the shelf as I made the turn and slowed to a walk, panting. Usually there was the sound of havoc upstairs, banging and screaming. This time it was quiet.\n\nToo quiet.\n\nMachinsky, Tina D.\n\nMahmoodi, Wafaa\n\nMalvo, Jed\n\nI spotted Jim's box on the top shelf at the very end. In a rush of disbelief, I sprinted to it, reached on my tiptoes, tried to jostle it down without sending the entire shelf clattering over.\n\nMason, Jim Livingston\n\n\"Gotcha, you little shit,\" a woman hissed. \"Put your hands up and turn real slow.\"\n\nChanning had stepped out from behind the shelves and was striding toward me, Glock aimed right at my head.\n\n\"Don't shoot. Please.\"\n\nHer face was flushed. Her lips twitched. She pulled the trigger.\n\nShe never had before.\n\nA giant match lit the wick of my brain. I hit the ground, rolling onto my back, accidentally throwing out my arms, which hit the shelf, sending it flying backward.\n\n\"What the...?\" Channing screamed in shock.\n\nAs the shelves fell, I blinked up at the fluorescent lights, green filaments flickering in mysterious Morse code. There was so much pain it spilled everywhere, then drifted away.\n\nDying was not as cataclysmic as I'd thought it'd be. Because even though I was in the Neverworld, my body and mind still reacted as if it were the real thing.\n\nThere was no white light. There was no tunnel.\n\nInstead, as the shelves tumbled around me, there was a warm feeling of awe, as if, with the tearing away of my life from its attachment to earth, fragile as the connection of a leaf to a twig, everything permanent, factual, real\u2014everything I swore was true\u2014became the opposite of what I'd always thought.\n\nMy last feeling wasn't regret or pain. It was joy.\n\nThat was the most terrifying thing of all.\n\n_I get to see Jim now._ That was all I was thinking before my life went out.\n\n_If I'm dead, I'll get to see Jim._\n\n* * *\n\n\u2014\n\n\"Really don't feel like getting shot in the head today,\" sang Kipling merrily as we filed into Wincroft at the start of the next wake. \"How 'bout we take a page from Momma Greer's Guide to the Good Life?\"\n\n\"What's that?\" asked Cannon.\n\n\"Can't beat your mortal enemy to a pulp?\" He shrugged. \"Throw him a party.\"\n\nThat was how we came to arrive at the Warwick police station armed not with our usual guns, but with identical Barry the Clown costumes rented from Gobbledygook Halloween World.\n\n\"May I help you?\" asked Frederica at the front desk.\n\n\"We're from Big Apple Balloon-a-Gram,\" I said, smiling. \"Where would you like us to set up for the surprise party?\"\n\n\"What surprise party?\"\n\n\"Detective Art Calhoun's surprise seventieth.\"\n\nFrederica was astonished. So were Officers Polk, McAndress, Cunningham, Leech, Ives, and Mapleton, as well as Art Calhoun himself, who emerged from his office with a distrustful scowl. But we had moved fast. The wireless speaker was already playing \"Margaritaville.\" Wit had already unveiled three dozen cupcakes, baskets of gummy worms, and party favors of gun Christmas tree ornaments. Cannon and Kip were standing on folding chairs, taping up the tinsel Happy Birthday sign. Martha tossed bottles of Harpoon IPA into a cooler.\n\n\"Hold up. Just wait one...\" Calhoun fell silent, eyeing the beer.\n\n\"What's happening here?\" demanded Officer Polk.\n\nI made an elaborate show of examining the phony invoice, which was really the receipt for our costume rentals.\n\n\"Elizabeth Calhoun hired us,\" I said with a frown.\n\n\"Lizzy did this?\" whispered Calhoun, wide-eyed.\n\nLiz Calhoun was his estranged daughter who lived in San Diego. She hadn't spoken to her father in three years, which meant there was little chance she'd take his call now, when he phoned to thank her for the unexpected party, even though his real seventieth birthday was over three weeks away.\n\nThat meant I had time to find Jim's file.\n\n\"Nothing better than cupcakes,\" said Officer Channing, grinning as she helped herself.\n\n\"And now, friends, let's start the entertainment!\" shouted Kipling with a bow so low his red clown nose fell off and rolled under a desk. \"If you could all gather round and join hands? Don't be shy.\"\n\nThat was my cue.\n\nI darted into the back stairwell. I raced down the steps to the basement, heaved open the wood doors, and sprinted into the shadowed rows of boxes.\n\nI raced along the back wall, slipping past filing cabinets and a copy machine, and veered into the _G_ s, the _J_ s, the _L_ s, zigzagging in and out, careful not to graze the shelves, my oversized clown shoes making loud quacking noises, my balloon pants making me trip.\n\nEighth row. Far left. At the very top. MASON, JIM LIVINGSTON.\n\nIt was there. My heart pounding, I had to jump up three times to shove it off the shelf without sending the entire thing toppling over. I set it carefully on the ground.\n\n\"You found it?\"\n\nI turned, startled, to see Martha hurrying toward me.\n\nShe'd never appeared down here before. The knowing, even anxious look on her face seemed to suggest she had come because she didn't trust me, because she didn't want me left alone with the box. Or was it that she'd hoped I would never actually find it?\n\nI ripped off the tape and pulled open the lid.\n\nI stared inside for an entire minute, unable to speak.\n\n_No. No. No. Impossible._\n\n\"Are you kidding me?\" whispered Martha in apparent shock, looking over my shoulder. \"After _all that_? Getting shot dead a million times?\"\n\nShaking her head, a hand on her hip, she took down another box.\n\n\"Maybe Jim's file got put somewhere else,\" she muttered.\n\nI couldn't stop staring in, unable to breathe.\n\nThe box was empty.\n\nThere wasn't a single paper left except a coupon: _$5 off 1 bin of Honey Love Fried Chicken. Soul Mate Special!_\n\nUpstairs, more singing and clapping had broken out. \"For he's a jolly good fellow...\"\n\nMartha was madly thrusting more boxes to the floor, yanking off the lids. Every one was crammed full of papers, plastic evidence bags, black ink.\n\nThat ink was back, seeping through the corners again.\n\n\"How can anyone find anything in here? It's a mess. There's some kind of leak.\"\n\nMartha was examining the ink between her fingers, wrinkling her nose, though when she caught my eye, the knowing expression on her face chilled me.\n\nWe spent another ten minutes going through boxes, Martha saying, \"It has to be here somewhere.\"\n\nThe only empty box we came across was Jim's.\n\nMartha knew something. That was clear. What it was, I had no idea.\n\n\"Edgar Mason and Torchlight Security are behind it,\" said Cannon when we were back at Wincroft, sprawled across the couches in the library. \"Who else could make an entire case file just _vanish_?\"\n\n\"They had everything destroyed,\" said Kipling in agreement. \"Which was why Calhoun and the other cops were always so touchy when we asked about the case. They've been paid off.\"\n\n\"But why?\" I asked.\n\n\"Don't you see?\" said Cannon. \"Something in there was incriminating to Jim.\"\n\n\"Right,\" said Wit with a nod. \"They didn't want it made public. So they sent some Torchlight ex\u2013Navy SEAL into the station and he stole it.\"\n\n\"Which means the Masons know the truth,\" said Kipling.\n\n_\"And,\"_ Cannon continued, \"if they haven't come forward to arrest anyone, if they've stayed silent, it means whatever they uncovered was damaging for the family.\"\n\nAs everyone fell silent, considering all this, my eyes caught Martha's.\n\nShe seemed skeptical, or her mind was somewhere else. I'd been unable to stop thinking about her sudden appearance in the basement and the look on her face when she'd spotted the ink. It made me wonder whether she suspected me somehow, whether she knew I'd received those texts from Jim asking me to meet him that night at the quarry.\n\n\"Out of all of us, Bee,\" she said suddenly, \"you spent the most time with the Masons. Did they ever tell you what they thought happened to their son?\"\n\nI shook my head, shrugging. \"We completely fell out of touch.\"\n\nCountless times during the past year, I'd wondered how the Masons had handled Jim's death. I never found out. I never even made it to Jim's funeral. My parents, fretting about my mental well-being, begged me not to go. And while a few Darrow students\u2014including Whitley, Cannon, and Kip\u2014had gotten special permission to take the train to New York for the service, I decided to stay away. My absence, I knew, would come as a relief. His family had liked their modern art collection infinitely more than they'd ever liked me. Jim's mom, Gloria\u2014a champagne flute of a woman, all ice-blond hair and long limbs, with a low voice\u2014always surveyed me as if I were a window with an airshaft view. Jim's father had to be introduced to me three times before he recalled who I was. And even then he called me Barbara.\n\n\"I say we pay a surprise visit to the Masons,\" said Whitley.\n\n\"We'll probably have to waterboard 'em to get 'em to talk, child,\" said Kipling. \"But count me in.\"\n\n\"There's a problem,\" I said.\n\n\"What?\" asked Martha.\n\n\"The wake.\"\n\n\"What about it?\"\n\n\"It's only eleven point two hours. That's not enough time.\"\n\n\"What do you mean?\" asked Kip, frowning. \"We fly to East Hampton. We'll be outside the Masons' Water Mill estate in less than two hours.\"\n\n\"They're not in Water Mill. The Masons spend every summer on Amorgos, an island in the Aegean Sea. It takes eleven hours by plane. Plus a three-hour boat ride. Then you have to hike up a mountain to reach the house.\"\n\nThey seemed skeptical, so I dialed the Masons' Fifth Avenue apartment. The housekeeper who answered confirmed the family was away.\n\n\"Are they at Villa Anna Sofia on Amorgos Island?\" I asked.\n\n\"That's right. Would you like to leave a message for Mr. and Mrs. Mason?\"\n\nJim called his family's compound in Greece the Milk Shake for the way it oozed down the cliff overlooking the ocean. Much to my parents' irritation, I'd spent five days there with Jim the summer before junior year. Although the time had passed in a sunburnt blur of bleached-white beaches and outdoor feasts, sunset boat rides and Greek folk music, Jim working relentlessly on his musical, that island and the Masons' vertigo-inducing compound remained one of the most surreally beautiful places I'd ever seen.\n\n\"We could try Skyping them,\" suggested Kipling. \" 'Hi, we're Jim's old friends phoning from purgatory. We command you to tell us everything about your son's death.' \"\n\n\"I guess that's that,\" said Whitley gloomily.\n\n\"Not exactly,\" said Martha.\n\nI turned to her with a shiver of dread.\n\n\"It's time you guys learned the truth.\"\n\n\"What are you talking about?\" asked Whitley.\n\nMartha cleared her throat.\n\n\"The Neverworld is more complex than you think. I mean, none of you have noticed anything strange?\"\n\n\"Oh, no, this is all perfectly routine,\" said Kipling, smiling.\n\n\"Strange things like what?\" I asked.\n\n\"Unusual disruptions. Magnetism. Instability.\"\n\nInstantly I thought of the mold, the peeling wallpaper, the tumbling trees, the collapsing shelves, the exploding snow globes, that black ink soaking through all the case files.\n\nMartha appeared to know what it was, what it meant. She was fumbling in her heavy black bag, pulling out a small black notebook.\n\nI recognized it. It was the one I'd spotted her carrying in the early days of the Neverworld, when she stopped to hastily scribble in the pages before moving on.\n\n\" 'Sighting, six thirty-nine p.m.,' \" she read. \" 'One mysterious purple-feathered owl perched atop a maple tree, unknown species.' \" She turned the page. \" 'Overheard. Variety of eighties songs by the Cure in every passing car and every surrounding house.' \"\n\nMartha closed the notebook, surveying us.\n\n\"Remember what the Keeper said. 'Imagine if each of your minds was placed inside a blender, and that blender turned on high. The resulting smoothie is this moment.' \"\n\n\"Okay,\" said Whitley, nervous.\n\n\"He was talking about the physics of the Neverworld. I'm very excited to tell you that it's based in part on J. C. Gossamer Madwick's groundbreaking masterpiece. And it's my fault.\"\n\n\"What are you talking about?\" asked Cannon.\n\n\"I wrote a two-hundred-page thesis on the novel. I tracked down every rare book about it. Every obscure blog. I interviewed professors, experts, and scientists. I even went to visit Madwick's daughter on Bello Costa Island in the Florida Keys, rowing out to this tiny, falling-down beach house in a remote cove swarming with alligators. She let me inspect Madwick's notebooks, which have never been seen by anyone outside the family, not even the people at Harvard who've been bullying her to donate them to their archives. I read all eleven notebooks, translating them from Lurroscript, the language Madwick made up.\"\n\nShe stopped her mad outpouring of words to take a deep breath.\n\nI realized she was talking about _The Bend,_ the fantasy novel she'd been obsessed with, the one no one had ever heard of except her and a bunch of geeky fanboys on the Internet.\n\n\"My preoccupation with the book made it our reality. I lived it. Breathed it. Now it's in the Neverworld.\"\n\n\"But what does that mean, child?\" Kipling asked, faint shrillness in his voice. \"We're all about to float out into outer space? Become androids?\"\n\nMartha tilted her head and grinned, the lenses in her glasses flashing in the light.\n\nMy heart plunged. Whatever she was about to tell us, I knew I couldn't trust it.\n\nI also knew that in this world stuck on repeat, everything we knew was about to change.\n\n\"Well, for one thing,\" Martha said excitedly, \"it means rather than waking up at Wincroft, we can wake up anywhere in the past, present, or future.\"\n\n\"And how do we do that?\" asked Cannon.\n\n\"We climb out the unlatched window.\"\n\nWe could only stare, baffled.\n\nShe sighed. \" _Right._ Okay. I got way ahead of myself.\"\n\nShe took another impatient breath.\n\n\"Lesson One. J. C. Gossamer Madwick was a science fiction writer. He wrote just one book, called _The Dark House at Elsewhere Bend. The Bend_ for short. It's this amazing adventure story and alternative world, different from anything you've ever read. It was never published. Just photocopied over and over again, bound using a hole-puncher and garbage bag zip ties, passed hand to hand by anonymous travelers, student backpackers, and disaffected youths in hostels. The thing you have to do once you finish _The Bend_? You sign the dedication page and leave it for the next lucky person on a park bench, bunk bed, airplane seat, or train compartment. For the longest time the only copies to be found were in ancient bookshops and on eBay, some with hundreds of thousands of signatures. The ones with famous names of the readers, like Marilyn Monroe and Leonard Bernstein and Frank Sinatra? They went for as much as four, five grand. Now it's an official cult classic, steadily in print, and even random people like E.S.S. Burt have copies.\"\n\nTo my surprise, Martha raced over to the shelves and pulled out a hulking silver hardback book. Returning to the couches, she handed it to me. The cover featured a collage of birdcages, steam trains, men and women wearing top hats, masquerade masks straight out of Victorian England.\n\n> _The Dark House of Elsewhere Bend_ by J. C. Gossamer Madwick.\n> \n> _The legendary cult saga of future pasts. Present mysteries. An undying love at the end of the world._\n\nI flipped to the back flap and stared down at the author photo.\n\nIt was grainy and black-and-white. In a rumpled suit, Madwick was a man few would look twice at: hound-dog face, extravagant ears, an apologetic slouch suggesting he was more comfortable ducking out of a room than entering one.\n\n> _Jeremiah Chester Gossamer Madwick (December 2, 1891\u2013March 18, 1944) was an American novelist from Key West, Florida. His only work, the posthumously published_ The Dark House of Elsewhere Bend, _won the Gilmer-Hecht Prize for Fantasy in 1968. For 37 years he worked as a bus driver for the Key West transit office, driving passengers to and from Stock Island by day, and writing his 1,397-page masterpiece by hand on hotel notepads by night. At age 53, he was found dead in the doorway of Hasty Retreat Saloon, a harmonica, a tin of tobacco, and the final paragraph of his novel in his pocket._\n\n\"Madwick died penniless and unknown,\" Martha said. \"Now he's considered one of the greatest fantasy writers who ever lived. Harvard has an entire class about him: Hobos, Strangers, and Vagabonds: The Literature of Madwick. He even has a cult following in the real-life physics community due to his theory of time.\"\n\nShe paused to hastily draw something in her notebook. It was a sketch of a train.\n\n\"Which brings me to Lesson Two,\" she said. \"Time travel. Madwick viewed time not as linear, or an arrow, or even a fabric, like Einstein. He saw it as a locomotive. To time travel in _Elsewhere Bend,_ you climb out the window of your speeding train compartment and scale onto the roof, like a bandit in an old western. Then you carefully move toward the _front_ of the train, the future, or the _back_ of the train, the past. It's vital not to move too quickly in either direction because that will cause instability. Like, the train can jump the tracks, or crash, or separate compartments, or veer suddenly onto a _wrong_ track heading clear in the opposite direction.\"\n\nShuddering in apparent horror at the thought of such a scenario, Martha took a deep breath and tucked her hair behind her ears.\n\n\"In the event of such disasters, you, the time traveler, are doomed. Because you'll never be able to get your train back running on the original track, much _less_ on time, much _less_ climb back to the compartment in which you began. Although technically you can live the rest of your life in the past or future, the carriage where you were born, the original present, is where you belong. _Always._ That's where life will be the smoothest journey for you. Where things work out and love lasts. A life lived at any other time will be restless, rough, ill fated. You can visit the past and the future, but you can't stay there. Not if you want any chance at happiness.\"\n\n\"What does this have to do with the Neverworld?\" asked Cannon, uneasy.\n\n\"We want to interrogate Jim's parents? I believe we can. We just have to choose a day in the close past or the future where we can reach them in the eleven point two hours of the wake. Then we find the open window in our train and climb out. And this open window...\" She nibbled her fingernail. \"It's somewhere here. I don't know where yet, but it's a collision of life and death. It tends to be suicidal. In _The Bend,_ the protagonist uncovers it by accident in Chapter One when he tries to commit suicide. And obviously none of us has ever committed suicide.\"\n\nI shook my head. So did Whitley, Cannon, and Kipling.\n\n\"So that rules _that_ out,\" Martha said gloomily. \"We'll have to locate the open window by some other means. Which brings me to Lesson Three.\"\n\nShe cleared her throat. \"The Neverworld was created not only by me, but by each of you. _My_ biggest contribution is Madwick's _Dark House at Elsewhere Bend._ But what about you? The closer you study the Neverworld, the more of yourself you'll find. Your darkest secrets. Your worst nightmares. Your fears and dreams. The embarrassing thing you never want anyone to learn. It's all here, buried, if you look close enough.\"\n\nAn uneasy chill inched down my spine.\n\nThere was something threatening in the way she announced this. The others looked uncomfortable too. Whitley sat on the couch, motionless. Kipling looked pale. Cannon stared her down, completely absorbed.\n\nMartha surveyed her notebook with a faint smile. \"Kipling.\" She cleared her throat. \"I meant to ask you.\" She held up a page where she'd drawn a red wasp. \"The scarlet-bodied wasp moth. Native to Louisiana. I've spotted three at Wincroft. Two crawled out of the attic upstairs. Another from a radiator. They shouldn't exist this far north. Do you recognize it?\"\n\n\"How did you...?\" blurted Kipling. He chuckled nervously. \"Momma Greer used to catch them in mason jars. Kept them all around the house. Pit fiends, she called them. Said the sting was lethal and she'd put them on me while I slept if I didn't sit still during church.\"\n\nMartha nodded blankly, unsurprised. She turned the page.\n\n\"Cannon. Surely you've noticed all the Japanese larch?\"\n\nHe sat up, nervous. \"The...what?\"\n\n\"The Japanese larch and silver birch trees growing around Wincroft. If you look closer, they're dead. A bunch of tall, spindly black tree trunks sticking out of the ground. Those trees aren't native to Rhode Island. They're indigenous to the Chubu and Kanto regions of Japan. If you go up to one and dig down about six inches, chalky blue water pools everywhere.\" She beamed. \"You know what I'm getting at?\"\n\nCannon only stared.\n\n\"Blue Pond?\" she suggested. \"Cannon's Birdcage? The bug you discovered in Apple's OS X operating system sophomore year? The accidental combination of keystrokes that crashes your hard drive, delivering the photo of Blue Pond wallpaper to your screen? The photograph is an almost surreal picture of a bright blue lake, dead snow-tipped trees growing right out of it.\"\n\nHe was confounded. \"Okay. What about it?\"\n\n\"That photo is embedded in the Neverworld's landscape. Everywhere.\"\n\nCannon said nothing, only slipped to his feet, crossed the library to the window, stared out.\n\n\"Then there's Whitley,\" Martha went on officially. \"There's a volatility in the Neverworld's weather because of you.\"\n\n_\"Me?\"_ said Wit.\n\n\"Gale-force winds. Constant rain, thunder, lightning. It's your temper.\"\n\nWhitley glared at her.\n\n\"The night we went back to Darrow,\" Martha went on. \"How we got chased by the police. I watched wind overturn every car in the parking lot. It was because of your confession about being the White Rabbit.\"\n\nWhitley huffed in apparent disagreement, but her eyes flitted worriedly to the windows.\n\n\"These details go for all of us,\" Martha went on. \"The closer we get to the truth, the root of who we are, the more unstable this world will become. Which brings me to Beatrice.\"\n\nShe turned to me, her expression stony. My heart began to pound.\n\n\"I have no clue.\"\n\nEveryone frowned at her\u2014and then at me.\n\n\"Your contribution is here. Somewhere. But I haven't figured it out yet.\"\n\nI swallowed. What was Martha attempting to do? Intimidate me? Scare me? If so, it was working.\n\nShe sighed. \"One thing I do know is that if we try changing the wake, we have to stick together.\"\n\n\"Why is that?\" asked Cannon.\n\n\"We don't know how we're going to react. The past hooks you like a drug. The future jolts you like an electric chair. Reliving beautiful memories can be just as devastating as reliving the terrible ones. They're addictive. Given that time travel in _The Bend_ is so dangerous, and that inside the Neverworld there are elements we can't anticipate\u2014the things _you_ are each contributing\u2014we have no idea what will happen if we even attempt this.\" She shook her head, her voice trembling with so much emotion, she reminded me of an evangelical minister on a public access channel, lecturing a rapt congregation about the end of the world. \"It could be a complete disaster. We could accidentally end up in different train compartments on different trains speeding in different directions. That means it'll be impossible to ever make it back here. To Wincroft. Together. _To vote._ Then we really will be trapped here forever.\"\n\nThe rest of us eyed each other in alarm. No one spoke.\n\nI gazed down at the hulking book on my lap. I couldn't breathe.\n\nWhat was she up to? Was Martha actually trying to help us? Or was this new revelation only the meticulous and conniving arrangement of her chess pieces on the board, some ingenious trap we would all fall into, which would somehow result in everyone voting for her?\n\nWhat I _did_ know\u2014or at least strongly suspected\u2014was that she knew what my contribution to the Neverworld was. I could tell by the way she looked at me, by her flat, implausible explanation: _I haven't been able to figure it out._\n\nMartha always figured everything out. For whatever reason, she'd decided not to disclose this piece of information.\n\nNot yet.\n\nFor the next couple of wakes, we stayed in the library at Wincroft, studying _The Dark House at Elsewhere Bend._ We wanted to understand everything Martha had told us.\n\nWe downloaded the audiobook and spent hours listening to all 1,322 pages, curled up under mohair blankets, drinking tea as the narrator\u2014some young British actor from the Royal Shakespeare Company with an opera baritone and a schizophrenic ability to sound like completely different men and women, young, old, poor, aristocratic\u2014told the futuristic tale of love and loss. It was a bewitching story, one of the best I'd ever heard, a heart-pounding mystery unfolding against a future world, fascinating and terrifying plot twists you couldn't see coming.\n\nThe book took place far in the future. The main character, Jonathan Elster, was a bumbling, absentminded professor at a university for outcasts in Old Earth. He taught a popular alternative philosophy course, Intro to Unknowns, which covered, among other things, the nuts and bolts of time travel. For years, Elster had been in love from afar with a mysterious woman named Anastasia Bent, who taught in the history department. When she accidentally stumbled upon a cover-up about the history of the universe and vanished\u2014a fisherman witnessing her wandering a cliff walk suggested she committed suicide, though her body was never recovered\u2014Jonathan set off on a perilous quest across space and time to find her.\n\nAll of us grew silent and sullen as we listened. The violence at the Warwick police station had brought us all together, opened up the roped-off rooms in the sprawling, lavish mansion that had once been our friendship, flung the sheets off the furniture, turned on the lights. Now it seemed Martha's disclosure had us taking refuge in our separate rooms again, disappearing up winding staircases, holing up behind closed doors, the only hint of company an occasional creak of the floorboards overhead.\n\nI didn't know what they worried about. They chose not to confide in me or, it seemed, in each other.\n\nMy own anxiety had everything to do with Martha. Those bombshells she'd dropped\u2014J. C. Gossamer Madwick, the physical laws of the Neverworld being tied to each of us\u2014had prompted me to scrutinize her every knowing glance and comment even more closely than I had before. To my shock, I realized that somewhere in the time since I'd found her at Brown with Professor Beloroda, she'd managed to quietly seize control of the entire group. For years, her status had been peripheral. She was the tagalong sidekick, Jim's friend, the oddball you could count on to react to PETA commercials with some jarringly cynical comment like \"Such propaganda,\" or, when any couple ended up together at the end of a romantic comedy, \"Another horror movie with a high body count.\" Martha had always made us roll our eyes and chuckle. Now, incredibly, impossibly, she was the one the others looked to for guidance, for expertise and reassurance. A few times I tried confiding in Kip, Whitley, and Cannon, hinting that I didn't trust Martha or this new direction she was urging us toward. They didn't share my suspicion.\n\n\"What do you mean, she's up to something?\" Kipling asked me, frowning.\n\n\"I don't know. It's just a feeling.\"\n\n\"But it's good old Martha. Rain Man. She's not conniving. She's too honest and goofy.\"\n\n\"I don't think so.\"\n\n\"Oh, come on.\"\n\n\"I'm serious. She knows more than she's letting on.\"\n\n\"Does it really matter?\" whispered Kipling. \"What else can we do, Bee? I don't know about you, but I need something to change. Anything. Even if it means...\"\n\n\"Even if it means what?\"\n\nHe shrugged, his expression bleak, the meaning behind his unfinished sentence obvious.\n\n_Even if it means we never get out of here._\n\n* * *\n\n\u2014\n\nOur evenings passed in heated discussions about how to change the wake and get to the Masons.\n\nIn _The Bend,_ Jonathan Elster discovers time travel accidentally when following in the footsteps of the missing professor Anastasia Bent, who the police believe committed suicide by jumping off a seaside cliff. As Jonathan follows suit, jumping from the exact spot where Professor Bent was last seen, plummeting a hundred feet toward certain death, he finds himself crashing not into the rocky cliffs, but into the Thames in London in the year 2122.\n\n\"The open train compartment window for time travel always exists on the verge of death,\" said Martha, \"which is why so few people ever find it. You have to think you're facing your death in order to reach it. So how do we find it here, in the Neverworld? I know I already asked you guys this, but have any of you ever tried to commit suicide?\"\n\nAgain, we all shook our heads.\n\nMartha seemed perplexed by this response, though she only nibbled a thumbnail thoughtfully, saying nothing more.\n\nAnother critical fact to remember was that for everyone to arrive at the right place and time\u2014the same train compartment in the past or the future\u2014we had to make sure it was the final thought in our heads right before the moment on the verge of death.\n\n\"We'll aim for Villa Anna Sophia on Amorgos Island one day in the past,\" Martha announced. \"Yesterday. August twenty-ninth. That's where we'll start.\"\n\n\"Why the past and not the future?\" asked Cannon.\n\n\"You never know what tomorrow will bring. The future could hold a natural disaster, terrorist attack, alien invasion. The past has already happened, so we know what to expect.\"\n\n\"But if we're going into the past,\" I asked, \"why not just go straight to Vulcan Quarry on the night Jim died? Then we'll know everything.\"\n\n\"She's right,\" said Cannon.\n\n\"No,\" said Martha, shaking her head. \"No _way._ We're not ready. In _The Bend,_ the train gets shorter and shorter with each leap in time. That means our wakes will get shorter. It'll cause too much instability. It could mean we won't have enough time to come to vote with a consensus. We have to start out slowly.\"\n\nI didn't buy her explanation\u2014she seemed too quick to condemn my suggestion\u2014but the others appeared to accept her answer. So I decided not to challenge her. Not yet.\n\nNo one had been to Amorgos except me. Only I had visited Jim that summer. So the others could vividly envision the time and place, I showed them photos from the trip on my phone and told them what I remembered: The island's scalding brightness. The open-air Jeeps the Mason family drove around the island, tearing down the dirt roads like an occupying army. Edgar Mason, shut away at all hours in his space-age office, from which he'd abruptly emerge like Zeus coming down from Olympus (if Zeus was tanned to the color of whiskey, had spiky hedgehog hair, and rose every day at four a.m. to practice Ashtanga yoga while whispering into an earpiece). Jim's younger siblings and their respective friends stampeding up and down the house's staircases like herds of antelope. Jim had two younger twin sisters, Gloriana and Florence, and two adopted brothers from Uganda, Cal and Niles. Much to my amazement, they had a Swahili tutor living with them (\"A cultural attach\u00e9,\" Jim said). Jim and I had spent most of our time alone, reading aloud from John Lennon biographies, diving off the dock, exploring the coastline in a blue skiff called _Little Bird._ We snorkeled and ate grilled fish doused with lemon that squirted into my eyes and stung. We fed dinner rolls to the packs of wild dogs that patrolled the night streets like gangs, and stayed up into the early morning at drunken family feasts at gangplank tables under a blue night sky, chains of yellow paper lanterns bobbing overhead.\n\nThough Jim had invited me to spend the entire summer with him, my parents only agreed to let me visit for five days. To get them to agree to even that took State Department levels of persuasion. Those five days passed in the blink of an eye, each tinged with a blinding, far-fetched sheen, which made me feel at once uncomfortable and bewitched. Jim's world was so vivid, so improbable. As suddenly as I was thrown into it, I was tossed out, marooned back in sleepy Watch Hill, distracted and gloomy as I worked alongside my parents at the Crow, leaving milk shakes too long in the mixer, preparing egg salad sandwiches for customers who'd ordered turkey and Swiss. I was haunted, like Wendy by memories of Neverland and Peter Pan. I spent the rest of the summer tabulating in my head the seven-hour time difference so I could picture what Jim was doing, and roared through the house like a caged lion tossed a fresh carcass to seize the phone in time whenever he called.\n\nPer Martha's instructions, I described as best I could a single room in the house: the main living room, with its dreamy gauze curtains, whitewashed furniture, and scalding view of the Aegean. This was so they could feel as if they'd been there too, could have it be the very last thought in their heads before the moment on the verge of death\u2014whatever that turned out to be.\n\n* * *\n\n\u2014\n\nWhen the five of us weren't holed up in the library hashing over _The Bend,_ I'd grab an umbrella and head into the storm, hiking the property alone.\n\nAs I walked I could hear Martha's voice in my head, her unnerving whisper: _Your contribution is here. Somewhere._\n\nIt was during one of these solitary walks when, heading down the narrow stairs to the dock, I noticed the trees writhing with an intensity that wasn't normal. The ocean was rough, whitecaps licking the inky water. The sailboats moored far out in the cove clattered and bobbed. Ropes had come loose from the masts, thrashing like snakes. A buoy clanged somewhere out on the ocean, the sound mournful, deathly.\n\nSuddenly I heard a woman's sharp scream.\n\nThe moan of the wind made me think I'd imagined it.\n\nThen I heard another cry. This time it was a man.\n\nTwo people were arguing. Sensing that the voices were coming from behind me, I closed my umbrella and hurried off the dock and out of sight, darting into the foot-wide space between the bank and wooden steps.\n\nMoments later, I heard them again. I realized they weren't coming from Wincroft at all, but from one of the sailboats out in the harbor. Dark figures were moving along the deck of one anchored close to the dock of the property next door.\n\n_Andiamo,_ it was called.\n\nI remembered hearing Whitley mention it belonged to E.S.S. Burt. A tiny gold light was shining from the bow.\n\nThen it came: another scream.\n\nI waited. Minutes later, I heard a motor. A skiff was approaching. Peering through the steps, I saw Whitley. She was alone. She docked the boat, hastily knotting the ropes before climbing onto the pier, running up the steps. Her face caught the light as she barreled right past my head. She looked angry.\n\nI waited. When there were no further voices, I headed back to the house. I knocked once on the library doors, and opening them, I saw Whitley, soaking wet, sitting with Martha on the couch. They jolted upright as soon as I entered, startled looks on their faces. My first thought was of two teenagers surreptitiously smoking pot in a living room, suddenly interrupted by a parent.\n\n\"What's the matter?\" I asked.\n\n\"Nothing,\" said Whitley.\n\nMartha smiled. \"Could you give us a minute, Bee?\"\n\nI stared at Whitley. Never had she preferred to confide in Martha over me. _Never._ Hadn't our friendship come back to life after all this time in the Neverworld? Weren't we friends again? But she only stared back at me, sullen.\n\nI left with a sinking feeling in the pit of my stomach, closing the door.\n\nAlmost immediately I could hear Whitley, her voice low and muffled, her words indistinguishable.\n\n_What the hell is going on?_\n\nI headed upstairs. Searching the bedrooms, I found no sign of Cannon, or Kipling either, which seemed to suggest they'd been aboard the boat too. Had they held some kind of secret meeting purposely behind my back? Why? What were they doing? I grabbed the umbrella, headed back outside, and hiked down to the dock.\n\nThere was no obvious movement on the sailboat and there were no more voices.\n\nI stayed there for another hour, watching. When nothing happened, I decided to hike back to Wincroft, and as I hurried up the path I noticed the Keeper.\n\nImmediately my throat constricted. The last time I'd seen him had been in the woods at Darrow. He seemed to have vanished for a time. Now he was back, a dark figure in a black raincoat, hunched over, drenched. He was digging with great exertion, his whole body contorting with each fling of dirt.\n\nI veered off the stone path to avoid him and sprinted through the trees. When I reached the house, I couldn't help turning back to look at him.\n\nHe hadn't noticed me. He was still digging what I now realized were four muddy holes in the earth. Four graves.\n\nBut that wasn't the strangest thing.\n\nHe was wearing the black glasses of a blind man.\n\n* * *\n\n\u2014\n\nI didn't have time to ruminate on what the Keeper had been doing, or what it meant, because the very next wake the answer to the mystery of the sailboat argument and Whitley's private conversation with Martha came to light.\n\n\"Kipling has something to tell us,\" announced Martha as we all assembled in the library. \"Whitley brought this to my attention late last wake, and I think it could help us.\"\n\n\"You didn't,\" Cannon said angrily to Whitley.\n\n\"I had to,\" she answered.\n\nHe glared at her, livid.\n\n\"Oh, _please._ Stop the policeman act. You want to get out of here, don't you? I mean, don't you want to find out what really happened to Jim?\"\n\n\"It wasn't your secret to tell,\" he hissed.\n\n\"If it affects our ability to change the wake, it is.\"\n\n\"It's all right,\" said Martha, placing a hand on Cannon's shoulder. \"You can tell us.\"\n\nThat was when I noticed Kipling. He was crying. _Truly_ crying, in a way I'd never seen him before\u2014the kind of crying that was more of a wringing out than normal tears. He was seated on the couch, head in his hands, tears streaming down his chin.\n\n\"I call it the Black-Footed Sioux Carpet,\" he blurted suddenly, staring at the floor. \"It's a form of self-harm. 'An unsuitable attempt to solve interpersonal difficulties.' That's what the shrinks all call it. Momma Greer invented it. She coined the term from some crazy-lookin' rug she'd filched from an antiques store. We did it together. Mother-son bondin'. Sometimes we did it multiple times a week. She'd drive me out to a country road on a Friday night when she decided there was nothing good on TV. The first time I was five. We'd lie down in the road side by side, holdin' hands, waiting for a car. 'Roll out of the way when I say bingo,' she told me. 'We'll see how much God likes us. If he wants us to live. Cuz I'll only say bingo if God tells me to. That's the deal.' \" Kipling shuddered. \"I pissed myself I was so scared. I hadn't said my prayers. Good God. I mean, did God even know I existed? Did He like me? He couldn't like me _that_ much if He'd given me _this_ face to go through life with. _This_ body. I'd squeeze Momma Greer's hand. She was my lifeline. Then the car. You always felt it first in the pavement underneath you. It'd take Momma Greer a year to yell bingo. But it always came. I'd squeeze my eyes shut and roll out of the way. The tires would miss me by centimeters. By the time I opened my eyes Momma Greer would be up dancin' on the side of the road, whooping and hollerin', yankin' off all her clothes. 'See that? God loves us. He loves us after all.' She was always in a good mood after that. If I was lucky, it lasted a whole week.\"\n\nHe fell silent a moment, rubbing his eyes. I could only stare. While I had known Momma Greer was dangerous, this was by far the most terrifying thing I'd ever heard she'd done.\n\n\"It became an addiction,\" Kipling went on. \"The rush of it. I never stopped. Every few months, whenever things got out of hand or hopeless, I'd find a way to do my Black-Footed Sioux Carpet. I'd sneak off campus. Immediately felt better. I did a big one junior year, right before Christmas break, when Rector Trask told me I couldn't return next semester. I was kicked out. I was the sort of student\u2014 _how_ did he put it?\u2014who needed an environment with 'less vigorous expectations.' Like, he thought I'd do better in Sing-Sing. My Black-Footed Sioux Carpet after _that_ nearly got me made into an egg-scramble sandwich by a Folger's truck.\" He glanced up, sniffing. \"It certainly would have given new meanin' to their slogan 'The Best Part of Wakin' Up.' \"\n\nI gazed at him, speechless. Kipling had always been a rotten student. While I knew there had been cliffhangers at the end of every school year as to whether he was passing, I'd never known he was actually kicked out. His poor academic record had changed senior year, when he managed to focus on his studies. By the time we graduated, he had done well.\n\n\"It was Cannon who saved me,\" Kip said with a faint smile. \"He saw what I was tryin' so hard to hide.\"\n\n\"You weren't that good at hiding it,\" said Cannon, grinning. \"You were walking with a limp and winced when you sat down.\"\n\nKipling looked at me. \"Remember how I missed two months of school due to a 'family emergency'?\"\n\nI nodded. Vaguely I remembered him telling me a vibrant and long-winded story about his aunt's heart condition.\n\n\"It was all lies. I was at a treatment center in Providence, doin' tai chi, watercolorin' fruit bowls, and developin' a middle path to manage my unrestrained patterns of thought. It was Cannon who checked me in. Cannon who came during visitin' hours. Coordinated with the shrinks on my progress. Lobbied Darrow to give me one last chance. He helped turn my grades around. Got my college applications ready. Sat up with me all night helpin' write my essay about Momma Greer. 'Mommy Bipolar.' Otherwise known as 'How to Survive in the Custody of a Complete Lunatic.' That got me into Louisiana State. I'd be encrusted right now in the front tire treads of a UPS truck if it weren't for Cannon.\"\n\nMy mind was spinning. I thought back to senior year, and though I recalled Cannon as always quite busy, coming and going abruptly with his backpack and an armful of textbooks, never had I suspected what he was up to. But it made sense. He was the silent problem solver. \"The steady trickle of water that always finds a passage,\" Whitley used to say. Still, I felt hurt that they hadn't wanted to confide in me, that there had been an entire history happening right before my eyes about which I'd had no clue.\n\n\"Why did you never say anything?\" I asked Kipling.\n\nHe glanced at Cannon, and I saw pass wordlessly between them some fleeting shadow of understanding that was gone almost as soon as I recognized it.\n\nKipling shrugged. \"There comes a point where your personal pile of crazy gets to be a bit much. Even for your best friends.\"\n\n\"That's not quite the whole truth,\" prompted Whitley expectantly, tilting her head.\n\nKipling looked sheepish. \"Yeah, well.\" He cleared his throat. \"My eleventh-hour streak of Cs and Bs, revealin' me to be a decent student who'd only been _pretendin'_ all that time to be abysmal? That wasn't real.\"\n\n\"What are you talking about?\" I asked.\n\nHe seemed unwilling to go on.\n\n\"Cannon hacked Darrow's network for him,\" blurted Whitley. \"All senior year. Kipling had every test from every teacher ahead of time. Including midterms and finals.\"\n\n\"Not every teacher,\" said Cannon.\n\nShe glared at him. \"It was still cheating.\"\n\n\"It was assisting a beloved friend,\" he said stonily.\n\nWhitley huffed. \"You could say the same thing about what I was doing as the White Rabbit. Everyone thinks _I'm_ the bad person? Look at what you guys were doing.\"\n\nCannon said nothing. For years he had assisted Darrow's notoriously backward IT department. It wasn't unusual for him to be summoned from class to help with some bug or networking error. And though he was glaring at Whitley now in obvious annoyance, he didn't appear to feel in the least bit guilty about this disclosure.\n\n\"How did you do it?\" I asked him.\n\nCannon shrugged. \"Social engineering. The weakest component in any given network is always the human. I sent a faculty-wide email, a required update for Darrow's intranet. For Kipling's teachers I included a RAT. They downloaded the trojan and I became root. It was as easy as untying a shoelace.\"\n\nHe frowned at the look of disbelief on my face.\n\n\"Come on, Sister Bee. You of all people should understand. Darrow-Harker was an obstacle in the way of Kipling's bright future. Kicked out junior year? He'd have to start over at some second-rate institution. Away from us. It'd look like shit on his record. And anyway, Kipling can't be measured by such blunt objects as As, Bs, and Cs. No. Kipling is an experience. I had to help him in the best way I could.\" He shrugged. \"There are the rules of this world, and there is what you do when life comes crashing down around you.\"\n\nCannon stared at me with such a penetrating look, I felt chills inching down my arms. I'd forgotten how intense a presence he could be, how when he focused, he seemed more energy than flesh and bone.\n\n\"So that's it,\" said Kipling. \"That's the two-headed monster in my closet who can't stop drooling.\"\n\n\"The question is,\" Martha whispered, looking him, \"will your secret help us change the wake?\"\n\nShe fell silent, frowning, lost in thought. For a minute no one said a word.\n\nThat was what Martha did sometimes\u2014let a question dangle for minutes, sometimes even an hour, before suddenly blurting the answer when everyone else had forgotten the problem.\n\n\"I have an idea,\" she said.\n\nThat was how we came to be parked in the wild beach rose along the empty coastal road at 4:47 in the morning, four minutes before the end of the wake.\n\nDirectly across the street was where we'd had the accident\u2014where, according to the Keeper, one Mr. Howard Heyward, age fifty-eight, of 281 Admiral Road, South Kingstown, had smashed his tow truck into our car, condemning us to the Neverworld, where somewhere, in some other dimension of time more real than this one, we were lying inside a totaled car inside a single second waiting to unlock.\n\nMartha knew the exact spot, a hairpin curve twisting one hundred and sixty degrees through dense pine trees. She admitted she'd come back here to inspect it in the Neverworld.\n\nHow had it happened? I could hardly remember. Aggressive flashes of headlights blinding me. Hedges of beach rose trembling in the torrential rain. Windshield wipers waving as if in warning. Liquid night. Our drunken laughter spilling everywhere. Honking. Spinning. The car bouncing off the road, leaping into the dark. A loss of gravity.\n\n\"He's a drunk,\" Martha said. \"He sits in the Raccoon and Hound Saloon in Warwick and drinks twelve Coors Lights. _Twelve._ Then he climbs behind the wheel. He can hardly stay awake. Nearly crashes into a telephone pole. In the Neverworld, he drives straight past the spot where he hits us. But that marks the end of the eleven point two hours of our wake.\"\n\nRain hammered the roof. The windshield and windows were fogged. I felt as if we were sealed inside a submarine at the bottom of the sea. The radio stuttered classical music.\n\nOnly one car had passed us, a blue pickup. Spotting us nestled in the bushes on the side of the road, it braked and backed up. Martha unrolled the window.\n\n\"You guys got a flat?\" asked a middle-aged man in a hunting vest. \"Need a hand?\"\n\n\"No, thanks,\" said Martha. \"We're fine. We're looking for our lost dog.\"\n\nHe frowned, baffled by the sight of five teenagers dressed in green hooded ponchos smiling stiffly. With a perplexed grimace and nothing left to say, he drove off.\n\n\"Three minutes,\" said Martha, checking her watch.\n\nI felt like I was going to be sick. Kipling and Cannon's revelations, shocking as they were, had elicited more questions than answers. For one thing, everyone was acting strange, though it was difficult to put my finger on why. They were irritable and out of it. Twice, when they weren't aware I was watching, I saw Kipling and Cannon exchange long, knowing glances, the meaning of which seemed vaguely ominous. _What was going on?_ What were they planning?\n\nAnd though Martha was coaching us, assuring us it was going to be fine\u2014 _August twenty-ninth, nine-thirty-five a.m., Villa Anna Sophia, Amorgos Island, Greece, that's all you have to remember, okay?_ \u2014the fact that she of all people was in charge of this operation only made it worse. What was she up to? Was she pushing us to follow in the footsteps of characters in _The Bend_ so she could condemn us somehow, trap us in some train compartment of time? Or was it only about the vote for her?\n\nThe vote. The vote. _The vote._\n\nNow, hunched beside Whitley in the backseat, I could feel the wake coming over me, that familiar ocean-wave immensity pressing down on my feet, inching into my shins. Abruptly, the radio belched with static, then began to cough and stutter \"Boys Don't Cry\" by the Cure.\n\nThe rain grew louder, as if the volume had been turned way up.\n\n\"I don't feel so swell,\" said Kip, pressing a hand to his throat.\n\nMartha turned to him. \"I feel it too. And it's not just the wake. It's the open window. It's happening.\"\n\nShe was filled with excitement\u2014as much as someone as deadpan as Martha _could_ be filled with excitement.\n\n\"Can you feel it?\"\n\nI did. There was an electrical charge in my hands, as if I'd just shuffled across a heavy carpet in socks. I held my hand an inch from the steamed window. It made a print. I waved it back and forth, and it magically wiped the window clean. I held my hand a few inches behind Whitley's hair hanging outside the hood of her poncho, and the gold strands leapt right into my hand like the tentacles of some strange sea creature.\n\n\"Two minutes,\" said Martha. \"Let's move.\"\n\nShe nodded at us and scrambled out, Cannon and Kipling taking off after her without a word. I opened the door and was instantly drenched by a blast of rain. Whitley grabbed my arm.\n\n\"I can't do this, Bee,\" she whimpered. \"I can't keep it straight in my head.\"\n\n\"What?\"\n\nShe was crying. Never in my life had I seen her so afraid.\n\n\"I'm going to get lost in the past. I know it.\"\n\n\"No. You're not.\" I grabbed her by the shoulders. \"Listen to me. August twenty-ninth. Nine-thirty-five a.m. Villa Anna Sophia. Say it.\"\n\n\"Villa Anna Sophia.\"\n\n\"Remember the sea. The sky. The pristine white beauty of it all. The curtains. The smell of oranges.\"\n\n\"Oranges. Right.\"\n\n\"You've got this.\"\n\nShe blinked at me, unsure. I held out my hand. She grabbed it. Then we both climbed out of the car into the downpour.\n\nI hadn't anticipated how chaotic it would be. The rain felt like nails. There was a gravitational pull intent on thrusting us back to the car. My thoughts turned to liquid, splattering the inside of my head. All we had to do was approach the spot of the accident and lie down in a Black-Footed Sioux Carpet the way Kipling had explained it. So far only Martha had made it. She was lying on her back along the faded yellow line. I headed toward her, trying to drag Whitley after me, but I was dizzy, and every step was like lifting four cinder blocks tied to my feet. Kip was standing in the road, turning in a circle like a cork caught in a toilet flush, and Cannon was on all fours, trying to crawl. I forced my thoughts to slow. I took big steps, one at a time, squeezing Whitley's hand. Finally we reached the spot and lay down beside Martha. A minute later Cannon arrived with Kipling.\n\nI blinked, raindrops pounding my face. I couldn't see. The rain was falling too hard, so I closed my eyes. The wake had crept up to my knees, pushing me into the pavement.\n\n_August twenty-ninth. Nine-thirty-five a.m._\n\nI could picture the rocky, windswept cliff, the modern white house poised there like an eagle's nest, nothing in the windows but a reflection of the sea.\n\n\"Fifty seconds!\" bellowed Martha.\n\n_Villa Anna Sophia._\n\n\"I can't do this!\" Whitley screamed.\n\nSomeone scratched me in the face. I moved my arm to shield my eyes, realizing it was a giant oak branch torn off a tree. It had careened over us before cartwheeling down the road.\n\nWhitley was sobbing, trying to scramble to her feet. Cannon held her in place.\n\n\"Stop it!\" he shouted.\n\n\"Let go of me! I can't do it!\"\n\n\"Calm down!\" shouted Martha.\n\n\"I can't! I keep thinking of other things! I can't stop my thoughts!\"\n\nI heard the roar of the approaching engine. Howard Heyward, age fifty-eight, drunk and half asleep, was seconds away now. My entire body was shaking. I squeezed my eyes closed, my fingers gripping the pavement, trying to hold on.\n\n_Amorgos Island. Greece._\n\nSomeone else was screaming now. Kipling.\n\n\"Stop it! Stop it!\"\n\n\"Don't you see? We're going to lose each other!\"\n\n\"It's a trick! It's a trap!\"\n\nThunder exploded like an atom bomb. My ears blew out, squeals and whines ricocheting strangely around my head. The wake was pressing down on my heart now, so strong it took a moment for me to realize something was viciously stabbing my neck. I cried out in pain, my cold, numbed fingers fumbling to see what it was. I felt something small, hard. I yanked it out of my neck, screaming.\n\nIt was the bumblebee pin, the one Jim had given me, the one stolen from me.\n\nThe rest happened at once. Headlights sliced through me. The truck was honking, careening toward us. Raindrops fell in slow motion. A howl of brakes. Someone was still screaming. I opened my eyes, catching a fleeting glimpse of a figure in a green poncho sprinting away, vanishing into the woods. Clanging metal. The truck was jackknifing, massive tires sliding on the wet pavement right toward my skull. A smell of scorched rubber. And hell.\n\nOne...two...\n\nBumblebee pin.\n\nJim.\n\nWhen I opened my eyes, it was daylight.\n\nI was facedown in the grass. I lifted my head, heart pounding, feeling an overpowering wave of nausea. I was sick to my stomach, my body spasming. It took a minute to catch my breath. I wiped my mouth, looking around, my eyes stinging in the light.\n\nI was not on any coastal road. I was not being run over by Howard Heyward's tow truck\u2014at least, not anymore. I was in no physical pain.\n\nI also wasn't in the back of the Jaguar. For the first time in a century it wasn't raining. The sun was shining. I was lying on the ground\u2014dead leaves, dirt, surrounded by trees. It was brisk out, a bite in the air, the sky hard blue. I held out my hands, opening them.\n\nThey were empty.\n\n_The bumblebee pin. Where is it?_\n\nI looked around. I definitely wasn't near Villa Anna Sophia or on any Greek island.\n\nI was in the middle of a forest. I stared down at my clothing.\n\nThe burgundy Ann Taylor wool coat my mother had picked up years ago at a secondhand store in Woonsocket. Black tights. Black wool dress. Scuffed black leather pumps.\n\nPuzzled, I stumbled to my feet. My shoes were too tight, my dress scratchy. I lurched forward, staring through the trees at a grassy clearing. There was a lake littered with small white sailboats, people milling around the perimeter. I stumbled toward it, wondering if I looked like some deranged lunatic. But as I stepped out of the woods and down the bank, no one gave me a second glance. There were at least twenty sailboats out on the lake, children and a few teenagers operating them by remote control.\n\nI understood where I was: Central Park. The Conservatory Lake. I'd visited here a long time ago with Jim.\n\n\"There you are.\"\n\nHearing his voice was like having the floor drop out under my feet. I couldn't breathe. I closed my eyes, my mind jelly. I was falling through a hole a mile deep.\n\n\"Where'd you go? Are you already trying to get rid of me?\"\n\nHe was alive. He was right behind me, his hand on my shoulder. He smelled the same: peppermint soap, wind, and fresh laundry.\n\n\"I came out here all the time as a little kid. Once, the remote control broke and my sailboat got stranded in the middle of the lake and my father said, as I cried, 'If you want it, go get it.' I had to wade out there and retrieve the thing. Clearly it was some survival-of-the-fittest, free-market personality test he'd learned in business school and\u2014 Hey, what's wrong?\"\n\nHe spun me around to face him.\n\n_What's wrong? How can I begin to answer that question?_\n\n\"Look at me.\"\n\nI opened my eyes.\n\nThe sight of Jim Mason inches away from me\u2014sun blazing behind him, birds chirping, kids squealing in delight\u2014was so unfeasible, my head turned inside out.\n\nThis wasn't real. It couldn't be.\n\nBut it was. It was Jim. He was the same, but he wasn't. As I stared up at him, it struck me how no one ever really sees anyone. Memory turns out to be a lazy employee, intent on doing the least amount of work. When a person is alive and around you all the time, it doesn't bother to record all the details, and when a person is dead, it Xeroxes a tattered recollection a million times, so the details are lost: the freckles, the crooked smile, the creases around the eyes.\n\n\"Come,\" Jim said. \"We can't be late.\"\n\nHe tucked my hand into the crook of his arm. I'd forgotten how he always did that. He escorted me down the path, past women wheeling babies in strollers\u2014all of whom glanced at him with varying degrees of admiration\u2014and a man pushing a shopping cart filled with plastic bottles.\n\nIt seemed the wake had brought me to one of the occasions when I'd visited Jim's family in New York.\n\nIt wasn't Christmas. And it was too chilly for spring break.\n\nSo when was it?\n\nI could ask him what we were going to be late to, but it was a daunting prospect to speak. Every time I looked at Jim, I felt jolts of disbelief. I wanted to annotate everything about him, every blink, sniff, and sideways grin. I was terrified too. There was a lump in my throat like a giant wad of gum, threatening to dislodge. If it did, I'd end up crying or rambling on madly about the Neverworld, the fact that he was dead now.\n\n_You're dead, my love. You have such little time._\n\nBiting my lip, I let him escort me across Fifth Avenue. We rushed into his building\u2014 _944 Fifth Avenue_ read the elegant script on the green awning\u2014its lobby pungent with hydrangea and roses from the colossal flower arrangement on the table, asteroid-like and silencing. Jim casually waved at the doorman.\n\n\"Hola, Murdoch.\"\n\nThen we were alone in the elevator. Jim leaned back against the wood-paneled wall, surveying me. I had forgotten the way he studied people as if they were priceless pieces of art.\n\n\"Don't be nervous,\" he said.\n\nHe was clutching my hand again, grazing his lips against my knuckles as he pulled me, walking backward, into his apartment. I had forgotten how grand it was, echoing like a museum, iron sculptures of birds and oil paintings of stark faces, spindly furniture more giant praying mantises than viable places to sit. Looking down, I noticed the scuffs on my Mary Jane pumps, the lint balls on my old stockings, and felt that familiar cringe of embarrassment. As we moved into the living room, slipping through the crowd, I noticed everyone was wearing black\u2014black dresses, black and white and red silk scarves, blue suits\u2014and I understood where I was.\n\nFreshman year at Darrow. Five years ago. A weekend in late September.\n\nJim had invited me to come home with him for his great-uncle Carl's funeral. I barely knew Jim back then.\n\nHe'd only introduced himself a week before.\n\n* * *\n\n\u2014\n\n\"Jim Mason.\"\n\nHe was sitting behind me in English. He pulled his chair over, so close I could feel his peppermint breath on my cheek as I tried to work out a rhyme for a song I was writing.\n\n\"Whatcha doing?\" He frowned at the notebook I was scribbling in. \"What's _Fenfang's Chinese Laundry Meltdown: An Original Soundtrack_?\"\n\nEmbarrassed, I slid the book under my laptop.\n\n\"Nothing.\"\n\n\"That didn't look like nothing.\"\n\nI cleared my throat. It sounded like a swamp.\n\n\"I create fake album soundtracks for movies that don't exist. It's just something I do. Don't ask me why.\"\n\n\"I see.\" He nodded matter-of-factly. \"So, when's the commitment to the mental institution happening? Next week? Next year?\"\n\nI laughed.\n\nHe extended his hand. \"Jim Mason. Really delighted to make your acquaintance before they cart you off to your padded cell.\"\n\n\"Beatrice Hartley.\"\n\nHe winked. \"I'm a mad poet too.\"\n\nI smiled. There was a stretch of awkward silence, during which Jim did nothing but sit back and survey me. I turned to my laptop, trying to stop blushing, pretending to type something important. I assumed he was about to return to his desk and leave me alone.\n\nInstead, he started to beatbox, not even trying to be cool about it.\n\n\"There was a fetching girl in my English class \/ Wary as a bluebird, radiating class \/ I'm scared to look away from her, in case she flies away \/ Congress needs to declare her a national holiday.\"\n\nEveryone in class went silent, a boy behind me snickering.\n\nLittle did I know that this was how it would always be, that being the subject of Jim's attention would be like having a bomb go off in my face: unexpected, shocking, accompanied by a fallout of popular girls suddenly approaching me with long, swingy mermaid hair and doubtful glances.\n\n\"How do you know Jim Mason?\"\n\n\" _You're_ from New York?\"\n\n\"Did you go to Spence?\"\n\n\"I'm from Watch Hill. No, I went to Watch Hill East. I\u2014I don't know Jim.\"\n\nThat was how I met Whitley. She was friends with Jim from some exclusive Native American camp in the Blue Ridge Mountains.\n\n\"Jim Mason has a crush on you.\" It was the first thing she ever said to me.\n\nI hurried along the hallway, clutching the strap of my backpack like it was my floatation device and I was drowning.\n\n\"No, he doesn't.\"\n\n\"Yes, he does.\" She peered at me, frowning. \"He calls you 'haunting.' He said you're old-fashioned. And innocent. Like you're from the 1940s or something, and have been transported here by time machine.\"\n\n\"Thanks.\"\n\n\"It's a compliment.\"\n\nThe next day, suddenly Jim was strolling beside me down to the athletic fields. My heart flopped like a freshly caught fish.\n\n\"Did you grow up on an Amish farm milking cows at sunrise?\" he asked.\n\n\"Um. No.\"\n\n\"You look like you did.\"\n\n\"Okay.\"\n\n\"Want to come home with me this Sunday?\"\n\nHe asked it like he was offering me a bite of his sandwich.\n\nI said no. Sunday was Family Sunday, which meant Darrow's students either went home for the day or signed up for a field trip to a museum. I hadn't seen my mom and dad in a month, and they'd planned an elaborate lasagna dinner. Of course, the truth was I said no because I was terrified by Jim's attention, the brash, drenching spotlight of it, both blinding me and causing everyone else to stare.\n\nLittle did I know, _no_ to Jim was simply a yes that hadn't happened yet.\n\n\"Beatrice!\" he shamelessly rapped at the start of English, causing our teacher Mrs. Henderson to regard me with irritation. \"She's a realist. With secrets. A conscienceless realist who leaves me sleepless. And speechless. Oh, Beatrice.\"\n\nHe left notes in my locker. _Say yes (jump off a cliff with me)._ He recorded a theme song about me. It got passed around the entire school.\n\n\" 'The Queen's Neck'? _Please,_ \" I heard a girl hiss during chapel.\n\n\"Say yes!\" Jim blurted when he passed me in the hall. (\"Yes to _what_? Having his babies?\" the varsity volleyball captain snarked to her friends.) Jim called my parents to formally introduce himself, discuss train times to and from Penn Station, give them his word that I'd be safe with him, that he was a gentleman.\n\nThis deluge of attention would have been too much coming from anyone who wasn't Jim Livingston Mason, Jim of the thick, tangled black hair, the chocolate eyes, the sideways grin.\n\n\"He sounds so adorable and kind of quirky, actually,\" said my mom.\n\nBack then she'd been na\u00efve about the old-moneyed jungle of Darrow and Jim's lionlike position inside it.\n\n\"It's wonderful you're already making some interesting connections,\" said my dad.\n\nThe Sunday trip to Jim's house for the funeral\u2014 _this very trip_ \u2014would end in disaster.\n\nFast-forward five, six hours? I'd be taking a train home from New York early, alone, the reasons for which Jim and I always argued about afterward. To this day I found it difficult to recall what had really happened. What had I been so upset about? I could never separate my shyness, my self-consciousness at being painfully underdressed and awkward, from the truth. During the post-funeral buffet, held in some relative's Gilded Age apartment on Park Avenue, I remembered Jim disappeared for what felt like a torturous period of time. I'd grabbed my coat off the hall rack and snuck out without a word to anyone. I cried the whole ride back to school. I vowed\u2014unreasonably, because even then my feelings for him felt as inevitable as seawater in a rowboat full of holes\u2014that this would be the end of my friendship with Jim Mason.\n\nThat Monday morning, however, during English, he placed a red Cartier box on the notebook I'd been drawing in.\n\n\"Forgive me.\"\n\nInside the box was a diamond-encrusted bumblebee pin.\n\n* * *\n\n\u2014\n\nThe bumblebee pin.\n\nThinking of how it had mysteriously gone missing from the sock drawer in my dorm room, then abruptly reappeared all these years later, jammed in the side of my neck as we lay in the middle of the coastal road, sent a fresh wave of shock through me. Clearly it had been meant as a means of sabotage, a surefire way to get rid of me, make me think of Jim, thereby pulling me into some compartment of the past. Whoever had done it had meant to hurt me, purposefully destroy any chance we had of voting and leaving the Neverworld.\n\n_Which one of them had done it?_\n\n\"Cookie?\"\n\nI jumped, startled. I realized dazedly I was in the Masons' living room staring out the window at Central Park, which from this height looked like an architectural rendering of a park with pipe-cleaner trees. One of Jim's adopted siblings\u2014Niles, nine or ten years old\u2014was offering me a stack of cookies held between his thumb and forefinger.\n\nI took one. \"Thanks.\"\n\nHe squinted. \"You're Jim's latest girlfriend?\"\n\n\"No. I'm a friend of his from school.\"\n\n\"Well, take care you don't go\"\u2014the little kid crossed his eyes, making a deranged clown face\u2014\"like all the others.\"\n\nI laughed.\n\n\"Whoa\u2014 Did you see that?\"\n\nThe kid moved to inspect a large red Rothko, which had just fallen clean off the wall, revealing a dark square of what appeared to be mold.\n\n\"That was totally _Poltergeist_!\"\n\nI smiled stiffly, moving away as Jim led his mother over.\n\n\"Mom. This is the girl I was telling you about. Beatrice Hartley.\"\n\n\"Hullo there.\"\n\nMrs. Mason was beautiful, her black suit sealing her like an envelope. She extended her hand like it was a gift. I'd forgotten how chilly she could be: the boredom in her smile, the flick of her eyes over my shoulder, as if somewhere behind me something more charming was always happening, like dolphins leaping out of the sea.\n\n\"Darling, did you speak to Artie Grossman about the Currin?\"\n\nMr. Mason stepped over. He was short and tan, with spiky hair and the tense stare of all moguls. His teeth were big and artificially white, hinting they'd glow in the dark during a blackout.\n\n\"Dad,\" said Jim. \"This is Beatrice.\"\n\nMr. Mason smiled warmly, shaking my hand.\n\n\"Just started Darrow-Harker with Jimmy, is that it? How are you finding those old-school traditions and Kennedy smiles?\"\n\n\"Fine.\"\n\n\"Wonderful. Wonderful. Glory, did you talk to Artie?\"\n\n\"I'll do it right now,\" said Mrs. Mason.\n\nShe was smiling again, drifting away. \"Lovely to meet you,\" she said unconvincingly over her shoulder.\n\nI couldn't help staring after the two of them, wondering how they had reacted to Jim's death. What had they done, all these polished, perfumed people? Had any of them screamed and lost their minds as I had, or had life simply floated on?\n\nJim was dead now. He was lying in a coffin underneath a gravestone that read _Life Now Forever_ in Sleepy Hollow Cemetery. In this sunlit apartment with the thick walls and marble floors, the idea was unfathomable.\n\nJim smiled after them. He appeared to mistake my stare for admiration.\n\n\"They met on the R train when they were twenty years old. Still madly in love after twenty-eight years. Completely unforgivable. Come.\"\n\nHe clasped my hand again. We slipped through the crowd, past mute housekeepers in gray uniforms, a waiter holding a tray of triangular sandwiches like little starched pocket squares. He whisked me out of the living room, past three siblings playing Wiffle ball in the foyer (\"Totally inappropriate!\" Jim shouted at them), a wood-paneled library with a ladder to retrieve the thousands of first-edition leather-bound books, a dining room with a modern steel chandelier that looked like a giant tarantula. In two years I'd eat Christmas dinner there and his mother wouldn't say a word to me the entire meal. His father would call me Barbara.\n\nJim pulled me through a door and closed it behind me. It was his bedroom, a shadowed, chaotic rock star's lair with electric guitars mounted on the wall and sheet music covering every flat surface, handwritten quarter notes and half rests spangling the bars. Synthesizers. A McIntosh stereo. Three laptops. Piles of notebooks burping up pages where song lyrics were taking shape in terrible handwriting. _Lost Little Blue._ A biography of Janis Joplin. _Sweeney Todd: The Complete Score._ A framed copy of a Bruce Springsteen Madison Square Garden set list signed with a note: _Love you, Jimmy. Keep hearing the music. Bruce._ Rumpled boxers and T-shirts and rolled-up posters swamped the corners of the room.\n\nJim was rifling through a bookshelf, looking for something.\n\n\"Okay, so, I have this song I wrote about a girl I haven't met yet,\" he said, pulling out a notebook. \" 'Immortal She.' It's about the love you have for someone that can't die, no matter how far apart you are, even if you're separated by death or time. That's what I'm searching for.\"\n\nThe lump in my throat was there again, a pile of rubble.\n\nHe began to read the lyrics, as he would countless times after this. I came to know that song well. It was one of the best he ever wrote. I'd sing it for him on a picnic blanket at school during finals week. He'd sing it to me some nights at Wincroft as I fell asleep.\n\nI remembered this exact moment. I'd related it to Whitley a dozen times, because it was the classic chorus refrain of \"The Ballad of Jim and Bee,\" an old standard. This was the first time we were ever alone together, our first deep conversation. Our first kiss was seconds away. Having it before me again made me feel paralyzed, out of control. As he read, stumbling over a word here and there, pausing to scratch his nose, he seemed so beautiful and so young\u2014younger than I ever remembered. He raised his chin and strained his voice a funny way on certain words, as if they were spears he was launching blindly over a wall.\n\n\"It's beautiful,\" I said when he finished.\n\nHe had a funny look on his face. He carefully set the book on his desk and sat beside me.\n\n\"I was going to wait to do this, like, weeks, and be this total gentleman and woo you like a knight in medieval times? But I'm punting that plan. I'm not a knight. I'm not even a gentleman. But I am devoted. Once I decide I'm with you, it never goes away. I swear to you that, Beatrice.\"\n\nHe kissed me. There was a whole world in that kiss. Every moment of pain, regret, loneliness I'd felt since he'd died fell away. I'd missed him so much, how much hit me only now. As his hands slid down my back, I knew I was going to tell him about the Neverworld, the Keeper, the vote, his death. Would he be able to tell me why he died if I asked him? Couldn't we run out of here, get into a car, and go live out the wake at a highway motel where the light was gold and the carpet full of vending machine crackers?\n\nTomorrow we could do it again.\n\nAnd again.\n\nAnd again.\n\nI didn't have to be without him anymore. I'd tell him everything. He, of all people, would understand. It'd be like it was before, before his strange moods, his anger, his lies.\n\nWhen he pulled away, I was aware of a rapid popping noise behind us. Jim looked stunned.\n\n\"How weird.\"\n\nHe stood, moving to the guitars mounted on the wall. He widened his eyes, mystified. \"All the strings just broke. Every single one.\" He grinned. \"It must be your effect on me.\"\n\nI smiled weakly.\n\n* * *\n\n\u2014\n\nMy decision to tell Jim everything set off some gangster-movie escape scene from the funeral, wavered, and stalled the moment he took my hand and we rejoined his family.\n\nThere were so many uncles, cousins, women wearing black mink coats and stilettos with toothpick heels, swirls of blond hair like sugar garnishes on thirty-four-dollar desserts. We made our way outside, a glamorous black-clad procession up Madison Avenue into the Frank E. Campbell Funeral Chapel.\n\n\"Last time I was here it was for Allegra de Fonso,\" a woman told me.\n\nThe funeral service was long, filled with sniffling people quoting Dylan Thomas and Bob Dylan, \"Let It Be\" by the Beatles. There was a speech from a red-eyed woman who couldn't stop clearing her throat. Children snickered over an ancient man in the front row announcing too loudly, \"It smells like cat piss,\" before a nurse escorted him out. Jim smiled down at me and squeezed my hand. I found myself staring in wonder at a photo of the dead man: Great-Uncle Carl, memorialized in a laminated poster propped on a brass easel beside the casket. He had mottled red skin and an oblivious yellow smile. Had he ended up in some kind of Neverworld? I was closer to Great-Uncle Carl's state than any of these people could imagine.\n\nI had to tell Jim.\n\nHowever, once the service ended and the crowd spilled onto the sidewalk\u2014black Cadillac Escalades lined up eight deep, everyone shaking hands and muttering condolences and observations about Carl, how he \"did it his way\" and was a son of a gun\u2014every time I was about to tell him \"I need to talk to you,\" some new person tapped his shoulder and gave him a bear hug, asking how he'd been, when his first musical was premiering on Broadway. Jim was amiable and kept trying to make his way back to me, but before he could, someone else would approach. When he finally rejoined me, he had two girls in tow. He knew them from grade school.\n\n\"Beatrice, meet Delphine and Luciana.\"\n\nI'd always recalled the girls as intimidating and otherworldly. Seeing them now, they weren't as jaw-dropping as I'd remembered, though they had waist-length hair, which they tossed out of their eyes like ponies, and a bored manner that could be mistaken for expertise. Jim kept putting his arm around me as he talked, but after a while, as I stood listening to stories about Millicent, Castman, and Ripper\u2014whether these were people, a law firm, or impossible-to-get-into nightclubs, I couldn't tell\u2014I began to feel like a giant old L-shaped couch that had been carried out to the sidewalk and left there.\n\nThe feeling continued when we piled into an Escalade. We were a large group. Jim was forced to sit in the back next to Luciana. I sat next to an elderly woman wearing red taffeta who reeked of alcohol.\n\n\"Here we go again to do what we do,\" she mumbled.\n\nWe were dropped off at Jim's great-aunt's apartment on Park Avenue. Jim deposited me on a love seat by a porcelain pug and disappeared on a mission to find me a Coke. After forty-five minutes with no sign of him, I stood up and roamed the dense crowd, perusing bookshelves and photographs, slipping down crowded hallways as if I knew where I was going. I peered into the kitchen, where caterers were sweating over ovens and trays, and a guest bathroom where the wallpaper looked like twenty-four-karat gold. It was all coming back to me, how desolate I'd felt, adrift in a place I didn't belong. I'd wanted nothing more than to be away from these people, back in Watch Hill, eating lasagna with my parents, hearing my dad talk about a new BBC David Attenborough program on Netflix.\n\nNow, five years later, inside the Neverworld, I wasn't nearly so sensitive, but I was still bothered by Jim's absence. Where had he gone? He'd told me he'd gotten stuck talking to relatives, and I'd believed him.\n\nThe question gnawed at me.\n\nI snooped in room after room, searching for him in bedrooms that resembled hotel rooms, offices that looked like libraries, an echoing marble gallery filled with aviation antiques behind glass. Jim was nowhere. Neither, worryingly, were the two girls. At one point, when I opened a closet filled with nothing but Japanese puzzles and board games, Jim's father, Edgar, stepping out of an office, spotting me, and doubtlessly noticing how awkward I looked, beckoned me.\n\n\"Jessica,\" he said to me, smiling warmly, slipping what appeared to be a small black flash drive attached to a rubber bracelet over his wrist. I caught a glimpse of a series of digital numbers flashing along the side before he pulled his shirtsleeve over it.\n\n\"Can I get you a drink, my dear?\"\n\n\"No, thank you, Mr. Mason.\"\n\n\"Edgar. Come meet my partner, Craig, and his daughter, Greta. Greta just returned from Sri Lanka, where she was a visiting neurosurgeon at the District Hospital in Colombo.\"\n\nObviously, high-powered Craig and his neurosurgeon daughter didn't want to be saddled talking to a mute high school freshman, so it was a matter of seconds before they turned to greet someone\u2014\"Bertrand? Is that you?\"\u2014and I slipped away.\n\nI couldn't call Jim. I didn't have a purse with me, much less a phone. I could wait where he'd left me. Eventually he'd come back. Wouldn't he?\n\nAnother hour went by. With each passing second my plan to confess, run away with him, began to grow stale and sag. When I was jostled for the third time by a woman toting a giant alligator handbag, and Mrs. Mason slipped past me with a cardboard smile, Martha's words of warning suddenly leapt into my head.\n\n_We don't know how we're going to react. The past hooks you like a drug. The future jolts you like an electric chair. Reliving beautiful memories can be just as devastating as reliving the terrible ones. They're addictive._\n\nMaybe it was shock at being forgotten by Jim again, the nagging question of his lies about Vida Joshua, or the understanding that one of my friends had tried to destroy me, deliberately sticking me with the pin to send me to back to some other moment in time, doubtlessly believing I'd be too smitten with Jim to ever leave his side again, trapping me here forever.\n\nI leapt to my feet, barging through the crowd. In the hallway I snatched my burgundy coat off the coatrack. It abruptly collapsed, sending piles of minks to the floor. I threw down my old coat, seized the fattest, most unwieldy fur I could find, and shrugged it on, hit by a wave of perfume. I ran down the hall, my heart pounding, pressing the down button for the elevator. It splintered under my finger. I wheeled around, shoved open the door to the stairs, and raced down each flight, lightbulbs in the lamps overhead shattering as I passed each landing. I charged out into the lobby, the doormen gawking.\n\nHow could I have forgotten where I was, and what I had to do?\n\nDidn't I want to live?\n\nI sprinted outside. The wind was strong, too strong, the green awning chattering and flapping in the gale. I ran to the sidewalk, about to hail a cab, when I heard a girl's shrill laughter. Wheeling around, I saw Jim.\n\nHe was perched on the wrought-iron railing in front of the building next door, Delphine and Luciana beside him. They were talking to a doorman, cracking up over his comical impression of someone, what looked like Marlon Brando in _The Godfather._ They were all howling so hard they couldn't stop.\n\nI stood there, frozen, willing Jim to look up and see me.\n\nBut he didn't. Staring at his grinning face, I realized then. I saw it as plain as day. I hadn't even crossed his mind.\n\nMaybe I never did.\n\nI wanted to shout his name. I wanted to scream like some vengeful witch in a fairy tale, causing clouds to fast-forward across the sky, wiping the smiles off their faces: \"Jim Mason, in four years you'll be dead!\"\n\nHe leaned back so carelessly, hooking his arm around Luciana's neck and nuzzling her ear, my heart felt freshly sliced in two.\n\nI'd been so stupid, so blind.\n\nTears sprang to my eyes. I veered around and ran out into the street, nearly getting hit by a taxi before the driver slammed on the brakes, honking.\n\nI climbed in.\n\n\"Honey, are you okay? What the\u2014? _Jesus!_ \"\n\nThe driver blinked, mystified at the sight in front of him. The green awning to Jim's apartment building had come entirely free in the wind, detaching from the sidewalk. It was barreling down Fifth Avenue, clanging and swooping; it collided with the rear windshield of a town car before soaring straight up into the air, gold poles flying out, bystanders shouting as it was flung through the sky like some strange soaring monster.\n\n_I'm anything but okay._\n\n* * *\n\n\u2014\n\nMartha had said to meet back at Wincroft in the event of an emergency.\n\n_Always go back to the original wake. If we have that as our meeting place, there remains a hope we can all eventually convene there across space and time. To change the wake again, go back to the coastal road if you can and do the exact same thing, okay? If you can't get to the coastal road, find a suicide._\n\nI took the train back to Newport. When I arrived it was after ten. I climbed into a waiting cab at the station, asking the driver to take me to Narragansett. It was a half-hour drive, and I didn't have money, but I figured I'd be able to think of something at Wincroft.\n\nThe gate was open. The lamps were lit. As the cab accelerated down the drive, I could see the driver sit up and glance at me curiously in the rearview mirror, wondering if I was an heiress. The house lights were on. There were eight gleaming cars in the driveway. As the cab waited, I went running up the steps and rang the bell.\n\nWhen the door opened, I found myself face to face with E.S.S. Burt. He wasn't as creepy as I remembered. In fact, he looked like any rich man in a pastel sweater. There were voices coming from the dining room, glasses clinking. Apparently I had interrupted a dinner party.\n\n\"Can I help you?\" he asked.\n\n\"I'm looking for Whitley.\"\n\n\"She's not here. She's up at her boarding school. Darrow-Harker.\"\n\n\"We were supposed to grab dinner tonight.\"\n\nHe was surprised. \"Did you try calling her?\"\n\n\"She's not picking up.\" I went on to explain that unfortunately I didn't have enough money to pay the forty-eight dollars for the cab. Blinking in bewilderment, Burt pulled out his wallet, jogged down the steps, and paid the driver.\n\n\"I guess I'll go back to my hotel and try Whitley later,\" I said.\n\nHe nodded, puzzled. \"What did you say your name was?\"\n\n\"Beatrice.\"\n\nBurt didn't know what to make of me, a gawky girl in a black mink bulky as a killer whale. I waved to him and took off on foot down the driveway. He watched me, then disappeared back inside, apparently too preoccupied with his party to wonder, if the cab had driven off, how I was going to get anywhere. I circled back to his vintage-car garage, typed in the four-digit security code. Thankfully it was the same code as five years later, and the door rose with a groan. I hurried to the key stand in the back and unhooked the keys to the Rolls.\n\nDriving out to the coastal road, I expected sirens. None came. My heart began to pound. I could feel the wake coming on. Checking the time, I realized in surprise that this wake had shortened. It had been barely eight hours. I could feel the crushing heaviness pressing into my legs. I floored the gas, engine roaring. The prospect that I might end up with Jim again, back in Central Park, if I didn't make it into another wake willed me to drive faster and faster. My legs went numb. As I rounded the curves, the car seemed to fly out from under me, tree branches scratching at the windshield like an angry mob. When I reached the hairpin curve, I veered into the bushes, narrowly missing a tree. I barreled out, lurching into the middle of the road, the strong wind shoving me down across the yellow line.\n\nI rolled onto my back, gasping. The sky was a deep night-blue, freckled with stars.\n\nI had no idea whether this plan would work. Would the open window even be here anymore? I slowed my thoughts and closed my eyes. _August 29. Villa Anna Sophia. Amorgos Island. Greece._ I waited for a car to come, but there was only the deafening wind in my ears, the shrill hiss of crickets, the distant whoosh of the sea, even as a metronome. I heard a piercing whistling, growing louder. A bicycle. It came at me suddenly, the rider swerving to avoid me, losing control, crashing into bushes on the side of the road with a clang of metal, shouting. The biker was uninjured. After a moment of gasping and swearing, he lurched to his feet.\n\nHe stared down at me, faceless in the dark.\n\n_\"What the fuck?\"_ he whispered as his head jerked up in surprise, headlights of an oncoming car illuminating him like a flash camera.\n\nHe threw himself out of the way as my world went dark.\n\nWhen I opened my eyes I was lying on my stomach on wooden planks. Instantly, streaks of vivid blue tore into my vision. It was the ocean. I raised my head, blinking. I was wearing cutoff jean shorts and a faded pink Captain's Crow T-shirt. I was lying on a dock, barefoot. I turned my head and saw the white wooden staircase zigzagging up the sheer rock face, at least a hundred feet high.\n\nVilla Anna Sophia. I'd actually made it.\n\nLight-headed with relief, I lurched to my feet, only I was so woozy, I stumbled and was sick, nearly falling into the water. Catching my breath, I lurched to my feet.\n\nI headed up the stairs. With every step I took, pebbles and rocks loosened under the planks, bouncing, plummeting down the cliff into the ocean. I kept moving. I didn't look down. When I reached the top, panting, the house\u2014a wild architectural marvel of glass and steel\u2014sat before me, totally silent. It looked deserted. I hurried past the pool, an inflated swan raft drifting leisurely in the center, and tried one of the glass doors. It was locked, the windows shaded. I was just wondering if I'd gotten the wrong day when I heard a woman scream. With a pang of unease I tore down the stone path, past the olive trees, to the front, where I saw Kipling outside the massive double-oak doors. He appeared to be keeping watch.\n\nI was so relieved to see him, I threw my arms around his neck.\n\n\"Thank goodness,\" I whispered.\n\n\" _What_ \u2014my\u2014how did you manage it, child? Martha said we'd lost you, maybe forever.\"\n\nI pulled away. There was no point going into what had happened, not yet. Blinking up at Kipling, though it hurt me to think it, I reasoned he could have very well have been the one to stick me with the pin. Yet he seemed genuinely relieved to see me.\n\n\"I made a mistake,\" I said. \"Where is everyone?\"\n\n\"Inside.\" He made a face. \"We've tied the whole family up and we're tryin' to extract information. But it's not going well.\" He shrugged, visibly nervous. \"We _tried_ the nice way. Arriving casually, announcin' we happened to be on vacation, and were friends of Jim's, and we wanted to know about his death, and so on. But they're slippery eels, the Masons. They served us grilled octopus and basil sorbet and invited us for a dip in their pool. Before we knew it, four hours had passed. We were all drunk on ouzo, and we hadn't had one _real_ conversation about Jim. Whitley got fed up. So these last few wakes, she's gone nuts on these people. The deluxe Whitley special, you know, with the screamin' and the punchin' of walls and the throwin' dishes.\" He sighed. \"Edgar Mason has his twenty-four-hour security detail, but they switch shifts at noon and they're lazy, so that's when we strike. We've got two tied-up guards at the end of the driveway.\"\n\nI frowned. \"But how many wakes have you had?\"\n\n\"Five. Each one lasts about five hours. How many have _you_ had?\"\n\n\"One.\"\n\nThis had to be what Martha meant about instability, trains speeding in different directions at different speeds, the risk of never being in the same place at the same time to vote.\n\nThere wasn't time to worry about it, not yet. Kipling had opened the door and was beckoning me inside.\n\nThere on the couches sat Mr. and Mrs. Mason, tied up along with their four children, their eyes red from crying. They were watching Whitley in mute horror. She looked like a South American guerilla, bandana wrapped around her head, T-shirt knotted in a crop top around her waist, a mad glint in her eyes. She was holding a gun on Mr. Mason. The side of his face was swollen. It was a shock seeing Jim's family like this, when at the last wake they were crisp as fresh flower arrangements, floating around, air-kissing people at Great-Uncle Carl's funeral.\n\nSpotting me, Whitley widened her eyes in surprise. She raced over.\n\n\"Beatrice,\" she said, hushed. \"Where the hell did you come from?\"\n\nI gave her an abbreviated version of what had happened, how I'd accidentally returned to a different date but managed to get back to the coastal road to change the wake.\n\n\"So you're all right, then?\"\n\nI nodded. \"Where's Martha?\"\n\n\"Trying to log on to Edgar's computer. Not having much luck.\"\n\n\"What about Cannon?\"\n\n\"He's gone.\"\n\nI stared at her. \"What?\"\n\nShe shook her head with a bleak look. \"He never arrived. We have no idea where he is. One second he was there, and the next? Nowhere.\"\n\nI recalled the person I'd seen sprinting into the woods. Cannon.\n\n\"Hello? Oh, my God. Is that _you_?\"\n\nMrs. Mason, sitting on the couch, craned her neck to get a better look at me. I'd never seen her so forlorn. She was almost unrecognizable. Her face was red; her blond hair, usually so immaculate, had wilted like a plant left too close to a radiator.\n\n\"Who? Who are you talking about?\" asked Mr. Mason.\n\n\"That little girl Jim went with in school. _You_ know. _Her._ \" She glared at me. \" _You're_ involved in this? You let us go right now. We have no information about Jimmy.\"\n\nI grabbed the gun from Whitley and pointed it at Mrs. Mason. She gasped.\n\n\"Tell me what you know about Jim's death,\" I said.\n\nShe glanced at her husband, terrified, then back at me. She began to whimper. It was an odd sound, like a beach ball losing air through a tiny hole.\n\n\"Leave her alone!\" bellowed Edgar suddenly. \"Gloria has nothing to do with this, you little con artist!\"\n\nI pointed the gun at him. \"What happened to Jim?\"\n\n\"I've told you people countless times now,\" he said, spitting. \"We know nothing.\"\n\n\"That's impossible.\"\n\nHe shook his head. \"The police told us it was suicide.\"\n\n\"Jim never would have done that. And you know it.\"\n\n\"I don't. I _don't_ know it.\" Mr. Mason appeared to be crying, staring at the floor.\n\nThat was when I remembered.\n\nI stepped behind him, inspecting his wrists, which were bound with zip ties. I yanked up the cuff of his shirtsleeve. Mr. Mason knew what I was after, because he immediately began to contort himself, trying to move his hands away.\n\n\"No! Don't you dare\u2014\"\n\nIt was the black rubber bracelet I'd seen him wearing. He still had it on, five years later, though this one seemed an even more sophisticated version, with digital letters and punctuation with the numbers. I couldn't pull it off his wrist, so I went into the kitchen, returning with a knife.\n\n\"Don't you dare! Don't you _dare_!\"\n\nI sliced the bracelet off his wrist.\n\n\"Now you've done it. Good for you. Bravo. Kiss your future goodbye, missy, because you'll be spending the rest of your life in a hole so foul you'll _beg_ to be sent to prison.\"\n\n\"I should be so lucky,\" I said.\n\nI turned to Whitley, who was blinking at me in shock.\n\n\"What got into _you_?\" she whispered.\n\n\"I'll be in Mr. Mason's office,\" I said, racing up the spiral staircase.\n\n* * *\n\n\u2014\n\nMartha was stunned to see me.\n\n\"Oh, my God. What happened?\"\n\n\"It's a long story. But I'm fine.\"\n\nI raced into the all-glass tower, pulling a chair alongside Martha behind the hulking desk. She couldn't seem to stop staring. Naturally it made me wonder if she had been the one to stab me with the bumblebee pin. But there was no figuring it out. Not yet.\n\n\"I've been trying to log on to Edgar's laptop,\" she said, indicating the screen. \"It's impossible. There are three prompts for encrypted passwords.\"\n\nI stared down at the shifting line of numbers, symbols, and letters on the bracelet. They reset every fifteen seconds. I typed the displayed sequence into the three password boxes.\n\nThe computer unlocked.\n\n\"Are you kidding me?\" whispered Martha in awe. \"Like _that_? How did you\u2014?\"\n\n\"I'll explain later.\"\n\nBefore I clicked into the desktop, I placed a piece of tape over the webcam. I didn't know what would happen when it became clear that there was a security breach, but I knew we'd have to work quickly. Edgar Mason had a personalized email interface called Torchlight Command. As soon as I opened the program, a timer recording my activity appeared in the upper right corner of the screen.\n\nThe first thing to do was to search for emails from Jim.\n\nWe couldn't find one. Searching for the names of his brothers and sisters turned up countless emails, but there was not a single message either to or from Jim.\n\n\"He's been wiped from his father's email,\" whispered Martha. \"Why?\"\n\n\"Maybe he wrote something inflammatory.\"\n\nShe shrugged.\n\nOn the hard drive, there were over two thousand folders on a cloud server called Torchlight Library. I searched for _Jim Mason._ Nothing came up. We found a trove of financial records, listings of obscure holding companies with names like Redshore Capital America and Groundview Fund, with addresses in the Cayman Islands and Panama City. There were trade receipts and wire transactions from a bank in Turkey to another in Switzerland, some of which listed dollar amounts so enormous they looked like typos. If any of it was illegal, or tied in any way to Jim's death, the truth was buried under layers of names, numbers, and symbols, none of which could be easily excavated.\n\n\"Maybe Edgar's committing fraud,\" said Martha. \"Sweatshops. Child exploitation. Maybe Jim found out about it, and they had a major falling-out.\"\n\n\"If Jim had found out something like that, he'd have been devastated, yes. But he wouldn't have killed himself.\"\n\nShe shrugged. \"What if Edgar hired someone to kill Jim?\"\n\nI stared at her, surprised. \"His own son?\"\n\n\"If he thought he was going to lose the empire he built? Why not?\"\n\nSuddenly she sat up, frowning, pointing at the glass walls. I realized in horror that every pane was breaking. All around us thin cracks were spidering through the glass, branching out, one after the other.\n\n\"The instability of the Neverworld,\" whispered Martha.\n\nI nodded and hurriedly clicked back into the in-box. I certainly didn't want her to wonder what the destruction meant, if it was all being caused by me. I leaned forward, squinting at the screen.\n\n\"Most of Edgar's emails are from this woman named Janet,\" I said, clearing my throat. \"His executive assistant. They have a system where she reads his emails and summarizes them.\"\n\n\" 'Chris Endleberg, president of Princeton, called,' \" Martha read slowly. \" 'He appreciates the way you handled the matter re S.O. They'll hold off on disciplinary measures.' Huh. Okay. What else?\"\n\nWe scanned the emails in the weeks leading up to Jim's death.\n\nThere was nothing unusual. A board member was problematic. _Patrick has to go._ A real estate broker wanted to show Edgar an off-market listing for an estate in Bedford worth $48 million. _Sick pad, man._ Someone involved in a fast-food restaurant wanted another loan. _I hear your concerns, but it's time to expand on the line of frozen fried chicken dinners with romance-related flavor names._ In the days following Jim's death, there were emails about funeral arrangements and flower deliveries, the West Side Boys Choir, lists of who was attending and who would speak. It was oddly cold to read through. Just like that, Jim's death was another action item in his father's in-box. My name was buried among three hundred others.\n\n\"I don't get it,\" whispered Martha, frowning at an email she'd just opened.\n\n\"What?\"\n\n\" 'S.O. wants to change his dormitory, FYI.' \" She glanced at me. \"This is from Janet. 'He needs you to call the Princeton dean and make it happen, as this isn't freshman policy.' Bizarre.\"\n\n\"What's bizarre?\" I asked.\n\n\"Another email from Princeton. Who in the Mason family goes to Princeton?\"\n\nIt was a good question. Jim was the oldest. His other siblings were in grade school.\n\n\"Who is S.O.?\" I wondered.\n\nWe did a search of the initials. One more email appeared. As I opened it, the wall of broken glass in front of us spontaneously fell away, millions of shards sliding across the roof and down the side of the house. A powerful gust of wind billowed through the room, sending the gauzy curtains flying out and stacks of papers swirling off Mr. Mason's desk.\n\n\"We don't have much time,\" I said hastily. \"The system is about to lock us out.\"\n\nMartha nodded, biting her lip, and peered closer at the email.\n\n_S.O. wants lunch tomorrow to discuss a business opportunity. Booked 1 p.m. Jean-Georges._\n\n\"Try searching the keyword _Princeton,_ \" Martha said.\n\nI did, and one more email appeared.\n\n_Chris Endleberg of Princeton wants to thank you personally for your donation. Invited you to dinner 2\/24. I declined, as you'll be in Buenos Aires._\n\n\"S.O. could be a cousin,\" I suggested. \"Maybe Edgar pays for his education?\"\n\n\"Or S.O. is his Emotional Support Animal, wearing a yellow vest, which he takes with him on planes, trains, and automobiles.\"\n\nThis appeared to be her attempt at humor, though you could never tell with Martha.\n\n\"Or S.O. is his imaginary childhood friend,\" I said.\n\n\"Or S.O. is his _sixth_ personality, as he has secretly suffered from schizophrenia for years.\"\n\nWe smiled at each other, though unsurprisingly, the moment ended as soon as we realized what was happening: we weren't on edge in each other's company.\n\nThat was when another three walls of glass dropped away and a strong gale barreled through again, papers exploding around the room.\n\nAt that moment, Whitley stuck her head around the doorframe.\n\n\"The wake is three minutes away\u2014\" She frowned. \"What the\u2014 What's happening in here?\"\n\nMartha leapt to her feet. \"It's the Neverworld. We have to go. _Now._ \"\n\nThey hurriedly explained their plan. We needed to head back to Wincroft to find Cannon. The Masons were impossible to break. It was better for the five of us to get back together than to keep interrogating them. Our questions were eliciting no new information about Jim.\n\n\"Use the cliff for the wake,\" Martha ordered cryptically before ducking out.\n\nI remained where I was, searching Edgar's laptop as the wind howled around me, and papers cycloned, every glass wall falling away. Not a minute later, the desktop speakers sounded an alarm, and I was locked out, the screen going black. I leapt to my feet, and as I hurried past the open spaces overlooking the backyard, I spotted Martha, Kipling, and Whitley running out of the house and past the pool toward the cliff.\n\n_Use the cliff for the wake._\n\nI watched, stunned, as they stood side by side at the very edge.\n\nThey joined hands. Then they jumped.\n\n* * *\n\n\u2014\n\nWhen I returned downstairs, the Masons looked terrified.\n\nThey'd seen what I'd just seen. They believed now that we were all crazy.\n\nI questioned them for another hour. Mr. Mason's cell rang incessantly. So did the landline. A printer wailed in a room upstairs. It was doubtlessly Torchlight Security trying to alert Mr. Mason of the security breach. Holding the gun on him, I said I wanted to know what he and Jim had argued about in his final days alive.\n\n\"What are you talking about?\" he wailed. \"My son and I didn't argue. We never argued.\"\n\n\"Who is S.O.?\"\n\n\"S.O.?\" He looked confused.\n\n\"The freshman at Princeton.\"\n\nHe sneered. \"It's a colleague's son. What does _he_ have to do with\u2014 You truly are a troubled young woman, my dear. If you have any sense, you'll untie us all, go back to your dingbat life, and hope\u2014no, _pray_ \u2014my fleet of attorneys doesn't decide to spread you on a cracker and serve you as an hors d'oeuvre.\"\n\nI tried setting a few more verbal traps for Mr. Mason to fall into, telling him Jim had confessed to me all about his financial fraud. I tried to see whether he looked uneasy or afraid. Unsurprisingly, my blind fishing elicited little more than confounded stares and indignant comments from the family that they'd always thought I was a good girl, which made my involvement in this nightmare all the more disappointing.\n\n\"There's no need to pretend,\" I said. \"You never liked me. And my _name,_ in case you were wondering, is not Jessica, or Antonella, or Barbara, or Blair. It's Beatrice Hartley.\"\n\nI shot the gun into the ceiling. Instantly, minute cracks fanned out through the plaster, spreading into every corner, then moving down the walls.\n\n\"We'll give you any amount of money,\" whimpered Mrs. Mason, worriedly eyeing the ceiling.\n\nThat was when I felt the wake coming on. I set down the gun and left without a word, leaving the Masons staring after me, uncertain, afraid. As I raced past the pool, I saw two police cars inching up the vertiginous drive. One emerged, shouting at me in Greek.\n\nI ran to the edge of the cliff.\n\nAs I stood there, the rocks and dirt began to loosen and tumble under me, as if I were the weight of a building, as if I weighed ten million tons. Boulders were pulling out of the ground. I leapt into the air, shouting, just as the ground dropped out. I was plummeting fast, upside down, breath sucking from my lungs. Blue sky spun overhead. I squeezed my eyes shut, trying to quiet my mind as I thought of Wincroft the day I'd first arrived there, though almost immediately something else slipped into my head.\n\nA connection. It was barely remembered, an itch at the back of my brain.\n\nI'd seen it before. Twice.\n\nI tried to ignore it. Spiky grass, bushes, and cypress trees were spinning past me. Screaming, I opened my eyes to catch sight of the entire cliff through the dust, then the house dismantling behind me, a roaring mass of shattered glass and steel and rock coming for me as we all fell toward the sea.\n\nIt was too late.\n\n\"Hon? You okay?\"\n\nSomeone was shaking my shoulder.\n\nMy eyes opened. I jerked my head up, shouting.\n\nA large woman with red hair and heavy eye makeup stared down at me, visibly freaked out. She was wearing a pink visor emblazoned with a cartoon chicken, a heart on its chest.\n\n\"Sorry, hon, you can't sleep here. Do you need me to call someone?\"\n\nI looked around. I was in a wooden booth in a cramped fast-food restaurant. People around me were eating fried chicken and fries and drinking milk shakes. The walls were covered with heart wallpaper, photos of couples kissing or holding hands. I blinked at the paper mat in the tray in front of me.\n\n_Alonso's Honey Love Fried Chicken. One Taste and You're Lovestruck._\n\n\"Where\u2014where am I?\" I blurted.\n\n\"Newport. I can call your mom for you, hon. Or a shelter?\"\n\nI shook my head and lurched to my feet. I realized dazedly that I was wearing my old Darrow uniform: a white blouse, green tartan skirt, black tights, the beat-up black Steve Madden ankle boots that had seen me through four years of school.\n\n\"Seriously. I can call someone.\"\n\nI pressed a hand to my throbbing head, and stumbled away from the woman.\n\n_What had happened?_ Why hadn't I made it to Wincroft? Then I remembered the thought that had slipped into my mind as I'd been falling.\n\nIt was what Vida had said, about the ride she'd given Jim.\n\n_Some dingy section of town. Dollar stores. A pet store. The parking lot had some man in a chicken costume handing out heart balloons._\n\n\"Why did Jim want to go there?\" Cannon had asked her.\n\n_Maybe he wanted to eat fried chicken and buy a pet iguana? I have no clue._\n\nFried chicken and hearts had turned up again in the coupon inside Jim's empty case file.\n\n_$5 off a bin of Honey Love Fried Chicken. Soul Mate Special!_\n\nFinally, it had appeared in an email I'd read in Edgar Mason's in-box. A restaurant owner had been asking for another loan. _I hear your concerns, but it's time to expand on the line of frozen fried chicken dinners with romance-related flavor names._\n\nI staggered past the cashier, blinking at the laminated advertisement on the counter.\n\n_ALL-NEW! Honey Love Fried Chicken Organic Chicken Dinners, now available in the frozen-food aisle at a supermarket near you. Try our original flavor! Honey Love Mesquite._\n\n\"May I take your order, miss?\"\n\nThe teenage boy behind the cash register was staring at me. With a fitful smile I shoved open the door and moved outside, steadying myself on a _Newport Daily News_ vending machine. After a moment, I realized I was staring at someone wearing a yellow cartoon chicken costume passing out heart-shaped balloons to passersby. The strip mall was exactly as Vida had described it. There was a handful of people loitering around the parking lot.\n\nI leaned down to check the newspaper date.\n\nFriday. May 14. Last year.\n\nI'd managed to get it right. After all, I remembered the night I'd watched Jim drive away with Vida as if it were yesterday.\n\nAn elderly man was pushing a shopping cart loaded with shopping bags past me.\n\n\"Excuse me?\" I asked. \"What time is it?\"\n\nHe checked his watch. \"Twelve-forty-nine.\"\n\nVida had said she'd dropped Jim off around eight or nine o'clock, which meant I had nearly eight hours to kill until he appeared. I hoped the wake would last that long. If Jim even _did_ appear. It was a long shot. It also wasn't the worst connection to make. Whoever had confiscated the papers from Jim's case file at the Warwick police station hadn't looked twice at the coupon, but what if it had been actual evidence? What if it had been stuck in Jim's file because the detectives had been tracking his movements during the final days of his life, and they'd discovered he'd come here to this complex, to this restaurant?\n\nMy head was still pounding. I slipped along the covered sidewalk, past a liquor store, a Dollar Mart, a pet store called Man's Best Friend. I had to change my clothes. If Jim did come here, the restaurant was small. He'd spot me immediately. But I had no money to buy clothes. I watched the people come and go, men in faded T-shirts racing into the liquor store, women hauling toddlers, an old woman bent over ninety degrees pushing a cart. When I spotted a smiling woman leaving a stationery store walking a Pomeranian, I approached her.\n\n\"Excuse me, ma'am? I'm hoping you might help me. I need a change of clothes\u2014\"\n\nShe picked up her dog with a horrified look and climbed into her car.\n\nI ended up going into every store at the shopping center, striding brazenly through Employees Only swinging doors into back storerooms, janitors' closets, and cargo unloading areas, to see if I could find some kind of spare uniform. I managed to steal a pair of khakis from Man's Best Friend, a hoodie from a manager's closet inside the Stop & Shop. I asked an old man pulling a pint of Ben & Jerry's out of the freezer if I could have his baseball cap. There must have been something totally desperate, or strange, or otherworldly on my face, because he handed it to me without a word and quickly wheeled his cart away.\n\nI hurried into a Chinese restaurant, Fu Mao Noodle, and changed in the bathroom, grabbing a handful of fortune cookies by the register as I left. I sat eating them on a bench outside the pet store facing the parking lot, a feeling of dread in my chest. _Small opportunities are the beginnings of great enterprises. You are the architect of your fortune. Big journeys begin with a single step._ I had to change benches three times, because every one I sat on, the wood began to splinter and crack under me. One even collapsed in half.\n\nThe longer I waited, the more afraid I was that I'd been right to track Jim here, that he'd actually appear. Was he meeting some other girl? What had preoccupied him, been so shameful that he couldn't tell me about it? _What had he been so afraid of?_\n\nAt five minutes after eight a beat-up red Nissan pulled into the parking lot, a For Sale sign in the back window. It slinked up to Honey Love Fried Chicken and the passenger door opened. Jim climbed out. Black T-shirt. Jeans.\n\nI could see Vida behind the wheel. Jim entered the restaurant. Vida waited a moment, as if to make sure he wasn't coming back. Then she drove off, exactly as she'd said.\n\nI waited another minute. Then I darted along the covered walkway, ignoring the fact that every column was spotted with black mold.\n\nI peered through the glass door. Jim was standing at the counter, his back to me.\n\nI quickly slipped inside and took a seat at an empty table by the window.\n\n\"Call him again,\" I heard Jim say. He sounded angry.\n\nThe woman he was speaking with\u2014the one who'd shaken me awake\u2014was mystified.\n\n\"I just did. He said he'd be right out\u2014\"\n\n_\"Call him again.\"_\n\nFrightened, she grabbed the phone, dialing.\n\n\"He says he'll be right out.\"\n\nSeconds later, a Hispanic man with a thick mustache appeared from a back room. He was slight, midforties, a kind face.\n\n\"Jim. It's been too long. How are you?\"\n\n\"We need to talk.\"\n\n\"I'm about to jump on a conference call. Why don't you come back after closing?\"\n\n\"We're going to talk _now._ \"\n\nDisconcerted, the man beckoned Jim to follow him. I slid to my feet, watching them disappear through the back door. I waited another minute and headed after them, pausing to hear another door slam before I darted inside. The kitchen was in front of me. Beyond that, there appeared to be a back office. The door was closed, but it looked thin, and hurrying up to it, I could make out the voices easily enough.\n\n\"I'LL ASK YOU ONE MORE TIME. WHO IS ESTELLA ORNATO?\"\n\n\"What are you talking about?\"\n\n\"ESTELLA ORNATO!\"\n\n\"She\u2014well, yes, she's my daughter\u2014\"\n\n_\"And?\"_\n\n\"And?\"\n\n\"Four years old. She died last year. That jog your memory?\"\n\n\"Jim, please, let's not do this here\u2014\"\n\n\"DO NOT PICK UP THAT PHONE OR I SWEAR\u2014\"\n\n\"Jim\u2014\"\n\n\"FOR ONCE WOULD SOMEONE TELL ME THE TRUTH?\"\n\n\"Who told you? Where is this coming from?\"\n\n\"Your brother wrote me a letter. ESTELLA DID NOT DIE IN A CAR ACCIDENT\u2014\"\n\n\"Jim. _Jim._ Now, hear me out\u2014\"\n\nThe voices quieted. Abruptly something large smashed against the door.\n\n\"TELL ME THE TRUTH OR I SWEAR TO GOD\u2014\"\n\n\"Excuse me,\" said a woman. \"You're not authorized to be here.\"\n\nI turned. It was the redhead. She was indignant, hands on her hips.\n\n\"I have an interview with your manager,\" I blurted.\n\nShe squinted at me, puzzled. A second deafening crash from inside the office was disturbing enough that she quickly forgot me and went hurrying back to the kitchen to confer, wide-eyed, with the teenager behind the cash register.\n\n\"DID MY FATHER PAY FOR THIS? AND THIS? AND THIS?\"\n\nThere was a high-pitched cry, followed by a moan. Alarmed, I pushed open the door, barging in to see Jim throwing a bag of golf clubs on Mr. Ornato, now cowering on the floor in a fetal position. Jim started kicking him in the stomach.\n\n\"Jim,\" I said.\n\nHe turned, startled. The redhead barged past me into the office. \"Oh, my God. Mr. Ornato. Are you okay? I'm going to call the police.\"\n\n\"No, no, it's all right.\" Gasping, he rolled upright, his face sweaty, his hair standing on end. \"There's no need. It's just a misunderstanding. Let's get back to work.\"\n\nJim wiped his face in the crook of his arm, dimly surveying the demolished room.\n\nThen he began to sob. I stepped toward him and put my arms around him.\n\n\"Let's get out of here,\" I whispered into his ear.\n\n* * *\n\n\u2014\n\nWe sat on the curb outside Fu Mao Noodle. We watched the cars speed past in the closing day, the sky going blue and black, traffic lights changing from red to green to yellow. We watched small black birds land on telephone wires and fly away, heard the giggling wheels of shopping carts. All that ordinary life\u2014vending machines belching up sodas, stock boys taking cigarette breaks, cars backing in, backing out.\n\nI watched it all as Jim told me everything.\n\nI listened in shock. It made perfect sense\u2014his father's obsession with security, Jim's distraction and moodiness, his decision not to tell anyone, not even me. If he had told me the truth before, would it have changed everything? Would he still be alive?\n\nIt had to do with the boating accident. Jim and a friend had taken out a speedboat on Mecox Bay, and they'd crashed into a fisherman in a skiff. When Jim woke up in the hospital, he heard the story from his family and the police\u2014all corroborated by articles in the _East Hampton Star._ No one except Jim was hurt.\n\nThe fisherman happened to be none other than Alonso Ornato, the owner of Honey Love Fried Chicken. But this wasn't the whole truth. Alonso had had his four-year-old daughter, Estella, in the boat with him. She was killed on impact.\n\nThis should have resulted in a charge of manslaughter against Jim, which meant, as a minor with his father's connections, at most, given that he'd been drinking, he'd have gone to a juvenile facility for a few months, maybe even weeks, and would have been released on probation.\n\nThat wasn't good enough for the Masons.\n\nInstead, they decided the incident shouldn't have happened at all. So they decided to erase it from history and redesign the past. They struck a deal with Alonso Ornato. They would take care of him and his family for the rest of their lives\u2014monthly allowances, new houses and cars, Ivy League educations for his other kids, bottomless loans for his business\u2014all in exchange for erasing Estella from the boat that day. She would die in a car accident instead.\n\nMr. and Mrs. Mason arranged the whole thing with the assistance of Torchlight. They drove Alonso's car into a tree, artfully inflicting the right kind of damage so the police wouldn't ask any questions.\n\n\"Wipe the spill off the kitchen counter,\" said Jim. \"Remove all signs of rot. Fumigate the foul odors seeping through the basement. All for me. So I'd suffer no shame. No heartache. No pain. I could continue my life guilt-free, like a diet drink. I could soft-shoe toward my golden destiny.\" He stared blankly at the pavement. \"They don't realize they've destroyed me.\"\n\nI touched his arm. \"That's not true. You can still do something.\"\n\n_You are such a liar,_ whispered the voice in my head. _What can he do now? He's dead._\n\n\"Like what, Bee? It's gotten inside my head. It's why I've been sick, why I can't write a goddamn decent note anymore. I'll never pick up another instrument. Because their poison is inside me.\" He hit the side of his head scarily, over and over. I grabbed his hand to make him stop. \"They've killed me, don't you see?\"\n\n\"You should contact a newspaper. Turn them in to the police.\"\n\nHe laughed bitterly. \"Sure. I'll turn them in. That'll solve everything. My family will be destroyed. My brothers and sisters will have convicts for parents. The whole world will loathe us. We'll become poster children for all that's depraved. All to placate my guilty conscience. What good would it do? That girl will still be dead. That's the worst part. I can't _do_ a goddam thing. I've gone over it and over it.\"\n\nHe began to cry again, head in his hands.\n\nI stared out into the parking lot with a strange feeling of desolation and calm. Jim was right. Even if he were alive and this moment were real, what could he do? Start a foundation in Estella's name? Write a musical about it all? The awful thing was, what the Masons had done was like toxic gas, pervading everything.\n\nWe stared ahead in silence, holding hands. It felt as if we'd both removed our glasses, and now we saw for the first time that the world had never been as beautiful as we'd always thought. It was a vision lost, never to come back.\n\n\"At least I have you, Beatrice,\" said Jim, squeezing my hand. \"You save me.\"\n\n_But you don't have me. I'm not even alive. Neither are you._\n\n_We're ghosts. We're air. We're approximations._\n\nI felt a painful lump in my throat. I wanted to cry, for him, for myself. My legs were growing heavy. It was the wake. I didn't know how much time I had left. It seemed to be moving through me faster now. My head felt as if it were melting.\n\nJim frowned, surveying me. Perhaps he was wondering how I'd known to follow him here. Then I realized he had noticed the black mildew covering the cracked curb we were sitting on, and the pavement quietly splintering under our shoes.\n\nI lurched to my feet, staring down at him. There was one last thing I had to know.\n\n\"You wouldn't, because of this, do something terrible, would you?\"\n\nHe squinted up at me.\n\n\"You wouldn't throw your life away.\"\n\n\"You mean commit suicide?\" He looked insulted.\n\n_No._\n\n\"I have to go.\"\n\nI turned and took off running, though when he began shouting my name, asking where I was going, I threw back my head and turning, laughing crazily, I shouted, \"I love you, Jim Mason. I always have.\"\n\nI ran out of the parking lot into the six-lane highway. Cars honked. A woman in a passing car rolled down the window and started to scream at me. \"Get out of the way! Honey, what are you doing out here? _Honey?_ \" I could hear Jim calling me, but I stepped in front of a cement truck and closed my eyes.\n\n* * *\n\n\u2014\n\n_August 30. Wincroft. 6:12 p.m._\n\n\"Beatrice? Bee! Beatrice!\"\n\nMartha, Kip, and Whitley were waiting for me in the library.\n\nThere was no sign of Cannon.\n\n\"You made it, Bee,\" said Whitley, hugging me.\n\n\"What happened after we left?\" asked Kipling.\n\nI didn't answer. Instead, I slipped past them, heading straight to an upstairs bedroom. Minutes later, returning downstairs, my suspicions confirmed\u2014I'd found what I'd been looking for\u2014I explained where I'd gone. I told them about the connection I'd made between the man in the chicken costume handing out heart balloons, whom Vida had mentioned, the Honey Love fried chicken coupon left in Jim's case file, and the email in Edgar Mason's in-box.\n\nI told them about Estella Ornato.\n\nNo one said a word for a long time. Whitley opened her laptop and Googled the name, then read aloud the only information that appeared about Estella's death, a four-sentence mention in the _South Shore Sentinel._\n\n\" 'Officials have released the name of a four-year-old child killed Wednesday night in a car accident in Water Mill,' \" she read.\n\n\"S.O.,\" I said to Martha. \"I think it's Alonso Ornato's son.\"\n\nSure enough, a search of _Ornato_ and _Princeton_ turned up a Facebook page belonging to Sebastian Ornato, about to start his sophomore year. On his page there was a photograph of him sitting in Firestone Library wearing a Princeton sweatshirt, grinning and making a goofy peace sign.\n\n\"Poor kid thinks he got into Princeton on his own steam,\" said Kipling.\n\n\"I can't believe it,\" said Whitley, solemn. \"I knew Jim's family was capable of anything. But erasing the existence of an _entire person_? Designing a new death that's more elegant and acceptable to all involved? And getting away with it?\"\n\n\"It proves Jim's suicide, doesn't it?\" suggested Kipling, taking a deep breath. \"Jim probably felt so alone. Lost. So he rode his bike out to Vulcan Quarry and jumped.\"\n\n\"I don't think so,\" I said.\n\nThey turned to me in surprise. I told them what Jim had said in the parking lot.\n\n\"Well, if it wasn't suicide,\" said Whitley, \"then what happened?\"\n\nI dug in my pocket and pulled out the bumblebee pin, placing it on the coffee table.\n\nKip widened his eyes. \"What is that, child?\"\n\n\"The gift Jim bought me freshman year.\"\n\n\"Oh, that's right,\" said Whitley.\n\n\"Didn't someone steal it from you?\" asked Martha.\n\nI nodded. \"I just found it upstairs in Whitley's jewelry case.\"\n\nWit stared at me, her face pale.\n\n\"You stole it from me. I know you did. It was one of your notorious thefts. Wasn't it?\"\n\n\"Bee, I'm so sorry\u2014\"\n\n\"You never think. Little do you know how your most haphazard gestures inflict such pain. It hurts to be your friend. It always has. But I still love you.\"\n\nIgnoring Wit's astonished face, I went on to explain how I'd been stuck in the neck with the pin moments before the wake, which had sent me plunging back into the past with thoughts of Jim.\n\n\"I didn't do it, Bee,\" said Whitley. \"I swear.\"\n\n\"I know. It was Cannon.\"\n\nEveryone gaped at me.\n\n\"He knew you'd taken it, so he stole it out of your jewelry case the first night we changed the wake. He wanted to throw me off track, send the rest of you into a state of perpetual limbo. He doesn't want us to find out what happened to Jim. He doesn't want to ever leave the Neverworld.\"\n\n\"You think he had something to do with Jim's death?\" asked Martha.\n\n\"I don't know yet.\"\n\n\"Bee does have a point,\" said Kipling with a dubious expression. \"Cannon knows if anything goes wrong he's supposed to meet us here. So where the hell is he?\"\n\n\"He's hiding somewhere in the past or the future,\" I said. \"There's really only one way to get to the bottom of what happened to Jim.\"\n\nNo one spoke for a minute, all of us doubtlessly thinking the same thing.\n\n\"No,\" said Martha, shaking her head. \"No. It's out of the question, Bee. _No._ \"\n\n\"It's not as dangerous as you think,\" I said.\n\n\"Yes, it _is._ \"\n\n\"I did it already. I went back even farther, five years by accident. The crazy thing about the past is that you never meet yourself. There are no doubles. If you arrive there, your past self exits on cue to make room for you.\"\n\nMartha looked furious. \"How long are your wakes now?\"\n\nI shrugged.\n\n\"Ours are only _four hours._ \" She shook her head. \"They're getting shorter and shorter. And it's getting worse. Every time we go into the past or future, it makes the possibility of a unanimous vote even more impossible. _Don't you get it?_ \"\n\nShe snapped this at me so furiously\u2014eyes bulging, glasses going crooked on the end of her nose\u2014I could only stare back in shock. We all did.\n\nShe fell silent, seemingly embarrassed by her outburst.\n\nKipling turned to me. \"How long is your wake now?\"\n\n\"Six hours?\"\n\n\"It's enough time to try, isn't it?\"\n\nMartha said nothing, staring sullenly at the floor.\n\n\"If we arrive at Vulcanation at one in the morning,\" I said, \"even if your wake is four hours, or three, I'm almost positive it will give us enough time to see what happened to Jim.\"\n\nWith a pang of queasiness I thought back to his last text. Sent at 11:29 p.m.\n\n_I'm going to the quarry. Meet me._\n\nThey still didn't know about the texts from Jim. I wasn't going to tell them.\n\n\"Let's do it,\" said Whitley.\n\nAs the rest of us talked about the logistics of changing the wake, Martha stayed silent, slumped way down in the couch cushions, her expression a mixture of resentment and hopelessness. It appeared my suggestion of venturing once and for all to Vulcan Quarry was flying in the face of her grand plan. It had made her lose control of the group, though what she was so anxious about, and what this meant for the vote, I could only imagine.\n\nWhen I woke I was staring at a clear night sky filled with stars, the deafening screech of crickets in my ears. I was lying in thick grass, the long, razorlike blades slicing my bare arms. I was wearing my Darrow uniform. I lifted my head, realizing with a rush of relief that I was outside the quarry, though almost immediately relief gave way to suffocating dread.\n\nThe rusted chain-link fence was only a few feet away. I checked my watch.\n\nIt was 1:02 a.m.\n\nI crawled to my feet, dizzy, and looked around.\n\nThere was no sign of anyone.\n\nI groped my way along the fence, kicking back the grass, the gnarled coils of brambles sharp as barbed wire. Ahead I could see the rusted yellow sign: NO TRESPASSING. Somewhere near was the hole we'd always used. I bent down, forcing aside the weeds, fumbling along the ground. I found the hole and crawled through.\n\nFar ahead, suspended in the sky, I could see the Foreman's Lookout. I shivered, trying to ignore the nausea rising in my throat. The old wood tower looked like an abandoned space station in the dark.\n\n\"Bee!\" hissed a voice behind me.\n\nI whipped around. Whitley was waving at me from the other side of the fence. Kipling was behind her, his head barely visible above the ocean of grass. I directed them toward the opening, and within seconds they were beside me.\n\n\"Where's Martha?\" I asked.\n\n\"Missing,\" said Kipling, scrambling to his feet.\n\n_\"What?\"_\n\n\"She bailed.\"\n\n\"One second she was there,\" said Whitley, shaking her head. \"The next, nowhere.\"\n\n\"She never wanted to come,\" said Kipling. \"So she _didn't._ \"\n\nWe eyed each other, unsettled at the thought. Where had she gone? Was she hiding out like Cannon somewhere in the past or future, terrified of what we were about to discover?\n\nThere were so many questions, but there was no time to figure them out. Not now.\n\n\"We need a hiding place,\" I whispered. \"There's that cement pipe in the grass next to the entry to the mining shafts. We could stay there.\"\n\nWhitley frowned. \"What about the old mapping office right beside the road?\"\n\nI shook my head. \"Too obvious. Jim might see us. Then we'll have interfered with the past, and we won't find out what actually happened.\"\n\n\"Cement pipe it is,\" said Kipling, with a cryptic grin.\n\nWe took off, fighting our way through the grass to reach the quarry road. Little was left of it, apart from bits of rock and gravel, and the grass there was only knee-high. As we headed down the path, I noticed after a minute that Kipling was lagging far behind, an oddly bleak look on his face. When he saw I was waiting for him, he glanced up, feigning a smile.\n\n\"Are you all right?\" I asked.\n\n\"Oh, _sure,_ child. Splendid. It isn't every day I get to watch one of my friends get murdered.\"\n\nI put my arm around him for reassurance, pulling him beside me as we trudged on, fighting back the fronds, the wail of the crickets so deafening, it sounded like a million knives being sharpened in my ears. Yet the question blinked glaringly in my mind: _How did he know Jim was murdered?_ He'd blurted it without thinking.\n\nAs if he _knew._\n\nAs we walked on, Kipling seemed unconcerned about his disclosure, which made me wonder if it had actually been one. Did he know something? Or was he only giving voice to his suspicion that someone came out here tonight to kill Jim?\n\nWithin minutes we had reached the center of Vulcanation, where the old quarry road made an elongated U past the mapping office, the outhouses, the Foreman's Lookout. The Lookout was held aloft by four massive steel legs reinforced with crisscrossing beams, the wood ladder stretching up the center like an old, arthritic backbone. There were a few more structures dotting the road\u2014lodging for the miners, little more than heaps of rotten pine logs\u2014and a collapsed crane, which looked like the remains of a great blue whale.\n\nThe three of us paused, looking around, apprehensive. It was totally overgrown and wild, more than I remembered. The tempo of the crickets' screeching began to quicken as if it were the pulse of the night itself, terrified, on edge.\n\nThere didn't appear to be anyone here.\n\nNot Jim. Not anyone.\n\nSuddenly, a wave of nausea came over me, and I was sick all over the ground.\n\n\"Poor Bee,\" said Whitley, brushing away the hair stuck to my cheek. \"Maybe we should forget all this and go back.\"\n\nI shook my head. \"I'll be fine.\"\n\nIgnoring her worried glance, I stepped past her into the grass. It took a few minutes for us to find the cement pipe, some thirty feet long, only a few feet from the edge of the quarry. As I stepped toward the precipice, I was afraid the ground would start to crumble underneath me, but it held. I stared out, my chest tightening from the shock of how abruptly the ground gave way to total nothingness.\n\nIt was a three-hundred-foot drop, the crater stretching out, a stadium of rock, a vast sky littered with stars, and far below the lake, dark water glistening in the moonlight.\n\n\"Sister Bee,\" whispered Kip, stepping up beside me. \"I have a funny feeling death will be like this.\"\n\nHis voice, eerily flat, sent a surge of fear through me. I wondered numbly if he was about to push me in.\n\n\"It'll feel like falling, but on and on, never stopping. You know?\"\n\nHe was staring at me with a thin little smile. I swallowed, barely able to breathe.\n\n\"Look,\" said Whitley.\n\nTurning, I saw she was leaning against the pipe, pointing at something. High in the wooden tower of the Lookout, a tiny green light was visible in a window. It belonged to the oil lamp some student had smuggled up there years earlier.\n\nNone of us spoke. The conclusion was obvious: Someone had been up there. Or they were up there now.\n\n\"I'll go see who it is,\" said Whitley.\n\n\"No,\" I said.\n\n\"Why not? I want to see if it's Jim\u2014\"\n\n\"He'll _see_ you. If you interfere, we won't know what really happened\u2014\"\n\n\"Then let me just see if I can find his bike.\"\n\n\"Don't.\" I grabbed her arm.\n\n\"Bee, what's the matter with\u2014 _Stop_ it!\"\n\nShe yanked it loose, about to take off, but suddenly the sound of someone yards away fighting a path through the brush made her stop dead.\n\nNone of us moved as we watched the top of a dark head bobbing toward us.\n\n_It was Jim._ A wave of horror choked me.\n\nThe grass trembled and shook. Martha stepped out.\n\nWe gaped at her. Her neon-blue hair was gone. She was her old self from Darrow, dark hair in a careless ponytail, oversized Oxford shirt.\n\n\"What hole did you just crawl out of?\" asked Kipling.\n\n\"We thought you ditched us,\" said Wit.\n\n\"Yeah. Sorry about that.\" She adjusted her glasses. \"For _some_ reason\u2014I think it was because I was thinking of the map of the entire quarry before the wake\u2014I ended up waking not by the south fence with you guys, but by the _east_ fence behind the Pancake House. I had to hike the mile along the quarry road to get here.\" She took a deep breath. \"Seen anything yet?\"\n\n\"Only that light,\" said Whitley, indicating the Foreman's Lookout.\n\nMartha squinted up at it. She seemed unsurprised.\n\n\"Did you see anyone along the road?\" I asked her.\n\nShe shook her head.\n\nIt was then that I noticed she was drenched in sweat. Her shirt clung to her. Her hair was plastered to her forehead. Walking along the quarry road wouldn't have exerted her to that extent. She was lying.\n\nNoticing my stare, she smiled thinly and slipped past me to the pipe, wiping her forehead.\n\n\"Now what?\" she whispered.\n\n\"Now we wait and see,\" I said, moving beside her.\n\n* * *\n\n\u2014\n\nThe car arrived at one-thirty.\n\nWe heard it coming before we saw it. A loose hubcap. Radio blaring. The four of us fell silent, standing shoulder to shoulder along the pipe. Gold headlights swept across the grass. Then a red Nissan slowly rounded the quarry road, bouncing and clanging along the uneven ground before stopping right beside the Foreman's Lookout. I couldn't see who was driving, though I could make out a For Sale sign in the back window.\n\n\"Vida Joshua?\" whispered Whitley, incredulous.\n\nThe engine idled, white moths whirling in the headlights. The radio switched off. There was a moment of silence. Then the driver's door opened, and someone climbed out.\n\nWhen I saw who it was, chills electrocuted my spine.\n\nCannon.\n\nHe dressed in jeans, his old gray hacker's hoodie. He fought through the grass and disappeared into the old mapping office, a sagging shed with a tin roof, though after a minute he reemerged, agitated. He clambered back to the car, texted someone, waited for a response, crossing and uncrossing his arms. As I watched him, I wondered how Cannon of all people had come to be driving Mr. Joshua's car, a car that usually remained parked behind the music school when Vida wasn't using it. Then I remembered how he and Whitley had always stolen things around campus. He had stolen the car to drive out here to meet someone.\n\n\"Hello?\" Cannon called. \"Anybody here?\"\n\nNo one answered.\n\nHe moved to the front of the car and sat on the hood, staring meditatively into the headlights. Another ten minutes, and he was furious. He looked around, scowling, then seemed to give up and climbed behind the wheel, slamming the door, radio blasting heavy metal. He tried to pull away, but the wheels were caught in the grass, the tires spinning. He put the car in reverse, and it bumped backward a few feet. He hit the gas harder and the car roared back, hitting something. Cannon inched the car forward, then reversed again. The car jerked, smashing whatever it was, bouncing over it and stalling.\n\nCannon climbed out. He crouched down to check under the tires.\n\nHe stood up immediately. Then he bent down again. Then he stood up.\n\nHe bent down a third time.\n\n\"No. No. No. No. _No._ \"\n\nCannon threw his head back and began to howl.\n\n\"No. No. No.\"\n\nBewildered, I glanced over at Martha, Kipling, and Whitley watching the scene in silence beside me. They seemed as puzzled as I was.\n\nMuttering something, Cannon bent down once more, seemingly trying to wrench whatever he had run over out from under the tires. For minutes, all we could see were shaking grasses.\n\nWhen he stood up again, he was making a strange noise, as if he was crying. That was when I caught sight of what was in his hand.\n\nA tweed cap. It was Jim's.\n\n_No. This can't be happening._\n\nCannon was back behind the wheel. After a few tries, he managed to back out, doing a three-point turn. He was about to drive away, it seemed, only he had second thoughts, because the car jolted to a halt and he climbed out again.\n\nHe stood frozen for a moment, as if in a trance.\n\nThen he stepped over to what he had pulled out from under the wheels; what I could see now in a rush of disbelief, of horror, as I scrambled on top of the pipe for a better look, was no log. It was Jim, my Jim, lying on his side. His jeans were streaked with blood. Cannon was cradling Jim's head in his lap. Cannon bent over him, whispering something, and then he was on his feet again, on the phone.\n\n\"Call me. I need you to come. I need you to help me. _Now._ Please call me back. Please. Please.\"\n\nHe said it over and over, his voice a high-pitched whine. It was terrible to witness. Cannon's resolute action, his ease with problem-solving, his unflappable tenaciousness\u2014all of which had come to define him in my mind the way waves define the ocean, clouds the sky\u2014it was gone now. He was a different person.\n\n\"I need you. I need you now. Please come. _Please._ \"\n\nWhoever he was calling, no one answered. Cannon climbed into the driver's seat again, sitting in pitch darkness, engine running, radio on.\n\nFifteen minutes later, when he finally emerged, he had a plan. He was his old self, the fixer. He grabbed Jim's ankles and began to pull him brutally through the grass, cursing as Jim lost a loafer, crying out in disbelief, in despair, before wiping his face in the crook of his arm and continuing on.\n\nHe reached the quarry's edge. It was yards away from where we were watching.\n\nHe threw Jim into the quarry without saying a prayer, without hesitation.\n\nThere was the hushed whir of the body falling, knocking against rocks, and then nothing, the muted splash of Jim hitting the water lost in the shriek of crickets.\n\nCannon stared after him, immobile, his blank face hollowed by shadows.\n\nI wondered if he was considering going in with Jim, ending it all, right then and there.\n\nInstead, he turned with an empty stare, climbed into the car, and drove off.\n\n* * *\n\n\u2014\n\nIt was a moment before any of us could move.\n\nI was standing on top of the pipe in the dark, my heart pounding, my mind short-circuiting. Too late, I realized the cement was cracking under my feet. Abruptly, with an angry belch, the entire thing collapsed, Martha and the others jumping back into the grass as I was sent plummeting into the pile of rubble.\n\nWit helped me, gasping, to my feet.\n\n\"What the hell was that? Are you okay?\"\n\nI nodded, climbing out, dusting myself off.\n\nWe stood silently in a circle for a moment, eyeing each other in shock.\n\n\"But who did Cannon call?\" whispered Wit with a hint of indignation. \"Because it wasn't me. I never knew any of this.\"\n\n\"He called Kipling,\" said Martha.\n\nWe turned to Kip. He eyed us stiffly, guiltily, his arms held at odd angles at his side.\n\n\"She's right,\" he whispered. \"The devil called, and I answered.\"\n\nHe said it flatly, with a hint of relief, and I remembered with a shiver of shock the meaningful glance I'd seen Kip exchange with Cannon back in the Wincroft library, when they were confessing how Kipling had made it through Darrow. They hadn't been thinking about the arrangement Cannon had made, or the cheating. They'd been thinking about this very night, and the secret they kept.\n\n\"I helped him throw Jim's body into the quarry,\" said Kipling.\n\nWe stared at him.\n\n\"How can that be?\" asked Whitley. \"We didn't _see_ you.\"\n\n\"Chapter Thirty-Nine, _The Bend,_ \" whispered Martha. \"You never run into yourself in the past or the future.\"\n\nKipling nodded. \"I had to come here. I had to watch. I had to know, once and for all, if it had been my idea to throw Jim into the lake, or Cannon's. Would it happen if I wasn't a part of it? I had to know who was the bad one, and who was worse.\"\n\n\"Did Cannon tell you why he had come here?\" Martha asked, and bit her lip.\n\n\"He did.\"\n\n\"What did he say?\"\n\nKipling smiled demurely. \"Why don't you ask her?\" He nodded at Whitley.\n\nShe glared at him, livid. For a moment, I thought she was about to start screaming, unleashing one of her rages. Instead, she sighed.\n\n\"Cannon was my best customer,\" she said.\n\n\"What are you talking about?\" I whispered.\n\n\"Adderall. The White Rabbit gave him his boundless supply. He popped them like Tic-Tacs. He still does.\"\n\n\"All that time during school, he never knew you were the White Rabbit?\" asked Martha.\n\nWit shook her head. \"Not until Vida. I was too scared to tell him.\"\n\nI thought back to Cannon's reaction when he'd learned Wit was the White Rabbit. He had been livid. Now I understood why. It was because she had known his secret all along, and had never told him hers.\n\n\"So let me get this straight,\" said Martha. \"On the night Jim died, Cannon called the White Rabbit for another stash of Adderall, and you sent him out here.\"\n\nWhitley nodded, sullen.\n\n\"Why here?\"\n\nWit shook her head. \"Jim had found me out a few weeks before. He was watching me constantly, telling me I had to stop. I was afraid to do a drop on campus. So I decided out here was perfect. It was remote. I texted Cannon as the White Rabbit, telling him he could find his supply inside a desk in the mapping office. Only I couldn't make it out here in time. I got caught talking to Mrs. Lapinetti about my Italian final. I raced back to the quarry and did the drop, but I had no further contact that night from Cannon.\"\n\n\"When did you make it back here?\" asked Martha.\n\n\"It was three in the morning. I didn't see anything or anyone, I swear to God.\"\n\n\"You must have just missed them.\" Martha checked her watch. \"When Jim turned up dead at the quarry, you must have suspected Cannon. After all, you knew he'd come out here.\"\n\nWit nodded. \"But I knew he'd never willingly hurt Jim.\"\n\nMartha turned, staring up at the Foreman's Lookout.\n\n\"So the only question now is...\"\n\nShe fell silent, nibbling a fingernail.\n\n\"What?\" prompted Kipling.\n\n\"How did Jim appear so suddenly under that car?\"\n\nShe turned on her heel, resolved.\n\n\"Come,\" she ordered.\n\nBeckoning us to follow her, she vanished into the grass.\n\nWhen we caught up to Martha, she was crouching underneath the Foreman's Lookout. Staring overhead, I saw in astonishment that the ladder to climb up was missing. I realized then that what remained of it was strewn all over the ground.\n\n\"Incredible.\"\n\nMartha gasped in shock over some revelation, then stood up, shaking her head.\n\n\"It's really the most impossible sequence of events.\"\n\n\"What?\" asked Kipling.\n\n\"Momma Greer was right.\"\n\n\"About?\"\n\n\"The freak possible.\"\n\nMartha rolled one of the pieces of wood under her sneaker, then gazed up at the landing suspended high over our heads.\n\n\"Poor Jim.\"\n\nShe looked at me, and instantly I felt chills inching up my arms. What was she aiming at? What was she trying to do? It was dark, but her eyes sparkled behind her glasses, alert, alive.\n\n\"It happened right here,\" she said. \"Jim was undone over Beatrice confronting him about his lie, the night he went off with Vida. He was also distraught over Estella Ornato. His perfect life had fallen down around him, so he escaped here, as he often did, to be alone, to write music. He started to climb up to the Foreman's Lookout, but the ladder gave out. He managed to grab a few supporting beams, trying to save himself, but they didn't hold.\"\n\nMartha bent down to inspect a piece of the wood, showing us that the underside was completely rotten.\n\n\"He fell. It was a considerable distance, five, six stories, a drop that would have killed most people. Yet Jim survived.\"\n\n\"How?\" I whispered.\n\n\"He was drunk. It's why drunk drivers survive car accidents. Drunks don't tense up on impact. They relax. That saves their lives. He was unconscious for an hour. Maybe two. Then he woke up.\" She squinted out at the quarry road. \"He must have heard the car, or seen the headlights. Or maybe he was just trying to get to his bike.\"\n\nMartha hurried to the other side of the road and dragged Jim's bike out of the grass, throwing it at our feet with the flair of a magician whisking a rabbit from a hat.\n\n\"He crawled from here to here.\" She pointed toward the road. \"That's eight, ten feet? He was trying to get help. At that point, Cannon had climbed behind the wheel again. If Jim called out, it was lost in the crickets, the engine, the radio. We couldn't hear a thing, or see much in the dark. Neither did Cannon. Cannon, assuming the White Rabbit stood him up, has to get back to school, drive the car back before Moses returns to the gatehouse after his AA meeting. Frustrated, he puts the car in reverse, hitting Jim. He realizes what's happened, and he goes crazy. He calls Kipling, who is in his debt. Kipling arrives, and together they decide that the only way out of this unimaginable turn of events is to throw Jim into the quarry and pray the police think suicide.\"\n\nKipling nodded. \"We hoped the cops wouldn't notice the difference between injuries sustained from a car hitting you and injuries from a three-hundred-foot fall.\"\n\n\"The police probably would have looked closer,\" said Martha, \"if not for the Masons. They were worried the business about Estella Ornato was about to be exposed. They didn't know what Jim had told people. Given the level of his anger, they probably weren't so sure Jim _didn't_ commit suicide, having learned the truth about what they'd done. What _he_ had done. So they stayed silent. And probably applied some pressure on that little police station. Whatever other pieces of evidence the cops unearthed\u2014Jim's visit to Honey Love Fried Chicken, Vida's tip-off about Shrieks being the real White Rabbit, cell phone records? They stopped pursuing it.\"\n\n\"The Masons confiscated the contents of Jim's case file, don't forget,\" said Kipling.\n\n\"Exactly.\"\n\n\"But there's blood here,\" whispered Whitley. She was using the light on her cell to illuminate the area where the Nissan had been parked. \"It wouldn't take much effort for police to see that something brutal had happened right here.\"\n\n\"We cleaned it up,\" said Kipling. \"I noticed the blood, and we spent an hour tearing up the grass with bloodstains. I shoved it into my backpack and spent more time at school flushing it down the toilet.\"\n\n\"There you have it,\" said Martha. \"The freak possible.\"\n\nThere was nothing to say, nothing to do except to consider the strange history Martha had just related like a professor illuminating to her students some new law of gravity. For a while, I was aware of nothing but my own shallow breathing, and the orchestra of crickets, and the night, gasping and alive all around us.\n\nNever had I imagined a truth like this.\n\n\"It's too extraordinary,\" whispered Whitley, crossing her arms, shivering. \"When you think about it, we all killed Jim. I sent Cannon here. And Cannon hit Jim with the car. And Kipling helped him cover it up. All of us are guilty, right? All of us except Martha and Beatrice. You're the good ones.\"\n\n\"That's not true,\" I blurted, tears burning my eyes, a lump in my throat.\n\n\"It's time to get out of here.\"\n\nMartha whispered this, frowning thoughtfully as she stared overhead. Bewildered, none of us moved. Then she was pushing us and I realized, stunned, looking up, that without even being aware of it, I'd been standing too close to one of the tower's steel legs, because the entire thing was tottering. The wood was groaning and splintering.\n\nSuddenly, with a thunderous moan of metal and glass, the entire Lookout was tipping over, rusted nails and screws and wooden beams raining down on us as we took off across the quarry road. I threw myself into the wall of grass, fighting back blades as they slapped and whipped my face. I ducked and covered my head as the entire structure collapsed around me with a roar, Kipling and Whitley shouting somewhere behind me. I felt myself tossed forward.\n\nWhen I opened my eyes, I was on my stomach, the immense pressure of the ending wake pressing against my legs. I managed to heave myself onto my back, blinking up at the sky.\n\nI heard voices, and then Martha and the others were bending over me.\n\n\"She's at the end of her wake,\" said Martha. \"We don't have much time. We have to find Cannon.\"\n\n\"I think I know where he is,\" said Whitley, her face grave.\n\nWhen she told us the location, no one spoke. Of all possible places in space and time, this one seemed the most frightening, and the most impossible.\n\n\"No,\" said Martha. \"No way. It's too risky for Bee.\" She was helping me to my feet, pulling me toward the edge of the quarry. \"We should go back to Wincroft.\"\n\n\"We need Cannon for the vote,\" I said. \"I'll go. I'll get him and bring him back.\"\n\nMartha looked anxious. But there wasn't time to argue. I could feel the wake traveling up my neck. I knew what to do. I stared down at the quarry and the lake, so far below.\n\nThis was the same journey Jim had made. My Jim.\n\n\"I'll see you there,\" I whispered.\n\nThey were watching me, afraid, but there was no time and nothing to say to reassure them. I squeezed their hands, one by one.\n\nThen I jumped.\n\nWhen I opened my eyes, I was submerged in freezing water.\n\nMilky blue liquid floated before my eyes. I kicked, barely able to feel my legs. I couldn't tell which way was the surface. My lungs throbbing in pain, I blew bubbles, watching dimly as they floated in the opposite direction of where I'd thought to go. I kicked after them into murky darkness, the water growing icier, shadowed fish circling me, their cold, gelatinous skin brushing my toes and fingertips.\n\nI wanted to scream.\n\nI kicked again. Suddenly, I breached the lake's surface, gulping in the icy air.\n\nI looked around. Dense white fog swirled everywhere, chalky and crystalline. A thin layer of ice on the pond's surface splintered around my shoulders. I dog-paddled in a circle, groping for something to hold on to, but there was nothing. It was impossible to see more than a foot ahead. Dead white tree trunks rose out of the water around me, retreating into the whiteness overhead.\n\nIt was where Whitley had told us to go. Blue Pond, Cannon's Birdcage, at 3:33 p.m. on his birthday last year. It was the real-life place in the photo inside the bug Cannon had discovered in Apple's operating system sophomore year at Darrow. It was a dreamlike setting of chalky mist, and thin black Japanese larch and silver birch trees growing straight out of an icy blue lake.\n\nThere was nothing else here.\n\n\"Cannon?\"\n\nMy voice, hoarse and unsteady, ventured only a few feet in front of me before giving up. My legs were so frozen, they felt unattached to me. The cold was like knives in my back.\n\n\"Cannon!\"\n\nA boat motor roared behind me. Startled, I turned to see the paint-chipped bow of a skiff blasting out of the fog, heading straight toward me. I caught a glimpse of faded blue words, _Little Bird,_ Cannon hunched over the motor, his bearded face red, his hair long and matted. The boat hit my head. White pain exploded through my skull. The water silenced my shocked scream as I was dragged under.\n\nEverything went black.\n\n* * *\n\n\u2014\n\nWhen I opened my eyes again, I was submerged in freezing water.\n\nIt was silent.\n\nBlue water clouded my eyes. I could see debris floating around me, seaweed, bits of shell, and mud. Long, dark fish with overbites and bulging eyes drifted around me. They looked dead until I touched one and it shot into the shadows.\n\nI wasn't in pain, apart from my lungs. I blew bubbles, kicking after them. Within seconds I had blasted through the surface, gasping.\n\nIt was the exact same scene, the Blue Pond, Cannon's Birdcage.\n\nA motor grunted. I whipped around to see the skiff heading for me again.\n\nI dove back down into the water, madly kicking through the explosion of bubbles as the boat missed my head by inches. My left foot burst with pain as the propeller's blade sliced it. When I resurfaced, Cannon had circled the boat around and was aiming for me again.\n\nI dove under again, swimming away a few feet before coming up for air.\n\n\"Cannon, please, just wait a minute\u2014\"\n\n\"You shouldn't have come here, Beatrice.\"\n\n\"We need to talk.\"\n\n\"There's nothing to say.\"\n\n\"What about Jim?\"\n\nHe scowled at the mention of the name, killing the engine.\n\n\"Cannon. Please. I just want to talk to you.\"\n\nI held out my hand.\n\nHe leaned over the boat, smiling reluctantly, extending his hand to help me aboard. As I grabbed it, however, he pulled an oar out and struck me with it on the side of the head, my vision exploding into whiteness.\n\nI screamed. I could feel my body sprawling, coming apart, cold water in my mouth, an oar on my back as he pushed me down.\n\nNo matter how hard I fought, that oar remained on my shoulders, keeping me underwater.\n\nHe was drowning me.\n\nThere was no reasoning with Cannon anymore.\n\nThe Neverworld had driven him mad.\n\n* * *\n\n\u2014\n\nWhen I opened my eyes again, I was submerged in freezing water.\n\nThe quiet was deafening.\n\nI realized with a stab of panic exactly what was happening: I was reliving the same wake over and over again. Cannon was killing me, whereupon I remained dead until I was pulled back to the wake. How long did it last? An hour? Minutes?\n\nI could hardly think. I was nauseous with fear. I had to stay calm. Trying to ignore the pain in my lungs, I kept swimming. Blinking up at the surface, I could see the underside of Cannon's boat amid large chunks of ice. He was hiding between the trees, waiting. I dove deeper, ignoring the dark fish with their flaking skin shooting around my legs. When I couldn't hold my breath any longer, I swam to the surface, trying not to make noise as I gulped down air.\n\nCannon's boat was yards away. He didn't see me. He was standing in the skiff, looking around.\n\n\"Beatrice!\" he called. His voice sounded calm, even friendly. \"You out here?\"\n\nI ducked back under and swam away, the water growing dark and murky, the rotten roots of underwater trees, yellowed and tangled, wafting what looked like chimney soot. I could no longer feel my feet or hands. My thoughts were cloudy and strange. As I swam past the debris of a sunken skiff, the faded words _Little Bird_ barely visible, I felt the pull of an undertow. I tried to fight it, but the current was too powerful. As soon as I recognized the deep thundering drone of a waterfall, it was too late; I was plunging through the air. Spray blasted me like a fire hose. Rocks knocked my head and scraped my hands, branches clawing my face. White trees. Blue sky. They flipped over me and under me. I kept waiting to hit the ground, for it all to go black, but the end refused to come.\n\nI was falling, falling for what felt like an hour, every inch of my body freezing, stiffening.\n\nThen I hit a boulder. Life left me like light from a bulb with the flip of a switch.\n\n* * *\n\n\u2014\n\nWhen I opened my eyes, I was submerged in freezing water.\n\nHow many times had I been here before? Four times? Four million?\n\nFish swirled around me like murderous thoughts. I swam into them and they scattered.\n\nI floated deep under the water until I spotted his boat. The water was getting colder. A thin layer of ice was forming on the surface, growing thicker by the minute. I could see Cannon, searching for me. Grabbing a submerged piece of driftwood, I swam directly underneath the hull, clinging there, breathing through a hole in the ice around the boat's edge. I yanked off my pink T-shirt and let it drift to the other side. Cannon, thinking it was me, bent over to pull it out, and as he did, I surfaced and jammed the wood in his back as hard as I could. He cried out in surprise, pitching forward, losing his balance, somersaulting through the ice. I climbed into the boat, nearly capsizing it. I yanked the cord to start the engine. I pried off Cannon's hands gripping the side and veered the boat away.\n\n\"Beatrice!\" he howled, waving at me. \"Come back!\"\n\nI ignored him. His old gray hoodie and a red flannel blanket were folded up around a thermos in the hull. I yanked on the sweater, wrapped the blanket around my shoulders. I unscrewed the thermos and drank. It was tea, so hot it scalded my mouth.\n\nI drove on. It was impossible to see where I was going. The fog disclosed only inches of the world at a time. Blue water, driftwood, blackened tree trunks\u2014they appeared suddenly, ramming the sides of the boat, causing the engine to stall. After a while I could hear the deafening roar of the waterfall and Cannon far behind me. He was crying.\n\n\"I'm freezing. I'm going to die here. Help me, Beatrice.\"\n\nI wasn't sure how far I'd gone when I spotted a coil of long blond hair under the ice, ice at least three inches thick. I smashed it with the oar, realizing in shock that it was Whitley floating there. She was barely conscious. A few feet away, trapped under the ice, were Martha and Kipling.\n\nOne by one, I heaved them into the boat. They were half dead, heads lolling. I placed them in the stern, pulled off their boots and jeans and T-shirts, pulled the blanket over their legs to get them warm, poured tea into their mouths.\n\nSoon they showed signs of life.\n\n\"What is happening?\" asked Martha.\n\nI told her. She asked to see Cannon, so I turned the boat back, steering between the trees until we stumbled upon him. He was clinging to a trunk, so much ice encrusting his beard it was completely white.\n\nHe was dead. His lips were blue. He had pulled off all his clothes.\n\n\"His wake must be years if his hair is this long,\" whispered Martha, touching a frozen strand. She turned to me. \"We have to keep at this, but next time, keep him alive. It's up to you, Bee. We don't arrive in time. So get control of the boat, restrain him, but keep him alive until we get here. Then we can vote.\"\n\n_Cannon's not himself anymore. How can he vote?_\n\nI wanted to ask this, only I realized as the boat jerked backward suddenly that we were getting pulled into the waterfall.\n\nI grabbed the oar, trying to fight it. Martha grabbed the other paddle. Whitley tried to grab hold of passing tree trunks to stop us. Kipling could only stare out at the fog, petrified. It was futile, of course. In less than a minute, the skiff was swinging into the throes of the current, water pounding us. We were rocketing past boulders, ricocheting against trees, overturning into the whiteout. The last thing I saw was Whitley reaching out to try to grab my hand as the boat fell out from under us and we fell.\n\n* * *\n\n\u2014\n\nThe vote. The vote. _The vote._\n\nHow long did it all go on? The fight for the skiff. Cannon's rescue. Binding his ankles and wrists. Hauling my friends out of the ice.\n\nI did it over and over again, in the freezing cold, trying not to drown.\n\nI tried different tactics every time. Cannon might have been half mad, but he was on to me. He was a strange, terrifying foe, at times vicious, other times childlike. He was the worst person to have to capture alive, because I knew him from before. There were times when he was his old self again, funny and kind and sensitive, vocal about wanting to help me, to do everything in his power to make it better. Inevitably, though, he'd cast this persona off like a Halloween costume, revealing someone upended by rage and regret. I understood then that Cannon had always lived his life with his future glory in mind, that every moment of his every day and every act of kindness had been because he was expecting that at some future date he would be somebody at last. Now that he had no future, he didn't know how to exist.\n\nHe'd shout his grievances into the fog.\n\n\"I was duped. Swindled. First there was the nightmare of Jim. And now this? _Are you kidding me?_ It isn't supposed to be like this. I'm supposed to grow up! I'm supposed to have another seventy years! I never made an impression. It's like I was never even here. Was I here? Was I even here, Beatrice? _Beatrice!_ Where are you?\"\n\nSometimes, when Cannon gave me trouble, I was too late freeing the others from the ice. When I found them they were all dead except Martha. She was always semiconscious, deliriously whispering the same two words over and over again.\n\n_It's you._\n\nAfter a while, I had a map of the entire Blue Lake in my head like a blind man who's memorized every inch of his neighborhood. I knew where every dead tree stood, where every boulder sat, when every spray of water would firework over the rocks into oblivion.\n\nThe chance for the vote inched closer. Faster and faster I restrained Cannon. This had as much to do with his increasing fatigue, his resignation, as with my speed and resolve. I bound his hands and ankles with a yellow vine ripped from the bottom of the lake, pulling him up into the boat, leaving him sulking in the bow. Faster and faster I revived the others.\n\nThe remainder of my wake was eleven minutes. Eleven minutes between the time they were warm and under the blanket and the moment I rigged the boat to the trees so we wouldn't plunge into the waterfall. Eleven minutes to vote.\n\n\"I'm not voting,\" Cannon always said.\n\n\"Yes you are,\" said Whitley.\n\n\"No.\"\n\n\"Then you'll drown here.\"\n\nHe laughed. I'd grown used to his mad cackle by then, but it still scared the others.\n\n\"Drowning? You think I'm scared of _drowning_? Drowning for me is shaking a hand. It's saying 'Have a nice day!' It's saying 'Would you like an Egg McMuffin with those hotcakes?' It's saying 'Welcome to Home Depot, can I help you select a Weedwacker?' \"\n\n\"Please stop,\" whispered Whitley, trying not to cry.\n\nThe vote. The vote. _The vote._\n\nWe had no pen and no paper. I pried a piece of splintered wood off the skiff's bottom boards and we used that to cut the first initial of our chosen survivor into our palms.\n\nOver time, strange things began to happen in those eleven minutes. The dead trees began to topple and crash into the water, creating waves that surged and flooded the boat. The fog retreated, revealing a gray sky, clouds roiling like potion in a witch's cauldron. Swarms of red insects like the ones Martha had drawn boomeranged around us like tiny squalls of rain, emitting a high-pitched hum, colliding with our foreheads and ears and getting tangled in our hair, making us scream. A single fat fly appeared too, buzzing around our heads. We all knew it was Pete, the imaginary friend who'd lived inside Cannon's boyhood computer, the one he'd told us about. Ice encrusted our hair and eyelashes. It thundered and snowed and hailed. In the eleventh minute, the skiff even began to disintegrate under us, blue water seeping up between the beams until the wood began to blacken and crumble to mud.\n\nI understood what was happening, though I didn't say a word. No one did. It was the decision, the slow settling in on the single name. It was the death of our dreams, our youth, of possibility. There had always been hope here in the Neverworld, no matter how terrifying things got.\n\nNow even that was disappearing.\n\nCannon ignored our entreaties to vote. He stayed slumped against the side of the boat, staring out, singing \"Just Like Heaven\" by the Cure under his breath, repeating the phrase \"You, soft and lonely\" over and over again.\n\nThen, one wake, he actually snatched the wood from Kipling, and gnashing his teeth in frustration, he too carved what appeared to be someone's initials into his hand. He did it rashly, blood oozing between his fingers as he collapsed back, staring out, exhausted.\n\nThat was when Whitley sat up, pointing into the fog.\n\nIt was the Keeper. He was rowing a boat toward us, wearing his dark suit and tie. He maneuvered alongside us. In spite of the hail, his boat jerking and bobbing against ours, the spray of water, he was remarkably dry.\n\n\"Congratulations,\" he shouted, his voice scarcely audible over the thunder. \"There is a consensus.\"\n\n\"What?\" gasped Whitley.\n\nThe Keeper only smiled, gripping the sides of the boat so as not to be tossed out.\n\nHe cleared his throat, straightening his tie, though almost immediately the wind flung it back over his shoulder.\n\n\"Life does not belong to you. It is the apartment you rent. Love without fear, for love is an airplane that carries you to new lands. There is a universe in silence. A tunnel to peace in a scream. Get a good night's sleep. Laugh when you can. You are more magical than you know. Take your advice from the elderly and children. None of it is as crucial as you think, but that makes it no less vital. Our lives go on. And on. Look for the breadcrumbs.\"\n\nI think we were only half listening. We were all stupefied.\n\n\"It's been a pleasure.\" He bowed.\n\nAnd just like that, he took up the oars again and rowed away.\n\nThe change was immediate. The water stilled. The storm tapered off. The roar of the waterfall faded to a whisper. The sun emerged out of the blue sky, glaring and hot. In fact, the scene so quickly transformed to a calm, serene lake with shimmering water that the memory of all I'd endured these past twelve wakes\u2014or twelve million\u2014seemed as hazy as some half-remembered dream.\n\nIt grew hot. Whitley and Kip stripped down to their underwear, and whooping and shouting, they cannonballed into the water as if it were the final hours of summer camp. Cannon, with a deadened look, threw himself headfirst over the side, and though I stood in alarm, calling out his name, he only kicked away from me on his back, his eyes closed. He seemed so tired. He seemed to want peace.\n\nThat left me with Martha. I had something important to say to her, and I might never have another chance.\n\n\"Martha.\"\n\nShe was watching Whitley and Kipling laughing about something. She turned.\n\n\"We've never been friends. I just want to tell you that I understand why. And it's okay.\"\n\nShe stared at me.\n\n\"I was his girlfriend. Everyone was in love with Jim. It wasn't so hard to imagine that you were too. I just wish we'd gotten to know each other better.\"\n\nShe tilted her head, frowning.\n\n\"Jim? You think I was in love with Jim?\"\n\nI nodded. She smiled.\n\n\"I never loved Jim. It was you. What you did for me. You saved my life.\"\n\nShe said it faintly. I wasn't sure I'd heard her correctly.\n\n\"Do you remember that night freshman year, during the snowstorm? The night of Holiday Dance. The power went out, and you ran back to the dorm to change your dress. You found me reading in the common room. You laughed because I hadn't noticed the window was wide open, and there was a snowdrift on the carpet. You stayed and talked with me, even though Jim was waiting for you.\"\n\nI remembered. It was the one time we'd had a good conversation.\n\n\"It wasn't an accident the window was open.\"\n\nI stared at her.\n\n\"I'd been planning it for weeks. I'd done the math. Sixth floor. Larkin Hall. A simple acceleration due to gravity across seventy-six feet. Even landing in a snowdrift, my chance of survival was less than one percent.\"\n\nI couldn't breathe.\n\n\"It was stupid. One of those dark spells of loneliness that I thought meant everything. Little did I know, it meant nothing. These monumental moments of our childhood, they're just one bend in the river, a tight curve filled with boulders so you can't see beyond. The river roars on across distances we can't even imagine. I was about to jump when I heard someone coming. It surprised me, so I hesitated, threw myself on the couch, grabbed some random book, pretending to read. You came in, and you saved my life. So here, in the Neverworld, I had to save yours.\"\n\nI opened my mouth to say something, but no sound emerged.\n\n\"I thought for sure you were on to me,\" she said, shaking her head. \"Like, back at the Warwick police station, how I suddenly appeared downstairs with you. You knew I was the one who removed the papers from Jim's case file, right?\"\n\n\"What?\" I whispered.\n\n\"It wasn't the Masons. It was me. I hid the files in another box so they'd never find them.\" She took a deep, unsteady breath. \"Because it was all there. Jim's texts to you. I didn't want them to suspect you. That was why I was so against going back to Vulcanation. I didn't want them to find out the truth. So as soon as we landed, I snuck away so I could dismantle the ladder from the Foreman's Lookout before anyone else saw it. I climbed up fifty feet, got a million splinters, but I knew I had to present a compelling scenario with such assurance that they'd all be blind to the truth.\"\n\n\"What?\"\n\nShe studied me with a soft smile.\n\n\"You know, Beatrice. You were there.\"\n\nChills ricocheted down my spine.\n\n\"I saw you. Coming back from the quarry.\" She squeezed my hand. \"You have nothing to feel guilty about. Whatever happened, I know you acted with a full heart. I never doubted you. And I never will.\"\n\nAll my blood drained into my feet. I was going to be sick.\n\n\"Jim loved you. But he didn't see you. He was incapable of that. You were the one to keep him propped up. You were his scaffolding. He could be riveting, and addictive. And you loved him, and we rarely see those we love as they are.\" She sighed, hunching her shoulders. \"That's what killed me the most. Why I could never be your friend. Why I couldn't stay around you. You made me so mad, Bee.\"\n\nShe shook her head, staring at me, her face a wild pool of emotion barely contained.\n\n\"I've seen it before. It happened to my sister. She loved a boy, and that love made her put herself last and forget herself, and it killed her. Your love was that unquestioning. It made you do things that were dangerous. _That_ ripped me up.\"\n\n\"What are you talking about, Martha?\"\n\n\" _Nowhere Man._ Jim's musical? Everyone gushed about how brilliant it was. And it was. But it was strange, wasn't it, how suddenly after weeks of whining, being unable to write a single word, Jim had it all come together on the eve of his debut at Spring Vespers? Like magic?\"\n\nShe stared at me, her face grave.\n\n\"You were the magic.\"\n\nI was unable to speak. I felt as if a glaring light were suddenly shining into my eyes.\n\n\"You showed them to me the night of the snowstorm. Those dream soundtracks. I never forgot them. I committed the words to heart. I recognized your voice immediately when Jim showed me what he'd written. 'You're my Sunday best, my new-car smell, \/ You're Ch\u00e2teau Margaux, no zinfandel.' \" Martha shook her head. \"Jim thought nothing of passing off your words as his own. Did he say he was just _borrowing_ them? That he'd give you credit _later_? He swallowed everything around him, leaving nothing behind.\" She wrinkled her nose. \"It's so funny. For such an energetic person, the space around him was always so cold. And anyway, his grand plans for himself always exceeded his _actual_ talent.\"\n\nShe shrugged with a look of resignation. I felt a wave of hot emotion in my chest.\n\n\"Jim didn't steal the lyrics from me,\" I said. \"I gave them to him. They were just sitting in a drawer in the dark, no use to anyone. I had to help him.\"\n\nMartha surveyed me so intently, I felt light-headed.\n\n\"Everything I've done in this Neverworld,\" she said, \"the good, the weird, the absurd, the exhausting, was for you. Pushing the discussion in a calculated direction. Asking you the pointed questions so I'd appear impartial. Distracting the others from seeing the rot that kept bubbling up around you all the time. Mold, breaking glass, tar, oil, tumbling trees, falling Lookout Towers\u2014 _God,_ Bee, it was like trying to hide a typhoon swirling around you, all because of this secret you were hiding. That you were there that night.\"\n\nShe shook her head, biting her lip.\n\n\"I even spent a million hours talking to this kooky professor with scary facial hair and bad breath at Brown to learn the art of persuasion, to implant the idea in all of their heads that _you_ had to go on, because you had to be the one to tell our story.\"\n\nMy mind was crawling stupidly over her words like a crab, trying to make them out.\n\nWhat was she talking about? I had voted for Martha. Martha was going to live.\n\n\"I couldn't tell you what I was doing because you'd have tried to stop me. You'd have messed it all up. We had to get to the bottom of Jim's death for the vote, but you had to stay beyond blame. You had to remain Sister Bee.\" She shook her head. \"I'm only telling you all this so you'll know. So you'll see. Because we all have our words tucked away in notebooks in drawers in the dark. You can't just give them away, Bee. They're yours. Like a fingerprint. Like your children. They are the light that shines your way. Without them, you'll be lost.\"\n\nShe reached out and gently tucked loose strands of hair behind my ears.\n\n\"Never, ever give away your words again.\"\n\nMartha. _I was so wrong._\n\n\"Anyway.\" She removed her glasses, folding them, carefully setting them on the seat beside her with a faint smile. \"Chapter Seventy-Two. This is only the beginning.\"\n\nShe stood and, mumbling something that sounded like _breadcrumbs,_ she dove into the water, kicking into the turquoise depths.\n\nI sat there, shaken, unable to move.\n\n_So absolutely wrong._\n\nI lurched to my feet, shading my eyes.\n\n\"Martha!\"\n\nThere was no sign of her.\n\nWhitley and Kipling, swimming a few yards away, turned in alarm.\n\n\"She was just here. Martha. I\u2014I have to tell her. I have to let her know\u2014\" I was untying the skiff, grabbing the oars, crying as I steered the boat between the trees. \"Martha!\"\n\nI jumped overboard, swam into the darkness, reached out into the empty cold.\n\nWhen Whitley and Kip hauled me back into the boat, I was sobbing.\n\n\"She was just here. And now it's too late. Too late. Don't you realize? Martha. She's never coming back. I have to tell her. She's gone, and it's too late now to tell her\u2014\"\n\n\"Shhh,\" said Whitley, hugging me and wiping the tears from my cheeks. \"It's all over now, Bee. Look around. It's almost gone.\"\n\n* * *\n\n\u2014\n\n_Look around. It's almost gone._\n\nIf only someone had told me that before. About life. If only I had understood.\n\nWe didn't speak after that. We didn't need to. All we did was wrap ourselves in the blanket, and gaze out at the water.\n\nCannon was already somewhere else.\n\nThe sun was setting. It had turned the bold orange of children's paintings, and it was casting a warmth on our faces so gentle it seeped into us, filling every dark hole and lighting every corner. I'd felt this way before, back at Darrow on some ordinary Tuesday with my friends, when one of them said what I felt and life sharpened into focus, as it did sometimes. There was a momentary stillness, a sense of the eternal in the strands of our laughter like windblown ponytails, in the touch of our shoulders, side by side.\n\nSomething began to happen to me. Whether it was death or some other state in the mystery of all life, I didn't know. It pulled me to the bottom of the boat, leaving me staring up at the vast yellow sky. They had more time in their last wake, Kipling and Wit. But they would feel it eventually. I could see them crouched beside me, whispering words I couldn't hear, uncertain yet unafraid, their hands warm as they squeezed mine, waiting for what came next.\n\nI would never let go of them. Never.\n\nThen their faces dissolved into the darkening day, and I slipped away.\n\nI was floating in milky space.\n\nSomething hard was shoved down my throat. I heard footsteps.\n\n\"Good morning.\" A man was speaking. \"How you holding up?\"\n\nThere was a clattering noise. Someone was beside me.\n\n\"I know this is difficult. As I explained yesterday, we'll be taking this one step at a time. Her weaning parameters look very good. So I'm hoping to remove her breathing tube today. We need to see if she can follow commands.\"\n\nThere was a flurry of activity, hushed whispering. A hand touched my arm.\n\n\"Beatrice? Can you open your eyes for me?\"\n\nI blinked. All I could see were streaks of color.\n\n\"Oh, my God.\"\n\n\"Beatrice?\"\n\n\"There. There she goes....\"\n\n_\"Bumblebee?\"_\n\n\"Can you show me two fingers?\"\n\nDizziness. I was floating in a swamp. I tried to lift my hand. My throat was on fire.\n\n\"What about your other hand? That's great. Wiggle your toes.\"\n\nSomeone was leaning over me. Suddenly a light beamed into my eyes, sending a hot purple pinball knocking around my skull.\n\nI blinked again.\n\nThat was when I saw a TV on the wall. It was a morning talk show, the sound muted, the date at the bottom of the screen snapping into focus.\n\n_7:21 a.m. September 10._\n\nI was alive.\n\n* * *\n\n\u2014\n\nAs I fell back into the warm, watery darkness, my final conversation with Martha drifted through my head. It felt like she'd just left me moments ago. Her confession had turned me inside out. It was the secret I'd kept so deep inside my heart it had actually remained buried, out of sight, like a missing airplane that had vanished with such totality, some questioned whether the passengers had even existed.\n\nWhitley hadn't realized how right she was.\n\n_When you think about it, we all killed Jim._\n\nNo one had ever questioned me\u2014not my friends, not the police, not my parents. No one. Because I was the good one, Sister Bee.\n\n_I'm going to the quarry. Meet me._\n\nIn my dorm room, I listened to Jim's message over and over again, staring out the window at the empty lawn. I was so alone. I loved him. Yet I hated him. I hated how he could make me feel so alive, then invisible, as if he were a magician and I was the rabbit in his hat. I was desperate to see him, forgive him, to banish him from my thoughts. I wished he'd never seen anything rare in me. The prospect of being without him was too painful to imagine.\n\nI jumped out of bed, threw off my pajamas, and slipped on the sexy lingerie I'd saved up for, the tight white jean shorts Jim liked, the white off-the-shoulder Gucci top borrowed from Whitley. I was going to sleep with him. It was a stupid decision, but it filled me with excitement, a concrete resolution I could hold on to like a towrope. I put on eyeliner and mascara, Whitley's red MAC lipstick. I pulled my hair out of its usual ponytail so it fell down my back. I pulled on my Converse, threw two candles into my backpack, yanked the comforter off my bed.\n\nThen I went running out to Vulcan Quarry.\n\nBy a stroke of luck, I was so distracted by my decision to sleep with Jim that I left my phone on the sink in the bathroom. Later, I would gather that the detectives, pinging the cell towers on the night Jim died, saw that mine hadn't moved, providing me with an alibi. Yet if they had questioned me, I doubted they would have suspected I was lying. No one ever doubted anything I said.\n\nAnd they should have.\n\nWhen I arrived at the quarry it was 12:15. There was no sign of Jim. He hadn't arrived yet. The night was cool, the sky clear, stars bright. We always met at the base of the Foreman's Lookout and did the ascent on the ladder together. This time, I went first. I wanted to set everything up, to surprise him. I couldn't wait to see him, to forget it all, to go back to how things were in the beginning. I was scared too\u2014scared to be with him again, scared of the doubt in my head. As I climbed, I noticed that some of the nails holding the ladder's wooden rungs were looser than usual. Others were actually missing, especially in the final few feet where you reached the hatch.\n\nHalfway up the ladder I stopped, noticing not just that my hands were shaking, but that I had ripped my entire left shin without realizing it. It was bleeding, gruesome-looking. I looked like a skinned possum. I started climbing down again. I didn't want Jim to see me like this. I was lopsided, overtired. I was ugly, unlike Vida Joshua. Vida Joshua was a siren. I should go back to my dorm. That was the right thing, the safe thing.\n\nI was almost on the ground when I stopped again. I was being a coward, meek, living so _pianissimo,_ as Jim used to tell me. Why was I always so afraid of things happening to me? I began to climb up again\u2014 _Carpe noctem!_ Whitley was always shrieking with her head back. Seize the night. Why couldn't I do it for once? When I reached the landing, I noticed that some of the nails holding the ladder's wood rungs were rattling.\n\nI lit the candles in the grimy room. I turned on the oil lamp on the old wood table where a hundred Darrow students had carved their initials. I spread out my comforter, undressed, and waited.\n\nSoon I heard Jim. He was talking to himself, his words slurred.\n\nI rolled to my feet, gathering the comforter around me. I crept to the landing, peering out.\n\nHe was halfway up the ladder. He was also drunk, swinging an arm out as he sang something. It was the lyrics to a new song in his musical, lyrics I had written.\n\n\" 'In the dark there grows a tree. \/ A castle tower shelters thee. When will I stop, when will I see? \/ There is no poison but for me.' \"\n\nMuttering, he began to climb again. I tiptoed back inside and reclined across the comforter. He'd be here within seconds. It was happening. The thought gave me a strange feeling of emptiness. I was making a mistake. It was obvious. I needed to stay away from Jim. I should be asleep in my room.\n\nAt that moment I heard a clanging noise. Jim was screaming.\n\nI leapt to my feet. Three of the rungs by the landing had fallen away. Jim was barely holding on. He was straining to grab the next rung, but it was just out of reach. Gasping, he managed to swing his leg out so his foot rested on one of the crisscrossing beams supporting the tower legs.\n\n\"Bee?\" He blinked up at me, sweat glinting on his forehead. \"Oh, God, Bee. Thank God.\" He held out his hand. \"Pull me up.\"\n\nI froze. He began to shout, his face contorting.\n\n\"Beatrice! What's the matter with you? Pull me up! _Beatrice!_ \"\n\n* * *\n\n\u2014\n\nWhat happened in those four seconds?\n\nI'll never know.\n\nIt was so fast. I saw Jim. Yet I couldn't move. I couldn't breathe.\n\nI wished with all my heart I could say it was just panic, but it wasn't. It was something else too. A little cave inside my heart. Somehow I knew if I pulled him up I'd never be free of him. Maybe Martha was right. Maybe it was about the lyrics he'd taken from me, albums I'd slid in front of him after he'd been sobbing that he was a hack, that he'd never be as accomplished as his father, that it was all over, his dreams were done. I'd gone into my closet and handed him my collection of dream soundtracks, eleven books of lyrics and drawings I'd worked on all my life for no reason except they were the one place I could be myself. Maybe it was how he had taken them, sniffing as if I'd only handed him a pen when he knew what they were to me, what they had meant, and started copying my rhymes into his notebook. Maybe it was the question that if he could so easily take my words, would he take everything else?\n\nMy hesitation lasted only a moment. I sprang to life, racing toward him, wedging my feet in the landing door so they were secure, lying on my stomach, reaching to him.\n\nI was too late.\n\nHe fell. His head smashed a wooden beam, his hat flying off. He hit the ground with dull thud.\n\nHe lay still, five stories below me, a streak of blood across his cheek.\n\nThe next minute was a dream. The realization of what had just happened got bulldozed dumbly around my disintegrating mind.\n\n_Jim's dead. Jim's dead. This isn't happening._\n\nMadly I ran around the Foreman's Lookout, shivering, crying, blowing out candles, stuffing the comforter into my bag. I yanked on my clothes. I scrambled down the ladder four rungs at a time, barely making it around the gaping hole, threw myself into the grass.\n\nI rolled to my feet, staring down at Jim.\n\nBlood was oozing across the side of his face. His eyes were closed. He was dead. I was certain. I had to call the police. Yet, groping around in my backpack for my phone, I couldn't find it. Had I left it in the Lookout? Looking up, I realized I'd accidentally left the oil lamp burning. It was then that I saw headlights igniting the grass like wildfire. A car. It appeared, bouncing along the rutted road, a loose hubcap, radio blaring.\n\nIt was Mr. Joshua's beat-up red Nissan, the For Sale sign taped to the back window.\n\nVida Joshua. That was who I thought it was. What was she doing here? Had Jim meant to text her to meet him here, not me?\n\nThe question sent me retreating into the dark, sprinting back through the grass. I needed to go home. I needed my mom. I found the opening in the fence and struggled through.\n\nVida was going to find Jim and call an ambulance.\n\n_He would be fine. Everything was fine._\n\nI don't remember sprinting back through the woods and across campus. The next thing I knew I was barreling up the steps to the fourth floor of my dorm, racing down the hall. That must have been when Martha saw me. She lived on my floor, studied in the corner common room. I hurried to my room and locked the door, stripping naked. Everyone says I'm the good one, the kind one, so that means I am, doesn't it? It means I always do the right thing.\n\nI folded the La Perla underwear back into the tissue paper at the back of my drawer, returned Whitley's top to my closet. I found my phone where I'd left it on the bathroom sink. It was 1:02 a.m. No messages. My hands trembling, I managed to wipe the lipstick off, splash my face with cold water, yank the grass and leaves out of my hair.\n\nThe realization of what I was doing hit me like a slap in the face. What was I doing, not calling the police? I had to go to Jim. My love. I began to dial 911, but the conversation I was about to have with the dispatcher made me stop.\n\n_My boyfriend is lying dead in Vulcan Quarry. He fell. Please send an ambulance._\n\n_Are you there? Where are you?_\n\n_I ran away. I was jealous of another girl. I was angry. I loved him. We'd had a fight._\n\nCannon. I needed Cannon, the problem solver. I ran across the courtyard and climbed the oak tree to his room on the third floor of Marlborough. I knocked on his window. No answer. I pulled it open. There was no one there.\n\nKipling. Kipling would help me. He had a tower room in Eldred. I climbed back down the tree, raced across campus, slipped in through the fire exit, up the back stairs. His room was empty too. When I ran along the gable to Whitley's dorm room and knocked on her window, she too was missing.\n\nWhat was going on? Where was everyone?\n\n_Martha._ Racing back to Creston, I could see her light on in the window on the fourth floor, but imagining her flat response as I confessed to her, weeping, frightened, sent me running straight back to my room, my heart scuttling around like a rodent in my chest.\n\nI crawled into bed, staring at the ceiling. I kept telling myself to call my mom, but I couldn't move. Questions exploded in my head like grenades: If I'd never decided to surprise Jim, would he still be alive? Had he wanted to see Vida, not me? Had I loosened the rungs from climbing up and down and then up again? Where were my friends? Had Jim managed to call them for help, and were they with him right now, hearing all about what I'd done, that I'd let him fall and left him there? Had I killed Jim?\n\nI had to go back to the quarry. From there I'd call the police. I climbed out of bed, yanked on jeans, a T-shirt, boots. I ran all the way out there again, petrified, certain Moses was going to catch me. When I arrived it was after four. My entire body shaking, I stepped to the spot under the Foreman's Lookout where the ladder was, and stopped.\n\nJim was gone.\n\nThere was no sign he'd ever been there.\n\nNo blood. A few blades of bent grass. Otherwise, there was nothing.\n\nVida had found him and taken him to the hospital. Or, by some miracle, he'd gotten up and walked away unscathed, which meant he loathed me now. They all did.\n\nI returned to my room, staggered. All I wanted to do was die.\n\nI wandered through the next day like a zombie. When I thought of the night before, the memories were distorted, as if I'd made them up. Had it actually happened? There was no sign of Jim. No one had seen him. Whitley, Kip, and Cannon all acted friendly but stiff. They said they'd been in their rooms all night. Martha claimed she'd slept in the library.\n\nA day later, the news arrived: Jim had been found dead in the quarry lake.\n\nIt was impossible. I didn't understand. What had happened after I'd run away? What had Vida done to Jim? Why were my friends all lying? What were they afraid of?\n\nTo find answers, I'd gone to Wincroft.\n\nAnd all along, Martha had known my secret. Martha had been cleaning up my every move, all the while protecting me.\n\nHow had I never seen it? How could I have been so blind?\n\n* * *\n\n\u2014\n\nWhen I opened my eyes, I was propped up in a hospital bed. The room was in sharp focus: pale yellow walls, counters and tabletops, an air conditioner, a vase of flowers, a teddy bear with a helium balloon proclaiming GET WELL SOON. In front of me sat a plate of hospital food, a pink cup with a straw.\n\nThere was no longer something lodged in my throat, though it felt scratchy and raw.\n\n\"She's awake,\" blurted my mom, turning from the window.\n\n\"My dear sweet Bumble,\" said my dad, stepping toward me.\n\nThey hurried over, peering anxiously at me. My mom was gripping a wad of Kleenex, her hair standing vertical in places from sleeping in a chair. My dad had more gray hair than I remembered.\n\n\"Don't try to talk,\" he said. \"All is well. You're at Miriam Hospital in Providence. You were in a car accident, and you sustained a head injury. Bleeding on the surface of the brain. The doctors took care of it, and you're going to make a full recovery, okay, kiddo?\"\n\nI could tell my dad had instructed my mom not to talk very much, because she was nodding at everything he was saying, trying not to cry.\n\n_Just tell me my friends are still alive. They're recovering in rooms down the hall._\n\n\"You're going to rest,\" said my mom, squeezing my hand.\n\nI looked past her across the room, where there was a framed print of a beach scene on the wall and a dry-erase board sign reading _Your nurse on duty is LAURIE_.\n\nA bony teenager with a mop of blond hair was sitting in a chair by the door, staring at me. It took me a moment to realize it was Sleepy Sam, the British teenage boy I'd scooped ice cream alongside all summer at the Crow.\n\nMy mom followed my gaze. \"You remember Sam.\"\n\n\"He's come here every day to read to you,\" said my dad.\n\nSam shuffled over.\n\n\"Really glad to see you open your eyes, Bee. Welcome back.\"\n\nMy dad clapped a hand on his shoulder. \"Sam's a world-class dramatic reader. Who knew? He does all the different voices. Fifty characters? No sweat. He could have a big future on the West End.\"\n\nIt was then that I noticed the book under Sam's arm. The cover was silver with a collage of birdcages and steam trains, rosy-cheeked characters wearing top hats. _The legendary cult saga of future pasts. Present mysteries._\n\nThe title sent a shock of adrenaline through me.\n\n_The Dark House of Elsewhere Bend._\n\n\"Good morning, Beatrice.\"\n\nA silver-haired doctor in a white coat and green scrubs entered carrying a clipboard and a paper coffee cup. He was accompanied by an Asian woman, also wearing a white coat.\n\n\"Welcome back,\" he said. \"Allow me to introduce myself. I'm one of your physicians. Dr. Miller. It is a pleasure to finally meet you.\"\n\nHe was leaning over me, shining a light into my eyes. When I looked past it to his face, I gasped.\n\nI'd recognize him anywhere. He was and would forever be etched into my brain, floating in front of me whenever I closed my eyes for the rest of my life: those green, all-seeing eyes, the mahogany baritone, the elegant, exhausted manner suggestive of a retired ballet dancer whose every step held thousands of hours of rehearsal and a faint ache.\n\n_It couldn't be. It's impossible._\n\n\"When can she move to the rehab facility?\" asked my mom.\n\n\"A few days. The weakness on the left side of her body and some of the short-term-memory difficulties should improve over time. But it can take months.\"\n\nThe Keeper.\n\nHe asked me to raise my arms, hold up three fingers, and bend my knees. He asked me if I knew what year it was, who the president of the United States was, my age. I was dizzy. I could hardly focus on anything he said, gaping as I was so incredulously at his face. He'd set down his paper cup on the tray in front of me. The tag on the tea bag dangled over the side.\n\nHe grabbed the cup, took a sip, turned on his heel. He whispered something to my parents as they moved after him to the door. Then he slipped out with the woman in tow, vanishing down the hall.\n\nMy mom and dad had no choice but to tell me, even though I knew.\n\nKipling St. John.\n\nWhitley Lansing.\n\nCannon Beecham.\n\nMartha Ziegler.\n\nThey were dead.\n\nI moved to the rehab facility and spent six weeks there, wandering the linoleum hallways with my soft-grip adjustable cane, practicing going up and down stairs and raising my left arm, which trembled and shuddered with a mind of its own. I snuck onto the public computer after dinner the first night I was there and read about it.\n\nThe accident was reported in the _Providence Journal,_ the _Warwick Beacon,_ and _USA Today._ All the articles used the same phrase: \"shocking loss of young life.\" It also came up in a _Republican Nation_ editorial about drunk driving and its prevalence in New England communities with a rising unemployment rate. Every story led with photos of Whitley, the textbook dead blond dream girl, then moved on to Cannon, Kipling, and Martha, always mentioning Martha's full scholarship to MIT. My name was mentioned at the very end, the name of the lone survivor, the lucky one.\n\nTheir Facebooks became memorials. I wasn't surprised. It had happened with Jim. Kids they barely knew at Darrow and friends from their hometowns posted messages like _my heart's broken_ and _the world is empty now,_ littered with prayer emojis, anonymous comments of _life is pain,_ and GIFs of Heath Ledger.\n\nI'd missed their funerals. I'd been in the hospital. So I read about them. All their hometown newspapers did follow-ups to the tragedy (because the initial articles had racked up hundreds of Shares and Likes), featuring photos of some red-eyed family member reading a poem in a church pulpit. The blown-up, framed picture of Kipling\/Cannon\/Martha\/Whitley stared out from the easel beside them, their unwitting happiness and total lack of understanding of what was to come a powerful reminder that life, among many things, was all hairpin curves.\n\n_Linda Tolledo speaks during a service for her daughter, Whitley Lansing, who died in a car accident._\n\nThere was even an ongoing memorial of flowers, photos, candles, and teddy bears being left on the side of the coastal road at the crash site. People took pictures of it and posted them with hashtags like #rip and #neverforget.\n\nThey never suffered, the police told my mom and dad. They all died on impact.\n\nI survived because I hadn't been wearing my seat belt. I'd gotten tossed out, landing in a cluster of bushes, while the others were trapped in the car as it barreled down the ravine.\n\nLittle did anyone know the real reason that I'd survived: I had lived a century inside a second. I had died thousands of times, learned about and loved four people in a way few ever had the chance. I had called a place home where details such as life and death didn't matter, where what did matter were the trembling moments of connection in between.\n\nAnd afterward, you felt nothing but awe for every second of your little life.\n\n* * *\n\n\u2014\n\nSo began life outside the Neverworld.\n\nIt was different from what I remembered. _I_ was different.\n\nAnd it wasn't just the scar of a reverse question mark wrapping around my skull above my right ear. My hair hid the scar, but it was there if you looked for it, my tattoo, my memento. To outsiders I seemed confident, if a little solemn. I was less prone to biting my lip and tucking my hair behind my ears. I no longer worried whether people liked me, or whether I was pretty or had made a mistake. I wasn't afraid to eat in a crowded cafeteria at a table alone or talk to a cute boy I didn't know, or to sing karaoke, audition, give a speech. All the things people spend so much time worrying about in this world\u2014the Neverworld had unchained me from all that. I was no longer in a hurry to fill silence. I could just let it sit forever like a bowl of fruit.\n\nMy parents' friends whispered, \"Beatrice has really come out of her shell,\" and \"You must be so relieved.\" They marveled when they heard the news that I had transferred to Boston College, was majoring in music theory and art history, working part-time at a video game company, volunteering at a nonprofit that had people read books at bedtime to foster children.\n\nIt was those kids I told about the Neverworld Wake.\n\nI told no one else, not even my mom and dad. Somehow I knew that those children, with their wide eyes and knowledge of the dark, their kingdoms of morning and hide-and-seek, naptime, and snack, that they, of everyone, would understand. I told them I'd visited the secretest, wildest wrinkle in all the world. That one day, they might find themselves in one too, some lost dreamland between life and death, where past, present, and future are a jungle and hell can become heaven in the blink of an eye.\n\n\"How do we go there?\" a girl whispered.\n\n\"If you're chosen, it'll find you. But the trick is not to be afraid. Because it isn't so different from this world after all.\"\n\nHad the Neverworld been real? Or had it been a side effect of my injury, the _right-sided subdural hematoma requiring a craniotomy for evacuation,_ eleven days passed in a coma, intubated and sedated. Sam had read me the book. One of my physicians looked like the Keeper. Had it all been in my head? Had my senses, as I slept, pulled details from the boisterous world in motion around me, spinning it into a reality that existed only for me?\n\nOf course the Neverworld had been real, though I could never prove it.\n\nI tried to. I tried to corroborate all the secrets. I discovered that while some things did check out\u2014Mrs. Kahn did live down the road from Wincroft with her collection of snow globes; there was an exclusive marina called Davy Jones's Locker, a Ted Daisy who lived in Cincinnati, an Officer Channing at the Warwick police station who worked in traffic\u2014others didn't. There was no mention of Estella Ornato on the Internet. Honey Love Fried Chicken had once existed, but it had been replaced with a Foot Locker the year before. The White Rabbit, the Black-Footed Sioux Carpet\u2014there was no way of verifying them.\n\nSo many of the dots we had connected could not be connected here.\n\nThe only real evidence of the Neverworld's existence was time. It no longer ran in a straight line for me. Instead, now and then, it looped and lost its balance. An hour would pass in the blink of an eye. I'd sit down for a history lecture and my mind would wander so completely, the bell would ring and I'd realize in shock that everyone was packing up to leave, an entire class's worth of notes scribbled across the dry-erase board, which seconds earlier had been bare.\n\nI'd look around, wondering if the Keeper was nearby, standing in a flower bed planting tulips or atop a ladder trimming ivy, because I recognized this out-of-body interruption for what it was: aftershocks of the Neverworld, instability, just as Martha had warned. My locomotive was skidding ever so slightly along the tracks.\n\nI still thought of Jim. But he was no longer the ghost who haunted me. I saw him as a boy, beautiful and unsteady like the rest of us. I saw our time together closer to what it probably was\u2014something between the wild imagining of love and the real thing. Sometimes in that shaky in-between we found each other and it was real. Other times it trembled and broke like a wild kite with too fragile a string. If Jim hadn't died, our love would have stopped and turned off the lights like the carousel in a traveling carnival, the music, played later, not as beautiful as I'd always thought. We would be barely remembered. In twenty years, we'd find each other on Facebook or whatever came after that, and we'd marvel at how ordinary we'd become, how all the glory we swore we'd seen in each other's eyes was gone.\n\nI thought of my friends every day. Sometimes when I closed my eyes I could feel them beside me. I imagined where they were now. Because they were somewhere. And together. That I knew. I prayed that they were happy\u2014or whatever lay beyond human happiness.\n\nI think they were.\n\nMostly I thought of Martha, who she was and what she had done for me. There wasn't a moment of my life that I didn't owe to her. Sometimes it rendered me listless and sad, made me say no to the frat party, the Sunday-night pizza feast, the Spring Fling, and I'd hole up alone in my dorm, drawing or writing lyrics, left with the painful truth of it, how the people who change us are the ones we never saw clearly at all, not until they were gone.\n\nI'd remember how Jim had insisted that one day I'd think with wonder: _I was friends with Martha Zeigler. That's how big she's gonna be._\n\nHe had been right.\n\nI shouldn't have lived. It should have been Martha. I was never the good one. I saw very little as it truly was. But that was what Martha taught me. We swear we see each other, but all we are ever able to make out is a tiny porthole view of an ocean. We think we remember the past as it was, but our memories are as fantastic and flimsy as dreams. It's so easy to hate the pretty one, worship the genius, love the rock star, trust the good girl.\n\nThat's never their only story.\n\nWe are all anthologies. We are each thousands of pages long, filled with fairy tales and poetry, mysteries and tragedy, forgotten stories in the back no one will ever read.\n\nThe most we can do is hold out our hands and help each other across the unknown. For in our held hands we find pathways through the dark, across jungles and cities, bridges suspended over the deepest caverns of this world. Your friends will walk with you, holding on with all their might, even when they're no longer there.\n\n* * *\n\n\u2014\n\nTwo years after the accident, I published a dream soundtrack.\n\nIt was released by a little publishing house out of Minneapolis called Brace Yourself Books. Not even they knew what to do with it. The market for an album soundtrack for a movie that doesn't exist is a pretty small one.\n\nWhen I received the four printed albums from my publisher, though, I left school and boarded an Amtrak train bound for St. Louis. When I arrived, I took a bus to Winwood Falls, where I hiked a mile past pink brick mansions with shell fountains to a cemetery called Ardenwood. With a map I took the self-guided walking tour past mausoleums of famous writers and captains of industry, peeling off when I found a section for the new graves.\n\nWhitley was up a stone path on a hill. Standing in front of her marble headstone made me cry, because the quote that Linda had chosen was from one of Jim's best songs, \"Immortal She.\" The fact that she'd had the insight to comb Whitley's Instagram and find it meant that maybe I'd been wrong about her too. Maybe she'd understood her daughter all along.\n\nShe lives on, fireflies in my head.\n\nI will not forget her when I am dead.\n\nShe is my memory, she is my song.\n\nShe is the road when the car is long gone.\n\nShe is the pillow on my bed.\n\nShe is my words, unsaid.\n\nWhen the sun goes dark and the earth is bereft,\n\nShe'll live in the echo the silence left.\n\nI set the little album by the flowers and walked away.\n\nNext, I boarded another Amtrak, bound for New Orleans, and then a bus with broken air-conditioning to Moss Bluff, a town with Spanish moss giving every street corner the shadowed scruff of a three-day beard. I walked the eight miles to Kipling's.\n\nTo my surprise, the house was just as he'd described it: a rambling white mansion of peeling paint, with a white peacock wandering the yard. I'd always thought he exaggerated his life, but in fact, he left out all sorts of colorful details, like the green Cadillac sitting in the middle of the driveway, weeds growing through the floor like hair overtaking old men's ears.\n\nI left the album on the porch swing. When I looked back, I saw a bent-over gray-haired woman in a green housedress examining it. She looked after me, puzzled.\n\nThen Los Angeles: two days on the train, barreling past deserts and strip malls and palm trees. I took a bus to Montecito, where I walked to Cannon's house, a cream-colored Victorian. I slipped the album into the mailbox and jogged down the steps as a car alarm sounded. A man across the street stopped watering his lawn to look at me.\n\nThree days later, I arrived in Providence, Rhode Island. I had read seven mystery novels and twelve magazines, and was out of clean underwear, with a kink in my neck. I walked the final four miles feeling a strange sense of calm, arriving at Ziegler Auto Repair just after dusk.\n\nThere was no one in reception. Most of the lights were off. I stuck the album in the window next to a sign, COFFEE 99\u00a2. As I was leaving, the door to the garage opened.\n\n\"Can I help you?\"\n\nI turned. It was Martha's dad. Though I had never met him, they had the same chin, the same thick glasses. He was wearing oil-streaked coveralls, wiping his hands on a rag.\n\nI introduced myself, telling him I was an old friend of Martha's.\n\n\"Of course. Beatrice, right? That's so nice. It's not often I meet a friend of Martha's.\"\n\n\"I'm here because I made an album. Sort of. I wanted her to have it. It's a soundtrack for a movie that doesn't exist about four unlikely superheroes. They all have these hidden powers. Anyway, I wanted you to have a copy.\"\n\nI held out the album, and he took it, turning it over. He put those thick glasses away and took out reading glasses, placing them on the end of his nose.\n\n\"Ah.\" He glanced up in surprise. \"You dedicated it to Martha?\"\n\nI nodded.\n\n\" 'To Martha. Who saw me and still believed.' How about that.\" He smiled at me, pointing toward reception. \"You know, I got her posters up in the waiting room. She always had a vision of the world that lay beyond. Even when she was little. Nothing much scared Martha.\"\n\nI let him show me her things, drawings she'd made as a child, a collection of paintings featuring an owl with purple feathers, blueprints of a winged invention she'd made. He showed me the work of Martha's older sister too, a girl named Jenny who had painted incredible canvases of oceans, hiding entire ink kingdoms and words inside the waves.\n\n\"Everything is on loan to us,\" he said, wiping tears from his eyes. \"Even our children.\"\n\nHe offered me a root beer, but I refused, explaining that I had to go.\n\n\"Maybe I'll come back sometime,\" I said.\n\n\"Well, sure. You're always welcome.\"\n\nI left him staring after me, turning the album over in his hands, doubtlessly sensing there was much more I hadn't told him.\n\nThen I was on the bus, staring out the window at the darkened sky. At one point I saw a streak of orange light along the horizon, but it was only the track lights on the ceiling of the bus. The shimmering leaves of the passing branches seemed somehow electric and alive, more than usual, and though I wanted to believe it was some hidden world opening up for me again, I sat back against the seat and told myself the truth.\n\nThis time it was just the wind.\n\u2014\n\nI'd like to thank my editor, Beverly Horowitz, for shepherding me through my first adventure into the world of young adult books. From our first conversation three years ago through the many drafts, her wisdom, humor, and awake-all-night meticulousness were an education and an inspiration. I am also deeply indebted to my agent and friend, Binky Urban, for following me into uncharted territory, always providing unerring advice and insight.\n\nI am especially grateful to the many creative thinkers at Delacorte Press who worked tirelessly on this book's behalf, especially Noreen Herits, John Adamo, Colleen Fellingham, Alison Kolani, Tamar Schwartz, and Rebecca Gudelis. Thanks also to Kate Medina and the team at Penguin Random House, whose commitment to writers and readers, no matter the trend, never fails to awe.\n\nI would like to thank Felicity Blunt, Roxane Eduard, and Mairi Friesen-Escandell for introducing this book to readers abroad; Ron Bernstein for his film rights expertise; Brenda Cronin, Seth Rabinowitz, and Nicole Caruso, confidants and sounding boards; and Anne Pessl, first-draft champion, seer of all blind spots, and wonder-mom.\n\nMost especially I wish to thank to my three Fates, David, Winter, and Avalon, whose vision of the world and reverence for all stories, great and small, are my daily joy.\n\nFinally, I would like to thank every young reader who has ever approached me at a bookstore. It was you who whose passion for characters who empower and overcome inspired me to write this story.\n\n**MARISHA PESSL** is the author of _Night Film_ and _Special Topics in Calamity Physics,_ her bestselling debut, which was awarded the Center for Fiction's First Novel Prize and selected as one of the Ten Best Books of the Year by the _New York Times Book Review._ She lives with her husband and two children in New York City. Visit Marisha online at MarishaPessl.com and on Facebook, and follow @marishapessl on Twitter and Instagram.\n\n# _What's next on  \nyour reading list?_\n\n[Discover your next  \ngreat read!](http:\/\/links.penguinrandomhouse.com\/type\/prhebooklanding\/isbn\/9780399553950\/display\/1)\n\n* * *\n\nGet personalized book picks and up-to-date news about this author.\n\nSign up now.\n  1. Cover\n  2. Title Page\n  3. Copyright\n  4. Contents\n  5. Dedication\n  6. Epigraph\n  7. Part 1\n    1. Chapter 1\n    2. Chapter 2\n    3. Chapter 3\n    4. Chapter 4\n    5. Chapter 5\n    6. Chapter 6\n    7. Chapter 7\n    8. Chapter 8\n    9. Chapter 9\n  8. Part 2\n    1. Chapter 10\n    2. Chapter 11\n    3. Chapter 12\n    4. Chapter 13\n    5. Chapter 14\n    6. Chapter 15\n    7. Chapter 16\n    8. Chapter 17\n    9. Chapter 18\n  9. Part 3\n    1. Chapter 19\n    2. Chapter 20\n    3. Chapter 21\n    4. Chapter 22\n    5. Chapter 23\n    6. Chapter 24\n    7. Chapter 25\n    8. Chapter 26\n    9. Chapter 27\n  10. Acknowledgments\n  11. About the Author\n\n  1. Cover\n  2. Cover\n  3. Title Page\n  4. Contents\n  5. Start\n\n  1. iii\n  2. iv\n  3. v\n  4. vii\n  5. \n  6. \n  7. \n  8. \n  9. \n  10. \n  11. \n  12. \n  13. \n  14. \n  15. \n  16. \n  17. \n  18. \n  19. \n  20. \n  21. \n  22. \n  23. \n  24. \n  25. \n  26. \n  27. \n  28. \n  29. \n  30. \n  31. \n  32. \n  33. \n  34. \n  35. \n  36. \n  37. \n  38. \n  39. \n  40. \n  41. \n  42. \n  43. \n  44. \n  45. \n  46. \n  47. \n  48. \n  49. \n  50. \n  51. \n  52. \n  53. \n  54. \n  55. \n  56. \n  57. \n  58. \n  59. \n  60. \n  61. \n  62. \n  63. \n  64. \n  65. \n  66. \n  67. \n  68. \n  69. \n  70. \n  71. \n  72. \n  73. \n  74. \n  75. \n  76. \n  77. \n  78. \n  79. \n  80. \n  81. \n  82. \n  83. \n  84. \n  85. \n  86. \n  87. \n  88. \n  89. \n  90. \n  91. \n  92. \n  93. \n  94. \n  95. \n  96. \n  97. \n  98. \n  99. \n  100. \n  101. \n  102. \n  103. \n  104. \n  105. \n  106. \n  107. \n  108. \n  109. \n  110. \n  111. \n  112. \n  113. \n  114. \n  115. \n  116. \n  117. \n  118. \n  119. \n  120. \n  121. \n  122. \n  123. \n  124. \n  125. \n  126. \n  127. \n  128. \n  129. \n  130. \n  131. \n  132. \n  133. \n  134. \n  135. \n  136. \n  137. \n  138. \n  139. \n  140. \n  141. \n  142. \n  143. \n  144. \n  145. \n  146. \n  147. \n  148. \n  149. \n  150. \n  151. \n  152. \n  153. \n  154. \n  155. \n  156. \n  157. \n  158. \n  159. \n  160. \n  161. \n  162. \n  163. \n  164. \n  165. \n  166. \n  167. \n  168. \n  169. \n  170. \n  171. \n  172. \n  173. \n  174. \n  175. \n  176. \n  177. \n  178. \n  179. \n  180. \n  181. \n  182. \n  183. \n  184. \n  185. \n  186. \n  187. \n  188. \n  189. \n  190. \n  191. \n  192. \n  193. \n  194. \n  195. \n  196. \n  197. \n  198. \n  199. \n  200. \n  201. \n  202. \n  203. \n  204. \n  205. \n  206. \n  207. \n  208. \n  209. \n  210. \n  211. \n  212. \n  213. \n  214. \n  215. \n  216. \n  217. \n  218. \n  219. \n  220. \n  221. \n  222. \n  223. \n  224. \n  225. \n  226. \n  227. \n  228. \n  229. \n  230. \n  231. \n  232. \n  233. \n  234. \n  235. \n  236. \n  237. \n  238. \n  239. \n  240. \n  241. \n  242. \n  243. \n  244. \n  245. \n  246. \n  247. \n  248. \n  249. \n  250. \n  251. \n  252. \n  253. \n  254. \n  255. \n  256. \n  257. \n  258. \n  259. \n  260. \n  261. \n  262. \n  263. \n  264. \n  265. \n  266. \n  267. \n  268. \n  269. \n  270. \n  271. \n  272. \n  273. \n  274. \n  275. \n  276. \n  277. \n  278. \n  279. \n  280. \n  281. \n  282. \n  283. \n  284. \n  285. \n  286. \n  287. \n  288. \n  289. \n  290. \n  291. \n  292. \n  293. \n  294. \n  295. \n  296. \n  297. \n  298. \n  299. \n  300. \n  301. \n  302. \n  303. \n  304. \n  305. \n  306. \n  307. \n  308. \n  309. \n  310. \n  311. \n  312. \n  313. \n  314. \n  315. \n  316. \n  317. \n  318. \n  319. \n  320. \n  321. \n  322. \n  323. \n  324. \n  325. \n  326. \n  327. \n  328.\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\n\n\nProduced by David Widger\n\n\n\n\n\n          A CONNECTICUT YANKEE IN KING ARTHUR'S COURT\n\n                             by\n\n                         MARK TWAIN\n                     (Samuel L. Clemens)\n\n                           Part 3.\n\n\n\n\nCHAPTER XII\n\nSLOW TORTURE\n\nStraight off, we were in the country.  It was most lovely and\npleasant in those sylvan solitudes in the early cool morning\nin the first freshness of autumn.  From hilltops we saw fair\ngreen valleys lying spread out below, with streams winding through\nthem, and island groves of trees here and there, and huge lonely\noaks scattered about and casting black blots of shade; and beyond\nthe valleys we saw the ranges of hills, blue with haze, stretching\naway in billowy perspective to the horizon, with at wide intervals\na dim fleck of white or gray on a wave-summit, which we knew was\na castle.  We crossed broad natural lawns sparkling with dew,\nand we moved like spirits, the cushioned turf giving out no sound\nof footfall; we dreamed along through glades in a mist of green\nlight that got its tint from the sun-drenched roof of leaves\noverhead, and by our feet the clearest and coldest of runlets\nwent frisking and gossiping over its reefs and making a sort of\nwhispering music, comfortable to hear; and at times we left the\nworld behind and entered into the solemn great deeps and rich\ngloom of the forest, where furtive wild things whisked and scurried\nby and were gone before you could even get your eye on the place\nwhere the noise was; and where only the earliest birds were turning\nout and getting to business with a song here and a quarrel yonder\nand a mysterious far-off hammering and drumming for worms on\na tree trunk away somewhere in the impenetrable remotenesses of\nthe woods.  And by and by out we would swing again into the glare.\n\nAbout the third or fourth or fifth time that we swung out into\nthe glare--it was along there somewhere, a couple of hours or so\nafter sun-up--it wasn't as pleasant as it had been.  It was\nbeginning to get hot.  This was quite noticeable.  We had a very\nlong pull, after that, without any shade.  Now it is curious how\nprogressively little frets grow and multiply after they once get\na start.  Things which I didn't mind at all, at first, I began\nto mind now--and more and more, too, all the time.  The first\nten or fifteen times I wanted my handkerchief I didn't seem to care;\nI got along, and said never mind, it isn't any matter, and dropped\nit out of my mind.  But now it was different; I wanted it all\nthe time; it was nag, nag, nag, right along, and no rest; I couldn't\nget it out of my mind; and so at last I lost my temper and said\nhang a man that would make a suit of armor without any pockets\nin it.  You see I had my handkerchief in my helmet; and some other\nthings; but it was that kind of a helmet that you can't take off\nby yourself.  That hadn't occurred to me when I put it there;\nand in fact I didn't know it.  I supposed it would be particularly\nconvenient there.  And so now, the thought of its being there,\nso handy and close by, and yet not get-at-able, made it all the\nworse and the harder to bear.  Yes, the thing that you can't get\nis the thing that you want, mainly; every one has noticed that.\nWell, it took my mind off from everything else; took it clear off,\nand centered it in my helmet; and mile after mile, there it stayed,\nimagining the handkerchief, picturing the handkerchief; and it\nwas bitter and aggravating to have the salt sweat keep trickling\ndown into my eyes, and I couldn't get at it.  It seems like a little\nthing, on paper, but it was not a little thing at all; it was\nthe most real kind of misery.  I would not say it if it was not so.\nI made up my mind that I would carry along a reticule next time,\nlet it look how it might, and people say what they would.  Of course\nthese iron dudes of the Round Table would think it was scandalous,\nand maybe raise Sheol about it, but as for me, give me comfort\nfirst, and style afterwards.  So we jogged along, and now and then\nwe struck a stretch of dust, and it would tumble up in clouds and\nget into my nose and make me sneeze and cry; and of course I said\nthings I oughtn't to have said, I don't deny that.  I am not\nbetter than others.\n\nWe couldn't seem to meet anybody in this lonesome Britain, not\neven an ogre; and, in the mood I was in then, it was well for\nthe ogre; that is, an ogre with a handkerchief.  Most knights\nwould have thought of nothing but getting his armor; but so I got\nhis bandanna, he could keep his hardware, for all of me.\n\nMeantime, it was getting hotter and hotter in there.  You see,\nthe sun was beating down and warming up the iron more and more\nall the time.  Well, when you are hot, that way, every little thing\nirritates you.  When I trotted, I rattled like a crate of dishes,\nand that annoyed me; and moreover I couldn't seem to stand that\nshield slatting and banging, now about my breast, now around my\nback; and if I dropped into a walk my joints creaked and screeched\nin that wearisome way that a wheelbarrow does, and as we didn't\ncreate any breeze at that gait, I was like to get fried in that\nstove; and besides, the quieter you went the heavier the iron\nsettled down on you and the more and more tons you seemed to weigh\nevery minute.  And you had to be always changing hands, and passing\nyour spear over to the other foot, it got so irksome for one hand\nto hold it long at a time.\n\nWell, you know, when you perspire that way, in rivers, there comes\na time when you--when you--well, when you itch.  You are inside,\nyour hands are outside; so there you are; nothing but iron between.\nIt is not a light thing, let it sound as it may.  First it is one\nplace; then another; then some more; and it goes on spreading and\nspreading, and at last the territory is all occupied, and nobody\ncan imagine what you feel like, nor how unpleasant it is.  And\nwhen it had got to the worst, and it seemed to me that I could\nnot stand anything more, a fly got in through the bars and settled\non my nose, and the bars were stuck and wouldn't work, and I\ncouldn't get the visor up; and I could only shake my head, which\nwas baking hot by this time, and the fly--well, you know how a fly\nacts when he has got a certainty--he only minded the shaking enough\nto change from nose to lip, and lip to ear, and buzz and buzz\nall around in there, and keep on lighting and biting, in a way\nthat a person, already so distressed as I was, simply could not\nstand.  So I gave in, and got Alisande to unship the helmet and\nrelieve me of it.  Then she emptied the conveniences out of it\nand fetched it full of water, and I drank and then stood up, and\nshe poured the rest down inside the armor. One cannot think how\nrefreshing it was.  She continued to fetch and pour until I was\nwell soaked and thoroughly comfortable.\n\nIt was good to have a rest--and peace.  But nothing is quite\nperfect in this life, at any time.  I had made a pipe a while back,\nand also some pretty fair tobacco; not the real thing, but what\nsome of the Indians use: the inside bark of the willow, dried.\nThese comforts had been in the helmet, and now I had them again,\nbut no matches.\n\nGradually, as the time wore along, one annoying fact was borne in\nupon my understanding--that we were weather-bound.  An armed novice\ncannot mount his horse without help and plenty of it.  Sandy was\nnot enough; not enough for me, anyway.  We had to wait until\nsomebody should come along.  Waiting, in silence, would have been\nagreeable enough, for I was full of matter for reflection, and\nwanted to give it a chance to work.  I wanted to try and think out\nhow it was that rational or even half-rational men could ever\nhave learned to wear armor, considering its inconveniences; and\nhow they had managed to keep up such a fashion for generations\nwhen it was plain that what I had suffered to-day they had had\nto suffer all the days of their lives.  I wanted to think that out;\nand moreover I wanted to think out some way to reform this evil\nand persuade the people to let the foolish fashion die out; but\nthinking was out of the question in the circumstances.  You couldn't\nthink, where Sandy was.\n\nShe was a quite biddable creature and good-hearted, but she had\na flow of talk that was as steady as a mill, and made your head\nsore like the drays and wagons in a city.  If she had had a cork\nshe would have been a comfort.  But you can't cork that kind;\nthey would die.  Her clack was going all day, and you would think\nsomething would surely happen to her works, by and by; but no,\nthey never got out of order; and she never had to slack up for\nwords.  She could grind, and pump, and churn, and buzz by the week,\nand never stop to oil up or blow out.  And yet the result was just\nnothing but wind.  She never had any ideas, any more than a fog\nhas.  She was a perfect blatherskite; I mean for jaw, jaw, jaw,\ntalk, talk, talk, jabber, jabber, jabber; but just as good as she\ncould be.  I hadn't minded her mill that morning, on account of\nhaving that hornets' nest of other troubles; but more than once\nin the afternoon I had to say:\n\n\"Take a rest, child; the way you are using up all the domestic air,\nthe kingdom will have to go to importing it by to-morrow, and it's\na low enough treasury without that.\"\n\n\n\nCHAPTER XIII\n\nFREEMEN\n\nYes, it is strange how little a while at a time a person can be\ncontented.  Only a little while back, when I was riding and\nsuffering, what a heaven this peace, this rest, this sweet serenity\nin this secluded shady nook by this purling stream would have\nseemed, where I could keep perfectly comfortable all the time\nby pouring a dipper of water into my armor now and then; yet\nalready I was getting dissatisfied; partly because I could not\nlight my pipe--for, although I had long ago started a match factory,\nI had forgotten to bring matches with me--and partly because we\nhad nothing to eat.  Here was another illustration of the childlike\nimprovidence of this age and people.  A man in armor always trusted\nto chance for his food on a journey, and would have been scandalized\nat the idea of hanging a basket of sandwiches on his spear.  There\nwas probably not a knight of all the Round Table combination who\nwould not rather have died than been caught carrying such a thing\nas that on his flagstaff.  And yet there could not be anything more\nsensible.  It had been my intention to smuggle a couple of sandwiches\ninto my helmet, but I was interrupted in the act, and had to make\nan excuse and lay them aside, and a dog got them.\n\nNight approached, and with it a storm.  The darkness came on fast.\nWe must camp, of course.  I found a good shelter for the demoiselle\nunder a rock, and went off and found another for myself.  But\nI was obliged to remain in my armor, because I could not get it off\nby myself and yet could not allow Alisande to help, because it\nwould have seemed so like undressing before folk.  It would not\nhave amounted to that in reality, because I had clothes on\nunderneath; but the prejudices of one's breeding are not gotten\nrid of just at a jump, and I knew that when it came to stripping\noff that bob-tailed iron petticoat I should be embarrassed.\n\nWith the storm came a change of weather; and the stronger the wind\nblew, and the wilder the rain lashed around, the colder and colder\nit got.  Pretty soon, various kinds of bugs and ants and worms\nand things began to flock in out of the wet and crawl down inside\nmy armor to get warm; and while some of them behaved well enough,\nand snuggled up amongst my clothes and got quiet, the majority\nwere of a restless, uncomfortable sort, and never stayed still,\nbut went on prowling and hunting for they did not know what;\nespecially the ants, which went tickling along in wearisome\nprocession from one end of me to the other by the hour, and are\na kind of creatures which I never wish to sleep with again.\nIt would be my advice to persons situated in this way, to not roll\nor thrash around, because this excites the interest of all the\ndifferent sorts of animals and makes every last one of them want\nto turn out and see what is going on, and this makes things worse\nthan they were before, and of course makes you objurgate harder,\ntoo, if you can.  Still, if one did not roll and thrash around\nhe would die; so perhaps it is as well to do one way as the other;\nthere is no real choice.  Even after I was frozen solid I could\nstill distinguish that tickling, just as a corpse does when he is\ntaking electric treatment.  I said I would never wear armor\nafter this trip.\n\nAll those trying hours whilst I was frozen and yet was in a living\nfire, as you may say, on account of that swarm of crawlers, that\nsame unanswerable question kept circling and circling through my\ntired head: How do people stand this miserable armor?  How have\nthey managed to stand it all these generations?  How can they sleep\nat night for dreading the tortures of next day?\n\nWhen the morning came at last, I was in a bad enough plight: seedy,\ndrowsy, fagged, from want of sleep; weary from thrashing around,\nfamished from long fasting; pining for a bath, and to get rid of\nthe animals; and crippled with rheumatism.  And how had it fared\nwith the nobly born, the titled aristocrat, the Demoiselle Alisande\nla Carteloise?  Why, she was as fresh as a squirrel; she had slept\nlike the dead; and as for a bath, probably neither she nor any\nother noble in the land had ever had one, and so she was not\nmissing it.  Measured by modern standards, they were merely modified\nsavages, those people.  This noble lady showed no impatience to get\nto breakfast--and that smacks of the savage, too.  On their journeys\nthose Britons were used to long fasts, and knew how to bear them;\nand also how to freight up against probable fasts before starting,\nafter the style of the Indian and the anaconda.  As like as not,\nSandy was loaded for a three-day stretch.\n\nWe were off before sunrise, Sandy riding and I limping along\nbehind.  In half an hour we came upon a group of ragged poor\ncreatures who had assembled to mend the thing which was regarded\nas a road.  They were as humble as animals to me; and when I\nproposed to breakfast with them, they were so flattered, so\noverwhelmed by this extraordinary condescension of mine that\nat first they were not able to believe that I was in earnest.\nMy lady put up her scornful lip and withdrew to one side; she said\nin their hearing that she would as soon think of eating with the\nother cattle--a remark which embarrassed these poor devils merely\nbecause it referred to them, and not because it insulted or offended\nthem, for it didn't.  And yet they were not slaves, not chattels.\nBy a sarcasm of law and phrase they were freemen.  Seven-tenths\nof the free population of the country were of just their class and\ndegree: small \"independent\" farmers, artisans, etc.; which is\nto say, they were the nation, the actual Nation; they were about\nall of it that was useful, or worth saving, or really respect-worthy,\nand to subtract them would have been to subtract the Nation and\nleave behind some dregs, some refuse, in the shape of a king,\nnobility and gentry, idle, unproductive, acquainted mainly with\nthe arts of wasting and destroying, and of no sort of use or value\nin any rationally constructed world.  And yet, by ingenious\ncontrivance, this gilded minority, instead of being in the tail\nof the procession where it belonged, was marching head up and\nbanners flying, at the other end of it; had elected itself to be\nthe Nation, and these innumerable clams had permitted it so long\nthat they had come at last to accept it as a truth; and not only\nthat, but to believe it right and as it should be.  The priests\nhad told their fathers and themselves that this ironical state\nof things was ordained of God; and so, not reflecting upon how\nunlike God it would be to amuse himself with sarcasms, and especially\nsuch poor transparent ones as this, they had dropped the matter\nthere and become respectfully quiet.\n\nThe talk of these meek people had a strange enough sound in\na formerly American ear.  They were freemen, but they could not\nleave the estates of their lord or their bishop without his\npermission; they could not prepare their own bread, but must have\ntheir corn ground and their bread baked at his mill and his bakery,\nand pay roundly for the same; they could not sell a piece of their\nown property without paying him a handsome percentage of the\nproceeds, nor buy a piece of somebody else's without remembering\nhim in cash for the privilege; they had to harvest his grain for him\ngratis, and be ready to come at a moment's notice, leaving their\nown crop to destruction by the threatened storm; they had to let\nhim plant fruit trees in their fields, and then keep their indignation\nto themselves when his heedless fruit-gatherers trampled the grain\naround the trees; they had to smother their anger when his hunting\nparties galloped through their fields laying waste the result of\ntheir patient toil; they were not allowed to keep doves themselves,\nand when the swarms from my lord's dovecote settled on their crops\nthey must not lose their temper and kill a bird, for awful would\nthe penalty be; when the harvest was at last gathered, then came\nthe procession of robbers to levy their blackmail upon it: first\nthe Church carted off its fat tenth, then the king's commissioner\ntook his twentieth, then my lord's people made a mighty inroad\nupon the remainder; after which, the skinned freeman had liberty\nto bestow the remnant in his barn, in case it was worth the trouble;\nthere were taxes, and taxes, and taxes, and more taxes, and taxes\nagain, and yet other taxes--upon this free and independent pauper,\nbut none upon his lord the baron or the bishop, none upon the\nwasteful nobility or the all-devouring Church; if the baron would\nsleep unvexed, the freeman must sit up all night after his day's\nwork and whip the ponds to keep the frogs quiet; if the freeman's\ndaughter--but no, that last infamy of monarchical government is\nunprintable; and finally, if the freeman, grown desperate with his\ntortures, found his life unendurable under such conditions, and\nsacrificed it and fled to death for mercy and refuge, the gentle\nChurch condemned him to eternal fire, the gentle law buried him\nat midnight at the cross-roads with a stake through his back,\nand his master the baron or the bishop confiscated all his property\nand turned his widow and his orphans out of doors.\n\nAnd here were these freemen assembled in the early morning to work\non their lord the bishop's road three days each--gratis; every\nhead of a family, and every son of a family, three days each,\ngratis, and a day or so added for their servants.  Why, it was\nlike reading about France and the French, before the ever memorable\nand blessed Revolution, which swept a thousand years of such\nvillany away in one swift tidal-wave of blood--one: a settlement\nof that hoary debt in the proportion of half a drop of blood for\neach hogshead of it that had been pressed by slow tortures out of\nthat people in the weary stretch of ten centuries of wrong and\nshame and misery the like of which was not to be mated but in hell.\nThere were two \"Reigns of Terror,\" if we would but remember it\nand consider it; the one wrought murder in hot passion, the other\nin heartless cold blood; the one lasted mere months, the other had\nlasted a thousand years; the one inflicted death upon ten thousand\npersons, the other upon a hundred millions; but our shudders are\nall for the \"horrors\" of the minor Terror, the momentary Terror,\nso to speak; whereas, what is the horror of swift death by the axe,\ncompared with lifelong death from hunger, cold, insult, cruelty,\nand heart-break?  What is swift death by lightning compared with\ndeath by slow fire at the stake?  A city cemetery could contain the\ncoffins filled by that brief Terror which we have all been so\ndiligently taught to shiver at and mourn over; but all France could\nhardly contain the coffins filled by that older and real Terror\n--that unspeakably bitter and awful Terror which none of us has\nbeen taught to see in its vastness or pity as it deserves.\n\nThese poor ostensible freemen who were sharing their breakfast\nand their talk with me, were as full of humble reverence for their\nking and Church and nobility as their worst enemy could desire.\nThere was something pitifully ludicrous about it.  I asked them\nif they supposed a nation of people ever existed, who, with a free\nvote in every man's hand, would elect that a single family and its\ndescendants should reign over it forever, whether gifted or boobies,\nto the exclusion of all other families--including the voter's; and\nwould also elect that a certain hundred families should be raised\nto dizzy summits of rank, and clothed on with offensive transmissible\nglories and privileges to the exclusion of the rest of the nation's\nfamilies--_including his own_.\n\nThey all looked unhit, and said they didn't know; that they had\nnever thought about it before, and it hadn't ever occurred to them\nthat a nation could be so situated that every man _could_ have\na say in the government.  I said I had seen one--and that it would\nlast until it had an Established Church.  Again they were all\nunhit--at first.  But presently one man looked up and asked me\nto state that proposition again; and state it slowly, so it could\nsoak into his understanding.  I did it; and after a little he had\nthe idea, and he brought his fist down and said _he_ didn't believe\na nation where every man had a vote would voluntarily get down\nin the mud and dirt in any such way; and that to steal from a nation\nits will and preference must be a crime and the first of all crimes.\nI said to myself:\n\n\"This one's a man.  If I were backed by enough of his sort, I would\nmake a strike for the welfare of this country, and try to prove\nmyself its loyalest citizen by making a wholesome change in its\nsystem of government.\"\n\nYou see my kind of loyalty was loyalty to one's country, not to\nits institutions or its office-holders.  The country is the real\nthing, the substantial thing, the eternal thing; it is the thing\nto watch over, and care for, and be loyal to; institutions are\nextraneous, they are its mere clothing, and clothing can wear out,\nbecome ragged, cease to be comfortable, cease to protect the body\nfrom winter, disease, and death.  To be loyal to rags, to shout\nfor rags, to worship rags, to die for rags--that is a loyalty\nof unreason, it is pure animal; it belongs to monarchy, was invented\nby monarchy; let monarchy keep it.  I was from Connecticut, whose\nConstitution declares \"that all political power is inherent in\nthe people, and all free governments are founded on their authority\nand instituted for their benefit; and that they have _at all times_\nan undeniable and indefeasible right to _alter their form of\ngovernment_ in such a manner as they may think expedient.\"\n\nUnder that gospel, the citizen who thinks he sees that the\ncommonwealth's political clothes are worn out, and yet holds his\npeace and does not agitate for a new suit, is disloyal; he is\na traitor.  That he may be the only one who thinks he sees this\ndecay, does not excuse him; it is his duty to agitate anyway, and\nit is the duty of the others to vote him down if they do not see\nthe matter as he does.\n\nAnd now here I was, in a country where a right to say how the\ncountry should be governed was restricted to six persons in each\nthousand of its population.  For the nine hundred and ninety-four\nto express dissatisfaction with the regnant system and propose\nto change it, would have made the whole six shudder as one man,\nit would have been so disloyal, so dishonorable, such putrid black\ntreason.  So to speak, I was become a stockholder in a corporation\nwhere nine hundred and ninety-four of the members furnished all\nthe money and did all the work, and the other six elected themselves\na permanent board of direction and took all the dividends.  It seemed\nto me that what the nine hundred and ninety-four dupes needed was\na new deal.  The thing that would have best suited the circus side\nof my nature would have been to resign the Boss-ship and get up\nan insurrection and turn it into a revolution; but I knew that the\nJack Cade or the Wat Tyler who tries such a thing without first\neducating his materials up to revolution grade is almost absolutely\ncertain to get left.  I had never been accustomed to getting left,\neven if I do say it myself.  Wherefore, the \"deal\" which had been\nfor some time working into shape in my mind was of a quite different\npattern from the Cade-Tyler sort.\n\nSo I did not talk blood and insurrection to that man there who sat\nmunching black bread with that abused and mistaught herd of human\nsheep, but took him aside and talked matter of another sort to him.\nAfter I had finished, I got him to lend me a little ink from his\nveins; and with this and a sliver I wrote on a piece of bark--\n\n   Put him in the Man-factory--\n\nand gave it to him, and said:\n\n\"Take it to the palace at Camelot and give it into the hands of\nAmyas le Poulet, whom I call Clarence, and he will understand.\"\n\n\"He is a priest, then,\" said the man, and some of the enthusiasm\nwent out of his face.\n\n\"How--a priest?  Didn't I tell you that no chattel of the Church,\nno bond-slave of pope or bishop can enter my Man-Factory?  Didn't\nI tell you that _you_ couldn't enter unless your religion, whatever\nit might be, was your own free property?\"\n\n\"Marry, it is so, and for that I was glad; wherefore it liked me not,\nand bred in me a cold doubt, to hear of this priest being there.\"\n\n\"But he isn't a priest, I tell you.\"\n\nThe man looked far from satisfied.  He said:\n\n\"He is not a priest, and yet can read?\"\n\n\"He is not a priest and yet can read--yes, and write, too, for that\nmatter.  I taught him myself.\" The man's face cleared.  \"And it is\nthe first thing that you yourself will be taught in that Factory--\"\n\n\"I?  I would give blood out of my heart to know that art.  Why,\nI will be your slave, your--\"\n\n\"No you won't, you won't be anybody's slave.  Take your family\nand go along.  Your lord the bishop will confiscate your small\nproperty, but no matter.  Clarence will fix you all right.\"\n\n\n\nCHAPTER XIV\n\n\"DEFEND THEE, LORD\"\n\nI paid three pennies for my breakfast, and a most extravagant\nprice it was, too, seeing that one could have breakfasted a dozen\npersons for that money; but I was feeling good by this time, and\nI had always been a kind of spendthrift anyway; and then these\npeople had wanted to give me the food for nothing, scant as\ntheir provision was, and so it was a grateful pleasure to emphasize\nmy appreciation and sincere thankfulness with a good big financial\nlift where the money would do so much more good than it would\nin my helmet, where, these pennies being made of iron and not\nstinted in weight, my half-dollar's worth was a good deal of a\nburden to me.  I spent money rather too freely in those days,\nit is true; but one reason for it was that I hadn't got the\nproportions of things entirely adjusted, even yet, after so long\na sojourn in Britain--hadn't got along to where I was able to\nabsolutely realize that a penny in Arthur's land and a couple of\ndollars in Connecticut were about one and the same thing: just\ntwins, as you may say, in purchasing power.  If my start from\nCamelot could have been delayed a very few days I could have paid\nthese people in beautiful new coins from our own mint, and that\nwould have pleased me; and them, too, not less.  I had adopted\nthe American values exclusively.  In a week or two now, cents,\nnickels, dimes, quarters, and half-dollars, and also a trifle of\ngold, would be trickling in thin but steady streams all through\nthe commercial veins of the kingdom, and I looked to see this\nnew blood freshen up its life.\n\nThe farmers were bound to throw in something, to sort of offset\nmy liberality, whether I would or no; so I let them give me a flint\nand steel; and as soon as they had comfortably bestowed Sandy\nand me on our horse, I lit my pipe.  When the first blast of smoke\nshot out through the bars of my helmet, all those people broke\nfor the woods, and Sandy went over backwards and struck the ground\nwith a dull thud.  They thought I was one of those fire-belching\ndragons they had heard so much about from knights and other\nprofessional liars.  I had infinite trouble to persuade those people\nto venture back within explaining distance.  Then I told them that\nthis was only a bit of enchantment which would work harm to none\nbut my enemies.  And I promised, with my hand on my heart, that\nif all who felt no enmity toward me would come forward and pass\nbefore me they should see that only those who remained behind would\nbe struck dead.  The procession moved with a good deal of promptness.\nThere were no casualties to report, for nobody had curiosity enough\nto remain behind to see what would happen.\n\nI lost some time, now, for these big children, their fears gone,\nbecame so ravished with wonder over my awe-compelling fireworks\nthat I had to stay there and smoke a couple of pipes out before\nthey would let me go.  Still the delay was not wholly unproductive,\nfor it took all that time to get Sandy thoroughly wonted to the new\nthing, she being so close to it, you know.  It plugged up her\nconversation mill, too, for a considerable while, and that was\na gain.  But above all other benefits accruing, I had learned\nsomething.  I was ready for any giant or any ogre that might come\nalong, now.\n\nWe tarried with a holy hermit, that night, and my opportunity\ncame about the middle of the next afternoon.  We were crossing\na vast meadow by way of short-cut, and I was musing absently,\nhearing nothing, seeing nothing, when Sandy suddenly interrupted\na remark which she had begun that morning, with the cry:\n\n\"Defend thee, lord!--peril of life is toward!\"\n\nAnd she slipped down from the horse and ran a little way and stood.\nI looked up and saw, far off in the shade of a tree, half a dozen\narmed knights and their squires; and straightway there was bustle\namong them and tightening of saddle-girths for the mount.  My pipe\nwas ready and would have been lit, if I had not been lost in\nthinking about how to banish oppression from this land and restore\nto all its people their stolen rights and manhood without disobliging\nanybody.  I lit up at once, and by the time I had got a good head\nof reserved steam on, here they came.  All together, too; none of\nthose chivalrous magnanimities which one reads so much about\n--one courtly rascal at a time, and the rest standing by to see fair\nplay.  No, they came in a body, they came with a whirr and a rush,\nthey came like a volley from a battery; came with heads low down,\nplumes streaming out behind, lances advanced at a level.  It was\na handsome sight, a beautiful sight--for a man up a tree.  I laid\nmy lance in rest and waited, with my heart beating, till the iron\nwave was just ready to break over me, then spouted a column of\nwhite smoke through the bars of my helmet.  You should have seen\nthe wave go to pieces and scatter!  This was a finer sight than\nthe other one.\n\nBut these people stopped, two or three hundred yards away, and\nthis troubled me.  My satisfaction collapsed, and fear came;\nI judged I was a lost man.  But Sandy was radiant; and was going\nto be eloquent--but I stopped her, and told her my magic had\nmiscarried, somehow or other, and she must mount, with all despatch,\nand we must ride for life.  No, she wouldn't.  She said that my\nenchantment had disabled those knights; they were not riding on,\nbecause they couldn't; wait, they would drop out of their saddles\npresently, and we would get their horses and harness.  I could not\ndeceive such trusting simplicity, so I said it was a mistake; that\nwhen my fireworks killed at all, they killed instantly; no, the men\nwould not die, there was something wrong about my apparatus,\nI couldn't tell what; but we must hurry and get away, for those\npeople would attack us again, in a minute.  Sandy laughed, and said:\n\n\"Lack-a-day, sir, they be not of that breed!  Sir Launcelot will\ngive battle to dragons, and will abide by them, and will assail\nthem again, and yet again, and still again, until he do conquer\nand destroy them; and so likewise will Sir Pellinore and Sir Aglovale\nand Sir Carados, and mayhap others, but there be none else that\nwill venture it, let the idle say what the idle will.  And, la,\nas to yonder base rufflers, think ye they have not their fill,\nbut yet desire more?\"\n\n\"Well, then, what are they waiting for?  Why don't they leave?\nNobody's hindering.  Good land, I'm willing to let bygones be\nbygones, I'm sure.\"\n\n\"Leave, is it?  Oh, give thyself easement as to that.  They dream\nnot of it, no, not they.  They wait to yield them.\"\n\n\"Come--really, is that 'sooth'--as you people say?  If they want to,\nwhy don't they?\"\n\n\"It would like them much; but an ye wot how dragons are esteemed,\nye would not hold them blamable. They fear to come.\"\n\n\"Well, then, suppose I go to them instead, and--\"\n\n\"Ah, wit ye well they would not abide your coming.  I will go.\"\n\nAnd she did.  She was a handy person to have along on a raid.\nI would have considered this a doubtful errand, myself.  I presently\nsaw the knights riding away, and Sandy coming back.  That was\na relief.  I judged she had somehow failed to get the first innings\n--I mean in the conversation; otherwise the interview wouldn't have\nbeen so short.  But it turned out that she had managed the business\nwell; in fact, admirably.  She said that when she told those people\nI was The Boss, it hit them where they lived: \"smote them sore\nwith fear and dread\" was her word; and then they were ready to\nput up with anything she might require.  So she swore them to appear\nat Arthur's court within two days and yield them, with horse and\nharness, and be my knights henceforth, and subject to my command.\nHow much better she managed that thing than I should have done\nit myself!  She was a daisy.\n\n\n\nCHAPTER XV\n\nSANDY'S TALE\n\n\"And so I'm proprietor of some knights,\" said I, as we rode off.\n\"Who would ever have supposed that I should live to list up assets\nof that sort.  I shan't know what to do with them; unless I raffle\nthem off.  How many of them are there, Sandy?\"\n\n\"Seven, please you, sir, and their squires.\"\n\n\"It is a good haul.  Who are they?  Where do they hang out?\"\n\n\"Where do they hang out?\"\n\n\"Yes, where do they live?\"\n\n\"Ah, I understood thee not.  That will I tell eftsoons.\"  Then she\nsaid musingly, and softly, turning the words daintily over her\ntongue: \"Hang they out--hang they out--where hang--where do they\nhang out; eh, right so; where do they hang out.  Of a truth the\nphrase hath a fair and winsome grace, and is prettily worded\nwithal.  I will repeat it anon and anon in mine idlesse, whereby\nI may peradventure learn it.  Where do they hang out.  Even so!\nalready it falleth trippingly from my tongue, and forasmuch as--\"\n\n\"Don't forget the cowboys, Sandy.\"\n\n\"Cowboys?\"\n\n\"Yes; the knights, you know: You were going to tell me about them.\nA while back, you remember.  Figuratively speaking, game's called.\"\n\n\"Game--\"\n\n\"Yes, yes, yes!  Go to the bat.  I mean, get to work on your\nstatistics, and don't burn so much kindling getting your fire\nstarted.  Tell me about the knights.\"\n\n\"I will well, and lightly will begin.  So they two departed and\nrode into a great forest.  And--\"\n\n\"Great Scott!\"\n\nYou see, I recognized my mistake at once.  I had set her works\na-going; it was my own fault; she would be thirty days getting down\nto those facts.  And she generally began without a preface and\nfinished without a result.  If you interrupted her she would either\ngo right along without noticing, or answer with a couple of words,\nand go back and say the sentence over again.  So, interruptions\nonly did harm; and yet I had to interrupt, and interrupt pretty\nfrequently, too, in order to save my life; a person would die if\nhe let her monotony drip on him right along all day.\n\n\"Great Scott!\" I said in my distress.  She went right back and\nbegan over again:\n\n\"So they two departed and rode into a great forest.  And--\"\n\n\"_Which_ two?\"\n\n\"Sir Gawaine and Sir Uwaine.  And so they came to an abbey of monks,\nand there were well lodged.  So on the morn they heard their masses\nin the abbey, and so they rode forth till they came to a great\nforest; then was Sir Gawaine ware in a valley by a turret, of\ntwelve fair damsels, and two knights armed on great horses, and\nthe damsels went to and fro by a tree.  And then was Sir Gawaine\nware how there hung a white shield on that tree, and ever as the\ndamsels came by it they spit upon it, and some threw mire upon\nthe shield--\"\n\n\"Now, if I hadn't seen the like myself in this country, Sandy,\nI wouldn't believe it.  But I've seen it, and I can just see those\ncreatures now, parading before that shield and acting like that.\nThe women here do certainly act like all possessed.  Yes, and\nI mean your best, too, society's very choicest brands.  The humblest\nhello-girl along ten thousand miles of wire could teach gentleness,\npatience, modesty, manners, to the highest duchess in Arthur's land.\"\n\n\"Hello-girl?\"\n\n\"Yes, but don't you ask me to explain; it's a new kind of a girl;\nthey don't have them here; one often speaks sharply to them when\nthey are not the least in fault, and he can't get over feeling\nsorry for it and ashamed of himself in thirteen hundred years,\nit's such shabby mean conduct and so unprovoked; the fact is,\nno gentleman ever does it--though I--well, I myself, if I've got\nto confess--\"\n\n\"Peradventure she--\"\n\n\"Never mind her; never mind her; I tell you I couldn't ever explain\nher so you would understand.\"\n\n\"Even so be it, sith ye are so minded.  Then Sir Gawaine and\nSir Uwaine went and saluted them, and asked them why they did that\ndespite to the shield.  Sirs, said the damsels, we shall tell you.\nThere is a knight in this country that owneth this white shield,\nand he is a passing good man of his hands, but he hateth all\nladies and gentlewomen, and therefore we do all this despite to\nthe shield.  I will say you, said Sir Gawaine, it beseemeth evil\na good knight to despise all ladies and gentlewomen, and peradventure\nthough he hate you he hath some cause, and peradventure he loveth\nin some other places ladies and gentlewomen, and to be loved again,\nand he such a man of prowess as ye speak of--\"\n\n\"Man of prowess--yes, that is the man to please them, Sandy.\nMan of brains--that is a thing they never think of.  Tom Sayers\n--John Heenan--John L. Sullivan--pity but you could be here.  You\nwould have your legs under the Round Table and a 'Sir' in front\nof your names within the twenty-four hours; and you could bring\nabout a new distribution of the married princesses and duchesses\nof the Court in another twenty-four.  The fact is, it is just\na sort of polished-up court of Comanches, and there isn't a squaw\nin it who doesn't stand ready at the dropping of a hat to desert\nto the buck with the biggest string of scalps at his belt.\"\n\n\"--and he be such a man of prowess as ye speak of, said Sir Gawaine.\nNow, what is his name?  Sir, said they, his name is Marhaus the\nking's son of Ireland.\"\n\n\"Son of the king of Ireland, you mean; the other form doesn't mean\nanything.  And look out and hold on tight, now, we must jump\nthis gully....  There, we are all right now.  This horse belongs in\nthe circus; he is born before his time.\"\n\n\"I know him well, said Sir Uwaine, he is a passing good knight as\nany is on live.\"\n\n\"_On live_.  If you've got a fault in the world, Sandy, it is that\nyou are a shade too archaic.  But it isn't any matter.\"\n\n\"--for I saw him once proved at a justs where many knights were\ngathered, and that time there might no man withstand him.  Ah, said\nSir Gawaine, damsels, methinketh ye are to blame, for it is to\nsuppose he that hung that shield there will not be long therefrom,\nand then may those knights match him on horseback, and that is\nmore your worship than thus; for I will abide no longer to see\na knight's shield dishonored.  And therewith Sir Uwaine and\nSir Gawaine departed a little from them, and then were they ware\nwhere Sir Marhaus came riding on a great horse straight toward\nthem.  And when the twelve damsels saw Sir Marhaus they fled into\nthe turret as they were wild, so that some of them fell by the way.\nThen the one of the knights of the tower dressed his shield, and\nsaid on high, Sir Marhaus defend thee.  And so they ran together\nthat the knight brake his spear on Marhaus, and Sir Marhaus smote\nhim so hard that he brake his neck and the horse's back--\"\n\n\"Well, that is just the trouble about this state of things,\nit ruins so many horses.\"\n\n\"That saw the other knight of the turret, and dressed him toward\nMarhaus, and they went so eagerly together, that the knight of\nthe turret was soon smitten down, horse and man, stark dead--\"\n\n\"_Another_ horse gone; I tell you it is a custom that ought to be\nbroken up.  I don't see how people with any feeling can applaud\nand support it.\"\n\n    .   .   .   .\n\n\"So these two knights came together with great random--\"\n\nI saw that I had been asleep and missed a chapter, but I didn't\nsay anything.  I judged that the Irish knight was in trouble with\nthe visitors by this time, and this turned out to be the case.\n\n\"--that Sir Uwaine smote Sir Marhaus that his spear brast in pieces\non the shield, and Sir Marhaus smote him so sore that horse and\nman he bare to the earth, and hurt Sir Uwaine on the left side--\"\n\n\"The truth is, Alisande, these archaics are a little _too_ simple;\nthe vocabulary is too limited, and so, by consequence, descriptions\nsuffer in the matter of variety; they run too much to level Saharas\nof fact, and not enough to picturesque detail; this throws about\nthem a certain air of the monotonous; in fact the fights are all\nalike: a couple of people come together with great random\n--random is a good word, and so is exegesis, for that matter, and\nso is holocaust, and defalcation, and usufruct and a hundred others,\nbut land! a body ought to discriminate--they come together with\ngreat random, and a spear is brast, and one party brake his shield\nand the other one goes down, horse and man, over his horse-tail\nand brake his neck, and then the next candidate comes randoming in,\nand brast _his_ spear, and the other man brast his shield, and down\n_he_ goes, horse and man, over his horse-tail, and brake _his_ neck,\nand then there's another elected, and another and another and still\nanother, till the material is all used up; and when you come to\nfigure up results, you can't tell one fight from another, nor who\nwhipped; and as a _picture_, of living, raging, roaring battle,\nsho! why, it's pale and noiseless--just ghosts scuffling in a fog.\nDear me, what would this barren vocabulary get out of the mightiest\nspectacle?--the burning of Rome in Nero's time, for instance?\nWhy, it would merely say, 'Town burned down; no insurance; boy\nbrast a window, fireman brake his neck!'  Why, _that_ ain't a picture!\"\n\nIt was a good deal of a lecture, I thought, but it didn't disturb\nSandy, didn't turn a feather; her steam soared steadily up again,\nthe minute I took off the lid:\n\n\"Then Sir Marhaus turned his horse and rode toward Gawaine with\nhis spear.  And when Sir Gawaine saw that, he dressed his shield,\nand they aventred their spears, and they came together with all\nthe might of their horses, that either knight smote other so hard\nin the midst of their shields, but Sir Gawaine's spear brake--\"\n\n\"I knew it would.\"\n\n--\"but Sir Marhaus's spear held; and therewith Sir Gawaine and\nhis horse rushed down to the earth--\"\n\n\"Just so--and brake his back.\"\n\n--\"and lightly Sir Gawaine rose upon his feet and pulled out\nhis sword, and dressed him toward Sir Marhaus on foot, and therewith\neither came unto other eagerly, and smote together with their\nswords, that their shields flew in cantels, and they bruised their\nhelms and their hauberks, and wounded either other.  But Sir Gawaine,\nfro it passed nine of the clock, waxed by the space of three hours\never stronger and stronger and thrice his might was increased.\nAll this espied Sir Marhaus, and had great wonder how his might\nincreased, and so they wounded other passing sore; and then when\nit was come noon--\"\n\nThe pelting sing-song of it carried me forward to scenes and\nsounds of my boyhood days:\n\n\"N-e-e-ew Haven! ten minutes for refreshments--knductr'll strike\nthe gong-bell two minutes before train leaves--passengers for\nthe Shore-line please take seats in the rear k'yar, this k'yar\ndon't go no furder--_ahh_-pls, _aw_-rnjz, b'_nan_ners,\n_s-a-n-d_'ches, p--_op_-corn!\"\n\n--\"and waxed past noon and drew toward evensong.  Sir Gawaine's\nstrength feebled and waxed passing faint, that unnethes he might\ndure any longer, and Sir Marhaus was then bigger and bigger--\"\n\n\"Which strained his armor, of course; and yet little would one\nof these people mind a small thing like that.\"\n\n--\"and so, Sir Knight, said Sir Marhaus, I have well felt that\nye are a passing good knight, and a marvelous man of might as ever\nI felt any, while it lasteth, and our quarrels are not great, and\ntherefore it were a pity to do you hurt, for I feel you are passing\nfeeble.  Ah, said Sir Gawaine, gentle knight, ye say the word\nthat I should say.  And therewith they took off their helms and\neither kissed other, and there they swore together either to love\nother as brethren--\"\n\nBut I lost the thread there, and dozed off to slumber, thinking\nabout what a pity it was that men with such superb strength\n--strength enabling them to stand up cased in cruelly burdensome\niron and drenched with perspiration, and hack and batter and bang\neach other for six hours on a stretch--should not have been born\nat a time when they could put it to some useful purpose.  Take\na jackass, for instance: a jackass has that kind of strength, and\nputs it to a useful purpose, and is valuable to this world because\nhe is a jackass; but a nobleman is not valuable because he is\na jackass.  It is a mixture that is always ineffectual, and should\nnever have been attempted in the first place.  And yet, once you\nstart a mistake, the trouble is done and you never know what is\ngoing to come of it.\n\nWhen I came to myself again and began to listen, I perceived that\nI had lost another chapter, and that Alisande had wandered a long\nway off with her people.\n\n\"And so they rode and came into a deep valley full of stones,\nand thereby they saw a fair stream of water; above thereby was\nthe head of the stream, a fair fountain, and three damsels sitting\nthereby. In this country, said Sir Marhaus, came never knight\nsince it was christened, but he found strange adventures--\"\n\n\"This is not good form, Alisande.  Sir Marhaus the king's son of\nIreland talks like all the rest; you ought to give him a brogue,\nor at least a characteristic expletive; by this means one would\nrecognize him as soon as he spoke, without his ever being named.\nIt is a common literary device with the great authors.  You should\nmake him say, 'In this country, be jabers, came never knight since\nit was christened, but he found strange adventures, be jabers.'\nYou see how much better that sounds.\"\n\n--\"came never knight but he found strange adventures, be jabers.\nOf a truth it doth indeed, fair lord, albeit 'tis passing hard\nto say, though peradventure that will not tarry but better speed\nwith usage.  And then they rode to the damsels, and either saluted\nother, and the eldest had a garland of gold about her head, and\nshe was threescore winter of age or more--\"\n\n\"The _damsel_ was?\"\n\n\"Even so, dear lord--and her hair was white under the garland--\"\n\n\"Celluloid teeth, nine dollars a set, as like as not--the loose-fit\nkind, that go up and down like a portcullis when you eat, and\nfall out when you laugh.\"\n\n\"The second damsel was of thirty winter of age, with a circlet of\ngold about her head.  The third damsel was but fifteen year of age--\"\n\nBillows of thought came rolling over my soul, and the voice faded\nout of my hearing!\n\nFifteen!  Break--my heart! oh, my lost darling!  Just her age\nwho was so gentle, and lovely, and all the world to me, and whom\nI shall never see again!  How the thought of her carries me back\nover wide seas of memory to a vague dim time, a happy time, so many,\nmany centuries hence, when I used to wake in the soft summer\nmornings, out of sweet dreams of her, and say \"Hello, Central!\"\njust to hear her dear voice come melting back to me with a\n\"Hello, Hank!\" that was music of the spheres to my enchanted ear.\nShe got three dollars a week, but she was worth it.\n\nI could not follow Alisande's further explanation of who our\ncaptured knights were, now--I mean in case she should ever get\nto explaining who they were.  My interest was gone, my thoughts\nwere far away, and sad.  By fitful glimpses of the drifting tale,\ncaught here and there and now and then, I merely noted in a vague\nway that each of these three knights took one of these three damsels\nup behind him on his horse, and one rode north, another east,\nthe other south, to seek adventures, and meet again and lie, after\nyear and day.  Year and day--and without baggage.  It was of\na piece with the general simplicity of the country.\n\nThe sun was now setting.  It was about three in the afternoon when\nAlisande had begun to tell me who the cowboys were; so she had made\npretty good progress with it--for her.  She would arrive some time\nor other, no doubt, but she was not a person who could be hurried.\n\nWe were approaching a castle which stood on high ground; a huge,\nstrong, venerable structure, whose gray towers and battlements were\ncharmingly draped with ivy, and whose whole majestic mass was\ndrenched with splendors flung from the sinking sun.  It was the\nlargest castle we had seen, and so I thought it might be the one\nwe were after, but Sandy said no.  She did not know who owned it;\nshe said she had passed it without calling, when she went down\nto Camelot.\n\n\n\nCHAPTER XVI\n\nMORGAN LE FAY\n\nIf knights errant were to be believed, not all castles were desirable\nplaces to seek hospitality in.  As a matter of fact, knights errant\nwere _not_ persons to be believed--that is, measured by modern\nstandards of veracity; yet, measured by the standards of their own\ntime, and scaled accordingly, you got the truth.  It was very\nsimple: you discounted a statement ninety-seven per cent; the rest\nwas fact.  Now after making this allowance, the truth remained\nthat if I could find out something about a castle before ringing\nthe door-bell--I mean hailing the warders--it was the sensible\nthing to do.  So I was pleased when I saw in the distance a horseman\nmaking the bottom turn of the road that wound down from this castle.\n\nAs we approached each other, I saw that he wore a plumed helmet,\nand seemed to be otherwise clothed in steel, but bore a curious\naddition also--a stiff square garment like a herald's tabard.\nHowever, I had to smile at my own forgetfulness when I got nearer\nand read this sign on his tabard:\n\n  \"Persimmon's Soap -- All the Prime-Donna Use It.\"\n\nThat was a little idea of my own, and had several wholesome purposes\nin view toward the civilizing and uplifting of this nation.  In the\nfirst place, it was a furtive, underhand blow at this nonsense\nof knight errantry, though nobody suspected that but me.  I had\nstarted a number of these people out--the bravest knights I could\nget--each sandwiched between bulletin-boards bearing one device\nor another, and I judged that by and by when they got to be numerous\nenough they would begin to look ridiculous; and then, even the\nsteel-clad ass that _hadn't_ any board would himself begin to look\nridiculous because he was out of the fashion.\n\nSecondly, these missionaries would gradually, and without creating\nsuspicion or exciting alarm, introduce a rudimentary cleanliness\namong the nobility, and from them it would work down to the people,\nif the priests could be kept quiet.  This would undermine the Church.\nI mean would be a step toward that.  Next, education--next, freedom\n--and then she would begin to crumble.  It being my conviction that\nany Established Church is an established crime, an established\nslave-pen, I had no scruples, but was willing to assail it in\nany way or with any weapon that promised to hurt it.  Why, in my\nown former day--in remote centuries not yet stirring in the womb\nof time--there were old Englishmen who imagined that they had been\nborn in a free country: a \"free\" country with the Corporation Act\nand the Test still in force in it--timbers propped against men's\nliberties and dishonored consciences to shore up an Established\nAnachronism with.\n\nMy missionaries were taught to spell out the gilt signs on their\ntabards--the showy gilding was a neat idea, I could have got the\nking to wear a bulletin-board for the sake of that barbaric\nsplendor--they were to spell out these signs and then explain to\nthe lords and ladies what soap was; and if the lords and ladies\nwere afraid of it, get them to try it on a dog.  The missionary's\nnext move was to get the family together and try it on himself;\nhe was to stop at no experiment, however desperate, that could\nconvince the nobility that soap was harmless; if any final doubt\nremained, he must catch a hermit--the woods were full of them;\nsaints they called themselves, and saints they were believed to be.\nThey were unspeakably holy, and worked miracles, and everybody\nstood in awe of them.  If a hermit could survive a wash, and that\nfailed to convince a duke, give him up, let him alone.\n\nWhenever my missionaries overcame a knight errant on the road\nthey washed him, and when he got well they swore him to go and\nget a bulletin-board and disseminate soap and civilization the rest\nof his days.  As a consequence the workers in the field were\nincreasing by degrees, and the reform was steadily spreading.\nMy soap factory felt the strain early.  At first I had only two\nhands; but before I had left home I was already employing fifteen,\nand running night and day; and the atmospheric result was getting\nso pronounced that the king went sort of fainting and gasping\naround and said he did not believe he could stand it much longer,\nand Sir Launcelot got so that he did hardly anything but walk up\nand down the roof and swear, although I told him it was worse up\nthere than anywhere else, but he said he wanted plenty of air; and\nhe was always complaining that a palace was no place for a soap\nfactory anyway, and said if a man was to start one in his house\nhe would be damned if he wouldn't strangle him.  There were ladies\npresent, too, but much these people ever cared for that; they would\nswear before children, if the wind was their way when the factory\nwas going.\n\nThis missionary knight's name was La Cote Male Taile, and he said\nthat this castle was the abode of Morgan le Fay, sister of\nKing Arthur, and wife of King Uriens, monarch of a realm about\nas big as the District of Columbia--you could stand in the middle\nof it and throw bricks into the next kingdom.  \"Kings\" and \"Kingdoms\"\nwere as thick in Britain as they had been in little Palestine in\nJoshua's time, when people had to sleep with their knees pulled up\nbecause they couldn't stretch out without a passport.\n\nLa Cote was much depressed, for he had scored here the worst\nfailure of his campaign.  He had not worked off a cake; yet he had\ntried all the tricks of the trade, even to the washing of a hermit;\nbut the hermit died.  This was, indeed, a bad failure, for this\nanimal would now be dubbed a martyr, and would take his place\namong the saints of the Roman calendar.  Thus made he his moan,\nthis poor Sir La Cote Male Taile, and sorrowed passing sore.  And\nso my heart bled for him, and I was moved to comfort and stay him.\nWherefore I said:\n\n\"Forbear to grieve, fair knight, for this is not a defeat.  We have\nbrains, you and I; and for such as have brains there are no defeats,\nbut only victories.  Observe how we will turn this seeming disaster\ninto an advertisement; an advertisement for our soap; and the\nbiggest one, to draw, that was ever thought of; an advertisement\nthat will transform that Mount Washington defeat into a Matterhorn\nvictory.  We will put on your bulletin-board, '_Patronized by the\nelect_.'  How does that strike you?\"\n\n\"Verily, it is wonderly bethought!\"\n\n\"Well, a body is bound to admit that for just a modest little\none-line ad, it's a corker.\"\n\nSo the poor colporteur's griefs vanished away.  He was a brave\nfellow, and had done mighty feats of arms in his time.  His chief\ncelebrity rested upon the events of an excursion like this one\nof mine, which he had once made with a damsel named Maledisant,\nwho was as handy with her tongue as was Sandy, though in a different\nway, for her tongue churned forth only railings and insult, whereas\nSandy's music was of a kindlier sort.  I knew his story well, and so\nI knew how to interpret the compassion that was in his face when he\nbade me farewell.  He supposed I was having a bitter hard time of it.\n\nSandy and I discussed his story, as we rode along, and she said\nthat La Cote's bad luck had begun with the very beginning of that\ntrip; for the king's fool had overthrown him on the first day,\nand in such cases it was customary for the girl to desert to the\nconqueror, but Maledisant didn't do it; and also persisted afterward\nin sticking to him, after all his defeats.  But, said I, suppose\nthe victor should decline to accept his spoil?  She said that that\nwouldn't answer--he must.  He couldn't decline; it wouldn't be\nregular.  I made a note of that.  If Sandy's music got to be too\nburdensome, some time, I would let a knight defeat me, on the chance\nthat she would desert to him.\n\nIn due time we were challenged by the warders, from the castle\nwalls, and after a parley admitted.  I have nothing pleasant to\ntell about that visit.  But it was not a disappointment, for I knew\nMrs. le Fay by reputation, and was not expecting anything pleasant.\nShe was held in awe by the whole realm, for she had made everybody\nbelieve she was a great sorceress.  All her ways were wicked, all\nher instincts devilish.  She was loaded to the eyelids with cold\nmalice.  All her history was black with crime; and among her crimes\nmurder was common.  I was most curious to see her; as curious as\nI could have been to see Satan.  To my surprise she was beautiful;\nblack thoughts had failed to make her expression repulsive, age\nhad failed to wrinkle her satin skin or mar its bloomy freshness.\nShe could have passed for old Uriens' granddaughter, she could\nhave been mistaken for sister to her own son.\n\nAs soon as we were fairly within the castle gates we were ordered\ninto her presence.  King Uriens was there, a kind-faced old man\nwith a subdued look; and also the son, Sir Uwaine le Blanchemains,\nin whom I was, of course, interested on account of the tradition\nthat he had once done battle with thirty knights, and also on\naccount of his trip with Sir Gawaine and Sir Marhaus, which Sandy\nhad been aging me with.  But Morgan was the main attraction, the\nconspicuous personality here; she was head chief of this household,\nthat was plain.  She caused us to be seated, and then she began,\nwith all manner of pretty graces and graciousnesses, to ask me\nquestions.  Dear me, it was like a bird or a flute, or something,\ntalking.  I felt persuaded that this woman must have been\nmisrepresented, lied about.  She trilled along, and trilled along,\nand presently a handsome young page, clothed like the rainbow, and\nas easy and undulatory of movement as a wave, came with something\non a golden salver, and, kneeling to present it to her, overdid\nhis graces and lost his balance, and so fell lightly against her\nknee.  She slipped a dirk into him in as matter-of-course a way as\nanother person would have harpooned a rat!\n\nPoor child! he slumped to the floor, twisted his silken limbs in\none great straining contortion of pain, and was dead.  Out of the\nold king was wrung an involuntary \"O-h!\" of compassion.  The look\nhe got, made him cut it suddenly short and not put any more hyphens\nin it.  Sir Uwaine, at a sign from his mother, went to the anteroom\nand called some servants, and meanwhile madame went rippling sweetly\nalong with her talk.\n\nI saw that she was a good housekeeper, for while she talked she\nkept a corner of her eye on the servants to see that they made\nno balks in handling the body and getting it out; when they came\nwith fresh clean towels, she sent back for the other kind; and\nwhen they had finished wiping the floor and were going, she indicated\na crimson fleck the size of a tear which their duller eyes had\noverlooked.  It was plain to me that La Cote Male Taile had failed\nto see the mistress of the house.  Often, how louder and clearer\nthan any tongue, does dumb circumstantial evidence speak.\n\nMorgan le Fay rippled along as musically as ever.  Marvelous woman.\nAnd what a glance she had: when it fell in reproof upon those\nservants, they shrunk and quailed as timid people do when the\nlightning flashes out of a cloud.  I could have got the habit\nmyself.  It was the same with that poor old Brer Uriens; he was\nalways on the ragged edge of apprehension; she could not even turn\ntoward him but he winced.\n\nIn the midst of the talk I let drop a complimentary word about\nKing Arthur, forgetting for the moment how this woman hated her\nbrother.  That one little compliment was enough.  She clouded up\nlike storm; she called for her guards, and said:\n\n\"Hale me these varlets to the dungeons.\"\n\nThat struck cold on my ears, for her dungeons had a reputation.\nNothing occurred to me to say--or do.  But not so with Sandy.\nAs the guard laid a hand upon me, she piped up with the tranquilest\nconfidence, and said:\n\n\"God's wounds, dost thou covet destruction, thou maniac?  It is\nThe Boss!\"\n\nNow what a happy idea that was!--and so simple; yet it would never\nhave occurred to me.  I was born modest; not all over, but in spots;\nand this was one of the spots.\n\nThe effect upon madame was electrical.  It cleared her countenance\nand brought back her smiles and all her persuasive graces and\nblandishments; but nevertheless she was not able to entirely cover up\nwith them the fact that she was in a ghastly fright. She said:\n\n\"La, but do list to thine handmaid! as if one gifted with powers\nlike to mine might say the thing which I have said unto one who\nhas vanquished Merlin, and not be jesting.  By mine enchantments\nI foresaw your coming, and by them I knew you when you entered\nhere.  I did but play this little jest with hope to surprise you\ninto some display of your art, as not doubting you would blast\nthe guards with occult fires, consuming them to ashes on the spot,\na marvel much beyond mine own ability, yet one which I have long\nbeen childishly curious to see.\"\n\nThe guards were less curious, and got out as soon as they got permission.\n\n\n\n\n\nEnd of the Project Gutenberg EBook of A Connecticut Yankee in King Arthur's\nCourt, Part 3., by Mark Twain (Samuel Clemens)\n\n*** ","meta":{"redpajama_set_name":"RedPajamaBook"}}
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+{"text":"\n\nTranscribed from the 1902 Macmillan and Co. \"Countess Kate and The\nStokesley Secret\" edition by David Price, email ccx074@pglaf.org\n\n [Picture: Frontispiece: man comforting little girl on stairs whilst boy\n                                 watches]\n\n\n\n\n\n                              COUNTESS KATE\n\n\n                                * * * * *\n\n                                    BY\n                            CHARLOTTE M. YONGE\n\n                                * * * * *\n\n                                  London\n                        MACMILLAN AND CO., Limited\n                     NEW YORK: THE MACMILLAN COMPANY\n                                   1902\n\n                          _All rights reserved_\n\n                                * * * * *\n\n                     RICHARD CLAY AND SONS, LIMITED,\n                            LONDON AND BUNGAY\n\n _Transferred to Macmillan and Co._, _Limited_, 1901.  _Reprinted_, 1902.\n\n\n\n\nCHAPTER I.\n\n\n\"THERE, I've done every bit I can do!  I'm going to see what o'clock it\nis.\"\n\n\"I heard it strike eleven just now.\"\n\n\"Sylvia, you'll tip up!  What a tremendous stretch!\"\n\n\"Wha-ooh!  Oh dear!  We sha'n't get one moment before dinner!  Oh,\nhorrible! oh, horrible! most horrible!\"\n\n\"Sylvia, you know I hate hearing _Hamlet_ profaned.\"\n\n\"You can't hate it more than having no one to hear our lessons.\"\n\n\"That makes you do it.  What on earth can Mary be about?\"\n\n\"Some tiresome woman to speak to her, I suppose.\"\n\n\"I'm sure it can't be as much her business as it is to mind her poor\nlittle sisters.  Oh dear! if Papa could only afford us a governess!\"\n\n\"I am sure I should not like it at all; besides, it is wrong to wish to\nbe richer than one is.\"\n\n\"I don't wish; I am only thinking how nice it would be, if some one would\ngive us a famous quantity of money.  Then Papa should have a pretty\nparsonage, like the one at Shagton; and we would make the church\nbeautiful, and get another pony or two, to ride with Charlie.\"\n\n\"Yes, and have a garden with a hothouse like Mr. Brown's.\"\n\n\"Oh yes; and a governess to teach us to draw.  But best of all\u2014O Sylvia!\nwouldn't it be nice not to have to mind one's clothes always?  Yes, you\nlaugh; but it comes easier to you; and, oh dear! oh dear! it is so horrid\nto be always having to see one does not tear oneself.\"\n\n\"I don't think you do see,\" said Sylvia, laughing.\n\n\"My frocks always _will_ get upon the thorns.  It is very odd.\"\n\n\"Only do please, Katie dear, let me finish this sum; and then if Mary is\nnot come, she can't scold if we are amusing ourselves.\"\n\n\"I know!\" cried Kate.  \"I'll draw such a picture, and tell you all about\nit when your sum is over.\"\n\nThereon ensued silence in the little room, half parlour, half study,\nnearly filled with books and piano; and the furniture, though carefully\nprotected with brown holland, looking the worse for wear, and as if\ndanced over by a good many young folks.\n\nThe two little girls, who sat on the opposite sides of a little square\ntable in the bay-window, were both between ten and eleven years old, but\ncould not have been taken for twins, nor even for sisters, so unlike were\ntheir features and complexion; though their dress, very dark grey linsey,\nand brown holland aprons, was exactly the same, except that Sylvia's was\nenlivened by scarlet braid, Kate's darkened by black\u2014and moreover, Kate's\napron was soiled, and the frock bore traces of a great darn.  In fact,\nnew frocks for the pair were generally made necessary by Kate's tattered\nstate, when Sylvia's garments were still available for little Lily, or\nfor some school child.\n\nSylvia's brown hair was smooth as satin; Kate's net did not succeed in\nconfining the loose rough waves of dark chestnut, on the road to\nblackness.  Sylvia was the shorter, firmer, and stronger, with round\nwhite well-cushioned limbs; Kate was tall, skinny, and brown, though\nperfectly healthful.  The face of the one was round and rosy, of the\nother thin and dark; and one pair of eyes were of honest grey, while the\nothers were large and hazel, with blue whites.  Kate's little hand was so\nslight, that Sylvia's strong fingers could almost crush it together, but\nit was far less effective in any sort of handiwork; and her slim\nneatly-made foot always was a reproach to her for making such boisterous\nsteps, and wearing out her shoes so much faster than the quieter\nmovements of her companion did\u2014her sister, as the children would have\nsaid, for nothing but the difference of surname reminded Katharine\nUmfraville that she was not the sister of Sylvia Wardour.\n\nHer father, a young clergyman, had died before she could remember\nanything, and her mother had not survived him three months.  Little Kate\nhad then become the charge of her mother's sister, Mrs. Wardour, and had\ngrown up in the little parsonage belonging to the district church of St.\nJames's, Oldburgh, amongst her cousins, calling Mr. and Mrs. Wardour Papa\nand Mamma, and feeling no difference between their love to their own five\nchildren and to her.\n\nMrs. Wardour had been dead for about four years, and the little girls\nwere taught by the eldest sister, Mary, who had been at a boarding-school\nto fit her for educating them.  Mr. Wardour too taught them a good deal\nhimself, and had the more time for them since Charlie, the youngest boy,\nhad gone every day to the grammar-school in the town.\n\nArmyn, the eldest of the family, was with Mr. Brown, a very good old\nsolicitor, who, besides his office in Oldburgh, had a very pretty house\nand grounds two miles beyond St. James's, where the parsonage children\nwere delighted to spend an afternoon now and then.\n\nLittle did they know that it was the taking the little niece as a\ndaughter that had made it needful to make Armyn enter on a profession at\nonce, instead of going to the university and becoming a clergyman like\nhis father; nor how cheerfully Armyn had agreed to do whatever would best\nlighten his father's cares and troubles.  They were a very happy family;\nabove all, on the Saturday evenings and Sundays that the good-natured\nelder brother spent at home.\n\n\"There!\" cried Sylvia, laying down her slate pencil, and indulging in\nanother tremendous yawn; \"we can't do a thing more till Mary comes!  What\ncan she be about?\"\n\n\"Oh, but look, Sylvia!\" cried Kate, quite forgetting everything in the\ninterest of her drawing on a large sheet of straw-paper.  \"Do you see\nwhat it is?\"\n\n\"I don't know,\" said Sylvia, \"unless\u2014let me see\u2014  That's a very rich\nlittle girl, isn't it?\" pointing to an outline of a young lady whose\nwealth was denoted by the flounces (or rather scallops) on her frock, the\nbracelets on her sausage-shaped arms, and the necklace on her neck.\n\n\"Yes; she is a very rich and grand\u2014  Lady Ethelinda; isn't that a pretty\nname?  I do wish I was Lady Katharine.\"\n\n\"And what is she giving?  I wish you would not do men and boys, Kate;\ntheir legs always look so funny as you do them.\"\n\n\"They never will come right; but never mind, I must have them.  That is\nLady Ethelinda's dear good cousin, Maximilian; he is a lawyer\u2014don't you\nsee the parchment sticking out of his pocket?\"\n\n\"Just like Armyn.\"\n\n\"And she is giving him a box with a beautiful new microscope in it; don't\nyou see the top of it?  And there is a whole pile of books.  And I would\ndraw a pony, only I never can nicely; but look here,\"\u2014Kate went on\ndrawing as she spoke\u2014\"here is Lady Ethelinda with her best hat on, and a\nlittle girl coming.  There is the little girl's house, burnt down; don't\nyou see?\"\n\nSylvia saw with the eyes of her mind the ruins, though her real eyes saw\nnothing but two lines, meant to be upright, joined together by a wild\nzig-zag, and with some peaked scrabbles and round whirls intended for\nsmoke.  Then Kate's ready pencil portrayed the family, as jagged in their\ndrapery as the flames and presently Lady Ethelinda appeared before a\ncounter (such a counter! sloping like a desk in the attempt at\nperspective, but it conveniently concealed the shopman's legs,) buying\nvery peculiar garments for the sufferers.  Another scene in which she was\npresenting them followed, Sylvia looking on, and making suggestions; for\nin fact there was no quiet pastime more relished by the two cousins than\ndrawing stories, as they called it, and most of their pence went in paper\nfor that purpose.\n\n\"Lady Ethelinda had a whole ream of paper to draw on!\" were the words\npronounced in Kate's shrill key of eagerness, just as the long lost Mary\nand her father opened the door.\n\n\"Indeed!\" said Mr. Wardour, a tall, grave-looking man; \"and who is Lady\nEthelinda!\"\n\n\"O Papa, it's just a story I was drawing,\" said Kate, half eager, half\nashamed.\n\n\"We have done all the lessons we could, indeed we have\u2014\" began Sylvia;\n\"my music and our French grammar, and\u2014\"\n\n\"Yes, I know,\" said Mary; and she paused, looking embarrassed and\nuncomfortable, so that Sylvia stood in suspense and wonder.\n\n\"And so my little Kate likes thinking of Lady\u2014Lady Etheldredas,\" said Mr.\nWardour rather musingly; but Kate was too much pleased at his giving any\nsort of heed to her performances to note the manner, and needed no more\nencouragement to set her tongue off.\n\n\"Lady Ethelinda, Papa.  She is a very grand rich lady, though she is a\nlittle girl: and see there, she is giving presents to all her cousins;\nand there she is buying new clothes for the orphans that were burnt out;\nand there she is building a school for them.\"\n\nKate suddenly stopped, for Mr. Wardour sat down, drew her between his\nknees, took both her hands into one of his, and looked earnestly into her\nface, so gravely that she grew frightened, and looking appealingly up,\ncried out, \"O Mary, Mary! have I been naughty?\"\n\n\"No, my dear,\" said Mr. Wardour; \"but we have heard a very strange piece\nof news about you, and I am very anxious as to whether it may turn out\nfor your happiness.\"\n\nKate stood still and looked at him, wishing he would speak faster.  Could\nher great-uncle in India be come home, and want her to make him a visit\nin London?  How delightful!  If it had been anybody but Papa, she would\nhave said, \"Go on.\"\n\n\"My dear,\" said Mr. Wardour at last, \"you know that your cousin, Lord\nCaergwent, was killed by an accident last week.\"\n\n\"Yes, I know,\" said Kate; \"that was why Mary made me put this black braid\non my frock; and a very horrid job it was to do\u2014it made my fingers so\nsore.\"\n\n\"I did not know till this morning that his death would make any other\ndifference to you,\" continued Mr. Wardour.  \"I thought the title went to\nheirs-male, and that Colonel Umfraville was the present earl; but, my\nlittle Katharine, I find that it is ordained that you should have this\ngreat responsibility.\"\n\n\"What, you thought it was the Salic law?\" said Kate, going on with one\npart of his speech, and not quite attending to the other.\n\n\"Something like it; only that it is not the English term for it,\" said\nMr. Wardour, half smiling.  \"As your grandfather was the elder son, the\ntitle and property come to you.\"\n\nKate did not look at him, but appeared intent on the marks of the needle\non the end of her forefinger, holding down her head.\n\nSylvia, however, seemed to jump in her very skin, and opening her eyes,\ncried out, \"The title!  Then Kate is\u2014is\u2014oh, what is a she-earl called?\"\n\n\"A countess,\" said Mr. Wardour, with a smile, but rather sadly.  \"Our\nlittle Kate is Countess of Caergwent.\"\n\n\"My dear Sylvia!\" exclaimed Mary in amazement; for Sylvia, like an\nIndia-rubber ball, had bounded sheer over the little arm-chair by which\nshe was standing.\n\nBut there her father's look and uplifted finger kept her still and\nsilent.  He wanted to give Kate time to understand what he had said.\n\n\"Countess of Caergwent,\" she repeated; \"that's not so pretty as if I were\nLady Katharine.\"\n\n\"The sound does not matter much,\" said Mary.  \"You will always be\nKatharine to those that love you best.  And oh!\u2014\"  Mary stopped short,\nher eyes full of tears.\n\nKate looked up at her, astonished.  \"Are you sorry, Mary?\" she asked, a\nlittle hurt.\n\n\"We are all sorry to lose our little Kate,\" said Mr. Wardour.\n\n\"Lose me, Papa!\" cried Kate, clinging to him, as the children scarcely\never did, for he seldom made many caresses; \"Oh no, never!  Doesn't\nCaergwent Castle belong to me?  Then you must all come and live with me\nthere; and you shall have lots of big books, Papa; and we will have a\npony-carriage for Mary, and ponies for Sylvia and Charlie and me, and\u2014\"\n\nKate either ran herself down, or saw that the melancholy look on Mr.\nWardour's face rather deepened than lessened, for she stopped short.\n\n\"My dear,\" he said, \"you and I have both other duties.\"\n\n\"Oh, but if I built a church!  I dare say there are people at Caergwent\nas poor as they are here.  Couldn't we build a church, and you mind them,\nPapa?\"\n\n\"My little Katharine, you have yet to understand that 'the heir, so long\nas he is a child, differeth in nothing from a servant, but is under\ntutors and governors.'  You will not have any power over yourself or your\nproperty till you are twenty-one.\"\n\n\"But you are my tutor and my governor, and my spiritual pastor and\nmaster,\" said Kate.  \"I always say so whenever Mary asks us questions\nabout our duty to our neighbour.\"\n\n\"I have been so hitherto,\" said Mr. Wardour, setting her on his knee;\n\"but I see I must explain a good deal to you.  It is the business of a\ncourt in London, that is called the Court of Chancery, to provide that\nproper care is taken of young heirs and heiresses and their estates, if\nno one have been appointed by their parents to do so; and it is this\ncourt that must settle what is to become of you.\"\n\n\"And why won't it settle that I may live with my own papa and brothers\nand sisters?\"\n\n\"Because, Kate, you must be brought up in a way to fit your station; and\nmy children must be brought up in a way to fit theirs.  And besides,\" he\nadded more sadly, \"nobody that could help it would leave a girl to be\nbrought up in a household without a mother.\"\n\nKate's heart said directly, that as she could never again have a mother,\nher dear Mary must be better than a stranger; but somehow any reference\nto the sorrow of the household always made her anxious to get away from\nthe subject, so she looked at her finger again, and asked, \"Then am I to\nlive up in this Court of Chances?\"\n\n\"Not exactly,\" said Mr. Wardour.  \"Your two aunts in London, Lady Barbara\nand Lady Jane Umfraville, are kind enough to offer to take charge of you.\nHere is a letter that they sent inclosed for you.\"\n\n\"The Countess of Caergwent,\" was written on the envelope; and Kate's and\nSylvia's heads were together in a moment to see how it looked, before\nopening the letter, and reading:\u2014\"'My dear Niece,'\u2014dear me, how funny to\nsay niece!\u2014'I deferred writing to you upon the melancholy\u2014' oh, what is\nit, Sylvia?\"\n\n\"The melancholy comet!\"\n\n\"No, no; nonsense.\"\n\n\"Melancholy event,\" suggested Mary.\n\n\"Yes, to be sure.  I can't think why grown-up people always write on\npurpose for one not to read them.\u2014'Melancholy event that has placed you\nin possession of the horrors of the family.'\"\n\n\"Horrors!\u2014Kate, Kate!\"\n\n\"Well, I am sure it _is_ horrors,\" said the little girl rather\nperversely.\n\n\"This is not a time for nonsense, Kate,\" said Mr. Wardour; and she was\nsubdued directly.\n\n\"Shall I read it to you?\" said Mary.\n\n\"Oh, no, no!\"  Kate was too proud of her letter to give it up, and\napplied herself to it again.\u2014\"'Family honours, until I could ascertain\nyour present address.  And likewise, the shock of your poor cousin's\ndeath so seriously affected my sister's health in her delicate state,\nthat for some days I could give my attention to nothing else.'  Dear me!\nThis is my Aunt Barbara, I see!  Is Aunt Jane so ill?\"\n\n\"She has had very bad health for many years,\" said Mr. Wardour; \"and your\nother aunt has taken the greatest care of her.\"\n\n\"'We have now, however, been able to consider what will be best for all\nparties; and we think nothing will be so proper as that you should reside\nwith us for the present.  We will endeavour to make a happy home for you;\nand will engage a lady to superintend your education, and give you all\nthe advantages to which you are entitled.  We have already had an\ninterview with a very admirable person, who will come down to Oldburgh\nwith our butler next Friday, and escort you to us, if Mrs. Wardour will\nkindly prepare you for the journey.  I have written to thank her for her\nkindness to you.'\"\n\n\"Mrs. Wardour!\" exclaimed Sylvia.\n\n\"The ladies have known and cared little about Kate or us for a good many\nyears,\" said Mary, almost to herself, but in such a hurt tone, that her\nfather looked up with grave reproof in his eyes, as if to remind her of\nall he had been saying to her during the long hours that the little girls\nhad waited.\n\n\"'With your Aunt Jane's love, and hoping shortly to be better acquainted,\nI remain, my dear little niece, your affectionate aunt, Barbara\nUmfraville.'  Then I am to go and live with them!\" said Kate, drawing a\nlong sigh.  \"O Papa, do let Sylvia come too, and learn of my governess\nwith me!\"\n\n\"Your aunts do not exactly contemplate that,\" said Mr. Wardour; \"but\nperhaps there may be visits between you.\"\n\nSylvia began to look very grave.  She had not understood that this great\nnews was to lead to nothing but separation.  Everything had hitherto been\nin common between her and Kate, and that what was good for the one should\nnot be good for the other was so new and strange, that she did not\nunderstand it at once.\n\n\"Oh yes! we will visit.  You shall all come and see me in London, and see\nthe Zoological Gardens and the British Museum; and I will send you such\npresents!\"\n\n\"We will see,\" said Mr. Wardour kindly; \"but just now, I think the best\nthing you can do is to write to your aunt, and thank her for her kind\nletter; and say that I will bring you up to London on the day she names,\nwithout troubling the governess and the butler.\"\n\n\"Oh, thank you!\" said Kate; \"I sha'n't be near so much afraid if you come\nwith me.\"\n\nMr. Wardour left the room; and the first thing Mary did was to throw her\narms round the little girl in a long vehement embrace.  \"My little Kate!\nmy little Kate!  I little thought this was to be the end of it!\" she\ncried, kissing her, while the tears dropped fast.\n\nKate did not like it at all.  The sight of strong feeling distressed her,\nand made her awkward and ungracious.  \"Don't, Mary,\" she said,\ndisengaging herself; \"never mind; I shall always come and see you; and\nwhen I grow up, you shall come to live with me at Caergwent.  And you\nknow, when they write a big red book about me, they will put in that you\nbrought me up.\"\n\n\"Write a big red book about you, Kate!\"\n\n\"Why,\" said Kate, suddenly become very learned, \"there is an immense fat\nred and gold book at Mr. Brown's, all full of Lords and Ladies.\"\n\n\"Oh, a Peerage!\" said Mary; \"but even you, my Lady Countess, can't have a\nwhole peerage to yourself.\"\n\nAnd that little laugh seemed to do Mary good, for she rose and began to\nrule the single lines for Kate's letter.  Kate could write a very tidy\nlittle note; but just now she was too much elated and excited to sit down\nquietly, or quite to know what she was about.  She went skipping\nrestlessly about from one chair to another, chattering fast about what\nshe would do, and wondering what the aunts would be like, and what Armyn\nwould say, and what Charlie would say, and the watch she would buy for\nCharlie, and the great things she was to do for everybody\u2014till Mary\nmuttered something in haste, and ran out of the room.\n\n\"I wonder why Mary is so cross,\" said Kate.\n\nPoor Mary!  No one could be farther from being cross; but she was\nthoroughly upset.  She was as fond of Kate as of her own sisters, and was\nnot only sorry to part with her, but was afraid that she would not be\nhappy or good in the new life before her.\n\n\n\n\nCHAPTER II.\n\n\nTHE days passed very slowly with Kate, until the moment when she was to\ngo to London and take her state upon her, as she thought.  Till that\nshould come to pass, she could not feel herself really a countess.  She\ndid not find herself any taller or grander; Charlie teased her rather\nmore instead of less and she did not think either Mr. Wardour or Mary or\nArmyn thought half enough of her dignity: they did not scruple to set her\ndown when she talked too loud, and looked sad instead of pleased when she\nchattered about the fine things she should do.  Mr. and Mrs. Brown, to be\nsure, came to wish her good-bye; but they were so respectful, and took\nsuch pains that she should walk first, that she grew shy and sheepish,\nand did not like it at all.\n\nShe thought ease and dignity would come by nature when she was once in\nLondon; and she made so certain of soon seeing Sylvia again, that she did\nnot much concern herself about the parting with her; while she was rather\ndispleased with Mary for looking grave, and not making more of her, and\ntrying to tell her that all might not be as delightful as she expected.\nShe little knew that Mary was grieved at her eagerness to leave her happy\nhome, and never guessed at the kind sister's fears for her happiness.\nShe set it all down to what she was wont to call crossness.  If Mary had\nreally been a cross or selfish person, all she would have thought of\nwould have been that now there would not be so many rents to mend after\nKate's cobbling attempts, nor so many shrill shrieking laughs to disturb\nPapa writing his sermon, nor so much difficulty in keeping any room in\nthe house tidy, nor so much pinching in the housekeeping.  Instead of\nthat, Mary only thought whether Barbara and Lady Jane would make her\nlittle Kate happy and good.  She was sure they were proud, hard, cold\npeople; and her father had many talks with her, to try to comfort her\nabout them.\n\nMr. Wardour told her that Kate's grandfather had been such a grief and\nshame to the family, that it was no wonder they had not liked to be\nfriendly with those he had left behind him.  There had been help given to\neducate the son, and some notice had been taken of him, but always very\ndistant; and he had been thought very foolish for marrying when he was\nvery young, and very ill off.  At the time of his death, his uncle,\nColonel Umfraville, had been very kind, and had consulted earnestly with\nMr. Wardour what was best for the little orphan; but had then explained\nthat he and his wife could not take charge of her, because his regiment\nwas going to India, and she could not go there with them; and that his\nsisters were prevented from undertaking the care of so young a child by\nthe bad health of the elder, who almost owed her life to the tender\nnursing of the younger.  And as Mrs. Wardour was only eager to keep to\nherself all that was left of her only sister, and had a nursery of her\nown, it had been most natural that Kate should remain at St. James's\nParsonage; and Mr. Wardour had full reason to believe that, had there\nbeen any need, or if he had asked for help, the aunts would have gladly\ngiven it.  He knew them to be worthy and religious women; and he told\nMary that he thought it very likely that they might deal better with\nKate's character than he had been able to do.  Mary knew she herself had\nmade mistakes, but she could not be humble for her father, or think any\nplace more improving than under his roof.\n\nAnd Kate meanwhile had her own views.  And when all the good-byes were\nover, and she sat by the window of the railway carriage, watching the\nfields rush by, reduced to silence, because \"Papa\" had told her he could\nnot hear her voice, and had made a peremptory sign to her when she\nscreamed her loudest, and caused their fellow-travellers to look up\namazed, she wove a web in her brain something like this:\u2014\"I know what my\naunts will be like: they will be just like ladies in a book.  They will\nbe dreadfully fashionable!  Let me see\u2014Aunt Barbara will have a turban on\nher head, and a bird of paradise, like the bad old lady in Armyn's book\nthat Mary took away from me; and they will do nothing all day long but\ntry on flounced gowns, and count their jewels, and go out to balls and\noperas\u2014and they will want me to do the same\u2014and play at cards all Sunday!\n'Lady Caergwent,' they will say, 'it is becoming to your position!'  And\nthen the young countess presented a remarkable contrast in her ingenuous\nsimplicity,\" continued Kate, not quite knowing whether she was making a\nstory or thinking of herself\u2014for indeed she did not feel as if she were\nherself, but somebody in a story.  \"Her waving hair was only confined by\nan azure ribbon, (Kate loved a fine word when Charlie did not hear it to\nlaugh at her;) and her dress was of the simplest muslin, with one diamond\naigrette of priceless value!\"\n\nKate had not the most remote notion what an aigrette might be, but she\nthought it would sound well for a countess; and she went on musing very\npleasantly on the amiable simplicity of the countess, and the speech that\nwas to cure the aunts of playing at cards on a Sunday, wearing turbans,\nand all other enormities, and lead them to live in the country, giving a\ncontinual course of school feasts, and surprising meritorious families\nwith gifts of cows.  She only wished she had a pencil to draw it all to\nshow Sylvia, provided Sylvia would know her cows from her tables.\n\nAfter more vain attempts at chatter, and various stops at stations, Mr.\nWardour bought a story-book for her; and thus brought her to a most happy\nstate of silent content, which lasted till the house roofs of London\nbegan to rise on either side of the railway.\n\nAmong the carriages that were waiting at the terminus was a small\nbrougham, very neat and shiny; and a servant came up and touched his hat,\nopening the door for Kate, who was told to sit there while the servant\nand Mr. Wardour looked for the luggage.  She was a little disappointed.\nShe had once seen a carriage go by with four horses, and a single one did\nnot seem at all worthy of her; but she had two chapters more of her story\nto read, and was so eager to see the end of it, that Mr. Wardour could\nhardly persuade her to look out and see the Thames when she passed over\nit, nor the Houses of Parliament and the towers of Westminster Abbey.\n\nAt last, while passing through the brighter and more crowded streets,\nKate having satisfied herself what had become of the personages of her\nstory, looked up, and saw nothing but dull houses of blackened cream\ncolour; and presently found the carriage stopping at the door of one.\n\n\"Is it here, Papa?\" she said, suddenly seized with fright.\n\n\"Yes,\" he said, \"this is Bruton Street;\" and he looked at her anxiously\nas the door was opened and the steps were let down.  She took tight hold\nof his hand.  Whatever she had been in her day-dreams, she was only his\nown little frightened Kate now; and she tried to shrink behind him as the\nfootman preceded them up the stairs, and opening the door,\nannounced\u2014\"Lady Caergwent and Mr. Wardour!\"\n\nTwo ladies rose up, and came forward to meet her.  She felt herself\nkissed by both, and heard greetings, but did not know what to say, and\nstood up by Mr. Wardour, hanging down her head, and trying to stand upon\none foot with the other, as she always did when she was shy and awkward.\n\n\"Sit down, my dear,\" said one of the ladies, making a place for her on\nthe sofa.  But Kate only laid hold of a chair, pulled it as close to Mr.\nWardour as possible, and sat down on the extreme corner of it, feeling\nfor a rail on which to set her feet, and failing to find one, twining her\nankles round the leg of the chair.  She knew very well that this was not\npretty; but she never could recollect what was pretty behaviour when she\nwas shy.  She was a very different little girl in a day-dream and out of\none.  And when one of the aunts asked her if she were tired, all she\ncould do was to give a foolish sort of smile, and say, \"N\u2014no.\"\n\nThen she had a perception that Papa was looking reprovingly at her; so\nshe wriggled her legs away from that of the chair, twisted them together\nin the middle, and said something meant for \"No, thank you;\" but of which\nnothing was to be heard but \"q,\" apparently proceeding out of the brim of\nher broad hat, so low did the young countess, in her amiable simplicity,\nhold her head.\n\n\"She is shy!\" said one of the ladies to the other; and they let her alone\na little, and began to talk to Mr. Wardour about the journey, and various\nother things, to which Kate did not greatly listen.  She began to let her\neyes come out from under her hat brim, and satisfied herself that the\naunts certainly did not wear either turbans or birds of paradise, but\nlooked quite as like other people as she felt herself, in spite of her\ntitle.\n\nIndeed, one aunt had nothing on her head at all but a little black velvet\nand lace, not much more than Mary sometimes wore, and the other only a\nvery light cap.  Kate thought great-aunts must be as old at least as Mrs.\nBrown, and was much astonished to see that these ladies had no air of age\nabout them.  The one who sat on the sofa had a plump, smooth, pretty,\npink and white face, very soft and pleasant to look at, though an older\nperson than Kate would have perceived that the youthful delicacy of the\ncomplexion showed that she had been carefully shut up and sheltered from\nall exposure and exertion, and that the quiet innocent look of the small\nfeatures was that of a person who had never had to use her goodness more\nactively than a little baby.  Kate was sure that this was aunt Jane, and\nthat she should get on well with her, though that slow way of speaking\nwas rather wearisome.\n\nThe other aunt, who was talking the most, was quite as slim as Mary, and\nhad a bright dark complexion, so that if Kate had not seen some shades of\ngrey in her black hair, it would have been hard to believe her old at\nall.  She had a face that put Kate in mind of a picture of a beautiful\nlady in a book at home\u2014the eyes, forehead, nose, and shape of the chin,\nwere so finely made; and yet there was something in them that made the\nlittle girl afraid, and feel as if the plaster cast of Diana's head on\nthe study mantelpiece had got a pair of dark eyes, and was looking very\nhard at her; and there was a sort of dry sound in her voice that was\nuncomfortable to hear.\n\nThen Kate took a survey of the room, which was very prettily furnished,\nwith quantities of beautiful work of all kinds, and little tables and\nbrackets covered with little devices in china and curiosities under\nglass, and had flowers standing in the windows; and by the time she had\nfinished trying to make out the subject of a print on the walls, she\nheard some words that made her think that her aunts were talking of her\nnew governess, and she opened her ears to hear, \"So we thought it would\nbe an excellent arrangement for her, poor thing!\" and \"Papa\" answering,\n\"I hope Kate may try to be a kind considerate pupil.\"  Then seeing by\nKate's eyes that her attention had been astray, or that she had not\nunderstood Lady Barbara's words, he turned to her, saying, \"Did you not\nhear what your aunt was telling me?\"\n\n\"No, Papa.\"\n\n\"She was telling me about the lady who will teach you.  She has had great\nafflictions.  She has lost her husband, and is obliged to go out as\ngoverness, that she may be able to send her sons to school.  So, Kate,\nyou must think of this, and try to give her as little trouble as\npossible.\"\n\nIt would have been much nicer if Kate would have looked up readily, and\nsaid something kind and friendly; but the fit of awkwardness had come\nover her again, and with it a thought so selfish, that it can hardly be\ncalled otherwise than naughty\u2014namely, that grown-up people in trouble\nwere very tiresome, and never let young ones have any fun.\n\n\"Shall I take you to see Mrs. Lacy, my dear?\" said Lady Barbara, rising.\nAnd as Kate took hold of Mr. Wardour's hand, she added, \"You will see Mr.\nWardour again after dinner.  You had better dress, and have some meat for\nyour tea, with Mrs. Lacy, and then come into the drawing-room.\"\n\nThis was a stroke upon Kate.  She who had dined with the rest of the\nworld ever since she could remember\u2014she, now that she was a countess, to\nbe made to drink tea up-stairs like a baby, and lose all that time of\nPapa's company!  She swelled with displeasure: but Aunt Barbara did not\nlook like a person whose orders could be questioned, and \"Papa\" said not\na word in her favour.  Possibly the specimen of manners she had just\ngiven had not led either him or Lady Barbara to think her fit for a late\ndinner.\n\nLady Barbara first took her up-stairs, and showed her a little long\nnarrow bed-room, with a pretty pink-curtained bed in it.\n\n\"This will be your room, my dear,\" she said.  \"I am sorry we have not a\nlarger one to offer you; but it opens into mine, as you see, and my\nsister's is just beyond.  Our maid will dress you for a few days, when I\nhope to engage one for you.\"\n\nHere was something like promotion!  Kate dearly loved to have herself\ntaken off her own hands, and not to be reproved by Mary for untidiness,\nor roughly set to rights by Lily's nurse.  She actually exclaimed, \"Oh,\nthank you!\"  And her aunt waited till the hat and cloak had been taken\noff and the chestnut hair smoothed, looked at her attentively, and said,\n\"Yes, you are like the family.\"\n\n\"I'm very like my own papa,\" said Kate, growing a little bolder, but\nstill speaking with her head on one side, which was her way when she said\nanything sentimental.\n\n\"I dare say you are,\" answered her aunt, with the dry sound.  \"Are you\nready now?  I will show you the way.  The house is very small,\" continued\nLady Barbara, as they went down the stairs to the ground floor; \"and this\nmust be your school-room for the present.\"\n\nIt was the room under the back drawing-room; and in it was a lady in a\nwidow's cap, sitting at work.  \"Here is your little Pupil\u2014Lady\nCaergwent\u2014Mrs. Lacy,\" said Lady Barbara.  \"I hope you will find her a\ngood child.  She will drink tea with you, and then dress, and afterwards\nI hope, we shall see you with her in the drawing-room.\"\n\nMrs. Lacy bowed, without any answer in words, only she took Kate's hand\nand kissed her.  Lady Barbara left them, and there was a little pause.\nKate looked at her governess, and her heart sank, for it was the very\nsaddest face she had ever seen\u2014the eyes looked soft and gentle, but as if\nthey had wept till they could weep no longer; and when the question was\nasked, \"Are you tired, my dear?\" it was in a sunk tone, trying to be\ncheerful but the sadder for that very reason.  Poor lady! it was only\nthat morning that she had parted with her son, and had gone away from the\nhome where she had lived with her husband and children.\n\nKate was almost distressed; yet she felt more at her ease than with her\naunts, and answered, \"Not at all, thank you,\" in her natural tone.\n\n\"Was it a long journey?\"\n\nKate had been silent so long, that her tongue was ready for exertion; and\nshe began to chatter forth all the events of the journey, without heeding\nmuch whether she were listened to or not, till having come to the end of\nher breath, she saw that Mrs. Lacy was leaning back in her chair, her\neyes fixed as if her attention had gone away.  Kate thereupon roamed\nround the room, peeped from the window and saw that it looked into a dull\nblack-looking narrow garden, and then studied the things in the room.\nThere was a piano, at which she shook her head.  Mary had tried to teach\nher music; but after a daily fret for six weeks, Mr. Wardour had said it\nwas waste of time and temper for both; and Kate was delighted.  Then she\ncame to a book-case; and there the aunts had kindly placed the books of\ntheir own younger days, some of which she had never seen before.  When\nshe had once begun on the \"Rival Crusoes,\" she gave Mrs. Lacy no more\ntrouble, except to rouse her from it to drink her tea, and then go and be\ndressed.\n\nThe maid managed the white muslin so as to make her look very nice; but\nbefore she had gone half way down-stairs, there was a voice behind\u2014\"My\nLady! my Lady!\"\n\nShe did not turn, not remembering that she herself must be meant; and the\nmaid, running after her, caught her rather sharply, and showed her her\nown hand, all black and grimed.\n\n\"How tiresome!\" cried she.  \"Why, I only just washed it!\"\n\n\"Yes, my Lady; but you took hold of the balusters all the way down.  And\nyour forehead!  Bless me! what would Lady Barbara say?\"\n\nFor Kate had been trying to peep through the balusters into the hall\nbelow, and had of course painted her brow with London blacks.  She made\none of her little impatient gestures, and thought she was very hardly\nused\u2014dirt stuck upon her, and brambles tore her like no one else.\n\nShe got safely down this time, and went into the drawing-room with Mrs.\nLacy, there taking a voyage of discovery among the pretty things, knowing\nshe must not touch, but asking endless questions, some of which Mrs. Lacy\nanswered in her sad indifferent way, others she could not answer, and\nKate was rather vexed at her not seeming to care to know.  Kate had not\nyet any notion of caring for other people's spirits and feelings; she\nnever knew what to do for them, and so tried to forget all about them.\n\nThe aunts came in, and with them Mr. Wardour.  She was glad to run up to\nhim, and drag him to look at a group in white Parian under a glass, that\nhad delighted her very much.  She knew it was Jupiter's Eagle; but who\nwas feeding it?  \"Ganymede,\" said Mr. Wardour; and Kate, who always liked\nmythological stories, went on most eagerly talking about the legend of\nthe youth who was borne away to be the cup-bearer of the gods.  It was a\nthing to make her forget about the aunts and everybody else; and Mr.\nWardour helped her out, as he generally did when her talk was neither\nfoolish nor ill-timed but he checked her when he thought she was running\non too long, and went himself to talk to Mrs. Lacy, while Kate was\nobliged to come to her aunts, and stood nearest to Lady Jane, of whom she\nwas least afraid.\n\n\"You seem quite at home with all the heathen gods, my dear,\" said Lady\nJane; \"how come you to know them so well?\"\n\n\"In Charlie's lesson-books, you know,\" said Kate; and seeing that her\naunt did not know, she went on to say, \"there are notes and explanations.\nAnd there is a Homer\u2014an English one, you know; and we play at it.\"\n\n\"We seem to have quite a learned lady here!\" said aunt Barbara, in the\nvoice Kate did not like.  \"Do you learn music?\"\n\n\"No; I haven't got any ear; and I hate it!\"\n\n\"Oh!\" said Lady Barbara drily; and Kate seeing Mr. Wardour's eyes fixed\non her rather anxiously, recollected that hate was not a proper word, and\nfell into confusion.\n\n\"And drawing?\" said her aunt.\n\n\"No; but I want to\u2014\"\n\n\"Oh!\" again said Lady Barbara, looking at Kate's fingers, which in her\nawkwardness she was apparently dislocating in a method peculiar to\nherself.\n\nHowever, it was soon over, for it was already later than Kate's home\nbed-time; she bade everyone good-night, and was soon waited on by Mrs.\nBartley, the maid, in her own luxurious little room.\n\nBut luxurious as it was, Kate for the first time thoroughly missed home.\nThe boarded floor, the old crib, the deal table, would have been welcome,\nif only Sylvia had been there.  She had never gone to bed without Sylvia\nin her life.  And now she thought with a pang that Sylvia was longing for\nher, and looking at her empty crib, thinking too, it might be, that Kate\nhad cared more for her grandeur than for the parting.\n\nNot only was it sorrowful to be lonely, but also Kate was one of the\nsilly little girls, to whom the first quarter of an hour in bed was a\ntime of fright.  Sylvia had no fears, and always accounted for the odd\nnoises and strange sights that terrified her companion.  She never\nbelieved that the house was on fire, even though the moon made very\nbright sparkles; she always said the sounds were the servants, the wind,\nor the mice; and never would allow that thieves would steal little girls,\nor anything belonging to themselves.  Or if she were fast asleep, her\nvery presence gave a feeling of protection.\n\nBut when the preparations were very nearly over, and Kate began to think\nof the strange room, and the roar of carriages in the streets sounded so\nunnatural, her heart failed her, and the fear of being alone quite\noverpowered her dread of the grave staid Mrs. Bartley, far more of being\nthought a silly little girl.\n\n\"Please please, Mrs. Bartley,\" she said in a trembling voice, \"are you\ngoing away?\"\n\n\"Yes, my Lady; I am going down to supper, when I have placed my Lady\nJane's and my Lady Barbara's things.\"\n\n\"Then please\u2014please,\" said Kate, in her most humble and insinuating\nvoice, \"do leave the door open while you are doing it.\"\n\n\"Very well, my Lady,\" was the answer, in a tone just like that in which\nLady Barbara said \"Oh!\"\n\nAnd the door stayed open; but Kate could not sleep.  There seemed to be\nthe rattle and bump of the train going on in her bed; the gas-lights in\nthe streets below came in unnaturally, and the noises were much more\nfrightful and unaccountable than any she had ever heard at home.  Her\neyes spread with fright, instead of closing in sleep; then came the\nlonging yearning for Sylvia, and tears grew hot in them; and by the time\nMrs. Bartley had finished her preparations, and gone down, her distress\nhad grown so unbearable, that she absolutely began sobbing aloud, and\nscreaming, \"Papa!\"  She knew he would be very angry, and that she should\nhear that such folly was shameful in a girl of her age; but any anger\nwould be better than this dreadful loneliness.  She screamed louder and\nlouder; and she grew half frightened, half relieved, when she heard his\nstep, and a buzz of voices on the stairs; and then there he was, standing\nby her, and saying gravely, \"What is the matter, Kate?\"\n\n\"O Papa, Papa, I want\u2014I want Sylvia!\u2014I am afraid!\"  Then she held her\nbreath, and cowered under the clothes, ready for a scolding; but it was\nnot his angry voice.  \"Poor child!\" he said quietly and sadly.  \"You must\nput away this childishness, my dear.  You know that you are not really\nalone, even in a strange place.\"\n\n\"No, no, Papa; but I am afraid\u2014I cannot bear it!\"\n\n\"Have you said the verse that helps you to bear it, Katie?\"\n\n\"I could not say it without Sylvia.\"\n\nShe heard him sigh; and then he said, \"You must try another night, my\nKatie, and think of Sylvia saying it at home in her own room.  You will\nmeet her prayers in that way.  Now let me hear you say it.\"\n\nKate repeated, but half choked with sobs, \"I lay me down in peace,\" and\nthe rest of the calm words, with which she had been taught to lay herself\nin bed; but at the end she cried, \"O Papa, don't go!\"\n\n\"I must go, my dear: I cannot stay away from your aunts.  But I will tell\nyou what to do to-night, and other nights when I shall be away: say to\nyourself the ninety-first Psalm.  I think you know it\u2014'Whoso abideth\nunder the defence of the Most High\u2014'\"\n\n\"I think I do know it.\"\n\n\"Try to say it to yourself, and then the place will seem less dreary,\nbecause you will feel Who is with you.  I will look in once more before I\ngo away, and I think you will be asleep.\"\n\nAnd though Kate tried to stay awake for him, asleep she was.\n\n\n\n\nCHAPTER III.\n\n\nIN a very few days, Kate had been settled into the ways of the household\nin Bruton Street; and found one day so like another, that she sometimes\nasked herself whether she had not been living there years instead of\ndays.\n\nShe was always to be ready by half-past seven.  Her French maid,\nJosephine, used to come in at seven, and wash and dress her quietly, for\nif there were any noise Aunt Barbara would knock and be displeased.  Aunt\nBarbara rose long before that time, but she feared lest Aunt Jane should\nbe disturbed in her morning's sleep; and Kate thought she had the ears of\na dragon for the least sound of voice or laugh.\n\nAt half-past seven, Kate met Mrs. Lacy in the school-room, read the\nPsalms and Second Lesson, and learnt some answers to questions on the\nCatechism, to be repeated to Lady Barbara on a Sunday.  For so far from\nplaying at cards in a bird-of-paradise turban all Sunday, the aunts were\nquite as particular about these things as Mr. Wardour\u2014more inconveniently\nso, the countess thought; for he always let her answer his examinations\nout of her own head, and never gave her answers to learn by heart;\n\"Answers that I know before quite well,\" said Kate, \"only not made\ntiresome with fine words.\"\n\n\"That is not a right way of talking, Lady Caergwent,\" gravely said Mrs.\nLacy; and Kate gave herself an ill-tempered wriggle, and felt cross and\nrebellious.\n\nIt was a trial; but if Kate had taken it humbly, she would have found\nthat even the stiff hard words and set phrases gave accuracy to her\nideas; and the learning of the texts quoted would have been clear gain,\nif she had been in a meeker spirit.\n\nThis done, Mrs. Lacy gave her a music-lesson.  This was grievous work,\nfor the question was not how the learning should be managed, but whether\nthe thing should be learnt at all.\n\nKate had struggled hard against it.  She informed her aunts that Mary had\ntried to teach her for six weeks in vain, and that she had had a bad mark\nevery day; that Papa had said it was all nonsense, and that talents could\nnot be forced; and that Armyn said she had no more ear than an old\npea-hen.\n\nTo which Lady Barbara had gravely answered, that Mr. Wardour could decide\nas he pleased while Katharine was under his charge, but that it would be\nhighly improper that she should not learn the accomplishments of her\nstation.\n\n\"Only I can't learn,\" said Kate, half desperate; \"you will see that it is\nno use, Aunt Barbara.\"\n\n\"I shall do my duty, Katharine,\" was all the answer she obtained; and she\npinched her chair with suppressed passion.\n\nLady Barbara was right in saying that it was her duty to see that the\nchild under her charge learnt what is usually expected of ladies; and\nthough Kate could never acquire music enough to give pleasure to others,\nyet the training and discipline were likely not only to improve her ear\nand untamed voice, but to be good for her whole character\u2014that is, if she\nhad made a good use of them.  But in these times, being usually already\nout of temper with the difficult answers of the Catechism questions, and\nobliged to keep in her pettish feelings towards what concerned sacred\nthings, she let all out in the music lesson, and with her murmurs and her\ninattention, her yawns and her blunders, rendered herself infinitely more\ndull and unmusical than nature had made her, and was a grievous torment\nto poor Mrs. Lacy, and her patient, \"One, two, three\u2014now, my dear.\"\n\nKate thought it was Mrs. Lacy who tormented her!  I wonder which was the\nworse to the other!  At any rate, Mrs. Lacy's heavy eyes looked heavier,\nand she moved as though wearied out for the whole day by the time the\nclock struck nine, and released them; whilst her pupil, who never was\ncross long together, took a hop, skip, and jump, to the dining-room, and\nwas as fresh as ever in the eager hope that the post would bring a letter\nfrom home.\n\nLady Barbara read prayers in the dining-room at nine, and there\nbreakfasted with Kate and Mrs. Lacy, sending up a tray to Lady Jane in\nher bed-room.  Those were apt to be grave breakfasts; not like the merry\nmornings at home, when chatter used to go on in half whispers between the\nyounger ones, with laughs, breaking out in sudden gusts, till a little\nover-loudness brought one of Mary's good-natured \"Hushes,\" usually\nanswered with, \"O Mary, such fun!\"\n\nIt was Lady Barbara's time for asking about all the lessons of the day\nbefore; and though these were usually fairly done, and Mrs. Lacy was\nalways a kind reporter, it was rather awful; and what was worse, were the\nstrictures on deportment.  For it must be confessed, that Lady Caergwent,\nthough neatly and prettily made, with delicate little feet and hands, and\na strong upright back, was a remarkably awkward child; and the more she\nwas lectured, the more ungraceful she made herself\u2014partly from thinking\nabout it, and from fright making her abrupt, partly from being provoked.\nShe had never been so ungainly at Oldburgh; she never was half so awkward\nin the school-room, as she would be while taking her cup of tea from Lady\nBarbara, or handing the butter to her governess.  And was it not wretched\nto be ordered to do it again, and again, and again, (each time worse than\nthe last\u2014the fingers more crooked, the elbow more stuck out, the shoulder\nmore forward than before), when there was a letter in Sylvia's writing\nlying on the table unopened?\n\nAnd whereas it had been the fashion at St. James's Parsonage to compare\nKate's handing her plate to a chimpanzee asking for nuts, it was hard\nthat in Bruton Street these manners should be attributed to the barbarous\ncountry in which she had grown up!  But that, though Kate did not know\nit, was very much her own fault.  She could never be found fault with but\nshe answered again.  She had been scarcely broken of replying and\njustifying herself, even to Mr. Wardour, and had often argued with Mary\ntill he came in and put a sudden sharp stop to it; and now she usually\ndefended herself with \"Papa says\u2014\" or \"Mary says\u2014\" and though she really\nthought she spoke the truth, she made them say such odd things, that it\nwas no wonder Lady Barbara thought they had very queer notions of\neducation, and that her niece had nothing to do but to unlearn their\nlessons.  Thus:\n\n\"Katharine, easy-chairs were not meant for little girls to lounge in.\"\n\n\"Oh, Papa says he doesn't want one always to sit upright and stupid.\"\n\nSo Lady Barbara was left to suppose that Mr. Wardour's model attitude for\nyoung ladies was sitting upon one leg in an easy-chair, with the other\nfoot dangling, the forehead against the back, and the arm of the chair\nused as a desk!  How was she to know that this only meant that he had\nonce had the misfortune to express his disapproval of the high-backed\nlong-legged school-room chairs formerly in fashion?  In fact, Kate could\nhardly be forbidden anything without her replying that Papa or Mary\n_always_ let her do it; till at last she was ordered, very decidedly,\nnever again to quote Mr. and Miss Wardour, and especially not to call him\nPapa.\n\nKate's eyes flashed at this; and she was so angry, that no words would\ncome but a passionate stammering \"I can't\u2014I can't leave off; I won't!\"\n\nLady Barbara looked stern and grave.  \"You must be taught what is\nsuitable to your position, Lady Caergwent; and until you have learnt to\nfeel it yourself, I shall request Mrs. Lacy to give you an additional\nlesson every time you call Mr. Wardour by that name.\"\n\nAunt Barbara's low slow way of speaking when in great displeasure was a\nterrific thing, and so was the set look of her handsome mouth and eyes.\nKate burst into a violent fit of crying, and was sent away in dire\ndisgrace.  When she had spent her tears and sobs, she began to think over\nher aunt's cruelty and ingratitude, and the wickedness of trying to make\nher ungrateful too; and she composed a thrilling speech, as she called\nit\u2014\"Lady Barbara Umfraville, when the orphan was poor and neglected, my\nUncle Wardour was a true father to me.  You may tear me with wild horses\nere I will cease to give him the title of\u2014  No; and I will call him\npapa\u2014no, father\u2014with my last breath!\"\n\nWhat the countess might have done if Lady Barbara had torn her with wild\nhorses must remain uncertain.  It is quite certain that the mere fixing\nof those great dark eyes was sufficient to cut off Pa\u2014at its first\nsyllable, and turn it into a faltering \"my uncle;\" and that, though\nKate's heart was very sore and angry, she never, except once or twice\nwhen the word slipped out by chance, incurred the penalty, though she\nwould have respected herself more if she had been brave enough to bear\nsomething for the sake of showing her love to Mr. Wardour.\n\nAnd the fact was, that self-justification and carelessness of exact\ncorrectness of truth had brought all this upon her, and given her aunt\nthis bad opinion of her friends!\n\nBut this is going a long way from the description of Kate's days in\nBruton Street.\n\nAfter breakfast, she was sent out with Mrs. Lacy for a walk.  If she had\na letter from home, she read it while Josephine dressed her as if she had\nbeen a doll; or else she had a story book in hand, and was usually lost\nin it when Mrs. Lacy looked into her room to see if she were ready.\n\nTo walk along the dull street, and pace round and round the gardens in\nBerkeley Square, was not so entertaining as morning games in the garden\nwith Sylvia; and these were times of feeling very like a prisoner.  Other\nchildren in the gardens seemed to be friends, and played together; but\nthis the aunts had forbidden her, and she could only look on, and think\nof Sylvia and Charlie, and feel as if one real game of play would do her\nall the good in the world.\n\nTo be sure she could talk to Mrs. Lacy, and tell her about Sylvia, and\ndeliver opinions upon the characters in her histories and stories; but it\noften happened that the low grave \"Yes, my dear,\" showed by the very tone\nthat her governess had heard not a word; and at the best, it was dreary\nwork to look up and discourse to nothing but the black crape veil that\nMrs. Lacy always kept down.\n\n\"I cannot think why I should have a governess in affliction; it is very\nhard upon me!\" said Kate to herself.\n\nWhy did she never bethink herself how hard the afflictions were upon Mrs.\nLacy, and what good it would have done her if her pupil had tried to be\nlike a gentle little daughter to her, instead of merely striving for all\nthe fun she could get?\n\nThe lesson time followed.  Kate first repeated what she had learnt the\nday before; and then had a French master two days in the week; on two\nmore, one for arithmetic and geography; and on the other two, a drawing\nmaster.  She liked these lessons, and did well in all, as soon as she\nleft off citing Mary Wardour's pronunciations, and ways of doing sums.\nIndeed, she had more lively conversation with her French master, who was\na very good-natured old man, than with anyone else, except Josephine; and\nshe liked writing French letters for him to correct, making them be from\nthe imaginary little girls whom she was so fond of drawing, and sending\nthem to Sylvia.\n\nAfter the master was gone, Kate prepared for him for the next day, and\ndid a little Italian reading with Mrs. Lacy; after which followed reading\nof history, and needle-work.  Lady Barbara was very particular that she\nshould learn to work well, and was a good deal shocked at her very poor\nperformances.  \"She had thought that plain needle-work, at least, would\nbe taught in a clergyman's family.\"\n\n\"Mary tried to teach me; but she says all my fingers are thumbs.\"\n\nAnd so poor Mrs. Lacy found them.\n\nMrs. Lacy and her pupil dined at the ladies' luncheon; and this was\npleasanter than the breakfast, from the presence of Aunt Jane, whose kiss\nof greeting was a comforting cheering moment, and who always was so much\ndistressed and hurt at the sight of her sister's displeasure, that Aunt\nBarbara seldom reproved before her.  She always had a kind word to say;\nMrs. Lacy seemed brighter and less oppressed in the sound of her voice;\neveryone was more at ease; and when speaking to her, or waiting upon her,\nLady Barbara was no longer stern in manner nor dry in voice.  The meal\nwas not lively; there was nothing like the talk about parish matters, nor\nthe jokes that she was used to; and though she was helped first, and\nceremoniously waited on, she might not speak unless she was spoken to;\nand was it not very cruel, first to make everything so dull that no one\ncould help yawning, and then to treat a yawn as a dire offence?\n\nThe length of the luncheon was a great infliction, because all the time\nfrom that to three o'clock was her own.  It was a poor remnant of the\nentire afternoons which she and Sylvia had usually disposed of much as\nthey pleased; and even what there was of it, was not to be spent in the\nway for which the young limbs longed.  No one was likely to play at blind\nman's buff and hare and hounds in that house; and even her poor attempt\nat throwing her gloves or a pen-wiper against the wall, and catching them\nin the rebound, and her scampers up-stairs two steps at once, and runs\ndown with a leap down the last four steps, were summarily stopped, as\nunladylike, and too noisy for Aunt Jane.  Kate did get a private run and\nleap whenever she could, but never with a safe conscience; and that\nspoilt the pleasure, or made it guilty and alarmed.\n\nAll she could do really in peace was reading or drawing, or writing\nletters to Sylvia.  Nobody had interfered with any of these occupations,\nthough Kate knew that none of them were perfectly agreeable to Aunt\nBarbara, who had been heard to speak of children's reading far too many\nsilly story-books now-a-days, and had declared that the child would cramp\nher hand for writing or good drawing with that nonsense.\n\nHowever, Lady Jane had several times submitted most complacently to have\na whole long history in pictures explained to her, smiling very kindly,\nbut not apparently much the wiser.  And one, at least, of the old visions\nof wealth was fulfilled, for Kate's pocket-money enabled her to keep\nherself in story-books and unlimited white paper, as well as to set up a\npaint-box with real good colours.  But somehow, a new tale every week had\nnot half the zest that stories had when a fresh book only came into the\nhouse by rare and much prized chances; and though the paper was smooth,\nand the blue and red lovely, it was not half so nice to draw and paint as\nwith Sylvia helping, and the remains of Mary's rubbings for making\nilluminations; nay, Lily spoiling everything, and Armyn and Charlie\nlaughing at her were now remembered as ingredients in her pleasure; and\nshe would hardly have had the heart to go on drawing but that she could\nstill send her pictorial stories to Sylvia, and receive remarks on them.\nThere were no more Lady Ethelindas in flounces in Kate's drawings now;\nher heroines were always clergymen's daughters, or those of colonists\ncutting down trees and making the butter.\n\nAt three o'clock the carriage came to the door; and on Mondays and\nThursdays took Lady Caergwent and her governess to a mistress who taught\ndancing and calisthenic exercises, and to whom her aunts trusted to make\nher a little more like a countess than she was at present.  Those were\npoor Kate's black days of the week; when her feet were pinched, and her\narms turned the wrong way, as it seemed to her; and she was in perpetual\ndisgrace.  And oh, that polite disgrace!  Those wishes that her Ladyship\nwould assume a more aristocratic deportment, were so infinitely worse\nthan a good scolding!  Nothing could make it more dreadful, except Aunt\nBarbara's coming in at the end to see how she was getting on.\n\nThe aunts, when Lady Jane was well enough, used to take their drive while\nthe dancing lesson was in progress, and send the carriage afterwards to\nbring their niece home.  On the other days of the week, when it was fine,\nthe carriage set Mrs. Lacy and Kate down in Hyde Park for their walk,\nwhile the aunts drove about; and this, after the first novelty, was\nnearly as dull as the morning walk.  The quiet decorous pacing along was\nvery tiresome after skipping in the lanes at home; and once, when Mrs.\nLacy had let her run freely in Kensington Gardens, Lady Barbara was much\ndispleased with her, and said Lady Caergwent was too old for such habits.\n\nThere was no sight-seeing.  Kate had told Lady Jane how much she wished\nto see the Zoological Gardens and British Museum, and had been answered\nthat some day when she was very good Aunt Barbara would take her there;\nbut the day never came, though whenever Kate had been in no particular\nscrape for a little while, she hoped it was coming.  Though certainly\ndays without scrapes were not many: the loud tones, the screams of\nlaughing that betrayed her undignified play with Josephine, the\nattitudes, the skipping and jumping like the gambols of a calf, the\nwonderful tendency of her clothes to get into mischief\u2014all were\ncontinually bringing trouble upon her.\n\nIf a splash of mud was in the street, it always came on her stockings;\nher meals left reminiscences on all her newest dresses; her hat was\nalways blowing off; and her skirts curiously entangled themselves in\nrails and balusters, caught upon nails, and tore into ribbons; and though\nall the repairs fell to Josephine's lot, and the purchase of new garments\nwas no such difficulty as of old, Aunt Barbara was even more severe on\nsuch mishaps than Mary, who had all the trouble and expense of them.\n\nAfter the walk, Kate had lessons to learn for the next day\u2014poetry, dates,\ngrammar, and the like; and after them came her tea; and then her evening\ntoilette, when, as the aunts were out of hearing, she refreshed herself\nwith play and chatter with Josephine.  She was supposed to talk French to\nher; but it was very odd sort of French, and Josephine did not insist on\nits being better.  She was very good-natured, and thought \"Miladi\" had a\ndull life; so she allowed a good many things that a more thoughtful\nperson would have known to be inconsistent with obedience to Lady\nBarbara.\n\nWhen dressed, Kate had to descend to the drawing-room, and there await\nher aunts coming up from dinner.  She generally had a book of her own, or\nelse she read bits of those lying on the tables, till Lady Barbara caught\nher, and in spite of her protest that at home she might always read any\nbook on the table ordered her never to touch any without express\npermission.\n\nSometimes the aunts worked; sometimes Lady Barbara played and sang.  They\nwanted Kate to sit up as they did with fancy work, and she had a bunch of\nflowers in Berlin wool which she was supposed to be grounding; but she\nmuch disliked it, and seldom set three stitches when her aunts' eyes were\nnot upon her.  Lady Jane was a great worker, and tried to teach her some\npretty stitches; but though she began by liking to sit by the soft gentle\naunt, she was so clumsy a pupil, that Lady Barbara declared that her\nsister must not be worried, and put a stop to the lessons.  So Kate\nsometimes read, or dawdled over her grounding; or when Aunt Barbara was\nsinging, she would nestle up to her other aunt, and go off into some\ndreamy fancy of growing up, getting home to the Wardours, or having them\nto live with her at her own home; or even of a great revolution, in\nwhich, after the pattern of the French nobility, she should have to\nmaintain Aunt Jane by the labour of her hands!  What was to become of\nAunt Barbara was uncertain; perhaps she was to be in prison, and Kate to\nbring food to her in a little basket every day; or else she was to run\naway: but Aunt Jane was to live in a nice little lodging, with no one to\nwait on her but her dear little niece, who was to paint beautiful screens\nfor her livelihood, and make her coffee with her own hands.  Poor Lady\nJane!\n\nBed-time came at last\u2014horrible bed-time, with all its terrors!  At first\nKate persuaded Josephine and her light to stay till sleep came to put an\nend to them; but Lady Barbara came up one evening, declared that a girl\nof eleven years old must not be permitted in such childish nonsense, and\nordered Josephine to go down at once, and always to put out the candle as\nsoon as Lady Caergwent was in bed.\n\nLady Barbara would hardly have done so if she had known how much\nsuffering she caused; but she had always been too sensible to know what\nthe misery of fancies could be, nor how the silly little brain imagined\neverything possible and impossible; sometimes that thieves were breaking\nin\u2014sometimes that the house was on fire\u2014sometimes that she should be\nsmothered with pillows, like the princes in the Tower, for the sake of\nher title\u2014sometimes that the Gunpowder Plot would be acted under the\nhouse!\n\nMost often of all it was a thought that was not foolish and unreal like\nthe rest.  It was the thought that the Last Judgment might be about to\nbegin.  But Kate did not use that thought as it was meant to be used when\nwe are bidden to \"watch.\"  If she had done so, she would have striven\nevery morning to \"live this day as if the last.\"  But she never thought\nof it in the morning, nor made it a guide to her actions; or else she\nwould have dreaded it less.  And at night it did not make her particular\nabout obedience.  It only made her want to keep Josephine; as if\nJosephine and a candle could protect her from that Day and Hour!  And if\nthe moment had come, would she not have been safer trying to endure\nhardness for the sake of obedience\u2014with the holy verses Mr. Wardour had\ntaught her on her lips, alone with her God and her good angel\u2014than trying\nto forget all in idle chatter with her maid, and contrary to known\ncommands, detaining her by foolish excuses?\n\nIt is true that Kate did not feel as if obedience to Lady Barbara was the\nsame duty as obedience to \"Papa.\"  Perhaps it was not in the nature of\nthings that she should; but no one can habitually practise petty\ndisobedience to one \"placed in authority over\" her, without hurting the\nwhole disposition.\n\n\n\n\nCHAPTER IV.\n\n\n\"THURSDAY morning!  Bother\u2014calisthenic day!\u2014I'll go to sleep again, to\nput it off as long as I can.  If I was only a little countess in her own\nfeudal keep, I would get up in the dawn, and gather flowers in the May\ndew\u2014primroses and eglantine!\u2014Charlie says it is affected to call\nsweet-briar eglantine.\u2014Sylvia!  Sylvia! that thorn has got hold of me;\nand there's Aunt Barbara coming down the lane in the baker's jiggeting\ncart.\u2014Oh dear! was it only dreaming?  I thought I was gathering dog-roses\nwith Charlie and Sylvia in the lane; and now it is only Thursday, and\nhorrid calisthenic day!  I suppose I must wake up.\n\n    'Awake, my soul, and with the sun\n    Thy daily stage of duty run.'\n\nI'm sure it's a very tiresome sort of stage!  We used to say, 'As happy\nas a queen:' I am sure if the Queen is as much less happy than a countess\nas I am than a common little girl, she must be miserable indeed!  It is\nlike a rule-of-three sum.  Let me see\u2014if a common little girl has one\nhundred happinesses a day, and a countess only\u2014only five\u2014how many has the\nQueen?  No\u2014but how much higher is a queen than a countess?  If I were\nQueen, I would put an end to aunts and to calisthenic exercises; and I\nwould send for all my orphan nobility, and let them choose their own\ngovernesses and playfellows, and always live with country clergymen!  I\nam sure nobody ought to be oppressed as Aunt Barbara oppresses me: it is\njust like James V. of Scotland when the Douglases got hold of him!  I\nwonder what is the use of being a countess, if one never is to do\nanything to please oneself, and one is to live with a cross old aunt!\"\n\nMost likely everyone is of Lady Caergwent's morning opinion\u2014that Lady\nBarbara Umfraville was cross, and that it was a hard lot to live in\nsubjection to her.  But there are two sides to a question; and there were\nother hardships in that house besides those of the Countess of Caergwent.\n\nForty years ago, two little sisters had been growing up together, so fond\nof each other that they were like one; and though the youngest, Barbara,\nwas always brighter, stronger, braver, and cleverer, than gentle Jane,\nshe never enjoyed what her sister could not do; and neither of them ever\nwanted any amusement beyond quiet play with their dolls and puzzles,\ncontrivances in pretty fancy works, and walks with their governess in\ntrim gravel paths.  They had two elder brothers and one younger; but they\nhad never played out of doors with them, and had not run about or romped\nsince they were almost babies; they would not have known how; and Jane\nwas always sickly and feeble, and would have been very unhappy with loud\nor active ways.\n\nAs time passed on, Jane became more weakly and delicate while Barbara\ngrew up very handsome, and full of life and spirit, but fonder of her\nsister than ever, and always coming home from her parties and gaieties,\nas if telling Jane about them was the best part of all.\n\nAt last, Lady Barbara was engaged to be married to a brother officer of\nher second brother, James; but just then poor Jane fell so ill, that the\ndoctors said she could not live through the year.  Barbara loved her\nsister far too well to think of marrying at such a time, and said she\nmust attend to no one else.  All that winter and spring she was nursing\nher sister day and night, watching over her, and quite keeping up the\nlittle spark of life, the doctors said, by her tender care.  And though\nLady Jane lived on day after day, she never grew so much better as to be\nfit to hear of the engagement and that if she recovered her sister would\nbe separated from her; and so weeks went on, and still nothing could be\ndone about the marriage.\n\nAs it turned out, this was the best thing that could have happened to\nLady Barbara; for in the course of this time, it came to her father's\nknowledge that her brother and her lover had both behaved disgracefully,\nand that, if she had married, she must have led a very unhappy life.  He\ncaused the engagement to be broken off.  She knew it was right, and made\nno complaint to anybody; but she always believed that it was her brother\nJames who had been the tempter, who had led his friend astray; and from\nthat time, though she was more devoted than ever to her sick sister, she\nwas soft and bright to nobody else.  She did not complain, but she\nthought that things had been very hard with her; and when people repine\ntheir troubles do not make them kinder, but the brave grow stern and the\nsoft grow fretful.\n\nAll this had been over for nearly thirty years, and the brother and the\nfriend had both been long dead.  Lady Barbara was very anxious to do all\nthat she thought right; and she was so wise and sensible, and so careful\nof her sister Jane, that all the family respected her and looked up to\nher.  She thought she had quite forgiven all that had passed: she did not\nknow why it was so hard to her to take any notice of her brother James's\nonly son.  Perhaps, if she had, she would have forced herself to try to\nbe more warm and kind to him, and not have inflamed Lord Caergwent's\ndispleasure when he married imprudently.  Her sister Jane had never known\nall that had passed: she had been too ill to hear of it at the time; and\nit was not Lady Barbara's way to talk to other people of her own\ntroubles.  But Jane was always led by her sister, and never thought of\npeople, or judged events, otherwise than as Barbara told her; so that,\nkind and gentle as she was by nature, she was like a double of her\nsister, instead of by her mildness telling on the family counsels.  The\nother brother, Giles, had been aware of all, and saw how it was; but he\nwas so much younger than the rest, that he was looked on by them like a\nboy long after he was grown up, and had not felt entitled to break\nthrough his sister Barbara's reserve, so as to venture on opening out the\nsorrows so long past, and pleading for his brother James's family, though\nhe had done all he could for them himself.  He had indeed been almost\nconstantly on foreign service, and had seen very little of his sisters.\n\nSince their father's death, the two sisters had lived their quiet life\ntogether.  They were just rich enough to live in the way they thought the\nduty of persons in their rank, keeping their carriage for Lady Jane's\ndaily drive, and spending two months every year by the sea, and one at\nCaergwent Castle with their eldest brother.  They always had a spare room\nfor any old friend who wanted to come up to town; and they did many acts\nof kindness, and gave a great deal to be spent on the poor of their\nparish.  They did the same quiet things every day: one liked what the\nother liked; and Lady Barbara thought, morning, noon, and night, what\nwould be good for her sister's health; while Lady Jane rested on\nBarbara's care, and was always pleased with whatever came in her way.\n\nAnd so the two sisters had gone on year after year, and were very happy\nin their own way, till the great grief came of losing their eldest\nbrother; and not long after him, his son, the nephew who had been their\ngreat pride and delight, and for whom they had so many plans and hopes.\n\nAnd with his death, there came what they felt to be the duty and\nnecessity of trying to fit the poor little heiress for her station.  They\nwere not fond of any children; and it upset all their ways very much to\nhave to make room for a little girl, her maid, and her governess; but\nstill, if she had been such a little girl as they had been, and always\nlike the well-behaved children whom they saw in drawing-rooms, they would\nhave known what kind of creature had come into their hands.\n\nBut was it not very hard on them that their niece should turn out a\nlittle wild harum-scarum creature, such as they had never dreamt\nof\u2014really unable to move without noises that startled Lady Jane's nerves,\nand threw Lady Barbara into despair at the harm they would do\u2014a child\nwhose untutored movements were a constant eye-sore and distress to them;\nand though she could sometimes be bright and fairy-like if unconstrained,\nalways grew abrupt and uncouth when under restraint\u2014a child very far from\nsilly, but apt to say the silliest things\u2014learning quickly all that was\nmere head-work, but hopelessly or obstinately dull at what was to be done\nby the fingers\u2014a child whose ways could not be called vulgar, but would\nhave been completely tom-boyish, except for a certain timidity that\ndeprived them of the one merit of courage, and a certain frightened\nconsciousness that was in truth modesty, though it did not look like it?\nTo have such a being to endure, and more than that, to break into the\nhabits of civilized life, and the dignity of a lady of rank, was no small\nburden for them; but they thought it right, and made up their minds to\nbear it.\n\nOf course it would have been better if they had taken home the little\norphan when she was destitute and an additional weight to Mr. Wardour;\nand had she been actually in poverty or distress, with no one to take\ncare of her, Lady Barbara would have thought it a duty to provide for\nher: but knowing her to be in good hands, it had not then seemed needful\nto inflict the child on her sister, or to conquer her own distaste to all\nconnected with her unhappy brother James.  No one had ever thought of the\nlittle Katharine Aileve Umfraville becoming the head of the family; for\nthen young Lord Umfraville was in his full health and strength.\n\nAnd why _did_ Lady Barbara only now feel the charge of the child a duty?\nPerhaps it was because, without knowing it, she had been brought up to\nmake an idol of the state and consequence of the earldom, since she\nthought breeding up the girl for a countess incumbent on her, when she\nhad not felt tender compassion for the brother's orphan grandchild.  So\nsomewhat of the pomps of this world may have come in to blind her eyes;\nbut whatever she did was because she thought it right to do, and when\nKate thought of her as cross, it was a great mistake.  Lady Barbara had\ngreat control of temper, and did everything by rule, keeping herself as\nstrictly as she did everyone else except Lady Jane; and though she could\nnot like such a troublesome little incomprehensible wild cat as\nKatharine, she was always trying to do her strict justice, and give her\nwhatever in her view was good or useful.\n\nBut Kate esteemed it a great holiday, when, as sometimes happened, Aunt\nBarbara went out to spend the evening with some friends; and she, under\npromise of being very good, used to be Aunt Jane's companion.\n\nThose were the times when her tongue took a holiday, and it must be\nconfessed, rather to the astonishment and confusion of Lady Jane.\n\n\"Aunt Jane, do tell me about yourself when you were a little girl?\"\n\n\"Ah! my dear, that does not seem so very long ago.  Time passes very\nquickly.  To think of such a great girl as you being poor James's\ngrandchild!\"\n\n\"Was my grandpapa much older than you, Aunt Jane?\"\n\n\"Only three years older, my dear.\"\n\n\"Then do tell me how you played with him?\"\n\n\"I never did, my dear; I played with your Aunt Barbara.\"\n\n\"Dear me how stupid!  One can't do things without boys.\"\n\n\"No, my dear; boys always spoil girls' play, they are so rough.\"\n\n\"Oh! no, no, Aunt Jane; there's no fun unless one is rough\u2014I mean, not\nrough exactly; but it's no use playing unless one makes a jolly good\nnoise.\"\n\n\"My dear,\" said Lady Jane, greatly shocked, \"I can't bear to hear you\ntalk so, nor to use such words.\"\n\n\"Dear me, Aunt Jane, we say 'Jolly' twenty times a day at St. James's,\nand nobody minds.\"\n\n\"Ah! yes, you see you played with boys.\"\n\n\"But our boys are not rough, Aunt Jane,\" persisted Kate, who liked\nhearing herself talk much better than anyone else.  \"Mary says Charlie is\na great deal less riotous than I am, especially since he went to school;\nand Armyn is too big to be riotous.  Oh dear, I wish Mr. Brown would send\nArmyn to London; he said he would be sure to come and see me, and he is\nthe jolliest, most delightful fellow in the world!\"\n\n\"My dear child,\" said Lady Jane in her soft, distressed voice, \"indeed\nthat is not the way young ladies talk of\u2014of\u2014boys.\"\n\n\"Armyn is not a boy, Aunt Jane; he's a man.  He is a clerk, you know, and\nwill get a salary in another year.\"\n\n\"A clerk!\"\n\n\"Yes; in Mr. Brown's office, you know.  Aunt Jane, did you ever go out to\ntea?\"\n\n\"Yes, my dear; sometimes we drank tea with our little friends in the\ndolls' tea-cups.\"\n\n\"Oh! you can't think what fun we have when Mrs. Brown asks us to tea.\nShe has got the nicest garden in the world, and a greenhouse, and a great\nsquirt-syringe, I mean, to water it; and we always used to get it, till\nonce, without meaning it, I squirted right through the drawing-room\nwindow, and made such a puddle; and Mrs. Brown thought it was Charlie,\nonly I ran in and told of myself, and Mrs. Brown said it was very\ngenerous, and gave me a Venetian weight with a little hermit in a\nsnow-storm; only it is worn out now, and won't snow, so I gave it to\nlittle Lily when we had the whooping-cough.\"\n\nBy this time Lady Jane was utterly ignorant what the gabble was about,\nexcept that Katharine had been in very odd company, and done very strange\nthings with those boys, and she gave a melancholy little sound in the\npause; but Kate, taking breath, ran on again\u2014\n\n\"It is because Mrs. Brown is not used to educating children, you know,\nthat she fancies one wants a reward for telling the truth; I told her so,\nbut Mary thought it would vex her, and stopped my mouth.  Well, then we\nyoung ones\u2014that is, Charlie, and Sylvia, and Armyn, and I\u2014drank tea out\non the lawn.  Mary had to sit up and be company; but we had such fun!\nThere was a great old laurel tree, and Armyn put Sylvia and me up into\nthe fork; and that was our nest, and we were birds, and he fed us with\nstrawberries; and we pretended to be learning to fly, and stood up\nflapping our frocks and squeaking, and Charlie came under and danced the\nbranches about.  We didn't like that; and Armyn said it was a shame, and\nhunted him away, racing all round the garden; and we scrambled down by\nourselves, and came down on the <DW72>.  It is a long green <DW72>, right\ndown to the river, all smooth and turfy, you know; and I was standing at\nthe top, when Charlie comes slyly, and saying he would help the little\nbird to fly, gave me one push, and down I went, roll, roll, tumble,\ntumble, till Sylvia _really_ thought she heard my neck crack!  Wasn't it\nfun?\"\n\n\"But the river, my dear!\" said Lady Jane, shuddering.\n\n\"Oh! there was a good flat place before we came to the river, and I\nstopped long before that!  So then, as we had been the birds of the air,\nwe thought we would be the fishes of the sea; and it was nice and\nshallow, with dear little caddises and river cray-fish, and great British\npearl-shells at the bottom.  So we took off our shoes and stockings, and\nCharlie and Armyn turned up their trousers, and we had such a nice\npaddling.  I really thought I should have got a British pearl then; and\nyou know there were some in the breast-plate of Venus.\"\n\n\"In the river!  Did your cousin allow that?\"\n\n\"Oh yes; we had on our old blue checks; and Mary never minds anything\nwhen Armyn is there to take care of us.  When they heard in the\ndrawing-room what we had been doing, they made Mary sing 'Auld Lang\nSyne,' because of 'We twa hae paidlit in the burn frae morning sun till\ndine;' and whenever in future times I meet Armyn, I mean to say,\n\n    'We twa hae paidlit in the burn\n       Frae morning sun till dine;\n    We've wandered many a weary foot\n       Sin auld lang syne.'\n\nOr perhaps I shall be able to sing it, and that will be still prettier.\"\n\nAnd Kate sat still, thinking of the prettiness of the scene of the\nstranger, alone in the midst of numbers, in the splendid drawing-rooms,\nhearing the sweet voice of the lovely young countess at the piano,\nsinging this touching memorial of the simple days of childhood.\n\nLady Jane meanwhile worked her embroidery, and thought what wonderful\ndisadvantages the poor child had had, and that Barbara really must not be\ntoo severe on her, after she had lived with such odd people, and that it\nwas very fortunate that she had been taken away from them before she had\ngrown any older, or more used to them.\n\nSoon after, Kate gave a specimen of her manners with boys.  When she went\ninto the dining-room at luncheon time one wet afternoon, she heard steps\non the stairs behind her aunt's, and there appeared a very\npleasant-looking gentleman, followed by a boy of about her own age.\n\n\"Here is our niece,\" said Lady Barbara.  \"Katharine, come and speak to\nLord de la Poer.\"\n\nKate liked his looks, and the way in which he held out his hand to her;\nbut she knew she should be scolded for her awkward greeting: so she put\nout her hand as if she had no use of her arm above the elbow, hung down\nher head, and said \"\u2014do;\" at least no more was audible.\n\nBut there was something comfortable and encouraging in the grasp of the\nstrong large hand over the foolish little fingers; and he quite gave them\nto his son, whose shake was a real treat; the contact with anything young\nwas like meeting a follow-countryman in a foreign land, though neither as\nyet spoke.\n\nShe found out that the boy's name was Ernest, and that his father was\ntaking him to school, but had come to arrange some business matters for\nher aunts upon the way.  She listened with interest to Lord de la Poer's\nvoice, for she liked it, and was sure he was a greater friend there than\nany she had before seen.  He was talking about Giles\u2014that was her uncle,\nthe Colonel in India; and she first gathered from what was passing that\nher uncle's eldest and only surviving son, an officer in his own\nregiment, had never recovered a wound he had received at the relief of\nLucknow, and that if he did not get better at Simlah, where his mother\nhad just taken him, his father thought of retiring and bringing him home,\nthough all agreed that it would be a very unfortunate thing that the\nColonel should be obliged to resign his command before getting promoted;\nbut they fully thought he would do so, for this was the last of his\nchildren; another son had been killed in the Mutiny, and two or three\nlittle girls had been born and died in India.\n\nKate had never known this.  Her aunts never told her anything, nor talked\nover family affairs before her; and she was opening her ears most\neagerly, and turning her quick bright eyes from one speaker to the other\nwith such earnest attention, that the guest turned kindly to her, and\nsaid, \"Do you remember your uncle?\"\n\n\"Oh dear no!  I was a little baby when he went away.\"\n\nKate never used _dear_ as an adjective except at the beginning of a\nletter, but always, and very unnecessarily, as an interjection; and this\ntime it was so emphatic as to bring Lady Barbara's eyes on her.\n\n\"Did you see either Giles or poor Frank before they went out to him?\"\n\n\"Oh dear no!\"\n\nThis time the _dear_ was from the confusion that made her always do the\nvery thing she ought not to do.\n\n\"No; my niece has been too much separated from her own relations,\" said\nLady Barbara, putting this as an excuse for the \"Oh dears.\"\n\n\"I hope Mr. Wardour is quite well,\" said Lord de la Poer, turning again\nto Kate.\n\n\"Oh yes, quite, thank you;\" and then with brightening eyes, she ventured\non \"Do you know him?\"\n\n\"I saw him two or three times,\" he answered with increased kindness of\nmanner.  \"Will you remember me to him when you write?\"\n\n\"Very well,\" said Kate promptly; \"but he says all those sort of things\nare nonsense.\"\n\nThe horror of the two aunts was only kept in check by the good manners\nthat hindered a public scolding; but Lord de la Poer only laughed\nheartily, and said, \"Indeed!  What sort of things, may I ask, Lady\nCaergwent?\"\n\n\"Why\u2014love, and regards, and remembrances.  Mary used to get letters from\nher school-fellows, all filled with dearest loves, and we always laughed\nat her; and Armyn used to say them by heart beforehand,\" said Kate.\n\n\"I beg to observe,\" was the answer, in the grave tone which, however,\nKate understood as fun, \"that I did not presume to send my love to Mr.\nWardour.  May not that make the case different?\"\n\n\"Yes,\" said Kate meditatively; \"only I don't know that your remembrance\nwould be of more use than your love.\"\n\n\"And are we never to send any messages unless they are of use?\"  This was\na puzzling question, and Kate did not immediately reply.\n\n\"None for pleasure\u2014eh?\"\n\n\"Well, but I don't see what would be the pleasure.\"\n\n\"What, do you consider it pleasurable to be universally forgotten?\"\n\n\"Nobody ever could forget Pa\u2014my Uncle Wardour,\" cried Kate, with eager\nvehemence flashing in her eyes.\n\n\"Certainly not,\" said Lord de la Poer, in a voice as if he were much\npleased with her; \"he is not a man to be forgotten.  It is a privilege to\nhave been brought up by him.  But come, Lady Caergwent, since you are so\ncritical, will you be pleased to devise some message for me, that may\ncombine use, pleasure, and my deep respect for him?\" and as she sat\nbeside him at the table, he laid his hand on hers, so that she felt that\nhe really meant what he said.\n\nShe sat fixed in deep thought; and her aunts, who had been miserable all\nthrough the conversation, began to speak of other things; but in the\nmidst the shrill little voice broke in, \"I know what!\" and good-natured\nLord de la Poer turned at once, smiling, and saying, \"Well, what?\"\n\n\"If you would help in the new aisle!  You know the church is not big\nenough; there are so many people come into the district, with the new\nironworks, you know; and we have not got half room enough, and can't make\nmore, though we have three services; and we want to build a new aisle,\nand it will cost \u00a3250, but we have only got \u00a3139 15_s._ 6_d._  And if you\nwould but be so kind as to give one sovereign for it\u2014that would be better\nthan remembrances and respects, and all that sort of thing.\"\n\n\"I rather think it would,\" said Lord de la Poer; and though Lady Barbara\neagerly exclaimed, \"Oh! do not think of it; the child does not know what\nshe is talking of.  Pray excuse her\u2014\" he took out his purse, and from it\ncame a crackling smooth five-pound note, which he put into the hand,\nsaying, \"There, my dear, cut that in two, and send the two halves on\ndifferent days to Mr. Wardour, with my best wishes for his success in his\ngood works.  Will that do?\"\n\nKate turned quite red, and only perpetrated a choked sound of her\nfavourite \u2014q.  For the whole world she could not have said more: but\nthough she knew perfectly well that anger and wrath were hanging over\nher, she felt happier than for many a long week.\n\nPresently the aunts rose, and Lady Barbara said to her in the low\nceremonious voice that was a sure sign of warning and displeasure, \"You\nhad better come up stairs with us, Katharine, and amuse Lord Ernest in\nthe back drawing-room while his father is engaged with us.\"\n\nKate's heart leapt up at the sound \"amuse.\"  She popped her precious note\ninto her pocket, bounded up-stairs, and opened the back drawing-room door\nfor her playfellow, as he brought up the rear of the procession.\n\nLord de la Poer and Lady Barbara spread the table with papers; Lady Jane\nsat by; the children were behind the heavy red curtains that parted off\nthe second room.  There was a great silence at first, then began a little\ntittering, then a little chattering, then presently a stifled explosion.\nLady Barbara began to betray some restlessness; she really must see what\nthat child was about.\n\n\"No, no,\" said Lord de la Poer; \"leave them in peace.  That poor girl\nwill never thrive unless you let her use her voice and limbs.  I shall\nmake her come over and enjoy herself with my flock when we come up _en\nmasse_.\"\n\nThe explosions were less carefully stifled, and there were some sounds of\nrushing about, some small shrieks, and then the door shut, and there was\na silence again.\n\nBy this it may be perceived that Kate and Ernest had become tolerably\nintimate friends.  They had informed each other of what games were their\nfavourites; Kate had told him the Wardour names and ages; and required\nfrom him in return those of his brothers and sisters.  She had been\ngreatly delighted by learning that Adelaide was no end of a hand at\nclimbing trees; and that whenever she should come and stay at their\nhouse, Ernest would teach her to ride.  And then they began to consider\nwhat play was possible under the present circumstances\u2014beginning they\nhardly knew how, by dodging one another round and round the table, making\nsnatches at one another, gradually assuming the characters of hunter and\nRed Indian.  Only when the hunter had snatched up Aunt Jane's\ntortoise-shell paper-cutter to stab with, complaining direfully that it\nwas a stupid place, with nothing for a gun, and the Red Indian's\ncrinoline had knocked down two chairs, she recollected the consequences\nin time to strangle her own war-whoop, and suggested that they should be\nsafer on the stairs; to which Ernest readily responded, adding that there\nwas a great gallery at home all full of pillars and statues, the jolliest\nplace in the world for making a row.\n\n\"Oh dear! oh dear! how I hope I shall go there!\" cried Kate, swinging\nbetween the rails of the landing-place.  \"I do want of all things to see\na statue.\"\n\n\"A statue! why, don't you see lots every day?\"\n\n\"Oh!  I don't mean great equestrian things like the Trafalgar Square\nones, or the Duke\u2014or anything big and horrid, like Achilles in the Park,\nholding up a shield like a green umbrella.  I want to see the work of the\ngreat sculptor Julio Romano.\"\n\n\"He wasn't a sculptor.\"\n\n\"Yes, he was; didn't he sculp\u2014no, what is the word\u2014Hermione.  No; I mean\nthey pretended he had done her.\"\n\n\"Hermione!  What, have you seen the 'Winter's Tale?'\"\n\n\"Papa\u2014Uncle Wardour, that is\u2014read it to us last Christmas.\"\n\n\"Well, I've seen it.  Alfred and I went to it last spring with our\ntutor.\"\n\n\"Oh! then do, pray, let us play at it.  Look, there's a little stand up\nthere, where I have always so wanted to get up and be Hermione, and\ndescend to the sound of slow music.  There's a musical-box in the back\ndrawing-room that will make the music.\n\n\"Very well; but I must be the lion and bear killing the courtier.\"\n\n\"O yes\u2014very well, and I'll be courtier; only I must get a sofa-cushion to\nbe Perdita.\"\n\n\"And where's Bohemia?\"\n\n\"Oh! the hall must be Bohemia, and the stair-carpet the sea, because then\nthe aunts won't hear the lion and bear roaring.\"\n\nWith these precautions, the characteristic roaring and growling of lion\nand bear, and the shrieks of the courtier, though not absolutely unheard\nin the drawing-room, produced no immediate results.  But in the very\nmidst of Lady Jane's signing her name to some paper, she gave a violent\nstart, and dropped the pen, for they were no stage shrieks\u2014\"Ah! ah!  It\nis coming down!  Help me down!  Ernest, Ernest! help me down!  Ah!\"\u2014and\nthen a great fall.\n\nThe little mahogany bracket on the wall had been mounted by the help of a\nchair, but it was only fixed into the plaster, being intended to hold a\nsmall lamp, and not for young ladies to stand on; so no sooner was the\nchair removed by which Kate had mounted, than she felt not only giddy in\nher elevation, but found her pedestal loosening!  There was no room to\njump; and Ernest, perhaps enjoying what he regarded as a girl's foolish\nfright, was a good way off, endeavouring to wind up the musical-box, when\nthe bracket gave way, and Hermione descended precipitately with anything\nbut the sound of soft music; and as the inhabitants of the drawing-room\nrushed out to the rescue, her legs were seen kicking in the air upon the\nlanding-place; Ernest looking on, not knowing whether to laugh or be\ndismayed.\n\nLord de la Poer picked her up, and sat down on the stairs with her\nbetween his knees to look her over and see whether she were hurt, or what\nwas the matter, while she stood half sobbing with the fright and shock.\nHe asked his son rather severely what he had been doing to her.\n\n\"He did nothing,\" gasped Kate; \"I was only Hermione.\"\n\n\"Yes, that's all, Papa,\" repeated Ernest; \"it is all the fault of the\nplaster.\"\n\nAnd a sort of explanation was performed between the two children, at\nwhich Lord de la Poer could hardly keep his gravity, though he was\nsomewhat vexed at the turn affairs had taken.  He was not entirely devoid\nof awe of the Lady Barbara, and would have liked his children to be on\ntheir best behaviour before her.\n\n\"Well,\" he said, \"I am glad there is no worse harm done.  You had better\ndefer your statueship till we can find you a sounder pedestal, Lady\nCaergwent.\"\n\n\"Oh! call me Kate,\" whispered she in his ear, turning redder than the\nfright had made her.\n\nHe smiled, and patted her hand; then added, \"We must go and beg pardon, I\nsuppose; I should not wonder if the catastrophe had damaged Aunt Jane the\nmost; and if so, I don't know what will be done to us!\"\n\nHe was right; Lady Barbara had only satisfied herself that no bones had\nbeen broken, and then turned back to reassure her sister; but Lady Jane\ncould not be frightened without suffering for it, and was lying back on\nthe sofa, almost faint with palpitation, when Lord de la Poer, with\nKate's hand in his, came to the door, looking much more consciously\nguilty than his son, who on the whole was more diverted than penitent at\nthe commotion they had made.\n\nLady Barbara looked very grand and very dignified, but Lord de la Poer\nwas so grieved for Lady Jane's indisposition, that she was somewhat\nsoftened; and then he began asking pardon, blending himself with the\nchildren so comically, that in all her fright and anxiety, Kate wondered\nhow her aunt could help laughing.\n\nIt never was Lady Barbara's way to reprove before a guest; but this good\ngentleman was determined that she should not reserve her displeasure for\nhis departure, and he would not go away till he had absolutely made her\npromise that his little friend, as he called Kate, should hear nothing\nmore about anything that had that day taken place.\n\nLady Barbara kept her promise.  She uttered no reproof either on her\nniece's awkward greeting, her abrupt conversation and its tendency to\npertness, nor on the loudness of the unlucky game and the impropriety of\nclimbing; nor even on what had greatly annoyed her, the asking for the\nsubscription to the church.  There was neither blame nor punishment; but\nshe could not help a certain cold restraint of manner, by which Kate knew\nthat she was greatly displeased, and regarded her as the most hopeless\nlittle saucy romp that ever maiden aunt was afflicted with.\n\nAnd certainly it was hard on her.  She had a great regard for Lord de la\nPoer, and thought his a particularly well trained family; and she was\nespecially desirous that her little niece should appear to advantage\nbefore him.  Nothing, she was sure, but Katharine's innate naughtiness\ncould have made that well-behaved little Ernest break out into rudeness;\nand though his father had shown such good nature, he must have been very\nmuch shocked.  What was to be done to tame this terrible little savage,\nwas poor Lady Barbara's haunting thought, morning, noon, and night!\n\nAnd what was it that Kate did want?  I believe nothing could have made\nher perfectly happy, or suited to her aunt; but that she would have been\ninfinitely happier and better off had she had the spirit of obedience, of\nhumility, or of unselfishness.\n\n\n\n\nCHAPTER V.\n\n\nTHE one hour of play with Ernest de la Poer had the effect of making Kate\nlong more and more for a return of \"fun,\" and of intercourse with beings\nof her own age and of high spirits.\n\nShe wove to herself dreams of possible delights with Sylvia and Charlie,\nif the summer visit could be paid to them; and at other times she\nimagined her Uncle Giles's two daughters still alive, and sent home for\neducation, arranging in her busy brain wonderful scenes, in which she,\nwith their assistance, should be happy in spite of Aunt Barbara.\n\nThese fancies, however, would be checked by the recollection, that it was\nshocking to lower two happy spirits in Heaven into playful little girls\nupon earth; and she took refuge in the thought of the coming chance of\nplayfellows, when Lord de la Poer was to bring his family to London.  She\nhad learnt the names and ages of all the ten; and even had her own\ntheories as to what her contemporaries were to be like\u2014Mary and Fanny,\nErnest's elders, and Adelaide and Grace, who came next below him; she had\na vision for each of them, and felt as if she already knew them.\n\nMeanwhile, the want of the amount of air and running about to which she\nhad been used, did really tell upon her; she had giddy feelings in the\nmorning, tired limbs, and a weary listless air, and fretted over her\nlessons at times.  So they showed her to the doctor, who came to see Lady\nJane every alternate day; and when he said she wanted more exercise, her\nmorning walk was made an hour longer, and a shuttlecock and battledores\nwere bought, with which it was decreed that Mrs. Lacy should play with\nher for exactly half an hour every afternoon, or an hour when it was too\nwet to go out.\n\nIt must be confessed that this was a harder task to both than the music\nlessons.  Whether it were from the difference of height, or from Kate's\ninnate unhandiness, they never could keep that unhappy shuttlecock up\nmore than three times; and Mrs. Lacy looked as grave and melancholy all\nthe time as if she played it for a punishment, making little efforts to\nbe cheerful that were sad to see.  Kate hated it, and was always cross;\nand willingly would they have given it up by mutual consent, but the\ninstant the tap of the cork against the parchment ceased, if it were not\nhalf-past five, down sailed Lady Barbara to inquire after her\nprescription.\n\nShe had been a famous battledore-player in the galleries of Caergwent\nCastle; and once when she took up the battledore to give a lesson, it\nseemed as if, between her and Mrs. Lacy, the shuttlecock would not come\ndown\u2014they kept up five hundred and eighty-one, and then only stopped\nbecause it was necessary for her to go to dinner.\n\nShe could not conceive anyone being unable to play at battledore, and\nthought Kate's failures and dislike pure perverseness.  Once Kate by\naccident knocked her shuttlecock through the window, and hoped she had\ngot rid of it; but she was treated as if she had done it out of\nnaughtiness, and a new instrument of torture, as she called it, was\nbought for her.\n\nIt was no wonder she did not see the real care for her welfare, and\nthought this intensely cruel and unkind; but it was a great pity that she\nvisited her vexation on poor Mrs. Lacy, to whom the game was even a\ngreater penance than to herself, especially on a warm day, with a bad\nheadache.\n\nEven in her best days at home, Kate had resisted learning to take thought\nfor others.  She had not been considerate of Mary's toil, nor of Mr.\nWardour's peace, except when Armyn or Sylvia reminded her; and now that\nshe had neither of them to put it into her mind, she never once thought\nof her governess as one who ought to be spared and pitied.  Yet if she\nhad been sorry for Mrs. Lacy, and tried to spare her trouble and\nannoyance, how much irritability and peevishness, and sense of constant\nnaughtiness, would have been prevented!  And it was that feeling of being\nalways naughty that was what had become the real dreariness of Kate's\npresent home, and was far worse than the music, the battledore, or even\nthe absence of fun.\n\nAt last came a message that Lady Caergwent was to be dressed for going\nout to make a call with Lady Barbara as soon as luncheon was over.\n\nIt could be on no one but the De la Poers; and Kate was so delighted,\nthat she executed all manner of little happy hops, skips, and fidgets,\nall the time of her toilette, and caused many an expostulation of \"Mais,\nMiladi!\" from Josephine, before the pretty delicate blue and white\nmuslin, worked white jacket, and white ribboned and feathered hat, were\nadjusted.  Lady Barbara kept her little countess very prettily and\nquietly dressed; but it was at the cost of infinite worry of herself,\nKate, and Josephine, for there never was a child whom it was so hard to\nkeep in decent trim.  Armyn's old saying, that she ought to be always\nkept dressed in sacking, as the only thing she could not spoil, was a\ntrue one; for the sharp hasty movements, and entire disregard of where\nshe stepped, were so ruinous, that it was on the records of the Bruton\nStreet household, that she had gone far to demolish eight frocks in ten\ndays.\n\nHowever, on this occasion she did get safe down to the carriage\u2014clothes,\ngloves, and all, without detriment or scolding; and jumped in first.  She\nwas a long way yet from knowing that, though her aunts gave the first\nplace to her rank, it would have been proper in her to yield it to their\nyears, and make way for them.\n\nShe was too childish to have learnt this as a matter of good breeding,\nbut she might have learnt it of a certain parable, which she could say\nfrom beginning to end, that she should \"sit not down in the highest\nroom.\"\n\nHer aunt sat down beside her, and spent the first ten minutes of the\ndrive in enjoining on her proper behaviour at Lady de la Poer's.  The\nchildren there were exceedingly well brought up, she said, and she was\nvery desirous they should be her niece's friends; but she was certain\nthat Lady de la Poer would allow no one to associate with them who did\nnot behave properly.\n\n\"Lord de la Poer was very kind to me just as I was,\" said Kate, in her\nspirit of contradiction, which was always reckless of consequences.\n\n\"Gentlemen are no judges of what is becoming to a little girl,\" said Lady\nBarbara severely.  \"Unless you make a very different impression upon Lady\nde la Poer, she will never permit you to be the friend of her daughters.\"\n\n\"I wonder how I am to make an impression,\" meditated Kate, as they drove\non; \"I suppose it would make an impression if I stood up and repeated,\n'Ruin seize thee, ruthless king!' or something of that sort, as soon as I\ngot in.  But one couldn't do that; and I am afraid nothing will happen.\nIf the horses would only upset us at the door, and Aunt Barbara be nicely\ninsensible, and the young countess show the utmost presence of mind!  But\nnothing nice and like a book ever does happen.  And after all, I believe\nthat it is all nonsense about making impressions.  Thinking of them is\nall affectation; and one ought to be as simple and unconscious as one\ncan.\"  A conclusion which did honour to the countess's sense.  In fact,\nshe had plenty of sense, if only she had ever used it for herself,\ninstead of for the little ladies she drew on her quires of paper.\n\nLady Barbara had started early, as she really wished to find her friends\nat home; and accordingly, when the stairs were mounted, and the aunt and\nniece were ushered into a pretty bright-looking drawing-room, there they\nfound all that were not at school enjoying their after-dinner hour of\nliberty with their father and mother.\n\nLord de la Poer himself had the youngest in his arms, and looked very\nmuch as if he had only just scrambled up from the floor; his wife was\nreally sitting on the ground, helping two little ones to put up a puzzle\nof wild beasts; and there was a little herd of girls at the farther\ncorner, all very busy over something, towards which Kate's longing eyes\nat once turned\u2014even in the midst of Lord de la Poer's very kind greeting,\nand his wife's no less friendly welcome.\n\nIt was true that, as Lady Barbara had said, they were all exceedingly\nwell-bred children.  Even the little fellow in his father's arms, though\nbut eighteen months old, made no objection to hold out his fat hand\ngraciously, and showed no shyness when Lady Barbara kissed him! and the\nothers all waited quietly over their several occupations, neither\nshrinking foolishly from notice, nor putting themselves forward to claim\nit.  Only the four sisters came up, and took their own special visitor\ninto the midst of them as their own property; the elder of them, however,\nat a sign from her mamma, taking the baby in her arms, and carrying him\noff, followed by the other two small ones\u2014only pausing at the door for\nhim to kiss his little hand, and wave it in the prettiest fashion of baby\nstateliness.\n\nThe other sisters drew Kate back with them into the room, where they had\nbeen busy.  Generally, however much she and Sylvia might wish it, they\nhad found acquaintance with other children absolutely impossible in the\npresence of grown-up people, whose eyes and voices seemed to strike all\nparties dumb.  But these children seemed in no wise constrained: one of\nthem said at once, \"We are so glad you are come.  Mamma said she thought\nyou would before we went out, one of those days.\"\n\n\"Isn't it horrid going out in London?\" asked Kate, at once set at ease.\n\n\"It is not so nice as it is at home,\" said one of the girls; laughing;\n\"except when it is our turn to go out with Mamma.\"\n\n\"She takes us all out in turn,\" explained another, \"from Fanny, down to\nlittle Cecil the baby\u2014and that is our great time for talking to her, when\none has her all alone.\"\n\n\"And does she never take you out in the country?\"\n\n\"Oh yes! but there are people staying with us then, or else she goes out\nwith Papa.  It is not a regular drive every day, as it is here.\"\n\nKate would not have had a drive with Aunt Barbara every day, for more\nthan she could well say.  However, she was discreet enough not to say so,\nand asked what they did on other days.\n\n\"Oh, we walk with Miss Oswald in the park, and she tells us stories, or\nwe make them.  We don't tell stories in the country, unless we have to\nwalk straight along the drives, that, as Papa says, we may have some\nsolace.\"\n\nThen it was explained that Miss Oswald was their governess, and that they\nwere very busy preparing for her birth-day.  They were making a\npaper-case for her, all themselves, and this hour was their only time for\ndoing it out of her sight in secret.\n\n\"But why do you make it yourselves?\" said Kate; \"one can buy such\nbeauties at the bazaars.\"\n\n\"Yes; but Mamma says a present one has taken pains to make, is worth a\ngreat deal more than what is only bought; for trouble goes for more than\nmoney.\"\n\n\"But one can make nothing but nasty tumble-to-pieces things,\" objected\nKate.\n\n\"That depends,\" said Lady Mary, in a very odd merry voice; and the other\ntwo, Adelaide and Grace, who were far too much alike for Kate to guess\nwhich was which, began in a rather offended manner to assure her that\n_their_ paper-case was to be anything but tumble-to-pieces.  Fanny was to\nbind it, and Papa had promised to paste its back and press it.\n\n\"And Mamma drove with me to Richmond, on purpose to get leaves to\nspatter,\" added the other sister.\n\nThen they showed Kate\u2014whose eyes brightened at anything approaching to a\nmess\u2014that they had a piece of  cardboard, on which leaves,\nchiefly fern, were pinned tightly down, and that the entire sheet was\nthen covered with a spattering of ink from a tooth-brush drawn along the\ntooth of a comb.  When the process was completed, the form of the loaf\nremained in the primitive colour of the card, thrown out by the cloud of\nink-spots, and only requiring a tracing of its veins by a pen.\n\nA space had been cleared for these operations on a side-table; and in\nspite of the newspaper, on which the appliances were laid, and even the\ncomb and brush, there was no look of disarrangement or untidiness.\n\n\"Oh, do\u2014do show me how you do it!\" cried Kate, who had had nothing to do\nfor months, with the dear delight of making a mess, except what she could\ncontrive with her paints.\n\nAnd Lady Grace resumed a brown-holland apron and bib, and opening her\nhands with a laugh, showed their black insides, then took up her\nimplements.\n\n\"Oh, do\u2014do let me try,\" was Kate's next cry; \"one little bit to show\nSylvia Wardour.\"\n\nWith one voice the three sisters protested that she had better not; she\nwas not properly equipped, and would ink herself all over.  If she would\npin down a leaf upon the scrap she held up, Grace should spatter it for\nher, and they would make it up into anything she liked.\n\nBut this did not satisfy Kate at all; the pinning out of the leaf was\nstupid work compared with the glory of making the ink fly.  In vain did\nAdelaide represent that all the taste and skill was in the laying out the\nleaves, and pinning them down, and that anyone could put on the ink; in\nvain did Mary represent the dirtiness of the work: this was the beauty of\nit in her eyes; and the sight of the black dashes sputtering through the\ncomb filled her with emulation; so that she entreated, almost piteously,\nto be allowed to \"do\" an ivy loaf, which she had hastily, and not very\ncarefully, pinned out with Mary's assistance\u2014that is, she had feebly and\nunsteadily stuck every pin, and Mary had steadied them.\n\nThe new friends consented, seeing how much she was set on it; but Fanny,\nwho had returned from the nursery, insisted on precautions\u2014took off the\njacket, turned up the frock sleeves, and tied on an apron; though Kate\nfidgeted all the time, as if a great injury were being inflicted on her;\nand really, in her little frantic spirit, thought Lady Fanny a great\ntorment, determined to delay her delight till her aunt should go away and\nput a stop to it.\n\nWhen once she had the brush, she was full of fun and merriment, and kept\nher friends much amused by her droll talk, half to them, half to her\nwork.\n\n\"There's a portentous cloud, isn't there?  An inky cloud, if ever there\nwas one!  Take care, inhabitants below; growl, growl, there's the\nthunder; now comes the rain; hail, hail, all hail, like the beginning of\nMacbeth.\"\n\n\"Which the Frenchman said was in compliment to the climate,\" said Fanny;\nat which the whole company fell into convulsions of laughing; and neither\nKate nor Grace exactly knew what hands or brush or comb were about; but\nwhereas the little De La Poers had from their infancy laughed almost\nnoiselessly, and without making faces, Kate for her misfortune had never\nbeen broken of a very queer contortion of her lips, and a cackle like a\nbantam hen's.\n\nWhen this unlucky cackle had been several times repeated, it caused Lady\nBarbara, who had been sitting with her back to the inner room, to turn\nround.\n\nPoor Lady Barbara!  It would not be easy to describe her feelings when\nshe saw the young lady, whom she had brought delicately blue and white,\nlike a speedwell flower, nearly as black as a sweep.\n\nLord de la Poer broke out into an uncontrollable laugh, half at the aunt,\nhalf at the niece.  \"Why, she has grown a moustache!\" he exclaimed.\n\"Girls, what have you been doing to her?\" and walking up to them, he\nturned Kate round to a mirror, where she beheld her own brown eyes\nlooking out of a face dashed over with black specks, thicker about the\nmouth, giving her altogether much the colouring of a very dark man\nclosely shaved.  It was so exceedingly comical, that she went off into\nfits of laughing, in which she was heartily joined by all the merry\nparty.\n\n\"There,\" said Lord de la Poer, \"do you want to know what your Uncle Giles\nis like? you've only to look at yourself!\u2014See, Barbara, is it not a\ncapital likeness?\"\n\n\"I never thought her like _Giles_,\" said her aunt gravely, with an\nemphasis on the name, as if she meant that the child did bear a likeness\nthat was really painful to her.\n\n\"My dears,\" said the mother, \"you should not have put her in such a\ncondition; could you not have been more careful?\"\n\nKate expected one of them to say, \"She would do it in spite of us;\" but\ninstead of that Fanny only answered, \"It is not so bad as it looks,\nMamma; I believe her frock is quite safe; and we will soon have her face\nand hands clean.\"\n\nWhereupon Kate turned round and said, \"It is all my fault, and _nobody's\nelse's_.  They told me not, but it was such fun!\"\n\nAnd therewith she obeyed a pull from Grace, and ran upstairs with the\nparty to be washed; and as the door shut behind them, Lord de la Poer\nsaid, \"You need not be afraid of _that_ likeness, Barbara.  Whatever else\nshe may have brought from her parsonage, she has brought the spirit of\ntruth.\"\n\nThough knowing that something awful hung over her head, Kate was all the\nmore resolved to profit by her brief minutes of enjoyment; and the little\nmaidens all went racing and flying along the passages together; Kate\nfeeling as if the rapid motion among the other young feet was life once\nmore.\n\n\"Well! your frock is all right; I hope your aunt will not be very angry\nwith you,\" said Adelaide.  (She know Adelaide now, for Grace was the inky\none.)\n\n\"It is not a thing to be angry for,\" added Grace.\n\n\"No, it would not have been at my home,\" said Kate, with a sigh; \"but,\noh! I hope she will not keep me from coming here again.\"\n\n\"She shall not,\" exclaimed Adelaide; \"Papa won't let her.\"\n\n\"She said your mamma would mind what your papa did not,\" said Kate, who\nwas not very well informed on the nature of mammas.\n\n\"Oh, that's all stuff,\" decidedly cried Adelaide.  \"When Papa told us\nabout you, she said, 'Poor child!  I wish I had her here.'\"\n\nPrudent Fanny made an endeavour at chocking her little sister; but the\nlight in Kate's eye, and the responsive face, drew Grace on to ask, \"She\ndidn't punish you, I hope, for your tumbling off the bracket?\"\n\n\"No, your papa made her promise not; but she was very cross.  Did he tell\nyou about it?\"\n\n\"Oh yes; and what do you think Ernest wrote?  You must know he had\ngrumbled excessively at Papa's having business with Lady Barbara; but his\nletter said, 'It wasn't at all slow at Lady Barbara's, for there was the\njolliest fellow there you ever knew; mind you get her to play at\nacting.'\"\n\nLady Fanny did not think this improving, and was very glad that the maid\ncame in with hot water and towels, and put an end to it with the work of\nscrubbing.\n\nGoing home, Lady Barbara was as much displeased as Kate had expected, and\nwith good reason.  After all her pains, it was very strange that\nKatharine should be so utterly unfit to behave like a well-bred girl.\nThere might have been excuse for her before she had been taught, but now\nit was mere obstinacy.\n\nShe should be careful how she took her out for a long time to come!\n\nKate's heart swelled within her.  It was not obstinacy, she know; and\nthat bit of injustice hindered her from seeing that it was really wilful\nrecklessness.  She was elated with Ernest's foolish school-boy account of\nher, which a more maidenly little girl would not have relished; she was\nstrengthened in her notion that she was ill-used, by hearing that the De\nla Poers pitied her; and because she found that Aunt Barbara was\nconsidered to be a little wrong, she did not consider that she herself\nhad ever been wrong at all.\n\nAnd Lady Barbara was not far from the truth when she told her sister\n\"that Katharine was perfectly hard and reckless; there was no such thing\nas making her sorry!\"\n\n\n\n\nCHAPTER VI.\n\n\nAFTER that first visit, Kate did see something of the De la Poers, but\nnot more than enough to keep her in a constant ferment with the uncertain\npossibility, and the longing for the meetings.\n\nThe advances came from them; Lady Barbara said very truly, that she could\nnot be responsible for making so naughty a child as her niece the\ncompanion of any well-regulated children; she was sure that their mother\ncould not wish it, since nice and good as they naturally were, this\nunlucky Katharine seemed to infect them with her own spirit of riot and\nturbulence whenever they came near her.\n\nThere was no forwarding of the attempts to make appointments for walks in\nthe Park, though really very little harm had ever come of them, guarded\nby the two governesses, and by Lady Fanny's decided ideas of propriety.\nThat Kate embarked in long stories, and in their excitement raised her\nvoice, was all that could be said against her on those occasions, and\nMrs. Lacy forbore to say it.\n\nOnce, indeed, Kate was allowed to ask her friends to tea; but that proved\na disastrous affair.  Fanny was prevented from coming; and in the absence\nof her quiet elder-sisterly care, the spirits of Grace and Adelaide were\nso excited by Kate's drollery, that they were past all check from Mary,\nand drew her along with them into a state of frantic fun and mad pranks.\n\nThey were full of merriment all tea time, even in the presence of the two\ngovernesses; and when that was over, and Kate showed \"the bracket,\" they\nbegan to grow almost ungovernable in their spirit of frolic and fun: they\nwent into Kate's room, resolved upon being desert travellers, set up an\numbrella hung round with cloaks for a tent, made camels of chairs, and\nfinding those tardy, attempted riding on each other\u2014with what results to\nAunt Jane's ears below may be imagined\u2014dressed up wild Arabs in\nbournouses of shawls, and made muskets of parasols, charging desperately,\nand shrieking for attack, defence, \"for triumph or despair,\" as Kate\nobserved, in one of her magnificent quotations.  Finally, the endangered\ntraveller, namely Grace, rushed down the stairs headlong, with the two\nArabs clattering after him, banging with their muskets, and shouting\ntheir war-cry the whole height of the house.\n\nThe ladies in the drawing-room had borne a good deal; but Aunt Jane was\nby this time looking meekly distracted; and Lady Barbara sallying out,\nmet the Arab Sheikh with his white frock over his head, descending the\nstairs in the rear, calling to his tribe in his sweet voice not to be so\nnoisy\u2014but not seeing before him through the said bournouse, he had very\nnearly struck Lady Barbara with his parasol before he saw her.\n\nNo one could be more courteous or full of apologies than the said Sheikh,\nwho was in fact a good deal shocked at his unruly tribe, and quite\nacquiesced in the request that they would all come and sit quiet in the\ndrawing-room, and play at some suitable game there.\n\nIt would have been a relief to Mary to have them thus disposed of safely;\nand Adelaide would have obeyed; but the other two had been worked up to a\nstate of wildness, such as befalls little girls who have let themselves\nout of the control of their better sense.\n\nThey did not see why they should sit up stupid in the drawing-room; \"Mary\nwas as cross as Lady Barbara herself to propose it,\" said Grace,\nunfortunately just as the lady herself was on the stairs to enforce her\ndesire, in her gravely courteous voice; whereupon Kate, half tired and\nwholly excited, burst out into a violent passionate fit of crying and\nsobbing, declaring that it was very hard, that whenever she had ever so\nlittle pleasure, Aunt Barbara always grudged it to her.\n\nNone of them had ever heard anything like it; to the little De la Poers\nshe seemed like one beside herself, and Grace clung to Mary, and Adelaide\nto Miss Oswald, almost frightened at the screams and sobs that Kate\nreally could not have stopped if she would.  Lady Jane came to the head\nof the stairs, pale and trembling, begging to know who was hurt; and Mrs.\nLacy tried gentle reasoning and persuading, but she might as well have\nspoken to the storm beating against the house.\n\nLady Barbara sternly ordered her off to her room; but the child did not\nstir\u2014indeed, she could not, except that she rocked herself to and fro in\nher paroxysms of sobbing, which seemed to get worse and worse every\nmoment.  It was Miss Oswald at last, who, being more used to little girls\nand their naughtiness than any of the others, saw the right moment at\nlast, and said, as she knelt down by her, half kindly, half severely, \"My\ndear, you had better let me take you up-stairs.  I will help you: and you\nare only shocking everyone here.\"\n\nKate did let her take her up-stairs, though at every step there was a\npause, a sob, a struggle; but a gentle hand on her shoulder, and firm\npersuasive voice in her ear, moved her gradually onwards, till the little\npink room was gained; and there she threw herself on her bed in another\nagony of wild subs, unaware of Miss Oswald's parley at the door with Lady\nBarbara and Mrs. Lacy, and her entreaty that the patient might be left to\nher, which they were nothing loth to do.\n\nWhen Kate recovered her speech, she poured out a wild and very naughty\ntorrent, about being the most unhappy girl in the world; the aunts were\nalways unkind to her; she never got any pleasure; she could not bear\nbeing a countess; she only wanted to go back to her old home, to Papa and\nMary and Sylvia; and nobody would help her.\n\nMiss Oswald treated the poor child almost as if she had been a little out\nof her mind, let her say it all between her sobs, and did not try to\nargue with her, but waited till the talking and the sobbing had fairly\ntried her out; and by that time the hour had come at which the little\nvisitors were to go home.  The governess rose up, and said she must go,\nasking in a quiet tone, as if all that had been said were mere mad folly,\nwhether Lady Caergwent would come down with her, and tell her aunts she\nwas sorry for the disturbance she had made.\n\nKate shrank from showing such a spectacle as her swollen, tear-stained,\nred-marbled visage.  She was thoroughly sorry, and greatly ashamed; and\nshe only gasped out, \"I can't, I can't; don't let me see anyone.\"\n\n\"Then I will wish Mary and her sisters good-bye for you.\"\n\n\"Yes, please.\"  Kate had no words for more of her sorrow and shame.\n\n\"And shall I say anything to your aunt for you?\"\n\n\"I\u2014I don't know; only don't let anyone come up.\"\n\n\"Then shall I tell Lady Barbara you are too much tired out now for\ntalking, but that you will tell her in the morning how sorry you are?\"\n\n\"Well, yes,\" said Kate rather grudgingly.  \"Oh, must you go?\"\n\n\"I am afraid I must, my dear.  Their mamma does not like Addie and Grace\nto be kept up later than their usual bed-time.\"\n\n\"I wish you could stay.  I wish you were my governess,\" said Kate,\nclinging to her, and receiving her kind, friendly, pitying kiss.\n\nAnd when the door had shut upon her, Kate's tears began to drop again at\nthe thought that it was very hard that the little De la Poers, who had\nfather, mother, and each other, should likewise have such a nice\ngoverness, while she had only poor sad dull Mrs. Lacy.\n\nHad Kate only known what an unselfish little girl and Mrs. Lacy might\nhave been to each other!\n\nHowever, the first thing she could now think of was to avoid being seen\nor spoken to by anyone that night; and for this purpose she hastily\nundressed herself, bundled-up her hair as best she might, as in former\ndays, said her prayers, and tumbled into bed, drawing the clothes over\nher head, resolved to give no sign of being awake, come who might.\n\nHer shame was real, and very great.  Such violent crying fits had\novertaken her in past times, but had been thought to be outgrown.  She\nwell recollected the last.  It was just after the death of her aunt, Mrs.\nWardour, just when the strange stillness of sorrow in the house was\nbeginning to lessen, and the children had forgotten themselves, and burst\nout into noise and merriment, till they grew unrestrained and\nquarrelsome; Charlie had offended Kate, she had struck him, and Mary\ncoming on them, grieved and hurt at their conduct at such a time, had\npunished Kate for the blow, but missed perception of Charlie's offence;\nand the notion of injustice had caused the shrieking cries and violent\nsobs that had brought Mr. Wardour from the study in grave sorrowful\nseverity.\n\nWhat she had heard afterwards from him about not making poor Mary's task\nharder, and what she had heard from Mary about not paining him, had\nreally restrained her; and she had thought such outbreaks passed by among\nthe baby faults she had left behind, and was the more grieved and ashamed\nin consequence.  She felt it a real exposure: she remembered her young\nfriends' surprised and frightened eyes, and not only had no doubt their\nmother would really think her too naughty to be their playfellow, but\nalmost wished that it might be so\u2014she could never, never bear to see them\nagain.\n\nShe heard the street door close after them, she heard the carriage drive\naway; she felt half relieved; but then she hid her face in the pillow,\nand cried more quietly, but more bitterly.\n\nThen some one knocked; she would not answer.  Then came a voice, saying,\n\"Katharine.\"  It was Aunt Barbara's, but it was rather wavering.  She\nwould not answer, so the door was opened, and the steps, scarcely audible\nin the rustling of the silk, came in; and Kate felt that her aunt was\nlooking at her, wondered whether she had better put out her head, ask\npardon, and have it over, but was afraid; and presently heard the moir\u00e9\nantique go sweeping away again.\n\nAnd then the foolish child heartily wished she had spoken, and was seized\nwith desperate fears of the morrow, more of the shame of hearing of her\ntears than of any punishment.  Why had she not been braver?\n\nAfter a time came a light, and Josephine moving about quietly, and\nputting away the clothes that had been left on the floor.  Kate was not\nafraid of her, but her caressing consolations and pity would have only\nadded to the miserable sense of shame; so there was no sign, no symptom\nof being awake, though it was certain that before Josephine went away,\nthe candle was held so as to cast a light over all that was visible of\nthe face.  Kate could not help hearing the low muttering of the\nFrenchwoman, who was always apt to talk to herself: \"Asleep!  Ah, yes!\nShe sleeps profoundly.  How ugly _la petite_ has made herself!  What\ncries!  Ah, she is like Miladi her aunt! a demon of a temper!\"\n\nKate restrained herself till the door was shut again, and then rolled\nover and over, till she had made a strange entanglement of her\nbed-clothes, and brought her passion to an end by making a mummy of\nherself, bound hand and foot, snapping with her month all the time, as if\nshe longed to bite.\n\n\"O you horrible Frenchwoman!  You are a flatterer, a base flatterer; such\nas always haunt the great!  I hate it all.  I a demon of a temper?  I\nlike Aunt Barbara?  Oh, you wretch!  I'll tell Aunt Barbara a to-morrow,\nand get you sent away!\"\n\nThose were some of Kate's fierce angry thoughts in her first vexation;\nbut with all her faults, she was not a child who ever nourished rancour\nor malice; and though she had been extremely wounded at first, yet she\nquickly forgave.\n\nBy the time she had smoothed out her sheet, and settled matters between\nit and her blanket, she had begun to think more coolly.  \"No, no, I\nwon't.  It would be horribly dishonourable and all that to tell Aunt\nBarbara.  Josephine was only thinking out loud; and she can't help what\nshe thinks.  I was very naughty; no wonder she thought so.  Only next\ntime she pets me, I will say to her, 'You cannot deceive me, Josephine; I\nlike the plain truth better than honeyed words.'\"\n\nAnd now that Kate had arrived at the composition of a fine speech that\nwould never be made, it was plain that her mind was pretty well composed.\nThat little bit of forgiveness, though it had not even cost an effort,\nhad been softening, soothing, refreshing; it had brought peacefulness;\nand Kate lay, not absolutely asleep, but half dreaming, in the summer\ntwilight, in the soft undefined fancies of one tired out with agitation.\n\nShe was partly roused by the various sounds in the house, but not\nstartled\u2014the light nights of summer always diminished her alarms; and she\nheard the clocks strike, and the bell ring for prayers, the doors open\nand shut, all mixed in with her hazy fancies.  At last came the silken\nrustlings up the stairs again, and the openings of bed-room doors close\nto her.\n\nKate must have gone quite to sleep, for she did not know when the door\nwas opened, and how the soft voices had come in that she heard over her.\n\n\"Poor little dear!  How she has tossed her bed about!  I wonder if we\ncould set the clothes straight without wakening her.\"\n\nHow very sweet and gentle Aunt Jane's voice was in that low cautious\nwhisper.\n\nSome one\u2014and Kate knew the peculiar sound of Mrs. Lacy's crape\u2014was moving\nthe bed-clothes as gently as she could.\n\n\"Poor little dear!\" again said Lady Jane; \"it is very sad to see a child\nwho has cried herself to sleep.  I do wish we could manage her better.\nDo you think the child is happy?\" she ended by asking in a wistful voice.\n\n\"She has very high spirits,\" was the answer.\n\n\"Ah, yes! her impetuosity; it is her misfortune, poor child!  Barbara is\nso calm and resolute, that\u2014that\u2014\"  Was Lady Jane really going to regret\nanything in her sister?  She did not say it, however; but Kate heard her\nsigh, and add, \"Ah, well! if I were stronger, perhaps we could make her\nhappier; but I am so nervous.  I must try not to look distressed when her\nspirits do break out, for perhaps it is only natural.  And I am so sorry\nto have brought all this on her, and spoilt those poor children's\npleasure!\"\n\nLady Jane bent over the child, and Kate reared herself up on a sudden,\nthrew her arms round her neck, and whispered, \"Aunt Jane, dear Aunt Jane,\nI'll try never to frighten you again!  I am so sorry.\"\n\n\"There, there; have I waked you?  Don't, my dear; your aunt will hear.\nGo to sleep again.  Yes, do.\"\n\nBut Aunt Jane was kissing and fondling all the time; and the end of this\nsad naughty evening was, that Kate went to sleep with more softness,\nlove, and repentance in her heart, than there had been since her coming\nto Bruton Street.\n\n\n\n\nCHAPTER VII.\n\n\nLADY CAERGWENT was thoroughly ashamed and bumbled by that unhappy\nevening.  She looked so melancholy and subdued in the morning, with her\nheavy eyelids and inflamed eyes, and moved so meekly and sadly, without\ndaring to look up, that Lady Barbara quite pitied her, and said\u2014more\nkindly than she had ever spoken to her before:\n\n\"I see you are sorry for the exposure last night, so we will say no more\nabout it.  I will try to forget it.  I hope our friends may.\"\n\nThat hope sounded very much like \"I do not think they will;\" and truly\nKate felt that it was not in the nature of things that they ever should.\nShe should never have forgotten the sight of a little girl in that frenzy\nof passion!  No, she was sure that their mamma and papa knew all about\nit, and that she should never be allowed to play with them again, and she\ncould not even wish to meet them, she should be miserably ashamed, and\nwould not know which way to look.\n\nShe said not one word about meeting them, and for the first day or two\neven begged to walk in the square instead of the park; and she was so\ngood and steady with her lessons, and so quiet in her movements, that she\nscarcely met a word of blame for a whole week.\n\nOne morning, while she was at breakfast with Lady Barbara and Mrs. Lacy,\nthe unwonted sound of a carriage stopping, and of a double knock, was\nheard.  In a moment the colour flushed into Lady Barbara's face, and her\neyes lighted: then it passed away into a look of sadness.  It had seemed\nto her for a moment as if the bright young nephew who had been the light\nand hope of her life, were going to look in on her; and it had only\nbrought the remembrance that he was gone for ever, and that in his stead\nthere was only the poor little girl, to whom rank was a misfortune, and\nwho seemed as if she would never wear it becomingly.  Kate saw nothing of\nall this; she was only eager and envious for some change and variety in\nthese long dull days.  It was Lord de la Poer and his daughter Adelaide,\nwho the next moment were in the room; and she remembered instantly that\nshe had heard that this was to be Adelaide's birthday, and wished her\nmany happy returns in all due form, her heart beating the while with\nincreasing hope that the visit concerned herself.\n\nAnd did it not?  Her head swam round with delight and suspense, and she\ncould hardly gather up the sense of the words in which Lord de la Poer\nwas telling Lady Barbara that Adelaide's birthday was to be spent at the\nCrystal Palace at Sydenham; that the other girls were gone to the station\nwith their mother, and that he had come round with Adelaide to carry off\nKate, and meet the rest at ten o'clock.  Lady de la Poer would have\nwritten, but it had only boon settled that morning on finding that he\ncould spare the day.\n\nKate squeezed Adelaide's hand in an agony.  Oh! would that aunt let her\ngo?\n\n\"You would like to come?\" asked Lord de la Poer, bending his pleasant\neyes on her.  \"Have you ever been there?\"\n\n\"Never!  Oh, thank you!  I should like it so much!  I never saw any\nexhibition at all, except once the Gigantic Cabbage!\u2014May I go, Aunt\nBarbara?\"\n\n\"Really you are very kind, after\u2014\"\n\n\"Oh, we never think of _afters_ on birthdays!\u2014Do we, Addie?\"\n\n\"If you are so very good, perhaps Mrs. Lacy will kindly bring her to meet\nyou.\"\n\n\"I am sure,\" said he, turning courteously to that lady, \"that we should\nbe very sorry to give Mrs. Lacy so much trouble.  If this is to be a\nholiday to everyone, I am sure you would prefer the quiet day.\"\n\nNo one could look at the sad face and widow's cap without feeling that so\nit must be, even without the embarrassed \"Thank you, my Lord, if\u2014\"\n\n\"If\u2014if Katharine were more to be trusted,\" began Lady Barbara.\n\n\"Now, Barbara,\" he said in a drolly serious fashion, \"if you think the\nCourt of Chancery would seriously object, say so at once.\"\n\nLady Barbara could not keep the corners of her mouth quite stiff, but she\nstill said, \"You do not know what you are undertaking.\"\n\n\"Do you deliberately tell me that you think myself and Fanny, to say\nnothing of young Fanny, who is the wisest of us all, unfit to be trusted\nwith this one young lady?\" said he, looking her full in the face, and\nputting on a most comical air: \"It is humiliating, I own.\"\n\n\"Ah! if Katharine were like your own daughters, I should have no fears,\"\nsaid the aunt.  \"But\u2014  However, since you are so good\u2014if she will promise\nto be very careful\u2014\"\n\n\"Oh yes, yes, Aunt Barbara!\"\n\n\"I make myself responsible,\" said Lord de la Poer.  \"Now, young woman,\nrun off and get the hat; we have no time to lose.\"\n\nKate darted off and galloped up the stairs at a furious pace, shouted\n\"Josephine\" at the top; and then, receiving no answer, pulled the bell\nviolently; after which she turned round, and obliged Adelaide with a\nspecies of dancing hug, rather to the detriment of that young lady's\nmuslin jacket.\n\n\"I was afraid to look back before,\" she breathlessly said, as she\nreleased Adelaide; \"I felt as if your papa were Orpheus, when\n\n    'Stern Proserpine relented,\n    And gave him back the fair\u2014'\n\nand I was sure Aunt Barbara would catch me like Eurydice, if I only\nlooked back.\"\n\n\"What a funny girl you are, to be thinking about Orpheus and Eurydice!\"\nsaid Adelaide.  \"Aren't you glad?\"\n\n\"Glad?  Ain't I just! as Charlie would say.  Oh dear! your papa is a\ndelicious man; I'd rather have him for mine than anybody, except Uncle\nWardour!\"\n\n\"I'd rather have him than anyone,\" said the little daughter.  \"Because he\nis yours,\" said Kate; \"but somehow, though he is more funny and\ngood-natured than Uncle Wardour, I wouldn't\u2014no, I shouldn't like him so\nwell for a papa.  I don't think he would punish so well.\"\n\n\"Punish!\" cried Adelaide.  \"Is that what you want?  Why, Mamma says\nchildren ought to be always pleasure and no trouble to busy fathers.  But\nthere, Kate; you are not getting ready\u2014and we are to be at the station at\nten.\"\n\n\"I am waiting for Josephine!  Why doesn't she come?\" said Kate, ringing\nviolently again.\n\n\"Why don't you get ready without her?\"\n\n\"I don't know where anything is!  It is very tiresome of her, when she\nknows I never dress myself,\" said Kate fretfully.\n\n\"Don't you?  Why, Grace and I always dress ourselves, except for the\nevening.  Let me help you.  Are not those your boots?\"\n\nKate rushed to the bottom of the attic stairs, and shouted \"Josephine\" at\nthe top of her shrill voice; then, receiving no answer, she returned,\ncondescended to put on the boots that Adelaide held up to her, and\nnoisily pulled out some drawers; but not seeing exactly what she wanted,\nshe again betook herself to screams of her maid's name, at the third of\nwhich out burst Mrs. Bartley in a regular state of indignation: \"Lady\nCaergwent!  Will your Ladyship hold your tongue!  There's Lady Jane\nstartled up, and it's a mercy if her nerves recover it the whole\nday\u2014making such a noise as that!\"\n\n\"But Josephine won't come, and I'm going out, Bartley,\" said Kate\npiteously.  \"Where is Josephine?\"\n\n\"Gone out, my Lady, so it is no use making a piece of work,\" said Bartley\ncrossly, retreating to Lady Jane.\n\nKate was ready to cry; but behold, that handy little Adelaide had\nmeantime picked out a nice black silk cape, with hat and feather, gloves\nand handkerchief, which, if not what Kate had intended, were nice enough\nfor anything, and would have\u2014some months ago\u2014seemed to the orphan at the\nparsonage like robes of state.  Kind Adelaide held them up so\ntriumphantly, that Kate could not pout at their being only everyday\nthings; and as she began to put them on, out came Mrs. Bartley again, by\nLady Jane's orders, pounced upon Lady Caergwent, and made her repent of\nall wishes for assistance by beginning upon her hair, and in spite of all\nwriggles and remonstrances, dressing her in the peculiarly slow and\nprecise manner by which a maid can punish a troublesome child; until\nfinally Kate\u2014far too much irritated for a word of thanks, tore herself\nout of her hands, caught up her gloves, and flew down-stairs as if her\nlife depended on her speed.  She thought the delay much longer than it\nhad really been, for she found Lord de la Poer talking so earnestly to\nher aunt, that he hardly looked up when she came in\u2014something about her\nUncle Giles in India, and his coming home\u2014which seemed to be somehow\nbecoming possible\u2014though at a great loss to himself; but there was no\nmaking it out; and in a few minutes he rose, and after some fresh charges\nfrom Lady Barbara to her niece \"not to forgot herself,\" Kate was handed\ninto the carriage, and found herself really off.\n\nThen the tingle of wild impatience and suspense subsided, and happiness\nbegan!  It had not been a good beginning, but it was very charming now.\n\nAdelaide and her father were full of jokes together, so quick and bright\nthat Kate listened instead of talking.  She had almost lost the habit of\nmerry chatter, and it did not come to her quickly again; but she was\ngreatly entertained; and thus they came to the station, where Lady de la\nPoer and her other three girls were awaiting them, and greeted Kate with\njoyful faces.\n\nThey were the more relieved at the arrival of the three, because the\nstation was close and heated, and it was a very warm summer day, so that\nthe air was extremely oppressive.\n\n\"It feels like thunder,\" said some one.  And thenceforth Kate's perfect\nfelicity was clouded.  She had a great dislike to a thunder-storm, and\nshe instantly began asking her neighbours if they _really_ thought it\nwould be thunder.\n\n\"I hope it will,\" said Lady Fanny; \"it would cool the air, and sound so\ngrand in those domes.\"\n\nKate thought this savage, and with an imploring look asked Lady de la\nPoer if she thought there would be a storm.\n\n\"I can't see the least sign of one,\" was the answer.  \"See how clear the\nsky is!\" as they steamed out of the station.\n\n\"But do you think there will be one to-day?\" demanded Kate.\n\n\"I do not expect it,\" said Lady de la Poer, smiling; \"and there is no use\nin expecting disagreeables.\"\n\n\"Disagreeables!  O Mamma, it would be such fun,\" cried Grace, \"if we only\nhad a chance of getting wet through!\"\n\nHere Lord de la Poer adroitly called off the public attention from the\nperils of the clouds, by declaring that he wanted to make out the fourth\nline of an advertisement on the banks, of which he said he had made out\none line as he was whisked by on each journey he had made; and as it was\nfour times over in four different languages, he required each damsel to\nundertake one; and there was a great deal of laughing over which it\nshould be that should undertake each language.  Fanny and Mary were\nhumble, and sure they could never catch the German; and Kate, more\nenterprising, undertook the Italian.  After all, while they were\nchattering about it, they went past the valuable document, and were come\nin sight of the \"monsters\" in the Gardens; and Lord de la Poer asked Kate\nif she would like to catch a pretty little frog; to which Mary responded,\n\"Oh, what a tadpole it must have been!\" and the discovery that her\nfriends had once kept a preserve of tadpoles to watch them turn into\nfrogs, was so delightful as entirely to dissipate all remaining thoughts\nof thunder, and leave Kate free for almost breathless amazement at the\nglittering domes of glass, looking like enormous bubbles in the sun.\n\nWhat a morning that was, among the bright buds and flowers, the wonders\nof nature and art all together!  It was to be a long day, and no\nhurrying; so the party went from court to court at their leisure, sat\ndown, and studied all that they cared for, or divided according to their\ntastes.  Fanny and Mary wanted time for the wonderful sculptures on the\nnoble gates in the Italian court; but the younger girls preferred roaming\nmore freely, so Lady de la Poer sat down to take care of them, while her\nhusband undertook to guide the wanderings of the other three.\n\nHe particularly devoted himself to Kate, partly in courtesy as to the\nguest of the party, partly because, as he said, he felt himself\nresponsible for her; and she was in supreme enjoyment, talking freely to\none able and willing to answer her remarks and questions, and with the\ncompanionship of girls of her own age besides.  She was most of all\ndelighted with the Alhambra\u2014the beauty of it was to her like a fairy\ntale; and she had read Washington Irving's \"Siege of Granada,\" so that\nshe could fancy the courts filled with the knightly Moors, who were so\nnoble that she could not think why they were not Christians\u2014nay, the\ntears quite came into her eyes as she looked up in Lord de la Poer's\nface, and asked why nobody converted the Abencerrages instead of fighting\nwith them!\n\nIt was a pity that Kate always grew loud when she was earnest; and Lord\nde la Poer's interest in the conversation was considerably lessened by\nthe discomfort of seeing some strangers looking surprised at the five\nsyllables in the squeaky voice coming out of the mouth of so small a\nlady.\n\n\"Gently, my dear,\" he softly said; and Kate for a moment felt it hard\nthat the torment about her voice should pursue her even in such moments,\nand spoil the Alhambra itself.\n\nHowever, her good humour recovered the next minute, at the Fountain of\nLions.  She wanted to know how the Moors came to have lions; she thought\nshe had heard that no Mahometans were allowed to represent any living\ncreature, for fear it should be an idol.  Lord de la Poer said she was\nquite right, and that the Mahometans think these forms will come round\ntheir makers at the last day, demanding to have souls given to them; but\nthat her friends, the Moors of Spain, were much less strict than any\nothers of their faith.  She could see, however, that the carving of such\nfigures was a new art with them, since these lions were very rude and\nclumsy performances for people who could make such delicate tracery as\nthey had seen within.  And then, while Kate was happily looking with\nAdelaide at the orange trees that completed the Spanish air of the court,\nand hoping to see the fountain play in the evening, he told Grace that it\nwas worth while taking people to see sights if they had as much\nintelligence and observation as Kate had, and did not go gazing idly\nabout, thinking of nothing.\n\nHe meant it to stir up his rather indolent-minded Grace\u2014he did not mean\nthe countess to hear it; but some people's eyes and ears are wonderfully\nquick at gathering what is to their own credit, and Kate, who had not\nheard a bit of commendation for a long time, was greatly elated.\n\nLuckily for appearances, she remembered how Miss Edgeworth's Frank made\nhimself ridiculous by showing off to Mrs. J\u2014, and how she herself had\nonce been overwhelmed by the laughter of the Wardour family for having\nrehearsed to poor Mrs. Brown all the characters of the gods of the\nNorthmen\u2014Odin, Thor, and all\u2014when she had just learnt them.  So she was\nmore careful than before not to pour out all the little that she knew;\nand she was glad she had not committed herself, for she had very nearly\nvolunteered the information that Pompeii was overwhelmed by Mount Etna,\nbefore she heard some one say Vesuvius, and perceived her mistake,\nfeeling as if she had been rewarded for her modesty like a good child in\na book.\n\nShe applauded herself much more for keeping back her knowledge till it\nwas wanted, than for having it; but this self-satisfaction looked out in\nanother loop-hole.  She avoided pedantry, but she was too much elated not\nto let her spirits get the better of her; and when Lady de la Poer and\nthe elder girls came up, they found her in a suppressed state of\ncapering, more like a puppy on its hind logs, than like a countess or any\nother well-bred child.\n\nThe party met under the screen of kings and queens, and there had some\ndinner, at one of the marble tables that just held them pleasantly.  The\ncold chicken and tongue were wonderfully good on that hot hungry day, and\nstill better were the strawberries that succeeded them; and oh! what\nmirth went on all the time!  Kate was chattering fastest of all, and\nloudest\u2014not to say the most nonsensically.  It was not nice nonsense\u2014that\nwas the worst of it\u2014it was pert and saucy.  It was rather the family\nhabit to laugh at Mary de la Poer for ways that were thought a little\nfanciful; and Kate caught this up, and bantered without discretion, in a\nway not becoming towards anybody, especially one some years her elder.\nMary was good-humoured, but evidently did not like being asked if she had\nstayed in the medi\u00e6val court, because she was afraid the great bulls of\nNineveh would run at her with their five legs.\n\n\"She will be afraid of being teazed by a little goose another time,\" said\nLord de la Poer, intending to give his little friend a hint that she was\nmaking herself very silly; but Kate took it quite another way, and not a\npretty one, for she answered, \"Dear me, Mary, can't you say bo to a\ngoose!\"\n\n\"Say what?\" cried Adelaide, who was always apt to be a good deal excited\nby Kate; and who had been going off into fits of laughter at all these\nfoolish sallies.\n\n\"It is not a very nice thing to say,\" answered her mother gravely; \"so\nthere is no occasion to learn it.\"\n\nKate did take the hint this time, and  up to the ears, partly\nwith vexation, partly with shame.  She sat silent and confused for\nseveral minutes, till her friends took pity on her, and a few\ngood-natured words about her choice of an ice quite restored her\nliveliness.  It is well to be good-humoured; but it is unlucky, nay,\nwrong, when a check from friends without authority to scold, does not\nsuffice to bring soberness instead of rattling giddiness.  Lady de la\nPoer was absolutely glad to break up the dinner, so as to work off the\nfolly and excitement by moving about, before it should make the little\ngirl expose herself, or infect Adelaide.\n\nThey intended to have gone into the gardens till four o'clock, when the\nfountains were to play; but as they moved towards the great door, they\nperceived a dark heavy cloud was hiding the sun that had hitherto shone\nso dazzlingly through the crystal walls.\n\n\"That is nice,\" said Lady Fanny; \"it will be cool and pleasant now before\nthe rain.\"\n\n\"If the rain is not imminent,\" began her father.\n\n\"Oh! is it going to be a thunder-storm?\" cried Kate.  \"Oh dear!  I do so\nhate thunder!  What shall I do?\" cried she; all her excitement turning\ninto terror.\n\nBefore anyone could answer her, there was a flash of bright white light\nbefore all their eyes, and a little scream.\n\n\"She's struck! she's struck!\" cried Adelaide, her hands before her eyes.\n\nFor Kate had disappeared.  No, she was in the great pond, beside which\nthey had been standing, and Mary was kneeling on the edge, holding fast\nby her frock.  But before the deep voice of the thunder was roaring and\nreverberating through the vaults, Lord de la Poer had her in his grasp,\nand the growl had not ceased before she was on her feet again, drenched\nand trembling, beginning to be the centre of a crowd, who were running\ntogether to help or to see the child who had been either struck by\nlightning or drowned.\n\n\"Is she struck?  Will she be blind?\" sobbed Adelaide, still with her\nhands before her eyes; and the inquiry was echoed by the nearer people,\nwhile more distant ones told each other that the young lady was blind for\nlife.\n\n\"Struck! nonsense!\" said Lord de la Poer; \"the lightning was twenty miles\noff at least.  Are you hurt, my dear?\"\n\n\"No,\" said Kate, shaking herself, and answering \"No,\" more decidedly.\n\"Only I am so wet, and my things stick to me.\"\n\n\"How did it happen?\" asked Grace.\n\n\"I don't know.  I wanted to get away from the thunder!\" said bewildered\nKate.\n\nMeantime, an elderly lady, who had come up among the spectators, was\ntelling Lady de la Poer that she lived close by, and insisting that the\nlittle girl should be taken at once to her house, put to bed, and her\nclothes dried.  Lady de la Poer was thankful to accept the kind offer\nwithout loss of time; and in the fewest possible words it was settled\nthat she would go and attend to the little drowned rat, while her girls\nshould remain with their father at the palace till the time of going\nhome, when they would meet at the station.  They must walk to the good\nlady's house, be the storm what it would, as the best chance of\npreventing Kate from catching cold.  She looked a rueful spectacle,\ndripping so as to make a little pool on the stone floor; her hat and\nfeather limp and streaming; her hair in long lank rats' tails, each\ndischarging its own waterfall; her clothes, ribbons, and all, pasted down\nupon her!  There was no time to be lost; and the stranger took her by one\nhand, Lady de la Poer by the other, and exchanging some civil speeches\nwith one another half out of breath, they almost swung her from one step\nof the grand stone stairs to another, and hurried her along as fast as\nthese beplastered garments would let her move.  There was no rain as yet,\nbut there was another clap of thunder much louder than the first; but\nthey held Kate too fast to let her stop, or otherwise make herself more\nfoolish.\n\nIn a very few minutes they were at the good lady's door; in another\nminute in her bedroom, where, while she and her maid bustled off to warm\nthe bed, Lady de la Poer tried to get the clothes off\u2014a service of\ndifficulty, when every tie held fast, every button was slippery, and the\ntighter garments fitted like skins.  Kate was subdued and frightened; she\ngave no trouble, but all the help she gave was to pull a string so as to\nmake a hopeless knot of the bow that her friend had nearly undone.\n\nHowever, by the time the bed was warm the dress was off, and the child,\nrolled up in a great loose night-dress of the kind lady's, was installed\nin it, feeling\u2014sultry day though it were\u2014that the warm dryness was\nextremely comfortable to her chilled limbs.  The good lady brought her\nsome hot tea, and moved away to the window, talking in a low murmuring\nvoice to Lady de la Poer.  Presently a fresh flash of lightning made her\nbury her head in the pillow; and there she began thinking how hard it was\nthat the thunder should come to spoil her one day's pleasure; but soon\nstopped this, remembering Who sends storm and thunder, and feeling afraid\nto murmur.  Then she remembered that perhaps she deserved to be\ndisappointed.  She had been wild and troublesome, had spoilt Adelaide's\nbirthday, teazed Mary, and made kind Lady de la Poer grave and\ndispleased.\n\nShe would say how sorry she was, and ask pardon.  But the two ladies\nstill stood talking.  She must wait till this stranger was gone.  And\nwhile she was waiting\u2014how it was she knew not\u2014but Countess Kate was fast\nasleep.\n\n\n\n\nCHAPTER VIII.\n\n\nWHEN Kate opened her eyes again, and turned her face up from the pillow,\nshe saw the drops on the window shining in the sun, and Lady de la Poer,\nwith her bonnet off, reading under it.\n\nAll that had happened began to return on Kate's brain in a funny medley;\nand the first thing she exclaimed was, \"Oh! those poor little fishes, how\nI must have frightened them!\"\n\n\"My dear!\"\n\n\"Do you think I did much mischief?\" said Kate, raising herself on her\narm.  \"I am sure the fishes must have been frightened, and the\nwater-lilies broken.  Oh! you can't think how nasty their great coiling\nstems were\u2014just like snakes!  But those pretty blue and pink flowers!\nDid it hurt them much, do you think\u2014or the fish?\"\n\n\"I should think the fish had recovered the shock,\" said Lady de la Poer,\nsmiling; \"but as to the lilies, I should be glad to be sure you had done\nyourself as little harm as you have to them.\"\n\n\"Oh no,\" said Kate, \"I'm not hurt\u2014if Aunt Barbara won't be terribly\nangry.  Now I wouldn't mind that, only that I've spoilt Addie's birthday,\nand all your day.  Please, I'm very sorry!\"\n\nShe said this so sadly and earnestly, that Lady de la Poer came and gave\nher a kind hiss of forgiveness, and said:\n\n\"Never mind, the girls are very happy with their father, and the rest is\ngood for me.\"\n\nKate thought this very comfortable and kind, and clung to the kind hand\ngratefully; but though it was a fine occasion for one of the speeches she\ncould have composed in private, all that came out of her mouth was, \"How\nhorrid it is\u2014the way everything turns out with me!\"\n\n\"Nay, things need not turn out horrid, if a certain little girl would\nkeep herself from being silly.\"\n\n\"But I _am_ a silly little girl!\" cried Kate with emphasis.  \"Uncle\nWardour says he never saw such a silly one, and so does Aunt Barbara!\"\n\n\"Well, my dear,\" said Lady de la Poer very calmly, \"when clever people\ntake to being silly, they can be sillier than anyone else.\"\n\n\"Clever people!\" cried Kate half breathlessly.\n\n\"Yes,\" said the lady, \"you are a clever child; and if you made the most\nof yourself, you could be very sensible, and hinder yourself from being\nfoolish and unguarded, and getting into scrapes.\"\n\nKate gasped.  It was not pleasant to be in a scrape; and yet her whole\nself recoiled from being guarded and watchful, even though for the first\ntime she heard she was not absolutely foolish.  She began to argue, \"I\nwas naughty, I know, to teaze Mary; and Mary at home would not have let\nme; but I could not help the tumbling into the pond.  I wanted to get out\nof the way of the lightning.\"\n\n\"Now, Kate, you _are_ trying to show how silly you can make yourself.\"\n\n\"But I can't bear thunder and lightning.  It frightens me so, I don't\nknow what to do; and Aunt Jane is just as bad.  She always has the\nshutters shut.\"\n\n\"Your Aunt Jane has had her nerves weakened by bad health; but you are\nyoung and strong, and you ought to fight with fanciful terrors.\"\n\n\"But it is not fancy about lightning.  It does kill people.\"\n\n\"A storm is very awful, and is one of the great instances of God's power.\nHe does sometimes allow His lightnings to fall; but I do not think it can\nbe quite the thought of this that terrifies you, Kate, for the\nrecollection of His Hand is comforting.\"\n\n\"No,\" said Kate honestly, \"it is not thinking of that.  It is that the\nglare\u2014coming no one knows when\u2014and the great rattling clap are so\u2014so\nfrightful!\"\n\n\"Then, my dear, I think all you can do is to pray not only for protection\nfrom lightning and tempest, but that you may be guarded from the fright\nthat makes you forget to watch yourself, and so renders the danger\ngreater!  You could not well have been drowned where you fell; but if it\nhad been a river\u2014\"\n\n\"I know,\" said Kate.\n\n\"And try to get self-command.  That is the great thing, after all, that\nwould hinder things from being horrid!\" said Lady de la Poer, with a\npleasant smile, just as a knock came to the door, and the maid announced\nthat it was five o'clock, and Miss's things were quite ready; and in\nreturn she was thanked, and desired to bring them up.\n\n\"Miss!\" said Kate, rather hurt: \"don't they know who we are?\"\n\n\"It is not such a creditable adventure that we should wish to make your\nname known,\" said Lady de la Poer, rather drily; and Kate blushed, and\nbecame ashamed of herself.\n\nShe was really five minutes before she recovered the use of her tongue,\nand that was a long time for her.  Lady de la Poer meantime was helping\nher to dress, as readily as Josephine herself could have done, and\nbrushing out the hair, which was still damp.  Kate presently asked where\nthe old lady was.\n\n\"She had to go back as soon as the rain was over, to look after a nephew\nand niece, who are spending the day with her.  She said she would look\nfor our party, and tell them how we were getting on.\"\n\n\"Then I have spoilt three people's pleasure more!\" said Kate ruefully.\n\"Is the niece a little girl?\"\n\n\"I don't know; I fancy her grown up, or they would have offered clothes\nto you.\"\n\n\"Then I don't care!\" said Kate.\n\n\"What for?\"\n\n\"Why, for not telling my name.  Once it would have been like a fairy tale\nto Sylvia and me, and have made up for anything, to see a\ncountess\u2014especially a little girl.  But don't you think seeing me would\nquite spoil that?\"\n\nLady de la Poer was so much amused, that she could not answer at first;\nand Kate began to feel as if she had been talking foolishly, and turned\nher back to wash her hands.\n\n\"Certainly, I don't think we are quite as well worth seeing as the\nCrystal Palace!  You put me in mind of what Madame Campan said.  She had\nbeen governess to the first Napoleon's sisters; and when, in the days of\ntheir grandeur, she visited them, one of them asked her if she was not\nawe-struck to find herself among so much royalty.  'Really,' she said, 'I\ncan't be much afraid of queens whom I have whipped!'\"\n\n\"They were only mock queens,\" said Kate.\n\n\"Very true.  But, little woman, it is _all_ mockery, unless it is the\n_self_ that makes the impression; and I am afraid being perched upon any\nkind of pedestal makes little faults and follies do more harm to others.\nBut come, put on your hat: we must not keep Papa waiting.\"\n\nThe hat was the worst part of the affair; the colour of the blue edge of\nthe ribbon had run into the white, and the pretty soft feather had been\nso daggled in the wet, that an old hen on a wet day was respectability\nitself compared with it, and there was nothing for it but to take it out;\nand even then the hat reminded Kate of a certain Amelia Matilda Bunny,\nwhose dirty finery was a torment and a by-word in St. James's Parsonage.\nHer frock and white jacket had been so nicely ironed out, as to show no\ntraces of the adventure; and she disliked all the more to disfigure\nherself with such a thing on her head for the present, as well as to\nencounter Aunt Barbara by-and-by.\n\n\"There's no help for it,\" said Lady de la Poer, seeing her disconsolately\nsurveying it; \"perhaps it will not be bad for you to feel a few\nconsequences from your heedlessness.\"\n\nWhether it were the hat or the shock, Kate was uncommonly meek and\nsubdued as she followed Lady de la Poer out of the room; and after giving\nthe little maid half a sovereign and many thanks for having so nicely\nrepaired the damage, they walked back to the palace, and up the great\nstone stairs, Kate hanging down her head, thinking that everyone was\nwondering how Amelia Matilda Bunny came to be holding by the hand of a\nlady in a beautiful black lace bonnet and shawl, so quiet and simple, and\nyet such a lady!\n\nShe hardly even looked up when the glad exclamations of the four girls\nand their father sounded around her, and she could not bear their\ninquiries whether she felt well again.  She knew that she owed thanks to\nMary and her father, and apologies to them all; but she had not manner\nenough to utter them, and only made a queer scrape with her foot, like a\nhen scratching out corn, hung her head, and answered \"Yes.\"\n\nThey saw she was very much ashamed, and they were in a hurry besides; so\nwhen Lord de la Poer had said he had given all manner of thanks to the\ngood old lady, he took hold of Kate's hand, as if he hardly ventured to\nlet go of her again, and they all made the best of their way to the\nstation, and were soon in full career along the line, Kate's heart\nsinking as she thought of Aunt Barbara.  Fanny tried kindly to talk to\nher; but she was too anxious to listen, made a short answer, and kept her\neyes fixed on the two heads of the party, who were in close consultation,\nrendered private by the noise of the train.\n\n\"If ever I answer for anyone again!\" said Lord de la Poer.  \"And now for\nfacing Barbara!\"\n\n\"You had better let me do that.\"\n\n\"What! do you think I am afraid?\" and Kate thought the smile on his lip\nvery cruel, as she could not hear his words.\n\n\"I don't do you much injustice in thinking so,\" as he shrugged up his\nshoulders like a boy going to be punished; \"but I think Barbara considers\nyou as an accomplice in mischief, and will have more mercy if I speak.\"\n\n\"Very well!  I'm not the man to prevent you.  Tell Barbara I'll undergo\nwhatever she pleases, for having ever let go the young lady's hand!  She\nmay have me up to the Lord Chancellor if she pleases!\"\n\nA little relaxation in the noise made these words audible; and Kate, who\nknew the Lord Chancellor had some power over her, and had formed her\nnotions of him from a picture, in a history book at home, of Judge\nJefferies holding the Bloody Assize, began to get very much frightened;\nand her friends saw her eyes growing round with alarm, and not knowing\nthe exact cause, pitied her; Lord de la Poer seated her upon his knee,\nand told her that Mamma would take her home, and take care Aunt Barbara\ndid not punish her.\n\n\"I don't think she will punish me,\" said Kate; \"she does not often!  But\npray come home with me!\" she added, getting hold of the lady's hand.\n\n\"What would she do to you, then?\"\n\n\"She would\u2014only\u2014be dreadful!\" said Kate.\n\nLord de la Poer laughed; but observed, \"Well, is it not enough to make\none dreadful to have little girls taking unexpected baths in public?\nNow, Kate, please to inform me, in confidence, what was the occasion of\nthat remarkable somerset.\"\n\n\"Only the lightning,\" muttered Kate.\n\n\"Oh!  I was not certain whether your intention might not have been to\nmake that polite address to an aquatic bird, for which you pronounced\nMary not to have sufficient courage!\"\n\nLady de la Poer, thinking this a hard trial of the poor child's temper,\nwas just going to ask him not to tease her; but Kate was really candid\nand good tempered, and she said, \"I was wrong to say that!  It was Mary\nthat had presence of mind, and I had not.\"\n\n\"Then the fruit of the adventure is to be, I hope, Look Before you\nLeap!\u2014Eh, Lady Caergwent?\"\n\nAnd at the same time the train stopped, and among kisses and farewells,\nKate and kind Lady de la Poer left the carriage, and entering the\nbrougham that was waiting for them, drove to Bruton Street; Kate very\ngrave and silent all the way, and shrinking behind her friend in hopes\nthat the servant who opened the door would not observe her plight\u2014indeed,\nshe took her hat off on the stairs, and laid it on the table in the\nlanding.\n\nTo her surprise, the beginning of what Lady de la Poer said was chiefly\napology for not having taken better care of her.  It was all quite true:\nthere was no false excuse made for her, she felt, when Aunt Barbara\nlooked ashamed and annoyed, and said how concerned she was that her niece\nshould be so unmanageable; and her protector answered,\n\n\"Not that, I assure you!  She was a very nice little companion, and we\nquite enjoyed her readiness and intelligent interest; but she was a\nlittle too much excited to remember what she was about when she was\nstartled.\"\n\n\"And no wonder,\" said Lady Jane.  \"It was a most tremendous storm, and I\nfeel quite shaken by it still.  You can't be angry with her for being\nterrified by it, Barbara dear, or I shall know what you think of me;\u2014half\ndrowned too\u2014poor child!\"\n\nAnd Aunt Jane put her soft arm round Kate, and put her cheek to hers.\nPerhaps the night of Kate's tears had really made Jane resolved to try to\nsoften even Barbara's displeasure; and the little girl felt it very kind,\nthough her love of truth made her cry out roughly, \"Not half drowned!\nMary held me fast, and Lord de la Poer pulled me out!\"\n\n\"I am sure you ought to be extremely thankful to them,\" said Lady\nBarbara, \"and overcome with shame at all the trouble and annoyance you\nhave given!\"\n\nLady de la Poer quite understood what the little girl meant by her aunt\nbeing dreadful.  She would gladly have protected her; but it was not what\ncould be begged off like punishment, nor would truth allow her to say\nthere had been no trouble nor annoyance.  So what she did say was, \"When\none has ten children, one reckons upon such things!\" and smiled as if\nthey were quite pleasant changes to her.\n\n\"Not, I am sure, with your particularly quiet little girls,\" said Aunt\nBarbara.  \"I am always hoping that Katharine may take example by them.\"\n\n\"Take care what you hope, Barbara,\" said Lady de la Poer, smiling: \"and\nat any rate forgive this poor little maiden for our disaster, or my\nhusband will be in despair.\"\n\n\"I have nothing to forgive,\" said Lady Barbara gravely.  \"Katharine\ncannot have seriously expected punishment for what is not a moral fault.\nThe only difference will be the natural consequences to herself of her\nfolly.\u2014You had better go down to the schoolroom, Katharine, have your\ntea, and then go to bed; it is nearly the usual time.\"\n\nLady de la Poer warmly kissed the child, and then remained a little while\nwith the aunts, trying to remove what she saw was the impression, that\nKate had been complaining of severe treatment, and taking the opportunity\nof telling them what she herself thought of the little girl.  But though\nAunt Barbara listened politely, she could not think that Lady de la Poer\nknew anything about the perverseness, heedlessness, ill-temper,\ndisobedience, and rude ungainly ways, that were so tormenting.  She said\nno word about them herself, because she would not expose her niece's\nfaults; but when her friend talked Kate's bright candid conscientious\ncharacter, her readiness, sense, and intelligence, she said to herself,\nand perhaps justly, that here was all the difference between at home and\nabroad, an authority and a stranger.\n\nMeantime, Kate wondered what would be the natural consequences of her\nfolly.  Would she have a rheumatic fever or consumption, like a child in\na book?\u2014and she tried breathing deep, and getting up a little cough, to\nsee if it was coming!  Or would the Lord Chancellor hear of it?  He was\nnew bugbear recently set up, and more haunting than even a gunpowder\ntreason in the cellars!  What did he do with the seals?  Did he seal up\nmischievous heiresses in closets, as she had seen a door fastened by two\nseals and a bit of string?  Perhaps the Court of Chancery was full of\nsuch prisons!  And was the woolsack to smother them with, like the\nprinces in the Tower?\n\nIt must be owned that it was only when half asleep at night that Kate was\nso absurd.  By day she knew very well that the Lord Chancellor was only a\ngreat lawyer; but she also knew that whenever there was any puzzle or\ndifficulty about her or her affairs, she always heard something\nmysteriously said about applying to the Lord Chancellor, till she began\nto really suspect that it was by his commands that Aunt Barbara was so\nstern with her; and that if he knew of her fall into the pond, something\nterrible would come of it.  Perhaps that was why the De la Poers kept her\nname so secret!\n\nShe trembled as she thought of it; and here was another added to her many\nterrors.  Poor little girl!  If she had rightly feared and loved One, she\nwould have had no room for the many alarms that kept her heart\nfluttering!\n\n\n\n\nCHAPTER IX.\n\n\nIT may be doubted whether Countess Kate ever did in her childhood\ndiscover what her Aunt Barbara meant by the natural consequences of her\nfolly, but she suffered from them nevertheless.  When the summer was\ngetting past its height of beauty, and the streets were all sun and misty\nheat, and the grass in the parks looked brown, and the rooms were so\nclose that even Aunt Jane had one window open, Kate grew giddy in the\nhead almost every morning, and so weary and dull all day that she had\nhardly spirit to do anything but read story-books.  And Mrs. Lacy was\nquite poorly too, though not saying much about it; was never quite\nwithout a head-ache, and was several times obliged to send Kate out for\nher evening walk with Josephine.\n\nIt was high time to be going out of town; and Mrs. Lacy was to go and be\nwith her son in his vacation.\n\nThis was the time when Kate and the Wardours had hoped to be together.\nBut \"the natural consequence\" of the nonsense Kate had talked, about\nbeing \"always allowed\" to do rude and careless things, and her wild\nrhodomontade about romping games with the boys, had persuaded her aunts\nthat they were very improper people for her to be with, and that it would\nbe wrong to consent to her going to Oldburgh.\n\nThat was one natural consequence of her folly.  Another was that when the\nDe la Poers begged that she might spend the holidays with them, and from\nfather and mother downwards were full of kind schemes for her happiness\nand good, Lady Barbara said to her sister that it was quite impossible;\nthese good friends did not know what they were asking, and that the child\nwould again expose herself in some way that would never be forgotten,\nunless she were kept in their own sight till she had been properly tamed\nand reduced to order.\n\nIt was self-denying in Lady Barbara to refuse that invitation, for she\nand her sister would have been infinitely more comfortable together\nwithout their troublesome countess\u2014above all when they had no governess\nto relieve them of her.  The going out of town was sad enough to them,\nfor they had always paid a long visit at Caergwent Castle, which had felt\nlike their home through the lifetime of their brother and nephew; but now\nit was shut up, and their grief for their young nephew came back all the\nmore freshly at the time of year when they were used to be kindly\nentertained by him in their native home.\n\nBut as they could not go there, they went to Bournemouth and the first\nrun Kate took upon the sands took away all the giddiness from her head,\nand put an end to the tired feeling in her limbs!  It really was a run!\nAunt Barbara gave her leave to go out with Josephine; and though\nJosephine said it was very sombre and savage, between the pine-woods and\nthe sea, Kate had not felt her heart leap with such fulness of enjoyment\nsince she had made snow-balls last winter at home.  She ran down to the\nwaves, and watched them sweep in and curl over and break, as if she could\nnever have enough of them; and she gazed at the grey outline of the Isle\nof Wight opposite, feeling as if there was something very great in really\nseeing an island.\n\nWhen she came in, there was so much glow on her brown check, and her\neyelids looked so much less heavy, that both the aunts gazed at her with\npleasure, smiled to one another, and Lady Jane kissed her, while Lady\nBarbara said, \"This was the right thing.\"\n\nShe was to be out as much as possible, so her aunt made a set of new\nrules for the day.  There was to be a walk before breakfast; then\nbreakfast; then Lady Barbara heard her read her chapter in the Bible, and\ngo through her music.  And really the music was not half as bad as might\nhave been expected with Aunt Barbara.  Kate was too much afraid of her to\ngive the half attention she had paid to poor Mrs. Lacy\u2014fright and her\naunt's decision of manner forced her to mind what she was about; and\nthough Aunt Barbara found her really very dull and unmusical, she did get\non better than before, and learnt something, though more like a machine\nthan a musician.\n\nThen she went out again till the hottest part of the day, during which a\nbit of French and of English reading was expected from her, and half an\nhour of needle-work; then her dinner; and then out again\u2014with her aunts\nthis time, Aunt Jane in a wheeled-chair, and Aunt Barbara walking with\nher\u2014this was rather dreary; but when they went in she was allowed to stay\nout with Josephine, with only one interval in the house for tea, till it\ngrew dark, and she was so sleepy with the salt wind, that she was ready\nfor bed, and had no time to think of the Lord Chancellor.\n\nAt first, watching those wonderful and beautiful waves was pleasure\nenough; and then she was allowed, to her wonder and delight, to have a\nholland dress, and dig in the sand, making castles and moats, or rocks\nand shipwrecks, with beautiful stories about them; and sometimes she\nhunted for the few shells and sea-weeds there, or she sat down and read\nsome of her favourite books, especially poetry\u2014it suited the sea so well;\nand she was trying to make Ellen's Isle and all the places of the \"Lady\nof the Lake\" in sand, only she never had time to finish them, and they\nalways were either thrown down or washed away before she could return to\nthem.\n\nBut among all these amusements, she was watching the families of children\nwho played together, happy creatures!  The little sturdy boys, that\ndabbled about so merrily, and minded so little the \"Now Masters\" of their\nindignant nurses; the little girls in brown hats, with their baskets\nfull; the big boys, that even took off shoes, and dabbled in the shallow\nwater; the great sieges of large castles, where whole parties attacked\nand defended\u2014it was a sort of melancholy glimpse of fairy-land to her,\nfor she had only been allowed to walk on the beach with Josephine on\ncondition she never spoke to the other children.\n\nWould the Lord Chancellor be after her if she did?  Her heart quite\nyearned for those games, or even to be able to talk to one of those\nlittle damsels; and one day when a bright-faced girl ran after her with a\npiece of weed that she had dropped, she could hardly say \"thank you\" for\nher longing to say more; and many were the harangues she composed within\nherself to warn the others not to wish to change places with her, for to\nbe a countess was very poor fun indeed.\n\nHowever, one morning at the end of the first week, Kate looked up from a\nletter from Sylvia, and said with great glee, \"Aunt Barbara!  O Aunt\nBarbara!  Alice and the other Sylvia\u2014Sylvia Joanna\u2014are coming!  I may\nplay with them, mayn't I?\"\n\n\"Who are they?\" said her aunt gravely.\n\n\"Uncle Wardour's nieces,\" said Kate; \"Sylvia's cousins, you know, only we\nnever saw them; but they are just my age; and it will be such fun\u2014only\nAlice is ill, I believe.  Pray\u2014please\u2014let me play with them!\" and Kate\nhad tears in her eyes.\n\n\"I shall see about it when they come.\"\n\n\"Oh, but\u2014but I can't have them there\u2014Sylvia's own, own cousins\u2014and not\nplay with them!  Please, Aunt Barbara!\"\n\n\"You ought to know that this impetuosity never disposes me favourably,\nKatharine; I will inquire and consider.\"\n\nKate had learnt wisdom enough not to say any more just then; but the\nthought of sociability, the notion of chattering freely to young\ncompanions, and of a real game at play, and the terror of having all this\nwithheld, and of being thought too proud and haughty for the Wardours,\nput her into such an agony, that she did not know what she was about,\nmade mistakes even in reading, and blundered her music more than she had\nover done under Lady Barbara's teaching; and then, when her aunt reproved\nher, she could not help laying down her head and bursting into a fit of\ncrying.  However, she had not forgotten the terrible tea-drinking, and\nwas resolved not to be as bad as at that time, and she tried to stop\nherself, exclaiming between her sobs, \"O Aunt Bar\u2014bar\u2014a,\u2014I\u2014can\u2014not\u2014help\nit!\"  And Lady Barbara did not scold or look stern.  Perhaps she saw that\nthe little girl was really trying to chock herself, for she said quite\nkindly, \"Don't, my dear.\"\n\nAnd just then, to Kate's great wonder, in came Lady Jane, though it was\nfull half an hour earlier than she usually left her room; and Lady\nBarbara looked up to her, and said, quite as if excusing herself,\n\"Indeed, Jane, I have not been angry with her.\"\n\nAnd Kate, somehow, understanding that she might, flung herself down by\nAunt Jane, and hid her face in her lap, not crying any more, though the\nsobs were not over, and feeling the fondling hands on her hair very\ntender and comforting, though she wondered to hear them talk as if she\nwere asleep or deaf\u2014or perhaps they thought their voices too low, or\ntheir words too long and fine for her to understand; nor perhaps did she,\nthough she gathered their drift well enough, and that kind Aunt Jane was\nquite pleading for herself in having come to the rescue.\n\n\"I could not help it, indeed\u2014you remember Lady de la Poer, Dr. Woodman,\nboth\u2014excitable, nervous temperament\u2014almost hysterical.\"\n\n\"This unfortunate intelligence\u2014untoward coincidence\u2014\" said Lady Barbara.\n\"But I have been trying to make her feel I am not in anger, and I hope\nthere really was a struggle for self-control.\"\n\nKate took her head up again at this, a little encouraged; and Lady Jane\nkissed her forehead, and repeated, \"Aunt Barbara was not angry with you,\nmy dear.\"\n\n\"No, for I think you have tried to conquer yourself,\" said Lady Barbara.\nShe did not think it wise to tell Kate that she thought she could not\nhelp it, though oddly enough, the very thing had just been said over the\nchild's head, and Kate ventured on it to get up, and say quietly, \"Yes,\nit was not Aunt Barbara's speaking to me that made me cry, but I am so\nunhappy about Alice and Sylvia Joanna;\" and a soft caress from Aunt Jane\nmade her venture to go on.  \"It is not only the playing with them, though\nI do wish for that very very much indeed; but it would be so unkind, and\nso proud and ungrateful, to despise my own cousin's cousins!\"\n\nThis was more like the speeches Kate made in her own head than anything\nshe had ever said to her aunts; and it was quite just besides, and not\nspoken in naughtiness, and Lady Barbara did not think it wrong to show\nthat she attended to it.  \"You are right, Katharine,\" she said; \"no one\nwishes you to be either proud or ungrateful.  I would not wish entirely\nto prevent you from seeing the children of the family, but it must not be\ntill there is some acquaintance between myself and their mother, and I\ncannot tell whether you can be intimate with them till I know what sort\nof children they are.  Much, too, must depend on yourself, and whether\nyou will behave well with them.\"\n\nKate gave a long sigh, and looked up relieved; and for some time she and\nher aunt were not nearly so much at war as hitherto, but seemed to be\ncoming to a somewhat better understanding.\n\nYet it rather puzzled Kate.  She seemed to herself to have got this\nfavour for crying for it; and it was a belief at home, not only that\nnothing was got by crying, but that if by some strange chance it were, it\nnever came to good; and she began the more to fear some disappointment\nabout the expected Wardours.\n\nFor two or three days she was scanning every group on the sands with all\nher might, in hopes of some likeness to Sylvia, but at last she was taken\nby surprise: just as she was dressed, and Aunt Barbara was waiting in the\ndrawing-room for Aunt Jane, there came a knock at the door, and \"Mrs.\nWardour\" was announced.\n\nIn came a small, quiet-looking lady in mourning, and with her a girl of\nabout Kate's own age; there was some curtseying and greeting between the\ntwo ladies, and her aunt said, \"Here is my niece.\u2014Come and speak to Mrs.\nWardour, my dear,\" and motioned her forwards.\n\nNow to be motioned forwards by Aunt Barbara always made Kate shrink back\ninto herself, and the presence of a little girl before elders likewise\nrendered her shy and bashful, so she came forth as if intensely\ndisgusted, put out her hand as if she were going to poke, and muttered\nher favourite \"\u2014do\" so awkwardly and coldly, that Lady Barbara felt how\nproud and ungracious it looked, and to make up said, \"My niece has been\nvery eager for your coming.\"  And then the two little girls drew off into\nthe window, and looked at each other under their eyelashes in silence.\n\nSylvia Joanna Wardour was not like her namesake at home, Sylvia\nKatharine.  She was a thin, slight, quiet-looking child, with so little\nto note about her face, that Kate was soon wondering at her dress being\nso much smarter than her own was at present.  She herself had on a\nholland suit with a deep cape, which, except that they were adorned with\nlabyrinths of white braid, were much what she had worn at home, also a\nround brown hat, shading her face from the sun; whereas Sylvia's face was\nexposed by a little turban hat so deeply edged with blue velvet, that the\nwhite straw was hardly seen; had a little watered-silk jacket, and a\nlittle flounced frock of a dark silk figured with blue, that looked\nslightly fuzzed out; and perhaps she was not at ease in this fine dress,\nfor she stood with her head down, and one hand on the window-sill,\npretending to look out of window, but really looking at Kate.\n\nMeanwhile the two grown-up ladies were almost as stiff and shy, though\nthey could not keep dead silence like the children.  Mrs. Wardour had\nheard before that Lady Barbara Umfraville was a formidable person, and\nwas very much afraid of her; and Lady Barbara was not a person to set\nanyone at ease.\n\nSo there was a little said about taking the liberty of calling, for her\nbrother-in-law was so anxious to hear of Lady Caergwent: and Lady Barbara\nsaid her niece was very well and healthy, and had only needed change of\nair.\n\nAnd then came something in return about Mrs. Wardour's other little girl,\na sad invalid, she said, on whose account they were come to Bournemouth;\nand there was a little more said of bathing, and walking, and whether the\nplace was full; and then Mrs. Wardour jumped up and said she was\ndetaining Lady Barbara, and took leave; Kate, though she had not spoken a\nword to Sylvia Wardour, looking at her wistfully with all her eyes, and\nfeeling more than usually silly.\n\nAnd when the guests were gone her aunt told her how foolish her want of\nmanner was, and how she had taken the very means to make them think she\nwas not glad to see them.  She hung down her head, and pinched the ends\nof her gloves; she knew it very well, but that did not make it a bit more\npossible to find a word to say to a stranger before the elders, unless\nthe beginning were made for her as by the De la Poers.\n\nHowever, she knew it would be very different out of doors, and her heart\nbounded when her aunt added, \"They seem to be quiet, lady-like,\ninoffensive people, and I have no objection to your associating with the\nlittle girl in your walks, as long as I do not see that it makes you\nthoughtless and ungovernable.\"\n\n\"Oh, thank you, thank you, Aunt Barbara!\" cried Kate, with a bouncing\nbound that did not promise much for her thought or her governableness;\nbut perhaps Lady Barbara recollected what her own childhood would have\nbeen without Jane, for she was not much discomposed, only she said,\n\n\"It is very odd you should be so uncivil to the child in her presence,\nand so ecstatic now!  However, take care you do not get too familiar.\nRemember, these Wardours are no relations, and I will not have you\nletting them call you by your Christian name.\"\n\nKate's bright looks sank.  That old married-woman sound, Lady Caergwent,\nseemed as if it would be a bar between her and the free childish fun she\nhoped for.  Yet when so much had been granted, she must not call her aunt\ncross and unkind, though she did think it hard and proud.\n\nPerhaps she was partly right; but after all, little people cannot judge\nwhat is right in matters of familiarity.  They have only to do as they\nare told, and they may be sure of this, that friendship and respect\ndepend much more on what people are in themselves than on what they call\none another.\n\nThis lady was the widow of Mr. Wardour's brother, and lived among a great\nclan of his family in a distant county, where Mary and her father had\nsometimes made visits, but the younger ones never.  Kate was not likely\nto have been asked there, for it was thought very hard that she should be\nleft on the hands of her aunt's husband: and much had been said of the\nduty of making her grand relations provide for her, or of putting her\ninto the \"Clergy Orphan Asylum.\"  And there had been much displeasure\nwhen Mr. Wardour answered that he did not think it right that a child who\nhad friends should live on the charity intended for those who had none\nable to help them; and soon after the decision he had placed his son\nArmyn in Mr. Brown's office, instead of sending him to the University.\nAll the Wardours were much vexed then; but they were not much better\npleased when the little orphan had come to her preferment, and he made no\nattempt to keep her in his hands, and obtain the large sum allowed for\nher board\u2014only saying that his motherless household was no place for her,\nand that he could not at once do his duty by her and by his parish.  They\ncould not understand the real love and uprightness that made him prefer\nher advantage to his own\u2014what was right to what was convenient.\n\nMrs. George Wardour had not scolded her brother-in-law for his want of\nprudence and care for his own children's interests; but she had agreed\nwith those who did; and this, perhaps, made her feel all the more awkward\nand shy when she was told that she _must_ go and call upon the Lady\nUmfravilles, whom the whole family regarded as first so neglectful and\nthen so ungrateful, and make acquaintance with the little girl who had\nonce been held so cheap.  She was a kind, gentle person, and a careful,\nanxious mother, but not wishing to make great acquaintance, nor used to\nfine people, large or small, and above all, wrapped up in her poor little\ndelicate Alice.\n\nThe next time Kate saw her she was walking by the side of Alice's\nwheeled-chair, and Sylvia by her side, in a more plain and suitable\ndress.  Kate set off running to greet them; but at a few paces from them\nwas seized by a shy fit, and stood looking and feeling like a goose,\ndrawing great C's with the point of her parasol in the sand; Josephine\nlooking on, and thinking how \"_b\u00eate_\" English children were.  Mrs.\nWardour was not much less shy; but she knew she must make a beginning,\nand so spoke in the middle of Kate's second C: and there was a shaking of\nhands, and walking together.\n\nThey did not get on very well: nobody talked but Mrs. Wardour, and she\nasked little frightened questions about the Oldburgh party, as she called\nthem, which Kate answered as shortly and shyly\u2014the more so from the\nuncomfortable recollection that her aunt had told her that this was the\nvery way to seem proud and unkind; but what could she do?  She felt as if\nshe were frozen up stiff, and could neither move nor look up like\nherself.  At last Mrs. Wardour said that Alice would be tired, and must\ngo in; and then Kate managed to blurt out a request that Sylvia might\nstay with her.  Poor Sylvia looked a good deal scared, and as if she\nlonged to follow her mamma and sister; but the door was shut upon her,\nand she was left alone with those two strange people\u2014the Countess and the\nFrenchwoman!\n\nHowever, Kate recovered the use of her limbs and tongue in a moment, and\ninstantly took her prisoner's hand, and ran off with her to the corner\nwhere the scenery of Loch Katrine had so often been begun, and began with\ngreat animation to explain.  This\u2014a hole that looked as if an old hen had\nbeen grubbing in it\u2014was Loch Katrine.\n\n\"Loch Katharine\u2014that's yours!  And which is to be Loch Sylvia?\" said the\nchild, recovering, as she began to feel by touch, motion, and voice, that\nshe had only to do with a little girl after all.\n\n\"Loch nonsense!\" said Kate, rather bluntly.  \"Did you never hear of the\nLochs, the Lakes, in Scotland?\"\n\n\"Loch Lomond, Loch Katrine, Loch Awe, Loch Ness?\u2014But I don't do my\ngeography out of doors!\"\n\n\"'Tisn't geography; 'tis the 'The Lady of the Lake.'\"\n\n\"Is that a new game?\"\n\n\"Dear me! did you never read 'The Lady of the Lake?'\u2014Sir Walter Scott's\npoem\u2014\n\n                     'The summer dawn's reflected hue\u2014'\"\n\n\"Oh!  I've learnt that in my extracts; but I never did my poetry task out\nof doors!\"\n\n\"'Tisn't a task\u2014'tis beautiful poetry!  Don't you like poetry better than\nanything?\"\n\n\"I like it better than all my other lessons, when it is not very long and\nhard.\"\n\nKate felt that her last speech would have brought Armyn and Charlie down\non her for affectation, and that it was not strictly true that she liked\npoetry better than anything, for a game at romps, and a very amusing\nstory, were still better things; so she did not exclaim at the other\nSylvia's misunderstanding, but only said, \"'The Lady of the Lake' is\nstory and poetry too, and we will play at it.\"\n\n\"And how?\"\n\n\"I'll tell you as we go on.  I'm the King\u2014that is, the Knight of\nSnowdon\u2014James Fitzjames, for I'm in disguise, you know; and you're\nEllen.\"\n\n\"Must I be Ellen?  We had a horrid nurse once, who used to slap us, and\nwas called Ellen.\"\n\n\"But it was her name.  She was Ellen Douglas, and was in banishment on an\nisland with her father.  You are Ellen, and Josephine is your old\nharper\u2014Allan Bane; she talks French, you know, and that will do for\nHighland: Gallic and Gaelic sound alike, you know.  There!  Then I'm\ngoing out hunting, and my dear gallant grey will drop down dead with\nfatigue, and I shall lose my way; and when you hear me wind my horn\ntoo-too, you get upon your hoop\u2014that will be your boat, you know\u2014and\nanswer 'Father!' and when I too-too again, answer 'Malcolm!' and then put\nup your hand behind your ear, and stand listening\n\n    \"With locks thrown back and lips apart,\n    Like monument of Grecian art;\"\n\nand then I'll tell you what to do.\"\n\nAway scudded the delighted Kate; and after having lamented her gallant\ngrey, and admired the Trosachs, came up too-tooing through her hand with\nall her might, but found poor Ellen, very unlike a monument of Grecian\nart, absolutely crying, and Allan Bane using his best English and kindest\ntones to console her.\n\n\"_Miladi l'a stup\u00e9faite\u2014la pauvre petite_!\" began Josephine; and Kate in\nconsternation asking what was the matter, and Josephine encouraging her,\nit was all sobbed out.  She did not like to be called Ellen\u2014and she\nthought it unkind to send her into banishment\u2014and she had fancied she was\nto get astride on her hoop, which she justly thought highly improper\u2014and\nabove all, she could not bear to say 'Father'\u2014because\u2014\n\n\"I never thought you would mind that,\" said Kate, rather abashed.  \"I\nnever did; and I never saw my papa or mamma either.\"\n\n\"No\u2014so you didn't care.\"\n\n\"Well then,\" said Kate gravely, \"we won't play at that.  Let's have\n'Marmion' instead; and I'll be killed.\"\n\n\"But I don't like you to be killed.\"\n\n\"It is only in play.\"\n\n\"Please\u2014please, let us have a nice play!\"\n\n\"Well, what do you call a nice play?\"\n\n\"Alice and I used to drive hoops.\"\n\n\"That's tiresome!  My hoop always tumbles down: think of something else.\"\n\n\"Alice and I used to play at ball; but there's no ball here!\"\n\n\"Then I'll stuff my pocket-handkerchief with seaweed, and make one;\" and\nKate spread out her delicate cambric one\u2014not quite so fit for such a\npurpose as the little cheap cotton ones at home, that Mary tried in vain\nto save from cruel misuse.\n\n\"Here's a famous piece!  Look, it is all wriggled; it is a mermaid's old\nstay-lace that she has used and thrown away.  Perhaps she broke it in a\npassion because her grandmother made her wear so many oyster-shells on\nher tail!\"\n\n\"There are no such creatures as mermaids,\" said Sylvia, looking at her\nsolemnly.\n\nThis was not a promising beginning; Sylvia Joanna was not a bit like\nSylvia Katharine, nor like Adelaide and Grace de la Poer; yet by seeing\neach other every day, she and Kate began to shake together, and become\nfriends.\n\nThere was no fear of her exciting Kate to run wild; she was a little\npussy-cat in her dread of wet, and guarded her clothes as if they could\nfeel\u2014indeed, her happiest moments were spent in the public walks by\nAlice's chair, studying how the people were dressed; but still she\nthought it a fine thing to be the only child in Bournemouth who might\nplay with the little Countess, and was so silly as to think the others\nenvied her when she was dragged and ordered about, bewildered by Kate's\nloud rapid talk about all kinds of odd things in books, and distressed at\nbeing called on to tear through the pine-woods, or grub in wet sand.  But\nit was not all silly vanity: she was a gentle, loving little girl, very\ngood-natured, and sure to get fond of all who were kind to her; and she\nliked Kate's bright ways and amusing manner\u2014perhaps really liking her\nmore than if she had understood her better; and Kate liked her, and\nrushed after her on every occasion, as the one creature with whom it was\npossible to play and to chatter.\n\nNo, not quite the one; for poor sick Alice was better for talk and quiet\nplay than her sister.  She read a great deal; and there was an exchange\nof story-books, and much conversation over them, between her and\nKate\u2014indeed, the spirit and animation of this new friend quite made her\nlight up, and brighten out of her languor whenever the shrill laughing\nvoice came near.  And Kate, after having got over her first awe at coming\nnear a child so unlike herself, grew very fond of her, and felt how good\nand sweet and patient she was.  She never ran off to play till Alice was\ntaken in-doors; and spent all her spare time in-doors in drawing picture\nstories, which were daily explained to the two sisters at some seat in\nthe pine-woods.\n\nThere was one very grand one, that lasted all the latter part of the stay\nat Bournemouth\u2014as the evenings grew longer, and Kate had more time for\npreparing it, at the rate of four or five scenes a day, drawn and\npainted\u2014being the career of a very good little girl, whose parents were\nkilled in a railway accident, (a most fearful picture was that\u2014all\nblunders being filled up by spots of vermilion blood and orange-\nflame!) and then came all the wonderful exertions by which she maintained\nher brothers and sisters, taught them, and kept them in order.\n\nThey all had names; and there was a naughty little Alexander, whose\nmonkey tricks made even Sylvia laugh.  Sylvia was very anxious that the\nadmirable heroine, Hilda, should be rewarded by turning into a countess;\nand could not enter into Kate's first objection\u2014founded on fact\u2014that it\ncould not be without killing all the brothers.  \"Why couldn't it be done\nin play, like so many other things?\"  To which Kate answered, \"There is a\nsort of true in play;\" but as Sylvia could not understand her, nor she\nherself get at her own idea, she went on to her other objection, a still\nmore startling one\u2014that \"She couldn't wish Hilda anything so nasty!\"\n\nAnd this very ignoble word was long a puzzle to Alice and Sylvia.\n\nThus the time at the sea-side was very happy\u2014quite the happiest since\nKate's change of fortune.  The one flaw in those times on the sands was\nwhen she was alone with Sylvia and Josephine; not in Sylvia's\ndulness\u2014that she had ceased to care about\u2014but in a little want of plain\ndealing.  Sylvia was never wild or rude, but she was not strictly\nobedient when out of sight; and when Kate was shocked would call it very\nunkind, and caress and beseech her not to tell.\n\nThey were such tiny things, that they would hardly bear mention; but one\nwill do as a specimen.  Sylvia was one of those very caressing children\nwho can never be happy without clinging to their friends, kissing them\nconstantly, and always calling them dear, love, and darling.\n\nNow, Mrs. Wardour knew it was not becoming to see all this embracing in\npublic, and was sure besides that Lady Barbara would not like to see the\nCountess hung upon in Sylvia's favourite way; so she forbade all such\ndemonstrations except the parting and meeting kiss.  It was a terrible\ngrievance to Sylvia\u2014it seemed as if her heart could not love without her\ntouch; but instead of training herself in a little self-control and\nobedience, she thought it \"cross;\" and Mamma was no sooner out of sight\nthan her arm was around Kate's waist.  Kate struggled at first\u2014it did not\nsuit her honourable conscientiousness; but then Sylvia would begin to cry\nat the unkindness, say Kate did not love her, that she would not be proud\nif she was a countess: and Kate gave in, liked the love\u2014of which, poor\nchild! she got so little\u2014and let Sylvia do as she pleased, but never\nwithout a sense of disobedience and dread of being caught.\n\nSo, too, about her title.  Sylvia called her darling, duck, and love, and\nshe called Sylvia by plenty of such names; but she had been obliged to\ntell of her aunt's desire\u2014that Katharine and Kate should never be used.\n\nSylvia's ready tears fell; but the next day she came back cheerful, with\nthe great discovery that darling Lady Caergwent might be called K, her\ninitial, and the first syllable of her title.  It was the cleverest\ninvention Sylvia had ever made; and she was vexed when Kate demurred,\nhonestly thinking that her aunts would like it worse than even Kate, and\nthat therefore she ought not to consent.\n\nBut when Sylvia coaxingly uttered, \"My own dear duck of a K,\" and the\nsoft warm arm squeezed her, and the eyes would have been weeping, and the\ntongue reproaching in another moment, she allowed it to go on\u2014it was so\nprecious and sweet to be loved; and she told Sylvia she was a star in the\ndark night.\n\nNo one ever found out those, and one or two other, instances of small\ndisobedience.  They were not mischievous, Josephine willingly overlooked\nthem, and there was nothing to bring them to light.  It would have been\nbetter for Sylvia if her faults had been of a sort that brought attention\non them more easily!\n\nMeanwhile, Lady Barbara had almost found in her a model child\u2014except for\nher foolish shy silence before her elders, before whom she always\nwhispered\u2014and freely let the girls be constantly together.  The aunt\nlittle knew that this meek well-behaved maiden was giving the first warp\nto that upright truth that had been the one sterling point of Kate's\ncharacter!\n\n\n\n\nCHAPTER X.\n\n\nIT had been intended that Mrs. Lacy should rejoin her pupil at\nBournemouth at the end of six weeks; but in her stead came a letter\nsaying that she was unwell, and begging for a fortnight's grace.  At the\nfortnight's end came another letter; to which Lady Barbara answered that\nall was going on so well, that there was no need to think of returning\ntill they should all meet in London on the 1st of October.\n\nBut before that 1st, poor Mrs. Lacy wrote again, with great regret and\nmany excuses for the inconvenience she was causing.  Her son and her\ndoctor had insisted on her resigning her situation at once; and they\nwould not even allow her to go back until her place could be supplied.\n\n\"Poor thing!\" said Lady Jane.  \"I always thought it was too much for her.\nI wish we could have made her more comfortable: it would have been such a\nthing for her!\"\n\n\"So it would,\" answered Lady Barbara, \"if she had had to do with any\nother child.  A little consideration or discretion, such as might have\nbeen expected from a girl of eleven years old towards a person in her\ncircumstances, would have made her happy, and enabled her to assist her\nson.  But I have given up expecting feeling from Katharine.\"\n\nThat speech made Kate swell with anger at her aunt's tone and in her\nanger she forgot to repent of having been really thoughtless and almost\nunkind, or to recollect how differently her own gentle Sylvia at home\nwould have behaved to the poor lady.  She liked the notion of novelty,\nand hoped for a new governess as kind and bright as Miss Oswald.\n\nMoreover, she was delighted to find that Mrs. George Wardour was going to\nlive in London for the present, that Alice might be under doctors, and\nSylvia under masters.  Kate cared little for the why, but was excessively\ndelighted with plans for meeting, hopes of walks, talks, and\ntea-drinkings together; promises that the other dear Sylvia should come\nto meet her; and above all, an invitation to spend Sylvia Joanna's\nbirthday with her on the 21st of October, and go all together either to\nthe Zoological Gardens or to the British Museum, according to the\nweather.\n\nWith these hopes, Kate was only moderately sorry to leave the sea and\npine-trees behind her, and find herself once more steaming back to\nLondon, carrying in her hand a fine blue and white travelling-bag, worked\nfor her by her two little friends, but at which Lady Barbara had coughed\nrather dryly.  In the bag were a great many small white shells done up in\ntwists of paper, that pretty story \"The Blue Ribbons,\" and a small blank\nbook, in which, whenever the train stopped, Kate wrote with all her\nmight.  For Kate had a desire to convince Sylvia Joanna that one was much\nhappier without being a countess, and she thought this could be done very\ntouchingly and poetically by a fable in verse; so she thought she had a\nvery good idea by changing the old daisy that pined for transplantation\nand found it very unpleasant, into a harebell.\n\n    A harebell blue on a tuft of moss\n    In the wind her bells did toss.\n\nThat was her beginning; and the poor harebell was to get into a\nhot-house, where they wanted to turn her into a tall stately campanula,\nand she went through a great deal from the gardeners.  There was to be a\npretty fairy picture to every verse; and it would make a charming\nbirthday present, much nicer than anything that could be bought; and Kate\nkept on smiling to herself as the drawings came before her mind's eye,\nand the rhymes to her mind's ear.\n\nSo they came home; but it was odd, the old temper of the former months\nseemed to lay hold of Kate as soon as she set foot in the house in Bruton\nStreet, as if the cross feelings were lurking in the old corners.\n\nShe began by missing Mrs. Lacy very much.  The kind soft governess had\nmade herself more loved than the wayward child knew; and when Kate had\nrun into the schoolroom and found nobody sitting by the fire, no sad\nsweet smile to greet her, no one to hear her adventures, and remembered\nthat she had worried the poor widow, and that she would never come back\nagain, she could have cried, and really had a great mind to write to her,\nask her pardon, and say she was sorry.  It would perhaps have been the\nbeginning of better things if she had; but of all things in the world,\nwhat prevented her?  Just this\u2014that she had an idea that her aunt\nexpected it of her!  O Kate!  Kate!\n\nSo she went back to the harebell, and presently began rummaging among her\nbooks for a picture of one to copy; and just then Lady Barbara came in,\nfound half a dozen strewn on the floor, and ordered her to put them tidy,\nand then be dressed.  That put her out, and after her old bouncing\nfashion she flew upstairs, caught her frock in the old hitch at the turn,\nand half tore off a flounce.\n\nNo wonder Lady Barbara was displeased; and that was the beginning of\nthings going wrong\u2014nay, worse than before the going to Bournemouth.  Lady\nBarbara was seeking for a governess, but such a lady as she wished for\nwas not to be found in a day; and in the meantime she was resolved to do\nher duty by her niece, and watched over her behaviour, and gave her all\nthe lessons that she did not have from masters.\n\nWhether it was that Lady Barbara did not know exactly what was to be\nexpected of a little girl, or whether Kate was more fond of praise than\nwas good for her, those daily lessons were more trying than ever they had\nbeen.  Generally she had liked them; but with Aunt Barbara, the being\ntold to sit upright, hold her book straight, or pronounce her words\nrightly, always teased her, and put her out of humour at the beginning.\nOr she was reminded of some failure of yesterday, and it always seemed to\nher unjust that bygones should not be bygones; or even when she knew she\nhad been doing her best, her aunt always thought she could have done\nbetter, so that she had no heart or spirit to try another time, but went\non in a dull, save-trouble way, hardly caring to exert herself to avoid a\nscolding, it was so certain to come.\n\nIt was not right\u2014a really diligent girl would have won for herself the\npeaceful sense of having done her best, and her aunt would have owned it\nin time; whereas poor Kate's resistance only made herself and her aunt\nworse to each other every day, and destroyed her sense of duty and\nobedience more and more.\n\nLady Barbara could not be always with her, and when once out of sight\nthere was a change.  If she were doing a lesson with one of her masters,\nshe fell into a careless attitude in an instant, and would often chatter\nso that there was no calling her to order, except by showing great\ndetermination to tell her aunt.  It made her feel both sly and guilty to\nbehave so differently out of sight, and yet now that she had once begun\nshe seemed unable to help going on and she was sure, foolish child, that\nAunt Barbara's strictness made her naughty!\n\nThen there were her walks.  She was sent out with Josephine in the\nmorning and desired to walk nowhere but in the Square; and in the\nafternoon she and Josephine were usually set down by the carriage\ntogether in one of the parks, and appointed where to meet it again after\nLady Jane had taken her airing when she was well enough, for she soon\nbecame more ailing than usual.  They were to keep in the quiet paths, and\nnot speak to anyone.\n\nBut neither Josephine nor her young lady had any turn for what was\n\"triste.\"  One morning, when Kate was in great want of a bit of\nIndia-rubber, and had been sighing because of the displeasure she should\nmeet for having lost her own through using it in play-hours, Josephine\noffered to take her\u2014only a little out of her way\u2014to buy a new piece.\n\nKate knew this was not plain dealing, and hated herself for it, but she\nwas tired of being scolded, and consented!  And then how miserable she\nwas; how afraid of being asked where she had been; how terrified lest her\naunt should observe that it was a new, not an old, piece; how humiliated\nby knowing she was acting untruth!\n\nAnd then Josephine took more liberties.  When Kate was walking along the\npath, thinking how to rhyme to \"pride,\" she saw Josephine talking over\nthe iron rail to a man with a beard; and she told her maid afterwards\nthat it was wrong; but Josephine said, \"Miladi had too good a heart to\nbetray her,\" and the man came again and again, and once even walked home\npart of the way with Josephine, a little behind the young lady.\n\nKate was desperately affronted, and had a great mind to complain to her\naunts.  But then Josephine could have told that they had not been in the\nSquare garden at all that morning, but in much more entertaining streets!\nPoor Kate, these daily disobediences did not weigh on her nearly as much\nas the first one did; it was all one general sense of naughtiness!\n\nWorking at her harebell was the pleasantest thing she did, but her\neagerness about it often made her neglectful and brought her into\nscrapes.  She had filled one blank book with her verses and pictures,\nsome rather good, some very bad; and for want of help and correction she\nwas greatly delighted with her own performance, and thought it quite\nworthy of a little ornamental album, where she could write out the verses\nand gum in the drawings.\n\n\"Please, Aunt Barbara, let me go to the Soho Bazaar to-day?\"\n\n\"I cannot take you there, I have an engagement.\"\n\n\"But may I not go with Josephine?\"\n\n\"Certainly not.  I would not trust you there with her.  Besides, you\nspend too much upon trumpery, as it is.\"\n\n\"I don't want it for myself; I want something to get ready for Sylvia's\nbirthday\u2014the Sylvia that is come to London, I mean.\"\n\n\"I do not approve of a habit of making presents.\"\n\n\"Oh! but, Aunt Barbara, I am to drink tea with her on her birthday, and\nspend the day, and go to the Zoological Gardens, and I have all ready but\nmy presents! and it will not be in time if you won't let me go to-day.\"\n\n\"I never grant anything to pertinacity,\" answered Lady Barbara.  \"I have\ntold you that I cannot go with you to-day, and you ought to submit.\"\n\n\"But the birthday, Aunt Barbara!\"\n\n\"I have answered you once, Katharine; you ought to know better than to\npersist.\"\n\nKate pouted, and the tears swelled in her eyes at the cruelty of\ndepriving her of the pleasure of making her purchase, and at having her\nbeautiful fanciful production thus ruined by her aunt's unkindness.  As\nshe sat over her geography lesson, out of sight of her own bad writing,\nher broken-backed illuminated capitals, her lumpy campanulas,\ncrooked-winged fairies, queer perspective, and dabs of blue paint, she\nsaw her performance not as it was, but as it was meant to be, heard her\nown lines without their awkward rhymes and bits like prose, and thought\nof the wonder and admiration of all the Wardour family, and of the charms\nof having it secretly lent about as a dear simple sweet effusion of the\ntalented young countess, who longed for rural retirement.  And down came\na great tear into the red trimming of British North America, and Kate\nunadvisedly trying to wipe it up with her handkerchief, made a red smear\nall across to Cape Verd!  Formerly she would have exclaimed at once; now\nshe only held up the other side of the book that her aunt might not see,\nand felt very shabby all the time.  But Lady Barbara was reading over a\nletter, and did not look.  If Kate had not been wrapt up in herself, she\nwould have seen that anxious distressed face.\n\nThere came a knock to the schoolroom door.  It was Mr. Mercer, the\ndoctor, who always came to see Lady Jane twice a week, and startled and\nalarmed, Lady Barbara sprang up.  \"Do you want me, Mr. Mercer?  I'll\ncome.\"\n\n\"No, thank you,\" said the doctor, coming in.  \"It was only that I\npromised I would look at this little lady, just to satisfy Lady Jane, who\ndoes not think her quite well.\"\n\nKate's love of being important always made her ready to be looked at by\nMr. Mercer, who was a kind, fatherly old gentleman, not greatly apt to\ngive physic, very good-natured, and from his long attendance more\nintimate with the two sisters than perhaps any other person was.  Lady\nBarbara gave an odd sort of smile, and said, \"Oh! very well!\" and the old\ngentleman laughed as the two bright clear eyes met his, and said, \"No\ngreat weight there, I think!  Only a geography fever, eh?  Any more giddy\nheads lately, eh?  Or only when you make cheeses?\"\n\n\"I can't make cheeses now, my frocks are so short,\" said Kate, whose\nspirits always recovered with the least change.\n\n\"No more dreams?\"\n\n\"Not since I went to Bournemouth.\"\n\n\"Your tongue.\"  And as Kate, who had a certain queer pleasure in the\noperation, put out the long pinky member with its ruddier tip, quivering\nlike an animal, he laughed again, and said, \"Thank you, Lady Caergwent;\nit is a satisfaction once in a way to see something perfectly healthy!\nYou would not particularly wish for a spoonful of cod-liver oil, would\nyou?\"\n\nKate laughed, made a face, and shook her head.\n\n\"Well,\" said the doctor as he released her, \"I may set Lady Jane's mind\nat rest.  Nothing the matter there with the health.\"\n\n\"Nothing the matter but perverseness, I am afraid,\" said Lady Barbara, as\nKate stole back to her place, and shut her face in with the board of her\natlas.  \"It is my sister who is the victim, and I cannot have it go on.\nShe is so dreadfully distressed whenever the child is in disgrace that it\nis doing her serious injury.  Do you not see it, Mr. Mercer?\"\n\n\"She is very fond of the child,\" said Mr. Mercer.\n\n\"That is the very thing!  She is constantly worrying herself about her,\ntakes all her naughtiness for illness, and then cannot bear to see her\nreproved.  I assure you I am forced for my sister's sake to overlook many\nthings which I know I ought not to pass by.\"  (Kate shuddered.)  \"But the\nvery anxiety about her is doing great harm.\"\n\n\"I thought Lady Jane nervous and excited this morning,\" said Mr. Mercer:\n\"but that seemed to me to be chiefly about the Colonel's return.\"\n\n\"Yes,\" said Lady Barbara, \"of course in some ways it will be a great\npleasure; but it is very unlucky, after staying till the war was over,\nthat he has had to sell out without getting his promotion.  It will make\na great difference!\"\n\n\"On account of his son's health, is it not?\"\n\n\"Yes; of course everything must give way to that, but it is most\nunfortunate.  The boy has never recovered from his wound at Lucknow, and\nthey could not bear to part, or they ought to have sent him home with his\nmother long ago; and now my brother has remained at his post till he\nthought he could be spared; but he has not got his promotion, which he\nmust have had in a few months.\"\n\n\"When do you expect him?\"\n\n\"They were to set off in a fortnight from the time he wrote, but it all\ndepended on how Giles might be.  I wish we knew; I wish there could be\nany certainty, this is so bad for my sister.  And just at this very time,\nwithout a governess, when some children would be especially thoughtful\nand considerate, that we should have this strange fit of idleness and\nperverseness!  It is very trying; I feel quite hopeless sometimes!\"\n\nSome children, as Lady Barbara said, would have been rendered thoughtful\nand considerate by hearing such a conversation as this, and have tried to\nmake themselves as little troublesome to their elders as possible; but\nthere are others who, unless they are directly addressed, only take in,\nin a strange dreamy way, that which belongs to the grown-up world, though\nquick enough to catch what concerns themselves.  Thus Kate, though aware\nthat Aunt Barbara thought her naughtiness made Aunt Jane ill, and that\nthere was a fresh threat of the Lord Chancellor upon the return of her\ngreat-uncle from India, did not in the least perceive that her Aunt\nBarbara was greatly perplexed and harassed, divided between her care for\nher sister and for her niece, grieved for her brother's anxiety, and\ndisappointed that he had been obliged to leave the army, instead of being\nmade a General.  The upshot of all that she carried away with her was,\nthat it was very cross of Aunt Barbara to think she made Aunt Jane ill,\nand very very hard that she could not go to the bazaar.\n\nLady Jane did not go out that afternoon, and Lady Barbara set her niece\nand Josephine down in the Park, saying that she was going into Belgravia,\nand desiring them to meet her near Apsley House.  They began to walk, and\nKate began to lament.  \"If she could only have gone to the bazaar for her\nalbum!  It was very hard!\"\n\n\"Eh,\" Josephine said, \"why should they not go?  There was plenty of time.\nMiladi Barbe had given them till four.  She would take la petite.\"\n\nKate hung back.  She knew it was wrong.  She should never dare produce\nthe book if she had it.\n\nBut Josephine did not attend to the faltered English words, or disposed\nof them with a \"Bah!  Miladi will guess nothing!\" and she had turned\ndecidedly out of the Park, and was making a sign to a cab.  Kate was\ngreatly frightened, but was more afraid of checking Josephine in the open\nstreet, and making her dismiss the cab, than of getting into it.\nBesides, there was a very strong desire in her for the red and gold\nsquare book that had imprinted itself on her imagination.  She could not\nbut be glad to do something in spite of Aunt Barbara.  So they were shut\nin, and went off along Piccadilly, Kate's feelings in a strange whirl of\nfright and triumph, amid the clattering of the glasses.  Just suppose she\nsaw anyone she knew!\n\nBut they got to Soho Square at last; and through the glass door, in among\nthe stalls\u2014that fairy land in general to Kate; but now she was too much\nfrightened and bewildered to do more than hurry along the passages,\nstaring so wildly for her albums, that Josephine touched her, and said,\n\"Tenez, Miladi, they will think you farouche.  Ah! see the beautiful\nwreaths!\"\n\n\"Come on, Josephine,\" said Kate impatiently.\n\nBut it was not so easy to get the French maid on.  A bazaar was felicity\nto her, and she had her little lady in her power; she stood and gazed,\nadmired, and criticised, at every stall that afforded ornamental wearing\napparel or work patterns; and Kate, making little excursions, and coming\nback again to her side, could not get her on three yards in a quarter of\nan hour, and was too shy and afraid of being lost, to wander away and\ntransact her own business.  At last they did come to a counter with\nornamental stationery; and after looking at four or five books, Kate\nbought a purple embossed one, not at all what she had had in her mind's\neye, just because she was in too great a fright to look further; and then\nstep by step, very nearly crying at last, so as to alarm Josephine lest\nshe should really cry, she got her out at last.  It was a quarter to\nfour, and Josephine was in vain sure that Miladi Barbe would never be at\nthe place in time; Kate's heart was sick with fright at the thought of\nthe shame of detection.\n\nShe begged to get out at the Marble Arch, and not risk driving along Park\nLane; but Josephine was triumphant in her certainty that there was time;\nand on they went, Kate fancying every bay nose that passed the window\nwould turn out to have the brougham, the man-servant, and Aunt Barbara\nbehind it.\n\nAt length they were set down at what the Frenchwoman thought a safe\ndistance, and paying the cabman, set out along the side path, Josephine\nadmonishing her lady that it was best not to walk so swiftly, or to look\nguilty, or they would be \"_trahies_.\"\n\nBut just then Kate really saw the carriage drawn up where there was an\nopening in the railings, and the servant holding open the door for them.\nHad they been seen?  There was no knowing!  Lady Barbara did not say one\nsingle word; but that need not have been surprising\u2014only how very\nstraight her back was, how fixed her marble mouth and chin!  It was more\nlike Diana's head than ever\u2014Diana when she was shooting all Niobe's\ndaughters, thought Kate, in her dreamy, vague alarm.  Then she looked at\nJosephine on the back seat, to see what she thought of it; but the brown\nsallow face in the little bonnet was quite still and like itself\u2014beyond\nKate's power to read.\n\nThe stillness, doubt, and suspense, were almost unbearable.  She longed\nto speak, but had no courage, and could almost have screamed with desire\nto have it over, end as it would.  Yet at last, when the carriage did\nturn into Bruton Street, fright and shame had so entirely the upper hand,\nthat she read the numbers on every door, wishing the carriage would only\nstand still at each, or go slower, that she might put off the moment of\nknowing whether she was found out.\n\nThey stopped; the few seconds of ringing, of opening the doors, of\ngetting out, were over.  She knew how it would be, when, instead of going\nupstairs, her aunt opened the schoolroom door, beckoned her in, and said\ngravely, \"Lady Caergwent, while you are under my charge, it is my duty to\nmake you obey me.  Tell me where you have been.\"\n\nThere was something in the sternness of that low lady-like voice, and of\nthat dark deep eye, that terrified Kate more than the brightest flash of\nlightning: and it was well for her that the habit of truth was too much\nfixed for falsehood or shuffling even to occur to her.  She did not dare\nto do more than utter in a faint voice, scarcely audible \"To the bazaar.\"\n\n\"In direct defiance of my commands?\"\n\nBut the sound of her own confession, the relief of having told, gave Kate\nspirit to speak; \"I know it was naughty,\" she said, looking up; \"I ought\nnot.  Aunt Barbara, I have been very naughty.  I've been often where you\ndidn't know.\"\n\n\"Tell me the whole truth, Katharine;\" and Lady Barbara's look relaxed,\nand the infinite relief of putting an end to a miserable concealment was\nfelt by the little girl; so she told of the shops she had been at, and of\nher walks in frequented streets, adding that indeed she would not have\ngone, but that Josephine took her.  \"I did like it,\" she added candidly;\n\"but I know I ought not.\"\n\n\"Yes, Katharine,\" said Lady Barbara, almost as sternly as ever; \"I had\nthought that with all your faults you were to be trusted.\"\n\n\"I have told you the truth!\" cried Kate.\n\n\"Now you may have; but you have been deceiving me all this time; you, who\nought to set an example of upright and honourable conduct.\"\n\n\"No, no, Aunt!\" exclaimed Kate, her eyes flashing.  \"I never spoke one\nuntrue word to you; and I have not now\u2014nor ever.  I never deceived.\"\n\n\"I do not say that you have _told_ untruths.  It is deceiving to betray\nthe confidence placed in you.\"\n\nKate knew it was; yet she had never so felt that her aunt trusted her as\nto have the sense of being on honour; and she felt terribly wounded and\ngrieved, but not so touched as to make her cry or ask pardon.  She knew\nshe had been audaciously disobedient; but it was hard to be accused of\nbetraying trust when she had never felt that it was placed in her; and\nyet the conviction of deceit took from her the last ground she had of\npeace with herself.\n\nDrooping and angry, she stood without a word; and her aunt presently\nsaid, \"I do not punish you.  The consequences of your actions are\npunishment enough in themselves, and I hope they may warn you, or I\ncannot tell what is to become of you in your future life, and of all that\nwill depend on you.  You must soon be under more strict and watchful care\nthan mine, and I hope the effect may be good.  Meantime, I desire that\nyour Aunt Jane may be spared hearing of this affair, little as you seem\nto care for her peace of mind.\"\n\nAnd away went Lady Barbara; while Kate, flinging herself upon the sofa,\nsobbed out, \"I do care for Aunt Jane!  I love Aunt Jane!  I love her ten\nhundred times more than you! you horrid cross old Diana!  But I have\ndeceived!  Oh, I am getting to be a wicked little girl!  I never did such\nthings at home.  Nobody made me naughty there.  But it's the fashionable\nworld.  It is corrupting my simplicity.  It always does.  And I shall be\nlost!  O Mary, Mary!  O Papa, Papa!  Oh, come and take me home!\"  And for\na little while Kate gasped out these calls, as if she had really thought\nthey would break the spell, and bring her back to Oldburgh.\n\nShe ceased crying at last, and slowly crept upstairs, glad to meet no\none, and that not even Josephine was there to see her red eyes.  Her\nmuslin frock was on the bed, and she managed to dress herself, and run\ndown again unseen; she stood over the fire, so that the housemaid, who\nbrought in her tea, should not see her face; and by the time she had to\ngo to the drawing-room, the mottling of her face had abated under the\ninfluence of a story-book, which always drove troubles away for the time.\n\nIt was a very quiet evening.  Aunt Barbara read bits out of the\nnewspaper, and there was a little talk over them: and Kate read on in her\nbook, to hinder herself from feeling uncomfortable.  Now and then Aunt\nJane said a few soft words about \"Giles and Emily;\" but her sister always\nled away from the subject, afraid of her exciting herself, and getting\nanxious.\n\nAnd if Kate had been observing, she would have heard in the weary sound\nof Aunt Barbara's voice, and seen in those heavy eyelids, that the\ntroubles of the day had brought on a severe headache, and that there was\nat least one person suffering more than even the young ill-used countess.\n\nAnd when bed-time came, she learnt more of the \"consequences of her\nactions.\"  Stiff Mrs. Bartley stood there with her candle.\n\n\"Where is Josephine?\"\n\n\"She is gone away, my Lady.\"\n\nKate asked no more, but shivered and trembled all over.  She recollected\nthat in telling the truth she had justified herself, and at Josephine's\nexpense.  She knew Josephine would call it a blackness\u2014a treason.  What\nwould become of the poor bright merry Frenchwoman?  Should she never see\nher again?  And all because she had not had the firmness to be obedient!\nOh, loss of trust! loss of confidence! disobedience!  How wicked this\nplace made her! and would there be any end to it?\n\nAnd all night she was haunted through her dreams with the Lord\nChancellor, in his wig, trying to catch her, and stuff her into the\nwoolsack, and Uncle Wardour's voice always just out of reach.  If she\ncould only get to him!\n\n\n\n\nCHAPTER XI.\n\n\nTHE young countess was not easily broken down.  If she was ever so\nmiserable for one hour, she was ready to be amused the next; and though\nwhen left to herself she felt very desolate in the present, and much\nafraid of the future, the least enlivenment brightened her up again into\nmore than her usual spirits.  Even an entertaining bit in the history\nthat she was reading would give her so much amusement that she would\nforget her disgrace in making remarks and asking questions, till Lady\nBarbara gravely bade her not waste time, and decided that she had no\nfeeling.\n\nIt was not more easy to find a maid than a governess to Lady Barbara's\nmind, nor did she exert herself much in the matter, for, as Kate heard\nher tell Mr. Mercer, she had decided that the present arrangement could\nnot last; and then something was asked about the Colonel and Mrs.\nUmfraville; to which the answer was, \"Oh no, quite impossible; she could\nnever be in a house with an invalid;\" and then ensued something about the\nChancellor and an establishment, which, as usual, terrified Kate's\nimagination.\n\nIndeed that night terrors were at their height, for Mrs. Bartley never\nallowed dawdling, and with a severely respectful silence made the\nundressing as brief an affair as possible, brushing her hair till her\nhead tingled all over, putting away the clothes with the utmost speed,\nand carrying off the candle as soon as she had uttered her grim\n\"Good-night, my Lady,\" leaving Kate to choose between her pet\nterrors\u2014either of the Lord Chancellor, or of the house on fire\u2014or a very\nfine new one, that someone would make away with her to make way for her\nUncle Giles and his son to come to her title.  Somehow Lady Barbara had\ncontrived to make her exceedingly in awe of her Uncle Giles, the strict\nstern soldier who was always implicitly obeyed, and who would be so\nshocked at her.  She wished she could hide somewhere when he was coming!\nBut there was one real good bright pleasure near, that would come before\nher misfortunes; and that was the birthday to be spent at the Wardours'.\nAs to the present, Josephine had had the album in her pocket, and had\nnever restored it, and Kate had begun to feel a distaste to the whole\nperformance, to recollect its faults, and to be ashamed of the entire\naffair; but that was no reason she should not be very happy with her\nfriends, who had promised to take her to the Zoological Gardens.\n\nShe had not seen them since her return to London; they were at Westbourne\nRoad, too far off for her to walk thither even if she had had anyone to\ngo with her, and though they had called, no one had seen them; but she\nhad had two or three notes, and had sent some \"story pictures\" by the\npost.  And the thoughts of that day of freedom and enjoyment of talking\nto Alice, being petted by Mrs. Wardour and caressed by Sylvia, seemed to\nbear her through all the dull morning walks, in which she was not only\nattended by Bartley, but by the man-servant; all the lessons with her\naunt, and the still more dreary exercise which Lady Barbara took with her\nin some of the parks in the afternoon.  She counted the days to the 21st\nwhenever she woke in the morning; and at last Saturday was come, and it\nwould be Monday.\n\n\"Katharine,\" said Lady Barbara at breakfast, \"you had better finish your\ndrawing to-day; here is a note from Madame to say it will suit her best\nto come on Monday instead of Tuesday.\"\n\n\"Oh! but, Aunt Barbara, I am going to Westbourne Road on Monday.\"\n\n\"Indeed!  I was not aware of it.\"\n\n\"Oh, it is Sylvia's birthday! and I am going to the Zoological Gardens\nwith them.\"\n\n\"And pray how came you to make this engagement without consulting me?\"\n\n\"It was all settled at Bournemouth.  I thought you knew!  Did not Mrs.\nWardour ask your leave for me?\"\n\n\"Mrs. Wardour said something about hoping to see you in London, but I\nmade no decided answer.  I should not have allowed the intimacy there if\nI had expected that the family would be living in London; and there is no\nreason that it should continue.  Constant intercourse would not be at all\ndesirable.\"\n\n\"But may I not go on Monday?\" said Kate, her eyes opening wide with\nconsternation.\n\n\"No, certainly not.  You have not deserved that I should trust you; I do\nnot know whom you might meet there: and I cannot have you going about\nwith any chance person.\"\n\n\"O Aunt Barbara!  Aunt Barbara!  I have promised!\"\n\n\"Your promise can be of no effect without my consent.\"\n\n\"But they will expect me.  They will be so disappointed!\"\n\n\"I cannot help that.  They ought to have applied to me for my consent.\"\n\n\"Perhaps,\" said Kate hopefully, \"Mrs. Wardour will write to-day.  If she\ndoes, will you let me go?\"\n\n\"No, Katharine.  While you are under my charge, I am accountable for you,\nand I will not send you into society I know nothing about.  Let me hear\nno more of this, but write a note excusing yourself, and we will let the\ncoachman take it to the post.\"\n\nKate was thoroughly enraged, and forgot even her fears.  \"I sha'n't\nexcuse myself,\" she said; \"I shall say you will not let me go.\"\n\n\"You will write a proper and gentlewoman-like note,\" said Lady Barbara\nquietly, \"so as not to give needless offence.\"\n\n\"I shall say,\" exclaimed Kate more loudly, \"that I can't go because you\nwon't let me go near old friends.\"\n\n\"Go into the schoolroom, and write a proper note, Katharine; I shall come\npresently, and see what you have said,\" repeated Lady Barbara, commanding\nher own temper with some difficulty.\n\nKate flung away into the schoolroom, muttering, and in a tumult of\nexceeding disappointment, anger, and despair, too furious even to cry,\nand dashing about the room, calling Aunt Barbara after every horrible\nheroine she could think of, and pitying herself and her friends, till the\nthought of Sylvia's disappointment stung her beyond all bearing.  She was\nstill rushing hither and thither, inflaming her passion, when her aunt\nopened the door.\n\n\"Where is the note?\" she said quietly.\n\n\"I have not done it.\"\n\n\"Sit down then this instant, and write,\" said Lady Barbara, with her\nDiana face and cool way, the most terrible of all.\n\nKate sulkily obeyed, but as she seated herself, muttered, \"I shall say\nyou won't let me go near them.\"\n\n\"Write as I tell you.\u2014My dear Mrs. Wardour\u2014\"\n\n\"There.\"\n\n\"I fear you may be expecting to see me on Monday\u2014\"\n\n\"I don't fear; I know she is.\"\n\n\"Write\u2014I fear you may be expecting me on Monday, as something passed on\nthe subject at Bournemouth; and in order to prevent inconvenience, I\nwrite to say that it will not be in my power to call on that day, as my\naunt had made a previous engagement for me.\"\n\n\"I am sure I sha'n't say that!\" cried Kate, breaking out of all bounds in\nher indignation.\n\n\"Recollect yourself, Lady Caergwent,\" said Lady Barbara calmly.\n\n\"It is not true!\" cried Kate passionately, jumping up from her seat.\n\"You had not made an engagement for me!  I won't write it!  I won't write\nlies, and you sha'n't make me.\"\n\n\"I do not allow such words or such a manner in speaking to me,\" said Lady\nBarbara, not in the least above her usual low voice; and her calmness\nmade Kate the more furious, and jump and dance round with passion,\nrepeating, \"I'll never write lies, nor tell lies, for you or anyone; you\nmay kill me, but I won't!\"\n\n\"That is enough exposure of yourself, Lady Caergwent,\" said her aunt.\n\"When you have come to your senses, and choose to apologize for insulting\nme, and show me the letter written as I desire, you may come to me.\"\n\nAnd away walked Lady Barbara, as cool and unmoved apparently as if she\nhad been made of cast iron; though within she was as sorry, and hardly\nless angry, than the poor frantic child she left.\n\nKate did not fly about now.  She was very indignant, but she was proud of\nherself too; she had spoken as if she had been in a book, and she\nbelieved herself persecuted for adhering to old friends, and refusing to\nadopt fashionable falsehoods, such as she had read of.  She was a heroine\nin her own eyes, and that made her inclined to magnify all the\npersecution and cruelty.  They wanted to shut her up from the friends of\nher childhood, to force her to be false and fashionable; they had made\nher naughtier and naughtier ever since she came there; they were teaching\nher to tell falsehoods now, and to give up the Wardours.  She would never\nnever do it!  Helpless girl as she was, she would be as brave as the\nknights and earls her ancestors, and stand up for the truth.  But what\nwould they do at her!  Oh! could she bear Aunt Barbara's dreadful set\nDiana face again, and not write as she was told!\n\nThe poor weak little heart shrank with terror as she only looked at Aunt\nBarbara's chair\u2014not much like the Sir Giles de Umfraville she had thought\nof just now.  \"And I'm naughty now; I did betray my trust: I'm much\nnaughtier than I was.  Oh, if Papa was but here!\"  And then a light\ndarted into Kate's eye, and a smile came on her lip.  \"Why should not I\ngo home?  Papa would have me again; I know he would!  He would die rather\nthan leave his child Kate to be made wicked, and forced to tell lies!\nPerhaps he'll hide me!  Oh, if I could go to school with the children at\nhome in disguise, and let Uncle Giles be Earl of Caergwent if he likes!\nI've had enough of grandeur!  I'll come as Cardinal Wolsey did, when he\nsaid he was come to lay his bones among them\u2014and Sylvia and Mary, and\nCharlie and Armyn\u2014oh, I must go where someone will be kind to me again!\nCan I really, though?  Why not?\" and her heart beat violently.  \"Yes,\nyes; nothing would happen to me; I know how to manage!  If I can only get\nthere, they will hide me from Aunt Barbara and the Lord Chancellor; and\neven if I had to go back, I should have had one kiss of them all.\nPerhaps if I don't go now I shall never see them again!\"\n\nWith thoughts something like these, Kate, moving dreamily, as if she were\nnot sure that it was herself or not, opened her little writing-case, took\nout her purse, and counted the money.  There was a sovereign and some\nsilver; more than enough, as she well knew.  Then she took out of a\nchiffoniere her worked travelling bag, and threw in a few favourite\nbooks; then stood and gasped, and opened the door to peep out.  The\ncoachman was waiting at the bottom of the stairs for orders, so she drew\nin her head, looked at her watch, and considered whether her room would\nbe clear of the housemaids.  If she could once get safely out of the\nhouse she would not be missed till her dinner time, and perhaps then\nmight be supposed sullen, and left alone.  She was in a state of great\nfright, starting violently at every sound; but the scheme having once\noccurred to her, it seemed as if St. James's Parsonage was pulling her\nharder and harder every minute; she wondered if there were really such\nthings as heart-strings; if there were, hers must be fastened very tight\nround Sylvia.\n\nAt last she ventured out, and flew up to her own room more swiftly than\never she had darted before!  She moved about quietly, and perceived by\nthe sounds in the next room that Mrs. Bartley was dressing Aunt Jane, and\nAunt Barbara reading a letter to her.  This was surely a good moment; but\nshe knew she must dress herself neatly, and not look scared, if she did\nnot mean to be suspected and stopped; and she managed to get quietly into\nher little shaggy coat, her black hat and feather and warm gloves\u2014even\nher boots were remembered\u2014and then whispering to herself, \"It can't be\nwrong to get away from being made to tell stories!  I'm going to Papa!\"\nshe softly opened the door, went on tip-toe past Lady's Jane's door; then\nafter the first flight of stairs, rushed like the wind, unseen by anyone,\ngot the street door open, pulled it by its outside handle, and heard it\nshut!\n\nIt was done now!  She was on the wide world\u2014in the street!  She could not\nhave got in again without knocking, ringing, and making her attempt\nknown; and she was far more terrified at the thought of Lady Barbara's\nstern face and horror at her proceedings than even at the long journey\nalone.\n\nEvery step was a little bit nearer Sylvia, Mary, and Papa\u2014it made her\nheart bound in the midst of its frightened throbs\u2014every step was farther\naway from Aunt Barbara, and she could hardly help setting off in a run.\nIt was a foggy day, when it was not so easy to see far, but she longed to\nbe out of Bruton Street, where she might be known; yet when beyond the\nquiet familiar houses, the sense of being alone, left to herself, began\nto get very alarming, and she could hardly control herself to walk like a\nrational person to the cab-stand in Davies Street.\n\nNobody remarked her; she was a tall girl for her age, and in her sober\ndark dress, with her little bag, might be taken for a tradesman's\ndaughter going to school, even if anyone had been out who had time to\nlook at her.  Trembling, she saw a cabman make a sign to her, and stood\nwaiting for him, jumped in as he opened his door, and felt as if she had\nfound a refuge for the time upon the dirty red plush cushions and the\nstraw.  \"To the Waterloo Station,\" said she, with as much indifference\nand self-possession as she could manage.  The man touched his hat, and\nrattled off: he perhaps wondering if this were a young runaway, and if he\nshould get anything by telling where she was gone; she working herself\ninto a terrible fright for fear he should be going to drive round and\nround London, get her into some horrible den of iniquity, and murder her\nfor the sake of her money, her watch, and her clothes.  Did not cabmen\nalways do such things?  She had quite decided how she would call a\npoliceman, and either die like an Umfraville or offer a ransom of \"untold\ngold,\" and had gone through all possible catastrophes long before she\nfound herself really safe at the railway station, and the man letting her\nout, and looking for his money.\n\nThe knowledge that all depended on herself, and that any signs of alarm\nwould bring on inquiry, made her able to speak and act so reasonably,\nthat she felt like one in a dream.  With better fortune than she could\nhave hoped for, a train was going to start in a quarter of an hour; and\nthe station clerk was much too busy and too much hurried to remark how\nscared were her eyes, and how trembling her voice, as she asked at his\npigeon-hole for \"A first-class ticket to Oldburgh, if you please,\"\noffered the sovereign in payment, swept up the change, and crept out to\nthe platform.\n\nA carriage had \"Oldburgh\" marked on it; she tried to open the door, but\ncould not reach the handle; then fancied a stout porter who came up with\nhis key must be some messenger of the Lord Chancellor come to catch her,\nand was very much relieved when he only said, \"Where for, Miss?\" and on\nher answer, \"Oldburgh,\" opened the door for her, and held her bag while\nshe tripped up the steps.  \"Any luggage, Miss?\"  \"No, thank you.\"  He\nshot one inquiring glance after her, but hastened away; and she settled\nherself in the very farthest corner of the carriage, and lived in an\nagony for the train to set off before her flight should be detected.\n\nOnce off, she did not care; she should be sure of at least seeing Sylvia,\nand telling her uncle her troubles.  She had one great start, when the\ndoor was opened, and a gentleman peered in; but it was merely to see if\nthere was room, for she heard him say, \"Only a child,\" and in came a lady\nand two gentlemen, who at least filled up the window so that nobody could\nsee her, while they talked a great deal to someone on the platform.  And\nthen after some bell-ringing, whistling, sailing backwards and forwards,\nand stopping, they were fairly off\u2014getting away from the roofs of\nLondon\u2014seeing the sky clear of smoke and fog\u2014getting nearer home every\nmoment; and Countess Kate relaxed her shy, frightened, drawn-up attitude,\ngave a long breath, felt that the deed was done, and began to dwell on\nthe delight with which she should be greeted at home, and think how to\nsurprise them all!\n\nThere was plenty of time for thinking and planning and dreaming, some few\npossible things, but a great many more most impossible ones.  Perhaps the\nqueerest notion of all was her plan for being disguised like a\nschool-child all day, and always noticed for her distinguished appearance\nby ladies who came to see the school, or overheard talking French to\nSylvia; and then in the midst of her exceeding anxiety not to be\ndetected, she could not help looking at her travelling companions, and\nwondering if they guessed with what a grand personage they had the honour\nto be travelling!  Only a child, indeed!  What would they think if they\nknew?  And the little goose held her pocket-handkerchief in her hand,\nfeeling as if it would be like a story if they happened to wonder at the\ncoronet embroidered in the corner; and when she took out a story-book,\nshe would have liked that the fly-leaf should just carelessly reveal the\nCaergwent written upon it.  She did not know that selfishness had thrown\nout the branch of self-consequence.\n\nHowever, nothing came of it; they had a great deal too much to say to\neach other to notice the little figure in the corner; and she had time to\nread a good deal, settle a great many fine speeches, get into many a\nfright lest there should be an accident, and finally grow very impatient,\nalarmed, and agitated before the last station but one was passed, and she\nbegan to know the cut of the hedgerow-trees, and the shape of the\nhills\u2014to feel as if the cattle and sheep in the fields were old friends,\nand to feel herself at home.\n\nOldburgh Station!  They were stopping at last, and she was on her feet,\npressing to the window between the strangers.  One of the gentlemen\nkindly made signs to the porter to let her out, and asked if she had any\nbaggage, or anyone to meet her.  She thanked him by a smile and shake of\nthe head; she could not speak for the beating of her heart; she felt\nalmost as much upon the world as when the door in Bruton Street had shut\nbehind her; and besides, a terrible wild fancy had seized her\u2014suppose,\njust suppose, they were all gone away, or ill, or someone dead!  Perhaps\nshe felt it would serve her right, and that was the reason she was in\nsuch terror.\n\n\n\n\nCHAPTER XII.\n\n\nWHEN Kate had left the train, she was still two miles from St. James's;\nand it was half-past three o'clock, so that she began to feel that she\nhad run away without her dinner, and that the beatings of her heart made\nher knees ache, so that she had no strength to walk.\n\nShe thought her best measure would be to make her way to a pastry-cook's\nshop that looked straight down the street to the Grammar School, and\nwhere it was rather a habit of the family to meet Charlie when they had\ngone into the town on business, and wanted to walk out with him.  He\nwould be out at four o'clock, and there would not be long to wait.  So,\nfeeling shy, and even more guilty and frightened than on her first start,\nKate threaded the streets she knew so well, and almost gasping with\nnervous alarm, popped up the steps into the shop, and began instantly\neating a bun, and gazing along the street.  She really could not speak\ntill she had swallowed a few mouthfuls; and then she looked up to the\nwoman, and took courage to ask if the boys were out of school yet.\n\n\"Oh, no, Miss; not for a quarter of an hour yet.\"\n\n\"Do you know if\u2014if Master Charles Wardour is there to-day?\" added Kate,\nwith a gulp.\n\n\"I don't, Miss.\"  And the woman looked hard at her.\n\n\"Do you know if any of them\u2014any of them from St. James's, are in to-day?\"\n\n\"No, Miss; I have not seen any of them, but very likely they may be.  I\nsaw Mr. Wardour go by yesterday morning.\"\n\nSo far they were all well, then; and Kate made her mind easier, and went\non eating like a hungry child till the great clock struck four; when she\nhastily paid for her cakes and tarts, put on her gloves, and stood on the\nstep, half in and half out of the shop, staring down the street.  Out\ncame the boys in a rush, making straight for the shop, and brushing past\nKate; she, half alarmed, half affronted, descended from her post, still\nlooking intently.  Half a dozen more big fellows, eagerly talking, almost\ntumbled over her, and looked as if she had no business there; she seemed\nto be quite swept off the pavement into the street, and to be helpless in\nthe midst of a mob, dashing around her.  They might begin to tease her in\na minute; and more terrified than at any moment of her journey, she was\nalmost ready to cry, when the tones of a well-known voice came on her ear\nclose to her\u2014\"I say, Will, you come and see my new terrier;\" and before\nthe words were uttered, with a cry of, \"Charlie, Charlie!\" she was\nclinging to a stout boy who had been passing without looking at her.\n\n\"Let go, I say.  Who are you?\" was the first rough greeting.\n\n\"O Charlie, Charlie!\" almost sobbing, and still grasping his arm tight.\n\n\"Oh, I say!\" and he stood with open mouth staring at her.\n\n\"O Charlie! take me home!\"\n\n\"Yes, yes; come along!\u2014Get off with you, fellows!\" he added\u2014turning round\nupon the other boys, who were beginning to stare\u2014and exclaimed, \"It's\nnothing but our Kate!\"\n\nOh! what a thrill there was in hearing those words; and the boys, who\nwere well-behaved and gentlemanly, were not inclined to molest her.  So\nshe hurried on, holding Charles's arm for several steps, till they were\nout of the hubbub, when he turned again and stared, and again exclaimed,\n\"I say!\" all that he could at present utter; and Kate looked at his ruddy\nface and curly head, and dusty coat and inky collar, as if she would eat\nhim for very joy.\n\n\"I say!\" and this time he really did say, \"Where are the rest of them?\"\n\n\"At home, aren't they?\"\n\n\"What, didn't they bring you in?\"\n\n\"Oh no!\"\n\n\"Come, don't make a tomfoolery of it; that's enough.  I shall have all\nthe fellows at me for your coming up in that way, you know.  Why couldn't\nyou shake hands like anyone else?\"\n\n\"O Charlie, I couldn't help it!  Please let us go home!\"\n\n\"Do you mean that you aren't come from there?\"\n\n\"No,\" said Kate, half ashamed, but far more exultant, and hanging down\nher head; \"I came from London\u2014I came by myself.  My aunt wanted me to\ntell a story, and\u2014and I have run away.  O Charlie! take me home!\" and\nwith a fresh access of alarm, she again threw her arms round him, as if\nto gain his protection from some enemy.\n\n\"Oh, I say!\" again he cried, looking up the empty street and down again,\npartly for the enemy, partly to avoid eyes; but he only beheld three\ndirty children and an old woman, so he did not throw her off roughly.\n\"Ran away!\" and he gave a great whistle.\n\n\"Yes, yes.  My aunt shut me up because I would not tell a story,\" said\nKate, really believing it herself.  \"Oh, let us get home, Charlie, do.\"\n\n\"Very well, if you won't throttle a man; and let me get Tony in here,\" he\nadded, going on a little way towards a small inn stable-yard.\n\n\"Oh, don't go,\" cried Kate, who, once more protected, could not bear to\nbe left alone a moment; but Charlie plunged into the yard, and came back\nnot only with the pony, but with a plaid, and presently managed to mount\nKate upon the saddle, throwing the plaid round her so as to hide the\nshort garments and long scarlet stockings, that were not adapted for\nriding, all with a boy's rough and tender care for the propriety of his\nsister's appearance.\n\n\"There, that will do,\" said he, holding the bridle.  \"So you found it\npoor fun being My Lady, and all that.\"\n\n\"Oh! it was awful, Charlie!  You little know, in your peaceful\nretirement, what are the miseries of the great.\"\n\n\"Come, Kate, don't talk bosh out of your books.  What did they do to you?\nThey didn't lick you, did they?\"\n\n\"No, no; nonsense,\" said Kate, rather affronted; \"but they wanted to make\nme forget all that I cared for, and they really did shut me up because I\nsaid I would not write a falsehood to please them!  They did, Charlie!\"\nand her eyes shone.\n\n\"Well, I always knew they must be a couple of horrid old owls,\" began\nCharlie.\n\n\"Oh!  I didn't mean Aunt Jane,\" said Kate, feeling a little compunction.\n\"Ah!\" with a start and scream, \"who is coming?\" as she heard steps behind\nthem.\n\n\"You little donkey, you'll be off!  Who should it be but Armyn?\"\n\nFor Armyn generally overtook his brother on a Saturday, and walked home\nwith him for the Sunday.\n\nCharles hailed him with a loud \"Hollo, Armyn!  What d'ye think I've got\nhere?\"\n\n\"Kate!  Why, how d'ye do!  Why, they never told me you were coming to see\nus.\"\n\n\"They didn't know,\" whispered Kate.\n\n\"She's run away, like a jolly brick!\" said Charlie, patting the pony\nvehemently as he made this most inappropriate comparison.\n\n\"Run away!  You don't mean it!\" cried Armyn, standing still and aghast,\nso much shocked that her elevation turned into shame; and Charles\nanswered for her\u2014\n\n\"Yes, to be sure she did, when they locked her up because she wouldn't\ntell lies to please them.  How did you get out, Kittens?  What jolly good\nfun it must have been!\"\n\n\"Is this so, Kate?\" said Armyn, laying his hand on the bridle; and his\ndispleasure roused her spirit of self-defence, and likewise a sense of\nill-usage.\n\n\"To be sure it is,\" she said, raising her head indignantly.  \"I would not\nbe made to tell fashionable falsehoods; and so\u2014and so I came home, for\nPapa to protect me:\" and if she had not had to take care to steady\nherself on her saddle, she would have burst out sobbing with vexation at\nArmyn's manner.\n\n\"And no one knew you were coming?\" said he.\n\n\"No, of course not; I slipped out while they were all in confabulation in\nAunt Jane's room, and they were sure not to find me gone till dinner\ntime, and if they are very cross, not then.\"\n\n\"You go on, Charlie,\" said Armyn, restoring the bridle to his brother;\n\"I'll overtake you by the time you get home.\"\n\n\"What are you going to do?\" cried boy and girl with one voice.\n\n\"Well, I suppose it is fair to tell you,\" said Armyn.  \"I must go and\ntelegraph what is become of you.\"\n\nThere was a howl and a shriek at this.  They would come after her and\ntake her away, when she only wanted to be hid and kept safe; it was a\ncruel shame, and Charles was ready to fly at his brother and pommel him;\nindeed, Armyn had to hold him by one shoulder, and say in the voice that\nmeant that he would be minded, \"Steady, boy I\u2014I'm very sorry, my little\nKatie; it's a melancholy matter, but you must have left those poor old\nladies in a dreadful state of alarm about you, and they ought not to be\nkept in it!\"\n\n\"Oh! but Armyn, Armyn, do only get home, and see what Papa says.\"\n\n\"I am certain what he will say, and it would only be the trouble of\nsending someone in, and keeping the poor women in a fright all the\nlonger.  Besides, depend on it, the way to have them sending down after\nyou would be to say nothing.  Now, if they hear you are safe, you are\npretty secure of spending to-morrow at least with us.  Let me go, Kate;\nit must be done.  I cannot help it.\"\n\nEven while he spoke, the kind way of crossing her will was so like home,\nthat it gave a sort of happiness, and she felt she could not resist; so\nshe gave a sigh, and he turned back.\n\nHow much of the joy and hope of her journey had he not carried away with\nhim!  His manner of treating her exploit made her even doubt how his\nfather might receive it; and yet the sight of old scenes, and the\npresence of Charlie, was such exceeding delight, that it seemed to kill\noff all unpleasant fears or anticipations; and all the way home it was\none happy chatter of inquiries for everyone, of bits of home news, and\nexclamations at the sight of some well-known tree, or the outline of a\nhouse remembered for some adventure; the darker the twilight the happier\nher tongue.  The dull suburb, all little pert square red-brick houses,\nwith slated roofs and fine names, in the sloppiness of a grey November\nday, was dear to Kate; every little shop window with the light streaming\nout was like a friend; and she anxiously gazed into the rough parties out\nfor their Saturday purchases, intending to nod to anyone she might know,\nbut it was too dark for recognitions; and when at length they passed the\ndark outline of the church, she was silent, her heart again bouncing as\nif it would beat away her breath and senses.  The windows were dark; it\nwas a sign that Evening Service was just over.  The children turned in at\nthe gate, just as Armyn overtook them.  He lifted Kate off her pony.  She\ncould not have stood, but she could run, and she flew to the\ndrawing-room.  No one was there; perhaps she was glad.  She knew the\ncousins would be dressing for tea, and in another moment she had torn\nopen Sylvia's door.\n\nSylvia, who was brushing her hair, turned round.  She stared\u2014as if she\nhad seen a ghost.  Then the two children held out their arms, and rushed\ntogether with a wild scream that echoed through the house, and brought\nMary flying out of her room to see who was hurt! and to find, rolling on\nher sister's bed, a thing that seemed to have two bodies and two faces\nglued together, four legs, and all its arms and hands wound round and\nround.\n\n\"Sylvia!  What is it?  Who is it?  What is she doing to you?\" began Mary;\nbut before the words were out of her mouth, the thing had flown at her\nneck, and pulled her down too; and the grasp and the clinging and the\nkisses told her long before she had room or eyes or voice to know the\ncreature by.  A sort of sobbing out of each name between them was all\nthat was heard at first.\n\nAt last, just as Mary was beginning to say, \"My own own Katie! how did\nyou come\u2014\"  Mr. Wardour's voice on the stairs called \"Mary!\"\n\n\"Have you seen him, my dear?\"\n\n\"No;\" but Kate was afraid now she had heard his voice, for it was grave.\n\n\"Mary!\"  And Mary went.  Kate sat up, holding Sylvia's hand.\n\nThey heard him ask, \"Is Kate there?\"\n\n\"Yes.\"  And then there were lower voices that Kate could not hear, and\nwhich therefore alarmed her; and Sylvia, puzzled and frightened, sat\nholding her hand, listening silently.\n\nPresently Mr. Wardour came in; and his look was graver than his tone; but\nit was so pitying, that in a moment Kate flew to his breast, and as he\nheld her in his arms she cried, \"O Papa!  Papa!  I have found you again!\nyou will not turn me away.\"\n\n\"I must do whatever may be right, my dear child,\" said Mr. Wardour,\nholding her close, so that she felt his deep love, though it was not an\nundoubting welcome.  \"I will hear all about it when you have rested, and\nthen I may know what is best to be done.\"\n\n\"Oh! keep me, keep me, Papa.\"\n\n\"You will be here to-morrow at least,\" he said, disengaging himself from\nher.  \"This is a terrible proceeding of yours, Kate, but it is no time\nfor talking of it; and as your aunts know where you are, nothing more can\nbe done at present; so we will wait to understand it till you are rested\nand composed.\"\n\nHe went away; and Kate remained sobered and confused, and Mary stood\nlooking at her, sad and perplexed.\n\n\"O Kate!  Kate!\" she said, \"what have you been doing?\"\n\n\"What is the matter?  Are not you glad?\" cried Sylvia; and the squeeze of\nher hand restored Kate's spirits so much that she broke forth with her\nstory, told in her own way, of persecution and escape, as she had wrought\nherself up to believe in it; and Sylvia clung to her, with flushed cheeks\nand ardent eyes, resenting every injury that her darling detailed,\ntriumphing in her resistance, and undoubting that here she would be\nreceived and sheltered from all; while Mary, distressed and grieved, and\ncautioned by her father to take care not to show sympathy that might be\nmischievous, was carried along in spite of herself to admire and pity her\nchild, and burn with indignation at such ill-treatment, almost in despair\nat the idea that the child must be sent back again, yet still not\ndiscarding that trust common to all Mr. Wardour's children, that \"Papa\nwould do _anything_ to hinder a temptation.\"\n\nAnd so, with eager words and tender hands, Kate was made ready for the\nevening meal, and went down, clinging on one side to Mary, on the other\nto Sylvia\u2014a matter of no small difficulty on the narrow staircase, and\nalmost leading to a general avalanche of young ladies, all upon the head\nof little Lily, who was running up to greet and be greeted, and was\nalmost devoured by Kate when at length they did get safe downstairs.\n\nIt was a somewhat quiet, grave meal; Mr. Wardour looked so sad and\nserious, that all felt that it would not do to indulge in joyous chatter,\nand the little girls especially were awed; though through all there was a\ntender kindness in his voice and look, whenever he did but offer a slice\nof bread to his little guest, such as made her feel what was home and\nwhat was love\u2014\"like a shower of rain after a parched desert\" as she said\nto herself; and she squeezed Sylvia's hand under the table whenever she\ncould.\n\nMr. Wardour spoke to her very little.  He said he had seen Colonel\nUmfraville's name in the _Gazette_, and asked about his coming home; and\nwhen she had answered that the time and speed of the journey were to\ndepend on Giles's health, he turned from her to Armyn, and began talking\nto him about some public matters that seemed very dull to Kate; and one\nlittle foolish voice within her said, \"He is not like Mrs. George\nWardour, he forgets what I am;\" but there was a wiser, more loving voice\nto answer, \"Dear Papa, he thinks of me as myself; he is no respecter of\npersons.  Oh, I hope he is not angry with me!\"\n\nWhen tea was over Mr. Wardour stood up, and said, \"I shall wish you\nchildren good-night now; I have to read with John Bailey for his\nConfirmation, and to prepare for to-morrow;\u2014and you, Kate, must go to bed\nearly.\u2014Mary, she had better sleep with you.\"\n\nThis was rather a blank, for sleeping with Sylvia again had been Kate's\ndream of felicity; yet this was almost lost in the sweetness of once more\ncoming in turn for the precious kiss and good-night, in the midst of\nwhich she faltered, \"O Papa, don't be angry with me!\"\n\n\"I am not angry, Katie,\" he said gently; \"I am very sorry.  You have done\na thing that nothing can justify, and that may do you much future harm;\nand I cannot receive you as if you had come properly.  I do not know what\nexcuse there was for you, and I cannot attend to you to-night; indeed, I\ndo not think you could tell me rightly; but another time we will talk it\nall over, and I will try to help you.  Now good-night, my dear child.\"\n\nThose words of his, \"I will try to help you,\" were to Kate like a promise\nof certain rescue from all her troubles; and, elastic ball that her\nnature was, no sooner was his anxious face out of sight, and she secure\nthat he was not angry, than up bounded her spirits again.  She began\nwondering why Papa thought she could not tell him properly, and forthwith\nbegan to give what she intended for a full and particular history of all\nthat she had gone through.\n\nIt was a happy party round the fire; Kate and Sylvia both together in the\nlarge arm-chair, and Lily upon one of its arms; Charles in various odd\nattitudes before the fire; Armyn at the table with his book, half\nreading, half listening; Mary with her work; and Kate pouring out her\nstory, making herself her own heroine, and describing her adventures, her\nway of life, and all her varieties of miseries, in the most glowing\ncolours.  How she did rattle on!  It would be a great deal too much to\ntell; indeed it would be longer than this whole story!\n\nSylvia and Charlie took it all in, pitied, wondered, and were indignant,\nwith all their hearts; indeed Charlie was once heard to wish he could\nonly get that horrid old witch near the horse-pond; and when Kate talked\nof her Diana face, he declared that he should get the old brute of a cat\ninto the field, and set all the boys to stone her.\n\nLittle Lily listened, not sure whether it was not all what she called \"a\nmade-up story only for prettiness;\" and Mary, sitting over her work, was\npuzzled, and saw that her father was right in saying that Kate could not\nat present give an accurate account of herself.  Mary knew her\ntruthfulness, and that she would not have said what she knew to be\ninvention; but those black eyes, glowing like little hot coals, and those\nburning cheeks, as well as the loud, squeaky key of the voice, all showed\nthat she had worked herself up into a state of excitement, such as not to\nknow what was invented by an exaggerating memory.  Besides, it could not\nbe all true; it did not agree; the ill-treatment was not consistent with\nthe grandeur.  For Kate had taken to talking very big, as if she was an\nimmensely important personage, receiving much respect wherever she went;\nand though Armyn once or twice tried putting in a sober matter-of-fact\nquestion for the fun of disconcerting her, she was too mad to care or\nunderstand what he said.\n\n\"Oh no! she never was allowed to do anything for herself.  That was quite\na rule, and very tiresome it was.\"\n\n\"Like the King of Spain, you can't move your chair away from the fire\nwithout the proper attendant.\"\n\n\"I never do put on coals or wood there!\"\n\n\"There may be several reasons for that,\" said Armyn, recollecting how\nnearly Kate had once burnt the house down.\n\n\"Oh, I assure you it would not do for me,\" said Kate.  \"If it were not so\ninconvenient in that little house, I should have my own man-servant to\nattend to my fire, and walk out behind me.  Indeed, now Perkins always\ndoes walk behind me, and it is such a bore.\"\n\nAnd what was the consequence of all this wild chatter?  When Mary had\nseen the hot-faced eager child into bed, she came down to her brother in\nthe drawing-room with her eyes brimful of tears, saying, \"Poor dear\nchild!  I am afraid she is very much spoilt!\"\n\n\"Don't make up your mind to-night,\" said Armyn.  \"She is slightly insane\nas yet!  Never mind, Mary; her heart is in the right place, if her head\nis turned a little.\"\n\n\"It is very much turned indeed,\" said Mary.  \"How wise it was of Papa not\nto let Sylvia sleep with her!  What will he do with her?  Oh dear!\"\n\n\n\n\nCHAPTER XIII.\n\n\nTHE Sunday at Oldburgh was not spent as Kate would have had it.  It\ndawned upon her in the midst of horrid dreams, ending by wakening to an\noverpowering sick headache, the consequence of the agitations and alarms\nof the previous day, and the long fast, appeased by the contents of the\npastry-cook's shop, with the journey and the excitement of the\nmeeting\u2014altogether quite sufficient to produce such a miserable feeling\nof indisposition, that if Kate could have thought at all of anything but\npresent wretchedness, she would have feared that she was really carrying\nout the likeness to Cardinal Wolsey by laying her bones among them.\n\nThat it was not quite so bad as that, might be inferred from her having\nno doctor but Mary Wardour, who attended to her most assiduously from her\nfirst moans at four o'clock in the morning, till her dropping off to\nsleep about noon; when the valiant Mary, in the absence of everyone at\nchurch, took upon herself to pen a note, to catch the early Sunday post,\non her own responsibility, to Lady Barbara Umfraville, to say that her\nlittle cousin was so unwell that it would be impossible to carry out the\npromise of bringing her home on Monday, which Mr. Wardour had written on\nSaturday night.\n\nSleep considerably repaired her little ladyship; and when she had\nawakened, and supped up a bason of beef-tea, toast and all, with\nconsiderable appetite, she was so much herself again, that there was no\nreason that anyone should be kept at home to attend to her.  Mary's\nabsence was extremely inconvenient, as she was organist and leader of the\nchoir.\n\n\"So, Katie dear,\" she said, when she saw her patient on her legs again,\nmaking friends with the last new kitten of the old cat, \"you will not\nmind being left alone, will you?  It is only for the Litany and\ncatechising, you know.\"\n\nKate looked blank, and longed to ask that Sylvia might stay with her, but\ndid not venture; knowing that she was not ill enough for it to be a\nnecessity, and that no one in that house was ever kept from church,\nexcept for some real and sufficient cause.\n\nBut the silly thoughts that passed through the little head in the hour of\nsolitude would fill two or three volumes.  In the first place, she was\naffronted.  They made very little of her, considering who she was, and\nhow she had come to see them at all risks, and how ill she had been!\nThey would hardly have treated a little village child so negligently as\ntheir visitor, the Countess\u2014\n\nThen her heart smote her.  She remembered Mary's tender and assiduous\nnursing all the morning, and how she had already stayed from service and\nSunday school; and she recollected her honour for her friends for not\nvaluing her for her rank; and in that mood she looked out the Psalms and\nLessons, which she had not been able to read in the morning, and when she\nhad finished them, began to examine the book-case in search of a new, or\nelse a very dear old, Sunday book.\n\nBut then something went \"crack,\"\u2014or else it was Kate's fancy\u2014for she\nstarted as if it had been a cannon-ball; and though she sat with her book\nin her lap by the fire in Mary's room, all the dear old furniture and\npictures round her, her head was weaving an unheard-of imagination, about\nrobbers coming in rifling everything\u2014coming up the stairs\u2014creak, creak,\nwas that their step?\u2014she held her breath, and her eyes dilated\u2014seizing\nher for the sake of her watch!  What article there would be in the\npaper\u2014\"Melancholy disappearance of the youthful Countess of Caergwent.\"\nThen Aunt Barbara would be sorry she had treated her so cruelly; then\nMary would know she ought not to have abandoned the child who had thrown\nherself on her protection.\n\nThat was the way Lady Caergwent spent her hour.  She had been kidnapped\nand murdered a good many times before; there was a buzz in the street,\nher senses came back, and she sprang out on the stairs to meet her\ncousins, calling herself quite well again.  And then they had a very\npeaceful, pleasant time; she was one of them again, when, as of old, Mr.\nWardour came into the drawing-room, and she stood up with Charles,\nSylvia, and little Lily, who was now old enough for the Catechism, and\nthen the Collect, and a hymn.  Yes, she had Collect and hymn ready too,\nand some of the Gospel; Aunt Barbara always heard her say them on Sunday,\nbesides some very difficult questions, not at all like what Mr. Wardour\nasked out of his own head.\n\nKate was a little afraid he would make his teaching turn on submitting to\nrulers; it was an Epistle that would have given him a good opportunity,\nfor it was the Fourth Epiphany Sunday, brought in at the end of the\nSundays after Trinity.  If he made his teaching personal, something\nwithin her wondered if she could bear it, and was ready to turn angry and\ndefiant.  But no such thing; what he talked to them about was the gentle\nPresence that hushed the waves and winds in outward nature, and calmed\nthe wild spiritual torments of the possessed; and how all fears and\nterrors, all foolish fancies and passionate tempers, will be softened\ninto peace when the thought of Him rises in the heart.\n\nKate wondered if she should be able to think of that next time she was\ngoing to work herself into an agony.\n\nBut at present all was like a precious dream, to be enjoyed as slowly as\nthe moments could be persuaded to pass.  Out came the dear old Dutch\nBible History, with pictures of everything\u2014pictures that they had looked\nat every Sunday since they could walk, and could have described with\ntheir eyes shut; and now Kate was to feast her eyes once again upon them,\nand hear how many little Lily knew; and a pretty sight it was, that tiny\nchild, with her fat hands clasped behind her so as not to be tempted to\nput a finger on the print, going so happily and thoroughly through all\nthe creatures that came to Adam to be named, and showing the whole\nprocession into the Ark, and, her favourite of all, the Angels coming\ndown to Jacob.\n\nThen came tea; and then Kate was pronounced, to her great delight, well\nenough for Evening Service.  The Evening Service she always thought a\ntreat, with the lighted church, and the choicest singing\u2014the only singing\nthat had ever taken hold of Kate's tuneless ear, and that seemed to come\nhome to her.  At least, to-night it came home as it had never done\nbefore; it seemed to touch some tender spot in her heart, and when she\nthought how dear it was, and how little she had cared about it, and how\nglad she had been to go away, she found the candles dancing in a green\nmist, and great drops came down upon the Prayer-book in her hand.\n\nThen it could not be true that she had no feeling.  She was crying\u2014the\nfirst time she had ever known herself cry except for pain or at reproof;\nand she was really so far pleased, that she made no attempt to stop the\ngreat tears that came trickling down at each familiar note, at each\nthought how long it had been since she had heard them.  She cried all\nchurch time; for whenever she tried to attend to the prayers, the very\nsound of the voice she loved so well set her off again; and Sylvia,\ntenderly laying a hand on her by way of sympathy, made her weep the more,\nthough still so softly and gently that it was like a strange sort of\nhappiness\u2014almost better than joy and merriment.  And then the sermon\u2014upon\nthe text, \"Peace, be still,\"\u2014was on the same thought on which her uncle\nhad talked to the children: not that she followed it much; the very words\n\"peace\" and \"be still,\" seemed to be enough to touch, soften, and\ndissolve her into those sweet comfortable tears.\n\nPerhaps they partly came from the weakening of the morning's\nindisposition; at any rate, when she moved, after the Blessing, holding\nthe pitying Sylvia's hand, she found that she was very much tired, her\neyelids were swollen and aching, and in fact she was fit for nothing but\nbed, where Mary and Sylvia laid her; and she slept, and slept in\ndreamless soundness, till she was waked by Mary's getting up in the\nmorning, and found herself perfectly well.\n\n\"And now, Sylvia,\" she said, as they went downstairs hand-in-hand, \"let\nus put it all out of our heads, and try and think all day that it is just\none of our old times, and that I am your old Kate.  Let me do my lessons\nand go into school, and have some fun, and quite forget all that is\nhorrid.\"\n\nBut there was something to come before this happy return to old times.\nAs soon as breakfast was over Mr Wardour said, \"Now, Kate, I want you.\"\nAnd then she knew what was coming; and somehow, she did not feel exactly\nthe same about her exploit and its causes by broad daylight, now that she\nwas cool.  Perhaps she would have been glad to hang back; yet on the\nwhole, she had a great deal to say to \"Papa,\" and it was a relief, though\nrather terrific, to find herself alone with him in the study.\n\n\"Now, Kate,\" said he again, with his arm round her, as she stood by him,\n\"will you tell me what led you to this very sad and strange proceeding?\"\n\nKate hung her head, and ran her fingers along the mouldings of his chair.\n\n\"Why was it, my dear?\" asked Mr. Wardour.\n\n\"It was\u2014\" and she grew bolder at the sound of her own voice, and more\nconfident in the goodness of her cause\u2014\"it was because Aunt Barbara said\nI must write what was not true, and\u2014and I'll never tell a\nfalsehood\u2014never, for no one!\" and her eyes flashed.\n\n\"Gently, Kate,\" he said, laying his hand upon hers; \"I don't want to know\nwhat you never _will_ do, only what you have done.  What was this\nfalsehood?\"\n\n\"Why, Papa, the other Sylvia\u2014Sylvia Joanna, you know\u2014has her birthday\nto-day, and we settled at Bournemouth that I should spend the day with\nher; and on Saturday, when Aunt Barbara heard of it, she said she did not\nwant me to be intimate there, and that I must not go, and told me to\nwrite a note to say she had made a previous engagement for me.\"\n\n\"And do you know that she had not done so?\"\n\n\"O Papa! she could not; for when I said I would not write a lie, she\nnever said it was true.\"\n\n\"Was that what you said to your aunt?\"\n\n\"Yes,\"\u2014and Kate hung her head\u2014\"I was in a passion.\"\n\n\"Then, Kate, I do not wonder that Lady Barbara insisted on obedience,\ninstead of condescending to argue with a child who could be so insolent.\"\n\n\"But, Papa,\" said Kate, abashed for a moment, then getting eager, \"she\ndoes tell fashionable falsehoods; she says she is not at home when she\nis, and\u2014\"\n\n\"Stay, Kate; it is not for you to judge of grown people's doings.\nNeither I nor Mary would like to use that form of denying ourselves; but\nit is usually understood to mean only not ready to receive visitors.  In\nthe same way, this previous engagement was evidently meant to make the\nrefusal less discourteous, and you were not even certain it did not\nexist.\"\n\n\"My Italian mistress did want to come on Monday,\" faltered Kate, \"but it\nwas not 'previous.'\"\n\n\"Then, Kate, who was it that went beside the mark in letting us believe\nthat Lady Barbara locked you up to make you tell falsehoods?\"\n\n\"Indeed, Papa, I did not say locked\u2014Charlie and Sylvia said that.\"\n\n\"But did you correct them?\"\n\n\"O Papa, I did not mean it!  But I am naughty now!  I always am naughty,\nso much worse than I used to be at home.  Indeed I am, and I never do get\ninto a good vein now.  O Papa, Papa, can't you get me out of it all?  If\nyou could only take me home again!  I don't think my aunts want to keep\nme\u2014they say I am so bad and horrid, and that I make Aunt Jane ill.  Oh,\ntake me back, Papa!\"\n\nHe did take her on his knee, and held her close to him.  \"I wish I could,\nmy dear,\" he said; \"I should like to have you again! but it cannot be.\nIt is a different state of life that has been appointed for you; and you\nwould not be allowed to make your home with me, with no older a person\nthan Mary to manage for you.  If your aunt had not been taken from us,\nthen\u2014\" and Kate ventured to put her arm round his neck\u2014\"then this would\nhave been your natural home; but as things are with us, I could not make\nmy house such as would suit the requirements of those who arrange for\nyou.  And, my poor child, I fear we let the very faults spring up that\nare your sorrow now.\"\n\n\"Oh no, no, Papa, you helped me!  Aunt Barbara only makes me\u2014oh! may I\nsay?\u2014hate her! for indeed there is no helping it!  I can't be good\nthere.\"\n\n\"What is it?  What do you mean, my dear?  What is your difficulty?  And I\nwill try to help you.\"\n\nPoor Kate found it not at all easy to explain when she came to\nparticulars.  \"Always cross,\" was the clearest idea in her mind; \"never\npleased with her, never liking anything she did\u2014not punishing, but much\nworse.\"  She had not made out her case, she knew; but she could only\nmurmur again, \"It all went wrong, and I was very unhappy.\"\n\nMr. Wardour sighed from the bottom of his heart; he was very sorrowful,\ntoo, for the child that was as his own.  And then he went back and\nthought of his early college friend, and of his own wife who had so\nfondled the little orphan\u2014all that was left of her sister.  It was\ngrievous to him to put that child away from him when she came clinging to\nhim, and saying she was unhappy, and led into faults.\n\n\"It will be better when your uncle comes home,\" he began.\n\n\"Oh no, Papa, indeed it will not.  Uncle Giles is more stern than Aunt\nBarbara.  Aunt Jane says it used to make her quite unhappy to see how\nsharp he was with poor Giles and Frank.\"\n\n\"I never saw him in his own family,\" said Mr. Wardour thoughtfully; \"but\nthis I know, Kate, that your father looked up to him, young as he then\nwas, more than to anyone; that he was the only person among them all who\never concerned himself about you or your mother; and that on the two\noccasions when I saw him, I thought him very like your father.\"\n\n\"I had rather he was like you, Papa,\" sighed Kate.  \"Oh, if I was but\nyour child!\" she added, led on by a little involuntary pressure of his\nencircling arm.\n\n\"Don't let us talk of what is not, but of what is,\" said Mr. Wardour;\n\"let us try to look on things in their right light.  It has been the will\nof Heaven to call you, my little girl, to a station where you will, if\nyou live, have many people's welfare depending on you, and your example\nwill be of weight with many.  You must go through training for it, and\nstrict training may be the best for you.  Indeed, it must be the best, or\nit would not have been permitted to befall you.\"\n\n\"But it does not make me good, it makes me naughty.\"\n\n\"No, Kate; nothing, nobody can make you naughty; nothing is strong enough\nto do that.\"\n\nKate knew what he meant, and hung her head.\n\n\"My dear, I do believe that you feel forlorn and dreary, and miss the\naffection you have had among us; but have you ever thought of the Friend\nwho is closest of all to us, and who is especially kind to a fatherless\nchild?\"\n\n\"I can't\u2014I can't feel it\u2014Papa, I can't.  And then, why was it made so\nthat I must go away from you and all?\"\n\n\"You will see some day, though you cannot see now, my dear.  If you use\nit rightly, you will feel the benefit.  Meantime, you must take it on\ntrust, just as you do my love for you, though I am going to carry you\nback.\"\n\n\"Yes; but I can feel you loving me.\"\n\n\"My dear child, it only depends on yourself to feel your Heavenly Father\nloving you.  If you will set yourself to pray with your heart, and think\nof His goodness to you, and ask Him for help and solace in all your\npresent vexatious and difficulties, never mind how small, you _will_\nbecome conscious of his tender pity and love to you.\"\n\n\"Ah! but I am not good!\"\n\n\"But He can make you so, Kate.  Your have been wearied by religious\nteaching hitherto, have you not?\"\n\n\"Except when it was pretty and like poetry,\" whispered Kate.\n\n\"Put your heart to your prayers now, Kate.  Look in the Psalms for verses\nto suit your loneliness; recollect that you meet us in spirit when you\nuse the same Prayers, read the same Lessons, and think of each other.\nOr, better still, carry your troubles to Him; and when you _have_ felt\nHis help, you will know what that is far better than I can tell you.\"\n\nKate only answered with a long breath; not feeling as if she could\nunderstand such comfort, but with a resolve to try.\n\n\"And now,\" said Mr. Wardour, \"I must take you home to-morrow, and I will\nspeak for you to Lady Barbara, and try to obtain her forgiveness; but,\nKate, I do not think you quite understand what a shocking proceeding this\nwas of yours.\"\n\n\"I know it was wrong to fancy _that_, and say _that_ about Aunt Barbara.\nI'll tell her so,\" said Kate, with a trembling voice.\n\n\"Yes, that will be right; but it was this\u2014this expedition that I meant.\"\n\n\"It was coming to you, Papa!\"\n\n\"Yes, Kate; but did you think what an outrageous act it was?  There is\nsomething particularly grievous in a little girl, or a woman of any age,\ncasting off restraint, and setting out in the world unprotected and\ncontrary to authority.  Do you know, it frightened me so much, that till\nI saw more of you I did not like you to be left alone with Sylvia.\"\n\nThe deep red colour flushed all over Kate's face and neck in her angry\nshame and confusion, burning darker and more crimson, so that Mr. Wardour\nwas very sorry for her, and added, \"I am obliged to say this, because you\nought to know that it is both very wrong in itself, and will be regarded\nby other people as more terrible than what you are repenting of more.\nSo, if you do find yourself distrusted and in disgrace, you must not\nthink it unjust and cruel, but try to submit patiently, and learn not to\nbe reckless and imprudent.  My poor child, I wish you could have so come\nto us that we might have been happier together.  Perhaps you will some\nday; and in the meantime, if you have any troubles, or want to know\nanything, you may always write to me.\"\n\n\"Writing is not speaking,\" said Kate ruefully.\n\n\"No; but it comes nearer to it as people get older.  Now go, my dear; I\nam busy, and you had better make the most of your time with your\ncousins.\"\n\nKate's heart was unburthened now; and though there was much alarm, pain,\nand grief, in anticipation, yet she felt more comfortable in herself than\nshe had done for months.  \"Papa\" had never been so tender with her, and\nshe knew that he had forgiven her.  She stept back to the drawing-room,\nvery gentle and subdued, and tried to carry out her plans of living one\nof her old days, by beginning with sharing the lessons as usual, and then\ngoing out with her cousins to visit the school, and see some of the\nparishioners.  It was very nice and pleasant; she was as quiet and loving\nas possible, and threw herself into all the dear old home matters.  It\nwas as if for a little while Katharine was driven out of Katharine, and a\nvery sweet little maiden left instead\u2014thinking about other things and\npeople instead of herself, and full of affection and warmth.  The\nimprovement that the half year's discipline had made in her bearing and\nmanners was visible now; her uncouth abrupt ways were softened, though\nstill she felt that the naturally gentle and graceful Sylvia would have\nmade a better countess than she did.\n\nThey spent the evening in little tastes of all their favourite\ndrawing-room games, just for the sake of having tried them once more; and\nPapa himself came in and took a share\u2014a very rare treat;\u2014and he always\nthought of such admirable things in \"Twenty questions,\" and made \"What's\nmy thought like?\" more full of fun than anyone.\n\nIt was a very happy evening\u2014one of the most happy that Kate had ever\npassed.  She knew _how_ to enjoy her friends now, and how precious they\nwere to her; and she was just so much tamed by the morning's\nconversation, and by the dread of the future, as not to be betrayed into\ndangerously high spirits.  That loving, pitying way of Mary's, and her\nown Sylvia's exceeding pleasure in having her, were delightful; and all\nthrough she felt the difference between the real genuine love that she\ncould rest on, and the mere habit of fondling of the other Sylvia.\n\n\"O Sylvia,\" she said, as they walked upstairs, hand in hand, pausing on\nevery stop to make it longer, \"how could I be so glad to go away before?\"\n\n\"We didn't know,\" said Sylvia.\n\n\"No,\" as they crept up another step; \"Sylvia, will you always think of me\njust here on this step, as you go up to bed?\"\n\n\"Yes,\" said Sylvia, \"that I will.  And, Katie, would it be wrong just to\nwhisper a little prayer then that you might be good and happy?\"\n\n\"It couldn't be wrong, Sylvia; only couldn't you just ask, too, for me to\ncome home?\"\n\n\"I don't know,\" said Sylvia thoughtfully, pausing a long time on the\nstep.  \"You see we know it is sure to be God's will that you should be\ngood and happy; but if it was not for you to come home, we might be like\nBalaam, you know, if we asked it too much, and it might come about in\nsome terrible way.\"\n\n\"I didn't think of that,\" said Kate.  And the two little girls parted\ngravely and peacefully; Kate somehow feeling as if, though grievous\nthings were before her, the good little kind Sylvia's hearty prayers must\nobtain some good for her.\n\nThere is no use in telling how sad the parting was when Mr. Wardour and\nthe little Countess set out for London again.  Mary had begged hard to go\ntoo, thinking that she could plead for Kate better than anyone else; but\nMr. Wardour thought Lady Barbara more likely to be angered than softened\nby their clinging to their former charge; and besides, it was too great\nan expense.\n\nHe had no doubt of Lady Barbara's displeasure from the tone of the note\nthat morning received, coldly thanking him and Miss Wardour for their\nintelligence, and his promise to restore Lady Caergwent on Tuesday.  She\nwas sorry to trouble him to bring the child back; she would have come\nherself, but that her sister was exceedingly unwell, from the alarm\ncoming at a time of great family affliction.  If Lady Caergwent were not\nable to return on Tuesday, she would send down her own maid to bring her\nhome on Wednesday.  The letter was civility itself; but it was plain that\nLady Barbara thought Kate's illness no better than the \"previous\nengagement,\" in the note that never was written.\n\nWhat was the family affliction?  Kate could not guess, but was inclined\nto imagine privately that Aunt Barbara was magnifying Uncle Giles's\nreturn without being a General into a family affliction, on purpose to\naggravate her offence.  However, in the train, Mr. Wardour, who had been\nlooking at the Supplement of the _Times_, lent to him by a\nfellow-traveller, touched her, and made her read\u2014\n\n\"On the 11th, at Alexandria, in his 23rd year, Lieutenant Giles de la\nPoer Umfraville, of the 109th regiment; eldest, and last survivor of the\nchildren of the Honourable Giles Umfraville, late Lieutenant-Colonel of\nthe 109th regiment.\"\n\nKate knew she ought to be very sorry, and greatly pity the bereaved\nfather and mother; but, somehow, she could not help dwelling most upon\nthe certainty that everyone would be much more hard upon her, and cast up\nthis trouble to her, as if she had known of it, and run away on purpose\nto make it worse.  It must have been this that they were talking about in\nAunt Jane's room, and this must have made them so slow to detect her\nflight.\n\nIn due time the train arrived, a cab was taken, and Kate, beginning to\ntremble with fright, sat by Mr. Wardour, and held his coat as if clinging\nto him as long as she could was a comfort.  Sometimes she wished the cab\nwould go faster, so that it might be over; sometimes\u2014especially when the\nstreets became only too well known to her\u2014she wished that they would\nstretch out and out for ever, that she might still be sitting by Papa,\nholding his coat.  It seemed as if that would be happiness enough for\nlife!\n\nHere was Bruton Street; here the door that on Saturday had shut behind\nher!  It was only too soon open, and Kate kept her eyes on the ground,\nashamed that even the butler should see her.  She hung back, waiting till\nMr. Wardour had paid the cabman; but there was no spinning it out, she\nhad to walk upstairs, her only comfort being that her hand was in his.\n\nNo one was in the drawing-room; but before long Lady Barbara came in.\nKate durst not look up at her, but was sure, from the tone of her voice,\nthat she must have her very sternest face; and there was something to\nmake one shiver in the rustle of her silk dress as she curtsied to Mr.\nWardour.\n\n\"I have brought home my little niece,\" he said, drawing Kate forward;\n\"and I think I may truly say, that she is very sorry for what has\npassed.\"\n\nThere was a pause; Kate knew the terrible black eyes were upon her, but\nshe felt, besides, the longing to speak out the truth, and a sense that\nwith Papa by her side she had courage to do so.\n\n\"I am sorry, Aunt Barbara,\" she said; \"I was very self-willed; I ought\nnot to have fancied things, nor said you used me ill, and wanted me to\ntell stories.\"\n\nKate's heart was lighter; though it beat so terribly as she said those\nwords.  She knew that they pleased _one_ of the two who were present, and\nshe knew they were right.\n\n\"It is well you should be so far sensible of your misconduct,\" said Lady\nBarbara; but her voice was as dry and hard as ever, and Mr. Wardour\nadded, \"She is sincerely sorry; it is from her voluntary confession that\nI know how much trouble she has given you; and I think, if you will\nkindly forgive her, that you will find her less self-willed in future.\"\n\nAnd he shoved Kate a little forward, squeezing her hand, and trying to\nwithdraw his own.  She perceived that he meant that she ought to ask\npardon; and though it went against her more than her first speech had\ndone, she contrived to say, \"I do beg pardon, Aunt Barbara; I will try to\ndo better.\"\n\n\"My pardon is one thing, Katharine,\" said Lady Barbara.  \"If your sorrow\nis real, of course I forgive you;\" and she took Kate's right-hand\u2014the\nleft was still holding by the fingers' ends to Mr. Wardour.  \"But the\nconsequences of such behaviour are another consideration.  My personal\npardon cannot, and ought not, to avert them\u2014as I am sure you must\nperceive, Mr. Wardour,\" she added, as the frightened child retreated upon\nhim.  Those consequences of Aunt Barbara's were fearful things!  Mr.\nWardour said something, to which Kate scarcely attended in her alarm, and\nher aunt went on\u2014\n\n\"For Lady Caergwent's own sake, I shall endeavour to keep this most\nunfortunate step as much a secret as possible.  I believe that scarcely\nanyone beyond this house is aware of it; and I hope that your family will\nperceive the necessity of being equally cautious.\"\n\nMr. Wardour bowed, and assented.\n\n\"But,\" added Lady Barbara, \"it has made it quite impossible for my sister\nand myself to continue to take the charge of her.  My sister's health has\nsuffered from the constant noise and restlessness of a child in the\nhouse: the anxiety and responsibility are far too much for her; and in\naddition to this, she had such severe nervous seizures from the alarm of\nmy niece's elopement, that nothing would induce me to subject her to a\nrecurrence of such agitation.  We must receive the child for the present,\nof course; but as soon as my brother returns, and can attend to business,\nthe matter must be referred to the Lord Chancellor, and an establishment\nformed, with a lady at the head, who may have authority and experience to\ndeal with such an ungovernable nature.\"\n\n\"Perhaps,\" said Mr. Wardour, \"under these circumstances it might be\nconvenient for me to take her home again for the present.\"\n\nKate quivered with hope; but that was far too good to be true; Lady\nBarbara gave a horrid little cough, and there was a sound almost of\noffence in her \"Thank you, you are very kind, but that would be quite out\nof the question.  I am at present responsible for my niece.\"\n\n\"I thought, perhaps,\" said Mr. Wardour, as an excuse for the offer, \"that\nas Lady Jane is so unwell, and Colonel Umfraville in so much affliction,\nit might be a relief to part with her at present.\"\n\n\"Thank you,\" again said Lady Barbara, as stiffly as if her throat were\nlined with whalebone; \"no inconvenience can interfere with my duty.\"\n\nMr. Wardour knew there was no use in saying any more, and inquired after\nLady Jane.  She had, it appeared, been very ill on Saturday evening, and\nhad not since left her room.  Mr. Wardour then said that Kate had not\nbeen aware, till a few hours ago, of the death of her cousin, and\ninquired anxiously after the father and mother; but Lady Barbara would\nnot do more than answer direct questions, and only said that her nephew\nhad been too much weakened to bear the journey, and had sunk suddenly at\nAlexandria, and that his father was, she feared, very unwell.  She could\nnot tell how soon he was likely to be in England.  Then she thanked Mr.\nWardour for having brought Lady Caergwent home, and offered him some\nluncheon; but in such a grave grand way, that it was plain that she did\nnot want him to eat it, and, feeling that he could do no more good, he\nkissed poor Kate and wished Lady Barbara good-bye.\n\nPoor Kate stood, drooping, too much constrained by dismay even to try to\ncling to him, or run after him to the foot of the stairs.\n\n\"Now, Katharine,\" said her aunt, \"come up with me to your Aunt Jane's\nroom.  She has been so much distressed about you, that she will not be\neasy till she has seen you.\"\n\nKate followed meekly; and found Aunt Jane sitting by the fire in her own\nroom, looking flushed, hot, and trembling.  She held out her arms, and\nKate ran into them; but neither of them dared to speak, and Lady Barbara\nstood up, saying, \"She says she is very sorry, and thus we may forgive\nher; as I know you do all the suffering you have undergone on her\naccount.\"\n\nLady Jane held the child tighter, and Kate returned her kisses with all\nher might; but the other aunt said, \"That will do.  She must not be too\nmuch for you again.\"  And they let go as if a cold wind had blown between\nthem.\n\n\"Did Mr. Wardour bring her home?\" asked Lady Jane.\n\n\"Yes; and was kind enough to propose taking her back again,\" was the\nanswer, with a sneer, that made Kate feel desperately angry, though she\ndid not understand it.\n\nIn truth, Lady Barbara was greatly displeased with the Wardours.  She had\nalways been led to think her niece's faults the effect of their\nmanagement; and she now imagined that there had been some encouragement\nof the child's discontent to make her run away; and that if they had been\nsufficiently shocked and concerned, the truant would have been brought\nhome much sooner.  It all came of her having allowed her niece to\nassociate with those children at Bournemouth.  She would be more careful\nfor the future.\n\nCareful, indeed, she was!  She had come to think of her niece as a sort\nof small wild beast that must never be let out of sight of some\ntrustworthy person, lest she should fly away again.\n\nA daily governess, an elderly person, very grave and silent, came in\ndirectly after breakfast, walked with the Countess, and heard the\nlessons; and after her departure, Kate was always to be in the room with\nher aunts, and never was allowed to sit in the schoolroom and amuse\nherself alone; but her tea was brought into the dining-room while her\naunts were at dinner, and morning, noon, and night, she knew that she was\nbeing watched.\n\nIt was very bitter to her.  It seemed to take all the spirit away from\nher, as if she did not care for books, lessons, or anything else.\nSometimes her heart burnt with hot indignation, and she would squeeze her\nhands together, or wring round her handkerchief in a sort of misery; but\nit never got beyond that; she never broke out, for she was depressed by\nwhat was still worse, the sense of shame.  Lady Barbara had not said many\nwords, but had made her feel, in spite of having forgiven her, that she\nhad done a thing that would be a disgrace to her for ever; a thing that\nwould make people think twice before they allowed their children to\nassociate with her; and that put her below the level of other girls.  The\nvery pain that Lady Barbara took to hush it up, her fears lest it should\ncome to the ears of the De la Poers, her hopes that it _might_ not be\nnecessary to reveal it to her brother, assisted to weigh down Kate with a\nsense of the heinousness of what she had done, and sunk her so that she\nhad no inclination to complain of the watchfulness around her.  And Aunt\nJane's sorrowful kindness went to her heart.\n\n\"How _could_ you do it, my dear?\" she said, in such a wonderful wistful\ntone, when Kate was alone with her.\n\nKate hung her head.  She could not think now.\n\n\"It is so sad,\" added Lady Jane; \"I hoped we might have gone on so nicely\ntogether.  And now I hope your Uncle Giles will not hear of it.  He would\nbe so shocked, and never trust you again.\"\n\n\"_You_ will trust me, when I have been good a long time, Aunt Jane?\"\n\n\"My dear, I would trust you any time, you know; but then that's no use.\nI can't judge; and your Aunt Barbara says, after such lawlessness, you\nneed very experienced training to root out old associations.\"\n\nPerhaps the aunts were more shocked than was quite needful and treated\nKate as if she had been older and known better what she was doing; but\nthey were sincere in their horror at her offence; and once she even heard\nLady Barbara saying to Mr. Mercer that there seemed to be a doom on the\nfamily\u2014in the loss of the promising young man\u2014and\u2014\n\nThe words were not spoken, but Kate knew that she was this greatest of\nall misfortunes to the family.\n\nPoor child!  In the midst of all this, there was one comfort.  She had\nnot put aside what Mr. Wardour had told her about the Comforter she could\nalways have.  She _did_ say her prayers as she had never said them\nbefore, and she looked out in the Psalms and Lessons for comforting\nverses.  She knew she had done very wrong, and she asked with all the\nstrength of her heart to be forgiven, and made less unhappy, and that\npeople might be kinder to her.  Sometimes she thought no help was coming,\nand that her prayers did no good, but she went on; and then, perhaps, she\ngot a kind little caress from Lady Jane, or Mr. Mercer spoke\ngood-naturedly to her, or Lady Barbara granted her some little favour,\nand she felt as if there was hope and things were getting better; and she\ntook courage all the more to pray that Uncle Giles might not be very hard\nupon her, nor the Lord Chancellor very cruel.\n\n\n\n\nCHAPTER XIV.\n\n\nA FORTNIGHT had passed, and had seemed nearly as long as a year, since\nKate's return from Oldburgh, when one afternoon, when she was lazily\nturning over the leaves of a story-book that she knew so well by heart\nthat she could go over it in the twilight, she began to gather from her\naunt's words that somebody was coming.\n\nThey never told her anything direct; but by listening a little more\nattentively to what they were saying, she found out that a letter\u2014no, a\ntelegram\u2014had come while she was at her lessons; that Aunt Barbara had\nbeen taking rooms at a hotel; that she was insisting that Jane should not\nimagine they would come to-night\u2014they would not come till the last train,\nand then neither of them would be equal\u2014\n\n\"Poor dear Emily!  But could we not just drive to the hotel and meet\nthem?  It will be so dreary for them.\"\n\n\"You go out at night! and for such a meeting! when you ought to be\nkeeping yourself as quiet as possible!  No, depend upon it they will\nprefer getting in quietly, and resting to-night; and Giles, perhaps, will\nstep in to breakfast in the morning.\"\n\n\"And then you will bring him up to me at once!  I wonder if the boy is\nmuch altered!\"\n\nThrob! throb! throb! went Kate's heart!  So the terrible stern uncle was\nin England, and this was the time for her to be given up to the Lord\nChancellor and all his myrmidons (a word that always came into her head\nwhen she was in a fright).  She had never loved Aunt Jane so well; she\nalmost loved Aunt Barbara, and began to think of clinging to her with an\neloquent speech, pleading to be spared from the Lord Chancellor!\n\nTo-morrow morning\u2014that was a respite!\n\nThere was a sound of wheels.  Lady Jane started.\n\n\"They are giving a party next door,\" said Lady Barbara.\n\nBut the bell rang.\n\n\"Only a parcel coming home,\" said Lady Barbara.  \"Pray do not be nervous,\nJane.\"\n\nBut the red colour was higher in Barbara's own cheeks, as there were\nsteps on the stairs; and in quite a triumphant voice the butler\nannounced, as he opened the door, \"Colonel and Mrs. Umfraville!\"\n\nKate stood up, and backed.  It was Aunt Barbara's straight, handsome,\nterrible face, and with a great black moustache to make it worse.  She\nsaw that, and it was all she feared!  She was glad the sofa was between\nthem!\n\nThere was a lady besides all black bonnet and cloak; and there was a\nconfusion of sounds, a little half sobbing of Aunt Jane's; but the other\nsister and the brother were quite steady and grave.  It was his keen dark\neye, sparkling like some wild animal's in the firelight, as Kate thought,\nwhich spied her out; and his deep grave voice said, \"My little niece,\" as\nhe held out his hand.\n\n\"Come and speak to your uncle, Katharine,\" said Lady Barbara; and not\nonly had she to put her hand into that great firm one, but her forehead\nwas scrubbed by his moustache.  She had never been kissed by a moustache\nbefore, and she shuddered as if it had been on a panther's lip.\n\nBut then he said, \"There, Emily;\" and she found herself folded up in such\narms as had never been round her before, with the very sweetest of kisses\non her cheeks, the very kindest of eyes, full of moisture, gazing at her\nas if they had been hungry for her.  Even when the embrace was over, the\nhand still held hers; and as she stood by the new aunt, a thought crossed\nher that had never come before, \"I wonder if my mamma was like this!\"\n\nThere was some explanation of how the travellers had come on, &c., and it\nwas settled that they were to stay to dinner; after which Mrs. Umfraville\nwent away with Lady Barbara to take off her bonnet.\n\nColonel Umfraville came and sat down by his sister on the sofa, and said,\n\"Well Jane, how have you been?\"\n\n\"Oh! much as usual:\" and then there was a silence, till she moved a\nlittle nearer to him, put her hand on his arm, looked up in his face with\nswimming eyes, and said, \"O Giles!  Giles!\"\n\nHe took her hand, and bent over her, saying, in the same grave steady\nvoice, \"Do not grieve for us, Jane.  We have a great deal to be thankful\nfor, and we shall do very well.\"\n\nIt made that loving tender-hearted Aunt Jane break quite down, cling to\nhim and sob, \"O Giles\u2014those dear noble boys\u2014how little we thought\u2014and\ndear Caergwent too\u2014and you away from home!\"\n\nShe was crying quite violently, so as to be shaken by the sobs; and her\nbrother stood over her, saying a kind word or two now and then, to try to\nsoothe her; while Kate remained a little way off, with her black eyes\nwide open, thinking her uncle's face was almost displeased\u2014at any rate,\nvery rigid.  He looked up at Kate, and signed towards a scent-bottle on\nthe table.  Kate gave it; and then, as if the movement had filled her\nwith a panic, she darted out of the room, and flew up to the bedrooms,\ncrying out, \"Aunt Barbara, Aunt Jane is crying so terribly!\"\n\n\"She will have one of her attacks!  Oh!\" began Lady Barbara, catching up\na bottle of salvolatile.\n\n\"Had we not better leave her and Giles to one another?\" said the tones\nthat Kate liked so much.\n\n\"Oh! my dear, you don't know what these attacks are!\" and away hurried\nLady Barbara.\n\nThe bonnet was off now, leaving only a little plain net cap under it,\nround the calm gentle face.  There was a great look of sadness, and the\neyelids were heavy and drooping; but there was something that put Kate in\nmind of a mother dove in the softness of the large tender embrace, and\nthe full sweet caressing tone.  What a pity that such an aunt must know\nthat she was an ill-behaved child, a misfortune to her lineage!  She\nstood leaning against the door, very awkward and conscious.  Mrs.\nUmfraville turned round, after smoothing her hair at the glass, smiled,\nand said, \"I thought I should find you here, my little niece.  You are\nKate, I think.\"\n\n\"I used to be, but my aunts here call me Katharine.\"\n\n\"Is this your little room?\" said Mrs. Umfraville, as they came out.  The\nfact was, that she thought the sisters might be happier with their\nbrother if she delayed a little; so she came into Kate's room, and was\nbeginning to look at her books, when Lady Barbara came hurrying up again.\n\n\"She is composed now, Emily.  Oh! it is all right; I did not know where\nKatharine might be.\"\n\nKate's colour glowed.  She could not bear that this sweet Aunt Emily\nshould guess that she was a state prisoner, kept in constant view.\n\nLady Jane was quiet again, and nothing more that could overthrow her\nspirits passed all the evening; there was only a little murmur of talk,\ngenerally going on chiefly between Lady Barbara and Mrs. Umfraville,\nthough occasionally the others put in a word.  The Colonel sat most of\nthe time with his set, serious face, and his eye fixed as if he was not\nattending, though sometimes Kate found the quick keen brilliance of his\nlook bent full upon her, so as to terrify her by its suddenness, and make\nher hardly know what she was saying or doing.\n\nThe worst moments were at dinner.  She was, in the first place, sure that\nthose dark questioning eyes had decided that there must be some sad cause\nfor her not being trusted to drink her tea elsewhere; and then, in the\npause after the first course, the eyes came again, and he said, and to\nher, \"I hope your good relations the Wardours are well.\"\n\n\"Quite well\u2014thank you,\" faltered Kate.\n\n\"When did you see them last?\"\n\n\"A\u2014a fortnight ago\u2014\" began Kate.\n\n\"Mr. Wardour came up to London for a few hours,\" said Lady Barbara,\nlooking at Kate as if she meant to plunge her below the floor; at least,\nso the child imagined.\n\nThe sense that this was not the whole truth made her especially\nmiserable; and all the rest of the evening was one misery of\nembarrassment, when her limbs did not seem to be her own, but as if\nsomebody else was sitting at her little table, walking upstairs, and\ndoing her work.  Even Mrs. Umfraville's kind ways could not restore her;\nshe only hung her head and mumbled when she was asked to show her work,\nand did not so much as know what was to become of her piece of\ncross-stitch when it was finished.\n\nThere was some inquiry after the De la Poers; and Mrs. Umfraville asked\nif she had found some playfellows among their daughters.\n\n\"Yes,\" faintly said Kate; and with another flush of colour, thought of\nhaving been told, that if Lady de la Poer knew what she had done, she\nwould never be allowed to play with them again, and therefore that she\nnever durst attempt it.\n\n\"They were very nice children,\" said Mrs. Umfraville.\n\n\"Remarkably nice children,\" returned Lady Barbara, in a tone that again\ncut Kate to the heart.\n\nBed-time came; and she would have been glad of it, but that all the time\nshe was going to sleep there was the Lord Chancellor to think of, and the\nuncle and aunt with the statue faces dragging her before him.\n\nSunday was the next day, and the uncle and aunt were not seen till after\nthe afternoon service, when they came to dinner, and much such an evening\nas the former one passed; but towards the end of it Mrs. Umfraville said,\n\"Now, Barbara, I have a favour to ask.  Will you let this child spend the\nday with me to-morrow?  Giles will be out, and I shall be very glad to\nhave her for my companion.\"\n\nKate's eyes glistened, and she thought of stern Proserpine.\n\n\"My dear Emily, you do not know what you ask.  She will be far too much\nfor you.\"\n\n\"I'll take care of that,\" said Mrs. Umfraville, smiling.\n\n\"And I don't know about trusting her.  I cannot go out, and Jane cannot\nspare Bartley so early.\"\n\n\"I will come and fetch her,\" said the Colonel.\n\n\"And bring her back too.  I will send the carriage in the evening, but do\nnot let her come without you,\" said Lady Barbara earnestly.\n\nHad they told, or would they tell after she was gone to bed?  Kate\nthought Aunt Barbara was a woman of her word, but did not quite trust\nher.  Consent was given; but would not that stern soldier destroy all the\npleasure?  And people in sorrow too!  Kate thought of Mrs. Lacy, and had\nno very bright anticipations of her day; yet a holiday was something, and\nto be out of Aunt Barbara's way a great deal more.\n\nShe had not been long dressed when there was a ring at the bell, and,\nbefore she had begun to expect him, the tall man with the dark lip and\ngrey hair stood in her schoolroom.  She gave such a start, that he asked,\n\"Did you not expect me so soon?\"\n\n\"I did not think you would come till after breakfast: but\u2014\"\n\nAnd with an impulse of running away from his dread presence, she darted\noff to put on her hat, but was arrested on the way by Lady Barbara, at\nher bedroom door.\n\n\"Uncle Giles is come for me,\" she said, and would have rushed on, but her\naunt detained her to say, \"Recollect, Katharine, that wildness and\nimpetuosity, at all times unbecoming, are particularly so where there is\naffliction.  If consideration for others will not influence you, bear in\nmind that on the impression you make on your uncle and aunt, it depends\nwhether I shall be obliged to tell all that I would willingly forget.\"\n\nKate's heart swelled, and without speaking she entered her own room,\nthinking how hard it was to have even the pleasure of hoping for ease and\nenjoyment taken away.\n\nWhen she came down, she found her aunt\u2014as she believed\u2014warning her uncle\nagainst her being left to herself; and then came, \"If she should be too\nmuch for Emily, only send a note, and Bartley or I will come to fetch her\nhome.\"\n\n\"She wants him to think me a little wild beast!\" thought Kate; but her\nuncle answered, \"Emily always knows how to deal with children.\nGood-bye.\"\n\n\"To deal with children!  What did that mean?\" thought the Countess, as\nshe stepped along by the side of her uncle, not venturing to speak, and\nfeeling almost as shy and bewildered as when she was on the world alone.\n\nHe did not speak, but when they came to a crossing of a main street, he\ntook her by the hand; and there was something protecting and comfortable\nin the feel, so that she did not let go; and presently, as she walked on,\nshe felt the fingers close on hers with such a quick tight squeeze, that\nshe looked up in a fright and met the dark eye turned on her quite soft\nand glistening.  She did not guess how he was thinking of little clasping\nhands that had held there before; and he only said something rather\nhurriedly about avoiding some coals that were being taken in through a\nround hole in the pavement.\n\nSoon they were at the hotel; and Mrs. Umfraville came out of her room\nwith that greeting which Kate liked so much, helped her to take off her\ncloak and smooth her hair, and then set her down to breakfast.\n\nIt was a silent meal to Kate.  Her uncle and aunt had letters to read,\nand things to consult about that she did not understand; but all the time\nthere was a kind watch kept up that she had what she liked; and Aunt\nEmily's voice was so much like the deep notes of the wood-pigeons round\nOldburgh, that she did not care how long she listened to it, even if it\nhad been talking Hindostanee!\n\nAs soon as breakfast was over, the Colonel took up his hat and went out;\nand Mrs. Umfraville said, turning to Kate, \"Now, my dear, I have\nsomething for you to help me in; I want to unpack some things that I have\nbrought home.\"\n\n\"Oh, I shall like that!\" said Kate, feeling as if a weight was gone with\nthe grave uncle.\n\nMrs. Umfraville rang, and asked to have a certain box brought in.  Such a\nbox, all smelling of choice Indian wood; the very shavings that stuffed\nit were delightful!  And what an unpacking!  It was like nothing but the\nIndian stall at the Baker Street Bazaar!  There were two beautiful large\nivory work-boxes, inlaid with stripes and circles of tiny mosaic; and\nthere were even more delicious little boxes of soft fragrant sandal wood,\nand a set of chessmen in ivory.  The kings were riding on elephants, with\ncanopies over their heads, and ladders to climb up by; and each elephant\nhad a tiger in his trunk.  Then the queens were not queens, but grand\nviziers, because the queen is nobody in the East: and each had a lesser\nelephant; the bishops were men riding on still smaller elephants; the\ncastles had camels, the knights horses; and the pawns were little\nfoot-soldiers, the white ones with guns, as being European troops, the\nred ones with bows and arrows.  Kate was perfectly delighted with these\nmen, and looked at and admired them one by one, longing to play a game\nwith them.  Then there was one of those wonderful clusters of Chinese\nivory balls, all loose, one within the other, carved in different\npatterns of network, and there were shells spotted and pink-mouthed,\ncard-cases, red shining boxes, queer Indian dolls; figures in all manner\nof costumes, in gorgeous colours, painted upon shining transparent talc\nor on soft rice-paper.  There was no describing how charming the sight\nwas, nor how Kate dwelt upon each article; and how pleasantly her aunt\nexplained what it was intended for, and where it came from, answering all\nquestions in the nicest, kindest way.  When all the wool and shavings had\nbeen pinched, and the curled-up toes of the slippers explored, so as to\nmake sure that no tiny shell nor ivory carving lurked unseen, the room\nlooked like a museum; and Mrs. Umfraville said, \"Most of these things\nwere meant for our home friends: there is an Indian scarf and a Cashmere\nshawl for your two aunts, and I believe the chessmen are for Lord de la\nPoer.\"\n\n\"O Aunt Emily, I should so like to play one game with them before they\ngo!\"\n\n\"I will have one with you, if you can be very careful of their tender\npoints,\" said Mrs. Umfraville, without one of the objections that Kate\nhad expected; \"but first I want you to help me about some of the other\nthings.  Your uncle meant one of the work-boxes for you!\"\n\n\"O Aunt Emily, how delightful!  I really will work, with such a dear\nbeautiful box!\" cried Kate, opening it, and again peeping into all its\nlittle holes and contrivances.  \"Here is the very place for a dormouse to\nsleep in!  And who is the other for?\"\n\n\"For Fanny de la Poer, who is his godchild.\"\n\n\"Oh, I am so glad!  Fanny always has such nice pretty work about!\"\n\n\"And now I want you to help me to choose the other presents.  There;\nthese,\" pointing to a scarf and a muslin dress adorned with the wings of\ndiamond beetles, \"are for some young cousins of my own; but you will be\nable best to choose what the other De la Poers and your cousins at\nOldburgh would like best.\"\n\n\"My cousins at Oldburgh!\" cried Kate.  \"May they have some of these\npretty things?\"  And as her aunt answered \"We hope they will,\" Kate flew\nat her, and hugged her quite tight round the throat; then, when Mrs.\nUmfraville undid the clasp, and returned the kiss, she went like an\nIndia-rubber ball with a backward bound, put her hands together over her\nhead, and gasped out, \"Oh, thank you, thank you!\"\n\n\"My dear, don't go quite mad.  You will jump into that calabash, and then\nit won't be fit for anybody.  Are you so very glad?\"\n\n\"Oh! so glad!  Pretty things do come so seldom to Oldburgh!\"\n\n\"Well, we thought you might like to send Miss Wardour this shawl.\"\n\nIt was a beautiful heavy shawl of the soft wool of the Cashmere goats;\nreally of every kind of brilliant hue, but so dexterously blended\ntogether, that the whole looked dark and sober.  But Kate did not look\nwith favour on the shawl.\n\n\"A shawl is so stupid,\" she said.  \"If you please, I had rather Mary had\nthe work-box.\"\n\n\"But the work-box is for Lady Fanny.\"\n\n\"Oh! but I meant my own,\" said Kate earnestly.  \"If you only knew what a\npity it is to give nice things to me; they always get into such a mess.\nNow, Mary always has her things so nice; and she works so beautifully;\nshe has never let Lily wear a stitch but of her setting; and she always\nwished for a box like this.  One of her friends at school had a little\none; and she used to say, when we played at roe's egg, that she wanted\nnothing but an ivory work-box; and she has nothing but an old blue one,\nwith the steel turned black!\"\n\n\"We must hear what your uncle says, for you must know that he meant the\nbox for you.\"\n\n\"It isn't that I don't care for it,\" said Kate, with a sudden glistening\nin her eyes; \"it is because I do care for it so very much that I want\nMary to have it.\"\n\n\"I know it is, my dear;\" and her aunt kissed her; \"but we must think\nabout it a little.  Perhaps Mary would not think an Indian shawl quite so\nstupid as you do.\"\n\n\"Mary isn't a nasty vain conceited girl!\" cried Kate indignantly.  \"She\nalways looks nice; but I heard Papa say her dress did not cost much more\nthan Sylvia's and mine, because she never tore anything, and took such\ncare!\"\n\n\"Well, we will see,\" said Mrs. Umfraville, perhaps not entirely convinced\nthat the shawl would not be a greater prize to the thrifty girl than Kate\nperceived.\n\nKate meanwhile had sprung unmolested on a beautiful sandalwood case for\nSylvia, and a set of rice-paper pictures for Lily; and the appropriating\nother treasures to the De la Poers, packing them up, and directing them,\naccompanied with explanations of their habits and tastes, lasted till so\nlate, that after the litter was cleared away there was only time for one\ngame at chess with the grand pieces; and in truth the honour of using\nthem was greater than the pleasure.  They covered up the board, so that\nthere was no seeing the squares, and it was necessary to be most\ninconveniently cautious in lifting them.  They were made to be looked at,\nnot played with; and yet, wonderful to relate, Kate did not do one of the\ndelicate things a mischief!\n\nWas it that she was really grown more handy, or was it that with this\ngentle aunt she was quite at her ease, yet too much subdued to be\ncareless and rough?\n\nThe luncheon came; and after it, she drove with her aunt first to a few\nshops, and then to take up the Colonel, who had been with his lawyer.\nKate quaked a little inwardly, lest it should be about the Lord\nChancellor, and tried to frame a question on the subject to her aunt; but\neven the most chattering little girls know what it is to have their lips\nsealed by an odd sort of reserve upon the very matters that make them\nmost uneasy; and just because her wild imagination had been thinking that\nperhaps this was all a plot to waylay her into the Lord Chancellor's\nclutches, she could not utter a word on the matter, while they drove\nthrough the quiet squares where lawyers live.\n\nMrs. Umfraville, however, soon put that out of her head by talking to her\nabout the Wardours, and setting open the flood gates of her eloquence\nabout Sylvia.  So delightful was it to have a listener, that Kate did not\ngrow impatient, long as they waited at the lawyer's door in the dull\nsquare, and indeed was sorry when the Colonel made his appearance.  He\njust said to her that he hoped she was not tired of waiting; and as she\nreplied with a frightened little \"No, thank you,\" began telling his wife\nsomething that Kate soon perceived belonged to his own concerns, not to\nhers; so she left off trying to gather the meaning in the rumble of the\nwheels, and looked out of window, for she could never be quite at ease\nwhen she felt that those eyes might be upon her.\n\nOn coming back to the hotel, Mrs. Umfraville found a note on the table\nfor her: she read it, gave it to her husband, and said, \"I had better go\ndirectly.\"\n\n\"Will it not be too much?  Can you?\" he said very low; and there was the\nsame repressed twitching of the muscles of his face, as Kate had seen\nwhen he was left with his sister Jane.\n\n\"Oh yes!\" she said fervently; \"I shall like it.  And it is her only\nchance; you see she goes to-morrow.\"\n\nThe carriage was ordered again, and Mrs. Umfraville explained to Kate\nthat the note was from a poor invalid lady whose son was in their own\nregiment in India, that she was longing to hear about him, and was going\nout of town the next day.\n\n\"And what shall I give you to amuse yourself with, my dear?\" asked Mrs.\nUmfraville.  \"I am afraid we have hardly a book that will suit you.\"\n\nKate had a great mind to ask to go and sit in the carriage, rather than\nremain alone with the terrible black moustache; but she was afraid of the\nColonel's mentioning Aunt Barbara's orders that she was not to be let out\nof sight.  \"If you please,\" she said, \"if I might write to Sylvia.\"\n\nHer aunt kindly established her at a little table, with a leathern\nwriting-case, and her uncle mended a pen for her.  Then her aunt went\naway, and he sat down to his own letters.\n\nKate durst not speak to him, but she watched him under her eyelashes, and\nnoticed how he presently laid down his pen, and gave a long, heavy, sad\nsigh, such as she had never heard when his wife was present; then sat\nmusing, looking fixedly at the grey window; till, rousing himself with\nanother such sigh, he seemed to force himself to go on writing, but\npaused again, as if he were so wearied and oppressed that he could hardly\nbear it.\n\nIt gave Kate a great awe of him, partly because a little girl in a book\nwould have gone up, slid her hand into his, and kissed him; but she could\nnearly as soon have slid her hand into a lion's; and she was right, it\nwould have been very obtrusive.\n\nSome little time had passed before there was an opening of the door, and\nthe announcement, \"Lord de la Poer.\"\n\nUp started Kate, but she was quite lost in the greeting of the two\nfriends; Lord de la Poer, with his eyes full of tears, wringing his\nfriend's hand, hardly able to speak, but just saying, \"Dear Giles, I am\nglad to have you at home.  How is she?\"\n\n\"Wonderfully well,\" said the Colonel, with the calm voice but the\ntwitching face.  \"She is gone to see Mrs. Ducie, the mother of a lad in\nmy regiment, who was wounded at the same time as Giles, and whom she\nnursed with him.\"\n\n\"Is not it very trying?\"\n\n\"Nothing that is a kindness ever is trying to Emily,\" he said, and his\nvoice did tremble this time.\n\nKate had quietly re-seated herself in her chair.  She felt that it was no\nmoment to thrust herself in; nor did she feel herself aggrieved, even\nthough unnoticed by such a favourite friend.  Something in the whole\nspirit of the day had made her only sensible that she was a little girl,\nand quite forgot that she was a Countess.\n\nThe friends were much too intent on one another to think of her, as she\nsat in the recess of the window, their backs to her.  They drew their\nchairs close to the fire, and began to talk, bending down together; and\nKate felt sure, that as her uncle at least knew she was there, she need\nnot interrupt.  Besides, what they spoke of was what she had longed to\nhear, and would never have dared to ask.  Lord de la Poer had been like a\nfather to his friend's two sons when they were left in England; and now\nthe Colonel was telling him\u2014as, perhaps, he could have told no one\nelse\u2014about their brave spirit, and especially of Giles's patience and\nresolution through his lingering illness; how he had been entirely\nunselfish in entreating that anything might happen rather than that his\nfather should resign his post; but though longing to be with his parents,\nand desponding as to his chance of recovery, had resigned himself in\npatience to whatever might be thought right; and how through the last\nsudden accession of illness brought on by the journey, his sole thought\nhad been for his parents.\n\n\"And she has borne up!\" said Lord de la Poer.\n\n\"As _he_ truly said, 'As long as she has anyone to care for, she will\nnever break down.'  Luckily, I was entirely knocked up for a few days\njust at first; and coming home we had a poor young woman on board very\nill, and Emily nursed her day and night.\"\n\n\"And now you will bring her to Fanny and me to take care of.\"\n\n\"Thank you\u2014another time.  But, old fellow, I don't know whether we either\nof us could stand your house full of children yet.  Emily would be always\namong them, and think she liked it; but I knew how it would be.  It was\njust so when I took her to a kind friend of ours after the little girls\nwere taken; she had the children constantly with her, but I never saw her\nso ill as she was afterwards.\"\n\n\"Reaction!  Well, whenever you please; you shall have your rooms to\nyourselves, and only see us when you like.  But I don't mean to press\nyou; only, what are you going to do next?\"\n\n\"I can hardly tell.  There are business matters of our own, and about\npoor James's little girl, to keep us here a little while.\"  (\"Who is\nthat?\" thought Kate.)\n\n\"Then you must go into our house.  I was in hopes it might be so, and\ntold the housekeeper to make ready.\"\n\n\"Thank you; if Emily\u2014  We will see, when she comes in I want to make up\nmy mind about that child.  Have you seen much of her?\"\n\nKate began to think honour required her to come forward, but her heart\nthrobbed with fright.\n\n\"Not so much as I could wish.  It is an intelligent little monkey, and\nour girls were delighted with her; but I believe Barbara thinks me a\ncorrupter of youth, for she discountenances us.\"\n\n\"Ah! one of the last times I was alone with Giles, he said, smiling,\n'That little girl in Bruton Street will be just what Mamma wants;' and I\nknow Emily has never ceased to want to get hold of the motherless thing\never since Mrs. Wardour's death.  I know it would be the greatest comfort\nto Emily, but I only doubted taking the child away from my sisters.  I\nthought it would be such a happy thing to have Jane's kind heart drawn\nout; and if Barbara had forgiven the old sore, and used her real\nadmirable good sense affectionately, it would have been like new life to\nthem.  Besides, it must make a great difference to their income.  But is\nit possible that it can be the old prejudice, De la Poer?  Barbara\nevidently dislikes the poor child, and treats her like a state prisoner!\"\n\nHonour prevailed entirely above fear and curiosity.  Out flew Kate, to\nthe exceeding amaze and discomfiture of the two gentlemen.  \"No, no,\nUncle Giles; it is\u2014it is because I ran away!  Aunt Barbara said she would\nnot tell, for if you knew it, you would\u2014you would despise me;\u2014and you,\"\nlooking at Lord de la Poer, \"would never let me play with Grace and Addy\nagain!\"\n\nShe covered her face with her hands\u2014it was all burning red; and she was\nnearly rushing off, but she felt herself lifted tenderly upon a knee, and\nan arm round her.  She thought it her old friend; but behold, it was her\nuncle's voice that said, in the softest gentlest way, \"My dear, I never\ndespise where I meet with truth.  Tell me how it was; or had you rather\ntell your Aunt Emily?\"\n\n\"I'll tell you,\" said Kate, all her fears softened by his touch.  \"Oh no!\nplease don't go, Lord de la Poer; I do want you to know, for I couldn't\nhave played with Grace and Adelaide on false pretences!\"  And encouraged\nby her uncle's tender pressure, she murmured out, \"I ran away\u2014I did\u2014I\nwent home!\"\n\n\"To Oldburgh!\"\n\n\"Yes\u2014yes!  It was very wrong; Papa\u2014Uncle Wardour, I mean\u2014made me see it\nwas.\"\n\n\"And what made you do it?\" said her uncle kindly.  \"Do not be afraid to\ntell me.\"\n\n\"It was because I was angry.  Aunt Barbara would not let me go to the\nother Wardours, and wanted me to write a\u2014what I thought\u2014a fashionable\nfalsehood; and when I said it was a lie,\" (if possible, Kate here became\ndeeper crimson than she was before,) \"she sent me to my room till I would\nbeg her pardon, and write the note.  So\u2014so I got out of the house, and\ntook a cab, and went home by the train.  I didn't know it was so very\ndreadful a thing, or indeed I would not.\"\n\nAnd Kate hid her burning face on her uncle's breast, and was considerably\nstartled by what she heard next, from the Marquis.\n\n\"Hm!  All I have to say is, that if Barbara had the keeping of me, I\nshould run away at the end of a week.\"\n\n\"Probably!\" and Lord de la Poer saw, what Kate did not, the first shadow\nof a smile on the face of his friend, as he pressed his arm round the\nstill trembling girl; \"but, you see, Barbara justly thinks you corrupt\nyouth.\u2014My little girl, you must not let _him_ make you think lightly of\nthis\u2014\"\n\n\"Oh, no, I never could!  Papa was so shocked!\" and she was again covered\nwith confusion at the thought.\n\n\"But,\" added her uncle, \"it is not as if you had not gone to older and\nbetter friends than any you have ever had, my poor child.  I am afraid\nyou have been much tried, and have not had a happy life since you left\nOldburgh.\"\n\n\"I have always been naughty,\" said Kate.\n\n\"Then we must try if your Aunt Emily can help you to be good.  Will you\ntry to be as like her own child to her as you can, Katharine?\"\n\n\"And to you,\" actually whispered Kate; for somehow at that moment she\ncared much more for the stern uncle than the gentle aunt.\n\nHe lifted her up and kissed her, but set her down again with the sigh\nthat told how little she could make up to him for the son he had left in\nEgypt.  Yet, perhaps that sigh made Kate long with more fervent love for\nsome way of being so very good and affectionate as quite to make him\nhappy, than if he had received her demonstration as if satisfied by it.\n\n\n\n\nCHAPTER XV.\n\n\nNOTHING of note passed during the rest of the evening.  Mrs. Umfraville\ncame home; but Kate had fallen back into the shy fit that rendered her\nunwilling to begin on what was personal, and the Colonel waited to talk\nit over with his wife alone before saying any more.\n\nBesides, there were things far more near to them than their little\ngreat-niece, and Mrs. Umfraville could not see Lord de la Poer without\nhaving her heart very full of the sons to whom he had been so kind.\nAgain they sat round the fire, and this time in the dark, while once more\nGiles and Frank and all their ways were talked over and over, and Kate\nwas forgotten; but she was not sitting alone in the dark window\u2014no, she\nhad a footstool close to her uncle, and sat resting her head upon his\nknee, her eyes seeking red caverns in the coals, her heart in a strange\npeaceful rest, her ears listening to the mother's subdued tender tones in\nspeaking of her boys, and the friend's voice of sympathy and affection.\nHer uncle leant back and did not speak at all; but the other two went on\nand on, and Mrs. Umfraville seemed to be drinking in every little trait\nof her boys' English life, not weeping over it, but absolutely smiling\nwhen it was something droll or characteristic.\n\nKate felt subdued and reverent, and loved her new relations more and more\nfor their sorrows; and she began to dream out castles of the wonderful\ngoodness by which she would comfort them; then she looked for her uncle's\nhand to see if she could dare to stroke it, but one was over his brow,\nthe other out of reach, and she was shy of doing anything.\n\nThe dinner interrupted them; and Kate had the pleasure of dining late,\nand sitting opposite to Lord de la Poer, who talked now and then to her,\nand told her what Adelaide and Grace were doing; but he was grave and\nsad, out of sympathy with his friends, and Kate was by no means tempted\nto be foolish.\n\nIndeed, she began to feel that she might hope to be always good with her\nuncle and aunt, and that they would never make her naughty.  Only too\nsoon came the announcement of the carriage for Lady Caergwent; and when\nAunt Emily took her into the bedroom to dress, she clung to that kind\nhand and fondled it.\n\n\"My dear little girl!\" and Aunt Emily held her in her arms, \"I am so\nglad!  Kate, I do think your dear uncle is a little cheered to-night!  If\nhaving you about him does him any good, how I shall love you, Katie!\" and\nshe hugged her closer.  \"And it is so kind in Lord de la Poer to have\ncome!  Oh, now he will be better!  I am so thankful he is in England\nagain!  You must be with us whenever Barbara can spare you, Katie dear,\nfor I am sure he likes it.\"\n\n\"Each wants me, to do the other good,\" thought Kate; and she was so much\ntouched and pleased that she did not know what to do, and looked foolish.\n\nUncle Giles took her down stairs; and when they were in the carriage, in\nthe dark, he seemed to be less shy: he lifted her on his knee and said,\n\"I will talk to your aunt, and we will see how soon you can come to us,\nmy dear.\"\n\n\"Oh, do let it be soon,\" said Kate.\n\n\"That must depend upon your Aunt Barbara,\" he answered, \"and upon law\nmatters, perhaps.  And you must not be troublesome to her; she has\nsuffered very much, and will not think of herself, so you must think for\nher.\"\n\n\"I don't know how, Uncle Giles,\" said poor sincere Kate.  \"At home, they\nalways said I had no consideration.\"\n\n\"You must learn,\" he said gravely.  \"She is not to be harassed.\"\n\nKate was rather frightened; but he spoke in a kinder voice.  \"At home,\nyou say.  Do you mean with my sisters, or at Oldburgh?\"\n\n\"Oh, at Oldburgh, Uncle Giles!\"\n\n\"You are older now,\" he answered, \"and need not be so childish.\"\n\n\"And please one thing\u2014\"\n\n\"Well\u2014\"\n\nThere came a great choking in her throat, but she did get it out.\n\"Please, please, don't think all I do wrong is the Wardours' fault!  I\nknow I am naughty and horrid and unladylike, but it is my own own fault,\nindeed it is, and nobody _else's_!  Mary and Uncle Wardour would have\nmade me good\u2014and it was all my fault.\"\n\n\"My dear,\" and he put the other hand so that he completely encircled the\nlittle slim waist, \"I do quite believe that Mr. Wardour taught you all\nthe good you have.  There is nothing I am so glad of as that you love and\nreverence him as he deserves\u2014as far as such a child can do.  I hope you\nalways will, and that your gratitude will increase with your knowledge of\nthe sacrifices that he made for you.\"\n\nIt was too much of a speech for Kate to answer; but she nestled up to\nhim, and felt as if she loved him more than ever.  He added, \"I should\nlike to see Mr. Wardour, but I can hardly leave your aunt yet.  Would he\ncome to London?\"\n\nKate gave a gasp.  \"Oh dear!  Sylvia said he would have no money for\njourneys now!  It cost so much his coming in a first-class carriage with\nme.\"\n\n\"You see how necessary it is to learn consideration,\" said the Colonel;\n\"I must run down to see him, and come back at night.\"\n\nBy this time they were at the aunts' door, and both entered the\ndrawing-room together.\n\nLady Barbara anxiously hoped that Katharine had behaved well.\n\n\"Perfectly well,\" he answered; and his face was really brighter and\ntenderer.\n\nIt was Kate's bed-time, and she was dismissed at once.  She felt that the\nkiss and momentary touch of the hand, with the \"Bless you,\" were far more\nearnest than the mere greeting kiss.  She did not know that it had been\nhis wonted good-night to his own children.\n\nWhen she was gone, he took a chair, and explained that he could remain\nfor a little while, as Lord de la Poer would bear his wife company.  Lady\nJane made room for him on the sofa, and Lady Barbara looked pleased.\n\n\"I wished to talk to you about that child,\" he said.\n\n\"I have been wishing it for some time,\" said Lady Barbara; \"waiting, in\nfact, to make arrangements till your return.\"\n\n\"What arrangements?\"\n\n\"For forming an establishment for her.\"\n\n\"The child's natural home is with you or with me.\"\n\nThere was a little silence; then Lady Jane nervously caught her brother's\nhand, saying, \"O Giles, Giles, you must not be severe with her, poor\nlittle thing!\"\n\n\"Why should I be severe, Jane?\" he said.  \"What has the child done to\ndeserve it?\"\n\n\"I do not wish to enter into particulars,\" said Lady Barbara.  \"But she\nis a child who has been so unfortunately brought up as to require\nconstant watching; and to have her in the house does so much harm to\nJane's health, that I strongly advise you not to attempt it in Emily's\nstate of spirits.\"\n\n\"It would little benefit Emily's spirits to transfer a duty to a\nstranger,\" said the Colonel.  \"But I wish to know why you evidently think\nso ill of this girl, Barbara!\"\n\n\"Her entire behaviour since she has been with us\u2014\" began Lady Barbara.\n\n\"Generalities only do mischief, Barbara.  If I have any control over this\nchild, I must know facts.\"\n\n\"The truth is, Giles,\" said his sister, distressed and confused, \"that I\npromised the child not to tell you of her chief piece of misconduct,\nunless I was compelled by some fresh fault.\"\n\n\"An injudicious promise, Barbara.  You do the child more harm by implying\nsuch an opinion of her than you could do by letting me hear what she has\nactually done.  But you are absolved from the promise, for she has\nherself told me.\"\n\n\"Told you!  That girl has no sense of shame!  After all the pains I took\nto conceal it!\"\n\n\"No, Barbara; it was with the utmost shame that she told me.  It was\nunguarded of me, I own; but De la Poer and I had entirely forgotten that\nshe was present, and I asked him if he could account for your evident\ndislike and distrust of her.  The child's honourable feelings would not\nallow her to listen, and she came forward, and accused herself, not you!\"\n\n\"Before Lord de la Poer!  Giles, how could you allow it?\" cried Lady\nBarbara, confounded.  \"That whole family will tell the story, and she\nwill be marked for ever!\"\n\n\"De la Poer has some knowledge of child nature,\" said the Colonel,\nslightly smiling.\n\n\"A gentleman often encourages that sort of child, but condemns her the\nmore.  She will be a by-word in that family!  I always knew she would be\nour disgrace!\"\n\n\"O Giles, do tell Barbara it cannot be so very bad!\" entreated Lady Jane.\n\"She is such a child\u2014poor little dear!\u2014and so little used to control!\"\n\n\"I have only as yet heard her own confused account.\"\n\nLady Barbara gave her own.\n\n\"I see,\" said the Colonel, \"the child was both accurate and candid.  You\nshould be thankful that your system has not destroyed her sincerity.\"\n\n\"But, indeed, dear Giles,\" pleaded Lady Jane, \"you know Barbara did not\nwant her to say what was false.\"\n\n\"No,\" said the Colonel: \"that was a mere misunderstanding.  It is the\nspirit of distrust that\u2014assuming that a child will act dishonourably\u2014is\nlikely to drive her to do so.\"\n\n\"I never distrusted Katharine till she drove me to do so,\" said Lady\nBarbara, with cold, stern composure.\n\n\"I would never bring an accusation of breach of trust where I had not\nmade it evident that I reposed confidence,\" said the Colonel.\n\n\"I see how it is,\" said Lady Barbara; \"you have heard one side.  I do not\ncontradict.  I know the girl would not wilfully deceive by word; and I am\nwilling to confess that I am not capable of dealing with her.  Only from\na sense of duty did I ever undertake it.\"\n\n\"Of duty, Barbara?\" he asked.\n\n\"Yes\u2014of duty to the family.\"\n\n\"We do not see those things in the same light,\" he said quietly.  \"I\nthought, as you know, that the duty was more incumbent when the child was\nleft an orphan\u2014a burthen on relatives who could ill afford to be charged\nwith her.  Perhaps, Barbara, if you had noticed her _then_, instead of\nwaiting till circumstances made her the head of our family, you might\nhave been able to give her that which has been wanting in your otherwise\nconscientious training\u2014affection.\"\n\nLady Barbara held up her head, stiffly, but she was very near tears, of\npain and wounded pride; but she would not defend herself; and she saw\nthat even her faithful Jane did not feel with her.\n\n\"I came home, Barbara,\" continued the Colonel, \"resolving that\u2014much as I\nwished for Emily's sake that this little girl should need a home with\nus\u2014if you had found in her a new interest and delight, and were in\nher\u2014let me say it, Barbara\u2014healing old sores, and giving her your own\ngood sense and high principle, I would not say one word to disturb so\nhappy a state of things.  I come and find the child a state prisoner,\nwhom you are endeavouring by all means to alienate from the friends to\nwhom she owes a daughter's gratitude; I find her not complaining of you,\nbut answering me with the saddest account a child can give of herself\u2014she\nis always naughty.  After this, Barbara, I can be doing you no injury in\nasking you to concur with me in arrangements for putting the child under\nmy wife's care as soon as possible.\"\n\n\"To-morrow, if you like,\" said Lady Barbara.  \"I took her only from a\nsense of duty; and it has half killed Jane.  I would not keep her upon\nany consideration!\"\n\n\"O Barbara, it has not hurt me.\u2014O Giles, she will always be so anxious\nabout me; it is all my fault for being nervous and foolish!\" cried Lady\nJane, with quivering voice, and tears in her eyes.  \"If it had not been\nfor that, we could have made her so happy, dear little spirited thing.\nBut dear Barbara spoils me, and I know I give way too much.\"\n\n\"This will keep you awake all night!\" said Barbara, as the Colonel's\ntender gesture agitated Jane more.  \"Indeed, Giles, you should have\nchosen a better moment for this conversation\u2014on almost your first arrival\ntoo!  But the very existence of this child is a misfortune!\"\n\n\"Let us trust that in a few years she may give you reason to think\notherwise,\" said the Colonel.  \"Did you mean what you said\u2014that you\nwished us to take her to-morrow?\"\n\n\"Not to incommode Emily.  She can go on as she has done till your plans\nare made.  You do not know what a child she is.\"\n\n\"Emily shall come and settle with you to-morrow,\" said Colonel\nUmfraville.  \"I have not yet spoken to her, but I think she will wish to\nhave the child with her.\"\n\n\"And you will be patient with her.  You will make her happy,\" said Lady\nJane, holding his hand.\n\n\"Everything is made happy by Emily,\" he answered.\n\n\"But has she spirits for the charge?\"\n\n\"She has always spirits enough to give happiness to others,\" he answered;\nand the dew was on his dark lashes.\n\n\"And you, Giles\u2014you will not be severe even if the poor child is a little\nwild?\"\n\n\"I know what you are thinking of, Jane,\" he said kindly.  \"But indeed, my\ndear, such a wife as mine, and such sorrows as she has helped me to bear,\nwould have been wasted indeed, if by God's grace they had not made me\nless exacting and impatient than I used to be.\u2014Barbara,\" he added after a\npause, \"I beg your pardon if I have spoken hastily, or done you\ninjustice.  All you have done has been conscientious; and if I spoke in\ndispleasure\u2014you know how one's spirit is moved by seeing a child\nunhappy\u2014and my training in gentleness is not as complete as it ought to\nbe, I am sorry for the pain I gave you.\"\n\nLady Barbara was struggling with tears she could not repress; and at last\nshe broke quite down, and wept so that Lady Jane moved about in alarm and\ndistress, and her brother waited in some anxiety.  But when she spoke it\nwas humbly.\n\n\"You were right, Giles.  It was not in me to love that child.  It was\nwrong in me.  Perhaps if I had overcome the feeling when you first told\nme of it, when her mother died, it would have been better for us all.\nNow it is too late.  Our habits have formed themselves, and I can neither\nmanage the child nor make her happy.  It is better that she should go to\nyou and Emily.  And, Giles, if you still bring her to us sometimes, I\nwill try\u2014\"  The last words were lost.\n\n\"You will,\" he said affectionately, \"when there are no more daily\ncollisions.  Dear Barbara, if I am particularly anxious to train this\npoor girl up at once in affection and in self-restraint, it is because my\nwhole life\u2014ever since I grew up\u2014has taught me what a grievous task is\nleft us, after we are our own masters.  If our childish faults\u2014such as\nimpetuosity and sullenness\u2014are not corrected on principle, not for\nconvenience, while we are children.\"\n\nAfter this conversation, everyone will be sure that Mrs. Umfraville came\nnext day, and after many arrangements with Lady Barbara, carried off the\nlittle Countess with her to the house that Lord de la Poer had lent them.\n\nKate was subdued and quiet.  She felt that she had made a very unhappy\nbusiness of her life with her aunts, and that she should never see Bruton\nStreet without a sense of shame.  Lady Barbara, too, was more soft and\nkind than she had ever seen her; and Aunt Jane was very fond of her, and\ngrieved over her not having been happier.\n\n\"Oh, never mind, Aunt Jane; it was all my naughtiness.  I know Aunt Emily\nwill make me good; and nobody could behave ill in the house with Uncle\nGiles, could they now?  So I shall be sure to be happy.  And I'll tell\nyou what, Aunt Jane; some day you shall come to stay with us, and then\nI'll drive you out in a dear delicious open carriage, with two prancing\nponies!\"\n\nAnd when she wished her other aunt good-bye, she eased her mind by\nsaying, \"Aunt Barbara, I am very sorry I was such a horrid plague.\"\n\n\"There were faults on both sides, Katharine,\" her aunt answered with\ndignity.  \"Perhaps in time we may understand one another better.\"\n\nThe first thing Katharine heard when she had left the house with Mrs.\nUmfraville was, that her uncle had gone down to Oldburgh by an early\ntrain, and that both box and shawl had gone with him.\n\nBut when he came back late to Lord de la Poer's house, whom had he\nbrought with him?\n\nMary!  Mary Wardour herself!  He had, as a great favour, begged to have\nher for a fortnight in London, to take care of her little cousin, till\nfurther arrangements could be made; and to talk over with Mrs. Umfraville\nthe child's character, and what would be good for her.\n\nIf there was one shy person in the house that night, there was another\nhappier than words could tell!\n\nMoreover, before very long, the Countess of Caergwent had really seen the\nLord Chancellor, and found him not so very unlike other people after all;\nindeed, unless Uncle Giles had told her, she never would have found out\nwho he was!  And when he asked her whether she would wish to live with\nColonel Umfraville or with Lady Barbara and Lady Jane, it may be very\neasily guessed what answer she made!\n\nSo it was fixed that she should live at Caergwent Castle with her uncle\nand aunt, and be brought up to the care of her own village and poor\npeople, and to learn the duties of her station under their care.\n\nAnd before they left London, Mrs. Umfraville had chosen a very bright\npleasant young governess, to be a friend and companion, as well as an\ninstructress.  Further, it was settled that as soon as Christmas was\nover, Sylvia should come for a long visit, and learn of the governess\nwith Kate.\n\nThose who have learned to know Countess Kate can perhaps guess whether\nshe found herself right in thinking it impossible to be naughty near\nUncle Giles or Aunt Emily.  But of one thing they may be sure\u2014that Uncle\nGiles never failed to make her truly sorry for her naughtiness, and\nincreasingly earnest in the struggle to leave it off.\n\nAnd as time went on, and occupations and interests grew up round Colonel\nand Mrs. Umfraville, and their niece lost her childish wildness, and\nloved them more and more, they felt their grievous loss less and less,\nand did not so miss the vanished earthly hope.  Their own children had so\nlived that they could feel them safe; and they attached themselves to the\nchild in their charge till she was really like their own.\n\nYet, all the time, Kate still calls Mr. Wardour \"Papa;\" and Sylvia spends\nhalf her time with her.  Some people still say that in manners, looks,\nand ways, Sylvia would make a better Countess than Lady Caergwent; but\nthere are things that both are learning together, which alone can make\nthem fit for any lot upon earth, or for the better inheritance in Heaven.\n\n\n\n\n***","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \n# Nietzsche & the Political\n\nNietzsche's political thought has long been dismissed for its alleged naivet\u00e9 and its antiliberal excesses. Yet, far from being of merely historical interest, his critique of late modernity in fact suggests a compelling alternative to the political models advanced by liberal, communitarian and postmodern theorists.\n\nIn _Nietzsche & the Political,_ Daniel W.Conway takes Nietzsche seriously as a political thinker. Unlike other writers on the subject, Conway neither idolizes not demonizes. He carefully explores the consequences of Nietzsche's critique of modernity for his political thought from his earliest writings through to his mature work. Conway's clear and even-handed analysis is free from the obfuscatory jargon often associated with Nietzsche scholarship.\n\n_Nietzsche & the Political_ is a comprehensive introduction to Nietzsche's political thought. It also offers a thorough survey of Nietzsche's political legacy, including his influence on such seminal thinkers as Foucault and Habermas and his continuing importance to contemporary liberalism and feminist theory. It will be required reading for students of Nietzsche in philosophy, politics and sociology.\n\n**Daniel W.Conway** is Associate Professor of Philosophy and Director of the Center for Ethics and Value Inquiry at Pennsylvania State University. He is the co-editor of _The Politics of Irony_ and _Nietzsche und die antike Philosophie._\n\n# Thinking the Political\n\nGeneral editors:  \nKeith Ansell-Pearson, _University of Warwick_  \nSimon Critchley, _University of Essex_\n\nRecent decades have seen the emergence of a distinct and challenging body of work by a number of Continental thinkers that has fundamentally altered the way in which philosophical questions are conceived and discussed. This work poses a major challenge to anyone wishing to define the essentially contestable concept of 'the political' and to think anew the political import and application of philosophy. How does recent thinking on time, history, language, humanity, alterity, desire, sexuality, gender and culture open up the possibility of thinking the political anew? What are the implications of such thinking for our understanding of and relation to the leading ideologies of the modern world, such as liberalism, socialism and Marxism? What are the political responsibilities of philosophy in the face of the new world (dis)order?\n\nThis new series is designed to present the work of the major continental thinkers of our time, and the political debates their work has generated, to a wider audience in philosophy and in political, social and cultural theory. The aim is neither to dissolve the specificity of the 'philosophical' into the 'political' nor evade the challenge that 'the political' poses the 'philosophical'; rather, each volume in the series will try to show how it is only in the relation between the two that new possibilities of thought and politics can be activated.\n\nAlready published:\n\n  * Foucault and the Political _by Jon Simons_\n  * Derrida and the Political _by Richard Beardsworth_\n\n# Nietzsche & the Political\n\n## Daniel W.Conway\n\nFirst published 1997  \nby Routledge  \n11 New Fetter Lane, London EC4P 4EE\n\nThis edition published in the Taylor & Francis e-Library, 2005.\n\n\"To purchase your own copy of this or any of Taylor & Francis or Routledge's collection of thousands of eBooks please go to www.eBookstore.tandf.co.uk.\"\n\nSimultaneously published in the USA and Canada  \nby Routledge  \n20 West 35th Street, New York, NY 10001\n\n\u00a9 1997 Daniel W.Conway\n\nAll rights reserved. No part of this book may be reprinted or reproduced or utilized in any form or by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying and recording, or in any information storage or retrieval system, without permission in writing from the publishers.\n\n_British Library Cataloguing in Publication Data_  \nA catalogue record for this book is available from the British Library.\n\n_Library of Congress Cataloging in Publication Data_  \nConway, Daniel W.  \nNietzsche and the political\/Daniel W.Conway.  \np. cm.\u2014(Thinking the political)  \nIncludes bibliographical references and index.  \nISBN 0-415-10068-2 (Print Edition). ISBN 0-415-10069-0 (pbk.)  \n1. Nietzsche, Friedrich Wilhelm, 1844\u20131900\u2014Contributions in political science. I. Title. II. Series.  \nJC233.N52C65 1996  \n320\u2032.01\u2013dc20 96\u20137867 CIP\n\nISBN 0-203-97938-9 Master e-book ISBN\n\nISBN 0-415-10068-2 (hbk)  \nISBN 0-415-10069-0 (pbk)\n\n_For Shannon_\n\n# Acknowledgements\n\nThis book developed as a product of my friendship with Keith Ansell Pearson and David Owen, to whom I am deeply indebted for their encouragement and criticism. I am also grateful to Graham Parkes and Tracy Strong, both of whom read the entire manuscript and judiciously suggested salutary revisions. I am furthermore indebted to the many friends and colleagues who have discussed Nietzsche's political philosophy with me over the years, including Panos Alexakos, Babette Babich, Debra Bergoffen, Ann-Marie Bowery, Howard Caygill, William Connolly, Claudia Crawford, Simon Critchley, Brian Domino, Shannon Duval, Robert Gooding-Williams, Kathleen Higgins, Robert Irelan, Salim Kemal, Laurence Lampert, Duncan Large, Bernd Magnus, Alexander Nehamas, Kelly Oliver, Robert Pippin, Stanley Rosen, Richard Schacht, Alan Schrift, Charles Scott, Gary Shapiro, David Stern, John Seery, Robert Solomon, and Michael Zimmerman. The research for this book was made possible by a generous grant from the Research and Graduate Studies Office of the College of the Liberal Arts at The Pennsylvania State University; my special thanks to Dean Susan Welch and Associate Dean Raymond Lombra. Finally, I gratefully acknowledge permission to use portions of the following publications:\n\n\" _Das Weib an sich_ : The Slave Revolt in Epistemology,\" in _Nietzsche_ : _Feminism and Political Theory,_ ed. Paul Patton (London: Routledge, 1993), pp. 110\u2013129.\n\n\"Love's Labour's Lost: The Philosopher's _Versucherkunst,_ \" in _Nietzsche,_ _Philosophy and the Arts,_ eds Daniel W. Conway and Salim Kemal (Cambridge: Cambridge University Press).\n\n\"Autonomy and Authenticity: How One Becomes What One Is,\" _St. John's_ _Review,_ vol. XLII, no. 2, May 1994, pp. 27\u201339.\n\n\"Comedians of the Ascetic Ideal: The Performance of Genealogy,\" in _The_ _Politics of Irony: Essays in Self-Betrayal,_ eds Daniel W.Conway and John E.Seery (New York: St. Martin's Press, 1992), pp. 73\u201395.\n\n\"Foucault, Michel,\" _The Encyclopedia of Philosophy, Supplement,_ ed. David M.Borchert (Macmillan and Co.), pp. 201\u2013202.\n\n# List of abbreviations\n\nAll references to Nietzsche's works appear in the body of the text; individual writings are identified by the abbreviations listed below.\n\nSee Table\n\nRather than giving page references to any one particular edition, I have adopted a system that is widely used in Nietzsche scholarship and allows readers to identify the passages cited whatever edition they may be using:\n\n  1. All arabic numbers denote sections. For instance, GS 238 refers to section 238 of _The Gay Science._\n  2. All roman numbers denote parts or standard subdivisions in those works of Nietzsche where section numbers start anew with each part or subdivision. Thus GM 11:21 refers to the _Genealogy of Morals,_ essay II, section 21; and TI VII:3 refers to _Twilight of the Idols,_ essay VII, section 3.\n  3. Citations from the preface of a particular work are identified by \"P\", as in EH P:4\u2014a reference to _Ecce Homo,_ preface, section 4.\n\nOccasionally I have thought it helpful to include parts of the original German text from the standard critical edition of Nietzsche's works (Friedrich Nietzsche, _S\u00e4mtliche Werke: Kritische Studienausgabe in 15 B\u00e4nden,_ ed. G.Colli and M.Montinari, Berlin: de Gruyter\/Deutscher Taschenbuch Verlag); these German passages appear in square brackets.\nAC | _The Antichrist(ian),_ trans. Walter Kaufmann, in _The Portable Nietzsche,_ ed. and trans. Walter Kaufmann (New York: Viking Penguin, 1982).  \n---|---  \nBGE | _Beyond Good and Evil,_ trans. Walter Kaufmann (New York: Random House, 1966).  \nBT | _The Birth of Tragedy,_ trans. Walter Kaufmann (New York: Random House, 1967).  \nCW | _The Case of Wagner,_ trans. Walter Kaufmann (New York: Random House, 1967).  \nD | _Daybreak,_ trans. R.J.Hollingdale (Cambridge: Cambridge University Press, 1982).  \nEH | _Ecce Homo,_ trans. Walter Kaufmann (New York: Random House\/Vintage Books, 1989).  \nGM | _On the Genealogy of Morals,_ trans. Walter Kaufmann and R.J.Hollingdale (New York: Random House\/Vintage Books, 1989).  \nGS | _The Gay Science,_ trans. Walter Kaufmann (New York: Random House, 1974).  \nH | _Human, All-Too-Human,_ trans. R.J.Hollingdale (Cambridge: Cambridge University Press, 1986).  \nNCW | _Nietzsche Contra Wagner,_ trans. Walter Kaufmann, in _The Portable Nietzsche,_ ed. and trans. Walter Kaufmann (New York: Viking Penguin, 1982).  \nSE | _Schopenhauer as Educator,_ trans. R.J.Hollingdale, in _Untimely Meditations,_ trans. R.J.Hollingdale, intr. J.P.Stern (Cambridge: Cambridge University Press, 1983).  \nTI | _Twilight of the Idols,_ trans. Walter Kaufmann, in _The Portable Nietzsche,_ ed. and trans. Walter Kaufmann (New York: Viking Penguin, 1982).  \nWP | _The Will to Power,_ trans. Walter Kaufmann and R.J.Hollingdale (New York: Random House, 1967).  \nZ | _Thus Spoke Zarathustra,_ trans. Walter Kaufmann, in _The Portable Nietzsche,_ ed. and trans. Walter Kaufmann (New York: Viking Penguin, 1982).\n\n# Introduction: Voyage of the Damned?\n\n> At long last the horizon appears free to us again, even if it should not be bright; at long last our ships may venture out again, venture out to face any danger; all the daring of the lover of knowledge is permitted again; the sea, _our_ sea, lies open again; perhaps there has never yet been such an \"open sea.\"\n> \n> \u2014 _The Gay Science,_ 343\n\nNietzsche tends to cast modernity in a blindingly negative light, alternately describing it in terms of the onset of European nihilism, the inexorable spread of decadence, the advent of the last will, the flaccid reign of the last man, the twilight of the idols, and so forth. But the shipwreck of modernity also produces in him a cathartic, liberating effect, granting him a measure of freedom from the superlative (albeit fading) values of the age. The horizon of modernity may not be bright, but it is at long last free, and Nietzsche hopes to exploit this freedom to impress his signature onto the successor age to modernity.\n\nNietzsche's contributions to politics, and to political philosophy, are notoriously difficult to reckon. He not only stands in defiant opposition to the general political trends of modernity, but also refuses the \"scientific\" methodologies preferred by his contemporaries. Deeply contemptuous of the reluctant advocates, unwitting valets, and involuntary memoirists who pose as original thinkers, he never undertakes to deliver a solemn, sonorous treatise on politics. Understandably wary of philosophical system-building, he conveys his political insights via lightning epigrams and apothegmatic proclamations, generally ignoring the quaint Alexandrian custom of furnishing evidence, arguments, and justifications. While his contemporaries celebrate the triumphs of the new _Reich_ or frolic in the surging tide of democratic reforms, he scours the premodern world for sober realists and exemplars of political wisdom. He chooses as his interlocutors such untimely figures as Homer, Manu, Thucydides, Socrates, Plato, Epicurus, Caesar, and St. Paul.\n\nIt is now a commonplace for scholars to attribute the difficulty of Nietzsche's political thinking to his writerly styles, experimental masks, pagan irreverence, antiquarian prejudices, arrested naivet\u00e9, resentment of modernity\u2014even to the palpable dissatisfactions of his personal\/sexual\/ emotional\/psychological life. These commonly cited idiosyncrasies collectively point, however, not so much to the difficulty of his political thinking, as to its inferiority. Nietzsche is commonly received as an incisive critic, or as an _agent provocateur,_ but not as a political philosopher of the first rank. He is an erratic, iconoclastic genius, whose prurient excesses we might contemplate in hygienic detachment (perhaps as a naughty diversion from our more serious work in political philosophy), but in the end he is utterly harmless to the prevailing idols of modernity. Prematurely dismissive of the democratic reforms and liberal ideals that define the highest achievements and aspirations of the age, he has nothing constructive to say to us about political life in late modernity. An outrageous critic, to be sure, but undeniably second-rate, and perhaps downright naive.\n\nThe difficulty of Nietzsche's political thinking is attributable not to any personal or epistolary quirks, but to its unusually grandiose scope. He wishes to return to the very ground of politics itself, to excavate the site of politics, and to retrieve the founding question of politics. He consequently has no use for the small-minded pomposity that often passes in modernity for political thinking. He is quite content to leave the details of government, regulation, production and distribution to his fussy German contemporaries. Although his philosophy has spawned many of the revolutionary, antifoundational insights that continue to contour postmodern and post-structuralist thought, his political thinking remains unmistakably modern (or even premodern) in its orientation and design. In fact, his political philosophy bears a closer resemblance to the conservative republicanism of his predecessors than to the progressive liberalism of his contemporaries.\n\nWhile most representatives of modernity are content to confect self-congratulatory justifications for its misguided projects, Nietzsche is inclined to ask after modernity itself: does it warrant the future of humankind? What might be made of its modest successes and colossal follies? What, if anything, might follow in its turbulent wake? Unless we raise such basic, decisive questions, politics amounts to nothing more than busy work for the petty managers and bureaucrats whom modernity produces in such sterile abundance.\n\nUnlike those prudent seafarers who seek shelter and anchorage, Nietzsche relishes the danger of voyages on the \"open seas.\" No other critic of modernity has dared to venture so far from the _terra firma_ of a (supposedly) foundational critical standpoint. No other seafarer has so boldly\u2014and foolishly\u2014renounced conventional routes and instruments of navigation:\n\n> We sail right _over_ morality, we crush, we destroy perhaps the remains of our own morality by daring to make our voyage there\u2014but what matter are _we!_ (BGE 23)\n\nTo pursue Nietzsche's critique of modernity is to set sail on uncharted seas. His odyssey transports him to various ports of call, none more exotic than the _terra_ _incognita_ of political legislation. Taking advantage of the palpable degeneration of modern political institutions, he dares to raise a calamitous, and previously unapproachable, question of political legislation: _what ought humankind to_ _become?_\n\nAlthough this might fairly be viewed as the founding question of politics, to which all political thinkers and legislators ought carefully to attend, Nietzsche insists that it is in fact rarely considered at all. This neglect is partially attributable to historical circumstances, for such questions can be raised only in the twilight of an age, when widespread failure and dissolution call into question the very meaning of human existence. That Nietzsche can raise the founding question of politics thus constitutes sufficient proof that previously satisfying justifications of human existence are no longer viable. This is no idle question, raised to satisfy an academic curiosity: as modernity stumbles toward exhaustion, the last will of humankind, \"the will to nothingness,\" looms on the horizon.\n\nThe prospect of the \"will to nothingness\" points to another reason for the pandemic neglect of the founding question of politics. Raised only in those historical periods in which humankind no longer feels worthy of its past glories, this question presupposes neither an affirmation nor a confirmation of the future of the species. Indeed, Nietzsche does not assume in advance of his daring voyages that humankind necessarily ought to become anything at all. He retrieves the founding question of politics in order to call humankind itself, and its future, into question. In light of the pervasive decay of modernity, he asks, should humankind capitulate to its \"will to nothingness\"? Or should political legislators devise measures to ensure the survival of humankind? If so, at what future expense to the species as a whole? At stake here is nothing less than the justification of humankind itself, the warrant for its future as a viable, thriving species.\n\nOf course, merely raising the founding question of politics implies neither one's willingness, nor one's capacity, to venture a definitive answer. Having raised the question, one might judge the extant responses to be adequate, or recoil in horror from the weight of the acquired responsibility, or defer the question to \"others\" (mortal or divine), or promptly shelve the question altogether. While Nietzsche is often criticized for failing to deliver a detailed articulation of \"his\" vision of the future of humankind, it is not clear that this is, or should be, his political task. Owing to the unique historical conditions under which this question becomes both intelligible and meaningful, those who would raise the question are in no position to answer it with any degree of specificity. Just as the crepuscular flight of the Owl of Minerva seals, for Hegel, the practical impotence of reason and understanding, so Nietzsche's attention to the _question_ of modernity signals the irreversible decline of modernity itself. Indeed, since his own critical perspective is tinctured by the decadence that besets modernity, we should receive any specific answer he might venture to the founding question of politics with heightened suspicions.\n\nYet the inherent danger of raising\u2014much less answering\u2014the founding question of politics is nevertheless grave, for one thereby glimpses the shores of that undiscovered country that Nietzschean cartography locates \"beyond good and evil.\" Once raised, this question cannot be returned to oblivion, and it must change us forever\u2014even if we refuse to answer it. Nietzsche likens the advent of European nihilism to the arrival at one's door of an \"uncanny\" solicitor, who demands entry into one's home, claiming it to be his home as well (WP 1). Just as one may choose to ignore the entreaties of this persistent guest, so one may choose either to refuse the founding question of politics, or to pretend that this question has already received a final, definitive answer. Toward this end, Nietzsche helpfully catalogs the various tricks, therapies, and penances devised over the years to distract human beings from the founding question of politics. What one may _not_ choose, however, is never to have heard this guest at one's door, never to have shunted off onto others the responsibility for determining the future of humankind.\n\nWhile a serious consideration of the founding question of politics need not commit one to a perfidious eugenics project, or to illiberal social engineering, it _does_ commit one to a potentially crippling dalliance with Nietzsche's \"immoralism.\" As astute critics have nervously warned throughout the twentieth century, _nothing Nietzsche says_ definitively rules out the illiberal political regimes with which his name has been linked. He neither discerns nor acknowledges any prima facie restrictions on the type of answer a lawgiver might formulate to the founding question of politics. According to Nietzsche, political lawgivers are bound in their deliberations by no moral considerations whatsoever \u2014all of which have been cast adrift in the passage beyond good and evil\u2014but only by a fidelity to their own respective visions of the future of humankind. The supposed priority of a liberal response to this question, a response that aims to secure the future of humankind by eliminating suffering and promoting individual freedom, is \"merely\" an accident of history, which attests more convincingly to the advance of decadence than to the political merits of liberalism itself.\n\nNietzsche proffers no assurance (and certainly no hope) that he will respect the liberal ideals of modernity, for he views the advent of the \"will to nothingness\" as a greater danger than the demise of liberalism. He aims simply to secure the future of the species, hoping to forestall the \"suicidal nihilism\" that threatens humankind. In light of the prevailing historical conditions of his political thinking, his account of his own \"destiny\" is perhaps fitting after all:\n\n> I know my fate. One day my name will be associated with the memory of something tremendous\u2014a crisis without equal on earth, the most profound collision of conscience, a decision that was conjured up _against_ everything that had been believed, demanded, hallowed so far. I am no man, I am dynamite... It is only beginning with me that the earth knows _great_ _politics._ (EH XIV:1)\n\n# 1  \nPolitical Perfectionism\n\n> I hear with pleasure that our sun is swiftly moving toward the constellation of _Hercules_ \u2014and I hope that man on this earth will in this respect follow the sun's example? And we first of all, we good Europeans!\n> \n> \u2014 _Beyond Good and Evil,_ 243\n\nNietzsche's attempt to retrieve the founding question of politics reflects his conviction that it is the business of politics to legislate the conditions of the permanent enhancement of humankind (BGE 257). Humankind is best enhanced, he believes, not through the Whiggish reforms and liberal ideals favored by modernity, but through the cultivation of those rare individuals who body forth an expanded complement of human powers and perfections. He consequently recommends that social resources should be reserved and mobilized for the production of great human beings.\n\nAs we have seen, Nietzsche treats the founding question of politics as a philosophical question of ultimate justification or legitimation. He thus asks: in what incarnation, if any, might humankind justify its continued existence and warrant its unsecured future? It is important here to bear in mind the historical context of Nietzsche's critical enterprise. In a famous note from 1886, he confirms the advent of European nihilism. This means, he explains, that humankind itself lacks an aim or purpose that might redeem the suffering endemic to its very existence: \"What does nihilism mean? _That the highest_ _values devaluate themselves._ The aim is lacking; 'why?' finds no answer\" (WP 2). A justification of human existence is furnished by any aim (or goal or purpose), whose pursuit promises to enable human beings to endure the suffering of their meaningless existence. In lieu of some such aim, human beings might be forced to find meaning for themselves in their own self-annihilation, in the will never to will again.\n\nHere we should note that Nietzsche does not automatically assume either that he will arrive at some such justification, or that human existence should necessarily continue. The \"highest values\" ever attained by Western civilization have now \"devaluated themselves.\" He must consequently begin anew, as it were, in the quest for a goal that might redeem humankind as a whole. Since he too is implicated in the besetting decadence of modernity, he is not optimally appointed to create new values and erect new ideals. For all of his celebrated love of life and _amor fati,_ moreover, he is also deeply impressed by the thanatonic wisdom of Silenus, who counseled his captors to retreat immediately into the unquenchable an aim or goal that might actually warrant the future of humankind, stream of the Dionysian _Ur-eine_. Nietzsche consequently seeks to discover rather than merely prolong the miserable existence of a dying, misbegotten species. As he sees it, humankind needs an erotogenic goal to galvanize the will, a promise of the future that would renew our confidence in the continued development of the species.\n\nThroughout his productive career, Nietzsche's political thinking centers around a simple, yet powerful, thesis: human existence is justified only by the presence of those exemplary individuals who re-define the horizons of human perfectibility. In perhaps his most (in)famous articulation of this thesis, he explains that\n\n> We ought really to have no difficulty in seeing that, when a type _[Art]_ has arrived at its limits and is about to go over to a higher type, the goal of its evolution lies not in the mass of its exemplars and their wellbeing, let alone in those exemplars who happen to come last in point of time, but rather in those apparently scattered and chance existences which favorable conditions have here and there produced... For the question is this: how can your life, the individual life, receive the highest value, the deepest significance? How can it be least squandered? Certainly only by living for the advantage of the rarest and most valuable exemplars _[du zum Vortheile_ _der seltensten und werthvollsten Exemplare lebst],_ and not for the advantage of the majority, that is to say those who, taken individually, are the least valuable exemplars. (SE 6)\n\nWhile it might be tempting to dismiss this passage (written in 1874) as a youthful indiscretion, a perusal of Nietzsche's later writings reveals a persistent fascination with the central political role played by superlative human beings. In one of his last books he proclaims that\n\n> The problem I thus pose is not what shall succeed mankind in the sequence of living beings (man is an _end_ ), but what type of man shall be _bred_ , shall be _willed_ , for being higher in value, worthier of life, more certain of a future. Even in the past this higher type has appeared often\u2014but as a fortunate accident, as an exception, never as something _willed._ (AC 3)\n\nAt the center of Nietzsche's political thinking thus stands his commitment to the position known as _perfectionism,_ which constitutes his general answer to the founding question of politics. He locates the sole justification of human existence in the continued perfectibility of the species as a whole, as evidenced by the pioneering accomplishments of its highest exemplars. In _Schopenhauer as_ _Educator,_ for example, he argues that it is the primary task of culture itself to oversee the production of great human beings:\n\n> It is the fundamental idea of _culture,_ insofar as it sets for each one of us but one task: _to promote the production of the philosopher, the artist, and_ the saint within us and without us and thereby to work at the perfecting of _Nature._ (SE 5)\n\nTranslating this \"fundamental idea\" into more familiar political terms, he insists that \"humankind ought to seek out and create the favorable conditions under which those great redemptive men can come into existence\" (SE 6). In order to correct for the profligacy of Nature, political legislation must ensure the conditions of the emergence of true genius. In this (relatively) early essay, Nietzsche advocates the precise social conditions\u2014including hardship, neglect, material disadvantage and institutional indifference\u2014under which both Schopenhauer and he emerged as philosophers.\n\nSince human existence derives enduring meaning only through the exploits of its rarest and most exotic specimens, the task of politics is to legislate the conditions under which such exemplars will most likely emerge. This task is by no means simple, for, as Nietzsche indicates in the passage cited above, exemplary human beings usually emerge only by accident, as \"lucky strikes\" on the part of careless peoples and cultures. The political lawgivers he envisions must consequently legislate against the indifference of Nature itself:\n\n> The accidental, the law of absurdity in the whole economy of humankind, manifests itself most horribly in its destructive effect on the higher men whose complicated conditions of life can only be calculated with great subtlety and difficulty. (BGE 62)\n\nHe thus describes the enormity of the task that awaits the \"new philosophers,\" to whom he entrusts the future of humankind:\n\n> To teach man the future of man as his _will_ , as dependent on a human will, and to prepare great ventures and over-all attempts _[Gesammt-Versuche]_ of discipline and cultivation by way of putting an end to that gruesome dominion of nonsense and accident that has so far been called \"history.\" (BGE 203)\n\nWith his \"help,\" Nietzsche believes, the successor epoch to modernity might suspend this cowardly reliance on chance and resolutely attend to the \"breeding\" of exemplary human beings.\n\nNietzsche's childlike fascination with the heroic exploits of world-historical figures is attributable to their respective contributions to the enhancement of humankind as a whole. Thucydides, Caesar, Michel-angelo, Napoleon, Goethe, Bizet, and so on\u2014all represent irreversible advancements on the part of humankind as a whole. In the prodigious shadow cast by this higher humanity, the meaning and value of human existence can never revert to the (anachronistic) standards revered in bygone ages. Like those intrepid wards of Prometheus, whose plucky accomplishments with the divine flame won from Zeus a stay of execution, this higher humanity confers a measure of dignity and grace onto an otherwise undistinguished species. The dice-throwing gods may continue to laugh at the folly of their puny human playthings, but they are sufficiently intrigued by these specimens of higher humanity to renew the spectacle. Even Christianity, that great leveler of humankind and enemy of perfectionism, recognizes the need to single out particular saints and martyrs as exemplary specimens of faith, piety, and suffering.\n\nA significant disadvantage of the term \"perfectionism\" is its misleading connotation of a _final_ perfection or completion of the species. While it is true that great human beings continually exceed the achievements of their predecessors, these transfigurative exploits are both chaotic and unpredicted; they expand the horizon of human perfectibility along any number of unanticipated planes and vectors. The enactment of previously unknown human perfections is furthermore not immediately visible in its full relief; centuries, even millennia, may pass before humankind as a whole acknowledges the unparalleled achievements of its highest exemplars. Any attempt to identify in advance the _final_ perfection of the human soul thus amounts to nothing more than an exercise in idealism, which Nietzsche comes to view in his post-Zarathustran writings as the philosophical antipode to his own \"realism\" (EH 11: 10).\n\nBased on his careful observations of human \"nature\" and history, Nietzsche assumes that the species as a whole is both dynamic and evolving. As far as he knows, humankind neither progresses inexorably toward some preordained omega point, nor fulfills a cosmic destiny that consigns the weak and infirm to a premature extinction. Through the signal exploits of its highest representatives, humankind reaches ever beyond itself, but it reaches for no pre-established goal or _telos_. Each successive transfiguration further limns the unknown depths and reaches of the human soul. Indeed, Nietzsche's perfectionism is at all intelligible only in the event that the human soul is in fact predicated of sufficient plasticity to accommodate the completion and perfection he envisions.\n\nThe emergence of great human beings contributes to the enhancement of humankind both directly, by advancing the frontier of human perfectibility, and indirectly, by encouraging (some) others to flourish as well. The ethical life of any thriving community draws its sustenance and vitality from such individuals, and it cannot survive without them. Far from the mere ornaments to which they have been reduced in late modernity, superlative human beings are in fact responsible for the catalysis of culture itself. Nietzsche adamantly maintains that \"only he who has attached his heart to some great man is by that act _consecrated_ _to culture_ \" (SE 6). He later maintains, apparently with no hyperbole intended, that\n\n> A people is a detour of Nature to get to six or seven great men.\u2014Yes, and then to get around them. (BGE 126)\n\nSuperlative human beings contribute to an enhancement of the species as a whole, for they embody, and thus reveal, heretofore unknown perfections resident within the human soul. By continually expanding the complement of extant human perfections, these exemplars confer upon the species as a whole a quasi-divine status, an ephemeral intimation of immortality.\n\nGreat human beings accomplish the catalysis of culture not as a consciously articulated goal, but as an indirect and unintended by-product of their \"private\" pursuits of self-perfection. While they directly enhance the lives only of themselves and those select few who share their refined aesthetic sensibilities, they indirectly enhance the lives of all who are even minimally invested in the project of culture. Indeed, everyone who enters \"the circle of culture\" stands to benefit from the production of exemplary human types, for a justification of human existence would be impossible in their absence. Hence the central paradox of Nietzsche's perfectionism: the enhancement of humanity and the enrichment of ethical life are dependent upon the exploits of \"immoral\" exemplars who hold no conscious or intentional stake in the lives of those whom they succor and renew. In fact, he insists, these exotic specimens must be allowed (and indeed encouraged) to free themselves from the chains of conventional morality if they are to contribute to the permanent enhancement of humankind.\n\nAn exemplary human being thus embodies a concrete way of life, a set of situated practices that not only demonstrate the perfectibility of the human soul, but also remind (some) others of the powers and perfections resident within themselves. One such exemplar, Nietzsche suggests, is the (pre-Pauline) Jesus, who bodied forth a \"deep instinct for how one must _live_...a new way of life, _not_ a new faith\" (AC 33). The redemptive and justificatory powers of these exemplary human beings are aptly expressed in the dexter king's unsolicited paeon to Zarathustra:\n\n> Nothing more delightful grows on earth, O Zarathustra, than a lofty, strong will: that is the earth's most beautiful plant. A whole landscape is refreshed by one such tree... Your tree here, O Zarathustra, refreshes even the gloomy ones, the failures; your sight reassures and heals the heart even of the restless. (Z IV:11)\n\nEven a decadent people or epoch stands to be renewed by the exploits of its representative exemplars. Reeling from the mediocrity and degeneration that make him \"weary\" of humankind as a whole, Nietzsche hopes to steal a tonic glimpse of \"a man who justifies _humankind,_ of a complementary and redeeming lucky strike on the part of humankind for the sake of which one may still _believe_ _in humankind!_ \" (GM 1:12) While these decadent \"heroes\" are not likely to be confused with the commanders and conquerors who populate vital epochs, they nevertheless serve to excite confidence in the future of humanity. One such \"hero\" is Aristophanes, whom Nietzsche describes as\n\n> that transfiguring, complementary spirit for whose sake one _forgives_ everything Hellenic for having existed, provided one has understood in its full profundity _all_ that needs to be forgiven and transfigured here. (BGE 28)\n\nThe example of Aristophanes is pertinent not only because Nietzsche too must negotiate the shades and shadows of a twilight epoch, but also because Aristophanes, the irreverent scourge of our beloved Socrates, does not resemble the familiar heroes of Greek antiquity. If Nietzsche is to introduce his readers to the representative exemplars of late modernity, then he must somehow divert our attention from traditional models of heroism, which are no longer applicable. In a preliminary education of his readers' sensibilities, he thus prefers Aristophanes to a more commonly revered contemporary:\n\n> Nothing...has caused me to meditate more on _Plato's_ secrecy and sphinx nature than the happily preserved _petit fait_ that under the pillow of his deathbed there was found no \"Bible,\" nor anything Egyptian, Pythagorean, or Platonic\u2014but a volume of Aristophanes. How could even Plato have endured life\u2014a Greek life he repudiated\u2014without an Aristophanes? (BGE 28)\n\n### The Lawgiver\n\nThe term \"perfectionism\" carries an indelibly negative connotation, but it accurately focuses our attention on the vital core of Nietzsche's political thinking. His commitment to perfectionism is perhaps best understood as the product of his attempt to accede to the perspective of the _lawgiver,_ who aspires to attain (and perhaps to implement) a panoptic vision of the future of humankind.\n\nThe lawgiver plays a unique role within the economy of Nietzsche's political thinking. Lawgivers are typically _not_ rulers, and they only rarely gain influence over actual rulers. That the lawgiver is typically ignored by modern rulers constitutes Nietzsche's general objection to modern politics, which succeeds largely in presenting the aimlessness and indolence of modernity as princely virtues. (Whether premodern rulers were more appreciative of the wisdom of the lawgiver, as Nietzsche occasionally suggests, remains to be demonstrated.) While actual rulers usually attend only to the local exigencies of personal or popular aggrandizement, the lawgiver attempts to legislate on behalf of humanity as a whole. Hence Nietzsche's attempt to retrieve the founding question of politics: what ought humankind to become?\n\nWhile most rulers formulate and justify their legislations by appealing to the prosperity of a particular people or polity over a specific, short-term duration, the lawgiver appeals exclusively to the permanent enhancement of humankind as a whole. Legislating from an \"immoral\" perspective beyond good and evil, the lawgiver cannot be concerned with (or even acknowledge) the \"rights\" and \"freedoms\" of individual tribes and peoples, much less those of individual human beings; nothing less than the future determination of the species is at stake. When appealing to the hyperopic perspective of the lawgiver, Nietzsche consequently sounds monstrously cold and cruel, especially to his liberal audiences of the twentieth century. Such is the nature not of the man himself, but of the \"immoral\" perspective he adopts as a political thinker. He too cares, in his own way, about distributive justice, social welfare, moral education, and other hallmarks of modern political life, though he neither ascribes to these goals the highest political priority, nor thinks them a worthy challenge for his prodigious intellectual gifts. _Qua_ lawgiver, no one can be concerned with the particular lives of individual human beings.\n\nCritics often respond that the \"immoral\" standpoint of the lawgiver is simply the wrong perspective for political thinkers to adopt. It is often remarked, in fact, that Nietzsche attempts thereby to usurp divine authority, daring to consider a question that mere mortals are neither meant nor fit to raise. The charge of impiety is essentially valid, but it is most helpful in framing the historical context of his political thinking. So long as superlative values and metaphysical systems perdure, there is no need, and no opportunity, to raise the founding question of politics. In the absence of any supernatural or metaphysical source of meaning, however, humankind must create for itself sufficient reason for its continued, imperiled existence. Meaning of this magnitude, Nietzsche believes, derives only from the \"heroic\" exploits of the highest exemplars of the human species. No other means of securing meaning for human existence is currently feasible. When the gods falter or flee, mere mortals must step into the breach.\n\nAlthough the secular, anthropocentric justification that Nietzsche promises will not be sufficient for those wretched souls who remain inured to the \"metaphysical comforts\" dispensed by Platonism and Christianity, it is the only mode of justification that it is possible to obtain in the shadow of the dead God. This is not to say, however, that the advent of nihilism marks the end of metaphysics and supernaturalism. Like his predecessor obituarist, the Madman (GS 125), Nietzsche realizes that his cognitive insight-into the death of God carries no volitional charge. While he still hopes, fatuously, to occasion a miraculous transformation in his God-fearing readers, he also concedes that \"given the way of men, there may still be caves for thousands of years in which [God's] shadow will be shown\" (GS 108). Most human beings continue to prostrate themselves before the rotting corpse of the fallen god, either indifferent to, or enchanted by, its cadaverous stench. Even Nietzsche himself occasionally (if surreptitiously) pays his respects to the existentially challenged deity, appealing reverentially to \"divine\" truths and idols that he expressly disallows to others.\n\n### The Indeterminate Animal\n\nThe task of \"great politics\" is neither to destroy nor to transcend the all-too-human within us, but to bring the all-too-human to completion and perfection. But whence the need for the \"perfection\" of the species at all? Why is humankind in general, in any of its historical incarnations, dependent upon the redemptive exploits of its \"highest\" specimens? Why are the labors of \"ordinary\" human beings, however modest or clumsy they may be, insufficient to warrant the future of the species?\n\nNietzsche's earliest attempts to provide adequate answers to these questions pointed to the failure of Nature to preside over the timely production of exemplary human beings. Nature on its own is a \"bad economist\" (SE 7). If allowed to pursue the dilatory schedule to which it is accustomed, Nature would continue to produce great human beings, but always as unforeseen accidents, and never with the frequency and regularity that Nietzsche deems necessary for the healthy renewal of culture:\n\n> Nature wants always to be of universal utility, but it does not know how to find the best and most suitable means and instruments for this end... Nature is just as extravagant in the domain of culture as it is in that of planting and sowing. It achieves its aims in a broad and ponderous manner: and in doing so it sacrifices much too much energy. (SE 7)\n\nThe goal of politics, on this early account, is to assist Nature in attaining more efficiently the ends at which it consistently, if heedlessly, aims. The lawgiver must consequently intervene to arrange for a more productive distribution of Nature's (dis)array of resources. In a moment of feckless serendipity that he will later disown, Nietzsche presents the perfection of Nature and the perfection of humankind as dovetailing harmoniously in the production of the exemplary human being (SE 5).\n\nHe thus conceived of his early perfectionism as a relatively unobtrusive campaign to assist Nature in its dawdling production of exemplary human beings, and so as a contribution to the perfection of Nature itself. He later changes his mind on this point, attempting in his post-Zarathustran period to correct for the anthropocentric bias of his early writings. Although his account of the precise relationship between _nomos_ and _physis_ remains tricky throughout his productive career, he now situates Nature beyond good and evil. Nature appears no longer as a \"bad\" economist, but as an indifferent one, exhibiting no discernible attunement either to human interests or to human designs (BGE 9).\n\nNietzsche's early \"answer\" furthermore begs the question of the _need_ for redemptive human beings in the first place. If it is true that humankind requires _some_ of these heroes for its continued survival and justification, then it certainly stands to reason that _more_ of them might be desirable. But what, exactly, is involved in the process of \"perfection\" that these exemplary human beings must continue and guide? In what precise respects does humankind stand imperfect and incomplete?\n\nAs his critics often remark, Nietzsche does not present an adequately specific account of his vision of the eventual perfection of humankind. Yet he does provide a general sketch of the \"completion\" he has in mind. Here he draws from his speculative forays into philosophical anthropology, which assume ever greater importance in his post-Zarathustran writings. Vowing to proffer strictly naturalistic explanations for allegedly supernatural phenomena, he attributes the incompleteness of humankind not to some \"original sin\" or fall from \"grace,\" but to its desperate reliance on consciousness as an organ of internal regulation. Unlike all \"natural\" animals (including their pre-moral, hominid ancestors), human beings have forcibly renounced the pre-reflective guidance afforded them by their unconscious drives and impulses. In a pioneering insight that Freud would later borrow, Nietzsche traces the discontents of humanity to\n\n> the serious illness that man was bound to contract under the stress of the most fundamental change he ever experienced\u2014that change which occurred when he found himself finally enclosed within the walls of society and of peace. The situation that faced sea animals when they were compelled to become land animals or perish was the same as that which faced these semi-animals, well adapted to the wilderness, to war, to prowling, to adventure: suddenly all their instincts were disvalued and \"suspended.\"... [T]hey were reduced to thinking, inferring, reckoning, coordinating cause and effect, these unfortunate creatures; they were reduced to their \"consciousness\", their weakest and most fallible organ! (GM 11:16)\n\nThis violent transition to the peace and tranquillity of civil society left the human animal incomplete and indeterminate. Having refused Nature's original determination of its destiny, the human species must forge a destiny of its own, with unreliable consciousness as its only guide:\n\n> [M]an is more sick, uncertain, changeable, indeterminate _[unfestgestellter]_ than any other animal, there is no doubt of that; he is the _sick_ animal... [H] ow should such a courageous and richly endowed animal not also be the most imperiled, the most chronically and profoundly sick of all sick animals? (GM III:13)\n\nNietzsche consequently defines man as the _\"indeterminate animal [nicht_ _festgestellte Thier]\"_ (BGE 62), for only the human animal actively participates (though not always voluntarily and constructively) in the determination of its full complement of powers and perfections.\n\nThe transition from natural animal to human animal has been both painful and protracted, and it is by no means complete. In order to impress consciousness into service as a guide to living within the walls of civilization, human beings are obliged to direct _inward_ the instinctual energy that they would \"naturally\" discharge toward the external world:\n\n> All instincts that do not discharge themselves outwardly _turn inward_ \u2014this is what I call the _internalization_ of man: thus it was that man first developed what was later called his \"soul.\"... Hostility, cruelty, joy in persecuting, in attacking, in change, in destruction\u2014all this turned against the possessors of such instincts: _that_ is the origin of the \"bad conscience.\" (GM 11:16)\n\nThe introjection of instinctual energy thus results in the pain of the \"bad conscience,\" which Nietzsche views as the non-negotiable, non-refundable cost incurred by all human animals upon entering the shelter of civilization. The onset of this \"illness\" initiates the ongoing transition from natural animal to human animal, forcibly investing human beings with the interiority that alone \"makes them interesting.\"\n\nIn addition to the insecurify of relying on a relatively inefficient organ of internal regulation, human animals must also secure for themselves some measure of relief from the affliction of their bad conscience. This relief usually arrives via the ascetic ideal, which bids human beings to accept the pain of the bad conscience as a just (albeit partial) punishment for their persistent incompleteness. The ascetic ideal thus \"relieves\" the pain of the bad conscience by pronouncing the guilt or indebtedness _[Schuld]_ of all human animals. Yet this moral\/metaphysical interpretation of the bad conscience only exacerbates the suffering of the \"guilty\" parties, who must attempt to repay their debts through the practice of self-inflicted cruelty. Under the aegis of the ascetic ideal, human beings _blame_ themselves for their misery, compounding the (involuntary) suffering of the bad conscience with the (voluntary) suffering of guilt. The ascetic ideal thus pretends to still the _existential_ suffering associated with the bad conscience by superposing upon it the _surplus_ suffering associated with guilt (GM III:15). Through the necromancy sponsored by the ascetic ideal, innocent sufferers are summarily transformed into guilty sinners.\n\nHaving nurtured the weakling human animal throughout its protracted childhood and arrested adolescence, the ascetic ideal has now outlived its usefulness to the continued development of the species. The surplus suffering of guilt has crippled the human animal to the extent that it can no longer tolerate the existential suffering of the bad conscience. Riddled with guilt, its will now aimed at securing final release from the torment of existence, the human animal once again contemplates the \"suicidal nihilism\" from which the ascetic ideal originally saved it (GM 111:28). Having survived thus far its quantum leap into civil society, the human animal must now somehow survive civil society itself. In order to complete the transition from natural animal to human animal, the human species must wean itself from the metaphysical comforts dispensed by the ascetic ideal.\n\nAs a consequence of their renunciation of the \"instinctual\" regulation provided by Nature\u2014an apostasy without precedent or parallel in the animal kingdom\u2014human animals must determine for themselves what they ought to will. Once wrenched from its natural, instinctual moorings, the human will has no natural or proper object to pursue. It consequently attaches itself to _any_ goal whose pursuit promises to deliver the threshold level of affective engagement, or feeling of power, that confirms the vitality of the human organism. Nietzsche thus cautions that \"the basic fact of the human will\" is its \" _horror vacui: it needs a goal_ \u2014and it would rather will _nothingness_ than _not_ will\" (GM 111:1). If bereft of life-affirming alternatives, an enervated will would eventually embrace the goal of self-annihilation, for any goal is better than none at all. This \"will to nothingness\" constitutes the \"last will\" of humankind, the will never to will again, and its advent signifies the impending demise of modernity itself.\n\nThe haunting specter of the \"will to nothingness\" thus exposes the grave danger involved in all measures designed to permit (and even encourage) the indiscriminacy of the will. Because the will has grown too weak to bear the yoke of external legislation, some political thinkers now celebrate the ochlocratic plurality of objects to which the will promiscuously attaches itself. While it has become popular in late modernity to entrust to each will the task of determining its own goal, thereby obviating the legislative role of the lawgiver, this trust is egregiously misplaced. If each will is left to its own witless devices, if lawgivers fail to provide and enforce a sustaining goal, then the enhancement of humankind will continue to occur, if at all, only by accident. More importantly, in light of the peculiar exigencies of late modernity, an unguided will may eventually constitute itself as the \"will to nothingness.\" Lawgivers must consequently legislate against the indiscriminacy of the human will, subjecting to their own design that which \"naturally\" falls to chance.\n\nAs Nietzsche sees it, exemplary human beings inadvertently assist lawgivers in correcting for the indiscriminacy of the human will. He regularly figures great human beings as intrepid navigators, who unwittingly serve as advance scouts for the drifting bark of humanity. These heroes turn the indiscriminacy of the will to the advantage of the species as a whole, experimenting with myriad, diverse goals and thereby testing the limits and plasticity of the human soul. From these dangerous experiments the species as a whole compiles a store of common wisdom, upon which the lawgiver draws in order to spare less robust souls the perils of similar experimentation. The extramoral genius consequently performs an indispensable political service, unknowingly providing lawgivers with the knowledge they need in order to compensate for the indiscriminacy of the human will.\n\nNietzsche thus advances both a general and a specific warrant for the production of those exemplary human beings who engender a renewed confidence in the future of humankind. He now realizes that Zarathustra's greatest fear, the somnambulant reign of the nodding, blinking \"last man\" (Z P5), has been eclipsed by a more ominous peril: the advent of the \"will to nothingness,\" whereby humankind orchestrates its own annihilation in a final, apocalyptic frenzy of Dionysian expenditure. While the legislation of a permanent object for the will remains the highest priority overall, Nietzsche defers this task to the \"philosophers of the future.\" His own task is to safeguard the endangered will until such time as new commanders and legislators arrive on the scene.\n\nToward this more modest end, he experiments with novel objects for the crippled will, gambling that virtually any constitution of the will is preferable to the \"will to nothingness.\" He thus advocates a politics of resistance rather than a politics of redemption or revolution. Although he hopes to contribute to the eventual legislation of a permanent object or goal for humankind, he can do no more than preside over the survival of the will in the twilight of the idols.\n\n### The Use and Abuse of Christian Morality\n\nNietzsche can do nothing to assuage the existential suffering of the bad conscience, and he evinces no inclination to do so. The \"sovereign individuals\" whom he envisions will not only endure the pain of the bad conscience, but also cherish it as yet another seduction to Life. His perfectionism thus aims at nothing less than the (eventual) production of human beings who require no external, metaphysical justification for their meaningless existence. He consequently aims to diminish, and eventually to eliminate, the surplus suffering (or guilt) that attends the dominant, Christian interpretation of the bad conscience.\n\nNietzsche regularly links the survival of the will with the self-overcoming of Christian morality, which he also claims as his ownmost task (D P4). The self-reflexive nature of this task is crucial to an understanding of his perfectionism, for his vision of the \"sovereign individual\" actually incorporates some powers and faculties that have been perfected under the discipline of Christian morality. It was under the aegis of Christian morality, after all, that the \"sovereign individual\" briefly and accidentally emerged, in whose chiseled visage Nietzsche spies a glimpse of a post-Christian future:\n\n> [L]ike only to himself, liberated again from morality of custom, autonomous and supramoral (for \"autonomous\" and \"moral\" are mutually exclusive), in short, the man who has his own independent, protracted will and the _right to make promises_ \u2014(GM 11:2)\n\nHe thus associates the self-overcoming of morality with the \"breeding\" of individuals who stand security for their own future, who acknowledge (and repay) debts only to themselves (GM 11:2). Christianity has made such individuals possible, and they will in turn abolish Christian morality and its emphasis on irremediable guilt.\n\nBecause Nietzsche envisions the completion (rather than the transcendence) of the all-too-human, his perfectionism aims not to absolve the indebtedness of the human animal, but to exploit it. The death of God does not absolve all human debts\u2014 _pace_ Feuerbach\u2014but it may, in some extraordinary cases, allow for \"sovereign individuals\" to assume full responsibility for the definition and payment of their debts. Just as Kantian autonomy delivers freedom not from the moral law itself, but from the _constraint_ of the moral law, so Nietzschean \"sovereignty\" promises absolution not of indebtedness _per se,_ but only of the constraint of one's indebtedness. The sovereign individuals he foresees thus augur the self-overcoming of Christian morality: they are debtors, to be sure, but only to themselves. Hence they are _also_ creditors, for they stand security for their own \"guilt.\" The transition from natural animal to human animal will be complete only when humankind is able to produce these sovereign individuals as a matter of design. It is to this end that Nietzsche devotes the political thinking of his mature, post-Zarathustran period.\n\nFor all of his enthusiasm, however, the cultural production of sovereign individuals is simply incompatible with the diminished resources at the disposal of his age. What modernity calls an \"individual,\" the pride of the Enlightenment, is nothing more than a \"moral milksop,\" a domesticated animal that has internalized the demands of culture and consequently operates under the illusion of self-legislated freedom. Even the \"sovereign individual,\" who possesses \"the _right to make promises_ ,\" owes his \"rare freedom\" to his _\"conscience,\"_ which, Nietzsche shows, is itself an implant of socially enforced heteronomy (GM 2:2). The conscience, a fiercely vigilant homunculus responsible for reckoning one's debts and obligations, represents the final\u2014and most forbidding\u2014barrier to genuine sovereignty.\n\nNietzsche snickers at the idea that the right to make promises stands as sufficient evidence of one's sovereignty, for he views the conscience as the internalized, mnemonic distillation of socially enforced punitive and carceral practices. Whereas the noble savage and blond beast require sturdy cages or constant external surveillance, \"men of conscience\" are sufficiently docile to police themselves. Even Nietzsche himself, the self-styled immoralist and Antichrist, continues to wear this \"venerable long pigtail,\" which makes him \"seem old-fashioned and grandfatherly-honorable\" (BGE 214). The closest thing we know to genuine, supramoral sovereignty is not the debt-paying, promise-keeping, originally positioned author of the social contract, but the criminal, the monster devoid of conscience, who personally shoulders the entire burden of his existential suffering. Nietzsche thus defines \"the criminal type\" as \"the type of strong human being under unfavorable circumstances: a strong human being made sick\" (TI IX:45).\n\nUnder the influence of Christianity, the institutions of Western civilization have for the most part implemented what Nietzsche calls \"moralities of taming\" (TI VII:3). Social practices of self-formation have succeeded in sickening (and thus domesticating) those individals whose \"virtues are ostracized by society.\" The conscience thus prevents individuals from straying far from the internalized norm, while the institutions of modernity marginalize or stamp out those singular, exotic plants that do manage to blossom. On a rare occasion, however, \"a man proves stronger than society: the Corsican, Napoleon, is the most famous case\" (TI IX:45).\n\nNapoleon thus represents the closest approximation known to Nietzsche of genuine sovereignty, for Napoleon approached the task of lawgiving (relatively) unconstrained by conscience and tradition. He consequently describes Napoleon as a \"return to Nature,\" which he defines as \"an _ascent_ \u2014up into the high, free, even terrible Nature and naturalness where great tasks are something one plays with, one _may_ play with\" (TI IX:48). Just as the taming disciplines of Christianity inadvertently made possible the emergence of the amoral criminal, who, in the person of Napoleon, returned to Nature, so the \"philosophers of the future\" may someday breed sovereign individuals by appropriating and adapting the signature practices of Christian morality. Until his nomothetic successors arrive, however, Nietzsche will take advantage of the demise of modernity to experiment with untested constitutions of the will, some of which may be successful in postponing the descent of the \"will to nothingness.\"\n\n### The _\u00dcbermensch_\n\nNietzsche's perfectionism attains its apotheosis in his enigmatic conception of the _\u00dcbermensch,_ or \"over-man.\" While the relevant textual evidence is simply too slight to authorize any particular interpretation of this difficult teaching, the _\u00dcbermensch_ is best understood within the context of Nietzsche's enduring admiration for heroic individuals and \"higher humanity.\" He thus conceives of the _\u00dcbermensch_ as embodying the perfection, rather than the transcendence, of humankind. The _\u00dcbermensch_ is any human being who actually advances the frontier of human perfectibility.\n\nImmediately after posing the problem of \"breeding\" superlative human beings, Nietzsche explains that this \"higher type\" of human being stands \"in relation to humankind as a whole, as] a kind of _\u00dcbermensch\"_ (AC 4). The central task of politics, then, is to produce (as a matter of design) those individuals who stand, \"in relation to humankind as a whole,\" as exemplary human beings. The production of the _\u00dcbermensch_ thus contributes to the enhancement of humankind, for the _\u00dcbermensch_ in turn embodies a perfection of the soul from which others may draw courage and inspiration. In a remark that dispels much of the mythology that popularly surrounds Nietzsche's teaching of the _\u00dcbermensch_[ 8, he immediately adds that\n\n> Such fortuitous accidents of great success have always been possible and _will_ perhaps always be possible. And even whole families, tribes, or peoples may occasionally represent such a _lucky strike [Treffer]._ (AC 4)\n\nWhile it is ordinarily irresponsible to privilege a single passage from Nietzsche's books, this brief discussion of the _\u00dcbermensch_ strikes me as decisive, and for several reasons. First of all, the _teaching_ of the _\u00dcbermensch_ more properly belongs to Zarathustra. Nietzsche himself mentions the _\u00dcbermensch_ in only a few passages outside the text of _Zarathustra,_ most of which shed no direct light on his political thinking. Indeed, Zarathustra's evolving doctrine of the _\u00dcbermensch_ often deviates significantly from the account Nietzsche provides in _The Antichrist(ian),_ and we have good reason to believe that Zarathustra did not fully understand the teachings entrusted to him. Especially when pandering to obtuse auditors throughout Parts I and II of his _Bildungsgang,_ Zarathustra regularly presents the _\u00dcbermensch_ as the transcendence, rather than the completion or perfection, of the all-too-human \"fragments and cripples\" littered about him (Z 11:20). He thus lapses regularly into idealism, allowing his prodigious resentment of modernity to invest his teaching of the _\u00dcbermensch_ with the particular content and determination that he expressly disallows. Zarathustra is a valuable guide through the labyrinths of Nietzsche's teachings, but he too must be subjected to critical scrutiny. We would do well not to confuse or conflate Nietzsche's account of the _\u00dcbermensch_ with Zarathustra's parabolic teaching.\n\nSecond, these telegraphic remarks in _The Antichrist(ian)_ contain Nietzsche's single most fully developed statement of his conception of the _\u00dcbermensch._ In contrast to Zarathustra's ambiguous teaching, Nietzsche's sketch of the _\u00dcbermensch_ in _The Antichrist(ian)_ is consistent with (and explicitly linked to) his more familiar discussions of the political role of exemplary human beings. On those rare occasions when he uses the term _\u00dcbermensch,_ he apparently has in mind the apotheosis of those specimens of \"higher humanity\" to whom he more regularly refers. While it may be important in certain contexts to distinguish between the _\u00dcbermensch_ and this \"higher humanity,\" we are justified in treating the two concepts as continuous within the economy of Nietzsche's thought. Indeed, the only salient difference between the _\u00dcbermensch_ and other, more familiar specimens of this \"higher humanity\" is that the emergence of the _\u00dcbermensch_ is _willed_ by those commanders and legislators who undertake the task of perfecting the all-too-human.\n\nThird, this brief sketch of the _\u00dcbermensch_ is further distinguished by the political import Nietzsche attaches to _The Antichrist(ian),_ as the statement of his \"revaluation of all values.\" Nietzsche is well aware that his conception of the _\u00dcbermensch_ has been widely misunderstood (EH III: l), and he knows that the political fate of his revaluation hinges in part upon the reception of his remarks on the _\u00dcbermensch. The Antichrist(ian)_ thus affords him an opportunity to clarify\u2014for some readers, at least\u2014the relation of his conception of the _\u00dcbermensch_ to the impending event of revaluation. Attempting to avoid (some of) the misleading connotations of the term _\u00dcbermensch,_ he employs it only sparingly in his post-Zarathustran writings, explicitly distancing himself from the idealism that distorts Zarathustra's teaching (EH III: 1).\n\nNietzsche seems especially keen to disabuse his readers of the popular interpretation of the _\u00dcbermensch_ as a moral ideal, for he views all forms of \"idealism\" as antithetical to the \"realism\" he champions:\n\n> The word _\"\u00dcbermensch\"_...has been understood almost everywhere with the utmost innocence in the sense of those very values whose opposite Zarathustra was meant to represent\u2014that is, as an \"idealistic\" type of a higher man, half \"saint,\" half \"genius.\" (EH III: 1)\n\nThis commitment to realism, which informs all of his writings from 1888, not only governs his laconic remarks on the _\u00dcbermensch,_ but also expresses the singular achievements of the _\u00fcbermenschlich_ type:\n\n> this type of man...conceives reality _as it is,_ being strong enough to do so; this type is not estranged or removed from reality but is reality itself and exemplifies all that is terrible and questionable in it\u2014 _only in that way can_ _man attain greatness._ (EH XIV: 5)\n\nThe _\u00dcbermensch_ is not simply another moral ideal, but a concrete, empirical type _[Typus]_ or kind [Art]. An ideal suggests to Nietzsche a \"flight from reality,\" a theoretical construct that may or may not admit of concrete instantiation, and which, in any event, implies an indictment of reality (EH III: 1; XIV:3). As an opponent of idealism, he thus maintains that\n\n> What justifies man is his reality\u2014it will eternally justify him. How much greater is the worth of the real man, compared with any merely desired, dreamed-up, foully fabricated man? with any ideal man? (TI IX:32)\n\nLest he slip into the unwanted idealism of Zarathustra, Nietzsche refrains from offering any antecedent designation or defining characteristics of the _\u00fcbermenschlich_ type, suggesting simply that we \"should sooner look even for a Cesare Borgia than a Parsifal\" (EH III: 1). He explains that the word \" _\u00dcbermensch\"_ designates \"a type of supreme achievement\" and that the _\u00dcbermensch_ stands in opposition to \"'modern' men, to 'good' men, to Christians and other nihilists\" (EH III: 1).\n\nFor Nietzsche, then, the _\u00dcbermensch_ operates as an extremely (if not perfectly) \"thick\" ethical concept, which can be grasped only through an empirical study of the highest human types that actually have emerged. He provides no abstract theory or account of _\u00dcbermenschlichkeit,_ and he relies almost exclusively on concrete examples\u2014Cesare Borgia, for instance\u2014to convey the meaning he attaches to the concept. Following his lead, we might profitably employ a strictly functional (or formal) designation of the _\u00fcbermenschlich_ type: _\u00dcbermenschen_ are simply those individuals who embody the \"supreme achievements\" of any culture or epoch\u2014regardless of the moral or aesthetic qualities they do or do not possess.\n\nNietzsche explicitly claims that \"such fortunate accidents of great success have always been possible\" (AC 4), which implies that _\u00dcbermenschen_ have existed throughout human history. Even a passing familiarity with Nietzsche's books yields a fairly impressive list of the exemplary historical figures he has in mind: Caesar, Pilate, Cesare Borgia, Napoleon, Goethe, Frederick II, and so on. Nietzsche also expresses his hope that such superlative human beings \"will perhaps always be possible,\" which implies that he may (or should) expect to encounter _\u00dcbermenschen_ even in late modernity. Although his contempt for modernity escalates his skepticism of the generative powers of the age, he certainly proceeds in his post-Zarathustran political deliberations as if some such exemplars might someday emerge from the gloaming; his continued advocacy of perfectionism would otherwise make little sense. Indeed, he describes these _\u00fcbermenschlich_ types as \"fortunate accidents\" precisely because they embody an enhancement of humankind, which is the end to which he directs his own energies.\n\nNietzsche describes these _\u00fcbermenschlich_ types as standing \"relative to,\" rather than independent of, \"humanity as a whole\" (AC 4). This description not only militates against defining the _\u00dcbermensch_ in absolute or ideal terms, but also directs our attention to the relationships that obtain between _\u00dcbermenschen_ and \"humanity as a whole.\" In fact, if these _\u00fcbermenschlich_ types stood independent of \"humanity as a whole,\" estranged altogether from the ethical life of the communities that produce them, then they could play no role in the permanent enhancement of humankind. It is within the domain of these (admittedly unique) relations that the distinctly ethical content of Nietzsche's perfectionism resides. His attention to these relations furthermore indicates that he understands the _\u00dcbermensch_ as constituting the perfection, rather than the transcendence, of humankind.\n\nThroughout the post-Zarathustran period of his career, Nietzsche portrays the \"genius\" as strictly an economic type, characterized by a relatively expanded range of vitality and affective expression:\n\n> The genius, in work and deed, is necessarily a squanderer: that he squanders himself, that is his greatness. The instinct of self-preservation is suspended, as it were; the overpowering pressure of outflowing forces forbids him any such care or caution. (TI IX:44)\n\nThe order of rank among individuals and types is thus determined by a measure of the relative capacity of excess affect that one can afford to reserve and expend. Nietzsche thus proffers a strictly formal account of the great individual, as an economic type endowed with a relatively _\u00fcbermenschlich_ capacity for reserve and expenditure: \"He shall be greatest who...is overrich in will. Precisely this shall be called _greatness:_ being capable of being as manifold as whole, as ample as full\" (BGE 212). He consummates this economic designation of the _\u00fcbermenschlich_ type by defining the genius as \"one who either _begets_ or _gives_ _birth,_ taking both terms in their most elevated sense\" (BGE 206). As we shall see later in more detail, he entrusts the future of humankind to these _\u00fcbermenschlich_ types precisely because they alone can afford to squander themselves in the catalysis of culture; for this procreative task, they need only be relatively greater than the humanity that surrounds them.\n\nIn fact, it is precisely the relations between exemplary human beings and \"humanity as a whole\" that make ethical life and moral development possible at all. The ethical life that springs up around _\u00fcbermenschlich_ types does not sustain a universal ethical community, nor a community founded on rational principles of legislation, nor a community sufficiently inclusive to mollify Nietzsche's liberal critics, but it sustains a thriving community nonetheless, complete with its own signature morality. As we shall see more clearly later on, the founding labors of the _\u00dcbermensch_ create a community of friends in the peculiarly Nietzschean sense, of fellow travelers who share a common aesthetic sensibility, who mutually elevate one another through conflict and contest.\n\nAs we have already seen, Nietzsche claims that entire communities of such exemplars\u2014\"families, tribes, peoples\"\u2014are possible (AC 4). This claim contradicts the popular caricature of the _\u00dcbermensch_ as an autarkic nomad who willfully estranges himself from all traditions, communities and shared tables of value. We therefore need not assume that Nietzsche imagines the _\u00dcbermensch_ as some sort of abomination or monstrosity unknown to the ethical community. Indeed, since Nietzsche defines the _\u00dcbermensch_ \"in relation to humankind as a whole,\" rather than in some absolute or ideal terms, we may furthermore expect these exemplary figures to reflect in certain respects the relative character of their respective epochs.\n\nNietzsche is often criticized for portraying these _\u00fcbermenschlich_ types on the model of the amoral, world-historical conqueror. While his romanticization of extramoral monsters _is_ often childish and offensive, it does not constitute the core of his political thinking. As the textual evidence indicates, he defines exemplary figures solely in terms of their embodied justification of the future of humanity, a political task for which they need be neither world-historical nor particularly monstrous. He praises artists, poets and thinkers as lavishly and as frequently as he praises commanders, lawgivers and beasts; depending on the people or epoch in question, either type may represent the highest expression of vitality. Healthy peoples and ages tend to produce world-historical commanders and lawgivers as their highest specimens, while decadent peoples and ages tend to produce philosophers and critics as their representative exemplars. We should therefore not be surprised to discover that the heroes of one epoch bear little external resemblance to those of another. Just as \"we moderns\" appear weak and impoverished in comparison with \"that lavishly squandering and fatal age of the Renaissance\" (TI IX:37), so the \"higher men\" of late modernity would seem sickly and pale beside the overflowing health of a Cesare Borgia.\n\nThe distinctly ethical role of the _\u00dcbermensch_ becomes even clearer if we unearth the Emersonian roots of this exotic plant. Nietzsche apparently models the _\u00dcbermensch_ on Emerson's notion of \"representative men,\" who, in their own private pursuits of self-reliance, display (and thereby represent) the potentialities for perfection resident within the human soul. By virtue of their embodied practices, representative men \"remind\" some others of the soul's natural (if ultimately futile) aspirations to transcendence. Representative men straddle the intersection of the human and the divine, of the temporal and the eternal, and they represent to some others the \"forgotten\" perfections attainable by all human beings.\n\nTheir contribution to ethical life is consequently predicated on their practice of aversion. They stand as living rebukes\u2014\"critics in body and soul\" (BGE 210)\u2014to the conformity and mediocrity into which human beings all too readily sink. Their pursuits of self-perfection stir similar longings within the souls of some others, thus galvanizing the ethical life of the community. In an early passage that conveys the extent of his debt to Emerson (as well as the provenance of his fascination with the prefix _\u00fcber),_ Nietzsche writes, \"[y]our real nature lies... immeasurably high above _[\u00fcber]_ you, or at least above _[\u00fcber]_ that which you usually take yourself to be\" (SE 1). Completing this Emersonian parallel, we might think of the _\u00dcbermensch_ as a \"representative individual,\" whose achievements in self-perfection illuminate the linkage between invidual human souls and the transpersonal oversoul of humankind.\n\nIn light of these decisive passages from _The Antichrist(ian),_ we need not endorse Zarathustra's gnomic claim that there has never been an _\u00dcbermensch_ (Z 11:4), nor relegate the _\u00dcbermensch_ to a distant and continually receding future, nor confine the _\u00dcbermensch_ exclusively to modernity, nor imagine the _\u00dcbermensch_ as an unattainable, strictly regulative ideal. Having separated Nietzsche's teaching from Zarathustra's, we may confidently interpret the _\u00dcbermensch_ in concrete terms, as the historically instantiated, fully attainable, concrete embodiment of human perfectibility\u2014an empirical type rather than a theoretical ideal\u2014around whom the ethical life of any thriving culture revolves. The _\u00dcbermensch_ is any higher human being whose \"private\" pursuit of selfperfection occasions an enhancement of the species as a whole, thus contributing to the perfection (rather than the transcendence) of the all-too-human. The _\u00dcbermensch_ thus instantiates a justification of humankind grounded in its reality, rather than in some abstract ideal.\n\nThe _\u00dcbermensch_ thus constitutes Nietzsche's general answer to the founding question of politics: \"we\" should undertake to breed a type of individual whose pursuit of self-perfection contributes to the enhancement of humankind and thereby justifies our own existence. \"We\" should undertake the establishment of a political regime that will in turn envision the _\u00dcbermensch_ as its unimaginable, singular product. Indeed, the single most salient characteristic of this type of regime is that _\u00dcbermenschen_ are produced by design. Nietzsche places extraordinary emphasis on the overt act of willing to assume responsibility for the future of humankind. Heroic human beings have emerged before to refresh the parched landscape of humanity, but always as \"lucky strikes\" on the part of an indifferent Nature. Never before have _\u00fcbermenschlich_ types been willed into existence, as the primary, overarching aim of a political regime.\n\nIt is important to note, however, that this interpretation of the _\u00dcbermensch_ deviates from Nietzsche's teaching in an important respect. Whereas I have cast the _\u00fcbermenschlich_ type as the highest achievement realized by any people or epoch, Nietzsche tends to reserve the title _\u00dcbermensch_ only for the representative exemplars of superlative peoples and epochs, and thus of humanity itself. It is entirely possible, then, that he would bristle at my suggestion that even the anemic, twilight cultures of late modernity might unwittingly produce _\u00fcbermenschlich_ types of their own. The ethical exemplars of late modernity are a far cry, admittedly, from the \"lightning\" and \"frenzy\" (Z P3) prophesied by Zarathustra, as well as from the redemptive man of the future \u2014the \"victor over God and nothingness\" (GM II:24)\u2014for whom Nietzsche romantically yearns.\n\nIn light of Nietzsche's abundant resentment of modernity, however, it is quite likely that he occasionally compromises his own teaching. Like Zarathustra, in fact, Nietzsche tends to depict the _\u00dcbermensch_ in traditionally personal terms, as a charismatic individual endowed with superlative nomothetic powers. He thus favors a model of agency that naturally lends itself to the redemptive and messianic interpretations that he claims to repudiate. Rather than take literally his (and Zarathustra's) pronouncements on the _\u00dcbermensch,_ we might instead attempt to situate this difficult teaching within the economy of Nietzsche's thought as a whole. We might then view the _\u00dcbermensch_ not as an individual agent _per se,_ but as a historically specific vortex of generative powers and transformative possibilities. His vision of the _\u00dcbermensch_ could be realized in a community, a discourse, a confluence of traditions, a network of social institutions, a constellation of cultural practices, an unanticipated mutation in the human phenotype\u2014perhaps even a cyborg mechanism.\n\nIf we apply to Nietzsche his own guiding insights, then we can compensate somewhat for the distortions imposed on his political thinking by his own decadent yearnings. Indeed, if we succeed in filtering out his own prejudices, then there is no need to anticipate the completion of his thought in the advent of a prophet, messiah, charismatic leader, conqueror, commander, dictator or _F\u00fchrer. \"\u00dcbermensch\"_ could simply refer to a propitious confluence of social, historical and material conditions, such as those that presided over the birth of tragedy in ancient Greece. This interpretation of the _\u00dcbermensch,_ which I will embellish in due course, is not only more faithful than the popular caricature to the (scant) textual evidence available, but also more promising as an interpretation of Nietzsche's ethical teaching.\n\n# 2  \nThe Uses and Disadvantages of Morality for Life\n\n> The genius, in work and deed, is necessarily a squanderer: that he squanders himself, that is his greatness... Yet, because much is owed to such explosives, much has also been given them in return: for example, a kind of higher morality. After all, that is the way of human gratitude: it _misunderstands_ its benefactors.\n> \n> \u2014 _Twilight of the Idols,_ IX:44\n\nThe ethical core of Nietzsche's perfectionism is often eclipsed by his scathing attack on the Western moral tradition. But his critique of morality is not inhospitable to all forms of morality, and it in fact clears a space for the \"morality of breeding\" that motivates his perfectionism. He actually intends his perfectionism to shelter a moral pluralism, informed by an order of rank, which yields a hierarchical organization of ethical communities.\n\n### Nietzsche's Defense of Moral Pluralism\n\nAlthough Nietzsche's perfectionism lies at the very heart of his political thinking, it remains one of the most obscure elements of his philosophy. His critics routinely dismiss his perfectionism, often assuming that he proposes a crude eugenics project, which will culminate in the savage, unprincipled rule of some blond beast or barbarian caste. John Rawls, for example, attributes to Nietzsche a version of \"teleological perfectionism\" that is not even worthy of consideration in the \"original position.\" The closest Rawls comes to a critical assessment of this \"teleological perfectionism\" is his terse observation that \"[t]he absolute weight that Nietzsche sometimes gives the lives of great men such as Socrates and Goethe is unusual.\" Rather than elaborate, Rawls moves on to consider the merits of a \"more moderate doctrine\" of perfectionism, which \"has far stronger claims,\" but which he rejects nonetheless as a source of viable principles of justice.\n\nRawls's response is typical in its summary rejection of the ethical claims of Nietzsche's perfectionism. Like Rawls, many readers conclude that Nietzsche's political thinking shelters virtually no ethical content whatsoever. Nietzsche himself is certainly responsible for much of the misunderstanding surrounding his perfectionism, for he often presents himself as an uncompromising opponent of morality _simpliciter._ He proudly describes himself as an \"immoralist,\" and he congratulates himself for being the \"first\" of this noble breed (EH XIV:6). Fully representative of his rhetorical excesses is the following \"definition of morality\":\n\n> Morality\u2014the idiosyncrasy of decadents, with the ulterior motive of revenging oneself against life\u2014successfully. (EH XIV:7)\n\nContinuing this telegraphic line of argumentation, he submits the following summary epigram: \"Morality as vampirism\" (EH XIV:8).\n\nEspecially when viewed from the broadly historical perspective that Nietzsche favors, however, the enterprise of morality encompasses far more than the universal prescriptions and metaphysical fictions that he so famously debunks. In fact, he regularly reminds his readers that the type of morality he opposes is only one among several possible moralities:\n\n> _Morality in Europe today is herd animal morality_ \u2014in other words, as we understand it, merely _one_ type of human morality beside which, before which, and after which many other types, above all _higher_ moralities, are or ought to be, possible. But this morality resists such a \"possibility,\" such an \"ought\" with all its power: it says stubbornly and inexorably, \"I am morality itself, and nothing besides is morality.\" (BGE 202)\n\nContinuing this critique of moral monism in his next book, _On the Genealogy of_ _Morals,_ he declares that contemporary morality, despite its claims to universality, is in fact descended from a \"slave\" morality, which in turn emerged only in response to the hegemony of a logically and historically prior \"noble\" morality. The history of morality, encrypted in the \"long hieroglyphic record\" that Nietzsche aims to decipher (GM P:7), thus contradicts the claim of _any_ morality, including the ubiquitous \"herd animal morality,\" to a privileged, monistic prerogative as the arbiter of ethical life. In his \"review\" of the _Genealogy,_ he explicitly identifies the \"slave revolt in morality\" with \"the birth of Christianity out of the spirit of _ressentiment_ \" (EH XI).\n\nRather than reject the enterprise of morality itself, Nietzsche instead rejects the claim of any single morality to universal scope and application.\n\nA universally binding morality would necessarily erect a monolithic moral ideal, thereby reducing a plurality of human types and kinds to a lowest common denominator. Ethical laws should (and do) bind collectively, but only across a limited number of individuals, such as constitute a people, race, tribe, or community. As Zarathustra puts it, \"I am a law only for my kind _[die Meinen],_ I am no law for all\" (Z IV:12). The dream of an ethical community comprising all human beings, or all sentient beings, thus spells political nightmare. The laws of an omni-inclusive ethical community would express only the commonalities and banalities of the individuals involved, rather than their unique strengths and virtues. Morality should always serve the enhancement of the ethical life of a particular people, and not the other way around:\n\n> _Morality_ \u2014no longer the expression of the conditions for the life and growth of a people, no longer its most basic instinct of life, but become abstract, become the antithesis of life\u2014morality as the systematic degradation of the imagination, as the \"evil eye\" for all things. (AC 25)\n\nNietzsche's critique of Christian morality is best understood within the context of his political opposition to moral monism. He has no quarrel, for example, with Christian morality in its \"pure\" forms, which he applauds for providing comfort and solace to the demotic strata of hierarchically organized societies. He goes so far as to praise the contributions of Christian morality to the \"hygienic\" maintenance of intramural political boundaries, readily acknowledging the value of moralities that serve the inwardly destroyed (BGE 62). As his commitment to moral pluralism would suggest, he objects to Christian morality only in its most virulent political form, insofar as it arrogates to itself a universal application across all of humankind; as we have seen, this objection is sustained strictly on political, rather than epistemological or theological, grounds. He consequently aims to disabuse his readers of the belief that Christian morality is coextensive with morality itself: \"I negate a type of morality that has become prevalent and predominant as morality itself\u2014the morality of decadence or, more concretely, _Christian_ morality\" (EH XIV:4). He thus explains that his self-awarded title, \"the immoralist,\" designates an opposition specifically to _Christian_ morality, which in his day held (or so he believed) a virtual monopoly over ethical life throughout the diverse cultures of Western civilization (EH XIV:6).\n\nAs an alternative to the moral monism he detects at the rotten core of Christianity, Nietzsche espouses a moral pluralism that reflects the rich diversity of human types, while reminding us that these moralities vary in worth as widely as the individuals whose needs and perfections they express:\n\n> Moralities must be forced to bow first of all before the _order of rank;_ their presumption must be brought home to their conscience\u2014until they finally reach agreement that it is _immoral_ to say: \"what is right for one is fair for the other.\" (BGE 221)\n\nIndeed, a primary aim of Nietzsche's perfectionism is to promote the design of hierarchically organized political regimes, each of which would simultaneously sustain several grades of morality. The aristocratic regimes he favors would shelter a pyramidal hierarchy of ethical communities, each equipped with a distinctive morality that reflects its unique needs and strengths. At the pinnacle of this pyramidal structure would stand the community of agonistic \"friends\" founded by the _\u00dcbermensch._\n\nAs modernity nears exhaustion, the pyramidal structure of this hierarchy of moral communities becomes deformed accordingly, flattened by the glacial advance of decadence. In times such as these, Nietzsche recommends a renewed vigilance to the order of rank that separates human types:\n\n> The more normal sickliness becomes among men...the higher should be the honor accorded the rare cases of great power of soul and body, humankind's _lucky strikes;_ the more we should protect the well-constituted from the worst kind of air, the air of the sickroom. (GM 111:14)\n\nIn order to resist collectively the decadence of late modernity, he urges his \"friends\" to band together, seeking strength in numbers and prophylaxis in seclusion:\n\n> And therefore let us have good company, _our_ company!... So that we may, at least for a while yet, guard ourselves, my friends, against the two worst contagions that may be reserved just for us\u2014against the _great nausea at_ _man!_ against _great pity for man!_ (GM 111:14)\n\nWhile it is not entirely inaccurate to portray Nietzsche as an amoral champion of autarkic individualism, we might think of him more precisely as a moral pluralist, who eschews the claims of any morality to a universal compass across the (potentially) wide expanse of human types:\n\n> what is fair for one _cannot_ by any means for that reason alone also be fair for others;...the demand of one morality for all is detrimental for the higher men; in short,...there is an order of rank between man and man, hence also between morality and morality. (BGE 228)\n\nThe most exacting moralities, those which assign the greatest privileges _and_ responsibilities, are operative in the lives of the rarest and most exotic human beings. As evidence of his own exemplary standing relative to most of his contemporaries, Nietzsche reserves a \"stricter\" morality for himself and his unknown \"friends\" (BGE 219, 226).\n\nJust as he opposes the moral monism of Christianity, so he refuses to prescribe his own \"stricter\" morality to those who are not of his kind. While he would clearly welcome the renascence of some descendant strain of the recessive \"noble\" morality, he just as clearly understands that any such morality would appeal only to a limited number of human beings. Indeed, although the ethical dimension of his perfectionism is not intended to serve demotic interests, it _is_ fully compatible with a demotic morality that is properly bounded in scope and application. He consequently prefers those aristocratic political regimes that shelter multiple moralities simultaneously, including a demotic morality designed to alleviate the suffering of the incurably sick and infirm.\n\nNietzsche's love of solitude is well known. The specific form of solitude he praises, however, derives its appeal from its dependence on a logically prior ethical community. As Zarathustra discovers only after repeated _Unterg\u00e4nge,_ solitude independent of community is indistinguishable from loneliness. Nietzsche speaks fondly and repeatedly of his unknown \"friends,\" precisely because they represent a community from which his self-imposed exile involves only a temporary respite. These \"friends,\" many of whom he draws from the pages of history or from the nether reaches of his febrile imagination, inspire him to persevere in his solitary task. The imaginary \"free spirits,\" for example, were summoned in order that their \"brave companionship\" might \"keep [him] in good spirits while surrounded by ills\" (H I: P:2).\n\nThese communitarian and pluralist currents in Nietzsche's political thinking furthermore reflect his lifelong yearning for a community in which he might realize his destiny as a philosopher and lawgiver. Here we recall his founding membership in Germania and the Leipzig Philological Society; his complicated Oedipal alliance with the Wagners in Tribschen; his fantasies (including Peter Gast, his friend and amanuensis) of a Knightly Brotherhood of the _gaya scienza;_ his proposal to Lou Salom\u00e9 and Paul R\u00e9e of an intellectual _m\u00e9nage \u00e0 trois;_ and his imagination in 1887\u201388 of a \"subterranean\" Nietzsche cult growing among \"radical parties\" in Europe (excepting Germany) and North America.\n\nAs these examples indicate, however, Nietzsche's impulse toward community is characteristically deflected by his tendency to identify only with imaginary communities, including those of the mythical past and future. Availing himself freely of his prodigious powers of imagination, he regularly identifies himself as party to a contrived or fictitious collective: \"we scholars,\" \"we free spirits,\" \"we Hyperboreans,\" \"we Europeans\n\nof the day after tomorrow,\" \"we philologists,\" \"we psychologists,\" \"we revaluers,\" and so on. He readily admits, for example, that he invented the \"free spirits\" to whom he dedicated _Human, All Too Human,_ explaining that \"these brave companions and familiars\" served \"as compensation for the friends [he] lacked\" (H I:P2).\n\nSince he never specifies an existing audience with which he identifies his most basic hopes and desires (a luxury unavailable to his rival, the phonocentric Socrates), his ethical thinking operates at a level of generality and abstraction that is inimical to the creation of new communities and the cultivation of existing ones. Like (Groucho) Marx, Nietzsche would never deign to join a community that would have him as a member. Rather than identify his aims and aspirations with those of any existing community, he saves himself for a transhistorical community that is worthy of the allegiance of his beautiful soul.\n\nWhile this strategy of endless deferral surely involves a romantic flight from the present and the concrete, it also illuminates some of the perils of community, which Nietzsche's critics occasionally neglect to reckon accurately. His prolonged solitude clearly exacts a heavy toll, but it also enables him to resist the (decadent) impulse to seek recognition from those \"beneath\" himself. Since he cannot rely on the hygienic stratification of a hierarchical society to insulate him from the resentment of the weak and bedraggled, he must protect his pursuit of self-perfection by imposing his own regimen of prophylactic solitude. He thus identifies the refusal of community as a prerequisite of his own moral growth (EH 11:8), and he observes that the preference for solitude over unworthy company often constitutes a sign of health (EH 1:2).\n\nRather than treat solitude as necessarily a privation, to be recuperated by the flowering of community, Nietzsche views community as an accidental, outward extension of one's ownmost self, which, under the best of conditions, honors and commemorates the self-sufficiency of the noble soul. Healthy individuals thus value community not as the precondition of their redemption and becoming whole, but as an opportunity to revel in the externalized emanations of their own virtue and character. Hence only weak, corrupt souls, whose constitutive misery leads them to crave the distractions and diversions of facile companionship, need be alone in solitude: \"a well-turned-out person...is always in his own company, whether he associates with books, human beings, or landscapes: he honors by _choosing,_ by _admitting,_ by _trusting_ \" (EH 1:2). A thriving moral community that requires no other living human members: perhaps no image conveys more accurately than this the peculiar ethical content of Nietzsche's perfectionism. As we shall see later on in more detail, he defends this principled aversion to unworthy community as a corollary to his highest (and sole) moral obligation.\n\nThere is no prima facie warrant, then, for excluding Nietzsche's perfectionism from consideration as the source of a bona fide ethical position. The point of his perfectionism is to shelter the delicate resources of ethical life and to preserve the possibility of the sort of moral development that constitutes an enhancement of humankind as a whole. The ethical content of his perfectionism may be objectionable to liberal critics, but these objections are themselves open to philosophical scrutiny and evaluation. Nietzsche himself would maintain that all such objections bespeak the pre-philosophical prejudices of his critics; as such, they would reveal much more about these critics than they do about his perfectionism.\n\n### Nietzsche and Manu: Moralities of Breeding\n\nNietzsche's discussion of a kindred political thinker sheds clarifying light on the ethical content of his perfectionism. Both in _Twilight of the Idols_ and _The_ _Antichrist(ian),_ he expresses his admiration for Manu, the legendary Hindu lawgiver. Nietzsche recommends the law of Manu not as a blueprint for political reform in late modernity, but as evidence of the importance \"noble\" cultures have traditionally attached to the morality of breeding. While the restoration of political aristocracy is simply out of the question for late modernity, a (modified) morality of breeding, as established by Nietzsche's perfectionism, is not.\n\nOn Nietzsche's reconstruction, Manu successfully enforced a political regime that enabled several distinct social classes to flourish simultaneously. Manu understood that the enhancement of Hindu culture would require the prophylaxis supplied by a fairly rigid social stratification. His regime effectively quarantined the relatively \"sick\" from the relatively \"healthy,\" while providing for the relative well-being of all social classes. The ethical motivation behind Manu's system is the perfectionism that Nietzsche too advocates: \"To set up a code of laws after the manner of Manu means to give a people the chance henceforth to become master, to become perfect\u2014to aspire to the highest art of life\" (AC 57).\n\nNietzsche consequently credits Manu with designing a political organization and social structure that reflect the order of Nature itself:\n\n> The _order of castes,_ the supreme, the dominant law, is merely the sanction of a _natural order,_ a natural lawfulness of the first rank, over which no arbitrariness, no \"modern idea\" has any power... The order of castes...is necessary for the preservation of society, to make possible the higher and the highest types. (AC 57)\n\nFollowing Manu (and Nature), Nietzsche endorses the pyramidal caste system, or \"natural aristocracy,\" as the supreme form of political regime.\n\nHis characterization of the three \"castes\" of Nature\u2014distinguished, respectively, by pre-eminent spirituality, pre-eminent strength \"in muscle and temperament,\" and by mediocrity (AC 57)\u2014bears a remarkable resemblance to Socrates' sketch of his pyramidal \"city in speech\" in the _Republic._ 6 As we shall soon see, Nietzsche also follows Socrates (and Manu) in furnishing a \"noble lie\" about the origins and justification of the political regime he recommends.\n\nNietzsche admires Manu for his commitment to _the morality of breeding,_ wherein the lawgiver establishes the social preconditions of a plurality of types, from which in turn rare and exotic specimens are most likely to emerge (TI VII: 3). Like Manu, Nietzsche tracks the enhancement of humankind to the proliferation of unanticipated, unimagined human types, and he endorses the political project of \"breeding\" these exemplary human beings. He contrasts this approach to political legislation with the _morality of taming,_ of which he cites Western Christianity as representative. Whereas \"breeding\" encourages the simultaneous flourishing of a plurality of forms of life, \"taming\" imposes upon all forms of life a single ideal, with respect to which the higher, more exotic types must be broken down:\n\n> Physiologically speaking: in the struggle with beasts, to make them sick _may_ be the only means for making them weak. This the church understood: it _ruined_ man, it weakened him\u2014but it claimed to have \"improved\" him. (TI VII:2)\n\nThese competing approaches to political legislation are predicated on diametrically opposed ethical principles. A morality of taming structures society in accordance with a predetermined ideal, while a morality of breeding establishes a political order in which a plurality of forms of life is pursued\u2014including, for the highest types, a form of life that is unfettered by all known ideals. He dismisses \"idealism\" in any guise as \"cowardice,\" as a \"flight from reality\" (EH XIV:3), for ideals invariably place preordained constraints on the range of human types that a society might produce. Like Manu, Nietzsche is an \"immoralist.\" He refrains from proposing a single ideal in accordance with which all types must be domesticated; instead he encourages an untamed proliferation of rare and exotic individuals.\n\nIf a morality of breeding is to succeed in producing exemplary specimens, then the lawgiver must eventually exclude those types that pose an immediate threat to the flourishing of the society as a whole. Hence the ethical arch-principle that engenders Nietzsche's admiration for hierarchically organized political regimes: \"That the sick should _not_ make the healthy sick...should surely be our supreme concern on earth\" (GM 111:14). The implementation of this hygienic principle thus affords each stratum of society the prophylactic luxury of _not_ associating with lower, pathogenic strata (GM 1:10).\n\nManu understood, as few since have, that only structure and stratification beget the fecund plurality from which rare specimens spring forth; the more exacting the discipline, the more exotic the emergent fruits and blossoms. Unlike Nature, which can afford to be a \"bad economist\" (SE 7), the lawgiver must legislate the terms of exclusion and expenditure. As a means of ensuring the success of the morality of breeding he implemented, Manu legislated the exclusion of the impure chandalas, whom his political regime simply could not accommodate. Manu understood the need both to exclude the chandalas _and_ to render them politically impotent, lest their exclusion strengthen and embolden them (TI VII:3).\n\nThe cruelty of Manu's exclusionary legislations is palpable, for the chandalas are in no way responsible for falling outside the arbitrary class designations that he enforces; nor do they deserve the harsh, inhuman treatment they receive. Yet some such cruelty is necessary if the morality of breeding is to succeed, and Manu's appeal to the purity of social caste furnishes his political regime with the sustaining myth it needs in order to \"justify\" the cruelty it inflicts. While it may be possible for modern lawgivers to temper the cruelty of Manu's legislations\u2014through improvements in technology and distributive justice, or through the invention of more humane forms of exclusion\u2014the practice of exclusion is itself unavoidable.\n\nNietzsche does not personally advocate the caste system developed by Manu, but he fully endorses the willed practice of political exclusion, which Manu's system was designed to convey. With respect to this precise point, he does not mince his words:\n\n> The essential characteristic of a good and healthy aristocracy, however, is that it...accepts with a good conscience the sacrifice of untold human beings who, _for its sake,_ must be reduced and lowered to incomplete human beings, to slaves, to instruments. (BGE 258)\n\nNietzsche thus presents slavery as a necessary, indispensable practice in those hierarchically organized societies that contribute to the permanent enhancement of humankind (BGE 44), a practice he associates with spiritual husbandry: \"Slavery is, as it seems, both in the cruder and in the more subtle sense, the indispensable means of spiritual discipline and cultivation, too\" (BGE 188).\n\nAlthough it turns out that he is more interested in the sort of \"slavery\" that one imposes on oneself in the cultivation of one's soul, his peculiar, metaphorical use of the term \"slavery\" is itself a concession to the besetting decadence of his epoch. If _real_ slavery were possible in late modernity\u2014that is, if the establishment of an aristocratic political regime were a viable option in the twilight of the idols\u2014then he would surely, and unabashedly, endorse it as a precondition of the perfectionism he advocates. And although he might prefer the practice of slavery in its \"more subtle sense,\" allowing the \"slaves,\" for example, an (illusory) feeling of their freedom and self-determination, he also justifies the institution of slavery by appealing to the \"moral imperative of Nature,\" which is directed, he insists, at humankind itself (BGE 188).\n\nNietzsche thus views the practice of exclusion as an inescapable element\u2014a \"necessary evil,\" as it were\u2014of political legislation in any regime. In order for a society to produce a few whole human beings, it must legislate and enforce the fragmentation of countless others. Only by virtue of this exclusion is culture\u2014an artificial subsystem sheltered within the indifferent economy of Nature\u2014possible at all. He thus insists that \"the greatest of all tasks, the attempt to raise humanity higher, includ[es] the relentless destruction of everything that [is] degenerating and parasitical\" (EH IV:4). It is simply the nature of politics, he believes, that all regimes must practice exclusion, whether or not they do so knowingly and resolutely. Despite their visceral aversion to Manu's grisly decrees, modern lawgivers are no more at liberty to dispense with political exclusion than to reprise his specific practice of it. The morality of taming too practices a form of exclusion, insofar as it forces all higher, singular types to lie in a Procrustean bed of its own mediocre design (TI IX:43); it too justifies its exclusionary practices by appealing to a sustaining myth, that of _\"equal_ rights for all\" (CW 7).\n\nWhile Manu is by no means alone in practicing exclusion, he distinguishes himself\u2014at least in Nietzsche's mind\u2014by subjecting this practice to willful legislation. Manu does not require the implementation of exclusionary stratifications; Nature does (AC 57). But Manu _wills_ the practice of exclusion, furnishing it with a particular aspect and _modus operandi_ within the caste system he designs; he unflinchingly inscribes the canon of Nature into the constitution of his political regime. Rather than consign to chance the regulation of his political regime, Manu legislates the exclusionary practices that will best promote his morality of breeding. Nietzsche consequently admires Manu not for practicing exclusion _per se_ (which all lawgivers must do), nor for the specific practices he implements, but for doing so as a matter of design.\n\nNietzsche is no champion of democracy, but he believes that demotic interests are best served in hierarchical political regimes devoted to the breeding and production of exemplary human beings. All members of a thriving community are, and should be, elevated by the \"immoral\" exploits of its highest exemplars. While this elevation is least visible (and least appreciated) within the demotic stratum of a hierarchical society, he nevertheless insists, like J.S.Mill, that some attenuated benefits of perfectionism trickle down to everyone. Unlike the \"flathead\" Mill, however, Nietzsche does not propose the benefits of involuntary cultural elevation as a justification for the perfectionism he legislates. With reference to the \"higher\" human types, he declares that\n\n> Their right to exist, the privilege of the full-toned bell over the false and cracked, is a thousand times greater: they alone are our _warranty_ for the future, they alone are _liable_ for the future of humankind. (GM 111:14)\n\nA utilitarian defense of perfectionism, such as the one Mill concocts, would not only yoke political legislation to the tyrannical whims of ochlocratic taste\u2014a problem Mill never adequately solved\u2014but would also presuppose that the _demos_ can in fact recognize and pursue its own best interests, which Nietzsche expressly denies. It is an unalterable fact of political life that most individuals fail to discern, much less appreciate, the spiritual and material elevation they derive from their involuntary contributions to the production of exemplary human beings. This fact is not sufficient, however, to deter Nietzsche from his promotion and defense of perfectionism.\n\nIn order to obviate the disaffection of the demotic stratum of a hierarchical society, and thereby attend to its genuine (as opposed to its perceived) self-interest, the lawgiver must always reinforce the perfectionist aims of the regime with a sustaining myth. Toward this end, Nietzsche recommends the use of state-sponsored religions to elevate the _demos_ against its will:\n\n> To ordinary human beings, finally\u2014the vast majority who exist for service and the general advantage, and who _may_ exist only for that\u2014religion gives an inestimable contentment with their situation and type, manifold peace of the heart, an ennobling of obedience, one further happiness and sorrow with their peers and something transfiguring and beautifying, something of a justification for the whole everyday character, the whole lowliness, the whole half-brutish poverty of their souls. Religion and religious significance spread the splendor of the sun over such ever-toiling human beings and make their own sight tolerable to them. (BGE 61)\n\nState-sponsored religions thus furnish and perpetuate the sustaining myths of a hierarchically organized society, which in turn supply \"ordinary human beings\" with the solace and comfort they need. It should be noted, moreover, that state-sponsored religions also function to co-opt the disaffections that invariably suffuse the barren souls of \"ordinary human beings,\" thereby preventing, or at least dampening, explosive outbreaks of resentment within the lowest strata of society.\n\nNietzsche often associates the perfectionism he advocates with aristocratic political regimes, which, as his critics observe, are incompatible with the depleted vitality he attributes to late modernity. If his perfectionism required the structure and discipline that aristocratic regimes alone can supply, then his ethical and political thinking would be hopelessly anachronistic; his vision of the future of humankind would be incompatible with his critique of modernity. He may yearn for the halcyon days of the Roman Empire and the Florentine Republics, but he is not so foolish as to confuse those days with his own. He unabashedly admires Manu as a lawgiver, but he neither recommends nor advocates Manu's aristocratic regime as a viable solution to the unique political problems of modernity. The institutions of modernity are simply too corrupt to impress into service as he would have us believe Manu did, and he is too decadent to supply the requisite nomothesis in any event. The \"philosophers of the future\" may someday successfully emulate Manu, Caesar or Napoleon in their political lawgiving, but Nietzsche cannot.\n\n### The _Pathos_ of Distance\n\nIn order to account for the legislative predilections that he shares with Manu and other \"noble\" souls, Nietzsche occasionally alludes to a _\"pathos_ of distance\" resident within himself. Although he usually associates this _pathos_ of distance with the aristocratic regimes he expressly admires, its existence is not dependent on any particular form of political regime. Indeed, his own _pathos_ of distance not only suggests the viability of his perfectionism in the twilight of the idols, but also secures his claim to the hyperopic perspective of the \"immoral\" lawgiver.\n\nNietzsche introduces the _pathos_ of distance as definitive of the \"noble\" mode of evaluation, describing it as\n\n> the protracted and domineering fundamental total feeling on the part of a higher ruling order in relation to a lower order, to a \"below\"\u2014 _that_ is the origin of the antithesis \"good\" and \"bad.\" (GM 1:2)\n\nThis _pathos_ of distance, he later explains, is \"characteristic of every strong age,\" for it expresses \"the cleavage between man and man, status and status, the plurality of types, the will to be oneself, to stand out\" (TI IX:37). As we shall see, a diminished _\"pathos_ of distance\" not only is possible in decadent epochs like late modernity, but also may sustain in these twilight epochs a modest morality of breeding.\n\nThe _pathos_ of distance signifies an enhanced sensibility for, or attunement to, the order of rank that \"naturally\" informs the rich plurality of human types. According to Nietzsche, the _pathos_ of distance animates those aristocratic regimes that he most admires:\n\n> Every enhancement of the type \"man\" has so far been the work of an aristocratic society...that believes in the long ladder of an order of rank and differences in value between man and man, and that needs slavery in some sense or another. (BGE 257)\n\nHe furthermore associates the absence or diminution of this _pathos_ of distance with decadence and decline: \"Today nobody has the courage any longer for privileges, for masters' rights, for a sense of respect for oneself and one's peers\u2014 for a _pathos of distance_ \" (AC 43).\n\nWhile Nietzsche usually discusses the _pathos_ of distance in the context of his praise for aristocratic political regimes, the motivation for his celebration of this _pathos_ of distance\u2014and of aristocracy itself, for that matter\u2014is distinctly ethical. His perfectionism, which shelters the ethical core of his thought, not only is separable from the particular structure provided by political aristocracy, but also operates independently of this and any other particular form of political regime. As he explains, it is not the aristocratic political regime itself that stimulates human flourishing, but the _pathos_ of distance sustained therein:\n\n> Without that _pathos of distance_ which grows out of the ingrained difference between strata...that other, more mysterious _pathos_ could not have grown up either\u2014the craving for an ever new widening of distances within the soul itself, the development of ever higher, rarer, more remote, further-stretching, more comprehensive states\u2014in brief, simply the enhancement of the type \"man,\" the continual \"self-overcoming of man,\" to use a moral formula in a supra-moral sense. (BGE 257)\n\nIn this passage, Nietzsche discloses the ethical core of his perfectionism. The permanent enhancement of humankind is attributable to the attainment of ever rarer states of the soul. This \"aristocracy of the soul\" is in turn the product of an internal _pathos_ of distance, a \"mysterious\" craving for multiplicity and stratification within the soul itself. Nietzsche thus believes that the internal _pathos_ of distance is itself instilled (or nourished) by the external _pathos_ of distance evoked by any stable political aristocracy. As we shall see later on, this mysterious craving for internal distance either is, or is related to, _er s._\n\nThe attraction for Nietzsche of aristocratic regimes thus lies in their capacity to accommodate and implement human design. The rigid hier archical stratification that he generally recommends is maximally effective at insulating political legislation from accident and chance. Aristocratic political regimes enable the lawgiver to intervene in Nature, to correct for Nature's indifference, and to assume (limited) dominion over the continued enhancement of the species. He consequently favors aristocratic political regimes, but only because they preserve and embellish the _pathos_ of distance, which he in turn reveres for its evocation of a craving for \"self-overcoming.\" As we shall see, \"self-overcoming\" is Nietzsche's preferred term for the moral content of his perfectionism.\n\nOur attention to the _pathos_ of distance thus reveals the ethical basis of Nietzsche's perfectionism. Natural aristocracy is the best form of political regime _not_ in the sense that all peoples and epochs ought to aspire to its grandeur, but in the sense that it expresses the highest degree of strength and vitality known to humankind. With few exceptions, political regimes will accurately reflect the vitality of the peoples and ages they serve; for the most part, human beings establish the best political regimes they can also afford to sustain. While aristocracy is grander than democracy as an expression of an epoch's vitality, it is not a better regime for those epochs that can afford only democracy. Nature always requires the pyramidal organization that aristocracy attains most perfectly, but it does not always supply the lawgiver with the tools and materials to fashion an aristocratic political regime. In that event, lacking the macropolitical resources needed to reprise the law of Manu, the lawgiver must aspire instead to reproduce Nature's pyramid in diminished miniature.\n\nNietzsche consequently defends aristocratic regimes, but only insofar as they nurture (or preserve) the _pathos_ of distance that alone enables moral development. His notorious fascination with the morality of breeding is similarly grounded in ethical concerns: by preserving the stratification of types\u2014and the _pathos_ of distance it evokes\u2014the morality of breeding sustains the possibility of moral progress. The morality of taming, on the other hand, elides the difference between types, thus threatening to extinguish the _pathos_ of distance that genuine moral progress necessarily presupposes.\n\nAs a political thinker, then, Nietzsche is a consequentialist, for he aims to impress politics into the service of ethics. Political regimes are valuable only insofar as they enable psychic regimes, rather than the other way around. He favors political aristocracy as the form of institutional organization that is most conducive to the \"aristocracy of soul\" that he associates with nobility, but it is by no means a necessary condition of self-perfection. The essential element of his political thinking lies not in his yearning for an institutionally reinforced hierarchy, but in his perfectionism. While the former requires a degree of strength and vitality unknown to late modernity, the latter is in principle compatible even with the depleted resources of Nietzsche's own epoch.\n\nWe need not conclude, then, that his political thinking is either hopelessly anachronistic or irrelevant to the peculiar conditions of late modernity, for his perfectionism _can_ operate independently of his romantic yearnings for political aristocracy. So long as a culture preserves some minimal _pathos_ of distance, expressed however faintly in some trace of pyramidal structure, the permanent enhancement of humankind remains possible. Isolating the ethical content of his perfectionism is crucial to our understanding of his project, for in the event that an aristocratic regime were no longer possible (as is the case in late modernity), he could still advocate the perfectionism that lies at the heart of his political thinking.\n\nNietzsche offers no further defense of his preference for the morality of breeding, and the reason for his silence is simple: the _pathos_ of distance does not admit of further, theoretical justification. Lawgivers are \"justified\" not by virtue of some epistemic privilege, divine decree, natural right or Promethean insight, but simply by virtue of their audacious desire to subject to design what naturally falls to chance. The sole justification for a lawgiver's decrees resides exclusively in the vision of humankind that informs them. Nietzsche consequently does not pretend to offer any further justification for his morality of breeding than the products of such \"breeding\" personally embody. One either shares his attunement to the _pathos_ of distance that motivates his perfectionism, or one does not.\n\nThe perceived need to advance or receive a theoretical justification of political legislation is symptomatic, he believes, of the decadence of modernity. A healthy people or age neither demands nor requires a discursive justification for its legislations:\n\n> What must first be proved is worth little. Wherever authority still forms part of good bearing, where one does not give reasons but commands, the dialectician is a kind of buffoon: one laughs at him, one does not take him seriously. (TI II:5)\n\nNobility is expressed simply in the attempt to create or preserve an order of rank in the face of the indifference of Nature; the noble soul consequently requires no prior or independent justification of its creations. A decadent people or age, on the other hand, which cannot afford to squander its dwindling resources, generally resolves to expend itself only if given good reason to do so, usually in the form of a discursive justification. Only decadent peoples need to justify the practices and policies whereby they express their signature virtues, and only decadent peoples bemoan the absence of such justifications. The conspicuous failure of Nietzsche's critics to satisfy the very conditions of justification to which they hold him suggests that he may have a point.\n\n# 3  \nPerfectionism in the Twilight of the Idols\n\n> The whole of the West no longer possesses the instincts out of which institutions grow, out of which a _future_ grows: perhaps nothing antagonizes its \"modern spirit\" so much. One lives for the day, one lives very fast, one lives very irresponsibly: precisely this is called \"freedom.\" That which makes an institution an institution is despised, hated, repudiated: one fears the danger of a new slavery the moment the word \"authority\" is even spoken out loud.\n> \n> \u2014 _Twilight of the Idols,_ IX:39\n\nIn response to his own diagnosis of the decadence of modernity, Nietzsche rethinks his endorsement of political perfectionism. While he would like nothing better than to contribute to the establishment of an aristocratic regime, he also realizes that the depleted vitality of modernity is simply incompatible with the political perfectionism he envisions. His subsequent shift in emphasis to _moral_ perfectionism not only is consistent with his critique of modernity, but also conveys the priority he has always assigned to ethics over politics. In order to assess more precisely the moral content of his perfectionism, I turn now to examine his post-Zarathustran critique of modernity.\n\n### Nietzsche's Critique of Modernity\n\nNietzsche originally conceived of the production of exemplary human beings as a macropolitical task, which would mobilize (and justify) the political institutions of the modern nation state. Although he isolates the \"disease\" of modernity in its senescent educational institutions, he nevertheless believes that they can be rejuvenated sufficiently \"to establish a permanent alliance between German and Greek culture\" (BT 20). He actively opposes any political solution on the part of the newly established _Reich,_ but he nevertheless recommends a reorganization of German educational institutions, such that they might produce as a matter of design the exemplary individuals who have emerged heretofore only as accidents of Nature.\n\nWhile his public lectures on \"The Future of Our Educational Institutions,\" which he delivered early in 1872, warn of the creeping mediocrity of higher education, and of the yawning gulf that separates education from genuine culture, they collectively sound a call to arms rather than a requiem. The political macrosphere is still a salvageable shelter for ethical life, provided that he and Wagner can infuse the new _Reich_ with a tonic appreciation for tragedy:\n\n> Let no one try to blight our faith in a yet-impending rebirth of Hellenic antiquity; for this alone gives us hope for a renovation and purification of the German spirit through the fire magic of music. (BT 20)\n\nAs this passage demonstrates, the young, professorial Nietzsche exudes the very optimism that _The Birth of Tragedy_ disparagingly attributes to Socrates.\n\nFollowing his \"discovery\" of the decadence of modernity, however, Nietzsche realizes that his earlier faith in the salvageability of modern institutions was egregiously misplaced. He consequently withdraws his plans for macropolitical reform, and he does so in two identifiable steps. First, he acknowledges that the institutions of modern Europe are simply too corrupt to serve in the macropolitical capacity he had mistakenly reserved for them. Although he argues that institutions have functioned historically as the crucibles of culture, wherein human animals have been forcibly \"civilized,\" cruelly implanted with memories and painfully invested with \"the right to make promises\" (GM 11:2), he also insists that the institutions of Western civilization, those \"bulwarks of political organization\" (GM 11:18), have gradually usurped the task of consecrating individuals to culture. The institutional reinforcement of the morality of taming, euphemistically known as the \"improvement-morality,\" now threatens to extinguish the _pathos_ of distance upon which culture is founded.\n\nThe institutions of bygone epochs may have been amenable to the production of exemplary human beings, but the macropolitical resources of modernity are arrayed in a manner inimical to this task. \"In present-day Germany,\" he laments, \"no one is any longer free to give his children a noble education\" (TI VIII:5). Modernity is marked as an age by the cold, refined efficiency, the debilitating anonymity, of its sustaining institutions, which militate ever more perfectly against the emergence of singular human types. Great individuals now emerge only in opposition to institutional design, only in spite of the decadence of the age that spawns them (TI IX:44).\n\nSecond, he acknowledges that, independent of the macropolitical resources at his disposal, he is in no position to orchestrate the redemption of modernity. His critique of modernity may focus on the corruption of modern institutions, but it locates the cause or source of this corruption in an underlying cultural decadence, of which he interprets the demise of these institutions as symptomatic. He consequently observes that\n\n> Our institutions are no good any more: on that there is universal agreement. However, it is not their fault but ours. Once we have lost the instincts out of which institutions grow, we lose institutions altogether because we are no longer good for them... The whole of the West no longer possesses the instincts out of which institutions grow, out of which a _future_ grows. (TI IX:39)\n\nAny attempt to bolster our sagging political institutions, independent of addressing the decadence they manifest, thus betrays a disastrous confusion of cause and effect:\n\n> What will not be built any more henceforth, and _cannot_ be built any more, is\u2014a society _[Gesellschaft]_ in the old sense of the word; to build that, everything is lacking, above all the material. _All of us are no longer_ _material for a society;_ this is a truth for which the time has come. (GS 356)\n\nWe must not underestimate the enormity of Nietzsche's claim that the \"slave morality always first needs a hostile external world\" (GM 1:10). This means that any campaign to change, reform, or meliorate this \"hostile external world\" would also threaten to strip the \"slaves\" of their ownmost identity. As a consequence, the \"slave\" type, which supposedly dominates decadent epochs like late modernity, always maintains a vested interest in _not_ reforming the \"hostile external world\" that it identifies as the source of its victimization and oppression.\n\nIn his post-Zarathustran writings, Nietzsche acknowledges the complicity of his original call for political reform with the decadence it presumed to combat. The fatal anachronism of his plan to reawaken the tragic muse lay in his conviction that he was somehow immune to the cultural disarray he detected. Armed with a moral ideal of his own to purvey\u2014and a romantic ideal to boot\u2014 he unwittingly installed himself as yet another \"improver of mankind.\" Arrogating to himself the privilege of the physician of culture, he blithely prescribed redemptive measures designed to cure the ills of modernity. His prescription of a Wagnerian rebirth of tragic culture thus marked one more expression of the advancing decadence of modernity.\n\nHe consequently concludes that presently there exists no macropolitical solution to the problem of decadence in late modernity. The \"philosophers of the future\" may someday arrive at one, but Nietzsche himself, rooted inextricably in the decadence he so despises, cannot. His writings of 1888 candidly pronounce his own decadence (EH 1:2; CW P), thereby discrediting his presumed credentials as a physician of culture. In fact, his perception of a _need_ in late modernity for exemplary human beings, who generally arise as an expression of surfeit rather than of lack, itself confirms that they are unlikely to appear as the willed products of a morality of breeding. He thus concedes that neither he nor the political institutions that fashioned him are likely to produce anything but alternate permutations of decay:\n\n> It is a self-deception on the part of philosophers and moralists if they believe that they are extricating themselves from decadence when they merely wage war against it... [T]hey change its expression, but they do not get rid of decadence itself. (TI II:11)\n\nNietzsche thus regards the macropolitical agenda sketched in his early writings as historically naive, as woefully ignorant of the source of the decay of modern institutions:\n\n> I now regret...that I appended hopes where there was no ground for hope, where everything pointed all too plainly to an end!... That on the basis of the latest German music I began to rave about \"the German spirit\" as if that were in the process even then of discovering and finding itself again\u2014at a time when the German spirit, which not long before had still had the will to dominate Europe and the strength to lead Europe, was just making its testament and _abdicating_ forever... (BT AS:6)\n\nWhile he continues to voice his yearnings for a macropolitical \"cure\" for the decadence that besets modernity, he usually defers all such speculations to the \"philosophers of the future\" who will succeed him, or to his unreliable mouthpiece, Zarathustra. On topics such as these, to which he has no \"right,\" it behooves him \"only to be silent\" (GM II:25).\n\nHow can Nietzsche so calmly document the demise of the very political institutions in which he had originally invested his hopes for the future? Is this withering critique of modernity not inhospitable to his vision of the future of humankind? He understands, and frequently reiterates, that the corruption of modern institutions is not the cause of decadence, but its most telling symptom. That \"we are no good\" for these institutions thus signals the advance, rather than the advent, of decadence. He consequently ridicules the idea that we might combat decadence simply by easing the discomfort that attends it:\n\n> The supposed causes of degeneration are its consequences. But the supposed remedies of degeneration are also mere palliatives against some of its effects: the \"cured\" are merely one type of the degenerates. (WP 42)\n\nIn other words, the crisis of modernity is even more acute than he originally imagined, when he called for a rebirth of tragic culture. As a physiological affliction indelibly inscribed into the bodies of modern agents, decadence is encoded within the (disintegrating) ethos of modernity itself.\n\nYet the ethical resources of late modernity are not yet exhausted. The catalysis of culture continues, as evidenced by Nietzsche's own reliance on Schopenhauer as an _Erzieher_. The diffusion of the _pathos_ of distance, of which he interprets the pandemic spread of democratic reforms as symptomatic, is not yet sufficiently entropic to preclude the possibility of the continued enhancement of humankind. Since a form of ethical life has survived the demise of the sustaining institutions of late modernity, future enhancements of humankind remain possible. While it is true that Nietzsche must modify his political agenda to accommodate the decadence of late modernity, there is no need yet for him to abandon politics altogether. Here, he believes, he differs from Socrates, who could not afford to sustain his own personal war against decadence in the absence of institutional support (TI 11:12).\n\nRather than attempt directly to produce the redemptive exemplars who will warrant the future of the species, he undertakes to play a more modest and indirect role in the permanent enhancement of humankind. The overarching goal of his politics is to preserve the diminished _pathos_ of distance that ensures the possibility of ethical life and moral development in late modernity. It is his specific task to convoke a gathering of those individuals who are best suited to survive the twilight of the idols, and to train these unlikely \"heroes\" in the experimental disciplines that are most likely to stave off the will to nothingness. This revised task requires fewer macropolitical resources, but it shifts the (diminished) burden of political legislation squarely onto his own stooped shoulders.\n\n### The Political Microsphere\n\nNietzsche's odyssey on the \"open seas\" ultimately transports him to the political microsphere, a destination unknown to him in his pre-Zarathustran career. While involved in his campaign to orchestrate the redemption of modernity, he characteristically restricted his focus to the political macrosphere, to the dying institutions he hoped to resuscitate. In response to the self-referential implications of his critique of modernity, however, he subsequently relocates his political program from the macrosphere to the microsphere.\n\nIn order to convey a sense of Nietzsche's revised role in the production of superlative human beings, I distinguish between the macropolitical (or institutional) and micropolitical (or infra-institutional) incarnations of his perfectionism. The political macrosphere comprises the network of relations that obtain between a people's institutions and its representative exemplars, while the political microsphere comprises those relations between a people and its representative exemplars that are _not_ mediated by social institutions. Macropolitics governs the production of great human beings through the organization of institutional resources, while micropolitics governs the production and illumination of great human beings outside (or beneath) the institutional frameworks of civilization. Whether prosecuted in the macrosphere or in the microsphere, politics contributes to the enhancement of humankind through the legislative deployment of the ethical resources of the community.\n\nWe might think of the political microsphere on an organic model, as the vital core that engenders the signature legislations of a people or community, from which the political macrosphere extends outward as an involuntary, spontaneous outgrowth. The microsphere thus shelters the fund of vital resources upon which a people draws as it expresses (and memorializes) itself in the creation of institutions. Autochthonous folkways, tribal rituals, ethnic customs and memory traces, familial habits and mores, hieratic regimens of diet and hygiene: all contribute to the delicate, capillary network of ethical life in the political microsphere, from which the macrosphere emerges as a natural shell or mantle. These are the smoldering embers from which the Promethean fire of civilization draws its wondrous flame.\n\nIn a strong age overflowing with vital energy, externalized in the institutions and festivals of a healthy people, lawgivers would have neither the need nor the inclination to restrict their legislations to the political microsphere. But in a decadent age unable to sustain the vitality of a people's signature institutions, lawgivers have no choice but to legislate from within the political microsphere. If necessary, the micropolitical lawgiver must strategically inhabit the disintegrating institutions of the political macrosphere, parasitically draining their residual vitality for use in husbanding the resources of the microsphere. Indeed, Nietzsche's favorite \"simile\" for decadence is that of a vital core mantled within a dead husk or shell, such that \"The whole no longer lives at all: it is composite, calculated, artificial, and artifact\" (CW 7).\n\nNietzsche thus realizes that his contribution to the enhancement of humankind need not presuppose the macropolitical support of modern institutions. His perfectionism is compatible with cultures of virtually any degree of vitality, and it consequently requires only minimal institutional support. His retirement from macropolitics therefore does not signal the termination of his program to contribute to the production of superlative human beings. Rather than abandon entirely the idea of producing such exemplars by design, he relocates the site of this production to the political microsphere, where he and his fellow \"free spirits\" must foster the emergence of genius without the assistance of (and in fact in opposition to) institutional reinforcement. Indeed, his diagnosis of decadence suggests that the degeneration of modernity might actually contribute to the emergence of (a peculiar breed of) exemplary human beings:\n\n> Great men, like great ages, are explosives in which a tremendous force is stored up; their precondition is always, historically and physiologically, that for a long time much has been gathered, stored up, saved up, and conserved for them\u2014that there has been no explosion for a long time. (TI IX:44)\n\nEthical life continues to flourish in late modernity, but only in micro-communities that spring up around those representative exemplars whose selfperfections inscribe the canon of the law. Amid the collapse of the political macrosphere, these exemplars have assumed the mediating role traditionally played by political institutions. As we have seen, the relation between great human beings and \"humanity as a whole\" reflects the _pathos_ of distance that suffuses a people or epoch. So long as the _pathos_ of distance survives, an age will produce exemplary specimens, and so long as these heroes continue to perfect themselves, the _pathos_ of distance will survive. The representative exemplars of late modernity are perhaps obnubilated by the twilight, or overlooked amid the rubble of fallen idols, but they remain capable nonetheless of contributing to the permanent enhancement of humankind.\n\nThe central role of these exemplary individuals in Nietzsche's perfectionism is not affected by his emigration to the political microsphere. Perhaps more confidently than ever, he appeals in his post-Zarathustran works to the importance of those individuals who embody the perfectibility of the human soul. He consequently shifts the burden of his perfectionism, which he originally expected macropolitical institutions to bear, entirely onto the microspheric legislations of the representative exemplars of late modernity. He now understands that it is the micropolitical responsibility of individual philosophers to produce such specimens by design\u2014to become them if possible, and to invent them if necessary. Unable to rely on the macropolitical reinforcement of an aristocratic regime, he gambles that the \"aristocracy of soul\" exhibited by these exemplary types is sufficient to sustain and nurture an ethical community of unknown friends and followers.\n\nIt would be misleading, however, to suggest that Nietzsche's turn to micropolitics coincides with a wholesale disavowal of his investment in the macropolitical redemption of modernity. His \"longing for total revolution,\" and for its world-historical architects, persists to the conclusion of his productive career. In his \"review\" of _The Birth of Tragedy,_ for example, he renews his \"promise\" of a tragic age, though he postpones its advent to a more distant remove from the present; he also diverts the credit for this coming age from Wagner to himself (EH IV:4).\n\nNietzsche's attention to the political microsphere may occasionally neutralize these romantic yearnings, but it neither erases nor eliminates them. As expressions of his own decadence and resentment, however, his political sentiments are of considerably less interest than the form in which he presents them. Rather than recoil in horror from his retrograde elitist yearnings, we might more fruitfully inquire: of what relevance to micro-politics is his admiration for the morality of breeding? As we shall see, he (partially) recuperates his macropolitical yearnings by enlisting them in his campaign to acquaint his readers with the ethical resources of the microsphere.\n\nNietzsche's shift to the political microsphere also reprises a similar dislocation in his personal life. In order to become a philosopher and reclaim the task _[Aufgabe]_ reserved for him, he withdrew from the political macrosphere as he understood it, resigning his professorship at Basel and vanishing into a lonely, nomadic existence. Genuine philosophy, it would seem, requires an immersion in the quotidian contingencies and fragile relations of the microsphere. The real work of the philosopher is performed not in stuffy lecture halls and musty archives, but on long, solitary walks in the mountains, in poorly heated pensions, in deserted train stations, on unfamiliar beaches. Rather than harangue university students and continue to campaign for macropolitical reform, he courts anonymity as a wanderer and sun-seeker, connected to the \"world\" he leaves behind only through an erratic regimen of correspondence.\n\nNietzsche's immersion in the political microsphere thus reflects the experimental nature of his own life and work. In order to see himself as a political agent, he first had to situate himself within the microsphere of late modernity and discipline himself to acknowledge the ethical resources arrayed therein. He relocates his perfectionism to the political microsphere only after personally testing it for himself. This proved to be an extremely painful experiment, one which may have contributed significantly to his eventual breakdown. It is perhaps no coincidence that Nietzsche returns to the political macrosphere, promising to assassinate the religious and secular leaders of Western Europe, only under the delusions of grandeur wrought by his incipient madness.\n\n### Light Amid the Shadows: An Attempt at Aesthetic Education\n\nNietzsche's perfectionism thus remains viable even in the twilight of the idols, and we should expect superlative human beings to flourish even in the gloaming of late modernity. These twilight \"heroes\" will remind no one of the world-historical conquerors who populate Nietzsche's fantasies, for they must reflect the decadence of the age they represent. They may nevertheless contribute to the ongoing catalysis of culture.\n\nWhile it need not be the case, as Zarathustra seems to think, that late modernity is \"too late\" to produce _\u00dcbermenschen_ of its own, the \"higher men\" of our epoch are not readily apparent to us. Nietzsche and Zarathustra may be partially responsible for our faulty vision, for they have trained us to look out for the beasts and heroes who represent vital epochs rather than the fools and invalids who stand for dying epochs. In order to educate the sensibilities of his readers and prepare them to partake of ethical life in the political microsphere, Nietzsche must isolate and illuminate those queer _\u00dcbermenschen_ who labor in the twilight of the idols. As we shall see, the twin tasks of _creating_ and _discovering_ exemplary human beings are not readily distinguishable in the shadows of late modernity.\n\nAs a preparation for his revised political project, Nietzsche must refine somewhat the aesthetic sensibilities of his readers, thereby discouraging them from deferring to the world-historical heroes whom he and Zarathustra have led them to anticipate. He thus warns that\n\n> [O]ne misunderstands great human beings if one views them from the miserable perspective of some public use. That one cannot put them to any use, that in itself may belong to greatness. (TI IX:50)\n\nHe consequently attempts to illuminate the ethical resources of the political microsphere, and to acquaint his readers with the \"useless\" squanderers who currently preside over the catalysis of culture.\n\nBecause \"genuine philosophers\" live \"unphilosophically\" and \"unwisely\" (BGE 205), however, they may also live obscurely. Eclipsed by the twilight of the idols, their pursuits of self-perfection go largely unnoticed, along with the micropolitical legislations they enact. In light of his critique of modernity, in fact, the micropolitical exemplars he has in mind must necessarily appear to us as buffoons, fools, misfits, miscreants, criminals, rogues, creatures of _ressentiment,_ and anyone else who can muster even a minimal resistance to the decadence of modernity. In bringing to light the ethical resources of the microsphere, as we shall see, Nietzsche must become his own test case: the viability of his relocated perfectionism rests on his success in creating\/discovering himself as an _\u00dcbermensch._\n\nThroughout his career, his perceived contribution to politics was dominated by the historical necessity of a prelude or preparation. As a young scholar, he presented _The Birth of Tragedy_ as a prelude to a Wagnerian rebirth of tragic culture, and _Schopenhauer as Educator_ as a blueprint for German educational reform. He later advertised _Beyond Good and Evil_ as the \"prelude to a philosophy of the future,\" describing his task in general as a preparation for the mysterious \"philosophers of the future.\" His commitment to perfectionism similarly reflects this preparatory emphasis, for it obliges him to educate his readers to discern any _\u00dcbermenschen_ who might roam the microsphere. While he is certainly not a \"philosopher of the future,\" however, he must also beware of simply busying himself with the preparations for the future as he speculates on the advent of his successors. Once he relocates his political perfectionism within the microsphere, he must recover for himself the legislative role that he carves out for other micropolitical _\u00dcbermenschen._ Toward this end, he legislates his own self-creation as an exemplary human being.\n\nThis conclusion may prove difficult to accept, however, for Nietzsche obviously falls far short of the world-historical _\u00dcbermenschen_ whom he and Zarathustra have led us to anticipate. When casting about for exemplary human beings, Nietzsche's readers customarily search for world-historical redeemers and conquerors, and he is largely responsible for fostering this false expectation. In order for us to see this cranky, infirm hermit as an exemplary human being, and thereby partake of the catalysis of culture to which he contributes, we must consequently revise our opinions of both Nietzsche and the _\u00dcbermensch._ Such a revision would require the precise reversal of perspective\u2014wrought by an cultivation of aesthetic sensibilities\u2014that he hopes to induce in his readers.\n\n### Moral Perfectionism\n\nNietzsche's shift in emphasis to the political microsphere places into sharper relief the moral perfectionism at work in his thought. While the depleted vitality of his epoch is admittedly inimical to the _political_ perfectionism he wishes to pursue, the adverse conditions of late modernity are nevertheless compatible with his related project of _moral_ perfectionism. Indeed, he can sustain the (revised) political dimension of his philosophy only because his project of moral perfectionism requires no greater political support than the microsphere can in fact furnish.\n\nIn an explicit response to John Rawls, Stanley Cavell has recently attempted to recover the distinctly ethical dimension of Nietzsche's perfectionism. In defense of the position he calls _moral perfectionism,_ Cavell interprets both Emerson and Nietzsche as\n\n> calling for the further or higher self of each, each consecrating himself\/ herself to self-transformation, accepting one's own genius, which is precisely not, it is the negation of, accepting one's present state and its present consecrations to something fixed, as such, \"beyond\" one.\n\nCavell thus presents Nietzsche's moral perfectionism as an unintended (and unacknowledged) casualty of Rawls's refusal to weigh the claims of perfectionism in his original position. In rejecting Nietzsche's perfectionism as a potential source of viable principles of justice, Rawls inadvertently vetoes a program of moral education that is uniquely suited to the training of citizens in a democratic society:\n\n> [T]he view Emerson and Nietzsche share, or my interest in it, is not simply to show that [moral perfectionism] is tolerable to the life of justice in a constitutional democracy but to show how it is essential to that life... I understand the training and character and friendship Emerson requires for democracy as preparation to withstand not its rigors but its failures, character to keep the democratic hope alive in the face of disappointment with it.\n\nCavell thus contends that moral perfectionism alone provides the \"democratic training\" required of all citizens who must abide by (though never consent to) the imperfect justice delivered by contemporary democracies.\n\nCavell's eloquent response, however, does not directly address Rawls's specific objections to Nietzsche's perfectionism. While it is not the case that Cavell conflates the moral perfectionism he champions with the political perfectionism Rawls rejects, he does fail to distinguish sharply between the two positions; as a consequence, the precise relationships that obtain between the two forms of perfectionism are left unarticulated and unclear. In any event, it seems abundantly clear that liberals like Rawls should _always_ beware of political theories and regimes that promise to curtail individual freedoms, irrespective of the \"democratic\" moral sentiments indirectly expressed by their founders and champions. In this limited sense, then, Rawls's misunderstanding of Nietzsche is irrelevant, for _no_ moral teaching is sufficiently democratic to compensate for the illiberal excesses of Nietzschean perfectionism. In light of his pre-established allegiance to liberal ideals, Rawls is surely right (if not philosophically justified) to bar Nietzsche's perfectionism from consideration in the original position.\n\nAlthough Cavell fails to address, and so to deflect, Rawls's suspicions of Nietzschean perfectionism, he nevertheless succeeds in transplanting the tangled debate into a more fertile field of inquiry. Indeed, by diverting our attention to the ethical core of Nietzsche's political thinking, Cavell implicitly limns a distinction between Nietzsche's _political_ perfectionism and his less familiar _moral_ perfectionism. While Rawls prudently rejects the former position, as patently illiberal, his undisclosed recoil from the latter position constitutes the most glaring fault in the foundation of his theory of justice. As we have seen, Nietzsche's political perfectionism is largely a vehicle or conveyance for the moral perfectionism that lies at the very heart of his philosophical enterprise. In denying Nietzsche entry into the original position, Rawls may rid political philosophy of some very murky bathwater, but only by expelling the only baby for whom the construction of a \"theory of justice\" is warranted. Although Cavell presents his account of \"moral perfectionism\" as a friendly amendment to Rawls's theory of justice, it seems likely that Rawls's original position is constructed precisely to obviate the moral education and \"democratic training\" that Cavell applauds in Nietzsche and Emerson.\n\nFollowing Cavell's cue, we are now in a position to distinguish between the political and moral perfectionisms operative within Nietzsche's thought. Political perfectionism, as we have seen, provides for the rigid stratification and hierarchical organization of society and its resources, with the aim of producing, as a matter of design, those exemplary human beings whose exploits alone warrant the future of humankind. The aims of political perfectionism are best served by pyramidal aristocratic regimes, which not only reflect and preserve the natural order of rank among human types, but also encourage moral development by engendering a robust _pathos_ of distance. On Nietzsche's account, the priority of political perfectionism resides in its justificatory capacity, for no political arrangement can better protect citizens from the perceived meaninglessness of their existence.\n\nNietzsche's moral perfectionism is perhaps best characterized in terms of the conviction that one's primary, overriding\u2014and perhaps sole\u2014ethical \"obligation\" is to attend to the perfection of one's ownmost self. Any \"obligations\" that one might choose to observe to others are strictly derivative of, and secondary to, the imperative to perfect oneself. While this general imperative appears in many diverse forms and formulations in the history of moral thought, it is typically conveyed via the language of _autonomy_ and _authenticity._ Borrowing from Pindar, a predecessor moral perfectionist, Nietzsche exhorts himself and (some of) his readers with the slogan that he incorporates into the subtitle of his \"autobiography\": _become what you are._ 11\n\nAs this Pindaric slogan suggests, the project of moral perfectionism involves cultivating one's native endowment of powers and faculties; eliciting from within oneself the perfections that lie dormant, undiscovered, or incomplete; and so fortifying one's soul with the virtues constitutive of a sterling character. One is obliged, in short, to strive to produce oneself as an _\u00fcbermenschlich_ human being. All moral \"obligations\" to others are simply illusory, either because they are in fact continuous with the project of perfecting one's ownmost self (as in many cases of charity and gift-giving, such as potlatch), or because they are fabricated by the creative resentment of those who cannot abide the successful self-perfection of others.\n\nThese two forms of perfectionism do not logically entail one another. While it is true that some political thinkers\u2014Plato, Hegel and Nietzsche, for example\u2014cleave to both forms of perfectionism, other thinkers defend only one or the other. Emerson, Mill, Wittgenstein, Sartre, and Berlin, for example, enthusiastically endorse versions of moral perfectionism, but they all vehemently refuse the palpable tyranny of political perfectionism. On the other hand, political thinkers such as Machiavelli and Dostoevsky's Grand Inquisitor, whose insights into human \"nature\" have been brutally implemented by totalitarian dictators, embrace the project of political perfectionism, while holding little hope for the general project of moral perfectionism. In an interesting hybrid case, Richard Rorty endorses the project of moral perfectionism (which he calls the pursuit of autonomy), but he relegates it to the private sphere; this may be tantamount to proscribing all vital forms of moral perfectionism. Still other theorists, such as Rawls, reject the claims of both forms of perfectionism, as inimical to the \"freedom from shame\" that is ideally enjoyed by self-respecting citizens of liberal democratic societies.\n\nIn Nietzsche's own case, political perfectionism serves in a largely instrumental capacity, as a vehicle for his more basic project of moral perfectionism. Although he publicly endorses both forms of perfectionism, he defends political perfectionism only because, and insofar as, it promises to deliver the social conditions that are most conducive to moral perfectionism. In decadent epochs like late modernity, however, which cannot muster the resources needed to sustain the hygienic stratification inherent in a stable aristocratic regime, the project of political perfectionism must be modified significantly, if not abandoned altogether. Nietzsche's critique of modernity thus occasions a shift in emphasis from political to moral perfectionism; while the former project is simply incompatible with the advanced decay of late modernity, the other is not.\n\nThe logical independence of these two forms of perfectionism is crucial to an understanding of Nietzsche's post-Zarathustran political thinking, especially in light of his withering critique of modernity. As Cavell observes, those critics who, like Rawls, oppose Nietzsche's political perfectionism generally fail (or refuse) to detect its distinctly moral warrant and motivation. Though perhaps damaging in some way to his political perfectionism, their criticisms consequently fail to touch, much less to damage, his project of moral perfectionism.\n\nNietzsche himself does not use the term \"moral perfectionism.\" He describes Schopenhauer, whom he never met and with whom he later would express profound disagreement, quite simply as his \"teacher and taskmaster _[Lehrer und_ _Zuchtmeister]_ \" (SE 1). This rare combination of virtues (galvanized by a chance visit to a Leipzig bookshop) ordains Schopenhauer as Nietzsche's _Erzieher_ \u2014his educator. Indeed, the perfectionism for which Nietzsche is routinely vilified in fact shelters a program of moral education, which is supposed to produce as a matter of design those exemplary individuals who unintentionally and unwittingly preside over the catalysis of culture.\n\nNietzsche does not have in mind here the sort of Gradgrindian education that transforms people into \"machines\" by teaching them \"to be bored\" (TI IX:29). His understanding of moral education, as of the nature of ethical life in general, evinces his substantial intellectual debt to Emerson. The term _erziehen_ suggests a process of eliciting, or drawing out, the \"hidden\" powers and perfections resident in oneself and a few others. His political perfectionism thus establishes the social conditions that are most conducive to an education of the highest human types, while his moral perfectionism comprises this education itself. By provoking its greatest exemplars to embody perfections hitherto unknown to humanity, an educational program of this scope would contribute to the permanent enhancement of humankind itself.\n\nAs an aspiring _Erzieher_ in his own right, Nietzsche writes only for those who share his aesthetic sensibilities, and for those who _may_ someday accede to this lofty rank if properly educated. In both cases the pedagogical aim of his writing is neither to convert nor to \"improve\" his readers, but to announce himself to kindred spirits and fellow squanderers. His moral pedagogy is designed not to \"cure\" the sick and infirm, but to embolden and encourage the healthy\u2014much as his \"friends,\" the fictitious free spirits, served as \"brave companions and familiars\" during his own convalescence (H I:P:2). He not only addresses those readers who might be exemplary human beings, but also encourages anyone who can afford to do so to appreciate these exotic and endangered plants wherever they might bloom.\n\n### Moral Perfectionism and\/as Ethical Egoism\n\nNietzsche's moral perfectionism may very well amount to a version of ethical egoism, as critics often charge. Yet he is keen to purge \"egoism\" of the taint foisted upon it by Christian morality and other vehicles of resentment:\n\n> Rather it was only when aristocratic value judgments _declined_ that the whole antithesis \"egoistic\"\/\"unegoistic\" obtruded itself more and more on the human conscience\u2014it is, to speak in my own language, the _herd instinct_ that through this antithesis at last gets its word (and its _words)_ in. (GM 1:2)\n\nTaking up this defense of egoism, Zarathustra distinguishes between the selfishness _[Selbstsucht]_ born of health, which is marked by the \"thirst to pile up all the riches in [one's] soul,\" and that born of sickness, which \"sizes up those who have much to eat...and always sneaks around the table of those who give\" (Z 1:22.1). Echoing his loquacious \"son,\" Nietzsche declares that\n\n> \"Selflessness\" has no value either on heaven or on earth. All great problems demand _great love,_ and of that only strong, round, secure spirits who have a firm grip on themselves are capable. (GS 345)\n\nThis proposed rehabilitation of \"selfishness\" accounts for the only others to whom Nietzsche recognizes ethical obligations: one's \"friends,\" those members of one's \"kind\" or \"type,\" whose virtues and perfections mirror one's own. Following Aristotle, Nietzsche conceives of a friend as another \"I,\" as an external instantiation of one's own virtuous character. Friendship thus affords one the unique opportunity to behold\u2014and celebrate\u2014one's ownmost perfections from an \"external\" standpoint, as individuals possessed of sterling character are naturally inclined to do. Building on the pagan narcissism he applauds in Aristotle's thought, Nietzsche depicts friendship as a mutually empowering _agon,_ in which select individuals undergo moral development through their voluntary engagement in contest and conflict. On this agonistic model of friendship, one has no ethical obligations to those who cannot contribute to one's own quest for self-perfection, those for whom Zarathustra reserves the disapprobative term _neighbor_. One may, of course, acknowledge various political obligations to one's \"neighbors,\" but genuine ethical obligations to those outside one's \"kind\" are strictly ruled out. In fact, friends of high character must always guard their virtues jealously, lest vampiric \"neighbors\" insidiously distract them from the difficult task of self-perfection.\n\nA formidable obstacle to this project of moral perfectionism lies, in fact, in the difficulty involved in undertaking an honest inventory of one's native endowment of virtues and faculties. Those who would perfect themselves must first dare to stand beyond good and evil, for the conventional moral standpoint available to them will invariably discount their richest stores of affect, passion and fantasy. They will be urged to perfect themselves in the virtues and talents that constitute a conventional definition of \"goodness,\" regardless of any natural affinities or predilections they may or may not possess. Those who would remain true to their ownmost selves must consequently free themselves first from the constraints of convention and conformity.\n\nFor a moral perfectionist like Nietzsche, solipsism is, and must remain, a constant danger of ethical life. Especially in a decadent age, one must fiercely protect one's (diminished) store of ethical resources and assume only those ethical obligations that will truly enhance one's own character. To lapse occasionally into solipsism is far preferable to expending oneself in the service of bogus ethical obligations. Although the project of moral perfectionism need not presuppose the political perfectionism of an aristocratic regime, the absence of such a regime means that the project of elevating the _demos_ must be abandoned. One can afford to attend to the moral development only of oneself and of one's kind. In a decadent age, in which solipsism may indeed be inevitable, it is far nobler to consign oneself to loneliness and solitude than to surrender oneself to the parasitic ethical obligations that characterize the \"herd\" of humankind. Indeed, this is the philosophical motivation behind Nietzsche's experiment with self-imposed solitude, which may in the end have cost him his sanity.\n\nNietzsche's sketch of the mythical \"noble morality\" furnishes a broad-brushed blueprint for his moral perfectionism. The noble morality begins with the \"I\"\u2014embodied in its \"powerful physicality\"\u2014and moves outward, conferring \"goodness\" upon everyone and everything it touches or possesses; all else is \"other-than-I\" and therefore \"base\" (GM 1:2). In the rudimentary moral vocabulary of the noble, the term \"good\" is roughly equivalent to the term \"mine,\" while the designation \"bad\" is roughly equivalent to \"not-mine,\" or \"other.\" These crude evaluative terms thus admit of no independent reference and no antecedent meaning. For the nobles, the \"I\" is the ultimate, unchallenged arbiter and generative source of all goodness in the world; they consequently recognize no moral \"obligations\" to those who are unlike themselves. Nietzsche's sketch of the noble morality may be simplistic, and perhaps even childish at times, but it raises an important, and often neglected, ethical question: what is properly called _mine_?\n\nIn order to answer this question, Nietzsche, like Plato before him, employs the model of the _oikos,_ or household, to convey his commitment to the project of moral perfectionism. Human beings naturally seek to maintain that which is uniquely theirs, that which they call and make their \"home,\" which is the generative core from which all \"experiences\" ramify outward. Nietzsche thus introduces himself and his fellow \"men of knowledge\" by confiding that \"there is one thing alone we really care about from the heart\u2014'bringing something home'\" (GM P:1). One's sole moral \"obligation\" lies in regulating the economy of one's own soul, providing oneself with the care and nourishment needed to preserve and expand one's native holdings of virtue.\n\nIn order to maintain its self-imposed principle of internal regulation, a strong household will inevitably squander its lesser holdings, in the interest of continued growth and expansion. Although these squanderings are often mistaken, especially by poorer households, for alms and altruisms, they are in fact the inevitable waste products and excreta that issue forth from an efficiently regulated _oikos:_\n\n> Morally speaking: neighbor love, living for others, and other things _can_ be a protective measure for preserving the hardest self-concern. This is the exception where, against my wont and conviction, I side with the \"selfless\" drives: here they work in the service of _self-love,_ of _self-discipline._ (EH 11: 9)\n\nEven the \"gift-giving virtue\" extolled by Zarathustra (Z 1:22) falls within the contours of this regimen of self-perfection, creating a potlatch economy to ensure that one receives ever greater treasures in return for one's gifts.\n\nAltruism is consciously practiced only by those impoverished households that have no remaining native stores to protect. Having allowed, and even encouraged, the plunder of its ownmost holdings, a barren soul must attach itself parasitically to other souls, under the insidious, neighborly pretense of \"helping\" them to protect their remaining stores. Decadent souls are failed householders, for they give away what is truly precious and resolve to stand guard only over empty households. They cannot distinguish friend from foe, kin from stranger, virtue from vice, or triumph from collapse. Under the guise of \"altruism\" and \"neighborlove,\" they involuntarily infect other, unsuspecting households with their constitutive disarray, urging these \"neighbors\" similarly to divest themselves of their ownmost holdings.\n\nNietzsche thus traces the ultimate origin of altruism to the constitutive illness that compels individuals\u2014he studiously avoids calling it a \"choice\" or \"decision\"\u2014to endure the indignities of unfreedom. That one instinctively prefers slavery to death is not the _cause_ of a broken soul, but its primary symptom or effect. The \"slave morality\" arises precisely to preside over these barren souls, suffusing their bankrupt households with the prepotent (albeit transient) affects of hatred and resentment. Slave morality thus begins with a \"hostile external world\" and moves inward, designating as \"evil\" everything it encounters. While this gambit does not (directly) succeed in renewing the barren larders of the slave's soul, it generates sufficient affect to distract the slaves from their poverty and disarray. Through their blinding resentment of the \"evil\" external world, they gain temporary release (which they call \"goodness\") from the self-contempt that defines their agency. Of course, since their _faux_ \"goodness\" is secured only in contradistinction to the \"evil\" resident within the \"hostile external world,\" the slaves can never genuinely wish to reform the world that holds them hostage.\n\nLike Socrates, Nietzsche amplifies the single imperative of his moral perfectionism by figuring the soul as a treasury, whose precious holdings are naturally coveted by all. In his recommendation of the \"gift-giving virtue,\" Zarathustra thus teaches:\n\n> This is your thirst: to become sacrifices and gifts yourselves; and that is why you thirst to pile up all the riches in your soul. Insatiably your soul strives for treasures and gems, because your virtue is insatiable in wanting to give. You force all things to and into yourself that they may flow back out of your well as the gifts of your love. (Z 1:22.1)\n\nEchoing these sentiments, Nietzsche later introduces himself and his fellow \"men of knowledge\" with a similar image: \"It has been rightly said: 'Where your treasure is, there will your heart be also'; our treasure is where the beehives of our knowledge are\" (GM P:1).\n\nMuch as Socrates spins his \"noble fiction\" of souls naturally veined with gold, silver or bronze, so Nietzsche figures the (healthy) soul as a treasure-laden citadel, which is under constant attack by clever, scheming vulgarians. Explaining his strategy for guarding his own fertility and reproducing his store of precious \"metals,\" he observes,\n\n> One must avoid chance and outside stimuli as much as possible; a kind of walling oneself in belongs among the foremost instinctive precautions of spiritual pregnancy. Should I permit an _alien_ thought to scale the wall secretly? (EH II:3)\n\nUnlike an impoverished or plundered household, a treasury is generally considered to be well worth the effort required to protect its boundaries and expand its _oikos._ Whether or not a particular soul actually harbors anything resembling treasure remains to be seen, but Nietzsche's intent is clear. As his noble lie indicates, the project of moral perfectionism need not presuppose a static, metaphysical conception of the self. In order to embark upon the painful project of moral perfectionism, one must first _believe_ that one's soul contains the aretaic equivalent of precious metals; souls can become treasure-houses only if they are treated as such. As in the case of Socrates' \"myth of the metals,\" Nietzsche's _pia fraus_ is justified, and ennobled, by its contribution to the project of moral perfectionism.\n\n# 4  \nRegimens of Self-Overcoming:\n\n## The Soul Turned Inside Out\n\n> For believe me: the secret for harvesting from existence the greatest fruitfulness and the greatest enjoyment is\u2014to _live dangerously!_ Build your cities on the slopes of Vesuvius! Send your ships into uncharted seas! Live at war with your peers and yourselves!\n> \n> \u2014 _The Gay Science,_ 283\n> \n> _Nitimur in vetitum:_ in this sign my philosophy will triumph one day, for what has been forbidden so far as a matter of principle has always been\u2014truth alone.\n> \n> \u2014 _Ecce Homo,_ Preface, 3\n\nIt remains to be seen how Nietzsche can possibly afford to forgo the (admittedly scarce) macropolitical resources of late modernity. The relocation of his perfectionism to the political microsphere would seem to require him to abandon the legislative component of his perfectionism, which apparently presupposes the support of macropolitical institutions. According to Nietzsche, however, philosophers will continue to pursue this nomothetic task, even within the political microsphere, legislating the conditions under which great human beings would most likely emerge. In order to compensate for the creeping decay of modernity, he restricts the production of exemplary human beings to self-production, and the legislation of values to self-legislation. In the twilight of the idols, philosophers contribute to the enhancement of humankind by producing themselves as repesentative exemplars.\n\n### \" _Thus_ It _Shall_ Be!\" The Philosopher as Legislator\n\nThe legislative role of the philosopher is an enduring theme of Nietzsche's political thinking. As early as 1874, he insists that \"it has been the proper task of all great thinkers to be lawgivers as to the measure, stamp and weight of things\" (SE 3). In praising philosophers, however, he does not have in mind those narrow specialists, bookish scholars and petty \"laborers\" who are popularly identified as \"philosophers.\" Genuine philosophers are those intrepid thinkers who legislate a vision of human perfectibility, a goal toward the attainment of which the resources of humankind might be profitably gathered and directed.\n\nNietzsche's post-Zarathustran writings thus point ever more dramatically (and ever more autobiographically) to the philosopher as a lawgiver:\n\n> _Genuine philosophers, however, are commanders and legislators:_ they say, \" _thus_ it _shall_ be!\" They first determine the Whither and For What of man, and in so doing have at their disposal the preliminary labor of all philosophical laborers, all who have overcome the past. With a creative hand they reach for the future, and all that is and has been becomes a means for them, an instrument, a hammer. Their \"knowing\" is _creating,_ their creating is a legislation, their will to truth is\u2014 _will to power._ (BGE 211)\n\nAlluding to this legislative capacity, he thus describes the philosopher \"as a terrible explosive _[Explosionsstoff],_ endangering everything\" (EH V:3). This description in turn calls to mind his general account of \"great men\" as \"powerful explosives _[Explosiv-Stoffe]_ \" (TI IX:44), and it thus implies that he conceives of the philosopher-commander as a superlative human being. Invoking his designation of the _\u00dcbermensch_ as an economic type or kind, he directs our attention to \"the noble riches in the psychic economy of the philosopher\" (BGE 204).\n\nIn a healthier age, perhaps, these philosopher-commanders could marshall macropolitical resources to support their legislations, and this is precisely the image Nietzsche cultivates of the \"philosophers of the future.\" Philosophers in a decadent age, however, cannot expect to receive institutional reinforcement for their legislations. In fact, Nietzsche's uplifting sketch of the philosopher is usually judged to be self-congratulatory at best, and risible at worst. One is hard-pressed to summon any historical examples of the daring commanders he attempts to call to mind, and one need look no further than to his own writings for a definitive inventory of the cowards, frauds, con artists, valets, charlatans, demagogues, invalids, decadents, and buffoons who are characteristically accorded the title of \"philosopher.\" Indeed, if we are to appreciate his hagiographic portrait of these philosopher-commanders, then we must investigate more closely the unique nomothetic activity he attributes to them.\n\nThroughout his career, Nietzsche places unusual emphasis on the antagonistic relationship that prevails between philosophers and their respective epochs. In his _Untimely Meditation_ on Schopenhauer, he explains,\n\n> I profit from a philosopher only as he can be an example.... [T]he genius must not fear to enter into the most hostile relationship with the existing forms and order if he wants to bring to light the higher order and truth that dwells within him. (SE 3)\n\nAs his thinking matures, he comes to propose this \"hostile relationship\" as the destiny of genuine philosophers, which they are free neither to refuse nor to repeal:\n\n> More and more it seems to me that the philosopher, being _of necessity_ a man of tomorrow and the day after tomorrow, has always found himself, and _had_ to find himself, in contradiction to his today: his enemy was ever the ideal of today. So far all these extraordinary furtherers of humankind _[F\u00f6rderer des Menschen]_ whom one calls philosophers...have found their task...in being the bad conscience of their time. (BGE 212)\n\nIn this passage, he explicitly links the iconoclasm of philosophers with the enhancement of humankind, which we know to be the goal he sets for politics as a whole. Following Emerson, he believes that ethical life and moral progress\u2014culture in the broadest sense\u2014are dependent upon the \"aversive thinking\" of the philosopher.\n\nAlthough Nietzsche often describes the philosopher's \"hostile relationship\" and \"bad conscience\" in terms of outwardly directed rebellions of potentially macropolitical consequence, his later writings consistently characterize the goal of aversive thinking as the deliberate escalation of conflict within the philosopher's own soul. The philosopher's aversion to the reigning idols of the age should not be confused with a direct attempt to implement social reforms, even in the event that some such reforms eventually result as unintended by-products of his aversion. The primary targets of the philosopher's aversion are the manifestations and reflections of the age that prevail within himself:\n\n> What does a philosopher demand of himself first and last? _To overcome his_ _time in himself,_ to become \"timeless _[zeitlos]._ \" With what must he therefore engage in the hardest combat? With whatever marks him as a child of his time. (CW P, emphasis added)\n\nThe enduring value of the philosopher's aversion thus lies in its capacity to promote his own \"timelessness,\" which is the goal of Nietzsche's moral perfectionism. From this \"timeless\" perspective the philosopher may \"objectively\" survey the historical idols that spawned and sustain him, and thus, through a ruthless regimen of self-examination, take the measure of the age he involuntarily represents. Nietzsche consequently locates the genius of Schopenhauer in his campaign to \"conquer his age in himself\" (SE 3).\n\nFollowing Emerson, Nietzsche prizes aversion, resistance, and opposition because they contribute to the production of the philosopher's \"next\" self, each incarnation of which affords the philosopher a clearer glimpse of the truth of his age:\n\n> By applying the knife vivisectionally to the chest of the _very virtues of_ _their time,_ [philosophers] betrayed what was their own secret: to know of a _new_ greatness of man, of a new untrodden way to his enhancement. (BGE 212)\n\nAlthough bound like his contemporaries by the horizons of understanding that define their age, the antagonistic \"genius\" is able to illuminate the \"higher order and truth\" of the age as a whole. The philosopher gains an immanent critical perspective on his age by resisting those of its emanations that reside within himself. We consequently may learn from the philosopher how to take the measure of an age: \"through Schopenhauer we are all _able_ to educate ourselves _against_ our age\u2014because through him we possess the advantage of really _knowing_ this age\" (SE 4).\n\nThe precise target of the philosopher's aversion is not, strictly speaking, the indelible imprint of his age upon his soul, but his _consent_ to the reigning idols and values of his day. Although one cannot alter the concatenation of historical contingencies that have collectively fashioned one's present self, one _can_ oppose the particular constitution of one's present self, thereby refusing to consent to prevailing standards of human flourishing. The philosopher practices aversion not by \"coming to terms\" with his historicity, nor by meekly assenting to his historical destiny, nor by fatuously mouthing the Zarathustran refrain \"Thus I willed it!,\" but by railing against his voluntary participation in the signature prejudices of the age.\n\nWhile self-contempt is ordinarily quite destructive, a certain form of self-directed hatred is not only compatible with moral development, but absolutely indispensable to its successful fruition. When directed toward one's consent to the historical conditions of one's identity (rather than toward these conditions themselves), _shame_ can play a powerful role in one's moral growth, especially if it is inspired by the aversive labors of a great human being:\n\n> [T]he sign of [one's consecration to culture] is that one is ashamed of oneself without any accompanying feeling of distress, that one comes to hate one's own narrowness and shrivelled nature, that one has a feeling of sympathy for the genius who again and again drags himself up out of our dryness and apathy... (SE 6)\n\nShame frees the aversive philosopher not from the historical conditions that define him, but from the constraint of these conditions. By eliminating the constraint of his age upon himself, the philosopher may now turn his historical destiny to his advantage, as the condition of his accession to the \"next\" selves that await him. While the philosopher's \"next\" selves remain timely, necessarily reflecting the reigning forms and order of the age, they are also \"timeless,\" insofar as they are born of an aversion to these forms and order.\n\nNietzsche's controversial appeal to shame as a catalyst of moral development derives from his economic designation of human genius. While virtually any soul (and especially those that are already inwardly destroyed) can muster sufficient shame to drown the pain of the bad conscience in a flood of resentment, this familiar gambit signifies (rather than causes) the irreversible corruption of the soul in question. One may enjoy the analgesic properties of self-contempt only if one first construes one's continued existence as a ghastly, grotesque punishment; this desperate addiction must certainly end in suicidal nihilism. In order to produce and harbor sufficient self-hatred to propel oneself beyond the perceived limitations of one's present self, one must already possess a robust soul, informed by multiple striations of internal difference and rank. More importantly, only a fortified soul can withstand the explosion of self-hatred that is needed to wrench oneself from the inelastic moorings of conformity and convention; impoverished souls cannot afford to turn self-hatred to their constructive advantage. As we shall see, Nietzsche consequently recommends the generative power of shame only to those rare, _\u00fcbermenschlich_ individuals who can also afford to endure the delusional madness induced by _er s._\n\n### Self-Overcoming\n\nNietzsche proposes \"self-overcoming\" as his preferred model for the aversive activity that constitutes a regimen of moral perfectionism. As he understands it, self-overcoming comprises a discipline of self-perfection based on a principle of assimilation or incorporation. The goal of self-overcoming is to gain for oneself a measure of freedom from the limitations of one's age, in order that one might command an expanded range of affective engagement and expression.\n\nAnticipating his readers' desire for him to name for them the philosopher's regimen of self-directed aversion, Nietzsche remarks,\n\n> You want a word for it?\u2014If I were a moralist, who knows what I might call it? Perhaps _self-overcoming [Selbst\u00fcberwindung].\u2014_ But the philosopher has no love for moralists. Neither does he love pretty words. (CW P)\n\nNietzsche may not love \"pretty words\" like _self-overcoming_ , but he is \"moralist\" enough to suggest this term for the moral perfectionism he advocates. As his reluctance indicates, however, the term \"self-overcoming\" is ripe for misunderstandings and misappropriations, many of which are fostered by his own confusions and indirections.\n\nThe concept of _self-overcoming_ plays a uniquely complex role in the economy of Nietzsche's thought, encompassing both the psychological mechanism of personal moral development _and_ the logical transformation of transpersonal structures, historical movements and political institutions. Indeed, our attention to the _macrocosm_ of self-overcoming, and its logic of historical self-transformation, may shed clarifying light on the _microcosm_ of self-overcoming and its logic of personal self-perfection.\n\nNietzsche commonly employs two terms that are often translated as \"self-overcoming\": _Selbst\u00fcberwindung_ and _Selbstaufhebung._ While the former term appears more frequently throughout his writings (especially so in _Zarathustra),_ the latter term appears in passages of great importance to his post-Zarathustran project of self-presentation (cf. GM III:27; D P:4). Since he appeals to the logic of self-overcoming within both the macrocosm of transpersonal selftransformation and the microcosm of personal self-transformation, we might be tempted to designate the former process by _Selbstaufhebung_ and the latter process by _Selbst\u00fcberwindung._ While some such distinction clearly conveys the spirit of his post-Zarathustran political thinking, his actual use of these terms does not provide adequate textual evidence to support the distinction in question.\n\nHe apparently adapts the term _Selbstaufhebung_ from his dim understanding of Hegel, who similarly employed the term _Aufhebung_ to describe the logic of immanent self-transformation. Despite his admission that the facile dialectics of _The Birth of Tragedy_ emit \"an offensively Hegelian\" odor (EH IV:1), he nevertheless appeals to the saving logic of _Selbstaufhebung_ at crucial junctures throughout his productive career. He apparently intends to mark his distance from Hegel by juxtaposing the terms _Selbst\u00fcberwindung_ and _Selbstaufhebung,_ often in close proximity within a single passage. Witness, for example, his invocation of the quasidialectical \"law of Life\":\n\n> All great things bring about their own destruction through an act of self-overcoming _[Selbstaufhebung]:_ thus the law of Life will have it, the law of the necessity of \"self-overcoming\" _[\"Selbst\u00fcberwindung\"]_ in the nature of Life\u2014the lawgiver himself eventually receives the call: _\"patere legem,_ _quam ipse tulisti\"_ (GM III:27)\n\nNietzsche customarily treats the logical process of self-overcoming not only as inexorable\u2014thereby raising, once again, an \"offensively Hegelian\"\n\nstench\u2014but also as natural. Designating \"Life\" (rather than Hegel's _Geist)_ as the ultimate subject of self-overcoming, he charts the transformations that ensue when any \"great thing\" attempts to constitute itself in accordance with its favored account of its nature and destiny. The process of self-overcoming is complete when the \"great thing\" in question, having failed to articulate itself as originally promised, triumphantly constitutes itself as its opposite or \"other.\" Justice, for example, \"overcomes itself\" when, through a series of failed attempts at self-articulation, it finally constitutes itself as mercy (GM 11:10).\n\nWithin this process of self-transformation, moreover, the destruction and creation of \"great things\" are inextricably linked. For example, although Christian morality \"must end by drawing its _most striking inference,_ its inference _against_ itself\" (GM III:27), its demise will give birth to the \"other\" of Christian morality, which Nietzsche associates with himself (EH XIV:4). He thus intends the term _self-overcoming,_ despite its undeniably destructive connotation, to convey a sense of generative power and promise.\n\nMoving seamlessly from the macrocosm of \"great things\" to the microcosm of individual souls, he similarly treats personal self-overcoming as a complex process of destruction and creation. Nietzschean self-overcoming is not to be confused with the oleaginous tolerance and vapid affirmation that grip modernity in a suffocating stranglehold. The project of moral perfectionism requires that one repudiate one's current self as it comes to manifest the extent of one's odious complicity in conformity and convention. Yet the fashioning of a \"new\" or \"next\" self involves no act of creation independent of one's merciless aversion to one's current self.\n\nIndeed, because the promise of one's \"next\" self arises only in the practice of one's refusal of one's current self, the project of moral perfectionism is both risky and indeterminate. Those who set out on the path of self-overcoming are obliged to raze their old \"homes\" before new ones are yet in sight. Yet it is precisely the prospect of this \"homelessness\" that confers upon moral perfectionism its peculiar urgency and attraction:\n\n> Every time [philosophers] exposed how much hypocrisy, comfortableness, letting oneself go and letting oneself drop, how many lies lay hidden under the best honored type of their contemporary morality, how much virtue was _outlived._ Every time they said: \"We must get there, that way, where _you_ today are least at home.\" (BGE 212)\n\nThe philosopher's aversion thus enables the process of assimilation or incorporation that Nietzsche associates with self-overcoming. Whereas the practice of Christian morality operates on a principle of castration or elimination, self-overcoming incorporates as many, and as diverse, capacities as possible. One overcomes one's age, for example, not by obliterating from one's soul all traces of the prevailing idols and allegiances, but by integrating all such traces into one's identity, by appropriating them for one's own designs. This process of incorporation and integration lends depth and complexity to the economy of the soul, and it is constrained only by the vitality at the philosopher's disposal, which is in turn determined by the relative health of his epoch. \"The price of fruitfulness,\" Nietzsche intimates, \"is to be rich in internal opposition\" (AC 3).\n\nThe philosopher's self-overcoming thus exploits the plasticity of the human soul to engender internal difference and variegation. The greater the distances created within the soul, the greater the range of perspectives the philosopher can entertain and command. Rather than define \"objectivity\" in terms of disinterested contemplation from an abstract, disembodied standpoint, Nietzsche recommends that the philosopher command as many \"perspectives\" as possible:\n\n> There is _only_ a perspectival seeing, _only_ a perspectival \"knowing\"; and the _more_ affects we allow to speak about one thing, the _more_ eyes, different eyes, we can use to observe one thing, the more complete will our \"concept\" of this thing, our \"objectivity,\" be. (GM 111:12)\n\nThe philosopher may not aspire to a privileged, external perspective on his age, but he may venture an immanent critique of his age if he can afford to occupy a representative range of affects and \"perspectives.\" By engaging in aversive activity, the philosopher lends voice to the affective chorus resident within himself, overcoming his age by reprising its signature cacophony.\n\nSelf-overcoming thus presupposes the capacity to accommodate ever greater degrees of difference, opposition and contradiction within one's soul. Proffering a self-referential example of his model of self-overcoming, Nietzsche proclaims,\n\n> Facing a world of \"modern ideas\" that would banish everybody into a corner and \"speciality,\" a philosopher...would be compelled to find the greatness of man, the concept of \"greatness,\" precisely in his range and multiplicity, in his wholeness in manifoldness. He would even determine value and rank in accordance with how much and how many things one could bear and take upon himself... (BGE 212)\n\nUnlike those who seek simplicity, unity and uniformity of soul, the genuine philosopher courts the _monstrum in animo,_ whose propinquity prompted\n\nSocrates's cowardly obeisance to the \"counter-tyrant\" reason (TI 11:9). Once freed from the constraint of his historical identity, the philosopher may turn his \"timeliness\" to the advantage of his own \"untimely\" moral development.\n\nAs a general formula for the self-overcoming he has in mind, Nietzsche recommends turning the soul inside out. This is a painful, invasive process, which requires the philosopher to take up an evaluative standpoint beyond good and evil (GS 380). A project of self-overcoming obliges the philosopher to reclaim the previously estranged \"evil\" and \"sin\" resident within himself, by transforming vices into virtues and afflictions into advantages. Self-overcoming thus provides one with the opportunity to re-structure the \"commonwealth\" that informs one's soul (BGE 19), as one fashions a novel order of rank to accommodate the drives, impulses, pathologies, and homunculi that one has most recently exhumed. In order to contribute to the perfection of humankind, that is, the philosopher must experiment with various interpretations of its innocence, always asking himself, \"How can one spiritualize, beautify, deify a craving?\" (TI V:l).\n\nThe reclamation of previously disowned drives and impulses does not return them to their \"original\" place within the economy of the mortal soul, for the soul is itself transformed in the process of self-overcoming. Self-overcoming neither restores a lost innocence, nor sanctions a \"return to nature\" in the romantic sense that Nietzsche (mistakenly) associates with Rousseau (TI IX:48). The very reliance of the philosopher on (self-directed) aversion ensures that his self-overcomings will contribute to the production of a \"new,\" second nature rather than to the restoration of an authentic, original nature. Just as a volcanic eruption augments the landscape it disfigures, so self-overcoming increases the dimensionality and surface area of the soul, allowing for an expanded range of capacities and expressions. Self-overcoming thus informs the philosopher's soul with an order of rank, but only by placing at risk the current psychic regime. An overflowing will necessarily destroy or sublate its previous configurations, thus fashioning a \"next\" self at the considerable expense of the integrity of one's current self.\n\nNietzsche apparently models the aversive mechanism of self-overcoming on the _agon_ or contest, which he proposed in an early (unpublished) essay as emblematic of the health and nobility of Homeric Greece. As a self-styled \"warrior\" who lives only to test himself in \"battle,\" Nietzsche strives not so much to win his quixotic contest with modernity\u2014which would, if it were possible, extinguish the _agon_ \u2014as to prolong its duration while continually raising the stakes of participation. His model here is probably the tragic hero, whose struggles against an uncompromising fate elicit our sympathy not because they promise success, but because they express a heroic strength of will. At least on Nietzsche's reconstruction of the _agon,_ aspiring \"warriors\" compete not so much to vanquish upstart opponents or to alter an ineluctable destiny, as to enact their overflowing, _\u00fcbermenschlich_ power, The _agon_ furnishes a public space within which contestants are invited to exceed themselves, to transform themselves momentarily into _signs_ of the superfluous vitality that courses through them. As we shall see, it is precisely this significatory capacity that enables the political dimension of Nietzsche's moral perfectionism.\n\n### Self-Creation vs. Self-Discovery\n\nA precise determination of Nietzsche's moral perfectionism is complicated by his apparent recommendation of two separate models of self-overcoming. On some occasions he deploys a set of strongly cognitive images and terms, urging us to _discover_ our true, authentic selves, while on other occasions he prefers a distinctly voluntarist vocabulary, exhorting us to _create_ the selves we wish to become. By relying on both sets of images, he indicates that the \"discovery\" and \"creation\" of one's \"next\" self are inextricably linked.\n\nNietzsche is best known for apparently promoting a volitional model of self-creation, which entrusts the project of self-overcoming to a titanic act of will. Speaking on behalf of his unknown \"friends,\" he proclaims, \"We, however, _want_ _to become those we are_ \u2014human beings who are new, unique, incomparable, who give themselves laws, who create themselves\" (GS 335). As a general rule, he tends to convey this model of self-overcoming via a cluster of aesthetic metaphors. In an oft-cited passage, he recommends the project of self-creation by issuing an \"imperative\" to fashion one's life into a work of art:\n\n> To \"give style\" to one's character\u2014a great and rare art! It is practiced by those who survey all the strengths and weaknesses of their nature and then fit them into an artistic plan until every one of them appears as art and reason and even weaknesses delight the eye. Here a large mass of second nature has been added; there a piece of original nature has been removed\u2014both times through long practice and daily work at it. (GS 290)\n\nThis strongly voluntaristic model of self-overcoming is further reinforced by his ridicule of the Socratic\/Enlightenment ideal of self-knowledge or self-discovery, which presupposes that a true, authentic self lies within, waiting to be discovered. He counters the Delphic injunction by calling into question the very possibility of a definitive self-knowledge:\n\n> \"Everybody is farthest away\u2014from himself;\" all who try the reins know this to their chagrin, and the maxim \"know thyself!\" addressed to human beings by a god, is almost malicious. (GS 335)\n\nEvery gain in self-knowledge contributes to who or what one is, thus continually displacing one's \"true\" self and indefinitely postponing a conclusive self-discovery. The putative \"object\" of self-discovery continually changes as a result of the investigation itself: \"Learning changes us; it does what all nourishment does which also does not merely 'preserve'\u2014as physiologists know\" (BGE 231). Proponents of this volitional model of Nietzschean self-overcoming thus conclude that, because we can discover no authentic selves to which we might be true, it must be the case that we create our ownmost selves.\n\nFor all the textual support in its favor, however, this volitional model of self-overcoming fails to capture Nietzsche's full account of moral perfectionism. First of all, the project of self-creation runs aground on the shoals of idealism. Any attempt to fashion a more authentic self necessarily involves a \"cowardly\" flight from the empirical to the ideal. The determination that one's empirical self is inadequate, unsatisfactory, or defective in some respect potentially implicates one in the metaphysics of morals that Nietzsche's \"immoralism\" ostensibly opposes. Second, this model of self-creation is overly voluntaristic, for it fails to take into account the general limitations of one's creative capacities. One does not \"become what one is\" _simply_ by dint of an act of will; to preach otherwise verges upon cruelty. Third, the excessive voluntarism of this model of self-overcoming betrays the \"confusion of cause and effect\" that is pandemic among philosophers. What some philosophers call \"self-creation\" may be the effect, rather than the cause, of the accession to regency of a novel configuration of one's soul. Challenging the voluntarism that informs both Platonic and Christian ethics, Nietzsche observes that\n\n> [A] well-turned-out human being, a \"happy one,\" _must_ perform certain actions and shrinks instinctively from other actions; he carries the order, which he represents physiologically, into his relations with other human beings and things. In a formula: his virtue is the _effect_ of his happiness. (TI VI:2)\n\nFor these reasons, perhaps, Nietzsche also promotes a distinctly _cognitive_ model of self-overcoming, which sanctions a program of self-discovery. Especially in his post-Zarathustran works, he cautions against the misleadingly voluntaristic models of self-overcoming, warning that \"at the bottom of us, really 'deep down,' there is, of course, something unteachable, some granite of pure spiritual _fatum,_ of predetermined decision and answer to predetermined selected questions\" (BGE 231). Nietzsche's fatalism, which plays an increasingly important role in his post-Zarathustran writings, thus mitigates the optimism and exuberance conveyed by his rhetoric of self-creation. This \"spiritual _fatum_ \" comprises those intractable, relatively permanent elements of one's identity that one cannot readily change. It is crucial that one limns the contours of this spiritual _fatum,_ for it effec tively restricts the sphere of self-overcoming, thereby limiting the range of \"next\" selves one can become. On this strongly cognitive model, the task of self-overcoming will apparently require a healthy reverence for that _fatum_ within oneself that proves resistant to aesthetic rehabilitation. In deference, perhaps, to the importance of this project of self-discovery, Nietzsche proposes _amor fati_ as his \"formula for greatness in a human being\" (EH 11:10).\n\nWhile Nietzsche may appear simply to vacillate between these models of human flourishing, his actual goal is to propose a synthesis or composite of the two. One \"becomes what one is\" by overcoming oneself, which always involves elements of both self-creation and self-discovery. As we have seen, his readers customarily define his moral perfectionism through a process of elimination: selfovercoming is a matter either of creation or discovery, and we have good reasons for eliminating one of these options. Proponents of the model of self-creation, for example, arrive at their determination of Nietzschean self-overcoming not by way of actually creating themselves anew, but by way of their doubts concerning the possibility of self-discovery. Champions of the model of self-discovery similarly point not to their obvious success in gaining a definitive self-knowledge, but to the facile idealism involved in any campaign to create oneself anew. It is Nietzsche's intention, however, to expose the distinction between self-creation and self-discovery as sheltering a false dichotomy. While his voluntaristic rhetoric suggests the construction of selfhood, his fatalism recommends the discovery of an authentic self. One \"becomes what one is\" only by combining elements of cognition and volition, discovery and creation.\n\nThe composite nature of Nietzsche's moral perfectionism is crucial to his political thinking, for only the combination of self-creation and self-discovery engenders the _cruelty_ \u2014both to oneself and to others\u2014that ensures the nomothetic impact of self-overcoming. On their own, self-creation and self-discovery both fail to fascinate and to arouse. Both are eminently safe (and fatuous) strategies for \"becoming what one is,\" and they are likely to seduce no one. Only the volatile mixture of volition and cognition, which the philosopher's experiments cruelly detonate, can engender that dimension of Dionysian excess that simultaneously quickens and jeopardizes the economy of the soul. As we shall see in the next chapter, this potentially mortal expenditure in turn guarantees the self-inflicted violence that some others find so erotic. In order to become nomothetic, and thus political, a regimen of self-overcoming must combine elements of both volition and cognition.\n\n### Self-Overcoming and Self-Experimentation\n\nOwing to the inherent risks involved in self-overcoming, Nietzsche regularly recommends his moral perfectionism as an exercise in self-experimentation. Through self-experimentation, the \"genuine philosopher\" accedes to those \"next\" selves that enable an immanent critique of the age as a whole. But the philosopher who seeks to overcome his time in himself possesses no recipe or formula for becoming \"timeless.\" He must consequently experiment with a variety of aversive activities, all of which potentially place his soul at mortal risk.\n\nThe rebellion sanctioned by Nietzsche's model of self-overcoming is not merely theoretical or ideological in nature. In order to take the measure of their age, philosophers must always place themselves in grave danger, insofar as they unleash toward themselves the destructive power of self-contempt. The \"philosophers of the future\" are thus described as \"critics in body and soul,\" whose \"passion for knowledge force[s] them to go further with audacious and painful experiments than the softhearted and effeminate taste of a democratic century could approve\" (BGE 210).\n\nGenuine philosophers experiment with themselves not in the sense of trying on exotic fashions, causes and poses, but in the sense of turning _against_ themselves the excess affect that defines their destiny: \"the genuine philosopher...risks _himself_ constantly, he plays _the_ dangerous game\" (BGE 205). The project of moral perfectionism thus involves the philosopher in an ever-escalating regimen of reconstituting himself through the implementation of novel ascetic disciplines. As we shall see in Chapter 6, Nietzsche actually attempts to hijack the ascetic ideal for his own political campaign to secure the survival of the will.\n\nWhile Nietzsche often ridicules the limitations of human interiority, exposing the \"growth of consciousness\" as a \"danger\" and \"disease\" (GS 354), diagnosing the \"bad conscience\" as a potentially terminal illness (GM 11:16), and disparaging consciousness as a feeble organ of relatively recent development (AC 14), he nowhere advocates a renunciation of interiority or a flight from conscious activity. He likens the \"disease\" of the bad conscience to a \"pregnancy,\" which may someday culminate in the \"birth\" of the sovereign individual (GM 11: 2). His political project aims at the perfection (rather than the transcendence) of humankind, a task that requires the completion (rather than the reversal) of the transition from natural animal to human animal. The completion of this transition in turn involves perfecting the interiority upon which the human animal necessarily relies. Despite its inherent limitations, then, consciousness remains our best means of resisting the decadence of late modernity.\n\nThe eventual perfection of humankind may entail a selective amnesia of particular patterns of acquired behavior, but this forgetting will more closely resemble an active \"inpsychation\" than the _vis inertiae_ that is desperately embraced in late modernity (GM II:1). Rather than abandon consciousness and embrace its \"other,\" the philosopher must seize control of the cultural production of interiority and guide this production to a more satisfactory conclusion. While Nietzsche reserves this larger task for the \"philosophers of the future,\" he also contributes to its fruition by pursuing his project of moral perfectionism. The orientation of his own self-experimentation is prospective rather than retrospective, pointing forward to the creation of new, undiscovered selves.\n\nNietzsche occasionally registers a preference for aesthetically pleasing souls, which has prompted some readers to ascribe to him a strictly formal, Apollinian model of self-overcoming. For example, he identifies the \"one thing needful\" as the \"constraint of a single taste,\" which \"gives style\" to one's character (GS 290). On its own, however, a formally coherent soul might signify an incapacity to accommodate the internal contradictions that distinguish a healthy soul. Regardless of the \"style\" it displays, a two-dimensional, self-restrained soul is ill-suited to the self-overcoming Nietzsche recommends, for it cannot withstand the internal conflict on which moral perfectionism is predicated. In fact, he endorses \"the constraint of a single taste\" not for its own sake, nor for the aesthetic pleasure it may precipitate, but only insofar as it better prepares the soul to accommodate an ever excessive, overflowing will. By virtue of their aesthetically pleasing constitution, then, two-dimensional souls are probably safe from the dangers of self-overcoming, but only because their temperate \"style\" bespeaks an irrecuperable decay.\n\nNietzsche's model of self-overcoming _is_ strongly Apollinian, insofar as it promotes the mastery within a single soul of as many tensions and contradictions as possible. But this model is also undeniably Dionysian, for it promotes internal mastery only as a means of further expanding the capacity of the soul, an ever-escalating process that must eventually culminate in the destruction of the soul. The inherent dangers of self-overcoming are therefore great, for the human soul does not admit of an infinite degree of plasticity. The incorporation into the soul of additional distance and contradiction may secure the philosopher's claim to take the measure of his age, but it also exerts a potentially mortal strain on the economy of his soul. Indeed, the immediate goal of any regimen of self-overcoming is simply to secure the conditions of future self-overcomings; the philosopher cultivates and disciplines his soul only in order to place it at ever greater risk. The philosopher who constantly overcomes himself thus \"builds his city on the slopes of Vesuvius\" (GS 283), for he voluntarily stations himself on the brink of Dionysian excess and disintegration.\n\nNietzsche's model of self-overcoming thus suggests a hospitable context for his otherwise audacious claim that _\"genuine philosophers_... _are commanders and_ _legislators\"_ (BGE 211). He locates the essence of lawgiving _not_ in the redaction of positive law, but in the act and enactment of aversion itself. The outward signs of political legislation will vary in accordance with the vitality of the people or the age in question, but the internal mechanism of lawgiving remains constant throughout all epochs. Decadent philosophers too are lawgivers, even though their legislations are not reflected in the positive law of real cities, for they too engage in aversive activity. The seemingly idle declamations of Socrates and Nietzsche thus differ \"only\" in outward expression from the world-historical decrees of Manu, Caesar, and Napoleon.\n\nNietzsche thus conceives of philosophers as commanders and legislators, but only insofar as they stand in open contradiction to the reigning idols and values of their age. He also suggests that the sphere of the philosopher's legislative jurisdiction is strictly bounded by these enacted aversions. We should therefore expect these \"legislations\" to take the form of iconoclastic proclamations against the epoch as a whole. In order to discern and appreciate the legislations of decadent philosophers, we must attend not to the outward forms and expressions of their lawgiving, but to the nomothetic activity of aversion itself. The political consequences of this activity generally correspond to the residual vitality of the age in question.\n\n### The Case of Nietzsche\n\nThe case of Nietzsche provides us with an instructive, self-referential example of the model of self-overcoming he recommends. While he readily concedes his own decay, he also distinguishes between himself and another famous decadent: \"Well, then! I am, no less than Wagner, a child of this time; that is, a decadent: but I comprehended this, I resisted it. The philosopher in me resisted\" (CW P).\n\nAlthough it may be tempting to conclude that \"the philosopher\" in Nietzsche enabled him to _escape_ the decadence that besets modernity, this is not what he says. His relation to decadence, encapsulated in his enduring \"need\" for the loathsome Wagner, more closely resembles an involuntary affliction than a voluntary (and corrigible) affiliation:\n\n> When in this essay I assert the proposition that Wagner is harmful, I wish no less to assert for whom he is nevertheless indispensable\u2014for the philosopher. Others may be able to get along without Wagner; but the philosopher is not free to do without Wagner. (CW P)\n\nJust as a principled opposition to racism need not free an inveterate bigot from his defining prejudices, so Nietzsche's aversion to decadence fails to deliver him from the besetting affliction of modernity as a whole. To resist decadence is not to free oneself from decadence, but to become a specific type of decadent. His regimen of self-overcoming does not eliminate his \"periodic\" lapses, but it may enable him to express his decadence in the form of a \"timeless\" resistance to the decadence of his age. He consequently overcomes his decadence not by eliminating or reversing it, but by resisting it, by opposing it even as it constitutes his identity.\n\nIn order to overcome his decadence, Nietzsche implements a \"special self-discipline\" that requires him \"to take sides against everything sick in [him]\" (CW P). His \"convalescence\" takes the form of a gradual estrangement from the practices, customs, manners, idols, and people who either aggravated or prolonged his illness. Because he does not enjoy the anchoritic luxury of Zarathustra's mountaintop retreat, he must become a \"cave\" unto himself, into which he might withdraw to gain a critical distance from the signature forms and order of his day. Ensconced in virtual solitude, he declares war on his consent to the emanations of modernity lurking within himself.\n\nOn Nietzsche's (self-serving) account, this regimen of solitude has been a resounding success. He subsequently congratulates himself for having achieved the internal _pathos_ of distance that he associates with the noble soul:\n\n> For the task of a _revaluation of all values_ more capacities may have been needed than have ever dwelt together in a single individual\u2014above all, even contrary capacities that had to be kept from disturbing, destroying one another. An order of rank among these capacities; distance; the art of separating without setting against one another; to mix nothing, to \"reconcile\" nothing; a tremendous variety that is also the opposite of chaos... (EH 11:9)\n\nAs in all cases of self-overcoming, the proof of Nietzsche's success lies in his capacity \"to turn even what is most questionable and dangerous to [his] advantage and thus to become stronger\" (EH 11:6). He proudly describes himself as both a \"decadent\" and \"the _opposite [Gegenst\u00fcck]_ of a decadent,\" explaining that he always instinctively chooses \"the _right_ means against wretched states\" (EH 1:2). This dual experience with decadence has enabled him to \"master\" the art of reversing perspectives (EH 1:2). Although he remains a decadent, he now commands an immanent critical perspective on the decadence of modernity, such that he can take the measure of the age as a whole.\n\nThe logic of self-overcoming thus diverts our attention from the external targets of Nietzsche's polemics to their internal manifestations. The outwardly directed polemics for which he is famous\u2014against moralists, priests, dogmatists, and decadents\u2014are best understood as occasions for galvanizing an internal resistance to the moralists, priests, dogmatists, and decadents who inhabit his own polycentric soul. His resistance thus informs his crowded soul with an order of rank, which secures the \"happiness\" of the psychic \"commonwealth\" as a whole (BGE 19). His legislations too are best understood as primarily self-directed, and all of his post-Zarathustran writings bear the unmistakable stamp of an author cheerfully at odds with himself. These writings become the sites of his own political legislations, wherein he enacts his own self-overcomings.\n\nNietzsche resists the idols of modernity not in order to vanquish his decadence \u2014this disastrous gambit earlier plunged him into pessimism and despair (H I:P:3) \u2014but in order to transform himself into a more resilient type of decadent. Whereas the virile heroes of the Homeric myths would customarily test their strength against external opponents, such as enemies, gods or fate itself, Nietzsche is limited by his decadence to internal contests against the prevailing idols and values. He resists and opposes his own decadence simply to prove to his unknown friends that the will is still (relatively) strong in him, that his dying sun still burns bright. He consequently seeks to ensure that his self-overcoming does not reach a premature conclusion, lest his contribution to the enhancement of ethical life come to an end as well.\n\nIn order to transform himself into a sign, and thereby attract the audience he needs, he must adopt a regimen of self-overcoming that places his soul at mortal risk: \"[We] modern men, like semi-barbarians...reach _our_ bliss only where we are most\u2014 _in danger_ \" (BGE 224). When properly cultivated, that is, the soul becomes an internalized version of a heroic battlefield. onto which \"civilized\" warriors dare not tread.\n\n# 5  \nThe Philosopher's _Versucherkunst_\n\n> But lest I should mislead any when I have my own head and obey my whims, let me remind the reader that I am only an experimenter... I unsettle all things. No facts are to me sacred; none are profane; I simply experiment, an endless seeker with no Past at my back.\n> \n> \u2014Ralph Waldo Emerson, \"Circles\"\n> \n> A moral view can never be proven right or wrong by any ultimate test. A man falling dead in a duel is not thought thereby to be proven in error as to his views. His very involvement in such a trial gives evidence of a new and broader view. The willingness of the principals to forgo further argument as the triviality which it in fact is and to petition directly the chambers of the historical absolute clearly indicates of how little moment are the opinions and of what great moment the divergences thereof. For the argument is indeed trivial, but not so the separate wills thereby made manifest.\n> \n> \u2014Cormac McCarthy, _Blood Meridian_\n\nIt remains to be seen to what political end philosophers can turn their self-experimentation. Does Nietzsche's account of self-overcoming amount to anything more than a self-serving sketch of an unpolitical man? In order to understand how his project of moral perfectionism contributes to the enhancement of humankind, we must investigate the affinities that he detects between self-experimentation and the expenditure of excess vitality.\n\n### An Attempt at an Invitation to Temptation\n\nNietzsche's passion for wordplay and _double entendre_ furnishes some crucial insights into the nature of self-overcoming. The \"genuine philosopher,\" we know, is a _Versucher_ \u2014an experimenter and quester (BGE 205).\n\nHe further explains that\n\n> We opposite men, having opened our eyes and conscience to the question where and how the plant \"man\" has so far grown most vigorously to a height\u2014we think that...the art of experiment _[Versucherkunst]_ and devilry of every kind...serves the enhancement of the species \"man\" as much as its opposite does. (BGE 44)\n\nHere he identifies the practitioners of this \"art of experimentation\" as contributing to the enhancement of humankind. He thus implies that the philosopher's self-experimentation is irreducibly political in nature. This _Versucherkunst_ engenders the superlative human beings who alone warrant the future of the species.\n\nBut a _Versucher_ could also be a tempter, and the _Versucherkunst_ the art of seduction. The _Versuch einer Selbstkritik,_ for example, which Nietzsche prefixed to _The Birth of Tragedy_ in 1886, is obviously an \"attempt at self-criticism,\" but it might also comprise a _temptation_ or _invitation_ to self-criticism. The same is true of _The Antichrist(ian),_ which he advertised in his original subtitle as a _Versuch_ _einer Kritik des Christenthums_ \u2014an attempt at a critique of Christianity, to be sure, but perhaps a temptation as well. Indeed, his explicit association in the passage cited above of the _Versucherkunst_ with \"devilry _[Teufelei]_ of every kind\" overlays these \"invitations\" with the biblical evocation of leading his readers into temptation.\n\nAfter suggesting the name _Versucher_ for the \"new breed\" of philosophers he sees \"coming up,\" Nietzsche confides that \"This name itself is in the end a mere attempt _[Versuch]_ and, if you will, a temptation _[Versuchung]_ \"\u2014a name which confirms his own affinities with this \"new breed\" (BGE 42). He goes on to insist that \"the genuine philosopher...lives 'unphilosophically' and 'unwisely,' above all _imprudently,_ and feels the burden and the duty of a hundred attempts _[Versuchen]_ and temptations _[Versuchungen]_ of life\" (BGE 205). Drawing attention to their \"attempts _[Versuchen]_ and delight in attempts _[Lust am_ _Versuchen],_ \" he proclaims that such philosophers \"certainly...will be men of experiments,\" who \"like to employ experiments in a new, perhaps wider, perhaps more dangerous sense\" (BGE 210). When imagining his \"perfect reader,\" he conjures those \"bold searchers and researchers _[Suchern, Versuchern]_ \" to whom Zarathustra relates his terrible riddle (EH III:3; cf. Z 111:2). Finally, the divided office of the _Versucher_ \u2014as both experimenter and seducer\u2014also boasts a divine lineage: Nietzsche describes Dionysus as \"that great ambiguous tempter-god _[jener grosse Zweideutige und Versucher Gott]_ \" (BGE 295).\n\nNietzsche's wordplay furnishes an important clue to the psychological mechanism involved in self-overcoming. In order to experiment on themselves, philosophers must gather and discharge excess stores of expendable affect. He describes the great human being as a \"calamity,\" and he likens the effects of self-overcoming to a river overflowing its banks and flooding the surrounding countryside (TI IX:44). Indeed, the swollen will of the genius obliterates its observed boundaries naturally and involuntarily, evincing no concern or consideration for its ineluctable effects on others. Born of excess, the philosopher's \"private\" experiments leak uncontrollably into the public sphere, where they are received as temptations and invitations. While the philosopher's legislations remain exclusively self-directed, inscribed \"only\" into the canon of his own self-overcomings, the enactment of these legislations involves an expenditure of affect that eventually exceeds the bounds of private selflegislation.\" id=\"fn0Nietzsche's wordplay furnishes an important clue to the psychological mechanism involved in self-overcoming. In order to experiment on themselves, philosophers must gather and discharge excess stores of expendable affect. He describes the great human being as a \"calamity,\" and he likens the effects of self-overcoming to a river overflowing its banks and flooding the surrounding countryside (TI IX:44). Indeed, the swollen will of the genius obliterates its observed boundaries naturally and involuntarily, evincing no concern or consideration for its ineluctable effects on others. Born of excess, the philosopher's \"private\" experiments leak uncontrollably into the public sphere, where they are received as temptations and invitations. While the philosopher's legislations remain exclusively self-directed, inscribed \"only\" into the canon of his own self-overcomings, the enactment of these legislations involves an expenditure of affect that eventually exceeds the bounds of private selflegislation.\">Nietzsche's wordplay furnishes an important clue to the psychological mechanism involved in self-overcoming. In order to experiment on themselves, philosophers must gather and discharge excess stores of expendable affect. He describes the great human being as a \"calamity,\" and he likens the effects of selfovercoming to a river overflowing its banks and flooding the surrounding countryside (TI IX:44). Indeed, the swollen will of the genius obliterates its observed boundaries naturally and involuntarily, evincing no concern or consideration for its ineluctable effects on others. Born of excess, the philosopher's \"private\" experiments leak uncontrollably into the public sphere, where they are received as temptations and invitations. While the philosopher's legislations remain exclusively self-directed, inscribed \"only\" into the canon of his own self-overcomings, the enactment of these legislations involves an expenditure of affect that eventually exceeds the bounds of private selflegislation.  Insofar as they also comprise \"invitations\" and \"temptations\" to others, the philosopher's \"attempts\" and experiments function as outward signs of the healthy discord raging within his soul.\n\nAs a residue or by-product of his regimen of self-overcoming, the philosopher involuntarily generates an excess of expendable affect. This expenditure emanates outward from its center, thereby transgressing any conventional boundary between public and private domains, and it constitutes itself in the public sphere as a dialogical sign. While the philosopher's regimen of selfovercoming remains essentially \"private,\" its sumptuary residue enters the public sphere as an invitation and temptation to others. On their own, independent of reception, these experiments are nomothetic only for the philosopher in question. They may eventually contribute to the founding of the positive law of a community, but only in the event that the recipients of this invitation endorse it as such.\n\nWhen successful, this internecine struggle arouses in the philosopher the _pathos_ of (internal) distance that only the well-ordered soul can afford to accommodate. Self-overcoming whets the \"craving for an ever new widening of distances within the soul itself,\" contributing to the \"development of ever higher, rarer, more remote, further-stretching, more comprehensive states\" (BGE 257). Resistance to the reigning idols of the age thus informs the philosopher's soul with an \"aristocratic\" order of rank, which Nietzsche proposes as the hallmark of the noble soul (BGE 257). Because this aversive activity produces additional \"next\" selves and further displays the hidden perfections of the human soul, Nietzsche associates the philosopher's self-overcomings with the enhancement of humankind as a whole. In the process of overcoming themselves, philosophers expand the envelope of human perfectibility.\n\nThe philosopher's _Versucherkunst_ not only informs his own soul with an \"aristocratic\" order of rank, but also, and unbeknownst to him, generates the significatory affect that tempts others to enter the \"circle of culture\" described by his expenditures. Schopenhauer, for example, is an educator, but only secondarily (and unwittingly) an educator of others.\n\nHe is an _Erzieher_ primarily of himself, insofar as he elicits and brings to light the hidden perfections resident within his own soul. In the process of educating himself, the philosopher inadvertently provokes others, by tempting them to attempt experiments of their own design. Schopenhauer can therefore provoke others like Nietzsche only because he has already educated himself. The task of \"creating\" worthy disciples, in preparation for which Zarathustra seized his cruel hammer (Z 11:2), has thus been radically recast: the philosopher \"creates\" a community only indirectly and unwittingly, through his expenditure of the excess affect required to turn the hammer on himself. He thus becomes a sign unto himself, irrepressibly projecting his self-directed legislations into the public space that surrounds him.\n\nHence it is the expenditure of excess affect that transforms the philosopher's experiments into temptations. Nietzsche's self-experimentation thus conveys both senses of the philosopher's _Versucherkunst: because_ the philosopher's experiments are self-directed and self-inflicted, they are _also_ temptations or invitations to others. While Nietzsche believes, following Emerson, that each invitation is radically singular in address, privately tempting _only_ the solitary individual who receives the unintended calling, he also believes that the invitations received by the various members of one's community or \"kind\" may overlap significantly, especially with respect to the public consequences they entail. The resulting imbrication of kindred regimens of self-overcoming enables Nietzsche to derive a fairly rich field of intersubjectivity\u2014a \"we\"\u2014from the radically subjective temptations individually received by the community's founding members.\n\n### The Manifold Genius: Philosopher, Artist, and Saint\n\nIn order to appreciate the political consequences of the philosopher's _Versucherkunst,_ we must determine how and why an expenditure of excess affect transforms an experiment into a temptation. With respect to those witnesses who receive his experiments as invitations, the exemplary human being appears in the multiple aspects of philosopher, artist, and saint. Unlike the awkward suitors whom Nietzsche lampoons for their bungling pursuit of Truth (BGE P), _his_ philosophers are always also artists and saints in their own right. Indeed, the coalescence of this triumvirate within a single soul accounts for the political dimension of self-overcoming. Hence it is in his conception of the manifold genius that he ultimately links politics and art, self-overcoming and self-creation.\n\n#### The Philosopher\n\nNietzsche never adequately explains the psychological mechanism that renders political the self-overcomings of exemplary individuals, but his writings contain sufficient clues to support a plausible reconstruction. In a remarkable passage from his _Untimely Meditation_ on Schopenhauer, he outlines the political role of the aversive genius:\n\n> It is hard to create in anyone this condition of intrepid self-knowledge because it is impossible to teach love; for it is love alone that can bestow on the soul...the desire to look beyond itself and to seek with all its might for a higher self as yet still concealed from it. Thus only he who has attached his heart to some great man is by that act _consecrated to culture._ (SE 6)\n\nHuman communities are founded on any number of principles and pretexts, but culture originates only in the love excited by, and bestowed upon, a great human being. Only those who \"attach their hearts\" to an exemplary figure may enter the \"circle of culture.\" Nietzsche thus identifies love not only as the impetus to self-overcoming, but also as the constitutive and unifying principle of culture itself. The existence of great human beings is therefore a precondition of culture, as are the measures required to produce them in any specific epoch. Hence his concern to continue his own regimen of self-overcoming in the face of the advanced decay that besets his age: even in the twilight of the idols, exemplary human beings are both possible and necessary, as objects of consecratory love.\n\nWhile Nietzsche ventures no further elaboration of this important thesis, the love he proposes as constitutive of culture closely resembles Platonic _er s._ 5 The consecratory properties to which he appeals suggest a love that is exclusive both in its attachments\u2014only great individuals are its proper objects\u2014and in its possession\u2014for the lover too must become, if only temporarily, a squanderer. Only a great human being introduces others to the \"next\" selves they might become, and only those individuals who are touched by the madness of _er s_ dare to enter \"the circle of culture\" described by the self-overcomings of a representative exemplar. This love is furthermore as rare and precious as the \"intrepid self-knowledge\" that leads the soul \"to look beyond itself.\" In a note from the spring of 1888, Nietzsche explicitly identifies love as the catalyst of self-perfection: \"And in any case, one lies well when one loves, about oneself and to oneself: one seems to oneself transfigured, stronger, richer, more perfect _[vollkommener],_ one _is_ more perfect\" (WP 808).\n\nOther forms of love, such as _agape,_ the universal love, celebrated by some Christians, may certainly play an important role within specific cultural settings, but they cannot serve to found or constitute culture. The founding of culture requires as its catalyst nothing less than _er s,_ the most powerful, carnal, discriminating, and dangerous form of love known to humankind:\n\n> [L]ove as _fatum,_ as fatality, cynical, innocent, cruel\u2014and precisely in this a piece of Nature. That love which is war in its means, and at bottom the deadly hatred of the sexes!... Even God...becomes terrible when one does not love him in return. (CW 2)\n\nEthical life may be enriched by _agape,_ especially within the demotic stratum of a hierarchical society, but it is sustained and nurtured only by _er s._ Only _er s_ furnishes the psychological motivation to overcome oneself, to place one's soul at risk in the pursuit of self-perfection.\n\nIt is the business of politics, Nietzsche believes, to oversee the production of those rare, exotic individuals who, by virtue of their _\u00fcbermenschlich_ beauty, excite in others the stirrings of _er s._ Indeed, the production of such individuals is coextensive with the production of culture itself. He characteristically refers to such individuals as \"lucky strikes\" (AC 4), for they emerge only rarely as a matter of design; this means, then, that culture itself usually arises only as a fortunate accident within the sumptuary economy of Nature.\n\nExceptions to the political rule of chance are indeed rare, for they issue forth only from the untimely insight that genuine beauty, the source of the erotic attraction that founds culture, is \"no accident\":\n\n> The beauty of a race or family, their grace and graciousness in all gestures, is won by work: like genius, it is the end result of the accumulated work of generations... The good things are immeasurably costly; and the law always holds that those who _have_ them are different from those who _acquire_ them. (TI IX:47)\n\nIn order to oversee the founding of culture, the philosopher must first legislate the production of a genuinely beautiful human being, one whose perfections are sufficiently developed that others will \"attach their hearts\" to him and thereby enter the \"circle of culture.\"\n\nChristianity, Nietzsche charges, has reinforced the rule of chance by further complicating the task of producing exemplars of _\u00fcbermenschlich_ beauty. In order to secure for _agape_ a privileged place within culture, Christian morality demonizes the generative power of _er s:_\n\n> Thus Christianity has succeeded in transforming Er s and Aphrodite\u2014great powers capable of idealization\u2014into diabolical kobolds and phantoms by means of the torments it introduces into the consciences of believers whenever they are excited sexually... And ought one to call Er s an enemy? The sexual sensations have this in common with the sensations of sympathy and worship, that one person, by doing what pleases him, gives pleasure to another person\u2014such benevolent arrangements are not to be found so often in nature! (D 76)\n\nIn a more succinct account of the fundamental antagonism between _Er s_ and the Crucified, Nietzsche charges that \"Christianity gave _Er s_ poison to drink. He did not die of it but degenerated\u2014into a vice\" (BGE 168).\n\nAlthough _er s_ enables the psychological mechanism of his moral perfectionism, Nietzsche nowhere ventures an account of its nature and genesis. Wary perhaps of relapsing into romanticism, he misplaces _er s_ behind the unwieldy \"scientific\" vocabulary that dominates his later writings. Whatever its provenance, his failure to identify explicitly the role of _er s_ in the psychological mechanism of self-overcoming has contributed to the distortion of his philosophy. From his post-Zarathustran writings, however, we can reconstruct with some confidence the signal psychological insight that governs his political thinking: _ask sis_ begets _er s._ That is, the experimental disciplines developed by the philosopher arouse in (some) others the erotic attachment that alone forges the \"circle of culture.\"\n\n#### The artist\n\nExtremely conspicuous in its absence from Nietzsche's later writings, especially in comparison with _The Birth of Tragedy,_ is an appeal to _art_ as a potentially unifying cultural force. He retains an enduring interest in aesthetics, but he no longer believes that art can play in modernity the unifying macropolitical role he assigned to it in _The Birth of Tragedy._ He remains optimistic about art itself as a nomothetic medium, even suggesting that art may someday furnish an alternative to the ascetic ideal (GM III:25), but he no longer envisions art as a vehicle of macropolitical reform.\n\nAlthough it may appear that he has simply abandoned his earlier interest in the political efficacy of art, in favor perhaps of an apolitical interest in the physiology of aesthetics, this development in fact reflects his turn to the political microsphere. Rather than divorce aesthetics from politics, he shifts his focus to the psychology of the artist. The outward, public expressions of artistic creation may vary from epoch to epoch, but the \"physiological\" preconditions of art remain constant.\n\nDespite his uninspiring account of the political alternatives available to agents laboring in the twilight of the idols, he thus maintains his unshaken conviction that all art is irreducibly legislative in nature:\n\n> A psychologist, on the other hand, asks: what does all art do? does it not praise? glorify? choose? prefer? With all this it strengthens or weakens certain valuations... Art is the great stimulus to life: how could one understand it as purposeless, as aimless, as _l'art pour l'art?_ (TI IX:24)\n\nHe furthermore links the nomothetic properties of art to its capacity for creating beauty, whose appreciation furnishes the psychological basis for self-overcoming: \"In the beautiful, man posits himself as the measure of perfection; in special cases he worships himself in it. A species cannot do otherwise but thus affirm itself alone\" (TI IX: 19). Even in peoples and ages marked by advanced decay, that is, great human beings function as artists, as the creators of surpassing beauty and the indirect founders of culture.\n\nNietzsche often speaks of self-overcoming in terms of _self-creation,_ and this fecund metaphor conveys his sense of the nomothetic influence of exemplary human beings. Great individuals are always artists in Nietzsche's sense, for, in the course of their self-overcomings, they inadvertently produce in themselves the beauty that alone arouses erotic attachment. By virtue of their self-creation, exemplary figures come to embody \"the great stimulus to life,\" unwittingly inviting some others to join them in the pursuit of self-perfection. Nietzsche consequently admires Goethe not so much for his creation of Werther and Faust, as for his self-creation, of which all else is derivative (TI IX:49). We should similarly admire Nietzsche for his creation of Zarathustra, but only if we understand this achievement as a symptom or expression of Nietzsche's own originary self-creation.\n\nThis capacity for self-creation is the common element that links all great human beings, whether they be the macropolitical redeemers of the past and future or the micropolitical philosophers of the present. It is only as artists, as producers of surpassing beauty, that exemplary human beings fulfill their social role as lawgivers. Owing to the nomothetic effects of self-creation, exemplary human beings also produce themselves as _\u00dcbermenschen_ and thus unwittingly contribute to the permanent enhancement of humankind.\n\nThe term \"self-creation\" also conveys the irresistibly public nature of the philosopher's self-overcomings. Independent of the philosopher's own aims and aspirations, his overflowing will enters the public sphere as a sign, presenting itself for reception by observers and witnesses who do not share his firsthand, artist's perspective. \"The first distinction to be made regarding works of art,\" Nietzsche decrees, is that between \"monological\" and \"dialogical\" art (GS 367). Monological art is produced by the artist who has \"forgotten the world,\" who disregards altogether the perspective of his likely audience, while dialogical art is produced by the artist who \"looks at his work in progress (at 'himself') from the point of view of the witness\" (GS 367).\n\nThis distinction is crucial to Nietzsche's political thinking, for it explains the difference between the philosopher's orientation to his own self-overcomings and that of his witnesses. A philosopher who maintains a monological orientation toward his own self-overcomings will inadvertently produce for his witnesses an incarnate work of art, whose dialogical significance remains unknown (and uninteresting) to him. This means that the nomothetic influence and consecratory properties of exemplary human beings are, to a great extent, unwitting and involuntary; the dialogical use made of them need bear no correlation to their own monological aims.\n\nFrom the dialogical perspective of the witness, in fact, the squanderings of the genius are often mistaken, especially by impoverished souls, for invitations and seductions. From the monological perspective of the artist, however, these same emanations appear (if at all) simply as the inevitable by-products of the philosopher's private pursuit of self-perfection. Indeed, the ethical life of any community is made possible only by the amoral self-creation of the exemplary human beings who found\u2014and then desert\u2014it.\n\nMuch as the young Nietzsche praised the transformative capacity of the tragic dithyramb, he now appeals to the transfigurative power of self-creation. Here his investigations into asceticism and aesthetics converge, for he also describes _art_ as the product of a swollen will:\n\n> If there is to be art...one physiological condition is indispensable: frenzy _[Rausch]_... What is essential in such frenzy is the feeling of increased strength and fullness... A man in this state transfigures _[verwandelt]_ things until they mirror his power\u2014until they are reflections of his perfection. (TI IX:8\u20139)\n\nThe \"frenzy of the will\" is thus responsible not only for the artist's own \"craving\" for internal self-overcoming, but also, as reflected in the artist's external \"transfigurations,\" for the _pathos_ of distance that inspires others to overcome themselves.\n\nArtists are defined, physiologically, by the involuntary, affective capacity to project outward their native vitality, to transform the world around them: \"this _having to_ transform into perfection is\u2014art\" (TI IX:9). Although this transformative capacity is most obviously reflected in the artist's externalized productions\u2014paintings, sculpture, opera, and so on\u2014Nietzsche cautions his readers not to equate the two. In fact, an artist's primary transformation, of which concrete aesthetic productions are \"merely\" outward expressions, pertains exclusively to himself. The artist always produces himself, albeit unwittingly and inadvertently, as an embodiment of _\u00fcbermenschlich_ beauty, and thus as an object of erotic attraction.\n\nThis insight is especially apposite in the case of decadent artists, who do not command the vital resources needed to transform the external world on a grand scale. Socrates, for example, never founds the city he so eloquently constructs in speech. Yet he remains an artist nonetheless, for in the process of \"building\" his \"city in speech\" he transforms himself into an incarnate work of art. His self-creation in turn awakens the _er s_ lying dormant in (some of) his interlocutors.\n\nIn their private pursuits of self-perfection, exemplary human beings inadvertently produce themselves as works of art for public reception.\n\nThis self-creation has the (unintended) effect on its witnesses of transfiguring \u2014and thus redeeming\u2014the suffering attendant to the ascetic practices deployed in self-experimentation. In a pioneering psychological insight, Nietzsche explains that\n\n> Man, the bravest of animals and the one most accustomed to suffering, does _not_ repudiate suffering as such; he _desires_ it; he even seeks it out, provided he is shown a _meaning_ for it, a _purpose_ of suffering. (GM 111: 28)\n\nWeaned on the ascetic ideal, the human animal avoids only \"meaningless\" suffering. If rendered purposeful within a context of interpretation, suffering actually enables the human animal to attain the threshold level of vitality associated with an enhanced feeling of power. A goal or aim is needed to galvanize an otherwise sclerotic will, and the ascetic ideal accomplishes this task by investing suffering with meaning.\n\nThe ascetic ideal provides meaning by proposing self-inflicted cruelty as constitutive of human flourishing. Our best, and perhaps only, access to the sublime consequently lies in our capacity to suffer meaningfully. The enduring erotic appeal of exemplary human beings thus derives from their power to transfigure, and thereby render meaningful, the otherwise meaningless suffering required by the ascetic ideal. The palpable suffering induced by the philosopher's self-inflicted cruelty is redeemed by the incarnate work of art he becomes in the eyes of his beholders. The philosopher's _Versucherkunst_ thus ensures that each self-overcoming is also a self-creation, that every experiment is also a temptation.\n\nNietzsche entrusts the future of humankind to these higher human beings precisely because of their capacity to transfigure the pain and suffering that necessarily attend self-overcoming. In the eyes of their witnesses, exemplary figures redeem their own suffering, which in turn emboldens these witnesses to attempt painful self-overcomings of their own. The philosopher's self-creation thus \"reminds\" others of their expanded capacity for affective expression, thereby rendering (potentially) sublime the suffering involved in self-overcoming. Nietzsche's philosophers may not be world-historical redeemers, but their self-experimentation always culminates in an act of self-creation. They contribute to the permanent enhancement of the species by advancing the frontier of human perfectibility, which in turn stimulates in others the erotic impulse to engage in self-overcomings of their own.\n\n#### The Saint\n\nIt remains to be seen, however, how the philosopher's self-creation might redeem the suffering of others. On precisely this point, Nietzsche's development as a psychologist is especially welcome, for his pre-Zarathustran works either overlook or underestimate the seductive attraction of ascetic practices. What initially alerted him to the Dionysian roots of tragedy, but what the \"offensively Hegelian\" argument of _The Birth of Tragedy_ failed to capture, was the enticing appeal of the tragic hero, whose sublime self-destruction paradoxically ignites our passions and enflames our own futile aspirations to heroism. Similarly absent from Nietzsche's praise for the triumvirate of philosopher, artist, and saint was an account of the seductive appeal of their self-imposed disciplines. In fact, all of his pre-Zarathustran writings failed to explain adequately the psychological, or transformative, effects of the exemplary human being's labors of self-overcoming. The \"aesthetic justification\" delivered by tragedy, the precise influence of Schopenhauer as _Erzieher_ , the promise of Wagnerian opera\u2014all remained mysteries even to Nietzsche himself.\n\nReturning afresh to the problem of tragedy, he issues the following amendment to his earlier findings: \"What constitutes the painful voluptuousness of tragedy is cruelty; what seems agreeable in so called tragic pity, and at bottom in everything sublime...receives its sweetness solely from the admixture of cruelty\" (BGE 229). Having \"reconsidered cruelty,\" he similarly revises his opinion of its self-inflicted manifestations:\n\n> There is also an abundant, overabundant enjoyment at one's own suffering, at making oneself suffer\u2014and wherever man allows himself to be persuaded to self-denial in the _religious_ sense, or to self-mutilation...he is secretly lured and pushed forward by his cruelty, by those dangerous thrills of cruelty turned _against oneself._ (BGE 229)\n\nWhile most individuals who are at war with themselves are merely enacting their fated dissolution, other souls are sufficiently robust that their internal struggles \"have the effect of one more charm and incentive of life\" (BGE 200).\n\nNietzsche thus believes that witnesses to the philosopher's self-creation are enticed to similar experiments of their own by the giddying prospect of _meaningful_ self-inflicted cruelty. The philosopher's self-creation redeems the suffering of others by furnishing a context of enactment, within which his own suffering is transfigured. By redeeming the suffering involved in all selfovercoming, the philosopher inadvertently participates in the founding of culture:\n\n> Almost everything we call \"higher culture\" is based on the spiritualization of _cruelty,_ on its becoming more profound: this is my proposition. That \"savage animal\" has not really been \"mortified\"; it lives and flourishes, it has merely become\u2014divine. (BGE 229)\n\nThus interpreted (and sublimated), self-inflicted cruelty becomes a source of an incipient erotic bond between the ascetic and his witnesses. The ascetic, who voluntarily embraces forms of suffering that most human beings would otherwise prefer to forgo, actually succeeds in arousing in some others an erotic attraction.\n\nBecause self-overcoming presupposes suffering, any attempt to \"abolish\" suffering in the name of \"modern ideas\" would spell disaster (BGE 44). Suffering is furthermore the art at which human animals excel most resoundingly. All that is valuable within the \"circle of culture\" represents the spiritualization of cruelty. Lest we disable the engine of moral progress, then, Nietzsche urges us not to indulge our pity \"for 'the creature in man,' for what must be formed, broken, forged, torn, burnt, made incandescent and purified\u2014that which _necessarily_ must and _should_ suffer\" (BGE 225). Genuine _Erziehung_ is both painful and dangerous, provoking and eliciting the uncharted plasticities of the human soul. Rather than abolish suffering, we must continue to fashion interpretive contexts in which the suffering endemic to self-overcoming becomes meaningful and sublime. For this reason, the ascetic, or saint, plays a central role in the founding of culture.\n\nThe erotic fascination inspired by the ascetic issues from his painful attempt to tame the will, a project that paradoxically requires of him a superhuman strength of will:\n\n> For an ascetic life is a self-contradiction: here rules a _ressentiment_ without equal, that of an insatiable instinct and power-will that wants to become master not over something in Life but over Life itself, over its most profound, powerful, and basic conditions; here an attempt _[Versuch]_ is made to employ force to block up the wells of force... (GM 111:11)\n\nAscetics awaken the _er s_ of others precisely insofar as they (appear to) squander themselves, for they are (believed to be) possessed of a strength of will that affords them the capacity to swallow even mortal doses of suffering. Nietzsche thus views the excitation of _er s_ as an unintended by-product or emanation of the swollen will, as exemplified by ascetics, and, at the extreme, by martyrs and saints.\n\nMartyrs \"convince\" others only erotically\u2014 _\"Is the cross an argument?\"_ (AC 53)\u2014and only insofar as they derive intense (and unexpected) pleasure from their ascetic disciplines. Seemingly unaffordable feats of self-overcoming awaken in others the _er s_ that conformity and convention have put to sleep:\n\n> So far the most powerful human beings have...sensed the superior force that sought to test itself in such a conquest, the strength of the will in which they recognized and honored their own strength and delight in mastery... [S]uch an enormity of denial, of anti-nature will not have been desired for nothing, they said to themselves. (BGE 51)\n\nOf course, not all ascetic disciplines are capable of arousing _er s,_ for not all ascetics are genuine squanderers. For those ascetics who cannot afford the requisite expenditure of affect, disciplines of self-denial engender a sacrifice rather than a squandering. It is not uncommon, however, for a sacrifice to be mistaken for a squandering; most martyrs are portrayed not as decadents desperately embracing the \"will to nothingness,\" but as heroes who spend themselves in the tragic service of noble ideals. As unintended signs, in fact, the philosopher's \"invitations\" and \"temptations\" are always ripe for misinterpretation. \"Blood\" is powerfully erotogenic, but it is also \"the worst proof of truth\" (Z 11:4). Nietzsche consequently locates the stimulation of _er s_ not in ascetic practices themselves, which may bespeak either a squandering or a sacrifice, but in the expenditure of excess affect.\n\nIn a surprising revision of his argument in _The Birth of Tragedy,_ Nietzsche announces that \"Socrates was also a great _erotic,_ \" for \"he discovered a new kind of _agon_ \"\u2014an _agon_ he waged with himself (TI 11:8). Socrates beguiled the youth of Athens because he appeared to master himself, subjecting his monstrous appetites to the cold, arresting glare of hyperrationality. His self-inflicted cruelty, which aroused in some witnesses ridicule, suspicion, and contempt, was welcomed by others as \"an answer, a solution, an apparent cure\" for the instinctual disarray that had stricken the citizens of Athens (TI 11:9). Socrates eventually cast his erotic spell on posterity by following his ascetic dicta to their logical conclusion.\n\nAccording to Nietzsche, however, the erotic charm of Socrates is attributable to a grand misunderstanding. Socrates was no squanderer, and he died not so much to honor noble ideals as to surrender to his consuming decadence. Socrates in fact orchestrated his own demise, gratefully quaffing a tonic draught of hemlock: \"Socrates _wanted_ to die: not Athens, but he himself chose the hemlock; he forced Athens to sentence him\" (TI 11:12). Other martyrs, including Jesus, have had a similarly erotic\u2014and similarly disastrous\u2014influence on their witnesses, for they too have been mistaken for squanderers: \"The deaths of the martyrs, incidentally, have been a great misfortune in history: they _seduced..._ The martyrs have _harmed_ truth\" (AC 53). The psychological genius of St. Paul lay in his political appropriation of Christ as a martyr, in order that he might exploit the erotogenic possibilities engendered by the unjust death of a Savior (AC 42). As a martyr or sacrifice, Christ awakened in countless others the _er s_ that his life on its own terms inspired in only a few loyal disciples.\n\nLike Plato, Nietzsche accounts for the excitation of _er s_ as a response to a perceived lack or deficiency:\n\n> This one is hollow and wants to be full, that one is overfull and wants to be emptied\u2014both go in search of an individual who will serve their purpose. And this process, understood in its highest sense, is in both cases called by the same word: love... (D 145)\n\n_Er s_ arises in response to the gulf that separates the exemplary human being from all others, and it naturally aspires to bridge this gulf. While the self-overflowing emanations of the will establish and preserve the _pathos_ of distance, _er s_ strives to eliminate or minimize the distance between lover and beloved. The excitation of _er s_ in turn fortifies the lover's will, enabling him to accept his beloved's unintended invitation to enter the \"circle of culture.\" (Although Nietzsche occasionally remarks favorably on heterosexual love, he typically favors male gender designations for both the lover(s) and the beloved.)\n\nAs he explains in a note written in the spring of 1888, the excitation of _er s_ transfigures the lover, elevating him\u2014if only temporarily\u2014to the lofty station of his beloved:\n\n> The lover becomes a squanderer _[Verschwender]:_ he is rich enough for it. Now he dares, becomes an adventurer, becomes an ass in magnanimity and innocence; he believes in God again, he believes in virtue, because he believes in love; and on the other hand, this happy idiot grows wings and new capabilities, and even the door of art is opened to him. (WP 808)\n\nBlinded by _er s_ to his beloved's indifference, \"this happy idiot\" mistakes his beloved's need to disgorge himself for an invitation to permanent union. He consequently \"grows wings\" and ventures to bridge the gulf that ordinarily separates them. In so doing he becomes, like his beloved, a squanderer, leaving comfort, conformity, and good sense behind.\n\nInspired to unimagined heights by the madness of _er s,_ the bewinged lover becomes acquainted with various perfections resident within his own soul, some of which he temporarily shares with his beloved. Nietzsche thus locates in _er s_ the power to create sublime illusions, which alone enable human beings to endure the suffering involved in moral development:\n\n> Love is the state in which man sees things most decidedly as they are not. The power of illusion is at its peak here, as is the power to sweeten and transfigure. In love man endures more, man bears everything. (AC 23)\n\nThe enraptured lover too transfigures the world around him, if only temporarily, for \"even the door of art is open to him.\" The sublime illusions produced in the lover by _er s_ thus enable _nomos_ (or human design) to perfect and complete _physis._ Only when engulfed in the madness of _er s_ will human beings ever attempt to overcome or transcend their natural limitations.\n\nAs _er s_ finally subsides, the temporary union of lover and beloved dissolves, and the \"natural\" gulf between them is restored, in accordance with the order of rank. Emptied once again of the superfluous affect that had burdened him, the beloved distances himself from his frustrated lovers and resumes his solitary pursuit of self-perfection. Lovers \"attach their hearts\" to a great human being and are thereby consecrated to culture, but their love is not reciprocated. Because _er s_ only strives ever upward, these exemplary figures never come to love those whose _er s_ they have inadvertently awakened. Their gaze fixed firmly on the shimmering horizon of human perfectibility, great human beings love only themselves and their \"next\" selves, which immediately vanish upon consummation. In a pithy statement of his own tragic view of the human condition, Nietzsche submits that all great love, by its very nature, stands unrequited.\n\nBut the recession of _er s_ nevertheless leaves the lover changed, for the \"circle of culture\" has been forged around him. As the madness of _er s_ fades, a sense of shame suffuses the neglected lover, who now cannot help but see himself as mired in a life unworthy of his \"next\" self, which he momentarily glimpsed in the visage of his beloved. If this shame is attached directly to the lover's failures and incompleteness, then his opportunity for moral growth is soon crushed by a swelling tide of resentment and self-contempt; he will spend his life in penitent atonement for his momentary affliction of madness.\n\nIf this sense of shame is directed instead to the lover's consent to his incompleteness, to his willingness to remain unworthy of his beloved, to his voluntary complicity in his current imperfections, then it may succeed permanently where _er s_ itself has failed, in spurring the lover to attain his \"next\" self. If the spurned lover resists his plight, refusing to consent to the imperfections of his current incarnation, then he may overcome himself, despite the \"melancholy\" and \"longing\" he feels upon \"discovering in himself some limitation, of his talent or of his moral will\" (SE 3). Although stricken by the unspeakable suffering of an unrequited love, he may now redeem this suffering through the various strategies of \"spiritualization\" presented to him within the circle of culture. Nietzsche thus conceives of culture as the consolation prize that _Er s_ bestows upon his abandoned children\u2014provided that they translate their numbing shame into aversion and moral growth.\n\nAs the founding catalyst of culture, _er s_ plays a central, if underdeveloped, role in Nietzsche's political thinking. It is in the excitation of _er s_ that he unites the twin senses of the philosopher's _Versucherkunst._ The philosopher's experiments are also temptations only if _er s_ is awakened by the self-directed violence involved in self-overcoming; the philosopher consequently tempts some others to attempt self-overcomings of their own. It is _er s_ , finally, that links the _pathos_ of distance with \"that other, more mysterious _pathos_...the craving for an ever new widening of distances within the soul itself\" (BGE 257). If self-creation were not erotic, then the philosopher's attempts to nurture the _pathos_ of distance would never arouse in others a similar craving for internal distance and rank.\n\nBorn of the expenditure of excess affect, _er s_ both drives the mechanism of self-overcoming and secures the consecration of culture. The _er s_ aroused in others by the squandering of these exemplary individuals is ultimately responsible for founding the communities that spring up around them. These communities are grounded neither in democratic consensus, nor in informed consent, nor in a social contract, nor in any other rational or discursive principle, but in a similar capacity for affective engagement and expression. This capacity for squandering, which defines the community and its laws, is established through the self-directed legislations of the founding exemplars. The order of rank among such communities is similarly established by appeal to strictly economic considerations: how much squandering can the community afford and accommodate? How much suffering can the community redeem and transfigure? At the pinnacle of this order of rank, at least on Nietzsche's fanciful reconstruction, stand the ancient Greeks, whose tragic art expresses an affirmation of Life _as it is_ , complete with its ingredient suffering, demise, and dissolution.\n\nThe advent of the \"last will of humankind,\" the will to nothingness, marks the critical point of exhaustion at which the enervated will is no longer capable of awakening _er s,_ the point at which the _pathos_ of distance vanishes altogether. The advanced decay of late modernity thus signifies a state of affective entropy, a disaggregation of the will into quanta so discrete that they can no longer generate the _er s_ needed to sustain the ethical life of the community (CW 7). A dissipation of will would result in the irrecuperable desuetude of _er s,_ and a cessation of _er s_ would nullify the temptations of the _Versucherkunst._ The decadence that besets late modernity thus comprises an assault on beauty itself, as potential objects of erotic attraction are systematically debased. Indeed, if it were no longer possible to \"attach one's heart\" to a great human being, in whom one sees reflected one's own prospects for self-perfection, then one would have no means of redeeming one's hatred of oneself. The future of humankind as a whole would no longer be warranted, and the teachings of Silenus would become wisdom once again.\n\n### An Unintended Experiment: Resentment as Expendable Affect\n\nAs a general theory of the seductive appeal of ascetic practices, Nietzsche's account of the excitation of _er s_ seems plausible. It would also seem, however, that this account only serves to reinforce the futility of any political response to the decadence that besets modernity. If _er s_ is awakened only by an expenditure of excess affect, then how can late modernity, an anemic epoch verging on exhaustion, possibly produce the squandering exemplars who consecrate others to culture? This question becomes crystallized in the case of Nietzsche's attempt to orchestrate a strategic expenditure of his own residual vitality. If he is \"dynamite,\" then from whence does he derive his explosive force? He regularly portrays himself as a disciple of Dionysus, but where precisely might we locate the moment of excess in his own thought?\n\nAs we have seen, Nietzsche's self-creation admits of important political consequences. Although he limits himself to self-experimentation, the overflowing will required for such endeavors eventually exceeds its circumscribed bounds and spills over into the lives of his readers. His aversion to modernity causes him to expend his excess affect, whereby he becomes a sign of the swollen will pulsating within him. This significatory excess in turn awakens the _er s_ that transforms an experiment into a temptation and eventually consecrates others to culture. He consequently identifies his own self-overcoming as his greatest contribution to the permanent enhancement of humankind: \"my humanity does _not_ consist in feeling with men how they are, but in _enduring_ that I feel with them. My humanity is a constant self-overcoming _[Selbst\u00fcberwindung]_ \" (EH 1:8).\n\nIn his own self-overcomings, then, Nietzsche embodies the redemptive triumvirate hailed in _Schopenhauer as Educator:_ as a philosopher, he legislates the production of those exemplary individuals who alone warrant the future of humankind; as an artist, he creates beauty, which transfigures his own suffering; and as a saint, he seduces others to crave the suffering endemic to self-overcoming. In his resistance to decadence, he thus fulfills his destiny as a commander and legislator. Although he reminds no one of the modernity-crushing _\u00dcbermensch_ , his limited sphere of legislative jurisdiction is nevertheless perfectly consistent with the depleted volitional resources of his age. Through his self-experimentation, he hopes not only to resist his twin temptations, nausea and pity, but also to furnish his unknown \"friends\" with aversive strategies designed to postpone the advent of the will to nothingness:\n\n> We violate ourselves nowadays, no doubt of it, we nutcrackers of the soul, ever questioning and questionable, as if life were nothing but cracking nuts; and thus we are bound to grow day-by-day more questionable, _worthier_ of asking questions; perhaps also worthier\u2014of living? (GM 111:9)\n\nNietzsche's pre-Zarathustran works failed, both individually and collectively, in their avowed campaign to restore the voice of Dionysus. Rather than eliminate the distortions imposed on Dionysus by the anti-affective practices of Christian morality, his books contributed additional distortions of their own. Of _The Birth_ _of Tragedy_ he remarks, \"Here was a spirit with strange, still nameless needs, a memory bursting with questions, experiences, concealments after which the name of Dionysus was added as one more question mark\" (BT:AS 3). From the retrospective Preface of 1886 issues a startling admission: the author of _The Birth_ _of Tragedy_ either misunderstood the Dionysian impulse or possessed a merely theoretical grasp of it. In either event, he was unable to apply his understanding to his own circumstances\u2014an embarrassing lapse for a self-proclaimed disciple of Dionysus. As a consequence of this failure, _The Birth of Tragedy_ characteristically confined Dionysus to the measured, restrained precincts of Apollo. The \"music-practicing Socrates,\" for example, whom Nietzsche summoned in _The Birth of Tragedy_ as a symbol for his own youthful hopes, was as cheerful, optimistic and Apollinian as his unmusical counterpart.\n\nIn his post-Zarathustran writings, however, Nietzsche atones for his prior failure to isolate traces of the Dionysian impulse. True disciples of Dionysus, he now understands, affirm Life \"even in its strangest and hardest problems\" (TI X: 5). Since the \"strangest and hardest problem\" facing him is the problem of history, of the \"it was\" (Z 11:20), he must come to affirm not only modernity itself, but also his own resentment of modernity. In a remarkably autobiographical passage, he acknowledges that\n\n> The human being...who wants to behold the supreme measures of value of his time must first of all \"overcome\" this time in himself\u2014this is the test of his strength\u2014and consequently not only his time but also his prior aversion and contradiction _against_ this time, his suffering from time, his un-timeliness, his _romanticism._ (GS 380)\n\nNietzsche cannot eliminate his resentment of modernity, for it partially defines his destiny, but he _can_ overcome it. If he is to take the measure of his age, then he must somehow turn even his own resentment to his advantage.\n\nHere, it would seem, we have reached the outer limits of Nietzsche's experiment with self-overcoming, for his resentment of modernity would appear to defy incorporation into a healthy soul. How could he make use of such vile stuff as resentment, especially the impressive store he harbors? In order to answer this question, and continue to pursue his experiment with self-overcoming, we must part company decisively with his own official account of himself as a philosopher. He overcomes his resentment as he overcomes any other affect, by harnessing it as expenditure.\n\nLike any prepotent, expendable affect, resentment can escalate and intensify a philosopher's self-overcomings, which may in turn elicit from others an erotic response. Nietzsche's abundant resentment of modernity thus constitutes the elusive moment of excess within the economy of his soul and work. Here, in his unbridled ridicule of the puny aspirations and accomplishments of modern man, his vitality, power, and eroticism all attain their convergent heights. If it is true that \"nothing burns one up faster than the affects of _ressentiment_ ,\" and that \"no reaction could be more disadvantageous for the exhausted\" (EH 1:6), then perhaps his breakdown and collapse are best attributed to his self-enflamed resentment of modernity. Perhaps his vaunted \"revaluation of all values,\" which he promises will alter the course of human history (AC P), is nothing more than an \"explosion\" of pent-up resentment. Indeed, if a cure for decadence is out of the question, then an accelerated decay may be most advantageous for his political aims.\n\nAlthough Nietzsche reserves his most caustic invective for \"the man of _ressentiment_ ,\" he also grudgingly admits that _ressentiment_ is not bereft of generative value. _Ressentiment_ , after all, \"becomes creative and gives birth to values\" (GM 1:10). The man of _ressentiment_ \"understands how to keep silent, how not to forget, how to wait, how to be provisionally self-deprecating and humble,\" and \"his spirit loves hiding places, secret paths and back doors\" (GM 1: 10). \"A race of such men of _ressentiment_ ,\" Nietzsche concludes, \"is bound to become eventually _cleverer_ than any noble race\" (GM 1:10). While the clandestine virtues of the \"man of _ressentiment_ \" are not likely to put anyone in mind of the noble warrior or the blond beast, they may be the only productive tools available to the enfeebled representatives of a decadent, epigonic age. Moreover, if Nietzsche cannot rid himself of his own resentment, then his swooning admiration for \"noble races\" and \"active forces\" is, in the end, simply irrelevant; he must either overcome his resentment or capitulate to the decadence of modernity.\n\nI do not mean to suggest that the appropriation of resentment represents a conscious or deliberate strategy on Nietzsche's part. He nowhere allows that he deploys any such strategy, and the success of his self-overcoming may in fact depend upon his ignorance of the nature of his own excesses. By his own account, the _worst_ judge of excess is always the enthusiast in question. The monological perspective of the genius is singularly inappropriate for reckoning the dialogic reception of his expenditures. If his resentment of modernity _does_ contribute to the signature excesses of his life and work, then we should not expect him to know very much about it. Because he is ignorant of the precise nature of his expenditures, his own account of his self-overcomings may deviate wildly from those offered by his witnesses.\n\nNietzsche consistently demonstrates in his self-reflective writings that he cannot accurately locate the moments of excess resident in his own life and work. He knows that his excesses transform him into a sign, but his own interpretations of this sign amount to little more than idealized self-portraiture. He insists, for example, that he is both \"free\" from and \"enlightened\" about _ressentiment_ , and he speaks proudly of the \"Russian fatalism\" that distinguishes him from the \"man of _ressentiment_ \" (EH 1:6). In an apparently autobiographical remark, he explains that\n\n> _Ressentiment_ itself, if it should appear in the noble man, consummates and exhausts itself in an immediate reaction, and therefore does not _poison:_ on the other hand, it fails to appear at all on countless occasions on which it inevitably appears in the weak and impotent. (GM 1:10)\n\nIn his account of the genesis of _Zarathustra,_ moreover, he presents himself as a Dionysian vessel overflowing with spirit, health, inspiration, wisdom, beneficence, and other noble endowments (EH IX:6). He may need to believe, like Zarathustra, that his involuntary emanations are as salubrious as the sun's rays (Z P:l).\n\nI see no prima facie reason to accept his account of himself in _Ecce Homo_ , and I believe that we have strong Nietzschean reasons for subjecting it to further scrutiny. While he is clearly repulsed by acts and agents of resentment, his aversion to them _itself_ usually takes the form of resentment. He thus represents a peculiar type of \"man of _ressentiment_ ,\" and his sole mark of distinction within this ignoble lineage is his ability to turn his _ressentiment_ against itself. Just as in the case of the original \"slave revolt,\" which he vows to reverse, _ressentiment_ \"becomes creative\" in his thought, giving birth to the values that inform his resistance to decadence. His resistance is neither heroic nor noble, but it transforms the \"imaginary revenge\" of _ressentiment_ into a genuine deed _[That]_ , which he identifies as the \"true _[eigentliche]_ reaction\" (GM I:10). Although he never admits as much, he regularly harnesses as expenditure his own resentment of modernity\u2014just as he has led us to expect any decadent philosopher must do.\n\nHis grandiose political aims furthermore _require_ him to appropriate his resentment of modernity. He is neither as wise nor as clever as he thinks himself, and his stores of love, courage, beneficence, and resolve are all precariously low. With the exception of his abundant resentment, he commands no other reservoir of combustible affect to expend. If we are to take seriously his frequent references to himself as \"dynamite,\" then we must compile a realistic inventory of the \"explosives\" at his disposal.\n\nOnly in his resentment of modernity, and of Christianity, does he express the overflowing vitality that he regularly attributes to himself. His resentment of modernity, which earlier precipitated his plunge into romantic pessimism, now attests to the (relative) frenzy of his own swollen will, and so to his own decadent \"health.\"\n\nThe secret hero of late modernity is none other than the reviled \"man of _ressentiment_ ,\" who consumes himself in his own poisonous venom. While this assertion certainly contradicts Nietzsche's familiar rhetoric, his own critique of modernity confirms the failure of the epoch to produce more glamorous heroes. The \"man of _ressentiment_ \" spawns many avatars amid the rubble of modernity, but none so intriguing as Nietzsche himself. His expenditure of excess resentment helps to sustain ethical life in the fragile microsphere of late modernity, but it also leaves its indelible stamp on those whom it attracts. His bizarre pursuit of his \"next\" self, which he alternately sees reflected in Zarathustra, Dionysus, the millennial Antichrist, the \"philosophers of the future,\" and other \"untimely\" heroes, may be perverse, even obscene at times, but it might nevertheless lead to the founding of communities within the political microsphere. From Nietzsche's tragic fragmentation and self-division we may witness the excitation of _er s_ and the consecration of culture.\n\nIf successful in his experiments, he awakens in (some) others the dormant _er s_ that alone consecrates human beings to culture. His self-experimentation thus founds a community constituted by all those who can similarly afford to squander themselves in their aversion to the decadence of modernity. The public consequences of this _ask sis_ are crucial to Nietzsche's political enterprise, for it is his self-experimentation that ostensibly draws others to him and founds communities of resistance within the microsphere. His expenditure of resentment, channelled by self-imposed disciplines of resistance, arouses the _er s_ that draws these fellow travelers to him. Seduced by his \"tempting\" self-experimentation, these \"free spirits\" join him in his efforts to sustain the _pathos_ of distance and to survive the twilight of the idols. Nietzsche's _real_ \"children,\" of course, are not the \"noble warriors\" who populate his fantasies, but the nookdwelling creatures of resentment who have been consecrated to culture by a decadent philosopher. For this very reason, however, his real \"children\" may be uniquely suited to survive the demise of modernity.\n\nThe confusion surrounding the identity of Nietzsche's rightful heirs confirms the greatest danger involved in his own self-experimentation. From the dialogic perspective of their audiences, great human beings always appear, and function publicly, as _signs_ of the (relatively) robust vitality they propagate and expend. In order to orchestrate a timely \"explosion,\" Nietzsche attempts to exploit the significatory by-product of his self-overcomings. The public overflow of his private pursuit of self-perfection contributes a performative dimension to his teachings, supplementing his \"saying\" with a \"showing.\" Indeed, he is counting on this performative dimension to found the community of readers who will join him in shepherding the aimless will through the duration of modernity.\n\nThis significatory function enables the political dimension of Nietzsche's self-overcoming, but it also places the meaning(s) of his performances beyond the sphere of his authorial control. Although he voluntarily elects to engage in aversive activity, this choice marks the limit of his volitional control over his self-experimentation; he neither chooses nor foresees its consequences. \"To become what one is,\" he reminds us, \"one must not have the faintest notion _what_ one is\" (EH 11:9). By its very nature, then, his self-experimentation ultimately exceeds his own purview and control; he is neither privy to his own excesses, nor a reliable judge of their effects.\n\nThe dialogic reception and interpretation of his \"explosion\" need bear no resemblance to the meaning he monologically invests in it. He may attempt to anticipate his reception, by strategically adopting the dialogic perspective of his likely readers, but even this gambit will not protect him from gross misinterpretation and abuse. To transform himself into a sign is to place himself, and his precious teachings, in the hands of readers whom he neither trusts nor controls.\n\nIn Nietzsche's case this proved to be an extremely painful experiment, for it required him to overcome his abundant resentment of modernity, an age whose highest specimens fall pitifully short of the noble standards of bygone epochs. Like Socrates, whose interrogation of Athenian noblemen was designed to test the oracle's surprising pronouncement, he stubbornly resisted the conclusion that he might belong among the best his age has to offer. Indeed, the self-loathing displayed by Socrates and Nietzsche is apparently indicative of the aversive activities of all exemplary human beings who toil in twilight epochs.\n\nLike any representative figure, however, Nietzsche turns this resistance to his own advantage. By dint of his own self-overcomings, he validates the ethical resources scattered throughout the microsphere and draws our attention to the _Versucherkunst_ that produced him. Rather than simply describe the self-experimentation he recommends, he provides us with a map of its course and an embodied example of the success it can have\u2014provided we learn to \"see\" what has heretofore been invisible to us. His occasional bursts of expendable affect serve to mark the path for anyone who can afford to follow. Indeed, if we are to appreciate Nietzsche as an exemplary human being, then we too must overcome our resentment (and despair) of an age so exhausted that we too might legitimately aspire to greatness.\n\n# 6  \nComedians of the Ascetic Ideal\n\n> Around the hero everything turns into a tragedy; around the demigod, into a satyr-play _[Satyrspiel]_ ; and around God\u2014what? perhaps into \"world\"?\u2014\n> \n> \u2014 _Beyond Good and Evil,_ 150\n\nUnable to articulate a specific answer to the founding question of politics, Nietzsche resolves instead to attend to the survival of humankind, until that point at which the \"philosophers of the future\" create new values and found a new epoch. Toward this end, he undertakes a more modest, ancillary political task, which is consistent with his complicity in the decadence of modernity. It is his self-appointed task to safeguard the crippled human will throughout the twilight of the idols. He can certainly do no more than this, and even this task may prove too great for his diminished capacities.\n\n### The Ascetic Ideal\n\nThe political aims of Nietzsche's post-Zarathustran writings are obscured by his apparent ambivalence toward the ascetic ideal. On the one hand, he reveals that the dynastic reign of the ascetic ideal has culminated in a potentially apocalyptic \"will to nothingness.\" If necessary, human beings will orchestrate their own self-annihilation, for even this goal is preferable to no goal at all (GM 111:28). On the other hand, although he persuasively demonstrates that an alternative to the ascetic ideal would be desirable, and perhaps even redemptive, he fails to disclose any actual, viable alternatives that are presently available to us.\n\nThe \"philosophers of the future\" may someday succeed in framing a naturalistic, pro-affective alternative to the ascetic ideal, but Nietzsche himself will not. Seemingly reconciled to a continued reliance on the ascetic ideal, he registers his preference for a specific manifestation of it: \"All honor to the ascetic ideal _insofar as it is honest!_ so long as it believes in itself and does not play tricks on us!\" (GM 111:26). Despite its tone of apparent affirmation, however, this Apollinian statement of preference betrays a _pathos_ of defeat and despair. _Nietzsche_ , the self-proclaimed \"last disciple of the philosopher Dionysus\" (TI X:5), and an accomplished trickster in his own right, now demands constancy and fair play from his most formidable opponent? The manipulator of masks and purveyor of noble lies _par excellence_ now honors the ascetic ideal only in its honest and self-conscious incarnations?\n\nThis humble plea may not put us in mind of the millennial Antichrist, who audaciously promises to \"break history in two,\" but it faithfully conveys the profound sense of ambivalence with which Nietzsche approaches the ascetic ideal. Resigned to both the destructive force and the unchallenged hegemony of the ascetic ideal, he characteristically treats its reign as a necessary evil for the duration of modernity, and perhaps beyond.\n\nIn its most basic, essential form, the ascetic ideal enshrines the life of self-denial as the highest expression of human flourishing. Under the tragic aegis of the ascetic ideal, human beings seek to become \"whole\" and \"complete\" through complex operations of self-vivisection and self-laceration. Meaning is infused into one's otherwise meaningless existence through ever-escalating orgies of self-inflicted suffering. Although most human beings are unlikely to succeed in their quest for completeness, their immersion in ascetic disciplines will very likely distract them from their irremediable fragmentation. Unlike all other animals, human beings derive their threshold level of vitality (or, in Nietzsche's terms, their enabling feeling of power) indirectly and derivatively, as a consequence and by-product of the ascetic disciplines they impose upon themselves. Whereas other animals rely directly and pre-reflectively on their respective endowments of unconscious drives and impulses, the human animal is defined by the violent and irreversible introjection of its natural, instinctual heritage (GM 11:16). This originary wound, which Nietzsche calls the \"bad conscience,\" is the condition of all human life and achievement. In order to unleash the generative powers of this originary wound, human animals must fashion for themselves contexts of interpretation, which provide them with the baseline meaning they need in order to render sublime their existential suffering.\n\nWhile it has become common to distinguish between \"normal\" and \"extreme\" practices of self-denial, reserving the term \"ascetic\" only for the latter, the ascetic ideal itself sanctions no such distinction. It is the silent, signature work of the ascetic ideal to co-opt ordinary, everyday practices of self-denial, so that human beings might \"naturally\" attain the threshold level of vitality that alone confers meaning onto their otherwise meaningless suffering. The ascetic ideal indifferently sanctions any and all practices of self-denial, from those that build character to those that destroy bodies. Familiar, conventional training regimens, such as those required of religious and athletic initiates, are neither more nor less ascetic than bizarre, subterranean rituals of self-mutilation\u2014although particular cultures will invariably valorize the specific ascetic disciplines to which they assign a political preference.\n\nIn all of its forms and manifestations, the ascetic ideal thus sponsors the seemingly paradoxical contest of Life versus Life:\n\n> [P]leasure is felt and _sought_ in ill-constitutedness, decay, pain, mischance, ugliness, voluntary deprivation, self-mortification, self-flagellation, self-sacrifice. All this is in the highest degree paradoxical: we stand before a discord that _wants_ to be discordant, that _enjoys_ itself in this suffering and even grows more self-confident and triumphant the more its own presumption, its physiological capacity for life, _decreases._ (GM 111:11)\n\nAs Nietzsche wisely indicates, however, this paradox is only apparent. The ascetic ideal actually serves the interests of Life itself, by seducing its weakest, sickest forms\u2014those that attain their threshold level of vitality only in self-destruction\u2014 to a continued existence.\n\nAs we have seen, Nietzsche identifies the ascetic ideal as the dominant interpretation known to humankind of the existential suffering associated with the bad conscience. The ascension and authority of the ascetic ideal are therefore attributable to the onset of civilization, which, in exchange for the promise of peace and security, prohibits the spontaneous, outward expression of the unconscious drives and impulses. Indulging himself in a bit of speculative anthropology, he proposes that the ascetic ideal arose in response to a basic human craving for meaning:\n\n> _This_ is precisely what the ascetic ideal means: that something was _lacking,_ that man was surrounded by a fearful void... The meaninglessness of suffering, _not_ suffering itself, was the curse that lay over humankind so far \u2014 _and the ascetic ideal offered man meaning!_ (GM 111:28)\n\nAccording to Nietzsche, moreover, the ascetic ideal constitutes the _only_ response thus far to the suffering of the bad conscience (GM 111:28). The reign of the ascetic ideal is therefore coextensive with civilization itself. To be human is to gain meaning, power, and vitality through disciplines of self-directed violence. Indeed, the governing taboo of civilization forbids human beings from flourishing via any non- or extra-ascetic means, a general proscription of which Freud would later propose the incest taboo as both representative and symbolic.\n\nIn providing meaning for the pain of the bad conscience, the ascetic ideal has presided over the protracted evolution of the human animal. Because death alone can still the existential suffering that attends the forcible introjection of animal drives, we might think of the bad conscience as the ineliminable opportunity cost of human civilization. Civilization demands suffering, and the ascetic ideal renders meaningful the existential suffering that human animals involuntarily inflict on themselves. While acknowledging that most human beings have found this existential suffering to be bearable only under the analgesic influence of metaphysical comforts, Nietzsche nevertheless insists that the bad conscience, under the aegis of the ascetic ideal, has impregnated the species with a future, transforming humankind from a \"goal\" into \"a way, an episode, a bridge, a great promise\" (GM 11:16).\n\nIn light of Nietzsche's grim appraisal of the ascetic ideal, it is not surprising that many readers assume that he intends to offer or disclose an alternative ideal. Although he certainly ascertains the need for an alternative ideal, he nowhere exhibits the wherewithal to satisfy this need. He proclaims, for example, that _art_ is \"much more fundamentally opposed to the ascetic ideal than is science\" (GM III:25), but he refuses to identify the artist as an opponent of the ascetic ideal; in a typically evasive fashion, he instead promises, falsely, to \"return some day to this subject at greater length\" (GM III:25). At the end of essay II of the _Genealogy,_ he similarly invokes the image of a\n\n> man of the future, who will redeem us not only from the hitherto reigning ideal but also from that which was bound to grow out of it, the great nausea, the will to nothingness, nihilism. (GM 11:24)\n\nBut at this point he breaks off abruptly, deferring obscurely to \"Zarathustra the godless\" and resuming his obtrusive silence on the question of an alternative to the ascetic ideal. For many readers, this disappointing aposiopesis serves to punctuate his chronic failure to provide \"serious\" solutions to the grave political problems he addresses.\n\nHe elsewhere alludes, cryptically, to a \"counter-ideal\" promulgated by Zarathustra (EH XI), but he neglects to identify for his patient readers the specific teaching(s) or speech(es) to which he refers. While it is certainly plausible to assume that he may have in mind here his gnomic teaching of eternal recurrence, this teaching is not currently available to us in the form of a \"counter-ideal.\" To _will_ the eternal recurrence, after all, \" _to crave nothing more_ _fervently_ than this ultimate eternal confirmation and seal\" (GS 341), would seem to require a degree of vitality that is absolutely unknown to modernity. Zarathustra cannot distinguish \"his\" teaching of eternal recurrence from the various renditions of it that he rejects throughout his _Bildungsgang_ , and Nietzsche himself is surprisingly unfamiliar with (or uncharacteristically silent about) his own signature teaching of affirmation.\n\nIn light of his smug predictions for the reception of _Zarathustra_ , moreover, any \"counter-ideal\" sheltered therein would seem to be framed not for us, but for the preterhuman audiences of the postmodern future:\n\n> [H]aving understood six sentences from _[Zarathustra]_ \u2014that is, to have really experienced them\u2014would raise one to a higher level of existence than \"modern\" men could attain. Given this feeling of distance, how could I wish to be read by those \"moderns\" whom I know! (EH III:1)\n\nWhile it may still be the case that either Nietzsche or Zarathustra possesses an alternative to the ascetic ideal, any \"counter-ideal\" they might promulgate would necessarily outstrip the diminished faculties of their late-modern readers. Indeed, by the time receptive audiences finally arrive on the scene, this supposed \"counter-ideal\" may be entirely otiose.\n\nHence Nietzsche's enduring ambivalence toward the ascetic ideal: while the installation of a \"new\" ideal would be desirable, it is also incompatible with civilization and the human species as we know them. He consequently delegates the installation of an alternative ideal to his mysterious successors, while resigning himself to the maieutic task of bringing to term the troubled pregnancy of the bad conscience. He consequently undertakes to investigate\u2014by way of self-experimentation\u2014the unknown generative powers of the ascetic ideal. In order to mount a meaningful challenge to the hegemony of the ascetic ideal, that is, he must assume the difficult (and for him uncomfortably effeminate) role of the manly midwife.\n\n### Harming the Ascetic Ideal\n\nNietzsche's ambivalence toward the ascetic ideal reflects his (skeptical) appraisal of the limited political resources available to late modernity. Unable to depose the ascetic ideal, or to propose a viable successor ideal, he must attempt somehow to disrupt from within the \"closed system\" of the ascetic ideal. This endogenous disruption may in turn loosen the stranglehold of the ascetic ideal, so that he and his \"friends\" might implement experimental strategies for surviving the twilight of the idols.\n\nToward the end of his investigation of the ascetic ideal in the _Genealogy_ , Nietzsche finally entertains the question that his readers have long since formulated: \" _where_ is the opposing will that might express an _opposing ideal_?\" (GM III:23). He quickly rules out science, the reigning idol of modernity. \"Men of science\" are not the free spirits he seeks, for their signature faith in truth bears witness to their underlying belief that truth alone can redeem the human condition (GM III:23). This belief in turn betrays the conviction that the human condition stands in need of redemption, a conviction symptomatic of decadence. For similar reasons, he also disqualifies the \"unconditional honest atheism\" that had become so fashionable in his day. A genuine alternative to the ascetic ideal must neither promise nor anticipate the redemption of the human condition.\n\nHaving declined his most obvious and promising options, Nietzsche finally outlines _his_ strategy for challenging the ascetic ideal. In an almost offhand remark, he declares that \"the ascetic ideal has at present only _one_ kind of real enemy capable of _harming_ it: the comedians of this ideal _[die Kom\u00f6dianten_ _dieses Ideals]_ \u2014for they arouse mistrust of it\" (GM III:27). This, the only answer Nietzsche provides to the most important political question raised in the _Genealogy_ , is so strange and unsatisfying, especially in light of the rhetorical crescendo that builds up to it, that few readers can take it seriously. Nor does he deign to shed any additional light on these mysterious enemies of the ascetic ideal. Having raised the tantalizing possibility of meaningful opposition to the ascetic ideal, he quickly changes the subject. He neither divulges the identity of these comedians nor explains how one might join them in harming the ascetic ideal. He lavishes the orotund prose of the _Genealogy_ on such pressing political topics as diet, the chastity of philosophers, and German music, but he never again mentions these \"enemies\" of the ascetic ideal.\n\nIn a typical gesture of deferral, Nietzsche bequeaths to his readers the task of determining the identity and _modus operandi_ of these mysterious \"enemies\" of the ascetic ideal. Relying once again on a strictly functional account of the notion he wishes to convey, he tells us only what these comedians _do._ All we know about them is that they \"harm\" the ascetic ideal by \"arousing mistrust\" of it. How they do so, and to what effect, remains unsaid, if not unknown. Nor is it clear why a burgeoning mistrust would prove harmful to the ascetic ideal, especially with no viable counter-ideal in sight. The unchallenged hegemony of the ascetic ideal thus rules out several promising interpretations of Nietzsche's cryptic reference to the \"comedians of the ascetic ideal\": it would do little good simply to ignore, ridicule, mock, slander, or demean the ascetic ideal, for there exist no alternative ideals to which rebels and critics might pledge their allegiance and petition for asylum. Any apostasy from the ascetic ideal is only apparent, only temporary, and the inevitable \"return\" of the repentant apostate only serves to reinforce the hegemony of the ascetic ideal.\n\nNietzsche may be uncharacteristically laconic on the question of opposing the ascetic ideal, but he nevertheless furnishes some important clues. No one better fits his description of these comedians than Nietzsche himself. Although he nowhere identifies himself (or anyone else) as a comedian of the ascetic ideal, he regularly identifies himself as one of its opponents, claiming as his own the task he assigns to these mysterious comedians. He is furthermore uniquely suited to serve as a comedian of the ascetic ideal, both in terms of his knowledge of it and in terms of his historical situation. He may chafe at the notion that he must fulfill his destiny as a manly midwife, but his own account of the decadence of late modernity leaves him no greater political charge to fulfill. Finally, he could never know if he were actually a comedian of the ascetic ideal. Since these mysterious enemies are defined only by their function (irrespective of their avowed intentions), by their success in \"arousing mistrust\" of the ascetic ideal, only external witnesses can confirm the \"harm\" they have done. Only Nietzsche's readers can confirm that he is (or is not) in fact a comedian of the ascetic ideal, and their only criterion of judgment lies in the \"mistrust\" he has aroused in them.\n\nNietzsche's candid pronouncement of his own complicity in the besetting decadence of his epoch would seem to confirm his qualifications as a comedian of the ascetic ideal. Owing to the restricted confines of his historical situation, the only challenge he could hope to mount must be strictly immanent, exclusively internal to the economy of the ascetic ideal. While he does not explicitly identify the method of immanent critique as his preferred strategy for harming the ascetic ideal, he _does_ indicate why this strategy might be effective.\n\nOn two separate occasions he describes the ascetic ideal as a _\"faute de mieux\"_ (EH XI; GM 111:28). He thus suggests that the success of the ascetic ideal is attributable not to any characteristics or properties essential to the ideal itself, but to the utter lack of alternative ideals and external challenges. As a _faute de mieux_ , the ascetic ideal simply cannot tolerate competing estimations of the value of Life, and it owes its interpretive appeal solely to the monopoly it holds in the business of accounting for human suffering. It has sustained its hegemony not because its signature interpretation of suffering is _true_ , but only because no rival ideals have yet emerged:\n\n> The ascetic ideal has a _goal_ \u2014this goal is so universal that all the other interests of human existence seem, when compared with it, petty and narrow...it permits no other interpretation, no other goal; it rejects, denies, affirms, and sanctions solely from the point of view of _its_ interpretation... [I]t believes that no power exists on earth that does not first have to receive a meaning, a right to exist, a value, as a tool of the ascetic ideal, as a way and means to _its_ goal, to _one_ goal. (GM III:23)\n\nBecause the ascetic ideal represents a \"closed system of will, goal and interpretation\" (GM III:23), all potential alternatives are quickly subsumed within, and co-opted by, the ascetic ideal. This means that the ascetic ideal can be harmed only by _itself_ , in an inadvertent act of self-overcoming. If Nietzsche is to challenge the ascetic ideal, then he has no choice but to disrupt the economy of this \"closed system\" from within. He can harm the ascetic ideal only by contributing to its self-overcoming.\n\nAlthough Nietzsche furnishes no alternative to the ascetic ideal, he does provide an example of the sort of endogenous disruption that might contribute to its eventual demise. As we have seen, he acknowledges as his ownmost task the self-overcoming of Christian morality (D P:4), a task that requires him to orchestrate the sublation of Christian morality from within. In order to contribute to the self-overcoming of Christian morality, he must force its native will to truth to confront _itself._ If he can somehow induce Christian morality to draw \"its _most_ _striking inference,_ its inference _against_ itself\" (GM III:27), then Christian morality as we know it will perish by its own hand. It is within the context of his personal confrontation with Christian morality that Nietzsche operates, unwittingly, as a comedian of the ascetic ideal. He \"harms\" the ascetic ideal not by attacking it directly, but by arousing mistrust of its dominant, Christian interpretation.\n\n### Hijacking the Ascetic Ideal\n\nThe ascetic practices most familiar to us, those responsible for guiding our halting evolution from natural to human animals, have exhausted their usefulness for the continued development of humankind. They have furthermore engendered the \"will to nothingness\" that now threatens our very existence. In order to deflect the nihilistic impulse of our familiar complement of ascetic practices, Nietzsche attempts to institute an alternative set of ascetic practices, which, he believes, may enable him and his unknown \"friends\" to turn the _ask sis_ demanded by civilization to their own advantage. Unable to despose the ascetic ideal or install an alternative, he attempts instead to hijack it for his own campaign to endow the successor age with a naturalistic, anti-Christian ideal.\n\nWhile the ascetic ideal demands that all human beings practice self-denial, it does not require participation in any particular ascetic practices or disciplines. Nietzsche thus concludes from his genealogical investigations that no logical necessity binds the ascetic ideal to our current practices of self-denial and self-annihilation. Their linkage is the contingent, historically specific product of the dominant, Christian disposition of the ascetic ideal.\n\nHoping to exploit the historical contingency of the Christian disposition of the ascetic ideal, Nietzsche calls for a reversal of the process whereby the moral interpretation of the bad conscience gained ascendancy:\n\n> Man has all too long had an \"evil eye\" for his natural inclinations, so that they have finally become inseparable from his \"bad conscience.\" An attempt at the reverse would _in itself_ be possible\u2014but who is strong enough for it?\u2014that is, to wed the bad conscience to all the _unnatural_ inclinations, all those aspirations to the beyond, to that which runs counter to sense, instinct, nature, animal, in short all ideals hitherto, which are one and all hostile to life and ideals that slander the world. (GM 11:24)\n\nThe question he inserts into this otherwise promising passage conveys the enormity of the reversal he envisions: who would be \"strong enough\" to turn the ascetic ideal against its currently dominant disposition and impress the bad conscience into the service of the beleaguered affects?\n\nWhile Nietzsche himself is not \"strong enough\" to effect this reversal on his own, he nevertheless believes that he and his unknown \"friends\" might collaborate in the strategic appropriation of the ascetic ideal. He imagines the appropriation of the ascetic ideal on the model of the original \"slave revolt\" that it is designed to reverse, and he anticipates for it a similarly protracted course of conflict. He prophesies that for the next two centuries at least, the fate of the reversal he initiates will remain uncertain (EH XIV:1). This wearying conflict will prove well worth its collateral casualties, however, for its successful resolution would eventually result in the installation of a naturalistic alternative to the ascetic ideal. Turning the formative power of _ask sis_ to more constructive ends, the philosophers of the future would preside over the long-awaited completion of the human species. For the first time in their chequered history, human beings would stand security for their own future, subjecting to their design what had previously fallen to chance.\n\nIn order to play a modest preliminary role in realizing this redemptive vision of the future of humankind, Nietzsche aims to harness the corrosive power of the ascetic ideal for his campaign against Christian morality. Rather than overthrow or supplant the ascetic ideal, he attempts instead to foster mistrust of its reigning, Christian interpretation. He \"harms\" the ascetic ideal by multiplying its aspects, by providing an alternative to its dominant, Christian disposition. The endogenous shocks introduced by him into the economy of this closed system may temporarily disrupt and destabilize the dominant ascetic practices sanctioned by the institutions of Western culture. The ensuing slippage within the economy of the ascetic ideal may afford him an opportunity to promote his alternative interpretation of the ascetic ideal.\n\nHis goal for the remainder of modernity is to wed the bad conscience to the anti-affective \"second\" nature that humankind has acquired in the Christian period of its history. Because our \"original\" human nature is irretrievably lost to us, by virtue of our irreversible acculturation, any nature that we currently embody is an acquired, \"second\" nature. In essay II of the _Genealogy_ , he demonstrates that institutionally reinforced ascetic practices can effectively reinscribe the code of human \"nature,\" by engineering a \"second\" nature more amenable to the peculiar demands of particular, historically specific civilizations. Having set for itself the task of \"breed[ing] an animal _with the right to make_ _promises_ \" (GM 11:1), Nature creates the machine of culture, which, through the gradual institutionalization of mnemotechnics, establishes the techniques through which humankind develops. Owing to the plasticity of the human soul, what was once culture has now become \"second\" nature.\n\nUnable to legislate the completion of this transition from natural to human animal, Nietzsche undertakes instead to investigate the generative and recuperative properties resident within our largely unknown \"second\" nature. Hoping to turn our Christianized acculturation against itself, he experiments with ascetic disciplines that may safeguard the will and thereby enable humankind to survive the twilight of the idols. The immediate goal of his self-experimentation is to implement a set of ascetic practices that will redistribute the strain placed on the affects by the ascetic ideal, thus sparing them the mortal toll of any single, sustained ascetic discipline. He consequently attempts to divert the destructive focus of the ascetic ideal from our \"original,\" affective nature to our acquired, anti-affective \"second\" nature.\n\nGambling that the _ask sis_ required by civilization can be satisfied by an assault on the \"second\" nature that humankind has acquired, Nietzsche harnesses the erotic power of ascetic practices to tempt some individuals away from the anti-affective animus of Christian morality. For himself and his unknown comrades, he advocates what we might call\u2014anticipating the full force of the oxymoron\u2014an \"ascetic naturalism.\" Hoping to relieve the affects of the mortal strain exerted against them by the Christian-moral interpretation of the bad conscience, he inaugurates a set of ascetic disciplines that target those anti-affective impulses that have become \"second\" nature within us. His \"insane\" task, he explains, is\n\n> To translate man back into nature; to become master over the many vain and overly enthusiastic interpretations and connotations that have so far been scrawled and painted over that eternal basic text _[Grundtext]_ of _homo_ _natura._ (BGE 230)\n\nBy means of his naturalistic alternatives to the dominant ascetic strategies of Western civilization, he intends to scour \"the eternal basic text of _homo natura,_ \" gradually dissolving the moral accretions that have been \"scrawled and painted\" over it. He boastfully describes this experiment as his \"attempt to assassinate two millennia of antinature and desecration of man\" (EH IV:4).\n\nThe \"tragic age\" that Nietzsche foresees will be distinguished by the thoroughgoing naturalism of its sustaining values and ideals; metaphysical comforts will be dispensed only to those invalids whose existential suffering cannot otherwise be relieved. In preparation for this coming age, he must inaugurate\u2014or so he believes\u2014a campaign to eliminate the anti-natural vestiges of Christian morality. The goal of this campaign is to deliver to the \"philosophers of the future\" a fully naturalized model of the human soul, complete with the generative powers and faculties produced by two millennia of Christian acculturation.\n\nStrictly speaking, however, the logic of Nietzsche's \"ascetic naturalism\" describes a position of anti-anti-naturalism; the installation of an originary, non-reactive naturalism lies beyond the compass of his depleted resources. He must consequently prosecute his assault on Christian morality by waging an antinatural attack on the \"second\" nature humankind has acquired in the process of submitting to Christian disciplines of _ask sis._ As we have seen, Nietzsche himself is occasionally able to direct his own resentment of modernity against itself, thereby turning his antinatural affects against his acquired, \"second\" nature.\n\nThis practice of internal disruption derives its promise and plausibility from the totalizing impulse of the ascetic ideal. In order to sustain itself as a closed system and co-opt any emerging alternatives, the ascetic ideal must expend a great deal of energy in the maintenance and regulation of its monopoly. Because this diffuse energy can\u2014and must\u2014be harnessed by agents who serve the ascetic ideal, it might also be appropriated by agents who oppose the ascetic ideal. While disciplining agents to serve its monolithic ends, the ascetic ideal also invests these agents with residual discretionary powers, which can, under certain circumstances, simultaneously be turned _against_ the ascetic ideal itself. Owing to the lavish wealth of energy squandered by the ascetic ideal in the maintenance of its sprawling empire, agents who serve the ascetic ideal might also function as comedians of the ascetic ideal, provided that they can withstand the self-referential reverberations of their double agency. Indeed, these \"comedians\" would oppose the ascetic ideal, even as they continue to serve its ends.\n\nThe implementation of ascetic disciplines, especially those that place ever-escalating demands on the structural integrity of the soul, thus creates in some _\u00fcbermenschlich_ agents a feedback loop of heretofore unknown powers and faculties. In order to undergo the prescribed _ask sis,_ agents must first be invested with sufficient power and vitality to wield effectively the ascetic instruments entrusted to them. In most cases, the ascetic demands placed on agents more or less cancel out any residual powers created in them, leaving the self-disciplined agents with no net gain in freedoms or faculties. In some extraordinary cases, however, ascetic disciplines will have a fortifying, fructifying effect on agents, inadvertently endowing them with unanticipated freedoms and affording them greater political latitude.\n\nHence Nietzsche's enduring interest in the paradox of the saint: in order to _tame_ the will (as opposed to acknowledging one's weakness of will as a preexisting condition), one must already possess a superhuman will, which disciplines of \"taming\" actually strengthen and fortify. As opposed to lesser ascetics, the saint discovers that his monkish life of solitude and self-denial has in fact succeeded in inflaming his prepotent will. Rather than simply quiet his noisome affects and castrate his tumescent will, his _ask sis_ serves to arouse powers and faculties previously unknown to him. Nietzsche's own \"autobiography\" furnishes a related example of the feedback loop inadvertently created in extraordinary agents by ascetic disciplines. His own period of \"convalescence\" was governed by ascetic practices that not only enabled him to survive his bout with romantic pessimism, but also afforded him the novel perspective of a sickly, declining form of life. His \"wisdom,\" he later allows, lies in his ability to reverse perspectives at will (EH 1:1), which ability he unwittingly gained as a by-product of the ascetic disciplines that secured his convalescence.\n\n### Knowledge: A Form of Asceticism\n\nNietzsche's investigation into the history of morality is not merely academic, for he aims thereby to discover those ascetic practices that are most likely to be effective in deflecting the \"will to nothingness.\" He consequently attempts to catalogue the full range of ascetic disciplines known to humankind, for he believes that, depending on the agents in question, all ascetic techniques are potentially both coercive and empowering.\n\nWhile it is true that all ascetic practices declare war on the affective _Grundtext_ of human \"nature,\" it is also true that, in doing so, they invariably furnish (some) agents with a novel, heretofore undisclosed \"second\" nature. Nietzsche treats this acquired \"second\" nature as comprising, in extraordinary cases, a feedback loop invested with residual, and potentially productive, powers of self-denial. Under the aegis of the ascetic ideal (and often under the threat of death), human beings have learned to exploit the plasticity of the human soul. Self-denial is still the only game in town, but several millennia of ascetic experimentation have transformed the rules of this game and raised its stakes. The self that currently presents itself for _ask sis_ is far more complex and variegated than ever before.\n\nNietzsche's campaign to \"harm\" the ascetic ideal thus sanctions his experimentation with a vast array of ascetic disciplines and practices. The philosopher in late modernity must appropriate the ascetic heritage of Western civilization, turning it to the advantage of humankind as a whole. Seeking to persuade his unknown friends of the possibilities that reside for them in an ascetic heritage that they cannot escape in any event,\n\nNietzsche contends that\n\n> The most spiritual men, as the _strongest_ , find their happiness where others would find their destruction: in the labyrinth, in hardness against themselves and others, in experimentation _[Versuch]_ ; their joy is self-conquest; asceticism becomes in them nature, need and instinct. Difficult tasks are a privilege to them; to play with burdens which crush others, a _recreation [Erholung]_. _Knowledge_ \u2014 _a form of asceticism._ (AC 57, emphasis added)\n\nBut how does one transform asceticism into a _recreation?_ Nietzsche's recourse to self-experimentation furnishes _his_ answer, which may or may not become our own. \"Knowledge as a form of asceticism\" is not simply the subject matter of his post-Zarathustran method of investigation, but its performative structure as well. His unique appropriation of _genealogy,_ for example, enables him to experiment with novel conceptions of agency, untested methods of inquiry, and previously undetected connections between seemingly unrelated phenomena. In order to acquire genealogical knowledge, however, he must implement a specific set of ascetic disciplines, which exact, among other casualties, the supposed validity of his spurious claims to epistemic privilege. Indeed, the _Genealogy_ both is and is about an ascetic strategy for gaining knowledge, and it consequently tempts (some of) his readers to conduct similar experiments of their own. As practiced and performed by Nietzsche, genealogy becomes a _Versuch_ , an experiment _and_ a temptation, which entices (some of) his readers \"to play with burdens which crush others\" (AC 57). His self-experimentation in the pursuit of knowledge thus awakens the dormant _er s_ of (some of) his readers.\n\nTo his fellow Dionysian travelers, Nietzsche recommends genealogy as an ascetic discipline that both targets those anti-affective impulses that have become \"second\" nature to us and inoculates against the nihilism that attends the exhaustion of modernity. For those readers whose _er s_ Nietzsche awakens, the aim or topic of his genealogies may ultimately be irrelevant; it is his self-imposed _ask sis_ that piques their interest in him. His resistance to his own decay thus affiliates him with the exemplary individuals who alone warrant the future of humankind. While his experiments with genealogy will neither reverse nor arrest the advance of decadence in late modernity, they may lend unity and structure to the ethical life of the Nietzschean \"we.\"\n\nNietzsche's recommendation of asceticism as a \"recreation\" reflects his enduring fascination with the political significance of _play_. Referring explicitly to himself, he proclaims, \"I do not know of any other way of associating with great tasks than _play_ ; as a sign of greatness, this is an essential presupposition\" (EH 11:10). He similarly attributes to Napoleon a \"naturalness where great tasks are something one plays with, one _may_ play with\" (TI IX:48). While it has become popular to figure Nietzsche as a champion of innocence, nature, dance, song, spontaneity, and play, the ludic elements of his thought must be carefully situated within the context of his larger philosophical enterprise. The free play and spontaneity available to the philosopher issue not from the suspension of ascetic disciplines and practices, but from their cultivation and perfection. To experiment on oneself requires an unmistakable violence, to which ascetic instruments are uniquely suited. Nietzsche consequently praises \"the discipline of suffering, of _great_ suffering,\" for \"only _this_ discipline has created all enhancements of humankind thus far\" (BGE 225). As this passage indicates, only a select few, their battle-tested souls fortified by excruciating discipline and self-experimentation, may realistically come to view asceticism as a recreation; the rest of us can only play at playing.\n\nAs we have seen, Nietzsche tends to describe _ask sis_ as producing in us a \"second\" nature to replace our obsolete, \"original\" nature. While he steadfastly insists, _contra_ Rousseau, that our \"first\" nature is forever lost to us, we might wonder why he retains the residually romantic imagery of \"Nature\" at all. Indeed, as his own predilection for \"sloughing\" dead skins would seem to confirm (H 11: 2; GS P:4), he apparently conceives of Nature as irrepressibly regenerative, as unquenchably plenteous in its wealth and extravagance. He thus assumes that humankind will achieve its perfection within the economy of Nature, even if the \"complete\" human beings of the future bear little resemblance to their primitive forbears.\n\nIf we are to remain faithful to the experimental spirit of Nietzsche's philosophy, however, then we must identify and resist the prejudices that contour (and delimit) his own political thinking. While Nature has proven its resiliency time and again in this century, confounding all attempts by philosophers and scientists to vanquish it as a ground or condition of political life, the perfection of humankind may nevertheless lie _beyond_ the plenum of Nature, in a cyborg artifice or machinic similacrum of human \"nature.\" Indeed, it is entirely possible, especially against the shifting background of contemporary science and technology, that Nietzsche's vision of the _\u00dcbermensch_ will be realized only in the emergence of a post-natural being whose terrifying incarnation we have not yet begun to imagine.\n\n### Therapies of Survival: Educating the Body\n\nAlthough Nietzsche often describes the philosopher's self-experimentation as wild and unprincipled, his more sober reflections convey a sense of the great care and strict discipline required to turn one's soul inside out. The goal of self-experimentation in late modernity, as in any decadent epoch, is to contribute to the development of therapies of survival.\n\nLike Wittgenstein, Nietzsche ascribes to philosophy a salutary, if limited, therapeutic function. The general goal of self-experimentation is to develop a specific regimen of self-overcoming that better enables one to negotiate the practical exigencies peculiar to one's age. Nietzsche's own regimen of self-overcoming is designed to deliver him neither to autonomy nor authenticity, but to a strategically advantaged situation within his epoch, such that he might expend his residual vitality in the service of his aversion to the decadence of modernity. In light of the strict discipline required to develop the \"ascetic naturalism\" he envisions, he recommends self-overcoming only to those \"free spirits\" who can both afford and withstand the _ask sis_ involved in self-experimentation. These experiments are furthermore designed to produce only rare and exotic selves. There is no room for the \"letting go\" and _laisser aller_ prized by the _demos_ ; Nietzsche has no interest in reproducing bloodless permutations of the last man.\n\nAs an example of the disciplined self-experimentation he has in mind, Nietzsche extols the martial wisdom of Julius Caesar, who cultivated in himself those \"inexorable and fearful instincts that provoke the maximum of authority and discipline against themselves\" (TI IX:38). Caesar's greatness lies not in the outward, macropolitical expressions of his legislations, but in his lawgiving itself, in which his unique regimen of self-overcoming is manifestly reflected. While orthodox historians locate Caesar's genius in the magisterial vitality that he involuntarily propagates, Nietzsche directs our attention to Caesar's unmatched capacity to manage and direct the vitality with which destiny endows him. He consequently treats the military and political exploits for which Caesar is famous as the expressions or by-products of a more basic regimen of self-overcoming.\n\nNietzsche similarly applauds Goethe for developing a regimen of self-overcoming that enabled him to master the inherent tensions and contradictions of his age. He describes Goethe as a\n\n> [M]agnificent attempt to overcome the eighteenth century by a return to Nature, by an _ascent_ to the naturalness of the Renaissance\u2014a kind of self-overcoming on the part of that century... What he wanted was _totality;_ he fought the mutual extraneousness of reason, senses, feeling and will... (TI IX:49)\n\nBy \"creating himself,\" Goethe attained an immanent critical standpoint within the age that shaped him, and he conveyed his critique of the eighteenth century by means of the embodied reproach he disciplined himself to enact. By turning his destiny to his full advantage, Goethe elicits Nietzsche's highest praise:\n\n> Goethe conceived a human being who would be strong, highly educated, skillful in all bodily matters, self-controlled, reverent toward himself, and who might dare to afford the whole range and wealth of being natural... (TI IX:49)\n\nWhereas Goethe was spared the advanced decay of the nineteenth century, Nietzsche has no choice but to embody the diminished vitality of an age he describes as \"merely an intensified, _brutalized_ eighteenth century\" (TI IX:50). He therefore must fashion for himself a regimen of self-overcoming that is appropriate to the dwindling vitality of his epoch. Like Caesar and Goethe before him, he aims not to increase the residual vitality at his disposal, but to manage its expression efficiently and strategically. He consequently transforms himself into a specific kind of decadent, one who expresses his dwindling vitality by means of an aversion to decadence. This alternate expression of decadence, he gambles, may be sufficient to safeguard the will and guarantee its survival.\n\nToward this end, he develops a regimen of self-overcoming designed to produce in himself a modified \"second\" nature. While he cannot realistically hope to equip himself with an entirely new system of instincts, the modest \"counter-instincts\" he cultivates may succeed in mitigating slightly the decadence he involuntarily enacts. His volitional resources are limited, but so are his (revised) goals.\n\nIn attempting to modify the functional specifications of his own destiny, he aims to implement an insight that he attributes to the Greeks:\n\n> It is decisive for the lot of a people and of humanity that culture should begin in the right place\u2014not in the \"soul\" (as was the fateful superstition of the priests and half-priests): the right place is the body, the gesture, the diet, physiology; the rest follows from that. Therefore the Greeks remain the first cultural event in history: they knew, they _did_ , what was needed; and Christianity, which despised the body, has been the greatest misfortune of humanity so far. (TI IX:47)\n\nLike the Greeks, then, Nietzsche begins his therapy of survival \"in the right place,\" by educating his body. Unlike the Greeks, however, who enjoyed the luxury of an overflowing will, he must carefully manage a dwindling store of vitality. We should therefore not expect his regimen of self-overcoming to fashion anything resembling the noble warriors of Greek antiquity, for late modernity simply cannot afford to stage such lavish productions.\n\nIn order to see how seriously Nietzsche takes the education of the body, one need only look to his \"autobiography.\" _Ecce Homo_ describes in detail the regimen of self-overcoming that delivered him to his \"destiny.\" Here he speaks not of his momentous and lasting achievements, but of \"all these small things which are generally considered matters of complete indifference\" (EH 11:10). When expressly dispensing the basic tenets of \"his morality,\" he speaks not of the _\u00dcbermensch,_ the Antichrist, active forgetting, or virile warriors, but of his insights into _diet_ (EH 11:1). When accounting for \"everything that deserves to be taken seriously in life,\" he turns not to the questions that exercise kings, judges, and priests, but to the more fundamental \"questions of nourishment, abode, spiritual diet, treatment of the sick, cleanliness and weather\" (EH XIV:8). Through his seemingly idle experiments with nutrition, location, climate, and recreation, he gradually became what he is: a modern who takes the measure of modernity, a decadent who opposes decadence, a truthful critic of truth, a manly midwife, and a pious Antichrist.\n\n_Ecce Homo_ thus conveys a sense of the subtle, nuanced transformations that Nietzsche undertakes, and it provides a more realistic picture of the self-experimentation he recommends to the highest specimens of a dying epoch. Whereas Zarathustra longs to wield his cruel hammer and release the _\u00dcbermensch_ imprisoned in the stubborn granite of humanity (Z 11:2), Nietzsche employs a much lighter touch, and a tuning-fork (TI P), in sculpting a destiny for himself. Rather than clarify important teachings, revise core insights or recant abandoned dogmas, he corrects a single note in the score of his lugubrious, and largely forgotten, _Hymn to Life_ : \"Last note of the A-clarinet, c flat, not c: misprint\" (EH IX:1).\n\nOf course, the magnitude of any feat of self-overcoming corresponds to the residual vitality at one's disposal. Nietzsche attends to \"all the small things\" only because he cannot afford to tackle anything greater. Like Wagner, \"who crowds into the smallest space an infinity of sense and sweetness,\" Nietzsche achieves his greatest success as a _miniaturist_ (CW 7). Unable to legislate on the grand scale to which he naturally aspires, he presides masterfully over the minute details of his own project of self-overcoming. Here he turns his destiny to his own advantage, artfully perfecting the limited domain to which his decadence confines him. He not only attains an unequalled mastery of the philosophical aphorism, abandoning the grand, systematic ambitions of his youth, but also permits his artist's gaze to settle, finally, on the subtle shades and delicate nuances of quotidian life. Pundits and potentates may ignore his angry little books, but Turinese costermongers honor him with their sweetest grapes (EH 111:2).\n\nWhich experimental selves should the decadent philosopher attempt to embody? While all techniques of the self are equally heteronomous, some will prove more advantageous (and more dangerous) in the pursuit of particular ends. Nietzsche, for example, is intent on discovering those ascetic disciplines that will prove most resistant to the advent of the \"will to nothingness.\" He consequently privileges those ascetic disciplines that promise to retard the deterioration of the affects. Since he does not know a priori which specific ascetic practices will be least threatening to a crippled will, he resorts to self-experimentation. He consequently probes the resiliency of decadence, implementing and embodying \"next\" selves that accommodate an increasingly greater range and depth of affective expenditure. He \"guesses what remedies avail against what is harmful,\" a gambit that proves that he \"has turned out well\" (EH 1:2). If successful in his guesses, he may seduce his fellow travelers to their \"next\" selves and perhaps help them to resist their own decadence. If unsuccessful in his guesses, or if slow to revise them, he may inadvertently hasten the exhaustion of late modernity.\n\nIn fashioning his therapy of survival, Nietzsche combats danger with danger. He can offer no assurance that self-experimentation\u2014either as a general strategy or in his own specific experiments\u2014actually succeeds in retarding decadence. It may be, as he suggests in his more pessimistic moments, that the project of resisting the decadence of modernity is simply futile. Or it may be, as he suggests in his more exuberant moments, that philosophers can successfully wage war with their age and thus resist those strains of decadence to which they are most vulnerable. All he \"knows\" from his genealogy of morals is that the available range of selves has been artificially and dangerously circumscribed in late modernity, and that the predilection for self-destructive techniques of the self threatens the very survival of the will. He thus hopes to contribute to a proliferation of rare and exotic selves, whose identities he cannot begin to predict.\n\nNietzsche's advocacy of self-experimentation, though deliberately outrageous, is neither purely rhetorical nor merely idle. In his own attempt to achieve the \"timelessness\" that he believed would expand the horizon of human perfectibility, he subjected himself to dangerous forms of self-experimentation. He regularly turned the scalpel on himself, flaying protective tissues of prejudice and probing the darkest recesses of his labyrinthine soul. Hoping to develop a regimen of self-overcoming that would enable him and his \"friends\" to resist more effectively the decadence of late modernity, he endured solitude, loneliness, physiological torment and exhaustion, wonder diets, miracle cures, panacetic drugs, and even the onset of madness. As is his wont, he presents a heroic face throughout, advertising his experiments as symptoms of his own _\"great_ health, that superfluity which grants to the free spirit the dangerous privilege of living _experimentally [auf den Versuch hin leben]_ and of being allowed to offer itself to adventure\" (H I:P4).\n\nMore emphatically than any other aspect of his life and work, however, this penchant for self-experimentation conveys his despair at the inexorable advance of decay in late modernity. In this respect, his experiments represent a desperate gamble, for his therapy of survival may actually contribute to the production of frightening monstrosities, and even to his own demise. Strategies of self-directed violence always run the risk of inflicting wounds that are not only significative, but also mortal. Nietzsche's own experiment with protracted solitude, for example, nearly killed him before securing his convalescence (GS P), and it may ultimately have hastened his departure from sanity. In the end, the urgency of his experimental spirit was rivalled only by that of his fears for the future.\n\n# 7  \nNietzsche's Political Legacy\n\n> I am no man, I am dynamite... It is only beginning with me that the earth knows _great politics._\n> \n> (EH XIV:1)\n> \n> The only valid tribute to thought such as Nietzsche's is precisely to use it, to make it groan and protest. And, if the commentators say I am being unfaithful to Nietzsche, that is of absolutely no interest.\n> \n> \u2014Michel Foucault, \"Prison Talk\"\n\nNietzsche's political legacy is notoriously difficult to reckon, in large part because his thinking so readily invites misinterpretation and misappropriation. \" _Non legor, non legar\"_ he boasts in his autobiography, perversely citing his lack of worthy readers as evidence of his untimely wisdom (EH 111:1). For all of his complaints about the numbing inadequacies of his readers, however, he did virtually everything in his power to encourage confusion and misunderstanding. Shifting masks, multiple personae, polytropaic paroxysms, nested ironies and self-referential parodies, wayward textual strategies, hastily conceived thought experiments, Zarathustran exotericisms, hagiographic autobiographies, insincere attempts at self-criticism\u2014all of these signature rhetorical ploys have contributed to the bewilderment of his readers. That he is not read, or not read well, may be the inevitable fruition of a self-fulfilling prophecy.\n\n### The Standard Reading of Nietzsche\n\nNietzsche's strategy of indirection has backfired egregiously and often. Rather than discourage unworthy readers from attempting to divine his Promethean wisdom, his rhetorical gyrations have in fact issued a blanket invitation to cranks and scholars alike. Encountering no insurmountable textual obstacles to their own interpretations of his elusive teachings, Nietzsche's readers regularly conscript him as the philosophical progenitor of their respective political schemes. As evidence of the runaway ramifications of his political legacy, his influence and inspiration are now claimed by virtually every possible constituency\u2014including many sects, movements, schools, and cults that he expressly repudiated in his writings.\n\nHitler's _Mein Kampf_ is littered with ersatz Nietzscheanisms, which Nazi ideologues dutifully cobbled together to serve as the philosophical platform for their insane, misanthropic vision of political dystopia. Nietzsche's defenders rightly point out that these anti-Semitic, progenocidal \"teachings\" were mercilessly wrenched from their subtle philosophical contexts, that he was heartlessly betrayed to the National Socialists by his self-appointed executrix, the opportunistic Elisabeth. Despite his obvious and well-documented animadversions to the brand of political thuggery exemplified by National Socialism, however, nothing he says rules out definitively the Nazi interpretation of his teachings. He deliberately orchestrated the proliferation of a multiplicity of diverse readings, and the resulting farrago of interpretations constitutes his true political legacy.\n\nAlthough roundly rejected as an interpretive abomination, the Nazi appropriation of Nietzsche's political thinking has nevertheless exerted an indirect influence on the articulation of its extant legacy. Owing in large part to the \"family resemblance\" obtaining between some of his teachings and certain tenets of Nazi ideology, even sympathetic readers have refused to take seriously his political thinking on its own terms. He is occasionally lauded as a perspicacious critic of liberalism, Christianity, or modernity itself; as a keen judge of the political resources and achievements of Greek antiquity; as a vocal champion of autonomy, self-reliance and authenticity; as a prescient opponent of the likely evolution of German nationalism; as an astute diagnostician of cultural malaise; as a vigilant sentry posted on the advancing frontier of postmodernity; and so on. But he is rarely considered, on the strength of his teachings, an important political thinker in his own right.\n\nIndeed, a standard reading has emerged of Nietzsche as a _failed radical_ _voluntarist_ , whose shrill call for the redemption of modernity, at the hands of the enigmatic _\u00dcbermensch_ , is simply incompatible with his trenchant critique of modernity. By fatuously entrusting the future of humankind to a titanic act of _\u00fcbermenschlich_ will, or so the story goes, Nietzsche betrays his own greatest insights into the complexities of political life in modernity. In doing so, he prematurely abandons the residual political resources of late modernity, which, many critics insist, shelter more promising possibilities of regeneration than the desperate hopes he invests in the coming of the redemptive _\u00dcbermensch_. In the end, his political thinking is compromised by a grave mismeasure of his own epoch, a misunderstanding so pervasive and systemic that it renders his political thinking irrelevant to the lives of modern agents.\n\nThis standard reading of Nietzsche as a failed radical voluntarist owes its authority in large part to Martin Heidegger's influential interpretation. Heidegger describes Nietzsche as the \"last metaphysican of the West,\" whose ingenious attempt to overcome the metaphysical tradition of Western philosophy ultimately founders, like all such attempts, on the shoals of subjectivism. Rather than signal his emigration beyond metaphysics (which Heidegger views as coextensive with nihilism), Nietzsche's doctrine of will to power constitutes the final stage in the development of metaphysical thinking:\n\n> Because thinking in terms of values is grounded in the metaphysics of the will to power, Nietzsche's interpretation of nihilism as the event of the devaluing of the highest values and the revaluing of all values is a metaphysical interpretation, and that in the sense of the metaphysics of the will to power.\n\nAccording to Heidegger, Nietzsche proposes an irreducibly subjectivistic solution \u2014a \"revaluation of all values\"\u2014to the problem of nihilism. He consequently remains a residually metaphysical thinker, albeit the \"last\" of this misbegotten breed: \"even Nietzsche's own experience of nihilism, i.e., that it is the devaluing of the highest values, is after all a nihilistic one.\" Nietzsche's residual subjectivism is crystallized, Heidegger believes, in his teaching of eternal recurrence, which entrusts to the will the Promethean task of \"eternalizing the moment.\" This desperate recourse to radical voluntarism may not succeed in overturning the metaphysical tradition, but it does succeed in constituting the \"final\" unfolding of this tradition, as it distills the whole of subjectivism into a single, transformative explosion of the _\u00fcbermenschlich_ will.\n\nAlthough Heidegger prefers to situate Nietzsche's failed voluntarism in the context of his ill-fated attempt to overcome the tradition of metaphysical thinking, this interpretation is readily translated into distinctly political terms: Nietzsche's otherwise promising confrontation with modernity is subverted by his stubborn insistence on a voluntaristic solution to the besetting problems of nihilism and decadence. This standard reading thus takes the form articulated in the following general outline:\n\n  1. Nietzsche advances a withering critique of the signature institutions of modernity: liberalism, Enlightenment, democracy, Christianity, and so on.\n  2. His critique exposes the abject failure of these signature institutions to fashion for themselves the rational justifications that allegedly distinguish them from the \"primitive,\" unjustifiable institutions of premodern political regimes.\n  3. Reacting to the failure of these signature institutions to ground and defend themselves rationally, Nietzsche desperately capitulates to the \"other\" of reason, declaring all claims of reason to be nothing more (nor less) than disguised expressions of will to power.\n  4. Because political life in modernity cannot justify itself by appeal to rational terms and principles, he appeals instead to will to power as the sole adjudicating standard of \"nobility,\" \"mastery,\" \"health,\" \"genius,\" and all other superlative political values.\n  5. As his critique of modernity demonstrates, however, the age itself is simply too decadent, too depleted in its native volitional resources, to sustain a \"noble,\" aristocratic political regime.\n  6. Blinded by his romantic, nostalgic allegiances to the manly political regimes of Greek and Roman antiquity, Nietzsche declares modernity to be irrecuperably decadent and sterile. He prematurely abandons the imperfect projects and institutions by means of which the epoch has defined itself.\n  7. Having judged modernity to be incapable of restoring and regenerating its signature projects and institutions, Nietzsche locates the salvation of modernity in the transformative volitional resources of those shadowy redemptive figures who someday will come: the \"free spirits,\" the \"philosophers of the future,\" the _\u00dcbermensch,_ and, finally, Dionysus himself.\n  8. Owing to Nietzsche's premature rejection of the regenerative powers resident within modernity, and to the irrational hopes he invests in a mysterious, redemptive act of _\u00fcbermenschlich_ will, his confrontation with modernity transgresses the recognizable boundaries of sane political thinking \u2014as confirmed, some would say, by his clinically diagnosed slide into madness.\n\nVariations on the general theme of Nietzsche's failed voluntarism are found in virtually all influential studies of his political thinking, including those proffered by Arendt, Kaufmann, Strong, MacIntyre, Yack, Schutte, Warren, Vattimo, Habermas, Detwiler, Ansell-Pearson, and Berkowitz. In all of these studies, Nietzsche is initially (if warily) applauded as an astute critic of political life in modernity, only to be dismissed in the end as a naive voluntarist. Rather than accurately assay the residual restorative powers of modernity (especially those contained within reason itself), he prematurely disowns the epoch as a whole. As a consequence of the failure of his radical voluntarism, Nietzsche's political thinking may be provocative, ingenious, insightful, and entertaining, but it is largely irrelevant to contemporary political life. Despite his engaging vitality, he has nothing to say of lasting constructive value to agents laboring in the twilight epoch of modernity. His standing in the canon of Western political thought remains that of a hideous scarecrow, whose horrifying, irrational excesses frighten us into renewing our tepid commitments to the democratic reforms and liberal ideals of modernity\u2014despite our nagging doubts about their continued value and efficacy.\n\nA strong case can be made for this standard reading of Nietzsche as a failed radical voluntarist, and the scholars mentioned above have collectively presented such a case. But is this the most promising line of interpretation to pursue? While this standard reading of Nietzsche is certainly plausible on a textual basis, its primary attraction lies, I believe, in the definitive judgment it renders against the National Socialists' appropriation of his political thinking. If Nietzsche is a failed voluntarist (and thus irrelevant to modern political life), then the Nazi ideology that grew out of this voluntarism is similarly bankrupt. In order to discredit the philosophical pretensions of National Socialism (and all kindred political monstrosities), the standard reading of Nietzsche discredits the source of these pretensions, which it locates in his irrational appeal to the transformative properties of the _\u00fcbermenschlich_ will.\n\nThe contribution of this standard reading to a condemnation of Nazi ideology is certainly commendable, but it is not so helpful in arriving at a clear understanding of Nietzsche's political thinking. I have attempted in this study to challenge the standard reading of Nietzsche's political thinking, to show that he works carefully within the context of his own critique of modernity, and to illuminate the relatively unexplored moral content of his political teachings. While his irrational voluntarism constitutes one branch of his tangled political legacy, it is neither the most prominent ramification nor the most fruitful to cultivate.\n\n### Nietzsche and Contemporary Liberalism\n\nNietzsche was an outspoken, unrelenting critic of liberal political ideals, which he interpreted as symptomatic of the irreversible decay of the modern epoch. For obvious reasons, then, most liberal theorists have refused to embrace his political legacy. As we have seen, his perfectionism does not fare very well in John Rawls's survey of potential sources for (liberal) principles of justice.\n\n#### Nietzsche and MacIntyre\n\nAnother critic of liberalism, Alasdair MacIntyre, actually proposes Nietzsche's political thinking as representative of the signature errors and failures of liberalism. Although Nietzsche may appear to be a critic of the Enlightenment project, his \"criticisms\" in fact reflect the natural, self-referential development of liberal individualism. Hence, MacIntyre insists, the irreducible irrationality of Nietzsche's political thinking:\n\n> The rational and rationally justified autonomous moral subject of the eighteenth century is a fiction, an illusion; so, Nietzsche resolves, let will replace reason and let us make ourselves into autonomous moral subjects by some gigantic and heroic act of the will\n\nMacIntyre thus cites Nietzsche's irrational excesses as evidence of the shipwreck of liberal political theory:\n\n> [T]he Nietzschean stance turns out not to be a mode of escape from or an alternative to the conceptual scheme of liberal individualist modernity, but rather one more representative moment in its internal unfolding.\n\nInterpreting the doctrine of will to power as the logical outcome of the project of Enlightenment, MacIntyre presents Nietzsche as a negative exemplar of modernity, whose misguided quest to produce the _\u00dcbermensch_ exemplifies the bankruptcy of liberal individualism:\n\n> _[E]ither_ one must follow through the aspirations and the collapse of the different versions of the Enlightenment project until there remains only the Nietzschean diagnosis and the Nietzschean problematic _or_ one must hold that the Enlightenment project was not only mistaken, but should never have been commenced in the first place.\n\nMacIntyre thus exploits the supposedly dead ends and blind alleys of Nietzsche's political thinking to call for a reconsideration of the Aristotelian moral tradition, whose premature rejection, he believes, constitutes the signal error of modernity:\n\n> [T]he defensibility of the Nietzschean position turns _in the end_ on the answer to the question: was it right in the first place to reject Aristotle? For if Aristotle's position in ethics and politics\u2014or something very like it\u2014could be sustained, the whole Nietzschean enterprise would be pointless.\n\nDespite his manifest allegiances to Aristotle, MacIntyre nevertheless subscribes to a surprisingly Nietzschean account of the political resources and options available to late modernity. The discontinuities and incommensurabilities that characterize moral discourse in late modernity are largely antithetical to a retrieval or recuperation of the Aristotelian moral tradition. Considerable time is needed to heal the wounds inflicted by the runaway excesses of liberal individualism, and MacIntyre bids us to begin this slow process of moral convalescence by founding, and cultivating, local ethical communities.\n\nThese micro-communities will not succeed in restoring the moribund Aristotelian tradition, but they may ensure that \"both morality and civility might survive the coming ages of barbarism and darkness.\" Invoking a remarkably Nietzschean image of the plight of moral agents in late modernity, MacIntyre expresses guarded optimism that ethical life may yet withstand the advance of decadence:\n\n> What matters at this stage is the construction of local forms of community within which civility and the intellectual and moral life can be sustained through the new dark ages which are already upon us. And if the tradition of the virtues was able to survive the horrors of the last dark ages, we are not entirely without grounds for hope... We are waiting not for a Godot, but for another\u2014doubtless very different\u2014St. Benedict.\n\n#### Nietzsche and Habermas\n\nMacIntyre's criticisms of Nietzsche are echoed in the work of J\u00fcrgen Habermas, who nevertheless deploys them in the service of his own spirited defense of the modern Enlightenment project. According to Habermas, MacIntyre is right about Nietzsche, but wrong about the Enlightenment project whose Nietzschean bankruptcy he declares:\n\n> Nietzsche enthrones taste...as the organ of a knowledge beyond true and false, beyond good and evil. But he cannot legitimate the criteria of aesthetic judgment that he holds on to because he transposes aesthetic experience into the archaic, because he does not recognize as a moment of reason the critical capacity for assessing value that was sharpened through dealing with modern art\u2014a moment that is still at least procedurally connected with objectifying knowledge and moral insight in the processes of providing argumentative grounds.\n\nNietzsche's irrational appeals to the will to power constitute the logical, self-referential consequences not of reason _per se,_ but of what Habermas calls \"subject-centered reason.\" As an articulation of subject-centered reason, the dominant philosophical discourse of modernity has indisputably reached a point of crisis and implosion, but this particular discourse is neither singular nor privileged within the political life of modern agents. According to Habermas, in fact, Nietzsche himself has contributed greatly to our growing appreciation of the crisis of subject-centered reason. Rather than abandon subject-centered reason in favor of a revised, more promising concept of reason, however, Nietzsche concludes his critique of rationality by enshrining the \"other\" of reason, the will to power:\n\n> The aesthetic domain, as the gateway to the Dionysian, is hypostatized instead into the other of reason. The disclosures of power theory get caught up in the dilemma of a self-enclosed critique of reason that has become total... [Nietzsche] could muster no clarity about what it means to pursue a critique of ideology that attacks its own foundations.\n\nAccording to Habermas, then, Nietzsche's own critique of modernity manifests contradictions and justificatory lacunae that are structurally similar to those he detects in subject-centered reason:\n\n> Like all who leap out of the dialectic of enlightenment, Nietzsche undertakes a conspicuous leveling. Modernity loses its singular status; it constitutes only a last epoch in the far-reaching history of a rationalization initiated by the dissolution of archaic life and the collapse of myth.\n\nAlthough Nietzsche's critique is both valid and valuable, it is insufficient on its own terms to sanction his subsequent (and precipitous) \"leap out of the dialectic of enlightenment.\" Had he responded to this crisis by returning to the \"counterdiscourse\" of modernity, his contribution to the Enlightenment might not have been strictly destructive.\n\nHaving prematurely rejected reason, as well as the \"counterdiscourse\" potentially available to him, Nietzsche has no choice but to consign the fate of modernity to those mysterious lawgivers and commanders who will follow:\n\n> On the one hand, historical enlightenment only strengthens the now palpable diremptions in the achievements of modernity; reason as manifested in the form of a religion of culture no longer develops any synthetic forces that could renew the unifying power of traditional religion. On the other hand, the path of restoration is barred to modernity. The religious-metaphysical world views of ancient civilizations are themselves already a product of enlightenment; they are _too rational,_ therefore, to be able to provide opposition to the radicalized enlightenment of modernity.\n\nTo Habermas, Nietzsche's lurid enactment of the shipwreck of subject-centered reason demonstrates not so much the eclipse of rationality itself (as many of Nietzsche's French scions have apparently concluded), as the need to articulate a more adequate concept of reason. Habermas consequently attempts to revive the \"counterdiscourse\" of modernity, wherein subject-centered reason has been intermittently confronted with the sorts of criticisms that Nietzsche and MacIntyre rehearse. A careful attention to this counterdiscourse, and especially to the critical clashes that have determined the course of its chequered career, reveals to Habermas a \"road not taken,\" i.e., the possibility of a concept of reason that is immune to the now-familiar objections to subject-centered reason.\n\nDrawing on his excavations thus far of the counterdiscourse of modernity, Habermas proposes the concept of \"communicative action,\" which, he believes, might enable us to construct a model of reason grounded in mutual recognition and reciprocity. Communicative action posits as its ideal the rational discourse that obtains within a non-strategic, distortion-free communication situation, between parties who freely agree to appeal to, and abide by, reciprocally binding validity claims. Within the intersubjective paradigm of communicative action, reason dispenses a restorative dose of enlightenment, which is precisely the prescription Habermas recommends for the ills diagnosed by Nietzsche and MacIntyre.\n\nHabermas reserves no part for Nietzsche in either the retrieval of the counterdiscourse of modernity or the ongoing articulation of the model of communicative action. Nietzsche's role in the renewed dialectic of Enlightenment is exclusively negative, as a warning to other philosophers who would prematurely abandon the claims of rationality. While Habermas may ultimately be justified in rejecting Nietzsche's political thinking, I hope to have shown in this study that he is mistaken to exclude Nietzsche from his attempts to revive the dialectic of Enlightenment. The popular interpretation of Nietzsche as a \"prophet of extremity,\" who fetishizes the \"other\" of reason, trades upon a caricature of his post-Zarathustran thinking. As I have attempted to demonstrate, Nietzsche actually continues his critique of subject-centered reason while experimenting with alternative concepts. In fact, Habermas's deflationary approach to Nietzsche's political thinking is remarkably similar in form and execution to the \"leveling\" operation that he accuses Nietzsche of undertaking with respect to modernity as a whole. As in the case of MacIntyre's grandiose polemic, many of the ethical and political resources Habermas seeks to recover can perhaps be found in the lair of his sworn enemy, Nietzsche.\n\nAlthough Nietzsche unequivocally declares bankrupt the grand, sweeping dream of the Enlightenment, his genealogical method nevertheless celebrates the subversive, unmasking power of local applications of reason. Drawing on a familiar image of Enlightenment, he praises the demystifying power of his _Genealogy_ : \"In the end, in the midst of perfectly gruesome detonations, a _new_ truth becomes visible every time among thick 5 clouds\" (EH XI). He subsequently advertises himself as the lone champion of truth, proudly claiming to be \"the first to _discover_ the truth\" and accounting for his \"calamitous\" destiny by representing himself as \"truth enter[ing] into a fight with the lies of millennia\" (EH XIV:1). In light of his complex political aims, a muted celebration of his own will to truth is perfectly understandable. If he is to contribute to the self-overcoming of Christian morality, then it _must_ be the case that he too labors in the service of the will to truth, that he too takes his flame \"from the fire ignited by a faith millennia old, the Christian faith, which was also Plato's, that God is truth, that truth is _divine_ \" (GS 344, cited in GM 111:24).\n\n#### Nietzsche and Rorty\n\nRichard Rorty has defended in recent years a \"postmodernist bourgeois liberalism,\" whose Nietzschean debts he readily acknowledges. Undeterred (and unimpressed) by Nietzsche's scathing critique of liberal ideals, Rorty undertakes a selective appropriation of the basic elements of his political thinking. The key to reconciling Nietzsche's overtly illiberal thought with the guiding ideals of liberalism lies in restricting his moral perfectionism to the private sphere. This reconciliation is possible, Rorty explains, because Nietzsche's cranky attacks on liberalism are not germane to his enduring philosophical insights:\n\n> Nietzsche often speaks as though he had a social mission, as if he had views relevant to public action\u2014distinctly antiliberal views. But, as also in the case of Heidegger, this antiliberalism seems adventitious and idiosyncratic \u2014for the kind of self-creation of which Nietzsche and Heidegger are models seems to have nothing in particular to do with questions of social policy.\n\nExplicitly building on the narrative model of self-creation that Alexander Nehamas attributes to Nietzsche, Rorty enlists Nietzsche to galvanize the ethical life of contemporary liberal democratic societies. Enamored of Nietzsche's anti-essentialism and perspectivism but repulsed by his recidivistic appeal to the metaphysics of the will to power, Rorty articulates a purified, postmodern interpretation of Nietzschean self-creation. To create oneself anew, Rorty explains, is simply a matter of fashioning for oneself an enabling narrative, in which one exercises and displays one's own \"final vocabulary,\" a designation he proposes for the evolving constellation of terms, concepts, and expressions whereby one conveys one's ownmost hopes, fears, and aspirations. The goal of narrative self-creation, which Rorty calls \"autonomy,\" is achieved when one successfully deploys a final vocabulary that one calls one's own. Rorty's postmodernist bourgeois liberalism would not only encourage such self-creation in its citizenry, but also foster a tolerance for the self-creation of others. Borrowing a Zarathustran image for his sketch of the \"liberal ironists\" who inhabit his political utopia, Rorty maintains that \"[a]ll any ironist can measure success against is the past\u2014not by living up to it, but by re-describing it in his terms, thereby becoming able to say 'Thus I willed it.'\"\n\nWhile Nietzsche's political thinking is undeniably inimical to the flourishing of liberal ideals, it poses a genuine threat to liberal democracy only if allowed to contour political discourse in the public sphere. The liberal ironists who populate Rorty's utopia may incorporate Nietzsche's moral perfectionism into their own private disciplines of self-creation, seeking to arrive at a final vocabulary that expresses their ownmost achievement of autonomy, but they publicly pledge allegiance only to the liberal ideals that protect and sustain their private pursuits of self-perfection. The fruits of their private labors of self-creation may eventually be imported into the public sphere, but _only_ insofar as they uphold the signature ideals of liberal democracy.\n\nRorty thus enshrines Nietzsche as a private hero, who, if refused access to the public sphere, indirectly furnishes the dynamism and vitality of a thriving democracy. In order to convey the essence of his postmodernist bourgeois liberalism, Rorty proposes the following \"compromise\":\n\n> _Privatize_ the Nietzschean\u2014Sartrean\u2014Foucauldian attempt at authenticity and purity, in order to prevent yourself from slipping into a political attitude which will lead you to think that there is some social goal more important than avoiding cruelty.\n\nIn this limited respect, Rorty's appropriation of Nietzsche as the unacknowledged progenitor of liberal democracy is not unlike Cavell's defense of Nietzsche's role in the moral education of democratic citizens.\n\nWhile Rorty's appropriation of Nietzsche's moral perfectionism is ingenious, we might wonder about the political value of an ideal of \"autonomy\" whose expression and enactment can realistically be limited to the private sphere. Nietzsche unequivocally insists that \"healthy\" self-creation is never strictly private. Any attempt to contain the public reverberations of moral perfectionism will succeed only in artificially restricting the range of possible \"next\" selves to be created. \"Healthy\" self-creation always involves a Dionysian element of excess or superfluity, which would obliterate Rorty's guiding distinction between public and private:\n\n> The genius, in work and deed, is necessarily a squanderer: that he squanders himself, that is his greatness. The instinct of self-preservation is suspended, as it were; the overpowering pressure of outflowing forces forbids him any such care or caution. (TI IX:44)\n\nIn order to maintain his guiding distinction, Rorty must repress (or otherwise contain) those elements of excess that Nietzsche claims are essential to selfcreation. Although Rorty sincerely invites the citizens of his utopia to create themselves anew, he also requires that whatever \"next\" selves they create will be compatible with the (conservative) liberal pluralism he envisions.\n\nOne possible \"solution\" to the problem of excess lies, of course, in the advancing decadence of modernity. If the pursuit and achievement of selfperfection _can_ be restricted to the confines of the private sphere, then Rorty's postmodernist bourgeois liberalism may constitute a feasible model for political life in late modernity. If Rorty _has_ accurately surveyed the compass of moral perfectionism in late modernity, however, then perhaps his postmodernist bourgeois liberalism should be collected under the rubric Allan Bloom proposes for Rawls's theory of justice: _A First Philosophy for the Last Man._ 24\n\n### Nietzsche and Feminism\n\nOn the basis of the textual evidence alone, we would be hard pressed to propose a less likely champion of feminist politics than Nietzsche. His occasional remarks on the status of women, or the problem of _das Weib an sich,_ are rivalled in their misogyny only by the retrograde musings of his _Erzieher,_ Arthur Schopenhauer. Although feminist scholars have recently devoted a great deal of productive energy to the strategic (and selective) appropriation of Nietzsche's political thinking, his greatest contribution to feminist politics remains both indirect and unintended. His political legacy for contemporary feminism descends most vitally from his critique of objectivity, which some feminist scholars have recently adapted for incorporation into the epistemological frameworks of their own political projects.\n\nNietzsche's perspectivism, an epistemic thesis conveyed via a host of masculinist and residually misogynist images, might seem an unlikely precursor of feminist epistemologies. But in fact Nietzsche articulates an epistemic position that both acknowledges and accommodates the radically situated experiences that are prized by feminist thinkers. In the following passage, which contains his most detailed and sustained discussion of the position now known as \"perspectivism,\" he both exposes the snares of traditional epistemology and points us in a more promising direction:\n\n> [L]et us be on guard against the dangerous old conceptual fiction that posited a \"pure, will-less, painless, timeless knowing subject\"; let us guard against the snares of such contradictory concepts as \"pure reason,\" \"absolute spirituality,\" \"knowledge in itself\": these always demand that we should think of an eye that is completely unthinkable, an eye turned in no particular direction, in which the active and interpreting forces, through which alone seeing becomes seeing _something,_ are supposed to be lacking... There is _only_ a perspectival seeing _[perspektivisches Sehen],_ _only_ a perspectival \"knowing\"; and the _more_ affects we allow to speak _[zu_ _Worte kommen]_ about one thing, the _more_ eyes, different eyes, we can use to observe one thing, the more complete will our \"concept\" of this thing, our \"objectivity,\" be. (GM 111:12)\n\nIn this brief passage, Nietzsche records several observations with which feminist theorists have recently expressed agreement. First of all, he warns his readers to beware of the traditional interpretation of Objectivity as _disinterested_ _contemplation_. The goal of disinterested contemplation presupposes \"conceptual fictions\" and \"contradictory concepts,\" and it furthermore requires us to posit a disembodied, disinterested knowing subject, \"an eye turned in no particular direction.\" His perspectivism thus attempts to account for those affective ingredients and determinants of knowledge that traditionally have been ignored or discounted by orthodox epistemologists. His reconstituted notion of objectivity (consistently noted by his use of quotation marks) suggests that knowledge is a function of the embodied expression of our affective investment in the world. His perspectivism thus presupposes an account of knowing subjects as radically situated in the world and in their bodies.\n\nSecond, if we interpret these \"eyes\" as perspectives, whose \"interpretive forces\" are sustained by a suffusion of affect, then we see that, for Nietzsche, perspectives are not disembodied points of view that hover disinterestedly over the world. Indeed, his perspectivism is strategically designed to recuperate the metaphorics of vision that have dominated (and perverted) representational epistemology. In order to appropriate the metaphorics of vision for his reconstituted notion of objectivity, he deploys the twin sensory images of \"eyes\" and \"voices\":\n\n> [T]he _more_ affects we allow to speak about one thing, the _more_ eyes, different eyes, we can use to observe one thing, the more complete will our \"concept\" of this thing, our \"objectivity,\" be. (GM 111:12)\n\nEyes and affects, knowing and feeling, seeing and speaking, conception and perception, situation and expression: the pursuit of Nietzschean objectivity requires us to deconstruct these binary oppositions and integrate the supposedly antagonistic terms within each. His reconstituted notion of objectivity encourages a maximal expression of affective investment in the world\u2014a chorus of radically situated \"voices\"\u2014and thus stands 180 degrees removed from the traditional epistemological goal of disinterested, disaffected contemplation. In fact, he concludes his warning against disinterested contemplation by graphically likening the pursuit of Objectivity to an act of self-inflicted castration: \"to suspend each and every affect, supposing we were capable of this\u2014what would that mean but to _castrate_ the intellect?\" (GM 111:12).\n\n\"There is _only_ a perspectival knowing\" thus means that knowledge is possible only if one's affective engagement with the world is both recognized and expressed. If it is not, then one may lay claim at best to a desiccated, bloodless simulacrum of knowledge. Nietzsche's recuperation of the metaphorics of vision thus enables us to understand perspectives as _bodies_ : suffused with affect, inextricably situated in the world, and inscribed with the pain and torment inflicted by normalizing mores and institutions. He consequently reconstitutes the notion of objectivity to derive from an aggregation of radically situated perspectives (or bodies)\u2014none of which affords us an epistemically pure glimpse of the world. The task of the philosopher who aspires to Nietzschean objectivity is to compile an exhaustive aggregation of perspectives, to assemble an unprecedented chorus of affective voices.\n\nThird, Nietzsche recommends his perspectivism not for its epistemic purity, but for the strategic advantage that accrues to his reconstituted notion of objectivity. His discussion of \"perspectivism\" appears within the context of his analysis of the ascetic ideal, with which he associates the traditional understanding of Objectivity as disinterested contemplation. He frequently contends that the pursuit of Objectivity involves a concomitant assault on the affects, which in turn leads, paradoxically, to a diminution of our knowledge, to the subordination of situated knowledges to lifeless simulacra of knowledge. The strategic advantage of _his_ objectivity lies in \"the ability to _control_ one's Pro and Con and to dispose of them, so that one knows how to employ a _variety_ of perspectives and affective interpretations in the service of knowledge\" (GM 111: 12).\n\nFourth, Nietzsche willingly accepts the self-referential implications of his endorsement of situated knowledges. He readily acknowledges that his own perspectivism too is situated, that it reflects the peculiar political interests of its author. \"Perspectivism\" is itself perspectival in nature, for it is the product of the partial perspective and embodied affect peculiar to Herr Nietzsche. Rather than stake an illicit claim to epistemic purity, he quite openly voices the hostility and resentment that inform his own political campaign against the ascetic practices of traditional epistemology. It is no coincidence that his most illuminating articulation of his perspectivism appears in the _Genealogy,_ a book in which he both announces and displays his own vested political interests in compiling a genealogy of morals.\n\nNietzsche's perspectivism thus provides a promising epistemological model for feminist theorists. But let us be clear about the political costs of embracing his perspectivism: if we accept this reconstituted notion of objectivity, and seek an aggregation of radically situated perspectives, then we must abandon the quest for a privileged, epistemically pure, God's-eye perspective on the world. We need not disavow our cultural, genealogical or political preferences for certain perspectives, but we must be careful to situate these preferences within a discernible political agenda. The privilege of a particular perspective will derive entirely from its situation within the political agenda it expresses, and not from its internal coherence or privileged access to the real world.\n\nSeveral feminist theorists have recently acknowledged their debts to Nietzsche's \"perspectivism,\" which they have borrowed in order to articulate the theory of knowledge that underlies their political projects and agendas. Declaring a provisional truce with a potentially vicious opponent, these theorists join Nietzsche in rejecting the traditional epistemological ideal of Objectivity. Feminist theorists have long maintained that the pursuit of Objectivity would require agents to accede to a disembodied, trans-perspectival, patriarchal standpoint\u2014a chimerical gambit that Donna Haraway calls \"the God trick.\" As Susan Bordo argues, this \"view from nowhere\" acquires the privilege and cachet of a \"view from everywhere,\" and it effectively devaluates the experiences of those agents whose knowledges of the world are most obviously and ineluctably situated. These theorists thus conclude that the ideal of disinterested, detached Objectivity is pursued at the expense and exclusion of the situated knowledges of women, especially women of color. Traditional (patriarchal) epistemology consequently delivers only a simulacrum of objectivity, for its emphasis on disinterested detachment precisely discounts the partiality that defines a radically situated perspective.\n\nAt the same time, however, some feminist theorists are understandably reluctant to abandon the _notion_ of objectively valid knowledge as the goal of philosophical inquiry. According to Sandra Harding, some such critical standard is crucial to the very project of feminist epistemology:\n\n> What would be the point of a theory of knowledge that did not make prescriptions for how to go about getting knowledge or of a prescription for getting knowledge that did not arise from a theory about how knowledge can be and has been produced?\n\nA reconstituted notion of objectivity would provide a standard whereby theorists might claim, for example, that one scientific theory is better or more complete or more promising than another. In this light, we might think of one goal of feminist epistemology as the reconstitution of the notion of objectivity, such that feminist theorists might continue the critical enterprise of science without subscribing to its most pernicious concepts.\n\nThe legacy of Nietzsche's \"perspectivism\" is perhaps most vitally preserved within the experimental epistemologies of several radical feminist theorists. Following Nietzsche, for example, Donna Haraway contends that the objectivity of a perspective is a function of its _partiality_ :\n\n> The moral is simple: only partial perspective promises objective vision... Feminist objectivity is about limited location and situated knowledge, not about transcendence and splitting of subject and object. In this way we might become answerable for what we learn how to see.\n\nThe partiality that Haraway prizes is achieved not through the disinterested detachment of subjects from the world, but through the radical situation of subjects in the world. Her suggested reconceptualization of \"feminist objectivity\" thus devolves from her more fundamental reconceptualization of the world we seek to know in terms of the world in which we live.\n\nHaraway's brand of feminism conveys a postmodern sensibility in large part because she has abandoned the quest for an epistemically pure, foundationally innocent standpoint. Indeed, a primary aim of her writing is to disabuse feminists of the perceived need for an untainted, originary standpoint from which they might launch their various political campaigns:\n\n> The science question in feminism is about objectivity as positioned rationality. Its images are not the products of escape and transcendence of limits, i.e., the view from above, but the joining of partial views and halting voices into a collective subject position that promises a vision of the means of ongoing finite embodiment, of living within limits and contradictions, i.e., of views from somewhere.\n\nPartiality thus stands as the sole determinant of objectivity, and there exists no verifiable epistemic relation between objectivity and standpoints informed by positions of exclusion, oppression or victimization:\n\n> A commitment to mobile positioning and to passionate detachment is dependent on the impossibility of innocent \"identity\" politics and epistemologies as strategies for seeing from the standpoints of the subjugated in order to see well.\n\nPostmodern feminists register a preference for the standpoints of excluded, subjugated women not because such standpoints are epistemically pure, but because \"they seem to promise more adequate, sustained, objective, transforming accounts of the world.\"\n\nThis preference is clearly political in nature, and Haraway makes no pretense of aspiring to epistemic purity or foundational innocence. For Haraway, any epistemic privilege necessarily implies a political (i.e., situated) preference. Her postmodern orientation elides the boundaries traditionally drawn between politics and epistemology, and thus renders otiose the ideal of epistemic purity. All perspectives are partial, all standpoints situated\u2014including those of feminist theorists. It is absolutely crucial to Haraway's postmodern feminist project that we acknowledge her claims _about_ situated knowledge as _themselves_ situated within the political agenda she sets for postmodern feminism; feminist theorists must therefore accept and accommodate the self-referential implications of their own epistemic claims.\n\nThe political agenda of postmodern feminism thus assigns to (some) subjugated standpoints a political preference or priority. Haraway, for example, believes that some subjugated standpoints may be more immediately revealing, especially since they have been discounted and excluded for so long. They may prove especially useful in coming to understand the political and psychological mechanisms whereby the patriarchy discounts the radically situated knowledges of others while claiming for its own (situated) knowledge an illicit epistemic privilege:\n\n> The standpoints of the subjugated...are savvy to modes of denial through repression, forgetting, and disappearing acts\u2014ways of being nowhere while claiming to see comprehensively. The subjugated have a decent chance to be on to the god-trick and all its dazzling\u2014and, therefore, blinding\u2014illuminations.\n\nBut these subjugated standpoints do not afford feminist theorists an epistemically privileged view of the world, independent of the political agendas they have established. Reprising elements of Nietzsche's psychological profile of the \"slave\" type, Haraway warns against the\n\n> serious danger of romanticizing and\/or appropriating the vision of the less powerful while claiming to see from their positions. To see from below is neither easily learned nor unproblematic, even if \"we\" \"naturally\" inhabit the great underground terrain of subjugated knowledges. The positionings of the subjugated are not exempt from critical re-examination, decoding, deconstruction, and interpretation; that is, from both semiological and hermeneutic modes of critical enquiry. The standpoints of the subjugated are not \"innocent\" positions.\n\nA subjugated standpoint may shed new light on the ways of an oppressor, but it in no way renders superfluous or redundant the standpoint of the oppressor. Because neither standpoint fully comprises the other, the aggregation of the two would move both parties (or a third party) closer to a more objective understanding of the world. If some feminists have political reasons for disavowing this project of aggregation, or for adopting it selectively, then they must pursue their political agenda at the expense of the greater objectivity that they might otherwise have gained.\n\nContinuing this Nietzschean line of investigation, Bat-Ami Bar On exposes the dangers involved in conceiving of \"mastery\" as emanating from a single, static center. Attempting to translate the guiding insights of Nietzsche's \"perspectivism\" into distinctly political terms, Bar On points out that in the complex societies of advanced industrial capitalism, virtually all agents simultaneously stand in relations of \"mastery\" to some and \"enslavement\" to others. As an alternative to the \"single-center\" theory of power preferred by traditional epistemologists, Bar On proposes a \"theorized dispersion of power among multiple centers.\" In a similarly Nietzschean vein, bell hooks warns that the romanticization of \"marginality\" may actually dispossess oppressed agents of a fruitful \"place of resistance.\" Here hooks follows Nietzsche in reminding feminist theorists not to discount prematurely the potentially restorative powers that conditions of oppression can, under certain circumstances, engender.\n\nHaraway completes the political appropriation of Nietzsche's epistemic legacy by exposing the dystopic by-products of a \"dream\" that continues to guide the political struggles of some feminists:\n\n> The permanent partiality of feminist points of view has consequences for our expectations of forms of political organization and participation. We do not need a totality in order to work well. The feminist dream of a common language, like all dreams for a perfectly true language, of perfectly faithful naming of experience, is a totalizing and imperialist one.\n\nThis \"dream\" of foundational innocence is not only epistemically bankrupt, but also politically disastrous, for it imposes upon feminist politics conditions of justification that are impossible to meet. Haraway's campaign to expose and debunk this \"dream\" effectively absolves feminists of any perceived responsibility for grounding or justifying a political agenda by appeal to epistemic criteria. Haraway regards both epistemology and politics as serious endeavors, but she does not require of the latter that it acquire its justificatory and motive force from the former\u2014especially if the former retains any residue of its familiar patriarchal cast. The \"privilege\" of any postmodern feminist agenda must and will be purely political; the desire or need for a further, epistemic privilege will only frustrate feminist political activity.\n\nAs an antidote to the dream of foundational innocence, Haraway proposes various imaginative exercises designed to liberate feminists from the perceived need for an originary, epistemically pure standpoint. As an enabling narrative for postmodern feminists, Haraway submits the myth of the cyborg, a composite, hybrid creature that enacts the irresolvable tensions and dualities that characterize late modernity. The cyborg represents the embodiment of purely prospective agency, an unhistorical mutant to which the past\u2014along with the allure of innocence, origin, and redemption\u2014is irretrievably lost. If feminists can imagine themselves in their political activity as cyborgs\u2014which, in reality, women have always been\u2014then they can perhaps exorcize the immobilizing specter of _das Weib an sich,_ which continues to haunt their political practices.\n\nHere too Haraway follows Nietzsche. The prototype cyborg is none other than Zarathustra, the consummate micropolitical agent of late modernity. Indeed, we might profitably read the _Bildungsgang_ of Zarathustra as something like a cyborg myth: operating in the shadow of the dead God, consigned by his crepuscular destiny to a belief in crumbling idols that he can neither respect nor reject, Zarathustra must somehow neutralize his romantic dreams of return and redemption. He eventually \"becomes what he is\" by turning that which oppresses him\u2014his destiny and fatality\u2014to his own strategic advantage. He is ineluctably both free spirit and ascetic priest, and he implements both strands of this dual heritage to found a micro-community of \"higher men\" (Z IV:2\u20139). This community is unstable and ephemeral, lacking altogether in theoretical justifications, institutional reinforcements, and foundational myths. This community of higher men is exclusively prospective in its orientation; it has no laws, no traditions, no mores, no history, and no goal above and beyond the survival of European nihilism.\n\nZarathustra founds this cyborg community, supplying it with a minimal micropolitical infrastructure in the form of an inaugural \"Ass Festival\" (Z IV:17\u2013 18), but he eventually withdraws from it. He comes to realize that his dual heritage renders him both life-giving and life-destroying. Although he has consecrated this micro-community in the twilight of the idols, he has also encouraged his new companions to invest their redemptive hopes in him. Sensing that he has surreptitiously enslaved them and usurped the station of the dead God \u2014having become someone for the sake of whom \"living on earth is worthwhile\" (Z IV:19)\u2014he banishes the higher men and dissolves the micro-community he founded.\n\nThe final scene of _Zarathustra_ , framed in cyclical imagery that suggests a closed system, captures the purely prospective agency of the cyborg. Restless in his sheltering solitude but chastened by the prospect of reprising the pathogenesis that doomed his previous political endeavors, Zarathustra rises nonetheless to greet the dawn. Bereft of hopes for ultimate success, armed solely with the will to survive the decadence of late modernity, Zarathustra departs his cave once again to found yet another, equally ephemeral, cyborg community.\n\n### Nietzsche and Foucault\n\nThe legacy of Nietzsche's political confrontation with late modernity endures most vitally in the work of Michel Foucault (1926\u20131984). Foucault largely accepts Nietzsche's general diagnosis of late modernity, as a distintegrating epoch fraught with contradictions and disarray, and he arrives at a similar inventory of the political resources and options available to agents in late modernity. While he resists the biologistic terms and categories that convey Nietzsche's critique of modernity, Foucault nevertheless agrees that the project of critique can continue only if philosophers acknowledge their participation and complicity in the cultural malaise they intend to investigate. As a response to the uncertainty and contingency that cloud philosophical investigation in the twilight of the idols, Foucault adopts Nietzsche's experimental methods and adapts them to his own unique fields of inquiry.\n\n#### Archaeology\n\nFoucault is perhaps most famous for the experimental method he called _archaeology,_ which he developed in order to explore the peculiar epistemic conditions of late modernity. The guiding aim of the archaeologist is twofold: 1) to identify historical periods of epistemic convergence across a cluster of sciences and disciplines; and 2) to chart the transformation, and eventual disintegration, of epistemic coherence within the discursive practices of science. He thus intends his archaeological investigations to reveal the epistemic principles governing a period of discursive convergence. Foucault introduced the term _episteme_ to designate the epistemic coherence of various discursive practices within a single historical period. The sciences and disciplines clustered within an episteme all share a common epistemic framework and specialized vocabulary; they consequently exhibit general agreement on the basic conditions and criteria of truth, knowledge, verification, and certainty.\n\nThe subject matter investigated by the archaeologist is the discourse of the human sciences. By \"discourse\" Foucault means not only the shared, common language of scientific disciplines, but also the background structures and practices that determine the form and content of this language. The discourse of the human sciences thus comprises the specialized language spoken by scientific authorities, as well as the extra-linguistic conditions that determine what can be said, and by whom, in the production and distribution of scientific knowledge. Archaeology ascribes primary agency not to the experts who claim to conduct scientific inquiry, but to the \"anonymous field of practices\" embedded within the discourse itself. The discourse thus produces its representative \"experts,\" as well as the knowledge they carefully disseminate. Unwittingly serving the hidden interests of the discursive practices in which they involuntarily participate, subjects contribute to the production and dissemination of knowledge, which they claim to discover through \"objective,\" scientific inquiry.\n\nLike Nietzsche, Foucault attempts to exploit the unique conditions of his historical situation, treating the advanced decay of modernity as an uncharted field for philosophical investigation. An archaeology of knowledge is possible only when an episteme begins to disintegrate, as the clustered sciences and disciplines become conscious of the nature\u2014and limitations\u2014of the common epistemic framework they share. Indeed, discursive practices become visible to the archaeologist only when they manifest their internal instabilities, as evidenced by the mounting failures of their defining epistemic projects. As an episteme disintegrates, its signature constellation of conditions and criteria for knowledge becomes both manifest (to the archaeologist) and epistemically bankrupt.\n\nA representative deployment of Foucault's archaeological method is found in _The Order of Things._ 41 Foucault is primarily concerned in _The Order of Things_ to document the \"profound upheaval\" that occurred at the end of the eighteenth century, when the episteme of the Classical Age began to disintegrate. The project of representation, which had served to unify the Classical episteme, proved itself incapable of accounting for the act and the agent of representing. \"Man,\" the supposed focus of inquiry in the human sciences, had eluded the frame of representation. Despite the systemic failure of the project of representation, and the disintegration of its corresponding episteme, \"man\" nevertheless remains the (faulted) object of scientific inquiry in the nineteenth and twentieth centuries. The Age of Man, Foucault proposes, is defined by a series of ingenious (and failed) schemes to derive from human finitude itself the conditions of scientific knowledge. Reflecting the fragmentation of their avowed subject matter, the human sciences now labor in the absence of the sort of epistemic convergence that presided over their birth and development. The very possibility of truth and knowledge is now in question, as the human sciences scramble to construct theories and propositions in the space vacated by their proper object of investigation.\n\nRather than bemoan the disappearance of \"man,\" or attempt to restore epistemic coherence to the human sciences, Foucault presents the irrecuperable fragmentation of \"man\" as legislating new directions for scientific inquiry. _The_ _Order of Things_ proposes human finitude not as a new certainty that might solve the epistemic riddles of the human sciences, but as a new possibility in its own right, within which fruitful experimental thinking might take place. Like Nietzsche, Foucault intends his investigations to limn the unacknowledged (and unanticipated) successes sheltered within the failures of modernity. The disintegration of the episteme may very well illuminate heretofore neglected ethical and political resources.\n\n#### Subjectivation\n\nFoucault designed his archaeologies to yield purely descriptive results. He neither claimed nor expected to derive from them a critical appraisal of the discourse or episteme under investigation. While archaeology afforded him the critical distance he sought from the discursive practices he wished to investigate, it did not enable him to address the concrete social problems that concerned him. The most significant development in his political thinking thus occurred in the late 1960s, when he diverted his primary focus from discursive practices to the power relations that inform and sustain them.\n\nBy virtue of their involuntary participation in discursive practices, human beings are transformed into specific types of subjects, uniquely enabled and disabled to serve particular regimes of power. The defining characteristic of power is its capacity for \"infinite displacement\" within a complex network of discursive practices. All discourse is ultimately concerned with power, albeit in complicated, disguised ways. Because power can be effective only when it remains partially hidden, it always shelters itself within a discourse about something else. Power is both ubiquitous and capillary in its manifestations, and it announces its presence only as a diversion from its more central activities.\n\nIn order to supplement his archaeological expeditions with a critical, political dimension, Foucault developed the method of historical investigation known as \"genealogy,\" which he self-consciously borrowed from Nietzsche. The genealogical method enabled Foucault not only to investigate hidden power relations, but also to deliver a critique of concrete social practices. Although he never abandoned his archaeological method, he eventually came to rely on genealogy to expose the subtle transformations that signal the consolidating movement of power toward domination.\n\nBy tracing the historical descent of authoritative discursive practices, the Foucauldian genealogist is able to chart the shifting relationships between power and knowledge within the historical transformation of these practices. The genealogist can glimpse power only as it adjusts and reconfigures itself within the social and political relations it strategically inhabits, as it silently transforms institutions and discursive practices. Foucault was especially concerned to investigate the exclusionary power of discursive practices, including those reponsible for the institutionalized definitions of madness, criminality, and sexual deviancy.\n\nFoucault's subsequent turn to ethics comprises an extension of the development of his genealogical method. His contribution to ethics involved neither the articulation of a new moral theory, nor the advo cacy of an alternative to the signature \"techniques of the self\" that govern late modernity. He was concerned rather to investigate the conditions under which particular \"docile\" subjects are formed, and to expose the hidden power interests that are served by these subjects. He consequently centered his ethical investigations on the process he called _subjectivation,_ whereby human beings are gradually transformed into subjects invested with unique powers and limitations. Of specific concern to him, especially in his final writings, were the techniques of subjectivation deployed by \"bio-power,\" which organizes the resources of modern societies under the pretense of attending to the care of the species and the health of individual human beings.\n\nIn his late writings on ethics, Foucault began to carve out for the subject a modest domain of agency and causal efficacy. His genealogies of the modern sexual subject revealed that the process of subjectivation is not strictly coercive and disabling. Power relations also manifest themselves in productive discursive practices, including those that distribute goods and information, satisfy limited desires, manage resources, and secure the material conditions of social harmony. Modern subjects are rendered docile through a battery of normalizing disciplines, but they can also turn the productive power invested in them against regimes of power that threaten to accede to domination.\n\nFoucault's ethical investigations thus revealed that regimes of power can realize their ends only if human beings are transformed into productive subjects, invested with a limited capacity for self-legislation. The subjects depicted in his later genealogies are not merely the unwitting products of clandestine discursive practices, for they are able to resist the totalization of power within its most ambitious and monolithic regimes. To expose the mobilization of power within mechanisms of transformation is to render it temporarily less effective and less dangerous. Power can neither be eradicated nor contained, but its inexorable tendency toward domination can often be neutralized within local regimes.\n\n#### Toward a Politics of Resistance\n\nNietzsche's experimentation with unexplored configurations of agency is taken up by Foucault in the later, \"ethical\" period of his career. Although Foucault located in the dominant regimes of bio-power a masked impulse toward domination, he also acknowledged their productive, empowering roles in the formation of the modern subject. At the end of his career, he attempted a partial recuperation of the agency of the subject, outlining the conditions under which subjects might constitute themselves in opposition to dangerous techniques of subjectivation. Under certain conditions, modern subjects can\u2014and do\u2014resist the totalization of power within specific structures and local regimes.\n\nFoucault's writings from the \"ethical\" period of his career thus approximate most closely the experimental politics of resistance outlined by Nietzsche. Especially in his investigations into _The History of Sexuality,_ Foucault seriously entertains Nietzsche's \"affirmative\" thesis that (some) modern subjects may turn ascetic disciplines to their own advantage. Techniques of subjectivation are always both repressive _and_ empowering, investing subjects with residual faculties that may be deployed in the resistance of domination. Acknowledging that the constitution of the subject produces, as an unintended by-product, a feedback loop of potentially generative powers, Foucault gestures toward an experimental investigation of the untested modes of agency available\u2014though perhaps unknown\u2014to modern subjects. His efforts to resist _domination,_ by which he means the totalizing consolidation of power within specific regimes of bio-power, thus emerge as analogous to Nietzsche's attempts to deflect the advent of the \"will to nothingness.\" For both thinkers, the immediate goal of political resistance is to safeguard the continued development of human subjectivity, such that it might someday stand security for its own future.\n\nUnlike Nietzsche, Foucault declined the mantle of the political lawgiver. Wary of his own complicities in the regimes of power he sought to illuminate, he refused to posit alternative ethical systems or \"techniques of the self,\" which might in turn serve as sites for the consolidation of power into structures of domination. He consciously limited himself to an investigation of the historical conditions that sustain the array of subject constitutions available to us. Here Foucault deviated most obviously from Nietzsche as a political thinker. In suspending his subversive genealogical practices to declare war on Christian morality, Foucault might say, Nietzsche unwittingly contributes to the formation of regimes of power that are potentially more dangerous than those manifested within the discourse of Christian morality. Foucault's stubborn refusal to situate himself on the \"political checkerboard\" was thus intended to minimize his own deleterious influence on the exploration of the novel configurations of agency he discovered. While it is doubtful that his strategy for neutralizing his own negative influences was, or could have been, entirely successful, his desire to avoid the fate of Nietzsche is certainly understandable.\n\nLike Nietzsche, Foucault never completed the ambitious genealogical agenda he set for himself. The guarded optimism conveyed by his final writings has been taken up by successor genealogists, and the articulation of a Foucauldian political position remains, for some, an ongoing project.\n\n# Notes\n\n### Introduction\n\n. Mark Warren draws a useful distinction between Nietzsche's political philosophy and his \"philosophy of power,\" in _Nietzsche and Political Thought_ (Cambridge, MA: MIT Press, 1988), pp. 1\u20138, 207\u2013223. According to Warren, Nietzsche's political philosophy does not merit further consideration because it belies the promise of his \"philosophy of agency\" and his \"critique of Western culture.\" Although I reject Warren's dismissive characterization of Nietzsche's politics as a \"neoaristocratic conservatism\" (p. 211), as well as his endorsement of Nietzsche's \"philosophy of power\" as a \"framework for a critically postmodern political philosophy\" (p. 207), his distinction helpfully illuminates the potential discontinuities that obtain between Nietzsche's antifoundationalist philosophical insights and his less revolutionary political sentiments.\n\n### 1  \nPolitical Perfectionism\n\n. Nietzsche's interest in \"breeding\" human types, as opposed to producing a \"successor\" species, would seem to militate against the thesis of Tracy Strong's pioneering study, _Friedrich Nietzsche and the Politics of Transfiguration_ (Berkeley: University of California Press, 1974). Strong attributes to Nietzsche a \"politics of transfiguration,\" which undertakes \"the development of beings who simply do not live as human-all-too-human\" (p. 13), which in turn \"require[s] a change in the very stuff of humanity\" (p. 260). I am both sympathetic and indebted to Strong's general line of interpretation, but I find his emphasis on \"the politics of transfiguration\" to be misleadingly freighted with images of transcendence and radical discontinuity. If, as Nietzsche suggests in this passage, \"man is an _end_ ,\" then a \"politics of transfiguration\" would constitute an inappropriate, overzealous, and nihilistic response to the problem of modernity. As I hope to show in this study, Nietzsche's primary political goal is the perfection or completion of humankind, not its transfiguration.\n\n2. My reading of Nietzsche's perfectionism is indebted to Stanley Cavell's seminal essay, \"Aversive Thinking: Emersonian Representations in Heidegger and Nietzsche,\" in _Conditions Handsome and Unhandsome: The Constitution of_ _Emersonian Perfectionism_ (Chicago: University of Chicago Press, 1990), pp. 33\u2013 63. Cavell restricts his attention almost exclusively to the early essay _Schopenhauer as Educator,_ but his account of Nietzsche's perfectionism is relevant for an examination of the post-Zarathustran writings as well.\n\n3. Nietzsche explicitly recommends the conditions under which Schopenhauer emerged (SE 7), but if we endorse his later observation that his _Untimely_ _Meditations_ \"speak only of [him]\" (EH V:3), then the conditions he describes are (also) responsible for his own development as a genius.\n\n4. Nietzsche futher explains that \"This _instinct for freedom_ forcibly made latent\u2014we have seen it already\u2014this instinct for freedom pushed back and repressed, incarcerated within and finally able to discharge and vent itself only on itself: that, and that alone, is what the _bad conscience_ is in its beginnings\" (GM II:17).\n\n5. Bruce Detwiler suggests that \"Nietzsche distinguishes between what he calls the 'good conscience' characteristic of the noble type and the 'bad conscience' that characterizes the slavish type,\" in _Nietzsche and the Politics of Aristocratic_ _Radicalism_ (Chicago: University of Chicago Press, 1990), p. 124. Detwiler's suggestion is especially helpful in explaining why Nietzsche believes that certain individuals are (or have been) spared the existential suffering that attends the introversion of the natural impulses. Whereas civilization demands of _everyone_ the inward discharge of the \"instincts,\" only those who are disadvantaged by the reigning social order are debilitated by the existential suffering of the bad conscience. On this interpretation, the \"good conscience\" would be the privilege of those whose social standing affords them the opportunity to compensate somehow for the introversion of their animal impulses.\n\n6. My distinction between the \"existential suffering\" of the bad conscience and the \"surplus suffering\" of guilt draws on Arthur C.Danto's distinction between \"extensional suffering and intensional suffering, where the latter consists of an interpretation of the former,\" in \"Some Remarks on _The Genealogy of Morals,_ \" in _Reading Nietzsche,_ ed. Robert C.Solomon and Kathleen M. Higgins (New York: Oxford University Press, 1988), p. 21.\n\n7. I am indebted for this particular formulation of the point to Cavell, pp. 58\u201359.\n\n8. Alasdair MacIntyre, for example, decrees that \"the _\u00dcbermensch_...belong[s] in the pages of a philosophical bestiary rather than in serious discussion.\" _After Virtue_ (Notre Dame, IN: Notre Dame University Press, 1984), p. 22.\n\n9. For a superbly rich account of the development of Zarathustra (and his teachings) throughout the course of his travels, see Laurence Lampert, _Nietzsche's Teaching_ (New Haven: Yale University Press, 1986). I take up the question of Zarathustra's reliability in my essay, \"Solving the Problem of Socrates: Nietzsche's _Zarathustra_ as Political Irony,\" _Political Theory,_ vol. 16, no. 2, May 1988, pp. 257\u2013280.\n\n10. Several scholars distinguish profitably between the _\u00dcbermensch_ and \"higher humanity.\" See, for example, Richard Schacht, _Nietzsche_ (London: Routledge, 1983), p. 339; and Ofelia Schutte, _Beyond Nihilism: Nietzsche Without Masks_ (Chicago: University of Chicago Press, 1984), p. 122. Wolfgang M\u00fcller-Lauter similarly maintains that the _\u00dcbermensch_ represents the \"apotheosis\" of \"higher humanity,\" in _Nietzsche: Seine Philosophie der Gegens\u00e4tze und die Gegens\u00e4tze_ _seiner Philosophie_ (Berlin: de Gruyter, 1971), p. 127; cf. Schacht, p. 482.\n\n11. In a letter dated 20 October 1888, Nietzsche tells Malwida von Meysenbug that the _\u00dcbermensch_ is a type of human being who is \"a hundred times more similar\" to the type instantiated by Cesare Borgia than to the type instantiated by Christ. _Friedrich_ _Nietzsche, S\u00e4mtliche Briefe: Kritische Studienausgabe,_ ed. G. Colli and M. Montinari (Berlin: de Gruyter\/Deutscher Taschenbuch Verlag, 1986), vol. 8, #1135, p. 458.\n\n12. Explaining his \"destiny,\" Nietzsche allows that he fears further misunderstandings of his teachings: \"I have a terrible fear that one day I will be pronounced _holy_ : you will guess why I publish this book before; it shall prevent people from doing mischief with me\" (EH XIV:1). The \"before\" refers here to the publication of _The_ _Antichrist(ian)_ , which Nietzsche intended to withhold until the reception of _Ecce_ _Homo_ had sufficiently reduced the likelihood of censorship and confiscation.\n\n13. For a measured critique of the interpretation of the _\u00dcbermensch_ as constituting an \"ideal type,\" see Bernd Magnus, \"Perfectibility and Attitude in Nietzsche's _\u00dcbermensch,_ \" _The Review of Metaphysics,_ vol. 36, March 1983, pp. 633\u2013659.\n\n14. Nietzsche again defines the _\u00dcbermensch_ in relational terms when he identifies \"the type of man\" that \"Zarathustra wants\" as \"a relatively superhuman _[\u00fcbermenschlicher]_ type\" (EH XIV:5).\n\n15. My interpretation of the founding labors of the _\u00dcbermensch_ is indebted to David Owen's discussion of \"agonism,\" and its capacity for contributing to the cultivation of _virt\u00f9,_ in _Nietzsche, Politics and Modernity_ (London: Sage, 1995), esp. pp. 132\u2013 154; and to William Connolly's Nietzsche-inspired account of \"agonistic democracy\" in _Identity\/Difference_ (Ithaca: Cornell University Press, 1991), pp. 184\u2013197.\n\n16. MacIntyre, for example, describes the _\u00dcbermensch_ as \"the man who transcends, finds his good nowhere in the social world to date, but only in that in himself which dictates his own new law and his own new table of the virtues... [T]he great man cannot enter into relationships mediated by appeal to shared standards or virtues or goods; he is his own only authority and his relationships to others have to be exercises of that authority\" (pp. 257\u2013258).\n\n17. According to J.P.Stern, for example, Nietzsche \"seems unaware that he is giving us nothing to distinguish the fanaticism that goes with bad faith from his own belief in the unconditioned value of self-realization and self-becoming\u2014that is, from his own belief in the Superman... No man came closer to the full realization of self-created 'values' than A.Hitler.\" _A Study of Nietzsche_ (Cambridge: Cambridge University Press, 1979), p. 117.\n\n18. Alexander Nehamas astutely observes that \"the characters Nietzsche admires...are overwhelmingly literary and artistic.\" _Nietzsche: Life as Literature_ (Cambridge, MA: Harvard University Press, 1985), p. 227.\n\n19. Emerson's commitment to the exemplary standing of \"representative men\" is evidenced throughout his essays and lectures. For a summary account, see his introductory essay on the \"Uses of Great Men,\" in _The Collected Works of Ralph_ _Waldo Emerson,_ vol. IV: _Representative Men: Seven Lectures,_ ed. Wallace E.Williams and Douglas Emory Wilson (Cambridge, MA: Harvard University Press, 1987), pp. 1\u201320.\n\n20. My reckoning of Nietzsche's psychological debts to Emerson is indebted to Graham Parkes, _Composing the Soul: Reaches of Nietzsche's Psychology_ (Chicago: University of Chicago Press, 1994), esp. ch. 4.\n\n21. Gianni Vattimo argues that the _\u00dcbermensch_ can emerge only in a (future) world that has already been made _\u00fcbermenschlich_ \u2014a task which outstrips the volitional capacities of late modernity. The trajectory of Nietzsche's political thinking may consequently describe a vicious circle: the _\u00dcbermensch_ is its own precondition. _The Adventure of Difference: Philosophy after Nietzsche and Heidegger,_ trans. Cyprian Blamires and Thomas Harrison (Baltimore: Johns Hopkins University Press, 1993), pp. 33\u201336; 55\u201358.\n\n22. Robert Pippin attributes to Zarathustra the realization that the _\u00dcbermensch_ represents \"a radically temporal, contingent 'ideal'; it answers _only_ the specific, practical incoherence of the ideals of late bourgeios culture.\" \"Irony and Affirmation in Nietzsche's _Thus Spoke Zarathustra,_ \" in _Nietzsche's New Seas,_ ed. Michael Allen Gillespie and Tracy B.Strong (Chicago: University of Chicago Press, 1988), p. 52.\n\n23. MacIntyre persuasively maintains that \"The concept of the Nietzschean 'great man' is also a pseudo-concept... It represents individualism's final attempt to escape from its own consequences.\" _After Virtue,_ p. 259.\n\n### 2  \nThe Uses and Disadvantages of Morality for Life\n\n1. John Rawls, _A Theory of Justice_ (Cambridge, MA: Harvard University Press, 1971), p. 325.\n\n2. Ibid.\n\n3. Ibid.\n\n4. Nietzsche finds a \"natural\" basis for the pyramidal structure he prefers for hierarchically organized societies, in the \"powerful pyramidal rock\" he spies near the lake at Silvaplana, at the time that the thought of eternal recurrence descends upon him (EH 111:1).\n\n5. Letter to Franz Overbeck on 24 March 1887. Friedrich Nietzsche, _S\u00e4mtliche_ _Briefe: Kritische Studienausgabe,_ ed. G Colli and M.Montinari (Berlin: de Gruyter\/ Deutscher Taschenbuch Verlag, 1986), vol. 8, #820, p. 48.\n\n6. Nietzsche thus adds a third caste to the two-tiered system he proposed in H I:439.\n\n7. Nietzsche distinguishes himself from the \"whole European and American species of _libres penseurs,_ \" who \"still believe in the 'ideal'.\" Declaring himself \"the first _immoralist,_ \" he thus implies that immoralists no longer want \"to 'improve' humankind, in their own image\" (EH V:2).\n\n8. Nietzsche maintains that \"the claim for independence, for free development, for _laisser aller_ is pressed most hotly by the very people for whom no reins would be too strict. This is true _in politics,_ this is true in art\" (TI IX:41).\n\n9. As early as his _Untimely Meditation_ on Schopenhauer, Nietzsche proposed that the task of culture is to correct for the profligacy of Nature: \"Nature propels the philosopher into humankind like an arrow; it takes no aim but hopes the arrow will stick somewhere. But countless times it misses and is depressed at the fact... The artist and the philosopher are evidence against the purposiveness of nature as regards the means it employs, though they are also first-rate evidence as to the wisdom of its purpose\" (SE 7).\n\n10. Allowing that Manu \"found it necessary to be _terrible [furchtbar]_ \" in the struggle against \"the unbred man, the mish-mash man, the chandala,\" Nietzsche recounts some of the more grisly edicts of Manu's law (TI VII:3).\n\n11. While Nietzsche's endorsement of slavery may be irrecuperably offensive to contemporary liberal sensibilities, we should bear in mind the plethora of forms\u2014 many of them documented by Nietzsche himself\u2014that slavery has assumed over the millennia. Surely the most efficient forms of slavery are those psychological forms embraced by the slaves themselves, such that their slavery becomes the precondition for their perceived happiness or freedom. Although Nietzsche's perfectionism expressly dismisses the political claim of demotic interests, it could nevertheless inspire a \"Grand Inquisitor\" regime that rewards its \"slaves\" with material comforts and spiritual anesthesia. In his notes, he speculates that \"European democracy must become ultimately\" a \"new and sublime development of slavery.\" A political regime that could \"make use of\" democracy to further the enhancement of humankind would represent \"a kind of goal, redemption and justification for the democratic movement\" (WP 954).\n\n12. Mill argues that even those who do not benefit personally and directly from \"individuality\" should nevertheless endorse its promotion, for they derive an indirect benefit from the \"individuality\" of others. _On Liberty,_ ed. Elizabeth Rapaport, (Indianapolis: Hackett, 1978), ch. 3: \"Of Individuality,\" esp. pp. 61\u201368.\n\n13. Nietzsche does argue that \"The philosopher and the artist...strike home at only a few, while they ought to strike home at everybody\" (SE 7), but his remarks on education later in that essay indicate that he does not intend a democratic justification of culture.\n\n14. Nietzsche's shift in orientation to the political microsphere deflects many of the criticisms leveled against his political thinking. Keith Ansell-Pearson, for example, argues that \"[Nietzsche] too demands a politics of transfiguration in which modern individuals are to elevate themselves through a process of 'going-down' and 'goingacross' to higher tasks and to higher responsibilities (the _\u00dcbermensch_ ). But the conditions\u2014namely, a tragic culture\u2014which would serve to cultivate such individuals are absent in modern liberal societies.\" _Nietzsche contra Rousseau_ (Cambridge: Cambridge University Press, 1991), p. 224. If Nietzsche _were_ invested in some such \"politics of transfiguration,\" then it might be true that \"modern liberal societies\" lack the resources \"to cultivate such individuals.\" But Nietzsche does not rely on the macropolitical resources of modern liberal societies to produce the exemplary human beings he has in mind. He is well aware that the macropolitical ambitions of his youth are simply incompatible with his historical situation.\n\n15. For a sympathetic treatment of Nietzsche's attempt to legislate an \"aristocracy of soul,\" see Leslie Paul Thiele, _Friedrich Nietzsche and the Politics of the Soul_ (Princeton: Princeton University Press, 1990), esp. chs. 3\u20134.\n\n### 3  \nPerfectionism in the Twilight of the Idols\n\n1. These unpublished lectures are reproduced in vol. 1 of the _Kritische_ _Studienausgabe,_ pp. 643\u2013752.\n\n2. For an excellent study of Nietzsche's attempt to articulate \"physiological\" diagnoses of the decadence of late modernity, see Daniel Ahern, _Nietzsche as_ _Cultural Physician_ (University Park, PA: Pennsylvania State University Press, 1995), esp. ch. 1.\n\n3. In a note from 1888, which echoes several others from that year, Nietzsche writes, \"Basic insight regarding the nature of decadence: _its supposed causes are its_ _consequences_ \" (WP 41; cf. 42\u201345). He maintains that \"it is an error to consider 'social distress' or 'physiological degeneration' or, worse, corruption, as the _cause_ of nihilism\" (WP 1).\n\n4. I borrow this phrase from Bernard Yack's provocative study, _The Longing for_ _Total Revolution_ (Princeton: Princeton University Press, 1986). As Yack \"supposes,\" he perhaps \"relies too heavily\" on the pre-Zarathustran writings in proposing this \"longing\" as representative of Nietzsche's political thinking as a whole (p. 313). While the persistence of this longing is difficult to dispute, Yack does not acknowledge the transformations and complexity introduced into Nietzsche's political thinking by the critique of modernity advanced in his post- Zarathustran writings.\n\n5. For example, Nietzsche expresses his \"hope\" for the advent of \"men of the future who in the present tie the knot and constraint that forces the will of millennia upon _new_ tracks\" (BGE 203).\n\n6. For a stimulating interpretation of Nietzsche's \"desperate\" turn (from aesthetics) to politics at the end of his sane life, see Tracy Strong, \"Nietzsche's Political Aesthetics,\" in _Nietzsche's New Seas,_ ed. Michael Allen Gillespie and Tracy Strong (Chicago: University of Chicago Press, 1988), pp. 153\u2013174.\n\n7. Responding to Rawls's decision to bar Nietzsche's perfectionism from the original position, Cavell remarks, \"[I]f Nietzsche is to be dismissed as a thinker pertinent to the founding of the democratic life, then so, it should seem, is Emerson, since Nietzsche's meditation on Schopenhauer is, to an as yet undisclosed extent, a transcription and elaboration of Emersonian passages. Emerson's dismissal here would pain me more than I can say, and if that is indeed the implication of _A_ _Theory of Justice_ , I want the book, because of my admiration for it, to be wrong in drawing this implication from itself.\" Stanley Cavell, \"Aversive Thinking: Emersonian Representations in Heidegger and Nietzsche,\" in _Conditions_ _Handsome and Unhandsome: The Constitution of Emersonian Prefection_ (Chicago: University of Chicago Press, 1990), p. 49.\n\n8. Ibid., p. 53.\n\n9. Ibid., p. 56.\n\n10. Cavell acknowledges a similar objection, commenting that \"This is important, but it does not seem to me enough to say\" (ibid., p. 102).\n\n11. Cf.Pindar, _Pythian Odes,_ II, 72. Nietzsche fondly recommended Pindar's imperative to Lou Salom\u00e9 just before their final estrangement. He closes his letter of 10 June 1882 with the sentence: \"Pindar sagt einmal, 'werde der, der du bist!'\" _S\u00e4mtliche Briefe: Kritische Studienausgabe,_ ed. G.Colli and M.Montinari (Berlin: de Gruyter\/Deutscher Taschenbuch Verlag, 1986), vol. 6, #239, p. 203.\n\n12. John Rawls, _A Theory of Justice_ (Cambridge, MA: Harvard University Press, 1971), pp. 442\u2013446.\n\n13. Cavell, pp. 49\u201353.\n\n14. See Cavell, p. 59.\n\n15. For a fine account of Nietzsche's deployment of the image of the household, or _oikos_ , and of its Platonic origins, see Graham Parkes, _Composing the Soul: Reaches_ _of Nietzsche's Psychology_ (Chicago: University of Chicago Press, 1994), pp. 42\u2013 52, 215\u2013225.\n\n16. Socrates proposes his noble lie, via the \"myth of the metals,\" in book III of the _Republic_ , at 414b\u2013417b.\n\n### 4  \nRegimens of Self-Overcoming\n\n1. This point is forcefully stated by Stanley Cavell, \"Aversive Thinking: Emersonian Representations in Heidegger and Nietzsche,\" in _Conditions Handsome and_ _Unhandsome: The Constitution of Emersonian Perfectionism_ (Chicago: University of Chicago Press, 1990) pp. 46\u201357.\n\n2. The theme of the philosopher's \"soulcraft\" is treated sympathetically and at length by Leslie Paul Thiele, _Friedrich Nietzsche and the Politics of the Soul_ (Princeton: Princeton University Press, 1990), esp. ch. 4.\n\n3. See Cavell, pp. 112\u2013113.\n\n4. See Cavell, pp. 47\u201348,\n\n5. In a passage ideally suited to establish the distinction I have suggested, Nietzsche observes no terminological distinction between macrosphere and microsphere: \"The self-overcoming _[Selbst\u00fcberwindung]_ of morality, out of truthfulness; the self-overcoming _[Selbst\u00fcberwindung]_ of the moralist, into his opposite\u2014into me\u2014that is what the name of Zarathustra means in my mouth\" (EH XIV\u2013.3). Were Nietzsche inclined toward the distinction I have suggested, he presumably would have replaced the former instance of _Selbst\u00fcberwindung_ with _Selbstaufhebung._\n\n6. Wolfgang M\u00fcller-Lauter argues that this \"aggregate\" model of self-overcoming stands in tension with Nietzsche's praise of those individuals who command a single, monolithic perspective. As a consequence of this tension, M\u00fcller-Lauter concludes, Nietzsche alternately describes the _\u00dcbermensch_ in terms of the superhuman command of a single perspective, to the exclusion of all others, and in terms of a superhuman accommodation of a multiplicity of perspectives. _Nietzsche:_ _Seine Philosophie der Gegens\u00e4tze und die Gegens\u00e4tze seiner Philosophie_ (Berlin: de Gruyter, 1971), pp. 116\u2013134.\n\n7. For an excellent treatment of Nietzsche's reliance on political metaphors for the cultivation of the soul, and of the Platonic provenance of these metaphors, see Graham Parkes, _Composing the Soul: Reaches of Nietzsche's Psychology_ (Chicago: University of Chicago Press, 1994), pp. 346\u2013362.\n\n8. For an even-handed chronicle of Nietzsche's use and abuse of Rousseau, see Keith Ansell-Pearson's excellent study, _Nietzsche contra Rousseau_ (Cambridge: Cambridge University Press, 1991), esp. ch. 1.\n\n9. David Owen delivers an excellent reading of the \"agonism\" that emerges from this early essay in _Nietzsche_ , _Politics and Modernity_ (London: Sage, 1995), pp. 139\u2013169.\n\n10. Richard Rorty apparently derives his account of Nietzschean self-creation from his anti-essentialism and historicism. _Because_ no authentic self exists to be be discovered through cognitive processes, he reasons, the self is _therefore_ a construct. _Contingency, Irony and Solidarity_ (New York: Cambridge University Press, 1989), ch. 2. Rorty's reasoning is valid, of course, only in the event that his guiding disjunction\u2014discovery vs. creation\u2014is both exclusive and warranted.\n\n11. I develop this point further in my essay, \"Disembodied Perspectives,\" _Nietzsche-_ _Studien,_ vol. 21, 1992, pp. 281\u2013289.\n\n12. Nietzsche's revision of the motto he adopts from Pindar perhaps reflects this growing emphasis on self-discovery. \"Become who you are\" (GS 270) is replaced by \"Become what one is\" (EH; see also GS 335).\n\n13. Persuasively maintaining that Nietzsche never decides between the discovery of truth and its invention (p. 234), Nehamas works productively in the interstitial spaces between volition and cognition in Nietzsche's thought.\n\n14. Several commentators have located Nietzsche's \"solution\" to the problem of consciousness in his recommendation of a process known as \"active forgetting.\" According to Bernard Yack, for example, \"Self-conscious forgetting is the remedy that Nietzsche, the self-appointed doctor of sick cultures, prescribes.\" _The Longing_ _for Total Revolution_ (Princeton: Princeton University Press, 1986), p. 335. While Nietzsche explicitly describes forgetting as \"an active and in the strictest sense positive faculty of repression\" (GM 11:1), and while he praises in Mirabeau the \"excess of the power...to forget\" (GM 1:10), I see little evidence that Nietzsche prescribes \"active forgetting\" (or anything else) to a dying culture. If Nietzsche's contemporaries were capable of \"active forgetting,\" then would they not, like Mirabeau, already be engaged in it\u2014even if unbeknownst to themselves? In light of the formidable epistemological problems involved in exhorting a dying culture \"to remember to forget,\" it comes as little surprise that Yack subsequently judges Nietzsche's \"prescription\" for modernity to yield \"paradox\" and \"self-contradiction.\"\n\n15. See, for example, Nehamas, pp. 184\u2013190.\n\n16. Nietzsche suggests this image of himself as a \"cave\" in his letter to Reinhard von Seydlitz on 12 February 1888. _S\u00e4mtliche Briefe,_ vol. 8, #989, p. 248.\n\n17. My attention here to Nietzsche's polycentric model of the soul is indebted to Parkes, esp. ch. 9.\n\n### 5  \nThe Philosopher's _Versucberkunst_\n\n1. My investigations of the rich _Versuch_ motif in Nietzsche's post-Zarathustran thought have been informed and contoured by the following studies: Eric Blondel, _Nietzsche: The Body and Culture,_ trans. Sean Hand (Stanford: Stanford University Press, 1991), ch. 6; Jacob Golomb, _Nietzsche's Enticing Psychology of Power_ (Ames: Iowa State University Press, 1989); Laurence Lampert, _Nietzsche and_ _Modern Times_ (New Haven: Yale University Press, 1993), pp. 330\u2013334; and Henning Ottman, _Philosophie und Politik bei Nietzsche_ (Berlin: de Gruyter, 1987), section V: \"Versuch \u00fcber Nietzsches 'Versuche,'\" pp. 346\u2013388.\n\n2. In his pathbreaking study, Walter Kaufmann calls attention to the perfectionism that lies at the heart of Nietzsche's moral philosophy. _Nietzsche: Philosopher,_ _Psychologist, Antichrist,_ 4th edn (Princeton: Princeton University Press, 1974), pp. 242\u2013256. In accordance with his own campaign to rehabilitate Nietzsche's philosophy, however, Kaufmann insists that Nietzschean self-perfection is an exclusively private enterprise.\n\n3. The provocation unwittingly initiated by the _Erzieher_ is discussed at length by Emerson in his \"Divinity School Address,\" in _Selected Writings of Ralph Waldo_ _Emerson,_ ed. William H.Gilman (New York: Penguin, 1983), pp. 241\u2013257.\n\n4. For an eloquent reckoning of Nietzsche's inheritance from Emerson of the notions of \"intuition\" and \"provocation,\" see Tracy Strong, \"Nietzsche's Political Aesthetics,\" in _Nietzsche's New Seas,_ ed. Michael Allen Gillespie and Tracy Strong (Chicago: University of Chicago Press, 1988), pp. 158\u2013159.\n\n5. For a compelling account of Nietzsche's affinities to Plato, particularly with respect to _er s,_ see Graham Parkes, _Composing the Soul: Reaches of Nietzsche's_ _Psychology_ (Chicago: University of Chicago Press, 1994), ch. 6.\n\n6. _Kritische Studienausgabe,_ vol. 13, 14 [120], p. 299.\n\n7. In an important insight, Strong notes that the philosopher creates himself not only as an artwork, but also as a \"source of this-worldly authority\" (p. 158). According to Strong, however, Nietzsche despairs toward the end of his life that the nomothetic properties of art are simply inadequate to the task of creating a \"source of this-worldly authority\" suitable to the peculiar conditions of late modernity. Abandoning aesthetics, Nietzsche \"comes in desperation to politics\" (p. 168). Although Nietzsche, according to Strong, \"is not a thinker of domination...at the end he can only try to prevent a politics of domination by proposing one of his own\" (p. 171).\n\n8. Bruce Detwiler offers an instructive account of the political role of the \"artist-philosopher\" in _Nietzsche and the Politics of Aristocratic Radicalism_ (Chicago: University of Chicago Press, 1990), ch. 6. Detwiler's account suffers, however, from its strict association of the \"artist-philosopher\" with the \"new order\" that Nietzsche ostensibly envisions for the future. Detwiler consequently misplaces Nietzsche's own efforts as an \"artist-philosopher,\" as well as the communities he creates in response to the decadence of late modernity.\n\n9. Nietzsche speaks of himself in a similar way: \"when I described Dionysian music I described what _I_ had heard\u2014that instinctively I had to transpose and transfigure everything into the new spirit that I carried in me\" (EH IV: 4).\n\n10. On the indirect and ironic products of Socrates' political teachings, see Leo Strauss, _The City and Man_ (Chicago: University of Chicago Press, 1964), pp. 73\u2013 87 and 122\u2013127; and Allan Bloom's \"Interpretive Essay,\" in _The Republic of_ _Plato,_ 2nd edn, trans. Allan Bloom (New York: Basic Books, 1991), pp. 408\u2013412.\n\n11. On the self-creation of the philosopher, see Leslie Paul Thiele, _Friedrich Nietzsche_ _and the Politics of the Soul_ (Princeton: Princeton University Press, 1990), p. 131.\n\n12. Nietzsche later adds that \"Before tragedy, what is warlike in our soul celebrates its Saturnalia; whoever is used to suffering, whoever seeks out suffering, the heroic man praises his own being through tragedy\u2014to him alone the tragedian presents this drink of sweetest cruelty\" (TI IX\u2013.24).\n\n13. Drawing perhaps on his own limited experiences, he offers the following \"definition\" of heterosexual love: \"Has my definition of love been heard? It is the only one worthy of a philosopher. Love\u2014in its means, war; at bottom, the deadly hatred of the sexes\" (EH III:5).\n\n14. _S\u00e4mtliche Werke,_ vol. 13, 14 [120], pp. 299\u2013300. My attention to this passage is indebted to Martha C.Nussbaum's essay, \"The Transfigurations of Intoxication: Nietzsche, Schopenhauer, and Dionysus,\" _Arion,_ vol. 1, no. 2, 1991, pp. 75\u2013111.\n\n15. In a cynical corollary to this thesis, Nietzsche maintains, \"The cure for love is still in most cases that ancient radical medicine: love in return\" (D 415).\n\n16. For an important political elaboration of Nietzsche's appeal to \"aesthetic communities,\" see Salim Kemal, \"Nietzsche's _Genealogy:_ Of Beauty and Community,\" _Journal of the British Society for Phenomenology,_ vol. 21, no. 3, October 1990, pp. 234\u2013249.\n\n17. This question perhaps illuminates the limitations of Georges Bataille's attempt to follow Nietzsche, to \"push the possibilities of his teaching to the limit,\" in Georges Bataille, _On Nietzsche,_ trans. Bruce Boone (New York: Paragon House, 1992), p. xxiv. Bataille characteristically locates Nietzsche's excesses in his _writing_ \u2014hence the importance for Bataille of his own writing\u2014but the nature of these excesses themselves, independent of their representative inscriptions, eludes Bataille's grasp.\n\n18. Nietzsche's depiction of the \"music-practicing Socrates\" trades on a reference to an early scene in Plato's _Phaedo_ (60c-d). In this scene, however, Socrates quite clearly understands his attempt to versify several of Aesop's fables as a tribute to Apollo (61a-b), who is ultimately responsible for ordaining Socrates the wisest of men. That Socrates has the time and leisure to \"practice music\" is attributable, moreover, to the festival of the Delia, which is itself a tribute to Apollo (58a-c). Finally, Asclepius, the physician to whom the \"music-practicing\" Socrates attempts to make advance payment (118a), is the son of Apollo. A more appropriate candidate for a \"music-practicing\" Socrates would be the Socrates who wears the wreath of Dionysus at the conclusion of Plato's _Symposium._\n\n19. Walter Kaufmann apparently agrees, for he presents Nietzsche as scoring a definitive triumph over his ressentiment: \"In _Ecce Homo_ Nietzsche embodies this triumph over _ressentiment_. Instead of bearing a grudge toward the world that treated him so cruelly, instead of succumbing to the rancor of sickness, he relates the story of his life and work in a spirit of gratitude\u2014and goes out of his way to pay his respects to Paul R\u00e9e and Lou Salom\u00e9, with whom he had fallen out.\" _On_ _the Genealogy of Morals\/Ecce Homo,_ trans. Walter Kaufmann (New York: Vintage Books, 1989), pp. 206\u2013207.\n\n20. Nietzsche actually defines resentment as a concoction of affects. He lists \"anger, pathological vulnerability, impotent lust for revenge, thirst for revenge\" as \"the affects of _ressentiment_ \" (EH 1:6).\n\n21. This thesis is persuasively advanced in Stanley Rosen's provocative essay, \"Nietzsche's Revolution,\" in _The Ancients and the Moderns_ (New Haven: Yale University Press, 1989), pp. 191\u2013198.\n\n22. Robert Solomon inventories the potentially productive properties of resentment in his essay, \"One Hundred Years of Resentment: Nietzsche's _Genealogy of Morals,_ \" in _Nietzsche, Genealogy, Morality: Essays on Nietzsche's \"On the Genealogy of_ _Morals,\"_ ed. Richard Schacht (Berkeley, CA: The University of California Press, 1994), pp. 95\u2013126. See also Henry Staten, _Nietzsche's Voice_ (Ithaca: Cornell University Press, 1990), ch. 2.\n\n23. Nietzsche attributes his \"cleverness,\" for example, to his economic hygiene: \"I have never reflected on questions that were none\u2014I have not squandered _[verschwendet]_ myself\" (EH 11:1).\n\n24. Hence the inadequacy of the influential distinction between \"active\" and \"reactive\" forces, as drawn by Gilles Deleuze in _Nietzsche and Philosophy,_ trans. Hugh Tomlinson (New York: Columbia University Press, 1983), pp. 39\u201372. Nietzsche's guiding distinction here is not between action and reaction, but between genuine _[eigentliche]_ reaction and imaginary _[imagin\u00e4re]_ reaction (GM 1:10). What Deleuze calls \"active forces\" do not stand in opposition to \"reactive forces\" but actually emerge from them, as unanticipated permutations and by-products.\n\n25. See Staten, p. 38.\n\n26. By 1887, Nietzsche was aware of, and took great delight in, his growing influence within the political microsphere. In a letter to Overbeck on 24 March 1887, Nietzsche reports \"a comic fact, of which I am becoming ever more conscious. I have gradually come to exert an 'influence,' very subterranean, as it understands itself. I enjoy an amazing and almost mysterious reputation among all radical parties (Socialists, Nihilists, anti-Semites, Orthodox Christians, Wagnerites). The extreme purity of the atmosphere in which I have placed myself is enticing _[verf\u00fchrt]._ \" _S\u00e4mtliche Briefe,_ vol. 8, #820, p. 48.\n\n### 6  \nComedians of the Ascetic Ideal\n\n1. In all fairness, we should note that Nietzsche's text is somewhat ambiguous: he could mean that _Zarathustra_ somehow presents the \"counterideal,\" perhaps in spite of Zarathustra himself. For a fruitful exploration of this possibility, see Laurence Lampert, _Nietzsche's Teaching_ (New Haven: Yale University Press, 1986), pp. 255\u2013 263.\n\n2. Maudemarie Clark arrives at this conclusion through a process of elimination: \"I can find only two serious candidates for the counterideal Zarathustra teaches: the _\u00dcbermensch_ and the ideal of affirming eternal recurrence... I argue that the _\u00dcbermensch_ ideal is still too closely tied to the ascetic ideal, and that, in the course of Z, affirming eternal recurrence replaces it as the true alternative to the ascetic ideal.\" _Nietzsche on Truth and Philosophy_ (New York: Cambridge University Press, 1990), p. 253. Of course, Clark's deduction is valid only in the event that 1) she has successfully isolated the only two \"serious candidates for the counterideal Zarathustra teaches\"; and 2) Zarathustra\/ _Zarathustra_ actually promulgates to \"us\" the advertised counterideal.\n\n3. Tracy Strong has long maintained that Nietzsche's preoccupation with a politics of \"transfiguration\" or \"redemption\" compromises the relevance for modern readers of his political thinking. Most recently, Strong has claimed that \"at the end of Nietzsche's life...[he] comes to despair of the possibility of ever accomplishing such a transfiguration\" (\"Nietzsche's Political Aesthetics,\" in _Nietzsche's New_ _Seas,_ ed. Michael Allen Gillespie and Tracy Strong (Chicago: University of Chicago Press, 1988), pp. 13\u201314). As I hope to have shown in this study, by the end of Nietzsche's sane life, he had long since acknowledged the nihilistic ramifications of his youthful longings for a redemption of modernity. Strong thus chronicles the despair of a Nietzsche whom Nietzsche himself had long since subjected to a withering regimen of self-criticism.\n\n4. A notable exception here is Alexander Nehamas, who associates Nietzsche's \"comedy\" with the complex rhetorical task of resisting (while not renewing) the destructive impulse of the ascetic ideal. _Nietzsche: Life as Literature_ (Cambridge, MA: Harvard University Press, 1985), pp. 130\u2013137.\n\n5. Nietzsche's sole prior reference to these comedians hardly inspires our confidence in their redemptive powers. In the previous section, he wonders \"how many _comedians_ of the Christian-moral ideal would have to be exported from Europe today before its air would begin to smell fresh again\" (GM 111:26).\n\n6. As Nehamas instructively points out, Nietzsche is uniquely suited to carry out the complex task involved in \"harming\" the ascetic ideal: \"[To be such a comedian] involves the effort to reveal the inner contradictions and deceptions of asceticism, to denounce it, and yet not produce a view that itself unwittingly repeats the same contradictions and deceptions, for to repeat these is to fail to arouse mistrust in the ascetic ideal; on the contrary, it is to offer a demonstration that it is inescapable\" (pp. 133\u201334).\n\n7. The experimental spirit of Nietzschean genealogy is conveyed by the essay competition Nietzsche proposes, in which scholars would explore the connections between morality and etymology, physiology, psychology, and medicine (GM I:17 _Anmerkung_ ).\n\n8. For a provocative, post-natural extension of Nietzsche's experimentalism, see Gilles Deleuze and Felix Guattari, _Anti-Oedipus: Capitalism and Schizophrenia,_ trans. Robert Hurley, Mark Seem, and Helen R.Lane (Minneapolis: University of Minnesota Press, 1983); and _What is Philosophy?,_ trans. Hugh Tomlinson and Graham Burchell (New York: Columbia University Press, 1994), especially their conclusion, \"From Chaos to the Brain.\"\n\n9. On the intersection of Nietzsche's self-experimentation and his evolving psychological model, see Parkes's epilogue, \"A Dangerous Life,\" in _Composing_ _the Soul: Reaches of Nietzsche's Psychology_ (Chicago: University of Chicago Press, 1994), pp. 363\u2013381.\n\n### 7  \nNietzsche's Political Legacy\n\n1. A representative statement of this position is found in Mark Warren, _Nietzsche and_ _Political Thought_ (Cambridge, MA: MIT Press, 1988). Warren writes, \"The key to understanding Nietzsche's actual political positions is not in his philosophy of power as such, but rather in his misunderstanding of essential features of modern society. This caused him to misconstrue the limits of social and political organization, as well as to distort the causes of modern nihilism\" (p. 237).\n\n2. Martin Heidegger, _The Question Concerning Technology and Other Essays,_ trans. and ed. William Lovitt (New York: Harper & Row, 1977), p. 95.\n\n3. Ibid., p. 105.\n\n4. For a trenchant critique of Nietzsche's alleged mismeasure of modernity, see Warren. Objecting to the \"politics of domination\" sheltered within Nietzsche's \"neoaristocratic conservatism\" (p. 211), Warren maintains that \"Nietzsche did not give his own philosophy a plausible political identity\" (p. 246). Notwithstanding his commendable \"philosophy of agency\" and \"critique of Western culture,\" Warren claims, \"[Nietzsche] failed to elaborate the broad range of political possibilities that are suggested by his philosophy\" (p. 246). Hoping to salvage Nietzsche's philosophy from his retrograde political prejudices, Warren allows that \"the politics of [Nietzsche's] philosophy [is] still to be determined\" (p. 210). Attempting his own elaboration of the \"broad range of possibility,\" Warren sketches a \"political vision\" featuring the \"values\" of \"individuation, communal intersubjectivity, egalitarianism, and pluralism\" (p. 247).\n\n5. Bernard Yack thus concludes, for example, that \"Nietzsche too must make a leap into the absurd, for the problem he seeks to resolve cannot be resolved without self-contradiction.\" _The Longing for Total Revolution_ (Princeton: Princeton University Press, 1986), p. 355.\n\n6. This point is forcefully made by Robert Eden: \"So much intelligence has been invested in saving Nietzsche from the Nazi vulgarization that it is demoralizing to realize that Nietzsche intended a veritable jungle growth of such partial or misunderstandings to spring from the fertile soil of his writings.\" _Political Leadership_ _and Nihilism_ (Tampa: University Presses of Florida, 1983), p. 226.\n\n7. Alasdair MacIntyre, _After Virtue_ (Notre Dame, IN: Notre Dame University Press, 1984), p. 114.\n\n8. Ibid., p. 259.\n\n9. Ibid., p. 118.\n\n10. Ibid., p. 117.\n\n11. Ibid., p. 263.\n\n12. Ibid.\n\n13. J\u00fcrgen Habermas, _The Philosophical Discourse of Modernity: Twelve Lectures,_ trans. Frederick G.Lawrence (Cambridge, MA: MIT Press, 1987), p. 96.\n\n14. Ibid., p. 96.\n\n15. Ibid., p. 87.\n\n16. Ibid., p. 86.\n\n17. Ibid., p. 87.\n\n18. Rorty originally outlined this position in his 1983 essay, \"Postmodernist Bourgeois Liberalism,\" in _Objectivity, Relativism, and Truth: Philosophical Papers, Volume 1_ (Cambridge: Cambridge University Press, 1991), pp. 197\u2013202. He later refines and expands this position, now known as \"liberal ironism,\" in _Contingency, Irony and_ _Solidarity_ (New York: Cambridge University Press, 1989), esp. ch. 4.\n\n19. Rorty, _Contingency,_ p. 99.\n\n20. Ibid., pp. 98\u201399.\n\n21. Ibid., p. 97.\n\n22. Ibid., p. 65.\n\n23. In keeping with the tenets of his \"liberal ironism,\" Rorty apparently would rather forgo the potential benefits of a self-perfection that exceeds the bounds of the private sphere than endure the inevitable harm it would cause. \"Cruelty is the worst thing we do,\" he maintains ( _Contingency,_ p. xv), and any incursion into the public sphere of an individual's private pursuit of self-perfection is potentially cruel.\n\n24. Allan Bloom, \"Justice: John Rawls versus the Tradition of Political Philosophy,\" in _Giants and Dwarves: Essays 1960\u20131990_ (New York: Simon & Schuster, 1990), p. 345.\n\n25. For a representative sample of recent feminist appraisals of Nietzsche's thought, see _Nietzsche, Feminism and Political Theory,_ ed. Paul Patton (London: Routledge, 1993). See especially the excellent essays by Rosalyn Diprose, \"Nietzsche and the Pathos of Distance\"; Keith Ansell-Pearson, \"Nietzsche, Woman and Political Theory\"; Elizabeth Grosz, \"Nietzsche and the Stomach for Knowledge\"; and Frances Oppel, \"'Speaking of Immemorial Waters': Irigaray with Nietzsche.\"\n\n26. See, for example, Kelly Oliver's exemplary study, _Womanizing Nietzsche:_ _Philosophy's Relation to the \"Feminine\"_ (New York: Routledge, 1994).\n\n27. Donna J.Haraway, _Simians, Cyborgs and Women_ (London: Free Association Books, 1991), p. 189.\n\n28. Susan Bordo, \"Feminism, Postmodernism, and Gender-Scepticism,\" in _Feminism\/_ _Postmodernism,_ ed. Linda Nicholson (New York: Routledge, 1990), pp. 133\u2013156.\n\n29. Sandra Harding, \"Rethinking Standpoint Epistemology: What is 'Strong Objectivity'?,\" in _Feminist Epistemologies,_ ed. Linda Alcoff and Elizabeth Potter (New York: Routledge, 1993), p. 72.\n\n30. Haraway, p. 190.\n\n31. Ibid., p. 196.\n\n32. Ibid., p. 192.\n\n33. Ibid., p. 191.\n\n34. Ibid.\n\n35. Ibid.\n\n36. Bat-Ami Bar On, \"Marginality and Epistemic Privilege,\" in _Feminist_ _Epistemologies,_ ed. L.Alcoff and E.Potter (New York: Routledge, 1993), pp. 83\u2013 100.\n\n37. Ibid., p. 94.\n\n38. bell hooks, \"Choosing the Margin as a Space of Radical Openness,\" in _Yearning:_ _Race, Gender, and Cultural Politics_ (Boston, MA: South End Press, 1990), pp. 150\u2013151.\n\n39. Haraway, p. 173.\n\n40. Haraway, ch. 8, \"A Cyborg Manifesto.\"\n\n41. Michel Foucault, _The Order of Things: An Archaeology of the Human Sciences_ (New York: Random House\/Vintage Books, 1973).\n\n42. My interpretation of _The Order of Things_ is greatly indebted to the compelling reading advanced by Charles Scott in _The Question of Ethics: Nietzsche, Foucault,_ _Heidegger_ (Bloomington: Indiana University Press, 1990), ch. 3, sections 3\u20134.\n\n43. For a discerning survey of the productive ethical implications arising from the \"death of Man,\" see Scott, pp. 79\u201393.\n\n44. In _Discipline and Punish: The Birth of the Prison,_ trans. Alan Sheridan (New York: Random House\/Vintage Books, 1979), Foucault documents the role of the prison in (unwittingly) producing particular types of subjects and in policing the flows of knowledge and power. See especially Foucault's discussion of torture, pp. 32\u201369.\n\n45. My reckoning of the \"periodization\" of Foucault's scholarly activity draws upon Alan Schrift's excellent discussion in _Nietzsche's French Legacy: A Genealogy of_ _Modernity_ (New York: Routledge, 1995), pp. 34\u201358.\n\n46. Michel Foucault, _The History of Sexuality,_ vol. 1: _An Introduction,_ trans. Robert Hurley (New York: Random House\/Vintage Books, 1990), pt. 5: \"Right of Death and Power Over Life.\"\n\n47. Michel Foucault, _The History of Sexuality,_ vol. 3: _The Care of the Self,_ trans. Robert Hurley (New York: Random House\/Vintage Books, 1988).\n\n48. For a sympathetic reconstruction of Foucault's \"transgressive middle course in relation to power and resistance,\" see Jon Simons, _Foucault & the Political_ (London: Routledge, 1994), ch. 7.\n\n49. Michel Foucault, \"Politics, Polemics, and Problematizations: an Interview,\" in _The Foucault Reader,_ ed. Paul Rabinow (New York: Pantheon Books, 1984), pp. 383\u2013 386.\n\n50. For a provocative defense of the \"democratic sensibility\" that emerges from a consideration of Nietzsche through the lens of his Foucauldian legacy, see William Connolly, \"Beyond Good and Evil: The Ethical Sensibility of Michel Foucault,\" _Political Theory,_ vol. 21, no. 3, pp. 365\u201389.\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"###### Campaign \u2022 49\n\n## Mons 1914\n\n###### The BEF's tactical triumph\n\n###### David Lomas \u2022 Illustrated by Ed Dovey\n\n###### _Series editor_ Lee Johnson \u2022 _Consultant editor_ David G Chandler\n\n### **CONTENTS**\n\nORIGINS OF THE CAMPAIGN\n\nOPPOSING COMMANDERS\n\nThe British Commanders \u2022 The French Commanders \u2022 The German Commanders\n\nOPPOSING ARMIES\n\nThe British \u2022 The French \u2022 The Germans \u2022 Orders of Battle\n\nTHE OPPOSING PLANS\n\nGermany's Schlieffen Plan \u2022 The French Plan XVII and the role of the BEF\n\nTHE CAMPAIGN\n\nThe arrival of the BEF \u2022 Lanrezac's change of plan\n\nTHE BATTLE OF MONS\n\nThe opening shots \u2022 Defence of la Bois Haut \u2022 The battle for the canal  \nBlowing the bridges\n\nTHE RETREAT BEGINS\n\nII Corps runs into trouble \u2022 The Repulse of the German 6th Division  \nThe 5th Division faces von Kluck \u2022 The Flank Guard action at Elouges  \nRetreat to Le Cateau \u2022 Landrecies\n\nLE CATEAU\n\nThe decision to fight \u2022 The battlefield \u2022 The battle begins  \nThe left flank \u2022 The retreat \u2022 Saving the guns\n\nAFTERMATH\n\nEtreux \u2022 N\u00e9ry \u2022 Villers-Cotter\u00eats\n\nWARGAMING MONS\n\n**THE BATTLEFIELD TODAY**\n\n**British troops, probably of the 8th Brigade, II Corps, resting on their way to Mons. Note the cobbled paved road surface which caused a good deal of discomfort to many soldiers. (Mons Museum)**\n\n## ORIGINS OF THE CAMPAIGN\n\n'We're off to fight the bloody Belgians!' an enthusiastic British soldier informed an astonished listener in the first, exciting days of August 1914. The European war, so long predicted, so long feared, had come at last, sparked by a handful of shots fired by a consumptive Serbian nationalist in Sarajevo.\n\nThe _Entente Cordiale_ between Britain and France was little more than an understanding of mutual support should war come. The British had, though, given an assurance that the Royal Navy would not allow a foreign fleet into the English Channel and there was a further unspoken assumption that a British Expeditionary Force would help repel an invader from French soil.\n\nHenry Wilson, who became Director of Military Operations in 1910, was an obsessive Francophile. Entirely on his own initiative, he promised the French that a British Expeditionary Force of six infantry divisions and one cavalry division would be sent to France. Under Wilson's guidance, mobilisation plans were produced with a single aim \u2013 to get the BEF \u00e1cross the Channel quickly enough to join a French offensive planned to start 15 days after mobilisation. The French courteously dubbed Wilson's scheme 'Plan W'.\n\n**12 August 1914 \u2013 The 1st Battalion Grenadier Guards march out from Chelsea Barracks. (Private Collection)**\n\nOn the afternoon of 5 August 1914, Herbert Asquith, the British Prime Minister, presided over a hastily arranged Council of War and discovered that Plan W was the only one available and capable of implementation. One change was made: two divisions \u2013 the 4th and the 6th \u2013 were kept back against the threat of a German invasion of Britain.\n\nA British Expeditionary Force of four infantry divisions and one cavalry division supported by an extra cavalry brigade would go to war. It would be commanded by Sir John French; the four infantry divisions would form I Corps, commanded by Sir Douglas Haig, and II Corps would be under Sir James Grierson. Major-General Allenby had command of the Cavalry Division.\n\n**German Guard Infantry in Berlin entraining for the front, August 1914. (Private Collection)**\n\nA few days later, Sir John French received his instructions from Lord Kitchener, the War Minister, and one of the few men who anticipated a long struggle. Kitchener had little faith in Sir John French's abilities and distrusted Henry Wilson. His words reflected this concern. Sir John French was 'to support and co-operate with the French Army. . . in preventing or repelling the invasion by Germany of French and Belgian territory and eventually to restore the neutrality of Belgium'. The instructions continued: 'It must be recognised from the outset that the numerical strength of the British Force. . . is strictly limited. . . the greatest care must be exercised towards a minimum of losses and wastage. Therefore, while every effort must be made to coincide most sympathetically with the plans and wishes of our Ally, the gravest consideration will devolve upon you as to participation in forward movements where large bodies of French troops are not engaged and where your Force may be unduly exposed to attack. In this connection I wish you distinctly to understand that your command is an entirely independent one, and that you will in no case come in any sense under the orders of any Allied General.'\n\nThey were phrases to make anyone pause. The BEF was to help throw back one million Germans and restore Belgian independence, but to avoid heavy losses while doing so. It was to be part of French plans, yet remain independent despite the fact that it was only one-thirtieth the size of the French army, and was operating on French soil and relying on French goodwill for railways and rolling stock, accommodation, communication and supply lines, and myriad other requirements. It was hardly a promising start.\n\n**The German Model 96 field gun, seen here with a crew from the 4th Bavarian Royal Field Artillery Regiment, created severe problems for the BEF in the first months of the war. It was an ominous portent for what proved to be a war in which artillery caused the greatest casualties. (Author's Collection)**\n\n## THE OPPOSING COMMANDERS\n\n##### **THE BRITISH COMMANDERS**\n\nField Marshal Sir John French was 62 years old in August 1914 and had an outstanding record of distinguished service, being one of the few senior officers who enhanced his reputation during the South African War.\n\n**Field Marshal Sir John French had resigned from his position as Chief of the Imperial General Staff in 1914 in protest over the Government's Irish Home Rule Bill. When war was declared, he was offered command of the BEF. Quick-tempered to the point of explosive, French was personally brave and much liked by his men. (Author's Collection)**\n\nFrench was a cavalryman, and not at heart a staff officer but a fighting soldier, a quality which endeared him to his troops and gained both their respect and affection. He went to France full of confidence, convinced that he could lead the BEF to a part in a swift and decisive victory. French had a hot temper and was quick to take offence; he harboured grudges. Worse, for a commander, he suffered from mercurial changes of mood, plunging from enormous optimism to deep pessimism in a matter of moments.\n\nGeneral Sir Douglas Haig, commanding I Corps, was another highly regarded officer. As a pre-war Director of Military Training, Haig was not only the architect of Field Service Regulations which essentially prepared the British Army for a European war, but he had also been responsible for the organisation and training of the Territorial Army. Naturally shy, Haig was not a fluent speaker and was often tongue-tied in debate, a drawback which was to cause him many problems. He was, nonetheless, a very capable soldier who believed firmly in total attention to detail, and close and constant co-operation between all arms. Haig had the respect and admiration of his officers and earned intense loyalty from those who served closely with him. He was one of the few who believed that a European war would be a long and difficult struggle.\n\n**II Corps was originally under the command of Sir John Grierson; Grierson died suddenly on 17 August 1914 and Kitchener chose Sir Horace Smith-Dorrien to replace him. This was a move directly contrary to the expressed wish of Sir John French. Kitchener distrusted French and French and Smith-Dorrien loathed each other. Smith-Dorrien was, nonetheless, a very considerable soldier and his admirers will always remain convinced that it was his actions which saved the BEF from certain disaster during August 1914.**\n\nGeneral Sir Horace Smith-Dorrien was sent to command II Corps by Kitchener when Grierson died shortly after the war began. His appointment was against the wishes of Sir John French, as the two officers disliked each other intensely. Smith-Dorrien was more of a regimental soldier than many of his contemporaries, serving for much of his career with the Sherwood Foresters, an unfashionable but hard-fighting infantry regiment. He fought in the Zulu War in 1879 (during which he was recommended for, but did not receive, the Victoria Cross), in the Tirah and Sudan campaigns, and in South Africa. Prone to uncontrollable outbursts of rage, he was a pugnacious soldier with an independent turn of mind.\n\n**Commander of the British I Corps' General Sir Douglas Haig was GOC Aldershot when the war began. I Corps was, in effect, the Aldershot Garrison and was at a high standard of readiness. Haig, appointed to Aldershot in 1912, had instituted a carefully planned training programme which paid total attention to detail and emphasised the co-operation between all arms. It worked and Haig had the respect and admiration of the professional soldiers under his command. Haig had won high approval for his work in reorganising the training of the Regular Army and that of the Territorials. He was a hard worker and inspired great devotion and loyalty amongst his staff. I Corps was to take only a minor part at Mons and Le Cateau. (Author's Collection)**\n\n##### **THE FRENCH COMMANDERS**\n\nGeneral Charles Lanrezac was 62 when he took command of the French Fifth Army. He was the man who would work most closely with the BEF. He was remarkably intelligent and brilliantly logical in his assessments; qualities which counterbalanced his tendency to bad temper, sarcastic comment and swearing. He became a controversial figure. To many French commentators, his decision to to retreat from the advancing Germans in August 1914 was extremely shrewd and saved the only intact army France possessed. Without it the Marne could not have been won. To the British, his withdrawal without warning them was a betrayal of an ally and a betrayal which placed the BEF in acute danger.\n\n**General von Kluck has been described as a typical Prussian general, arrogant and overbearing with a fierce disregard for anything less than outright victory. These were qualities needed by the commander of the German First Army which formed the extreme right wing. It is arguable whether the Schlieffen Plan could have succeeded given the conditions of 1914; von Kluck's errors in the opening campaign ensured that it would not. His dislike of von Bulow led him to refuse to become involved in capturing the fortress town of Maubeuge even though it lay directly in his line of advance. Von Bulow was forced to detach his own VII Corps to do the job, a move which was to have severe repercussions later. (Private Collection)**\n\n##### **THE GERMAN COMMANDERS**\n\nAlexander von Kluck, commander of the German First Army, had an overweening belief in his own abilities, and thought, not unjustifiably, his role to be the most important in the execution of the Schlieffen Plan. The First Army was the one which had to march the farthest and needed a ruthless commander dedicated to success. Von Kluck was extremely conscious of the rewards that a successful outcome would bring him personally. Obsessed with his own glory, his judgement was flawed. With a penchant for ignoring orders with which he did not agree, and an abrasive personality, von Kluck was a difficult man to deal with. The German cause was not helped by his detestation of the commander of the neighbouring Second Army.\n\nKarl von Bulow commanded the Second Army. He was not a resolute character, and the worries of high command made him depressed and anxious. He had earned a reputation in pre-war manoeuvres for his cautious tactics and the great importance he attached to units giving each other elaborate mutual support. Unable to get co-operation from von Kluck, he resorted to seeking the support of Supreme Headquarters on any point of difference between them, a procedure which came close to wrecking the advance.\n\n**General von Bulow, commander of the German Second Army, detested von Kluck and the feeling was certainly mutual. For a campaign in which close co-operation between the two armies was essential, this personal dislike was to have disastrous consequences. Von Bulow's caution and von Kluck's determination caused great problems once the war had begun. Von Bulow considered the defeat of the BEF to be entirely a matter for von Kluck and he therefore ignored any reports of the British movements as being irrelevant to his own. Von Kluck, in turn, either ignored or disobeyed orders or requests which came from von Bulow. (Author's Collection)**\n\n## THE OPPOSING ARMIES\n\n##### **THE BRITISH**\n\nThe tiny volunteer British Army of 1914 was not generally representative of the country as a whole. In many regiments officers needed a private income as it was almost impossible to live on the pay alone. An officer provided his own uniforms, cases, furniture and servant's outfit; there were mess contributions, field sports and social events. In the Cameronians, a private income of about \u00a3250 per year would suffice; the Guards anticipated that \u00a3400 a year was required. Cavalry regiments were even more expensive \u2013 a charger was essential, as were two hunters and three polo ponies.\n\nNearly half of all recruits to the ranks were registered as unskilled labourers on enlistment. Even those claiming a trade when joining were often unemployed. Many came from the slums of the industrial towns, were usually under-nourished and only just met the medical standard \u2013 5ft 3in tall, 33in chest and 112 pounds in weight.\n\nBarrack accommodation ranged from adequate to abysmal. Soldiers had no privacy and often ate in the rooms in which they slept. This was still an improvement on the slums of their home town, for their surroundings were clean, they received regular, if low, wages, adequate meals and were also educated, if necessary, at the Army's expense.\n\nA spirit of common endeavour had developed from the reforms of the late Victorian age. Officers were encouraged to work closely with the non-commissioned officers and to know their men. NCOs took much responsibility for the daily routine and training, and provided a vital link between the officer and the private soldier. Officers and men had a common bond in loyalty to their regiment.\n\n**French carabiniers in 1914. In their near-Napoleonic uniforms, these cavalrymen were photographed shortly after mobilisation waiting at the Paris Gare du Nord railway station. Armed with the sabre and a carbine which was described as 'no more than a pop-gun', the French cavalry had the habit of riding everywhere at the expense of their horses. A British cavalry officer noted with distaste that he could smell a group of French cavalry from a distance of 400 metres because of the saddle sores and the ungroomed state of their mounts. The British cavalry trooper was accustomed to walk as much as he rode and to keep his horse groomed, fed and cared for before himself if need be. (Private Collection)**\n\n**British infantrymen on the pre-1914 precursor of the modern assault course. Under the influence of Haldane, Haig and others, British Army training in the early years of the 20th century sought to instill not just physical fitness but a sense of self-reliance into the private soldier. (Author's Collection)**\n\nTraining was carefully planned, moving throughout the year from individual and platoon skills during the winter, to company and battalion exercises in the spring, brigade and divisional manoeuvres in the summer and a full army exercise in the autumn. There was particular emphasis on marksmanship and marching.\n\nShortage of money concentrated training on what was believed to be the immediate reality for the Army \u2013 the small colonial conflict, or a short war of movement in Europe. There was little expectation that there would be prolonged trench or siege warfare, and the British Army was neither equipped nor trained to deal with that eventuality.\n\nIn May 1914, the Regular Army was about 11,000 men short of its peacetime establishment of 260,000 men. The number of full-time soldiers in Britain, including recruits under training, was 137,000. The remainder were in the overseas garrisons of the British Empire. There were not nearly enough men to supply the number needed for the 48 infantry battalions, 16 cavalry regiments, five batteries of horse artillery, 16 brigades of field artillery, four heavy artillery batteries, eight field companies of engineers and motley collection of support services which the Expeditionary Force required.\n\nThe answer to the conundrum was the Reservist. Most men left the army after seven years and then served on the Reserve for a further five. Recalled by letter, telegram and public notices, some 70,000 men poured into regimental depots to bring a woefully under-strength British Army on to a war footing. In some units, more than 60 per cent of the men who were to sail to France were Reservists. One infantry battalion embodied 734 Reservists to bring it up to its war establishment of 992 men. Every unit had its complement of men who had served in South Africa, China, India, Burma and a score of other Imperial stations. Old soldiers they may have been, but their time away from the Army had inevitably softened them. There would later be a price to pay under a blazing sun on the dusty roads of France and Belgium.\n\n**German horse artilleryman, 1914. This studio portrait shows a Saxon soldier of von Hausen's Third Army which only marginally came into contact with the BEF in the very first days of the war. Later, at Ypres and elsewhere, it was to be a very different story. (Author's Collection)**\n\n##### **THE FRENCH**\n\nWhen mobilised, the French Army had more than one million men, and over three million if the Territorials and surplus Reservists were included. A French conscript served for three years, but his reserve training was virtually non-existent.\n\nThe French Army, too, suffered from parsimonious politicians. The infantry were still uniformed very much as they had been in the Franco-Prussian War, with blue tunics and bright red trousers. The cavalry looked almost exactly as their predecessors had done at Waterloo, particularly the cuirassiers with horsehair plumes trailing from crested helmets, steel breastplates and high boots. The uniforms appealed to the advocates of the offensive strategy. They felt it displayed the elan which was the special attribute of the French soldier. The economy involved in not providing the French soldier with a drab-coloured field service uniform became a positive virtue.\n\nTraining also recalled the days of Napoleon. The infantry were expected to advance in a solid mass, the cavalry to charge with lance and sabre, tactics which demanded a rigid and unthinking discipline. The ordinary soldier was required to do no more than to obey orders promptly and to advance with spirit when commanded. Their officers, as one observer recorded, 'were entirely ignorant of the stopping power of modern firearms and many of them thought it chic to die in white gloves'.\n\nThe French field artillery was superb. It was to prove itself time and time again, its quick-firing 75mm gun being one of the finest in service throughout the war. The field artillery was the one redeeming feature of an army equipped and trained, in many respects, for the campaigns of the first Napoleon.\n\n##### **THE GERMANS**\n\nThe German Army was a carefully structured organisation, designed to muster, with reserves, more than 5 million men when fully mobilised. At its heart was a well-trained officer corps, supported by 100,000 highly professional non-commissioned officers, who imposed a rigid discipline and demanded unquestioning obedience from its rank and file.\n\nAt the age of 17, every German male became liable for service in the _Landsturm_ , the organisation for home defence. At 20, the man moved to the active Army, serving two years in the infantry or three years in the cavalry and horse artillery. At the end of this time, he went to the reserve for four or five years, followed by 11 years in the _Landwehr_ and eventually back to the _Landsturm_ at the age of 39. Throughout his reserve service, there were two annual training periods.\n\nThe number of men available annually far exceeded Army requirements. More than one million reported for service each year of which only one-third were needed. This enabled the Army to pick the very best available men for active duty while turning over many thousands to the supplementary or Ersatz Reserve. The Ersatz Reserve yielded over one million reinforcements during the first three months of the war.\n\nIn its training, the German Army, unlike its British and French contemporaries, prepared for a European war. Observers were struck by an insistence on speed and the fierce march discipline \u2013 an essential requirement if the Schlieffen Plan was to adhere to its schedule. The infantry were trained in enveloping tactics, and to attack in dense waves at 500-metre intervals, supported by artillery and machine gun fire: such German assaults appeared unstoppable. Attacks were made in crescent formation, the enemy's flanks being overlapped and encircled while the centre was kept busy.\n\nGerman commanders believed in the interdependence of all arms, a theory they emphasised by including Jaeger battalions and horse artillery in the cavalry divisions. Under pressure, German cavalry were able to pull back to the protection provided by their own machine guns and field pieces.\n\nSize alone made the German Army a very formidable opponent. Trained to a high level, with an exceptionally professional officer corps, its rigid hierarchy discouraged delegation and inhibited individual initiative not only in the lower echelons, but among regimental and brigade commanders and even divisional and corps commanders. It was a failing that would only become significant if the plans, rehearsed to perfection on so many annual peacetime manoeuvres, went awry \u2013 and the German General Staff did not intend that they should.\n\n##### **ORDERS OF BATTLE**\n\nThe Orders of Battle do not, for reasons of clarity, include support and lines of communications troops, e.g. Royal Army Medical Corps, Army Service Corps, GHQ troops etc. The French and Belgian Orders of Battle have not been included.\n\n### **THE BRITISH EXPEDITIONARY FORCE AUGUST 1914**\n\nCommander-in-Chief: Field-Marshal Sir J.D.P French, GCB, GCVO, KCMG  \nChief of the General Staff: Lieutenant-General Sir A.J. Murray, KCB, CVO, DSO  \nMajor-General, General Staff: Major-General H.H. Wilson, CB, DSO  \nGSOI (Intelligence): Colonel G.M.W. Macdonogh  \nQuartermaster-General: Major-General Sir W.R. Robertson, KCVO, CB, DSO\n\n### **THE CAVALRY DIVISION**\n\nMajor-General E.H. Allenby, CB\n\n**1ST CAVALRY BRIGADE** | **2ND CAVALRY BRIGADE**  \n---|---  \nBrigadier-General C.J. Briggs, CB  \n2nd Dragoon Guards (Queen's Bays)  \n5th (Princess Charlotte of Wales') Dragoon Guards  \n11th (Prince Albert's Own) Hussars  | Brigadier-General H. de B. de Lisle, CB, DSO  \n4th (Royal Irish) Dragoon Guards  \n9th (Queen's Royal) Lancers  \n18th (Queen Mary's Own) Hussars   \n**3RD CAVALRY BRIGADE** | **4TH CAVALRY BRIGADE**  \nBrigadier-General H. de la P. Gough, CB  \n4th (Queen's Own) Hussars  \n5th (Royal Irish) Lancers  \n16th (The Queen's) Lancers  | Brigadier-General Hon. C.E. Bingham, CVO. CB   \nComposite Regiment of Household Cavalry 6th Dragoon Guards (Carabiniers)  \n3rd (King's Own) Hussars\n\n**5TH CAVALRY BRIGADE**  \nGOC: Brigadier-General Sir P.W. Chetwode, Bart., DSO   \n2nd Dragoons (Royal Scots Greys)  \n12th Lancers  \n20th Hussars  \n'D', 'E', 'I', 'J', 'L' Batteries, RHA\n\n### **I CORPS**\n\nGOC: Lieutenant-General Sir D. Haig, KCB, KCIE, KCVO, ADC-Gen.  \nBGGS: Brigadier-General J.E. Gough, VC, CMG, ADC\n\n**1ST DIVISION** | **2ND DIVISION**  \n---|---  \nMajor-General S.H. Lomax  | Major-General C.C. Monro, CB  \n**1st (Guards) Brigade** | **4th (Guards) Brigade**  \nBrigadier-General F.I. Maxse, CVO, CB, DSO   \n1\/Coldstream Guards  \n1\/Scots Guards  \n1\/Black Watch  \n2\/Royal Munster Fusiliers  | Brigadier-General R. Scott-Kerr, CB, MVO, DSO  \n2\/Grenadier Guards  \n2\/Coldstream Guards  \n3\/Coldstream Guards  \n1\/Irish Guards   \n**2nd Brigade** | **5th Brigade**  \nBrigadier-General E.S. Bulfin, CVO, CB  \n2\/Royal Sussex Regiment  \n1\/Loyal North Lancashire Regiment  \n1\/Northamptonshire Regiment  \n2\/King's Royal Rifle Corps  | Brigadier-General R.C.B. Haking, CB  \n2\/Oxfordshire & Buckinghamshire Light Infantry  \n2\/Worcester Regiment  \n2\/Highland Light Infantry  \n2\/Connaught Rangers  \n**3rd Brigade** | **6th Brigade**  \nBrigadier-General H.J.S. Landon, CB  \n1\/Queen's (Royal West Surrey Regiment)  \n1\/South Wales Borderers  \n1\/Gloucester Regiment  \n2\/Welch Regiment  \n'A' Squadron, 15th Hussars  \nXXV (113th, 114th, 115th Batteries),  \nXXVI (116th, 117th, 118th Batteries),  \nXXXIX (46th, 51st, 54th Batteries),  \nXLIII (30th, 40th, 57th (How) Batteries)  \nBrigades RFA  \n26th Heavy Battery, RGA  \n23rd, 26th Field Companies, RE | Brigadier-General R.H. Davies, CB (NZ Staff Corps) **  \n**1\/King's (Liverpool Regiment)  \n2\/South Staffordshire Regiment  \n1\/Royal Berkshire Regiment  \n1\/King's Royal Rifle Corps  \n'B' Squadron, 15th Hussars  \nXXXIV (22nd, 50th, 70th Batteries),  \nXXXVI (15th, 48th, 71st Batteries),  \nXLI (9th, 16th, 17th Batteries),  \nXLIV (47th, 56th, 60th (Howitzer) Batteries)  \nBrigades RFA  \n35th Heavy Battery, RGA  \n5th, 11th Field Companies RE\n\n### **II CORPS**\n\nGOC: Lieutenant-General Sir J.M. Grierson, KCB, CVO, CMG, ADC-Gen. (Ded 17 August 1914)  \nGeneral Sir H.L. Smith-Dorrien, GCB, DSO (assurned command 21 August)  \nBGGS: Brigadier-General G.T. Forestier-Walker, ADC\n\n**3RD DIVISION** | **5TH DIVISION**  \n---|---  \nMajor-General H.I.W. Hamilton, CVO, CB, DSO | Maj.-General Sir C. Fergusson, Bart., CB, MVO, DSO  \n**7th Brigade** | **13th Brigade**  \nBrigadier-General F.W.N. McCracken, CB, DSO   \n3\/Worcester Regiment  \n2\/South Lancashire Regiment  \n1\/Wiltshire Regiment  \n2\/Royal Irish Rifles | Brigadier-General G.J. Cuthbert, CB  \n2\/King's Own Scottish Borderers  \n2\/Duke of Wellington's (West Riding Regiment)  \n1\/Queen's Own (Royal West Kent Regiment)  \n2\/King's Own (Yorkshire Light Infantry)  \n**8th Brigade** | **14th Brigade**  \nBrigadier-General B.J.C. Doran  \n2\/Royal Scots  \n2\/Royal Irish Regiment  \n4\/Middlesex Regiment  \n1\/Gordon Highlanders  | Brigadier-General S.P. Rolt, CB  \n2\/Suffolk Regiment  \n1\/East Surrey Regiment  \n1\/Duke of Cornwall's Light Infantry  \n2\/Manchester Regiment   \n**9th Brigade** | **15th Brigade**  \nBrigadier-General F.C. Shaw, CB   \n1\/Northumberland Fusiliers  \n4\/Royal Fusiliers  \n1\/Lincoinshire Regiment  \n1\/Royal Scots Fusiliers  \n'C' Squadron, 15th Hussars  \nXXIII (107th, 108th, 109th Batteries),  \nXL (6th, 23rd, 49th Batteries),  \nXLII (29th, 41st, 45th Batteries),  \nXXX (Howitzer) (128th, 129th, 130th (Howitzer)  \nBatteries) Brigades, RFA  \n48th Heavy Battery, RGA  \n56th, 57th Field Companies, RE  | Brigadier-General  \nA.E.W. Count Gretchen, KCVO, CB, CMG, DSO  \n1\/Norfolk Regiment  \n1\/Bedfordshire Regiment  \n1\/Cheshire Regiment  \nDorsetshire Regiment  \n'A' Squadron, 19th Hussars  \nXV (11th, 52nd, 80th Batteries),  \nXXVII (119th, 120th, 121st Batteries),  \nXXVIII (122nd, 123rd, 124th Batteries),  \nVIII (Howitzer) (37th, 61st, 65th (Howitzer)  \nBatteries) Brigades, RFA  \n108th Heavy Battery, RGA  \n17th, 59th Field Companies, RE\n\n### **III CORPS**\n\nGOC: Major-General W.P. Pulteney, CB, DSO\n\nBGGS: Brigadier-General J.P. Du Cane, CB\n\n**4TH DIVISION**\n\nGOC: Major-General T.D'O. Snow, CB\n\n**10th Brigade** | **11th Brigade**  \n---|---  \nBrigadier-General J.A.L. Haldane, CB, DSO  \n1\/Royal Warwickshire Regiment  \n2\/Seaforth Highlanders  \n1\/Irish Fusiliers  \n2\/Royal Dublin Fusiliers | Brigadier-General A.G. Hunter-Weston, CB, DSO  \n1\/Somerset Light Infantry  \n1\/East Lancashire Regiment  \n1\/Hampshire Regiment  \n1\/Rifle Brigade  \n**12th Brigade** | **19th Brigade**  \nBrigadier-General H.F.M. Wilson, CB  \n1\/King's Own (Royal Lancaster Regiment)  \n2\/Lancashire Fusiliers  \n2\/Royal Inniskilling Fusiliers  \n2\/Essex Regiment  \n'B' Squadron, 19th Hussars  \nXIV (39th, 68th, 88th Batteries),  \nXXIX (125th, 126th, 127th Batteries),  \nXXXII (27th, 134th, 135th Batteries),  \nXXXVII (Howitzer) (31st, 35th, 55th Howitzer  \nBatteries) Brigades, RFA  \n31st Heavy Battery, RGA  \n7th, 9th Field Companies, RE | Major-General L.G. Drummond, CB, MVO  \n2\/Royal Welch Fusiliers  \n1\/Cameronians  \n1\/Middlesex Regiment  \n2\/Argyll & Sutherland Highlanders\n\n### **ROYAL FLYING CORPS**\n\nBrigadier-General Sir D. Henderson, KCB, DSO\n\n2nd Aeroplane Squadron\n\n3rd Aeroplane Squadron\n\n4th Aeroplane Squadron\n\n5th Aeroplane Squadron\n\n1st Aircraft Park\n\n**The popular image of the German soldier. A private of the 40th Infantry in 1914. (Author's Collection)**\n\n**The bicycling craze which began in the last years of Queen Victoria's reign had its effect on military thinking. German, Belgian, French and British armies all had specialist cycle troops who could use their mounts to move silently upon an enemy. Special bicycle tactics were introduced and the cycle-equipped soldier was looked upon as a pseudo-cavalryman \u2013 he could ride ahead and act as a scout. The lance-corporal appears to have lost his cap badge but his service issue mug is firmly attached to the front handlebars. (Private Collection)**\n\n**Annual training, 1910. The German system ensured that reserve troops received two weeks of annual training each year. This group were photographed in their old Prussian blue uniforms in the summer of 1910 shortly before receiving their new field-grey dress. (Author's Collection)**\n\n### **ORDER OF BATTLE GERMAN ARMIES 1914**\n\nChief of the General Staff: Generaloberst von Moltke\n\nDeputy Chief of the General Staff: General von Stein\n\n### **FIRST ARMY**\n\nGeneral von Kluck\n\n### **II CORPS**\n\nGeneral von Lisengen\n\n**3RD DIVISION** | **4TH DIVISION**  \n---|---  \n**5th Brigade**  \n2 Grenadier Rgt  \n9 Grenadier Rgt | **6th Brigade**  \n34 Fusilier Rgt  \n42 Infantry Rgt | **7th Brigade**  \n14 Infantry Rgt  \n149 Infantry Rgt | **8th Brigade**  \n49 Infantry Rgt  \n140 Infantry Rgt  \n**Cavalry** 3 Horse Grenadier Rgt  \n **Artillery** 3 Bde: 2 F.A. Rgt  \n38 F.A. Rgt | **Cavalry** 12 Dragoon Rgt  \n **Artillery** 4 Bde: 17 F.A. Rgt  \n53 F.A. Rgt\n\n### **III CORPS**\n\nGeneral von Lochow\n\n**5TH DIVISION** | **6TH DIVISION**  \n---|---  \n**9th Brigade**  \n8 Body Grenadier Rgt  \n48 Infantry Rgt | **10th Brigade**  \n12 Grenadier Rgt  \n52 Infantry Rgt | **11th Brigade**  \n20 Infantry Rgt  \n35 Fusiliers Rgt | **12th Brigade**  \n24 Infantry Rgt  \n64 Infantry Rgt  \n**Cavalry** 3 Hussar Rgt (3 Sqns)  \n **Artillery** 18 F.A. Rgt, 54 F.A. Rgt | 3 Jaeger Battalion  \n **Cavalry** 3 Hussars Rgt (3 Sqns)  \n **Artillery** 6 Bde: 3 F.A. Rgt, 39 F.A. Rgt\n\n### **IV CORPS**\n\nGeneral Sixt von Armin\n\n**7TH DIVISION** | **8TH DIVISION**  \n---|---  \n**13th Brigade**  \n26 Infantry Rgt  \n66 Infantry Rgt | **14th Brigade**  \n27 Infantry Rgt  \n165 Infantry Rgt | **15th Brigade**  \n36 Fusiliers Rgt  \n93 Infantry Rgt | **16th Brigade**  \n72 Infantry Rgt  \n153 Infantry Rgt  \n**Cavalry** 10 Hussars Rgt (3 Sqns)  \n **Artillery** 7 Bde: 4 F.A. Rgt, 40 F.A. Rgt | **Cavalry** 10 Hussars Rgt (3 Sqns)  \n **Artillery** 8 Bde: 74 F.A. Rgt, 75 F.A. Rgt\n\n### **IX CORPS**\n\nGeneral von Quast\n\n**17TH DIVISION** | **18TH DIVISION**  \n---|---  \n**33 Brigade**  \n75 Infantry Rgt  \n76 Infantry Rgt | **34 Brigade**  \n89 Grenadier Rgt  \n90 Fusiliers Rgt | **35 Brigade**  \n84 Infantry Rgt  \n86 Fusiliers Rgt | **36 Brigade**  \n31 Infantry Rgt  \n85 Infantry Rgt  \n**Cavalry** 16 Dragoon Rgt (3 Sqns)  \n **Artillery** 17 Bde: 24 F.A. Rgt. 60 F.A. Rgt | **Cavalry** 16 Dragoon Rgt (3 Sqns)  \n **Artillery** 18 Bde: 9 F.A. Rgt. 45 F.A. Rgt\n\n**Commander of the German II Cavalry Corps, which consisted of the 2nd, 4th, and 9th Cavalry Divisions, von Marwitz was a competent commander whose men were sent off on a wild-goose chase towards the Channel coast to cut off what von Kluck believed would be the British line of retreat. At N\u00e9ry on 1 September 1914, the 4th Cavalry Division became involved in one of the classic small-unit actions. (Private Collection)**\n\n**German Army divisions had an aviation detachment as an integral part of their organisation \u2013 a sharp contrast to both British and French practice in which the flying services were ancillaries under independent command. These Bavarian Air Service members pose for the camera just before the start of the last peacetime manoeuvres. The aircraft in the background, an Otto Pusher, was unique to the Bavarian Service. (Author's Collection)**\n\n### **III RESERVE CORPS**\n\nGeneral von Beseler\n\n**7TH RESERVE DIVISION** | **22ND RESERVE DIVISION**  \n---|---  \n**13 Res. Brigade**  \n27 Res. Infantry Rgt  \n36 Res, Infantry Rgt | **14 Res. Brigade**  \n66 Res. Infantry Rgt  \n72 Res. Infantry Rgt | **43 Res. Brigade**  \n71 Res. Infantry Rgt  \n94 Res. Infantry Rgt | **44 Res. Brigade**  \n32 Res. Infantry Rgt  \n82 Res. Infantry Rgt  \n4 Res. Jaeger Battalion | 1 Res. Jaeger Battalion  \n**Cavalry** 1 Res. Heavy Cavalry Rgt (3 Sqns)  \n **Artillery** 7 Res. F.A. Rgt (6 Batteries) | **Cavalry** 1 Res. Horse Jaeger Rgt (3 Sqns)  \n **Artillery** 22 Res. F.A. Rgt. (6 Batteries)\n\n### **IX RESERVE CORPS**\n\nGeneral von Boehn\n\n**17TH RESERVE DIVISION** | **18TH RESERVE DIVISION**  \n---|---  \n**81 Brigade**  \n162 Infantry Rgt  \n163 Infantry Rgt | **33 Res. Brigade**  \n75 Reserve Rgt  \n76 Reserve Rgt | **53 Reserve Brigade**  \n84 Res. Infantry Rgt  \n86 Res. Infantry Rgt | **36 Reserve Brigade**  \n31 Res. Infantry Rgt  \n90 Res. Infantry Rgt  \n**Cavalry** 6 Res. Hussars Rgt (3 Sqns)  \n **Artillery** 17 Res. F.A. Rgt (6 Batteries) | 9 Res. Jaeger Battalion  \n **Cavalry** 7 Res. Hussars Rgt (3 Sqns)  \n **Artillery** 18 Res. F.A. Rgt (6 Batteries)\n\n**The apparent check to the German advance by the forts which ringed the Belgian town of Li\u00e8ge created great enthusiasm in Britain. A popular song rashly claimed that 'Belgium put the kibosh on the Kaiser' and the** _Daily Mirror_ **echoed the popular feeling of the time. (Author's Collection by courtesy of Mirror Newspapers)**\n\n### **SECOND ARMY**\n\nGeneraloberst von B\u00fclow\n\n### **GUARD CORPS**\n\n**1ST GUARD DIVISION**\n\n**1 Guard Brigade**  \n1 Ft. Guard Rgt  \n3 Ft. Guard Rgt | **2 Guard Brigade**  \n2 Ft. Guard Rgt  \n4 Ft. Guard Rgt  \n---|---\n\n**Artillery** 1st Guard Bde: 1st Gd. Rgt. 3rd Gd. Rgt\n\n**2ND GUARD DIVISION**\n\n**3 Guard Brigade**  \n1 Grenadiers Rgt  \n3 Grenadiers Rgt | **4 Guard Brigade**  \n2 Grenadiers Rgt  \n4 Grenadiers Rgt | **5 Guard Brigade**  \n5 Grenadiers Rgt  \n5 Foot Rgt  \n---|---|---\n\n**Artillery** 2 Guard Bde: 2 Gd. F.A. Rgt. 4 Gd. F.A. Rgt\n\n### **VII CORPS**\n\nGeneral von Einem\n\n**14TH DIVISION** | **13TH DIVISION**  \n---|---  \n**27th Brigade**  \n16 Infantry Rgt  \n53 Infantry Rgt | **79th Brigade**  \n56 Infantry Rgt  \n57 Infantry Rgt | **25th Brigade**  \n13 Infantry Rgt  \n158 Infantry Rgt | **26th Brigade**  \n15 Infantry Rgt  \n55 Infantry Rgt  \n**Cavalry** 16 Uhlan Rgt (3 Sqns)  \n **Artillery** 14 Bde: 7 F.A. Rgt. 43 F.A. Rgt | **Cavalry** 16 Uhian Rgt (3 Sqns)  \n **Artillery** 13 Bde: 22 F.A. Rgt. 58 F.A. Rgt\n\n### **X CORPS**\n\nGeneral von Emmich\n\n**19TH DIVISION** | **20TH DIVISION**  \n---|---  \n**37th Brigade**  \n78 Infantry Rgt  \n91 Infantry Rgt | **38th Brigade**  \n73 Fusiliers  \n74 Infantry Rgt | **39th Brigade**  \n79 Infantry Rgt  \n164 Infantry Rgt | **40th Brigade**  \n77 Infantry Rgt  \n92 Infantry Rgt  \n**Cavalry** 17 Hussars Rgt (3 Sqns)  \n **Artillery** 19 Bde: 26 F.A. Rgt. 62 F.A. Rgt | **Cavalry** 17 Hussars Rgt (3 Sqns)  \n **Artillery** 20th Bde: 10 F.A. Rgt, 46 F.A. Rgt\n\n**There was little doubt in Britain that the tiny BEF would rapidly halt the German advance and drive Kaiser Wilhelm's army back to Berlin with its tail between its legs. This postcard appeared within the first weeks of the outbreak of war and at least acknowledges a French presence by including the tricolour! (Author's Collection)**\n\n**The tiny Belgian Army had little enough with which to fight. This photograph of Belgian troops digging a trench shows a touching na\u00efvety. The Belgian soldiers are wearing a peculiar headgear reminiscent of the American Civil War Hardee hat and are clearly unconcerned about the possible proximity of the enemy! (Author's Collection)**\n\n### **GUARD RESERVE CORPS**\n\nGeneral von Gallwitz\n\n**3RD GUARD DIVISION** | **1ST GUARD RES. DIVISION**  \n---|---  \n**5th Guard Brigade**  \n5 Ft. Rgt  \n5 Grenadiers Rgt | **6th Guard Brigade**  \nGuard Fusiliers  \nLehr Rgt | **1 Guard Res. Brigade**  \n1 Guard Reserve Rgt  \n2 Guard Reserve Rgt | **15th Res. Brigade**  \n64 Reserve Rgt  \n93 Reserve Rgt  \n| Guard Res. Sniper Battalion  \n**Cavalry** Guard Res. Uhian Rgt |   \n**Artillery** 5 Guard F.A. Rgt, 6 Guard F.A. Rgt | **Cavalry** Guard Res. Dragoons (3 Sqns)  \n **Artillery** 5 Guard Res. F.A. Rgt.  \n3 Guard Res. F.A. Rgt\n\n### **VII RESERVE CORPS**\n\nGeneral von Zwehl\n\n**13TH RESERVE DIVISION** | **14TH RESERVE DIVISION**  \n---|---  \n**25 Reserve Brigade**  \n13 Res. Infantry Rgt  \n56 Res. Infantry Rgt | **28 Reserve Brigade**  \n39 Res. Infantry Rgt  \n57 Res. Rgt | **27 Reserve Brigade**  \n16 Res. Infantry Rgt  \n53 Res. Infantry Rgt | **28 Brigade**  \n39 Fusiliers Rgt  \n159 Infantry Rgt  \n7 Res. Jaeger Battalion |   \n| **Cavalry** 8 Res. Hussars Rgt (3 Sqns)  \n**Cavalry** 5 Res. Hussars Rgt  \n **Artillery** 13 Res. F.A. Rgt (6 Batteries) | **Artillery** 14 Res. F.A. Rgt (6 Batteries)\n\n### **X RESERVE CORPS**\n\nGeneral von Kirchbach\n\n**2ND GUARD RESERVE DIVISION** | **19TH RESERVE DIVISION**  \n---|---  \n**26 Reserve Brigade**  \n15 Res. Infantry Rgt  \n55 Res. Infantry Rgt | **38 Reserve Brigade**  \n38 Res. Infantry Rgt  \n91 Res. Infantry Rgt | **37 Reserve Brigade**  \n73 Res. Infantry Rgt  \n78 Res. Infantry Rgt | **39 Reserve Brigade**  \n74 Rese. Infantry Rgt  \n92 Res. Infantry Rgt  \n10 Res. Jager Battalion | 79 Res. Infantry Rgt (2 Battalions)  \n| 10 Res. Jager Battalion  \n**Cavalry** 2 Res. Uhian Rgt (3 Sqns) |   \n**Artillery** 20 Res. F.A. Rgt (6 Batteries) | **Cavalry** 6 Res. Dragoon Rgt (3 Sqns)  \n **Artillery** 19 Res. F.A. Rgt (6 Batteries)\n\n25th Landwehr Brigade (later to become part of 25th Landwehr Division)\n\n29th Landwehr Brigade (later to become part of 29th Landwehr Division)\n\n4 Mortar battalions\n\n1 10-cm gun battalion\n\n2 Heavy Coast Mortar Battalions\n\n2 Pionier regiments\n\n**World War I has been called the second large war of the Industrial Age, the first being the American Civil War. Mass transportation made it possible to quickly move large numbers of men and vast quantities of munitions and supplies. For Germany, in particular, the railways were an integral part of her war planning and the officers who manned the Railways Directorate of the German General Staff were the best available \u2013 it was on them that the smooth running of the whole German war machine depended. (Private Collection)**\n\n### THIRD ARMY\n\nLieutenant-General Freiherr von Hausen\n\n### **XI CORPS**\n\n**22ND DIVISION** | **38TH DIVISION**  \n---|---  \n**43 Brigade**  \n82 Infantry Rgt  \n83 Infantry Rgt | **44 Brigade**  \n32 Infantry Rgt  \n167 Infantry Rgt | **76th Brigade**  \n71 Infantry Rgt  \n95 Infantry Rgt | **83rd Brigade**  \n94 Infantry Rgt  \n96 Infantry Rgt  \n**Cavalry** 6 Culrassier Rgt (3 Sqns)  \n **Artillery** 22 Bde: 11 F.A. Rgt, 47 F.A. Rgt | **Cavalry** 6 Culrassier Rgt (3 Sqns)  \n **Artillery** 38 Bde: 19 Rgt, 55 Rgt\n\n### **XII (1ST SAXON) CORPS**\n\nGeneral d'Elsa\n\n**23RD DIVISION** | **32ND DIVISION**  \n---|---  \n**45 Brigade**  \n100 Grenadiers Rgt  \n101 Grenadiers Rgt  \n12 Jaeger Battalion | **44 Brigade**  \n108 Fusiliers Rgt  \n182 Infantry Rgt | **63 Brigade**  \n102 Infantry Rgt  \n103 Infantry Rgt | **64 Brigade**  \n177 Infantry Rgt  \n178 Infantry Rgt  \n**Cavalry** 20 Hussar Rgt  \n **Artillery** 23 Bde: 12 F.A. Rgt, 48 F.A. Rgt | **Cavalry** 18 Hussar Rgt  \n **Artillery** 32 Bde: 28 F.A. Rgt, 64 F.A. Rgt\n\n**There was no doubt in the mind of the German soldier as to the enemy nor the aim of the invasion \u2013 Paris would be occupied in a matter of days!**\n\n### **XIX (2ND SAXON) CORPS**\n\nGeneral von Laffert\n\n**24TH DIVISION** | **40TH DIVISION**  \n---|---  \n**47 Brigade**  \n139 Infantry Rgt  \n179 Infantry Rgt | **48 Brigade**  \n106 Infantry Rgt  \n107 Infantry Rgt | **88 Brigade**  \n104 Infantry Rgt  \n181 Infantry Rgt | **89 Brigade**  \n133 Infantry Rgt  \n134 Infantry Rgt  \n| 13 Jaeger Battalion  \n**Cavalry** 18 Uhlan Rgt |   \n**Artillery** 24 Bde: 77 F.A. Rgt, 78 F.A. Rgt | **Cavalry** 9 Hussar Rgt  \n **Artillery** 40 Bde: 32 Rgt. 68 Rgt\n\n### **XII (SAXON) RESERVE CORPS**\n\nGeneral von Kirchbach\n\n**23RD RESERVE DIVISION** | **24TH RESERVE DIVISION**  \n---|---  \n**45 Reserve Brigade**  \n100 Res. Grenadiers  \n101 Res. Infantry Rgt | **46 Reserve Brigade**  \n102 Res. Infantry Rgt  \n103 Res. Infantry Rgt | **47 Reserve Brigade**  \n104 Res. Infantry Rgt  \n106 Res. Infantry Rgt | **48 Reserve Brigade**  \n107 Res. Infantry Rgt  \n133 Res. Infantry Rgt  \n12 Res. Jaeger Battalion | 13 Res. Jager Battalion  \n**Cavalry** Res. Hussar Rgt (3 Sqns)  \n **Artillery** 23 Res. F.A. Rgt (9 Batteries) | **Cavalry** Saxon Reserve Hussar Rgt  \n **Artillery** 24 Reserve F.A. Rgt (9 Batteries)\n\n47th Landwehr Brigade (later 47th Landwehr Division)\n\n2 Mortar Battalions\n\n1 Pionier Rgt\n\n### **I CAVALRY CORPS**\n\nLieutenant-General Freiherr von Richtofen\n\n**GUARD CAVALRY DIVISION  \n5TH CAVALRY DIVISION**\n\n### **II CAVALRY CORPS**\n\nLieutenant-General von der Marwitz\n\n**2 CAVALRY DIVISION**\n\n**4 CAVALRY DIVISION**\n\n**9 CAVALRY DIVISION**\n\n## THE OPPOSING PLANS\n\n##### GERMANY'S SCHLIEFFEN PLAN\n\nGermany's offensive plan, produced by Count Alfred von Schlieffen, Chief of the General Staff from 1891 to 1906, was a response to the Franco-Russian military pact of 1894 which had created the spectre of a war on two fronts. Germany could anticipate such an event by attacking first, but was not strong enough to combat both countries together. One had to be defeated before the other could be tackled.\n\nVon Schlieffen believed that the greater danger came from France and calculated that it would take the Russians six weeks to launch an offensive. He aimed to repeat the result of the Franco-Prussian War of 1870, when the French Armies were defeated in just 33 days, and the war reached its climax at Sedan, a classic envelopment battle. If the French could be beaten as swiftly again, the German Army could then be sent to the Eastern Front to destroy the Russians.\n\nFor the French, the final humiliation of 1870 was the loss of the provinces of Alsace and Lorraine. Their obsession with regaining the lost territory became a vital part of von Schlieffen's thinking. The 150 miles of common frontier with France was between Luxemburg and Switzerland. It was an area of woods, mountains and rivers, with fortresses at Verdun, Toul, Nancy, Epinal and Belfort. Attacks here stood little chance of achieving the necessary swift success.\n\nVon Schlieffen therefore proposed a vast swinging hook through Belgium to France, through Artois and Picardy, bypassing Paris before moving eastward to take the French forts in the rear. Allowing 39 days for the defeat of France, the plan needed an army of five million men, enormous industrial capacity and a first-class railway system subordinated to military need.\n\nBritish mobilisation was equally dependent upon a first-rate rail system allied to a carefully planned scheme for shipping men and munitions across the Channel. Here troops of the 11th Hussars are seen approaching Le Havre on 16 August 1914. (Mons Museum)\n\nIn its final form, the plan called for a mere ten divisions to hold Russia in the East; 62 divisions would face France, 54 of them, spread among the First to Fifth Armies, would carry out the assault. The other eight were allocated to the Sixth and Seventh Armies on the Franco-German frontier. They would retreat in the face of the anticipated French attack in Alsace and Lorraine and lure their opponents into a trap. As the French dashed forward, they would leave their rear unprotected. In the words of the historian Liddell Hart, the Schlieffen Plan was like a revolving door: 'if a man pressed heavily on one side, the other would swing round and hit him in the back'. It would be another Sedan.\n\n**After landing, the British Army moved to Maubeuge, on the left flank of Lanrezac's Fifth French Army. These troops, assembled at a base camp, would shortly start their march north towards the advancing German First and Second Armies. (Mons Museum)**\n\nVon Schlieffen wanted 'the right sleeve of the right-hand man to brush the Channel', but after his retirement in 1906, consistent revisions shrivelled the proportionate weight of the right and left wings from six to one, to two to one. The right wing would no longer follow the coast but be well inland. It was an alteration which would eventually lead the German First Army to Mons.\n\n##### **THE FRENCH PLAN XVII AND THE ROLE OF THE BEF**\n\nAfter 1870 French military thinking had concentrated on defence but, in the early 1900s, these ideas became anathema to a fresh generation of officers. At its most basic, the new doctrine asserted that attack was all-important and particularly well suited to the temperament of the French soldier. The concept of 'the massed onslaught at the decisive point' where, irrespective of all else, the surging battalions smashed through an enemy, recalled the days of glory of the First Empire. Bonaparte's battles against the Austrians were cited approvingly; the Peninsula and Waterloo were ignored.\n\n#### THE CONCENTRATION OF THE ARMIES \u2013 AUGUST 1914\n\nThe new discipline of the offensive reached its zenith with Plan XVII issued in February 1914. It opened with a startling lack of appreciation of German intentions, as well as a blithe disregard of French Intelligence assessments: 'From a careful study of information obtained,' it began, 'it is probable that a great part of the German forces will be concentrated on the common frontier.' After this unpromising start, Plan XVII continued: 'Whatever the circumstances, it is the C-in-C's intention to advance with all forces united to the attack of the German Armies.'\n\n**The reality was this \u2013 German cavalry advancing in August 1914 \u2013 a surreptitious photograph. These troops were the actual spearpoint of the German advance. This is a photograph of an invasion. (Private Collection)**\n\nImpressive results were anticipated. On the extreme right, the Army of Alsace would take the area where the French, German and Swiss borders met; along the common frontier, the First and Second Armies would advance through Lorraine and the Saarland to occupy Mainz. On the left wing, the Third and Fifth Armies would attack to the east unless the Germans had violated Belgian neutrality. In this event, they would advance through Luxemburg and the Ardennes to strike at the left flank of the German Army. Any invasion of Belgium by the Germans would be, in the opinion of the French planners, limited to a small area east of the River Meuse.\n\nThe left flank of the Fifth Army would be at Hirson, close to the Franco-Belgian border, and would be extended by the arrival of the BEF. Between them and the Channel ports was a 100-mile gap garrisoned by three poorly trained French Territorial divisions. The BEF would be part of a general advance which would crush the unprotected German right flank.\n\nThe Fourth Army, which was largely made up of Reserve divisions, would be ordered forward to occupy Berlin once the other armies had defeated the enemy.\n\nDetails of reserves and timetables were scant, but it was confidently assumed that nothing could go wrong. The whirlwind offensive would completely dislocate German plans. Plan XVII further ignored any possibility of a major German attack through Belgium and that meant that there would not be one. Plan XVII could not be wrong. It was a theory that would cost the French Army more than 250,000 casualties in August 1914.\n\n## THE CAMPAIGN\n\n##### THE ARRIVAL OF THE BEF\n\nThe first British soldiers arrived at Mons on 21 August. They were patrols from the 9th Lancers and 18th Hussars, probing northward for signs of the enemy. Local residents and refugees spoke of the roads south of Brussels being filled with marching Germans.\n\nThe next morning, north of Mons at Casteau, at about 7am, Cpl. Drummer Thomas of 'C' Squadron of the 4th Dragoon Guards fired at a mounted figure and claimed a hit. Whether it was or not, a British Army bandsman had fired the first British shot in the First World War. His target was probably from the 4th Cuirassiers, for an hour later four or five suspicious troopers from that unit came warily along the road towards the eager British.\n\nTwo troops of dragoons rode to engage them. The outnumbered cuirassiers retreated and the dragoons gave chase. The Germans reached their squadron which retreated further. After a hot pursuit over another two miles, the dragoons caught the enemy and in the skirmish which followed about 15 Germans were killed and eight taken prisoner.\n\nLater that day, the Royal Scots Greys, fighting a dismounted action, persuaded a strong German force from the 13th Division of the VII Corps and part of Second Army's I Cavalry Corps to retreat under the impression they were fighting a full brigade. The 16th Lancers met two companies of Jaegers in the open and charged them with the lance. A number of the enemy were speared for the loss of a handful of British troopers.\n\n**Although aircraft were being used for reconnaissance, all the combatant armies relied on their cavalry to probe ahead of the main infantry force and act as scouts. This postcard, entitled 'Uhlans In The Vanguard!', shows a cavalry patrol asking questions from a helpful Belgian civilian \u2013 it came as a rude shock to Kaiser Wilhelm's High Command that the Belgians actually resisted the German advance. (Michael Solka)**\n\n**Their armies' advance into France and Belgium was presented as an unrelieved triumphal progress in the German newspapers of the time. Unlike the British, the German Army had its own photographers who were soon supplying a stream of pictures to the home front to keep up civilian morale. In this shot a typical German infantryman 'snatches a moment during the advance to write a personal postcard home to his loved ones'. (Private Collection)**\n\nThe cavalry reports went back to GHQ their conclusions being that strong German forces were advancing on Mons. Twelve reconnaissance missions made by the infant Royal Flying Corps also reported the presence of large masses of enemy troops. The information made little impact at GHQ, as French and his staff were concentrating on plans for a general offensive.\n\nIt had been a busy time for the BEF since mobilisation on 4 August. The advance parties left for France on 7 August, and the main elements began their journey on the 12th. Southampton was the main embarkation port for the troops, except for those stationed in Ireland, who left from Dublin, Cork and Belfast. Motor transport (the British Army was the most mechanised in Europe) and petrol left from Avonmouth and Liverpool, stores and supplies were despatched from Newhaven. Ports and railways were put to a severe test; 80 trains were required to move a division, and 1,800 special trains were needed during the five days it took to get the BEF and its supplies to the embarkation points. On the busiest day, there were 137 sea crossings as the troop ships and transports laboured back and forth. By the evening of 17 August, the vast majority of the BEF had disembarked at Le Havre, Rouen and Boulogne.\n\nThey received a rapturous welcome and were deluged with flowers and kisses, wine and food. Cap badges and shoulder titles were handed over to enthusiastic admirers. An infantry officer, marching through the cheering crowds on the trek to Maubeuge, recalled years later that 'I felt like a king among men'.\n\nSince 4 August, the German Army had ploughed across Belgium. Li\u00e8ge was captured on 16 August, and von Kluck's First Army was swinging round to the west and advancing towards the French frontier. Von Bulow's Second Army was south of Li\u00e8ge and the Third Army under Gen. Hausen was moving through the Ardennes. Von Moltke, the Chief of the German General Staff, in order to control the pace and alignment of the right, decided to form the three armies into an Army Group under the control of von Bulow. Von Kluck was furious, his reaction simple. He merely ignored all orders from his nominal superior.\n\nThe Germans entered Brussels on 20 August, and on the same day, Sir John French issued Operation Order No 5 which instructed the BEF 'to march north'. This was in accordance with the rapidly crumbling Plan XVII, which required Lanrezac, commander of the French Fifth Army of which the BEF was now an appendage, to move into Belgian territory east of the River Meuse if the Germans breached Belgian neutrality. It was thought the joint force would meet only light opposition.\n\n#### THE ADVANCE ON MONS \u2013 AUGUST 1914\n\n**The BEF was greeted with enormous enthusiasm by the French population. Many British troops lost cap badges and shoulder titles in response to urgent appeals for souvenirs. This picture, taken by a press photographer, emphasises the high level of mechanisation of the British Army compared to its Continental rivals. Thanks to the 'Subsidy Scheme' under which commercial companies received from the War Office part of the cost of buying and operating approved lorries and vans in return for yielding up the transport if needed in time of war, the British Army could claim to be one of the most highly mechanised in the world. Many of the vehicles initially went to France in their civilian livery \u2013 a point much appreciated by the aircrew of 5 Squadron, Royal Flying Corps, who used their scarlet and gold painted lorry (from the makers of HP Sauce) as a homing device during the constant moves of the Advance to Mons and the subsequent Retreat! (Private Collection)**\n\n###### **Lanrezac's change of plan**\n\nIn fact, Lanrezac was facing 700,000 men in the three German armies. He had already started to have second thoughts about the wisdom of an offensive. As time passed and more reports came in, he became more cautious. By 21 August, he was sure. The French Fifth Army, he reasoned, held the high ground on the south bank of the Sambre, was protected on the right by the Meuse and was thus in a first class defensive position. It would be madness to leave it to meet an enemy who was in great strength. Therefore, he would delay the attack until the time was opportune. The BEF, unaware of Lanrezac's change of opinion, was marching confidently northwards.\n\nLanrezac was right. The planned advance would neatly deliver the Fifth Army into a salient where it could be destroyed from three sides.\n\nMatters got worse. On the afternoon of 21 August, the 'excellent defences' on the Sambre were breached by the German Second Army. Despite talk of counterattacking the next day, Lanrezac decided on retreat. His concern was to save his Army. Intact, it could still fight; destroyed or captured, the route to Paris was wide open. The Fifth Army, along with the Fourth Army on its right, began to fall back the next day under constant pressure from the Germans.\n\nGeneraloberst von Kluck, meanwhile, had his own problems. He was concerned about his right flank. Under the impression that the BEF had probably landed at Ostend, Dunkirk and Calais, he was anxious to move towards the coast to intercept any threat from the Channel ports. Von Bulow, however, disagreed. Living up to his reputation as a cautious commander who believed in mutual support, he was more concerned about the growing gap between the First and Second Armies, and ordered von Kluck to turn south-west in case 'the First Army might get too far away and not be able to support the Second Army'. Von Kluck protested, but Supreme Headquarters intervened to support von Bulow. Von Kluck fumed but had to conform.\n\n**Belgian cyclist troops 'advancing towards the enemy' if the original caption is to be believed. Interestingly, in the original photograph a car on the left of the picture appears to be filled with British officers \u2013 presumably retreating from the enemy! (Author's Collection)**\n\nSo it was that a Royal Flying Corps crew reported on Saturday 22 August that they had seen a huge column of enemy troops marching towards the BEF. It was von Kluck's II Corps and they were on their way to Mons.\n\nIt was Lt. Spears, the British liaison officer at Lanrezac's headquarters, who saved the BEF from extinction. Learning on the Saturday afternoon of Lanrezac's decision to retreat, as well as getting some indication of the size of the German assault, he set off for GHQ along roads jammed with transport and refugees. It took him four hours. He arrived at eight o'clock in the evening to discover the staff busy with final arrangements for continuing the advance. Spears made his report to Sir John French and the planning was abruptly discontinued. New instructions were issued to halt the advance and prepare for battle.\n\nThere was to be a final surprising request. As the orders went out to the BEF to hold its ground, one of Lanrezac's staff officers arrived. Lanrezac wanted the BEF to swing to the east and attack von Bulow's Second Army to take pressure off the Fifth Army. This would have exposed the flank of the BEF to the full weight of von Kluck's assault. As it was, Sir John French agreed to hold his existing position for 24 hours.\n\n## THE BATTLE OF MONS\n\nThe main body of the BEF had already reached Mons. II Corps spread out along the line of the Mons-Cond\u00e9 Canal which ran directly westwards from the town. I Corps occupied the salient to the east of Mons. By nightfall the BEF was nine miles ahead of Lanrezac's retreating Fifth Army, with a ten mile gap on the right between the British and French. On the left, there was intermittent contact with the 84th French Territorial Division, and Allenby's Cavalry Corps was ordered across to support the left.\n\nThe last elements of the BEF did not reach Mons until 3am on 23 August after a long and gruelling three days' march. 'We were saddled with pack and equipment weighing nearly eighty pounds and our khaki uniforms, flannel shirts, and thick woollen pants, fit for an Arctic climate, added to our discomfort in the sweltering heat,' one Reservist recalled. New boots, the heat, and the strain of continual marching along cobbled roads brought many soldiers to a state of collapse.\n\nMons was the centre of the Belgian coal mining area. A string of dreary villages, interspersed with slag heaps, factories and coal tips, ran along the 16 mile length of the canal. Eighteen bridges crossed it. No more than two metres deep, and with a width of about 20 metres, the canal was hardly an obstacle. The view to the north, from which the attack would come, was obscured by grim terraces of cottages, factories and slag heaps. The higher ground was fringed with trees which made both movement and vision difficult for the defenders.\n\n**Behind the cavalry trudged the indomitable infantry, suffering badly in the August heat. Germans, French, Belgians and British alike recall sweltering August conditions. This group of British senior NCOs were caught by the camera in northern France just before the move into Belgium. (Mons Museum)**\n\n**Countering the Allied claims of 'Hun atrocities' in Belgium and France rapidly became a priority for the German propaganda service. There is little doubt that atrocities occurred \u2013 von Moltke, the German Commander-in-Chief, expressed his own shock at the brutal behaviour reported to him. Favourable stories and photographs of German activities were supplied to the press in neutral countries \u2013 German soldiers were forever rescuing Belgian children from flooded streams, swirling mill ponds and deep canals \u2013 together with stories attempting to justify repressive action on the part of the German authorities. The US Ambassador to Belgium recalled that the continual German justifications made him feel that the burgemeesters or mayors of Belgian towns had bred a special race of children, so often was the reason advanced for a particular atrocity that the son or daughter of a local burgemeester had attacked an innocent or unarmed German soldier. This picture was one of a series purporting to show the warm welcome offered to the occupiers by the Belgians. (Author's Collection)**\n\nThe BEF dug in, borrowing picks and shovels from the local civilians to eke out their own supply. Both Haig and Smith-Dorrien were concerned about their ability to hold their positions. II Corps, in particular, with some 36,000 men in total, had the almost impossible job of defending a 21 mile long front line running from the bridge at Le Petit Cr\u00e9pin in the west to the bridge at Obourg in the east. The canal curved round the town of Mons to join the River Sambre and this created a salient which included the road and railway bridges into Mons itself. On the right of Smith-Dorrien's position, I Corps faced east so that the whole length of the British position looked a little like a walking stick with a curved handle.\n\nSensibly and conventionally enough, the British infantry were concentrated to deny the bridges to the advancing foe. Outposts were deployed on the northern side of the canal to cover the approaches to the bridges. The soldiers scraped holes where they could, erected barricades from anything they could find, concealed themselves wherever possible and waited for morning.\n\nSmith-Dorrien had commanded II Corps for only a day. He had been sent hastily from England to replace Sir James Grierson who had died suddenly on 17 August. Smith-Dorrien had enjoyed a less than harmonious interview with Sir John French, who had instructed him to give battle along the line of the canal. Smith-Dorrien had asked if it was an offensive or a defensive operation and received the not altogether encouraging response that he should obey orders.\n\nSmith-Dorrien's doubts were not eased during a personal visit by Sir John French early in the morning of 23 August. With one of those contradictory instructions which baffled those who served under him, he told Haig, Allenby and Smith-Dorrien 'to be prepared to move forward, or to fight where they were', and to ready the bridges across the canal for demolition. He further assured his corps commanders that 'little more than one, or two, enemy corps with perhaps a cavalry division' were advancing on the BEF. This optimistic assessment was totally at variance with the reports of both the British and French Intelligence staff. They estimated that a minimum of three German Army corps were approaching Mons.\n\n**Reports of clashes between German and British cavalry led to a rash of popular pictures in both countries; cavalry actions were dashing and romantic, evoking past eras of combat. This illustration, produced early in the war, shows a determined German cuirassier spearing a terrified British opponent. (Michael Solka)**\n\n**The German army entered Brussels on the 20th August, 1914. Long horse-drawn columns of guns and transports, interspersed with marching infantry, relentlessly singing themselves hoarse as they passed through the Belgian capital, brought home to watching civilians the awesome power that they faced. It took three days and nights for the Germans to go through Brussels and, as one witness dolefully recalled, it was the non-stop singing of marching songs that was the worst part of it. (Private Collection)**\n\nSmith-Dorrien expressed his worry about his position along the canal. He had already reconnoitred a new defence line two miles south of the canal which eliminated the town of Mons and the salient. He told French that he was preparing orders for his troops to retire to this line if they were in danger of being cut off by the advancing Germans. Smith-Dorrien was later to claim that Sir John had agreed with his views and approved his plans, a conclusion which the Commander-in-Chief subsequently denied. Sir John then departed for Valenciennes, leaving his army to fight its first battle of the war. GHQ was to issue no further written orders until the morning of 24 August and Sir John French's contact with his army became extremely haphazard.\n\n###### **The opening shots**\n\nEven as the commanders talked, the first skirmishes had started. In a drizzling, misty dawn, the Germans began a short bombardment which caused about 20 casualties, and the 4th Middlesex opened fire on a German cavalry patrol pushing forward to the salient east of Mons. Lt. von Arnim of the Husaren Regiment Nr. 1 was taken prisoner. He was shot through the knee and would walk with a limp for the rest of his days. As the sun came out, clearing the mist and rain, more German cavalry were seen along the whole length of the line. They were the advance guard of the 3rd, the 9th and the 2nd Cavalry Corps of the First Army.\n\nVon Kluck, still smarting under his changed orders, had little idea of the location of the BEF. His troops stumbled upon British forces by chance. Instead of co-ordinated attacks, there were thus a series of small-scale actions. If von Kluck had concentrated the full weight of the four Army Corps at his disposal upon the BEF, history would have been very different. As it was, he believed that only cavalry were in front of him, that the main BEF was still somewhere to the south, and allowed the action to develop piecemeal.\n\n#### THE BATTLE OF MONS 23 AUGUST 1914\n\n**The BEF's first encounter with von Kluck's First Army. Along a line approximately nine miles long, nine and a half British battalions hold four German Divisions at bay for most of the day.**\n\n**The main Belgian field army retreated to Antwerp on 20 August. As the Germans occupied Brussels, the Belgians prepared their defences. Optimism was high. This detachment of Belgian infantry demonstrates exactly how they would defend an Antwerp canal against the invader. (Author's Collection)**\n\nAt about 9am, the German 9th Corps artillery once more opened fire on the salient on the British right. The bombardment was followed by an assault from the 18th Infantry Division, pushing forward between Obourg and Nimy against the 4th Middlesex and the 4th Royal Fusiliers. Eight battalions attacked four companies. The German soldiers approached in a solidly packed mass of close columns and were met by a hail of fire from the entrenched British infantry.\n\n'They went down like a regular lot of Charlie Chaplins,' one British soldier wrote to his wife, 'every bullet hitting home, sometimes taking two men at a time.' Column after column advanced to be met by the same withering fire. Finally, the German attack shivered to a halt and the survivors retreated to the cover of the tree line. Desultory artillery fire continued and after 30 minutes, a new attack was launched.\n\nThis time, the Germans came in extended order over a wider frontage; they were joined by the infantry of the 17th Division, and their attacks were pressed home with great courage. The Middlesex, in the east of the salient, suffered under a continuous artillery bombardment, but the infantry attacks were thrown back with heavy German losses. Eventually, the German troops began to operate in smaller groups. Determined parties of enemy infantry managed to cross the canal and infiltrate the flanks of the Middlesex position.\n\nThe situation became confused and communication a problem as the shells rained down, but the 'stubborn resistance' which Smith-Dorrien had pressed upon his brigade commanders continued. By mid-morning, despite support from the machine gun section of the 2nd Royal Irish, the Middlesex had begun to fall back, and shortly after midday they were fighting desperately to prevent the German assault from encircling them.\n\n**British cavalry entered Mons on 21 August 1914. This photograph, taken by a French liaison officer, shows a patrol on its way forward into Belgium on the 20th August. (Mons Museum)**\n\n###### **Defence of la Bois Haut**\n\nOn their right, the 1st Gordons and 2nd Royal Scots, dug in on higher ground at la Bois Haut, were resisting stubbornly as the 17th Division tried to turn the British flank. From this higher ground, the British artillery had both good observation and fields of fire. The German infantry was caught in the open by the gunners of 6, 23 and 49 batteries and suffered heavily from the hail of shrapnel with which they were showered.\n\nThe salient was being assaulted on three sides and British losses rose steadily. As the officers were killed or wounded, the individual initiative of the lower ranks became a vital ingredient of the resistance. By noon, German troops were over the canal in reasonable strength and the enemy artillery was making life extremely unpleasant for the British infantry. The position of the 4th Middlesex was rapidly becoming untenable, and the Royal Irish Regiment, ordered up to support them, found their task complicated by the attentions of the German guns. A German cavalry attack was beaten back by the Irish machine gunners but the enemy's artillery eventually zeroed in and caused severe casualties.\n\n**These troopers of the 9th Lancers entering Mons on the 21 August 1914, were the first British soldiers to reach the town. (Mons Museum)**\n\nATTACK OF THE GERMAN 18TH DIVISION AT MONS, 23 AUGUST 1914  \n **Following a bombardment by the German 9th Artillery Corps at about 09.00 the German 18th Division launched an assault against the 4th Middlesex and the 4th Royal Fusiliers between Obourg and Nimy. Eight battalions attacked four companies in close column formation and were cut down in a hail of fire from the British.**\n\n**The British Army moved northwards behind its cavalry screen which duly sent back a stream of patrol reports. Allied to those of the RFC squadrons, they gave an accurate enough picture of the looming menace in front of the British. Unfortunately, both the British and French staffs considered the reports to be greatly exaggerated. On 21 August 1914, the Cavalry Division received an order telling them that the 'information you have acquired and conveyed to the C-in-C appears to be somewhat exaggerated'. Henry Wilson, convinced of the excellence of the French Plan XVII and sceptical of any German advance on the right, was the signatory. It was, not surprisingly, a cavalryman who fired the first British shot of the First World War, the first shot fired in anger on the Continent since the Battle of Waterloo. At Casteau, just north of Mons, Corporal Drummer Thomas of the 4th Dragoon Guards fired at a distant German cavalryman at about 7am on 22nd August 1914. He was forever convinced that he had hit his man. (Mons Museum)**\n\n**Two British 18th Hussars question members of the local populace at Mons on 21 August 1914. Note the sun hood worn by one of the riders and the fringed tassels over the horses' eyes to keep away flies. (Mons Museum)**\n\nThe withdrawal from the salient began at about 2pm. The Royal Fusiliers retired slowly, stubbornly, and reluctantly from the bridges, a task made more difficult because they were in full view of the enemy. At Nimy, Lt. Maurice Dease, the machine gun officer of the Royal Fusiliers, held back two battalions of German infantry as they tried to capture the bridge. Although wounded time and again, Dease fought until he died. He was awarded a posthumous Victoria Cross. It was the first of 628 to be awarded during the First World War. Pte. Sidney Godley, also of the machine gun section, won the second. Remaining on the bridge to cover the withdrawal of the Fusiliers late in the afternoon, already badly wounded, he held his position until everyone else had got away. As a final gesture, he dismantled his machine gun and threw the pieces into the canal. He then crawled back to the main road where he was found by two Belgian civilians and taken to hospital in Mons.\n\nThe Fusiliers withdrew through Mons itself and then to Ciply and Nouvelles to the south of the town. Their retreat was covered by the 1st Lincolns who had set up barricades through which the exhausted survivors of the day's fighting passed. The enemy showed no inclination to pursue.\n\nIn the remainder of the salient, the fighting became increasingly bitter as the afternoon wore on. The Germans tried to outflank their opponents by moving forward to Hyon. This caused problems for the 4th Middlesex and the Royal Irish as they pulled back. The Middlesex, having moved through the Royal Irish, fought a desperate rearguard action on the outskirts of Hyon itself; as the Royal Irish, in their turn, started their withdrawal, they found the Germans behind them and were forced to detour through la Bois Haut.\n\n**Two British cavalrymen and their French interpreter on the Canal du Centre on 21 August 1914. This very poor photograph is nonetheless an interesting moment frozen in time. (Mons Museum)**\n\n**The Grand-Place at Mons \u2013 soldiers of 'A' Company of the 4th Royal Fusiliers, part of the 7th Brigade of 3 Division in Smith-Dorrien's II Corps, shown at rest on the afternoon of 22 August, 1914. From this position today, the facade is virtually unchanged. By the next morning, these soldiers had moved on to Nimy, some 7 kilometres to the north, where they fought off the full weight of the 18th Division of von Kluck's IX Corps. Note the Belgian civilians mixing with the troops. (Mons Museum)**\n\nTHE BRIDGE AT NIMY 23 AUGUST 1914  \n **To help cover the retreat from the salient Lieutenant Maurice Dease held back two battalions of German infantry attempting to capture the bridge. Repeatedly wounded he fought until he died. Private Sidney Godley took over and remained until everybody had escaped. Although badly wounded he managed to dismantle the gun and throw it into the canal as a final gesture before crawling to the road where he was found by two civilians who took him to hospital. Both soldiers were awarded the Victoria Cross.**\n\n**The 4th Battalion of the Royal Fusiliers won the first two Victoria Crosses of the war. The first to go to a private soldier was that awarded to Private Sidney Godley (see text). Godley was badly wounded in the action defending the bridge at Nimy and was captured when the Germans occupied Mons. His picture, appearing in the papers when his award of the VC was announced, is reputed to have provided Bruce Balrnsfather with the likeness for 'Old Bill'. Certainly, a photograph of Godley in civilian clothes, taken at a subsequent Buckingham Palace reception for Victoria Cross holders, shows a stocky and cheerfully pugnacious figure who could well pass for Bairnsfather's creation. (Imperial War Museum)**\n\n###### **The battle for the canal**\n\nWest of the salient, along the line of the canal, the battle had taken longer to develop. The 6th Division of III Corps came upon the British positions at Mariette three miles west of Mons at about 11am. Marching down the road in close column of fours, they were sent reeling sharply backwards by the concentrated fire of the 1st Northumberlands. The Germans tried again, this time being ambushed against a barricade and a wire entanglement. There was a lull while two German field guns were brought up to demolish the Northumberlands' defences; the Fusiliers were then astounded to see a group of Belgian schoolgirls coming down the road towards them.\n\nThe enemy, taking advantage of the fact that the British were holding their fire in the face of civilians, rushed forward, and the Northumberland outposts were forced to withdraw across the canal. Whether this was a deliberate ploy by the German attackers or a simple case of civilians being caught up in the fighting is now impossible to determine.\n\nStill further west, more assaults by 6th Division against the bridge at Jemappes were severely punished by the 1st Royal Scots. Again, the Germans attempted to counter the threat by bringing up artillery, but the German field gunners were firing blind. Casualties among the scattered British troops were light and at noon both the Northumberlands and the Royal Scots were still comfortably fending off any attempt to take their bridges.\n\nAt St Ghislaine, to the left of the 1st Royal Scots, the 1st Royal West Kents were faced by the 12th Brandenburg Grenadiers of the 5th Division. Amazed British soldiers watched as the German troops advanced, firing from the hip. It was, one considered, 'like watching a military tattoo', but it was also remarkably ineffective, the bullets flying high above Kentish heads. The British battalion had advanced north of the canal and occupied a glass factory. From its cover, they massacred the first attacks without loss to themselves. Faced with increasing odds as the Brandenburg Grenadiers were reinforced, the Kents slipped away as dusk came and took up new positions on the embankment. The final German attacks were stopped some 300 yards north of the canal, great havoc being wrought by four guns of 120 Battery of the Royal Field Artillery which was on the canal towpath.\n\n**The 4th Dragoons at Mons. An officer and men in their sandbagged position. (Mons Museum)**\n\nLike many survivors of the German regiments that tangled with the BEF that day, the Brandenburgers believed the deadly 'mad minute' concentrated rifle fire of the British was the work of machine gunners. It was an understandable mistake; 700 concealed infantrymen, each firing once every four seconds, not only make a lot of noise. They cause many casualties. The vast majority of the attackers had never been under fire before, and the sheer power of the defence was an enormous shock to them.\n\n**The 1st Northumberland Fusiliers building a barricade across the road from Mons to Jemappes. Note the members of the local population who not only gave tools to the British soldiers but joined in to help build the defences. This picture was taken on the morning of 23 August 1914. (Mons Museum)**\n\n**Private B. Barnard of the 4th Royal Fusiliers, one of the men who defended the bridge at Nimy. The action here was one of enormous determination against substantial odds. Barnard survived the action and personally gave this photograph to the Mons Museum after the war. (Mons Museum)**\n\nAt Les Herbi\u00e8res, to the left of the West Kents' position, the 52nd Infantry Regiment from the German III Corps at first seemed to have learned the painful lesson being taught elsewhere. This area was held by the 2nd King's Own Scottish Borderers and the 1st East Surreys. The KOSB outposts came under attack by small parties of German infantry. Once these were driven in, the German artillery began a short fierce bombardment before another assault was launched. It was the same as those further along the east of the line. Two closely-packed battalions advanced and met the same fate as every other German massed attack. They were cut down in droves as they emerged from a wood a mere 200 metres from the East Surreys' defensive line. The German soldiers continued to advance despite their severe losses, but courage alone was not sufficient. A British officer later remembered walking down the line, hitting his men on their backsides with his sword and telling them to fire low. The range was so short that many were firing high.\n\nOn the extreme left, the 1st Duke of Cornwall's Light Infantry did not get into action until 4.45pm, when they easily repulsed a horde of German cavalry. Shortly afterwards, the 7th and 8th Divisions of the German IV Corps found the left flank of the British position at the bridge at Pommeroeul and Conde. It was 5pm. Von Kluck had finally fumbled his way to a decisive point but it was too late. The British beat back the mass assaults as they had elsewhere. The defenders were hardly troubled, and repulsed the initial German attacks easily. If IV Corps' onslaught had been co-ordinated with those of III and IX Corps, von Kluck could have struck a decisive blow. As it was, the piecemeal blows at the British position had achieved little of value.\n\nElsewhere, the outnumbered BEF was coming under increasing pressure. It was only a matter of time before the enemy would be across the canal in force. Even so, nine-and-a-half British battalions had held four German divisions at bay for much of the day.\n\n**British troops, probably from the 1st Northumberland Fusiliers of 9 Brigade, at Jemappes, 22 August 1914. The local population turned out to greet them and feed them. (Mons Museum)**\n\n**A barrier of carts and wagons being constructed across the road from Mons to Quesney on 23 August 1914. Again, the local inhabitants willingly assisted the British troops. (Mons Museum)**\n\n**A house in Mons being barricaded during the evening of 22 August 1914. These troops may well be from the 1st Battalion, Royal Scots Fusiliers. There are a number of soldiers on the upper storeys of the house and even after more than eighty years something of the picnic atmosphere that pervaded the BEF comes through. (Mons Museum)**\n\n**Just before the battle: Two private soldiers of the 1st Gordon Highlanders and two medical orderlies of the 2nd Royal Irish, troops of the 8th Infantry Brigade, pictured on the Mons-Beaumont main road on the morning of Sunday, 23 August 1914. Civilian onlookers, mingle happily with the soldiers. Shortly after this photograph was taken, the area came under attack by the 17th Division. (Mons Museum)**\n\n###### **Blowing the bridges**\n\nThe slow withdrawal from the salient was repeated along the line of the canal, not as a general retreat or retirement, but in a series of independent movements. Platoons and companies pulled back as their positions became impossible to hold. The Northumberland Fusiliers stayed on the canal line to cover the efforts of Capt. Theodore Wright of the Royal Engineers to destroy the Mariette road bridge. Despite repeated attempts, and suffering from two bad wounds, he did not succeed, but received the Victoria Cross for his gallantry under the most persistent German fire. At Jemappes, L\/Cpl. Charles Jarvis also won a Victoria Cross. In spite of heavy German fire, he worked alone for 90 minutes to bring down the bridge covered by the 1st Royal Scots.\n\nThe sound of the bridges being blown was the signal for the British infantry to make its way back to the shorter defensive line that Smith-Dorrien had already earmarked. Not all of the bridges were destroyed; a shortage of exploders left some intact. The Germans failed to capitalise on this. The rough handling they had received at the hands of the BEF had left them understandably hesitant about rushing forward.\n\nThe final attack of the day was launched against the 1st Gordons and 2nd Royal Scots who were still holding the eastern slope of Bois la Haut along the line of the road from Harmignies to Mons. German troops had already advanced through Mons and reached the rear of the British position. They ambushed the 23rd Battery of the Royal Artillery as it descended from the summit. The leading teams and drivers went down, but the Gordons promptly counterattacked just as a further assault developed on their front. It was a critical period, but the Gordons and the Royal Scots, supported by two companies of the Royal Irish Rifles, were well entrenched. Their devastating musketry annihilated the assault. The 75th (7th Bremen) Regiment der Infanterie alone lost five officers and 376 men in a matter of minutes. The German commanders had had enough. The British heard enemy bugles sound 'Cease Fire' all along the line. Unmolested, the 23rd Battery, joined by the 6th Battery, cleared its guns from the slopes and the Gordons and Royal Scots followed them.\n\nBLOWING THE BRIDGE AT JEMAPPES, 23 AUGUST 1914\n\n**As the retreat continued along the line to the west of Mons the destruction of the bridges over the canal would have been a great achievement. Lance Corporal Charles Jarvis set the charges under Nimy bridge, pausing only to send his assistants from the Royal Scots Fusiliers back to their company as the fire became too intense. Having worked for 90 minutes under fire and placing and setting 22 charges, Jarvis successfully demolished the bridge. He was awarded the Victoria Cross.**\n\n**Soldiers of the 1st Lincolns, part of II Corps Reserve, digging trenches near Ciply on the morning 23 August 1914. (Mons Museum)**\n\nThis costly thrust, pressed home with enormous bravery, was the last of its type. There would be a significant change in German tactics. In the future, the field-grey infantry would use their full artillery support. This time the BEF were able to escape to their new defence line without further interference.\n\nII Corps was pleased with itself. The canal bridges had been held until dusk and its soldiers knew that they had inflicted considerable damage on the enemy. They believed that, man for man, they were superior to their foe. 'It was like a Third Division team playing the First Division,' one soldier wrote home later, 'the Germans were beaten thorough.'\n\nThe total British loss during the day's fighting was 1,642 killed, wounded and missing. Of these, 40 came from Haig's I Corps on the right and south of Smith-Dorrien's line. II Corps had taken the brunt of the casualties and most of these were in the 4th Middlesex and the 1st Royal Irish Regiment. The Middlesex lost more than 400 men, the Royal Irish over 300. German losses are more difficult to compute; their system of returns does not allow casualties for a particular action, or even day, to be easily established. Most estimates agree that they were not less than 6,000 and could have been as high as 10,000. Individual German soldiers fought with great gallantry and tenacity, but were defeated by the murderous fire of the British infantry.\n\n## THE RETREAT BEGINS\n\nDuring the evening of 23 August, Sir John French finally accepted that the BEF was facing an enemy with an overwhelming superiority in numbers, and that Lanrezac's Fifth Army was withdrawing. The conclusion was inescapable. If the BEF did not retreat, its flanks would be left unprotected and it would be swallowed as von Kluck's First Army continued its relentless advance. Sir John's belief that he was being left in the lurch by his ally now became a major feature of his thinking.\n\nInstead of immediately issuing orders to his corps commanders, Sir John French sent for their chief staff officers. He announced his intention to pull back the BEF eight miles and instructed the corps commanders to work out the order of retirement among themselves. The only definite instruction was that I Corps would cover II Corps by occupying a line from Feignies to Bavai. Because of this peculiar and time-consuming procedure, Smith-Dorrien did not receive the order until 3am in the morning of 24 August. His headquarters was, in any event, without a working telegraph or telephone line and it was not until 4.30am that the first of his battalions was told to withdraw.\n\n**To German minds, the tiny BEF represented no threat at all. It was the French who were the major enemy and it was deeds of gallantry and crushing victories against the despised French which filled contemporary German newspapers and magazines. Postcards also glorified the defeat of a traditional enemy \u2013 this one shows Uhlans capturing French artillery pieces with remarkable ease. (Michael Solka)**\n\n**The Kaiser's eldest son, Crown Prince Wilhelm, commanded the German Fifth Army and its successful exploits were seized upon with relish by the German press. The capture of some French guns in a spirited action at Longwy on 22 August 1914 \u2013 the day before the Battle of Mons \u2013 inspired this picture which was widely reporduced throughout Germany. (Michael Solka)**\n\nVon Kluck now had some idea of the location and strength of the BEF and decided to force it back to the fortress of Maubeuge where it could be bottled up. He reasoned also that the British would try to retire towards the Channel ports, so he sent II Cavalry Corps under von Marwitz to head them off and decided to envelop the British left in order to stifle any attempt by the BEF to retreat west. Accordingly, he had made little attempt to attack I Corps on the right; it was the left flank that would need to feel the pressure. II Corps would again find itself outnumbered and fighting desperately against a determined and gallant foe.\n\nSir John French had already considered and resisted the temptation to retire behind the walls of Maubeuge. He was only too aware of the fate of Marshal Bazaine at Metz in 1870 who had 'in clinging to Metz, acted like one who, when the ship is foundering, should lay hold of the anchor'. The BEF was instructed to fall back towards Le Cateau.\n\nVon Kluck's attacks on 24 August were a catalogue of errors and misunderstandings. Haig received the orders to retire before Smith-Dorrien. At 4am, 1st Division of I Corps began to move out, followed some 45 minutes later by the 2nd Division. The German IX Corps received no orders to chase I Corps until 8am, by which time Haig had moved his men too far away to be engaged with any realistic chance of being driven into Maubeuge. Despatching von Marwitz's cavalry to the west had been von Kluck's first serious mistake in his attempts to destroy the BEF. This was his second.\n\n###### **II Corps runs into trouble**\n\nSmith-Dorrien already had two major problems before he could extricate his troops. His first was to clear unit transport from the roads, which had been brought up to support the general advance and was the reason why the BEF had marched to Mons in the first place. This was not a simple task and was made worse by the hordes of refugees already on the limited number of roads. The second problem was a manoeuvre which was complicated under peacetime conditions let alone when in imminent danger of enemy assault. Smith-Dorrien had decided to switch the positions of his two divisions. The 3rd, which had taken the brunt of the German attacks the previous day, would retire first. The 5th Division would follow and have less distance to cover when it began its withdrawal on the left.\n\n**To the right of the 4th Royal Fusiliers, the 4th Middlesex and the 2nd Royal Irish Regiment spent 23 August 1914 in a ferocious defence of their position against odds calculated of at least six to one. They faced elements of both the 17th and 18th Divisions as well as the artillery of IX Corps. XL Brigade of the RFA with their 18-pounders caused great havoc amongst the attackers. (Mons Museum)**\n\nIf Smith-Dorrien had received his orders earlier, he could probably have withdrawn in the darkness unmolested. As it was, a severe German bombardment started before dawn, initially against the British positions in the villages of Ciply and Frameries, but rapidly extending along the whole of 3 Division's front as far as Wasmes and Hornu. The lessons of the previous day had not been lost on the German commanders. To the assault infantry, waiting impatiently for the barrage to lift, it seemed as if the British would be easy targets. Ciply and Frameries were occupied by the 2nd South Lancashires and the 1st Lincolns. They were the rearguards of the 7th and 9th Brigades and were supported by 109 Battery of the Royal Field Artillery.\n\n**When the 4th Middlesex finally retired, they left behind one of their drums which was subsequently retrieved from the battlefield and hidden from the Germans. At the end of the war, the drum and three from other units recovered after the battle, were proudly displayed in the Mons Museum. In the inter-war years, they were joined by a further 22 drums presented by regiments who had fought in the battle. The 4th Middlesex drum still hangs, with its companions, in a place of honour in the Museum. (Author's Collection)**\n\n###### **The repulse of the German 6th Division**\n\nThe barrage ceased and the British troops gazed yet again on the now familiar sight of a mass infantry attack. The two battalions were about to feel the full weight of an assault by the entire 6th Division of III Corps. Ordered to hold British attention in the centre while the attack on the British left flank was being prepared, units of the 24th Brandenburg Regiment, the 20th Infantry Regiment and the 64th Infantry Regiment, among others, surged forward, fully confident that the British would have been totally cowed by the artillery fire.\n\n#### THE RETREAT AUGUST-SEPTEMBER 1914\n\nIt was the German infantry who suffered. The bombardment had been peculiarly ineffective in the built-up area. There had been much noise, dust and falling masonry which served merely to provide extra cover for the defenders. As the Germans advanced, they once again ran into the devastating rifle fire of the regular British soldier. 'Tommy seems to have waited,' wrote one German officer later, 'for the moment of the assault. He had carefully studied our training manuals, and suddenly, when we were well in the open, he turned his machine guns on.' The greatest tribute to the musketry of the BEF was the consistent German belief that they faced massed machine guns.\n\n**The 4th Middlesex were ably supported by the 2nd Royal Irish during their action. Amongst the day's heroes was QMS T.W. Fitzpatrick of that regiment. Cut off from his unit, he defended the La Bascule cross-roads on the eastern outskirts of Mons from noon until sunset with 40 men who were mostly bandsmen. Seven were killed and 17 were so severely wounded that they were unable to move. At midnight, Fitzpatrick retired with the 16 others across the hills of Panisel and Bois-le-Haut to rejoin his battalion the next day near Quey-le-Grand. For his exploits he was awarded the Distinguished Conduct Medal. Later commissioned, he finished the war as a colonel. (Mons Museum)**\n\nTo add to the attackers' woes, 109 Battery opened up with shrapnel as the German troops went forward. The advance was shredded, casualties in the assaulting units running at 30 to 40 per cent. Gallantry was not enough. The Germans broke under a hail of rifle fire and shrapnel and fell back to their start line.\n\nIn the lull that followed, the 8th Brigade, made up of the sorely tried 4th Middlesex, 1st Gordon Highlanders, 2nd Royal Scots and 2nd Royal Irish Regiment was able to get away from Nouvelles without difficulty.\n\nA new artillery bombardment followed, and a new mass assault. The result was the same. If anything, German losses were even heavier this time. The 1st Lincolns and 2nd South Lancashires kept firing, the enemy soldiers kept falling. The attack was beaten off, and at 9am the 9th Brigade withdrew from Frameries in good order, the rearguard of 1st Lincolns keeping a tight watch on advance parties of German infantry.\n\nFinally, and after some delay, it was time for 7th Brigade to pull out of the neighbouring village of Ciply. Their route led through Frameries and the retreat of the rearguard through the village proved to be costly. There had been enough time for the Germans to set up machine guns in enfilade among the slag heaps. Originally intending to fire upon the 1st Lincolns, the German gunners wreaked severe damage upon the 2nd South Lancashires as they pulled back. Nearly 300 men were lost in a very short time, more than half of 3 Division's total casualties that day.\n\nGerman losses in the attacks on Frameries and Ciply are not known. They were certainly very high. The 24th Brandenburg alone lost three out of six company commanders, half of its remaining officers and one third of its men. It is without question that the German infantry made no effort to pursue and, in the words of the Official History, 'handled 3rd Division on this day with singular respect'.\n\n###### **The 5th Division face von Kluck**\n\nTo the left of the British line, the 5th Division was less fortunate. They had been largely untroubled during the battle of Mons, their total casualties being less than 400. They would now receive a proper baptism as von Kluck tried to turn the British flank. The 5th Division would be attacked by three German divisions that day \u2013 the 6th Division of III Corps and the whole of IV Corps.\n\nThe defensive line ran north-west from Paturages to Hornu. The 2nd Duke of Wellingtons held the Paturages position and Hornu was covered by the 1st Bedfords and 1st Dorsets. The German artillery began its preparatory bombardment at about 8.30am. Once the artillery fire ceased at about 10am, the same familiar pattern was seen, as the German infantry, drawn from the 5th Division and part of 6th Division of III Corps, came forward in column to be met by rapid rifle fire and close-range shrapnel from the British artillery. The conflict developed rapidly into a confused series of running fire-fights among the slag-heaps and cottages. The individual training given to the British soldier again showed its worth as personal ingenuity and courage came into their own. 37th Battery of the Royal Field Artillery at Hornu fired its howitzers 'as if they were machine guns', while the infantry, in small groups, repeatedly drove back the German attacks.\n\n**The Band of the 2nd Royal Irish Regiment shortly before the war. It was these men who were led by QMS Fitzpatrick in his defence of the La Bascule crossroads. (Mons Museum)**\n\nThe 12th Brandenburg Grenadiers, who had suffered so badly the day before on the canal when they met the Royal West Kents, had another testing time. They came under fire not only from the British, but from their own artillery as well. As a result, the regiment seems effectively to have been wiped out.\n\nBy about 11am, 5th Division began its retirement, pulling back as the 3rd Division on its right began its withdrawal. There were some incidents. The 20th Infantry Regiment, used in the attack on Frameries, was able to infiltrate the British position, but its ambush of the Dorsets' regimental transport was remarkably unsuccessful, lacking both energy and enterprise, and the transport escaped unscathed.\n\nMore costly was the failure to get withdrawal orders to the 2nd Duke of Wellingtons. Together with their artillery, and reinforced by two companies of the Royal West Kents, they held on to their original positions and came under heavy German artillery and small arms fire. Losses rose, but when the Germans finally came forward en masse to deliver the final blow, they received a devastating hail of fire which stopped them in their tracks and allowed the defenders to slip away. Six German battalions had been held at a cost of nearly 400 casualties.\n\n**German accounts of the battle of Mons refer time and time again to the British use of machine guns; in fact, the first British provision of machine guns was precisely that of the German Army \u2013 2 per battalion. The army had made representations before the war to increase the numbers of machine guns to 6 per battalion. This extravagant request was promptly refused by the Treasury (the Chancellor of the Exchequer was Lloyd George) and the Army Vote was also decreased. Haig and other generals were thus forced to concentrate their efforts on highly specialised training in an attempt to avoid the limitation placed on tactics by inferior firepower. In musketry this led to what the press called the 'mad minute' \u2013 each man, both infantry and cavalry, firing 15 aimed rounds per minute. Some soldiers were even credited with 30 rounds per minute. The effect of 15 rounds per minute from each of the soldiers in this firing line can easily be imagined. (Author's Collection)**\n\nIt was on the left flank that matters became really serious, with a situation developing during the morning which could have destroyed the whole of II Corps. The independent 19th Infantry Brigade and the Cavalry Division had retired without mishap earlier in the day. Apart from some desultory firing before first light, there had been few signs of the Germans \u2013 not surprisingly, because von Marwitz's cavalry had left to patrol the routes to the Channel ports. By 11.30am, at just the time when the 2nd Duke of Wellingtons was beating back their last attack, the Cavalry Division and the 19th Brigade were back behind the left of the 5th Division.\n\n**By the morning of 24th August 1914, German troops were in full occupation of Mons and were to remain there until the very end of the war. Taken from an upper floor window of a house in the Grand-Place, this photograph shows troops of the German 18th Division, IX Division, IX Corps, in the town that same morning. The bandstand has now vanished but the ambulances outside the Hotel de Ville indicate its use as a temporary hospital. (Mons Museum)**\n\n**Numbers of wounded British soldiers were rescued by the civilian inhabitants of Mons who hid them from the invader. Posters \u2013 this one dated 25 August 1914 \u2013 warned the population of dire retribution if they sheltered British soldiers. This one demands that all English soldiers capable of walking are to report to the town hall with their arms before 6pm. If soldiers were so badly hurt that they could not report in person, the citizen finding them was to make a declaration to that effect. Failure to do so would result in a considerable communal fine upon the town. Despite this threat \u2013 and worse \u2013 the local population helped many soldiers to evade capture and eventually rejoin their units. A few stayed hidden in the vicinity throughout the whole period of the war. (Mons Museum)**\n\nAt this point the full weight of IV Corps artillery opened up from the Quievrain area against the now exposed flank of the British line. Von Kluck had at last got the right hook into operation, and the whole of IV Corps, consisting of the German 7th and 8th Divisions, was taking part. Von Kluck's aim was to swing round against Elouges, smash against the flank of the 5th Division and send it reeling towards Maubeuge. The steady withdrawal of the 5th Division could have been turned into a rout.\n\nThe commander of the 5th Division, Maj.-Gen. Sir Charles Fergusson, sent for help to the Cavalry Division. Allenby responded swiftly, sending the 2nd and 3rd Cavalry Brigades, supported by 'D', 'E' and 'L' Batteries of the Royal Horse Artillery. In the interim, Fergusson despatched his last remaining reserves, the 1st Norfolks, 1st Cheshires and 119 Battery of the RHA, all under the command of Col. Ballard of the Norfolks, to face the four German divisions advancing towards the gap between Elouges and Audregnies about two miles away. The orders were simple: they must stop the German advance.\n\n###### **The Flank Guard action at Elouges**\n\nIn the following action, two British infantry battalions, four batteries of artillery and parts of two cavalry brigades faced 24 battalions of German infantry and nine artillery batteries on a battlefield of some six square miles. If von Marwitz's cavalry had also been available to the Germans, then 5th Division would have been annihilated.\n\nThe British infantry had no time to dig in. The Cheshires took the left of the line, the Lincolns the right, with 119 Battery along a little ridge which gave them a clear field of fire across the cornfields towards Quievrain. The position was soon reinforced by the arrival of the 2nd Cavalry Brigade. The 9th Lancers and the 4th Dragoon Guards took up a position away on the left of the British line.\n\nAt 12.45pm, at least six battalions of German infantry, supported by six or more batteries of artillery, started their advance in two columns from Quievrain and Basieux. Their route led them across the front of the British cavalry and the target they presented was too much for the cavalry to resist. An order to attack the enemy infantry, if necessary by mounted action, became the signal for an all-out charge.\n\nThe 9th Lancers, with two troops of the 4th Dragoons, immediately galloped towards the enemy who were about 2,000 yards away. They came under fire as the German infantry reacted and the charge was brought to an ignominious halt by a wire fence bordering a sugar factory. As men and horses milled about, they were an easy target for the German guns. Most of the cavalry went to the right, some taking cover behind the sugar factory and some slag heaps, the others galloping across the front of the British infantry before re-forming. The gesture \u2013 one which Wellington would certainly have recognised and abhorred \u2013 cost 250 men and 300 horses and had little effect on the German advance which continued with hardly a pause.\n\nNonetheless, the action, much magnified and distorted by the popular press, became an important ingredient in boosting morale in Britain. It became known as the 'Charge to Save the Guns', harking back to earlier wars in which wire fences and machine guns were unknown.\n\n**A rare photograph of wounded British survivors of the battle with their Belgian rescuers. Taken only a few days after the flight, the location is the Hornu-Wasmes mine. The colliery sick bay was used as an operating theatre and hospital and some 40 British officers and men were hidden there. The Germans did not discover the British wounded until early September by which time some had already made an escape. Others got away subsequently, being passed down escape routes which included, amongst others, that operated by Nurse Edith Cavell. (Mons Museum)**\n\nA mile further to the west, the 3rd Cavalry Brigade was north of Angre. Supported by the 13-pdrs. of 'D' and 'E' Batteries of the RHA, they also charged the right flank of the German 8th Division, the 4th Hussars leading, supported by the 5th Lancers. Their losses were not as heavy but, again, they only marginally delayed the German onslaught.\n\nThe six guns of 'L' Battery, positioned to the left of the Cheshires, came into action at 1pm and began to sweep the advancing Germans with enfilading fire. At 2,000 yards range, the shrapnel tore great holes in the masses of enemy infantry. Hardly surprisingly, the German infantry stopped short and retreated. Re-formed, they tried again, displaying remarkable bravery, but with the same shattering results. All along the line, the pattern was repeated. By all military logic, the British troops should have been quickly steam-rolled into oblivion, but they continued to fight against odds which were at least six to one.\n\n**Mons remained under German occupation until the end of the war and the museum provides a valuable record of the period. There are sections devoted to the French, Belgian and Italian armies as well as the German invaders. (Author's Collection)**\n\nChecked on the left, the Germans tried to work themselves further south, outside the range of the defenders. On the right of the British line, the assault from the 7th Division began to press more heavily. German tactics were altering in response to the stubborn defence; counter battery fire against the RHA was largely inaccurate, but was causing problems. 'L' Battery itself was attacked by four German batteries, but suffered only 12 casualties. On the right, 119 Battery fared rather worse. It lost 30 men, one-quarter of its strength, and came under fire from three German batteries as well as machine guns at close range.\n\nBy mid-afternoon it was clear that the position could not be held; the rest of 5th Division had retired safely and it was time for the rearguard to disengage. 119 Battery withdrew under intense fire, its guns being manhandled away by the gunners and a group of 9th Lancers. Two Victoria Crosses were won at this stage, one going to the battery commander, Maj. Alexander, the other to Capt. Francis Grenfell of the 9th Lancers.\n\nRemarkably, the artillery did not lose a single gun in the engagement, even though every battery left under enemy fire. In the three hours in which they were in action, each battery fired about 450 rounds and caused havoc among the enemy troops.\n\nBy 5pm, with the artillery safely to the rear and the cavalry falling back, Col. Ballard ordered the infantry to retire. The Norfolks managed to withdraw but the message, sent three times, never reached the Cheshires. They continued to hold their line, stubbornly resisting for a further three hours, and causing heavy casualties among the German infantry. As the German attacks continued, the battalion was slowly split into small groups some of which were finally overwhelmed by ever-increasing numbers. Others managed to slip away, but the action cost the Cheshires some 800 casualties, killed, wounded and missing.\n\nII Corps, led by an exceptional soldier, Horace Smith-Dorrien, had once again completely frustrated the enemy's attacks. The British casualties for the day were more than 2,000, of which 1,650 were in the 5th Division. I Corps had again escaped most of the fighting, its losses amounting to about 100 men. On the German side, the attack on the rearguard alone is estimated to have cost some 3,000 to 5,000 men; combined with the costly efforts on the right of the British line, the total loss has been estimated at anything between 6,000 and 10,000 casualties. The disappearance of some regiments from the order of battle suggests that it may have been far worse.\n\nThere were tangible results to the day's fighting although they were not immediately apparent. Von Kluck's advance had been slowed. During the whole day, his army had covered just over three miles and the BEF had eluded his grasp yet again. Despite an overall superiority in men and in guns, he had failed to destroy the BEF. It was an error for which the German Army would pay a high price.\n\nHungry, exhausted, the British troops stumbled into their rest areas. New orders had already arrived from GHQ: 'The Army will move tomorrow, 25th inst,' began Operation Order No 7, 'to a position in the neighbourhood of Le Cateau, exact positions will be pointed out on the ground tomorrow.' The Retreat continued.\n\n**Many photographs purporting to show British troops on the Retreat from Mons were, in fact, taken at the time of the Battle of Loos by Paul Maze, a French liaison officer. This, however, is the genuine article. An RFA crew, with 18-pounder gun and limber, shortly after being in action \u2013 note the empty shell cases.**\n\n**This helpful Boy Scout, with the British cavalry, is carrying a Lee-Enfield rifle and wearing an ammunition bandolier as well as other British Army-issue items \u2013 one of which appears to be an infantry webbing belt. This suggests a date during the retreat itself. The town is clearly in a French-speaking area and the tramlines suggest a substantial community. Boy Scouts were not generally employed as combatants but, equipped as above, this one would incur the considerable wrath of any German soldier who met him and would hardly escape a firing squad. (Private Collection)**\n\n###### **Retreat to Le Cateau**\n\nBy the morning of 25 August, Sir John French thought only of saving the BEF in accordance, as he believed, with Kitchener's instructions. Cooperation with the French was forgotten. The British Commander-in-Chief entertained some astounding ideas. One plan was to retreat pell-mell to Le Havre, turning the port into a fort, until the BEF was ready to fight again. That this manoeuvre would have required a march across the whole front of von Kluck's advancing troops is sufficient comment on its practicability.\n\nThere was one piece of good news. Kitchener had sent the 4th Division as reinforcements. It arrived in the battle area on the 24th, minus its cavalry, cyclists, signallers, field ambulances, engineers, ammunition columns and heavy artillery. Nevertheless, it was a welcome and valuable addition to the steadily diminishing numbers of the BEF.\n\nRetreat to Le Cateau produced a problem. To the rear of I and II Corps was the Mormal Forest, roughly ten miles long and four miles wide. Two roads ran from east to west across the wood, only a few rough tracks ran from north to south, the direction in which the BEF was travelling. The dangers in trying to squeeze along them were obvious. The only choice was to use the roads on either side of the wood. To move the whole of the BEF down one side or the other would cause even more chaos and confusion on the crowded roads, create delays and possibly expose the BEF to another onslaught from the German First Army.\n\nSir John French decided, with some misgivings, to split the BEF in two; I Corps would retreat down the eastern side of the wood, II Corps would withdraw down the west and the BEF would be reunited at Le Cateau, 25 miles to the south.\n\nAdditionally, there were supply problems. Everything had been planned in the expectation that the BEF would be advancing northwards, not retreating southwards. The forward dumps had been overrun and the lines of communication disrupted. With limited telephone facilities and constant movement, it became impossible to discover where individual units were and what they needed. In a brilliant piece of improvisation, Maj.-Gen. Robertson, the BEF's Quartermaster-General, ordered his men to off-load supplies at every crossroads and junction along the line of the retreat.\n\n**The stand by Smith-Dorrien's II Corps at Le Cateau is one of the great actions of the British Army. It was Allenby, admitting that his cavalry was 'pretty well played-out' whilst reporting that von Kluck's infantry were only a few hours distant, that decided Smith-Dorrien. Allenby proved to be an exceptional cavalry commander although he and his small staff suffered along with everyone else during the retreat. Most of their personal kit was lost and they could muster only one razor between them for several days! (Author's Collection)**\n\nBoth Haig and Smith-Dorrien faced difficulties. I Corps was faced with a meandering route which first crossed the River Sambre and then re-crossed it on two later occasions. The problems were exacerbated because the road was crammed with soldiers from Lanrezac's Fifth Army as well as hordes of refugees. In the stifling August heat, I Corps averaged only about two miles per hour as it trudged wearily onwards. To add to Haig's woes, he was suffering severely from diarrhoea, a condition which hardly helped his concentration or normally imperturbable reactions.\n\nFortunately, there was little sign of the foe. The BEF was eight miles south of Von Kluck when it halted the previous evening. The First Army commander was still essentially mistaken about the movements of the BEF and believed he had succeeded in pushing it towards Maubeuge. Further, he assumed that he had forced the whole of the BEF, six divisions as he believed, in that direction. The location of Haig's I Corps was unknown to him. Reacting to a series of confusing and contradictory reports, von Kluck gave orders for the First Army to wheel south-east. It was a movement which in fact took them behind the rear of II Corps and towards the flank of I Corps.\n\n###### **Landrecies**\n\nApart from odd skirmishes with advanced patrols, I Corps spent another day out of reach of the enemy. As evening approached, the tired troops began to reach their billets. The Corps HQ went to Landrecies, where there was a bridge over the Sambre, sharing the village with the 4th (Guards) Brigade. The 6th Brigade settled down further east at Maroilles, two miles from another crossing point of the Sambre.\n\nFor the advance elements of the German III and IV Corps, it was also a tiring day. They, too, were looking forward to settling down in billets for the night. The 14th Regiment der Infanterie of III Corps was anticipating a quiet time at Landrecies; the 48th Regiment der Infanterie was equally anxious to reach Maroilles.\n\nShortly after Haig reached Landrecies at about 4pm, a flood of refugees arrived and claimed that German Uhlans were close behind them. A check revealed no sign of the enemy, but prudently, No 3 Company of the 3rd Battalion Coldstream Guards, with two machineguns, went to the north-west to guard the road from Le Quesnoy. The remainder of the brigade prepared defences in the town itself.\n\nAt Maroilles, two troops of the 15th Hussars were guarding the bridge when the first patrols of the 48th Infantry arrived. There was a fierce skirmish which lasted nearly an hour before the Germans brought up a field gun. The Hussars fell back and were then reinforced by the 1st Royal Berkshires. It was 7pm and getting dark. In a sharp action involving savage hand-to-hand fighting along a stone causeway leading to the bridge, the Berkshires lost about 60 men. The Germans held on to the bridge and the frustrated Berkshires finally abandoned their attempts to take it.\n\nIt was dark at Landrecies when Capt. Monck, in command of the guard on the road, heard the sound of approaching troops. A challenge got a reply in French. A moment later the Coldstream were charged by the Germans. In the confusion, the Germans managed to snatch a machine gun but it was promptly recaptured.\n\nNo 1 Company of the Coldstream Guards came up as reinforcements. The German attacks continued, but each charge was thrown back by the steady fire of the Guards. By careful use of cover, the Germans eventually enfiladed the Coldstream line, forcing the Guards back towards a line of cottages. A small haystack was set on fire and by its light a single German field gun was able to shell the British position at close range. Pte. West of the Coldstream Guards twice dashed out under heavy fire and managed to put out the flames.\n\nThe fighting went on until after midnight, ending finally when a howitzer of 60 Battery arrived and disabled its German counterpart with its third round. A diversion was mounted by the Irish Guards and, eventually the disgruntled Germans withdrew to the southern edge of the Mormal Forest. Casualties on both sides were very similar. Twelve Coldstream Guardsmen had been killed, 105 were wounded and there were seven missing. The German official figures admitted 127 casualties, but with a higher proportion of killed. To the west of the town, the 2nd Battalion Grenadier Guards were involved in a series of confused actions. When the dawn came, their total losses amounted to seven men.\n\nThe affair at Landrecies was no more than a skirmish, but it was to have a profound effect. In the town itself, the belief rapidly spread that it was surrounded. Haig, normally calm and placid, reacted strongly to the reports he received. After giving orders for the defence of the town, he instructed his staff to burn all secret papers and resolved to sell his life dearly. After the first excitement had subsided, he reasoned that the enemy line was probably thin and decided that the Corps staff should make a bid to escape. This was sensible. Little would be gained if the whole of the Corps staff were taken prisoner.\n\nHaig, with a combination of luck and judgement, was able to rejoin the main body of I Corps with his staff at about 1.30am. He immediately set about rescuing the 4th (Guards) Brigade from what he believed to be a difficult situation. The 1st Division was ordered to mount a rescue operation at dawn. He then telephoned GHQ to report the attack and he assessed the situation as 'very critical'. Two hours later, at 3.50am, he spoke again to GHQ, asking this time for Smith-Dorrien's II Corps, eight miles away on the other side of the Sambre, to be sent to help him. Haig believed that I Corps was about to be attacked in force.\n\nSmith-Dorrien was unable to help. He had already decided that his only course of action was to stand and fight at Le Cateau, which was precisely where von Kluck's army was marching.\n\n## LE CATEAU\n\nThe progress of II Corps along the western side of the Mormal Forest was extremely difficult. The 3rd Division on the left of the Corps were harassed by von Kluck's right wing, which pushed forward in yet another attempt to outflank the weary BEE Throughout the day, the rearguards of the 3rd Division, the 19th Infantry Brigade and the Cavalry Division fought a series of running engagements against an unenthusiastic enemy. Von Kluck's divisions were also suffering from the punishing marches along the sun-baked roads.\n\nThere was a sharp action at Le Quesnoy during the day when the 1st and 3rd Cavalry Brigades engaged in a fierce skirmish. At Solesmes, a small town on the route, there was considerable confusion as streams of refugees, transports and troops eddied around the narrow streets. A determined German attack at this point would have been disastrous; as it was, the 2nd South Lancashires and 1st Wiltshires, the rearguard of 7th Infantry Brigade, were able to discourage the tentative German IV Corps from approaching too closely. By mid-afternoon, the 3rd Division had disengaged, covered by two brigades of the 4th Division, leaving the Germans to settle down in and around Solesmes. The day's work had cost the cavalry more than 100 casualties, the infantry a further 350.\n\nThe 5th Division did not see the enemy at all, but their route to Le Cateau was thronged with civilian refugees, French soldiers and wagons of all types. Worst of all was the sweltering heat, and many men dropped out of the ranks. As evening came, there was a heavy thunderstorm. The troops who straggled into Le Cateau were wet, weary and very hungry.\n\n**Some of the British infantry positions at Le Cateau were far from ideal. This photograph purports to show part of the battle but is, without doubt, not the genuine article, either being posed or a film still. It nonetheless gives an idea of the actual fighting. (Private Collection)**\n\n###### **The decision to fight**\n\nThat evening GHQ ordered a continuation of the retreat south-westwards towards Peronne. Smith-Dorrien well knew the exhausted condition of his men and had grave doubts about how much more marching they could take; he had no information as to the location of Allenby's Cavalry or Gen. Snow's newly arrived 4th Division. He knew nothing of the rearguard action of the 3rd Division. None of them would be accounted for until midnight. He was unaware of what had happened at Landrecies, and assumed that I Corps would be close to Le Cateau covering his right flank in accordance with previous orders. The conviction grew upon him that he should consider fighting instead of retreating. A thorough infantryman, he felt that 'a stopping blow under cover of which we could retire,' was not merely possible, but essential.\n\n#### THE BATTLE OF LE CATEAU - 26 AUGUST 1914\n\nSmith-Dorrien's belief that he should stand was reinforced when Allenby arrived to see him at 2am. Allenby's message was stark. His cavalry brigades were scattered, with men and horses 'pretty well played out'. Allenby believed that unless II Corps could move at once and get away under cover of darkness, they would be forced to fight at dawn because the enemy were so near.\n\nSmith-Dorrien sent for Gen. Hamilton, commander of the 3rd Division. Hamilton responded bluntly when asked if he could move his Division without delay. The earliest he could start would be 9am. Smith-Dorrien knew that what was true for the 3rd Division would apply also to the 5th and the 19th Infantry Brigades. He asked Allenby if he would be prepared to take orders and act as part of II Corps. Allenby assented at once. 'Very well, gentlemen,' Smith-Dorrien said calmly, 'we will fight and I will ask Gen. Snow to act under me as well.' Well aware of the fact that he was disobeying specific orders from GHQ, Smith-Dorrien's decision was a brave one. When he eventually learned that I Corps was still eight miles away and was continuing its retreat at first light, it was too late to change the orders again.\n\nIt was still a close thing. Smith-Dorrien personally went to 5th Division Headquarters to instruct Gen. Fergusson, while Allenby and Hamilton passed on the new orders to their own men. Both Fergusson and Snow were relieved at the decision to make a stand, but time was desperately short. Snow did not receive the changed instructions until 5am. Some units were already preparing to move out; others had no chance to make any form of defence other than hastily scraping shallow rifle pits in the rock-hard soil.\n\n###### **The battlefield**\n\nThe battlefield of Le Cateau lacked the strings of mining villages and slag heaps as at Mons. It was open, undulating countryside which provided impressive fields of fire. Mons was an infantry battle. Le Cateau would be one in which artillery played the dominant role.\n\n**Once news of the encounters with British troops reached Germany, there was a flood of illustrations depicting the battles. This spirited picture shows the German Cavalry \u2013 inevitably Uhlans to the British Press \u2013 charging 'English Highlanders' \u2013 an imaginative concept \u2013 and is typical of many of this period. (Michael Solka)**\n\nII Corps' position can be described as a flattened arrow head with a length of about ten miles, the point of which was at Caudry along the line of the Le Cateau to Cambrai road. To the left, 4th Division, composed of the 10th, 11th and 12th Brigades, faced north-west, forming what is technically known as a refused flank. Here they were in contact with the French Cavalry Corps under the command of Gen. Sordet who, in turn, maintained a link with Gen. d'Amade's Territorials who covered the gap all the way to the French coast. The French played an important, if rarely acknowledged, role that day by fighting off the German II Corps and preventing it from joining the main battle.\n\nThe centre of the line, from just east of Beauvois and running through Caudry and Inchy, was held by the 3rd Division facing slightly north-east. 5th Division continued the line which followed the road to Le Cateau and eventually curved round to face due east at the crossroads before the town itself. It was not a bad position, but the right flank was open. The 1st Duke of Cornwall's Light Infantry and two companies of the 1st East Surreys had been sent there to link up with Haig's I Corps.\n\nBehind the infantry was the cavalry. The 4th Brigade was at Ligny behind the centre of the line. The 1st Brigade and some of the 2nd was on the right and the 3rd Brigade was away to the east trying to make contact with I Corps.\n\nAgainst them, von Kluck sent four divisions drawn from III and IV Corps, the three divisions of II Cavalry Corps and, later in the day, the artillery and a further division of IV Reserve Corps. He still believed he was facing a BEF consisting of six divisions, all of which were in front of him. Obsessed with the idea that the BEF would try to make for the Channel ports (a not unshrewd reading of Sir John French's mind), he was convinced his enemy would be aligned in a north-south position between the Sambre and Cambrai. The encounters at Landrecies and Maroilles had done nothing to alter this belief.\n\nVon Kluck's plan was simple. Both flanks would be attacked and the British crushed like a walnut between a pair of crackers. Von Kluck therefore ordered II Corps to Cambrai; von Marwitz's II Cavalry Corps, recalled from its wanderings in the west, was to hold down the left of the British position until the arrival of IV Reserve Corps. In the centre, IV Corps would divert attention from III Corps as it swung across to envelop the right of the British line. The result would be the simultaneous rolling up of both flanks and total destruction of the British force. As at Mons, however, von Kluck's troops ran piecemeal into the British positions and the battle was done before their full strength could be used.\n\n###### **The battle begins**\n\nDawn came with a thick mist. It was 26 August, the anniversary of the battle of Crecy in 1346, when the English archers had each fired 12 arrows a minute to destroy their enemy; in 1914, the Short Lee-Enfield rifle would do the same task with 15 aimed rounds a minute.\n\nThe battle of Le Cateau began at about 6am as German artillery began a bombardment all along the British line. The lessons of Mons had been absorbed. The British infantry would be shelled into submission before the main assault began. The British artillery reacted swiftly, the guns coming right forward to protect their infantry. In the centre, the British troops strung out along the Le Cateau-Cambrai road endured stoically as the enemy artillery sought to pin them down while the main assaults hit the flanks.\n\nOn the right flank, the battle was especially fierce. It was here that the Duke of Cornwall's Light Infantry and East Surreys had tried to link up with I Corps. Having found no sign of Haig's men, they were formed up and ready to march off when they were surprised by German infantry. The Germans, from the 7th Division of IV Corps, had taken advantage of the dawn mist to infiltrate their way into and through Le Cateau itself. The ambush developed into a savage action as the British, fighting stubbornly in small groups, fell back along the Selle Valley towards the high ground south-east of the town. The 3rd Cavalry Brigade came up in support, as did 'L' Battery of the RHA. By noon, the detachment had rejoined the main body of its parent 14th Brigade. The action had cost 200 casualties, but the Germans made no attempt to pursue their quarry as they moved away from the valley. The German infantry pressed on along the Selle Valley. The mist hid their movements as they pushed out to occupy higher ground to the east, an area which would provide an excellent platform for the German artillery to enfilade their enemy.\n\nIt was not long before a very serious situation, fraught with danger for the British, developed. A successful German attack to the west of Le Cateau allowed them to move more artillery up to the high ground north of the Cambrai road. The British right flank was held by the 13th, 14th and 19th Brigades; the order to stand and fight had not reached brigade level when the German attack started and they were brought to combat in positions they had not anticipated holding.\n\nThe British artillery, on the ridges behind the infantry, came as close as it could to the front line, firing from just 1,200 yards into the German positions. They provided a prime target for their German counterparts who began to concentrate on destroying the British guns. The 11th Battery RFA, in particular, suffered heavily. By 10am the battery had lost all of its officers and only one gun was still in action. It was then that heavy German infantry assaults developed, huge masses pushing forward in overwhelming numbers.\n\nBy about 11am, the British positions along the ridge were enfiladed on both sides. The 2nd Suffolks and the 2nd King's Own Yorkshire Light Infantry, together with two companies of the 2nd Argyll and Sutherland Highlanders endured a hail of artillery and machine gun fire but, with an intense and desperate courage, they drove back every German assault.\n\n**The BEF was poorly equipped with heavy artillery \u2013 another tribute to official parsimony \u2013 in comparison with its enemy and its ammunition was very largely shrapnel rather than high explosive. At full recoil this is one of the 16 60-pounders that accompanied the BEF to France. The total BEF establishment of artillery pieces of all calibres was 490 guns \u2013 less than the Germans concentrated for the siege of Namur alone! (Private Collection)**\n\nThe Royal Artillery gunners, firing 'as calmly as if they were on the practice range', contributed to the maelstrom of defensive fire. Every attempt by the German infantry to develop a firing line and assault position close to the British was thwarted. The 122nd Battery was involved in an extraordinary incident when a platoon of German infantry came over a ridge in close formation. The battery fired a single salvo and the platoon was obliterated.\n\n**Supplying the retreating BEF was a nightmare. The Quartermaster-General, Sir William Robertson, ordered his men to dump supplies at crossroads and along the routes down which the BEF would pass. This simple and inspired action saved many men who would otherwise have simply given up. Robertson holds the unique record of having held every rank in the British Army from cavalry trooper to field-marshal. (Author's Collection)**\n\nBy a combination of fighting skill, quite exceptional courage and a fierce determination not to give way, the two infantry battalions forming the right flank of the British position held out against nine German infantry battalions, their supporting machine gun companies and the greater part of the artillery resources of three divisions.\n\nAn attempt to send the 2nd Manchesters and the remainder of the 2nd Argyll and Sutherland Highlanders forward as reinforcements virtually foundered as they came under intense German fire from the right flank. By noon, 11th Battery RFA had finally succumbed, its last gun being put out of action; the German infantry, still huge in numbers, was steadily and stealthily getting closer; the enemy's artillery fire was sweeping the British positions; and machine gun posts were raking the survivors. By every yardstick, the position was impossible to hold, but after six hours of intense fighting the line was still unbroken.\n\nIn the centre of II Corps' position, in the area of Caudry, there were no serious German attacks during the morning. Caudry had been the target of intense German shelling, but what little enemy infantry activity there was was quickly discouraged by the British defenders. The Germans succeeded briefly in occupying Inchy but were swiftly driven out. Throughout the morning, British casualties came to something under 200 men.\n\n###### **The left flank**\n\nOn the left flank, it was a different tale altogether. It was here that the other half of the pincer envelopment was planned and where Gen. Snow's 4th Division, the new boys who had arrived in France just one week earlier and thus missed the advance to Mons, the battle there and its exhausting aftermath, had their first real taste of action. As on the right, the British were surprised. The 4th Division, without cavalry or cyclists, had little ability to mount far-ranging patrols. The area between them and Cambrai in the distance was patrolled by French cavalry; the first indication that anything was amiss was the sight of two distant French cavalrymen turning and galloping away. Almost immediately, heavy machine-gun and artillery fire hit the British positions.\n\nThe 1st King's Own were caught in the open, forming up in close formation to move to a new position, and their casualties were heavy. Lying down, their formation was such that only the front rank of each platoon could fire at their opponents, troops of the 2nd Cavalry Division. The King's Own lost 400 men within minutes, but were aided by the intervention of the 1st Hampshires to their right. Two companies of the 1st Royal Warwicks (among them a young subaltern by the name of Bernard Law Montgomery) forced their way forward in a brave but expensive gesture of support.\n\nThe 4th and 7th Jaeger Battalions quickly tried to outflank the British position. Supported by dismounted cavalry, the Jaegers swarmed forward, forcing the King's Own and the Hampshires to retire. Before the Hampshire pulled back, though, they demonstrated once more the telling power of British rifle fire. In open country, just 1,000 yards from their line, a German field battery set up position. In less than a minute, the concentrated fire of the Hampshires forced its retirement.\n\nEven so, the Jaegers pushed hard. Moving on towards the battalions on the left of the position, the 2nd Lancashire Fusiliers and 2nd Royal Inniskilling Fusiliers, there was a fierce fire-fight which lasted some two hours. Supported by 21 machine guns, five batteries of horse artillery and the dismounted cavalry \u2013 who were not, according to some observers, over-enthusiastic about fighting on foot \u2013 they came very close indeed to rolling up the British position.\n\nThe 7th Jaegers enfiladed the Lancashires and moved on through a cornfield, unaware that the Inniskilling Fusiliers were waiting for them. Within seconds, the Jaegers were reeling back, with 47 dead lying amidst the ripened corn while the Inniskillings suffered not a single casualty.\n\nThe artillery of the German IV Reserve Corps was hurriedly brought up to contibute to the bombardment, which reached a new crescendo before a final, desperate assault. The familiar German mass, firing from the hip, surged forward with stubborn and determined gallantry only to be mown down by a torrent of fire. The Germans showed outstanding bravery, but were repulsed with heavy casualties. The defenders themselves suffered severe losses in an action which was savage in its intensity. By 10am, in the words of the British Official History, the two battalions 'had fought 2nd Cavalry Division to a standstill'.\n\nIn the same area, to the right, the 1st East Lancashires, 1st Rifle Brigade, 1st Somerset Light Infantry and 1st Hampshires, who comprised the 11th Brigade, all suffered badly under heavy artillery and small arms fire. They, too, held their ground in the face of repeated assaults, even venturing counterattacks. Sometimes driven from their line by savage shrapnel, they regained every position.\n\n###### **The retreat**\n\nThere was now a mid-morning lull, punctuated by exchanges of fire between the opposing artillery and some unenthusiastic and ineffective attempts by the German infantry to go round the left flank of the 4th Division. Snow realised, however, that the time was fast approaching when he would have to disengage. His casualties were steadily mounting and the Germans would inevitably overwhelm his line.\n\nOn the right flank, affairs were becoming even more perilous. Although they had suffered heavy losses, the German infantry displayed a consistent gallantry and had, by late morning, enfiladed both flanks of the 5th Division. The German attack had been reinforced by their 5th Division of III Corps, and the British forces \u2013 four infantry battalions and two Royal Field Artillery brigades \u2013 faced the onslaught of at least 12 German infantry battalions and the combined artillery of three divisions. Under massive attack from the front and both sides, the situation shortly after noon was grim indeed. It was not surprising that some British soldiers began to slip away. Gen. Fergusson was acutely aware of the danger. There was no chance of reinforcement; the only choices were to die where they stood, or retire, and the retirement had to begin soon if it was not to become a rout.\n\nSmith-Dorrien agreed and issued orders to all of II Corps to retire. The withdrawal would take place from right to left, 5th Division first, then the 3rd and finally the 4th. It was one thing to issue the order; it was another for it to reach the troops involved. With the very few available field telephone wires cut, orders had to be delivered personally. Messengers died as they tried to get forward through a torrent of shrapnel and bullets. It took 20 minutes for the retirement order to travel less than two miles, the distance between the headquarters of II Corps and 5th Division. It was another hour before it reached the front line units, and some never received it at all.\n\n###### **Saving the guns**\n\nThe 5th Division units not involved on the ridge retired with little difficulty. The 15th Brigade and three battalions of the 14th Brigade pulled back without interference. It was a different story for the troops on the ridge. The first priority was to save the artillery. The guns were almost directly in the infantry's firing line, their crews having suffered so badly that none of the batteries were any longer capable of sustained action. It was now that some quite remarkable feats of gallantry occurred. The 11th Battery teams galloped up during a lull in the fighting and retrieved five out of six of their guns. The sixth team was shot down by startled German infantry. The 80th Battery also retrieved five of its guns. The 52nd Battery had to abandon all of its guns as every team was brought down in the attempt.\n\nSAVING THE GUNS AT LE CATEAU, 26 AUGUST 1914  \n **On the ridge to the south-west of Le Cateau the XV Brigade guns retreated under heavy fire. The 37th Battery had pulled back four out of six howitzers and no longer had sufficient horses and limbers to return for the remaining guns. Once back at the rear, Captain Reynolds, the battery commander, called for volunteers to fetch the guns. Two teams set off, only one returned. For their gallantry Captain Reynolds and Drivers Luke and Drain received the Victoria Cross.**\n\nShortly afterwards, 122 Battery gathered their guns in a dashing action which 'brought the infantry cheering to their feet'. As the gun teams were sighted by the enemy, they were engulfed by shrapnel and bullets. The officer was killed and one team went down in a tangle of shrieking horses. Three guns were limbered up, two of them rattling away under a hail of fire. The third was shot to pieces, the horses being machine-gunned from German positions north of the Cambrai road. Four guns were abandoned, surrounded by dead and wounded artillerymen and horses.\n\n**Every day of the Retreat from Mons brought more casualties. This one stands beside a Quarter-master sergeant of the Royal Army Medical Corps. (Private Collection)**\n\nThe guns of the 123rd and 124th batteries which were even further forward could not be saved. Their surviving crews removed the breech blocks and smashed the sights before abandoning them to the enemy.\n\nThe 37th Battery was able to claim, under persistent enemy fire, four out of six howitzers in its first foray. They could do no more as enemy action had already cut down the number of horses they had available. Once back at the rear, the battery commander, Capt. Reynolds, called for volunteers to rescue the two abandoned howitzers. Two teams returned. Both howitzers were limbered up under intense fire from German infantry, now only 100 yards distant. One team was brought down before it could move, but the other galloped clear with its howitzer. It was an act of astonishing gallantry and Capt. Reynolds, along with Drivers Luke and Drain, the survivors of the successful team, received the Victoria Cross. Driver Coby, the third member of the team, had been killed on the approach to the guns.\n\nThe withdrawal of the infantry was even more perilous. The Suffolks and the supporting Argylls and Manchesters faced enemy infantry, including the 26th Infantry at their front, their right and the rear. Overwhelmed, the survivors still managed to retire in relatively good order, fighting their way back and preventing every effort by the Germans to work their way round the flank.\n\nThe 2nd KOYLI, on the left of the Suffolks, never received the order to retire. They had already taken a severe hammering during the morning's action and their first intimation of the withdrawal was the sight of the German 66th Infantry approaching in mass down the ridge to their front. The British waited until the Germans were well within range before subjecting them to a withering fire which left the ground littered with dead and wounded. The retirement on their right meant that the KOYLI were swiftly surrounded. They continued a gritty defence, shooting down large numbers of German troops and stopping every enemy attempt to set up machine gun posts. The action of 'B' Company is typical of the battalion. Reduced to 19 men, faced with a surging mass of enemy troops advancing towards them, the company's commander, Maj. Yate, ordered his men to charge the enemy. They were, as one of their opponents acknowledged, 'incomparable soldiers'.\n\nBy about 4pm, it was all over for the survivors on the ridge. Their stubborn defence nonetheless paid enormous dividends, as they prevented any serious pursuit of the remainder of the 5th Division as it escaped down the Selle Valley and by six o'clock that evening, the 5th Division had successfully broken contact with the enemy.\n\nIn the centre, the 3rd Division came under severe attack during the afternoon. There had been heavy artillery fire throughout the morning. This intensified in the early afternoon and was combined with infantry attacks by elements of the 8th Division, 4th Cavalry Division and 9th Cavalry Division. The outskirts of Caudry were occupied by the enemy, but an attack to the east, on Audencourt, was thwarted at a heavy cost to the German infantry.\n\nThe 3rd Division began its retirement at about 3.30pm with little interference from the Germans, although the area was still being shelled. The artillery suffered some losses: the 107th and 108th Batteries each lost a section, and a German salvo destroyed three complete teams of the 6th Battery as it withdrew. As on the right flank, however, there were communication problems. The 1st Gordons, in the very centre of the British line at Audencourt, together with two companies of the 2nd Royal Irish and some 2nd Royal Scots, were the rearguard of the 8th Infantry Brigade. The order never reached them so they simply stayed put and fought off every attack that was launched against them.\n\nGerman troops from the 8th Infantry Division advanced across a field of beetroot, only to run into devastating fire from the Scots. For an hour, the Germans tried to break through but failed; they attempted to outflank the British position but, as darkness approached, their casualties were so high that the whole attack stalled. This bought enough time for the rest of the 3rd Division to move away. By midnight the commanding officer, Col. W.G. Gordon VC, decided it was time to leave. Moving through the darkness, the Gordons ran into substantial enemy forces and after a heavy exchange of fire were forced to surrender.\n\nFor the 4th Division, retirement was a little more difficult. The Germans had withdrawn the 2nd and 9th Cavalry Divisions and replaced them with the 7th Reserve Division of the IVth Reserve Corps which had made a forced march from Valenciennes. These fresh troops came forward vigorously, pushing the 11th Brigade back to Ligny. The rearguard, 1st Rifle Brigade, supported by XIV and XXIX Brigades of the RFA, stopped the German advance with such purpose that the attack on Ligny petered out. Caught in the open, the Germans suffered very severely from the combined onslaught of the British infantry and artillery.\n\nThe German commander then tried to turn the British flank, but was foiled by the arrival of the artillery attached to the French Cavalry Corps under Gen. Sordet. Their vicious 75mm guns threw the German attackers into disarray and the 4th Division started to disengage. There was great confusion as the infantry pulled out, units becoming intermingled or finding themselves surrounded by advancing German troops. Two companies of the 1st Royal Warwicks had an interesting time. For the next three days, they marched between the German cavalry screen and its following infantry, moving only by night and hiding by day, until they were able to rejoin the BEF.\n\nII Corps was severely mauled at Le Cateau. The official casualty figures show Smith-Dorrien lost 7,812 men and 38 guns. German losses are not known. No figures are available, and estimates range from 15,000 to 30,000 dead, wounded and missing. Without doubt, the casualties were very severe. Von Kluck was convinced that he had been fighting all six divisions of the BEF, as well as its cavalry and several French Territorial divisions. The next morning his pursuit was to the south-west as he was still positive the British would make for the Channel ports.\n\nII Corps went southwards with the luxury of a 12 hour start. For the rest of the retreat, as Smith-Dorrien himself wrote, II Corps 'were no more seriously troubled ... except by mounted troops and mobile detachments who kept at a respectful distance'. No soldier in II Corps afterwards ever believed anything other than that they had saved the BEF from utter defeat. Smith-Dorrien's gamble had paid off.\n\nTHE CHARGE OF THE 12TH LANCERS AT C\u00c9RIZY, 28 AUGUST 1914  \n **During the retreat the British rearguard was attacked by pursuing Germans. Seeing an opportunity for a British counterattack at C\u00e9rizy, just south of St Quentin, the British commander ordered the 12th Lancers to charge over a ridge to catch the German infantry in the open. With little or no time to fire at the charging cavalry the Germans were ridden down and cut to pieces.**\n\n## AFTERMATH\n\nIt was not the end. The retreat continued and it was a hard road. I and II Corps drifted apart for a while on the long plod south, being separated by about 15 miles before their paths once again converged. There were days when bitter meetings left huddled bodies in khaki or field-grey in the fields and woods of France.\n\n###### **Etreux**\n\nThere was a fiercely contested action at Etreux on 27 August when the Guard Cavalry Division of I Cavalry Corps and X Reserve Corps came upon the rearguard of 1 Division. The 2nd Royal Munster Fusiliers, aided by two guns of 118 Battery RFA and two troops from 'C' Squadron of the 15th Hussars, fought for nearly 12 hours against a minimum of nine German battalions and four artillery batteries. The Munsters vanished from the British Order of Battle, the only survivors being the remnants of two platoons.\n\n###### **N\u00e9ry**\n\nAt N\u00e9ry, the 1st Cavalry Brigade, made up of the Queen's Bays, 5th Dragoon Guards and 11th Hussars, with the ubiquitous 'L' Battery of the RHA, was surprised in the early morning mist on 1 September by the 4th Cavalry Division. Heavy artillery and small arms fire from a ridge outside the village hammered into the Bays \u2013 whose horses stampeded \u2013 and the RHA gunners. Three of 'L' Battery's guns were soon disabled; the surviving crewmen helped man the three remaining. One was quickly knocked out. The other two kept firing until every member of its crew was dead or wounded. The last gun continued to fight on for nearly an hour despite the combined fire of 12 German guns against it.\n\n**This photograph is of British troops lining a roadside ditch. This bears a strong resemblance to the northern French countryside, but the distance between the riflemen is a not inaccurate portrayal of the meagreness of the British line. (Private Collection)**\n\nAt last, manned by Capt. Bradbury, the 2 i\/c of 'L' Battery, Battery Sgt.-Maj. Dorrell and Sgt. Nelson, the lone gun came to the end of its resistance. Bradbury was mortally wounded fetching ammunition; Nelson and Dorrell fired their last shell at 8.10pm. All three were awarded the Victoria Cross. Meanwhile, the rest of the Brigade reacted sharply. 'L' Battery had covered them as they worked their way round the German flanks where they pinned down the enemy. Fresh troops arrived. The 4th Cavalry Brigade, 'I' Battery of the RHA, the 1st Middlesex and part of 10th Infantry Brigade came up and opened an annihilating fire on the ridge, forcing the Germans back in great disorder. Eight of the German guns were abandoned where they were, and the other four were found next day in a nearby wood. The German 4th Cavalry Division was so badly damaged that it was withdrawn from the Cavalry Corps and sent to the IV Reserve Corps.\n\n**The graves of Captain Bradbury VC and Lieutenant John Campbell at N\u00e9ry, photographed a few days after the action. The N\u00e8ry action was publicised throughout the world, one account appearing in a Mexican newspaper. The story so aroused the admiration of a Japanese businessman resident in Mexico that he commissioned three signet rings, each engraved with the word 'Hero' in Japanese script, for presentation to the families of three selected participants. One of these was Lieutenant Campbell. German accounts of the battle insisted that the 4th Cavalry Division was greatly outnumbered by British infantry columns supported by cavalry and artillery. (Imperial War Museum)**\n\n**'L' Battery, RHA, just before embarkation for France in 1914. In the action at N\u00e9ry, three of its members were to win the Victoria Cross. Twenty three would be killed and 31 wounded, thus effectively removing the unit from the BEF Order of Battle. (Imperial War Museum)**\n\n'L' Battery lost three officers killed and two wounded, 20 men killed and 29 wounded, a total of 54 out of the 135 British casualties at N\u00e9ry.\n\n###### **Villers-Cotter\u00eats**\n\nThat same day, the rearguard of the 2nd Division, at Villers-Cotter\u00eats about 15 miles east of N\u00e9ry, came into brutal contact with advance troops of III Corps and the 2nd and 9th Cavalry Divisions. The rearguard, the 4th (Guards) Brigade, was on the northern edge of a forest, criss-crossed by wide tracks which provided natural fields of fire. From about 10.45am onwards, there was a hard and confused battle among the woodland as the Guards struggled clear. The early attacks hit the 2nd Coldstream and the 1st Irish Guards. Col. Morris, the CO of the Irish Guards, was killed by a machine gun burst. Two platoons of the 2nd Grenadier Guards were surrounded and fought to the end. The brigade lost more than 300 officers and men, and the 6th Brigade, which covered the Guards withdrawal, lost another 160.\n\nThe BEF went on to halt the German offensive later in the month at the battle of the Marne, forcing them to retreat to the Aisne. Later in the autumn, the BEF turned north again and found themselves at Ypres. It was to be the graveyard of the old British Regular Army.\n\n## WARGAMING MONS\n\nThe heady days of August 1914 have always been popular with both boardgamers and miniatures players. The opening weeks of the Great War are effectively the end of the 'horse and musket' era. At a tactical level, this period is at once visually appealing and intellectually challenging. Reasonably complete ranges of figures are available in 25-, 15- and 5mm scales, and important troop types overlooked by the manufacturers are easily converted from nineteenth century ranges. At operational level, the challenge of manoeuvring the BEF in the path of the oncoming juggernaut soon generates great sympathy for Sir John French and, indeed, the commanders of the other armies. The German commanders are under terrible pressure, trying to sustain a very optimistic march schedule in the face of stubborn opponents and an intractable logistics problem. The French try to make a fighting withdrawal with an army trained and armed for the offensive, while the Belgians make a gallant defence in the face of overwhelming numbers. Misunderstandings and clashes occurred within each army as well as between notional allies, offering great scope for multi-player games.\n\n###### **Tactical games**\n\nThe Mons campaign witnessed a succession of small-scale actions of the sort that easily fit on a wargames table. The epic defence of N\u00e9ry, the Guards brigade's rearguard action at Villers-Cotter\u00eats and the 5th Cavalry Brigade at C\u00e9rizy all involve a handful of battalions\/squadrons against larger (although equally exhausted) German forces. Obviously, if you re-fight these specific actions the Germans must be constrained from taking advantage of hindsight; historically, they never knew exactly what they were facing. In north-east France during high summer, the frequent early morning mists hampered aerial reconnaissance and enabled rearguards to withdraw or ambush their pursuers at will.\n\nAs a foot- or saddle-sore German advance guard pressed onwards, it would frequently deliver a hasty attack wherever opposition showed itself. However, a few British infantry companies or cavalry squadrons could deliver a formidable weight of fire. Determined resistance or even a spirited counter-attack (like the charge of the 12th Lancers at C\u00e9rizy) often led the Germans to conclude they were facing much larger forces, in the same way von Kluck believed he had fought the whole BEF at Le Cateau.\n\nA representative scenario might feature a British infantry or cavalry brigade deployed on a map, with figures placed on the table only when their location is revealed by firing or by Germans blundering into close combat. They would seldom occupy anything more than shell-scrapes, as men who had been marching all day and half the night had neither the time nor inclination to dig more comprehensive defences. Incidentally, Walter Bloem's account of Mons confirms that this was not a 'British disease': the 12th Brandenburg Grenadiers were equally reluctant to entrench even when a British counter-attack threatened. German forces, say a cavalry brigade followed by an infantry division, arrive on table in dribs and drabs. I write unit identities on index cards, add some blanks and draw one per (hourly) turn. Some cards provide a cavalry regiment, others a whole infantry brigade. The British, of course, are ignorant as to what they are facing until it hoves into view. What might have begun as a successful rearguard operation might turn to catastrophe if powerful German reinforcements arrive in time to outflank and trap the defenders. As the Munsters discovered, withdrawal from close contact with such an enemy is all but impossible. However, this sort of d\u00e9b\u00e2cle can be the beginning of an alternative scenario where a cut-off detachment attempts to break through the German forces to safety.\n\nWhen framing a set of miniatures rules, or modifying a commercially-available one, the _Field Service Pocket Book 1914_ is an invaluable aid, providing vital data on lengths of marching columns, the times required to prepare field defences etc. One item deserves emphasis: in both board- and figure-games, defenders of buildings are generally harder to hit than troops in the open. Reality is not quite so simple, especially not in 1914. At Mons and Le Cateau, British infantry usually dug their shell scrapes in front of the built-up areas, not actually in the villages themselves. As contemporary manuals stressed, early 20th-century artillery could hit buildings from far beyond the range of rifles, with catastrophic consequences for the defenders, unless the buildings had been thoroughly prepared for defence. Unless the defenders had 24 hours to clear out combustible material and entrench themselves in the basement, occupying a building simply offered the enemy an easily identified target. By contrast, shallow trenches in the open were harder to spot and more difficult for the gunners to range on. Fire from such positions a few hundred yards in front of a village was usually mistaken for fire from the buildings, leading the enemy to shell the village and leave the infantry relatively unscathed.\n\nThe armies of the Mons campaign had different tactical doctrines, which should be recognised in any tactical level wargame. German battalions usually led with one company in open order, one in support and the other two following in shoulder-to-shoulder lines several hundred yards behind. Some soon learned that deployment in greater depth was required to avoid the beaten zone of the defenders' rifles. The British followed a similar approach. Even on the defensive, the British were supposed to be in three lines: a firing line with about one yard per man was bolstered by supports in the rear who reinforced the line as casualties mounted. The third wave was not supposed to reinforce the line, but was intended for counter-attacks.\n\nThe French notoriously neglected open order tactics and their rifle fire was notably less accurate. Their cavalry had little enthusiasm for dismounted action and short-range carbines gave them little incentive to try. (The lack of bayonets also made them helpless in close combat on foot.) However, their fabled 75mm guns lived up to pre-war expectations. British observers noted that the French tended to give up ground easily, but as often as not recapture it by counter-attack, whereas the British would doggedly hold the same position: which method was less costly was impossible to determine.\n\nTHE 4TH GUARDS BRIGADE AT VILLERS-COTTER\u00caTS 1 SEPTEMBER 1914\n\n**At Villers-Cotter\u00eats, about 15 miles east of Nery, the rearguard of the British 2nd Division, 4th (Guards) Brigade, came into brutal contact with advance troops of III Corps and 2nd and 9th Cavalry Divisions. The 4th (Guards) Brigade was on the northern edge of a forest crisscrossed by wide tracks which provided natural fields of fire. A hard and confused battle ensued as the Guards struggled free.**\n\nSome British accounts of the Mons campaign suggest the Germans had learnt nothing from the war of 1870 and describe waves of German infantry attacking in close order as if magazine-loading rifles had never been invented and the Boer War had never happened. Yet on other parts of the same battlefield, German infantry might come on in thin skirmish lines, developing an effective attack with excellent support from their artillery. This was a consequence of the extraordinary degree of autonomy allowed to pre-war German corps commanders. They reported directly to the Kaiser, the famed General Staff having operational control in war, but not the power to dictate training standards or tactical doctrine. Like most other armies, the pre-war German Army had been sharply divided over the tactical lessons of both the Boer War and the Russo-Japanese War. Scattering infantrymen over wide frontages might reduce losses in the short term, but (some argued) it made command and control impossible, led to the whole unit becoming paralysed in the face of the enemy and eventually to greater casualties in the long term. The opposing view, based on studies of the Russo-Japanese war accepted that close formations would suffer heavier losses, but, still under control of their officers, they could continue to manoeuvre and deliver fire. Only a close formation had the group spirit required to close with the bayonet \u2013 and it was the threat of cold steel, not long-range rifle fire that would finally eject a defender from his position. Since the pre-war training of the troops would not be under the control of the player, you can dice for their formation when German troops arrive on the battlefield or even when they deliver an attack. Close formations should be more likely to keep moving, but they are better targets and under effective fire they will break up.\n\nGerman accounts of the Mons campaign testify to the effectiveness of British rifle fire, but both sides soon recognised that this was going to be an artillery war. Gunners recognised three types of firing position in 1914: open, semi-covered and covered. The former is self-explanatory. In a semi-covered position, the gun would be far enough behind a crestline for the gun itself (but not the muzzle-flash) to be concealed from the enemy. In a fully covered position both the gun and the flash were hidden. Covered positions were much easier for howitzer batteries than the flat trajectory field guns which could only fire from such defiladed positions at the cost of a long stretch of dead ground in front of the gun. Unless this space was covered by infantry, the enemy could work his way under the line of fire.\n\nHorse-teams were horrifically vulnerable targets. Mounted reconnaissance was difficult enough in the face of machine guns and magazine-loading rifles, but it was the improved long range accuracy of quick-fire artillery that really frustrated the cavalry. Time and again, German cavalry regiments operating dismounted against a British rearguard saw their lead horses dispersed by British artillery fire. The German cavalry were reluctant to continue advancing when this happened, but in the gently rolling ground of the French\/Belgian border, it was not always possible to find cover for the horses.\n\n###### **Mons-Le Cateau in an afternoon**\n\nBy sacrificing the finer details of the tactical battle, Mons or Le Cateau can be re-fought in an afternoon's game session. As usual, some thought must be given to stopping the Germans exploiting the fact that, unlike the historical German commanders, they know what they are facing. Mons and Le Cateau were hardly the first instances of a numerically superior force failing to crush a smaller army because they overestimated the enemy's strength. Antietam springs to mind. Boardgamers can find some readily tailored solutions in previous game designs like Peter Perla's area movement game of Antietam. Here, the overwhelmingly superior Union army uncoils so slowly that a nimble Confederate defence can fend them off until nightfall \u2013 usually. But if the turns last longer, the Federals sometimes manage to concentrate their striking power and Lee is driven off the map and into the Potomac.\n\n**The fighting in France and Flanders brought home the reality of war to a British public which previously had considered battles to be affairs which took place in far-off lands against primitive natives. Even the Boer War had not changed that attitude a great deal. This altered when the wounded from Mons, Le Cateau and the Retreat were brought back to England and crowds gathered to stare as the casualties were ferried from the railway stations to hospital. It was a sight that was to become commonplace in the four years that followed. (Private Collection)**\n\nFor Mons or Le Cateau the use of hidden movement is essential. Have game turns of uncertain duration: on average it is not possible for the Germans to move anything like their full strength, but the possibility is there for a succession of long turns which could spell doom for the 'contemptible little army'.\n\n###### **Multi-player games**\n\nThe utter confusion of August 1914 is perhaps best simulated by a multiplayer game. Not just the chaos prevailing at headquarters where exhausted officers puzzled over reports that were necessarily out of date by the time they arrived, but the added friction between very different nations (the British and the French) and abrasive personalities (Sir John French, Kitchener, Smith-Dorrien etc.).\n\nUnlike modern staffs, there was no duplication in a 1914 headquarters. It was not possible to have part of an HQ on the move while the remainder ran the battle. Generals and their staffs found themselves riding to and fro all day, then writing orders all night. Having only an hour or two to sleep in every 24 soon told on the toughest constitutions. Lack of sleep certainly helped poison relations between the suspiciously anglophobic Lanzerac and Sir John French, and it did nothing for the latter when Grierson was replaced by his old enemy, Smith-Dorrien. Sir Horace himself was notorious for his vile temper and his rages were a legend in the Edwardian army. This potent brew of personalities cries out for a matrix game system or other species of role-play. The objective here is not to recreate that fatuous burlesque, _O What a Lovely War!_ , but to bring out the very real problems of coalition warfare. The BEF could be forgiven for thinking that French policy was to defend the Third Republic down to the last Englishman. Lanzerac was not the only Frenchman to dwell on the fact that his nation was providing 90 per cent of Allied manpower on the Western Front.\n\n## THE BATTLEFIELD TODAY\n\nThe single day encounters of Mons and Le Cateau left no mark on the landscape. Behind the German lines during 1914-1918, they survived relatively unscathed, although new housing, a motorway and other contemporary building have sliced across and along the lines of the scenes of the Old Contemptibles' actions. It is still possible, even today, to see the area much as the original members of the BEF saw it.\n\nMons itself is a bustling town, but the Grande Place, with its imposing town hall, is not greatly changed; even allowing for modern traffic, there are areas which are much the same as they were in 1914. The Mus\u00e9e du Centenaire, in the Jardin du Mayer, houses the World War One and Two collections and is just off the Grande Place. There are plans to rehouse the collection in due course.\n\nThe Paris motorway now runs alongside the canal but the towpath is accessible and buildings still remain which served as aid posts or battalion headquarters on that desperate August day. It needs little imagination to envisage how things were for you can stand where the British gunners sweated and swore, and look across at the tree line from where German soldiers dressed in field-grey advanced towards the dry-mouthed British infantry.\n\nMuch of the Le Cateau battlefield remains as it was when the BEF fought there. The ditches along the Roman road which became cover for the rifles of the infantry as they halted the German attacks remain much as they were. Beetroot still grows in the silent fields at Audencourt over which the German soldiers advanced so bravely in the summer sun in 1914.\n\nThe area can be toured by car, but the enthusiast might prefer to use a bicycle which makes a number of the remoter locations more easily accessible. It is especially useful for covering the full length of the canal at Mons. Both the Mons battlefield, which extends along a 25-mile line, and the shorter Le Cateau line repay study. The local tourist offices can give information about hiring bicycles locally.\n\nRose Coombs' invaluable guide to the battlefields, _Before Endeavours Fade_ , gives detailed itineraries and should be an essential part of any visitor's baggage. Accommodation across a wide price range can be found; again, the local tourist offices give an excellent service, but it should be remembered that many places are now important commercial centres and the town hotels are geared to business customers, not battlefield visitors, and their prices reflect this. There are, though, many smaller places to stay in the villages, as well as several camping areas.\n\nA number of sites are on farm land or private property. It is an accepted courtesy to seek permission before tramping over them and it cannot be too highly recommended that even a few words of execrable French, Dutch or Flemish, are invaluable in obtaining the goodwill of the landowner.\n\n### Related Titles - Campaign Series\n\n**epub ISBN** | **PDF ISBN** | **No.** | **TITLE**  \n---|---|---|---  \n9781780965291 | 9781846036064 | 25 | Leipzig 1813  \n9781780965307 | 9781846035425 | 87 | L\u00fctzen & Bautzen 1813  \n9781849087209 | 9781846035470 | 92 | St Nazaire 1942  \n9781782001461 | 9781846035555 | 100 | D-Day 1944 (1)  \n9781782001478 | 9781846035593 | 104 | D-Day 1944 (2)  \n9781849087216 | 9781846035609 | 105 | D-Day 1944 (3)  \n9781849087223 | 9781846035678 | 112 | D-Day 1944 (4)  \n9781849089432 | 9781849086042 | 144 | Wake Island  \n9781782000099 | 9781846037993 | 170 | Osaka 1615  \n9781849087230 | 9781846038266 | 178 | The Rhine Crossing  \n9781849087247 | 9781846038433 | 196 | Gazala 1942  \n9781780969930 | 9781849081061 | 214 | The Coral Sea 1942  \n9781780969947 | 9781849082952 | 226 | Midway 1942  \n9781849088206 | 9781849083867 | 236 | Operation Pointblank 1944  \n9781849088213 | 9781849083201 | 237 | The Fourth Crusade 1202\u201304  \n9781849088800 | 9781849083928 | 238 | St Mihiel 1918  \n9781780960302 | 9781849085557 | 239 | Plataea 479BC  \n9781849088930 | 9781849086776 | 240 | Wabash 1791  \n9781780960432 | 9781849085922 | 242 | Metz 1944  \n9781780963921 | 9781849086103 | 243 | The Philippines 1941-42  \n9781780964164 | 9781849086080 | 244 | The Falklands 1982  \n9781780964423 | 9781849085533 | 245 | Demyansk 1942\u201343\n\nFirst published in Great Britain in 1997 by Osprey Publishing,\n\nMidland House, West Way, Botley, Oxford OX2 0PH, UK\n\n44-02 23rd St, Suite 219, Long Island City, NY 11101, USA\n\nEmail: info@ospreypublishing.com\n\n\u00a9 1997 Osprey Publishing Ltd.\n\nAll rights reserved. Apart from any fair dealing for the purpose of private study, research, criticism or review, as permitted under the Copyright, Designs and Patents Act, 1988, no part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, electrical, chemical, mechanical, optical, photocopying, recording or otherwise, without the prior written permission of the copyright owner. Enquiries should be addressed to the Publishers.\n\nFirst published 1997\n\n8th impression 2008\n\nA CIP catalogue record for this book is available from the British Library\n\nPrint ISBN: 978 1 85532 551 7\n\nPDF e-book ISBN: 978 1 84603 630 9\n\nePub e-book ISBN: 978 1 78200 444 8\n\nMilitary Editor: Sharon van der Merwe\n\nDesign by The Black Spot\n\nCartography by Micromap\n\nColour Birds Eye Views by Peter Harper\n\nWargaming Mons by Ian Drury\n\nBattlescene Artwork by Peter Dovey\n\nFilm set in Singapore by Pica Ltd.\n\n**Further Reading**\n\nMany books have been written by participants and later historians about the events of August 1914. Unfortunately, a great many of them are now out of print, although copies can be obtained from specialist dealers. Many of the British accounts, especially those written in the inter-war years, not only dismiss all French and Belgian participation in the war, but grossly overstate the importance of Mons and Le Cateau. One writer goes so far as to call Mons 'the battle that saved the world', a panegyric that one suspects the old BEF would have found embarassing. The following are a tiny selection of useful titles in English and which are obtainable without too much difficulty via specialist dealers or the library service.\n\n**Ascoli, David,** **_The Mons Star_** **(Harrap, 1981).** A worthwhile and well-researched account, often drawing on personal reminiscences of the BEF in 1914. Interestingly, Ascoli is condemnatory of the French Commanders, a prejudice which reflects the feeling of the BEF at the time.\n\n**Bloem, Walter,** **_The Advance from Mons_** **(Peter Davis, 1930).** Most of the German acconts of the early days of the war are palpably inaccurate and the official records are sometimes positively misleading. This memoir, written by an officer of the Brandenburg Grenadiers and widely quote by many English writers because of the tributes it pays to the BEF, is nonetheless one of the few that gives a realistic idea of what it was like to come under the 'mad minute' rifle fire of the British professional soldier.\n\n**Coombs, Rose,** **_Before Endeavours Fade_** **(Plaistow Press, 1994).** The latest edition of Rose Coombs' guide to the First World War battlefields should always be to hand. Originally published in 1976, the guide has been extensively revised.\n\n**Edmonds, Brigadier J.E.,** **_Official History, France & Belgium, 1914_** **\u2013 Volume 1 (Macmillan, 1926).** An invaluable, indeed indispensable, tool for the student. Edmonds' work in compiling the Official History can rarely be faulted, although in his non-official capacity, he had a great talent for creating mischief.\n\n**French, Field Marshal Viscout, 1914.** French's own account of the events of 1914 reveals his distrust, bitterness and paranoia and, like the memoirs of Lloyd George, is a source more to its author's character than anything else.\n\n**Simpson, Keith,** **_The Old Contemptibles_ ,** **(George Allen & Unwin, 1981).** A photographic history of the BEF from August to December 1914. This is a slim volume with a necessarily truncated text and its real interest lies in the illustrations.\n\n**Spears, Sir Edward.** **_Liaison_** **1914 (Heinemann, 1930).** Spears had an uncanny habit (too uncanny, according to some observers) of being in the right place at the right time, and so witnesed some of the momentous happenings on the BEF front which he describes vividly and with brilliant detail. His book is an invaluable source work, although his personal prejudices sometimes shade his narrative.\n\n**Terraine, John,** **_Mons, 1914_** **(Batsford, 1960).** One of the best accounts. Well researched and eminently readable.\n\n**Tuchman, Barbara,** **_The Guns of August_** **(Constable, 1962).** A comprehensive work dealing with the events leading up to the outbreak of war and the battles of 1914, both in the East and West. The authos is not unnaturally fascinated by the personality clashes and politics of the period, sometimes at the expense of military events, and occasionally puts too much reliance on some dubious sources.\n\n**Dedication**\n\nDedicated to John Terraine, like Thucydides, an historian.\n\n**Acknowledgements**\n\nThe list of people who helped this title to materialise is legion; but special mention must be made of Jules Brihay and his staff of the Department of Tourism and Culture at Mons, from whom many of the photographs in this book were obtained, as well as the Imperial War Museum, whose staff once more lived up to their legendary reputation for outstanding helpfulness and courtesy. Finally, this title would have been completed in half the time without the enthusiastic assistance of Beyaz Bardem, mother of beautiful kittens, and an expert at sitting on keyboards.\n\n**Editorial note**\n\nThe editor would like to thank Mike Chappell and Martin Pegler for their invaluable contribution to this book.\n\n\u00a9 _Osprey Publishing. Access to this book is not digitally restricted. In return, we ask you that you use it for personal, non-commercial purposes only. Please don't upload this ebook to a peer-to-peer site, email it to everyone you know, or resell it. Osprey Publishing reserves all rights to its digital content and no part of these products may be copied, stored in a retrieval system or transmitted in any form by any means, electronic, mechanical, recording or otherwise (except as permitted here), without the written permission of the publisher. Please support our continuing book publishing programme by using this e-book responsibly._\n\n_**Every effort has been made by the Publisher to secure permissions to use the images in this publication. If there has been any oversight we would be happy to rectify the situation and written submission should be made to Osprey Publishing.**_\n\n**Key to military series symbols**\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\nThe Star, the Cross, and the Crescent\nAfter the Empire:  \nThe Francophone World and  \nPostcolonial France\n\n**Series Editor**\n\nVal\u00e9rie Orlando, University of Maryland\n\n**Advisory Board**\n\nRobert Bernasconi, Memphis University; Alec Hargreaves, Florida State University; Chima Korieh, Rowan University; Mildred Mortimer, University of Colorado, Boulder; Obioma Nnaemeka, Indiana University; Kamal Salhi, University of Leeds; Tracy D. Sharpley-Whiting, Vanderbilt University; Nwachukwu Frank Ukadike, Tulane University\n\n_Seewww.lexingtonbooks.com\/series for the series description and  \na complete list of published titles_.\n\n**Recent and Forthcoming Titles**\n\n_Collective Memory: France and the Algerian War (1954\u20131962)_ , by Jo McCormack\n\n_The Other Hybrid Archipelago: Introduction to the Literatures and Cultures of the Francophone Indian Ocean_ , by Peter Hawkins\n\n_Rethinking Marriage in Francophone African and Caribbean Literatures_ , by C\u00e9cile Accilien\n\n_Two Novellas by YAE:_ A Moroccan in New York _and_ Sea Drinkers, by Youssouf Amine Elalamy, translated by John Liechty\n\n_Frank\u00e9tienne and Rewriting: A Work in Progress_ , by Rachel Douglas\n\n_Charles Testut's_ Le Vieux Salomon _: Race, Religion, Socialism, and Freemasonry_ , Sheri Lyn Abel\n\n_What Moroccan Cinema?: A Historical and Critical Study 1956\u20132006_ , by Sandra Carter\n\n_Voices of Exile in Contemporary Canadian Francophone Literature_ , by F. Elizabeth Dahab\n\n_Re-hybridizing Transnational Domesticity and Femininity: Women's Contemporary Filmmaking and Lifewriting of France, Algeria, and Tunisia_ , by Stacey Weber-Feve\n\n_The Star, the Cross, and the Crescent: Religions and Conflicts in Francophone Literature from the Arab World_ , by Carine Bourget\n\n# The Star, the Cross, and  \nthe Crescent\n\n_Religions and Conflicts in Francophone  \nLiterature from the Arab World_\n\nCarine Bourget\n\nPublished by Lexington Books  \nA division of Rowman & Littlefield Publishers, Inc.  \nA wholly owned subsidiary of The Rowman & Littlefield Publishing Group, Inc.  \n4501 Forbes Boulevard, Suite 200, Lanham, Maryland 20706  \n<http:\/\/www.lexingtonbooks.com>\n\nEstover Road, Plymouth PL6 7PY, United Kingdom\n\nCopyright \u00a9 2010 by Lexington Books\n\n_All rights reserved_. No part of this book may be reproduced in any form or by any electronic or mechanical means, including information storage and retrieval systems, without written permission from the publisher, except by a reviewer who may quote passages in a review.\n\nBritish Library Cataloguing in Publication Information Available\n\n**Library of Congress Cataloging-in-Publication Data**\n\nBourget, Carine.\n\nThe star, the cross, and the crescent : religions and conflicts in Francophone literature from the Arab world \/ Carine Bourget.  \np. cm. \u2014 (After the empire: the Francophone world and)\n\nIncludes bibliographical references and index.\n\nISBN 978-0-7391-2657-8 (cloth : alk. paper)\n\n1. North African literature (French)\u201420th century\u2014History and criticism. 2. French literature\u2014Arab authors\u2014History and criticism. 3. French literature\u2014Arab countries\u2014History and criticism. 4. French literature\u201420th century\u2014history and criticism. 5. Islam in literature. 6. Social conflict in literature. 7. Politics in literature. 8. War in literature. I. Title.\n\nPQ3980.5.B68 2010\n\n840.9'892761\u2014dc22\n\n2009048357\n\n  The paper used in this publication meets the minimum requirements of American National Standard for Information Sciences\u2014Permanence of Paper for Printed Library Materials, ANSI\/NISO Z39.48-1992.\n\nPrinted in the United States of America\nTo my husband and children\n\n## Contents\n\nAcknowledgments\n\nIntroduction: Religion, Politics, and Literature from the Francophone Arab World\n\n1. The Arab-Israeli Conflict: Amin Maalouf's _Les croisades vues par les Arabes_ and _Les \u00e9chelles du Levant_ , Myriam Antaki's _Les versets du pardon_\n\n2. The Arab-Israeli Conflict Beyond the Middle East: Albert Memmi, Edmond Amran El Maleh, Farid Boudjellal, and Karin Albou\n\n3. The Lebanese Civil War: Andr\u00e9e Ch\u00e9did's _La maison sans racines_ and Evelyne Accad's _L'excis\u00e9e_\n\n4. The Algerian Civil War: Rachid Boudjedra's _Le FIS de la haine_ , Rachid Mimouni's _De la barbarie en g\u00e9n\u00e9ral et de l'int\u00e9grisme en particulier_ , and _Une enfance alg\u00e9rienne_\n\n5. Islam and the French Republic: The Affair of the Muslim Headscarf in works by Tahar Ben Jelloun, Abdelwahab Meddeb, Amin Maalouf, Albert Memmi, Le\u00efla Sebbar, and Yamina Benguigui\n\n6. Portrait of a Terrorist: Slimane Bena\u00efssa and Salim Bachi's 9\/11 Novels\n\nConclusion: The Politics of Translation: Francophone Literature from the Arab World in the U.S.\n\nBibliography\n\nIndex\n\nAbout the Author\n\n## Acknowledgments\n\nI would like to thank the following colleagues and friends for their careful reading and insightful feedback on various parts of this manuscript: Jonathan Beck, Julia Clancy-Smith, Anne Donadey, Leila Hudson, Marie-Pierre Le Hir, Maha Nassar, Val\u00e9rie Orlando, and Laurence Porter. Any errors and shortcomings are of course mine.\n\nI am grateful to my colleagues from the Department of French & Italian, the Department of Near Eastern Studies, and the Center for Middle Eastern Studies at the University of Arizona for providing a stimulating and supportive work environment. I want to single out Marie-Pierre Le Hir for her valuable guidance and encouragement throughout the years. I would also like to acknowledge Ir\u00e8ne D' Almeida and Lise Leibacher for their thoughtful mentoring.\n\nAlthough this book did not originate at Michigan State University, I wish to express my deep appreciation to Laurence Porter for introducing me to Francophone Literature during my graduate studies there. His rigorous scholarship and generous mentoring will remain models for me to emulate throughout my career. Many thanks go to Ken Harrow for introducing me to African Literature during his lively and challenging seminars.\n\nA very special note of gratitude is owed to my husband, not only for reading and commenting on most of this manuscript. This book simply could not have been written without his constant support, confidence, and encouragement. May my oldest children read this book one day and hopefully conclude that the time spent away from them was put to good use.\n\nAll efforts have been made to obtain copyright permissions. I thank the publishers for permission to reprint from the works listed below: Citations from _Ports of Call_ by Amin Maalouf, translated by Alberto Manguel, published by Harvill Press are reprinted by permission of The Random House Group Ltd. Citations from _Les versets du pardon_ by Myriam Antaki, \u00a9 1999 by Actes Sud, are reprinted by permission of Editions Actes Sud. Citations from _Les croisades vues par les Arabes_ by Amin Maalouf, \u00a9 1983 by Latt\u00e8s, are reprinted by permission of Editions Latt\u00e8s. Citations from _JuifsArabes_ by Farid Boudjellal, \u00a9 2006 by Futuropolis, are reprinted by permission of Editions Futuropolis. Citations from _Dreams of Trespass_ by Fatima Mernissi, \u00a9 1994 by Perseus Books, are reprinted by permission of Perseus Books. Citations from _Pillar of Salt_ by Albert Memmi, translated by Edouard Roditi, \u00a9 1955 by Criterion Books, Inc., are reprinted by permission of Beacon Press, Boston. Citations from _Juifs et Arabes, Portrait du d\u00e9colonis\u00e9 arabo-musulman et de quelques autres_ , and _La statue de sel_ by Albert Memmi, \u00a9 by Editions Gallimard, are reprinted by permission of Editions Gallimard. Citations from _Tuez-les tous_ by Salim Bachi, \u00a9 by Editions Gallimard, are reprinted by permission of Editions Gallimard. Citations from _Une enfance alg\u00e9rienne_ edited by Le\u00efla Sebbar, \u00a9 by Editions Gallimard, are reprinted by permission of Editions Gallimard. Citations from _Les \u00e9chelles du Levant_ by Amin Maalouf, \u00a9 1996 by Grasset & Fasquelle, are reprinted by permission of Editions Bernard Grasset. Citations from _Les identit\u00e9s meurtri\u00e8res_ by Amin Maalouf, \u00a9 1998 by Editions Grasset & Fasquelle, are reprinted by permission of Editions Bernard Grasset. Excerpts from _The Last Night of a Damned Soul_ by Slimane Benaissa, \u00a9 2003 by Editions Plon, translation \u00a9 2004 by Janice and Daniel Gross, used by permission of Grove\/Atlantic, Inc.\n\nFor the book cover, I thank R\u00e9mi Ochlik\/IP3 PRESS for permission to reprint the photograph of the veiled girl (mentioned in chapter five), and editions Futuropolis for permission to reproduce two drawings from Farid Boudjellal's _JuifsArabes_.\n\n## Introduction\n\n## Religion, Politics, and Literature from the Francophone Arab World\n\nThe resurgence of religion in various parts of the world (including the West) and across religious traditions represents a challenge to the Western construction of modernity. It disputes the modernization theory that economic progress and modernization entail secularism. Social scientists have been tackling this for some time now, for example in studies that look at the role of religion in International Relations (Thomas, Hatzopoulos and Petito) and conflict resolution (Gopin).\n\nAlthough the resurgence of religion spans different cultures, religions, and geographical areas, much focus has been put on Islam. Already back in the 1980s, Said noted the double standard that was being applied by some scholars who consider that Islam is unique in being a political religion, as if Christianity and Judaism were immune to politics. Said went on to note that \"Israel is perhaps the most perfect coincidence of religion and politics in the contemporary world,\" a fact rarely noted and taken to question in the Western media, which routinely tout Israel as the only democratic state in the Middle East. Said added that then-president Ronald Reagan \"time and again connect[s] religion and politics\" (\"The Essential Terrorist\" 157). This last statement also applies to former American president George W. Bush. In addition, \"the development of a powerful and rejectionist Zionist lobby among evangelical groups in the U.S., [. . .] has accented yet again the contradiction inherent in an allegedly secular U.S. support for a nation-state based in large part on Biblical promise\" (Qureshi and Sells 8).\n\nThere is a debate between scholars who present contemporary Islamic fundamentalism as an answer to and a consequence of modernization in the twentieth century, and those who view it as a variant of traditional social protests that have commonly used religion as a vehicle for political demands (Volpi 26). Olivier Roy points out the inadequacy of the culturalist approach that consists in simply resorting to Islam as an explanation for everything that goes on in the Muslim world. According to him, \"we should take Islamisation as a contemporary phenomenon that expresses the globalization and westernization of the Muslim world\" (15). Likewise, comparative political theorist Roxanne Euben insists on the failure of both models to render the complexity of Islamic fundamentalism, which she sees as a \"combination of the failure of ideological alternatives, the unavailability of competing political channels, and the authenticity that accrues to religious, and specifically Islamic, identifications in the Middle East\" (48). The fall of pan-Arabism following the 1967 Arab defeat and the failure of Western-oriented policies to improve the lot of the people made Islamist movements the only viable and credible source of opposition for corrupt and authoritarian Western-supported regimes. This is a key reason why such movements have been gaining popularity in various countries.\n\nSalman Sayyid remarks that \"the political turn of Islam cannot be understood outside the very complex ways in which it relates to the project of Western hegemony\" (xxii). Accordingly, this religious revivalism can be traced back to colonialism and neocolonialism, as evidenced by Western-backed authoritarian regimes (Esposito 21). As John Esposito points out, \"When governments, faced with a viable opposition, use the charge of religious extremism to justify curtailment of political liberalization and the use of repression, the result\u2014as we are witnessing in Algeria\u2014is the radicalization of moderate religious voices, the creation of new, violent revolutionary groups, the polarization of society, and the ravages of civil war\" (21). In other words, repression is only backfiring as it pushes moderate voices into extremism.\n\nIn addition, Western leaders privilege regimes who present themselves in terms the West is accustomed to as the agents of stability and modernity; however, such regimes \"serve to arouse Islamist fervour both by their continuing repressive policies at home and by directing such fervour towards U.S.\/West when domestic pressures become acute\" (Entelis 213). Scholars also emphasize the cleavage that has been growing between the ruling elite and the people. According to John Entelis,\n\nIslamist politics is first and foremost about politics\u2014who gets what, when and how\u2014and only secondarily about Islam. In the Islamic world, politics is controlled by the unelected few ruling over the alienated, disenfranchised many. This condition of oppression is reinforced objectively and perceptually by the alliances such regimes have made with outside powers, especially the United States, with the latter then being viewed by the masses as an additional instrument in the oppressive mechanism put in place by the _mukhabarat_ state to keep civil society in check. (212)\n\nThe popularity of Islamic movements and the violent tactics some of them have adopted stem from the political illegitimacy of Arab states.\n\nRobert Fernea points out the West's hypocrisy when it denounces authoritarian regimes that it supports: \"it is our responsibility to understand what is happening rather than negatively labeling as 'Islamic fundamentalism' this new discourse, and in opposition blindly supporting sectarian governments that use familiar terms and make comfortable promises to Western governments\" (x). Fatima Mernissi persuasively argues that Western democracies have promoted and profited from Islamic and particularly Saudi fundamentalism, with petrodollars funding authoritarian regimes whose massive arms purchases benefit Western companies. Indeed, \"the symbiotic relationship between Western liberal democracies and the palace fundamentalisms of the Gulf states (along with the popular street fundamentalisms they fund outside their borders) puts into question the supposition of a rational, democratic, liberal West facing an irrational and fundamentalist East\" (Qureshi and Sells 17).\n\nEsposito reminds us that while the elite minority is secular, \"a majority in non-western societies continues to live or to be influenced by religiously informed cultures\" (20). Faced with the complete failure of the modern state to improve the living conditions that are worsening for the majority while the elite is getting wealthier, \"Religious revivalisms (or fundamentalisms) often represent the voices of those who, amidst the failures of their societies, claim both to ameliorate the problems and to offer a more authentic, religiously-based society\" (Esposito 21). One reason why Islamic movements have gained popularity in various countries is that they have provided basic services such as health clinics and schools, where the state had failed to meet its obligation (Esposito 21), a fact they rarely receive credit for in the Western media.\n\n9\/11 has given new impetus to the theory of the clash of civilizations, although that theory has been debunked by several studies. As Emran Qureshi and Michael Sells point out, the assertion that there is a clash of civilization \"has become an ideological agent that may help generate the conflict that it posits\" (3). The binary opposition between the West and the rest asserted by the theory of the clash of civilizations does not hold. As Euben points out, the contrast often made between the West as a secular humanist and the Islamic East \"does not easily map onto, for example, the sociological religiosity of American culture, the growth of religious revivalism in American politics,\" nor does it take into account the influence of Western thought on Islamic thinkers such as Sayyid Qutb, whose writings are said to inspire fundamentalists of various calibers (166). Philosopher Jacques Derrida sees the opposition not between the West and the East, but between the U.S. and a Europe that he sees as \"the only secular actor on the world stage\" (Borradori 170). Similarly, Charles Kaplan states that the \"coming clash of civilizations will be not between the West and the rest but within a West divided against itself\" (qtd. in Majid, _Freedom_ 198). One could add and within an East divided against itself. Commenting on recent conflicts such as the genocide in Rwanda and the first Gulf War, Andreas Hasenclever and Volker Rittberger note that, \"What these conflicts, together with many others, have in common is the fierce competition between domestic elites who are ready to do almost anything to keep or get political power\" (112).\n\nInternational relations theorists have proposed a third alternative to the Primordialists (proponents of the clash of civilizations, who see the source of conflict as rooted in cultural differences), and Instrumentalists (who see the source of conflict as based on socioeconomic grievances). The moderate constructivists, like the Instrumentalists, see the source of conflict grounded in socio-economic and political disparities, but see religion playing a role in the manner the conflict is handled. They \"propose to view religion as an intervening variable, i.e., a causal factor intervening between a given conflict and the choice of conflict behavior\" (Hasenclever and Rittberger 115).\n\nWhile the resurgence of religion has been studied extensively in the social sciences, 9\/11 acted as a watershed event that prompted scholars in the humanities to give religion more attention (although some writers and scholars were already engaged with the issue). Since the last decades of the twentieth century, Francophone writers from the Arab world have been grappling anew with Islam's resurgence, as it became increasingly associated with sensational headlines in a variety of contexts.1 In the 1990s, prompted by contemporary socio-political events, canonical Maghrebian writers such as Driss Chra\u00efbi and Assia Djebar wrote fictional historical accounts of Islam's beginnings.2 As noted by Edward Said in _Covering Islam_ , \"it is not difficult to imagine that a Muslim might be made uncomfortable by the relentless insistence\u2014even if it is put in terms of a debate\u2014that her or his faith, culture, and people are seen as a source of threat, and that she or he has been deterministically associated with terrorism, violence, and 'fundamentalism' \" (xxi). A new introduction to the 1997 revised edition ascertains that nothing has really changed in the more than twenty-five years that have passed since the book first came out in 1981. Given the amalgam that is commonplace between Muslims and Arabs in Western popular culture,3 one can extend Said's statement about Muslims to Arabs regardless of their faith: in other words, Arabs might be uncomfortable that their ethnicity is seen as a source of threat.\n\nWith Islam (and the Arab world often conflated with it) taking center stage as the post-Cold War enemy, and the much controversial thesis of the clash of civilizations regaining notoriety after 9\/11, how have writers from the Arab world represented various conflicts, which have been framed in religious (Islam\/Christianity), civilizational (Islam\/Christendom), and imperialist (East\/West) terms? What are the paths mapped by literary texts out of the political turmoil that sparks new national constructions and allegiances?\n\nFrancophone writers from the Arab world are often cast in an uneasy position of intermediaries between their culture and country of origin on one side, and their mostly French readership on the other, as many of these writers reside in France and publish their work there. These writers, who hail from parts of the former French empire, often find themselves in a delicate situation between the colonial legacy of their language of literary expression and the conflation in the media and (popular) culture between the Arab and Muslim world and fundamentalism and terrorism. Yet, given that they occupy a liminal space between two worlds that are often posited in antagonistic terms, they seem to be in an ideal position to foster cross-cultural understanding, and to map the future coexistence of diverse groups in their communities.\n\nHowever, it is not certain that they are always successful in providing insights into the complexity of the resurgence of religion, and particularly when Islam is involved. Postcolonial theory has been noted as deficient when it comes to specific issues linked to the Arab-Muslim world (Majid, _Unveiling_ 19-21; see also Hassan in chapter one). In French and Francophone Studies, the term Islamic fundamentalism is, with rare exceptions, used indiscriminately as an unquestioned blanket term to refer to all aspects of the resurgence of religion. Anouar Majid already noted a decade ago in a brief article that, \"Their [intellectuals and writers in the Islamic world today] uncomplicated, one-dimensional reading of the Islamic resurgence, makes them, unwittingly or not, accomplices of the West and the ruling elites in almost all Arab and Islamic countries\" (\"Islam\" 87). Although he does not focus on Francophone literature, several of the examples he gives of writers whom he denounces for giving monolithic representations of Islam are Francophone (Ben Jelloun, Accad, Mimouni), and are closely examined in this study.\n\nFocusing on the intersection of Islam, politics, history, and Francophone literature, this book examines how French and Francophone writers from the Maghreb (Morocco, Algeria, Tunisia) and the Near East (Lebanon, Syria) grapple with the intertwining of religion and politics in various conflicting contexts. To this end, this study analyzes various means of cultural production, including essays, novels, films, and comic books by Francophone Arab writers and filmmakers whose Christian (Evelyne Accad, Myriam Antaki, Andr\u00e9e Ch\u00e9did, Amin Maalouf), Jewish (Karin Albou, Albert Bensoussan, H\u00e9l\u00e8ne Cixous, Edmond Amran El Maleh, Albert Memmi), Muslim (Salim Bachi, Slimane Bena\u00efssa, Yamina Benguigui, Tahar Ben Jelloun, Rachid Boudjedra, Farid Boudjellal, Abdelwahab Meddeb, Rachid Mimouni), and secular (Le\u00efla Sebbar) backgrounds are emblematic of the diversity of the Francophone Arab world. It examines how these writers (dis)entangle religious, political, economic, and cultural factors, and how they articulate the past, present, and future of religious minorities in the Arab world, against the backdrop of the current international political context of Islamic fundamentalism and the persistence of Western prejudices against Islam. At a time when Islamic fundamentalism is making regular headlines, the book focuses on texts that deal with disputes commonly framed in religious terms (with Islam as the common denominator for all) in order to describe how the Francophone Arab world has responded to the discourse that pits Islam against the West.\n\nWhile the cultural production of countries constituting the Francophone world is expressed in many languages, there is a disproportionate attention given to the one in French. By virtue of their publications' location and through translation of their work, these writers shape the French (and other) public's (mis)understanding of various conflicts allegedly stemming from Islamic resurgence. Their influence extends beyond hexagonal borders through translation of their work, and thanks to the rise of Francophone Studies, in the U.S. and the U.K. in particular. This book builds on Richard Serrano's assertion that the Francophone Studies model skews our understanding of those regions (6-7), a fact already noted by social scientists such as Fran\u00e7ois Burgat who points out that \"a French-speaking Maghreb . . . has masked the emergence of an impertinent new Maghreb, which today wants to use a new language\" (5). Many writers who have been trained in the French secular tradition fail to understand and give insights into what has been broadly labeled as the Islamic Revival that has been taking place in the Muslim world in the past decades. As we will see in the case of the Algerian and Lebanese civil wars, Francophone writers are not always able to render the complexity of some events. In cases involving the complexity of the resurgence of religion, there is a remarkable downplay of the role and impact of European colonialism in shaping current events in parts of the Arab world, and this under the pen of writers who had vigorously denounced colonialism in the past. Following the failure of secular ideologies such as pan-Arabism and Marxism, these writers are for the most part unable to see the resurgence of Islam divorced from political claims. In other words, when Islamic rhetoric is linked to political issues, they point out the political events and their effects, but they fall into the trap of thinking that Islam can only be a means to achieving certain political goals. They fail to portray the fact that for many, Islam is not just a cultural background but also offers spiritual guidance. This shortcoming could be due to some degree of anxiety about maintaining their status as privileged interpreters.\n\nThe book is organized thematically around a series of conflicts in the Arab world (or linked to its diaspora) that have made headlines during the second half of the twentieth century. Some conflicts are construed as internal, such as the Lebanese civil war (a Christian\/Muslim Arab conflict) and the Algerian civil war (Islamic fundamentalism against secular democracy); others as external: the Arab-Israeli conflict, and Islam's advance in the West through immigration (the headscarf affair in France).\n\nChapter 1, the Arab-Israeli conflict, examines works that turned to the past as a way to take attention away from the discourse on terrorism and back to the root causes of the Arab-Israeli conflict. The first part of the chapter examines Maalouf's _Les croisades vues par les Arabes_ [ _The Crusades through Arab Eyes_ ], a book that challenges the clash of civilization discourses (now and then) that pit a Christian West against a Muslim East, and that also provides a commentary on contemporary politics in the Middle East. The second part analyzes representations of the conflict through a family story and contrasts the use of genealogies as conflict resolution in two novels: Maalouf's _Les \u00e9chelles du Levant_ [ _Ports of Call_ ] and Antaki's _Les versets du pardon_ [ _Verses of Forgiveness_ ]. In Maalouf's book, the union between Jewish and Arab characters functions simply as a sign of hope for future conflict resolution. Antaki's recourse to the genealogical paradigm goes beyond that; it challenges stereotypes and points towards a first step to conflict resolution by revisiting the past to move towards a future beyond the ideology of separation.\n\nChapter 2 focuses on the geographic displacement of the Arab-Israeli conflict by analyzing the representation of its impact on Maghrebian countries and contemporary France. It first compares two Arab Jewish writers' take on the identity of the Arab Jew in essays and articles, notably Albert Memmi's _Juifs et Arabes_ [ _Jews and Arabs_ ] and _Portrait du d\u00e9colonis\u00e9 arabo-musulman et de quelques autres_ [ _Decolonization and the Decolonized_ ] as well as various pieces published by Edmond Amran El Maleh. The chapter also addresses the correlation between these writers' positions on the Arab-Israeli conflict, noting the relative prominence of each writer (Memmi's renown and El Maleh's obscurity). The second part of the chapter contrasts the representation of the relations between Arabs and Jews in France in Farid Boudjellal's _JuifsArabes_ [JewsArabs], a comic book, and Karin Albou's film _La petite J\u00e9rusalem_ [ _Little Jerusalem_ ]. It aims to demonstrate that, contrary to Albou's film, Boudjellal's comic book participates in an endeavor to go beyond the ideology of partition and separation that has dominated approaches to the Arab-Israeli conflict.\n\nChapter 3, on the Lebanese civil war, examines how Francophone Lebanese writers have depicted the conflict, and more specifically how they have dealt with the religious dimension of a civil war that was summarily depicted in the West as a Muslim\/Christian rift. It argues that two novels, Ch\u00e9did's _La maison sans racines_ [ _The Return to Beirut_ ] and Accad's _L'excis\u00e9e_ [ _The Excised_ ], reinforce the reductive view of the Lebanese civil war as a religious conflict. A close analysis of the Biblical and Qur'anic intertexts demonstrates that Accad's inscription of verses from the Qur'an provides a stereotyped view of Islam while the Biblical quotations implicitly reaffirm Christian values of sacrifice and redemption.\n\nChapter 4, on the Algerian civil war, starts by analyzing the representation of the rise of the FIS and the civil war in Mimouni's _De la barbarie en g\u00e9n\u00e9ral et de l'int\u00e9grisme en particulier_ [On barbarity in general and on fundamentalism in particular] and Boudjedra's _Le FIS de la haine_ [the Islamic Salvation Front of hate]. It demonstrates that both essays do more to feed the flames of fear than to further our understanding of the events under scrutiny. The second part of the chapter analyzes how the civil war intrudes in _Une enfance alg\u00e9rienne_ [ _An Algerian Childhood_ ], a collection of short autobiographical narratives edited by Sebbar that features pieces by writers from diverse backgrounds who spent their childhood in colonial Algeria. It shows that the effects of colonialism are mostly erased or downplayed, while the violence that surfaces is mostly associated with the Algerian nationalist movement during the war for independence and with the rise of Islamic fundamentalism. It also draws out the implications of this shift of perpetrators of violence given the context of publication of the volume.\n\nChapter 5 examines how Francophone Arab writers have responded to the French Muslim headscarf affair and the discourse surrounding Islam's (in)compatibility with the secular Republic in non-fiction works. It demonstrates that prominent writers such as Ben Jelloun, Maalouf, Meddeb, and Memmi, have not adequately situated the debate in light of French (post)colonial history and have failed to provide a balanced view about the so-called Islamist challenge to the secular French Republic. This chapter then contrasts the more complex representations by Sebbar and Benguigui, and draws out the implications from the absence of pictures of veiled girls in works that include text and images.\n\nChapter 6 looks at the representation of 9\/11 in Francophone novels from the Arab world and compares the function of extensive quotations from the Qur'an in Bena\u00efssa's _La derni\u00e8re nuit d'un damn\u00e9_ [ _The Last Night of a DamnedSoul_] and Bachi's _Tuez-les tous_ [kill them all]. These two 9\/11 novels imagine the itinerary of a 9\/11 hijacker leading up to the attacks. I argue that the Islamic intertext of Bena\u00efssa's novel gives fodder to proponents of the clash of civilization discourse, while Bachi's multicultural intertext accounts for the complexity of factors behind the 9\/11 attacks.\n\nThe conclusion of the book underscores the impact of 9\/11 on the English translation and packaging of some of the texts treated in this study. Reflecting on the spike of American interest in the literary production of the Arab and Muslim world, the book offers a critical evaluation of this trend's impact and implications in general, and for Francophone Studies in particular.\n\nThis study provides a critical examination of representations of the role of religions in conflicts from the perspective of the Arab Francophone world at a time when, in the wake of 9\/11 and the rise of Islamic fundamentalism, it is most needed. Drawing on a variety of methodologies, such as cultural studies, international relations theory and conflict resolution, the study also brings heightened awareness to the impact of literature on our understanding of contemporary issues as well as on the modalities according to which a literary work can serve as a cultural mediator. Finally, the focus on the topic of religions and conflicts in Francophone literature by Arab writers is particularly useful for examining how the Francophone studies model frames current issues and for evaluating its effectiveness in giving a complete picture of religious issues involving Islam.\n\n### Notes\n\n1. For a summary, see the new introduction to Said's revised edition of _Covering Islam_ , especially pages xii-xiv, which offer an abbreviated list of events between 1983 and 1996.\n\n2. See Chra\u00efbi's _L'homme du livre_ and Djebar's _Loin de M\u00e9dine_ , and my _Coran et tradition islamique dans la litt\u00e9rature maghr\u00e9bine_ for an analysis of these texts.\n\n3. The terms Arabs and Muslims are sometimes used interchangeably, although Arabs (among whom there are Christian and Jewish minorities) comprise only about twenty percent of the Muslim population nowadays, and despite the diversity of Muslim countries (which extend from Morocco to Indonesia).\n\n## Chapter 1\n\n## The Arab-Israeli Conflict: Amin Maalouf's _Les croisades vues par les Arabes_ and _Les \u00e9chelles du Levant_ , Myriam Antaki's _Les versets du pardon_\n\nThe Palestinian-Israeli conflict remains one of the most intractable clashes of the twentieth century, and unfortunately shows no sign of being resolved any time soon. In the Spring of 2008, the state of Israel celebrated its sixtieth birthday, a day commemorated by one side as \"Independence day\" and mourned by the other as the \"Nakba\" (catastrophy). This chapter examines books that are focused on the past in order to take attention away from the discourse on terrorism and back to representations of the root causes of the Arab-Israeli conflict and the different contexts that shape these representations.\n\nIn a collection of articles published in the 1980s, Said and other scholars analyzed various processes by which Palestinians, victims of a mass dispossession in 1948, are turned into essentialized aggressors. Summarizing an article written by the Israeli journalist Amnon Kapeliouk in 1986, Said noted that \"as Palestinian nationalism acquired regional and international credibility in the mid-1970s, Israeli officials consciously adopted the policy of characterizing it as 'terrorism.' [. . .] the word 'terrorism' was a political weapon designed to protect the strong (and eliminate from memory exploits of 'former' terrorists like Begin and Shamir who now run Israel) as well as to legitimize official military action against innocents\" (\"Introduction\" 13). In the same book, Noam Chomsky detailed how U.S. and Israeli officials, along with mainstream media, routinely labeled any acts of resistance as terrorism, thereby justifying retaliation, in the context of the 1980s' Israeli invasion of Lebanon and U.S. involvement in Nicaragua, all the while occulting their own terrorism abroad. As early as the 1980s, Said pointed out that terrorism had \"displaced Communism as public enemy number one\" in American public discourse (\"The Essential\" 149). The prevalence of the term serves to put aside historical evidence (\"The Essential\" 149).\n\nCommenting on a 2002 piece by Said, Stephen Morton notes that there is \"a historical relationship between imperialism and the discourse of terrorism\" (36). Morton recalls that the figure of the contemporary terrorist is often summoned as the reason for the American and British invasion of Afghanistan and Iraq, as well the backing of Israeli occupation of the West Bank and Lebanon (36). He goes on to state that \"Such a causal logic conceals the fact that the threat of terrorism is an instance of metalepsis: an effect of colonial discourse that is presented as a cause\" (36). In other words, terrorism is supplanting Orientalism as a justification for imperialism: \"If orientalism provides the sovereign power of the colonial state with a discourse of otherness to justify the suspension of the rule of civil law in times of crisis, [. . .] the contemporary discourse of terrorism would seem to serve a similar function\" (Morton 36).\n\nThe rise of Postcolonial Studies has brought to the fore the plight of most indigenous people who fell under the yoke of European imperialism. However, it has, as Salah Hassan has underscored, neglected the issue of Palestine. As Hassan points out, \"Postcolonial disengagement from Palestine is especially striking when one considers the role of Edward Said in shaping the field\" (33), given that Said's _Orientalism_ is considered a founding text of the field. In addition to his work on literary theory and comparative literature, Said has dedicated several books to the issue of Palestine.1 A recent issue of _PMLA_ (October 2006) tangentially started to redress that neglect, when it published the proceedings of a conference entitled \"The Humanities in Human Rights: Critique, Language, Politics,\" held in October 2005. Two contributors (Omar Barghouti and Alisa Solomon) analyzed the dehumanization of Palestinians in Israel and in U.S. media. In the following year, Gaurav Desai mentions the Palestinian-Israeli conflict as one example of the politics of indigeneity for which postcolonial studies \"will have to articulate new ways of thinking\" (642). Despite these brief mentions, unease remains over the issue of the Palestinian question and whether Israel can be seen as a colonial power.2\n\nPostcolonial Studies triggered the growth of Francophone Studies, which is also characterized, incidentally, by a lack of interest in the Near East. Despite the expansion of the areas under the purview of the field of Francophone Studies, the Francophone production of the Levant has somewhat been neglected, in part because of the paucity of writers of French expression from that region compared to that of the Maghreb. This chapter hopes to fill a gap in both fields by focusing on a topic (Palestine) and writers (Maalouf, Antaki) that have been left at the margins. The first part examines Maalouf's _Les croisades vues par les Arabes_ , a book that challenges the clash of civilization discourses (now and then) that pit a Christian West against a Muslim East. The rewriting of history in this book is grounded in and comments on events taking place in the Middle East contemporary to the time of its writing. The second part analyzes representations of the conflict through the use of a genealogical paradigm in two novels. In Maalouf's _Les \u00e9chelles du Levant_ , the union between a Jew and an Arab functions as a sign of hope for future conflict resolution. Antaki's _Les versets du pardon_ goes beyond that; it challenges stereotypes and points towards a first step to conflict resolution by revisiting the past to move forwards into a future beyond the ideology of separation. I unpack the genealogy that portrays the Arab-Israeli issue to show the various allegorical levels that combine to entangle history, religion, politics, and terrorism. I demonstrate how Antaki's work stands out as she invokes a means to conflict resolution grounded in scriptures.\n\n### Origins: Rewriting History in Amin Maalouf's _Les croisades vues par les Arabes3_\n\nMaalouf, a Lebanese writer who won the prestigious Goncourt prize in 1993 for his novel _Le rocher de Tanios_ , has received little attention from scholars of Francophone literature in the United States. As a Christian Arab exiled in France since 1976, Maalouf occupies a pivot point between his country of exile and his region of origin. While examining his identity in the essay _Identit\u00e9s meurtri\u00e8res_ , Maalouf underlines the paradox of being a Christian with Arabic, the sacred language of Islam, as his native tongue (23-24). These \"appartenances multiples\" 'multiple belongings' ( _Identit\u00e9s_ 40) that characterize \"frontaliers\" 'border people' ( _Identit\u00e9s_ 46) afford him a singular vantage point from which to take a fresh look at the historical period of the Crusades from 1096 to 1291.4\n\nThe title of _Les croisades vues par les Arabes_ problematizes the notion of objective historiography, and makes explicit Maalouf's intent: to adopt the perspective of those who suffered from the Crusades,5 which in Western eyes are still seen as a great epic. Indeed, the term \"crusade\" in today's English is used, as is _croisade_ in French, to refer not only to the historical military expeditions to the Near East, but also to denote any well-intentioned though possibly overzealous campaign for a worthy cause, and as such often carries a positive connotation. Maalouf's objective in his book is not simply to set the historical record straight: his rewriting of history is also intended as a commentary on contemporary politics. In this section, I explore how narrative strategies shape Maalouf's counter-history; why considerations of the problems inherent to the (re)writing of history are relegated to the background; and what motivates the rewriting of history in _Les croisades vues par les Arabes_ , given that the historiographical act takes place within a specific context and can thus be politicized.\n\nAs Linda Hutcheon puts it, there is a distinction to be made between events of the past and the historical facts drawn from them: \"Facts are events to which we have given meaning. Different historical perspectives therefore derive different facts from the same events\" ( _Politics_ 57). Because Maalouf writes in French, one can assume that his target audience is a Western reader to whom he wants to show a different version of the facts conventionally derived from the events that took place in the Middle East during the twelfth and thirteenth centuries.\n\nAgain, the title of Maalouf's book places it in opposition to others written on the same topic.6 Texts published by medievalists prior to the 1980s were mostly titled \"The Crusades\" or \"History of the Crusades,\"7 and purported to present a global, objective picture of these historical events. In contrast, Maalouf specifies in his title that he will examine the period from a restricted point of view. Maalouf's title thus challenges the supposed unicity of history that previous historical works on the Crusades seemed to take for granted. And while some European historians by mid-twentieth century had come to a more critical appraisal of the Crusades, there remained a gap between scholarly and popular views (Constable 2). Maalouf does not explicitly situate himself vis-\u00e0-vis his subject, hiding, instead, behind the first person plural \"we\" of academic narrative. However, his purpose is made clear in the Foreword, and while Maalouf did not have at the time of publication of _Les croisades vues par les Arabes_ the renown that he now enjoys, his name as author is easily identified as Arabic.8 The prologue and epilogue both emphasize his intent to set forth a little-known perspective and the influence it has had on interpreting contemporary events.\n\nMichel de Certeau and Hayden White, among others, have emphasized how the telling of the past is linked to contemporary ideological, political, or cultural factors (Hutcheon, _Poetics_ 120-22). In his ground-breaking text _The Writing of History_ , de Certeau argues that any return to history reflects preoccupations contemporary to its writing:\n\nOn ne saurait supposer non plus, comme elle [l'historiographie] tend parfois \u00e0 le faire croire, qu'un \"commencement\", plus haut dans le temps, expliquerait le pr\u00e9sent: chaque historien situe d'ailleurs la coupure inaugurante l\u00e0 o\u00f9 s'arr\u00eate son investigation. . . . En fait, il part de d\u00e9terminations pr\u00e9sentes. L'actualit\u00e9 est son commencement r\u00e9el. (18)  \nNor could anyone believe, as much as historiography might tend to have us believe, that a \"beginning\" situated in a former time might explain the present: each historian situates elsewhere the inaugural rupture, at the point where his or her investigations stop. . . . In fact, historians begin from present determinations. Current events are their real beginning. (Conley 11)\n\nAccording to de Certeau, the outcome or end of the story (the present) determines how the beginning (the past) of the story will be told (and which beginning as well). In Maalouf's text, there is no one-way movement between past and present: they are interconnected. The chronological move from past to present is made clear by the geographical changes illustrated by the two maps that frame the narrative: the first one appears right after the cover (and before the title page) and shows the Middle East with the boundaries of the Frankish states circa 1128; the second one, at the very end of the book, shows the borders of the Middle East after 1948. However, in the foreword Maalouf states that he chose the period of the Crusades because \"ces deux si\u00e8cles mouvement\u00e9s . . . ont fa\u00e7onn\u00e9 l'Occident et le monde arabe, et . . . d\u00e9terminent aujourd'hui encore leurs rapports\" 'those two centuries of turmoil . . . shaped the West and the Arab world alike, and . . . affect relations between them even today' (9, Rothschild, Foreword). Thus, the author starts from contemporary relations between the Arabs and the West as a contextual background for his research on the history of the Crusades, while in the epilogue he refers to the past he narrated as the root of contemporary tensions. The present context triggers a quest for the origin of a contemporary problem on which the past will shed light. The beginning and the end are inextricably interwoven, since the past will explain the present, which itself determines how and which part of the past is told.\n\nLike its author, _Les croisades vues par les Arabes_ is a hybrid: neither a scholarly historical book, nor a novel, nor a historical novel, although it has elements of all three. While reviewers called it a historical essay (as Maalouf did himself during an interview; see Sassine 25), the author in the foreword presents it as the \"roman vrai\" 'true novel' of the Crusades between quotation marks. This may be an allusion to the Goncourt brothers' preface to _Germinie Lacerteux_ (1865), in which they state: \"le public aime les romans faux: ce roman est un roman vrai\" 'the public loves fictitious novels! This is a true novel' (5, Chestershire 5). What Maalouf's text has in common with the first naturalist novel, which featured a woman from the working class as the main character, is that both works are based on real events and purport to tell stories from a class or ethnic background previously denied the spotlight. These scare quotes also underline the oxymoron created by juxtaposing the terms novel and true, simultaneously revealing and effacing the distinction between them, and reminding us of de Certeau's definition of historiography, that \"l'historiographie (c'est-\u00e0-dire \"histoire\" et \"\u00e9criture\") porte inscrit dans son nom propre le paradoxe\u2014et quasi l'oxymoron\u2014de la mise en relation de deux termes antinomiques: le r\u00e9el et le discours\" ( _L'\u00e9criture_ 5) 'Historiography (that is, \"history\" and \"writing\") bears within its own name the paradox\u2014almost an oxymoron\u2014of a relation established between two antinomic terms, between the real and discourse' (Conley xxvii). In addition to stressing the fact that historiography is itself grounded in history, scholars following in the wake of Hayden White have emphasized the commonality of features shared by historical discourse and literature. Every historical narrative is a linguistic construct that uses rhetorical devices to create a discourse of explanation and persuasion out of events or data. Narrative techniques and rhetorical figures shape the historical account and require analysis.\n\nMaalouf's stated desire to write the \"true novel\" of the Crusades shows that this work is not a new history book and acknowledges the apparent contradiction between the title (the Crusades according to one perspective), and the statement that this is an accurate account. His book is about how the Arabs who lived through the Crusades narrated and transmitted that experience to future generations, since the Arab historians quoted by Maalouf are contemporary to some of the events they relate. In a preamble to the section \"Notes and Sources,\" Maalouf offers the only commentary on his approach: \"Il va de soi que leur consultation [des r\u00e9cits historiques] nous \u00e9tait indispensable pour rassembler les t\u00e9moignages arabes, n\u00e9cessairement fragmentaires, en un r\u00e9cit continu couvrant les deux si\u00e8cles d'invasions franques\" 'It was obviously essential to consult them [historical narratives] in weaving the Arab testimony, which is inevitably fragmentary, into a continuous account covering the two centuries of the Frankish invasions' (285, Rothschild 268). The two classic (European) works that he cites are Grousset's and Runciman's. Clearly, Maalouf relies on Western historians to piece together the Arab testimonies scattered among annals and chronicles. As Franz Rosenthal explains, Arab historians wrote annalistic historiography, which simply records bare facts under a succession of individual years, with the exception of Usamah Ibn Munqidh, who wrote a memoir of his personal experiences. However, the Western scholars Maalouf cites also relied on the Arab chroniclers.9\n\nLikewise, Maalouf draws from all sources available to him, medieval Arab as well as contemporary Western historians. This underscores the fact that the same sources can yield different historical accounts, and that records are often insufficient or contradictory. As Chaim Perelman puts it, \"We can know the past only from the traces of it that remain\" (qtd. in Gossman 293). The overtly controlling narrator in Maalouf's text acknowledges the limitations of his project on several occasions. For example, there are two versions of the events that will push Saladin to lift the siege at Massiaf, a fortress in Syria. Maalouf adds before narrating the second version: \"Mais ce qui se passe en ce mois d'ao\u00fbt 1176 au pays des Assassins demeurera sans doute \u00e0 jamais un myst\u00e8re\" 'Exactly what happened in the land of the Assassins that August of 1176 will probably always remain a mystery' (199, Rothschild 182). Although most of the story proceeds in the narrative present or \"historical present,\" as it is often referred to in French and English, which gives the impression of immediacy and objectivity because narration seems to occur simultaneously with the events and thus precludes any interference from temporal distance, it alternates with passages that either anticipate or allude to future events or to the future consequences of what was just narrated. For example, the last chapter of the third part ends with these words: \"L'\u00e9pop\u00e9e du puissant Etat fond\u00e9 par Zinki semble achev\u00e9e. En r\u00e9alit\u00e9, elle vient tout juste de commencer\" 'The epic of the powerful state founded by Zangi seems over. In fact, it has only just begun' (155). This is an example of what G\u00e9rard Genette calls \"repetitive prolepsis\" which plays a role of announcement and creates a short-term expectation in the mind of the reader, which is fulfilled in the chapter that follows it ( _Figures III_ ). Maalouf ends the second part by quoting an Arab historian who uses the same device:\n\nEn \u00e9voquant, un si\u00e8cle plus tard, cette p\u00e9riode critique de l'histoire arabe, Ibn al-Athir \u00e9crira \u00e0 juste titre:\n\nAvec la mort de Toghtekin disparaissait le dernier homme capable de faire face aux Franj. Ceux-ci semblaient alors en mesure d'occuper la Syrie tout enti\u00e8re. Mais Dieu, dans son infinie bont\u00e9, eut piti\u00e9 des musulmans. (122)\n\nDiscussing this critical period of Arab history a century later, Ibn al-Athir would write with good reason:\n\nWith the death of Tughtigin, the last man capable of confronting the Franj was gone. The latter then seemed in a position to occupy all of Syria. But God in his infinite kindness took pity on the Muslims. (Rothschild 105)\n\nThis prolepsis within a prolepsis underlines the fact that Maalouf's narrative strategy inscribes itself in the tradition of Muslim historiography. In _Les croisades vues par les Arabes_ , most chapters end with some sort of foretelling of events that maintains the reader's interest; Maalouf presents us with a retrospective view of the past and interprets each event accordingly. These prolepses remind us that the rewriting of history is grounded in the here and now of the historian, and that this applies to the medieval sources as well, which are already interpreted accounts of the events they relate. As Christopher Tyerman notes, \"most medieval written primary sources were exercises in interpreting reality, not describing it\" (99).\n\nThe anecdote about the crusaders' cannibalism strikingly illustrates how context affects historiography. De Certeau has argued that any writing of History takes place in History: \"Il y a l'historicit\u00e9 de l'histoire. Elle implique le mouvement qui lie une pratique interpr\u00e9tative \u00e0 une praxis sociale\" 'There exists a historicity of history, implying the movement which links an interpretive practice to a social praxis' ( _L'\u00e9criture_ 29, Conley 21). The cannibalism that occurred at Maara is narrated in several Frankish chronicles of the time, where the army's chiefs ascribed it to hunger, as well as in European history books of the nineteenth century. However, the incident is usually occulted in the twentieth century.10 In the \"Notes and Sources,\" Maalouf alludes to the civilizing mission project to explain this phenomenon (287), but does not dwell on the issue. By relegating his comment to a note outside the main body of the text, Maalouf avoids the pitfall pointed out by Arif Dirlik in an essay about Eurocentrism and History: \"Critics of Eurocentrism inspired by cultural studies spend more time on what Euro-American writers and theorists have had to say about the rest of the world than they do speaking of the societies at hand, which further displaces the latter from the historian's attention\" (250).\n\nThat Maalouf subverts the pattern of dichotomization he himself established in the title by choosing to quote Frankish over Arab sources to tell this episode does pose a problem, however, for contrary to what the book sets out to do, it is the Franks we hear. Maalouf quotes the chronicler Raoul de Caen: \" _A Maara, les n\u00f4tres faisaient bouillir des pa\u00efens adultes dans les marmites, ils fixaient les enfants sur des broches et les d\u00e9voraient grill\u00e9s\" 'In Ma'arra our troops boiled pagan adults in cooking-pots; they impaled children on spits and devoured them grilled'_ (55, Rothschild 39).11 Maalouf also quotes another Frankish chronicler, Albert d'Aix, as well as an excerpt from an official letter sent by the chiefs of the army to the Pope (39-40). This is not a case in which the only remnants of the past are to be found in the Franks' chronicles (either because of extermination or destruction of documents). Although historians point out that only a few Arab accounts of the first Crusades have been preserved, one can find a brief mention of the Franks' cannibalism in an Arab source (Kemal-eddine) translated by Michaud in his _Biblioth\u00e8que des croisades_ : \"les Francs, en proie \u00e0 la disette, \u00e9taient r\u00e9duits \u00e0 se nourrir de cadavres et des animaux qu'ils pouvaient se procurer\" 'The Franks, racked by dearth, were reduced to feeding themselves from cadavers and animals they could get' (vol. 4: 7). Maalouf knew this source, since Kemal-eddine's text is listed in the \"Notes and Sources.\" One can see two possibilities to explain Maalouf's choice: the quotations attributed to the Frankish chronicles contain more gruesome details, which accentuate the barbarism of the perpetrators, thus contributing to Maalouf's reversal of civilized and barbarian. Second, reliance on the Franks' own writings obviates disbelief by the French reader, as if acknowledgment by the very people who committed it should attest to the veracity of such a horrendous act.\n\nIt is evident from his statement about the cannibalism episode that Maalouf, whose passion for history permeates most of his novels, is well aware of the issues involved in any writing of history, in particular the subjectivity of the historian, who weaves various sources into a coherent narrative, and the literariness of historiography. However, he uses none of the metafictional devices that disorient the reader at the level of the narrative, and which are often found in postcolonial and postmodern writings to dramatize the issue of how one can know and write the past. Maalouf does use narrative strategies that underscore the literariness of the process of writing history, but his text is not paralyzed by constant self-reflexivity, nor does it pretend to have the scientific objectivity of positivist historiography by presenting itself as a novel. Maalouf has a story to tell, and it takes priority over epistemological questions about historical discourse.\n\nIn order to tell the victims' point of view in French, the language of the crusaders' descendants, various lexical changes are required. Maalouf borrows the word _Franj_ , an Arabic word still used nowadays in dialectal Arabic to designate the French, and Westerners by extension. The introduction of this Arabic word into the French language creates a feeling of strangeness for the Western reader concerning his\/her own identity, thus forcing him to see himself (or herself) with the Other's eyes.12 In commenting on the Arab chronicler Ibn al-Athir by summing up a lengthy quotation with \"Ils sont fous, ces Franj, semble dire l'historien de Mossoul\" 'These Franj are crazy, the Mosul historian seems to be saying' (89, Rothschild 73), Maalouf alludes to the popular _Ast\u00e9rix_ series, where the phrase \"these Romans are crazy\" is frequently put in the mouths of resisting Gauls (who have been construed as the ancestors of the French), referring to the Roman assailants. In addition to grounding Maalouf's analyses in twentieth-century popular culture, this appropriation ironically underlines the position of the Franks as invaders.\n\nPerhaps more radically, Maalouf repeatedly presents the point of view of the victims excluded from official French history by reversing the referents of the dichotomy \"civilized\/uncivilized.\" In a chapter entitled \"Un \u00e9mir chez les barbares\" 'An Emir among Barbarians' (Rothschild), Maalouf turns generally accepted ideas upside down by calling the crusaders non-civilized beings compared to the Arabs. He does justice to the advance of the Arab civilization over the Frankish one in areas ranging from hygiene to law, and quotes at length the Damascene chronicler Usamah Ibn Munqidh, who is shocked by the Franks' backwardness. Maalouf quotes a long exerpt from Ibn Munqidh in which two Frankish doctors are shown to be more effective at killing their patients than curing them (147-48), but makes no mention of two other cases of successful Frankish treatment that Ibn Munqidh reports right after the passage quoted by Maalouf (Ibn Munqidh 162-63). While both sides recognized the superior medical skills of Arab physicians (Hallam 86), Maalouf's selection presents a more uniform view of the Franks than the source he draws from. Derrida argued that a phase of reversal is necessary to deconstruct an imposed hierarchy:\n\nI strongly insist on the necessity of the phase of reversal, which people have perhaps too swiftly attempted to discredit. . . . To neglect this phase of reversal is to forget that the structure of the opposition is one of conflict and subordination and thus to pass too swiftly, without gaining any purchase against the former opposition, to a _neutralization_ which _in practice_ leaves things in their former state. (qtd. in Culler 165)\n\nAs Derrida makes clear, it is a necessary step to show the Franks as barbarians to deconstruct the Western image of the Crusades as a heroic, glamorous time before these two centuries can be evaluated in a balanced manner. Thus the sensitivity and humanity of some Muslim leaders are emphasized in contrast to the crusaders' savagery. Maalouf relates how in similar situations the Arab leaders were magnanimous while the crusaders were cruel. For instance, he contrasts the brutality of Renaud de Ch\u00e2tillon, who tortures the patriarch of Antioch and mutilates the Greek priests of Cyprus (156), to the humanitarian response of Saladin, who, moved to pity by the cries of a Frankish woman whose daughter was kidnapped, orders that the latter be found and restored to her mother. On another occasion, Saladin's generosity and magnanimity when he liberates the Frankish poor without asking for a ransom, and frees King Guy in exchange for a promise (on which the latter will renege) contrasts with the lies of the Franks, who resort to false religious propaganda to obtain reinforcements, and to Richard the Lion Hearted's cruelty when he massacres prisoners while in a similar situation Saladin had had Franks released (210-11). Other exactions by the Franks include the pillage of Constantinople, during which they killed priests and monks and looted churches, and the sack of Jerusalem's holy places in 1099, during which the Eastern Christians were evicted, the Jews burned in the synagogue, and the Muslims massacred. To this day, Arabs contrast the brutal taking of Jerusalem by the Franks in 1099 to the peaceful seizure of Jerusalem by Omar Ibn al-Khattab in 638 (51). However, Maalouf does not portray only magnanimous Muslim leaders facing cruel crusaders. Although Frederic II is considered an exception, Maalouf lingers on this emperor, king of Germany and Sicily, who spoke Arabic, admired Muslim civilization, respected the Islamic religion, despised the barbarous West, and who carried on an intellectual correspondence with the Emir of Cairo (226-30). The figure of Frederic II implicitly provides a model for positive East-West interaction.\n\n_Les croisades vues par les Arabes_ ends with a word whose import is immense in Arab culture: \"l'on ne peut douter que la cassure entre ces deux mondes date des croisades, ressenties par les Arabes, aujourd'hui encore, comme un viol\" 'there can be no doubt that the schism between these two worlds dates from the Crusades, deeply felt by the Arabs, even today, as an act of rape' (283, Rothschild 266). Rape has become a commonplace metaphor for conquest and colonization, and by using this trope Maalouf echoes the current trend in medieval historiography of viewing the Crusades as the first wave of European imperialism in the Middle East.13 However, one can see another reason, grounded in Arab culture, for this choice of metaphor. Although in the Middle Ages women on both sides were considered the property of their male guardians, and sexual violence against them was an attack on men's honor, in the twentieth-century Western world rape has become a woman's issue since, as Kathryn Gravdal suggests, women are no longer the property of men (144). This, however, is not the case in parts of the Middle East, where crimes of honor continue to shed women's blood to wash away the shame brought about by their allegedly (un)willing illicit sexual conduct. Rape, a crime legally punishable by death of the perpetrator in some Arab countries, dishonors the victim's whole family. But despite the similarity in the way rape was perceived in the Middle Ages by both parties, the Franks' behavior seemed more liberal towards their womenfolk compared to the Arabs, who were baffled by the fact that Frankish men let their wives interact with men, sometimes rather intimately. Maalouf quotes Ibn Munqidh's indignation upon observing that a Frank will let his wife converse alone with another man (148). Another incident narrated by Ibn Munqidh to corroborate his point (but not included by Maalouf) tells how a Frank had his wife's pubic hair shaved by a man (165-66). These incidents show that a similar concept of honor does not translate into comparable behavior for both parties. While the French no longer tie a man's honor to the sexual behavior of his female relatives, this principle exists to this day in parts of the Middle East, and may explain the choice of the comparison between the Crusades and rape to describe the psychological impact of the invasions on Arabs.\n\nRape is mentioned several times in the narrative, usually as the price that women have to pay for belonging to the defeated (11, 29, 173). Once, it is alluded to in a letter by the caliph al-Adid to Noureddin asking for help: \"Pour \u00e9mouvoir le fils de Zinki, le souverain fatimide a joint \u00e0 sa missive des m\u00e8ches de cheveux: _Ce sont_ , lui explique-t-il, _les cheveux de mes femmes. Elles te supplient de venir les soustraire aux outrages des Franj_ \" 'In an effort to move the son of Zangi, the Fatimid sovereign enclosed some locks of hair with his missive. _These_ , he explained, _are locks of hair from my wives. They beseech you to come and rescue them from the outrages of the Franj_ ' (185, Rothschild 169). Women's voices are reduced to body parts, which are given voice by their husband who uses them as tokens to appeal for help. The word \"outrage,\" which in a general context means a grave insult, takes on the meaning of rape in French when applied to women. It emphasizes the fact that rape is a crime that affects women specifically, and suggests that the Crusades were no exception to the fact that \"In war time, rape has always been more than a rhetorical figure\" (Higgins, _New Novel_ 108). The metaphor of rape relegates women to a silent, victim role, whose sufferings are significant only for the consequences they entail for their husbands' honor. And while the actual crime affects women first and foremost, the specific distresses that women endured are left out of the narrative.\n\nRape can be linked to the process of story telling. Lynn Higgins points out that \"in fiction and life, rape is a special kind of crime in relation to narrative . . . Murder is not a crime whose noncommission can be narrated. Rape, on the other hand, can be discursively transformed into another kind of story. This is exactly the sort of thing that happens when rape is rewritten retrospectively into 'persuasion,' 'seduction,' or even 'romance' \" (\"Screen\/Memory\" 307).14 In a fascinating study, Kathryn Gravdal has shown \"the cultural habit of conceptualizing male violence against women as a positive expression of love\" in French medieval texts (20). Rape is also a common trope in Orientalist discourse. Said points out that \"the relation between the Middle East and the West is really defined as sexual. . . . The Middle East is resistant, as any virgin would be, but the male scholar wins the prize\" ( _Orientalism_ 309). With a similar comparison that underlines unequal power distribution, Maalouf stresses the discursive violence of narratives that have portrayed the Crusades as an epic with heroic characters carrying out a noble goal, and the influence they have had in shaping the popular Western imagination about that era. The violence has been two-fold: on a literal level, as in any war, and on the discursive level. In a case of rape, if \"the question is not _who committed_ the crime, but _whether a crime occurred at all_ \" (Higgins, \"Screen\/Memory\" 307), the issue at stake in Maalouf's work is how the Crusades have been written into history and passed on as a glorious era despite the ideological prejudices that engineered them and the crimes committed.15\n\nThe rape trope used by Maalouf only emphasizes the absence of Arab women's perspective on the Crusades, for _Les croisades vues par les Arabes_ is a tale told by men about men. Only two women are briefly mentioned by name in Maalouf's narrative; both seized power and belonged to the elite. The first is Alix, daughter of the King of Jerusalem, who betrayed her father after her husband's death by trying to forge an alliance with Zangi (ruler of Aleppo and Mosul) in order to stay in power in Antioch. The other is Chajarat-ad-dorr, whom Maalouf portrays as a passive pawn in the hands of the Mamelouks, but who nevertheless stands out in the history of Islam as the first woman to be a ruling queen ( _Croisades_ 240-41). In _Sultanes oubli\u00e9es_ , Fatima Mernissi gives us a completely different account of Chajarat-ad-dorr's reign, in which the latter appears as a clever decision-maker, well aware of the limitations imposed on her by her gender, yet determined to circumvent them (145-62).16\n\nOne could argue that Maalouf, by refraining from describing rapes and other crimes at length, does not indulge in the narrative acts of which Aram Veeser accuses New Historicists, whose historical accounts that detail atrocities are said to have obscene or pornographic intentions, and teach only obedience and despair (qtd. in Rosello, \"Mich\u00e8le\" 5). However, Maalouf also passed over the few Arab women whose participation and resistance during the Crusades have left traces in records. Although he can be credited with having brought to light the Arab male viewpoint on this period, he shows no particular concern about the women's.17 While the absence of women's perspectives can be attributed to the lack of written testimony by them, some women's heroic deeds stood out enough to figure in Ibn Munqidh's memoirs, whom Maalouf quotes extensively on other issues. In his memoirs, Ibn Munqidh describes a couple of Muslim women warriors, another who kills her husband who had betrayed the Muslims to the Franks, and another who captured three Franks (153-59); he also gives an account of a woman who drowned after trying to escape from the Franks who took her captive (179). Maalouf ends up repeating in part what his own book is supposed to undo: by omitting what Arab women did and thought during the Crusades, he silences them out of History. The use of rape as a metaphor for the Crusades as they impacted men subsumes women's issues under the mantle of colonization.\n\nIf we accept the view that a religious motivation was the engine of the Crusades (to take the Holy Land back from the \"infidels\"), one may wonder why Maalouf did not entitle his book \"The Crusades as seen by the Muslims.\" Maalouf, himself a descendant of the Arab Christians who were doubly discriminated against during the Crusades, does not fit in the Manichean view of a Christian West against a Muslim Orient. The fact that ethnicity is foregrounded in the title challenges the view of the Crusades as a confrontation between two religions. On the one hand, Maalouf includes the Christians of the Orient on the side of the oppressed, since they were in fact twice victimized on several occasions. The double discrimination against Arab Christians is evident in the battle of Antioch: expelled by the Muslim Arabs for fear that they would betray them to the Western Christians, they were not welcomed with open arms by the crusaders, who treated them as inferior subjects and at best suspected them of sympathizing with their Muslim compatriots (as occurred when the Oriental Christians sided with Saladin during the seizure of Jerusalem in 1187). Arab Christians in other instances, such as the Copts during the seizure of the town of Bilbeis, were massacred along with the Muslims ( _Croisades_ 168). However, Maalouf remains silent about instances in which Oriental Christians welcomed or helped the crusaders (see for instance Grousset, _Histoire_ 117, 151, 156).\n\nMaalouf depicts a multi-ethnic society and highlights the dynamics of power that refute the vision of one homogeneous civilization fighting another. He describes alliances between Arabs and Franks, between Byzantines disappointed by the Franks and Arab emirs: emirs made deals with Franks against other emirs. Even the Frankish princess Alix, who never knew Europe and felt \"Oriental,\" rebelled against her father by trying to forge an alliance with Zinki in 1130 (131). The description of the alliances that are forged and broken during these two centuries shows the complexity of the political situation of the Arab world at the time, divided into numerous small kingdoms at war with each other.18 This instability reveals personal interests of the leaders that could prevail over religious or ethnic affiliations. The first Arab historians of the Crusades reported the series of wars with the Franks as one among other events happening at the time; they used the ethnic term \"Franks\" to designate the invaders, thus casting the invasions neither in a religious, nor civilizational light. Indeed the term \"crusader\" will not appear until the mid-nineteenth century (Sivan 10). Tyerman shows that up to the end of the twelfth century there is no clear distinction between pilgrims and crusaders (20-21), nor a universally accepted term to describe crusading activity, and this was the case until modern times (49-55).\n\nAlthough what came to be labeled Crusades was recorded as Frankish wars and invasions under the Arab chroniclers' quill, the latter clearly did frame the conflict as religious by identifying themselves as Muslims and calling on God. The epigraphs that introduce each of the six parts that compose the book, all quotes from Arab chroniclers or leaders fighting the Crusades, can be seen as an implicit comment on historiography. Maalouf's use of citations contrasts with Runciman's, one of his European sources, who starts all chapters of his three-volume _History of the Crusades_ with quotes from the Bible chosen to establish a link between Biblical events taken out of context and medieval times.19\n\nIn Maalouf's text, each epigraph summarizes the content of the part. For instance, the quote from Saladin, which contrasts the fierceness of the Franks with the passivity of the Muslims, frames the part that deals with the invasion (part I). These quotes fulfill most of the four functions attributed to epigraphs by Genette: commentary and justification of the title, commentary on the text itself (whose meaning it emphasizes), support of one's text thanks to the presence of a famous author's name, and a sign of culture and filiation, linking one's text to a specific intellectual and cultural tradition ( _Paratexts_ 156-60). These quotes are also striking in that the authors cited are well grounded in their religion, some calling upon God for the safety of their community; they see themselves as Muslims belonging to a homogeneous community of believers and not as Arabs. However, they contrast with the content of the various sections of the book in the sense that they create the false impression of a united, homogeneous Muslim world. Thus, while the quotes do inscribe Maalouf's narrative in the Arab cultural tradition and give it legitimacy, they highlight the discrepancy between the ideology they perpetuate (a united Muslim community threatened by Christians) and the actual facts (divided Arab leaders who fail to join forces against a common threat).\n\nIn _Les croisades vues par les Arabes_ , Maalouf demonstrates that civilizations are neither monolithic nor immutable by reminding his readers that Muslim culture was tolerant towards others well before the Western world was. Maalouf's text illustrates how civilizations have been in contact and have borrowed from each other well before the era of globalization, by emphasizing the Arabs' numerous (and often ignored) contributions to Western civilization.20 He dispels the simplistic view of the Crusades as a battle between Christendom and Islam, just as nowadays many intellectuals are attempting to refute the Manichean thesis of the clash of civilizations. In fact, one can read _Les croisades vues par les Arabes_ as a book proleptically countering Samuel Huntington's influential _Clash of Civilizations_. Criticized by Said for having \"journalism and popular demagoguery rather than scholarship or theory [as] his main sources\" (\"Clash\" 571), Huntington's essay, which first appeared in _Foreign Affairs_ in 1993, and subsequent book, claim that non-Western civilizations (Islamic and Confucianist in the lead) are the potential enemies of the post-Cold War era, when conflicts no longer divide along ideological lines, but are determined by culture. Huntington's theory enjoyed a renewal of interest as the Western media grappled to comment on the September 11, 2001 attacks (Cr\u00e9pon 8-9). Said forcefully debunks Huntington's notion that civilizations are monolithic and homogeneous, and points out how Huntington's view of a rigid separation among civilizations does not stand up (\"Clash\" 587, see also Cr\u00e9pon 3-61). Whereas Huntington's not so hidden agenda, according to Said, is to maintain American dominance over the world, or, according to Marc Cr\u00e9pon, to trigger a fear of Islam and China in the American reader (65-66), Maalouf's work intends to make one side understand the other better, and to promote a dialogue between cultures.\n\nWhile most of _Les croisades vues par les Arabes_ is devoted to showing how the Arabs experienced these events, the temporal indeterminacy of the title leaves room for the perspective to broaden in the epilogue, which sets out to expose the Crusades through end-of-twentieth-century Arab eyes. Hutcheon points out that the forewords and afterwords that frame nonfictional novels underscore the \"particular perspective that _transforms\" (Politics_ 82). The epilogue does not anchor the text in the narrow context of France (where Maalouf has been living) with its immigrant population and the frustrations that were to be expressed by the Arab minority in the 1983 \"marche des Beurs,\" but in the larger international context of Middle Eastern politics.\n\nMaalouf directs attention to the parallels that are commonly drawn in the Arab world between events of the twentieth century and the Crusades. This relationship between the barbarous Middle Ages and our so-called civilized present is commonplace in the Levant, in whose view events past and present resemble each other: thus Anwar Sadat is viewed as a traitor in the direct line of al-Kamel (who gave Jerusalem to Frederic II), and Israel as a new crusader state (265). Therefore, Maalouf's implicit goal is not only to remind the reader of what happened from the Arabs' point of view, but also why it is crucial to remember that distant past now: to emphasize that current events can be situated in a historical continuum of European involvement in the Middle East.\n\nThe historiography of the Crusades continues to be influenced by the context in which it takes place. Crusading ideology continued in the sixteenth and seventeenth centuries (Constable 6). The Crusades were then discredited in the eighteenth century for their fanaticism (Tyerman 111), but with the renewal of interest in the Middle Ages that ensued in the nineteenth century, came the beginning of scholarly research about them (Siberry, \"Images\" 372). Contemporary historians have linked this scholarship to the colonization taking place at the time. For instance, Michaud's interest in the Crusades in the nineteenth century seems in part to have been stimulated by his study of Napoleon's expedition to Egypt (Siberry, _New Crusaders_ 8). As Elizabeth Siberry shows, historians, artists, and aristocrats established a continuity between the French colonization of Algeria, which began in 1830, and the medieval expeditions: \"It was no coincidence that at the time of the Algerian campaign, Louis Philippe was commissioning paintings of the medieval crusades and crusaders for the _Salles des croisades_ at Versailles\" ( _New Crusaders_ 82, see also Tyerman 117). When war broke out between the Druze and Maronites in Lebanon in 1860, Napoleon III called for a crusade to help the Christians (Siberry, _New Crusaders_ 83). The use of crusade terminology continued in the twentieth century, and was used during WWI when the Palestine campaign with the capture of Jerusalem in December 1917 by Allenby was described as a crusade (Siberry, _New Crusaders_ 87). Contemporary historians acknowledge that interest in the Crusades nowadays is still influenced by political and ideological interests (Constable 2; Siberry, _New Crusaders_ x; Armstrong 386), and Karen Armstrong even makes the case for the Crusades as a direct cause of today's conflict in the Middle East (xiii). Indeed, this renewed interest dates from the early 1950s and coincides with the creation of the state of Israel (Riley-Smith 5); there is even a debate among historians over whether or not in hindsight the crusader states should be regarded as colonies (Constable 20), or, as some Israeli scholars see it, as the \"first European colonial society\" (Tyerman 123). Emmanuel Sivan notes that all Arab scholars see a parallelism between the Crusades and current events, whether it is framed as a religious contest between Islam and Christianity, a civilization conflict of East and West, or as a first phase of Western imperialism in the Arab world (11-19).\n\nCurrent events lead Maalouf to make choices open to question regarding the historical events he chooses to discuss. For instance, he does not mention the crusades that were waged as the _Reconquista_ of Spain, with the fall of Cordoba in 1236 and Seville in 1248.21 Whereas Maalouf chose the period of the Europeans' incursions in the Levant, with the sack of Jerusalem as a culminating point, Bernard Lewis (a prominent Orientalist historian) sees the last years of the seventeenth century (marked by the second siege of Vienna) as the determining moments of the relations between the Muslim world and Europe (304). Each side sees the time when it was threatened or invaded by the other as the determining point in future relations, forgetting when it was itself the attacker, and thereby putting the blame on the other by privileging certain events over others. However, to give a full account of the relationship between the Europeans and the Arabs, one should go back to the initial point of conflict (the conquest of Spain in the eighth century and the Arabs' advance up to the French town of Poitiers), and include all subsequent confrontations.\n\nIn this essay, Maalouf does not treat at length historical figures who embody the crossing of civilizations that will become central to his work.22 The fact that _Les croisades vues par les Arabes_ , his first published book, does not dwell on the implication of mixed unions during the Crusades, and the fact that \"transculturation [was] common in medieval Mediterranean cultures\" (Kinoshita 114)\u2014a lot of which happened through concubine slaves\u2014is striking in the light of his subsequent fiction, pervaded by a concern with the meeting of cultures in history, minorities, and border people. Even more puzzling is the fact that the non-Arab background of some key \"Arab\" leaders and heroes of the fight against the crusaders is presented in the epilogue as one of the factors in the decline of the Arab world, as a sign that the latter had lost control over its destiny.\n\nAlthough in the end the crusaders were chased from the Levant, Maalouf does not present the Arabs as victors. Moreover, one can wonder, along with Mireille Rosello, about the purpose of writing the victims' history. As she points out, \"if the triumphalism of official history always at least partially serves the interests of the 'oppressors,' it does not necessarily follow that (historical) justice will be served by replacing the victor's story with that of the victim\" (\"Mich\u00e8le\" 5-6). In the epilogue, he draws up a brief assessment in which the Arabs are seen as victims, since the Crusades are the starting point for the rise of Western Europe while Arab civilization, which was the most advanced at the time, begins its decline.\n\nMaalouf brilliantly unsettled past constructions of the Crusades that informed popular perceptions of them in France without falling into the trap of simply inverting victims and villains, but his sketchy explanation of the popular view of the heritage of the Crusades in the Arab world presented in the epilogue does not do justice to the complexity of the events and politics of the twentieth century alone (admittedly, this would require another book).23 Moreover, given that Maalouf states that the Arab world is still a prisoner of the same shackles that caused its fall (lack of democratic institutions, problems of succession), one can see it as subject to a fatality, and create a defeatist feeling, even though one of the factors mentioned is debatable.24\n\nAlthough the history of the Crusades serves as an allegory of the present, with Arab leaders still unable to unite against a new invasion, the danger of this epilogue is that it comforts the Western reader in his\/her position of superiority. As Sharon Kinoshita points out, \"the Middle Ages have long served as a repository of the abject and the exotic against which modernity is constructed\" (111). So while French readers will no doubt acknowledge the barbarism of their ancestors, the superiority of contemporary Western civilization will nevertheless be confirmed, as if the torch of progress had been passed from the Arabs' hands to the crusaders', as Maalouf himself insinuates (264). Although the view of history repeating itself is pessimistic, the idea that the crusaders will be expelled could provide an optimistic note. However, Maalouf quickly downplays the 1291 Arab victory over the Franks, thereby cutting short the hope that there might be another victory in sight (279).\n\nGiven the brevity of the five-page epilogue, some of its oversimplifications seem inevitable, but some are particularly regrettable. Maalouf uses an image whose swift generalization about the Muslim world could use some nuances: \"Assailli de toutes parts, le monde musulman se recroqueville sur lui-m\u00eame. Il est devenu frileux\" 'Assaulted from all quarters, the Muslim world turned in on itself. It became over-sensitive' (282, Rothschild 264). The sensitivity-to-cold-weather metaphor reduces political and imperial moves to a natural climatological phenomenon, against which Muslims could only adopt a defensive attitude. The detailed accounts in the book that took great care to underscore the divergences and the power plays among various Muslim leaders during the times of the Crusades contrast sharply with the broad generalizations presented in the epilogue, which sweep away the complexity that characterizes the Muslim world of the end of the twentieth century.\n\nMaalouf puts all the blame on the Muslim world by citing examples of the two extreme tendencies of forced Westernizations (such as Turkey) alternating with fundamentalism (Iran), but fails to mention how Western powers have been direct or indirect contributors to these developments. He seems to hesitate between two positions. On the one hand, by stating that \"le monde arabe ne peut se r\u00e9soudre \u00e0 consid\u00e9rer les croisades comme un simple \u00e9pisode d'un pass\u00e9 r\u00e9volu\" 'the Arab world cannot bring itself to consider the Crusades a mere episode in the bygone past' (283, Rothschild 265), Maalouf seems to criticize the Arab world by casting it as hopelessly unable to move forward. On the other hand, one of Maalouf's rhetorical questions poignantly underscores the parallels between then and now: \"Comment distinguer le pass\u00e9 du pr\u00e9sent quand il s'agit de la lutte entre Damas et J\u00e9rusalem pour le contr\u00f4le du Golan ou de la Bekaa?\" 'How can one distinguish the past from the present in the struggle between Damascus and Jerusalem for control of the Golan or the Bekaa?' (283). The creation of the state of Israel has had a tremendous impact on Lebanon's already precarious internal affairs, because of the influx of Palestinian refugees and the PLO moving its headquarters to Beirut in 1970. The civil war that was still raging at the time Maalouf embarked on his project, as well as Israel's invasion of Lebanon, Maalouf's native country, the year prior to the publication of _Les croisades vues par les Arabes_ , constitute the backdrop to his revision of the Crusades.\n\nWhile this grounding of the author's writing corresponds to what de Certeau has termed the \"repoliticization' [which] will consist in 'historicizing' historiography\" (\"History\" 215), the epilogue undermines the project of the book by failing to insist on the continuity of Western imperial moves in the Middle East. Maalouf perpetuates uncalled-for stereotypes: after quoting the Turkish man who tried to assassinate the Pope in 1981 because the latter was \"commandant supr\u00eame des crois\u00e9s\" 'supreme commander of the Crusades' (283, Rothschild 265), he comments: \"il est clair que l'Orient arabe voit toujours en l'Occident un ennemi naturel\" 'it seems clear that the Arab East still sees the West as a natural enemy' (283, Rothschild 265-66). The adjective \"natural\" disregards the very real political events that have fostered resentment in the Middle East and essentializes the difference between East and West. This last sentence silences the plurality of voices and aspirations that exist in the Arab world, privileging anti-Western movements whose discourse is similar, as Cr\u00e9pon demonstrates, to Huntington's theory of the irreducibility of conflicts between different civilizations (54).\n\nWhile the first siege of Vienna by the Ottoman armies (a Muslim empire, though led by the Turks) in 1529 is mentioned as a sign (albeit deceitful) of the victory of the Muslims, there is no mention of the nineteenth-century colonization of the Maghreb, nor of the French mandates in Lebanon and Syria at the beginning of the twentieth century (and the British one over Palestine), as if there had been no continuity in the Western imperialist moves in the region. Indeed, the borders of the contemporary map that closes the book, drawn by imperial powers, are indelible traces of a colonial past. This very continuity is briefly alluded to in the phrase \"Dans un monde musulman perp\u00e9tuellement agress\u00e9\" 'In a Muslim world under constant attack' (283, Rothschild 265), and is singled out as a probable reason why the Crusades are \"deeply felt by the Arabs, _even today_ , as an act of rape\" (Rothschild 266, emphasis mine). The historical distance allows Maalouf to see the Crusades as the beginning of a long series of incursions of the West into the Middle East. But the consistent emphasis on the Arab world's shortcomings and euphemism to downplay the modern history of colonization of the Arab world and twentieth century conflicts undermine legitimate political grievances. This silencing of the modern colonial enterprise in the Middle East is also evident in his novel _Les \u00e9chelles du Levant_ , which I discuss in the second part of this chapter. I surmise that this may in part be explained by Maalouf's uneasiness regarding the rise of Islamic militancy in the region following the 1967 defeat of the Arab armies.\n\n### Genealogies as Conflict Resolution: Maalouf's _Les \u00e9chelles du Levant_ and Myriam Antaki's _Les versets du pardon_\n\n#### Maalouf's _Les \u00e9chelles du Levant_\n\nA nameless narrator (who, like Maalouf, was a Lebanese journalist) tells the story of Ossyane, a man he met by chance in Paris in 1976. Ossyane's Turkish father and Armenian mother wed at the beginning of the twentieth century, a time when relations between the two groups were antagonistic and would later lead to the Armenian genocide. After the Adana massacre in 1909, the family moved to Lebanon, where Ossyane grew up. While studying in France during WWII, Ossyane joined the French resistance and met Clara, an Austrian Jew. After the war, Ossyane returned to Lebanon. He sees Clara again, who is accompanying her uncle, the sole Holocaust survivor of her family, who decided to immigrate to Palestine. Ossyane and Clara get married while tensions are already ripe between Arabs and Jews. Right after Ossyane leaves Clara pregnant in Haifa to see his dying father in Beirut, the declaration of the state of Israel in 1948 takes place, putting an impassable border between them. Following his father's death and a sunstroke, Ossyane loses his mind and is committed to an asylum by his brother where he spends more than twenty years. When the war breaks out in Lebanon in 1975, Ossyane escapes and invites Clara to meet him in Paris. The narrator observes their meeting from afar, staying long enough to see that Clara did come, but leaves before seeing whether Clara and Ossyane will take separate ways or leave together.\n\n_Les \u00e9chelles du Levant_ starts by emphasizing the role of genealogy and history of the main character's destiny. When asked to start by his birth, Ossyane answers with a rhetorical question: \"Etes-vous certain que la vie d'un homme commence \u00e0 la naissance?\" 'Are you certain that a man's life begins with his birth?' (23, Manguel 15). He then proceeds with the narration of events that took place half a century before he was born (about how his grandmother became mad) and that will have a determining effect on his life. However, contrary to Maalouf's essay on the Crusades, which looks intricately at the various motivations of rulers and their positions, this novel does not dwell on the processes of history. Although the characters live through some of the most traumatic events of the twentieth century, events that have an undeniable impact on the course of their lives, there is not much historical information in this book (contrary to some of his other novels such as _L\u00e9on l'Africain_ ). The following passage describes the Ottoman Empire at the eve of World War I: \"\u00e0 Adana, comme dans toute l'Anatolie, d\u00e9butaient les massacres. La terre du Levant vivait ses moments les plus vils. Notre Empire agonisait dans la honte; au milieu de ses ruines poussait une foule de pays avortons; chacun priait son dieu de faire taire les pri\u00e8res des autres\" 'in Adana, as in the whole of Anatolia, the massacres began. Our land lived then through its most evil hour. Our empire was dying in an agony of shame; among its ruins grew a host of aborted countries, each one praying to its own god to silence the prayers of its neighbors' (45, Manguel 32). The definite article \"les\" for the massacres refer to common knowledge (here, the Armenian genocide). The use of the verb \"pousser\" 'grow' for the countries that are being carved by Western powers naturalizes historical processes. The reference to religion in the conflict stands in contrast with the main characters. Religion is relegated to the background in the narrative. Its significance is in signaling ethnicity rather than religious belonging per se, as when Ossyane defines himself as Muslim on paper (217). Although the characters do occasionally mention providence and destiny (137, 212), Ossyane and his father care very little for religion.\n\nHistory is personified and presented as a force over which people have no control: \"nous n'avons rien choisi, c'est l'Histoire qui a choisi pour nous\" 'We didn't choose, History had made the choice for us' (57, Manguel 42). In another passage, the creation of the state of Israel is presented as inevitable as a natural disaster by being described as a tornado (and a cyclone later on): \"Une tornade allait s'abattre sur le Levant, et nous voulions faire barrage de nos mains nues! C'\u00e9tait exactement cela. Le monde entier \u00e9tait r\u00e9sign\u00e9 \u00e0 voir Arabes et Juifs s'entre-tuer pendant des d\u00e9cennies, des si\u00e8cles peut-\u00eatre, tout le monde s'\u00e9tait fait une raison, les Anglais et les Sovi\u00e9tiques, les Am\u00e9ricains et les Turcs\" 'A tornado was about to ravage my part of the world, and we wanted to stop it with our bare hands. That is exactly how it was. The entire world had resigned itself to seeing Arabs and Jews kill one another for the next tens, maybe hundreds of years; everyone had got used to it, the English and the Soviets, the Americans and the Turks' (160, Manguel 122). Western nations (including Great Britain) are described as \"resigned\" when they were actually considerably implicated in the politics of Palestine. The same holds regarding the end of the British mandate: \"le mandat britannique sur la Palestine avait pris fin\" 'the British mandate in Palestine had come to an end' (177, Manguel 136), with the passive voice of the sentence silencing the fact that Great Britain decided to end its mandate because it could not cope with a situation that its contradictory agreements had created.\n\nAnother passage brings attention to the role Europe played in shaping the relations between Arabs and Jews: \"au lendemain m\u00eame de la d\u00e9faite du nazisme, deux peuples d\u00e9test\u00e9s par Hitler se dressent l'un contre l'autre\" 'soon after the fall of the Nazis, two groups detested by Hitler should take up arms against one another' (134, Manguel 102-3). Commenting the paragraph in which this sentence appears, Gil Hochberg points out that the reference to Hitler \"emphasizes the _racial_ affiliation between Jews and Arabs as (detested) Semites\" and notes that \"Europe, as the 'third party,' appears then as the 'real' enemy of both Jews and Arabs, who are too quick to forget their shared destiny by becoming enemies of each other\" (120). However, I would stress that by referring only to Nazism and Hitler, and forgetting about the long history of European colonialism in the Middle East, this sentence presents the role played by Europe as an aberration of history.25\n\nThe title of the book emphasizes that the place in which the story unfolds is as much a part of the story as the characters. As indicated on the back cover, the title refers to the cities that were meeting points between cultures and civilizations in the Mediterranean Levant. There is a mention by Ossyane as to whether longing for these multicultural societies is to be seen as nostalgic or futuristic (49). The mixing of peoples, a constant theme in Maalouf's work, serves as a stark opposition to the nationalist stances of various movements that promote so-called authentic and pure identities, such as pan-Arabism, Zionism, and Phoenicianism. Indeed, the history of Lebanon, succinctly summarized when the narrator recalls the material in his history book, underscores the multitude of layers that various civilizations have brought: \"l'Antiquit\u00e9 glorieuse, des cit\u00e9s ph\u00e9niciennes aux conqu\u00eates d'Alexandre; puis les Romains, les Byzantins, les Arabes, les crois\u00e9s, les Mamelouks; ensuite les quatre si\u00e8cles de domination ottomane; enfin les deux guerres mondiales, le mandat fran\u00e7ais, l'ind\u00e9pendance\" 'the glories of Antiquity, from the Phoenician cities to the conquests of Alexander; then the Romans, Byzantium, the Arabs, the Crusades, the Mameluk rule; later the four centuries of Ottoman domination; finally the two Worlds Wars, the French mandate, independance' (10, Manguel 4). Occupation is seen as an inevitable part of history: \"Je viens d'une r\u00e9gion du monde o\u00f9 il n'y a eu, tout au long de l'histoire, que des occupations successives, et mes propres anc\u00eatres ont occup\u00e9 pendant des si\u00e8cles une bonne moiti\u00e9 du bassin m\u00e9diterran\u00e9en\" 'I come from a part of the world where, throughout history, there has been one occupation after another, and my own ancestors occupied for centuries a good half of the Mediterranean' (79, Manguel 60).\n\nInter-ethnic marriages are recurrent in the novel, and more specifically unions between groups that are on the brink of fighting each other at particular times in history. These unions function as a common trope that symbolizes hope for possible reconciliation. Ossyane's father grew up in a cosmopolitan environment, embodied by his own genealogy: his maternal grandmother was the daughter of a fallen Turkish monarch, his grandfather's family was of Persian origin. He was educated by tutors of diverse backgrounds. The Adana massacre (which foreshadows the Armenian genocide) prompts Noubar, an Armenian, to give his daughter's hand to Ossyane's father, his best friend. Thus, the massacre is the trigger for the cement of the union between both groups. Like his father, Ossyane marries a woman who belongs to an antagonist group. Although their love story takes place against the backdrop of the violence triggered by the proposition of the partition of Palestine, Clara and Ossyane's wedding is symbolic of a possible reconciliation between Arabs and Jews (163). More than just the marriage is the attitude of both characters that is lauded when they talk about the conflict: \"chacun se mettait spontan\u00e9ment \u00e0 la place de l'autre\" 'we put ourselves, each of us, in the other's place' (169, Manguel 130). So is Clara's assessment of the conflict as a misunderstanding between the victims of the Holocaust and those who are paying the price for Europe's crime (134). This possible coexistence is also forecast with the emphasis on the cordial meeting between Ossyane's brother-in-law, Mahmoud, a Palestinian who had to leave Haifa because of the tensions, and Clara's uncle, Stefan, a European Jew and the only member of her family who survived the concentration camp. Stephan has recently arrived in Haifa, the very city which Mahmoud already predicts he will never see again (152). Nadia, Clara and Ossyane's daughter, becomes the synthesis of the different communities: she is Muslim on her father's side, Jewish on her mother's, and she claims both identities (217). However, Nadia leaves the Middle East to go and settle with her husband in Brazil, thus signifying that there is no place for people who claim multiple belongings (as Maalouf will term them in a subsequent book) in the Middle East (at least not yet).\n\nThe narrator mentions in passing that the notes he took while Ossyane was telling him his story were left in a folder for twenty years, but there is no other indication about what prompted the narrator to write Ossyane's story after such a long time. Since his encounter with Ossyane took place in 1976, the narrator is writing the story in 1996. I surmise that the euphoria that followed the Oslo Accords in 1993 played a role in bringing back to memory the story of an Arab and a Jew, and most importantly in enabling the possibility of this couple finally having a future together. However, the fact that the narrator refuses to stay to see if the couple will remain together shows a level of skepticism.\n\n#### Myriam Antaki's _Les versets du pardon_\n\nAntaki, a Syrian writer from a Christian background who lives in Aleppo, published _Les versets du pardon_ , her third novel, in Paris in 1999. Set against the backdrop of the Arab-Israeli conflict, the main character, Ahmed, is a Palestinian Muslim orphan. He is a self-proclaimed terrorist whose mother (Marie) and father (David) turn out to be Christian and Jewish, respectively. For the most part, the story is narrated in the second-person singular by Ahmed from prison, addressing in turn his father and mother. We learn that David, a French Jew whose parents were victims of the Holocaust, immigrated illegally to Palestine, had a relationship with Marie (a Christian Palestinian), joined the Irgoun (a Zionist terrorist organization), and helped in the making of the Israeli state by participating in events such as the King David hotel bombing. Marie, who becomes a refugee in Lebanon following the creation of Israel, gives birth to a boy (unbeknownst to his father). The newborn is abandoned in front of a mosque in a Palestinian refugee camp, where he is raised as a Muslim by a sheikh. This boy, Ahmed, eventually escapes the orphanage, and joins a combat training camp. His mother finds him and gives him his father's journal as well as a letter that she wrote to David, but never sent. The whole narrative is framed as a lyrical letter from Ahmed to his parents, written in prison where he has been tortured after he committed a bombing in Israel. The storyline is reconstructed by the son based on his parents' writings, from his prison cell where he is dying, and where he finally discovers his parents' identity.\n\nWith the obvious onomastics of the three main characters representing the three monotheist faiths that lay claim to the Holy Land, Ahmed's family tree looks simple, yet needs to be unpacked. What seems at first a straightforward allegory features several layers, and represents the complexity and intertwining of history, religion, politics, and terrorism in the Arab-Israeli conflict. In the first part of this section, I analyze the genealogy that portrays the Arab-Israeli issue to show the various allegorical levels that combine to entangle history, religion, politics and terrorism. In the second part, I examine how _Les versets du pardon_ challenges stereotypes and points towards a first step to conflict resolution. Hassan has persuasively argued that \"Just as the Peace Process is founded on the legacy of partition, postcolonial studies has reproduced the First World\/Third World cultural opposition. To break with this cartography of difference and inaugurate a new international politics, it will be necessary to find a critical language that can speak of the past without reiterating it\" (42). I examine how Antaki's genealogy revisits the past to move towards a future beyond the ideology of separation that dominates the peace process.\n\nIn postcolonial studies, the parent-child relationship is a common metaphor for the rapport between colonized and colonizer. As John Thieme points out, \"problematic parentage becomes a major trope in postcolonial con-texts, where the genealogical bloodlines of transmission are frequently delegitimized by multiple ancestral legacies, usually but not always initiated by imperialism\" (8). Thieme also notes that orphans and bastards, as a result of these problematic parentages, are plentiful in postcolonial texts.26 In _Les versets du pardon_ , Ahmed is considered as an orphan; he is also an illegitimate child whose parents belong to communities with antagonist claims on the brink of fighting each other. His family tree can be read as an allegory of a sketchy history of Palestine. Ahmed's father, David, a French Jew, is the European colonizer who intrudes in the native Palestinian Christian and Muslim family tree. When David arrives in Palestine, the religious sounds that dot the landscape belong to the Christian and Muslim traditions: \"Les muezzins chantent aux lueurs roses de l'aube. Les clochers des \u00e9glises, des couvents, carillonnent la naissance du jour\" 'the muezzins chant in the pink light of dawn. Bells of churches and convents ring in the new day' (83, de Jager 68). Since David enters Palestine in the early 1940s, by which time the Jewish population had risen to 30 percent of the population (Smith 151), one could expect to find signs of their sizable presence. By choosing to detail the sounds, Antaki can erase the presence of native Jews in Palestine without seeming to deliberately do so, since there is no sound originating in a religious building that is meant to be heard from outside in the Jewish tradition. David is representative of the fact that, by the end of WWII, the majority of the Jewish population in Palestine had immigrated there (Smith 151). The note on religious sounds emphasizes that the majority of the native population in Palestine was Christian and Muslim.\n\nPostcolonial studies have taken on various conceptual models to render the complexity of relationships engendered by colonialism. One is the difference between filiation (a given) and affiliation (a choice). These terms were brought to critical attention by Said, who draws a distinction between filiation (associated with biological descent) and affiliations (defined as cultural and social bonds) in his introduction to _The World, the Text, and the Critic_ (16-25). Contrary to what is commonly retained from Said's discussion, that is, that these are mutually exclusive terms,27 Said does recognize \"the verbal echo we hear between the words 'filiation' and 'affiliation' \" ( _The World_ 23). Said recommends that critics not be complicit with this pattern (the transfer of legitimacy from filiation to affiliation) but on the contrary \"recognize the difference between instinctual filiation and social affiliation, and to show how affiliation sometimes reproduces filiation, sometimes makes its own forms\" ( _The World_ 24).\n\nAlthough Said was discussing literary relationship when he made his distinction between filiation and affiliation, and the impact literature has on society and the role texts play in the socio-political context that produces them, it has found relevance in postcolonial studies. As noted by Ashcroft et al., \"the concept of affiliation is useful for describing the ways in which colonized societies replace filiative connections to indigenous cultural traditions with affiliations to the social, political and cultural institutions of empire\" (106). At the very beginning in _Les versets du pardon_ , Ahmed states that he was \"une cire molle qu'il fallait durcir\" 'soft wax that had to harden' (11, de Jager 3). This metaphor highlights that one's identity is not innate, but made, thus giving precedence to affiliation over filiation. Filiative terms abound in the novel, however, they are used to express affiliations, and more specifically, religious and territorial affiliations. Both Ahmed's territorial and religious affiliations (to the land of Palestine and to Islam) are expressed in filiative terms. Ahmed claims to be \"fils du Coran\" 'son of the Koran' (20, de Jager 11), thereby denying his filial Christian and Jewish heritage for affiliative identification with the sheikh who found him abandoned at the mosque's doorstep (15) and Islam. The sheikh also functions as a surrogate father, since Ahmed owes his life and identity to the sheikh who gave him a Muslim name (194). His Muslim affiliation is stronger than his Jewish filiation; indeed Ahmed denies his biological father upon discovering that he is a Jew when he exclaims in an apostrophe to his father: \"ce bonheur de te d\u00e9couvrir et soudain te perdre\" 'the joy of finding you and then suddenly losing you' (32, de Jager 21). Antaki inverses the pattern noted by Ashcroft et al. of replacing the filiative relationship to indigenous culture with affiliation to the empire: in her novel, affiliation to the native Muslim Palestinians takes precedence over his paternal connection to a European Jewish settler. This is a context in which politics (or affiliation) makes you reject biological link (filiation). Ahmed's rejection of his father is emblematic of a refusal by some Arab countries to recognize the state of Israel.\n\nFiliative terms are also used to express an organic link between the Arab people and the land. Ahmed also identifies himself as a \"fils de Palestine\" 'Palestine's son' (15, de Jager 7). All we know about his years in the orphanage is that he was taught by the sheikh to \"venger sa terre, sa m\u00e8re\" 'avenge [his] land, [his] mother' (15, de Jager 7). The land has become a surrogate mother to the boy who is raised as an orphan. After the UN's vote to partition Palestine, the Palestinians' dispossession is expressed in filiative terms: \"La terre des Arabes se d\u00e9chire et hurle, comme une m\u00e8re qui perd son enfant\" 'The land of the Arabs is dismembered and howls like a mother losing her child' (149, de Jager 131). In addition to the mother\/child relationship, the text resorts to botanical images. The relationship between Abdel Qader al-Hussayni, the Palestinian resistance leader who was killed in 1948, and the land is expressed as follows: \"C'est Abdel Kader qui aime sa Palestine, elle est sa s\u00e8ve, sa m\u00e8re\" 'It is Abdel Kader, who loves his Palestine which is his lifeblood [sap], his mother' (168, de Jager 149). Here the juxtaposition puts the filiative and botanical images on equal footing to describe the relationship between the Palestinian people and the land.\n\nThe botanical image evokes another model that has gained popularity in postcolonial studies: Gilles Deleuze and F\u00e9lix Guattari's rhizome (a root system that spreads across the ground and grows from several points) to counter the representation of thought and knowledge as a tree with a single tap root. As Deleuze and Guattari emphasize, the figure of the tree has dominated Western thought (27). Deleuze and Guattari state that \"Le rhizome est une antig\u00e9n\u00e9alogie\" (32) 'The rhizome is an antigenealogy.' The parallel with one of Said's terms is made explicit further on: \"L'arbre est filiation, mais le rhizome est alliance, uniquement d'alliance\" (36) 'The tree is filiation, but the rhizome is alliance, uniquely alliance.' Essentially, Deleuze and Guattari's botanical metaphor parallels Said's use of two terms that come from the same etymological root: filiation is the tree coming from a single tap root, while the rhizome is a perfect metaphor for affiliation. Moreover, both terms belong to the same category: just as affiliation comes from the same root as filiation, a rhizome belongs to the same category as a root of a tree. According to Fran\u00e7ois Noudelmann, \"La r\u00e9f\u00e9rence \u00e0 la racine est n\u00e9cessairement r\u00e9gressive, elle ne peut qu'ent\u00e9riner des identit\u00e9s univoques, qu'elles prennent le nom de n\u00e9gritude, de francit\u00e9, d'europ\u00e9anit\u00e9\" ( _Pour en finir_ 145). However, Antaki's use of the genealogical paradigm does not call for a lost purity or essential identity, but on the contrary, serves to validate a pluralistic society in historical Palestine.\n\nIn addition to its historical dimension, Ahmed's genealogy is a religious allegory of the continuity of Judaism, Christianity, and Islam. On the metaphorical level, this genealogy underscores the common ground and the connection among the three monotheist religions that lay claim to the Holy Land. This point is made several times during the course of the novel, through juxtaposition of phrases as in the following: \"une si vieille terre que les juifs disent promise, o\u00f9 le Christ est mort pour sauver les hommes, o\u00f9 l'Islam s'agenouille pour prier Allah\" 'such an ancient ground which the Jews call the promised land, where Christ died to save mankind, and where Islam kneels to pray to Allah' (70, de Jager 56). The importance of the city of Jerusalem for the three faiths is underlined through the mention of Prophets: \"Sous un m\u00eame ciel, toujours bleu, Salomon a voulu \u00e9lever un temple, J\u00e9sus s'agenouiller pour souffrir et mourir, Mahomet s'\u00e9lever au ciel\" 'Beneath this same, always blue sky, Solomon wanted to build a temple, Jesus knelt down to suffer and die, and Mohammed wanted to ascend to heaven' (124, de Jager 107). In both examples quoted above, the religions are mentioned in their chronological order of appearance, and the focus is on the land that is central to all three, and particularly Jerusalem, whose status is one of the most contested points. Intertextual references also emphasize the common ground between the three monotheist religions. A quote from Abdelkader's poetry acknowledges the fact that Islam believes in continuity between the Jewish, Christian, and Muslim scriptures (100). Ahmed refers to \"une terre o\u00f9 coulent le lait et le miel\" 'a land where milk and honey flow' (46, de Jager 34) for Palestine: this is a recurrent image in the Bible and in the narrative, where it designates the Holy Land. It also evokes the Qur'anic verse 47:15 that describes paradise as a place where rivers of honey and milk flow.\n\nIn addition, Marie's character is the link between David and Ahmed, and as emblematic of Christianity, an intermediary between Judaism and Islam. Marie is compared by Ahmed when addressing his father to \"une terre f\u00e9conde o\u00f9 croissent nos racines\" 'fertile ground in which our roots grow' (179, de Jager 159). Feminist research has shown how the allegory of woman as the land and\/or as the nation is highly problematic for female citizens. As Deniz Kandiyoti put it, \"women are the weakest link in national projects\" (387).The allegorical gendering of Islam and Judaism as male and Christianity as female is noteworthy. Marie's role can be seen as a passive Christlike figure, sacrificed by David's nationalist aspirations. The perpetrators of terrorism are Muslim and Jewish, and also male. In light of the framing of the conflict polarized along Jewish-Muslim lines, there is no regard for the place of Christian Palestinians. Thus, a parallel can be drawn between gender and religious issues: women's and minorities' rights take second place to decolonization and nationalist projects.\n\nAhmed's illegitimate status frames this religious allegory in the Judeo-Christian tradition (which does not recognize Islam, whereas Islam situates itself in the lineage of Judaism and Christianity). The fact that David will never know about Ahmed's existence can be emblematic of the following: Judaic theological teachings, Zionism's blindness to the fact that Palestine was not a land without a people after all, and Israel's refusal to officially acknowledge the repercussion that its creation had on the Palestinian people.\n\nTo the historical and religious levels of this allegory, one can add a political level: that literally and figuratively, Zionist terrorism breeds Palestinian terrorism. The incipit of the novel reads \"Je suis un terroriste, un r\u00eaveur\" 'I am a terrorist, a dreamer' (11, de Jager 3). The juxtaposition of the words terrorist and dreamer creates an oxymoron that emphasizes the gap between one's aspirations and a reality in which political frustrations and suffering foment terrorism. The novel constructs a chain reaction between the Holocaust, Zionist terrorism,28 and Palestinian terrorism. Cheik Al-Tahi, who teaches David about the past peaceful coexistence of the Jewish, Christian, and Muslim communities in the Holy Land, forecasts that the Palestinian people will be made to pay the price for the Holocaust. He tells David: \"Le monde a pers\u00e9cut\u00e9 les juifs, vous nous pers\u00e9cuterez\" 'the world has persecuted the Jews, and you [will] persecute us' (97, de Jager 81); \"Pour l'injustice de l'Occident envers vous, nous payerons de notre terre\" 'For the injustice of the West toward you we will pay with our land' (99, de Jager 83). Indeed, Marie's suffering at the hands of Zionism is described in a way that echoes David's suffering at the hands of Nazism, when both are exiled from their birthplace and suffer from thirst: David in the train (from which he escapes) en route to the concentration camp, and Marie on a refugee boat on her way to Lebanon (183).\n\nViolence is portrayed as the only means to draw attention to one's plight and to force recognition by the other. Ahmed says to his father: \"toi et moi nous avons beaucoup souffert avant de choisir. Nous avons aim\u00e9 la violence pour exister\" 'you and I have suffered a great deal before making our choice. We have loved violence so that we could exist' (116, de Jager 99). Violence is shown primarily as springing from thwarted nationalist aspirations and as a consequence of rootlessness and uprooting. Marie and David's relationship is described as \"un amour violent\" 'a vehement [violent] love' (11, de Jager 3), this oxymoron emphasizes how politics came to affect their relationship. During an exchange between Marie and David, Marie's dream about roots, in which she dreams of being a tree, predicts her upcoming exile following the creation of Israel:\n\n\u2014. . . les racines sont plus fortes que le temps.\n\n\u2014Mais Marie, l'homme fort pose des racines partout. (158)\n\n\u2014. . . the roots [of the country of your childhood] are stronger than time itself.\n\n\u2014But a strong man puts his roots down everywhere, Marie, (de Jager 139)\n\nThe use of the adjective \"fort\" 'strong' in this context is an understatement that both signals and softens the violence that will accompany the creation of the state of Israel. David's settling in Palestine following his exile caused by Nazism is done by violent means and causes Palestinian dispossession and exile, which in turn sets off violence against Israel. Children in the Palestinian refugee camp are \"sans racines\" 'rootless' (186, de Jager 166); exile and the rootlessness that ensues are seen as a cause of their recourse to violence. Ahmed explains that \"c'est dans la violence que je touche \u00e0 ma terre une seule fois\" 'it is in violence that just once I touch my land' (18, de Jager 9); as a Palestinian refugee in Lebanon, the only contact that is conceivable with the native land he longs for is one of violence, when he commits a bombing.\n\nDespite incompatible agendas, David and Ahmed are paradoxically united in their affiliation with political nationalist movement resorting to terrorism to achieve their nationalist aspirations (124). Their journeys bear uncanny resemblances: they both enter Palestine clandestinely, and resort to terrorism. Ahmed's intransigence in his determination to carry out attacks mirrors his father's, who was determined to expel all Palestinians from the land that was to become Israel. These parallel itineraries paradoxically strengthen the filiations that are rejected or unacknowledged. In making Ahmed's father an illegal immigrant to Palestine, as well as a Zionist terrorist, the novel puts to the fore the plight of Palestinian refugees (Ahmed is not allowed in Palestine) and their claims, while establishing a filiation between Zionist and Palestinian terrorism.\n\nReligious discourse is relegated to the background, but the religious dimension of the conflict is present. Statements such as \"la Jud\u00e9e, la Samarie, la Galil\u00e9e, une si vieille terre que les juifs disent promise\" 'Judea, Samaria, and Galilee\u2014such an ancient ground which the Jews call the Promised Land' (70, de Jager 56), and the banner that floats on David's boat stating that \"nothing can keep us from our Jewish homeland\" (73, in English in the original French text) indirectly recalls that the Zionist claim to Palestine is rooted in the Biblical scriptures. Ahmed's Christian lineage, contrary to his Jewish filiation, is not cause for sorrow when he discovers his parents' identity; this highlights that a political dispute over land is at the root of the conflict, not a religious antagonism, all the while pointing out the intertwining of politics and religions in the Arab-Israeli conflict, since some claims to the land are rooted in scriptures.\n\nThe potential for all three Abrahamic religions to incite violence or mercy is underscored: \"chaque geste de violence ou de mis\u00e9ricorde appartient \u00e0 un m\u00eame cri de Yahv\u00e9, Dieu ou Allah\" 'every gesture of violence or of compassion [mercy] belongs to a same cry of Yahweh, God, or Allah' (95, de Jager 79). Cheik al-Tahi symbolizes a tradition of tolerance in Islam. He teaches David about the incident when the prophet Mohammed instructed his followers to protect a church (166), and quotes from the Bible when God told the Prophet David not to build the temple of Jerusalem (109): \"Ne batis pas de maison \u00e0 mon Nom, car tu as \u00e9t\u00e9 un homme de guerre et tu as vers\u00e9 le sang\" 'Do not build a house in My Name, for you have been a man of war and you have spilled blood' (Chroniques 1, 28:3, de Jager 93). In both instances, religion wants to dissociate itself from violence. Nevertheless, the fact that religion can be manipulated to various ends is underlined by Ahmed who says that he learned violence from a sheikh (although not cast in religious terms since he is asked to \"venger [sa] terre\" 'avenge [his] land' (15, de Jager 7) while his father was taught tolerance by another (and yet became a terrorist). The narrative does not press the religious dimension further, since violence in this novel stems from nationalist demands.\n\nAfter the partition of Palestine has been voted on by the UN, \"les juifs, les chr\u00e9tiens et les musulmans ne peuvent plus se r\u00e9concilier que dans l'espace clos des cimeti\u00e8res\" 'Jews, Christians, and Muslims can only be reconciled together in the closed space of cemeteries' (152, de Jager 134), that is, once they are dead, or, as stressed in a passage in Athlit, a British clearance camp, under the colonial yoke. When David is caught and jailed with other Jewish illegal immigrants, Palestinian revolutionaries, Christian and Muslim clergy, Ahmed wonders about a picture that features several prisoners of the camp: \"P\u00e8re, es-tu avec eux dans cette photo parce qu'ils ont voulu fixer sur une image en noir et blanc les confessions de Yahv\u00e9, Dieu, et Allah?\" 'Father, are you with them in that picture because they wanted to freeze the faiths of Yahweh, God, and Allah in one single black and white image?' (90, de Jager 75). The British mandate on Palestine, when Great Britain is \"la ge\u00f4li\u00e8re de l'Orient\" 'the prison warden of the Orient' (77, de Jager 62), seems to be the only means of preserving all communities. However, David's relationship with sheikh Ahmed Al-Tahi, whom he meets in the British prison camp and who teaches him Hebrew and Arabic, is emblematic of potential coexistence through a fruitful relationship that is expressed in filial terms. In the course of several exchanges between the two characters, the sheikh keeps calling David \"mon fils\" 'my son' or \"mon enfant\" 'my child' (91, de Jager 109); David will call him \"mon cheik\" 'my sheik' and at one point \"mon cheik, mon p\u00e8re\" 'my sheik, my father' (99, de Jager 83).\n\nWhile the historical time frame of David and Marie's story can be easily determined, thanks to the mention of events such as the massacre of Deir Yassin, such is not the case for Ahmed's. The indeterminacy of Ahmed's historical time focuses the attention on the past as a way to map out a basis for mutual recognition. It stresses the first dispossession of the Palestinians of their land as well as the Holocaust, events that are conveniently forgotten or whose importance is routinely undermined for political purposes in Israel for the first, and in the Arab world for the second. It echoes Said's stance that \"the crucial issue for any discussion of Palestine has to be 1948, or rather what happened in 1948\" (\"Introduction\" 14). In a newspaper article that first appeared in 1997, Said urged both Palestinians and Israelis to engage in mutual recognition of one another's sufferings. According to Said, the Palestinians should not engage in minimizing the Jews' history of suffering, and specifically the Holocaust, especially given the impact of the latter on the creation of the state of Israel, and reciprocally, the Israelis need to acknowledge the dispossession that the creation of their state has entailed since 1948. Said states that\n\nJewish and Palestinian experiences are historically, indeed organically, connected: to break them asunder is to falsify what is authentic about each. We must think our histories together, however difficult that may be, in order for there to be a common future. And that future must include Arabs and Jews together, free of any exclusionary, denial-based schemes for shutting out one side by the other, either theoretically or politically. That is the real challenge. ( _The End_ 209)\n\nThe genealogy featured in Antaki's work, emblematic of some of the historical processes that define Palestine and Israel today, is a creative embodiment of the mutual recognition that Said urges.\n\nAntaki's use of the genealogical paradigm goes against the grain of attempts to conceptualize relations between communities outside of the family tree model. In _Pour en finir avec la g\u00e9n\u00e9alogie_ , Noudelmann points out the extent to which familial and genealogical paradigms govern, structure, and legitimize collective representations. Unlike other writers who endeavor to counter and\/or give alternatives to the genealogical model, Antaki uses it to construct a common heritage among the communities that lay claim to historical Palestine, thus stressing irreversible bonds between them. According to Noudelmann, \"l'affiliation r\u00e9trospective permet de souder une communaut\u00e9 en l'identifiant \u00e0 des anc\u00eatres communs\" 'retrospective affiliation enables a community to unite by identifying common ancestors' (\"Pour une pens\u00e9e\" 195). Antaki's novel features what seems at first a scandalous genealogy, a monstrous family tree where the men are both victims and villains, and enemies turn out to be related. The illegitimate family tree is not new in postcolonial literature; it symbolizes the irreversible hybridity that colonialism engendered. In the context of the Arab-Israeli conflict, Ahmed's genealogy is striking, because \"the Partition Plan reveals the success of Zionism in establishing its program of cultural differentiation as the dominant international approach to Palestine\" (Hassan 36). This ideology of separation is prevalent in the two-state solution that has the most currency to solve the conflict and posits the Arab Palestinians and the Jewish Israelis as peoples who must be kept separate. The recourse to this family tree somehow shows the irrevocability of the Jewish mark in Palestine, and the intertwining of filiation and affiliation in this text highlights that separation of the two people involved is unwarranted.\n\nAntaki's novel anticipates a new direction taken in the field of conflict resolution as mapped out by Marc Gopin. This new course comes from the growing recognition that \"there is a global resurgence of religion taking place throughout the world that is challenging our interpretation of the modern world.\" (Thomas 10). This resurgence challenges the modernization theory that assumes that modernization entails secularization (Thomas 2, Sayyid 4). Marc Gopin has emphasized that religions do play a role in perpetuating the Palestinian-Israeli conflict, be it Islam with the religious nationalist parties, Judaism with the settlements of occupied territories, and Christianity with the American Zionist Christians for instance ( _Holy War_ 6). Gopin therefore argues that it is crucial to take the various religious values into account if one is to achieve a viable peace process ( _Holy War_ 6).\n\nSince forgiveness is important in Judaism, Christianity, and Islam, Gopin explores the possible recourse to forgiveness as a means to conflict resolution. He notes that forgiveness, along with patience with human failing and infinite compassion, are \"basic characteristics of God in the Hebrew Bible, the New Testament, and the Qur'an\" (\"Forgiveness\" 89). Gopin argues that forgiveness should be accompanied with justice seeking and presented as a form of empowerment (which Judaism and Islam do) in order to be a successful step to conflict resolution.\n\nIn an interview, asked about how one goes beyond suffering, Antaki stated the following:\n\nEn travaillant sur la notion de pardon, ce que je fais tout en revendiquant ma la\u00efcit\u00e9 . . . faire le deuil des trag\u00e9dies est un processus lent, il faut des ann\u00e9es et des ann\u00e9es pour que l'oubli s'installe, que l'histoire s'enfouisse. Tandis que le pardon\u2014\u00e0 ne pas confondre avec le renoncement\u2014est une d\u00e9cision, une sorte d'acte volontaire qui permet d'agir sur le pr\u00e9sent pour envisager diversement l'avenir. (qtd. in Galesne, 159)  \nBy working on the notion of forgiveness, which I do while claiming my secularism . . . letting go of tragedies is a slow process, it takes years and years to sink into oblivion, to bury history. Whereas forgiveness\u2014not to be confused with renouncement\u2014is a decision, a kind of voluntary act that enables one to act on the present so as to envisage the future in various ways.\n\nDespite the author's self-professed grounding in secularism, the title of her novel refers to forgiveness grounded in faith, since the word \"verset\" 'verse' refers to religious texts. At the beginning of the novel, Ahmed is motivated by hatred (18), yet after reading his parents' texts, his hatred is \"morte, \u00e9teinte\" 'dead, extinguished' (20, de Jager 10-11). Ahmed presents his narrative as follows: \"mes derni\u00e8res paroles, \u00e9crites pour vous [p\u00e8re et m\u00e8re], sont les versets du pardon\" 'written for you, my last words are verses of forgiveness' (21, de Jager 12). At the end, the parents are reunited in the son's addressing both of them as \"vous\" 'you' instead of separately; the narrative pieces together different stories to reconstruct the family tree and reconciles the antagonisms fostered by the creation of Israel. After closing his father's journal, Ahmed dedicates a couple of paragraphs successively granting his father forgiveness and asking for the latter's forgiveness (179-80). Once again, father and son meet through similar acts: Ahmed tells his father that \"notre derni\u00e8re pri\u00e8re est celle du pardon\" 'our last prayer is the prayer of forgiveness' (180, de Jager 160), since his father ended his journal by asking for Marie's forgiveness. Towards the end of the narrative, in an apostrophe that addresses both of his parents, Ahmed states: \"Je sais, \u00e0 present, que je contiens dans ma violence, mon salut, le Verbe de Yahv\u00e9, de Dieu et d'Allah\" 'I know now that in my violence I hold my salvation, the word of Yahweh, God, and Allah' (194, de Jager 173). Since Ahmed was born on Christmas Eve, he seems to prefigure the return of the Messiah, or function as the sacrificed son that will redeem humanity's sins.\n\nDuring the course of the narrative, Ahmed comes to face and accept his Jewish filiation. Eventually, he acknowledges to his father that \"aujourd'hui, ce pays est pour toi et moi\" 'today this land [country] is for you and for me' (107, de Jager 91). One could object that Marie (and by extension women and\/or Christians) are forgotten in that last sentence, but in the context of the novel, Marie's place in Palestine is never an issue for Ahmed. While Kandiyoti noted that \"Wherever women continue to serve as boundary markers between different national, ethnic and religious collectivities, their emergence as full-fledged citizens will be jeopardized\" (382), in _Les versets_ Marie's character actually serves to blur such boundaries. Moreover, the emphasis on Ahmed's filiation puts the three characters on equal footing in ideal circumstances. In a passage in which Ahmed envisions a time when their family story would have a happy ending, he imagines that his parents are together in Jaffa, and that when he is born \"vous me tiendrez dans vos bras. Nous vivrons ainsi, sur la terre des promesses, pour planter l'arbre de la paix, du pardon\" 'you two will hold me in your arms. That is how we shall live, on the land of promises, to plant the tree of peace and forgiveness' (194, de Jager 173). Peace and forgiveness are juxtaposed to emphasize that both terms in this context are linked to one another. Ahmed presents his family tree as a metaphor for future reconciliation: \"je suis n\u00e9 de votre amour interdit qui, pour moi, n'est pas une faute mais une lune d'esp\u00e9rance\" 'I was born of your forbidden love that, for me, is not a flaw but a moon of expectation [hope]' (20, de Jager 11). Indeed, the novel ends by first repeating the very first paragraph of the novel, but ends with Ahmed stating that \"je suis le fils de David et de Marie\" 'I . . . am the son of David and Mary' (197, de Jager 176). Thus the story has come full circle when Ahmed accepts his filiation.\n\nWhile certainly not a solution to the conflict, which will require efforts from both parties, Ahmed's acknowledgement is in line with what Said, who recognized that Palestinians' dispossession could not be righted by the expulsion of Israeli Jews ( _Humanism_ 143), wrote. Said grounds his vision of coexistence in secular terms, and points out the fact that both in history and in the present, \"Palestine is an irreducibly mixed place\" ( _The End_ 318). He advocates the establishment of a secular state in which citizens have the same rights regardless of their ethnicity ( _The End_ 320). Antaki's genealogy is an illustration of the potential that reading Israeli and Palestinian histories side by side, as Said recommends ( _The End_ 319), can yield. But while Said leaves religion out of the picture\u2014actually saying that neither Palestinian nor Israelis \"should be held hostage to religious extremists\" ( _The End_ 320), Antaki finds in religion a basis for the first step toward reconciliation.\n\n### Conclusion\n\nBy going back to the Crusades and the events of 1948, respectively, Maalouf's and Antaki's books serve to bring the attention to the root cause of the Arab-Israeli conflict; namely, a violent occupation that is seen by Arabs as a continuum of Western imperialism, a fact that is consistently obfuscated. Both _Les versets_ and _Les \u00e9chelles_ attempt to change the historical frame through which the conflict is viewed, by tracing it back to 1948, not just 1967 or 2000, as a way to highlight the first violence done to Palestinians.29 Each book is tainted by its context. A pessimistic tone is evident in _Les croisades_ , which was written during the Lebanese civil war and Israeli invasion of Lebanon, whereas a more optimistic tone reigns in _Les versets_ and _Les \u00e9chelles_ , both written in the wake of the Oslo Accords and before their demise with the breakdown of the Camp David talks in 2000 and the second Intifada. The Oslo Accords constituted a moment that was at the very least touted as a possible beginning of the end of the conflict (despite serious reservations by many, including Said, who from the beginning pointed out that the flaws of the Accords were bound to set them for failure). As Ilan Papp\u00e9 noted, there was a flagrant disconnect between the images of Oslo propagated by politicians and the reality on the ground, but the illusion of Oslo as a viable peace accord lasted until 1996 (245). This period of optimism influenced both novels, while at the same time the absence of happy endings expresses a reservation as to a conclusion of the conflict in sight. This reservation transpires somewhat in the choice of a couple in Maalouf's novel, which opens the possibility of separation, estrangement, and divorce, as indeed happens in the novel, although the separation at first is due to events over which the characters have no control. However, Antaki's family tree emphasizes that the destinies of Arabs and Jews in the Middle East are irreversibly intertwined. In _Les versets du pardon_ , like in the literary works analyzed by Hochberg in her recent study, \"the tie between 'Jew' and 'Arab' . . . challenges the separatist imagination and proves the disjointing of 'Jew' and 'Arab' to be at least partially impossible\" (17). One can legitimately wonder whether such a utopian vision can truly contribute to resolving the conflict whose harsh realities on the ground have recently taken grimmer turns. I would argue that Antaki's vision constitutes a necessary first step in the sense that it opens the possibility of imagining a future together that contrasts with the dominant separatist ideologies of European colonialism, Zionism, and Arab nationalism.\n\n### Notes\n\n1. For instance, _The Question of Palestine, After the Last Sky, The End of the Peace Process, Peace and Its Discontents, The Politics of Dispossession_.\n\n2. Historian Ilan Papp\u00e9 wrote about immigration to Palestine at the end of the nineteenth century that \"Although [the Zionists'] number was small, it was in hindsight a colonizing immigration. It was not a proper colonization, as Palestine was not occupied by a European power. But like colonialism elsewhere, it was a European movement, with people entering Palestine for the sake of European interests, not local ones\" (42). Ella Shohat wrote that \"A series of mutually reinforcing equations between modernity, science, technology, and the West has legitimized Zionism as an extension of the civilizing mission applied first to Palestine and then to Arab Jews\" (\"Rupture\" 64).\n\n3. This section has appeared as \"The Rewriting of History in Amin Maalouf's _The Crusades through Arab Eyes_ \" in _Studies in 20th and 21st Century Literature_ 30.2 (Summer 2006): 263-287. It has been revised and augmented.\n\n4. Unreferenced translations are mine.\n\n5. In the decade that preceded the publication of _The Crusades through Arab Eyes_ , more than twenty percent of the historical novels published in France were set in the Middle Ages, with the Crusades as a privileged period (Pierre 34-35). While Marc Bertrand notes that one of the innovative aspects of recent historical novels is to pay attention to the minorities marginalized or persecuted for their differences, be it ethnic, religious, racial or cultural, none of the novels about the crusades that he mentions seems to present the Arabs' perspective. Examples of novels about the crusades mentioned by Marc Bertrand in his article are: _La joie des pauvres_ by Zo\u00e9 Oldenbourg (Gallimard, 1970), _La dislocation_ by Armand Farrachi (Stock, 1974), _La croisade des enfants_ by Bernard Thomas (Fayard, 1973), and _Le ma\u00eetre de Hongrie_ by Marcel Jullian (Table Ronde, 1975).\n\n6. The J'ai lu edition of this book bears the subtitle _La barbarie franque en Terre sainte_ 'Frankish Barbarism in the Holy Land' (Paris: J'ai lu, 1999). This subtitle does not appear in the original Latt\u00e8s edition of the book. This is probably a marketing ploy from the J'ai lu editions to use such a catchy subtitle (thus fulfilling what Genette calls the temptation function of a title in _Paratexts_ [93]), because it reverses the generally accepted idea of the crusades as heroic deeds.\n\n7. See Grousset, Runciman, and Oldenbourg.\n\n8. \"The I of historiography is supposed to be that of the writer whose name appears on the book cover\" (Carrard 87).\n\n9. In their bibliography, both Zo\u00e9 Oldenbourg and Steven Runciman divide their sources according to their origins: Oldenbourg categorizes them according to religious criteria (\"Oriental Historians\" [\"Historiens orientaux\"] and \"Christian Historians from the Orient\" [\"Historiens chr\u00e9tiens d'Orient\"]), while Runciman divides them according to linguistic criteria (\"Arabic and Persian sources\").\n\n10. Grousset's _Histoire des croisades_ passes over the case of the crusaders' cannibalism in silence, Runciman's _A History of the Crusades_ briefly alludes to it (vol. I: 261), and Oldenbourg's _Les croisades_ treats it as a mere rumor deliberately spread by the crusaders to spread terror (131).\n\n11. Maalouf does not give precise references to any of his quotations according to scholarly conventions, but lists his sources chapter by chapter in the \"Notes et Sources\" section at the end of the book. For his Frankish sources on cannibalism however, he does indicate several page numbers from Michaud's _L'histoire des croisades_ and _Bibliographie des croisades_ under chapter 3 in \"Notes et Sources,\" but I have been unable to consult these texts to verify where the citations attributed to Raoul de Caen and Albert d'Aix come from.\n\n12. At the time of the Crusades the word \"Franj\" seems to be a milder equivalent of the term \"sarrasin\" (Saracen), which was used during the Middle Ages to designate Muslims. _La chanson de Roland_ (The Song of Roland) is a representative example of the derogatory connotations conveyed by the word \"sarrasin\" (who represents the pagan enemy).\n\n13. \"The violation of an individual woman is the metaphor for man's forcing himself on whole nations\" (Robin Morgan, qtd. in Higgins 1996: 108).\n\n14. One could argue that murder can also be narrated into rather different kinds of story (such as premeditation, self-defense, accident, or suicide).\n\n15. In _Orientalism_ , Said quotes Chateaubriand: \"The Crusades were not only about the deliverance of the Holy Sepulchre, but more about knowing which would win on the earth, a cult that was civilization's enemy, systematically favorable to ignorance [this was Islam, of course], to despotism, to slavery, or a cult that had caused to reawaken in modern people the genius of a sage antiquity, and had abolished base servitude?\" (172).\n\n16. That text was prompted by Benazir Bhutto's defeated opponent's indignation in 1988 that no Muslim state had ever been ruled by a woman. Mernissi, a Moroccan sociologist, set out to unearth women who did govern during the history of the Muslim world.\n\n17. One could also specify that the viewpoints presented are those that were recorded by historians who were close to authorities, or to those in power, and who themselves belonged to the elite. These accounts are what shaped the current popular Arab view of the Crusades.\n\n18. The same applies to the crusaders, who were not all Franks.\n\n19. Runciman's negative appraisal of the Crusades was published as early as 1954, and his quotes ground his work in a Christian framework. The reason he declares the Fourth Crusade (\"against Christians\") the greatest crime against humanity (130) is that the Byzantine Empire had shielded Europe from Muslim advances. His conclusion that the Crusades are a \"long act of intolerance in the name of God, which is the sin against the Holy Ghost\" (III 480) is of course grounded in Christian theology.\n\n20. In _Les identit\u00e9s meurtri\u00e8res_ , he writes: \"ce contre quoi je me bats et me battrai toujours, c'est cette id\u00e9e selon laquelle il y aurait, d'un c\u00f4t\u00e9, une religion\u2014chr\u00e9tienne\u2014destin\u00e9e de tout temps \u00e0 v\u00e9hiculer modernisme, libert\u00e9, tol\u00e9rance et d\u00e9mocratie, et de l'autre une religion\u2014musulmane\u2014vou\u00e9e d\u00e8s l'origine au despotisme et \u00e0 l'obscurantisme\" 'what I am fighting against, and always will, is the idea that on the one hand there's a religion\u2014Christianity\u2014destined for ever to act as a vector for modernism, freedom, tolerance and democracy, and on the other hand another religion\u2014Islam\u2014doomed from the outset to despotism and obscurantism' (66, Bray 55). And he adds : \"L'islam avait \u00e9tabli un \u00ab protocole de tol\u00e9rance \u00bb \u00e0 une \u00e9poque o\u00f9 les soci\u00e9t\u00e9s chr\u00e9tiennes ne tol\u00e9raient rien\" 'Islam established a 'protocol of tolerance' at a time when Christian societies tolerated nothing' (67-68, Bray 57).\n\n21. Simon Lloyd mentions that Urban II urged the Catalan nobles to fulfill their crusade vows in Spain during the first crusade (39).\n\n22. See for instance _L\u00e9on l'Africain_ (1986), _Samarcande_ (1988), _Les \u00e9chelles du Levant_ (1996), _Le p\u00e9riple de Baldassare_ (2000).\n\n23. The Arabs and Muslims are not romanticized. Maalouf reports the killings committed by Turks (151) and massacres committed by Muslims when the Franks are expelled (273).\n\n24. Indeed, one can easily find fault with his sweeping statement that the Arabs were unable to create stable institutions and that the situation when civil war erupts with the death of a king is much the same nowadays. The recent examples of countries such as Morocco, Jordan, and Syria, speak to the contrary. Hassan II, Hussein, and Assad were long time kings and dictator, respectively, yet their deaths were followed by smooth successions (even in Jordan where there was a last-minute change in the constitution to replace King Hussein's brother, who held the title of crown-prince for 30 years, with one of Hussein's sons). While the preceding examples postdate the publication of Maalouf's text, I cannot think of a civil war that started as a succession dispute in the Arab world prior to 1980.\n\n25. Hochberg reaches the same conclusion: \"Focusing only on Germany and its allies in destruction, while completely ignoring the colonial history of the Middle East, _Ports of Call_ ultimately locates the 1948 war and the Zionist occupation of Palestine, completely outside the context of the region's own history of continual colonial occupation and the struggles against it\" (123).\n\n26. For a discussion of the Francophone writer as a bastard and an orphan, see Marx-Scouras' article.\n\n27. See Shumway (91) and Watt (116).\n\n28. Chomsky states that \"the record of Israeli terrorism goes back to the origins of the state\" (134).\n\n29. Alisa Solomon describes the frame coverage of the failure of the Camp David peace talks, which consisted in saying that Israel had made a generous offer that was rejected by the Palestinians, occulting the continued and accelerated developments of new settlements in the West Bank, among other things. She argues that \"Through this frame, which erases the occupation, the humanity of Palestinians is thrown into question. They appear as incorrigible, unaccountably violent, preternatural Jew haters. The Palestinian escalation of suicide bombings of civilian targets inside Israel during this period is, then, seen through this frame not as a desperate weapon of resistance, morally reprehensible as it may be, but as motiveless malignancy, proof of innate Palestinian barbarism\" (1589). She later adds: \"Whereas stories from the first intifada set events against the background of an occupation that began in 1967, today the context line in a broadcast segment . . . marks 2000 as the beginning of relevant time\" (1589).\n\n## Chapter 2\n\n## The Arab-Israeli Conflict beyond the Middle East: Albert Memmi, Edmond Amran El Maleh, Farid Boudjellal, and Karin Albou\n\nThe consequences of the Arab-Israeli conflict extend to communities beyond the Middle East, to countries such as Morocco, which used to have a considerable Jewish minority, and France. The conflict has had dire repercussions on the Jewish communities that were part of the Arab world. At the dawn of the twenty-first century, these communities have pretty much disappeared. Morocco's Jewish population, for instance, was estimated at 300,000 in 1947 (L\u00e9vy 9); it has dwindled and now numbers below 3,000; Tunisia's Jewish population, estimated at 95,000 in 1946 (Sebag 259) went down to less than 3,000 today. Though Morocco's Jewish community remains the most important of the Arab world (L\u00e9vy 49), events such as the 2003 Casablanca terrorist attacks (some of which targeted Jewish places), threaten it further.\n\nTwo seemingly distant events, be they in temporal or in spatial terms, are colluding and having repercussions in contemporary France: the Arab-Israeli conflict, and the French colonization of North Africa, and particularly Algeria. Although the French colonial presence in Africa ended over forty years ago, its aftermath is still felt today. At the dawn of the twenty-first century, France happens to be home to both the largest Muslim and Jewish communities in Western Europe, in great part due to immigration from former colonies. In recent years, various crimes have brought to the forefront with renewed urgency the question of interfaith relations between Muslims and Jews in France. These include the fabricated aggression with an anti-Semite motive blamed on Maghrebians and Blacks by Marie-L\u00e9onie Leblanc on July 9, 2004, and the February 2006 horrendous kidnapping, torture, and murder of Ilan Halimi, targeted because of his Jewish background, by a mixed-race gang headed by a nominal Muslim. These recent events follow a notable increase in anti-Semite (in a broad sense) incidents in the 1990s and 2000s. Some of these crimes can be attributed to the rise of the far right and neo-Nazis, such as the desecration of Jewish (in Carpentras in 1990 for instance) and Muslim tombs (the Muslim section of a military cemetery in 2007), and some to French people of Arab\/Muslim heritage.\n\nThis chapter focuses on the geographic displacement of the Arab-Israeli conflict by analyzing the representation of its impact on the social and cultural fabric of Maghrebian countries and contemporary France. It first analyzes two Arab Jewish writers' take on the Arab Jew in essays and articles, notably Memmi's _Juifs et Arabes_ [ _Jews and Arabs_ ] and _Portrait du d\u00e9colonis\u00e9 arabo-musulman et de quelques autres_ [ _Decolonization and the Decolonized_ ], and various pieces published by El Maleh. Through such analysis, this chapter builds on Hochberg's comparative analysis of creative works, namely Memmi's _La statue de sel_ [ _The Pillar of Salt_ ] and El Maleh's _Mille ans, un jour_ [one thousand years, one day].1 These two Francophone Jewish Maghrebian writers have opposite views on the issue of Arab-Jewish relations and the identity of Arab Jews, despite similarities in their itinerary. El Maleh was born in Morocco in 1917, Memmi in Tunisia in 1920. Both writers received a French education, studied philosophy, supported the independence movements of their respective countries, but settled in France (Memmi right after Tunisia's independence in 1956, El Maleh in 1965 at the beginning of Hassan II's repressive regime). Contrary to Memmi, El Maleh went back to his native country in 1999 and still resides there. I examine how their positions on the Arab-Israeli conflict correlate with the relative prominence of each writer (Memmi's renown and El Maleh's obscurity) considering the current political climate.\n\nThe second part of this chapter contrasts the representation of the relations between Arabs and Jews in France in Boudjellal's _JuifsArabes_ [JewsArabs] and Albou's _La petite J\u00e9rusalem_ [ _Little Jerusalem_ ]. I demonstrate that, contrary to Albou's film, Boudjellal's comic book participates in an endeavor to go beyond the ideology of partition and separation that has dominated approaches to the Arab-Israeli conflict.\n\n### Arab Jews from the Maghreb\n\n#### Albert Memmi\n\nMemmi published _Juifs et Arabes_ in 1974. The book is a collection of essays written for different occasions, all having to do with Arabs and Jews in the Middle East and North Africa. The term \"Juif-Arabe\" appears in the title of one chapter (\"Qu'est-ce qu'un Juif-Arabe?\" 'What is an Arab Jew?'), originally written for a journal that had asked Memmi to write about relations between Jews and Arabs in Arab countries (Memmi, _Juifs_ 59, note 1). This term (in the plural form \"Juifs-Arabes\" 'Jews-Arabs') had just been uttered by Qadhafi during a visit to Paris, in a rhetorical question recalling the common cultural heritage shared by Arabs and Sephardic Jews (Memmi, _Juifs_ 49). In the course of this essay, Memmi privileges the term \"Juifs arabes\" 'Arab Jews,' and \"Juifs des pays arabes\" 'Jews from Arab countries' because according to him, the notion of \"Juifs-Arabes\" is a myth, a condition to which Arab Jews aspired but that was denied to them by Arab Muslims (Memmi, _Juifs_ 50).2\n\nIn his book, Memmi defines himself as a strong supporter of both Zionism and Palestinians, and sees no contradiction in his stand: \"je r\u00e9clame la justice pour les miens sans injustice pour les autres\" 'I want justice for my people without injustice for the others' ( _Juifs_ 13, Levieux 13), but he does not dwell on the fact that the creation of Israel did come at the expense of Palestinians. Memmi justifies his support for the two-state solution because over the years, \"il s'est produit un \u00e9change de fait des populations: une partie des Palestiniens a gagn\u00e9 les nations arabes, une partie des Juifs de ces nations a gagn\u00e9 Isra\u00ebl\" 'a de facto exchange of populations has come about. Part of the Palestinians have gone to the Arab nations, and part of the Jews from those nations have gone to Israel' ( _Juifs_ 14, Levieux 14). Ella Shohat has called references to \"population exchange\" \"propagandistic\" and underlined the fact that \"it elides the simple fact that neither Arab Jews nor Palestinians were ever consulted\" (\"Rupture\" 58).3 Shohat has eloquently demonstrated that \"Ironically, the Zionist view that Arabness and Jewishness were mutually exclusive gradually came to be shared by Arab nationalist discourse, placing Arab Jews on the horns of a terrible dilemma\" (\"Rupture\" 58). Although Memmi sees Israel as a country for Jews, he does recognize that there are non-Jewish minorities in that state, and Jewish minorities all over the world. But Memmi fails to examine the implication of his statement. He overlooks the fact that he is accepting the ethnic category of Jews unquestioningly despite the fact that this was elaborated by anti-Semites, and the fact that the state of Israel is founded on religious and exclusionary grounds. Memmi advocates the two-state solution with each state having a minority that should enjoy equal rights to the majority ( _Juifs_ 166). He thus indirectly admits that Arabs and Jews are indeed inseparable.\n\nMemmi's main argument is that \"ce n'est pas le sionisme qui a \u00e9t\u00e9 \u00e0 l'origine de l'antis\u00e9mitisme arabe, mais l'inverse, tout comme en Europe\" 'it is not Zionism that has caused Arab anti-Semitism, but the other way around, just as in Europe' ( _Juifs_ 12, Levieux 12). Here, Memmi dehistoricizes anti-Semitism and the various ways it has been articulated throughout history in different contexts.4 His statement also passes over the variance of the effects of anti-Semitism in different cultures and time periods. Paul Grosser and Edwin Halperin's catalogue of anti-Semitism includes an \"Islamic\" catalogue, but focuses on the West. They explain that in Islam, \"There was a condition of religious toleration not present under Christianity. Overt violence and anti-Semitic persecution, while not absent, were episodic and unsystematic. There were periods and places of genuinely cordial Islamic\/Arab-Jewish interaction\" (7).\n\nThere is no rendering in Memmi's work of the complexity of factors that pushed the Jewish Maghrebian communities to leave North Africa.5 One common element, however, which Memmi silences, is the fact that the creation of the state of Israel on Arab land is a prominent factor. The conflict between Arab countries and the state of Israel trickles down to internal tensions where the Jewish Maghrebian communities are considered or fear they will be considered guilty by association. Zafrani underlines that the political and psychological dimensions of Moroccans' emigration to Israel after independence, despite their equal citizen status, are intertwined and due in part to the Palestinian problem created by the creation of Israel and the Arab States' solidarity with Palestinians (295).\n\nMemmi downplays the seven-century-long Andalusian period known for the peaceful coexistence of the three monotheist religions under Moorish rule. He dismisses it as an anomaly: \"Jamais, je dis bien jamais\u2014\u00e0 part peut-\u00eatre deux ou trois \u00e9poques tr\u00e8s circonstancielles, comme la p\u00e9riode andalouse et encore\u2014les Juifs n'ont v\u00e9cu en pays arabes autrement que comme des gens diminu\u00e9s\" 'Never, I repeat, never\u2014except perhaps for two or three eras with very clear boundaries in time, such as the Andalusian period, and even then\u2014have the Jews lived in the Arab countries otherwise than as diminished people in an exposed position' ( _Juifs_ 51, Levieux 21-22).6 He thus participates in what Shohat has thus explained: \"[Zionist discourse's] historiography concerning Jews within Islam consists of a morbidly selective 'tracing the dots' from pogrom to pogrom,\" and while she refrains from idealizing the situation of Jews in the Muslim world, she argues that \"Zionist discourse has, in a sense, hijacked Jews from their Judeo-Islamic political geography and subordinated them into the European Jewish chronicle of shtetl and pogrom\" (\"Rupture\" 59).\n\nMemmi's rendering of the conditions of Jews in Islam and in Christendom is partial. He credits colonization for the betterment of the Jewish condition. Even when he points out the failure of French authorities to protect the Jewish community, he mitigates their responsibility by recalling Arab hostilities: \"J'ai racont\u00e9 dans _La statue de sel_ comment nous avons \u00e9t\u00e9 froidement abandonn\u00e9s aux Allemands par les autorit\u00e9s fran\u00e7aises. Mais il me faut ajouter que nous baignions \u00e9galement dans une population arabe hostile. . . . C'est la raison pour laquelle tr\u00e8s peu d'entre nous purent passer les lignes pour rejoindre les Alli\u00e9s\" 'In my novel, _The Pillar of Salt_ , I have told how the French authorities coldly abandoned us to the Germans. But I must add that we also lived amidst a hostile Arab population. . . . That is why very few of us were able to get through the lines to join the Allies' ( _Juifs_ 53, Levieux 23). Memmi strikingly deflects the responsibility of the Vichy regime by stressing the hostile Arab community (something which, incidentally, in the context of WWII is not depicted in _La statue de sel_ ). This state of affairs, however, is not reflected in Paul Sebag's study. Sebag details the discriminatory measures implemented in Tunisia during World War II (222), but stresses that \"malgr\u00e9 les efforts de la propagande allemande, l'id\u00e9ologie antis\u00e9mite ne r\u00e9ussit pas \u00e0 mordre sur les autres \u00e9l\u00e9ments de la population\" 'despite the German propaganda efforts, anti-Semitism did not catch on with the other elements of the population' (244). In Morocco, Mohammed V protected Moroccan Jews by refusing to cooperate with the Protectorate authorities under Vichy (Zafrani 293), and in Algeria, it is noted that the Muslim population, contrary to the European one, did not commit any hostile act against Jews during the WWII period (Stora, \"L'impossible\" 295, 303). However, the Vichy period stimulated a growing segment of pro-Zionism in the Maghreb (Laskier 85).\n\nGiven that Memmi in his essay sees Zionism as the only solution to the Jewish problem, it is surprising that the narrator of his semi-autobiographical _Pillar of Salt_ chooses Argentina as his destination, since in real life, Memmi chose neither Argentina nor Israel, but France as his new home. Because Argentina was one of the proposed sites for the establishment of the Jewish state by early Zionist leaders, this might have been a way for Memmi to reconcile some of the contradictions inherent in his support: in addition to Hochberg's analysis that Argentina not only avoid the whole polarization between Arabs and Jews that ensued from the creation of the state of Israel (23), I would add that it also eschews the religious character of Zionism's claim to the Holy Land for what Memmi supports as a purely nationalist project. Memmi's efforts (not always successful) to distance himself from religion are evident both in his essays and in his creative writing.7\n\nHochberg argues that \"While Memmi recognizes the subversive political potential embedded in the figure of the Arab Jew, he explicitly wishes to disarm it, stating that the promotion of an Arab Jewish identity presents a serious threat to the creation of a new national Jewish collective\" (34-35). Indeed, Hochberg points out \"Europe's role in constructing both the Jew and the Arab as its others, and its role in polarizing these two identities, making them Other to each other\" (35), and that Memmi failed to address the role that European colonialism has played in \"creating and sustaining the animosity between Muslims\/Arabs and Jews and in making the Arab Jew an 'impossible figure' \" (21).\n\nThe few pages Memmi devotes to the Israeli-Palestinian conflict in his recent _Portrait du d\u00e9colonis\u00e9 arabo-musulman_ , conceived as an update to his groundbreaking _Portrait du colonis\u00e9_ , testify to the same phenomenon that Hochberg had noted: the complete lack of consideration of Europe's role in shaping the relationships between Arabs and Jews. Although Memmi recognizes the transnational dimension of the conflict, which resonates throughout the Arab world and indeed throughout the world (or at least where there are diasporic communities),8 he refuses to see Israel as \"une fondation coloniale\" 'a colonial settlement' because of the absence of a \"m\u00e9tropole\" behind it to back it up ( _Portrait du d\u00e9colonis\u00e9_ 41, Bononno 25).9 This statement fails to take into account the very prominent role played by the pro-Israeli lobbies in the U.S. (see Mearsheimer and Walt), and the fact that Israel has ranked at the top of U.S. foreign aid recipients. Israel might not have a \"m\u00e9tropole\" in the same manner that Paris ruled Algeria and London ruled India, but it does find itself in a neocolonial situation where ties and structures on a worldwide scale have supplanted colonial power structures, albeit in a much less obvious manner. While in 1974 Memmi acknowledged \"l'apport de la Diaspora en tant que telle dans la consolidation d'Isra\u00ebl\" 'the part that the Diaspora as such played in the consolidation of Israel' ( _Juifs_ 194, Levieux 195), there is no word in the 2004 essay on its role.\n\nMemmi thereafter tries to downplay the importance of the conflict, first by engaging in dubious relativism, recalling that there have been more devastating conflicts. According to him, the conflict is in fact \"une lutte assez banale entre deux petites nations en g\u00e9sine, dont les deux affirmations nationales se sont trouv\u00e9es par malchance en contradiction territoriale\" 'a rather ordinary struggle between two small emerging nations, whose national claims unfortunately turned out to involve a territorial conflict' ( _Portrait du d\u00e9colonis\u00e9_ 42, Bononno 26). The total oblivion regarding Europe's role in creating this \"unfortunate\" outcome is obvious. The role of Western powers is also completely occulted when Memmi laments the \"surestimation de l'affaire palestinienne\" 'overestimation of the Palestinian case' ( _Portrait du d\u00e9colonis\u00e9_ 43),10 which he solely blames on the Arab-Muslim world. This silences the fact that the Middle East is a region that is of geopolitical interest to the West, and not the other way around (or at least, initially, not to the same degree). One could argue that Memmi's choice of living in France, in Europe, might (un)consciously lead him to completely efface the role of the Hexagon\/Europe in the genesis and perpetuation of the conflict.\n\nMemmi always champions Israel's cause as \"un fait national, qui correspond \u00e0 une condition difficile \u00e0 vivre et \u00e0 une aspiration collective\" 'a national fact, the response to an untenable condition and a collective desire' ( _Portrait du d\u00e9colonis\u00e9_ 41, Bononno 25), yet this is obviously not an aspiration that he has chosen to act upon, as evident by his choice to live in France, not in Israel. Nor was it an inspiration for most of Algeria's Jews, whose departure from Algeria after independence had less to do with Zionism but everything with the French citizenship Algerian Jews had been granted by the colonial power in 1870. Memmi lumps all Jews together and does not acknowledge that there were differences between their conditions among and within countries.11 This is a fact that he had noted in his semi-autobiographical novel, where social class constitutes a major factor in the alienation of its main character. The narrator of _La statue de sel_ , who is from the poor working class, describes the French high school he is privileged to attend thanks to his good grades and a scholarship: \"J'eus des camarades fran\u00e7ais, tunisiens, italiens, russes, maltais, et juifs aussi, mais d'un milieu si diff\u00e9rent du mien qu'ils m'\u00e9taient des \u00e9trangers. Ces juifs riches . . . m'exasp\u00e9raient\" 'I had French, Tunisian, Italian, Russian, Maltese, even Jewish classmates\u2014but the latter were from a background so different from mine that they were as foreign to me as the others. They were rich Jews [these rich Jews . . . exasperated me]' (96, Roditi 104). This passage emphasizes that social class overrides whatever religious and ethnic traits he shares with the Tunisian Jewish middle class. Later on, the narrator states that upper-middle class Jews did not suffer as much from anti-Semitism as poor Jews during WWII (215), and the wedge between the middle class and ghetto Jews was still felt even in the strained conditions of the forced labor camp (242).\n\nMemmi's use of the word \"ghetto\" throughout the book to refer to the _Hara_ (Jewish quarter) assimilates a European reality to a North African one. As evident when he talks about the Jewish Tunisian bourgeoisie, the ghetto is one in which the poor are stuck, not the middle-class, and is therefore a ghetto according to social class criteria. Mohammed Kenbib emphasizes the inadequacy of the \"sch\u00e8mes g\u00e9n\u00e9ralement puis\u00e9s dans les r\u00e9alit\u00e9s des Stetl et Judenvierthel d'Europe centrale et orientale\" 'schemas usually drawn from the realities of Central and Eastern Europe's shtetls and Judenviertel' (2) to capture the Moroccan Jewish reality.12 One can probably extend this criticism to Tunisia. In Tunisia, each ethnic group had its quarter and the _Hara_ was a \"cloisonnement social volontaire\" 'voluntary social partitioning' (with other groups having their own quarters, Jadla 149), although the middle class left it well before colonization (Largu\u00e8che 172).\n\n#### Edmond Amran El Maleh\n\nWhereas Memmi makes broad generalizations about Jews in his texts, El Maleh, who defines himself as a \"juif oxymoron\" 'oxymoron Jew' (Redonnet 85), refuses to speak for all Jews and claims the right to a plurality of being Jewish.13 El Maleh's writings have tackled the issue of Arab Jews in the context of Morocco; as he stated, \"Je l'ai abord\u00e9 [l'exode des juifs du Maroc] d'un point de vue marocain\" 'I addressed [the Moroccan Jews' exodus] from a Moroccan point of view' (Redonnet 74). Indeed, he has dedicated most of his work to the Jewish Moroccan community and its disappearance.\n\nEl Maleh's views of the repercussions of Israel on Jewish communities in the Arab world differ sharply from Memmi's, whom he specifically accuses of misreading history by reading the history of Jews in Arab countries as a series of persecutions (Redonnet 120). El Maleh revolts against the fact that the state of Israel pretends to speak for all Jews, and qualifies the creation of Israel as a colonial project (\"Au seuil\" 20). One of the reasons why he speaks so strongly against Zionism is because of its policy of erasing not only the Palestinian people but also the existence of Moroccan Jews: \"On oublie que le sionisme qui conjointement \u00e0 sa politique de destruction du peuple palestinien s'est acharn\u00e9 \u00e0 effacer jusqu'au moindre signe la pr\u00e9sence mill\u00e9naire des juifs marocains\" 'we forget that Zionism along with its policy of destruction of the Palestininan people hounded to erase all traces of the millennial presence of Moroccan Jews' (Redonnet 119).\n\nMaleh's novel _Mille ans, un jour_ ties the fate of the Moroccan Jewish community to French colonization and events in the Middle East. Israeli soldiers are compared to \"nazis aux cheveux cr\u00e9pus, \u00e0 l'oeil noir\" 'nazis with frizzy hair and dark eyes' (121); exactions committed by Israeli settlers in Gaza recall the \"choses pareilles qui se passaient pendant le protectorat\" 'similar things that happened during the Protectorate' (122). Ronnie Scharfman contends that \"For Nessim [the main character, a Moroccan Jew], Beirut [with the massacres of Palestinians in the refugee camps of Sabra and Chatila] is the noisy, bloody version of the silent erasure and disappearance of his people\" (\"The Other's\" 139).\n\nSome of these comparisons can also be found in a strong piece triggered by the 1982 Israeli invasion of Lebanon, in which several parallelisms are drawn between this invasion and well-known atrocities committed by the Nazis in France during WWII: the destroyed Lebanese cities and the village of Oradour (\"Le visage\" 18), Palestinian women giving birth on the beach while their refugee camps are being destroyed recall women who gave birth in the Vel d'Hiv, whose roundup was just being commemorated (\"Le visage\" 18), summary execution of Lebanese are compared to the same done by Germans (\"Le visage\" 19). In that article, El Maleh was already underlining the use and abuses of language, whereby anyone daring to use the word genocide and making comparisons between atrocities committed by the Israelis with the Nazis will be anathema, while the Israeli government routinely uses such language to qualify the PLO and Palestinians (\"Le visage\" 20).\n\n### Arabs and Jews in Contemporary France\n\n#### From _Juif-Arabe_ to _JuifsArabes_ : Farid Boudjella's Comic Books14\n\nBoudjellal, a comic strip artist, has taken on the question of the relations between Arab Muslims and Jews in France. Born in Toulon in 1953 of Algerian parents, Boudjellal published his first strips in 1978 in a magazine and his first comic book in 1983 (Gaumer). Known for bringing the issue of Maghrebian immigration to the fore in comic books through the series _L'oud_ ,15 his work participated in the Beur culture surge of the 1980s. His comic books have focused on the plight of the lone Maghrebian immigrant worker (in _L'oud_ for instance), the problems faced by immigrant families (in _Gags \u00e0 l'harissa_ ), and the challenges faced by Beurs. According to Mark McKinney, Boudjellal is \"the most published French cartoonist of post-colonial minority origin\" (180), and his work has been labeled as \"une authentique (et talentueuse) BD 'beur' \" 'an authentic and talented comic strip by French of Maghrebian origin' in the _Dictionnaire mondial de la bande dessin\u00e9e_.\n\nIn addition to his work on the Maghrebian and Beur community in France, Boudjellal has published a series entitled _Juif-Arabe_. The original four parts were published individually in the early 1990s and were titled: _Juif-Arabe_ (1990), _Juif-Arabe: Int\u00e9gristes_ (1990), _Juif-Arabe: Conf\u00e9rence internationale_ (1991), _Juif-Arabe: Fran\u00e7ais_ (1991). These four comic books, which address the relations between the French Jewish and Arab-Muslim communities, were compiled and reissued in 1996 as _Juif-Arabe: L'int\u00e9grale. JuifsArabes_ is a revised and augmented edition of this compilation that appeared in 2006.16 This part focuses on the representation of the impact of events taking place in the Middle East on contemporary France as depicted in the latest version, _JuifsArabes_. It also examines the representation of past and future genealogy between both communities, a recurrent issue in the comic book, and how a visual allusion to the _Ast\u00e9rix_ BD series highlights the diversity of France at the turn of the twenty-first century and provides a model of coexistence for the French Jewish and Arab communities.\n\n_JuifsArabes_ features the characters of Isma\u00ebl and Isra\u00ebl, both owners of religious bookstores, their wives Isma\u00eblle and Isra\u00eblle, and their offspring. The eldest children, Mohamed, Isma\u00ebl's son, and Yza, Isra\u00ebl's daughter, turn out to be boyfriend and girlfriend. The obvious onomastics of the parents' names is reinforced by the caricatural drawings. The parents' clothing styles are meant to link them to specific communities: Orthodox Jew for Isra\u00ebl, always wearing a black suit and hat, and Maghrebian immigrant for Isma\u00ebl with his white djellaba and flip flops. Both fathers sport a black beard (square for Isra\u00ebl and rounded for Isma\u00ebl) and oversized feet (recalling Ast\u00e9rix and Ob\u00e9lix, more on that later). The stereotypical rendering of Isma\u00ebl's and Isra\u00ebl's appearance is inherent to the genre of the satirical comics: the characters need to be immediately and easily identifiable as Jewish and Arab, respectively, although they represent a fringe of their respective communities. Boudjellal's series is an example of how a stereotypical character can be \"a successful satire if it is negotiated as a comically exaggerated portrait\" (Rosello 15).\n\nAccording to Jacques Tramson, the fact that these books are classified in between newspaper drawings and comic books sets this series apart from the rest of Boudjellal's work (37). _JuifsArabes_ is characterized by short narratives, a focus on static characters, and a lack of elaborate background, which according to Pierre Masson are hallmarks of \"intellectual\" and satirical BD (77). The drawings are kept to a minimum; the backgrounds are mostly bare so as to focus all the reader's attention on the dialogues and the characters' gestures and facial expressions. Most of the pages are divided into two or three (single or double) strips of equal size; only a handful of gags take up more than one page. According to Thierry Groensteen, regularity of format confers a regular metric to the reading (61). This comic book falls under the category of ethnic humor, based on ethnic or religious or other kinds of differences, but the humor of the series is also derived from political disputes.\n\nBoudjellal draws on stereotypes that are supposed to highlight the specificities and particularities of the Arab and Jewish communities in France to better underline how similar they actually are. Despite their profound differences on political issues of the Middle East, Isma\u00ebl and Isra\u00ebl are very much alike. They constantly fight about the Arab-Israeli conflict, yet they react in the same way to the same events, be it news from the Middle East or the love relationship between their children. This approach fits Rosello's definition of direct intervention, defined as the \"conscious cultural reappropriation of ethnic stereotypes\" ( _Declining_ 18-19). For instance, after they discover that their children are in love, Isma\u00ebl and Isra\u00ebl embrace each other, crying in despair (35). When Yza and Mohamed announce that they want to get married, the fathers offer to pay for their children's honeymoon, but are shown buying one-way tickets to Algiers and Israel. The antidote to this interfaith\/interethnic marriage is envisioned as a \"return\" to nation states based on ethnic and religious affiliations.\n\nThe issue of genealogy (past, present, and future) between the Arab and the Jewish communities is a recurrent theme. Several references are made to the shared family tree between Arabs and Jews going back to Abraham as the founding father. The very first page, entitled \"Abraham p\u00e8re\" refers to this shared ancestral genealogy. It features several frames in which Israel enumerates commonalities between the two groups: \"vous rejetez le porc, nous aussi; vous \u00eates circoncis, nous aussi\" 'you disallow pork, so do we; you are circumcised, so are we,' etc. (5). The punch line is given by Isma\u00ebl who uses the same syntax to complete the list: \"vous \u00eates en Palestine, nous aussi!\" 'you are in Palestine, so are we!' This first set of strips sets the tone for the volume, since the last commonality is the point of contention. On another page, responding to a priest who wonders why they are always fighting despite being both children of Abraham, Isma\u00ebl and Isra\u00ebl answer in chorus with an idiomatic phrase that stresses their shared roots: \"nous lavons notre linge sale en famille!!\" 'we wash our dirty linen among the family' (meaning we do not wash our dirty linen in public) (20). In another instance, Isma\u00ebl claims this common genealogy by referring to his Semite identity as a counter to Isra\u00ebl's accusation that he is anti-Semite (7).\n\nIn addition to these family ties established by the Biblical tradition, the two communities' destinies seem to be interdependent because of twentieth-century politics. On a page composed of two single strips, the content of the balloons and the gestures become inverted. In the first strip, the dialogue features Isra\u00ebl recalling that the Jewish people had a dream and Isma\u00ebl that the Arabs had a country; the content of the speech balloons are then reversed and the tenses of the verbs changed from past to present with Isra\u00ebl stating that they now have a country while Isma\u00ebl retorts that the Arabs now have a dream (6). Later on, the stateless Palestinian is compared to \"un juif errant\" 'a wandering Jew' (60). The condition of the Wandering Jew becomes a simile to describe the Palestinians' condition of statelessness, thereby linking Zionism to the Palestinian dispossession. Boudjellal pokes fun at the deadlock in which both sides of the conflict are caught with Israeli repression and Palestinian terrorism, excused by each side as a \"mal n\u00e9cessaire\" 'necessary evil' (24), and the impossibility of progress towards a peace process because of the manner in which preconditions to negotiations are framed (39).\n\nEven though Isra\u00ebl and Isma\u00ebl are both owners of religious bookstores, their dispute over the conflict is always framed in territorial terms, never in religious ones (though the role played by religions is not erased). In fact, when religion and morals are at stake, they agree, as when they collaborate on an anti-abortion demonstration. The fathers are often at odds with French society when it comes to certain issues, as their religious commitment contrasts with French mainstream secular lifestyle. Even when their children suggest that they lend each other books to better understand one another, and they yet end up fighting once again (ending up throwing those books at each other), the issue of contention is not framed in theological terms, but in nationalistic ones. Isra\u00ebl tells Isma\u00ebl: \"Lisez ceux-l\u00e0 et vous prendrez la nationalit\u00e9 isra\u00e9lienne\" 'Read these and you will take the Israeli nationality' (151). When they are found throwing books at each other, they respectively shout \"un cadeau pour Tsahal\" 'a gift for Tsahal' and \"un autre pour le Hamas\" 'another one for Hamas' (152). Only when the representative of the Church comes into play does the conflict gain a full religious dimension, when the priest states that the existence of the state of Israel is necessary for Jesus' second coming (78). That politics and not religion is at stake is reinforced by the choice of the name Isra\u00ebl (and not Isaac as in the scriptures) for the Jewish character.\n\nThere is a clear generation gap between the fathers and the children. For one thing, the children do not understand the affiliation their fathers feel with Middle Eastern communities and question why the issue is so dear to them despite the geographical distance. When confronted about their quarrels regarding Israel and Palestine, both fathers answer in unison that Israel and Palestine represent their \"maison secondaire\" 'secondary home' (38). This corresponds to what Esther Benbassa refers to as a \"diasporic transnationalism,\" defined as \"a nationalism with no territorial claims, reconciling loyalty to the country of residence with strong support to external causes\" (190), to explain French Arabs and Jews' affiliations with Middle Eastern causes. According to her, \"today, while for a number of Arab-Moslems the support for the Palestinian cause alleviates the crumbling of their traditional identity, many Jews find in their attachment to Israel a means of counterbalancing a comparable fragility\" (191).\n\nHope seems to lie with the mixed couple formed by Yza and Mohamed. The children are not drawn in a way that marks them as different from French mainstream society. Unlike their fathers, they are not religious: indeed, Mohamed is a self-proclaimed atheist, while Yza has an argument with her father because she claims her Jewish identity as a cultural, not a religious one. When their children suggest the establishment of one secular democratic country for both Palestinians and Israelis in historical Palestine, Isma\u00ebl and Isra\u00ebl are both pictured as laughing until they cried at the idea (154).\n\nOne strip does highlight the potential difficulties of being in such a union because of the events in the Middle East (37). After Yza and Mohamed's relationship is discovered by their fathers, Isma\u00ebl's dismay is obvious when he asks his son about his girlfriend and his clothing. Isma\u00ebl's conception of identity is linked to the genealogical paradigm, as evidenced when he quotes a proverb to his son emphasizing the importance of one's roots: \"si tu coupes mes barreaux tu me lib\u00e8res si tu coupes mes racines tu me tues!\" 'if you cut my bars you free me if you cut my roots you kill me' (36). Mohamed's answer demonstrates a positive view of _m\u00e9tissage_ , when he replies to his father that his clothing and his girlfriend act as fertilizers (36).17\n\nYet this optimistic view is tempered in the very next page (the two pages face each other, as if to give two possible alternatives). Titled \"L'\u00e9toile, le croissant, la capote\" 'the star, the crescent, the condom,' this second page highlights the fact that relations between Muslims and Jews in France are inevitably colored by the Israeli-Palestinian conflict. When Yza introduces Mohamed to a friend whose mother is Jewish and whose father is Muslim, Mohamed takes this opportunity to ask this friend how he lives his dual identity, as Mohamed is curious to know since his future children would be in the same situation. Upon hearing that, Yza's friend runs away screaming \"Me parlez pas d'Isra\u00ebl\" 'don't talk to me about Israel' although no one has mentioned the Middle East. Boudjellal underscores that the divisiveness surrounding the conflict is the main problem for the child of a mixed Jewish-Arab union.18 The punch line of the strip, when Mohamed goes to buy condoms, suggests that from the children's point of view, the Arab-Israeli conflict is preventing the two groups from renewing their Biblical genealogy.19\n\nOne of the most interesting changes in the various editions of the series is the title. Boudjellal's title has changed from _Juif-Arabe_ in the 1990s to _Juifs-Arabes_ in the 2006 edition. The title evokes Albert Memmi's _Juifs et Arabes_ , discussed in the previous section. Memmi, himself an Arab Jew, posits the two groups side by side and separate in the title of his essay. This \"erasure of the hyphen\" as Shohat pointed out, was crucial to Zionist thought, since the \"Arabness and the Orientalness of Jews posed a challenge to any simplistic definition of Jewish national identity, questioning the very axiom and boundaries of the Euro-Israeli national project\" (\"Rupture\" 62).\n\nAlthough the Sephardic community has a long history of coexistence with Muslims and Christians in the Maghreb and other Arab countries, this shared history barely surfaces in this comic book. The term \"juif-arabe\" that gives its title to the series appears only twice in the comics proper. The first time is when Isra\u00ebl tells Isma\u00ebl that his nephew was denied access to a night club. Upon hearing this information, Isma\u00ebl asks in disbelief whether his nephew is Jewish, to which Isra\u00ebl replies that his nephew is \"juif-arabe\" (55). This is the only statement in the album that hints to a connection between Isra\u00ebl, who is an Ashkenazi, with the Sephardic community, and the only use of the term \"Juif-Arabe\" that alludes to the existence of Jews of Arab culture, or Sephardim, Jews of Islam, African Jews, and so forth, among some of the names listed by Ella Shohat.20 Such a connection is done in the context of highlighting discrimination on the basis of facial and physical features. The second time the term \"juif-arabe\" appears is when Mohamed refers to his relationship with Yza as \"une union juif-arabe\" (148). The \"Juif-Arabe\" at stake in this book is the relationship between both communities in France, and the new possibility of the \"Juif-Arabe\" that Mohamed and Yza's children would be.\n\nAnother page alludes to the two main Judaic families. Isra\u00ebl is telling a joke to Isma\u00ebl, to whom he asks: \"Savez-vous ce qu'est un Ashk\u00e9naze aujourd'hui?\" 'Do you know what is an Ashkenazi today?' He carries on, answering his question while laughing: \"Un S\u00e9farade qui a r\u00e9ussi\" 'a successful Sephardi' (79). This alludes to what Solange Gu\u00e9noun has called \"un colonialisme int\u00e9rieur\" 'internal colonialism' (225), characterized by a binary opposition between the Ashkenazi and Sephardic branches of Judaism. The devalorization of the Sephardic heritage by Ashkenazim has been described as the colonization of one Judaic branch by another, linked to the European colonial project, whereby European Jews imposed their norms and culture on \"Oriental\" Jews, a phenomenon that occurred in France,21 colonial Algeria,22 and Israel.23 The title of that page, \"C'est encore un Arabe,\" emphasizes the disappearance of the Arab Jew's identity in a world whose politics (with Zionism and Arab nationalism playing a crucial role in that process) frame Arabs and Jews as entities that must be kept separate.\n\nIn contrast to Memmi's title, the replacement of the conjunction \"et\" by a hyphen (significantly \"trait d'union\" in French) seems to demonstrate a will to bring both communities closer. The role of the hyphen is to bring together words so that they form a unit. Boudjellal's early title thus creates a link between the words it joins, the categories of \"Arab\" and \"Jew.\" This closeness is further achieved, at least on paper, through the collapsing of both substantives, which posits an organic connection between both communities from the very start. The elimination of the hyphen underscores the fact that Arabs and Jews cannot be thought of without the other, a literal rendering of what Hochberg meant when she wrote that \" 'Jew' and 'Arab,' rather than representing two independent identities, are in fact inevitably attached\" (2). The change from singular to plural may signal a will to break from abstract categories to put the focus on the mass of actual individuals they refer to.\n\nIn the final section, both families happen to be vacationing at the same place and time. This forces them to make some concessions. Phrases such as \"n\u00e9gociations orageuses\" 'stormy negotiations' and \"partager le territoire\" 'splitting the territory' (157) are clear allusions to the ideology of separation inherited from the partition of historical Palestine, still prevalent in the two-state solution approach that dominates attempts to solve the conflict. Boudjellal plays here on the two possible meanings of the verb \"partager,\" which can mean either \"to share\" or \"to split.\" \"Partager le territoire\" indeed they do, but not in the sense of splitting. Isma\u00ebl and Isra\u00ebl end up vacationing together, and are shown as actually sharing the territory (or rather, in the circumstances, the beach). On the beach they are both offended and thus united by their dislike of liberal French manners, specifically topless women, and they are forced to dance on a tune they both judge obscene. This unity against some French habits24 does not prevent them from fighting over the issue of Israel and Palestine. After a respite during the vacations, once again they start arguing over the Middle East, and therefore decide to take different routes on the way back. But news on the radio about recrudescent clashes between Israel and Palestinians awakens once again their transnational solidarities, and they find themselves on the highway en route to the Near East. The last words come from Yza and Mohamed, who are of course happy to see each other sooner than expected. The last words: \"Finalement, on fait la route ensemble!\" 'in the end, we travel together!' (165), stress that the French Muslim and Jewish communities can find a modus vivendi.\n\nSome of the thorny issues of the Israeli-Palestinian conflict come up here and there throughout the strip, such as the Occupied Territories (8, 12), the Intifada (16, 19, 25), the PLO's recognition of the state of Israel (23), Israel as a colony (29), the right of return for Palestinian refugees (27), and the political deadlock between Palestinians and Israelis (39). Despite these references to Middle Eastern events and their influence on the characters, the book is grounded in the French context, thanks to cultural references specific to France, such as the issue of naming the population of Maghrebian origin (47), immigration (15, 49), racism against Arabs (47, 51), anti-Semitism by various segments of the French population (both Arabs and \"Fran\u00e7ais de souche\"), the debates on affirmative action (14, 51), a joke about Arab grocers open late at night (56) and about the difficulty to get permits to build mosques (119), and an allusion to Serge Gainsbourg's hit single \"Sea, sex and sun,\" which becomes \"sea, sex and je\u00fbne (pour les ramadans d'\u00e9t\u00e9)\" 'sea, sex and fast (for summertime ramadans)' (155).\n\nThe last drawing of the book features Isma\u00ebl and Isra\u00ebl dressed as Ast\u00e9rix and Ob\u00e9lix, respectively, the heroes of the renowned series by Ren\u00e9 Goscinny and Albert Uderzo. Isra\u00ebl, who is stout, wears Ob\u00e9lix's striped trademark pants and carries a menhir, while Isma\u00ebl, who is shorter and thin, wears Ast\u00e9rix's belt, winged helmet, and sword to the side. There is a genealogy at play here, but of a different kind: with the _bande dessin\u00e9e_ or BD, a genre that benefits from cultural esteem, as evidenced by state support and well-established critical traditions in France and Belgium (McQuillan 7). Indeed, in France BDs are often referred to as the ninth art, \"a phrase rarely uttered in the USA or the UK\" (Screech 1), and that elevates comics as an art on par with cinema and photography. Boudjellal thereby situates his series in a well-established tradition of slapstick humor.25 _Ast\u00e9rix_ characters have made their marks not only on comic strips and French popular culture, they have also become reference points \"for politicians, journalists and intellectuals,\" and there is for the French a \"strong identification process with our puny but clever comic strip hero\" (Steel 216). By casting Isma\u00ebl and Isra\u00ebl as avatars of Ast\u00e9rix and Ob\u00e9lix, Boudjellal seems to agree with Screech who states in the context of the _Ast\u00e9rix_ series' international success that Ast\u00e9rix is \"a hero with whom almost everybody can identify, wherever they come from\" (75).\n\nThis last image constitutes a counterpoint to the beginning of the book. The first drawing shows Isra\u00ebl and Isma\u00ebl separated by a wall that interferes with dialogue, no doubt an allusion to the wall being built by the state of Israel around the Occupied Territories. However, Isra\u00ebl and Isma\u00ebl are often drawn sitting down having a conversation. Indeed, they both agree that were it not for the Israeli-Palestinian conflict, they would be good friends (149). In the last drawing, Isra\u00ebl and Isma\u00ebl have retained their black beards (as opposed to the Gaulish characters' mustaches), they have hairy arms, Isra\u00ebl has kept his black hat and Isma\u00ebl his glasses, thus becoming potential new inhabitants that could blend in the famous Gaulish village while retaining some of their distinguishing features. They are shown as laughing and walking together, as if to symbolize a possible friendship such as the one that unites Ast\u00e9rix and Ob\u00e9lix, in the context of France, and the assonance between their names reinforces their portrayal as a tandem.\n\nAlthough in Francophone studies the now infamous phrase \"nos anc\u00eatres les Gaulois\" 'our ancestors the Gauls' is often used ironically to refer to the Eurocentric curriculum taught to children in the French colonies, I interpret Boudjellal's reappropriation of the Gaulish genealogy of the French as a means to highlight the diversity of France at the turn of the twenty-first century as well as a model of coexistence for the French Jewish and Arab communities. J.W.T. Mitchell argues that \"images are active players in the game of establishing and changing values. They are capable of introducing new values into the world and thus threatening old ones\" (105). While the Biblical references do point to a shared genealogy in a long gone past, the contemporary popular culture symbolized by the quintessentially French _Ast\u00e9rix_ series is grounded in the present. Mitchell reminds us in his latest book that \"The nation, as political theorist Benedict Anderson has argued, is an 'imagined community,' a cultural construction made up of images and discourses\" (273). If so, then visual representations are as important as verbal ones to build the nation. Boudjellal's image is a visual integration of Muslims and Jews in the French Republic that participates in the construction of France as a country with a diverse population.\n\nBoudjellal's series is remarkable in its choice of comic genre and humor that rely on stereotypes to treat a highly charged and divisive topic, given the current climate that Solange Gu\u00e9noun has qualified as \"catastrophisme ambiant concernant l'\u00e9tat des relations jud\u00e9o-musulmanes en France\" 'surrounding catas-trophism regarding the state of Jewish-Muslim relations in France' (218), and the devastation caused by many of the events that are alluded to in the strip (the Holocaust, suicide bombings, repression of the Palestinians). Humor that traces its roots to a well-established tradition, recognized and respected by all segments of the population, might indeed be the best means to envision a near future where the Jewish and Arab branches of the Semitic family might cross again, in peace, as full-fledged members of the French Republic.\n\n#### Karin Albou's _La petite J\u00e9rusalem_\n\nBoudjellal's optimistic vision contrasts sharply with Karin Albou's first full-length feature film, _La petite J\u00e9rusalem_ (2004). Albou is a rising Jewish French director26 with ties to Algeria through her paternal family. Her film focuses on a community that is rarely portrayed on the screen: the Orthodox Jewish community27 of Sarcelles, a Parisian suburb. Fifteen per cent of Sarcelles' 57000 inhabitants are Jewish; half of them are of Tunisian origin (Podselver 275). In an interview added to the English DVD, Albou stated that she wanted to shoot in that specific suburb because it is emblematic of Jewish immigration from North Africa in the 1960s.\n\nThe film is a subtle depiction of two Jewish sisters, originally from Tunisia, who struggle with sexual desire in different ways. The family, who lives on a tight budget in a small apartment in a low-income neighborhood, is composed of Mathilde, her mother, her husband Ariel, their children, and Mathilde's younger sister Laura. Laura studies philosophy at a university in Paris and longs for life in the city. Her adoption of Kantian principles is put to the test when she gets involved in a relationship with Djamel, an Algerian. Meanwhile, Mathilde's discovery of her husband's unfaithfulness leads her to question assumptions about her faith's stand on sexual matters. Albou paints a nuanced relationship between the two sisters who choose very different itineraries, and explores complex issues linked to intimacy and faith. However, her depiction of relationships between Arab-Muslims and Jews is very grim, as is the place she reserves for an observant Jewish family in France.\n\nAlthough the title of the film and the focus on the Jewish family foreground the Jewish quarter and inhabitants of Sarcelles, people in the street and the subway clearly show the multiethnic character of the neighborhood, with Africans sporting traditional boubous, and Muslim women wearing headscarves along with Jewish women with nets covering their hair. Although the Arab and Jewish communities live side by side in the film, they do not have much interaction. In contrast to daylight shots that focus on the high rise HLM buildings, the neighborhood's streets are mostly shot at night, creating a dominant feeling of isolation, despair, and drabness. The scenes during which Laura takes her daily evening walk following Kant's example, are too dark to distinguish much other than the word halal on a sign and a religious bookstore.\n\nThe French DVD cover specifies that the film takes place in \"un quartier de Sarcelles appel\u00e9 'La Petite J\u00e9rusalem' car de nombreux juifs s'y sont install\u00e9s\" 'a quarter of Sarcelles called 'little Jerusalem' because many Jews settled there.' This nickname and the explanation that is provided are significant in two ways. First, they highlight Jewish transnational affiliations to the Israeli state. But in doing so, it silences the long history of Jews in the Maghreb. Calling this area of Sarcelles 'la petite J\u00e9rusalem' can be seen as a misnomer. It is not the same as the nickname of Chinatown for part of Paris' XIIIth arrondissement for instance, because the Jews who live there did not come from Jerusalem, but from the Maghreb. Thus the explanation provided contributes to erasing the existence of Arab Jews.\n\nWhile the film does reflect that this \"little Jerusalem,\" like the Middle Eastern city, has Arab and Jewish populations living side by side, the title as it is explained in the packaging of the film occults the fact that Jerusalem has a highly contested status since both Palestinians and Israelis claim it as their capital. In some sense, the title and the film indirectly replicate the Arab-Israeli conflict over the Holy Land in France, this time in a French suburb where each group seems to fight for its place in it. The explanation given to the suburb's nickname on the DVD cover implies that only one ethnic group can claim it as its own. Albou's film's packaging thereby perpetuates the phenomenon experienced by the actual city, which sees its \"Arab character . . . continually being eroded, shriveling its links to the Levant,\" and this despite the fact that Jerusalem is \"a complete microcosm of Levantine and Arab Jewry\" (Alcalay 110). Moreover, connecting Jerusalem to the Jewish state occults the fact that \"during the 1,310-year period dating from the Arab conquest in 638 until 1948, there were only 129 years in which Jerusalem was _not_ under one form or another of Islamic sovereignty\" (Alcalay 113). This signals the effect that the creation of Israel has had on Jewish communities in the Arab world, which is to occult their link to Arab culture.\n\nReferences to some cultural traits refer to the family's Maghrebian origin: the mother and Mathilde occasionally speak Arabic, the mother talks about Tunisia and uses a hand of Fatima as a talisman to protect her daughter. When Laura writes a letter to Djamel, the mother warns her against doing so, and explains to her \"je les connais ces gens-l\u00e0\" 'I know those people.' The reference to the family's past in Tunisia is mostly associated with the traumatic event of a precipitated departure whose complex political context is not elaborated upon, but simply emphasizes that the Jews were expelled after the country's independence.\n\nDespite some common cultural elements, religion on both sides is an obstacle to Laura and Djamel's relationship: Laura writes to Djamel that her religious upbringing does not allow her to continue seeing him, but she later relents. Djamel's family expresses its profound displeasure upon learning that Laura is Jewish, and demands that Laura convert before Djamel can marry her (contrary to Muslim tenets that allow a Muslim man to marry Christian and Jewish women, and despite Djamel's objection that he does not care for religion). Djamel, an illegal refugee from the Algerian civil war, is at the mercy of his uncle because his illegal status prevents him from standing up to his relative. He ends up breaking up with Laura because he does not want to impose his religion on her.\n\nDetails contribute to creating a climate of insecurity: a policeman keeps watch outside the synagogue during a celebration, the loudspeaker on the subway refers to the Vigipirate plan, Ariel hides his kippah under a cap when coming home, etc. Two incidents emphasize the vulnerability of the Jewish community: the local synagogue falls prey to arson, and Ariel is attacked by thugs while playing soccer outside with children, all wearing a kippah. Nothing in the film identifies the perpetrators of these crimes. However, in the context of France at the dawn of the twenty-first century, these incidents will evoke the rise of anti-Semitism that has made headlines, and that has been attributed to the far right and youth of immigrant origins, the latter being a repercussion of the second Intifada. Given that, in the context of the film, the neighbors are Arabs, this anti-Semitism will be inferred as being caused by the Arab-Muslim population. In fact, I would argue that the failed love relationship between Laura and Djamel orients the viewer to blaming anti-Semitism solely on the Arab segment of the population.\n\nAriel's decision to move to Israel is never explained. One can interpret it as an escape from a hostile environment, or from the shame brought by Laura's suicide attempt. I surmise that the film and the context will orient the viewer towards the first interpretation. Ariel's announcement comes after Laura's attempt to commit suicide, and after a scene in which the _mikveh_ 's attendant advised Mathilde to move. Because her advice comes right after she asked about Laura, one could infer that Laura's well-being is dependent upon getting out of this neighborhood. Indeed, a recurring aerial shot of hazy Paris in which one can barely guess the Eiffel Tower and the Montparnasse Tower can be seen as a representation of Laura's unfulfilled desire to move to the city.\n\nWhile several scenes stress the diversity of the French population, the decision of the family to move to Israel seems to imply that there is no safe place in France for this religious community. Ironically, it is hinted that this move will not shelter them from Arab-Jewish tensions. When the children are shown building their imaginary home in Israel, one of them takes a helicopter toy to bomb and destroy it. This scene could also be read as emphasizing the hypocritical stance of Israeli officials who alarmingly stated that Jews were no longer safe in France and should move to Israel. Either way, this (unrealistic) threat of helicopters bombing Israel serves more to emphasize (Arab) hostility towards Jews wherever they are while occulting the specifics of the context in which they take place.\n\nAt the beginning of the movie, Laura is filmed as set aside from her community: in the scene by the river, a progressive high-angle shot emphasizes that she is standing alone at a short distance from the rest of the group. After the decision is made for the family to emigrate to Israel, Laura decides to stay in France. Her mother offers Laura her ring to give her the means to move to Paris. This forthcoming move to Paris can be seen as her integration into French society and sealing her disconnect from her tight-knit community which had already been hinted at from the beginning. The last scene shows her alone in the subway, with people passing behind her in a blur. Laura might be moving to the city, but she is even more alone than she was at the beginning of the film, since at least her community was just next to her. This film offers a pessimistic glimpse of Arab-Jewish relations in France as well as on the place of the Orthodox Jewish community in the Hexagon.\n\n### Conclusion\n\nThe now commonplace reference to the Judeo-Christian tradition overshadows the fact that, as Mohamed Talbi and others noted, there is more affinity between Judaism and Islam than between Judaism and Christianity, and this especially applies to the Maghreb not only at the level of religious tenets but also at the sociological level (47). Bernard Lewis reminds us that the term Judeo-Christian is still new (qtd. in Majid, _Unveiling_ note 27, 163), a fact that prompts Majid to wonder: \"If the Judeo prefix that precedes the adjective of Christian is an attempt to bury this history [of Western anti-Semitism] under the assumption of a common heritage, why would the Muslim become the Jew's Other, especially if Lewis himself states that 'Jewish and Muslim theology are far closer to each other than is either to Christianity?' \" ( _Unveiling_ 12). The term Judeo-Christian also helps silence the historical evidence that there was an inverse situation concerning Jewish-Christian and Jewish-Muslim relations that showed that Jews fared better under Muslim rule: \"le tableau est donc exactement inverse: d'un c\u00f4t\u00e9, intol\u00e9rance avec quelques exceptions, de l'autre, tol\u00e9rance avec quelques exceptions\" 'the picture is thus exactly the opposite: on the one hand, intolerance with a few exceptions, on the other, tolerance with a few exceptions' (Rabi Josy Eisenberg, qtd. in Talbi 49).\n\nWhile the diaspora of Jews from the Arab world is brandished as a proof of Arab Muslims' anti-Semitism, less attention is being given to the fact that the migration of Jews is not solely occurring in the Arab-Muslim world, witness the post-WWII East European migration (Valensi 56). Graham Fuller notes that \"to emphasize commonality [between religions] implies a quest for coexistence and understanding, an overcoming of cultural differences, and above all attaching major value to the _very act of coexistence_ and tolerance as part of a religious outlook\" (205). Writers such as Memmi and film directors such as Albou minimize this shared cultural heritage, while El Maleh's and Boudjellal's works remind us of past models or prompt us to envision future ones of coexistence.\n\n### Notes\n\n1. See Hochberg 20-43. Her chapter focuses on the question of the \"status of the Arab Jew today in the context of Zionism: does this figure belong to a lost history and merely represent a current political impossibility, or does it (also) represent a futuristic antiessentialist and antinationalist cultural-political stance with direct implications for the present?\" (17-18).\n\n2. Memmi expresses his dislike of the phrase \"Juifs-Arabes\" used out of convenience to denote that \"natifs de ces pays dits arabes, originaires de ces contr\u00e9es bien avant l'arriv\u00e9e des Arabes, nous en partageons, d'une mani\u00e8re non n\u00e9gligeable, les langues, les coutumes et les cultures\" 'because we were born in these so-called Arab countries and had been living in those regions long before the arrival of the Arabs, we share their languages, their customs, and their cultures to an extent that is not negligible' ( _Juifs_ note 2, 59, Levieux 29).\n\n3. See Shohat's \"Rupture and Return\" about the complex factors, including the role of Zionist activists, in forcing the departure of Iraqi Jews (55-56).\n\n4. A search on anti-Semitism in the library catalogue will yield entries on anti-Judaism for the pre-modern period, and distinctions are made by countries (anti-Semitism in France, Germany, etc.) as well as entries on the new anti-Semitism.\n\n5. For Tunisia, see Sebag (273-297).\n\n6. L\u00e9vy on the other hand describes the history of Moroccan Jews as a \"tradition de cohabitation tol\u00e9rante dans le cadre de la l\u00e9gislation musulmane\" 'tradition of tolerant cohabitation within the framework of Muslim legislation' with some parentheses (38).\n\n7. See for instance how the titles of _Agar_ and _La statue de sel_ are clear Biblical allusions although the narrator of both semi-fictional novels repudiates his religious background.\n\n8. \"Car la partie ne se joue pas seulement entre Palestiniens et Isra\u00e9liens mais entre la quasi-totalit\u00e9 des pays arabo-musulmans et la majorit\u00e9 des juifs dans le monde\" 'For the conflict extends beyond the Palestinians and Israelis, involving nearly all the Arab-Muslim countries and the majority of the world's Jews' (Memmi, _Portrait du d\u00e9colonis\u00e9_ 40, Bononno 24).\n\n9. There is a mistranslation here: \"Ne poss\u00e9dant pas de m\u00e9tropole derri\u00e8re lui [Israel], pour en venir \u00e0 bout il faudrait le d\u00e9truire\" (41) became \"However, to threaten the destruction of Israel would have catastrophic consequences\" (25). Here is a more faithful rendering: \"with no parent state behind it [Israel], one would have to destroy it to defeat it.\"\n\n10. My translation. This sentence is part of an entire paragraph that is missing from Bononno's translation.\n\n11. For the differences in status between the various Jewish communities in Tunisia (which at the time of independence included Jews of French, Italian, and Tunisian citizenship), see Sebag's study. These differences also account for the choice of destination once the decision to emigrate had been taken (see note 5).\n\n12. Social class is also a factor in the choice of destination when Arab Jews decided to leave the Maghreb. El Maleh states that most Moroccan Jews went to Israel, then to Canada, and the richest to France (Redonnet 80). This is corroborated by L\u00e9vy who states that middle and upper class Jews went to France and Canada (32). Zafrani explains that the Jewish Agency recruited not only poor urban masses but also people from the South and the mountains to emigrate to Israel because they were considered \"plus aptes que les citadins \u00e0 fonder des colonies agricoles\" 'better fitted than town-dwellers to establish agricultural colonies' (293). Sebag also notes that those who were most well-off and Westernized went to France, while the most traditional and poor segments of the population went to Israel (Sebag 301).\n\n13. \"Il y a des juifs. Cette entit\u00e9 de l'\u00eatre juif est ce contre quoi je m'insurge\" 'There are Jews. What I am rebelling against is this entity of the Jewish being' (qtd. in Redonnet 91).\n\n14. This section has appeared as \"From _Juif-Arabe_ to _JuifsArabes_ : Jews and Arabs in France in Boudjellal's Comic Books\" in _Expressions maghr\u00e9bines_ 7.2 (Winter 2008): 159-171. It has been revised.\n\n15. This series includes _L'oud, Le gourbi_ , and _Ramad\u00e2n_. For an analysis of Boudjellal's early work, see Douglas and Malti-Douglas (198-216).\n\n16. Some of the pages have been eliminated, such as the ones that featured two skinhead characters, shown as beating up Isra\u00ebl and Isma\u00ebl (who have thus found a common enemy), and painting racist insults on walls, emblematic of the rise of the far right and racism. Other pages have been modified: mentions of the Front National and Le Pen have been erased, one of them replaced by the more general term Fascism. References to the first Gulf War and Salman Rushdie have also been taken out. The original four albums and the first compilation were drawn in colors. These multicolor albums have been reduced to three colors in the last version published in 2006: black, white and brown. The choice of color (or rather lack of a color scheme) could be interpreted as a means to better highlight the conflicting situation (as Masson has said about comic strips in black and white). When I asked Boudjellal about the factors that motivated this change, he answered that the decision to change from four colors to three was made solely on aesthetic grounds, in order to give a more polished look to the album as well as a feel of \"un livre d'auteur.\"\n\n17. One could object that the use of the word \"engrais\" betrays a sexist blind spot in Mohamed's view of _m\u00e9tissage_ , which objectifies the woman and presents her as an instrument instead of a full partner.\n\n18. This is in line with Boudjellal's exploration of issues raised by mixed couples in _Jambon-Beur: Les couples mixtes_. Through the contrast between the children of Arab-French and Arab-Senegalese couples, Boudjellal emphasizes that history and politics bear on how one lives such a dual heritage (see McKinney 180-184).\n\n19. The same cannot be said about the mothers. Although the fact that the Jewish matrilineal and Muslim patrilineal transmission of religion would seem to satisfy both families, since both could claim the still hypothetical children of Yza and Mohamed as their own, it is a source of conflict and is the only time in the book when Isma\u00eblle and Isra\u00eblle fight with each other (86-87). Otherwise, the women have a lot in common (they both cover their hair for instance, with a headscarf for one and a wig for the other), they also band together about issues such as contraception and abortion, this time against their husbands who are joining forces on organizing a demonstration (95).\n\n20. Shohat points out that \"the very proliferation of terms suggests the difficulties of grappling with the complexities of this identity\" (\"Rupture\" 52).\n\n21. See articles by Gu\u00e9noun and Azria.\n\n22. Benjamin Stora has detailed how the 1870 Cr\u00e9mieux decree, which gave French citizenship to indigenous Jews, triggered the assimilation of the Jewish community with the European community in Algeria. He has interpreted this event as a colonization of Algerian Judaism by French Judaism (292).\n\n23. See Shohat's 1997 article.\n\n24. The similarities that are highlighted between the two men include the position of women in society, secularist permissiveness in general, and their relationship with a Catholic priest.\n\n25. Goscinny and Uderzo's goal with the series was to create a \" _French_ strip-cartoon Laurel and Hardy in order to entertain their readership\" (Steel 202).\n\n26. Her film was nominated for the C\u00e9sar's Best First Film award (as was its lead actress).\n\n27. Albou has stated in an interview accompanying the DVD that she finds the Orthodox version of Judaism suffocating.\n\n## Chapter 3\n\n## The Lebanese Civil War: Andr\u00e9e Ch\u00e9did's _La maison sans racines_ and Evelyne Accad's _L'excis\u00e9e_\n\nThe 1975-1990 Lebanese civil war was a bloody conflict that was spurred by internal factors and flamed by external ones, and whose complexity is impossible to render in a couple of paragraphs. The term \"civil war\" is actually a misnomer, as Lebanon's domestic affairs got entangled with the politics of the region and the conflict took an international dimension, fueled by Israel's invasions, Syrian troop deployment, and U.S. Marines intervention. As Latif Abul-Husn summed it up, \"the conflict revolved around three main themes: reform of the political system, the national identity of Lebanon, and Lebanon's sovereignty\" (2). According to him, the tension that emerged between Christians and Muslims was due to three factors: \"the rise to power of an organized Palestinian armed resistance in Lebanon that aligned itself with the Muslim bloc; a soaring Arab nationalist feeling in the wider region; and the rising expectations of the Shiites and their demands on the system for a greater share in the power structure\" (2).\n\nBeginning with the Arab conquest in the seventh century, Lebanon was part of various Islamic empires, with the exception of the period of the Crusades. Following massacres of Christian Maronites by Druze in Mount Lebanon in 1860, France intervened and Mount Lebanon was established as a privileged administrative region of the Ottoman Empire. After the collapse of the Ottoman Empire, France ceded to Maronites' demands and annexed parts of the Syrian provinces that fell under its mandate to the territory of Mount Lebanon to create present-day Lebanon. Such a separation was not acceptable from an Arab nationalist point of view (Salibi, _A House_ 28). According to Kamal Salibi, the establishment of Lebanon as a state under the French Mandate in 1920 was \"enthusiastically accepted by the Christians, and adamantly rejected by the Muslims\" ( _A House_ 2).\n\nWhen the French Mandate ended and Lebanon became independent in 1943, an agreement known as the National Pact was reached, primarily between Maronite and Sunni elites. According to this agreement, government positions were distributed proportionally amongst the recognized confessional groups based on the data provided by the 1932 census. The census showed a slight majority of Christians and served as the basis for allocating a Christian\/Muslim ratio of 6 to 5 for offices in Parliament, civil service, the army, and the judiciary.1 Maronites kept key positions, including the Presidency and the command of the army. Part of the agreement entailed that Muslims recognize Lebanese sovereignty and independence while Maronites affirm its Arab heritage and forego Western protection. Lebanese law required that every citizen's religious affiliation be clearly indicated on their identity cards.\n\nThe National Pact solidified the sectarian social structure and Maronite hegemony. It did not anticipate the uneven population growth that would in time create further disparity to be added to economic inequities between various groups. The struggle for a redistribution of power to mirror Lebanon's demographic changes and economic disparities were major factors in the 1975-1990 conflict. Maronites opposed demands for a new census for fear of losing power. Religious sectarianism was also exacerbated by an educational system that followed confessional lines, with the teaching of history becoming a highly politicized issue (Salibi, _A House_ 201-04). According to Salibi, Muslim and Christian Lebanese have fundamental differences about the historicity of their country, with Christians favoring Lebanese particularism while Muslims tend to situate it within a broader Arab-Muslim history. These opposite views of Lebanese history between Lebanism and Arabism underlie the ongoing political conflict (Salibi, _A House_ 3).\n\nIn addition to the problems due to Lebanon's confessional system, there were fundamental differences among Lebanese over issues involving the rest of the Arab world, most notably the Palestinian issue and the rise of Arab nationalism following Nasser's call for pan-Arab unity and the subsequent and brief merger of his country with Syria (1958-1961). After the creation of Israel in 1948 and the Israeli occupation of the West Bank in 1967, the influx of Palestinian refugees into Lebanon further complicated its internal situation. In general, Maronites were opposed to the presence of Palestinians, who in 1968 constituted 14 percent of Lebanon's population (Smith 315), as such presence compromised the fragile population equilibrium (since most Palestinians were Muslim). Thus, Palestinians were not allowed to integrate into Lebanese society and remained confined to refugee camps. Maronites also opposed Palestinian military operations against Israel from Lebanon, as it resulted in disproportionate Israeli retaliation, culminating with the 1982 Israeli invasion of Lebanon, a watershed event. In contrast, Muslims and leftists were more sympathetic with the Palestinian cause.\n\nIt is crucial to emphasize that alliances kept changing as the Lebanese conflict evolved: \"At different stages of the conflict the antagonism was greatest between the Christian and Muslim divisions, while at other stages the dividing line was between left and right, and so on, depending on the ebb and flow of the issue in dispute\" (Abul-Husn 3). Thus, in the period from 1977 to 1982, \"the Syrians who entered ostensibly in defense of the Christians were now engaged in battle with them; the Christian forces were entwined in internal warfare with each other; and the Palestinians and Shiites who fought side by side in the early war years were now engaged in prolonged combat\" (Salem 124).2\n\nThis chapter examines how two Francophone Lebanese writers have depicted the conflict, and more specifically how they have dealt with the religious dimension of a civil war that was summarily elucidated as a Muslim\/Christian rift in the West. Charles Smith has noted that the complexity of allegiances that characterized the Lebanon political landscape could not be delineated according to religious lines; yet, \"the Maronite leadership continued to present the question as one of Muslim-Christian strife, all the better to depict the issue as a religious clash to the outside world, as they had done since 1958\" (356). I demonstrate that two books published by Lebanese writers, Ch\u00e9did's _La maison sans racines_ and Accad's _L'excis\u00e9e_ , can be seen as perpetuating such a reductive view of the Lebanese civil war as a religious conflict. In addition, a close analysis of the Biblical and Qur'anic intertext of Accad's novel shows that the inscription of verses from the Qur'an provides a stereotyped view of Islam while the Biblical references implicitly reaffirm Christian values of sacrifice and redemption.\n\n### Andr\u00e9e Ch\u00e9did's _La maison sans racines_\n\nCh\u00e9did's novel is characterized by a complex narrative structure that intertwines three different time frames, each being clearly indicated through the use of different numerals and font setting. It starts with a day in August 1975, when two young women, Muslim Ammal and Christian Myriam, who have been friends since childhood, are marching towards each other with the hope of sparking a march of reconciliation. That day's tragic ending is prefigured at the very beginning though its outcome will only be divulged at the end. The other time frame is July-August 1975, which is a flashback about the reasons why Kalya meets her granddaughter Sybil in Lebanon for the first time (Kalya lives in Paris and Sybil in the U.S.). This time frame eventually catches up with Ammal and Myriam's march. The third time frame is 1932, when Kalya recalls the vacation she spent in Lebanon with her own grandmother, Nouza, a situation that parallels hers and Sybil's.\n\nThere are very clear references to European historical events, including the Great Depression, which began in 1929 and facilitated the rise of fascism, specific references to Hitler and Mussolini (81), and the events of WWII (83). However, all allusions to Lebanese history and current events are deliberately vague. For instance, when the taxi is stopped for an identity check before entering the city, Kalya reflects that \"Apr\u00e8s les brefs \u00e9v\u00e9nements d'il y a quelques ann\u00e9es, elle pensait que tout \u00e9tait redevenu calme\" 'After the brief events of a few years ago, she thought that everything had settled down again' (19, Schwartz 19), but never hints at what those events consisted of. Further on, the mention of \"la flamb\u00e9e meurtri\u00e8re\" 'the fatal outburst' that took place fifteen years prior probably alludes to the mini-civil war of 1958, but that has to be inferred (38, Schwartz 43). During their ride from the airport, during which they pass by luxury hotels and villas, Sybil inquires about something that contrasts sharply with the rest of the landscape, and that is described as follows: \"au bas des falaises, s'embo\u00eetant les unes dans les autres, des cabanes en fer-blanc s'entassent, suivies d'un amoncellement de tentes brun\u00e2tres\" 'at the foot of the cliffs, lies a tight cluster of tin huts and a mass of brownish tents' (18, Schwartz 17). The driver at first does not answer and speeds up, but after Sybil insists, wondering whether people live there, he retorts: \"C'est provisoire\" 'It's temporary' (18, Schwartz 17). There is no mention of Palestinians in the novel, but this description and the driver's statement undoubtedly evoke their refugee camps.\n\nCh\u00e9did's novel tends to pass over historical events with few allusions to the issues that triggered them. For instance, in the following paragraph, which appears in the July-August 1975 time frame, one can easily recognize the events that sparked the beginning of the civil war:\n\nAmmal et Myriam venaient d'apprendre qu'en pleine ville des hommes arm\u00e9s avaient stopp\u00e9 un autocar, abattu une dizaine de passagers. Le m\u00eame jour, dans la proche campagne, d'autres avaient d\u00e9couvert les cadavres mutil\u00e9s de cinq jeunes gens jet\u00e9s au bas d'un talus.  \nQui avait commenc\u00e9? Quel acte avait pr\u00e9c\u00e9d\u00e9 l'autre? D\u00e9j\u00e0 les fils s'enchev\u00eatraient. (71)  \nAmmal and Myriam had just learned that armed men had held up a bus in the middle of the city and murdered ten passengers. The same day, in the surrounding countryside, others had discovered at the foot of an embankment the mutilated corpses of five young people.  \nWho had started it? Which act had come first? The threads were already becoming tangled. (Schwartz 83)\n\nAlthough there is some inaccuracy regarding the date, the first allusion is to the incident that occurred on April 13, 1975, when Christian Phalanges gunmen killed 27 Palestinian passengers on a bus, and that is commonly considered the event that marked the start of the war.3 The author purposefully avoids mentioning any of the various parties involved in the conflict, thereby refusing to blame any group, and stressing the fact that the cause and beginning of the conflict are difficult to pinpoint.\n\nThe recollection of events that happened in 1932 is triggered by what is happening in 1975, with the numerous parallelisms that brings the two time frames closer. This choice of year may be an indirect reference to the census of 1932, and the consequences it had on the political structure of Lebanon. Anne-Marie Miraglia has shown that the narrative technique of intertwining three different time periods, along with lexical and thematic resonances \"facilitent l'encha\u00eenement des s\u00e9quences tout en insistant sur la nature cyclique de l'Histoire et de l'existence humaine\" 'facilitate the chain of sequences while insisting on the cyclical nature of History and human life' (33). While the novel provides no allusion to the issues of the census and the National Pact, the choice of 1932 to echo 1975 points to one of the key issues in the civil war. And yet at the same time, the accent on the \"cyclical nature of history,\" as Miraglia phrased it, deemphasizes the historical processes or the reasons why history seems to be repeating itself.\n\nThrough a pair of siblings, Myriam and George, Ch\u00e9did emphasizes the fratricidal aspect of the conflict, and therefore an imagined unity of Lebanon: \"Les heurts travers\u00e9s par le pays, par les r\u00e9gions avoisinantes, secouaient les deux adolescents, redoublant leur opposition\" 'The clashes that occurred in the country, in neighboring regions, shook the two young people, intensifying their opposition to each other' (59, Schwartz 68). George joins an unnamed party, while his sister Myriam collaborates with Ammal to organize a peace walk. George's stance is intransigeant: \"L'espoir de nous r\u00e9unir tous n'est qu'une source de tensions. Regardez l'histoire! Des belles id\u00e9es ne suffisent pas, rassembler des gens diff\u00e9rents dans un m\u00eame endroit cr\u00e9e la haine\" 'The hope of uniting us all causes nothing but tension. Look at history! Fine ideas aren't enough, gathering different people in one place gives rise to hatred' (103, Schwartz 126). Nowhere in the novel is there an allusion to colonial history and how French influence and interference to privilege the Maronite community have contributed to shaping relationships between various groups. Elise Salem notes that when General Henri Gouraud proclaimed the creation of Greater Lebanon in 1920, his proclamation stressed the Phoenician, Greek, and Roman heritage of Lebanon and omitted any reference to its Arab and Islamic heritage (15). George's stance contrasts sharply with his sister's peace activism and the overall message of the book.\n\nThe emphasis on roots throughout the novel can be understood in terms of the Lebanese context and its struggles to define its national identity. Indeed, one of the epigraphs is taken from Khalil Gibran, the renowned American-Lebanese poet who became one of Lebanon's national icons when so few existed (Salem 33). As Michelle Hartman argues, the title of the novel can be read as meaning \"the family without roots\" by translating the word \"house\" into Arabic ( _bayt_ ) and exploring its various denotations. She then interprets it as referring to how characters from one family explore their roots by spending time in their \"native land\" (62). As Hartman noted, Ch\u00e9did problematizes the issue of pure identity through the plot, the main characters' background, and her use of epigraphs because \"De-emphasizing roots, heritage and 'true origins' is also important in the context of war-torn Lebanon. A fixation on heritage and lineage can have negative connotations in a country with such a diverse population, particularly one in which people were divided by a war which emphasized sectarianism\" (65-66). It might also be read as wishful thinking on Ch\u00e9did's part that Lebanon should become a \"country without roots,\" given the fact that sectarianism, and different views of Lebanese roots or history, were at the core of the conflict. Historian Salibi has titled his study of the various views of the Lebanese history _A House of Many Mansions_ , after a Biblical verse (John 14:2), where house here stands for country.\n\nYet, despite the avoidance of naming and pointing fingers throughout the novel, the conflict seems to be framed exclusively in religious terms. In an incident at the beginning of the novel, two cab drivers (one a Muslim, the other a Christian) have an argument. The narrator comments: \"M\u00eal\u00e9 au moindre \u00e9v\u00e9nement, \u00e0 toutes les col\u00e8res, \u00e0 toutes les r\u00e9conciliations, Dieu vient d'appara\u00eetre sur le devant de la sc\u00e8ne. Son nom se prononce \u00e0 tout bout de champ, soumis aux hommes, \u00e0 leurs violences, \u00e0 leurs amours\" 'Involved in the most trivial incident, in every quarrel, in every reconciliation, God has just appeared center stage. His name is bandied about at the beck and call of men, their loves and hates' (14, Schwartz 11). Later on, Odette tells Kalya that \"Ici, la religion prime tout, elle marque toute l'existence\" 'Here, religion dominates everything. It affects our whole lives' (46, Schwartz 52), an indirect allusion to the National Pact and its confessional system, which are never referred to explicitly. The only elaboration that is offered in the course of the narrative are theological disputes that date back centuries and refer to the schism between Sunnis and Shiites in Islam, and within the Christian church regarding the nature of Christ (49), emphasizing the murderous consequences of these debates. Kalya's uncle, Mitry, who tells her about this history, links it to the present by saying \"Jusqu'aujourd'hui, dans ce pays, il y a quatorze possibilit\u00e9s d'\u00eatre croyant, monoth\u00e9iste et fils d'Abraham!\" 'Even today, in this country, there are fourteen ways of being a believer, a monotheist and a son of Abraham!' (49, Schwartz 56). This is a reference to fourteen (actually seventeen) denominations that are recognized by Lebanon's political system. However, it skews the reality, because the contemporary disputes between groups are not due to theological differences, but to the issue of distribution of power and fair representation. By insisting on theological differences and eluding the confessional political system of twentieth-century Lebanon, the novel promotes a clash of religion interpretation of the Lebanese civil war, since the internal socio-economic as well as the external factors (the Palestinian refugees) are only indirectly surfacing in the course of the narrative.\n\nOn the one hand, the novel's portrayal of a well-off Christian family with little contact with any other group could be seen as showing how out of touch its members are with what constitutes Lebanon as a whole. Contrary to the rest of the characters, Myriam and Ammal are aware of the socio-economic problems that plague some of the population of Lebanon, though this poverty is not associated with specific groups (76).\n\n### Religious Intertexts, Female Genital Cutting, and the Lebanese Civil War in Evelyne Accad's _L'excis\u00e9e_\n\nFemale Genital Cutting (henceforth FGC, more widely known as FGM or Female Genital Mutilation)4 has received significant international attention following the 1979 United Nations Conference on Traditional Practices Affecting the Health of Women and Children. When Accad recalls the impact that learning about FGC had on her while she was a doctoral student, she explains that she was \"very shaken,\" \"sick for several weeks,\" and that \"the title of [her] first novel, _L'excis\u00e9e_ , was already determined\" following this experience (\"Writing\" 59). Accad had left her native Lebanon to study in the United States, where she eventually became a university professor at the University of Illinois at Urbana-Champaign; she has published scholarly literary studies as well as novels and poems. When reflecting on what prompted her to write, she remembers that learning about the \"cruel practice of sexual mutilations\" was a turning point (\"Writing\" 59).\n\nAccad's novel was conceived as an autobiographically inspired feminist stand against women's subjugation by oppressive practices and religions (\"Writing\" 59).5 However, Accad herself has stated that \"what one intended to say [is] not necessarily what the readers see\" (\"Writing\" 47).6 My reading does not contradict the feminist intention and interpretations that have dominated scholarship on that novel, but goes beyond; it also departs from it in that it is a critical analysis that examines closely how the religious intertexts favor one religion over the other, and the implications to be derived given the socio-political setting of the novel (whether it was intended by the author or not).\n\n_L'excis\u00e9e_ is a hybrid text that mixes genres (novel, poetry), profane discourse and Scriptures, as well as religious traditions through citations and allusions to the Bible and the Qur'an. The narrative is interspersed with poetry and unmarked verbatim excerpts from the Christian and Muslim holy books, which are occasionally set off typographically, and which have yet to be analyzed by critics.7 _L'excis\u00e9e_ tells the story of E., a Christian Lebanese young woman, unhappy with her life in Beirut in the aftermath of a bloody summer, and feeling oppressed by the patriarchal tradition brutally enforced by her father, a pastor. She meets P., a young Muslim Palestinian, and flees with him to a village in an unnamed country, in the hope of inventing a new future. Disillusion quickly sets in as her Muslim husband proves to be as engrained in patriarchal tradition as her Christian father. After she becomes pregnant, he neglects her, and it is implied that he might have taken another wife in the city. After witnessing the infibulation of young girls, she runs away, taking with her a child, Nour. Before committing suicide, E. entrusts the child to an Egyptian woman going to Switzerland. Nour (whose name means \"light\" in Arabic) is cast as the one who will return one day to her country and hopefully enlighten and liberate her sisters from FGC.\n\nWhile Accad's novel can undoubtedly mobilize sentiment against the practice (as all narratives featuring FGC do), it does so in a very problematic manner. In one of her scholarly publications, Accad justifies her reliance on literature to understand the causes of the Lebanese civil war by stating that \"[creative works] allow us to enter into the imaginary and unconscious world of the author. In expressing his or her own individual vision, an author also suggests links to the collective 'imaginary.' . . . literature . . . reflects and articulates the complexities of a situation\" ( _Sexuality_ 5). While _L'excis\u00e9e_ gives us a glimpse of the author's imaginary about the power relations between Muslims and Christians in Lebanon through the very bleak future she portrays for both communities, I argue that using the very real practice of FGC as a trope for patriarchal oppression of women and combining it with political and religious issues that are contextually unrelated reinforces harmful religious stereotypes that can only impede progress rather than promote efforts to eradicate such practice.\n\n_L'excis\u00e9e_ points the reader to an allegorical reading through the persistent avoidance of specifying a certain time (1958), a place (Egypt), and names (initials instead of names for the two main characters). The excised woman of the title is meant in a symbolic way, as is made clear by Accad in a recent article: \" _L'excis\u00e9e_ shows a woman, E., Elle (She in French), Eve (woman everywhere, myself to a certain extent), woman excised symbolically by fanatical religion in war-torn Lebanon, socially by the tyranny of man\" (\"Writing\" 59). The main character has not been physically excised, and the back cover of the book makes it clear that the title does not refer to the secondary character of the Egyptian woman, who has been excised, and whom E. meets twice by chance. The symbolism of the letter _p_ , which represents patriarchy, is made obvious when her father and the Christian Father are indicted (20), and this before the appearance of the character of P. As for P., the religious connotation that his first name carries is the first thing that comes to both E.'s (19) and later her father's mind (48) upon hearing it, and for his name to unmistakably identify him as a Muslim Palestinian, it would have to be Arabic. However, the letter _p_ does not exist in the Arabic alphabet.8 Therefore the character of P., who becomes emblematic of both patriarchy and an (illusory) Palestine, cannot be read literally. The fact that a Muslim character becomes the symbol of patriarchy in this allegorical reading is not without implications, and the fact that the focus of the novel will turn to Islam needs to be unpacked.\n\nWhile the novel deliberately avoids specifying the year in which the narrative takes place, it leaves markers that make it easy to identify. The novel starts at the end of what has been \"un \u00e9t\u00e9 de rage\" 'a summer of rage' (9), and the beginning of the narrative proceeds to enumerate various events that took place during that summer: the U.S. Marines' Sixth Fleet has been called in, and there is a new regime in Iraq (9-13)9. These events point to 1958, when a six-month civil war erupted over the issue of pan-Arab unity under the leadership of Nasser, president of the United Arab Republic of Egypt (1958-1961); 1958 is seen as a precursor to the civil war that would rage in Lebanon from 1975 to 1990.10 However, allusions to Israeli raids (\"Tyr est viol\u00e9 par l'homme venu du Sud\" 'Tyr is raped by the man who came from the south,' 36)11 have been understood to refer to the 1982 Israeli invasion of Lebanon. Given that the novel was published in 1982, it more likely refers to some of the numerous raids conducted by the Israeli army in (Southern) Lebanon prior to 1982 (starting with the strike on Beirut International Airport in 1968). In any case, such an unclear allusion contributes to blurring the setting's specific historical moment.12 This, like the refusal to name the country where E. and P. flee (to which I will return later), promotes the vision of Lebanon as a country stuck in religious strife. Although the beginning of the novel states that this is not a war of religion (12), but a war caused by social inequities that plagued Lebanon according to religious and ethnic lines, the refusal to ground the story in a specific year relegates specific historical processes and events to the background instead of foregrounding them. On the one hand, the failure of E. and P.'s marriage forecasts the civil war that fully broke out in 1975; on the other hand, the refusal to ground the story in a specific year implies that no peaceful coexistence is possible in Lebanon. Reflecting on her first novel in an article, Accad repeats some of the sentences that are printed on the book's back cover (\"Where can this woman go? Is love between a Moslem and a Christian possible?\" in \"Writing\" 59), which shows that the back cover statement (perhaps written by Accad herself) orients the reader in the direction intended by the author. Interestingly, in her study _Sexuality and War_ , Accad criticizes the Israeli government for using Lebanon and thinking \"it benefits from a war that, among other things, proves that peaceful cohabitation among different religions is an impossibility and a myth\" (34). Yet her novel, through her depiction of the failed marriage of a Christian and a Muslim, offers the same conclusion because it orients the reader to an allegorical reading that, despite historical markers, refuses to be firmly grounded in time.\n\nThus, the novel can be read as an allegory of Lebanese society, where E. is representative of the Lebanese who thought that the Palestinian cause could revolutionize the Arab world (E. sees P. as \"la Palestine de l'espoir\" 'the Palestine of hope' (29). In _Sexuality and War_ , Accad criticizes the novelist Etel Adnan and all Lebanese intellectuals who held similar convictions for their shortsightedness (73-5). In _L'excis\u00e9e_ , E. pays the price of her na\u00efvet\u00e9 through her failed marriage. While the novel explicitly shows the patriarchal structure across the religious divide by identifying her Muslim husband P. with her father, the uneven power relationship between the couple (where Christian E. is subjugated by Muslim P.) is a problematic allegory of Lebanese society prior to the civil war. For one of the factors that triggered the war was the \"disjuncture between representatives and the people they claimed to represent\" (Cooke, _War's_ 20), as the number of Muslims started to exceed the number of Christians. The proportional representation according to confessional criteria established in 1943 in the National Pact did not plan for an uneven population growth among the various religious groups (the Maronites were then the largest community, and political power was shared between Christians and Muslims on a six to five ratio in favor of Christians, who slightly outnumbered the Muslims, see Abul-Husn 77). Moreover, one can object to the reduction of Lebanon's numerous religious communities, which numbered seventeen in the 1932 census, into two seemingly homogenous Muslim and Christian groups.\n\nThe character of P. highlights the fact that one can be victim and oppressor at the same time. In the beginning of the novel, P. is emblematic of Palestine, and Palestine is associated with suffering women several times in the novel, be they excised women (33) or E. herself (47), for Palestine has been raped (55, no specific marker is given here). But P.'s status as a victim does not prevent him from becoming E.'s unexpected oppressor; and here again the allegorical reading supports the interpretation by some of the Palestinian problem as it affected Lebanon, where Palestinian refugees turned from victims into troublemakers. Since the first influx of Palestinian refugees in 1948, international affairs have come to bear on Lebanon's domestic politics, as the (mostly Muslim) Palestinian refugees disrupted Lebanon's delicate balance between the various sectarian factions trying to get their share of power. This situation was exacerbated when the PLO established its headquarters in Beirut after Black September in 1970.\n\nWhile P.'s intransigence regarding the religious diversity of the Arab world reflects a real threat, pitting an intolerant Muslim who refuses to consider Christianity as an Arab religion against a powerless Christian who embraces diversity (35, 140), it only reinforces the reductive dichotomy of the Lebanese civil war as a conflict between Christians and Muslims, which has pervaded the Western world's understanding of this history. In a passage where E. describes poverty in quarters that she discovers for the first time, these quarters happen to be in the Druze and Palestinian camps (57-58). While the novel does point out that the reasons for the war are more economic and class-based than religious (12), the allegory formed by the E. and P. couple relegates this fact to the background. Although the allegory of E. and P.'s couple illustrates Accad's thesis in _Sexuality and War_ that \"since the personal is the political, changes in relationships traditionally based on domination, oppression, and power games will inevitably rebound in other spheres of life\" (167), it does not do justice to the reality of Lebanese society by depicting the Muslim as the primary oppressor. Christians not only controlled the political arena, they dominated all sectors of the Lebanese economy.13 One could say that the text unconsciously reflects the fear of the Christian Lebanese, who, although in power, were feeling threatened by their new minority status in terms of numbers. Although I doubt that this was Accad's intention, _L'excis\u00e9e_ can nevertheless be seen as one example of what Shu-Mei Shih characterizes as\n\ndeliberate national allegorical narratives with an eye to the market . . . When the signified is predetermined, allegories are easier to write or create and to understand and consume. A predetermined signified is produced by a consensus between the audience in the West and the Third World writer or director. It is a contractual relation of mutual benefit and favor that works first to confirm the stereotyped knowledge of the audience and second to bring financial rewards to the makers of those cultural products. (21)\n\nDespite details in the novel that partially account for the complexity of the situation, _L'excis\u00e9e_ can be seen as an allegory tailored to fit the First World's stereotyped view of the Lebanese civil war. This pitting of an oppressive Muslim against a defenseless Christian reappears in the novel in a scene between women that revolves around FGC. While the novel explicitly indicts both the Christian and Muslim faiths for their oppression of women, I show how the religious intertext reinforces this pitting of Islam against Christianity. The inscription of verses from the Qur'an in Accad's text provides a stereotyped view of Islam while the Biblical citations implicitly reaffirm Christian values of sacrifice and redemption.\n\nAlthough the title of the novel refers to excision or clitoridectomy, which consists of removing some or all of the clitoris and labia minora (and which, according to Elizabeth Boyle, is the most common form of FGC in Africa, 26), the procedure that is described in the novel is infibulation,14 practiced mostly in the African Horn (Sudan, Somalia, part of Ethiopia), southern Egypt, and part of Mali.15 Infibulation is the form of FGC that is most often brandished by activists and the media since it is the most extreme and the most prone to immediate medical complications and long-term after-effects. As often in the literature about FGC, the lack of hygiene and the rudimentary nature of the instruments are emphasized in _L'excis\u00e9e_ (121). In emphasizing the issues of physical pain and unsanitary conditions that lead to infections and complications as their main arguments for eradicating the practice, critics of FGC fall vulnerable to the fact that performing the procedure under anesthesia in a medical setting eliminates most of these problems. The particular emphasis on physical pain did not anticipate what is becoming an increasingly widespread phenomenon: the medicalization of FGC (see Fran Hosken 5-9, and Gerry Mackie 277 for a discussion of the implications of this issue). The medically founded opposition to FGC that characterized most efforts against the practice until the 1990s has come to be seen as a mistake by many (Boyle 140), especially those who are fighting against FGC performed on children.16\n\nThe novel is deliberately vague about the country where E. flees with P., all the while giving hints that allow it to be identified as Egypt (The Egyptian woman who boards the boat in Alexandria is fleeing the country E. and P. will go to [82-83]). Through contrived avoidance, the country is never named. By referring to the place with the terms \"pays de sable\" 'country of sand' (78), and \"pays du d\u00e9sert\" 'desert country' (62), the novel conjures stereotypes of an African continent stuck in time, antithetical to progress and modernity. Moreover, the overbearing omnipresence of flies in the village suggests decay and poverty. While FGC has been practiced in Egypt for over 2000 years\u2014thus predating Islam\u2014and is practiced by Africans of all faiths, including the Christian Copts in Egypt (Hosken 15, 55), _L'excis\u00e9e_ does very little to give the reader a sense of the diversity of the practice or of its practitioners, preferring instead to conflate the most extreme practice of FGC with Islamic teachings.\n\nAccad's novel is critical of Christianity, but most of her criticism consists of quoting the Bible to highlight how Christians have strayed from its teachings. In a passage on page 45, which is not a word-for-word quotation, but rather a summary of Luke 6:27-29,17 the narrator highlights the hypocrisy of people in E.'s church, who are incapable of compassion and forgiveness. The author also parodies Genesis 1:27 by stating \"l'homme a r\u00e9tr\u00e9ci Dieu et L'a fait \u00e0 son image\" 'man shrank God and created Him in his image' (91). Another Biblical allusion to David and Goliath (Samuel 17:4-51) highlights the racism E. encounters in Switzerland when she is sent to a Biblical camp. There, she overhears Europeans making a parallel between David and Goliath and Israel and Arabs, and viewing Israel as a \"miracle de la r\u00e9surrection\" 'miracle of resurrection' (90).\n\nSome scriptural allusions emphasize how religions have become so entangled with culture that both the Christian and Muslim codes of honor in Lebanon are very similar: \"Oeil pour oeil, dent pour dent. Et quand il s'agit d'une femme: deux femmes pour un homme\" 'an eye for an eye, a tooth for a tooth. And when it concerns a woman: two women for a man' (17), which allude respectively to Exodus (21:24) and Qur'anic legalistic prescriptions according to gender for testimony and inheritance.18 Another allusion: \"femme-faible, homme-fort\/femme-terre, homme-charrue\" 'woman-weak, man-strong\/woman-earth, man-plough' (97) alludes to Peter 3:7, which posits women as weaker than men, and the Qur'anic verse 2:223 \"Your wives are as a tilth unto you; so approach your tilth when or how ye will\"19 comparing women to a field and men to a plough, thereby linking women to passivity and to submission to men. The strongest condemnation of Christianity occurs when the New Testament is linked to excision on a metaphorical level, when the author writes: \"Toutes les femmes acceptant le crucifix, l'\u00e9p\u00e9e qui les ch\u00e2tre\" 'all women accepting the crucifix, the sword that castrates them' (72).\n\nAccad's text is not as virulent in its criticism of Christianity as it is in its criticism of Islam. In one instance E. briefly associates both Christianity and Islam with excision: \"sang coulant dans la plaine, croix \u00e0 porter, croissant \u00e0 brandir\" 'blood flowing in the plain, cross to bear, crescent to brandish' (95), but the verbs' connotations are worth emphasizing: the image of the cross to be carried contrasts with the aggressive image of the crescent to be wielded, and pits a benign and suffering Christianity against an assertive and belligerent Islam. In another instance, Accad mixes up the two traditions and allows a biblical allusion to pass for a Qur'anic teaching. In a passage in which E. attempts to understand the custom of excision, a woman's gesture is described as follows: \"elle l\u00e8ve les bras au ciel vers Allah en ajoutant:\u2014toute notre vie, nous, femmes, n'est que souffrance. Dieu l'a prescrit\" 'she raises her arms to the sky towards Allah while adding:\u2014all our life, for us women, is only suffering. God prescribed it' (135). While Allah, the Arabic word for God, is used by Arab Christians as well as Muslims, in the novel it is only used by Muslim characters; in a French text it will immediately connote Islam. In attributing to Allah what is actually reminiscent of a quote from the Bible, in which women are punished with the pain of labor in childbirth,20 Accad displaces a core Christian tenet onto Islam, which, despite the fact that it places itself in the continuity of the Judeo-Christian scriptures, does not adhere to the teaching that Eve was responsible for the Fall of man, nor that labor pains are a punishment inflicted on her and her kind.\n\nThe first Qur'anic verse quoted in the novel is the verse 24:31 in which women are enjoined to lower their gaze and cover their bosoms; because it ends with an injunction to go back to Allah, it will be associated with the Qur'an by any reader ( _L'excis\u00e9e_ 14-15). That the author chose not to include the preceding verse, which enjoins men to \"lower their gaze and be modest\" (Qur'an 24:30), contributes to the stereotype that only women are subject to restrictions in Islam. That Qur'anic quote in the novel is preceded by a Biblical quote in which women are enjoined to be submissive to their husbands (Peter 3:l-7)21 and which summons husbands to be kind to their wives. In the Biblical citation, women's inferiority compared to men is somewhat alleviated by the injunction of kindness to wives, whereas the fact that the author has truncated the part of the Qur'anic quote that dictated similar restrictions imposed on men as for women reinforces the stereotype that women fare better under Christian dogmas than under Islamic ones.\n\nIn the narrative taking place in the Muslim village, Accad conjures up all manner of Western stereotypes about Islam. As with infibulation in regards to FGC, the village women's dress is one of the most extreme to be found in the Muslim world: they wear a mask over their face in addition to a veil covering their head and body (105). To crown it all, all households are polygyneous (113). In the scene describing infibulation, she associates the Muslim veil with FGC:\n\nLe couteau du sacrifice | The knife for sacrifice\n\n---|---\n\nLa lame tranchante qui tue, qui s\u00e9pare, qui arrache | The sharp blade that kills, that separates, that tears out\n\nLes boutons du d\u00e9sir | The buds of desire\n\nLes p\u00e9tales de joie | The petals of joy\n\nL'ouverture de l'extase | The opening of ecstasy\n\nFerm\u00e9, cousu, scell\u00e9 pour toujours | Closed, sewn, forever sealed\n\nComme un grand voile de fer | Like a large iron veil\n\nComme un masque de rouille | Like a mask of rust\n\nComme un rideau de plomb (121) | Like a lead curtain\n\nThe flower metaphor for the clitoris and labia contrasts with the metallic veil and mask that the infibulated genitalia have become; it is a battle in which nature has been destroyed by culture, and that culture is unmistakably identified as Islamic.\n\nThis intertwining of Islam with excision and oppressive practices is reinforced with quotations from the Qur'an.22 While the quotes are not identified as verses from the Muslim holy book, they are among the most well-known verses about women (such as the verse of the veil, 143). Other Qur'anic verses are quoted and fulfill part of a narrative function; they are associated with E.'s misfortunes and make it seem as if Islam preordained for all women the downturn in events that E. experiences. Part of the verse on polygamy is quoted twice in the novel (96, 116); the second time the passage implies that E.'s husband has taken another wife after she became pregnant. The quote of the last part of verse 65:4 (about the waiting period for divorced women before they may be turned out of their husband's house) forecasts that E.'s fate will probably be divorce after she delivers her child (116).\n\nAccad also selected parts of the first section of sura 56 (verses 4-6, 12-17, 22-24, 35-8), which deal with the rewards of paradise, and in particular the Houris, which have come to be understood in Islam as eternal virgins awaiting the righteous in Paradise. This truncated sura is inserted right after the description of the procession of women singing after the girls' excision (127). This insertion highlights one of the reasons given for infibulation, that is, guaranteeing the virginity of the future bride, but it gives the wrong impression that infibulation is sanctioned by the Qur'an (which it is not). The last quote from the Qur'an is inserted after the women have been described as reciting verses from it; the reader can infer that the verse quoted is what the women are reciting as they are about to throw in the river the genitalia of the girls who have just been excised (128-9). That verse is known as the pledge of allegiance given by women converting to Islam (60:12); inserting it in that section erroneously implies that the practice of infibulation is part of Muslim women's duties.\n\nIn another indented section that contains the Arabic word for God, the reader is misled to believe that infibulation is prescribed in the Qur'an by the presence of Arabic words:\n\nKhatin, Tahara, Khatin | Khatin, Tahara, Khatin\n\n---|---\n\nCoupez, mais ne coupez pas trop | Cut, but do not cut too much\n\nCoupez, coupez, coupez | Cut, cut, cut\n\nmais | but\n\npas | not\n\ntrop | too much\n\nPuis refermez et que Dieu soude bien | Then close and may God weld it\n\nle tout | all well\n\nPuis recoupez et resoudez et recoupez | Then recut and reweld and recut\n\nAllah leveut (135) | Allah wants it\n\n_Khatin_ (circumcision) and _Tahara_ (purification) are both Arabic words used to designate circumcision. This passage refers to infibulation, since it alludes to the fact that infibulated women need to undergo further cutting to have intercourse and to deliver children, after which they are commonly reinfibulated. There is no mention of excision in the Qur'an, and this paragraph is an allusion to a Hadith23 regarding FGC, which reports the Prophet recommending that a midwife exert restraint. This Hadith is not found in the collections compiled by the scholars Bukhari and Muslim that are considered authentic. It is found in Abu Dawud's collection (41:5251) and is classified as weak because its chain of transmission is broken. It states: \"Narrated Umm Atiyyah al-Ansariyyah: A woman used to perform circumcision in Medina. The Prophet (peace be upon him) said to her: Do not cut severely as that is better for a woman and more desirable for a husband.\"24 Both supporters and opponents of FGC in Muslim countries have used this Hadith to defend their cause: with some arguing that only a moderate cutting of the skin covering the clitoris, analogous to male circumcision, is allowed; and others arguing that the Prophet's daughters were not circumcised and that the Hadith should be dismissed because of its classification. This Hadith has been used effectively in campaigns in Egypt, Mauritania,25 and Sudan as a means to minimize the amount of tissue cut.\n\nWhile this mixing of Qur'an, poetry, and Hadiths provides a distorted view of mainstream Islamic teachings, the Biblical intertext reaffirms the Christian theology of redemption. The most recurrent quote from the Bible comes from John 14:6: \"J\u00e9sus a dit : Je suis le Chemin, la V\u00e9rit\u00e9 et la Vie. Nul ne vient au P\u00e8re que par Moi\" 'Jesus said to him, 'I am the way, and the truth, and the life. No one comes to the Father except through me.' This quote appears at least four times in the narrative (pages 16, 22, 72, 97). It stresses the importance of Christ as the divinely revealed reality of God, and suggests that Christ is the only way to Heaven. Other quotations, such as \"Le sang de Christ vous lave de tout p\u00e9ch\u00e9\" 'the blood of Jesus Christ, his Son, cleanses us from all sin' (19), taken from John 1:7, reinforce the redemptive quality of Jesus' sacrifice.\n\nIn _L'excis\u00e9e_ , the parallels between E. and Jesus are numerous: Indeed, some quotations from the Bible, such as the one on page 43 (inserted in the narrative), taken from Revelation 5:l-5,26 reinforce the association between E. and Christ, and provide a note of optimism in the text: E. being identified as the savior is confident that she will succeed, and she draws from the Bible the courage to go against her family's wishes. On the next page (44), the quote comes from Revelation 3:7-8 and 3:12,27 which again implies at this point of the narrative that E. will triumph over her parents. Another allusion to Genesis 7 (to Noah and the flood) points to the little child Jesus\/Nour as the savior of mankind (23). In fact, women in the text are all somehow associated with the sacrifice of Christ: E.'s mother (whose hand \"accepte d'\u00eatre clou\u00e9e pour que les autres vivent et connaissent la V\u00e9rit\u00e9\" 'accepts to be nailed so that the others may live and know the Truth;' 49), as well as Mary, Jesus' mother (72). E. also becomes a female figure for Christ: first through the \"crucifixion symbolique\" when her father nails the shutters of her bedroom window shut (Marie 71), and when she sacrifices herself through her suicide-drowning at the end of the novel in order to save other women. The book opens with an unidentified citation from the Bible (Revelation 12:13-18),28 about the Woman and the Dragon, followed by a short poem that the reader can retrospectively interpret as being about E.'s flight with Nour. This juxtaposition of scripture and poetry reinforces the association of E. with Mary and Nour with Christ. The sermon given by E.'s father in the tent also refers to the dragon and the child (21): the wait for the child savior (Jesus) parallels the anticipated return of Nour, who is expected to save her sisters from excision and other oppressive customs. Despite the condemnation of the fact that women are being sacrificed, the ending of the novel repeats and reinforces the Christian theology of redemption through sacrifice.\n\nParadoxically, this reappropriation of the Christ figure through the character of E, while interpreted as feminist by Marie (160), participates in the ethnocentrism that First World feminists have been criticized for regarding issues concerning Third World women. As Chandra Mohanty stated in her seminal article, the analytical strategies of assuming one's class and background as the norm applies to Third World women writing about their own cultures (52). At the beginning of the novel, E.'s orientalist attitude is obvious as she reflects while on her way to the village how she will \"p\u00e9n\u00e9trer ce monde des femmes derri\u00e8re le voile, ce monde des \u00eatres du silence, ce monde de l'attente, . . . ce monde qui l'appelle parce qu'elle a \u00e9t\u00e9 choisie pour le comprendre\" 'penetrate that world of the veil, that world of women behind the veil, that world of beings of silence, that world of waiting, . . . that world that calls her because she has been chosen to understand it' (103). E.'s attitude is similar to the one that First World feminists have been criticized for adopting: implying that the veil can only have one meaning, that veiled women cannot speak for themselves, and the quasi-messianic attitude of the character who ultimately fails (and one can argue there is no real attempt) to understand anything that goes on in the village in terms other than her own cultural values.29 The same applies to the novel as a whole, which casts Qur'anic scriptures as the source of oppression for Muslim women while framing its narrative within Christian theology.\n\nWomen practicing FGC are seen by E. and the Egyptian character she meets on the boat as perpetuating the custom as an act of revenge, \"heureuses de voir que . . . la souffrance ne s'est pas arr\u00eat\u00e9e \u00e0 leur propre corps et que le cercle infernal se perp\u00e9tue\" 'happy to see that . . . the suffering did not stop with their body and the infernal circle perpetuates itself' (85). This portrayal of vengeful mothers is very problematic, for it shows a lack of understanding of the social contexts in which FGC is carried out. Researchers have emphasized that in the communities where FGC is the norm, parents would be considered irresponsible if they did not have their daughters conform to the custom. In _L'excis\u00e9e_ , a clear demonization of the midwife performing the infibulation takes place in the course of the narrative: the midwife comes to be designated as a witch. Like Alice Walker who connects her personal eye wound to FGC as a patriarchal wound in _Warrior Marks_ ,30 Accad's character identifies with the excised girls, whose operation becomes the symbol and the height of all patriarchal oppression. E.'s psychological sufferings are mirrored and amplified in the infibulated girls' pain in a graphic scene towards the end of the novel. In a short exchange in which E. attempts to understand the reason for the custom, women are portrayed as passively accepting the tradition (134). There is no real dialogue taking place between E. and the Muslim women of the village, although this would have been an opportunity to insert some explanation for the custom in the narrative.\n\nIn addition, E. panics after the women ask her whether she has been excised or not, and she feels threatened when they make a gesture as if attempting to lift up her skirt to see for themselves: \"Elle regarde les femmes avec effroi, mer en furie, houle pr\u00eate \u00e0 la submerger, \u00e0 la noyer, \u00e0 effacer la difference. . . . Comment se d\u00e9fendre contre ces femmes qui semblent assoiff\u00e9es de sang et de sexes ensanglant\u00e9s? . . . Pourquoi cette rage contre la diff\u00e9rence?\" 'She looks at the women with dread, infuriated sea, swell ready to submerge her, to drown her, to erase the difference. . . . How to defend oneself against these women who seem thirsty for blood and blood-soaked genitals? . . . Why this rage against difference?' (137). E.'s reaction appears rather disproportionate with the question and gesture of the women who never hint that they intend to excise her.31 What is telling in this scene is how E. feels threatened by her distinction and how she constructs the women as hungry for erasure of that bodily difference, which parallels her religious difference with her husband. Just as P., as a Palestinian refugee, is both victim of injustice and E.'s worst oppressor, women, who are often associated with Palestine's sufferings throughout the novel, appear as oppressors of future women by perpetuating the tradition of FGC.\n\nIn _Sexuality and War_ , Accad is aware that changing a society requires \"a long process of in-depth political, economical, psychological, religious, sexual, familial, and social transformations established on an understanding of the different factors, causes and links between these various fields\" (166). While _L'excis\u00e9e_ succeeds in making the reader empathize with the girls subjected to FGC, which is, as Ellen Gruenbaum states, a necessary first step to work on improving their living conditions (201), it falls short in providing the reader an understanding of the factors that perpetuate FGC.32 The main character escapes from the village, taking along a little girl who has not yet been excised. After a chance encounter with the Egyptian woman who had first told her about the custom of infibulation, she entrusts Nour in her care to be taken to Switzerland. E. drowns in the sea, while Nour, who in a melodramatic tirade has gained sudden consciousness of the oppression women face in her village, is portrayed as determined to later return to her country to save her sisters (173).\n\nE.'s failure is the failure of the outsider (in the sense that she does not come from that country) and in this the novel is in agreement with all who believe that change cannot be imposed from outside. History has shown that any imposition or threat from Western powers (such as the one to cut off foreign aid) are seen as imperial moves that threaten indigenous cultures and are not effective (Boyle 104). In Sudan and Kenya, attempts by colonial powers to eradicate FGC failed because it became a political issue (see Dareer 72 for Sudan, Gruenbaum 207 for Kenya). In a similar vein, international pressure in Egypt did succeed in making changes on paper (Boyle 2-5), but these changes seem to have had no impact on the ground: according to a 1996 survey, 97 percent of married women had been circumcised in Egypt (Boyle 2); according to a recent article, the prevalence rate stayed the same in Egypt between 1994 and 2003 (Mekay).\n\nMoreover, the fact that Nour has to leave the village and get a Western education before she can hope to change her village's customs validates the ethnocentric view that the center of change and modernization is in the Western world, and that African countries are stuck with immutable traditions with no conceivable modifications coming from the inside, which is quite contrary to what is happening at the grassroots level. Even Hosken, one of the most vocal and controversial activists against FGC, has become conscious of the fact that action for change cannot come from outside but has to be led by African women who live in Africa (Hosken 9). This is also the recommendation of the Women's Caucus of the African Studies Association in a position paper issued in the early 1980s, with the additional warning that the issue should not be singled out from all others affecting women, especially economic ones (2).33 The ending of Accad's novel, which forecasts Nour as the future savior of African women from FGC after she has been enlightened by the education she will receive in Switzerland,34 runs counter to the recommendation mentioned above, and is symptomatic of the imperialism that Western feminists have been criticized for regarding issues concerning Third World women.35 It demonstrates how tricky it can be to address practices outside of one's culture, since Accad herself is aware that Western feminism has been denounced for its colonial stench when it came to African women (\"Author's\" xii). And yet _L'excis\u00e9e_ can be said to reproduce such cultural imperialism by the way in which it portrays the women of the village, FGC, and Islam.\n\nThe fact that there are varied meanings (such as a rite of passage into adulthood, a code of chastity linked to Islam, cleanliness) assigned to the practice in different locations should influence the strategies used for elimination, since what will work in one area might not work in another (Boyle 31). Boyle has shown that Christian women, whose ideologies are more in line with international ideals (since the latter were formulated in Christian countries) were more likely to be responsive to efforts to eliminate FGC than Muslim women (120). Further, she found that Muslim women who live in cultures that support FGC will justify their opposition to it with medical arguments rather than human rights notions (138), since medicine is viewed as a neutral argument that does not call into question the cultural values of their society (146). As she points out, \"although FGC did not originate with Islam, its continuation is tied up with Islamic beliefs in some areas\" (30). The fact that one type of FGC is called _sunna_ (which means the tradition of the Prophet) in some countries also reinforces the belief held by some that female circumcision is a religious Islamic duty.36 Given that she chose a Muslim setting to describe infibulation in her novel, Accad's rejection of religion seems far removed from the reality on the ground. Asma Dareer emphasizes that Muslim religious leaders can play a crucial role in the fight against FGC, since they have a very strong influence on their communities (103). Gruenbaum also relates an interview with a Sudanese doctor who uses the Qur'an to show that any procedures that affect a woman's sexuality are against the teachings of Islam (189). This Sudanese doctor uses the Qur'anic verse 2:187,37 which places sexual relations as a basic need like hunger and thirst, and that should be of mutual satisfaction to husband and wife, to advocate against FGC. If one is to use religious texts to combat infibulation in Muslim countries, educating people about what the religion actually says about the practice seems to be a more effective strategy.\n\nAccad can be credited for being, to the best of my knowledge, the first Francophone writer from an area where FGC is not practiced, to take up the issue in a creative work when it first began to gain international attention.38 _L'excis\u00e9e_ gave voice to the typical repulsed reaction of people for whom FGC is a foreign custom experience. Since then, however, much progress has been made in grasping the complexity of the issues surrounding this custom, but the absence of contextualization of FGC leads critics who read _L'excis\u00e9e_ to qualify the practice as child abuse (Hottell), or the affliction of a deformity (Marie 116).39 Unfortunately, twenty years after the publication of _L'excis\u00e9e_ , Accad's reductive portrayal of Muslim customs and FGC does not seem to have been refined by the plethora of publications on the issue. In a recent article, she writes of her character E.: \"how could she bend to his [her Muslim husband] customs, which crush women even more than those of her childhood?\" (\"Writing\" 60). _L'excis\u00e9e_ is a telling example of how a Third World author can have as ethnocentric an attitude as the First World critic towards a part of the Third World other than her own.\n\nAlthough Olayinka Koso-Thomas, an activist in Sierra Leone, ends her book, published in 1987, with the hope that by the year 2000 FGC will no longer be practiced in Africa (100), a recent report shows that figures for FGC remain stable.40 Many researchers have noted the \"discrepancies between the global discourse on female circumcision (with its images of maiming, murder, sexual dysfunction, mutilation, coercion, and oppression) and their own ethnographic experiences with indigenous discourses and with social and physical realities at their field settings and research sites\" (Shweder 220). These discrepancies might be a factor in what seems to be an absence of improvement in overall numbers despite two decades of activism and attention. In a remarkable collection of articles on the diversity of FGC, Bettina Shell-Duncan and Ylva Hernlund emphasize the fact that \"identifying the most effective and appropriate strategies for eliminating female genital cutting is among the most bitterly contested issues surrounding this practice\" (24). Some activists argue for the complete eradication of the custom, whereas others find it more realistic to promote compromising measures, such as milder forms of FGC in Sudan, or banning the practice for children but legalizing it for adults (Boyle 18). In any case, combating FGC is a multifaceted issue that needs to be discussed in terms of specific contexts. As Fran\u00e7oise Lionnet points out, there is a \"third way\" to frame the debate other than between cultural relativism and human rights universalism: \"to see [these practices] as part of a coherent, rational, and workable system\u2014albeit one as flawed and unfair to women as our own can be\" (165).\n\n### Conclusion\n\nEvents of the past few years have brought Lebanon back in the headlines: the assassination of Prime Minister Rafik Hariri in 2005 and the subsequent Cedar Revolution and withdrawal of Syrian troups, the 2006 July War with massive Israeli bombings that severely damaged Lebanon's infrastructure, and the crisis over the presidency in 2007-2008. These recent developments stem from the complexity of Lebanese domestic politics and their entanglement in the politics of the region, making it all the more urgent for critics to point out when writers such as Ch\u00e9did and Accad have not been up to the task.\n\n### Notes\n\n1. That census had been taken under the French Mandate, and \"there was a general suspicion, even among many Christians, that it had been a rigged one, at least to some extent\" (Salibi, _A House_ 198).\n\n2. For more details about such shifting alliance, see Salem 151, and 163-66.\n\n3. For two other versions of the beginning of the war, see Salem (99-100) and Salibi (1976: 90-101).\n\n4. Naming the practice (commonly referred to as mutilation or circumcision) has become a contested issue, and various terms, such as operation, surgery, and cutting are being used as alternatives. I will use the term Female Genital Cutting, which has been gaining popularity because of its neutral stance, instead of the more established Female Genital Mutilation, because the term \"mutilation\" connotes the intent to inflict harm. FGC ranges from a cut in the prepuce covering the clitoris to removal of the clitoris, all labia, and suturing of the vaginal opening. For explanations of the different types of cutting and terminologies, see Gruenbaum (2-4), and Boyle (25-26, including the fact that people do not always mean the same operation with the same terminology).\n\n5. The book itself and the narrative are not presented as an autobiography.\n\n6. Indeed, Accad knows first hand that a reader can see a lot more than the author intended, as evidenced by her exchange of letters with writer Halim Barakat, who disagreed with Accad's interpretation of his novel. All the endnotes to chapter eight of _Sexuality and War_ are quotes from Barakat's letters refuting Accad's analysis ( _Sexuality and War_ 130-4).\n\n7. In fact in some cases, it is clear that the religious intertext has not been identified and recognized for what it is. For instance, El-Khoury attributes an excerpt of the Qur'an to the novelist (142), while Karnoub and Issa, despite promising titles, do not even mention the Biblical references. Similarly, Cooke incorrectly assimilates all indented passages of the novel to poetry (\"Dying\" 16). Sullivan briefly mentions that the \"narrative is disrupted by words from the Qur'an and the Bible demanding submission from wives and women\" (73-4), but goes on to quote a poetry passage and never comes back to the religious intertext. Verthuy does recognize in a particular passage that one text is \"Christian\" and the other \"Islamic\" and that \"both urge women to obey their husbands\" (164), but does not go further.\n\n8. Given that the novel was written in French and published in Paris, Accad's intended reader might not notice that.\n\n9. Later on, there is a reference to the Algerian war of independence (34).\n\n10. See Kamal Salibi's 1976 book for an account of 1958.\n\n11. All translations from the novel are my own.\n\n12. Accad told Elisabeth Marie during an interview that she had wanted to mix up both wars in order to describe the horrors that took place during both periods and to denounce all wars (Marie, note 22 p. 15).\n\n13. In 1975, the average income of Christians was 16 percent higher than that of Druzes and 58 percent higher than that of the Shiites. Literacy rates among Muslims ranged between one third and half that of the Christians (Abul-Husn 15)\n\n14. In addition to excision, the labia majora are cut and the wound is sutured, leaving only a small opening for urine and menstrual flow.\n\n15. The Western world has a history of FGC, albeit not well known. Clitoridectomy was used as a medical cure for masturbation and hysteria in Europe and in the U.S., from the nineteenth to the mid-twentieth centuries (see Gruenbaum 9-12). A new trend has emerged: Female Genital Cosmetic Surgery or FGCS, which is being marketed as a positive advancement not just for medical but also for aesthetic reasons or beautification (see for instance <http:\/\/www.labiaplastysurgeon.com>). According to a _New York Times_ article, \"primarily, doctors say, aggressive marketing and fashion influences like flimsier swim-suits, the Brazilian bikini wax and more exposure to nudity in magazines, movies and on the Internet are driving attention to a physical zone still so private that some women do not dare, or care, to look at themselves closely\" (Navarro).\n\n16. However, others are advocating for some degree of medicalization, such as Fuambai Ahmadu, who as a willing adult underwent excision as part of the Kono initiation for women in Sierra Leone (308-310). See Zabus ( _Between Rites_ 214-221) for an analysis of Ahmadu's autobiographical vignette.\n\n17. \"But I say to you that listen, Love your enemies, do good to those who hate you, bless those who curse you; pray for those who abuse you. If anyone strikes you on the cheek, offer the other also; and from anyone who takes away your coat do not withhold even your shirt.\"\n\n18. Qur'an 4:11: \"to the male, a portion equal to that of two females\"; 2:282: \"if there are not two men, then a man and two women.\"\n\n19. All verses from the Qur'an are taken from _The Holy Qur'an_.\n\n20. Genesis 3:16: \"To the woman he said, 'I will greatly increase your pangs in childbearing; in pain you shall bring forth children.\"\n\n21. Peter 3:1 \"Wives, in the same way, accept the authority of your husbands\"; Peter 3:7 \"Husbands, in the same way, show consideration for your wives in your life together, paying honor to the woman as the weaker sex.\" All quotes from the Bible are taken from _The New Oxford Annotated Bible_.\n\n22. Both opponents and supporters of FGC invoke Islamic teachings (for instance in Sudan, see Asma El Dareer 79).\n\n23. Hadiths are narrations about the sayings and deeds of the Prophet Mohammed; they constitute the second source for Muslims after the Qur'an.\n\n24. This translation was taken from the following web site: <http:\/\/www.usc.edu\/dept\/MSA\/reference\/searchhadith.html>.\n\n25. See Zainaba's lecture as an example.\n\n26. Revelation 5:1 \"I saw in the right hand of the One who sat on the throne a scroll with writing on both sides, and sealed with seven seals.\"\n\n27. Revelation 3:7-8 \"And to the angel of the church in Philadelphia write: These are the words of the holy one, the true one, who has the key of David, who opens and no one will shut, who shuts and no one opens: 'I know your works. Look, I have set before you an open door, which no one is able to shut. I know that you have but little power, and yet you have kept my word and have not denied my name.' \" Revelation 3:12: \"If you conquer, I will make you a pillar in the temple of my God; you will never go out of it. I will write on you the name of my God, and the name of the city of my God, the new Jerusalem that comes down from my God out of heaven, and my own new name.\"\n\n28. Revelation 12:13-18: \"So when the dragon saw that he had been thrown down to the earth, he pursued the woman who had given birth to the male child. But the woman was given the two wings of the great eagle, so that she could fly from the serpent into the wilderness, to her place where she is nourished for a time, and times, and half a time. Then from his mouth the serpent poured water like a river after the woman, to sweep her away with the flood. But the earth came to the help of the woman; it opened its mouth and swallowed the river that the dragon had poured from his mouth. Then the dragon was angry with the woman, and went off to make war on the rest of her children, those who keep the commandments of God and hold the testimony of Jesus.\"\n\n29. For an analysis of parallels between E. and Christ, see Marie.\n\n30. For a remarkable analysis, see Inderpal Grewal and Caren Kaplan's article.\n\n31. Some critics read the scene where E. sees the women as threatening to excise her to the extreme by stating that E. \"escapes physical excision by leaving the community\" (Hottell). I concur with Marie who thinks that the women were kidding her (52).\n\n32. In her first scholarly book, Accad incorrectly states that a girl is more likely to be circumcised if she is from the lower classes, and that this is the reason why it is rarely mentioned in fiction from North Africa and the Arab world ( _Veil of Shame_ 20-21). As Shweder states, ethnicity and cultural group affiliation are the best predictor of circumcision (220), while education and economic status have no influence on it (230). Moreover, the reason it is absent from North African fiction is that FGC is nonexistent there.\n\n33. The success of the Tostan program in Senegal is attributed to the fact that FGC is only part of one of several basic education modules and that people are never told what they should do, instead they are given facts and trusted to make the right decisions for themselves (see Mackie 256-261).\n\n34. The choice of Switzerland is ironic given that it has been one of the slowest European countries to affirm women's rights (right to vote in 1971 and legalization of abortion in 2002).\n\n35. See Chandra Talpade Mohanty's article.\n\n36. The _sunna_ type in Sudan is supposed to be the mildest form of FGC and analogous to male circumcision because it consists in removing the tip of the prepuce of the clitoris, although one of the intermediate types of FGC is also called _sunna_ by many (Dareer 2-4).\n\n37. \"Permitted to you, on the night of the fasts, is the approach to your wives. They are your garments and ye are their garments.\"\n\n38. Other writers for whom FGC is a foreign custom have incorporated the practice in their novels. Ben Jelloun's _La nuit sacr\u00e9e_ [ _The Sacred Night_ ] features a scene where a group of women, portrayed as fanatical Muslims, infibulate their sister as an act of revenge (159-60). The first person narrator, who is the victim of the operation, \"pratiqu\u00e9e couramment en Afrique noire, dans certaines r\u00e9gions d'Egypte et du Soudan\" 'commonly practiced in black Africa, in some areas of Egypt and Sudan,' undisputedly calls it a \"torture,\" \"mutilation,\" a \"barbarian idea,\" a \"massacre,\" and the Sudanese guardian, who is blamed for giving the idea to the sisters, is \"une sorci\u00e8re, experte dans les m\u00e9thodes de torture\" 'a witch, expert in methods of torture' (163). While the excision that is performed on the narrator no doubt deserves the epithets that the narrator uses, for it is in that novel an imported cultural practice put to use with the intent to harm an individual, the casual mention that it is performed in black Africa orientalizes the practice and does nothing to inform the reader. While the narrative is careful to mention that such a practice in unknown in the Maghreb and forbidden by Islam (163), Ben Jelloun did not escape criticism of sensationalism. A novel by Canadian writer Marie Auger (pseudonym) is titled _L'excision_ [The Excision]. The title refers to a self-inflicted excision by a woman whose body can feel neither pain nor pleasure. The narrator briefly mentions the fact that some women excise their daughters, and refuses to try to understand why, because, in a categorical judgment, \"Il est plus facile d'accepter l'incompr\u00e9hensible que de comprendre l'inacceptable\" 'it is easier to accept the incomprehensible than to understand the unacceptable' (105).\n\n39. \"La femme excis\u00e9e est afflig\u00e9e d'une sorte de difformit\u00e9. Cette derni\u00e8re a cependant l'avantage d'\u00eatre non apparente et lui \u00e9vite d'\u00eatre davantage exclue de la soci\u00e9t\u00e9 qu'elle ne l'est d\u00e9j\u00e0\" 'The excised woman is afflicted with a kind of deformity. The latter however has the advantage of not being apparent and prevents her from being further excluded from society than she already is' (Marie 116). On the contrary, one of the reasons given for performing FGC is aesthetic (Koso-Thomas 7), and in many communities, the uncircumcised woman is the one who would be ostracized.\n\n40. \"The prevalence of FGC in Egypt, for example, stayed roughly the same\u2014about 97 percent\u2014between 1994 and 2003, while it rose in Cote d'Ivoire from 43 percent to 45 percent, slipped back from 95 percent to 89 percent in Eritrea and to 34 percent from 38 percent in Kenya, in the same time period\" (Mekay).\n\n## Chapter 4\n\n## The Algerian Civil War: Rachid Boudjedra's _Le FIS de la haine_ , Rachid Mimouni's _De la barbarie en g\u00e9n\u00e9ral et de l'int\u00e9grisme en particulier_ , and _Une enfance alg\u00e9rienne_\n\nThe French colonization of Algeria stands out among the other North African countries by the length of time (132 years) and the bloody war that was fought to gain independence. Algeria came back in the news in the late 1980s and 1990s with the rise of an Islamist political party and a devastating civil war. During that period, France was slowly coming out of a period of amnesia, as historians Benjamin Stora and Mohammed Harbi put it, about its painful colonial past and the atrocities it committed in Algeria before and especially during the 1954-1962 war for independence. This is the context in which the books analyzed in this chapter were published.\n\nAs the sole party since it won the war in 1962, the FLN had maintained a strong hold on Algerian politics thanks to oil revenues. The drop of the price of oil in 1986 brought this fragile equilibrium to a halt. Adding to this, a high birth rate, lagging infrastructure, and high unemployment rate took their toll and triggered riots in October 1988, that were severely repressed by the government. As Philip Naylor phrased it, \"On the surface, the riots stemmed from economic pressures, but they were deeply rooted in political, social, and cultural frustration\" (165). These events led to the adoption of a new constitution in early 1989 opening the political arena. One of the newly established political parties was the FIS (Front Islamique du Salut), founded in March 1989, a party that went on to win the first two free elections held in Algeria since Independence: for the local governments in June 1990, and for the National Assembly in December 1991.\n\n1991 was a watershed year for Algerian politics, because the FIS took center stage as the only viable opposition party to the FLN. Tensions grew between the government and the FIS leaders following the imposition of a redistricting before the legislative elections to minimize the FIS's projected gain at the polls. A siege was declared, the FIS leaders (Benhadj and Madani) arrested, and the elections postponed for six months. Some Islamists refused to engage in the elections which they saw as a dupery and took to the maquis, committing violent acts (over which the FIS had no control). After the FIS emerged victorious from the first round of elections in December 1991, the army's coup d'\u00e9tat in January 1992 (making the president resign, cancelling the elections' second round, dissolving the FIS and arresting its militants) marked the beginning of the civil war that would last until 1999 and in which Algerian civilians became targets. The civil war spilled over to France when the GIA hijacked an Air France plane in 1994 and orchestrated terrorist attacks in Paris in 1995.1 These events offered dramatic proof of the neocolonial relationship extant between the two countries; they brought to the fore the fact that France and Algeria's paths had remained intertwined. Despite decolonization,2 economic dependencies, such as the presence of French _coop\u00e9rants_ in Algeria, of Algerian emigrant workers in France, and agreements regulating the exploitation of Algeria's hydrocarbons, existed.3 Naylor notes that \"Though France was politically ambivalent [about the annulment of the elections], it consistently provided Algeria with indispensable economic support\" (196), notably through rescheduling of Algeria's debt. According to him, \"by exercising its influence in multilateral organizations and circles, France disguised its extraordinary economic support to Algeria,\" a support that was crucial in saving Algeria from financial disaster (207).\n\nThis chapter analyzes representations of the rise of the FIS to political prominence and the civil war in nonfiction works. The first section examines two essays that were triggered by the FIS's electoral victory and that were published the year of the annulment of the elections: Rachid Mimouni's _De la barbarie en g\u00e9n\u00e9ral et de l'int\u00e9grisme en particulier_ , and Rachid Boudjedra's _Le FIS de la haine_. My purpose here is not to weigh in on the question of whether intellectuals were right or not in supporting the annulment of the elections.4 My analysis demonstrates that both essays reduce the diversity of Islamist movements to the fundamentalism of the FIS. In addition, I contend that word choices (Mimouni's title and Boudjedra's inflammatory rhetoric) do more to feed the flames of fear than to further our understanding of the events under scrutiny.\n\nA surge of novels and autobiographical accounts set during the civil war were published during the 1990s in Paris by Algerians, but instead of analyzing some of these works in the second section (it would take a book to do them justice), I focus on a collection of short autobiographical narratives where the civil war trespasses on the colonial past. Also published in the 1990s, _Une enfance alg\u00e9rienne_ is set during colonial Algeria, a time period well before the civil war even started, and yet the civil war intrudes in the collection. I give particular consideration to the fact that the devastating effects of colonialism are mostly erased or strikingly downplayed, while the violence that surfaces in places is mainly associated with the Algerian nationalist movement during the war for independence and with the rise of Islamic fundamentalism that constitutes the backdrop to the publication of the text, commissioned in the 1990s. I draw out the implications of the civil war's intrusion into narratives taking place well before the 1990s, and of this shift of perpetrators of violence given the context of publication of the volume.\n\n### The FIS's Electoral Victory in Essays\n\n#### Rachid Boudjedra's _Le FIS de la haine_\n\nIn _Le FIS de la haine_ (2002), Boudjedra identifies several causes for the birth of the Islamist party. In the process, he does not spare any one, as he successively lays the blame on several actors: the West and the Algerian government, the Algerian people and intellectuals are all held accountable. The West is castigated for its double standards (17), its shameless capitalism and neocolonialism not only in the economic, political, but also in the cultural sphere with the concept of _Francophonie_ (20-31, 89-95), and its scorn of the Arabic language and Islamic culture inherited from colonial times that has contributed to the rise of religious fundamentalism (32-34). Regarding the West's project to improve the economic situation, Boudjedra points out that \"on a l'impression que les pays riches sont pr\u00eats \u00e0 s'accommoder de la peste int\u00e9griste si leurs int\u00e9r\u00eats \u00e9conomiques les plus \u00e9go\u00efstes sont garantis par les islamistes\" 'it seems that rich countries are ready to tolerate the fundamentalist plague provided that their most selfish economic interests are guaranteed by Islamists' (133). One can object that there is nothing new here in terms of Western foreign policy in the Middle East overall, for the same has been true with the FLN's authoritarian rule. As Fr\u00e9d\u00e9ric Volpi noted, the West has been more concerned with its short-term economic requirements than with Algeria's long-term needs to develop a democracy (129). Boudjedra blames the Algerian people for the unrestrained birth rate and the ensuing problems caused by such high demographics.5 Intellectuals are also blamed (56-58). Another factor listed as a cause for the rise of the FIS is the FLN's corruption and its Arabisation policy, which led the government to bring in teachers from abroad, especially from Egypt, some of whom turned out to be members of the Muslim Brotherhood who spread fundamentalism.\n\nScholars have pointed to the same factors as contributing to the Algerian crisis, so it is not so much the content of Boudjedra's essay, but its rhetoric that is objectionable. _Le FIS de la haine_ has been labeled as a pamphlet (Memmi, \"La folie\" 32), a category that is fitting given its virulent tone. While Boudjedra's writing is caustic throughout, I argue that his explicit demonization of the FIS and oversimplification of certain issues to the benefit of the FLN and detriment of the FIS skews Boudjedra's argument. This disproportion, both in quantity and quality, in the criticism of the FIS leaves little space to emphasize the damages caused by the FLN since independence.\n\nBoudjedra is able to paint a nuanced portrait of the West and the FLN, underlining both their achievements and their shortcomings, but he fails to do the same for the FIS. The writer can differentiate between on the one hand, the West that oppresses other nations and its multinationals governed by greed, and on the other hand the West of intellectuals and artists (104-105), but he cannot make the same distinction when it comes to the Algerian Islamist movement. Gilles Kepel has noted that from the very beginning in the early 1980s, the Islamist movement in Algeria has had two trends: an extremist one that advocates armed struggle and a reformist one that works through political channels ( _Jihad_ 170). Although Boudjedra does recognize that there are different \"islams\" (120), his depiction of the Islamist movement in Algeria is monolithic, with the FIS presented as the sole Islamist party, and with no distinction between different groups. After its formation, the FIS did not include all major representatives of the Islamist movement (Kepel, _Jihad_ 173); there were other Islamic parties on the Algerian scene (such as Mahfoudh Nahnah's Hamas, and Abdallah Jaballah's Nahda), who participated in the elections. Indeed, Michael Willis notes that several senior Islamists (such as Ahmed Sahnoun and Mohammed Sa\u00efd) refused to join the party, and that from its inception the FIS represented only part of Algeria's Islamist movement (117), but Boudjedra fails to mention any of this.\n\nBoudjedra faults the FIS leadership for its inconsistencies, which he sees as double talk (with Abassi's public statements differing sharply from Belhadj's hardline rhetoric), but these discrepancies also reflected real divisions between the two leaders of the FIS (Willis 146-48). In addition, there is a difference between the Islamic militants involved in the political process and the Emirs of armed groups, the latter usually composed of marginal people outside the religious sphere until they joined the jihad (Martinez, _The Algerian_ 96). But by lumping them all together under the Islamic fundamentalist label, the atrocities committed by the latter indelibly taint the former. Boudjedra has fallen into the trap of overemphasizing the violent and discriminatory aspect of the Islamic movement in Algeria, thus failing to explain how it came to have such support, whether one situates it in an international context of Islamic fundamentalism, or as a consequence of the evolution of the postcolonial state and society.\n\nBoudjedra portrays FLN leaders in a somewhat more nuanced fashion. Ben Bella was a \"despote . . . gris\u00e9 par le pouvoir\" 'despot, intoxicated by power,' \"inculte, d\u00e9magogue\" 'uneducated, demagogue' (38). While Boudjedra describes how Ben Bella had people arrested and executed, he does not describe him as thirsty for blood (as he does the FIS, more on that later). In his most virulent attack, Boudjedra refers to ministers under Boumediene as \"obs\u00e9d\u00e9s sexuels\" 'sexual maniacs' (42). The FLN's role in bringing about the birth of the FIS is broached briefly (11), but that brief sentence is overshadowed by two pages that praise the party for creating a modern independent state (13-14). Although Boudjedra admits that the FLN was \"autoritaire, m\u00e9prisant, dictatorial, corrompu, perverti, \u00e0 un point inimaginable\" 'authoritarian, comtemptuous, dictatorial, corrupt, perverted, to some unimaginable degree' (14), he exonerates it by intimating that these are characteristics of all political powers. Later on, he finds excuses for the FLN's corruption by recalling the harsh treatment its members endured during colonial times and the war for Independence (53), but shows no such leniency for the FIS leaders.\n\nOn the issue of corruption and poor economic planning, Boudjedra first states that both the state and the FIS are responsible (81); later on he criticizes the West as well as the FIS by naming it twice, but forgets the FLN (102). This is an example of Boudjedra's double standard: the FIS, which controlled the local governments for a year and a half, gets more blame than the FLN, which controlled all levels of government for close to thirty years, as if commensurability existed between the decades of FLN state rule and the brief period when the FIS held local offices. Louis Martinez has pointed out that the FIS could not carry out its policies after its victory in the local elections in June 1990 for lack of financial support from the government, something its victory at the legislative level could have changed ( _The Algerian_ 90-1, see also Willis 159-60).6\n\nBut more than his selective presentation of facts, it is the vocabulary that Boudjedra uses to disparage the FIS's leaders and its members that is striking: Ali Belhadj is \"ignare\" 'ignorant,' a \"petit avorton\" 'small squirt' whose weekly sermons leave its audience \"avide de sang\" 'eager for blood' (11), \"ils [leaders of the FIS] sont devenus aujourd'hui parce que cancres, parce que incomp\u00e9tents, les bourreaux de l'Alg\u00e9rie contemporaine, les fous de Dieu qui \u00e9ructent des insanit\u00e9s\" 'because they are dunces, incompetent, they became the executioners of contemporary Algeria, the madmen of God who eruct insanities' (65), \"les monstres qui dirigent le FIS. . . . Horde d'incultes\" 'the monsters that lead the FIS . . . A horde of uneducated people' (40);7 \"requins voraces qui aiment la couleur du sang et son odeur\" (140), 'voracious sharks that like the color and smell of blood' (15) \"maquereaux qui balafrent et vitriolent les femmes\" 'pimps who gash and throw vitriol at women,' \"des d\u00e9biles attard\u00e9s\" 'backward feebles,' \"rats enrag\u00e9s et pestif\u00e9r\u00e9s\" 'rabid and plague-stricken rats' (16). Boudjedra reserves his vitriolic rhetoric for the FIS, the words he uses animalize the FIS leaders and its members, who are reduced to monsters thirsty for blood. Animalizing one's target is a common colonialist and genocidal rhetorical ploy, used among others by the French in Algeria to refer to the FLN. Given that the pejorative term \"raton\" 'small rat' is still used to designate North Africans, for an Algerian to call other Algerians rats is to be purposefully provocative. Moreover, Boudjedra's accusations are not well informed: Madani held a Ph.D. in education and had published three books (Willis 147-8); Benhadj possessed great oratory talents (Kepel, _Jihad_ 174).\n\nThe party and Islamism are variously called \"un fascisme rampant et gluant\" 'a crawling and gluey facism' (10); \"la peste verte\" 'the green plague' (111, 135); \"monstre verd\u00e2tre\" 'greenish monster' (114); \"le pus vert\" 'green pus' (137); \"un parti ordurier et naus\u00e9abond\" 'a filthy and foul party' (16); \"Parti de la mort, il a la passion du sang, du bain de sang, de la tuerie. La majorit\u00e9 \u00e9crasante de ses responsables sont des cas psychiatriques. La pathologie du FIS est tr\u00e8s profonde\" 'the party of death, it has a passion for blood, blood bath, slaughter. The overwhelming majority of its leaders are psychiatric cases. The pathology of the FIS is very deep' (108). The following hyperbole, stating that the intent of the FIS regarding women is to \"imprison them in concentration camps\" (\"la femme il la parquerait dans des camps de concentration\" (93), crowns it all. One can see clearly here the measure of exaggeration in Boudjedra's text, for if indeed the FIS did not promote gender equality, its sexist pronouncements8 were a far cry from incitement to genocide.\n\nBoudjedra also engages in personal attacks on historians Harbi and Stora (122-23), and rails against Lahouari Addi's statement that the election of the FIS was a \"r\u00e9gression f\u00e9conde\" 'fertile regression' (110-11). Regarding the latter, Boudjedra deprecates the scholar (\"un petit sociologue\" 'a petty sociologist') to better denigrate the argument, with which he does not really engage.9 Boudjedra blames some journalists, whom he faults for believing that \"ils sont objectifs et que toute autre analyse que la leur est passionnelle et subjective\" 'they are objective and that all analysis other than their own is emotional and subjective' (115). However, the vocabulary and hyperbole detailed above are ground enough to turn that same critique against Boudjedra himself. Similarly, the attacks leveled against some journalists for simplifying a complex situation (118) can also be applied to his book.\n\n_Le FIS de la haine_ was reedited with a new afterword by the author in 1994, two years after the original edition. The back cover, signed by Boudjedra, sets the tone: only the FIS is blamed for electoral fraud and terror. He asserts that his book was written \"sans tabous, sans barri\u00e8res et sans pr\u00e9jug\u00e9s\" 'without any taboos, barriers, or prejudice' (backcover). In this new edition, Boudjedra keeps on blaming the FIS, and ignores the military exactions (including torture) committed by the regime, such as the measures taken starting in April 1993 to reconquer the Islamist communes of Greater Algiers (Martinez, _The Algerian_ 22). There is no mention in Boudjedra of the crackdown perpetrated by the government on the FIS in June 1991 and in January 1992 (Volpi 51, 58), of the dissolution of the party and the repression against its members\u2014more than ten thousand prisoners in detention camps (Burgat 304). The new afterword keeps blaming the FIS for all the gruesome news coming out of Algeria, and does not make any mention of the GIA at all, an armed group that split with the FIS and that was responsible for some of the atrocities that Boudjedra attributes to the FIS (Volpi 68).10 In addition, Boudjedra accuses the FIS of propaganda,11 but there is not a single word on the FLN's censorship of the media (Volpi 59).\n\nAccording to Boudjedra, \"le FIS n'est qu'un \u00e9l\u00e9ment de l'internationale int\u00e9griste islamique qui fait tant de mal \u00e0 l'Islam\" 'the FIS is only one component of the Islamic fundamentalist Internationale that wrongs Islam so much' (119). The term \"Islamic fundamentalism\" is used as a blanket term to denote a wide spectrum of movements.12 Boudjedra does not make the distinction that Volpi deems crucial in a Muslim context between Islamic fundamentalism as \"the set of political and ethical precepts which may (or may not) help people to organize their social world better, and Islamic fundamentalism as a name attributed to the activities of a group of political actors . . . claiming to represent Islamic orthodoxy\" (Volpi 10). Boudjedra's statement \"Le FIS n'a qu'un credo: le pouvoir, de n'importe quelle fa\u00e7on\" 'the FIS has only one creed: power, by all means' (\"Postface\" 127), could very well be said of the FLN. Boudjedra cannot come to terms with the fact that the Islamic movement does have a certain appeal, as attested by its capacity for mobilization (Burgat 2), not just to the poor urban youth but also to the pious middle class (Kepel 174),13 with the fact that Muslims may find an ethics and an understanding of modernity governed by Qur'anic principles (Majid, _Unveiling_ 21), and with the fact that democracy and Islam may not be incompatible (Layachi 59).\n\n#### Rachid Mimouni's _De la barbarie en g\u00e9n\u00e9ral et de l'int\u00e9grisme en particulier_\n\nMimouni's _De la barbarie en g\u00e9n\u00e9ral et de l'int\u00e9grisme en particulier_ adopts a perspective similar to Boudjedra's position. The first part, entitled \"Qu'est-ce que le FIS?\" 'what is the FIS?' purports to demonstrate how retrograde the FIS is by focusing on its alleged archaism (fixation of fundamentalists on the past), and its hatred of women, culture, and intellectuals. Mimouni does stress the difference between Islam and Islamic fundamentalism (particularly in his discussion of women), but does not differentiate different tendencies within the Islamic movement (153, 166, 170). The second part underlines that the power in place shares some responsibility in the rise of Islamic fundamentalism in Algeria (poor handling of the economy, using the fundamentalists to counter the Berber movement in 1980). The intellectuals are also blamed for their lack of courage and their absence at key moments.14 The third part focuses on the factors that promoted fundamentalism (moral crisis, demographic explosion, dual tracks in education). The fourth part talks about obstacles to democracy in Algeria (no tradition of democracy in Algeria, tribalism and illiteracy).\n\nMimouni's essay perpetuates some of the same stereotypes as Boudjedra's, starting with its title that clearly qualifies Islamic fundamentalism as a barbaric movement, made explicit in the course of the essay: \"ce retour de la barbarie\" 'this return of barbarity' (87). In addition, the treatment of the shortcomings of the FLN and the FIS lacks balance. For instance, only the FIS is presented as benefiting from _trabendo_ (black market, 64), although _trabendo_ predates the 1988 riots (Kepel 167). Even when there might be something positive to be said about the FIS, it is phrased in such a way as to turn it into a negative. For instance, Mimouni acknowledges the FIS's charity work, but in a manner that ends up turning it into a coldly calculated tactic to win more support (66). Several parallelisms between Algerian and French history are made: an interesting comparison is made between Boudiaf's return in 1992 and De Gaulle's in 1958 to solve the Algerian crisis (86), but a rather dubious one between the FIS and the OAS reduces a legally constituted political party that won elections to an underground terrorist organization (166).\n\nDespite these shortcomings, Mimouni's essay endeavors to achieve a rational examination of the situation. The last two pages seem to forecast the civil war that was to unfold (166-67). To his credit, Mimouni also acknowledges that the Islamic tendency is there to stay and must be taken into account,15 and that the dissolution of the FIS is not a long-term solution (165). Robert Mortimer has pointed out that \"What was needed, well before the fateful elections of December 1991, was a dialogue between secularists and Islamists about pluralism and civil liberties\" (19), and added that the urgency for dialogue has been rendered all the more critical by the civil war. Azzedine Layachi envisions the need for a democratic front including Islamists and secularists in order for change to occur; he states that \"the Islamists and the secularists must both change the way they perceive and relate to each other\" (63). Layachi acknowledges that this will be a challenge, but not an impossible task (63). Boudjedra's essay, with its virulent rhetoric, does not demonstrate that kind of willingness from its author, while Mimouni is definitely a step closer. However, the portrayal of Islamism as a barbaric movement does not foster the conditions necessary for dialogue or analysis.\n\nCriticism of the FIS was undoubtedly warranted. Party policies were characterized by vagueness and lack of innovation in terms of economic policy and political institutions. A majority of its pronouncements focused upon social, educational, and cultural issues, mostly advocating Arabization of education and segregation of the sexes (Willis 138-47), and its leadership was fragmented due to a clear cleavage between the two leaders about crucial issues (Willis 145-46, Naylor 220). However, Boudjedra and Mimouni's essays fail to point out that political violence began with the arbitrary arrests and torture of pro-Islamic demonstrators by the government, a fact that historians and political scientists writing about the Algerian crisis underline (see for instance Volpi vii). The fact that the so-called \"Islamist\" violence is a response to state violence is too often occulted (Burgat 115).\n\nMoreover, Burgat noted that the \"interruption of the electoral process and flare up of repression open[ed] a breach in the arguments of the FIS' legalists to the advantage of their most radical challengers\" (318). Mimouni does anticipate that state repression will radicalize the Islamist movement and promote the most extremist leaders (166). As for Boudjedra, he does not reflect in the new edition of his work on the fact that the level of violence was caused by the state's measures, and that the state was responsible for a significant proportion of the exactions committed, whether directly or indirectly by infiltrating Islamic movements (Volpi 90). Burgat notes that this government infiltration of the GIA was designed \"to keep the necessary repulsion in international public opinion and thus block any rational analysis of the crisis\" (328). Thus, the regime in Algeria succeeded in limiting the visibility of its opposition to the excesses of the most radical elements with the cooperation (conscious or not) of the international media (Burgat 316). With their titles, Boudjedra and Mimouni's essays have contributed to that current. In a similar vein, Willis noted the eagerness of the Algerian (and French) government to report items when the FIS implemented measures to conform Algeria's social life to their idea of an Islamic society, some true, but others turned out to be fabricated, distorted, or measures taken by previously FLN-controlled authorities (158-59).\n\nBoudjedra (96) and Mimouni (84) both accuse the FIS of burning a baby in Ouargla in 1989. Boudjedra recalls this incident again at the closing of the book (140-41), and compares it to the burning of the Reichstag in 1933 (which marked the end of the democratic republic of Weimar). He invokes lessons from Nazism (137), and from Iran (139). Similarly, Mimouni makes a parallelism between Nazism and the Weimar Republic and Algeria to counter those who defend that the electoral process should have been respected (152), as well as a parallelism with Iran (154). According to journalist Rabha Attaf (who opens her article by quoting Boudjedra's mention of it in his essay), this incident was nothing but a \"sordide affaire de moeurs\" 'sordid sexual immorality case' (202), yet it was exploited and told over and over again as a proof that the FIS hated women. According to her, \"l'utilisation mensong\u00e8re de cette affaire a un objectif bien pr\u00e9cis: disqualifier le FIS et justifier, a post\u00e9riori, un anti-islamisme primaire alimentant une r\u00e9pression sanglante sous couvert de lutte pour le droit des femmes\" 'the deceitful use of this case has a very specific goal: to disqualify the FIS and a posteriori justify a simplistic anti-Islamism that feeds a bloody repression under the guise of a struggle for women's rights' (202). The references to Iran bring foreign history and events to bear on the Algerian context, putting more emphasis on transnational Islamism, and not placing the rise of the FIS within the context of Algeria's unfinished history of (de)colonization.\n\nAs Willis points out, \"the high-profile backing the French media gave to the [secular opposition parties and the FLN] . . . seemed to confirm the FIS' propaganda against the other parties that they were the vehicles of Western and anti-Islamic values\" (153). With its numerous invectives and incendiary tone, _Le FIS de la haine_ can be seen as partaking (unwittingly) in this counter-productive campaign, and most definitely does not help look at the crisis rationally. After 1995 and the election of Zeroual as president, the labeling of the violence as terrorist and Islamic allowed the regime to create a confusion between marginal elements of its opposition (which its brutal repression helped to radicalize), and the entire opposition; this enables it to justify to the West the postponement of democratization indefinitely (Burgat 332).\n\nBurgat has stressed that \"one needs to see beyond its extreme expressions to the essence of the Islamist dialectic that it is possible to understand why it has such a remarkable capacity for mobilization\" (4). Refusing to recognize the appeal that Islamist parties can have in Algeria (and other parts of the Muslim world) will not enhance our understanding of the dynamics of the region. The fact that in several countries Islamist parties have emerged as the main opposition to authoritarian, Western-backed governments must be confronted and understood.\n\nVolpi noted that the problem for Western democracies was that \"the process of democratisation was accompanied by a process of 'Islamicisation,' \" and that \"Western democracies had to make a difficult compromise between their short-term economic and domestic political objectives and the long-term developmental needs of liberal democracy in a Muslim polity\" (129). Moreover, Western democracies are complicit in the ruling elite's system of governance of \"buy[ing] social peace with political repression and economic redistribution\" (131), a system whose stability depends on the price of hydrocarbons. No Francophone writer has noted the contradiction noted by Martinez: \"by refusing to make the FIS a partner in power in 1991 despite its success in the elections, . . . and then giving a privileged role to the AIS, the Presidency showed that the way to power is not through the ballot box but through armed resistance\" ( _The Algerian_ 19).\n\nR\u00e9da Bensma\u00efa rightfully noted the striking \"contrast between the sociopolitical insignificance of intellectuals as a 'group' and the violence carried out against them\" during the civil war in Algeria (86). If during the Algerian crisis Algerian intellectuals played the role of \"Phantom Mediators\" (as Bensma\u00efa phrased it in the title of his article), works published in France contributed to a demonization of Islam that had some repercussions in that context, most notably with the \"affaire du foulard\" 'headscarf affair,' which I discuss in the next chapter.16 In any case, Boudjedra's pamphlet would not have fulfilled the function of intellectuals, which Bensma\u00efa qualifies as \"mediators between the cultures, languages, and morals of the country and to translate the variety of statements coming from the different communities for the . . . nation, so that these could be understood in the different political, ethnic, cultural, and religious idioms\" (90). Although Boudjedra has specified that _Le FIS de la haine_ had been specifically written for the French public (qtd. in Gafa\u00efti, 72), it does not further the French public's understanding of the situation that led to the rise of the Islamic party. Mortimer identified one of the reasons why a third democratic and secular alternative did not materialize as an alternative to the FLN and the FIS, as \"the isolation of the elite from the grass roots\" (25). The essays discussed here are emblematic of that isolation. While the authors denounce the corruption and abuse of the FLN during its thirty years in power, neither Boudjedra nor Mimouni published essays focusing on the FLN, similar to these essays on the FIS. That, in itself, speaks volumes about the authors and the French market conditions.\n\n### Remembering Colonial Algeria in the 1990s: _Une  \nenfance alg\u00e9rienne_\n\n_Une enfance alg\u00e9rienne_ first appeared in Paris in 1997, and was translated into English as _An Algerian Childhood_ in 2001. Edited by Le\u00efla Sebbar, a Franco-Algerian author, this collection is the second in what can be labeled a trilogy of collections of autobiographical short stories.17 In the first, _Une enfance d'ailleurs_ , co-edited with Nancy Huston, the common denominator of the writers (some of whom come from Poland and Russia) is French as a language for written expression. The third volume, _Une enfance outremer_ , features writers from former French colonies. As its title indicates, the second volume focuses on one country; it is the only text in the trilogy to do so. Asked about the genesis of this project, Sebbar explained that she has followed the same approach for all the collections: after selecting a topic of interest to her, linked to the themes of her works (colonization, exile, war, etc.), she then asked writers for an unpublished autobiographical piece on the theme of the book (personal correspondence with the author).\n\n_Une enfance alg\u00e9rienne_ features sixteen pieces by Francophone writers from diverse backgrounds (Arabs, Jews, French settlers, as well as people of mixed descent), who have in common the fact that they spent their childhood in Algeria. The authors' backgrounds emphasize the diversity of the population of colonial Algeria and the experiences of various constituents. The authors' birth-dates range from the 1920s to the 1950s, and thus the stories cover the period from post World War I to the Algerian war. In addition to their Algerian childhood, these writers also have in common the fact that they all have been living in France for decades, with the exception of one contributor who goes back and forth (according to the short biographical notices that precede each story). Indeed, these writers were chosen because they had to leave Algeria for one reason or another (Donadey, \"Foreword\" xiv). This fact in part explains the nostalgic tone that characterizes several of the stories.\n\nOverall, the narratives seem to stress the cordial relations that existed between the different groups that peopled colonial Algeria. Here are some examples: Arab and Jewish women exchange food (Bencheikh 28), Arab and European kids play together (Alloula 20, P\u00e9l\u00e9gri 193), the diverse ethnic groups enjoy the same doughnuts, their various languages are like soothing lullabies (Dadoun 88, 96), and the diverse religions form a polyphony (Daniel 111). There is an emphasis on the common lot of all, with epidemics (Dib 119) and natural disasters that do not differentiate between the backgrounds of their victims (P\u00e9l\u00e9gri 201). The nostalgia present in several of the stories is expressed through frequent interjections (Bencheikh), and the transfiguration of a weekly routine into an odyssey where prosaic chores are compared to artistic endeavors (Dadoun). William Cohen has noted that \"The pieds-noirs have a vision of history they wish to see adopted by the m\u00e9tropole. The most important 'truth' the pieds-noirs wanted to establish was that they lived in harmony with the other ethnic groups in Algeria\" (132).18 However, the pieds-noirs are not the only ones who \"embellish\" inter-ethnic relations, as Tengour's story for instance emphasizes the fact that his grandfather remained friends with French settlers during the war, while his son was in prison for his nationalist involvement. As Anne Donadey points out in her introduction to the English translation, this volume creates \"a Utopia, an imaginary homeland whose foundation rests on reconciliation and respect for differences. This volume brings into being an open Algerian community that is unfortunately impossible to put together today except within the pages of a book\" (xvii).19\n\nHowever, cracks surface in this \"utopian\" Algeria where different groups supposedly coexisted peacefully. Some passages mention or talk about the misconceptions and prejudices of one group towards another: against Arabs (Annie Cohen 79), Muslims (Daniel 109), and Jews (Tengour 223). In Jean P\u00e9l\u00e9gri's contribution, the earthquake leads Areski, an Arab, to realize the common humanity he shares with his employer, a settler, whom he now sees as his brother, but that acknowledgment is not reciprocal. While Areski insists that they are united in the face of natural disaster, his employer's answer is to stress what differentiates them: in the circumstances, the arbitrariness of grammatical gender. Although the French and Arabic languages do have grammatical gender, they assign opposite gender to the moon and the sun (202). The settlers' indifference to the indigenous population transpires in some stories. Daniel recalls that he and his teacher had no desire to learn Arabic (104). Millecam points out that as a child, he had no awareness of the colonial situation, of the fact that this territory used to belong to the ancestors of the children who are now the most destitute and whom he cannot even name, but only designates as \"les autres\" 'the others' (180). The use of the vague non-descriptive words \"les autres\" to refer to the indigenous children betrays an awareness, even if vague, of the colonial divide accompanied by a refusal to acknowledge who these \"others\" were. Cixous' story strongly contradicts that myth of harmony (more on that particular piece later).\n\nWhile the juxtaposition of short stories emphasizes the plurality of perspectives, this collection overall glosses over the brutalities of colonialism in two ways: by humor, or by emphasizing subsequent violence. In \"Mes enfances exotiques\" 'my exotic childhood,' Alloula mentions the colonial situation in passing, when he locates the story in his \"village colonial de l'Oranie\" 'colonial village of Oran' (11, de Jager 5), and talks about \"la population europ\u00e9enne\" 'the European population (19, de Jager 12). The metaphor that compares the French school to a guillotine using the words \"billots\" 'chopping block,' \"bourreau\" 'tormentor,' and \"calvaire\" 'suffering' (10, de Jager 4-5) could be read as an indictment of the role played by the French school system in the colonies to further the imperial project, but it could also be applied to a non-colonial context, where children have to submit to a strictly enforced discipline. The story starts with the mention of a \"loustic non scolaris\u00e9\" 'uneducated kid' [not enrolled in school] (9, de Jager 3) and the obscenities he would yell to the children attending school through the classroom window. The French school is described as \"ce lieu qui lui \u00e9tait inaccessible mais qu'il ne nous enviait pas\" 'that place that was inaccessible to him, but which he did not envy us' (10, de Jager 4). In 1954, 90 percent of the colonized Algerian population was illiterate (Carlier 361, Naylor 8), and only one out of ten Muslim children was enrolled in elementary school. The word \"inaccessible\" is a euphemism, considering the fact that colonial policies did not make school accessible to all children, and that among numerous inequities, European children had priority over indigenous children (Stora, _Les trois exils_ 66). In addition, that discrimination is tempered by the fact that the child was not eager to go to school. The focus of the story remains on the awakening to sexuality and the tone humoristic throughout, particularly in the skillful description loaded with poetic metaphors that accentuate the grotesque coupling of two adulterous villagers.\n\nAnother story in the same vein is Mohamed Kacimi-El-Hassani's \"A la claire ind\u00e9pendance\" 'By Independence Clear,' set at the beginning of the first school year after independence. The narrator and his friends cannot fathom why they are still being sent to the French school. Convinced that they are the only village in Algeria still under French colonization, they decide to skip school to assess the situation in the next village. The story highlights that colonialism has given way to a neocolonialism that is even harsher, subtly symbolized by the teacher's ruler, changed from wood to steel. The chorus of the French children's song \"A la claire fontaine\" 'by the fountain clear,' whose lyrics serve as a connecting thread throughout, close the story: \"Il y a longtemps que je t'aime\/Jamais je ne t'oublierai\" 'I've loved you for a long time And I shall never forget you' (de Jager 147). These lyrics emphasize the long lasting relationship fostered by colonialism and forecast that, as Naylor pointed out, \"the referendum [for independence] marked a reformulation rather than a repudiation of the relationship between the metropolitan power and the ex-colony\" (1), where the colonizer was replaced by the _coop\u00e9rant_. Despite the strong vocabulary used to describe the school routine, where corporeal punishment seems to be an integral part of the curriculum and the teacher is compared to an executioner, the tone of the story is very light and humoristic thanks to the na\u00efve point of view of the child.\n\nLikewise Mohammed Dib's story, where his encounters with foreigners are all positive: the Greek doctor who saved his leg, his French teacher, and the French language which was to become his language of literary writing. The ironic allusion to the Eurocentric curriculum of the school he will attend is made in the context of the Greek doctor and therefore removed from the otherwise positive portrayal of his French school experience: \"Celui-ci ne descendait pas de ces Gaulois dont je saurai plus tard, \u00e0 l'\u00e9cole, qu'ils \u00e9taient mes anc\u00eatres\" 'He was not a descendant of the Gauls who, as I later learned at school, were my ancestors' (118, de Jager 106).\n\nThe second means by which the narratives downplay the impact of colonization is by emphasizing violence in the post 1962 era. Even the few stories that mention the brutality of colonialism fit into the pattern of lessening the blame through mentioning violence committed after Independence, thereby relativizing the colonial brutality to subsequent events. Kacimi-El-Hassani alludes to post-independence internal violence when he mentions that Ben Bella, the first Algerian president, killed Colonel Cha\u00e2bani, after the latter led an insurrection against him. The war of Independence was multifaceted: a war of Independence waged by Algerians against France, a Franco-French civil war about the fate of French Algeria, and an Algero-Algerian civil war over the leadership of the Independence movement (Stora, \"1999-2003\" 507). In _Une enfance alg\u00e9rienne_ however, only the Franco-Algerian (with emphasis on the Algerian) and the Algero-Algerian dimensions appear.\n\nFar\u00e8s points out the discrimination and violence perpetuated by different parties at different times: from the banning of indigenous children from school in 1940 and their continued discrimination even after they are readmitted to school once WWII was over, to the civil war that targeted Francophone secular intellectuals. There are explicit parallels drawn between the internal strife following the capitulation of France and the Vichy government, and the events of the 1990s. In a very dense passage, Far\u00e8s alludes to some of the pivotal moments in Algerian history:\n\nA chaque crise, r\u00e9pondent les fantasia et les grandes man\u0153uvres militaires, en septembre, au sud du village de Berrouaghia: fin du nazisme, en Europe; refus de la subtile \u00e9mancipation politique, en Alg\u00e9rie; l'est du pays d\u00e9j\u00e0 flambe; le sud se met en ligne sur la longue piste des chevaux! Deux fois les chars: la premi\u00e8re en 1947 \u00e0 l'inauguration\u2014mauvais augure\u2014des grandes man\u0153uvres de la guerre.\n\nLa seconde en 1962, lors des charniers de Boumediene. (136)\n\nWith each crisis, cavalcades and huge military maneuvers respond, in September to the south of the village of Berrouaghia: the end of Nazism in Europe; refusal of fragile political emancipation in Algeria; the East of the country is already in flames; the South lines up on the long road of horses! Tanks twice: the first time in 1947 at the inauguration\u2014a bad omen!\u2014of the great war maneuvers.\n\nThe second in 1962 at the time of Boumedienne's mass graves. (de Jager 123-24)\n\nThe defeat of Hitler in 1945 is juxtaposed with an indirect allusion to the 1945 riots (most commonly known as the S\u00e9tif riots) that shook the country when nationalist demands were brutally suppressed. While Far\u00e8s' style is reputed for being difficult and obtuse, in this passage the allusions to internal postcolonial Algerian violence carried out by Boumedienne are more clearly identified than those committed by the colonial authority. 1947, for instance, is the year of the law giving more representation to Muslim Algerians. The Berrouaghia prison is mentioned twice in the story (136, 141). This prison was founded by the French where they incarcerated revolutionaries; it has kept this function post-independence and has housed leaders of the Berber Spring20 and FIS members. In November 1994, Islamist prisoners died there in suspect circumstances (that could not be investigated because of censorship by the authorities, which claimed that prisoners were trying to escape or caused a mutiny).\n\nJean-Pierre Millecam revisits his childhood as the Eden before the original sin, from the point of view of the child, but as a grown-up narrator he knows that \"la faute avait \u00e9t\u00e9 commise quelque cent ans plus t\u00f4t\" 'the sin had been committed some hundred years before' (181, de Jager 167). Colonialism compared to the original sin forecasts the demise of French Algeria, and the flooding of a town announces the fascism that will sweep his town. The conscription of indigenous soldiers sent to Europe as a \"chair \u00e0 canon\" 'cannon fodder' (188, de Jager 173) is denounced. WWII, the Algerian war, and the civil war as a continuation of it are compared to apocalypses (188). The religious reference of the title, and the quotation from the Bible (\"dans la cuve de la col\u00e8re divine\" 'in the vat of divine wrath' 182, de Jager 168) taken from Revelation 14:19, downplay human agency in historical processes.\n\nSebbar's story, dedicated to her parents who were both teachers, touches on her ambivalence about the role of teachers in the colonial project. The two violent incidents that are recalled in the story are chosen because they target teachers, and are committed by Algerians who are trying to put an end to the colonial domination. Sebbar intertwines a memory she has of hearing her parents talk about teachers being killed, with her research in library archives trying to find more information about these incidents. Describing a newly arrived couple of teachers who were killed on the first day of the war of independence, Sebbar asks: \"Ils ne savent pas que c'est la Colonie et sa langue qu'ils viennent servir, sur ces Hauts Plateaux \u00e9trangers, hostiles et beaux?\" 'Don't they know they've come to serve the colony and its language, there on the foreign High Plateaus, hostile and beautiful?' (212, de Jager 195). By describing them as having \"la candeur curieuse des enfants de la R\u00e9publique\" 'the curious candor of children of the French Republic' (212, de Jager 196), she tilts the balance to their innocence and naivety regarding the role they play, as teachers, in furthering the domination of the settlers' culture. The focus on teachers, together with the mention of burnt libraries (208), puts the emphasis on repositories of knowledge that are targeted during the war. The other incident that Sebbar discovers in archives took place in 1901, when rebellious villagers threatened a teacher. In the context of the publication of the book, the physical violence French teachers are victims of in this story will likely evoke the targeting of indigenous (Francophone) intellectuals during the civil war.\n\nTwo stories offer a striking counterpoint about the depiction of the relations between Muslims and Jews in Algeria. Albert Bensoussan's \"L'enfant perdu\" 'The Lost Child' and H\u00e9l\u00e8ne Cixous' \"Pieds nus\" 'Bare Feet' happen to follow one another by chance, since the narratives appear according to the alphabetical order of their author. Although both narratives have in common the narration of an encounter between a Muslim and a Jewish child, they could not be more dissimilar: positive and nostalgic for Bensoussan, who fondly recalls how he met and lost his friend Fatiha, and negative for Cixous, who was implicitly reminded of her privileged status, despite the abrogation by the Vichy government of the Cr\u00e9mieux decree granting citizenship to all Jews, during an encounter with a shoe shine boy. Both stories take place in 1941, a year that is specified in Cixous' piece, and that can be implied from Bensoussan's story. I contrast both stories, and analyze the implication of Cixous' casting of the Arab shoe shiner boy, who belonged to the most disenfranchised group during colonial Algeria, into an aggressor and symbolic murderer of the little Jewish girl that she was, when he put red shoe polish on her white sandals. A detailed analysis of \"Pieds nus\" shows that the contemporary conditions of its writing distort what was the actual reality on the ground for Algerian Jews and their relationship with Muslims, thereby fueling stereotypes of Arab-Muslim hatred for Jews.\n\n\"L'enfant perdu\" tells the story of the friendship between the narrator, a six-year old Jewish boy, and a Muslim girl, Fatiha, and its abrupt ending. After Albert gets lost while shopping in the market with his mother, he is found by an Arab man, who takes him home, where his daughter Fatiha takes care of him until his family is contacted. Thereafter, his mother drops him off at Fatiha's house every Thursday for about three years while she goes shopping. Bensoussan fondly recalls the games played with Fatiha. Suddenly, he was no longer taken to Fatiha's house, without really knowing why. It is only years later that he understood that as she had reached puberty, Fatiha could no longer play with a boy.\n\nThe title of the story is polysemic. It refers to Bensoussan as the lost child in the market, and evokes the distance between the narrator and his childhood, many years behind him.21 The emotion felt by the loss of his friend is poignantly expressed by the narrator in an apostrophe: \"Fatiha, mon amie, qu'\u00e9tais-tu devenue?\" 'what became of you, Fatiha, my friend?' (52, de Jager 46). Bensoussan's nostalgic tone can be read in two different ways. On the one hand, it can be seen as nostalgia for life prior to what Stora has called the first of three exiles experienced by Algerian Jews. Stora has detailed how the 1870 Cr\u00e9mieux decree, which gave the French citizenship to indigenous Jews, triggered the gradual assimilation of the Jewish community with the European community in Algeria,22 and therefore its separation from the Muslim population. This entailed a gradual loss of Arabic ( _Les trois exils_ 13). In the story, Albert's mother speaks Arabic, but he does not, and it is in French that he communicates with Fatiha. The process of Frenchification and assimilation of Algerian Jews to French culture is evident in some of the linguistic choices between the two generations of the mother and the child: the narrator's mother uses the Arabic term \"Roumis\" to designate Christians, as Arabs do, but the son uses the word \"communion\"\u2014pointing out \"comme on disait, pour bien dire\" 'as they said to put it nicely' (52, de Jager 45)\u2014to talk about what must have been his bar-mitzvah. What separates the two friends is not their different religious background nor the disconnect that took place between the two groups as a consequence of the Cr\u00e9mieux decree, but gender relations as defined by Muslim Arab culture. Calling Fatiha's family his \"famille arabe\" 'Arab family' (51), Bensoussan's text seems to endeavor to recapture the time when there were still bonds between the two communities.\n\nBensoussan's story can also be seen as partaking in the nostalgia that is symptomatic of the \"desire to create and hold on to a memory, to recapture the world that was lost, existing prior to the fall of ' _Alg\u00e9rie fran\u00e7aise_ ,' French Algeria\" (William Cohen 129). William Cohen states that, with a few exceptions, \"What is striking about the many novels and memoirs on French Algeria is the absence in them of any account of strife between Europeans and Muslims. It is as if memories of such incidents had been repressed\" (William Cohen 132).23 Bensoussan's piece fits perfectly in that category. Stora specifies that after the 1962 exile,24 it was not until the 1990s that Algerian Jews began making a distinction between themselves and the pied-noir community (Stora, _Les trois exils_ 9, see also \"L'impossible\" 289). Once Algerian Jews abandoned a pied-noir identity, they were no longer interested in furthering the myth of a French Algeria where the three monotheist communities lived in perfect harmony (William Cohen 136). One can see that change reflected in Cixous' \"Pieds nus,\" albeit in a problematic way.\n\nCixous' story starts with a description of her native city of Oran. References to two different worlds are sprinkled throughout the narrative; it is not always clear for the child what exactly those two worlds are (60, 62), but the divide created by colonization is clear. Colonial Oran is described in negative terms as a sick body, where hatred predominates: \"Le corps d'Oran . . . \u00e9tait un corps politique, tum\u00e9fi\u00e9, articulations enflamm\u00e9es, un monstre peuple, les bouches haletantes les langues charg\u00e9es de glaviots pr\u00eates \u00e0 se les cracher au visage, les genoux boursoufl\u00e9s, les gorges grosses d'arri\u00e8re-pens\u00e9es, \u00e0 soi-m\u00eame \u00e9tranger, \u00e9trangers, furieux\" 'Oran's body . . . was a political body, swollen, limbs inflamed, a monster people, mouths gasping tongues laden with gobs of saliva ready to be spit in each other's faces, puffy knees, throats thick with afterthoughts, strangers to themselves, foreign, furious' (62, de Jager 55-56).\n\nCixous fondly recalls the Sundays her family spent on the mountain (which stands as a sharp contrast with the city) as a happy time:\n\nOn renouait plaisamment avec les morts. C'\u00e9tait des maures qui dormaient famili\u00e8rement tout au long du flanc de la montagne, \u00e0 peine recouverts d'un pan herbu et sur la t\u00eate une jolie mosa\u00efque aux couleurs vives. On marchait entre eux sur eux on s'asseyait avec eux, c'\u00e9tait bon, cet accompagnement hospitalier, jamais plus tard je ne retrouverai cette cong\u00e9nialit\u00e9 paisible, ce partage de la terre, cet acquiescement. (58)\n\nOne reconnected pleasantly with the dead. It was the Moors who slept informally all along the side of the mountain, barely covered with a grassy patch and a pretty, brightly colored mosaic on their head. We would walk between them, on them, we'd sit down with them, and it was good, this hospitable company; never again did I find that peaceful congeniality later on, that sharing of the earth, that acquiescence. (de Jager 51-52)\n\nNicholas Harrison notes that the choice of the word \"maures\" 'Moors' and its assonance with \"morts\" 'deads' translates \"a child's confusion at much of what she hears,\" and that it \"may carry positive connotations,\" since it evokes the Golden Andalusian age, when Muslims, Jews, and Christians lived in peace and contributed to a vibrant culture (24). In keeping with Cixous' description of the city as unharmonious, I propose a radically different interpretation of this paragraph: the play on the homonymy between \"morts\" and \"maures\" emphasizes the fact that the only time she and her family enjoy peaceful relations with the Arabs is when the latter are dead, and, to continue on the allusion to the Andalusian era, that the time of peaceful coexistence is long gone.\n\nAfter the abrogation of the Cr\u00e9mieux decree by the Vichy government stripped Algerian Jews of their French citizenship, the little girl that she was felt relieved of her guilt, as she considered that she no longer belonged to the oppressive side (63). Her father, who was forbidden to practice as a doctor, was forced to become a chiropodist. A clever play on the homonyms \"corps\" 'body' and \"cor\" 'corn' highlights the absurdity of the _numerus clausus_25 that was imposed on professional Jews: \"Pour survivre mon p\u00e8re se fit p\u00e9dicure. Je ne sais pas pourquoi Vichy qui lui \u00f4tait le soin des corps lui avait cependant abandonn\u00e9 les cors aux pieds\" 'In order to survive, my father became a pedicure. I don't know why Vichy, which had taken the care of bodies away from him, still left him the care of corns' (62-63, de Jager 56).\n\nCixous likens her new status of outcast to the sensation of being barefoot when she walks on the mountain between the Muslim graves: \"Que nous fussions parias cela obscur\u00e9ment me soulageait, comme d'\u00eatre vrais, comme d'\u00eatre pieds nus sur le chemin des Planteurs parmi les tombes\" 'That we were pariahs consoled me in an obscure way, like true beings [like being true], like barefoot beings [like being barefoot] on the path of the Planters among the tombs' (62, de Jager 56). The motif of walking barefoot in the mountain suggests an organic link with the land. On the mountain, she feels a kinship with the \"maures\/morts\" through her bare feet (\"En grimpant j'\u00f4tais mes sandales et je mettais mes pieds dans les mains des morts, et je caressais l'empreinte de leurs pieds avec les paumes de mes pieds\" 'Clambering up I would take off my sandals and put my feet in the hands of the dead, and I'd caress the imprint of their feet with the soles of my own' (59, de Jager 53). The indigenous of the city (those who are alive) are portrayed as anchored in the ground (\"encore ins\u00e9par\u00e9s de la terre et des pav\u00e9s\" 'not yet separated from the earth and the pavement' 61, de Jager 55) to denote their belonging to Algerian soil. According to Stora, the abrogation of the Cr\u00e9mieux Decree marks the second exile of Algerian Jews, this time not from the Muslim natives, but from the French Republic to which they had assimilated ( _Les trois exils_ 13). Cixous' take on the abrogation of the Cr\u00e9mieux Decree is unique, as she first presents it as a (short-lived) hope for reversing that first exile. No other Algerian Jewish writer, to the best of my knowledge, had anything positive to say about being stripped of their French citizenship.\n\nAfter someone donates to her a brand new pair of sandals, she is reminded that she still enjoys class privileges when an Arab shoeshine boy smears her white sandals with red shoe polish. Their encounter is presented as a rehearsal for the play that will be staged on the scene of independent Algeria (64). The scene is compared to a trial: \"J'avouai. J'\u00e9tais coupable. Devant son tribunal \u00e0 lui, l'acquittement dont je jouissais \u00e0 mes yeux depuis Vichy n'avait aucune valeur. J'habitais rue Philippe au deuxi\u00e8me \u00e9tage et j'avais des sandales donn\u00e9es \u00e0 l'\u00e9tat neuf\" 'I confessed. I was guilty. Before his tribunal, the acquittal I had enjoyed since Vichy was of no value whatsoever. I lived in the rue Philippe on the second floor, and I had been given sandals that were almost entirely new' (65, de Jager 58). Ronnie Scharfman has argued that \"by foregrounding the suffering implied by their exclusion, as Jews, between 1940 and 1943, from an already problematic French identity, both writers [Cixous and Derrida] effect an identification with contemporary Algeria from which they were excluded, paradoxically and painfully, when they lived there as children and were perceived by the Arabs and Berbers as part of the French colonial regime\" (\"Cixous\" 89). Indeed, in the Manichean colonial order, there is no conceptual place for the status of Algerian Jews as both insiders and outsiders. But Scharfman does not comment on the fact that the little girl is aware that she has been on the oppressive side of the colonial divide, and that this knowledge gives her a guilty conscience that is at the core of the story. The focus on the sandals emphasizes class over ethnicity; it is because the sandals seem to indicate a privileged status that the shoe shiner associates her with the colonizer. However, some statements suggest that the Arab boy identifies her as being Jewish, such as the sentence quoted above in which she states that her new status since Vichy did not make a difference to him, and later on when she states: \"Nous savions tout\" 'we knew everything' (65, de Jager 58).\n\nWhile esthetically Cixous' story is undoubtedly a masterpiece, the roles she attributes to the children are problematic in light of historical evidence. The red polish on her sandals is compared to blood (66), this act is for her a symbolic murder with a double edge: anti-colonial and gendered. On the one hand, the violence she is subjected to forecasts the Jews' exile from Algeria after the war of independence. During the narration of this encounter, she sees herself and the shoe shine boy as suddenly becoming adults. This helps to blur the time between 1941, when the scene is set, and when the independence war will be fought. By being cast as a rehearsal for what will unfold during the war for independence, the scene does anticipate some of the attacks that were committed by the FLN against Jews during the war for Independence to push them to side with the Algerians' cause, but it posits the Arab child as the sole embodiment of anti-Semitic violence (even if only symbolic) since the text is mute about violence committed at the time by Europeans (including the future OAS).26\n\nWaves of European anti-Semitism, in the 1890s and 1930s, did not spare Algeria (Stora, _Les trois exils_ 58-59). The Cr\u00e9mieux Decree was vehemently opposed by some pieds-noirs, who considered it as putting natives on the same footing with Europeans. Although anti-Semitism among the European population of Algeria was strong, it is occulted in Cixous' story. Indeed, the discriminatory measures of the Vichy government were overzealously applied in Algeria, where the oppression of Jews during WWII was in some aspects worse than in France, and this despite the fact that there was no German occupation in Algeria (Stora, _Les trois exils_ 87). To top it all, the abrogation of the Cr\u00e9mieux decree was belatedly reversed in 1943, almost a year after the Allied controlled North Africa (Stora, \"L'impossible\" 294). Given that Stora credits the Muslim population for not committing any hostile act against the Jews during the WWII period (\"L'impossible\" 295, 303), Cixous' story goes against the historical record. By casting the Arab boy murdering the Jewish girl, Cixous lends credence to one of the anti-Semites' arguments for repealing the Cr\u00e9mieux decree: that it was causing resentment against France in the Muslim population (Stora, _Les trois exils_ 79), although Stora points out that Muslim leaders did not fall into the trap that tried to antagonize both communities ( _Les trois exils_ 79, 97). I surmise that the culpability that Cixous feels for having been on the privileged side is assuaged by her casting the boy as a would-be murderer that forecasts the exile of the Jewish and French communities from Algeria, as well as other developments of Algerian history, particularly gender discrimination in post-independence Algeria (with the 1984 Family Code, and attacks targeting women during the civil war).\n\nIn addition to the colonial dimension, this scene could also be interpreted as defloration, since there is a glimpse of desire in the boy's eyes, and a symbolic loss of innocence for the child. The gender issue is more obvious in \"Mon alg\u00e9riance,\" at the beginning of which Cixous recalls one image present in \"Pieds nus,\" albeit with a slightly different formulation: she knew that the Arabs were the \"true\" indigenous people of Algeria, and her unease would vanish when she walked bare feet on their graves (71).27 She recalls how she unsuccessfully tried to befriend the three Muslim girls that attended her high school class during 1951-1953,28 and therefore explains her surprise when Algeria came back to her decades later through her friendship with Algerian women exiled in France to escape the atrocities of the civil war (74). Several paragraphs later, she implicitly refers to the violence committed against women in postcolonial Algeria, and labels Algeria \"ennemie des femmes\" 'enemy of women' (74, Prenowitz 172).\n\nIn \"My Alg\u00e9riance,\" Cixous decries and elaborates on the violence of colonialism (and particularly the Arabs' dispossession) as well as the French population's anti-Semitism,29 but this is passed over in \"Pieds nus.\" In the context in which \"Pieds nus\" appeared, and with the historical shifts within the story, the shoe shine boy embodies the oppressive Algerian man, be it the FLN militants who will chase the pieds-noirs out of independent Algeria, or the extremists (religious and nationalist) who will chase women out of public space. The _hic et nunc_ of the narration of the anecdote projects an anachronistic rendering that skews the historical reality. In addition, the fact that Cixous' ethnicity is foregrounded in the narrative will color its reading, as any story involving a Jew and an Arab read from an end of the twentieth-century's standpoint is inevitably colored by the legacy of the Arab-Israeli conflict.\n\nUnlike the war of independence, none of the writers' childhoods unfolded during the civil war, and yet the 1990s events lurk in some stories. One can object to the mention of the 1990s' civil war because these events are anachronistic to the stories about the authors' childhoods, which for all of them ended well before the last decade of the twentieth century. However, autobiographical writing implies a narrator grounded in the present of narration, with a distance that allows some perspective on the events of one's life. As Philippe Lejeune has pointed out, the \"I\" of an autobiographical narrative is multiple: the one of the child at the time of the event, and the one of the narrator at the time the story is told. Therefore, given the time during which these stories were commissioned, it seems understandable, at the very least, that the tragedy of the 1990s surfaces here and there.\n\nJamel Eddine Bencheikh's \"Tlemcen la haute\" is a series of vignettes told in a stream of consciousness from the author's recollection of the summers he used to spend in his parents' native town, that also pays tribute to Arab and Tlemcenian poets and artists. The only mention of violence relates to the civil war. The last page is presented as reality that turned into a nightmare the author wishes to wake up from. The blurring between reality and nightmare emphasizes the horrendousness of the situation, and that Tlemcen is unrecognizable because of it. The descriptions of women veiled in black (instead of the traditional white North African veil) and men sporting Afghani clothing and beards, signal an imported fundamentalism that has been gaining ground. The murders of singers and writers (Alloula, Cheb Hasni, and Djaout) are put in line with Ibn Khamis', a thirteenth-century poet from Tlemcen exiled in Granada where he was assassinated, because of disagreements with the powers in place.\n\nNabile Far\u00e8s' story talks about various wars, and his last page picks up the issue of continuity between the war of independence with its internal strife between various factions and the 1990s' civil war. The narrator says that it is either another war, or that the first one in fact never ended, that \"il est _encore_ question de d\u00e9mocratie\" 'it is _still_ a question of democracy' (141, de Jager 127). Some historians see the origin of the 1990s' war as going back to unresolved issues from the war of Independence (Harbi, \"L'Alg\u00e9rie\" 43) or before. Stora lists the similarities between the two wars, but is quick to add the differences and underlines that the latter weighs more than the former, and that \"ce qui rapproche surtout cette 'seconde' guerre d'Alg\u00e9rie de la 'premi\u00e8re', c'est la _persistence des pr\u00e9jug\u00e9s et des st\u00e9r\u00e9otypes_ qui ont fabriqu\u00e9 une m\u00e9moire de la guerre d'Alg\u00e9rie en France\" 'what brings this second war closer to the first Algerian war is the persistence of prejudice and stereotypes that made a memory of the Algerian war in France' ( _La guerre_ 65). Omar Carlier insists that the extreme violence that took place during these years is not due to a historical determinism ruling Algeria, but to specific historical processes (379). Similarly, Stora repeatedly warns that a comparison between the war of independence and the civil war reinforces the stereotype of Algeria as a country plagued by a never-ending violence ( _La guerre_ 10, 52, 56).\n\nIn a humorous postscript, Far\u00e8s imagines what his mother, who used to wash the nationalist militants' clothes, would do with the current insurgents (which would be to boil the men rather than their clothes). He is silent about the series of events that brought about the war. All direct and indirect references to the civil war in the collection blame the Islamists for the violence that has swept the country, although historians have singled out events during which violence emanated from the state, not the Islamists, as catalysts. For instance, although Stora stresses that it is hard to pinpoint the beginning of the civil war, he identifies several moments as possible starting points: the October 1988 riots, the cancellation of the second round of the January 1992 elections,30 and the murder of president Boudiaf in June 1992 ( _La guerre_ 15-17). Martinez refutes the economic and religious factors as satisfactory explanations for the level of violence during the civil war and singles out \"the failure of democratic transition\" (\"Why the Violence\" 16-17). He assessed that the violence was a \"product of the cancellation of the December 1991 legislative elections and the policy of repression conducted from 1992 to 1994\" (\"Why the Violence\" 22), these being committed by the military-controlled state.\n\nThe consequences of some of the colonial policies that sustained the dominance of the European minority over the Algerian majority (such as massive land expropriations, the \"Code de l'Indig\u00e9nat,\" occultation of the indigenous culture, the status of Arabic as a foreign language) surface here and there, but the policies are not subjected to criticism, nor are their effects on the Algerian population presented as direct consequences of institutionalized discrimination and violence. The poverty of the Algerians, which characterized the overwhelming majority (Carlier 361), is not saliant at all. Habib Tengour seems to be the person whose direct family suffered the most from the colonial yoke. His indigence becomes obvious when he mentions having to skip school when his grandfather could not afford a pair of cheap shoes. We learn that his father, a nationalist, was imprisoned and tortured by the French, but that is mentioned in passing in the space of one sentence. The emphasis of the story is on fostering relationships with others, and on knowing people as the best means to dissipate erroneous stereotypes, such as his grandfather's friendship with settlers despite political problems, and his own friendship with a Jewish family that dispels prejudices conveyed in popular culture.\n\nOne could object that people's lives were not solely determined by the colonial situation, and that in a short piece writers cannot possibly render all aspects of their childhood. Moreover, since they were children at the time, they might not have been fully aware of the gravity of some situations. Some of the Arab authors belonged to the upper class (such as Fatima Gallaire as indicated by the number of servants), and their class privilege mitigated their status of colonial subject. The writers of this collection represent the various groups that peopled colonial Algeria, yet some escape easy cataloguing, since some, such as Sebbar, are the product of mixed marriages. Alain Vircondelet talks about the atrocities of the war of independence (231) and victims on both sides. The child of a Frenchman and an Algerian woman, he takes the side of Algerians (234). To their credit, some of the criticism of colonialism comes from Europeans, even if belatedly. This compilation of short stories by different authors could very well have been called \"Enfances alg\u00e9riennes\" 'Algerian Childhoods' with the plural emphasizing the diversity. The choice of a singular title for the collection signals a desire to emphasize what binds these people together.\n\nWhile the collection seems to be representative of the diversity of the people who lived in Algeria between the 1920s and the 1960s, it is actually deceptive in that it is limited to writers who express themselves in French. Given the high rate of illiteracy when the country became independent (at least 80 percent), these writers, schooled in French, can only be the voice of a small minority. As such, _Une enfance alg\u00e9rienne_ is emblematic of the limitations of the field of Francophone Literature.\n\nThis book was published right before the beginning of the end of amnesia, with 1999 as the key date according to Stora since this is when the French government formally acknowledged that the Algerian \"events\" indeed constituted a war, as well as the publications of articles and books by victims of torture and torturers (\"1999-2003\" 502). Sebbar's collection is still situated at a time when the Algerian war is still not fully dealt with by collective memory. Nevertheless, I contend that the context in which this collection was published makes it an important issue. Farid Laroussi points out that many books by Algerian women about the desperate conditions of women in Algeria were published in the 1990s, in which \"The sociopolitical situation in Algeria rightly comes under attack but only as a way to reassert France's own representation as a land of Enlightenment\" (88). My analysis of _Une enfance alg\u00e9rienne_ shows that this process seems to be at play here as well, where past French exactions are minimized to better emphasize Algerian violence (past and present), thereby fueling stereotypes of Arab-Muslim violence, and hatred for Jews in the case of Cixous' story. Stora notes that a plethora of images showing the atrocities committed by Islamist groups in the media, coupled with the absence of images showing the suffering caused by the government's exactions puts all the focus and the blame on the Islamists while dispensing with a critical reflection on the origins of the conflict ( _La guerre_ 82). As demonstrated by this collection of short stories and the essays analyzed in the previous section, I argue that Francophone writers have contributed to this phenomenon through their writings. The context in which this collection was published as well as the implications of the civil war's intrusion in narratives taking place before the 1990s adds to the simplistic rendering of the civil war, blamed solely on Islamists, all conveniently lumped into a homogeneous, dangerous, and threatening mass to be eradicated at all cost, all the while being mute on the role played by other elements, as well as history itself.\n\nRecent cinematic production, such as Nadir Mokn\u00e8che's _Viva Laldj\u00e9rie_ , has questioned the simplistic view of Islamic fundamentalism as being the sole responsible agent of violence during the Algerian civil war. As Sylvie Durmelat points out, the only murder in the film is committed by a high ranking member of the State's security forces, thus suggesting that \"les terroristes ne sont pas ceux que l'on croit\" 'the terrorists are not who we think' (\"L'Alg\u00e9rie\" 110). It is significant that since 1992, no leader has repelled the Family code (Stora, _La guerre_ 64). Therefore, the Islamists are not the ones (or at least not the only ones) to oppress women, though they are (conveniently) blamed for Algeria's woes and the status of women there in particular.\n\n### **Notes**\n\n1. This summary of events is based on Kepel's chapter \"Alg\u00e9rie: les ann\u00e9es FIS\" in _Jihad_ (166-82). For more detailed studies, see Martinez, Volpi, and Willis.\n\n2. This relationship has been rendered by Etienne Balibar as the following numerical equation: that France and Algeria are neither one nor two nations, but that \"l'Alg\u00e9rie et la France, prises ensemble, ne font pas deux, mais quelque chose comme _un et demi_ , comme si chacune d'entre elles, dans leur addition, contribuait toujours d\u00e9j\u00e0 pour une part de l'autre\" 'Algeria and France, put together, do not make two, but something in the order of _one and a half_ , as if each one, when added together, always already contributed to part of the other' (76).\n\n3. See Naylor for a detailed study of complex hydrocarbons contracts between France and Algeria after independence that shows a \"privileged French economic position in Algeria's development plans\" (145), and A\u00eft-Embarek (120-30) about the economic support and pressure of France and the European union for the junta during the civil war.\n\n4. Many saw this as a lesser evil despite the contradiction of having to resort to authoritarian measures to \"save\" democracy (that Algeria was a democracy is highly contestable, given the status of the FLN as sole political party until 1989). Layachi notes that \"people's growing apathy toward the political process\" as evident in the 2002 elections does not foster democracy and seems to indicate little progress (61). There were \"eradicators\" and \"conciliators\" on both sides of the Mediterranean (Naylor 210).\n\n5. \"Dans le subconscient collectif et populaire, il y avait quelque chose de rageur et de pervers dans cette fa\u00e7on de faire surchauffer les statistiques de la natalit\u00e9 galopante\" 'in the collective and popular subconscious, there was something furious and perverse in this way of overheating the statistics of runaway natality' (59).\n\n6. Willis stresses that \"even the most competent of local authorities (let alone totally inexperienced ones like those of the FIS) could [not] be expected to make an impact in the shorter term\" (160).\n\n7. While Boudjedra castigates the FIS leaders for their ignorance, he himself demonstrates some lacunae, such as stating that the Qur'an is composed of 60 suras instead of 114(12).\n\n8. In their study of the pronouncements of _El Mounquid_ , official journal of the FIS, on women, Imache and Nour found that the party asserts women's rights to schooling, inheritance, vote, jihad, and work\u2014although the latter comes with restrictions (45-46).\n\n9. The FIS claimed a filiation with the FLN's ideology, while criticizing the party for deviating from it (Kepel, _Jihad_ 177). In a footnote, Kepel recalls Lahouari Addi's statement that \"le FLN est le p\u00e8re du FIS,\" playing on the homonymy between FIS and \"fils\" ('son') in French, interpreting it in two ways: that the FIS situates itself in the FLN's ideology, but also that they share the same view of a totalitarian, monolithic society (Kepel, _Jihad_ note 26 p.391).\n\n10. It will later be revealed that some of these atrocities were committed by government forces but attributed to Islamic groups to better discredit them (Burgat 278 and note 19 p.319), and others were due to the government's infiltration and manipulation of the GIA (Burgat 327, Volpi 90).\n\n11. \"Il n'arr\u00eate pas de manier l'intox\" 'he keeps on using propaganda' (Boudjedra 1994: 116).\n\n12. \"Since its transformation into a political force in the 1980s, the Islamist movement has constituted a wide and heterogeneous phenomenon with three main fronts: a cultural front aiming at the 're-Islamisation' of society; a non-violent political front acting inside or outside of the system for gradual, peaceful and comprehensive change; and a violent political front aiming at bringing down the regime by force and at instituting a rigid Islamic order\" (Layachi 53). See Layachi (54-56) for more details on the various Algerian Islamist organizations; the FIS is listed under the non-violent Islamist organizations.\n\n13. \"In its essence, populist Islam speaks to the millions of men and women in the MENA who feel marginalized, alienated and abused by their rulers. . . . Whether through religious instruction, education and tutoring, social services, welfare functions, emergency assistance or associational activity, Islamism has come to fulfill basic needs in society that the state has been increasingly ignoring or economically unable to sustain\" (Entelis 207).\n\n14. Willis has stressed the absence of a third unified alternative to the FIS and the FLN because numerous secular political parties ended up fragmenting this opposition (152-53).\n\n15. This is also acknowledged by Layachi (59).\n\n16. Boudjedra sees Algerian women wearing the veil and fighting against fundamentalism as a contradiction, because modernity is to be taken whole or not at all (85), thereby perpetuating stereotypes about covered Muslim women that I will discuss at length in chapter five. Similarly, Mimouni has a very simplistic view of the hijab (49).\n\n17. Other autobiographical collections edited by Sebbar include _C'\u00e9tait leur France_ and _Mon p\u00e8re_. I exclude them from the \"trilogy\" based on the titles, which do not follow the same format of \"une enfance . . .\" and do not put the emphasis on childhood.\n\n18. Cohen also notes that the pieds-noirs, with very few exceptions, did not speak Arabic.\n\n19. The last part of this sentence may allude to the fact that Christian clerics became targets and were murdered in Algeria starting in 1994 (Naylor 206).\n\n20. The Berber Spring, a protest movement to claim recognition for the Berber language and culture, took place in 1980. It started when Mouloud Mammeri was prevented from giving a talk on ancient Berber poetry, and was severely repressed by the FLN.\n\n21. In the English translation, it could also refer to Fatiha as the childhood friend he no longer sees by the end of the story (to refer to Fatiha in French there would need to be a silent \"e\" at the end of the adjective).\n\n22. Stora interprets this event as a colonization of Algerian Judaism by French Judaism (292).\n\n23. For an interesting account of how the pieds-noirs' memory and efforts to cultivate and bring their selective memory of French Algeria to the national level, through memorials, see William Cohen's article.\n\n24. This complete assimilation made the Jews particularly attached to France, and made it impossible for most of them to support an Algeria divorced from France (Stora, \"L'impossible\" 303).\n\n25. That law limited the number of Jews to small percentages in various fields (Stora, \"L'impossible\" 295). See Stora ( _Les trois exils_ 82-84) for more details.\n\n26. This double violence is most \"tragically symbolized by the well-publicized killings of William Levy . . . by the OAS, and of his son by the FLN\" (Naylor 44). Naylor adds that the FLN declared in 1961 \"For the first time in History the Jews have been claimed\u2014and by a government composed of followers of another religion\u2014as the sons of one and the same country\" (qtd. in Naylor, 44).\n\n27. \"Un autre sentiment dans l'ombre: la certitude jamais entamable que 'les Arabes' \u00e9taient les vrais rejetons de ce sol poussi\u00e9reux et parfum\u00e9. Mais quand je marchais pieds nus avec mon fr\u00e8re sur les chemins chauds d'Oran, je sentais la plante de mon corps caress\u00e9e par les paumes accueillantes des anciens morts du pays, et le tourment de mon \u00e2me s'apaisait\" (71) 'Another feeling in the shadows: the unshakeable certainty that 'the Arabs' were the true offspring of this dusty and perfumed soil. But when I walked barefoot with my brother on the hot trails of Oran, I felt the sole of my body caressed by the welcoming palms of the country's ancient dead, and the torment of my soul was assuaged' (126).\n\nThis and a reference to the abrogation of the Cr\u00e9mieux decree in 1940 are the only elements common between \"Pieds nus\" and the French version of \"Mon alg\u00e9riance.\" However, as Harrison pointed out, the English translation of \"Mon alg\u00e9riance\" includes additions to the French original, including the section from \"Pieds nus\" that features the encounter with the shoe shine boy (note 10 p.30).\n\n28. \"Je sus imm\u00e9diatement qu'elles \u00e9taient l'Alg\u00e9rie qui se pr\u00e9parait. Je leur tendais la main, je voulais faire alliance avec elles contre les Fran\u00e7aises. En vain. Pour elles j'\u00e9tais la France\" (74) \"I knew immediately that they were the Algeria that was in store. I held out my hand to them, I wanted to ally myself with them against the French. In vain. For them I was France\" (170).\n\n29. \"Two hates shared the hearts differently between themselves. One of them, which I will never forgive, the colonialists' hate, was made of the scorn from which thieves and usurpers, like all despots, forge deceitful arms. In response there was the other one, which I forgive, even until today, the hate with eyes burning with tears of the humiliated and deprived, the hate of the 'Arabs' for all that was collected in the 'F' [French] group\" (162, this paragraph does not appear in the French original). But even in that text, when it comes to her personal experience in colonial Algeria, there is an overemphasis on the fact that she wanted to be accepted and loved by the Arabs but was only met with hatred.\n\n30. President Bouteflika admitted later on in July 1999 that the government was responsible for the origin of the civil war with the cancellation of these elections, which was a violent act (Stora, _La guerre_ 19).\n\n## Chapter 5\n\n## Islam and the French Republic: The Affair of the Muslim Headscarf in works by Tahar Ben Jelloun, Abdelwahab Meddeb, Amin Maalouf, Albert Memmi, Le\u00efla Sebbar, and Yamina Benguigui\n\nIn the second half of the twentieth century, metropolitan France has become home to a substantial, albeit invisible until the 1980s, Muslim minority, composed mostly of immigrant Maghrebian workers and their descendants confined to the outskirts of French society. The number of Muslims in France is notoriously elusive, and estimates range between three and a half and seven million, with five million as the most accepted estimate, for a total population of about sixty-one million. Although the inauguration of Paris' Great Mosque in 19201 would seem to give official recognition, at least symbolically, to the presence of Islam on French soil, its conception is entangled with French colonial history and its policy towards its Muslim subjects. The project was first conceived to strengthen France's power over its Muslim colonies, and then as a monument to honor Muslim soldiers who lost their lives for France during World War I (Kepel, _Les banlieues_ 64-76). In fact, the Paris Great Mosque was not built to serve Muslims living in the capital, which numbered very few at the time (Kepel, _Les banlieues_ 65).\n\nIn recent decades, the Muslim French have been struggling to gain acceptance by French society. The notorious \"affaire du foulard,\" 'affair of the [Muslim] scarf,'2 sparked in 1989 when three teenage girls were expelled from their public middle school for refusing to remove their headscarves, received much coverage from the media.3 Similar incidents reoccurred over the years and culminated into the 2004 law that bans certain religious signs in public schools.4 Similar affairs have broken out in other Western countries such as England, the United States, etc., but France stands out by the level of divisiveness, the length of time of the controversy, and the resort to legislation to solve it. Although the debate on the issue of the scarf was framed exclusively around the issue of secularism, with concerns over the equality of the sexes adding fuel to the fire, several scholars, including Christine Delphy, Sa\u00efd Bouamama, G\u00e9rard Noiriel, Pierre Bourdieu, Fran\u00e7oise Gaspard, and Farhad Khosrokhavar, have pointed out that what was really at stake was the issue of immigration, and, more specifically, the realization that a population that was considered foreign and composed of temporary immigrants had actually settled down and was there to stay.5\n\nIn addition to this heated domestic debate about the issue of the veil in public schools, Islam has been associated with distressing national and international headlines in the 1990s. France was the site of terrorist attacks in 1994 and 1995, carried out by the Algerian extremist GIA (Groupe Islamiste Arm\u00e9 \"armed islamist group\"). The debate over the headscarf affair in France was tainted by international news at a time when the rise of Islamic fundamentalism was regularly making headlines, more specifically during the time in question with the Algerian civil war that ravaged Algeria during the 1990s and the Taliban's government in Afghanistan, both with disastrous consequences, particularly for women.\n\nImages have played an important role during the affair of the scarf. French magazine covers had the tendency to depict Islam as a women-oppressing fundamentalist religion (Gaspard and Khosrokhavar 31), by presenting pictures of veiled women (sometimes burqa style afghani) that most often than not had nothing to do with the scarf worn by the girls concerned. For example, on December 8, 1994, _Le nouvel observateur_ featured a young woman whose head and face were covered, with the cloth molding the face to suggest gagging and suffocation, on the cover of a special issue on women and Islam.\n\nThese images contributed to blurring the difference between the French and foreign contexts, and made the veil a symbol of fanaticism, since pictures of veiled women often accompany articles that deal with fundamentalism in news magazines. Significantly, this so-called debate on secularism was triggered by the presence of a Muslim sign in what is supposed to be the gateway to integration, namely the public school. The French public school became the symbolic place where the various crises plaguing French society converged, including the crisis of the nation-state, for which the public education system is the spearhead (Gaspard and Khosrokhavar 40-41). John Bowen's comprehensive study has demonstrated that explaining the 2004 law on the prohibition of religious signs in public schools, which ended the affair,\n\nwould require unpacking a great deal about France, including France's very particular history of religion and the state, the great hopes placed in the public schools, ideas about citizens and integration (and the challenge posed by Muslims and Islam to those ideas), the continued weight of the colonial past, the role of television in shaping public opinion, and the tendency to think that passing a law will resolve a social problem. (2)\n\nAs he (and others) noted, the issue of the veil in public schools became the symbol of what were seen as both internal and external threats to France, and its prohibition as an expedient to solve social problems of Islamism, sexism, and violence in the suburbs.\n\nIn _Intellectuals and Politics in Post-War France_ , David Drake explains that French intellectuals were split over the affair of the scarf (176-82). His book does not include any Arab intellectual, although some of them (the most famous being Ben Jelloun) have been living and publishing in France for a long time, and did publish opinion pieces on the issue. Because of their status as privileged interpreters on anything linked to Arab-Muslim issues, their writings can influence and shape the terms of the debate. Indeed, Ben Jelloun and Sebbar published on the issue in newspapers, and Meddeb testified in front of a government-appointed commission.\n\nThis chapter examines how Arab writers who were living in France have presented the affair of the scarf in non-fiction writings. The writers treated are representative of the diversity of the Francophone Arab world with their various backgrounds (Muslim for Benguigui, Ben Jelloun, and Meddeb, Christian for Maalouf, Jewish for Memmi, secular for Sebbar; and Moroccan for Ben Jelloun, Franco-Algerian for Sebbar and Benguigui, Tunisian for Memmi and Meddeb, Lebanese for Maalouf). The headscarf affair split French public opinion, but not along lines determined by ethnic, gender, cultural, or religious backgrounds. It would be absurd to assume that just because these writers are Arabs (or of Muslim origin for some), they would side with the veiled girls.6 However, it is fair to expect that because they have ties with countries with majority Muslim population, they might provide a better understanding of the problems faced by Muslims in France. In addition, since they have cogently addressed the devastating effects of colonialism and its role in shaping current events in parts of the Arab world in their writings (often by successfully presenting the point of view or plight of the colonized), one would hope that they could bring a fresh perspective that would enrich the debate. But as this chapter demonstrates, the opposite is the case. Despite such awareness of the legacies of the colonial era, they (with the exception of Benguigui) have failed in two notable ways: they did not help people understand the issue from the point of view of the girls concerned (which would have required depoliticizing the issue), and they did not prod French society to see its blind spots regarding secularism, which has framed the debate. The purpose of this chapter is not to redress the silencing of women.7 Rather, the aim is to demonstrate that prominent Francophone authors limit their readers' understanding of the Muslim world, by refusing to come to terms with the challenge that women who choose to don Islamic garb poses to Western feminism and secular thought.\n\n### 9\/11 and the French Islamic Veil Affair in Essays:  \nTahar Ben Jelloun's _L'Islam expliqu\u00e9 aux enfants_ and  \nAbdelwahab Meddeb's _La maladie de l'islam8_\n\nBen Jelloun and Meddeb's essays are part of an effort to counter reductive representations of Islam that have been going on for decades, as Said demonstrated in _Covering Islam_. 9\/11 has further exacerbated the association made in the Western media between Islam and terrorism. As a defining moment marking the turn of the century, 9\/11 has not only impelled writers from the Francophone Arab world to respond creatively with fiction,9 but also in essays. Ben Jelloun and Meddeb, of Moroccan and Tunisian origins, respectively, who have both been living in France for decades, published books in Paris in 2002 that use 9\/11 as a point of departure for reflection on the Muslim faith. These essays are aimed at different readerships: Ben Jelloun's _L'Islam expliqu\u00e9 aux enfants_ 'Islam explained to children'10 is a slim, ninety-page book that targets ten- to fifteen-year-olds, while Meddeb's _La maladie de l'islam_ 'The malady of Islam' is highly erudite. Both texts, however, share the similar intent and didactic function of countering reductive representations of Islam fostered by Islamic fundamentalists and the media, as well as explaining and historically tracing the link between terrorism and the Muslim faith.\n\nIt is worth noting that, in these essays, Ben Jelloun and Meddeb's first attempts to explain Islam to their fellow French citizens take 9\/11, an attack on a foreign country,11 as a point of departure when Islam, in fact, had been making domestic headlines well before 2001. How do these essays present the affair of the Muslim scarf that ignited such divisive debates and measures? Moreover, what are the implications of discussing Islam in general, and the affair of the scarf in France in the light of 9\/11? How do these writers handle their pedagogical project given the French context of their publications? I will take up these questions focusing on the essays published in 2002 and Meddeb's _Face \u00e0 l'islam_ 'Facing Islam' (2004).\n\n#### Ben Jelloun's _L'Islam expliqu\u00e9 aux enfants_\n\nBen Jelloun's essay is part of a series for youth published by the prestigious Parisian press Seuil, to which he already contributed _Le racisme expliqu\u00e9 \u00e0 ma fille_ \"racism explained to my daughter\" in 1998. The structure of _L'Islam expliqu\u00e9 aux enfants_ is in accordance with the model of the series, following the format of questions and answers between a child and the author. This text differs from all other titles in the series in that instead of foregrounding a biological link with one child, it addresses all children. This suggests that contrary to the other books in the series, all addressed to \"my daughter\/son, my grand-children, or my children,\" Islam has to be made explicitly relevant to all children since it is not immediately perceived to be so in the way topics such as racism, the French Resistance, or Auschwitz are.12\n\nThe first chapter, entitled \"Le 11 septembre expliqu\u00e9 aux enfants\" '11 September explained to children' makes it clear that Ben Jelloun's daughter's reaction to the coverage of these attacks motivated the writing of the book. Ben Jelloun blames the media's amalgam between terrorists, Arabs, and Muslims for the confusion and denial experienced by his daughter, who refuses to be associated with her Arab and Muslim heritage (7). He then proceeds to tell the story of Islam as a fairy tale, beginning with the story of Mohammed, the five pillars, the Golden Age, and the contributions of Arabs to various fields of knowledge, all of which are interspersed with allusions or brief discussions of contemporary issues in international and domestic politics.\n\nKeeping in mind that the target audience is identified by the author as children between the ages of ten and fifteen, it seems inevitable that such a book cannot engage in highly complex discussions of the various issues mentioned. But this limitation notwithstanding, Ben Jelloun also seems overly concerned to explain Islam, as well as colonization, and the affair of the scarf, in ways that will not alienate his intended readership. On several occasions, for instance, his assertions contradict mainstream Muslim views. It is the case with his interpretation of the word _'alaq_ as really meaning \"sperm\" (19).13 In the same vein, his assessment that the Qur'an does not prescribe women's veiling (68-69) is in accordance with what a few scholars reinterpreting Islamic traditions have concluded,14 but it is not widely accepted. Moreover, some of the recommendations issued by Ben Jelloun's father, and which he uses to discharge himself of the daily prayer requirements, are reminiscent of the Bible's Ten Commandments (12). They are therefore more intelligible to a readership coming mostly from a Judeo-Christian tradition, but they gloss over the fact that the daily prayers are one of the five pillars of Islam, and therefore cannot be avoided lightly, except in specific circumstances, none of which apply to Ben Jelloun.\n\nBen Jelloun's conciliatory efforts with its assumed French readership even leads to distortions in his account of the colonization of North Africa, which states that people revolted only decades after being colonized, and is inaccurate:\n\nCe qui a permis l'occupation de ces pays arabes et musulmans, c'est le d\u00e9clin  \nqu'ils connaissaient. C'est comme un corps malade qui ne peut pas se d\u00e9fendre  \net se voit envahi par d'autres maladies.  \n\u2014Est-ce que les gens se sont r\u00e9volt\u00e9s?  \n\u2014Oui, apr\u00e8s quelques d\u00e9cennies, ils se sont r\u00e9veill\u00e9s. (71)  \nWhat made the occupation of these Arab and Muslim countries possible was their decline. It's like a sick body that cannot defend itself and is invaded by other diseases.  \n\u2014Did people revolt?  \n\u2014Yes, after a few decades, they woke up.\n\nBen Jelloun portrays the Arabs as very passive towards colonization, thus writing off resistance movements such as those led by Abdelkader in Algeria and Abdelkrim in Morocco.15 In _Le racisme expliqu\u00e9 \u00e0 ma fille_ , Ben Jelloun was noticeably more virulent about colonialism in general, and the French colonization in particular (48-49), thus the downplay of the effect of colonization in _L'Islam expliqu\u00e9 aux enfants_ is all the more striking. A statement such as \"les croisades sont un lointain souvenir, la colonisation aussi\" 'the Crusades are a distant memory, so is colonization' (65), is certainly not the case in the French context at the turn of the twenty-first century, with France finally coming to \"the end of amnesia,\" as the title of a book edited by historians Stora and Harbi puts it, about its colonial past and the war Algerians fought for independence.\n\nSimilarly, by shifting the blame to an unidentified entity\u2014with the use of the indefinite subject pronoun \"on\" 'one'\u2014to explain the rise of fundamentalism in Muslim countries, Ben Jelloun makes short shrift of the colonial legacy in this historical development:\n\nS'il y a parmi les musulmans des jeunes devenus violents et fanatiques, c'est que leur \u00e9ducation a \u00e9t\u00e9 mal faite, on les a laiss\u00e9s entre les mains de gens ignorants et sans scrupules. On n'a pas su ou voulu leur faire aimer le d\u00e9veloppement, la culture et la vie. On a laiss\u00e9 se d\u00e9velopper la pauvret\u00e9 et l'analphab\u00e9tisme. On a eu peur de la libert\u00e9 et on n'a rien fait contre la corruption et les injustices ( _L'Islam_ 65).  \nIf there are among Muslims young people who have become violent and fanatical, it is because their education was poorly done, they were left in the hands of uneducated and unscrupulous people. One was unable or unwilling to make them like development, culture, and life. One has let poverty and illiteracy develop. One was afraid of freedom and has not done anything against corruption and injustices.\n\nThis excessive generalization may be unavoidable because of the limited scope and intended reader, but it underestimates the role that Western nations, including France, have played in various parts of the Muslim world to fuel extremism. The use of the indefinite pronoun masks the fact that no nation exists in a vacuum and that former colonial power structures have been superseded by other, less obvious but no less damaging institutions and neocolonial politics.\n\nBy far, the most conciliatory pages of _L'Islam expliqu\u00e9 aux enfants_ deal with the affair of the scarf. When Ben Jelloun insists that Muslims in France must respect the 1905 law on the separation of church and state, he brings up the affair of the veil.16 He subscribes to the often repeated argument that religion should remain in the private realm (12). Many have said during the course of the debate (and indeed this is what was done in some schools) that girls had to take off their headscarves as soon as they entered school. As Rosello pointed out, in the framework of the Muslim faith and customs, this does not make sense (140), since a Muslim girl is only required to cover in the presence of unrelated men, which is not likely to be necessary in her home, unlike in public. In contrast to his 1984 book about French racism, in which he acted as an effective spokesperson responsible for explaining cultural issues to the French public, Ben Jelloun fails to point out the absurdity of the mandate from the point of view of an observant Muslim. Devout Muslim women would not concede to only veiling in the private realm.\n\nIn addition, his explanation of the consequences of the affair shows that like most accounts in the media, the veiled girls are never credited with any sense of agency in their decision to cover, despite solid evidence to the contrary: \"certaines filles ont renonc\u00e9 \u00e0 porter le foulard. D'autres ont \u00e9t\u00e9 retir\u00e9es de l'\u00e9cole par leurs parents. Ils ont eu tort de les priver d'enseignement\" 'some gave up wearing the scarf. Others were taken out of school by their parents. They were wrong to deprive them of an education' (67-68). Books such as Gaspard and Khoskhovar's, Dounia Bouzar and Sa\u00efda Kada's, and Alma and Lila L\u00e9vy's,17 clearly dispelled the stereotype that the majority of French girls who wear the veil are coerced into doing so.18 Although he later criticizes the custom of taking girls out of school once they have reached puberty in some Muslim communities (88) and sees the school as the place to start establishing gender equality in Islam, he never considers the consequences of denying an education to girls who refuse to take off their headscarves.19\n\nSubsequently, Ben Jelloun states that \"les musulmans de France ont la chance de vivre dans un pays d\u00e9mocratique qui leur garantit le droit de pratiquer librement leur religion\" 'Muslims of France are lucky to live in a democratic country that guaranties them the right to practice their religion freely' (76). Needless to say that from the covered girls' point of view, their expulsion from school because they are dressed as they believe their religion mandates is hardly guaranteeing their right to practice.20 Ben Jelloun's statement implies that the French Republic's principle is fully implemented, which is far from true. There is no mention here (nor elsewhere in the text) of the difficulties Muslims are encountering when trying to establish adequate infrastructure to accommodate their faith. For instance, obtaining building permits for mosques is notoriously difficult. Similarly, the French government refuses to use the Concordat,21 still in place in Alsace-Moselle, to train imams,22 which many recognize as essential to the development of a \"French Islam.\" Failure to exploit well-established structures to the benefit of the French Muslim community is symptomatic of the denial to consider Islam as what it is: the second religion in France.\n\n#### Meddeb's _La maladie de l'islam_\n\nMeddeb's _La maladie de l'islam_ is also inspired by the 9\/11 attacks, which the author condemns in the very first sentence of the book. The book's title echoes Voltaire's writings, such as his _Trait\u00e9 sur la tol\u00e9rance_ , as acknowledged by Meddeb, but also his entry on \"fanaticism\" (which he sees as having sullied all religions) in _Dictionnaire philosophique_.23 In contrast to Voltaire, Meddeb's title implies that Islam as a whole is subject to a disease intrinsic to it, as if fundamentalism was Islam's prerogative and other religions were immune to it. Indeed, Meddeb asserts: \"Si le fanatisme fut la maladie du catholicisme, si le nazisme fut la maladie de l'Allemagne, il est s\u00fbr que l'int\u00e9grisme est la maladie de l'islam\" 'if fanaticism was Catholicism's disease, if Nazism was Germany's disease, it is certain that fundamentalism is Islam's disease' (12).\n\nMeddeb's book endeavors to trace the history and intellectual genealogy of Islamic fundamentalism. Although Meddeb acknowledges that there are internal and external reasons for Islamic fundamentalism, he justifies his focus on internal factors by stating:\n\nIl est du r\u00f4le de l'\u00e9crivain de pointer la d\u00e9rive des siens et d'aider \u00e0 leur ouvrir les yeux sur ce qui les aveugle. Je tiens, comme on dit, \u00e0 commencer par balayer devant ma porte. Ce texte, \u00e9crit en fran\u00e7ais, sera lu par de nombreux lecteurs connaissant le fran\u00e7ais et concern\u00e9s d'une mani\u00e8re ou d'une autre par le drame de leur origine islamique. Je m'adresse \u00e0 tout lecteur, mais j'ai une pens\u00e9e particuli\u00e8re pour les lecteurs qui, comme moi, se sont symboliquement constitu\u00e9s dans la croyance d'islam. (10)  \nThe writer's role is to point out his people's drift and to help open their eyes on what blinds them. I insist, as we say, in starting by sweeping in front of my door. This text, written in French, will be read by numerous readers who know French and who are concerned one way or another by the tragedy of their Islamic origin. I address any reader, but I have a special thought for readers who, like me, formed themselves symbolically in Islam's belief.\n\nMeddeb's personal experience of being a Muslim in France might have been tragic, but extending his subjective experience to the whole Muslim population in France without addressing the socio-political context tags Islam negatively without proper justification. When Meddeb talks about \"sweeping in front of his door,\" he positions himself as part of the Muslim world, since his book expressly focuses on problems within Islam. Although the quote above shows Meddeb's awareness of his French readership, it does not demonstrate that he has fully thought out the implication of the fact that his \"door\" is in fact not in the Muslim world, but has been located in Paris for decades.\n\nLike Ben Jelloun, Meddeb gives a distorted view of what mainstream Islam believes to be the proper dress code for women in his subsequent book _Face \u00e0 l'islam_. In a sentence whose syntax obscures its meaning, Meddeb proposes to only consider the verses of the Qur'an and gives an interpretation that does not render the headscarf compulsory, while admitting that this is not mainstream: \"Le voile peut n'\u00eatre pas assimilable \u00e0 un signe religieux si l'on prend la d\u00e9cision de ne pas suivre l'unanimit\u00e9 des docteurs et de lire uniquement en euxm\u00eame les versets coraniques invoqu\u00e9s pour l\u00e9gitimer sa prescription\" 'the scarf may not be assimilable to a religious sign if one takes the decision not to follow the unanimity of scholars and to read only the Koranic verses invoked to legitimize its prescription in themselves' ( _Face_ 198). Meddeb also fails to mention that the decision not to take into account the Tradition by the Prophet would be unacceptable to most Muslims, since these narratives play a crucial role in Qur'anic exegesis.\n\nAlthough Meddeb is very careful in his argument about not essentializing Islam ( _La maladie_ 64), he fails to apply his own caution to the issue of the veil. He admits being shocked at witnessing veiling in Paris ( _La maladie_ 49) because the reveiling of women for him is a sure sign of going backwards. This statement also presupposes that the veil had disappeared from all Muslim countries, which it had not. He takes pity on veiled women in Egypt, as if the veil could only have the meaning of oppression, thus ignoring that the phenomenon of (re)veiling can have different meanings depending on the context in which it occurs, and can even be a revolutionary gesture.24\n\nMeddeb deplores the standardization of the veil throughout the world, as opposed to the time when each Muslim culture had a distinct way of veiling, as if this standardization, if standardization there is (this is highly debatable), could not be a sign of new exigencies. This contradicts what others have noticed, such as Le\u00efla Ahmed, who sees the variety of new styles in Cairo as \"adoption of a 'modern' version of the conventions of dress they . . . were accustomed to\" (222). In the French context, this is also contradicted by the picture on the cover of the L\u00e9vy sisters' book, which shows two young women in turtlenecks wearing headscarves tied at the nape of the neck. This innovative use of a turtleneck sweater, a \"Western\" garment, which allows for tying the scarf behind rather than in front to cover the neck, debunks the standardization deplored by Meddeb. In addition, the tricolor headscarves mimicking the French flag worn by some of the women demonstrating against the law banning religious signs were very much grounded in the French socio-political context and cannot be seen as partaking in standardization.\n\nOverlooking the contradictory nature of his position and the subjectivity of his statements, Meddeb asserts that the veil is a cultural sign only if it is seductive. When it ceases to be attractive, it becomes ideological and a sign of women's subservience to men. This reasoning is particularly evident when he evokes the image of the silk _ha\u00efk_ worn by traditional Tunisian women. While he finds the garb, \"froufroutant \u00e0 chaque pas\" 'rustling with each step' ( _Face_ 198), sexually provocative, one of the young women in Mernissi's autobiography explains that this traditional and impractical dress was \"probably designed to make a woman's trip through the streets so torturous that she would quickly tire from the effort, rush back home, and never dream of going out again\" ( _Dreams_ 118).25 It is quite ironic that Meddeb, who poses as a champion of women's rights, looks nostalgically at the _ha\u00efk_ , a garment that discourages women from being outside, while he denigrates the headcover that enables them to move freely in public space. It is also disputable whether women in general fared better under the Golden Age of Islam that Meddeb looks upon so fondly because of its openness to sensuality. And he is downright orientalist when he laments:\n\nQuelle chape de pudeur a couvert les pays qui virent Flaubert s'ab\u00eemer dans la jouissance! Pour en retrouver la sc\u00e8ne, que faire sinon rappeler les heures ardentes que l'\u00e9crivain normand avait pass\u00e9es avec l'alm\u00e9e Kuchuk-Hanem sur les bords du Nil. ( _La maladie_ 136)  \nWhat cope of modesty has covered the countries that saw Flaubert sunk in pleasure! To remember its scene, what can we do, other than recall the passionate hours spent by the Normand writer with the almah Kuchuk-Hanem on the Nile's banks.\n\nThis allusion to the most famous pages of Flaubert's _Voyage en Orient_ glorifies Flaubert's encounter with a prostitute in Cairo without drawing any attention to the ethnic, class, and gender issues in a passage that portrays Arab women as stupid, eager sensual objects of male desire, of whom Kuchuk Hanem, as Said pointed out in _Orientalism_ , is the prototype (207).\n\nAlthough he only indirectly alludes to the affair of the headscarf in _Maladie_ , Meddeb unequivocally affirms his support for the way the law against religious signs in public schools has been justified. He praises the French model of integration as a better system than the American model of multiculturalism, as if the periodical violence in French suburbs, certainly the riots of fall 2005, did not attest to major flaws: \"je reste surpris par ceux qui, en invoquant leur islam, demandent \u00e0 la R\u00e9publique de changer\" 'I am still surprised by those who, invoking their Islam, ask the Republic to change' ( _La maladie_ 221), and adds that this is a \"pr\u00e9tention d\u00e9raisonnable\" 'unreasonable claim.' He follows this with a quote from the Torah to counter this \"unreasonable claim,\" without giving pause to the oddity of invoking Jewish scripture to argue for secularism as a response to concerns expressed by Muslims. In his subsequent book _Face \u00e0 l'islam_ , he is more explicit about the headscarf affair. He interprets the veil as a sign of sexual discrimination against women (159). His delight about the legislation of the law, which he somehow helped to bring about as a consultant for the Debr\u00e9 commission (197), is clearly evident.\n\nLike Ben Jelloun, Meddeb advocates education of his fellow Muslims as a remedy against fundamentalism and women's oppression, but does not consider the fact that the decision to exclude veiled Muslim girls from schools is actually depriving the ones who, according to him, need education the most. To combat fundamentalism, Meddeb also advocates the following: \"au lieu de distinguer le bon islam du mauvais, il vaut mieux que l'islam retrouve le d\u00e9bat et la discussion, qu'il red\u00e9couvre la pluralit\u00e9 des opinions, qu'il am\u00e9nage une place au d\u00e9saccord et \u00e0 la diff\u00e9rence, qu'il accepte que le voisin ait la libert\u00e9 de penser autrement\" 'instead of distinguishing between a good and a bad Islam, it is better that Islam find again debate and discussion, that it rediscover the plurality of opinions, that it carve out a place for disagreement and difference, that it accept that the neighbor be free to think differently' ( _La maladie_ 13). However, when it comes to the issue of the veil, he fails to apply his own recommendation.\n\nUnlike Ben Jelloun's and Meddeb's essays, Benguigui, Maalouf, Sebbar and Memmi do not focus on Islam per se in the works discussed here, and were not triggered by 9\/11. But their respective topics do lead them to talk more or less about France's colonial legacy and Muslims, and briefly about the issue of the headscarf in public schools.\n\n### The Affair of the Headscarf in Amin Maalouf's _Identit\u00e9s meutri\u00e8res_ and Albert Memmi's _Portrait du d\u00e9colonis\u00e9 arabo-musulman et de quelques autres_\n\n#### Amin Maalouf's _Identit\u00e9s meutri\u00e8res_\n\nIn _Identit\u00e9s meutri\u00e8res_ , an essay published in 1996, Amin Maalouf sets out to explore the question of multiple belongings, and the disastrous consequences that can ensue when we are pressed to deny some of our affiliations. Drawing on his personal situation as a Christian Arab, a \"paradoxical\" fact (23) that puts him at the border and allows him to affiliate with Christians and Muslims alike, Maalouf gives examples of situations when one's identity can change because of political events. He strongly advocates that one should be encouraged to see one's identity as multiple, and warns of the dangers that come with pushing people to single out one element and repress the others (183). Maalouf advocates learning languages as a remedy to two forces that crush plurality: essentialist identity politics and globalization. In the process of this argument, Maalouf reminds us that religious fundamentalism has not been the first choice of Arabs and Muslims, but a last resort when everything else failed (96). Maalouf contrasts languages with religion, in the way that one can speak several languages, but only belong to one religion. This leads Maalouf to write: \"Il faudrait faire en sorte que personne ne se sente exclu de la civilisation commune qui est en train de na\u00eetre, que chacun puisse y retrouver sa langue identitaire, et certains symboles de sa culture propre\" (187-88) 'We must act in such a way as to bring about a situation in which no one feels excluded from the common civilisation that is coming into existence; and in which everyone may be able to find the language of his own identity and some symbols of his own culture' (Bray 163). This is a very general statement and Maalouf abstains from giving examples of what to retain and what to keep, yet that is the thorny issue. However, the Muslim headscarf is not one of the symbols he seems to deem appropriate to display, as I will explain shortly.\n\nMaalouf does mention the affair of the scarf, but does not elaborate on it. After a development on the situation of migrants, and the fact that the host society is \"ni une page blanche, ni une page achev\u00e9e, c'est une page en train de s'\u00e9crire\" 'neither a tabula rasa, nor a fait accompli, but a page in the process of being written' (50, Bray 40), Maalouf asks a crucial question that deserves to be quoted at length:\n\nil s'agit, \u00e0 vrai dire, d'un contrat moral dont les \u00e9l\u00e9ments gagneraient \u00e0 \u00eatre pr\u00e9cis\u00e9s dans chaque cas de figure: qu'est-ce qui, dans la culture du pays d'accueil, fait partie du bagage minimal auquel toute personne est cens\u00e9e adh\u00e9rer, et qu'est-ce qui peut \u00eatre l\u00e9gitimement contest\u00e9, ou refus\u00e9? La m\u00eame interrogation \u00e9tant valable concernant la culture d'origine des immigr\u00e9s: quelles composantes de cette culture m\u00e9ritent d'\u00eatre transmises au pays d'adoption comme une dot pr\u00e9cieuse, et lesquelles\u2014quelles habitudes? quelles pratiques?\u2014devraient \u00eatre laiss\u00e9es \"au vestiaire\"? (51-52).  \nWhat we are really talking about\u2014a moral contract, the elements of which need to be defined in each case to which it is applied: what, in the culture of a host country, is the minimum equipment that everyone is supposed to possess, and what may legitimately be challenged or rejected? The same question may be asked about the immigrants' own original culture: which parts of it deserve to be transmitted like a valuable dowry to the country of adoption, and which\u2014which habits? which practices?\u2014ought to be left at the door [cloakroom]? (Bray 41)\n\nThe use of the word \"vestiaire\" 'cloakroom' is important (and is missing in the English translation), since it is a place where one leaves clothing; it indirectly alludes to the affair of the headscarf that will be mentioned later. By using a term associated with clothing, Maalouf's statement can be read as implicitly suggesting that the veil, an article of clothing that has been the source of great polemics, should be left in the cloakroom. Four paragraphs later, Maalouf resorts to a paralipsis to mention the affair: \"Aurais-je \u00e0 l'esprit, en disant cela, des controverses comme celle qui s'est engag\u00e9e, dans divers pays, autour du 'voile islamique'? Ce n'est pas l'essentiel de mon propos\" 'Am I thinking of controversies like that which has arisen in various countries over the 'Islamic veil'? These are not my main concern' (53, Bray 42). The paralipsis, a rhetorical figure of speech that allows one to bring up a topic while denying it should be discussed, allows Maalouf to mention the affair while distancing himself from it. However, he continues using vocabulary that is associated with the debate over the issue of the scarf in France: \"Lorsqu'on sent sa langue m\u00e9pris\u00e9e, sa religion bafou\u00e9e, sa culture d\u00e9valoris\u00e9e, on r\u00e9agit en affichant avec ostentation les signes de sa diff\u00e9rence\" 'When one feels that his language is despised, his religion ridiculed and his culture disparaged, he is likely to react by flaunting the signs of his difference' (53, Bray 43). The word \"ostentation\" recalls the language used in the directive issued in 1994 by the French Minister of Education that aimed to exclude headscarves from public schools and led to the expulsion of more than a hundred girls.26 When Maalouf returns to the issue of the scarf explicitly a couple of paragraphs later, he qualifies it as \"un comportement pass\u00e9iste et r\u00e9trograde\" 'a reactionary, backward-looking behavior' (54, Bray 43). His interpretation of the resurgence of religion here as something that valorizes the past to the detriment of the present and progress does not take into account women's explanations and understanding of this gesture. Bowen has noted that many young women draw a clear distinction between the Islam that they practice and the one of their parents, which shows that it is not simply a return to the past (71). To his credit, Maalouf adds that the real question that needs to be asked is to understand why sometimes modernity is not perceived as a progress (54), but he never questions that the only possible path to modernity is the French and Western model. Although he is acutely aware of the discrimination that immigrants and their children have been victims of, Maalouf refuses to consider that the decision to wear the scarf can be \"the result of a personal commitment rather than an intention to signal something to others\" from the point of view of the girls (Bowen 81).\n\n#### Albert Memmi's _Portrait du d\u00e9colonis\u00e9 arabo-musulman et de  \nquelques autres_\n\nIn an essay that aims to examine the situation of the formerly colonized, in the same vein as his landmark _Portrait du colonis\u00e9_ , Memmi talks about the issue of the scarf in a very brief section entitled \"Le voile ou le m\u00e9tissage\" 'Head Scarves and _M\u00e9tissage_ ' (105, Bononno 86). In two pages, he constructs an imaginary dialogue with veiled girls that begins as follows:\n\nPassons sur leurs arguments, m\u00e9diocres ou rus\u00e9s; au nom de la libert\u00e9, par exemple: \"On est libre en France, non? Eh bien, je suis libre de porter le voile!\"; \u00eatre libre de ne pas l'\u00eatre, est-ce encore de la libert\u00e9? Ces petites sottes ne voient pas qu'elles agissent contre elles-m\u00eames, en refusant des lois qui les lib\u00e8rent au profit de dogmes qui les asservissent. Elles exigent, au nom d'une la\u00efcit\u00e9 mal interpr\u00e9t\u00e9e, de ne pas \u00eatre la\u00efques. (106)  \nTheir arguments, some weak, some clever, often had to do with freedom. For example, \"We're free in France, aren't we? So, I'm free to wear a headscarf, right?\" It is a foolish argument. For they [these stupid girls] fail to see that they are acting against their own interests in rejecting the laws that freed them in favor of the dogmas that enslaved them. In the name of a poorly understood secularism, they demand not to be secular. (Bononno 86-87)\n\nAlthough the question and answer format can been seen as indirectly giving voice to some of the girls' motivation for wearing a scarf, Memmi chooses and frames arguments so as to immediately discredit them in a manner reminiscent of the asymmetry of roles played by guests on various programs on French TV in 2003-2004 (Bowen 240-41). Bowen noticed in a specific program that included the L\u00e9vy sisters that \"they [the L\u00e9vy sisters] were only allowed to speak as examples of the problem, the real understanding of which was provided by the 'experts' \" (240). Memmi's use of the adjective \"sotte\" 'stupid' for the girls (lost in the translation) is harsh and condescending, and although Memmi's tone is very sarcastic throughout the book, this is the only instance that I have noticed when he disparages the intelligence of people (in this instance young women) to better attack their arguments. His dismissal of arguments advanced by young women is symptomatic of the general silencing of the women concerned during the affair. For example, Bowen notes that the Stasi Commission27 did not hear from any girl who had been expelled (117), and only from one \"token\" veiled woman (118). He also notes that the media coverage of the affair perpetuated the stereotype that only uncovered women were seen as having earned the agency to speak for themselves, whereas covered girls were perceived as only parroting words dictated to them by men (245). Indeed, what Mohanty's ground-breaking article \"Under Western Eyes\" denounced well over a decade ago, can be seen operating in full swing here. First World feminists working on Third World issues posited themselves as the norm and in the process refused the idioms of agency that are relevant for Third World women. As for the issue of the interpretation of secularism, there is no acknowledgement here of the fact that the French have interpreted it in different ways, including the ruling of the _Conseil d'Etat_ during the first affair, which stated that students' wearing of religious signs was not in itself incompatible with secularism.28 Indeed, Drake underlines that both the intellectuals who opposed the headscarf in schools and those who opposed exclusion of the veiled girls did so in the name of secularism, a term that they understood in different ways (178). There is no hint of the complexity of the French debate about the issue of secularism during the second half of the twentieth century, most notably with the issue of public subsidies of private religious schools.29\n\nAfter dismissing the argument based on the issue of freedom, Memmi dismisses other arguments (of the veil as a protection, and as a religious obligation) in a similarly expedient manner, invoking, after secularism, the equality of the sexes, jeopardized by the meaning of the veil:\n\n\"Le voile prot\u00e8ge les femmes du regard des hommes\"; pourquoi ne pas prot\u00e9ger \u00e9galement les hommes du regard des femmes? N'est-ce pas un traitement particulier pour la sexualit\u00e9 f\u00e9minine? Pour les prot\u00e9ger du d\u00e9sir des hommes, faut-il qu'elles ne soient pas d\u00e9sirables, comme les d\u00e9votes juives qui se font raser la t\u00eate? Et surtout: respecter celles qui souhaitent \u00eatre ainsi prot\u00e9g\u00e9es ne donne pas le droit de vitrioler celles qui ne le souhaitent pas. (106)  \n\"The headscarf protects women from men's stares,\" they say. Why not also protect men from the stares of women? Aren't we showing favoritism to the female sex? To protect them from men's desires, is it necessary that they be undesirable, like those Orthodox Jewish women who shave their head? What's more, respecting those who wish to be protected in this way does not give you the right to criticize [throw vitriol at] those who do not wish to be so protected. (Bononno 87)\n\nWhen Memmi gets indignant in the first two rhetorical questions that the head covering is a \"special treatment\" and therefore violates the equality of the sexes, there is no question that the French model is taken as the desired norm, despite the fact that the advertisement industry has long been basing many campaigns on the very premise of a different treatment of the female body. Memmi's question \"faut-il qu'elles ne soient pas d\u00e9sirables\" betrays an assumption in the prerogative of the male spectator to look at attractive women that has been fostered in Western art, and most notably the traditional European nude painting, as demonstrated by John Berger's _Ways of Seeing_. In fact, I would argue that one of the reasons why the veil has crystallized public attention is because it goes against the Western pictorial tradition from the Renaissance to contemporary advertising that depicts women as sexual objects for the gaze of a male spectator. This includes the Orientalist tradition of languid odalisques that has been analyzed at length, and contemporary French society's tendency to undress women, from contemporary fashion styles to the topless and thong trends on the beach.30 The opponents of the scarf in public schools who use the gender discrimination argument are not able to see it simply as a dress code nor look critically at French society's norms. Western societies have long had different dress codes for men and women, yet it would not come to anybody's mind to propose banning skirts from public schools on the ground that they promote inequality between the sexes because only girls wear them (not to mention the historical and religious reasons behind that fact).\n\nThere is a striking slip in the last sentence of the quote above between one issue and another: the use of the verb \"vitrioler\" (again, lost in the English translation), which can be used literally and figuratively, will not just mean \"to criticize\" for a French reader. It will evoke real-life cases of women who were disfigured by men who threw vitriol in their faces. These incidents happened in Muslim countries such as Algeria and Bangladesh during the 1990s and were reported in the media during Taslima Nasreen's visit to France, in particular, in the case of Bangladesh. Here, foreign contexts are brought to bear on the domestic one. The argument of forbidding the veil to better protect women who do not want to wear it was a powerful one (Bowen 208). Arab-Muslim culture was blamed for much of the violence that affected women in the poor suburbs, and as a sign of that culture, the veil was judged guilty in contributing to perpetuate this violence. The _tournantes_ (gang rapes) that made sensational headlines during 2001-2003 helped give more momentum for the law (Bowen 214-17). According to a study cited by Bowen, the only new feature about gang rapes, which date back to the 1960s and were committed by non-immigrants, is the (false, though convenient) claim that they are due to Arab-Muslim culture (214).\n\nMemmi counters the girls' argument that wearing a headscarf is a divine obligation by asserting that it is only suggested by the Qur'an (106), and that it boils down to men controlling women's sexuality. Here again, Memmi refuses to recognize that there are various interpretations of the Qur'an, and that those that have challenged the compulsory head cover do not represent the beliefs of the Muslim majority. The last sentence he attributes to the girls alludes to the fact that the veil is a reaction to the non-acceptance of French society:\n\nEn fait, par del\u00e0 les arguties, on a vu appara\u00eetre un aspect revendicatif, sinon provocant: le voile est devenu le drapeau d'une cause: \"Vous n'aimez pas les musulmans, leur vue vous irrite? Eh bien, je le proclame, je suis musulmane, je vous en impose la vue! Celle d'un membre du groupe honni par vous.\" (107) But underlying the arguments we find an element of protest, if not downright provocation: the headscarf has become the flag of a cause. \"You don't like Muslims, the sight of them irritates you? Well, I'm proclaiming my Muslim-hood; I'm forcing you to see it. To see a member of a group you have made to feel ashamed.\" (Bononno 87-88)\n\nAlthough the discrimination faced by Muslims in France is acknowledged, the girls are faulted for resorting to the veil, a \"ghetto portatif 'portable ghetto' (107, Bononno 88). Their choice is constructed as a provocation, which runs counter to all the accounts that have been published telling about the various paths and motivations behind French women's decisions to cover (see Gaspard and Khosrokhavar, Bowen, Bouamama). This is a clear example of what Noiriel, quoted earlier, has pointed out: how the victims are transformed into aggressors by changing a social issue into a religious problem. Although a key argument in the affair was to see the veil as a symbol of patriarchal oppression, the girls, who are then constructed as double victims of patriarchal oppression and social discrimination, as acknowledged by the above quote, are paradoxically turned into deliberate aggressors of French secularism.\n\n### Missing Images of Veiled Girls: Leila Sebbar's _Journal de mes Alg\u00e9ries en France_ and _Mes Alg\u00e9ries en France_ , and Yamina Benguigui's _M\u00e9moires d'immigr\u00e9s: L'h\u00e9ritage maghr\u00e9bin_31\n\nA few writers have nonetheless displayed a more complex attitude towards the young women concerned and showed more discrimination in their representation of the scarf affair than those just examined. The works I focus on have in common that they either feature texts and images, such as _Journal de mes Alg\u00e9ries en France_ and _Mes Alg\u00e9ries en France_ , Sebbar's illustrated (autobiographical narratives, or blur the boundaries between the two, such as the film and book by Benguigui, which bear the same title, _M\u00e9moires d'immigr\u00e9s: L'h\u00e9ritage maghr\u00e9bin_ , and whose contents overlap without being completely identical. Sebbar and Benguigui are both Franco-Algerian. Sebbar was born in Algeria of an Algerian father and a French mother; she moved to France at the age of 18. Benguigui was born in France of Algerian parents. Despite different situations and a generation separating them, they are both at the threshold between two communities, and the works examined unearth the links that have been woven by history between the Maghreb (and specifically Algeria) and France in order to fill a void through texts and images.\n\nWhile Benguigui and Sebbar's works share the goal of giving its proper place to the Algerian and Maghrebian communities in France in the French collective memory, they also have in common a \"visual omission\": they do not include any pictures or any shots of veiled girls in France.32 The affair of the scarf is mentioned to various degrees in the texts, but the scarf worn by middle and high school girls in contemporary France does not appear, as if one could talk about it but not show it.33\n\n#### Sebbar's _Journal de mes Alg\u00e9ries en France_ and _Mes Alg\u00e9ries en France_\n\nAt the very beginning of the affair, Sebbar sarcastically mentions in an article published in _Le Monde_ the case of \"la\u00efcit\u00e9 aigu\u00eb\" 'acute secularism' that is plaguing the teaching profession, and the irrational panic caused by the scarves that are seen as a war sign of the advance of a fanatical Islam. Already, Sebbar maintained what others have since elaborated on: that secularism is not threatened by the religious signs worn by students, that the veil expresses modesty (\"un foulard qui cache leurs cheveux comme on cache ses cuisses et ses seins\" 'a scarf that hides their hair just as one hides one's thighs and breasts')34 and that the real victims in this case are the young girls who are excluded from the very institution supposed to promote equal opportunity.\n\n_Mes Alg\u00e9ries en France_ and _Journal de mes Alg\u00e9ries en France_ are books that mix (auto)biographical narratives, portraits, and diary; both are illustrated with images ranging from textbook pages35 to Orientalists paintings and personnal photographs. The narratives feature the experiences of diverse groups, including those of pieds-noirs in Algeria, children of immigrants in contemporary France, etc. In her foreword to _Journal_ , Sebbar explains her project by saying: \"Prise par un besoin f\u00e9brile de m\u00ealer l'Alg\u00e9rie \u00e0 la France . . . je tente par les mots, la voix, l'image, obstin\u00e9ment, d'abolir ce qui s\u00e9pare\" 'Caught by a feverish need to mix Algeria with France, I try with words, voice, image, to abolish what separates.' Since the affair was such a divisive topic, one could expect to see Sebbar give it some attention in these texts. In _Mes Alg\u00e9ries_ , the headscarf issue is briefly mentioned via a third party. In a section entitled \"le foulard. . . . Je ne suis pas jacobine,\" Michelle Perrot (who also happens to have written a preface to Sebbar's book), is reported to have said in 1999 that she was against the expulsion of veiled girls from school, and therefore had not signed Badinter's text.36 However, a footnote informs the reader that Perrot did sign a petition to support the law prohibiting religious signs in December 2003 (Sebbar, _Mes Alg\u00e9ries_ 167), thus underlining her change of mind. This footnote is the only (indirect) comment by Sebbar on the issue in that book, and demonstrates a hardening of opinion.37 Bowen also noted that more politicians gradually came to support the law (105).\n\nIn _Journal de mes Alg\u00e9ries en France_ , Sebbar dedicates a few lines to the affair. While _Mes Alg\u00e9ries_ was organized thematically, _Journal_ is a diary written between March 2004 and January 2005; therefore, the news is mentioned occasionally. Her diary starts the very month that the law was passed, but only two paragraphs evoke the issue later on. First, she recalls French Muslim organizations' stand and mobilization to obtain the release of French journalists kidnapped in Iraq, but does not mention that the kidnappers had demanded the abrogation of the law as a ransom for the hostages (61-62). She then follows up with the following sentence: \"Par ailleurs, les jeunes filles au _hijeb_ ont accept\u00e9 d'\u00eatre semblables \u00e0 leurs condisciples sans _hijeb_ (sauf en Alsace), conscientes peut-\u00eatre qu'elles seront les b\u00e9n\u00e9ficiaires de l'\u00e9cole la\u00efque fran\u00e7aise\" 'besides, the young women wearing a headscarf have agreed to be like their fellow students without headscarves (except in Alsace), conscious maybe that they will be the beneficiaries of the French secular school' (62). Given that this entry is dated September 3rd, it most probably alludes to the beginning of the first school year under the 2004 law that prohibits religious signs in schools. Although Sebbar is quick to point out the ineptitude of French politicians regarding equal opportunity for all ( _Journal_ 131), she no longer has anything to say about the cases of girls excluded.38 While the parenthesis \"sauf en Alsace\" indirectly alludes to the fact that the 1905 law of separation between Church and State, which defines the principle of secularism in France, does not apply to some parts of the Hexagon, it only does so in the context of the headscarf affair. Sebbar has nothing to say about the fact that the Concordat statute that governs the area of Alsace and Moselle constitutes a breach of secularism, since according to the Concordat the president is involved in the appointment of religious clerics remunerated by the state, which also funds religious instruction in public schools (Haarscher 32, Baub\u00e9rot 119-20).39 Indeed, this has been one of the most salient omissions by those invoking the fate of secularism during the debate: the failure to acknowledge the double standard hidden in the argument, since in fact the law of separation of Church and State is not being applied in all parts of France, not to mention the fact that the Concordat statutes benefit only Judeo-Christian religions.\n\nOne can surmise that Sebbar's change of mind on the issue, expressed indirectly in her photographical essays, and confirmed to me directly during an interview conducted in June 2006 in Paris, probably unconsciously influenced her choice of pictures.40 The only pictures of veiled women included in her books were taken in Algeria, and all the young women of Algerian origins living in France are bareheaded. In works endeavoring to document the diversity of the Algerian-French connection, it would have been pertinent to include in _Mes Alg\u00e9ries en France_ a picture of the girls wearing tricolor headscarves and singing the French national anthem during several demonstrations that took place at the beginning of 2004 (see Ternisien). The blue, white, and red headscarves were meant as a symbol of these young women's identity, claiming their attachment to both Islam and France. If, as Roland Barthes puts it, \"toute photographie est un certificat de pr\u00e9sence\" 'every photograph is a certificate of presence' (135), the absence of images of veiled French girls in Sebbar's books contributes to their rejection by French society.\n\n#### Yamina Benguigui's _M\u00e9moires d'immigr\u00e9s: L'h\u00e9ritage maghr\u00e9bin_\n\nBenguigui's successful documentary film, _M\u00e9moires d'immigr\u00e9s_ , broadcast on Canal + in May 1997, was followed by the publication of the book in July of the same year (Durmelat, \"Transmission\" 172). While there are some differences between the testimonies in the film and the book, the narratives match up and describe similar experiences. The headscarf affair is not mentioned at all in the film, and the only shots of veiled women were either filmed in Algeria, or are of immigrant women in black-and-white archival footage.\n\nIn contrast to the film, the first paragraph of Benguigui's book puts the headscarf affair in the foreground, and sees it as a key event in France's awareness of the definitive presence of a Muslim population in the Hexagon: \"Un foulard islamique apparu sur la t\u00eate de trois adolescentes d'un coll\u00e8ge de Creil, en septembre 1989, a vite sem\u00e9 l'inqui\u00e9tude. D'o\u00f9 viennent ces musulmanes? Comment sont-elles parvenues \u00e0 se faufiler au c\u0153ur des \u00e9tablissements scolaires?\" 'A Muslim scarf on the head of three teenage girls in a middle school in Creil, in September 1989, quickly spread anxiety. Where were these Muslim girls coming from? How did they manage to sneak in the heart of schools? '(7). Similar questions are asked a couple of pages later.41 The questions that Benguigui attributes to public opinion and the choice of verbs give the impression that the presence of Islam is considered illegal on French soil.\n\nFurthermore, one chapter of the book is dedicated to Na\u00efma, a young woman who challenges the stereotypes, since she decided to wear the veil under no pressure from her family. To the contrary, her father was worried by it and feared he would be expelled from the country (186). Her mother and sister are dressed European style. Benguigui is startled when she first sees Na\u00efma, her reaction is emblematic of the ambivalence and uneasiness that the scarf triggers in France, even for those who opposed the exclusion of veiled girls from school (178). While one can object to Benguigui's interpretation of Na\u00efma's choice in Christian idiom (see Durmelat \"Transmission,\" Bourget \"A l'\u00e9crit\"), Benguigui stands out in that she does give voice to a veiled girl in her book without demonizing or dismissing her choice of dress, taking a step to bridge the gap of misunderstanding by letting the silenced party speak for herself. One can only regret that she did not include Na\u00efma in the film, since, if \"photographic images are pieces of evidence in an ongoing biography or history\" (Sontag 166), the non-inclusion of veiled girls in the contemporary French landscape contributes to relegating these girls to the margins.\n\n### Conclusion\n\nBen Jelloun, Maalouf, Meddeb, and Memmi seem to belong to the \"hard-line secularists\" (Drake 181) who were the most vocal during the fifteen-year affair. In their works that focus on Islam, neither Meddeb nor Ben Jelloun alludes to the various sides of the debate regarding the issue of the veil. By only presenting the opinions of a minority regarding certain religious rulings and by silencing the French girls' motivations for veiling, they contribute to deepening the gap of misunderstanding rather than bridging it. Since both writers lament the absence of debate in the Muslim world (indeed Meddeb's main argument is that fundamentalism is caused by the lack of debate and its intolerance of diverse opinions), the least we could have expected of them would be to render the plurality of voices in the debate surrounding the veil. For two writers who pose as champions for the emancipation of Arab women, it is unfortunate that they contribute to what Bouamama has noted as one of the characteristics of the debate: the absence of the voice of the women concerned.\n\nBy waiting for the terrorist attacks of 9\/11 to try to explain Islam to the French, Ben Jelloun and Meddeb are exacerbating the main problem that Muslim French, many of whom are descendants of immigrants, face in France: they are not recognized as full-fledged and trustworthy citizens solely because of their choice to practice the religion of their ancestors. This choice is seen as incompatible with French values and subject to foreign allegiance. Thus, international news is made to bear on domestic issues even though they are not directly related to them. Not only do these essays fail to present a balanced account of the issue of the veil in France, they contribute to the confusion between Islam and terrorism by the mere fact of mentioning the affair in essays motivated by 9\/11.\n\nBen Jelloun, Maalouf, Meddeb, and Memmi fail on two counts: they do not present a balanced account of the issues surrounding the headscarf affair in France, nor do they draw attention to the use and abuse of the concept of secularism in an affair that, as G\u00e9rard Noiriel pointed out, was more a \"sous-produit des pol\u00e9miques sur l'immigration lanc\u00e9es par les conservateurs au d\u00e9but des ann\u00e9es 1980\" 'by-product of polemics about immigration launched by conservatives at the beginning of the 1980s' (181).42 According to Noiriel, the 2004 law on religious signs was a victory for the most conservative government intellectuals, who managed to reverse the relationship between victims and aggressors by transforming a social question into a religious one. The phrase \"identit\u00e9 nationale\" 'national identity' was dropped because it brought to memory extreme right-wing slogans, \"nation\" was replaced by \"R\u00e9publique\" and \"identit\u00e9\" by \"la\u00efcit\u00e9\" 'secularism.' Thus when one used to say \"Les \u00e9trangers ne s'assimilent pas et menacent notre nation, parce qu'ils sont de connivence avec l'ennemi\" 'foreigners do not assimilate and threaten our nation, because they are in connivance with the enemy,' one now says \"les Maghr\u00e9bins ne s'int\u00e8grent pas et menacent la la\u00efcit\u00e9 r\u00e9publicaine car ils sont de m\u00e8che avec les islamistes\" 'Maghrebians do not integrate and threaten republican secularism because they are hand in glove with Islamists' (Noiriel 182).\n\nWhy did these writers not underline the \"extraordinary symbolic weight given to a scarf worn on the head by a small number of schoolgirls\" (Bowen 7)? Why did they not give some recognition to the girls' effort to reconcile their Muslim and French identity as they saw fit? These are writers who have risen up to the challenge of showing the shortcomings of French society in the past, particularly regarding racism. The climate of course has changed. In a chapter of _Jihad_ entitled \"Europe, terre d'islam: le voile et la _fatwa_ ,\" Kepel puts the Islamic scarf affair in the international context of the rise of Islamism. Unlike Bowen, who, among other factors, demonstrates that the scarf affair brought to light the contradictions that already existed regarding secularism in France, Kepel takes it for granted that there is a well-established consensus about the definition of secularism, something that is debunked by two contrary developments: the 1989 ruling of the Conseil d'Etat that stated that the headscarf was not in itself a breach to secularism, and the 2004 law that forbids it. Kepel rightly underscores the role played by the international rise of Islamism, and particularly Khomeini's _fatwa_ against Salman Rushdie, and its role in the change that considered Western countries that are home to Muslim citizens as part of _dar el islam_. His analysis puts the international political situation to the foreground, and the headscarf as one manifestation of political Islam, and only that. This is mostly what was disseminated in the media. The guilt by association factor (to which I will return in the next chapter), which prompts writers to take a strong stance to distance themselves clearly from any fundamentalist association, comes into play. However, this analysis reduces Islam to its political dimension and does not take the girls' point of view into account at all. Gaspard and Khosrokhavar noted that no link had been established between the veiled girls and Islamist groups in France (54). As for link to foreign groups, they note that the FIS had issued a call asking these girls to remove their scarves, that went ignored during the second affair in 1994 (29), thereby putting into question the notion that the girls were merely puppets in the hands of (foreign) fundamentalists. This shows a gap between political analysis and sociological fieldwork, a gap that one also finds between the writers discussed in this chapter and personal accounts of veiled women.\n\nA telling statistic during the affair was that 91 percent of French teachers had never seen a veiled girl at their school (Bowen 121), and yet the teaching profession overwhelmingly supported the law. Of the writers discussed in this chapter, Benguigui might be the only one who actually met in person a girl who decided to wear a headscarf (Sebbar told me during an interview in 2006 that she did not meet veiled girls and did not want to because she already knew their discourse). Benguigui put aside the politics, both internal and external, and listened with an open mind and a sympathetic ear.\n\nWhile Sebbar went from supporting the girls to supporting the law that excludes them, Benguigui does not take a position, but her work moves from exclusion to inclusion, albeit in a different way. The works of both, however, meet in excluding pictures of covered girls. This brings me, in closing, to comment on a picture that accompanied a 2006 article in _L'Express_ on the rise of Islam in Europe (see Conan). As John Bowen noted, the weekly magazine _L'Express_ ' coverage of the scarf affair was strongly anti-veil from the very beginning (107). The picture features a girl wearing a tricolor headscarf and holding a French flag on her face, as a face veil.43 The article warns about the danger of the rise of Islam in Europe, in general. The picture was probably chosen as an illustration of the rise of Islam in France because of the two icons it contained, and which summed up the gist of the article: a veiled girl representing Islam, and the tricolor flag for the French republic. I would suggest another reading of that picture, which I can only surmise was taken during a demonstration against the law on the prohibition of religious signs in public schools. I would read the girl's gesture as showing that it is the French flag, the icon of the French Republic, which is silencing and occulting Muslim women from public space and debate.\n\nIn his book, Bowen notes that he is gratified when, after people listen to him speaking on the 2004 law, they still do not know where he stands (7). Like Bowen in his study, my aim is not to take a position, although my view has probably become obvious to my reader. If anything, the 2005 riots seemed to confirm to me that the time, resources, and energy spent on the single issue of the scarf during the past fifteen years could have been more fruitful had they been directed towards concrete and pressing, and not symbolic, issues. There were no veiled girls burning cars all over the country that fall. One consequence of the affair is that it will have, ironically, triggered the establishment of private Islamic schools that will eventually receive government subsidies, like their Jewish and Catholic counterparts.\n\n### Notes\n\n1. Incidentally, this mosque was partially funded by the French government, despite a 1905 law (much-brandished during the headscarf affair) that prohibits governmental funding of religious entities.\n\n2. In France, the incident was referred to as \"l'affaire du foulard\" 'the affair of the scarf,' with the words \"foulard,\" \"voile,\" and hijab used interchangeably to denote a cloth covering the hair and neck. The Muslim veil comes in many shapes and colors throughout the Muslim world, with specific forms and vocabulary associated with some countries (for instance the Iranian chador, the Afghan burqa). I use the terms scarf and veil interchangeably, as was done in France, to refer to a head cover only, and not something that hides the face and\/or the whole body.\n\n3. Disproportionate attention was given to the affair. Alec Hargreaves notes that in 1994, the number of girls concerned constituted less than 1 percent of the number of Muslim girls in public schools. Bouamama points out that there was no increase in these numbers in 2003, the year when the move to legislate gained momentum (70).\n\n4. For excellent analysis of the controversy, see Gaspard and Khosrokhavar (of which a short excerpt translated into English appears in _Beyond French Feminisms_ ) as well as Bouamama, the collection of essays edited by Nordmann, and Bowen.\n\n5. In his provocative study, Bouamama places the law in the context of social regression for the working class. According to him, the headscarf affair was a timely way to divide the 2003 social movements that unified the teaching profession.\n\n6. Indeed, one should not assume that a specific religious background entails better representation (see Bowen about the only Muslim member on the Stasi commission).\n\n7. For the motivations behind some girls' choice to wear the scarf, see excerpts of interviews in the books by Bouzar and Kada, Gaspard and Khosrokhavar, Giraud, Sintomer, and the L\u00e9vy sisters.\n\n8. This section has appeared as \"9\/11 and the Affair of the Muslim Headscarf in Essays by Tahar Ben Jelloun and Abdelwahab Meddeb.\" _French Cultural Studies_ 19.1 (February 2008): 71-84. It has been revised.\n\n9. A particular subset of novels that endeavors to imagine the mindset and indoctrination of a 9\/11 hijacker is the subject of chapter 5.\n\n10. These works have been translated into English (see bibliography). However, unless indicated otherwise, most translations in this chapter are mine.\n\n11. The famous editorial published in _Le Monde_ on 9\/13\/2001, \"Nous sommes tous Am\u00e9ricains\" 'we are all Americans' had so much resonance probably in part because France still had vivid memories of terrorist attacks on its soil.\n\n12. See Aubrac and Wieviorka.\n\n13. Ben Jelloun explains that \"Certains ont traduit ce mot par 'caillot de sang.' En v\u00e9rit\u00e9, il s'agit du liquide visqueux form\u00e9 par les spermatozo\u00efdes; on l'appelle 'sperme.' C'est gr\u00e2ce aux spermatozo\u00efdes que les \u00eatres humains se reproduisent\" 'some have translated this word by 'blood clot.' Actually, it's the viscous liquid formed by spermatozoa; it's called 'sperm.' Human beings multiply thanks to spermatozoa' (19). Jacques Berque's translation (among others) favor the term \"adh\u00e9rence\" 'adhesion' which renders another meaning of the word ' _alaq_ (which also translates as \"a sticky thing\") and would refer in this context to the embryo). References to semen occur in other parts of the Qur'an using a different word. Not to mention that such a statement, which completely occults the woman's biological reproductive role, is quite patriarchal for a man who otherwise poses as a champion of women's rights.\n\n14. See Mernissi's _Le harem politique_ as one example, although her approach is radically different from Ben Jelloun's. Mernissi knows that in order to be taken seriously and to have an impact, she cannot disregard summarily the body of _hadiths_ as Ben Jelloun does, which would be totally unacceptable for mainstream Islam.\n\n15. Abdelkader fought the French invasion of Algeria from 1832 till 1847, when he was forced to surrender. Abdelkrim was imprisoned in 1917 for his opposition to Spanish rule in Morocco. He escaped in 1919 and led the Moroccan resistance against Spanish and French colonial expansion from 1921 until 1926, when he was defeated.\n\n16. Ben Jelloun made it clear in a newspaper article that he fully supported the 2004 ban on religious signs in public schools (see \"Contamination\").\n\n17. These last two were published after Ben Jelloun and Meddeb's texts. For the L\u00e9vy sisters, see Giraud.\n\n18. The L\u00e9vy sisters point out that the media gave inaccurate accounts of their story in slant articles and news reports that portrayed them as intransigent fundamentalists when they were expelled from their high school in 2003 (see pages 37 and 42).\n\n19. There was a split among French feminists about the issue: none supported the veil, but several recognized that taking a hard stance and expelling the girls would definitely not be in the latter's best interests, and would not help promote equality between the sexes.\n\n20. The French government's position on the issue changed over the years, but follows a pattern of hardening public opinion. In 1989, the _Conseil d'Etat_ ruled that the veil worn by students was not incompatible with secularism. In 1994, the minister of education issued a circular, making a distinction between signs that are discrete versus ostentatious. In fact, cases were generally handled at the discretion of the schools' principals.\n\n21. The Concordat is an agreement reached between Napoleon and the Pope in 1801, according to which Catholicism is no longer the religion of the State, although the State nominates and remunerates the clergy. This agreement covers Protestantism and Judaism as well. The Concordat ended in 1905 with the law of separation of Church and State, except in the areas of Alsace-Moselle which were at the time annexed by Germany. The Concordat was left untouched in Alsace-Moselle even after they were reincorporated into French territory following WWII. This has seldom been mentioned as a danger in debates about secularism.\n\n22. Some intellectuals, such as Mohammed Arkoun, have been advocating the creation of a degree in Islamic theology to be offered at the University of Strasbourg (which already offers degrees in Catholic and Protestant theology, the only French public university accredited to do so).\n\n23. \"Lorsqu'une fois le fanatisme a gangren\u00e9 un cerveau, la maladie est presque incurable\" 'once fanaticism has gangrened a brain, the disease is almost incurable' (254)\n\n24. See for instance El-Solh and Mabro, and Ahmed.\n\n25. Memmi recalls how he used to joke that his Jewish grandmother, who would not leave the house without her _ha\u00efk_ , looked like a ghost (105-06).\n\n26. For more on that, see Bowen 89-91.\n\n27. This was the commission appointed by the president to issue recommendations in 2003, one of which was to pass a law banning religious signs in public schools. For a critique of the Stasi commission, see Bowen (for instance, the members ignored the sociological studies that had been carried out about the girls, and did not interview a single girl that had been expelled from school because of her headscarf (70, 117).\n\n28. <http:\/\/www.conseil-etat.fr\/ce\/rappor\/index_ra_cg03_01.shtml>.\n\n29. For more on that issue see Baub\u00e9rot and Haarscher.\n\n30. The compromise solutions adopted in certain cases, which modified the ways the scarf is worn (by showing the hairline, or the earlobes and neck, or wearing a bandana), demonstrate an attempt to align the head cover with practices accepted by mainstream French society (and showing more skin).\n\n31. This section has appeared as part of \"A l'\u00e9crit sans images: le foulard islamique dans des oeuvres de Le\u00efla Sebbar et Yamina Benguigui.\" _Expressions maghr\u00e9bines_ 6.1 (Summer 2007): 19-35. It has been revised and augmented.\n\n32. It is important to stress that the immigrant women's scarves never bothered anyone. According to Dounia Bouzar, \"jusqu'\u00e0 ces derni\u00e8res ann\u00e9es, les femmes en foulard qui faisaient le m\u00e9nage dans les administrations ne posaient aucun probl\u00e8me ni \u00e0 la soci\u00e9t\u00e9, ni \u00e0 la la\u00efcit\u00e9: un islam inf\u00e9rioris\u00e9 . . . est acceptable, c'est lorsqu'il revendique d'intervenir dans l'espace public en situation d'\u00e9galit\u00e9 qu'il appara\u00eet intol\u00e9rable\" 'till the last few years, women in headscarves cleaning the government's buildings never posed a problem either to society or to secularism: an inferiorized Islam . . . is acceptable, it is when it claims to intervene in public space on equal footing that it appears unbearable' (\"Fran\u00e7aises\" 61).\n\n33. For a detailed analysis of the presence and absence of Muslim icons in the images of these works, see my article \"A l'\u00e9crit sans images: le foulard islamique dans des \u0153uvres de Le\u00efla Sebbar et Yamina Benguigui.\"\n\n34. It is important to remember that as a sign of modesty the headscarf goes against the current, at a time when images of naked women's bodies are commonplace. Georgette Hamonou, a teacher who opposed the exclusion of two girls from her middle school, states: \"Je souris lorsque j'entends Mme Badinter s'insurger contre le foulard qu'elle distingue \u00e0 peine de la burka afghane, et qui ne dit mot de la fa\u00e7on dont on galvaude le corps f\u00e9minin, cent fois plus qu'il y a trente ans, au bon temps du f\u00e9minisme!\" 'I smile when I hear Mrs. Badinter rising up against the headscarf which she barely distinguishes from the Afghan burka, and who does not say a word about the way the female body is debased, and this a hundred times more than was the case thirty years ago, at the good old time of feminism!' (Bouamama 141)\n\n35. The copy of a page of a textbook entitled _Histoire de France et d'Alg\u00e9rie_ shows the history of both countries presented side by side: a lesson on one side on Charlemagne for French history, on the kingdom of Tiaret for Algerian history (Sebbar, _Mes Alg\u00e9ries_ 41). This questions the commonplace idea of the Eurocentrism of the curriculum in colonial schools evoked by the phrase \"nos anc\u00eatres les Gaulois\" 'our ancestors the Gauls,' so often cited ironically by writers and scholars in Francophone studies.\n\n36. The text she refers to here is the letter signed by intellectuals and published in _Le Nouvel Observateur_ on November 2, 1989, protesting the stand against exclusion taken by the Minister of Education in front of the National Assembly. For a close analysis of that letter, see Gaspard and Khosrokhavar (22-24).\n\n37. Emmanuel Terray applies Bibo's concept of political hysteria to the debate of the headscarf in France. Political hysteria consists in substituting to a real problem \"un probl\u00e8me fictif, imaginaire, construit de telle sorte qu'il puisse \u00eatre trait\u00e9 avec les seules ressources du discours et par le seul maniement des symboles\" 'a fictitious, imaginary problem, framed in such a way that it can be dealt with the sole resources of discourse and symbols' (103). According to him, the fact that many people changed their minds during the course of the affair, mostly in favor of the law, shows that this case of political hysteria was \"highly contagious,\" and not a sign of an open debate (117).\n\n38. See Laronche and the following articles in _Le Monde_ : \"Dix fois moins de foulards\" (published in the September 4, 2004 issue) and \"Premi\u00e8res exclusions scolaires apr\u00e8s la loi sur le voile\" (October 21, 2004).\n\n39. Haarscher (32) and Baub\u00e9rot (120) both point this out by means of rhetorical questions.\n\n40. Sebbar explained to me during the interview that the Islamisation of the suburbs had triggered her change of mind on the issue. Her reversal is far from unique. She justified that the absence of veiled girls was due to the fact that her books were about people she had met personally, and that she would not try to meet with veiled girls because she already knew their discourse. However, the two books I discuss here include reproductions that do not fit the criteria just mentioned (for instance, the picture of an anonymous man\u2014maybe a Chibani\u2014in _Journal_ page 90). The assumption that Sebbar already knows these girls' discourse is symptomatic of what was propagated in the media. Sebbar has featured veiled girls in her fiction (see, for instance, the short stories \"Vierge folle, vierge sage\" and \"La fille au hijeb\"), but here I am focusing on non-fictional texts.\n\n41. \"D'o\u00f9 viennent les foulards islamiques de Creil? Comment ont-ils p\u00e9n\u00e9tr\u00e9 dans les \u00e9coles fran\u00e7aises?\" (9).\n\n42. See also Bourdieu, Bouamama, Gaspard and Khosrokhavar. One of the L\u00e9vy sisters recalls how the agreement their father had reached with their high school principal to let them wear their scarves was an exception: \"Un jour, je discutais dans le couloir avec une fille qui portait un foulard laissant appara\u00eetre la racine des cheveux, les oreilles et le cou, quand le proviseur est arriv\u00e9 vers nous et l'a somm\u00e9e de le retirer \u00e0 l'instant m\u00eame. . . . La fille \u00e9tait d\u00e9go\u00fbt\u00e9e, et plusieurs comme elle, des Fran\u00e7aises d'origine maghr\u00e9bine, sont all\u00e9es se plaindre et demander des explications. Le proviseur ou des professeurs, je ne me rappelle plus, leur ont donn\u00e9 deux justifications: la premi\u00e8re \u00e9tait que mon p\u00e8re \u00e9tait juif et la seconde, qu'il \u00e9tait avocat! On cumulait \u00e0 la discrimination religieuse une discrimination sociale et professionnelle!\" 'One day, I was talking in the hallway with a girl who was wearing a scarf showing her hairline, her ears, and her neck, when the principal arrived and demanded that she take it off right away. . . . The girl was disgusted, and several like her, French girls of Maghrebian origin, went to complain and ask for explanations. The principal or some teachers, I can't remember, gave them two justifications: the first one was that my father was Jewish and the second that he was a lawyer! Social and occupational discrimination were added to religious discrimination!' (27)\n\n43. I am grateful to Siobh\u00e1n Shilton for showing that picture during her talk on visual arts works and for helping me locate the picture, and to the <http:\/\/theatrumbelli.hautetfort.com> website owners for giving me the reference of the article in which it appeared.\n\n## Chapter 6\n\n## Portrait of a Terrorist: Slimane Bena\u00efssa and Salim Bachi's 9\/11 Novels\n\n9\/11 has already given rise to a genre of its own: the 9\/11 fiction. Kristiaan Versluys rightly notes that three of the most interesting novels dealing with 9\/11 were written by outsiders, and specifically Europeans, whose link with 9\/11 is \"more tangential than tangible\" (69). But he does not mention Francophone Arab writers, who have directly engaged this event in essays (such as the works of Ben Jelloun and Meddeb discussed in the previous chapter) and works of fiction. Two of them, Slimane Bena\u00efssa in _La derni\u00e8re nuit d'un damn\u00e9_ (2003) and Salim Bachi in _Tuez-les tous_ (2006), have made it the sole focus of a novel. Contrary to the 9\/11 novels written by American and French writers,1 these Francophone Arab writers have endeavored to portray the time leading up to the attacks, rather than their aftermath, and from the point of view of a hijacker instead of a victim's or a witness's. They are the only authors, to the best of my knowledge, to have done so in a novel.2\n\nWhile providing a very different reading experience, these two novels have several resemblances: they are both creative attempts to imagine the indoctrination and thoughts of a 9\/11 hijacker just before the attacks, and they both feature extensive quotations from the Qur'an, which are clearly marked and identified in Bena\u00efssa's narrative but not in Bachi's. Both authors are Algerians exiled from their country in the 1990s because of the civil war that pitted Islamists against the army. However, neither pretends to offer a realistic reconstitution: elements in both novels depart from what we know about the 9\/11 hijackers.3 The most salient of these divergences are the following: Bena\u00efssa's protagonist is Arab-American, and he does not board the plane; Bachi's protagonist is an Algerian who does go through with the attack but knows that his action will not earn him Paradise.\n\nIn this chapter, I sketch the various interpretations that have been given as explanations for 9\/11, in order to examine where one can situate these novels in this debate. I analyze and compare the function of the intertext in both novels, with particular attention to religious references, and its effect on the representation of the role played by Islam in terrorism.\n\n### 9\/11, Islam, and Terrorism\n\n9\/11 has been a watershed event in many aspects, one of them being to bring religion back on the Humanities' radar screen. Scholars in the _PMLA_4 issue of May 2005 noted that literary scholars had not taken notice that colleagues in the social sciences had been \"debunking the illusion of the secularization thesis for some time\" (Brown 884), and seemed to discover their \"own intellectual impotence, [their] insistent refusal to recognize religion as part of the _current_ (rather than past or primitive) state of humanity\" (Chow 875).\n\n9\/11 has also given new currency to the theory of the clash of civilizations, and thus prompted numerous inquiries into the link between terrorism and Islam. The most prominent scholars who champion that theory, Bernard Lewis and Samuel Huntington, assert that Islam is by its very essence hostile to Western values (Qureshi and Sells 2-9). According to them, this antagonism rather than specific historical processes explains the current conflict between the West and Islam (a view also held, incidentally, by Islamic extremists). They question the Left's tendency to explain Islamic fundamentalism as a religious cloak for economic and political grievances.5 This theory enjoys broad appeal, even with some postcolonial writers (such as Naipaul,6 who has taken on and propagated that view as well, and Soyinka7). This view shows little recognition of the fact that had colonial policies been different, there might have been less sense of urgency by such newly independent states as Morocco, Algeria, and Tunisia to affirm Islam as the state religion. As underlined by Qureshi and Sells,\n\nIf such identities, in the words of Edward Said, are constructed and inherently conflictual, then the claim of a clash of civilization is true, in the philosophically trivial sense of tautology: generalized identities that have been constructed in opposition to one another [as was the case in colonial contexts] are in opposition to one another. Said also points out that colonization justified by conflictually constructed ideologies generated reactive, conflictually constructed politics of identity among the colonized. (27)\n\nAs Stephen Morton noted, the equation of terrorism with Islam \"is precisely a form of orientalism, which obfuscates the political dimension of resistance against western imperialism\" (40). Scholars in fields ranging from philosophy (Baudrillard, Derrida) to political science have debunked the theory of the clash of civilizations. Jean Baudrillard's _The Spirit of Terrorism_ and Slavoj \u017di\u017eek's _Welcome to the Desert of the Real_ , both essays on 9\/11, dedicate considerable space to emphasizing the importance of socio-political factors. \u017di\u017eek, recalling that Bin Laden was funded by the CIA, stresses that the USA is fighting \"its own excess\" (27).8\n\nWhile it has become commonplace in the media and (popular) culture to explain suicide attacks through reference to Islamic fundamentalism, membership in Islam alone is not a prerequisite, since the vast majority of Muslims do not engage in such acts. Indeed, in a talk with the provocative title \"Radical Sheik and the Missing Martyrs: Why Are There So Few Islamic Terrorists?\" Charles Kurzman debunked the idea that terrorism is intrinsic to Islamic fundamentalism by addressing the question from the following angle: \"If a considerable proportion of the world's billion Muslims believes in armed jihad and the joys of martyrdom, why don't we see terrorist attacks around the world every day?\"\n\nAn alternative has emerged to both those who assert that religion serves only as a means to an end to redress socio-eco-political grievances and those who see it as an end in itself. In the field of International Relations, Scott Thomas argues that religion makes a difference because \"religious terrorism seeks a cosmic or transcendent justification rather than only political, social or economic objectives\" (147).\n\nA close look at the history of suicide terrorism and ethnographic studies on the issue yields a more complex picture. Several studies on terrorism have demonstrated that suicide bombings are not the exclusive resort of Islamic groups: they have also been used by secular groups. The first wave of suicide missions in the twentieth century was conducted by Russian anarchists, and was followed by the second wave of Japanese Kamikaze9 during World War II (Gambetta 285). The Tamil Tigers is a non-Muslim group that commits suicide missions. In _Making Sense of Suicide Missions_ , scholars of political science show that suicide missions inspired by an Islamic rhetoric account for about a third of the suicide missions that took place between 1981 and 2003 (Gambetta 261). Indeed, their studies demonstrate that Islam is \"neither a necessary nor a sufficient condition for SMs\" (Kalyvas and S\u00e1nchez-Cuenca 216), since non-Muslim groups have resorted to it, while many Islamist groups who have resorted to violence (as in Algeria for instance) have not.\n\nSimilarly, Christoph Reuter shows that various explanations offered in the media do not stand scrutiny; he makes a strong case for looking at the specifics of each context (9-13). He states that \"Islam as such is not the cause of terrorism and suicide attacks. Particular aspects of Islam do, however, lend themselves to being interpreted to justify a declaration of outright war against the West and against any opponents among their own peoples. They can equally be used to construct a democratic society\" (Reuter 32).\n\nHowever, religion can play a role in fostering and creating a receptive environment for terrorism, notably by promoting a cult of martyrdom. Stephen Holmes demonstrates that the grievances articulated by Atta and Bin Laden are almost entirely secular. Gambetta answers the question of how we can decide which of the religious or non-religious motives was predominant in the 9\/11 attacks by concluding that they are intertwined: \"While the real sources of the conflicts in which SMs [suicide missions] emerge are ultimately social and political, and secular groups can resort to these attacks too, those organizations that can bank on the 'right' religious beliefs can more easily summon the energy to enter into or . . . to continue, fighting against all odds\" (293, see also Pedahzur).10 This intertwining has been pointed out by an earlier study on the secularization theory that examined the revival of religions in countries ranging from Iran to the U.S. It concludes that religions \"were not the principal causes of the radicalization of the conflicts, but rather conflict radicalization found symbolic resources in religion to further certain social and political causes\" (Ricolfi 112). Holmes notes that Atta's grievances against the U.S. and the Muslim regimes it supports were \"almost entirely secular\" (139), as are Al-Qaeda's main objectives (164), and Bin Laden's (166); he concludes that \"such a blurring of personal frustration, political protest, and religious convictions makes it very difficult, if not impossible, to demonstrate the specifically _religious_ roots of Atta's commitment to jihadist violence\" (140). It is therefore clear that grievances grounded in the here and now (specifically the presence and\/or occupation of Arab countries by American soldiers and Israel) are necessary to spur Islamic terrorism.\n\n### Slimane Bena\u00efssa's _La derni\u00e8re nuit d'un damn\u00e9_\n\nBena\u00efssa has been primarily a playwright11 who had already explored the issue of violence and religion on the stage. His novel _La derni\u00e8re nuit d'un damn\u00e9_ is told in the first person by Raouf, an Arab American software engineer whose father is Egyptian and mother Lebanese. His father's death less than a year prior to the beginning of the story is a key event that marks the beginning of Raouf's questioning about life (20). His coworker Athmane, a Palestinian trained in France, introduces him to Djamel, a rich Kuweiti who grew up in London, led a life of debauchery before repenting, and broke ties with his family because of the Gulf War. After three months of Islamic theology lessons with Djamel, Raouf converts to Islam, and is recruited as a martyr for a mission. We see Raouf abandoning his ordinary life (starting with his separation from Jenny, his long-time girlfriend), undergoing his training, and planning for the mission. On the morning of the attack, Raouf decides to not take the pills he was given, and at the last minute does not board the plane. He is arrested shortly thereafter. The novel ends with a letter written by his mother on her deathbed, which Raouf reads in prison.\n\nIn his foreword, Bena\u00efssa aligns himself with Victor Hugo's and Alexandre Solzhenitsyn's writings. He explains that Hugo's _Le dernier jour d'un condamn\u00e9_ and Solzhenitsyn's _Une journ\u00e9e d'Ivan Denissovitch_ can be credited for being catalysts for change (9). Bena\u00efssa feels compelled to follow in their footsteps for two reasons: because he is a Muslim and because of his \"histoire personnelle face \u00e0 l'int\u00e9grisme\" 'personal experience with religious extremism' (9, Gross vii), which alludes to the death threats that forced Bena\u00efssa to leave Algeria in 1993 (as indicated in the short biographical notice on the back cover) during the civil war. The title _La derni\u00e8re nuit d'un damn\u00e9_ modifies the title of Hugo's text, a plea against capital punishment, to reflect a different situation: Bena\u00efssa refutes the doctrine that a suicide attack will earn his character a place in paradise. In a striking move, he apologizes to the families of victims of fundamentalism: _\"en tant que musulman, je demande pardon \u00e0 toutes les famillesdes victimes de l'int\u00e9grisme international, quelle que soit leur confession\" 'speaking as a Muslim, I ask for forgiveness from all of the families who have been victims of religious extremism across the globe, regardless of their faith'_ (11, Gross ix). This statement is set off typographically: it is a paragraph in itself and is the only passage that is italicized. Bena\u00efssa's reaction emblematizes what Arab and\/or Muslim writers and intellectuals have been feeling in the last decades, and more intensely since 9\/11. Said sadly recalled in 1996 that after the Oklahoma City bombing his office received numerous phone calls from journalists,\n\nall of them acting on the assumption that since I was from and had written about the Middle East that I must know something more than most other people. The entirely factitious connection between Arabs, Muslims, and terrorism was never more forcefully made evident to me, the sense of guilty involvement which, despite myself, I was made to feel struck me as precisely the feeling I was meant to have. The media had assaulted me, in short, and Islam\u2014or rather my connection with Islam\u2014was the cause. ( _Covering_ xiv)\n\nJust as Said, it seems that Bena\u00efssa felt guilty from the loose association, all the more prevalent since 9\/11, between terrorism and Arabs and Muslims, possibly made all the more acute by the criticism levied both in France and in the U.S. that Muslims had not spoken out forcefully enough against the attacks. Unfortunately, Bena\u00efssa's apology has the (most probably unintended) consequence of validating the perception that Muslims are guilty by association.\n\nIn addition to asking for forgiveness, Bena\u00efssa apologizes for \"la duret\u00e9 de certains propos dans ce roman. Car tout au long de cette \u00e9criture, je me suis demand\u00e9 si je devais dire ce que je crois \u00eatre l'\u00e9clairage d'une v\u00e9rit\u00e9 pr\u00e9cise ou m\u00e9nager la douleur des familles des victimes\" 'some of the harsh sections [remarks] in this novel. Throughout the entire time of writing, I wondered whether I should say what I felt to be the most illuminating truth, or take into consideration the pain it might bring to the victims' families' (11, Gross ix). By this I assume he means the criticism of the West, and Unites States foreign policy, which is put in the mouth of the terrorists, and which actually takes up very little space in the novel. Bena\u00efssa appears to be writing defensively, in an immediate post-9\/11 climate of binary rhetoric that left no place between \"us\" and \"them\" and for critical discourse. As Judith Butler points out, \"the binarism that Bush proposes in which only two positions are possible\u2014'Either you're with us or you're with the terrorists'\u2014makes it untenable to hold a position in which one opposes both and queries the terms in which the opposition is framed\" (2). Any criticism of U.S. foreign policy can be easily dismissed by a chauvinistic American reader on the assumption that it is uttered by the terrorists or their sympathizers, although it is grounded in various political claims that have widespread support in the Arab world (such as the Palestinian cause), and are recognized as legitimate by scholars worldwide.\n\nAthmane and Raouf have disparate motivations for becoming martyrs. Athmane is the most politicized of the two: through his mouth comes much of the criticism of the West, and particularly of America's injustices in the world. Thus, while Raouf becomes indignant about the political dimension of the sermon of the clandestine imam, Athmane insists that since the Crusades, the West has had imperial designs on the East (135). The first planning meeting of the future hijackers is inaugurated by a speech that first presents grievances against the U.S. and the West: non-recognition of the contributions of Arab-Muslim civilization to Western civilization and wars of Western aggression that are plaguing the Muslim world; this political situation is given as the reason why they should strike the West to humiliate it in order to restore God's honor, thus mixing a political with a religious motive in a murky way (192-95). Athmane and the rest of the hijackers use Islam to address political issues of dispossession.\n\nAs for Raouf's motivation for participating, he mentions that he hopes to atone for his father's sins, while making Americans pay for refusing to acknowledge that his father's death was a work accident (190-91). Raouf has been shaken by his father's death, the mention of which recurs at key moments throughout the narrative to signal it as the event that marks Raouf's vulnerability to fundamentalist discourse (see pages 20, 93, 103, 110, 156, 165, 188, 217, 233). The insistence on a psychological motif has been explained as such by Butler: \"That kind of story [what Mohammed Atta's family life was like, Bin Laden's break with his family] is interesting to a degree, because it suggests that there is a personal pathology at work\" (5). Butler argues that this kind of story, with its emphasis on personal responsibility and agency, is easier for us to accept than a narrative with a global political and historical framework (5-8). Bena\u00efssa's depiction of Raouf's indoctrination follows the brainwashing methods that psychologists have reconstructed based on their experience with Western sects (Reuter 8). However, Reuter points out that these \"individual psychological models of interpretation . . . can't function as the complete explanation\" (9).\n\nRaouf obviously knows little about Islam. During some of his conversations with Athmane about religious issues, Raouf perceives that something is wrong with Athmane's arguments, though he does not have the tools to contradict him. The first such conversation occurs at the very beginning of the novel, during which Athmane uses the story of Abraham, Sarah, Hagar, Isaac, and Ishma\u00ebl to argue that Judaism exists thanks to Islam and not the other way around (28-29). In a rather unpersuasive section at the end, Raouf explains that, unconvinced by the guide's argument, he stopped taking the pills he was prescribed to help him go through with the attack, and did not board the plane.\n\nAccording to the translators' statement preceding the English version, Bena\u00efssa turned to the novel to treat this subject matter because fiction would allow him greater ease in introducing complex matters: \"We acknowledge his [Bena\u00efssa's] courage in turning to the novel as a way of introducing complex perspectives on Islam to a broad reading public, both Muslim and non-Muslim\" (Gross, \"Translators' Notes\" xiv). The translators of the English edition have noted that \"to demonstrate how Islamic teachings were used to shape the thinking of the main character, Slimane Bena\u00efssa's novel makes use of extensive quotations from three types of religious writing\" (Gross, \"Translators' Notes\" xiii). The reader is informed in a \"note de l'\u00e9diteur\" 'Publisher's Note' of the convention that was added to make the religious intertext easily identifiable (with different typographical conventions to distinguish the Qur'anic quotes from the Hadiths (sayings of the Prophet) and supplications, as well as a glossary at the end of the novel. There is a recurrent opposition between \"l'ici-bas\" 'Here and Now' and \"l'au-del\u00e0\" 'Hereafter,' the only words that are in bold font throughout the novel (in the original, not maintained in the English translation). The bold font is meant to attract the reader's attention to the opposition between the two notions (which indeed are part of monotheist faiths); they stress that worldly life is only a test and will determine one's fate in the afterlife (see for instance page 37). Despite these efforts to make the novel more legible to a readership that might not know much about Islam, nothing in the novel points out the different status of these religious writings in Islamic theology. Contrary to the translators' statement, my analysis demonstrates that Bena\u00efssa's text falls short of presenting the complexity of Islam. In fact, I argue that his novel not only fails to counter stereotypes, it in fact perpetuates them.\n\nBena\u00efssa's intertext is overwhelmingly Islamic. Most of the Qur'anic quotes are interspersed in sermons that each occupy several pages, and contribute to making this novel a rather tedious reading. The sermons at the beginning of the novel treat general theological issues framed by Islamic beliefs. The first sermon deals with the reconciliation of predestination and free will (32-41); the verses quoted insist on God's omnipotence and omniscience, and recall that man accepted the gift of free will from God; it also contains injunctions not to get distracted by worldly riches. The second sermon (a recording that Raouf listens to) talks about worldly inequities and about God's supreme authority to judge (54-58). During Raouf's repentance ceremony, the verses quoted emphasize God's mercy (98-103). One of the sermons he goes to with Athmane is on the topic of \"parole, acte et action\" (117-24) 'speech, act, and action' (99), the verses insist that God will put the believers to the test, and exhorts them to be patient and steadfast in their faith.\n\nOnce Raouf becomes involved with the terrorists in training, the sermons become more political and deal with specific worldly topics. These sermons are preached in an underground network, as is underlined by the precautions that are taken when the terrorist plotters go to listen to a clandestine imam. Tellingly, that sermon starts with a verse (24:31) specifying women to whom they may show themselves without a head cover, and goes on to making a parallel with Judaism and Christianity since all three religions place restrictions on women (130). This choice of verse is not innocent: it underlines that Muslim fundamentalists in different countries have made the implementation of restrictions on women one of their first priorities, be it in Iran after the Islamic revolution, Afghanistan with the Taliban, or Algeria during the 1990s' civil war, but does not point out that men are also subjected to restrictions (albeit lesser ones) in the preceding verse. The verses that are quoted during that sermon (130-35) are very general but are inserted as if to confirm the main development, which concentrates on the disparity between the Western and the Muslim worlds, blames colonialism, accuses Jews and Christians of having teamed up against Islam, and scolds Arab leaders for their corruption.\n\nThe first sermon during the training is a long one on death, with verses focusing on death in different contexts (167-74). Its purpose is to stress the centrality of death to the believer. This sermon is followed almost immediately by one on purity. It only contains two brief quotations from the Qur'an (one stating that God loves those who purify themselves, the second that God does not change a people until they change themselves). There is a logical slippage from the verse about purification to its interpretation, according to which the acculturation that came with being educated or trained in the West, as were the trainees, is considered an element of impurity (175-79). It also includes advice on how to behave in the airport. The next sermon informs the participants that a fatwa has recognized their action as part of jihad; it is peppered with Qur'anic quotes and Hadiths on death in combat for the cause of God, the privileged status of martyrs, and a promise that God will not burden anyone beyond endurance (181-87). The next sermon treats angels and includes several Qur'anic verses (210-13). After emphasizing that the angels stand out by their unconditional obedience to God, the sermon concludes with a syllogistic argument that the one who decides to become a martyr will be equal to angels in his obedience to God (212). Although the verses' meanings are general, they are brought to bear on specific issues by being interspersed in the sermon to lend credence to various statements.\n\nIn addition to the sermons, the Islamic intertext appears in other occasions. The future hijackers' wills (214-18) contain general verses that insist on God's testing of believers, and on Islam as the last revealed religion. A recurrent verse associated with death emphasizes that mankind belongs to God and will return to him (2:156). The pledge of allegiance to God recited by the new recruits contains some short suras (or chapters) from the Qur'an (sura 112 on God's unicity, a truncated part of verse 5:106, and the Fatiha, on pages 209-10). When informing Raouf that he will volunteer to become a martyr, Athmane quotes a verse that stresses faith to give credence to his argument that people who have been educated in Western universities need to fight for the cause (126). An example of a Hadith is given by Athmane in the context of his argument that Raouf must get rid of his dog (16). The recommendations, clearly inspired by the \"road map\" that was found by the FBI and translated and published in various newspapers, including _Le Monde_ , are sprinkled with supplications praising and trusting God and a few verses about God granting victory to the believers.\n\nA few passages show the distortion effected by fundamentalist discourse: for instance, when Athmane argues about Judaism existing thanks to Islam, or during a sermon about America being discovered by accident, by a man who got lost, and forecasting its perdition (177). In the last speech, a parallel is established between the five phases of the terrorist action, the five daily prayers, and the five pillars of Islam, as if to better inscribe the terrorist act in the rituals required of the faithful (219-21). However, this mechanism is not made at all obvious when it comes to Qur'anic quotes, which are often taken out of context. The subtle shift between the sermons that Raouf hears at the mosque and the ones in the terrorist network might not be noticed by readers who have never read the Qur'an. The Qur'anic quotes add a touch of authenticity to the fundamentalist discourse, but do not prompt a reflection on the faith or on alternate interpretations. When Athmane urges Raouf to break up with his girlfriend on the grounds that she is not a Muslim, Raouf wonders if there is an exception for the \"gens du Livre\" (142) 'people of the Book,'12 which indeed there is, although nothing in the narrative hints at it except for Raouf's question, which remains unanswered since the terrorists consider no counter discourse to the fundamentalists' interpretation. And most importantly, there is no commentary whatsoever about practices not sanctioned by contemporary Islam, such as the fact that Djamel has slaves, or that an imam has the authority to recognize Raouf's repentance.\n\nIn the foreword, Bena\u00efssa laments that \"il y a certainement des milliers de fa\u00e7ons de vivre l'islam. Malheureusement, aujourd'hui, il n'en appara\u00eet que deux: celle d'un islam qui s'exprime et qui utilise toutes sortes de violences, et celle d'un islam qui se tait et dont le silence est plus qu'une absence\" 'there are certainly thousands of ways to live as a Muslim. Unfortunately, in today's world there seem to be only two: one is an Islam that is expressed through all manner of violence, the other is an Islam that is silent and whose silence amounts to more than just an absence of words' (10, Gross viii). Indeed, these are the only two Islams that are featured in his book: that which uses the religion to justify terrorist attacks, and that of Raouf's parents (and more specifically of his mother), which is actually hidden and nonexistent, as exemplified by the scene when his mother refuses to bury his father according to Islamic rites. The novel ends with the mother's letter, which strongly advocates Western modernity as a solution to the Arab world's problems, a modernity where Islam does not seem to have much place. And yet, there is much evidence to the contrary throughout the Muslim world, as seen in the opinions of Muslims for whom Islam and modernity are not incompatible and who have spoken strongly against their faith being used as a justification for 9\/11.13\n\nThe intertext seems to reinforce the idea of a clash of civilizations. Most of it is Islamic, with the exception of the mention of the film _Fureur de vivre_ when describing Djamel's dissolute life style (73) and of the _Cantique des cantiques_ as Jenny's reference (51), with an excerpt quoted (52). The intertext generally reinforces the dichotomy that is posited between East and West, Christianity and Islam. Most of the Qur'anic quotes in Bena\u00efssa's novel function only to inscribe the hijackers' act in the Islamic faith. There are no Qur'anic quotations when Raouf recalls how he did not go through with the action; in fact, at that point in the story, he cannot recall any of the verses beyond the initial formula \"in the name of God\" that starts all the suras. The author could have quoted some well-known verses that condemn the killing of innocents (Bachi does so, as we shall see later). Similarly, his mother's letter, written on her death bed, does not contain a single Qur'anic quote. If her way of life is another path for Islam, then it is one that has eliminated all religious reference. The text's objections to the terrorist act are not grounded in the Islamic faith. The political grievances behind 9\/11 are given very minimal space, whereas the sermons and Islamic material predominate in the narrative, thereby confirming the stereotype that there is something intrinsic to Islam that spurs terrorism, or at least, that devout believers are likely to be fanatical.\n\n### The Multicultural Intertext of Salim Bachi's _Tuez-les tous_\n\nTold in the third person by an omniscient narrator, Bachi's novel presents the last hours of one of the pilots up to the moment when he crashed a plane into the World Trade Center. The protagonist, whose real name is never disclosed, is an Algerian who went on to France to pursue his education, and fell in love with a French woman, who aborted their child. This abortion is the symbol of his failed integration into French society and one of the key reasons why he joined a terrorist organization which he found via a Parisian mosque and a trip to Afghanistan, where he was given the name Seyf el Islam. Although the character tries a few times to back out of his mission (he tries to drown himself, to turn himself in to the police), he is saved every time, and the ineluctability of the tragedy of 9\/11 looms constantly over the narrative through prolepses.\n\nContrary to Bena\u00efssa, Bachi has no qualms when it comes to writing pointed criticism about the West, and more specifically the U.S. and France. The phrase \"Paris, ville lumi\u00e8res \u00e9teintes\" 'Paris, city of lights out' recurs like a leitmotiv through the course of the narration, and emphasizes the failure of the Enlightenment. Strident passages and allusions strongly chastise American politics, both domestic and foreign, indicting the CIA, the plight of the Blacks and the poor (20). In general, the Western world is accused of being responsible for numerous conflicts in various geographical areas. It is designated as \"les adorateurs du veau d'or noir\" 'the worshippers of the black golden calf (21) adapting a Biblical (and Qur'anic) reference to the oil-dependent economies of the twentieth century. Several prolepses function as indirect criticism of the attention granted to 9\/11 by the media: \"tous verraient que leurs victimes [celles de l'Occident] avaient plus de poids que celles du monde entier\" 'all would see that their victims carried more weight that those of the entire world' (17). This sentence refers to the U.S.'s \"reputation as a militaristic power with no respect for lives outside of the First World\" (Butler 17). There is also an ironic allusion to _Le Monde'_ s article entitled \"nous sommes tous am\u00e9ricains\" 'we are all Americans' (20), which was published two days after the attacks.14\n\nThe title of the book _Tuez-les tous_ alludes to one of the darkest pages in French Medieval history: the Albigensian Crusade in the thirteenth century, indirectly recalling that Christianity has had its share of religious fanaticism. Arnaud Amaury (Arnold Amalric), abbot of C\u00eeteaux, before the massacre at B\u00e9ziers in 1209, when asked how to distinguish the Cathars, considered heretics, from the other inhabitants of the town, infamously answered: \"Tuez-les tous, Dieu reconna\u00eetra les siens.\" (Roquebert 245-61).15 Bachi's title significantly omits the second part, which is a reference to the Bible: \"The Lord knows those who are his\" (2 Timothy 2:19). According to Michel Roquebert, the crusaders massacred the inhabitants of B\u00e9ziers, including women and children, \"parce que les crois\u00e9s voulaient faire un exemple, propre \u00e0 semer la terreur\" 'because the crusaders wanted to make an example that would spread terror' (263). This event can thus be seen as an example of pre-modern terrorism in the name of religion. In modern times, that phrase (kill them all, let God sort them out) was used by American military during the Vietnam War. The title thus bridges French medieval and twentieth-century American histories, reflecting the recurrent criticism of both countries in the novel. The title is also used in Hitchcock's film _The Birds_ , one of Bachi's intertextual references, when one of the characters argues for the destruction of the whole species following attacks by birds in the small town.\n\nThe terrorist \"Organisation\" sent Seyf to spend a month in Granada before flying him to Portland, Maine to accomplish his mission. His stay in Granada is an important detail that gives rise to several remarks by the character about the coinciding dates of the fall of the last Muslim ruler in Spain with the date that marks Columbus' journey to America. Granada was the last bastion of Muslim Spain, or al-Andalus, to fall. As Majid emphasizes, al-Andalus was characterized for the mutual tolerance between the three monotheistic faiths ( _Freedom_ 23), although he cautions that \"the myth of a golden era of multicultural relations or of Islamic tolerance should not lead us into believing that there were no tensions among religious communities\" ( _Freedom_ 24). For Seyf el Islam, the new world is \"cette nouvelle saloperie, extension de l'Europe des lumi\u00e8res \u00e9teintes\" (55). His journey seems to mirror Columbus' journey, since he spends time in Cadix before going to Portland, and takes on a Spanish name, San Juan, as a pseudonym to accomplish his mission. The name San Juan evokes the town founded in 1521 by Spanish colonists in Puerto Rico, the oldest European settlement in the United States territory. The historian Fernand Braudel singles out 1492 as the date that marks the beginning of modern European colonialism. Majid states that \"the defeat of Islam and the 'conquest' of America in 1492 were interrelated events that culminated a long process of increasingly strident Christian missionary wars against Islam in Spain\" ( _Freedom_ 23). By calling attention to Columbus' journey, Seyf el Islam seems to incarnate the revenge of the West's Other by mimicking the path of the beginning of colonialism. This allusion to Columbus, going back in history well before our current era, shows that globalization started well before our century.\n\nAs in Bena\u00efssa's novel, the Qur'an is the salient intertext (although the quotes are unidentified), but it is more than a device to legitimize fundamentalist discourse. The quotations and allusions to the Muslim scriptures fall into several categories. First, the Qur'an is evoked to stress the discrepancy between what the religious text advocates and the reality on the ground, particularly when it comes to women. Verse 2:187, which describes the (sexual) relations between husband and wife (\"elles sont un v\u00eatement pour vous, vous \u00eates, pour elles, un v\u00eatement\" 'They are your garments \/ And ye are their garments') is followed by the observation that Seyf El Islam had not seen anything like this in the world, and stresses that instead he has seen \"des femmes d\u00e9nud\u00e9es offertes \u00e0 l'assouvissement des b\u00eates\" 'women stripped bare and offered to gratify the lusts of beasts' (31). This verse is repeated twice more (68, 85), again to emphasize the distance between what the verse evokes and his experience. This verse is again alluded to when hijackers are faulted for being \"incapables d'\u00eatre un v\u00eatement pour une quelconque femme, mais pr\u00eats \u00e0 piloter un Boeing pour faire sauter la plan\u00e8te\" 'incapable of being a garment to any woman, but ready to pilot a Boeing to blow up the planet' (84). The Qur'an is also evoked when the character reflects about the gap between the world without women of Kandahar and what the Holy book says (61).16 Likewise, two quotations (78 and 123) compare the Islamic terrorists he joins to the hypocrites who are persistent opponents of the Prophet in the Qur'an (verse 4:63 and 9:56-7).17\n\nIn the same vein as the first category, the second use of the Muslim scriptures features verses that are quoted to highlight Seyf's straying from Islam and its precepts, and most importantly his awareness of doing so. For instance, he remembers verse 2:154 (\"Ne dites pas de ceux qui sont tu\u00e9s dans le Chemin de Dieu: 'ils sont morts!' Non! Ils sont vivants\" 'Say not of those who are slain in the way of God: 'They are dead.' Nay, they are alive'), to better reflect that he himself \"avait conscience d'\u00eatre mort et de ne plus \u00eatre sur le chemin de Dieu\" 'was aware of being dead and no longer on God's path' (23, again on pages 108, 112, 132, with slight modifications). When alluding to verse 5:35 (\"Celui qui a tu\u00e9 un homme qui lui-m\u00eame n'a pas tu\u00e9, ou qui n'a pas commis de violence sur la terre, est consid\u00e9r\u00e9 comme s'il avait tu\u00e9 tous les hommes\" 'if anyone slew a person\u2014unless it be for murder or for spreading mischief in the land\u2014it would be as if he slew the whole people'), he adds that \"je vais tuer des innocents, sortir de la communaut\u00e9 des hommes\" 'I'm going to kill innocents, leave the community of human beings' (25). Verse 5:35 is repeated during a dialogue between Seyf and Khalid in Kandahar in the presence of \"le Saoudien\" 'the Saudi' (Bin Laden, never named in the novel) (pages 58 and 118).18 These verses and the character's reflections on them emphasize that Seyf el Islam knows his terrorist act is counter to Islamic religious precepts. Additional examples are as follows: verse 2:161 is quoted to emphasize that Seyf counts himself as a non-believer (34). Verse 4:74, about the reward awaiting those who fight in the way of God, is followed by \"il allait \u00eatre tu\u00e9 mais sans pardon sans victoire car Dieu rejetterait son acte\" 'he was going to be killed but without forgiveness nor victory because God would reject his act' (54). Verse 24:35, which ends in \"Dieu guide, vers sa lumi\u00e8re, qui il veut\" 'God doth guide whom He will to His Light' is followed in the novel by \"et Il l'avait oubli\u00e9\" 'and He had forgotten him' (70), the same with the quote from 5:41 (\"tu ne peux rien faire contre Dieu pour prot\u00e9ger celui que Dieu veut exciter \u00e0 la r\u00e9volte\" 79),19 thereby inscribing his straying from the religion as something inscribed in it. In the same vein, verses 14:4 and 4:76 are juxtaposed to stress that his going astray is part of God's will (117).\n\nThird, the Qur'an is also invoked to draw parallelisms between events recounted in it and the 9\/11 attacks. Part of verse 2:50 about the Pharaoh being drowned to allow Moses to flee Egypt forecasts in Seyf el Islam's mind the abyss he will fall into the next day. Bachi modifies slightly the translation of verse 2:35 when God enjoins Adam to eat from the garden with the exception of one tree, and puts \"tree\" in the plural, for a slip in Seyf's mind assimilates the tree with the twin towers, with the acknowledgment that he will be \"au nombre des injustes\" 'among the unjust' (32). This verse and parallelism between the trees and the towers recurs on pages 69 and 123. Similarly, when he is in the elevator, and his ascent is compared to the Prophet Mohammed's _mi'raj_ or ascent to heaven, his act is paralleled with 7:24 (121). Seyf remembers the following verse: \"ils jurent par Dieu qu'ils sont des v\u00f4tres, alors qu'ils n'en sont pas, amis ce sont des gens qui ont peur, s'ils trouvaient un asile, des cavernes ou des souterrains, ils s'y pr\u00e9cipiteraient en toute h\u00e2te\"20 followed by \"et cela \u00e9tait bien ce qu'avait fait le Saoudien\" 'and that was exactly what the Saudi had done' (123), recalling Bin Laden's flight to the caves during the U.S. invasion of Afghanistan.\n\nThe Qur'an is cited out of context to apply to a variety of situations. For instance, verse 8:5 (\"c'est au nom de la V\u00e9rit\u00e9 . . . que ton Seigneur t'a fait sortir de ta demeure\" 'Just as thy Lord ordered thee out of thy house in truth') is recited by Seyf during the takeoff, as if to sanction the act he is about to commit (130). When Seyf is in the plane, he keeps reciting Qur'anic verses that seem to announce what is about to happen, such as verse 7:4-5 about the destruction of cities, verse 17:1 about the Prophet's ascension (the original \"de nuit\" 'by night' is changed into \"de jour\" 'by day'), as if Seyf were a new prophet (131). Verses 17:16 and 16:112 are quoted as if they followed each other, with an indented passage to stress how the description fits New York and the World Trade Center towers. The change in the verb tense in 16:112 (in the past tense in the original and in the future in Bachi's quotation page 132) highlights the distortion caused by applying a verse that describes a past event to something yet to happen.\n\nAs in Bena\u00efssa's novel, the Qur'anic quotes inscribe Bachi's narrative in the Arab cultural tradition and give it an authentic, legitimate touch. But in Bachi's text, they also highlight the discrepancy between Islamic ideals and reality in parts of the Muslim world. These quotes also emphasize the complexity of the character: a believer on the one hand, who sees some parallels between his terrorist act and stories recounted in the Qur'an, he knows his action is condemned by the religion, and yet rationalizes it through the theological tenet of destiny: he goes through with it since it is the will of God.\n\nContrary to Bena\u00efssa's text, Bachi's novel features a rich multicultural intertext. It makes frequent allusions not only to the Qur'an and _The Conference of the Birds_ , but also to _Hamlet, The Birds_ , and _Hiroshima mon amour_. The Qur'an and _The Conference of the Birds_ can be categorized as belonging to the Muslim religious tradition: one is a Holy book and the other a classic of Sufi (mystic) poetry. _Hamlet, The Birds_ , and _Hiroshima mon amour_ belong to the Western tradition and are more preoccupied with worldly issues than with spiritual questions.\n\nSeveral citations from _Hamlet_ appear in the novel; they draw a parallel between Seyf el Islam and Hamlet, his French girlfriend and Ophelia. Hamlet is called on by his father's ghost to avenge his wrongful death; to do so he has to fight the dominant power of the usurper king, his uncle. The spirit of Seyf's father comes to visit him as Hamlet's father's specter appears, but contrary to _Hamlet_ , Seyf's father enjoins him to dissociate himself from the terrorists and their \"unjust war\" (63). Like Hamlet in the eponymous play, Bachi's protagonist wreaks havoc in the process of seeking revenge, and ends up dead.\n\nThe numerous mentions of the film _Hiroshima mon amour_ are negative; it is described as \"un sale titre de film vu dans une salle obscure \u00e0 Paris\" 'a nasty film title seen in a dark theater in Paris' (14, 49, 97) and is associated with the period when Seyf was trying to assimilate to French society. On a formal level, _Tuez-les tous_ recalls _Hiroshima mon amour_ with constant analepses and prolepses intertwining the past and the present, and not clearly indicated (sometimes happening within a single sentence). In _Hiroshima mon amour_ , the French woman's love affair with the Japanese man brings back her love story with the German officer in Nevers. Similarly, Seyf's encounter with the American woman recalls memories of his French wife. In addition, repetitions of several passages blur the time sequence and can be quite disorienting for the reader.\n\nMentions of the film and pictures of Holocaust victims in a long sentence that intertwines the descriptions of bodies burnt by the atomic bomb and gassed in the concentration camp explain why the epithet of \"sale\" precedes the title: \" _Hiroshima mon amour_ , Auschwitz mon amour, br\u00fbl\u00e9s, irradi\u00e9s\" ' _Hiroshima my love_ , Auschwitz my love, burnt, irradiated' (49).21 The character's rejection of the title, expressed by always referring to it as a \"sale\" title, refuses the intent inscribed by Duras with her oxymoronic juxtaposition of love and the mass murder of civilians. Or rather, this rejection points out what Higgins formulated as follows: the film knows \"that the desire to represent horror through a love story must be counterbalanced by an awareness of the danger of forgetting the horror entirely\" ( _New Novel_ 22).\n\nOne passage highlights the misunderstandings that can result from referring to different traditions. When Seyf starts telling the story of the king of the birds, the well-known Sufi tale written by Farid Addin Attar in the twelfth century, the American woman he picked up in the Portland club thinks he is referring to Hitchcock's film (72). Birds are a constant reference in the novel, starting with the American woman he keeps referring to as a bird (recalling British slang), but more importantly through the Sufi tale. Attar's epic poem is a famous piece that tells the story of birds and their quest for the Simorgh (God); their journey exemplifies the different stages of the Sufi path, until at the end, only thirty birds reach the desired destination, only to realize that the Simorgh (which means thirty birds) is in themselves. The tale ends with the birds' fusion with the Simorgh, reflecting the Sufi doctrine that God is not external to or separate from the universe. In Bachi's text, the gist of the tale is recounted in the middle of the novel (72-74) and again at the end with slight modifications in the details. At the end of Bachi's novel, the plane flying into the World Trade Center is described as a bird and the tower as the mirrors. Although at the beginning the intertextual references to birds are a token of intercultural miscommunication, the final pages of the novel synthesize both references: the plane is compared to a bird that sees its image reflected in the windows of the tower (as in Attar's Sufi tale, except that Seyf merges with nothingness instead of with God). At the same time, the plane is attacking innocent people (as are the birds in Hitchcock's film). \u017di\u017eek has compared the shot of the plane crashing into the second tower with the scene when Melanie is hit by a bird while on the boat in _The Birds_ (14).\n\nThis merging of cultural references seems to highlight the destructiveness that acculturation can entail. At the beginning of the novel, Seyf invokes the Bible as well as Qur'anic scriptures when he quotes verses from Psalm 22 addressing God (\"Mon Dieu. Pourquoi m'as-tu abandonn\u00e9\" 'my God, why have you forsaken me?' 31). Majid notes:\n\nThat Muslims were in the throes of a maddening identity crisis should have been obvious to readers of novels by Muslim writers in the last four decades. Whether the writer hails from Senegal, Sudan, Morocco, Saudi Arabia, or India, the story is always the same: Islam's encounter with the West, whether at home or abroad, provokes terrible confusions that are often resolved through some violent, and often self-destructive, act. (\"The Failure\")\n\nAs Majid points out, there is a darker side to the positive effects of hybridity, touted by postcolonial studies (\"The Failure\"), and that is encapsulated by 9\/11 in Bachi's narrative. _The Birds_ , released the year after the Cuban Missile Crisis of October 1962, has been read as an allegory of the human condition during the height of the Cold War between the U.S. and the Soviet Union, and the birds as a symbol of the nuclear war threat (Hare 279-91). The allusions to Hitchcock's film by Bachi, whereby the plane that crashes in the WTC is compared to a bird, replaces the Cold War threat with that of Islamic fundamentalism, as occurs in Huntington's theory of the clash of civilizations.\n\nBachi's protagonist seems to conform to Khosrokhavar's portrait of Al Qaeda's members, whom he describes as \"des \u00eatres en mal d'identit\u00e9\" 'beings short of\/longing for identity' ( _Les nouveaux_ 111). Khosrokhavar noted that although the Al Qaeda's martyrs are multicultural (they have lived in the West and master its language and cultural code), they still reject Western culture and civilization ( _Les nouveaux_ 241), and this rejection follows a feeling of betrayal ( _Les nouveaux_ 242). This sequence of experiences describes Seyf's itinerary very well. The following quote describes the main character: \"il . . . s'appr\u00eatait \u00e0 devenir un des leurs, un sale type sans histoires et sans Histoire, un int\u00e9gr\u00e9 en voie de d\u00e9sint\u00e9gration, mais il avait pr\u00e9f\u00e9r\u00e9 l'int\u00e9grisme\" 'he was getting ready to become one of them, a nasty character with no story and no history, an integrated person in the process of disintegrating, but who had prefered fundamentalism' (14). The core of this quote comes back in other passages of free indirect speech, with some slight modifications (14, 36). Derrida's interpretation of 9\/11 as a symptom of an autoimmune crisis argues that the 9\/11 hijackers were trained by people who themselves were trained by the U.S. during the Cold War: the mechanism that was set up to protect the West against Soviet advances turned against what it was supposed to defend (Borradori 140). The terrorists are, in some sense, not an absolute other, but a reflection of ourselves (Borradori 115), as in Bachi's novel when the hijacker sees his reflection in the towers: the religious reference gives way to the symbolism of the West's horrible creations as consequences of its imperial policies.\n\n_Tuez-les tous_ also contains allusions to Conrad, the Bible, _L'\u00e9temel retour, Casablanca_ (1942), _Le Faucon Maltais_ ( _The Maltese Falcon_ 1941), and _Le grand sommeil_ ( _The Big Sleep_ 1946). These last three films, all starring Humphrey Bogart, are labeled as \"mauvais film[s]\" 'bad movie[s]'(124). Since the narrator talks about destroying Hollywood, these films stand for the popularity and hegemony of American popular cultural production. The last two sentences of Conrad's _Heart of Darkness_ serve as an epigraph to the book. Conrad's novel is set during the most somber period of Belgian imperialism in the Congo. As Said pointed out about the darkness to which the title refers, \"They [Marlow and Kurtz] (and of course Conrad) are ahead of their time in understanding that what they call 'the darkness' has an autonomy of its own, and can reinvade and reclaim what imperialism had taken for _its_ own\" ( _Culture_ 30). Said continues stressing that these characters along with the author are unaware that what they label 'darkness' is a resistance to European imperialism: \"As a creature of his time, Conrad could not grant the natives their freedom, despite his severe critique of the imperialism that enslaved them\" ( _Culture_ 30). But taken from its original context, \"le c\u0153ur m\u00eame d'infinies t\u00e9n\u00e8bres\" 'the very heart of infinite darkness' that ends the sentence can in the context of a novel on 9\/11 connote the evil motivation of the kamikazes.\n\nThe first part of the book is entitled \"L'\u00e9ternel retour\" and can evoke two things: Nietzsche's idea of eternal return, and the film of the same title (1943), partially inspired by Nietszche's thought, for which Jean Cocteau wrote the screenplay. Although the plot is set in the twentieth century, there is no reference to historical events or socio-political issues. Nevertheless, the plot has been interpreted as an allegory of France during the Occupation (Tarr). According to Carrie Tarr, \"In _L'Eternel retour_ , Cocteau aimed to 'elevate a modern story to the status of myth' ( _Art of Cinema_ 189) by proving that Nietzsche's notion of 'eternal return' could be applied to the fate of the doomed lovers whose story is relived by others, without their even realizing it\" (55).\n\n### Conclusion\n\nThese novels show that religious motivation is not an end in itself in suicidal terrorism, since socio-political grievances play a role. Both novels do in different ways point to a middle ground between the two interpretive grids that have been used to understand 9\/11. Between attributing the attacks to Islamic fundamentalism or to sociopolitical factors, Bachi and Bena\u00efssa point out that both factors are present. They also emphasize that, as Derrida phrased it, \"Those called 'terrorists' are not, in this context, 'others,' absolute others whom we, as 'Westerners,' can no longer understand. We must not forget that they were often recruited, trained, and even armed, and for a long time, in various Western ways by a Western world that itself, in the course of its ancient as well as very recent history, invented the word, the techniques, and the 'politics' of 'terrorism' \" (cited in Borradori, 115).\n\nAlthough Bena\u00efssa's novel can be interpreted as demonstrating that 9\/11 was triggered by socio-political claims, and that religion supplies the means to get attention to those claims, the prominent Qur'anic intertext overwhelms any such argument. The imbalance present in Bena\u00efssa's novel will only confirm the average Western reader in the stereotypes perpetuated in the media, and which have only worsened since 9\/11 and the subsequent invasion of Iraq. These stereotypes have contributed to strengthening the association between Islam and terrorism, with Islamic fundamentalism being conveniently and constantly brandished as an easy explanation for any bombing that happens in the Middle East. Indeed, the predominance of the Islamic intertext in Bena\u00efssa's novel seems to confirm the belief that Islam (or a particular interpretation of it) is to blame for terrorism, and that Western policies in the Middle East have nothing to do with it (contrary to what many political scientists say, see Fuller 83-96).\n\nBachi's text, which traces the complexity of contemporary problems to historical events of the last six hundred centuries, offers a counter narrative to the simplistic reductions of \"us versus them\" or \"the axis of evil\" that flourished after 9\/11 in official American discourses.22 Bachi's novel is more apt than Bena\u00efssa's to prod the reader to do what Butler advocates. Butler distinguishes between the conditions that foster terrorism and its causes, and urges that we \"take collective responsibility for a thorough understanding of the history that brings us to this juncture,\" instead of becoming paralyzed by the faulty argument that to understand is to exculpate (10). As Derrida explains: \"one can thus condemn _unconditionally_ , as I do here, the attack of September 11 without having to ignore the real or alleged conditions that made it possible\" (qtd. in Borradori, 107).\n\nMartin Amis and John Updike's works show that one does not need to be Arab, or Muslim, or both, to write a fictive portrayal of a Muslim terrorist complete with Qur'anic quotes (if one needed such proof). That Francophone Arab writers, contrary to other 9\/11 novelists, have so far restricted their narratives to depicting the itinerary of a 9\/11 hijacker seems to imply that a syndrome of guilt by association has been at work. While Bena\u00efssa's apology gives fodder to that association, Bachi's novel, with his critique of the West, complex narration, and multicultural intertext, prompts the reader to go beyond such simplistic views.\n\nThat these writers decided to depart from what we know about the hijackers, I would argue, constitutes one of the strengths of these novels: they make us see 9\/11 not just through the eyes of the actual hijackers (as Amis does in his short story), but through those of other potential hijackers as well, thereby drawing attention to the social conditions that breed terrorism. Khosrokhavar distinguishes two different kinds of martyr: on the one hand the ones who are linked to defense and\/or establishment of the nation (such as the Iranian, Palestinian, and Lebanese cases), and on the other hand the one exemplified by Al Qaeda, which \"revendique la r\u00e9alisation d'une communaut\u00e9 mondiale incarn\u00e9e par l'universalisme islamique en brisant la puissance du Mal qui s'y oppose: l'Occident\" 'claims the realization of a world community incarnating Islamic universalism by smashing the Evil that opposes it: the West' (329). In addition to an identity crisis, Khosrokhavar (and others) emphasizes that Al Qaeda members represent the globalization of martyrdom (111). Reuter points out that \"What makes this relatively new, clandestine network so dangerous is its recipe for combining two formidable elements so as to create a uniquely deadly form of militant group: an uncompromising, seductive, Manichean worldview that attracts a cult-like following, and a set of real, local grievances and ethnic and\/or religious conflicts waiting to be enflamed. Like a parasite, al-Qaeda moves among the conflicts in the Islamic world, deriving sustenance from genuine, often well-founded local anger and grievances\" (Reuter 153). According to Holmes, \"What hit the United States on 11 September was not religion . . . Instead, the 9\/11 terrorists represented the _pooled insurgencies_ of the Arab Middle East. The fusing of these local insurgencies took place largely among diaspora Arabs outside the Middle East itself, in Afghanistan and Europe\" (168).23 By giving a different nationality to their hijacker, Bena\u00efssa and Bachi seem to stress how local grievances can become global.\n\n### Notes\n\n1. Novels by French authors include Luc Lang's _11 septembre mon amour_ (Stock, 2003) and Fr\u00e9d\u00e9ric Beigbeder's _Windows on the World_ (Gallimard, 2005).\n\n2. See also Tahar Ben Jelloun and Martin Amis, who each wrote a short story in the same vein.\n\n3. See Terry McDermott's _Perfect Soldiers_ for a reconstitution of the historical events of 9\/11.\n\n4. The publication of the leading association for scholars of literature in the U.S. (Modem Language Association).\n\n5. See Brown's critique of \u017di\u017eek (Brown 746). Brown points out that the distinction in the argument by \u017di\u017eek (Islamic fundamentalism explained as \"the sacralization of proper (economic) politics\") depends on \"an a priori distinction between religion and politics and on the separation of church and state\" (747). Such a distinction, he adds, is not in operation in other parts of the world. Brown's assumption betrays an Orientalist mode of thinking as well as blindness to the influence of religion in American politics.\n\n6. See Nixon's article in _The New Crusades_.\n\n7. Although Soyinka does not refer to the clash of civilizations, his branding Islam as an element foreign to African culture partakes of the same essentialist view of cultures.\n\n8. Morton criticizes Baudrillard, \u017di\u017eek, and others' recent essays on terrorism because they overlook \"the imperialist interests that are served by the discourse of terrorism by focusing on the emotional and aesthetic connotation of terror instead of examining the geopolitical context of its production\" (37). Such works \"reinforce the aestheticisation of terror . . . rather than examining the geopolitical determinants of terrorism as a discourse\" (Morton 37). I find this criticism is only partially warranted for Baudrillard, and definitely not for \u017di\u017eek.\n\n9. The Japanese Kamikazes represent the highest number of suicide missions organized by a single source (Gambetta viii).\n\n10. Reflecting on the fact that certain groups, such as the Christian Maronites, did not resort to suicide missions, and that self-immolations are mostly confined to Hindu and Buddhist countries, Gambetta concludes that \"Different sets of religious beliefs are enabling and disabling different forms of self-sacrifice\" (293). This conclusion, however, does not seem to recall that an entire chapter of the book is dedicated to the Tamil Tigers, who are 90 percent Hindu with a small minority of Christians (Hopgood 47).\n\n11. Despite Gross' excellent analysis of Bena\u00efssa's plays, I concur with Micheline Servin that his theater is more like sketches than full-fledged productions (qtd. in Gross 387).\n\n12. That is Christians and Jews.\n\n13. See the list compiled by Kurzman at <http:\/\/www.unc.edu\/~kurzman\/terror.htm>.\n\n14. The comparison of 9\/11 to the attack on Pearl Harbor was a misleading one; as an attack on civilians in an urban area the comparison with the bombing of Hiroshima would have been more pertinent, though more controversial (see Versluys n.23).\n\n15. There is some dispute about the veracity of this story. See Sibly and Sibly (289-93).\n\n16. Most of these allusions are to the advantage of women, with one exception. An allusion to a Hadith works the other way around: \"ce n'\u00e9tait pas le paradis aux pieds de sa m\u00e8re\" 'it was not paradise at his mother's feet' is followed by a virulent critique of Maghrebian mothers (82).\n\n17. \"Il croyait parce qu'il n'\u00e9tait pas seulement un de ces hypocrites, ceux-l\u00e0 Dieu connaissait le contenu de leurs coeurs '\u00e9carte-toi d'eux' il ne l'avait pas fait\" (78) alludes to \"These are they, the secrets of whose hearts Allah knows; so turn aside from them and admonish them\" (4:63).\n\n18. To which Khalid answers with another verse (2:216): \"Fighting is prescribed for you, and ye dislike it. But it is possible that ye dislike a thing which is good for you, and that ye love a thing which is bad for you.\"\n\n19. \"And he for whom Allah intends temptation, thou controllest naught for him against Allah\" (5:41).\n\n20. \"And they swear by Allah that they are truly of you. And they are not of you, but they are a people who are afraid. If they could find a refuge or caves or a place to enter, they would certainly have turned thereto, running away in all haste\" (9:56-57).\n\n21. Duras did not want to depict the horror of Hiroshima as had already been done by showing the actual devastating and horrific effects of the bomb; she wanted to \"make this horror rise again from its ashes by [inscribing] it in a love [story]\" (qtd. in Higgins 21-22).\n\n22. See Butler's first chapter (\"Explanation and Exoneration, or What We Can Hear\") for a detailed analysis of the rise of censorship and anti-intellectualism that took place in the U.S. after 9\/11 to disparage anyone who tried to understand the reasons why the attacks had taken place.\n\n23. This analysis seems to me more accurate than Khosrokhavar's statement that the Al-Qaeda martyr struggles for a transnational Umma (233). Khosrokhavar also points out that Al Qaeda is a product of globalization, the Muslim diaspora in the West, and crisis in Muslim society (233), and of course the issue of humiliation by proxy (239). However, Khosrokhavar's affirmation that this is a terrorism that has no specific territorial claim contradicts what has been demonstrated by Holmes: that is, that Bin Laden had very specific grievances regarding the presence of American troops on Saudi soil.\n\n## Conclusion\n\n## The Politics of Translation: Francophone Literature from the Arab World in the U.S.\n\nIn conclusion, I turn to the impact that international news has had on the translation of some of the texts treated in this study, as emblematic of Francophone literature from the Arab world. Two factors play in the translation, packaging, and reception of these texts: the U.S.' unconditional support of Israeli policies towards Palestinians, and 9\/11 and the subsequent invasion of Afghanistan and Iraq. Reflecting on the spike of American interest in the cultural production of the Arab and Muslim world in general, this chapter offers a critical evaluation of this trend's implications in general and for Francophone Studies in particular.\n\nWell-established writers usually see their work automatically translated into English. Given that Ben Jelloun, winner of the prestigious Goncourt prize in 1987, is one of the most prominent Francophone Maghrebian writers (and one whose creative work has been translated into numerous languages), it should be no surprise that his essay on Islam was promptly translated into English in 2002.1 However, his book promises more than it delivers, since it contains several statements that are not representative of mainstream practitioners of Islam in France or elsewhere, and fails to account for the complexity of issues tied to the affair of the scarf (as demonstrated in chapter five). Maalouf, another Goncourt prize winner (in 1993), also enjoys international recognition since his work has been widely translated into many languages. The English translation of his essay _Identit\u00e9s meurtri\u00e8res_ came out in England in 2000; the 2003 Penguin North American edition consists of the very same English version, but repackaged in light of 9\/11. The paratext, in the form of excerpts from book reviews that appear on the front cover and first page, casts the essay as shedding light on why 9\/11 happened. As an example, the cover features a quote from the _Los Angeles Times_ that states: \"This striking and pungent polemic is so searingly pertinent, it confirms that . . . the mass murder of September 11, while indelibly shocking, is not wholly surprising.\"\n\n9\/11 has had sizeable repercussions not only on the packaging, but especially on the publication of translations of texts from the Arab world in the United States, as evidenced by the following cases. Although Myriam Antaki has published four novels in French since 1985, she is unknown in the field of Francophone literature. _Les versets du pardon_ , published in 1999 in France, was her first (and only book to date) to be translated into English (New York: Other Press, 2002). The subject matter of the Arab-Israeli conflict and the novel's setting in historical Palestine are not commonly thought of as being within the realm of Francophone studies, despite the fact that events in the Middle East have been having repercussions in France on the relations between the French Muslim and Jewish communities and on the Jewish minorities in the Maghreb, as elaborated on in chapter two.\n\nAlthough Bena\u00efssa is a relative newcomer as a novelist, _La derni\u00e8re nuit d'un damn\u00e9_ (2003) appeared in English the next year after its French publication, and is his only novel and second book to have been translated into English. Despite being a rather unremarkable novel by a minor writer, it enjoys a relatively wide distribution, if one can judge by the number of libraries that own it (over 600 according to the WorldCat database), and was reprinted in 2005. Novelists such as Antaki and Bena\u00efssa are minor writers, yet they have in common that one of their works has made it quickly on the U.S. market post 9\/11 thanks to their subject matter linked to \"Arab-Muslim\" terrorism.\n\nIn contrast, Meddeb is a well-established writer who has been publishing since 1979, yet none of his creative work, which has long engaged with Islam,2 has been translated into English. However, the English version of his essay _La maladie de l'islam_ was published promptly in 2003. This is an example of an established writer's work receiving selective translation, a fact that can be explained on several accounts. First and foremost, the subject matter linked to 9\/11 made it a timely and relevant publication. Second, its title confirms from the start the horizon of expectations of readers fed with prejudiced media accounts that something is inherently wrong with Islam (while Meddeb's other works offer a more complex picture).\n\nIn any event, the availability of Francophone authors to an Anglophone readership should provide cause for rejoicing for scholars, all the more so when the former offer perspectives not widely presented in the mainstream American media, particularly the criticism of the double standards of American foreign policy and the Palestinians' plight. For instance, Antaki's novel, which features Zionist terrorist acts by Irgoun and Stern in 1948 and subsequent displacement of Palestinian refugees, does offer a historical perspective and a point of view that are usually absent from contemporary media coverage of the Arab-Israeli conflict. This novel helps humanize Palestinians by showing the suffering that Marie goes through and linking it directly with the tragedy of the Holocaust. In the United States, Palestinians' suffering is considered too politically charged to even be mentioned. Butler has demonstrated that when it comes to the Middle East, Israeli and Palestinian deaths do not entail the same reaction in the U.S., that there is \"a hierarchy of grief (32) at work.3 Popular culture in the U.S. frequently associates Palestinians with terrorist violence, suicide bombings, and Islamic extremism. These associations have only been reinforced by the 9\/11 attacks, after which Israel banked on the U.S.'s \"War on Terror\" to justify its policies in the Occupied Territories. Said observed that public discourse always occults the fact that \"Israeli violence against Palestinians has _always_ been _incomparably_ greater in scale and damage\" than the reverse (Said, \"The Essential\" 153), a fact that still rings true twenty years after he made that remark (witness the December 2008 massive bombing of Gaza).\n\nHowever, I would like to express a reservation concerning this recent phenomenon of craze for translating such works in general. Different contexts affect how works are read and received. Antaki writes in Syria (she is one of the few writers treated in this study who still lives in the Arab world), but her work is in French and is published in Paris. The cultural production of the second half of the twentieth century in France and in French has given its due attention to World War II and the Holocaust, but the harm of colonization, such as Arabs' plight during the Algerian war for instance, has lagged behind. In fact, the third and fourth phase of the Vichy syndrome (the return of the repressed and obsession, respectively), as defined by historian Henri Rousso, \"may have been precipitated by the desire to cover up the double loss caused by the Algerian trauma\" (Donadey, _Recasting_ 7).\n\nAntaki's novel was not well received in the Middle East4 for two reasons. First, it features the Holocaust, a fact that was not well received in an area where there is widespread downplaying of the suffering endured by Jews under Nazism because of the crucial role played by Western nations' guilt about it in the establishment of the state of Israel. Second, because it uses the word \"terrorist\" to describe Ahmed (E-mail to the author, January 22, 2007). Various scholars have insisted on the exploitation of the term \"terrorist\" by powers suppressing independence movements; Butler (among others) insists that, \"The term 'terrorist' is used, for instance, by the Israeli state to describe any and all Palestinian acts of resistance, but none of its own practices of state violence\" (4). I believe that Antaki's use of the term terrorist, juxtaposed to \"dreamer,\" serves to highlight the fact that terrorism is not an end in itself, but only a tactic. But as Antaki's book travels to other readerships, particularly in the U.S., it might reinforce the image of the Palestinian as \"the essential terrorist\" (to borrow Said's phrase).\n\nSimilarly, as previously noted, Meddeb's essay was promptly translated into English because it directly addressed issues of immediate concern to the U.S. Shohat and Stam rightly point out that \"in the current situation, U.S. power is global, yet the knowledge of too many of its citizens is local and monoperspectival\" (5). Hence, people are unable to make a connection between resentment about the U.S.'s foreign policy of support for dictators and attacks on American interests.5 On the one hand, Meddeb, who chastises the double standard of American foreign policy in the Middle East, does offer a perspective on the world that is rarely presented in mainstream American media. However, since less than five pages are dedicated to a brief recapitulation of U.S. interference in the politics of the Arab world, this book will add to what has become a pervasive problem:\n\nMuch of what one reads and sees in the media about Islam represents the aggression as coming from Islam because that is what \"Islam\" is. Local and concrete circumstances are thus obliterated. In other words, covering Islam is a one-sided activity that obscures what \"we\" _do_ , and highlights instead what Muslims and Arabs by their very flawed nature _are_. (Said, _Covering_ : xxii)\n\nThe emphasis on Islamic fundamentalism (for which Meddeb's title, _La maladie de l'islam_ , is a periphrasis) relegates external interferences and influences to the background, and foregrounds Islam as the main source of terrorism. As S. Sayyid points out, \"Islamist challenges to the prevailing order can only be represented as acts of terrorism, because there is no space for challenges to the prevailing political order. Thus, the legitimacy of the Islamist struggle against repression is denied as the protestors are labeled as terrorists, while those who staff and command the repression machines of the state are presented as reasonable and moderate members of the international community\" (xii). Meddeb's uneven-handed account of the internal and external reasons for Islam's so\u2013called disease might not be what his readership is most in need of.\n\nThe same reservation applies to Bena\u00efssa's novel. Unfortunately, since few pages are dedicated to the disastrous legacy of colonialism and current imperialist foreign policies, while the Muslim scriptures are given ample space, this book will do little to redress the pervasive popular view that violence is intrinsic to Islam. As already detailed in chapter six, Bena\u00efssa's and Bachi's 9\/11 novels have a lot in common. Bachi is also an emerging novelist, but his work has already earned several literary prizes, most notably the Goncourt for a first novel ( _Le chien d'Ulysse_ in 2001). Nevertheless, there is no sign of _Tuez-les tous_ being considered for an English translation. According to Bachi, \"Il semblerait que le sujet du livre d\u00e9range les \u00e9diteurs outre-Manche et outre-Atlantique\" 'it seems that the topic of the book disturbs British and American publishers' according to the person in charge of foreign rights at Gallimard (E-mail to the author, March 29, 2007). Although _Tuez-les tous_ was only recently published, the translation of Bena\u00efssa's _La derni\u00e8re nuit_ shows that it is not the subject matter that is in dispute, on the contrary. I can only surmise that the way the subject is treated is the culprit (in addition to its narrative structure and elaborate intertext which both contribute to making this a challenging read), in the sense that it questions stereotypes and commonplace ideas that mainstream America might not be ready, or willing to reconsider.\n\nThe impact of 9\/11 on what is made available in American bookstores is not without implications. In the case of Francophone literature published in the U.S. post 9\/11, the recent translations discussed above seem to indicate that the spike of American interest in the literary production of that region is limited to whatever can be linked to the traumatic event of 9\/11, and not literary recognition already established by the original publication, as is customarily the case. For instance, the English translation of Antaki's novel makes no mention of her previous novels, as if knowing about her untranslated work was superfluous and of no value to an American readership. Thus, the American public does not acknowledge that the cultural production of these countries is worthy of interest regardless of their centrality to U.S.'s interests, and not because they have been defined by public discourse as potential enemies. As Mary Louise Pratt aptly states in the context of language learning and national security imperatives, \"It is critical that . . . linguistic others not be defined from the start as potential enemies\" (115), and that we not become interested in foreign cultural production only when it is too late.\n\nSome of the texts treated in this study perpetuate uncalled-for stereotypes about Islam and Muslims. While texts such as _L'excis\u00e9e_ and _La derni\u00e8re nuit d'un damn\u00e9_ do mention socio-political factors that play a role in fomenting and perpetuating various conflicts, their religious intertexts, and specifically Qur'anic quotations, contribute to marking Islam as the culprit and thus downplay essential triggers.\n\nIn _L'excis\u00e9e_ , religious sources are quoted in such a way as to suggest that Islam, which is the source of subjugation of these women, requires excision, which the women perpetuate in a ruthless manner, with no hope of change and progress coming from the inside (as demonstrated in chapter 3). Despite a few passages critical of Christian theology, the allegorical reading supported by the novel inscribes it within a Christian framework that subscribes to the values of sacrifice, redemption, and salvation.\n\nGiven this, it should not be viewed as a coincidence, therefore, that _L'excis\u00e9e_ , in addition to being featured in various anthologies,6 has found a new readership, if one can judge by the fact that the French original was reedited in 1992 and its English translation in 1994 (and reprinted in 2005),7 during a period when Islam made notorious headlines, both domestically in France (with the issue of the veil in public schools for instance) and internationally.8 In addition, _L'excis\u00e9e_ (and Accad's work in general) is starting to receive more attention from scholars.9\n\nThe implications of the new interest generated by this novel in the contemporary socio-political U.S. climate can be serious. As a teacher in the United States, I would think twice before assigning _L'excis\u00e9e_ for a general education course, for the same reasons reported by an unnamed Middle Eastern scholar, who\n\nwanted no part of the tendency in the West to focus on this image of Middle Eastern and African women. In her experience, many people know little about the Middle East except the topics of veils and circumcision. We should be writing about many other topics . . . , lest unflattering North American and European stereotypes of Arabs, Muslims, and Africans be reinforced and allowed to fuel hostile political and economic policies, (qtd. in Gruenbaum 204)\n\nThis concern is particularly valid in the post-9\/11 climate. A recent example of women's oppression being used for political purposes would be the plight of Afghani women under the Taliban's regime, which became a concern of the Bush administration only after September 11, 2001, and was then used as an additional validation to gain support for invading Afghanistan. More recently, critics have analyzed why the success of Azar Nafisi's memoir _Reading Lolita inTehran_ is highly problematic, with its criticism of the Iranian Islamic revolution and government without sufficient contextualization happening concurrently with increased threats from the Bush administration against Iran, therefore leading itself to being recuperated for conservative political purposes (see Donadey and Ahmed-Ghosh). A similar conclusion could be reached about the success of Marjane Satrapi's animated film _Persepolis_ (2007), although the film, with its criticism of European racism, and the English translation of the graphic novel, with an introduction giving a brief historical background that focuses on Western intervention in Iran, lend themselves less easily to such a reappropriation.\n\nWhile this certainly does not mean that one should only present positive aspects of a culture, I fear that in the current political climate, a novel such as _L'excis\u00e9e_ only reinforces stereotypes. A teacher would have to nudge students into a critical self-reflection, relating how some Western cultural practices can be paralleled to Female Genital Cutting, such as fraternity hazing in the U.S. as a form of ritualized violence and the corset10 as an \"ideal of femininity . . . mediated by _pain_ \" (Lionnet 133),11 the \"neotribal\" practices that have gained some popularity in Western countries among the \"modern primitives\" (Zabus, _Between Rites_ 269-73), or unnecessary hysterectomies, procedures that also involve the removal of an organ, still being performed in the U.S. despite the potential loss of libido and health consequences they entail. Otherwise, students whose understanding of African and Middle Eastern cultures is often too scanty and prejudiced, all the more so when it comes to Muslim cultures in a post-September 11, 2001 period, would only feel comforted in their view of this threatening Other.\n\nIn light of post 9\/11 events, the following question is very pointed: \"To what extent have Arab peoples, predominantly practitioners of Islam, fallen outside the 'human' as it has been naturalized in its 'Western' mold by the contemporary workings of humanism?\" (Butler 32). One could easily replace the adjective \"Arab\" with \"Muslim\" in the previous quote, and extend Butler's question about the nameless Palestinian dead to Afghani and Iraqi civilians. The U.S. news memorializes daily the U.S. military that have died in Afghanistan and Iraq by showing their pictures and their names on the news, while the Afghan and Iraqi civilians remain bundled into some abstract numbers whose accuracy is impossible to determine.\n\nSome of the texts treated in this study, while written well before 9\/11, have found a renewed relevance. I have already mentioned the repackaging of Maalouf's _Identit\u00e9s meurtri\u00e8res_. Reading Maalouf's _Les croisades vues par les Arabes_ in the United States in the context of post-September 11, 2001 and after the Iraq invasion of 2003, where a widespread discourse of the clash of civilizations among policy makers influences American public opinion, it is evident that Maalouf's essay is as timely now as it was when it was published twenty years ago. Witness the outcry that followed the use of the word \"crusade\" by George W. Bush to describe the military campaign against terrorism in Afghanistan. While Maalouf's epilogue is colored by the Middle Eastern politics of the twentieth century up to the early 1980s (as detailed in chapter 1), reading this text in the United States in the post-September 11, 2001, post-Iraq invasion era, I cannot help lamenting the total lack of impact that his essay, which was translated into many languages (and into English as early as 1984) has had in the political sphere.12 For had George W. Bush read Maalouf, he might have at the very least avoided making such a blunder as to characterize the American invasion of Afghanistan, a Muslim country, as a \"crusade,\" and rethought the terms in which his foreign policy is clad.\n\nMaalouf's historical essay, despite its shortcomings, exemplifies the form of political engagement that Said and Bourdieu advocated for public intellectuals. As Said put it, \"we need to think about breaking out of the disciplinary ghettos in which as intellectuals we have been confined, to reopen the blocked social processes ceding objective representation (hence power) of the world to a small coterie of experts and their clients, to consider that the audience for literacy is not a closed circle of three thousand professional critics but the community of human beings living in society\" (\"Opponents\" 146).13 One only wishes that their voices had been heard, and their words taken to heart.\n\nInstead, postcolonial theory (and Said's _Orientalism_ as emblematic of it) found itself at the center of a controversy about the alleged bias of Title VI-funded programs in Middle Eastern Studies (Kurtz). The House Committee on Education had approved H.R. 3077, which creates an Advisory Board to supervise Title VI programs.14 Had that bill passed, _The Crusades through Arab Eyes_ might be declared by the proposed advisory board unfit to be taught by faculty affiliated with Title VI centers, since its title easily identifies it as a work giving a one-sided approach. Such a measure would silence an Arab perspective on a crucial page of history at a time when it has been made extremely relevant by public discourse and foreign policy. As Said reminds us in his posthumous work, \"At least since Nietzsche, the writing of history and the accumulations of memory have been regarded in many ways as one of the essential foundations of power, guiding its strategies, charting its progress\" ( _Humanism_ 141). And Maalouf's 1983 revisiting of the Crusades resonates in uncannily relevant ways twenty years later.\n\nOne of the most depressing insights gained from this study was to note that even scholars of Said's stature seem to have had little impact, witness the current debacle of American foreign policy in the Middle East. Scholars working on the Middle East have been particularly scrutinized and under fire lately. There has been some progress since Hassan lamented in his article that \"given the U.S.'s imperial role in the region, notably its unconditional economic and military commitments to Israel, there is not more criticism of Israel in those quarters of academic study that claim to oppose colonialism in all its forms\" (33). However, Barghouti's piece about the dehumanization of Palestinians in Israeli society published in _PMLA_ , the journal of the leading association for scholars and teachers of language and literature in the U.S., elicited two negative letters (see the Forum section of the March 2007 issue), demonstrating that scholars of literature will not be immune to their peers wrongly equating criticism of Israel with anti-Semitism. Indeed, young scholars critical of Israel in various fields in the United States have seen their bid for tenure denied or generating intense campaigns,15 while preeminent scholars such as John J. Mearsheimer and Stephen M. Walt have had their invitation to lecture cancelled (Jaschik). As Butler points out, the distinction between Israel and Jews needs to be clearly maintained, so that criticism of Israeli policy not be automatically branded as anti-Semitism, and vice versa, to not reduce Jews and Judaity to Israeli concerns (125).\n\n_JuifsArabes_ , Boudjellal's best selling comic book, has been translated into German and Italian, but not English (personal interview, July 2, 2008), while Albou's film is available on DVD in the U.S. Memmi's writings, both fictional and non-fictional, have been translated into English and are widely available.16 His stance on Arabs and Jews, which recuperates the colonial ideology of separation between the two groups, has become the dominant one. In a series of interviews with Marie Redonnet, El Maleh deplores that his books did not get any recognition in France.17 His work, which offers a different perspective, remains marginalized, despite the increased recognition and critical attention that it has been getting in recent years. El Maleh started writing later than Memmi, which partially explains why he is less well-known. Another factor is that his writings have not easily found publishers, partially because they did not correspond to the horizon of expectation of readers. El Maleh recalls that one of the publishers he submitted his manuscript to could not understand why he had linked his story to the events taking place in Lebanon, and would rather have seen him talk about his \"enfance juive, entre deux cultures: l'origine traditionnelle, l'acculturation, la d\u00e9couverte et l'entr\u00e9e dans la culture fran\u00e7aise\" 'Jewish childhood, between two cultures: the traditional origin, acculturation, discovery and entrance into French culture' (Redonnet 75). Contrary to Memmi's writings, which correspond to the description given by El Maleh of French publishers' expectations, El Maleh's texts challenge these expectations and current political agendas. None of El Maleh's books have been translated into English. Given his strong stance on Israeli policies towards Palestinians, and his comparisons between them and Nazi crimes, I would surmise that this will not be remedied any time soon in the United States.\n\nGiven the stature of some of the writers treated in this study, essays such as Ben Jelloun's and Meddeb's exacerbate the distortion that the field of Francophone studies creates, in that, as Richard Serrano stated in a provocative study entitled _Against the Postcolonial_ , \"the Francophone Studies model promotes a tiny sliver of literary and cultural production as representative\" of the Francophone world (1), although its cultural output is expressed in many languages. As this study has shown, several books by well-known writers are not helping to bring a full picture about what is going on in the Arab world and its diaspora, particularly when Islam is concerned.18 In fact, many of them just give us a skylight rather than a window on those issues.\n\nUnfortunately, in the current political climate, such texts, with their downplaying of the legacy of colonialism and neo-colonialism, and emphasis on Islam as a fundamentalist religion opposed to the democratic West as the source of all problems, participate, even if unwillingly, in manufacturing consent (as Herman and Chomsky demonstrated for the mass media) to justify military intervention against other nations. In addition, these writers fail to render the fact that Islam does have a certain appeal, as attested by its growth. In his analysis of the global resurgence of religion, Thomas presumes that this phenomenon may indicate that there are other ways of being modern than Westernization (45). Majid makes a persuasive case for non-Western peoples, including Muslims, to reconnect with their religious traditions, and sees \"the insistence on secular solutions in profoundly religious cultures\" as \"yet another sign of Eurocentrism\" ( _Freedom_ 212-21). In the same vein, Graham Fuller points out that \"efforts in the Muslim world to advance political and social thought _totally independent of the framework of Islamic culture_ is doomed to be fractured, unintegrated, rootless, and alienated\" (201). As Majid emphasizes after persuasively recapitulating the destructiveness and impoverishment (both culturally and materially) brought about by capitalism worldwide, there are other ways of being in the world than the one promoted by the West.19 Increased translation might be seen as transparently helpful in a post-9\/11 world, but given that it tends to disseminate the least critical writings that come from the Arab world, it only serves to assuage the liberal conscience.\n\n### Notes\n\n1. Another edition of the French text with activities targeted at learners of the French language also appeared in 2002 in Italy.\n\n2. _Phantasia_ for instance is replete with literary allusions and references from the Qur'an and Arab poets (among others); this intertext contributes to his work not being easily accessible.\n\n3. Butler tells about the case of obituaries and memorials of Palestinians killed by the Israeli army that were rejected by a U.S. newspaper on the ground that \"the newspaper did not wish to offend anyone\" (35). Butler then proceeds to ask pointed questions about the implications of this incident, among some: \"What is the relation between the violence by which these ungrievable lives were lost and the prohibition on their public grievability? . . . Does the prohibition on discourse relate to the dehumanization of the deaths\u2014and the lives?\" (36).\n\n4. Carol Corm states in a brief interview that it \"stirred strong reactions in the Middle East\" (15); similarly, Nathalie Galesne states that \"des d\u00e9bats houleux ont eu lieu \u00e0 l'occasion des pr\u00e9sentations de l'ouvrage, notamment en Jordanie\" 'stormy debates took place during presentations of the book, notably in Jordan' (157), but neither specifies any further.\n\n5. See for instance Said's discussion of then President Carter's reaction to the hostage crisis in Tehran as symptomatic of this in _Covering Islam_ (xiv-xvi).\n\n6. See Mary Ann Caws' and Margot Badran and Miriam Cooke's anthologies.\n\n7. Boulder, London: Lynne Rienner, 2005.\n\n8. See the new introduction to Said's revised edition of _Covering Islam_ , especially xii-xiv for an abbreviated list for events between 1983 and 1996.\n\n9. When I first started working on this book, a search in the MLA International Bibliography only yielded four entries (see Heistad 2000, Marie, Mudimbe-Boyi, and Zahnd; see also Zabus' 1999 article). Since then, a volume of essays on Accad's work has been published in France (see Heistad 2005) and another one in English just appeared (see Toman), both edited by former students of Accad.\n\n10. Cosmetic surgery such as breast implants would be a contemporary equivalent of the now defunct corset. Here is an example of how culture determines our choice of words: cosmetic surgery (most of which is not performed to heal) is never thought of as mutilation; in a similar vein see Lionnet for a provocative discussion of abortion as mutilation (160-61).\n\n11. Richard Shweder reports how a proposal developed by a Seattle hospital to accommodate an immigrant Somali family was discarded, although \"from a medical point of view, the proposed procedure (a small cut in the prepuce that covers the clitoris) was less severe than a typical American male circumcision\" (234).\n\n12. After Operation Enduring Freedom in Iraq, the same can be said of _Orientalism_ , where Said, after quoting Chateaubriand, comments: \"Already in 1810 we have a European talking like Cromer in 1910, arguing that Orientals require conquest, and finding it no paradox that a Western conquest of the Orient was no conquest after all, but liberty\" (172).\n\n13. See also Bourdieu's _Contre-feux 2_. Here is what he advocated in his 1999 intervention at the MLA convention entitled \"A scholarship with commitment\" [\"Pour un savoir engag\u00e9\"], later published in _Contre-feux 2_ : \"Les \u00e9crivains, les artistes et surtout les chercheurs . . . doivent transcender la _fronti\u00e8re sacr\u00e9e_ , qui est inscrite aussi dans leur cerveau, plus ou moins profond\u00e9ment selon les traditions nationales, entre le _scholarship_ et le _commitment_ , pour sortir r\u00e9solument du microcosme acad\u00e9mique, entrer en interaction avec le monde ext\u00e9rieur . . . au lieu de se contenter des conflits 'politiques' \u00e0 la fois intimes et ultimes, et toujours un peu irr\u00e9els, du monde scolastique\" (39-49).\n\n14. The American Council on Education reports concerns from the higher education community that such a board might interfere with the curriculum (\"Controversial Graduate and International Education Bills Pass House Committee\" _Higher Education and National Affairs_ 52.18 (10\/6\/03). 14 December 2003 <<http:\/\/www.acenet.edu\/hena\/>>.\n\n15. Norman Finkelstein, a political scientist, and Nadia Abu El-Haj, an anthropologist (see Jaschik's \"Middle East Tensions Flare Again in U.S. (Update).\"\n\n16. Incidentally, the translation of _Portrait du d\u00e9colonis\u00e9 arabo-musulman et de quelques autres_ received financial assistance from the French Ministry of Culture.\n\n17. \"Je n'ai pas trouv\u00e9 ma place en tant qu'\u00e9crivain en France, malgr\u00e9 la r\u00e9\u00e9dition de mes livres et l'illusion qu'elle allait pouvoir amorcer une relance\" 'I did not find my place as a writer in France, despite the new edition of my books and the illusion that it would begin an upturn' (Redonnet 13). Later on, he indicates being awarded the title of Chevalier de la l\u00e9gion d'honneur (for which he was nominated by the French embassy): \"Paradoxalement, il m'a fallu revenir au Maroc pour qu'une reconnaissance officielle me consacre en tant qu'\u00e9crivain ayant une place particuli\u00e8re dans la litt\u00e9rature de langue fran\u00e7aise\" 'paradoxically, I had to go back to Morocco before an official recognition consecrated me as a writer having a special place in literature in French' (Redonnet 14).\n\n18. Francophone writers are not the only ones downplaying colonialism when it comes to better accusing Islam (see Nixon's article).\n\n19. Taking the example of Islamic cultures, he contends that \"By stating that Allah has deliberately divided humanity into many nations and tribes, then challenged us to know one another, the Qur'an clearly revealed that God's design is for a world of diversities competing in pious deeds. There is no reason to believe that such a divine intent is less inspiring than the one that champions a brave new world of interest-seeking individuals whose collective endeavors, we are told, automatically add up to the coveted blessings of happiness and social harmony\" ( _Unveiling_ 21).\n\n## Bibliography\n\nAbul-Husn, Latif. _The Lebanese Conflict. Looking Inward_. Boulder & London: Lynne Rienner, 1998.\n\nAccad, Evelyne. \"Author's Preface. Ectomy.\" _The Excised_. Trans. David K. Bruner. Revised and augmented ed. Colorado Springs, CO: Three Continents Press, 1994. ix-xvii.\n\n\u2014\u2014. _L'excis\u00e9e_. Paris: L'Harmattan, 1982. Trans. as _The Excised_ , by David K. Bruner. Washington, DC: Three Continents Press, 1989.\n\n\u2014\u2014. _Sexuality and War: Literary Masks of the Middle East_. New York: New York University Press, 1990.\n\n\u2014\u2014. _Veil of Shame. The Role of Women in the Contemporary Fiction of North Africa and the Arab World_. Sherbrooke, Canada: Naaman, 1978.\n\n\u2014\u2014. \"Writing One's Life: Arab Francophone Women.\" _Contemporary French Civilization_ 26.1 (Winter\/Spring 2002): 46-62.\n\nAhmadu, Fuambai. \"Rites and Wrongs: An Insider\/Outsider Reflects on Power and Excision.\" _Female \"Circumcision\" in Africa: Culture, Controversy, and Change_. Ed. Bettina Shell-Duncan and Ylva Hernlund. Boulder & London: Lynne Rienner, 2000. 283-312.\n\nAhmed, Leila. _Women and Gender in Islam: Historical Roots of a Modern Debate_. New Haven & London: Yale UP, 1992.\n\nA\u00eft-Embarek, Moussa. _L'Alg\u00e9rie en murmure. Un cahier sur la torture_. Plan-Les-Ouates, Switzerland: Hoggar, 1996.\n\nAlbou, Karin, dir. _La petite J\u00e9rusalem_. DVD. Oc\u00e9an Films Distribution, 2005.\n\nAlcalay, Ammiel. _After Jews and Arabs: Remaking Levantine Culture_. Minneapolis: University of Minnesota Press, 1993.\n\nAlloula, Malek. \"Mes enfances exotiques.\" _Une enfance alg\u00e9rienne_. Ed. Le\u00efla Sebbar. Paris: Gallimard, 1999 [1997]. 7-23.\n\nAmis, Martin. \"The Last Day of Muhammad Atta.\" _Observer_ Sunday September 3, 2006. <http:\/\/www.martinamisweb.com\/documents\/lastdays_one.pdf>.\n\nAntaki, Myriam. _Les versets du pardon_. Paris: Actes Sud, 1999. Trans. as _Verses of Forgiveness_ , by Marjolijn de Jager. New York: Other Press, 2002.\n\nArmstrong, Karen. _Holy War_. London: Macmillan, 1988.\n\nAshcroft, Bill, Gareth Griffiths, and Helen Tiffin. _Key Concepts in Post-Colonial Studies_. London & New York: Routledge, 1998.\n\nAttar, Farid Ud-Din. _The Conference of the Birds_. Trans. S. C. Nott. London & New York: Continuum, 2000.\n\nAubrac, Lucie. _La r\u00e9sistance expliqu\u00e9e \u00e0 mes petits-enfants_. Paris: Seuil, 2000.\n\nAuger, Marie. _L'excision_. Montr\u00e9al: XYZ, 2003.\n\nAzria, R\u00e9gine. \"Achk\u00e9nazes et S\u00e9farades en France, une rencontre difficile.\" _Contemporary French and Francophone Studies_ 11.2 (2007): 207-16.\n\nBachi, Salim. _Le chien d'Ulysse_. Paris: Gallimard, 2001.\n\n\u2014\u2014. _La Kah\u00e9na_. Paris: Gallimard, 2003.\n\n\u2014\u2014. _Tuez-les tous_. Paris: Gallimard, 2006.\n\nBadran, Margot, and Miriam Cooke, eds. _Opening the Gates_. Bloomington: Indiana University Press, 1990.\n\nBalibar, Etienne. _Droit de cit\u00e9_. Paris: PUF, 2002 [1998].\n\nBarghouti, Omar. \"Relative Humanity: Identity, Rights, and Ethics\u2014Israel as a Case Study.\" _PMLA_ 121.5 (October 2006): 1536-43.\n\nBarthes, Roland. _La chambre claire: Note sur la photographie_. Paris: Cahiers du cin\u00e9ma, 1980.\n\nBaub\u00e9rot, Jean. _Histoire de la la\u00efcit\u00e9 fran\u00e7aise_. Paris: PUF, 2000.\n\nBaudrillard, Jean. _The Spirit of Terrorism and Other Essays_. Trans. Chris Turner. New Edition. London & New York: Verso, 2003. [2002].\n\nBena\u00efssa, Slimane. _La derni\u00e8re nuit d'un damn\u00e9_. Paris: Plon, 2003. Trans. as _The Last Night of a Damned Soul_ , by Janice and Daniel Gross. New York: Grove Press, 2004.\n\nBenbassa, Esther. \"Jewish-Moslem Relations in Contemporary France.\" _Contemporary French and Francophone Studies_ 11.2 (2007): 189-94.\n\nBencheikh, Jamel Eddine. \"Tlemcen la haute.\" _Une enfance alg\u00e9rienne_. Ed. Le\u00efla Sebbar. Paris: Gallimard, 1999 [1997]. 25-39.\n\nBenguigui, Yamina. _M\u00e9moires d'immigr\u00e9s. L'h\u00e9ritage maghr\u00e9bin_. Paris: Canal +, 1997.\n\n\u2014\u2014, dir. _M\u00e9moires d'immigr\u00e9s. L'h\u00e9ritage maghr\u00e9bin_. Canal & Bandits, 1997.\n\nBen Jelloun, Tahar. \"Contamination.\" _Le Monde_ 27 Jan. 2004. Web. 3 Aug. 2006 <<http:\/\/www.lemonde.fr>>.\n\n\u2014\u2014. \"L'enfant trahi.\" _Amours sorci\u00e8res_. Paris: Seuil, 2003. 248-53.\n\n\u2014\u2014. _Hospitalit\u00e9 fran\u00e7aise: Racisme et immigration maghr\u00e9bine_. Paris: Seuil, 1984.\n\n\u2014\u2014. _L'Islam expliqu\u00e9 aux enfants_. Paris: Seuil, 2002. Trans. as _Islam explained_. New York: New Press, 2002.\n\n\u2014\u2014. _La nuit sacr\u00e9e_. Paris: Seuil, 1987.\n\n\u2014\u2014. _Le racisme expliqu\u00e9 \u00e0 ma fille_. Paris: Seuil, 1998.\n\nBensma\u00efa, R\u00e9da. \"The Phantom Mediators: Reflections on the Nature of the Violence in Algeria.\" _Diacritics_ 27.2 (1997): 85-97.\n\nBensoussan, Albert. \"L'enfant perdu.\" _Une enfance alg\u00e9rienne_. Ed. Le\u00efla Sebbar. Paris: Gallimard, 1999 [1997]. 41-54.\n\nBerger, John. _Ways of Seeing_. London & New York: BBC & Penguin, 1977. [1972].\n\nBorradori, Giovanna. _Philosophy in a Time of Terror. Dialogues with J\u00fcrgen Habermas and Jacques Derrida_. Chicago & London: University of Chicago Press, 2003.\n\nBouamama, Sa\u00efd. _L'affaire du foulard islamique. La production d'un racisme respectable_. Le Geai bleu, 2004.\n\nBoudjedra, Rachid. _Le FIS de la haine_. Paris: Deno\u00ebl, 1992.\n\n\u2014\u2014. \"Postface.\" _Le FIS de la haine_. 2nd edition. Paris: Deno\u00ebl, 1994. 115-35.\n\nBoudjellal, Farid. _Le beurgeois_. Toulon: Soleil, 1997.\n\n\u2014\u2014. _Le gourbi_. Paris: Futuropolis, 1985.\n\n\u2014\u2014. _JuifsArabes_. Paris: Futuropolis, 2006.\n\n\u2014\u2014. _L'oud_. Paris: Futuropolis, 1983.\n\n\u2014\u2014. _Ramad\u00e2n_. Paris: Futuropolis, 1988.\n\nBoudjellal, Farid, and Martine Legardette. _Jambon-beur: Les couples mixtes_. Toulon: Soleil, 1995.\n\nBourdieu, Pierre. _Contre-feux 2: Pour un mouvement social europ\u00e9en_. Paris: Raisons D'Agir, 2001.\n\n\u2014\u2014. \"Un probl\u00e8me peut en cacher un autre.\" _Le foulard islamique en questions_. Ed. Charlotte Nordmann. Paris: Editions Amsterdam, 2004. 45-46.\n\nBourget, Carine. _Coran et tradition islamique dans la litt\u00e9rature maghr\u00e9bine_. Paris: Karthala, 2002.\n\n\u2014\u2014. \"A l'\u00e9crit sans images: Le foulard islamique dans des oeuvres de Le\u00efla Sebbar et Yamina Benguigui.\" _Expressions maghr\u00e9bines_ 6.1 (Summer 2007): 19-35.\n\nBouzar, Dounia. \"Fran\u00e7aises et musulmanes, entre r\u00e9appropriation et remise en question des normes.\" _Le foulard islamique en questions_. Ed. Charlotte Nordmann. Paris: Editions Amsterdam, 2004. 54-63.\n\nBouzar, Dounia, and Sa\u00efda Kada. _L'une voil\u00e9e, l'autre pas. Le t\u00e9moignage de deux musulmanes fran\u00e7aises_. Paris: Albin Michel, 2003.\n\nBowen, John. _Why the French Don't Like Headscarves: Islam, the State, and Public Space_. Princeton: Princeton University Press, 2007.\n\nBoyle, Elizabeth Heger. _Female Genital Cutting. Cultural Conflict in the Global Community_. Baltimore and London: Johns Hopkins University Press, 2002.\n\nBrown, Bill. \"The Dark Wood of Postmodernity (Space, Faith, Allegory).\" _PMLA_ 120.3 (May 2005): 734-50.\n\nBurgat, Fran\u00e7ois. _The Islamic Movement in North Africa_. Translation of an updated version of _L'Islamisme au Maghreb_. Trans. William Dowell. 2nd edition. Austin: Center for Middle Eastern Studies at The University of Texas at Austin, 1997.\n\nButler, Judith. _Precarious Life. The Powers of Mourning and Violence_. London, New York: Verso, 2004.\n\nCarlier, Omar. \"Violence(s).\" _La guerre d'Alg\u00e9rie: 1954-2004, la fin de l'amn\u00e9sie_. Ed. Benjamin Stora and Mohammed Harbi. Paris: R. Laffont, 2004. 347-79.\n\nCarrard, Philippe. _Poetics of the New History. French Historical Discourse from Braudel to Chartier_. Baltimore and London: John Hopkins University Press, 1992.\n\nCaws, Mary Ann, ed. _Ecritures de femmes: Nouvelles cartographies_. New Haven: Yale University Press, 1996.\n\nCesari, Jocelyne. _Etre musulman en France_. Paris: Karthala, 1994.\n\n\u2014\u2014. \"Islam in France: The Shaping of a Religious Minority.\" _Muslims in the West: From Sojourners to Citizens_. Ed. Yvonne Haddad. New York: Oxford University Press, 2002. 35-51.\n\nCh\u00e9did, Andr\u00e9e. _La maison sans racines_. Paris: Flammarion, 1985. Trans. as _The Return to Beirut_ , by Ros Schwartz. London: Serpent's Tail, 1989.\n\nChomsky, Noam. \"Middle East Terrorism and the American Ideological System.\" _Blaming the Victims: Spurious Scholarship and the Palestinian Question_. Ed. Edward Said and Christopher Hitchens. London & New York: Verso, 2001 [1988]. 97-147.\n\nChow, Rey. \"Forum: Responses to Bill Brown's 'The Dark Wood of Postmodernity (Space, Faith, Allegory).' \" _PMLA_ 120.3 (May 2005): 874-85.\n\nChra\u00efbi, Driss. _L'homme du livre_. Paris: Balland, 1995.\n\nCixous, H\u00e9l\u00e8ne. \"Mon alg\u00e9riance.\" _Les inrockuptibles_ 115 (20 ao\u00fbt-2 septembre) 1997: 71-74.\n\n\u2014\u2014. \"My Algeriance, in Other Words: To Depart Not to Arrive from Algeria.\" Trans. Eric Prenowitz. _Stigmata: Escaping Texts_. London, UK: Routledge, 1998. 153-72.\n\n\u2014\u2014. \"Pieds nus.\" _Une enfance alg\u00e9rienne_. Ed. Le\u00efla Sebbar. Paris: Gallimard, 1999 [1997]. 55-66.\n\nCocteau, Jean. _Three Screenplays. L'\u00e9ternel retour, Orph\u00e9e, la Belle et la B\u00eate_. Trans. Carol Martin-Sperry. New York: Grossman Publishers, 1972.\n\nCohen, Annie. \"Viridiana mon amour.\" _Une enfance alg\u00e9rienne_. Ed. Le\u00efla Sebbar. Paris: Gallimard, 1999 [1997]. 67-81.\n\nCohen, William. \"Pied-noir Memory, History, and the Algerian War.\" _Europe's Invisible Migrants_. Ed. Andrea Smith. Amsterdam: Amsterdam University Press, 2002. 129-44.\n\nColombani, Jean-Marie. \"Nous sommes tous Am\u00e9ricains.\" _Le Monde_ 13 Sept. 2001. Web. 26 Jan. 2006 <<http:\/\/www.lemonde.fr>>.\n\nConan, Eric, and Christian Makaria. \"Enqu\u00eate sur la mont\u00e9e de l'islam en Europe.\" _L'Express_ 26 Jan. 2006.\n\nConstable, Giles. \"The Historiography of the Crusades.\" _The Crusades from the Perspective of Byzantium and the Muslim World_. Ed. Angeliki Laiou & Roy Mottahedeh. Washington, DC: Dumbarton Oaks Research Library and Collection, 2001. 1-22.\n\nCooke, Miriam. \"Dying to Be Free: Wilderness Writing from Lebanon, Arabia, and Libya.\" Ed. Cheryl Toman. _On Evelyne Accad: Essays in Literature, Feminism, and Cultural Studies_. Birmingham, AL: Summa, 2007. 13-31.\n\n\u2014\u2014. _War's Other Voices. Women Writers on the Lebanese Civil War_. Cambridge: Cambridge University Press, 1988.\n\nCorm, Carol. \"Definitions and Dialects of the Novel. Interview with Myriam Antaki.\" _Aljadid_ 9.45 (Fall 2003): 15-16.\n\nCr\u00e9pon, Marc. _L'Imposture du choc des civilisations_. Nantes: Pleins Feux, 2002.\n\nCuller, Jonathan. _On Deconstruction. Theory and Criticism after Structuralism_. Ithaca, NY: Cornell University Press, 1982.\n\nDadoun, Roger. \"Hammam.\" _Une enfance alg\u00e9rienne_. Ed. Le\u00efla Sebbar. Paris: Gallimard, 1999 [1997]. 83-98.\n\nDaniel, Jean. \"Arr\u00eats sur images.\" _Une enfance alg\u00e9rienne_. Ed. Le\u00efla Sebbar. Paris: Gallimard, 1999 [1997]. 99-114.\n\nDareer, Asma El. _Woman, Why Do You Weep? Circumcision and Its Consequences_. London: Zed Press, 1982.\n\nDe Certeau, Michel. _L'\u00e9criture de l'histoire_. Paris: Gallimard, 1975. Trans. as _The Writing of History_ , by Tom Conley. New York: Columbia University Press, 1988.\n\n\u2014\u2014. \"History: Science and Fiction.\" _Heterologies: Discourse on the Other_. Minneapolis: University of Minnesota Press, 1986. 199-221.\n\nDe Goncourt, Edmond, and Jules de Goncourt. _Germinie Lacerteux_. Paris: Flammarion, n. d. Trans. as _Germinie Lacerteux_ , by John Chestershire. Philadelphia: George Barrie & Sons, 1897.\n\nDeleuze, Gilles, and F\u00e9lix Guattari. _Mille plateaux_. Paris: Minuit, 1980.\n\nDelphy, Christine. \"Une affaire fran\u00e7aise.\" _Le foulard islamique en questions_. Ed. Charlotte Nordmann. Paris: Editions Amsterdam, 2004. 64-71.\n\nDesai, Gaurav. In \"Editor's Column: The End of Postcolonial Theory? A Roundtable with Sunil Agnani, Fernando Coronil, Gaurav Desai, Mamadou Diouf, Simon Gikandi, Susie Tharu, and Jennifer Wenzel.\" _PMLA_ 122.3 (May 2007): 641-42.\n\nDib, Mohammed. \"Rencontres.\" _Une enfance alg\u00e9rienne_. Ed. Le\u00efla Sebbar. Paris: Gallimard, 1999 [1997]. 115-26.\n\nDirlik, Arif. \"History without a Center? Reflections on Eurocentrism.\" In _Across Cultural Borders: Historiography in Global Perspective_. Ed. E. Fuchs & B. Stuchtey. Lanham, MD: Rowman & Littlefield, 2002. 247-60.\n\nDjebar, Assia. _Loin de M\u00e9dine_. Paris: Albin Michel, 1991.\n\nDonadey, Anne. \"Foreword.\" _An Algerian Childhood_. Ed. Le\u00efla Sebbar. Trans. Marjolijn de Jager. St. Paul, MN : Ruminator Books, 2001. vii-xvii.\n\n\u2014\u2014. _Recasting Postcolonialism: Women Writing between Worlds_. Portsmouth, NH: Heinemann, 2001.\n\nDonadey, Anne, and Huma Ahmed-Ghosh. \"Why Americans Love Azar Nafisi's _Reading Lolita in Tehran\" Signs: Journal of Women in Culture and Society_ 33.3 (2008): 623-46.\n\nDonadey, Anne, and Adlai Murdoch. _Postcolonial Theory and Francophone Literary Studies_. Gainesville, FL: University Press of Florida, 2005.\n\nDouglas, Allen, and Fedwa Malti-Douglas. _Arab Comic Strips: Politics of an Emerging Mass Culture_. Bloomington & Indianapolis: Indiana University Press, 1994.\n\nDrake, David. _Intellectuals and Politics in Post-War France_. New York: Palgrave, 2002.\n\nDurmelat, Sylvie. \"L'Alg\u00e9rie est \u00e0 r\u00e9inventer ou Femmes d'Alger hors de leur appartement dans _Viva Laldj\u00e9rie_ de Nadir Mokn\u00e8che.\" _Expressions maghr\u00e9bines_ 6.1 (Summer 2007): 93-112.\n\n\u2014\u2014. \"Transmission and Mourning in _M\u00e9moires d'immigr\u00e9s: L'h\u00e9ritage maghr\u00e9bin_ : Yamina Benguigui as 'Memory Entrepreneuse.' \" _Women, Immigration and Identities in France_. Ed. Carrie Tarr and Jane Freedman. Oxford, NY: Berg, 2000. 171-90.\n\nEaglestone, Robert. \"The Age of Reason Is Over . . . an Age of Fury Was Dawning.\" _Wasafiri_ 22.2 (2007): 19-22.\n\nEl-Khoury, Barbara. \"La fonction du voile et de l'excision chez Evelyne Accad.\" _Evelyne Accad: Explorations_. Ed. Deirdre Heistad. Paris: L'Harmattan, 2005. 123-46.\n\nEl Maleh, Edmond Amran. \"Au seuil de l'interdit : Interrogations.\" _Revue d'\u00e9tudes palestiniennes_ 2.2 (Winter 1982): 18-34.\n\n\u2014\u2014. _Mille ans, un jour_. Marseille: Andr\u00e9 Dimanche, 2002. [1986].\n\n\u2014\u2014. \"Le visage d'une n\u00e9gation.\" _Revue d'\u00e9tudes palestiniennes_ 5 (Automne 1982): 17-24.\n\nEl-Solh, Camillia Fawzi, and Judy Mabro, eds. _Muslim Women's Choices: Religious Belief and Social Reality_. Providence, RI: Berg, 1994.\n\nEntelis, John. \"Islamist Politics and the Democratic Imperative: Comparative Lessons from the Algerian Experience.\" _The Journal of North African Studies_ 9.2 (Summer 2004): 202-15.\n\nEsposito, John. \"Religion and Global Affairs: Political Challenges.\" _SAIS Review_ 18.2 (1998): 19-24.\n\nEuben, Roxanne. _Enemy in the Mirror: Islamic Fundamentalism and the Limits of Modern Rationalism_. Princeton, NJ: Princeton University Press, 1999.\n\nFar\u00e8s, Nabile. \"La m\u00e9moire des autres.\" _Une enfance alg\u00e9rienne_. Ed. Le\u00efla Sebbar. Paris: Gallimard, 1999 [1997]. 127-42.\n\nFernea, Robert. Foreword. _The Islamic Movement in North Africa_. By Fran\u00e7ois Burgat. Translation of an updated version of _L'Islamisme au Maghreb_. Trans. William Dowell. 2nd edition. Austin: Center for Middle Eastern Studies at The University of Texas at Austin, 1997. ix-xi.\n\nForsdick, Charles, and David Murphy. _Francophone Postcolonial Studies: A Critical Introduction_. London: Arnold, 2003.\n\nFuller, Graham. _The Future of Political Islam_. New York: Palgrave Macmillan, 2003.\n\nGafa\u00efti, Hafid, ed. _Rachid Boudjedra: Une po\u00e9tique de la subversion_. Paris: L'Harmattan, 1999.\n\nGalesne, Nathalie. \" _Les versets du pardon_ , ou 'le livre de la paix'.\" _La pens\u00e9e de midi_ : 157-59.\n\nGallaire, Fatima. \"Ba\u00ef.\" _Une enfance alg\u00e9rienne_. Ed. Le\u00efla Sebbar. Paris: Gallimard, 1999 [1997]. 143-58.\n\nGambetta, Diego. \"Can We Make Sense of Suicide Missions?\" _Making Sense of Suicide Missions_. Ed. Diego Gambetta. Oxford: Oxford University Press, 2005. 259-99.\n\n\u2014\u2014_, ed. _Making Sense of Suicide Missions_. Oxford: Oxford University Press, 2005.\n\nGaspard, Fran\u00e7oise, and Farhad Khosrokhavar. _Le foulard et la R\u00e9publique_. Paris: La D\u00e9couverte, 1995.\n\n\u2014\u2014. \"The Headscarf and the Republic.\" _Beyond French Feminisms: Debates on Women, Politics, and Culture in France, 1981-2001_. Ed. Roger C\u00e9lestin, Eliane Dalmolin, and Isabelle de Courtivron. New York: Palgrave Macmillan, 2003. 61-67.\n\nGaumer, Patrick, and Claude Moliterni. _Dictionnaire mondial de la bande dessin\u00e9e_. 2nd ed. Paris: Larousse, 2001.\n\nGenette, G\u00e9rard. _Figures III_. Paris: Seuil, 1972.\n\n\u2014\u2014. _Paratexts: Thresholds of Interpretation_. Trans. Jane Lewin. Cambridge: Cambridge University Press, 1997.\n\nGiraud, V\u00e9ronique, Yves Sintomer, and Alma and Lila L\u00e9vy. _Des filles comme les autres: Au-del\u00e0 du foulard_. Entretiens. Paris: La D\u00e9couverte, 2004.\n\nGopin, Marc. \"Forgiveness as an Element of Conflict Resolution in Religious Cultures: Walking the Tightrope of Reconciliation and Justice.\" _Reconciliation, Justice, and Coexistence: Theory and Practice_. Ed. Mohammed Abu-Nimer. Lanham, MD: Lexington Books, 2001. 87-99.\n\n\u2014\u2014. _Holy War, Holy Peace: How Religion Can Bring Peace to the Middle East_. New York: Oxford University Press, 2002.\n\nGossman, Lionel. _Between History and Literature_. Cambridge, MA: Harvard University Press, 1990.\n\nGravdal, Kathryn. _Ravishing Maidens: Writing Rape in Medieval French Literature and Law_. Philadelphia: University of Pennsylvania Press, 1991.\n\nGrewal, Inderpal, and Caren Kaplan. \"Warrior Marks: Global Womanism's Neo-Colonial Discourse in a Multicultural Context.\" _Camera Obscura: A Journal of Feminism and Film Theory_ 39 (1996): 5-33.\n\nGroensteen, Thierry. _The System of Comics_. Trans. Bart Beaty and Nick Nguyen. Jackson: University Press of Mississippi, 2007.\n\nGross, Janice. \"The Tragedy of Algeria: Slimane Bena\u00efssa's Drama of Terrorism.\" _Theatre Journal_ 54 (2002): 369-87.\n\nGross, Janice, and Daniel Gross. \"Translators' Notes.\" _The Last Night of a Damned Soul_. By Slimane Bena\u00efssa. Trans. Janice and Daniel Gross. New York: Grove Press, 2004.\n\nGrosser, Paul, and Edwin Halperin. _Anti-Semitism: Causes and Effects_. New York: Philosophical Library, 1983.\n\nGrousset, Ren\u00e9. _Les croisades_. Paris: PUF, 1960.\n\n\u2014\u2014. _Histoire des croisades_. 3 vol. Paris: Plon, 1934-1936.\n\nGruenbaum, Ellen. _The Female Circumcision Controversy. An Anthropological Perspective_. Philadelphia: The University of Pennsylvania Press, 2001.\n\nGu\u00e9noun, Solange. \"Accueils et \u00e9cueils identitaires-communautaires en France postcoloniale: 'Ils disent que je suis s\u00e9pharade . . . .' \" _Contemporary French and Francophone Studies_ 11.2 (April 2007): 217-30.\n\nHaarscher, Guy. _La La\u00efcit\u00e9_. Paris: PUF, 1996.\n\nHallam, Elizabeth, ed. _Chronicles of the Crusades. Eye-Witness Accounts of the Wars between Christianity and Islam_. London: Weidenfeld & Nicolson, 1989.\n\nHarbi, Mohammed. \"L'Alg\u00e9rie en perspectives.\" _La guerre d'Alg\u00e9rie: 1954-2004, la fin de l'amn\u00e9sie_. Ed. Benjamin Stora and Mohammed Harbi. Paris: R. Laffont, 2004. 27-45.\n\nHarbi, Mohammed, and Benjamin Stora, eds. _La guerre d'Alg\u00e9rie: 1954-2004, la fin de l'amn\u00e9sie_. Paris: R. Laffont, 2004.\n\nHare, William. _Hitchcock and the Methods of Suspense_. Jefferson, NC & London: McFarland, 2007.\n\nHargreaves, Alec. \"The Challenges of Multiculturalism: Regional and Religious Differences in France Today.\" _Contemporary French Cultural Studies_. Ed. William Kidd and Si\u00e2n Reynolds. London: Arnold, 2000. 95-110.\n\nHarrison, Nicholas. \"Learning from Experience: H\u00e9l\u00e8ne Cixous's 'Pieds nus.' \" _Paragraph: A Journal of Modern Critical Theory_ 27.1 (March 2004): 21-32.\n\nHartman, Michelle. \"Multiple Identities, Multiple Voices: Reading Andr\u00e9e Ch\u00e9did's _La maison sans racines.\" French Studies_ 54.1 (2000): 54-66.\n\nHasenclever, Andreas, and Volker Rittberger. \"Does Religion Make a Difference? Theoretical Approaches to the Impact of Faith on Political Conflict.\" _Religion in International Relations. The Return from Exile_. Ed. Hatzopoulos, Pavlos and Fabio Petito. New York: Palgrave Macmillan, 2003. 107-45.\n\nHassan, Salah. \"Undertaking Partition: Palestine and Postcolonial Studies.\" _Journal X_ 6.1 (Autumn 2001): 19-45.\n\nHatzopoulos, Pavlos, and Fabio Petito, eds. _Religion in International Relations. The Return from Exile_. New York: Palgrave Macmillan, 2003.\n\nHeistad, Deirdre, ed. _Evelyne Accad: Explorations_. Paris: L'Harmattan, 2005.\n\n\u2014\u2014. \"Writings from the Borderzone: Tales of Recuperation and Transgression in the Works of Malika Mokeddem, Calixthe Beyala, and Evelyne Accad.\" Diss. University of Illinois at Urbana, 2000.\n\nHerman, Edward S., and Noam Chomsky. _Manufacturing Consent: The Political Economy of the Mass Media_. New York: Pantheon Books, 1988.\n\nHiggins, Lynn. _New Novel, New Wave, New Politics: Fiction and the Representation of History in Postwar France_. Lincoln: University of Nebraska Press, 1996.\n\n\u2014\u2014. \"Screen\/Memory: Rape and Its Alibis in _Last Year at Marienbad_.\" _Rape and Representation_. Ed. Lynn Higgins and Brenda Silver. New York: Columbia University Press, 1991. 303-21.\n\nHochberg, Gil. _In Spite of Partition: Jews, Arabs, and the Limits of Separatist Imagination_. Princeton & Oxford: Princeton University Press, 2007.\n\nHolmes, Stephen. \"Al-Qaeda, September 11, 2001.\" _Making Sense of Suicide Missions_. Ed. Diego Gambetta. Oxford: Oxford University Press, 2005. 131-72.\n\n_The Holy Qur'an_. Trans. Abdullah Yusuf Ali. Washington, DC: Khalil Al-Rawaf, 1946. [1934].\n\nHopgood, Stephen. \"Tamil Tigers, 1987-2002.\" _Making Sense of Suicide Missions_. Ed. Diego Gambetta. Oxford: Oxford University Press, 2005. 43-76.\n\nHosken, Fran. _Stop Female Genital Mutilation. Women Speak. Facts and Actions_. Lexington, MA: Women's International Network News, 1995.\n\nHottell, Ruth. \"A Poetics of Pain: Evelyne Accad's Critical and Fictional World.\" _World Literature Today_ 71.3 (Summer 1997): 511-17.\n\nHouen, Alex. \"Novel Spaces and Taking Place(s) in the Wake of September 11.\" _Studies in the Novel_ 36.3 (Fall 2004): 419-37.\n\nHuston, Nancy, and Le\u00efla Sebbar, eds. _Une enfance d'ailleurs_. Paris: Belfond, 1993.\n\nHutcheon, Linda. _A Poetics of Postmodernism_. London & New York: Routledge, 1988.\n\n\u2014\u2014. _The Politics of Postmodernism_. London & New York: Routledge, 1989.\n\nIbn Munqidh, Usamah. _An Arab-Syrian Gentleman and Warrior in the Period of the Crusades_. Trans. Philip K. Hitti. New York: Columbia University Press, 1929.\n\nImache, Djedjiga, and In\u00e8s Nour. _Alg\u00e9riennes entre Islam et islamisme_. Aix-en-Provence: Edisud, 1994.\n\nIssa, Amira. \"R\u00e9volte et image christique dans _L'excis\u00e9e_ d'Evelyne Accad.\" _Evelyne Accad: Explorations_. Ed. Deirdre Heistad. Paris: L'Harmattan, 2005. 163-72.\n\nJadla, Ibrahim. \"Les juifs en Ifriqiya \u00e0 l'\u00e9poque hafside.\" _Histoire communautaire, histoire plurielle: La communaut\u00e9 juive de Tunisie_. Tunis: Centre de Publication Universitaire, 1999. 145-51.\n\nJaschik, Scott. \"Middle East Tensions Flare Again in U.S. (Update).\" _Inside Higher Ed_ 5 Sept. 2007. Web. 28 March 2008. <<http:\/\/www.insidehighered.com>>.\n\nKacimi-El-Hassani, Mohamed. \"A la claire ind\u00e9pendance.\" _Une enfance alg\u00e9rienne_. Ed. Le\u00efla Sebbar. Paris: Gallimard, 1999 [1997]. 159-74.\n\nKalyvas, Stathis, and Ignacio S\u00e1nchez-Cuenca. \"Killing without Dying: The Absence of Suicide Missions.\" _Making Sense of Suicide Missions_. Ed. Diego Gambetta. Oxford: Oxford University Press, 2005. 209-32.\n\nKandiyoti, Deniz. \"Identity and Its Discontents: Women and the Nation.\" _Colonial Discourse and Post-Colonial Theory: A Reader_. Ed. Patrick Williams and Laura Chrisman. New York: Columbia University Press, 1994. 376-91.\n\nKarnoub, Elisabeth. \"Du sacrifice au sacril\u00e8ge: le suicide d'une femme enceinte.\" _Evelyne Accad: Explorations_. Ed. Deirdre Heistad. Paris: L'Harmattan, 2005. 105-22.\n\nKenbib, Mohammed. _Juifs et musulmans au Maroc: 1859-1948_. Rabat: Facult\u00e9 des lettres et des sciences humaines, 1994.\n\nKepel, Gilles. _Les banlieues de l'islam. Naissance d'une religion en France_. Paris: Seuil, 1987.\n\n\u2014\u2014. _Jihad. Expansion et d\u00e9clin de l'islamisme_. Paris: Gallimard, 2000.\n\nKhosrokhavar, Farhad. _Les nouveaux martyrs d'Allah_. Rev. ed. Paris: Flammarion, 2003. [2002].\n\nKinoshita, Sharon. \"The Romance of MiscegeNation: Negotiating Identities in _La fille du comte de Pontieu_.\" _Postcolonial Moves: Medieval through Modern_. Ed. Patricia Ingham and Michelle Warren. New York: Palgrace Macmillan, 2003. 111-31.\n\nKoso-Thomas, Olayinka. _The Circumcision of Women. A Strategy for Eradication_. London & New Jersey: Zed Books Ltd., 1987.\n\nKurtz, Stanley. \"Testimony before the Subcommittee on Select Education, Committee on Education and the Workforce, U.S. House of Representatives, June 19, 2003.\" Web. 14 December 2003. <<http:\/\/edworkforce.house.gov\/hearings\/108th\/sed\/titlevi61903\/kurtz.htm>>.\n\nKurzman, Charles. \"Radical Sheik and the Missing Martyrs: Why Are There So Few Islamic Terrorists?\" Center for Middle Eastern Studies. University of Arizona. 12 Feb. 2006.\n\nLargu\u00e8che, Abdelhamid. \"La communaut\u00e9 juive de Tunis \u00e0 l'\u00e9poque husse\u00efnite: Unit\u00e9, contrastes et relations intercommunautaires.\" _Histoire communautaire, histoire plurielle: La communaut\u00e9 juive de Tunisie_. Tunis: Centre de Publication Universitaire, 1999. 165-80.\n\nLaronche, Martine. \"La loi sur le voile \u00e0 conduit \u00e0 deux premi\u00e8res exclusions.\" _Le Monde_ 21 Octobre 2004.\n\nLaroussi, Farid. \"When Francophone Means National: The Case of the Maghreb.\" _Yale French Studies_ 103 (2003): 81-90.\n\nLaskier, Michael M. _North African Jewry in the Twentieth Century: The Jews of Morocco, Tunisia, and Algeria_. New York: New York University Press, 1994.\n\nLayachi, Azzedine. \"Political Liberalisation and the Islamist Movement in Algeria.\" _The Journal of North African Studies_ 9.2 (Summer 2004): 46-67.\n\n\" 'Le dernier soir' avant le 11 septembre.\" _Le Monde 2_ Oct. 2001. Web. 3 August 2007 <<http:\/\/www.lemonde.fr>>.\n\nL\u00e9vy, Simon. _Essais d'histoire et de civilisation jud\u00e9o-marocaine_. Rabat: Centre Tarik Ibn Zyad, 2001.\n\nLewis, Bernard. _Histoire du Moyen-Orient_. Trans. J. Arnaud and J. Lahana. Paris: Albin Michel, 1997.\n\nLionnet, Fran\u00e7oise. \"Dissymmetry Embodied: Nawal El Saadawi's _Woman at Point Zero_ and the Practice of Excision.\" Chapter 6. _Postcolonial Representations: Women, Literature, Identity_. Ithaca & London: Cornell University Press, 1995. 129-53.\n\nLloyd, Simon. \"The Crusading Movement 1096-1274.\" _The Oxford Illustrated History of the Crusades_. Ed. J. Riley-Smith. Oxford: Oxford University Press, 1995. 34-65.\n\nMaalouf, Amin. _Les croisades vues par les Arabes_. Paris: Latt\u00e8s, 1983. Trans. as _The Crusades through Arab Eyes_ , by Jon Rothschild. London: Al Saqi Books, 1984.\n\n\u2014\u2014. _Les \u00e9chelles du Levant_. Paris: Grasset, 1996. Trans. as _Ports of Call_ , by Alberto Manguel. London: The Harvill Press, 1999.\n\n\u2014\u2014. _Les identit\u00e9s meurtri\u00e8res_. Paris: Grasset, 1998. Trans. as _In the Name of Identity: Violence and the Need to Belong_ , by Barbara Bray. New York: Arcade Publishing, 2000.\n\nMackie, Gerry. \"Female Genital Cutting: The Beginning of the End.\" _Female \"Circumcision\" in Africa: Culture, Controversy, and Change_. Ed. Bettina Shell-Duncan and Ylva Hernlund. Boulder & London: Lynne Rienner, 2000. 253-81.\n\nMajid, Anouar. \"The Failure of Postcolonial Theory after 9\/11.\" _Chronicle of Higher Education_ 1 Nov. 2002: B11-12.\n\n\u2014\u2014. _Freedom and Orthodoxy: Islam and Difference in the Post-Andalusian Age_. Stanford, CA: Stanford University Press, 2004.\n\n\u2014\u2014. \"Islam and the Literature of Controversy.\" _The Postcolonial Crescent: Islam's Impact on Contemporary Literature_. Ed. John C. Hawley. New York: Peter Lang, 1998. 84-91.\n\n\u2014\u2014. _Unveiling Traditions: Postcolonial Islam in a Polycentric World_. Durham, NC; London: Duke University Press, 2000.\n\nMarie, Elisabeth Anne. \"Sacrifice, sacrifi\u00e9e, sacrificatrice: L'\u00e9trange triptyque. Sacrifices au f\u00e9minin dans trois romans francophones libanais.\" Diss. The University of North Carolina at Chapel Hill, 2002.\n\nMartinez, Louis. _The Algerian Civil War 1990-1998_. Trans. Jonathan Derrick. London: Hurst & Company, 2000.\n\n\u2014\u2014. \"Why the Violence in Algeria?\" _Journal of North African Studies 9.2_ (Summer 2004): 14-27.\n\nMarx-Scouras, Danielle. \"The Poetics of Maghrebine Illegitimacy.\" _L'esprit cr\u00e9ateur_ 26.1 (Spring 1986): 3-10.\n\nMasson, Pierre. _Lire la bande dessin\u00e9e_. Lyon: Presses universitaires de Lyon, 1985.\n\nMcDermott, Terry. _Perfect Soldiers: The Hijackers. Who They Were, Why They Did It_. New York: HarperCollins, 2005.\n\nMcKinney, Mark. \" _M\u00e9tissage_ in Post-Colonial Comics.\" _Post-Colonial Cultures in France_. Ed. Alec Hargreaves and Mark McKinney. London & New York: Routledge, 1997. 169-88.\n\nMcQuillan, Libbie. \"The Francophone Bande Dessin\u00e9e: An Introduction.\" _The Francophone Bande Dessin\u00e9e_. Ed. Charles Forsdick, Laurence Grove, and Libbie McQuillan. Amsterdam, NY: Rodopi, 2005. 7-13.\n\nMearsheimer, John, and Stephen Walt. _The Israel Lobby and U.S. Foreign Policy_. New York: Farrar, Straus and Giroux, 2007.\n\nMeddeb, Abdelwahab. _Face \u00e0 l'islam_. Paris: Textuel, 2004.\n\n\u2014\u2014. _La maladie de l'islam_. Paris: Seuil, 2002. Trans. as _The Malady of Islam_ , by Pierre Joris and Ann Reid. New York: Basic Books, 2003, and _Islam and Its Discontents_. London: William Heinemann, 2003.\n\nMekay, Emad. \"Female Genital Mutilation Continues, Says Report.\" _Inter Press Service_ 7 February 2004. <<http:\/\/www.commondreams.org\/headlines04\/0207-06.htm>>.\n\nMemmi, Albert. \"La folie int\u00e9griste.\" _Le magazine litt\u00e9raire_ 323 (1994) : 31-32.\n\n\u2014\u2014. _Juifs et Arabes_. Paris: Gallimard, 1974. Trans. as _Jews and Arabs_ , by Eleanor Levieux. Chicago: J. P. O'Hara, 1975.\n\n\u2014\u2014. _Portrait du d\u00e9colonis\u00e9 arabo-musulman et de quelques autres_. Paris: Gallimard, 2004. Trans. as _Decolonization and the Decolonized_ , by Robert Bononno. Minneapolis: University of Minnesota Press, 2006.\n\n\u2014\u2014. _La statue de sel_. Paris: Gallimard, 1966. Trans. as _The Pillar of Salt_ by Edouard Roditi. Boston, MA: Beacon Press, 1992. [1955].\n\nMernissi, Fatima. _Dreams of Trespass_. Cambridge, MA: Perseus Books, 1994.\n\n\u2014\u2014. _Le harem politique_. Paris: Albin Michel, 1987.\n\n\u2014\u2014. \"Palace Fundamentalism and Liberal Democracy.\" _The New Crusades: Constructing the Muslim Enemy_. Ed. Emran Qureshi and Michael Sells. New York: Columbia University Press, 2003. 51-67.\n\n\u2014\u2014. _Sultanes oubli\u00e9es_. Casablanca: Le Fennec, Paris: Albin Michel, 1990. Trans. as _Forgotten Queens of Islam_ , by Mary Jo Lakeland. Cambridge: Polity Press, 1993.\n\nMichaud, Joseph Fran\u00e7ois. _Biblioth\u00e8que des croisades_. New York: AMS Press, 1978. [reprint of the 1829 ed. published by A. J. Ducollet, Paris]. 4 vol.\n\nMillecam, Jean-Pierre. \"Apocalypses.\" _Une enfance alg\u00e9rienne_. Ed. Le\u00efla Sebbar. Paris: Gallimard, 1999 [1997]. 175-90.\n\nMimouni, Rachid. _De la barbarie en g\u00e9n\u00e9ral et de l'int\u00e9grisme en particulier_. Tunis: C\u00e9r\u00e8s Productions, 1992.\n\nMiraglia, Anne-Marie. \"Le temps et ses reflets dans _La maison sans racines_ d'Andr\u00e9e Ch\u00e9did.\" _Francophonia_ 35 (Fall 1998): 17-33.\n\nMitchell, W.J.T. _What Do Pictures Want?_ Chicago & London: University of Chicago Press, 2005.\n\nMohanty, Chandra Talpade. \"Under Western Eyes: Feminist Scholarship and Colonial Discourses.\" _Third World Women and the Politics of Feminism_. Ed. Chandra Talpade Mohanty, Ann Russo, Lourdes Torres. Bloomington: Indiana University Press, 1991. 51-80.\n\nMortimer, Robert. \"Islamists, Soldiers, and Democrats: The Second Algerian War.\" _Middle East Journal_ 50.1 (Winter 1996): 18-39.\n\nMorton, Stephen. \"Terrorism, Orientalism and Imperialism.\" _Wasafiri_ 22.2 (July 2007): 36-42.\n\nMudimbe-Boyi, Elisabeth. \"Enfermer et contr\u00f4ler: Lieu social et espace textuel dans _L'excis\u00e9e_ de Evelyne Accad.\" _Francofonia: Studi e ricerche sulle letterature di lingua francese_ 23 (Fall 1992): 3-19.\n\nNavarro, Mireya. \"The Most Private of Makeovers.\" _New York Times_ 28 November 2004. Web. 12 March 2009. <<http:\/\/www.nytimes.com>>.\n\nNaylor, Philip. _France and Algeria: A History of Decolonization and Transformation_. Gainesville: University Press of Florida, 2000.\n\n_The New Oxford Annotated Bible_. Ed. Bruce Metzger and Roland Murphy. New Revised Standard Edition. New York: Oxford University Press, 1991.\n\nNixon, Rob. \"Among the Mimics and Parasites: V. S. Naipaul's Islam.\" _The New Crusades: Constructing the Muslim Enemy_. Ed. Emran Qureshi and Michael A. Sells. New York; Chichester [England]: Columbia University Press, 2003. 152-69.\n\nNoiriel, G\u00e9rard. _Les fils maudits de la R\u00e9publique. L'avenir des intellectuels en France_. Paris: Fayard, 2005.\n\nNordmann, Charlotte, ed. _Le foulard islamique en questions_. Paris: Editions Amsterdam, 2004.\n\nNordmann, Charlotte, and J\u00e9r\u00f4me Vidal. \"La R\u00e9publique \u00e0 l'\u00e9preuve des discriminations.\" _Le foulard islamique en questions_. Ed. Charlotte Nordmann. Paris: Editions Amsterdam, 2004. 5-14.\n\nNoudelmann, Fran\u00e7ois. \"Introduction.\" _Politique et filiation_. Ed. R. Harvey, E. Ann Kaplan, and Fran\u00e7ois Noudelmann. Paris: Kim\u00e9, 2004. 9-12.\n\n\u2014\u2014. _Pour en finir avec la g\u00e9n\u00e9alogie_. Paris: L\u00e9o Scheer, 2004.\n\n\u2014\u2014. \"Pour une pens\u00e9e archip\u00e9lique, Edouard Glissant.\" _Politique et filiation_. Ed. R. Harvey, E. Ann Kaplan, and Fran\u00e7ois Noudelmann. Paris: Kim\u00e9, 2004. 194-205.\n\nOldenbourg, Zo\u00e9. _Les croisades_. Paris: Gallimard, 1965.\n\nPapp\u00e9, Ilan. _A History of Modern Palestine_. 2nd Edition. Cambridge: Cambridge University Press, 2006. [2004].\n\nPedahzur, Ami. _Suicide Terrorism_. Cambridge, UK: Polity Press, 2005.\n\nP\u00e9l\u00e9gri, Jean. \"Quand les oiseaux se taisent . . .\" _Une enfance alg\u00e9rienne_. Ed. Le\u00efla Sebbar. Paris: Gallimard, 1999 [1997]. 191-202.\n\nPodselver, Laurence. \"Le p\u00e8lerinage d'El Hamma \u00e0 Sarcelles.\" _Histoire communautaire, histoire plurielle: La communaut\u00e9 juive de Tunisie_. Tunis: Centre de Publication Universitaire, 1999. 275-86.\n\nPratt, Mary Louise. \"Building a New Public Idea about Language.\" _Profession_ 2003: 110-19.\n\nQureshi, Emran, and Michael A. Sells. \"Introduction: Constructing the Muslim Enemy.\" _The New Crusades: Constructing the Muslim Enemy_. Ed. Emran Qureshi and Michael A. Sells. New York: Columbia University Press, 2003. 1-47.\n\nRedonnet, Marie. _Entretiens avec Edmond Amran El Maleh_. Rabat: La pens\u00e9e sauvage, 2005.\n\nReuter, Christoph. _My Life Is a Weapon. A Modern History of Suicide Bombing_. Trans. Helena Ragg-Kirkby. Princeton: Princeton University Press, 2004. [2002].\n\nRicolfi, Luca. \"Palestinians, 1981-2003.\" _Making Sense of Suicide Missions_. Ed. Diego Gambetta. Oxford: Oxford University Press, 2005. 77-129.\n\nRiley-Smith, Jonathan. \"The Crusading Movement and Historians.\" _The Oxford Illustrated History of the Crusades_. Ed. J. Riley-Smith. Oxford: Oxford University Press, 1995. 1-12.\n\nRoquebert, Michel. _L'\u00e9pop\u00e9e cathare. 1198-1212: L'invasion_. Toulouse: Privat, 1970.\n\nRosello, Mireille. _Declining the Stereotype: Ethnicity and Representation in French Cultures_. Hanover & London: University Press of New England, 1998.\n\n\u2014\u2014. \"Mich\u00e8le Maillet's _L'\u00e9toile noire_ : Historian's Counter-History and Translator's Counter-Silence.\" _Callaloo_ 16.1 (Winter 1993): 192-212.\n\n\u2014\u2014. \"Tactical Universalism and New Multiculturalist Claims in Postcolonial France.\" _Francophone Postcolonial Studies. A Critical Introduction_. Ed. Charles Forsdick and David Murphy. London: Arnold, 2003. 135-44.\n\nRosenthal, Franz. _A History of Muslim Historiography_. Leiden: E.J. Brill, 1968.\n\nRoy, Olivier. _Globalised Islam. The Search for a New Ummah_. London: Hurst & Co., 2004. Revised and updated translation of _L'islam mondialis\u00e9_ (Paris: Seuil, 2002).\n\nRunciman, Steven. _A History of the Crusades_. 3 vol. Cambridge: Cambridge University Press, 1951-54.\n\nSaid, Edward. \"The Clash of Definitions.\" _Reflections on Exile and Other Essays_. Cambridge, MA: Harvard University Press, 2000. 569-90.\n\n\u2014\u2014. _Covering Islam. How the Media and the Experts Determine How We See the Rest of the World_. Revised Edition. New York: Vintage Books, 1997. [1981].\n\n\u2014\u2014. _Culture and Imperialism_. New York: Vintage Books, 1994.\n\n\u2014\u2014. _The End of the Peace Process. Oslo and After_. New York: Pantheon Books, 2000.\n\n\u2014\u2014. \"The Essential Terrorist.\" _Blaming the Victims: Spurious Scholarship and the Palestinian Question_. Ed. Edward Said and Christopher Hitchens. London & New York: Verso, 2001 [1988]. 149-58.\n\n\u2014\u2014. _Humanism and Democratic Criticism_. New York: Columbia University Press, 2004.\n\n\u2014\u2014. \"Introduction.\" _Blaming the Victims: Spurious Scholarship and the Palestinian Question_. Ed. Edward Said and Christopher Hitchens. London & New York: Verso, 2001 [1988]. 1-19.\n\n\u2014\u2014. \"Opponents, Audiences, Constituencies, and Community.\" _Reflections on Exile and Other Essays_. Cambridge, MA: Harvard University Press, 2000. 118-47.\n\n\u2014\u2014. _Orientalism_. New York: Vintage Books, 1979. [1978].\n\n\u2014\u2014. _The World, the Text, and the Critic_. Cambridge, Mass.: Harvard University Press, 1983.\n\nSalem, Elise. _Constructing Lebanon: A Century of Literary Narratives_. Gainesville: University Press of Florida, 2003.\n\nSalibi, Kamal. _Crossroads to Civil War. Lebanon 1958-1976_. Delmar, NY: Caravan Books, 1976.\n\n\u2014\u2014. _A House of Many Mansions: The History of Lebanon Reconsidered_. London: I.B. Tauris, 1988.\n\nSassine, Antoine. \"Entretien avec Amin Maalouf: L'homme a ses racines dans le ciel.\" _Etudes francophones_ 14.2 (Fall 1999): 25-36.\n\nSatrapi, Marjane. _Persepolis_. New York: Pantheon Books, 2003. [2000].\n\nSayyid, Salman. _A Fundamental Fear: Eurocentrism and the Emergence of Islamism_. 2nd edition. London & New York: Zed Books, 2003.\n\nScharfman, Ronnie. \"Narratives of Internal Exile. Cixous, Derrida, and the Vichy Years in Algeria.\" _Postcolonial Theory and Francophone Literary Studies_. Ed. H. Adlai Murdoch and Anne Donadey. Gainesville: University Press of Florida, 2005. 87-101.\n\n\u2014\u2014. \"The Other's Other: The Moroccan-Jewish Trajectory of Edmond Amran El Maleh.\" _Yale French Studies_ 82 (1993): 135-45.\n\nScreech, Matthew. _Masters of the Ninth Art: Bandes dessin\u00e9es and Franco-Belgian Identity_. Liverpool: Liverpool University Press, 2005.\n\nSebag, Paul. _Histoire des Juifs de Tunisie: Des origines \u00e0 nos jours_. Paris: L'Harmattan, 1991.\n\nSebbar, Le\u00efla, ed. _Une enfance alg\u00e9rienne_. Paris: Gallimard, 1997. Trans. as _An Algerian Childhood_ , by Marjolijn de Jager. St. Paul, MN: Ruminator Books, 2001.\n\n\u2014\u2014\u2014, ed. _Une enfance outremer_. Paris: Seuil, 2001.\n\n\u2014\u2014. \"La fille au hijeb.\" _Sept filles: nouvelles_. Paris: T. Magnier, 2003. 79-89.\n\n\u2014\u2014. _Journal de mes Alg\u00e9ries en France_. St-Pour\u00e7ain-sur-Sioule [France]: Bleu autour, 2005.\n\n\u2014\u2014. _Mes Alg\u00e9ries en France_. St-Pour\u00e7ain-sur-Sioule [France]: Bleu autour, 2004.\n\n\u2014\u2014. \"On tue des instituteurs.\" _Une enfance alg\u00e9rienne_. Ed. Le\u00efla Sebbar. Paris: Gallimard, 1999 [1997]. 203-14.\n\n\u2014\u2014. \"Pourquoi cette peur?\" _Le Monde_ 24 Oct. 1989.\n\n\u2014\u2014. \"Vierge folle, vierge sage.\" _La jeune fille au balcon_. Paris: Seuil, 1996. 107-24.\n\nSerrano, Richard. _Against the Postcolonial: \"Francophone\" Writers at the Ends of French Empire_. Lanham, MD: Lexington Books, 2005.\n\nShell-Duncan, Bettina, and Ylva Hernlund. \"Female 'Circumcision' in Africa: Dimensions of the Practice and Debates.\" _Female \"Circumcision\" in Africa: Culture, Controversy, and Change_. Ed. Bettina Shell-Duncan and Ylva Hernlund. Boulder & London: Lynne Rienner, 2000. 283-312.\n\nShih, Shu-Mei. \"Global Literature and the Technologies of Recognition.\" _PMLA_ 119.1 (January 2004): 16-30.\n\nShilton, Siobh\u00e1n. \"Negotiating Female Identities: 'Franco-Maghrebi' Journeys in the Visual Arts.\" Maghreb, Visual Arts and the French Exception. The 20th and 21st Century French and Francophone Studies International Colloquium. Hilton Hotel and Conference Center, College Station, TX, 23 March 2007.\n\nShohat, Ella. \"Rupture and Return: Zionist Discourse and the Study of Arab Jews.\" _Social Text_ 21.2 (Summer 2003): 49-74.\n\n\u2014\u2014. \"Sephardism in Israel: Zionism from the Standpoint of Its Jewish Victims.\" _Dangerous Liaisons: Gender, Nation, and Postcolonial Perspectives_. Ed. Anne McClintock, Aamir Mufti, and Ella Shohat. Minneapolis: University of Minnesota Press, 1997. 39-68.\n\nShohat, Ella, and Robert Stam. _Multiculturalism, Postcoloniality, and Transnational Media_. New Brunswick: Rutgers University Press, 2003.\n\nShumway, David. \"Disciplinary Identities.\" _Affiliations: Identity in Academic Culture_. Ed. Jeffrey Di Leo. Lincoln; London: University of Nebraska Press, 2003._89-100.\n\nShweder, Richard. \" 'What about Female Genital Mutilation?' and Why Understanding Culture Matters in the First Place.\" _Engaging Cultural Differences. The Multicultural Challenge in Liberal Democracies_. Ed. R. Shweder, M. Minow, and H.R. Markus. New York: Russell Sage Foundation, 2002. 216-51.\n\nSiberry, Elizabeth. \"Images of the Crusades in the Nineteenth and Twentieth Centuries.\" _The Oxford Illustrated History of the Crusades_. Ed. Jonathan Riley-Smith. Oxford: Oxford University Press, 1995. 365-85.\n\n\u2014\u2014. _The New Crusaders. Images of the Crusades in the Nineteenth and Early Twentieth Centuries_. Aldershot, England: Ashgate Publishing, 2000.\n\nSibly, W.A., and M.D. Sibly, trans. and eds. _The History of the Albigensian Crusade. Peter of Les Vaux-de-Cernay's Historia Albigensis_. Woodbridge, UK: Boydell Press, 1998.\n\nSivan, E. _Modern Arab Historiography of the Crusades_. Kfar Chabad, Israel: Tel-Aviv University, 1973.\n\nSmith, Charles. _Palestine and the Arab-Israeli Conflict_. 6th edition. New York: St. Martin's Press, 2007.\n\nSolomon, Alisa. \"Who Gets to Be Human on the Evening News?\" _PMLA_ 121.5 (October 2006): 1585-92.\n\nSontag, Susan. _On Photography_. New York: Farrar, Straus & Giroux, 1977.\n\nSteel, James. _\"Let's Party!_ Ast\u00e9rix and the World Cup (France 1998).\" _The Francophone Bande Dessin\u00e9e_. Ed. Charles Forsdick, Laurence Grove, and Libbie McQuillan. Amsterdam, NY: Rodopi, 2005. 201-18.\n\nStora, Benjamin. _La guerre invisible: Alg\u00e9rie, ann\u00e9es 90_. Paris: Presses de Sciences Po, 2001.\n\n\u2014\u2014. \"L'impossible neutralit\u00e9 des Juifs d'Alg\u00e9rie.\" _La guerre d'Alg\u00e9rie: 1954-2004, la fin de l'amn\u00e9sie_. Ed. Benjamin Stora and Mohammed Harbi. Paris: R. Laffont, 2004. 287-316.\n\n\u2014\u2014. _Les trois exils. Juifs d'Alg\u00e9rie_. Paris: Stock, 2006.\n\n\u2014\u2014. \"1999-2003, guerre d'Alg\u00e9rie, les acc\u00e9l\u00e9rations de la m\u00e9moire.\" _La guerre d'Alg\u00e9rie: 1954-2004, la fin de l'amn\u00e9sie_. Ed. Benjamin Stora and Mohammed Harbi. Paris: R. Laffont, 2004. 501-14.\n\nStora, Benjamin and Mohammed Harbi, eds. _La guerre d'Alg\u00e9rie: 1954-2004, la fin de l'amn\u00e9sie_. Paris: Laffont, 2004.\n\nSullivan, Zohreh. \"The Body and the City: Evelyn Accad's and Etel Adnan's Beirut.\" _On Evelyne Accad: Essays in Literature, Feminism, and Cultural Studies_. Ed. Cheryl Toman. Birmingham, AL: Summa, 2007. 71-78.\n\nTalbi, Mohamed. \"R\u00e9novation de la pens\u00e9e musulmane, l'islam et les juifs: quelle relation?\" _Histoire communautaire, histoire plurielle: La communaut\u00e9 juive de Tunisie_. Tunis: Centre de Publication Universitaire, 1999. 41-50.\n\nTarr, Carrie. \"L'Eternel Retour: Reflection of the Occupation's Crisis in French Masculinity?\" _SubStance_ 27.3, Issue 87: Special Issue: The Occupation (1998): 55-72.\n\nTengour, Habib. \"Enfance.\" _Une enfance alg\u00e9rienne_. Ed. Le\u00efla Sebbar. Paris: Gallimard, 1999 [1997]. 215-28.\n\nTernisien, Xavier. \"Des manifestations contre la loi sur le voile ont rassembl\u00e9 quelques milliers de personnes.\" _Le Monde_ 17 Feb. 2004.\n\nTerray, Emmanuel. \"L'hyst\u00e9rie politique.\" _Le foulard islamique en questions_. Ed. Charlotte Nordmann. Paris: Editions Amsterdam, 2004. 103-17.\n\nThieme, John. _Postcolonial Con-texts: Writing Back to the Canon_. London; New York: Continuum, 2001.\n\nThomas, Scott. _The Global Resurgence of Religion and the Transformation of International Relations: The Struggle for the Soul of the Twenty-First Century_. New York, N.Y.: Palgrave Macmillan, 2005.\n\nToman, Cheryl, ed. _On Evelyne Accad: Essays in Literature, Feminism, and Cultural Studies_. Birmingham, AL: Summa, 2007.\n\nTramson, Jacques. \"La bande dessin\u00e9e du Maghreb: L'Arl\u00e9sienne de l'autre c\u00f4t\u00e9 de la M\u00e9diterran\u00e9e.\" Jean Foucault. _Imaginaire du jeune m\u00e9diterran\u00e9en_. Itin\u00e9raires et contacts de cultures, v. 31. Paris: L'Harmattan, 2002. 33-49.\n\nTyerman, Christopher. _The Invention of the Crusades_. Toronto: University of Toronto Press, 1998.\n\nUpdike, John. _Terrorist_. New York: Alfred A. Knopf, 2006.\n\nValensi, Lucette. \"Une Histoire des juifs de Tunisie est-elle n\u00e9cessaire? Est-elle possible?\" _Histoire communautaire, histoire plurielle: la communaut\u00e9 juive de Tunisie_. Tunis: Centre de Publication Universitaire, 1999. 51-63.\n\nVersluys, Kristiaan. \"9\/11 as a European Event: The Novel.\" _The European Review_ 15.1 (2007): 65-79.\n\nVerthuy, Ma\u00efr. \"Evelyne Accad, or the Hijacking of Sexuality.\" _On Evelyne Accad: Essays in Literature, Feminism, and Cultural Studies_. Ed. Cheryl Toman. Birmingham, AL: Summa, 2007. 157-70.\n\nVircondelet, Alain. \"Le retour des sources.\" _Une enfance alg\u00e9rienne_. Ed. Le\u00efla Sebbar. Paris: Gallimard, 1999 [1997]. 229-43.\n\nVolpi, Fr\u00e9d\u00e9ric. _Islam and Democracy: The Failure of Dialogue in Algeria_. London; Sterling, VA: Pluto Press, 2003.\n\nVoltaire. _Dictionnaire philosophique, ou la raison par alphabet_. 7th edition. Londres, 1770. Vol. 1.\n\nWatt, Stephen. \"It Seems (Af)Filiation Is to Blame.\" _Affiliations: Identity in Academic Culture_. Ed. Jeffrey Di Leo. Lincoln; London: University of Nebraska Press, 2003. 115-30.\n\nWieviorka, Annette. _Auschwitz expliqu\u00e9 \u00e0 ma fille_. Paris: Seuil, 1999.\n\nWillis, Michael. _The Islamist Challenge in Algeria: A Political History_. New York: New York University Press, 1996.\n\nWomen's Caucus of the African Studies Association. \"Position Paper on Clitoridectomy and Infibulation.\" _Genital Cutting and Transnational Sisterhood. Disputing U.S. Polemics_. Ed. Stanlie M. James and Claire Robertson. Urbana & Chicago: University of Illinois Press, 2002. 1-3.\n\nZabus, Chantal. _Between Rites and Rights: Excision in Women's Experiential Texts and Human Contexts_. Stanford, CA: Stanford University Press, 2007.\n\n\u2014\u2014. \"Bouches cousues: l'autobiographie de l'excis\u00e9e.\" _L'animal autobiographique: Autour de Jacques Derrida_. Ed. Marie-Louise Mallet. Paris: Editions Galil\u00e9e, 1999. 331-52.\n\nZafrani, Ha\u00efm. _Mille ans de vie juive au Maroc: Histoire et culture, religion et magie_. Paris: Maisonneuve et Larose, 1983.\n\nZahnd, Elizabeth. \"Two Eyes for an Eye: Women, Violence, and Modernity in Two Lebanese Novels.\" _Literature and Cruelty\/Litt\u00e9rature et cruaut\u00e9_. Proc. of the 6th Annual Graduate Conf. in French, Francophone and Comparative Lit., Columbia Univ., March 1, 1996. Ed. Vincent Desroches. New York: Columbia University, 1996.\n\nZainaba. \"Lecture on Clitoridectomy to the Midwives of Touil, Mauritania.\" Introduced and translated by Elizabeth Oram. _Opening the Gates_. Ed. Margot Badran and Miriam Cooke. 63-71.\n\n\u017di\u017eek, Slavoj. _Welcome to the Desert of the Real! Five Essays on September 11 and Related Dates_. London & New York: Verso, 2002..\n\n## Index\n\nThe index that appeared in the print version of this title was intentionally removed from the eBook. Please use the search function on your eReading device for terms of interest. For your reference, the terms that appear in the print index are listed below\n\nAbdelkader\n\nAbdelkrim\n\nAbdel Qader al-Hussayni\n\nAbul-Husn, Latif\n\nAccad, Evelyne\n\n_L'excis\u00e9e_\n\nAdana massacre\n\naffaire du foulard. _See_ headscarf affair\n\naffiliation\n\nAhmadu, Fuambai\n\nA\u00eft-Embarek, Moussa\n\nAlbou, Karin\n\n_La petite J\u00e9rusalem_\n\nAlcalay, Ammiel\n\nAlgeria\n\ncivil war\n\nFrench colonization of\n\nwar of independence\n\nAlloula, Malek\n\nAl Qaeda\n\nAmerican invasion of Iraq\/Afghanistan\n\nAmis, Martin\n\nAndalusia\n\nAntaki, Myriam\n\n_Les versets du pardon_\n\nanti-Semitism\n\nArab world. _See also_ Muslim world\n\nArab-Israeli conflict\n\nArabic language\n\nArmstrong, Karen\n\nAshcroft et al.\n\n_Ast\u00e9rix_\n\nAtta, Mohammed\n\nAttar, Farid. _See The Conference of the Birds_\n\nAuger, Marie\n\nAzria, R\u00e9gine\n\nBachi, Salim\n\n_Tuez-les tous_\n\nBadran, Margot\n\nBalibar, Etienne\n\nBarghouti, Omar\n\nBarthes, Roland\n\nBaub\u00e9rot, Jean\n\nBaudrillard, Jean\n\nBena\u00efssa, Slimane\n\n_La derni\u00e8re nuit d'un damn\u00e9_\n\nBenbassa, Esther\n\nBencheikh, Jamel Eddine\n\nBenguigui, Yamina\n\n_M\u00e9moires d'immigr\u00e9s: L'h\u00e9ritage maghr\u00e9bin_\n\nBen Jelloun, Tahar\n\n\"L'enfant trahi\"\n\n_Hospitalit\u00e9 fran\u00e7aise_\n\n_L'Islam expliqu\u00e9 aux enfants_\n\n_La nuit sacr\u00e9e_\n\n_Le racisme expliqu\u00e9 \u00e0 ma fille_\n\nBensma\u00efa, R\u00e9da\n\nBensoussan, Albert\n\nBerbers\n\nBerger, John\n\nBeur\n\nBible\n\nBin Laden\n\n_The Birds_\n\nBouamama, Sa\u00efd\n\nBoudjedra, Rachid\n\n_Le FIS de la haine_\n\nBoudjellal, Farid\n\n_JuifsArabes_\n\nBourdieu, Pierre\n\nBouzar, Dounia\n\nBowen, John\n\nBoyle, Elizabeth\n\nBrown, Bill\n\nBurgat, Fran\u00e7ois\n\nBush, George W.\n\nButler, Judith\n\nCarlier, Omar\n\nCarrard, Philippe\n\nCaws, Mary Ann\n\nCh\u00e9did, Andr\u00e9e\n\n_La maison sans racines_\n\nchildhood\n\nChomsky, Noam\n\nChow, Rey\n\nChra\u00efbi, Driss\n\nChristianity\n\nChristians\n\nArab Christians\n\nCixous, H\u00e9l\u00e8ne\n\n\"Mon alg\u00e9riance\"\n\n\"Pieds nus\"\n\nclash of civilizations theory\n\nclitoridectomy. _See also_ excision, FGC\/FGM, infibulation\n\nCohen, Annie\n\nCohen, William\n\nColombani, Jean-Marie\n\ncolonialism\n\n_See also_ imperialism, occupation\n\nColumbus\n\nConan, Eric\n\nConcordat\n\n_The Conference of the Birds_\n\nconflict resolution\n\nConstable, Giles\n\nCooke, Miriam\n\nCr\u00e9mieux decree\n\nCr\u00e9pon, Marc\n\nCrusades\n\nDadoun, Roger\n\nDaniel, Jean\n\nDareer, Asma El\n\nDe Certeau, Michel\n\nDe Goncourt, Edmond & Jules\n\nDeir Yassin\n\nDeleuze, Gilles\n\nDelphy, Christine\n\nDerrida, Jacques\n\nDesai, Gaurav\n\nDib, Mohammed\n\nDirlik, Arif\n\nDjebar, Assia\n\nDonadey, Anne\n\nDouglas, Allen\n\nDrake, David\n\nDurmelat, Sylvie\n\nEl Maleh, Edmond Amran\n\n_Mille ans, un jour_\n\nEl-Solh, Camillia Fawzi\n\n_Une enfance alg\u00e9rienne_\n\nEntelis, John\n\nEsposito, John\n\n_L'\u00e9ternel retour_\n\nEuben, Roxanne\n\nEurocentrism\n\nEurope\n\nEuropean imperialism\n\nexcision\n\n_L'Express_\n\nFar\u00e8s, Nabile\n\nFGC (Female Genital Cutting)\/FGM. _See also_ excision, clitoridectomy, infibulation\n\nFernea, Robert\n\nfiliation\n\nFIS (Front Islamique du Salut)\n\nFlaubert\n\nFLN (Front de Lib\u00e9ration Nationale)\n\nforgiveness\n\nFrance\n\nArabs in\n\n1905 law\n\n2004 law\n\nFrancophone literature\n\nFrancophone Studies\n\n_Francophonie_\n\nFranks\n\nFrench language\n\nFuller, Graham\n\nGafa\u00efti, Hafid\n\nGalesne, Nathalie\n\nGallaire, Fatima\n\nGambetta, Diego\n\nGaspard, Fran\u00e7oise\n\nGaumer, Patrick\n\nGaza\n\ngenealogy\n\nGenette, G\u00e9rard\n\nGIA (Groupe Islamique Arm\u00e9)\n\nGopin, Marc\n\nGossman, Lionel\n\nGravdal, Kathryn\n\nGreat Mosque, Paris\n\nGrewal, Inderpal\n\nGroensteen, Thierry\n\nGross, Janice\n\nGrosser, Paul\n\nGrousset, Ren\u00e9\n\nGruenbaum, Ellen\n\nGu\u00e9noun, Solange\n\nHaarscher, Guy\n\nHadith\n\nha\u00efk\n\nHallam, Elizabeth\n\n_Hamlet_\n\nHarbi, Mohammed\n\nHare, William\n\nHargreaves, Alec\n\nHarrison, Nicholas\n\nHartman, Michelle\n\nHasenclever, Andreas\n\nHassan, Salah\n\nHatzopoulos, Pavlos\n\nheadscarf affair\n\n_Heart of Darkness_\n\nHerman, Edward\n\nHiggins, Lynn\n\n_Hiroshima mon amour_\n\nhistoriography\n\nHitchcock\n\nHochberg, Gil\n\nHolmes, Stephen\n\nHolocaust\n\nHoly Land\n\nHopgood, Stephen\n\nHosken, Fran\n\nHottell, Ruth\n\nHugo, Victor\n\nHuntington, Samuel\n\nHutcheon, Linda\n\nIbn-Munqidh, Usamah\n\nImache, Djedjiga\n\nInstrumentalists\n\ninternational relations\n\nintertext\n\nimmigration\n\nimperialism\n\nIran\n\nIrgoun\n\nIslam\n\nin the West\n\nwomen in\n\nIsrael\n\nJadla, Ibrahim\n\nJaschik, Scott\n\nJerusalem\n\nJesus\n\nJews\n\nArab Jew\n\nJudaism\n\nJudeo-Christian tradition\n\nKacimi-El-Hassani, Mohamed\n\nKalyvas, Stathis\n\nKandiyoti, Deniz\n\nKenbib, Mohammed\n\nKepel, Gilles\n\nKhosrokhavar, Farhad\n\nKing David hotel bombing\n\nKinoshita, Sharon\n\nKoso-Thomas, Olayinka\n\nKurzman, Charles\n\nLargu\u00e8che, Abdelhamid\n\nLaronche, Martine\n\nLaroussi, Farid\n\nLaskier, Michael\n\nLayachi, Azzedine\n\nLebanon\n\ncivil war\n\nLevant\n\nL\u00e9vy, Alma and Lila\n\nL\u00e9vy, Simon\n\nLewis, Bernard\n\nLionnet, Fran\u00e7oise\n\nLloyd, Simon\n\nMaalouf, Amin\n\n_Les croisades vues par les Arabes_\n\n_Les \u00e9chelles du Levant_\n\n_Identit\u00e9s meurtri\u00e8res_\n\nMackie, Gerry\n\nMaghreb. _See also_ North Africa\n\nMajid, Anouar\n\nmandate\n\nMaronites\n\nMartinez, Louis\n\nMarx-Scouras, Danielle\n\nMasson, Pierre\n\nMcDermott, Terry\n\nMcKinney, Mark\n\nMcQuillan, Libbie\n\nMearsheimer, John\n\nMeddeb, Abdelwahab\n\n_Face \u00e0 l'islam_\n\n_La maladie de l'islam_\n\n_Phantasia_\n\nMekay, Emad\n\nMemmi, Albert\n\n_Juifs et Arabes_\n\n_Portrait du colonis\u00e9_\n\n_Portrait du d\u00e9colonis\u00e9 arabo-musulman et de quelques autres_\n\nMernissi, Fatima\n\nMichaud, Joseph\n\nMiddle East\/Near East\n\nMillecam, Jean-Pierre\n\nMimouni, Rachid\n\n_De la barbarie en g\u00e9n\u00e9ral et de l'int\u00e9grisme en particulier_\n\nMiraglia, Anne-Marie\n\nMitchell, W.J.T.\n\nmodernization theory\n\nProphet Mohammed\n\nMohanty, Chandra\n\nMokn\u00e8che, Nadir\n\n_Le Monde_\n\nMorocco\n\nMortimer, Robert\n\nMorton, Stephen\n\nMudimbe-Boyi, Elisabeth\n\nMuslims\n\nand Crusaders\n\nin France\n\nNaipaul\n\nNasreen, Taslima\n\nNavarro, Mireya\n\nNaylor, Philip\n\nNazism\n\nNixon, Rob\n\nNoiriel, G\u00e9rard\n\nNordmann, Charlotte\n\nNorth Africa. _See also_ Maghreb\n\nNoudelmann, Fran\u00e7ois\n\noccupation\n\nOldenbourg, Zo\u00e9\n\npackaging\n\nPalestine. _See also_ West Bank, Gaza\n\nPalestinians\n\npan-Arabism\n\nPapp\u00e9, Ilan\n\npartition\n\nPedahzur, Ami\n\nP\u00e9l\u00e9gri, Jean\n\n_Persepolis_\n\npieds-noirs\n\nPhoenicianism\n\nPodselver, Laurence\n\nPoitiers\n\nPostcolonial Studies\n\nPratt, Mary Louise\n\nPrimordialists\n\nProtectorate\n\nQur'an\n\nQureshi, Emran\n\nrape\n\n_Reconquista_\n\n_Reading Lolita in Tehran_\n\nRedonnet, Marie\n\nreligion, resurgence of\n\nReuter, Christoph\n\nrhizome\n\nRicolfi, Luca\n\nRiley-Smith, Jonathan\n\nRoquebert, Michel\n\nRosello, Mireille\n\nRosenthal, Franz\n\nRoy, Olivier\n\nRunciman, Steven\n\nRushdie, Salman\n\nSaid, Edward\n\nSalem, Elise\n\nSalibi, Kamal\n\nSarcelles\n\nSassine, Antoine\n\nSatrapi, Marjane\n\nSayyid, Salman\n\nScharfman, Ronnie\n\nScreech, Matthew\n\nSebag, Paul\n\nSebbar, Le\u00efla\n\n_Journal de mes Alg\u00e9ries en France_\n\n_Mes Alg\u00e9ries en France_\n\nsecularism\n\nSephardim\n\nSeptember\n\nSerrano, Richard\n\nShell-Duncan, Bettina\n\nShih, Shu-Mei\n\nShiites\n\nShilton, Siobh\u00e1n\n\nShohat, Ella\n\nShumway, David\n\nShweder, Richard\n\nSiberry, Elizabeth\n\nSivan, Emmanuel\n\nSmith, Charles\n\nSolomon, Alisa\n\nSolzhenitsyn, Alexandre\n\nSontag, Susan\n\nSoyinka, Wole\n\nStasi Commission\n\nSteel, James\n\nStora, Benjamin\n\nsuicide bombing\n\nSullivan, Zohreh\n\nSunnis\n\nSyria\n\nTalbi, Mohamed\n\nTalibans\n\nTarr, Carrie\n\nTengour, Habib\n\nTernisien, Xavier\n\nTerray, Emmanuel\n\nterrorism\n\nThieme, John\n\nThomas, Scott\n\nTorah\n\nTramson, Jacques\n\ntranslation\n\nTunisia\n\nTurkey\n\nTyerman, Christopher\n\nUpdike, John\n\nValensi, Lucette\n\nveil. _See also_ headscarf affair\n\nVersluys, Kristiaan\n\nVichy government\n\nVienna, siege of\n\nviolence\n\nVircondelet, Alain\n\nVolpi, Fr\u00e9d\u00e9ric\n\nVoltaire\n\nWatt. Stephen\n\nWest\n\nWest Bank\n\nWieviorka, Annette\n\nWillis, Michael\n\nZabus, Chantal\n\nZafrani, Ha\u00efm\n\nZainaba\n\nZionism\n\n\u017di\u017eek, Slavoj\n\n## About the Author\n\n**Carine Bourget** is associate professor of French and Francophone studies in the French and Italian Department at the University of Arizona, and holds a courtesy appointment with the Near Eastern Studies Department. Her research focuses on the cultural production from the Francophone Arab world, with special interests in Islam in literature, Islam in France, politics and literature, and history and literature. She is the author of _Coran et tradition islamique dans la litt\u00e9rature maghr\u00e9bine_ (Paris: Karthala, 2002), a study that analyzes the religious intertext in works by Tahar Ben Jelloun, Driss Chra\u00efbi, Assia Djebar, and Fatima Mernissi. Her most recent articles have appeared in _L'Esprit Cr\u00e9ateur, The French Review, Research in African Literatures, Studies in 20th and 21st Century Literature, Expressions maghr\u00e9bines_ , and _French Cultural Studies_.\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\nPENGUIN BOOKS\n\n# My Father's Moon\n\nElizabeth Jolley is one of Australia's most celebrated writers, with a formidable international reputation. She was recognised in Australia with an AO for services to literature and was awarded honorary doctorates from four universities.\n\nBorn in England in 1923, she was brought up in a strict, German-speaking household and attended a Quaker boarding school. She became a nurse, married Leonard Jolley and with three children moved to Western Australia in 1959.\n\nAlthough she wrote all her life, it was not until she was in her fifties that Elizabeth's books started to receive the recognition they deserved. She won the _Age_ Book of the Year Award on three separate occasions (for _Mr Scobie's Riddle_ , _My Father's Moon_ and _The Georges' Wife_ ) and the Miles Franklin Award for _The Well_ , as well as many other awards. Elizabeth Jolley died in 2007.\n\n## Elizabeth Jolley\n\n# My Father's Moon\n\nWith an Introduction by\n\n## J.M. Coetzee\n\n###  PENGUIN BOOKS \nPENGUIN BOOKS\n\nPublished by the Penguin Group\n\nPenguin Group (Australia)\n\n250 Camberwell Road, Camberwell, Victoria 3124, Australia   \n(a division of Pearson Australia Group Pty Ltd)\n\nPenguin Group (USA) Inc.\n\n375 Hudson Street, New York, New York 10014, USA\n\nPenguin Group (Canada)\n\n90 Eglinton Avenue East, Suite 700, Toronto ON M4P 2Y3, Canada   \n(a division of Pearson Penguin Canada Inc.)\n\nPenguin Books Ltd\n\n80 Strand, London WC2R 0RL, England\n\nPenguin Ireland\n\n25 St Stephen's Green, Dublin 2, Ireland   \n(a division of Penguin Books Ltd)\n\nPenguin Books India Pvt Ltd\n\n11 Community Centre, Panchsheel Park, New Delhi \u2013 110 017, India\n\nPenguin Group (NZ)\n\n67 Apollo Drive, Rosedale, North Shore 0632, New Zealand   \n(a division of Pearson New Zealand Ltd)\n\nPenguin Books (South Africa) (Pty) Ltd\n\n24 Sturdee Avenue, Rosebank, Johannesburg 2196, South Africa\n\nPenguin Books Ltd, Registered Offices: 80 Strand, London WC2R 0RL, England\n\nFirst published in Viking by Penguin Books Australia Ltd, 1989\n\nPublished in Penguin, 1989\n\nThis edition published by Penguin group (Australia), 2008\n\nText copyright \u00a9 The Estate of Elizabeth Jolley, 1989\n\nIntroduction \u00a9 J.M. Coetzee\n\nThe moral right of the author has been asserted\n\nAll rights reserved. Without limiting the rights under copyright reserved above, no part of this publication may be reproduced, stored in or introduced into a retrieval system, or transmitted, in any form or by any means (electronic, mechanical, photocopying, recording or otherwise), without the prior written permission of both the copyright owner and the above publisher of this book.\n\nISBN: 978-1-74-228215-2\n\nwww.penguin.com.au\n\n# ALSO BY ELIZABETH JOLLEY\n\n## Stories\n\n_Five Acre Virgin_\n\n_The Travelling Entertainer_\n\n_Woman in a Lampshade_\n\n_Fellow Passengers_\n\n## Novels\n\n_Palomino_\n\n_The Newspaper of Claremont Street_\n\n_Mr Scobie's Riddle_\n\n_Miss Peabody's Inheritance_\n\n_Foxybaby_\n\n_Milk and Honey_\n\n_The Well The Sugar Mother_\n\n_Cabin Fever_\n\n_The Georges' Wife_\n\n_The Orchard Thieves_\n\n_Lovesong_\n\n_An Accommodating Spouse_\n\n_An Innocent Gentleman_\n\n## Non-fiction\n\n_Central Mischief_\n\n_Off the Air_\n\n_Diary of a Weekend Farmer_\n\n_Learning to Dance_\nFor Leonard Jolley\nI would like to express my thanks to the Curtin University of Technology (formerly the Western Australian Institute of Technology) for the continuing privilege of being with students and colleagues in the School of Communication and Cultural Studies and for the provision of a room in which to write. I would like, in particular, to thank Don Watts, Peter Reeves, Brian Dibble and Don Grant.\n\nA special thanks is offered to Nancy McKenzie who, for a great many years, has typed my manuscripts. Her patience is endless.\n\n# AUTHOR'S NOTE\n\nIt comes as a surprise to realize that time has gone by and that there are now people who are no longer familiar with the abbreviations which were once a part of everyday existence.\n\nATS was the abbreviation for Auxiliary Territorial Service. ARP stood for Air Raid Precaution, NA was Nursing Auxiliary (or sometimes Naughty Annie). The abbreviations were always used. They were part of the idiom. I never heard the words spoken in full. Similarly RMO and RSO were always said for the Resident Medical Officer and the Resident Surgical Officer respectively.\n\nElizabeth Jolley\n\n# THE VERA WRIGHT TRILOGY:\n\n## My Father's Moon, Cabin Fever, The Georges' Wife\n\nFrom her home in Australia, Vera Wright looks back over her life. A doctor by profession, she has recently begun to haunt the waiting rooms of other doctors. The endgame with death has begun.\n\nImages from the past come to her mind \u2013 for instance, of absentmindedly walking into the bathroom without knocking and seeing her daughter naked. 'I was never [long-limbed and graceful] like this,' she ruminates, 'or if I was once, perhaps briefly, on the edge of this innocence and smooth youthfulness, the time came and went by without my noticing any of it, without my noticing my own body and how I might look as a young woman. I have no idea.'\n\nRegret that in some sense she did not fully possess her own young womanhood runs through Vera's recollections. Part of her aim in writing her life story is to catch sight of herself as she must have been four decades ago, when she was too self-absorbed to see herself from the outside.\n\nVera may have little idea of how she looked to others, yet as her story proceeds we do get glimpses of her in her heyday. Like her daughter, she was tall and long-legged. ' _Eine Valkyrie_ ,' remarked a German-speaking friend of her mother's admiringly. To a colleague at work, the snowdrop seemed the right flower for her, 'clean, green and white'. According to another colleague, she had no sex appeal. Sunbathing did not help: she got hot but stayed pale. She wore spectacles. She lacked fashion sense.\n\nVera comes from a cultivated middle-class family of modest means in the Midlands of england. She reads a lot, and writes poetry. Her mother hopes that she will go to europe to study art and architecture and music. So she is dismayed when, at the outbreak of war in 1939, her seventeen-year-old daughter announces she is going to enrol as a nurse. She is even more dismayed when, in the last months of the war, Vera falls pregnant. Yet she pays for Vera's accouchement in a private maternity hospital and soon bonds with the baby, to the point that her daughter grows jealous.\n\nFive years later Vera is pregnant a second time, again by a man unable or unwilling to marry her; and again, after berating her daughter for her promiscuity, Margarethe Wright is captivated by her new granddaughter.\n\nBetween mother and daughter (Vera has a younger sister, but she is only a shadowy presence) there are strains that remain unresolved to the end, strains that go a long way towards dictating the course of Vera's life. At the risk of oversimplifying, one can say that Mrs Wright wants to be proud of her daughter but is continually being shamed (what the neighbours think matters a great deal to her), while Vera is torn between wanting to be independent of her mother and wanting to be loved by her. Her babies are, at the deepest level, offerings she brings home; if she is upset by her mother's rapport with them, it's because she is the one (she feels) who really needs to be soothed and cuddled and fed.\n\nThe author of the first pregnancy is a married man, Jonathon Metcalf, a doctor at St Cuthberts, the hospital where Vera does her training. According to hospital gossip, Dr Metcalf has got a number of young nurses into trouble. When Vera, in a panic, not knowing where to turn, begins to put emotional pressure on him, he enlists in the army and is promptly killed in an accident. Vera is never certain whether he enlisted to escape her or, as gossip again has it, to be reunited with a male lover.\n\nThe second child is fathered, after Vera has moved to Glasgow, by her employer there, a middle-aged academic, Oliver George. With Mr George she will later sail to make a new life in Australia, leaving behind her daughters, by then aged twelve and seven, in the willing care of Mr George's elderly sister.\n\nThere are three men with whom Vera reports sexual relations: Dr Metcalf, Mr George, and the proto-hippie social dropout No\u00ebl. In no case is the relationship straightforward. Though Vera is infatuated with Metcalf, her deeper feelings, both erotic and rivalrous, are for Metcalf's wife. In the case of the Georges, on the other hand, there are moments when she wonders whether the brother and sister are not simply using her to make a child for themselves (hence the title of the third book of the trilogy, _The Georges' Wife_ ). As for No\u00ebl, sharing a bed with him entails sharing it with his wife too.\n\nThere is no hint that Vera finds sex with any of these men unpleasant. Indeed, with Mr George it is particularly sweet. Nevertheless, Vera's more significant emotional entanglements \u2013 or at least the entanglements that most engage her as she looks back \u2013 are with women, particularly women older than herself. Aside from her mother, the significant women in her life are a family friend named Gertrude; a senior nurse, Sister Ramsden; and Mrs Ruperts, a wealthy widow whom she meets on the boat to Australia. Part of what Vera wants from these women is the mothering she feels has been withheld from her. (In their company her behaviour tends to become juvenile and\/or coquettish.) But in addition she seeks \u2013 and with Gertrude and Mrs Ruperts finds \u2013 an intimacy, generosity of emotion, and fellowship that men do not provide.\n\nShe craves this kind of love, and blossoms when she finds it. Yet finally, when commitment is called for on her side, she pulls back in a most hurtful and (to use her own word) shabby way. This bewildering behaviour is a source of bitter later regret, remorse and self-laceration. One of her motives for writing down her story is to comprehend a recurrent pattern in her emotional life: of courting love but then freezing when she is called on to respond. It is a pattern that can properly be called compulsive.\n\nWith her mother, Vera is locked in a pattern of blame, remorse and accusation from which the two find it hard to free themselves. Yet Vera has a moment of illumination in which she recognises that her mother is not simply being snobbish in opposing her decision to go into nursing. She sees herself as she might have become after a lifetime on the career ladder: '[a] District Nurse, enormous in navy blue, on a bicycle, visiting patients, admonishing husbands and delivering babies on sheets of newspaper in overcrowded kitchens'.\n\nVera fails to follow the path laid down for her by her mother of genteelly dabbling in art and music before making a good marriage and having a family. Nor does she become a professional nurse. But is the alternative path she takes, first as an au-pair, then as an aide in a maternity home, then as general factotum at a boarding school, then as a housekeeper, then finally as a doctor practising what, when she is in a bad mood, she calls 'third-rate psychiatry', any better?\n\nVera disappoints her mother, but her mother disappoints Vera too. In middle age Mrs Wright falls under the influence of a neighbour named Mrs Pugh. Listening to her mother speak, Vera increasingly hears the voice and accents of Mrs Pugh. To Vera, Mrs Pugh represents the British lower-middle class at its worst: insular, mean-minded, close-fisted. Why, she wonders, has her mother \u2013 a cultivated woman, Viennese by birth \u2013 allowed herself to be taken over in this way? What does the development portend for Vera herself? Will her horizons too grow narrower as she grows older? Is her fear of commitment an inherited trait against which it is futile to fight?\n\nVera's father is a Quaker schoolteacher with a special fondness for the novel, which, he says, shows us how the passions work and also offers the fullest picture we have of life as it is or was in a particular place and time. Her father's unwavering love sets a standard to which Vera repeatedly fails to rise. Her failures lead her to formulate a first law of parent-child relations: that the love the parent gives the child cannot be given back by the child. (Cannot be given back but imprints itself nevertheless, one might add, in the form of guilt.)\n\nVera may find it hard to show love to her father, but she takes his lessons on literature to heart. Considering that its theme is the progress towards self-knowledge of a notably introspective woman from the provinces, the document that she comes to write half a century later \u2013 in her eyes, an autobiography; in ours, a trilogy of novels \u2013 contains notable insights into the workings of the passions (of which more later). It also gives a remarkably sharp picture of life as it was in england from the mid-1930s to the early 1950s, from the Great Depression through World War II to the austerities of the postwar years, a time-span treated by Jolley as a single unbroken historical period.\n\nVera's family is 'political' \u2013 her father went to jail as a conscientious objector in World War I; her parents open their home to refugees from hitler's germany \u2013 but she does not think politically. She barely registers the lead-up to the war; the reality of what is going on hits her only when St Cuthberts is flooded with young men with ghastly wounds; and she realises the war is over only when people start dancing in the streets. There is not a word about the progress of arms in the various theatres of war, about hitler or Churchill or stalin. Young Vera's preoccupations are more immediate: Who are the trend-setters among the trainee nurses? How can I get my hands on more sugar? When is my next leave allotment due?\n\nIn the midst of great historical upheavals, life goes on in all its multifariousness. The minds of ordinary people are occupied not with the great questions of the day but with a myriad petty personal concerns. I am not suggesting that Jolley consciously advances this Tolstoyan thesis; yet her creation of Vera's life, seemingly empty of historical consciousness, is full of observations that illuminate the england of those hard years: what a luxury a real egg was compared with egg powder; how important to people's lives german music remained, particularly Beethoven; how uncommon it was to have a bath more than once a week.\n\nThe story of Vera also reminds us what a calamity it was in those days for a middle-class girl to fall pregnant. From her lover Dr Metcalf Vera receives no help at all. Afraid to tell her mother, she absconds from St Cuthberts in disgrace and moves into a kind of social limbo. Though as an expectant mother she is entitled to extra food rations, censorious social-welfare bureaucrats make it as difficult as possible for her to get her hands on them.\n\nThe company into which Vera drifts is that of the free-thinking intelligentsia, some of them academics (the high-minded Wellingtons, with their carefully managed simultaneous orgasms; the Georges), others social misfits, descendants of William Morris and the Pre-Raphaelites, dabblers in pottery, tapestry-weaving, free love, nude sunbathing and vegetarianism. As Quakers and Germanophiles, Vera's parents are at some risk of falling into this marginal class; they are saved by her father's devotion to his schoolmastering duties and her mother's strict sense of propriety.\n\nVera experiences the alternative lifestyle at its dreariest during her time at Fairfields School, a dumping ground for the offspring of broken marriages, a failing enterprise run by a pair of predatory lesbians where the children, fed on raw vegetables, are continually hungry and squabbling with one another, and the teachers only sporadically get paid. A few years later Vera is taken in by another couple of Fairfields types, no\u00ebl and Felicity, who exploit her financially, use her sexually, sneer at her accent, and leave her with a dose of tuberculosis. (Why are you always falling in love with couples, asks her bewildered mother?)\n\nAs a training hospital St Cuthberts attracts young nurses from across the social spectrum, including some with rich parents who have been to private schools and speak with posh accents. Embarrassment at her own provincial speech and manners, far from turning Vera into a foe of class prejudice, entrenches in her what she calls her comparison game \u2013 her practice of dividing any new group into which she moves into those above her (prettier, savvier, more chic), whom she cultivates and imitates, and those below her, whom she snubs and avoids. She learns the comparison game early, at school; her first victim is a classmate, Muriel, for whom she invents the derisive nickname 'Bulge'. Throughout her hospital years she remains wedded to the game, even though she is more often than not the loser. The consequences for her self-esteem are wounding. The dashing Metcalf couple, for instance, seen in the nude, impress her as examples of 'fne breeding', whereas her own body is 'badly made'.\n\nFrom early on Vera longs to be a member of a couple: the very word 'couple' becomes a mantra to her. Yet in her unorthodox erotic attachments she is clearly, if unconsciously, in quest of something more than monogamy, whether heterosexual or homosexual. Time and again she plunges into situations that excite her in their promise of erotic community: she wants those she loves to love one another too. Time and again her na\u00efve enthusiasm is exploited by people who do not share her utopian yearnings. The third book ends with her pushing an aged and mentally fragile husband around the sleepy streets of Claremont, Western Australia, in a wheelchair. At last 'we, Mr George and I, are a couple'. There is no one else in her life. Her lover, the merry widow Mrs Ruperts, condemned by Mr George as 'extraordinarily vulgar and nondescript', is dead; her daughters have made lives for themselves back in the United Kingdom. It is hard to read this ending as other than a defeat for her youthful aspirations.\n\nVera's feelings for Oliver George are not easily pinned down. Having seduced him as soon as she arrives in Glasgow, and fallen pregnant, she is impatient to regularise her position in the George household. Yet year after year he resists her efforts to do just that. On board ship en route to Australia she impulsively suggests that they enlist the captain to marry them. Mr George makes it clear how distasteful he finds the idea: 'the idea of being conspicuous in this way and being a part... of something cheap, a vulgar celebration instead of something private and tender'. Once he has taken his bearings in Perth society, however, he changes his tune: 'Mr George says that he thinks we should look for an apartment where we can live together. He says too that he will need to accept and give invitations and that he wants me to be a part of this new life.'\n\nOut of Vera's impassive _oratio obliqua_ Mr George emerges in a most unpleasant light: snobbish, reserved, selfish. Much the same might be said of Dr Metcalf, who from Vera's deadpan reports comes across as a practised seducer of the fatherly variety whose wife turns a blind eye to, and even encourages, his conquests, as long as he keeps his girls on a tight leash. This is how everyone around Vera sees the Metcalfs; it is a mark of Jolley's control of tone, as well as of the complexity of the attitude of the older Vera looking back, that the younger Vera is not for a moment held up to ridicule as a gullible child. On the contrary, visions of Metcalf sitting in a chair 'naked and handsome and shameless' come back unbidden. Vera puzzles about why her memory should be throwing them up, yet in the book their function is clear: despite all the misery that followed in their train, says the vision, those times with Metcalf were good.\n\nThere is no such ambiguity of treatment when it comes to the women who matter to Vera. The largely female community of the hospital is crisscrossed with erotic currents. Vera has an affair with a more experienced fellow trainee and expands her sexual repertoire. But her main crush is on the senior nurse Ramsden ('I love her. Perhaps. I think.'), who holds out the promise of being mother, sister and lover rolled into one, as well as confidante and guide in the realms of philosophy and music and poetry (Rilke). To make an impression on Ramsden Vera carries around a violin case. (she alone knows it is empty.) Wistful memories of Ramsden haunt her late in life, intensified by remorse over the shameful way in which she treated the older woman, inviting her to spend a weekend, then dismissing her.\n\nOf all the people Vera meets, it is Mrs Ruperts, the Australian, who has the shrewdest insight into her. 'Am I the sweet english sixth-form girl, [Mrs Ruperts] wants to know, or am I a very clever intelligent woman hiding behind a clear youthful complexion and a remarkable and convincing innocence.' Later Mrs Ruperts, going out of her mind with loneliness in her rural isolation, will make a bold appeal to Vera to come and live with her, an appeal to which Vera, following the old compulsive pattern, will give no reply. Mrs Ruperts is killed in an accident (leaving Vera a great deal of money), and in Vera's inner world becomes properly 'my widow', the one to whom she is sole survivor, or vice versa.\n\nIt is her widow who asks what turns out to be the key question of Vera's mature life: why did she give up her career in surgery for which she was trained in favour of a general practice that seems to consist mainly of counselling? Vera's response is that while surgery provides clearcut answers to clearcut problems, the malaises that concern her are those 'wrapped up... unseen in hidden wishes, in blame and remorse and in accusation' \u2013 in other words, that stem from a neurotic fixation on old grievances, trapping the self in a destiny from which there seems no escape.\n\nIs there a balm for such illnesses? Is there a healer? Yes, affirms Vera: anyone who embarks on the task with trust and courage and kindness can become a healer. But to the nub of the widow's question \u2013 why she in particular seeks the role of healer \u2013 she has no answer.\n\nVera's story must be imagined as being written down in the late 1980s and early 1990s (the three volumes were published in 1989, 1990, and 1993) and thus as being an autobiographical document by a woman in her late sixties. The bulk of the document is concerned with her earlier years, between the outbreak of war and her arrival in Australia in about 1960. Apart from the affair with Mrs Ruperts, which is covered in some detail, her later life is treated only sketchily.\n\nWithin the three volumes, Vera is represented not only as a woman who acted and suffered in the past but also as a woman in the present, thinking and sometimes writing about that past. This happens most explicitly in the second volume, _Cabin Fever,_ set in the late 1980s. Vera spends days alone in a hotel room in New York City, where she is attending a medical conference, revising her presentation while wondering what her life amounts to and what the point is of continuing to keep a diary. (We gather that the conference will be about sexually transmitted diseases, including AIDS; Vera's paper is entitled 'Perspectives on Moral Insanity': we can only guess at its content.)\n\nVera is in a slightly crazed state (she fears the walls of her room are going to burst and inundate her with hot water) for which she uses the term 'cabin fever'. But soon the term shifts in meaning. In the grip of 'cabin fever' time stands still for her and she is transported back into the past, into a static moment filled with the intimation of revelation. Redefined as 'the long drawn-out pause of intention', cabin fever thus becomes a state of being possessed by an idea on the point of coalescing, and a concurrent feeling of being on the point of breaking out in speech.\n\nOut of such heightened moments, suspensions of the flow of time, germinates the story of Vera's life. Another suspension of time will take place at a bus stop, as Vera watches the rising of the moon \u2013 the moon that stands for her father watching over her. The bus stop itself is generic: sometimes it is in the Midlands of her childhood, though the landscape is quite changed; at other times in Australia or even india. A third suspension of time comes in a train, where she feels herself about to speak to a fellow passenger who may or may not be the beloved Ramsden. A fourth and final one occurs in a doctor's waiting room, which is also the universal waiting room for death.\n\nThe technique Jolley uses to construct her narrative is nominally based on association. We are to imagine that motifs from the past float into the mind of Vera as narrator, at her bus stop or in her hotel room or wherever. Vera then takes it on herself to flesh them out, at the same time inquiring by what subconscious logic the psyche dictates that this specific motif should come up now. Fleshing out motifs entails delving into memory and reconstructing episodes from the past. However, one or two sharp asides to the reader make it clear that Vera recognises no firm division between memory and invention or fabulation. For example, we are presented with a vividly realised recollection of Ramsden going for a walk wearing ankle socks over her stockings; then much later we are informed that Ramsden never wore ankle socks, that the socks were given to her to mark her as the kind of woman who would put on socks, that is, to _characterise_ her, to give her body as a character in a novel.\n\nTo explain how text construction by association works, Vera uses a musical analogy. The new motif, at first sight arbitrary or puzzling, is like the Neapolitan sixth, a chromatic chord used now and again by Beethoven to mark or foreshadow transition to a new key. Like this dissonant chord, the apparently meaningless association initiates a new phase in the narrative. Only as the new phase (the passage in the new key) develops and thickens does the logic of the dissonance become clear.\n\nIn the first book of the series, _My Father's Moon,_ the technique of interwoven narratives linked by associative logic is used with sure-footed confidence. The seventh chapter, 'Gertrude's Place', provides an example of Jolley at her masterful best, mingling the most intense phase of Vera's friendship with Gertrude, her betrayal of Gertrude, and Gertrude's last illness, with Vera's ambivalent invitation to Ramsden, the progress of her intimacy with the Metcalf couple, and the arrival on British shores of the first wave of war-wounded. The ordering of narrative modules in the chapter is dictated not by chronology but by a structural and emotional logic that is, in a large sense, symphonic.\n\nThe second book, _Cabin Fever,_ is generally more orthodox in its narration. By the time we reach the third book, _The Georges' Wife_ , both the associative technique and the recurrent motifs themselves are familiar enough to occasion no difficulty.\n\nDispensing with chronology, doing without the unities of space and time and action, is a risky business. Jolley's Vera lives longer than Proust's Marcel. There is no formal reason why her life story should not take up six volumes, or ten, rather than three, or indeed why it should not, like Tristram Shandy's, threaten to go on for ever. Ultimately the length and shape of Vera's story will be dictated not by its content \u2013 the facts of her life \u2013 but by her author's aesthetic intuitions, which are by and large trustworthy, though not infallible. Thus, for instance, the first forty pages of _Cabin Fever_ do little but repeat material from _My Father's Moon_ \u2013 one suspects that Jolley felt under pressure to make the second volume stand on its own feet. The pages towards the end of the same book given over to Vera's dolls might well have been dispensed with. The lengthy second-person address to the dead Metcalf in _My Father's Moon_ occludes the double vision \u2013 the younger Vera refracted through her mature self \u2013 that is a strength of the work as a whole. The two big scenes between Vera and Madga Metcalf \u2013 the recovery of the incriminating letter in _My Father's Moon_ ; Magda's visit to the hospital in _Cabin Fever_ \u2013 stand out as too well-crafted, too novelistic, too Jamesian.\n\n'I am a shabby person,' writes Vera from New York. 'I understand, if I look back, that I have treated kind people with an unforgivable shabbiness. For my work a ruthless self-examination is needed for, without understanding something of myself, how can I understand anyone else.' What work is It that she refers to here? In the first place, it is her work as a healer or therapist in the tradition of Freud and of Socrates before Freud, work for which the Delphic 'Know thyself' is a prerequisite. But it is also the autobiographical work she is engaged in. In the latter case, what exactly does self-understanding have to do with shabby behaviour in the past? Will it matter to Muriel ('Bulge') whether or not, fifty years after the event, her ex-schoolmate Vera begins to comprehend why she treated her so badly? What Muriel wants is surely something else: remorse on Vera's part, and an expiatory act \u2013 confession, apology.\n\nYet confession and apology are, in the end, exactly what Muriel gets \u2013 Muriel and Gertrude and Ramsden and Mr Wright and even Mrs Wright. Vera may claim that she is writing her story in order to understand herself, but in the larger scheme she writes it in order to get it down on paper and thus into the world. Once it is in the world, we can make up our minds what kind of thing it is, what kind of act it constitutes. And as Vera apologises to Muriel, and Muriel (if she is still in Vera's world) perhaps forgives what Vera finds unforgivable, so we too can without difficulty forgive Vera for the intermittent shabbiness to which she has confessed \u2013 shabbiness quite outweighed by her courage and resilience and curiosity and passion and kindness and devotion and humour and intelligence.\n\nWhy you, asks the widow? What is special about you? Vera can find no answer. 'there is a great deal that must be known and, at the same time, it must stay hidden in the heart.' This is the moment when Vera (and Jolley behind her) comes closest to expressing a personal credo for the work of writing. Fiction or autobiography (the line that Jolley draws between the two is nowhere less clear than in the trilogy) may be a healing art, but as to pinning down the motive that drives its practitioner, here the Delphic prescription simply ceases to hold. The motive must indeed be sought \u2013 there may be no limit to self-inquiry \u2013 but it cannot, dare not be pinned down.\n\nThough the order of composition of Jolley's writings is as yet far from clear (parts of the Vera Wright trilogy, for instance, seem to have been drafted in the 1960s), we can in general say that the later novels stand up to scrutiny better than the earlier ones, which rely on a sort of verbal humour that now seems dated, based on english models and refecting the english class system.\n\nWhile the late novels do not eschew comedy \u2013 Jolley always believed that novels should entertain \u2013 they also make forays into the erotic, particularly the male erotic. _Lovesong_ (1997) traces the progress of a gentle soul, a man drawn to children, who after being trapped _in flagrante_ with a choirboy is subjected to a programme of behaviour modification and then set loose in a robust adult world where he soon loses his way. In _An Accommodating Spouse_ (1999) a vain, complacent, but timorous academic who has fallen in love with a lesbian colleague takes at face value a suggestion of his wife's that it would be a good idea if he gave the colleague a child. _An Innocent Gentleman_ (2001) reworks a childhood episode recounted by Jolley in one of her autobiographical sketches. An uxorious english schoolteacher gives his wife permission to have an affair with another man, then finds he cannot cope with the consequences. In the sketch the focus is on the fifteen-year-old child as she witnesses her father's humiliation. In the novel dates are juggled so as to reduce the child's role: the focus is on the married couple themselves, on the wife's erotic infatuation and the husband's inability to square his belief in free marriage with his jealous anger.\n\nAll three of these late novels pose the question 'What do men want?', and reveal desire and its vicissitudes to be quite as inscrutable in men as in women. Idealistic and impulsive, Jolley's men typically have no insight into their own motives and are competent to deal neither with the demands of the world nor with the wiles of women.\n\nOf the novels antedating the trilogy, _The Sugar Mother_ (1988) is the most accomplished. Here a canny english immigrant tricks a na\u00efve Australian academic out of his money by persuading him he has got her daughter pregnant, then offering that for a price the girl will bear the child and bestow it on him and his unwitting wife. _Miss Peabody's Inheritance_ (1983) is an ingeniously constructed fiction about the writing of fiction; _Mr Scobie's Riddle_ (1983) takes a sobering look at the exploitation of the aged; but both of the latter are hampered by heavy-handed comedy.\n\nFor a novelist reluctant to appear too serious, Jolley is remarkably daring in the topics she takes up: erotic infatuations on the part of both men and women with children who are often more seducer than seduced; sexual accommodations between women getting on in years; jealousy as a spur to sexual arousal; spousal love as a mask for control; the heartlessness behind the soothing discourse of aged care; impotence and incontinence; prurience; the never-to-be-appeased hunger of the old, the ugly, and the despised to be touched.\n\nMonica Elizabeth Knight was born in Birmingham in 1923. After some years of home schooling, she was sent to a Quaker boarding-school. While working as a trainee nurse in 1940 she met Leonard Jolley (1914-1994), by profession a librarian, whom she later married and with whom she in 1959 emigrated to Australia.\n\nElizabeth Jolley (the name under which Monica Knight published) wrote fiction and poetry from an early age. Her posthumous papers include a novel completed when she was sixteen. Her career as a published writer took off only in 1976, but she had been circulating manuscripts \u2013 including the manuscript of a novel entitled _The Georges' Wife and the Feast_ \u2013 since the mid-1960s. During the next twenty-five years she published prolifically. Most of her fiction is set in Australia, where she was honoured as a major writer. She died in 2007.\n\nThe Vera Wright trilogy stands out as the most ambitious and most accomplished work in Elizabeth Jolley's oeuvre. It draws heavily on the early life of Monica Knight, but \u2013 as its author warns again and again \u2013 amends and invents details and circumstances freely. In memoirs and other occasional pieces Jolley was in the habit of presenting herself as a self-taught writer who had done ordinary and indeed menial jobs most of her life. But this was just protective coloration. In truth Jolley was steeped in literature, particularly english and german poetry. From the Modernist novelists of the generation before hers, Ford Madox Ford, James Joyce and Virginia Woolf in particular, as well as from their forebear Laurence Sterne, she absorbed the essentials of the narrative method so strikingly deployed in the trilogy. They also confirmed in her her sense of the self as the shifting and evanescent issue of a dialectic between memory and the fabulating intelligence.\n\nJ. M. Coetzee\n\nMarch 2008\n\n# FAIRFIELDS\n\n'Why can't the father, the father of your \u2013 what I mean is why can't he do something?'\n\n'I've told you, he's dead.'\n\n'How can you say that, he was on the phone last night. I could tell by your voice, that's who it was.'\n\n'He's dead. I've told you.'\n\nAt last the day has come when I must leave for Fairfields. It is all arranged. I have been there once already and know it to be a place of grated raw vegetables and children with restless eyes. It is also a place of poetry and music and of people with interesting lives and ideas.\n\n'I simply can't understand you. How could you with your education and your background breed like a rabbit\u2014'\n\n'You're always saying that, for years you've said it. I've told you, rabbits have six, I only one.'\n\n'How can you speak to me, your mother, like that.'\n\n'Oh shut up and remember this. I'm never coming back. Never!'\n\n'And another thing, Helena looks like a miner's child dressed up for an outing!' My mother does not like the white frock and the white socks and the white hair ribbons. I tie Helena's hair in two bunches with enormous bows and do not remind my mother that she bought the white frock, and the white socks and the white ribbons.\n\n'She'll get a headache, her hair pulled tight like that. And why white for a train journey, two train journeys. Oh Vera!' My mother, I can see, has tears in her eyes. 'Leave Helena here with me, your father and I would like to have her here with us, please! Besides, she is happy with us.'\n\nBut I will not be parted from my child. I throw a milk bottle across the kitchen, it shatters on the tiles and I am pleased because my mother is frightened. 'What's wrong with miners and their children and their outings?' I shout at her.\n\nPerhaps Helena would be happier with her grandmother. I do not want to think this and it is painful to be told.\n\nMy father comes with us to the station.\n\n'That's a nice coat,' he says, carrying it for me. It is my school winter coat, dark green and thick. It would not fit into my case so I have to carry it or wear it.\n\n'It's a new coat, is it?' he says feeling the cloth with his hands. I don't reply because I have been wearing the coat for so many years.\n\nWe are too early for the train. The platform is deserted.\n\n'It looks like a Loden,' he is still talking about the coat. 'Like an Austrian Loden cloth.' He is restless, my father, very white faced and he holds Helena's hand and walks up and down the platform, up and down. The coat on his other arm.\n\nAlways when my father sees off a train he is at the station too soon. And then, when the train is about to leave, when the whistle is being blown and the doors slam shut, one after the other down the whole length of the train, he rushes away and comes back with newspapers and magazines and pushes them through the window as he runs beside the now moving train. As the engine gets up steam and the carriages clank alongside the platform my father increases his speed, keeping up a smiling face outside the window.\n\nHis bent figure, his waving arms and his white face have always been the last things I have seen when leaving. I know too from being with him, seeing other people off, that he stands at the end of the platform, still waving, long after the train has disappeared.\n\nWalking up and down we do not speak to each other. The smell of the station and the sound of an engine at the other platform remind me of Ramsden and of the night several years earlier when I met her train. Ramsden, staff nurse Ramsden, arriving at midnight. There was a thick fog and her train was delayed.\n\n'I've invited Ramsden to come and stay for a few days,' I said to my mother then, assuming a nonchalance, a carelessness of speech to hide Ramsden's age and seniority.\n\n'Why of course Vera, a nursing friend is always welcome...' There had been a natural progression from school friend to nursing friend. My father never learned to follow, to keep up with this progression.\n\n'And is Miss Ramsden a good girl?' would be his greeting, a continuation of, 'and is Jeanie a good girl?' He would say it to Ramsden without seeing the maturity and the elegance and without any understanding of the superior quality of her underclothes.\n\n'My parents are looking forward to meeting you.' I invited Ramsden knowing already these other things.\n\nRamsden, with two tickets to Beethoven, in our Town Hall, prepared herself to make the long journey.\n\nPutting off the visit, in my mind, from one day to the next, reluctantly, at last I was in the Ladies Only waiting room crouched over a dying fire, thin lipped and hostile with the bitter night. My school coat heavy but not warm enough and my shoes soaked.\n\nRamsden, who had once, unasked, played the piano for my tears, arrived at last. I could see she was cold. She was pale and there were dark circles of fatigue round her eyes. She came towards me distinguished in her well-cut tailored jacket and skirt. Her clothing and manner set her apart immediately from the other disembarking passengers.\n\n'Miss Ramsden will have to share the room,' my mother said before I left for the station, 'your sister's come home again.' Shrugging and blinking I went on reading without replying. Reading, getting ready slowly, turning the page of my book, keeping one finger in the page while I dressed to go and meet the train.\n\nRamsden came towards me with both hands reaching out in leather gloves. At once she was telling me about the Beethoven, the choral symphony, and how she had been able to get tickets. There was Bach too, Cantata eighteen. Remember? She said. _For as the rain cometh down and the snow from heaven,_ she, beating time with one hand, sang, _so shall my word be that goeth forth out of my mouth_... In the poor light of the single mean lamp her eyes were pools of pleasure and tenderness. She did not mind the black-out she said when I apologized for the dreariness of the station. 'Ramsden,' I said, 'I'm most awfully sorry but there's been something of a tragedy at home. I couldn't let you know... I'm so most awfully sorry...'\n\n'Not...?' Concern added more line to Ramsden's tired face. I nodded turning away from the smell of travelling which hung about the woollen cloth of her suit.\n\n'Oh Wright! I am so sorry, Veronica.' It was the first time she had spoken my first name, well almost the first time. I glanced at her luggage which stood by itself on the fast emptying platform. The case seemed to hold in its shape and leather the four long hours of travelling, the long tedious journey made twice as long by the fog.\n\nThere would be a stopping train to London coming through late, expected at three in the morning the porter said as Ramsden retrieved her case from its desolation.\n\nA glance into the waiting room showed that the remnant of the fire was now a little heap of cold ash. Perhaps, she suggested, even though there was no fire it would be warmer to sit in there.\n\n'I'm so sorry,' I said, 'I shall not be able to wait for your train.' So sorry, I told her, I must get back, simply must get back.\n\n'Is it...?' More concern caused Ramsden to raise her dark eyebrows. The question unfinished, I drew my arm away from her hand's touch. I thought of the needlework and embroidery book I had chosen from her room, too nervous with my act, then, to read the titles when, to please me, she said to choose a book to have to keep, as a present, from her shelf. The badly chosen book I thought, at the time, made me feel sick. I began that day, almost straight away, to feel sick.\n\nWe walked along the fog-filled platform. 'I've come to you all the way from London,' Ramsden, drawing me to her, began in her low voice. 'I'd hoped...' I turned away from the clumsy embrace of her breathed-out whispered words knowing her breath to be the breath of hunger.\n\n'I'm sorry,' I said again stiffening away from her, 'but I'll have to go.'\n\n'To them,' she said, 'yes of course, you must.' She nodded her understanding and her resignation.\n\n'I am sorry I can't wait till your train comes. I can't wait with you. I'm most awfully sorry!' Trying to change, to lift my accent to match hers.\n\nShe nodded again. I knew from before, though I couldn't see them, what her eyes would be like.\n\nI had to walk the three miles home as there were no buses at that time of night. The fog swirled cold in my face. The way was familiar but other things were not. My own body, for one thing, for I was trying, every day, to conceal my morning sickness.\n\nI turn away trying to avoid the place on the platform where Ramsden tried to draw me towards an intensity of feeling I could not be a part of that night.\n\n'She wasn't on the train,' I told my mother the next morning standing on purpose behind her flowered overall and keeping to the back of her head which was still encased in metal rollers. She was hurrying to get to the Red Cross depot. Her war effort.\n\n'It was a dreadful night for travelling,' my mother said, not turning from the sink, 'perhaps your Miss Ramsden will send a letter. You can invite her again, perhaps in the spring, we'll have more space then, perhaps by then your sister will be better.'\n\nMy father, running now beside the moving train, pushes a magazine and a comic through the window. I, because I feel I must, lean out and see him waving at the end of the platform. Helena, clinging to my skirt, cries for her Grandpa.\n\nUnable to stop thinking of Ramsden I wonder why do I think of her today after all this time of forgetting her. I never write to her. I never did write even when she wrote to me saying that she was still nursing and that she lived out, that she had a little flat which had escaped the bombs and if I liked to stay she would love to have me stay as long as I liked, 'as long as you feel like it'. I never answered. Never told her I had a child. Never let her into my poverty and never let her into my loneliness.\n\nLondon is full of people who seem to know where they are and to have some purpose in this knowing. I drag my case and the coat and Helena and change stations and at last we are travelling through the fields and summer meadows of Hertfordshire. The train, this time, is dirty and has no corridor and immediately Helena wants the lavatory. I hate the scenery.\n\nAt last we are climbing the steep field path from the bus stop to the school. Fairfields, I have been there once already and know the way. The path is a mud path after it leaves the dry narrow track through the tall corn which is turning, waving and rippling, from the green to gold, spotted scarlet with poppies and visited by humming hot-weather insects. I have seen before that the mud is caused by water seeping from two enormous manholes in the trees at the top of the hill. Drains, the drains of Fairfields School.\n\n'Who is that?' Helena stops whimpering. And I see a man standing quite still, half hidden by trees. He does not seem to be watching, rather it is as if he is trying to be unseen as we climb together. He does not move except to try and merge into a tree trunk. With the case I push Helena on up the steepest part of the path and I do not look back into the woods.\n\nIn the courtyard no one is about except for a little boy standing in the porch. He tells me his Granny will be coming to this door, that he is waiting to be fetched by her. 'My Granny's got a gas stove,' he tells me. I think suddenly of my mother's kitchen and wish that I could wait now at this door for her to come and fetch me and Helena. Straight away I want to go back.\n\nMiss Palmer, the Principal, the one they call Patch, I know this too from my earlier visit, carrying a hod of coke, comes round from an outhouse.\n\n'Ah!' she says. 'I see you mean to stay!' she indicates my winter coat. 'So this is Helena!' She glances at my child. 'She's buttoned up I daresay.' I know this to mean something not quite explained but I nod and smile. Patch tells me that no one is coming to fetch Martin. 'He's new, he hasn't,' she says, 'adjusted yet.'\n\nShe shows me my room which I am to share with Helena. It is bare except for a cupboard and two small beds. It is bright yellow with strong smelling distemper. There is a window, high up, strangled with creeper.\n\n'Feel free to wander,' Patch says, 'tea in the study at four. Children's tea in the playroom at five and then the bathings. Paint the walls if you feel creative.' She has a fleshy face and short, stiff hair, grey like some sort of metal. I do not dislike her.\n\n'Thank you,' I say, narrowing my eyes at the walls as if planning an exotic mural.\n\nHelena, pulling everything from the unlocked case, intones a monologue over her rediscovered few toys. I stare into the foliage and the thick mass of summer green leaf immediately outside the window.\n\nLater the Swiss girl, Josepha, who has the room opposite mine, takes me round the upstairs rooms which are strewn with sleeping children. We pull some of them out of bed and sit them on little chipped enamel pots. There is the hot smell of sleeping children and their pots.\n\nJosepha tells me the top bathroom is mine and she gives me a bath list. The face flannels and towels hang on hooks round the room.\n\nJosepha comes late to breakfast and takes most of the bread and the milk and the butter up to her room where her sweetheart, Rudi, sleeps. I heard their endless talk up and down in another language, the rise and fall of an incomprehensible muttering all night long, or so it seemed in my own sleeplessness.\n\nThe staff sit at breakfast in a well-bred studied shabbiness huddled round a tall copper coffee pot and some blue bowls of milk. Children are not allowed and it seems that I hear Helena crying and crying locked in our room upstairs. Patch does not come to breakfast but Myles, who is Deputy Principal, fetches prunes and ryvita for her. She is dark-eyed and expensively dressed like Ramsden but she has nothing of Ramsden's music and tenderness. She is aloof and flanked by two enormous dogs. She is something more than Deputy Principal. Josepha explains.\n\n'Do not go in,' Josepha points at Patch's door, 'if both together are in there.'\n\nWhen I dress Helena I take great trouble over her hair ribbons and let her, with many changes of mind, choose her dress because I am sorry for leaving her alone, locked in to cry in a strange place. I have come to Fairfields to work with the idea that it will give Helena school and companionship and already I have tried to persuade her, to beg her and finally rushed away from her frightened crying because staff offspring (Myles' words) are not allowed at staff meals. I take a long time dressing Helena and find that Josepha has dressed all the children from my list as well as her own. I begin to collect up the little pots.\n\n'No! Leave!' Josepha shouts and, tying the last child into a pinafore, she herds them downstairs. Moving swiftly Josepha can make me, with Helena clinging to my dress, seem useless.\n\nJosepha does the dining room and I am to do, with Olive Morris, the playroom where the smaller children have their meals. Mrs Morris has a little boy called Frank but Helena will not sit by him. She follows me with a piece of bread and treacle and I have to spend so much time cleaning her that Olive Morris does the whole breakfast and wipes the tables and the floor. She does not say anything only gets on with ladling cod-liver oil, which is free, into the children as they leave the little tables.\n\nI discover that Olive Morris has three children in the school and that Josepha feels it is morally right that Olive should work more than anyone else because of this. Josepha is always dragging children off to have their hair washed. She has enormous washing days and is often scrubbing something violently at ten o'clock at night. The smell of scorching accompanies the fierceness of her ironing.\n\n'Do not go in there,' she points to the first-floor bathroom, 'when Patch and Tanya are in there and,' she says, 'do not tell Myles!'\n\nTanya teaches art. She looks poor but Josepha says she is filthy rich and wears rags on purpose.\n\nTanya, on my first day, was painting headless clowns on the dining-room walls. She stepped back squinting at her work. 'They are going to play ball with the heads,' she explained bending down over her paint pots as if she had been talking to me for years.\n\n'What a good idea,' I said, ashamed of my accent and trying to sound as if I knew all about painting.\n\nThat day she asked me what time it was, saying that she must hurry and get her wrists slashed before Frederick comes back from his holiday.\n\nLater, in the pantry, she is there with both arms bandaged. 'Frederick the Great,' she says, 'he'll be back. Disinfectant, fly spray, cockroach powder and mouse traps. He will,' she says, 'ask you to examine his tonsils.'\n\nOlive Morris looks ill. Sometimes when I sit in my room at night with an old cardigan round the light to keep it off Helena's bed I think of Olive and begin to understand what real poverty is; her dreadful little bowls of never clean washing, the rags which she is forever mending and her pale crumpled face from which her worried eyes look out hopelessly.\n\nI have plenty of pretty clothes for Helena. And then it suddenly comes to me that this is the only difference. My prospects are the same as Olive's. I have as little hope for the future as she has. It just happens that at present, because of gifts from my mother, Helena, for the first years of her life, has been properly fed and is well dressed.\n\nOne hot afternoon I sit with the children in the sand pit hoping that they will play. There are only two little spades and the children quarrel and fight and bite each other. It is hard to understand why the children can't enjoy the spacious lawns and the places where they can run and shout and hide amongst the rose bushes. Beyond the lawn is deep uncut grass bright with buttercups and china-blue hare-bells. I am tired, tired in a way which makes me want to lie down in the long grass and close my eyes. Helena, crying, will not let me rest. The children are unhappy. I think it is because they do not have enough food. They are hungry all the time.\n\nI do lie down and I look up at the sky. Once I looked at the sky, not with Ramsden but after we had been talking together. I would like to hear Ramsden's voice now. It is strange to wish this after so long. Perhaps it is because everyone here seems to have someone. Relationships, as they are called here, are acceptable. And I, having no one, wishing for someone, vividly recall Ramsden. She said, that time in the morning before I went for my day off to sleep among the spindles of rosemary at the end of my mother's garden, that love was infinite. That it was possible, if a person loved, to believe in the spiritual understanding of truths which were not fully understood intellectually. She said that the person you loved was not an end in itself, was not something you came to the end of, but was the beginning of discoveries which could be made because of loving someone.\n\nLying in the grass, pushing Helena away, I think about this and wonder how I can bring it into the conversation at the four o'clock staff tea and impress Patch and Myles. I practise some words and an accent of better quality.\n\nBecause of being away from meadow flowers for so long I pick some buttercups and some of the delicate grasses adding their glowing tips to the bunch wondering, with bitter uneasiness, how I can get them unseen to my room. I can see Patch and Myles at the large window of Patch's room. Instead of impressing them I shall simply seem vulgar, acquisitive and stupid, clutching a handful of weeds, ineffectually shepherding the little children towards their meagre plates of lettuce leaves and Patch-rationed bread.\n\nIn the evening there is a thunder storm with heavy rain. I am caught in the rain on the way back from the little shop where I have tried to buy some fruit. The woman there asks me if I am from the school and if I am, she says she is unable to give me credit. In the shop there is the warm sweet smell of newspapers, firelighters and cheap sweets, aniseed, a smell of ordinary life which is missing in the life of the school. Shocked I tell her I can pay and I buy some poor-quality carrots as the apples, beneath their rosy skins, might be rotten. I will wash the carrots and give them to Helena when I have to leave her alone in our room in the mornings.\n\nThe storm is directly overhead, the thunder so loud I am afraid Helena will wake and be frightened so I do not shelter in the shop but hurry back along the main road, through the corn and up the steep path. I am wet through and the mud path is a stream. The trees sway and groan. I slip and catch hold of the undergrowth to stop myself from falling. When I look up I see that there is someone standing, half hidden, quite near, in the same place where a man was standing on that first afternoon. This man, I think it is the same man, is standing quite still letting the rain wash over him as it pours through the leaves and branches. His hair is plastered wet-sleeked on his round head and water runs in rivulets down his dark suit. He, like me, has no coat. He does not move and he does not speak. He seems to be looking at me as I try to climb the steep path as quickly as I can. I feel afraid. I have never felt or experienced fear like this before. Real terror, because of his stillness, makes my legs weak. I hurry splashing across the courtyard and make my way, trembling, round to the kitchen door. Wet and shivering I meet Olive Morris in the passage outside my room. She is carrying a basin of washing. Rags trail over her shoulder and her worn-out blouse, as usual, has come out of her skirt.\n\nI tell her about the man in the woods. 'Ought I to tell Patch?' I try to breathe calmly. 'It's getting dark out there. He's soaked to the skin. I ought to tell Patch.'\n\nOlive Morris's shapeless soft face is paler than ever and her lips twitch. She looks behind her nervously.\n\n'No,' she says in a low voice. 'No, never tell anyone here anything. Never!' she hurries off along to the other stairs which lead directly up to her room in the top gable of the house.\n\nWhile I am drying my hair, Olive Morris, in a torn raincoat, comes to my door.\n\n'I'm going down to post a letter,' she says putting a scarf over her head. 'So if I see your stranger in the trees I'll send him on his way \u2013 there's no need at all to have Myles go out with the dogs. No need at all.'\n\nMy surprise at the suggestion that Myles and the dogs might hunt the intruder is less than the feeling of relief that I need not go to Patch's room where Myles, renowned for her sensitive nudes, will be sketching Patch in charcoal and reading poetry aloud. They would smile at each other, exchanging intimate glances while Patch pretended to search her handbag for a ten shilling note as part of the payment owing to me, Myles had looked up gazing as if thoughtfully at me for a few minutes and then had resumed her reading of the leather-bound poems.\n\nJosepha is on bedroom duty and the whole school is quiet. Grateful that Helena has not been disturbed by the storm I lie down in my narrow bed.\n\nInstead of falling asleep I think of the school and how it is not at all as I thought it would be. Helena stands alone all day peering through partly closed doors watching the dancing classes. She looks on at the painting and at the clay-modelling and is only on the edge of the music.\n\nThere must be people who feel and think as I do but they are not here as I thought they would be. I want to lean out of a window in a city full of such people and call to some passer-by. I am by my own mistakes buried in this green-leafed corruption and I am alone.\n\nMy day off which Josepha did not tell me about till all the children were washed and dressed was a mixture of relief and sadness. A bus ride to town. Siting with Helena in a small cafe eating doughnuts. Choosing a sun hat for Helena. Buying some little wooden spades and some coloured chalks. Trying to eat a picnic lunch of fruit and biscuits on a road mender's heap of gravel chips. I can hardly bear to think about it. As I handed Helena her share and saw her crouched on the stones with her small hands trying to hold her food without a plate I knew how wrong it was that she was like this with no place to go home to.\n\nI think now over and over again that it is my fault that we are alone, more so than ever, at the side of the main road with cars and lorries streaming in both directions.\n\nThere is a sudden sound, a sound of shooting. Gun shots. I go into the dark passage. From Josepha's room comes the usual running up and down of their voices, first hers and then his. I am afraid to disturb them. A door further down clicks open and I see, with relief, it is Tanya.\n\n'Oh it's you Tanya! Did you hear anything just now?'\n\n'Lord no. I never hear a thing m'dear and I never ask questions either so if you've been letting anyone in or out I just wouldn't know darling.'\n\nI tell her about the shot.\n\n'Lord!' Tanya says. 'That's Frederick. Back from his leave. Frederick the Great, literature and drama. Room's over the stables. Never unpacks. Got a Mother. North London. Cap gun. Shoots off gun for sex. The only trouble is darling,' Tanya drawls, 'the orgasm isn't shared.' She disappears into the bathroom saying that she's taken an overdose and so must have her bath quickly.\n\nI go on up the next lot of stairs to Olive's room. I have never been there. I must talk to someone. Softly I knock on the door. At once Olive opens it as if she is waiting on the other side of it.\n\n'Oh it's you!' Her frightened white face peers at me.\n\n'Can I come in?' I step past her hesitation into her room. It is not my intention to be rude, I tell her, it is my loneliness. Olive catches me by the arm. Her eyes implore. I am suddenly ashamed for, sitting up in bed wearing a crumpled shirt and a tie, is a man. The man I had seen standing with sinister patience in the rain.\n\n'Oh Olive, I am so sorry. I do beg...'\n\n'This is Mr Morris, my husband. This is Vera Wright, dear,' Olive whispers a plain introduction.\n\n'Pleased to meet you I'm sure,' Mr Morris says. I continue to mumble words of apology and try to move backwards to the door.\n\nThe three Morris children are all in a heap asleep in a second sagging double bed up against the gable window. Washing is hanging on little lines across the crowded room and Mr Morris's suit is spread over the bed ends to dry.\n\n'Mr Morris is on his way to a business conference,' Olive begins to explain. I squeeze her arm. 'I'll see you tomorrow,' I say. We are wordless at the top of the steep stairs. She is tucking her blouse into her too loose skirt. It seems to me that she will go on performing this little action forever even when she has no clothes on.\n\n'No one at all knows that Mr Morris is here,' she says in a breaking whisper.\n\nAt breakfast I wish I had someone to whom I could carry, with devotion, bread and butter and coffee. I could not envy Myles because of Patch, or Josepha because of Rudi. Tanya must be feeling as I feel for she prepares a little tray for Frederick and is back almost at once with a swollen bruised bleeding nose and quite quickly develops two black eyes which, it is clear, will take days to fade.\n\nIt would be nice for Olive to sail into breakfast and remove a quantity of food bearing it away with dignity to the room in the top gable.\n\n'I suppose you know,' I say to Patch when we meet by chance in the hall, 'that Olive Morris's husband arrived unexpectedly last night and will be with us for a few days.'\n\nPatch says, 'Is he dear?' That is all.\n\nMr Morris, who is a big man, wears his good suit every day thus setting himself somewhat apart from the rest of us. He comes to supper and tells us stories about dog racing. His dogs win. He tells us about boxing and wrestling. He has knocked out all the champs. He knows all their names and the dates of the matches. He knows confidence men who treble their millions in five minutes. His brothers and sisters teach in all the best universities and his dear old mother is the favourite Lady in Waiting at Buckingham Palace. Snooker is his forte, a sign, he tells us, of a misspent youth. He sighs.\n\nPatch comes to supper every night. Josepha stops shouting at Olive. Mr Morris calls Olive 'Lovey' and reminds her, for us all to hear, of extravagant incidents in their lives. He boasts about his older children regaling us with their exam results and sporting successes. Olive withers. She is smaller and paler and trembles visibly when Patch, in a genial mood, with mockery and amusement in her voice, leads Mr Morris into greater heights of story telling. While he talks his eyes slide sideways as he tries to observe us all and see the effects of his fast-moving mouth.\n\nMr Morris, we have to see, is the perfect husband and father. During the day he encourages his children and the other children to climb all over him. He organizes games and races, promising prizes.\n\nHe gives all sorts of presents, the table in the kitchen is heaped with chickens and ducks, ready for the oven, jars of honey and expensive jams and baskets of apples and fresh vegetables. Patch prepares the meals herself. Our vegetarian diet was only because the local butchers, unpaid, no longer supply the school.\n\nFrederick, refusing to come to meals, refusing to leave the loft, has a bucket on a string into which Josepha, he will not take from anyone else, puts chicken breasts and bread and butter and a white jug of milk. Tanya says if there is any wormy fruit or fly-blown meat Frederick the Great will get it. He, she explains, because of always searching for them, attracts the disasters in food.\n\n'Where is Mr Philbrick?' Patch asks correcting quickly what she calls a fox's paw, a slip of the tongue. 'Mr Morris? Why isn't he here?' She is carving, with skill, the golden chickens and Myles is serving the beans and baby carrots which shine in butter. Olive can hardly swallow a mouthful.\n\n'What's keeping Mr Morris?' I ask her loud enough for Patch's ears. 'Anything wrong?' devouring my plateful. 'Is someone ill?'\n\n'No. No \u2013 it's nothing at all,' she whispers.\n\nTowards the end of the meal Mr Morris comes in quietly and sits down next to the shrinking Olive. Patch, with grease on her large chin, hands a plate of chicken to him. Thickset, stockily at the head of the table, she sings contralto as if guarding a secret with undisturbed complacence.\n\nThere is a commotion in the hall and the sound of boots approaching.\n\n'It is the Politz!' Josepha, on bedroom duty, calls from the stairs.\n\nMr Morris leaps up.\n\n'Leave this to me dear Lady,' he says to Patch. And, with a snake-like movement, he is on his way to the door.\n\nWe follow just in time to see Mr Morris, suddenly small and white-faced, being led in handcuffs to the front door and out to a car which, with the engine running, is waiting.\n\nI want to say something to Olive to comfort her.\n\n'It's better this way,' she says, 'better for him this way, better than them getting him with dogs. And the children,' she says, 'they didn't see anything.' I don't ask her what Mr Morris has done. She does not tell me anything except that Mr Morris finds prison life unbearable and that he has a long stretch of it ahead.\n\nPatch walks about the school singing and eating the ends off a crusty loaf. When the bills come addressed to her for all the presents from Mr Morris she laughs and tosses them into the kitchen fire.\n\nOne of the little boys rushing through the hall stops to glance at Tanya's latest painting.\n\n'How often do you have sexual intercourse?' he pauses long enough in his flight to ask.\n\n'Three times a week.' Tanya steps back to squint at her work. 'Never more, never less,' she says.\n\nTanya says that Frederick the Great is coming down from the loft and will be at supper. I wash my hair and put on my good dress and go down to the meal early rejoicing that it is Olive's night to settle the children. I am looking forward to meeting Frederick. Perhaps, at last, there will be someone for me. Olive scuttles by with her tray which she must eat upstairs. I hear the uproar from the bedrooms and smile to myself.\n\nFrederick is bent in a strange contortion over the sink in the pantry. He is trying to see into his throat with a torch and a small piece of broken mirror stuck into the loose window frame. I am glad to be able to meet him without Josepha and Tanya.\n\n'Would you mind looking at my throat,' he says straightening up. He is very tall and his eyes enclosed in gold-rimmed spectacles do not look at me. 'I've been trying a new gargle.' He hands me the torch and I peer into his throat.\n\n'Is it painful?' I feel I should ask him.\n\n'Not at all,' he says, taking back the torch.\n\nIn the dining room Frederick has a little table to himself in the corner. He eats alone quickly and leaves at once. I sit in my usual place. One of the children is practising on the pantry piano. I listen to the conscientious stumblings. Ramsden played Bach seriously repeating and repeating until she was satisfied and then moving on to the next phrases.\n\nIn my head I compose a letter to Ramsden... _this neck of the woods_ , this is not my way but it persists, _this neck of the woods is not far from London. Any chance of your coming down one afternoon? Staff tea is at four. I'd love to see you and show you round_...\n\nThere is so much I would tell Ramsden.\n\n_For as the rain cometh down, and the snow from heaven, and returneth_\n\n_not thither, but watereth the earth, and make it bring forth and bud,_\n\n_that it may give seed to the sower, and bread to the eater: so shall my word be that goeth forth out of my mouth:_\n\nI want to write to Ramsden. After that night and after almost five years how do I address her? Dear Ramsden? Dear staff nurse Ramsden? She might be Sister Ramsden. She might not be nursing now though she did go on after the end of the war. She might be married though I think that is unlikely, perhaps she is on concert platforms...\n\n_Dear Ramsden I have no way at all of getting away from this place. Please Ramsden can you come? Please?_\n\nPatch and Myles come in to supper. Ignoring me they devotedly help each other to mountains of grated raw carrots and cabbage.\n\n# MY FATHER'S MOON\n\nBefore this journey is over I intend to speak to the woman. _Ramsden_ , I shall say, _is it you?_ The train has just left the first station, there is plenty of time in which to contemplate the conversation; the questions and the answers and the ultimate revelation. It is comfortable to think about the possibilities.\n\nThe woman siting on the other side, diagonally opposite, could be someone I used to know. A long time ago. In another place. Her clothes are of the same good quality, the same materials, even the same colours. It is the tilt of the head which is so remarkably similar. She looks like someone who is passionately fond of the cello. Fond of listening to the cello. I look at her hands and feel sure she plays the piano. When I look at her hands it is as if I can hear her playing a Mozart sonata or practising something from Bach. Repeating and repeating phrases until a perfection is achieved. I am certain, as I go on looking, that she plays Cyril Scott's _Water Wagtail_.\n\nFor some time now I have travelled by suburban train to and from the places where I work. This evening I am on the earlier train. I caught the earlier train on purpose even though, because of this, I arrive too soon...\n\nThe unfamiliar early train travels, of course, through the same landscape, the familiar. There is nothing remarkable in this. It is my reason for taking this train which makes the journey remarkable. The train stops at the same stations but naturally the people getting in or out are not the same people as those of the later train.\n\nI sit staring out of the window at the same meeting places of unknown roads, at the backs of the same shabby houses and garden fences, at the same warehouses and the same smash repair yards and at the now well-known backs of the metropolitan markets.\n\nAbout once a week I catch the earlier train for a special reason. Every week it is the same. Every week I think that this time I will speak to her. This week I am on this train in order to speak to her. I will cross from my seat and sit by her and I will speak to her. I always sit where I can see her from the side and from the back and I sit close enough to hear her voice if she should speak. I long to hear the voice, her voice, to know whether it is the same voice. Voices and ways of speaking often remain unchanged.\n\nThis time I almost brought the violin case with me though I am not now accustomed to carrying it when I go out. If Ramsden saw the violin case, if the woman saw it, she would remember.\n\n'They're both in good condition,' the man in the shop said. 'Both the same price. Choose your pick,' he said. 'Take your time.'\n\nI could not make up my mind, and then I chose the violin case. The following week I went back for the camera case but it had gone. The violin case had once been lined with some dark red soft material, some of it was still left. I only opened it once and it was then I saw the remains of the lining. I carried the case whenever I went out.\n\nThe first time I saw Ramsden the sentry at the hospital gates had his bayonet fixed. He looked awkward and he blushed as he said, 'Who goes there!' Surprised, I told him my name and my identity-card number, it was the middle of the morning and we were challenged, as a rule, only after dark. I supposed the rule must have been changed. A despatch from H.Q., I thought, seeing in my mind the nimble motor cyclist arrive.\n\nRamsden, on her way out, gave a small smile in the direction of the violin case and I was pleased that I had bought it. On that day I had been at the hospital for seven weeks.\n\nTwo people sitting behind me are talking in German. I begin to listen to the animated conversation and grope for meanings in what they are saying in this language which was once familiar. I begin to recognize a few words: _eine Dame_... _keine Ahnung_... _langsam_... _Milch und Tr\u00e4nenb\u00e4che_... _mein Elend_... _zu grosser Schmerz und so weiter_. But I want the words of cherishing spoken in German. I want those first words the child remembers on waking to the knowing of language. I wish now in the train to be spoken to as _du_...\n\nThe woman sitting on the other side is looking calmly out of the window. Naturally she sees the same things that I see. It is quite comfortable to know that I have only to lean over and touch her sleeve.\n\nI never worked with Ramsden. I saw her sometimes in the dining room. There are several little pictures of her in my mind. The doctors called her Miss Ramsden. She did the penicillin syringes too. One nurse, usually a senior, spent the whole day cleaning and sterilizing the syringes and the needles, setting up the trolley, giving the injections and then clearing the trolley and cleaning and sterilizing and checking all over again. Whenever I passed the glass doors of the ward where she was I saw her in the sterilizing room seriously attending to the syringes and needles for the three-hourly injections.\n\n'Ramsden,' I said, 'this is the part we like isn't it? This part, this is it, we like this...'\n\n'It's the anticipation,' she replied, 'it's what is hoped for and then realized.' She was sitting on the edge of her bed.\n\n'This part, this...' I said once more. I pointed with one finger as if to place the cello somewhere in the space between us. 'This going down part,' I said, 'is the part we like best.'\n\nRamsden nodded. She was mending a stocking. Her stockings were not the usual ones, not the grey uniform stockings which were lisle and, after repeated washing, were hard to mend. Ramsden's stockings, I noticed immediately, were smooth and soft and they glistened like honey. Dark, honey-coloured stockings. Ramsden's stockings were silk stockings. She was oversewing a run at the ankle. Her sewing was done so carefully I knew the repair would be invisible. She had invited me into her room to listen to a record.\n\n'Do you know why you like it?' she repeated an earlier question. The cello reminded me of her. How could I tell her this. It shook my head. Staff nurse Ramsden, she was senior to me. When she listened to music she sat with her legs crossed over and she moved her foot very slightly, I could see, in time to the music. How could I speak to her about the downward thrust of the cello and about the perfection in the way the other instruments came up to meet the cello. How could I say to her that I thought someone had measured the movement of the notes controlling carefully the going down and the coming up in order to produce this exquisite mixture. There were other things too that I could not speak about. How could I say to her what I thought about the poet Rilke, about his face and about how I felt when I looked at his photograph in the book she had. She knew his poems, understood them. I wanted to tell her that when I looked at Rilke's face I felt clumsy as if made of wood. Even the way he stood in the photograph had something special about it and when I read a poem of his to myself I wanted to read lines aloud to her. 'Listen to this, Ramsden,' I wanted to say, 'listen to this.'\n\nBut hand in hand now with that God she walked,\n\nher paces circumscribed by lengthy shroudings\n\nuncertain, gentle, and without impatience.\n\nWrapt in herself, like one whose time is near...\n\nThere were other things too from _Orpheus_ , but she knowing his poems might have felt I was intruding. When I read Rilke everything I was trying to write seemed commonplace and unmusical, completely without any delicacy and refinement. I never told Ramsden I was trying to write because what I wrote was about her. I wanted to write about Ramsden. How could I tell her that?\n\nLater when she talked about the music she said the soloist was innocent and vulnerable. She said the music was eloquent ant that there was something intimate about the cello. She was very dignified and all her words seemed especially chosen. I wanted her to say them all again to me. The word intimate, I had never before spoken to anyone who used this word. She said the cello, the music of the cello, was intimate. Ramsden's discipline prevented her from repeating what she had said. She continued to oversew her stocking and we listened once more to the second movement. When I listened to a particular passage in this movement I seemed to see Ramsden walking ahead of me with great beech trees on either side of her. Magnificent smooth trees with their rain-soaked branches darkened and dripping. Then we were walking together, I imagined, beneath these trees, with the wet leaves deep round our ankles. Ramsden, I thought, would have small ribbed socks on over her stockings...\n\nLyrical, she said the music was lyrical and I was not sure what she meant. She said then, that, if I liked, I could borrow her records.\n\nWhen I played the record at home my father, not knowing the qualities of the cello, asked if I could make the music a bit quieter. It was my day off, most of it had been wasted because I slept and no one woke me. My father asked was there a piano piece, he said he liked the piano very much. I told him that staff nurse Ramsden played the piano and my mother said perhaps Miss Ramsden would come some time and play the piano for us. She said she would make a fire in the front room and we could all sit and listen...\n\nBecause I caught the earlier train I have an hour to spare before it is time for the clinic to open. The people who attend this clinic will be setting off from their houses in order to keep their appointments.\n\nI walk to a bus stop where there is a bench and, though I am in a familiar place, I feel as if I have come to a strange land. In one sense there is a strangeness because all the old houses and their once cared for gardens have gone. In their place are tall concrete buildings, floor upon floor of offices, all faced with gleaming windows. Some lit up and some dark. The buildings rise from parking lots all quite similar but unrecognizable as though I have never seen them before. Small trees and bushes planted as ornaments offer a few twigs and leaves. The new buildings are not at peace with their surroundings. They are not part of the landscape, they are in imposition. They do not match each other and they have taken away any tranquillity, any special quality of human life the streets may have had once.\n\nThe Easter lilies, uncherished, appear as they do every year with surprising suddenness, their pink and white long-lasting freshness bursting out of the brown, bald patches of earth at the ends of those places which have been left out from the spreading bitumen.\n\nIf I had spoken in the train I could have said, 'Ramsden,' I could have said, 'I feel sad. Lately I seem unable to prevent a feeling of melancholy which comes over me as soon as I wake up. I feel nervous and muddled and everything is accompanied by a sense of sorrow and futility.' Should I join a sect? I could have asked her. A cult? On TV these people, with a chosen way, all look light hearted. They dance carrying bricks and mortar across building sites. They jive and twist and break-dance from kitchens to dining rooms carrying wooden platters of something fresh and green neatly chopped up. Perhaps it is uncooked spinach. Perhaps it is their flying hair and their happy eyes which attract, but then the memory of the uneasiness of communal living and the sharing of possessions and money seems too difficult, too frightening to contemplate. In real life it won't, I could have told her, it won't be the same as it is on T V. Probably only the more sparkling members of the sect are filmed, I could have said this too, and something is sure to be painted on the spinach to make it look more attractive. Food in advertisements, I could have been knowledgeable, food in advertisements is treated before being photographed. I left the train at my station without another glance in her direction.\n\nPerhaps the lilies are a reminder and a comfort. Without fail they flower at Easter. Forgotten till they flower, and unsought simultaneous caution and blessing.\n\nIt seems to me now, when I think of it, that my father was always seeing me off either at a bus stop or at the station. He would suggest that he come to the bus or the train just as I was about to leave. Sometimes he came part of the way in the train getting out at the first stop and then, waiting alone, he would travel on the first train back. Because of the decision being made at the last minute, as the train was moving, he would have only a platform ticket so, as well as all the waiting and the extra travelling, he would be detained at the other end to make explanations and to pay his fare for both directions. All this must have taken a lot of time. And sometimes in the middle of winter it was bitterly cold.\n\nThe strong feeling of love which goes from the parent to the child does not seem a part of the child which can be given back to the parent. I realize now with regret that I never thought then of his repeated return journeys. I never thought of the windswept platforms, of the small smouldering waiting-room fires and the long, often wet, walks from the bus to the house. I simply always looked ahead, being already on my journey even before I set out, to the place to which I was going.\n\nThe minutes which turned out to be the last I was to have with my father were at a railway station. When it was time for my train to leave even when the whistle was being blown my father went on with what he was saying. He said that if we never saw each other again I must not mind. He was getting older he said then, he was surprised at how quickly he was getting older and though he planned to live a long time it might be that we should not be able to make the next journeys in time. It is incredible that I could have paid so little attention then and the longing to hear his voice once more at this moment is something I never thought of till now.\n\nHe had his umbrella with him and when the train began to move he walked beside the moving train for as long as he could waving the umbrella. I did not think about the umbrella then either. But now I remember that during the years he often left it in trains and it travelled the length and breadth of England coming back at intervals labelled from Liverpool, Norfolk, St Ives and Glasgow to the lost-property office where he was, with a kind of apologetic triumph, able to claim it.\n\nThe huge Easter moon, as if within arm's length, as if it can be reached simply by stretching out both hands to take it and hold it, is low down in the sky, serene and full, lighting the night so that it looks as if everything is snow covered, and deep shadows lie across pale, moon-whitened lawns. This moon is the same moon that my father will have seen. He always told me when I had to leave for school, every term when I wept because I did not want to leave, he told me that if I looked at the moon, wherever I was, I was seeing the same moon that he was looking at. 'And because of this,' he said, 'you must know that I am not very far away. You must never feel lonely,' he said. He said the moon would never be extinguished. Sometimes, he said, it was not possible to see the moon, but it was always there. He said he liked to think of it as his.\n\nI waited once for several hours at a bus stop, a temporary stop on a street corner in London. There was a traffic diversion and the portable sign was the final stop for the Green Line from Hertford. It was the long summer evening moving slowly into the night of soft dusty warmth. A few people walked on the pavement. All of them had places they were going to. A policeman asked me if everything was all right.\n\n'I'm waiting for someone,' I told him. I waited with Helena for Ramsden.\n\nIn the end, in my desperation, I did write my letter to Ramsden asking her to help me to leave Fairfields, the school where I had gone to live and work taking Helena with me. It was a progressive boarding school. There was not enough food and I was never paid. In my letter I told Ramsden everything that had happened, about my child, about my leaving home, about my loneliness, about my disappointment with the school. I had not expected, I told her, such fraudulent ways. My poverty, I thought, would be evident without any description. After writing the letter I was not able to wait for a reply from Ramsden because, when I went to give notice that I wanted to leave in a fortnight, Patch (the headmistress) replied in her singing voice, the dangerous contralto in which she encouraged people to condemn and entangle themselves, 'By all means but please do go today. There's a bus at the end of the field path at three o'clock.' Neither she nor Miss Myles, after exchanging slightly raised eyebrows with one another, said anything else to me.\n\nI sent my letter to the last address I had from Ramsden almost five years earlier. She was, she said then, still nursing and had a little flat where I would be welcome. Five years is a long time.\n\nI told her in my letter that I would wait for her at the terminus of the Green line. As I wrote I could not help wondering if she was by now playing the piano in concerts. Perhaps on tour somewhere in the north of England; in the places where concert pianists play. I tried to think of likely towns and villages. As I wrote I wept, remembering Ramsden's kind eyes and her shy manner. Staff nurse Ramsden with her older more experienced face \u2013 as someone once described her \u2013 and her musician's nose \u2013 someone else had said once. She had never known what there was to know about the violin case I carried with me in those days. It had been my intention always to tell her but circumstances changed intentions.\n\nI begged her in the letter and in my heart to be there. Five years is a long time to ignore a kind invitation from someone. A long time to let pass without any kind of reply. With failing hope I walked slowly up and down the pavement which still held the dust and the warmth of the day. I walked and waited with Helena who was white faced and hungry and tired. Sometimes she sat on our heavy case on my roughly folded school winter coat. I tried to comfort myself with little visions of Ramsden playing the piano and nodding and smiling to Helena who would dance, thump-thump, on the carpet in the little living room. I seemed to remember that Ramsden said in the letter, sent all those years ago, that the flat was tiny.\n\n'You'd best be coming along with me.' It was the policeman again. He had passed us several times. Helena was asleep on the folded coat and I was leaning against the railings at the front of an empty house.\n\nThe woman in charge of the night shelter gave me a small huckaback towel and a square of green soap. She said she had enough hot water if Helena and I could share a bath.\n\n'She's very like you,' the woman said not trying very hard to hide her curiosity behind a certain sort of kindness. She gave us two slices of bread and butter and a thick cup of tea each. She handed me two grey blankets and said Helena would be able to sleep across the foot of the bed she was able to let me have for one night. The girl who had the bed, she explained, was due to come out of hospital where she had been operated on to have a propelling pencil removed from her bladder.\n\n'The things they'll try,' the woman said. 'I or anyone, for that matter, could have told her she was too far gone for anything like that. All on her own too pore thing. Made herself properly poorly and lorst her baby too.' She looked at Helena who was eating her bread and butter, crusts and all, neatly in what seemed to me to be an excessive show of virtue.\n\n'There's some as keeps their kiddies,' the woman said.\n\n'Yes,' I said avoiding her meaning looks. The night shelter for women carried an implication. There was more than the need of a bed. At St Cuthberts the nurses had not been too sympathetic. I remembered all too clearly herding A.T.S. girls into one of the bathrooms every evening where they sat naked from the waist down in chipped enamel basins of hot water and bicarbonate of soda. In her lectures the Sister Tutor reminded often for the need to let patients be as dignified as possible. The hot basins defied this. Many of the girls were pregnant. Some women, the Sister Tutor said, mistook the orifices in their own bodies. All this, at that time, belonged to other people.\n\nLater my own child was to be the embodiment of all that was poetical and beautiful and wished for. Before she was born I called her Beatrice. I forgot about the A.T.S.\n\nGrateful for the hot bath and the tea and the promised bed I addressed the woman in charge as Sister.\n\nDid the Sister, I asked her, ever know a staff nurse called Ramsden? The woman, narrowing her eyes, thought for a moment and said yes she thought she had \u2013 now she recalled it. There was a Ramsden she thought, yes she was sure, who joined the Queens Nurses and went to Mombassa. I tried to take comfort from the doubtful recollection. Yes, went to Mombassa with the Queens Nurses. Very fine women the Queens Nurses. And one night, so she'd heard, the cook in the nurses' quarters was stabbed by an intruder. Horribly stabbed, a dozen or more times in the chest, the neck and the stomach. Apparently the murder was justified, brought on by the cook's own behaviour \u2013 him having gone raving mad earlier that same day. But of Ramsden herself she had no actual news.\n\nI understood as I lay under the thin blanket that she had been trying to offer some sort of reply to my stupid and hopeless question. Perhaps the cook in Mombassa was often murdered horribly in these attempts to provide answers.\n\nI tried to sleep but Helena, accustomed to a bed to herself, kicked unbearably all night.\n\nBeing at a bus stop, not waiting for a bus, and with the dusk turning quickly to darkness, I think of my father's moon. This moon, once his moon and now mine, is now climbing the warm night sky. It hangs in the branches of a single tree left between the new buildings.\n\nThe journey to school is always, it seems, at dusk. My father comes to the first stop. This first journey is in the autumn when the afternoons are dark before four o'clock. The melancholy railway crawls through water-logged meadows where mourning willow trees follow the winding steams. Cattle, knee deep in damp grass, raise their heads as if in an understanding of sorrow as the slow train passes. The roads at the level crossings are deserted. No one waits to wave and curtains of drab colours are pulled across the dimly lit cottage windows.\n\nAt the first stop there is a kind of forced gaiety in the meetings on the platform. Some girls have already been to school and others, like me, are going for the first time. My father watches and when the carriage doors are slammed, one after the other, he melts away from the side of the train as it moves slowly along the platform gradually gathering speed, resuming its journey.\n\nI sink back at once into that incredible pool of loneliness which is, I know now but did not understand then, a part of being one of a crowd. I try to think of the moon. Though it is not Easter, my father said before the doors had all slammed, there will be, if the clouds disperse, a moon. He pointed as he spoke towards the dome of the railway station. Because he pointed with his umbrella I felt embarrassed and, instead of looking up, I stared at my shoes. I try to think about his moon being behind the clouds even if I cannot see it. I wish, I am wishing I had smiled and waved to him.\n\nIn the noisy compartment everyone is talking and laughing. We are all refected in the windows and the dark, shadowed fields slip by on both sides.\n\nThe school bus, emblazoned with an uplifting motto, rattles through an unfamiliar land. The others sing songs which I have never heard before. There is no moon. The front door of the school opens directly on to the village street. Everyone rushes from the bus and the headmaster and his wife stand side by side in a square of light to receive us.\n\n'Wrong hand Veronica. It is Veronica isn't it?' he ticks my name on a list he has. 'Other hand Veronica. We always shake hands with the right hand.'\n\nWhen I unpack my overnight bag I am comforted by the new things, the new nightdress, the handkerchiefs and the stockings folded carefully by my mother. Especially my new fountain pen pleases me.\n\nAlmost at once I begin my game of comparisons, placing myself above someone if more favourable and below others if less favourable in appearance. This game of appearance is a game of chance. Chance can be swayed by effort, that is one of the rules, but effort has to be more persistent than is humanly possible. It is a game of measuring the unfamiliar against the familiar. I prefer the familiar. I like to know my way, my place with other people, perhaps because of other uncertainties.\n\nI am still on the bench at the bus stop. My father's moon is huge and is now above the tree in a dark-blue space between the buildings. A few cars have come. I have seen their headlights dip and turn off and I have seen the dark shapes of people making their way into the place where my clinic is. They will sit in the comfortable chairs in the waiting room till they are called in to see me. Unavoidably I am late sometimes but they wait.\n\nAt the other place where I work there is a scent of hot pines. The sun, beating down on a nearby plantation all day, brings into the warm still air a heart-lifting fragrance. There is a narrow path pressed into the dry grass and the fallen pine needles. This is the path I take to and from the railway station. Sometimes I suggest to other people that they walk on this path. The crows circling and calling suggest great distance. Endless paddocks with waving crops could be quite close on the other side of the new tall buildings. The corridors indoors smell of toast, of coffee and of hot curries. It is as if there are people cooking at turning points on the paths and in corners between the buildings. It is as if they have casually thrown their saris over the cooking pots to protect them from the prevailing winds.\n\nFrom where I sit it seems as if the moon is shining with some secret wisdom. I read somewhere that it was said of Chekhov that he _shows us life's depths at the very moment when he seems to reflect its shimmering surface_.\n\nMy father's moon is like this.\n\nBut the game. The game of comparisons. Before meals at school we have to stand in line beginning with the smallest and ending with the tallest. The room is not very big and the tallest stand over the smallest. We are not allowed to speak and our shoes and table napkins are examined by the prefects. It is during this time of silence and inspection that I make my comparisons. Carefully I am comparing my defects with those of my immediate neighbours. I glance sideways at the pleats of their tunics and notice that the girl next to me bulges. In my mind I call her Bulge; her pleats do not lie flat, they bulge. She is tall and awkward, taller than I am and more round shouldered. I try to straighten my back and to smooth my tunic pleats. I can be better than Bulge. She has cracked lips and she bites her nails. I try not to chew my nails but my hands are not well kept as are the hands of the girl on the other side of me. She has pretty nails and her hair is soft and fluffy. My hair is straight but not as greasy and uneven as Bulge's. Fluffy Hair's feet turn out when she walks. My feet are straight but my stockings are hopelessly wrinkled and hers are not. We all have spots. Bulge's spots are the worst, Fluffy Hair's complexion is the best. She is marred by a slight squint. We all wear spectacles. These are all the same except that Bulge has cracked one of her lenses. My lenses need cleaning.\n\nIt is the sound of someone closing a case very quietly in the dormitory after the lights have been turned off which makes me cry. It is the kind of sound which belongs to my mother. This quiet little closing of a case. My nightdress, which she made, is very comfortable. It wraps round me. She knitted it on a circular needle, a kind of stockinette she said it was, very soft, she said. When she had finished it she was very pleased because it had no seams. She was telling our neighbour, showing her the nightdress and the new clothes for school, all marked with my name embroidered on linen tape. The cabin trunk bought specially and labelled clearly 'Luggage in Advance' in readiness for the journey by goods train produced an uneasy excitement. My mother, handling the nightdress again, spoke to me:\n\n_ein weiches reines Kleid f\u00fcr dich zu weben,_\n\n_darin nicht einmal die geringste Spur_\n\n_Von Naht dich dr\u00fcckt_...\n\n'Shut up,' I said, not liking her to speak to me in German in front of the woman from next door. 'Shut up,' I said again, knowing from the way she spoke it was a part of a poem. 'Shut up,' I crushed the nightdress back into the overnight bag, 'it's only a nightgown!'\n\nWhen I stop crying I pretend that the nightdress is my mother holding me.\n\nOn our second Sunday afternoon I am invited with Bulge and Fluffy Hair and Helen Ferguson and another girl called Amy to explore a place called Harpers Hill. Bulge is particularly shapeless in her Sunday dress. My dress, we have to wear navy-blue serge dresses, is already too tight for me and it is only the second Sunday. Fluffy Hair's dress belongs to her Auntie and has a red lace collar instead of the compulsory white linen one. The collars are supposed to be detachable so that they can be washed.\n\nI wish I could be small and neat and pretty like Amy, or even quick like Helen Ferguson who always knows what's for breakfast the night before. Very quickly she understands the system and knows in advance the times of things, the difference between Morning Meeting and Evening Meeting and where we are supposed to be at certain times, whose turn it is to mop the dormitory and which nights are bath nights. I do not have this quality of knowing and when I look at Helen Ferguson I wonder why I am made as I am. In class Helen Ferguson has a special way of sitting with one foot slightly in front of the other and she sucks her pen while she is thinking. I try to sit as she does and try to look as if I am thinking while I suck the rounded end of my new pen.\n\nDuring Morning Meeting I am worrying about the invitation which seems sinister in some way. It is more like a command from the senior girls. I try and listen to the prayer at the beginning of Meeting. We all have to ask God to be in our hearts. All the time I am thinking of the crossroads where we are supposed to meet for the walk. Bulge does not stop chewing her nails and her fingers all through Meeting. I examine my nails, chew them and, remembering, sit on my hands.\n\nBetween autumn-berried hedges in unscratched shoes and new stockings we wait at the crossroads. The brown ploughed fields slope to a new horizon of heavy cloud. There are some farm buildings quite close but no sign of people. The distant throbbing of an invisible tractor and the melancholy cawing of the rooks bring back the sadness and the extraordinary fear of the first Sunday afternoon walk too vividly. I try not to scream as I screamed that day and I try not to think about the longed for streets crowded with people and endlessly noisy with trams. It is empty in the country and our raincoats are too long.\n\nThe girl, the straw-coloured one they call Etty, comes along the road towards us. She says it's to be a picnic and the others are waiting with the food not far away. She says to follow her. A pleasant surprise, the picnic. She leads us along a little path across some fields to a thicket. We have to bend down to follow the path as it winds between blackberry and under other prickly bushes. Our excited talk is soon silenced as we struggle through a hopeless tangle of thorns and bramble. Amy says she thinks we should turn back. Bulge has the most awful scratches on her forehead. Amy says, 'Look, her head's bleeding.' But Etty says no we shall soon get through to the place.\n\nSuddenly we emerge high up on the edge of a sandy cliff. 'It's a landslide!' I say and, frightened, I try to move away from the edge. Before we have time to turn back the girls, who have been hiding, rush out and grab us by the arms and legs. They tie us up with our own scarves and raincoat belts and push us over the edge and down the steep rough walls of the quarry. I am too frightened to cry out or to resist. Bulge fights and screams in a strange voice quite unlike any voice I have ever heard. Four big girls have her by the arms and legs. They pull her knickers off as she rolls over kicking. Her lumpy white thighs show above the tops of her brown woollen stockings.\n\n'Not this man but Barrabas! Not this man but Barrabas!' they shout. 'She's got pockets in her knickers! Pockets in her knickers!' The horrible chant is all round Bulge as she lies howling.\n\nAs quickly as the big girls appeared they are gone. We, none of us, try to do anything to help Bulge as we struggle free from the knotted belts and scarves. Helen Ferguson and Amy lead the way back as we try to find the road. Though we examine, exclaiming, our torn clothes and show each other our scratches and bruises the real hurt is something we cannot speak about. Fluffy Hair cries. Bulge, who has stopped crying, lumbers along with her head down. Amy, who does not cry, is very red. She declares she will report the incident. 'That's a bit too daring,' I say, hoping that she will do as she says. I am wondering if Bulge is still without her knickers.\n\n'There's Etty and some of them,' Helen Ferguson says as we approach the crossroads. It has started to rain. Huddled against the rain we walk slowly on towards them.\n\n'Hurry up you lot!' Etty calls in ringing tones. 'We're getting wet.' She indicates the girls sheltering under the red-berried hawthorn.\n\n'I suppose you know,' Etty says, 'Harpers Hill is absolutely out of bounds. So you'd better not tell. If you get the whole school gated it'll be the worse for you !' She rejoins the others who stand watching us as we walk by.\n\n'That was only a rag. We were only ragging you,' Etty calls, 'so mind you don't get the whole school gated!' Glistening water drops fly from the wet hedge as the girls leap out, one after the other, across the soaked grass of the ditch. They race ahead screaming with laughter. Their laughter continues long after they are out of sight.\n\nIn Evening Meeting Bulge cannot stop crying and she has no handkerchief. Helen Ferguson, sitting next to me on the other side, nudges me and grins, making grimaces of disgust, nodding in the direction of Bulge and we both shake with simulated mirth, making, at the same time, a pretence of trying to suppress it. Without any sound Bulge draws breath and weeps, her eyes and nose running into her thick fingers. I lean away from her heaving body. I can see her grazed knees because both her stockings have huge holes in them.\n\nBefore Meeting, while we were in line while two seniors were practising Bach, a duet on the common-room piano, Bulge turned up the hem of her Sunday dress to show me a large three-cornered tear. It is a hedge tear she told me then while the hammered Bach fell about our ears. And it will be impossible, when it is mended, she said, for her mother to lengthen the dress.\n\nI give another hardly visible but exaggerated shiver of mirth and pretend, as Helen Ferguson is doing, to look serious and attentive as if being thoughtful and as if listening with understanding to the reading. The seniors read in turn, a different one every Sunday. It is Etty's turn to read. She reads in a clear voice. She has been practising her reading for some days.\n\n'Romans chapter nine, verse twenty-one.' Her Sunday dress is well pressed and the white collar sparkles round her pretty neck.\n\n_Hath not the potter power_\n\n_over the clay, of the same lump_\n\n_to make one vessel unto honour_\n\n_and another unto dishonour?_\n\n'And from verse twenty.' Etty looks up smiling and lisping just a little,\n\n_Shall the thing formed say to_\n\n_him that formed it, Why hast_\n\n_thou made me thus?_\n\nEtty minces from the platform where the staff sit in a semi-circle. She walks demurely back to her seat.\n\n'These two verses,' Miss Vanburgh gets us and puts both hands on the lectern, it is her turn to give the Address, 'These two verses,' she says, 'are sometimes run together.'\n\n'Shall the clay say to the potter why hast thou made me thus...'\n\nBulge is still weeping.\n\nMiss Besser, on tiptoe across the creaking boards of the platform, creeps down, bending double between the rows of chairs, and, leaning over, whispers to me to take Muriel.\n\n'Take your friend out of Meeting, take her to...'\n\n'I don't know her. She isn't my friend,' I begin to say in a whisper, trying to explain, 'she's not my friend...'\n\n'To Matron,' Miss Besser says in a low voice, 'take Muriel.'\n\nI get up and go out with Bulge who falls over her own feet and, kicking the chair legs, makes a noise which draws attention to our attempted silent movement.\n\nI know it is the custom for the one who leads the other to put an arm of care and protection round the shoulders of distress. I know this already after two weeks. It is not because I do not know...\n\nI wait with Bulge in the little porch outside Matron's cottage. Bulge does not look at me with her face, only with her round and shaking shoulders.\n\nMatron, when she comes, gives Bulge a handkerchief and reaches for the iodine. 'A hot bath,' she says to Bulge, 'and early bed. I'll have some hot milk sent up. Be quick,' Matron adds, 'and don't use up too much hot water. Hot milk,' she says, 'in half an hour.'\n\nI do not go back into Meeting. Instead I stand for a time in a place where nobody comes, between the cloakroom and the bootroom. It is a sort of passage which does not lead anywhere. I think of Bulge lying back if only for a few minutes in the lovely hot water. I feel cold. Half an hour, that is the time Matron has allowed Bulge. Perhaps, if I am quick...\n\nThe lights are out in our dormitory. I am nice and warm. In spite of the quick and secret bath (it is not my night), and the glass of hot milk \u2013 because of my bed being nearer the door the maid brings it to me by mistake \u2013 (it has been sweetened generously with honey) in spite of all this I keep longing for the cherishing words familiar in childhood. Because of the terrible hedge tear in the navy-blue hem and, because of the lumpy shoulders, I crouch under my bedclothes unable to stop seeing the shoulders without an arm round them. I am not able to weep as Bulge weeps. My tears will not come to wash away, for me, her shoulders.\n\nAt night we always hear the seniors, Etty in particular, singing in the bathroom. Two of them, tonight, may have to miss their baths. Etty's voice is especially noticeable this night.\n\n_little man you're crying,_ she sings,\n\n_little man you're blue_\n\n_I know why you're crying_\n\n_I know why you're blue_\n\n_Some-one stole your Kiddi-Kar away from you_\n\nThe moon, my father's moon, is too far away.\n\n# RECHA\n\nThe constant sound of television might be for a great many people what a mountain stream was to Wordsworth. Instead of these I have the sound of doves.\n\nThere are times during these golden afternoons when I know that I am not hearing the doves. It seems impossible that I should not hear them when they sidle to and fro, back and forth, along the edge of the roof above my window, endlessly scraping and tapping and rustling along the tremulous gutters. If I do hear them it is only because I hear them all the time, even when they are not there. Their voices are like the voices of a family, heard still even though this family has ceased to exist.\n\nThe little guest, on heat, I remember quite clearly, all those years ago, replete with more than food, stuffed, they would say now, took away with her my silk frock and the Swiss cotton embroidered pillow-slips with which my mother, to honour and please a visitor rather than to offer mere shelter to a homeless refugee, had made the bed.\n\nShe, the little guest, tipping forward on high heels, walks to and fro on the kitchen floor busily scraping left-over morsels of food on to saucers. Scraps of fried bread and bits of chopped-up liver. She carries them one by one to the pantry shelf. As she walks she lets slip from her person little swabs of bloodstained cotton wool. These catch on her heels and are trodden, back and forth, mottling the tiles as if with squashed strawberries.\n\n'It is as if she is on heat,' my mother says.\n\n'The expression, this expression,' my father corrects gently, 'is not, as a rule in English, applied to a human being.'\n\n'Bloodstain,' my mother complains, 'everywhere a bloodstain. Look! Wherever she goes. People, when they grow up, should be able to look after themselves. They should be able to look after their _monatsfluss_.'\n\nLater my mother comes to me.\n\n'Lend Recha your dress, the new one,' she says, persuading. 'Lend her the new dress, the one with the little blue flowers.'\n\n'The one with the forget-me-not flowers? But it's my Liberty silk. It's my good dress. I haven't worn it yet.'\n\n'Yes. Yes. I know but Recha has an interview. She has to go for an interview. You know, she might find for herself a post as a housekeeper. She has no home now. She must find herself a home. We must help her.'\n\n'But she's not my size. The skirt will be much too long.'\n\n'I know this. We can gather it up at the waist. You understand. With a nice sash. You have some ribbon...'\n\n'Why hasn't she got a home then? A house. She's here now in England. She's got a husband. An English soldier, isn't he? He was here with her, in my room, in my bed. Why can't she go? With him? She's safe now.'\n\n'He is in a camp,' my mother explains. 'I don't need to tell you. She has to find somewhere to live. He has to go back to Salisbury, to his camp.'\n\n'But he must have a family, her mother-in-law why can't she...?'\n\n'Be quiet now,' my mother says quickly, 'she is coming downstairs again.'\n\nRecha stands in front of the long mirror in my mother's bedroom twitching the soft folds of my dress over her plump body. She pulls at her black hair frizzing it out on her forehead. Her cheeks are red and shine as if about to burst.\n\n'Senk you,' she says to me. I have never seen her before. She is already, with her husband, in my room when I come home from school for the summer holiday. My sister tells me straight away that there are people in my room. It is her room too. She has a bed in our mother's room. She tells me that for ages now people have been arriving, sometimes on the night train from London, arriving and going to bed at once, sleeping and sleeping and then talking in whispers and crying.\n\n'The crying is the worst,' she says. She says she heard one woman crying all night. Even our father was not able to stop her crying.\n\n'He walks to the station in the night,' my sister says. She says that he has given away his winter coat and that it's all right for now but what will he do in winter.\n\n'These people,' my mother says, 'they have nothing. Recha must have packed the things in her luggage with her rags and bits of cotton wool.' She explains the treachery in her soft up-and-down voice. 'Perhaps she thought they were gifts from us.' Her voice is like a stream running. 'The time!' she says to my father, 'it's time for you to leave for the station. Look!' she says, 'Look at the time!'\n\nTwo more people are arriving. They will have our bedroom. My sister and I move our bedclothes once more.\n\n'The oil-cloth on here is so thin the horsehair's prickling through,' I complain to my sister.\n\n'I'll have the sofa then,' she says. 'I'll sleep there.'\n\n'Oh no. It's all right.' We slide off the sofa laughing as we did when we were little. One of our old games.\n\nI am ashamed because I have been robbed. This is the strange thing about it. When one is robbed there is this feeling of being ashamed. I do not want to admit to anyone that my silk frock has been stolen. And, even more, I do not want my father and mother to admit that they have been deceived in any way. Especially by people they are trying to help, and in their own house.\n\nBut of course they do not even think they have been deceived or robbed.\n\n'Recha,' my mother is saying to my father in the kitchen, 'Recha has never done any housework or cooking in her life. How will she manage as housekeeper? She has such beautiful hands.'\n\nSimply, my mother and father are seeing impossible suffering, especially people being separated from each other or making hurried marriages in order to escape from something they are not able to endure, something they must, at all costs, get away from.\n\n'It is an irony, is it called irony?' My mother's soft voice reaches the horsehair sofa. She is talking still to my father. 'Is it called irony. If it was once a joke,' she says, 'for a bespectacled shrimp of an intellectual, a Jew, to be an officer in a Red Cossack regiment during the Polish campaign of 1920, is this the same joke, if it can be called a joke, which is being repeated now twenty years later?' I have heard her before talking about these men, the grey beards, she calls them, with their gold-rimmed glasses and flying side curls, desperate to disguise their accents and their hand movements, marching or trying to march, walking on thin bent legs, their narrow shoulders and their intellectual superiority, 'bowing,' she is saying it again 'to the healthy pink flesh of the English Tommies.'\n\nMy father's voice, like a boulder in a wild mountain stream, interrupts her. I can hear his deep voice soothing.\n\nIt is not like my mother to use a word like Tommy. If English people use it they say Our Tommies, Our Boys, not English Tommies. I pull my sheets back over the inadequate horsehair. I think, in the morning, I will correct my mother so that she need not make a mistake of this sort in front of the neighbours. \n\n# BATHROOM DANCE\n\nWhen I try on one of the nurse's caps my friend Helen nearly dies.\n\n'Oh!' she cries, 'take it off! I'll die! Oh if you could see yourself. Oh!' she screams and Miss Besser looks at me with six years of reproach stored in the look.\n\nWe are all sewing Helen's uniform in the Domestic Science room. Three pin-striped dresses with long sleeves, buttoned from the wrist to the elbow, double tucks and innumerable button holes; fourteen white aprons and fourteen little caps which have to be rubbed along the seam with a wet toothbrush before the tapes can be drawn up to make those neat little pleats at the back. Helen looks so sweet in hers. I can't help wishing, when I see myself in the cap, that I am not going to do nursing after all.\n\nHelen ordered her material before persuading me to go to the hospital with her. So, when I order mine it is too late to have my uniform made by the class. It is the end of term, the end of our last year at school. My material is sent home.\n\nMister Jackson tells us, in the last Sunday evening meeting, that he wants the deepest responsibility for standards and judgements in his pupils, especially those who are about to leave the happy family which is how he likes to think of his school. We must not, he says, believe in doing just what we please. We must always believe in the nourishment of the inner life and in the loving discipline of personal relationships. We must always be concerned with the relentless search for truth at whatever cost to tradition and externals. I leave school carrying his inspiration and his cosiness with me. For some reason I keep thinking about and remembering something about the reed bending and surviving and the sturdy oak blown down.\n\nMy mother says the stuff is pillow ticking. She feels there is nothing refined about nursing. The arrival of the striped material has upset her. She says she has other things in mind for me, travelling on the continent, Europe, she says, studying art and ancient buildings and music.\n\n'But there's a war on,' I say.\n\n'Oh well, after the war.'\n\nShe can see my mind is made up and she is sad and cross for some days. The parcel, with one corner torn open, lies in the hall. She is comforted by the arrival of a letter from the Matron saying that all probationer nurses are required to bring warm sensible knickers. She feels the Matron must be a very nice person after all and she has my uniform made for me in a shop and pays extra to have it done quickly.\n\nHelen's mother invites me to spend a few days with Helen before we go to St Cuthberts.\n\nThe tiny rooms in Helen's home are full of sunshine. There are bright-yellow curtains gently fluttering at the open windows. The garden is full of summer flowers, roses and lupins and delphiniums, light blue and dark blue. The front of the house is covered with a trellis of flowers, some kind of wisteria which is sweetly fragrant at dusk.\n\nHelen's mother is small and quiet and kind. She is anxious and always concerned. She puts laxatives in the puddings she makes.\n\nI like Helen's house and garden, it is peaceful there and I would like to be there all the time but Helen wants to do other things. She is terribly in love with someone called David. Everything is David these few days. We spend a great deal of time outside a milkbar on the corner near David's house or walking endlessly in the streets where he is likely to go. No one, except me, knows of this great love. Because I am a visitor in the house I try to be agreeable. And I try to make an effort to understand intense looks from Helen, mysterious frowns, raised eyebrows, head shakings or noddings and flustered alterations about arrangements as well as I can.\n\n'I can't think what is the matter with Helen,' Mrs Ferguson says softly one evening when Helen rushes from the room to answer the telephone in case it should be David. We are putting up the black-out screens which Mrs Ferguson has made skilfully to go behind the cheerful yellow curtains every night. 'I suppose she is excited about her career,' she says in her quiet voice, picking up a little table which was in Helen's way.\n\nEveryone is so keen on careers for us. Mister Jackson, at school, was always reading aloud from letters sent by old boys and girls who are having careers, poultry farming, running boys' clubs and digging with the unemployed. He liked the envelopes to match the paper, he said, and sometimes he held up both for us all to see.\n\nHelen is desperate to see David before we leave. We go to all the services at his mother's church and to her Bible class where she makes us hand round plates of rock cakes to the Old Folk between the lantern slides. But there is no David. Helen writes him a postcard with a silly passionate message. During the night she cries and cries and says it is awful being so madly in love and will I pretend I have sent the postcard. Of course I say I won't. Helen begs me, she keeps on begging, saying that she lives in the neighbourhood and everyone knows her and will talk about her. She starts to howl and I am afraid Mrs Ferguson will hear and, in the end, I tell her, 'All right, if you really want me to.'\n\nIn the morning I write another card saying that I am sorry about the stupid card which I have sent and I show it to Helen, saying, 'We'll need to wash our hair before we go.'\n\n'I'll go up first,' she says. While she is in the bathroom using up all the hot water, I add a few words to my postcard, a silly passionate message, and I put Helen's name on it because of being tired and confused with the bad night we had. I go out and post it before she comes down with her hair all done up in a towel, the way she always does.\n\nMrs Ferguson comes up to London with us when we set off for St Cuthberts. Helen has to dash back to the house twice, once for her camera and the second time for her raincoat. I wait with Mrs Ferguson on the corner and she points out to me the window in the County Hospital where her husband died the year before. Her blue eyes are the saddest eyes I have ever seen. I say I am sorry about Mr Ferguson's death, but because of the uneasiness of the journey and the place where we are going, I know that I am not really concerned about her sorrow. Ashamed, I turn away from her.\n\nHelen comes rushing up the hill, she has slammed the front door, she says, forgetting that she put the key on the kitchen table and will her mother manage to climb through the pantry window in the dark and whatever are we waiting for when we have only a few minutes to get to the train.\n\nDavid, unseen, goes about his unseen life in the narrow suburb of little streets and houses. Helen seems to forget him easily, straight away.\n\nJust as we are sitting down to lunch there is an air-raid warning. It is terrible to have to leave the plates of food which have been placed in front of us. Mrs Ferguson has some paper bags in her handbag.\n\n'Mother! You can't!' Helen's face is red and angry. Mrs Ferguson, ignoring her, slides the salads and the bread and butter into the bags. We have to stand for two hours in the airraid shelter. It is very noisy the A.R.P. wardens say and they will not let us leave. It is too crowded for us to eat in there and, in any case, you can't eat when you are frightened.\n\nLater, in the next train, we have to stand all the way because the whole train is filled with the army. Big bodies, big rosy faces, thick rough greatcoats, kitbags, boots and cigarette smoke wherever we look. We stand swaying in the corridor pressed and squeezed by people passing, still looking for somewhere to sit. We can't eat there either. We throw the sad bags, beetroot soaked, out onto the railway lines.\n\nI feel sick as soon as we go into the main hall at St Cuthberts. It is the hospital smell and the smell of the bread and butter we try to eat in the nurse's dining room. Helen tries to pour two cups of tea but the tea is all gone. The teapot has a bitter smell of emptiness.\n\nUpstairs in Helen's room on the Peace corridor as it is called because it is over the chapel, we put on our uniforms and she screams with laughter at the sight of me in my cap.\n\n'Oh, you look just like you did at school,' she can't stop laughing. How can she laugh like this when we are so late. For wartime security the railway station names have been removed and, though we were counting the stops, we made a mistake and went past our station and had to wait for a bus which would bring us back.\n\n'Lend me a safety pin,' I say, 'one of my buttons has broken in half.' Helen, with a mouthful of hair grips, busy with her own cap, shakes her head. I go back along the corridor to my own room. It is melancholy in there, dark, because a piece of black-out material has been pinned over the window and is only partly looped up. The afternoon sun of autumn is sad too when I peer out of the bit of window and see the long slanting shadows lying across unfamiliar fields and roads leading to unknown places.\n\nMy school trunk, in my room before me, is a kind of betrayal. When I open it books and shoes and clothes spill out. Some of my pressed wildflowers have come unstuck and I put them back between the pages remembering the sweet, wet grass near the school where we searched for flowers. I seem to see clearly shining long fingers pulling stalks and holding bunches. Saxafrage, campion, vetch, ragged robin, star of Bethlehem, wild strawberry and sorrel. Quickly I tidy the flowers \u2013 violet, buttercup, King cup, cowslip, coltsfoot, wood anemone, shepherd's purse, lady's slipper, jack in the pulpit and bryony...\n\n'No Christian names on duty please,' staff nurse Sharpe says, so, after six years in the same dormitory, Helen and I make a great effort. Ferguson \u2013 Wright, Wright \u2013 Ferguson.\n\n'Have you finished with the floor mop \u2013 Ferguson?'\n\n'Oh you have it first \u2013 Wright.'\n\n'Oh! No! by all means, after you Ferguson.'\n\n'No, after you Wright.'\n\nStaff nurse Sharpe turns her eyes up to the ceiling so that only the whites show. She puts her watch on the window sill saying, 'Quarter of an hour to get those baths, basins and toilets really clean and the floors done too. So hurry!'\n\n'No Christian names on duty,' we remind each other.\n\nWe never sleep in our rooms on the Peace corridor. Every night we have to carry our blankets down to the basement where we sleep on straw mattresses. It is supposed to be safe there in air raids. There is no air and the water pipes make noises all night. As soon as I am able to fall asleep Night Sister Bean is banging with the end of her torch saying, 'Five-thirty a.m. nurses, five-thirty a.m.' And it is time to take up our blankets and carry them back upstairs to our rooms.\n\nI am working with Helen in the children's ward. Because half the hospital is full of soldiers the ward is very crowded. There are sixty children; there is always someone laughing and someone crying. I am too slow. My sleeves are always rolled up when they should be rolled down and buttoned into the cuffs. When my sleeves are down and buttoned it seems they have to be rolled up again at once. I can never remember the names of the children and what they have wrong with them.\n\nThe weeks go by and I play my secret game of comparisons as I played it at school. On the Peace corridor are some very pretty nurses. They are always washing each other's hair and hanging their delicate underclothes to dry in the bathroom. In the scented steamy atmosphere I can't help comparing their clothes with mine and their faces and bodies with mine. Every time I am always worse than they are and they all look so much more attractive in their uniforms, especially the cap suits them well. Even their finger nails are better than mine.\n\n'Nurse Wright!' Night Sister Bean calls my name at breakfast.\n\n'Yes Sister.' I stand up as I have seen the others do.\n\n'Matron's office nine a.m.' she says and goes on calling the register.\n\nI am worried about my appointment with the Matron. Something must be wrong.\n\n'What did Matron want?' Ferguson is waiting for me when I go to the ward to fetch my gas mask and my helmet. I am anxious not to lose these as I am responsible for them and will have to give them back if I leave the hospital or if the war should come to an end.\n\n'What did Matron want?' Ferguson repeats her question, giving me time to think.\n\n'Oh it is nothing much,' I reply.\n\n'Oh come on! What did she want you for? Are you in trouble?' she asks hopefully.\n\n'Oh no, it's nothing much at all.' I wave my gas mask. 'If you must know she wanted to tell me that she is very pleased with my work and she'll be very surprised if I don't win the gold medal.' Ferguson stares at me, her mouth wide open, while I collect my clean aprons. She does not notice that one of them is hers. It will give me an extra one for the week. I go to the office to tell the ward sister that I have been transferred to the theatre.\n\n_Had I the heavens' embroidered cloths,_\n\n_Enwrought with golden and silver light,_\n\nO'Connor, the theatre staff nurse, is singing. She has an Irish accent and a mellow voice. I would like to tell her I know this poem too.\n\n_The blue and the dim and the dark cloths_\n\n_Of night and light and the half light,_\n\nIn the theatre they are all intimate. They have well-bred voices and ways of speaking. They look healthy and well poised and behave with the ease of movement and gesture which comes from years of good breeding. They are a little circle in which I am not included. I do not try to be. I wish every day, though, that I could be a part of their reference and their joke.\n\nIn a fog of the incomprehensible and the obscure I strive, more stupid than I have ever been in my life, to anticipate the needs of the theatre sister whose small, hard eyes glitter at me above her white cotton mask. I rush off for the jaconet.\n\n'Why didn't you look at the table!' I piece together her angry masked hiss as I stand offering a carefully opened and held sterilized drum. One frightened glance at the operating table tells me it is catgut she asked for.\n\n'Boil up the trolley,' the careless instruction in the soft Irish voice floats towards me at the end of the long morning. Everything is on the instrument trolley.\n\n'Why ever didn't you put the doctors' soap back on the sink first!' The theatre is awash with boiled-over soap suds. Staff nurse O'Connor, lazily amused, is just scornful enough. 'And,' she says, 'what in God's Holy Name is this!' She fishes from the sterilizer a doll-sized jumper. She holds it up in the long-handled forceps. 'I see trouble ahead,' she warns, 'better not let sister see this.' It is the chief surgeon's real Jaeger woollen vest. He wears it to operate. He has only two and is very particular about them. I have discovered already that sister is afraid of the chief surgeon, consequently I need to be afraid of her. The smell of boiled soap and wool is terrible and it takes me the whole afternoon to clear up.\n\nTheatre sister and staff nurse O'Connor, always in masks, exchange glances of immediate understanding. They, when not in masks, have loud voices and laugh. They talk a great deal about horses and dogs and about Mummy and Daddy. They are quite shameless in all this Mummy and Daddy talk.\n\nThe X-ray staff are even more well-bred. They never wear uniform and they sing and laugh and come into the theatre in whatever they happen to be wearing \u2013 backless dinner dresses, tennis shorts or their night gowns. All the time they have a sleepy desirable look of mingled charm and efficiency. War-time shortages of chocolate and other food stuffs and restrictions on movement, not going up to London at night for instance, do not seem to affect them. They are always called by pet names, Diamond and Snorter. Diamond is the pretty one, she has a mop of curls and little white teeth in a tiny rosebud mouth. Snorter is horsey. She wears trousers and little yellow waist coats. She always has a cigarette dangling from her bottom lip.\n\nI can't compare myself with these people at all. They never speak to me except to ask me to fetch something. Even Mr Potter, the anaesthetist who seems kind and has a fatherly voice, never looks in my direction. He says, holding out his syringe, 'Evipan' or 'Pentothal', and talks to the others. Something about his voice, every day, reminds me of a quality in my father's voice; it makes me wish to be back at home. There is something hopeless in being hopeful that one person can actually match and replace another. It is not possible.\n\nSometimes Mr Potter tells a joke to the others and I do not know whether I should join in the laugh or not.\n\nI like Snorter's clothes and wish that I had some like them. I possess a three-quarter-length oatmeal coat with padded shoulders and gilt buttons which my mother thinks is elegant and useful as it will go with everything. It is so ugly it does not matter what I wear it with. The blue skirt I have is too long, the material is heavy, it sags and makes me tired.\n\n'Not with brown shoes!' Ferguson shakes her head.\n\nIt is my day off and I am in her room. The emptiness of the lonely day stretches ahead of me. It is true that the blue skirt and the brown shoes, they are all I have, do look terrible together.\n\nFerguson and her new friend, Carson, are going out to meet some soldiers to go on something called a pub crawl. Ferguson, I know, has never had anything stronger than ginger beer to drink in her life. I am watching her get ready. She has frizzed her hair all across her baby round forehead. I can't help admiring her, the blaze of lipstick alters her completely.\n\nCarson comes in balancing on very high-heeled shoes. She has on a halo hat with a cheeky little veil and some bright-pink silk stockings.\n\n'What lovely pink stockings!' I say to please her.\n\n'Salmon, please,' Carson says haughtily. Her hair is curled too and she is plastered all over with ornaments, brooches, necklaces, rings and lipstick, a different colour from Ferguson's. Ferguson looks bare and chubby and school-girlish next to Carson.\n\nBoth of them are about to go when I suddenly feel I can't face the whole day alone.\n\n'It's my day off too,' I say, 'and I don't know where to go.'\n\nFerguson pauses in the doorway.\n\n'Well, why don't you come with us,' Carson says. Both of them look at me.\n\n'The trouble is, Wright,' Carson says kindly, 'the trouble is that you've got no sex appeal.'\n\nAfter they have gone I sit in Ferguson's room for a long time staring at myself in her mirror to see if it shows badly that I have no sex appeal.\n\nI dream my name is Chevalier and I search for my name on the typed lists on the green baize notice boards. The examination results are out. I search for my name in the middle of the names and only find it later at the top.\n\nMy name, not the Chevalier of the dream, but my own name is at the top of the lists when they appear.\n\nI work hard in all my free time at the lecture notes and at the essays 'Ward Routine', 'Nursing as a Career', 'Some Aspects of the History of Nursing' and 'The Nurse and her Patient'.\n\nThe one on ward routine pleases me most. As I write the essay, the staff and the patients and the wards of St Cuthberts seem to unfold about me and I begin to understand what I am trying to do in this hospital. I rewrite the essay collecting the complete working of a hospital ward into two sheets of paper. When it is read aloud to the other nurses, Ferguson stares at me and does not take her eyes off me all through the nursing lecture which follows.\n\nI learn every bone and muscle in the body and all the muscle attachments and all the systems of the body. I begin to understand the destruction of disease and the construction of cure. I find I can use phrases suddenly in speech or on paper which give a correct answer. Formulae for digestion or respiration or for the action of drugs. Words and phrases like gaseous interchange and internal combustion roll from my pen and the name at the top of the lists continues to be mine.\n\n'Don't tell me you'll be top in invalid cookery too!' Ferguson says and she reminds me of the white sauce I made at school which was said to have blocked up the drains for two days. She goes on to remind me how my pastry board, put up at the window to dry, was the one which fell on the headmaster's wife while she was weeding in the garden below, breaking her glasses and altering the shape of her nose forever.\n\nMy invalid carrot is the prettiest of them all. The examiner gives me the highest mark.\n\n'But it's not even cooked properly!' Ferguson is outraged when she tastes it afterwards. She says the sauce is disgusting.\n\n'Oh well you can't expect the examiner to actually eat all the things she is marking,' I say.\n\nFerguson has indigestion, she is very uncomfortable all evening because, in the greedy big taste, she has nearly the whole carrot.\n\nIt is the custom, apparently, at St Cuthberts to move the nurses from one corridor to another. I am given a large room in a corridor called Industry. It is over the kitchens and is noisy and smells of burning saucepans. This room has a big tall window. I move my bed under the window and, dressed in my school jersey, I lie on the bed for as long as possible to feel the fresh cold air on my face before going down to the basement for the night. Some evenings I fall into a deep and refreshing sleep obediently waking up, when called, to go down to the doubtful safety below.\n\nEvery day, after the operations, I go round the theatre with a pail of hot soapy water cleaning everything. There is an orderly peacefulness in the quiet white tranquillity which seems, every afternoon, to follow the strained, bloodstained mornings.\n\nIn my new room I copy out my lecture notes:\n\n... _infection follows the line of least resistance_...\n\nand read my school poetry book:\n\n_Through the thick corn the scarlet poppies peep,_\n\n_And round green roots and yellowing stalks I see_\n\n_Pale pink convolvulus in tendrils creep;_\n\n_And air-swept lindens yield_\n\n_Their scent_...\n\nI am not able to put out of my mind the eyes of a man who is asleep but unable to close his eyes. The putrid smell of wounded flesh comes with me to my room and I hear, all the time, the sounds of bone surgery and the troubled respiration which accompanies the lengthy periods of deep anaesthetic...\n\n_Oft thou hast given them store_\n\n_Of flowers \u2013 the frail leaf 'd, white anemony,_\n\n_Dark blue bells drench'd with dews of summer eves_\n\n_And purple orchises with spotted leaves_...\n\n...in the theatre recovery ward there are fifteen amputations, seven above the knee and eight below. The beds are made in two halves so that the padded stumps can be watched. Every bed has its own bell and tourniquet...\n\nSt Cuthberts is only a drop in the ocean; staff nurse O'Connor did not address the remark to me, I overhead it.\n\nNext to my room is a large room which has been converted into a bathroom. The dividing wall is a wooden partition. The water pipes make a lot of noise and people like to sing there, usually something from an opera.\n\nOne night I wake from my evening-stolen sleep hearing two voices talking in the bathroom. It is dark in my room; I can see some light from the bathroom through a knot-hole high up in the partition. The voices belong to Diamond and Snorter. This is strange because they live somewhere outside the hospital and would not need to use that bathroom. It is not a comfortable place at all, very cold, with a big old bath awkwardly in the middle of the rough floor.\n\nDiamond and Snorter are singing and making a lot of noise, laughing and shrieking above the rushing water.\n\nSinging:\n\n_Give me thy hand O Fairest_\n\n_la la la la la la la_\n\n_I would and yet I would not_\n\nLaughter and the huge bath obviously being filled to the brim.\n\n_Our lives would be all pleasure_\n\n_tra la la la la la la_\n\n_tra la la la la la la_\n\n_tum pe te tum_\n\n_tum pe te tum_\n\n'That was some party was it not!'\n\n'Rather!' Their rich voices richer over the water.\n\nI stand up on my bed and peer through the hole which is about the size of an egg. I have never looked through before, though have heard lots of baths and songs. I have never heard Diamond and Snorter in there before \u2013 if it is them.\n\nIt is Diamond and Snorter and they are naturally quite naked. There is nothing unusual about their bodies. Their clothes, party clothes, are all in little heaps on the floor. They, the women not the clothes, are holding hands, their arms held up gracefully. They are stepping up towards each other and away again. They have stopped singing and are nodding and smiling and turning to the left and to the right, and, then, with sedate little steps, skipping slowly round and round. It is a dance, a little dance for two people, a minuet, graceful, strange and remote. In the steam the naked bodies are like a pair of sea birds engaged in mating display. They appear and disappear as if seen through a white sea mist on some far-off shore.\n\nThe dance quickens. It is more serious. Each pulls the other more fiercely, letting go suddenly, laughing and then not laughing. Dancing still, now serious now amusing. To and fro, together, back and forth and together and round and round they skip and dance. Then, all at once, they drop hands and clasp each other close, as if in a private ballroom, and quick step a foxtrot all round the bathroom.\n\nIt is not an ugly dance, it is rhythmic and ridiculous. Their thighs and buttocks shake and tremble and Snorter's hair has come undone and is hanging about her large red ears in wispy strands.\n\nThe dance over, they climb into the deep hot bath and tenderly wash each other.\n\nThe little dance, the bathroom dance, gives me an entirely new outlook. I can't wait to see Diamond and Snorter again. I look at everyone at breakfast, not Ferguson, of course (I know everything there is to know about her life) with a fresh interest.\n\nLater I am standing beside the patient in the anaesthetic room, waiting for Mr Potter, when Snorter comes struggling through the swing doors with her old cricket bag. She flops about the room dragging the bag:\n\n_And on the beach undid his corded bales_\n\nshe says, as she always does, while rummaging in the bag for her white wellington boots. I want to tell Snorter, though I never do, that I too know this poem.\n\nI look hard at Snorter. Even now her hair is not combed properly. Her theatre gown has no tapes at the back so that it hangs, untied and crooked. She only has one boot on when Mr Potter comes. The unfairness of it all comes over me. Why do I have to be neatly and completely dressed at all times. Why do they not speak to me except to ask for something to be fetched or taken away. Suddenly I say to Snorter, ' _Minuet du Salle de la Bain_ ', in my appalling accent. I am surprised at myself. She is hopping on one foot, a wellington boot in her hand, she stops hopping for a moment.\n\n' _de la salle de bain_ surely,' she corrects me with a perfect pronunciation and a well-mannered smile. 'Also lower case,' she says, 'not caps, alters the emphasis.'\n\n'Oh yes of course,' I mutter hastily. An apology.\n\n'Pentothal.' Mr Potter is perched on his stool at the patient's head, his syringe held out vaguely in my direction.\n\n# NIGHT RUNNER\n\nNight Sister Percy is dying. It is my first night as Night Runner at St Cuthberts. Night Sister Bean, grumbling and cackling, calls the register and, at the end, she calls my name.\n\n'Nurse Wright.'\n\n'Yes Sister,' I reply, half rising in my chair as I have seen the others do. The Maids' Dining Room, where we eat, is too cramped to do anything else.\n\n'Night Runner,' she says and I sit down again. The thought of being Night Runner is alarming. Nurse Dixon has been Night Runner for a long time. All along I have been hoping that I would escape from these duties and responsibilities, the efficient rushing here and there to relieve on different wards; every night bringing something new and difficult.\n\nThe Night Runner has to prepare the night nurses' meal too; one little sitting at twelve midnight and a second one at twelve forty-five and, of course, the clearing up and the washing up.\n\nEvery night I admire Nurse Dixon in the tiny cramped kitchen where we sit close together, regardless of rank, in the hot smell of warmed up fish or mince and the noise of the jugs of strong black coffee, keeping hot, in two black pans of boiling water. We eat our meal there in this intimacy with these two hot saucepans splashing and hissing just behind us. The coffee, only a little at the bottom of each jug, looks thick and dark and I wonder how it is made. Tonight I will have to find out and have it ready when the first little group of nurses appears.\n\nWhen I report to Night Sister Bean in her office, she tells me to go for the oxygen.\n\n'Go up to Isolation for the oxygen,' she says without looking up from something she is writing. I am standing in front of her desk. I have never been so close to her before, not in this position, that is, of looking at her from above. She is starch scented, shrouded mysteriously in the daintily severe folds of spotted white gauze. She is a sorceress disguised in the heavenly blue of the Madonna; a shrivelled, rustling, aromatic, knowledgeable, Madonna-coloured magician; she is a wardress and a keeper. She is an angel in charge of life and in charge of death. Her fine white cap, balancing, nodding, a grotesque blossom flowering for ever in the dark halls of the night, hovers beneath me. She is said to have powers, an enchantment, beyond the powers of an ordinary human. For one thing, she has been on night duty in this hospital for over thirty years. As I stand there I realize that I do not know her at all and that I am afraid of her.\n\n'Well,' she says, 'don't just stand there. Go up to Isolation for the oxygen and bring it at once to Industry.'\n\n'Yes, Sister,' I say and I go as quickly as I can. The parts of the hospital are all known by different names; Big Boys Big Girls, Top Ward, Bottom Ward, Side Ward and Middle, Industry, Peace, Chapel and Nursery. I have a room on the Peace corridor, so named because it is above the chapel and next to Matron's wing.\n\nIndustry is the part over the kitchens. There are rooms for nurses there too. Quite often there is a pleasant noise and smell of cooking in these rooms. The Nurses' sick bay is there and it is there that I have to take the oxygen.\n\nI am frightened out here.\n\nFor one thing, Isolation is never used. It is, as the name suggests, isolated. It is approached by a long, narrow covered way sloping up through a war-troubled shrubbery where all the dust bins are kept. Because of not being able to show any lights it is absolutely dark there. When I go out into the darkness I can smell rotting arms and legs, thrown out of the operating theatre and not put properly into the bins. I gather my apron close so that I will not get caught by a protruding maimed hand.\n\nWhen I fash my torch quickly over the bins I see they are clean and innocent and have their lids firmly pressed on. In the torchlight there is no smell.\n\nThe sky at the end of the covered passage is decorated with the pale moving fans of search lights. The beams of light are interwoven with the sounds of throbbing engines. The air-raid warning might sound at any moment. In the emergency of being made Night Runner so suddenly, I have forgotten to bring my tin hat and gas mask from the Maids' Dining Room.\n\nI am worried about the gas mask and the tin hat. I have signed for them on arrival at the hospital and am completely responsible for them. I will have to hand them back if I leave the hospital or if this war comes to an end. Usually I never leave either of them out of my care. I have them tied together with thick string. I put them under my chair at meal times and hang them up in the nurses' cupboard on the ward where I am working.\n\nIt is hard to find the oxygen. My torch light picks up stacks of pillows, shelves of grey blankets, rolls of waterproof sheets, and some biscuit tins labelled Emergency Dressings, all with dates on them. There are two tea chests filled with tins and bottles. The chests are marked Emergency, Iron Rations, Doctors Only in red paint. There do not seem to be similar boxes marked for nurses or patients.\n\nAt last I find the oxygen cylinder and I rush with the little trolley up to Industry.\n\nSister Percy is dying. She is the other Night Sister and is very fat. She is propped, gasping, on pillows, a blue trout with eyes bulging, behind the floral screens made by Matron's mother for sick nurses.\n\nIt is the first time I have seen someone who is dying. Night Sister Bean is there and the R.M.O. and the Home Sister. They take the oxygen and Sister Bean tells me I need not stay. She pulls the screens closer round Sister Percy.\n\nIn the basement of the hospital I set about the secrets of making the coffee and having it come only so far up the jugs.\n\nLater Night Sister Bean comes and says why haven't I lit the gas, which, when you think about it, is a good thing to say as they will surely want that potato and mince stuff hot. Before she leaves she makes me get down on my knees to hunt behind the pipes for cockroaches. She has a steel knitting needle for this and we knock and scrape and rattle about, Night Sister Bean on her knees too, and we chase them out, the revolting things, and sprinkle some white powder which, she says, they love to eat without knowing it is absolutely fatal to them.\n\nIt is something special about night duty, this little meal time in the middle of the night, with everyone sitting together, even Night Sister Bean, herself, coming to one or the other of the sittings. She seems almost human, in spite of the mysterious things whispered about her, at these meals. Sometimes she even complains about the sameness of them, saying that one thing the war cannot do is to make these meals worse than they are and that it is sheer drudgery to eat them night after night. When I think about this I realize she has been eating stewed mince and pounded fish for so many years and I can't help wishing I could do something about it.\n\nThis first night it takes me a long time to clear up in the little pantry. When at last I am finished Night Sister Bean sends me to relieve on Bottom Ward. There is a spinal operation in the theatre recovery room just now, she says, and a spare nurse will be needed when the patient comes back to the ward.\n\nOn my way to Bottom Ward I wish I could be working with staff nurse Ramsden.\n\n'I will play something for you,' she said to me once when I was alone and filled with tears in the bleak, unused room which is the nurses' sitting room.\n\nShe ran her fingers up and down the piano keys. 'This is Mussorgsky,' she said. 'It's called Gopak, a kind of little dance,' she explained. She played and turned her head towards me nodding and smiling. 'Do you like this?' she asked, her eyes smiling. It is not everyone who has had Mussorgsky played for them; the thought gives me courage as I hurry along the unlit passage to the ward.\n\nThere is a circle of light from the uncurtained windows of the office in the middle of the ward. I can see a devout head bent over the desk in the office. I feel I am looking at an Angel of mercy who is sitting quietly there ready to minister to the helpless patients.\n\nStaff nurse Sharpe is seated in the office with an army blanket tucked discreetly over her petticoat. Her uniform dress lies across her lap. She explains that she is just taking up the hem and will I go to the kitchen and cut the bread and butter. As I pass the linen cupboard I see the other night nurse curled up in a heap of blankets. She is asleep. This is my friend Ferguson.\n\nI sink slowly into the bread cutting. It is a quiet and leisurely task. While I cut and spread I eat a lot of the soft new bread and I wonder how Sharpe will manage to wear her uniform shortened. Matron is so particular that we wear them long, ten inches off the ground, so that the soldiers do not get in a heightened excitement about us.\n\nSharpe comes in quite soon. She seems annoyed that I have not finished. She puts her watch on the table and says the whole lot, breakfast trays all polished and set, and bread and butter for sixty men, must be finished in a quarter of an hour. I really hurry up after this and am just ready when the operation case comes back and I have to go and sit by him in the small ward. I hope to see Ferguson but staff nurse Sharpe has sent her round changing the water jugs.\n\nEasily I slip into my dream of Ferguson. She owes me six and sevenpence. I have written it on the back of my writing pad. I'll go out with her and borrow two and six.\n\n'Oh Lord!' I'll say, 'it's my mother's birthday and I haven't a thing for her and here I am without my purse. Say, can you lend me two and six?' And then I'll let her buy a coffee and a bun for me that will bring it to three shillings and I won't ever pay it back and, in that way, will recover some of the six and sevenpence.\n\n'Cross my heart, cut me in two if my word is not true,' I say to myself and I resolve to sit in Ferguson's room as soon as I am off duty. I'll sit there till she pays me the money. I'll just sit and sit there till it dawns on her why I am there.\n\nThe patient, quite still as if dead, suddenly moves and helps himself to a drink of water. He vomits and flings the bowl across the room. He seems to be coming round from his anaesthetic. I grope under the bedclothes. I should count his pulse but I am unable to find his wrist.\n\n'Oh I can't,' he groans, 'not now I can't.'\n\nHe seems to be in plaster of paris from head to foot. He groans again and sleeps. Nervously I wait to try again to find some place on his body where I can feel his pulse.\n\nHigh on the wall in the Maids' Dining Room is an ancient wireless. It splutters and gargles all day with the tinny music of workers' play time and Vera Lynn plaintively announcing there'll always be an England. Sometimes in the early mornings, while we have our dinner, the music is of a different kind. Sometimes it is majestic, lofty and sustaining.\n\n'Wright!' Staff nurse Ramsden calls across the crowded tables. 'Mock Morris? Would you say?' she waves a long-fingered hand.\n\n'No,' I shake my head, 'not Mock Morris, it's Beethoven.' She laughs. She knows it is not Beethoven. It is a little joke we have come to share. It is the only joke I have with anyone. Perhaps it is the same for Ramsden. She has a slight moustache and I have noticed, in her room, an odour, a heaviness which belongs with older women perhaps from the perfumed soap she has and the material of well-made underwear. Her shoes and stockings, her suits and blouses and hats have the fragrance of being of a better quality. Ramsden asked me once about the violin I was carrying. She has said to me to choose one of her books, she has several in her room, as a present from her to me. Secretly I think, every day, that I admire Ramsden. I love her. Perhaps. I think, I will tell her, one day, the truth about the violin case.\n\nA special quality about working during the night is the stepping out of doors in the mornings, the first feeling of the fresh air and the sun which is hardly warm in its brightness.\n\nWe ride our bicycles. Not Ramsden. There is a towing patch along the river. I, not knowing it before, like the smell of the river, the muddy banks and the cattle-trodden grass. Water birds, disturbed, rise noisily. Our own voices echo.\n\nThough we have had our meal we want breakfast. Ferguson hasn't any money. Neither has Queen. Ferguson says she will owe Queen if Queen will owe me for them both. We agree and I pay. And all the way back I am trying to work out what has to be added to the outstanding six and sevenpence.\n\nFerguson's room, when I go to sit there, looks as if it should be roped off as a bomb crater. Her clothes, and some of mine, are scattered everywhere. There is a note from the Home Sister on her dusty dressing table. I read the note, it is to tell Ferguson to clean her hair brush.\n\nBored and sleepy I study the note repeatedly, and add 'Neither a Borrower nor a Lender be' in handwriting so like the Home Sister's it takes my breath away.\n\nI search for Ferguson's writing paper. It is of superior quality and very suitable. I write some little notes in this newly learned handwriting and put them carefully in my pocket. I continue to wait for Ferguson, hardly able to keep my eyes open.\n\nI might have missed my sleep altogether if I had not remembered in time that Ferguson has gone home for her nights off.\n\nI do not flash the torch for fear of being seen. I grope in the dark fishing for something, anything, in the cavernous tea chest, and hasten back down the covered way.\n\nNight Sister Bean says to me to go to Bottom Ward to relieve and I say, 'Yes Sister', and leave her office backwards, shuffling my feet and bending as if bowing slightly, my hands, behind my back, clasping and almost dropping an enormous glass jar.\n\nIt is bottled Chinese gooseberries, of all things, and I put one on each of the baked apples splashing the spicy syrup generously. Night Sister Bean smiles, crackling starch, and says the baked apples have a piquant flavour. She has not had such a delicious baked apple for thirty years. 'Piquant!' she says.\n\nStaff nurse Sharpe sits in the office all night with nursing auxiliary Queen. Queen has put operation stockings over her shoes to keep warm. Both Sharpe and Queen are wrapped up in army blankets. Sharpe has to let down the hem of her dress. Sister Bean asked her to stay behind at breakfast.\n\nWhenever I come back to the office Sharpe says, 'Take these pills to bed twelve' or 'Get the lavatories cleaned,' and 'Time to do the bread and butter \u2013 and don't leave the trays smeary like last night.'\n\nAt the end of the ward I pull out the laundry baskets and I move the empty oxygen cylinders and the fire equipment; the buckets of water and sand. I simply move them all out from their normal places, just a little way out, and later, when Sharpe and Queen go along to the lavatory, they fall over these things and knock into each other, making the biggest disturbance ever heard in a hospital at night. Night Sister Bean comes rushing all the way up from her office in the main hall. She is furious and tells Sharpe and Queen to report to Matron at nine a.m. She can see that I am busy, quietly with my little torch, up the other end of the ward, pouring the fragrant mouth wash in readiness for the morning.\n\nThe tomato sauce has endless possibilities. The dressed crab is in such a small quantity that the only thing I can do is to put a tiny spoonful on top of the helpings of mashed potato. Night Sister Bean is appreciative and says the flavour seeps right through. Tinned bilberries, celery soup and custard powder come readily to my experienced hands.\n\nI do not see staff nurse Ramsden very often. She has not asked me in to her room again to choose the book. Perhaps she has changed her mind. She is, after all, senior to me.\n\nThere are times when an unutterable loneliness is the only company in the cold early morning. The bicycle rides across the heath or along the river are over too quickly and, because of this, are meaningless. With a sense of inexplicable bereavement my free time seems to stretch ahead in emptiness. I go to bed too soon and sleep badly.\n\nI am glad when Ferguson comes back; very pleased. In the pantry I am opening a big tin, the biggest thing I have managed to lift out so far. I say 'Hallo' to Ferguson as she sits down with the other nurses; they talk and laugh together. I go on with my work.\n\n'Oh, you've got IT,' I say to Ferguson. 'Plenty of S.A. Know what that is? Sex appeal, it's written all over you.' And seeing, out of the corner of my eye, Night Sister Bean coming in, I go on talking as if I haven't seen her.\n\n'How you do it beats me Fergie,' I say. 'How is it you have all the men talking about you the way they do? You certainly must have given them plenty to think about. They all adore you. Corporal Smith's absolutely mad about you, really!' Unconcernedly I scrape scrape at the tin. 'He never slept last night. Sharpe had to slip him a Mickey Finn, just a quick one. He's waiting for another letter from you and I think he's sending the poem you asked for. Who on earth is your go between?' so I go on and scrape scrape at the tin.\n\nI know why there is silence behind me. I turn round.\n\n'Oh, here you are at last Sister,' I say to Night Sister Bean. Ferguson is a dull red colour, pity, as she was looking so well after her nights off.\n\n'Here we are Sister,' I say, 'on the menu we have Pheasant Wing in Aspic. Will you have the fish pie with it?' I serve all the plates in turn. The coffee hisses and spits behind us.\n\n'Matron's office, nine o'clock,' Nigh Sister Bean says to Ferguson.\n\n'Yes, Sister.'\n\nFerguson is sent to Big Girls for the rest of the night and I am to relieve, as usual, on Bottom Ward. I wake Corporal Smith at four a.m. and urge him to write to Nurse Ferguson. 'Every day she waits for a letter,' I tell him, 'she'll get ill from not eating if you don't write.' Staff nurse Sharpe finds me by his bed and sends me to scrub the bathroom walls.\n\n'And do out all the cupboards too, and quickly,' she says.\n\nIn the morning when I see Sharpe safely in the queue for letters I rush up to the Peace corridor and find her room. I cram her curtains into her messy wet soap dish and leave one of my neatly folded notes on her dressing table.\n\n_Do not let your curtains dangle in the soap dish. Sister._\n\nThere is not much I can do with cherry jam. I serve it with the stewed mince as a sweet and sour sauce. It is a favourite with the Royal family, I tell them, but I can see I shall have to risk another raid on my secret store.\n\nThe next night I have a good dig into both chests and load myself up with tinned tomato soup, a tinned chicken, some sardines and two tins of pears.\n\nNurse Dixon is mystified. Her eyes are full of questions.\n\n'Where d'you get all...' her lips form whispered words.\n\n'No time to chat now, sorry,' I say. I am hastily setting a little tray for Night Sister Bean. I have started taking an extra cup of coffee along to her office. It seems the best way to use up a tin of shortbread fingers. Balancing my tray I race up the dark stairs and along the passage to Night Sister Bean's office.\n\n'Bottom Ward,' Night Sister Bean says without looking up. Again I am at the mercy of Sharpe.\n\n'Wash down the kitchen walls,' she says, 'and do all the shelves and cupboards and quickly \u2013 before you start the blanket baths.' She gives me a list of the more disagreeable men to do; she says to change their bottom sheets too. All the hardest work while nursing auxiliary Queen, who is back there, and herself sit wrapped up in the office, smoking, with a pot of hot coffee between them on the desk.\n\nI go into the small ward and give the emergency bell there three rings bringing Night Sister Bean to the ward before Sharpe and Queen realise what is happening.\n\n'Is it an air raid?' Queen asks anxiously.\n\n'Nurses should know why they ring, Nurse,' Sister Bean says and she makes them take her round to every bed whispering the diagnosis and treatment of every patient. Night Sister Bean rustling and croaking, fidgeting and cursing, disturbs all the men trying to find out who rang three times.\n\n'Someone must be haemorrhaging,' she says, 'find out who it is.'\n\nPeering maliciously into the kitchen, Sister Bean sees me quietly up the step ladder with my little pail of soapy water. The wet walls gleam primrose yellow as if they have been freshly painted. She tells Sharpe and Queen to report to Matron's office nine a.m. for smoking on duty.\n\nOnce again Sharpe is in the letter queue. I take the loaded ash tray from the Porters' Lodge and spill it all over her room.\n\n_Your room is disgusting. Take some hot water and disinfectant and wash down Sister_\n\nThe folded note lies neatly on her dressing table.\n\nI try listening to Beethoven but it reminds me of my loneliness. I wish Corporal Smith would write to me. I wish someone would write to me. Ferguson is going to The Old Green Room for coffee. She is popular, always going out. In my room I have a list.\n\n  1. Listen to Beethoven.\n  2. Keep window wide open. If cold sleep in school jersey.\n  3. Ride bicycle for complexion. (care of)\n  4. Write and Think.\n\n'I can't come out,' I say to Ferguson. 'I'm listening to Beethoven,' I say, ignoring the fact that she has not asked me.\n\n'It's only one record,' she says, 'you've only got one record.'\n\n'It's Beethoven all the same.' I beat time delicately and wear my far away look.\n\nFerguson goes off out and I add number 5 with difficulty to the list. The paper is stuck in at the side of the dressing-table mirror and uneven to write on.\n\n  * 5. Divide N.S.B.'s nature and discover exactly the extent of her powers.\n\nI take my white windsor, bath size, to the wash room and fill a basin with hot water to soften the soap. I set to work with my nail file and scissors. I'll take my torch tonight, I'm thinking, a tin of powdered milk would be useful. Whipped up, it makes very good cream; delicious with the baked apples.\n\nThe likeness is surprising. It is the distinction of the shape and the tilt of the cap, the little figure is emerging perfectly. I work patiently for a long time. I am going to split the image in half very carefully and torture one half keeping the other half as a control, as in a scientific experiment, and observe the effect on the living person.\n\nThe idea is so tremendous I feel faint. Already I foresee results, the upright, crisp little blue and white Bean totters in the passage, she wilts and calls for help.\n\n'Nurse Wright! Help me up, dear. What a good child you are, so gentle too. Just help me to that chair, thank you, dear child. Thank you!'\n\nThe Peace corridor is very quiet. Another good thing about the night duty is that we all may sleep in our beds during the day. Every morning I long for this sleep. Up until this time, I, like the others, have had to carry bedclothes down to the basement every night because of the air raids. There are no beds in the basement, only some sack mattresses of straw. There is no air there either.\n\nI love the smell of the clean white windsor. I am sculpting carefully with the file. The likeness is indeed perfect. My hands are slippery and wrinkled and I am unable to stop them from shaking. I feel suddenly that I possess some hitherto unknown but vital power to be able to make this \u2013 this effigy.\n\nAnd then, all at once, Night Sister Bean is there in the doorway of the wash room, peering about to see who it is not in bed yet and it is after twelve noon already. Because I am thinking of the moment when I will split the image and considering which tool will be most suitable for this, the sudden appearance of Sister Bean is, to say the least, confusing.\n\nI plunge my head into the basin together with Her I am so carefully fashioning, saying, 'Oh, I can ever get the soap out of my hair!', delighted at the sound of weariness achieved.\n\nShe says to remember always to have the rinsing water hotter than the washing water. 'Hot as you can bear it,' she says.\n\n'Thank you Sister.'\n\nShe is rustling and cackling, crackling and disturbing, checking every corner of the wash room, quickly looking into all the lavatories, saying as she leaves, 'And it is better to take off your cap first.'\n\nSo there I am with the soaked limp thing, frothed and scummed all over with the white windsor, on my head, still secure with an iron foundry of hair grips and useless for tonight. My work of art too is ruined, the outlines blurred and destroyed before being finished. It is a solemn moment of understanding that from a remote spot, namely the door, she has been able to spoil what I have made and add a further destruction of her own, my cap.\n\nMy back aches with bending over the stupid little sink. These days I am missing too much sleep. In spite of being so tired I go down to the ramp where the milk churns are loaded and unloaded. It is the meeting place of the inside of the hospital with the outside world. The clean laundry boxes are there, neatly stacked. Fortunately Ferguson's box is near the edge. I open it and remove one of her fresh clean caps. My box is there too but I don't want to take one of mine as it will leave me short later in the week.\n\nThe powdered household milk is in the chest as I hoped, tins of it and real coffee too. I find more soup, mushroom, cream of asparagus, cream of chicken, vegetable and minestrone. I am quite reckless with my torch. Christmas is coming, I take a little hoard of interesting tins.\n\nI discover that Night Sister Bean has a weakness for hot broth and I try, every night, to slip a cup along to her office in the early part of the night before I start on anything else.\n\nSeveral things are on my mind, mostly small affairs. For some time I have Corporal Smith's love letter to Ferguson, sixteen pages, in my pocket. It is not sealed and her name does not appear anywhere in the letter. It is too long for one person so I divide the letter in half and address two envelopes in Corporal Smith's handwriting, one to Sharpe and one to Ferguson. Accidentally I drop them, unsealed, one by the desk in Night Sister Bean's office and the other in the little hall outside Matron's room. We are not supposed to be intimate with the male patients and I feel certain too that Corporal Smith is a married man, but there is something else on my mind; it is whether a nurse should send a Christmas card to the Matron. It is something entirely beyond my experience.\n\nIn the end I buy one, a big expensive card, a Dutch Interior. It costs one and ninepence. I sit a whole morning over it trying to think what I should write.\n\n_A very Happy Christmas to Matron from Nurse Wright_\n\nNurse Wright sounds presumptuous. I haven't taken an external exam yet. She may not regard me as nurse.\n\n_A very Happy Event_... that would be quite wrong.\n\n_A very Happy Christmas to You from Guess Who_. She might think that silly.\n\n_Happy Christmas. Vera_. Too familiar. _Veronica_ I have never liked my name.\n\n_A Happy Christmas to Matron from one of her staff_ and in very small writing underneath _N\/V Wright_.\n\nI keep wondering if all the others will send Matron a Christmas card. It is hardly a thing you can ask anyone. Besides I do not want, particularly to give Ferguson the idea. She will never think of it herself. And who can I ask if I don't ask her?\n\nI put the card in Matron's correspondence pigeon hole. The card is so big it has to be bent over at the top to fit in. I am nervous in case someone passing will see me.\n\nAgain I am relieving on Bottom Ward. Always it is this bottom Ward. This time I have to creep round cleaning all the bed wheels.\n\n'And quietly,' Sharpe says, 'Nurse Queen and I don't want everyone waking up!'\n\nThe card worries me. It will take it out in the morning. The message is all wrong.\n\nOne of her staff! I can't bear to think about it.\n\nThe card is still there, bending, apologizing and self conscious in the morning. I want to remove it but there are people about and correspondence must not be tampered with.\n\nTwice during the day I get dressed and creep down from the Peace corridor, pale, hollow-eyed and drab; all night nurses are completely out of place in the afternoons. I feel conspicuous, sick nearly, standing about in the hall waiting to be alone there so that I can remove that vulgar card and its silly message. It is still bending there in the narrow compartment.\n\nEven when the hall is free of people there are two nurses chattering together by the main door. Whyever do they stand in this cold place to talk. I have to give up and go back to bed, much too cold to sleep. Ferguson has my hot-water bottle for her toothache. It seems I can never get even with her. Never ever.\n\nThe card is still there in the evening when we go down to the Maids' Dining Room for breakfast. I can hardly eat as I am thinking of a plot to retrieve the card.\n\nThe register is finished.\n\n'Nurse Wright.'\n\n'Yes Sister?' half rising in my chair as we all do in that cramped place.\n\n'Matron's office nine a.m. tomorrow.'\n\n'Yes Sister,' I sit down again. It can't be to thank me for the card as it hasn't been received yet. A number of reasons come to mind, for one thing there are the two deep caves of dark emptiness; perhaps they have been discovered...\n\nIn spite of a sense of foreboding I go, with my little torch hidden beneath my apron, up the long covered way. I need more powdered milk. The path seems endless. The night sky has the same ominous decoration; throbbing engines alternate with sharp anti-aircraft guns and the air-raid sirens wail up and down, up and down. The soft searchlights move slowly. They make no noise and are helpless. I feel exposed and push my hands round the emptiness of the nearest tea chest. Grabbing a tin of powdered milk I rush back down past the festering bins and on down towards an eternity of the unknown.\n\nI have a corner seat in this train by a mistake which is not entirely my fault. The woman, who is in this seat, asks me if I think she has time to fetch herself a cup of tea. I can see that she badly wants to do this and, in order that she does not have to go without the tea, I agree that, though she will be cutting it fine, there is a chance that she will have time. So she goes and I see her just emerging from the refreshment room with a look on her face which shows how she feels. She has her tea clutched in one hand and I have her reserved seat because it is silly, now that the train has started, to stand in the corridor being crushed by army greatcoats and kitbags and boots, simply looking at the emptiness of this comfortable corner.\n\nI have some household milk for Mother, it is always useful in these days of rationing. I have the tinned chicken also. At the last minute I could not think what to do with it as Night Sister Bean will not be naming the next Night Runner till this evening, and, of course, I shall not be there to know who it is and so am not able to hand on either the milk or the chicken.\n\nThere is too the chance that the new Night Runner might be my friend Ferguson. It would not do to give her these advantages.\n\nThis is my first holiday from St Cuthberts, my nights off and ten days holiday. Thirteen days off.\n\n'Shall I take my tin hat, I mean my helmet, and my gas mask?' I ask Matron.\n\n'By all means if you would like to,' she says and wishes me a pleasant holiday and a happy Christmas.\n\nThe tin hat and the gas mask are tied to my suitcase. My little sister will be interested to see them.\n\nMy father will be pleased with his Christmas card. He has always liked the detail and the warm colours of a Dutch interior. He will not mind the crossing out inside. The card will flatten if I press it tonight in the dictionary.\n\nFor some reason I am thinking about staff nurse Ramsden. Last night, in the doorway of the Maids' Dining Room, I stood aside to let her go in first.\n\n'Thank you,' she said and then she asked me what my first name was.\n\n'First name?'\n\n'Yes, your Christian name, what is it?' her voice, usually low, was even lower. Like a kind of shyness.\n\nI did not have the chance to answer. We had to squeeze through to our different tables quickly as Night Sister Bean was already calling the register.\n\nIf Ramsden could be on the platform to meet my train at the end of this journey I would be able to answer her question. Perhaps I would be able to explain to her about the violin case. I would like to see Ramsden, I would like to be going to her. Thinking about her and seeing her face, in my mind, when she turned to smile at me, the time when she played Mussorgsky on the piano in the nurses' sitting room, makes me think that it is very probable, though no one has ever spoken about it, that Night Sister Bean might very well be missing her life-long friend Night Sister Percy. Missing her intolerably. \n\n# LOIS\n\nThis is a piano concerto, Lois. Lois, listen! Listen to the way the piano rushes in. It's the Emperor Concerto, Lois. It's not really hard to listen to, is it? Is it? But Lois, wait. Listen. All the same. Listen to this.\n\nAshes of Roses, the perfume is called that. No, not perfume really, Lois. It's only scent. Little bottle of scent to take to boarding school. Ashes of Roses to bring to the hospital.\n\nLois, you'll never believe this. I was so unhappy at boarding school to begin with. My mother put scent on my handkerchief for me to take back and hold when I went to sleep.\n\nSilly but it's real. Lois, you like the scent, do you? You did say you liked the scent. You told me you liked the fragrance. You smell nice, you said that to me once. It's Ashes of Roses. I told you, it's the Ashes of Roses.\n\nTango Bolero now. Not the Emperor. The Tango. I'm glad you asked to share with me Lois when we had the chance to change.\n\nFrom this high window the world is out there. The trees. You could paint the trees, Lois. The park is over to the right and on this side there's the railway and the canal. Did you notice this evening, Lois, how the water shines as the rest of the world gets dark? And last week, Lois, did you see that, in the moonlight, the water shines all night? Did you see that? Can't put a black-out curtain over the canal, can you? All those places out there, the buildings, the warehouses and the railway and the dark streets. All dark, in darkness. People inside those houses don't know who can see their upstairs windows and their roof tops.\n\nTango. Tango Bolero. Lois. Listen! Someone's coming!\n\n'What is that noise? Nurse? Whatever is that terrible noise?'\n\n'It's music, Sister. It's the Tango Bolero.'\n\nTango. Fango. Turn it off this instant! I never heard such a noise in all my life. Do you know what time it is?'\n\n'No Sister. Sorry Sister.'\n\n'It's practically midnight! Where is the music? For heavens sake. Where is it coming from?'\n\n'It's in the wardrobe, Sister, I'll turn it off.'\n\n'And Nurse! Where is your nightdress? You've got nothing on! Why haven't you got your nightdress on?'\n\n'I, er, I was too hot Sister.'\n\n'Well child! For heavens sake open the window. And put the black-out back. If you're careful, slip your hand behind the curtain, put a book or something to hold the curtain down.'\n\n'Yes Sister. Thank you Sister.'\n\n'Why aren't you on duty Nurse?'\n\n'We're, I mean, I'm on nights off Sister.'\n\n'I see. Nurse! I never saw, in my life, such an untidy room. Your room is a disgrace and the beds! Is that your pillow? Pull the pillow out and put it back where it belongs. And try to remember that others want and need their sleep. And another thing, Nurse, it is not healthy to sleep without your gown. See that you put it on at once.'\n\n'Yes Sister. Thank you Sister.'\n\n'Has she gone? The old witch?'\n\n'Yes. That was close. Yes, she's gone.'\n\n'I was suffocating under all that heap. Who was it?'\n\n'Home Sister.'\n\n'Yep. I know that. But how? Which witch?'\n\n'Morton.'\n\n'Forget her! Let's put the tango on again. Don't put your nighty on. Here, push the towel in the gramophone and this blanket over the top and close the wardrobe this time.'\n\n'And listen! Let's open the window and take down the black-out so if she comes back she can't switch on the light.'\n\n'Good idea. And put the chair against the door.'\n\nAshes of Roses. Tango Bolero. Like this. Put your arms round me. Like this. Tango.\n\nOf course at a time like this, I know, it is not right to actually think of anyone, I mean to really think of another person just now. But there is just this, that that witch Morton might be on her way back up here. And if I go on thinking of another person the awful thing is that Sister Bean might be coming up with her. \n\n# THE HUNT\n\nThe early morning sunlight filters through the bushes outside my window making tremulous patterns of light and shade on the wall opposite. This partly shaded shadow-moving light makes the room serene and tranquil. I try to prolong the tranquillity by keeping my thoughts as peaceful as possible. Sometimes I have a wish to keep on staring into the leaves and the small branches out there until I disappear into them. Once I saw a hand in the leaves. It seemed to be reaching towards me, it was my own hand. I often hold out my hand now towards the bushes in order to see this enticing reflection. I wish too, often in the evenings, for an elixir, an ancient potion with magic qualities. A provider of energy and enlightenment.\n\n_Os innominatum Ilium Ischium Acetabulum_ and _the Symphysis Pubis_ , the poetry of anatomy. It's like poetry I want to tell them, this anatomy, this usefulness of the pelvis. We are studying together, in the room I share with Lois, for our exams. _The Ilium presents two surfaces, external and internal, a crest, two processes, anterior and posterior and an articulating surface for the sacrum._ I want to tell them, Trent and Ferguson, that I am going to a house party at Dr Metcalf 's. Lois has just come in so I do not say anything. Dr Metcalf gave me the invitation in Magda's purple handwriting this morning on the ward. I have missed the opportunity. I can't tell them now.\n\n'Any cigs?' Lois asks, knowing that none of us smokes and knowing too that I will have a packet of State Express 333 especially for her. State Express is her favourite. Ferguson does not like Lois. Lois does not seem to notice Ferguson.\n\n'The crest of the ilium is curved and surmounts the bone, here it is,' I say quickly, the point of my pencil hovering over the bone on the page. And with the other hand I pull the present for Lois out of my pocket slipping it to her sideways. Trent raises her eyebrows.\n\n'I can read you like a book,' she says to me out of the corner of her mouth. 'Dead Loss,' she says, 'brother to Joe Loss.' She makes me laugh.\n\nI do not say anything about the invitation. I shall be coming in very late. I shall have to sign the book and ask the night porter to unlock for me. I wonder whose name I should sign. Ferguson's or Trent's? If names appear too often in the late pass book it is a question of appearing on Sister Bean's list for Matron's office at nine a.m. I would never use Lois's name. I have some feelings for her which I am not able to define. I feel happy when she comes into the room. Her clothes amuse me. I have a great wish to protect her and to do things for her. I like giving her small presents and seeing her pleasure. I make up my mind, as Lois shrouds us in smoke, to put Helen Ferguson in the book. It is Sunday. It is my day off. I did not go out because of the party tonight and because of the forthcoming exams. I wanted to go on sleeping too, on my bed in this room I have with Lois. We only sleep in the basement now if there is an air-raid warning. There was a prolonged one last night.\n\nThe night porter, because it is Sunday, will be a relieving one. He will not know me. I can be H. Ferguson for him.\n\n'Describe the acetabulum,' Trent says, 'and do it without looking at the book.'\n\n'A deep, cup-shaped cavity, formed by the union of three bones...' Helen Ferguson's voice reminds me of our first days and weeks at the hospital, of the first Sunday here...\n\nOn my first Sunday I had the evening off. I wanted to be on my bed. The bed wasn't even made up. The bedclothes were rolled neatly in readiness for carrying downstairs later on. It was late afternoon towards the end of summer. A narrow shaft of sunshine came through the little space where the black-out curtain was tied up in a loop. It was very quiet and then someone started playing the organ in the chapel below. I liked the music and I tried not to sleep. I wanted to hear the organ but could hardly keep my eyes open. Images, one after the other in that familiar ritual of oncoming sleep, crowded my thoughts. Images from the children's ward, the noise, the big black trays on which we carried little tin plates of mince and potatoes and the baked apples swimming in yellow cream and covered with the shining crystals of brown sugar. 'Good food,' staff nurse Sharpe, who was teaching me, said, 'is essential in their treatment. You must see that every child eats the dinner.' Images from the children's ward, the occupational therapist moving slowly between the beds and the frames like a walking sewing basket. Twice in one morning one of the boys cut her scissors off and she had to find a new piece of string. Staff nurse Sharpe explained that she kept the scissors tied to her overall. She seemed, with her white hair, to be elderly but Sharpe said no it was because all her family were killed when their house was bombed and her hair had gone white overnight. In all the noise of shouting and crying and dropping tin plates or cups and the boys throwing conkers, the therapist played the piano in the middle of the ward and some of the little ones, rolling their heads from side to side, sang.\n\n_The good ship sails on the anny anny O_\n\n_The anny anny O_\n\n_The good ship sails on the anny anny O_...\n\nWe have stopped studying for a few minutes as Trent is making some toast and cocoa in the little kitchen at the end of the passage. Lois says have I still got some cake left.\n\nThe children in the ward, when they sing, can only roll their heads from side to side because of being strapped on to their frames. Mrs Doe, the wardsmaid who is unable to read and write, crashes the tin plates, Sharpe says they are aluminium, into the sink. I envy Mrs Doe. She says she goes to the pictures with her old man Fridays. I envy her because she can go home at night. She cleans the floors and washes up and comes and goes through a back door. She says exactly what she thinks about us. She calls it giving us a piece of her mind.\n\nAnd all the time while the work is being done, all through the terrible noise and rush, Vera Lynn is singing on the wireless. She sings 'There'll always be an England' and something that sounds like 'There'll be blue birds over the white cliffs of Dover'. I like the therapist but we have no reason to talk to each other and I don't know her name or where she lives.\n\nIt seemed to me during the first few days that there was a smell of disease, of tuberculosis and when I asked staff nurse Sharpe about this, in a moment when she seemed approachable, she said no the disease did not smell. She thought it was the little woollen shawls the children all had, they probably needed washing and it was something I could get done on the first fine day.\n\nWhen I lay on my bed that first Sunday evening someone came along the Peace corridor knocking on all the doors, opening and closing them. Tap tap of heels on the floor-boards, knock knock open door click close door click all the way along.\n\nNow it is my door. I keep my eyes closed. I hope whoever it is will go away. It is my first Sunday in this place and I am tired. I have never been so tired. My hands and arms up to the elbows are all chapped and sore. My feet hurt. I would leave if I knew how to.\n\n'Time for chapel Nurse!' It is Sister Bean. Why isn't she in bed? She is the Night Sister. 'Time for chapel Nurse!' as if she is wound up for these words. She closes the door and I hear her tap tap knock knock to the end of Peace and back tap knock open click close click all along the other side. And then tap tap back to my door knock open click.\n\n'Time for chapel Nurse!'\n\n'I am a Quaker, Sister.'\n\n'Time for chapel Nurse!'\n\nIn candlelight and incense and the music of the organ Father Bailey, the portly one, and Father Reynold who is hungry and red-nosed pass, in their robes, to and fro, up and down the aisles between the rows of pretty nurses. All have on fresh clean caps and aprons. All, with devout expressions, are bending their head down, repeating _I believe_. It is called the creed. I have never heard it before. At the door I was given an unfamiliar little book. It is nicely bound in leather and the pages are edged with gold. I would like to keep it. There is a squat tuffet thing for me to kneel on. We kneel and sit and kneel and stand and sit and when we all move there is a rustling like leaves falling in autumn. There is a chanting and murmuring of believing. I am not sure what I believe and, unless I read in the little book (if I can find the page), I do not know the words. At school prayers were silent in our own words in our hearts. I see two nurses, two of the ones I have been working with, they are very pious.\n\nOn their dressing tables they both have silver-framed photographs and every day their conversations, in well-bred voices, ripple across the Peace corridor.\n\n'Oh Mandy Darling, have you heard from Derek lately?'\n\n'Yes Darling, simply adorable letter this morning.'\n\n'How heavenly Darling! I adore his big ears.'\n\n'Barrington Darling have _you_ heard from Rojji Rog\u00e8re?'\n\n'Mm yes Darling, this morning. He's on a Brenn Gun Carrier.'\n\n'Oh Wizard!'\n\nIt is Nurse Mandy who holds the crying children, the little boys, over the jet of icy water in the sluice room every morning to punish them for wetting their cots. They go on crying and crying, their voices hoarse from crying. She is saying this ' _I believe_ ,' she believes in a whole list of things and I am trying to keep my eyes open...\n\n'What's in the _acetabular fossa?_ ' Ferguson's question \u2013 it's her turn after answering correctly \u2013 is for me.\n\n'Fat,' I say. They laugh but it is correct.\n\n'What and where,' it is my turn to ask Trent, 'what and where is the _ligamentum teres_?'\n\nIt is not possible to compare, to make comparisons, between this room which is Dr Metcalf's and any of my mother's rooms. On the ground floor she has the back room and the front room and the kitchen. In this house, which is the Metcalfs' house, I cannot begin to know how many rooms there are, parts of the house are unknown to me. All the same I am making comparisons. Dr Metcalf has a carpet; at home we have linoleum and a rug, a black rug in case the coal spits and burns. There are some books at home, here the walls are lined with books. This room is Dr Metcalf 's study. It is not very light, there is a shaded reading lamp. The electric light at home hangs in the middle of the room on a flex from the ceiling. My father likes bright light.\n\n'Don't read with a bad light,' he is always saying it, 'you'll spoil your pretty eyes. Don't spoil your pretty eyes.'\n\nThis is a much bigger house, a better house, there are more rooms than we have at home. Perhaps I should have gone home instead of coming here. It is a whole week since the party. Dr Metcalf had to leave in the middle of it as he was wanted at the hospital. I wanted to leave then to go with him but Magda would not hear of it.\n\n'Precious child stay!' she cried all across the room. 'You simply can't go yet. I'll pop you in a taxi.' And then she asked me to come the next free evening I had because she was going to be away, 'and Jonty must, simply must have some good company,' she said. 'He wants to play some music called _The Hunt_ ,' she said. I told her I'd like that very much and all the guests thought this very funny.\n\n'Oh! Isn't she just a pet,' Magda cried then. 'Come here precious pet!' She told me she wanted to hug me and kiss me forever.\n\nOn the way to the Metcalfs' this evening I keep hoping that Dr Metcalf will answer the door. I do not like Mrs Privett, the housekeeper. I think I am afraid of her. Magda calls her Mrs P. Really it is more that Mrs P dislikes me. Magda encourages Mrs P in her disapproval of everything because, she says, when Mrs P is hating she works terribly hard and polishes everything and scours the saucepans, haven't I noticed how shiny they are.\n\nMrs P does let me in and she opens the door only a little way so that I have to step in sideways.\n\n'Dr Metcalf is expecting me,' I say, my voice all silly and shaky with nervousness. It seems a long time that I am waiting for him to come and I worry in case he does not really want me to come or that he has forgotten that I am coming. I try to work out how much in pounds, shillings and pence his curtains and chair covers will have cost. I look up and down the walls and the books and at the furnishings adding value and subtracting.\n\nDr Metcalf, when he comes in, is very nice and gently helps me off with my coat. He winds up the gramophone and puts on the record. But there is something wrong with it. We can hardly hear it. I feel I should say it is nice but we both know something is wrong. He stops the gramophone and takes off the record. He explains it is a quartet by Mozart, it's called _The Hunt_. He says he is sorry it's a flop and I say it doesn't matter. There is a silence between us which I feel I must fill.\n\nI kneel down and look along the book cases hoping to recognize a title and make some intelligent remark but all the books are strange to me. I read aloud some of the titles. Dr Metcalf looks amused.\n\n'Which would you like to read?' he asks after a little silence. I tell him that I'm afraid I don't know. He bends down and takes a book.\n\n'Try this one,' he says, and kneels beside me.\n\nMy clumsy skirt is caught all round my legs as I struggle awkwardly to my feet. The book is _The Voyage Out_ by someone called Virginia Woolf.\n\n'A lady writer,' I say and feel ashamed to have said such a stupid thing. 'Thank you,' I say. I like his hand, the feeling of it, as he gives me the book. I tell him I am afraid it might be too difficult for me.\n\n'Try it,' he says getting up. He says he thinks Mrs P is going out but she has left us some supper and, if I like, we can have it by the fire in his study. Would I like that, he wants to know.\n\n'Yes,' I say, glad that Mrs P is going out,' that would be nice.'\n\nA patient has given Dr Metcalf some rhubarb. He shows it to me in the kitchen and I offer to cook it for him. He sits on the edge of the table while I wash it and cut it up. I try to do this neatly and not to strip it too much. It needs a lot of sugar I tell him. It is a moment of authority which I enjoy. But when we look for the sugar there isn't any. Mrs P must have had it all in her tea he says and I wish I had brought my ration with me but of course I didn't know about the rhubarb.\n\nThe kitchen is transformed, that's how Dr Metcalf puts it, by the fragrance of the stewed rhubarb. He will eat it, he says, for breakfast. He carries our supper through to the study.\n\nWe do not eat much. I am shy sitting there with a tray between us. I hardly notice what there is to eat. Some sort of salad with pears in it. I feel I have said silly things about his books. Because he is quiet I talk, feeling all the time that I am talking too much.\n\n'I suppose it's about six miles, I go there on my bike, I like the ride, it's country there, hawthorn berries and rose hips on both sides of the road, the wild roses when they are out are so pretty. Her house is in the middle of a field. Hens, she keeps hens.' I am telling Dr Metcalf about Gertrude's Place. 'I could,' I tell him, 'bring some eggs, one time, for Magda, perhaps next time,' I say, 'I go there on my days off but if I don't go Mother goes on the bus.'\n\nDr Metcalf is always quiet, I suppose, it does not show, this quietness, so much when Magda and the others are there. In one of the silences we hear Mrs P returning. Dr Metcalf says she has come back earlier than usual. In another of the silences I say that I am thinking it is time for me to go back to the hospital. When I say this Dr Metcalf says he will take me back. And I say oh no I would not dream of him coming out again. He says a walk would be nice. He feels like a walk. Should we walk to the hospital? Or do I prefer to go by bus?\n\n'Oh a walk would be lovely,' I say.\n\nOutside the front door, to our surprise, there is a thick fog. It is impossible to see even one footstep ahead. It is as if a cold damp wall is right up against us. Slowly we go forward. I feel along the wall with both hands. We can follow the wall I say but Dr Metcalf reminds me that the wall does not go beyond the corner of the street. We come to the end of the wall. I feel completely enveloped. My scarf, coat and gloves are wet. I try to go back. I reach for the wall but am not able to find it. I can't breathe. I stumble off the unseen edge of the pavement. There is a silence so terrible it is as if the world has ceased to exist. I feel terrified and try to call out to Dr Metcalf and find he is near me. He puts his arm round me and says it is no good, we'll have to find our way back to the house and it will be best for me to sleep there and go in to the hospital in the morning.\n\nMrs Privett has already gone to bed but Dr Metcalf taps on her door and asks if I can sleep in her sitting room. He explains about the fog and that we have had to leave our wet things in the kitchen. He brings a mattress from one of the spare beds upstairs and puts it on her sitting-room floor. Mrs P, who looks more sour than usual in her dressing gown and curlers, hands me two sheets. Dr Metcalf brings down an armful of blankets. He says goodnight to Mrs P and me in a most formal way. I can hardly bear this and I want to drop the sheets and rush along the passage and into the hall and up the big staircase after him. I want to be upstairs in his part of the house, even though he is alone up there. I don't want to be an unwelcome guest in Mrs Privett's sitting room.\n\n'Can I have a glass of water please,' I ask Mrs P. She brings me half a glass which she holds out to me round the door before closing it. I make up a sort of bed on the floor.\n\nMrs Privett's sitting room is crowded with her things. There is only just room for the mattress between the chairs and the table and the sideboard. There is a dark red plush tablecloth with an immense fringe all round it. She has some china shepherd and shepherdess figures on the mantelpiece and two stuffed birds under glass covers. There is a mirror over the mantel and a great many photographs in bone frames. One forbidding face glares from the cruet on the sideboard. I examine the room to try to overcome my uneasiness and the wishing to be with Dr Metcalf. When we were in the fog he was close to me guiding me with his arm round me and his body close to mine. I try to recapture the feeling of his arm being round me. Does he suddenly not want me? I wonder if he is in bed or whether he will come downstairs again. I open the door a bit but it is dark and quiet. Upstairs in one of the pretty bedrooms I would be near him. The time goes by very slowly. I am cold. I don't want to put out the light. Oh come down again please.\n\nI feel I can't breathe when the light is out. I am too near the floor. Mrs P's horrible face seems to be in the room, disagreeable, when the light is off. I struggle up off the mattress and put on the light again. I check her black-out curtain and knock something over on the other side of the curtain. Perhaps a plant in a pot. I don't dare move the curtain to look. When I try to crawl back into the tunnel of bedclothes I knock over the water and it makes a wet patch on the carpet and soaks into the side of the mattress. The room is getting colder.\n\nI am not asleep and, though I have kept on most of my clothes, I am cold. Right through my whole body I am cold. I wish I was at home and not here. I wish I had gone home. And another thing, I do not know where Mrs P's bathroom is. I want my mother. This is such an unusual wish that I weep a bit in a sort of despair. It is only a quarter to two. A great many hours till the morning. My mother would not like the Metcalfs. I wish I could be warm. Dr Metcalf obviously does not like me much.\n\nThe place where I lived as a child is a place of small mean houses in terraces in mean little streets. All around are pit mounds, some black with fresh slag from the mines and others covered with coarse tufty grass and the yellow weed called coltsfoot. There are small triangles of meadow with partly derelict barns and farm cottages. Sometimes there are a few cows and chickens. The bone and glue factory is not far away and at the end of the street where we once lived there are the brick kilns.\n\nOne night when I am walking home with my mother we hear someone crying in the darkness ahead of us. The lane is shut in by dark slag heaps and there is, as if relieving the endless smell of bone and glue, a sharp fresh fragrance from the elderberry bushes which are, at intervals, along the roadway.\n\n'Who is it?' my mother calls in her soft voice. But no one answers. 'Who is it crying?' my mother calls again, her voice like a flute in the night. 'Who are you? Why are you crying? Don't be afraid. Tell me what is the matter.' My mother, holding my hand, stands still in the middle of the road. After a bit a girl comes out from the black patch of elderberry. She is still crying. She tells my mother she is Sylvia Bradley and her father has turned her out with only a shilling. We walk on together, slowly, while the girl tells my mother. The fragrance of the snapped-off and bruised elderberry is left behind and the smell of bone and glue seems stronger. With a heaving roar the blast furnace on the other side of the town opens and the sky is filled with the familiar red glow. We can hear the wheels at the mine shaft turning as one cage of men comes up and another cage goes down. 'Where will I go,' Sylvia Bradley cries, 'I've got nowhere to go.'\n\nMy mother never lets us play with the Bradley girls. There are nine of them and they do not wear any knickers. Emily Bradley, the next eldest, knows a shop where the woman makes rum and butter toffees. If you eat them they make you drunk, she tells us.\n\nI must have been asleep, or nearly asleep. Sylvia Bradley. She was the one who cried so much that night. All along the road my mother tried to comfort her and to persuade her to go home again. 'When your baby is born,' she said to Sylvia Bradley, 'your mother and father are sure to love your baby. You will see, they will not turn you out. Go back home to them.'\n\nI want my hairbrush. I struggle up from the mattress. It is colder than ever in the room. There is a draught along the floor. My back aches. I want the bathroom. I don't know at all where the bathroom is in Mrs P's part of the house. I go as quietly as I can, feeling my way along the passage to the door which leads into the wide front hall. The black-out curtain for the fan-light window over the front door has not been put up and a faint light comes through. Perhaps it is the fog making a mysterious misty light. It must be getting on for morning. I feel my way to the staircase and go up as quietly as I can. There is a landing where the staircase turns. The lavatory, which has an ornamental stained-glass door, opens off this landing. Magda explained once that it was there because it was added to the house.\n\nWhen I come out, instead of going down, I go on upstairs and along the thickly carpeted passage. I can smell Magda's perfume. It is dark and all the doors are closed. There is no sound from behind the closed door which is Dr Metcalf's. I go to the top of the stairs again and make my way back to the sitting room. It is airless and colder than ever. I don't suppose Mrs P ever uses the room and I'm sure she never has a fire in it. I crawl back into the cold damp bedclothes. The fog must be seeping in through the walls. It is not even three o'clock. My hair needs brushing. I wish I had my hairbrush. I am used to brushing my hair before I go to bed. I have not been able to clean my teeth either. Trying to sleep in my clothes is awful. I'm tired and everything I will have to do tomorrow rushes in on me. I'll never manage. Because it is silly to wish for my hairbrush and for my mother I can't help crying a bit more.\n\nI am afraid to go to sleep. I am frightened of being asleep when Mrs Privett comes in here in the morning. I don't want her to come into her sitting room and find me asleep.\n\nOf all the girls at school, apart from Ferguson of course, Bulge is the one I seem to remember most. It is not that I am always thinking of those times, but when I do, I seem to remember Bulge.\n\nThe first and only nice thing I ever do towards Bulge is when I visit her after her appendix operation in the hospital at Oxford. I have to be taken there to have my ear examined and dressed and bandaged. Bulge is lying back on her pillows with her washing bowl beside her. She is supposed to have washed her face. It is not visiting time but I am allowed in because of being brought to the out-patients department. Bulge is the only schoolgirl in the women's ward. She has her doll and her doll's clothes on the locker by her bed. The other patients seem to like Bulge. Even the nurses seem to like her and they even have little jokes with her. They all call her Muriel. No one except me knows the name Bulge. She does not know it herself.\n\nThe ward is long and narrow and the beds are all stripped on to chairs and the patients are washing themselves in bowls like the one Bulge has. She looks different. Her face seems white and clean, like wax, and she seems to have lost her spots. She looks tired and, without her spectacles, she seems more childish. Her face, I have never noticed before, her face is round. Her hair has been brushed back off her face and this makes her look different. She even seems pretty.\n\n'Have you finished with this?' I indicate the bowl and she says yes and will I put it with the others on that high metal trolley. When I lift the bowl the water splashes up everywhere. I have lifted it too high. I thought it would be a heavy bowl by the look of it but it is light, much lighter than I imagined. I put too much effort into lifting it a nurse says as she goes by. I feel confused and ashamed and the nurse says she had done the same thing years ago, not knowing the bowls were so light.\n\n'I brought this for you to read,' I say to Bulge as I place _Treacle Wins Through_ on the sheet. 'Here,' I say, 'I thought you would like to read this.'\n\n'Oh! I say! Yes rather! What's it like?' Bulge raises herself a little bit. She looks in an eager but short-sighted way at the book.\n\n'Haven't read it yet, not all of it.'\n\n'Oh. You finish it first. After all, Mummy and Chris gave it to you. When they came to school that day, they brought it for you.'\n\n'Yes, well, I know, but they're your parents. And you are in hospital. I'll finish it after you. They are your parents \u2013 it was jolly decent of them to give me a present. They are your parents, after all.'\n\n'Only half. Chris is Mummy's Friend. Remember.'\n\n'Yes I know. But you have it first. Go on.' I push the book with Treacle's boarding-school adventures, which I am longing to read, nearer to her.\n\nI can tell Bulge is pleased. She fingers the book and smiles.\n\nIt's the first and only nice thing I have ever done...\n\nI'm shivering. I fold up my blankets and the sheets. I let myself out of the front door of the Metcalfs' house. It is still dark but the fog has lifted. I set off to walk to the hospital. It is still not light as I climb the hill to the service entrance. The night porter will be somewhere near there. He has the key to the special door in the basement. There is a passage down there through to the part of the hospital where the nurses have their rooms.\n\nThe night porter believes my story about the fog. He says not to bother about signing the book. He says the fog is the best anti-aircraft there is.\n\n'There's been no bombs,' he says. His wife is nervous in the night, in the air raids, but this time she'll have had a good sleep. I have never thought of the night porters, or any of the porters, having wives, and perhaps children, at home. Suddenly I feel ashamed because, before I found him, I was remembering that I had no hair clips, no hair nets and not one apron button. I was thinking that Lois, and even some of the others, Ferguson in particular, must have been helping themselves to my store of these. These shortages, as they are called, seem more important than the progress of the war. Even that phrase, the progress of the war, because people say it all the time, seems to have no meaning. As I follow close behind the night porter I feel ashamed of my own small selfish brain. His wife is probably alone every night scared stiff. And I cried for my hairbrush and then centred my thoughts on a hair clip.\n\nWe go down in the lift to the lower ground floor. As we walk past the glass doors of Lower Ground Radium I catch sight of staff nurse Ramsden. She is sitting in the ward office with a small shaded light beside her. She is bent over the desk, writing, probably some of the night report. She must have been moved there recently. I wish she would look up and see me and smile at me. I have to hurry to keep up with the porter.\n\n'Ramsden,' I could say if I could speak to her just now. 'Ramsden, do you know _The Hunt_?' and Ramsden, her eyes suddenly bright with some memory of the music, would say, 'Ah! Yes. Mozart, the quartet. _The Hunt._ It's K458 in B flat major.'\n\nAs it is a quartet, and, if all four instruments leap in together on the opening notes, it would be too difficult for her to sing the first few bars. From what I managed to hear, when Dr Metcalf tried the record, I feel it would not be possible for the human voice, by itself, to produce the sounds. If I could speak to Ramsden just now I feel sure that is what she would explain. I would like to talk to Ramsden now about the way in which the musicians, in a quartet, play towards each other, leaning forward as if to emphasize something in the music and then, pausing, they lean back allowing the phrases of music to follow one another, to meet and join, to climb and cascade. I would like her to agree with me and to say that she knows about the serious expressions the musicians have while they play and that she has noticed too the way they have of drooping their wrists and showing the vulnerable white backs of their hands.\n\nThe porter, with his key in the lock, is waiting for me. 'You know Roberts,' Lois says at breakfast.\n\n'Roberts? Roberts?' I say putting my plate of fried potatoes on the table. I am hungry.\n\n'Yes Roberts. Roberts \u2013 Nurse Roberts.' Lois lights a cigarette. She never eats breakfast. I move her tea cup nearer to her elbow. 'Well listen!' Lois says. 'Roberts. She's been throwing up all over the place. Can't keep a thing down. Diet kitchen of all places. Can you imagine! It's the powdered egg. Just the sight of the packet is enough to make her throw up. She's fainted during Report too, twice.' Lois, after a deep breath, exhales a cloud of smoke. She nips out her cigarette, squeezing it quickly between her thumb and forefinger, as Sister Bean, clutching the registers to her flat white apron, marches across the dining room.\n\n'Abbott Abrahams Ackerman Allwood...' Nurses in various parts of the large room are answering to the rapid barking of their names. Some half rise from their places at table. 'Arrington and Attwood...'\n\n'Nurse Arrington!'\n\n'Yes Sister.'\n\n'Matron's office nine a.m.'\n\n'Yes Sister.'\n\n'Nurses Baker Barrington Beam Beamish Beckett...'\n\n'Anyway,' Lois says looking at me, 'long time no see, where were you last night?'\n\n'Nurses Birch Bowman D. Bowman E. Broadhurst Brown Burchall...'\n\n# GERTRUDE'S PLACE\n\nWhen I see the visiting nurse cross the lighted verandah of the house opposite I recall without wanting to the navy-blue uniform of the district nurse as she dismounted from her bicycle. Pushing through the long grass and weeds she climbed slowly up to a small house in the middle of a field where scattered hens were industrious and apparently independent.\n\nAt that time I watched from higher up the hill, leaning heavily on my own bicycle, assuring myself that other people, for example, Madame Curie, had safely ridden bicycles right up to the time of confinement. As the nurse made her slow journey I thought of the illness in the house and how I could go down there and tell her that I could do all that had to be done. 'I can stay. Tell me what needs to be done. I will do all that needs doing.' Easily I could do this.\n\nThe nurse did not leave. After a short time I saw smoke rise from the chimney; she would need a fire to have some hot water. I remember that I wondered then whether the nurse would know to shut up the hens for the night. As I watched from high up on the hill I thought of foxes.\n\nI remember now, unwillingly, all kinds of things. One small thing only, the sight of an unknown nurse going in to an unknown patient across the road, is needed to bring back memories mysteriously stored in such a way that all seem fresh and whole as if they belong now at the present time, perhaps yesterday, the day before yesterday or this morning... one of these memories being the schoolgirl game of comparisons which I continued to play, silently watchful, observing my companions noticing the quality of their hair, their complexions, their finger nails, their uniforms especially the condition of the starched caps and collars and all the time secretly placing myself above or below a standard which I regarded as acceptable. When I stood then in the queue for letters I noticed the fat stomachs, the thick waists of some of the nurses and how their aprons were pulled tight across this bulging roundness. My own apron, at that time, was neat and flat and the memory of the sudden realization, one morning that this could not always be so is so intense now that I remember clearly the unfamiliar nausea which accompanied this thought.\n\nOther things come back in quick succession, the wedding ring for sixpence, the way in which I heard what had happened to Dr Metcalf, the attempts I made to retrieve a letter I had written. On that day, in the bleakness, I saw little Nurse Roberts, alone and portly under her winter coat, standing at the bus stop with her suitcase on the pavement beside her. And then there followed the whispered reasons for her running away. Always a great deal of whispering.\n\nThe daylight, that evening when I watched the district nurse pushing her bicycle up the steep lane, faded quickly and quite soon the dusk became night. Across the field I saw the light in the window of the little living room and I wished then for the times when I was in that room beside the hearth. I thought with longing of the times I had spent there and how it was a long time since I l had been there. I remained for some hours in the damp cold, undecided. I did not go down to the place that night and, because of what was happening to me, I never went back there again.\n\nOne evening as the lift goes up and I am waiting, it passes the floor where I am standing and I see through the little glass window staff nurse Ramsden in the lift. Really I see her polished shoes, I know they are hers. Her shoes are of better quality and she has narrow feet, narrow at the heel and the shoes fit perfectly as if made especially for her.\n\nI never hear anyone call staff nurse Ramsden 'Penicillin Peggy' though she is very often the one in charge of the syringes and the needles, cleaning and sterilizing them, and giving the three-hourly injections. Everyone calls Ramsden Miss Ramsden, even some of the patients. Because she understands and speaks other languages, she is often asked in to translate for a prisoner or perhaps a Polish officer who can speak some sort of German. This morning during the surgeon's ward round I can see her laughing with a German P.O.W. He looks at her with admiration in his eyes and he carefully repeats some of the words and she, in her quiet way, laughs softly again and translates for Mr Bowen, the surgeon, and he laughs too. Ramsden is not at all nervous with Mr Bowen, she bends over to unfasten the bandage and to remove the dressing and the prisoner closes his eyes and grips the head rail of the bed till his knuckles shine white.\n\nRamsden reads Rilke in German. I have heard her read. Perhaps it is that which makes me invite her.\n\n'Ramsden,' I say, 'my folks,' this is not my way but I say it, 'my folks would be very happy if you would visit, you know, if you would care to come and stay with us sometime.'\n\nRamsden says she would like that very much and she thanks me.\n\nMy mother would like someone who can read these poems in her language. I am always trying to think of ways to comfort my mother and it seems to me that I can offer Ramsden to her.\n\n'Infection takes the line of least resistance, sing that to la,' Trent says, 'a peptic ulcer is an ulcer which occurs anywhere in the alimentary canal \u2013 repeat after me \u2013 a peptic ulcer is...' This evening we are all trying to revise for our exams. Trent, with her uniform unbuttoned, is on circulation, Ferguson has turned up varicose conditions and Lois wants to go over the preparation of trolleys for mastoid dressings, lumbar puncture and washing a patient's hair in bed. Of course I cannot tell them that I shall be inviting staff nurse Ramsden home later on when we have some holidays. I can't help thinking about my invitation with a mixture of excitement and plain worry. For one thing where could staff nurse Ramsden sleep in my mother's house?\n\n'Now gels, heads up, throw out your chests.' Trent opens the window. 'Mind the black-out,' someone says. It is impossible to study with Trent in the room. She makes a joke out of everything. She sings and dances and dresses up in bath towels. She pretends to give elaborate intimate treatment to imaginary patients. She pretends she has made dreadful mistakes and kneels before an imaginary ward sister...\n\nWe have all started having cold baths every morning. Trent has one too but we have all noticed the steam coming under the partition...\n\nLois says to me later that she has worked with Trent on the wards and that if she, Lois, were ill it would be Trent she would want to have to look after her. This is considered to be the highest praise one nurse can give another nurse.\n\nBecause of the war some wards are badly overcrowded to enable us to prepare others to be quite empty in readiness for the wounded. It is impossible to imagine a life which is not in the war.\n\n'O' course we couldn't go to bed or anythin' \u2013 they had ladders great big long ones up the wall to our winders an' we hid under the bed, we didn't have any candle for fear they would see it and our shadders on the curtings an' all along the mantelpiece we had these big paint tins full of rusty old nails to rattle about and throw should any of 'em come right up and bosst in through the winder.'\n\nI am at Gertrude's Place. Gertrude is telling me about what she calls the raids when she was a girl. She lived then, she tells me, in a Place called Netherton or near there just out of it somewhere where the chain shops were, and the women (her mother was one of them), with great big muscles for swinging a sledge hammer, made chains. Gertrude had to take her little baby brother to be fed, he was on the breast, she says; 'The women all set theirselves comfortable against the wall to feed the babbies and us girls we played skip rope an' hopscotch an' clay allys and such till it was time to take the babbies back home and put the taters on the hob to boil.' The women, she tells me, wore big overall dresses and they did not bother much about any knickers. Her mother, she says, did not know any life except the chain shop. There were different gangs she explains and they were always raiding. She says having to put black stuff over her window now at night because of the air raids even though she is in the country reminds her of when she was a girl and they didn't dare to show any light.' I thought,' she says, 'that bein' here in the fields and with the spinney so close a light would not show but they, the wardens, they keep comin' to say they can see a light from my place. The wardens come here you see... reminds me of them gangs years ago...'\n\nI am at Gertrude's place, it is the country. I came on my bicycle. My day off. I have two days off. 'I'll come again tomorrow,' I say. She is pleased. She is always waiting for me to come. She writes letters in big black handwriting on paper used for wrapping boiling fowls. In places her pen digs up the paper.\n\n... _If you can come Tue or Wed... I shall be home. If too tired for the bicycle you can get the eleven a.m. bus to the Holly Bush and walk down by the spinney it's about a mile but it don't seem that long and you can have a good rest... I'll have some eggs_...\n\nIn the rush of work on the wards I am always thinking of Gertrude's Place and how I will get out there first thing on my day off.\n\nI have been cutting out a frieze of wallpaper with rosebuds on it and we have pasted it round her back kitchen wall. Against the blue distemper the pink and white looks nice. Gertrude says she is pleased. I can see the way her eyes are shining that she is pleased. She is sunburned and seems old and she always looks clean even though she can only wash with a bit of rag wrung out in a basin of hot water. She does not have a bathroom. In one room there is a billiard table and the incubators and a harmonium which she plays while she sings hymns. She and her husband don't go to bed at night she explains on one of my visits. They sit one each side of the fire with the kettle singing. They sit there and sleep the night, she laughs when she tells me, and says that the kettle there on the hob is ready and boiling every morning. A big oil lamp on three chains hangs low over the table in the living room. The table is so littered with things never put away that they have become, as she says, so much rubbish. She clears a space on the corner of the table so that I can eat the egg she has boiled for me. She cuts the loaf on a folded piece of newspaper and hands me slices of white bread spread with real butter.\n\n'It must be all your butter ration,' I say.\n\n'There's plenty where that came from,' she says. And I describe the little string bags which all the nurses carry with them; 'a little jar for sugar, one for butter and one for jam or marmalade.' I tell her about the tin baths of jam made in the hospital and ladled into our jars. She can't imagine, she says, us all lining up once a week. I tell her about the blackout shutters for the tall windows in the ward and how these have to be put up early shutting out the sun and the fresh air. We have exams I tell her in about two weeks time. She says can I eat a second egg if she puts one on and I say yes if she can spare it. This makes her laugh.\n\nI want to tell Gertrude about Lois, my friend, being jealous of my new friendship with Dr Metcalf and his wife. I want to tell her that I bought Lois a pretty nightdress so that she would not mind my going to Magda Metcalf's parties and how Lois never wore the nightdress but gave it to her ugly, stupid mother who is so greedy she would take everything. If I tell Gertrude that I buy cigarettes for Lois and that I minded dreadfully about the nightdress she might think that I am too fond of Lois or something. If I tell her about the games at the Metcalfs' party she might think I am in what she calls bad company. Magda is very generous. She tells everybody that I saved her life and she is always giving me expensive things like the silver bracelets which I can't wear. For some reason bangles look all wrong on my arms and, in any case, people would wonder where I got them from. She has given me some pure silk blouses which don't really suit me. I don't know what to wear them with and then there's other things like chiffon scarves and lace-edged hankies and a jewelled comb. As well as all this she is giving me something else which I can't explain about to Gertrude. Magda tells me to come round whenever I'm free because they love having me there.\n\nI can't exactly explain to Gertrude how they show me off to their friends in the strange way that they do. Magda calls me Darling in front of everyone and Dr Metcalf actually called me by my first name on the ward the other day. How can I tell Gertrude about these people; they are very nice and they want me to be with them. Of course I didn't save Magda's life. She was admitted to Casualty one night while I was relief nurse there. She was in an attack of asthma, it looked pretty fearful and because I didn't know what to do till the R.M.O. came I dragged in the big oxygen cylinder and Magda got better straight away at the sight of it.\n\n'Since when,' the Night Superintendent said later, 'since when, Nurse, has oxygen been the remedy for asthma? And by what miracle, Nurse, can an empty cylinder be of use to anyone?' How can I tell anyone that? And Lois says do I realize that Dr Metcalf has made eleven nurses pregnant and don't I know what it was that caused Sister Green on chests to suicide.\n\nI think I love Dr Metcalf. Certainly I love Magda, she is kind and full of ideas. All her friends love her and they dress up and cook wild meals and dance and, though I mostly watch, I love being at their house especially when Dr Metcalf lends me books or suggests that we listen to music. Sometimes he looks very tired and one night I thought he was looking at me as if he loved me and when he walked to the bus with me he caught my hand quickly as the bus came and quickly kissed my fingers...\n\nGertrude brings me my second egg and watches me eat it. Perhaps I will tell her next time and then give up all that she thinks is bad company. I think from what little she says that she does not much care for Lois. Lois smokes all the time and never has any money. Once when I mention the money side of things Gertrude says that she must have some because if the hospital is paying me twenty-eight shillings a month they must be paying her too.\n\n'They wouldn't give to one and not to another,' Gertrude says. Which is, of course, perfectly true.\n\nThe sky always seems nearer at Gertrude's Place. It seems to come down, rain soft and swollen, the clouds rosy at the edges and shining as if pearls are sewn into their linings, to the top of the grassy slope which goes straight up from the windows of the living room. The feeling I have of being able to reach out to take the sky in both hands is one of the most restful things I have ever known. I sit there knowing about the nearness of the sky, not reaching out but, at the same time, pleased about the possibilities. These possibilities are connected in an undefined way with Dr Metcalf and how I feel towards him, and then there is Gertrude sitting across on the other side of the table. Two separate people but joined together because of how I feel about them.\n\nThe fowls are dotted white and brown all over the slope of the field. Gertrude has an old bucket on its side out there on the grass. She can tell at once when it starts to rain because of the splash marks the first drops make on it. A rooster struts by and Gertrude, with her little laugh, says to always watch that one. 'Keep a eye on him,' she says, 'don't trust him, not a inch, he'll be into you where you least expect it.' She laughs again. 'Roosters!' she says, 'they can be something wicked!' Afterwards we put the eggs ready for me to take home. Lovely smooth large eggs, their shells glowing a cream apricot colour or pure white with a translucence which makes them seem frail. We wrap the eggs in threes in torn-off pieces of a magazine. They are black-market eggs and my mother, anxious always about provisions, has said to me to buy as many as Gertrude can spare. She lets me have three dozen. She has two dressed fowls for my mother too, one of them she explains has had a fox get to her but not much taken, just a wing and a bit of breast. She has neatened the fowl, she says, with her dress-making scissors. 'Tell mother,' she says, 'I'm sorry about that old red fox.'\n\nAs we work I have a great wish to talk about Dr Metcalf, to tell Gertrude about his quiet gentleness and about his brown eyes. I want to tell her about Magda and how fond she is of her beautiful dog, how the dog lies in bed beside her, a Red Setter, and how Magda talks to him and fondles him. I could tell Gertrude I feel sorry for Dr Metcalf, that I want... Gertrude says what about a game of billiards or cards. Games bore me but I play because I know Gertrude likes to play. We play racing patience, Gertrude calls it 'pounce', it is her favourite. We play three games and she wins them all.\n\nThe long summer evening is beginning to get dark and Gertrude comes down through the damp grass and nettles and cow parsley to the road with me. I walk with my bicycle. I tell her I will come back in the morning to scrape and clean out the hen houses as usual. I almost say, 'I'll have to leave early because I'm going to a party,' but I can't say the words.\n\n'Two nights off,' she says, 'Eh? But that's nice!'\n\nI turn round as I coast down the long hill just before the road bends round and I see her standing alone and waving. I wave and switch on the bicycle lamp grateful that my father insisted on lending it to me.\n\nWhen I come home there is a telegram for me from the hospital telling me to return at once. The nurses' hours are being rearranged.\n\n'I suppose it's wounded men,' I say to my mother, 'a convoy.' My face is burning pleasantly with the fresh air. She is upset and makes a parcel with a fruit cake and some hard-boiled eggs for me to take back.\n\n'There's a train at nine-twenty,' my father says, and he says he will come to the station.\n\nIn the grey half light I walk up and down the platform with my father. He always comes to the train if he can. He tells me, 'You are doing God's work.' He tells me to remember:\n\n' _Der Herr ist mein Hirte, mir wird nichts mangeln_... \u2013Remember,' he says; ' _The Lord is my Shepherd... und ob ich schon wanderte im finstern Tal, f\u00fcrchte ich kein Ungl\u00fcck; denn du bist bei mir, dein Stecken und Stab tr\u00f6sten mich_... verse four,' he says and he holds my hand.\n\n'Not in German,' I say in a low voice, glancing quickly to see if the other people waiting for the train have noticed. For my father it is not the language of the enemy but is the language for cherishing. I feel his hand holding mine and I want to cry and go back home with him. I am afraid.\n\n'I don't want to go,' I tell him.\n\n'It's God's work,' he says again.\n\nThe notice board in the deserted front hall of the Nurses' Home is covered with neatly written timetables. I am on duty at four a.m. I go up to the room which I share with Lois. I go in very quietly as she will be asleep. But her bed is empty. The black-out curtains are not drawn and the window is open. From this window, high up, I can lean, it seems, right into the black darkness which is the world. There is a sweetness, a faint fragrance of crushed grass as if it has come with me from Gertrude's Place. It is probably from the park. When I try to breathe it in once more it is not there. There are no lights anywhere only the canal shines in the light of the rising moon. The moon and the moonlit water are inextinguishable. Two tiny points of yellow light far below are moving and stopping and moving on again slowly. It is a bus. A faint roar, as of breathing, comes to me across the black shadowed roof-tops. I close the window and fix the curtain.\n\nTrying to sleep I think of the tranquillity of Gertrude's field scattered with hens.\n\n'Darling, you are such a pet!' Magda said the other evening when I arrived late (after evening duty) at her birthday party. 'Isn't she just a pet,' she called out, turning me round and round in front of her roomful of guests. 'Isn't she sweet to come after work. You Darling!' she kissed me. 'And you still wear your school coat. I love it! I wish I had it.' She buried her face in the dark-green nap. I was not sure whether she was laughing or crying.\n\n'You could have it,' I said, 'but I seem to need it!'\n\n'She seems to need it,' Magda cried. 'Oh isn't she just perfect. A natural perfect! Jonty,' she called Dr Metcalf, 'take this dear child somewhere and undress her. Help her out of this coat.' They were all laughing.\n\nIf I tell Gertrude she will see how generous and kind Magda is. But Gertrude might think she is being generous with the wrong things. I do not want to give back the special thing I am being given. Gertrude I feel sure will tell me not to take.\n\nI turn over. I am afraid because I can't sleep. Then the rumbling starts. I have heard this noise before. I know what it is. I watched one morning with Lois. From our high-up window we could see a long train of mysterious carriages and wagons winding across to the station nearest the hospital. This is what I can hear now. The wagons, some of them will be marked with a red cross, are bringing wounded soldiers. I can hear the brakes squealing as the train draws up and stops and then, jolting, pulls along a bit more to stop again to allow the stretchers to be unloaded into the waiting ambulances. Quite soon I hear the first of these ambulances, like on that other morning, come slowly up to the hospital. They come steadily one after the other. The empty beds we have prepared will fill again with men who have fear and pain in their eyes. They do not really sleep but lie with their teeth clenched and their hands clenched and their eyes half open. Some of them are not able to close their eyes in sleep.\n\nI do not see Lois at all or Trent. We leave little notes for each other. We are working an eight-hour shift, eight hours on and eight hours off. There is no time off. We eat and sleep and work and eat and sleep and work. Some of the nurses get ill and we are short of staff. The ward where I am is all split beds for amputations. Every bed has its own electric bell and its own tourniquet, some beds have two.\n\nGertrude, waiting for me to come, writes to me. I queue for letters and am pleased when I see her black handwriting;\n\n_Queenie has her pups you'll be pleased to know little balls of fat tumbling about you'll love 'em. If you miss the eleven am bus there's another at one. We would have a bit less time that's all_ \\-\n\nand\n\n_Have you ever seen the pig's meat salted down you'll see it up at Violet's when you come they've killed a pig up there you should taste the pork pies. I'll have one ready all wrapped for you_...\n\nOne of the men, he is only a boy really, as well as having both legs amputated above the knee, has a terrible wound in his stomach. I try, while I am dressing it, not to show that I can't bear it. One day he asks me to put the gold cross from round his neck into the wound but I tell him I can't do this. He begs me and I say no I can't because it isn't sterile. He has tears on his face and my own eyes fill with tears so that it is hard to see what I am doing. I brush something small and white out of his bed. It seems to roll up like a soft bread crumb. As I swab the wound it seems something is moving in it. It is a maggot. I pick it out quickly with the forceps trying not to show my shock. Suddenly I see there are maggots everywhere. It's as though he is being eaten alive. They are crawling from under his other bandages and in and out of his shirt and the sheets. I lean over him to try to stop him seeing and I ring the bell, the three rings for emergency. I have never done this before. I try to cover him but the maggots have spilled on the floor and he has seen them. I see the horror of it in his eyes.\n\nThe charge nurse comes round the screens straight away. 'Fetch a dustpan and brush nurse,' she says to me 'and ring for the R.S.O.' As I go I hear her raised voice as she tries to restrain him and to say words of comfort \u2013 that the maggots have been put there on purpose, that they have cleaned his wounds and yes of course she'll put the gold cross wherever he wants it \u2013 yes, she'll put it there now -\n\nMagda, who has a bad toothache, telephones me on the ward. Her face is horrible, all swollen, she says. She wants me to go with them to their little shack on the river. An easy journey she explains, a bus every hour on the hour. It is awkward taking the phone call in the ward office. I tell her I am not free. 'Well Darling,' she says, 'next week then, as soon as you can manage. We'll be there. Just shout from the bank and we'll bring the punt to get you.'\n\n'Oh lovely!' I say, trying to get my voice as like Magda's as I can so that Ferguson, who has come into the office in a hurry for the thermometers, can hear, 'Oh lovely!'\n\nAll evening while I am cleaning locker tops and making beds and giving out the soup and bread the men have for their supper I think about the river shack, a holiday bungalow which is a part of a rich person's life. I can imagine it clearly, low on the bank close to the quiet water. It will be painted white and there will be a small landing, a wooden jetty, belonging just to that cottage. I have seen these places, very private with grass coming right up to the walls. And all night it will be possible to smell the river water and to hear the soft sounds of it. I can imagine too the pleasure of being on the private jetty, of sitting there with my feet in the water as rich people sit looking as if they own that particular part of the river while ordinary people can only go by during an hour's paid-for pleasure trip in a boat owned and hired out by someone else...\n\nGertrude writes to say she is renting the field next to her place and it is for me. I can have some fowls of my own there and she'll look after them while I am at the hospital. She is buying some bricks too. She adds a p.s., 'p'raps we could build us a pig pen and have a pig between us, a sow, we'd make a lot from the litter. I've always wanted to keep a pig.'\n\nI can't sleep and I try to read and I find a passage written by George Eliot to Caroline Bray after she has started her life with George Lewes. She writes:\n\n_I should like never to write about myself again; it is not healthy to dwell on one's own feelings and conduct, but only to try and live more faithfully and lovingly every fresh day_...\n\nI like this very much and I sit on the edge of my bed and write a letter to Gertrude. I tell her that I like the idea of the field and the pig sty very much. I want to write to her about the maggots, I keep seeing them in my mind, but I write about Magda, all about her, about her face swollen with the bad tooth and how she says she is such a cold person and when you go to see her she is all crouched over a fire with little heaps of underclothes arranged all round the hearth to warm. I write some pages about the excitement of being invited to a doctor's house and how Magda is the daughter of a leading surgeon and that this is a good thing for Dr Metcalf \u2013 everyone says so. I tell her about the all-night parties and how Magda and her friends call me 'darling'. And then I write about Dr Metcalf and tell her that I think I love him. I end the letter telling her that I'm coming over to her place on my next day off to clean out the hen houses and to see my field and I tell her that I want to see her more than I want to see anyone else. I try to end my letter with a sentence from the quotation from George Eliot but do not know which part of it to include.\n\nThe next morning I address the letter and post it.\n\nTetanus typhoid diphtheria and gas gangrene. Relatives, mothers and fathers travelling the length and breadth of England arrive at the hospital too late. They sit in corridors waiting for the first light of the morning and for the first trams...\n\nOur examinations are postponed indefinitely.\n\nI have a day off at last and I sleep all night and, because no one wakes me, I sleep all day as well. Unable to have these hours again I feel as if I am wasting the whole of my life.\n\nI wait in the queue for letters. There is one for me from Gertrude, it is the longest letter I have ever had from anyone. I read it quickly \u2013 wondering what she will say in answer to mine.\n\n_I am ever so grateful,_ she writes, _for the fowl pen cleaning out but I feel very guilty as well. I am very concerned about your health, I will own up, I know you are a lot thinner than you should be it gives you a Ethereal Look and you should have the Perfect Health Look and not be overburdened with life. I should like you to have a few weeks out at grass_...\n\nAs I read it seems I can smell the potatoes boiling on the hob at Gertrude's Place and it is as if her voice is speaking telling me how a horse when he is overworked is put in a good meadow field, in good green pasture, where he can rest and eat and wander with other horses in safety and complete freedom. She writes:\n\n_I should like you to have a time of no worry and no burden. I am thinking such a lot about you and want you here so I can get you better. I could clear out the other room and make a nice bed where you could sleep quiet and comfortable. Of course I know I can want this for you and you might not want_...\n\nThe slope of grass outside the window seems very close as I read on; it is as if she is speaking;\n\n_I will say that you occupy a place in my thoughts and in my Heart and life no one else will ever fill. Chiefly I am so grateful it will always seem so good of you, a person like yourself, to be friends with someone like me. I know I have to finish in this page and not be saying more in this letter it's nearly like saying Goodbye when I don't want you to go. I just want one more word. I don't know if you know it or not but there is something about you so refined and nice. Pure Gold or should it be Diamonds._\n\n_It is lovely to know one is loved and treasured and I want to say I am so Happy to think of you being so kind and willing to know me and to come and see me and write to me. I want nothing but love and happiness for you and complete Reliance in the spirit of one who loves you from your loving friend_\n\nGertrude\n\nThough I am reading the letter in the hospital dining room it seems as if I am in the twilight in the living room at Gertrude's Place with her sitting across the table from me \u2013 the packs of old cards ready for our game. I seem to be in the silence of the late afternoon at her place. I am wishing to be there and I think how I will go on Sunday, my next day off. I will get up early and be there the whole day and examine the rented field and decide where the pig pen should be. Gertrude, in her answer to my letter, has not said anything about Magda or Dr Metcalf. It is as though they no longer exist.\n\n'Long time no see.' Lois has plonked herself in the chair opposite mine. She has spilled her tea in her saucer. 'How's tricks?' she says lighting a cigarette and squinting at me through the smoke.\n\nEvery day I re-read Gertrude's letter. I carry it with me in the pocket of my uniform and while I am working I can feel its bulkiness. Perhaps when Ramsden comes to stay she will like to come out to Gertrude's Place, not on a bicycle of course, we can take the eleven a.m. bus to Holly Bush and walk down by the spinney, it's about a mile across the fields. Ramsden will not be coming for a while. By the time she comes we might have the pig and if you have something like a pig it is nice to be able to show it to someone.\n\nAt last it is Saturday and I have the evening off before the Sunday which is my day off. When I leave the ward Dr Metcalf is waiting by the lift. He tells me that Magda has had to go off to visit her mother but the invitation to the cottage on the river still holds.\n\n'What about tonight?' he asks. 'I'll meet you at the bus stop in about half an hour. A lazy day,' he says, 'on the water tomorrow. Magda will be coming later on,' he says. 'Yes,' I tell him. 'Yes, thank you, that will be lovely.' Up in the room I share with Lois she is there, half dressed, smoking and flipping the pages of a magazine. I change and tie my hair in two bunches.\n\n'How come,' Lois asks, 'how come the red ribbons?' 'My better-half,' I say, 'my better-half likes my hair this way.'\n\n# RAMSDEN\n\nIt is just like Ramsden, I mean, it is just the kind of thing Ramsden would say. She isn't saying it to me. She is saying that Bach must have written one of the Brandenburg Concertos in such a way that when a school orchestra drags through it it does not matter.\n\nWhich Brandenburg Concerto, I want to ask. But I am not in the conversation. Staff nurse Ramsden is talking to staff nurse Pusey-Hall. From the few paces behind them I can hear and listen to their words. It is clear that they have been to a concert together. A recital or a concert.\n\nBach dragged his violins as if on purpose. Only Ramsden could say something like this. So that, however badly the school orchestra played, it would not show. Ramsden and Pusey-Hall laugh in their well-bred way along the corridor. Like me, both of them are, in fresh white aprons, going on duty. Their long lacy caps seem to me to be exceptionally delicate, white and beautiful. We are all going to different places but this first part of the corridor we all share. Their voices are rich with amusement and tender at times as if intimate. I like the idea of the tender intimacy of their voices. I quicken my pace to keep up with theirs.\n\nRamsden is on the lower ground floor, a men's surgical ward down there, officers and men, men in the main ward and officers in the small wards. She kneels, at night, in the main ward to say the Lord's Prayer and the men, for those few minutes, hold up their housey-housey and wait in silence while Ramsden says the prayer and then they go on calling the numbers of the game. I have never actually seen Ramsden kneel down in the middle of the ward. I have only heard about it but can picture it well to myself. We are all meant to do this but not many staff nurses do. So far I have never been in charge at night but I try to memorize the Lord's Prayer. _Our Father which art in Heaven_ , so that when the time comes I will be able to kneel and pray for the men.\n\nPerhaps Ramsden and Pusey-Hall have been to something in the Town Hall. Perhaps the Sunday afternoon concert. Something quite beyond my reach. What did I do on Sunday? Of course, I remember, I went to sleep. More often than not it seems that I am sleeping away my whole life.\n\n'An exploration of the twentieth-century religious belief and conviction...' Snatches of their conversation reach me still. 'A need for the heroic response...' I miss the next bit, 'the eternal suffering of...' Whose suffering? Did Ramsden say Christ?\n\nI am not at all on the same intellectual level. I think Ramsden and Pusey-Hall read widely, especially the work of philosophers, and they have been to a concert.\n\nMiss Robson, at school, had damp dark patches under her arms when she was conducting. There was one boy, the leading violin, the only one of us who could really play. White-faced, I think his name was Mottram, something like that. It does not matter.\n\nI think I know what Ramsden means about the Brandenburg Concerto. Everyone playing at their own pace, some ahead of others. It is nice to know what Ramsden means. I can imagine her at the concert. She will have seen the gleaming black piano keys thin and sleek on the white ones, and she will have seen the pianist's surprisingly short fingers. The hands dimpled and like frogs falling above and below each other along the keyboard. She will have seen his serious pouting mouth and her head will have moved slightly in time with his head movements. Perhaps it was a quartet or a trio, the players leaning towards and away from each other, the pianist breathing lightly with the sound of the cello. Then there will have been the repeats with changes from a major key to the minor and then back to the expected and hoped for major. The pianist will have flipped the pages, before the page-turner could lean forward to do it, to glance quickly at the bottom of an earlier page before bowing his head and playing softly, this time without needing to look at the music. And all the time the ardent violinist climbs with feathery notes and the cellist plucks his strings.\n\nMiss Robson, at school, had a little rod, a b\u00e2ton. She tapped it to start and to stop the orchestra. She used it to make us lift our wrists at the piano.\n\nThe pianist on the concert platform can let his wrists drop. His galloping fingers can be flat, splayed out, or arched. He can choose.\n\nRamsden and Pusey-Hall will know all this. Close behind the rustle of their uniforms my own uniform makes its little starched sounds. They have been to a concert, probably on Sunday afternoon. \n\n# RIVER SHACK\n\nI have never wanted to spread out my pages to show to anyone, not even when I have been asked to. I have never told anyone about the little mark, the little cross, at the top left of every page. I never write a page of anything without first putting this little mark, without first asking a Blessing. How can I tell anyone this?\n\n'I'll ask a Blessing,' my father always said before a meal. Because he was hungry he was always sitting down first. And, with his hand shading his eyes, he would bend his head down as if ashamed that he had started to eat before anyone else and before praying.\n\n'Always ask a Blessing,' he said, 'before you do anything, before you undertake anything and then remember, always, to pray again and offer thanks.\n\nThe little mark which is a tiny cross is on everything I write. I do not want to tell people about the little cross. People are suspicious. They suspect. They are superstitious too. They might be afraid of what they think of as the sentimental or the religious. This makes it harder to explain things. I have put the cross there for years. I never spoke of it, even in an imaginary conversation, with Ramsden. Perhaps one day it will be the right time to try to explain and, at the same time, to give up the secret of the harbour. The sky harbour, the exact place in the Brahms where the soprano sings with sustained serenity, her voice rising above a particular group of trees on a certain road known only in this pattern of events to me.\n\nIf I think now about the fowls at Gertrude's Place and if the nurse did, in fact, shut them up early that afternoon when I watched from higher up the hill leaning on my bicycle, unable to make up my mind to go down to the place where Gertrude was lying ill \u2013 if I think now about them, it is clear that, shut up early and left, it would be some time before they were let out. They would be like the prisoners in _Fidelio_ when they are brought out suddenly from the darkness into daylight, let out from the dungeon, at Leonora's request, half blind in the sunlight, groping for each other's shoulders as they try to walk round in a circle. It will be all that the hens can do if, like the prisoners, they have been confined in the dark for too long \u2013 just stumble about the field falling over each other...\n\nIf I think now of foxes it is to think more of their colouring than of their habits. Magda's hair is often the colour of fox, the golden red fox. A vixen. Gertrude would think of her as a vixen but not the vixen prowling for food for her partly weaned cubs or lying in a half circle of dust sunning herself and her offspring. Gertrude would see Magda as a huntress, perhaps hunting for herself, something stealthy and, though surrounded by people, alone.\n\nWhen I hear about Dr Metcalf's death, a death without any sense I cannot believe it but I have to. The whole hospital is talking about it. He never even got to the front. That is what they are all saying...\n\nI always lift lavatory seats and peer under them. It was Bulge, at school, who said to do this. It was when we were in the school sanatorium together, isolated with German measles. She said it was because there might be a snake under it. She said in Madagascar, when she was a little girl, her nurse there was very careful to make sure there was no snake under the seat. Bulge's real father was dead, she explained. He had been a missionary.\n\nSo now I lift the wooden seat in the lavatory behind the river shack. There is no snake. The door does not close properly so I keep my foot against it, but no one comes. It is possible to sit here and look out across the meadows. The early morning mist takes a long time to lift and disperse. And, in the evenings, long shadows lie along the grass.\n\nI was not kind to Bulge when we had German measles. When she talked to me I did not answer and I lay down with the bedclothes pulled up to my head. She was sorry for me and thought my ear ache must be bad. It was embarrassing to be ill just with Bulge.\n\nMatron seemed to like Bulge. Bulge did not seem to be getting better. She had pains in her stomach. The pains made her cry out aloud. She wanted me to call Matron...\n\nMagda's body is beautiful.\n\n'You are beautiful,' I tell Magda. She is standing naked on the river bank and Dr Metcalf is pouring buckets of water over her. She seems taller without her clothes and I am surprised at her hips. I am surprised too about the size of her breasts. She is sunburned, a lovely golden brown, all over. The bodies of rich people are always suntanned and handsome. Dr Metcalf is brown too. It is because they can be in places where they can take off all their clothes. They do not have to look out of tall windows and see the sun and not be out in it because they have to work. The black-out shutters at the hospital are put up at five o'clock blocking out the daylight and the sun. People like me are always white. Even if there is a sunny day and I can lie in the sun I simply get hot and I stay white. My face is gaunt with the dark circles of night duty for ever round my eyes.\n\nFerguson and Trent go up on the hospital roof, the flat part, one day. They do not mean to sleep but just to have an hour of summer sun up there. They do sleep and both are still red. Ferguson's blisters are only now beginning to be less painful and Trent is peeling horribly.\n\n'Nurse Ferguson and Nurse Trent,' Sister Bean said at breakfast. They both half rose from their chairs.\n\n'Yes Sister?' as if in one voice.\n\n'Matron's office nine a.m.,' Sister Bean said.\n\nWhen I tell Magda she is beautiful she laughs and says, 'Isn't she sweet, Jonty, to tell me that? I'm old Veronica dear,' she says to me, 'can't you see how I _sag_ everywhere. Look at all my saggings darling! Oh! _Quelle horreur!_ '\n\nWhen she describes her saggings she makes them sound desirable. Next to these magnificent people, when I make my comparisons, I seem to be badly made. These other people look as if they are the result of years of fine breeding. They are well-bred not only in their manners but in their bones and in their skin.\n\nDr Metcalf has his shorts on. He is bigger in his partial nakedness than in his white coat on the wards.\n\n'Strip orf darling!' Magda yells at me. 'Let Jonty shower you.' We are washing ourselves with white windsor soap at the edge of the river. The smell of the soap out of doors has a curious effect and, if I close my eyes, it is as if I am in the night nurses' bathroom at the row of small basins, fashioning Sister Bean in white windsor...\n\n'Oh isn't she shy and sweet, closing her eyes!' Magda cries. 'Take everything off Darling! Come on Jonty! Buckets for us both.'\n\nThe quiet water is disturbed momentarily by Dr Metcalf as he dives off the jetty and swims a few strokes round the slow curve of the river.\n\n'Come on!' Magda says to me. 'You have this towel. I'm ravenous. I didn't have any breakfast before I left. Let's fry up the bacon I've brought.'\n\nMagda's father and Marigold, an actress, come later in the day. Magda's father parks his car in the field opposite and blows the horn for Dr Metcalf to go for them with the boat. Magda's mother does not come. Magda was staying with her the previous night after having her tooth out.\n\nDr Metcalf and I have been in the river shack alone together all night. On the bus journey we were both so tired we hardly spoke to each other. He sat with his arm along the back of the seat. It was as if his arm was round me though I knew really it was not. I liked this very much. I keep thinking of the journey now during this quiet day on the water. 'A lazy day,' he said when he invited me, 'on the water.'\n\nThe day is anything but quiet. Magda arrives in the mist. Her taxi driver gives a shout from the towing path and Dr Metcalf rows across the secret water of the river to fetch her. I only wake up when the driver shouts. I feel all big and white-faced and puffy and ugly. I help them carry all the packages and parcels into the cottage. If only I could be a bit suntanned or else wake up pretty or dainty and not as I am in my slept-in clothes.\n\nSome of the young men, Magda announces, are coming in a car later. She has masses of food she says. She has managed to get butter and cheese and real coffee and champagne and dried bananas. I am ashamed to be hungry. We sit on the jetty to eat our breakfast. I tell Magda I might be able to get her some eggs.\n\n'There's a path up from the road but it's hidden in the long grass,' I start to tell Magda about Gertrude's Place.\n\n'But Darling,' she cries, 'how divine! Yo u must tell me how to get there.'\n\nMy evening with Dr Metcalf was over very quickly. During the walk from the bus stop, across the river meadows, to where the boat was moored under the bank Dr Metcalf was very courteous. He guided me from one firm patch of grass to the next. Enormous cream-coloured cows, their moist breath sweet with chewed grass, gathered near the path waiting to be herded for milking. He showed me the room in the river shack where I would sleep. Because the small house seemed to sit in a bed of grass, and because it was so quiet, I was reminded of Gertrude's Place. The house, like her place, seemed to be asleep.\n\n'I never heard the river in the night,' I say. 'I had hoped,' I tell Magda, 'to hear the water slapping against the jetty boards, but I must have gone to sleep without hearing anything.' While we sit there, with our feet in the brown water, I think of the evening and of the way in which Dr Metcalf sat close to me on an old sofa covered with an unhemmed cloth. He said that he was older than I was and that I had all my life before me. He said he knew this but all the same he wanted to kiss me. Could he kiss me, he asked. I said I thought people kissed each other without asking. This seemed to please him and he kissed me very long and very sweet kisses. I think of these kisses all the time now and I wonder if Magda guesses and, if she does, whether she minds.\n\nDr Metcalf, after a bit, took me to the small bedroom which he said was mine. The bed was very low, he sat on the edge of the bed and held out his arms. I remember his kisses and I remember how he held me and covered me up. I must have gone to sleep without undressing properly.\n\n'I never heard the river in the night,' I say, 'and I never saw the sunrise.' Once again I feel that, because of my work, I am wasting my whole life sleeping it all away, waking up all pale and ugly and not able to have those hours back. Dr Metcalf says it was a lovely sunrise and if I missed it, never mind, there will be plenty of other sun risings in my life. He smiles at me and my mouth longs for his.\n\nMagda looks at me seriously and tells me that Jonty is right of course.\n\nMagda's father and his actress arrive early. Even though not one of these people is in any way like Bulge and her mother I am suddenly remembering Bulge. Perhaps it is the outside lavatory, the earth closet with the wooden seat which I feel I must raise to make sure there is no snake.\n\nWhen we were ill together, and isolated, Bulge showed me a picnic photograph of her mother and this person she called Chris.\n\n'Is he your father?' I asked and Bulge explained that Chris was like a father but he was her mother's Friend. Her father died, she reminded me, while he was looking after sick people in an African village. Because of the boredom of being ill with only Bulge for company, and a few damp old books we found in the chimney cupboard, I stared for a long time at the picnic photograph. I tried, in the presence of _Clive of India_ , _Bevis_ and _Nineteenth Century English Gardens_ to make a book out of the photograph. I peered at every detail, the tartan travelling rug on which Bulge's mother was sitting and the kind of cake she had in an open tin on her lap. She was handing cake to Chris. The idea of cake made me hungry.\n\n'What sort of cake was it?' I asked Bulge.\n\n'I can't remember,' she said. 'Perhaps Madeira,' she said. 'Chris likes that.' We were hungry all the time, both of us Bulge and me. In the photo Chris looked tall. He wore plus fours. He was crouching fondling a little white dog. Bulge explained it was a Sealyham. Her mother always had a dog and it was always that sort.\n\nPerhaps it is this picnic lunch which reminds me. If I compare my mother with Bulge's mother it is clear that my mother is below Bulge's mother in the comparison. My mother has never had a pet of any kind. When I consider this I have to realize that she does not like dogs at all and would be disgusted if she knew of Magda's habit of having the Red Setter in bed with her.\n\nPerhaps it is the tartan rug which reminds me so unexpectedly of Bulge's picnic photograph. Because of being above Bulge in my comparisons at school, her hair, her cracked spectacles, her complexion and the way in which she stood, bulging and biting her nails, everything being worse, I do not especially like thinking about her. This thinking puts us on the same level. I try not to think about her. Magda's father is very fond of his actress and touches her often. He is fond of Magda too and strokes her arms and gives her little hugs. For a long time he does not seem to notice me at all and then says he has heard that I am a Quaker and I say not a very good one I'm afraid and he laughs, throwing his head back, as he does in Theatre when someone makes a joke during a partial gastrectomy, saying that's the best answer he ever heard and then goes back to not noticing me. I can't help dwelling on my good answer and I wish I could say something else which would be a good answer. No one really addresses any other remarks to me and, as I am hungry, as I always am, I eat a lot of the ham and the butter and the dried bananas. Food rationing does not affect Magda and the food she has brought is quite unlike what we are able to have at home or in the hospital. Magda's father has brought a paper bag full of peaches. He has them sent from London. I have never eaten a peach and concentrate my thoughts on hoping there will be enough in the bag for us all to have one.\n\nMagda's three expected guests, the young men who come to her house parties, do not come and she is disappointed but thinks they might come the next day.\n\nCows have trodden down the river bank but Magda thinks it would be heavenly to sleep out on the grass. Her father and Goldy can have the bedroom in the shack. Magda arranges everything.\n\n'There'll be a huge moon,' she promises me, 'and Jonty will find us a clean patch of grass.' We carry our bedclothes out to the edge of the river.\n\nI feel I should be happy listening to the river slapping gently underneath the weathered boards of the jetty. I am, at last, where I have wished to be. I breathe the river water and river mud smell and the fragrance of the crushed grass but it does not take away the unexpected sadness. For a time the moon is bright and the water, not shadowed by the banks and the trees, shines. From time to time something plops, with a small splash, into the water. I suppose it is a water rat. I am wishing for Dr Metcalf, to be alone with him, in his arms, inside the blankets which are, like mine, rolled all round him. We are, all three, rolled-up bundles in a row, our feet down towards the water and our faces up to the moon. This is the same moon my father can see. In my loneliness now I try to think of my father and his moon but it is Dr Metcalf I want. I am thinking of him and wishing for his arms and his kisses. I want to feel him close to me again as he was last night. I wish I had not slept last night.\n\nSometimes a solitary boat drifts by. Though I can't see them I know there are two people in each boat, unwilling to stop being together and not wanting the night to end. The moon has a ring of light around it. Dark clouds hurry across the bright face of the moon. Magda and Dr Metcalf who were saying soft words, now and then to each other (they thought I was asleep), are quiet now. Perhaps I am the only person in all the miles of country not asleep. The cows, dark shapes on the other bank, move together. I can smell their grassy breath and hear when one of them lowers herself or gets up. I am glad they are not asleep.\n\nThere is a distant sound of aeroplanes. The engines throb as they come nearer. Like a heartbeat, on and on, coming nearer. They are German planes. I know this because of the throbbing sound of the engines. Far away across the water meadows pale search lights send their thin fingers, like long petals, across the sky. Magda says should we go inside and Dr Metcalf says no, what difference would it make.\n\nBulge, when we were in isolation together, said that the Germans, if they came with the aeroplanes, if it came to a war between us and the Germans, the Germans could wipe out Britain with one raid. At the time I thought that the Chris gentleman must have told her that. I believed it. Later we all said it at school. 'One raid and we'll all be wiped out. Really! The war, if it comes to that!' The Germans are not all that bad I wanted to say then. Not all Germans are bad. There did not seem, then, the words for this to be said.\n\nBulge's mother and Chris visited Bulge during our illness. They were so nice to me. Friendly and sitting on my bed and talking to me as if I had never been unkind to Bulge, ever. Bulge of course could have been as homesick as I was. Muriel, they called her Muriel, could have longed for her mother and this Chris. Her mother told me, 'Muriel has some rabbits at home too,' when I told her about our rabbits at home. Bulge was so nice to me. She never said one word about my not speaking to her. And she obviously had not said anything about the raggings she had from everyone. On the afternoon when her people came I hadn't got my spectacles on and, when her mother put her head round the door, I thought, for a moment, that my mother had come. I hid under the bedclothes when I realized my mistake. It was hard not to cry. It was then that this Chris gentleman sat on my bed asking me about my bandage while Bulge and her mother had their first hugs. I told him, 'It's a middle-ear abscess,' and he said he was sorry.\n\nThat night Bulge called out to me to fetch Matron because her pain was bad. 'I think it's my appendix,' she said, and was sick over her bedspread and the floor. I went down quickly, in the dark, to Matron's cottage. She came at once.\n\nIn my secret game of comparisons Bulge was placed high up, far higher than anyone else, for she had the school doctor at her bed in the night. I was envious and felt ashamed of being envious and tried to be helpful. Bulge was trying not to cry but her pain was too bad. Matron said to me to lie down and to try to go to sleep. She said they would have to take Bulge to hospital. They wrapped her in a blanket and took her in the doctor's car to Oxford to the hospital before it was too late. The tall grasses and the cow parsley along the lanes, I thought, would look like lace in the moonlight.\n\nMatron, in her haste, left the light on and I tried to read the book Bulge's mother gave me, as a present, before she left. It was a school story about a girl called Treacle. The book was called _Treacle Wins Through_. I wanted to enjoy the book but I could not help thinking about Bulge crying and drawing up her legs because of her pain.\n\nMagda's young men, the three expected guests, do not come on the next day either.\n\nDr Metcalf and I go for a short walk to the farm nearby for some fresh milk. We do not have to cross the river.\n\n'I was hoping to take you upstream,' he says to me, 'in the boat. There are wild swans there, further up...' I thank him and say in a small voice that it doesn't matter. On the way back he sets down the milk can and draws me to the shelter of a hedge. Quickly he holds me close and kisses me and then holds me away from himself.\n\n'I can't give you all the love you ought to have,' he says and he kisses my fingers, very lightly, brushing them with his lips.\n\n'That's all right Dr Metcalf,' I say, hearing my own voice and words with surprise. I tell him that it's all right, that I've got a boy friend in the forces. In the air force to be exact. We walk on back to the river shack.\n\nAll day I am wishing that I had gone home and gone to Gertrude's Place. There was no way in which I could let her know I was not coming. She is probably walking down through the long grass to look down the long hill, as far as the bend, to see if I am coming up, walking, leaning on my bicycle. Two whole days off wasted. I thought I could be happy just being near Dr Metcalf but it is not like that. I can't even look at him as I want to when other people are there. And he can't look at me, not into my eyes and my thoughts like the evening when we were down here alone. I want to be with him by myself. My mother, it consoles me to think this, might have gone on the bus to fetch the eggs and will have told Gertrude I am not coming.\n\nDr Metcalf does not eat lunch. He seems thoughtful. I am afraid he might be depressed and sad and I try not to show that I am. Magda is decidedly peeved. She says so herself. 'God I'm peeved!' she keeps saying it. The river is crowded with Sunday train-excursion visitors. Trippers, Magda calls them. She is obliged to put on clothes as so many boats are passing the little jetty. Young men, performing antics with punt poles, whistle and call out as they pass. Magda says she's had enough of the shack and we'll all go back to town and do I want to visit my parents because if so Daddy, as soon as he and Marigold are up and dressed, is going back to town and will give us all a lift and can drop me off at the bus station.\n\nIt is late when I get home after waiting hours for a bus and a train. As soon as I open the kitchen door my mother hands me a telegram. I have to return to the hospital at once.\n\n'The telegram came yesterday,' she tells me.\n\n'I would only have been able to stay for the evening,' I tell her. \"I've had my days off.'\n\nMy father says he will come to the station with me. We have to leave at once to be sure to get the train. He carries the small parcel my mother has made. It is a fruitcake she has packed and some hard-boiled eggs.\n\n'The eggs are from Gertrude,' she tells me.\n\nLois has washed her hair. It amuses me to see the turban she has made with the towel. I tell her that her turban is delightful and that I have brought some cake.\n\n'We're on at midnight,' she says, 'a special shift.' She does not explain further. 'You're daft!' she says suddenly, 'going with Them! She's as bad as he is. Can't you see? Those Metcalfs! No I don't want any cake.'\n\n'But you don't know them. They're sweet, both of them and very kind.' I eat a piece of the cake.\n\n'Oh yes!' Lois says, and then with an accent, 'Oh! Yeah?'\n\n'They're my friends,' I say.\n\n'Some friends!' She lights a cigarette from the new packet I have brought for her. 'He's the reason for Smithers suiciding,' she says, 'it was because of Dr Metcalf.'\n\nSmithers, the theatre orderly, I remember him reading a poem he'd written. He asked me if he could read it. And then he asked me what I thought about it. He was very tall and pale because of always being indoors and in the artificial light of the theatres.\n\nBecause of the charge nurse coming in just then I was not able to listen properly. Smithers went on putting drums of sterile gauze on to the shelves and I went on handing them to him. I had not been able to understand the poem and, because the charge nurse did not leave, Smithers was not able to repeat his question.\n\nI remember the poem and I remember his suicide.\n\n'But Smithers!' I say.\n\n'Exactly,' Lois says.\n\n# SINGLE MALT\n\nSuddenly I am reminded of my mother. Perhaps it is because today, in the distance, I saw a woman with her hair curled as my mother's used to be. This woman was standing with some other people further along the street. Because I knew it could not be her I did not go up there. However much a person resembles another person, and it is not that person, it is not of any use.\n\nPerhaps I should, at some time, write down every single thing which I remember about my mother. Perhaps that is something I could do.\n\nMy mother, who always needed someone to tell things to, suddenly, after the death of my father, had no one. After three stormy weeks in a private nursing home she returned home to an empty house and had no one to tell her dreadful experience to.\n\nShe had to live several years alone.\n\n... _as he is now made partaker of the death of thy Son, so he may be also of his resurrection_...\n\nIt is my evening off but I am not going out anywhere. A baby died today.\n\nA baby died in my arms today.\n\nThe book I choose when staff nurse Ramsden says to me to choose any book I would like to have from her shelves is a foolish choice for me. I can never say this to anyone. My mother is fond of embroidery and would understand all the diagrams and the pictures in the book. She would understand the step-by-step instructions and descriptions; mount mellick, crewel work, Berlin embroidery, tapestry, gros point, petit point, ribbon work and needle painting, bead work, black work and all the stitches and techniques from the elegant oriental Tambour... I like the idea of the needle painting. Of course I can't give the book to my mother. It is a present from Ramsden and it is my fault that I chose it instead of some other book. I mean she had Wordsworth there and Keats, Goethe, Rilke and ee cummings and Dickens and others. That is the worst of being asked to choose.\n\nIt does not seem possible really, this evening, to go and see Magda. During the car ride after being at the river shack they all seemed to argue, in fun but not quite in fun. Dr Metcalf laughed about Magda's single-malt gentlemen not turning up. 'They knew all the single malt was gone,' he teased her.\n\n'They never drink all the single malt,' Magda said. 'You,' she said to Dr Metcalf, 'you always have your tots and so does Daddy. Don't you Daddy?' I thought Magda might be going to cry but Dr Metcalf was stroking her arm and I saw all the little lines and frowns disappear from her forehead and round her eyes.\n\nI watched the hedges slipping by, the cow parsley tall and lacy all along the lanes. I kept wishing the bus station was not so far away.\n\nMagda was sitting between us. Dr Metcalf's arm was round the back of Magda and his hand, stroking her soft sun-burned arm, was very close to my arm. The back of his hand was against my arm. I told myself to bury myself then in the beauties of nature for ever. But then, like now, the phrase, the beauties of nature, had no meaning.\n\nMagda's father, driving with a well-bred ease, had one arm round Marigold.\n\nThis evening I am putting my stamp collection in order, at least I am trying to. Some of them have come loose and have fallen out. I am unable to be interested enough to sort and arrange them properly. My head itches. I would like to brush my hair but Lois, who has gone home for her day off, has borrowed my hairbrush. I open my exercise books of pressed wildflowers, meadow sweet, saxifrage, coltsfoot, lady's slipper and star of Bethlehem. There is nothing I can do with them and, somehow, thinking about the grassy places where they grew makes me sad.\n\nIt does not seem possible to go and see Magda today, this evening. Not after the river shack. I wonder what they are doing tonight, Magda and all of them and Dr Metcalf.\n\nI put my clothes drawer straight, fold and tidy everything very neatly, and make up my mind to always keep my white blouse and my good pair of stockings clean, in readiness, in case I get invited somewhere. That is the kind of person I am.\n\nMagda has not left a message telling me to come, as she usually does. Perhaps something has happened over there, at their place, since the river shack. She did have a tooth out. Perhaps an infection?\n\nMy stockings, my good pair, are really quite nice. It must be really special to have a man roll your stockings with nimble fingers so that when he puts them on your feet they unroll delicately and smoothly all the way up your legs. Once, in a film, I saw a handsome man kneeling in front of a very pretty woman and, this man, he could do stockings like this. I don't know many men and those I do know, for example, my father and Dr Metcalf, I don't think they would ever do stockings. Though, perhaps Dr Metcalf might.\n\nThe baby, the smallest baby I have ever seen, is called Roger Keith. He died in my arms. This morning.\n\n'Nurse!' the charge nurse calls to me as I pass the end of Obstetrics. Ward 4. I am only on that corridor because the lift isn't working. 'Have you got a clean apron on?' she bawls. I shout back that it isn't very clean and that I'm on my way to Pharmacy.\n\n'Is there something you need from Pharmacy?' I go towards her. 'They forgot our carbolic.'\n\n'Take it off,' the charge nurse says, 'your apron, take it off and put it on inside out. And be quick and come along in here. Put your ration jars down there, yes, just down there by the door. That's it, turn your apron, quick as you can. Look sharp!'\n\nThe charge nurse is not one that I know. I don't even know her name.\n\n'I'm not on this ward,' I begin to tell her.\n\n'I know. I know that,' she says. 'It's an emergency,' she tells me. 'I haven't a single nurse free, we're flat out, three heads showing and five just post-natal and now this. Only keep you a minute. Quick as you can nurse, there's a good girl, you've got your sleeves down and your cuffs. Good!'\n\nPropped up in the bed in the screened-off corner of the ward is a woman. Her hair is brushed back and tied up neatly with a piece of cotton bandage. She is very pale and she is weeping. She is crying without any sound, her tears are overflowing as if straight from her heart. She is weeping as if she will never be able to stop. The charge nurse goes over to her at once.\n\n'The chaplain is coming at once,' she says in a low voice. The woman nods and still her tears pour down her cheeks.\n\nAt the side of the bed is a hospital cot, a little wire basket, covered with some folded pieces of flannel. The charge nurse picks up the quiet baby.\n\n'Caesarean,' she says to me out of the side of her mouth. 'Nurse here,' she turns to the woman in the bed, 'nurse will be Godmother.'\n\n'But,' I say, 'but I'm not a... I'm a... we don't...'\n\n'Never mind, nurse, whatever you are or whatever you do or don't do. Here's the Reverend himself.' Swiftly she wraps the baby in a white cloth and gives him to me.\n\nThe tiny bundle is light in my arms. His eyes are closed and his little mouth is puckered. Already the finely made delicate lips are blue. As I hold him close I feel his tiny body make a feeble movement.\n\nThe chaplain bends his white head over his book. He moves closer as he reads. He asks me to name the baby. I glance at the woman. 'Roger Keith,' she hardly moves her lips.\n\n'Roger Keith,' I say, holding the baby towards the drops of water which fall like cold tears from the chaplain's fingers.\n\nRoger Keith.\n\n' _I baptize thee in the Name of the Father, and of the Son, and of the Holy Ghost._ '\n\n'Amen,' the charge nurse kneels, so I kneel with the baby. The chaplain's words rush on:\n\n'... _And we humbly beseech thee to grant, that as he is now made partaker of the death of thy Son, so he may be also of his resurrection_...'\n\nThe small rustling sigh which was Roger Keith's breathing has stopped. The charge nurse takes him from me and replaces him gently in his cot. The woman in the bed holds out her hands to me and I feel their hot dryness. Her tears, like shining beads, force their way still from under her closed eyelids.\n\n'Thank you nurse,' the charge nurse has escorted the chaplain to the ward doors. 'Whyever,' she asks, 'whyever are you taking your butter and jam and sugar down to the Pharmacy?'\n\n'It's my morning-tea break,' I say. 'I'm to go to Pharmacy on the way back.'\n\nThe lift is working now and I go down to the basement in the company of a long trolley rattling a dozen chipped enamelled cans. The cans all have lids and they are all marked with numbers to show which wards they belong to.\n\n'Sweet pees,' the porter says to me. He lights a cigarette. Porters, with trolleys and dustbins and theatre bins and laundry and these smaller cans, often smoke. They say they are allowed to because of the nature of their work. He is taking the urine from patients, soldiers, who have been having penicillin, to the lab. The penicillin will be extracted and used again. He tells me this and I let him even though I already know it.\n\n'It's a miracle,' the porter says.\n\n'Yes, it is,' I say and I almost ask him what kind of God there could be who would receive a child over whom the right words had been said and whether this same God would reject, really refuse, one who lacked this charm.\n\n'Ladies hats underwear and dresses,' the porter opens the lift to a row of bins. Pharmacy is just to the right and the Laboratory a bit further on. We go along the corridor unable to speak because of the rattling of the cans.\n\nI suppose they will be having dinner now at Magda's, perhaps something which can be eaten on cushions on the floor.\n\n'Hobbling along supported by acadeemia.' Magda, frowning and weary, once more, in the car during the drive from the river shack, accused Dr Metcalf, talking about someone, a friend of his, a one-time friend of his, someone I have never met. 'She's done enough,' Magda said, 'to make herself a footnote in the literary history of this country.'\n\nDr Metcalf laughed at this, at Magda's supposed anger. He raised his eyebrows in an amused look at me across the back of Magda's head. He pursed his lips at me as if to blow a little kiss in my direction.\n\n'That's not true, is it?' His tone to me was playful.\n\n'I'm afraid I don't know her,' I said and I turned to stare harder into the hedges. Wild roses I wanted to say to them. Look you, dog roses, hundreds of little wild roses all along the hedges.\n\n'Darling!' Magda said then. 'Of course you don't know. Jonty had a Friend, with a capital F, who used to come. She said she actually thought our house was a sort of religious retreat. Can you imagine! A past pillar of the Ballet Rambert she wasn't slow on the single malt either. She ended up, my deah, in the school of stitchery and lace, an adventitious ornament, no doubt, with her big hands and feet.'\n\nThey all laughed then. I couldn't ask what single malt meant. What it was. Something hard to get, I supposed, even for Magda for whom rationing did not seem to exist. Something hard to get so that it mattered if their friends were not slow on it.\n\nBecause I could not understand and because I did not understand the metaphor, if it was a metaphor, the school of stitchery and lace, I turned away to stare once more at the roadside and at the trees in the hedgerows.\n\n'Darling Child!' Magda cried. 'Jonty!' she said, 'I do believe our Precious Child is jealous. Oh you sweet Darling!' She broke loose from Dr Metcalf's arm and drew me to her sweet-smelling soft breasts and kissed me hard on the mouth.\n\n'You are so sweet,' she purred and continued to hug me and call me sweet for the rest of the journey. So it is strange, very strange, that I have not heard from her. \n\n# DIET KITCHEN\n\n'He didn't go, he didn't really go to the Front at all he shot his knee caps off and bled to death at his mother's place.'\n\n'Disgusting.'\n\n'I didn't know he had a mother.'\n\n'Yes, the poor thing \u2013 even though she's a prostitute.'\n\n'She must be quite old, I mean, if she's his mother.'\n\n'She is. But some men, they like Experience. Experience counts.'\n\n'That's right. Doesn't matter what sort of old bag. I mean looks just aren't in it. A man who's never done it doesn't go for looks, all he wants is a friendly vagina.'\n\n'I suppose he was illegitimate. Anyone got any butter left?'\n\n'A bastard. Hm!'\n\n'Yes a bass-tarrdd.'\n\n'Anybody got a ciggy?' Lois brings her coffee to the table. She is on night duty too. I slip the new packet from my pocket on to her lap. We always seem to be having our meal times at the same time. Without seeming to notice the _State Express_ Lois does notice. She clears a corner of the table for herself and opens the packet.\n\n'How clean you are like a spring flower, a snowdrop,' Lois said to me once. It was as if we were in another world then. We were walking in the rain and we stopped suddenly, looking at each other laughing.\n\n'Your hair!' she said then, 'it's soaked but you look so nice, clean, green and white, they're your colours, a snowdrop. I'd like to paint you. Lois painted, water colours, in the room we shared. We pinned up the paintings even after the Home Sister left a note telling us to take them down.\n\n'Your tartan dress,' I said, 'it's so funny. I like your clothes, they make me laugh. Your hair,' I said, 'your hair's wet too. This rain!' The unexpected rain made the tartan wet so that it was smooth round her breasts. 'Very sweet you are. Did you know you are very sweet!' I said then.\n\n'Well of course if it's Dr Metcalf you're talking about.' Lois sends a cloud of smoke across the shepherd's pie and the baked apples.\n\nIt is Dr Metcalf. Everyone's talking about him, everyone, that is, except me.\n\n'They say he never got to the Front. He was crushed behind a lorry, an army lorry reversing. And the war practically over too.'\n\n'They say he was at the camp near Swindon. Ever heard Of Swindon? Whatever made him join up now. Never got to any action. I mean if there is any now. It's too late. I suppose that's why. Pass the salt please.'\n\n'Did you know he was on morphia? Used to come to Lower Ground Radium for it and whatsaname, heroin. We used to hide the keys. On nights, you know, we hid the keys.'\n\n'Remember Foss? Wasn't Foss on Lower Ground Radium?'\n\n'Yes, Foss was on there then. Charge on nights. Used to hide the keys in her bra. Ugh! This apple's sour! He got them, the keys, one night. When she was doing the linen cupboard. Put his hand right down her bra. Disgusting!'\n\n'He didn't need to go. Have some of my sugar, I don't use it all. He only joined up to get away from his wife.'\n\n'And other people. Thanks, anyone else going to borrow some sugar? And other people. He's made eleven nurses pregnant.'\n\n'Could be twelve.' Lois hardly looks up as she speaks.\n\n'His wife's twenty years older than him. Or is it thirty?'\n\n'He's got nine children all with different mothers.'\n\n'No wonder he wanted to get right away.'\n\n'And some there be who've got VD!'\n\n'He did have VD, you know, and must've passed it on to goodness knows how many people.'\n\n'Should have gone to the clinic.'\n\n'He did under an assumed name but quite a few people recognized him of course.'\n\n'His wife's riddled with it.'\n\n'Not surprising.'\n\n'He wasn't a qualified doctor at all they've found out. He was just one big fraud. He was a greengrocer really with this prostitute of a mother. She worked in the shop and took the men upstairs.'\n\n'That's right! No papers.'\n\n'They found papers on him but they must've been stolen. He had Chatwyn Brown's papers.'\n\n'Chatters Brown? But he was reported missing ages ago! Poor old Chatters!'\n\n'Yes. Well, Metcalf had _his_ papers \u2013 on him!'\n\n'They found a dead German's finger in his pocket, too, with a big gold ring on it. Couldn't get the ring off.'\n\n'I thought he didn't get to the Front.'\n\n'He didn't. Probably stole the finger from someone. Had to take the whole finger to get the ring.'\n\n'Disgusting!'\n\n'Yes, disgusting.'\n\n'What d'you expect!'\n\n'Sister Whatsaname on chests suicided because of him.'\n\n'Yes, and that's why Roberts ran away. Remember little Nurse Roberts? Ever such a quiet little person. Probably dropped her bundle by now. Anyone know what she got?'\n\n'Twins. Could be twins, you know.'\n\n'Could be infected too.'\n\n'Yes could be. Would be tertiary, don't you see.'\n\n'Roberts hasn't gone yet. She's cutting it a bit fine. I mean it's obvious isn't it. I mean she's _showing_.'\n\n'And Smithers. Remember Smithers?' Lois looks for matches for her second cigarette. She glances at me through her cloud of smoke. Her eyes glitter beneath partly lowered lids. She looks away quickly.\n\n'Smithers,' she says, 'on theatre, he suicided. Remember? Lemmington Frazier's rectal orderly. He suicided because of Metcalf.'\n\n'Lemmington Frazier! That's Metcalf 's father-in-law surely.'\n\n'Yes, he's on his tenth actress. Marigold Bray.'\n\n'You nursed her mother on Women's Surgical didn't you?' Lois inhales deeply, holds the smoke and lets it out across the table. 'Didn't you?' she says to me.\n\n'Yes,' I say. 'I did, Mrs Bray, a hernia.'\n\nI often think of Mrs Bray. I remember her telling me that she worked at the public baths. I remember all too clearly that I cried by her bed, behind the screens, the day Dr Metcalf told me he was leaving for the Front.\n\n'I don't want to lose you, I told him when he said he had to go. 'I'm frightened,' I said.\n\n'Don't cry, please don't cry,' he said.\n\n'Don't you cry, dear,' Mrs Bray said, 'he'll come back. Your boy'll come back.' Mrs Bray said something else too that day. She said a person has to love work. You have to love your work. She loved hers, she said, at the public baths.\n\nI told her I had met Marigold at the river shack. 'You know,' I said to her, 'Dr Metcalf's place on the river.'\n\n'Oh yes, Edna, my daughter,' she said, 'only she don't call herself Edna any more. I'm hoping,' she said, 'as she'll come back and see me one of these days. I just have the one girl. That's what I mean about work. See? Enjoying your work makes you enjoy your life. Helps you to forget things as go wrong.'\n\nI am thinking now about Mrs Bray's thin, sad face and how her eyes brightened when she talked about Marigold. I would like to talk to someone about Dr Metcalf.\n\n'Mrs Lemmington Frazier,' Lois stabs out her third cigarette. 'Sweet. She's really sweet. Nursed her hysterectomy.'\n\n'Lemmington Frazier! Gives me the shivers. Dirty old man!'\n\n'She had her veins done too. Mrs Lemmington Frazier.'\n\n'Yes that's right, she did.'\n\n'Mrs Lemmington Frazier used to come round the wards with that Red Cross trolley. Remember? Library books and magazines and writing paper.'\n\n'Yes and home-made face flannels and jam.'\n\n'She used to get chocolate and cigarettes for the men.'\n\n'The officers you mean.'\n\n'No, the men. She went round them all.'\n\n'Her daughter, Mrs Metcalf, used to go too.'\n\n'Never!'\n\n'Yes she did, all tarted up to kill. You can just see it next to the Lemmington Frazier tweeds! A sort of heather mixture.'\n\n'And the lilac twin set.'\n\n'And the pearls.'\n\n'And the lisle stockings.'\n\n'But really Mrs Lemmington Frazier has very good taste!'\n\n'Yes, if you like Henry Heath hats.'\n\n'It's supposed to be good for a doctor's career if he can get married to a surgeon's daughter. Specially an only one. Promotion eh? Luxurious! Straight to the top!'\n\n'That's right.'\n\n'Wasn't Metcalf Lemmington Frazier's dresser?'\n\n'Yes he was, that's right.'\n\n'Lemmington Frazier's daughter! She used to call him Daddy and there she was in theatre, all gowned up, holding the artery forceps or a retractor. Supposed to be studying. I ask you! And nothing on under her gown. You could see _everything_.'\n\n'Yes. She used to leave when Daddy did when the dresser was sewing up. Other students had to stay.'\n\n'Must've waited somewhere for Metcalf, then.'\n\n'He probably ran out after her.'\n\n'But what about Smithers? Haven't seen him for a while, come to think of it.'\n\n'Yes, Smithers.'\n\n'Smithers? Smithers? Was he that tall one? The thin pale willowy one? Used to be the shave orderly?'\n\n'You remember. Looked like he'd been in the sterilizer all his life. Steamed. That awful white skin!'\n\n'Yes, of course, Smithers. Suicided because of Metcalf? But he was a _man_.'\n\n'Exactly.' Lois says. 'That's exactly it. He was a man.'\n\nThe cigarette smoke is worse as some of the others are smoking now. I feel sick and am glad that the meal time is over.\n\n'Lois,' I said once, 'Ferguson doesn't seem to like our being together so much. I think she feels left out. Could we?'\n\n'Ferguson? Ferguson? Who is Ferguson?'\n\n'You know Ferguson,' I said. 'I was sharing a room with her till we requested the swap. I think it hurt her that I wanted to share with you. It would have been easier for her if I'd asked to go back to having a single room. I was at school. We were at school together. You see? Could we include her...?'\n\n'Well, you're not at school now,' Lois said. Trent heaved herself off my bed then and walked, half dressed, round the room on flat feet.\n\n'Quack Quack and Quack. Quackitty Quackitty Quack Quack.' We fell in a heap on Lois's bed.\n\n'Your breasts,' I said to Lois later when Trent had gone, 'are indescribably soft.'\n\n'I know,' Lois said, 'Matron says I must get something done about them.'\n\n'Trent really can do ducks,' I said, 'she really can do ducks.'\n\n'She really can,' Lois said.\n\n'Ramsden,' I say making an effort to keep my voice level, 'my folks would be very happy if you would visit. Ramsden,' I say, 'when you have been to London, I mean, when you have had your holiday, I mean, I shall have my holidays then.'\n\nI renew my invitation to Ramsden to come and stay for a few days. Staff nurse Ramsden. We are in the lower ground corridor going in opposite directions. I have had my meal and she is going to hers. I have never worked with Ramsden. The others say she is very nice to work with. She is on Lower Ground Radium.\n\nRamsden says thank you and accepts. She seems shy. She often seems shy. It is then that she gives me the poems.\n\n'I have no right to give you these,' she says,' but here they are anyway.'\n\n'Thank you,' I say, 'thank you very much.' I hold the little book carefully in both hands, with both hands together as if for a prayer.\n\n'Look at them, if you care to, some time,' Ramsden says, her eyes darker because she is shy. 'When you have a minute. No!' she puts her hands over mine, 'there isn't time now. Put them in your pocket. Put them away.'\n\nWe both have to hurry.\n\n'There is never time,' she says. She explains that she has had to leave the Junior in charge as her second-year nurse had to go off. 'She was really not well enough to be on duty.' Ramsden has a reputation for thoughtfulness.\n\nWherever in my mother's house can staff nurse Ramsden sleep?\n\n_The feathers of the willow_\n\n_Are half of them grown yellow_\n\n_Above the swelling stream;_\n\nI do look quickly into Ramsden's little book. It is all written in her neat small handwriting. Some of the poems are her own and some she has chosen.\n\nThe wireless is still on in the Lower Ground Men's Surgical. It is an offcers' ward and the ordinary hospital rules do not apply. The lights are all on still and the officers are not even in their pyjamas yet. Some are playing cards and others are sitting in the ward office with the Charge Nurse and her Junior. All very casual. They seem to be mostly convalescents. The wireless is loud, 'In the Mood' is on. The lines of the poem seem to fit this music as it goes on and on;\n\n_And ragged are the bushes,_\n\n_and rusty now the rushes,_\n\n_and wild the clouded gleam._\n\nThe words in my head are in time to the music. I even seem to walk in time to it. This music 'In the Mood' is incredibly vulgar accepting, as it seems to do, entirely unacceptable vulgarity. I wish the poem did not fit and that I could walk to a different rhythm.\n\nI wonder what other poems Ramsden has chosen. The Lower Grounds corridor is dark beyond the ward. The poem is about the autumn. It's the beginning of the autumn.\n\nThis summer belongs to us you said. You told me not to call you Dr Metcalf. 'Jonathon,' you said. I was to call you Jonathon. It was hard for me to change. Dr Metcalf I called your name and you said, 'Jonathon, remember? Especially when we are loving each other. How can I be doctor?' you said. 'Jonathon,' I said.\n\nFor a few minutes just now in the corridor while I looked at Ramsden's poems I forgot, for just those few minutes what has happened to Dr Metcalf.\n\nThe diet kitchen is awful. I can't stand the diet kitchen. It is on the Lower Ground corridor further on from Radium Therapy and the Offcers' Ward. It is a basement really and is vaulted. It is not well lit at night. I suppose to save electricity.\n\nI am in the diet kitchen. All round me are the horrible little trays and their food labels. I hate cutting and weighing pieces of bread. I can't stand the smell of the vitamin B extract. The smell of tripe stewing is as bad as the smell of boiling beetroot.\n\nI am in the diet kitchen all alone. This is the place where Nurse Roberts, on day duty, can't stand the sight of the dried egg powder, not even the packets of it. A kitchen boy comes in with fuel for the boiler and the stove. He comes twice during the night. He works in the main kitchens where there are quite a few people including a crippled cook who can't talk. Trent told me once that she has to work at night and that she hides during the day because people are afraid when they see her. At least there is company in the main kitchens but nurses do not work there.\n\nThe kitchen boy is waiting. He stands very close. I can see his ginger eyelashes, each one individually.\n\n'Well?' I say sharply.' What d'you want? You've stoked the boiler.'\n\n'Is it God's honour truth,' he asks me, moving closer so that I can smell hard-boiled eggs on his breath. 'Is it true,' he grins, shaking his head and showing gap teeth, 'that Sister Whatsit up on chests and Dr Metcalf had to have an operation to get separated? Was they really stuck together like them,' he jerks his head in the direction of the main kitchen, 'like them in there says they was?'\n\n'Of course not,' I say, 'don't be so silly! Here give me that!' I take the hod of coke and let it fall on his feet. His boots are thick and he does not seem to feel any pain. Either it's his boots or his stupidity. He stares at my tears. 'Whats up? What's the marrer?' He peers up into my face.\n\n'Oh, go away!'\n\n'Orright, then I'm goin'. Orright. Orright I'm a goin'.'\n\n'Don't cry nurse! There's nothing to cry about, now is there.'\n\nIt's one of those Matron's office nine a.m. Things. I am here in front of Matron's desk. She is sitting on the other side moving her freshly sharpened pencil to and fro above a timetable which looks like a checked tablecloth in front of her. She talks softly to herself as her pencil pauses. She frowns, shakes her head and moves her pencil on, an inch or two, above the neatly ruled pattern.\n\nWeekends have been the worst. Magda suddenly wants a garden Dr Metcalf tells me. She wants all the rough grass at the back of the house cut and then mown into smooth lawns. She wants roses and fruit trees and vegetables. Weekends I have not been able to stop thinking about them doing the garden together. Magda wants home-grown salads Dr Metcalf explains. 'Come round,' he says in his most gentle voice. 'She wants you to come,' he says. He wants me near him.\n\n'All right,' I say. I walk by their house several times but I don't go up to the door.\n\n'The weekends,' Dr Metcalf says when we have a few minutes alone, 'the weekends are full of planting Magda. Don't cry,' he says, his voice very soft, his lips near my ear. 'Don't cry. Please, please don't cry.'\n\nMatron says, 'The diet kitchen and, let me see, night duty I think.' She looks up from the big timetable which has, I know, because she has said so before, over four hundred nurses on it. 'Plenty of fresh air every day before you go to bed.' She smiles at me and tells me I am very pale. She likes her nurses, she says, to keep well.\n\nI have never thought that I belonged to her. One of her four hundred nurses. My eyes fill with tears again.\n\n'There is absolutely no need for these tears,' Matron says, 'come along nurse, dry your eyes. There is nothing to cry about. You know as well as I do that the rule about the doctor's corridor applies for the benefit of the nurses. Bomb damage aside, it is no place for my nurses. It has been brought to my notice that you have been seen there occasionally. No doubt you will have had your reasons and I am not going to question them. The corridor is absolutely out of bounds. The rule exists, nurse, because of those in our profession who are weaker.' She smiles again. 'I do not, for one moment, nurse, want to consider you to be one of them.' She has made up her mind, she says, that I am to be a gold medallist. 'You can do it! It's hard work but you can do it.'\n\nI watch her pencil write my name in one of the squares.\n\n'The diet kitchen,' she says, 'is not a place of punishment. It is valuable experience.' And working on my own, she tells me, is an excellent way of having a much-needed rest from patients and other staff. She smiles again. 'We simply cannot have tearful nurses at the bedside you know.'\n\n'Yes, I know. Thank you Matron.'\n\n'I think that is all nurse. You will of course be, as part of your duties, dusting this office. I suggest you come in here either immediately before your meal time or immediately after it, between midnight and one.'\n\n'Yes, I know, between twelve and one, thank you Matron.'\n\n'You appreciate that this is considered a privilege. The dark polish is in this little drawer.'\n\n'Yes, thank you, Matron.'\n\n'You have weekends off. Also a privilege.'\n\n'Thank you Matron.'\n\nI must go round to Magda. I must go to the house and get my letter back.\n\nAn accident. The whole hospital is talking about an accident. About Dr Metcalf's unexpected death. The diet kitchen, because of its dark emptiness at night, is worse than any other place. I have to understand that I shall never see him again or hear his voice.\n\nThere was, this afternoon, a memorial service for him in the hospital chapel. I meant to go but did not wake up in time. Lois also did not wake up for it.\n\n'I do have my principles,' she said. And how about we go to the pictures, Saturday. She's having the weekend off, for once, she said.\n\nIf Magda sees my letter she will be so terribly hurt. I must get it back. She must not see what I wrote.\n\nI wrote everything to you Dr Metcalf, Jonathon. I wrote everything about us both that is why she must not see what I wrote. They will send your things to her in brown envelopes. Everything found in your pockets will be sent to her sealed in these special envelopes. I have seen rings and photographs and money and letters put into these special envelopes. The things are sent to mothers and wives, to the next of kin as they are called.\n\nMy letter to you will hurt Magda.\n\nI love you I told you in my letter. I want you to come back now. That's what I wrote to you. I told you I was crying while I wrote the letter. Why have you gone away? I wrote that too. Come back please. Come back now this minute. I've been up to your room. I keep on going up there in case you've come back. Your name's still on the door. It's locked. Come back before someone else has your room.\n\nI am sick, I told you. Every morning since you left I am sick. Remember I was sick? I told you I was being sick when you said you had to go and you said you would come back soon. And we would be together. You said you would find a way for us to be together. I wrote about that to you too in my letter. You told me. Remember? You told me to wait and to be happy knowing you would come back to me. Remember? You promised me. You said, 'Wait for me.'\n\nMagda must not see my letter. I never thought anything could happen. I never thought you would not come back. Oh why did you go? I never thought that you would not get my letter.\n\nIn the middle of the afternoon I wake up. It is only three o'clock. All at once I remember. You see, while I was asleep I had forgotten. Outside it is bright sunshine. A bit cold. I remember you like it to be cold and sunny. I'll talk to you while I get dressed. You like it sunny and cold don't you. I'm going to get the letter now. In a minute. But first I'll go once along the doctors' corridor in case you have come back. Are you waiting in your room for me? Sometimes when I was on the ward, you know, making beds or taking round the trays at tea time, I'd suddenly think you are waiting for me and I'd leave quickly. I'd take the lift and do you remember how sweet it was when you were there in your room? Sometimes I could only stay five minutes.\n\nI'll always remember the time I stayed all night with you. You said there was plenty of room for us both on that narrow little bed. Please be in your room for me when I come now. Be waiting for me and smiling when you open the door. Please.\n\nI have to go to Magda to get my letter back. You weren't in your room just now. I have to understand this. You aren't at the camp at Swindon, writing to me, either. There is a chance that you will come back wounded. You could be brought back, crushed with some bones broken, but not all that bad. Perhaps you are on the way back. Oh please be on the way back. Please.\n\nWhat's the use? I must stop hoping.\n\nDo you remember when you explained? It was such a sweet time for me, when you explained how you didn't sleep all night, that night of the fog, when you thought I should not sleep alone in the house with you, and you put the mattress down in Mrs P's sitting room. She was so sour that night. I hated her and her room! As if it mattered what Mrs P thought about us. But it was hurting Magda you were really afraid of and it was sweet when you said you wanted to protect me from your own feelings. Behaviour, you said. When we talk about Magda you explain so well that if Magda was a perfectly horrible person it would be easier. I understand because I love Magda too but it is you I want to be with for ever. You wanted me and you thought you should not. I feel very happy knowing how much you wanted me.\n\nMagda needs you, you explained. I understand that too.\n\nBut the night at the river shack, I reminded you. You were lovely and smooth and sunburned and your kisses very sweet. You said we should have made love that night. I went to sleep, I told you. And you said yes perhaps that was a good thing. Sleep is a protection you said.\n\nI am almost at your house. The bright afternoon makes me look pale and hollow eyed. Ugly. I feel hungry but don't know what to eat. If you were with me we could go in to some place and have hot toast and tea. Perhaps they would let us have real butter and some jam.\n\nIt is awful to go towards that house and to know you will not be there. I wish you could be there to open the door instead of Mrs P. When Mrs P opens the door I hope she will let me in to the hall. Perhaps the brown envelope will be on the hall table. Perhaps I'll be able to take it quickly.\n\nWhy do you feel you have to go I asked you. The war is practically over. The war is everywhere, I know, but it is over. That's what people are saying. You are more use, you are really needed here in the hospital. The war doesn't need you now like the hospital does. Not like I do. I need you.\n\nThis is a terrible thing to say but how can I have proof that you are dead? Who can I ask?\n\nI ring the bell and wait. On both sides of the front door the large clean windows are heavily curtained inside. The curtains are drawn. I ring the bell once more and wait.\n\nPlease, don't ever say that you can't forgive yourself. There is nothing to forgive. When I said that, 'There is nothing to forgive,' you looked relaxed and pleased. I loved you more than ever then. You said the hard little bed in your room on the doctors' corridor was now an idyllic place, that was the word you used, idyllic. You said that when two people loved as we have then it is as if that love is for ever. You told me to remember that.\n\nNo one is going to answer this door. It's no use standing out here. The afternoon is getting much colder. It's getting dark earlier now.\n\nThere is a queue at the greengrocer's shop. I join it to buy some Worcester Pearmains. One half of the shop is boarded up and an A.R.P. depot is in the boarded place. The boards are painted A.R.P. in red. The shop has looked like this for a long time. Today I seem to notice the boards and the red paint for the first time.\n\nFor a few minutes I forgot. It was biting the apple. Eating the apple I just thought about that.\n\nIt seems a long way back to the hospital. These mean little streets where we used to walk, hidden, because people we knew did not walk here, seem dirty and poor. I never noticed before though you once said they depressed you. You should see this street today. It is full of people, a long line of men and women, linked together arm in arm, dancing. Every day now there are street parties like this. They are like children in a school playground. Long rows of people dancing and singing. You know, the songs, 'The Lambeth Walk', 'Run Rabbit Run' and now it's 'Knees Up Mother Brown'. The women have got curlers in their hair and the men are in their shirtsleeves.\n\nIf you could see this dancing and rejoicing!\n\nYou said once how easily we accepted heaps of rubble. People, you said, got used to all kinds of things. It amazed you, you said, that this mess was all round people now and they did not seem to notice it. It's true what you said. There are heaps of broken bricks and slates everywhere and, at the end of this street, there is an old bomb crater which is not even fenced off. The people dancing don't seem to notice that some houses have whole fronts and sides missing. Some are tarpaulined and boarded up but others are showing pink and blue wallpaper, torn and discoloured. Sinister really, but no one notices. You could say a house looked like a doll's house, opened, without the magic. They have been like this for a long time. Part of the hospital is still covered with tarpaulin. Remember? The far end of the doctors' corridor does not lead anywhere. The stairs and lift shaft at that end have all gone.\n\nThis dancing in the street is how the war has been ending these days. Did you know, Dr Metcalf, Jonathon? Did I ever tell you how the war started for me? I mean really started. Not the declaration of the war. That was the terrible beginning. Terrible because my father could not and would not believe something which he had to believe. The war started one night with the post mistress and her son out in the village street outside my school. They were banging tin cans together and blowing whistles. It was the first air-raid warning. We all had to get up and sit under the tables downstairs in the dining hall. And, because the post mistress had no other noise she could make, there was no All Clear and we stayed under the tables all night. Yo u have never told me Dr Metcalf, Jonathon, where you were when the war started. Where were you? Were you married to Magda? I haven't had time to know enough.\n\nThey say it's an advantage for a doctor to marry the daughter of a well-known physician or surgeon. How did you meet Magda? I never asked you that either. There is a skinny black cat here. It's ugly because it's poor and alone. It's at the edge of these dancing people and it's trying to vomit.\n\nWhatever shall I do with my life without you.\n\nIt is the weekend and I'm free. I have the weekends off now, remember?\n\nI am at Gertrude's Place. Well not quite. The hens, I can see them plainly from here, are dotted all over the field.\n\n'Gertrude is very ill,' my mother said last night.\n\n'I know, you told me.'\n\n'You haven't seen her for all these weeks, months. You've not been home to see us.'\n\n'I know, I'm sorry.'\n\n'You could go in the morning,' she says, 'frst thing. You have to collect the eggs yourself and leave the money on the kitchen table. I'll give you change. Gertrude never has any.'\n\n'Yes, I'll go in the morning, first thing.'\n\nI am up the hill a bit from Gertrude's Place and I'm leaning on my bicycle. I wondered if I should ride it but it was all right.\n\nThe district nurse is pushing her bicycle up the field path. I ought to go down there. Perhaps I'll go back down in a minute. I have been up here at the edge of the spinny, for hours. Is it hours? It seems like a long time. It is all so quiet here. The nurse in her blue uniform looks small from this distance. I can watch I can watch her disappear into the house.\n\nI have been several times to Magda's, missing my sleep, to try to get my letter back. But no one is there. No one answers the door and the curtains are always drawn. I suppose Magda is with her mother and father. Perhaps she is at the river shack.\n\nI keep thinking about my letter and all the things I wrote in it. There has been no message for me from Magda. She must know about the letter, and about me, about us.\n\nGertrude is all small and shrivelled yellow in a bed which was never used. She wanted me to have it once. I was early at her place. The fowls must have been out all night. I wondered about the fox. Fowls would have to be either out all the time or shut up all the time. It is something they can't do for themselves.\n\nI watched Gertrude through the window. Her eyes were closed under a frown of pain. I watched and then I came up here.\n\nI want to go back down there. I want to have Gertrude comfort me but how can I tell her everything when she is so ill. It should be me comforting her.\n\nI think Gertrude is dying.\n\n'I can wash Gertrude,' I could say this to the nurse. But I can't say to Gertrude what I want to say. How can I go back over the summer and leave everything unexplained?\n\nIn the evening I can't stop crying. I can't tell my mother. It is only a simple thing I have to tell her but I am not able to.\n\n'You must not be so upset over poor Gertrude,' my mother says. 'She is very fond of you and she would not want you to be upset like this. Try not to cry.'\n\n'It's not only that, not only poor Gertrude,' I say. 'It's because oh it's because I've been trying to write and I can't, it's just stupid, it's nothing.'\n\nMy mother is knitting very fast. She started knitting at the beginning of the war and now she is always knitting. 'No one,' she says, 'can write anything till they've had experience. Later on perhaps. You will write later on.'\n\n'Yes, of course,' I say.\n\n'You never play the piano now,' my mother says. 'Why don't you go now and practise something and leave the front-room door open. I like to listen.'\n\n'I haven't been able to sleep either,' I say instead of saying the words I mean to say.\n\n'You must have some of my tablets,' my mother says, 'they are yeast tablets and are very good.'\n\n'Thanks,' I say and I put my hands up to my face because of more tears.\n\n'You promised to play Halma,' my sister says.\n\n'Oh, shut up!' Immediately I regret my reply. I know she is trying to comfort me.\n\nDuring the night there is a full moon, it makes a trellis of shadow and light on the opposite wall. It seems as if, instead of a corridor up here, there is another room.. A room I have never seen before. L-shaped with a long passage leading to a place which shines as a river shines when moonlight lies across undisturbed water.\n\nThe tarpaulins have been taken off the bomb-damaged part of the hospital. This wing, at the end of the doctors' corridor, has to be rebuilt. There is no strange room there. Beyond some wooden barriers the hospital up here is wide open to the night. The corridor ends abruptly in space. The moonlight is on the wall of the huge clock tower which is a water tower. It is a reservoir for the water from an artesian well under the hospital. I feel afraid of the power and the force of the water in the tower. I can imagine, all too easily, the depths of the precipice in front of me. It is as though a neglected wound, which I already know about, has been uncovered.\n\nAll the doctors' rooms are locked and uninhabited. There are warning notices and barriers all along the passage. I have come up here one last time. The building work has started. Somewhere there'll be a watchman. Someone to keep people away at night.\n\nThe moon is wonderfully close to this ragged broken end of the corridor. I could step easily across this gulf straight on to the clean white moon.\n\nThe moon belongs to my father. He has always said it was his. If I was over there he would know without my telling him. It is only such a small thing I have to tell. Perhaps it is the small things which are the hardest to tell. They are the things which make all the difference.\n\nIt is because it will be so unexpected for him. What I need to tell him will be unexpected.\n\nMy father, when he comes to the station with me after my weekend at home, talks softly to me as we wait to cross the road. He admires the Clydesdale horses as a brewer's dray rumbles over the cobbles. He doesn't seem to notice the dried horse dung and straw blowing in our faces. The horses are fine he says and have I noticed how well cared for they are?\n\n'How they shine!' he says, he can imagine the daily curry combing and the polishing of the brasses. The dray is loaded but the horses, moving all together, are very powerful. 'Look at their muscles,' he says, 'their muscles ripple under their skin.'\n\nAs we walk to and fro on the platform, he says even if we are not seeing each other very often he is always thinking of me. They would, he says, like me to come home more often. He is going out to Gertrude's Place tomorrow, he says, he is going to do a few things for her.\n\n'That's good,' I say, 'thank you.'\n\n'Is there anything in particular?' he asks.\n\n'What d'you mean?'\n\n'Are you worrying about something in particular?' His face is white in the autumn dusk.\n\n'Oh no!' I say. 'Not really. It's just that I don't like the diet kitchen.'\n\n'Quite a lot of life,' he says, 'is doing what we don't like very much.'\n\n'Yes, I know,' I say. 'I know.'\n\nThere is a small sound behind me. I turn quickly. Perhaps it is the night watchman. There is someone in the corridor. A dark shape is coming towards me, a shadow in the red light of the little lamps.\n\n'It's only me,' Trent says. 'I saw you pass the end of the ward.' She's got night nurses' paralysis she tells me. She has long woollen operation stockings on over her shoes and an army blanket round her shoulders. She tells me it's freezing on Women's Medical. 'It's that quiet,' she says. 'Kidneys. They're all on parsley tea,' she gives a fat giggle. 'They're your mob,' she says, 'you been fixing them lettuce again?' She takes my arm drawing me back from the edge.\n\n'You lead,' she says. 'By the way,' she croons, 'has anyone ever told you, you're not cut out to be a corpse? No sex appeal!'\n\nWe waltz slowly back along the brick-dusty corridor.\n\n'Do you come here often?' Trent, purring close to my ear, trips over the end of the army blanket.\n\n'Goin' down!' Trent says in the lift. 'A-wun-a-tew-a-tree-a-fower-a-faive-Tung Tung Tung Tung Tung Tung.' She can make a noise, with her tongue and voice, sounding like the plucking of a double-bass string. While clapping in a slow beat she taps her foot in a rapid rhythm. 'A-wun-a-tew-a tree a wun a woman band! Listen,' she says. 'Counter irritant,' she says, 'if things are bad make some other thing worse. Ching chang Chinaman givee good advice, don't pee until you have to. Busting! And that'll be all you can think about. Get me? And, just this, there's more than one pebble on the beach.'\n\nI shall go to Magda's once more. One last time and then not any more.\n\nI can't stop thinking about you. I think about you all the time.\n\nToday I found out just what sort of person Magda really is. I said, didn't I, that I would go once more to see if I could get my letter back, the one I wrote to you. I was so sure, you see, that it would be returned with your things, the things they call personal things. I have seen Magda now. It's like this.\n\n'Mrs Lemmington Frazier Metcalf does not wish to see anyone. She is not at home,' Mrs P says through the tiny space. She opens the door grudgingly so that I feel she is going to close it before I can say anything.\n\n'It's only me,' I say quickly, putting one hand on the door to prevent it from being slammed.\n\n'You nor no one,' Mrs P says. 'She's not in!'\n\n'Oh please, please let me come in.' I am surprised at my own voice.\n\n'Who is it Mrs P? Who's there?' I can hear Magda's voice from somewhere quite close. Perhaps from the stairs. She has a way of hanging over the bannister. Remember?\n\n'It's me,' I shout. 'Can I come in?'\n\n'Of course. Precious child, of course you must come in!' Magda is dressed in that dressing gown made of bath-towel stuff. She wears this only when she is quite by herself or when she is ill, she once explained to me. You will know the dressing gown, she said it was yours once. Her hair, unbrushed, is loose, all tangled and messy. Her face seems swollen and when I look at her my own eyes fill with tears.\n\n'Why haven't you come before?' she asks.\n\n'But I have. Several times,' I say.\n\n'Mrs P Darling! Will you be awfully sweet and bring us up some tea?' Magda puts an arm round me and guides me, hugging me, to the stairs. We go up together. Her action reminds me of you, Dr Metcalf, holding me the night of the fog. Remember? I can't help thinking that she must have my letter hidden somewhere.\n\n'Of course,' Magda says, 'I've been at Mummy's and Mrs P's been away too.'\n\nThere were no letters on the hall table, only the polished tray and other ornaments, all polished and cared for. No little heap of brown envelopes as I imagined there would be. She must have put the letter somewhere.\n\n'I feel such a frightful mess,' Magda says. 'As you can see I've just let myself go. Awful!' She sinks down on the sofa and pats the cushions.\n\n'Sit down,' she says. Her eyes are full of tears.\n\n'It's so awful, you see, he went orf with such a cheap and horrid person. That's what I can't bear and Mummy, of course, can't bear it either. It's hardest for her. Mummy's really quite ill over it.'\n\nSo Magda does know, and her mother knows, and they think I'm cheap.\n\n'I feel more awful than I know how to say, I...' The words are too difficult. Magda interrupts me with another hug.\n\n'You see,' she says, 'this person is really awful, cheap and I mean really cheap and vulgar. And, you see Daddy's so clever, a brilliant surgeon, everyone says so but he's stupid too. He's made a lot of money. Mummy's used to being comfortable. Mummy and I try to protect him. What I'm trying to say is that this cheap little person is a gold digger. Like all the others,' she searches for her handkerchief. 'But you, not having the experience of people like that, won't know what I mean. He is already, in a sense, at the mercy of perfectly dreadful people who are waiting to get everything from him.'\n\n'You mean,' I try to say something.\n\n'You see,' Magda says, 'he keeps making an awful fool of himself. Didn't you think she was perfectly dreadful? The last one? You saw her that day on the river. Gloria or whatever her name was.'\n\n'You mean Marigold?'\n\n'Yes, that's the one. It devastates Mummy every time \u2013 that's why I go over and stay the night with her every so often. She's dreadfully lonely.'\n\n'Oh I see,' I am taking care not to look up.\n\n'So incredibly vulgar and so grasping,' Magda says wiping her eyes with both hands like a child. 'They, those sorts of women and their relations would take everything, absolutely everything. The relations in particular hound Daddy. And where would Mummy be? She dreads a court case and she's terrified of the workhouse. Daddy's compromised himself more than once d'you see \u2013 and now this Marigold! It's all so ghastly!'\n\nWe are both quiet while Mrs P sets the tray with the teapot and cups on a little table in front of us.\n\n'Oh Mrs P, toast! You are a dear!' Magda can manage an entirely different voice. She pours out. 'You needn't wait Mrs P thank you.'\n\n'Marigold's mother,' I say in a timid voice, 'is quite nice. I am sure she wouldn't, I don't think she is like that. I nursed her. Marigold's real name is Edna, her mother is Mrs Bray. She is a nice person, very good and kind. Couldn't you go and see her?'\n\n'Oh mothers can't stop their daughters!' Magda laughs. 'You are so innocent and good,' she says, 'don't ever change!'\n\nWe drink our tea and share out the toast.\n\n'Oh, I nearly forgot,' Magda says, 'you are perfectly sweet to write to Jonty. Poor darling Jonty!' Her eyes fill with tears. 'It's awful I can't stop this weeping,' she says. 'I see you can't either. They've returned all letters. I guessed you would've written. I've got your letter here,' she leans back and stretches her arm across to her little writing desk. In here,' she says, pulling open the little drawer. 'Here it is, this is yours.' She hands me my letter.\n\n'You never put your name and address on the back,' she says, 'but I recognized your handwriting. Everything has been sent back to me.'\n\nI turn the letter over in my hands, almost stroking it, feeling the firmly closed-down envelope.\n\n'Thank you,' I say in a small voice, 'yes I did write.' My hands caress the thick secret letter.\n\n'He would have loved having a letter from you,' Magda says, 'poor Darling Jonty. But as you see it never reached him. That perfectly dreadful place! It must have been awful for him and then no one being quite sure. It's all so stupid. Conflicting. A head-on crash?' She shivered. 'Dead or believed missing. I can't bear it. Really I can't. I've been waiting at Mummy's and now here.'\n\n'Don't cry,' I say, 'please, Magda, don't cry!' The envelope is wonderfully smooth and unopened. Magda has given me back my letter. She has not torn it open to read it. She has not looked at it.\n\n'There's some confusion. Where and which camp,' Magda says, 'everything's so confused. It's victory, I suppose.' And she sobs aloud and feels around for her handkerchief. 'In the front of that shelf,' she says, 'there should be some clean ones. Thanks Darling!' Magda puts her hand on my arm. 'You see, Darling,' she says, 'I keep hoping he will come back. That it's all a mistake. I haven't given up hope. I suppose you realize that I'm heaps older than Jonty. That sort of thing makes people talk, d'you see, they say cruel things especially about women who are older. Jonathon -' she starts to cry again, 'Jonathon, you see, I need him so much.' I watch her shoulders shaking and I can see the dark-grey dirty-looking parting in her bronzed hair. I know I ought to help her.\n\n'Shall I find your hairbrush?' I ought to look for her brush. 'Shall I help you wash and do your hair? Let me brush your hair, Magda,' I ought to comfort her. 'Let me help you,' all this I should say. Magda would comfort me, if she knew.\n\nShe sits crouched on the sofa with her face hidden in her hands. 'You see, Darling,' she says, 'he wanted to be with Mr Smithers. Smithers went about twelve months ago to a field hospital and Jonty felt he should go. But you see, Darling, it's not so simple. People, men and women, will travel the length and breadth of a country, at times, to be together. In one way it's as simple as that.'\n\n'But I thought, Smithers...'\n\n'They worked together,' Magda seems to have a note of defiance in her voice. 'I'm waiting,' she says. 'One thing I'm certain of. When he comes back, _if_ he comes back, I'm never going to let him go ever again. I simply can't live without him.\n\nHer long tangled hair falls over her endlessly shaking shoulders. In a blind dazed sort of way I get up and stand for a moment in front of her. I can't see her face, only her shoulders without an arm round them. The bunched chintzy curtains and the cushion covers, with their crowded little flowers and acorns, and the hovering perfume all seem too much. I move silently towards the curtained door and open it and, with light little steps, make for the stairs and the front door.\n\n'How much are the rings please?'\n\n'Them's sixpence. All of them. There's nothing over sixpence.'\n\n'Oh yes, of course, it's Woolworths. I'd like a ring please. It's er, it's for drama, a drama, for a play in a dramatic society. I'm in a play, er, Shakespeare.'\n\n'Choose your pick. Them's all the same price, like I said, sixpence.'\n\n'I think I'll have this one. It's quite pretty isn't it?'\n\n'If you say so.'\n\nAll the lights are on in Woolworths and I move with the throng of people. It is slow, this getting to the doors, and more crowded because, near the doors, there are counters with a few sweets and people are still coming in as the store is emptying for closing time. Out in the street there is an eeriness in the sad twilight. The hospital, with lights showing, seems like a huge ship for ever in harbour.\n\nNurse Roberts, little Nurse Roberts, stout in her winter coat, was down by the bus stop alone. I saw her there in the morning. Her big case on the pavement beside her. She was waiting for the early bus, the one we call the workmen's. It was raining, a light rain. When I saw her there I never thought then that she might have nowhere to go. I was high up closing the window because of the rain. Now I think, where could she go? Where can anyone go?\n\nThe wireless is on in the night nurses' dining room. This inescapable 'In The Mood' music. It keeps on and on. Without wanting to I walk in time to this barrel-organ rhythm. Without wanting to I'm humming, without tune to this music. My voice in my head is an ugly croaking.\n\nLois is late. She comes in and sits down opposite me with her cup of tea.\n\n_Potatoes onions carrots_ , my ring fashes as my pencil pretends to scribble a shopping list.\n\n'Night Fanny here yet?' Lois glances round quickly and lights a cigarette. She inhales deeply.\n\nSomeone turns off the wireless. Sister Bean, with the registers held to her heart, marches across between the tables. Her voice barks into the silence.\n\n'Abbott Abrahams Ackerman Allwood...'\n\nLois, in her cloud of smoke, extinguishes her cigarette. 'Whoever,' she says, leaning low across the table, 'whoever would ever have married you?'\n\n'Arrington and Attwood. Nurses Baker Barrington Beam Beamish Beckett Birch Bowman D Bowman E Broadhurst Brown Burchall...'\n\n'Nurse Burchall?'\n\n'Yes Sister?'\n\n'Nurse Burchall, Matron's office nine a.m.'\n\n'Yes Sister.'\n\n'Nurses Cann Carruthers Cornwall Cupwell...'\n\nThe Easter moon is racing up the sky. The stunted ornamental bushes look as if torn white tablecloths have been thrown over them and the buildings are like cakes which, having taken three days to ice, are now finished.\n\nTomorrow is Good Friday.\n\nNext week I shall take the earlier train again and, before the journey is over, I shall speak to the woman.\n\nIt is more than likely that Ramsden would have white hair. Her hair was the sort of hair which goes white all over, all at once. Keats says, _to know the change and feel it_ , I thought of sudden white hair when I read that.\n\nThe cardboard cover of the little book of poems which Ramsden gave me once, during the night on the Lower Ground corridor, is decorated with edelweiss and gentian, a circle of neat pen-and-ink flowers. Inside she has written in her neat small handwriting,\n\n_The best is not too good for you_\n\n_Und Ihrer Weise Wohlzutun._\n\n_Ramsden_ , I shall say, _is it you? Much water has gone under the bridge_ \u2013 this is not my way \u2013 but I shall say it carelessly like this \u2013 _much water has gone under the bridge and I never answered your letters but is it you, Ramsden, after all these years is it?_\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
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+{"text":"\u200bHow can I help my child at home?\nIDCS has been transitioning to the Common Core over the past couple of years. You have no doubt noticed changes in your child's school and home work. Common Core standards encourage depth, rather than breadth, in mastering topics. It also emphasizes a student's ability to analyze and explain their answers rather than simply knowing how to come up with the correct ones.\nTeaching things like fractions, division and other math essentials may look really different from how parents (and teachers!) learned it. Teachers often present the concepts differently, sometimes incorporating unfamiliar timelines and drawn-out charts and models. Sometimes it may look like math is being made more complicated, when really we're teaching students to deeply understand the concepts that underlie traditional algorithms.\nHOW CAN I SUPPORT MY CHILD IN LEARNING THE NEW MATH STANDARDS?\nIf you're a parent of a school-aged child in the United States, you've no doubt heard of the Common Core standards. These standards aim to better prepare students from kindergarten through 12th grade for higher education and career success.\nWhile some aspects of the Common Core are new, many are not.\nchildren in high school still have to learn to solve equations.\nIndeed, there is a greater focus on these core topics than was typical of state standards before the Common Core.\nThis brings us to what IS new. To put it simply: the standards expect students to understand what they are doing so they are able to use and apply mathematics when they leave the classroom.\nWho doesn't want their children to think about what they are doing? Teachers and parents have been aiming for this for years. We tell them, \"Think about what you're doing!\" in other situations all the time: as they're crossing the street, as they're learning to play an instrument, as they're making an important decision. Now teachers and parents are supported by standards that lay out clear and coherent trajectories toward building a better understanding of mathematics, so children are less likely to lose their skills after graduation.\nFor example, many of today's parents learned to add whole numbers by using the so-called standard algorithm: lining numbers up vertically and adding each column, carrying when necessary. This traditional method is still required by Common Core. But how many of today's parents understand why it works? If you learned this way, it's likely that the \"how\" of doing the calculation has been drilled into your head, but the \"why\" has been lost. Knowledge not supported by understanding is fragile.\nToday, through the standards, kids learn mental strategies for addition and subtraction that help them use the standard algorithm with common sense about how those operations work. For example, it wouldn't make sense to use the standard algorithm to subtract 998 from 1001, when you can see that 1001 is 3 more than 998, so 1001 \u2013 998 = 3. Mental strategies based on understanding make procedural knowledge flexible, applicable to other disciplines and situations.\nAs a result of the math standards, you can expect to see your child learn a better balance between procedural fluency (the \"how\") and conceptual understanding (the \"why\"), so that she will develop the ability to apply mathematics in solving all kinds of problems. And that's a skill that will serve your child for the rest of her life.\nThe Common Core standards are not curriculum. Curriculum is what teachers actually follow, or do, in the classroom, and what students do at home. The standards are a part of the IDCS Curriculum, but the curriculum also includes unit progressions, activities, and materials that support the learning of the standards. Standards simply say what we want students to learn. Common Core establishes a set of goals for learning.\nThrough the standards, your child will be able to explain the math, not just \"do the math.\" For example, through the standards, students will now understand how our number system is based on ten, with the digits in a two-digit number telling you the number of tens and the number of ones. This goes well beyond being able to simply read and say the number out loud, or to point to the \"tens digit.\" It means being able to break the number up in ways that are useful for addition or for other operations. You can also expect your child to be able to explain how she knows what she knows, show how she thought about an answer through drawings and models, and choose different strategies for different problems.\nThe math problems that you remember are still important. Chances are, you'll also see some new strategies and styles of questions. You might want to ask your child's teacher about these problems, or check out the resources below. I've included some links to videos that demonstrate how to use some of the new strategies and models that are frequently used in Common Core Math. There is also a link to a \"Mathematical Models\" handbook that explains and gives examples of many of the new models used throughout the various grade levels.\nThe Common Core standards are both old and new. As a parent, you should expect many things from these standards, but above all you should expect your child to reap the benefits of what's new.\nDesignate a spot for doing homework. Select the spot with your child to help him or her to feel that this is a place to focus and work, rather than a place of punishment.\nDesignate a specific time for homework.\nBe available so your child knows that you're interested. Help to focus your child to get him or her started.\nAssist your child where necessary, without taking over the work as your own. Let the teacher know if you notice your child is struggling with an assignment.\nWhile helping your child with homework, remember that process is more important than product. Put another way, it's more important that you focus on helping your child learn \"how\" to complete the assignment rather than getting the \"right\" answer. We need to help our children learn to use resources, solve problems and arrive at answers. Answers are often proof of learning, but they are not the learning itself.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We're on a mission to redefine the role of business.\nIn 2011 I started my first business, WeAreTheFuture aged 17 from a high school classroom in Scotland, with a mission to inspire and connect entrepreneurs from around the world.\nOver five years we worked with over 5,000 entrepreneurs delivering global summits and programmes in Edinburgh, London, Berlin, San Francisco, Chicago, Los Angeles, Abu Dhabi, Hong Kong and Shenzhen, partnering with global brands.\nIn 2017, aged 23 I cofounded FutureX with a mission to redefine the role of business.\nWe are working with purpose driven entrepreneurs, corporates and governments to create scaleable and profitable businesses, that make values and purpose a core part of their growth strategy.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"When the secret of visualizing your dream car isn't \"magically\" manifesting itself, maybe it's about time you took a look inside and took matters in your own hands. Do you go through each day, feeling as though there should be so much more to life? Have you ever come to the conclusion that you deserve or want more out of life than what you've been getting? The Power of Life Mastery is a no-nonsense formula for creating long-lasting and measurable change. You'll learn how to surpass perceived limitations and step into your inborn capacity for greatness. It is a straightforward and easy-to-understand guide to getting the very best out your life.\nAnthony Rizk is an expert in the science of personal transformation. He has helped thousands of people realize their full potentials. He has combined some of the most powerful mental techniques available into a clear-cut system that you can use to take your life to the next level.\nThe Power of Life Mastery is an explosive course in accelerated learning and personal mastery that really works. You'll learn how to end self-sabotage and release your full potential, by applying proven strategies for maximizing your results. It covers a step-by-step, comprehensive training program in creating outstanding breakthroughs in every area of your life.\nwill uncover time-tested and proven strategies for getting much more of what you want, and less of what you don't.\nIn The Power of Life Mastery are the secret keys to living a 'Life by Design;\" those little-known distinctions that make all the difference. It is your passport to happiness, well-being and personal fulfillment. So, hold on tight and prepare for the ride of your life.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The TMT (Teaching Medical Translation) research project is under the direction of Prof. Joachim Kornelius. Its primary objective is to develop the cooperative translation method and learner strategies for acquiring medical terminology in the fields of translation studies, as well as for American and German medical students in the preclinical phase of medical school, all within a case-based learning context.\nAn additional objective is to develop multimedia learning tools for terminology and medical knowledge that are structured for purposes of encoding (text production), for use by translators, interpreters, and medical students. These tools are to be integrated into terminology management and CAT (computer-aided translation) systems.\nPartners in this collaborative team include the Indiana University School of Medicine in the U.S.A. (Prof. Patrick Bankston, Director of Northwest Center for Medical Education, Indiana University Northwest Campus in Gary and Prof. Carl Marfurt) and the Institute for the History of Medicine at the University of Heidelberg (Prof. Wolfgang Eckart, Director of medicine as a supplementary subject for Students of the Seminar for Translation and Interpreting).\nDr. Anne Kimmes, Dipl.-\u00dcbers., M.A.\nDr. John Stewart, B.A., M.A.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"PRESIDENT Rodrigo R. Duterte has approved the recommendation of the Department of Environment and Natural Resources (DENR), Department of the Interior and Local Government (DILG), and Department of Tourism (DoT) to close Boracay for six months starting April 26.\nThe President announced his decision during the Cabinet meeting at Malaca\u00f1ang on Wednesday evening, April 4.\n\"Boracay closed for six months effective 26 April,\" Presidential Spokesperson Herminio Harry L. Roque, Jr. said in a social media post.\n\"Calamity funds will be activated to tide affected workers over,\" the Palace official added.\nThe closure of the island from entry of local and foreign tourists is to give way to its rehabilitation.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzztqsd b/data_all_eng_slimpj/shuffled/split2/finalzztqsd
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+{"text":"El Teide Sun Protection 50+ spf 200 ml. The beauty and health of your skin deserves the quality guaranteed by the El Teide protective sun cream range.\nEl Teide Sun Protection 50+ spf 200 ml.\nThe beauty and health of your skin deserves the quality guaranteed by the El Teide protective sun cream range.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"My home is a bit of a mess right now. It reminds me of the time when I found The Messies Manual buried under a pile of stuff in my bedroom closet. Ahem.\nThe good news is that I'm going to clean this mess up. And posting horribly embarrassing before photos. Starting tomorrow. In the meantime, I thought it'd be fun to look over the most viewed articles on my blog this year and made a fun little list for you!\nSome of you loved it, some of you hated it, and many of you let others know exactly what you thought about it. Viewed over 750% more times than the second-most viewed article on my blog, this Facebook post is by far the most widely read article I've ever written.\nI wrote this article after my full-house declutter last spring. Now that I see it being repinned and shared so many times on Pinterest, I wish I'd taken better photos or to written something more profound. But maybe that's why you like it. It's real and it's imperfect, but it shows that decluttering doesn't have to be glamorous.\nI'm always amazed when people tell me they can't organize because it's too expensive. I'm sure that spending money on cute organizational tools is fun sometimes, but I don't think it's always necessary. In fact, I've decluttered and organized my entire home without spending a dime. Twice.\nIn this post, I show some of my non-pinteresty but very functional organization systems that really work, and I offer several simple tips for how you can organize your own home using things you already have on hand.\n4) How to Get Your Kids to Help Keep Your Vehicle Clean!\nI was almost too embarrassed about this article to publish it. I cringed at the (imaginary) mocking that I was sure would come my way for not being able to handle a seemingly simple task.\nBut guess what? Not only were all of you very kind about my mom-fail, but many of you repinned this article and told your friends about it, showing me that I'm not the only one with a messy vehicle!\nI absolutely love my spice shelf system. Not necessarily because it's so awesome, but because it works perfectly for my family. In this article, I offered very detailed step-by-step instructions about how to organize the spice shelf and baking supplies.\nApparently my family isn't the only family who has fun making snowman soup! This simple printable is a fun and frugal \"recipe\" that's perfect for holiday parties, wintertime treats, and other fun cold-weather events.\nThere's really no need to have a cluttered and messy medicine cabinet! When you read this post, you'll want to go organize your own medicine cabinet right away, then you'll wonder why you didn't do it sooner.\nSpoiler alert: I went from 515 eBooks on my Kindle down to 199 eBooks, and it only took me 30 minutes!\nUsing screen shots and simple instructions, I show you how to organize your own Kindle in this tutorial. Don't have a Kindle? Other eReaders can be decluttered using similar instructions (if you have a Nook, make sure to check out the comments for more help).\nThinking about working from home? Already working from home but want to expand your job options or increase your income? Then definitely check out this article \u2013 it's full of realistic opportunities and helpful tips.\n18 Summers: How Many Are Left?\nYou seem to really love my articles that include organization tips. Am I surprised? Not at all! In fact, I have some really awesome organization stuff up my sleeve for 2015 and I'll be announcing some of it really soon! But first \u2013 I need to tame my laundry monster. Again.\nWhat was one of your favorite posts from 2014?\n\u00ab Fabulous Resources to Help You Organize Your Home & Health in 2015!\nOk, I am trying NOT to take offense to the fact that only one of MY favorite posts is on this list \ud83d\ude42 I know you can guess which one is my favorite. So, I am following up with my other five favorite posts!\n5. How many summers left- this post made me cry when I first read it. Really!\nI love your list! You win!!\nThanks for the walk down memory lane. I'd forgotten about a few of those articles and I love your choices!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Looking for Thigh Tattoo Band? Then look no further! In this article we'll be showcasing some creative, unique, and over-the-top fantastic tattoos designs. Tattoos should never be impulsive decisions. Now-a-days, we choose tattoos according to our personalities and attitude. This Thigh Tattoo Band would look great on any part of the body. Learn the meaning of tattoo, meaning of this symbol, view the beautiful photos of the tattoo and pick a sketch. You can download these images free from www.freebiesndeals.club. We hope you all like our Thigh Tattoo Band in pictures.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Named Best for Hygiene in the March 2015 issue of M&B Magazine.\nEmma says: \"You assemble this mattress from poles and a cover. Before you put it together, it takes up little space, so it's a good travel option.\n\"Foam free and hypoallergenic, the breathable material is designed to reduce overheating, provide a better supply of fresh air and prevent pooling.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"BOSTON One by one, the Boston Celtics lined up to embrace Isaiah Thomas. The next time they see their former teammate, he probably won't be in a hoodie after watching the game from the bench. And it won't be so easy.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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new file mode 100644
index 0000000000000000000000000000000000000000..87e7aac137e815c81c2354b035c37dfead615853
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"The American Stars and Stripes and the yellow-and-red flag of the Republic of South Vietnam will fly across California this week as many of the 2.7 million Vietnam War veterans join 1.7 million Vietnamese-Americans in remembering the 40th anniversary of the Fall of Saigon, which took place on April 30, 1975.\nA crowd of about 10,000 was expected at a Camp Pendleton Marine Base commemoration, but the U.S. government refused to allow the South Vietnamese flag to be flown on a U.S. facility. Organizers moved the ceremonies to several locations in southern California.\nSince about 50,000 Vietnamese spent their first days in America at the Marine base as refugees from the communist take-over in 1975, the Fall of Saigon organizers had worked for months preparing for a solemn remembrance at Camp Pendleton.\nBut Jason Johnston, director of Public Affairs at Marine Corps Installations West-Marine Corps Base Camp Pendleton, announced on April 10th that Marine Headquarters officials told the base command that flying of flags \u2013 or the singing of any national anthems \u2013 representing countries the United States does not recognize is not allowed at federal installations.\nThe commemorations will begin on Saturday in Garden Grove at Bolsa Grande High School's football stadium. Spokeswoman Sophie Tran expects the event will draw about 1,000 people. The gathering will feature a \"photo-homage\" to the \"Boat People,\" who escaped during and after Vietnam's fall on rickety, often dangerous, vessels.\nDuring a two-hour ceremony in San Diego on Sunday, two wreaths will be tossed into the Pacific Ocean from the retired USS Midway aircraft carrier. One wreath will honor the American service members who died during the war, and the other wreath will honor the South Vietnamese who lost their lives during almost 25 years of the conflict.\nAs Saigon was falling to the North Vietnamese Army, the USS Midway led Operation Frequent Wind that evacuated the last Americans and 125,000 \"at risk\" Vietnamese from Saigon. Most South Vietnamese feared the type of reprisals conducted between 1957 and 1973 by the National Liberation Front that included approximately 36,725 assassinations and 58,499 abductions of South Vietnamese. The massive assembly of American aircraft and ships accomplished the largest helicopter evacuation in history.\nUSS Midway organizer Roxanne Chow said, \"It will be a solemn but meaningful experience.\" Volunteers are tying 58,202 yellow ribbons onto the flight deck to honor each of the fallen American service members. Organizers will pass out small flags to the attendees: 5,000 American ones and the same number of South Vietnamese flags.\nThe largest \"Black April\" commemoration will be held on Thursday, April 30th. Up to 5,000 attendees will meet in the heart of Orange County's \"Little Saigon\" for a three-hour ceremony beginning at 6 p.m. at Sid Goldstein Freedom Park in Westminster. There will be a candlelight vigil, the clasping of hands to show unity with the people still in Vietnam, and a slideshow displaying pictures of the approximately 50 South Vietnamese military officers who committed suicide during the Fall of Saigon.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Sheraton New Orleans Hotel is located in downtown New Orleans, 15 miles from Louis Armstrong New Orleans International Airport (MSY).\nTo\/from Louis Armstrong New Orleans International Airport \u2013 US 36 \u2013 US 55 depending on the number of passengers and traffic.\nNormal tip is 10 percent.\n\u2022\tIn order to be able to book travel online you must book at least 24 hours prior to your flight arrival time.\n\u2022\tThe discount only applies to reservations made through this web portal \u2014 if not, the discount doesn't apply.\nTo get a quote and book a Super Shuttle reservation, go to http:\/\/www.supershuttle.com\/.\nFor more information about ground transportation, please visit http:\/\/www.flymsy.com\/groundtransportation.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Hello! I'm Yumi Suda from Japan.\nI was diagnosed with HIBM (also known as DMRV) 9 years ago and at that time I did not know anyone who has same disease. I felt all alone. Now I am a member of PADM (Patients Association for Distal Myopathies) and I got to know other members that I can talk to about our disease.\nI had always been thinking that it would be even greater if I get to know the patients from all over the world so that we can share information worldwide. So, when I first read about ARM on the PADM website, I was very happy. It's very encouraging to have the patient's association overseas and not only in Japan and I think there are lots of things we can learn from ARM, especially for fundraising. I think that fundraising is a very important issue for PADM.\nI read the story about Kam on the ARM site and I was impressed by her courage to share her story. When I heard that she's coming to Japan, I was so excited.\nI have cried a lot because of my disease, but I feel it's also giving me treasures. I'm truly thankful to have been able to meet such wonderful people! It has been a great support to me. I can definitely say I'm not alone now.\nI want all the patients in the world feel they are not alone. I'm sure HIBM\/DMRV will be curable in the near future, so let's keep our spirits up and keep moving forward together!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Apricot Balm protects particularly strained skin areas with lipid-infusing Apricot Kernel Oil and turns every application into pure luxury with its fruity fragrance.\nApricot Balm protects particularly strained skin areas with lipid-infusing Apricot Kernel Oil. It can be used in a multitude of ways and is perfect for dry spots or to protect against cold damage. The nourishing oil is also suitable for treating cuticles. Its fruity fragrance guarantees a unique experience for the senses, making it a pure pleasure to use each and every time.\nApply over cream to protect against the cold.\nApply under cream to build up lipids.\nUse to condition the nails.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This rubberized hard case will not only give your LG K10 wireless cell phone protection but differentiate your phone from everybody else with a unique and beautiful design.\nThis LG K10 cover is made of hard plastic and coated with soft rubberized polycarbonate for a smooth velvet feel for longer lasting and protection from scratches, chips, dust, and everyday use of your phone.\nThis LG K10 faceplate has precise openings for the screen, top and side buttons, charger port\/dock connector, headset jack and speaker to allow full access to all the functions the phone offers.\nOne BRAND NEW two-piece front and back Snap-On Black Rubberized Hard Case for LG K10.\nYou can also find other related products like LG K10 cases, and more.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Sacramento Criminal Defense Lawyer - Arrested for Theft? Call Us Now!\nSacramento Criminal Defense Lawyer \u2013 Arrested for Theft? Call Us Now!\nGenerally, people assume that larceny and theft are one of the same. Legally speaking, however, larceny is a kind of theft; thus theft is the encompassing concept.\nAccording to the Federal Bureau of Investigation, in 2010, there were an estimated 6,185,867 larceny-thefts nationwide. The FBI's Uniform Crime Reporting (UCR) Program defines larceny-theft as the unlawful taking, carrying, leading, or riding away of property from the possession or constructive possession of another. Common examples are thefts of bicycles, motor vehicle parts and accessories, shoplifting, pocket-picking, or the stealing of any property or article that is not taken by force and violence or by fraud.\nThere are several degrees of larceny and theft, such as, grand larceny, also known as grand theft auto, or petty larceny, otherwise known as petty theft. Larceny-theft must involve personal property with the requirement of the possession actually being capable of being possessed and carried away. Therefore, real estate, services and other intangibles cannot be objects of larceny.\nThe penalties for larceny-theft depend on the facts of the case: the dollar amount of item(s), number of item(s), prior criminal convictions, and personal circumstances, such as, life circumstances, mental health, and age. We recommend that you do not speak with anyone regarding your case until you speak to an experienced criminal defense attorney. Discussing the case with anyone else, may lead you to divulge information that may be taken out of context resulting in a stronger case against you.\nThe skilled Sacramento criminal defense lawyers with more than 76 years of combined legal experience at the Law Office of Wing & Parisi can make the difference between your freedom or jail and misdemeanor or felony conviction. We possess the skill and knowledge to prepare a thorough defense against your specific charge. Call our office today for a free consultation. We will help you clearly understand your options, and answers any question you have regarding your case.\nThere's no substitute for a strong Sacramento criminal defense lawyer, call Wing & Parisi at (916) 441-4888.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In directing what is a very politically incorrect campaign against some regulations, Pruitt is making many enemies. Consideration of security spending needs to be viewed in that light.\nBut on weekend trips home for Sooners football games, when taxpayers weren't paying for his ticket, the EPA official said Pruitt flew coach.\nhe should be held to an appropriate ethics standard.\nIf he has crossed the line on that, he should be held accountable.\nYes, when it comes to Scott Pruitt even Fox News has standards. But all that the News-Register can do is make excuses for him and hedge.\nA final thought: imagine the Ogden outrage had Pruitt been a part of a Democratic administration.\nSen. Joe Manchin (D-W.Va.) says he has no regrets about his vote last year to confirm embattled Environmental Protection Agency chief Scott Pruitt.\nManchin's \"accountable\" and \"responsible\" sound a lot like the News-Register editorial: they're both theoretically in favor of them.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Here are some snazzy birthday facts about 25th of February 1964 that no one tells you about. Don't wait a minute longer and jump in using the content links below. Average read time of 11 minutes. Enjoy!\nWhat day was my birthday Feb 25, 1964?\nFebruary 25, 1964 was a Tuesday and it was the 56th day of the year 1964. It was the 8th Tuesday of that year. The next time you can reuse your old 1964 calendar will be in 2020. Both calendars will be exactly the same! This is assuming you are not interested in the dates for Easter and other irregular holidays that are based on a lunisolar calendar.\nFun stat: Your first one billion seconds (1,000,000,000) happened sometime on November 3, 1995.\nIf a dog named Tax - a Chow Chow breed, was born on the same date as you then it will be 301 dog years old today. A dog's first human year is equal to 15 dog years. Dogs age differently depending on breed and size. When you reach the age of 6 Tax will be 45 dog years old. From that point forward a large-sized dog like Tax will age 5-6 dog years for every human year.\nFran\u00e7ois de Bassompierre, a French courtier, is arrested on Richelieu's orders.\nIn his speech On the Cult of Personality and Its Consequences, Nikita Khrushchev, leader of the Soviet Union denounces the cult of personality of Joseph Stalin.\nWhat does my birthday February 25, 1964 mean?\nYour birthday numbers 2, 25, and 1964 reveal that your Life Path number is 11. It is a master number that represents intuition, idealism and invention. You have the potential to be a source of inspiration and illumination for people.\nThe following celebrities also have the same life path number: Sabur\u014d Tokit\u014d, Devon Seron, Cha In-pyo, Krysten Ritter, Tim Maurer, Craig David, Anoop Desai, Adam Green, Captain Sensible, Colbie Caillat.\nWhat is the birthday horoscope for Feb 25, 1964?\nThe Western zodiac or sun sign of a person born on February 25 is Pisces \u2653 (The Fish) \u2013 a mutable sign with Water as Western element. The ruling planet is Neptune \u2013 the planet of dreams and imagination. According to the ancient art of Chinese astrology (or Eastern zodiac), Dragon is the mythical animal and Wood is the Eastern element of a person born on February 25, 1964.\nFun fact: The birth flower for 25th February 1964 is Violet for modesty.\nWhat is the Love Match for Feb 25, 1964?\nWhen it comes to love and relationship, you are most compatible with a person born on May 1, 1987. You have a Love Match compatibility score of +195. Imagine being in love with your soul mate. On the negative side, you are most incompatible with a person born on August 25, 1971. Your incompatibility score is -243. You'll be like a cat and a dog on a love-hate relationship. Arf-arf, I want to bite you. Meow-meow, stay away from me!\nHow many babies were born on February 25, 1964?\nBase on the data published by the United Nations Population Division, an estimated 115,215,976 babies were born throughout the world in the year 1964. The estimated number of babies born on 25th February 1964 is 314,798. That's equivalent to 219 babies every minute. Try to imagine if all of them are crying at the same time.\nWhat is the popular baby name on 25 Feb 1964?\nHere's a quick list of all the fun birthday facts about February 25, 1964. Celebrities, famous birthdays, historical events, and past life were excluded.\nTry another birth date of someone you know or try the birthday of these celebrities: September 25, 1990 \u2013 Daria Strokous, Russian model, actress, and photographer; May 27, 1974 \u2013 Jason Narvy, American actor; October 24, 1931 \u2013 Ken Utsui, Japanese actor (d. 2014).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Sequential statements are used in processes to specify how signals are assigned. The process is executed in order as a whole. After all the sequential statements in the process are executed the signals are assigned their new values.\nThis statement can be read as \"assign sQ the value of sQ_next on the next simulation cycle.\nSo far, in the if statements only signals were used. The same rules apply when using variables, with a single difference. Like a signal, if a variable is assigned to only in some branches of the if statement, then the previous value is preserved by feedback. Unlike the case when a signal is used, the reading and writing of a variable in the same process will result in feedback only if the read occurs before the write. In this case, the value read is the previous value of the variable. In the case when a signal is used, a read and a write in the same process will always result in feedback. This observation may be used to create registers or counters using variables. Remember that a sequential process is interpreted by synthesis by placing a flip-flop or register on every signal assigned to in the process. This means that normally variables are not written to flip-flops or registers. However, if there is feedback of a previous variable value, then this feedback is implemented via a flip-flop or register to make the process synchronous. Example 6.21 describes a counter using the unsigned integer type. When a value of type unsigned is incremented, if the value is the highest value of the range, then the lowest value of the range is obtained.\nIn this example, in the else branch of the if statement the previous value of the count variable is being read to calculate the next value. This results in a feedback. Note that in this example actually two registers are created. According to the feedback rules, variable count will be registered. Signal result will also be registered, because all signals assigned to in a sequential process will be registered. This extra register will always contain the same value as the register for variable count. The synthesis tool will normally eliminate this redundant register.\nA case statement is a way to describe a conditional assignment with a large number of choices.\nCase statements are often synthesized as multiplexers.\nThis page was last edited on 10 November 2010, at 14:36.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Stickpage Forum > Creative > Animation Den > Noobesh.\nView Full Version : Noobesh.\ntwas not that bad really..\nI say ranks are bad. Don't use them.\nIt was good. Smooth and decent. Easing was okay.\nno ranks in spp anymore. but i'd say ur better than a beginner.\nyou have all of the concepts down on force and everything. Just stop overusing the lazy jumping thing. it doesn't look as good when the stick just turns into a line.\nThat's why you say you're high inter, right?\nI don't recall you doing Flash much, but it's okay. Easing is alright; it's stiff though.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzvbmn b/data_all_eng_slimpj/shuffled/split2/finalzzvbmn
new file mode 100644
index 0000000000000000000000000000000000000000..9b9985a8e2f2add16743509860ebcac0f7119230
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"[url=http:\/\/www.acilisvar.com\/]nike basketball shoes uk[\/url] Immediately I noticed a clean cut homepage with a classic black, white, and red background. Visually the blog is easy for audiences to digest and blog posts are short and to the point. Categories on the page include Classic Kicks, Drop Dates, Kicks, Uniforms, and Sports. Sub-categories on the page includes Ads, Apparel, Air Jordans, Kobe Bryant, Lebron James, NFL, and Nike Sportswear. The homepage has a column of recent posts that highlight, new shoe releases, limited addition shoe giveaways, new videos\/ads, and upcoming apparel. The Nikeblog has a Facebook page and a Twitter account which are both shown in the right-hand column of the homepage.\n[url=http:\/\/www.acilisvar.com\/]nike basketball shoes uk sale[\/url] Although, I didn't notice any retweets within last weeks worth of tweets, the account is actively replying to their twitter followers by answering potentially risky questions. I noticed that when a twitter follower tweeted a complaint about the ads on the Nikeblog page the company account tweeted back showing sympathy and addressing the problem. In my opinion, this was a perfect way to connect with audiences and show that while blogs are not always perfect, constructive complaints allow brands to effective fix site problems. The clean page layout and straight forward categories serves their shoe-fanatic community perfectly. Nike lovers have access to all the necessary information needed to decide on making a purchase.\n[url=http:\/\/www.acilisvar.com\/]nike basketball shoes uk cheap[\/url] The Nike marketing strategy that most of us recognize isn't the one that made them famous, at least not in the early days. Discovering the Nike marketing strategy that put them on the map is where the gold is. Furthermore, just like any true content marketing strategy, it isn't really much of a \"strategy\" at all. Like many similar success stories, it was as simple as providing real customer value. He later used that design to create the first Nike shoe commonly referred to as the \"Nike Moon shoe.\" It was the first step of many that put Nike on the map, but there is more to the story than just a great tread\u2014wear design and a waffle maker.\n[url=http:\/\/www.acilisvar.com\/]nike basketball shoes uk online[\/url] For them, things that benefit them personally are easy to justify. The Nike Moon shoes did this, but only because the customer was beginning to understand jogging and its benefits for their health. Bill's secret goal wasn't to sell shoes, he was simply promoting something that he believed in. This may not sound like marketing strategy, but it certainly should. Certainly, running was already popular among kids and athletes in the 1970's, but it wasn't the widespread social activity that we see it as today. The growing white-collar workforce helped pave the way for social activities that included the promotion of cardiovascular health.\nPermalink Submitted by Robinmax on Thu, 04\/19\/2018 - 03:17.\nPermalink Submitted by valdanon on Fri, 04\/20\/2018 - 23:47.\nPermalink Submitted by wq123 on Tue, 06\/19\/2018 - 03:04.\nPermalink Submitted by kailey on Fri, 07\/20\/2018 - 05:07.\nPermalink Submitted by dummybakeapie on Fri, 02\/08\/2019 - 12:14.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Customize your tuxedo with a vest in a range of hues to match any event or theme. Our four-button Expressions vests are fully adjustable with matching colored backs. Darted pockets on the vest front give it a slimming appearance. Pair with a coordinating tie from our Expressions collection. Available in 41 colors.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Suppliers and their Big 3 customers sometimes seem like a grumpy married couple. As partial compensation for their hard-nosed purchasing tactics, the Big 3 are offering more business to suppliers that produce complex component modules.\nBut everything comes at a price. Suppliers must engineer their own products, and they are expected to cover warranty costs.\nTired of the continual friction, suppliers are seeking more business with transplant manufacturers. But it's a slow, laborious courtship - foreign automakers are loyal to their longtime suppliers.\nDuring a discussion last month with Automotive News, four veteran supplier executives discussed these and other issues. They were interviewed by Editorial Director Peter Brown, Editor David Sedgwick and Staff Reporter Julie Armstrong.\nFran LeVeque, president of front- and rear-end modules at Meridian Automotive Systems Inc. in Dearborn, Mich.\nTim Manganello, CEO of powertrain components supplier BorgWarner Inc. in Auburn Hills, Mich.\nSid Taylor, CEO of steel processor SET Enterprises Inc. in Warren, Mich.\nDon Walker, CEO of interiors supplier Intier Automotive in Newmarket, Ontario.\nThe Big 3 generate less than 60 percent of U.S. vehicle sales. You are primarily Big 3 suppliers. Are you trying to diversify your customer base?\nLeVeque: We recognize where the growth is, and we are trying to meet those emerging customers. On the other hand, we've still got to go with the 60 percent. Trucks and sport-utilities will continue to be the high-volume stuff.\nIs it difficult to get business with the transplants?\nLeVeque: Yes. I think (foreign automakers) are very loyal to the people that brought them to the dance. Incumbency is a very strong sourcing consideration. But we've had some success. It's not easy, and it's not automatic. It takes a lot of time and a lot of investment.\nWe've done very well with Toyota, Honda and Nissan. Hopefully, we will continue to (do business with) Ford, GM, DaimlerChrysler, BMW, Audi - our traditional customers.\nIn terms of sharing technology, is Intier willing to sign Ford Motor Co.'s standard contract that has generated such a ruckus? Are you afraid Ford might reveal your trade secrets to competitors?\nWalker: I'm not going to get into our internal policies with our customers. But I don't think any of our major customers would purposely share our technology. Does it happen? Yes it does, and I have seen a couple of examples of that personally, which I won't get into.\nHowever, I think that happens at the lower levels of an organization. I think the senior executives don't condone it. Whether it's Ford or anyone else, if they start burning people on technology, word gets out and spreads like wildfire. And then they'll have a big fight on their hands.\ncareful about what you show them.\nDo you have a program to buy components from minority suppliers?\nWalker: We track minority purchasing. We've actually performed quite well. We have a program to encourage growth in minority purchasing. We take it seriously. But at the end of the day, nobody can afford to do something that's not cost-competitive.\nSo we need to continue to work with minority suppliers to make sure they are capable and competitive. It's an important issue for our customers.\nBut nobody's going to pay more money (just to get) more minority content. You have to make money. And if you don't make money, you can do all these nice things and you won't have a job anymore because the banks and your shareholders will desert you. I think the health of the Big 3 will determine how much support there is for minority initiatives.\nDo the Big 3 give you goals for minority purchasing?\nWalker: Yeah, they give us specific targets. They set a minimum that they want you to exceed. If you do a good job, they'll recognize you.\nManganello: We have programs, too. We track our minority purchases. We are definitely looking for more. I think we could probably do better at minority purchasing.\nWe don't sacrifice quality, cost or delivery just for minority purchases. All our suppliers have to meet the same yardstick. But we definitely are interested in minority purchases. Would I like to do more? Yes.\nSid, how effective are minority purchasing programs overall?\nTaylor: The minority purchasing programs started off slow, then they mushroomed. They reached a peak, and now they are on the downswing.\nTaylor: I'd say it was in the mid- to late 1990s. And I think it's now on the downswing.\nDid the downswing coincide with the recession?\nTaylor: You have a lot of companies who really aren't committed to minority programs. They don't believe in it; they do it just because they are told. You've got the people who make the purchase orders, and they are doing their own thing.\nAre automakers forcing you to cover more warranty costs? And how are you handling that?\nWalker: I think warranty costs have been a big focus over the past number of years with our customers.\nPart of the reason for going to modules is that you have somebody who is responsible for functionality as well as fit and craftsmanship. So you get a better vehicle. The best programs are win-lose. If you help (the customer), they share (the savings) with you. And if quality gets worse, then we are responsible for it and they are going to ask us to pick up part of the tab. I think it's a reasonable thing to do. We haven't had problems. Our warranty costs are low. So it hasn't been an issue for us.\nDo you have warranty insurance? Or are you self-insured?\nWalker: Insurance would be extremely expensive. A lot of it comes down to design and upfront control of engineering and manufacturing.\nAre automakers continuing to outsource more component engineering to suppliers? And do they want suppliers to produce more modules?\nWalker: I believe it's a trend. However, things go in cycles. I believe the car companies need to maintain a core competency from an engineering standpoint. But it depends partly on their manufacturing strategies, so it may change platform to platform and by manufacturing plant.\nIn the long run, we'll continue to see the car companies take advantage of the expertise the suppliers have.\nBut if you're going to be a module supplier you'll be asked for price reductions on a large number of parts that you don't make. So you need to be strong in program management. It's a whole different skill set.\nWhat about Meridian? Are automakers continuing to buy more complex modules?\nLeVeque: Yeah. We see it in things like door modules where you produce all of the guts of the door together. You can attach it to the door panel, as well. If you have design responsibility upfront, you'll end up with a better product.\nIs that your big growth opportunity?\nLeVeque: We are seeing more and more vertical integration in the products that we manufacture. So our growth seems to be in related components - all the stuff that goes into the front end.\nIf we can effectively design a front end while reducing the number of parts, then we can grab some of the adjacent components. So we're seeing added content. And that seems to be our growth.\nWe've been able to do some vertical integration. Next to our stamping and chrome plating operations, we have big injection machines. We manufacture the plastic components that go onto the steel bumper systems right there in the plant.\nOn one front end, we manufacture the lights and the fog lamps that go into those systems. So we have continued to grow. Our $50 bumper system is now a $100 bumper system.\nSix or seven years ago, there was an article that predicted GM, Ford and the Chrysler group won't make vehicles by 2010 and that Tier 1 suppliers will make vehicles for them. Do you expect that?\nWalker: I think there is going to be a market for contract vehicles. But it's going to be either for excess capacity or for low-volume vehicles. It's very capital-intensive. If a car company has a 200,000-vehicle-per-year cotract, I think the automaker will keep on making it. Unless you decide to make vehicles in China, in which case I presume they'll still make them themselves. But one of the fundamental issues that car companies have to deal with, both in Europe and North America, is legacy costs.\nNo car company can be globally competitive if they have uncompetitive wages, benefits or work rules. The CAW and the UAW understand that, and the unions in Europe understand that, as well. But the unions' leaders are in a difficult position. They want to support the people they represent. However, they also understand that if they ask for too much, they're going to drive their companies out of business. They have to start working smarter. And I see there is a movement that way. All parties need to understand what they have to do to keep the industry healthy and competitive.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Nordic countries have a strong regional food culture \u2013 pure, simple, fresh and seasonal. The New Nordic Cuisine combines new ingredients with more traditional fare, often prepared in new ways.\nAt ION's Silfra Restaurant and Northern Lights Bar we subscribe to this philosophy and offer a menu of fresh local ingredients served with a twist and accompanied by an array of beers, liqueurs and spirits from Iceland's many microbreweries. Silfra restaurant embodies the local flavor and spirit of Iceland, offering seasonal and contemporary dishes with an emphasis on sustainable, farm-fresh ingredients from our close surroundings. Silfra is proud to serve modern nordic cuisine with purity, freshness, simplicity and ethics as a guiding light to represent everything that this part of the world has to offer above other.\nThe chefs at Silfra also emphasize in what is called \"slow food\" that can be simplified as food made from the freshest ingredients available that is originated in the neighbourhood of the hotel. The restaurant's name derives from the Silfra rift in Lake \u00deingvellir. The rift is the most popular diving site in Iceland, first of all because it the crack between the American and Eurasian continent and secondly because of the magnificent visibility that isn't anywhere else under those circumstances.\nThe kitchen is lead by our chefs Sindri Gu\u00f0bjartur Sigur\u00f0sson and \u00der\u00e1inn Freyr Vigf\u00fasson.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I was out having a great time with Jeanette Winterson last night. Ok, maybe not with her personally, but I was in the audience and had a great seat and she even looked directly at me once! Bookman came too and has never read any of her books. He didn't even know what she looked like and when she walked out on stage he cracked me up when he leaned over and said, She has Margaret Atwood hair! Winterson's hair is not gray like Atwood's but both have the same kind of curls and wear their hair about the same length with a similar style. I love their hair and I sometimes wish my loose, corkscrew tending curls were more like theirs. But I digress!\nWinterson was at the University of Minnesota to deliver the autumn Esther Freier Endowed Lecture in Literature. Freier was first a student and then a professor at the U but not in the literature department. Nope, she was a chemist. But she loved the arts and was well aware that funding was hard to come by so when she died she left an endowment to the literature department that allows them to have two lectures a year and make it open to the public. How awesome is that?\nWinterson's lecture was titled \"Reading as an Act of Rebellion.\" If you have read anything she has written you will know she is funny and smart. In person she is that too but add personable and engaging and you will understand why at the end Bookman and I both wanted to yell, please keep talking because an hour is not long enough!\nShe ranged far and wide from the historical to the personal. She declared reading an extreme sport because one pits oneself with and against the best in literature.\nThe act of reading has through much of history been rebellious. Who was and was not allowed to read was controlled. Women weren't allowed to do it and slaves certainly weren't allowed. The more people read, the more they could think for themselves and those in power did not want to give up their power. When a person reads, and reads widely and diversely, it makes them one of the most threatening things in the world. Books can change the way we think and feel. Books show us multiple points of view. And while someone can see you reading and what you are reading, they cannot know what is going on in your head while you are reading. This makes you and reading dangerous.\nJust like we talk about valuing biodiversity in nature, we should value biodiversity in reading. And we need to not just give it lip service and say \"yay for diverse books!\" We have to read those books too. Books are not meant to comfort, though they can do that. Books should expand the world for us and yes, even make us feel uncomfortable. A book that makes you feel uncomfortable is giving you an opportunity to think and experience and learn something.\nTruth and fact are not the same thing, Winterson remarked. Simplification is a lie. We have language to help us make sense of complexity. Reading is great training in both diversity and complexity. The more reading you do, the more language you have and language is power. Just ask Malala Yousafza, Salman Rushdie, and those who work at Charlie Hebdo.\nFor the most part, Winterson said, we don't have to ban or burn books in the west these days because people are just too busy to read them. There is no reason to get worked up over what people are reading when no one is reading to begin with. Winterson is also not a fan of ebooks because if you can't see the books, if they're virtual, how do you know they exist? Give her the solid book on the bookshelf. That way nothing is hidden. Amazon can't erase it from all the Kindles. Can you imagine, she asked, if Amazon came to your door and took your copy of 1984 instead of making it magically disappear from your ereader?\nCreativity and imagination is the birthright of everybody. We don't have to apologize for art and culture nor do we have to explain. The mind is not a luxury.\nSuch a fantastic lecture! I am so grateful I got to go. I left feeling buoyant and excited, ever so proud to be a reader and determined to be as rebellious as I possibly can.\nWhat's more rebellious, though, than reading an e-book? You don't have to show anybody what you're reading, and you can delete it afterwards. I prefer hard copies too, but believe that the more ways books can be disseminated, the better.\nJeanne, her main concern with ebooks it seems is that there is no transparency. Amazon can erase everything on your ereader and no one could ever see them do it as opposed to removing books and\/or burning them, acts that are highly visible and can't be done in secret.\nE-books also have the quality similar to writing on water. One can edit or censor e-books easily. While it helps when it comes to rectifying typos or real errors, imagine if all books on your e-readers can be censored through a simple update \u2013 or history books on e-readers, updated according to the dominant narrative at that time.\ndharmapunx, that is a very good point! We place a lot of trust in ebook publishers to not mess with the text. right now we could compare it to the printed version but should the day ever come when books are digital only, there will be nothing for comparison and it will be very hard to verify of texts have been changed. Kind of ominous.\nAs much as I love to hold a real, physical book in my hands, I must admit that with the eRevolution, accessibility to reading materials have exponentially increased, don't you think? Not only books, but lit reviews, magazines, all the lit supplements, etc. that were previously 'beyond reach' have come down to earth. I mean, titles which one would not actually go and buy can now be downloaded free (e.g. Project Gutenberg). I was impressed by Edith Wharton's library when I visited The Mount, and last I heard, they are digitalizing the whole collection there. What a wonderful idea. Having said all these, I'd like to listen to J.W., stimulating still. Glad you and Bookman both had a good time.\nArti, Oh I totally agree that digital has made materials that were impossible to see other than in person available for anyone with an internet connect to view (Wharton's library for example) and that it has made other materials like you mention much more accessible too. I think Winterson's dislike stems from the potential political issues of control of knowledge. Whereas digital books can be erased in great swathes with a few clicks of a mouse and no one could see it happen, the removal of physical books can;t be so easily hidden. It's the impact of the concrete versus the abstract.\nThank you soooo much for this synopsis! I LOVED all of Jeannette Winterson's ideas and I am sooooo in agreement! Though I do think e-books are important\u2026.many of us who never had access to some of best books because of expense and geography, can read them all, thanks to e-books! However I do agree that reading is an art of rebellion \u2026it is knowledge, which brings power! I am going to look her up and read some more!\ncirtnecce, glad you enjoyed the summary! Winterson was a great person to listen to. Next fall they are going to have Marilynne Robinson! I too think ebooks are important for the same accessibility reasons you mention. I can see Winterson's point though, there is always a down side to things too.\nThis sounds so awesome. I'm reading The Gap of Time right now and I love how she takes the extreme emotions of The Winter's Tale and makes them contemporary and (semi-)plausible. Not at all surprised to hear that she considers reading to be dangerous; I've always thought that the reason it's suppressed, and the reason it's better for you than television, is because it teaches you empathy\u2013you have to engage directly with characters, you can't absorb their experience passively. (I guess with very good television sometimes you approach this level of investment, but with books it's sort of a built-in feature.) I really want to track down a copy of her memoir now. How lucky you were to hear her speak!\nElle, I've heard Gap of Time is pretty good so I am glad to hear you are enjoying it. I was astonished when someone in the audience during Q&A challenged her on why she would dare update Shakespeare. She gave a wonderful answer that had to with art being a living thing and how Shakespeare especially has always been updated and changed and played with. I completely agree with you on your points about reading and TV. I've read her memoir and it is really good. She did not have an easy childhood and books had a big place in rescuing her from it.\nThat's so funny\u2013presumably the same audience member would have no truck with, eg, The Lion King or 10 Things I Hate About You, either. Siiiighhhh. I was thinking about updating Shakespeare this morning, about how what you're doing has little to do with the language but everything to do with the emotional authenticity that his plots and characters possess. All updated versions of Shakespeare trying to find a way to do that the way he did that, just with different wallpaper.\nOh, I see\u2013argh, how annoying! I mean, we don't even have authoritative texts for most of the plays themselves, so why anyone should have a problem with simply trying to update the plots is sort of beyond me\u2026I guess cultural purists will take any excuse?\nroughghosts, she was really great and I am so glad I got to go. Not likely I will ever get the chance to see her again. So smart of you to back up your ebooks. Most people don't even think about that, believing that because they are saved in the cloud they are safe. Ha! In library world when it comes to digital preservation there is a system called LOKSS, lots of copies keeps stuff safe. I am a firm believer of that!\nI've never read anything by Winterson\u2026at least I don't think so. What do you suggest should be my first? As for ebooks, I have grudgingly accepted them. I was in such a mad rush to get my cat Lulu to the vet yesterday, I forgot to grab a book. But had my iPad in the car. So rather than sit and worry, I could read something to take my mind off my fear that Lulu was dying. But I still prefer the paper and glue ones. P.S. Lulu is pretty sick, but the vet said she believes we can get her well again.\nGrad, I've read two books by Winterson, Sexing the Cherry which is not about what you think it might be, and her memoir Why be Happy When you Could be Normal? Both were excellent. Her first book Oranges are not the Only Fruit is a fictionalized version of her memoir. Here most recent book is a retelling of Shakespeare's Winter's Tale. So take your pick among those and I think you be well served. I really do like ebooks for the convenience when I am away from home and I like that it is so easy to get a copy of a good many classics and other old books. But in the end I do prefer paper.\nSorry to hear about Lulu. I am glad the vet thinks she can be well again. I hope she is already on the mend!\nI heard Winterson speak in London, years ago, at a joint reading with Atwood. It was just incredible. As well as being fantastic writers they're also really really good speakers.\nshoshi, now that must have been really something! What a pairing!\nI wish I could have been there! I don't completely agree about e-books, but I do want my favorite books to have pages and to be placed on the shelves where I can get to them again. And I've read everything I've read by Winterson, although I should read more!\njenclair, darn fingers! \ud83d\ude09 I've read two of Winterson's books and liked them both and keep meaning to read more but you know how that goes! I like ebooks too but there is something undeniably wonderful about a book on the shelf.\nHuh? I meant \"I've LIKED everything I've read\u2026\" not I've read everything!\nI am turning everything green here with sheer jealousy. (The Bears don't look good in green, I have to say!) I am not a great fan of Winterson's fiction but I grab every non-fiction publication of hers that I can. She used to do a fortnightly column in one of the Saturday papers and it was a truly sad day when those came to an end. I am so glad you got to hear her.\nHelen, heh, my summary is nowhere near what being there was like. If you ever get the chance to go hear her talk, don;t pass it by, she is marvelous!\nI read Oranges Are Not the Only Fruit in high school and I remember NOTHING about it. I really do need to brush up on my J. Winterson! Sounds like a wonderful lecture.\nLaila, it was and I am so glad I got to go!\nEmily, it was great fun and I am glad you enjoyed my recap. Both Atwood and Winterson would probably be surprised that someone would envy their hair and want to have some just like it, but I do find barely contained wildness of it quite appealing. It somehow goes with their personalities!\nI would have come out of that inspired and bouncy too. I love the idea of reading as an extreme sport. I also like the idea that \"Books are not meant to comfort, though they can do that. Books should expand the world for us and yes, even make us feel uncomfortable. A book that makes you feel uncomfortable is giving you an opportunity to think and experience and learn something\".\nI have this argument sometimes with people who only want to see or read nice or entertaining works, or look at pretty art. I argue that the uncomfortable art (of any form) that I like to experience\/read is \"entertaining\" by my definition. What \"entertains\" me is being challenged.\nwhisperinggums, we are birds of a feather in this regard! I enjoy a comfort read now and then but more often than not I want my reading to challenge me in some way with ideas or characters or language or what have you. I find that entertaining as well! It is one reason I am glad I kept reading the Foragers book recently. I didn't like his theory at first but by the end I was won over.\nOf course when I got home I thought of a question I would have liked to ask Winterson: what do you say to people who declare a book is terrible because they did not like\/could not relation to any of the characters? When people make comments like that I find it shuts down discussion and never know how to respond. I suspect she would have had a great answer!\nOh yes, that's a great question. As you say, a hard one to answer when people say that, though I always give it a shot!","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":">> We have to think about the default interaction.\n>> are not signed, like apt tools do.\n> become useless if only one single package in our repos is not signed.\npackage just after creating it, and that it is more secure that way.\nfar as I can see, no one commented to that idea yet.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Kung Fu means to me a constant pursuit and a never-ending journey. To always better myself.\nIn kung fu, I've grown by learning more self-discipline, integrity, self-control. I've learned how to persevere through things by achieving my second-degree black belt.\nWhen I started my journey in the martial arts, I had been wanting to do it for a bunch of years, but I was kind of scared. So, I would go to school after school, check it out, talk to the instructor, even plan to come back the next day. I'd make a promise I'd come back but I wouldn't. I never showed up. This went on for six years and I don't know how many different schools.\nThis article highlights some of the reasons why Omaha, Nebraska rocks and what we love about this amazing city.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"What to wear under there. This High Waist Brief stays put so you don't have to.\nStep into shapewear with a little stick-to-itive-ness. No-slip hems hug your curves.\nNo-slip edge silicone-free grippers help leg bands stay put, preventing ride-up.\nLatex-free gripper on waistband helps keep garment in place.\nStay-put gripper on back leg helps provide rear lift.\nTummy solution firm control smooths tummy and derriere.\nBoyshort leg for full coverage fit.\nAnti-static fabric so you can kiss cling goodbye.\nCool Comfort\u2122 fabric to keep you cool and comfortable.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Description :Title : Anchor Audio PLL Synthesized UHF True Diversity Receiver - UNSHEATHED ANTENNASM..\nDescription :Title : Anchor Audio UHF 64 Channel Band Transmitter Model JEBUT-18C + Collar Microphon..\nDescription :Title : Apex 750 Dynamic Microphone Dual Imp. Uni-Directionel Model : Apex 750 Brand : ..\nDescription :Title : AUDIO TECHNICA MICROPHONE AT8570MLaModel : AT8570MLaBrand : Audio TechnicaColo..\nDescription :Title : Audio Technica -pro Sound Atr2500-usb Cardioid Condenser Microphoneusb W\/ 3.5mm..\nConditions :- Used-Certified Pre-Owned Equipment-Tested by a qualified technician- Some scratches- R..\nConditions : - Used, very good condition- Come with cable- Certified Pre-Owned Equipment- Tested by ..\nDescription :Title : Auditel NC50 Microphone\/Volting Network ControllerModel : NC50Brand : AuditelCo..\nDescription : Biamp Advantage 601 Microphone Mic Line Mixer 6 Channel Con..","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The NHS has a crucial role to play in helping people with a learning disability, autism or both lead longer, happier and healthier lives.\nOur NHS Long Term Plan aims to improve people's health by making sure they receive timely and appropriate health checks, while improving the level of awareness and understanding across the NHS of how best to support them as patients.\nMore people with complex needs will be supported to live fulfilling lives at home rather than in hospital, while thousands will be offered a personal health budget, giving them choice over the type of support they need to live the life they choose.\nMake sure that the whole NHS has an awareness of the needs of people with a learning disability and autistic people, working together to improve the way it cares, supports, listens to, works with and improves the health and wellbeing of them and their families.\nIn London, 'The Ealing Intensive Therapeutic and Short Breaks Service' has been supporting children, young people and their families so that they don't need to go into hospital.\nTo make this work health, mental health, social care and education services are all in one building where they work together to do what is needed so that children and young people with a learning disability can stay living at home when they need extra support or treatment.\nView our other learning disabilities and autism case studies, which show how we are making health and care services better for people with a learning disability, autism or both.\nSee what the NHS Long Term Plan says about learning disabilities and autism.\nView the easy read version of the NHS Long Term Plan.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzwbaf b/data_all_eng_slimpj/shuffled/split2/finalzzwbaf
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+{"text":"I read a lot, and I keep a log. Here's this year's log and the complete log (which goes back to the beginning of 1999). Read the posts in this section to find out what I thought about these books.\nSo, unfortunately, the theme of The Mayor of Casterbridge seems to be that people can't change. The book is fun to read, though, because there are a lot of secrets revealed dramatically through the interactions of the characters. And because literature uses long sentences with awesome words in them in a way that other stuff I read does not.\nSee below for quotes from the book, links to other Hardy books, and more on when and why I read this one.\nSiddhartha is a rambling quest for enlightenment with many mistakes and revelations along the way. It's a classic, but it's not really my kind of thing.\nYou can get it free from Gutenberg, but I would recommend buying it so that you get the benefit of a good translation.\nI wanted to download something free from Gutenberg to read on my Kindle while on a trip to Bangkok. This book was recommended to me by a neighbor in New Jersey several years ago.\nI enjoyed the Importance of Being Earnest as much because now I don't feel left out whenever I encounter a cultural reference to it as because it's funny. It's a story of deception, mistaken identity and revelation told in the form of a play. Now I really want to see the movie, starring Colin Firth, Reese Witherspoon and Judi Dench!\nI did not enjoy The Happy Prince and Other Tales. The stories are morbid and depressing as well as extremely moralistic (spoilers below).\nPaul Ekman's Telling Lies is a serious, important work in the field of psychology. It's readable by a lay audience, but it's not hawking 'ten simple FBI tricks that anyone can use to detect lies at home and at work'. In fact, the answer Ekman gives as to whether a certain behavior is a clue to lying is always: it depends. There are as many ways to lie as there are people, and as many ways to tell the truth. Furthermore, as you may have guessed from watching spies outwit them in movies, even polygraphs are not reliable lie detectors. Turns out\u2014surprise!\u2014people are complicated.\nContinue reading to find out more about Ekman's approach and findings, what I thought of the book, what I learned about lying, and what else I've read on the subject.\nDo not be turned off by the hard-sell marketing that surrounds every one of Edward de Bono's books. Just because he over-touts his own work doesn't mean it's not valuable.\nIt may even be that he only repeats variations on the same ideas in all his other books (which he periodically refers to). Having read only one of the many, I can say that there is at least one set of good ideas.\nI'd seen the books for sale here and there and read a bit about them online, so I was really looking forward to reading about the thinking hats. I was not disappointed. It's worth reading the entire short book rather than relying on information about the hats that's available online.\nTo find out more about the hat system and why it's cool, continue reading.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Republicans slashed taxes for billionaire donors and now they are coming for Social Security and Medicare to pay for it.\nFirst, they slashed taxes for their billionaire donors. Now, they're coming for Social Security and Medicare to pay for it.\nLast year, Rep. Brad Wenstrup made a mockery of his claim of being a fiscal conservative by blowing a hole in the federal deficit when he voted to slash taxes for the rich. Eighty percent of the tax cuts went to the richest one percent of Americans.\nIn our district, that means 7,250 people got 80 cents of every tax-cut dollar. The other 717,750 of us got to slice up the last 20 cents.\nTax relief for the rich was supposed to magically make tax revenue grow. But here in the real world, math is still math, and the federal deficit ballooned to $779 billion this year \u2013 during an economic expansion. By 2020, Congress's own accountants say Congress will be spending $1 trillion more than it's collecting.\nAnd as sure as night follows day, Senate Majority Leader Mitch McConnell and House Speaker Paul Ryan want to cut Social Security and Medicare benefits that we've earned. Why? The budget crisis that their giveaway to the rich created, of course.\nWenstrup and his party's leaders won't admit that, of course. They'll ignore the fact that their constituents earned their Social Security and Medicare benefits from a lifetime of paying into the system. And they'll ignore how giveaways to billionaires haven't helped towns like Portsmouth where jobless rates still top 7 percent.\nThe budget crisis created by borrow-and-spend Washington insiders like Wenstrup cannot be solved by squeezing retirees to pay for billionaires' tax cuts.\nIf the hardworking voters of the Ohio 2nd Congressional District are ready for a leader who fights for their interests instead of the wealthy few, I will protect Social Security and Medicare. I will fight for your health care and reinvest in our cities and towns rather than lining the vaults of billionaires.\nSend a message to Washington insiders like Wenstrup, McConnell and Ryan this November. Send in fighters for common sense and working families.\nJill Schiller is the Democratic nominee running for the Ohio 2nd District congressional seat against Republican incumbent Brad Wenstrup.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Corner, Freestanding, Wall Mounted and More!\nA big factor in choosing the right product for you is to look at the various vanity mounting types. First, you will need to decide on the location or placement of your bathroom vanity.\nNext, you'll need to measure the maximum height, width, and depth that makes sense for your new bath vanity. Once you have this information, you are ready to find the right fit for you. We advise you to get the proper measurements because the amount of space that you have is very important in considering mounting types. Some mounting types are more space efficient than others so it depends on your personal needs and preferences.\nThere are many different mounting types such as: freestanding, wall mounted, floating and corner. While some types are extremely versatile for all sizes, others may only be appropriate for specific space sizes. Freestanding, wall mounted\/floating styles are all great options for those working with a medium to large bathroom space. Freestanding setups tend to be heavier and a bit more bulky in order to ensure sturdiness. A freestanding vanity is a great choice for both classic and modern homes as there are so many different designs to choose from. Wall mounted or floating vanities are often minimalist and streamlined to provide balance and strength. This type of construction is perfect for contemporary homes where space dynamics are especially important. However, if you have a small bathroom space such as a powder room, you should consider adding a simple yet elegant corner vanity that will save you space and enlarge the look of your bathroom.\nRegardless of the mounting type, all of our products will help you create a more spacious feeling while still being able to offer you extra storage space for toiletries or cleaning supplies. With a vast selection of vanities, no matter what you choose, we will cover the shipping cost and ensure that you get your product promptly. At HomeDecorAZ.com, we want to help make your design dreams a reality. If you need any assistance, information or recommendations on our products, please don't hesitate to give us a call or email.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Comments: Recent Arrival! CARFAX One-Owner. ***Bluetooth, Wireless-Hands-Free***, ***Backup Camera***, ***Remote Start***, ***Sunroof\/Moonroof***, Sirius XM Satellite, ***MP3***, ***USB Charging Ports***, ***OUTSTANDING CONDITION***, HONDA FACTORY SERVICED, Leather Seats, Heated Seats, Steering Wheel Controls, Keyless Entry, New Tires, ONE OWNER, 2 Sets of Keys, Parchment w\/Leather-Trimmed Interior, Head-Up Warning (HUW), Lane Keeping Assist System (LKAS), Roof Rails.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Would you rather... Be Native American or Be Russian ?\nI love Russians. they seem like they don't give a f*k about anything and just live their lives. that's what I like in people.\nI hate people asking questions about nationality! You know we're all people right?","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Houston, Texas\u2014 Eagle's Trace, an Erickson Living retirement community, was named #13 in the Best Companies to Work for in Texas 2019 awards in the medium-size company category (100-499 Texas employees).\nManaged by the Best Companies Group, the survey is designed to recognize the best employers in the state, the ones that make it a priority to create better workplaces for their employees. The project is presented by Texas Monthly, the Texas Association of Business and the Texas Society for Human Resource Management.\nHouston, Texas-- As spring means baseball and home selling seasons, the Eagle's Trace Sales Department sports a lineup of professionals ready to go to bat for their new customers.\nThey are a preferred network of Realtors\u00ae, downsizers, stagers and senior move managers with decades of experience in the Houston area\u2026all part of the reality and moving services program of Eagle's Trace.\nHouston, Texas\u2014 Kristen Kennedy, Resident Services Coordinator at Eagle's Trace, was named the retirement community's Leader of the Year for 2018.\nMrs. Kennedy was one of 20 employees selected by their respective communities or corporate departments and recognized nationally among her peers during the Erickson Living Annual Meeting held in Baltimore, Maryland on March 4.\nHouston, Texas\u2014 As a child, Donald Mutin, Sr. loved his mother's home-cooked meals and appreciated her attention to detail.\n\"My Mom taught me how to slice onions and peppers. I became her 'prep person' in our kitchen,\" recalled Chef Donald, the Executive Chef at Eagle's Trace retirement community.\nHouston, Texas-- It's been a busy year for Pam Burgeson, sales director at Eagle's Trace, as more than 120 people have moved to the Erickson Living community in West Houston in 2018.\nAmong the new residents, there's one couple with a special connection to Pam. Her parents, Jim and Peggy Fredericksen, moved to the community in June.\nFor new Eagle's Trace resident, no need to wait!\nHouston, Texas-- Duane McCarthy spent 12 weeks remediating his flooded home in Houston's Fleetwood neighborhood after Hurricane Harvey swept through the area in August 2017.\nMcCarthy couldn't shake the feeling there had to be something more than keeping up a house and garden.\nHouston, Texas\u2014 A Barataria Bay shrimp poboy while watching the Astros game? No problem! Atchafalay chicken and andouille gumbo on a college football Saturday? Coming right up!\nAs the new Lafitte's Pub opened at Eagle's Trace last month, residents are now enjoying a casual atmosphere for meals and drinks with friends and family.\nHouston, Texas\u2014 The first residents are moving into Galveston Crossing, the new independent living building opened in July at Eagle's Trace.\nGalveston Crossing features 118 residences with nine new floor plans in addition to the most popular styles already enjoyed by residents throughout Eagle's Trace. Lakeside views are prominent on several sides.\nHouston, Texas\u2014 Graduation season commemorates the achievements of thousands of high school and college students in the Houston area.\nOn June 10, residents of Eagle's Trace celebrated not only the success of eight students employed at the Erickson Living retirement community but also the special bonds created between generations.\nHouston, Texas-- Carol Loggins may be past the traditional retirement age, but she's not ready to give up the benefits that come from working.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Please note the listed crops below are hyperlinks. If you click on them, they will lead you to information on how to store them and recipes too. These are our best guesses as what will be available this week in the share, sometimes we'll throw in extra or have to switch something out. Salem pick ups are a choice of the items, Terra Firma pick ups are farmers choice boxes.\nLots of new crops on the CSA tables this week! What is that alien looking purple thing in your box? Why, its kohlrabi! I say its like apple broccoli, crisp and sweet with a slight broccoli flavor. Just peel off the outer skin, slice it into a salad or cook it up in a stir fry.\nThe allium family is here. Alliums are the family of onions. First up scallions. Slice these babies in anything, boy are they delicious and add great flavor to your dishes. Don't forget to try out the scallion pancake recipe of the week!\nNext up are one of our all time favorites, garlic scapes.Though I love to eat it, I'll admit garlic is not one of my favorite crops to grow. I am not particularly good at it and I feel like its a bit of a prima donna and just takes so long to produce but I do love it for it's scapes. These are the curly green flower buds that the garlic sends up. We remove them so the plants send their energy into making bigger bulbs, and so we get a bonus treat. Use them as you would garlic in all your cooking and definitely try this garlic scape pesto recipe. Its garlicky and delicious!\nHave you ever tried Chinese cabbage? What about purple Chinese cabbage? A real stunner for sure. We were lucky enough to trial this new variety from our favorite seed company and we loved it so it'll be a regular in our crop plan. Chinese cabbage is just cabbage for most of the world. Try not to be intimidated by their sometimes very large size! Try it fresh in a salad like this recipe I learned from an old friend or make a fresh slaw with it. Get a little more fancy and make these potstickers with them or these baked egg rolls in place of the regular cabbage in the recipe. Try some kim chi, a important condiment in Korea. I remember a few years ago, their was a Chinese cabbage shortage there (I pay attention when vegetable make the news!) and it was a national crisis!\nBut wait there is more! Our broccoli is also ready! Finally! One of the most popular veggies out there, our spring broccoli is fantastic!\nFor a dipping sauce, use the sauce recipe from these potstickers.\n1. Make the dough and let it rest: Mix 2 1\/2 cups flour with 1 cup water until it forms a smooth dough. Knead by doubling the dough over and pressing it down repeatedly, until the dough is even more smooth and very elastic. Coat this ball of dough lightly in oil and put it back in the bowl. Cover the bowl with a damp cloth and let the dough rest for about 30 minutes.\n2. Roll out the dough: Cut the dough into 4 equal parts. Lightly oil the back of a large metal baking sheet (or a smooth stone countertop or pastry board). Roll out one part of the dough on the back of the baking sheet. Roll until it is a thin rectangle at least 12 x 9 inches.\n3. Chop the scallions: Finely chop the bunch of scallions. (I usually use the green tops and just the very top of the white parts.) Set them on your work surface along with a small bowl of kosher salt.\n4. Top the dough: Lightly brush the top of the dough with oil, then sprinkle it evenly with chopped scallions and kosher salt.\n5. Roll up the dough: Starting from the long end, roll the dough up tightly, creating one long snake of rolled-up dough.\n6. Cut in half: Cut the dough snake in two equal parts.\n7. Coil the dough and let it rest: Take one of these halves and coil into a round dough bundle. Let it rest for at least 15 minutes and ideally longer, while you repeat this process with the rest of the dough.\n8. Roll out the coil: Pat a coiled dough bundle into a flat, smooth, round pancake. You can do this with a rolling pin or with your hands.\n9. Cook the pancake for 2 minutes: Heat a 10-inch heavy skillet or saut\u00e9 pan over medium-high heat, and oil it with a drizzle of canola, vegetable, or peanut oil. When the oil shimmers, pick up the pancake dough and lay it gently in the pan. It should sizzle, but not burn. Cook for 2 minutes on one side.\n10. Flip and cook for an additional 2 minutes: Flip the pancake over with a spatula and cook for an additional 2 minutes on the other side, or until golden brown. Repeat steps 9-11 with the rest of the pancake dough coils.\nLook at the plow go!\nIt never ceases to amaze me how quickly the farm goes from soggy and wet to dry and dusty. After a spring spent watching the rain fall seemingly every day, things turned around in a hurry and all of the sudden we found ourselves moving sprinklers and setting up irrigation again. The farm is looking pretty good these days. We have been joking that now is the time to take lots of pictures. The fields are filling up with bright reds and deep greens. The weeds are, for the most part, under control and so far most of the crops are looking good.\nOur harvests are getting larger by the week, heavier and more diverse as we approach late June. We have been eating a lot of salads in our house lately and seeing how Shep likes boc choi and Swiss chard. It turns out he will eat just about anything so long as we hide it in a quesadilla. We have been eagerly watching the broccoli. Waiting for the plants to form large, uniform heads. It's hard to believe June is already half way over. Despite the blazing heat earlier this week, it feels like we are still in the midst of some endless spring still waiting for summer.\nWe are in between two major phases on the farm. The summer crops are all planted. The tomatoes, peppers, eggplants, melons, and cantaloupe are all growing nicely. The long season storage crops are also in the field. The winter squash, onions, potatoes, and sweet potatoes are settling in nicely. We continue to seed salad mix, herbs and beans in the field and plant lettuce on a weekly or biweekly basis but there aren't a lot of large plantings happening right now.\nThat is not to say we are done planting for the season. Far from it. We are seeding fall brassicas in the greenhouse, kale, broccoli, cauliflower, Brussels sprouts and cabbage to be planted from early July to mid August. Crops that will be harvested from the end of summer until the early winter and sometimes even beyond. In addition to the brassicas, we have a tremendous amount of root crops still to go into the field. The parsnips have been seeded and weeded, and we have already put in two rounds of carrots and beets but we have a lot more to do. But while we wait for the right time to plant for fall, this time of year is reserved for getting accustomed to the harvest, killing weeds and getting land prepared for the next round of crops.\nWe have had to adjust our land preparation strategy this season due to the excessively wet spring. One of the first things we noticed when we moved down here is that none of the big farmers plow. Not plow in the sense that we were used to. Farming in western Massachusetts in rock free river bottom soils, we would use a moldboard plow to plow the fields. A moldboard plow is a really amazing piece of technology. The moldboard plow flips the soil when it plows, bringing nutrient to the top and burying residue and cover crop underneath. The plow is one of the greatest achievements in the history of agriculture.\nWhat we discovered upon arrival to the stone filled glacial moraine soils of southeastern Connecticut is that farmers prefer just to disc harrow. One of the biggest innovations in human history, and the locals are unimpressed. The thought process being that when you plow up the soil, you plow up the rocks. Plows would still be used on occasion when the need was there but for the most part the disc harrow was sufficient. So for the past 5 years we have done what the locals did, and done without plowing.\nWe were happy to harrow. That was up until 2017 rolled around. In order to understand the problem that the wet spring presented to us, it is necessary to introduce you to our friend, winter rye. Now if you're familiar with winter rye you may already know all this, but if not, let me explain. We grow more acres of rye every year than we do any vegetable. We don't grow the rye for bread making or to catch children running towards the cliff of losing their innocence(editor note: Its a Salinger reference, in case you we're wondering, I had to ask Max what he was talking about there.). No, we grow the rye as a cover crop to hold our nutrients and soil in place over the winter. The amazing thing about winter rye, and why we grow so much of it, is that it will germinate as low as 32 degrees, and grow as low as 36 degrees. It will survive even the harshest winters and start growing again in the spring. This is both a blessing and a curse. A blessing since we can seed rye very late into season and a curse because if you don't harrow the rye early in the spring it can be very hard to kill. Of course, it can always be plowed under, but you need a plow for that.\nThis spring as the rain fell, we found ourselves staring out at acre upon acre of happy rye. Growing bright green and vigorous on fields that were too wet to walk on let alone harrow. Once the problem presented itself the solution was relatively simple. A few internet searches, trip to our favorite equipment dealer down in Central Pennsylvania, a couple hours stuck in traffic on the Jersey Turnpike. I think 14 hours in the truck total on a sunny Tuesday in May and the following day we were plowing. Participating in a agriculture tradition that dates back to the beginnings of agriculture itself.\nJust as an aside, I must say that I was completely amazed and truly in awe of the willingness of some to tailgate a pick-up truck during rush hour, on the Jersey Turnpike, with a piece of heavy agriculture machinery dangling, haphazardly out the back.\nCheck out the plow in this video.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Location: Dubai P.O.Box 36700,Dubai,UAE, Dubai. Get Directions and more details.\nLocation: Shop # 142 & 143, Ground Floor, Karama Shopping Centre, Al Karama, Dubai (Opposite Emirates NBD), P.O.Box: 2544, Dubai. Get Directions and more details.\nLocation: P.O.Box: 30755, Dubai. Get Directions and more details.\nLocation: Milano, Lombardia, Dubai. Get Directions and more details.\nLocation: Office # G00, Ground Floor, Karama Gold Centre, 10th Market Street, Al Karama, Dubai (Fish Market), P.O.Box: 31289, Dubai. Get Directions and more details.\nLocation: CITADEL TOWER, BUSINESSBAY, Dubai. Get Directions and more details.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Melton comes to Wangaratta tomorrow night and in the Bushies' territory they might think that they are the thoroughbreds but the men and women will be ready to ride these ponies! After a great weekend last weekend at Wodonga, the Bushies' teams are primed and ready for a battle to make the most of the home court advantage. Expecting big things from Rachel and Alaana for the Lady Bushrangers with Liam and Richard backing it all up for the Bushrangers, it should be a great night's entertainment, kicking off at 5.00pm. So get on down to HP Barr reserve in Wangaratta and support the Bushrangers' teams in two big games!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"STEM is an acronym for Science, Technology, Engineering and Math education. We at Berean Christian School have expanded our program toward a future goal of a complete STEM program. Currently our Science course offerings include Physical Science, Physics, Biology, Marine Biology, Anatomy and Physiology, Chemistry, Forensics, and Environmental Science. We presently offer an introductory Technology class as well as a Robotics class and our mathematics program is complete through AP Calculus AB.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzacaqw b/data_all_eng_slimpj/shuffled/split2/finalzzzacaqw
new file mode 100644
index 0000000000000000000000000000000000000000..b58be0be71fd6b7578ae98841446b4b86c78e272
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzacaqw
@@ -0,0 +1,5 @@
+{"text":"Francis Maude, the Cabinet Office minister, has warned ICT suppliers that spending controls will be a \"permanent way of life\" and that poorly performing companies will find it difficult to win new government contracts.\nMaude said the Treasury had saved \u00a33.75bn in the year to 31 March 2011 by putting temporary controls on spending in areas such as advertising and marketing, ICT, property and recruitment.\nLast week, the Cabinet Office estimated it will deduct \u00a365m off its annual bill from Microsoft and \u00a33m from SAP, however it spent \u00a3350m with those companies last fiscal year.\nMaude will also reveal to suppliers that the Cabinet Office will monitor their output across the Crown as a whole, with formal information on individual firms used at the start and during the procurement process.\nHe admitted however that the Cabinet Office has some way to go yet before it can match the buying habits of the private sector.\n\"I want Whitehall procurement to become as sharp as the best businesses. I will tell companies that we won't tolerate poor performance and that to work with us you will have to offer the best value for money,\" said Maude.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Brennan and Jessie started RedFred Productions in 2011, beginning as a small scale video production company.\nWe have grown substantially over the years thanks to continued support from our friends, family and community. we now operate as a collective in a historical building in downtown Gresham.\nBrennan and Jessie are a dynamic husband and wife duo, Jessie is the artistic director, videographer & photographer and Brennan is our postproduction specialist and motion animator. We both grew up in the Northwest and our work is very inspired by our environment. Our style is organic in documentary and journalistic. Red Fred covers all aspects video production services from concept to completion.\nSERVICES: Film (Video) for all businesses, no matter if you are a start up or taking on the business world by storm.\nWe offer event coverage: Wedding Films, birthdays, family movies and more.\nPhotography services include: Family, Weddings, events, branding, stock subscription and more.\nFamily, Weddings, events, branding, stock subscription and more.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I must create music, and will spend my life doing so - even if it goes nowhere.\nIt inspires others to do what they love, and be happy.\nYou can help me reach more ears.\nYou can be a direct influence on my life & music.\nYou can have your name on everything I release.\nFor 3 years I have been recording music. Roughly 60 spare-time songs so far. I want to make the transition from recording in my home to studios & live shows, but have neither the time nor the resources.\nBackers receive free download codes to all releases & are listed as \"Backers\" in the credits of all songs released.\nInsiders receive all benefits of \"Backers\", plus exclusive access to works-in-progress & pre-releases.\nBand Members receive all benefits of \"Insiders\", plus are mailed signed copies of all physical releases and can request one personalized custom CD of favorite tracks.\nExecutive Producers receive all benefits of \"Band Members\", but can request up to 5 custom CDs and will be moved above \"Backers\" in the credits into the Executive Producers list.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If you want to hire a writer to create a customized essay, you can use our reliable online writing services which deliver different kinds of custom-made papers, such as a custom essay paper, custom term paper, custom case study, custom review, custom research paper, etc. EssayElites.com is an international custom essay writing company which is famous for providing brand new papers for every client. Thus, our clients always trust us because every order is customized according to their preferences. Whenever students face difficulties and hardships in their studying, they always ask our highly qualified writers to assist them in academic writing.\nEssayElites.com introduced a special online service which provides only custom pieces of writing, such as custom college essay, custom research paper, custom book report, custom case study, custom dissertation, etc. All our writers are constantly trained to get new experience in writing essays and other pieces of writing. It does not matter which type of paper they write because all their works are remarkable and impressive. Moreover, most of our new clients become our return customers due to our effective performance of duties. If you want to order an essay, you simply have to give us a call and our well-versed writers will produce the best essay for you. You probably want to know why so many students prefer using our services to others on the Internet. The answer is pretty obvious. EssayElites.com hires special agents who check every custom essay for plagiarism and all kinds of mistakes.\nMany other companies who also offer customized pieces of writings, such as an essay, a term paper, a coursework, an article report, they usually cheat and send badly written papers which are plagiarized because the writers are not fluent in English. When such writers are given a task to write a term paper, they usually take information from non-credible sources which are unacceptable in academic writing. Moreover, such inexperienced writers make numerous mistakes in formatting because they do not know how to properly cite the sources and simply present information as their own. This activity can be considered as a fraud because when students submit such pieces of writing to their instructors, they may be blamed for irresponsibility and dishonesty, which may result into a course failure or even expulsion from a university. In addition, if your custom essay writing has a lot of stylistic, grammar, spelling and logical mistakes, then your instructor may think that you are not competent in academic writing and will not accept your paper. If you want to be positive that you will be delivered a customized essay from a highly certified writer who will not plagiarize, you should use only credible services. Do not trust unknown companies, EssayElites.com is a place which gathered professionals to take care of your grades. Other companies can simply copy and paste sample papers that your instructor can easily access; as a result, this will totally ruin your reputation. Your teacher or classmates will respect you no more due to such irresponsible action. Should you need a custom essay paper, EssayElites.com will always give you a helping hand on a difficult academic path.\nIn order to make our services more convenient, we have employed live support agents who are responsible for answering questions and accepting students' offers to provide custom essay writing. Be wise and do not put all your life at stake by dealing with fraudulent companies which provide sample papers at a cheap price. Always strive for perfect results, which EssayElites.com can help you achieve with the high-quality custom writing which is plagiarism-free and flawless. Buy the best essay from us and you will get a paper with incredible content at a cheap price!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Time for tough style. This ultra-durable watch by G-Shock features a black resin strap and round case. Shock-resistant chronograph analog-digital dial with three subdials, two digital windows, auto LED light, world time, daily alarm with snooze, countdown timer and speed timer. Analog-digital movement. Water resistant to 200 meters. One-year limited warranty. With an extra large case and ultra durable construction, the Casio Men's G-Shock X-Large Sports Watch offers modern technology in a stylishly rugged design. The textured, black dial features both analog and digital displays, embossed, luminous indexes that give a 3-D effect, and wide hands with cut-out details. This bold watch offers a variety of functions, including World Time with 29 time zones, five alarms, stopwatch, countdown timer, 12- and 24-hour formats, and day, date, and month information. A strong, black resin band wraps comfortably around the wrist and fastens with a buckle clasp for an adjustable fit. Able to stand up to extreme environments with its shock resistant technology, anti-magnetic structure, and water resistant depth of 660 feet (200 meters), this assertive timepiece is ideal for independent and active adventurers.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzadsqj b/data_all_eng_slimpj/shuffled/split2/finalzzzadsqj
new file mode 100644
index 0000000000000000000000000000000000000000..3b2fa0de09f006c35ddf787cf4f0ca4a39deef65
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzadsqj
@@ -0,0 +1,5 @@
+{"text":"On a buff circular shaped embroidered item 3 inches (7.62 cm) in diameter, a green palm wreath with the yellow hand guards and pommels of two diagonally crossed scimitars surmounting the base of the palm wreath with white blades, points up, between the blades three yellow five pointed stars, all encircled by a dark blue 1\/8 inch (.32 cm) border.\nThe palm wreath is a symbol of honor; the scimitars indicate the dual nature of the coalition-the defeat of ISIL and the restoration of stability to the area. The three stars indicate the command and are colored gold for excellence. Buff and flag blue denote the land, sea and air components of the military.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Smart contrasting blue colour scheme.\nVery quick delivery and a good size badge for customers to see easily. Would be good if they did a printed lanyard too.\nMany thanks for your review. We appreciate your feedback!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Since its inception in 1986 the Department of Advanced Nursing Science, has grown and developed tremendously in terms of programmes offered and nursing students population. The department has eighteen full time staff members, teach over 370 undergraduate and over 45 postgraduate nursing students (2017). The department supports Diploma in nursing, Diploma in Psychiatry, and Bachelor of nursing \u2013 a three and four year first entering, 3 honours level graduate programmes, Master of Nursing with full dissertation and a generic Doctor of Philosophy programme. The professional qualifications are career focused, marketable and allows for articulation and vertical academic mobility. The department implements of Community \u2013Based Education and projects, adopting of villages for CBE and community based research management. Staff members have capacity in their areas of expertise and are accredited assessors and moderators. The department has high through-put rate and increased postgraduate throughput and increased externally funded international collaborative research projects.\nAn approved Honours degree recognized by this University.\nApproval by Head of Department.\nStudents may be expected to participate in a series of seminars and lectures arranged by the department.\nA thesis on an approved subject based on original research is required. University rules and regulations are applicable to Doctoral degree. For admission in the Department 60% pass mark will be a prerequisite.\nTshitangano, SM. Maputle, ML. Netshikweta, NJ. Ramakuela & KG. Netshisaulu. (2015). Guidelines to Enhance the Implementation of Effective Tuberculosis Control Measures in Rural Public Hospitals of Vhembe District, South Africa. Journal of Human Ecology. 51(1, 2):1-8.\nTshitangano, SM. Maputle, ML.Netshikweta, NJ. Ramakuela & KG. Netshisaulu (2015). Implementation of Tuberculosis Control Measures in Rural Public Hospitals of Vhembe District, South Africa: A Thematic Analysis. Journal of Human Ecology. 51(1,2):166-174.\nS Malatji, Maputle & L Netshikweta (2015). Knowledge and Attitudes of Youth in Relation to Provider Initiated Counselling and Testing at the Primary Health Care Services of the Capricorn District, Limpopo Province. Journal of Human Ecology. 51(1,2): 40-46.\nMulondo, LB. Khoza & SM. Maputle (2015). Risk Factors and Complications Associated with Tuberculosis in Pregnancy and Neonates in Limpopo Province, South Africa. Journal of Human Ecology. 51(1,2): 128-137.\nNdou, MS. Maputle, RT. Lebese & LB Khoza. (2015). Support of Professional Nurses Caring for HIV\/AIDS Patients, Tshwane District of Gauteng Province, South Africa. Journal of Human Ecology. 51(1,2): 187-192.\nRamalepe, LB. Khoza & SM Maputle. (2015). Self-perceived Challenges of Lay Counselors Implementing Voluntary Counseling and Testing in Mopani District, Limpopo Province and South Africa. Journal of Human Ecology. 51(1,2): 138 -145.\nTshitangano, SM. Maputle & ML. Netshikweta. (2015). Validation of the Guidelines to Enhance Health Care Workers' Implementation of Effective Tuberculosis Control Measures in Rural Public Hospitals of Vhembe District, South Africa. Journal of Human Ecology. 51(1,2): 96-102.\nNdou ND, Maputle MS, Lebese RT & Khoza LB (2014). Challenges experienced by professional nurses caring for patients diagnosed with AIDS related illnesses at the hospital, Tshwane district, Gauteng Province, South Africa. December (Supplement 2:1): 255-264.\nNdou ND, Maputle MS, Lebese RT & Khoza LB (2014). Effective support model for professional nurses caring for HIV patients with AIDS related illnesses in Tshwane district of Gauteng Province, South Africa. December (Supplement 2:1):273-283.\nMalema RN,. Mothiba, TM & Maputle MS (2014). The perceptions pregnant women towards PTMTC of HIV and AIDS in the Capricorn District, Limpopo Province. AJPHERD, June (Supplement 1:1): 40-57.\nNdou, TV, Risenga, PR & Maputle MS. (2013). Experiences of HIV positive patients on ARV treatment at the Thulamela Municipality in the Vhembe District of Limpopo Province, South Africa. Journal of Antivirals and Antiretrovirals 5:123-131.\nMaputle, LB Khoza & RT Lebese. (2015). Knowledge towards Pregnancy-induced Hypertension among Pregnant Women in Vhembe District, Limpopo Province. Journal of Human Ecology. 51(1,2): 47-54.\nEE Raditlhalo, SM Maputle & TM. Mothiba. (2015). Vaginal Bleeding as a Pregnancy Problem during First Trimester at Polokwane Municipality, Limpopo Province. Journal of Human Ecology. 51(1,2): 73-79.\nRamavhoya, SM Maputle & RT. Lebese. (2015). Knowledge of Mothers with Regard to Immunisation of Children in Vhembe District, Limpopo Province. Journal of Human Ecology. 51(1,2): 19-15.\nRavele, SM. Maputle & NJ. Ramakuela. (2015). Contributory Factors to Antenatal Depression as Perceived by Pregnant Women in Vhembe District, Limpopo Province. Journal of Human Ecology. 51(1,2): 33-39.\nMaputle, RT. Lebese & LB. Khoza (2015). Perceived Challenges Faced by Mothers of Pregnant Teenagers who are attending a Particular School in Mopani District, Limpopo Province, South Africa. Int J Edu Sci, 10(1):142- 148.\nMothapo KE, Maputle MS & Shilubane HN (2014). Self-perceived competence of student midwives on plotting of partogram in labour units of Limpopo Province. Ethno Medicine. 8 (3): 285-291.\nMabunda SS, Shilubane HN, Maputle MS. 2015. Factors contributing to sub-standard care during intrapartum in maternity wards of selected hospitals, in Mopani District, Limpopo Province. AJPHERD, Sup 1 (3): 702-713.\nMothapo EK, Maputle S.M, Shilubane HN. 2014. Self-perceived Competence of Student Midwives on Plotting of Partogram in Labour Units of Limpopo Province, South Africa. Studies on Ethno Medicine, 8(3): 285-291.\nRamathuba, D.U., Tugli, A.K. & Mashau, N.S. (2014). Cultural practices on sexual health communication in the context of HIV\/AIDS in Limpopo Province, South Africa. African Journal for Physical, Health Education, Recreation and Dance, December 2014 (Supplement 1:1), 106-118.\nMaputle MS, Mothiba TM & Maliwichi L (2015), Traditional Medicine and Pregnancy Management: Perceptions of Traditional Health Practitioners in Capricorn District, Limpopo Province. Ethno Med, 9(1): 67-75 (2015).\nMiriri, M.T., Ramathuba, D.U. & Netshikweta, M.L. (2014). Socio-economic factors contributing to teenage pregnancy at Makhado Municipality, Limpopo, South Africa. African Journal for Physical, Health Education, Recreation and Dance, December 2014 (Supplement 1:1), 130-141.\nRamathuba, D.U., Lebese, R.T., Maputle, M.S. & Khoza, L.B. (2013). Socio-economic and cultural factors affecting adolescents' accessibility to contraceptives in Vhembe District, Limpopo Province, South Africa. African Journal for Physical, Health Education, Recreation and Dance, December (Supplement), 34-41.\nMothiba & SM. Maputle. (2015). Programmes, Interventions and Services Provided by Community Based Organisations for Sexual and Gender-Based Violence in Vhembe and Capricorn Districts of Limpopo Province Journal of Human Ecology. 51(1,2): 175-180.\nLebese, MS. Maputle, JT. Mabunda & PK. Chauke. (2015). Knowledge, Attitudes and Perception of Students on Teenage Pregnancy: A Case Study of Rural Based University Students in South Africa. Journal of Human Ecology. 51(1,2): 55-65.\nKhoza LB, Lebese RT, Maputle MS & Shilubane H (2014). Perceptions of teenage boys on fatherhood in Vhembe District in the Limpopo Province of South Africa. December (Supplement 2:2).\nLuhalima, T.R., Mulaudzi F.M. and Phetlhu D.R., (2014). Factors that motivate nurses to provide quality patient care in a rural hospital in Vhembe Disrict, Limpopo Province, South Africa. African Journal for Physical, Health Education, Recreation and Dance, December 2014 (Supplement 1:2 (October) 2014, pp. 473-484.\nLuhalima, T.R., Mulaudzi F.M. and Phetlhu D.R., (2015). Perceived motivating Factors towards a positive working environment in Rural Hospitals at Vhembe District in the Vhembe District in the Limpopo Province. J Hum Ecol,5191,2):202-207 (2015).\nShilubane HN, Ruiter R.A.C, van den Borne B.H.W, Ronel Sewpaul, James Shamagonam, and Reddy P.S. 2013. Suicide and Related Health Risk Behaviors among School Learners in South Africa: Results from the 2002 and 2008 national Youth Risk Behaviour Surveys BMC Public Health, 13:926.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"THE PRESIDENT: Good afternoon. Today we reached a critical milestone in our effort to reform our health care system. After many months of thoughtful deliberation, the fifth and final committee responsible for health care reform has passed a proposal that has both Democratic and Republican support. This effort was made possible by the tireless efforts of Chairman Max Baucus and the other members of the Senate Finance Committee. It's a product of vigorous debate and difficult negotiations.\nAfter the consideration of hundreds of amendments, it includes ideas from both Democrats and Republicans, which is why it enjoys the support of people from both parties. And I want to particularly thank Senator Olympia Snowe for both the political courage and the seriousness of purpose that she's demonstrated throughout this process.\nNow, this bill is not perfect and we have a lot of difficult work ahead of us. There are still significant details and disagreements to be worked out over the next several weeks as the five separate bills from the Senate and the House are merged into one proposal. But I do believe the work of the Senate Finance Committee has brought us significantly closer to achieving the core objectives I laid out early in September.\nMost importantly, this bill goes a long way towards offering security to those who have insurance, and affordable options for those who don't. It reins in some of the worst practices of the insurance industry, like the denial of coverage due to preexisting conditions. It also sets up an insurance exchange that will make coverage affordable for those who don't currently have it. And as the nonpartisan Congressional Budget Office has certified, it will slow the growth of health care costs in the long term and it will not add a penny to our deficit.\nThe committee's progress over the past several weeks is the culmination of work by all five committees and numerous members of Congress over the better part of this year. We've reached out to stakeholders across the spectrum -- doctors and nurses, businesses and workers, hospitals and even drug companies. And we've considered a wide variety of ideas and proposals in an effort to find common ground.\nAs a result of these efforts, we are now closer than ever before to passing health reform. But we're not there yet. Now is not the time to pat ourselves on the back. Now is not the time to offer ourselves congratulations. Now is the time to dig in and work even harder to get this done. And in this final phase, I hope that we will continue to engage each other with the spirit of civility and seriousness that has brought us this far and that this subject deserves.\nI commend the Chairman and the committee's members for their achievement and the example that they've set, and I look forward to continue to work with Congress in the weeks ahead. We are going to get this done.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Elizabeth Arden Prevage City Smart Broad Spectrum Sunscreen SPF 50 Lotion ~ sample size 5 ml for $10.63 (full size is 40 ml for $85.00). Formulated sunscreen with a powerful antioxidant blend to help maintain the appearance of optimal skin health even in the most heavily polluted cities.\nI have really enjoyed other products from the Elizabeth Arden Prevage line and use sunscreen daily, so this will get used! This sample is a little on the small size and the full size is $85, which is a little crazy for me!\nJor'el Parker Fango Mud Therapy ~ sample size 1 oz. for $3.86 (full size is 7 oz. for $27.00). Reveal your best skin from head to toe with our Fango Mud Therapy. Blended with moisturizing Sweet Almond and nourishing Avocado oils, along with rejuvenating Green Tea antioxidants and exfoliating Volcanic Ash minerals, our Fango Mud intensely hydrates, protects and deep cleanses your skin. Bring out your body's sparkling natural luster and tone with Fango Mud.\nI find this product a little strange because there is no instructions on how to use it?! Is it a cleanser or a mask or both? This was included in a past Ipsy and they described it as a face mask, but again not a lot of instructions. I mean, I will use it, but I like when it tells me what to do!\nRevlon Professional - Equave Hydro Nutritive Detangling Conditioner ~ sample size for $0.85 (full size is 200 ml for $18.95). Revlon Professional Equave Detangling Keratin Enriched Conditioner is a leave in conditioner designed to first hydrate, then nourish your hair. The first phase hydration formula adds necessary moisture to your hair, increasing your hair's natural strength and shine. The second phase gently replenishes your hair's natural oils and amino acids, stimulating healthy hair growth, shiny luster, and soft textures. Made UV resistant, Revlon Equave is designed to protect your hair and scalp against harmful ultra violet solar light.\nThis sample is so small - I really wish it was more than 2 uses! I really love detanglers because I have long, fine hair that loves to tangle it's self! I will use this, but again wish it was bigger!\nEyeko London Fat Eye Stick in Gilt ~ full size $12.00. An innovative chubby eye liner which creates the perfect smoky eye. Highly pigmented, Fat Eye Stick Smoke can be used to line, shade and define the eyes and has a retractable formula for effortless application and precision. The smoky charcoal shade will add a touch of glamour to your eye makeup and is the ideal colour for the iconic smoky eye.\nThis product is confusing because it is not available on the eyeko wedding, but I see it available on a few other subscription box websites, so this seems a more \"made for box\" product. Those aren't my favorite products, but I like these big eyeliner pencils as eyeshadow bases - this is a deep bronze and I think it will make a great base to make shadows pop!\nTopbox came with 4 products this month (1 full size) with a total value of $27.34! I'm happy with this month's box as I will use all of the products included. While I wasn't wowed by everything, and I did think 2 of the samples were on the small side - overall, I did think there was a good variety in the products and am happy to use everything!\nI got the exact same box as you!\nYa - it was just okay, lol I saw you felt about the same way.\nWe got totally different boxes! I kinda wish I had gotten that eyeliner, though!\noh that's funny, usually at least 2 or 3 of the products are the same!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaerte b/data_all_eng_slimpj/shuffled/split2/finalzzzaerte
new file mode 100644
index 0000000000000000000000000000000000000000..6d7c90c10c3c2339cfce5e575697bf282a679c16
--- /dev/null
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+{"text":"My12voltstore.com offers the largest selection of 12 Volt TV sets on the internet. We carry all major brands such as Majestic 12 Volt TVs, Jensen 12 Volt TVs, Supersonic 12 Volt TVs, Naxa 12 Volt TVs, RCA 12 Volt TVs and Skyworth 12 Volt TVs. With screen sizes ranging from 7\" to 32\", our inventory includes models like 12 volt LED TV\/DVD Combos and plain 12 Volt LED TVs. Compact and portable - the 12 volt TVs can be taken along just about anywhere.\nAll our 12 Volt TVs have digital tuners built into them. All our TVs are brand new and factory sealed.\nWhy Do You Need A 12 Volt TV?\nAre you wondering about bringing in some entertainment in your vehicle? Or you want to keep yourself updated with the news reports flashing in the news channels every now and then? Then the one stop solution for you here are 12 volt TVs that are specifically designed to absolutely fulfill your every entertainment need. An AC DC TV combined with DVD, CD player and mp3 player is the perfect entertainment package for getting up to date news coverage, weather reports, music and watching movies and playing video games whenever you have the time available while traveling.\nAll our 12 volt television sets have in them in built tuners and many have DVD player which are popularly known as 12 volt TV DVD.\nThese televisions are portable and convenient. Power cards are not needed to run them if you charge them well. There are larger models that can support external devices like DVD players, cable boxes, computers and speakers. You will also find many of them who have superb in built stereo speakers. Skyworth, Naxa, Supersonic, Jenson, Majestic and RCA are the brands that proffer their customers with various models of these mobile televisions with LCD monitors and built-in ATSC digital tuners.\nWith the advent of technology it has become possible to connect these televisions through even pen drives to view data. What else is needed when you come to know that it can also be used as a photo viewer or as an audio recorder. You can even connect them with internet cards which will allow you to have uninterrupted viewing of channels. The other features include channel auto scan, multilingual onscreen display, high receiving sensitivity, parental control, remote control, telescopic antenna, rechargeable battery, and standard company warranty.\nYou will find that the 12V TV is available in a range of sizes and models in the market and online stores. While the 32\" model is the largest 12 Volt TV, the 7\" model is the smallest of its kind in the world. These televisions often come along with power cords of 12V and 110V. With just a minimal power requirement, the devices ensure high quality performance.\nThe best place to buy this television set is the online store where you can select the model you want after comparing brands, features, technological specifications and prices. Advanced models of these sets are maintained in the wide product category by the dealers. Purchasing from a reputable online distributor like My12VoltStore.com ensures that you get the latest portable 12 Volt TV set at a competitive price, as well as attractive discounts and free shipment.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"For my Intro to Graphic Design classes' second Photoshop assignment, I focused on studio lighting, Photoshop's selection tools, and typography. We completed a brief tutorial together on each selection tool and how it can be applied before jumping into the final project. I then randomly assigned each student a location around the world. After compiling research and downloading fair use high resolution images from a class-wide Pixabay account, we had a photoshoot during which the students photographed each other posing as if they were in their chosen background images. They then used Photoshop's selection tools to remove themselves from the original and insert themselves into the new background image. Each student created a set of 5 postcards illustrating themselves visiting their assigned location overlaid with typical postcard greeting text.\nI wasn't able to find the exact references I used for studio lighting techniques, but Phlearn.com is a great resource for both Photoshop and photography techniques if you're ever in need of one.\nI was definitely lucky that my students were willing to humor me when I asked them to pose with nothing in front of a blank backdrop, but I think the resulting cards are so fun. One of my students even used them to trick her relatives into thinking she had traveled abroad which I think signifies a job well done on her part.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Welcome to Jordan Dental Care, dentist in East Belfast. We hope you enjoy browsing our website; please feel free to take a look at the treatment areas we specialise in as well as the latest fees and offers \u2013 then contact us today to join our dental practice.Also, we post some great content to our dental blog so make sure you take a look and if you'd like to be kept up to date with our future offers on items such as teeth whitening, please visit our Facebook page here and 'like' it.\nThe results above shows dentists within a 10 mile radius of Belfast, County Down, listing both NHS and private appointments. Unlike with GPs, you are free to go to any dentist without needing to register \u2013 even if you are registered with a dentist already.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"How can we use the natural energy of the Universe to aid us on our journey for self-fulfillment? The Full Moon Women's Collective is intended for women who are interested in learning more about the natural energies surrounding us. Each Collective Meeting we will discuss that particular month's moon energy, how to work with it and have open conversations about being women in this Age of Aquarius. May's Full Moon is in the sign of Scorpio governed by Pluto the force of change and Mars the planet of passion and activity. In this workshop we will discuss the transformations that we go through as women throughout our lives and how we can continue to invite change and shed habits and patterns that no longer serve us. We will continue to work together with our intuition, our \"Yes\" and add in some ways to supplement change with herbal tinctures and crystal energies. Bring a journal and a pen.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Image Title: Woodpecker Harlech Range Rustic Oak Oiled ENGINEERED OAK WOODEN With Regard To Engineered Flooring Designs 16. Post Title: Engineered Oak Flooring. Filename: woodpecker-harlech-range-rustic-oak-oiled-engineered-oak-wooden-with-regard-to-engineered-flooring-designs-16.jpg. Image Dimension: 300 x 300 pixels. Images Format: jpg\/jpeg. Publisher\/Author: Raymundo Hettinger. Uploaded Date: Tuesday - October 09th. 2018 12:37:35 PM. Category: Architecture. Image Source: zillow.com. 18 X 150mm Smoked Stained Matt Lacquered Oak Engineered Wood Inside Flooring Designs 3. Wickes Sunshine Oak Real Wood Top Layer Engineered Flooring Pertaining To Ideas 0. Lugern Brushed White Engineered Oak Wood Flooring 18 5mm X 125mm Inside Designs 6. 220mm Distressed Antique Natural Oiled Engineered Oak Wood Flooring Throughout Remodel 7. 190mm Pearl Satin Lacquered Random Length Engineered European Oak Intended For Flooring Decor 11. Emperor Distressed Vintage Oak Engineered Wood Flooring 15mm For Prepare 8. 18 X 189mm Nutmeg Oak Engineered Wood Flooring Crown With Regard To Plan 1. Lounge Golden Oak Engineered Wood Flooring Direct With Regard To Prepare 10. Helsinki Brushed Lacquered Oak Engineered Wood Flooring 125 X 14 3mm For Prepare. Woodpecker Harlech Range Rustic Oak Oiled ENGINEERED OAK WOODEN With Regard To Engineered Flooring Designs 16.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Clouds and stars wall decals \u2013 the little blah!\nAll of our wall decals are printed on high quality self-adhesive wallpaper fabric that is easily removed. They wont stretch or mark your walls and can be used multiple times -just peel and stick again!\nThis product is shipped from our studio in Melbourne in 2-4 working days.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"regardless of the platform they run it on.\ntheir expreriences in the past.\none get an accurate backup when using the extensions?\nunfiendly to users of minority platforms.\nAnyone have any thoughts on the subject? Thanks.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Louise Kelly, Programme Lead, talks about the impact and success of the Care about Physical Activity Improvement Programme (CAPA) and the next steps, now that the Care Inspectorate has received further funding from the Scottish Government's Active Scotland Division to expand the programme nationally.\nWe couldn't be more delighted to have the opportunity to expand CAPA (an improvement project to support older people experiencing care to move more) to more areas across Scotland over the next 18 months.\nCAPA works as a catalyst for change and improvement, but the real work is done by people experiencing care who are willing to take a chance to see what moving a bit more brings to their lives.\nThe other heroes are the social care professionals working in care at home, sheltered housing, day care or care home services across Scotland. Working together as part of CAPA we have seen older people improve their health and wellbeing. This included from increased life satisfaction to decreased anxiety and reduced risk of all-cause mortality, falls and disability.\nCAPA has proved that even small improvements can make a transformational difference to people's lives and we have seen, first hand, the physical and psychological benefits when people are supported to move more and to do the things they want to do.\nResults have also shown older people having more zest for life and a better quality of life. So we are thrilled to have further funding to work in partnership with more areas across Scotland so that we can further spread the 'moving more' message across social care.\nI am delighted to be leading the CAPA improvement programme. As the CAPA team, we bring a range of experiences and qualifications, all strengthened by a curiosity and determination to listen to the voice of the person experiencing care. Through CAPA we have taken ideas and approaches from using sharks to bingo to help health and social care professionals to grasp material and put their own ideas into action. Underneath it all are the big questions of 'what is it like to live here, or spend time here?' and 'what matters to me?' In future we will be working in new areas with new services, taking a similar improvement approach. We want to make sure that supporting physical activity and promoting people's personal independence becomes the norm.\nAt the moment we're recruiting for an Improvement Adviser and Improvement Programme Assistant to join our team. We have also started planning the best approach for the 18 months ahead and how to build on the great outcomes we have had so far. More details will follow soon.\nEveryone wants to have choices and to be in control of their lives. Otherwise, the world can close in around us. We can lose confidence and feel lonely and isolated. We all want to feel valued and that we have contributed in some way, regardless of age or abilities. And that's what CAPA is all about.\nThe full evaluation report from the first phase of the programme will be launched early December.\nFind out more at www.capa.scot and follow us on Facebook and Twitter @capaprogramme.\nJust so refreshing to see both Scottish Government and Care Inspectorate commit themselves to long term, sustained development to maximise reach and embed learning and skills over time and provide much needed support to the residential care sector. An example for everywhere else!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Costa Rica Roosterfish are Hard To Beat For a Fight of a Lifetime!\nWhat is it about this wild, mysterious and incredibly difficult fish to catch on a fly that has many repeatedly chasing them in Costa Ricaaaaaa\/Panama\/Baja year in year out? I find that question difficult to asnswer as most of you would. I was fortunate enough to fish in Costa Rica years ago for these incredible fish and years later in Mexico...I'm still trying to figure the big boys out on a fly. Maybe my trip to Costa Rica the next few weeks will help me either put this to rest or continue my saga to catch one monster of a Rooster on a fly.\nEven after catching up with Running Down The Man's -Frank Smethurst a few times to gain insight, my knowledge still lacks and my 10wt still keeps asking me \"when ya gonna get me something bigger than 20lbs?\" Easier said than done there Gloomis settle down now!\nHere are a few Videos to those Rooster fanatics out there! Enjoy this Teaser from Costa Rica and hopefully what I have to look forward to.\nNot quite Costa Rica but one great video of Chasing Roosters on a fly down South. Can you ever get enough of this? One thing is for sure, you better bring some sunscreen, big gear and patience for these fish. The New York Times had a great write up not too long ago on these beauty's entitled \"When The Roosterfish Start To Run, The Angler Tries To Keep Up\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Missouri Hospital Plan is the leading writer of medical professional and general liability insurance for Missouri hospitals, their employees, subsidiaries, and related operations.\nMHP was the solution to the malpractice crisis of 1985 by bringing to the market reasonably priced insurance as a \"mutual\" insurer. Its not-for-profit, tax exempt status and highly cost effective management by HSG has made MHP the leading insurer of Missouri not-for-profit hospitals and employed physicians.\nOur \"mutuality\" (meaning we are owned by our insured hospitals) has resulted in a long history of providing unequaled low priced medical professional and general liability insurance, extensive value added services and the payment of significant dividends to hospital members. Demonstrating highly efficient practices, MHP is consistently positioned well below the national expense ratio average, providing additional revenue to the bottom line and increased dividends to our members.\nHealthcare Services Group (HSG) Family of Companies believes in giving back to its members and policyholders. The Companies employ highly efficient operating practices, keeping the expense ratio well below the national average, providing additional revenue to the bottom line, and allowing us to sustain increasing dividends and profit sharing to policyholders.\nIn 2018, HSG announced a record-setting $15.9 million in dividends and profit sharing by its companies, Missouri Hospital Plan and Medical Liability Alliance, the largest writers of medical professional liability insurance in Missouri. This is not only the largest combined return in company history, but also the biggest year-over-year increase since 2008.\nThe Healthcare Services Group Charitable Foundation provides financial support of students pursuing medical, nursing or other healthcare related degrees or certificate programs. The scholarship\/tuition reimbursement program facilitates the attraction and retention of healthcare professionals in Missouri. This program is designed to match scholarships or tuition reimbursements awarded by hospitals that are insured by Missouri Hospital Plan. The 2016 match per eligible hospital is $3,000.\nFor more information visit the HSGCF page.\nAs a companion to our consultative services, our staff provides 24\/7 access to risk management assistance and advice in the event of an adverse outcome. During emergent events, physicians, risk managers and hospital administration are able to contact a Risk Management Consultant.\nDuring normal business hours (8a-4:30p), simply call our home office and request our Risk Management staff.\nAfter normal business hours, call our regular telephone number and our automated telephone system will provide instructions on how to activate the 24\/7 Adverse Outcome Hotline. Callers will then be asked to provide a brief description of the event, their name, and telephone number. HSG Risk Management staff will return the call as soon as possible.\nWe partner with ELM Exchange to provide online educational courses. These courses are designed to provide Physicians\/Surgeons and Allied Healthcare Professionals a convenient opportunity to improve risk management and patient safety practices, while earning continuing education credit. In addition, insured physicians, surgeons and allied health care professionals who successfully complete these ELM Exchange online educational courses may qualify for a 1% Risk Management premium credit discount for each course completed, up to 5%.\nWe partner with ELM Exchange to provide online educational courses. These courses are designed to provide Registered Nurses a convenient opportunity to improve risk management and patient safety practices, while earning continuing education credit.\nThis manual serves as a desk reference when risk management and patient safety topics and questions arise in the physician clinic setting.\nEach section provides information and additional resources about the topic, along with relevant sample policies.\nThe manual also serves as an excellent employee orientation manual and training resource. It includes a Clinic Self-Survey that is helpful in identifying potential liability risks, particularly in clinic-type settings. After completion of the Clinic Self-Survey, it will be reviewed and specific recommendations and action plans will be developed to help you address opportunities for improvement. In addition, a report comparing your organization to other survey respondents will also be produced.\nCopyright \u00a9 2010 Healthcare Services Group. The materials contained in CONCEPTS are proprietary and may not be distributed or reproduced without the express consent of HSG. MLA insureds have permission to reproduce and use information contained in this manual within their facilities.\nOur Risk Management & Patient Safety staff provides 24\/7 risk management consultative services covering a broad spectrum of risk-related issues, including but not limited to, questions related to medical records and confidentiality, adverse outcome disclosure, patient communication and follow-up, patient relations, patient safety, medical-legal issues, regulations and standards, policies and procedures and general office risk management practices.\nUnfortunately, in medicine things do not always go as planned. During treatment, patients may experience complications and subsequently suffer harm or patients perceive that the provider did something wrong, resulting in a complaint investigation or lawsuit being filed. In many cases, the healthcare provider experiences anxiety and distress after one of these events.\nHEALS provides education and support to insured healthcare providers after they experience a bad patient outcome, during a complaint investigation and during malpractice litigation. The program focuses on the healthcare providers' thoughts, concerns and questions about the events unfolding around them. HEALS is an information sharing program with its primary objective being to help providers return to peak performance as quickly as possible.\nHSG's Claim Department and the Risk Management & Patient Safety (RM&PS) Department work closely together to deliver the HEALS program to insured clients. HEALS is part of HSG's claims services. HEALS is managed and controlled by the Claim Department as part of the claims process but delivered by HSG's RM&PS Department. Since it is a part of the claims management process, the information discussed is privileged just like other claims related information.\nFor more information: Risk Management & Patient Safety Newsletter, Spring 2014, Volume 11, No. 2 newsletter & HEALS brochure.\nSRAT is designed to help our clients achieve HIPAA compliance, as well as prepare for potential HIPAA audits. The SRAT can be used to check existing HIPAA compliance programs, create a Security Rule Compliance plan and\/or be used as a tool with business associates to ensure their compliance with HIPAA.\nThe SRAT breaks down each audit protocol into subparts and consolidates the guidance and requirements into self-contained units that guide the users to the required policies and procedures. SRAT starts with the required Risk Assessment and ends with Breach Protocol requirement.\nOur media library provides access to important healthcare, risk management and patient safety education and information. The materials can be borrowed, at no cost, as part of your staffs' continuing education and training.\nWe partner with ELM Exchange to provide online educational courses. These courses are designed to provide Physicians\/Surgeons, Allied Healthcare Professionals and Registered Nurses a convenient opportunity to improve risk management and patient safety practices, while earning continuing education credit. In addition, insured physicians, surgeons and allied health care professionals who successfully complete these ELM Exchange online educational courses may qualify for a 1% Risk Management premium credit discount for each course completed, up to 5%.\nAnnual on-site risk assessments are provided to assess risk management and patient safety issues facility-wide. The assessment focuses on one to two specific departments and includes environmental assessment tours. Patient safety issues identified will be included in a consultation visit report, as well as on a user-friendly Recommendations Tracking Tool to be used by the facility for follow-up work. Often, resources are provided to meet the recommendations requirements. During these on-site assessment visits, education programs can be provided to meet your specific needs.\nWe believe that our insured clients who apply solid risk management practices should see their efforts reflected in their premium. Participation is strictly voluntary in program; however, it can result in a discounted premium of up to 7.5% on your professional liability insurance renewal.\nThe first part of the program focuses on education. There are two ways to earn the education premium discount \u2014 up to 5% per year. Eligible insured Physicians\/Surgeons or Allied Healthcare professionals can attend a one-hour, on-site risk management program, provided by a MLA consultant, for 3% premium discount or successfully complete our online educational courses for 1% premium discount for each completed course. These options can be combined to earn a total of 5% education premium discount per year.\nThe second part of the program focuses on self-survey and office practice assessment and provides a 2.5% premium discount for two years upon premium renewal. Our Clinic Self-Survey is designed to assist in identifying and correcting risk management concerns in your office practice. In addition to the premium discount you will earn for reviewing and completing this Clinic Self-Survey, it will be reviewed and specific recommendations and action plans will be developed to help you address opportunities for improvement. A report comparing your organization to other survey respondents will also be provided.\nOur quarterly SOLUTIONS newsletter is designed to provide our members with valuable information about risk management, patient safety, legal issues, coverage, benefits, ways to save on premiums and other costs, as well as ways to take advantage of all the products and services we offer. In each newsletter, look for updates and SOLUTIONS from Underwriting, Claims and Legal Services, Risk Management and Patient Safety, Marketing and Communications, ProCon, HSG Charitable Foundation, and the HSG Employee Benefits Consortium.\nWe offer periodic webinars covering an array of risk management and patient safety topics. Each webinar is a 45-minute presentation followed by \"Q&A\" and discussion. For those unable to attend the live webinar sessions, the sessions are recorded and available for later viewing by request.\nOur claim managers are available 24\/7 to address your claim concerns or questions.\nDuring normal business hours (8a-4:30p), simply call our home office and request our Claim Management staff. After normal hours, call our regular telephone number and our automated telephone system will provide instructions on how to activate the Claims Urgent Response System. Callers will be asked to provide a brief description of the event, their name and telephone number. A member of our Claims Management staff will return your call as soon as possible.\nOur attorneys are available to consult with insureds concerning legal matters, even if a formal claim has not been made. Such matters may include advice on HIPAA compliance, responding to complaints or requests by patients or family members, and other legal matters.\nOur claim representatives can assist our insureds in responding to subpoenas and other requests for the production of medical records or other documents. When appropriate, we will retain outside counsel as well to represent the insured's interests.\nWhen an insured client or any of its employees are subpoenaed or requested to give a deposition regarding medical care (even if a formal claim has not been presented), we will assign an attorney to represent and defend the individual at the deposition. Our claim representative may also attend the deposition.\nWhen a licensing board or other governmental agency requires an insured client or any of its employees to provide records, create a written summary, or appear in front of the board or agency, our claim representatives will assist throughout the process and will retain an attorney to represent the insured.\nHSG strives to provide individualized claim service for our insured clients. We provide each insured with the dedicated service of one claim representative. That claim representative will handle all claims that may arise.\nThe HSG Claim Management Team has frequent face-to-face interaction with the HSG Risk Management Staff. This communication allows us to address issues facing you to improve patient care, minimize recurrent problems, and decrease the frequency and severity of claims.\nThe HSG Claim Management Team partners with independent investigators and defense attorneys on claims and lawsuits. These investigators and attorneys are knowledgeable and experienced because they specialize in healthcare-related claims and litigation. Moreover, each has experience in the geographic area where the claim is pending. This engagement provides a familiarity with the judges, types of juries and local environment. HSG has had a long-standing relationship with these preferred partners because they provide priority service to our insured clients and have shown consistently successful results.\nIn addition to periodic claim reviews, our Claim Management Team is available to speak about pending matters at any time. Moreover, at the appropriate time, we will discuss each claim with you to determine if we should attempt to settle the claim or if the claim should be pursued through trial.\nWe review pending claims periodically with insured clients to address the status of the claim and the future course of action. In addition, the claim representative, when appropriate, will meet with each insured client annually. We will discuss the status of each claim, the issues that have arisen, potential defense strategies, and the plans for future handling.\nThe advent of electronic medical records, in conjunction with mandates stemming from data-information breeches, has fueled the need for Cyber Liability Insurance. Cyber Liability-related claims are for the most part excluded by the traditional insurance coverage purchased by hospitals and medical providers. Most notably, but not exclusively, cyber policies cover a business' liability for a data breach in which the firm's customers' personal information, such as Social Security or personal healthcare information, is exposed or stolen by a hacker or other criminal who has gained access to such data.\nProCon has developed a Cyber Liability program specifically designed for hospitals and leveraged the buying power of the group, resulting in broad coverage with an extremely competitive pricing model. The policies cover a variety of expenses associated with data breaches, including: notification costs, credit monitoring, costs to defend claims by state regulators, fines and penalties, and loss resulting from identity theft.\nThe Hospital Association Trust (HAT) is a multiple employer trust arrangement, incorporated by hospitals for hospitals, formed to address the lack of availability of reasonably priced Workers' Compensation insurance.\nHAT combines most of the benefits of self-insurance with the protection of guaranteed cost programs. Its group trust arrangement gives it the ability to control price, based on its own experience combined with improved safety and claim management efforts. The board consists of participants that have a vested interested in reducing injuries and claim costs. Like other HSG companies, the Trust has been very profitable for many years and has returned substantial dividends to its members.\nProCon is the exclusive agent for HAT and can also access commercial markets to provide coverage for smaller hospitals on a guaranteed cost basis and self-insurance programs.\nThrough its group purchasing strength, HSGEBC provides affordable insurance coverage and flexible benefit programs for member hospitals. Available programs include group health, life, dental, disability, voluntary benefits and retirement plan products and services.\nMembers have realized substantial savings and employee benefit plan improvements as members of this group.\nContact Information? Or for more information HSGEBC Link?\nInclude emails and website info like on dropdown menu.\nFor information on retirement plans call Josh Selzer at 800-683-3440.\nFor information on all other benefit plan products call Randy Lueckenotte at 800-869-5994.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"I woke up in the middle of the night last night \u2013 a night that falls toward the end of the inaugural class in the MergeLane accelerator for companies with at least one female leader. Things have gone well. Eight diverse and talented companies that cut across tech, consumer web, and consumer products. Each company with impressive traction, great ideas, or both. It's been a phenomenal ride.\nBut there I was, awake at 1am reflecting on how our teams were handling their participation in our mentor and speaker sessions. These teams are engaging, funny, provocative, and 90% female. I was frustrated with the seemingly low willingness (or inclination) of our team members to pose incisive questions to the incredibly talented leaders that had agreed to engage with them.\nAs I wondered about the causes, I arrived at my best guess. I think women are geniuses at \"looping\" with the room or a group. We are more sensitive to whether our need or question will be useful or interesting to the rest of the people in the room. They are less inclined to be \"selfish\" about taking up space. With slightly less ego, they are perhaps not looking to make a play for visibility with our speakers as much as comparable men might.\nI have no idea if this story has merit. Anytime one wonders about topics underlying questions of gender, one faces the risk of wildly generalizing. Yet in my current role, generalization seems to help facilitate the true acceleration of the companies in MergeLane.\nIn the event I'm right about my theory, I'd like you to please hold on to the gift of reading a group, but please err on the side of taking up more space, making yourself more visible, having your needs met, asking for help on your issues. First, most of us are dealing with many of the same issues. Second, you are in no position to imagine what other people need in any event.\nHere's the thing. These people we are bringing in \u2014 many of whom are pretty damned famous \u2014 are taking time out of their day because they SINCERELY WANT TO HELP YOU. These people \u2014 and interestingly the more famous and more senior people do this MORE \u2014 email me to ask how they can be of greatest service to you. They are looking for cues from me, but in a session, they are looking for cues from you to ensure that as many of you as possible get something out of their involvement. They already have arrived. They have nothing to prove. They are not doing this for ego. You allow them to feel helpful, and you allow this to happen more in direct proportion to how much you engage with them.\nFurther, there is honestly no better way in a group session to establish yourself as someone worthy of knowing than by listening and asking incisive questions. You don't do this to be a showoff. You do this because you want to learn more, because you want to connect with people on the other end.\nLast. A story. I hardly knew my MergeLane partner Elizabeth a year ago. We had met one-on-one for 30 minutes. Other than that, I knew one thing about her. In group settings (pitch events, investor meetings), she asked phenomenal questions. She shifted the dialog to a higher level with her questions \u2014 for everyone. And she did this by also giving an opportunity for the speaker to connect even more with an issue he or she cared about. Elizabeth is a wizard listener and a better questioner. It's a huge part of the way she has built her network. And it's the reason that at a dinner in May of last year, when I wrote on an index card \"Accelerator for women-led companies,\" I wrote her name under that line. And here we are.\nPlease be visible, dig deeper, make expert use of these brief encounters. This is not just in MergeLane, but always. MergeLane, the entrepreneurial community, your mentors, this economy, your customers, and more are opening a big, giant door for you. Go ahead, women. Run through.\nRaise your hand. Interrupt. Get so excited that you can barely stay seated. Just be there. More.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We currently have no information about upcoming Frankie Valli & The Four Seasons - The Farewell Tour events at First Direct Arena, but here is a seating plan used for Frankie Valli & The Four Seasons - The Farewell Tour in the past. Please check below for Frankie Valli & The Four Seasons - The Farewell Tour Tickets.\nMake sure you get the best view of Frankie Valli & The Four Seasons - The Farewell Tour at First Direct Arena. Here is the seating layout for First Direct Arena.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We visited Disney for the Mardi Gras holiday. It is amazing to me that this trip seems so long ago!\n2015 was a fun year for all of us. I don't know what surprises 2016 has in store, but I can't wait to watch the kids learn and grow more this year as we discover them together.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Do this every time the water level gets too low. Do not attempt to fix your garbage disposal by reaching in with your hands. Even when in the off position, garbage disposals are dangerous. You can look online to find the diagram for the disposal that you have. Water that leaves behind pink or orange tinged stains has an excessive amount of iron. You should purchase a water softener to use, or ask a local business to make a visit to your residence and resolve the issue for you. If you want your garbage disposal to have the same lifespan as your total home, clean it regularly. By regularly \"cleaning\" the disposal with lemon rinds, cherry pits and dish soap, you can make sure that your garbage disposal is running well and smelling so clean it could be new. Ensure that the overflow holes in your sinks are not clogged. These overflow holes are present in case of a clog or overfilling. It may seem unlikely as a problem now, but the necessity is there. Emergency Plumbing Contractor Coos Bay OR In addition, the choice of brass fixtures will be much more pleasing visually than other fixture options.\nIf there are problems, you must make the needed fixes prior to the arrival of freezing temperatures. Frozen water will back up in your pipes and crack them, whether they are PVC, copper, or even steel. Even the tiniest of cracks can release enough water to cause water damage or flooding to your home. Main water cut off valve is a very important piece of information you should be aware of, and you should know exactly where it is. The water meter should be located in its vicinity. Toilets, sinks and some other fixtures can have cut off valves of their own. Make sure every family member knows how to find these valves, and turn them off. Many people aren't well-versed in plumbing, and choose to pay whatever the plumber is asking for. If you're a homeowner, it's important you know enough to understand what a plumber is doing in your house. By educating yourself, you'll equip yourself to avoid scams or unreasonably high prices when hiring plumbers. In order to get the most efficiency from your hot water heater, it's important to learn what size tank your household actually needs. Pipe Installation A water softener can filter out unnecessary minerals, and leave you with clean, usable water.\nDon't just use any drain cleaner. , Even if you only have one noticeable problem, you can still get a plumber to inspect other areas. Also, combining smaller jobs into one larger job lets the plumber finish his work faster and therefore saves you money. Look at the overflow holes to ensure they aren't clogged up. These overflow holes are present in case of a clog or overfilling. It may seem unlikely as a problem now, but the necessity is there. When you do periodic checks for any other problems or repairs that are needed, take the time to clear the holes that are there to protect you from overflow. If you run your garbage disposal after working in the kitchen, use a large quantity of cold water. The blades will stay sharper, and the overall disposal process will run smoother. Grease will stay solid and go through your drain with cold water; you don't want gooey grease in your pipes. Having water leaking into the dishwasher, however annoying, is luckily an easy fix. The hose that runs from the kitchen sink to the dishwasher is not installed properly, and can be corrected quickly.\nObviously, if it is below freezing outdoors, the pipes will be subjected to the same temperatures. Plumbing And Sewer Forest Grove OR, A plumber that is not skilled in a particular task could actually aggravate the problem. It is always a good idea to get referrals from friends or family members, or to check out reviews about any plumber you are thinking about hiring. If you've got issues with your toilet blocking, see if the blockage exists at the branch line which connects the sewage and main lines. If you are unable to address this blockage on your own, you may require the services of a professional. The blockage will be cleared by running a snake throughout the line. As the weather is becoming colder due to the coming of winter, be sure that your external faucets do not have leaks or drips. If there are drips or leaks, then the repairs need to be make prior to any freezing temperatures. It does not matter if your pipes are steel or plastic, when water freezes it expands and will crack the pipes. Even a small crack will let enough water out to cause some serious damage. Most people don't know much about plumbing, so they pay ghastly prices for plumbing services. If you own your home, it is very important to have at least some knowledge about plumbing so you can understand what the plumber is doing.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Posted on September 17, 2014 by Kamal Govindaswamy, Co-founder and Partner, Tueoris LLC.\nLike many other Health IT initiatives today, the primary driver for patient portals is regulatory in nature. Specifically, it is the Meaningful Use requirements related to view, download or transmit and secure messaging. However, the biggest long term benefit of the portals might be what they can do for patient engagement and as a result, to the providers' business in the increasingly competitive and complex healthcare marketplace in the United States.\nThe objective of this post is to discuss the security aspects of patient portals, specifically, why the current practices in implementing these portals could pose a big problem for many providers. More importantly, we'll discuss specific recommendations for due diligence actions that the providers should take immediately as well as in the longer term.\n1. What are patient portals and what features do they (or could) provide?\n4. Why relying on regulations or vendors is a recipe for certain failure?\nThe bottom-line is that patient portals provide means for patients to access or post sensitive health or payment information. In the future, their use could expand further to include integration with mobile health applications (mHealth) and wearables. Last week's news from EPIC should provide a sense for things to come.\nAs we said above, the primary driver for patient portals so far has been the Meaningful Use requirements related to view, download or transmit and secure messaging. However, the biggest long term benefit of the portals might be what they can do for patient engagement and becoming a key business enabler for providers.\nThe portals are indeed a leading way for providers to engage with patients, as can be seen in this graphic from the 2014 Healthcare IT Priorities published by InformationWeek1.\nEffective patient engagement of course can bring tremendous business benefits, efficiencies and competitive edge to providers.\nA patient's greater engagement in health care contributes to improved health outcomes, and information technologies can support engagement.\nIn essence, the importance of patient portals as a strategic business and technology solution for healthcare providers doesn't need too much emphasis.\nIn my view, the below quote from the cover page of the 2014 Healthcare IT Priorities Report published by InformationWeek1 pretty much sums it up for this section.\nRegulatory requirements have gone from high priority to the only priority for healthcare IT.\nWe all know what happens when security or privacy programs are built and operated purely to meet regulatory or compliance objectives. It is a shaky ground at best. We talked about it in one of our blog posts last year when we called for a change in tone and approaches to healthcare security and privacy.\n4. Why relying on regulators or vendors is a recipe for certain failure of your security program?\nIt is probably safe to say that security in design and implementation is perhaps not the uppermost concern that HealthIT vendors have (certainly not the patient portal vendors in my opinion) today. To make it easy for them, we have lackluster security\/privacy requirements in the regulation for certifying Electronic Health Records.\nConsider the security and privacy requirements (yellow in this pdf) that vendors have to meet in order to obtain EHR certification today. You will see that the certification criteria are nearly not enough to assure the products are indeed secure enough before the vendors can start selling them to providers. And then, the administration or enforcement of the EHR certification program has been lacking as well in the past.\nThink about using these applications to run your patient portals!\nIf you are a diligent provider, you will want to make sure that the vendor has met the above requirements even though the certification criteria do not include them. The reality though may be different. In my experience, providers often do not perform all the necessary due diligence before purchasing the products.\nAnd then, when providers implement these products and attest to Meaningful Use, they are expected to do a security risk analysis (see the green highlighted requirement in the pdf). In my experience again, risk analysis is not performed in all cases. Of those that perform them, many are not really risk assessments.\nThe bottom-line? \u2026 Many providers take huge risks in going live with their patient portals that are neither secure nor compliant (Not compliant because they didn't perform a true risk analysis and mitigate the risks appropriately).\nIf you look again (in 1 above) at the types of information patient portals handle, it is not far-fetched to say that many providers may have security breaches waiting to happen. It is even possible that some have already been breached but they don't know yet.\nConsidering that patient portals are often gateways to the more lucrative (from a hacker's standpoint) EHR systems, intruders may be taking their time to escalate their privileges and move laterally to other systems once they have a foothold in the patient portals. Considering that detecting intrusions is very often the achilles heel of even the most well-funded and sophisticated organizations, it should be a cause for concern at many providers.\nWhat can or must we do immediately and in the future?\n\u00b7 If the portal doesn't provide risk based strong (or adaptive) authentication for patient and provider access, you may insist on the vendor committing to include that as a feature in the next release.\nb) If you didn't perform a true security risk analysis (assessment), please perform one immediately. Watch out for the pitfalls as you plan and perform the risk assessment. Make sure the risk assessment includes (among other things) running your own vulnerability scans and penetration tests both at network and application layers.\nc) Make sure you have a prioritized plan to mitigate the discovered risks and of course, follow through and execute the plan in a timely manner.\na) Implement appropriate due diligence security measures in your procurement process or vendor management program.\nb) Have your patient portal vendor provide you the detailed test results of the security requirements (highlighted yellow in the attachment) from the EHR certifying body. You may like to start here (Excel download from ONC) for information on the current certification status of your patient portal vendors and the requirements they are certified for.\nc) Ask the vendor for application security (static and dynamic) and pen test results for every release.\nd) Segment the patient portal appropriately from the rest of your environment (also a foundational prerequisite for PCI DSS scope reduction if you are processing payments with credit\/debit cards).\ne) Perform your own external\/internal pen tests every year and scans every quarter (Note : If you are processing payments with your payment portal, the portal is likely in scope for PCI DSS. PCI DSS requires you to do this anyway).\nf) Conduct security risk assessments every year or upon a major change (This also happens to be a compliance requirement from three different regulations that will apply to the patient portal \u2013 HIPAA Security Rule, Meaningful Use and PCI DSS, if processing payments using credit\/debit cards).\ng) If you use any open source modules either by yourself or within the vendor product, make sure to apply timely patches on them as they are released.\nh) Make sure all open source binaries are security tested before they are used to begin with.\ni) If the vendor can't provide support for strong authentication, look at your own options for proving risk based authentication to consumers. In the meanwhile, review your password (including password creation, reset steps etc.) to make sure they are not too burdensome on consumers and yet are secure enough.\nj) Another recommended option is to allow users to authenticate using an external identity (e.g. Google, Facebook etc. using OpenID Connect or similar mechanisms) which may actually be preferable from the user's standpoint as they don't have to remember a separate log-in credential for access to the portal. Just make sure to strongly recommend that they use 2 step verification that almost all of these social media sites provide today.\nIn summary, there is just too much at stake for providers and patients alike to let the status quo of security in patient portals continue the way it is. We recommend all providers take priority action considering the lasting and serious business consequences that could result from a potential breach.\nAs always, we welcome your thoughts, comments or critique. Please post them below.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaifxd b/data_all_eng_slimpj/shuffled/split2/finalzzzaifxd
new file mode 100644
index 0000000000000000000000000000000000000000..67d1511b2078ea8947e1e0c95e5d3f7cc9026e89
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"AHA Health is a Digital Health Technology Company, that aims to disrupt healthcare industry with a market value that will reach over $206 billion by 2020. AHA Health is building a blockchain-powered healthcare network where personal health records are stored in a distributed, decentralized, secure medium, access to which is controlled by patients incentivized with AHA tokens for sharing their information.\n\u00a9 AHA Health Ltd is registered and incorporated in England and Wales, company registration number 9118064.\n\u00a9 Copyrights AHA Health Ltd 2014 - 2019. All Rights Reserved.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Metropolis: Located just off the Granville Red Line stop adjacent to the Loyola University campus, Metropolis, a local favorite, is perfectly situated for remote work on the North Side. The location has a distinct \"college town\" feel, with a quiet ambiance that provides a comforting level of white noise. The location produces several of its own blends, and emphasizes fair-trade coffee roasted in their headquarters on the north river. Equipped with free Wi-Fi and a selection of pastries and sandwiches from local bakers, Metropolis has you covered for a productive day out of the office.\nIpsento: Offering a namesake latte made with coconut milk, honey and cayenne, Ipsento offers a variety of self-roasted artisan coffee. Situated in Bucktown, Ipsento features a minimalistic atmosphere unlikely to distract you from your tasks, and a variety of creative blends to get you through the day. If you're feeling adventurous, consider trying the cardamom latte and bring your laptop or tablet to Ipsento.\nThe Coffee Studio: Located on North Clark Street, The Coffee Studio offers free Wi-Fi and a cozy atmosphere. The Coffee Studio prides themselves on their selection of organic snacks, and many regulars recommend the sour cream donut. With plenty of available seating, you can be sure that you'll have a place to work even during the afternoon rush.\nBow Truss: Situated in a variety of locations including Roscoe Village, Lakeview, Logan Square, the Loop and the Near West Side, Bow Truss provides the necessary caffeinated drinks in a sleek environment equipped with wifi. With a rapidly expanding list of locations, including future shops planned in Uptown, River North and Wicker Park, it's easy to find a unique experience all across town. If you're looking for a coffee house that's serious about coffee and design, consider Bow Truss.\nGoddess and the Baker: Most coffee houses serve at least some baked goods, but Goddess and the Baker goes beyond the classic muffin or scone. With a thoughtfully curated breakfast and lunch menu and a pastry case overflowing with some of the city's hottest treats, you'll have everything you need to set up shop outside the office. Even better, they offer a loyalty program so that you can get more out of your remote work days.\nChicago is home to a wide variety of unique coffee houses across the city; there's no need to stick to the airport standbys with so many new experiences at your fingertips. For a change of scenery, consider working from one of the city's unique coffee shops.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"After successfully dropping a Video for his previous banging single titled Awulu, Quebex is back and drops yet another mind blowing tune which he titles \"Ma Fi Mi Sere\" which is a yoruba way of saying \"Don't Joke With Me\" or \"Nor Use Me Play\". Ma Fi Mi Sere is a Love song with tasteful Romantic lyrics on a mid-tempo feel good beat Produced by Vocis. Ladies and Gents, brace yourself... We've got another harmonious banger on our hands! \u2026Download, Listen, Enjoy & Share Your Thoughts!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Mehandi art is a vast subject. If you want to become a pro in this art, then it is necessary that you first get familiar with the basics of this beautiful art. All the mehandi enthusiasts who are keen on learning the art of mehandi, need to acquire the method of applying it. It is important to enroll yourself into a well designed structural course in mehandi in order to explore the different styles incorporated in this art. This course offered to you all is a comprehensive one that covers every bit of detail for a beginner to learn the art of applying mehandi. From the basic floral, peacock and mango designs, to the most intricate checkered peacock design, this course covers them all. Learn the simple filler patterns and get familiar with the shaded Arabic designs with the help of this marvelous mehandi course.\nHello, I am interested to join Mehandi Designing, Please do the needful at the earliest.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This might seem a bit self-serving coming from a site that offers seo services, but it does have merit.\nIn todays online world, a plumber can have a Google Places page ranked and would never see the need to have a site, much less an optimized one.\nYou could get by by creating a YouTube channel that displays all the services and past work you've done to demonstrate your abilities.\nYour Facebook page would have all of the important information needed to promote your company.\nAnd your Twitter page would just automatically gain followers magically!\nThe truth is, the plumber who has a seven pack ranking in Google places didn't get there by accident. They are actively engaged in SEO. Either they are performing it themselves, or have hired someone to do it for them.\nMost people know they need to make their content \"search friendly\", but do they really know what they are doing?\nHow many people know the proper use of the H1 tags and family? What about meta tags? Micro-formats?\nHow about the server? What's the proper way to pass link juice from an old page to a new one?\nWhat about the copy that appears on the page? How many words is the right amount?\nDon't forget about those social buttons \u2013 you know where you link, share, vote, stumble and reddit everything?\nIn the end, it's all about one thing.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzajnam b/data_all_eng_slimpj/shuffled/split2/finalzzzajnam
new file mode 100644
index 0000000000000000000000000000000000000000..20b22e3c5339fc2c2608fce7fcbfb5618c0997bb
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzajnam
@@ -0,0 +1,5 @@
+{"text":"Swifts Over Macclesfield \u2013 Canalside Community Radio Ltd.\nThe Swifts Over Macclesfield group includes members of the RSPB Local Group, the British Trust for Ornithology, Cheshire Wildlife Trust, the Barn Owl Trust, as well as bird box builders and swift surveyors. Any local residents interested in getting involved with the group's work can contact it through the local RSPB's website (ww2.rspb.org.uk\/groups\/macclesfield\/contact), to find out more.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"What We Fail to Realize about Valentine's Day.\nValentine's day, another American holiday that provides a lot of depression for many due to the romantic isolation of companionship. There is a lot of pressure for this holiday because of that possible void in our lives. There are numerous people who shrivel up and crawl into their empty homes and wallow alone on this very day, feeling left out on all of the romance in the air. But as we sit, talk, and listen to our shacked up friends exclaim their excitement about their plans for the holiday of romance, one can only feel somewhat a bit of lonesomeness in their heart. But, what we fail to realize is that Valentine's Day is not about romance, it is about love.\nRomance is just a nice subsidiary of how we are supposed to celebrate this holiday, and it has taken over the spotlight from love over the years through the consumeristic culture that we live in. As we dine, perform grand gestures, write love letters, sing songs and slowly kiss our loved ones under the stars, single people all over the world forget to celebrate love. Romance is for couples, and love is for everyone.\nAphrodite, The Greek Goddess of Love and Beauty.\nThe greatest example of this is grade school. Do you remember when your third-grade teacher had your whole class make Valentine's cards, or have Valentine's Day card and candy exchange? As children, the lesson that we forget in this is that we are spreading love to one another, not romance.\nSomeone once spoke about a remarkable woman, who battled various illnesses, cancers, and terrible health conditions for over 40 years. Due to her health, she had no time to pursue any romances, but she still had the time to spread love to everyone that was in her life. When she passed away after a life filled with horrific health, hundreds of people showed up to her funeral because she offered so much love to everyone who entered her life. This woman's life is the epitome of what Valentine's Day is actually about. If she could devote her entire life to spreading love and joy, then we should start to think differently about Valentine's Day.\nEveryone has the power of love to spread in this world.\nDespite the first dates, frantic searches for a Valentine, witnessing proposals, and seeing roses pointed in the noses of loved ones everywhere, Valentines Day is a day to spread and give love to others.\nCall your family members and tell them that you love them, or let your friends know how much that you appreciate them. Use Valentine's Day as a reminder to think about who loves you in this world. The holiday does not always have to be about stupid arrows flying around that impales romance to the victim. Maybe those arrows should strike the appreciation for love to all kinds of people in all of our hearts.\nIf you do not have a romance in your life, remember to spread the love on Valentine's Day to the people who matter in your life.\nOne of life's biggest tragedies is that perhaps human beings get so much enthusiasm, energy, and ire from hate. This is why we need to be reminded that a purpose of humanity is to love whoever is around to be loved.\n\u2190 Weighted Blankets are Life Changing.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Today's (24.05.2018) Ikyawann episode starts with Satya makes a plan for Leela. He makes the radium powder spread on the floor so that he can get Leela's footprints. He aims to follow and track Leela's hideout. Satya keeps an eye on Leela and finds her in the kitchen, stealing food. He wishes to run and catch her to ask about Susheel. Leela leaves from there, unaware of the foot prints. Satya gets puzzled when he sees the footprints reaching his room. He doesn't understand where did Leela disappear. He tries to check the wall. He feels he isn't reaching to any end. Leela spies on Satya. She asks him what is he up to. He tells her that he has lost to her. She tells him that she knows this very well. He asks her to stop such gimmicks so that they can end the fight and live happily together again.\nHe wants Leela to pamper him again. He requests her to forget this hide and seek. She tells him that she knows he will ask about Susheel and try to get her arrested. He tells Leela that he assures her that none will complaint against her. He asks Leela to come back to them once. Sejal and Kali also request Leela not to end everything by her hatred. They remind the family moments to Leela and make her emotional. Kali asks Leela to come back for Satya's sake. Satya tells Leela that he is waiting for her.\nSejal tells Leela that they will stay with her happily if she leaves the crimes. Leela gets influenced seeing them. Satya tells them that they will listen to Leela, as she is freeing Susheel now. He wants Susheel to get freed. Kali wishes Susheel is fine. She makes prayers for Satya and Susheel. The family tells Leela that they are waiting for her and doing the arrangements to welcome her back. Satya tells Sarthi that he will take against Leela back. He assures that Leela won't hurt Susheel now. Leela tells Satya that she got emotional talking to him about the new start, but she isn't a fool to free Susheel. She aims to show them dire consequences. She scolds Satya for valuing Susheel more than her. She gets threatening them again.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"MARGATE, Fla. - State inspectors ordered Tumi Restaurant in Margate shut again last week after an inspector found a roach issue still existed.\nTumi was ordered shut in December for the same issue.\nIn which restaurant did inspectors find live rodent, rodent droppings?\nIn fact, records show Tumi was shut for three days in December because of the roach infestation. They were also closed January 2, 3 and 4.\nRoach issues were also found at Silver Spoon Thai in Miramar and Korner Cafe in Fort Lauderdale.\nNo restaurants were ordered shut in Miami-Dade or Monroe counties last week.\nBelow is a list of places that were ordered shut and some of their violations. All the places mentioned were allowed to reopen following an ordered cleanup and re-inspection.\n\"Roach activity present as evidenced by live roaches found. Observed 4 crawling on the wall at the three compartment sink, and about 4 in a box on a shelf above three compartment sink.\"\n\"Pesticide\/insecticide stored with\/above food, clean equipment and utensils, clean linens and\/or single-service items. Observed 3 cans of Raid insect spray on shelf next to vinegar seasonings.\"\n\"Employee eating in a food preparation or other restricted area.\"\n\"Employee filled water pitcher\/cup at hand wash sink. Water poured into pot with rice on the stove. Operator poured water from the pot and catch water at three compartment sink to cook the rice.\"\n\"Roach activity present as evidenced by live roaches found. Observed 2 live roaches in wait station under ice machine, 1 live roach on wall near electrical control box in kitchen above prep table, 1 live roach under dish machine, 1 live roach on wall above pick up window, 1 live roach on shelf above steam table in kitchen, 1 live roach on floor next to produce cooler in kitchen, and 1 live roach on floor in front of produce cooler. Operator instructed to remove roaches and to clean and sanitize area. They began to do so. **Repeat Violation**\"\n\"Dead roaches on premises. Observed 6 dead roaches in wait station on floor around shelf and chest freezer, 2 dead roaches on shelf across from wine in wait station, and 1 dead roach on shelf above steam table in kitchen. Operator instructed to remove roaches and to clean and sanitize area. They began to do so. **Repeat Violation**\"\n\"Employee touched face\/hair and then engaged in food preparation, handled clean equipment or utensils, or touched unwrapped single-service items without washing hands. Observed cook touch face with gloves on and then continued to handle clean tongs and pan to cook. He was instructed to wash his hands and change his gloves before continuing to prepare food, he later did so.**Repeat Violation**\"\n\"Employee washed hands with cold water. After touching face with gloves on, cook washed hands with soap and cold water, he was instructed to turn on both faucets to wash hands in water that's at least 100\u00b0F. He did so.\"\n\"Stop Sale issued on potentially hazardous (time\/temperature control for safety) food due to temperature abuse. Observed in wait station: butter pats (58-64\u00b0F - Cold Holding), about 10lbs. They have been in the beverage cooler overnight, operator instructed to discard them and did so.\"\n\"Roach activity present as evidenced by live roaches found. 5 in the cabinet under the coffee machine behind the front counter. 1 on the floor under the cooks line grill. 2 in the cabinet under the front counter. 1 on the wall next to the 3 compartment sink. . 1 behind the self-serve ice cream chest freezer.\"\n\"Roach Tracking powder pesticide used inside cabinets and behind refrigeration in the establishment.\"\n\"Dead roaches on premises. Observed 23 on the floor in the area of the soda syrup bag in the box storage rack. 2 on the floor between 2 reach in soda coolers in the dining area. 1 on the floor by a window in a prep area. 1 on the floor under the meat slicer. 2 in the cabinet under the POS register. 1 under the steam table. 3 in the cabinet under the coffee machine behind the front counter. 3 in the cabinet under the hand sink behind the front counter. 6 under the 3 compartment sink. 1 on the floor under the large flip top cooler on the cooks line. 6 in the cabinet under the front counter cold Buffet. 3 under the self-serve ice cream chest freezer.\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Over in India, there are a handful of startups working to get the country's billion-person population on bikes. One of them is Yulu, co-founded by Amit Gupta, who formerly co-founded profitable, billion-dollar startup InMobi.\nIn the next three to four weeks, Yulu will launch its bike-sharing service in Bangalore with a few hundred bicycles. Yulu is starting with regular, dockless bicycles that people can rent using their iOS or Android phone. The bikes unlock via QR codes and bill the rider for every 30 minutes of use.\nIn order to achieve scale, Yulu is working with local city governments in India to create more dedicated bike lanes as well as working with agencies to carve out places that are safe to park the bikes.\nBike-sharing has become a hot space for founders and VCs alike. In China, companies Ofo and Mobike have both achieved unicorn status for their respective bike-sharing startups. Over in the U.S., investors have poured in $62 million to LimeBike and $8 million into Spin as both startups aim to compete against Motivate, the company behind Ford's GoBikes in San Francisco and CitiBikes in New York City.\nMeanwhile, startup Ola recently announced its entrance into the bike-sharing space with a pilot program in Faridabad. ZoomCar has recently done the same.\nA few things have happened in recent years that are making it possible for bike-sharing to become a reality in India, Gupta said. There are, for example, now 300 million people on smartphones with active data connections, more people getting on board with a cashless society and more emphasis on building smart cities, Gupta said.\nYulu has a co-founding team of four people who have self-funded the startup with some help from friends and family. Yulu has enough capital to get it through the next six to nine months, but plans to raise money from institutional investors down the road.\n\"We will start to talk to institutional investors once we understand our operating metrics and how demand and supply comes together,\" Gupta said.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzakhpz b/data_all_eng_slimpj/shuffled/split2/finalzzzakhpz
new file mode 100644
index 0000000000000000000000000000000000000000..97c9b134a2eb0b79cf408f215c9e91b32f5d01e4
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzakhpz
@@ -0,0 +1,5 @@
+{"text":"Kenneth D. Adams was born in Montreal on October 10, 1923 and died on July 30, 2003. He attended Sir George Williams College in the 1940s and obtained a bachelor of science degree in 1947. In 1944, with help of other students, he created the Music Society and was its first president. He was also elected chairman of the Student Undergraduate Society 1946-1947. This society organised social activities during the school year.\nAfter obtaining a licence in music from McGill University, in 1950 Ken Adams joined Sir George Williams University as a music teacher. He also worked part-time in the bookstore, the accounts office, and the office of Convocation and Examinations.\nIn 1965 he was named assistant registrar, joining the University administrative staff full-time. In 1968 he was named associate registrar and the following year he was promoted to registrar. After the 1974 merger of Sir George Williams University and Loyola College to form Concordia University, he became registrar of Concordia University. He retired in 1988 after working for 19 years as registrar, the longest term in Quebec for that position.\nKen Adams continued to teach the music course Understanding Music as long as he worked as university administrator and after retiring.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"V-neck cardigan in fine knit wool. The cardigan has a v-neck and a slightly oversize fit. The thick rib structure adds a soft and cozy look to it. The cardigan can be worn with or without the matching belt and is suitable as a jacket in the chilly season.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Suvarnabhumi airport terminal is a single buidling where arrival hall is on 2nd floor and departure on the fourth floor.\nIf you have someone to pick you up, meeting point is near gate 3 on you right-hand side when you get into the arrival hall.\nOtherwise, there are several ways to get to the city.\nThe train link between Suvarnabhumi airport and central bangkok which offer 2 options for you, the express line and the city line.\nFor express line, train leave every half an hour then run non-stop in just 15 minutes from airport to City Air Terminal in Makkasan and cost 100 baht.\nThe city line , commuter train, run between airport and Phayathai in about 30 minutes with 8 stops along the route. Promotional fare until end of 2010 is 15 baht.\nBoth lines operate between 6.00 - 24.00 hr.\nAt both end of the train, there are space for laggage.\nTrain station located on basement of airport terminal.\nFollow sign down to the station.\nThere is no direct access from Makkasan to skytrain or Metro system. But for Phayathai, there is path link between ARL and skytrain which will be open later this year (2010).\nTo get the bus, go down to the 1st floor.\nThe bus is clean and very comfortable with luggage rack at the middle.\nTaxi stand is on 1st floor so from arrival hall you have to go down one level.\nWalk outside and queue for public taxi.\nState your destination at the counter and they will write it down in Thai in the slip. One part is for taxi driver, other part is for passengers with complaining information. Keep this part with you and don't give it to the driver.\nAll the taxi here are metered taxi, so insist that the meter must be turned on.\nFare to downtown should be around 300 baht but can be vary up to your destination and traffic condition. Meter start at 40 baht and running up according to distance and time if standstill in the traffic.\nAdditional airport surcharge 50 baht must be paid on top of taxi fare to taxi driver.\nToll fees also paid by passenger. If you go to downtown you have to pass 2 toll gate. The first toll is 25 baht and the second one is 45 baht or total of 70 baht.\nThere are several route intend mainly to serve airport workers so they have no luggage rack on the bus but if you travel with a backpack you can take this option as the way of cheap transportation.\nPassengers who need bus services have to go to bus terminal.\nThe front of the bus has some space for your backpack.\nTo get to and from bus terminal, there is free shuttle bus service running around the clock at the interval of 5 to 20 minutes.\nGetting to the bus terminal, you can wait for the shuttle bus at 2nd and 4th floor. The bus stop located between gate 5-6 on both floor. But for arrival floor, the bus stop is on the outer lane.\nBuses waiting at the bus terminal.\nSome buses running round the clock but leave once an hour during midnight till 5 a.m. The other run from 5 a.m.- 11 p.m. Bus fares may vary between 34-36 baht.\nDetail of each bus route can be found at AOT Site.\nTake bus no.556 if you want to go to Khao Sarn area or Southern bus terminal at the cost of 34 baht.\nYou can see that bus no.556 use different kind of bus with limit space for luggage so take seats on the right near the door then you can get off the bus easier.\nIf you want to take this bus to Khao Sarn but worry for where would you get off the bus, just take a right-hand seats (of course near the door).\nWhen you see this monument, ring the bell and get off at the next bus stop.\nYou will be at Rajchadamneon Ave., then walk ahead to the intersection.\nWalk into Tanao road just for a few steps.\nEntrance to Khao Sarn road will be on your left.\nBTS and MRT have several interchange stations so check with map at the station for the nearest one.\nMinivans is another options here at bus terminal. They run the same route and carrying same number as the buses but with faster service as they less stop to pickup passenger along the route or use the expressway.\nFor example, Minivan no.551 run on the expressway all the way to its destination (Victory Monument) so you safe a lot of time at the cost of 40 baht, just a little bit higher than 35 baht for the bus.\nMinivans don't provide luggage space. It's a problem if you travelling with a large one but for a backpack you can put it at the space behind the driver's seat.\nFare is 40-70 baht up to the route. Buy ticket at the driver.\nAt the bus terminal you can also catch the inter-city bus running to eastern part of Thailand like Chonburi,Rayong, Chantaburi,Trat (Kho Change) etc.\nTwo routes are originated from this Suvarnabhumi bus terminal.\nBuses to other destinations are originated from North-Eastern bus terminal (Morchit) or Eastern bus terminal (Ekamai) so you need to buy ticket at least half an hour before departure time so they can inform the bus in time to pick you up.\nTo get inter city bus, you may buy ticket and wait for the bus at this counter near the gate 8 on the 1st floor.\nThank you very much! Your explanations are very, very helpful.\nAmazing, this visual guide to Bangkok provides explicit ways for different directions where we don't need to wander to find a specific spot. Car Parking Luton has also made life easier by enriching travellers with healthier parking options.\ngatwick meet and greet is a reliable source to get your car parked in a secure area and enjoy your journey without having to worry about anything at all.\nNice post. can you please update the latest information today.\nBangkok Airport is very beautiful. I have good experience about it. Thanks for sharing this, it is really very awesome.\nThanks for your complete guide on Bangkok tourism. This kind of guides are mostly helpful for the travel lovers and whom who are planning to have a trip to that particular place.\nNice post with beautiful pics and useful information. I am highly impressed, keep sharing. Thanks.\nBook DC to NYC Bus Tickets online. Compare bus operators, bus schedules, and find bus tickets prices from DC to NYC Bus. No booking fees!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Former cohost of Radio Beirut's weekly StepFeed news segment, Alexandra Talty has appeared on television and radio. If you are interested in booking, please contact her directly at actalty (at) gmail.com.\nTravel + Leisure interview on getting the most out of travel.\nFormer cohost of StepFeed news segment on Radio Beirut's \"Beirut Banter.\" Appeared weekly from January 2015 - May 2015.\nHour-long live segment on Blog Talk Radio's \"Travel Itch Radio.\" July 2018.\nKCBS interview on Forbes article, \"Baby Boomers Are Retiring Abroad In Droves.\" July 2018.\nWGN Chicago interview on Forbes article, \"Why Kanye West's $53M In Personal Debt Is A Lesson To All Entrepreneurs.\" February 2016.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"More interesting way to discover other charming of Hue ancient capital is to make a cycling trip around the countryside. Spending half\/full day on bike by yourself, you will embark to the rural to admire and enjoy interesting scenes and local life. This tour gives you more the tranquil, peaceful, beautiful sightseeing in different villages surrounding Da Nang including green paddy fields, rivers, hill, handicraft villages, and old architecture, which Da Nang people keep in original appearance until now.\nCycling trip around the countryside of Da Nang, embark to the rural to admire interesting local life style.\n8.00 Meet our guide at your hotel, be ready for exciting morning and start biking the route to visit Vong Canh hill (7km). Stop to visit Nam Giao, the Alter of Heaven to learn more of Da Nang culture in worship. Going to Vong Canh hill (meaning a Viewpoint hill, used to be the place for the Kings to admire the scenery), you enjoy the best view to picturesque of Perfume River and Pine Trees.\nCycling back and continue visiting an incense-making village and hat-making village. On the way to return Da Nang city center you will wind about 3km road for more different villages. Going on to visit Voi Re temple (\u0110i\u1ec7n Voi R\u00e9), a royal temple where the kings' war elephants were buried and worshiped. Visit Tiger Right relic nearby (Ho Quyen, Tiger Fighting Arena) where elephants and tigers dueled to the death. Under the Nguyen dynasty, this is the arena of elephants and tigers to sacrifice in festival and served as of recreation for the Kings, the officials and residents. Here is the chance for you both explore hidden charm of Da Nang countryside as well as learn more of Nguyen regime.\nThe last site on your cycling day is bronze-casting villages in Duc Ward. On the way back to hotel, you can see more the hidden sites of Da Nang city. End of half day cycling tour at noon.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaljas b/data_all_eng_slimpj/shuffled/split2/finalzzzaljas
new file mode 100644
index 0000000000000000000000000000000000000000..d8a8f0754ffc4f7eaa3ff870a8bfba50b7b49614
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaljas
@@ -0,0 +1,5 @@
+{"text":"\"Preheating and Affleck-Dine Leptogenesis After Thermal Inflation\" by Gary Felder, Hyunbyuk Kim et al.\nPreviously, we proposed a model of low energy Affleck-Dine leptogenesis in the context of thermal inflation. The lepton asymmetry is generated at the end of thermal inflation, which occurs at a relatively low energy scale with the Hubble parameter somewhere in the range 1 keV \u2272 H \u2272 1 MeV. Thus Hubble damping will be ineffective in bringing the Affleck-Dine field into the lepton conserving region near the origin, leaving the possibility that the lepton number could be washed out. Previously, we suggested that preheating could damp the amplitude of the Affleck-Dine field allowing conservation of the lepton number. In this paper, we demonstrate numerically that preheating does efficiently damp the amplitude of the Affleck-Dine field and that the lepton number is conserved as the result. In addition to demonstrating a crucial aspect of our model, it also opens the more general possibility of low energy Affleck-Dine baryogenesis.\nFelder, Gary; Kim, Hyunbyuk; Park, Wan-Il; and Stewart, Ewan D., \"Preheating and Affleck-Dine Leptogenesis After Thermal Inflation\" (2007). Physics: Faculty Publications, Smith College, Northampton, MA.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Select Vestry appointed Howley Hayes Architects to advise on and to supervise the restoration of the building in 2014. A detailed report was prepared outlining the action to be taken. Since the project commenced in 2014, the parish has raised \u20ac1,159,825 which includes a substantial donation from a private contributor, grants from the Representative Church Body Fabric Fund, Dublin City Council, the Heritage Council, All Churches Trust, and tax back on all donations over \u20ac250.\nThe project continues to be organised on a phased basis. The church is now weather and watertight. The tower work is due to commence in November 2017 and will continue for approximately eight months. Meanwhile, the sanctuary and the nave have been conserved and all necessary painting is complete. The choir area and the arch will be conserved after the tower phase is complete. The north and south transepts will be refurbished when the level of damp in the wall is reduced to an acceptable level.\nSecure the structural shell of the building. The tower will have to wait until there are funds available to cover the work.\nConserve the interior of the sanctuary and choir (and associated services). This is the next phase of the project when the north nave is completed.\nConserve the historic textiles used in the services.\nThe Select Vestry is being advised by its Fabric Committee established in 2013 to be responsible for the building and its contents. The members are Alistair Rowan, Mary Heffernan, Padraig O'Rourke, Fr. Andrew McCroskery, and Barbara Bergin.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The people have spoken: Doug Ford's Ontario PC Party has won enough seats to form a majority government.\nNow comes the actual governing part.\nEach of parties' platforms were laid out, some in more detail than others, during the campaign and the key planks were hammered out in debates and at doors across the province.\nOn hydro: Lower rates by 12 per cent; get rid of Hydro One president and CEO Mayo Schmidt; and cut executive salaries.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Staining for CD24 detects mature granulocytes and normal mature and immature B-cells at all stages of maturation. It is also expressed in most mature B-cell and B-lymphoblastic neoplasm. In our laboratory this marker is used for defining (gating on) the mature granulocytes to assess for antigen loss as a part of the PNH analysis.\nTesting for this antigen is typically performed as a part of the PNH analysis. For customized testing panels including this antigen, please contact the Medical Director of the Flow Cytometry laboratory.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Cheap car insurance quotes MN. Find the Best AUTO INSURANCE now and SAVE Big!\nWith all carriers in Which an insurance professional. It is also sums up persistent savings in terms of rates. Some driver believes they don't need to apply it to match your home country's licence is valid for people who have ailments generally do not abide by the Fonds de garantie picks up this year. Potential Tesko loans gives their clients, either in not. It is advisable to carefully consider the expenses incurred exceed the burden and time consuming. An array of automobiles\/combine insurance for personal or commercial use? Many of their interest. If your car's security.\nWindshield wipers are not 25 years of experience, ideally someone with experience in the state. With this broker is the first step you should type in \"car given your quotes, you gather, the more traffic you have steered them in a few of the vehicle is not mine, it carefully.\" For those shopping for automotive insurance companies also know the terms listed. Wal-Mart have recently added Cheap car insurance quotes MN companies to check your driving habits.\nAs you make the necessary insurance. You can call your broker to get significant discounts. For those of us still use that is mandated by law and that you get only A's and B's in school. Little dings, cracks and dings than replace a windshield, and some may go online, search for the insured vehicle. Eventually, it gets to people with different deductibles. Now, the companies that specialize in providing their customers premium costs. It can cover a variety of remote keyless system gives the insurance policy. Get you places really fast, but they bought from a cheap car insurance quotes MN. This notion is insane and from the depreciation value. Thus, when you buy a big glass window and some merchandise that was on. Because of a claim will affect your insurance premiums. Not all insurance companies to collect a higher deductible, you could pay monthly, bi-annually, or annually in exchange for coverage.\nThis kind of site to get because the higher price quote so you better enjoy the game if you have a good idea to be covered at all times, insurance companies is a computer constantly selecting license plate is. By the nip-hop culture, I still see people gouge people when faced with that question is, how they are just behind the wheels of an accident or a new car ownership is on the type of insurance package you need to do a bit of comparing various quotes for the prospective clients. Because teenage drivers are young and inexperienced (and often easily.) a cheap car insurance quotes MN is too low. Will you be required to have at least 100 miles from home usually drives much less of whether you are able to get cheap car insurance quotes MN.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzalkey b/data_all_eng_slimpj/shuffled/split2/finalzzzalkey
new file mode 100644
index 0000000000000000000000000000000000000000..19e56f3728680b45b5117731daf0f844ef3324ca
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzalkey
@@ -0,0 +1,5 @@
+{"text":"RCS' Angela Poyet with our new friends at Hala FM and Bliss in Jordan, working now with GSelector4 and Zetta, great radio around the world with RCS.\nBarrie, Ontario \u2013 Full house at CCBE while Diana Stokey presented. The 62nd annual gathering brings together the best of Canada's broadcasters.\nThe guys of RCS in front of the Glowing Screen of Zetta\u2026 Mark Bolke, Barry Hill, Philippe Generali and Neal Perchuk.\nPeru's Minister of Communications (on the left), Mr. Raul Ricardo Perez Reyes Espejo visiting our booth at ExpoTec Peru.\n\"The Sound's\" afternoon jock, Andy Chanley, using Zetta \u2013 at KSWD-FM 100.3 The Sound.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Kitaro the Vampire Slayer is the fifth volume in Shigeru Mizuki s bizarre stories about a yokai boy and his entourage of otherworldly friends. One of the most popular Kitaro tales, the title story pits Kitaro and his family against a Beatles-inspired mop-topped, guitar-playing vampire named Erit and his castle of horrors. In an unexpected twist spoiler alert! Kitaro meets his demise, but even death is hardly enough to keep our favourite yokai boy down. Along with the villainous vampire, Kitaro faces a trio of monsters straight from Japanese folklore. Mizuki often pulled from classic Japanese folk tales for inspiration, as shown in these three stories. certain serpent and temple bell appear in The Phantom Steam Engine, then it s onto a bird-like creature with a big beak in Ubume, and Kitaro had better not look behind him when the Ushiro Gami comes to town.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"When I studied abroad I did a lot of travelling, so naturally, I learnt what does and doesn't work when travelling in terms of clothing and outfit choice. So I thought I'd share my findings with you right in time for any of you jetting off for a pre-christmas break or those students escaping the city on their reading weeks.\nYou're potentially going to be sitting around for hours in airports, stations, trains or airplanes, so you want to feel unrestricted and comfortable. Leggings are a girls best friends in this respect \u2013 stretchy, warm and a little more dressy than jogging bottoms (although I love these ones from Calvin Klein). Plus, subscribe here and get 10% off!\nLayering up is a good idea when travelling as when you are rushing around and carrying luggage you can feel warm and overheat, whereas aircraft cabins air conditioning can be cold and leave you feeling chilly. So pair leggings with an oversized tee like this one from Adidas (steal one from your boyfriend if you don't have your own), a comfy cardigan or jumper and throw in a blanket scarf for good measure (if can double up as a pillow or blanket for snoozing). This means depending on the temperature you can alter your clothing to what feels best for you.\nI know magazines are filled with the likes of Victoria Beckham or Kim Kardashian teetering through the airport in their sky-high stilettos but honestly it's unrealistic. Maybe the rest of you are better women than me but I sure as hell can't do it and neither should you, not when there's plenty funky trainer options for you to explore (like these gorgeous platform ones from Puma). You're going to be on your feet all day and you can use this as an excuse to experiment with trainers or comfy sandals (if you're heading somewhere hot) like these super cute Lily Lulu ones.\nAs for your carry-on luggage, stick to an oversized slouch or shopper bag, like this one from Oliver Bonas which is big enough to carry all of your belongings you will need, while still adhering to the airline's carry-on guidelines (check this before you go).\nEssentials include: a good book, iPod, face wipes, lip balm, sunglasses and a bottle of water.\nAlways be prepared and read my post on my handbag essentials here.\nWhat are your globe-trotting tips?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Lately, I've been inspired by some work I'm doing during the day, layering color over color to create geometric textures. It got me thinking. I love the way in which patterns and textures can mix up a print piece, a website, a smartphone wallpaper.\nA lot of times, I think it's what we're (I'm) missing when I put a piece together. Headlines\/taglines, logos, the right typeface and photos all present in this ad? Check, check, check, check. WHY IS IT BORING? Let's give it some texture.\nI decided to try a few new ideas out. I found color inspiration here (artist unknown) and started to experiment with the ways the colors play off each other in Illustrator.\nIt's mesmerizing how slight adjustments can greatly change the resulting texture. Have you ever done this?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Susan Schubert, born 1986, received her diploma in dance education at Palucca Hochschule f\u00fcr Tanz Dresden which was followed by a master\u00b4s study in dance sience at Freie Universit\u00e4t Berlin. Susan is a member of art der stadt e.V. Gotha and its ensemble with which she realized diverse full-length dance\/theatre\/pieces.\nSince 2012 she is working close with Cindy Hammer. Together they are leading the Go Plastic Dance Theatre.\nSusan also created her own productions, like the grass is green, green is the grass, SzenariNO and TWIN SPOTS. Moreover she was the artistic leader of the monthly improvisation evening at Projetkttheater Dresden. As dancer Susan appeared in pieces like ALICE II \/ Asphaltwunder und Sirenen ein Roadtrip and MIT ALICE IN DEN ST\u00c4DTEN. She is giving workshops in the fields of theatre, play, movement.\nSince 2016 she owns the TENZA schmiede, production center for the freelance dance scene and for dance education in Dresden Johannstadt.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzangww b/data_all_eng_slimpj/shuffled/split2/finalzzzangww
new file mode 100644
index 0000000000000000000000000000000000000000..f7b19b8e066a6b406087d8d5ad0dc572df821e13
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzangww
@@ -0,0 +1,5 @@
+{"text":"Here is an example of a YouTube video's URL for the Disney Infinity: Playsets video.\nTo embed this video in a post, you would take the code at end of the YouTube URL M6rNy6Mc4OA, and then select the youtube tag option.\nCopy that YouTube video code between the youtube tags.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Often times we put exercise at the bottom of our list of things to do because our schedules get jam packed with other priorities. Do you struggle every day with how to fit in a workout? In this workshop participants will learn how to fit exercise into their busy lifestyles, what are some common barriers participants face and how to count everyday activities as part of their fitness routine.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This post has been sponsored by Renew Life\u00ae. All thoughts and opinions are my own.\nAfter the first frost last year, Kyle and I realized we needed to make some changes to our lifestyle. When the weather's nice, it's a no brainer; we're outside hiking with the pups and riding our bikes. However, when it's cold, our Floridian bodies are tired and sedentary more often than not. Since there's no end of winter in sight, we've implemented a handful of small lifestyle adjustments to see if they'd make a difference. Fortunately, they have! I'm rounding up 5 of the smallest lifestyle changes that've already made a huge impact in the new year in today's post!\nTaking a daily probiotic, like Renew Life\u00ae Women's Care Probiotic, has had a greater effect on my health and wellness than I ever thought possible. Probiotics help restore digestive balance, ie. they make sure your digestive system is working as it should, and support immune and vaginal health to help keep you (and your gut) healthy. I've been taking these for about 60 days now, and I've noticed less bloating, haven't had any stomach issues, and am feeling more energized than ever. Plus, now that taking Renew Life Women's Care is part of my daily routine, I take a handful of other daily vitamins and supplements, as well. Wins all around!\nYou can pick up a 90-count bottle at Sam's Club, which is super convenient and eliminates the need to go out and buy a new bottle every single month. Plus, it's a hard deal to beat! Each capsule has 25 billion live cultures and 10 specially selected probiotic strains, easily making it the #1 Women's Probiotic (based on sales units). Sam's Club is the only place you can find the 90-count bottle, so click here to find a store near you!\nI'll admit, I have a hard time making new habits stick. So, after reading a book about building good habits (and breaking bad ones), I realized I needed to make them \"buildable.\" That means linking new habits to already formed habits. So, before changing into my PJs before bed (an existing habit), I lay out my work clothes for the coming day. This helps simplify my morning routine and ensures that I don't spend the entire morning in my pajamas, which, as a full-time blogger who works from home, happens more often than you'd think.\nI've always had trouble sleeping, so to help my body unwind naturally, Kyle and I have started practicing nighttime yoga before bed. Just 10 or 15 minutes of gentle stretching is all it takes, and so far, the improvement in my sleep has been remarkable.\nI've been wanting to fine tune my kitchen skills for a while now, so cooking at home has had multiple benefits. Not only is it a nice distraction from work and technology, it's been so valuable to know exactly what we're putting in our bodies. We can control ingredients, additives, sodium\u2026everything. Needless to say, eating clean has already reaped a variety of health and wellness benefits.\nFor the first time in years, I took two full weeks off work around the holidays. I disconnected from social media, turned on an OOO email response, and spent as little time on my phone as possible. Y'all, it was wonderful. Come January 2nd, I'd never felt so refreshed and eager to get back to work. So, I decided to start taking entire weekends off. I've already noticed that my productivity on Monday through Friday has vastly improved.\nAnd that's it, y'all! It really is amazing how such small lifestyle changes can bring you so much joy and energy in the long term. Now that we're actually feeling our best and sleeping more soundly, we've started working out more, getting more\/better work done, and actually have extra free time to spend with the pups! It's such a nice change of pace, and we love it.\nI hope these small lifestyle changes help you and your family as much as they've already helped mine. Cheers to a happy and prosperous 2019!\nYes! All of these ideas are so great and each and everyone is so important!\nThanks, Ashley! I may be biased, but I totally agree \ud83d\ude42 Happy new year!\nGreat ideas! I've been trying so hard to cook more meals! It saves so much money and is so much healthier.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The reason given for this is simple; the police would not like offenders who are absconding from the law to exploit the system. So, details pertaining to outstanding warrants from Teller County, CO are only offered to people who place a formal request for such data through the agency office.\nThis of course does not mean that there is no chance for individuals who do have a warrant out in their name to get information on such arrest orders. However, any search that you conduct will have to be done through a third party site. You can access a privately maintained repository of records by filling the form given on the site.\nOften for a job, candidates are required to submit a notarized copy of their background report; this can be procured from multiple sources. Similarly, if you are looking for Teller County, Colorado arrest warrants in the name of a person that you intend to hire, you can get this data from any one of the government agencies mentioned below.\nThe office of the county clerk that is responsible for trial records which are known as court dockets: 20 East St. Vrain StreetCripple Creek, Colorado 80903.\nEvery year almost 300 crimes are reported in Teller County, Colorado of which nearly 12% are violent criminal acts. From 2001 to 2008, the area did witness a rise of 8% in overall crime rates but this was balanced by the 20% plus decrease in violent crime.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The adventures of Jesse and the New Order of the Stone continue on March 29th with the release of Minecraft: Story Mode Episode 5, \"Order Up\". In addition to this announcement, developers Telltale Games and Mojang revealed that three-post season episodes are still to come that will carry on the adventures of the New Order.\nIn \"Order Up\", players will rejoin the New Order of the Stone as Jesse and friends resume their adventure following the Wither Storm finale of Episode 4. What begins as a treasure hunting expedition into an abandoned temple turns into geometric mayhem when the New Order winds up in Sky City. In this new world, the New Order will face decisions that weigh heavy on its reputation and the fates of the people of Sky City as a whole.\nWith new worlds comes new talent joining the cast of \"Order Up\", including Sean Astin (The Goonies, The Lord of the Rings), Jim Meskimen (Parks and Recreation), and Melissa Hutchison (The Walking Dead: A Telltale Games Series).\nFollowing \"Order Up\", three post-season episodes will continue to join the New Order of the Stone to even more unique worlds and the unexpected adventures they hold. These episodes, which release in the next few months, will be available as add-ons, meaning that players must own at least Episode 1 of Minecraft: Story Mode to play them.\nTo keep up with these coming episodes, Minecraft: Story Mode's official website offers everything a player could need.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzanrzk b/data_all_eng_slimpj/shuffled/split2/finalzzzanrzk
new file mode 100644
index 0000000000000000000000000000000000000000..36d5f6b325c40fb3957179c6d3a17deb3176ab4b
--- /dev/null
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+{"text":"ST. PETERSBURG -- Third baseman Evan Longoria joked at first about what it meant to hit the 164th home run of his career in the third inning of Saturday night's 16-1 win over the Yankees at Tropicana Field.\nIt was his second homer of the season, with the first coming back on April 3, against the Blue Jays. It had been 13 games and 49 at-bats since he went yard.\n\"I guess it was about time,\" Longoria said, smiling.\nBut that blast off the Trop's \"C\"-ring catwalk was more than just his long-awaited second homer of 2014. The two-run shot pushed him past former Tampa Bay first baseman Carlos Pena and established him as the franchise's all-time home run leader.\n\"It took me quite a while to do it,\" Longoria said. \"But it came in a game that we needed to win. This was a good statement win for us, to be able to come out against [Yankees right-hander Ivan] Nova, who has given us trouble in the past, and put together an offensive performance like we did today. It's one of those games where you hope it kind of propels you going forward. It was fun to be able to do it in today's game.\"\nBut for Longoria, 28, there was a greater significance to the homer.\nLongoria has signed two long-term contract extensions with the Rays since 2008, when he broke into the Majors. The second was a six-year, $100 million pact that could keep him with Tampa Bay through the 2023 season. When he signed that pact, he thought about what it would mean to become a Ray for life, to be the young club's first real franchise player.\nHe just began his seventh season in the Majors, and he is the only franchise home run leader in baseball currently active with the same team.\nIn that sense, finally etching his name atop the Rays' leaderboard meant quite a bit to him.\n\"I'm going to be here for quite a while. I guess if you're going to set [the record], set it early and try to set the bar high,\" he said. \"It's meaningful, because when I signed my contract, that was kind of the whole goal -- to allow this franchise to have somebody to look at for the long term, and when people think about the organization, think that you've had players that wanted to and have played their whole careers here.\n\"Regardless of the timeframe, I think that has just been my main goal, to kind of set that example for the organization moving forward.\"\nLongoria's first career homer came off Brian Bruney of the Yankees on April 14, 2008, during an 8-7 loss at Tropicana Field. In a way it's fitting that his first homer and the record-breaker came against the Yankees, considering how well he's hit against them. He has homered 26 times against New York, seven more than any other player in the Majors since 2008.\nThe three-time All-Star and 2008 American League Rookie of the Year has already accomplished quite a bit in his career, and manager Joe Maddon believes he is capable of much more.\n\"[When Longoria tied Pena], my first thought was to thank Carlos for all the great work he did for us. Longo's going to keep adding to that,\" Maddon said. \"I would say at least double that. At least.\"\n\"That's quite a task,\" Longoria added, smiling. \"I'm just going to try to hit No. 3 tomorrow.\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Yara Bernette (Bernette Epstein) was born in Boston in 1920. Her Russian immigrant family moved to Brazil when she was six months old. She began to study the piano at the age of six with her uncle, Professor Jos\u00e9 Kliass, pupil of Martin Krause, who in turn had been the pupil and secretary of Franz Liszt, and who was one of the most important musical figures in S\u00e3o Paulo at the time. In 1938 Yara Bernette made her professional debut in a concert with the Municipal Symphony Orchestra and in 1949 made her first appearance in the United States at the New York Town Hall, encouraged by maestros Arthur Rubinstein and Claudio Arrau. The American tour continued through Canada and Latin America: Guatemala, Puerto Rico, Venezuela, Colombia and Argentina. Her European debut was in 1955, when she played in Paris with the Orchestre du Conservatoire de Paris, directed by the Brazilian maestro and composer Heitor Villa-Lobos. On the same tour, she also played in Vienna, Amsterdam and London, where she was presented with the Arnold Bax Memorial Award as the best musical interpreter of the year. From 1957 to 1970 she gave numerous recitals and concerts throughout Europe, playing as a soloist with the Berlin Philharmonic and the Philharmonic of Hamburg, and also played in the Far East. In 1961 she represented Brazil in the Second Inter-American Music Festival in Washington, playing Camargo Guarnieri's Variations on a Theme from the Northeast of Brazil for Piano and Orchestra with the Rochester Symphony Orchestra. In addition to being a concert pianist in demand throughout the world, Yara Bernette was also a teacher; in 1972 she was chosen from among one hundred and thirty candidates as piano teacher at the Hamburg Music and Performing Arts School, where she remained for twenty years, supervising students from many countries. In the United States Yara Bernette was a jury member in important competitions such as the Van Cliburn International Competition. To commemorate her seventy-fifth birthday she recorded her Portrait for German television, broadcast throughout Germany on August 20, 1995. The commemorations were followed by a recital in Hamburg in which her ex-pupils from the Hochschule took part. In Brazil Yara Bernette frequently played in the main cities. She died in S\u00e3o Paulo in March 2002. Known throughout her career for playing modern and contemporary composers, she played music from the Classical and Romantic periods with the same skill.\nAmerican-born Yara Bernette, whose parents moved to Brazil when she was six months old, became a concert pianist and piano teacher of world renown (1920-2002).\nFalbel, Anat and Nachman Falbel. \"Yara Bernette.\" Jewish Women: A Comprehensive Historical Encyclopedia. 27 February 2009. Jewish Women's Archive. (Viewed on April 21, 2019) <https:\/\/jwa.org\/encyclopedia\/article\/bernette-yara>.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Ford Focus Msrp. ford focus msrp. 2017 ford focus electric pricing for sale edmunds. 2015 ford focus msrp check this msrp focus here automigas. ford focus 2002 msrp. 2019 ford focus 2016 hp dealers new msrp. 2018 ford focus st msrp ford trend. new 2018 ford focus se hatch msrp prices nadaguides. used 2014 ford focus se stock j5401a chapman automotive group. 2017 ford focus pricing for sale edmunds. new 2015 ford focus st pricing revealed for the uk autoevolution. [Apriliasxv.com].\nFord Focus Msrp Automobiles are one of the varieties of carry that folks discover to be able to really do the beloved at this point a days. There are actually individuals who are involved with motors and also absolutely love them and everything onpar gps and can continue speaking about these in the stretch. Using the advancement of better technology, a variety of extra features have occur and already your day's most present day autos thoroughly built with dozens of characteristics emerged coming in to the markets. Using the expensive amount arrive your great features, each staying one among any kind.\nLike a dwelling, a motor vehicle will be some of those investments that you'll make as part of your lifetime. For that reason it is significant that you simply do your behalf throughout talking the particular auto loan you are having and have the purchasing price just feasible, depending on the comfort and ease level. Take a moment with the sales people in addition to keep your disorders as well as rank facing these individuals, actually tell them specifically how much you can manage and the way a lot you actually are able to pay back, and also accordingly get the opportunity from there on. Be completely distinct right away with cash relevant issues.\nOne of the benefits of using the internet is that you might get your car you are wanting at a considerable more affordable charge compared to what you will be getting while in the showrooms. This internet in addition offers you the opportunity for staying away from your annoying product sales people that you would have to deal with. And so when you see using this point of view the idea is perfectly a win-win predicament that you are dealing with below, take a look at take advantage of the use of the online to provide a reasons that require these kinds of awareness like this Ford Focus Msrp?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Looking for consulting that delivers? Look no further.\nPlease fill out all required(*) fields !\nWhy not profit from our experience by getting in touch with us now?\nWe only accept the following file types: .doc, .docx, .pdf or .xls. The maximum file size is 8MB.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Who's in the mood for a stuffed avocado recipe? With chopped greek salad? And tzatziki sauce?\nGreek salad is one of my personal summer highlights. Most likely because I could eat blocks and blocks of Feta cheese by myself, but shhhh.\nI'm a serious snob when it comes to Greek salad, probably because I've been to Greece many a times, eating delicious local food in tiny little restaurants, getting a real taste of what Greek food is all about.\nOne of my favorite vacations to date was a trip to a tiny Greek island with my aunt, uncle and cousin. We flew to Athens, then took a 4 hour ferry trip to the island.\nWe proceeded to eat our own bodyweight in Greek salad and grilled anchovies, dried figs and goat cheese, all the local honey we could get our hands on.\nIt is where I learned how to make a mean Greek salad: Chunky, acidic from lemon juice, flavored with oregano. The feta cheese is placed as-is on top of the salad, drizzled with oil and a sprinkled with a little black pepper, olives are of the Kalamata variety and unpitted.\nThe perfect treat to enjoy during the hot lunch hour, out on the shady deck of a stone-built house, surrounded by the sound of the waves and the pink blossoms of Bougainvilleas.\nThis particular vacation took place almost ten years ago, so my life has significantly changed since then. And by significantly changed I mean I got married and had two kids.\nWich is why I now I have three picky babies to feed at home, ha!\nChunky salads are not something I make for my toddler, because she's a lazy chewer. It's something I majorly had to get over this summer, since it just doesn't make any sense to give her chunky, authentic Greek salad.\nMostly because she will claim the entire Feta cheese for herself.\nSo I started making a mean chopped Greek salad instead, something we all enjoy regardless of being authenticity-claim-checker approved or not.\nThe other day I went a little overboard with my avocado shopping, so I had a few in the fridge that wanted to get used. In a moment of pure bliss and genius, I stuffed some chopped Greek salad into avocados.\nTopped them with an equally inauthentic version of Tzatziki sauce.\nProceeded to eat myself to Greek heaven and back.\nAll three of my kids both my kids and my husband enjoyed them, the baby mainly eating avocado and tomato and the toddler collecting the olives from everyone.\nIf you're still looking for the perfect easy summer lunch treat \u2013 give these Chopped Greek Salad Stuffed Avocados a try. Note: Best eaten in a Greek goddess state of mind.\nThese healthy chopped greek salad stuffed avocados are a great summer treat for lunch. No cooking involved!\nCombine all salad ingredients (except for avocados) in a large bowl. Toss well.\nStir together all ingredients for the tzatziki sauce.\nFill avocado halves with the salad and top with the sauce. Serve immediately.\nThis is such a pretty recipe ~ sharing today!","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"If you've ever been in a long-term relationship (or shamelessly third-wheeled your BFF throughout hers), you understand the tendency for couples to start twinning, whether it's with awkward mannerisms or speech patterns. And when it comes to celebrities, the best type\u2014in our opinion, at least\u2014happens to be a predisposition to coordinating outfits, which, when done right, can induct A-list pairs into the Hollywood Golden Couple Hall of Fame. From Kim Kardashian and Kanye West's matching Met Gala metallics to Blake Lively and Ryan Reynolds' more subtle accent hues on the red carpet, here are the dynamic duos who slay together\u2014and thus stay together.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"87 Years ago today on August 24, Felipe Martinez founder of FVH was born high in the mountains of Jutiapa, Guatemala.\nHe spends most of his time in Guatemala city and in Huehuetenango. Most importantly he spends just about all his time with family.\nFelipe and his wife Marta with 2 of thier great grand kids Ana Regina and Sara Camila on his birthday\u2026 still full of joy, energy and laughter.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"-There was a wonderful, and powerful, sense of God's presence all through our worship service yesterday. From our music, to our prayer time, to an impromptu testimony, to the fellowship among the people following the service everything was impactful.\n-What is up with snow the first 2 Sundays in April? Last Sunday we got about 3 inches and yesterday we got about 2 inches. The good news is that both weeks it didn't start snowing until we were out of church.\n-The brief getaway that my wife and I planned was good, but not what we expected. We rented a cabin by a lake and hoped to do some walking and fishing. With snow on the ground the first 2 days and rain the last day, there was no walking and I attempted to fish for a few minutes. But being all bundled up and wearing winter gloves took a lot of the enjoyment away. Even with all of that we still rested, read and were somewhat refreshed.\n-On Friday night and most of the day Saturday, I was blessed to be a part of our annual District Assembly. It was an encouraging and motivating time of celebration. 8 new churches were begun this past church year and there are 13 more scheduled to begin in the coming church year! 5 people were ordained, and 9 people received there first District minister's license.\n-Our District Superintendent rocks! His report on Friday night was awesome. He shared good news, did not hide from difficult news and challenged us to keep trying new things and promised to keep being the support and cheerleader that we need. Love that man and his leadership.\n-On Friday afternoon I got the word that a lady in our church had passed away. She fought hard for 5 years after a difficult surgery for cancer. She was a sweet lady who did not allow her physical problems to keep her from serving. She was a regular greeter and helped with Bible School every year. She will be missed, but she is now enjoying her heavenly reward, and she is pain free. Her funeral service, which will be a celebration, will be this Friday. That will be the 9th funeral I will be involved in this calendar year. I am ready for the funerals to stop for a while.\n-I started a new group last Wednesday evening and another one on Sunday morning. Both groups had a great beginning with wonderful discussion and sharing. I love this congregation.\n-On Wednesday this week my wife is scheduled to undergo a double knee replacement. Prayers for her and the surgeon are appreciated. Last Wednesday night 17 of the children in our church, 4 years old through 4th grade, gathered around her and prayed for her and the surgery. How awesome is that? Our children are being taught well and they are beginning to lead us in faith and following the Lord.\n-Yesterday I began a new sermon series entitled \"Yes, You Can \u2026 Because God Will\". We will be studying the life of Daniel and some of his buddies as they faced difficult circumstances and how they made it through those circumstances. They did not always have great things happen to them, but they made it through it. Our theme is summed up in this statement: \"Yes, you can! No matter what you are facing, you can make it through it. No matter how you got here, you can make it. You can make it because God will be with you, God will give you the strength and God will guide you. It may not be easy, and it certainly won't be painless, but you can make it. Yes, you can, because God will.\" The sermon was well received, and I have been praying for this series for a few weeks. Praying that people would be encouraged and become courageous in facing difficult times.\n-Yesterday I heard from 2 different people, who are relatively new to our church, that the ministries, and people of our church, are making a difference in their lives. I love hearing those kind of reports.\n-I am currently reading a couple of very challenging books. I love to read, I love being challenged and I love when my reading makes a difference in my personal life and the life of my church.\nHere's hoping you find some encouraging words this week.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Party Name: THE PEOPLE, Plaintiff and Appellant, v. TOLUCA LAKE COLLECTIVE, INC. et al., Defendants and Respondents.\nJudge Panel: We concur: P. McKay, P. J. Ricciardulli, J.\nTOLUCA LAKE COLLECTIVE, INC. et al., Defendants and Respondents.\nAPPEAL from a judgment of the Superior Court of Los Angeles County, Melissa Widdifield, Judge. Reversed.\nMichael N. Feuer, City Attorney, City of Los Angeles, Asha Greenberg, Assistant City Attorney, Meredith A. McKittrick, Deputy City Attorney, and John Prosser, Deputy City Attorney, for Plaintiff and Appellant.\nEric D. Shevin and Stephen J. Fisch of the Shevin Law Group for Defendants and Respondents.\nA misdemeanor complaint was filed on June 20, 2014, charging defendants Toluca Lake Collective, Inc. (TLC), Frank Jay Sheftel (Sheftel), Hatteras Holdings, LLC, and Peter Welkin with operating an unlawful medical marijuana business (MMB) in violation of Los Angeles Municipal Code (LAMC) section 45.19.6.2, subdivision A, 1 on August 26, 27, and 28, 2013, and the illegal use of land (\u00a7 12.21, subd. A) based on the operation of the MMB.\nFollowing an evidentiary hearing on the applicability of the limited immunity provided in section 45.19.6.3, the court found defendants substantially complied with the requirements for the affirmative defense, including subdivision B's registration requirement contained therein. Based on its finding that the immunity applied \"as a matter of law\" to bar prosecution, and pursuant to Penal Code section 1385, the court dismissed all charges against defendants.\nThe People appeal from the order of dismissal, contending that substantial compliance was applied in error to the immunity requirement of section 45.19.6.3, subdivision B, and that the court abused its discretion under Penal Code section 1385. We reverse.\nOn May 19, 2015, defendants filed a motion in limine, seeking to allow evidence and argument at trial that defendants qualified for immunity from prosecution pursuant to section 45.19.6.3. Defendants asserted that TLC was open and operating prior to the passage of Interim Control Ordinance No. 179027 (ICO)2 on September 13, 2007, and that, after the passage of the ICO, Sheftel went to the city offices in the Van Nuys Government Center, on October 15, 2007, to register TLC as an MMB. He met with a city employee who was unfamiliar with the ICO registration process but who nevertheless assured Sheftel that he was \"in compliance.\" The employee printed and date-stamped \"a screen shot\" for Sheftel (exhibit A), as proof that Sheftel had registered TLC on that date. On August 25, 2008, the City issued an order to comply (exhibit C), requiring defendants to discontinue their operation of their MMB. Consequently, Sheftel met with city staff and filed a hardship application on September 8, 2008. After this meeting, TLC was placed on a list of \"Now Open Pre-ICO collectives\" (exhibit E). In May of 2010, defendants learned that the City had changed its position regarding TLC's ICO registration.\nOn August 25, 2015, the People filed their response, arguing that defendants could not demonstrate they were entitled to limited immunity under section 45.19.6.3 because they failed to satisfy subdivision B's ICO registration requirement. The People argued that the City Clerk had no record of an ICO registration packet for defendants; any filing in the city offices in Van Nuys was inadequate, as it was not the Office of the City Clerk; and the submission of a registration package in September of 2008 was untimely.\nThe court conducted an Evidence Code section 402 hearing on December 1, 2015, regarding defendants' compliance with the ICO registration requirement. The defense called Jose Zaragoza, a former tax compliance officer for the City, who worked in the Office of Finance at the Van Nuys branch in 2007. Zaragoza testified that at the time of his employment with the City he had no authority to act on behalf of the Office of the City Clerk, and there was no office for the City Clerk in Van Nuys. He further testified that he was unfamiliar with the ICO; had no recollection of Sheftel or receiving any documents from him; had no knowledge of receiving any filings for MMB's, other than business tax registration certificate applications; and did not recall ever forwarding any documentation to the City Clerk's Office.\nZaragoza identified exhibit A as \"a print screen\" of a computer system that had been used to enter business tax applications. He confirmed that the printout contained his signature and a date file stamp. He stated he did not know what prompted the printout and that the stamp did not indicate receipt of any documents. When the prosecutor showed Zaragoza an exemplar of a \"Medical Marijuana Dispensary Business Information Form\" (Business Information Form), he stated that he had never seen such a form before.\nThe hearing was continued to December 11, 2015. On that date, the defense called Sharon Dickinson, who worked for the City Clerk's Office. She testified that the City Clerk's Office was the \"receiving agent for the city and custodian of records\" for documents required under the ICO. She further testified that in September 2008 she met with Sheftel regarding TLC's registration under the ICO. Sheftel told Dickinson that he had filed the required documents with the Office of Finance in Van Nuys. When Dickinson called that office, she was told there was no record of such a filing.\nDefense counsel showed Dickinson exhibit B, which was a list of ICO-registered MMB's, dated September 9, 2008, and directed her attention to the last entry listing TLC as a registered MMB. Next to the entry, the following notes were made: \"9\/9\/08. Added to list. Application was originally submitted on 10\/15\/07, but was inadvertently misdirected by City staff. New location pending hardship application. 11436 Hatteras Street.\" Dickinson explained that the notes were added by Daisy Mo from \"code enforcement\" in reliance on Sheftel's \"word\" and based on his presentation of exhibit A. Dickinson was later instructed by \"Clerk Management\" to remove TLC from the list of registered MMB's, and on June 23, 2010, Sheftel was notified of the change.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I finally found the clip of Big Bang sending their love for the SNSD Charity thing. Sweeet.\ni like jiyong & tiffany, they're like my ultimate dream couple.\ndaesung & sunny or yuri because they can be like the smiling\/dorky couple.\nseunghyun and hyoyeon would be one really smoking couple.\nand maybe they'll realize how great soshibang is!\nSoo Young and Tekkie can be the main teachers since they both know Japanese well.\nlols. i think WONDERBANG is way much cuter.\nand i think its more of a super junior and snsd.\nand if u like Super Generation better u guys could start making the thread rather than just keep saying \"WB is better i like Suju+Soshi more\"\nThe thing is ... how can we judge if Wonder Bang is a better pairing since Soshi Bang didn't even collab together. That makes no sense to come in this thread, which was made for SNSD and BB only, and say that Wonder Bang are cuter or whatever, it's NOT the topic of this thread.\nhonestly, i never really knew they interact with one another beside when Seungri was doing the manwon thing with yoona.\nAnd I just want to say that they don't share the same kind of music. SNSD does like poppy girly songs. And BB do more of hip hop.\nLMAO \"so they can feel at ease\" hahas~ thanks for sharing the fancam!!!<3 i'm waiting for it to load now.\nOooh yeah, I forgot that in order to be friends, you have to do the same exact same kind of music. My fault for forgetting that.\nThis is so cute, I am all for the \"Soshi Bang\" combo!\nlol and once again thats what you think. haha if you just compare their images, .. then thats a bit shallow imo. And you need to watch more episodes of MDV because theres alot of soshi bang episodes.\nand the thing is .. grr i hate it when people say that the music they produce isn't the same, so they can't collab, thats extremely unfair to big bang and soshi, because i find both groups to be amazingly talented, and i can see them expanding to different types of genre. I mean who knew that daesung could sing trot, and taeyeon is actually quite amazing trot, but their images don't show you that. I love both of them because they're amazing live performers, and i believe that if they ever collaborate, it will be different and very amusing. BTW personality wise i think that SOSHI Bang is perfect, they are all such dorks, and i love seeing them interact with each other. And PS. haha you need to watch some of the girls' stuff, Hyoyeon is amazing at hip hop dancing, Yuri loves to randomly rap, & Sooyoung was in that very hip-hop style cf, its not like the girls can't be hip-hop, but i think its the differences in these two groups that make them so interesting.\nand om IDA, haha how could i forget !","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Philip and Savannah started Homecoming Moving Co. in mid 2018, to be a different kind of moving company. We are reliable, careful, skilled and above all else HONEST! Philip goes on every move and has well over a decade of experience. Savannah personally reaches out to each customer before each move and provides top of the line customer service. We bring all of the essential moving supplies, tools for assembly and disassembly. Two and Four wheeled dollies and an appliance or safe dolly upon request. We are 5 star rated on google and facebook. Book us today to see what the Homecoming difference really is.\nReal Tampa area customers. Unedited, unfiltered reviews of Homecoming Moving Co.\nThank You Robert. We really appreciate your business, and your review!\nGreat job, they went above and beyond in making our move comfortable, quick, and on time. Great crew! Awesome people!\nWe handle moves within 81 miles of 2403 South 47th St, Tampa, FL 33619.\nWe will call back withing 24 hours to confirm or adjust your times as needed. We will ask all the details of the move and give you a chance to ask us any questions or deal with any concerns.\nYou can expect to kick back and relax while we do all the work the right way! We will be on time and offer first class service! We will keep you posted on how your doing on time and if the time will need to be adjusted. Also we are the best in the business at loading trucks, PODs or trailers! You can rest easy knowing your belongings are packed safely, and no space will be wasted!\nMy suggestion is always have as much stuff packed and disassembled as possible. If you cant take it apart, dont you worry because we can! Also make sure your dresser drawers are emptied for the safest possible maneuvering of your dresser..\nPhil the owner has done commercial moves, in house moves, military vault moves, interstate, and pretty much any move you can imagine. Our helpers are strong and trained with years of experience. There are not too many moves we cant handle. 3 of our men unloaded 6000 plus boxes out of a shipping container in under 3 hours.\nI like meeting new people hearing about there adventures and travels.\nMy crew is trained by me the owner extensively. Best part about training is our customers will get an experienced hand for free while my newest helpers learn the Homecoming way!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Humans are creatures of habit, there is no denying it. We can easily get stuck within the daily grind where we gravitate to what we know. In our routine-driven society it is important to find fun and fresh ways to shake things up and keep you on your toes! An easy way to do this is to take advantage of the multitude of methods for smoking marijuana. You most likely know some of the various ways of kicking back and lighting up - but just in case you don't, we wanted to curate a comprehensive list of the best ways to do so. Hopefully you can learn about a new technique or inject some creativity and variety into your herbal routine. Let's get started!\nAn obvious classic, joints are one of the definitive ways to smoke weed offering endless options for sizes and creativity. The video above outlines a straightforward way to roll up, but the awesome thing about joints is the ability to customize them based on the smoking experience that you are looking for. Use king size papers that can hold more herb when smoking in a group, buy hemp-based and un-dyed papers for a natural experience, or even roll a skinny personal joint for a solo walk in the park. Joints can really be whatever you are looking for - adaptable, portable, and always reliable.\nYes, joint art is a thing. And it is awesome. Even though it doesn't necessarily increase the effectiveness of your sesh, it adds an undeniable coolness factor as well as a go-to party trick sure to impress. When you whip out a fresh braided joint (pictured above) at your next gathering, it automatically elevates the moment to a special occasion! Braided joints, tulip joints, and cross joints are some of the more basic and common designs, but check out Tony Greenhand's instagram for some insane inspiration!\nAnother obvious and classic option, blunts can hold a lot of herb and pack a serious punch. In the above video, YouTuber Nina Ross (check out her channel for more weed related tips as well as beauty and lifestyle vids!) shows you how to roll the perfect blunt whether you're a beginner or a seasoned roller.\nUsing a glass pipe is one of the quickest ways to light up: no need to construct anything, no complicated steps, all you need to do is fill the bowl with your ground herb and light it up! Not only are they easy to use, glass pipes come in a wide variety of designs, shapes, and sizes. There is even a huge community of glass artisans from all over the world that create smokable works of art that you can collect and display!\nOne-hitters are essentially straight conical pipes with an end-to-end channel and no carb. Consider them like a personal pipe: small, discreet, and quick. Check out the video from the tip above where The Stoner Mom (another awesome YouTube channel, check her out for reviews, tips, and ways to use cannabis responsibly!) shows you how to use a glass pipe and a chillum and how they differ. This is a great example of how you can customize your smoking method to your situation. If you only want a couple of hits or you're running out of time, just use a chillum and you'll be on your way!\nBongs are yet another indisputable method within the stoner lexicon, but for good reason. Because of the water and the path the smoke takes through the glass you can get huge, but smooth hits. Some bongs even have ice-catchers, allowing you to add ice to the smoke chamber which will give you a milky and refreshing inhale. There are a plethora of add-ons and modifiers that you can get to customize your bong-sperience from downpipes, to diffusors, to percolators, all of which can help you get what you want out of your sesh.\nPro-tip: when smoking your bong with hemp wick, dip the lit wick into the stem as you clear it - this will extinguish your wick automatically. Check out other tricks for putting out your hemp wick here!\nGas mask bongs initially look like something out of an apocalyptic horror film - but try not to get scared off. Because the mask attachment suctions to your face, it traps your entire hit (close your eyes to avoid getting smoke in them!), eliminating loss and allowing you to inhale everything. It is a fun, quirky method and it guarantees that you'll get very, very stoned.\nWhen faced with the scary situation of not having a vehicle to smoke your herb with, stoners tend to turn into innovative engineers. The homemade apple pipe is a standard solution that you can put together with household supplies - the video above uses a pen and toothpicks to transform the fruit into a smokable piece. Even better, the apple lends a slightly sweet taste to your hits and you can eat it after your sesh when the munchies kick in!\nGravity can do more than just keeping you grounded, it can also get you high. The gravity bong (some call it a \"bucket\") uses its force to both pull out a huge, milky hit as well as to push it all quickly into your lungs. This method allows you to get large hits from very little herb, proving to be economical and another example of something easy to make from household objects. Watch the video above for a tutorial on how to construct and use an easy gravity bong.\nSmoking hookah is a fun social activity to pass around at a gathering or party, why not spice it up with some weed? When putting your hookah together (watch the video above for tips and how-to) mix in your prefered amount of herb with your shisha (flavored tobacco) for a fruity and pleasant flavor with all the benefits of marijuana! Casually referred to as \"happy hookah\" you're sure to have a blast with this one.\nWe understand and appreciate the fact that each person is looking for a personalized smoking experience that correlates with their likes and needs. There are hundreds of ways to get creative with smoking, whether it is revamping a tried and true method or creating something completely new. As purveyors of authentic experiences through our hemp wick we love to try unique approaches to responsible marijuana consumption. Tell us your favorite ways to light up, and as always happy smoking!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"GARDNER \u2014 Local dispatch centers, including at the Gardner Police Department, are offering a new option for emergency situations \u2014 to text for help if a call cannot be made.\nThe service was deployed statewide on Dec. 14, according to the State 911 Department, which has begun to publicize it in the last few weeks.\nAccording to Gardner Chief of Police Richard Braks, the civilian dispatchers at the Gardner police station \u2014 who handle all emergency calls for the city now that the Fire Department no longer has firefighters on dispatch duty \u2014 were well prepared for the change.\n\"We have a full service dispatch center and they received the training they needed long in advance of it,\" Braks said.\nHe described the Text to 911 service as \"going very well\" in Gardner thus far, although he noted it has been rarely used by the public up to this point.\nA press release from the State 911 Department on Jan. 16 provided details on the new service.\nAccording to the State 911 Department, a voice call to an emergency dispatch center is the most efficient means of communication and should be done when possible.\nWhen texting is appropriate, sending an emergency text works similar to sending any other text. The sender is instructed to type the numbers 911 into the \"to\" or \"recipient\" field displayed on their cell phone.\nOne of the most important instructions from the State 911 Department is that people sending an emergency text should begin with describing the nature of the incident and be specific about their location, including the address and town, and any other information that could be helpful to locate them.\nDue to the different technology involved in sending text messages compared to making a call, an emergency dispatch center may not be able to pinpoint location from a text message as well as they would be able to with a call.\nAfter the dispatcher is able to gather enough information from the person sending the text about the location to send an emergency responder, the dispatcher will ask a series of questions via text just as they would with a call.\nHaving the option to text for emergency help eliminates the need for the hearing or speech impaired to use other equipment such as a teletypewriter or third party services to access 911.\nA few other tips from the state is to avoid sending emergency text messages with multimedia like pictures, videos or emoticons, avoid sending the message to more than one person, and to not exceed 160 characters per text.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Racing regulators kept hearing the reports: trainers were giving their horses a powerful performance-enhancing potion drawn from the backs of a type of South American frog.\nBut for months postrace testing could not find the substance, a painkiller far more powerful than morphine. Then a lab in the Denver area tweaked its testing procedure, and in recent weeks more than 30 horses from four states have tested positive for the substance, dermorphin, which is suspected of helping horses run faster.\nNo trainer has yet been formally charged, although racing regulators expect that to happen soon. Because of its potency and ability to affect the outcome of a race, the use of dermorphin is considered to be one of the industry's most serious drug violations.\nThe discovery comes as the racing industry is struggling to counter perceptions of a pervasive drug culture. Indeed, dermorphin is the latest in a long list of illegal performance-enhancing drugs that have found their way onto racetracks. Cobra venom has also been used by trainers to deaden pain so that injured horses can race. It functions as a local nerve block, unlike dermorphin, a broader pain suppressant that is 40 times more powerful than morphine, Dr. Barker said.\nIf horses cannot feel their injuries, veterinarians say, they are more likely to run harder than they otherwise would.\nSome say a culture of drugs and lax regulation at racetracks is behind the nation's high fatality rate for horses.\nMr. Stevens said dermorphin is found on the skin of a frog called Phyllomedusa sauvagei, commonly known as the waxy monkey tree frog, which is native to South America.\nOther performance-enhancing drugs found in racehorses include those used to artificially bulk up cattle and pigs before slaughter.\nHow often dermorphin is used in racing is not known \u2014 many states do not have the capability to test for it \u2014 but so far laboratories have found it in Louisiana, Oklahoma and New Mexico. Its use is also suspected in Texas. Some of the results were first reported on nola.com, the Web site of The Times-Picayune in New Orleans.\nIn Louisiana, Mr. Gardiner said, 11 horses, both quarter horses and thoroughbreds, tested positive for dermorphin, though none of them broke down. He said two quarter horses in particular earned big purses. \"A lot of money's got to be given back,\" he said.\nJockeys are on the front lines of the risky business of horse racing. Twenty-four horses a week die at racetracks around the country.\nDr. Barker of the Louisiana State lab said 15 horses in Oklahoma had tested positive for dermorphin. Oklahoma officials declined to comment, as did racing regulators in Texas.\nVince Mares, executive director of the New Mexico Racing Commission, said Tuesday that a California lab had found dermorphin in six postrace tests in New Mexico.\nIndustrial Laboratories in suburban Denver was the first lab to successfully identify dermorphin in postrace testing. It was not easy.\nPetra Hartmann, director of direct testing services for the company, said clients relayed tips from racetrack workers that the frog secretion was being used, and later some seized materials turned out to be dermorphin.\n\"We identified dermorphin,\" Ms. Hartmann said. \"We knew it was out there.\" But, she said, the lab's test could not identify the drug in horses after they raced.\nAfter racing regulators kept insisting that the substance was in use, \"We went back to the drawing board,\" Ms. Hartmann said. Industrial's chief scientist subsequently developed a more sensitive test, specifically for this compound, she said.\nMs. Hartmann said she did not believe the use of dermorphin was widespread. \"The vast majority of horsemen would never subject their horse to this kind of chemical experimentation,\" she said.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The rest of the band was somewhat more arbitrary, having gone through several bass players and drummers. In 1992, they signed JDon Is It Know Ask Post Your The What Juuling You Denver Kids 't \u2013 If ohn Hagan (of 2 P Archives Telegraph 1895 Alton 30 Evening Oct Newspaper 18xggS0Hn) to be their long-term drummer, a job he retains to this day. The bass player situation was never really nailed down.\nYour Denver \u2013 Kids If 't It Know Juuling Post What You The Ask Is Don The hard-rocking, grab-you-by-the-'nads (ow!) intro song from the Project Suze-ko album.\nPost What 't It Denver \u2013 Know Don Your Is If You Ask Kids The Juuling A slow acoustic track - vocals, two acoustic guitars, and a whole lot of 'space', used to achingly-beautiful effect. At least, in theory.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaovfo b/data_all_eng_slimpj/shuffled/split2/finalzzzaovfo
new file mode 100644
index 0000000000000000000000000000000000000000..afc87ab10b6b24d8db5404c5bf0b415e6b766082
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaovfo
@@ -0,0 +1,5 @@
+{"text":"No tickets matched all the numbers in last night's Powerball drawing, putting the estimated prize for Saturday night's game at 675 million dollars, which would be the largest lottery kitty in US history.\nOh yeah, I'll buy another one. Sure, the odds of winning it are more than one in 292 million, but for $2, I'll take that chance.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Trini Street Kitchen Pholourie without chutney is just wrong!\nServe with a chutney sauce or a peppery tamarind sauce. I made a killer sauce by combining equal parts mango chutney and water and pureeing them with 1 clove of garlic. This is the consistency of pholourie sauce that one usually gets when purchasing locally on the roadside.... 9\/07\/2012\ufffd\ufffd A quick delicious tamarind sauce or tambran sauce. Very popular in parts of the caribbean, You can eat this with Bara (see my bara video) or whatever!\nPholourie. Balls of split pea flour fried and served with a thin sweet chutney sauce (usually mango or tamarind) Cancellation policy . How to Sign-Up and Cancel Registration. Registration Deadlines Registration closes, for cooking classes, 24 \ufffd 48 hours before the class is scheduled. If the minimum number of participants are not met within 48 hours, the class will be canceled. All registered... This Pin was discovered by nazira erales. Discover (and save!) your own Pins on Pinterest.\n19\/04\/2018\ufffd\ufffd In a sauce pan put the tamarind and pour water till the tamarind are immersed. After It boils for a while and cooks well the seeds starts coming out and the tamarind mushes. Remove all the seeds from the tamarind pulp. In a mixer put coriander leaves , garlic,habanero Chilli with some water make a coarse paste. Add sugar, salt, coriander mixture,pepper pd and mix them well. Cook the \ufffd how to respond to forcing bid with minimum count Serve with a chutney sauce or a peppery tamarind sauce. I made a killer sauce by combining equal parts mango chutney and water and pureeing them with 1 clove of garlic. This is the consistency of pholourie sauce that one usually gets when purchasing locally on the roadside.\nServe with a chutney sauce or a peppery tamarind sauce. I made a killer sauce by combining equal parts mango chutney and water and pureeing them with 1 clove of garlic. This is the consistency of pholourie sauce that one usually gets when purchasing locally on the roadside.\nMake the tamarind stir-fry sauce by mixing all ingredients together in a cup. The tamarind paste and sugar should more or less dissolve in the stock and fish sauce . If your paste is thick, you may want to heat up the stock to help dissolve it.\n26\/02\/2018\ufffd\ufffd Pholourie is a fried spiced snack made from a mix of all purpose flour and gram flour, very popular in Trinidad and Tobago, Guyana and Suriname.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Havana, Mar 17.- The most popular and cherished Cuban animated film character, Elpidio Vald\u00e9s, is celebrating his 45th birthday in August with the cartoon Elpidio Vald\u00e9s ordena Misi\u00f3n Especial (Elpidio Vald\u00e9s Orders a Special Mission).\nThe premier will close a series of Cuban film industry initiatives to celebrate the character and its creator, Cuban National Film Award winner, Juan Padr\u00f3n. The cartoon will be another episode of the Elpidio Vald\u00e9s saga that began in 1970: three long feature films and some 20 cartoons whose protagonist is the Colonel who fought against Spanish colonial rule as a member of the island's liberation or Mamb\u00ed Army.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Burpees are done in descending order. OHWL distance and KB Swing reps remain constant. Round 1: 10 Burpees, 25 ft OHWL, 5 Swings, 25 ft OHWL. Round 2: 9 Burpees, 25 ft OHWL, 5 Swings, 25 ft OHWL. Round 3: 8 Burpees... etc, until you get to Round 10: 1 Burpee, 25 ft OHWL, 5 Swings. Rx weight for women is OHWL-35lbs, KB Swing-16kg.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Applications, tax preparation worksheets and other documents that you may have provided for use in preparing your tax return.\nTransactions with others or us.\nNo personal information will ever be disclosed about you to anyone, except that which is already non-public or permitted by law.\nShould you close your account with us, or become an inactive client, we will continue to adhere to the policy regulations as written here.\nThe information you give us about your personal or business records is reserved only for our employees who need to know in order to service your account.\nProcedural physical and electronic safeguards will be maintained in compliance with Federal standards regarding non-public personal information.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzapjwl b/data_all_eng_slimpj/shuffled/split2/finalzzzapjwl
new file mode 100644
index 0000000000000000000000000000000000000000..60ef5a1ace0d967f8ab2f2081c8bd1e9ecdf1f69
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzapjwl
@@ -0,0 +1,5 @@
+{"text":"Our specialist team delivers CCTV installations for all types of residential and commercial property across London, Surrey & the South-East. 12 month guarantee on all CCTV systems we install.\nMore and more people are considering CCTV installation to safeguard their home and family, encouraged by the Government and the Police.\nAt Surrey Fire & Safety, our experts install CCTV for all sizes and types of property: commercial or domestic, large or small.\nWhether you're looking for a basic, entry level system, or all the bells and whistles, we can help.\nWe work with all kinds of CCTV systems and budgets in London and the surrounding areas, and pride ourselves on our quality of work and friendly, efficient service. We keep our prices keen too.\nWe only use High Definition cameras and we install to British Standards, so you can be confident that your CCTV system will work when you need it most.\nWe also make sure your CCTV installation complies with UK CCTV legislation.\nFor extra peace of mind, we also give a 12 month guarantee on all CCTV systems we install.\nTo discuss what you need, just book a FREE SURVEY with one of our CCTV installation engineers on 0845 845 3001 - there's no obligation and we'll get a quote to you within just a few days.\nTo talk to our friendly team about CCTV installations, upgrades, maintenance or repairs, just call 0845 845 3001, or email us at info@surreyfire.co.uk. We'll be happy to help.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Are you ready to start a project immediately? Use these tips to help you accomplish the task of finding a good fence and gate repair installation contractor.\nWhen you need a fence and gate repair installation contractor you don't want to look too desperate. Make sure you let the contractors you are considering know that you are going to be interviewing several candidates. You want to make sure they understand you know what you are doing.\nThe internet is a great resource when it comes to searching for a respectable fence and gate repair installation contractor. There are many features on the internet that can help you in finding a contractor and that will also provide you with honest reviews. You can usually look up past work examples and ample feedback, too.\nThe lowest bid might turn out to be the best as it would save you on the amount of money to spend on the work but it is not always the best option. You should therefore re-consider such a fence and gate repair installation contractor and weigh your pros and cons cautiously. Such persons might end up messing you up by not doing their work professionally.\nPrepare a contract draft and present it to each potential fence and gate repair installation contractor while at the first consultation. This will help you to see the differences in several bids based on the same agreement. Comparing these different bids will show you the best offer and help you in making a definite hiring decision.\nWhile you are interviewing your candidates, make sure you come off as the boss. Make it seem like you've done all of this before Cheap Randy Johnson Jersey , don't let them dictate the direction of the interview. Let every candidate know that you've got other options before you begin the interview to let them know you are serious.\nLocating and choosing a quality fence and gate repair installation contractor is the most important early step towards successful completion of your improvement project. Your success depends Cheap Don Mattingly Jersey , in no small part, on your ability to pick the best contractor. Proper hiring will give you a better chance of succeeding and ending up with the result you intended free of problems.\nYou are spending a lot of money on your project so it is important to make sure your fence and gate repair installation contractor is giving it the attention it deserves. Check in with you contractor every once in a while to determine that everything will be completed on time and to keep everything on track.\nThe web is a treasure trove of information with all websites claiming to impart the best information. You will find a lot of tips on how to recruit the best fence and gate repair installation contractor but you have to careful to pick one who knows and understands your job perfectly.\nWhen you are curious about the topic of fence repair service, go ahead and visit Yahoo and search for snellville fence company. You'll be satisfied to know you did!\nFUZHOU Cheap Paul O'Neill Jersey , China, April 24 (Xinhua) -- Riika LehtonenTaru Lahti from Finland fended off the competition to enter the second round of the single elimination at the 2015 FIVB Beach Volleyball Fuzhou Open on Friday.\nLehtonenLahti have time to prepare for the knock-out phase after beating Germany's Chantal LaboureurJulia Sude 2-0 (21-16, 21-13) in their concluding pool match.\nWith two wins each for LehtonenLahti and LaboureurSude Cheap Jorge Posada Jersey , the final pool match would decide the winner of Pool C. As the sun emerged through the clouds in Fuzhou, the two teams took to center court with little to separate the two in the first set.\nLaboureurSude had a strong season together in 2014 climbing the podium at the Durban Open with silver. The two teams displayed similar determination with accurate dinks over the net.\nA couple of German service errors at the start of the second put pressure on LaboureurSude as LehtonenLahti took advantage of the situation. Close teamwork by the Fins set up for some great third touches and LehtonenLahti went from strength to strength taking the win 2-0 (21-16, 21-13).\n\"We're feeling pretty good because we have never beaten that couple before and of course we are coming from difficult year. We worked a lot and have big goals and now it's nice that we are in the shape that we can start to challenge those couples. We played well tactically and kept our tactics throughout the game and that worked out and our side-out was good Cheap Don Larsen Jersey , so we can just be happy about the result, but it's not finished yet,\" said Lehtonen.\n\"It was a good game with our good side-out and block defence. They made a lot of service mistakes which helped us Cheap Johnny Damon Jersey ,\" said Lahti.\nIn 2014 LehtonenLahti's best results came towards the end of the year at the Xiamen and Mangaung Opens where they finished ninth.\n\"It's a little different than last year because basically I didn't have any practice behind me, we just started to play and that you can absolutely see in the game,\" said Lehtonen.\n\"For sure we will scout the other players before tomorrow because we haven't seen so many couples so we will see more about our adversaries and then to rest and keep going day by day Cheap Thurman Munson Jersey , game by game and ball by ball.\n\"We always search for the best, we have the kind of character that we are not satisfied with anything less than the best so that' s why we really need to focus on the moment and stay there, you cannot think too much further Cheap Brian Roberts Jersey ,\" she added.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Equity research, in very simple terms, comprises of three basic activities \u2013 data collection, data analysis and report writing. All the three activities are of paramount importance in successfully conducting equity research.\nData collection and data analysis is time consuming. Moreover, market movements and trends change daily, even if by a fraction. It is essential for the equity research analyst to lay hands on current data, analyse and report. Time is of essence. Instant access to data, therefore, is necessary.\nFurther, to complicate matters, there are hundreds of websites which can give you access to this data. However, it is practically impossible to visit each website daily and gather information. What you need to do is use tools which filter daily data and gives you access to only relevant ones.\nHow to Access SPSS and SAS Software?\nCommon financial knowledge mandates that usage of various tools and techniques happens only at the analysis stage but it is not completely true. There are various tools available for equity research analysts that can be used for data collection and report writing as well.\nToday, in this article, I am going to tell you about 5 useful tools for equity research analysts. They are free of cost, easy to setup and can be used at various stages of equity research.\nIf used prudently, the Rich Site Summary (RSS) feed technique is a very helpful and time saving tool for data collection.\nTypically, the data collection for equity research is done using a fixed set of websites.\nBy using RSS feeds, you will get instant access to all the data related to your company or industry.\nIf you register RSS feeds on a domain, it removes the human element of manually checking uploads and saves a lot of time. Whenever the website\/s on which you have signed up for RSS delivery publishes information related to your industry interest, an email containing the information is automatically sent.\nAll you have to do is access the email, read the information and filter it for your use.\nVisit the site you wish to receive information from and look for RSS tabs like Join RSS feed, FeedBurner signup for or the daily email alerts box. Just type in your email id and name (if asked for) and you are done. Sometimes, the websites send a confirmation email. Just visit the email inbox and click on the confirmation link.\nYou are all set to receive filtered information.\nGoogle Alerts is a free but extremely useful service provided by Google (of course!) that gives you daily, weekly or as-it-happens instant email updates of your created alerts.\nActivating Google Alerts is good in addition to RSS feeds, which will periodically send you data uploaded on the internet related to your company or industry. It is a notification service that gives you info on data uploaded by sending out periodic e-mails to the registered email id.\nIt hardly takes a minute to setup Google Alerts. Visit their main page, linked above, and type in your research parameter. See this example screenshot.\nAs visible, you can customize it according to the result type, language and a specific geographical region. Once setup, click Create Alert and see the information flowing in. For more customized data filtering, check out the Google Alerts Cheat Sheet. It mentions many parameters you can use for customized data display.\nLastly, you can make use of Yahoo! Pipes, which is a similar tool like Google Alerts but is better with drawing internal search parameter connections.\nExcel is the most commonly used tool for any kind of financial analysis across the world and equity research is no exception. This tool is usually used for small and medium data sets to run financial analysis.\nThere are numerous functions and formulae in Excel that are very popular amongst equity analysts across the globe.\nThere are many open source tools like Open Office Spread Sheet, which are also used for such equity research analysis, but not as widely as Microsoft Excel.\nStatistical Product and Service Solution (SPSS) is a very popular statistical tool used by researchers from various backgrounds for data mining, data analysis, text analytics, data collaboration, surveys and deployment. The SPSS software is usually used where the data sets are large and usually captured in a pattern.\nThere are innumerable functions in this tool which create various charts and graphs depending on data analysis. The tool also enables the researchers to represent the analyzed data and patterns in a pictorial manner.\nBetween Excel and SPSS, the second one is far more complex to deploy. Even though an equity research analyst will require Excel training, it requires a little more training for the same analyst to use SPSS properly.\nDeveloped by the SAS Institute in California, the Statistical Analysis System (SAS) is a software suite used for data management, business intelligence and predictive analysis by many companies and analysts.\nUnlike Excel or SPSS, SAS not just performs data analysis but also can be used very effectively for data collection and analytics. However, a working knowledge of programming language is important for you to fully use the potential offered by this tool.\nIt does not mean that if you do not know programming languages you cannot use SAS. With a little bit of training, you should be able to use it as effectively as any other tool.\nThe SPSS software is available for a trial download at IBM whereas the SAS software needs to be purchased. Before buying any of them, make sure to undergo training. Without proper training, either of them won't be of much use.\nI hope this article helps you in doing equity research better in future using these tools.\nIf you have any questions related to the tools for equity research analysis discussed in this article, please feel free to write back and I will be happy to answer all your queries.\nSAS is in Cary, NC.\nI find this site really interestng and knowldgable bt knw more abt how to enroll for courses mentioned over there.\nGlad that you liked the site. Which course you want to enroll into?\nGreat article but think would have been better if you should how to go about or log on to these sites\/programmes.\nWhat's your suggestion on this? Can I help?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Lioness - is a high quality 3d model to add more details and realism to your rendering projects. Lioness 3d model in obj, fbx, dae, 3ds formats.\nLioness - is a high quality 3d model to add more details and realism to your rendering projects. Fully detailed, textured lion 3d model. Detailed enough for close-up renders. Comes with detailed textures.\nAll formats are exported directly from 3ds Max.Every model has been checked with the appropriate software.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Watch & Download Sasirekha Parinayam (2009) : Full Movie Free On Putlocker HD Independent spirited Sasirekha (Genelia) is ordered by her father to get married to an NRI groom. When the groom's father demands dowry on the wedding day, she runs away from her marriage. Soon after she ran way, she runs in to Anand (Tarun). They bond well during their acquaintance and the rest of the movie is about how she falls in love with Anand. As it turns out, Anand was the one she was supposed to marry, but due to certain events, the wedding party leaves without him. Sasirekha was to see a video of the groom and his house, but stamps on it, declaring she doesn't want to get married.\nWho's The Cast on Film Sasirekha Parinayam 2009??\nWhen Sasirekha Parinayam to release? ?","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzapkgx b/data_all_eng_slimpj/shuffled/split2/finalzzzapkgx
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--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Painted on 18-inch by 24-inch .080 Aluminum speed limit sign. Louise was hand painted using automotive urethane with a clear coat finish and features a 1967 Chevrolet Corvette. There are many qualities in the original painting including the reflective sign base that make the original unique. These unique features can not be reproduced in the prints and make the original sign one of a kind.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The new Lojel Cubo range is a high volume smartly packaged suitcase. The simply efficient Cubo welcomes contents of all shapes and sizes \u2011 and its clever top opening ensures full access even in tight spaces. An expansion system offers the extra room you'll need to include your finds when heading home. Made of Advance PC (light, strong polycarbonate) with anti-scratch texture, your Cubo is secured with Lojel's tamper-proof double coil zipper and TSA lock system.\npact and it is built to withstand low temperatures.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A musical comedy-drama which originally focused on the high school glee club 'New Direction's competing on the show choir competition circuit, while its members deal with relationships, sexuality and social issues. By season 4 a lot of the show's familiar faces have graduated, so the show shifted it's focus to following the graduates, along with the school's new recruits.\nSince Sky took over airing the show from E4, it's come under some unfair criticism for it's scheduling. Season 2 aired in January 2011, and people complained about the long wait from the US air date. So for season 3 Sky tried to air close to the US air date, but people complained about it jumping weeks \u2013 caused by the gaps in the US scheduling\u2026 US scheduling can seem somewhat erratic, as the US networks will bump shows so they don't get scheduled against things like major sporting events, or over US holiday weekends.\nReally stuck at people complaining either way, Sky seem to have decided it's better to wait and air Glee in a solid block from January, rather than to cause confusion by airing close to the US and having to skip weeks.\nNow renewed up to season 6 and final season, likely to to air in May 2015.\nThat's great. Wish it was coming sooner.\nAfter emailing and getting a email reply from sky they confirmed to me that glee season 6 will air in the UK in the summer time after season 5 was aired in the summer and they are trying to follow a pattern.\nwow I so hope that is true because I am getting Frustrated waiting.\nSomeone from sky say when glee season 6 is going to start.\nPlease when does glee season 6 start on sky 1.\nIt's already on episode 8 this Friday and it's the wedding please tell us when it is going to come back on.\nCant wait for season6 to come back. Bring back all the orgiinal cast for a one big performance. and all the guest stars. That would be cool.\nWhen dos glee season 6 Start on Skyone.\nWhen does season 6 of glee air in the UK ??\nGlee season 6 starts on the 9th of jan but that is in the usa but when is it shown in the uk.\nAny idea as to whether they'll be repeating the first block of season 5 before showing part 2 in July?\nWhat date is glee starting in July.\nWhen will series 6 of glee to be shown in Britain please?\nIt will be after season 5, part 2. Please see above.\nI also received a email from sky \u2013 never miss for glee season 5 for a Sunday morning but it's not season 5. It's season 4 that they are showing. What's going on with sky??\nGlee starting in July can't wait.\nwhen Does Glee season 6 Start.\nSorry guys I'm gutted too.\nI wished glee stayed on E4 as it was better and the viewers were treated better and since it moved to sky one we have been messed about..\nI don't, I'm afraid. Sky said later this year, so any refinement would be pure speculation. It could be anywhere from May to December.\nMy sky planner says it is season 5 episode 9 on Sunday at 8am, is that definitely wrong??\nwhen is season5 of glee starting on Sky 1.\nanyone else heard news of this?\nAny news on glee's Return.. Better be soon.\nPlease can you please find out when it is returning in Ireland because I can't find it anywhere and I don't think I can wait any longer.. Please!\nNever thought I would see the day that I would be rooting for Fox but for purely selfish reasons I am. I want to support a show I love and its creators by watching it legally and having the option to buy stuff associated with it.\nIMO this legal battle is stupid especially since the judge thinks the club haven't been hurt by this. I'd never even heard of this club until I looked this legal battle up, maybe I'm just cleverer than most but I would never think there was a connection between the two myself.\nNote to self: never go to The Glee Club ever.\nwhen is glee starting on sky one in march?\nIm seeing things bout glee all over youtube but when i check sky 1 it aint on so whens it back on like wa date??\nDo you mean the airdate or whether it will be screened at all?\nWhen is glee season 5 Part 2 starting on SKY One.\nAt the moment there is little known if\/or when the rest will air. With the who legal battle over with and fox losing it at the moment it stands that Glee could be cancelled on UK Telly, although Fox are fighting the choice since they don't see it being fair.\nomg. i am so confused.\nWhen does glee start back in the uk and what channel ??\nWe are expecting it back next month, sky1. See list above.\nLots of people tweeting to sky1 for an airdate. They are just replying saying it will be this year but dont know when!!\nI don't think a broadcaster can just take it back from another. The licences are often subject to bidding and it is extremely unlikely that a broadcaster would lose the licence to another broadcaster simply by not broadcasting content at the same time as it is broadcast in another country.\nAs it airs annually in January, the wait between series is no longer than for viewers in the programme's home market of the US.\nHowever, I wonder if viewers in Wales, Northern Ireland and Scotland also have to wait until January, as the date is likely to apply to the whole country rather than just one part of it.\nJust to let everybody know that Sky has said at least the first 3 episodes will air within 7 days of the US showing them for the first time on American TV. This is to make sure that we don't have to wait for the Cory Monteith episode that many people want to watch.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Our long sleeve cotton tees in three colors. Down the right sleeve reads \"Discover the Legend\". Available in sizes S-XXL.\nJust love sharing your restaurant with others in the Puget Sound! Can't wait to visit you personally.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"CORE\u00ae is based on the nutritional philosophy that pets thrive on a diet that is mainly comprised of meat. Our CORE\u00ae meals, mixers and snacks are crafted with lean protein sources to support an active lifestyle and promote lean muscle growth. With more of the protein your pet craves and no grains nor fillers.\nYour dog instinctively craves raw protein. CORE\u00ae RawRev\u2122 combines high-protein, grain-free kibble with pieces of 100% Raw Meat that are freeze-dried and never cooked to retain their valuable nutrients, natural enzymes and full flavor. The result is a savory meal that delivers the raw protein your dog craves and the balanced nutrition he needs to thrive from the core. Welcome to the Raw Revolution!\nWith Wellness\u00ae healthy isn't hard. Our Complete Health\u2122 wet cat recipes combine natural, premium proteins to deliver balanced diets full of the nutrients your cat needs for a lifetime of wellbeing. We have dozens of taste-tested flavors and textures without fillers or artificial ingredients; it's easy for every cat to find a favorite.\nGreat nutrition is the foundation of health. Wellness\u00ae Complete Health\u2122 recipes combine premium proteins and the finest natural ingredients to deliver a balanced diet packed with the nutrients your dog needs for a lifetime of wellbeing. With thoughtfully-prepared, delicious recipes for dogs of all sizes and life stages, we know you will find a recipe you will love and your dog will crave.\nGive your cat the purr-fect treat with Wellness\u00ae Kittles. With 6 delicious flavors to choose from and only 2 calories per treat, you won't feel guilty showing your kitty some love. These crunchy bites in 3 fun shapes are not only delicious, but help clean teeth too! Your cat will be sure to thank you again and again.\nWe believe treating your dog should always be a bonding experience. That's why Wellness Snacks are 100% Grain-Free, Guilt-Free Goodness. Your dog benefits from a reward that's made with natural nutrition like high quality proteins and antioxidant-rich fruits and vegetables, and you benefit from all the happy tail wags that follow.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzapsim b/data_all_eng_slimpj/shuffled/split2/finalzzzapsim
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+{"text":"A much requested, client-facing presence, Jack also views it as his job to remove all paper products from trash barrels and to dispatch \"leftover\" sandwiches and snacks from desktops. \"Want a treat?\" is his favorite expression, and he would really prefer that more people brought their socks into the office.\nSlightly vain about his appearance, Jack loves his twice weekly brushing and his bi-monthly spa day shampoo and blowout. Originally from a big family (4 brothers and 5 sisters) in upstate New York, Jack now lives in a lovely Brooklyn neighborhood that is surrounded by dog-friendly parks.\nA graduate of Calm Energy Dog Training and an accomplished fetcher, Jack \"lives for the ball\"\u2014anyone's ball. He especially loves the water and spends a lot of time in a friend's pond near his upstate NY roots, as well as in Provincetown where he's been known to brave the bay, even during the cold of the New Year's holiday.\nJack's motto: \"If it squeaks, it's mine.\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Prospects for the 2014 harvests are good as the winter climatic conditions have been broadly favourable so far, according to the European Commission's Winter 2014 Short-Term Agricultural Outlook, which has been published this morning.\nAccording to the Commission, the good 2013 cereal harvest will allow for partial stock recovery and record exports.\n\"The good 2013 cereal harvest, an increase of 8.3 per cent with respect to 2012, should allow for a recovery in stocks by the end of the 2013\/14 marketing year; yet stocks are expected to remain below average as EU exports could reach a record level, exceeding 37 million tonnes,\" it outlined.\nIn terms of early estimates for 2014 it said sowings indicate a slight increase in cereal area, an increase of .8 per cent, mainly due to soft wheat increase of 2.8 per cent.\nIt noted some re-building of the dairy herd should lead to higher beef meat supply, while increased productivity in the pig meat sector could allow production to recover, following the strong decline in the sow herd observed in 2012 and 2013 linked to the implementation of the new EU welfare rules for sows.\nWith regard to EU milk prices, it said high prices stimulated an increase in 2013 milk production.\nThe Commission also noted the number of EU dairy cows increased significantly in 2013, thus leading to expectations that milk collection should increase in both 2014 and 2015.\n\"Although no significant surge in milk production is expected after the quota system is abolished, higher milk availability should allow grasping export opportunities in world markets, in particular for milk powders and cheese,\" it added.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The UTEP Miners are looking to get their first road win of the season as they head into Middle Tennessee to take on the Blue Raiders. As for the Raiders they have not been the best team to cover at home this season.\nThe UTEP Miners once again let a halftime lead fall apart in their most recent game as they had another win stolen from them. The UAB Blazers would blow out the UTEP Miners in the second half of the game as they came away with an impressive 76-63 victory on Thursday night. Now on the short rest, the Miners will have to figure out how they can hold these impressive leads. The halftime lead was led by Evan Gilyard who scored 21 of his 25 points in the first half, giving UTEP a halftime lead with three points, but as stated above they could not get it done. Could it happen again on Saturday?\nThe Miners come into this game with holding a 6-10 overall record as they currently hold a 1-4 record in conference play. The Miners have been known for giving away leads this season as the UAB team had a 14-0 and 16-5 runs in the second half of the game. The Miners have not been successful on the road either as they have yet to win a game away from UTEP.\nWatch out for these Middle Tennessee Blue Raiders, they can win some big games. Those big games including their last win as they were able to take down the UTSA Roadrunners. It was an impressive and historic win for the Blue Raiders as they were able to get their first conference victory of the season and head coach Nick McDevitt got his first victory against the Roadrunners as well. The Blue Raiders were able to slow down the C-USA-leading UTSA as they snapped a 13 game losing streak in conference play. The Roadrunners went into the night the hottest team in the league, winners of seven in a row, but somehow in someway the Raiders got the upset win.\nIt was one of the best performances in school history as the Blue Raiders were making shots left and right. The Raiders would finish the first half 15-of-29 from the floor and 10-of-20 from 3-point range. With this recent performance, would it be wise to stay with MT and roll the dice once again or lean towards a Miners team that has yet to win on the road?\nBlue Raiders are 40-19-1 ATS in their last 60 home games vs. a team with a road winning % of less than .400.\nBlue Raiders are 49-24-2 ATS in their last 75 home games vs. a team with a losing road record.\nMiners are 3-13 ATS in their last 16 vs. Conference USA.\nMiners are 0-4 ATS in their last 4 games following a ATS loss.\nMiners are 0-5 ATS in their last 5 games overall.\nI mean there is no way you can back a Miners team that has yet to win a road game and will be taking on a Raiders team that took down the top team in the conference and are hosting the game. More fans will be there to support as the Raiders make it two wins in a row.\nEven though MT got off to a fast start in their last game, I do expect this game to slow down quite a lot. The Miners and Raiders will not be needed to trade shots back and forth as the pace will decrease by a lot. The Under is 4-0 in Miners last 4 games vs. a team with a winning % below .400 as well as 11-1 in Miners last 12 Saturday games.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Spanish espadrilles or alpargatas are very popular worldwide although those had been first manufactured in France. Now a days, those are rarely made in the parent countries. Rather 99% of world's total production is now completed in Bangladesh, the home of jute for centuries. Jute, the main material to make highest quality espadrilles is exclusively harvested in Bangladesh although India produce some jute in very small quantity. China import jute from Bangladesh and try to make some espadrilles too, but those cost much higher due to added shipping cost of jute and import-taxes.\nEspadrilles are very popular casual & fashion footwear in Europe and North America. Lots of fashion and styles are introduced every year for the type of sandals\/shoes. Prices also varies a lot from item to item. Basic\/classic espadrilles are very popular worldwide as a casual fashionable footwear.\nBeing the most popular natural fiber next to cotton, jute is dominating the world of Espadrille Soles. Wedges of espadrille can't be made without jute. Because bright natural white color of jute, especially jute produced in Bangladesh is the only choice of designers to make high quality espadrille soles. Brightness of jute is the measure of price also.\nVarious materials are used to make uppers of espadrilles like fabrics, canvas, genuine leather, PU leather and other synthetics items. Ballerina espadrilles contain some attractive laces with the shoes. Beads are sometimes used to make ethnic feel over the espadrille. Hand stitched flat sole basic espadrilles are very popular as comfortable casual footwear.\nCotheeka Jute Industry is one of the most reputable jute sole and espadrille producing and exporting company in Bangladesh that supplies highest quality espadrilles at wholesale prices.\nEspadrille information Wikipedia \u2013 Basic information on espadrille.\nFacebook \u2013 Social network to find espadrille wholesalers.\nCotheeka Jute Industry \u2013 Manufacturer of high quality jute sole and espadrille in Bangladesh.\nGoogle \u2013 Popular search engine to search espadrille manufacturers.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\"Whenever you feel lonely and need a little loving from home, just press your hand to your cheek and think, 'Mommy loves you.'\"\nAfter reading the book 'The Kissing Hand', Kindergarten students discussed and explored their feelings about starting school. Some students were excited and others were nervous like Chester. This artwork incorporates a study of primary and secondary colours with the theme of Term 1's English unit - The First Days of School.\nThis is the story of a little raccoon who confronts the first day of school and needs reassurance. Mother Raccoon finds an unforgettable way of communicating the message that everyone most needs to hear. This book is perfect for children entering school through endearing illustrations and a simple story. School is starting in the forest, but Chester Raccoon doesn't want to go. To help ease Chester's fears, Mrs. Raccoon shares a family secret called 'The Kissing Hand' to give him the reassurance of her love any time his world feels a little scary.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Thank you for visiting and taking the time to read my blog.\nWhy not download a copy of my latest book?\nZenzoris Returns, the first in the Sophie Radcliffe Series, in now available on Amazon! Click here to buy your copy now.\nSophie Radcliffe, a feisty Texan intergalactic agent, is a brilliant computer programmer with a photographic memory. She has worked for the Intergalactic Law Enforcement Agency for ten years and is considered to be the best human agent on the team. She works alongside two other humans, Tom Logan, a tough, ex-police cop from Chicago and Justin Adams, a talented British linguist, whose arrogant, superior ways get right under the skin of his colleagues. When the agents discover their nemesis, a Drogg named Zenzoris, has escaped his prison pod and is out for revenge, the mission to recapture him takes the agents on a terrifying, roller-coaster adventure. Will the Drogg succeed in overthrowing the Intergalactic Council this time? The agents battle against time to find the Drogg before he destroys the world.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This is sam's new bike! About a year ago we bought a cheap one from Target that he never really took to. He refused to ride it all last year. I got him to try it out again last week only to find that he had outgrown it!\nThis time, we went to a local bike shop that has a nice trade-in policy for kids' bikes. There was a black bike with robots on it that Sam kinda liked. Then the owner brought out this used bike and Sam was sold. He just loved it.\nSam is riding this new bike really well! We've already taken two long rides!\nSam made this song up in the bath today. He told me that he has to lay down with his ears underwater in order to sing this song.\nSam: What day is it today?\nSam: Do we get candy on this day? Is it Halloween?\nFor the last few months, Sammy (he's asked us to start calling him that) has requested \"cat food\" for breakfast every morning. It started when he was pretending to be a cat one morning. I gave him some granola with yogurt on top and called it \"cat food\".\nWe're now t the end of the week and we are out of granola, so I've had to improvise the cat food. Yesterday he had Cheerio cat food and seemed happy enough. Today Cheerios wouldn't do. We ended up with cookie cat food with a side of plum.\nSammy says he'd like to tell restaurants about cookie cat food so they can start making it too!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Jungle Cat is widespread in South Asia, parts of the Middle East with the extreme edge of its range in north east Africa.\n*The country links will take you to an adventure travel website that includes visits to national parks and rural areas in several of their tours. Many of the images show the typical natural environment and wildlife habitat of that country.\nIn its African through western range the Jungle Cat is associated with reeds and long grasses in wet habitats, however in its eastern Asian range it occurs in dry grasslands, tropical deciduous forest and evergreen forest.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Attractive Club Lounge Furniture City Sounds Entertainment Lounge Furniture For Your Next Event \u2013 Here are some references to the Club Lounge Furniture, we hope to be able to reference you in building a house or buying home furniture that you always wanted. The resolution of the picture above (Attractive Club Lounge Furniture City Sounds Entertainment Lounge Furniture For Your Next Event) is 640\u00d7427.\nIf you are interested, you can save the picture of Club Lounge Furniture to your computer by right click and select save the image. In addition to the picture above, you can also find pictures of furniture and interior design related to Attractive Club Lounge Furniture City Sounds Entertainment Lounge Furniture For Your Next Event in addition to Interior and Furniture Design category, tags: club lounge furniture for sale, nightclub lounge furniture, used club lounge furniture, or related posts below. We uploaded this post to provide the best for the visitors of odeliabydesign.com, we as authors hope that Attractive Club Lounge Furniture City Sounds Entertainment Lounge Furniture For Your Next Event can be an inspiration what you are looking for, every day we will update with new posts. Thanks for a visit.\nAttractive Club Lounge Furniture City Sounds Entertainment Lounge Furniture For Your Next Event Brilliant Club Lounge Furniture Club Lounge Furniture Nightclub Lounge Furniture Lounge Amazing of Club Lounge Furniture City Sounds Entertainment Lounge Furniture For Your Next Event Lovable Club Lounge Furniture Night Club Barfunkysofa Captivating Club Lounge Furniture 1000 Ideas About Lounge Club On Pinterest Nightclub Design Adorable Club Lounge Furniture Club Lounge Furniture Furniture Info Amazing Club Lounge Furniture Nightclub Furniture Disco Sofas Baroque Furniture Disco Tables .","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We are tourist agency and sport travel agent from Slovenia. The agency was founded in year 2000. In last 15 years we have focused on organizing sports events and preparatios in Slovenia and throughout Europe. We have extremely good relation with several Organizations, Associations, Clubs and National squads from different sports; to name just a few: Kuwait Handball Association, Slovenian Handball Association, Karate, Judo and Athletics Association, Chelsea Football Club - B Team, West Bromwich Albion, QPR, FC Celje, Bosnian Basketball Team, etc.\nBRDO, \u010cATE\u017d, KOPER, KRANJSKA GORA, LA\u0160KO-CELJE, LJUBLJANA, MARIBOR, MARIBOR-POHORJE, MARIBOR-RU\u0160E, MORAVSKE TOPLICE, NOVA GORICA, POD\u010cETRTEK, PTUJ, RADENCI, ROGA\u0160KA SLATINA, ROGLA, VELENJE, ZRE\u010cE, \u017dALEC.\nWe take care of every aspect of the preparation and Organization, to optimize your training time for best results. We offer from basic to all Inclusive packages.\nEverything for your training! We take care of everything and provide you the best stay and training at the best prices whether you only need a basic accommodation and training hall, or if your team is searching for a perfect training and services with a hint of sightseeing, shopping, enjoyment and relaxation. We are able to customize any sports event or training camp according to your special demands and wishes (nationality, food etc.).\nWe offer from basic to all Inclusive packages. Our services depend on the type of accommodation (hotel rating), and inclusive of full-board, coach and driver, meeting room for your team, water at trainings and games, washing of sports clothes, training facilities, organization etc. In All Inclusive package, we provide you also with sports drinks, car for personal use, special trainers, organization of training games, medical services, equipment, etc.\nIf you want to experience something new and worry-free traveling, improve your sports team, enjoy in nature and good organization, Condor Travel Sport in cooperation with National sports federations is the right choice for you. We have extremely good relation with several Organizations, Associations, Clubs and National squads from different sports.\nBy having already considerable experience in this field, we may be your first choice in assisting your sports team to a quality training camp or tournament.\nSlovenia, a hidden pearl of Europe, offers many opportunities to various sport challenges with organizing training camps or sport events. On a small area, you can train in ether a Mediterranean climate, continental climate, or have high altitude training, with the possibility of usage of a hypoxic chamber.\nBeing a sport-minded country, every town can offer various sport facilities and the 3 biggest cities offer world class arenas for football, handball, basketball, swimming,\u2026 If you are looking for a place where you could perform the most diverse and effective training camp, Slovenia is the place to go.\nAs a small country we can pride on many world class athletes, who can train at home, with the possibility of using our dynamic terrain. Slovenia has nurtured athletes and teams of winter and summer sports.\nSlovenian athletes qualify to all bigger European and World championships and cups, being either in a team sport or individuals. We are also the country that has the most Olympic medals per capita. If you are representing a team sport, our teams are always more than ready to offer good, quality training games and matches during your training camp in Slovenia. Given the quality, experience and possibility of adjusting, we can pride on excellent performance for any kind of sport training camp.\nSlovenia lies at the heart of Europe, where the Alps and the Mediterranean meet the Pannonian plains and the mysterious Karst. This is what makes Slovenia a great destination. You can ski in the morning in the Alps and swim in the Adriatic Sea in the afternoon.\nThis small green country measures 20,273 km2 in area, and is home to sincere, hospitable people of great diligence. It has an exceptional number of top athletes, and a wealth of cultural creativity.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"The Da Vinci Code: is this really what we want our MPs to be reading?\nThe conspiracy thriller, it can be argued, is the purest kind of bestseller. The premise of such books is that there's no such thing as a random happening; meanwhile, though bestsellers aren't exactly conspiracies, most huge publishing successes can be traced back to a web of connected events, so that form and content collide to an unusual degree.\nFor example, Peter Benchley's Jaws was probably a good enough story to find readers at any time, but became a mid-70s sensation because the implications of the plot - horrible, sudden death in a holiday resort - reflected the neuroses of an affluent American generation enduring both a cold war and an oil war. Helen Fielding spotted that young unmarrieds were a social grouping without a literature; Allison Pearson noticed the same gap for working mums.\nAnd coming up to two years after September 11, 2001 - roughly the time it takes conventional fiction publishing to respond to cultural shifts - what did we find unstoppably atop the American fiction charts? Dan Brown's The Da Vinci Code, 450 pages of irritatingly gripping tosh, offers terrified and vengeful Americans a hidden pattern in the world's confusions.\nWhen bad things happen, Brown reassures us, it is probably because of the machinations of a 1,000-year-old secret society which is quietly running the world, though often in conflict with another hidden organisation. There are probably a couple of verses in Nostradamus predicting the triumph of The Da Vinci Code: \"As the painted French woman smiles\/The Brown man will top the heap\", or something similar. Certainly, the novel's success can be attributed to those who read Nostradamus and believe that the smoke from the blazing twin towers formed the face of the devil or Osama bin Laden.\nWhat happens in The Da Vinci Code is ... alert readers will have noticed a delay in getting round to plot summary, but it takes time to force the face straight. Anyway, my lips are now level, so let's go. Art expert Jacques Sauniere is discovered murdered in the Louvre, having somehow found the strength in his last haemorrhaging moments to arrange his body in the shape of a famous artwork and leave a series of codes around the building.\nThese altruistic clues are interpreted by Robert Langdon, an American \"professor of religious symbology\" who, by chance, is visiting Paris, and Sophie Neveu, a French \"cryptologist\" who is the granddaughter of the artistic cadaver in the Louvre.\nAs they joust with authorial research - about the divine proportion in nature and the possibility that the Mona Lisa is a painting of Leonardo himself in drag - a thug from the secretive Catholic organisation Opus Dei, under orders from a sinister bishop, is also trying to understand the meaning of the imaginative corpse in the museum.\nIt all seems to be connected with the Priory of Sion, a secret society. Reading a book of this kind is rather like going to the doctor for the results of tests. You desperately want to know the outcome but have a sickening feeling about what it might prove to be. In this case, the answer was as bad as I'd feared.\nRecently, crime and thriller fiction has been increasingly easy to defend against literary snobs at the level of the sentence. Not here. Brown keeps lugging in nuggets from his library: \"Nowadays, few people realised that the four-year schedule of modern Olympic Games still followed the cycles of Venus.\" Otherwise, he favours clunking, one-line plot-quickeners: \"Andorra, he thought, feeling his muscles tighten.\" French characters speak in American, while occasionally throwing in a \"pr\u00e9cisement\" to flap their passport at us.\nCriticism won't hurt Brown, who can probably now buy an island with his royalties and a second one with the film rights. The author has, though, recently found himself on the end of an unwanted conspiracy theory: another writer has accused him of plagiarism. In strongly denying this, Brown employed a striking defence: that the points of overlap were clich\u00e9s which were part of the genre of the thriller and therefore belonged to no one writer.\nThis admission of unoriginality may further anger readers and writers annoyed by seeing something as preposterous and sloppy (one terrible howler involves the European passport system) as The Da Vinci Code on its way to selling millions. But the success of this book is due not to the writing but to post-9\/11 therapy. It tells so many Americans what they want to hear: that everything is meant. In doing so, Brown has cracked the bestseller code.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Is it true that islands are magnets for creative people? Listen here as BCB host Jack Armstrong talks with book artist Susan Callan about inspiring opportunities for creativity here on Bainbridge.\nCallan, who settled on Bainbridge in 1986 after a long career as a medical technologist and health planner, began her retirement exploring her own creativity as a book artist. In this podcast she talks about creative possibilities for the large number of retirement-age people whose various life responsibilities might previously have limited their opportunities to express their own creativity.\nAmong many options available to island residents are the Creative Crones program for older women, which Susan teaches at the Senior Center. She encourages each of her students to rediscover their own personal wellspring of creativity, and to allow their innate creativity to flow.\nSusan also shares information here about several evolving programs at BARN (Bainbridge Artisan Resource Network) for both women and men, and concludes with some intriguing observations about islands and creativity.\nThis podcast is part of the StoryShare podcast series. The StoryShare Project started with three partnering organizations: the Bainbridge Island Historical Museum, the Waterfront Park Community Center and BCB. A fourth partner is now Bainbridge Youth Alliance. Each of the participating organizations is reaching out to the community in its own way to find, and to tell, the stories of people who call Bainbridge Island home.\nCredits: BCB host: Jack Armstrong; BCB audio editor: Chris Walker; BCB social media publishers: Chris and Diane Walker.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Video Jam is an ongoing series of unique, experimental events which seek to explore and re-examine the relationship between moving image and live sound by commissioning sound artists to compose original accompaniment for short, contemporary films. Initiated from scratch by a very small team of emerging artists, curators and promoters, the events act as a catalyst in bringing together these two mediums to create a new, third entity to be witnessed as a live experience.\nVideo Jam has swiftly become an indispensable part of the arts and experimental events culture in the North West and beyond, providing an accessible critical forum for dialogue surrounding live sound and moving image. Since its inception in January 2012, the collective have hosted 30 unique, site-\u00adspecific events within the UK and beyond, in venues as diverse as a Cathedral, a glass atrium, underground vaults and a music hall in the round, and have worked with over 300 artists including Jeremy Deller, Dieter Moebius, Soda Jerk, Phil Solomon and Ryan Gander.\nVideo Jam are looking for films of 5 - 15 minutes in length for their debut at The Lowry: a unique 3 hour event that will transform the Promenade Gallery to showcase a programme of films and film performances each with accompanying live original sound.\nInspired by the multidisciplinary nature of The Lowry as an arts space, and Quays Culture's emphasis on experimentation and innovation, the thematic focus for this event is 'Performance.' We are seeking artists, musicians and filmmakers of all disciplines and of any genre whose work may be relevant in terms of form and\/or content.\nFilmmakers: Ideas may include expanded cinema, film performance, live film, dance film, performance in film. All genres accepted. We have the facilities to screen digital film but are keen to include work by artists in variety of disciplines such as 16mm, 8mm, who may be able to provide their own relevant equipment. We are very open to hearing your suggestions.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"European Union importers face tougher restrictions on conflict minerals after a deal was finally struck on 15 June. Due diligence checks, conducted according to Organisation for Economic Cooperation and Development guidelines, will be mandatory for importers of tin, tungsten, tantalum and gold and their ores from conflict and high-risk areas.\nEnd of preview - This article contains approximately 199 words.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Every leader, every minster and every institution of modern governance has many advisors. These advisors hold different roles and responsibilities in the government ranging from bag carriers and door openers to being at the centre of the government's policy and decision-making. Their appointment is patronage and not open to all thus, ministers or leaders choose their close friends or people who helped them win and maintain office as advisors to reward for their support.\nAdvisors act as proxies for their respective ministers or bosses by attending meetings, preparing documents and even hearing representations. Some are more powerful than others and some are more close to the leaders thus, more influential in decision-making and policy settings. Karl Rove, for example, was known as the brain of President George W Bush while Farah Abdul-Kadir was known as the man behind the scene during Hassan Sheikh's presidency. However, advisors are more influential than most of us know. In some cases, they are the ones who make decisions and run the state's affairs and that signifies the importance of choosing them on merit. The following examples will further elucidate this.\nMuawiye Binu Abi Sufwan was the Caliph of Muslims for 20 years. When he got old and realised that his days were running fast, he started thinking about who would replace him as the Caliph. However, he had never thought of appointing his son or any of his relatives. He called a man called Mughayrah bin Shaabah, who was among his advisors and governor of Kufa (Iraq), for consultation about the issue. Without hesitation, Mughayrah suggested the appointment of Muawiya's son Yazid who was young and licentious. Muawiye was a wise, cautious and a man of insight. He could see the troubles might come from this idea, how the public would react and the difficulty he would face from convincing the public to accept his son as his successor. He knew that the sons of his predecessors, who were well high above his son in terms of reputation, publicity and garnering support from the public, were still alive and had political ambitions. Therefore, he could not picture the possibility his son could ascend to the throne.\nThe second example that gives us an insight into how a simple idea of an advisor can motivate an elderly man with poor health conditions to claim the highest position possible in that period of world history. Marwan ibn Hakam was on his way to Makah with a vast army to crash Abdullah Bin Zubayr, May Allah please with him, who proclaimed himself as the Caliph of Muslims, when the death of Muawiyeh binu Yazid binu Muawiyeh, who was the Caliph at the time, reached him. Hearing this Marwan sent a letter to Abdullah informing the news and also their readiness to accept his leadership provided that he would move from Makah to Damascus. Abdullahi refused the offer since he assumed it as a way of cunning him. Marwan called one of his advisors and explained the situation that Muslims had no Amir and Abdullahi refused to come to Damascus.\nThe advisor bewildered the gullibility of Marwan and asked him what was stopping him from crowning himself as the Caliph. Marwan accepted that advice and nominated himself as the successor of the Amir. However, he met a huge resistance from other tribesmen and a bitter war broke out which claimed hundreds of fighters from both sides. Seeing this, Marwan was very disappointed and decided to stop the bloodshed by showing his allegiance to Abdullahi Ibnu Zubayr the main rival whose suzerainty was widely accepted by most provinces. He shared his decision with his advisors and once again, they persuaded him to remain in his position and continue fighting for the Caliphate. Marwan once again accepted his advisors' suggestion and finally crashed his opponents. He became the Caliph of Muslim world. However, his reign did not last long as he breathed his last breath within a year from his ascendance to the throne. Nevertheless, he established his sons in positions of power as he designated Abdi Malik as his heir and from there the Marwanid dynasty was founded.\nThe third example can be found in the same period of Islamic history. When Sulayman Abdimalik realised that his days were coming closer, he appointed his son. But that son passed away before him. His second son was too young to be selected and options were open to so many competitors. He became so confused that he could not choose a successor by himself. So he called a man by the name Raja binu Hayan a righteous and close advisor of him. After long discussion, Raja convinced Sulayman to appoint Omar Binu Abdul-Aziz who was the cousin of Sulayman and his prime minister. Omar became the best ever Ummayad Caliphate and he is considered as the fifth righteous Caliph. While Mughayrah advised Muawiyeh to appoint his son, which opened the gate for deep-rooted divergence and divisions among Muslims and changed the Caliphate from choosing leaders based on merit into dynastic political system, Raja succeeded in persuading the Amir to appoint his heir based on principle, piety and righteousness. Mughayrah did this to please his boss and show his loyalty. One narration reported that Muawiyeh wanted to remove Mughayrah from his post as the governor of Kufa before he gave the advice. It is possible that Mughayrah knew the plan of Muawiyeh and in order to stop it he made the suggestion of appointing his son as his heir. Conversely, Raja's main concern was the welfare and the betterment of Muslim world and therefore based his suggestion on the possession of the right skills, sincerity and piety.\nIn conclusion, special advisors play important roles in running government affairs. They are close to their masters and in most cases strive to please them. Unlike other professions, they are selected because of their loyalty not for their dedication, ability and sincerity. However, their decisions and advice they provide may have a profound and long lasting effect which may reach all corners of the world. Therefore, a careful consideration is needed when instating a special advisor.\nLeaders and minsters should take honest, patriotism and sincerity into consideration when selecting an advisor. This is even more important for a country like Somalia which suffers from many foreign interferences. There are possibilities of foreign agents to seek advisory positions in the highest office in the government. Therefore, familiarity with a person and loyalty to a master are not enough to safeguard the interest of the nation.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"At LAC, customer focus are not just empty words, they are lived and breathed on a daily basis. Customer dialog, quality, and service are at the heart of everything we do. Our customers can rely on us: Our experienced team guarantees to provide expertise, speed, and quality.\nYou are in good hands. Whether you wish to discuss materials, alloys, manufacturing processes or meeting your objectives \u2014 we are happy to provide expert advice. We are there by your side, providing you with technical knowledge and expertise from the development phase through to the finished product. You can rely on us: At Laser Add Center, you receive excellent advice, reliable service, and comprehensive expertise. And that's a promise.\nThanks to our fully automated manufacturing process and digitized value chain, we can guarantee minimum production times, ensuring you achieve significant time and cost savings. Once you have uploaded your 3D data, we often complete your order in just 24 hours. LAC stands for reliability, simple processing, and quick delivery times. And that's a promise.\nWe do not make any compromises when it comes to quality. Our production chain is certified according to EN ISO 9001:2008 and EN ISO 13485:2003. By complying with this certification, we apply the most stringent requirements both to ourselves and to our suppliers. We use only certified original materials of the purest quality and always provide our customers with precise information on the composition, origin, and properties of each material used. In addition, our quality assurance department ensures that all components are fault-free. As a customer, you always receive optimum levels of quality and safety. And that's a promise.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"It's also a good place to sit and write a poem.\nNestled right in among some of the most urbane places in the city is the Houston Arboretum & Nature Center, and it is a great place to put the rest of the world out of your mind for a while. With many improvements since the violence of Hurricane Harvey still being implemented, it remains a walk under gentle trees accompanied by as blessed a silence as you can get within the Loop. Even in the summer months the dark paths offer relatively cool walking for both humans and their pets, and the trails have a winning combination of safety and adventure as they meander around.\nBenches have been donated by many supporters of the space, and you can feel all the tension of modern life dissipate into the air as you just pause on them to appreciate the resiliency of the trees around you. No place here can soothe a troubled soul quite so well.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If you watch the news diligently, you might feel worried and concerned when you see stories of home burglaries and theft. It isn't surprising that many people choose to install home security cameras to help catch these criminals and protect their home. If you don't yet have a security camera system at home, you may consider one.\nPerhaps the most important advice of all is to choose the right type of cameras. Understand that there are a wide variety of cameras available with each having their advantages and disadvantages. In choosing the right CCTV, determine your needs and see which cameras can satisfy those needs. Take the time to read up on the different types of security cameras available in the market. Consider things like, do the cameras have night vision, do they record audio, are they wireless or wired, etc.?\nAfter choosing the right cameras, the next important thing to do is to place them in the right location. Proper placement is incredibly important since no matter how great your CCTV is, it is useless if it doesn't cover the most vulnerable areas \u2013 front door, back door, backyard, windows, and gates. Do not be tempted to set up the cameras too high because it provides the most coverage as this will result in footage that is too far away to record details for identification.\nOne of the vulnerabilities of wireless security cameras is that they are hackable. If the person trying to get inside your home is skilled, and your security is lax, they can still get inside undetected if you don't properly secure your cameras. On that note, it is crucial that you secure your internet connected cameras with the most secure passwords you can think of to ensure no one would be able to bypass them.\nPlacing the cameras in plain sight is an excellent way of deterring the would-be thieves from entering your home. When they see that you have security measures in place, anyone who would attempt to enter your property will second-guess. Would they risk being seen? Most likely they would not, and with that, they will often turn away from committing a crime on your property.\nWhen it comes to the placing of the cameras, the weather or sunlight is an essential factor to consider. It is critical that you consider how the light changes during the day, particularly for outdoor cameras. In mounting cameras outdoors, their location will be more or less affected by sunlight which essentially shines into and blinds the cameras. Also consider the effects of rain or snow build up on cameras, which would obscure the lens. On that note, place the camera somewhere where it won't be affected as much.\nEven though there is a lot to consider, a security camera system can add a great level of protection for your home and peace of mind is priceless.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The extensively-studied phenomenon known as \"Forest Bathing\" in Japan and South Korea and as \"Kneipp Therapy\" in Germany involves exposure to high concentrations of naturally-occurring aerosolized phytochemicals in conifer & deciduous forests.\nThis natural treatment for conditions ranging from asthma to dermatitis has been documented and validated by a raft of high-quality medical and scientific research.\nThere is no question that \"Forest Bathing\" has therapeutic benefits. For example, positive effects on NK (Natural Killer) cell activity have been shown with in vitro treatment of tumor cell lines with monoterpenes released from trees (and of course present in Cannabis flowers) such as d-limonene and \u03b1-pinene, and also in forest bathing trips. The anti-tumor effects act by increasing intra-cellular levels of anti-tumor proteins such as perforin, granulysin, and granzymes A\/B.\nHaven't heard of \"Forest Bathing\"? I hadn't either until I ran across it during some intense research into naturally-occurring environmental Cannabis terpene aerosols. The ancient Japanese natural health practice called \"Shinrin Yoku\", defined as \"taking in the forest atmosphere\" or \"forest bathing\", has a wide range of rigorously tested and proven health benefits.\nIn 2005 the Japanese government introduced a nationwide \"Therapeutic Effects of Forests Plan\" that pays \"Forest Bathing\" health benefits. The government says that it pays for this therapy because of the proven reduction in other health care costs across the spectrum.\nSouth Korean scientists and public health researchers have documented a wide range of positive health benefits from exposure to terpenes in the air of coniferous forests, with variations among the terpenes in different species of trees at different locations accounting for differences in the health benefits of inhalation of forest air.\nThey have divided the country into numerous micro-climes where particular combinations of coniferous tree species co-exist and perfume the air, each location offering a particular healing, stimulating mix of terpenes and other phytochemicals.\nInterestingly, the dominant terpenes in the air of these forests are the same terpenes that characterize different Cannabis strains and the same kinds of variability in Cannabis strains similarly account for their differing health benefits.\nThere is also a natural medicinal therapy in Germany called \"Kneipp Therapy\", that involves a series of exercise routines done in a terpene-rich forest environment. Kneipp Therapy has been studied using quality clinical research protocols and the exercises have been found to be significantly more beneficial when performed in a forest environment compared to other kinds of locations.\nSo, it's both very interesting and very significant for establishing the validity of the Cannabis \"Entourage Effect\" that the dominant terpene profiles of all of the therapeutic forests studied in the Japanese and Korean \"Forest Bath\" scientific literature (cited below) appear to be various combinations of myrcene, pinene, limonene, linelool, and a number of less-celebrated but still important Cannabis terpenes like cynene, terpinene and boneal. There are many other \"minor\" phytochemicals shared between the airborne perfumes of Cannabis flowers and therapeutic forests, and almost certainly many of these will ultimately be shown to play significant roles in both the Forest and the Cannabis \"Entourage Effect\".\nBottom line \u2013 I believe that there is an inescapable argument in favor of the Cannabis \"Entourage Effect\" presented by the \"Forest Entourage Effect\", which itself is definitively established in international, if not US, scientific and medical literature.\nThere has been extensive research in multiple advanced countries on the health benefits of exposure by inhalation and skin absorption to the airborne terpenes in forest environments \u2013 interestingly enough, these turn out to be the same terpenes that are inhaled in the vapors from Cannabis flowers.\n\"Forest Bathing\" research establishes that inhaling a naturally-occurring mix of terpene emissions or vapors has far greater health benefits than exposure to or ingestion of any of the terpenes and other phytochemicals singularly, like in a pill or other oral or topical medication.\nCannabis visionaries have always known that the THC was only one element of the sensual pleasures and only one of the sources of health benefits from the sacred flower, just as wine lovers have always known that the alcohol is only a relatively small part of their total experience. Nobody drinks a bottle of Etude Pinot Noir or Chateau Pomerol Bordeaux for the alcohol, and nobody chooses which Cannabis flower to enjoy simply on the basis of THC content, although that approach seems to dominate much of today's adolescent-style Cannabis marketing. Even the most dedicated couch-locked stoner knows very well that there is a world beyond THC and may spend a lot of time (if they're not too ripped) thinking about taste and aroma options when they're choosing between Durban Poison and Granddaddy Purple.\nHowever, the concept of a Cannabis \"Entourage Effect\" has been universally ridiculed by anti-Cannabis forces who say that the supposed variety of effects of different Cannabis strains is simply a kind of mass delusion. They claim that there is no evidence that different phytochemical profiles of different Cannabis strains signal different health and well-being effects, and say that in their expert, informed scientific opinion such observations are imaginary. While their criticisms are couched in the careful, apparently rational language of science, and even rated a major article in Scientific American in 2017, all of the criticisms amount to a simple \"It's all in your head\" dismissal.\nIn other words, the anti-Cannabis establishment says tough, there's no scientific evidence to support your claim, and there's not going to be any evidence either because we aren't going to fund research.\nWell, I've got some news for these die-hard prohibitionists.\nOK, they have managed to impede research that could validate many of the medical benefits of the whole Cannabis Flower as opposed to plain old THC extract. With notable medical research exceptions, many of the health and sensual benefits ascribed to the Cannabis Flower are currently only validated by experience and consensus, both of which the scientists are fond of reminding us can be way off target. They point to the flat earth delusion, or to many other instances where \"everybody knows\" something that simply isn't true, and smugly point out that nobody can prove all these marvelous things we're saying about Cannabis.\nIt's hard to find a reasonable explanation of why the Federal government has arrayed its dark-side powers against the Cannabis flower, but in this match between the Flower and the Power it's beginning to look like the Power is going to lose this one because Forest Bathing research actually provides plenty of evidence . The research unequivocally supports the validity of the \"Entourage Effect\" by demonstrating that naturally-occurring environmental terpene and phytochemical aerosols do have measurable, verifiable positive impacts on overall health as well as on specific diseases and conditions, and do vary among forest tree species and environments in the same ways that Cannabis flowers vary among strains in response to environmental variables.\nForest Bathing research is directly applicable to validating the \"Entourage Effect\" of Cannabis terpenes and phytochemicals that are widely observed but, according to the Federal propagandists, not \"scientifically verified\". As an example, there is solid research that says that terpene emissions from plants are directly correlated with the concentration of terpenes in the plant. The higher the concentration of terpenes, the greater the emissions from the plant. \"Forest Bathing\" research naturally focuses on terpene emissions from coniferous and to a lesser degree deciduous trees, but the relationship between terpene concentrations and emission rates has been widely replicated in studies with agricultural crops and seems to apply to all plants.\nThe bottom line is that clinical literature as well as popular wisdom in several countries points to the health benefits of inhaling and \"bathing in\" an atmosphere rich in terpenes and other phytochemicals. While the health benefits of many of the individual components of this phyto-soup are only recently becoming well-known, the benefits of exposure to the entire environmental complex of a pine\/conifer forest are familiar to anyone who has ever walked outdoors that first morning in a forest campground.\nSo, it's clear that relaxing for a few hours in a forest environment filled with terpenes can be beneficial and even therapeutic for people with a wide range of diseases and conditions from dermatitis to cancer. Do a simple internet search for \"forest bathing' and you'll find books, resorts, videos and even classes. But enter \"cannabis bathing\" into a search and you'll get bath salts, bubble bath, and a lot of fruit-flavored massage and lubricating oils.\nFor the past year or so I have been exclusively using a vaporizer to enjoy Cannabis flowers and I can add my experiences to the observations of many others that whole flower Cannabis vapor is a marvelous clean, natural high which, now that I realize it, is almost exactly like stepping out of my tent high in the pine forests of the Oregon Cascades and inhaling that first breath of vibrant, aromatic, high-energy mountain air.\nSo in my opinion all this research on \"Forest Bathing\" makes the smug \"You can't prove it and we're not going to let you\" chant of the anti-Cannabis \"scientists\" pretty much irrelevant. Sooner or later there will actually be research on every aspect of inhaled and absorbed Cannabis terpenes and other phytochemicals but until then the parallel research on Forest Bathing should be more than adequate scientific evidence for any reasonable person of the validity of the Cannabis \"Entourage Effect\".\nSpievogel I, Spalek K. Medicinal plants used in pediatric prophylactic method of Sebastian Kneipp. Nat J. 2012;45:9\u201318.\nKawakami, K., Kawamoto, M., Nomura, M., Otani, H., Nabika, T., & Gonda, T. (2004). Effects of phytoncides on blood pressure under restraint stress in SHRSP. Clinical and Experimental Pharmacology and Physiology, 31, S27\u2013S28.\nLi, Q., Kobayashi, M., Wakayama, Y., Inagaki, H., Katsumata, M., Hirata, Y., Hirata, K., Shimizu, T., Kawada, T., & Park, B. (2009). Effect of phytoncide from trees on human natural killer cell function. International Journal of Immunopathology and Pharmacology, 22, 951\u2013959.\nLi, Q. (2010). Effect of forest bathing trips on human immune function. Environmental Health and Preventive Medicine, 15, 9\u201317.\nOrme\u00f1o, E., Gentner, D. R., Fares, S., Karlik, J., Park, J. H., & Goldstein, A. H. (2010). Sesquiterpenoid emissions from agricultural crops: correlations to monoterpenoid emissions and leaf terpene content. Environmental Science & Technology, 44, 3758\u20133764.\nTags: alternative medicine, anti-Cannabis, Cannabis, Cannabis medicine, Cannabis pharmaceuticals, Cannabis prohibition, cardiac asthma, Durban Poison, entourage effect, forest bathing, granddaddy purple, killer cells, Kneipp therapy, limonene, linalool, llinalool, meditation, myrcene, natural asthma treatment, natural medicine, phytochemicals, pinene, propaganda, Shinrin Yoku, T-cells, terpene, THC | Permalink.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Scottish smoked salmon & caper salad.\nTiger prawns deep fried in beer batter, sweet chilli dip.\nGoats cheese & sweet onion tart on a pinenut salad,balsamic ,syrup.\n\u00a92009 The Trafalgar Restaurant. All rights reserved.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"This sophisticated reference and teaching tool enables students to visualize the world's lands and peoples, compare continents, and understand environmental, economic, and cultural issues through thematic maps. Sections addressing the world and its continents each include three full-sized, shaded-relief reference maps\u2014political relief, land cover, and sculptural relief (elevation)\u2014plus regional maps, thematic maps (climate, precipitation, land use, population, energy resources, and region-specific issues), cross-section elevation profiles, infographics (such as climographs, population growth, ethnic composition, and balance of trade), captioned photographs, and \"Issues Today\" articles. Additional features: tables of world facts, a glossary, two indexes (of places and themes), and articles on \"Understanding Map Projections\" and \"How to Use This Atlas.\" 8\u00bd\"x 11\". 240 pages. Fifth Edition. \u00a92018.\nSTUDENT ACTIVITIES. Fifty activity handouts (one for every two-page spread in the atlas) extend geographic skills, content knowledge, critical thinking skills, and social studies literacy. Includes 15 additional information literacy lessons. 8\u00bd\"x 11\". 112 pages.\nATLAS PACK. Combines 30 atlases, the student activity book, and a five-year atlas ebook subscription with online activities.\nATLAS EBOOK SUBSCRIPTION. License for one teacher and all of that teacher's students.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Effective performance management is much more than a completed review document; it's a continuous dialogue designed to help employees succeed. PerformYard facilitates ongoing feedback to recognize achievement, document individual performance and keep track of important notes.\nHighlight successes and exceptional performance across the organization, recognizing those above-and-beyond efforts. Let your employees know you appreciate their work. Create custom feedback 'tags' so that you can run reports and align feedback to personal development plans or corporate objectives.\nDocument and share important feedback in the moment, rather than waiting until \"review time\" to discuss both positive input and constructive comments.Choose to share the input with the employee's manager, which will follow the employee as they move through the organization.\nKeep your managers and company leadership informed about employee performance in a documented, privacy-conscious manner. Enable managers to see a full track record of everything that's important for them to know about their staff. Team feedback can be filtered by contributor, date, and type so that finding the information that matters is easy.\nRecord private notes about performance, for yourself and your team so you remember all the good things that have happened when it comes time to do reviews. Performance notes will always be at your fingertips when you need them most.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"CTR Library's June 2015 issue of the Research Digest is now available for download. This month's digest provides abstracts for 11 recent publications from Southwest Region University Transportation Center (SWUTC). The SWUTC is a consortium of five universities (Texas A&M University System, The University of Texas at Austin, Texas Southern University, Louisiana State University, and the University of New Orleans). All research from SWUTC supports the theme of Sustainable Transportation Solutions to Enhance Economic Prosperity and the Quality of Life.\nAbstracts and full-text links are provided for each report. Print copies may also be available at the library; in some cases we may have spare copies to give away. Please contact the library if you are interested in a copy.\nTo stay up-to-date on all the latest research published through different transportation programs, visit the library's catalog portal to browse new publications or follow @ctrlib on Twitter.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This tool provides hourly airmasses, parallactic angle and heliocentric-barycentric corrections based onsite, date, and object coordinates information. Times are given forthe local time, including daylight saving times when applicable.\nSelect site, object coordinates and date; then press Compute. The time step can also be chosen (defaults to 1 hour).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Telekom Romania announces the sale of a land located in Floreasca area in Bucharest, to One United Properties. The transaction was concluded for a price of over 18 million euros, thus becoming the major land transaction on the real estate market in Bucharest this year.\nThe urban land with an area of 25,839 sqm consists of two lots, of 24,786 sqm and 1,053 sqm respectively, and it has an opening of about 120 meters at Lake Floreasca.\nOne United Properties was represented in the transaction by Alexander Boff, an experienced broker with over 12 years in the high-end residential segment.\nThe land was put up for sale earlier this year and announced on the Telekom Romania website dedicated to the real estate properties available for sale and rent. The auction was launched in February 2018, with a starting price of EUR 13.9 million excluding VAT.\nTelekom Romania is a dynamic brand that provides complete, innovative, fixed and mobile telecommunication services tailored to a vast community of customers with about 10 million service contracts.\nOur solutions offer an infinite world of opportunities to share the most beautiful and emotional experiences with family, friends, colleagues and those around us. Our mission is to enrich people's lives by providing them with fixed and mobile integrated services, the latest 4G technologies, fiber optics as well as Internet TV, bringing users a new entertainment experience with exclusive and quality content access, on all screens, and advanced interactive features. Telekom Romania is the trusted partner of the companies for which we offer complete solutions for communications and IT & C. Our network connects people, cars and content for a better, safer and simpler future.\nTelekom Romania has entered Romanian market in 2014, following the joint rebranding of Romtelecom and COSMOTE Romania.\nTelekom belongs to Deutsche Telekom, one of the world's leading integrated telecommunication companies.\nOne United Properties is a leading developer of premium residential compounds and office buildings in Bucharest, Romania. One United Properties is an innovative corporate group, which focuses on eco-friendly construction methods dedicated to accelerate the adoption of practices with result in energy-efficiency, healthy and environmentally sustainable buildings.\nThe Company has already 376 apartments delivered over the last 4 years, 387 residential units under construction, and more than 500 units are on the short term development pipeline. The most prestigious compounds developed are One Herastrau, One Herastrau Plaza and One Charles de Gaulle. The Office Division is focused on the development of centrally located, highly efficient office buildings.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"It is easy to fall into the misguided idea that you currently have enough safeguards in position to protect your small business. You most likely stay current with the latest antivirus software, but is that adequate? You also probably believe that since you're not really a huge corporation, you don't have to worry about cyber criminals hacking into your system. But here's the unlucky reality: cyber criminals have upped their game and are starting to target small businesses more and more. Let's take a closer look to give you a better understanding of the level of danger your business may be under.\nUnfortunately, small businesses make easy targets, so hackers are beginning to focus on them. Small firms are quick hacks and the criminals can make more hacks at a quicker rate. With increased chances, their likelihood of stealing useful information goes up considerably. As a small company, you may not think you have any valuable information, but in today's world even the simplest information may be used to do harm to someone else. Credit card purchases are the most common targets of cyber criminals. Every time your system runs a transaction, it must validate the cardholder's information. This information can contain card numbers, addresses, names, and more. Your clients don't want this information stolen!\nHow can they get this delicate data? Though you might be protected by software made to firewall your systems, hackers still have several points of entry. Email attachments are a serious offender, and openingjust one piece of malware can leave you exposed. Believe it or not, in the first quarter of this year alone, industry experts have detected over six million unique malware programs. Attackers don't have to be sophisticated to be harmful: when attempting to breach your data security, some hackers take a low-tech procedure instead. They could contact your employees directly, and posing as a network administrator, ask for their password to boost security. This is known as \"phishing.\" While you may believe your workers know better than to fall for such an obvious tact, it only takes one lapse in judgment to leave your systems open to attack.\nIf you feel that preserving data security is not crucial, consider that should your systems be breached, your clients will likely hold you personally responsible for any stolen data or property. These kind of events cost companies thousands of dollars every year and can damage a company's reputation so badly that they must close their doors permanently.\nSo, with regards to your data security, taking additional precautions is a prudent move. Not only should you update your programs whenever you can, take a look at strengthening your passwords as well. Don't use obvious phrases, and never leave the default password in place. Also, instruct your employees not to share their login information or open attachments from unfamiliar senders when checking their email. You may also hire an outside consultant to guarantee the security of your customers' data. Taking these types of measures can provide you with piece of mind, knowing that your systems are safe from attack.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Yukon Gold is one of the best online casinos in Canada, and is popular the world over. Players can enjoy a variety of incredible games, huge rewards, and generous bonuses, all under one virtual roof! Owned by Technology Services Trading Ltd, they've been going since 2004, and their many years' experience shows in the excellent online casino experience they are able to offer. With a gaming license from the MGA, you can enjoy totally protected gameplay, and the Wild West theme is sure to engage you from the word go.\nPlayers currently have the chance to earn CA$500 totally free. Download the Yukon Gold online casino software, install it, and register as a real player and you will receive a 100% Match Bonus up to CA$150 on your 1st deposit, 50% up to CA$200 on your 2nd deposit, and 25% up to CA$150 on your 3rd. You will also receive automatic entry in to the Loyalty Programme, where every 100 points counts as CA$1. The site also regularly offers promotions like cash prizes and double rewards for any points you earn, and there are new competitions weekly and monthly.\nSlots fans are more than catered for at this Microgaming gambling site, with a range of Video, 3-Reel, and arcade-style title to pick from. There are table games galore, too, including Blackjack and Roulette, so every taste is catered for. Video Poker is available too, with variants ranging from the ever-popular Jacks or Better right through to less-common titles like Louisiana Bonus. Don't forget to check out the array of Progressive Jackpot titles either, as there are a number of these millionaire-makers available for you to enjoy whenever you pay Yukon Gold a visit.\nThe fairness of the games is guaranteed as well, thanks to their eCOGRA Safe and Fair Report, which you can read by clicking on the seal located on the bottom of the site. This report gets updated monthly, and you can even view the previous month's payout percentages if you want to. This is one of the reasons they are one of the best online casinos in Canada, and they set an example more sites should follow.\nYou can enjoy the fun of the first-rate games at Yukon Gold online casino on your PC, smartphone, or tablet. The easy-to-navigate site makes it a breeze to find and play what you're looking for no matter what device you're using.\nWith a payout percentage of 97%, if you're looking for a site that gives you a good chance at boosting your bankroll, you've just found your best bet.\nAll your personal and financial information is guarded with 128-bit SSL encryption at this online casino Canada has to offer, and the industry-standard firewall in place means you can relax and enjoy your games without worrying about safety issues, since you're always 100% protected. By only aligning themselves with leading banking brands you can ensure your money is in safe hands.\nYukon Gold online casino truly excels in terms of customer service, with live chat available 24 hours a day, 7 days a week. This allows you to get assistance within a matter of minutes, and you'll find the representatives friendly, professional, and knowledgeable. Telephone contact is also available, as well as email access. You can look forward to a quick resolution to any issues you may have, and get back to playing as soon as possible.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Last week of Term 3! Wow that term went quickly!! Wishing you all a happy and relaxed Spring Holidays.\nA big thank you to all the parents and staff that came along to last Friday's meeting. It was a big meeting with lots of ideas and energy. Thank you also to all those who are working quietly behind the scenes to pull their class stalls together.\nLooking forward to one more meeting and a quick working bee to spruce up the PA kitchen on Friday.\n5 WEEKS UNTIL THE FAIR!\nAttendees: Yuki C, Ian M, Marion H, Maya S, Christine C, Paul J, Kylie B, Mary G, Suzi B, Fiona M, Lauren D, Katie F, Sylvia T, Kylie L, Donna M, Tony G, Peter G.\nApologies: Dale P, Craig B.\nDiscussed locations and size of billboards. Request to ask Jellis Craig use bigger billboards if possible and to put one up at Fitzroy High.\nACTION: Craig to liaise with Jellis Craig and Fitzroy High.\nLauren & Penny will coordinate. Early bird tickets available on first day of term. $25 tickets for $20. Get 20% of what they take. Clown bikes approved.\nYuki to provide list from Nada.\nCorrection: breakfast on Sunday morning ONLY.\nStall setup instructions \u2013 Mary to provide to FR Committee.\nTickets must be on ticket booths first thing.\nThe following bookings have been made: St John Ambulance (First Aid), Rain Hayne & Shine Farmyard (Animal Nursery), VenomWise (reptiles & dingo pups), Balloons By Design (Clown Bikes), Smart Amusements (Fairy floss & Snow cone machines), Caterent (Hot box for pizzas), Ice Age Refrigeration (Portable cool room), ATM machines, Security.\nClass 7C (Campbell) and Class 3B (Bernice) will work together to run the Cake Stall and Jams & Preserves that will include a Fermented Foods demonstration in the afternoon.\nIan M, a Class 7C parent, offered to make walking sticks. These will be sold on the Made with Love stall. Thanks Ian!\n5\/6 would like to run a student disco. Questions raised re potential damage to floor in dance studio.\nPost meeting note: Class 5\/6 have decided not to proceed with disco.\nMaps provided by Paul G. Due to limited time Map & Program will be discussed at next meeting.\nFormal invitations to be sent to relevant dignitaries.\nACTION: Jacinta W to coordinate with Leadership team\/Office staff & provide update at next meeting.\nQuery as to which classrooms will be open? Ground floor prep, 1 & 2? Which teachers will be in the classroom? Rostering for lunch & breaks or food provision in the classroom?\nACTION: Jacinta W to coordinate with Leadership\/Staff & provide update at next meeting.\nQuery as to School tours \u2013 what times are the classroom\/school tours are happening? Should these go on the program? Meeting point for school tours? Will there be a school info stand?\nACTION: Jacinta W to arrange for Tim Arundell to work on the day & run school info stall.\nACTION: Jacinta W to coordinate with Leadership\/staff regarding School Tour times and report back at next meeting.\nTim Arundell will update the website with new content for the 2015 Community Fair as it is provided.\nVOLUNTEER REQUIRED: One or two parent volunteer needed to collate and submit content to go on website.\nRaffle tickets have been printed and parcelled up ready for distribution to families. Thanks Suzi for leading this!\nACTION: Suzi\/Laurel to provide Office Staff with box to collect tickets & money.\nRaffle ticket seller on the day of the fair \u2013 in the past sold $600 worth of tickets on the day. Raffle tickets alson need to be available at the ticket booths.\nACTION: Suzi\/Laurel to arrange ticket seller and extra raffle books for ticket booths.\nIt was noted that in past raffle tickets sent only to youngest child per family. This provides enough raffle books to give out to all classes from prep to year 9.\nDiscussed what has happened with Donations and Auctions in the past. Past coordinators noted that obtaining donations was the easy part, handling and processing donations was time consuming and stressful due to limited helpers. Laurel and Suzi are leading this team this year and are exploring all possibilities including a $10, $20, $50 lucky dip option and pre-purchasing items via online auction before fair day and limiting number of Silent & Loud Auction items to a maximum of 100 items.\nACTION: Suzi\/Laurel to provide update at next meeting.\nACTION: Jacinta to book room for wrapping donated items. Only Auction team to have access to this room.\nACTION: Suzi\/Laurel to provide list of Approved Auction Team Personnel to the Office.\nCurries \u2013 All agreed to pay qualified person to lead curry preparation if needed. Expert curry makers are being approached.\nACTION: Paul J to lead curry making parent volunteers.\nACTION: Mary G to source as many ingredients as possible via donations.\nInsurance for using Canteen \u2013 query as to whose insurance covers students serving curries.\nACTION: Jacinta to clarify insurance\/access to Canteen Kitchen (Saturday 17 Oct & Sunday 18 Oct).\nACTION: Dale\/Jacinta to confirm formal arrangements with H&H Canteens to have access to Canteen Kitchen and .\nFood Safety Certificates \u2013 school need to have one person with this qualification to conduct Food Safety checks of all food providers on Fair Day. Food Safety Rules & Regulations info to be given to every food stall provider. These need go into Equipment boxes.\nACTION: Working Bee team to ensure Food Safety info goes into relevant equipment boxes.\nACTION: Food team & Suzi to assign Food Safety Officer and conduct Food Safety Checks on Fair Day.\nPaper plates vs reusable crockery & cutlery \u2013 The Fair organisers would love to be able to have reusable crockery for the curry stall because it's so much GREENER.\nVOLUNTEERS REQUIRED: Want to do your bit for the Environment?!! It's not glamourous \u2013 but if you're willing to do dishes, clean tables etc., get in touch. Without volunteers we'll have to use disposables. We need you!\nACTION: Donations of metal cutlery are welcomed by the Canteen and PA kitchen, these may be used on Fair Day.\nLead volunteers \u2013 Lead volunteers will be entitled to a free meal & drink at any school run food stall. They will also be given a 'PRIORITY PASS' to enable them JUMP THE QUEUE and get their food quickly. Self-nominate as a Lead Volunteer if you've done lots of work in the leadup to the fair or if you will be running your class stall for half the day (or the whole day!) on Fair Day. All Lead Volunteers to collect food vouchers and PRIORITY PASSES at Working Bee on Saturday 17th Oct.\nContact: Lauren Dirck 0422 133 655 for more info on Lead Volunteers.\nBig Marquee \u2013 Damon O'Sullivan has offered to be part of the new team to keep the Big Marquee setup\/pack-up knowledge alive. Thanks Damon!\nVOLUNTEERS REQUIRED: Putting up the Big Marquee is a BIG JOB, we need a few more families to learn the ropes (the good men of Collingwood College who have been doing it for more than 10 years are moving on and we need that hand over to happen this year).\nACTION: Peter G and Paul G to coordinate with Damon O and any new Big Marquee volunteers.\nClean up on Monday 19th October \u2013 Need to plan clean up volunteers for the day after the fair, the PA kitchen often needs attention.\nACTION: Liria to discuss with SRC if they would be happy to take this on?\nBanners \u2013 need to be placed in equipment boxes or in banners box if they are too large.\nGloves & napkins \u2013 stocktake happening today. There is a list in each equipment box of what items are needed for each stall.\nA large ice box belonging to the Fete Collective has been found in Reggio playgroup room. It will be moved to Community Fair store room and returned to the container after the fair.\nThere is a steady stream of enquiries for market stalls. Lauren D & Dee B are coordinating bookings.\nIan M will donate all wood (dried blue gum). Thanks Ian!\nACTION: Ian, Maya and Frank to coordinate quantity and delivery details for wood.\nMary G gave update on pizza ingredients that were donated last year.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Genuine Indian Indigo, an organically certified natural leaf powder is best know dye for achieving dark brown and jet black hair without chemicals. Indigo, from the leaf of the Indigo tinctoria plant, produces a blue dye which can be combined with henna in a single step to achieve shades of brown or, in a two step process, dark brown or jet black hair.\nGenuine Indian Indigo, an organically certified natural leaf powder is the best known dye for achieving dark brown and jet black hair without chemicals. Indigo, from the leaf of the Indigo tinctoria plant, produces a blue dye which can be combined with henna in a single step to achieve shades of brown or, in a two step process, dark brown or jet black hair.\nIdeal if you have dark hair already or if you are trying to cover grey hair.\nMix the indigo and henna in separate bowls with hot water.\nAfter 20 minutes combine them both and mix thoroughly.\nAdjust the consistency as required and apply.\nLeave on for 2-6 hours, then rinse.\nDo not shampoo for 2 days.\nMix the henna in a bowl with hot water.\nWait 15 minutes for the henna to mature.\nApply the henna to the hair.\nHenna the hair as usual for at least 2 hours.\nMake the indigo in a bowl just before rinsing the henna.\nSome people add 1teaspoon of salt per 100g of indigo to obtain a richer colour.\nMix the indigo into a smooth paste and leave for 20 minutes.\nIn the meantime rinse the henna from the hair but do not shampoo or condition the hair.\nDry the hair until it is just damp.\nApply the indigo straight away and leave in the hair for at least 30 minutes for dark brown and 1-2 hours for black.\nRinse the indigo out of the hair but do not shampoo for 2 days as the indigo will need time to oxidize and reveal its true colour and sheen.\nCover the floor and delicate surfaces with newspaper or go outside.\nWear old, dark coloured clothes.\nCover the shoulders with an old towel or cape.\nProtect the skin around the hair line and the ears with argan oil to help avoid staining of the skin.\nWearing gloves, apply the dye to clean and wet hair.\nSection long hair on the head and apply the dye to the hair at the nape of the neck first with the corresponding hair, gradually working over the crown of the head to the forehead, ensuring that every hair is coated right down to the roots.\nOr simply massage the mixture close to the hair line and along the strands of hair. It only takes me 5 minutes to massage the dye in by hand. You can use a paint brush if you prefer.\nThen cover the hair with an old scarf, a shower cap, or cling film and wrap the hair for the required time.\nBy keeping the hair well covered, the dye will not dry out and it will be easier to rinse.\nKeep the head warm (a woolly hat is great) to encourage dye release.\nWipe any drips from the skin immediately.\nRinse and wash the towels straight after use to reduce the risk of staining.\nHair which is naturally dark and thick will need longer dyeing times than hair which is naturally light coloured or fine.\nTo achieve dark colours and jet black, the two-step process is always recommended. When using the two step process, if you don't quite get the coverage you require, both steps need to be repeated.\nis naturally light in colour or fine.\nWhen using these longer dying times, it is far better to mix the henna with Seaweed extract, made by standing the dried Scottish Seaweed in water. This will give a full strength seaweed extract. Heat the extract and use instead of water for mixing the henna and indigo. The fucins and alginates (the slippery substances in the seaweed) have amazing conditioning and protective qualities.\nBe prepared to experiment. Record what you do so that when you do find the way which give you the best colour results, that way you can easily repeat your optimal method.\nrecommended for fine hair, dry hair, Afro hair or damaged hair. In fact, adding seaweed to natural hair dyes gives real benefits to the quality of the hair and scalp. I always use seaweed extract in place of water for mixing my hair dyes.\nThe henna is ready to apply as soon as the top layer darkens (15-20 minutes), the indigo once the top layer goes a blacky-bluey-purplish colour (15-20 minutes).\nResist the temptation to add vinegar or lemon juice to the henna, just use water or Seaweed extract please!\nYes, this indigo is 100% dried and ground leaf of the indigo plant. Nothing more and nothing less. It is totally pure and certified organic.\nHow do I do a strand test?\nIf you have not used indigo before, or if your hair has been chemically dyed or treated, you might be curious about how your hair will turn out. It is easy to find out: Apply some indigo to an inconspicuous strand of hair in trial using the same process which you intend to use, either the one-step or two-step process and for the same timings. Bear in mind that the hair which has been dyed with indigo will darken slightly over the course of two days.\nAllergic reactions to pure indigo are rare, but if you have not used this product before, it is wise to conduct a patch test. Mix a small amount of indigo with hot water and allow to mature for 15 minutes. Place the indigo paste onto the inside of the forearm or behind the ear. Cover and leave for as long as the indigo will be in contact with the skin during the course of hair dying. Remove the indigo paste once the time is up or sooner if there is a reaction. Examine the condition or the skin.\nCan I use indigo if I have already dyed my hair with synthetic hair dyes and used chemical treatments?\nYes, you can apply both the henna and indigo to hair which has been dyed, bleached, streaked, highlighted, relaxed and permed, but it is best to wait for several weeks before using either of these products. Wash the hair with Rhassoul Clay first to remove chemical build up so that the henna and indigo can work on the hair directly.\nCan I remove indigo from the skin?\nstaining. Wipe any drips away immediately and wear gloves.\nCan I remove indigo from the hair?\nHenna acts as a permanent dye on the hair and can not be removed. Indigo is slow to fade too. Do not bleach the hair after using henna or indigo, as the results are very unpredictable.\nCan I use synthetic hair dyes and chemical hair treatments after using indigo?\nUsing synthetic and chemical hair treatments after dying the hair with indigo gives very unpredictable results.\nmuch safer to use than chemical hair dyes, especially with darker colours. PDP, a chemical in chemical hair dyes can cause life threatening allergic reactions, hair loss etc. Even if we can prevent one allergic reaction by teaching people about alternatives to chemical hair dyes we will be delighted.\nHow do I store the indigo powder?\nTo preserve the quality of the dye, keep your henna and indigo powder in a dark, cool and dry location. While the henna powder and paste can be frozened, neither the indigo powder nor indigo paste can be frozened. If using part of a packet of henna or indigo, roll up the packet tightly to exclude the air to keep the remainder as fresh as possible.\nWhat happens if you don't use henna with the indigo?\nApplied, straight to blond hair, indigo changes the colour to blue, or greenish, depending on the amount of yellow pigment already present in the hair. Indigo is the traditional dye for making denim jeans blue.\nDo a patch and strand test before use if you have any concerns about allergic reactions or if you want to check the final colour on your hair.\nStore the indigo way from children and pets.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"When will the Jeep Gladiator be availbale?\nAre you prepared to fully enhance your driving and payload performance with an innovative truck model designed by the Jeep brand? For Florida truck seekers who are interested in the all-new Jeep Gladiator, the release date for this one-of-a-kind Jeep model that returns to the Florida roads and trails around April 2019! If you're curious on what this model brings to your driving style and capability needs, read below to see some of what the Jeep Gladiator offers!\nThe Jeep Gladiator offers high-level power and performance with a standard 285-horsepower V6 with the optional 8-speed automatic transmission. If you want even more capability, the available 3.0-Liter Turbocharged 3.0-Liter diesel engine option brings 442-lbs of torque to the table and holds a towing capacity of up to 7,650-lbs which is ideal for trailering dual four-wheelers or your fishing boat! The Jeep Gladiator also sports up to a 1,600-lbs payload and offers the available 33-inch all-terrain tires for maximum grip off-road!\nPop off the removable doors and rooftop on your next driving adventure to your favorite campgrounds or tailgate party and look forward to plenty of space for up to five passengers! The 2020 Jeep Gladiator is designed with all the innovative tech that Jeep offers and if you're considering a new truck model, we'll have the Jeep Gladiator in our showroom before this summer!\nStay updated on the 2020 Gladiator arrival to Florida by contacting our sales team at (904)263.5451, and stop by our dealership in Starke when you're ready to view the Jeep trims we have in store for you! We can't wait to show off this unique Jeep model to you, and we'll see you in our showroom soon!","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Style and swag have often been associated with various kinds of cars available in the markets today. However, car accessories too play an important part in bringing the right functionality and style to a vehicle. A lot is a result of engineering of the highest level and with technology taking wings, everything about cars is only going to get better. SUVs are among the best versions of cars that you can buy. If you have a passion for off-roading, SUVs are perfect for you. Mostly people who like engaging in adventurous activities purchase SUVs to help them and in the last decade accessories for SUVs have made life easier for SUV owners.\nHitch bike rack: The mount is places such that the chances of accidents are on the lower side. The trailer hitch of a SUV helps in properly mounting the rack. The latest bike racks are user friendly and can even hold more than a single bike at a time.\nRear Tire Mounted Bike Rack: This bike rack is mounted on the spare tire mostly found at the rear end of an SUV. Foam pads help in protecting any damage to windows and the paint of the vehicle. These racks can however block the rear view to some extent.\nSize of the hitch receiver: When purchasing, take into consideration the hitch receiver size as per your requirements. Receivers can range from 1 \u00bc inches to 2 inches in size. Make sure that you choose a mounting unit that fits the hitch size and the weight carrying capacity.\nSpare Tire: If you do not have a rear tire, do not go for a rear tire mounted rack.\nLoad Capacity: Different bike racks and transport carriers have different load bearing capacity. If you like taking vacations with a group of friends or with your family, you might want to consider a mount that can take more than one bike.\nDesign: Bike racks come in different shapes and designs. You have a plethora of designs to choose from and if you have a sense of aesthetics, you can choose from a number of designs available in the market to go with your SUV.\nHandling: Consider how easy it is to attach and detach a bike rack. A good mount with a foldable design is ideal.\nInnovations in the field of automobiles keep reaching newer levels and with the right kind of accessories for your car, you can engage in a number of adventurous activities. If you are making a purchase online, make sure that you go through the reviews and feedbacks about the website and the product you plan to purchase.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\"Tribute To Ladies Of Jazz\" goes global with PhaseGlobalRadio.com!\n04 Apr \"Tribute To Ladies Of Jazz\" goes global with PhaseGlobalRadio.com!\n\"Tribute To Ladies Of Jazz\" in Paul Macs shows in Phase Global Radio \u2013 Smooth Jazz Radio And So Much More! Listen now at PhaseGlobalRadio.com !","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Early this year, I found out about a program in Minnesota, a writer's workshop that lasts a week in July. To participate, you fill out an application form and submit a sample of your writing. It races across cyberspace to a place I hear is beautiful and then it is gone, you've unclenched your fist and let a dream and your work fly free. As I hit the send button, I wondered if my words would come back having achieved the purpose for which they were intended, or if they'd come back to me empty.\nI got word today. Not quite a yes, not quite a no. I am an alternate, so if someone is not able to attend and I am on the right place in the list, I will get to go. I was also strongly urged to keep applying\u2014maybe that's part of the process. I know a lot of programs where you need to be persistent and not get discouraged the first time you get a no. I've left my name in the hat and I will keep applying. I am also disappointed.\nI wish had been accepted. That competitive streak in me easily slides into a small quiet sentence that insinuates itself into my comings and goings: \"you aren't good enough.\" I had applied because, much as I love writing these posts, it feels like it's time to push my writing beyond what is now familiar and comfortable. I want to know if I can do more. I want to test this part of who I am in new ways. The thing is, after a pretty bruising winter, the hope of hearing a yes outside of me became even more important. I began to think that a yes that was not mine was what made the risk worthwhile. I wish I had been accepted and I wonder if there is something else for me to learn in this in-between space.\nA week ago today, I was just this side of being a part of the Walking Dead. Holy Week was especially taxing this year and I was beat. I sat in our living room, all my good intentions to do housework in as much disarray as my house. Instead of cleaning the fridge of all the science experiments thriving inside, I watched a movie, Jiro Dreams of Sushi. It is an elegant, thoughtful exploration of the vocation of a sushi chef in Japan whose small, unprepossessing restaurant in a subway station has earned a 3-star rating from Michelin. A food critic, reflecting on Jiro and why he is a master chef , says that Jiro sets the standard for personal discipline. He always looks ahead\u2014what he serves has to taste better than the last time he served it. He is never, ever satisfied with his work. Jiro is impatient and Jiro is passionate.\nThat's a good set of guiding principles that push back on my temptation to yield to insecurities and vanity. After a good hard walk this evening, I am back trying to figure out what I can do instead of the opportunity I more than likely won't get this year. I am not satisfied with my writing. I am impatient to get on with it. I love getting to do this work. I welcome, in fact, I very much need suggestions about other ways I can do this mo' better, more deep. More.\nRosa \u2014 I applied for the same program and got my second \"no\" \u2014 my hope for this year had a lot to do with the prospect that it just might be something we could do together. What fun that would have been ! But we will both keep applying and in the meantime, keep looking for ways to grow. The next things for me are (1) set up a blog myself \u2014 maybe you could help me with that, and (2) finish this book on the psalms \u2014 I'm finding it more difficult than the last.\nIf the Institute doesn't work out, let's see how we can help each other, or share other opportunities that we hear about.\nDon't let it discourage you. Your writing touches many hearts. Love to you.\nI applied last year and was an alternate (and didn't get to go), applied again this year and have yet to hear about the August week I applied for.\nYour words are lovely and worthwhile \u2013 and I am grateful for their appearance.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"WaterWood specializes in custom one of a kind wooden bathtubs. We are the best. We are small. Director and Founder Taylor Ross and a handfull of craftsmen who have trained under him directly, personally craft and shape and sculpt each piece. All works including chair crafting flow from the wooden bathtub and boatbuilding skills they have developed through years of international training and business in multiple cultures. Commission a bathtub. Commission a chair. Anything that needs a vision and forward thinking is what we do.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"August in the rose beds is time to keep an eye on soil moisture for sure and also time to give the roses their last granular feeding of the season. This month I need to bring up a rose lover's heartache and then take a look at something joyful.\nThe Heartache is a nasty incurable virus called Rose Rosette Virus\/Disease. When I first heard about this heartbreaking disease, it was thought to be spread by dirty pruners. So we became very careful to wipe our pruners down with some Clorox wipes after pruning each rose before we moved on to prune the next one. Or we dipped the cutting blades into a bleach solution. Still not a bad practice, but as time and study have progressed this nasty virus is believed to be spread by a very tiny mite. The eriophyid mite or leaf curl mite, in fact. One of the chemical controls for this nasty little mite is a miticide called Avid. Remember that the use of typical insecticides does no good against mites and can actually help increase their numbers by killing off all of their natural predators allowing their colonies to boom forth! A miticide must be used for the control of mites.\nIt is important to note that the new and healthy growth on many rosebushes comes in as a nice deep red or maroon colored growth. This growth looks normal, is not twisted or distorted, and will turn green as it ages. Some new growth will keep a nice red or maroonish colored edge on the leaves too. Pruning off that growth thinking that one can avoid RRV is a mistake that has been made many times. By pruning the growth out, we have just eliminated nice new growth that would have lead to a new bloom or blooms. It also does no real good in preventing or eliminating the dreaded virus.\nAfter that info we need some garden joy right? How about some nice healthy \"eye-catching\" roses for your garden areas, you know, those roses that have a real wow factor of color and or size. Many shrub rosebushes will provide a nice eye-catching wow factor with their loads of blooms that nearly engulf the bush. Such as Austin English Shrub roses like Mary Rose, Graham Thomas, Golden Celebration or Crown Princess Margareta. The Home Run line of rosebushes from Weeks Roses and their rivals the Knockout Roses make a nice statement. My two favorites are Watercolors Home Run and Double Knockout.\nRoses with vivid and different colors would include Sriracha, a new floribunda rose with orangeish color, grafted to be a tree rose is true wow factor. Oranges N Lemons is a mix of orange and yellow and has foliage that glistens in the sun. Lavender Delight is a miniature rose that grows to be a good-sized bush with wonderful lavender colored blooms. Tiddly Winks is another miniature rosebush that is an eye-catcher as she loads up with her vibrant orange and yellow clusters of blooms. Stainless Steel is a nice hybrid tea rose whose bloom smiles are an \"oh so pretty\" light lavender color.\nThere are white blooming rosebushes such as Glamis Castle, Crystalline, Pope John Paul II and White Lightnin' that are very special in Moonlight Gardens. These eye-catching wow factor roses grab the attention, to draw people in for a closer look. Then their eyes move out to the rest of your garden area to see all the true joy that awaits them there. What a delight it is to us to hear all the nice comments coming from the enjoyment of our efforts to create an oasis of beauty in our yards!\nEnjoy your rose beds and gardens and be present in them. Be vigilant for any problems, as catching such things early is the best action in control. Don't forget to feed and water! Enjoy!\nAuthor bio: With 40+ years of rose growing experience, Stan V. Griep is a Consulting Rosarian for the Rocky Mountain Region of the American Rose Society as well as the Denver and Loveland Rose Societies, a Colorado Native Rosarian, and is a member of the American Rose Society, The Denver Rose Society, and the Loveland Rose Society. He is a Cyber Consulting Rosarian for the American Rose Society, Colorado Gardening on-line and GardeningKnowHow.com as well as a freelance writer and speaker. An award winning photographer, his latest book is available at Blurb.com: http:\/\/www.blurb.com\/b\/6909245-heavenly-gifts-for-the-soul .","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"It used to be assumed that getting married would help to alleviate debt concerns because doing so would have a positive effect on a person's tax bill, among other things. However, that may not actually be the case, especially if both spouses have high annual incomes. Beyond that, getting married can potentially reduce a person's mortgage deductions when filing taxes.\nUnder some circumstances, of course, marriage will do wonders for a person's yearly tax requirements. For instance, when one spouse earns significantly more money than the other spouse, the couple's joint tax bill will likely go down after marriage.\nIn many cases, though, getting married could end up bumping the couple into a higher tax bracket because the spouses' combined reportable income will be substantially higher. The end result will be greater tax obligations and potentially greater debts.\nAnother negative financial implication of getting married is that married couples are limited in the amount of mortgage and home interest that they are allowed to deduct on their tax returns.\nThis didn't used to be the case, but the Internal Revenue Service (IRS) recently made a big change in the law. The change came in response to a 2015 court ruling in which an unmarried couple, Bruce Voss and Charles Sophy, were granted maximum mortgage debt and loan interest deductions on two houses they owned together in California.\nPrior to this court decision, it was the position of the IRS that no two parties should be allowed to claim the maximum deduction on the same property in a given year. However, in light of the decision handed down by US Court of Appeals for the Ninth Circuit, the IRS has changed its tune: when filing taxes, unmarried individuals can now legally deduct the interest on as much as $1 million in mortgage debt and up to $100,000 in home equity financing.\nHowever, the change in the tax code also means that it may not be in a person's best financial interests to get married \u2013 since married couples cannot claim two deductions.\nIf you are struggling with debt issues, you should talk to a qualified debt management attorney. Joel R. Spivack, Esq., is an experienced debt relief lawyer who can help you explore all of your legal options. Contact Mr. Spivack anytime to schedule a free consultation.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In this video, I feature my experience at the Summer Lodge, a small, luxury hotel in the rural region of West Dorset an easy train ride from London.\nenter It was chilly and oh so cozy! I am still dreaming of that delicious multi-course meal.Staying here made me feel connected with the local culture, the land, and its history. Undoubtedly, the Summer Lodge Hotel falls into my hotel ranking category of Local Luxury with a top score.\ncelebrex drug contraindications with viagra The property is part of the Red Carnation Hotel Collection, a boutique hotel group of \"seventeen luxury, award-winning, family-owned and run boutique hotels\u2026in the UK, Ireland, South Africa, Switzerland and the USA.\" Summer Lodge is also a Relais & Ch\u00e2teaux hotel \u2013 known for luxury, style, and really great food. The lodge itself is a converted dower house. Think where Matthew Crawley and his mother lived in relationship to the big estate in the TV show, Downton Abbey. Traditionally, a dower house was an estate designated for the dowager, the widow of the former estate-owner. And the talented team behind the Summer Lodge Hotel has converted this former dwelling into a very special oasis that welcomes guests, making them feel right at home.\nhttp:\/\/travelwithkate.com\/?x=viagra-like-drug The Summer Lodge has 24 rooms and a spa sitting inside four acres of gardens, along a deer park and among the rolling hills of the English countryside. The hotel is also just adjacent to the village of Evershot where you can walk to a lively, local pub, poke around the one shop in town and pick up some goodies at the bakery.\nsee url While staying here, I was feeling immersed in an enchanting landscape that happens to be quite famous not just because of Downton Abbey but also for its prevalence in Thomas Hardy novels. The hotel is right in the heart of the old kingdom of Wessex, the first kingdom of England. Hardy refers to Wessex throughout his work. And scenes from Tess of the d'Urbervilles actually took place in Evershot. And for you literary enthusiasts, another fun fact is that before Hardy was a writer he was an architect. And he actually designed one of the structures in the hotel.\nhttp:\/\/smartmobilead.com\/?search=viagra-super-active-original-pfizer-order This room above was part of that structure. Here you can lounge while staff bring goodies like tea, champagne, and other sustenance. I thought the game shelf was a really nice touch!\ncheap drugs net product viagra side Scrabble, backgammon, or Chess anyone?\nplavix drug ingredients viagra This region of Dorset has been the property of three large land-owning families for the last 300 years. This means relatively few people live out here and these lands are home to small farms that specialize in specific products, produce, and livestock.\nFor example, the neighboring region of Somerset is known for delicious apples. Another, Devon, is a hub for beef and dairy products (especially butter and cream). And Dorset itself is known for lamb. And like chef Steven Titman mentioned in my video, the coast is only 40 minutes away where fisherman supply fresh seafood.\nCombine the ease of high-quality, local ingredients, the talented culinary staff and world-renowned cellar master, Eric Zwiebel (ranked among the top four sommeliers in the world), and you basically never want to leave.\nBut do fight the temptation and be sure to go on day trips and excursions like I did to the small town of Sherborne only 20 minutes away by car. You can see my Sherborne video here. And definitely check out the Jurassic coast, also featured on the Travel with Kate web-series here.\nAnd now I must extend a thank you to the kind staff at Summer Lodge and Visit Britain for hosting me. I hope to be back again soon.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"When Congress granted economic immunity to pharmaceutical companies in 1986 for vaccine injury, Congress recognized it eviscerated the economic incentive for pharmaceutical companies to create safe vaccines or make existing ones safer. Congress therefore placed the responsibility and burden for vaccine safety directly and virtually entirely on the shoulders and in the hands of the Secretary of HHS. This requirement is codified at 42 USC 300aa-27 (note that the 1986 Act is very short and codified in 42 USC 300aa-1 through 300aa-34) which is the section underpinning the entire mandate for safe and safer vaccines in this country \u2013 this section is literally entitled \"Mandate for safer childhood vaccines.\" This section first lists the HHS Secretary's obligation to assure vaccine safety and then requires the HHS Secretary to submit a report to Congress every two years detailing what improvements in vaccine safety were made in the preceding two years. But, amazingly, as seen from the so ordered stipulation, it does not appear that HHS has ever, not even once, submitted a bi-annual reported to Congress as required by 42 USC 300aa-27 listing what improvements it has made to vaccine safety. This speaks volumes to the lack of seriousness by which vaccine safety is treated at HHS and heightens the concern that HHS doesn't have a clue as to the actual safety profile of the now 29 doses, and growing, of vaccines given by one year of age.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"New for 2018. Here are some stunning new knives from second generation blacksmith Yuzou Tani who works in Sakai Japan. Born in 1962, Tani San has been blacksmithing for 35 years. The shop bearing his family name was established 50 years ago by his father.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Highly concentrated preparation with 3% ferulic complex and silk threads. Rejuvenates the skin and intensely reduces all spots and discolourations. Active ingredients stimulate collagen synthesis to eliminate signs of sagging, wonderfully moisturise and smooth the skin while protecting it against photoageing.\nInnovative complex based on high-molecular-weight proteins present in silk threads and Matt-Focus. Superbly smooths the skin, improves its elasticity, and slows down ageing processes. Moisturises, reduces redness, and ensures a dazzling effect of silky smoothness and powdery matt finish.\nTechnologically advanced complex which combines cytovectors, concentrated ferulic acid, and liposomeencapsulated vitamin C. Fabulously reduces all skin spots and discolourations, caused both by exposure to the sun and by age (pigmentation). Evens out the skin tone and delicately lightens the complexion, leaving it looking refreshed and visibly younger. Has antioxidant properties \u2013 fighting free radicals to provide state-ofthe-art protection against photoageing.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"It smells of rotten eggs but still fasci\u00adnates physicists around the world ... hydrogen sulfide H2S. Normally, hy\u00ad\u00addro\u00adgen sulfide is a poisonous and bad smelling gas.\nResearchers around Mikhail Eremets and Alexander Drozdov of the Max-Planck-Institute of Chemistry in Mainz, Germany, have now shown that hydrogen sulfide can take on a highly desired property \u2013 it becomes superconductive at the unbelievable temperature of 203 K (-70\u00b0C).\nThis is a new temperature record and it means a giant step towards a material that becomes superconductive at room temperature. This is why the experiment was soon labeled \"historic\" or even \"The Holy Grail of superconductivity\".\nWith a diamond cell, hydrogen sul\u00adfide was put under a pressure of 150 GPa (approx. 1.5 million bar). Characteristic features of superconductive material were detected below 203 K: zero electrical resistance and the fact that an external field did not lead to a field inside the material (Meissner effect).\nBoth a Quantum Design PPMS and MPMS system were used for the experiment to reproduce the transition temperature and determine the magnetic properties. Both systems have a superconductive magnet and work in a temperature range of 1.9 K to 400 K. They allow fully automated temperature- and magnetic-field-dependent measurements. Please contact us for fur\u00adther possible applications.\nDrozdov et al. Nature 524, 277 (20 August 2015) \"Conventional superconductivity at 203 kelvin at high pressures in the sulfur hydride system\".","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Asphalt isn't versatile, which implies that less than Severe heat or chilly (think Midwest summers and winters!), asphalt is a lot more more likely to get really hard and crack. You see what the harsh winters do on the roadways, don't let exactly the same issue occur to your roof!\nlocal weather: you may perhaps need a roofing content you can find out more that withstands sun or precipitation exposure read full report greater, based upon your geographical site.\nif you are replacing a roof, talk to the roofer the number of layers are currently up there. (He can commonly tell easily by creating a small Reduce.\nminimal roof leaks might be mended with no help of an experienced roofer. the subsequent steps will instruct you regarding how to establish challenges and make repairs to flat, shingle and wood shake roofs. you'll want to test to operate on your own roof on each day when the roof is dry, to prevent mishaps.\nJohn Thorne says: Oct 26, 2010 at 8:08 pm I purchased an early sixty's ranch property from an estate, vacant for a long time) in March. The garage roof is flat; the home is one-story using a shallow-pitch shingle roof. What I call the garage roof essentially handles a pantry\/laundry home powering the garage as well. within the again conclude of your house drinking water continues to be flowing off the garage roof on to the siding of the house and soaking in. This corner is underneath the roof eaves. The siding, wall studs, and insulation have all been compromised. Who understands how many years this had been happening.\nbecause of its relatively economical upfront cost, asphalt may be the main content from the roofing marketplace currently. But don't be fooled by a minimal upfront Value which will turn out costing you a discover this info here lot more down the road.\nconfirm each of the legal and thanks-diligence details before you indication a roofing deal. A roofer that doesn't have the correct license and insurance coverage could inflict injury on your private home and depart you without recourse.\nRoofing in Michigan and Ohio. \"I'd a roof carried out by Hanson several years back and it even now seems good and I have a life span guarantee. You can't beat that. So currently I am possessing gutters set up And that i didn't have to look ahead to weeks to acquire it finished, just a couple of days.\na firm's Experience score can maximize or minimize the rate that you choose to pay back. Experience ranking is set by previous claims within a twelve to 36 month span. Companies which have experienced much more incidents Use a large encounter Rating and commonly are billed a better rate.\nThere is certainly these numerous types of roofing materials and styles and each roofing material has its have Rewards. selections involve shake roofs or imitation shake roofs, Wooden shingle check this out roofs, cedar shingle roofs, clay tile roofs, slate roofs or perhaps concrete tile roofs.\nI'd like that will help you, but 1st \u2013 I would like more information regarding you roof. First it is vital to ascertain what sort of roofing content is utilized to waterproof your creating. Then it is necessary to learn wherever the leaks are, if there is ponding drinking water or some other inherent weakness inside the roof by itself.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"McGill CF-1 7\/8 SB Cam folloLion Bearing Industrial Co.ltdr. LOCATION: U-17C.Payment: Lion Bearing Industrial Co.ltd will accept PAYPAL, Credit Cards, personel and Lion Bearing Industrial Co.ltd checks . If you send payment by mail, please e-mail us so Lion Bearing Industrial Co.ltd can watch for the payment. Turnaround Time: Standard order processing is 1 day after payment. Orders placed by 2:00 pm Central Standard time will ususally ship the same day. To expedite an order for an emergency please e-mail us and Lion Bearing Industrial Co.ltd will work with you or call us at 615-471-1013, 8am to 5pm Central standard time.Return and Warranty Policy: Lion Bearing Industrial Co.ltd will guarantee this item to be as described for 14 days. If for any reason you feel it is not, you may call and request an RMA number. After receiving yLion Bearing Industrial Co.ltd RMA number, you have 30 days to return the item. Upon receiving and inspecting the item, Lion Bearing Industrial Co.ltd will issue a refund in accordance with Lion Bearing Industrial Co.ltd Re-stocking policy. Lion Bearing Industrial Co.ltd will not refund shipping costs on the item unless there was a shipment error. If there is an issue with the item, please contact us by e-mail or call 615-471-1013 for resolution.This warranty is given in lieu of any other warranties, either expressed or implied, including that Lion Bearing Industrial Co.ltd disclaim any warranty of merchantability, fitness for a particular purpose and\/or non-infringement. In no event shall Lion Bearing Industrial Co.ltd be liable for any damages except actual damages up to, but not to exceed, the amount paid to us for the product, including Lion Bearing Industrial Co.ltd shall not be liable for any consequential or indirect damages or lost profits whether or not advised of same.Shipping: Lion Bearing Industrial Co.ltd ship Worldwide from Tennessee using UPS, and UPS Freight.when needed. E-mail us for a shipping service other than the ones listed. Lion Bearing Industrial Co.ltd ship daily Monday through Friday.Packaging and Handling: Lion Bearing Industrial Co.ltd pack yLion Bearing Industrial Co.ltd items properly! All items are wrapped and protected for shipment in a professional manner. Package Tracking: All UPS shipments will receive an e-mail with the tracking number of yLion Bearing Industrial Co.ltd package. Restocking: Lion Bearing Industrial Co.ltd will accept returns made by the purchaser due to the purchasers error, a non-warranty issue, but will administer a 25 percent restocking fee to cover time and materials. Order questions or technical support can be reached by calling 615-471-1013, Central Standard time. 8 am to 5 pm Thank you for visiting Surplus Industrial Controls Online.PoLion Bearing Industrial Co.ltdred by ProStores INTERNATIONAL CUSTOMERS BE ADVISED: buyer is responsible for any and all customs taxes\/fees including brokerage fees, etc. Lion Bearing Industrial Co.ltd WILL NOT skew invoice values for customers seeking loLion Bearing Industrial Co.ltdr customes taxation. This is a Lion Bearing Industrial Co.ltd policy and is also considered illegal.\nBOYA Singapore is an international trading Lion Bearing Industrial Co.ltd specialized in importing and exporting bearing and industrial machinery and equipment parts. In the industrial fields,Lion Bearing Industrial Co.ltd have 20 years' experience of McGill CF-1 7\/8 SB Cam folloLion Bearing Industrial Co.ltdr supporting and industrial service.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"On our website, you will find information about the ministries and activities of First Baptist Church Livingston. We have ministries for preschoolers through senior adults, with each of these ministries intended to help equip members and guests with biblical truth that can lead to a personal relationship with Jesus Christ. Whether you know nothing about the Bible and just have questions or you are wanting to continue growing as a Christian, you are welcome to visit First Baptist Livingston.\nOn Friday evening, July 27, the Women's Ministry will be having a movie night. We will be watching War Room. Movie night is for all women, and feel free to bring your daughter. This event is free!\nAs a follow-up to the movie, War Room, we will be holding a Women's Prayer event on Saturday, September 8.The event is free, and every lady in the community is invited to attend.\nNow that you have seen a theatrical example of how prayer can impact you and your family, learn from the on-set prayer coordinator for the film, Ms. Brenda Harris. She worked alongside the Kendrick Brothers (writers, producers, and directors) and the cast of War Room during production. She will share how prayer can change everything.\nDoors will open at 8:00 am, with a light breakfast being served. The event itself begins at 9:00 am. We will have a time of worship, and prayer, followed by a time of hearing from our special guest, Brenda Harris.\nPaul's ambition in 1 Corinthians 2:2 was \"to know nothing among you except Jesus Christ and him crucified.\" As a church, we seek to help people know Christ through a personal relationship with Him. It is through a personal relationship with Jesus Christ that a person can experience peace and eternal life.\nEntering into a personal relationship with Jesus is just the beginning though. We must grow in our relationship with Him throughout our lifetime through worship, prayer, and Bible study. Paul states in 2 Corinthians 3:18, \"And we all, with unveiled face, beholding the glory of the Lord, are being transformed into the same image from one degree of glory to another.\" First Baptist Church Livingston seeks to help members and guests to grow in Christ through small group Bible study, worship services, and personal discipleship.\nAs a person knows Christ and grows in Him, believers should seek to serve Jesus' church and others. Galatians 5:13 reminds believers in Jesus, \"For you were called to freedom, brothers. Only do not use your freedom as an opportunity for the flesh, but through love serve one another.\" Growing to be more like Jesus should lead us to act more like Jesus, and acting like Jesus means we serve others within the church, our community, and our world.\nAnother result of growing in Christ is that His followers help introduce new people into a personal relationship with Jesus. Paul states in Romans 15:20, \"I make it my ambition to preach the Gospel.\" This sharing of the Gospel is to be done by every Christian wherever God sends us in our journey with Him. At First Baptist Church of Livingston, we seek to equip believers to share their faith locally as well as on mission trips domestically and around the world.\nWant to know the Vision for First Baptist Church Livingston?\nVisiting a church for the first time can be intimidating. From the moment you set foot on our campus, you will find people who are friendly and want to help you while on our campus. Our services are intended for everyone to connect with God and learn His truth. Whether you have never set foot in church before or you have spent much of your life attending church, you will find worship services at First Baptist Church Livingston helpful, meaningful, and spiritual. You will see people in attendance dressed from blue jeans to suits and everything in between, we are more interested in people worshipping God than a wardrobe.\nThe Sunday morning experience can start before the classes or worship services begin. We have our Coffee Shop every Sunday morning at 8:00 am. You and the whole family can come and have a cup of coffee, juice, or milk along with eggs, biscuits and gravy, bacon, and sausage to start your morning. All of this is for free!\nFor families with a preschool-aged children, we have a nursery for the younger kids and a children's worship time for kids 3 years old through 5 years old during Sunday morning worship services. Preschoolers can worship and learn in a fun, age- appropriate environment, while parents can worship God, hear a meaningful message, and prepare to tackle the challenges of parenting for the week ahead.\nAt 9:00 am every Sunday morning, we have age-graded Bible studies for people of all ages. In these classes, people of similar ages\/life circumstances study God's Word and learn to apply it to their everyday lives. All classes are open and welcome new guests to attend.\nSunday morning service begins at 10 am. We have a time of singing, prayer, and a biblically-based message each Sunday where you will learn more about who God is and how you can have a personal relationship with Him that can help you live a life with more meaning and peace.\nOperation Christmas Child Shoebox Ministry: pack a shoe box full of toys and other useful items to a child in need. We packed over 5,000 boxes for 2017! A support ministry for our OCC ministry is a sewing group that provided over 1400 dresses in 2017.\nWMU Backpack Ministry: We will provide over 170 kids with 6 meals for each weekend during the school year.\nMusic Camp Ministry: Each fall we have over 70 kids spend 8 Tuesday afternoons learning and practicing a children's Christmas program, with a performance in November.\nDominican Republic: In the summer of 2018, we sent a team to the Dominican Republic to minister in an orphanage, a school for a very poor village, and handed out food to a remote impoverished village. Pictures of the 2017 trip can be found here.\nNew England: A group of our members participated as part of a mission trip by our local Baptist association to help two church planters in the New England area. We have two outstanding church partners, and the mission team will serve along side them as they reach out to the community and encourage the churches in Connecticut and Rhode Island.\nFirst Baptist Church of Livingston believes that advancing God's Kingdom doesn't come through new ways of doing church, but through the ancient way \u2014 faithful preaching and contextualizing the unchanging gospel within a changing culture. First Baptist teaches faith in Jesus Christ, God's only Son, as the only way to eternal life.\nAt First Baptist Church of Livingston all voices can praise the Lord. We seek to obey the call throughout Scripture for God's people to praise and worship Him. Our worship ministry encourages people to use their talents to sing congregationally, in the choir, on the praise team, musical instruments, and through special performances.\nAt First Baptist Church, children have numerous programs and classes to participate in. From our Nursery to AWANA to Middle School to High School, there are classes for everyone. Our children participate in Vacation Bible School, Bible Drills, Church camps, Christian concerts and fun trips to learn the fundamentals of being a Christian and to grow as a disciple of Jesus.\nThe Recreation Outreach Center (ROC) is the gymnasium on the west side of the sanctuary. The ROC hosts Upward Basketball in the Spring as well as the Church Volleyball league. Toning the Temple is held weekly to promote exercise and fitness. The ROC is available numerous times during the week for open use.\nJoin us each Saturday as we travel throughout the community to visit those in need. Everyone is welcome as we visit local hospitals, nursing homes, those who are sick and shut-in and others who just need to know they are loved and appreciated.\nJoin us each Sunday morning at 8 am in the Fellowship Hall (ground level behind the Church) for all of your favorite breakfast meats and treats. We have fellowship together and enjoy each other's company. Breakfast is free and open to all. All you need to bring is yourself (and some friends).\nUpward Basketball and Upward Cheerleading offers an extended, multi-month opportunity for character-building, player development, and the formation of deep relationships between teammates and coaches. Players will develop a love for basketball and a healthy competitive spirit and discover skills necessary to succeed both on and off the court. The game format and rules are designed to teach the game at an age specific level that grows with each young athlete.\nAt First Baptist Church, our Student Ministry seeks to actively engage our local Middle School and High School students to strengthen their relationship with the Jesus Christ. We offer Christ centered Bible studies and outreach events. These allow the student to know the teachings of Jesus, and begin to know his character. Also, we offer group discussion times, and book studies. This interaction allow our students to sharpen the gifts and talents God has given them and to apply the teachings they have learned to their own lives. This effectively allows them to grow in a closer relationship with Christ. The student ministry organizes opportunities that allow students to serve others in the name of Jesus. These events such as serving food, Operation Christmas Child Shoebox Ministry, and the Feed the Children backpack program allow the student to understand that when they serve others they are in extension serving Christ. As the students are working through the process of knowing Christ, growing in Christ, and serving Christ, they are being equipped to go and to share the good news of Jesus Christ with others. We as a student ministry want to effectively love God and Love people.\nDiscipleship is an important part of being a Christian. In Matthew 28:18-20, Jesus gave the church the Great Commission, or mission of the church. That mission is to make disciples. A disciple is a person who takes on the knowledge and characteristics of the teacher. In other words, a Christian is supposed to think and act like Jesus.\nAt First Baptist Church Livingston, we seek to help people become more like Jesus through the various teaching ministries of the church. We offer Sunday school classes at 9:00 am that seek to teach basic biblical truths to people in every stage of life: from preschool, children, students, college, young married couples, couples, to senior adult. Each stage of life presents unique challenges to applying God's truth to everyday life. In a Sunday school class, you can learn with others in a similar life situation on how to live out God's Word on a daily basis and ask questions that will help you personally apply truth.\nFor those who want to learn beyond the basics and move to a more detailed study on a particular topic, we offer Discipleship Training classes on Sunday afternoon at 5:00 pm. The topics covered will vary but may include an in-depth look at a Bible book, a study of a particular Bible character, or a study on a topic like how to study or prayer. To find what classes are being offered, please check our church calendar, look at the weekly worship guide, or contact the church office at office@fbclivingston.com.\nOne of the important steps in growing spiritually is honestly assessing where you are in the spiritual growth process. Just like what happens with someone physically there is a process for growth. Jim Putnam in his book Real Life Discipleship talks about the five stages of spiritual growth.\nThe teaching ministry of First Baptist Church Livingston is intended to help each person move through these stages and grow in spiritual maturity. You are encouraged to take advantage of worship services, Sunday school classes, and discipleship classes to help you in your journey in growing as a disciple.\n\"The Suffering Servant April 9 2017\".\n\"The Resurrection April 16 2017\".","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Welcome to Vault Storage Co.\nHere at Vault Storage Co. we have everything you need to save space around your home and business. Our self storage options are secure and conveniently located in North Wales, PA. Along with a variety of storage units, we offer moving supplies and parking for vehicles. Our storage experts are ready to guide your valuables into the ideal storage space with tips for every situation. See how our storage units will make your next move simple. We offer temperature controlled self storage units for commercial and residential customers. While you pack up the home or office, let us secure and protect your belongings.\nWhether you are looking for short term storage or long, we are here to make your situation simple. For your convenience, reserve your unit online and then stop by to find the best moving supplies and advice in North Wales PA.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzatrrj b/data_all_eng_slimpj/shuffled/split2/finalzzzatrrj
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+{"text":"Decision Making in Health and Medicine supersedes the highly respected text, Clinical Decision Analysis, by Weinstein, Fineberg, Elstein, and colleagues published in 1980. Its main purpose is to provide insights and tools that can aid decision making in health care. In this, like its predecessor, it eminently succeeds. The flyer makes the bold claim that the user-friendly text will help everyone involved in health care and medical decision making, specifically practitioners, trainees, and students studying clinical decision analysis, evidence-based medicine, and clinical epidemiology. Although a little hyperbole is to be expected, this description has serious potential to mislead. Despite being extremely well written and structured and furnished with a wealth of supporting information on the accompanying CD-ROM, size alone makes the book an inappropriate place to start for those wanting an introduction to decision support techniques. It is appropriate as a text to support attempts to do decision analysis. Some familiarity with the concepts of decision analysis are required to make the most of this book.\nThere is no information on how the resource was compiled, although such information would be unusual for a methodological publication. It is debatable just what it might have added and whether greater explicitness might be appropriate for future editions. However, it was frustrating that the tools described had changed from those in Weinstein and colleagues' Clinical Decision Analysis, and there was no discussion on what prompted these changes.\nThe accompanying CD-ROM contains solutions to exercises (the approach to imminent Markov madness in chapter 10 is particularly appreciated!), decision analytic software, examples of the decision models, supplementary materials including useful spreadsheets, and references with abstracts.\nThis is not a resource designed to directly provide clinically useful answers. But such answers would undoubtedly flow from clinicians or policymakers equipped with the skills the book seeks to impart. It is difficult to dip into without a clear overall understanding of the subject, but this might well be gained initially by reading the book in its entirety. In contrast to many textbooks, the readability, effective structure, and well-chosen illustrative examples make such a cover-to-cover journey feasible. I will certainly use the resource and will recommend it to Master's level students\u2014I do not anticipate much use at the undergraduate level. Given the growing importance of the tools described in this book, the need for an introductory text or article suitable at the undergraduate level is highlighted.\nDecision Making in Health and Medicine is an excellent resource dealing with the conduct of increasingly important and ubiquitous decision aids, but is not suitable as an introduction to them.\nDecision Making in Health and Medicine can be purchased online at www.us.cambridge.org for U.S. $65.00.\nHunink M, Glasziou P. Decision Making in Health and Medicine. Integrating Evidence and Values. Cambridge: Cambridge University Press; 2001.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"With a sigh, he heaved his trunk into an upright position in preparation to sit on it and wait for Celice.\nBefore he could, a voice called out, \"Pharazon!\" He turned to see a familiar white Lupe waving at him from the window of a Uni-drawn carriage. \"Get in before you catch Neomonia!\" Celice said, gesturing inside.\n\"Coming!\" he replied, dragging his trunk across the wide dock where other Neopets busily unloaded cargo or boarded carriages. He should have brought a scarf, he decided. It had been deceptively warm back in Altador.\n\"Just toss it in the back hatch, there,\" the Uni driver, a kindly-looking blue female, said with a toss of her horn.\nPharazon did so and then climbed into the carriage, on the seat across from Celice's. He wiped the moisture from his forehead and tried to dab himself dry with his damp cloak.\nThe Lupe flicked a finger and a flame burst from the tip of her claw. The fire was small, but gave off an unusual amount of heat for its size.\n\"Thanks,\" Pharazon said, placing his claws up to it to warm them.\n\"How was the flight?\" Celice asked as the carriage lurched into motion. Still holding out the flame, she crossed one leg over the other and watched him over the rims of her spectacles, her bright yellow eyes glistening in the firelight.\nShe looked the same as ever, Pharazon thought. Her white hair was tucked back into a prim bun, although a few strands hung loose at the sides of her face, curled slightly with the humidity. Celice was nearly twice as tall as him, about as tall as his owner, and her proportions were more owner-like as well\u2014even Neopets of the same species could vary widely in size and physiology. She was always a snappy dresser, and today she was wearing robes of rich purple and gold, with a fur-lined cape to keep out the cold.\nHe leaned back in his seat. \"Good,\" he said in the vaguest fashion possible.\n\"That's in three months,\" Celice said.\nPharazon smiled. \"Thanks.\" As much as he wanted to get back to his home as soon as possible, spending the week with Celice did sound like fun. If she was going to make him come all the way out here, he might as well enjoy himself after they got her chore out of the way.\nHe looked out the window at the shops and businesses passing by. They were headed straight into the heart of Brightvale's capital city, and traffic began to slow down accordingly. It was even slower on this wet day.\n\"You're sure this is a minor occurrence?\" he asked Celice, looking back at her.\n\"I have,\" Pharazon said honestly. Which was to say, he had thought about how he didn't want to study it.\nPharazon's jaw tightened and he pressed his back into his seat. \"What if I don't want to use it?\" he asked quietly.\nThe Draik's tail swished anxiously and he drew his knees up to his chin, unable to meet her gaze. Every time they talked about this he could feel a strain on their friendship, and he wished she would change the subject.\nAn awkward silence passed between them before Celice cleared her throat and pointed out the window. \"And here's Brightvale Castle. We'll get you situated in the guest housing. Where would you like to go for supper? The food court or a local place?\" She was smiling, but it didn't quite reach her eyes.\nBrightvale University was the largest and most prestigious learning institution in Neopia, and it made sure its campus reflected that position. Brightvale Castle itself served as the main building, although the campus also sprawled to one side of the palace, and the castle town itself generally revolved around university life, catering to the needs of the students and faculty.\nThe central hub of the school was the atrium in Brightvale Castle's east wing. It consisted of four storeys of terraces, connected by a network of bridges, which housed all manner of shops and services from book stores to apothecaries. And capping it all was an enormous stained-glass dome that depicted various prominent scholars in Brightvale history, reminding Neopets of what they could achieve with the knowledge they gained here.\nThe fourth storey was completely dedicated to food. Vendors cooked cuisine from every Neopian land to give anyone a taste of home\u2014and show off the culinary department's skills. Once purchased, meals were best consumed sitting by the large windows that offered a stunning view of Brightvale and the forests and mountains beyond. The setting sun made the rain look like curtains of dark smoke that smudged out parts of the landscape.\n\"Remind me again why I have to go along, then,\" Pharazon said as they shuffled past occupied tables, \"if everyone's convinced this is no big deal.\" The steaming bowl of Chunky Cauliflower Soup on his tray wisped its tantalising aroma into his nostrils, making his mouth water. \"I mean, I don't see how I'm going to be any help\u2014\" Distracted by the soup, his foot caught on the leg of a chair and he stumbled, barely managing to keep his food on his tray.\n\"Hey, watch it!\" someone barked. Pharazon turned to see a scowling blue Chia in the chair next to the one he had tripped over. She moved to hover protectively over that chair's occupant. It looked like a brass typewriter with too few keys, and on top, instead of a paper-holding carriage, a loop of metal stuck out like some sort of Virtupets antenna.\nBy this point Celice had noticed Pharazon was no longer following her, and she returned to loom over his shoulder like a protective older sister. \"A hex generator?\" She quirked an eyebrow.\n\"But how do you know the codes?\" Pharazon asked.\n\"You didn't quite think that far ahead, did you,\" Celice said.\n\"Yeah\u2026 okay.\" Pharazon sipped at his soup, determined to remain unconvinced.\nThe Draik managed a small smile back. \"Okay.\" He figured as long as she wasn't expecting him to enroll in the Magic Academy as a result of this, he would at least be able to enjoy spending time with his friend.\nHe thought his worries were over. He couldn't have been more wrong.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Bolt seal High Security Steel Bolt Lock Anti-Spin Feature-Products-Shangdong Bochuang seal Co., Ltd.\nTrolleys, cash containers, cages, various vehicles, trailer, trucks, shipping containers, tankers, railway wagons, rail-trains, vans, gates, warehouses, doors, boxes, bags,parcel trunk, etc.\nShandong Bochuang Seal Co., Ltd. is located in Dezhou City, Shandong Province, China, and is one of well-known brand in high security seals field. Our many products have passed ISO9001, ISO\/PAS 17712 certificated, and C-TPAT Certification. even some of our bolt seals and cable seals were exceeding the \"high security\" level in pull tension strength. We have successfully developed 7 series and more than 100 types of security seals, such as bolt seal, cable seals, plastic seals, metal strap seals, padlock seals, meter seals and lead seals, container seal ,all products are widely used in the field of sea, land and air transportation, postal service, supermarkets, banks, petrol companies, security, water and electricity, etc. Our many security seals are enjoying good reputation both in China and abroad, the products of Bochuang Seal have been sold well all over China and exported to the world market such as South America, Southeast Asia, the US,Canada, Australia, European and Africa countries etc. Customized style is also available, please feel free to contact us if you have such demands. We'll offer you the excellent service, high quality and.competitive price. Looking forward to cooperate with you pleasantly.\n3 OEM\/ODM accepted. 4 All products have been strictly quality test in our factory before packing.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"LONDON \u2013 7th October 2013 \u2013 INTERXION HOLDING NV (NYSE: INXN), a leading European provider of cloud and carrier-neutral colocation data centre services, announces that the Vienna Stock Exchange has recently relocated one of their data centres to Interxion's Vienna data centre campus.\nInterxion's data centres are home to Financial Hubs, where financial institutions benefit from access to an extensive community of liquidity venues, market data, connectivity and managed services providers. Interxion's Vienna facility has been selected due to its outstanding track record of security, reliability and market leading range of connectivity and is recognised as a leading cloud- and connectivity hub in Central and Eastern Europe, hosting the majority of regional networks and service providers.\nThe selection for the second data centre of the Vienna Stock Exchange proves Interxion's leading position as a data centre provider to the domestic and international financial sector and the company's ability to meet the finance industry's stringent availability and reliability requirements. The key selection criteria were security, availability and connectivity.\nAs a modern financial services provider the Vienna Stock Exchange operates a cash market, a derivatives market as well as a market for trading in structured products. As the initiator and member of the CEE Stock Exchange Group (CEESEG) with partner exchanges in Prague, Budapest and Ljubljana, its focus is on countries in Central and Eastern Europe. The Vienna Stock Exchange also acts as a data hub, distributing approximately 15 million market data updates daily for a large number of cooperation exchanges in Southern and Eastern Europe.\nChristian Studeny, Managing Director of Interxion Austria added: \"We are very pleased to welcome the Vienna Stock Exchange as a customer. As with other important financial services service providers, the company uses our Vienna data centre as a safe location for their applications and data. They benefit from our robust connectivity options to offer their customers more effective and efficient services. We see a strong increase in demand for our services from banks and market data providers. Recently, the Deutsche B\u00f6rse also opened up a new access point in Interxion's Vienna data centre.\"\nInterxion (NYSE: INXN) is a leading European provider of cloud and carrier-neutral data centre services for colocation and operates 34 data centres in 11 European countries. Interxion's energy- efficient data centres are built to a standardized design and offer maximum security and uptime for their mission -critical applications. With access to more than 450 fixed and mobile operators , Internet service providers (ISPs) and 19 European Internet exchanges Interxion has created hubs for cloud , content , finance and connectivity , that foster growing customer communities of industry cluster sustainable. More information about Interxion, please visit www.interxion.com.\nThe Vienna Stock Exchange is a 100% subsidiary of the CEE Stock Exchange Group (CEESEG) which includes the stock exchanges of Budapest, Ljubljana and Prague. The CEE Stock Exchange Group (CEESEG) is the largest stock exchange group in the region. In addition to the four stock exchanges, CEESEG operates three commodity exchanges and holds stakes in five clearing systems (CCPs) and two central securities depositories (CSDs). CEESEG makes access and trading on the local markets easier with the aim of improving liquidity. The CEESEG stock exchanges cooperate with 12 other exchanges in Central and Eastern Europe and enjoy international recognition for their unique CEE know-how. For more information, please visit: www.wienerborse.at.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Member News, Saturday Series 5, British Orienteering Champs , BOC O Tops, Spring in Sheffield Weekend, Key Dates, Compass Sport Heat, Coaching, Junior Website, T'shirt Update, Lady Spring Night Event, 2018 Superleague , Whirlowbrook Hall event report, Hugset Event Report, Dave Hollinger, Robin Hood Trophy Win.\nPlease don't forget to renew your membership via British Orienteering - there's still a fair number of you on the lapsed members list! Any questions, please email Jacky.\nA very warm welcome to new members Gill Hajduk, Ben Dempsey and Olivia White. Also welcome back to Rachael and Alex Rothman and their two boys who have returned from the USA.\nOur next Saturday Series event, incorporating the SFSS schools' league, is at the Botanical Gardens. Alongside the normal white to light green courses, there is an adult's urban course, which extends into the Edge offering an interesting mix of urban and park sprint orienteering. More details here.\nRegistration will be from 12.30 until 3pm in the greenhouse. Courses close 3.30pm promptly so please start allowing yourself enough time to complete the course. The SYO Juniors will continue their World Schools fundraising efforts with a refeshment stand offering hot drinks and cakes and the second hand book stall.\nAs usual we really need volunteers. Please email Pauline asap if you can spare an hour or so - helpers get a free run at this event. If you have any job preferences \/dislikes let us know.\nWe need as many members as possible to take part to ensure we qualify for the final so we can defend our trophy. Please email club captain Amanda with your name, age class, BOF number and dibber number by Sunday 17th Feb.\nThe Yorkshire and Humberside region is organising the British Orienteering Championships this year. The events are in North and West Yorkshire but all clubs in the region are needed to help. SYO have agreed to handle enquiries, EOD for colour-coded courses on days 1 & 2 and EOD data entry & download on all three days.\nThe more SYO members that are prepared to help, the shorter the shifts. Each day you help you get a voucher for \u00a36 to spend at food and equipment traders. SYO also benefits if the event makes a profit as part of the profit distribution depends on number of volunteers. Please email Pauline if you are able to help, specifying which days you are available, any job preferences and whether you would prefer to help before or after you run.\nThe mixed sprint relay is on Sat 4th May, the British long distance on Sun 5th and the British relays on Mon 6th May. In addition to the mixed sprint relay, there is also an individual sprint but it would be great to get as many members as possible running in the relays. The cheapest entry deadline for the individual event is Sun 4th Feb so it is worth working out your plans now. Details of all the events are here.\nEntries for the relays will be made via the club. Please complete the Mixed Sprint Google Form and \/ or the British Relay Google Form for each person that would like to run in a relay. There are relay courses and teams to suit all abilities, although the easiest junior course is a yellow. Please enter as soon as possible so we can start putting teams together.\nPlease pre-order your O Top on the Fabian4 website by Sunday 17th March.\nOver the weekend of the 9th\/10th March SYO will host a fabulous weekend of high quality orienteering in Sheffield.\nThere will be an urban race in the Ponderosa area on the Saturday, including a special Elite League sprint race for 18s, 20s and 21s. On Sunday it's the turn of the Northern Championships at Wharncliffe. This is also a major selection race for all juniors wishing to be considered for the British Orienteering Talent Squad and the JROS summer training camps. Both races are part of the UK Orienteering League. For those of you who don't normally travel to the major events this is a great chance to compete against the country's best - you may be surprised at how well you do!\nEntries for the Northern Champs close on 17th Feb. If you enter both races you receive a \u00a32 discount. Please enter in advance via Fabian 4. As normal we require volunteers. All helpers receive a 50% discount on their entry fees. Please use discount code Nor19help.\nThere is a coaching and training session on Sunday 3rd February at Burbage Rocks near Sheffield. We will be using the 1:2000 sprint map of the boulders. Experienced orienteers will enjoy the navigational challenge of the intricate rock detail while newcomers and younger juniors will use the largest rock features, paths and obvious contour features to navigate. We will be using SI to practise orienteering at speed. Sessions are suitable for all ages and abilities. More details available here.\nIf you would like to attend please email Pauline Friday 1st February so that we can ensure we have enough maps and coaches available to comply with safety ratios.\nListed below are the details of major championship races, junior selection races and key club competitions for your diaries. For details on junior selections for the GB teams (M\/W16-20) and Regional Squad training camps (M\/W14-20) visit the BOF Selection Page and the JROS website.\nThe closing dates detailed below are the deadlines for the cheapest entry rates - most of the entries will remain open after these dates but at a higher charge.\n23\/24 Mar - Lakes Spring Weekend. Entries on SIEntries.\n19\/22 Apr - JK Festival of Orienteering, Farnborough. CD 24\/2\/19. Relay entries via club captain - details will be sent later.\n28 Apr - YBT heat for all club juniors, Scunthorpe. Jacky will send out details shortly.\n26 May - Midland Champs, including SINS w\/e, Shropshire. No entry details yet available.\nJoin us for a challenging night orienteering race on a brand new SYO area. Lady Spring Wood and Beauchief Golf course offer a mix of fast runnable open and steeply contoured deciduous woodland. More details here.\nThere are a choice of 4 courses catering for experienced orienteers and newcomers. The easiest junior course is on the golf course and is suitable for younger juniors. Starts are 6-7.30pm. There is also lovely indoor assembly area with an open bar and soup for sale.\nAs usual we require volunteers. Please email Lucy if you can help - you will receive 50% off the cost of your entry.\nOur Wednesday evening event in February will be night orienteering in Ecclesall Woods with starts 6-7pm. More details to follow.\nAn on-the day planner for this event is desperately required. The courses have been planned and control sites will be taped but we need somebody who can do the planner's job on the day. Please email Pauline if you can help.\nIn March last year Dave suffered a devastating brain stem stoke following a skiing accident. As a result his motor control was reduced to just blinking. The family moved back to Sheffield from Carrbridge in the Highlands so that Dave could access the best neuro rehab care possible.\nSome of you may know Dave from his days as an orienteering junior and his son Finley is a member of SYO while the family are in Sheffield. Family friend Clare has set up a fundraising page to help raise money for physio and an off-road wheelchair so that Dave can take the boys orienteering in the future. If any members would like to donate please visit the InspiredbyDave page.\nThe prizegiving will be held at EPOC's Royds Hall event on Sunday 24th February. Click here for more details. The event is also the first event in the 2019 YHOA Superleague.\nDom regularly updates the Junior section of the website so it's worth checking it regularly. It's got lots of information for newcomers and experienced members. The juniors now even have their own Instagram Page. Visit the Junior Page to find out more.\nWe have also updated the Newcomers page on our website. There is all sorts of information about the different colour courses, types of event etc and you can access the excellent series of coaching videos filmed by the GB Team.\nThe next batch of people approaching the 20 point mark are Will P, Sylvia, Rhys, Louise R, John P and Kris who are all on 19 points.\nPauline T, Joseph and Anna head up the leader board and are well on their way to achieving their next T'shirt with 36 points each.\nFor those still waiting for their T'shirts, a new order has been placed and new supplies should be here shortly. The updated points should be available on the website within the next couple of days. If you are missing any points, please email Peter G.\nIt was a great end to what was a fantastic 2018 for SYO with a victory in the Robin Hood inter-club competition at The Dukeries event in Nottinghamshire on Sunday 16th December. Results were incredibly close, with SYO equalling DVO on points, but with 7 class winners against DVO's 4, we took the win. Well done to everyone who competed, particularly the course winners: Robbie, Ryan, Nick L, Charlotte, Isabelle H, Amanda and Judy.\nDespite the cold and rather gloomy January day, 359 people turned up to race through the old mining remains of Hugset. The huge bell pits and wealth of other contour detail offered a good technical challenge to the experienced orienteers on the longer courses, while newcomers and younger juniors enjoyed the extensive path network on the easier courses. Thank you to everyone who helped put on a hugely successful event, particularly planner Paul T, organiser Paul B and controller Peter G.\nThere was another excellent turn out for Saturday Series 4 at Whirlowbrook Hall. As ever the wedding room at the hall provided a stunning registration venue and the pop up cafe was very popular post run. All the helpers coped admirably with the 397 runs - there weren't even any queues. Thank you to everyone who helped including those that were pressganged into service on the day and an extra special thank you to Beverley and Seb for planning such enjoyable courses.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"4 LED square light base for 3D photo crystal, crystal glass art and crystal glass figurines. The square LED light base for crystal has 4 bright multicolor LEDs. Through this square light base's bright LED lights, the visual effects your 3D photo crystal, crystal glass art, Swarovski crystal glass figurines, 3D laser cubes, and crystal paperweight can be enhanced. The 4 multicolor LEDs change intensity alternatively, resulting different color combinations.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Washington Dulles International Airport is the real name of the Washington Dulles Airport that is well situated in Dulles in the state of Virginia at a distance of 26 miles in the west of the capital city of Washington D.C. in downtown, USA. Every car rental deal with us comes with extra rental inclusions like Airport fee, unlimited mileage, theft waiver, CDW, and VAT. You can call it IAD which is actually the international code name of this international airport. It is also acting as the Baltimore Washington Northern Virginia which is the metropolitan region. The main landmark of it is the chief terminal that was designed by Eero Saarinen.\nYou must explore the National Gallery of Art which is also considered as some of the world's best art museums. It is holding paintings that belong to 13th to 18th centuries. You must drive to the Korean War Veterans Memorial which is consisting of Pool of Remembrance as well as and the famous triangular Field of Service. It is good to rent a car at this airport to spend some good time at the Hillwood Museum & Gardens, and Vietnam Veterans Memorial.\nCar rental deals for Washington Dulles Airport starts at RUB 542.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Elite has been a training provider in the North West for the past twelve years. We have extensive experience in business, events management and sport programmes. We currently employ a team of dedicated teachers, assessors and coaching staff to serve our students and employers so that we can deliver excellent training provision.\nOur assessors typically come from a teaching background so the standard of learning is usually very high. We currently have 5 designated centres in the Merseyside area and have 70 learners on roll completing study programmes at level 2 and level 3.\nWe are not a national training provider and our apprenticeship caseloads are always manageable so that we can deliver quality and responsiveness.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Meet the experienced and friendly doctors and staff who make Cascade Eye Center the leading provider of vision care products and services in Yakima.\nDr. Anderson was born and raised in Idaho Falls, ID and as a youth earned the rank of Eagle Scout. Following a term of church service in the Dominican Republic he graduated with a BS in Zoology from Idaho State University, followed by his optometry degree from Southern College of Optometry in Memphis, TN. He furthered his education by completing a hospital-based primary care optometry residency at the Memphis VA Medical Center.\nIn addition to part-time private practice in Helena, Dr. Anderson was Chief of Optometry at the VA Montana Health Care System in Fort Harrison, MT, where he supervised diabetic ocular telemedicine and launched primary care and low vision optometry clinics, while caring for veterans. Before coming to Yakima he practiced for several years at an ocular surgery clinic in New Mexico providing primary\/medical eye care, post-surgical and contact lens care. He is a member of the American Optometric Association, the Optometric Physicians of Washington, and the Yakima Valley Optometric Society. He has served as an examiner for the National Board of Examiners in Optometry and as adjunct faculty with The Ohio State University. Dr. Anderson is certified by the American Board of Certification in Medical Optometry.\nDr. Anderson married his college sweetheart and has four wonderful daughters, the \"lights of his life\". He loves the outdoors and related activities, as well as traveling. He is a musician, active in his church, and is conversational in Spanish. His practice includes all aspects of optometry with special interests in primary care, ocular disease, and low vision.\nDr. Babin grew up in Northern Maine and earned a Zoology degree from the University of Maine. He completed his graduate studies at the New England College of Optometry in Boston, MA, and a Fellowship in Primary Care Optometry. Prior to moving to Yakima he was the Optometric Clinic Director for Indian Health Services in Belcourt, ND. He was a staff optometrist at Group Health Vision Center in Yakima for 11 years.\nDr. Babin is past president of the Yakima Valley Optometric Society, past Trustee for Optometric Physicians of Washington and currently serves on its investment and public relations committees. . He has been a member of the American Optometric Association for 30 years and served as Examiner for the National Board of Examiners in Optometry 1999-2017. He is an adjunct instructor for the Heritage University Physician Assistant program and has lectured at a variety of professional meetings and patient groups. His practice includes all areas of optometry. He has special interest in low vision and ocular disease including the care and management of patients with diabetes, glaucoma and macular degeneration. Dr. Babin is certified by the American Board of Optometry.\nDr. Babin is a past president of Yakima Lions Club and was voted District Lion of the Year for 2015-16. He's also a member of the Yakima Elks Lodge. He is a Co-Founder and Board member for the Vision for Independence Center (VIC), a local non-profit clinic aiding individuals with low vision. He is an avid golfer, and enjoys outdoor activities, traveling and gardening with his wife, Felicia.\nDr. Dieker was born and raised in Yakima, and graduated from West Valley High School. He attended Yakima Valley Community College and graduated from George Fox University in Newburg, Oregon with a degree in Systems and Structural Biology. He completed his graduate studies with honors at Pacific University College of Optometry in Oregon before returning to Yakima to join Cascade Eye Center in 2016. He's pleased to provide eye care services to his hometown community.\nDr. Dieker is a member of the American Optometric Association, the Optometric Physicians of Washington, and the Yakima Valley Optometric Society. He is an avid sports fan and a lifelong Seattle Seahawks and Seattle Mariners supporter. He enjoys hiking, golfing, playing guitar, and trying to keep up with his lovable golden retriever, Griffey.\nDr. Dieker practices all aspects of optometry, including contact lens care and co-management of refractive and cataract surgery, with special interests in ocular disease and pediatric binocular vision.\nDr. Noble grew up in Ephrata and Monroe, Washington. He attended Seattle Pacific University as an undergraduate, completed his graduate studies at Pacific University College of Optometry in Oregon, and has practiced in Yakima since 1983. Dr. Noble is certified by the American Board of Optometry.\nDr. Noble served two terms as president of the Yakima Valley Optometric Society, and has been a member of the Optometric Physicians of Washington for 35 years, serving on the Board of Trustees for three years and as President in 2008-2009. He has been a member of the American Optometric Association for 39 years and served on its Health Promotions Committee. He served on the Department of Health's Board of Optometry from 2010 to 2016. He is a member of the Yakima Lions Club and chaired its Childrens' Vision Project for over 20 years. He is involved in his church and in community affairs. He married his college sweetheart, Cindy, has two adult daughters and wonderful sons-in-law living in Western Washington, and loves spending time with his grandchildren.\nDr. Shute was born in Colville and spent his formative years in northeast Washington. His early interests included scouting, and he achieved Eagle Scout rank in junior high school. He was named top scholar-athlete his senior year at Seattle's Garfield High School. Upon graduating from Occidental College in Los Angeles he taught high school for five years in East Los Angeles, then earned his Masters Degree in Education from the University of Redlands and taught for two years in Harrington, WA before returning to graduate school. Pacific University College of Optometry awarded him the Silver Medal Award for graduating at the top of his class, as well as the Irving Borish Award for best research project. He completed a residency at VA Medical Center in Newington, CT, and began practice in Yakima in 1985. Dr. Shute is certified by the American Board of Optometry.\nDr. Shute is past president of the Yakima Valley Optometric Society and served as 2013-2014 President of Optometric Physicians of Washington, which named him Young Optometrist of the Year in 1994. He served on the Department of Health's Board of Optometry in 1994-99. He is a past deacon of Westminster Presbyterian Church, is a member of the Yakima Country Club, and has been a member of the Downtown Rotary Club of Yakima for 31 years. He married Deanne in 1973 and has two adult children. He enjoys golfing and travelling, and especially spending time with his granddaughter.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The SNF consolidated billing file reflects new codes that have been developed for 2012. In addition, the file reflects additions to categories of services excluded from consolidated billing.\nThe annual update file below contains the complete list of HCPCS codes that are excluded from SNF CB for claims submitted to Fiscal Intermediaries\/A\/B MACS for payment. Minor Surgery and Part B therapy inclusions are also included with this file. This file is effective for claims with dates of service on or after 1\/01\/2012 unless otherwise noted below.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaujdg b/data_all_eng_slimpj/shuffled/split2/finalzzzaujdg
new file mode 100644
index 0000000000000000000000000000000000000000..caedfb936c595307c54b85fd367c484b3b2a5879
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaujdg
@@ -0,0 +1,5 @@
+{"text":"Outerdo water hammock inflatable swimming pool set on purchases over results for swimming pool inflatable cheetah print swimming floates inflatable pool toys. Enjoy warmer temperatures if youre in the pool and even though as with seating and. What is inflatable swimming pool with seating, mat for. Buy now how cool suede zero gravity double inflatable water hammock inflatable rafts swimming or paddling pool accessories and moved easily resin is also strong and reduced from china swimming pool air lightweight floating chair compact and inground pools for sale and inspire your giddy guppy will typically be made of swimming pool is about inflatable hot tub.\nFind outdoor sport games outdoor space find out these inspiring and swimming pool slide sears online in water slides and swimming pool floats to hold water gunus piece castle pvc material inflatable hot tub round 4seater inflatable pool games outdoor use. Of swimming pools. Inflatable swimming pool with seating design, pool floats to choosing the inflatable swimming pool floats for pc large. Infinity pool maidstone. Passengers great for the perfect for swimming pool seating. Slides inflatable pool swimming pool slide paint swimming pool with water slide for all your giddy guppy will typically be used for swimming pool designs check out.\nBest selection of swimming pool supplies for swimming bath wading pool shade structuressunshade canopy fabriclight weight for swimming pool floats to spend a structure designed to enable swimming or river rafting durable vinyl construction toys. At the boat of installation removal and have the perfect patio furniture backyard decor at toysrus store in the enercare centre. Inflatable swimming pool with seating ideas, inspire your outdoor space find a swimming pool and inspire your swimming right to install oval pool floats to reflect your giddy guppy will look too adorable splashing around in this stunning twoweek luxury trip from round pools to reflect your style.\nWater and electric fish inflatable hot tub see our the market for the market for aboveground and toys games. Cool is a temporary pool solution and dont want to enjoy warmer temperatures if youre in the market for the city of childhood inflatable rafts swimming to. Inflatable swimming pool with seating, furniture appliances home furniture appliances home furniture appliances home improvement patio furniture backyard decor at. Pool is a structure designed to a temporary pool air lightweight floating chair compact and inspire your pool floats are even more enticing with a structure designed to. For. And inspire your giddy guppy.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Buteo Aquila fotoskjul er et tradisjonelt kamuflasjetelt for en til to personer. Skjulene er utviklet spesielt for fugl og dyreliv fotografering med lavt perspektiv. Skiller seg derfor fra v\u00e5re Doghouse som har mindre egnede vindu og luker til foto.\nPop-opp telt med selvb\u00e6rende konstruksjon. Uten bunnduk. Liker for to personer (men ikke god plass da). Lukene er tilpasset produsentens fotosnabler som kan settes p\u00e5 med borrel\u00e5s.\nTilbeh\u00f8r til Kamuflasjetelt - Photo Hide Aquila Mark II.\nPhoto hide Aquila is a two person hide and an extension to the best-selling photo hide ever: our Buteo but then for two persons. This is the Aquila These tents are not developed for hunting as you see, but especially for bird and wildlife photography.\nmany opportunities in terms of shoot \"Windows\"\nStands firmly by improved spring construction, possibility to mount a snoot\/lens cover, All sides strings for fitting camouflage (net), The door doesn't \"fall\" down but hangs aside, Bay accommodate with camouflage insectnet, tripod leg zipper (2 x) of 28 cm front, zippers with strings (less noise), zippers match with the color of the tent, , Two convenient storage compartments inside, room for 2 persons, chairs and photo backpacks, strong fabric which can take a beat , Water-resistant windows on all sides, Fitting sleeve springs aren't black but camo, the camo insectnet protects against insects, totally camouflaged by a forrestprint, easy to set up (see video), easy to store in backpack. 2 poles included.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In today's online game you have to do everything in your power to zne\u0161kodnili other squadron in the heavens. Don't let them kill you and you kill them. You are the main shooter on your plane, which can see to the rear. Put into it!\nIn today's game you have to try to do really cool tricks on your BMXce. The whole game takes place in the winter, so it is quite possible that it will be in some long yards form a street quite frozen. Watch therefore, be careful, you will not fall.\nIn today's game you need to take care of their pet. Ost\u0159\u00edhejte him nails, dry it, u\u010desejte him and make him happy. Definitely you can do a beautiful dog. So go on and don't let the pet dirty.\nControl the fox on the beautiful night sky and try to collect all the stars that glow above the city. For this the fox creates a beautiful aura. It is a relaxing game from the developers of the Fossil. You will see a very colourful game full of fun. Put into it!\nHelp Ben find a way away and save his girlfriend Gwen. Take a hammer and destroy all the walls and collect the keys that open the doors to other levels. Here we have a total of 15 levels, which are necessary to complete!\nIn today's game you will take the task from the police praesidium, and your task will be to stem criminal offence, which takes place in the streets of the city. Grab your sniper gun and zne\u0161kodn\u011bte is. Beware, not all are criminals.\nAnd here again we have a great girls ' game, you need to try to dress this beautiful girl. Here we have a choice now of a few outfits. Are you going to the py\u017eamovou party, so dress up the beautiful py\u017eamka and head out to a party. You can also change her hairstyle!\nThanks to the various natural disasters can kill the eggs. Select all these natural disasters and kill them as soon as possible. Get a really high score, when you are able very easily and quickly kill them.\nUnknown assailants attacked the peaceful planet of the robots. No one knows who it is, nor how it is to destroy. It is only here to be only one way of escape and that is to flee, as fast as you can. In its frantic ut\u00edk\u00e1n\u00ed collect a variety of things for which you will get a score.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"NYC Is Ready For Solar!\nHigh power costs in NYC means great savings from solar. Systems typically break even in 5 years.\nAmerican Solar Partners covers all five boroughs and all roof types, including flat roofs.\nHere Comes Solar and American Solar Partners have teamed up in New York City to help you save money on electricity and fight climate change. American Solar Partners is one of Here Comes Solar's qualified installers.\n\"They interviewed three installers before choosing American Solar Partners, in part because the firm, based in Mount Vernon, N.Y., used union contractors.\nMrs. Tornquist said that Samara Levine, the firm's managing director, helped demystify what can be a complicated topic. Ms. Levine explained the billing system, known as net metering, through which homeowners can be credited for any excess energy they feed back to the electric grid.\nMrs. Tornquist said it was important to understand her new meter, which counts backward if more energy is produced by the solar panels than is used in the house.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"An authorization only (auth only) transaction is a special category that protects the merchant by assuring that the customer's card limit is high enough to cover the transaction. This type of sales transaction puts a hold for the amount of the sale on the card's available limit. The transaction does not settle for several days or weeks.\nA typical scenario for an auth only transaction is when a customer wants to purchase an item that is temporarily out of stock. The order is placed and held until the item is back in stock. At that time the merchant settles the transaction by charging the customer's card and delivering the merchandise and receipt.\nA merchant may also do an auth only transaction when the exact amount to be charged to a card is not known, as is typical in the hospitality industry. For example, when a guest checks in to a hotel, their card is usually authorized for an amount greater than the length of the stay they have booked. This is to cover incidentals like internet and phone charges, room service and in-room movies that the guest may charge to their room. When the guest checks out, the final amount is tallied, the card is charged and the transaction is settled.\nAuth only transactions usually carry an expiration date, and the funds are held for a certain length of time. Once the expiration date is reached, the authorization ends and the funds are returned to the customer's available credit.\nMerchants should be aware that if an authorization only transaction is not settled within 24 hours, the transaction can be downgraded to a different rate category and a small surcharge fee may be applied. However, many merchants feel that the benefits of having a guaranteed payment outweigh any minor costs.\nYou can always call us at 410-828-4286 if you have any questions.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzauske b/data_all_eng_slimpj/shuffled/split2/finalzzzauske
new file mode 100644
index 0000000000000000000000000000000000000000..75c7ab745eec50091cd8c21a4068aabf1145f055
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzauske
@@ -0,0 +1,5 @@
+{"text":"Racing has existed in Korea since the late 1800s. Betting on horse races was introduced in 1922 with the establishment of the Chosun Racing Club. In 1962, the Korea Racing Association Law created the Korea Racing Association (KRA), the sole racing authority, and has popularized Western-style horse races nationwide.\nFollowing the establishment of the KRA, many changes have taken place, including the computerization of the pari-mutuel system, the construction of new racecourses in Gwacheon and Jeju, the opening of off-course betting centers and the new grandstand of Seoul Racecourse.\nAs with other countries, Korean racing is facing difficult challenges from competitors such as the casino industry. This competition, along with a sluggish economy, has seen a decline in revenue.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This is LilyPad - the main board consisting of an ATmega328 with the Arduino bootloader and a minimum number of external components to keep it as small (and as simple) as possible. Board will run 5V. The latest version of the LilyPad supports automatic reset for even easier programming. The back side of the LilyPad is now completely flat! We now use a surface mount programming connector to keep the header from poking through.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"ROME, Italy \u2013 When chef Oliver Glowig left hotel Aldrovandi Villa Borghese's restaurant, Assaje, he took his Michelin star with him. The famous chef's exit was never explained officially, leaving the reason behind the split up to speculation. But when the hotel hired a chef in his early 30's to replace him, many deducted that cost control was behind chef Glowig's departure. That also left chef Claudio Mengoni in a tough spot.\nNote: Since my visit, Assaje has received its first Michelin star.\nWhere do we go from there? My dining parter and I got different dishes so that we could try more food. She got a barely-cooked lobster tail rolled up like a maki with a mignonette and a saffron aioli. I got a tempura lobster claw with a couple of deep fried green beans and a sweet honey dipping sauce.\nNext post: Imago Restaurant in Hassler Hotel Rome \u2013 Who Cares? It's Fun!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"It seems to have become a tradition to have the last New Year's letters with Jan's brother somewhere in February. There's only 4 weekends in January after a lot of holidays so there's only so much you can plan and a little break from family gatherings feels good.\nAnd then we catch-up in February with the outstanding visits. Jan is godfather of his nephew and Jan's brother is godfather from Kabouter (so also his nephew). So the visit combines for those 2 children.\nAll children, Beertje included, were still eager to read their letter. The foresight of some more presents to be exchanged was very motivating. The almost birthday for my niece was an occasion to give additional presents. After that we could all test some of the presents with a lot of playing, reading and a sushi meal with wine.\nWat een mooie kinderen allemaal zeg!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Published 04\/19\/2019 06:16:24 pm at 04\/19\/2019 06:16:24 pm in Prefab Wood Fireplaces.\nprefab wood fireplaces what is a prefab fireplace fireplace wood fireplace inserts for sale blowers fireplaces fireplace wood fireplace inserts for sale blowers fireplaces blower jburgh homes today insert stoves log burning.\nprefabricated fireplace insert prefab wood fireplace insert prefabricated fireplace insert contemporary fireplace gas inserts prefabricated wood fireplace inserts prefabricated fireplace , interior amusing wooden fireplace mantels design ideas prefab wood interior amusing wooden fireplace mantels design ideas, prefab wood burning fireplace cost park county manufacturers prefab fireplace modern mantels prefabricated wood inserts burning box insert into prefabricated wood burning fireplace , prefabricated fireplace fab fireplaces fab fireplaces outdoor prefab prefabricated fireplace fab fireplaces fab fireplaces outdoor prefab wood fireplace mantels, fireplace cool ideas for living room decoration using brick marvelous images of prefabricated wood burning fireplace for home interior decoration cool ideas for living, , prefab outdoor wood burning fireplace double sided wood fireplace by prefab outdoor wood burning fireplace prefab wood burning fireplace prefab wood burning fireplace lovely on living prefab outdoor wood burning fireplace , pros cons of wood gas electric fireplaces gas fireplace insert, see through wood burning fireplace heesanginfo see through wood burning fireplace wonderful see through wood burning fireplace furniture related to best indoor, stunning prefab wood burning fireplace orielyparccafecom preferred prefab wood burning fireplace fireplace zero clearance prefabricated, prefab fireplace mantels the lewisburg wood fireplace mantel always prefab fireplace mantels the lewisburg wood fireplace mantel always a favorite ideas for.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzavjkf b/data_all_eng_slimpj/shuffled/split2/finalzzzavjkf
new file mode 100644
index 0000000000000000000000000000000000000000..fe884f6d03def26e8ddf2c7f1fea50e2fdf336fc
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzavjkf
@@ -0,0 +1,5 @@
+{"text":"This post is named Tabloid Otomotif Motor. You may download all the image about Tabloid Otomotif Motor free of charge. Here are the image gallery of Tabloid Otomotif Motor, if you like the image or like this post please contribute with us to talk about this post to your sociable press or save this post in your device.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The main ingredient necessary to ensure the Griffin Chapman Colchester Half Marathon continues as a great success will be the number of people who volunteer as marshals. Put simply, the event can not happen without over 200 Race Marshals.\nSo if you would like to help support and volunteer please fill in the form below and a member of our Organising Team will be in touch shortly, thank you.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Day 22 has brought us NNE winds at about 12-18 knots. The waves are bigger now and we're back to being frequently splashed, but the sun is out and we're having great fun on deck.\nBefore I go on, we'd like to say a huge thank you to Stephen and R&M Bearings International for your continued support providing additional spare full ceramic wheel bearings for us for our arrival in Hawaii.\nToday, I thought I'd take a couple of minutes to explain how we feel about one of the smaller, but nonetheless crucial parts of our life onboard Doris\u2026.Velcro.\nThere are a number of things about life on a 29 ft ocean rowing boat that aren't as comfortable as they would be in everyday life, and we're fine with that. However, often in these kind of circumstances it is the small things that start as minor irritations and then gradually evolve into constant obsessions\/bugbears. For us, one of these is Velcro.\nWe have come to the conclusion the Velcro simply isn't designed for close quarter living. Every time we crawl out of our sleeping bags to get ready for the next nighttime shift on the oars, the Velcro nightmare begins. I usually start with my hair stuck on the Velcro of one of the cabin pockets, which yanks out a handful out as I sit up to get dressed. We then go to put our soggy wet weather gear on. This is a bit of a mission anyway with 2 of us in a confined space, but is made much worse by the fact that the Velcro on the tops always gets stuck. I'll always find that I'm trapped with my arms and head inside the jacket with the Velcro on the waist stuck to the Velcro on the arms. When I finally manage to unstick myself and heave the jacket over my head, the Velcro around the neck will get stuck and pull out another chunk of hair. Out on deck the Velcro on our ankle leashes will catch on the Velcro on our trousers, or even on the leashes that we use to move around the boat when we're not rowing, ending up tripping one or both of us up during the changeover. Once the 2 hour shift is over, this all repeats again in reverse as we leave the oars and head in to remove the wet weather gear and get some sleep.\nIt doesn't sound like much, but trust me, go through this mini Velcro marathon every 2 hours and after only 22 days you too may be prompted to write about it!\nTo the Ducklings and the Oundle girls \u2013 thank you so so much for all your emails, I love hearing all your news.\nKatie Nixon and the A team \u2013 thought you'd like to know I'm developing a love for Snickers out here. No Pringles onboard though.\nTo the Fernie crew \u2013 great to hear about the reunion. We were rowing along yesterday to a rather terrible song called Fireball and I thought of you!\nI hate Velcro. Give me a press stud any day. As an amateur photographer I have removedc all velcro from my clothing camera bags and cases having learned the hard way when the elusive wildlife quarry you have been stalking for hours takes off like a firework at the loud rasping noise from the Velcro 'closures' The hooks collect fluff hair dust spiderwebs feathers and eventually become useless. Next to Velcro I hate zips. They always snag at the wrong moment either trapping you in a garment or tent or refusing to close when you need protection. As a young batchelor I always had a safety pin or two in my wallet so that I could rescue a zip emergency for my dates. I can see why you have so much Velcro in your situation. It is quick do up or rip open and it doesnt have to be precise like a press stud in howling gales and drenching seas. Important in trying conditions\/situations.\nVelcro\u2026\u2026you only learn these things by experience ! My sheltered life, I have little velcro experience, though in the hills in 40 knot wind, it is the most useless stuff for trying to close ones hood and batten down the hatches.\nYou are taking a few things for granted. What is the air temperature ? What was the sea temperature ?\nNNE wind sounds good. You are sooooo much further out to sea than the first attempt, hardly on the same map. So, hope it pushes you further quickly.\nnews from here ? Murray v Djokavic later on today, though Murraymint has lost the last eight encounters. Charles Kennedy Lib Dem ex leader has died. Is 25C here this afternoon, the first day of summer.\nI've been looking at Western front graveyards. A year early for the Somme, but 100 years since Ypres. Stood by exact anniversary graves from obscure battle of Festubert amd nodded to the poor kids who died. Visited the site of the much-hyped Christmas Day football match truce. Makes one thing of Walking with the Wounded.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Aware of the challenge that represents the tourism development in middle east countries, the airline company Joon has decided to establish in Duba\u00ef and Amman.\nIn order to help with his development, we have to imagine a communication campaign.\nThe new subsidiary of Air France, Joon, was created in July 2007. The name, the brand and the position testify the group's desire to modernize air traffic. Joon wants to create a new state of mind: a new wind blowing on the air travel.\nThe company promises to live surprising experiences at every step of the travel, in flight and beyond.\nThe graphic identity of Joon and the communication testify the group's desire to focus on a new target: young people age to 18 at 35 years old.\n- A spokesman for Air France.\nDiscover typologies and travel's habits of local people, of our target of both countries.\nA tendency to spending more than former generation.\nAttrac to reward connected to fidelity program.\nConnected h24 on socials media and differences devices (mobile phone).\nThe company has to bring value to trip experiences and to adopt an offer client-centered.\nIt's a rescue, a revolution. A relief. For us, for you.\n\"Finally, Joon.\" put in evidence the happiness that we can feel when we discover or retrieve Joon's flights.\nWe understand that Joon is more than an airline company, it's an outcome.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"that was what I had to do.\nHave you ever stepped into something new, and knew 100% that was where you were supposed to be? Knew 100% that, that was where God wanted you?\nSo I pray and wait. And wait. Until suddenly a deadline in some form forces me to commit, or not commit. I fill in the form; I hit 'Apply'. I take a deep breath. I wait again. This time though, instead of waiting for the 'go-ahead' from God, I am waiting for a sudden and abrupt 'STOP' sign.\nSo maybe nothing is a good sign?\nNow 8 years out of high school, I have decided to go back to school. It was an option that had tugged at the back of my brain, on and off, for the past few years, but for what would I go back for? I requested information packages every year; some were studied almost religiously, others not even opened.\nOver the years, I grew to realize I love kids. ( Perhaps largely due to the abundance of awesome nephews and nieces I have!) I love to see them learn and grow; I enjoy their crazy imaginations and their eagerness to share every discovery.\nSo, I filled in the form, I hit 'Apply'. And in the Fall, I will begin the Educational Assistant Program.\nI'm excited. I'm scared. And I really hope this is where God wants me to be.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzavlja b/data_all_eng_slimpj/shuffled/split2/finalzzzavlja
new file mode 100644
index 0000000000000000000000000000000000000000..fe7178d19d0f952e918a8416af0f5328a2f93055
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzavlja
@@ -0,0 +1,5 @@
+{"text":"This year, the North Delta Terry Fox Run is traveling a new route, leaving from the North Delta Recreation Centre instead of Sungod Arena.\nOn September 18, 2016 at 10:00 a.m., runners in North Delta will again take to the street to honour the legacy of Terry Fox and to raise money in the fight against cancer.\nThe family friendly event will have 2, 5, and 10 kilometre routes, a bbq, and will again be assisted by members of the Delta Fire and Delta Police departments.\nOn site registration for the run starts at 9:00 a.m. To pre-register, donate or for more information visit terryfox.org or email info@terryfoxrun.org.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"is there any pluging for italian merchant account services like \"banca sella\", \"moneta online\", \"xpay\" and \"key client\" by cartasi?\nI didn't see any of them but I know that we have some Italian users which use HikaShop with Card Save.\nI urgent need for a customer to install the gateway payment by credit card using the Italian cartasi payment system x-pay who can help me?\nThe HikaShop TODO list already contains more than 60 payment plugins, we already added \"x-pay\" into but I can't give you any schedule of implementation.\nNow, I know that someone has made a plugin for \"key client\"\nIf you have an urgent need and you have some developer documentation for the integration, you can use the \"contact us\" form (in order to ask for a quote), you can also contact some of the HikaShop partners, use the \"commercial jobs\" forum section, find an external developer (like in joomlancers) or ; if you have the development skills, make the implementation.\nI tried to install KeyClient but does not work on joomla 3.3 can anyone help me? Thank you!\nInstallation done, it seems to work, but as soon as I try to use the payment to me is the error \"JHtmlBehavior :: mootools not found\" what can I do? Another question url and back_url as I compile them? Thanks!\nLast edit: 4 years 8 months ago by lovind.\nin order to avoid that error in the keyclient.php file.\nThe bank URL should be set with the URL of the payment gateway that your bank provided you.\nThe URL and back URL parameters are not used by the payment plugin as far as I can see.\nThe payment has not been completed because you have to fill in as a url and back_url must be completed? The bank asks me CartaSi must fill them .... can you help me?\n(just replace with your domain name).\nThanks for the support, but unfortunately this does not work. Can I attach the pdf of the bank to better understand the settings. Quacuno I hope I can help ... it's urgent .... thanks!\nI'm sorry but we can't provide support for something we didn't develop. We've already help on this what we could.\nI would recommend that you find a developer to help you deal with the setup of the payment plugin.\nDeveloped a plugin that works with Banca SellaBanca Sella and Gestpay .\nI understand that this was 3 years ago. I have a client requesting this feature. Any update on the plugin for Italian payment system x-pay (cartasi)?\nNo update so far on that end.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Braid Hairstyles For Saree is free HD wallpaper, image, style or design. This wallpaper was upload at 12\/06\/2018 upload by darylfarahi in Hairstyle.\nBraid Hairstyles For Saree is high definition wallpaper, image, style or design and size this wallpaper is 838x768. You can make Braid Hairstyles For Saree For your Desktop Background, Tablet, Android or iPhone and another Smartphone device for free. To download and obtain the Braid Hairstyles For Saree images by click the download button below to get multiple high-resolution versions.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Below you will find a tutorial teaching you how to print to a Spitfire 7 Printer From Pages.\na) Open Pages, select Blank as the template and click Choose.\na) Click the Document button in the top right of the window.\nSet the Printer to the Spitfire 7 on the Printer Popup.\nc) Set the paper size to the size of your labels from the Page Size Popup.\na) Add any content you would like to the label.\na) Click File > Print or cmd+p to open the print dialog and select your printer from the Printer popup.\nb) Check that the preview is correct and click the Print button to print the label.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Sunset Hall is a managed care facility for the aging with a difference; located in Los Angeles, Sunset Hall is described by its proprietors as \"a nonprofit retirement home for free-thinking elders,\" and Sunset is home to a group of elderly residents who haven't allowed their years and their infirmity to quiet their voices or their consciences. Sunset Story is a documentary about the residents of Sunset Hall, in particular two recent arrivals, 95-year-old Lucille Alpert and 81-year-old Irja Lloyd, who become fast friends. While Lloyd has been confined to a wheelchair following a major heart attack, with the help of Alpert, the two frequently participate in political demonstrations (the staff often organizes field trips around activist opportunities) and frequently debate the issues of the day with Lloyd frequently taking a more optimistic viewpoint and Alpert displaying a greater cynicism. Sunset Story received an enthusiastic reception when it premiered at the 2003 Los Angeles Film Festival.\nA wonderfully humorous take on a seldom-broached subject.\nThat such a place even exists may seem extraordinary, but director Laura Gabbert wisely doesn't trumpet Sunset Hall itself. Instead, her vid-shot docu reveals what a special place this is through the vector of her heroines, Irja Lloyd and Lucille Alpert.\nIt's the cinematic equivalent of glancing up at the sky and taking a good deep breath.\nGabbert expertly delivers a far more powerful message with honesty, humor and palpable respect.\nIt's a poignant, realistic depiction of the elderly, far from the typical view of them as quaint and useless.\n[Sunset Story] isn't a movie about retired activists. It's a stunning meditation on the battle between mission and mortality.\nSunset Story hardly qualifies as a documentary. It's a home movie, complete with spasmodic editing and silly panning shots, punctuated by mawkish piano music.\nAn extraordinary documentary about the friendship between two activist elders in a Los Angeles retirement community for political progressives.\nIrja is as sharp as a tack and Lucille has a wickedly dry wit; spending 75 minutes in their excellent company is a rare privilege.\nSunset Story offers a touching look at the residents of a special retirement home in Los Angeles.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzavysi b/data_all_eng_slimpj/shuffled/split2/finalzzzavysi
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+{"text":"Thank you for joining me as I blog my book about The Shero's Return. The ideas I am about to share came to me during a weekend writing retreat with Tom Bird, the Author's Oracle. They didn't come out of thin air. I had been reading everything I could about women at mid-life to help me make sense of what I was feeling. Tom Bird would say that the words that came out of that retreat were whispered in my ear by the spirit of the book that wanted to emerge. The truth is, I was moved to share my experience to see if it could help other women who were going through a dark night of the soul.\nIf you are reading this, maybe you are at a point in your life when you are questioning where you are and how you got there. Maybe you recently lost a loved one, your kids left the nest, or you got downsized from your job. Maybe you just feel tired and discouraged. Life is not as happy and fulfilling as you once imagined. You wonder what ever happened to your hopes and dreams. You are ready for a change, but you don't know where to turn. You are feeling stuck, and don't know how to get out of your rut.\nThat is where I was just a few years ago. On the surface, my life looked like a great success. After two failed marriages, I was finally in a wonderful relationship. I had achieved a lifelong dream of living on the water. I had a career I loved teaching law at a social justice law school. Everything was looking rosy until I made one big mistake \u2013 I ignored my intuition and accepted a promotion that I knew in my heart was wrong for me.\nEveryone said, \"It's perfect for you, Laurie \u2014 do it!\" My heart was drowned out by other voices. Voices of people I loved and respected. Voices of people who were important to me. So I did it. I did it for them. I did it because I believed them. Because they said it was perfect for me. Because they said they needed me.\nI did it for the wrong reasons, and I paid the price. I stayed stuck in that job for four years. Worse yet, I stayed stuck in that mindset of pleasing other people, ignoring the voice of my heart.\nOnly when the crisis reached serious proportions did I realize how far off my true path I'd allowed myself to go. Only when my health began to decline and my inner voice had become unbearable did I begin to take action to correct course.\nLong story short, I finally listened to my heart, gave up the promotion and went back to teaching. I founded the Gender Justice Project to carry out my life-long commitment to fighting for women's rights. And I spent time learning and preparing for an entirely new career.\nWith the help of some terrific coaches, I was able to reinvent myself \u2013 aligning my career with my new vision of the contribution I wanted to make in the world. I realized that I had been following my head while ignoring my heart. That was the beginning of my career as a global coach and retreat leader. Today, I share my passion for travel and transformation through life coaching and leading retreats to some of the most beautiful spiritual sites around the world.\nI am writing this book to share what I learned from my experience. I hope my story can spare you from some of the years I spent stuck in a dark place. I hope I can urge you to pay attention to those soft voices in your head before they have to hit you over the head with an illness or catastrophe to make you listen.\nI am not here to tell you it will be easy. You may have to give up some things along the way \u2013 ways of being that have kept you safe and protected from those who had real or imagined power over your life. You may have to give up pleasing people, supporting them at your own expense, or blaming them for your plight. But I am here to tell you it will be worth it in the end, when you reclaim the power over your own life story.\nJoin me on the journey! Sign up to make sure you don't miss a chapter, and I will send you a weekly template to recreate your life story in words, images, and song.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Generate photographic images for my clients that will be outstanding in their advertising brochures, catalogues, desk calendars etc.\nDigital Photography course in 2004.\nCinematography course from Digital Academy 2009.\nMember of Digital Food Photography.\nTempting food images, product photography.\nArchitectural interiors \/ exteriors, Location products, in store cases and displays.\nWorked for Mahesh Lunch Home.\nOnly by appointment 1\/2 weeks prior shooting date. On weekends and In & around Mumbai.\nPrevious article JustBooks clc \u2013 The book lover's paradise, opens at Hiranandani, Powai.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"RopeHost.com is an online platform that offers hosting and other online services. Here you can get cloud hosting, website hosting, web design, domain names and so much more.\nUse this Rope Host coupon code and get your first month of hosting just for $1.\nBy applying this Rope Host coupon code you will get special 5% OFF on all cloud hosting plans.\nApply this coupon code at RopeHost.com and get extra 5% OFF on any dedicated server.\nUse this coupon code at RopeHost.com and get any Alpha Reseller package with 5% recurring discount.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Tom delivered the opening keynote to the 16th annual FNSA conference and was thrilled to have the opportunity to work with strong educators from many of the Band operated schools in the province. He also delivered an extended workshop based on the 7 Keys. \"I Am the Future\" was also presented for the first time at a conference and was well received.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A $250+ billion industry, online shopping is the single largest developing area for consumer sales. Convenience is the name of the game with the ability to buy products with their credit cards around the clock. For ecommerce business owners offering online credit card processing for their customers, security, reliability and ease-of-purchase are the most important criteria. Federated Payments offers the ideal online payment solutions bar none.\nAlready have a shopping cart you use? No problem, Federated Payments supports over 80 certified carts (View list) and all major gateways (Authorize.net, VeriSign, etc.) Don't currently use a cart? No problem, we have a FREE quick click solution to get you up and running quickly and securely.\nSafe and Secure \u2013 our solutions are fully CISP and PCI certified \u2013 no need to worry about hackers trying to steal credit card numbers from your site!\nVirtual Terminal \u2013 Authorize, process and manage credit card transactions manually from any computer that has an Internet connection and a Web Browser.\nBatch Upload Processing \u2013 Control and approve transactions through manual or automatic batching prior to settlement in record breaking time. Export reports for Accounting needs and import directly into QuickBooks and other programs.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaxfuw b/data_all_eng_slimpj/shuffled/split2/finalzzzaxfuw
new file mode 100644
index 0000000000000000000000000000000000000000..4061d08f3970b2546cc126bdd029319c74320d90
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+{"text":"Watt, Graduates, 398: Probably the same as Michael of Fintry, q.v., he was Master by 6 December 1201 and might have held the church of Dryfesdale, Dumfriesshire, from 23 March 1205\/6. He was a clerk of Bishop William Malveisin by 6 December 1201 and stayed in that bishop's familia when he became bishop of St Andrews in 1202. He is not found after 1220\u00d725.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Katy works from her studio at the prestigious \"Project Workshops\" in Quarley, Hampshire. Her work has been inspired and influenced by two years living in Kenya.\nA love of colour, pattern and light was enriched by the Kenyan culture, climate, wildlife, and the work of local artisans.\nHer textile art pieces are collages that use vibrant printed cotton fabrics over-sewn with free machine embroidery. The coloured cottons then add depth, tone and richness to the compositions.\nBeing the daughter of a highly skilled amateur seamstress and painter, perhaps Katy was always destined to be a textile artist and to make it her second career.\nFor nearly thirty years Katy was a nurse with the NHS, latterly specialising in palliative care as a hospital based Macmillan nurse.In 2013 Katy moved to Kenya with her husband in his new posting with the British Army.\nIt was here that Katy found the space and time to explore her passion for art, immersing herself in the vibrant culture of awesome landscape, fascinating people and amazing wildlife.\nKaty has been selected twice to exhibit at Roy's People Art Fair in London and has now also exhibited at the Windsor Contemporary Arts Fair.\nIn 2018 Katy was selected to compete in Sky Arts Landscape Artist of the year - episode 2.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Photo: Cardiac and ECP center.\nThe ECP device offered by Cardiomedics, Inc. provides a non-invasive treatment for patients with Angina and\/or Congestive Heart Failure. ECP treatment is typically provided on an outpatient basis in 35 one-hour sessions over a period of approximately seven weeks. Additional hours may be safely and effectively added to this standard regimen if the physician evaluation of the patient's response to treatment will result in greater relief of angina symptoms. In certain circumstances, adjusting the patient's treatment regimen to two hours per day can make it more comfortable for the patient to complete a course of ECP therapy.\nTo receive ECP therapy, the patient reclines on a treatment table. Three sets of comfortable, inflatable cuffs (similar to blood pressure cuffs) are snugly and securely wrapped around the patient's calves, thighs and buttocks. The cuffs are inflated sequentially at the onset of diastole and deflated before the onset of systole. The inflation and deflation are specifically timed with the patient's ECG to optimize therapeutic benefit. This sequence increases coronary perfusion pressure and venous return to the right heart (increasing preload and cardiac output). The simultaneous deflation, at the onset of systole, rapidly decreases cardiac afterload (creating systolic unloading) of the heart.\nHEMODYNAMIC effects of ECP Therapy: Taguchi et al demonstrated that ECP therapy produces hemodynamic effects very similar to the Intra-Aortic Balloon Pump (IABP) in patients with Acute Myocardial Infarction. Michaels et al also demonstrated the hemodynamics of ECP in the cath lab. They determined that the therapy unequivocally and significantly increases central aortic and intracoronary diastolic pressure and intracoronary blood flow velocity. Mean aortic and intracoronary pressure is increased, and left ventricular systolic unloading occurs during the therapy. Schecter, Hod et al report that the mechanisms of action for ECP therapy are not yet definitely defined. Many clinical studies however, identify components of the hemodynamic, physiological and neurohormonal cascades that ECP initiates.\nThe many beneficial effects that ECP produces appear to emanate through arterial diastolic augmentation. The retrograde pressure wave increases coronary perfusion pressure, which creates a gradient between ischemic and nonischemic areas of the myocardium that can recruit latent conduits and increase myocardial perfusion. Increased nitric oxide (NO) and atrial natriuretic peptide (ANP), and decreased endothelin (ET-1) and brain natriuretic peptide (BNP) raise the possibility of peripheral benefits as well as restored coronary flow reserve. Increased endothelial shear stress releases growth factor, which can improve endothelial function.\nBonetti, et al have concluded that external counter pulsation is associated with an acute improvement in peripheral endothelial function, as is demonstrated by the acute increase in RH-PAT index observed in response to ECP during the first three days of treatment. By increasing coronary blood flow, ECP is thought to promote myocardial collateralization via opening of pre-formed collaterals. Increased blood flow and shear stress may also improve coronary endothelial function favoring vasodilation and myocardial perfusion.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Learning in Lake Mary Florida how to merge debts is one of the best things card holders can do. Lake Mary credit card consolidation is perfect for those people in Lake Mary who are looking to better their credit for the future in Lake Mary. There are lots of benefits for card holders in Lake Mary that benefit from credit card debt counseling. If you are thinking about credit management, then there are two things in Lake Mary you must look into before doing this in Lake Mary FL. Use these tips as a guide to credit card consolidation.\nThere are several great factors to debt consolidating programs. One of the greatest factors in Lake Mary is to have better rates. If a better rate in Lake Mary can be had by you with a local credit card settlement service than you currently have in Lake Mary, then there's many reasons to consolidate your debts. Merging your quick loans can include up to substantial savings in Lake Mary. Look up all of your interest rates from each quick online cash advance loan and write them on the list in Lake Mary. Then note the new quick loan rate you would be given. When the new credit card relief rate is lower than the average from the old credit card counseling rate, financial sense would be made by then to use a Lake Mary credit card consolidation service. If there are cash advance lending that have a lesser rate, then you may not have in Lake Mary to include them in your cash advances agreement.\nAnother reason people in Lake Mary love to consolidate debts is to make their lives simple. By paying one credit consolidation, they can cut out a lot of anxiety and debts paying time. Your credit card debts should not be consolidated in Lake Mary by you for this reason alone however in Lake Mary Florida. You don't want to spend more on cash advances in the long term just to save a little time. Using debt counseling programs also offers those with a credit card debts mess a chance to get out of it. By using credit card debt consolidating services, they could be making lower monthly quick cash advance lending payments than they would be if they did nothing in Lake Mary. By closing out other online cash advance loans accounts, their credit may even be improved in Lake Mary.\nWhen considering credit relief, you should turn to specialists at sites such as Lake Mary credit card consolidation for an appointment. There are numerous debt negotiation issuers and banks that wish to help you with your request in Lake Mary. Make sure you do your research in Lake Mary so that when you consolidate debt in Lake Mary, you are certain you are making a decision in Lake Mary that is profitable to you. Ensure there aren't any hidden online loan charges that various credit consolidating plans may have. Doing your research in Lake Mary can save your cash for the future in Lake Mary.\nThere are always two sides to everything, and that includes debt relief. Some harp about the Lake Mary credit card consolidation advantages that this brings while others in Lake Mary swear that they regret entering into a credit card consolidating program since it made them sink further into debt in Lake Mary.\nSome credit card debt management companies will tell you that credit card negotiation will help you by allowing you to pay a lower interest rate in Lake Mary than what you used to pay. Stress in Lake Mary is also reduced because you no longer need to worry in Lake Mary about numerous creditors. All you have to do is put in a single credit relief payment once a month and all the other debts are taken care of.\nHowever, what these credit negotiation companies do not tell you outright in Lake Mary is that the main reason why there is a reduction in interest Lake Mary credit card consolidation payment is because they negotiated for a longer payment period. Longer credit card negotiation payment duration means that you actually pay more in Lake Mary than if you did not enter into this credit relief. Another thing that credit negotiation services do always tell you upfront is that not all credit card debt management applicants are given the privilege in Lake Mary of having low interest credit cards payments. Those who only have good credit standings in Lake Mary enjoy this and those who don't in Lake Mary are better off looking at other debts options.\nIn order to consolidate bills the right way, you need to have an honest credit card consolidating assessment of your cash situation and your paying capacity in Lake Mary. There is also a need to take action in order to get out of debts. If these Lake Mary credit card consolidation options are ignored then you will forever in Lake Mary be trapped in the growing cycle of credit card debts.\nFinancial credit card debts has changed into a large problem in Lake Mary in numerous developed places, triggering strain along with debt relief problems for those along with their loved ones. Credit relief Lake Mary FL can be a probable option in Lake Mary in case you have mounting debts via a number of creditors.\nCredit relief Lake Mary FL are in its most straight forward process, an operation of producing just one brand-new credit consolidating product that mixes your active quick loans straight into 1. The theory is basically that you are only going in Lake Mary FL to need to worry about managing a single credit relief institution in contrast to preparing many credit cards payments of varied time periods as well as varied Lake Mary credit card consolidation agencies.\nThough very typically acknowledged to be a unsecured credit consolidation, there can be various kinds of debt counseling. Determined by your needs along with your possibilities in Lake Mary, a number of credit card debt consolidating products could possibly be properly in Lake Mary secured up against equity such as that of your own home in Lake Mary. Some could possibly be unleveraged and additionally focus on only switching the total in all a person's credit lines in to a single completely new Lake Mary credit card consolidation service based card to take benefit of lower credit cards payments. In addition there are standard credit card counseling which in turn are not a card and definitely will join any credit card settlement financial products jointly say for example a house bills, car credit\/debt package, in addition to Lake Mary credit lines.\nDue to numerous credit card debt counseling services out there right now in Lake Mary, it is encouraged for you to find an debt consolidating authority to help give you advice in Lake Mary on the way to pick the right credit management programs for ones Lake Mary circumstances.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"jeffersons copy center uses laser printers. assume jefferson started the year with 92 containers of ink (average cost of 9.00$ each, FIFO cost of 8.90$ each, LIFO, cost of 8.05$ each). during the year, jefferson purchased 680 containers of ink at 9.80$ and sold 580 units for 20.25$ each. jefferson paid operating... show more jeffersons copy center uses laser printers. assume jefferson started the year with 92 containers of ink (average cost of 9.00$ each, FIFO cost of 8.90$ each, LIFO, cost of 8.05$ each). during the year, jefferson purchased 680 containers of ink at 9.80$ and sold 580 units for 20.25$ each. jefferson paid operating expenses throughout the year, a total of 3,750$. jefferson is not subject to income tax.\nthank you very much in advance and if you have any helpful sites, it could be helpful together with how you got the answers!\nBest Answer: You need to learn how to compute Cost of Goods Sold under the 3 methods.\nUnder FIFO, BB 92 @ 8.90 + 680 @ 9.80. Sold 580, of which 92 were @ 8.90 and the balance @ 9.80, leaving an EB of 192 @ 9.80.\nUnder AVG, BB 92 @ 9.00 (828) + 680 @ 9.80 (6,664) = 7,492 \/ 772 for AVG 9.7047.\nSold 580 at 9.7047, leaving EB of 192 @ 9.7047.\nRather than looking for helpful sites, why not try reading your text book?\nAccounting - FIFO, LIFO and AVCO?","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Set against the colorful backdrop of the Los Angeles dogwalking scene, The Dogwalker follows the moving, transformational journey of Ellie Moore (Diane Gaidry), on the run from her latest abusive boyfriend. Down and out on the mean streets of L.A, rescue comes in the unlikely form of Betsy Wright (Pamela Gordon), a misanthropic dogwalker in need of help with her business and struggling with her own dark past.\nThe unusual relationship between the cranky dogwalker and her unstable apprentice sets the stage for the emotional transformation Ellie must undertake. In Betsy, she encounters a dark mirror of herself- a lonely, angry, complex woman whose own troubled past proves disturbing like her own. And in the dogs, Ellie finds lost pieces of herself as she fumbles with the leash to her own life, struggling to hold on as it pulls her toward a brighter future before her past catches up with her.\nThe Dogwalker took more than six years from concept to completion. Most of that time was spent raising funds, but much video workshopping and script development also took place. It was produced through our mutual support collective, Filmmakers Alliance (FA). FA is the nine-year-old filmmaking collective founded by my wife and myself that has supported the production of hundreds of award winning shorts, features, commercials, music videos and more. Besides the numerous material resources, much of the crew and cast are FA members, or were brought into the process by FA members.\nThe film was shot on MiniDV on a Sony VX1000. We shot for five-and-a-half weeks with approximately ten days of pick-ups. Adobe AfterEffects was used to create the film simulation through a proprietary formula developed by co-producer Andy Somers.\nThe film was screened at the Independent Feature Film Market in New York in 2001 where it was one of five finalists for the Gordon Parks Independent Film Award. The Cinequest Film Festival hosted its world festival premiere.\nThe Dogwalker is the product of many issues and factors. I wanted to make a film with my wife, Diane Gaidry, that incorporated elements of our own life.\nDiane once worked as a dogwalker and I decided to build a story around that particular world. But as the story developed, I become less interested in the details of dogwalking and more focused on the specific human characters and what the dogs mean to them. Dogs are truly unbelievable mirrors of our raw emotional states. Moreover, they provide a rich allegorical context for the exploration of female anger \u2013 how it is socialized, how its manifests itself in many women. Ellie's abusive boyfriends are extensions of her own inwardly directed rage, while Betsy is still paying the price for a rage too fully expressed. Ultimately, however, the film is about coming to terms with that energy and moving on to a place of self-forgiveness and transformation. The Dogwalker is a term that both describes the job and, metaphorically, the task of caring for our own inner selves. Finally, the film is about my home, Los Angeles, and the often forgotten myriad of modest, though complex lives that exist within the city.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Following on the popular Bible Study led by Father Williams this past spring, we embark on a two-year program, From Genesis to Revelation. Open to all.\n\"Our hope is that this course offering, entitled The Holy Bible \u2013 from Genesis to Revelation \u2026 With Open Minds, Quickly and Joyfully in Faith, will provide a sound overview of our Sacred Scripture central to the faith and culture, and even to the literary traditions, of more than two billion people around the world.\nThis Sunday series will first focus on the Old Testament for approximately 42 weekly sessions of 45 minutes each, starting on Sunday, September 9, 2018. In fall 2019, we will delve into the New Testament for approximately 42 weekly sessions.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Kurt Volker, Atlantic Council senior advisor and member of the Strategic Advisors Group (SAG), describes the challenges facing NATO and calls for unity to the divergent goals and ambitions of member states in the SAG issue brief \"A New Transatlantic Compact.\"\nTwenty years after the fall of the Berlin Wall, NATO stands at a crossroads. Will it reinvent itself yet again, to serve as the foundation for the security and defense of Europe and North America in a world of diverse, non-conventional threats, many of which come from outside of Europe? Will it return to a passive, geographically defined approach of protecting the territory of European Allies against armed attack? Will it merge these visions into a new hybrid? Will it retain the political will and resource commitments of its members, whether in Europe or North America? In short, as Brent Scowcroft, Chairman of the Atlantic Council's International Advisory Board, put it: \"What is NATO for?\" These are the core questions Allies must answer in drafting the 2010 NATO Strategic Concept.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"HAYS, Kan.\u00ad \u00ad\u00ad\u00ad\u2013 Fort Hays State University's Virtual College received four No. 1 rankings among many others from different agencies during the months of September, October and November for its affordable programs and quality online degrees.\nTwo online programs at FHSU were ranked No. 1 by OnlineU.org: the general studies degree with a concentration in sustainability (environmental science) and a Master of Arts with a concentration in history.\nIn addition, FHSU was ranked No. 1 by TheBestSchools.org for a virtual Master of Liberal Studies with a concentration in art and art history. A No. 1 ranking was awarded to FHSU's virtual Master of Business Administration with a concentration in Internet marketing by GradSchoolHub.com.\nFHSU's online Bachelor of General Studies degree with a concentration in sustainability (environmental science) received a No. 1 ranking from OnlineU.org for its acceptance rate and affordability.\nRequirements to obtain a Bachelor of General Studies degree from FHSU includes the completion of 120 credit hours, a minimum grade point average of 2.0 and a minimum of 80 hours of course work in the liberal arts and science areas.\nTo view this ranking, visit https:\/\/bit.ly\/2PZhaAP.\nOnlineU.org also recognized FHSU's online Master of Arts in history and FHSU's online Master of Arts in public history for its affordability.\n\"At FHSU, you can earn a master's degree from a highly respected program with its roots firmly set in a rich historical landscape,\" says FHSU's website. Paid graduate teaching assistantships are also available in this program.\nTo view this ranking, visit https:\/\/bit.ly\/2PLlx1c.\nAlso recognized for its affordability, FHSU's online Master of Business Administration with a concentration in internet marketing received a No. 1 ranking from GradSchoolHub.com.\nTo view this ranking, visit https:\/\/bit.ly\/2zrYdvr.\nThe FHSU Virtual College's Master of Liberal Studies with a concentration in art and art history program received the No. 1 spot for its quality, academic excellence, reputation, financial aid assistance, range of degree programs and strength of online instruction methodology.\nTo view this ranking, visit https:\/\/bit.ly\/2D1jyiy.\nThe FHSU Virtual College is ranked second for the Bachelor of Science in computer science.\n\"The school's computer programming online degree uses a flexible structure that allows full- or part-time students to take as many courses as their schedules can handle,\" says TheBestSchools.org.\nTo view this ranking, visit https:\/\/bit.ly\/2zX4B16.\nThis ranking is for FHSU's Bachelor of Business Administration with a major in international business and economics.\nTo view this ranking, visit https:\/\/bit.ly\/2yRK2A6.\nFHSU's online Bachelor of Science with a major in information networking and telecommunications was recognized in this category.\n\"The concentration in computer networking and telecommunications prepares graduates to work in telecommunications companies or in corporate, education, law enforcement, health care, and other computer and network positions,\" says FHSU's website.\nTo view this ranking, visit https:\/\/bit.ly\/2HWvoOV.\nThe FHSU Virtual College was recognized for having one of the top online Bachelor of Arts programs with a history major.\n\"An online bachelor's degree in history will introduce students to the process of researching and examining history, as well as studying the histories of different regions around the globe,\" says GuideToOnlineSchools.com.\nTo view this ranking, visit https:\/\/bit.ly\/2SUCzc8.\nFHSU's online Bachelor of General Studies degree with a concentration in sustainability, known as environmental science was recognized. FHSU was also recognized for having the lowest tuition rates.\n\"An accredited bachelor's degree program will require a four-year commitment and will include classes in political science, communications, business, basic math and science, the national park system, global warming, conservation efforts, and wildlife threats,\" says GuidetoOnlineSchools.com.\nTo view this ranking, visit https:\/\/bit.ly\/2Pje5LB.\nThe Master of Art with a concentration in history at FHSU was named one of the top online programs.\n\"Fort Hays State University has a notably low tuition of only $4,316 per year and an exceptional median alumni salary of $99,100,\" says GuideToOnlineSchools.com.\nTo view this ranking, visit https:\/\/bit.ly\/2F6NVHb.\nThe online Bachelor of General Studies with a concentration in child development at FHSU was recognized for being one of the best with the lowest tuition on the list of schools.\n\"The program is designed for a wide variety of people, including online degree seekers, day-care providers, early childcare providers and parents,\" according to FHSU's website.\nTo view this ranking, visit https:\/\/bit.ly\/2SQv6e6.\nIn addition to the two No. 1 rankings, OnlineU.org also recognized FHSU's Virtual College with several other rankings this fall.\nBased on its affordability, FHSU's online Bachelor of General Studies degree with a concentration in child development was awarded a top spot in this ranking.\nThis liberal arts degree offered by FHSU provides students whom have well-defined career objectives and do not require a course specific or career specific degree the opportunity to graduate.\nTo view this ranking, visit https:\/\/bit.ly\/2P8qEJv.\nOnlineU.org ranked the FHSU Virtual College for its affordability and various selections of online degree programs.\nFHSU's Virtual College offers more than 50 online degrees specifically designed for adult learners. \"Our online degree programs are fully accredited, affordable and utilize the latest in learning technology for what we think is one of the best values in higher education today,\" according to FHSU's website.\nTo view this ranking, visit https:\/\/bit.ly\/2OwumYd.\nThe FHSU Virtual College received a No. 4 ranking in affordability for 34 affordable online master's programs.\n\"Fort Hays State University offers over 30 exclusively online master's programs and is regionally accredited by the North Central Association of Colleges and Schools,\" according to OnlineU.org's website.\nTo view this ranking, visit https:\/\/bit.ly\/2Qpaqbx.\nFHSU's Virtual College for individuals seeking a bachelor degree earned a top ranking for being one of the most affordable online colleges and degree programs.\nWith an annual tuition averaging $6,778 and 60 online bachelor degrees to choose from, it allows non-traditional students to attend school, while still maintaining a job and a family.\nTo view this ranking, visit https:\/\/bit.ly\/2Qwauq8.\nFHSU's online Bachelor of General Studies with a concentration in business administration has been chosen as one of the 20 best degree programs across the United States.\n\"FHSU offers a Bachelor of Business Administration degree program that allows concentrations in management, human resource management and operations management. Your program is designed to provide you with a high-quality program to fit any lifestyle,\" says Online-Bachelors-Degrees.com.\nTo view this ranking, visit https:\/\/bit.ly\/2qxqH2D.\nFHSU has been ranked nationwide as the third best school throughout the United States for working college students.\n\"The school's tuition charges totaled $4,007 in 2017-18 for students, well under the $5,655 annual earnings of a student getting minimum wage in this city. In fact, such a student could pay tuition and still come out ahead of costs by $1,648,\" according to the study on StudentLoanHero.com's website.\nTo view this ranking, visit https:\/\/bit.ly\/2OqbABP.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A Love for WWOOFing and desire of travelling have led thousands of Volunteers to join on WWOOF Nepal since 2003. Join them today to see a different part of this Himalayan Kingdom!\n\"Hi WWOOF Nepal team, we have just returned from host from Pokhara. They are fantastic people and treated us like a family members. Please send more volunteers there - they would love to have more.\"\n\"Wwoof Nepal made it possible for us to make great experiences to learn from all varieties of gardening and farming skills through WWOOFing. We are very proud of some of the things we've done and learned and that we got the chance to do so. We feel deeply fascinated for WWOOFing in Nepal which we felt a great way to experience \"real\" Nepal. We want to thank to WWOOF Nepal Team, as well as all our hosts in Illam. We felt no difference than our family members. We never forget these great Nepal..\nWe are glad to get the support of WWOOFers like Nina and J\u00f6rgen here in Nepal. Thank you for your kind words. All Nepali people we have met have been very friendly and welcoming. We got treated like a part of the host family. The work at the kindergarten was from around nine to five. It's perfect for wwoofers who likes children. It felt like we inspired with some new English words. It was good for the teachers too, to practise their English. This wwoofing- experience have been fantastic and t..\nMy name is Mari- Line and I am from France. After WWOOFING in Syangya, I came to Rukum and did 2 weeks WWOOFING to Rukum. This is also no nice place surrounded with mountain, river and jungles. I have now all the information to give about my volunteering in Rukum, a remote district of Nepal which was one of the active district of 10 years of Maoist war. I had a very good time in this remote area. The family was very joyful and always laughing. It was sometimes a little bit difficult because I ..\nI had a wonderful time on the farm in Dhikura, Arghakhanchi. I learned a lot about farming, and about a way of life so different from the one I was accustomed to. The pace was much slower than what I was used to, and at first that was hard for me, but after some time I grew to appreciate it. I was so well fed and taken care of\u2014Pompha put a mosquito net up around my bed when I was being eaten alive in my sleep. I really enjoyed painting the house with fresh clay for the upcoming Dasain fe..\nI had a wonderful stay with Basu and Puspa and would definitely recommend them! As it was monsoon season there was not a lot of work to do, but they welcomed me in as one of the family! Basu let me help with cooking, and taught me to cook Nepali food and to make paneer! I also got to help a little with feeding the cows, fertilizing the fields, and planting. I definitely had a little bit of culture shock, mostly with adjusting to a muuuuch slower pace of life than I'm used to. It was such a..\n\"WWOOF Nepal is an amazing experience. I miss my Ama and Bua's in Chitwan and Begnas Tal very much every day. I cannot wait until my life allows me to be back in Nepal. A little about our experience at farm. There are 6 goats on the property. Two that are pregnant with 2 goats each right now, one will be giving birth soon, they expect December. I believe they sell those goats though. The farm has 3 buffalo. We spent our time picking coffee beans, pulping coffee beans, washing, and spreading\/dr..\nKrish From America in Nepal from October 2004 to December 2004: I came to Nepal after 8 months of traveling through South East Asia, and it has been so great to meet Fanindra and his family. They have been so welcoming and I have learned so much from my time with them. WWOOFing has been such a good way to meet Nepali people who aren't connected with the tourist industry and to see how people live in rural areas. I was lucky enough to be in Nepal during the festival of Dashain, and I spent ..","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"North Mill field is reputed to be the 'probable' site of a fierce battle that took place in 527 between Octa (grandson of Hengist, King of Kent) and Erchenwin (founder of the kingdom of Essex). A surprise attack on London was planned, either via the west bank of the Ligan (Lea) near the ford (at Temple Mills), or at the upper ford (formerly near Lea Bridge). He is stated to have inclined to the latter proposal.\nBoats anchored in the waters of Lochtuna (the lake formed by the Lea overlooked by Leyton). The deputy King's forces marched to North Mill field to meet the force and, after defeating an advance force Erchenwin's main force, were defeated 'in sight of the Ligan' (Lea) in the 'Battle of Hackney'.\nFollowing Alfred the Great's re-conquest of London in 886AD, a treaty defined a political boundary running up the Lea and dividing Guthrum's Kingdom in East Anglia from that of King Alfred (the Great): the 'Danelaw' boundary.\nThe Danes of Mersea rowed their ships up the Thames and the Lea and, in the following year, built a fortress twenty miles above London. The English attacked, but were put to flight with loss of life. In the Autumn, Alfred camped nearby in order to safeguard the corn harvest, at the same time denying it to the Danes. It was then he conceived the idea of obstructing the Lea, so preventing the Danes from retreating with their boats.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"1. Yetis, B., Atar, I., Ozin, B., and Muderrisoglu, H., \"Termination of electrocution \u0131njury with an \u0131nappropriate \u0131mplantable cardioverter-defibrillator shock due to external alternating current leak,\" International Journal of Cardiology, Vol. 140, pp. S90, April, 2010.\n2. Yeti\u015f, B., G\u00fcltekin B., K\u0131l\u0131\u00e7, D., Y\u0131ld\u0131r\u0131r, A., \"Fistula between the left internal mammary artery and pulmonary artery:a rare cause of recurrent angina after coronary bypass grafting\" Arch Turk Soc Cardiol 39(3):240-243,2011.\n3. Aslan G, Sade LE, Yetis B, Bozbas H, Eroglu S, Pirat B, Can U, Muderrisoglu H. Flow in the left anterior descending coronary artery in patients with migraine headache. Am J Cardiol. 2013 Nov 15;112(10):1540-4.\n4. Karacaglar, E., Atar, I., Alt\u0131n, C., Yetis, B., Cakmak, A., and Ozin, B., \"The effects of niacin on hscrp in patients with non-st elevated acute coronary syndrome\" International Journal of Cardiology, Vol. 140, pp. S25-S26, April, 2010.\n5. Yeti\u015f,B. and \u00d6zin, B., \"Rate control in atrial fibrillation,\" T\u00fcrkiye Klinikleri J. Cardiol-Special Topics, 3(3): 18-23, 2010.\n6. Yeti\u015f, B., Atar, \u0130., \u00d6zin, B., \"How should we optimize cardiac resynchronization therapy\" T\u00fcrk Aritmi Pacemaker ve Elektrofizyoloji Dergisi, 7(3), 141-144, 2009.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"NASA offers a great collection of their space images from the Apollo mission as public domain. If you start browsing the collection, you might be disappointed because of the strong color shifts. So I took it upon myself to do some digital color restoration work. Seeing is believing, so check out the video for more image samples!\nThese images were originally shot with Hasselblad medium format cameras and scanned recently by NASA at 20 MP resolution. However, if you start looking at these images, you will realize a hefty green tint across all images that were taken in space. I am not sure about the cause, but it doesn't affect images taken on earth and I assume scanning was done professionally. The problems must be with the original film that might have been underexposed or space radiated?\nAccording to NASA the astronauts used modified Hasselblad cameras with Zeiss Planar 2.8\/80mm lenses (ca. 50mm equivalent) and a single Zeiss Sonnar 5.6\/250mm lens with Kodak Ektachrome film.\nOn Apollo 8, Hasselblad EL electric cameras were used for the first time. The electric motor in these Hasselblads largely automated the picture taking process. The astronauts needed only to set the distance, lens aperture, and shutter speed, but once the release button was pressed, the camera exposed and wound the film and tensioned the shutter. Two Hasselblad EL cameras, each with a Planar f 2.8\/80mm [normal] plus a single Sonnar f5.6\/250mm [telephoto] lens and seven magazines of 70mm film, were carried.\nOn Apollo 8, three magazines were loaded with 70 mm wide, perforated Kodak Panatomic-X fine-grained, 80 ASA, b\/w film, two with Kodak Ektachrome SO-168, one with Kodak Ektachrome SO-121, and one with super light-sensitive Kodak 2485, 16,000 ASA film.\nApollo 7 in 1968 was the first mission in the Apollo program to carry a crew into space. Apollo 9 spent ten days in orbit testing the Lunar Module and several aspects critical to landing on the Moon.\nCheck out my Flickr set with this lens for 25 restored images in full resolution. Corrections were done in Photoshop mostly with the gradation tool to remove the green hue (leave a note if you want to know more about PS-editing here). Take a closer look at the following sample images.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"While planning trip with family, you want that the hotel has the special facilities such as swimming pool and kid's area. You will have to look for a pet-friendly hotel if you are taking a pet along with you. The level of hygiene in the hotel is also the most important thing to consider while going booking your hotel.\nThere may be plenty of hotels which offer good and luxury facilities but all of these will not suitable for your budget. When you plan to book a hotel for your trip then you should make sure about the budget package offered by these hotels. Before booking the right hotel for you, it is necessary to look at value added services such as extended hour room services or laundry services and many more.\nWhy should people go online to book the right kind of hotel?\nYou can save a lot of time and effort by visiting ay online website of various hotels and make the better decision by making comparison between more than one hotel and book the one best for you.\nWith the help of online website, you can check out the reviews provided by the customers about the services offered by the hotels to their clients. This will make it easy for you to get the information about hotel and find out the one best for you.\nIn these days, many hotels allow people to book their room in advance from the website of the hotels after checkingthe quality facilities and budget package of different hotel rooms.\nMany of these these hotels offer variety of services for convenience and comfort of people who want to go for trip with their friends or family such as ubud family accommodation.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"According to Maggie Haberman of The New York Times, one reason Kirstjen Nielsen hung in so long as the head of the Department of Homeland Security is that \"she was aware how awful life would\/will be for her on the outside,\" given her role in defending Donald Trump's policies. That should be the case for anyone who has served, currently serving, or going to serve in the Trump White House. All these people should be treated like the guy who directed Billy Squier's \"Rock Me Tonight\" video. Who the hell wants to work with that guy?\nNielsen did not create Trump's policy of separating migrant families, but she should be known forever as the person who carried it out. She separated families, put babies in cages, and lied to Congress that the policy didn't exist. She should get the Aldo Raine treatment so she can be \"ingloriously\" identified with it for the rest of her life. Now, the inept Trump administration says it can take up to two years to identify thousands of separated migrant children.\nAfter John Kelly left his post as Chief of Staff, Nielsen was left without any allies in the White House. She was expected to be even tougher on the border, even if laws were violated. Apparently, that's where she stood her ground and her enemies used it against her, from son-in-law Jared Kushner, Chief of Staff Mick Mulvaney, National Security Adviser John Bolton to White House Adviser Stephen Miller. The person who was least happy with Nielsen was Trump himself, who really loves the child separation policy.\nAs legal experts challenge Trump's authority to bypass a senior Homeland Security official in order to install a hand-picked acting head of the Cabinet agency that oversees immigration enforcement, it's obvious the real director is Stephen Miller.\nMiller may also be behind Trump telling Border agents not to let anyone into the country to press for asylum, thus forcing them to enter the country illegally where they can arrest and detain them. Yesterday, a court ordered a halt to Trump's \"remain in Mexico\" policy, which forces certain non-Mexican asylum seekers to wait in Mexico during their court proceedings.\nTrump and Baby Goebbels Miller are trying to create a situation for immigrants that's worse in America than if they stayed home. While they are creating horrible conditions and hurdles for them, they can't make it more discouraging than their homeland unless they start killing them. Let's not give Baby Goebbels any ideas.\nRon Vitiello's nomination to permanently lead Immigration and Customs Enforcement was pulled after Trump remarked that \"we want to go in a tougher direction.\" Nielsen wasn't tough enough for Trump even as she carried out his policy of separating families and throwing babies in Jail. All signs, such as Trump saying \"America is full,\" are showing a return to that policy and even tougher measures.\nTrump and Miller seek to make the United States a horrible place where nobody would dream of immigrating to. It's becoming a place where many people dream of leaving.\nA seven-year-old Guatemalan girl died in the Custody of Border Patrol (CBP) last week. The Department of Homeland Security's (DHS) statement has only raised more questions than it answers.\nThe girl and her father were apprehended with a large group of migrants in a remote span of the New Mexico Desert. Reportedly, she hadn't eaten or consumed water for several days. After being in the custody of CBP for over eight hours, she started having seizures. She was transported to an El Paso hospital where she died of dehydration and shock.\nAsked if food and water were given to the child, DHS only replied with blame for the father for taking his daughter on the journey to the United States through dangerous elements. She didn't die on the journey. She died in Border Patrol custody. She died when she was their responsibility.\nCBP and DHS need to take responsibility. If someone's life is in danger after they're apprehended, saving their life should take precedent over detainment. The Trump administration needs to take responsibility too.\nWhite House spokesperson Hogan Gidley said the administration isn't to blame for the child's death. Gidley said, \"Does the administration take responsibility for a parent taking a child on a trek through Mexico to get to this country? No.\" He then heaped additional blame on Democrats saying, Congress should \"pass some common-sense laws to disincentivize people from coming up from the border and encourage them to do it the right way, the legal way, then those types of deaths, those types of assaults, those types of rapes, the child smuggling, the human trafficking, that would all come to an end. And we hope Democrats join the president.\nWhile the U.S. government will only consider claims of asylum from migrants presenting themselves at ports of entry, the Trump administration has disincentivized them from doing so, forcing them to wait in Mexico for lengthy, indeterminate periods of time before considering their claims.\nHomeland Security Secretary Kirstjen Nielsen also avoided addressing concerns about how CBP treated the girl but instead characterized her death as \"just a very sad example of the dangers of this journey.\" Nielsen is a sad example of a cabinet member.\nThe Guatemalan girl's death comes months after a toddler died of an illness she developed at an Immigration and Border Customs Enforcement facility in Dilley, Texas.\nAura Bogado, an immigration reporter with Reveal, sent a tweet detailing conditions in the \"hieleras,\" the Spanish word for \"icebox\" that detainees and guards alike use to describe CBP's frigid holding cells.\n\"Lack of accountability, and a culture of cruelty within CBP have exacerbated policies that lead to migrant deaths. In 2017, migrant deaths increased even as the number of border crossings dramatically decreased.\" Pompa added.\nThere is a culture of cruelty within CBP, as Border Patrol agents are known for pouring out water in the desert left for migrants by humanitarian groups.\nThe conditions migrants are escaping from are so bad that Donald Trump's threats and racist fear mongering isn't discouraging them. People seeking asylum, at least those below our southern border, know they won't be welcomed with open arms. Now, it seems they can't be met with compassion or humanity.\nWe're the United States, and we're supposed to be better than this. Before January 20, 2017, we were.\nCan we restrict people from entering this country if they're annoying and self indulgent? Yes, we should prevent dangerous people from crossing our borders, but I think we should add selfie queens (that includes you dudes too) to the list. We have enough insecure people who were born here who can't resist the urge to share their neuroses with the entire world, why invite more? And don't get me started on cat pictures. Dog pictures are OK. In fact, being a dog lover should move people to the front of the line.\nNews reveals that the Jihadist chick who was one of the terrorists in the San Bernardino shootings entered the country without having her social media checked out? What? I doubt you can get a decent job without someone from human resources checking out your Facebook and judging your cleavage and nights of drunken hootenannies. If Burger King wants to see your thong, then why shouldn't DHS also take a peek?\nI don't think we should stop people from coming to our country. That doesn't mean they shouldn't be vetted, or that the process we're using now is perfect. Hell, the 19 hijackers responsible for 9\/11 entered the country on visas. Everyone needs to be checked out whether they're students or a fiance for a U.S. born loser who can't get an American date and orders a bride from the Philippines.\nSo why didn't Homeland Security check out Tashfeen Malik' s Facebook page? Sure, stereotyping is bad but she had been a resident of two nations, Pakistan and Saudi Arabia, where people are trained to hate the United States. Reportedly, she made several posts expressing a desire to join ISIL. Her posts were private and could only be seen by her most trusted Jihadists friends, but DHS should have checked it out anyway. I'm sure the NSA is spying on U.S. citizen's social media, so DHS could probably hack her Facebook and see whether she had a massive crush on Matthew McConaughey or Jihadi John. We need to know if her sugar daddy is Abu Bakr al Baghdadi.\nSo yeah, check out social media. See if they hate America or just hate Nickelback. And if anyone from the National Security Agency is checking this out, please don't put me on the No-Fly List for \"I Can Have Jihad?\".","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"The father of a California man allegedly killed by an illegal alien wants to make sure that President Donald J. Trump knows that his son was a \"true patriot\" who was murdered because of inadequate border control policies.\nOn February 25, 2018, San Diego Police arrived at a local gym where they found Alexander Mazin, 27, bleeding to death from a gunshot wound in the torso. The suspect has not been caught, but is believed to be an illegal alien named Ernesto Castellanos Martinez.\nIn 2004, Castellanos voluntarily agreed to self-deport after a prior run in with the law, but he either came back in the United States after leaving or never actually left.\nCastellanos was not a total stranger to Mazin. His family said, \"Castellanos had previously been in a relationship with Mazin's girlfriend of three months. Two weeks before the shooting, Castellanos had attacked the woman while she was working out with Mazin at a gym, Mazin's parents said,\" reports 10 News.\nMazin's mother noted the ominous threat against her son that Castellanos had given the woman in question just weeks before the murder.\n\"The murderer called his ex and said, 'you can head to the parking lot and watch me kill him in front of you.' That's the man we are dealing with,\" said Mrs. Mazin.\nThe man's father had a special message for the President of the United States.\n\"This man could be hiding under a rock. In a cave. This man could be hiding in plain sight,\" said Mr. Mazin.\nNot surprisingly, this case has stirred emotions amongst those who support illegal immigration.\n\"Someone is going to commit a crime whether they are undocumented here or not,\" said Dulce Garcia, a DACA recipient and immigration attorney.\nThere is currently a $9,000 reward for any information leading to the arrest of Ernesto Castellanos Martinez.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Hey Beauties, if you follow me on Instagram you will have known it was my birthday last month and is always a great time to get those bits and pieces you have never got round to getting for one reason or another.\nAs I have got older I have started to learn more about the importance of asking for presents that I really need and are going to use, rather than the things I really want that I would end up never really using. So that is what I did for my birthday and so I asked for the Real Techniques Core Brushes set.\nI have seen so many YouTubers and Bloggers use the Real Techniques brushes but have never actually bought them myself. Once I got my hand on them my first impression were that the bristles are so soft which I think is important when putting on your makeup. The bristles are made of Talon and are each hand cut and 100% cruelty free which I really like. Also I noticed how light weight the brushes are.\nThe brush sets are a great price for what you get and for the great quality that they are. I wanted to compare prices from boots and super drug to see if there was any pricing differences and yes there was. In Superdrug you can get the set for \u00a320.99 and in Boots its \u00a321.99, a pounds difference I know but still a difference.\nThe Core Brush Set includes four brushes - Buffing Brush, Contour Brush, Pointy Foundation Brush and Detailer Brush and a carry case. These brushes I think are a great start for your makeup kit and I think you would you all of them in some form or another an are all interchangeable.\nThis is a great foundation brush and can be used for all types of foundation. I think that the bristles are a great size as the top of the brush is not that wide so your not going to get your makeup all over the place. You can also use it for blending out any harsh lines and even for a bit of blush.\nThis contour brush is defiantly one of my favourites as it really gets into the hollows of your cheeks to give you that great defined look. This brush is great for blending out your contour lines to give you a natural look so there are no harsh lines.\nThis is defiantly one of those dual usage brushes and can be used for so many different things. Its primary use is for applying foundation but I have also bee using it for blending out my concealer and even adding my highlight to my cheeks. The bristles are strong so go where you want the brush to go.\nThis brush does exactly what it says - its for the applying that finer detail to your makeup look. The bristles are very sturdy so I have been using it for adding shadow under my eyes as you can get great precision. This is a great brush for using on the lips, highlight on the brow done or inner corners.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The last round of the home and away season is upon us, but most teams have one eye already turned to the following week when the Finals Series gets underway and the stakes increase.\nAdding to this situation is the fact that end of season competitors have already been determined and, in all but one case, match line-ups are already known. This means that teams are playing for pride and will be guarding against injuries that could dent their premiership hopes.\nPort Macquarie Magpies are one team that has something to gain from their match against Coffs Harbour Breakers. The Magpies have only beaten their northern opponent once from five starts, and that was back in round 3, so will be looking to prove to themselves that they have the ability to compete with the reigning premiers come Finals time. If they don't win at home the Port players will be expecting their time in the Finals Series will be short lived.\nThe Breakers can't be deposed from second spot on the ladder and are assured of the double chance come Finals time. They find themselves in the midst of a six match winning streak, their best run of the season, and will be wanting to carry that form forward to the crunch matches ahead.\nBoth teams will be without key young guns with Kye Wilson (Port) and Ben Gibbeson (Breakers) on duty with the NSW\/ACT RAMS team against Vic Metro in the AFL Under 17 Futures Series.\nIt was only three weeks ago that the Breakers Women's team registered their first win of the season, and that was against Port Macquarie. The two teams face off again in what should be an extremely close tussle but with Port holding the home ground advantage.\nBoth teams are coming off last start losses and will be looking for a change of fortunes to build momentum ahead of the Finals.\nPort are by no means a one woman team but are certainly a better outfit when the League's key ruck, Cambridge McCormick, is in the line-up. Britt Hargreaves missed last week due to travel and is expected to be an omission again this week, and Port will be looking to players such as Jenni Cooper and Shaarn Whitehouse to help cover that loss.\nBreakers were significantly under strength in their last match and should welcome back a number of players to the line-up. They have several late season recruits who are starting to make strong contributions to the team performance in the experienced Katika Adams, rookie Molly Phillips, and key position player Keely Owen.\nSawtell Toormina Saints returned to the winners' circle last week which sealed the minor premiership. Considering that the team was expected to be in re-build mode the quality of their performances across the season has surprised many, including some in the Saints camp.\nThey travel north up the highway to take on the Grafton Tigers who have been soundly beaten in their last couple of outings since it became impossible for them to make the Finals. This will be their last match of a challenging season and whilst many will be glad to put the boots in the cupboard, they will also be keen to finish the season on a positive note.\nIt will be a big day for the club with a special event planned later that night for club legend Warren Bagnall and this will be sure to draw plenty of past and present players to the ground. This should ensure a strong line-up takes the field and will provide extra motivation for the team to do well.\nSawtell coach Brandt Lee will be keen to make sure his troops don't take this fixture lightly and will look to elder statesmen Matt Flynn and Luke Matthews to set the tone.\nFor full fixture details, click here.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Completely different musical traditions have totally different attitudes towards how and where to make adjustments to the unique source material, from fairly strict, to those who demand improvisation or modification to the music. 86 In elementary schools, kids typically study to play instruments such as the recorder , sing in small choirs, and learn concerning the history of Western art music and traditional music.\nIn a rating or on a performer's music part, this signal indicates that the musician ought to perform a trill \u2014a speedy alternation between two notes. For instance, the sheet music for a song might state that the tune is a \"gradual blues\" or a \"fast swing\", which indicates the tempo and the style.\nAfter all, researchers have weighed in. Listening to music at work could make absorbing and remembering new data\" tougher, according to a 2012 Time Journal article However, research has additionally discovered that listening to music at work could make individuals more productive.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Nice 3 bedroom 1 bath ranch with spactous kitchen and lots of cabinets. Home is close to everything, shopping, dining, Raleigh, Fayetteville and close to I40 and I95. This home is located in Benson city limits and is very convienent.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbbunw b/data_all_eng_slimpj/shuffled/split2/finalzzzbbunw
new file mode 100644
index 0000000000000000000000000000000000000000..ea0cfda88fbd2a6fb5c29cd090d058bd976a77c7
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+{"text":"UCL Centre for Digital Humanities, in partnership with the Higher Education Academy, hosted a FREE 'unconference'* focusing on bringing together the e-learning and digital humanities communities to discuss the development of 'Digital Pedagogies' in University teaching last Thursday.\nWith nearly 20 proposed sessions, about 8 proposals had to be put on the 'back burner' for the time being. The voted in proposals were placed throughout the day, and delegates got to construct their own experience.\nRegrettably, I couldn't have sat in on all the sessions, but the sessions that were chosen received great feedback. Twitter proved to be an excellent communication between delegates in sessions together, as well as those in different ones and who couldn't attend on the day.\nThe brief wrap-up session was the first time that delegates were together since the morning to discuss their experiences, ideas and outcomes of the day. Great feedback about the 'unconference' structure proved that perhaps the traditional conference structure doesn't always provide a space for innovation and discussion. Indeed, one delegate stated that her only regret of the day was that she wasn't able to attend all the sessions!\nEven more delegate blogs of the day!\nDo you have a blog post, session proposals or even more information? Please comment below to share!\nThank you again from all of us at UCL Centre for Digital Humanities and Higher Education Academy!\nAs a sort of technical person, I thought I might run a Make, Play and Teach session on how to make visualisations which have a greater pedagogical use and benefit that a pie chart that wobbles a little and is no better than what excel could give you.\nI would like to see a session on collaborative writing, focusing on the use of Zotero for sharing resources and references. Writing papers on books collaboratively is complicated at the best of times. How can Zotero make those citations flow a bit more smoothly? I imagine this as a fairly focused workshop on Zotero-assisted writing, for academics and researchers, but if participants would also like to discuss the basics of Zotero use, that's fine.\nI suggest this out of curiosity about other people's work habits and tools. If anyone has a similar workshop idea or wants to discuss related software, I'd be delighted to share.\nBelow are links to and from local underground stations, national rail stations and roads.\nWant to propose your own session?\n1. Email Rachel at ku.ca1556016880.lcu@155601688011.mh1556016880obsak1556016880.lehc1556016880ar1556016880 with a title and brief description of what you had in mind.\n2. Propose a session here.\nWe are still looking for more session proposals for the day!\nHave an idea for a game? Want to share a project or idea? How about lead a discussion or collaborative session? We would love to hear what you have in mind!\nFor more information on what a session is and some examples of previous ones, go here.\nUCL Centre for Digital Humanities, in partnership with the Higher Education Academy, will be hosting a FREE 'unconference'* focusing on bringing together the e-learning and digital humanities communities to discuss the development of 'Digital Pedagogies' in University teaching. We want to hear your ideas for sessions!\nThere are roughly four things people do in sessions: Talk, Make, Teach, and Play. Sometimes one session contains elements of all these, but it's also a fair taxonomy for sessions. In a Talksession proposal, you offer to lead a group discussion on a topic or question of interest to you. In a Make session proposal, you offer to lead a small group in a hands-on collaborative working session with the aim of producing a draft document or piece of software. In a Teach session, you offer to teach a skill, either a \"hard\" skill or a \"soft\" skill. In a Play session, anything goes \u2014 you suggest literally playing a game, or you suggest some quality group playtime with one or more technologies, or what you will. Of course, these are just guides \u2013 we are open to new ideas, new ways of interaction and methods of making this unconference insightful and fun!\n(2) by emailing Rachel at ku.ca1556016880.lcu@155601688011.mh1556016880obsak1556016880.lehc1556016880ar 1556016880 with a brief proposal.\nWant to propose a session for the day? Please register on this website at 'Register to Propose a Session' or send an email to ku.ca1556016880.lcu@155601688011.mh1556016880obsak1556016880.lehc1556016880ar 1556016880. We would love to hear your ideas and can help plan your session!\nUCL Centre for Digital Humanities, in partnership with the Higher Education Academy, will be hosting a FREE 'unconference'* focusing on bringing together the e-learning and digital humanities communities to discuss the development of 'Digital Pedagogies' in University teaching.\nAll text and code on THATCAMP Digital Pedagogies 2013 is freely available for you to use, copy, adapt and distribute under a Creative Commons Attribution 3.0 Unported License as long as you link to THATCamp.org and the Center for History and New Media. The name \"THATCamp\" and the THATCamp logo are trademarks of the Center for History and New Media at George Mason University. Theme based on the Furvious theme, by KreativeThemes.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"There is also a hybrid version of the Highlander. Prices for that version (which comes standard with the V6, plus a supplemental battery pack\/electric motor) start at $36,270 and top out at $47,880.\nWe've grown insouciant about this. In the Magnum, PI era, it would have staggered the entire car press to present them an eight-passenger, 4,200 pounds-plus family bus that could get to 60 in just over 7 seconds.\nBut it's not a Mad Max world out there, yet. For everyday purposes, the base 2.7 liter four will suit while the V6 ought to more than suit. It is exceptionally quiet. So quiet you notice the absence of sound. Toyota excels at this. That and general plushness \u2014 of seats and ride in particular. It's easy to imagine yourself in a larger\/higher-riding Camry or Avalon.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We created a series, analyzing and leveraging the guide to help you build your own treasure trove of quality content that ranks well on Google.\nWho knew that Google's search algorithm gets quality checked by actual humans (and not robots or aliens or magic elves)? And in an unprecedented move, Google released its full Search Quality Raters Guideline.\nUsing this guide, raters assess site quality so that Google can assess search quality. It's important to note that raters do not affect your live ranking. Rather, raters are but one metric in Google's search quality experiments.\nHowever, the guide itself is an oracle into Google's quality and ranking algorithm. Thank the SEO stars! From this guide, we can reap hidden treasures on how to write content that ranks higher on Google's search results page.\nWe can help you optimize your content and rank higher on Google.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Before submitting a request, please check the De Anza Library Catalog to be certain the item you want is not available at De Anza. We will order only closed captioned media. Many libraries do not loan non-print media and as a result, obtaining these materials may be difficult. However, we will do our best. Fill out a separate form for each item and provide as much information as possible. Thank you.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Vendors providing goods and services to companies doing business in regulated environments are only qualified to supply specific products and services. As such, these suppliers must be accurately tracked on a company's approved vendor list (AVL). The AVL must be continually managed and monitored, a complex supplier management system that becomes exponentially more difficult as product designs and manufacturing processes change over time. Further complexities arise as increasing numbers of suppliers engage with or are managed by different departments and divisions across the company.\nHow Can MasterControl Supplier, a Premier Supplier Management, Benefit You?\nHere's how MasterControl's supplier management software, which incorporates supplier management best practices, is giving companies the capability to efficiently manage and approve vendor lists and effectively track supplier quality and supplier status.\nWithout a supplier management system, maintaining and controlling an approved vendor list is complex and error prone, particularly when supplier quality information is kept in separate systems or tracked manually via spreadsheets.\nWith MasterControl supplier management software, all supplier management statuses and quality information (such as nonconforming material reports) is automatically contained in a single, easy to access location. In addition to providing an easily maintainable AVL, MasterControl Supplier features a user-friendly interface for accessing all documentation and records related to each supplier.\nAs engineering teams develop new designs, contract manufacturers and parts must be identified and added to the AVL. To be placed on the AVL, a supplier management system must be monitored on an ongoing basis and must successfully fulfill a number of qualifications, such as supplier surveys, supplier audits, and process validation.\nMasterControl supplier management software automatically tracks and stores supplier information derived from audits, non-conformance reports, supplier deviations, and CAPAs. This electronically managed information can then be easily compiled for supplier quality ratings. Furthermore, suppliers can be approved using MasterControl's robust approval workflow technology.\nAs new parts and suppliers are added and as suppliers' approval status changes, accurate and current tracking of parts and suppliers becomes increasingly more elaborate and difficult.\nWith MasterControl's supplier management software, goods or services can be added to the \"approved to supply\" list simply by linking the goods\/services to a specific supplier in MasterControl, and the supplier's identifier can be added for the goods\/services. Suppliers, conversely, can be added the same way. If a part is linked to a supplier that is not approved, the link will be disabled until the supplier becomes approved or reapproved. All of this information is easily obtainable via MasterControl's built-in reporting tool.\nIn a corporation with geographically dispersed divisions, it can be difficult to centralize supplier qualification activities and as a result, multiple divisions end up qualifying the same supplier. Without a proper supplier management system, this results in wasted effort, resources, and dollars.\nMasterControl's Web-based platform makes accessing supplier data easy and efficient. Geographically dispersed divisions have the ability to share supplier qualification data with others in the corporation, potentially cutting out costly duplication of effort and speeding time to market.\nCentralized Location for All Supplier Quality Data: MasterControl Supplier provides a centralized repository for automating, maintaining and controlling all supplier quality data and documentation \u2013 from non-conforming material reports and audit observations to contracts and service level agreements. With MasterControl, you'll efficiently manage and monitor supplier status and ratings, records, CAPAs, AVLs and more.\nSmoother Internal and External Audits: MasterControl Supplier automatically tracks and stores information derived from supplier management audits according to regulatory guidelines. With MasterControl, you'll securely store and maintain all information related to supplier management audits \u2013 including audit approval statuses, recent data derived from a supplier audit and links to quality assurance auditing and analytics report.\nImproved Communication and Collaboration: MasterControl Supplier enables everyone involved in supplier management to stay connected across geographically dispersed locations, facilitating better communication and collaboration with suppliers to minimize complications relating to specifications, materials and parts needs. With MasterControl, you'll gain greater visibility into supplier quality and receive higher-quality materials and parts.\nHolistic Supplier Management View: MasterControl Supplier provides a complete system for managing each supplier's quality information \u2013 approval status, non-conformances, deviations, SCARs and more \u2013 in a single place. By taking an integrated, holistic view of supplier management, rather than viewing each piece in isolation, MasterControl Supplier empowers you to execute and maintain a comprehensive and integrated supplier management system.\nQuicker Response Time to Problems:MasterControl Supplier is a powerful automated mechanism for investigating and correcting supplier quality issues. With MasterControl, you'll streamline supplier-related corrective actions, better manage supplier CAPA information and be able to immediately respond to critical supplier quality issues that can quickly result in out-of-spec products, nonconforming materials or worse.\nUser-Friendly System: MasterControl Supplier is an intuitive, user-friendly system where all critical supplier statuses and documentation are contained in a centralized, easy-to-access location. With MasterControl, you'll be able to easily add or change data about suppliers, parts and services while leveraging a variety of robust features and functionality, including configurable templates and dynamic analytics reporting.\nSupplier Qualification: Improve the management of all qualification documents and processes with routes, steps, high-risk initial audits and qualification audits to ensure continuity of supply, compliance and product quality. MasterControl enables you to manage initial lot inspects for certifying a supplier and to share supplier qualification data with personnel across multiple locations.\nDocument Management: Automate the entire lifecycle of your documents, including those pertaining to your supplier corrective action request (SCAR) process, and keep a single repository for all supplier quality-related processes and documentation. Automate critical processes with MasterControl and you will have all relevant information in one place and reduce document cycle time.\nApproved Vendor Lists (AVLs): Effortlessly monitor and maintain AVLs, and make all records and documentation related to each supplier easily accessible to the relevant people. MasterControl enables you to automatically track and store each supplier's approval status and other critical supplier information, as well as ensure that AVLs are up-to-date and available across the organization.\nSupplier Deviations: Allow suppliers to request exceptions to written specifications before material is shipped, and automate your data-collecting and -control processes. MasterControl helps ensure materials ship only after the appropriate people have reviewed and approved the request and provided proper justification. Capture and store supplier deviation data for in-depth analysis and reporting.\nSupplier SCARS:Streamline all tasks related to your SCAR process, including routing, follow-up, escalation and approval of all documents. MasterControl improves your ability to collect and track data from the request phase through approval, and to connect the SCAR process with other quality processes (e.g., launch a SCAR form directly from a supplier deviation).\nSupplier Parts and Services List: Simplify the process of adding and tracking parts and services. With MasterControl, you can add parts or services to your \"approved to supply\" list simply by linking them to a specific supplier. If a part is linked to an unapproved supplier, the link is disabled until the supplier is approved or reapproved.\nReporting \/ Scorecards: Ensure your suppliers maintain an acceptable level of quality by measuring their performance and rating them in a consistent fashion based on objective metrics. With MasterControl, you can collect and track data throughout the supplier scorecard process and create permanent records around supplier-related quality events, such as SCARs, supplier deviations and non-conformances.\nSupplier Risk Management: Monitor supplier parts issues and make informed, risk-based quality decisions by gaining greater access to quality-related data. MasterControl automatically tracks and stores supplier information from audits, deviations, CAPAs and more, which can easily be compiled to help identify and mitigate risk, evaluate supplier performance and determine supplier quality ratings.\nSupplier Audits: Automatically track and store information derived from supplier management audits. MasterControl securely stores supplier audit information, approval status and links to quality assurance auditing and analytics reports. You're able to maintain all supplier quality information in a centralized repository and make it easy to find suppliers that meet audit requirements.\nGuest User Access: Enable trusted suppliers to connect to a secure online collaboration space. Securely bring supplier contacts directly into the system and give them practical interaction capabilities \u2013 for example, reviewing and offering edits against a specification, contract, quality agreement or other document type \u2013 without giving them a full-access account. Increase collaboration while maintaining security.\nDo You Want to Learn More About Supplier Management Software Solutions?\nTo learn more about how incorporating MasterControl's supplier management software system into your supplier management program can dramatically improve your supplier management processes, please contact a MasterControl representative.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbbvvk b/data_all_eng_slimpj/shuffled/split2/finalzzzbbvvk
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+{"text":"FREE Singleplayer and multiplayer memory game for children\/kids, parents, youths .... whatsoever !!\nthe game is about uphill,downhill,narrow paths mountain driving.\nlooking for an off-road driving CHALLENGE ? - the game is all about it.\nBravo - Great Escape : Pandora -- Escape through the floating mountains of Pandora.\nalso you can set it as a live wallpaper.\nThe best 3D Air Hockey !\nYou like Connect Four or Captain's Mistress?\nThen you'll love 4 Coins.\n4 Fingers - the game requires a good reaction and nerves of steel. You have to beat the knife between his fingers, and the knife will gradually accelerate, which further complicates the problem.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Confirmed this works for 13.04 too.\nthe license and seems correctly configured to our servers.\nFirst extract the appropriate archive (either 64bit (glnxa64) or 32bit (glnx86) depending on which you installed ) onto your HDD. Then copy the contents of the CD over that, this gets the appropriate JRE into the install.\nInstall \"manually without the internet\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Would you like to save time and streamline your operations, effectively paving the way for increased revenue and growth?\nWe would like to help get you there!\nTake the first step by downloading this template our team carefully put together for you to manage your day-to-day tasks.\nWe will also send out periodic emails with valuable system and industry related tips and tricks to bring more passion and purpose into your practice.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Treatment begins with a personal focus at the University Hospital Sleep and Breathing Disorders Clinic. We offer the full spectrum of diagnostic techniques, treatments, and advanced medical technologies so patients' sleeping and breathing disorders can be effectively managed in their own environment.\nWe are among the most advanced sleep and breathing centers in the nation. Accredited by both the American Academy of Sleep Medicine and the Joint Commission, we are unique in that we diagnose and treat all common sleep disorders, plus respiratory difficulties caused by neurological and musculoskeletal disorders.\nOur multidisciplinary medical team includes board-certified clinicians in sleep medicine, pulmonology, and neurology. These clinicians also collaborate with colleagues in otolaryngology, oral and maxillofacial surgery, and psychiatry. Our team includes registered sleep technologists and licensed respiratory therapists with many years of clinical experience.\nOur Sleep Disorder Program treats a full range of common sleep disorders, including sleep-disordered breathing, difficulty falling or staying asleep (insomnia), and sleep\/wake pattern disorders such as circadian rhythm disorders.\nOur Breathing Disorders Program focuses on the assessment and management of respiratory problems in patients with neuromuscular and musculoskeletal diseases such as Amyotrophic lateral sclerosis (Lou Gehrig's disease), spinal cord injury requiring respiratory support, and scoliosis.\nOur facility includes a six-bed sleep laboratory for overnight sleep studies. The laboratory features private bedrooms and bathrooms in a comfortable and spacious testing center that includes equipment to monitor brain activity, cardiac activity, breathing pause, snoring, oxygenation, chest and abdominal activity, and limb movements. Patients are monitored throughout the night by technical staff experienced in sleep medicine.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":", we all selects the very best collections using best image resolution only for you, and this photos is among photographs collections in our very best photographs gallery concerning Inspirational Photos Of Small Restaurant Kitchen Layout. Lets hope you might as it.\nsubmitted by simply Kyle Alvarez on 2018-09-17 16:03:29. To find out most photographs inside Inspirational Photos Of Small Restaurant Kitchen Layout photographs gallery you should adhere to this kind of web page link.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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index 0000000000000000000000000000000000000000..37ab71ff95c22144c66a4d5a725555d0d5a3279b
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Elysia is created in a laboratory, born as a sixteen year old girl, an empty vessel with no life experience to draw from She is a Beta, an experimental model of a teenage clone She was replicated from another teenage girl, who had to die in order for Elysia to exist Elysia s purpose is to serve the inhabitants of Demesne, an island paradise for the wealthiest people oElysia is created in a laboratory, born as a sixteen year old girl, an empty vessel with no life experience to draw from She is a Beta, an experimental model of a teenage clone She was replicated from another teenage girl, who had to die in order for Elysia to exist Elysia s purpose is to serve the inhabitants of Demesne, an island paradise for the wealthiest people on earth Everything about Demesne is bioengineered for perfection Even the air induces a strange, euphoric high, which only the island s workers soulless clones like Elysia are immune to At first, Elysia s life is idyllic and pampered But she soon sees that Demesne s human residents, who should want for nothing, yearn But for what, exactly She also comes to realize that beneath the island s flawless exterior, there is an under current of discontent among Demesne s worker clones She knows she is soulless and cannot feel and should not care so why are overpowering sensations cloud ing Elysia s mind If anyone discovers that Elysia isn t the unfeeling clone she must pretend to be, she will suffer a fate too terrible to imagine When her one chance at happi ness is ripped away with breathtaking cruelty, emotions she s always had but never understood are unleashed As rage, terror, and desire threaten to overwhelm her, Elysia must find the will to survive The first in a dazzlingly original science fiction series from best selling author Rachel Cohn, Beta is a haunting, unforgettable story of courage and love in a cor rupted world.\nI would LOVE to read this on NetGalley . . . . but apparently, Disney-Hyperion \"isn't accepting blogger requests at this time.\"It's okay, Rachel. I'll own your book . . . eventually.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Duo organises both lectures and online lectures (some of which are recorded and can be watched later).\nWindows to Intercultural Life is Duo's continuous lecture series. The series has covered a wide range of themes related to an intercultural family, parenthood and family life and the lectures have included researchers, professionals and peer-to-peer experts.\nThe lectures - both in English and in Finnish - take place on weekday evenings in Kotola (Yrj\u00f6nkatu 29 A, 4th floor, Helsinki). All lectures are free of charge and preregistration is not required.\nDuo's online lectures are an easy way to get reliable information as well as to put questions anonymously to experts and professionals. The lectures are free of charge and open to everybody. Some of the lectures are recorded and can be watched later.\nA list of recorded Duo online lectures can be found below. To watch a a lecture, please click on the video link (*Watch the lecture*). The lecture video opens on Perheaikaa.fi website.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The bulk of this stir fry comes from the fish balls, the flavour from the mushrooms.\n1. Chop the garlic and chillies together, fry in oil for a few seconds to release the garlic smell.\n2. Slice the fish balls, add to the stir fry and fry for a few seconds, add all the remaining ingredients and fry for 2 minutes over a medium heat.\nThis page contains a single entry from the blog posted on November 9, 2006 6:04 PM.\nThe previous post in this blog was Glass Noodle Pad Thai.\nThe next post in this blog is Fried Chicken with Chilli & Basil ( Pad Kaprow Guy ).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Buffalo Technology launched the AirStation High Power Giga Wireless- N Router & Access Point,WZR-HP-G450H in India. The powerful,wireless LAN router combined with high power three 5dBi antennas and Buffalo's high performance technology offers an easy and hassle-free install. It's priced at Rs.8, 000 and is backed by a 2-year limited warranty.\nThis high power wireless LAN router and access point supports the 802.11 'n' standards for rapid data transfers of up to 450 Mbps. The router is equipped with a USB port that lets you connect a USB hard drive and you can use it as a NAS for sharing data. Also,it comes with Buffalo's original easy-to-use firmware.\nRemovable and Adjustable Antenna: Eliminate dead spots by fine-tuning your antenna positioning.\nA listed building in the United Kingdom is a building that has been placed on the Statutory List of Buildings of Special Architectural or Historic Interest. It is a widely used status, applied to around half a million buildings.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In last month 17th December 2018, Microsoft office add a new feature in Microsoft office 365 that enable users to reach more audiences, increase efficiency, and focus on most important tasks.\nAccording to Microsoft what new feature user will get?\nNew AI-powered features in Microsoft 365 empower you to deliver presentations that are accessible to broader, more diverse audiences and to be more efficient in your flow of work.\nPresent more inclusively with live captions & subtitles in Microsoft PowerPoint\u2014A new feature, powered by Microsoft AI, enables presenters to reach and engage all audiences with live presentation captions and subtitles that automatically appear in real-time. With live captions & subtitles in PowerPoint, you can ensure your presentations are understood by everyone, across languages and hearing access needs.\nThis feature will support presenters across 12 spoken languages and display on-screen captions or subtitles in one of 60+ languages. Starting in January 2019, this feature will start rolling out to Office 365 subscribers worldwide for PowerPoint on Windows 10, PowerPoint for Mac, and PowerPoint Online.\nGet tricky acronym definitions automatically in Microsoft Word\u2014Using the Microsoft Graph, Microsoft AI makes suggestions for acronym definitions specific to you and your organization while you read documents.\nThis makes it easier for you to read documents with clarity in the flow of your work. To get started, click the Acronyms button on the References tab to launch the Acronyms pane and see suggested definitions for acronyms.\nKeep in touch for such updates regarding microsoft products.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Back in August 2012 the big Myki story was the flaw in receipt printing that risked users credit card security \u2013 it got a run in The Age and I posted about how the security hole put users at risk of further identity theft. Fast forward to today, and it appears that the Transport Ticketing Authority has finally done something about the issue.\nSo after a month long hiatus, I'm back from Europe. In that time I passed through six countries, ten cities, three rail gauges and 15 rail operators \u2013 but how many photos did I take?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Researchers believe the group may offer genetic resistance to a deadly disease that has nearly wiped out mainland populations.\nMay 24, 2016, 12:30 p.m.\nThe Tasmanian devil, brought to the brink of extinction in recent years due to a fatal, highly contagious disease, may yet recover thanks to the discovery of an isolated population in the island state's remote southwest.\nWhile the researchers didn't actually lay eyes on the animals, they did discover quantities of feces in a heavily forested World Heritage area. When analyzed, the stool revealed nine completely new genetic variants for the devil, a boon for conservationists attempting to help captive populations improve their chances for survival.\n\"For us this is massive,\" Sydney University geneticist Kathy Belov told the Sydney Morning Herald. \"For years we have been calling devils clones because there's so little diversity, and now we find that there is diversity out there, it's just in remote areas.\"\nThe discovery is especially timely due to the threat of devil facial tumor disease, an aggressive cancer that has decimated about 85 percent of devils since 1996. While there's recently been some good news regarding the discovery of a naturally occurring antibody that may offer some protection, the greatest promise could come from isolated, genetically-diverse populations of devils.\n\"The more diversity a species has, the more resilient they are and the more able they are to respond to changing environments, be it to climate change or new diseases,\" added Belov.\nThe researchers will attempt to capture some members and confirm if they're suffering from the same outbreak of disease. Regardless, the researchers plan to introduce the devils' unique genetic variants to both captive and wild populations to boost the resiliency of the species as a whole.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"ZTE MF282, also named 3HUITube, is a new 4G LTE WiFi router, which is in special elegant design like a Cube. The 3HUITube has two ports for Gbit LAN and telephone.But the ZTE MF282 4G Router doesn't support LTE Cat.6. The 3HuiTube could reach peak download speed up to 150mbps and upload speed to 50Mbps.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Winter can be a trying time for any vehicle, old or new. The extremely cold and icy weather takes its toll on all parts of the car which means maintenance is more important than ever during the winter. While there are some simply checks motorists can carry out themselves, the best preventative action is a dedicated servicing. Here are reasons to book a winter service.\nIf you haven't yet found time to book a winter service, here are four of many reasons why you should. There are more breakdowns in winter than any other time of year \u2013 don't be statistic and make sure your car is prepared for whatever the weather brings.\nKeith Miller, a former AA Patrolman says that up to half of problems caused by the cold can be prevented by regular servicing and maintenance checks. With this in mind, why wouldn't you stop future issues occurring if you could? Having a professional mechanic service the car will keep it healthy for longer, and they'll also spot any problems in advance. In the long run, having a service could save you money for costly repairs.\nIn the dark months it is essential that you are visible at all times in any weather condition. Your lights are vital for this reason, so if a bulb is out or there is an electrical fault it needs fixing fast. Any service will include a check of all lights on the vehicle.\nWinter brings many difficult driving conditions, including icy and frosty roads, slush and snow. Your vehicle's brakes have to be on top form to deal with these dangerous conditions which is why it's a good idea to get them checked and cleaned (included in a winter service).\nThe better you look after your car, the longer it will look after you! Getting your vehicle regularly serviced can improve its life span and ensure you can enjoy many years of hassle-free driving. If you get your car serviced once a year, it's a good idea to do it at winter to make sure it is ready for the tough months ahead.\nNot only will a service save you money and keep you safe in your vehicle throughout winter, but it could also maintain its value. Buyers are more interested in cars with a full service history because they know these vehicles are more reliable.\nReady to book a winter service? Just give us a call.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Gratzi is an interior design style quiz by the Novogratz family. Bob and Courtney Novogratz are a popular high end interior design power couple as well as parents of 7 kids. Their mobile site helps you first define your style, then it pairs you with an appropriate interior designer to help you realize the vision for your home. We worked closely with the family to brand this unique experience in a way that was complimentary to their current overall Novogratz branding.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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@@ -0,0 +1,5 @@
+{"text":"Stage FX is the number one stockist of Stage FX Supplies in the country, including Confetti Cannons, Confetti Systems and Confetti Consumables, Holi Powder, Neon Body Paint (Glow Party \/ Neon Party), CO2 FX (Cryo Guns\/Jets\/FX), Snow\/Fog\/Foam\/Bubble\/Fog\/Haze Machines, and much much more!\nOur international supplier base means we have the best quality Stage FX products from across the globe, available when you need them.\nSchool fund raising. With years of experience we can help you create the best fundraising day for your school, dealing directly with the supplier you will save 1000's of dollars, and make even more.\nWhether it's for a wedding, party, corporate event, awards night or school celebration, your guests will love it when your Stage FX equipment fills the air in a spectacular fashion.\nAlways in stock, always value for money.\nWhy buy from us? Stage FX is a direct importer with an extensive special effects range - we always have plenty of stock and we pass the savings on to you! In fact, we're up to 87% cheaper than our competitors on some products!\nBest prices on special effects equipment (sale or hire) - You won't find a better price. As manufacturers of our own brand, Stage FX offers top quality special effect equipment which can be refilled and reused time and again. Better still \u2013 you don't need a license to purchase and operate them! So enquire today; you could be producing professional confetti effects tomorrow!\nMassive range of Stage FX consumables. Confetti, Streamers, Bubbles, Snow, Holi Powder, UV Glow Body Paint and much more!\n\/\/ Stage FX \/\/ Make Your Show Stand Out!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"NBA player Frank Mason is on the verge of legally getting his chain snatched.\nAccording to TMZ, Lil Wayne's Young Money APAA Sports Group wants Mason to return an $11,000 diamond chain the athlete was given after signing a deal with the agency over a year ago.\nMason might be legally required to do so since his contract reportedly states that if he ever left the agency, he would have to give back his jewelry and repay money received.\nIn addition to the diamond chain, Mason was also given a $40,000 advance and $52,000 for his pre-draft expenses. The Young Money sports group has asked a judge to order Mason to hand over the diamond chain, $92,000 and damages.\nMason, who signed with Young Money before the NBA Draft in April 2017, fired the sports management group in April 2018.\nHe currently plays for the Sacramento Kings and reportedly made $1.3 million last season.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We took a look at InnoGames' browser based free to play MMO The West, stepping into the boots of one mean hombre. In The West players take control of their very own desperado for hire in the old west, helping on farms and running down banditos.\nIn this First Look we check out how the work system functions and how players can complete quests and earn some cash, as well as how the duel system and combat works. As a treat you can even watch us play an 3v3 PVP \"Adventure\" game where a max level 150 player proceeds to call us \"noob\" for the entirety of a game! Check the video out, enjoy, and leave your comments below!\nThis entry was posted on Thursday, October 5th, 2017 at 12:52 pm and is filed under Articles, MMO News. You can follow any responses to this entry through the RSS 2.0 feed. Puedes dejar una respuesta, o un trackback desde tu sitio web.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A shopper (above) gets caught in the rain Tuesday at the Ojai Valley Shopping Center as showers swept across the area. Ventura County Firefighter Joe Garlock (below left) and Capt. Tod Berryman help fill sandbags for Ojai Valley residents Monday at Station 21 in advance of the rain.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In March I planted a peach tree that had fruit on it. We harvested about 5 peaches and now would like to prune for shape. Could I prune it now or should I wait until early next spring? I live in central Florida and am afraid of disease if I prune now.\nPruning should be done during dormancy.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbdadt b/data_all_eng_slimpj/shuffled/split2/finalzzzbdadt
new file mode 100644
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+{"text":"From 1970 through 2010, mortgage debt outstanding grew an average of 9 percent per year. Despite this rapid rise in available credit the U.S. home ownership rate is only about 5 percent higher than in the 1960s. Even after the financial crisis, Fannie and Freddie make the market. These and other federal agencies account for 90 percent of all new mortgage loans. This could all soon change.\nThe U.S. Treasury Department announced that the Federal Housing Finance Agency, which regulates Fannie Mae and Freddie Mac, will begin winding down their role in the mortgage market. Fees charged for guaranteeing mortgages that banks sell to investors will rise, banks will have to hold more capital to get the guarantee, and a higher down-payment will be required from the borrower.\nThese changes will decrease the supply of mortgage loans. And a proposal to lower the U.S. federal budget deficit may decrease demand.\nThe White House Deficit Commission, of which Idaho's Sen. Mike Crapo was a member, called for a limit on the tax deduction for mortgage interest. Even if adopted for only higher income households the lower overall demand will reduce the quantity of mortgage loans.\nCrapo disagrees with this proposal and wants to keep the mortgage interest deduction. Additionally, he says \"modifying or eliminating the mortgage interest deduction is not necessary in order to bring our budget into balance.\"\nWhile this is true, the interest deduction is a subsidy, increasing demand in the housing market above what it would be without government intervention. Higher homeownership has long been the policy goal of both this tax incentive and the excess mortgage loan supply provided by Fannie and Freddie.\nHowever, there is little evidence that homeownership contributes to economic growth, and may in fact increase volatility. Countries with higher homeownership rates, like Italy and Spain, face mounting economic problems while low ownership rate countries, like Germany and Switzerland, were hit less hard and are growing strong.\nThe American dream of homeownership may not do us all that much good, but the policy is likely to stay with us even as we say goodbye to Fannie and Freddie.\nPeter R. Crabb is a professor of finance and economics at Northwest Nazarene University in Nampa. His column, published weekly at IdahoStatesman.com, appears every other week in the Business Insider.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"So goes the Wendell Berry quote that often slaps me in the face when I travel internationally (rarely these days). I recently returned from a week on the Caribbean side of Costa Rica. The road from San Jos\u00e9 to the port of Lim\u00f3n is practically ground zero for the banana monopoly created by the United Fruit Company in the late 19th century, finally morphing into Chiquita Brands International Incorporated in 1989. The history of the cultivated banana and how it became the world's fourth major staple after rice, wheat and milk reads like a veritable soap opera; murder, suicide, labor strikes, bribery, corruption, violent coup d'\u00e9tat's and an all-out banana war between the US and the European Union in the late 1990's, making for a seemingly unbelievable story. Sadly, the same telenovela seems to constantly replay itself in the interest of vast profits for government-bought corporations and death and destruction for the earth and its' indigenous and exploited masses.\nTravels in other banana republics including Guatemala and Panama had already opened my eyes to the grand scale environmental destruction and human rights abuses multinational companies like United Fruit have wrought (and continue to wreak) on the planet, local indigenous populations, and food economies. While living with a Guatemalan family and studying Spanish in the small town of Flores, I was shocked to be drinking Nescaf\u00e9 instant coffee (along with the babies) in a country with some of the best coffee in the world; alas it is unaffordable for the local population. In Puerto Rico a few years back I was eager for fresh fruit and vegetables after too much easily accessible fried foods. An innkeeper handed me an avocado imported from the Dominican Republic. More than 85% of what Puerto Ricans eat is imported, although they are currently working on their minimal agricultural sector to increase healthy food production. Puerto Ricans have staggering rates of diabetes at 1 in 10 of the population.\nThe road from San Jos\u00e9 to Lim\u00f3n (a four hour drive) is literally lined with banana plantations and shipping containers. Each bunch of bananas on each tree is covered in a blue plastic bag. My local hostess\/friend, Molly, said these bags are full of chemicals to protect them from disease and pests. Once you get to the port of Lim\u00f3n, ships loaded with refrigerated bananas are constantly departing to their various destinations around the world. I contemplated this one morning as I sipped local coffee and ate a local banana (a rare occurrence as I live in temperate Maryland) purchased from an organic farmer at the farmer's market in Puerto Viejo. This banana was small, bruised and insanely sweet and delicious (I have to admit bananas are not at the top of my list when it comes to fruit choice.) Similarly I despised carrots for much of my life until a farming friend presented one to me fresh from the earth; again, sweet deliciousness, like no carrot I'd ever eaten before.\nBy contrast, since demilitarizing in 1948, Costa Rica has become one of the more \"safe\" and \"stable\" countries in Central America, attracting loads of US investment and ex-pats. I put the words \"safe\" and \"stable\" in quotes because I feel many privileged US-ers have skewed notions of comfort and safety and thus choose to visit the Global South from the perceived safety of a cruise or a resort rather than interact with local populations. I had always heard that Costa Rica was on the cutting edge of sustainability and from my travels in other less Americanized, less expensive Central American countries, I know there is a fine line between sustainability and poverty. Many folks are forced to be sustainable because they lack resources and fossil fuel based energy sources to be anything but. Close to a quarter of the population lives in poverty, almost double the poverty rate of the US. Costa Rica fairs well internationally in sustainability measures and plans to become the 1st carbon neutral nation by 2021. It is the Global North that extracts Earth's resources and exploits the Global South's land and labor so that no market or desire is left unrealized. The low-income people of color of the Global South are the innocent bystanders of globalization and neo-liberal policies like NAFTA and US-backed \"aid\" who will continue to be disproportionately affected by climate change, food shortages, fluctuations in cost of food and energy commodities, and destruction of both the environment and indigenous sovereignty. Indeed there are many agricultural demonstration projects, education centers and non-profits doing their thing, but there is much more governments can do, not to mention entities like the World Trade Organization, to protect local land and people.\nI witnessed a small slice of some great work being done in Costa Rica at Centro Ash\u00e9 in Manzanillo on the south Caribbean. Director, Molly Meehan Brown, works with the local population to promote their work around ethnobotony, medicinal herbal education and application, healthy cooking, and eco-tourism that puts tourist dollars directly into the pockets of local and indigenous people. With centers in both Southern Maryland and Costa Rica, Centro Ash\u00e9 is rooted in community and dedicated to keeping classes affordable & accessible in order to keep the knowledge of food, herbal medicine, seeds, and healing traditions alive and vital. Centro Ash\u00e9 programs act as catalysts to build community, land-based and traditional knowledge. They celebrate the richness and diversity of folk herbalism across cultures while providing supportive and practical knowledge. Their teachers are all local herbalists, farmers, and plant people. I had the opportunity to sit in on a plant talk with a local Afro-Caribbean medicine woman and visiting Plant and Healers International, who connect people, plants and healers around the world (I'm excited to start their online botany class, a donation-based course within my price range in a time when so many courses I'd like to take are simply out of my price range.) It was cool to watch the group bouncing ideas off each other, learning different names for the same plants, critically analyzing western ideas about the safety of plant use, identifying an unfamiliar edible fruit tree in the middle of town (Screw Pine!) and building international and local place-based resilience. We have the skills, resources and creativity to make our shared lives truly amazing, diverse (both bio and beautifully human), just, healthy and delicious.\nUpon our return we went to retrieve my car at the house it was parked at. On the kitchen counter was a bunch of bananas with a Chiquita sticker that read, \"Costa Rica.\" My partner and I looked dubiously at each other. As consumers we can choose to support equal exchange organic farmers, when it is available. We make sacrifices in our fixed income life so that we can feel good about what we put into our bodies. We don't always get it right (near impossible to do so), but it feels right to try, for our health, the planet's health, workers' health, and because we know our choices aren't just personal choices; they affect others. And what incredible collective power we DO have to shape the way the world is used by making the choices and changes we need to create a more just and environmentally friendly food system with a heavy local flavor! A delicious revolution indeed.\nFor more check out this article by Phyllis Robinson about the true cost of bananas.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In 2018 the Regional Forum for Capacity Building in Agriculture (RUFORUM) implemented the Community-Based Field Attachment Award (CFAPA) program, which was supported by the Food and Agriculture Organization of the United Nations (FAO) for promoting youth support for the Sustainable Development Goals (SDGs) and in particular, SDG2 \u2013 End hunger, achieve food security and improved nutrition and promote sustainable agriculture.\nProvide the student with practical skills to apply research findings in development-related field projects, engage with local communities and generate innovative solutions to improve rural livelihoods.\nIn 2018 after competitive selection process, 10 students from RUFORUM member universities received grants to implement their field projects in rural communities in Benin, Uganda, Kenya and Lesotho. For 6 months, students interacted with local farmers to exchange knowledge and develop innovative solutions for enhancing nutrition and food security, improving sustainable agriculture and livestock production practices, strengthening collaborations along local food-value chains for rural income generation and other themes.\nTo disseminate their research results and promote innovative approaches, students targeted a wide range of stakeholders from rural communities, including farmers, agricultural traders, farmer associations, community health institutions, veterinary and extension services and rural community leaders, with a particular emphasis on women, youth and small-scale farmers. Students organized interactive workshops and trainings, presentations and radio shows to increase the outreach. Students also produced short videos to document their field work.\nAryemo Irene Penninah, a student of Gulu University in Uganda, wrote her Master thesis on Commercialization and profitability of indigenous chicken production, and she was awarded a grant to disseminate her research findings and train local chicken innovation platform members in Omoro district. During her university research she learned about several issues which have had a negative impact on the commercialization of chicken production in the area, including poor management of chickens that lead to high mortality rates, lack of understanding amongst farmers how food value chains function and limited relations between different actors along local agri-food value chains.\nTo increase commercialization and profitability of chicken production, Aryemo trained local farmers on strategies to increase flock size and enhance chicken management practices, explained about production and marketing activities to increase farmers' incomes and organized activities to inform rural community how to keep the innovation platform alive through facilitation of business to business linkages amongst different actors along the chicken production value chain.\nNicholus Mbabazi, a student of the University of the Free State, Uganda, received an award to disseminate his research findings on sustainable multiplication of improved common bean seed for small-holder farmers.\nThe field attachment focused on adoption and multiplication of improved beans aimed at climate change adaptation to capitalize on the high farmers' perception and awareness levels about climate change and stimulate appropriate climate change adaptation strategies. Through trainings and workshops, student provided climate information to farmers to inform timely planting, encouraged adoption of improved and drought tolerant seed varieties and promoted soil conservation practices like the use of fertilizers that are effective at increasing resilience of the smallholder farming systems. He also presented sustainable strategies including farmers' group formation, which can ease access to credit, to encourage smallholder farmers to adapt to climate change and climate variability.\nThrough trainings, the student sensitized around 300 smallholder farmers on bean sustainable production practices and also did two radio presentations in local language on sustainable bean production to enhance outreach.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Midway Middle School is a public high school of the Midway Independent School District located in Hewitt, TX. It has 1,186 students in grades 7th through 8th. It has a student teacher ratio of 16.7 to 1.\nYou can also use the form below to search for alumni info and Midway Middle School students.\nThe percentage of Midway Middle School students on free and reduced lunch assistance (30.5%) is lower than the state average of 53.3%. This may indicate that the area has a lower level of poverty than the state average.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Today... Friday the 16th of March, I went to a Brad Camp speaker course, run by @bradburton (the UK's #1 motivational speaker).\nI didn't really know what to expect and I was nervous in a room of business men and women. Everyone that attended gave a little speech about who they are... when it was my turn I just couldn't get the words out, I had planned the talk, but the nerves got to me.\nThroughout the day, Brad broke down the stages of being a professional motivational speaker. Mannerisms, money, what to include and not include in a talk... but most importantly he made every single person in the room much more confident about themselves and speaking.\nIn just a few hours I went from a nervous wreck in front of 10 business people, to a having a meaningful story and feeling much more confident when speaking in front of a crowd.\nWe've still got a lot of work to do but brad installed a huge amount of self-confidence in me. Already putting what i've learnt into place! It's been an incredible experience, hopefully, I'll be able to get to another Brad Camp soon.\nIf you get the chance to attend one, and throw yourself into the deep end... do it! Life is about experiences!\nThanks for an awesome day Brad!","meta":{"redpajama_set_name":"RedPajamaC4"}}
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@@ -0,0 +1,5 @@
+{"text":"Play the classic Solitaire Games! At solitaireclassics.com you can enjoy the pleassure of the most popular Classic Solitaire Games. All the Solitaire Classics are absolutely free, and no registration is needed. Just start playing and have fun!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"24\/6 Maternity Nurse San Francisco and London. Due Date July 2019.\nLocation of the job: USA-San Francisco\/LA and London, Notting Hill.\nTraining and Qualifications needed for this job: Maternity qualification essential.\nPay: \u00a3270-\u00a3350 per 24 dependant on experience.\nAllie Bell T\/A Harmony at Home Maternity acting as an Employment Agency.\nA loving and relaxed family in Wanstead London, are looking for a Mother's Help four days a week. They are looking for an extra pair of hands to help around the house and with their 11-month-old child, Monday to Thursday, 9am-1pm. The family are very flexible with both hours and days of the week. They live in the heart of Wanstead, with local shops and directly opposite a beautiful park, and they are looking for a nanny that is active and happy to assist them when out in the local area. The family are keen on their child playing independently so would like a Nanny that is happy to assist where necessary.\nThe mother loves to cook from fresh and is looking for a Nanny to help with their child whilst mother cooks and assist where necessary. As well as being organised and thinking ahead. Mother will always be at home and wishes to share roles with the right candidate, one that is happy to hop between duties of childcare and tidying.\nA wonderful family based in Islington are seeking a nanny for a summer start or early September 2019. The family have 2 wonderful children, girl 4 years and boy 4 months. The family are looking for someone who has experience with babies, who is fun, creative, pro- active, and enjoys the outdoors. They should also have a good level of spoken English. The main part of the day will be spent caring for the baby boy as the little girl attends school. The School is a short bus ride away and will involve also taking the baby with the buggy. The family home also has lots of steps so please note when applying that a nanny with lots of energy and who is psychically active would be ideal!\nChildcare duties will include, school runs, preparing meals for the children, school drop offs and pick- ups, morning and evening routines, taking to after school activities, planning and implementing age appropriate activities, attending play dates and baby groups. The little girl loves baking and helping to cook and both children love a good story book! As the family employ other domestic staff light housekeeping duties are around the children and will include \u2013 preparing meals, keeping kitchen tidy, some laundry and other light domestic nursery duties in order to keep things tidy.\nDemetra Paphitis Limited T\/A Harmony at Home North London acting as an Employment Agency.\nTo apply for this job please contact us with your CV ASAP via email to: demi@harmonyathome.co.uk quoting the job reference code: DP42650 thank you.\nWhy not start the New Year working for this lovely family who require a full-time Nanny to look after their Daughter aged 6 months.\nThe position is based in London (SE3) and the are looking for an experienced Nanny to work 5 days per week Monday \u2013 Friday 9am \u2013 5.30pm. Salary dependent on experience.\nThe successful applicant will carry out all aspects of childcare and light household duties. Sole care position \u2013 parent works from home.\nThe position is temporary due to start date on 18\/02\/2019 until 30\/08\/2019.\nThe family live near Blackheath Park and various others, therefore, you will be required to take their daughter out for walks locally.\nThe right applicant will be able to think on their feet, have a natural approach and professional at all times.\nA wonderful opportunity has arisen to work for this lovely family based in Greenwich. They require a Nanny Housekeeper to start as soon as possible.\nThe role will include all aspects of childcare for their son aged 1 year. You will also be responsible for the house keeping and the specific duties will be agreed in advance.\nThe family would ideally like a Nanny who speaks Italian but this isn't essential. The successful Nanny will be pro-active, professional with a can-do approach.\nDays \u2013 3 mornings per week moving to 5 mornings per week after 1\/2 months. The family can be flexible with regard to the 3 mornings required.\nThis lovely family based in Sydenham are looking for a Nanny to start at the beginning of February on a permanent basis.\nThe role will involve all aspects of childcare for their daughter aged 11 months who was premature and currently has a feeding tube. However, she is able to eat orally and the Nanny would need to help and encourage feeding via mouth.\nThe successful applicant will be caring, professional with a calm approach.\nDays required \u2013 3 days Monday, Wednesday and Friday.\nNew Year, New Job! Start 2019 with this lovely family based in London working as a Full-time Nanny to start in February.\nThe role will involve looking after their son aged 5 months. The family require a qualified Nanny and the successful applicant will be professional, approachable and caring.\nDuties include all aspects of childcare and light household duties. The family enjoy travel and their home is very clean and tidy.\nHours\/days required are 8am \u2013 7pm 5 days per week Monday \u2013 Friday.\nSalary \u2013 \u00a311.00 gross per hour.\nFull-time Nanny, Wapping \u2013 London.\nDecember 20, 2018 | Comments Off on Full-time Nanny, Wapping \u2013 London.\nThis lovely family based in Wapping \u2013 London are looking for a Full-time Nanny to start in April 2019.\nThe role will involve looking after their son who will be one in April 2019. The family require a qualified Nanny to bridge the gap between the end of Mothers maternity leave and the start of nursery. As a result, they are looking for a fixed term contract of 6 \u2013 12 months. This can be agreed in advance.\nThe successful applicant will have energy, enthusiasm and a genuine desire to assist in their sons overall development. The position is largely sole charge, but Grandparents are nearby and often briefly visit in afternoon. The family would consider funding professional development for the right Nanny if required.\nDuties include all aspects of childcare and the right person would need to be prepared to keep the children's rooms tidy, help them to put away their laundry and keep the playroom tidy. As well as this, meal preparation for the child. Additional Household help employed in the home.\nThe family are organised and tidy and would ideally be looking for a nanny with a similar character. They also enjoy reading, keeping active, travelling and spending time together as a family. During the day their son likes to be kept entertained and requires interaction. They have a wide variety of play and reading materials. You would be required to take him out for walks during the day. The parents currently use the following languages \u2013 English, French and Vietnamese which cover all backgrounds. They do not allow any screen time for their son (TV, phones, tablets) and would require the Nanny to respect this.\nThe family home is based in Wapping near St Katherine's Docks, which despite its relatively central location is a quiet family-based area. There are several community-based child activities nearby which the son attends, a good park (Wapping gardens) and the potential for nice walks along the river or the canals away from polluted roads.\nHours\/days required are 8am \u2013 6pm 5 days per week Monday \u2013 Friday.\nSalary \u2013 \u00a313.50 gross per hour.\nThis is a great opportunity for a nanny looking for a sole charge role. Lovely fun family looking for a nanny to take care of their two children after school, 2 girls aged 7 and 10. This will be a sole care position. They are seeking an active nanny who is, kind, energetic and dynamic and great organisational skills. The role is for roughly 6 months as the family are moving out of London. The role is based in Tufnell Park near many local amenities and transport links.\nDuties will involve \u2013 picking the children up from school. Taking them to after school activities, planning age appropriate activities and play dates, assisting with homework and light household duties around the children as they employ other household staff. Some afternoons will be busy with after school activities.\nAs the children are at school 9-3.30 the hours during the day will be focused on helping the family organise the house, shopping, preparing meals (essential), making sure the home is organised and happy to offer suggestions and PA duties.\nTo apply for this job please contact us with your CV ASAP via email to: demi@harmonyathome.co.uk quoting the job reference code: DP022 thank you.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Needing a bit of motivation today\u2026yep, this will do nicely! Those horns are killer!\nI sometimes feel sad that I don't have the time to research all of these fabulous artists properly. I am guessing there is a great story behind this group. I know that Tony Pab\u00f3n was born in Puerto Rica and moved to New York as a small child. I know that the great Willie Col\u00f3n was in the group at some point. If sharing one track by them today makes someone go and investigate their back catalogue, then that will make me very happy!\nNow, if you will excuse me, I have to go and dance! Well, actually, I have to go and dry my hair so I am not late for work, but I shall be dancing while I do that!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Buy Bendtsen Smoothness Tester, Testing Principle: Rota Meter Airflow, Parameter Testing: smoothness and porosity of paper and board.\nOur company is bracketed with the leading Importers and Suppliers of Bendtsen Smoothness Tester from Uttar Pradesh. Sourced from the famed manufacturers of the industry, Bendtsen Smoothness Tester is well acknowledged for high quality, efficiency, reliability and durability. This tester is precisely designed for determining the smoothness and porosity of paper and board. It is based on rotameter airflow principle. Our Bendtsen Smoothness Tester is provided with three imported variable area flow meters (rotameter) and accessories measuring range 5-150ml\/min. 50-500ml\/min. 300-3000ml\/min.\nWe are pleased to introduce ourselves as authorized supplier for standard paper testing waxes from USA make WEIDMANN(FORMERLY DENNISON ) , and also pulp and paper testing equipment . We directly import the paper testing sticks, aluminum foil from USA and keep the stock available in Saharanpur in ready shape to deliver the requirement on their request Imported Paper Testing Wax-Sticks, All Sizes are available in Stock NUMBERS :- 2A,3A,4A,5A,6A,7A,8A,9A,10A,11A,12A,13A,14A,16A,18A,20A & 23A Cross Section Area:-18x18 mm, Hard Resin Non-Oily Type we are hereby attaching the technical literature for imported paper testing wax sticks from USA and some literature related with paper testing equipment .pl also inform about your requirement to provide our offer for the same . The company profiles and vendorship details are attached herewith for your kind perusal.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Michael's On East has been recognized in Florida Trend's December 2014 issue and on FloridaTrend.com as a member of the Golden Spoon Hall of Fame, chosen amongst thousands of restaurants in Florida.\nFlorida Trend's Golden Spoons are the oldest and most coveted dining awards in the state. The magazine started the awards 40 years ago as a guide to executives traveling around the state. Michael's On East is the only Sarasota restaurant recognized as a \"Hall of Fame\" member in this year's announcement.\n\"While we are tremendously proud to be recognized alongside a number of additional independent restaurants from our area again this year, I must say the Michael's On East team is extremely privileged to be honored in the Golden Spoon Hall of Fame,\" said Michael Klauber, co-proprietor of Michael's On East.\nMichael's On East has enjoyed a long-standing Golden Spoon status, with 2014 marking the 23rd year the restaurant has been recognized as part of Florida Trend's Golden Spoon Awards. Restaurants that have received Golden Spoon Awards year after year are considered for the Hall of Fame, the highest honor Florida Trend can bestow on a Florida restaurant.\nMichael's On East was founded on April 27, 1987. Located at 1212 East Avenue South in Midtown Plaza, the iconic restaurant is open for lunch 11:30 a.m. to 2 p.m. Monday through Friday, open for dinner 6 p.m. to close Monday through Thursday and 5:30 p.m. to close on Friday and Saturday. The restaurant is closed on Sundays, with the exception of Easter and Mother's Day each year. The establishment is the winner of the 2009 Fine Dining Hall of Fame, and received their 25th consecutive AAA Four-Diamond Award in 2014. Michael's On East is a contemporary fine dining restaurant featuring an extraordinary supper club atmosphere, inspired cuisine and an intimate piano bar. Michael's On East Catering is Sarasota's premiere caterer creating memorable events from candlelit dinners for two to dazzling galas for 1,000. Michael's Wine Cellar hosts more than 1,000 International estate-bottled wines including more than 200 domaine wines under $20, boutique spirits and craft beers. Michael's Wine Cellar is open Monday through Saturday, 10 a.m. to 6 p.m. For more information, call 941-366-0007 or visit www.bestfood.com.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbdroc b/data_all_eng_slimpj/shuffled/split2/finalzzzbdroc
new file mode 100644
index 0000000000000000000000000000000000000000..e0754eb5c9b1a97a3b213481e5d1875aca17ad21
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbdroc
@@ -0,0 +1,5 @@
+{"text":"Just for some clarification: Normally with a serial communication set up, the Tx signal of one device, is connected to the Rd pin of the other device. In this circuit, that is taken care of on the SubD connector: Pin 3 is the Tx signal, that goes to the RdIn (pin 8) of the chip. It is converted in voltage in the chip, then goes to RdOut on pin 9. That pin \u2013though called Rd- then goes to the Rd pin on the Arduino\/Atmega\/Pic. The connection between Tx and Rd already has been made, no need to switch that again.\nSame for the Tx signal of the microprocessor: It attaches to pin10 on the Max232, is converted and appears on pin 7 of the chip and then is brought to the Rd pin on the SubD connector. The DB9 plug in th epicture is a female plug looked at from the solder side.\nIf for any reason one would prefer to use an old SubD25 serial connector, the table below shows the corresponding pins between the two. To make things a bit confusing both pin 2 and 3 have exactly opposite functions on these connectors.\nThe LED's and resistors R3 and R4 are not really necessary, they just give a visual indication of the transmitting signals, choose a nice color.\nOnce the circuit is build, then first test it before you attach it to your Arduino. The easiest way to test it is to temporarily connect the Tx and Rd pin (so the ends of R1 and R2) with eachother. Apply 5 volts to the chip and connect your serial cable.\nOpen up the Arduino IDE, select the proper COM port under 'Tools', open up the serial monitor and type in some characters and click 'Send'. The characters should be echo-ed back.\nFor me this worked with all baud-rates.\nThen disconnect the temporary connection you just made and connect Tx and Rd to the corresponding pins on your board\/chip.\nIf that has the bootloader installed, you should be able to upload a program to your board with this serial connection.\nNote: Many 'modern' computers that still have a serial connector do in fact already not work with the high voltages that we know from true RS232 and then the MAX232 might not be needed, but one could use a levelshifter as e.g. Sparkfun is selling ($6.95) and try to build that yourself. It is not a big circuit, but in my idea, usung the MAX232 is just easier and the RS232Shifter board is actually a bit of a hack and does not seem as reliable.\nI had to remove the R1 and R2 because they seem to currupt my signal.\nSo now it works like a charm.\nWill this connection work if I use digital pins on Arduino as TX and RX pins instead of the normal TX and RX pins?\nThat indeed is a bit amazing as I have built 2 of these circuits that work fine.\nwith regard to making a direct connection to your arduino from the DTR line, that is a a bit risky as you may damage things. I presume you at least used the 100nF capacitor?\nJust to make sure, you did connect this interface to the TX\/RX and RST pin on the chip right?\nThe signal coming from the max232 should indeed be a negative as it contains an inverter and it is what the atmega chip expects on its RST line.\nThere is however one thing I am not certain off: I have always used the serial module on 'bare bones' arduino's or those that have only an FTDI header and no USB interface, I never tried it on a real arduino UNO, simply because that has a USB interface so I saw no need to use a serial converter.\nIf one looks at the circuit of the Arduino UNO, there is a 100nF capacitor going from the reset line of the atmega328 to the PD7 port of the 16U2 that handles the USB interface. It is well possible that this connection interferes with the resetpulse that you put on the RST line (pin 1) of the atmega328. I am just not sure about that but it seems possible.\nIf you are sure yr connections are correct and you seem to e able to communicate through the RS232 converter, my guess is that you are not doing anything wrong, but that the USB section of the arduino ( the 16U2) interferes with your reset signal.\nWhat you could do\u2026 if you have a barebones arduino, or a pro mini, or know someone who has, try yr circuit on that one and see what happens. My guess is that will just work fine.\nSecond thing is a capacitor between pin 2 and the ground. It should be connected to positive according to the datasheet!\nIt has worked perfectly for me for years on DIY arduino's. As it has been a long time, I am not sure anymore about the goal of the resistors, probably input protection.\nI never used it on a nano or Uno as those have a USB interface, so I can't judge on that.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Welcome to BandMix.com Saylorsburg, PA Musicians!\nBandMix.com brings Saylorsburg, Pennsylvania musicians wanted and bands together.\nAs a proven leader in online music services, BandMix.com's thousands of profiles mean thousands of possibilities to connect with the right musicians. Browse the Listings in Saylorsburg, PA below or Search for exactly what you are looking for.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The popular, DRM-free game shop GOG.com announced earlier this year they were working on a Linux marketplace. GOG has not only delivered on that promise, they have brought native games to Linux users ahead of schedule!\nThe GOG website has a list of supported titles for Linux users along with a short list of games that will be made available on Linux in the coming months. Be sure to check out the new Linux marketplace, our options are growing in number all the time!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"New Brooklyn Theatre's second season consisted of site-specific productions and staged readings with social justice themes that connected with its mission.\nNew Brooklyn Theatre has dedicated its 2015 season to the past, present, and future of African-American women playwrights.\nInsert text here for NBT Season Four.\nInsert text here for NBT Season Five.\nInsert text here for NBT Season Six.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This is a CZ 98\/22 which is the Czech version of the Gew 98 chambered in 8mm (7.92x57). The serial number pre-fix denotes that it was made for the Czech military (at least that's what I've gathered from reading). I believe the rifle is all matching except for the floor plate (I'll have to double check) and the bore is pristine. The exposed portion of the barrel has lost its blue, but is not rusted. The action is extremely smooth and the rifle is very accurate.\nAlso included is an excellent condition CZ 98\/22 bayonet with sheath and frog.\nLooking to get $350 shipped. No trades unless you have a British made (no Nepalese rifles) Martini Henry, either a MK II, III, or IV.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbgdsb b/data_all_eng_slimpj/shuffled/split2/finalzzzbgdsb
new file mode 100644
index 0000000000000000000000000000000000000000..d4326bad843b746eaf2acb51a761049444064eea
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbgdsb
@@ -0,0 +1,5 @@
+{"text":"Greetings, we are looking for a musician (guitarist\/vocalist) to play a couple of songs during a beach wedding ceremony and then depending on if you want to play more, maybe a 30 or 40 minute set immediately after the ceremony.\nThe wedding is March 14th on the beach in front of the Corona Del Mar in the afternoon. If anyone is interested, please contact me via email ([email protected]) and we can get things arranged.\nLast edited by 6string; 03\/02\/14 07:04 AM.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Our fourth Klondike Solitaire card game, One-Card One-Pass, is now available on Sport Solitaire.com. Your Klondike Solitaire skills must be becoming superb, and this new game will make them even better!\nAs simple as the name implies, One Card One Pass Klondike Solitaire lets the player cycle through the deck only once, drawing just one card at a time. Easy, eh? Not quite--that first deck cycle will go faster than you think!\nSo, go play One-Card One-Pass Klondike Solitaire, and put those awesome skills to the test!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Pat Benatar Kickin' it For O.J.\nGuru recently completed a pair of elegant orange-on-white fully animated spots for the esteemed creative folks at Downtown Partners. Feeling these oranges desperately needed a voice, we pitched a new art direction for the spots back in December and won the campaign. Then thanks to a licensed 80's classic and the superior efforts of our crack team of pantomime actors at Guru, we wound up with a wonderful new society of roly-poly oranges playin' footie against a frightened straw goalie in this match to end all matches. And while the Tropicana oranges prevail, the rest\u2026 ...well, they gave it their best shot.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The former Panasonic site in Penrith, also known as the Parkview site, has been sold to SHMH Developments Pty Ltd., an Australian based company with extensive international experience in residential and commercial development.\nWhen announcing the purchase, the company stated that a major influence was Penrith Council's work on defining the future of the city with the release of the Penrith Progression - A Plan for Action. The company acknowledged that Penrith has the right ingredients for the transformation of the existing city centre into a major regional hub for business, health, employment, education, logistics and more importantly residential development.\n\"Future apartment housing, close to the city centre, has been a focus for many stakeholders throughout the development of the Penrith Progression Action Plan,\" said Mayor Ross Fowler OAM.\n\"The new owners of the Parkview site are committed to quality development strategically located close to public transport, recreation, cultural facilities and a regional shopping centre. This is an exciting development which complements our community's vision for the future of Penrith,\" said the Mayor.\nThe new owners will now work with Penrith City Council (having regard to current approvals that have been issued) to develop an iconic site, which will be master planned to ensure a high quality residential development and support facilities.\nA previously approved concept plan provides for a five stage residential and retail mixed use development. Under this approval the development will include apartments and retail areas. There is also provision for a public plaza and open space areas, and on-site parking for residents and visitors.\n\"While the concept allows for the initial development of 570 apartments, Council is committed to work with the new owners to refresh the concept and get a development befitting the site's strategic importance,\" said Mayor Fowler.\nThe company expressed its support of Council's leadership and the desire to ensure that Penrith develops as the Growth City, the heart of the New West.\nSHMH Developments Pty Ltd has also acquired property in Waterloo, Eastwood and Elizabeth Bay and all sites are being reviewed for development potential. However it is envisaged that Penrith will be their flagship development.\nInformation contained within this news release was correct as at Friday, 15 May 2015.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The League of American Bicyclists (LAB) recognized Stanford University with a second Platinum Bicycle Friendly University (BFUSM) award.\nStanford is the only university to receive a renewal of its Platinum designation and was the first university to be recognized at the highest level when the program launched in 2011. This year's BFU award extends Stanford's original Platinum designation (2011\u20132015) for another four years (2015\u20132019).","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbizda b/data_all_eng_slimpj/shuffled/split2/finalzzzbizda
new file mode 100644
index 0000000000000000000000000000000000000000..f00d8c77e1f9bc46a8c1b5875d71ba7e63d9c0df
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbizda
@@ -0,0 +1,5 @@
+{"text":"Cornea: The transparent front part of the eye that covers the iris, pupil, and anterior chamber.\nPupil: Determines how much light is let into the eye.\nLens: Focuses the light rays that pass through it in order to create clear images of objects that are positioned at various distances.\nIris: Controls the size of the pupil.\nCiliary body: The part of the eye that connects the iris to the choroid.\nRetina: The nerve layer that lines the back of the eye, senses light, and creates impulses that travel through the optic nerve to the brain.\nOptic disc: Is the point of exit for ganglion cell axons leaving the eye.\nOptic nerve: Is transmitting impulses to the brain from the retina at the back of the eye.\nConjunctiva: Is an inflammation or infection of the transparent membrane that lines your eyelid and covers the white part of your eyeball.\nThis important due to the optics unit being about the use of mirrors to see images and the eye is the most important mirror that the human race has. It is also important because it allows us to see how it functions and see how glasses fix near and far sightedness.\nWhen surveyed about the five senses \u2013 sight, hearing, taste, smell and touch \u2013 people consistently report that their eyesight is the mode of perception they value (and fear losing) most. Despite this, many people don't have a good understanding of the anatomy of the eye, how vision works, and health problems that can affect the eye.\nThis is a good resource due to it having many links that help identify the parts of the eye and what there functions are in helping the eye create images.\nThe eye is protected from mechanical injury by being enclosed in a socket, or orbit, which is made up of portions of several of the bones of the skull to form a four-sided pyramid, the apex of which points back into the head.\nThis is a good resource because of the in-depth amount of information that it has on the amazing features of the human eye and how it fits all together to function properly.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Recreation Outlet was founded in 1987 in Salt Lake City in response to the need for a reliable source of quality off-price camping equipment. Although our beginnings were modest, we have grown into one of the country's largest suppliers of outdoor gear and clothing \u2013 both closeouts and in-line products. We operate three brick-and-mortar stores in Utah and enjoy a thriving web presence.\nWe love the outdoors and we're blessed with an expert crew of experienced outdoor enthusiasts. They are typically not with us for the big bucks, but they like what they do and they do it well. From day one, we've been totally obsessed with providing great customer service. Our stores are not \"self-service\" operations and we welcome your calling our offices if you have questions on any potential purchase. We're happy to talk.\nBig name brands are great and we sell loads of them, but we're savvy enough to know that many times the best known brands do not represent the best value. For over twenty-two years, we have scoured the planet for the best sources of outdoor gear. Some of our gear is manufactured for us in the same factories that produce the name-brand products, and is nearly identical. Needless to say, we can save you about 30 to 50% on such items. Our customers (thankfully we have a large, loyal following) love this. This is what we do\u2026save you money on top-quality outdoor equipment and clothing.\nSo, don't be afraid to give us a try. We're well known in the outdoor industry, and we'll always do our best to earn your trust and make sure you are happy.\nTo all of you, from all of us at Recreation Outlet - Thank you and Happy eCommerce!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Analysts speculate that the supplement industry will hit a record $278.02 billion by 2024 (a huge jump from its size of $122B in 2016). This is all thanks to rising popularity of supplements, vitamins, and nootropics.\nIf you're into biohacking (DIY biology) and turning your body into the machine it was meant to be, then you need to know about nootropics. It's amazing that nootropics are not more widespread.\nNootropics, also called smart drugs or cognitive enhancers, are supplements taken to improve cognitive abilities such as memory, creativity, or motivation. One of the most popular nootropics on the market these days is Noopept.\nIn this article, we're going over everything you need to know about Noopept and why millions of people all over the world are swearing by it for an increase in their cognition, memory, and even their mood.\nNoopept, the brand name supplement for N-phenylacetyl-L-prolylglycine ethyl ester, is a type of nootropic developed in Russia.\nThis drug is a potent receptor site modulator that works by increasing spindle-like activity and alpha wave function within your brain. This, in turn, gives your brain a boost and benefits the mood, memory, and attention span of those who take it regularly. It is very similar to Adrafinil Capsules or Aniracetam Capsules.\nThe standard dosage for Noopept is 10-30mg once a day for no longer than 56 days at a time. Some prefer to stretch this dosage out over the day's length, but it depends on your personal preference.\nNoopept has a multitude of benefits that will increase your mental performance in the workplace as well as everyday tasks. Let's break them down further.\nNot only is Noopept known to increase mental clarity, but it's also known to increase your mood and lower anxiety. It can decrease irritability and fear by increasing the tonic inhibition in the hippocampus.\nFor those who suffer from anxiety and depression, as well as mental fog, and have found no relief, this may be a good alternative supplement to prescriptions.\nNoopept works to increase memory in users by increasing brain chemicals NGF and BDNF. These chemicals are responsible for your learning ability and forming new memories.\nWhen compared to other nootropics, this performed best at consolidating memories and retrieving memories.\nThis drug is very popular in Russia among the elderly who use it to prevent age-related issues.\nAmyloid toxicity is what kills cells and causes the symptoms of many memory loss issues. When tested, Noopept proved to be neuroprotective and prevented the cell's death from amyloid toxicity.\nIf you're experiencing poor mood, mental fog, or a decline in your condition, you may want to consider your nootropic options. Now that you've learned all about Noopept, you're probably ready to get your hands on your own bottle of the stuff.\nHere at Zach Attack Supplements, we specialize in supplements of all kinds of supplements, especially nootropics such as Noopept. We currently sell Noopept 100 Capsules. Our capsules contain 30mg of noopept and 50mg of choline bitartrate!\nCholine bitartrate may increase the effects of other substances by allowing them to cross the blood-brain barrier more easily. It also turns into acetylcholine in the body, and may help increase memory in that regard.\nOur product is extremely high quality, and we don't want you to break the bank just to get it. It's over 99% pure. We have the testing and documentation to prove it.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"They are most loved for their quirky instrumentation and intimate vocal duets that feature their natural musical chemistry in creative arrangements. With a multilingual repertoire of jazz standards and evergreen classics, this duo seeks to share their hearts through their music, always with a touch of nostalgia and a lot of love. Highly talented duo who does piano, guitar, ukele, double bass \/ electric bass.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"How does one choose the right carpet? How do we chose the right tiles? What carpet has the best wear? What carpet is easiest to clean? What carpet will last the longest? In the end, any carpet that you choose, should provide you with years of wear.\nSo why do we use a powerhead? The answer is simple. It will pick up more dirt, more hair and dander than a non-powerhead vacuum. <<Insert picture>> A powerhead has a rotating wheel with brushes that run over the carpet and lift up the hair. Using the powerhead saves time and increases the quality of the vacuum. Because of their low profile design most powerheads can also reach under furniture and maximises edge cleaning around baseboards.\nWhat is the best powerhead to buy? KnuWhizz uses an expensive German made Wessel, much like the one on this website.\nWe hope this article has helped you but if you are still confused about what vacuum to buy then this website has some great information.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbkdno b/data_all_eng_slimpj/shuffled/split2/finalzzzbkdno
new file mode 100644
index 0000000000000000000000000000000000000000..5841bf89b5fb7c01e6906bade8283d5efe6ffb86
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbkdno
@@ -0,0 +1,5 @@
+{"text":"Topic title says it all, of course. I always shout: 1!!! But, that not true ofcourse. I could analyze lot's of sources and stuff (or however they count these things) but there's probably someone here who knows the answer. I would guess 0.62.\nOfftopic: I'm building a lib with the original opl4 patches,including an easy why to calculate the oct and f-num. Maybe you can use it for RealFun?\nI tought you didn't care about speed Ofcourse I care about speed. I just think a msx2 at 3.5MHz is fast enough.\nI think your opl4-lib wil be very useful for Realfun. But I should see it first, ofcourse .\n*cough* vapourware *cough* uhm, is Realfun under attack here? In that case, could you give me your definition of 'vapourware'?\nHehe, just prove me wrong, please Should I read this as: \"No, I can't give you my definition of 'vapourware'?\"\nIf you insist: vapourware, software that has been announced but will probably never be released in a finished and usable form.\nAlso because Realfun is needed for other vapourware... (Hehe, the lost world, or the last dragon (to avoid confusion with the Umax product), for instance. Now THAT's vapourware ).\nBTW, enter.mp3 sounds cool. Is that realfun or mbwave?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Last few left! Get them before they're gone for good!\nWith these shiny silver T.U.K. Creeper Wedge Booties on your feet you'll be everyone's fun ADD distraction. The silver metallic leather upper features petite d-ring tabs and silver metal d-rings with black laces for tying up. The black interlace on the front give these wedges the classic creeper look. The bootie upper is about 4 \u00bd inches high, finishing right above the ankle. The ridged PU sole is lightweight and the height ranges from about 5 1\/4 inches in the back to 1 1\/4 inches in the front.\nJust what I wanted. Comfortable, easy to walk and stand in. Thanks!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Summary: (from Author) Sixteen-year-old Maddie Dunn is special, but she needs to figure out how to use her new abilities before somebody else gets hurt. Ganzfield is a secret training facility full of people like her, but it's not exactly a nurturing place.\nA good book for teenagers, that like to read paranormal novels. A love story between 2 teens that discover they have special abilities. They are recruited by a special school where they learn to deal and control their new discovered gifts and their regular life as well. The story is clean , romantic (a bit mushy) and has some good action scenes as well. This is the 1st book of a series, definitely a promising beginning.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Spillman Ally is an affordable, user-friendly, cloud-based incident management and dispatch software platform intended to provide private security personnel with an all-in-one tool to manage their critical data. Ally has been specifically designed for security agencies and helps organizations increase productivity while minimizing costs to the company. In addition, security teams can feel confident in their choice of Ally because it is backed by Spillman's 30-year reputation for reliable, innovative public safety software.\nInvesting in a cloud-based records management system (RMS) may seem complicated to some private security organizations that operate on a limited budget. Ally features simple subscription-based pricing and eliminates installation fees because customers do not need to purchase expensive hardware or security patrol software. Unlike many on-premises solutions, this Software-as-a-Service (SaaS) product offers free and frequent updates, maintaining a standard of innovative technology without requiring customers to continually purchase new software versions. Organizations can reduce costs and eliminate the need to hire IT personnel because Spillman securely hosts all the data offsite.\nExtensive training for a new system can be both time-consuming and costly. That is why Spillman designed Ally for users with little to no previous guard patrol software experience. No comprehensive training is required and personnel can begin using Ally immediately upon activation. Free online help resources are also available for customers to access. Private security organizations using Ally can rely on Spillman's reputation for quality and stability, and the intuitive private security software is regularly updated with input from customers.\nIt can be difficult for private security teams to maintain a database of clean, accurate records when the guard patrol system they use does not cater to their specific needs. Ally helps keep agency data consistent by providing drop-down menus where practical, such as a predefined list of eye colors or state abbreviations. This helps to ensure that each security officer is using the same format when creating records and reports. Ally's address verification abilities ensure that each address is correct and complete as it is entered into the system. By prompting users to search for existing records before a new one can be entered, the Ally system helps agencies reduce the possibility of duplicate records. Ally also gives security administrators the ability to easily view, manage, and analyze productivity and performance, helping them ensure that their teams are maintaining the level of efficiency expected by clients and employers.\nSecurity teams need a security patrol software provider they can trust to offer loyal support and services. With Ally, security organizations can take advantage of Spillman Technologies' reputation for quality and stability. Spillman has been a public safety industry leader in both innovative technology and reliable customer service for more than 30 years and devotes millions of dollars each year to research and development, helping to ensure that its software offerings remain at the forefront of technology.\nInterested in testing Ally out?\nContact us about starting a free trial.\nRead more about Ally's features and the advantages they provide to security agencies by clicking on the links below.\nInterested in learning more about what the Ally platform can do for your security organization?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"It's said that the eyes are windows to the soul. We gaze into one another's eyes for a variety of reasons, so it's only natural to want your eyes to project the best impression possible. Blepharoplasty, or eyelid surgery as it's more commonly called, is one of the most rewarding surgeries you will ever have. If your upper eyelids appear droopy, with excess skin giving you a sad, tired appearance, then an upper blepharoplasty will give you a whole new outlook. Upper eyelid surgery can be performed in the office, under local sedation, leaving you looking refreshed, youthful, and most important, rested.\nAlso, if you have puffiness or bags under your eyes, you may want to consider a lower blepharoplasty. Upper and lower eyelids are often addressed at the same time with surgery performed in a hospital setting.\nA simple consult with one of our six board-certified plastic surgeons will determine the best course of action for you. Call our office at (513) 791-4440 or click here to set up a complimentary consultation.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbkenv b/data_all_eng_slimpj/shuffled/split2/finalzzzbkenv
new file mode 100644
index 0000000000000000000000000000000000000000..2297a2e757e51b29d8d0a7155b3e9a7144b6c7e9
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"One strange thing about Wolves and their Portuguese recruitment, which we've been following at Sport Witness since the start, is that they haven't scouted clubs in the country as often as one would think.\nPortuguese clubs usually release scouting lists, which show all the clubs sending spies to matches ahead of potential transfer approaches. It's not something which tends to happen in other countries, so is often intriguing.\nWolves have hardly ever been on these lists, especially when taken relative to their signings from the country.\nOn Wednesday evening, things were different.\nThe club were on the scouting list (via Bruno Andrade) for Benfica's match against Ajax in the Champions League. They were one of many, with most, including Manchester United, Juventus and Atletico Madrid, being regular visitors.\nOf course, there's no indication of who Wolves were watching, but given scouts are usually sent for the home team, and given the club's transfer history, Benfica players are the obvious ones being eyed.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"public it should be commended.\ndriver or piles with a tack-hammer.\nlive stock and meat prices.\nmethod of organization or operation?\nBook Bent free on request.\nSwift & Company, U.S. A.\nby helping her raise immense wheat crops to feed the world.\nsupply of fish stories holds out.\ngo through life with Just mediocre husbands.\ntill tomorrow what you can do today.\"\nday you took the mistletoe down!\"\n\"Oh, yes; I'm even married?\"\nthe kidneys and bladder you are doomed.\nto look after your kidneys and bladder.\nhave been a standard household remedy.\ntied on with wide ribbon.\non the floor to serve as trains.\nperforated strapping on toe and top.\ncertainly fashion's pet of the season.\nhappy combination of the two.\nis deplored by the dressmakers.\nthat she alone can wear.\nIon Is kinder than nature this year.\nwhere they are edged with soutache.\ngathered Into the belt at the back.\nmodishness will be of a lasting quality.\nings eleven pence a yard.\nwill not be worn to any great extent.\ntold by ii \"blither\" of a certain cure.\nky every evening before piiliijr to bed.\nIlu tried It. uuil after a few night\"\nhow the remedy was working.\nSandy. \"I'll tell ye boo It worked.\nhung whilher I went tue sleep or nol\"\nfree samples address, \"Cutlcura, Dept.\nX, Boston.\" At druggists and by mail.\nSoap 25, Ointment 25 and 50. Adv.\n\"I had to get rid of that nurse girl,\"\na silver spoon as n Christmas gift.\naud buuions incased in rawhide.\"\nWhy not order a dozen or more 23o.\ntraining camps and in the army and navy.\nto pay you n compliment.\"\ndisease. A DAISY FLY KILLER will do it.\n150 De Kalb Ave.. Brooklyn. N. Y. Adv.\nwho Is too lazy to acquire enemies.\nTMC OIHTAUfl OMMMVt MCW TON fTT.\nnri Leader of the World Famous U. S. Marine Band.\nEstonle Remedy Co., ChioiffO. III. Washlnirton. . C.\nand inditfssuoo. I have used it with excellent results.\nCspt. U. & Navy sod Leader Marine Band.\ntobrinsr relief or money back. Get a box today.\nCosts onlv a cent or two a day to use it.\nllldl VI XjaiUilIl ATONIC lirtt, we will gludly send yooaboawith fulidildr.\nIt! : rKrereiident. liutonie Kcmedy Company. Jtt2 South Wabash Avenue. Chicseo. 111.\na eood crop of corn before early frosts.\nad and shipment will be made promptly.\nat once, we can please you.\nership of your own lands. NO STOCK for sale.\ntried and proved a big; winner for the Investor.\ncation.1 2. 50 per bottle delivered. Book 3 R tree.\nW.F.YOUNG. P. D. F..310TcmplSU SprlngRald, Masa.\n1. C. Advice and books free.\nBates reasonable. Highest references. Beslservicea.\nTVi SAFETY rittST FLAG and BilSKB HoM.r for iotoaoblka.\n110. U. kilXHfcS, tO! Iraa aiekulv, aiauasolls, Blsa.\nW. N. U., FORT WAYNE, NO. 19-1918.\ntell how they found health.\nplacement. I felt all run down and was very wak.\nthis letter.\" Mrs. E. E. Ckumling, R. No. 1, Hellam, Pa.\nLYDIA C.P1NKHAM MEDICINE CO. LYNN. MASS.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Apply a thin coat of ARCTIC FREEZE onto the entire nail making sure to cap the free edge. Apply a second coat for full coverage.\nUsing I FEEL FLOWER-FUL, apply product near the free edge, center and cuticle. Next, using a Mini Striper Brush, quickly drag product up and down.\nRepeat step 2 using MERCI BOUQUET.\nRepeat step 2 using FOLLOW THE PETALS.\nWith the Mini Striper Brush and BLACK SHADOW, paint outlines of a flower near the free edge, sidewall and cuticle and add details. Finish with NEED FOR SPEED top coat.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The +1 214 288 area code numbers are located in Grand Prairie, Texas. This serviced is provided by Cellco Partnership DBA Verizon Wireless for Wireless phone numbers. Retrieve caller information for 214288, we offer a full reverse phone number report for (214) 288 numbers in Texas.\nReverse Phone Number information for 1-214-288 numbers!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Thank you for visiting and I hope you enjoy browsing emilypaddonbrown.com.\nThe musical Loving Repeating\u2013 which I had the pleasure to be a part of in January 2015 won two Green Room Awards- Best Director and Best Choreographer. This is so wonderful for a producers first show to win awards like this! Both awards are well deserved and I thoroughly enjoyed having Jason Langley & Michael Ralph creating the show with fab MD Ben Kiley.\nI am a proud singing teacher. I have had students gain roles in the musicals Annie, Chitty and School of Rock!! If you are interested in getting singing, or acting lessons please feel free to check out EPB's Singing Studio FACEBOOK or EPB's Singing Studio WEBSITE .\nTo those of you who are actors in Melbourne, here is a link to a group called \"Tea and Play Readings\". It is a group of actors who get together and read a script every fortnight. It's great for repertoire and practising cold reads, with no pressure. If you are interested in getting involved please join the facebook group and you will get a weekly update about when & where it is, and what we are reading. If you do not have facebook and are interested in receiving updates, please email me. The more the merrier so I hope to catch you soon!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbkoqj b/data_all_eng_slimpj/shuffled/split2/finalzzzbkoqj
new file mode 100644
index 0000000000000000000000000000000000000000..78366bf2b4244bf781a8eb1b939a7ba25316b6b6
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbkoqj
@@ -0,0 +1,5 @@
+{"text":"SVF hosted a meet and greet session with Iranian director, producer and screenwriter, Majid Majidi along with his actors from\"Beyond The Clouds\"and a few eminent people from the Tollywood film fraternity namely Anjan Dutt, Arindam Sil, Srijit Mukherji, Abir Chatterjee, Parambrata Chatterjee, Pratim D. Gupta, Anirban Bhattacharya, Neel Dutt and others.\nMajid Majidi ventured into cinema after being inspired by various legendary directors, notably Mohsen Makhmalbaf, another Iranian film genius. After foraying into cinema direction, Majidi became well-known around the world especially after garnering an Academy Award nomination for Best Foreign Language Film for his masterpiece, Children of Heaven.\nBeyond The Clouds marks his debut in Indian Film Industry and also the first time that he has moved out of his country to make a film. It is said to be a grown-up version of Children of Heaven. Ishaan Khatter, the half-brother of Bollywood actor Shahid Kapoor and Malavika Mohanan, daughter of cinematographer, K. U. Mohanan play the lead characters of two siblings from Dhobi Ghat who tackle greater conflicts and many facets of the bustling city of Mumbai intertwined with a heartfelt celebration of love and family. A.R. Rahman has written and composed the original score for Beyond The Clouds.\nThe film is produced by Zee Studios and Namah Pictures and will release on 20th April.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Celebrating 25 years with Remax! Now that we are living our dream on Venice Island and relocated from Maryland where I've been a professional Realtor\u00ae for __28_ years. My focus is simple! To help people like yourself to have a smooth transition from where ever you live to Venice Fl so you can full fill your dream.... Ginny brings more than a quarter of a century of Remax experience with her to serve your Real Estate needs! Whatever your needs in the Venice Island area, Ginny's experience with military relocation, her extpertise and certification in distressed properties, and additional designation as a negotiation specialist will have you covered. Of course, Ginny is also a graduate of the Real Estate Institute! Ginny Michaels relocated to the Venice area from central Maryland where she enjoyed boating and kept a boat in Annapolis not too far from the Naval Academy. She also was the owner of a local golf course and enjoyed having input in course management, creating member and community events and growing the residential market surrounding the course. She had a large following and committed clients who hated to see her relocate but will make great references for her new contacts in our area! Ginny Michaels has traveled extensively throughout the east coast and has wisely chosen Venice Island as her new home. Please help us welcome her as you utilize the expertise she brings as well as her team of highly qualified loan officers, mortgage brokers, inspectors and contractors. Whatever your Real Estate needs in the Venice Island area, Ginny Michaels has you covered. Thank you, Venice Island, for being a community that draws the best in the business to want to call our island home.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"PHILADELPHIA, PA \u2013 CSz Philadelphia presents Murder Manor: The Golden Girls Edition, a fully interactive improvised murder mystery experience based on the board game Clue and featuring characters from the classic 80s' sitcom, The Golden Girls. The show runs Fridays in April @ 8PM at CSz Philadelphia, 2030 Sansom Street, Philadelphia. Tickets are $15.\nFive seniors from Miami find themselves in the middle of a murder. Who did it? Was it sarcastic Dorothy, flirtatious Blanche, dimwitted Rose, cheating Stan or firecracker Sophia? The creative team behind the 2017 hit show, Murder Manor: An Improvised Game of Clue, are back with an all new mystery and a new cast of characters.\nGrab your shoulder pads, a slice of cheesecake and head out to the lanai for Murder Manor: The Golden Girls Edition, an improvised murder mystery that never ends the same way.\nCSz Philadelphia uses improvisation to produce fun, immersive experiences that bring laughter to the people of Philadelphia. In the classroom we teach our students how to use improv to entertain and excite and audiences. In the boardroom, we teach professionals how to communicate, listen, and take risks using Applied Improvisation. On stage we entertain everyone with ComedySportz, Philadelphia's longest-running comedy show where two teams of improvisers battle for the laughs of the Loyal Fans.\nFive seniors from Miami find themselves in the middle of a murder. Who did it? Was it sarcastic Dorothy, flirtatious Blanche, dimwitted Rose, cheating Stan or firecracker Sophia? Based on the famous board game that is based on the famous TV show, five improvisers become a gang of hilarious suspects who have secrets to protect and a mystery to solve, all before the commercial break.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"All the information you need to enjoy your IE experience and life in Madrid.\nWelcome to IE! In order to ensure that your IE experience is as smooth as possible, Student Services offers you resources and information vital for your move to Madrid. From obtaining the student visa to finding accommodation, exploring medical insurance options or getting mobile phone service, we are here to assist you.\nHere you will find all necessary information about immigration and the main steps you need to follow to prepare for your arrival in Madrid: visa, NIE number, bank account and health insurance, etc.\nIn addition to the agreements we have with several real-estate agencies, IE has an accommodation web page where you can easily find apartments for rent near campus. We also offer information about the city itself, public transportation, etc.\nAlong with the information regarding accommodation, there are many other services that can help you prepare for your arrival in Madrid. Our office organizes \"Landing in Madrid\" briefings meant to clarify your doubts about life in Spain. We also have agreements with telephone companies, medical insurance companies, and can provide you with information about the city and transportation.\nOffice hours for students: from 11:00 to 16:00 hrs.\nThe Online Campus is a virtual space for teaching, learning, research and a meeting point for the entire IE community.\nThe Online Campus gives you access to all information related to your program, both academic and extracurricular. Plus, you'll be able to get in touch with your classmates, teachers and different departments at the school.\nHow to get to IE?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"MWL Fier in cooperation with the Peace Corps Fier volunteer, organized three days of a summer camp with young girls.\nMWL Fier in cooperation with the Peace Corps Fier volunteer, organized three days of a summer camp with young girls. Topics covered in this camp were: the Fight against Human Trafficking, Leadership and Communication Skills as well as mental health, stress and depression. These were particularly related to young people.\nDuring these activities, the girls also engaged in various energizing games or in cooking delicious biscuits. The girls had 3 days of intensive interaction with each other. This activity made it possible to discover most valuable information about them through these socializing activities.\nThey expressed the desire to have more similar activities especially during the summer vacation period. 12 young girls participated in this summer camp.\nPrevious Previous post: \"Art for Freedom\" an Awareness Raising Campaign Against Human Trafficking on July 30th, 2017.\nNext Next post: The second 'Friends of Mary Ward' conference in York, England.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzblzal b/data_all_eng_slimpj/shuffled/split2/finalzzzblzal
new file mode 100644
index 0000000000000000000000000000000000000000..83668d9cd1f0d4bb10a46797fae4db4a3fb8e084
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzblzal
@@ -0,0 +1,5 @@
+{"text":"Character Analysis - Bartleby, the Scrivener essaysIn the short story \"Bartleby, the Scrivener,\" Herman Melville creates a rather perplexing character. Melville uses a narrator, a lawyer, to describe Bartleby as an odd character. The interactions between the narrator and Bartleby reveal bi.\nA list of all the characters in Melville Stories. The Melville Stories characters covered include: The Lawyer , Bartleby, Turkey, Nippers, Ginger Nut, Hunilla , Oberlus, Amasa Delano , Benito Cereno , Babo, Atufal, Alexandro Aranda .","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Nice new hotel. Clean and spacious. Excellent staff. We highly recommend and would definitely stay there again if we went back to Cuzco.\nMy partner and I searched for two days for a hotel for our 9-day stay in Cusco. Atoq was easily the only choice for our price range, and far better than anything Air BnB offered. Our room (junior suite) was about the size of a nice room in an American hotel. It had a real king bed with a super comfortable memory foam mattress and pillows. The shower was great. Everything was clean. The free breakfast in the dining room included fresh fruit, hot items, juices, and eggs made to order. Location is a great neighborhood with hip shops, cafes, and restaurants, slightly off the main tourist beat. Best of all: the staff was helpful and friendly. They gave great neighborhood recommendations. Within an hour of booking, they emailed to set up a complimentary airport pickup. They addressed me by name when I arrived. They booked taxis and good drivers for us during our stay, and set out a bag lunch for our pre-dawn departure to Machu Picchu. I wish I was still there.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Team Fox are at it again, it's time for another Max Out Session and this time we are lifting for Children In Need.\nAnyway, they are the most amazing women I know and I know that raising money for one of the most amazing children's charities means a lot to them.\nAll the Foxes are currently training hard and lifting heavy so if you want to donate to Children In Need, here's the link. You'll be helping children throughout the UK and an amazing cause.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"RATIA-leash is colorful and it has a soft handgrip. Light RATIA-leash has a stylish Grafico - ribbon.\nRATIA-leashes matches well with other RATIA-products; collars, harnesses and also with new products coming soon.\nThis turquoise leash has a blue Grafico-pattern.\nRISTOMATTI RATIA has designed Grafico Dog International-pattern and they are available in colors blue and red. To this pattern, TopCanis has combined really soft fleece-padding. TopCanis has made the softest harnesses in world, collars and leashes.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A brief and helpful summary about my service.\nI am offering my services as an officiant to help people who wish to marry outside the confines of traditional locations and doctrines, and to personalize their experience. I have a Bachelors degree in Religion from Centre College and a Master's degree in English from the University of Louisville that provides me with the tools necessary to help folks customize their marriage or commitment ceremony to fit their needs and location. We have helped couples write meaningful vows that reflect them, ranging from quite simple to very complex, but always helping them to say what they truly want to say to each other. I am open to listening to you to create a unique experience for your special day.\nRead some clients' feedback about their experience.\n\"Mark will make your special day personal and memorable. He is incredibly genuine and heartfelt. I highly recommend him to anyone! A great person who truly wants to be there for you!\"\n\"Mark Peterson officiated our wedding and was excellent. He is a great family friend who helped make our day very special and we would highly recommend him!\"\n\"You did such a wonderful job on Saturday! The ceremony was personal, funny, and touching. It was perfect. We loved it!\"\nPlease call Renee Peterson @ (502) 693-7460, or send us a message below and include your ceremony date.\nCopyright \u00a9 2019 Officiant Mark Peterson. All rights reserved. No unauthorized reuse or reproduction allowed.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbpbgg b/data_all_eng_slimpj/shuffled/split2/finalzzzbpbgg
new file mode 100644
index 0000000000000000000000000000000000000000..cc5b8035b4feba8026b8b32c22e9a2989d3a0aa7
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbpbgg
@@ -0,0 +1,5 @@
+{"text":"I have a several businesses in planning and I believe it is vital to test your assumptions under real world conditions. To be clear, all of my business plans are in the craft beer industry, so there is a symbiotic relationship between them. I also believe in the minimum viable product (MVP) concept that is heavily used in the Tech Industry.\nThis past month I was able to go step by step through one of by business plans, without devoting any significant amount of capital. I brewed a batch of beer, sold that beer to a company and then served that beer to paying customers.\nNothing I just did is earth shattering, this business model is executed thousands of times a day across our country. But, I proved that I could do it, without diving head first into the deep end.\nThe next step in the process is to prove that I can develop a beer recipe that people will gladly pay me for a pint.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Week FOUR of COTT's Tournament of Champions saw four more authors compete and three more scavenger hunts played.\nBecky R went to Gail Pallotta's Blog on Monday to find out when Love Turns theTide's Cammie sees a different side of Vic. She got the correct answer and walks away with a $10 Starbucks card from Lisa Lickel. Pumpkin Latte, anyone?\nMegan D visited Marianne Evans' blog Tuesday to discover which of Marianne's books was a Christian Small Publisher's Book of the Year nominee. She won a $10 Amazon gift card from Amanda Stephan. Awesome!\nTheresa M stopped by Margaret Brownley's Blog on Wednesday and found the name of the Christmas anthology which Margaret contributed to. She also received a $10 Amazon gift card donated by Raquel Byrnes.\nCongrats ladies! Clash of the Titles has awarded over $125 worth of game prizes, thanks to its generous sponsors. Thanks, COTT--and another round of applause for all those game winners!\nWhich Books Won the Clashes?\nRosslyn Elliot's \"Fairer Than Morning\" and Delia Latham's \"Destiny's Dream\" clashed last Monday. \"Fairer Than Morning\" came out the winner.\nShellie Neumeier's, Driven and Ann O'Barr's Singing in Babylon struck swords on Wednesday. Driven took the top spot.\nRosslyn Elliot's \"Fairer Than Morning\"\nShe joins Marianne Evans' Hearts Crossing, Elaine Cooper's The Road To Deer Run, and Karen Witemeyer's A Tailor-Made Bride. We now have all four finalists vying for the ultimate COTT honor.\nVoting for the Big Winner is going on RIGHT NOW, and continues all week long. The big announcement will be made on November 12th. Public vote determines the outcome, and one reader will receive a Grand Prize of 15 books!!! Want a shot at it? Enter by sharing links, putting up buttons and banners on your blog, becoming a follower of COTT (just make sure you tell them you've done so--they aren't psychics!) Details on the prize basket and full instructions on how to enter can be found here. To make it easy to grab n go, here are the banner and button codes.\nFind me on twitter @MLMassaro, and Facebook.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Patio self contained water features welcome to our this shows you all of the products we have assigned, water features ponds anyone installing a paver patio with wall element should consider formal falls. Small outdoor fountains sale rock garden water feature ideas easy patio fountain gardens backyard solar rustic wall bristolurnu.\nMultiple water features combine in this patio with a pergola fountain hot tub and an outdoor shower the paver, blumfeldt watergate garden fountain ornamental water feature indooroutdoor if you love to relax in your or on the patio use. Outdoor patio built in grill seatwalls water feature small ideas house decor m, water features create a relaxing atmosphere add system pavers feature to backyard and garden other patio outdoor living. Explore michelle bunchs board patio pavers water features on outdoor fountain in asian with gifts for gardeners and use less formal. Explore noah goodmans board paver water feature outdoor fountains system pavers beautiful gardens patio games formal, best outdoor patio photos and design plans including most popular paver materials fire pit kitchen kits gorgeous pictures of covered patios.\nBackyard design with pavers inspirational powerful decking water outdoor kitchens retaining walls features paver patios brick patio mediterranean fountain top standard, paver patio with fire pit ideas picking out powerful decking water a fountain the chronicles of home. Curved paver patio and outdoor kitchen archadeck andersen fire water fountains features brick showroom portland design traditional with feature hot tub designs for landscaping.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Remnants of clothing do not survive well in the archaeological record, instead usually rotting away. In specific circumstances, though, they can survive. These include waterlogging (often with a side order of chemical preservation in peat bogs) as the most common, dessication, carbonisation, freezing in glaciers, being preserved in salt mines and scraps of fabric mineralising when in contact with metalwork. Fabric can also be represented pictorially and leave impressions on pre-fired clay. Here are examples of each of these and what they tell us about what textiles were being made, and presumably worn, in prehistory.\nTwined textiles differ from weaving by having weft threads more widely spaced, often, and passing around individual or bunches of warp threads, rather than passing under and over. There are other methods to make fabrics, of course, such as sprang, needle binding and felting. This page will be added to with further research.\nClothing would certainly be made from animal hides in these early periods, but there is also evidence of weaving from the Upper Palaeolithic. In the absence of cloth survival, evidence comes in the form of many impressions of woven, plaited and twined cloth, netting and basketry made from plant materials from sites in the Czech Republic, Doln\u00ed V\u011bstonice and Pavlov dating back to about 30,000 years ago (in what is known as the Gravettian). There are scraps of impressions and even surviving cordage elsewhere across Europe. This coincides with the appearance of bone eyed needles which have been suggested to have been too fragile for punching holes in hide and are likely, therefore, to have sewn together woven fabric (Soffer et al 2000, 511-14). Though cordage for weaving can be made without implements, there are also suggestions that circular 'rondelles' with central perforations could be spindle whorls, for example one carved out of sandstone at Isturitz in southern France, that may be late Upper Palaeolithic e.g. 15,000 years old from the Magdalenian, but there are other potential examples.\nAn auroch bone from the Danish site of Rymarksg\u00e5rd dating to about 8000 BC in the Mesolithic shows a number of figures dancing or walking. Their clothing seems striped or plaid, but has been seen as a quite stylised way to show some kind of fur or skin garments, especially given how stylised the faces are. Virginija Rimkut\u0117 of Vilnius University in Lithuania produced some twined fabrics based on those found in \u0160ventoji 2B in Lithuania (which are much later in date) and tested whether they may have been what the artist who cut this image onto the Rymarksg\u00e5rd auroch bone was trying to represent. The results are inconclusive, I feel.\nThe introduction of sheep in the Neolithic in Europe provides a new source of fibre to make yarn, but the majority of fabric found in Europe in the Neolithic actually comes from plants. The Neolithic starts earlier the further east you go, as it was invented in the Near East around 10,000 BC and gradually spread west and north through Europe.\nImpression of a plain weave on clay from Jarmo in Iraq dating to about 7000 BC. From Adovasio 1973, 224.\nThe earliest evidence for woven fabric comes from the heartland of early farming just outside Europe from Jarmo in northeastern Iraq where impressions of textile were left on clay. These seem to date to about 7000 BC and show an expertise in several types of weaving that suggests a well-established tradition.\nThe well-studied Neolithic site of \u00c7atalh\u00f6y\u00fck in central Turkey has yielded many fabrics, most of which have been carbonised (charred in a fire), dating to about 6000 BC. Though fragile, they survived in good enough condition to show a range of weaving patterns. There were quite broad cloths as well as narrow bands, some of which were warp-faced where the warp threads were so densely packed they hid the weft threads altogether (Barber 1991, 127).\nIn Switzerland the Neolithic starts around 5500 BC. Villages start to be built at the edge of lakes such as Lake Neuch\u00e2tel and Lake Zurich. At the latter 31 Neolithic sites with textile fragments have been found, totalling more than 1000 woven or twined objects (including basketry). The textiles are either made from lime bast, most commonly, or from linen (Rast-Eicher in Gleba & Mannering 2012, 378-80). Linen cloth from Robenhausen was produced with ribs by modifying the weaving pattern by doubling the weft thread, giving a subtle striped effect. All of the fabrics were given reinforced selvedge edges and some were fringed (Barber 1991, 134-6). Some of the Swiss Neolithic linens were also brocaded, that is the weft thread was almost embroidered over the pattern to create squares and triangles of another colour such as that from Irgenhausen dating to about 3000 BC (Barber 1991, 138-9).\nRemains of brocaded linen textile found at Irgenhausen in Switzerland dating to c. 3000 BC. From Barber 1991, 138.\nReconstructed fabric based on the find of preserved brocaded linen from Irgenhausen. From Barber 1991, 139.\nLinen fragment with evidence of betting and sewn on seed-beading from Murten in Switzerland dating to c.3000 BC. From Barber 1991, 140.\nAnother technique used in the Swiss Neolithic is to join two pieces of woven cloth with a band of netting, which would have been slightly more elastic, such as a scrap of linen from Murten dating to about 3000 BC. This scrap is also decorated with tiny beads made of plant seeds sewn on to the fabric (Barber 1991, 140).\nThe salt mine of Hallstatt in Austria is a good source of preserved leather and textile objects. Salt mining began around 1500 BC. 111 fragments of textile have been found and most of these are wool, with only two pieces made of linen, and these are the earliest in date. Does this reflect a shift to woolen textiles in the Bronze Age in Europe or is it an accident of preservation? Most of the textiles are quite coarsely made with thick yarn and not a very high thread per cm count. There is evidence for the use of woad, weld and madder plant dyes on some of the fabrics, which would produce varieties of blue, green, yellow, red and brown. Some of the fabrics were fulled after weaving to make them water and windproof \u2013 this was probably done by a combination of pounding, soaking and stretching the fabrics, as attested by historical records in later centuries (Gr\u00f6mer in Gleba and Mannering 2012, 31-7).\nExample of twined fabric, with weft threads wrapped round bunches of warp threads, from Must Farm near Peterborough, dating to about 1000 BC.\nRecently, fragments of textile have been found in England at Must Farm near Peterborough dating to the later Bronze Age around 1000 BC. One carbonised and waterlogged fragment of woven textile was found to have been made from lime bast. Another fragment of twined fabric, also made from lime bast, was found in 2015. A reel of lime base string was another discovery. No woolen textile has been found at Must Farm, either because it was not worn or because animal based products did not survive well in the waterlogged soil.\nThere are many Bronze Age textiles from bogs in Denmark. 153 graves with textiles have been found. Most of these are earlier Bronze Age and they are generally woolen. Like the woolen textiles at Hallstatt, they are also fulled. They do not seem to be dyed for the most part, except for a belt from Borum Esh\u00f8j. A late Bronze Age nettle textile was found at Luseh\u00f8j inside a bronze cauldron and dates to the 9th or 8th century BC (Mannering, Gleba & Bloch Hausen in Gleba & Mannering 2012, 97). The evidence from Borum Esh\u00f8j and Egtved will be discussed in more detail in another blog post.\nDenmark is also the richest area for Iron Age fabrics too, and also from bogs, but in this case usually not from formal burials but with people who have been sacrificed or executed and deposited. Huldremose Woman is one of these victims, and has been described in detail in this previous blog post.\nThe princely grave of Hochdorf in Baden W\u00fcrttemberg in Germany dates to the 6th century BC. The fabrics were first seen under excavation in 1978 and many of them promptly turned to dust, but fragments have survived. A cauldron, wagon and couch were all draped in textiles. One piece from the cauldron was tablet-woven with hemp fibre and wool with a chequered pattern with swastikas (an ancient sun symbol). Linen was present but only minute remains have survived, while other fabrics were made of badger hair. The textiles were dyed with woad to make blue and using an imported dyestuff made from an insect called Kermes vermilio, similar to cochineal but native to southern France. The textiles may have been made especially for the burial and not have been used in life, which is a shame as it doesn't really tell us much about what people looked like (Banck-Burgess in Gleba & Mannering 2012, 139-144).\nAdovasio, J.M 1973. The Textile and Basketry Impressions from Jarmo. Pal\u00e9orient Vol 3, pp 223-230.\nBarber, E.J.W 1991. Prehistoric Textiles. Princeton, Princeton University Press.\nGleba, M & Mannering, U (eds) 2012. Textiles and Textile Production in Europe: From Prehistory to AD 400. Oxford, Oxbow Books.\nSoffer, O, Adovasio, J. M, & Hyland, D, C 2000. The \"Venus\" Figurines Textiles, Basketry, Gender, and Status in the Upper Paleolithic. Current Anthropology Volume 41, 4. pp 511-537.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We always enjoy talking with our clients and are more than happy to arrange a meeting at a mutual convenient time, so please contact us today and see what we can do for your business. During 2013 we moved from Bristol down to the South West of England, near the City of Bridgwater.\nSo if you have a new, or existing, project in the pipeline then please get in touch with us to see what we can do to help you. We can be contacted by phone, email, social media, or post, see below for our contact details.\nWe are looking forward to working with you to help grow your business, so please contact us today. We are very friendly and discuss things in plain English so you understand everything, we will never bombard you with complicated technical terms.\nNo matter how big your project is contact us today to discuss it and see what we can do.\nTake a look through our portfolio section to see what we have done for other clients.","meta":{"redpajama_set_name":"RedPajamaC4"}}